Physical Models for Quantum Wires, Nanotubes, and Nanoribbons 9814877913, 9789814877916

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Part I: Semiconductor Quantum Wires
Chapter 1: Size Effects on Polar Optical Phonon Scattering of One-Dimensional and Two-Dimensional Electron Gas in Synthetic Semiconductors
1.1: Introduction
1.2: Theory
1.3: Conclusion
Chapter 2: Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures
Chapter 3: Plasmon Dispersion Relation of a Quasi-One-Dimensional Electron Gas
Chapter 4: Size Effects in Multisubband Quantum Wire Structures
4.1: Introduction
4.2: Model
4.3: Monte Carlo Code
4.4: Results
4.5: Conclusions
Chapter 5: Impurity Scattering with Semiclassical Screening in Multiband Quasi-One-Dimensional Systems
5.1: Introduction
5.2: Model
5.3: Dielectric Constant Matrix
5.4: Impurity Scattering Rate
5.5: Conclusions
Chapter 6: Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures
6.1: Introduction
6.2: Model
6.3: Results
6.4: Conclusion
Chapter 7: Intersubband Population Inversion in Quantum Wire Structures
Chapter 8: Intersubband Resonant Effects of Dissipative Transport in Quantum Wires
8.1: Introduction
8.2: Model
8.3: Monte Carlo Method
8.4: Results
8.5: Conclusion
Chapter 9: Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires
Chapter 10: Transient Simulation of Electron Emission from Quantum-Wire Structures
10.1: Introduction
10.2: Model
10.3: Scattering Rates
10.4: Monte Carlo Simulation
10.5: Results
10.6: Conclusions
Chapter 11: Carrier Capture in Cylindrical Quantum Wires
Chapter 12: Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures
12.1: Introduction
12.2: Model
12.3: Spatial Velocity Oscillations
12.4: Conclusion
Chapter 13: Transient and Steady-State Analysis of Electron Transport in One-Dimensional Coupled Quantum-Box Structures
13.1: Introduction
13.2: Electronic Model
13.3: Scattering Model
13.4: Transport Model
13.5: Results and Discussion
13.5.1: Time-Dependent solutions
13.5.2: Steady-State Solutions
13.6: Concluding Remarks
Chapter 14: Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires
14.1: Introduction
14.2: Electronic Band Structure
14.3: Acoustic-Phonon Scattering
14.4: Results
14.4.1: Short Modulation Period Quantum Wires
14.4.2: Long Modulation Period Quantum Wires
14.5: Conclusion
Chapter 15: Antiresonant Hopping Conductance and Negative Magnetoresistance in Quantum-Box Superlattices
15.1: Introduction
15.2: Electronic Model
15.3: Transport Model
15.4: Antiresonances and Resonances in Hopping Transport
15.5: Conclusion
Chapter 16: Oscillatory Level Broadening in Superlattice Magnetotransport
Chapter 17: Breakdown of the Linear Approximation to the Boltzmann Transport Equation in Quasi-One-Dimensional Semiconductors
Chapter 18: Optic-Phonon-Limited Transport and Anomalous Carrier Cooling in Quantum-Wire Structures
18.1: Introduction
18.2: Electronic Properties and Scattering Rates
18.3: Boltzmann Equation
18.4: Solution of the Boltzmann Equation
Chapter 19: lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping” in Quantum Wires
19.1: Introduction
19.2: Model
19.3: Optical Gain Analysis
19.4: Conclusions
Chapter 20: Superlinear Electron Transport and Noise in Quantum Wires
20.1: Introduction
20.2: Model and Method
20.3: Results and Discussion
20.4: Conclusions
Chapter 21: Importance of Confined Longitudinal Optical Phonons in Intersubband and Backward Scattering in Rectangular AlGaAs/GaAs Quantum Wires
Chapter 22: Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs Quantum Wires
22.1: Introduction
22.2: Model
22.3: Results and Discussions
22.4 Conclusion
Chapter 23: Hole Scattering by Confined Optical Phonons in Silicon Nanowires
Part II: Carbon Nanotubes and Nanoribbons
Chapter 24: Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes
Chapter 25: Joule Heating Induced Negative Differential Resistance in Freestanding Metallic Carbon Nanotubes
Chapter 26: Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power in One-Dimensional Conductors
Chapter 27: High-Field Electrothermal Transport in Metallic Carbon Nanotubes
27.1: Introduction
27.2: Model
27.2.1: Electric and Energy Flow
27.2.2: Electron-Phonon interaction
27.2.3: Electron-Phonon Heat Exchange
27.2.4: Heat Flow Diagram
27.3: Results
27.3.1: Low-Field Regime
27.3.2: High-Field Regime
27.3.3: Heat Flow
27.3.4: Hot Phonons
27.3.5: Electrical Power vs Length
27.4 Conclusions
Chapter 28: Atomic Vacancy Defects in the Electronic Properties of Semi-metallic Carbon Nanotubes
28.1: Introduction
28.2: Model
28.3: Results and Discussions
28.4: Conclusion
Chapter 29: Chirality Effects in Atomic Vacancy–Limited Transport in Metallic Carbon Nanotubes
29.1: Introduction
29.2: Model
29.3: Results and Discussion
29.4: Conclusion
Chapter 30: Vacancy Cluster–Limited Electronic Transport in Metallic Carbon Nanotubes
30.1: Introduction
30.2: Model
30.3: Results and Discussions
30.4: Conclusion
Chapter 31: Vacancy-Induced Intramolecular Junctions and Quantum Transport in Metallic Carbon Nanotubes
31.1: Introduction
31.2: Computational Methods and Models
31.3: Results and Discussion
31.4: Conclusions
Chapter 32: On the Sensing Mechanism in Carbon Nanotube Chemiresistors
32.1: Introduction
32.2: Results and Discussion
32.3: Methods
32.3.1: Fabrication and Design of the Chemiresistor
32.3.2: SWNT Preparation and Deposition on Silicon Substrate
Chapter 33: Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons
33.1: Introduction
33.2: Model and Methods
33.3: Results and Discussions
33.4: Conclusion
Chapter 34: Controllable Tuning of the Electronic Transport in Pre-designed Graphene Nanoribbon
34.1: Introduction
34.2: Model
34.3: Results and Discussions
34.4: Conclusion
Chapter 35: Quantum Conduction through Double-Bend Electron Waveguide Structures
Chapter 36: Quantum Ballistic Transport through a Double-Bend Waveguide Structure: Effects of Disorder
36.1: Introduction
36.2: Numerical Method
36.3: Results and Discussion
36.3.1: Ideal Structures
36.3.2: Systems with Disorder
36.3.3: Multiple Double-Bend Structures
36.4: Conclusions
Chapter 37: Quantum Transport through One-Dimensional Double-Quantum-Well Systems
Chapter 38: Cascaded Spintronic Logic with Low-Dimensional Carbon
38.1: Introduction
38.2: Results
38.2.1: Device Structure and Physical Operation
38.2.2: Edge Effects and Operation Temperature
38.2.3: Switching Behaviour
38.2.4: Logic Gates and System Integration
38.3: Discussion
38.4: Methods
38.4.1: Hubbard Tight-Binding Hamiltonian
38.4.2: Diagonalization and the Secular Equation
38.4.3: Mean-Field Approximation
Index

Citation preview

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons

edited by

Jean-Pierre Leburton

Published by Jenny Stanford Publishing Pte. Ltd. 101 Thomson Road #06-01, United Square Singapore 307591

Email: [email protected] Web: www.jennystanford.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Layout Copyright © 2024 by Jenny Stanford Publishing Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4877-91-6 (Hardcover) ISBN 978-1-003-21937-8 (eBook)

Contents

Preface



Part I: Semiconductor Quantum Wires

1. Size Effects on Polar Optical Phonon Scattering of One-Dimensional and Two-Dimensional Electron Gas in Synthetic Semiconductors J.-P. Leburton 1.1 Introduction 1.2 Theory 1.3 Conclusion 2. Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures S. Briggs, B. A. Mason, and J.-P. Leburton 3. Plasmon Dispersion Relation of a Quasi-One-Dimensional Electron Gas Jin Wang and J.-P. Leburton 4.

Size Effects in Multisubband Quantum Wire Structures S. Briggs and J.-P. Leburton 4.1 Introduction 4.2 Model 4.3 Monte Carlo Code 4.4 Results 4.5 Conclusions

5. Impurity Scattering with Semiclassical Screening in Multiband Quasi-One-Dimensional Systems Yilin Weng and J.-P. Leburton 5.1 Introduction 5.2 Model

xv

3 4 6 15 19

29

39 40 42 48 49 56 59 60 61

vi

Contents

5.3 Dielectric Constant Matrix 5.4 Impurity Scattering Rate 5.5 Conclusions

6. Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures S. Briggs and J.-P. Leburton 6.1 Introduction 6.2 Model 6.3 Results 6.4 Conclusion 7. Intersubband Population Inversion in Quantum Wire Structures S. Briggs, D. Jovanovic, and J.-P. Leburton 8. Intersubband Resonant Effects of Dissipative Transport in Quantum Wires D. Jovanovic, S. Briggs, and J.-P. Leburton 8.1 Introduction 8.2 Model 8.3 Monte Carlo Method 8.4 Results 8.5 Conclusion 9. Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires Dejan Jovanovic, Jean-Pierre Leburton, Khalid Ismail, Jeffrey M. Bigelow, and Marcos H. Degani

62 67 75 77 78 79 80 86 89

97 98 99 104 105 110 113

10. Transient Simulation of Electron Emission from Quantum-Wire Structures S. Briggs and J.-P. Leburton

123



124 125 128 133

10.1 Introduction 10.2 Model 10.3 Scattering Rates 10.4 Monte Carlo Simulation

Contents



10.5 Results 10.6 Conclusions

11. Carrier Capture in Cylindrical Quantum Wires N. S. Mansour, Yu. M. Sirenko, K. W. Kim, M. A. Littlejohn, J. Wang, and J.-P. Leburton 12. Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures D. Jovanovic and J.-P Leburton 12.1 Introduction 12.2 Model 12.3 Spatial Velocity Oscillations 12.4 Conclusion

134 139 143

153 154 154 156 159

13. Transient and Steady-State Analysis of Electron Transport in One-Dimensional Coupled Quantum-Box Structures 161 H. Noguchi, J.-P. Leburton, and H. Sakaki 13.1 Introduction 162 13.2 Electronic Model 164 13.3 Scattering Model 166 13.4 Transport Model 170 13.5 Results and Discussion 171 13.5.1 Time-Dependent Solutions 171 13.5.2 Steady-State Solutions 174 13.6 Concluding Remarks 177 14. Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires J. E. Stacker, J.-P. Leburton, H. Noguchi, and H. Sakaki 14.1 Introduction 14.2 Electronic Band Structure 14.3 Acoustic-Phonon Scattering 14.4 Results 14.4.1 Short Modulation Period Quantum Wires

181 182 184 186 190 190

vii

viii

Contents





14.4.2 Long Modulation Period Quantum Wires 14.5 Conclusion

15. Antiresonant Hopping Conductance and Negative Magnetoresistance in Quantum-Box Superlattices Yuli Lyanda-Geller and Jean-Pierre Leburton 15.1 Introduction 15.2 Electronic Model 15.3 Transport Model 15.4 Antiresonances and Resonances in Hopping Transport 15.5 Conclusion

193 195 197 198 198 200 203 207

16. Oscillatory Level Broadening in Superlattice Magnetotransport 211 Yu. B. Lyanda-Geller and J.-P. Leburton 17. Breakdown of the Linear Approximation to the Boltzmann Transport Equation in Quasi-One-Dimensional Semiconductors S. Briggs and J.-P. Leburton 18. Optic-Phonon-Limited Transport and Anomalous Carrier Cooling in Quantum-Wire Structures J.-P. Leburton 18.1 Introduction 18.2 Electronic Properties and Scattering Rates 18.3 Boltzmann Equation 18.4 Solution of the Boltzmann Equation 19. lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping” in Quantum Wires J.-P. Leburton

19.1 Introduction 19.2 Model 19.3 Optical Gain Analysis 19.4 Conclusions

219

229 230 231 235 237 251 252 253 256 259

Contents

20. Superlinear Electron Transport and Noise in Quantum Wires R. Mickevičius, V. Mitin, U. K. Harithsa, D. Jovanovic, and J.-P. Leburton 20.1 Introduction 20.2 Model and Method 20.3 Results and Discussion 20.4 Conclusions

21. Importance of Confined Longitudinal Optical Phonons in Intersubband and Backward Scattering in Rectangular AlGaAs/GaAs Quantum Wires W. Jiang and J.-P. Leburton 22. Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs Quantum Wires W. Jiang and J.-P. Leburton 22.1 Introduction 22.2 Model 22.3 Results and Discussions 22.4 Conclusion

23. Hole Scattering by Confined Optical Phonons in Silicon Nanowires Mueen Nawaz, Jean-Pierre Leburton, and Jianming Jin



261

262 263 266 274

277

287 288 289 295 304 307

Part II: Carbon Nanotubes and Nanoribbons

24. Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes Marcelo A. Kuroda, Andreas Cangellaris, and Jean-Pierre Leburton

319

25. Joule Heating Induced Negative Differential Resistance in Freestanding Metallic Carbon Nanotubes 331 Marcelo A. Kuroda and Jean-Pierre Leburton

ix

x

Contents

26. Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power in One-Dimensional Conductors Marcelo A. Kuroda and Jean-Pierre Leburton

341

27. High-Field Electrothermal Transport in Metallic Carbon Nanotubes 353 Marcelo A. Kuroda and Jean-Pierre Leburton 27.1 Introduction 354 27.2 Model 355 27.2.1 Electric and Energy Flow 355 27.2.2 Electron-Phonon Interaction 357 27.2.3 Electron-Phonon Heat Exchange 361 27.2.4 Heat Flow Diagram 362 27.3 Results 364 27.3.1 Low-Field Regime 364 27.3.2 High-Field Regime 366 27.3.3 Heat Flow 368 27.3.4 Hot Phonons 371 27.3.5 Electrical Power vs Length 373 27.4 Conclusions 374 28. Atomic Vacancy Defects in the Electronic Properties of Semi-metallic Carbon Nanotubes Hui Zeng, Jun Zhao, Huifang Hu, and Jean-Pierre Leburton 28.1 Introduction 28.2 Model 28.3 Results and Discussions 28.4 Conclusion 29. Chirality Effects in Atomic Vacancy–Limited Transport in Metallic Carbon Nanotubes Hui Zeng, Huifang Hu, and Jean-Pierre Leburton 29.1 Introduction 29.2 Model 29.3 Results and Discussion 29.4 Conclusion

379

380 381 383 392 395 396 397 398 404

Contents

30. Vacancy Cluster–Limited Electronic Transport in Metallic Carbon Nanotubes Hui Zeng, Jean-Pierre Leburton, Huifang Hu, and Jianwei Wei 30.1 Introduction 30.2 Model 30.3 Results and Discussions 30.4 Conclusion

31. Vacancy-Induced Intramolecular Junctions and Quantum Transport in Metallic Carbon Nanotubes Hui Zeng, Jun Zhao, Jean-Pierre Leburton, and Jianwei Wei 31.1 Introduction 31.2 Computational Methods and Models 31.3 Results and Discussion 31.4 Conclusions

407

408 409 410 414 417

418 420 423 431

32. On the Sensing Mechanism in Carbon Nanotube Chemiresistors 435 Amin Salehi-Khojin, Fatemeh Khalili-Araghi, Marcelo A. Kuroda, Kevin Y. Lin, Jean-Pierre Leburton, and Richard I. Masel 32.1 Introduction 436 32.2 Results and Discussion 438 32.3 Methods 445 32.3.1 Fabrication and Design of the Chemiresistor 445 32.3.2 SWNT Preparation and Deposition on Silicon Substrate 446

33. Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons 449 Hui Zeng, Jean-Pierre Leburton, Yang Xu, and Jianwei Wei 33.1 Introduction 450 33.2 Model and Methods 451 33.3 Results and Discussions 453 33.4 Conclusion 458

xi

xii

Contents

34. Controllable Tuning of the Electronic Transport in Pre-designed Graphene Nanoribbon 463 Hui Zeng, Jun Zhao, Jianwei Wei, Dahai Xu, and J.-P. Leburton 34.1 Introduction 464 34.2 Model 465 34.3 Results and Discussions 467 34.4 Conclusion 471 35. Quantum Conduction through Double-Bend Electron Waveguide Structures T. Kawamura and J.-P. Leburton

475

36. Quantum Ballistic Transport through a Double-Bend Waveguide Structure: Effects of Disorder 485 T. Kawamura and J.-P. Leburton 36.1 Introduction 486 36.2 Numerical Method 487 36.3 Results and Discussion 491 36.3.1 Ideal Structures 491 36.3.2 Systems with Disorder 498 36.3.3 Multiple Double-Bend Structures 502 36.4 Conclusions 504 37. Quantum Transport through One-Dimensional Double-Quantum-Well Systems T. Kawamura, H. A. Fertig, and J.-P. Leburton

509

38. Cascaded Spintronic Logic with Low-Dimensional Carbon 521 Joseph S. Friedman, Anuj Girdhar, Ryan M. Gelfand, Gokhan Memik, Hooman Mohseni, Allen Taflove, Bruce W. Wessels, Jean-Pierre Leburton, and Alan V. Sahakian 38.1 Introduction 522 38.2 Results 523 38.2.1 Device Structure and Physical Operation 523

Contents



38.3 38.4

Index

38.2.2 Edge Effects and Operation Temperature 525 38.2.3 Switching Behaviour 527 38.2.4 Logic Gates and System Integration 527 Discussion 530 Methods 533 38.4.1 Hubbard Tight-Binding Hamiltonian 533 38.4.2 Diagonalization and the Secular Equation 533 38.4.3 Mean-Field Approximation 534



539

xiii

Preface

He who becomes the slave of habit, who follows the same routes every day, who never changes pace, who does not risk and change the color of his clothes, who does not speak and does no experience, dies slowly. —Martha Medeiros “A Morte Devagar”

The concept of quantum wire emerged in the early 1980s when advances in lithography and fabrication techniques enabled chargecarrier confinement in multiple spatial directions in solid state nanostructures. As ultimate conductors, quantum wires allow electronic carriers to move only along their length, restricting their perpendicular motion into quantum modes as in electromagnetic wave guides. Very early, it was anticipated that the reduction of the carrier phase space into one-dimensional subbands would quench scattering of electrons by impurities and lattice vibrations to enhance carrier transport. New effects associated with the confinement of phonon modes as well as phonon-mediated intersubband scattering combined with structural size reduction opened research avenues to improve the quantum wire functionality and their technological integration into densely packed electronic systems. Later on, the emergence of carbon nanotubes and graphene nanoribbons broadened the field of one-dimensional structures to include molecular semiconductors with anticipated applications in ultra-large electronic integration. This book is a compilation of articles on the theoretical framework developed in the early days of the quantum wire emergence into semiconductor technology. The first part of the book describes impurity and phonon-limited transport within semi-classical as well as quantum approaches to understand the role of confinement on carrier scattering. It emphasizes the use of Monte Carlo techniques

xvi

Preface

to investigate high-field multisubband transport in one-dimensional structures, for which new effects such as anomalous carrier cooling, intersubband resonance, and optic-phonon-induced population inversion are predicted. It addresses topics on the effects of a periodic potential along quantum wires as well as the quantization of the phonon modes resulting from the two-dimensional confinement. The second part of the book is devoted to the theoretical formalism utilized for analyzing carrier transport in one-dimensional molecular structures by capitalizing on models developed for conventional quantum wire structures. It focuses on high-field transport and dissipation in metallic carbon nanotubes by solving the coupled Boltzmann and heat diffusion equations to explain the onset of negative differential resistance in free standing nanotubes. Finally, quantum transport in the presence of atomic defects in graphene nanoribbons and one-dimensional electronic waveguides is presented within the non-equilibrium Green’s function formalism to end up with an elegant proposal for an all-carbon efficient spin logic. Jean-Pierre Leburton Spring 2023

Part I

Semiconductor Quantum Wires

Chapter 1

Size Effects on Polar Optical Phonon Scattering of One-Dimensional and Two-Dimensional Electron Gas in Synthetic Semiconductors

J.-P. Leburton

Coordinated Science Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois 61801, USA [email protected]

The total scattering rate and the transition probability for electronphonon interaction in 1-D and 2-D semiconductor materials are calculated in taking into account the finite dimensions of the structure. Although noticeable, size effects on the scattering rate are generally small, with more pronounced features for 1-D structures than for 2-D structures. For 2-D layers, our theory agrees with recent experimental results whereas it contradicts the previous theory predicting large size effects and mass-independent electron-phonon scattering rates. In 1-D structures singularities in the phonon Reprinted from J. Appl. Phys., 56(10), 2850–2855, 1984. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1984 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

4

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

emission rate appear as a natural consequence of the 1-D density of states. However, for high energy the 1-D emission rate is found smaller than the corresponding 3-D rate. An additional consequence of the confinement is the quenching of the phonon absorption rate.

1.1 Introduction

The remarkable developments of fine line lithography and the new epitaxial technologies of molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) have added astonishing possibilities to the fabrication of new artificial materials. Alternating structures of lattice matched III-V semiconductor compounds, as for example AlxGa1–xAs are presently fabricated with typical thicknesses of 10 nm or less and with accurately controlled doping. In these artificial materials, the layer dimensions are comparable to the de Broglie wavelength λe and electrons are confined in the ultrafine structure. Transport is essentially two dimensional (2-D) and presents new characteristics not exhibited in bulk semiconductors. For example, the mobility parallel to the plane of the layer teas been shown to be improved drastically by modulation doping of the structures and values as high as 106 cm2/V s have been achieved [1]. Recently, an alternative structure based on the confinement of electrons in a “wire” semiconductor has been proposed to improve further the conducting properties of the materials [2]. In these one-dimensional (1-D) III-V synthetic materials, the electron gas is quantized in two transverse directions and the charge carriers can only move in the longitudinal direction. GaAs-AlAs quantum well wires have already been realized by Petroff et al. [3] and other structures based on field effects promise to be feasible in the near future [4, 5]. Because of the limited number of available final states during the scattering process (forward or backward scattering), the 1-D electron mobility is seemingly enhanced and its value has been estimated to be far beyond 106 cm2/V s [6]. The obvious potential of these new structures for high-speed devices makes the knowledge of their transport parameters desirable. Although transport theory in 1-D semiconductor structures is still in its infancy, calculations of the most common scattering mechanisms impurity [6, 7], acoustical

Introduction

phonon [8], and optical phonon [9] have already been performed. However, interaction with polar optical phonons (POP) merits special attention; the large energy exchange between carriers and lattice vibrations influences considerably transport at room temperature. Moreover, the POP interaction is a major scattering mechanism responsible for hot electron effects [10]. So far, the dependence of the POP scattering on the “wire” cross section has not been investigated. Yet, wire parameters appear to be technologically controllable, and owing to device applications, it is of importance to appreciate realistically the confinement effects on the electronPOP interaction. In this chapter we derive the total POP scattering rate in 1-D semiconductor structures and discuss the influence of size effects on the electron-phonon interaction. Furthermore, we reconsider the corresponding 2-D transition probability. 2-D electron-POP interaction represents an important limitation on the high-speed performance of high-electron-mobility transistor (HEMT). Although experimental data show a weakening of the electron-phonon interaction in 2-D structures compared to 3-D materials [11, 12] theoretical models predict a larger scattering rate [13]. This discrepancy has been attributed to the screening of the electron-POP interaction by high-density carriers in 2-D structures. However, we believe that the discrepancy is partially due to crude approximations previously used for the 2-D calculation, e.g., qzLz ≪ 1 (strictly 2-D gas), where qz is the transverse component of the phonon wave vector and Lz is the layer width [14, 15]. Hence, Ridley, introducing the momentum-conservation approximation (MCA) has obtained an expression for the 2-D scattering rate which is mass independent and varies as 1/Lz [16]. Prior to these results, Price obtained a m law for tire 2-D collision rate, the same as for 3-D electron gases [17]. He also has shown that the size effects are weaker than predicted by Ridley. In the following, we clarify this question by calculating exactly the phonon scattering rate for 2-D electron gases. Our theory generally agrees with experimental data from Chiu et al. [12] and shows that size effects lower the POP scattering rate. Screening effects, however, are not considered here and likely account for further lowering of the electron-POP interaction as claimed by Shah et al. [11]. For 1-D electron gas, size effects are shown to have important consequences on the POP emission rate.

5

6

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

1.2 Theory In this chapter, we consider the extreme quantum limit (EQL); i.e., only the lowest subband of the 1-D or 2-D system is occupied. This case is the most interesting for transport problems in practice; moreover, its generalization to higher subband interactions is straightforward. Also, as mentioned in the introduction, screening effects will not be considered here. In the infinite well approximation, the 2-D and 1-D electron wave functions have the well-known expressions

k 2- D Ò =

k 1- D Ò =

e

eik x x Lx

ik r

A

2 py sin Ly Ly

with the corresponding energies

and

E2- D =

E1 - D =

2 pz sin (1.1a) Lz Lz

2 k2 2m

+

2 pz sin , Lz Lz

2p 2

2mL2z

(1.1b)

, (1.2a)

h2k x2 h2p 2 Ê 1 1ˆ + Á 2 + 2 ˜ , (1.2b) 2m 2m ÁË Lz L y ˜¯

in the effective mass approximation. Here A is the layer area, k  is the wave vector component parallel to the layer, Lx is the “wire” length, and we assume that Lx ≫ Ly, Lz (see Fig. 1.1). In the semiclassical transport theory, the transition probability for electron-POP interaction is expressed in the Born approximation by Fermi’s golden rule.

where

e e 2 Ê 2p ˆ W { a } ( k , k ¢ ) = Á ˜ M { a } d ÈÎE ( k ¢ ) - E ( k ) ± hw ˘˚ , (1.3) Ë h ¯

e  e - ph k , N + 1 / 2 ± 1 / 2Ò M { a } = · k ¢ , Nq + 1 / 2 ± 1 / 2 H q

is the matrix element for electron-phonon interaction, e(a) stands for emission (absorption) process, Nq is the phonon occupation

Theory

number, and q is the 3-D phonon wave vector with frequency w. The Fröhlich Hamiltonian is He - ph =



Â(C / q)(a e q

iqz

q

- aq+ e - iqz ), (1.4)

where aq+ and aq are the phonon creation and annihilation operators, respectively, and È 2p Ê 1 1 ˆ˘ C = i Í e2 hw Á - ˜ ˙ Ë •  s ¯ ˙˚ ÍÎ V



1/ 2

. (1.5)

Figure 1.1 Schematic representation of a (a) 2-D and (b] 1-D structure.

Here, V is the crystal volume and • and  s are the optical and static dielectric constants, respectively. Using the wave functions in Eqs. (1.1a) and (1.1b), we obtain the matrix elements for 2-D and 1-D electron systems

(

{e}

M2-a D =  C / q)( Nq + 1 / 2 ± 1 / 2)1/2 d k

¢ || - k|| ± q||

)

F (qz Lz , (1.6a)

Ê ˆ {e} M1-a D =  Á C / q)( Nq + 1 / 2 ± 1 / 2)1/2 d k - k ¢ ± q F (q y L y ˜ F (qz Lz ) . ¢ x x Ë ¯ x (1.6b) The F function is defined as Lz

z F (qz Lz ) = 2 dze - iqz sin2 p z =

Lz

Ú 0

Lz

Ê -iqz Lz ˆ sin(qz Lz / 2) , exp Á Ë 2 ˜¯ (qz Lz / 2) p - (qz Lz / 2) 2

p2

2

(1.7)

7

8

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

and similarly for F (q y L y ). The transition probabilities for 2-D and 1-D electron gases, per unit area and unit length, respectively, are

(

)

2 e W2{-aD} k|| , k||¢ = Lz | C | ÊÁ Nq + 1 ± 1 ˆ˜ I2-D q|| , Lz d k 2

4p  Ë

)

(

2¯

2

¢ || - k|| ± q||

dÈ Î

(

) ( )

E k|| ∓ q|| - E k|| ± w ˘ ˚

,

(1.8a) and

(

{e}

)

W1-aD k x , k ¢x =

Lz L y | C |2 Ê 1 1ˆ Á Nq + 2 ± 2 ˜¯ I1- D qx , L y , Lz d k x - k ¢x ± q x d ÈÎE (k x ∓ q x )- E (k x )± w ˘˚ , 4p 2  Ë

)

(

(1.8b) where



) Ú dq

(

I2- D q|| , Lz =



and

+•

z

-•

| F(qz Lz )|2 q　2 + qz2

(1.9a)

+•

I1–D(qx, Ly, Lz) =

Ú dq

-•

y

| F (q y L y )|2 I2- D (qx , q y , Lz ). (1.9b)

The first integral I2-D can be calculated explicitly (see Appendix):

È1 Ê ˘ 1 ¸Ô p ÏÔ 1 S2 ˆ Lz Ì 2 ÈÎ1 - G ( S )˘˚ + Í + Á 2 - 2 G S , ˙ ( ) ˜ 2 2˝ 2 ÔS S + p2 ¯ ˙˚ S + p ˛Ô ÍÎ 2 Ë Ó (1.10)

(

)

I2- D q|| , Lx =

where

with

G ( S ) = ÈÎ1 - exp ( -2S )˘˚ / 2s ,

The simple expression

S = q Lz / 2 .

I2- D 

p

(1.11) Lz q ˆ Ê q Á 1 + 4 ˜¯ Ë constitutes a very good approximation, having an error of approximately 5% over the range of values of q shown in Fig. 1.2. This expression is similar to Price’s interpolation formula except for the coefficient of Lz/4 [17].

Theory

Figure 1.2 (a) Normalized 2-D integrals I2–D as a function of the normalized phonon wave vector q. Dashed line: strictly 2-D electron gas (Lz = 0). Solid lines: quasi-2-D electron gas with Lz increasing from L0 (1) to 5 L0 (5). L0 = h / 2mw . (b) 1-D integral I1–D as a function of the longitudinal component qx of the phonon wave vector. Numbers between parentheses indicate the wire dimension in units of L0, i.e., (n,m) stands for Ly = nL0, Lz = mL0.

9

10

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

The second integral I1–D has been performed numerically. The two expressions have been calculated for GaAs. Figure 1.2a shows the 2-D integral (10) as a function of the longitudinal component of the phonon wave vector with the transverse dimension Lz as parameter. The 1/ q relation derived by Holonyak et al. [14] is also shown. This relation is obtained in the limit Lz → 0 (strictly 2-D approximation). It should be noticed that for finite values of Lz, the 2-D transition probability is always smaller than the strictly 2-D expression. At small values of q , the 2-D probability is almost independent of the layer size. In this case, the function F achieves its maximum value, i.e., F ≃ 1. For large q , size effects are noticeable and nearly one order of magnitude separates the 1/ q relation from the lowest Eq. (1.5) of our curves [18]. Interactions between the confined electron wave function and the transverse component qz of the 3-D phonon field occur because the quantum well provides the required momentum qz. In the limit qz → 0, the transverse interaction is transferred to the quantum well under a translation motion which yields a maximum electron-phonon interaction. On the contrary, at short wavelengths, the transverse phonon field oscillates very rapidly over the confinement distance Lz which results in a vanishing interaction. Therefore, size effects reduce the transition probability with respect to the strictly 2-D approximation. Moreover, the larger Lz, the smaller the electron-phonon interaction. This effect is enhanced in 1-D structures because of the twofold influence of the transverse dimensions Ly and Lz. Although the dependence of the 1-D transition probability on q is smoother than the corresponding 2-D expression, size effects are more important in the 1-D case (Fig. 1.2b). Hence, for q = qz = 1/ L0, with L0 = 2mw / h , (i.e., Qz = Q = 1 in the figures); I2–D is barely reduced by a factor of 2 whereas I1–D decreases by nearly a factor of 5 when the transverse dimensions Ly,Lz increase from L0 to 5L0. It should be noticed that the strictly 1-D approximation leads to a divergent 1-D transition probability which obviously is unphysical and thus pointless. I1–D for rectangular “wire” structures is also shown in Fig. 1.2b. An important rule of thumb is that size effects on rectangular wires are almost identical if the sum S = Lz + Ly is constant; for example, curves (1,3) and (2,2) on Fig. 1.2b are very similar. Other combinations, for example, (1,5), (2,4), and (3,3), though not shown on the figure, satisfy the same rule.

Theory



The total scattering rate in either case is written as

| C |2 V2- D Ê 1 1ˆ È ˘ Á Nq + 2 ± 2 ˜¯ df dqq I2- D q , Lz d ÎE k ∓ q - E ( k ) ± w ˚ , 4p 2  Ë

( ) ( ) Ú Ú k ) ( (1.12a) 1

{e} t 2-a D

=



1

{e} t 1-a D

(k ) x|

=

| C |2 V1- D Ê 1 1ˆ Á Nq + 2 ± 2 ˜¯ dqx I1- D qx , L y , Lz d ÈÎE (k x ± qx ) - E (k x ) ± hw ˘˚ , 4p 2 h Ë

Ú

)

(

(1.12b) where f is the angle between k  and q and V2–D (V1–D) is the volume of the layer (of the “wire”). Here distinctions should be made between emission and absorption. Because of energy and momentum conservation, allowed phonon wave vectors have different expressions depending on the interaction process. In 2-D systems the emission process has

2mw , (1.13a)  whereas for the absorption process, the phonon wave vector is determined by

q±e = k cosf ± k2cos2f -



q±a = -k cos f ± k2cos2f +

2mw , (1.13b) 

In the latter q-a is forbidden; in the former both q±e are permitted but for 0 < f < f max and 2p – f max < f < 2p, with

fmax = arccos hw / E and E = h2k　2 / 2m .

Similarly, for 1-D systems we obtain

and

q±e = kx ± kx2 - (2mw / h ) (1.14a)

q±a = -kx ± kx2 + (2mw / h ) (1.14b)

Finally, the total 2-D scattering rate has the following expression: fmax

aw = ( Nq + 1) e t 2- D ( E ) p 1

Ú df 0

I2- D ÈÎq+e (E ,f )˘˚ q+e (E ,f ) + I2- D ÈÎq-e (E ,f )˘˚ q-e (E ,f ) E cos2f - 1 hw

,

(1.15a)

11

12

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas



1 t 2a- D (E )

=

p I2a- D ÈÎq+a (E ,f )˘˚ q+a (E ,f ) aw . (1.15b) Nq df p E 2 0 cos f + 1 hw

Ú

For the 1-D rate we obtain similarly 1

t 1e- D (E )

=

I1e- D ÈÎq+e (E )˘˚ + I1e- D ÈÎq-e (E )˘˚ aw , (1.16a) ( Nq + 1) 2p E -1 hw 1

t 1a- D (E )

=

I1- D ÈÎq-a (E )˘˚ aw , (1.16b) Nq 2p E +1 hw

where a is Fröhlich’s coupling constant given by [19]

a=

Ê 1 1ˆ 1 e2 / hw . (1.17) 2 h / 2mw ÁË •  s ˜¯

Figure 1.3 shows the scattering rates for 2-D and 1-D GaAs structures. In Fig. 1.3a the 2-D emission rate exhibits the well-known steplike behavior at E = hw characteristic of the 3-D density of states. Although noticeable, size effects in 2-D structures are generally small; the variation of the scattering rate is barely a factor of 2 when Lz increases from L0 to 5 L0. This weak influence of the size effect stems from the fact that the main contribution of the I2-D integral to the total rate occurs at q = 1/L0, where I2-D presents a weak Lz dependence (see Fig. 1.2a). From Eq. (1.11) we obtain a useful expression for the scattering rate at the onset of phonon emission and absorption:

1

t 2e- D (E

= É )

 paw ( Nq + 1)

1 t 2a- D (E

= 0)

 paw Nq

1 , (1.18a) 1 + Lz / 4L0

1 , (1.18b) 1 + Lz / 4L0

having an error of less than 5%. Notice the through a.

m dependence

Theory

Figure 1.3 Phonon absorption and emission scattering rates as a function of the electron energy for GaAs at 300 K. (a) 2-D structures with Lz as a parameter increasing from L0(2) to 5L0(5). (b) 1-D structures with Lz = Ly = L0 (1,1) to Lz = Ly = 5L0 (5,5) as parameters. Dashed line: 3-D case.

It is also observed that for thick layers the 2-D scattering rate is smaller than the 3-D rate. This effect is mainly pronounced for

13

14

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

absorption and, as mentioned previously, results from the weakening of the transverse component of the electron-phonon interaction. Without being directly related to the carrier mobility (neither is the momentum relaxation time for POP), the scattering rate generally indicates the POP influence on the mobility. Therefore, our results show a reasonable agreement with recent experimental observation by Chiu et al. [12] who find a 2-D POP limited mobility higher than for 3-D electrons. From a general standpoint, our theory contradicts Ridley’s conclusions on the importance of size effects as well as on the strength of the POP scattering in 2-D gas. In our opinion this discrepancy comes from Ridley’s MCA which overestimates the role of the confinements. Dimensionality effects are enhanced in 1-D structures (see Fig. 1.3b). The phonon emission rate exhibits a singularity at E = hw as a consequence of the 1-D electron density of states. Thus, strong energy relaxation via phonon emission is expected to characterize transport in 1-D structures. However, as seen from the figure, this mechanism is limited to a short range of energy and at higher carrier energy the emission rate drops below the corresponding 3-D rate. This behavior is particularly noticeable for wider transverse dimensions and seems to preclude the possibility of important velocity runaway [20]. On the other hand, because of the double influence of confinement, size effects are larger than in 2-D structures. Hence, the scattering rates for both absorption and emission are reduced by a little less than a factor of 3 when the transverse dimensions Ly, Lz increase from L0 to 5L0. Moreover, an additional consequence of the reduced dimension of the structure is the large difference between the emission and absorption processes. This is particularly observable in the absorption rate which even for the thinnest 1-D wire is always lower than the corresponding 3-D rate. This special effect is characteristic of the difference in the tail of the 1-D density of states for phonon emission (~ 1 / E - hw ) and absorption (~ 1 / E + hw ) . Again, our theory yields lower scattering rates than those predicted by Ridley’s model [9]. For Ly = Lz = 5L0, for example, his model gives a 1-D emission rate similar to our (1,1) curve, whereas, the corresponding absorption rate is almost equal to the bulk value, which is approximately three times higher than our results.

Conclusion

1.3 Conclusion Size effects on the unscreened electron-POP interaction have been discussed in the EQL. For a 2-D electron gas, a rather weak dependence of the POP scattering rate on the layer thickness has been determined. For large layer thickness, the 2-D POP scattering rate is shown to be smaller than the corresponding 3-D rate. Generally, our results agree with experimental data and the Price model. In 1-D systems, size effects and the influence of the density of states are more pronounced than in 2-D systems. low absorption rates, singularities in the emission rates, and the possibility of a large velocity runaway are the characteristics of the 1-D electron POP interaction.

Acknowledgments

This work was supported by the Army Research Office and the Joint Service Electrical Program. The author would like to acknowledge the fruitful discussions with and encouragements of Professor K. Hess and Dr. J. Shah. He is also indebted to K. Kahen for critical comments and Mrs. E. Kesler for typing the manuscript.

Appendix

The function θ (Z) = F2(Z) can be written as

Ê 1 3 1 3 1 1 1 1 1 ˆ q ( Z ) = sin2 Z Á 2 + + + + ˜, 4p (p + Z ) 4p (p - Z ) 4 (p + Z )2 4 (p - Z )2 ˜¯ ÁË Z (1.A1)

such that the I2-D integral

I2- D

L = z 2

+•

dZq( Z )

ÚS

2

-•

is now a sum of five simpler integrals

I2- D

L = z 2

+ Z2

5

(1.A2)

 J , (1.A3) i =1

i

15

16

Size Effects on Polar Optical Phonon Scattering of 1-D and 2-D Electron Gas

where

+•

+•

Ê sin2 Z 1 cos 2Z ˆ dZ Á 2 + ˜. 2 2 Z ( S + Z ) S -• Ë Z 2( S + Z ) 2( S 2 + Z 2 ) ¯ -• (1.A4) J1 =

Ú

dZ sin2 Z 2

2

2

=

1

Ú

2

The first term J11 in the brackets is equal to π, the second one J12 is equal to π/2S and the third one J13 is equal, to (π/2S)e–2S.

3 4p

J{ 2 } = 3

+•

+•

dZ sin2 Z

Ú (p ± Z )(S

2

-•

+ Z2 )

=

Ú

3 1 dZ sin2 2 4p p + S 2 (1.A5) -•

Z ˆ p Ê 1 + ± . ZÁ Ë p ± Z S 2 + Z 2 S 2 + Z 2 ˜¯

The first terms J21 and J31 and the third terms J23 and J33 vanish by symmetry. The second terms J22 and J32 are equal to (p / 2S )(1 - e -2S ) according to J1. J{ 4 } = 5



1 4

+•

dZ sin2 Z

Ú (p ± Z ) (S 2

-•

2

+ Z2 )

=

1 1 2 4 ( S + p 2 )2

+•

Ú dZ sin

-•

Ê 2 S +p p -S 2p Z ˆ ZÁ + + 2  2 ˜. 2 2 S +Z S + Z2 ¯ Ë p ± Z (p ± Z ) 2

2

2

2

2

(1.A6)

Again J41, J51, J44, and J54 vanish by symmetry; whereas, except for a multiplication, J42 and J52 are equivalent to J11 in the same way J43 and J53 are equivalent to J22 and J32. After the summation (1.A3) we obtain finally Eq. (1.10).

References

1. H. L. Störmer, A. G. Gossard, and W. Wiegemann, Appl. Phys. Lett. 39, 912 (1981). 2. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

3. P. M. Petroff, A. C. Gossard, R. A. Logan, and W. Wiegemann, Appl. Phys. Lett. 41, 635 (1982).

4. A. B. Fowler, A. Harstein, and R. A. Webb, Phys. Rev. Lett. 48, 196 (1982). 5. W. J. Skocpol, L. D. Jackel, E. L. Hu, R. E. Howard, and L. A. Fetter, Phys. Rev. Lett. 49, 951 (1982).

References

6. H. Sakaki, Proceedings of the International Symposium on GaAs and Related Compounds, 1981, Oiso, Japan, Inst. Phys. Phys. Soc. Conf. Ser. (Bristol, 1982), p. 251. 7. J. Lee and H. N. Spector, J. Appl. Phys. 54, 3921 (1983). 8. V. K. Arora, Phys. Rev. B 23, 5611 (1981).

9. F. A. Riddock and B. K. Ridley, Proceedings of the 5th International Conference on Electronic Properties of 2-D Systems, Oxford, England (1983). 10. K. Hess in Physics of Non Linear Transport in Semiconductors, edited by D. K. Ferry, J. R. Barker, and J. C. Jacoboni (Plenum, New York, 1980), p. 1. 11. J. Shah, A. Pinczuk, H. L. Störmer, A. C. Gossard, and W. Wiegemann, Appl. Phys. Lett. 42, 55 (1983). 12. L. C. Chiu, S. Margalit, and A. Yariv, Jpn. J. Appl. Phys. 22, L82 (1983). 13. K. Hess, Appl. Phys. Lett. 35, 484 (1979).

14. N. Holonyak, Jr., R. M. Kolbas, W. D. Laidig, B. A. Vojak, K. Hess, R. D. Dupuis, and P. D. Dapkus, J. Appl. Phys. 51, 1328 (1980). 15. J. B. Roy, P. K. Basu, and R. B. Nag, Solid State Commun. 40, 491 (1981). 16. B. K. Ridley, J. Phys. C 15, 5899 (1982). 17. P. J. Price, Ann. Phys. 133, 217 (1981).

18. We limit our values of quantum well width to 5L0 in order to respect the EQL. For larger values of Lx, the separation between quantized levels becomes smaller than hw ~ kT (300 K) and intersubband scatterings take place. 19. R. P. Feynman, Statistical Mechanics: A Set of Lectures (Benjamin, New York, 1972), p. 221. 20. W. Fröhlich, Proc. R. Soc. London Ser. A 188, 532 (1947).

17

Chapter 2

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

S. Briggs,a,b B. A. Mason,c and J.-P. Leburtona,b aBeckman

Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA bDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA cDepartment of Physics and Astronomy, University of Oklahoma, 440 West Brooks, Room 131, NH, Norman, OK 73019, USA [email protected]

We calculate the polaron self-energy in quantum-wire structures. We use the Fock approximation and consider the interaction of electrons and polar optic phonons in GaAs wires of different sizes at 300 K and solve for the self-energy iteratively. The result S(E, k) is presented as a function of both E and k. We compare ImS with a simple first-order calculation using Fermi’s “golden rule” to investigate the importance Reprinted from Phys. Rev. B, 40(17), 12001–12004, 1989. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1989 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

20

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

of higher-order quantum effects. A constant broadening of the density of states has been included in Fermi’s “golden rule” and is successful in reproducing the correct scattering rate. The investigation of quasi-one-dimensional (quasi-1D) artificial structures is a rapidly growing field chiefly stimulated by the considerable advances in the fabrication of highly confined electronic systems. New quantum phenomena have been observed at temperatures above 4.2 K [1]. With these new developments, transport with dissipation [2] and electron-phonon interaction is a subject of recent attention. In III–V compounds, high-energy transport is essentially limited by interactions with polar optic phonons (POP’s). In general, phonon scattering rates in transport simulations are computed according to Fermi’s “golden rule” [3]

2p | M( k i , k f )|2d k i - k f ± q (2.1) h ¥ d ( E ( k i ) - E ( k f ) ± hw ) ,

W (k i , k f ) =

where W(ki, kf) is the transition probability from an initial electron state ki to a final state kf, M(ki, kf) is the corresponding matrix element for the transition, E(ki) is the initial electron energy, E(kf) is the final electron energy, q is the phonon wave vector, and hw is the phonon energy. Fermi’s “golden rule” is only valid if the scattering rate is low enough so that the scattering events are spatially and temporally independent. W(ki, kf) is then integrated over all final states to obtain the total scattering rate as a function of the initial electron energy. This introduces a density of states term D(Ei ± hw),

D(E i ± hw ) =

ˆ 1Ê m* Á h Ë 2(E i ± hw ) ˜¯

1/ 2

(2.2)

with m* the effective mass. The functional dependence of D(E) is a consequence of the semiclassical expression of energy conservation which ignores any quantum correlations between scattering events. Clearly, this term diverges for E(ki) = hw. A more accurate description of the electron-phonon interaction considers the self-energy S(k, E) which can be calculated using the Fock approximation [4]. This approximation is a general method which includes higher-order quantum effects (including collisional broadening) in the electron-phonon interaction. In 3D

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

calculations the strong q dependence of the matrix element makes a self-consistent treatment of POP’s difficult [5]; however, the slowly varying q dependence in 1D systems make the problem tractable. In this Rapid Communication, we include the correct q dependence in the matrix element and employ an iterative method to solve the Fock approximation self-consistently. This approach gives the full k and E dependence of the self-energy S exactly. We perform this calculation for several different sized GaAs quantumwire structures where the confinement is due to an infinite square well in one direction and a semi-infinite triangular potential in the other direction. The calculation is performed at 300 K and considers a single subband with inelastic POP scattering. We then compare our results for ImS with the rates obtained from Fermi’s “golden rule.” In 1D systems, the Fock approximation for the self-energy S is given by S (k , E ) =



Ú

ge2 (k - q) dq 2p E - hw - e (q) - S(q , E - hw )

ga2 (k - q) dq , + 2p E + hw - e (q) - S(q , E + hw )

Ú

(2.3)

where ge2 and ga2 are the electron-phonon coupling constants corresponding to mission and absorption processes which include the 1D form factors, k and q are scalars corresponding to the components of the electron and phonon wave vectors parallel to the wire, and e is the electron dispersion relationship (assuming the effective-mass approximation). This equation is then solved selfconsistently for S. Because of the divergence in the denominator of Eq. (2.3), bruteforce integrations (which must be done numerically) introduce significant round-off error. To circumvent this problem, we make two assumptions that allow us to solve a simpler integral analytically then iterate to solve the resulting analytic equation for S. We note that if g2 is a constant, S is a function of only E (Ref. [6]) and Eq. (2.3) can be integrated analytically. By replacing g2(k – q) in Eq. (2.3) by a value independent of q, g′2(k), we still preserve much of the structure of g2. Our second simplification is to replace S(q, E – hw) with S(k, E – hw). With these assumptions, Eq. (2.3) becomes

21

22

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

S 0 (k , E ) = g '2 (k , E )

Ú

¥ dq

1 E - hw - ( h q / 2m*) - S 0 (k , E - hw ) 2 2

,

(2.4)

which can be integrated analytically (for the sake of clarity, we have omitted a corresponding term for phonon absorption). Since g2 varies by several orders of magnitude over the q range of interest, it is of vital importance to choose a relevant value for g′2(k, E). We define

where

g¢2 (k , E ) = g2 (k - q - (E )) + g2 (k + q - (E )), q - (E ) =

[2m *(E - hw )]1/2 . h

(2.5a)

(2.5b)

The rationale behind this choice of g′2 is the following: When we evaluate S(k, E) on the mass shell to obtain S(E),

Ê (2m * E )1/2 ˆ S (E ) = S Á , E ˜ = S(k , e (k )), (2.6) h Ë ¯

then g2(k, E) becomes equal to the classical coupling constant, which has the form g2(qp+) + g2(qp–) where qp± are the classical phonon wave vectors associated with forward and backward emission. We solve Eq. (2.4) including absorption and emission over an energy range of –30–80 meV and use 150 k values to obtain S0(k, E) at room temperature. We can then interpolate between k values to obtain S0(E). Equation (2.4) converges after approximately 30 algebraic iterations; its output is then used as input to a numerical integration program which solves Eq. (2.3) exactly. The output from the analytic stage of the calculation is very close to the final result, therefore Eq. (2.3) converges after only two interations. In Fig. 2.1, we show the imaginary part of the self-energy along the classical energies E = e(k). The confinement conditions correspond to a square-well width of 135 Å and a triangular gate field of 120 kV/cm. (We have also computed S for a square-well width of 215 Å and a triangular gate field of 29 kV/cm.) The three curves shown are S0 [i.e., the result of iterating Eq. (2.4)], S1 [the result obtained by substituting S0 into the integrand of Eq. (2.3)], and S which is obtained by repeated iterations on Eq. (2.3). As can be seen

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

from Fig. 2.1, ImS0 has the correct shape but exceeds the final result by about a factor of 2. The main discrepancies between ImS0 and ImS are due to the specific choice of g′2 which is independent of q in S0. In particular we use g′2 = 0 below E = hw because there is no way to define a classical phonon wave vector for emission in this region. This results in a sharp peak at E = hw without broadening below the phonon energy.

Figure 2.1 Convergence of ImS(E). Dotted line: S0 [the result of iterating Eq. (2.4)]. Dashed line: S1 [the result of inserting S0 into Eq. (2.3)]. Solid line: S [the result after two iterations on Eq. (2.3)].

In Figs. 2.2a and b, we show the results for ReS and ImS plotted as functions of both k and E. Although we calculate S over an energy range of –30–80 meV and a k range from 1 ¥ 103 cm–1 to 1 ¥ 108 cm–1 the plot covers a smaller area due to convergence problems near the borders. The energy axis ranges linearly from zero to 140 meV and the k axis varies logarithmically from 1 ¥ 105 cm–1 to 2 ¥ 107 cm–1. The plots reveal two branches in the self-energy, an electronlike branch and a phononlike branch. The sharp discontinuity just above E = hw is particularly noticeable in ReS (for ImS the discontinuity is just below the emission threshold and not visible in Fig. 2.2) and corresponds to the phononlike branch. The electron branch starts near E = hw and curves towards larger k and E, roughly following E = e(k). The jagged peaks in ImS along this branch are artifacts from the finite mesh and plotting package employed. The large peaks in both the real and imaginary parts occur at the phonon energy;

23

24

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

however, they do not occur for the same k value nor do they lie along E = e(k). The peak in ReS corresponds to the minimum value of the denominator in Eq. (2.3) while g2(k – q) reaches its peak value. From ReS we can see that there are states away from the classical regime but the lifetime of these states is short due to the large value of ImS. Further information above the excitation spectrum can be obtained from the spectral density r(k, E),

r (k , E ) =

-2Im S(k , E )

[E - e (k ) - Re S ]2 + (Im S )2

. (2.7)

Figure 2.2 Plot of S(k, E). The energy axis ranges linearly from 0 to 140 meV and k ranges logarithmically from 1 ¥ 10 cm–1 to 2 ¥ 107 cm–1. (a) ImS(k, E); (b) –ReS(k, E). The phononlike branch at E = hw and the electronlike branch along E = e(k) are visible in both plots.

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

r(k, E) is interpreted as the probability of the electron to have the energy E while in state k [7]. Figure 2.3 is a contour plot of r as a function of both k and E. It is obvious from the plot that r is negligible except along the classical dispersion E = e(k). Although there are states away from this dispersion, they have short lifetimes due to the high scattering rate. The electron can thus be described in classical (quasiparticle) terms.

Figure 2.3 Contour plot of the spectral density r(k, E). Energy ranges linearly from 0 to 140 meV and k ranges linearly from 0 to 6 ¥ l06 cm–1. The isolated peaks in r are due to the finite mesh employed in the plot. The inset shows a perspective view of r from the same viewing angle.

In Fig. 2.4 we compare the self-consistent scattering rate 2ImS/h, with the rate obtained from Fermi’s “golden rule” and a modified golden rule calculation where a broadening factor is used to “fudge” the quantum broadening effects in the final density of states. We perform a convolution of the density of states with a Gaussian [8],

(E ) =

Ê m *ˆ Á ˜ hG 2p Ë 2 ¯ 1

1/ 2

Ú

•

-E

e-E ¢

2

/2G 2

1

E¢ + E

dE ¢ , (2.8)

where the amount of broadening G is set equal to 2.5 meV. As can be seen, both classical rates are good approximations away from the peak. The broadening smooths out the divergence in the scattering rate at the peak and allows for a much better fit there; it has a negligible effect on the rates away from the peak.

25

26

Self-Consistent Polaron Scattering Rates in Quasi-One-Dimensional Structures

Figure 2.4 Comparison of S with rates obtained from Fermi’s “golden rule.” Solid line: 2ImS/h. Dashed line: rates from Fermi’s “golden rule.” Dotted line: Fermi’s “golden rule” using a density of states broadened by 2.5 meV.

We have computed S and r and find that the spectral density function is essentially independent of the external potential for the different conditions considered. The self-energy is a function of the confinement, but its general shape remains unchanged. In addition, a good agreement can still be obtained between ImS and the scattering rate from Fermi’s “golden rule” by use of a 2.5 meV broadening factor. In conclusion, we have calculated the self-energy S(k, E) for the electron-phonon interaction in 1D systems. Although S has a phononlike branch and shows significant features away from the classical dispersion, the spectral function is essentially zero away from the noninteraction energies which implies that the system can be described adequately in terms of a quasiparticle model. We have compared ImS to the rates from Fermi’s “golden rule” and find the semiclassical result to be an excellent approximation away from the peak; a constant broadening factor in the final density of states accounts for the general feature of quantum effects in the scattering rates at the peak. We thank Karl Hess for helpful discussions. This work was supported by the National Science Foundation under grant no. NSFECS-85-10209, and the Joint Services Electronics Program.

References

References 1. K. Ismail, D. Antoniadis, and H. Smith, Appl. Phys. Lett. 54, 1130 (1989). 2. S. Datta and M. McLennan, in Nanostructure Physics and Fabrication, edited by M. A. Reed and W. P. Kirk (Academic, Boston, 1989). 3. E. M. Conwell, High Field Transport in Semiconductors (Academic, New York, 1967).

4. D. Pines, in Polarons and Excitons, edited by C. G. Kuper and G. D. Whitfield (Plenum, New York, 1963), p. 155. 5. K. Kim, B. Mason, and K. Hess, Phys. Rev. B 36, 6547 (1987).

6. Y. C. Chang, D. Z.-Y. Ting, J. Y. Tang, and K. Hess, Appl. Phys. Lett. 42, 76 (1983). 7. G. D. Mahan, Many-Particle Physics (Plenum, New York, 1981).

8. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988) [in the reference, Eq. (2.5) should show the correct limits of integration, not –• to +•].

27

Chapter 3

Plasmon Dispersion Relation of a Quasi-One-Dimensional Electron Gas

Jin Wanga and J.-P. Leburtonb aDepartment

of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 1406 West Green Street, Urbana, Illinois 61801-2991, USA bThe Beckman Institute for Advanced Science and Technology, 405 North Mathews Avenue, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

The plasmon dispersion relation of a quantum wire is derived by solving Maxwell’s equations for an anisotropic dielectric waveguide including retardation effects. In the long-wavelength limit, and for the extreme quantum limit, the group velocity of the one-dimensional electron gas is found to be finite and given by the Fermi velocity. There has been increasing interest in one-dimensional electron systems (1DES’s) since quantum-wire structures have been fabricated with GaAs surrounded by AlxGa1–xAs by Petroff et al. [1]. From a fundamental viewpoint, 1DES’s are important since they Reprinted from Phys. Rev. B, 41(11), 7846–7849, 1990.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1990 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

30

Plasmon Dispersion Relation of a Quasi-1D Electron Gas

constitute one of the simplest many-body systems of interacting fermions with properties basically different from three-dimensional particle systems. For this reason, a number of theoretical papers have attempted to describe the dielectric response and the collective excitations of 1DES’s (Refs. [2–4]) in relation to the electronic properties of quasi-1D metal or linear chains of organic conductors [5]. The emergence of low-dimensional artificial semiconductor structures has stimulated further work in this direction. In a recent experiment Demel and coworkers [6, 7] have investigated the farinfrared (FIR) response of a multiple 1D semiconductor structure at low-temperature measurements and interpreted the FIR resonances as caused by the lateral quantization of the 2D plasmon modes. Meanwhile, a theoretical model of screening effects and elementary excitations in artificial 1DES’s has been provided by Das Sarma and Lai [8], who calculated the dielectric functions ϵ(q, w) for single quantum wires and 1D superlattices within the BohmPines random-phase approximation (RPA). These authors obtained the plasmon dispersion relation by solving the standard equation ϵ(q, w) = 0. As a consequence of this model, however, the plasmon group velocity diverges logarithmically with the wave vector q Æ 0 [9] as well as with the radius of the quantum wire r0 Æ 0. In this chapter, we derive the plasmon dispersion relation by considering the collective excitation as an induced electromagnetic wave which obeys Maxwell’s equations and satisfies continuity conditions at the boundary between the confined structure and the surrounding materials [10]. Generally, 1DES’s are laterally and electrostatically confined at the interface of a modulation-doped structure. For the sake of simplicity, the physical system investigated here is a GaAs quantum wire of cylinder geometry embedded inside an AlxGa1–xAs material for which the 1DES eigenstates are given by a simple 2D harmonic oscillator [11]: h— 2 1 + m *( x 2 + y2 )W2 . (3.1) 2m * 2 Here m* is the effective mass, W is the oscillator characteristic frequency, and —2 is the three-dimensional Laplace operator. The eigenfunctions are given by

H0 = -

Plasmon Dispersion Relation of a Quasi-1D Electron Gas



fnmk z (r ) =

1 L

y n ( x )y m ( y )e - ikz z . (3.2)

In the self-consistent-field approximation [2], the self-consistentfield dielectric function ϵSCF for GaAs is given by

SCF ·v ¢ | VS| v Ò , (3.3) =1 ·v ¢ | V| v Ò



where n is a collective label for the (n,m,kz) indices, ϵ is the GaAs dielectric constant, and VS is the self-consistent potential. Starting from the single-particle Liouville equation, we get the potential induced by a test charge, K 0 (q | r - r ¢ |) e2 2p 0L

Ú

VS = dr ¢



¥

 v ,v ¢

·v ¢ | r1| v Òfv* (r ¢ )fv ¢ (r ¢ ),

(3.4)

where K0(x) is the zeroth-order modified Bessel function of the second kind with argument x; q is the z component of the wave vector for the Fourier component of the potential, r1 is the operator describing the perturbation of the density matrix, and ϵ0 is the dielectric constant in the vacuum. At 0 K, if we assume the extreme quantum limit (EQL), i.e., all the electrons occupy the ground state, the 1D dielectric function is given by

where

SCF e2 =1· K 0 ÒF (q ,w ), (3.5) 2p 0 

F (q , w ) =

 k ,s



f0 (E k+q ) - f0 (E k ) E k+q - E k - hw 2

2

q ˘ È m * w ˘ (3.6) È ÍkF - 2 ˙ - Í hq ˙ m* Î ˚ Î ˚ = 2 ln 2 2 ph q È q ˘ Èm *w ˘ + k Í F 2 ˙ Í hq ˙ Î ˚ Î ˚

31

32

Plasmon Dispersion Relation of a Quasi-1D Electron Gas

with the Fermi wave vector kF. The form factor

·K 0 Ò =

1 2

Ú

•

0

e -q t , (3.7) t + 1 / 2h 2

dt

with h = m *W/h, is obtained after some tedious algebraic and integral manipulations [11]. At 0 K, these results are justified if h2kF2 / 2m* < hW . For high temperatures and a large radius of the quantum wire, higher energy levels need to be considered [12]. But this condition, which also involves broadening effects, is beyond the scope of this chapter. In order to derive the dispersion relation of a plasmon wave, we consider the GaAs quantum wire as a cylindrical dielectric waveguide embedded inside AlxGa1–xAs material with a dielectric constant ϵ′. For a wave traveling along the z direction, the electric field E and the magnetic field B must satisfy the Maxwell equations [13]

—¥E= -

—¥H=

∂B , (3.8a) ∂t

∂D + J, (3.8b) ∂t

with B = m0H and D = ϵ0ϵE for GaAs (D = ϵ0ϵ′E for AlxGa1–xAs). Here we assume that an induced current flows along the z direction with unit vector z ; the current density is given by J = 0 c (∂Ez / ∂t )z inside the waveguide, where c is the polarization coefficient. Therefore, for  GaAs the dielectric function  becomes a tensor, È 0 0 ˘  Í ˙  = Í0  0 ˙ , (3.9) ÍÎ0 0 z ˙˚  where ϵz = ϵ + c, and we can redefine D¢ =  ◊ E. In the region outside the waveguide, the current density J is identically zero. Hence we can solve Eq. (3.8) separately in the inside and outside regions.

Plasmon Dispersion Relation of a Quasi-1D Electron Gas



Inside the core, we assume the following form for the E field:

È E J (a r ) ˘ Í r0 1 ˙ E = ÍEf 0 J1 (a r )˙ eiwt - iqz , (3.10) Í ˙ Î Ez0 J0 (a r )˚ where Jn is the Bessel function of the first kind, q and a are the z and radial components of the wave vector for the E field, respectively, and Er0, Ef0, and Ez0 are the magnitudes of the radial, angular f, and z components of the field. From the Maxwell equations, we obtain the displacement current

D¢ =

1

w m0 2

— ¥ (— ¥ E), (3.11)

 and from the constitutive relation D¢ =  ◊ E, we derive the matrix equation

˘ È E ˘ È0˘ Èw 2 m00 - q2 0 -iqa Í ˙ Í r0 ˙ Í ˙ Í 0 0 w 2 m00 - (q2 + a 2 ) ˙ ÍEf 0 ˙ = Í0˙ . Í ˙ 2 2˙Í 0 w m00 - a ˙˚ Î Ez0 ˚ ÍÎ0˙˚ iqa ÍÎ (3.12) In order to obtain nontrivial solutions, the determinant of the matrix on the left-hand side must vanish, which yields the dispersion relations for longitudinal waves,

w2 =

a 2 + q2z , (3.13) m00z

and that for transverse waves, w2 = (a2 + q2)/m0ϵ0ϵ. For plasmon oscillations, only the longitudinal waves are relevant. The solutions of the electromagnetic fields are found to be

˘ È Í J (a r ) ˙ 1 Í ˙ ˙ eiwt - iqz , (3.14) E = E0 Í 0 Í ˙ Í-i a J (a r )˙ 0 ÍÎ z q ˙˚

33

34

Plasmon Dispersion Relation of a Quasi-1D Electron Gas



0 È ˘ Í ˙  w H = E0 Í 0 J1 (a r )˙ eiwt - iqz , (3.15) Í q ˙ Í ˙ 0 Î ˚

where E0 is a constant. Outside the 1DES, the wave is evanescent and must decay with the distance r away from the wire. We thus can choose the modified Bessel functions of the second kind, K0(br) and K1(br), where b is the radial decay constant. The solution is found to be



È ˘ Í K (br ) ˙ Í 1 ˙ iwt - iqz E = E0¢ Í 0 , (3.16) ˙e Í ˙ Íi b K ( b r )˙ ÍÎ q 0 ˙˚

0 ˘ È Í ˙  ¢ w H = E0¢ Í 0 K 1 ( b r )˙ eiwt - iqz , (3.17) Í q ˙ Í ˙ 0 Î ˚

with w2 = (q2– b2)/m0ϵ0ϵ′, and E′0 is a constant. At the wire boundary r = r0, the normal component of D, the tangential component of E and H must be continuous. For nontrivial solutions of E0 and E′0, we obtain the dispersion relation

a J0 (a r0 ) b K 0 ( b r0 ) + = 0. (3.18) z J1 (a r0 )  ¢ K 1 ( b r0 )

When the current density J = 0, this dispersion relation results in the solution of the classical dielectric waveguide [13]. Notice that Eqs. (3.13) and (3.17) describe retardation effects which are consequently included in Eq. (3.18). The longitudinal plasmon modes are obtained by solving Eqs. (3.5) and (3.18) with ϵSCF = ϵZ. Figure 3.1 shows a comparison between the dispersion relation derived from the relation ϵSCF = 0 and our results for two different confinement conditions. For our

Plasmon Dispersion Relation of a Quasi-1D Electron Gas

Figure 3.1 Calculated plasmon dispersion relation of a quantum wire. The solid line describes our numerical result, while the dashed line results from ϵSCF = 0. (a) The radius r0 is 70.7 Å and the Fermi wave vector kF is 1.0 ¥ 106 cm–1, (b) r0 is 353 Å and kF is 5.0 ¥ 106 cm–1.

35

36

Plasmon Dispersion Relation of a Quasi-1D Electron Gas

model, we have chosen m* = 0.067me, ϵ = 13.2 for the inside material, and ϵ′ = 12.51 for the surrounding material. The results are, however, not very sensitive to the difference between ϵ and ϵ′, so that the waveguide approximation seems to be justified. From Fig. 3.1 we can see as q Æ 0 the two curves are somewhat different, but start to converge as q Æ •. This is due to the constraints imposed by the Maxwell equations and the dispersion relation (3.18). At small radius r0 and long wavelength, the modified Bessel functions of the second kind, K0 and K1, and the Bessel functions J0 and J1 are all positive. Therefore in order to obtain a solution from Eq. (3.18), the dielectric function ϵz must be negative. The longer the wavelength, the larger the absolute value of ϵz and, consequently, the larger the deviation from the ϵSCF = 0 solution. For large q, the ratio between the modified Bessel functions approaches unity while the ratio between J0 and J1 is of the same order. So the two curves begin to converge and Eq. (3.18) reduces to the standard self-consistent-field result ϵSCF = 0. In fact, as we can see from Fig. 3.1b, the two curves are almost identical as the radius of the wire increases. Numerically, we always find the solutions of Eq. (3.18) in pairs, but we have eliminated the solution corresponding to single-particle excitation (i.e., the production of an electron-hole pair) [14], which is irrelevant and unphysical in this case. Another feature of our model is the existence of multiple solutions to Eq. (3.18) because of the multivalued Bessel functions J0 and J1. This result is inherent to the cylindrical geometry assumed in our model. The higher-order solutions correspond, however, to unusually large carrier concentrations and are irrelevant for the EQL condition considered here. The slope of the plasmon dispersion curve gives the wave group velocity, which is an important characterization of the quantum wire. As we indicated earlier, the ϵSCF = 0 derivation provides a group velocity vg which diverges logarithmically as q Æ 0 [8]. In our case, however, the dispersion relation [Eq. (3.18)] is satisfied for the negative pole of c in the long-wavelength limit; therefore we find vg = h(kF + q)/m* as q Æ 0, i.e., the Fermi velocity [15]. Strictly speaking, this result is only valid at T = 0 K; any broadening effect in the polarization will result in an r0-dependent cutoff frequency for plasmon modes.

References

In summary, we derived the plasmon dispersion relation of onedimensional electron gas embedded in a host material assuming the collective oscillations are confined in a cylindrical dielectric waveguide. This approximation does not seem to limit the validity of the model to more realistic configurations since the III–V compound semiconductors of interest the results are not very sensitive to the difference between the dielectric constants of the guiding region and the surrounding material. At T = 0 K and in the long-wavelength limit, our waveguide model provides a finite group velocity, given by the Fermi velocity.

Acknowledgments

This work was supported in part by the U.S. Joint Services Electronics Program through Contract No. N00014-84-C-0149 and through National Science Foundation grant no. DMR-86-12860 of the Materials Research Laboratory at the University of Illinois at Urbana-Champaign. One of us (J.W.) is indebted to Edwin Kan for technical assistance.

References

1. P. M. Petroff, A. C. Gossard, R. A. Logan, and W. Wiegman, Appl. Phys. Lett. 41, 635 (1982). 2. P. F. Williams and A. N. Bloch, Phys. Rev. B 10, 1097 (1974). 3. M. Apostol, Z. Phys. B 22, 279 (1975).

4. W. I. Friesen and B. Bergersen, J. Phys. C 13, 6627 (1980). 5. J. Lee and H. N. Spector, J. Appl. Phys. 57, 366 (1985).

6. T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. B 38, 12 732 (1988). 7. T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Superlatt. Microstruct. 5, 287 (1989). 8. S. Das Sarma and W. Y. Lai, Phys. Rev. B 32, 1401 (1985).

9. Similar results are obtained in two-dimensional systems, where ϵSCF = 0 gives w pl ~ q , while inclusion of boundary conditions yields finite group velocity. See, T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).

37

38

Plasmon Dispersion Relation of a Quasi-1D Electron Gas

10. K. W. Chiu and J. J. Quinn, Phys. Rev. B 9, 4724 (1974).

11. Yilin Weng and J. P. Leburton, J. Appl. Phys. 65, 3089 (1989).

12. In the course of printing this Brief Report, we notice that this effect has been recently discussed in the framework of the standard selfconsistent-field approach, ϵSCF = 0, of Q. Li and S. Das Sarma, Phys. Rev. B 40, 5860 (1989). 13. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

14. D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964).

15. Although of different origin, this result is consistent with Apostol’s results, where Tomonaga’s model for a one-dimensional many-fermion system was used, and the plasmon dispersion relation was calculated by use of Shevchik’s technique.

Chapter 4

Size Effects in Multisubband Quantum Wire Structures

S. Briggs and J.-P. Leburton

Coordinated Sciences Laboratory and Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, USA [email protected]

We present a Monte Carlo simulation of multisubband quasi-onedimensional GaAs-AlxGa1-xAs structures. The simulation includes polar-optical-phonon and inelastic acoustic-phonon scattering and investigates the effect of changing confinement on electron velocity. Even at low longitudinal fields, for confinements in the range 50– 230 Å, intersubband scattering has a profound effect on transport parameters. Under optimum conditions, differential mobility in excess of twice the bulk value at 300 K is obtained. Asymmetric geometry and high degree of confinement result in interesting differences from magnetotransport. In addition, resonance analoReprinted from Phys. Rev. B, 38(12), 8163–8170, 1988.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1988 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

40

Size Effects in Multisubband Quantum Wire Structures

gous to the magnetophonon effect occurs when the separation between lowest subbands is equal to the polar-optical-phonon energy.

4.1 Introduction

In recent years there has been increasing interest in the investigation of quasi-one-dimensional (1D) structures. Progress in epitaxial technologies combined with the emergence of new fine-line— patterning techniques have made the fabrication of artificial lowdimensional structures feasible in the near future [1–7]. From a fundamental viewpoint, the interest in 1D systems has been motivated by the fascinating consequences of the localization theory, which predicts that in weakly disordered structures no extended state exists and, consequently, the 1D conductivity goes to zero at low temperature [8, 9]. At intermediate temperature and finite transverse confinement, however, the “particle-in-a-box” picture seems more realistic and the concept of a quasi-1D system is a natural extension of the ultraconfined two-dimensional (2D) electron gas. Sakaki suggested that semiconductor quantum wire structures could be the basis for very fast transport processes [10]. Because of the reduction of phase space, the number of available final states during the scattering process (only forward or backward scattering) is very limited and results in the enhancement of the 1D mobility with respect to the bulk value. Theoretical investigations of the electronic properties of semiconductor wire structures have recently been accomplished [11–13]. For III-V compounds, calculations of the most important scattering mechanisms, i.e., impurity [10, 14], acoustic [15], and optical phonons [16], show the importance of significant size effects which have been confirmed by Monte Carlo simulation [17]. In modulation-doping structures, ionized-impurity scattering is vanishingly small, which suggests 1D mobility above 1 × 106 cm2/V s [18]. However, because of the high level of quantization, most of the transport models assume the extreme quantum limit (EQL) and neglect the influence of multiple subbands (one-subband models) [17, 19]. This approximation is precarious when the spacing

Introduction

between subbands is comparable to kT, as recently discussed by Das Sarma and Xie for low temperature in Si [20], or when hot-electron effects induce intersubband scattering. Although these processes are analogous to longitudinal magnetotransport phenomena [21, 22], they occur with a variety of features which are specific to artificial 1D systems. For instance, the two transverse confining potentials are, in general, independent; this provides a situation completely different from the harmonic-oscillator-like spectrum of energy levels resulting from the cylindrical symmetry of the magnetic field. The different character of the two types of wave functions and the unequally spaced energy spectrum of the artificial 1D structures significantly influences the intersubband transitions. Furthermore, energy-level spacing of the order of the optical-phonon energy or the thermal energy at room temperature can, in principle, be achieved with artificial confinement; this anticipates resonance conditions superior to magnetotransport. Therefore, in order to assess realistically the transport properties of quantum wires, the development of a multisubband transport model is desirable. The chapter presents a Monte Carlo simulation of a multisubband quasi-1D GaAs-AlxGa1-xAs structure. Because of the small electron effective mass, quantization effects are more pronounced than in Si. Our purpose is to study the effect of varying confinement on the transport properties of the 1D system. We specifically focus on lattice scattering and analyze the influence of intersubband transitions on the transport properties. In the first approximation, we limit our simulation to the Γ valley in GaAs and neglect high-energy processes such as intervalley scattering and real-space transfer. In addition, we neglect nonparabolicity corrections in the energy dispersion relation. Our model involves up to seven subbands, which allows a realistic simulation of confinement conditions in the range 50–230 Å. As the exact position of electronic states is not of primary importance for this work, elementary confining-potential profiles have been assumed. This assumption does not limit the validity of our model, as the real transport features may merely be shifted but not altered with respect to our simple electronic model.

41

42

Size Effects in Multisubband Quantum Wire Structures

4.2 Model Our model consists of a GaAs-AlxGa1-xAs quantum well (QW) with a perpendicular gate electrode and a triangular electrostatic potential. At present, we assume a geometry similar to the V-groove field-effect-transistor (FET) structure proposed by Sakaki [10] (Fig. 4.1a). This device configuration may not be suited for highspeed transport due to interface states between the insulator and the active region resulting from processing which may degrade the mobility. However, the V-groove wire characterized by a quantum. well in the y direction and a triangular potential in the z direction offers, in principle, the largest degree of confinement which can be controlled by external transverse electric fields (gate fields) Fz. In addition, the quantizations resulting from the square well and the triangular potential are rather different and result in various interesting features in the transport characteristics. The model is flexible enough to also simulate a multiwire gate FET structure of the type proposed by Warren and Antoniadis [23, 24]. In the present simulation all electrons remain in the Γ valley and all subbands are parabolic. We assume that electrons will transfer to threedimensional (3D) states (primarily by intervalley scattering or real-space transfer [25]) before they reach 400 meV above the well bottom. The program currently has no routine for these processes, which limits the maximum electron energy that we can model. If an electron reaches an energy of 400 meV, we stop the simulation of its path and choose a new electron to replace it. We are thus limited to longitudinal fields (Fx) of 1 kV/cm or less and typically 500 V/cm to avoid electron runaway at 300 K. At lower temperatures we can go to somewhat higher fields because of the lack of phonon absorption. We have run simulations at 300 and 77 K with well widths Ly in the range 250–50 Å. Wider wells require too many levels for an accurate simulation; however, this is not a serious constraint, as there are few electrons at the higher energies anyway. To save memory in the code, the lowest energy we consider is 100 meV above the bottom of the well, which sets a lower limit on the range of confinement conditions we can model. Our transverse electric fields Fz range from an upper limit of 200 down to 20 kV/cm and are subject to the same restrictions as those on well widths.

Model

Figure 4.1 (a) Schematic representation of a FET 1D quantum wire [after Sakaki (Ref. 10)]. (b) A representation of the confining potentials and twoelectron energy levels and wave functions in the wire structure.

For the y direction we calculate wave functions in the infinitesquare-well approximation. The z wave functions are computed using a variational approach, with exponentially damped polynomials as the trial wave functions. We compute the first three y and z wave functions and combine them to obtain nine wave functions. Then, the total wave functions Ψ are given by

Yi , j

È2˘ =Í ˙ ÍÎ L y ˙˚

1/ 2

È ip y ˘ -a z j ck , j z k , i = 1, 2,3, j = 1, 2,3 (4.1) sin Í ˙e j ÍÎ L y ˙˚ k =1

Â

43

44

Size Effects in Multisubband Quantum Wire Structures

where aj and ck,j are determined by a variational calculation. The corresponding energies (Fig. 4.1b) are functions of the longitudinal electron k vector and are given by E i , j (k ) =

h2 k 2

+ Ei + E j , i = 1, 2,3, j = 1, 2,3 2m* with Ei, the square-well energy, calculated from

Ei =



( hp i )2 2m* L2y

(4.2a)

, i = 1, 2,3.

(4.2b)

Although Ej, the triangular potential energy, is obtained from a variational calculation, a useful approximation is [26] 1/3

2/3 2/3 È h2 ˘ Ê 3 1ˆ ˆ Ê E j = Í * ˙ Á p qF ˜ Á j - ˜ , j = 1, 2,3 . (4.2c) ¯ Ë 4¯ ÍÎ 2m ˙˚ Ë 2 These approximations are good if the energy level lies deep in the well. The higher y levels should be spaced more closely as they approach the top of the QW, and the z levels more closely as screening flattens out the triangular potential. The y = 2, z = 3 and y = 3, z = 3 states are omitted because of memory constraints in the Monte Carlo code. For most confinement conditions of interest, these two levels are above the edge of the GaAs-AlxGa1-xAs barrier and can be neglected. Although the lowest-energy level is clearly E1,1, the ordering of the higher subbands depends on the confinement conditions. To avoid confusion, the subbands will be numbered with a single subscript ν, which will range in order of increasing energy from 1 for the lowest subband to 7 for the highest subband. The y and z quantum numbers will, in general, not be used. Currently, we consider only polar-optical-phonon (POP) and inelastic acoustic-phonon scattering. The 1D transition probabilities from an electron state ki in initial subband ν to a state kf in final subband μ are calculated according to Fermi’s golden rule as



(

)

Wn , m ki , k f =



2p d h ki - k f ± qx

+• +•

Ú Ú dq dq y

-• -•

z

( ( )

)

(

| M3D ,n , m qx , q y , qz |2

)

1 1ˆ Ê ÁË Nq + 2 ± 2 ˜¯ d Eu k f - En (ki ) ± hw q ,

(4.3)

Model

where qx is the longitudinal- and qy, qz the transverse-phonon wave vectors, respectively. Nq is the phonon occupation number with the ± corresponding to phonon emission or absorption. The double integral over qy and qz represents the calculation of the 1D matrix elements M1D,n,m(qx) from the normal 3D matrix elements. Here we consider only bulk- (i.e., 3D) phonon modes and neglect 1D and surface modes. This does not introduce significant error as long as the confinement is not excessively high, i.e., less than 50 Å. The POP dispersion relation is assumed to be a constant, which makes Nq and the energy-conservation d function independent of q; therefore they can be factored out of the double integral. However, the nonconstant dispersion relation for acoustic phonons makes computation of the integral considerably more complex. The numerical integration routine has qx as an input parameter, which we vary to obtain 1D matrix elements. For acoustic phonons we evaluate the integral for qx in the range 7 × 104 < qx < 1. 5 × 107 cm–1. The acoustic-phonontransition probabilities are essentially independent of qx for smaller values, while 1.5 × 107 cm–1 is the largest possible momentum exchange. For POP transitions the integral is evaluated for 1 × 100 < qx < 1 × 107 cm–1. The dk term in Eq. (4.3) represents conservation of longitudinal momentum, with the ± sign corresponding to phonon absorption or emission, respectively, and is used to select a value for qx and the corresponding matrix element. Scattering rates are computed by integrating the transition probabilities over the final electron k states. The total scattering rate for POPs or acoustic phonons from the initial state ki in subband ν is then given by

ln (ki ) =

7

ÂÚdk m =1

f

2p m 1 | M1D ,n , m (qx ) |2 d ki - k f ± qx 2 , (4.4) h h kf

(

)

where we have transformed the energy-conservation δ function into a wave-vector-conservation d function. For each 1D subband the wave-vector-conservation d function reduces the integral to a sum over the four possible final states corresponding to forward or backward emission or absorption. A broadening of the phonon energy, h/tPOP, was used to smooth out the divergence in the scattering rate due to the final density of states. The physical justification for this broadening is that electrons

45

46

Size Effects in Multisubband Quantum Wire Structures

encounter a broadened final density of states as they are scattered by phonons distributed within a finite energy range. This approximation should be distinguished from the explicit broadening in the density of states as considered by Das Sarma and Xie [20]. Because of its long tail, a Lorentzian-broadening factor allows for unrealistic phonon energies; this is especially noticeable at low temperatures. To avoid this unrealistic distribution, a Gaussian profile was used to drastically reduce the number of POPs with energy substantially different from 36 meV. In the first approximation this replaces the 1/kf factor in Eq. (4.4) by its convolution with a Gaussian distribution: t POP 1 1 ¨ kf (2m* )1/2 2p



+•

È (E ¢ - hw LO )2 ˘ 1 dE ¢. ˙ 2 ) [ E ¢ + E k = 0 ) - En (k f )]1/2 ˙˚ POP n(

Ú exp ÍÍÎ- 2(h / t

-•

(4.5)

A similar broadening factor was also used for acoustic-phonon scattering. We consider energies from 100 to 400 meV above the bottom of the well. This energy range is divided into 400 intervals of uniform size to compute the scattering rates. For each energy interval in each subband, we consider forward and backward emission and absorption for both POPs and acoustic phonons to each possible final subband. This gives us 56 (2 × 2 × 2 × 7) possible independent scattering mechanisms for each initial state. This number is then multiplied by the number of initial subbands (seven) and the number of points in the energy mesh (400) to obtain the total number of rates stored in the code. This large number of scattering rates is the primary limitation on the maximum number of subbands in the code since the storage required is proportional to the square of the number of subbands. These scattering rates are saved in files and used as input to the Monte Carlo code. The rates show a large number of peaks; each peak is proportional to the density of final states and corresponds to an emission or absorption to the bottom of a subband (Fig. 4.2a). The large single peaks are due to POP scattering, the small peaks in pairs are acoustic phonon absorption and emission pairs to subband bottoms. These peaks make the velocity and distribution functions (Fig. 4.2b) sensitive to the energy separation between subbands, particularly between the first and second subbands. Below the POP emission threshold, rates are somewhat smaller than in bulk, while at high energies the large

Model

number of subbands enhances the rates with respect to the bulk value. The rates drop above 344 meV because we no longer allow for POP absorption beyond that point.

Figure 4.2 (a) Total (POP and acoustic phonon) scattering rates for a quantum wire with Ly = 135 Å and Fz = 120 kV/cm at 300 K. Peaks correspond to an emission or absorption to the bottom of a subband. Large peaks represent POP transitions; small double peaks are due to acoustic phonons. For clarify, only the first three subbands are shown, with the heaviest line representing the lowest subband and the lightest representing the second subband. (b) Electrondistribution function for the same quantum wire.

47

48

Size Effects in Multisubband Quantum Wire Structures

4.3 Monte Carlo Code We run a steady-state single-particle Monte Carlo code. An electron is placed at an arbitrary position in the bottom subband and undergoes 6 × 103 scattering events to eliminate any effect of the initial conditions before we begin collecting statistics on it. Although the velocity converges after less than 4 × 104 scattering events, we simulate 4 × 105 (not including the 6 × 103 needed to achieve steady state) events to obtain better convergence on the distribution functions. If an electron goes above our maximum energy (assumed to have scattered to a 3D state), it is lost to the simulation and a new electron is injected in the bottom subband. While the energy range for computing scattering rates is split up into 400 intervals, the Monte Carlo code has a resolution of δE = 0.075 meV, which corresponds to 4000 intervals on the same range. We compute k(E) at the start of the program for each interval and save it in a table for later use in evaluating free-flight times. Because of the large number of peaks in the rates, normal methods for computing free-flight times are inefficient. Using constant or piecewise-constant scattering rates with self-scattering would have introduced a very large percentage of self-scattering events. An iterative gamma method [27] was tried, but the sharpness of the peaks meant that a large number of iterations were needed to find an appropriate gamma. Instead, a direct integration method was used. For a given subband ν, if r is a uniformly distributed random number on [0,1], then t



Ú (

)

-lnr = ln kn (t ¢ ) dt ¢ , 0

(4.6)

where t is the time of the free flight, kn (t) is the momentum as a function of time in subband n, and ln (kn) is the scattering rate as a function of momentum for that subband. The program approximates the integral as a sum and moves the electron in uniform energy steps of size δE from an initial energy Ei until a final energy Ef is found such that Ef



-lnr =

 l (k n

E = Ei

n

(E )) Dt (E ) ,

(4.7a)

Results

with

and

Dt (E ) =

h Dkn (E ) eFx

Dkn (E ) = kn (E ± d E ) - kn (E ) ,

(4.7b) (4.7c)

the ± depending on whether the electron is accelerating or decelerating. In 3D simulations it is virtually impossible to store k(E) in tabular form because of the large number of possible k values. In 1D systems there are only two scalar kn values for each energy and each n (tabulated earlier by the program) and, hence, Dkn is essentially a look-up function which avoids the time-consuming square-root computation required to determine kn (E). Moreover, in three dimensions, Dt is a complicated function of k, involving squares and square-root computations, which typically prohibits direct-integration algorithms in Monte Carlo codes. For 1D systems, however, direct integration compares favourably with other methods. The electron state is sampled after every free flight to obtain the before-scattering distribution nb,ν(k) for the νth subband. The correct distribution nν(k) is obtained from [28]

nn (k ) = Cn

nb ,n (k ) ln (k )

,

(4.8)

where Cn is a normalizing constant required to preserve the relative electron population in each subband. This distribution is saved and used to calculate all the quantities of interest.

4.4 Results

In general, velocities in quasi-1D systems are higher than in bulk (3D) GaAs, despite the fact that scattering rates exhibit large singularities and background values on the average comparable to bulk rates. We attribute this increase to the reduction in phase space of the 1D system. Because of the lack of transverse scattering and the 1/q dependence of the electron-POP interaction, the scattering is more strongly forward peaked than in three dimensions. Although acoustic phonons were included in this simulation, they do not play an important role and simply tend to reduce the velocities by about 10%.

49

50

Size Effects in Multisubband Quantum Wire Structures

Figure 4.3 summarizes the effects of various confinement conditions on the velocity. The degree of confinement is represented by the energy of the first subband relative to the bottom of the well. Although this is a simplification and ignores the fact that confinement (and velocity) is a function of two variables (Ly and Fz), it demonstrates the primary effects on velocity. From a general standpoint, when the structure is in the EQL (i.e., when only the lowest subband is occupied), increasing confinement reduces the velocity. This is in agreement with the size effects predicted by several authors [16, 17]. In this case the overlap integrals (form factors) in the matrix elements approach unity and enhance the scattering rates. However, this also occurs under less restricted conditions below the EQL when increasing confinement does not affect the separation between the first and second subbands, as can be seen from curve a in Fig. 4.3, where Fz is varied from 100 to 200 kV/cm while holding Ly at 200 Å. For these conditions, the first-excited subband is the y = 2, z =1 subband, and since Ly remains constant the separation between the bottom and first-excited subbands remains 42 meV. As confinement increases, the velocity at Fx = 500 V/cm decreases from 6.11 × 106 to 5.63 × 106 cm/s.

Figure 4.3 Variation of velocity with confinement for Fx = 500 V/cm and T = 300 K. Confinement is expressed as the position of the bottom subband relative to the well bottom. In curves a (○ ) and c (□), Fz is varied while Ly remains constant, while in curve b (D) Ly is varied and Fz is constant. Curve a shows confinement increasing without affecting the separation between the bottom two subbands. In curves b and c the separation between the bottom two subbands does change as a function of confinement.

At low confinement, upper subbands play a significant role because more final states (more final subbands) are available

Results

for scattering and velocities are lower than the EQL values. Consequently, increasing confinement towards the EQL increases velocity as intersubband scattering is suppressed. In the low-energy portion of curve b, as the bottom subband moves from 128 to 138 meV (Ly is varied from 165 to 135 Å while holding Fz fixed at 120 kV/cm), the separation between the lowest subband and the firstexcited subband is increased from 62 to 93 meV. This increases the fraction of carriers in the lowest subband from 80% to 92% (for Fx = 500 V/cm) and enhances the velocity from 6.89 × 106 to 8.08 × 106 cm/s. These two mechanisms—increasing overlap integrals and decreasing influence of upper subbands—explain the general trends as confinement is varied. For instance, under conditions resulting in the first-excited state being the y = 2, z = 1 state, increasing Fz does not change the separation between the ground-state and first-excited subbands; consequently, velocity drops as the overlap integrals increase. However, if L is decreased, the separation increases and results in velocity increase as long as the first-excited state remains the y = 2, z = 1 state, even though the overlap integrals approach unity. This is in contradiction with other 1D results, where only one subband is considered [17]. If Ly is reduced sufficiently, the firstexcited state will become the y = 1, z = 2 state instead of the y = 2, z = 1 state; further reductions cause the velocity to decrease as the overlap integrals increase without reducing the influence of scattering to the second subband. The last four points on curve b of Fig. 4.3 show this effect as Ly is lowered from 135 to 105 Å and the velocity decreases to 7.42 × 106 cm/s. The peak in velocity (near 138 meV on curve b in Fig. 4.3) is a tradeoff between increasing overlap integrals and approaching the EQL. When there is a crossover between the first-excited y state and the first-excited z state, increasing one confinement parameter increases the overlap integrals without reducing the influence of intersubband scattering (the other excited state does not move relative to the ground state and the separation remains the same). On the other hand, decreasing one confinement parameter decreases the overlap integrals, but only at the expense of moving away from the EQL (the corresponding excited state drops relative to the bottom state). This tradeoff condition is achieved for Ly = 135 Å and Fz = 120 kV/cm. The peak has a first-excited state at 93 meV relative to the ground state and 92% of the carriers are in the bottom subband for Fz = 500 V/cm.

51

52

Size Effects in Multisubband Quantum Wire Structures

It is clear from the above discussion why the velocity in curve a is lower than in curves b or c. While the latter curves represent a tradeoff between the conflicting effects of overlap integrals and intersubband scattering, the former curve has both high overlap integrals due to high confinement in the z direction and high intersubband scattering due to low confinement in the y direction. While these two mechanisms explain the general trends in velocity as confinement is varied, there are several other important effects to consider. The first is due to the asymmetry between the y confinement and the z confinement. Since we assume an infinite square well, the y wave functions do not spread out for higherenergy levels like the z wave functions do. This tail in the z wave functions lowers the overlap integrals in the scattering rates and tends to enhance the mobility in subbands with z index greater than 1 with respect to subbands with y index greater than 1. Hence, systems with the second subband corresponding to y = 1, z = 2 tend to have higher mobilities than structures with the second subband corresponding to y = 2, z = 1 due to the higher mobility in the second subband. In addition, the y energy levels increase as i2 [Eq. (4.2b)], while the z levels are roughly proportional to j2/3 [Eq. (4.2c)]. Curve c in Fig. 4.3 shows the same trends as curve b, but in this case Fz is varied from 80 to 160 kV/cm while Ly is held fixed at 135 Å. Initially, the second subband is the y = 1, z = 2 state and as Fz exceeds 120 kV/cm the second subband becomes the y = 2, z = 1 state. Although the variation of bottom subband energies in curve c is greater than in curve b, the corresponding velocity variation is smaller, for two reasons. First, the separation between the first and second subbands is less sensitive to confinement (from 72 to 93 meV versus 62 to 93 meV in curve b) because of the sublinear dependence of the energy on the z quantum number j. In addition, at low Fz the second subband is a z = 2 state with weak intersubband scattering even when the separation between subbands is small. The velocity-field relation for the confinement condition giving the highest velocity (Ly = 135 Å, Fz = 120 kV/cm is shown in Fig. 4.4a, along with the equivalent relation for bulk GaAs. The velocity at Fz = 500 V/cm corresponds to a mobility of 16160 cm2/V s, which is over twice the bulk value of 8000 cm2/V s. An important effect is the dependence of the differential mobility on the field. At low fields it is over twice the bulk value, but converges toward the

Results

bulk value at high fields. We attribute this effect to intersubband scattering (breakdown of the EQL) at high fields. The inset of Fig. 4.4 shows the average electron energy in units of kT as a function of Fx. For low fields the electron energy is roughly equal to kT and most of the electrons are near the bottom of the first subband. At higher fields hot electrons become significant, with higher scattering rates due to the presence of other subbands (i.e., they are not in the EQL), which reduces their differential mobility. The velocity-field curve for the same confinement conditions at 77 K is shown in Fig. 4.4b. Again, at low fields differential mobility is higher in the 1D system than in the bulk, while at high fields the velocities approach the same value.

Figure 4.4 Velocity-field relation for highest-velocity confinement condition (Ly = 135 Å and Fz = 120 kV/cm) compared to bulk values [after Haase (Ref. 29)] at (a) 300 K and (b) 77 K. The inset in (a) shows the average electron energy in units of kT as a function of the longitudinal field.

53

54

Size Effects in Multisubband Quantum Wire Structures

Finally, when the separation between first and second subbands is near hwPOP a resonance condition can occur with the electron jumping back and forth between the first and second subbands. This can be seen clearly in the distribution functions near and at resonance. In Fig. 4.5a the separation between the first two levels is 28 meV and the structure is below the resonance condition. (The sharp narrow peaks in the distribution functions are artifacts of the method used, corresponding to peaks in the scattering rates). The first-subband population falls off slightly at 122 meV and more dramatically at 138 meV. 122 meV is the onset of POP absorption to the third subband, while 138 meV is the onset of emission to the first subband. The distribution function of the second subband at a given energy is not significantly different from that of the first subband. 60% of the electrons are in the first subband, with 30% in the second subband, and the remainder primarily in the third subband. In Fig. 4.5c the separation is 44 meV and the system is above resonance. 71% of the electrons are in the first subband, 24% in the second, and 5% in the third. In Fig. 4.5b the separation is 36 meV and the two subbands are in resonance. The first-subband population shows marked decreases at the onset of emission to both the first and second subbands, while the second subband shows a decrease at the onset of emission to the second subband. More importantly, the second-subband population is almost 3 times that of the first-subband population for the same energy and almost as great as the first-subband population at the bottom of the subband. 55% of the electrons are in the first subband and 38% are in the second subband, with the remaining 7% being primarily in the third subband. We observe a weak dependence of the velocity on confinement at this point, which may be analogous to the magnetophonon effect [21, 22]. This resonance effect may also appear in 2D systems, but not so dramatically, since the density of states is constant instead of singular. As an example of the flexibility in the structure due to the asymmetry between y and z directions, in Fig. 4.5b the bottom three subbands are the y = 1, z = 1, 2, 3 states, and not only is the second subband in resonance with the first, but it is also very near resonance with the third subband. For the equivalent case with the lowest three subbands being the z = 1, y = 1, 2, 3 states, when the first two subbands are in resonance (Fz = 150 kV/cm and Ly = 215 Å), the second and third subbands are separated by 61 meV.

Results

Figure 4.5 Distribution functions for Fx = 500 V/cm and 300 K near the POP resonance condition. The heaviest line is for the second subband and the lightest for the third subband. (a) System below resonance. (b) System at resonance. (c) System above resonance. For clarity, only the lowest three subbands are shown.

55

56

Size Effects in Multisubband Quantum Wire Structures

4.5 Conclusions We have presented a multisubband Monte Carlo model of quantum wire systems including POP and acoustic-phonon scattering mechanisms. The transport parameters are not a simple function of the confinement conditions and result primarily from the respective influence of two factors: the magnitude of the wave-function— overlap integral in the transition probability and the relative influence of the upper (mainly second) subbands. The asymmetry between y and z states is significant in determining the relative importance of the above factors. Under optimum conditions, mobilities near twice the bulk value at room temperature are anticipated. Resonances similar to the magnetophonon effect with enhanced second-subband populations occur when the subband separation is hwPOP. The ability to achieve, in principle, energy-level spacing comparable to kT at room temperature makes artificial 1D structures well suited for comparison to magnetically confined systems. Quantum wires present additional important features which differ from longitudinal magnetotransport. The absence of azimuthal degeneracy and the flexibility in varying independently the y and z confinement provide a useful means of controlling externally the transport parameters. In the FET configuration, negative-differential transconductance can be obtained by simply changing the gate voltage to modify the subband spacing. Similarly, transitions from one to two dimensions could be investigated by gradually turning off the external field. Finally, owing to the finite well depth, negativedifferential resistance though real-space transfer, or intervalley scattering, is expected to occur under strong longitudinal fields.

Acknowledgments

We are indebted to Dan Jovanovic for technical assistance. This work was supported by the U.S. National Science Foundation under grant no. NSF-CDR-85-22666 and the Joint Services Electronics Program. Part of the computation was performed using the resources of the National Center for Supercomputing Applications (NCSA) of the University of Illinois.

References

References 1. P. Petroff, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982). 2. T. J. Thornton, M. Pepper, H. Ahmed, D. Andrews, and G. J. Davies, Phys. Rev. Lett. 56, 1198 (1986).

3. K. Kash, A. Scherer, J. Worlock, H. Craighead, and M. Tamargo, Appl. Phys. Lett. 49, 1043 (1986). 4. H. Temkin, G. Dolan, M. Panish, and S. Chu, Appl. Phys. Lett. 50, 413 (1987). 5. P. Petroff, J. Cibert, A. Gossard, G. Dolan, and C. Tu, J. Vac. Sci. Technol. 5, 1204 (1987). 6. K. Ishibashi, K. Nagata, K. Gamo, S. Namba, S. Ishida, K. Murase, M. Kawabe, and Y. Aoyagi, Solid State Commun. 61, 385 (1987).

7. T. Hiramoto, K. Hirakawa, Y. Iye, and T. Ikoma, Appl. Phys. Lett. 51, 1620 (1987). 8. D. Thouless, Phys. Rev. Lett. 39, 1167 (1977).

9. N. Giordano, W. Gilson, and D. Prober, Phys. Rev. Lett. 43, 725 (1979). 10. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

11. S. Laux and F. Stern, Appl. Phys. Lett. 49, 91 (1986).

12. K. Wong, M. Jaros, and J. Hagon, J. Vac. Sci. Technol. 5, 1198 (1987). 13. S. Das Sarma and Wu-yan Lai, Phys. Rev. B 32, 1401 (1985). 14. J. Lee and H. Spector, J. Appl. Phys. 54, 3921 (1983). 15. V. Arora, Phys. Rev. B 23, 5611 (1981).

16. J. P. Leburton, J. Appl. Phys. 56, 2850 (1984). [In this paper, the 1D absorption rate in Eq. (16b) should be corrected to include forward scattering corresponding to the q+ value in Eq. (14b).]

17. A. Ghosal, D. Chattopadhyay, and A. Bhattacharyya, J. Appl. Phys. 59, 2511 (1986). 18. G. Fishman, Phys. Rev. B 34, 2394 (1986).

19. P. Yuh and K. Wang, Appl. Phys. Lett. 49, 1738 (1986).

20. S. Das Sarma and X. Xie, Phys. Rev. B 35, 9875 (1987).

21. R. Peterson, in Transport Phenomena, Vol. 10 of Semiconductors and Semimetals, edited by R. K. Willardson (Academic, New York, 1975), p. 221. 22. R. Stradling and R. Wood, J. Phys. C 1, 1711 (1968).

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Size Effects in Multisubband Quantum Wire Structures

23. A. Warren, D. Antoniadis, H. Smith, and J. Melngailis, IEEE Electron. Dev. Lett. EDL-6, 294 (1985). 24. Warren, D. Antoniadis, and H. Smith, Phys. Rev. Lett. 56, 1858 (1986).

25. K. Hess, H. Morkoç, H. Shichijo, and B. Streetman, Appl. Phys. Lett. 35, 469 (1979). 26. F. Stern, CRC Crit. Rev. Solid-State Sci. 499 (1974). 27. R. Yorston, J. Comput. Phys. 64, 177 (1986).

28. Jacobini and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).

29. M. Haase, V. Robbins, N. Tabatabaie, and G. Stillman, J. Appl. Phys. 57, 2295 (1985).

Chapter 5

Impurity Scattering with Semiclassical Screening in Multiband Quasi-OneDimensional Systems

Yilin Weng and J.-P. Leburton

Department of Electrical and Computer Engineering and Material Research Laboratory, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, USA [email protected]

We present a calculation of screened impurity scattering in multiband GaAs one‐dimensional systems. The influence of background impurities is extended to a domain outside the wire within a phenomenological model. Screening effects are evaluated in the semiclassical Thomas–Fermi approximation. A feature of our model is the consideration of level degeneracy which enhances carrier occupation and screening in higher subbands. For confinement conditions corresponding to subband separation equal to the Reprinted from J. Appl. Phys., 65(8), 3089–3097, 1989. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1989 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

60

Impurity Scattering with Semiclassical Screening

thermal energy, we found that this effect equalizes the different scattering rates in each subband.

5.1 Introduction

Double confined structures have recently stimulated some interest because of the expectation of new fundamental effects as well as the possibility for new device applications [1–3]. The weak level of quantization achieved with the actual artificial structures permits the observation of one-dimensional (1D) effects at low temperature [4–6] only where impurity scattering is a dominant mechanism for carrier relaxation. Sakaki has theoretically studied the Coulomb scattering of electrons confined in an ultrafine GaAs wire structure [7]. He predicted that the low temperature mobility of quantized electrons along a well designed wire can be much higher than 106 cm2/Vs. The case considered, however, impurities located at a fixed distance outside the wire Lee and Spector [8, 9] used a model similar to Sakaki’s to theoretically compute the impurity limited mobility for both background and remote impurities later on, Fishman proposed an analytical approach of mobility in quasi one-dimensional systems [10] by taking into account the finite temperature and static screening in the Coulomb interaction effects. His model is, however, limited to an extreme quantum limit, i.e., only the lowest subband is occupied by carriers DasSarma and Xie calculated low temperature dc transport properties in quasi-1D silicon inversion layers including intersubband scattering [11]. The research reported here is the evaluation of screened impurity scattering in multiband, 1D GaAs systems. Technological obstacles for highly confined structures are expected to be overcome in the near future so that subband spacing of several meV’s can realistically be achieved; 1D effects will then be observable at temperature above liquid helium, with only the few first subbands being populated. For this reason, we deal with a two-band model which shows the essential features of multiband scattering. In addition for the temperature of interest, we assume that the electron’s phase coherence length ll = (Dlτl)1/2, where τl is the phase coherence time and Dl the elastic diffusion constant, is in

Model

general shorter than the impurity mean free path, simply because other collision events such as phonon scattering or thermalization processes occur between two collisions with impurities. In these conditions, we ignore quantum interference effects and treat collision processes semiclassically. For the sake of simplicity, we assume that our 1D system has a square parallelepiped symmetry. Actually, our model can handle any size wires of rectangular cross section with multiple subbands, which would be suitable for highfield transport when energetic carriers populate higher subbands. However, the square parallelepiped symmetry offers the advantage of being characterized by an energy level (azimuthal) degeneracy which enhances the subband carrier occupation and screening effects at high energy. Only the background impurity (BI) scattering resulting from the screened Coulomb potential in many band 1D system is discussed in this chapter. The influence of remote impurities is assumed to be negligible if the spacer layer of the 1D modulation structures is relatively thick [12]. We have extended the BI influence to a region outside the wire and introduce an empirical cutoff factor to describe the limited range of BI. In Section 5.2, we introduce our model of the multiband quasi-1D system. The dielectrical constant matrix and the impurity scattering rate are derived and discussed in Sections 5.3 and 5.4, respectively. Finally, Section 5.5 is devoted to our conclusion.

5.2 Model

The model used here consists of a basic wire structure whose electronic properties are, for mathematical simplicity, described in terms of the wave functions of a simple harmonic oscillator (SHO). The latter wave functions constitute a better representation of the electronic states than the sinusoidal electron wave function currently used since the tail effect in the confining barrier is considered in the SHO approach. We limit our model to the first two energy levels resulting from the confinement along the x and y directions. The consideration of higher subbands will complicate the model without adding essential new physical features to the problem. Thus, the first two normalized wave functions in the x direction are

y 1 ( x ) = (a x / p )1/4 e -(a x x

2

/2 )

,

(5.1)

61

62

Impurity Scattering with Semiclassical Screening

y 2 ( x ) = (2a 3x / p )1/4 xe -(a x x



2

/2 )

,

where αx is Gaussian distribution factor and given by

*

a x = m*w x /  ,

(5.2) (5.3)

where m = 0.068 m0 is the electron effective mass, wx is the oscillator frequency, and  is the Planck constant divided by 2p. We assume a similar expression for the y direction, so that the various subbands energies are given by

1ˆ 1ˆ Ê Ê E xy = Á n + ˜ w x + Á m + ˜ w y , Ë Ë 2¯ 2¯

(5.4)

where n and m are quantum numbers. As mentioned in Section 5.1, only the square geometry where one Gaussian distribution factor needs to be determined (ax = ay = a) will be considered. Hence, the multiband case is reduced to a two-band degenerate system with

E1 = w

(m = n = 0) ,

E2 = 2w (m = 1, n = 0 and m = 0, n = 1).

For 77 K the two-subband approximation is thus valid if w ≥ kBT ª 6 meV. This corresponds to a spatial extent of the wave function of 〈x〉 ≤ 97 Å. At 300 K this approximation becomes w ≥ 25 meV and 〈x〉 ≤ 47 Å.

5.3 Dielectric Constant Matrix

The electrostatic potential f produced by an external source is related to the charge density r by Poisson’s equation

where

—2f = - ( r /  s ) , r = rext + rind

(5.5)

is the sum of the external charge density rext and the induced charge density rind, and es = 12e0 is the GaAs dielectric constant. The induced charge density rind is a function only of the local potential seen by the electrons, as in the three dimensional Thomas–Fermi model [13]. The semiclassical formalism used here gives a simple physical picture of screening effects in terms of temperature and electron concentration. This approach, however, provides an incom-

Dielectric Constant Matrix

plete representation of the scattering processes because it neglects intersubband screening in quantized systems. Fortunately, as we discuss in Section 5.4, intersubband scattering is negligible compared with intrasubband scattering even in the conditions of subband degeneracy considered here. For one-dimensional electron gas (1DEG) in the semiclassical approximation, we have rind = r (f ) - r (0) = -e



ÂDn g (r ), j

j

j

(5.6)

where Dnj = nj(f) – nj(0) is the variation of the linear electron density in the jth band, r is two-dimensional position vector, and g(r) is the two-dimensional electron probability density. At the first order of the potential we obtain rind ª -e



dnj

 df j

Ú

j

f j g j (r ) ,

f j = drg j (r )f(r ) ,



(5.7) (5.8)

where f j is the average potential for the jth band. The external charge density due to one impurity is given by

rext = ±

Ê x 2 y2 ˆ e exp Á - 02 - 02 ˜ d ( x - x0 )d ( y - y0 )d ( z ) , s Ë lx lx ¯

(5.9)

where the exponential factor accounts phenomenologically for the limited range of influence of the outside impurities which are either neutralized by the outside carriers or overcome by remote impurities which are not treated in this paper. The ± sign accounts for the type (donor or acceptor) of impurity lx and ly are cutoff parameters in the x and y directions, respectively, d(r) is the delta function, and r0(x0, y0) is the impurity position. We can define the screening parameter Sj, which is given by [14]

Sj =

e dnj e nj ª 2 s df j 2 s kBT

(5.10)

in the Debye approximation. Here kB is the Boltzmann constant and T is the temperature. For the two-band case n = n1 + n2

is the electron concentration per unit length of the wire, measured in cm–1, and

63

64

Impurity Scattering with Semiclassical Screening

n2 / n2 = 2e -( w /kBT ) ,



(5.11)

where 2 is the degeneracy factor resulting from the overlap of the electronic states in the x and y direction. To simplify, we assume that each band has the same temperature T. Substituting Eqs. (5.7), (5.9), and (5.10) into (5.5), we obtain

Â

—2f = 2 

S j f j g j (r ) ±

j

Performing a Fourier transform in the z direction [15], we get Ê ∂2 ˆ ∂2 + - q2 ˜ j g (r ) = r g (r ) , Á 2 2 ∂y Ë ∂x ¯

where and

Ú

Â

rq (r ) = 2



(5.13)

j q (r ) = f (r , z ) etqz dz





È Ê x 2 y2 ˆ ˘ e exp Í- Á 02 + 02 ˜ ˙ d ( x - x0 )d ( y - y0 )d ( z ) s Í ÁË l x l y ˜¯ ˙ Î ˚ (5.12)

S j j qj g j (r ) ±

j

È Ê x 2 y2 ˆ ˘ e exp Í- Á 02 + 02 ˜ ˙ d ( x - x0 )d ( y - y0 ) s Í ÁË l x l y ˜¯ ˙ Î ˚ (5.14)

The Green’s function is given by [16]

G(r , r ¢ ) = K 0 (q | r - r |)/ 2p ,

(5.15)

where Kn(x) is the modified Bessel function of the second kind of order n and argument x. Therefore, the solution of Eq. (5.13) can be written as the integral of the products of ρq(r) and the Green’s function:

Ú



e

ÂS j Údr G(r , r¢)g (r )  

j q (r ) = dr ¢G (r , r ¢ ) ÈÎ- rq (r )˘˚ = -2

j qj

j

È Ê x 2 y2 ˆ ˘ exp Í- Á 02 + 02 ˜ ˙ Í ÁË l x l y ˜¯ ˙ Î ˚ The average potential on the ith band is

j

G(r , r0 )

s

(5.16)

Dielectric Constant Matrix

ÂS j

j i (q , r0 ) = drj q (r ) g j (r ) = -2

Ú





2p

(

Ú Ú

)

K 0tj = dr drK 0 q r - r ¢ gi (r ¢ )g j (r ) ,



j

K 0tj

È Ê x 2 y2 ˆ ˘ exp Í- Á 02 + 02 ˜ ˙ Ni (q , r0 ), Í ÁË l x l y ˜¯ ˙ Î ˚

with r0 = | r0|, and

j qj (q , r0 )

Ni (q , r0 ) = drgi (r ) K 0 (q | r - r0 |)

Ú

Reorganizing (5.17), we find

 j



e 2p s

(5.17)

(5.18)

(5.19)

È Ê x 2 y2 ˆ ˘ Ê K 0ij ˆ e exp Í- Á 02 + 02 ˜ ˙ Ni (q , r0 ) Á d ij + 2S j ˜ j j (q , r0 ) = ± 2p ¯ 2p s Í ÁË l x l y ˜¯ ˙ Ë Î ˚ (5.20)

and we define the dielectric constant as

ij = d ij + S j (K 0tj / p ) ,

(5.21)

which for the two band case is the general element of the dielectric constant matrix

È K 011 Í1 + S1 p  =Í Í K 021 Í S1 p Î

K 012 ˘ ˙ p ˙ K 022 ˙ 1 + S2 ˙ p ˚ S2

(5.22)

According to Eq. (5.18), we know that K 0 ij = K 0 ji . However, Sj is not equal to Sj, since the carrier density is different in each band, hence, the e matrix is not symmetric. After some algebraical and integral manipulation, we obtain •



where

K 0ij =

1 dte - qt T , 2 t + (1 / 2a ) ij

Ú 0

T11 = 1,

T12 = T21 =

4ta + 1 , 4ta + 2

(5.23)

65

66

Impurity Scattering with Semiclassical Screening



T22 =

2ta (2ta + 1) + 3 / 4 (2ta + 1)2



(5.24)

The curves of the static dielectric constant for a typical 1D configuration are shown in Figs. 5.1a and b as a function of wavevector q. Two different temperatures T equal to 77 and 300 K with a total carrier density n equal to 105 and 106 cm–1, respectively, are considered. The thermal energy kBT for each case is almost equal to the SHO energy separation ℏω so that the two-band approximation is valid. For instance, with T = 77 K, kBT = 6.6 meV, we have ℏω = 5.61 meV. As can be seen in Figs. 5.1a and b, the diagonal elements, e11 and e22, which are single-band dielectric constants, are much greater than off-diagonal interband dielectric constants e12 and e21 • e11 (default line) exceeds e22 (dashed line) since n1 is greater than n2. However, because of the twofold degeneracy [factor of 2 in Eq. (5.11)], the difference is not significant. Also we found e12 (chaindot line) approximately equals e21 (chain-dash line) in Eq. (5.22) since K 012 = K 021 and n ª n2 thus S ª S 2 . As expected from the properties of the 1D system, the dielectric constants are logarithmic divergent when q goes to zero [see Eq. (5.23)]. In addition, the screening effect disappears when the wave vector goes to infinite with e11 and e22 approaching unity and e12 and e21 approaching zero. On the other hand, the screening effect increases with carrier density n but the difference between eii and eij decreases. Finally, we should point out the decrease of the dielectric constant with the temperature resulting from the definition of the S parameter [Eq. (5.10)]. The corresponding results for the inverse dielectric constant are shown in Figs. 5.1c and d. Because e11 is greater than e22, the inverse -1 -1 -1 dielectric constant 11 (11 π 1 / 11 ) is less than 22 . We found

ij-1 (i π j ) is negative but stillles is absolute value than ij-1 . When q -1 -1 -1 goes to zero, 11-1 , 22 , 12 , and 21 approach

S2 S1 S1 S2 , ,, and S1 + S 2 S1 + S 2 S1 + S 2 S1 + S 2

-1 respectively. Since S1 is greater than S2, 22 is still greater than till -1 -1 -1 11 and 12 greater than 21 , both in absolute value for small q. When q approaches infinity, ii-1 goes to one and ij-1 to zero.

Impurity Scattering Rate

Figure 5.1 Dielectric constants and inverse dielectric constant matrix elements as a function of the wave vector. (a) and (c) T = 77 K, E = 5.61 meV, the impurity density n is equal to 105 cm–1. (b) and (d) T = 300 K, E = 22.44 meV, n is equal to 106 cm–1.

5.4 Impurity Scattering Rate Before calculating the scattering rate, we evaluate the average Coulomb potential. Solving the matrix Eq. (5.20) with Eq. (5.12), the potential becomes

67

68

Impurity Scattering with Semiclassical Screening

Ê x 2 + y2 ˆ e exp Á - 0 2 0 ˜ 2p s l ¯ Ë



j i (q , r0 ) =



| j i (q , r0 )|2 =

Â j

-1 ij N j (q , r0 )

(5.25)

-1 -1 ij il N j Nl ,

(5.26)

for the square case l x = l y = l . The square of this expression is given by where

N j (q , r0 ) =



with

e2

4p 2 s2

exp( -2

Â

)

i, j

•

Ú



J2 =

4a t + 4a t + 1



J3 =

4a t + 4a t + 1



l

2

1 Ê 1 ˆ dtJ J exp( -q2t - {r02 / 4[t + (1 / 4a )]})/[t + Á ], Ë 4a ˜¯ 2 0 (5.27)

J1 = 1,



r02

x02

Ê 1 ˆ 2 , 8a[t + Á ] Ë 4a ˜¯

(5.28)

y02

Ê 1 ˆ2 8a[t + Á ] Ë 4a ˜¯ Now we calculate the total scattering rate in the ith subband, 1 = t i (k )

ÂW (k , k ¢) ,

(5.29)

i

k

where k is the longitudinal wave vector. The transition probability in the ith band is given by the Fermi golden rule

with

Wi (k , k ¢ ) =

2p 

 | ·k | f | k ¢Ò | i

r

2

d (E - E ¢ ) ,

Ê eˆ · k | fi | k ¢Ò = - Á ˜ j i (q , r0 ) Ë Lz ¯

(5.30)

(5.31)

Here q = k – k ¢, k is the initial state, and k ¢ is the final state, and Lz is the length of the device. The summation over r0 represents the total scattering rate for a uniform distribution of impurities. We evaluate 〈| j i (q)|2 Òr , which is the average value of | j i (q)|2 for a uniform distribution of impurities, i.e.,

Impurity Scattering Rate

·| j i (q ) |2 Òr =



 | j (q , r ) |

(5.32)

2

i

0

r

Because ij-1 does not depend on r0, ·| j i (q ) |2 Òr can be defined as follows: ·| j i (q ) |2 Òri =



e2

4p 2 s2

Â jl

Ê r02 ˆ -1 -1 ij il ·exp Á -2 2 ˜ N j Nl Òr

with the form factor Fjl F jl = ·e



Ê r2 ˆ -2Á 0 ˜ Ël¯

N j Nl Ò r =

•

,

l ¯

Ë

•

•

(5.33) •

 I (cm -2 ) dte - q t due - q u dx0 dy0 4 t + (1 / 4a ) u + (1 / 4a ) 2

Ú

2

Ú

0

Ú

-•

0

È 1Ê 1 1 8ˆ ˘ exp Í- Á + + 2 ˜ r02 ˙ J j J l ÍÎ 4 Ë t + 1 / (4a ) u + 1 / (4a ) l ¯ ˙˚

Ú

-•

,

(5.34)

where  I is the impurity concentration per unit area. We can carry out three of these integrations. We find •



Ú

F jl =  I p dve - q v I jl (v ) ,





where



(5.35)

0

where n = t + u,



2

v

I11 =

dt

ÚR, 0

I12 = I11 -

v

1 Ê 1 ˆ 4 dt [ v+ + 1] 2 , 4a ÁË 2a ˜¯ l 2 R 0

Ú

v

1 Ê 1 ˆ 4 3 dt I22 = I11 [ v+ + 1] 2 + 2a ÁË 4a ˜¯ l 2 16a 2 R 0

Ú

v

dt

ÚR

3

0

8 8v 2 ˆ 1 Ê 1 Ê ˆ R = - t2 + t + vÁ1 + + + 1˜ Á ˜ Ë ¯ l l al ¯ 2a Ë al

(5.36) , (5.37)

is a quadratic polynomial of variable t. Therefore, it is easy to find the Iij’s solutions from any integral table and Eq. (5.35) can be performed numerically. We can rewrite Eq. (5.33) as

69

70

Impurity Scattering with Semiclassical Screening



or



·| j i (q ) |2 Òr =

e2 4p 2 s2

Â jl

-1 -1 il  jl F jl (q ) ,

·| j i (q ) |2 Òr =  I | xi ( l , q)|2 ,

(5.38)

where x is the average potential per impurity area density (unit V m2).

Figure 5.2 Square average Coulomb potentials per impurity area density in each subband for conditions T = 300 K. n = 106 cm–1, l = 320 Å, E = 22.44 meV. Inset shows the behavior of x Æ 0 at the origin for each subband. The peaks occur around 1 ¥ 104 cm-1 .

From Eq. (5.33) we now obtain the square average potentials per impurity area density shown in Fig. 5.2. It can be seen that the average potential vanishes at the origin and exhibits a peak which results from the assumption of a cutoff (l parameter) for the BI influence in our model. Lee, Spector, and Arora found a similar

Impurity Scattering Rate

behavior with a sine function model (SFM) for a two-dimensional system [17] in considering an abrupt cutoff of the background impurities, i.e., the impurity density is constant inside the 2D layer and zero outside the layer. The position of the maximum in x varies with the cutoff parameter l. The larger l, the bigger the x maximum closer to the origin of the q wave vector. However, except for very short l, x is only slightly sensitive to the cutoff parameter. Therefore, for the cylindrical symmetry considered here we have chosen an intermediate value of l = 320 Å. Finally we notice that x decreases with temperature since the screening has the inverse effect [Eq. (5.10)]. For the total scattering rate, we finally obtain

Â

| k efi k ¢ |2 =

r

e2 L2z

 r

| j i ( l , q , r0 ) |2 =

(

e2 L2z

(

)

 I cm -2 | xi ( l , q ) |2

)

e2 = NI cm -3 | xi ( l , q ) |2 , Lz

(5.39)

where we have implicitly expressed the dependence on l in the Fourier transform of the potential j i . By means of delta function properties we can readily derive the total intraband scattering rate 1/ti as

1 = t i (k )

ÂW (k , k ¢ ) = i

k

e2m*  k 3

NI [| xi ( l , 0) |2 + | xi ( l , 2k ) |2 ] , (5.40)

where k is the wave vector of the initial state and NI is the impurity density in the GaAs, typically 1015 cm–3. In Fig. 5.2, it is shown that |x(q)2| vanishes at q = 0 as a direct consequence of the screening and the finite range of BI influence, as discussed previously. Therefore, backward scattering only occurs for q = k – k ¢ = 2k. This is consistent with the energy-conservation law which expresses the initial state k and the final state k ¢ must lie on an equal energy-surface (EES) in the k space. Since the EES of 1DEG consists of the two discrete points on the k space in the z direction, the energy conservation law requires that an electron in the state k be scattered only to the state –k. Thus, Eq. (5.40) reduces to

1 e2m* = 3 NI [| xi (2k ) |2 t i (k )  k

(5.41)

71

72

Impurity Scattering with Semiclassical Screening

From a general standpoint, the scattering rate decreases with the carrier concentration since screening increases. In addition, we found that the rate is a weakly increasing function of l, which is understandable since for large cutoff parameter more ionized impurities, external to the wire, contribute to carrier scattering; however, the effect is less significant for remote impurities because of the decreasing influence of the Coulomb potential at large distance. In Fig. 5.3, we have plotted the scattering rate in the two bands at 77 K for two confinement conditions. The divergence at the origin is a consequence of the 1D density of states in the semiclassical approximation. The rates in the two subbands are comparable because in the confinement and temperature conditions considered here, screening is about the same for the two bands. This results from the degeneracy of the second and third subbands, which makes n1 approximately equal to n2. The rate 1/t1 of the lower band slightly exceeds the rate 1/t2 of the upper band at an intermediate value of the wave vector, because the overlap integral of the Coulomb matrix element is superior in the former case. In the conditions of nondegeneracy for the second subband, we would have obtained 1/t2 larger than 1/t1 since then n2 would have been smaller than n1. A similar behavior is obtained when the second energy level moves up with high confinement (lower curves); in this case the rate 1/t1 of the lower subband decreases due to the higher carrier occupancy and increasing screening at low energy. At higher temperature and carrier concentrations, we obtain the same trend (Fig. 5.4). The overall screening effect is more important than in the former case, but the difference between the rates of the two subbands increases with confinement as described above. As mentioned in Section 5.3, our semiclassical formalism does not account for intersubband screening. This is easily seen from Eq. (5.16) where, by considering the matrix element jq(r) between two subband levels, the screening term vanishes because of opposite parity of the wave functions. This absence of intersubband screening is an important drawback for a realistic description of scattering processes in multiband quantum wire, especially for scattering between the two degenerate subbands. In order to estimate the magnitude of this effect, we have computed the form factor for intersubband scattering in the degenerate level

2 F1221 = e-2(r0 / l ¢ )N1221 , 2

(5.42)

Impurity Scattering Rate

where

N1221

•

dt exp( -q2t - {r02 / 4[t + (1 / 4a )]}) 2 = x0 y0 (5.43) 32a [t + (1 / 4a )]3

Ú 0

which is similar to Eq. (5.27). The results are compared with the form factor Fjl for intrasubband scattering given by Eqs. (5.35)– (5.37). These form factors represent the magnitude of the bare Coulomb interaction [i.e., last term of Eq. (5.16)] between 1D electronic states and impurities. Figure 5.5 shows the form factors for inter- and intrasubbands with an energy separation E = 5.61 meV and λ = 320 Å. But the inset only shows the intersubband form factor for the different energy separations. As can be seen, intersubband scattering is negligible with respect to the intrasubband. The difference increases with level separation due to orthogonality of the wave functions.

Figure 5.3 Subband scattering rates as a function of the carrier wave vector of NI = 1015 cm–3, l = 320 Å, T = 77 K, n = 105 cm–1, E = 5.61 meV (right-hand vertical scale), E = 11.22 meV (left-hand vertical scale).

73

74

Impurity Scattering with Semiclassical Screening

Figure 5.4 Subband scattering rates as a function of the carrier wave vector for NI = 1015 cm–3, λ = 320 Å, T = 300 K, n = 106 cm–1, E = 22.44 meV (right-hand vertical scale) and 33.66 meV (left-hand vertical scale).

Figure 5.5 Comparison of inter- and intrasubband form factors with E = 5.61 meV, λ = 320 Å. Inset shows intersubband form factor for different E; from top to bottom, E is equal to 5.61, 11.22, and 22.44 meV, respectively.

References

5.5 Conclusions The degeneracy of the upper energy level in multisubband quantum wire enhances screening effects in higher subbands. For the conditions investigated here, where the energy separation is of the order of the thermal energy, screening and level degeneracy equalize the impurity scattering rates in the lower and upper subbands. For weaker confinement, we thus expect a higher rate in the lower subband than in the next few degenerate ones since the carrier occupation and screening will reduce the electron-impurity interaction in the upper subbands. The decreasing rate in upper bands is also predicted for unscreened impurity scattering where the carrier-impurity interaction is essentially determined by the overlap of the electronic wave functions in the interaction matrix element which decreases with higher quantized levels [8]. This effect is responsible for the weak intersubband scattering compared with intrasubband event. Finally, we would like to mention the weak but non-negligible influence of outside BI (described by the λ parameter) in the scattering rates.

Acknowledgments

This work was supported by the National Science Foundation grant no. DMR 86-12860 of the Material Research Laboratory. We are indebted to Dr. B. Mason for critical comments. The authors also would like to thank Hanyou Chu for his suggestions on the program.

References

1. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

2. P. Petroff, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1981).

3. A. Warren, D. Antoniadis, and H. Smith, Phys. Rev. Lett. 56, 1858 (1986). 4. T. P. Smith III, H. Aront, J. M. Hong, C. M. Knoedler, S. E. Laux, and H. Schmid, Phys. Rev. Lett. 59, 2802 (1987). 5. G. Timp, H. U. Baranger, P. deVegvar, J. E. Coumngham, R. E. Howard, R. Behringer, and P. M. Mankiewich, Phys. Rev. Lett. 60, 2081 (1988). 6. M. L. Roukes, A. Scherer, S. J. Allen, H. G. Craighead, R. M. Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett. 59, 3011 (1987).

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Impurity Scattering with Semiclassical Screening

7. H. Sakaki,  Proceedings of the International Symposium on GaAs and Related Compounds,  Oiso, Japan 1981, edited by T. Sugano (The Institute of Physics, Bristol, 1982), pp. 251–256. 8. J. Lee and H. N. Spector, J. Appl. Phys. 54, 3921 (1983). 9. J. Lee and H. N. Spector, J. Appl. Phys. 57, 366 (1985). 10. G. Fishman, Phys. Rev. B 34, 2394 (1986).

11. S. DasSarma and X. C. Xie, Phys. Rev. B 35, 9875 (1987).

12. J. J. Harris, C. T. Foxon, D. E. Lackhson, and K. W. J. Barnham, Superlattices and Microstructures 2, 563 (1986). 13. T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). 14. F. Stern and W. E. Howard, Phys. Rev. 163, 816 (1967). 15. K. Hess, Appl. Phys. Lett. 35, 484 (1979).

16. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

17. J. Lee, H. N. Spector, and V. K. Arora, J. Appl. Phys. 54, 6995 (1983).

Chapter 6

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

S. Briggs and J.-P. Leburton

Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

Transport properties of multi-subband quasi-one dimensional (1-D) structures are investigated with Monte-Carlo simulation. The confinement conditions are varied to produce resonant intersubband (polar) optic phonon scattering (RISOPS) during the transport process. The spatial degeneracy which occurs when the y- and z-energy levels overlap results in noticeable features in Reprinted from Superlattices Microstruct., 5(2), 145–148, 1989.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1989 Academic Press Limited Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

78

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

the subband carrier concentrations and the transport parameters. RISOPS is similar to Magnetophonon effects but exhibits novel features which unlike longitudinal magneto-transport, occur in absence of azimuthal symmetry.

6.1 Introduction

One-Dimensional (1D) artificial structures have become a rapidly emerging field in semiconductor physics. The ability to produce highly confined electronic systems with only one degree of freedom [1–5] has stimulated intensive research on the transport properties of these new structures [6–10]. In highly quantized 1D systems where the subband spacing is equal to the polar optic phonon (POP) energy wLO, the situation appears analogous to longitudinal magnetotransport with intersubband scattering causing magnetophonon resonance [11, 12]. However, unlike in magnetic fields, the spatial asymmetry in the confining potentials results in inequally spaced energy subbands. In addition, the confinement conditions which in principle can be achieved in 1D artificial structures are such that energy level spacing is greater than that observed in magnetically confined structures. Both these differences produce new features in the intersubband transport properties [13J. In this chapter, we investigate RISOPS in quantum wire structures. These RISOPS effects are similar to magnetophonon resonance but occur for irregularly spaced subband energies. More specifically, we investigate theoretically the two-band RISOPS when the two lowest bands are in resonance with the POP energy (Fig. 6.1a), and the degenerate RISOPS when there is a cross-over (degeneracy) of two upper bands in resonance with the bottom and (Fig. 6.1b). Because of the complexity of the system involving multiple subbands as well as a non-equilibrium electron distribution, we use a Monte Carlo simulation [14, 15] to investigate the resonant intersubband transport in 1D structures.

Model

Figure 6.1 (a) Schematic representation of simple (two subband) resonance when the position of the upper subband is varied around wLO with respect to the bottom subband. (b) Degeneracy (three subband) case with one upper subband held in resonance while the third subband is moved in and out of resonance with the bottom subband.

6.2 Model The model used in our simulation has been described in detail in a previous paper [13]; we only summarize the main features here. The confinement conditions are similar to the V-groove quantum wire FET suggested by Sakaki [6]. In the y-direction the electrons are confined with a GaAs-AlGaAs quantum well (QW) of width Ly. The z-confinement is achieved by a gate electrode perpendicular to the QW which forms a triangular electrostatic potential with an electric field Fz. The V-groove wire offers in principle the largest degree of confinement which can be controlled by external transverse electric fields (gate fields) Fz. For the y-direction we calculate wavefunctions in the infinite square well approximation while the z-wavefunctions are computed using a variational approach, with exponentially damped polynomials as the trial wavefunctions. We calculate the

79

80

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

first three y and z wavefunctions and combine them to yield the total electron wavefunctions for our simulation. In general, we include seven subbands in the model with POP and inelastic acoustic phonon scattering; however, we have also performed simulations in which we limit the model to three subbands (the y = 1, z = 1, y = 1, z = 2, and y = 2, z = 1 subbands) with POP scattering only. The fundamental reason for the simplification is it allows us to focus in on the specific processes of interest, namely intersubband optic phonon scattering between the bottom bands. Results from both models will be presented. In the present simulation all electrons remain in the Γ valley, all subbands are parabolic, and temperatures are 300 K. We assume that electrons will transfer to three dimensional (3-D) states before they reach 400 meV above bottom; this assumption limits the maximum electron energy that we can model. We compute 1-D matrix elements by integrating the normal 3-D matrix elements over the two transverse phonon wavevectors; these are used to obtain scattering rates according to Fermi’s golden rule. A small broadening is introduced at this point to smooth out the divergence in the rates due to the density of states. For each energy interval in each subband, we consider forward and backward, emission and absorption, for POPs and acoustic phonons to each possible final subband. Peaks in the scattering rate make the velocity and distribution functions sensitive to the energy separation between subbands, particularly between the first and second subbands. The Monte Carlo code is a steady-state single particle code using a direct integration technique for determining free-flight times. The electron is scattered 400,000 times with statistics collected after each event to obtain the quantities of interest.

6.3 Results

The Monte Carlo simulation predicts RISOPS when two subbands are separated by the optical phonon energy wLO. As we vary the confinement (Fig. 6.1a), the effect of phonon scattering changes as the subbands move in and out of resonance. Figure 6.3a shows the effect of changing confinement on the electron populations in subbands one and two for the seven band model. In this case, the second subband is the y = 2, z = 1 subband and the quantum well width, Ly, is varied from 245 Å to 195 Å. The confining field, Fz, is

Results

equal to 125 KV/cm, which places the y = 1, z = 2 subband over 90 meV above the bottom subband, where it has negligible effect. As Ly, changes, the electron population in band two is enhanced at the expense of the band one population. The peak in band two occupation occurs when the separation is wLO; at that point 59% of the electrons are in the first subband and 38% are in the second with the remainder in the other bands. The three band model gives essentially identical results except that band one has about 5% more electrons.

Figure 6.2 (a) Scattering mechanisms below resonance. The scattering forbidden region (F.R.) reduces the emission rate from band two and the effective absorption rate from band one. (b) The analogous situation occurs above resonance, where the scattering forbidden region reduces the effective emission rate from band two and the absorption rate from band one.

We attribute this effect to POP absorption from the bottom of band one to band two. Most electrons are at the bottom of a subband, where the density of states is the greatest, and in addition, the maximum scattering rate occurs for scattering to the bottom of a subband. Because of these two facts, the maximum transfer of

81

82

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

electrons from band one to band two is achieved when electrons at the bottom of band one can absorb a POP and scatter to the bottom of subband two.

Figure 6.3 (a) Subband electron occupation in simple resonance for the seven subband model. The horizontal axis represents the energy of subband two relative to subband one. (b) Total velocity and subband velocity for the same case. The inset shows velocity (omitting total velocity) for the three subband model.

Results

In addition to band population, we have investigated the effect of resonance on electron velocity. In general, subband velocity exhibits a minimum at resonance. Figure 6.3b illustrates the results of the seven band Monte Carlo code with the same simulation parameters as in Fig. 6.3a. The resonance effect is superimposed over a general downward trend in velocity as confinement is increased due to an increase of the overlap integrals in the scattering rate [9, 13, 15]. Band one velocity ranges from 7.0 × 106 cm/sec below resonance to 6.0 × 106 cm/sec near resonance and 6.1 × 106 cm/sec above resonance. Band two velocity initially shows a minimum of 4.9 × 106 cm/sec due to the scattering by POP emission of high momenta electrons. As resonance is approached, the emission scatters lower momenta electrons and the velocity shows a peak of 5.0 × 106 cm/sec. At resonance the velocity has a minimum of 4.5 × 106 cm/sec and above resonance it increases to 6.4 × 106 cm/sec. The total velocity shows the general downward trend, with a minimum at resonance. The velocities for the equivalent three band model are shown in the inset of Fig. 6.3b. The effects described are clearer in the three band case due to the absence of high momenta scattering between upper subbands. We attribute this velocity minimum to the influence of RISOPS on the scattering rate. Our explanation considers only absorption from band one to band two and emission from band two to band one. Although other scattering mechanisms are active, their scattering rates are monotonic functions of confinement throughout the resonance regime. Only absorption to band two and emission to band one show peaks in their rates at resonance and can explain a minimum in velocity. At resonance, the intersubband scattering rate is a maximum for both subbands and this tends to lower the velocity. Now, consider the case illustrated in Fig. 6.2a, where the subbands are above resonance. There is a region at the bottom of subband one in which intersubband scattering is forbidden. This reduces the intersubband scattering rate for subband one and consequently enhances the velocity. Also, scattering to the bottom of subband one from subband two is forbidden. This reduces the intersubband scattering rate in band two (the scattering rates show peaks when scattering to a subband bottom) and results in an increase of velocity. When the subbands are below resonance, as in Fig. 6.2b, the situation is

83

84

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

reversed. Electrons at the bottom of subband two cannot scatter to band one and electrons in subband one cannot scatter to the bottom of subband two. This enhances the velocity in both subband one and two with respect to the resonance case. However, near resonance, the emission from band two predominately scatters electrons with low momenta. Well below resonance, the emission scatters electrons with higher momenta and the velocity in subband two decreases. The overall effect is to lower scattering rates (and enhance velocity) away from resonance. As a special case of the RISOPS effect, we have investigated the situation where two upper subbands are not only in resonance with a third, but also degenerate with each other. Figure 6.1b schematically illustrates this effect when the third subband is brought into resonance with the first subband and degeneracy with the second subband. In this case, electrons are depleted from the lower resonant subband by both of the degenerate bands. Figure 6.4a shows the electron population by subband for this degenerate resonant case in the seven band model. Subbands one and two are in resonance and subband three is moved into and then out of resonance with band one. In this case, Ly is held fixed at 215 Å, and Fz is varied from 20 KV/cm to 38 KV/cm. The second subband, y = 2, z = 1, is in resonance with the y = 1, z = 1 level and the third, the y = 1, z = 2 subband is moved into and then out of resonance. As long as band three is well away from resonance, the populations for band one and two are qualitatively the same as for the simple resonance case. The population of the highest energy band (band two) is exceeds by 50% that of the lower energy band (band three). However, as band three moves into resonance, the population of band two drops by 25% and the population of band three is enhanced by 50%. As band three continues to increase in energy and move out of resonance, its population drops by over 50% and band two’s population returns to its prior value. Again, the three band model confirms these effects. The total band populations can be understood by considering the simple resonance case with the additional constraint that the total occupation of all three bands be conserved. As the third band moves into resonance, it depletes electrons from the bottom of the first band. Because of this, there are fewer electrons available to scatter to the second band; consequently its population drops.

Results

Figure 6.4 (a) Subband electron occupation for the degeneracy case. Subbands one and two are fixed in resonance while subband three is moved into and out of resonance. The horizontal axis is the energy of subband three relative to subband one. (b) Total and subband velocity for the same case. The inset shows the velocity (omitting total velocity) for the three subband model.

The velocity for the degeneracy condition is shown in Fig. 6.4b. The three band results shown in the inset are qualitatively the same

85

86

Resonant Intersubband Optic Phonon Scattering in Quasi-One-Dimensional Structures

as for the simple resonance (Fig. 6.3b); however the seven band results do not show clear minima structures. This surprising effect is due to multiple resonances with the upper subbands. For the given confinement conditions, there are permanent resonances between subbands one and three, subbands two and five, and subbands four and six. At the point where subbands one and two are in resonance, there are additional resonances between subbands three and four, subbands three and five, and subbands two and four. These additional resonances influence the distribution functions at high energy and obscure the effect on velocity caused by the degenerate three band resonance.

6.4 Conclusion

Both cases of RISOPS (i.e. single and degenerate resonances have the same qualitative effect on the subband electron occupation; the carrier drift velocity however, is qualitatively different in the two cases. This particular behavior is due to the asymmetry in the y and z confining potentials and is different from the magnetophonon effect to which RISOPS can be a priori compared. These varieties of features in the transport parameters make phonon resonance a qualitatively new effect in 1D artificial structures. Although the level of confinement we investigate is beyond current fabrication technology, lower degrees of confinement should show the same effects with higher subbands in resonance. It should be possible to experimentally observe RISOPS at intermediate temperatures, where the thermal broadening is small and the POP occupation number is non-zero.

Acknowledgment

We are indebted to Dr. E. A. Pisces for helpful encouragement. This work was supported by NSF grant no. NSF-CDR-85-22666 and the Joint Services Electronics Program. Part of the computation was performed using the resources of the National Center for Supercomputing Applications (NCSA) and the Materials Research Laboratory at the University of Illinois.

References

References 1. P. Petroff, A. Gossard, R. Logan, W. Wiegmann, Applied Physics Letters 41, 635 (1982).

2. A. Warren, D. Antoniadis, H. Smith, Physical Review Letters 56, 1858 (1986).

3. K. Kash, A. Scherer, J. Warlock, H. Craighead, M. Tamargo, Applied Physics Letters 49, 1043 (1986). 4. K. Ishibashi, K. Nagata, K. Gamo, S. Namba, S. Ishida, K. Murase, M. Kawabe, Y. Aoyagi, Solid State Communications 61, 385 (1987).

5. T. Hiramoto, K. Hirakawa, Y. Iye, T. Ikoma, Applied Physics Letters 51, 1620 (1987). 6. H. Sakaki, Japanese Journal of Applied Physics 19, L735 (1980). 7. V. Arora, Physical Review B 23, 5611 (1981).

8. J. Lee, H. Spector, Journal of Applied Physics 54, 3921 (1983).

9. J. P. Leburton, Journal of Applied Physics 56, 2850 (1984). (In this paper, the 1-D absorption rate in Eq. 16b should be corrected to include forward scattering corresponding to the q+ value in Eq. 14b.) 10. T. J. Thornton, M. Pepper, H. Ahmed, D. Andrews, G. J. Davies, Physical Review Letters 56, 1198 (1986).

11. R. Peterson in Transport Phenomena, edited by R.K. Willardson, Semiconductors and Semimetals, Vol. 10 (Academic, New York, 1975), p. 221. 12. R. Stradling, R. Wood, Journal of Physics C 1, 1711 (1968). 13. S. Briggs, J. P. Leburton, Physical Review B (in press).

14. C. Jacoboni, L. Reggiani, Reviews of Modern Physics 55, 645 (1983).

15. A. Ghosal, D. Chattopadhyay, A. Bhattacharyya, Journal of Applied Physics 59, 2511 (1986).

87

Chapter 7

Intersubband Population Inversion in Quantum Wire Structures

S. Briggs, D. Jovanovic, and J.-P. Leburton

Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

When the spacing between subbands in a quantum wire is equal to the optic phonon energy a situation occurs under longitudinal field conditions where the upper subband population is enhanced with respect to the lower subband. We focus our investigation on the case where a third, intermediate subband is located slightly below the upper subband. We use a Monte Carlo simulation and obtain a population inversion between the upper and intermediate subbands. Intersubband optical transitions with the possibility of far-infrared stimulated emission seem to be significant. Reprinted from Appl. Phys. Lett., 54(20), 2012–2014, 1989. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1989 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Intersubband Population Inversion in Quantum Wire Structures

With the continuous advance in fine line lithography, the field of one-dimensional (1D) systems has been growing rapidly for several years [1–5]. It is anticipated that the obstacles for strong confinement will soon be overcome and 1D effects will be observable at room temperature. Quantum wires with carrier confinement below 1000 Å have recently been achieved [6]. In III–V compounds at room temperature, transport is limited by polar optic phonon (FOP) scattering and Monte Carlo simulation shows that resonant intersubband optic phonon scattering (RISOPS) similar to magnetophonon resonance [7, 8] takes place in 1D structures [9].

Figure 7.1 Schematic of the two confinement conditions simulated. In the triangular potential structure (a) the y confinement is achieved with a quantum well of width Ly and the z confinement is obtained with a triangular potential due to a applied gate field Fz,(b) Parabolic potential with a quantum well width of Ly.

In this chapter we report the possible occurrence of population inversion in quantum wire structures under RISOPS conditions. In order to confirm the generality of this effect, we consider two different 1D systems characterized by different confinement conditions. In the first case (Fig. 7.1a), electrons are confined in the y direction by a GaAs-AlGaAs quantum well (QW) of width Ly while the z confinement is achieved with a gate electrode perpendicular to the QW which forms a triangular electrostatic potential with an electric field Fz. This triangular potential model is similar to the V-groove quantum wire field-effect transistor (V-FET) [10] or the modulation-doped GaAs-AlGaAs wire structures fabricated with the use of ion beam assisted etching [6]. In the second case (Fig. 7.1b) confinement is realized by a combination of a quantum well and a parabolic potential. This parabolic potential structure is similar to

Intersubband Population Inversion in Quantum Wire Structures

the GaAs-AlGaAs grating capacitor reported by Smith [11], where the triangular potential has been replaced by a quantum well. For the sake of consistency, we have kept the y variable for the quantum well confinement axis and assumed the parabolic confinement along the z direction.

Figure 7.2 Energy-level diagrams for the three simulations, y and z quantum numbers for each energy level are shown in parentheses along with the phonon energy hwLO, and the optical transition considered of energy hw. (a) Triangular potential with optical transition between the first excited y state and first excited z state, (b) Triangular potential with optical transition between the second and first excited z states, (c) Parabolic potential with optical transition between the third and second excited z states.

We focus our investigation on the case where an upper subband is separated from the bottom subband by the optical phonon energy hwLO (in resonance) and a third, intermediate subband is located slightly below the upper subband. In this case, a population inversion can be obtained between the upper and intermediate subbands. We consider three different ways of achieving this situation. In case A (Fig. 7.2a), we use the triangular potential structure with quantum well width Ly = 215 Å and confining field Fz = 23 kV/cm. This choice places the first excited y state (y = 2, z = 1) in resonance with the bottom subband (y = 1, z = 1) and the first excited z state (y = 1, z = 2) just below the first excited y state. Case B (Fig. 7.2b) uses the triangular potential structure with Ly = 150 Å and Fz = 10 kV/cm. The second excited z state (y = 1, z = 3) is in resonance with the bottom subband and the y = 1, z = 2 state is at an intermediate energy. The narrow quantum well places the excited y states at a high energy, where they have little effect on the simulation. In case C (Fig. 7.2c)

91

92

Intersubband Population Inversion in Quantum Wire Structures

we consider the parabolic potential structure with the separation between harmonic oscillator levels 12 meV and Ly = 150 Å. The parabolic potential places the fourth harmonic level (y = 1, z = 4) in resonance with the bottom level (y = 1, z = 1) and we examine transitions between the third (y = 1, z = 3) and fourth subbands. Again, the narrow quantum well width places the excited y states at a high energy, where they have little effect on the simulation. For each of these cases we have performed a Monte Carlo simulation of a multisubband system at 300 K including POP and acoustic phonon scattering [12]. In Fig. 7.3, the distribution functions for the three cases show a number of interesting features unique to 1D systems. First, the electron energy distribution in the lowest subband shows a definite bowing between the bottom of the subband and the phonon emission threshold. In addition the distribution function shows a peak at the phonon emission threshold. These effects can be explained in terms of an absence of angular randomization in 1D systems and nonlinearities in the Boltzmann transport equation at low fields [13]. There is also a strong intersubband effect due to RISOPS which causes an excess of carriers in the second subband. Because the third subband is not in resonance with the first, the distribution functions for these subbands are very similar. In case A (Fig. 7.3a) 27% of the carriers occupy the second subband and 16% the third subband. Case B (Fig. 7.3b) shows the largest inversion with 24% in the second subband and only 4% in the third subband. In case C (Fig. 7.3c) 15% of the carriers are in the second subband and 13% in the third subband. To assess the possibility of stimulated emission between the inverted subbands, we begin with the optical transition probability from an electron state ki to a state kf [14]

W (w , ki , kf ) =

p h2 e 2

1 m*2we0n 2 Aeff

¥ | · Y *f



| e ◊ — | Y i Ò | d (E f - E i - hw )d ki ...kf , 2

(7.1)

where e is the electric field polarization vector, hw the photon energy, e0 the permittivity of free space, n the refractive index, and Vc the crystal volume. We obtain the photon scattering rate lp (w)

Intersubband Population Inversion in Quantum Wire Structures

Figure 7.3 Distribution functions for the three simulations. (a) The triangular potential. Subband 1 is the bottom subband, 2 is the y = 2, z = 1 state, and subband 3 the y = 1, z = 2 state. (b) Triangular potential. Subband 1 is the bottom subband, 2 is the y = 1, z = 3 state, and subband 3 is the y = 1, z = 2 state. (c) Parabolic potential. Subband 1 is the bottom subband, 2 is the y = 1, z = 4 state, and subband 3 the y = 1, z = 3 state. For the sake of clarity, the y = 1, z = 2 subband has been omitted.

by summing W(w, ki, kf) over all electron initial and final states and replacing the sum over initial states with the electron linear density. After accounting for the total number of wires in the crystal, the total rate for one transition is

93

94

Intersubband Population Inversion in Quantum Wire Structures



lp (w ) =

p h2 e 2

1 A m we 0n eff *2

¥ | · Y f*

2

| e ◊ — | Y i Ò | d (E f - E i - hw )ne,i , 2

(7.2)

with the effective area, Aeff, being the square of the wire spacing and ne,i the linear electron density for the ith subband. Here we have made use of the normalization condition for the distribution function, f(k). This rate considers only absorption or emission; in the cases that follow we obtain total rates by considering both types of transitions between subbands. We also have calculated the scattering rate for electrons and find the rate to be several orders of magnitude less than the photon scattering rate. The optical matrix element has been calculated for the three cases in Fig. 7.2. For each case we assume a linear electron density of 2 ¥ 106 cm–1, a photon energy of 10 meV, and an effective area of 10–10 cm2 which corresponds to a wire spacing of 1000 Å. For the absorption coefficients, we assume a broadening in the electronic states of 1 meV. In case A the optical matrix element for the transition (y = 2, z = 1 Æ y = 1, z = 2) is zero because the e ◊ — operator only acts in one direction (along e) whereas the wave functions are orthogonal in two directions (both y and z). Case B has the potential for stimulated emission; the transition of interest is y = 1, z = 3 Æ y = 1, z = 2. To calculate the net stimulated emission rate we assume the photon polarization is along the z axis and replace ne,i in Eq. (7.2) with the population difference between subbands, ne,upper – ne,lower or 4 ¥ 105 cm–1. The scattering rate for photons in this structure is 4.4 ¥ 1012 s–1 which corresponds to an absorption coefficient a of 510 cm–1; this is weak but should make far-infrared (FIR) stimulated emission observable in highly packed quantum wire structures. In case C the population inversion occurs between the states y = 1, z = 4 and y = 1, z = 3. In contrast to the triangular potential structure where all intersubband transitions have different energies, optical transitions with the same w happen between any two adjacent levels and all electrons can participate in optical transitions. By summing Eq. (7.2) over all subbands, our estimate of the photon scattering rate is relatively large with a value of 8.7 ¥ 1013 s–1 for an a of 10000 cm–1. Therefore, we expect stimulated emission to be overwhelmed by the overall intersubband absorption.

References

In conclusion, we have used Monte Carlo simulation to investigate a population inversion due to RISOPS in quantum wire structures. We have calculated optical transition rates for three different confinement conditions and predict significant stimulated FIR emission in the triangular potential configuration and a very large absorption coefficient in the parabolic potential structure. The authors are indebted to N. Hoionyak, Jr., for helpful discussions. This work was supported by National Science Foundation grant no. NSF-ECS-85-10209 and the Joint Services Electronics Program. Part of the computation was performed using the resources of the National Center for Supercomputing Applications (NCSA).

References

1. P. Petroff, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982).

2. A. Warren, D. Antoniadis, and II. Smith, Phys. Rev. Lett. 56, 1858 (1986). 3. K. Kash, A. Scherer, J. Worlock, H. Craighead, and M. Tamargo, Appl. Phys. Lett. 49, 1043 (1986). 4. K. Ishibashi, K. Nagata, K. Gamo, S. Namba, S. Ishida, K. Murasc, M. Kawabe, and Y. Aoyagi, Solid State Commun. 61, 385 (1987). 5. G. Timp, H. U. Baranger, P. deVegvar, I. E. Cunningham, R. E. Howard, R. Behringer, and P. M. Mankiewich, Phys. Rev. Lett. 60, 2081 (1988).

6. M. L. Roukes, A. Scherer, S. J. Allen, H. G. Craighead, R. M. Ruthcn. E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett. 59, 3011 (1987).

7. Peterson, in Transport Phenomena, edited by R. K. Willardson, Semiconductors and Semimetals, Vol. 10 (Academic, New York, 1975), p. 221. 8. R. Stradiing and R. Wood, J. Phys. C 1, 1711 (1968).

9. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989). 10. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

11. T. P. Smith III, H. Arnot, J. M. Hong, C. M. Knoedler, S. E. Laux, and H. Schmid, Phys. Rev. Lett. 59, 2802 (1987). 12. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988).

13. S. Briggs and J. P. Leburton, Phys. Rev. B (to be published).

14. H. B. Bebb and E. W. Williams, in Transport and Optical Phenomena, edited by R. K. Willardson, Semiconductors and Semimetals, Vol. 8 (Academic, New York, 1972), p. 184.

95

Chapter 8

Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

D. Jovanovic, S. Briggs, and J.-P. Leburton

Beckman Institute for Advanced Science and Technology and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We investigate the effects of inelastic optical- and acoustic-phonon scattering in quasi-one-dimensional systems with up to 20 subbands. We use a direct-integration Monte Carlo method to study intersubband scattering effects when the subband spacing approaches resonance (nDE being the optical-phonon energy of 36 meV). Simulations performed on first-, second-, and third-order resonant structures at temperatures ranging from 150 to 300 K reveal the appearance of dissipative transport properties. Specifically, nonlinearities in Reprinted from Phys. Rev. B, 42(17), 11108–11113, 1990. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1990 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

the electron distribution induce velocity fluctuations analogous to the longitudinal magnetophonon effect and generate population inversions between adjacent subbands.

8.1 Introduction

Contemporary confinement capabilities are approaching the point where subband-dependent phenomena are experimentally realizable in quasi-one-dimensional (1D) systems [1–4]. Ismail et al. [5] recently reported on the observation of Shubnikov–de Haas–type oscillations in the longitudinal resistance of electrostatically confined 1D systems. This observation is important for two reasons. It shows that, despite inherent irregularities in the fabrication of a set of 100 parallel quantum wires, fluctuations in the confining potential are not a cause of significant quantum-effect smearing, and, second, oscillations persist at temperatures as high as 77 K. Physically, the presence of subbands in low-dimensional systems is a quantummechanical consequence of transverse confinement and can be expected to introduce subband-dependent effects in dissipative transport processes [6, 7]. These effects can become quite significant as the degree of confinement increases with improved technological resolutions. Specifically, when the intersubband spacing approaches kT at room temperature, novel quantum transport [8] and optical [9] properties are expected to occur. The purpose of this investigation is to discuss the effects of resonant intersubband optical-phonon scattering [10] (RISOPS) in 1D systems. Two subbands are defined to be in resonance when the energy spacing between them approaches the polar optical-phonon (POP) energy (hwLO = 36 meV for GaAs). Of specific interest are highorder resonances for which one or more nonresonant subbands lie between the two bands in resonance. Such structures are found to generate transport phenomena up to the third order configuration. As the spectrum of 1D subbands is passed through resonance, a velocity minimum similar to the longitudinal magnetophonon (LMP) effect occurs at both 150 and 300 K. For device applications, such an effect carries the potential for negative differential resistance. In addition, the resonant configuration is found to maintain a population inversion between the resonant subband

Model

and a lower adjacent subband. To investigate these phenomena, we perform single-carrier multisubband Monte Carlo simulations of an idealized GaAs-AlxGa1–xAs 1D structure. Inelastic polar optical and acoustic [11] phonons are taken as the dominant scattering mechanisms to enable simulations from 150 K to room temperature. Appropriate broadening of the 1D density of states is included to account for quantum effects in the electron-phonon interaction. Low confinement structures are simulated by allowing up to 20 subbands to be modeled at once. This is accomplished primarily by employing a recursive algorithm for computing harmonic-oscillator matrix elements and a novel “on-line” method for updating phasespace information in the Monte Carlo code.

8.2 Model

We model transverse confinement via an infinite squarewell approximation to a heterojunction quantum well and an electrostatic harmonic-oscillator potential. Although experimentally unrealistic, the square-well potential does not seriously affect the accuracy of our model since it is the energy eigenspectrum rather than the nature of the wave functions that determines the overall behavior of quasi-1D systems. By considering only weak harmonicoscillator potentials, the nature of the eigenspectrum is dominated by harmonic-oscillator eigenenergies and the influence of the highly confined square-well potential is minimized. Furthermore, the inclusion of the square-well potential, as opposed to a more realistic electrostatic quasitriangular potential, introduces a significant degree of computational simplicity into our code. Both of our potential profiles are distributed over many lattice constants, thereby allowing use of the effective-mass approximation. The simulated energy space extends 400 meV above the Γ-valley bottom since we assume that electrons occupy 3D states above this limit. A similar model [12] incorporating nonparabolicity and intervalley scattering demonstrates the validity of our approximations for small longitudinal fields (Fx £ 500 V/cm). Although important for studying the effects of disorder and localization, impurity scattering is neglected in the present simulation since these types of scattering events are rare in modulation-doped GaAs structures

99

100

Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

[13, 14]. The above approximations are primarily invoked to ease the computational requirements involved with simulating systems with up to 20 subbands. Wave functions for the transverse dimensions are solved analytically and the intersubband scattering rates are subsequently computed in the Born approximation. The quantumwell wave function reads

È2˘ y (z) = Í ˙ Î Lz ˚

1/ 2

Èp z ˘ sin Í ˙ , Î Lz ˚

(8.1)

where Lz represents the width of the well. The quantum number has been purposely omitted since we consider only one quantumwell state throughout our simulation. Intersubband scattering then occurs between harmonic-oscillator levels with analytical wave functions given by

È m *w ˘ fn ( y ) = Í 2˙ Î p h(n!) ˚

where

1/ 4

Hn (x )e -x

2

/4

,

x = (2m * w / h )1/2 y.

(8.2) (8.3)

Here, w is the proper frequency of the harmonic oscillator and n is the index of the quantum state. Electron transport is driven in the x direction by a longitudinally applied field, Fx, which gives rise to a weak field-dependent Airy-potential spectrum. By restricting our simulations to low fields, quantum effects arising from Fx are negligible and free-electron wave functions within the effectivemass approximation suitably describe electron transport. Therefore, the spectrum of subband energies is given by

E n (k x ) =

p 2 h2

2m * L2x

+ hw (n + 1 / 2) +

h2kx2 , 2m *

(8.4)

where kx is the longitudinal wave vector. Again, note that the index corresponding to the quantum-well eigenspectrum is explicitly omitted since only one subband is considered for this potential. From these wave functions and energy spectra, intersubband transition probabilities are computed for the inelastic POP interaction using the Frohlich polaron formalism [15], and for acoustic phonons via deformation-potential scattering. Recent investigation [16] on phonon dispersion in confined systems indicates that optic modes

Model

can be significantly quantized in quasi-1D structures. However, as we are interested in the first-order effects of transport processes, we neglect the particular influence of phonon confinement and restrict our study to bulk modes. Using Fermi’s golden rule, the transition probabilities are given by

2p | Mnn ¢ ( k , k ¢ )|2 d (E( k ¢ ) - E( k ) ± hw q ). (8.5) h Here, k and k′ represent the initial and final wave vectors and n and n′ represent initial and final subbands, respectively. The 1D matrix elements, Mnn′(k, k′), can be rewritten with a variable transformation as



where

S(k , k ¢ ) =

Mnn ¢ (q) = Vqd k

F (qz ) =

x

- k ¢x , ± q x

F (qz )Gnn ¢ (q y ),

sin(qz Lz / 2) ± iqz Lz /2 e , qz Lz / 2 p - (qz Lz / 2) p2

2

2

Ú

Gnn ¢ (qy ) = dyfn* ¢ ( y )fn ( y ) e

± iqy y

,

(8.6) (8.7) (8.8)

and Vq is a coupling constant dependent on the type of interaction. The ± in the above equations indicate phonon absorption ( – ) and emission ( + ). Total scattering rates for each subband are then computed by summing over the final states giving

Vol 1 = t n (E ) h(2p )2

Â Ú dq | M

2 nn ¢ (q )|

n¢

¥ d (E( k ¢ ) - E( k ) ± hw q ).



(8.9)

Of particular importance in simulating 1D multisubband systems at high temperatures is the issue of collision broadening. Specifically, the broadening of the phonon energy, dEph, is a strong impediment to the observation of quantum effects at room temperature. A thorough treatment would tackle this problem by self-consistently [17] computing the polaron self-energy. Calculations of ImS for a single subband, however, indicate that scattering rates computed with a prebroadening 1D density of states give a good estimate to those obtained by the self-energy method [18]. We compute these prebroadened densities of states by convolving their exact form with a Gaussian distribution and taking dEph = 2.3 meV at 300 K and dEph = 1.6 meV at 150 K.

101

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

The computation of the 1D matrix elements presents the major bottleneck in our simulations. This step involves computing a table of the 1D matrix elements as functions of qx. These are obtained numerically by a routine that performs a double integral of Eq. (8.6) over qz and qy. The input to this routine is a table of relevant values for qx that are varied over 3 orders of magnitude (7 ¥ 104 £ qx £ 1.5 ¥ 107 cm–1) for acoustic phonons and 7 orders of magnitude (1 ¥ 100 £ qx £ 1 ¥ 107 cm–1) for polar optical phonons. Limits on qx arise because below 7 ¥ 104 cm–1, the acoustic-phonon-dispersion relation is essentially independent of qx while 1.5 ¥ 107 cm–1 represents the maximum electron momentum limited by our energy space. A 20-subband system requires 420 such double integrations to fully model the effects of intersubband scattering. The computational overhead associated with these double integrations is greatly alleviated by the choice of a harmonic-oscillator confining potential that permits an efficient algorithm for computing the harmonic matrix elements, Gnn′(qy). Specifically, we generate Hermite polynomials recursively via Hn+1(x) = xHn(x) – nHn–1(x)

(8.10)

thereby introducing appreciable vectorizability into our code. Using this technique, we obtain a speedup of 30% over the nonvectorized code when run on a CRAY-2. Also, storage of the 2D matrix elements is facilitated by the reversibility of phonon emission and absorption processes between subbands [i.e., Mnn′(k, k′) = Mn′n(k′, k)]. The table of 1D matrix elements is then used as an input file to a routine that computes the total scattering rates in the manner specified by Eq. (8.9). The energy range for 1/tn(E) varies from 0 to 400 meV and is divided into 2400 intervals. Due to memory limitations, the finalstate information is suppressed at this stage and recomputed later in the Monte Carlo code. The total 1D scattering rates for a 20-subband second-order resonant system are shown in Fig. 8.1. Each line in Figs. 8.1a and b represents the total scattering rate for a particular subband, including both intrasubband and intersubband scattering processes, with the inset in Fig. 8.1a detailing the total rate (solid) for the first subband along with a decomposition into phonon emission (dash) and absorption (dash-dot) rates. All the rates show the characteristic peaking due to the 1D density of states with the large and smaller peaks corresponding to intrasubband POP emission

Model

Figure 8.1 (a) Total scattering rates for a 20-subband 1D system with Ly = 150 Å and a second-order resonant harmonic-oscillator energy spectrum (hw = 18 meV). The rates are computed for T = 300 K with a broadening energy dEph = 2.3 meV. The inset shows the emission (dashed) and absorption (dot-dashed) rates along with the total rate for the first subband. Intrasubband POP emission becomes dominant for energies hwLO above the subband bottom when final scattering states become available. (b) Total scattering rates for the same confinement as (a) but with T = 150 K and dEph = 1.6 meV. In both figures the origin of the energy scale is taken as the G-valley bottom.

103

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

and absorption, respectively. Secondary peaks that are smaller in magnitude occur at energies where intersubband transitions become possible. The effect of acoustic-phonon scattering is not noticeable due to the relatively weak coupling of the deformation-potential interaction. Differences between the rates at 300 and 150 K are accounted for by the phonon occupation number and broadening. Since both of these parameters decrease with temperature, the scattering rates at 300 K display broader peaks and a higher overall rate. Figure 8.1b shows that both phonon emission and absorption processes are still favorable at 150 K, despite the lower phonon occupation. This fact, coupled with the smaller broadening, implies that the effects of intersubband scattering will be more resolved at the lower temperature.

8.3 Monte Carlo Method

Once a table of scattering rates is obtained, a direct integration Monte Carlo solution to the Boltzmann equation is used to track electronic motion under the influence of Fx. In the direct integration method, the probability that an electron in subband n, which suffered a collision at time t = 0, scatters at time t is given by

- ln(r ) =

Ef

Ât

E = Ei

1 Dt (E ), ( E ) n

(8.11)

where r is a random number chosen from a uniform distribution. At invocation, the Monte Carlo program requests the longitudinal electric field, Fx, and an initial state for the electron. The electron is placed in the initial state and allowed to undergo 6000 scattering events to allow independence of its initial conditions. Statistics are collected for the next 1000000 scattering events to guarantee that the electron achieves steady-state conditions. After each free flight, a bin for the final wave vector, k′x, is updated and the electron is allowed to scatter by randomly choosing a scattering agent (acoustic or polar optic phonon). The final-state information is then computed for the chosen scattering process, a new initial wave vector is randomly chosen, and the electron is subsequently launched on another free flight. The computation of final-state information “on line” after each free flight is a departure from other multisubband Monte Carlo

Results

methods. This is necessary to accommodate the extensive memory requirements associated with simulating 20 subbands. Storing all the scattering-rate information in memory, as in the method proposed by Briggs et al. [8], would require 30 Mb in our simulation. By storing only the total rates, we reduce the memory utilization to 500 Kb at the expense of making 80 relatively simple computations after each scattering event. What makes Monte Carlo simulations of quasi-1D devices particularly affordable is the restricted phase space found in 1D systems. Since the outcome of a scattering event can be forward emission, forward absorption, backward emission, or backward absorption, and since there are only two possible wave vectors corresponding to each energy, all possible values of Dt(E) can be stored in a lookup table. This results in a significant speeding up of the integration over 2D and bulk simulations. The statistics generated by the Monte Carlo code are tables of before-scattering carrier concentrations, nb,n(E) are then obtained via [19]

fn (E ) =

1 t n (E ) nb,n (E ), g1D (E ) t 0

(8.12)

where g1D(E) is the 1D density of states and t0 is an appropriate normalization constant that preserves the relative carrier concentrations over the entire set of subbands. A typical set of distribution functions is shown in Fig. 8.2 for the same confinement conditions as in Fig. 8.1. Carrier velocities and other properties of interest are obtained by averaging over the distribution functions for each subband.

8.4 Results

We apply our simulation to study the effects of RISOPS on low field transport at 150 and 300 K, where the former temperature represents near optimal conditions for observing velocity oscillations analogous to the longitudinal magnetophonon (LMP) effect [20, 21]. In general, low field distribution functions at 300 K show a high degree of nonlinearity just below the POP emission threshold as indicated in Fig. 8.2a. This is attributed to the sharp peaking in the scattering rates for both absorption and emission, and the absence of angular randomization in 1D systems, both of which lead to depleted carrier

105

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

Figure 8.2 Distribution functions for the second-order resonant system with Fx = 50 V/cm at T = 300 K and (b) T = 150 K. Each curve represents the relative distribution function in a particular subband with the energy origin lying at the G-valley minimum. For clarity, only the first six subbands are shown.

populations in the subthreshold regions. As electrons approach the resonant energies, their tendency to scatter by POP increases an order of magnitude and, if the broadening is large enough, they

Results

scatter before reaching Eop, thereby causing subthreshold valleys in F(E). This effect is evident by comparing the distribution functions at 150 K with dEph = 1.6 meV (Fig. 8.2b) to those at 300 K with dEph = 2.3 meV (Fig. 8.2a). The larger broadening at 300 K allows electron scattering well below the emission threshold, causing the depletion to extend to lower energies. In addition, POP absorption is more likely at 300 K, which gives rise to secondary minima such as those 18 meV above the first subband in Fig. 8.2a. The peaks in the distribution functions just at Eop reflect POP absorptions from the bottom of the subband with a propagation of the peaking behavior to higher energies. Figure 8.3 demonstrates that at higher longitudinal fields, the electrons have more of a tendency to accelerate through the threshold regions without scattering, as evidenced by the relatively small subthreshold depopulation.

Figure 8.3 Second-order resonant distribution functions with the same parameters as in Fig. 8.2 except with Fx = 200 V/cm at 300 K.

Another interesting feature of 1D distribution functions is the occurrence of intersubband population inversions under RISOPS conditions. This phenomenon has been explored in a previous paper [9] and found to have significant potential for far-infrared stimulated emission. Population inversion occurs for the second-order resonant

107

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

configuration at 150 K (Fig. 8.2b) between the third (resonant) and second subbands with relative occupancies, Orel, of 19.9% and 19.4%, respectively. The effect also shows up for the third-order configuration at 150 K (subband 4, Orel = 14.0%; subband 3, Orel = 11.8%). It should be noted that although population inversions exist between the harmonic-oscillator subbands discussed here, they tend to be weaker than those in systems with irregularly spaced subbands [22]. This is due to the propagation of the resonant intersubband coupling to the upper equally spaced subbands resulting from the harmonic-oscillator potential. This is particularly noticeable at room temperature, where population inversions do not occur because the higher absorption rate transfers electrons to the upper subbands more effectively, thereby smoothing out intersubband population anomalies. In addition, the potential for measuring stimulated emission between harmonic-oscillator subband should be reduced due to optical reabsorption. The behavior of electron velocity with transverse harmonicoscillator confinement is shown in Fig. 8.4. Note that the general increase in carrier velocities for both temperatures is consistent with the trend predicted by Sakaki [23] since, as the intersubband energy separation increases, upper subbands become excluded from the scattering processes and a general increase in carrier velocity with confinement is observed. Although intrasubband scattering rates tend to increase with confinement [8], their effect on carrier velocities in the regime of Fig. 8.4 is offset by the reduction of intersubband POP scattering. The predominant features of Fig. 8.4 are the velocity fluctuations at resonant intersubband energy separations. These fluctuations, in the form of velocity minima, are a result of the strong coupling of the POP scattering mechanism to peaks in the 1D density of states (DOS). They are somewhat analogous to LMP oscillations of the longitudinal resistance observed in magnetically confined systems. The explanation of this phenomenon for magnetically confined systems depends on a nonmonotonic scattering rate and the dominance of the optical-phonon interaction over other mechanisms, both of which occur for electrostatically confined quasi1D systems in GaAs. At resonance, intersubband POP scattering is dominated by transitions between subband minima [10]. Since both the initial and final states in these processes correspond to peaks

Results

Figure 8.4 Electron velocity as a function of subband energy separation at (a) T = 300 K and (b) T = 150 K. Both sets of data are taken for a system with Ly = 150 Å and Fx = 50 V/cm. The arrows indicate the position of high-order RISOPS (i.e., nDE = hwLO).

109

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

in the 1D density of states, a significant increase in the overall POP scattering rate occurs and gives rise to a velocity minimum at resonance. Configurations above and below resonance generally have lower scattering rates due to the absence of the strong peakto-peak intersubband transitions. This effect has been examined previously [22] for an electrostatically confined two-band model that demonstrated the decrease of average kinetic energy in the lower band and the subsequent increase in average kinetic energy of the upper band when the confinement is passed through resonance. The system considered here is just a generalization of the twoband model and can be expected to behave in a similar manner. Figure 8.4a shows that this oscillatory behavior is significantly less pronounced at room temperature than at 150 K (Fig. 8.4b). The explanation is due to the larger DOS broadening at 300 K, which produces a smaller relative increase in the scattering rates under RISOPS conditions. Several interesting features of Fig. 8.4a are the split velocity minima near the first-order resonant configuration and the shift of the velocity minimum to a higher confinement energy for the third-order resonant configuration. A similar peak splitting was found in the transverse magnetoresistance by Warmenbol et al. [24], who attribute it to the separate contributions of LO emission and absorption to rxx in systems with relatively small broadening and low longitudinal fields. However, we believe that shift of the velocity minimum for the third-order configuration is due to poor resolution between resonant and near-resonant confinements, which is caused by the large broadening at room temperature coupled with the proximity between subbands (hw = 12 meV). For this reason we omit data for resonant configurations higher than third order.

8.5 Conclusion

With the aid of Monte Carlo simulation, we have demonstrated the emergence of intersubband dissipative transport in quasi1D systems and generalized RISOPS to high orders. For parabolic confining potentials, velocity fluctuations and intersubband population inversions were shown to be quite prevalent at 150 K and significantly reduced, yet still observable, at room temperature. Such phenomena are attributed to the strong coupling between resonant

References

subbands and other features unique to dissipative transport in parabolically confined quantized systems. Generally, these effects can be externally controlled via modulation of the confining potentials, which points to their application in semiconductor devices with nonlinear transport and optical characteristics.

Acknowledgments

This work was supported by the Joint Services Electronics Program under grant no. N00014-90-J-1270. All of the computations were performed using the resources of the National Center for Computational Electronics (NCCE) and the National Center for Supercomputing Applications (NCSA) at the University of Illinois.

References

1. P. Petroff, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982).

2. T. Hiramoto, K. Hirakawa, Y. Iye, and T. Ikoma, Appl. Phys. Lett. 51, 1620 (1987). 3. M. Roukes, A. Scherer, S. Allen, H. Craighead, R. Ruthen, E. Beebe, and J. Harbison, Phys. Rev. Lett. 59, 3011 (1987).

4. E. Colas, E. Kapon, S. Simhony, H. Cox, R. Bhat, K. Kash, and P. Lin, Appl. Phys. Lett. 55, 867 (1989). 5. K. Ismail, D. Antoniadis, and H. Smith, Appl. Phys. Lett. 54, 1130 (1989). 6. S. Datta and M. McLennan, in Nanostructure Physics and Fabrication, edited by M. Reed and W. Kirk (Academic, Boston, 1989). 7. T. Yamada and J. Sone, Phys. Rev. B 40, 6265 (1989).

8. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988).

9. S. Briggs, D. Jovanovic, and J. P. Leburton, Appl. Phys. Lett. 54, 2012 (1989). 10. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989). 11. V. Arora, Phys. Rev. B 23, 5611 (1981).

12. S. Briggs and J. P. Leburton, unpublished.

13. H. Stormer, A. Gossard, and W. Wiegemann, Appl. Phys. Lett. 39, 912 (1981). 14. G. Fishman, Phys. Rev. B 36, 7448 (1987).

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Intersubband Resonant Effects of Dissipative Transport in Quantum Wires

15. H. Fröhlich, H. Pelzer, and S. Zienau, Philos. Mag. 41, 221 (1950). 16. M. Stroscio, Phys. Rev. B 40, 6428 (1989).

17. K. Kim, B. Mason, and K. Hess, Phys. Rev. B 36, 6547 (1987).

18. S. Briggs, B. Mason, and J. P. Leburton, Phys. Rev. B 40, 12001 (1989). 19. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).

20. V. Gurevich and Y. Firsov, Zh. Eksp. Teor. Fiz. 47, 734 (1964) [Sov. Phys.– JETP 20, 489 (1965)]. 21. P. Vasilopoulos, P. Warmenbol, F. Peeters, and J. Devreese, Phys. Rev. B 40, 1810 (1989).

22. S. Briggs, D. Jovanovic, and J. P. Leburton, Solid State Electron. 32, 12 (1989); 32, 1657 (1989). 23. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

24. P. Warmenbol, F. Peeters, and J. Devreese, Solid State Electron. 32, 12 (1989); 32, 1545 (1989).

Chapter 9

Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

Dejan Jovanovic,a Jean-Pierre Leburton,a Khalid Ismail,b Jeffrey M. Bigelow,c and Marcos H. Degania,c aBeckman

Institute for Advanced Science and Technology and the Coordinated Science Laboratory, University of Illinois, Champaign-Urbana, Illinois 61820, USA bIBM Watson Research Center, Yorktown Heights, New York 10598 and Department of Electronics, Faculty of Engineering, Cairo University, Egypt cDepartamento de Fisica e Ciência dos Materias, Instituto de Fisica e Química de São Carlos, Universidade de São Paulo-Caixa Postal 369, 13560 São Carlos, São Paulo, Brazil [email protected]

We investigate the occurrence of resonant intersubband optic phonon scattering in an array of quantum wires at high temperatures. Selfconsistent solutions of the Schrödinger and Poisson equations with Reprinted from Appl. Phys. Lett., 62(22), 2824–2826, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

114

Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

a subsequent Monte Carlo analysis of intersubband scattering reveal a strong resonant polar optic phonon coupling between subbands at the gate bias indicated by the experimental data. The observation of resonant scattering is only possible because the quasi-1D form of Gauss’ law shows an opposite trend to that for 2D systems in the dependence of the eigenenergy spectrum on gate bias. The physics of one-dimensional (1D) systems has attracted much attention in recent years. Quantum effects such as conductance fluctuations, quantized conductance and electron interferences are now routinely observed in well-prepared high mobility samples [1]. Experimental investigations of these phenomena require cryogenic conditions and small geometries in order to provide subband separations in excess of kT and avoid phase-breaking mechanisms during transport. With increasing temperature, phonon scattering destroys the coherence of the electron wave, and thermal broadening of the carrier distribution function erases the resolution between adjacent quantized 1D levels. An interesting high-temperature effect is resonant intersubband optic phonon scattering (RISOPS) [2–5] which occurs when the energy separating two 1D subbands, DEmn (m and n are subband indices), is equal to the polar optic phonon energy, hwLO, (Fig. 9.1). The nature of RISOPS is somewhat analogous to the longitudinal magnetophonon effect proposed by Gurevich and Firsov [6] except that in quantum wires, confinement is achieved purely by structural or electrostatic means with no angular symmetry in the confinement. The effect is strengthened by the 1 / DE mn - hw LO dependence of the quasi-1D intersubband scattering rates which enhances momentum randomizing transitions between resonant subbands and results in localized minima of the longitudinal conductivity. Monte Carlo simulations of longitudinal transport in quantum wires [3] have indicated that resonant intersubband scattering is most pronounced around T = 150 K with decreasing intensity at both higher and lower temperatures. This behavior is a result of the tradeoff between the requirement of significant phonon absorption which is absent at low temperatures and thermal broadening which washes out the oscillatory behavior at high T.

Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

Figure 9.1 Schematic representation of resonant intersubband optic phonon scattering in a quantum wire. Resonance occurs when two subbands are separated in energy by hwLO.

Recently, Ismail [7] observed a negative differential transconductance (NDT) in the I–V characteristic of a set of ten parallel quantum wires (Fig. 9.3). The wires were 2 mm long and were fabricated by initially growing a modulation doped GaAs/AlGaAs heterojunction with MBE to confine the electrons in a plane parallel to the heterointerface. Confinement in the remaining transverse direction was performed by deep mesa etching with a nominal mesa width of 300 nm and a periodicity of 600 nm between wires (Fig. 9.2). A heavily diluted crystallographic wet etch was used to achieve nearly perfect control and uniformity over the wire array [8, 9]. Although the measurements reported here were conducted in the dark, the samples were also tested under strong illumination conditions, where the DX centers in the doped Al0.3Ga0.7As layer are fully saturated, and the same NDT was observed. The position of the NDT, however, shifted because of a threshold voltage shift due to the increase in carrier density. This rules out any connection between the NDT and trapping in the Al0.3Ga0.7As layer. The NDT effect is unusual in that it occurs over a large range of temperatures with maximum intensity around T = 175 K (Fig. 9.3, inset) which suggests a dissipative mechanism similar to RISOPS may be responsible for the NDT. Indeed, present experimental evidence supports the 1D nature of the effect since the Hall bars made of the same materials showed no evidence of NDT.

115

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Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

Figure 9.2 Experimental device cross section. Electrons are confined just below the Al0.3Ga0.7As-GaAs interface by the vertical heterojunction potential and the opposing transverse Schottky barriers.

Figure 9.3 Conductance-gate voltage characteristic of an array of 10 quantum wires at T = 150 K. The experimental results are displayed (solid line, left axis) along with results from Monte Carlo simulations incorporating only polar optical scattering (triangles, right axis). The inset shows the peak-to-valley ratio of the negative differential transconductance as a function of temperature (after Ref. [7]).

In the relatively wide system encountered here, the separation between successive 1D sublevels is generally no more than 3–4 meV which, given the optic phonon energy (hwLO = 36 meV in GaAs), should correspond to high order RISOPS transitions hwLO/ DEn,n–1 ≃9–12). One would therefore expect that, over the voltage range indicated, the small subband separation would result in a

Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

I–V characteristic exhibiting several NDTs. Instead, the presence of a single prominent NDT in the experimental data indicates that the role of resonant phonon scattering needs to be assessed on a rigorous basis, specifically for the device configuration shown in Fig. 9.2. In order to address the issue, we have performed a self-consistent 2D Schrödinger–Poisson simulation of the experimental device and examined the influence of intersubband polar optic scattering. We solve the Schrödinger equation using a split-time propagation scheme in which a trial wavefunction y(y, z) is propagated in imaginary time increments such that the pure ground state is eventually extracted. Excited states are maintained throughout the simulation by application of the Gram-Schmidt algorithm, and all levels are used to compute the charge density for the Poisson equation. The details of this approach have been previously discussed [10–12]. In the present structure, electrons are confined to the region just below the heterojunction by a quasitriangular potential in the z-direction and a quasiharmonic oscillator potential in the y-direction. The confining potentials are a result of the band-bending across the heterojunction and the back-to-back lateral Schottky barriers, respectively. From Fig. 9.4a, when VG is swept forward, the eigenenergy spectrum is lowered, and approaches the Fermi energy (0 eV on the vertical scale) resulting in a subsequent enhancement of the carrier population and a compression of the eigenenergies. The lateral compression is similar to the behavior observed by Laux and Stern [13] in the sense that lowering the Schottky barrier also opens the lateral quasiharmonic oscillator potential. The decrease in the separation of quasitriangular potential eigenenergies with increasing gate bias (and electron density) shows a trend opposite that for 2D inversion layers [14]. This effect is a pure expression of Gauss’s law in quasi-1D electrostatically confined systems where the lateral variation of the electrostatic potential induces a vertical expansion of the electron charge [12] (i.e., the vertical field decreases with gate bias). In light of these last results, we calculated the scattering rate between all resonant subbands due to LO phonon absorption. Since the source-drain bias is fixed at 2 meV [7], the electrons never reach the polar optic emission threshold by drift alone thereby excluding any hot electron effects. Instead, intersubband polar optic absorption

117

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Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

scatters the electron to higher subbands where subsequent emission is highly probable. For these intersubband scattering processes, a resonant subband structure corresponds to optimal momentum randomization and gives rise to conductance minima as pointed out by Briggs et al. [2, 3].

Figure 9.4 (a) Eigenenergies of the first 15 states as a function of gate bias (T = 150 K, dashed line: {1,n} eigenenergies; solid line: {2,n} eigenenergies). The arrows indicate relevant resonant transitions, (b) The density weighted coupling as a function of gate bias for resonant intersubband transitions. The various transitions are identified in the legend.

As the intersubband LO phonon absorption rates depend primarily on the scattering matrix elements between 1D subbands, we calculated form factors of the various resonant polar optic

Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

transitions [15] and weighted them by the carrier density of the lower interacting level: Mnn ¢ = -



Ú

•

En

d E g1D (E - E n ) f (E )

2

e iq^ ◊t ^ ¥ dq ^ ·n¢ | | nÒ , q^

Ú



(9.1)

where r^ = (y, z) and q^ = (qy, qz). The reason for introducing the density factor in Eq. (9.1) is that a strong resonant coupling may have little influence on the transport properties if the level occupancy is small. We use the population of the lower interacting level for the weight factor since carrier absorption is the dominant process which drives resonant scattering. The density weighted form factor was computed for the relevant resonant transitions over the experimental biasing range (Fig. 9.4b). The nomenclature used for representing the states reflects the number of antinodes of the wave functions in the vertical and horizontal directions. For example, the {2,1} state refers to the second quasitriangular state and the first quasiharmonic oscillator state. As a result of the matrix elements calculated in Eq. (9.1), the resonant intersubband coupling is strongest when it involves 1D subbands characterized by similar quantum numbers. For instance, the interaction between the {1,1}-{1,2} levels is much stronger than for the {1,1}-{1,10} transition. The most striking feature of Fig. 9.4b is the anomalously large coupling near VG = 0.2625 V for the {1,1}{2,1} resonant transition which combines both an exceptionally strong intersubband coupling (only the quasi triangular quantum number is changed) and a large level occupancy at the relevant bias. Moreover, other strong resonant transitions ({1,2}-{2,2} and {1,3}{2,3}) contribute to intersubband scattering in the vicinity of the resonant bias and tend to broaden the range of VG over which the NDT occurs. To assess the influence of resonant scattering on carrier transport, we have performed Monte Carlo simulations [3] of the present structure and superimposed the data over the experimental results in Fig. 9.3. Although the simulations taken into account only polar optical scattering, the two sets of data show excellent

119

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Intersubband Optic Phonon Resonances in Electrostatically Confined Quantum Wires

qualitative agreement. The quantitative discrepancy arises from the omission of background quasielastic scattering processes which reduce the experimental conductance and broaden the NDT. Additional broadening arises from the variation of the resonant gate bias over the array of wires. We therefore attribute the NDT to the mobility degradation caused by resonant intersubband optic phonon scattering mostly between the {1,1} and {2,1} subbands. The observation of this effect was possible because of the quasi-1D form of Gauss’ law in which increasing free charge is related to decreasing vertical field. In addition, the singular nature of the 1D density of states found in quantum wires strongly enhances RISOPS and the NDT effect over bulk and 2D configurations. This effect provides novel opportunities for investigating the electronic properties of quantum structures at high temperatures as well as the electron-LO phonon interaction in confined systems. This work was supported by the Joint Services Electronics Program.

References

1. Nanostructure Physics and Fabrication, edited by M. A. Reed and W. P. Kirk (Academic, New York, 1989). 2. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989).

3. D. Jovanovic, S. Briggs, and J. P. Leburton, Phys. Rev. B 42, 11108 (1990). 4. F. M. Peeters and J. T. Devreese, Semicond. Sci. Technol. 7, B15 (1992). 5. N. Mori, H. Momose, and C. Hamaguchi, Phys. Rev. B 45, 4536 (1992). 6. V. Gurevich and Y. Firsov, Sov. Phys. JETP 20, 489 (1965).

7. K. Ismail, Proceedings of the HAS Symposium on Science and Technology of Mesoscopic Systems, Nara, Japan (1991). 8. K. Ismail, M. Burkhardt, H. I. Smith, N. H. Karam, and P. A. Sekula-Moise, Appl. Phys. Lett. 58, 1539 (1991). 9. K. Ismail, S. Washburn, and K. Y. Lee, Appi. Phys. Lett. 59, 1998 (1991). 10. M. H. Degani, Appl. Phys. Lett. 59, 57 (1991).

11. M. H. Degani and J. P. Leburton, Phys. Rev. B 44, 10901 (1991).

12. D. Jovanovic and J. P. Leburton, IEEE Electron Device. Lett. EDL-14, (1993).

References

13. S. Laux and F. Stern, Appl. Phys. Lett. 49, 91 (1986). 14. F. Stern, Phys. Rev. B 5, 4891 (1972).

15. J. P. Leburton, J. Appl. Phys. 56, 2850 (1984).

121

Chapter 10

Transient Simulation of Electron Emission from Quantum-Wire Structures

S. Briggs and J.-P. Leburton

Beckman Institute for Advanced Science and Technology, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We model a GaAs quasi-one-dimensional quantum wire in an applied longitudinal field and focus on mechanisms of electron emission leading to real-space transfer from the wire. The Monte Carlo simulation assumes an initial electron distribution in the wire and calculates the time required for electrons to undergo nonequivalent intervalley scattering to three-dimensional states. The model includes multiple subbands, polar optic and acoustic phonons, intervalley scattering, and band-structure nonparabolicity. Results have been obtained for different confinement conditions as Reprinted from Phys. Rev. B, 43(6), 4785–4791, 1991.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1991 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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well as different temperatures. We find that the required time is a very strong function of the longitudinal field and ranges from 4 ns down to 1 ps for fields in the range of 100 V/cm to 8 kV/cm. The corresponding distances in the wire vary from 130 mm down to the submicrometer range.

10.1 Introduction

The physics and fabrication of quasi-one-dimensional (1D) artificial structures have experienced rapid progress in the past few years [1–9]. While early confined systems were limited to the observation of quantum effects at low temperature, 1D effects are now observable well above liquid-helium temperature. Quantum wires with carrier confinement below 1000 Å have recently been achieved and quantum features in the transconductance of 1D field-effect devices have been reported at 77 K [7]. Meanwhile, new approaches to fabricate quantum structures with two degrees of confinement by direct growth on vicinal GaAs substrates have demonstrated the feasibility of quantum-wire based lasers [8, 9], which raises issues concerning the dynamics and dissipation of nonequilibrium carriers in 1D systems. Above 77 K, transport is essentially limited by phonon scattering which is a strong impediment to the observation of quantum interference effects. Also, high-temperature operation is desirable and the interesting aspect of 1D transport is the reduction of transverse degrees of freedom which tends to limit scattering to forward and backward events [10]. Under these conditions, thermal effects become determinant and 1D transport is described in terms of a semiclassical Boltzmann formalism. Theoretical investigations of the electronic properties of semiconductor wire structures have been accomplished including calculation of the dominant scattering mechanisms [11–14] and transport simulations [15–17]. One serious unresolved issue is carrier emission from the quantum wire, since at sufficiently high electric fields, the carriers gain enough energy to overcome the confining barrier [15]. The escape mechanisms depend on the wire configuration and include real-space transfer effects [18] such as emission from a quantumwell confining potential or drift away from a heterojunction triangular confining potential. Carrier losses due to scattering

Model

include intravalley as well as intervalley phonon scattering to 3D states. If the losses due to these mechanisms are significant, either in terms of electron lifetimes in the wire or mean paths in the wire, then the whole concept of an infinitely long, semiclassical quantum wire is invalid. To date, no work has addressed this question.

10.2 Model

We simulate electron transport with a Monte Carlo technique which is based on a model consisting of a GaAs-AlxGa1–xAs quantum well (QW) and a perpendicular gate electrode with a triangular electrostatic potential. The model includes seven electronic subbands. Figure 10.1 shows this configuration, which is similar to the V-shaped-groove quantum-wire field-effect transistor [1] (VFET) or the modulation doped GaAs-AlxGa1–xAs wire structures fabricated using ion-beam-assisted etching [6]. The V-shaped-groove wire, characterized by a quantum well in the y direction and a triangular potential in the z direction, offers, in principle, the largest degree of confinement that can be controlled by external transverse electric fields (gate fields) Fz. Although different confinement configurations may be considered, the conclusions obtained with this configuration can be generalized to any kind of geometrical confinement.

Figure 10.1 Idealized structure used in the simulation. Confinement in the y direction is due to a quantum well of width Ly. Fz is the applied field in the z direction which gives rise to a triangular well.

125

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Transient Simulation of Electron Emission from Quantum-Wire Structures

For the y direction we calculate wave functions in the infinite square-well approximation while in the z direction we perform a variational calculation with exponentially damped polynomials as the trial wave functions. We compute the first three y and z wave functions and combine them to obtain nine wave functions. Then, the total wave functions Y are given by Y i,j( x ) = e



- ikx x

¥e

-z j z

1/ 2

È ip y ˘ sin Í ˙ ÍÎ Ly ˙˚

n ,jz

n

È2˘ Í ˙ ÍÎ Ly ˙˚ n= j

Âc

, i = 1, 2, 3, j = 1, 2, 3.

n =1



(10.1)

where zj and cn,j are determined by a variational calculation as well as the corresponding energy Ej; we also have assumed free electrons (plane waves) in the x direction. The corresponding energies of the subband bottoms are given by

E i,j =

( hp i )2

2m * L2y

+ E j , i = 1, 2, 3, j = 1, 2, 3.

(10.2)

These approximations are good if the energy level lies deep in the well. The higher y levels should be spaced more closely as they approach the top of the QW and the z levels spaced more closely as screening flattens out the triangular potential. The y = 2, z = 3 and y = 3, z = 3 states are omitted because of memory constraints in the Monte Carlo code. For most confinement conditions of interest, these two levels are above the edge of the GaAsAlxGa1–xAs barrier and can be neglected. Although the lowest energy level is clearly E1,1, the ordering of the higher subbands depends on the confinement conditions. To avoid confusion, the subbands will be numbered with a single subscript v, which will range in order of increasing energy from 1 for the lowest subband to 7 for the highest subband. The y and z quantum numbers will, in general, not be used. Because of the relatively high energy transport considered in this simulation, nonparabolic effects in the band structure are important and have been included in the model using the Kane approximation [19] with a nonparabolicity factor a = 0.67 eV–1. Then, the relation between E(kx), the electron energy in the subband, and kx, the longitudinal wave factor, is given by

Model



2 (k 2 + kx2 ) = [E(kx ) + E v ]{1 + a[E(kx ) + E v ]} 2m * v   = E T (k )[1 + a E T (k )]

with



and

kx2 =

2m * h2

2m *

E v (1 + a E v ),

(10.3b)

E(kx ){1 + a[E(kx ) + 2E v ]},

(10.3c)

kv2 =



(10.3a)

h2

k = (kx , kv ).

(10.3d)

Ev is the energy at the bottom of the subband; here we have defined the total energy relative to the quantum-well bottom as ET (k ) = Ev + E(kx). We express kv in terms of the energy and do not attempt to define ky and kz because only the magnitude of ky can be determined and kz is not even a constant of motion. In addition, it is important to note that k is defined as an ordered pair rather than a vector, the reason being it does not transform as a vector (i.e., it is not closed under addition or scalar multiplication). The E(kx) relation is now a function of not only the energy relative to the subband bottom but also the energy of the subband relative to the quantum well. The density of states is also now a function of both Ev and E(kx). For the satellite valleys, since the effective masses are large, the level splitting brought about by the confinement will be small; therefore we assume that only 3D states exist in the X and L valleys. We consider polar optic, acoustic, and intervalley phonon scattering mechanisms. At present, the 1D Monte Carlo code is not interfaced to a 3D transport code for electrons in the satellite valleys and our model does not include any mechanism for electron injection into the wire. Due to this approximation, our results are limited to the transient case where all electrons are in the wire at time t equals zero; we study the time evolution of the system as electrons escape to 3D states.

127

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Transient Simulation of Electron Emission from Quantum-Wire Structures

10.3 Scattering Rates In general, the transition rate from a wave vector k to k′ is given by Fermi’s golden rule [20]

2p | Hfi |2 d (E final - E initial ), (10.4) h where Hfi is the matrix element of the perturbing potential H between the initial and final states. The transition rate is modified due to nonparabolic effects in the conduction band, which introduces a term G(k, k′). This term represents the overlap integral between the periodic parts of the Bloch functions at k and k′ summed over the doubly degenerate final spin states and averaged over the initial spin states. For 3D cases, this is equal to [21]



with

W ( k , k¢ ) =

G( k , k¢ ) = (ak ak ¢ + ck ck ¢ cos b )2 , È 1 + a ET (k ) ˘ ak = Í ˙ Î 1 + 2a E T ( k ) ˚ È a ET (k ) ˘ ck = Í ˙ Î 1 + 2a E T ( k ) ˚

(10.5a)

1/ 2

,

1/ 2

,



(10.5b)

and b the angle between k and k′ [then cos b = (kxk′x + kyk′y + kzk′z)/|k||k′|]. In 1D systems the energy quantization replaces k by k in which case only kx and kv are known whereas ky and kz are undetermined; this makes b undetermined. To overcome this difficulty, we average cosb over all possible values of ky, kz, k′y, and k′z subject to the constraint kv2 = ky2 + kz2 and similarly for k′v. Although it may be argued that ky is a good quantum number with discrete values, fixing ky fixes kz as well; however, this kz value does not correspond to any physical quantity. The cosb term is derived assuming plane waves and it is not clear that either ky or kz are the corresponding quantities in the 1D form of cosb. In addition, using explicit values for ky and kz still requires an averaging process, since the signs of ky and kz are undetermined. Rather than making these assumptions, we simply average over kz and ky as well. This method of determining b is certainly an approximation to calculating the overlap between initial and final Bloch functions; however, this

Scattering Rates

seems the most reasonable approach given the small magnitude of ck . b is thus obtained as a function of kx, k′z, kv, and k′v, which are all known quantities. With this assumption, G(k , k ') becomes G(k , k ¢ ) = (ak ak ¢ )2 + 2ak ak ¢ ck ck ¢ ¥



kx kx¢ (kx2 + kv2 )1/2 (kx¢2 + kv¢2 )1/2

¥ (ck ck ¢ )2

Ê 2 ¢2 1 2 ¢2 ˆ ÁË kx kx + 2 kv kv ˜¯



(10.6a)



(10.6b)

(kx2 + kv2 )(kx¢2 + kv¢2 )

for intersubband transitions and

G(k , k ¢ ) = (ak ak ¢ )2 + 2ak ak ¢ ck ck ¢ ¥



kx kx¢ + kv2 (kx2 + kv2 )1/2 (kv¢2 + kv¢2 )1/2

¥ (ck ck ¢ )2

(kx + kx¢ + kv2 )2 (kx2 + kv2 )(kx¢2 + kv¢2 )

for intrasubband transitions. Currently, we consider polar-optic-phonon (POP) and inelasticacoustic-phonon scattering for 1D states within the G valley. In addition, we consider intervalley scattering from 1D states in the G valley to 3D states in the X or L valleys. For POP or acoustic phonons, the transition probabilities from an electron state kx in initial subband v to a state k′x in final subband m are calculated according to Fermi’s golden rule as Wv,m (kx + kx¢ )



=

2p G(k , k ¢ ) d ¢  k x - k x ,q x

¥

Ú Ú

+•

+•

-•

-•

dqy dqz | M3D,v,m (qx , qy , qz )|2 1 1ˆ Ê ¥ Á Nq + ± ˜ d ([E v + E(kx )] Ë 2 2¯ - [E m + E(kx¢ )] ± w q ),



(10.7)

129

130

Transient Simulation of Electron Emission from Quantum-Wire Structures

where qx is the longitudinal and qy, and qz the transverse phonon wave vectors, respectively. Nq is the phonon occupation number with the plus or minus sign corresponding to phonon emission or absorption. The double integral over qy and qz represents the calculation of the 1D matrix elements M1D,v,m (qx) from the 3D matrix elements [15]. Here we consider only bulk (i.e., 3D) phonon modes and neglect 1D and surface modes. This does not introduce significant error as long as the confinement is not excessively high, i.e., less than 50 Å [22]. The POP dispersion relation is assumed to be a constant, which makes Nq and the energy conservation d function independent of q; therefore they can be factored out of the double integral. However, the nonconstant dispersion relation for acoustic phonons makes computation of the integral considerably more complex. The numerical integration routine has qx as an input parameter, which we vary to obtain 1D matrix elements. For acoustic phonons, we evaluate the integral for qx in the range 7 ¥ 104 < qx < 1.5 ¥ 107 cm–1. The acoustic-phonon transition probabilities are essentially independent of qx for smaller values, while 1.5 ¥ 107 cm–1 is the largest possible momentum exchange. For POP transitions, the integral is evaluated for 1 < qx < 1 ¥ 107 cm–1. The dk term in Eq. (10.7) represents conservation of longitudinal momentum with the plus or minus sign corresponding to forward or backward phonon scattering, respectively, and is used to select a value for qx and the corresponding matrix element. Scattering rates are computed by integrating the transition probabilities over the final electron k states. The total scattering rate for POP’s or acoustic phonons from initial state kx in subband v is then given by 7

lv (kx ) =

with

2p

  |M

2 1 ,v,m (qx )|

m =1

¥

M (1 + 2a E T¢ ) 2

kx¢

dk

x

G(k , k ¢ )

- k ¢x ,q x



È 2m * ˘ | kx¢ | = Í 2 [E T¢ (1 + a E T¢ ) - E m (1 + a E m )]˙ Î h ˚

(10.8a)

1/ 2



(10.8b)

Scattering Rates

and

E T¢ = E T ± hw LO .



(10.8c)

Equations (10.8b) and (10.8c) represent conservation of energy and determine the final k state and the factor 2aE¢T in Eq. (10.8a) is due to nonparabolicity in the final density of states. For each 1D subband, conservation of energy gives four possible final states corresponding to forward or backward emission or absorption. A constant broadening factor was used to account for quantum corrections to the final density of states. Because of the divergence in the classical density of states at the bottom of a subband, we consider quantum correlations between scattering events and neglect other intracollisional effects such as the influence of the field or phonon lifetime. The density-of-states broadening was selected by fitting the scattering rates from Fermi’s golden rule to the imaginary part of the polaron self-energy obtained from a self-consistent solution to the Fock approximation and was found to be equal to 2.5 meV at 300 K and 1.6 meV at 77 K [23]. A similar broadening factor was also used for acoustic-phonon and intervalley scattering. For intervalley scattering, the transition rate between state kx in subband v of the G valley and state k′ in the X or L valley is given by Wv,X,L (kx , k ¢ )



=

2p d G(k , k ¢ ) ¢  k x - k x ,q x

¥

Ú

+•

-•

dqy dqz | M3D,v,X,L (qx , qy , qz )|2 1 1ˆ Ê ¥ Á Nq + ± ˜ d [E T (k ) - E X,L ( k ¢ ) Ë 2 2¯



(10.9)

± w X,L ], where EX,L(k′) is the total energy of the final state relative to the bottom of the quantum well (i.e., the bottom of the G valley) and hwX,L the intervalley phonon energy. Again, we assume 3D modes and a constant dispersion relation which allows us to factor Nq and the energy-conservation d function out of the integral. In addition,

131

132

Transient Simulation of Electron Emission from Quantum-Wire Structures

the form of the matrix element allows the integral to be separated into two independent single integrals over qy and qz without any qx dependence. Then, Sv,L(kx), the total rate to a specific L valley, is given by S v,L (kx ) =



2p | M1D,v (qx )|2 d k - k ¢ ,q x x x h

¥ k¢

ML* h2

(1 + 2a L EL ),

with the final k state k′ given by

1 1ˆ Ê ÁË Nq + 2 ± 2 ˜¯

(10.10a)

1/ 2

È 2m* ˘ | k¢ | = Í 2L EL (1 + a L EL )˙ ÍÎ h ˙˚

,

(10.10b)



EL = [E v + E(kx )] - D L ± hw L .

(10.10c)



lv (kx ) = 3S v , X (kx ) + 4S v ,L (kx ),

(10.10d)

and the final energy relative to the bottom of the L valley,

DL is the energy difference between L and G. A similar expression holds for Sv,X(kx). Then, the total intervalley scattering rate is given by

with the factors of 3 and 4 accounting for the number of satellite valleys. The scattering rates are shown in Fig. 10.2. For the sake of clarity, only the bottom two subbands are shown with a being the lowest subband and b the first excited state. The threshold for intervalley absorption to the L valley occurs at 262 meV and the onset of emission to the L valley begins at 318 meV. Absorption to the X valley starts at 450 meV and emission to that valley occurs above 510 meV. The rates for POP and acoustic phonons show a large number of peaks, each peak being proportional to the density of final states and corresponding to an emission or absorption to the bottom of a subband (the peaks at higher energy represent scattering to subbands not shown in the figure). These peaks make the velocity and distribution functions sensitive to the energy separation between subbands, particularly the first and second subbands. Although the scattering rates for POP and acoustic phonons are different for the different subbands, we can see that the rate for intervalley scattering is independent of the initial subband.

Monte Carlo Simulation

Figure 10.2 Scattering rates for a quantum-wire structure including POP, acoustic, and intervalley phonons. The confinement conditions are Ly = 135 Å, Fz = 120 kV/cm, and T = 300 K. Although the rates include transitions between all seven subbands, only the lowest two subbands are shown for the sake of clarity. The peaks in the rates are due to POP scattering to the bottom of a subband. Clearly visible are the thresholds for phonon absorption to the L valley at 262 meV, emission to the L valley at 318 meV, and absorption to X valley at 450 meV. The emission threshold to the X valley at 510 meV is not on the graph, but is included in the simulation.

10.4 Monte Carlo Simulation Because of the large number of peaks in the rates, normal methods of computing free flight times are inefficient. Using constant or piecewise constant scattering rates [24, 25] with self-scattering would have introduced a very large percentage of self-scattering events. Instead, a direct integration method was used. For a given subband v, if r is a uniformly distributed random number on [0,1], then

- ln r =

t

Ú l [k (t ¢)] dt ¢ , 0

v

v

(10.11)

where t is the time of the free flight, kv(t) is the momentum as a function of time in subband v, and lv(kv) is the scattering rate as a function of momentum for that sub-band. In 3D simulations it is virtually impossible to store k(E) in tabular form because of the large number of possible k values. In 1D systems, there is only one

133

134

Transient Simulation of Electron Emission from Quantum-Wire Structures

scalar kv value for each energy and each v (tabulated earlier by the program), therefore k(E) is simply a lookup function. Moreover, in 3D, dt is a complicated function of k, involving squares and squareroot computations, which typically prohibits direct integration algorithms in Monte Carlo techniques. For 1D systems, however, direct integration compares favorably with other methods. For a given initial electron energy and subband we follow an electron in the longitudinal field Fx until it undergoes an intervalley scattering, at which point it is lost to the simulation. The total time spent tracking that electron is saved and a new electron is started with the same initial conditions. This procedure is repeated until we have accumulated at least 400000 scattering events and at least 1000 intervalley scatterings. The electron lifetime in the quantum wire is then obtained by averaging the time required to undergo intervalley scattering.

10.5 Results

We have run Monte Carlo simulations at 300 and 77 K for two different confinement conditions. The first is a high-confinement condition characterized by Ly = 135 Å and Fz = 120 kV/cm. This condition places the bottom subband 138 meV above the quantumwell bottom and the first excited state 231 meV above the well bottom. The second case is a lower-confinement condition with Ly = 215 Å and Fz = 20 kV/cm, which places the bottom two levels 45 and 73 meV above the well bottom, respectively. We have varied the electron initial energy as well as the electron initial subband; we find only a small dependence of the electron lifetime on the energy and essentially no dependence on the initial subband. The lack of dependence on initial subband can be explained by noting that the intervalley scattering rate in Fig. 10.2 is the same for both subbands a and b. In Fig. 10.3, we present the electron lifetime in the highconfinement case as a function of initial energy for two different longitudinal fields. The temperature is 300 K, curve a shows the carrier lifetime in the presence of a longitudinal field of 1000 V/cm, and curve b is for a lower field of 100 V/cm. The lifetime is relatively

Results

independent of the initial energy for energies near the subband bottom and is about 150 ps at 100 V/cm and 50 ps at 1000 V/cm. Although the actual value of the lifetime at low energy depends on temperature, field, and confinement conditions, the different curves are all approximately parallel. At higher initial energies, the lifetime decreases at the threshold for intervalley scattering which occurs at 262 meV for phonon absorption to the L valley. At high longitudinal fields, the transition is smoother, but still occurs at roughly the same initial energy. The existence of two distinct energy regions in the curves can be understood as a trade-off between the intervalleyscattering time and the time required to reach equilibrium. We would expect the time required to scatter intervalley to be proportional to exp[–(E – Et)/kT], where Et is the threshold energy for intervalley scattering and E is some characteristic energy of the system. Normally, E is the initial energy, however, if the time is longer than that required to reach equilibrium, the system reaches equilibrium before undergoing an intervalley scattering and the time is independent of the initial energy.

Figure 10.3 Electron lifetime in quantum wires as a function of initial energy for two different longitudinal fields under high-confinement conditions at T = 300 K. a is for a longitudinal field of 100 V/cm, while b is for a longitudinal field of 1000 V/cm.

In Fig. 10.4 we show the influence of the longitudinal field on the time and distance required to undergo an intervalley scattering

135

136

Transient Simulation of Electron Emission from Quantum-Wire Structures

event. The inset in Fig. 10.4 shows an estimate of the distance an electron travels for a, high, and b, low, confinement at 300 K. The high-confinement curves are for a simulation where the initial electron energy is 140 meV while in the low-confinement case the initial electron energy is 50 meV. Varying the initial energy does not substantially alter the result except for very high initial energies (as can be expected from Fig. 10.3). These distances are only approximations as they are calculated assuming ·xÒ = ·vÒ·tÒ rather than explicitly calculating ·xÒ = ·vtÒ where x is the distance, v is the velocity, and t is the lifetime (the reason lies in the Monte Carlo code which keeps statistics on velocity and time independently and does not keep track of correlations between the two). This assumption limits the validity of the results to long lifetimes where the velocity is sufficiently randomized and electron runaway is not important. Accordingly, we do not present the distance for the 77-K case, or for longitudinal fields larger than 2 kV/cm (although the trend is clearly to shorter distances as the field increases). The distance in the lowconfinement case is much larger than that for the high-confinement case, even though electron velocities in high confinement are larger than in low confinement. The distance rises rapidly, peaks near 400 V/cm, and then drops off slowly. This behavior can be understood by considering the two extremes of infinite and zero field. At very large fields, the lifetime is so short that even though the acceleration is very high, the distance still tends towards zero. At zero field, electrons still scatter by intervalley phonons with some small, but finite probability. Since the average velocity is zero, the average distance, as previously defined, required to undergo an intervalley scattering is zero. Of course, a more meaningful concept for the zerofield case would be ·dÒ = ·vthtÒ, where vth is the thermal velocity. We estimate this to be approximately 44 mm under high confinement and 1150 mm for low confinement. These numbers are larger than one would estimate from extrapolation in the figure; however, vth is considerably larger than the average (drift) velocity at low field and the results would converge at higher fields. The influence of the longitudinal field on the lifetime is shown in Fig. 10.4 for high and low confinement at both 77 and 300 K. Curve a(b) is for high (low) confinement at 300 K; c(d) is high (low) confinement at 77 K. All curves show a very strong dependence on

Results

the longitudinal field. At lower fields, the effects of confinement and temperature are significant, but at higher fields these effects are washed out as the field dominates in determining the lifetime. The times (for the same field) are larger at 77 K because of the lack of phonon absorption, which tends to both lower the average energy and also raise the threshold for intervalley scattering. The 77-K results do not go below 1500 V/cm because the lifetime is over 10 ns and it requires too much CPU time to get an accurate estimate of the lifetime.

Figure 10.4 Electron lifetime and distance vs longitudinal field at 77 and 300 K for high- and low-confinement conditions, a and b are high and low confinement at 300 K, respectively, while c and d are the corresponding curves for 77 K. Assumptions used in calculating the distance limit its validity to 300 K and fields below 2 kV/cm.

The fraction of electrons above the scattering threshold for the four different cases is tabulated in Table 10.1 and supports the data in Fig. 10.4. The cases with shorter lifetimes have greater carrier concentrations above the threshold for intervalley scattering. As can be seen, the high-confinement condition at 300 K has a much shorter lifetime than the low-confinement case. This is due to the fact that under high confinement the average energy is high because of the location of the bottom subband relative to the L or X valley. However, at 77 K the trend is reversed and under low confinement a larger fraction of carriers is above the emission threshold.

137

138

Transient Simulation of Electron Emission from Quantum-Wire Structures

Table 10.1 Simulation conditions for the four cases shown in Figs. 10.4 and 10.5. The fraction is the fraction of carriers found above the threshold for intervalley scattering (262 meV at 300 K and 318 meV at 77 K) and the time is the average electron lifetime Temperature

Field

Fraction

Time

(K)

(V/cm)

%

(ps)

500

0.679

Case

Confinement

a

High

300

Low

77

b c d

Low

High

300 77

500

1500

1500

0.0338

0.00523

0.0602

84

1580 3985

926

Figure 10.5 presents a highly schematic picture of the distribution functions for the four cases. If we compare the two distribution functions at 300 K (a and b), it is apparent that the field does not play a major role since in both cases the high-energy slope of the distribution function is characterized by an electron temperature of approximately 400 K. Near equilibrium, detailed balance requires the distribution function to be insensitive to the confinement. However, at 77 K, hot carriers are much more important due to the lower lattice temperature and the higher longitudinal field. The corresponding electron temperatures are 260 K for the high-confinement case (c) and 550 K for the low confinement case (d). Therefore we suspect this difference is due to the dependence of the scattering rates on confinement and energy. At high fields, detailed balance is irrelevant and consequently the tail of the distribution function is more sensitive to the scattering rate. At high energy, where carriers undergo intervalley transfer, the effective POP scattering rate is stronger under high confinement than under low confinement due to the overlap integral in the matrix element [15]. As a result, the tail of the distribution function decreases more rapidly in the former case than in the latter, resulting in lower electron temperatures. This high-field effect results in longer electron lifetimes for the highconfinement case and can be seen at 300 K for fields above 3 kV/cm where curves (a) and (b) cross over.

Conclusions

Figure 10.5 Idealized distribution functions for the four cases shown in Fig. 10.4. The 300-K curves, a and b, are for a longitudinal field of 500 V/cm while the 77-K curves, c and d, are for a longitudinal field of 1500 V/cm.

10.6 Conclusions In conclusion, we have performed the first multi-subband quasi-1D simulation which includes intervalley scattering to 3D states. The electron lifetime and distance for escaping the wire are not only very strong functions of longitudinal field but also are dependent on the confinement conditions and temperature. Lifetimes ranging from 4 n down to 1 ps have been calculated at 77 and 300 K for two confinement conditions with distances ranging from 130 mm down to the submicrometer range. The computed distances are in the 10 mm range for fields below 2 kV/cm at 300 K, which demonstrates the feasibility of operating quantum wires as purely 1D structures without being concerned with intervalley scattering. On the other hand, under high-field conditions, electrons escape the wire quickly. This should be accompanied by a corresponding negative differential resistance in the I–V characteristics similar to real space transfer as they scatters from high mobility 1D states in the G valley to lower mobility 3D states in the L or X valleys.

139

140

Transient Simulation of Electron Emission from Quantum-Wire Structures

Acknowledgments The authors are indebted to Karl Hess for helpful discussions on real-space transfer. This work is supported by National Science Foundation grant no. NSF-CDR-85-10209 and the Joint Services Electronics Program.

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13. S. Laux and F. Stern, Appl. Phys. Lett. 49, 91 (1986). 14. J. Lee and H. Spector, J. Appl. Phys. 57, 366 (1985).

15. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988). In this reference, Eq. (10.4) should be corrected to agree with Eq. (10.8) in this chapter. 16. S. Das Sarma and X. C. Xie, Phys. Rev. B 35, 9875 (1987).

17. A. Kubasi, C. Chattopadhyay, and C. K. Sarkar, J. Appl. Phys. 65, 1598 (1989). 18. K. Hess, H. Morkoc, H. Shichijo, and B. G. Streetman, Appl. Phys. Lett. 35, 469 (1979).

References

19. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).

20. E. M. Conwell, High Field Transport In Semiconductors (Academic, New York, 1967).

21. W. Fawcett, A. D. Boardman, and S. Swain, J. Phys. Chem. Solids 31, 1963 (1970). 22. N. Mori and T. Ando, Phys. Rev. B 40, 6175 (1989).

23. S. Briggs, B. A. Mason, and J. P. Leburton, Phys. Rev. B 40, 12 001 (1989). 24. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983). 25. R. Yortson, J. Comput. Phys. 64, 177 (1986).

141

Chapter 11

Carrier Capture in Cylindrical Quantum Wires

N. S. Mansour,a Yu. M. Sirenko,a K. W. Kim,a M. A. Littlejohn,a J. Wang,b and J.-P. Leburtonb aDepartment

of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695-7911, USA bDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We present quantum mechanical calculations of electron capture rates in cylindrical quantum wires via polar-optical phonon scattering. The capture rate dependence on quantum wire radius and lattice temperature is investigated. An oscillatory behavior of the electron capture rate is observed as a function of the quantum wire radius at the temperatures considered in this study (20–300 K). However, the amplitude of these oscillations decreases significantly at large wire radii and high lattice temperatures. Reprinted from Appl. Phys. Lett., 67(23), 3480–3482, 1995. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1995 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Carrier Capture in Cylindrical Quantum Wires

The progress achieved in nanofabrication techniques has made possible the manufacturing of high-quality semiconductor quantum wires [1] (QWRs). In contrast to quantum well based structures which are utilized both in high-speed electronic devices and in semiconductor lasers, application of QWR structures are aimed mainly toward fabrication of advanced optoelectronic devices. Because of a narrowed density of states distribution due to additional spatial confinement, QWR lasers [2, 3] have the potential for superior performance, including higher optical gain, ultralow threshold current, and narrow spatial linewidth. While increased spatial confinement leads to enhanced optical gain for a given QWR, there is an inherent problem in QWR laser design associated with small wire cross sections. In order to achieve an optical gain comparable to that of conventional or quantum well lasers, the densely packed wire arrays or quantum well embedded QWR laser [4] should be utilized. The small cross-sectional area of QWRs also results in a reduced capture rate of carriers into an isolated wire and serves as a main limiting factor for modulation bandwidth [5]. Thus, a detailed analysis of the carrier capture process into a QWR is crucial for theoretical analysis and design of QWR laser devices for high-speed optical communication and data processing. The similar problem of carrier capture into quantum wells and superlattices has been studied in much detail. Quantum mechanical oscillations of capture rate with the well width were predicted [6] and observed experimentally [7]. It was established that longitudinal-optical (LO) phonon emission [8–10] constitutes the main mechanism for electron capture, though other processes are important at high impurity [11] or carrier [12] concentrations. However, to the best of our knowledge, there exist no corresponding treatments of carrier capture in QWRs. In this chapter, we present the quantum mechanical calculation of LO-phonon assisted electron capture in a cylindrical QWR. In cylindrical coordinates r = (r, j, z), we choose the quantum wire to occupy the region r £ R. The potential is taken to be –V0 inside the wire and zero in the barrier; we assume a constant electron effective mass m* throughout the whole structure. Electron wave functions Y(r) are characterized, in addition to wave vector kz and azimuthal quantum number m, by the discrete quantum number n = 1, 2,...

Carrier Capture in Cylindrical Quantum Wires

for the states localized in the vicinity of the wire, or by a quantum number k for unbound states according to the following equations:

Y k z ,m,n = L –1/2exp( ikzz+imj)l(ymn(r),

(11.1)

Y k z ,m,k =(L S )–1/2exp(ikzz + imj)ymk(r),

(11.2)

where L and S are the normalization length and cross-sectional area of the sample structure. The corresponding eigenenergies are given by



E k z ,n =

h2 h2 2 2 ( -k wmn ( -k mn + kz2 ) ∫ + kz2 ) - V0 , (11.3) 2m * 2m *

E k z ,k =

h2 h2 (k 2 + kz2 ) - V0 . (11.4) (k 2 + kz2 ) ∫ 2m * 2m * w

2 Thus, mn = h2k mn / 2m * is the ionization energy of the sub-band (mn); the variables kwmn and kw (the subscript “w” stands for wire) are defined by the right-hand sides of Eqs. (11.3) and (11.4). The transverse wave function of the bound states can be written as



y mn ( r ) =

C mn ÏK m (k mn R ) J m (k wmn r ), Ì p R2 ÔÓ J m (k wmn R )K m (k mn r ),

For unbound states, we have

r R.

(11.5)

r > R,

(11.6)

where Cmn, cmk, and dmk are specified by the continuity and normalization conditions. Applying the boundary conditions at r = R gives the following characteristic equation for the ionization energies of the bound states:

¢ ¢ k wmn J m (k wmn R )K m (k mn R ) = k mn K m (k mn R ) J m (k wmn R ). (11.7)

We now consider the electron capture to a QWR as mediated by emission of LO phonons via polar-optical phonon interaction. In general, the “net capture’’ rate is determined by a difference between the capture and emission from the QWR. However, when carriers are effectively removed from the QWR due to recombination (i.e., no bottleneck), the probability of escape (after the initial capture) is much smaller than that for further relaxation to the bottom

145

146

Carrier Capture in Cylindrical Quantum Wires

of sub-bands. This is a consequence of higher density of states in low-dimensional structures near the subband offset compared to the three-dimensional states, as discussed in Ref. [11]. Thus, the “net capture’’ and capture rates are essentially the same in our calculations. Taking the Fröhlich Hamiltonian for electron-phonon coupling and applying the Fermi golden rule, we arrive at the following probability for electron transition from an extended state (kz, m, k) to a bound state (kz¢ , m¢ , n) assisted by emission of a LO phonon with momentum Q = (q, qz) and energy hw0: WkQ mk Æk ¢ m ¢ n = z

4p 2e2w 0 (nph + 1) * Q2SL

z



¥ | · Y k ¢ ,m ¢ ,n (r )| exp(Q ◊ r ) | Y k z ,m,n (r )Ò |2 z

¥ | d (E k z ,m,k - E k ¢ ,m¢ n - hw 0 ). z

(11.8)

Here, nph= [exp(hw0/T) –1]–1 is a phonon occupation number and T is a lattice temperature in the unit of energy; also, 1/ϵ* = 1/ϵ•– 1/ϵ0, where ϵ0 and ϵ• are the static and high-frequency dielectric constants of the crystal, respectively. The overall capture rate W is obtained by summation of the elementary capture rates [given in Eq. (11.8)] over all possible phonon momenta as well as initial and final electron states weighted with the electron distribution function f(E) for the extended states:

W =2

 Â

k z k ¢z

,k,Q m,m ¢ ,n

f (E k z ,m,k ) WkQ mk Æk ¢ m ¢ n . (11.9) z

z

Since captured electrons rapidly lose energy due to subsequent phonon emission,11 we omitted the Pauli factor 1 – f ª 1 in Eq. (11.9). Finally, assuming a Maxwellian distribution with a temperature T for the initial electron states and using Eqs. (11.1)–(11.9), the total capture rate can be written as W = 8 pw 0 (nph + 1)



Ê k 2k 2 ˆ ¥ exp Á - 2 z ˜ Ë kT ¯

e2 kT p R 2 * T S mm¢ n

ÂÚ

Ú

•

0

(

qdq q M kT K mk ,m ¢ n

• dk

0

)

2

È ˘ kT2 kT2 ¥Í 2 + . 2˙ 2 2 ÍÎ q + (K + kz ) q + (K - kz ) ˙˚

z

kT

•

kdk

kmin

kT2

Ú

(11.10)

Carrier Capture in Cylindrical Quantum Wires



Here the dimensionless transition matrix element is given by q -1 Mmk ,m ¢ n = R

Ú

•

0

rd ry m ¢ n ( r ) J|m-m ¢|(qr )y mk ( r ), (11.11)

2 kT = 2mT /h is a thermal wave vector; the condition k 2 ≥ kmin ∫ 2 2 max (0, 2m * w 0 /h - k m ¢ n - kz ) assures that an electron would find a final state at subband m′ n after emission of energy hw0 and implies 2 2 2 that K 2 ∫ k m ¢ n + k + kz – 2m*w0/h is positive. The dependence of the capture rate on temperature and other parameters of the problem is specified by the factors and dimensionless integrals in Eq. (11.10). The latter is extremely sensitive to the subband positions in QWR and is responsible for a resonant behavior of the capture rates with respect to the wire radius. The factor pR2/S indicates that the capture rate of an electron spread over the normalization area S is proportional to the ratio of the wire cross-section and the total area of the sample. In order to obtain a physically significant measure of capture rate independent of arbitrarily large normalization area S , the combination S W has to be considered. The latter has a dimensionality “cm2/s” and is analogous to an interface capture velocity of dimensionality “cm/s” introduced for the description of carrier capture into quantum wells [11].

Figure 11.1 Subband ionization energies ϵmn vs wire radius R. The solid bars on the horizontal axis mark the radii at which new confined states emerge (i.e., ϵmm = 0).

147

148

Carrier Capture in Cylindrical Quantum Wires

For numerical calculations, we choose a GaAs QWR with an AlGaAs barrier of height V0 = 300 meV. The sub-band energies as a function of QWR radius are shown in Fig. 11.1. For QWR radii smaller than 33 Å, there is only one confined state corresponding to quantum numbers m = 0 and n = 1. As the wire radius R increases, this lowest subband becomes deeper and new energy states become confined. The dependencies of capture rate on the wire radius R is shown in Figs. 11.2 and 11.3, with left (right) vertical axes corresponding to parameter W(S W) measured in 1/s (cm2/s) and normalization area chosen to be S = 1 mm2. Figure 11.2 depicts the partial contributions of the four lowest subbands at temperature T = 20 K which is well below the phonon energy hw0 ª 450 K. As in the case of carrier capture in a quantum well [6–8, 10, 11], two sets of resonances appear corresponding to subband ionization energies ϵmn = 0 and ϵmn = hw0 (denoted by solid and dashed bars on the horizontal axis, respectively).

Figure 11.2 Partial contributions of four lowest subbands to capture rates W (left-hand side) and S W (right-hand side) vs wire radius R at temperature T = 20 K. Normalization area S = 1 mm2. The solid and dashed bars on the horizontal axis denote the wire radii where the subband ionization energies ϵmn become 0 or hw0, respectively.

The first set of peaks (at ϵmn = 0) occurs when, with the decrease of the radius R, the bound state mn leaves the wire and becomes the virtual resonant extended state [6]. An electron in such a resonant

Carrier Capture in Cylindrical Quantum Wires

state has an enhanced probability to be observed in the vicinity of the wire, leading to a drastic increase of the overlap integral and total capture rate in Eqs. (11.10) and (11.11). The second set of resonances (at ϵmn = hw0) corresponds to subband positions with optical phonon energy below the barrier [11]. The sharp increase of capture rate is due to an opening of a new transition channel from three-dimensional states to subband mn. Such transitions are very effective because of a large density of one-dimensional final states and the small required phonon momenta Q [cf., factor 1/Q2 in Eq. (11.8) for the electron-phonon interaction].

Figure 11.3 Capture rate W (left-hand side) and S W (right-hand side) vs wire radius R at three lattice temperatures. The solid and dashed bars on the horizontal axis denote the wire radii where the subband ionization energies ϵmn become 0 or hw0, respectively.

In Fig. 11.3, we plot capture rates to the QWR for three different temperatures, T = 20, 77, and 300 K. As in the case of capture to a quantum well [9, 11], both sets of resonances are very sharp for T = 20 K; they become more broadened at T = 77 K, and at room temperature only the first set of resonances (ϵmn ª 0) produces maxima, while the second set (ϵmn ª hw0) is practically obscured. The slight rise of the peak value at ϵmn = 0 with a decrease in temperature is due to narrowing of the electron distribution function and better matching of the resonance condition. This effect, however, is expected to diminish because of energy level broadening [9]. We note that the temperature dependence of the phonon occupation

149

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Carrier Capture in Cylindrical Quantum Wires

number does not play an essential role in calculated capture rates (which are proportional to nph + 1), since nph is considerably smaller than 1 even at 300 K. Our final remark concerns the magnitude of the calculated values. We note that obtained capture rates to QWR per unit wire area, S Wwire/pR2, is of the same order of magnitude as the calculated capture rates to quantum well [10, 11] per unit well area, L Wwell/a, where a is the width of the well. This means that, in general, the “efficiency” of capture by quantum well and QWR are comparable, after renormalization for different cross-sectional areas. In summary, we examined the effect of the wire radius and lattice temperature on the electron capture rate by LO phonons into cylindrical QWRs. Oscillations of a quantum mechanical origin were found as a function of QWR radius and have been interpreted in terms of positions of the wire subbands. This work was supported, in part, by the Office of Naval Research and U.S. Army Research Office.

References

1. Y. Arakawa, Solid-State Electron 37, 523 (1994); D. Bertram, B. Spill, W. Stoltz, and E. O. Göbel, ibid. 37, 591 (1994); T. Kono, S. Tsukamoto, Y. Nagamune, F. Sogawa, M. Nishioka, and Y. Arakawa, Appl. Phys. Lett. 64, 1564 (1994). 2. E. Kapon, in Quantum Well Lasers, edited by P. S. Zory (Academic, Boston, 1993), pp. 461–500.

3. E. Kapon, D. M. Hwang, and R. Bhat, Phys. Rev. Lett. 63, 430 (1989); S. Y. Hu, M. S. Miller, D. B. Young, J. C. Yi, D. Leonard, A. C. Gossard, P. M. Petroff, L. A. Coldren, and N. Dagli, Appl. Phys. Lett. 63, 2015 (1993); R. D. Grober, T. D. Harris, J. K. Trautman, E. Betzig, W. Wegscheider, L. Pfeiffer, and K. West, ibid. 64, 1421 (1994). 4. S. C. Kan, D Vassilovski, T. C. Wu, and K. Y. Lau, Appl. Phys. Lett. 62, 2307 (1993). 5. K. Y. Lau, in Quantum Well Lasers, edited by P. S. Zory (Academic, Boston, 1993), pp. 217–275. 6. J. A. Brum and G. Bastard, Phys. Rev. B 33, 1420 (1986).

7. P. W. M. Blom, C. Smit, J. E. M. Haverkort, and J. H. Wolter, Phys. Rev. B 47, 2072 (1993); B. Deveaud, A. Chomette, D. Morris, and A. Regreny, Solid

References

State Commun. 85, 367 (1993); K. Muraki, Y. Takahashi, A. Fujiwara, S. Fukatsu, and Y. Shiraki, Solid-State Electron. 37, 1247 (1994).

8. C. S. Lent, L. Liang, and W. Porod, Appl. Phys. Lett. 54, 2315 (1989); M. Babiker, A. Ghosal, and B. K. Ridley, Superlattice Microstruct. 5, 133 (1989); G. Weber and A. M. Paula, Appl. Phys. Lett. 63, 3026 (1993); Y. Lam and J. Singh, ibid. 63, 1874 (1993). 9. T. Kuhn and G. Mahler, Physica Scripta 38, 216 (1988); Solid-State Electron 32, 1851 (1989).

10. N. S. Mansour, K. W. Kim, and M. A. Littlejohn, J. Appl. Phys. 77, 2834 (1995). 11. D. Bradt, Yu. M. Sirenko, and V. Mitin, Semicond. Sci. Technol. 10, 260 (1995). 12. P. Sotirelis and K. Hess, Phys. Rev. B 49, 7543 (1994); B. K. Ridley, ibid. 50, 1717 (1994)

151

Chapter 12

Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures

D. Jovanovic and J.-P Leburton

The Beckman Institute for Advanced Science and Technology and The Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801, USA [email protected]

A 1D Monte Carlo simulation of a multisubband quantum wire indicates the presence of spatial velocity oscillations dependent on the strength of a longitudinally applied electric field. This effect results from the quasi-coherent relaxation of the electron ensemble upon reaching the threshold for polar optical phonon emission. Although these oscillations have been predicted for bulk semiconductors, the peaked scattering rates and lack of angular randomization existing in quasi-1D systems make these structures Reprinted from Superlattices Microstruct., 11(2), 141–143, 1992. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1992 Academic Press Limited Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures

ideally suited for detecting this phenomenon. Both the field and temperature dependence are explored.

12.1 Introduction

Although fabrication technologies associated with the realization of quantum wires are maturing rapidly, many quantum phenomena are as yet undetected due to the high levels of confinement and process control required for their exposure. As processing issues are resolved, new effects will undoubtedly emerge and provide applications for quantum wires in novel devices. In an effort to uncover these phenomena, we have performed Monte Carlo simulations of quasi-one dimensional (1D) structures for various temperatures and biasing conditions. Even though this approach is inherently semiclassical in nature, it is extremely useful for studying the interactions between electrons and various scattering mechanisms in highly confined structures. It has long been predicted [1] that the dispersionless nature of the optical phonon interaction in zincblende crystals may lead to oscillations of carrier velocity at low temperatures. Matulionis et al. [2, 3] quantified the strength of this effect via Monte Carlo simulation and demonstrated that very weak oscillations of the carrier velocity should occur in bulk materials. In contrast to bulk, quantum wires are ideally suited to expose these oscillations. The sharp peaking of the scattering rates and the lack of angular randomization enable coherent transport over relatively long distances thereby allowing the electron ensemble to travel in phase over several periods.

12.2 Model

For our device, we simulate a modulation doped GaAsAlGaAs wire structure similar to that fabricated by Ismail et al. [4] (Fig. 12.1). The transverse electronic confinement in this structure is approximated by a combination of harmonic oscillator and triangular potentials. The resulting set of eigenvalues and eigenfunctions are computationally tractable in addition to being adequate approximations for the real potential in the device. Electrons are essentially free in the

Model

longitudinal direction and are driven by a longitudinally applied electric field, Fx.

Figure 12.1 A cross section of the MODFET based structure simulated here. Confinement is achieved by etching a grating into the AlGaAs cap layer (after Ismail et al., Ref. [4]).

The modulation doped channels are characterized by a lack of impurity scattering and surface roughness. In addition, the lack of structural confinement in our device implies that the phonon modes will be essentially bulk-like. Bulk polar optical (POP), acoustic deformation potential (AP), and unscreened piezoelectric (PZP) phonons are therefore the only scattering agents considered in our simulation. Although debatable, the unscreened piezoelectric interaction introduces a conservative picture of the oscillatory phenomena investigated here. One key feature that determines the transport characteristics of quantum wires is the sharp peaking behavior of the scattering rates. This results from the density of final states which, for quasi-1D structures has a 1 / E dependence. At low temperatures (T < 77 K), the overall transport properties are determined predominantly by POP emission. This is indeed the case as AP, PZP, and POP absorption scattering rates are at least 3 orders of magnitude smaller than POP emission at 30 K. In our analysis, the dynamic properties of the carrier distribution are investigated by running an ensemble of electrons through a spatially dependent Monte Carlo code and collecting statistics as a function of the distance down the channel. To treat the issue of carrier injection into the channel, we assume that the electrons are initially in a bulk-like source region obeying Maxwell-Boltzmann statistics

155

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Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures

at a given temperature. The initial states in the channel are chosen stochastically from the right half (positive velocity) of the initial distribution and are then allowed to evolve under the influence of the longitudinal field and scattering rates. Aspects concerning the specific implementation of the Monte Carlo simulation may be found elsewhere [5, 6].

12.3 Spatial Velocity Oscillations

The relatively low probability of phonon scattering in the region below the POP energy (hwpop) indicates that carrier transport will be essentially ballistic up to the POP emission threshold. At this point carriers will emit a POP and relax to the subband minima. Figure 12.2 shows the carrier velocity for a 100,000 electron ensemble at 77 K and 30 K with a longitudinal field Fx = 100 V/ cm. The non-zero initial velocity reflects the forward component of the initial Maxwell-Boltzmann velocity distribution since these are the only carriers which are injected into the active region. As the electrons drift under the influence of the field they overshoot their steady state velocity several times in a manner analogous to an underdamped oscillator. The period of oscillation, xc, can be derived classically by recognizing that quasi-ballistic transport occurs only for energies below the POP energy, hwpop, so that

xc =

hw pop eFx

.

(12.1)

For Fx = 100 V/cm, xc = 3.6 mm, which correlates well with the data in Fig. 12.2. These oscillations persist for several periods until spatial dephasing and randomization by the acoustic interactions cause the electrons to converge onto a steady-state quasi-1D distribution. Comparison of Fig. 12.2a and b shows a weakening of the effect with increasing temperature manifested by a smaller oscillatory amplitude at 77 K. This is due to the enhanced probability of phonon absorption at higher temperatures resulting in a degradation of the ballistic behavior in the subthreshold region and therefore a shorter distance required for the electrons to achieve steady state.

Spatial Velocity Oscillations

Figure 12.2 Electron velocity vs distance in a quasi-1D channel at (a) 30 K and (b) 77 K with Fx = 100 V/cm.

157

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Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures

Figure 12.3 Transport structure disappears for both (a) low fields (Fx = 1 V/cm) and for (b) high fields (Fx = 104 V/cm) at 30 K.

In addition, the oscillations occur only for a limited range of Fx as evidenced in Fig. 12.3. For low fields (Fx < 1 V/cm), transport is dominated by acoustic deformation-potential and piezoelectric

References

scattering both of which cause a rapid randomization of the carrier distribution before electrons are allowed to drift up to the POP emission threshold. The saturation value of the velocity reflects a complete randomization of the electron ensemble for which the average kinetic energy is hwpop/2 above the subband minima. High field transport (Fx > 104 V/cm) is characterized by electrons drifting through the emission peak with little coherence in their relaxation to the subband minima. The apparent runaway of the electron velocity in Fig. 12.3b results from a lack of bandstructure and intervalley scattering in our model.

12.4 Conclusion

With the aid of Monte Carlo simulation, we have demonstrated the emergence of a novel phenomenon associated with the electronphonon interaction in quantum wires. Specifically, the nature of transport and scattering in quasi-1D channels allows strong oscillations of the carrier velocity to occur over relatively long distances. The effect was found to occur for a limited range of longitudinal fields and to diminish with increasing temperature.

Acknowledgments

This work was supported by JSEP under grant no. N00014-90-J-1270. All of the computations were performed using the resources of the National Center for Supercomputing Applications (NCSA) and the National Center for Computational Electronics (NCCE) at the University of Illinois.

References

1. W. Shockley, Bell Syst. Tech. J., 30, 990 (1951).

2. A. Matulionis, J. Pozela, and A. Reklaitis, Phys. Stat. Sol., (a) 31, 83 (1975).

3. A. Matulionis, J. Pozela, and A. Reklaitis, Phys. Stat. Sol., (a) 35, 43 (1976).

159

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Electron-Phonon Interaction and Velocity Oscillations in Quantum Wire Structures

4. K. Ismail, D. A. Antoniadis, and H. I. Smith, Appl. Phys. Lett., 54, 1130 (1989).

5. D. Jovanovic, S. Briggs, and J. P. Leburton, Phys. Rev. B, 42, 11108 (1990). 6. S. Briggs and J. P. Leburton, Phys. Rev. B, 38, 8163 (1988).

Chapter 13

Transient and Steady-State Analysis of Electron Transport in One-Dimensional Coupled Quantum-Box Structures

H. Noguchi,a J.-P. Leburton,a,b and H. Sakakia

aResearch Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan and Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan bBeckman Institute for Advanced Science and Technology, University of Illinois, 405 North Mathes Avenue, Urbana, IL 61801, USA [email protected]

We investigate electron transport in one-dimensional coupled quantum-box (1D-CQB) structures at room temperature by using an iterative technique for solving the time-dependent Boltzmann equation. The scattering rates in the mini-Brillouin zone are characterized by several large peaks reflecting the singularities of the 1D density of states and the features of the miniband structure. As a result of Bragg reflection, the momentum distribution function Reprinted from Phys. Rev. B, 47(23), 15593–15600, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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deviates significantly from a displaced Maxwellian, with carrier accumulation at the miniband edges. Under the condition of suppression of optic-phonon scattering, the time evolution of the distribution function, and the electron velocity under high electric field undergo damped Bloch oscillations with a period of a few picoseconds. In the steady-state analysis, we found that the carrier mobility is a strong function of the structure confinement and periodicity parameters.

13.1 Introduction

In the past few years, quantum microstructures [1–3] (QMSs) such as quantum wires and quantum boxes (QBs) have attracted much attention because they manifest fundamental phenomena such as quantum interferences [4] and single-electron effects [5]. The observation of these quantum effects is presently limited to very low temperatures because of the requirement of a dissipationfree environment and the difficulty of achieving a confinement dimension below 100 nm with energy-level separation in excess of a few meV. It is, however, anticipated that continuous improvement and innovation in nanostructure technology will provide new opportunities to realize QMSs with feature sizes smaller than 500 Å [6], and expand the research activities on QMSs to high temperatures. One of the most remarkable features of QMSs is their flexibility to geometrical confinement and design to achieve arbitrary spectra of electronic states and they provide new windows for technological innovation. Recently, Sakaki [3] proposed a nanostructure consisting of a one-dimensional chain of identically coupled QBs (1D-CQBs) (Fig. 13.1a). This new periodic structure can be arbitrarily designed according to the confining potential and its periodicity. The dispersion relation and the density of states (DOS) are shown in Fig. 13.1b. For certain values of the 1D-CQB parameters, it is possible to suppress polar optical-phonon (POP) scattering which is a dominant dissipation process at room temperature. Namely, if the miniband width eb is smaller than the POP energy hwPOP (= 36 meV in GaAs), intraminiband POP scattering does not occur. Furthermore, if the minigap width eg exceeds hwPOP, interminiband scattering is also prohibited. This condition is expressed as

Introduction



eb < hwPOP , eg > hwPOP . (13.1)

With the suppression of POP scattering, it is expected that the transport performance of 1D-CQBs will be significantly enhanced, even at room temperature. Moreover, strong nonparabolicity of the miniband shape, and the short periodicity of the mini-Brillouin zone, are well suited for strong nonlinear effects such as negative resistance and Bloch oscillations [7].

Figure 13.1 (a) Schematic illustration of 1D-CQBs. (b) Dispersion relation and density of states of 1D-CQBs. Numbers on the miniband denote the index (m, n) as defined in the text. The folded miniband locates much higher than the minibands considered in the text. The dashed arrow in the DOS indicates a prohibited process of the absorption of optical phonon.

In this chapter, we study electron transport in 1D-CQB structures at lattice temperature TL = 300 K by using an iterative technique applied to the semiclassical Boltzmann equation. In Section 13.2, we describe our electronic model, and in Section 13.3 we calculate the scattering rates for polar optical and acoustic deformation-potential phonons in the nanostructure. In Section 13.4, we formulate the

163

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transport theory and solve numerically the Boltzmann equation. Finally, our results on steady-state and transient transport are reported in Section 13.5.

13.2 Electronic Model

In modeling the electronic properties, we assume that the 1D-CQB structure consists of a system of square GaAs QBs separated by a AlxGa1–xAs barrier as shown in Fig. 13.1a. In this 1D system, electrons are strictly confined in the xy plane of the wire and its motion along the wire (z direction) is modulated by the periodic potential. Therefore, the electron energy can be written as e(k) = eSL(kz) + emn,



(13.2)

where eSL( kz) is the energy dispersion along the z axis (SI denotes superlattice) and emn is the quantized energy in the quantum-wire cross section (xy plane). We assume an infinite square-well potential in the xy plane. The energy resulting from the confinement in this plane is written as

e mn =

(p h )2

2meff L2

(m2 + n2 ) m, n = 1, 2,. . .

(13.3)

where m and n are indices assigned to the minibands split by the confinement, meff is the effective mass, and L is the width of the square well. The energy dispersion along the z direction and the miniband width are calculated by a Kronig–Penney model [8]. The periodic potential is set so that the upper miniband folded at the Brillouin-zone edge (see Fig. 13.1b) is located high enough to neglect the electron distribution even at high temperatures. In the present model, we restrict our analysis on interminiband scattering to the first four minibands that we refer to the index (m,n) being (1,1), (1,2), (2,1), and (2,2) since for the confinement and potential period considered here most of the electrons populate in the lowest miniband. For the sake of simplicity in forthcoming calculations the miniband structure is approximated by cosine shape,

eSL (kz ) =

eb (1 - cos kz dSL ), 2

(13.4)

Electronic Model

where eb is the miniband width, and dSL is periodicity of the potential. Then, the density of states (DOS) is expressed as

D(e ) =

 {p d

SL

m,n

-1

(e - e mn )(e mn + e b - e )} .

(13.5)

This indicates that the DOS has singular points at the top (e = emn + eb) and the bottom (e = emn) of each miniband as shown in Fig. 13.1b. The wave function can be written as

where

y(k, r) = z(kz, z)Lmn(x, y),

(13.6)

mp np 2 sin x sin y. (13.7) L L L z(kz,z) is Bloch function expressed by using the nearly free-electron model [9] as



L mn ( x , y ) =

È z(kz , z ) = Í ÍÎ

ÂC (k , z )e G

iGz

z

G

= CNFE (kz , z )e

ik z z

˘ ik z ˙e z ˙˚

,

(13.8)

where G = N2p/dSL (N = 0, ±1, ±2. . .). We take the simplest case N = 0 and – 1 with CNFE(kz,z) expressed as

CNFE (kz , z ) =

Èg ˘ e - ig z/2 cos Í z - i r ˙ . 2 dSL cosh 2r Î ˚ 1

(13.9)

1 Here, g = 2p/dSL, r = ln( 1 + Dk 2 + Dk ), Dk = (h2/2meffU)[ kz2 – 2 (kz – g)2], and U is the potential barrier height. At temperatures above 100 K, polar optical-phonon (POP) and deformation-potential (DP) acoustic-phonon scattering are dominant in III–V compound materials. Here, we make the usual assumptions that the POP energy has no dispersion and is given by the bulk phonon energy hwPOP = 36 meV, while DP scattering has a linear qz dependence. The latter assumption is essential to account for dissipation of the electron energy when the POP scattering is completely inhibited. To avoid complexity, confined and interface phonon modes are not taken into account, since the model and its interaction Hamiltonians are still under argument. However, it is

165

166

Transient and Steady-State Analysis of Electron Transport

justified to use the bulk model when the QB size is larger than 100 Å [10], or the content of Al in the AlxGa1–xAs barrier is small. This point will be discussed in detail later. Ionized impurity scattering is neglected since we can practically eliminate it by introducing a modulation doping technique into an edge quantum wire–like structure [1, 11]. Interface roughness (IF) scattering [12] is also an important process in QMS. Its effect depends strongly on the height and lateral correlation length of the interface roughness, as briefly described in Sec. III. In the main body of the text, however, we neglect IFR scattering assuming an ideal case in which uniform boxes are periodically coupled with a negligible fluctuation. Another important scattering process in the SL structure which we did not take into account is the Umklapp process in which phonons with large wave-vector scatter carriers from one minizone to the others. Although it leads to an underestimate of the scattering rates, we did not take account of this process because of the limitation of computer memories.

13.3 Scattering Model

In this section we derive the expression for the transition probability of the DP and POP scattering in the case of CNFE(kz , z) = 1. Following the expression in Ref. [13], the scattering probability can be written as Pj± ( k , k ¢ ) =

 V F (q)| ·k ¢ |e

iq◊r

j

| k Ò |2

q



j = DP,POP =



C j2

¥ d {e( k ¢ ) - [e( k ) ± hw j ]}



where

2p h

C j2 2p h

(13.10)

I j± ( k , k ¢ )d {e ( k ¢ ) - [e ( k ) ± hw j ]},

Ú ¥ Ú dRL

I j± ( k , k ¢ ) = dQFj(Q ,| kz¢ - kz |) * i(Q◊R ) m ¢ n ¢ (R )L mn (R )e

.

(13.11)

Scattering Model

Here, Q = (qx, qy), R = (x, y;), V is the volume of the system, and hwj is the phonon energy with the signs of absorption (+) and emission (–). CDP and CPOP are constant values expressed as Eq. (13.7) in Ref. [13]. F(q) is defined as 1 and nq± / q2 (nq± is phonon occupation number) for DP and POP scattering, respectively. Using Eq. (13.7), the sum of the scattering rates with arbitrary function f(k) can be written as

Âf (k ¢)P

± j (k , k ¢) =

k'

=



Lz 2p

C j2

Â Ú dk f(k ¢)P ¢ z

m ¢ n ¢±

Â

4p h m¢ n ¢± k 2

± j (k , k ¢)

Sa (ki )I j±

(13.12)

i

¥ f(m¢n¢ ki ),

where ki is one of the solutions of e(k′) – [e(k) ± hwj] = 0. The coefficient a(ki) is derived from integration of the energy-conservation term and is written as -1

˘ Èe a (ki ) = Í b dSL | sin ki dSL|˙ . (13.13) Î2 ˚



I j± can be written as [14] IDP =



and

where

± PPOP =

nq± (2p )2

4p 2

1 1 (1 + d mm ¢ )(1 + d nn ¢ ) 2 2 L

ÚÚ

(13.14)

2

ds x ds y

| G mm ¢ (s x )|2| G nn ¢ (s y )|2 s x2

+ s y2

ÈL ˘ + Í | ki - kz |˙ p Î ˚

2

,

(13.15)

Ï Ô4p 2 , s = 0, m¢ = m, Ô Ô | G mm ¢ (s )|2 = Ìp 2 , s = ±m¢ ± m, otherwise; Ô  (13.16) 2 m+m ¢ cos sp } Ô 1 (16m¢ms ) {1 - ( -1) , ÔÓ 2 {s 2 - (m¢ + m)2 }2 {s 2 - (m¢ - m)2 }2

167

168

Transient and Steady-State Analysis of Electron Transport

for DP and POP scattering, respectively. These expressions are easily expanded when CNFE(kz , z) is expressed as Eq. (13.9). In Fig. 13.2a we show the total scattering rate t( k )-1 = S k ¢ , ± Pj± ( k , k ¢ ) for the case where eb = 30 meV, dSL = 90 Å, and L = 200 Å. The intraminiband POP scattering does not occur in this case since the optical-phonon energy hwPOP (= 36 meV) exceeds the miniband width. The short-dashed and long-dashed lines denote the scattering rates for DP and POP scattering, respectively. The solid line indicates the sum of the scattering rate for DP and POP scattering. We set the zero energy at the bottom of the (1,1) miniband. The lateral confinement L results in an energy separation De12 = 42 meV between the bottoms of (1,1) and (1,2) minibands. The energy range spanned by the miniband is indicated by horizontal arrows. The scattering rate is zero between 30 meV [top of the (1,1) miniband] and 42 meV [bottom of the (1,2) miniband] since there is a real minigap with no electronic states in this energy range. This minigap also causes the suppression of interminiband POP scattering for electrons in the bottom of the (1,1) miniband (hatched area). This is because electrons cannot reach the (1,2) miniband by absorbing a POP. Only DP scattering is active over this energy range. As lateral confinement increases, the energy range in which POP scattering is suppressed becomes wider, and finally, complete suppression of POP scattering is achieved when Eq. (13.1) is satisfied. The feature of the DP scattering is quite similar to that of the DOS since acoustic-phonon scattering is regarded as a quasielastic process at small qz. In Fig. 13.2a the separation between the singular points of the miniband edges reflects the qz dependence of the phonon energy. Note that the DP scattering rate is nearly 1012 sec–1 which is still smaller than the POP rate, but much larger than the bulk value. This is due to large form factor resulting from strong confinement. The importance of the DP scattering will be emphasized in Section 13.5. Figure 13.2b shows the total scattering rate for eb = 60 meV (>hwPOP), dSL = 90 Å, and L = 200 Å where intraminiband POP scattering occurs for electron energy greater than 36 meV. Most of the electrons which have accelerated to this energy easily emit an optic phonon and relax to lower energy. Moreover, the presence of several peaks caused by the interminiband POP scattering prohibit electrons from moving away to higher energy, suggesting that the

Scattering Model

coherent electron transfer by high electric field may be impossible for this structure configuration as will be discussed later.

Figure 13.2 Calculated scattering rates of electrons in 1D-CQBs for (a) eb = 30 meV and (b) eb = 60 meV. Here, L = 200 Å, dSL = 90 Å, and TL = 300 K. Shortdashed, long-dashed, and solid lines denote the scattering rates for DP, POP, and the sum of these two processes, respectively. The arrows indicate the miniband width.

Here, we briefly estimate the scattering rate caused by interface roughness along the wire direction. By applying a Gaussian-type autocorrelation function [12], we obtain the scattering rate without screening as

1 / t IFR (kz ) =

32h2p 4 p LD 2

m * L6e bdSL | sin kz dSL|

exp( -kz2 L2 ),

(13.17)

where D is the amplitude and L is the lateral correlation length of the roughness. In Fig. 13.2a we plot 1/tIFR for the (1,1) miniband taking

169

170

Transient and Steady-State Analysis of Electron Transport

D = 5 Å and L = 150 Å, that is, a 5% fluctuation of the wire width. For very small energy, 1/tIFR is comparable to the DP scattering rate, and it decreases rapidly as the electron energy increases. Therefore, IFR scattering is not important for these parameters as long as we consider the transport under high electric field. In a steady-state lowfield analysis, IFR scattering (µL–6) cannot be neglected especially when the wire width L decreases. However, we do not take account of this effect since it will become sufficiently small as the processing technology gets refined, while phonon scatterings remain as an unavoidable process.

13.4 Transport Model

In studying transport phenomena at high temperatures, we cannot employ the relaxation time approximation because of strong inelastic scattering processes such as POP scattering. Therefore, we directly solve the semi-classical time-dependent Boltzmann transport equation (BTE) [15],

where

df ef ∂f È ∂f ˘ + = , dt h ∂kz ÍÎ ∂t ˙˚coll

È df ( k ) ˘ Í dt ˙ = Î ˚coll

 {[1 - f (k )] f (k ¢)P(k ¢ , k ) k¢

- [1 - f ( k ¢ )] f ( k )P( k , k ¢ )}.

(13.18) (13.19)

Here, f(k) is the electron distribution function, and P(k, k′) is the scattering probability from the state with wave vector k to the final state with k′ and calculated in Eq. (13.10). F is the applied electric field and the other symbols have their usual meanings. One popular method to solve Eq. (13.18) is the Monte Carlo method [16] which simulates numerically the stochastic processes of electron interaction with lattice vibrations and crystal defects. Although a recent Monte Carlo simulation of electron transport in quantum wires [17] has been shown to be very powerful for the highfield effect, memory requirement and running time for computers are quite severe to study the structural dependence of transport in

Results and Discussion

1D-CQB structures. Therefore, we employed the iterative method formulated by Rees [18]. Because of the singularities in the DOS at the miniband edges of 1D-CQB shown in Fig. 13.1b, numerical differentiation of the distribution function on the left-hand side of Eq. (13.18) must be handled with care because it might give an unstable solution. Therefore, we transformed Eq. (13.18) into the equivalent integral equation

where

fn+1 (kz ) =

Ú

•

0

gn (kz - eFt ¢ / h )e - Ft ¢ dt ¢ ,

(13.20)

È ∂f ( k ) ˘ gn (kz ) = Í n z ˙ + Ff n (kz ). (13.21) Î ∂t ˚coll

Here, we introduce a self-scattering process expressed as Ffn(kz), which adds no influence to the original equation. Starting from the initial distribution function f0, we can obtain the (n + 1) th iterative solution fn+1 from the nth approximate solution of Eqs. (13.20) and (13.21) by introducing fn into the right-hand side of Eq. (13.20). A steady-state solution is obtained when the iterative process converges. Within this scheme, the self-scattering constant F determines the convergence rate of the solution. Moreover, Rees [18] showed that when F is chosen to be large enough, each iterative step corresponds to the time evolution of the distribution function with time step 1/F. Thus, we can obtain both steady- and transient-state distribution functions simultaneously by solving Eqs. (13.20) and (13.21) and using Eqs. (13.10–13.16). First, we obtain the scattering probability and store them into memories as a lookup table. Next the iterative procedure is carried out using the lookup table. Finally the distribution function is used to calculate the electron velocity and mobility.

13.5 Results and Discussion

13.5.1 Time-Dependent Solutions In this section we show the distribution function obtained from the time-evolution analysis by solving the Boltzmann equation

171

172

Transient and Steady-State Analysis of Electron Transport

iteratively. In Figs. 13.3a and 13.3b, we show the time variation of the distribution function at a low electric field (F = 100 V/cm) and its deviation from the initial Maxwell distribution function. Here, we use the same parameters as in Fig. 13.2a where optical-phonon scattering is partially suppressed around the bottom of the first miniband. The horizontal axis denotes the wave vector along the kz direction normalized to half of the reciprocal lattice vector of the SL. The iterative procedure was performed at 1/F = 20 fsec, and the solutions were picked up at every 0.4 psec.

Figure 13.3 (a) Time evolution of the distribution function fn and (b) the difference between fn and the Maxwellian distribution function f0. The parameters used here are eb = 30 meV, L = 200 Å, and F = 100 V/cm. The results are picked up at every 0.4 psec. The horizontal axis denotes the wave vector along the z direction normalized by half of the minizone period, (c) Time evolution of the distribution under a high electric field. Here, eb = 30 meV, L = 150 Å, and F = 1000 V/cm. Short-dashed and long-dashed lines denote the distribution at T = 2.4 and 4.4 psec, respectively, (d) Electron velocity for three different electric fields (F = 200, 1000, and 5000 V/cm) as functions of time. The parameters for solid lines are the same as in Fig. 13.3c except for the electric field. The dashed line is for eb = 60 meV, L = 200 Å, and F = 1000 V/cm. Other parameters used in (a–d) are TL = 300 K, dSL = 90 Å, Nd = 5 ¥ 105 cm–1, and 1/F = 20 fsec.

Results and Discussion

The feature of the nonequilibrium distribution function is clearly different from the displaced Maxwellian shown in dashed line in Fig. 13.3a. Here, the displacement Dkx = eFt/h is given by setting the relaxation time t = 2 psec and F = 100 V/cm. After a few picoseconds the distribution function shows two shoulders at kz = ±0.3p/dSL and a rapid decrease at the origin of the minizone. The shoulders correspond to the peak in the scattering rate at 6 meV above the first miniband edge shown in Fig. 13.3a, and which is due to the scattering by POP absorption to the bottom of the second miniband. Therefore, electrons with wave vector |kz| < 0.3p/dSL are easily accelerated since they suffer neither POP absorption nor POP emission, which results in a depletion of carriers around the origin. Above 0.3p/dSL, interminiband scattering with carrier exchange between the minibands occurs quite often, resulting in a small shift from the initial distribution. Thus, the abrupt change in the scattering rate causes a nonuniform deformation of the electron distribution, and accumulation of electrons at energies where the scattering rate has singularities (see Fig. 13.3b). A similar behavior has recently been obtained in quantum-wire structures [19]. Another important difference in the displaced Maxwellian is an increase of the electron distribution in the negative kz region. This effect is caused by the reflection of electrons at the minizone edge due to the periodicity of the 1D-CQB structure. Since the electrons reflected to the other zone boundary tend to have negative velocity, the net current decreases correlatively. Thus, the periodicity introduced in the 1D system which causes the formation of minibands has somewhat of a negative effect on the transport properties. In Fig. 13.3c, we show the time evolution of the distribution function under the condition where POP scattering is suppressed. Here, dSL = 90 Å, Nd = 5 ¥ 105 cm–1, L = 150 Å, F = 103 V/cm, and 1/F = 20 fsec. The initial distribution function is Maxwellian and the data were picked up after every 0.4 psec. The figure shows the electron acceleration by the high electric field with a time shift of the center of the distribution function accompanied by a broadening. After 2.4 psec, the center of the distribution function is reflected to the opposite zone boundary because of the periodicity of the mini-Brillouin zone. Note also that as time goes on, the distribution

173

174

Transient and Steady-State Analysis of Electron Transport

function at the miniband edges becomes comparable with that at the zone center. This indicates that the distribution function tends to spread out over all the miniband width which is quite different from a displaced Maxwellian distribution. Another important feature of the carrier dynamics in the miniband structure is seen after 0.8 psec when many electrons are accelerated to the negative mass region which induces a drop in the electron velocity. In Fig. 13.3d we show the variation of the drift velocity (v) = Sv(k)f(k)/Sf(k) as a function of time for three values of the electric field. The conditions for the solid lines are the same as Fig. 13.3c. Bloch oscillations are clearly seen with the frequency 2ph/eFdSL. The damping of the oscillations is due to a degradation of the coherence of the electron distribution, i.e., the electrons are not reflected simultaneously at the minizone edges, which tends to spread out the distribution over all the mini-Brillouin zone. This is the reason why the final velocity at the convergence shows almost zero value. On the other hand, when the POP scattering is not suppressed, Bloch oscillations cannot be seen even at F = 103 V/cm because of the high scattering probabilities. The dashed line in Fig. 13.3d shows the electron velocity for the case of eb = 60 meV, L = 200 Å, dSL = 90 Å, Nd = 5 ¥ 105 cm–1, F = 103 V/cm, TL = 300 K, and 1/F = 20 fsec, in which both interminiband and intraminiband POP scattering processes are present. The velocity converges to a constant value before reaching the negative mass region since frequent absorption and emission processes of optic phonons lead to efficient randomization of carriers with different energies and prohibit the coherent carrier transport. From these results we conclude that negative velocity and Bloch oscillations may be realized even at room temperature as long as the condition for quenching of POP scattering is satisfied.

13.5.2 Steady-State Solutions

Next we discuss the steady-state analysis under low electric fields. In Fig. 13.4a, we show the low-field mobility calculated as functions of the cross-sectional width L of the wire. The squares and solid circles correspond to the mobility for the case of a wide miniband (eb = 60 meV) and a narrow miniband (30 meV), respectively. Dashed and dotted lines indicate the mobility component for these two cases that would be determined by the DP scattering only. Here, we set dSL

Results and Discussion

= 90 Å, Nd = 5 ¥ 105 cm–1, and F = 100 V/cm. The upper horizontal scale denotes the energy generation De = e12 – e11.

Figure 13.4 (a) Calculated mobility at TL = 300 K as a function of the crosssectional width L of the wire. Squares and solid circles denote the total mobility for eb = 60 and 30 meV, respectively. Dotted and dashed lines indicate the mobility determined by DP scattering. Here, dSL = 90 Å, Nd = 5 ¥ 105 cm–1, F = 100 V/cm, and TL = 300 K. (b) Schematic illustration of 2D-CQBs.

In the case of eb = 30 meV where the intraminiband POP scattering is suppressed, the minigap width eg = De – eb is larger than hwPOP as long as L is smaller than 159 Å. Therefore, the total mobility for L < 159 Å is not affected by POP scattering, but is mainly determined by DP scattering. When L is set larger than 159 Å, the POP contribution comes in. Indeed, when L = 200 Å, the large dip of the mobility appears. This is caused by optic-phonon resonance between the two minibands and resembles to the well-known magnetophonon resonance [20] whose band separation originates from the Landau splitting. The POP scattering is strongly enhanced when the opticphonon energy equals the energy separation De between the bottoms of the minibands where due to the singularities, the joint

175

176

Transient and Steady-State Analysis of Electron Transport

DOS is maximum. The rapid recovery of the mobility with decrease of L indicates the effective reduction of the POP scattering. Although the mobility feature clearly shows the effect of suppression of POP scattering, the mobility amplitude itself is not large as compared to the bulk value (m = 6000 cm2/Vs at 300 K) or other QMS structures. Three factors limit the mobility to low values: first, the acoustic-phonon scattering is enhanced by the strong confinement (size effect [21]) in the wire which brings its contribution to the level comparable to the POP scattering rate. The second factor is the miniband structure itself; in order to suppress intraminiband POP scattering, one needs to choose a narrow miniband. This, in turn, causes an increase of the effective mass by a factor of 3 to 4 times larger than the bulk value at the G point and reduces the mobilities. This explains why the mobility for the case of eb = 60 meV is larger than that of eb = 30 meV. Moreover, the effect of a negative mass region in the upper part of the miniband also decelerates the electron motion. The third factor is Bragg reflection at the minizone edges; once electrons reach the minizone boundaries, electrons are reflected back to the opposite zone boundaries with negative wave vector. Therefore, these shifts in the electron distribution by the electric field result in a total loss in the total momentum. To achieve the highest mobility, one must optimize the structure parameters L, eb, and dSL which influence these three factors. However, the highest value of the mobility is always limited by the POP scattering except in the case where POP scattering is completely suppressed as discussed earlier. Therefore, reducing the acousticphonon scattering is the most important challenge to improve the mobility under the suppression of POP scattering. Among the three mobility limiting factors discussed above, the second and third ones cannot be avoided since they derive from the essential characteristics of the miniband. However, the first factor, or the confinement-induced enhancement of DP scattering can be reduced if one employs 2D- or 3D-CQBs (super crystals) as shown in Fig. 13.4b. Since the confinement in these structures is weaker than in the 1D case, the electron-phonon interaction can, in principle, be reduced. The band structure is almost the same as 1D case except that new minibands are formed along the other directions. In addition, the energy separation between the upper minibands is still maintained, or in some cases can become larger since the resulting

Concluding Remarks

minigap can be far larger than the energy separation between quantized levels. The details of the analysis of 2D- and 3D-CQBs will be discussed in a forthcoming publication. Before concluding, we would like to point out two important considerations which have been omitted in the present model. One is the broadening of the DOS caused by scattering and roughness of the interface [22]. The broadening removes the singularities in the DOS which play quite an important role in the determination of the distribution function. Moreover, since band tailing resulting from the DOS broadening may appear in the minigap, residual optical-phonon scattering may still be active even when eg > hwPOP. Another important effect is the consideration of confined and interface phonon modes [23]. When the well width dw decreases below 40 Å, the nature of dominant phonon modes is affected by confinement. The acoustic-phonon branch is folded at q = p/dSL generating many new phonon modes around q = 0. The opticalphonon branches split to several discrete energy levels with no dispersion (confined phonon modes) and localized modes at the heterointerfaces (interface phonon modes). These effects change the electron-phonon interaction Hamiltonian as well as the energy exchange involved during the interaction. However, these effects will not qualitatively change our conclusions as long as GaAs-like POP modes are concerned. This is because the whole set of GaAslike modes affected by the confinement have almost the same energy (between hwTO = 33.3 meV and hwLO = 36 meV) and almost the same consequences as bulk phonons [24]. Thus, the condition of suppression of optical-phonon scattering [Eq. (13.1)] is still maintained. Note that AlAs-like interface phonon modes become important with decreasing the well width [25]. Since they have higher energy than that of GaAs-like modes, the condition, Eq. (13.1), has to be changed to realize complete suppression of POP scattering.

13.6 Concluding Remarks

We have presented a transport analysis in 1D-CQB structures at room temperature by using an iterative technique applied to the time-dependent Boltzmann equation. Transient solutions of the distribution function predict the occurrence of Bloch oscillations in

177

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Transient and Steady-State Analysis of Electron Transport

the picosecond regime. In the steady-state analysis, the mobility in 1D-CQBs is shown to depend strongly on wire sizes and periodicity. The important requirement to realize long-lasting Bloch oscillations and high-mobility systems at high temperatures is to set up the condition in which DP scattering in CQB systems is also suppressed. We believe that this requirement can be satisfied by manipulating the miniband structures, including the possible use of 2D and 3D coupled quantum-box structures.

Acknowledgments

One of us (J.P.L.) would like to acknowledge partial support from ARO under grant no. DAAL 03-91-G-0052. J.P.L. is also indebted to Hitachi LTD for supporting his research work during his stay at RCAST. This work is partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, and also partly by the JRDC through the ERATO quantum wave project.

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20. H. Noguchi, T. Takamasu, N. Miura, and H. Sakaki, Phys. Rev. B 45, 12 148 (1992). 21. J. P. Leburton, Appl. Phys. Lett. 56, 2850 (1984).

22. J. P. G. Taylor, K. J. Hugill, D. D. Vvedensky, and A. MacKinnon, Phys. Rev. Lett. 67, 2359 (1991).

23. M. V. Klein, IEEE J. Quantum Electron. QE-22, 1760 (1986); G. Fasol, M. Tanaka, H. Sakaki, and Y. Horikoshi, Phys. Rev. B 38, 6056 (1988); N. Mori and T. Ando, Phys. Rev. 40, 6175 (1989); D. J. Mowbray, M. Cardona, and K. Ploog, Phys. Rev. B 43, 1598 (1991). 24. T. Tsuchiya and T. Ando, Semicond. Sci. Technol. 7, B73 (1992).

25. H. Noguchi, T. Takamasu, N. Miura, J. P. Leburton, and H. Sakaki, in Phonons in Semiconductor Nanostructures, NATO Advanced Research Workshop, edited by J. P. Leburton, J. Pascual, and C. Sotomayor Torres (Kluwer Academic, Boston, in press).

179

Chapter 14

Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

J. E. Stacker,a J.-P. Leburton,a H. Noguchi,b and H. Sakakib

aBeckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA bResearch Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan [email protected]

We investigate acoustic-phonon scattering in quantum wires subject to a periodic potential along the propagating direction. A technique for modeling the electronic structure of the periodic system is introduced using the imaginary time propagation method. The acoustic-phonon scattering rate is evaluated by taking Umklapp processes into account. We found Umklapp processes can cause a significant increase in intersubband scattering but is negligible for intrasubband scattering. Overall, the exact treatment of the electron Reprinted from J. Appl. Phys., 76(7), 4231–4236, 1994. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1994 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

dispersion relation improves the acoustic-phonon limited mobility compared to earlier estimates [4].

14.1 Introduction

In recent years, the advancement of nanofabrication techniques has considerably improved the feasibility of quantum structures with multiple degrees of confinement. Quantum wires in particular are expected to exhibit interesting new effects among which mobility enhancement which are due to the reduction of the phase space restricting the number of final scattering states and thereby reducing the scattering rate. These high mobilities could be utilized in highspeed device applications [1, 2]. The possibility of further increasing mobility by suppression of polar optical-phonon (POP) scattering has been suggested by Sakaki by modulating periodically the potential along the propagating direction of the quantum wire [3]. The widths of the miniband (Eb) and the minigaps (Eg) resulting from the periodic potential can be modified by changing the period and amplitude of the potential. If the miniband width is less than the POP energy (Eb < hwLO), intraminiband POP scattering. can not occur. Also, if the minigaps are greater than the POP energy (Eg > hwLO), no interminiband POP scattering can take place. Consequently, the electron mobility is essentially limited by deformation-potential acoustic-phonon (DAP) scattering which is much weaker than POP scattering. Recently, Noguchi et al. investigated theoretical transport in periodically modulated quantum well wires (QWW) by using an iterative technique based on the nearly free-electron model and normal (N) DAP processes [4]. In the conditions of suppressed POP scattering, they found DAP limited mobilities slightly higher than the GaAs bulk mobility with the structure of the DAP scattering quite similar to the density of states (DOS). In this analysis, we are interested in a more complete model of the DAP scattering rate in periodically modulated quantum wire structures. The quantum-mechanical treatment of phonon

Introduction

scattering in low dimensional systems has received some attention recently [5]. In the single electron picture, and by assuming that the electron-phonon interaction is localized in a finite region of space, one phonon process, i.e., real phonon emission or absorption, can be treated within the distorted-wave Born approximation also containing elastic-scattering processes [6]. It has been shown in this case, that the average rate for phonon scattering can be calculated by the Fermi’s golden rule within the Born approximation provided that the elastic processes such as the coherent transmission and reflection through the periodic potential are calculated exactly by other methods. Therefore, we first calculate the contribution of the elastic processes in the quantum transmission through the periodic potential by using the imaginary time propagation (ITP) method [7] for solving Schrödinger equation, which results in the electronic band structure of the quantum wire. This method allows us to accurately model the band separation and dispersion relation for the propagating states as well as justifying the use of the Fermi’s golden rule in the calculation of phonon scattering. In this framework, we derive the DAP scattering rate beyond the single-component approximation by including the total q-vector dependence in the scattering matrix element throughout the calculation [8]. Owing to the periodic nature of the potential, we also include the contribution of Umklapp processes to the total DAP scattering rate [9]. We consider two types of periodically modulated quantum wire structures. In the first case, we investigate the DAP scattering rate in 150 ¥ 150 Å one-dimensional coupled quantum box structures with short period (90–180 Å) (Fig. 14.1a) and compare our results with those of Noguchi et al. [4]. In the second case, we investigate the contribution of Umklapp processes in the DAP scattering of a 200 ¥ 200 Å 1D structure modulated by a periodic cosinelike potential with rather long period (500–1000 Å) (Fig. 14.1b). This kind of structure which could be realized by deposit of a periodic gate structure on a quantum wire is more accessible to the present technology. However, because the amplitude of the potential is weaker than in the previous case, our analysis will be limited to low temperature (T = 77 K).

183

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

Figure 14.1 (a) Periodic square potential along the propagating direction of the quantum wire. (b) Periodic cosine potential along the propagating direction of the quantum wire.

14.2 Electronic Band Structure In our investigation, we assume the electron energy in the quantum wire is given by Eijk(kx) = Ek(kx) + Ei + Ej,

(14.1)

where Ek(kx) results from the 1D periodic potential and describes the dispersion of the minibands. Ei and Ej are the energy levels due to the transverse confinement of the quantum wire. We use the imaginary time propagation method to solve for the 1D energy bands and associated 1D wave functions [10]. The method is flexible enough to model all kinds of periodic potentials and uses a split-operator scheme where the kinetic operator and potential operator propagate the wave function according to

y( x , t + Dt )  e

- iV ( x ) D t /2 - iK E ( x ) D t / h - iV ( x ) D t /2

e

e

y ( x ,t )

(14.2) + O( Dt ) for each successive time step. V(x) describes the potential and KE(x) is the kinetic-energy operator which is given by 3

Electronic Band Structure



KE( x ) = -

h2 Ê ∂2 ˆ . (14.3) 2m * ÁË ∂x 2 ˜¯

Figure 14.2 (a) Dispersion relation of infinite square potential with an amplitude of 245 meV and period of 90 Å. (b) Dispersion relation of infinitely periodic cosine potential with an amplitude of 12 meV and period of 500 Å.

The ITP method requires a reasonable initial guess for the wave functions and propagates these eigenvectors until convergence is achieved [7, 11, 12]. Once the wave functions have been calculated, the energy values can be computed using

185

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

Ek(kx) = Úy(x)*H(x)y(x)dx, (14.4)



where H(x) is the 1D Hamiltonian of particles in the periodic potential. Because of the exponential dependence of the propagating operator, this method is extremely fast, but it necessitates discretization of k space and large amounts of memory to store the wave functions. In order to minimize calculations, we assumed the quantum wire to be infinitely periodic [10]. The accuracy of the method was verified by comparing with the results of a Kronig–Penney model for an infinite set of periodic quantum wells. An example of the dispersion relation of a periodic square-well structure and periodic cosine potential, as calculated using the ITP method, is shown in Fig. 14.2. The square-well structure is similar to a periodic GaAs-Al0.3Ga0.7As structure with a barrier height of 245 meV and a period of 90 Å with 12 Å barriers and 78 Å quantum wells (see Fig. 14.2a). In this structure, the energy gap between minibands is practically constant up to high energies, thus preserving the mini-band structure well above the barrier. On the other hand, the cosine potential has a 12 meV amplitude and 500 Å period (see Fig. 14.2b). For this kind of structure, the ground state is relatively well localized and the miniband structure begins to deteriorate at energies above the cosine potential. These features for both types of periodic potentials remain when one changes the amplitude and periodicity.

14.3 Acoustic-Phonon Scattering

In 1D systems, the deformation-potential acoustic-phonon scattering rate is given by [13] 1



t kx

=

V0l 2p dkx¢ (2p )3 h

Ú ÚÚ dx

y

¥ dqz | M(kx , kx¢ , qy , qz )|2

d (E k ¢ - E k ± hu us q),

where the matrix element is calculated according to



Ú

Ú



M(kx , kx¢ , qy , qz ) = Vq2 dx y *( x )y ( x )e ± iq x x dy f*( y ) ¥ f( y )e

± iq y y

and the constant Vq is given by [14]

Ú

dz x*( z ) x( z )e ±iqz z ,

(14.5)



(14.6)

Acoustic-Phonon Scattering



Ê hDa2 ˆ Vq = i Á ˜ Ë 2V0l rus ¯

1/ 2

)

(

q1/2 Nq + 1 / 2 ± 1 / 2

1/2

. (14.7)

The parameter Da is the deformation potential for acoustic phonons, r the material mass density, us the speed of sound in the material of interest, Nq the phonon occupation number, and the ± sign is for phonon emission and absorption, respectively. The infinite square-well model used for the double transverse confinement simplifies the calculations since the analytical integrations over the y and z directions in the matrix element are straightforward [15]. Because of the periodic nature of the longitudinal potential, the wave function in the propagating direction takes on the form of a Bloch function y ( x ) = U( x )e ik x x . Therefore, the integration in the x direction becomes

Ú dx y *( x )y ( x )e

± iq x x

= dk

x

- k ¢x ± q x

Ú dx U *( x )U( x ), (14.8)

where one only integrates over one period of the longitudinal potential. In this model, we go beyond the DAP single-component approximation [16] and consider the full integration over the three components of the DAP wave vector [8]. After multiplication and factoring, the energy conservation delta function for phonon absorption in Eq. (14.5) reads d ( E k - E k ¢ + hu s q )



=

1 Ê Ek ¢ - Ek hus ÁË hus

2 ˘ ˆ ÈÊ E - E k ˆ 2˙ q˜ d ÍÁ k ¢ q . ˙ ¯ ÍË hus ˜¯ Î ˚

(14.9)

The q vector can be broken into its components, i.e., q = (qx2 + qy2 + qz2 )1/2 , argument of the delta function can be solved for qz such that d ( E k - E k ¢ + hu s q )



where



=2

E k ¢ - E k È d (qz - Q1/2 ) + d (qz + Q1/2 ) ˘ Í ˙, ( hus )2 ÍÎ 2Q1/2 ˙˚

(14.10)

˘ ÈÊ E - E ˆ 2 k Q = ÍÁ k ¢ - qx2 - qy2 ˙ . (14.11) ˜ ÍË hus ¯ ˙ Î ˚

187

188

Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

This form of the delta function allows us to eliminate the integration over qz in Eq. (14.5). The same method is used for phonon emission. The remaining equation only involves a double integration over qy and kx¢ . The limits of integrations are finite since Q in Eq. (14.11) cannot be negative, i.e.,

ÈÊ E - E ˆ 2 ˘ 2˙ ¢ k k k - ÍÁ k ¢ ( ) x x ÍË hus ˜¯ ˙ Î ˚

1/ 2

ÈÊ E - E ˆ 2 ˘ 2˙ ¢ k ( k k ) £ qy £ + ÍÁ k ¢ x x ÍË hus ˜¯ ˙ Î ˚

1/ 2

,

(14.12)

where qx = kx¢ – kx has been substituted into Eq. (14.11). The range of integration on kx¢ is limited by the equation 2



Ê Ek ¢ - Ek ˆ 2 ¢ ÁË hu ˜¯ ≥ (kx - kx ) . (14.13) s

For U processes which allow for the electrons to scatter to final states outside of the first Brillouin zone, the final kx vector can be represented by

K x¢ = kx¢ ± nG , (14.14)



where kx¢ is the k vector within the first Brillouin zone, n an integer, and G the reciprocal-lattice vector. Here, E( K x¢ ) = E( kx¢ ) because of the periodic dispersion relation. A scattering schematic including U processes is shown in Fig. 14.3. The formalism for calculating the DAP U-processes scattering rate is essentially unchanged compared with the normal DAP scattering process except for the x component of the matrix element of Eq. (14.6), which becomes

Ú dx y *( x )y( x )e = dk

¢ x -k x

±qx

± iq x x

Ú

dx U*( x ) U( x )e ± inGx .

(14.15)

For large values of nG, the integral of Eq. (14.15) decreases toward zero and the relevant Umklapp processes are limited to a few Brillouin zones. Therefore, the range of integration over kx¢ in Eq. (14.5) can be reduced to only a few extended Brillouin zones (Fig. 14.3).

Acoustic-Phonon Scattering

Figure 14.3 Scattering schematic showing allowed regions of scattering for acoustic phonons including those due to Umklapp processes.

In general, the contribution of the Umklapp processes is limited to the first few Brillouin zones, and the DAP energy exchanged during scattering events is relatively small (of the order of 10%) compared with the thermal energy of the electrons. Therefore, we can calculate the mobility due to DAP scattering in the quasielastic approximation and use the linear response. Here we are mostly concerned by nondegenerate carriers at high temperature as in Ref. [4]. We obtain [17]

∂f ˘ È m = Í-e t rel (kx ) 0 ui2 (kx )dkx ˙ / ∂E Î ˚

Ú

(Ú f dk ) , (14.16) 0

x

where f0 is the Boltzmann distribution, ui(kx) the carrier velocity, and

V 1 = 0l dk ¢ S (k , k ¢ )(1 - cosq ) t rel (kx ) 8p 3

Ú

=

1 t kx

(1 - cosq ).

(14.17)

For forward scattering, q = 0 and thus the momentum scattering rate goes to zero. For backward scattering, q = p and the momentum scattering rate equals twice the backward-scattering rate.

189

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

14.4 Results 14.4.1 Short Modulation Period Quantum Wires In the following, we apply our model to a periodic square potential structure similar to that studied by Noguchi et al. [4]. The nomenclature used for identifying the different minibands reflects the miniband created by the periodic potential and the potential energy due to the transverse confinement. For example, the (2,1,2) miniband refers to the second miniband associated with the first and second energy levels of the transverse confinement. For a 150 ¥ 150 Å QWW and a 90 Å period with 12 Å barriers, 78 Å quantum wells, and 245 meV barrier height, the (1,1,1) miniband lies at roughly 70 meV and is 30 meV wide, and the (2,1,1) miniband is at 145 meV and is approximately 100 meV wide (see Figs. 14.2a and 14.4]. Also, the (1,1,2) and (1,2,1) minibands are located at 150 meV, and the (2,1,2), (2,2,1), and (1,2,2) minibands are all located at approximately 225 meV (Fig. 14.4). Figure 14.4 also shows the total DAP scattering rate for this structure from 60 up to 260 meV.

Figure 14.4 DAP scattering rates for a 150 ¥ 150 Å infinite square quantum wire with periodic square-well potential (245 meV amplitude and 90 Å period with 12 Å barrier and 78 Å well) along the propagating direction. The two curves represent scattering at 300 K with U processes (solid) and N processes only (dashed). The horizontal bars represent the position and width of the minibands.

Results

The scattering rate at T = 30 K (Fig. 14.4) for N processes is slightly lower than those reported by Noguchi et al. when the difference in the deformation-potential constant is taken into consideration (We chose a deformation potential of 6.7 eV instead of 11 eV used in Ref. [4]). This slight difference is most likely due to the more exact modeling of the band structure and the rigorous treatment of the DAP scattering rate. With U processes included, the DAP scattering rate increases slightly compared to the scattering rate in N processes. In Fig. 14.4, the gaps in the scattering rate reflect the gaps in the miniband structure, and usually, there is a small energy exchange between electrons and phonons. The peaks in the scattering rates are due to the very high DOS at the miniband edges which leads to a larger number of possible final states for scattering compared to the rest of the band. However, the peaks in the scattering rate do not occur right at the band edge because electrons located at a band edge will either scatter toward the middle of the band where the DOS is lower, or they will not scatter at all because there are no final states in the minigap. Because of the overlap of minibands and interminiband scattering, peaks in the scattering rate occur throughout the energy range. The mobility for N processes was calculated to be 26,700 cm2/Vs, which is about four times higher than those obtained by Noguchi et al. [4]. This increase is partly due to the difference in the DP constant. Other contributions include the rigorous modeling of the dispersion relation which reduces the DOS at the Brillouin zone edge and the full treatment of the matrix element which eliminates divergence and reduces the peaks in the scattering rate at k = 0 and k = p/d. By including U processes, we found a mobility of 21,100 cm2/Vs, which reflects a reduction of 20% on the N-DAP mobility. The 90 Å periodic square-well structure was also simulated for a 200 ¥ 200 Å QWW which causes the subbands to become more closely spaced. As seen in Fig. 14.5, POP scattering suppression no longer exists, and the contributions of each miniband make the scattering profile quite complicated. The mobility for N processes was calculated to be 41,100 cm2/Vs, whereas it is 32,500 cm2/Vs with U processes included, which also show a reduction of approximately 20% on the N-DAP mobility. The overall increase of the mobility compared to the previous case is due to the weaker confinement of the structure being investigated.

191

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

Figure 14.5 DAP scattering rates for a 200 ¥ 200 Å infinite square quantum wire with periodic square-well potential (245 meV amplitude and 90 Å period with 12 Å barrier and 78 Å well) along the propagating direction. The two curves represent scattering at 300 K with U processes (solid) and N processes only (dashed). The horizontal bars represent the position and width of the minibands.

In Fig. 14.6, the QWW is 150 ¥ 150 Å and the period of the structure is doubled to 180 Å which causes the energy levels to drop lower in the periodic wells and be more localized. Therefore, the minibands and gaps are smaller with the (1,1,1) miniband at approximately 60 meV, the (2,1,1) miniband at 95 meV, the (3,1,1) miniband at 150 meV, and the (4,1,1) miniband at 220 meV. The (1,1,2) and (1,2,1) minibands begin at 140 meV, and the (1,2,2) miniband is at 210 meV with subsequent minibands (2-4,1,2), (24,2,1), and (2-4,2,2) being spaced the same as minibands (2-4,1,1). As shown in Fig. 14.6, by doubling the period of the structure, the desired suppression of POP scattering no longer exists. The total DAP scattering rate is calculated for an energy range of 50–275 meV (Fig. 14.6), and the characteristics of the scattering rate in normal DAP processes do not significantly change when compared with the 90 Å structure. For this structure, the calculated mobility is 3300 cm2/Vs for N processes and 2190 cm2/Vs when U processes are included, which is a reduction of more than 33% on the N-DAP mobility. The overall decrease of the mobility compared to the previous cases is due to the increase of the miniband effective mass resulting from the thicker QW barrier in the longer period structure.

Results

Figure 14.6 DAP scattering rates for a 150 ¥ 150 Å infinite square quantum wire with periodic square-well potential (245 meV amplitude and 180 Å period with 24 Å barrier and 156 Å well) along the propagating direction. The two curves represent scattering at 300 K with U processes (solid) and N processes only (dashed). The horizontal bars represent the position and width of the minibands.

14.4.2 Long Modulation Period Quantum Wires In this section, we investigate the influence of Umklapp process on the scattering rate in a long period quantum wire modulated by a cosine periodic potential. The structure cross section is 200 ¥ 200 Å with a period of 500 Å. The minibands and gaps are greatly reduced compared to the short period structures, so our analysis will be limited to liquid-nitrogen temperature (T = 77 K). The minibands under consideration here cover an energy range of 30–120 meV. Figure 14.7 shows the miniband structure and the contribution to the DAP scattering rate of the (1-6,1,1), (1-5,1,2), (1-5,2,1), and (1,2,2) minibands. The long period structure creates a nearly continuous miniband structure, but the scattering profile is very similar to that shown in the short period structures. One noticeable difference is the large increase in the U-process scattering at the miniband edges. With U processes, the long period and close proximity of minibands allows for a large number of scattering events to the high DOS band edges. The mobility is 357,000 cm2/Vs for N processes, and 295,000 cm2/Vs when U processes are included. These mobilities show

193

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

approximately a 20% reduction in the mobility when U processes are included. The large increase in mobility compared to the short period structures is due to relaxed confinement and weaker modulation.

Figure 14.7 DAP scattering rates for the first six minibands of a 200 ¥ 200 Å infinite square quantum wire with a periodic cosine potential (12 meV amplitude and 500 Å period) along the propagating direction. The two curves represent scattering at 77 K with U processes (solid) and N processes only (dashed). The horizontal bars represent the position and width of the minibands.

In Fig. 14.8, we doubled the period of the cosine structure to 1000 Å and kept all other parameters the same. The DAP scattering rate is shown for electrons initially in the (2-6,1,1) minibands which cover an energy range of 35–55 meV. The (1,1,1) miniband is not shown due to the fact it is highly confined and not useful for transport. Due to the small energy range, no other subbands overlap, and thus intersubband processes are not included in the scattering rate. In Fig. 14.8, the scattering rate for N processes tails off much quicker compared to Fig. 14.7. This is due to the very small q vector needed to scatter in the first Brillouin zone since the scattering rate is proportional to q at very small values of q. Figure 14.8 also demonstrates that U processes become more significant in longer period structures. This is a direct effect of doubling the period of the potential which decreases the size of the Brillouin zones. Hence, twice as many final states will be available for scattering, which

Conclusion

makes a significant contribution to the scattering rate as determined by Eq. (14.15). Also, since the band edges provide a larger number of final states, the edges of the bands will show a larger increase caused by the doubling of the dispersion relation.

Figure 14.8 DAP scattering rates for the second through the sixth miniband of a 200 ¥ 200 Å infinite square quantum wire with a periodic cosine potential (12 meV amplitude and 1000 Å period) along the propagating direction. The two curves represent scattering at 77 K with U processes (solid) and N processes (dashed). The horizontal bars represent the position and width of the minibands.

14.5 Conclusion We have investigated acoustic-phonon scattering rates in periodically modulated quantum wires, and showed that the exact profile of the electron dispersion relation is important in the determination of the DAP scattering rates. We have also shown that Umklapp processes made a minor contribution to intrasubband scattering, but is very important in intersubband scattering and causes a decrease in the carrier mobility in the two sets of periodic systems, square wells and cosine potentials. By doubling the period of the structure, it was found that the influence of Umklapp processes is increased for long period structures. Finally, we would like to emphasize that in the case of POP suppression due to the miniband structure of periodic wires as proposed by Sakaki [3], our analysis predicts

195

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Acoustic-Phonon Limited Mobility in Periodically Modulated Quantum Wires

an enhancement of the DAP limited mobility compared with our previous results [4]. This mobility enhancement is partially due to the more exact determination of the miniband structure and could be used for implementing high-speed quantum wire devices.

Acknowledgments

One of us (J. S.) thanks Dr. Tetsuo Kawamura, Dr. Dan Jovanovic, and Dr. Paul Sotirelis for their valuable input. This work is supported by the ARO grant no. DAAL03-91-6-0052.

References

1. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

2. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988). 3. H. Sakaki, Jpn. J. Appl. Phys. 12, 1735 (1989).

4. H. Noguchi, J. P. Leburton, and H. Sakaki, Phys. Rev. B 47, 15593 (1993). 5. F. Sols, Ann. Phys. 214, 386 (1992). 6. F. Sols, Ann. Phys. 214, 414 (1992).

7. R. Kosloff and H. Tal-Ezer, Chem. Phys. Lett. 126, 223 (1986). 8. R. Mickevicius and V. Mitin, Phys. Rev. B 48, 17194 (1993).

9. J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge University Press, Cambridge, 1972). 10. J. M. Bigelow, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1993. 11. M. Degani and J, P. Leburton, Phys. Rev. B 44, 10901 (1991).

12. D. Jovanovic and J. P. Leburton, Phys. Rev. B 49, 7474 (1994).

13. D. Jovanovic and J. P. Leburton, Monte Carlo Device Simulation: Full Band and Beyond, edited by Karl Hess (Kluwer Academic, Boston, 1991), p. 196. 14. B. K. Ridley, Quantum Processes in Semiconductors (Oxford Science, Oxford, 1988), p. 296. 15. J. P. Leburton, J. Appl. Phys. 56, 2850 (1984).

16. D. Jovanovic, S. Briggs, and J. P. Leburton, Phys. Rev. B 42, 11108 (1990). 17. K. Hess, Advanced Theory of Semiconductor Devices (Prentice-Hall, Englewood Cliffs, NJ, 1988).

Chapter 15

Antiresonant Hopping Conductance and Negative Magnetoresistance in Quantum-Box Superlattices

Yuli Lyanda-Geller and Jean-Pierre Leburton

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

Electric current in a one-dimensional chain of quantum boxes is characterized by a set of zero minima (antiresonances) due to the momentum selection rule for the interaction of acoustic phonons with Wannier–Stark (WS) localized electrons or due to the folded phonon gaps. Antiresonances manifest also in the conduction of superlattices in the presence of an external source of phonons. Negative magnetoresistance of superlattices in high electric fields is predicted. We also clarify general questions of boundary conditions in the hopping regime. Reprinted from Phys. Rev. B, 52(4), 2779–2783, 1995. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1995 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Antiresonant Hopping Conductance and Negative Magnetoresistance

15.1 Introduction Transport in quantum microstructures (QMS) manifests such fundamental phenomena as quantum interference and electron localization. QMS can be designed to achieve arbitrary spectra of electronic states and are flexible to geometrical confinement providing alternative windows for technological innovation. In particular, transport [1–4] and magnetotransport [5–7] in superlattices (SL) have been the subject of intense investigations. With magnetic field perpendicular to the layers SL realize a high degree of confinement of the electron gas and are well suited for simulating transport in quantum-box superlattice (QBSL) recently proposed to control phonon scattering [8, 9]. In the present chapter, we investigate the conductance mechanisms in QBSL and predict a set of minima (antiresonances) in the current-voltage characteristics, due to the momentum selection rule for the interaction of acoustic phonons with the WS localized electrons. We also discuss the onset of antiresonances, due to the gaps in folded phonon spectrum and analyze resonant hopping of electrons between localized states, due to elastic and optical phonon scattering. We find that owing to the electron localization in high electric field the longitudinal magnetoresistance is negative over substantial range of magnetic field. The effects we study are the manifestation of quantum confinement. We suggest that not only electron, but also phonon confinement can be observed in conductance experiments. In addition, we discuss critical issues on the importance of boundary conditions imposed on the electron chemical potential in the description of the nonequilibrium hopping conduction in QMS.

15.2 Electronic Model

In one-dimensional (1D) QBSL the electron motion is modulated by the periodic potential in the z direction of the chain and is strictly confined in the xy plane (Fig. 15.1). Consider the Wannier representation for the electron wave functions of QBSL,

ynmqv = Lnm (x, y) uq (z – vd),(15.1)

Electronic Model

where Lnm is the transverse wave function, m and n are the integer numbers, d is the period of QBSL, and uq (z – vd) is the wave function of an eigenstate Eq in a separate quantum well v. The Hamiltonian in the presence of the uniform electric field F || z is

H0 = (Eq + ϵnm – eFdv) dvv¢ + Dq (dv, v¢+1 + dv, v¢–1),

(15.2)

Dq is the tunneling matrix element (we consider only the tunneling between neighboring QB) and ϵnm is the transverse energy. Without essential loss of generality we discuss mostly the states corresponding to the lowest longitudinal mode. In this way, we set E1 = 0 and omit the index q. In analogy with conventional crystals [10], the Hamiltonian (15.2) in zero electric field results in a band spectrum with a width 4D. If the potential drop over the period of a structure, eFd, exceeds the collisional broadening of the electron levels h/t, the electronic subband splits into WS ladder of localized states. The corresponding eigenvalues are Enma – ϵnm – eF da, and the wave functions are given by



Y nma =

ÂJ

n -a y nmn

n

(15.3)

,

(15.4)

where Jk(2D/eFd) are the Bessel functions of order k (Ref. [11]) and a is the Stark diagonal representation index. Therefore, the band conduction breaks down and electrons move in the z direction only by hopping from one well to the other, due to scattering; direct tunneling between the WS states for q = 1 manifests in the electron spectrum and is taken into account in Eqs. (15.3, 15.4). As for tunneling between states q = 1 and the WS states of higher minibands [4], one can diagonalize the Hamiltonian for several minibands in electric field and, thus, include the effect of direct interband tunneling in the q component wave functions (see Ref. [12]). Then only the scattering-assisted hopping results in a current.

Figure 15.1 Schematic QBSL with WS localization in the magnetic-field confinement.

199

200

Antiresonant Hopping Conductance and Negative Magnetoresistance

There is an essential difference between conventional SL and QBSL. SL electrons are characterized by a continuous spectrum in the direction transverse to its axis and the energy conservation law can always be satisfied for any scattering process. For instance, in elastic scattering by impurities, the energy conservation is the result of an interchange between longitudinal and transverse energies [4, 13], whereas phonon-assisted hopping is accompanied by a partial energy transfer to the lattice. On the other hand, in QBSL the transverse degree of freedom is characterized by a discrete spectrum ϵmn and electron hopping exists only when the transverse energy spacing is equal to the separation between the WS levels or differs from the latter by the phonon energy. It follows then that elastic and optical phonon scattering should manifest in a number of resonant peaks in the current. We note that elastic and optic phonon-assisted resonances are known to influence electron transport in double quantum well structures with transverse 2D continuous spectra [14]. However, the only background mechanism for conduction in SL is scattering by acoustic phonons. We are going to demonstrate that acoustic-phonon-assisted hopping in QBSL is characterized by a set of antiresonant minima. These minima are related to the quasi-1D character of the phonon propagation in QBSL, which dramatically affects both the phonon spectrum and the electronphonon interaction.

15.3 Transport Model

In this section, we discuss the electric current in QBSL as a hopping current between the WS localized states, which is due to electron scattering of phonons, impurities, and various irregularities of super-lattice structure. We describe the hopping conductance in QBSL, by the following transparent formula:

jz = e

Â

a ,a ¢ ,N,N ¢

a ,a ¢ ( za ,N - za ¢ ,N¢ )WN,N ¢ ,

(15.5)

where za,n – za¢,n¢ is the electron displacement (the hopping length) upon the scattering (a, N Æ a ¢, N ¢), N stands for the set of indices a ,a ¢ describing the transverse states, WN,N¢ is the scattering probability. For elastic processes (due to impurities, surface or interface roughness), the scattering probability is

Transport Model



a ,a ¢ a ,a ¢ 2 WN,N ¢ | VN,N ¢ | d ( Ea N - Ea ¢ N ¢ )( fa N - Ea ¢ N ¢ ),

(15.6)

a ,a ' where VN,N' is the scattering matrix element (|V|2 accounts for the number of defects and their correlation), faN is the nonequilibrium electron distribution function. For phonon-assisted hopping,



a ,a ¢ WN,N ¢ =

Â|C

q ^ qz

a ,a ¢ 2 N,N ¢ | d ( Ea N

- Ea ¢ N ¢



+ hw q )[( fa N - fa ¢ N¢ )Nq - fa ¢ N¢ (1 - fa N )],

(15.7)

a ,a ¢ CN,N¢ is the electron-phonon scattering amplitude, wq and Nq are the phonon frequency and the occupation number. The phonon occupation number, in a general case, is a nonequilibrium quantity and has to be determined by rate equation. Equation (15.7) accounts for spontaneous emission, emission, and absorption of phonons. As far as we know, a formula similar to Eq. (15.5) was first derived by Luttinger [15], and later on this approach was applied to conventional hopping in crystals [16] and conductance of SL [3, 17]. However, it turns out that the derivation and applicability of Eq. (15.5) for hopping between localized states in QMS are nontrivial. In this paper, apart from the application of Eq. (15.5) to the QBSL, we would like to emphasize critical features regarding this formula and hopping regime in microdevices. One important observation is that the nonequilibrium character of the system in the hopping regime results only from the boundary conditions imposed on the electron distribution function. The nonequilibrium state of phonons in our case plays a minor role and in what follows, we use the Planck functions for phonon distribution. Now, if we interchange (aN) ¤ (a′ N′) in the second term between brackets of Eq. (15.5), we obtain

jz = e



 z  (W aN

=e

a ,a ¢ N,N ¢

a ,N

Âz

a ¢ N¢

a ,N Icoll ( fa N ),

aN

)

- WNa¢¢Na ,



(15.8)

where Icoll is the collisional integral, which determines the evolution of the electron distribution function in the rate equation,



∂fa N = Icoll ( fa N ). (15.9) ∂t

201

202

Antiresonant Hopping Conductance and Negative Magnetoresistance

At first glance, we have absolutely surprising results: In steady state, all the electron processes form a closed cycle, the collisional integral vanishes, and the current vanishes also. The solution of this puzzle is the following: The nonequilibrium factor in our system is the electric field. However, in Eq. (15.9), it determines only the equilibrium parameters, namely, the energies and wave functions of the localized states. In order to calculate the current, we are to take into account the difference of the chemical potentials between the left and right contact as the result of the applied voltage. Correspondingly, the electron distribution function in the contacts is determined by the boundary conditions, i.e., by the chemical potential, rather than by Eq. (15.9). If we assume, that the distribution of electric field in SL or QBSL is uniform, the potential drop over any period is the same, and the boundary conditions may be imposed to one cell of the structure. Then, in the simplest case, there is no need to solve Eq. (15.9), since the electron distribution function, which determines the hopping current (15.8), is given by the boundary conditions for the nearest neighbor quantum wells. These electron distribution functions, in the mean energy gain approximation, are taken in equilibrium with the chemical potential varying from one cell to another and the electron concentration being kept constant. Actually this method was conventionally used for hopping conductance [3, 16, 17]. In general, distribution of the field may be nonuniform. Then a selfconsistent calculation of the electron distribution functions with boundary conditions at contacts and current given by (15.8, 15.9) is required. We notice that the self-consistent procedure may be important even for uniform electric field when heating is essential. Evaluating the current as due to hopping processes, we explicitly take into account the perfect structure of a system, include direct tunneling between wells in a spectrum and wave functions of localized states and consider impurities, fluctuations, and inelastic scattering as a perturbation. This procedure in many cases may be more adequate for the description of microstructures than the calculation of the current due to direct tunneling. It is especially important when the electric field in a device cannot be considered as a perturbation, for instance, in the regime of the negative differential conductance.

Antiresonances and Resonances in Hopping Transport

15.4 Antiresonances and Resonances in Hopping Transport We use now Eq. (15.5) for the hopping current in QBSL and demonstrate its peculiar features. The scattering matrix element in the basis of eigenfunctions (15.4) has the form

iq ^ r^ a ,a ¢ CN,N | N ¢Ò·n | e iqz z| n Ò , (15.10) ¢ = Vq Jn -a Jn -a ¢ · N | e

where q is the transferred momentum. The last multiplier in (15.11) is the matrix element, which is diagonal in Wannier index; within Wannier basis (15.1) only, the intrawell scattering [18] is taken into account. If we assume that the chain structure is invariant under the coordinate inversion transformation z ´ –z, then the matrix element of the Fourier component is given by [19]

·n | e iqz z| n Ò = e iqz dn , (15.11)

and the summation over v (Ref. [21]) in (15.11) leads to

iq ^ r^ a ,a ¢ 2 2 | CN,N | N ¢Ò |2 ¢ | =| Vq| | · N | e

q d ˆ (15.12) Ê 4D ¥ Ja2 -a ¢ Á sin z ˜ . Ë eFd 2 ¯

2p n result in vanishing electron d transition between different wells a [Ja–a′ (0) = 0 at a π a′]. Assume first that the dispersion of acoustical phonons is described by a constant speed of sound s and that phonons propagate in the z direction. Then the energy conservation law given by d function in Eq. (15.7) determines qz. At electric fields F = nF0, n is the integer number, and



We see that phonons with qz =

F0 = 2phs/ed2,

(15.13)

phonons are ineffective. The scattering probability (15.12) and the hopping current (15.5) vanish. This effect takes its origin in Bragg reflection: the transfer of phonon momentum equal to the momentum of reciprocal lattice does not change the longitudinal electron state. We have a specific momentum selection rule for the scattering of the WS localized electrons. The difference between SL with the continuous transverse spectrum and QBSL with the discrete one is very important here, and only the discrete spectrum results in

203

204

Antiresonant Hopping Conductance and Negative Magnetoresistance

a single value of qz. If there are no other acoustic phonons in the structure, this means the absence of the background current and the appearance of zero minima in the conductance, i.e., antiresonant effect. (The accuracy of this zero is determined, as usual, by the width of levels and the accuracy of d function approximation.) Let us notice, that there are also additional zero minima of the scattering probability [Eq. (15.12)] when the argument of the Bessel function Jk(x), k π 0 coincides with one of its roots xi π 0. However, in contrast to the momentum selection rule mentioned above, these minima are unlikely to be observed: if we consider the transitions between next-nearest neighbor wells [3], the total current will be determined by Bessel functions of different orders. Correspondingly, the superposition of these transitions leads to nonzero current. Let us consider the possibility for observation of the momentum selection rule. This effect is the result of the 1D propagation of phonons. It will be observable experimentally if either the density of states of 1D phonons is essential or the transverse scattering form factor given by

F (q ^ ) = | · N | e iq^ r^ | N ¢Ò |2 (15.14)



F (q ^ ) µ e - lBq^ , (15.15)



eFd/s  q^ ~ lB-1 (15.16)

has a sharp maximum at q^ = 0. For instance, such a maximum is realized in SL in magnetic field, when the transverse modes are Landau levels. At the moment, this experimental geometry is the best way to simulate a quantum-box superlattice. In strong electric field, when the WS quantization is present, the 1D array of quantum boxes with magnetic- and electric-field-controlled discrete spectra are realized. If the energy difference between the initial and final WS levels satisfies the antiresonant condition, electron hopping occurs between partially filled Landau levels with the same quantum number and the form factor is 2 2

where lB is the magnetic length. Consequently, the energy conservation law takes the form eFd  hsqz, if the inequality or

(eFd)2/ms2 ≫ hwc, (15.17)

Antiresonances and Resonances in Hopping Transport

where wc is the cyclotron frequency, is satisfied. One sees that in high electric fields, only the qz component of the phonon momentum is relevant and therefore hopping current reaches a minimum. We note that transitions to the next nearest QB states with the same number of Landau level are also in antiresonance. However, transitions between different Landau levels result in finite current. Its amplitude is much weaker than the one of the current, due to the transition to the nearest box, because phonon emission at low temperatures and eFd < hwc requires a large hopping distance. Let us notice that one can also eliminate the transverse component of the phonon momentum by using an external source of phonons with a given direction of propagation. If the signal related to external phonons is extracted from the total current, the zero minima at the certain voltages can be found. In Fig. 15.2, we present the current calculated taking into account the transitions between nearest QB for the case of magnetically confined QBSL (Fig. 15.2a) and in the case of an external source of phonons [22] (Fig. 15.2b). The two figures are remarkably similar. The range of electric fields eFd ≫ h/t turns out to be very interesting because of the negative magnetoresistance (Fig. 15.2a). The physical origin of this effect in our case is that in the electricfield-induced localization regime, the current is proportional to the scattering probability, in contrast to Drude current at small electric fields, which is proportional to the scattering time and inverse proportional to scattering probability. The latter in the case of transition between zero Landau levels is proportional to the “density of states:”

Ú dq

^e

- l2Bq2^

= 2p/lB2 ,

(15.18)

and increases with magnetic field. Correspondingly, the current in the WS localization regime also increases and we see (Fig. 15.2a) that the bigger the magnetic field the bigger the hopping current. We predict that this negative magnetoresistance will be observed until the probability 1/t becomes the order of eFd/h with increasing magnetic field. Then the magnetic field destroys the WS localization, and the magnetoresistance becomes positive.

205

206

Antiresonant Hopping Conductance and Negative Magnetoresistance

Figure 15.2 (a) Hopping current in the conditions of the partial occupation of the zeroth Landau level. F0 = 2.17 kV/cm. Curve 1: B = 6 T; Curve 2: B = 12 T. (b) Hopping current induced by an external source of phonons.

Consider now the effect of folded acoustical phonons [23] on conductance. The dispersion relation for folded phonons in the periodic potential is of the same form as the Kronig–Penney dispersion relation for electrons and spectrum can be considered as linear only at small while at the Bragg plane the wave velocity is zero and a gap arises. Obviously, if the energy conservation law Eq. (15.7) requires that hwq falls into the gap, the current vanishes, and an antiresonant plateau is to be observed. Note that if the growth direction of a structure is (001), the gaps in the phonon spectrum are only at the Brillouin zone edge qz = ±p/d and its center qz = 0. Equations (15.7, 15.11) determine the conditions for the current vanishing in the vicinity of the gaps. The current breakdown is not abrupt at qz = 0, but at qz = ±p. This is another manifestation of the

Conclusion

selection rule. Note that current peaks may arise due to the phonon “defects,” which levels fall inside the folded phonon gap. These phonon modes come from the potential fluctuations [24]. We would like to emphasize that the resonance and antiresonance features discussed in the present analysis are to be observed in hopping magnetoconductance experiments. For instance, the resonance, due to LO phonon scattering occur in the conductance, not in the resistance, and the absence of scattering will result in zero of the conductance, not in zero of the resistance. The claim on zero resistance in the absence of scattering in Ref. [25] is incorrect; the reason is that for h/t < eFd (and, naturally, in the absence of scattering) the electron states are localized, not delocalized, irrespective of an absolute value of electric field. Only for h/t > eFd, in the conditions of the band conduction, the LO phonon resonance will result in a resistance maximum. In order to observe antiresonances and resonances, it is necessary to study the SL magnetoconductance in the regime of WS localization. Evidence for the scattering-assisted resonant tunneling in SL in magnetic field was reported in Ref. [5]. However, in these experiments, optical excitation plays an essential role and results in conductivity mechanisms varying from sample to sample. Different electron and hole subbands contribute to the conductivity and it appears difficult to extract the acoustic-phonon spectrum. In another experiment [6], magnetoresistance in a quantizing magnetic field was studied in the band conduction regime, without WS localization. Finally, in Ref. [7], hopping conduction in magnetoresistance was investigated in the presence of strong disorder. The fact that no current oscillations were observed was due to the nature of localization, which was the result of disorder, but not of the electric field.

15.5 Conclusion

We have shown that the influence of quantum confinement on the electron and phonon states results in antiresonances in the hopping conductance of quantum-box superlattices. The momentum selection rule for the interaction of phonons with WS localized electrons as well as the folded phonon gaps are to be observed in

207

208

Antiresonant Hopping Conductance and Negative Magnetoresistance

the magnetoconductance. We predict negative magnetoresistance in high electric fields and point out the importance of boundary conditions imposed on the electron distribution in the description of the nonequilibrium character of the system. Note added in proof. Recently, O. Raichev and F. Vasko [Phys. Rev. B 50, 12199 (1994)] considered the effect of an interference of matrix elements of transitions in a double-quantum-well system, which leads to minima of scattering probability. These minima are similar to minima in hopping current in superlattices due to one of the effects (Bragg reflection), which are considered in the present chapter.

Acknowledgments

This work was supported by NSF under grant no. ECS 91-08300 and by the US Joint Service Electronics Program under contract no. N00014-90-J-1270.

References

1. F. Capasso, F. Beltram, D.L. Sivco, A.L. Hutchinson, S.-N. G. Chu, and A.J. Cho, in Resonant Tunneling in Semiconductors, edited by L.L. Chang et al. (Plenum Press, New York, 1991). 2. L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). 3. R. Tsu and G. Dohler, Phys. Rev. B 12, 680 (1975).

4. R.F. Kazarinov and R.A. Suris, Fiz. Tekh. Poluprovodn. 6, 148 (1972) [Sov. Phys. Semicond. 6, 120 (1972)]. 5. W. Muller, H.T. Grahn, K. von Klitzing, and K. Ploog, Surf. Sci. 305, 386 (1993). 6. N. Noguchi, T. Takamasu, N. Miura, J.P. Leburton, and H. Sakaki, in Phonons in Semiconductor Nanostructures, Vol. 236 of NATO Advanced Study Institute, Series B: Physics, edited by J.P. Leburton, J. Pasqual, and C. Sotomayor (Plenum, New York, 1992), p. 471.

7. M. Lee, N.S. Wingreen, S.A. Solin, and PA. Wolff, Solid State Commun. 89, 687 (1994). 8. H. Sakaki, Jpn. J. Appl. Phys. 28, L314 (1989).

9. H. Noguchi, J.P. Leburton, and H. Sakaki, Phys. Rev. B 47, 15593 (1993). 10. G. Wannier, Elements of Solid State Theory (Cambridge University Press, London, 1959).

References

11. H. Fukuyama, R.A. Bari, and H.C. Fogedby, Phys. Rev. B 8, 5579 (1973).

12. Yu.B. Lyanda-Geller and I.L. Aleiner, Abstracts of the VII International Conference on Superlattices, Microstructures and Microdevices, Banff, Canada 1994, edited by S. Charbonneau and D. J. Lockwood (National Research Council, Ottawa, 1994); and (unpublished). 13. J. Leo and A.H. Macdonald, Phys. Rev. Lett. 64, 817 (1990).

14. D.Y. Oberli, Jagdeep Shah, T.S. Damen, J.M. Kuo, and J.E. Henry, Appl. Phys. Lett. 56, 1239 (1990). 15. J.M. Luttinger, Phys. Rev. 112, 739 (1958).

16. A. Miller and E. Abrahams, Phys. Rev. 120, 745 (1960).

17. D. Calecki, J.F. Palmier, and A. Chomette, J. Phys. C 17, 5017 (1984). 18. B. Laikhman and D. Miller, Phys. Rev. B 48, 5395 (1993).

19. The coordinate matrix element in the Wannier representation 〈sv| [Eq. (15.1)] is related to the matrix element in the basis of the Bloch functions: 〈sv\z\s′v′〉 = Skk’eid(kv–k′v′) ·sk|z|s′k′Ò; ·sk|z|s′k′Ò =

d ss¢ i

∂ ∂ * ¢ ss ¢ d kk ¢ + Wss usk dz , usk is the k d kk ¢ , where W k = us,k ∂k ∂k

Ú

Bloch amplitude (Ref. [20]). In our case, when s = s′, the structure is symmetric with respect to z ´ –z and spin effects are not essential ·v|z|v′Ò = dvv′ vd, the coordinate matrix element is diagonal, and ·n | eiqz z | n Ò = eiqz dn . We note that the interwell scattering in Wannier basis taken into account (Ref. [3]) does not change the Bessel function factor in Eq. (15.11).

20. L.D. Landau, Statistical Physics, Part II, Theory of Condensed States, edited by E. N. Lifchitz and L. P. Pitaevskii (Pergamon Press, New York, 1980).

21. I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980). 22. We assume that the frequency of phonon source covers a wide range of spectrum or can be tuned. 23. C. Colvard, R. Merlin, M.V. Klein, and A.C. Gossard, Phys. Rev. Lett. 45, 298 (1980).

24. J. Hori, in Spectral Properties of Disordered Chains and Lattices, International Series of Monographs on Natural Philosophy, edited by T. der Haar (Pergamon, Oxford, 1968). 25. V.M. Polyanovskii, Fiz. Tekh. Poluprovodn. 17, 1801 (1983) [Sov. Phys. Semicond. 17, 1150 (1983)].

209

Chapter 16

Oscillatory Level Broadening in Superlattice Magnetotransport

Yu. B. Lyanda-Gellera,b and J.-P. Leburtona aBeckman

Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA bPhysics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected]

We demonstrate the existence of oscillations in the magnetic peak positions of Stark-cyclotron resonances (SCRs) due to acoustic phonon scattering in confined superlattices and investigate the oscillations of level broadening in the Wannier–Stark localization regime as a function of the magnetic field due to acoustic phonons. Manifestation of these effects in recent experiments is discussed. It is well known that in the absence of external fields, the periodicity of superlattices (SL) leads to the splitting of the conduction band into several minibands separated by minigaps. The presence of an external electric field F along the SL growth axis, however, breaks Reprinted from Solid State Commun., 106(1), 31–34, 1998. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1998 Elsevier Science Ltd Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

212

Oscillatory Level Broadening in Superlattice Magnetotransport

down the system periodicity and localizes electron states along that direction, thereby splitting the minibands into a Wannier–Stark (WS) ladder, for which the states in neighboring quantum wells are separated in energy by eFd, where e is the electron charge and d is the SL period [1]. WS ladders can be resolved experimentally if the level broadening of the localized states G is less than eFd/h. For G > eFd/h state overlap still exists in SLs, resulting in ohmic conductance. In the opposite case electrons are localized in space within the distance D/eFd where D is the miniband width and the miniband conductance breaks down: electrons move in the direction of the field by hopping from one localized state to another by scattering. If a magnetic field B is simultaneously applied along the SL growth axis (Fig. 16.1), the in-plane states split into Landau levels, with wavefunctions localized on the scale of the magnetic length l = (h/eB)1/2. This configuration realizes a quantum box superlattice [2, 3] with discrete spectrum. In the present work, we discuss novel oscillatory effects in such a system due to acoustic phonon scattering.

Figure 16.1 Schematic representation of the SCR and acoustic phonon emission in superlattice; n and v are the Landau and the SL well index, correspondingly. Inset: formfactor characterizing the scale of transverse phonon wavevectors q^ ~ l–1.

Whereas in bulk crystals at low temperatures acoustic-phononassisted scattering leads to the background hopping current which is a monotonic function of electric or magnetic field, in confined SLs, acoustic-phonon-assisted tunneling current turns out to be characterised by a set of maxima and minima [2, 3]. These maxima

Oscillatory Level Broadening in Superlattice Magnetotransport

and minima are related either to the Bragg reflection of electrons or to the folded spectrum of acoustic phonons in SLs. In the present chapter we will only consider the continuous spectrum of acoustic phonons in SLs:

wq = sq,

(16.1)



q dˆ Ê D C(qz ) = J12 Á sin z ˜ ,  Ë eFd 2 ¯

(16.2)

dveFd = dnhwc.

(16.3)

where s is the sound velocity and q is the phonon wavevector. In this case minima and maxima in the electron hopping conductance in SLs as a function of the electric or the magnetic field are due to electron Bragg interferences. Owing to the nature of the WS states and the periodic electronic potential, the probability of the nearest neighbor hopping is determined by the oscillating formfactor where J1 denotes the Bessel function of first order, D is the bandwidth at F = 0. As shown in [2, 3], when magnetic and electric fields satisfy the inequality (eFd)2 ≫ hwcm*s2, the transverse component of the phonon wavevector q^ which is approximately equal to the inverse magnetic length l–1, is much smaller than the total phonon wavevector as well as its longitudinal component qz (Fig. 16.1 inset). Therefore, energy conservation is to a good accuracy given by the relation eFd = hsqz and the hopping probability becomes an oscillating function of the electric field. For (eFd2)/(2hs) = pp, where p is the integer the hopping probability becomes zero, resulting in antiresonances in the hopping current. Recently antiresonant hopping in a 20-period magnetically confined GaAs/AlAs SLs was observed by Nogaret et al. [5] in the WS regime at low temperatures. Let us turn our attention to the influence of acoustic phonon scattering on the SCR, which is due to elastic tunneling between WS levels in neighbouring QWs. SCR was recently demonstrated experimentally by Canali et al. [6] and occurs with impurity-assisted hopping when the energies of the initial and the final states coincide, i.e.:

Here wc = eB/m* is the cyclotron frequency, dv and dn are the variation of Stark ladder and Landau level indices, respectively. It is important to emphasize that scattering by impurities (or SL

213

214

Oscillatory Level Broadening in Superlattice Magnetotransport

imperfections) is the only elastic tunneling mechanism in SCR because coherent tunneling between initial and final states with orthogonal wavefunctions is forbidden. Experimentally, the external voltage is fixed at a particular bias value corresponding to the regime of WS localization and, accordingly, hopping conductance. At this fixed bias the current variation with magnetic field reveals one or more current maxima which correspond to the SCR condition eFd = dnhwc. When the values of the magnetic fields achieving the current maxima are plotted experimentally as a function of external bias, the experimental points align (as we shall see, approximately) along straight lines in agreement with equation (16.3). However, deviations of the peak positions away from the straight lines also show pronounced oscillations, especially for dn = 3. In order to analyze this oscillatory effect one should keep in mind that electron transport and energy relaxation in confined SLs are characterized by all electron transitions between energy levels involving impurities, acoustic and optic phonons. Depending on the SL level spacing, different contributions may become more or less important. For instance, in steady state, impurity scattering from one quantum well to another is followed by an intrawell energy relaxation which is due to inelastic phonon scattering (Fig. 16.1). In magnetically confined SLs, impurity-assisted hopping between the WS–Landau levels (v, n) Æ (v + 1, n + dn), where dn = n – n′, is followed by intrawell phonon scattering (v + 1, n + dn) Æ (v + 1, n″). After this process the next interwell elastic impurity-assisted tunneling (v + 1, n″) Æ (v + 2, n″ + dn″) occurs and this chain of events continues until the electron reaches the contact electrode. In steady state elastic and inelastic processes form a complete set of transitions which determines the current [2, 3]. Thus, at low temperatures, SL intrawell phonon scattering is necessarily important for SL transport, even in the case when the transfer of electrons between wells is essentially due to impurity hopping. Consider for simplicity the situation when acoustic phonon transitions between neighboring quantum wells are weak. Then tunneling is mostly elastic due to SL imperfections. Peak positions of the elastic tunneling current would be along straight lines, if the level broadening G of the SL states which describes the SCR profile by the Lorentzian

Oscillatory Level Broadening in Superlattice Magnetotransport



P (B, F, G) = G/((eFd – dnhwc)2 + G2)

(16.4)

is magnetic- and electric-field-independent. However, acoustic phonon scattering leads to energy relaxation within wells and thereby to a contribution Gac to G which oscillates with magnetic field. Indeed, the leading acoustic phonon contribution to Gac is due to transitions between nearest Landau levels within the same SL well. These transitions cause also the intrawell energy relaxation of electrons, which is a necessary dissipative process for determining the electric current. The probability of spontaneous emission of phonons, which is the main process at low temperatures, reads (see, for instance, [3]) n Wn,n -1 =



2p h q

 | V | | ·n | e ^ qz

¥

q

2

iqa ra

| n - 1Ò |2

q dˆ D sin z ˜ d ( hw c - hw q ), Ë eFd 2 ¯

Ê J02 Á



(16.5)

where r is the electron coordinate, J0 is the Bessel function of zero order, Vq is the Fourier-component of the acoustic phonon scattering potential, |Vq|2 = hwqD2/2prs2, | ·n | e iq^ r^ | n - 1Ò |2 is the transverse scattering formfactor. In strong magnetic field for which hwc ≫ m*s2, we may again neglect q^ compared to qz and therefore assume qz ~ q. Hence, energy conservation reads hwc = hqzs or qz = wc /s and by using this value of qz in the argument of J0 in equation (16.5) we see that the scattering probability oscillates with B, leading to oscillations in G with the period given by DB = 2pcsm*/ed. For GaAs/ AlGaAs SLs with d = 84 Å the magnetic field period is DB = 1.5 T. SCR magnetic field peak position at a fixed bias is determined by the solution of the equation dP(B, F, G(B))/dB = 0. We have determined these magnetic field peak positions by using the microscopically calculated proportionality coefficient between n 2 G ac ~ hWnn -1 and J0 equal to 2.0 meV at B = 5 T (as calculated from [2, 3]) and a total G = Gac + Ginh = 10 meV (taken from [6]), where Ginh is the magnetic-field-independent inhomogeneous level broadening. This value agrees with the observation [6, 7] that G is caused mainly by inhomogeneous broadening due to monolayer fluctuations. We found (Fig. 16.2) periodic deviations of peak positions for dn = 3 in rather good agreement with the experimental data.

215

216

Oscillatory Level Broadening in Superlattice Magnetotransport

Figure 16.2 Magnetic field peak positions with bias for dn = 3. Solid linetheory; dots-experiment [6].

We notice that not only acoustic phonon transitions within one SL well are important, but that there is also a competition between elastic and inelastic tunneling between wells. In particular, when two energy levels in neighboring SL wells are close, acoustic phononassisted transitions become important. On the other hand when these levels coincide, transitions between these levels require zero phonon energy and wavevectors and acoustic phonon transitions vanish, replaced by elastic impurity assisted processes. Hence, energy conservation for the interwell transition (v, n) Æ (v + 1, n + 1) reads eFd – lhwc = hqs and the formfactor given by equation (16.2) also oscillates with magnetic field. The period of peak oscillations due to interwell phonon transitions (n Æ n + l) is given by

DB = 2psm*/ed(dn – l).

(16.6)

Oscillations determined by dn = 3 and l = 2 which are relevant to experiment [6] are characterized by the same period as the contribution to peak deviations resulting from intrawell energy relaxation discussed above, in good agreement with experimental data. Therefore, interband tunneling contribute to magnetic field peak deviations as well. Note, that we do not extend our model to the experimental data at B > 8 T, because 3hwc exceeds 36 meV and corresponds to the

References

onset of optical phonon transitions which may contribute both to the resonant current and the level broadening. Hence resonances in SLs may occur due to optic-phonon-assisted tunneling, when the initial and final states differ by the optic phonon energy hW:

1ˆ 1ˆ Ê Ê ÁË n + 2 ˜¯ hw c + n eFd ± hW = ÁË n¢ + 2 ˜¯ hw c + n ¢eFd ,

(16.7)

where the small dispersion of optic phonons is neglected. These opticphonon-assisted resonances were recently observed in triple barrier resonant tunneling structures [4]. Due to the high inhomogeneous broadening of the electron levels contribution of optic phonons possibly affect the peak positions at energies lower than 36 meV (Fig. 16.2). Analysis of different contributions into deviations of SCR peaks requires further experimental and theoretical study. In conclusion, in this chapter we have discussed a novel effect due to acoustic phonon scattering which leads to oscillatory deviation of SCR peaks in confined superlattices. These oscillations manifests in recent experiments [6] and have the same origin—Bragg reflection of electrons—as the anti-resonant effect recently directly observed in the current–voltage characteristics of magnetically confined superlattices [5]. We also point out the possible contribution of optic phonons to the observed current in [6] and the fact that the oscillations in SCR curves for different dn may provide a powerful scattering spectroscopy tool for phonon dispersion in SLs.

Acknowledgments

We acknowledge discussions with F. Capasso, L. Eaves, A. Nogaret and J.C. Portal. This work was supported by ARO under grant no. DAAH04-95-0190.

References

1. Wannier, G., Elements of Solid State Theory, Cambridge University Press, London, 1959. 2. Lyanda-Geller, Yu.B. and Leburton, J.P., Phys. Rev., B52, 1995, 2779.

3. Lyanda-Geller, Yu.B. and Leburton, J.P., Semiconductor Science and Technology, 10, 1995, 1463.

217

218

Oscillatory Level Broadening in Superlattice Magnetotransport

4. Wirner, C, Awano, Y., Futatsugi, T., Yokoyama, N., Nakagawa, T., Bando, H. and Muto, S., Appl. Phys. Lett., 69, 1996, 000.

5. Nogaret, A., Eaves, L., Main, P., Henini, M, Maude, D., Portal, J.C., Molinari, E., Beaumont, S. to be published.

6. Canali, L., Lazzarino, M., Sorba, L. and Beltram, F., Phys. Rev. Lett., 76, 1996, 3618.

7. Capasso, F., Mohammed, K. and Cho, A., J. Quant. Electron., 22, 1986, 1853.

Chapter 17

Breakdown of the Linear Approximation to the Boltzmann Transport Equation in Quasi-One-Dimensional Semiconductors

S. Briggs and J.-P. Leburton

Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We investigate the validity of the linear approximation to the Boltzmann transport equation (BTE) in a one-dimensional quantum-wire structure. We model a single-subband wire at 300 K with polar optic-phonon scattering. The results of the linearized BTE calculation are analyzed for consistency and compared to the results of a Monte Carlo simulation. The linear approximation is found to be inconsistent for fields as low as 50 V/cm for two different confinement conditions. The Monte Carlo results predict a large deviation from a Maxwellian distribution for even lower fields. Reprinted from Phys. Rev. B, 39(11), 8025–8028, 1989.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1989 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

220

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

We attribute this breakdown to the low scattering rate below the phonon emission threshold and the lack of angular randomization in one-dimensional systems. The investigation of quasi-one-dimensional (1D) artificial structures is rapidly emerging as a new field of semiconductor physics [1–7]. While presently achievable confinement allows for observation of quantization effects only at low temperature [6, 7] technological obstacles for higher confinement are expected to be rapidly overcome and allow for the observation of 1D effects with liquid nitrogen and at room temperature. In these conditions, thermal effects will become determinant and 1D transport is described in terms of semiclassical Boltzmann formalism. Because of the particular confinement configuration of 1D artificial systems, the absence of angular randomization during scattering with defects and lattice vibrations makes the carrier distribution extremely sensitive to external perturbations and causes appreciable deviations from the equilibrium relatively rapidly. In this rapid communication we investigate the validity of the linear approximation for the Boltzmann transport equation (BTE) and compare our results with a Monte Carlo simulation. We consider a quantum-wire structure with confinement conditions similar to the V-groove quantum-wire field-effect transistor suggested by Sakaki [1]. The exact configuration is not essential here provided that electronic properties are quantized along the two transverse directions. In the y direction the electrons are confined in a GaAs-AlxGa1-xAs quantum well (QW) of width Ly. The z confinement is achieved with a gate electrode perpendicular to the QW which forms a triangular electrostatic potential with an electric field Fz. We only mention the salient features of the model, which has been discussed elsewhere [8]. In the present investigation we consider a single subband at 300 K with polar optic-phonon (POP) scattering only. We compute 1D matrix elements by integrating the normal 3D matrix elements over the two transverse phonon wave vectors; these are used to obtain scattering rates according to Fermi’s “golden rule.” In the 1D homogeneous system, the BTE is expressed as [9] 1/ 2

È 2E ˘ eF Í * ˙ Îm ˚

∂f = IC , ∂E

(17.1)

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

with F the longitudinal field, E the energy, and f(E) the distribution function. For positive electron momentum k, the collision integral, IC, is given by

È1 1˘ IC+ = - Í + ˙ f + (E ) + dE ¢ ÈWe+ E - E ¢ + Wa+ E - E ¢ ˘ f + E ¢ Î ˚ Ît a t e ˚  (17.2) + dE ¢ ÈWe- E - E ¢ + Wa- E - E ¢ ˘ f - E ¢ , (k > 0), Î ˚

(

Ú

Ú

(

)

(

) ( ) ( )

) ( )

with a similar equation for IC- , i.e., for k < 0. Wa±,e are the transition probabilities for forward and backward phonon absorptions and emission, respectively, and f ±(E) is the distribution function evaluated for k > 0 and k < 0. Considering only POP scattering, IC+ becomes

È Nq N q N q + 1 Nq + 1 ˘ + Nq + 1 + IC+ = - Í + + - + + f (E + hw w) ˙ f (E ) + + te t e ˙˚ t a+ ÍÎ t a t a Nq + 1 Nq Nq  f (E + hw ) + + f + (E - hw ) + - f - (E - hw ), (17.3) + ta te te ± where t a,e are the inverses of the transition probabilities for forward and backward absorption or emission, respectively, and hw is the phonon energy. In order to simplify the equation, we write



with

f ± (E ) = f0 (E )[1 ±

E0 =

EE0

kT

g ± (E )] ,

(17.4a)

t 2 [eF a ]2 , m Nq

(17.4b)

g ± (E )  1 .

(17.5)

where the ± signs refer to k > 0 and k < 0, f0 is the Maxwellian + distribution, and 1 / t a = 1 / t a + 1 / t a . In the linear approximation we assume f is only a small perturbation from f0, i.e.,

EE0

kT

Then, substituting Eq. (17.4a) into Eq. (17.3) and using detailed balance yields

221

222

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

1 = g + (E ) + a 

- ag [

ta

t e+

Èt ˘ ta + t g (E ) - b Í a+ g + (E + hw ) - a- g - (E + hw )˙ te ta ÍÎ t a ˙˚

g + ( E - hw ) -

ta

t e-

g - (E - hw )],

(17.6a)

a = e hw / kT ,

(17.6b)



g = (1 - hw / E )1/2 ,

(17.6d)



1 = Nn gn (E ) + An gn+1 (E ) + En gn-1 (E ) ,

(17.7)



1 = Mg(E ),

(17.8)



b = (1 + hw )1/2 ,

(17.6c)

where, in agreement with the linear approximation, we have neglected the ∂g / ∂E term on the left-hand side of Eq. (17.1). A similar expression can be obtained with g–exchanged for g+. To solve Eq. (17.6), we partition the energy range into regions of width hw. g(E) is then divided up into separate functions, gn(E) for each energy interval n. Then, Eq. (17.6) can be written in matrix form as

where the gn vectors are of length 2, corresponding to g+ and g–, and the 2 ¥ 2 matrices Nn , An, and En are all functions of a, b, g, and ta,e. If we then use the fact that g0(E) = 0 [corresponding to f(E) =0 for E < 0] and also assume that for some particular J energy interval, gm(E) = 0 for m ≥ J (i.e., the perturbation is negligible above a certain energy), we obtain the following matrix equation for g: where 1 is a 2J-column vector of 1’s, M is a 2J ¥ 2J matrix, and g is a 2J-column vector. This equation is inverted and solved for g(E) and then for f(E). Since the perturbation is linear in the field we only show the right-hand side of Eq. (17.5) for one field, F = 50 V/cm, in two different confinement conditions (Fig. 17.1). The low confinement condition corresponds to Ly = 215 Å and Fz = 20 kV/cm in the V-groove structure. In this case, the assumption of one subband is questionable since some of the upper subbands are populated [10]. However, we do not see any reason to assume that the introduction of a few (two or three) upper bands will make the system more linear (of course, in extremely low confinement, many subbands

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

are present and intersubband scattering will randomize the carrier momentum sufficiently and restore linearity to the system). The high confinement condition is achieved with Ly =135 Å and Fz = 120 kV/cm and has essentially only one band occupied; hence the single subband approximation is justified. In this case the perturbation is smaller than for low confinement because of the strong scattering due to size effects. With only a few subbands in the range of one POP, carrier transport is limited by the strength of the POP interaction which is an increasing function of the overlap integral between electronic states in the matrix elements [8]. The results show that for fields as low as 50 V/cm the linear approximation is no longer valid since Eq. (17.5) is no longer fulfilled, especially at low confinement. We attribute this breakdown of the BTE linear approximation to the low scattering rate below the phonon emission threshold and the lack of transverse momentum exchange in 1D scattering. Because of this, electron scattering reduces to essentially forward scattering [8] and electrons behave quasi ballistically even at low fields.

Figure 17.1 Perturbation from the Maxwellian distribution as calculated in the linear approximation at F = 50 V/cm and T = 300 K for two confinement conditions. Notice for the linear approximation to be valid, ( EE0 / kT )G(E )  1.

We have simulated the same system using a Monte Carlo [11] technique which has been discussed at length previously [8]; however, unlike our previous simulations, one million scatterings

223

224

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

were considered to insure convergence at the low fields considered here. Also, for the sake of comparison with the BTE solution, the Monte Carlo simulation has been restricted to one subband. Figure 17.2 shows f obtained by the Monte Carlo simulation in low confinement conditions for three different longitudinal fields: 50, 100, and 200 V/cm. f +(E) and f –(E) are the distributions for k > 0 and k < 0, respectively. For comparison, we also show the BTE results at 50 V/cm. The Monte Carlo and BTE results agree only qualitatively, with much more structure in the Monte Carlo data. Figure 17.3 presents the Monte Carlo simulation for high confinement conditions at 50, 100, and 200 V/cm as well as the BTE results for 50 V/cm.

Figure 17.2 Distribution functions for both k > 0 and k < 0 from BTE and Monte Carlo simulation for the low confinement case. For the sake of clarity, the functions have been multiplied by arbitrary constants to separate the plots. From the top down the plots are (1) Monte Carlo, F = 200 V/cm; (2) Monte Carlo, F = 100 V/cm; (3) Monte Carlo, F = 50 V/cm; and (4) BTE, F = 50 V/cm. We omit the Monte Carlo behavior at E = hw due to numerical error caused by the large scattering rate.

As can be seen from the figures, the Monte Carlo simulation shows non-Maxwellian behavior for both confinement conditions and all three fields. We attribute this primarily to two features characteristic of 1D structures. First, because of absence of angular randomization in 1D systems, parallel momentum can only be exchanged in discrete amounts. Consequently, an absorption followed by an emission generally produces no net momentum exchange since the final

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

state after the two scatterings is the same as the initial state. For example, an electron with zero energy that absorbs a POP is very likely to immediately emit another POP because of the large rate at the emission threshold and return to the original k state. Second, because of the large variation of the scattering rate around the POP emission threshold (a factor of 50 difference) due to the 1 / E - hw dependence of the emission rate, electron transport shows different behavior at low and high energies.

Figure 17.3 Distribution functions for both k > 0 and k < 0 from BTE and Monte Carlo simulation for the high confinement case. For the sake of clarity, the functions have been multiplied by arbitrary constants to separate the plots. From the top down the plots are (1) Monte Carlo, F = 200 V/cm; (2) Monte Carlo, F = 100 V/cm; (3) Monte Carlo, F = 50 V/cm; and (4) BTE, F = 50 V/cm. We omit the Monte Carlo behavior at E = hw due to numerical error caused by the large scattering rate.

Below the emission threshold (E < hw), electrons behave quasiballistically, especially at high fields (200 V/cm) where f+ becomes flat. This can be interpreted very simply by noting the low scattering rate in the prethreshold region. Only weak POP absorptions scatters electrons to high energy followed by a high probability for POP emission to the initial state. The large number of carriers resulting from POP emission to the bottom of the band accelerate through this region. The large peaked scattering rate at and above the phonon emission threshold (E > hw) dramatically reduces the number of electrons at high energy and prevents ballistic

225

226

Breakdown of the Linear Approximation to the Boltzmann Transport Equation

acceleration in that region; high-energy electrons (well above hw) result essentially from phonon absorption over the emission threshold. The sharp decrease in f – just below the emission threshold can be understood in similar terms. Because electrons with k < 0 are deaccelerating in the field, the large number of electrons that scatter to E = 0 cannot contribute to the electron population just below the emission threshold. Electrons enter this region either by deaccelerating from higher energy or by scattering. In the former case the large peak in scattering rates at E = hw prevents electrons above the emission threshold from deaccelerating into the region below the threshold. In the latter case, scattering into the prethreshold region occurs only by phonon emission from a region near E = 2 hw. However, high-energy carriers with negative k are negligible and scattering from electrons with positive k is improbable because the scattering is strongly forward peaked. The large peak in f – above threshold is due to POP absorption from the high electron population at the origin. The Monte Carlo simulation also shows a strong coupling between energy points separated by hw. This is evident in the quasiperiodic nature of the distribution functions as shown in Figs. 17.2 and 17.3. This is especially prevalent at low fields (F = 50 V/cm) and low energy where the electron energy does not change significantly during each free flight between scattering events. In particular, we attribute the decrease in f + below the phonon emission threshold at low field to the coupling by two-phonon processes (forward absorption followed by backwards emission or backward absorption followed by forward emission) with f –, which is strongly depleted in the subthreshold region. This quasiperiodicity is due to the strong influence of POP scattering which weakens at high field (200 V/cm) and would disappear if other strong scattering mechanisms were present, typically electron–electron. However, in 10 systems single-subband electron–electron interaction seems to have no influence on transport processes. Energy and momentum conservation requires electrons to exchange energy and momentum without altering the direction of motion. Since electrons are indistinguishable, the electron–electron interaction seems totally ineffective as a randomizing process. Therefore, the lack of randomization by electron–electron and the weakness of the acoustic phonon interaction, interface-roughness scattering

References

for good heterointerfaces [12], and ionized impurity scattering in modulation-doped structures suggests substantial deviations from a Maxwellian distribution for very low applied field in 1D systems. In conclusion, we have shown the linear approximation to the BTE is invalid for fields as low as 50 V/cm at room temperature. Monte Carlo simulation shows the distribution functions to be highly nonMaxwellian at low fields, which is in agreement with the breakdown of the linear approximation. We attribute this effect to the abrupt change in the scattering rate at the POP emission threshold and the nonrandomizing nature of POP scattering in 1D systems. The authors are indebted to Karl Hess for helpful discussions. This work was supported by National Science Foundation grant no. NSF-CDR-85-22666 and the Joint Services Electronics Program. Part of the computation was performed using the resources of the National Center for Supercomputing Applications (NCSA).

References

1. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

2. A. Ghosal, D. Chattopadhyay, and A. Bhattacharyya, J. Appl. Phys. 59, 2511 (1986).

3. P. Petrol, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982).

4. T. Hiramoto, K. Hirakawa, Y. Iye, and T. Ikoma, Appl. Phys. Lett. 51, 1620 (1987).

5. A. Warren, D. Antoniadis, and H. Smith, Phys. Rev. Lett. 56, 1858 (1986). 6. T. P. Smith III, H. Arnot, J. M. Hong, C. M. Knoedler, S. E. Laux, and H. Schmid, Phys. Rev. Lett. 59, 2802 (1987). 7. J. Alsmeier, Ch. Sikorski, and U. Merkt, Phys. Rev. B 37, 4314 (1988). 8. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988).

9. F. Blatt, in Solid State Physics, Advances in Research and Application, edited by F. Seitz and D. Turnbull (Academic, New York, 1957), Vol. 4, pp. 214–217. 10. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989). 11. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).

12. H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, Appl. Phys. Lett. 51, 1934 (1987).

227

Chapter 18

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling in Quantum-Wire Structures

J.-P. Leburton

Beckman Institute for Advanced Science and Technology, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We present an analytical derivation of the distribution function of a one-dimensional (1D) electron gas at high temperature. The quenching of carrier-carrier scattering and the predominance of polar-optic-phonon scattering make the system nonergodic and provide a “jagged” profile to the distribution which results in carrier cooling below the thermal energy. In GaAs quantum wires, this anomalous effect, which is characterized by extremely high carrier mobility, is predicted to occur above TL = 150 K. These conclusions are in excellent agreement with results of Monte Carlo simulations. Reprinted from Phys. Rev. B, 45(19), 11022–11030, 1992. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1992 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

230

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

18.1 Introduction In the past few years, the physics of reduced dimensionality systems has experienced a rapid development, mainly stimulated by the considerable advance in fine-line lithography and crystal-growth techniques [1–6]. Quasi-1D systems, or quantum wires, have recently been investigated for the transport properties associated with universal conductance fluctuations and quantum interference phenomena [7]. At high temperature, however, strong phonon scattering prohibits long-range coherence arising from the wave nature of the electrons, and carrier motion can be described in terms of particle dynamics. In this context the transport properties of 1D systems are essentially different from 2D and 3D systems [8]. Among other factors, the reduction of momentum space to a single dimension (even with multiple transverse modes), has profound consequences on carrier statistics because it limits the number of available final scattering states [9]. Significant mobility enhancement has been predicted for both low- and hightemperature 1D transport [10, 11]. Detailed Monte Carlo simulation of hot 1D carrier transport at room temperature shows significant deviations from the Maxwellian distribution at intermediate electric fields (F ~ 100 V/cm) [12]. Pronounced structures at multiples of the longitudinal-optic (LO) polar-optic-phonon (POP) energy E = nhw are predicted in the distribution function (DF) which exhibits a “jagged” profile once POP absorption is appreciable, i.e., above T = 100 K [13]. It has been shown in previous works that this singular behavior is enhanced in the case of resonances between the POP energy and the 1D subband separation [14], and induced population inversion between adjacent off-resonance subbands [15]. Although these effects have only been established by numerical simulations, the existence of non-Maxwellian jagged DF compatible with the Boltzmann equation has not yet been demonstrated. In this paper, we derive an analytical solution of the Boltzmann equation which is consistent with Monte Carlo simulation. More than providing a confirmation of earlier numerical results, our analysis reexamines the popular concept of Maxwell distribution [16] and the ergodicity of 1D systems in the presence of POP scattering. In particular, we show that the 1D carrier mobility can exceed its bulk value by almost an order of magnitude at room temperature, and the electron system undergoes a cooling below the thermal energy.

Electronic Properties and Scattering Rates

18.2 Electronic Properties and Scattering Rates In this analysis we assume that simple confinement configurations in quantum wires arise from elementary GaAs-AlxGa1–xAs potential wells which are decoupled along the two transverse y and z directions. Electrons are free to move along the x direction with energies and wave functions given by

and



E ij(kx ) =

h2kx2 + Ei + E j 2m

y ij(kx , r ) =

1 Lx

with i , j = 1,..., N

e ik x x xi ( y )f j( z ),

(18.1)

(18.2)

where kx and Lx are the wave vector and the wire length in the longitudinal direction and xi(y) and fj(z) are the transverse wave functions which correspond to the quantized energy levels Ei and Ej, respectively. N is the maximum number of levels considered in a particular confinement situation. The remaining term in Eq. (18.1) is the kinetic energy of the particle resulting from the free-electron component of the overall electronic wave function [Eq. (18.2)]. The advantage of dealing with elementary configuration is the decoupling of the electronic y and z wave functions, which facilitates the computation of the scattering rates. In modulation-doped GaAs structures, ionized-impurity (II) scattering is insignificant at high temperature, and acoustic-phonon 1 (AP) scattering (t ac ª 1010-11 s-1 ) becomes inefficient compared to 1 POP scattering (t POP ≥ 1012 s-1 ) once electric fields F significantly exceeds 10 V/cm [17]. In addition, carrier-carrier scattering vanishes for intrasubband binary processes. From the 1D conservation laws, one obtains

p12 p22 p1¢2 p2¢2 + = + , 2m 2m 2m 2m p1 + p2 = p1¢ + p2¢ ,

(18.3a)

(18.3b)

where p1, p2 and p1¢ , p2¢ are the 1D electron momenta, respectively, before and after interaction. It can be seen that carriers simply exchange energies and momenta during collisions. For indistinguishable particles, this process is irrelevant [12].

231

232

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

Consequently, intrasubband thermalization is suppressed. Therefore, electron-POP interaction is the only mechanism for carrier-energy dissipation and momentum randomization. The general form of the electron-phonon Hamiltonian reads [18]

He-ph =

 C (a e q

q

q

i qr

aq

- aq e - iqr ), (18.4)

where and aq are the phonon creation and annihilation operators and Cq determines the electron-phonon coupling strength given by [19]

È 2p 2 È 1 1 ˘ ˘ C q = -i Í e hw Í - ˙ ˙ ÍÎ Vol Î • 0 ˚ ˙˚

1/ 2

1 . (18.5) q

Here Vol is the wire volume and ϵ• and ϵ0 are the optical and static dielectric constants, respectively, w (@36 meV) is the POP frequency. The 1/q dependence of Cq implies that collisions involving small exchanges of momentum are the most favorable for the POP interaction. Although recent work has demonstrated the importance of treating confined phonon modes in highly quantized systems [20], only bulk modes are treated here. This is a good approximation as long as well widths are not narrower than 50 Å. In the semiclassical limit, the expression, for the 1D transition probability Svv′(kx, kx¢ ) derived from Fermi’s golden rule, reads

with

S vv ¢ (kx , kx¢ ) =

2p h

Â| M q^

¢ 2 vv ¢ ( k x , k x ; q )|

¥ d [E v¢ (kx¢ ) - E v (kx ) ± hw ],

Ê Mvv ¢ (kx , kx¢ q ) = Á Nq Ë

1 1ˆ + ± ˜ 2 2¯

Ú

(18.6)

1/ 2

C qd k

¥ dyxi*¢ ( y )xi ( y )e

Ú



¢ x k x ±qx

± iq y y

¥ dzfi¢ * ( z )f j( z )e ± iqz z .

(18.7)

The indices i, i ¢, j, and j ¢ refer to the quantum numbers of the transverse wave functions which together determine the subband indices v and v ¢. M(kx, kx¢ ; q) represents the electron-phonon matrix elements for a transition from an initial kx,v state to the final kx¢ , v ¢

Electronic Properties and Scattering Rates

state mediated by a phonon with wave vector q. This matrix element is summed up over all transverse components q^ to provide the 1D transition probability. The ± sign in the prefactor of Eq. (18.7), and the energy-conserving d function [Eq. (18.6)], account for emission (+) and absorption (–) of phonons between subbands with indices v′(i′,j′) and v(i,j). We can transform the energy d function into wavevector-conserving d functions and obtain d [E v¢ (kx¢ ) - E v (kx ) ± w ]



with

=

m



2

∓ K vv ¢

∓ ∓ ¢ [d (kx¢ - K vv ¢ ) + d ( k x + K vv ¢ )],

(18.8)

1/ 2

È 2m 2˘ ± K vv ¢ = Í 2 ( E v - E v ¢ ± hw ) + k x ˙ Îh ˚

and Ev = Ei + Ej,



E v (kx ) = E v +

(18.9) (18.10a)

h2kx2 , (18.10b) 2m

and similarly for Ev′( kx¢ ). If we call backward (forward) scattering events which do (not) reverse the carrier momentum, the first term on the right-hand side (RHS) of Eq. (18.8) corresponds to forward (f) [backward (b)] scattering if kx is positive (negative). The second term corresponds to backward (b) [forward (f)] for the same condition (Fig. 18.1). The transition probability becomes

with

  ¢ ¢ S vv ¢ (k - kx¢ ) = I vv ¢ (| k x - k x|)[d ( k x - K vv ¢ )  + d (kx¢ + K vv ¢ )]

∓ ¢ Ivv ¢ (| k x - k x|) =

2p 

Â| M q^

2 ¢ vv ¢ ( k , k x ; q )|

(18.11)

m ∓  K vv ¢ 2



(18.12)

and depends on the absolute value of qx = kx – kx¢ since Cq is proportional to 1/|q|. In particular, if we consider only the lowest subband and neglect intersubband transitions v = v′ = (1,1),

233

234

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

S11 (kx - kx¢ ) = S(a,e )(| kx - kx¢ |) with

= I(a,e )(| kx - kx¢ |) ¥ [d (kx¢ - K  ) + d (kx¢ + K  )],

2mw ˘ È K ∓ = Íkx2 ∓  ˙˚ Î



(18.13)

1/2

. (18.14)

Figure 18.1 Schematics representing the four fundamental POP scattering processes in 1D systems POP emission and POP absorption with both forward and backward processes from an initial kx state to a final kx¢ state.

Here, the subscripts a and e stand for phonon absorption and emission, respectively. The total scattering rate from an initial state with energy E(kx) is given by

lv [E(kx )] =

ÂS

k ¢x , v ¢

¢ vv ¢ ( k x , k x ).

(18.15)

The summation over the final states kx¢ ,v′ takes into account all possible forward (f) and backward (b) intersubband transitions along with those occurring within the same subband v.

Boltzmann Equation

18.3 Boltzmann Equation In the following, we will assume that we deal with a nondegenerate electron gas interacting with POP’s only. In order to simplify the analysis, we consider the extreme quantum limit, i.e., where only the lowest subband is occupied, and neglect intersubband transitions which would considerably complicate our derivation without introducing new physical effects. Such a situation arises in extremely confined 1D systems as in field-effect transistor quantum wires realized with a V-groove structure, with moderate longitudinal electric fields (F ≲ 500 V/cm) to prevent electron excitation to the upper subbands. In steady-state and spatially homogeneous systems, the Boltzmann equation takes the following form: eF d ± f (kx ) = - f ± (kx )  dk x

Â[S (k

+



Â[S (k f a

¢ x

f a

x

- kx¢ ) + S ab (kx - kx¢ ) + S ef (kx - kx¢ ) + S eb (kx - kx¢ )]

k ¢x

- kx ) f ± (kx¢ ) + S ab (kx¢ - kx ) f ∓ (kx¢ ) + S ef (kx¢ - k ) f ± (kx¢ ) + S eb (kx¢ - k ) f ∓ (kx¢ )],

k ¢x

(18.16)

where f ±(kx) is the distribution function for momentum hkx = ± 2mE and the superscripts identify the f- and b-scattering processes. The first sum on the RHS of Eq. (18.16) corresponds to outscattering events involving all POP a, e, f, and b processes. The second sum corresponds to all POP inscattering events. Because of the energy conservation (d function in the S terms) the summation over kx¢ is reduced to only one term, i.e., L eF d ± f (kx ) = - x [Iaf (kx ∓ K + ) + Iab (kx ± K + ) + Ief (kx ∓ K - )  dkx 2p

Lx f [Ia (K ∓ kx ) f ± (K - ) + Iab (K - ± kx ) f ∓ (K - ) 2p + I f (K + ∓ k ) f ± (K + ) + I b (K + ± k ) f ∓ (K + )], (18.17) x e x e + Ieb (kx ± K - )] f ± (kx ) +

where Lx is the wire length. From Eqs. (18.13) and (18.14) [since I(|kx – k′x|) only depends on the absolute value of the difference between the two wave vectors] we notice that

Lx Iaf (kx  K + ) Lx Ief (K +  kx ) 1 = = f , 2p 2p 1 + Nq Nq t+

(18.18a)

235

236

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

Lx Iaf (kx ± K + ) 2p Ieb (K + ± kx ) 1 = = b , (18.18b) 2p 1 + Nq Nq Lx t+



where the ± sign is for kx positive or negative, respectively. We obtain similar expressions with K– that we call t -f and 1 / t -b , respectively. By changing variables and setting E = h2kx2 / 2m, Eq. (18.17) reads

È 1 + Nq Nq ˘ ± d ± f (E ) = - Í + ˙ f (E ) + (1 + Nq ) t+ ˚ dE Î t È f ± (E + w ) f ∓ (E + w ) ˘ È f ± (E - w ) f ∓ (E - w ) ˘ + N + + Í ˙ Í ˙, q t +f t +b t -f t -b ÍÎ ˙˚ ÍÎ ˙˚ (18.19) È 2E ˘ ±eF Í ˙ Îm˚

1/ 2

where the bare POP rate (i.e., without the phonon occupation numbers 1 + Nq and Nq) can be written as

1

t ±f,b

=

1

1

1/ 2 t 0f,b± (E ) È E ˘ 1 ± Í hw ˙ Î ˚

.

(18.20)

The square root accounts for the profile of the 1D density of states and 1 / t -f,b0 is a form factor containing the 3D phonon matrix element Vol 1 È m ˘ = Í ˙ f,b t 0± (E ) (2p )2 h2 Î 2hw ˚ 1



1/ 2

Ú

¥ dq ^ | M(| kx ± K ± |;q ^ )|2 ,



(18.21)

where Vol is the wire volume. From the total scattering rate, we have also the relation

1 1 1 = f + b , t± t± t±

(18.22)



1 1 1 = + , t 0± t 0f ± t 0b±

(18.23)

with

which yield the useful relation

1

t 0f,b+ (E )

=

1 t 0f,b- (E

+ hw )

.

(18.24)

Solution of the Boltzmann Equation

In Fig. 18.2, we show the total POP scattering rate for absorption and emission with f and b contributions for the ground subband of a quantum wire formed at the heterojunction of a V-groove structure [10] with a quantum well of 135 Å well width and a triangular confining potential with F = 120 kV/cm [11].

Figure 18.2 Total POP scattering rate for the confinement conditions described in the text. The influence of upper subbands are omitted solid lines. Total emission (1/te) and absorption (1/ta) rates, respectively; dot lines denote forward (f) processes; dashed lines denote backward (b) processes. Inset shows normalized matrix element 1/t0 for electron-POP interaction summed over the transverse POP wave vectors. Upper curve represents forward scattering; lower curve represents backward scattering; dots denote expression (1 / 2)[1 - E / (E + hw )] used in Eq. (18.37).

18.4 Solution of the Boltzmann Equation Because of the discrete nature of optic phonons, POP scattering redistributes carrier energy between the intervals nhw £ E £ (n + 1) hw with n = 0, 1, 2, ..., etc. without causing intrainterval randomization. Moreover, as we emphasized in Section 18.2, intercarrier scattering is absent in the single-subband process, so carriers cannot thermalize. Thus, the system is not ergodic at equilibrium, since particles are unable to cover all points in phase space [21]. In these conditions the DF function has no a priori specific profile reminiscent of the Maxwellian distribution, as would be expected from the linear-

237

238

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

response theory, or the electron-temperature model [16]. In the first-energy interval 0 £ E £ hw, POP emission and inscattering absorption are prohibited, so Eq. (18.19) reads [12]

Nq d ∓ f ± (E ) ± (1 + N ) f (E ) = ∓ W+ (E ) dE

È f ± (E + w ) f ∓ (E + w ) ˘ ¥Í + ˙ f W+b (E ) ˙˚ ÍÎ W+ (E )



(18.25)

for 0 £ E £ hw, and

Nq ˘ ± È 1 + Nq d ∓ + f (E ) = ∓ Í ˙ f (E ) ± (1 + Nq ) dE Î W- (E ) W+ (E ) ˚



È f ± (E + w ) f ∓ (E + w ) ˘ + ¥Í ˙ f W+b (E ) ˙˚ ÍÎ W+ (E ) È f ± (E - w ) f ∓ (E - w ) ˘ ± Nq Í + ˙ f W-b (E ) ˙˚ ÍÎ W- (E )

for nhw £ E £ (n + 1)hw and n ≥ 1, with

È 2E ˘ W±f,b (E ) = eF Í ˙ Îm˚

1/ 2



(18.26)

1/2

È E ˘ t 0f,b± (E ) Í ± 1˙ h w Î ˚

,

(18.27)

which obeys the relation W+f,b (E - hw ) = W-f,b (E ) [see Eq. (18.24)]. We can translate Eq. (18.26) into the energy interval 0 £ E £ hw, by the transformation E Æ E + nhw, and obtain Nq È 1 + Nq ˘ ± d ± f (E + nw ) = ∓ Í + ˙ f (E + nw ) dE Î W- (E + nw ) W+ (E + nw ) ˚





È f ± [E + (n + 1)w ] f ∓ [E + (n + 1)w ] ˘ + ± (1 + Nq ) Í ˙ f W+b (E + nw ) ˚˙ ÍÎ W+ (E + nw ) È f ± [E + (n - 1)w ] f ∓ [E + (n - 1)w ] ˘ ± Nq Í + ˙ f W-b (E + nw ) ˙˚ ÍÎ W- (E + nw ) (18.28)

for n = 0, 1, 2, ..., etc. The first term within the first set of large parentheses and the two terms within the last set of large parentheses on the RHS vanish for n = 0.

Solution of the Boltzmann Equation

Let us define the symmetric fs and antisymmetric fa parts of the distribution function: 1 fsn (E ) = [ f + (E + nhw ) + f - (E + nhw )], 2 1 + n fa (E ) = [ f (E + nhw ) - f - (E + nhw )], 2 and let us call



(18.29a)

(18.29b)

Wnf,b = W-f,b (E + nhw ) = W+f,b [E + (n - 1)hw )].

We obtain the following equations: 1 + Nq n d n fs (E ) = f (E ) dE Wn a

È 1 1 ˘ + Nq Í f - b ˙ fan -1 (E ) ÍÎ Wn Wn ˙˚ Nq n f (E ) Wn+1 a





(18.30)

(18.31a)

È 1 1 ˘ + (1 + Nq ) Í f - b ˙ fan +1 (E ), ÍÎ Wn+1 Wn+1 ˙˚



d n 1 f (E ) = [(1 + Nq ) fsn (E ) - Nq fsn -1 (E )] dE a Wn 1 + [(1 + Nq ) fsn+1 (E ) - Nq fsn (E )], Wn+1

with fs-1 = fa-1 = 0. This yields the following (exact) sum rules [22]:



•

d dE

Â

d dE

•

fan (E ) = 0 or

n=0

Â

•

Âf

n a (E ) =

n=0

fsn (E ) = -

n=0

•

È 1

 ÍÍÎW

n+1

n=0

-

1 f Wn+1

J = const,

+

1 ˘ b ˙ Wn+1 ˙˚

¥ [Nq fan (E ) + (1 + Nq ) fan+1 (E )] •

=-

ÂW n=0

2 b n+1

[Nq fan (E )

+ (1 + Nq ) fan+1 (E )],



(18.31b)

(18.32a)

(18.32b)

239

240

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

by using Eqs. (18.22) and (18.27). Under moderate or lower fields (F £ 500 V/cm) the electron-phonon interaction is still strong enough to establish a detailed balance between energy intervals so that we can assume

n n+1 Nq fa,s (E ) = (1 + Nq ) fa,s (E ).

(18.33)

We note, in passing, that this equation would be satisfied by a Maxwellian and a linearly “displaced” Maxwellian distribution. However, Eq. (18.33) with the sum rule (18.32a) imposes fa to be constant on each interval so that we obtain

J . 1 + Nq

fa0 (E ) =

(18.34)

By making use of Eqs. (18.32a) and (18.33) we can transform Eq. (18.32b) and get d dE



4 JNq d 0 fs (E ) = dE (1 + Nq )2

with



n+1

Â

where we used



Â

2 È Nq ˘ =Í ˙ b Î 1 + Nq ˙˚ n=0 Wn+1 Í





È Nq ˘ 0 Í ˙ fs (E ) n=0 Í Î 1 + Nq ˙˚ •

or



n

•

2J



(18.35)

n

•

È Nq ˘ 1 , Í ˙ b 1 N + q˙ Î ˚ Wn+1 (E ) n=0 Í

Â

•

È Nq ˘ 1 + Nq = Í ˙ Î 1 + Nq ˙˚ n=0 Í

n

Â

-1

hw È ˘ Nq = Íexp - 1˙ . kT Î ˚

The solution fs0 (E ) is then given by fs0 (E ) = fs0 (0) •

4 JNq

(1 + Nq )2

È Nq ˘ ¥ Í ˙ n=0 Í Î 1 + Nq ˙˚

Â

n E

dE

0

b Wn+1 (E )

Ú

,



(18.36a)

Solution of the Boltzmann Equation



@ fs0 (0) -

4 JNq (1 + Nq )

2

E

dE

0

W1b (E )

Ú

,

(18.36b)

where we dropped the high-order terms of the rapidly convergent n series since n



È Nq ˘ - nhw / kT Í ˙ =e ÍÎ 1 + Nq ˙˚

b and 1 / Wn+1  1 / W1b . Here fs(0) is an integration constant which is determined by the condition f–(hw) = 0 because, in the low-field limit, there is a net depletion of carriers with negative momentum just below the POP emission threshold [12]. In this momentum range, the depopulation due to carrier deceleration is not balanced by a corresponding carrier repopulation by POP scattering (Fig. 18.3). This situation is unique to 1D systems because of the absence of both angular randomization and carrier-carrier scattering, which otherwise would result in a finite population for p ≥ - 2mhw .

Figure 18.3 Schematics of the 1D carrier dynamics in energy space indicating the subthreshold carrier depletion in the negative momentum region. The double arrows represent the carrier drift while the solid line represents the carrier scattering by POP emission. The breakdown of the inversion symmetry f(t) = f (–k) by the electric field is obvious.

241

242

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling



Under this condition, fs becomes fs (E ) =

Ï J Ô Ì1 + g 1 + Nq Ô Ó

È È x ˘1 / 2 ˘ dx Í1 - Í ˙ ˙ E/hw Í Îx + 1˚ ˙ Î ˚

Ú

1

¥

J = 1 + Nq

1 ¸ ˝ x(x + 1) ˛

ÏÔ È 2˘ ˙ Ì1 + 2g ln Í1 + 2 ˙˚ ÍÎ ÓÔ



(18.37)

È È E ˘1/2 ˘ ¸Ô - ln Í1 + Í ˙ ˙˝ ÍÎ Î E + hw ˚ ˚˙ Ô˛ with g = 2Nqhw/(1 + Nq)eFvct0 and

vc = 2hw / m for 0 £ E £ hw .



(1 / 2t 0 )[1 - E /(E + hw )]

Here x is the integration variable and we have approximated 1 / t 0b in Eq. (18.21) by to fit the energy dependence of the backward-scattering matrix element which is a rapidly decreasing function of energy (Fig. 18.2 inset). We use Eq. (18.33) to obtain fs(E) at high energy E > hw and the normalization condition nL =



•

ÂÚ n=0

hw

0 1/2

È 2mw ˘ =Í ˙ Î h ˚

dE D(E + nhw ) fs (E + nhw ) J ( A + g B ), p (1 + Nq )



(18.38)

where D(E) is the 1D density of states, to determine J. Here nL is the linear carrier concentration, and A and B are coefficients dependent on temperature but independent of field and confinement.

2 A= 1 + Nq

•

È Nq ˘ Í ˙ n=0 Í Î 1 + Nq ˙˚

Â

n

n +1,

(18.39a)

Solution of the Boltzmann Equation

n

•

È Nq ˘ B= Í ˙ n=0 Í Î 1 + Nq ˙˚ 1 dx ¥ 0 x +n

Â



Ú

Ú

1

x

ds s( s + 1)

È È s ˘1/2 ˘ ¥ Í1 - Í ˙ ˙. ÍÎ Î s + 1 ˚ ˙˚



(18.39b)

Figure 18.4 Energy-distribution function of a single-band 1D electron gas with a longitudinal field F = 100 V/cm at T = 300 K. The quantum-wire structure results from confinement in a quantum well with width Ly = 135 Å and a triangular potential in a field Fz = 120 kV/cm, which places the ground state at 140 meV above the conduction-band edge [11]. Solid line denotes Monte Carlo simulation dots, which represent the analytic results of Eq. (18.37).

Figure 18.4 shows the profile of fs(E) for F = 100 V/cm, compared with the Monte Carlo simulation (POP scattering only) for the same confinement conditions as in Fig. 18.2. The agreement is very good except below E = hw, where the dip in the solid curve is an artifact of the Monte Carlo model which is caused by the quantum broadening of the density of states included in the scattering rates [23]. The distribution function is characterized by “evenly spaced” peaked

243

244

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

structures reminiscent of the 1D density of states. This jagged profile in the carrier distribution is naturally possible at high temperature where POP absorption is significant to replicate the peaked structures at multiples of the POP energy E = nhw. The particular shape of fs(E) shows that despite the “detailed balance” condition (18.33) between energy intervals, the Maxwellian distribution is not the solution of Eq. (18.16). In the limit F = 0 [24], the distribution function is given by f0 ( E ) =



p È h ˘ nL B ÍÎ 2mw ˙˚

1/ 2

ÏÔ È È È E ˘1/2 ˘ ¸Ô 2˘ Í1 + Í ln ¥ Ìln Í1 + ˙ ˙ ˙˝ 2 ˙˚ ÍÎ Î E + hw ˚ ˙˚ Ô˛ ÔÓ ÍÎ

(18.40)

as a result of Eqs. (18.37) and (18.38) in the zero-field limit (for g Æ 0 and J Æ 0). The fact that the latter distribution [Eq. (18.40)] is not a Maxwellian distribution is a manifestation of the nonergodicity of the 1D system that accommodates a solution different from the Maxwellian at equilibrium. Actually, it is easily verified that the Maxwell-Boltzmann distribution is also a solution of Eqs. (18.16– 18.19) for F = 0, but this solution is not unique, since any function which satisfies the detailed balance [Eq. (18.33)] is also a solution of Eqs. (18.12–18.19). In this respect it should be noticed that our results have been obtained by neglecting all scatterings events except POP scattering. As we mentioned earlier, this situation occurs in practical cases when F 10 V/cm and it is conceivable that at equilibrium (F = 0) a Maxwellian distribution is established as a result of the combined effects of POP, AP, and II scattering, which rules out the solution (18.40). At finite fields, however, AP and II scattering become rapidly negligible, and the distribution converges towards the actual profile (Fig. 18.4). Thus, under the influence of an electric field, the Maxwellian symmetry is more rapidly destroyed than in 2D or 3D systems where the electric field only “displaces” the equilibrium distribution without drastically changing its nature. One can, therefore, regard the onset of this effect as the manifestation of a pronounced organization of the system in the k (momentum) space under the joint influence of the electric field and POP scattering, which drives the electron gas toward a more stable configuration

Solution of the Boltzmann Equation

than that given by a Maxwellian profile. This is particularly noticeable if one evaluates the average carrier energy given by ·E Ò =



1 nL

•

ÂÚ n=0

hw

0

dE D(E + nhw )(E + nhw )

C +g D , ¥ fs (E + nhw ) = hw A+g B

(18.41)

which is plotted in Fig. 18.5a as a function of electric fields for different temperatures. Here C and D are temperature-dependent coefficients.



2 1 C= 3 1 + Nq •

È Nq ˘ 3/2 Í ˙ (n + 1) , 1 + N Í ˙ q n=0 Î ˚

Â

È Nq ˘ D= Í ˙ n=0 Í Î 1 + Nq ˙˚

Â

¥

Ú

1

0

n

•

n

dx x + n

Ú

1

x

ds s( s + 1)

È È s ˘1/2 ˘ ¥ Í1 - Í ˙ ˙. ÍÎ Î s + 1 ˚ ˙˚



(18.42a)

(18.42b)

It is easily seen that above TL = 150 K, ·EÒ is smaller than the thermal energy kTL /2 over an increasing range of electric fields [25]. The big dots are the results of a Monte Carlo simulation which includes the influence of AP scattering at TL = 300 K. The agreement between the two approaches is excellent except for F < 50 V/cm where the predominance of AP scattering contributes to heating the system before the transition toward the jagged profile characterized by a low average energy. The figure also shows that the cooling from the thermal energy increases with the temperature, which indicates the dominant influence of POP absorption. This phenomenon is a high-temperature effect, which is therefore different from the decrease of the thermal (Maxwellian) energy ·EÒTh discussed by previous authors [26–29] and which occurs in 3D systems at low temperature as a result of POP emission processes only. We wish also to point out that the decrease of ·EÒ below the thermal energy does not violate Joule heating since the energy gained from the field is

245

246

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

directly transferred to the lattice without thermalizing the electron gas in the absence of carrier-carrier collision.

Figure 18.5 (a) Average carrier energy as a function of electric fields for four temperatures. Dots denote Monte Carlo simulation with AP and POP scattering at 300 K. Thin lines denote thermal energy kT/2. (b) POP mobility for the same temperatures.

Solution of the Boltzmann Equation

The organization of the distribution into a jagged profile is therefore an increasing function of temperature which occurs under highly dissipative conditions. This has important consequences for the transport properties of 1D systems, as seen below. From Eqs. (18.37) and (18.38), it is straightforward to calculate the 1D current I =e



•

ÂÚ n=0

hw

0

dE D(E + nhw )v(E + nhw )

2eJw ¥ fa (E + nhw ) = , p

as well as the carrier mobility

m=

(1 + Nq )2 e t 0 , B + Ag -1 m Nq



(18.43)

(18.44)

which is shown as a function of electric field in Fig. 18.5b. At high temperature the 1D mobility exceeds the bulk value by a factor of 4 to 5 over a large range of electric fields. For instance, for F = 100 V/cm at T = 300 K, Eq. (18.44) yields m ~ 3.8 ¥ 104 cm2/Vs (Monte Carlo simulation yields m = 3 ¥ 104 cm2/Vs with both POP and AP scattering [11]) which is several times larger than the bulk value (m = 8 ¥ 103 cm2/Vs). This high value results from the fact that the distribution function only depends on the backward-scattering rate (1/tb) [Eq. (18.36)], which is much weaker than the f rate (Fig. 18.2 inset). In this transport process controlled by strong POP emission and absorption, the 1D thermal energy is converted into a drift motion of carriers which boosts the mobility. From Fig. 18.5b, it can be seen that this effect also increases with temperature, contrary to bulk or 2D transport. In conclusion, we want to emphasize that the cooling effect and the mobility enhancement are unique features of 1D transport at high temperature which manifests the absence of ergodicity in the electron system. Specifically, phonon absorption is the important process which is responsible for the existence of jagged structures in the distribution and results in improved transport performances. From a general standpoint, the structures in the distribution function are relatively robust and persist in multiband systems where the basic features of this effect contribute to enhanced POP-assisted intersubband resonances similar to the magneto-phonon effect, and set on population inversion between subbands [14, 15].

247

248

Optic-Phonon-Limited Transport and Anomalous Carrier Cooling

Acknowledgments The author would like to thank S. Briggs and D. Jovanovic for fruitful discussions. He is indebted to Mrs. C. Willms and G. Miller for technical assistance. This work is supported by the Joint Service Electronic Program and the National Center for Computational Electronics at the University of Illinois.

References

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2. A. Warren, D. Antoniadis, and H. Smith, Phys. Rev. Lett. 56, 1858 (1986). 3. K. Kash, A. Scherer, J. Worlock, H. Craighead, and M. Tamargo, Appl. Phys. Lett. 49, 1043 (1986).

4. T. Hiramoto, K. Hirakawa, Y. Iye, and T. Ikoma, Appl. Phys. Lett. 51, 1620 (1987). 5. M. Tsuchiya, J. M. Gaines, R. H. Yan, R. J. Simes, P. O. Holtz, L. A. Colden, and P. M. Petroff, Phys. Rev. Lett. 62, 466 (1989). 6. E. Colas, E. Kapon, S. Simhony, H. M. Cox, R. Bhat, K. Kash, and P. S. D. Lin, Appl. Phys. Lett. 55, 867 (1989).

7. Nanostructure Physics and Fabrication edited by M. A. Reed and W. P. Kirk (Academic, New York, 1989). 8. J. P. Leburton and D. Jovanovic, Semicond. Sci. Technol. (to be published). 9. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

10. H. Sakaki, in Proceedings of the Ninth International Symposium on GaAs and Related Compounds, Oiso, Japan, 1981, edited by T. Sugano (IOP, Bristol, 1982), p. 251. 11. S. Briggs and J. P. Leburton, Phys. Rev. B 38, 8163 (1988). 12. S. Briggs and J. P. Leburton, Phys. Rev. B 39, 8025 (1989).

13. J. P. Leburton, S. Briggs, and D. Jovanovic, Superlattices Microstruct. 8, 209 (1990). 14. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989).

15. S. Briggs, D. Jovanovic, and J. P. Leburton, Appl. Phys. Lett. 54, 2012 (1989). 16. K. Hess, in Physics of Non-Linear Transport in Semiconductors, edited by D. K. Ferry, J. R. Barker, and C. Jacoboni (Plenum, New York, 1980).

References

17. S. Briggs (unpublished).

18. C. Kittel, Quantum Theory of Solids (Wiley, New York, 1987).

19. H. Fröhlich, H. Pelzer, and S. Zienau, Philos. Mag. 41, 221 (1950). 20. M. Stroscio, Phys. Rev. B 40, 6428 (1989).

21. This point was brought to my attention by P. Kocevar.

22. These sum rules are relatively general; similar results for spatially nonhomogeneous transport have been obtained by Mahan et al. for optic-deformation-potential phonon scattering; see, e.g., G. D. Mahan, J. Appl. Phys. 58, 2242 (1985); G. D. Mahan and G. S. Canright, Phys. Rev. B 35, 4365 (1987). 23. S. Briggs, B. A. Mason, and J. P. Leburton, Phys. Rev. B 40, 1200 (1989). 24. K. Hess, Phys. Rev. B 10, 3374 (1974).

25. Without loss of generality we can assume the electron concentration is weak enough (nL ≲ 106/cm) and does not change the phonon temperature during the process. 26. V. V. Paranjape and T. P. Ambrose, Phys. Lett. 8, 223 (1964).

27. Z. S. Gribnikov and V. A. Kochelap, Zh. Eksp. Teor. Fiz. 58, 1046 (1970) [Sov. Phys. JETP 31, 562 (1970)]. 28. R. L. Peterson, Phys. Rev. B 2, 4135 (1970).

29. L. E. Vorobev, V. S. Komissarov, and V. I. Stafev, Fiz. Tekh. Poluprovodn. 7, 88 (1973) [Sov. Phys. Semicond. 7, 59 (1973)].

249

Chapter 19

lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping” in Quantum Wires

J.-P. Leburton

Beckman Institute for Advanced Science and Technology Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We present an analysis of the optical gain for stimulated emission between subbands in quasi-one-dimensional (quasi-1D) structures. The population inversion is induced by a new pumping mechanism involving phonon absorption during transport in quantum wires. The population inversion threshold for far infrared stimulated emission is calculated in a generic GaAs quantum wires laser structure and shows a quite reasonable minimum value of Dn = 3 ¥ l05/cm at l = 100 mm. For longer wavelengths, free carrier absorption is the main limitation to stimulated emission. Reprinted from J. Appl. Phys., 74(2), 1417–1420, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping”

19.1 Introduction Recently, Briggs et al. have predicted the onset of population inversion between subbands during transport in quantum wires at high temperature [1]. This anomalous effect is induced by polar optic phonon (POP) absorption, which acts as the pumping mechanism between two resonant subbands separated by the longitudinal optic (LO) phonon energy E = hwLO [2, 3]; population inversion, however, occurs between two closely spaced off-resonant subbands (DE < hwLO). This new phenomena finds its origin in the absence of carrier thermalization in quasi-1D systems where the carrier distribution deviates drastically from the Maxwellian distribution, even in moderate electric fields [4]. In particular, the occupation of upper subbands is significantly enhanced by resonant POP scattering for which the joint 1D density of states is maximum at the 1D subband edges (k = 0). The resulting distribution has a “jagged” profile characterized by sharp peaks at multiple of the POP energy E = nhwLO (Fig. 19.1). If an intermediate subband lies between the resonant subbands, its population can be significantly less than in an upper subband even at room temperature. Therefore, unlike conventional pumping mechanisms that require carrier injection at high energy [5], population inversion arises as a natural consequence of carrier transport in quasi-1D systems. Typically, population inversion by “phonon pumping” would arise between subbands separated by at most 20 meV, which is particularly interesting for far-infrared (FIR) stimulated emission. An important condition for the observation of this new effect is the realization of highly confined 1D systems with well-controlled and reproducible wire structures. The recent fabrication of quantum wires by direct growth on patterned GaAs vicinal substrate offers a promising and convenient approach to obtain dense and regular arrays of 1D structures suited for FIR lasers [6–9]. This letter addresses the issue of optical gain for stimulated emission obtained by “phonon pumping” in structurally confined quantum wires.

Model

Figure 19.1 Schematic profile of the distribution function f(E) in quantum wires subject to a moderate electric field (F~100 V/cm). Population inversion arises between the resonant levels Eres and the lower levels E3, E2, and E1. Among the possible transitions (dashed lines), only the Eres Æ E2 shows appreciable oscillator strength and population inversion to lead to stimulated emission.

19.2 Model Among the different 1D confinement shapes obtained by regrowth on patterned substrate [6–8], the sawtooth-like array of GaAs quantum wire structures obtained by growth on patterned (100) substrate proposed by Colas et al. [6] appears particularly attractive because of its modeling simplicity and its potential for high density packing. A schematic cross section of the structure is shown in Fig. 19.2a. The two-dimensionally confined wire regions are bounded by thinner quantum wells, and the confining layers of wide gap AlGaAs are modulation doped, which provides carriers in the 1D region. The array density is determined by the period of the perturbation on the substrate while the shape, the thickness and the relative dimension of the wires are controlled by the growth temperature and rate. Scanning electron microscopy (SEM)

253

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lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping”

photographs, however, reveal rounded corner on the lower side of the quantum wires [10]. In the simplified version of a FIR laser, we assume electrical contacts are achieved on the front and back of the structure and optical confinement is realized by metallic walls on the lateral sides, top and bottom of the device.

Figure 19.2 (a) Schematic structure for FIR lasing device. Metallic plates for optical confinement are represented on the top, bottom, front, and back sides of the structure. Light emission occurs partially through the contact layers, (b) Schematic quantum wire cross section; the x and y axis are represented with the structure orientation.

In order to model the electronic properties of the system, we assume the wire cross section resembles a portion of a circle with radius a and angular aperture Q (Fig. 19.2b). The two-dimensional Schrödinger equation in cylindrical coordinates reads

-

1 ∂ 1 ∂2 ˆ h2 Ê ∂2 + + y ( r ,q ) = Ey ( r ,q ). (19.1) 2m * ÁË ∂r 2 r ∂r r 2 ∂q 2 ˜¯

For the sake of simplicity, we ignore the influence of the thinner quantum well links, and consider infinite potential barriers for the wire states; this simplifies the boundary conditions for y considerably:

Model



Y(a,q) = Y(r,0) = Y(r,Q) = 0.

The eigen functions and energies are given by y ( r ,q ) =

2 r J v n ( x v nm r / a) sin(vnq ), (19.2a) a F | J vn +1 ( x vnm )| E nm =



h2 x v2nm 2ma2

, (19.2b)

where J vn are the Bessel functions of order vn = np/Q and x vnm is the mth zeros of J vn . We choose Q = 8p/9, which corresponds to the experimental results [10] and consider two wire configurations with a = 237 and a = 283 Å, which lead to POP resonances (see Table 19.1). In the former configuration, POP resonance occurs between the E11 and E12 levels, and population inversion (PI) arises between E12 and E31 separated by ~5 meV (the E21 state is too close to the ground state E11 to result in PI); in the latter case, the POP resonance occurs between E11 and E41 with PI between E41 and E12 (~9 meV) and E41 and E31 (~14 meV). Table 19.1 Calculated Enm energy levels (in meV) for the two quantum wire cross considered

a = 237 Å a = 283 Å

E11

E21

E31

E12

E41

E22

16.2

30.0

47.3

52.4

68.0

77.4

11.4

21.1

33.2

36.7

47.7

54.3

The gain for stimulated emission between subbands with a population inversion DN reads

g( l ) =

T2 DN e2 fif , (19.3) 2 2 2 A 2n0cm0 1 + 4p c T2 ( l -1 - l0-1 )2

where l0 = hc/Eif is the wavelength at resonance for intersubband energy separation, Eif, c is the speed of light, h is the Planck constant, T2 is the intersubband scattering time, A = 4pa2/9 is the wire cross section, and n is the index of refraction. The oscillator strength is given by

fif = 2m0 E if /2 | · f | e i r Òi |2 , (19.4)

 are the polarization where m0 is the free electron mass, e and r and position vectors, respectively, and |iÒ and |fÒ are the initial and

255

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lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping”

final states during the optical transition. With our configuration, one obtains

m r fif = 0 ( x v2nm - x v2¢n m¢ ) vn m vn ¢ m¢ m* a

2

c (e i i x Inn ¢

c 2 + e i i y Inn ¢) ,

(19.5)

where vnm and v n¢ m¢ are the initial and final states, and

c,s In,n ¢ =

2 Q

Ú

Q

0

Ê npq ˆ Ê n¢pq ˆ Ïcos q ¸ dq sin Á sin Á ˝ . (19.6) Ì Ë Q ˜¯ Ë Q ˜¯ Ósin q ˛

For the E12 Æ E31 transition in the thinner (a = 237 Å) wire, the oscillator strength for x and y polarization are fifx = 5.7 ¥ 10-3 and fify = 0.18, respectively. If we assume DN = 105 cm–1 and T2 = 2 ¥ 10–13 s, the gain at the peak value gx(l0) = 0.22 cm–1 and gy(l0) = 7.0 cm–1. These small values indicate that the nature of the states involved in the transition is not very favorable for stimulated emission. On the other hand, for the E41 Æ E31 transition in the thicker (a = 283 Å) wire, fifx = 31.65 and fify = 0.98, with gx(l0) = 860 cm–1 and gy(l0) = 27 cm–1, respectively, which is much more encouraging especially for x-polarization emission. From a general standpoint, optical transitions between electronic states where both n and m quantum numbers change are relatively weak, while transitions between states that conserve norm are much stronger. This selection rule for intersubband transitions was already emphasized by Briggs et al. in their original paper [1]; this is the reason why we focus on the E41 Æ E31 transition in the wider wire.

19.3 Optical Gain Analysis

In order to assess the potential of quantum wires for FIR stimulated emission, we now analyze the optical losses in the device configuration shown in Fig. 19.2 in a way similar to Borenstain and Katz [11]. Optical confinement is realized by the metallic walls on the sides of the device, and the active region consisting of the array of quantum wire structures of thickness ta is comprised between two cladding layers of intrinsic GaAs layers of effective mode thickness t i = l / 2n ~ 10 - 15 mm. The total width of the structure is taken as w = l / 2n. The most important losses for optical frequencies below

Optical Gain Analysis

the band gap are free carrier absorption and absorption by phonon interactions, which are calculated by using the general form of the complex dielectric constant (w ) = (n - ik )2



Ê ˆ (19.7) w r2 w2 - w2 = • Á 1 + 2 L 2 T ˜ ÁË w L - w + iwg ph w (w - ig pl ˜¯

with a = 2wk(w)/c. Here, wL, wT, and wpl are the frequencies of the LO phonons, TO phonons, and the plasmons, respectively; gph and gpl are the damping constants of the phonons and plasmons. Given the wire orientation (Fig. 19.2), appreciable gain for x polarization will build up in the perpendicular direction compatible with the device structure, i.e., along the wire direction. At lasting threshold, gain equals the losses, which for FIR excitation, can be expressed by the following relation

l t A A 1 Ê 1ˆ gth (w 0 ) = a qwi + i a i + a m + c a c + ln Á ˜ . (19.8) Aeff Aeff ta li L Ë R¯

Here gth is the gain at threshold, L is the cavity length and R the reflectivity; aqwi, am, ai, and ac are the absorption coefficients for the quantum wires, the modulation doped, the intrinsic, and the contacting layers, respectively. Aeff is the quantum wire effective cross section in the active layer, i.e., ti ¥ li/number of quantum wires. Since the radiations emitted by intersubband emission are polarized perpendicular to the direction of the wires, interaction with longitudinal 1D plasmons is forbidden so we assume aqwi~0. Therefore, losses by plasmon absorption take place only in the modulation-doped AlGaAs and the GaAs contacting layers; the absorption in the intrinsic region is due to phonon only ϵ(wpI = 0). By neglecting the last term in Eq. (19.8), we can express the population inversion at threshold DNth



Ê ˆ l t e2 DNth = Aeff Á a m + c a c + i a i ˜ / fif T2 , (19.9) ll t a ¯ 2n0cm0 Ë

Here we used Aeff = 10–10 cm2, wL = 291.5 cm–1, wT = 269.2 cm–1,

wpl = Nv e2/m * 0s /2p c , where Nv is the carrier concentration per unit volume ϵs = 10.06, gph = 2.3 cm–1, and gpl = e/2pmm*c = 28 cm–1 with m = 5 ¥ 103 cm2/Vs.

257

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lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping”

In Fig. 19.3, we show DNth for the E41Æ E31 transition as a function of the carrier concentration, Nd, in the modulation doped layer for x and y polarizations, and for different values of the length ratio lc/li. Here we have assumed an aspect ratio ti/ta = 50, the contact layers have a thickness ta and the doping concentration nc = 1018/cm3. As expected from the values of the oscillator strengths, DNth for y polarization is almost two orders of magnitude higher than for the x polarization, and increases rapidly with Nd. It is also seen that DNth become independent of absorption in the contact layers for lc/li < 3 ¥ 10–3. However, the important result here is the relatively low population inversion at threshold, DNth = 2–3 ¥ l05/cm for Nd ~ 1016/ cm3, which is quite reasonable. Monte Carlo simulation by Briggs et al. have indicated the possibility of achieving 24% carrier occupation in the resonant upper subband against 4% only in the intermediate subband in certain configurations [1], which would require an overall concentration of 1–1.5 ¥ 106/cm to lead to stimulated emission.

Figure 19.3 Inverted population at threshold between E41 and E31 levels (DE = 14.5 meV) in the thicker wire (a = 283 Å) for both x and y polarizations, tl/ta = 50. Solid: lc/li = 10–2, short dashed: lc/li = 3 ¥ 10–3; long dashed: lc/li = 10–3.

Conclusions

In order to analyze the gain dependence on the photon energy, we have plotted DNth as a function of the wavelength in Fig. 19.4. Here, lc/li = 3 ¥ 10–3. For Nd = 1016/cm3, the two upper curves are roughly identical with a minimum around 100 mm. The large increase at short wavelengths is due to phonon absorption as l approaches lT [Eq. (19.7)] and is less pronounced for constant ta when ti decreases with l. The increase of DNth at long wavelengths is due to plasmon absorption in the modulation doped layer as seen by comparison with the Nd = 0 (no plasmon) curve, which decreases monotonically with l. Therefore, it appears that high gain at long wavelengths requires a serious reduction of free carrier absorption in the quantum wire adjacent layers. It should be pointed out that, according to Fig. 19.1 the population inversion at threshold is more easily achieved at long wavelength since the distribution is the smallest before the threshold for POP emission. However, this criterion is not absolute since long wavelength PI could be limited by the broadening of the 1D levels.

Figure 19.4 Inverted population at threshold between E41 and E31 levels as a function of the energy separation. Solid: ta = 0.25 mm, Nd = 1016 cm3; long dashed; tl/ta = 50, Nd = 1016 cm3; short dashed; ta = 0.25 mm, Nd = 0.

19.4 Conclusions In conclusion, we have presented an analysis of the optical gain for intersubband stimulated emission in quantum wires. Although the device structure and the wire cross section investigated here were

259

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lntersubband Stimulated Emission and Optical Gain by “Phonon Pumping”

rather idealistic, our analysis indicates that substantial optical gain can be achieved by the new phonon pumping mechanism. The real advantage of this mechanism over conventional carrier injection is its natural occurrence as a manifestation of the nonequilibrium carrier statistics in quasi-1D systems, which requires moderate longitudinal field to induce population inversion [4].

Acknowledgments

This work was supported by NSF under grant no. ECS-9108300. The author is indebted to Dr. E. Colas at Bellcore and Dr. M. Kushner for many useful discussions.

References

1. S. Briggs, D. Jovanovic, and J. P. Leburton, Appl. Phys. Lett. 52, 2012 (1989). In this article, the driving fields is 100 V/cm. 2. S. Briggs and J. P. Leburton, Superlattices Microstruct. 5, 145 (1989).

3. D. Jovanovic, S. Briggs, and J. P. Leburton, Phys. Rev. B 42, 11108 (1990). 4. J. P. Leburton, Phys. Rev. B 45, 11022 (1992).

5. S. J. Allen, G. Brozak, E. Colas, F. DeRosa, P. England, J. Harbison, M. Helm, L. Florez and M. Leadbeater, Semicond. Sci. Technol. 7, B 1 (1992). 6. P. M. Petroff, K. Ensslin, M. Miller, S. Chalmers, H. Weman, J. Merz, H. Kroemer, and A. C. Gossard, Superlattices Microstruct. 8, 35 (1990).

7. E. Colas, E. Kapon, S. Simhony, H. M. Cox, R. Bhat, K, Kash, and P. S. D. Lin, Appl. Phys. Lett. 55, 867 (1989).

8. E. Colas, S. Simhony, E. Kapon, R. Bhat, D. M. Hwang, and P. S. D. Lin, Appl. Phys. Lett. 57, 914 (1990). 9. S. S. Chang, S. Ando, and T. Fukui, Surf. Sci. 267, 214 (1992). 10. E. Colas (private communication).

11. S. I. Borenstain and J. Katz, Appl. Phys. Lett. 55, 654 (1989).

Chapter 20

Superlinear Electron Transport and Noise in Quantum Wires

R. Mickevičius,a V. Mitin,a U. K. Harithsa,a D. Jovanovic,b and J.-P. Leburtonb

aDepartment of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202, USA bBeckman Institute for Advanced Science and Technology and the Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA [email protected]

We have employed a Monte Carlo technique for the simulation of electron transport and noise (diffusion) in GaAs rectangular quasione-dimensional quantum wire structures at low temperatures. It is demonstrated that with the heating of electron gas the efficiency of acoustic phonon scattering decreases and the mobility increases. The increase of electron mobility appears as a superlinear region on velocity-field dependence. It is shown that electron noise increases in the superlinear region. The transition from superlinear transport Reprinted from J. Appl. Phys., 75(2), 973–978, 1994.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1994 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Superlinear Electron Transport and Noise in Quantum Wires

to the regime close to electron streaming with a further increase of electric fields is reflected on the diffusivity-frequency dependence by the appearance of a separate peak at the streaming frequency. The electron streaming regime which takes place at higher fields causes the collapse of the diffusion coefficient (noise spectral density) to the streaming frequency.

20.1 Introduction

Low-dimensional semiconductor structures and particularly quasione-dimensional (1D) structures of quantum wires (QWI) have recently attracted much attention due to the possibilities of achieving very high electron mobilities [1] and low-noise performance at low temperatures with possible technological applications. In spite of the fact that electric noise is a crucial device characteristic because it sets lower limits to the accuracy of any measurement, there are almost no studies on electron noise in QWIs at low temperatures. The reason is that electron transport and noise in quantum wires at low and intermediate temperatures, and at intermediate electric fields are primarily controlled by acoustic-phonon scattering. It is common practice to treat acoustic-phonon scattering within elastic or quasi-elastic approximations (see, e.g., Refs. [2–5]) as in bulk materials. However, it has been shown earlier [6] that acousticphonon scattering in low-dimensional structures and particularly in QWIs is essentially inelastic and is far more important for electron energy and momentum relaxation than in bulk materials. Correct treatment of acoustic-phonon scattering in low-dimensional structures requires full consideration of uncertainty of momentum conservation (quasi-conservation) [6]. Due to peculiarities of acoustic-phonon scattering in QWIs, electron kinetic and noise parameters may be considerably different from those in bulk materials. This is due to the dependence of the 1D density of states on electron energy whereby the acoustic-phonon scattering rate decreases with an increase in electron energy, excluding the region close to the subband bottom [6]. As a result, the efficiency of electron scattering by acoustic phonons should decrease with the electron heating by electric fields. At rather low electric fields, where electrons are still below the optical-phonon energy, it

Model and Method

is expected that the decrease in the efficiency of acoustic-phonon scattering might lead to an increase in electron mobility [7]. In real structures, however, acoustic-phonon scattering may be either too weak, resulting in an electron runaway towards the optical-phonon energy, or too strong so that electrons cannot be heated at all up to very high electric fields. It seems, however, that there should be optimum structure parameters for which an increase in electron mobility can be quite pronounced. The aim of the present chapter is to get an insight into electron transport and noise in rectangular QWIs at low temperatures for a wide range of electric fields through the study of velocity-field and energy-field characteristics as well as electron noise spectral density. We developed a Monte Carlo code which efficiently includes electron scattering by acoustic as well as by optical phonons and allows simulation of electron transport and noise in 1D structures.

20.2 Model and Method

We consider rectangular GaAs QWIs with several different cross sections embedded into AlAs material. We assume that the electron gas is nondegenerate with electron concentration of the order of 105 cm–1 or less. Electron scattering by confined longitudinal optical (LO) phonons and localized surface (interface) optical (SO) phonons [8] as well as by bulklike acoustic phonons [6] has been taken into account. Ionized impurities are assumed to be located sufficiently far from the QWI so that their influence on the electron motion inside the wire is negligible. Our program incorporates all subbands occupied by electrons, but for the present structure parameters only the first two or three subbands are relevant to the electron transport at low temperatures. The transition probabilities are given by

2p | M(kx , kx¢ , q )|2 h (20.1) ¥ d [(kx¢ ) - (kx ) - D ± hw q ].

W (kx , kx¢ , q ) =

Here, M(kx, kx¢ ,q) represents the matrix element for a transition from the initial kx state to the final kx¢ state mediated by a phonon with wave vector q, and Dϵ is the intersubband separation energy. The ± sign in the energy-conserving d function accounts for the

263

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Superlinear Electron Transport and Noise in Quantum Wires

emission (+) and absorption (–) of phonons. The total scattering rate from an initial state with energy ϵ is then given by l( k x ) =



V

(2p )

3

Â Ú dq W ( k kx¢

¢ x , k x , q ),

(20.2)

where the summation over the final kx¢ states takes into account all the possible intrasubband and intersubband transitions. The 1D electron scattering by LO and SO phonons is described in more detail in Ref. [8]. Electron scattering by acoustic phonons is usually treated as an elastic mechanism, which is not true for low-dimensional structures. Let us take a closer look at electron interaction with acoustic phonons in a rectangular QWI. The acoustic-phonon scattering rate is given by [6]

l( ) =

Ea2kBT

4p ru  2

2 2

m * dqy dqzG(qy , qz ) , (20.3) 2  ∓ uqT - D

Ú

where Ea is the acoustic deformation potential, r is a material density, u is a sound velocity in the material, and qT = qy2 + qz2 is a transverse component of the phonon wave vector. The function G(qy, qz) is a form factor which represents the uncertainty of momentum conservation in low-dimensional structures. The above equation as obtained in Ref. [6] assumes that qx ≪ qT and huqT ≪ kBT. If the phonon energy under the delta function is neglected [5] one gets the elastic approximation:

l( ) =

Ea2kBT

ru h Ly Lz 2 2

1 1 m* (1 + d jj¢ )(1 + d ll¢ ), (20.4) 2( - D ) 2 2

where indices j and l denote the initial subband, j′ and l′ denote the final subband, and Ly and Lz are transverse dimensions of the QWI. One can see that the elastic approximation of Eq. (20.4) yields two unphysical divergencies of the scattering rate: (i) when the electron energy goes to zero after scattering and (ii) when spatial dimensions of a QWI approach zero. Both these divergencies disappear within the inelastic approach of Eq. (20.3) [6]. In order to demonstrate the differences between the elastic and inelastic models we have plotted the scattering rate versus electron energy for the two different models in Fig. 20.1. Note that within the elastic approximation the absorption and emission rates are equal. One can see the elastic approximation highly overestimates the scattering rate in the low-energy region.

Model and Method

Figure 20.1 Acoustic-phonon scattering rate vs electron energy for elastic and inelastic models of electron-phonon interaction. The emission and absorption rates are equal within the elastic approximation. The temperature and cross section are indicated on the figure.

Computationally, the main differences between 1D simulations and 2D and bulk models is that phase space reduction lifts some of the complexities associated with computation of angular scattering. For instance, due to the limited number of final scattering states, the total rates can be stored in the memory, thereby eliminating lengthy computations during free-flight loops and final state selection [9]. We have developed a novel and very efficient procedure for random selection of acoustic-phonon energy involved in scattering. The essence of this procedure is that we first numerically perform a von Neumann procedure for a set of random numbers and tabulate the phonon energy as a function of a random number. Actually, we have solved the following equation:

r

Ú

qTmax

qTmin

dqT F (qT ) =

Ú

qT*

qTmin

dqT F (qT ), (20.5)

with respect to the unknown upper integration limit qT* for a set of 100 random values of r ranging from 0 to 1. Here the function F(qT) is an intergrand of Eq. (20.3) and represents the scattering probability dependence on the transverse components of the phonon wave vector. For a set of uniform random numbers (r1, r2, ..., r100) we

265

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Superlinear Electron Transport and Noise in Quantum Wires

obtain the set of solutions of Eq. (20.5) (qT*1 , qT*2 , ..., qT*100 ) (which are transverse components of the phonon wave vector) distributed according to the probability F(qT) of electron scattering by the acoustic phonon with a given transverse component qT. Since the transverse component of the phonon wave vector is directly related to the phonon energy within this approach [10], one can find the desired phonon energy for each value of r. The tables of such values have been separately calculated for a set of electron energies for the emission and absorption of the acoustic phonon. Hence, the random choice of energy of the phonon involved in a scattering event in the Monte Carlo procedure is just the generation of the random number r and the selection of the corresponding phonon energy value from the appropriate table. This procedure essentially speeds up the Monte Carlo simulation. As the transient process under near-streaming conditions lasts a very long time and electrons have to undergo a great number of scattering events before reaching the stationary regime, the conventional ensemble Monte Carlo simulation becomes essentially inefficient. We have developed an ensemble Monte Carlo technique [11] which permits a quick relaxation to steady state in the presence of long-lasting transient processes. The essence of this technique lies in the choice of the initial electronic state. We choose every next electron randomly from the trajectory of the previous electron so that we approach the stationary initial distribution function in three to four iterations. This technique has been used for velocity autocorrelation function calculations. All stationary characteristics have been calculated by the single-particle Monte Carlo technique, averaging over one electron trajectory [12].

20.3 Results and Discussion

The results presented here are obtained for T = 30 K. Similar results have been obtained for temperatures T = 20 and 77 K, and the characteristics are qualitatively the same as at 30 K. Figure 20.2 shows the electron drift velocity as a function of applied electric field. For thick QWIs the superlinear region appears on the velocity-field dependence at an electric field of the order of 10 V/cm. With a decrease in the thickness of the QWI,

Results and Discussion

the superlinear region shifts towards higher electric fields. The superlinear dependence disappears for the QWI with a cross section of 40 ¥ 40 Å2. As we have already pointed out in Section 20.1, the superlinear dependence is a purely 1D effect caused by the reduction of the efficiency of acoustic-phonon scattering as the electron gas is heated. Figure 20.3 shows the mean electron energy plotted versus the electric field for the same QWIs. In 1D structures the thermal equilibrium electron energy corresponding to one degree of freedom is equal to kBT/2, which at T = 30 K is 1.3 meV. This equilibrium energy establishes well in our simulations, indicating that acoustic-phonon scattering is treated correctly. One can clearly see that electron heating starts at higher electric fields in thin QWIs. This is due to the fact that the rate of electron scattering by acoustic phonons increases when the thickness of a quantum wire decreases [within the elastic approximation (20.4) the rate of scattering is inversely proportional to the cross section of a QWI].

Figure 20.2 Electron drift velocity as a function of applied electric field for three different cross sections of the quantum wire. Dashed lines show the drift velocity estimated from the low-field mobility. Wire cross sections and lattice temperatures are indicated on the figure.

In thick QWIs electrons with the energy of the order of or higher than kBT easily escape from acoustic-phonon scattering and run away up to the optical phonon emission threshold. A further increase

267

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Superlinear Electron Transport and Noise in Quantum Wires

in electric field leads to a near saturation of the drift velocity at a streaming value vs = hw/m * = 2.2 ¥ 107 cm/s, where hw is the LO phonon energy [11]. This behavior is related to the electron transition from the superlinear to streaming regime. Under streaming conditions, acoustic-phonon scattering can no longer hold back low-energy electrons from reaching the optical-phonon scattering threshold to emit a phonon, in which case the electrons travel to the subband bottom, repeating this process again and again [13]. In the case of ideal streaming (infinitely high electron-optical phonon scattering rate and the absence of any other scattering mechanism) electrons oscillate in k space for an indefinite period of time, which leads to permanent oscillations of the electron drift velocity and mean energy [14]. In real conditions both the penetration through the optical-phonon scattering threshold and the acoustic-phonon scattering randomizes the phase of the electron ensemble, and even though each electron under certain conditions continues oscillating, the mean parameters approach stationary values. This problem has been investigated in Refs. [2, 14].

Figure 20.3 Mean electron energy as a function of applied electric field for the same quantum wires as in Fig. 20.2.

To examine the superlinear region in more detail we have calculated the electron diffusion coefficient as a function of electric

Results and Discussion

field in Fig. 20.4. The superlinear region is reflected on difiusivity-field dependence as a broad maximum. The maximum on the diffusivity curve is well pronounced for thick QWIs and almost disappears for a 40 ¥ 40 Å2 QWI. Electron scattering by acoustic phonons in this thin QWI is so strong that it prevents electron heating by the electric field up to very high values where optical phonon emission starts to dominate and electrons enter the streaming regime. The decrease in diffusivity at higher electric fields indicates the transition from superlinear electron transport to electron streaming.

Figure 20.4 Electron diffusion coefficient as a function of electric field for the same quantum wires as in Fig. 20.2.

The dependence of electron mobility on electric field can be easily extracted from velocity-field dependence (Fig. 20.2). We have estimated the low-field electron mobility for three different QWIs. The low-field mobility values of 1.5 ¥ 104, 7 ¥ 104, and 1.5 ¥ 105 cm2/Vs are obtained for QWI cross sections of 40 ¥ 40, 150 ¥ 100, and 150 ¥ 250 Å2, respectively. Almost the same mobility values are derived from Einstein’s relationship m = eD/kBT, where D is the lowfield diffusion coefficient (Fig. 20.4). Figure 20.5 shows the electron distribution functions at different electric fields for a 250 ¥ 150 Å2 QWI. One can see that the distribution function in the intermediate energy region is flattened

269

270

Superlinear Electron Transport and Noise in Quantum Wires

and extended up to the optical phonon energy. There is no electron penetration beyond the optical-phonon energy in the given fields, so that the distribution function is cut off at the phonon energy. At lower electric fields the cutoff energy in the electron distribution coincides with the lowest SO phonon energy, which is equal to 34.5 meV. At higher electric fields a fraction of the electrons penetrate beyond the SO phonon energy. However, the electron penetration beyond the LO phonon energy is very weak. The point is that electron scattering by LO phonons in this thick QWI is much stronger than electron scattering by SO phonons [8]. The distribution function at 20 and 200 V/cm has a steep slope at low energies, which can be characterized by a very low temperature (lower than the equilibrium temperature). There are not enough electrons accumulated at the subband bottom to assure the electron cooling effect. It has been shown in Ref. [11] that a simplified treatment of acoustic-phonon scattering leads to the cooling effect due to stronger scattering at the low-energy region and thus to the excess accumulation of electrons at the subband bottom after they have emitted optical phonons.

Figure 20.5 Electron distribution function vs electron energy for three different electric fields. The quantum wire cross section is 150 ¥ 250 Å.

In the near-streaming regime, electrons have “a long memory” since their trajectory is periodically repeated. A powerful technique

Results and Discussion

to recover this memory and to reveal the transition from a diffusive to a streaming regime is the analysis of the current density autocorrelation function [14]. Here we deal with a conservative electron ensemble with a spatially uniform electron concentration. Therefore, fluctuations of current density arise merely from the fluctuations of electron velocity. That is the reason why, instead of analyzing the current density autocorrelation function, we prefer to consider the velocity autocorrelation function given by C(T) = ·dv(t)dv(t + T)Ò ,

(20.6)

where the angular brackets stand for an average over time t, and dv(t) = v(t) – vd is the deviation from the drift velocity vd at time t. The dependence of autocorrelation function on delay time T contains information on all characteristic times of the given conservative electron system. Figure 20.6 shows autocorrelation functions plotted versus the delay time and calculated for different electric fields corresponding to the ohmic electron transport (zero electric field), the superlinear electron transport regime (20 V/cm), and the near-streaming regime (200 V/cm). The characteristic decay time increases as the electric field increases from 0 to 20 V/cm, reflecting the decrease in acoustic-phonon scattering efficiency responsible for a mobility increase at those fields. The negative autocorrelator which appears at 20 V/cm turns to damping oscillations when electrons approach the streaming regime (200 V/cm). The oscillation period coincides with the period of electron motion in k space:

t s = 2hw m */eE + t op , (20.7)

where top is the effective optical-phonon emission time above optical-phonon energy. The characteristic oscillation decay time is mainly defined by the acoustic-phonon scattering rate since electron penetration into, the active region where they can emit optical phonons is negligible at those electric fields. With a further increase of the electric field the efficiency of the acoustic-phonon scattering rate decreases (since electrons spend less and less time in the low-energy region) while their penetration into the active regions gets deeper. Therefore, at sufficiently high electric fields, electron streaming is controlled by penetration through the LO phonon emission threshold rather than by acoustic-phonon scattering. At very low electric fields, in the absence of acoustic-phonon scattering,

271

272

Superlinear Electron Transport and Noise in Quantum Wires

electron streaming is controlled by the lowest-energy SO phonons. Even in this idealized case the streaming oscillations are slightly damped due to the electron penetration beyond the SO phonon energy.

Figure 20.6 Electron velocity autocorrelation function vs delay time at different electric fields for the same quantum wire as in Fig. 20.5.

The calculated autocorrelation functions have been used to calculate the frequency dependencies of the electron diffusion coefficient (velocity noise spectral density) related with the autocorrelation function through the Wiener–Khintchine theorem:

D(w ) =

Ú

•

0

dT e - iw TC(T ). (20.8)

The results are presented in Fig. 20.7. On the vertical scale we plotted the normalized diffusion coefficient, which in fact is merely the normalized noise power spectral density of the conservative electron system. The relationship between the spectral density and diffusion coefficient is given by [15]

S(w) = 4D(w) . (20.9)

The frequency dependence of the diffusion coefficient at zero field has a Lorentzian shape (1 + w2t2)–1, where t is effective scattering time (practically the same Lorentzian dependence is

Results and Discussion

obtained for 20 V/cm, i.e., at the maximum of diffusivity). The critical frequency wc = t–1 increases effectively with the onset of optical-phonon scattering. The effective time of electron scattering by optical phonons is determined primarily by the electric field, i.e., it is equal to the streaming time [the first term of the time given by Eq. (20.7)], because electron penetration into the active region is still negligible [the second term in Eq. (20.7) may be neglected]. At higher electric fields when electron streaming takes through the electron diffusive motion, the peak related to the streaming frequency fs = 1/ts separates from the Lorentzian diffusivity-frequency dependence (see Fig. 20.7). Additional peaks also appear on frequencies that are

Figure 20.7 Electron diffusion coefficient as a function of frequency at various electric fields for the same quantum wire as in Fig. 20.5.

multiples of the streaming frequency. The appearance of these peaks indicates the transition to the streaming regime. Note that critical Lorentzian frequency wc increases as electrons enter the streaming regime. With a further increase in electric field, the peak related to the streaming increases while the plateau of constant diffusivity is going down. The simulation of this regime in the presence of two very different characteristic times (streaming period is much less than the acoustic-phonon scattering time) is very complicated and does not provide good accuracy. In order to study pure streaming

273

274

Superlinear Electron Transport and Noise in Quantum Wires

we have calculated the diffusivity-frequency dependence, ignoring acoustic-phonon scattering. In that case almost all diffusivity (noise) collapses to the streaming frequency and frequencies which are multiples of the streaming frequency.

20.4 Conclusions

We have simulated the electron transport and noise (diffusivity) in rectangular GaAs/AlAs quantum wires at low and intermediate electric fields for T = 30 K by the Monte Carlo technique. Due to heating of the electron system, the efficiency of electron scattering by acoustic phonons decreases and the mobility increases. The decrease of acoustic-phonon scattering efficiency is reflected by the velocity autocorrelation function, which has a longer decay time in that electric-field region. The increase in mobility is also reflected by the velocity-field dependence as a superlinear region. The appearance of enhanced mobility and the corresponding range of electric fields strongly depend on the thickness of a QWI. The point is that the rate of electron scattering by acoustic phonons increases with a decrease in the cross section of a QWI. With a further increase in electric field, acoustic-phonon scattering can no longer hold back electrons from being heated by electric fields. As a result, electrons start to run away even from the subband bottom and the streaming regime is realized. The transition from diffusive electron transport regime to streaming has been studied by analyzing the frequency dependencies of the diffusion coefficient (noise spectral density). It is demonstrated that this transition is reflected on the latter dependence by the appearance of a separate peak at the streaming frequency. The electron thermal noise almost completely collapses to the streaming frequency and its higher harmonics in the pure streaming regime. Similar results have been obtained for temperatures of 20 and 77 K.

Acknowledgments

The work of R. Mickevičius, V. Mitin, and U. K. Harithsa is supported by the National Science Foundation under grant no. ECD 92-46559 and by Army Research Office under grant 29541EL, and the work of

References

D. Jovanovic and J. P. Leburton has been supported by Joint Services Electronics Program under grant no. N00014-90-J-1270.

References

1. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980).

2. D. Jovanovic and J. P. Leburton, Superlattices Microstruct. 11, 141 (1992). 3. B. K. Ridley, J. Phys. C 15, 5899 (1982).

4. B. K. Ridley, Rep. Prog. Phys. 54, 169 (1991).

5. F. Comas, C. Trallero Giner, and J. Tutor, Phys. Status Solidi B 139, 433 (1987).

6. V. Mitin, R. Mickevičius, and N. Bannov, Proceedings of the International Workshop on Computational Electronics, Leeds, 1993, pp. 219–233 (unpublished).

7. S. D. Beneslavski and V. A. Korobov, Sov. Phys. Semicond. 20, 650 (1986). 8. R. Mickevičius, V. Mitin, K. W. Kim, M. A. Stroscio, and G. J. Iafrate, J. Phys. Condens. Matter 4, 4959 (1992).

9. D. Jovanovic and J. P. Leburton, in Monte Carlo Device Simulation: Full Band and Beyond, edited by K. Hess (Kluwer, Dordrecht, 1991), p. 191. 10. Within a more elaborate approach where the qx components are not neglected, the phonon energy is uniquely related to qx, so that one can use this relationship to generate a phonon energy for each value of r.

11. R. Mickevičius, V. Mitin, D. Jovanovic, and J. P. Leburton, in Proceedings of the International Workshop on Computational Electronics, Urbana, II, 1992, p. 293 (unpublished). 12. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983). 13. W. Shockley, Bell Syst. Tech. J. 30, 990 (1951).

14. V. Mitin and C. M. Van Vliet, Phys. Rev. B 41, 5332 (1990).

15. J. Pozela, in Hot Electron Diffusion, edited by J. Pozela (Mokslas, Vilnius, 1981), p. 210.

275

Chapter 21

Importance of Confined Longitudinal Optical Phonons in Intersubband and Backward Scattering in Rectangular AlGaAs/GaAs Quantum Wires

W. Jiang and J.-P. Leburton

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 60801, USA [email protected]

The important role of confined longitudinal optical (LO) and surface optical (SO) phonons is investigated for different types of individual scattering processes in AlGaAs/GaAs quantum wires. Electron wave function tailing due to finite barrier height has been properly taken into account. We demonstrate that for highly confined wires structures Ly = Lz = 40 Å, forward and backward scattering are dominated by SO phonons. For 80 Å ¥ 80 Å structures, forward scattering is still predominately by SO phonons while backward Reprinted from J. Appl. Phys., 74(3), 2097–2099, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

278

Importance of Confined Longitudinal Optical Phonons

scattering is dominated by confined LO phonons. Finally, for 150 Å ¥ 150 Å, confined phonons control both forward and backward scattering. However, we demonstrate that confined LO phonons play a dominant role in intersubband transitions even in highly confined structures, and that it has the most significant effect on the backward scattering in quantum wires of Ly = Lz > 80 Å. Recently, investigations on carrier scattering and carrier capture in one-dimensional (1D) heterostructures have driven much attention because of the expected new size effects as well as potential device applications [1–9]. As in two-dimensional (2D) systems, it is suggested [5] that in 1D quantum wire structures confined longitudinal optical (LO) and surface optical (SO) phonon modes arise due to the sharp discontinuity in dielectric constant near the heterointerfaces. It is also demonstrated [5, 10] that the electron interaction with SO phonons can dominate over that with confined LO phonons in highly confined quantum wires. This point, however, needs to be further investigated, since the conclusion is drawn either from extreme quantum conditions [5] wherein the infinite wall model is assumed and only the ground state is considered, or from analysis of comparison of confined LO and SO phonons based on total scattering rate [10]. The purpose of the present article is to investigate the important role that confined LO and SO phonons interplay in different individual scattering processes. We show that in highly confined quantum wires, confined LO phonons still play a dominant role in intersubband scattering, although SO phonons dominate intrasubband scattering. We also indicate that electron interaction with confined LO phonons is larger than with SO phonons in backward scattering processes for quantum wire width Ly = Lz > 80 Å. We consider a rectangular single GaAs quantum wire embedded in Al0.3Ga0.7As. The electron-phonon scattering rate is estimated from the well known Fermi golden rule:



W

{ }(k ¢ , k ) = 2p | M { } | e a

e a

2 d [E( k ¢ ) - E( k ) ± w ],  (21.1)  where e(a) stands for phonon emission (absorption), E(k) is the electron energy that takes the following form:



E i,j(kx ) =

2 2 2 (k + kyi + kzj2 ),  2m x

(21.2)

Importance of Confined Longitudinal Optical Phonons

where kyi and kzj are quantized electron wave vectors along the two

{} e

transverse y and z directions, respectively, and M a is the matrix element for the electron-optical phonon interaction:

{ } = ·k ¢ , N e a

1 1 1 1 (21.3) ± | He-ph| k , Nq + ± Ò , 2 2 2 2 where Nq is the phonon occupation number, He–ph is the electronconfined LO or SO phonon-interaction Hamiltonian in a rectangular AlxGa1–x As/GaAs quantum wire, which is given in Refs. [5, 10], k and k′ are the electron wave vectors for the initial and final states, respectively. The electron wave functions are obtained by solving the finite barrier Schrödinger equation with an assumption that the potential and the electron wave function can be decoupled along two transverse directions of the quantum wire and that electrons are free to propagate in the longitudinal direction. The total scattering rate of confined LO or SO phonons is calculated by summing over all the possible final subbands and electron wave vectors. We consider three quantum wires with square cross sections. The calculation with finite barrier model indicates that 1, 3, and 8 subbands exist in 40 Å ¥ 40 Å, 80 Å ¥ 80 Å, and 150 Å ¥ 150 Å quantum wires, respectively. The parameters related to these quantum wires are listed in Table 21.1. In Table 21.1, (1,2) and (2,1), (1,3) and (3,1), (2,3), and (3,2) are degenerate states due to the symmetry of the wires. M

q

+

Table 21.1 Parameters related to the three quantum wires Sample

L y = L z( Å )

Subband index

1

40

[1,1]

2 3

80

150

[1,1],[1,2],[2,1]

[1,1],[1,2],[2,1],[2,2],[1,3],[3,1],[2,3],[3,2]

It was demonstrated previously [10] that for GaAs type of phonons the high-frequency symmetric branch (s+) and antisymmetric branch (a+) give rise to much higher scattering rate than the low frequency branches (s– and a–). Therefore, we only need to concentrate on these high-frequency branches. Shown in Figs. 21.1a and b are the inter- and intrasubband scattering rates of confined LO and SO phonons for a 80 Å ¥ 80 Å quantum wire. It

279

280

Importance of Confined Longitudinal Optical Phonons

is seen clearly that the intrasubband scattering rates of SO phonons are larger than that of confined LO phonons. However, (1,2)(2,1) Æ (1,1) and (1,1) Æ (1,2)(2,1) intersubband transitions are completely dominated by confined LO phonons since SO phonon transitions are forbidden by selection rules. The intersubband transition between the degenerate (1,2) and (2,1) states are not affected by this rule. Therefore, energy relaxation toward the ground state is essentially dominated by LO modes in this wire.

Figure 21.1 Intra- and intersubband scattering rates as a function of electron initial energy in a square GaAs/Al0.3Ga0.7As quantum wire of Ly = Lz = 80 Å. (a) Confined LO phonons, (b) SO phonons. T = 300 K.

Importance of Confined Longitudinal Optical Phonons

A comparison of forward and backward scattering rates between confined LO and SO phonons are illustrated in Fig. 21.2a for a 40 Å ¥ 40 Å quantum wire. One can see that SO phonons play a dominant role in both the forward and the backward scattering processes for this cross section. A comparison between the total rate of

Figure 21.2 Forward (FW) and backward (BW) scattering rates as a function of electron initial energy in a square GaAs/Al0.3Ga0.7As quantum wire of Ly = Lz = 40 Å. (a) Confined LO phonons (solid line) and SO phonons (dashed line), (b) Bulk phonons (dashed line) and summation of confined LO and SO phonons (solid line). T = 300 K.

281

282

Importance of Confined Longitudinal Optical Phonons

confined LO and SO phonon modes, and bulk phonons are shown in Fig. 21.2b. It is noticeable that the forward and the backward scattering rates of bulk phonons are in good agreement with those obtained by summing over the confined LO and SO phonons. This conclusion is consistent with the sum rule demonstrated previously for total scattering rates by different authors [10–12].

Figure 21.3 Forward (FW) and backward (BW) scattering rates as a function of electron initial energy in a square GaAs/Alo0.3Ga0.7As quantum wire of Ly = Lz = 80 Å. (a) Confined LO phonons (solid line) and SO phonons (dashed line), (b) Bulk phonons (dashed line) and summation of confined LO and SO phonons (solid line). T = 300 K.

Importance of Confined Longitudinal Optical Phonons

In Fig. 21.3a, we show the rates of forward and backward scattering for a 80 Å ¥ 80 Å quantum wire. It can be seen clearly that SO phonon still dominates the forward scattering. However, the backward scattering is dominated by electron-confined LO phonon interaction. Similar to Fig. 21.2b, Fig. 21.3b reflects again the validity of the sum rule in both the forward and the backward scattering and this characteristic is also seen to occur in Fig. 21.4b.

Figure 21.4 Forward (FW) and backward (BW) scattering rates as a function of electron initial energy in a square GaAs/Al0.3Ga0.7As quantum wire of Ly = Lz = 150 Å. (a) Confined LO phonons (solid line) and SO phonons (dashed line), (b) Bulk phonons (dashed line) and summation of confined LO and SO phonons (solid line). T = 300 K.

283

284

Importance of Confined Longitudinal Optical Phonons

Shown in Fig. 21.4a are the energy dependence of forward and backward scattering for a 150 Å ¥ 150 Å quantum wire. It is shown quite strikingly that the backward scattering rate of confined LO phonons is about two orders of magnitude higher than that of SO phonons. However, for the case of forward processes both confined LO and SO makes approximately the same contribution to the total scattering rate with the rate of confined LO being a little bit higher than that of SO phonons for electron energy below 200 meV. In summary, we have studied the role played alternatively by confined LO and SO phonons in forward, backward, intra- and intersubband scattering in quantum wires. We demonstrated that confined LO phonons play a dominant role in intersubband transitions although SO phonons dominate intrasubband transitions in highly confined quantum wires. In addition, we showed that confined LO phonon is responsible for backward scattering for quantum wire width Ly = Lz > 80 Å. Therefore, for very narrow quantum wires, such as Ly = Lz < 60 Å, momentum relaxation and carrier mobility are essentially determined by scattering with SO modes. We would like to acknowledge Dr. Register for helpful discussions. This work is supported by National Science Foundation under grant no. NSF ECS 91-08300.

References

1. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980). 2. J. P. Leburton, J. Appl. Phys. 56, 2850 (1984).

3. S. Briggs, D. Jovanovic, and J. P. Leburton, Appl. Phys. Lett. 54, 2012 (1989). 4. U. Bockelmann and G. Bastard, Phys. Rev. B 42, 8947 (1990).

5. K. W. Kim, M. A. Stroscio, A. Bhatt, R. Mickevicius, and V. V. Mitin, J. Appl. Phys. 70, 319 (1991). 6. R. Mickevicius, V. V. Mitin, K. W. Kim, M. A. Stroscio, and G. J. Iafrate, J. Phys. Condens. Matter 4, 4959 (1992).

7. P. Selbmann and R. Enderlein, Superlattices Microstruct. 12, 219 (1992). 8. H. Leier, A. Forchel, B. E. Maile, G. Mayer, and J. Hommel, Appl. Phys. Lett. 56, 48 (1990).

References

9. J. Christen, M. Grundmann, E. Kapon, E. Colas, D. M. Hwang, and D. Bimberg, Appl. Phys. Lett. 61, 67 (1992). 10. W. Jiang and J. P. Leburton, J. Appl. Phys. 74, 1652 (1993). 11. N. Mori and T. Ando, Phys. Rev. B 40, 6175 (1989). 12. L. F. Register, Phys. Rev. B 45, 8756 (1992).

285

Chapter 22

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs Quantum Wires

W. Jiang and J.-P. Leburton

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 60801, USA [email protected]

We report on the calculation of the total scattering rate in finite barrier GaAs/AIGaAs quantum wires based on the interaction Hamiltonian of confined longitudinal optical (LO) phonon and surface (SO) phonon modes. With multisubband processes being properly taken into account, our calculation indicates that for GaAs type of phonons the high-frequency symmetric (s+) branch plays an important role among all the other SO phonon branches; it can even dominate over confined LO phonons in highly confined quantum wires as observed by Kim et al. [22]. Our results also demonstrate Reprinted from J. Appl. Phys., 74(3), 1652–1659, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

that the total contributions of confined LO and SO phonon scattering resemble closely to GaAs bulk LO phonon scattering. Selection rules between intersubband transitions for SO modes suggest the possibility of a bottle-neck effect for carrier relaxation in square wires compared with rectangular wires.

22.1 Introduction

In the past few years, a great deal of effort has been devoted to the growth and microfabrications of one-dimensional (1D) semiconductor structures [1–3]. These wire-like structures have received considerable attention in both fundamental and applied physics because of the discovery of new physical phenomena and their potential applications in high-speed and efficient optoelectronic devices [2–7]. In order to have a better understanding of the physical processes involved in these devices, a clear picture of carrier scattering and carrier relaxation in 1D quantum wire structures must be established. Theoretically, an accurate determination of the scattering and the relaxation of carriers directly depends upon . the modeling of the electron-phonon interaction and upon the effects of confinement on both carriers and phonons. In undoped polar media, the scattering by polar optical phonons (POP) plays a dominant role at room temperature. Several theoretical approaches have been used to model optical phonon modes and their interaction with electrons. It is well known that in bulk ionic crystals the electron-longitudinal optical (LO)-phonon interaction can be described by the Fröhlich Hamiltonian, the derivation of which is based on the dielectric continuum model, also referred to as the macroscopic model. When applied to ionic slabs or quantum wells this model shows that the presence of dielectric interfaces produces confined LO phonons as well as surface optical (SO) phonons which are localized in the vicinity of the interfaces and decay exponentially into the material [8–11]. This approach was further extended by Stroscio [12] to describe confined LO phonons, SO phonons, and their interaction with electrons in rectangular quantum wires. More recently, an alternative, microscopic approach has been proposed by Huang and Zhu [13, 14] to deal with phonon modes in quantum well and quantum wires structures. Their results are mostly consistent with those of the macroscopic approaches mentioned above.

Model

A wide range of new features due to carrier confinement effects in quantum wires have been studied extensively over the past few years with bulk LO phonon approximation [15–19]. It was also demonstrated that the electron interaction with confined LO phonons shows a weakening when compared to that with bulk LO phonons [20], and indicates that the electron interaction with SO phonons can be larger than with confined LO phonon in certain circumstances [21–23]. Experimental evidence of phonon confinement [24] and the enhancement of the electron-SO-phonon interaction [25] has also become available recently. This issue is essentially important for transport in realistic wires where the subband separation is of the order of the thermal energy. In particular, the recent demonstration of intersubband optic phonon resonances when the optic phonon matches the subband separation and induces drop in the wire conductance needs to be rexamined on the basis of a realistic phonon model [26]. In contrast to the intensive studies on confined LO phonon modes and SO phonon modes in 1D heterostructures [12, 14, 27, 28], little work has been reported [22, 23] on the scattering rates in quantum wires involving the confined and interface optical phonons. Kim et al. [22] have calculated intrasubband scattering rates for quantum wires with square cross sections and infinite potential wells. In their calculation, only the ground state of electrons and lowest confined LO phonon mode were considered. In this chapter, we extend Kim and Stroscio’s treatment to more realistic conditions: we consider finite potential wall, multisubband processes, and higher confined LO phonon modes for rectangular and square quantum wires. By using the same electronic model, we also compute the total scattering rate of bulk LO phonons. We show that the total scattering rate calculated by summing over the contributions of all the possible confined LO and SO phonons is similar to the total rate obtained with GaAs bulk LO phonons.

22.2 Model

In this analysis, we consider a rectangular single GaAs quantum wire with a dielectric constant, ϵ1(w), embedded in Al0.3Ga0.7As with a dielectric constant, ϵ2(w). To compute the electron wave functions in this structure, we assume that the potential and the

289

290

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

electron wave functions can be decoupled along the two transverse y and z directions and that the electrons are free to propagate in the x direction. The wave functions in the two transverse directions are obtained separately by solving the finite barrier Schrödinger equation with properly matched boundary conditions along each direction, which gives rise to sine and cosine functions inside the wire with exponentially decaying tails outside the wire. Plane waves are used to describe the free propagation along the longitudinal direction. The electron energy has the well-known parabolic form in the effective mass approximation:

h2 2 2 (k + kyi + kzj2 ), (22.1) 2m x where kyi and kzj are quantized electron wave vectors along the two transverse y and z directions, respectively, and kx is the electron wave vector along x direction; m is the effective mass of GaAs. In this work we are mostly interested in the semiclassical expression of the scattering rates which are relevant for hightemperature transport of dilute electron gas [16, 17] or high-energy carrier relaxation as investigated in femtosecond spectroscopy in quantum structure [29], for instance. Quantum corrections to the present picture only reduce and broaden the divergence in the rates due to the singularity in the density of states without altering the global rate profile as a function of the carrier energy [30]. The transition probability of electron-POP-interaction is estimated in first-order perturbation theory by the Fermi golden rule: E i,j(kx ) =



e 2p | M {a } |2 d [E( k ¢ ) - E( k ) ± hw ], (22.2) h where k and k′ are the electron wave vectors for the initial and final states, respectively, w is the phonon frequency, e(a) stands for {e } phonon emission (absorption), and M a is the matrix element for the electron-optical-phonon interaction:





e

W {a } ( k ¢ , k ) =

e

M { a } = · k ¢ , Nq +

1 1 1 1 ± | He-ph| k , Nq + ± Ò , 2 2 2 2

(22.3)

where q is the phonon wave vector with frequency w, Nq is the phonon occupation number, and He-ph is the Hamiltonian for electron-

Model

optical-phonon interaction. The total scattering rate t–1 is obtained by summing over all the possible final subbands and electron wave vectors. In 3D systems, the Fröhlich Hamiltonian for the electron interaction with bulk LO phonons is written as 3D HLO =



ÂV e q

q

- iq◊r

(aq + a-+q ),

(22.4)

where aq and a-+q are the phonon annihilation and creation operators and È e2 hw Ê 1 1 ˆ˘ Vq = Í ˜˙ 2Á ÍÎ 20 Wq Ë (• ) (0) ¯ ˙˚



1/ 2

,

(22.5)

where W is the volume of the crystal, ϵ(•) and ϵ(0) are the highand low-frequency dielectric constants, respectively, and ϵ0 is the permittivity of vacuum. By applying appropriate electronic boundary conditions to the 3D Fröhlich Hamiltonian in the two transverse directions, Stroscio [12] has derived an interaction Hamiltonian for 1D rectangular quantum wires,

1D HLO = 2a ¢

+

 qx

Â

Ï Ô Ê mp ˆ Ê np ˆ Ô cos Á y ˜ cos Á z Ë Lz ˜¯ Ë Ly ¯ iq x x Ô e [ A(qx ) + A+ ( -qx )] Ì 1/ 2 2 2˘ Ôm=1,3,5, . . . n=1,3,5, . . . È Ê ˆ Íq2 + mp + Ê np ˆ ˙ Ô Í x ÁË Ly ˜¯ ËÁ Lz ¯˜ ˙ Ô Î ˚ Ó

Â

Â

Ê mp ˆ Ê np cos Á y ˜ sin Á Ë Lz Ë Ly ¯

m=1 ,3,5, . . . n=2,4 ,6 , . . . È

+

Â

Â

Â

ˆ z˜ ¯

1/2 2˘

2

[ A(qx ) + A+ ( -qx )]

Ê ˆ Íq2 + mp + Ê np ˆ ˙ Í x ÁË Ly ˜¯ ÁË Lz ˜¯ ˙ Î ˚ Ê mp ˆ Ê np sin Á y ˜ cos Á Ë Lz Ë Ly ¯

m=2,4 ,6 , . . . n=1 ,3,5, . . . È

2

ˆ z˜ ¯

1/2 2˘

Ê ˆ Íq2 + mp + Ê np ˆ ˙ Í x ÁË Ly ˜¯ ÁË Lz ˜¯ ˙ Î ˚

[ A(qx ) + A+ ( -qx )]

291

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Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

¸ Ô Ô Ô + + [ A(qx ) + A ( -qx )]˝ , 1/ 2 2 2˘ Ô m=2,4 ,6 , . . . n=2,4 ,6 , . . . È Ê ˆ Íq2 + mp + Ê np ˆ ˙ Ô Í x ÁË Ly ˜¯ ÁË Lz ˜¯ ˙ Ô Î ˚ ˛  (22.6)

Â

where

Â

Ê mp sin Á Ë Ly

ˆ Ê np y ˜ sin Á Ë Lz ¯

ˆ z˜ ¯

1/2

ÏÔ e2 hw l1 Ê 1 1 ˆ ¸Ô a¢ = Ì Á ˜˝ ÓÔ 20 Ly Lz L Ë 1 (• ) 1 (0) ¯ ˛Ô

,

ϵ1(•) and ϵ1(0) are the high- and low-frequency dielectric constants inside the quantum wire, Ly and Lz are the wire widths along the y and z directions, L is the wire length, wl1 is the frequency of the confined LO phonons in the quantum wire, m and n are the phonon mode numbers in the y and z directions, qx is the phonon wave vector along the x direction, A+(–qx) and A(qx) account for the phonon creation and annihilation operators, which satisfy the following equations:

A+ ( -qx )| N- q x ,m,n Ò = ( N- q x ,m,n + 1)1/2 | N- q x ,m,n + 1Ò , A(qx )| Nq x ,m,n Ò = ( Nq x ,m,n )1/2 | Nq x ,m,n - 1Ò.

(22.7) (22.8)

Stroscio et al. [22, 31] have also calculated SO phonon modes in rectangular quantum wires by using approximations similar to those used for the rectangular dielectric waveguides [32]. The interaction Hamiltonian for SO modes can thereby be written as follows [22]:

1D = HSO

 qx

Ê h ˆ ( -e )F(qx , y , z )e iq x x Á Ë 2w ˜¯

+ ( -qx )], ¥ [ ASO (qx ) + ASO

1/ 2



(22.9)

+ where ASO (qx ) and ASO(–qx) are the creation and annihilation operators for SO phonons, respectively, and F(qx,y,z) is the potential for the SO phonon modes. For symmetric-symmetric SO mode, this potential is given by the following expressions [22]:

Model

cosh( b z ) Ï cosh(a y ) Ô cosh(a L / 2) cosh( b L / 2) , | y | £ Ly / 2, | z | £ Lz / 2, y z Ô Ô cosh(a y ) b L /2 - b|z| , | y | £ Ly / 2, | z | ≥ Lz / 2, e z e ÔÔ 2 cosh( a / ) L y FS (qx , y , z ) = CS Ì Ô a L /2 -a| y| cosh( b z ) , | y | ≥ Ly / 2, | z | £ Lz / 2, Ôe y e cosh( b Lz / 2) Ô Ô a Ly /2 -a| y| b Lz /2 - b|z| e e e , | y | ≥ Ly / 2, | z | ≥ Lz / 2, ÔÓe



(22.10)

where CS is the normalization constant which is given in Ref. [11]. For antisymmetric-antisymmetric SO mode: cosh( b z ) Ï cosh(a y ) Ô cosh(a L / 2) cosh( b L / 2) , | y | £ Ly / 2, | z | £ Lz / 2, y z Ô Ô sin(a y ) b Lz /2 - b|z| e e , | y | £ Ly / 2, | z | ≥ Lz / 2, ÔÔsgn( z ) sinh(a Ly / 2) F A (qx , y , z ) = C A Ì Ô sinh( b z ) a L /2 -a | y| , | y | ≥ Ly / 2, | z | £ Lz / 2, Ôsgn( y )e y e sinh( b Lz / 2) Ô Ô a Ly /2 -a | y| b Lz /2 - b|z| e e e , | y | ≥ Ly / 2, | z | ≥ Lz / 2, ÔÓsgn( yz )e



(22.11)

where sgn(argument) is a sign function which takes the sign of the argument, CA is a normalization constant which can be obtained by using the same method as in Ref. [11] for deriving CS. CA satisfies the following relation: L-1C A-2 = 01 (• )



2 2 w l1 - w t1

(w

2

2 2 - w l1 )

[sinh(a Ly / 2)sinh( b Ly / 2)]-2

È a Ê sinh(a Ly )sinh( b Lz ) Lz ˆ - sinh(a Ly )˜ Í Á ab a ¯ ÍÎ 2 Ë 2

2 2 ˆ˘ - w t2 w l2 b 2 Ê sinh(a Ly )sinh( b Lz ) Ly sinh( L ) ( )   b + • ˙ 0 2 z ˜ Á 2 2 ab b (w 2 - w t2 ) ¯ ˙˚ 2 Ë

+

[sinh(a Ly / 2)]-2

ÏÔ È sinh(a Ly ) Ê sinh(a Ly ) Ly ˆ ˘ ¸Ô w2 - w2 ¥ Ìa 2 Í + b2 Á - ˜ ˙ ˝ + 02 (• ) 2l2 2t2 2 ab ab b ¯ ˙˚ Ô (w - w t2 ) Ë ÓÔ ÍÎ ˛ [sinh( b Lz / 2)]-2

293

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Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

Ê sinh( b Lz ) Lz ˆ ˘ Ô¸ w2 - w2 ÔÏ È sinh( b Lz ) ¥ Ìb 2 Í + a2 Á - ˜ ˙ ˝ + 02 (• ) 2l2 2t2 2 ab a ¯ ˙˚ Ô˛ ab Ë (w - w t2 ) Ô ÍÎ Ó Ê 2(a 2 + b 2 ) ˆ  Á (22.12) ˜, ab Ë ¯

where the subscript l and t represent the longitudinal- and transverseoptical phonon modes. In Eqs. (22.10–22.12), the parameters a and b are defined by the following expressions:

Èa 2 + b 2 - qx2 = 0 Í ÍÎ a Ly = b Lz .

(22.13)

ϵ1(w)tanh(aLy/2) + ϵ2(w) = 0

(22.14)



The dispersion relation for the frequency can be written as



ϵ1(w))coth(aLy/2) + ϵ2(w) = 0



for symmetric-symmetric SO modes and

(22.15)

for antisymmetric-antisymmetric SO modes. The frequencies of symmetric-symmetric and antisymmetric-antisymmetric SO phonon modes can be calculated from Eqs. (22.13–22.15) along with the Lyddane–Sachs–Teller relation. Figure 22.1 shows the frequencies of the SO modes as a function of aLy for GaAs type of phonons in GaAs/ Al0.3Ga0.7As rectangular quantum wires.

Figure 22.1 Dispersion of symmetric (s±) and antisymmetric (a±) SO phonon modes as a function of aLy in a GaAs/Al0.3Ga0.7As rectangular quantum wire. The ± superscripts indicate the high- and low-energy modes, respectively.

Results and Discussions

It was demonstrated previously [33] that there exist two types of phonon modes, GaAs type and AlAs type, in the Al0.3Ga0.7As systems. In this paper, we concentrate on the GaAs type of phonons because the Ga concentration is high in the system we are considering. The relevant material parameters are listed in Table 22.1 [33]. Table 22.1 Relevant material parameters in GaAs/AlGaAs quantum wires Parameters

GaAs

AlxGa1–xAs

Dielectric constants ϵ(w)

10.89

10.89 – 2.73x

GaAs type LO phonon hwLO (meV)

GaAs type TO phonon hwTO (meV)

36.25

33.29

36.25 – 6.55x + 1.79x2

33.29 – 0.64x – 1.16x2

22.3 Results and Discussions In previous works, the calculations of transition rates in quantum wires were based essentially on the infinite wall approximation with electron completely confined inside the quantum wire [22]. Our calculation properly incorporates the tailing of electron wave functions into the barrier for a practical finite potential height in GaAs/AlxGa1–xAs quantum wire structures. A comparison of the intrasubband scattering rates between these two models is shown in Figs. 22.2a and b for confined LO phonons in quantum wires with square cross sections. The peak at the emission threshold is the well-known result of the singularity in the 1D density of final scattering states as described by the semiclassical theory [15]. The intrasubband scattering rates calculated with the infinite wall model in Fig. 22.2a are basically the analogs of Kim et al. [22] except that higher phonon modes, i.e., m, n = 1, 2, . . ., 12 are included in our calculation. In Fig. 22.2a one can see that the scattering rates increase monotonically as the wire widths decrease from 1000 to 40 Å. Shown in Fig. 22.2b are the scattering rates calculated with the finite potential well model. It is noticeable that the scattering rates increase as the wire widths decrease from 1000 to 160 Å, a feature resembling closely to that in Fig. 22.2a. However, as the wire widths continue to decrease from 160 to 40 Å, the scattering rates start to decrease because of the penetration of the electron wave function into the barrier as we pointed out earlier. Consequently, the infinite

295

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Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

wall model is only valid for quantum wire width up to 160 Å; for narrow quantum wire, this model considerably overestimates the confined phonon scattering.

Figure 22.2 Intrasubband LO-phonon emission rates for the ground state as a function of the electron initial energy Ex in the longitudinal direction of square quantum wires. (a) Infinite wall model for electrons; starting from top to bottom, the wire widths are 40, 80, 120, 160 (solid lines), 200, 300, 500, and 1000 Å (dashed lines). (b) Finite wall model for electrons; starting from top to bottom, solid lines correspond to 160, 120, 80, and 40 Å wire widths, respectively; dashed lines represent 200, 300, 500, and 1000 Å wire widths, respectively. T = 300 K.

Results and Discussions

According to the interaction Hamiltonian discussed previously, numerical calculations have been performed on four different quantum wire structures. The parameters related to the four quantum wires are shown in Table 22.2. Table 22.2 Parameters related to the four quantum wires Sample

Ly(Å)

Lx(Å)

No. of subbands

A1

80

80

3

B2

106.8

300

12

A2 B1

54

150

160

150

3

8

The quantum wire widths were chosen such that the two highly confined structures (Al and A2) and the two less confined structures (Bl and B2) have approximately the same ground-state subband bottoms, respectively. The results for the four quantum wires are illustrated in Figs. 22.3–22.6. In Fig. 22.3a we show the scattering rates as a function of the electron initial energy for a square quantum wire of 80 Å ¥ 80 Å. The calculation with finite barriers indicates that there are only three subbands, E11(kx), E12(kx), and E21(kx) in the wire, with the last two being degenerate owing to the symmetry of the wire. The subband bottoms are located at E11(0) = 83 meV and E12(0) = E21(0) = 202 meV, respectively. The threshold for the scattering rates of both bulk LO phonon (solid line) and confined LO phonon (dotted line) occurs at the subband bottom of the ground state. It can be seen clearly that the energy dependence of the scattering rates exhibits sharp peaks due to the divergence of the 1D density of states at the subband bottom. The peaks a, b, c, d, and e are interpreted as being intrasubband absorption to the lowest subband E11(kx), intrasubband emission to E11(kx), intersubband absorption to the degenerated subband E12(kx) and E21(kx), intrasubband absorption to E12(kx) and E21(kx), intrasubband emission to E12(kx) and E21(kx), respectively. Comparing the results of these two phonon models, one notes that the scattering rates of the confined LO phonons are the analogs of those of bulk LO phonons with the former lying consistently lower than the latter.

297

298

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

Figure 22.3 Scattering rates in a square GaAs/Al0.3Ga0.7As quantum wire (Ly = Lz = 80 Å), (a) Total scattering rates as a function of electron initial energy. Solid line: bulk LO phonons; dotted line: confined LO phonons; dash dot line: SO phonons; dashed line: summation of all the confined LO and SO modes, (b) Intra- and intersubband scattering rates as a function of electron initial energy. Dotted line: intrasubband transitions; solid line: E12(kx) to E11(kx) transitions; dashed line: E11(kx) to E12(kx) transitions. T = 300 K.

The scattering rates of SO phonons are plotted with dash dotted lines. In this highly confined quantum wire, the scattering rates of s+ phonons is higher than that of confined LO phonons. For GaAs type of SO phonon modes considered here, the contribution to the total

Results and Discussions

scattering rate comes mostly from the high-frequency symmetric SO phonon branch (s+ on the curve). The contributions of the lowfrequency symmetric SO phonon branch (s–) as well as the high- and low-frequency antisymmetric SO phonon branches (a+ and a–) are relatively small. The reason for this behavior can be understood as follows. It is known that the SO phonon scattering rate is proportional to the square of the normalization factor CA (CS) [see Eqs. (22.2), (22.3), (22.9–22.11)], which is approximately proportional to 2 (w 2 - w t1 ) [Eq. (22.12)]. From examination of Fig. 22.1 and Table 22.1 we see that the frequencies of s– and a– branches are very close to wt1, while those of s+ and a+ branches are relatively far away from wt1, giving much higher scattering rate for high-frequency branches than for low-frequency branches. Moreover, the respective parities of the phonon modes determine scattering selection rules between electronic states. Hence, intrasubband transitions which are generally stronger than intersubband transitions in quantum wires are forbidden for antisymmetric modes, which enhances s+(s–) phonon scattering compared to a+(a–) phonon scattering. This is easily seen in Fig. 22.3a where a± phonon scattering is only effective above 200 meV, i.e., at the onset of intersubband scattering. The dependence of normalization constant CA(CS) on the phonon frequency and the selection rule for a± modes are the main reason for s+ phonon scattering dominating over other interface modes. The scattering selection rules mentioned above have a profound effect on individual electronic transitions. For instance, by comparing the scattering rates of the different modes on Fig. 22.3a, one notices that the peak c due to intersubband absorption on the curve of confined or bulk LO phonons vanishes for s± modes precisely due to the selection rule between electronic states of different parities. It should be noted that the reverse process, i.e., intersubband phonon emission from the excited subband to the bottom subband is also forbidden, although the peak corresponding to this transition is not directly seen in Fig. 22.3a because of the smooth density of final 1D states away from the subband bottom. More details of the individual transitions due to confined LO phonon scattering are shown in Fig. 22.3b. It can be seen that the rate of intersubband phonon emission from the excited state {1,2} to the ground state {1,1} is about 4 ¥ 1011 s–1 (solid line), which is relatively low compared to the rate for intrasubband phonon emission of (1,1 Æ 1,1). In addition,

299

300

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

as indicated earlier SO phonon scattering, which plays an important role in the overall relaxation rate, is forbidden for the (1,2 Æ 1,1) transition. Therefore, hot carriers can be trapped in the excited state {1,2} for a relatively long time compared to the intrasubband relaxation time before relaxing to the ground states. This may lead to a bottle-neck effect in the carrier relaxation for this highly confined quantum wire with square cross section. Another interesting feature also emerges from our results in Fig. 22.3a: the summation extending over the scattering rates of both confined LO phonon modes and all the surface phonon modes as indicated by the dashed line is in good agreement with the electronGaAs bulk LO-phonon scattering rate. This result demonstrates that the reduction of electron-confined LO-phonon scattering is compensated by the enhancement of the electron-SO-phonon scattering. This conclusion agrees with the sum rule demonstrated by Mori and Ando [11] within the dielectric continuum model for single and double heterostructures and generalized by Register [34] with a fully microscopic phonon model for heterostructures of arbitrary geometry. However, it should be noted that the sum rule derived in these previous works applies only to form factors within the scattering rates. Exact compensation between interface and confined LO-phonon scattering is predicted only when differences in SO-phonon mode eigenfrequencies can be neglected. Thus, the variation in the interface frequencies from the bulk (Fig. 22.1) may account for the small discrepancy between the total rate obtained here and that for GaAs bulk LO phonons. Shown in Fig. 22.4a is the scattering rate as a function of energy for a rectangular quantum wire of 54 Å ¥ 160 Å. The scattering rates of the energy dependence reflect again the behavior of 1D density of states, and this characteristic is also seen to occur in Fig. 22.5 and Fig. 22.6. Comparing Fig. 22.4a with Fig. 22.3a, one can see that the onset of the scattering rates in Fig. 22.4a starts at the same energy as in Fig. 22.3a. Although there are three subbands in both the square and the rectangular wire, there are more transition peaks in Fig. 22.4a than in Fig. 22.3a because the change of the wire shape from square to rectangular lifts the degeneracy of the second and third subband in the former. Figure 22.4a indicates clearly that the antisymmetric SO phonon scattering has completely vanished due to the selection rule. It can also be seen that the electron interaction

Results and Discussions

with SO phonons is still dominant over that with confined LO phonon in most of the energy range of the figure, although this effect is not as pronounced as in Fig. 22.3a. In addition, like in Fig. 22.3a, the

Figure 22.4 Scattering rates in a rectangular GaAs/Al0.3Ga0.7As quantum wire (Ly = 54 Å, Lz = 160 Å). (a) Total scattering rates as a function of electron initial energy. The meaning of the different line types is the same as in Fig. 22.3a. (b) Intra- and intersubband scattering rates as a function of electron initial energy. Dotted line: intrasubband transitions; solid line: E13(kx) to E11(kx) transitions; dashed line: E12(kx) to E11(kx) transitions; dash dot line: E13(kx) to E12(kx) transitions. T = 300 K.

301

302

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

sum rule is also seen to hold approximately for the rectangular wire. Similar to Fig. 22.3b, Fig. 22.4b gives more insights of individual transitions due to confined LO phonons. A careful examination of hotcarrier relaxation processes in Fig. 22.4b indicates that the phonon emission rates for the (1,3 Æ 1,2) and the (1,3 Æ 1,1) transitions are around 3.2 ¥ 1011 – 5 ¥ 1011 s–1. Therefore, the scattering rate for hot electrons to relax from the {1,3} state to lower subbands is about 8 ¥ 1011 s–1. The scattering rate for the (1,2 Æ 1,1) transition is of the same magnitude and very close to the ground-state intrasubband scattering rate of confined LO phonons. Besides, SO phonons also contribute to the relaxation toward the ground state (1,3 Æ 1,1) relaxation. Hence, the bottle-neck effect mentioned in Fig. 22.3a is reduced significantly by changing the wire shape from square to rectangular.

Figure 22.5 Total scattering rates as a function of electron initial energy in a square GaAs/Al0.3Ga0.7As quantum wire (Ly = Lz = 150 Å). The meaning of the different line types is the same as in Fig. 22.3a. T = 300 K.

Shown in Figs. 22.5 and 22.6 are the energy dependence of the scattering rates for the two wider quantum wires of 150 Å ¥ 150 Å and 106 Å ¥ 300 Å. It should be noted that the rates in these two wires are higher than those in smaller ones. This feature, at least for the square wire, is consistent with what has been shown in Fig. 22.2b for intrasubband scattering, where the POP emission rate is larger for wire width of 150 Å than for that of 80 Å. Unlike the case

Results and Discussions

in highly confined wires, electron-confined LO-phonon scattering plays a dominant role over the entire energy region in both of these less confined wires, which reduces the importance of the bottle-neck effect. Finally, it is worth emphasizing that the sum of the reduced electron-confined LO-phonon interaction and of the enhanced electron-SO-phonon interaction gives rise to a total interaction which is equivalent to the interaction with bulk LO phonons. This result is best exhibited in Fig. 22.5 and is roughly valid in Fig. 22.6 and demonstrates that in quantum wires the total scattering rate calculated by using electron-GaAs bulk LO-phonon interaction is still a very good estimate of the overall POP scattering processes, should the confinement on the carriers be properly taken into account. Therefore we do not expect significant changes in the transport parameters when they are calculated with confined phonon modes and interface modes instead of bulk modes [19]. However, this conclusion does not apply to physical processes which result from particular individual scattering events where rates of confined LO and SO phonons deviate substantially from those of bulk phonons. This is especially the case for intersubband transitions where the SOphonon scattering is forbidden, leading to a situation of relaxation bottleneck in narrow quantum wires.

Figure 22.6 Total scattering rates as a function of electron initial energy in a rectangular GaAs/Al0.3Ga0.7As quantum wire (Ly = 106 Å, Lz = 300 Å). The meaning of the different line types is the same as in Fig. 22.3a. T = 300 K.

303

304

Confined and Interface Phonon Scattering in Finite Barrier GaAs/AlGaAs QWRs

22.4 Conclusion We have discussed different size effects of electron-confined LOphonon scattering in GaAs/AlGaAs quantum wires by comparing the results obtained with infinite wall and finite wall models for the carriers. In contrast to the case of the infinite wall model, where the scattering rates increase as the wire widths decrease, intrasubband scattering rates based on the finite wall model show a maximum value for quantum wire widths Ly = Lz ≃ 160 Å, and decrease for lower value of these parameters. In the case of square and rectangular quantum wires with finite potential heights, we have also demonstrated that for GaAs type of SO phonons, the s+ phonon branch plays an important role among all the other phonon branches. For highly confined quantum wires, the s+ phonon scattering can even dominate over the confined LO-phonon scattering as it was observed by Stroscio et al. [22]. Moreover, we suggested the possibility of a bottle-neck effect in the carrier relaxation when SO-phonon intersubband scattering is forbidden due to the selection rule in transitions between subbands of different symmetries. We also indicated that the bottle-neck effect may be reduced by changing the wire shape. Finally, we have shown that the electron-bulk LO-phonon scattering resembles closely to the total contribution of the electronconfined LO-phonon and SO-phonon scattering with multisubband processes taken into account. This result agrees with the sum rule proposed previously by Mori and Ando in 2D systems. This analysis of the scattering rates in quantum wires. provides a significant improvement on more simplistic approaches based on bulk phonons or infinite barrier models for calculating the relaxation constants and transport properties in quantum wires.

Acknowledgments

We would like to acknowledge Dr. M. Stroscio and Dr. Register for helpful discussions. Special thanks are due to Dr. von Allmen for reviewing part of the manuscript and Dr. Bigelow for helping the preparation of the graphs. This work is supported by the National Science Foundation under grant no. NSF ECS-91-08300 and the Army Research Office under grant no. DAAL-03-91-6-0052.

References

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Chapter 23

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

Mueen Nawaz,a Jean-Pierre Leburton,a and Jianming Jinb

aBeckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 and Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA bDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 1406 W. Green St., Urbana, Illinois 61801, USA [email protected]

The authors provide a theoretical analysis of the influence of confinement on the hole-optical phonon interaction in freestanding Si nanowires. Their model based on deformation potential optical phonons shows that in narrow quantum wires, hole scattering is less frequent than when calculated with a bulk (continuous) phonon spectrum. In addition, scattering by confined optical phonons results in dissimilar rates for backward and forward events, in contrast with continuous phonons. Reprinted from Appl. Phys. Lett., 90, 183505-1–183505-3, 2007. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2007 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

308

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

In quantum wires (QWRs), the free motion of carriers is restricted to one dimension (1D), whereas there is a twofold quantum confinement in the other transverse directions. This situation limits available states for scattering, which tends to increase carrier mobility [1]. At the same time, quantum confinement enhances the interaction between carriers and lattice vibrations, which increases their scattering rates [2]. Transport properties in 1D semiconductors are thus the result of a subtle interplay between the reduced density of states and the enhanced carrier-phonon interaction [3]. In 2003, Cui et al. reported a higher than bulk hole mobility in silicon (Si) nanowires of 2.5 nm radius [4]. Those results tend to support the dominant influence of the reduced density of states over the enhanced carrier-phonon interaction. However, a recent theoretical study by Kotlyar et al. on the electron-phonon interaction in Si QWRs reached the opposite conclusion and found a lower electron mobility [5]. Far from being academic, this question has important technological consequences since Si QWRs can be used as active channels in nano-field-effect transistors [4]. Recently they have also been shown to be effective as DNA detectors [6]. In this chapter, we investigate hole scattering by optical phonons through the deformation potential interaction, which is one of the main mechanisms for dissipation at high field and at room temperature [7]. We consider freestanding cylindrical silicon nanowires of radius R, length L, and with the z axis along the QWR direction (see inset, Fig. 23.1). We assume an infinite cylindrical well potential profile for the holes (i.e., V = 0 inside the wire and V = • on the radial edges). Solving Schrödinger’s equation inside the nanowire provides the following hole wave functions:

y n,m =

e ik z z e inf J n (kn,m r ) p R2 LJ n¢2 ( jn,m )

,

(23.1)

where Jn(x) and J′n(x) are the nth order Bessel function and its derivative, respectively. kn,mR = jn,m is the mth zero of the nth order Bessel function. L is the length of the nanowire. The hole 2 eigenenergies are given by E = h2 (kz2 + kn,m )/ 2m*, where m* is the hole effective mass.

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

It has been shown that the effective mass of holes in silicon QWRs grown in the [100] direction can vary dramatically as the radius shrinks [8]. Furthermore, the band structure of nanowires of radius 2.6 nm and higher is not far removed from that of bulk silicon [9]. Additionally, it has been shown that once surface reconstruction has been taken into account, the hole effective masses are bounded above by the 0.49m0 mass that we assume for the radii in question [10]. In this work, the smallest nanowire treated has a radius of 3.5 nm.

Figure 23.1 Variation of the minimum heavy hole energy within a subband as a function of radius for the first six least energetic subbands. The dashed curve represents a light hole subband, and the solid curves represent heavy hole subbands. Inset: Geometry of the nanowire.

In Fig. 23.1 we plot the first few QWR subband edges where we distinguish between heavy and light hole subbands. We note that the lowest two subbands are heavy hole subbands that remain separated by an energy difference larger than kBT at room temperature for radii less than 5.2 nm. At larger radii, the separation between the subbands decreases dramatically and approaches a continuum, as is to be expected in bulk materials. Therefore for QWRs with a 5 nm radius or less, we can assume extreme quantum limit and neglect intersubband scattering at room temperature—restricting ourselves to the lowest heavy hole band. This assumption is strengthened by decreasing the effective mass. We obtain the square of the matrix elements for hole-optical phonon interaction given by [11]

309

310

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

Ê h | D |2 ˆ | ·y n ¢ ,m ¢| HOP| y n,m Ò |2 = Á ˜ Ë Mw OP ¯ 2

Ê ˆ 2I1 ¥Á ¢ ˜ ¢ Ë J n ¢ ( jn ¢ ,m ¢ ) J n ¢ ( jn,m ) ¯



Ê 1 1ˆ ¥ Á Nq ± ˜ d k ¢ , k + q . Ë 2 2¯ z z z

(23.2)

Here HOP = u ◊ D [12], where D is the vector deformation potential and is assumed to be isotropic. In deriving this result, a continuous phonon model was assumed, where u assumes a plane wave form. M is the mass of the nanowire, Nq is the phonon occupation number, and qz is the longitudinal component of the phonon wave vector. hwOP = 63 meV is the optical phonon energy [13]. For bulk Si (continuous) phonons, I1 is a geometry dependent form factor given by

I1 =



n kctz =

Ú

1

0

J n ¢- n (q^ Rx ) J n ¢ ( jn ¢ ,m ¢ x ) J n ( jn.m x )xdx ,

(23.3)

where q^ is the transverse component of the phonon wave vector. Using Fermi’s golden rule for the transition probability, and integrating over all the possible phonon states and final hole states, one obtains the scattering rate in extreme quantum limit (m = n = 0 in the lowest subband) [14] 2 | D |2 2m * Ahrw OP

00 I00



Ú =

1

0

00 I00

Ez ± hw OP

J04 ( j0,0 x )xdx

[ J0¢ ( j0,0 )]4

1 1ˆ Ê ÁË Nq + 2 ± 2 ˜¯ ,

= 4.8365,

(23.4a)

(23.4b)

where A is the cross-sectional area and Ez = h2kz2/ 2m * . The ±hwOP term in Eq. (23.4a) is either for phonon absorption or emission. For comparison, the bulk scattering rate for holes with wave functions y bulk = e ik ◊ r / V is given by

n bulk =

| D |2 m * 2m * p h2 r hw OP

E 1 1ˆ Ê ± 1 Á Nq + ± ˜ , Ë hw OP 2 2¯

(23.5)

which for 1D intrasubband rate [Eq. (23.4)] yields the following ratio:

Hole Scattering by Confined Optical Phonons in Silicon Nanowires



n bulk n nanowire

=

ˆ Am * w OP Ê E ± 1˜ . 00 Á hw ¯ 2p hI00 Ë OP

(23.6)

Through the A factor, this ratio shows a R2 dependence and at E ª 0; where only phonon absorption occurs, it is less than 1 for R £ 4.9 nm, confirming that in the extreme quantum limit, hole scattering by optic phonons is enhanced in QWRs compared to bulk materials, in agreement with Kotlyar et al. [5]. Lattice vibrations in confined geometries exhibit different spectra than in bulk materials [15]. Indeed, the presence of near surfaces or interfaces imposes boundary conditions to the phonon system, which results in a discrete number of standing modes [16]. Hence, interaction with charge carriers is restricted to discrete phonon wavelengths, which tends to reduce the scattering rate compared with bulk phonons characterized by a continuous spectrum. This is especially true for intrasubband scattering in QWRs, for which the form factor is maximum in the limit of long phonon wavelengths [17]. The situation arises in the Si valence band, which, unlike the six X-valley conduction band with anisotropic effective mass, has single valley G25 symmetry with isotropic mass, which favors intrasubband scattering. In the hydrodynamic continuum approximation, the components (ur, uf, uz) of the ion displacement u are given by [18, 19]

ucm = Cstm e isf e iqz z Fcm (r ).

(23.7)

c represents the coordinates r, f, or z. m refers to any of the three possible phonon modes (longitudinal and two transverse modes, denoted henceforth by LO, TO1, and TO2). The expressions for Fcm are given in Table 23.1. The s and t indices label the phonon modes, and Cst is the phonon normalization constant [12, 17]. The vector u is continuous at the center of the cylindrical nanowire and is subject to the condition that — ◊ u = 0 inside the cylinder for longitudinal modes, which leads to the boundary condition: uf(R) = uz(R) = 0. For transverse modes, — ◊ u = 0 inside and ur(R) = 0. These boundary conditions lead to an infinite set of solutions qst and q′st corresponding to the zeros of Js(qstr) and J′s(q′str), respectively. The square of the interaction matrix element is

311

312

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

cf M(qz ) = | ·y n ¢ ,m ¢| HOP | y n,m Ò |2

=

( Nq + (1/2) ± (1/2))| Cstm |2 d k ¢ ,k z

z +qz

( J n¢ ¢ ( jn ¢ ,m ¢ ) J n¢ ( jn.m ))2

È Ê Í 2 2 m 2 mÁ  ¥ Í4I0,1 , Dz g d n ¢ ,n+s + | D | h d n ¢ ,n+s -1 Á Í Ë Î



2

ˆ ˆ Ê s Ê s ¥ Á m I0,0 + I1,1 ˜ + d n ¢ ,n+s+1 Á m I0,00 - I1,1 ˜ ¯ ¯ Ë jst Ë jst Ii,j =



Ú

1

0

2ˆ ˘

(23.8a)

˜˙, ˜¯ ˙ ˙˚

m J n ¢ ( jn ¢ ,m ¢ x ) J n ( jn.m x ) Js(i )( js,t x )x jdx ,

(23.8b)

Table 23.1 Fcm (r ) for each c component (ur, uf, uz) and each of the m phonon modes ucm

LO

TO1

TO2

z–

Js(qstr)

−q′st/qz)Js(q′str)

0

f– r–

s/(qzr)Js(qstr)

(s/q′str)Js(q′str)

−i(qst/qz)J′s(qstr)

J′s(qstr)

−iJ′s(q′str)

−i(s/qstrJs(qstr)

 = D + iD (components of D), and J (i )( x ) is the ith derivative with D x y s m with respect to x. js,t refers to either the ith zero of the sth order Bessel function, or the zero of its derivative (Table 23.2). The Kronecker deltas impose selection rules which greatly reduce the actual calculations. The expressions for gm and hm are given in Table 23.2. m Table 23.2 gm, hm, and corresponding js,t zeros

gm

hm

LO

1

m [ js,t /(Rqz )]2

TO2

0

TO1

m [ js,t /(Rqz )]2

1

1

m js,t

Bessel

Bessel derivative Bessel

We readily derive the expression for the total scattering rate by using Fermi’s golden rule summing over all final hole states and all phonon modes:

Hole Scattering by Confined Optical Phonons in Silicon Nanowires



n kcfz =

L m* 2h2

 q st

Ê M( q + ) + M( q - ) ˆ z z Á ˜. w h E ± Ë z OP ¯

(23.9)

Note that L in Eq. (23.9) cancels as the matrix element terms m M(qz± ) contain the crystal volume in their denominator via Cs,t , and ± 2 1/ 2 qz = ±(2m *(Ez ± hw OP )/h ) - kz . The ± superscript refers to the square root sign, whereas the inner ± denotes phonon absorption or emission. The sum over the phonon modes is finite, terminating at qst = p/2a0, as a higher qst would imply that the wavelength of the atomic displacement a0 is shorter than the interatomic distance. Figure 23.2 shows the ratio of the confined phonon emission scattering matrix elements normalized to the bulk (continuous) phonon emission scattering matrix element as a function of qz. In contrast with bulk phonons, the forward and backward scattering rates are actually different for the same qz. At large qz, all curves converge to unity and the more rapid the convergence is the larger the nanowire radius. We observe that the normalized value of the scattering rates remains less than unity, indicating a weaker scattering than obtained with the bulk (plane wave) phonon model.

Figure 23.2 Ratio of the square of the scattering matrix elements for confined phonons and continuous phonons in nanowires with different radii as a function of qz.

Figure 23.3a shows the normalized scattering rates for confined phonon emission (i.e., n kcfz / n kctz ) in nanowires of different radii as a function of the hole kinetic energy along the QWR. For all QWR

313

314

Hole Scattering by Confined Optical Phonons in Silicon Nanowires

radii, the normalized rates are smaller than unity and decrease with smaller radii, again indicating that a weaker scattering is expected in the presence of confined phonons. The effect weakens at large radii, where the normalized rates approach unity, converging toward the continuous phonon values as expected. In Fig. 23.3b we display the ratio between forward and backward scattering rates for different radii calculated with the confined phonon model as a function of the hole kinetic energy along the QWR. The diagram shows a stronger tendency for holes in smaller nanowires to backscatter than in wider nanowires.

Figure 23.3 (a) Ratio of the confined phonon emission rate to the continuous phonon emission rate in nanowires of different radii. (b) Ratio of the forward emission scattering rate to the backward emission scattering rate with confined phonons as a function of hole energy for different QWR radii.

In conclusion, we calculated the hole scattering rates for deformation potential interaction with confined optic phonons in freestanding QWRs, and found considerably lower rates than if calculated with a continuous phonon spectrum, especially at small radii. In addition, we showed that confined phonons favor backward scattering over forward scattering in narrow QWRs. This work was supported by the Network for Computational Nanotechnology under NSF grant no. EEC-0228390 and SRC-CSR grant no. G00513. The authors are indebted to M. Stroscio for his helpful discussions.

References

References 1. H. Sakaki, Jpn. J. Appl. Phys., Part 2 19, L735 (1980). 2. J.-P. Leburton, J. Appl. Phys. 56, 2850 (1984).

3. J.-P. Leburton, Phys. Rev. B 45, 11022 (1992).

4. Y. Cui, Z. Zhong, D. Wang, W. U. Wang, and C. M. Lieber, Nano Lett. 3, 149 (2003).

5. R. Kotlyar, B. Obradovic, P. Matagne, M. Stettler, and M. D. Giles, Appl. Phys. Lett. 84, 5270 (2004). 6. J. Hahm and C. M. Lieber, Nano Lett. 4, 51 (2004).

7. While acoustic phonons are important in the scattering of carriers, this chapter focuses on scattering by deformation potential optical phonons, and acoustic phonon scattering is the subject of future research. 8. Y. Zheng, C. Rivas, R. Lake, K. Alam, T. B. Boykin, and G. Klimeck, IEEE Trans. Electron Devices 52, 6 (2005). 9. J. Wang, A. Rahman, A. Ghosh, G. Klimeck, and M. Lundstrom, IEEE Trans. Electron Devices 52, 7 (2005). 10. T. Vo, A. J. Williamson, and G. Galli, Phys. Rev. B 74, 045116 (2006). 11. F. Szmulowicz, Phys. Rev. B 28, 5943 (1983).

12. M. Stroscio, M. Dutta, D. Kahn, and K. Kim, Superlattices Microstruct. 29, 405 (2001). 13. J. M. Hinckley and J. Singh, J. Appl. Phys. 76, 4192 (1994).

14. Note that the scattering rate decreases with decreasing mass. Hence, by using heavy hole mass, our treatment places an upper bound on the scattering rate. As shown by Vo et al. (Ref. [10]), the actual mass is smaller. 15. M. A. Stroscio and M. Dutta, Phonons in Nanostructures (Cambridge University Press, Cambridge, UK, 2001), pp. 60–90. 16. W. Jiang and J.-P. Leburton, J. Appl. Phys. 74, 1652 (1993).

17. M. Nawaz, MS thesis, University of Illinois at Urbana-Champaign, 2005. 18. N. C. Constantinou and B. K. Ridley, Phys. Rev. B 41, 10622 (1990). 19. H. P. W. Gottlieb, J. Acoust. Soc. Am. 81, 5 (1986).

315

Part II

Carbon Nanotubes and Nanoribbons

Chapter 24

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

Marcelo A. Kuroda,a,b Andreas Cangellaris,c and Jean-Pierre Leburtona,c aBeckman

Institute, University of Illinois at Urbana-Champaign, Illinois 61801, USA bDepartment of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA cDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Illinois 61801, USA [email protected]

We show that the local temperature dependence of thermalized electron and phonon populations along metallic carbon nanotubes is the main reason behind the nonlinear transport characteristics in the high bias regime. Our model is based on the solution of the Boltzmann transport equation considering both optical and Reprinted from Phys. Rev. Lett., 95, 266803-1–266803-4, 2005. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2005 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

zone boundary phonon emission as well as absorption by charge carriers. It also assumes a local temperature along the nanotube, determined self-consistently with the heat transport equation. By using realistic transport parameters, our results not only reproduce experimental data for electronic transport but also provide a coherent interpretation of thermal breakdown under electric stress. In particular, electron and phonon thermalization prohibits ballistic transport in short nanotubes. Carbon nanotubes (CN) are one-dimensional (1D) nanostructures that have stimulated broad research interest because of their unique electrical versatility as semiconductors or metals, depending on their chirality [1]. From a technological viewpoint, their remarkable electrical and mechanical properties make them promising materials for applications in high performance nanoscale electronic and mechanical devices [2, 3]. Among these properties, the interrelation between electronic and thermal transport in these quasi-1D structures is particularly interesting. Early experiments on nonlinear transport in metallic single walled nanotubes (m-SWNTs) using low resistance contacts revealed current saturation at the 25 mA level, a fact attributed to the onset of electron backscattering by high energy optical (OP) and zone boundary (ZB) phonons in the high bias regime [4]. More recently, series of independent high-field transport measurements on various length m-SWNTs demonstrated the absence of current saturation by achieving currents over 60 mA in short samples (≲55 nm), which was interpreted as ballistic transport along the CN [5, 6]. Thermal breakdown and burning under high electric stress were also observed in SWNT. In the interim, the electrical conductance of multiwalled nanotubes (MWNTs) under high bias has shown a steplike decrease caused by the successive burning of the CN outer shells [7, 8]. In these experiments CNs burned unexpectedly at midlength under stress even on a substrate and on the presence of a back gate in a field effect device geometry [9]. Despite various attempts to model these systems [4–6], up to now no coherent interpretation has been proposed that reconciliates heat dissipation and electronic transport and describes thermal effects in m-CNs under electric stress. In this chapter, we show that the nonlinear characteristics of metallic CNs find their origin in the nonhomogeneous Joule heating along the nanotube, which is caused by the thermalized distribution

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

of electrons scattered by high energy phonons, even in short m-SWNT. We specifically show that Joule heating (temperature) is maximum at CN midlength. Moreover, owing to the 1D nature of the structure, Joule heating increases drastically at constant current with the CN length, resulting in thermal breakdown at lower electric fields than in shorter CNs. Our model is based on the Boltzmann transport equation with OP and ZB phonon scattering and solved selfconsistently with the heat transfer equation, providing a coherent interpretation of electric and thermal transport in m-SWNTs, in agreement with experimental data [5, 6]. In particular, we show that the high current level in short CNs is not due to ballistic transport but to reduced Joule heating. We use the linear dispersion relation of electronic states around the Fermi level [ϵ(k) = ±huFk] [10], where uF is the Fermi velocity. Thus, the Boltzmann equation reads

eF a ∂ f a ( ) = C ph ( I ,T ). (24.1) h k Here f a(ϵ), F, e, and Cph are the distribution function for the a energy branch, the electric field, the electron charge, and the electron-phonon collision integral, respectively. The index a denotes the energy branches with positive (+) and negative (–) Fermi velocity in the first (1) and second valleys (2) of the m-CN. In metallic systems high electron density and strong intercarrier scattering thermalizes the electron distribution. We therefore assume that the electron distribution function f a(ϵ) obeys Fermi-Dirac statistics with a local electronic temperature Tel(x): uF ∂x f a ( ) +



f a ( ) = 1 / {1 + exp[( - Fa )/ kBTel ( x )]},



Fa

(24.2)

where is the quasi Fermi level of branch a. As a result, the collision a integral C ph also depends on the position. We neglect acoustic phonons that are only relevant in the low bias regime and consider the contribution of high density OP and ZB phonons [11, 12] as playing a central role in energy dissipation in the high bias regime. As illustrated in Fig. 24.1a, the different processes (interbranch and intrabranch) considered in Cph include both the emission and absorption of these phonons with energy (hwop ª 0.2 eV) much larger than thermal fluctuations at room temperature. Each of these phonons contributes to the collision integral as follows:

321

322

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

Figure 24.1 (a) Scattering processes considered in the calculations. The intravalley (left) and intervalley (right) transitions with emission and absorption of optical phonons are included. (b) IV characteristics for different constant temperatures along the tube. Dashed line: asymptotic behavior for all temperatures in the high bias regime.

a ( I ,T ( x )) = C ph

 i,b

i ÔÏ Re b a Ì { f (k )[1 - f (k - q)] ÔÓ p

- f a (k )[1 - f b (k - q)]}

+

Rai b { f (k )[1 - f a (k + q)] p ¸Ô f a (k )[1 - f b (k + q)]}˝ , Ô˛



(24.3)

where the index b runs over the two branches and two valleys, i stands for OP and ZB phonons, and Rei (Rai ) is the phonon emission (absorption) rate. Hence the first (second) two terms in Eq. (24.3) correspond to processes involving the emission (absorption) of a phonon limited by Pauli exclusion principle. For instance, the first term describes a process in which an electron scatters from a state

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

in branch b with momentum k to a state in branch a with momentum k – q by emitting a phonon. In all these processes both total energy and momentum are conserved. The emission and absorption rate coefficients are given by

Rai (TL ) =

Rei (TL ) =

Nq ti

1 1 , (24.4) t i exp( hw / kBTL ) - 1

=

Nq + 1 ti

= Ra exp( hw / kBTL ), (24.5)

where TL is the lattice temperature and 1/ti stands for the bare scattering rate for OP and ZB phonons that we assume to be independent of carrier energy in a first approximation. In computing the collision integral, we make the key “ansatz” that electrons and lattice are in local thermal equilibrium [i.e., TL = TL(x) = Tel(x)] [13]. We define the electron density as

na =

1 p

Ú

+•

- k ( Ec )

f a (k )dk , (24.6)

where Ec is the bottom of the conduction band, and the current as I = e(n+ – n–)uF,

(24.7)



(24.8)

where the + (–) index corresponds to the branches with positive (negative) Fermi velocity. Then, integrating Eq. (24.1) over the momentum for each branch, and properly accounting for different branches, we obtain



uF¶x(n+ – n–) = 0,

u F ∂ x ( n+ + n- ) -

2eF = 2 dkC ph ( I ,T ( x )) p (24.9)  = 2C ph ( I ,T ( x )).

Ú

Equation (24.8) is the expression of the current conservation ± ± in the system, in which by symmetry F1 = F2 and with charge + neutrality in the CN yields F(1,2) = -F(1,2) . Integrating Eq. (24.9) over the length of the nanotube L and assuming equal electron densities at the contacts, we find the voltage drop VDS along the nanotube to be

VDS = -

p e

Ú

L/2

- L/2

dxC ph ( I ,T ( x )). (24.10)

This equation implicitly depends on the current and the temperature profile along the nanotube. In order to obtain both of

323

324

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

these quantities, the equation must be solved self-consistently with the heat transport equation. As a particular case, it is interesting to compute the IV relation from Eq. (24.10) by assuming only OP phonon scattering at constant temperature [i.e., T(x) = T0] in the CN. The results are plotted in Fig. 24.1b, where the high bias regime exhibits an asymptotic behavior independent of temperature which is given by

VDS ( I ) =



-k

1 L ( I - Iwop ), (24.11) G0 uFt op

where Iwop = ew op / p is the threshold current corresponding to the onset of electron backscattering by OP phonons [4], and G0 = 2e2/h is the quantum conductance. We should point out that Eq. (24.11) is not consistent with the current interpretation of electrons accelerated ballistically in the electric field until acquiring enough energy to emit a phonon, but rather results from the imbalance between the population of the energy branches with positive and negative Fermi velocities. When considering also ZB phonons, the voltage drop in the m-SWNT is a linear combination of expressions similar to Eq. (24.11), and it still results in a threshold current, only with a more complicated expression. When heating effects become relevant, thermal dissipation is taken into account self-consistently with Eq. (24.10). We consider two mechanisms for heat dissipation: (i) diffusion through the supporting substrate (if the CN stands on one) and (ii) flow through the contacts. Hence, by defining DT = T(x) – T0 (where T0 is the temperature of the substrate and leads), the heat equation becomes [14] d 2 DT

+ gDT = q*, (24.12) dx 2 where k is the thermal conductivity, g is the coupling coefficient with the substrate, and q* is the power dissipated per unit volume. Here we make the usual approximation that process (i) is proportional to the local temperature difference between CN and substrate. In our calculations both the thermal conductivity and the coupling coefficient are assumed to remain constant along the tube. The coefficient g is given [14] by g = ksub/(td), where ksub, t, and d are the thermal conductivity of the substrate, the diameter of the nanotube,

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

and the thickness of substrate, respectively. We also assume that the power is homogeneously generated along the CN and given by Joule’s law: q* = jF,



(24.13)

where j = I/A is current density through the effective cross section A and the electric field F is given by F = |VDS/L|. Then the solution for the temperature profile is given by

DT ( x ) =

q* È cosh( Gx ) ˘ Í1 ˙ , (24.14) g LS Î cosh(GL / 2) ˚

where G = g / k . Different scenarios can take place depending on the value GL (see the inset of Fig. 24.2). On the one hand, diffusion through the substrate is negligible for GL ≪ 1 and the temperature profile exhibits a parabolic shape. On the other hand, for GL ≫ 1, heat basically dissipates through the substrate and the temperature is almost constant along the CN. This latter situation occurs in long tubes strongly coupled to the substrate. In all cases, the highest temperature point is at the middle of the tube.

Figure 24.2 Temperature difference between the middle of the tube and the leads as a function of the current for different CN lengths [5]. Inset: temperature profile along the NT for different values of GL.

In our calculations we use T0 = 300 K. The energies of OP and ZB phonons are hwop = 0.20 eV and hwzb = 0.16 eV, respectively. We use the standard accepted value for the thermal conductivity k (30 W/

325

326

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

cmK) [15] and g = 1011 W cm–3 K–1. Figure 24.2 shows the temperature difference in the middle of the tube [DT(0)] as a function of I for different CN lengths, corresponding to the data of Ref. [5]. The longer the nanotube, the faster the rise in temperature as the threshold current is overcome. This is due to the fact that dissipation occurs over a longer distance while heat removal mainly takes place at the contacts in 1D structures. Estimates for the breakdown temperature correspond to 800°C [16]. Therefore short tubes are expected to carry larger currents before thermal breakdown. As shown in Fig. 24.3, the results for the IV characteristics are in good agreement with the experimental data [5, 6]. Deviations in the low bias regime

Figure 24.3 Comparison between theoretical and experimental IV characteristics for different CN lengths: (a) Ref. [5] and (b) Ref. [6].

are mainly due to the absence of acoustic phonon scattering in our model. For the sake of simplicity, we assume relaxation times for OP and ZB phonons with equal values, which are t = (13 ± 2) fs for the first [5] and t = (6.9 ± 1.5) fs for the second [6] set of experimental data. The difference between the two values could be due to the fact that CNs may have different diameters with different phonon spectra (breathing modes) [17] in each case and contact quality. Nevertheless, the obtained mean free paths (between 6 and 10 nm) are consistent with previous experimental estimates [5, 6]. For long tubes (i.e., for L = 700 nm in the first case [5] and for L ≥ 1000 nm in

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

the second case [6]) we need to increase the relaxation times to (27 ± 3) fs to fit the experimental data. Since at the same current level, the temperature is considerably higher in long CNs, these longer times may be associated with the emergence of nonlinear thermal effects not taken into account in Eq. (24.12), or to the temperature and geometry dependence of the thermal conductivity at room temperature or higher in m-SWNTs, which remains an open issue [18, 19]. Finally, Fig. 24.4 shows the IV characteristics obtained for CNs of different diameters and lengths, assuming that both the thermal conductivity and the relaxation time are equal among tubes. Despite this strong assumption, the relevant issue to emphasize here is the weak dependence of the IV characteristics on the size of the CN. Appreciable deviations can be observed only in the 700 nm tube (dashed line) for small diameters (~1 nm). This weak relation is consistent with the interpretation of Collins et al. [7, 8] for the breakdown under electrical stress observed in MWNTs, where different layers in a MWNT (separated by about 0.4 nm) carry similar currents in the high bias regime. The breakdown of successive carbon layers produces approximately constant diminutions of the current in the high bias regime because IV characteristics are geometry independent. Moreover, the highest temperature arises at the CN midlength (inset of Fig. 24.2) and therefore electrical breakdown is, as experimentally observed, expected to take place there as well.

Figure 24.4 IV characteristics for different tube lengths and diameters. Results are practically independent of the diameter for short tubes (100 and 300 nm); appreciable deviations appear for the smallest diameter (1 nm) in the 700 nm tube.

327

328

Nonlinear Transport and Heat Dissipation in Metallic Carbon Nanotubes

In conclusion, we have shown that the consideration of a thermalized electron distribution in local equilibrium with a nonhomogeneously heated lattice through OP and ZB scattering determined self-consistently by the current level accounts for the nonlinear IV characteristics of the m-SWNTs in the high bias regime. The magnitude of the temperature variation as a function of the CN length is consistent with the occurrence of thermal breakdown at midlength for long CN under electrical stress. While the dependence of thermal conductivity on temperature still remains under investigation, our self-consistent model provides a coherent picture of the onset of thermal effects with electronic transport in m-SWNT.

Acknowledgments

The authors are indebted to E. Pop and H. Dai for technical comments. This work was supported by the Beckman Institute for Advance Science and Technology and NSF-Network of Computational Nanotechnology.

References

1. R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998).

2. P. L. McEuen, M.S. Fuhrer, and H. Park, IEEE Trans. Nanotechnol. 1, 78 (2002).

3. Ph. Avouris, J. Appenzeller, R. Martel, and S. Wind, Proc. IEEE 91, 1772 (2003). 4. Z. Yao, C. L. Kane, and C. Dekker, Phys. Rev. Lett. 84, 2941 (2000).

5. A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, and H. Dai, Phys. Rev. Lett. 92, 106804 (2004).

6. J. Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. Üstünel, S. Braig, T. A. Arias, P. W. Brouwer, and P. L. McEuen, Nano Lett. 4, 517 (2004). 7. P. G. Collins, M. Hersam, M. Arnold, R. Martel, and Ph. Avouris, Phys. Rev. Lett. 86, 3128 (2001). 8. P. G. Collins, M. S. Arnold, and P. Avouris, Science 292, 706 (2001).

9. R. S. Muller and T.I. Kamins, Device Electronics for Integrated Circuits (John Wiley & Sons, New York, 2002), 3rd ed.

References

10. R. A. Jishi, D. Inomata, K. Nakao, M. S. Dresselhaus, and G. Dresselhaus, J. Phys. Soc. Jpn. 63, 2252 (1994). 11. R. Saito, T. Takeya, T. Kimura, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 57, 4145 (1998).

12. V. Perebeinos, J. Tersoff, and Ph. Avouris, Phys. Rev. Lett. 94, 027402 (2005). 13. This ansatz is relatively satisfied here as the temperature gradient is not very important because, as shown in Fig. 24.2, in short CNs the temperature does not increase as drastically with current as in long CNs, and for the latter the temperature profile is relatively flat except at the edge, with minor consequences.

14. C. Durkan, M. A. Schneider, and M. E. Welland, J. Appl. Phys. 86, 1280 (1999). 15. J. Che, T. Cagin, and W. A. Goddard III, Nanotechnology 11, 65 (2000).

16. M. Radosavljevic, J. Lefebvre, and A. T. Johnson, Phys. Rev. B 64, 241307 (2001). 17. B. J. LeRoy, S. G. Lemay, J. Kong, and C. Dekker, Nature (London) 432, 371 (2004).

18. T. Yamamoto, S. Watanabe, and K. Watanabe, Phys. Rev. Lett. 92, 075502 (2004).

19. J. X. Cao, X. H. Yan, Y. Xiao, and J. W. Ding, Phys. Rev. B 69, 073407 (2004).

329

Chapter 25

Joule Heating Induced Negative Differential Resistance in Freestanding Metallic Carbon Nanotubes

Marcelo A. Kurodaa,b and Jean-Pierre Leburtona,c

aBeckman Institute, University of Illinois at Urbana-Champaign, Illinois 61801 bDepartment of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA cDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Illinois 61801, USA [email protected]

The features of the IV characteristics of metallic carbon nanotubes (m-CNTs) in different experimental setups are studied using semiclassical Boltzmann transport equation together with the heat dissipation equation to account for significant thermal effects at high electric bias. The model predicts that the shape of the m-CNT characteristics is basically controlled by heat removal mechanisms. In particular, the authors show that the onset of negative differential Reprinted from Appl. Phys. Lett., 89, 103102-1–103102-3, 2006. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2006 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Joule Heating Induced Negative Differential Resistance

resistance in freestanding nanotubes finds its origins in strong transport nonlinearities associated with poor heat removal unlike in substrate-supported nanotubes. Since their discovery [1], carbon nanotubes (CNTs) have captured the attention of both the scientific and technological communities because of their mechanical stability as well as their high thermal and electrical capabilities [2] not usually seen in other materials. Among their outstanding properties is the peculiarity to behave as metals or semiconductors depending on their chirality [3, 4]. Their capability of carrying large current densities at room temperature make them prominent materials for field effect transistors [5, 6] and interconnects [7] in high speed nanoscale electronics. The first measurements of the IV characteristics on individual metallic carbon nanotubes [8] (m-CNTs) were performed on substrate-supported tubes in a configuration similar to a field effect transistor. The output characteristics resembled those obtained in semiconductors as they exhibited linear dependence on voltage in the low field regime, and saturation at I ~ 25 mA under high biases. Later measurements have shown that in short m-CNTs, the saturation level could be overcome, but a nonlinear behavior still persists in substrate-supported configurations [9, 10]. In recent experiments performed on m-CNTs standing freely across a trench, current levels in the high bias regime were a few times smaller than those in nanotubes supported by substrates while negative differential resistance (NDR) was reported [11]. This behavior, indicating significant heat production/ dissipation in m-CNT devices, was interpreted in terms of the onset of a nonequilibrium optical phonon population. In this chapter we show that the nonlinearities in the IV characteristics of m-CNTs find their origins in Joule heating and the efficiency in heat removal from the m-CNT. We determine the temperature profile along nanotubes by solving the Boltzmann transport equation simultaneously with the heat transfer equation. We specifically demonstrate that while the nonlinear behavior in substrate-supported tubes (SSTs) (Fig. 25.1a) emerges due to a strong imbalance between carrier distribution functions with positive and negative Fermi velocities, the characteristics in freestanding tubes (FSTs) (Fig. 25.1b) essentially arise from an inhomogeneous selfheating effect.

Joule Heating Induced Negative Differential Resistance

Figure 25.1 Experimental setups: (a) substrate-supported tube (SST) and (b) freestanding tube (FST). (c) IV characteristics for 800 nm long nanotube with different values of coupling coefficient g. (d) Temperature in the middle of the tube DT(0) as a function of the current I for different g.

The distance between subbands in m-CNTs using tight-binding calculations can be estimated as DEmet ª 6g0a/D, where a = 0.14 nm, g0 ª 3 eV, and D is the diameter of the tube [12]. Hence, for transport purposes, the electronic structure of small diameter m-CNTs close to the Fermi level can be well described by linear dispersion relations E(k) = ±huF(k – kF) with a Fermi velocity uF close to 8.107 cm/s. This approximation is valid as long as temperature satisfies that kBT ≪ DEmet and the bias voltage V < DEmet. The current along the nanotube is naturally defined as I = euF(n+ – n–),

n+

(n–)

(25.1)

where is the electron density with positive (negative) Fermi velocity. Similar to Ref. 13, we assume that carrier populations are described by Fermi statistics with different quasi-Fermi level E Fa depending on the sign a of the Fermi velocity. Using the zeroth moment of the Boltzmann equation, the electric field F satisfies

u F ∂ x ( n+ + n- ) -

2eF = 2C ph ( I ,T ). p

(25.2)

333

334

Joule Heating Induced Negative Differential Resistance

Here C ph ( I ,T ) is the momentum integral of the collision integral considering only the contribution to scattering involving high energy optical phonons (hw ~ 0.18 eV). In our model, we use the ansatz that phonons are in local thermal equilibrium with electrons, and their occupation number follows the Bose–Einstein statistics [14]. The nonequilibrium high field conditions set the system in a diffusive regime inducing inhomogeneous heating. Hence we account for the local temperature variation by solving the heat production/ dissipation equation:

-

d Ê dT ˆ k + g (T - T0 ) = q*, dx ÁË dx ˜¯

(25.3)

in which k, g, T0, and q* are the thermal conductivity, the thermal coupling to the substrate, the substrate temperature, and the heat dissipation per unit volume, respectively. The first term on the left hand side (LHS) of Eq. (25.3) corresponds to the heat flowing through the leads; the second, to the flow driven through the substrate or any surrounding medium. The heat produced locally is assumed to obey Joule’s law q* = IF/ pDt, being t the effective thickness of the tube (t ª 0.34 nm). We use k(T) = kRTTRT/T for T ≳ 300 K [11, 15]. In this equation, kRT is the thermal conductivity at a reference temperature TRT (k ~ 20 W/cm K at 300 K). An estimate of the thermal coupling constant g of nanotubes standing on SiO2 substrates [13] is 1011 W/cm3 K. The explicit dependence of the temperature on the local field requires Eqs. (25.2) and (25.3) to be solved simultaneously, leading to a nonlinear second order differential equation for the temperature. Neglecting any thermal contact resistance (which might be questionable in short nanotubes), we set the boundary conditions as T(±L/2) = T0, where L is the m-CNT length, assuming that both contact leads have the same temperature as the substrate. In Fig. 25.1c we compare the performances of an 800 nm long m-CNT assuming different g values. The parameters used in the calculations are k = 20 W/cm K, t =22 fs, and hw = 0.18 eV. For the smallest g values, the m-CNT thermal coupling to the substrate is vanishingly small (less than 10% of the power generated along the tube is dissipated in the high bias through the substrate) and NDR is observed in the IV characteristics. As we increase g to ~1011 W/ cm3 K saturation is observed. If we further increase g, the saturation is overcome but deviations from the linear regime still arise. Our

Joule Heating Induced Negative Differential Resistance

results clearly indicate that in the low bias regime, thermal effects can be neglected, but the high bias regime is basically governed by the m-CNT capability for heat removal. Hence, the features of the electrical characteristics depend not only on the tube’s length but also on g. In Fig. 25.1d we plot the temperature difference DT(0) = T(0) – T0 at the m-CNT midlength as a function of the current for the same set of g values. Our model predicts that the larger the coupling coefficient g, the higher current in the m-CNT. We obtain similar results for nanotubes of different lengths. In Fig. 25.2, we consider the situation of an 800 nm long m-CNT in two thermal coupling regimes, i.e., (i) strongly coupled (SST with g = 1012 W/cm3 K, left hand side of Fig. 25.2) and (ii) weakly coupled [FST with g = 109 W/cm3 K, right hand side (RHS) of Fig. 25.2] to the substrate. In the SST, the IV characteristic exhibits a nonlinear behavior with high resistance at high fields (Fig. 25.2a LHS). In this case, the substrate acts as a heat sink, providing efficient dissipation of the heat produced along the tube while keeping both the electric field and power dissipated per unit volume low. This efficient heat removal reduces the temperature rise in the m-CNT, thereby favoring relatively high current levels while maintaining a quasiuniform temperature profile along the tube (except at the edges), as shown in the LHS of Fig. 25.2b. The LHS of Fig. 25.2c displays the profile of the (Fermilike) carrier distributions at the CNT midlength for electrons with positive and negative Fermi velocities f + and f –) at the different points indicated along the IV curve in Fig. 25.2a (LHS). No significant heating is observed before the separation between the two distribution quasi-Fermi levels becomes comparable to the optical phonon energy 5.eps Hence, the nonlinear behavior observed in this kind of configuration is basically attributed to the smooth onset of optical phonon scattering [8, 13], despite minor thermal broadening induced by Joule heating on the carrier distributions. In the FST, the IV characteristic exhibits a NDR at relatively low biases (Fig. 25.2a RHS), and the current level is lower than in the previous case of strong thermal coupling. Because of poor heat removal, temperature rises rapidly with external biases manifesting a pronounced maximum at CNT midlength (RHS of Fig. 25.2b). One notices the different scales on the voltage and temperature axis between the SST (LHS) and FST (RHS). Simultaneously, the Fermi distributions f + and f – remain weakly separated but experiencing

335

336

Joule Heating Induced Negative Differential Resistance

significant heating even at low voltages as shown in Fig. 25.2c (RHS) for the set of points along the IV characteristic on Fig. 25.2a (RHS). The fast rise of temperature, consequence of the poor heat removal, locally broadens the electron distributions, and therefore enhances the scattering locally even at low bias. The rapid onset of nonuniform T profile induces nonlinearities in the electric field, which result in a current diminution in order to satisfy the boundary conditions on the applied voltage at the CNT contacts.

Figure 25.2 Features comparison of 800 nm long m-CNT strongly (left) and weakly (right) coupled to the substrate: (a) IV characteristics, (b) temperature profile DT(x) as a function of the voltage V, and (c) Fermi distribution at midlength for each of the carrier branches at different points indicated in the IV characteristics. The distance between dashed line corresponds to the optical phonon energy.

In Fig. 25.3 we compare the results of our model for FSTs with the experimental data of Ref. [11], for which a good agreement is observed for all CNT lengths. The physical parameters used in the

Joule Heating Induced Negative Differential Resistance

calculations are hw = 0.18 eV and kRT = 20 W/cm K. The relaxation time (t ~ 30 fs) used to compute the characteristics increases with diameter consistently with theoretical predictions [16] and lies within previous experimental estimates [8–10]. Our model also assumes the presence of a contact resistance (Rc ~ 20 kW).

Figure 25.3 Comparison between the IV characteristics for FSTs of different lengths calculated with our model (lines) and experimental data (symbols) of Ref. [11].

In conclusion our model shows that the high field transport properties of m-CNTs are strongly controlled by the onset of thermal effects, which can be altered by modifying the experimental setup. Using realistic parameters, our model is able to reproduce the IV characteristics in both substrate-supported and freestanding nanotubes. However, the nonlinear behavior observed in each of the systems is caused by different phenomena affecting the carrier distribution. In the former, the transport features are due to an imbalance in the carrier population with positive and negative Fermi velocities associated with strong scattering and high current levels. In the latter, the NDR is attributed to strong scattering enhancement due to the broadening of the carrier distribution with low current level.

Acknowledgments

This work was supported by the Beckman Institute and Network for Computational Nanotechnology under NSF grant no. ECC-0228390.

337

338

Joule Heating Induced Negative Differential Resistance

References 1. S. Iijima, Nature (London) 354, 56 (1991).

2. R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998).

3. J. W. Mintmire, B. I. Dunlap, and C. T. White, Phys. Rev. Lett. 68, 631 (1992). 4. R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Appl. Phys. Lett. 60, 2204 (1992). 5. S. J. Tans, M. H. Devoret, H. Dai, A. Thess, R. E. Smalley, L. J. Geerligs, and C. Dekker, Nature (London) 386, 474 (1997).

6. R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, Appl. Phys. Lett. 73, 2447 (1998). 7. P. L. McEuen, M. S. Fuhrer, and H. Park, IEEE Trans. Nanotechnol. 1, 78 (2002). 8. Z. Yao, C. L. Kane, and C. Dekker, Phys. Rev. Lett. 84, 2941 (2000).

9. A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, and H. Dai, Phys. Rev. Lett. 92, 106804 (2004).

10. J. Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. Üstünel, S. Braig, T. A. Arias, P. W. Brouwer, and P. L. McEuen, Nano Lett. 4, 517 (2004). 11. E. Pop, D. Mann, J. Cao, Q. Wang, K. Goodson, and H. Dai, Phys. Rev. Lett. 95, 155505 (2005).

12. L. C. Venema, J. W. Janssen, M. R. Buitelaar, J. W. G. Wildoer, S. G. Lemay, L. P. Kouwenhoven, and C. Dekker, Phys. Rev. B 62, 5238 (2000).

13. M. A. Kuroda, A. Cangellaris, and J.-P. Leburton, Phys. Rev. Lett. 95, 266803 (2005).

14. Here we neglect acoustic phonon effects in a first approximation, and as a result, nonequilibrium phenomena between optical and acoustic phonon populations. By identifying the temperature profile obtained from Eq. (25.3) with the optical phonon temperature in the transport equation [Eq. (25.2)], our model overestimates the nanotube temperature. However, this approximation has minor effects (at most a slight renormalization of the scattering rate values) on the onset of the nonlinear transport (saturation and NDR) as seen in Figs. 25.1 and 25.3 while it sets an upper limit on the maximum temperature reached in the nanotubes. Inclusion of nonequilibrium phonon population in our self-consistent scheme will be the subject of a forthcoming analysis.

References

15. M. A. Osman and D. Srivastava, Nanotechnology 12, 21 (2001).

16. M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 95, 236802 (2005).

339

Chapter 26

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power in One-Dimensional Conductors

Marcelo A. Kurodaa and Jean-Pierre Leburtonb aDepartment

of Physics and Beckman Institute, University of Illinois at Urbana-Champaign, Illinois 61801, USA bDepartment of Electrical and Computer Engineering and Beckman Institute, University of Illinois at Urbana-Champaign, Illinois 61801, USA [email protected]

In one-dimensional conductors with linear E-k dispersion (Dirac systems), intra-branch thermalization is favored by elastic electronelectron interaction in contrast with electron systems with a nonlinear (parabolic) dispersion. We show that under external electric fields or thermal gradients, the carrier populations of different branches, treated as Fermi gases, have different temperatures as a consequence of self-consistent carrier-heat transport. Specifically, in the presence of elastic phonon scattering, the Wiedemann–Franz Reprinted from Phys. Rev. Lett., 101, 256805-1–256805-4, 2008. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2008 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

342

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

law is restricted to each branch with its specific temperature. In addition, thermoelectric power vanishes due to electron-hole symmetry, which is validated by experiment. Within the last decades, one-dimensional (1D) conductors such as nanowires, nanotubes, and molecular chains have become experimentally accessible [1]. While electron populations at low temperatures in ideal 1D systems are predicted to behave as Tomonaga–Luttinger liquids [2, 3], transport experiments in 1D conductors have revealed various behaviors depending on the temperature range and quality of the samples. Indeed, at low temperatures, conductance quantization [4], signatures of Tomonaga–Luttinger liquid [5, 6], and Wigner crystallization [7] have been observed. However, the experimental realization of such systems is tremendously challenging and still requires further unambiguous confirmation. As temperature is increased, the features of Tomonaga–Luttinger liquids are smeared out by thermal broadening, and carriers behave as Fermi gases [8]. In this regime, electron transport, ranging from ballistic to diffusive, has been successfully described by semiclassical approaches, such as Landauer–Büttiker formalism [9], direct solution of the Boltzmann equation [10, 11], and Monte Carlo simulations [12]. Paradoxically, these approaches often neglect electron-electron (e-e) interaction as well as self-consistent heat transport regulating the energy carried by electrons. In this chapter, we show that in 1D conductors with linear energy dispersion (Dirac system), energy and momentum conservation favors elastic inter-branch e-e scattering, in contrast to 1D systems with nonlinear (parabolic) dispersion [13]. As a consequence, the fermion populations in different branches are not in thermal equilibrium, and are characterized by two different temperatures, even in the lowest electric fields due to the mutual influence between carrier and heat transport. Our self-consistent analysis of electro-thermal transport of 1D Dirac systems shows that the ratio between thermal and electrical conductivity is proportional to the branch temperature (Wiedemann–Franz law). The thermoelectric power (TEP) in 1D conductors vanishes because of electron-hole symmetry. The band structure of 1D Dirac systems is given by E±(k) = ±huFk

(26.1)

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

where uF is the Fermi velocity and k is the wave vector along the 1D z-direction. The ± sign refers to two different energy branches which present a constant density of states. We set k = 0 at the branches crossing (Dirac point). In these systems, binary elastic e-e collisions are grouped in three classes of processes: i.e., intra-intra, intra-inter, and inter-inter-branch scattering, depending on whether the initial and final states remain in the same (intra) branches or change (inter) branches with collisions. Hence, we consider scattering from the initial state |k1,h1; k2,h2Ò to the final state |k′1, h′1; k′2,h′2Ò, where k and h indicate the wave vector and sign of the branch’s Fermi velocity (h = +, –), respectively (Fig. 26.1a). We assume for simplicity that none of these bands is degenerate and that there is only one valley, but the analysis can be easily extended to degenerate branches and multiple valleys. Momentum and energy conservations read k1 + k2 = k¢1 + k¢2

Eh1 (k1 ) + Eh2 (k2 ) = Eh¢ (k1¢ ) + Eh¢ (k2¢ ). 1

2

(26.2) (26.3)

Because of the proportionality between E and k, these two equations are linearly dependent for intra-intra-branch transitions (i.e., all the electrons states belonging to the same branch). For example, for intra-intra-branch |k1, +; k2, +Ò Æ |k′1, +; k′2, +Ò scattering, multiple values of k2 and k′2 satisfy energy and momentum conservation, given an arbitrary pair of values for k1 and k′1 (Fig. 26.1b). If any inter-branch transition takes place, Eqs. (26.2) and (26.3) become linearly independent (as in particle systems with nonlinear E-k dispersion). Figure 26.1c shows the two possible cases of inter-inter-branch scattering for which the first electron transition is |k1, –Ò Æ |k′1, +Ò. The solid (dashed) arrow to the left corresponds to the exchange (symmetric) scattering for which k2 = k′1 and k′2 = k1 (k2 = –k1 and k′2 = – k′1) which is totally inefficient for thermalization [13]. Intra-inter band transitions occur if and only if the wave vector of the state scattering to the different band is at the Dirac point (i.e., k = 0) as illustrated in Fig. 26.1d for |k1, +; k2, +Ò Æ |k′1, +; 0, –Ò. Therefore, the number of collisions involving only intra-branch scattering processes are roughly Nk (number of k-states) larger than the number of inter-intra or inter-inter-branch transitions. Since Nk is proportional to the number of atoms in the system (Nk ≫ i), the former process is much more likely to occur than the later ones.

343

344

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

Figure 26.1 Scattering diagrams for elastic electron-electron interaction: (a) Feynmann diagram; (b) intra-intra-branch scattering; (c) inter-inter-branch scattering ; (d) intra-inter-branch scattering.

As intra-intra-branch scattering is more efficient than interintra or inter-inter-branch scattering, carrier populations in different branches behave as independent Fermi gases with specific temperatures. In the presence of an electric field F, an imbalance arises (different quasi-Fermi level mh) between carrier populations with different Fermi velocity signs with a nonzero current flow. Consequently the distribution functions in each (thermalized) branch reads

fh (E ) =

1 (26.4) exp[(E - mh )/ kBTh ] + 1

where Th = Th(z) is the local electronic temperature of the branch h. Because the quasi-Fermi levels are far away from the band edges and the thermal broadening is much smaller than the band width, the carrier densities are independent of the respective temperatures due to the constant density of states. Since all the carriers share the same group velocity, the current is proportional to the quasi-Fermi level difference [10], i.e.,

I = euF (n+ - n- ) = gcG0

( m+ - m- ) , (26.5) e

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

where G0 = e2/(ph) is the quantum conductance. The factor gc accounts for the band degeneracy (spin degeneracy has already been considered). In Eq. (26.5), the current can be interpreted as the superposition of the electron current (I+) and hole current (I–) I± = ±



gc e I ( m ± - m ) = , (26.6) ph 2

where m ∫ (m+ – m–)/2 is the effective Fermi level of the system. Similarly, the branch energy flow per unit length with respect to the effective Fermi level is U± = ±



gc ph

Ú

•

-•

(E - m )[ f ± (E ) - Q( -E + m )]dE

g È p 2 (kBT± )2 ( m + - m - )2 ˘ =± c Í + ˙, p h ÍÎ 6 8 ˙˚

(26.7)

where Q(x) is the Heaviside step function [14]. It must be emphasized that current and heat flows are independent of the magnitude of the Fermi velocity due to the 1D nature of the system [15]. Moreover, if T+ = T–, no net heat can be transferred by carriers since U ∫ U+ + U– = 0 (independently of the current level). In the high temperature semiclassical regime, the distribution function in each branch follows the stationary Boltzmann transport equation (BTE) which after making use of Eq. (26.1) reads

± uF[¶zf± (E) + eF¶Ef± (E)] = ¶tf±|coll, (26.8)

where ¶t f±|coll is the collision integral accounting for carrier scattering. We solve this equation using the method of moments; i.e., we multiply the BTE by (E – m)m, where m is the moment index, and integrate over energy, for which we obtain 1 ph coll Ê ˆ - e Á F - ∂z m ± ˜ = ±  Ë ¯ e gc 0, ±



for

IÊ 1 ˆ ±∂zU ±  Á F - ∂z m ± ˜ = 1coll , ± (26.10) Ë ¯ 2 e

0th and 1st moment equations, respectively. • gc ∫ (E - m )m ∂t fh|coll dE is the generalized m-th -• p hu F moment of the collision integral for each branch. The left-hand side mcoll ,h

the

(26.9)

Ú

345

346

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

of Eq. (26.9) is the measured effective field (F′ ∫ F – ¶zm±/e) [16], and we note that in general, the expressions for ℱm,h depend on both the local temperatures and current level (or difference between quasiFermi levels). This dependence couples the equations describing the heat flow [Eq. (26.10)] for each carrier population. We solve both equations for the field and the temperatures profiles by using the current level as a parameter. In the case of 1D metals, the main scattering process (besides e-e interaction) is scattering with phonons [17]. We compute the collision integral for this mechanism as ∂Fh ∂t

=

coll

 {R (q)[ f e

q ,h '

h ¢ (+ )(1 - fh ( E ))

- fh (E )(1 - fh ¢ (- ))] + Ra (q)[ fh ¢ (- ) (26.11) ¥ (1 - fh (E )) - fh (E )(1 - fh ¢ (+ ))]}

where ϵ± = E ± hwq and q labels the phonon wave vector. The prefactor Re (Ra) is the phonon emission (absorption) rate. The dominant contribution in the low bias regime is that of longitudinal acoustic phonons (hwq = hus |q| ≪ |m+ – m–|). For phonon energies much smaller than the lattice temperature TL (hwq ≪ kBTL) and us /uF ≪ 1, collisions are assumed to be elastic [i.e., ϵ+ = ϵ– = E in Eq. (26.11)] and using the deformation potential approximation, we set Re(q) = Ra(q) µ TL [18]. We then define the mean free path lac ∫ uF/Re. Contributions of any elastic scattering to Eq. (26.11) vanish when electron states belong to branches with the same Fermi velocity sign (even if the magnitude of the velocity is different), provided that their distributions functions have the same quasi-Fermi level. By subtracting and adding the 0th moments of the BTE [Eq. (26.9)] for the two carrier branches, we obtain

and



¶z(m+– m–) = 0 I = gcG0lacF ¢,

(26.12)

(26.13)

respectively. Equation (26.12) expresses the local current conservation in the system (I µ m+ – m–). Equation (26.13) is nothing but Ohm’s law for which the conductivity is inversely proportional to the lattice temperature (lac µ TL-1 ) but independent of the carrier temperatures T±.

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

Using Eq. (26.6), we define the branch electrical conductivity as sh ∫ gcG0lac/2. We emphasize that owing to the electron-hole symmetry, Eq. (26.13) does not depend on the thermal gradient. Therefore, TEP in 1D (Dirac) conductors vanishes, which is in agreement with recent experiments on metallic carbon nanotubes that exhibit much smaller TEP than that of semiconducting carbon nanotubes [19]. In the presence of acoustic phonon scattering, the 1st moment equation [Eq. (26.10)] for each branch reads

∂zU ± =

IF ¢ 1  (U + U - ). (26.14) 2 lac +

Using Eqs. (26.7) and (26.12), the LHS of Eq. (26.14) is proportional to ¶zT±. This variation in the carrier temperature profiles is attributed to the Joule heating (first term on the RHS) and the inter-branch carrier scattering due to the carriers thermal imbalance (second term). Combining Eq. (26.14) for the different branches, we establish the heat flow conservation

¶z(U+ + U–) = IF ¢

(26.15)

which shows that all the heat is transported by the electrons and results in an inhomogeneous temperature profile. This is consistent with our approximation of elastic phonon scattering for which no energy gained by the carriers from the external field is transferred to the lattice (i.e., the lattice temperature remains constant). The energy production or dissipation in the system couples the two carrier temperatures T± [Eq. (26.14)]. Furthermore, Eq. (26.15) validates the assumption of the two temperature model for the electronic population in each branch. Indeed, if T+(z) = T–(z), the LHS of Eq. (26.15) vanishes, which is inconsistent with a nonzero current. Substituting Eqs. (26.7) and (26.15) into Eq. (26.14), we obtain for the net heat flow U = –k+¶zT+– k–¶zT–

(26.16)

in which we define the carrier thermal conductivity of the branch h as



kh =

lac gcp kB2Th 6h

=

lac gc Gth (26.17) 2

347

348

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

where Gth is the quantized thermal conductance [20]. The ratio between thermal and electrical conductivity for each branch kh



s hTh

=

p 2kB2 3e2

= L (26.18)

obeys Wiedemann–Franz law with their specific carrier temperature where L is the Lorenz number [16]. In the case of isothermal lattices, lac and F are constant, and by assuming ideal contacts (i.e., completely absorbing and thermalizing) T±(∓L/2) = T0, the solution of Eq. (26.14) is

T± ( z ) 3 = 1+ 2 T0 p

2

È 1  2z / L ˘ 1 ± 2z / L Ê I ˆ Á ˜ Í1 + ˙ 2lac / L ˚ 2lac / L Ë IT0 ¯ Î

(26.19)

for which IT0 ∫ ( gc ekBT0 )/(ph ) is the current associated to the lattice temperature. The temperature of the carriers with positive (negative) Fermi velocity has a maximum at z = lac = 2 (z = –lac/2). In the quasiballistic limit, lac ≫ L, the carrier temperature profiles T±QB ( z ) have a linear z-dependence

2

T±QB ( z ) 3L Ê I ˆ Ê 2z ˆ (26.20) =1+ 2 Á ˜ Á1 ± ˜ . T0 L¯ 4p lac Ë IT0 ¯ Ë

In this regime, the temperature difference T± – T0 is small even for I ~ IT0 because of the ratio L/lac ≪ 1. In the diffusive regime (lac ≪ L), the carrier temperature profiles have a parabolic shape, and the maximum temperature difference is of the same order as T0 even for small current levels. In Fig. 26.2, we display the temperature and heat flow profiles corresponding to carrier populations with positive Fermi velocity [T+(z, I) and U+(z, I)] for ratios lac/L = 20, 1, and 0.25, respectively. The electron-hole symmetry in this case is expressed by the fact that T+(z) = T–(– z) and U+(z) =–U–(–z). If a lattice temperature gradient exists along the conductor [TL(z) = T0 – DTz/L] and no current flows, the carrier temperature profiles T± satisfy 2

Ê DT ˆ L T± ( z )/ T0 = Á 1 ± + ˜ 2T0 ¯ L + l0 Ë

2 2 ÈÊ ˆ Ê ˆ ˘ Í 1 - DT z - 1 ± DT ˙ (26.21) 2T0 ˜¯ ˙ T0 L ˜¯ ÁË ÍÁË Î ˚

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

for which we have assumed lac(z)TL(z) = lac(0)T0 ∫ l0T0 and the symmetric boundary conditions T±(∓L/2) = TL(∓L/2). For DT ≪ T0, Eq. (26.21) becomes

DT (2z  l0 ) (26.22) 2 ( L + l0 )



T± ( z ) = T0 -



T+ ( z ) - T- ( z ) = DT

and the carrier temperature difference reads,

l0 . L + l0

(26.23)

In the case of ballistic transport (l0/L ≫ 1), the temperature difference remains T+ – T– = DT along the conductor, while for diffusive transport (l0/L ≪ 1), the temperature difference between branches is negligible.

Figure 26.2 Temperature (T+) and heat flow (U+) profiles for carriers with positive Fermi velocity as a function of the current for: l0 = 20L (a), (b), l0 = L (c), (d), and l0 = 0.25L (e), (f).

349

350

Restricted Wiedemann–Franz Law and Vanishing Thermoelectric Power

It is important to emphasize that Eqs. (26.9) and (26.10) are valid both in low and high field regimes. However, here we only consider elastic scattering and thereby the results are valid in the linear regime (i.e., low current levels and hwq ≪ kBTL). At high current levels, other scattering processes (e.g., optical phonon scattering) have to be included in the collision integrals. In conclusion, because of the effective intra-branch carrier thermalization in the high temperature regime, electron populations in the different branches of 1D Dirac conductors behave as independent Fermi gases (with their respective temperatures) out of thermal equilibrium as a consequence of the electro-thermal flow. In the presence of elastic (acoustic phonon) scattering, the carrier population in each energy branch follows the Wiedemann–Franz law. The TEP coefficient in 1D conductors vanishes as a result of the electron-hole symmetry.

Acknowledgments

One of the authors (M. K.) is indebted to E. Fradkin for providing valuable discussion and acknowledges the support of the Department of Physics at the University of Illinois, Urbana-Champaign.

References

1. J. Hu et al., Acc. Chem. Res. 32, 435 (1999). 2. J. Solyom, Adv. Phys. 28, 201 (1979).

3. J. Voit, Rep. Prog. Phys. 58, 977 (1995).

4. A. Yacoby et al., Phys. Rev. Lett. 77, 4612 (1996).

5. M. Bockrath et al., Nature (London) 397, 598 (1999).

6. S. V. Zaitsev-Zotov et al., J. Phys. Condens. Matter 12, L303 (2000). 7. V. Deshpande et al., Nat. Phys. 32, 314 (2007).

8. T. Hertel et al., Phys. Rev. Lett. 84, 5002 (2008). 9. J.-Y. Park et al., Nano Lett. 4, 517 (2004).

10. M. A. Kuroda et al., Phys. Rev. Lett. 95, 266803 (2005). 11. M. Lazzeri et al., Phys. Rev. B 73, 165419 (2006).

12. A. Javey et al., Phys. Rev. Lett. 92, 106804 (2004). 13. J. P. Leburton, Phys. Rev. B 45, 11022 (1992).

References

14. The Heaviside function is introduced to eliminate the artificial divergence arising from the extension of the bottom of the conduction bands to infinity in the integration.

15. S. Datta, Electronic Transport in Mesoscopic Systems (University Press, Cambridge, 1995). 16. J. M. Ziman, Principles of the Theory of Solids (Cambridge University, Cambridge, England, 1972), 2nd ed.

17. The consideration of impurity scattering (negligible at high temperatures) in our model would result in a renormalized elastic mean free path.

18. K. Hess, Advanced Theory of Semiconductor Devices (IEEE Press, New York, 1999). 19. J. P. Small et al., Phys. Rev. Lett. 91, 256801 (2003). 20. L. G. C. Rego et al., Phys. Rev. Lett. 81, 232 (1998).

351

Chapter 27

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

Marcelo A. Kurodaa and Jean-Pierre Leburtonb aDepartment

of Physics and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA bDepartment of Electrical and Computer Engineering, Department of Physics and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected], [email protected]

We describe the electrothermal transport in metallic carbon nanotubes (m-CNTs) by a semiclassical approach that takes into account the high-field dynamical interdependence between charge carrier and phonon populations. Our model is based on the selfconsistent solution of the Boltzmann transport equation and the heat equation mediated by a phonon rate equation that accounts for the onset of nonequilibrium (optical) phonons in the high-field regime. Given the metallic nature of the nanostructures, a key ingredient of the model is the assumption of local thermalization of charge Reprinted from Phys. Rev. B, 80, 165417-1–165417-10, 2009. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2009 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

354

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

carriers. Our theory remarkably reproduces the room-temperature electrical characteristics of m-CNTs on substrate and free standing (suspended), shedding light on charge-heat transport in these onedimensional nanostructures. In particular, the negative differential resistance observed in suspended m-CNTs under electric stress is attributed to inhomogeneous field profile induced by self-heating rather than the presence of hot phonons.

27.1 Introduction

Since their discovery [1], carbon nanotubes (CNTs) have been subject of intense research for both scientific interest and technological purpose. Their thermal, mechanical, and electrical properties [2] make them prominent candidates for broad range of applications in nanotechnology [3]. The electronic structure of these one-dimensional (1D) macro-molecules exhibits semiconducting or metallic (m) features depending on their chirality [4]. The latter have been proposed for interconnect applications because they can support currents densities larger than copper without suffering electromigration [5, 6]. Recent experiments on individual m-CNTs in the high-field regime [7–11] have revealed strong nonlinear transport characteristics, such as current saturation and negative differential resistance (NDR), which have been attributed to the emergence of optical phonon (OP) populations out of equilibrium (hot phonons) [10, 12]. Additionally, the large electrical power dissipated in the high-bias regime have fostered further studies of the electrothermal transport in these structures [13–18]. As a consequence numbers of theoretical and experimental works have attempted to quantify both the electron-phonon (e-p) and phononphonon (p-p) coupling interactions in m-CNTs [19–22]. Prominent among the theoretical approaches for investigating high-field transport in m-CNTs at room temperature are the Boltzmann transport equation (BTE) [23, 24], Monte Carlo simulations [9], Landauer–Büttiker formalism [10] based on the Fermi’s golden rule [12] that successfully described transport in nanoscale semiconductors. However one of the major drawbacks of these approaches is the absence of electron-electron (e-e) interaction, despite the metallic character of the system, and the

Model

contribution of the charge carriers to thermal transport. In this work, which is applicable to any metallic 1D system, we describe the system dynamics as the interplay between carrier, acoustic phonon (AP), and OP populations. Our self-consistent approach describing electrothermal transport accounts for the onset of phonon and carrier populations out of equilibrium and provides a comprehensive picture of the carrier flow and heat exchange in the nanostructures. Our model is in excellent agreement with experiments on individual m-CNTs in both substrate supported (SS) and free-standing (FS) configuration at room temperature.

27.2 Model

We describe the high-field transport in m-CNTs by accounting for both electric and thermal transport through the 1D structure. While the former originates from the charge-carrier populations only, both electron and phonon populations contribute to heat transfer in a conjugated manner.

27.2.1 Electric and Energy Flow

In small diameter m-CNTs, the higher conduction and lower valence subbands lie far above or below the Fermi level so that the charge flow arises solely from the contribution of the conducting bands crossing the Fermi level (here taken as the zero-energy level), which are well described by a linear E–k dispersion,

E±(k) = ±huFk, (27.1)

where uF is the Fermi velocity (uF ª 8 ¥ 107 cm/ s in CNTs) (Ref. [4]) and the +(–) sign corresponds to the forward (backward) carrier branches. Each of these branches has a degeneracy 2gc (where the factor of 2 indicates the spin degeneracy and gc = 2 for m-CNTs). Due to the efficient intrabranch e-e interaction [25] carrier populations are thermalized and described by Fermi-Dirac distribution functions. In the presence of an external electric field F, the local quasi-Fermi levels of forward and backward populations separate (m+ π m–), and the carrier distributions in each branch are not necessarily in local thermal equilibrium with each other [T+(x) π T–(x)], as illustrated in Fig. 27.1. The electric current in m-CNTs reads [23]

355

356

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

( m+ - m- ) , (27.2) e where e is the electron charge and G0 = e2/(ph) is the quantum conductance. Here we point out that the quasi-Fermi-level difference does not necessarily coincide with the drain-source voltage, i.e., |m+ – m–| £ |eVds|, where the equality only holds in the absence of back scattering. In other words carriers injected from the contacts are immediately thermalized by the effective e-e interaction. Hence, even if eVds is larger than the separation between subbands, contribution of conducting subbands can be neglected provided that the local quasi-Fermi levels in each branch lie far away from the upper or lower subband (|Es– m±| ≫ kBT±), as illustrated in Fig. 27.1. We also recognize that quasi-Fermi levels m±(x) are position dependent but their difference remains constant for current conservation [23].

I = gcG0

Figure 27.1 (a) Schematics of the forward and backward quasi-Fermilevel profiles as a function of the distance along the m-CNT. (b) Forward and backward distribution functions at position x = x0. (c) Band structure at x = x0. Contributions of upper and lower subbands are neglected as they lie far from the active scattering area (shaded region).

Our model neglects the presence of a band gap arising due to a Mott insulating behavior observed in m-CNTs [17, 26]. The characteristic energy gap for typical m-CNTs is on the order of roomtemperature fluctuations corresponding to a current level of the order of 4 mA in Eq. (27.2). The high-field regime, which is the main focus of this work, corresponds to large quasi-Fermi-level separation and/or thermal broadening of the carrier distribution for which the linear bands represent a good depiction of the electronic structure. In this metallic system, the distribution function f± (E, x) for each branch obeys the stationary BTE expressed as ± uF[¶x f±(E, x) + eF¶E f±(E, x)] = ¶t f±|coll, (27.3)

Model

after using the linear dispersion [Eq. (27.1)]. Here, x denotes the position along the m-CNT and the right-hand side of the equation is the collision integral. The local temperatures and quasi-Fermi levels of the thermalized distributions can be determined using the method of moments (from here on we omit the variable x for sake of brevity). We have recently shown that the electric field F and carrier temperature T± profiles fulfill the relations [25] -eF = ±



and

±

ph coll 0, ± (27.4) gc

gcp kB2T± IF ∂xT± ∓ = 1coll , ± , (27.5) 3 2

respectively. The absence of thermal gradients in Eq. (27.4) asserts the lack of thermoelectric power in 1D systems with linear dispersion [25]. In Eqs. (27.4) and (27.5), kB is the Boltzmann constant and

mcoll ,± =

gc p hu F

Ú

•

-•

∂t f ± |coll (E - m )m dE (27.6)

is the mth moment of the collision integral accounting for carrier scattering. We express the boundary conditions for T±(x) in terms of the transmission coefficient through the contacts t, such that

T±2 (  L / 2) = tT02 + (1 - t )T2 (  L / 2), (27.7)

where T0 is the temperature at the contacts. The value of t ranges between 0 and 1. The latter corresponds to perfect contacts that inject carriers from the leads without any reflection. After obtaining the F profile [Eq. (27.4)], the net drain-source bias is computed as

Vds = IRc +

Ú

L/2

- L/2

F ( x )dx , (27.8)

where Rc is the contact resistance which in the case of perfect contacts is 1/(gcG0) [27].

27.2.2 Electron-Phonon Interaction

The main contribution to the collision integrals [Eq. (27.6)] in m-CNTs, other than e-e interaction that thermalizes the branch distributions, is scattering with lattice vibrations (phonons). The collision integrals read

357

358

High-Field Electrothermal Transport in Metallic Carbon Nanotubes



h ¢ ,h ∂t fh|h ¢ ,a - in = Rem, a fh ¢ (+a )[1 - fh ( E )] h ,h ¢ - Rab, a fh ( E )[1 - fh ¢ (-a )], h ,h ¢ ∂t fh|h ¢ ,a - out = Rem, a fh ( E )[1 - fh ¢ (-a )]

(27.9)

h ¢ ,h - Rab, a fh ¢ (+a )[1 - fh ( E )],

(

)

(27.10)

h ,h ¢ h ,h ¢ where ϵ±a = E ± hwa and Rem, denotes the emission a Rab,a (absorption) scattering rate for the a phonon. Equation (27.9) [Eq. (27.10)] is the collision integral for a phonons where the emission process scatters an electron from the carrier branch h¢ (h) to the carrier branch h (h¢). The rates in Eqs. (27.9) and (27.10) have different functional forms depending on the phonon branch involved in individual scattering events. The total contribution to carrier scattering with phonons is



∂t fh|ph =

Â∂ f |

t h h ¢ ,a - in

a ,h ¢

-

Â∂ f |

a ¢ ,h ¢

t h h ¢ ,a ¢- out . (27.11)

For any given phonon wave vector q and energy hwa(q), the initial and final carrier wave vectors are determined by using momentum and energy conservation during collision. For instance with the Fermi energy located at the Dirac point [k = 0 in Eq. (27.1)], the conservation relations are written as

k = k¢ ± q, (27.12)

Eh(k) = Eh¢(k¢) ± hwa(q), (27.13)

where the + (–) sign denotes phonon emission (absorption) process from the k state in the branch h to the k¢ state in the branch h¢. In our model we consider AP scattering that dominates transport in the low-bias regime and scattering with high energy zone-boundary (ZB) and OPs that induces the electric transport nonlinearities in high field. The different mechanisms are depicted in Fig. 27.2.

Figure 27.2 Phonon scattering mechanisms: (a) interbranch AP, (b) interbranch OP, and (c) intrabranch OP.

Model

The AP populations are considered to be in thermal equilibrium with the lattice as their energies are smaller than thermal fluctuations at room temperature. In addition, we assume that carrier collisions with APs are elastic (i.e., no energy is exchanged with the lattice but only between carrier populations) due to the small ratio between the sound and Fermi velocity (us/uF ≲ 1/40). Under these assumptions, the AP contribution to the collision integral vanishes when the initial and final states have the same Fermi velocity sign, and emission and absorption rates become equal. Therefore, the collision integral due to interbranch AP scattering is given by

∂t f ±|AP =

ÂR h¢

AP [ f  ( E ) - f ± ( E )] (27.14)

with RAP = ( Z A2 kBT )/( rMus2 h2uF ) by using the deformation-potential approximation where T = T(x) is the local lattice temperature, rM is the linear mass density, and ZA ~ 5 eV [8]. The first two moments of the collision integral for AP scattering read [25]

0AP ,± = -

1AP ,±

gc ( m + - m - ) p hu F

ÂR h¢

h¢ AP , (27.15)

g p k2 h ¢ (27.16) = ∓ c B (T+2 - T-2 ) RAP . 6uF h¢

Â

Here we point out that any elastic backscattering, e.g., scattering with impurities, would have given similar expressions with their specific rates, provided that the latter do not depend on carrier energies. The OP emission and absorption scattering rates are written as

and

h ,h ¢ Rem, a =

Na (q) + 1

h ,h ¢ Rab, a =

t aep

Na (q) t aep

(27.17)

, (27.18)

respectively. In Eqs. (27.17) and (27.18), t aep and Na(q) are the bare relaxation time and the occupation number for the a phonon,

359

360

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

respectively. The experimental estimate for the former parameters is about 30 fs (Refs. [8, 9]) and is an order of magnitude smaller than those values obtained by first-principle calculations [19]. We distinguish between interbranch (h π h¢) and intrabranch (h = h¢) OP scattering, as depicted in Fig. 27.2 and consider two interbranch (ZB and OP1) and one intrabranch (OP2) phonon modes interacting with carriers in m-CNTs [24]. Their energies are assumed to be constant and given by hwZB = 0.16 eV (ZB phonon) and hw OP1 = hw OP2 = 0.20 eV (interbranch and intrabranch OP), respectively. If the OPs are in thermal equilibrium with the lattice, Na(q) = Neq =1/{exp[hwa(q)/kBT] – 1} as given by the Bose–Einstein distribution. Otherwise, the occupation number Na(q) depends on the local temperatures of carriers and lattice as well as the current level. At steady state, the occupation number of the a phonon is obtained by using the following rate equation:

∂t Na (q)|aep +∂t Na (q)|app = 0. (27.19)

In Eq. (27.19) we neglect phonon diffusion because of the small OP group velocity. The first term in Eq. (27.19) accounts for the generation/annihilation of OP due to e-p scattering and is given by

h ,h ¢ ∂t Na (q)|aep = Rem, a fh (+a )[1 - fh ¢ ( E )]

h ¢ ,h - Rab, a fh ¢ ( E )[1 - fh (-a )],

(27.20)

where E is determined by using Eqs. (27.12) and (27.13). The second term in Eq. (27.19) describes the decay of OPs into APs and is expressed as

∂t Na (q)|app = -

1

t app

[Na (q) - Neq ], (27.21)

where OP decay time t app has been estimated to be 1–7 ps from Raman studies in m-CNTs [20, 28, 29]. In addition, decay times have been found to vary as the inverse of T [21, 28]. For intrabranch (h = h¢) OP scattering, the phonon wave vectors are q = ± w OP2 /uF, independently of the initial or final states. The moments of the collision integral become

0, ± 2 = 0, (27.22) OP

Model

1, ± 2 = OP



gc2 ( hw OP2 )2 p huFt OP2

È ˘ Í ˙ (27.23) NOP2 (q) Í NOP2 (q) + 1 ˙ ¥Í ˙. Í exp Ê hw OP2 ˆ - 1 exp Ê hw OP2 ˆ - 1 ˙ Á k T ˜ Á k T ˜ Í ˙ Ë B ± ¯ Ë B ± ¯ Î ˚

For interbranch scattering with ZB and OP1 modes, the phonon wave vector q depends on the initial and final electron states. Consequently, Na(q) is obtained from Eq. (27.19) and depends on the carrier and lattice temperatures as well as the quasi-Fermilevel separation (current). For t app π 0 and T+ π T–, the moments of the collision integrals [Eq. (27.6)] for interbranch OP scattering do not have an analytical expression and are obtained numerically. In the Appendix, we derive the expressions for the moments of the collision integral corresponding to interbranch scattering with OPs when t app = 0 = 0 and T+ = T–.

27.2.3 Electron-Phonon Heat Exchange

Combining the first moment equation for forward and backward carrier populations [Eq. (27.5)], the heat production/dissipation satisfies qel + qlat = IF ∫ q , (27.24) where

qel =

gcp kB2T+ g p k 2T ∂xT+ - c B - ∂xT- (27.25) 3 3

is the amount of heat carried by the electrons and

qlat ∫

1 2p

ÂÚ a

•

-•

wa (q) ∂t Na (q)|aep dq (27.26)

is the amount of energy removed from the carriers for the OP populations. In addition, since heat transport is neglected in Eq. (27.21), qlat denotes the rate (per unit length) of energy transferred to the lattice (APs) by OP decays. This value determines the local lattice heating and depends on the current and the carrier and lattice

361

362

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

temperatures. We assume that lattice (AP) heat transport is diffusive and follows the Fourier’s law for which the heat equation reads qap + qsub = qlat . (27.27) The first term on the left-hand side of Eq. (27.27) accounts for the heat carried by the AP population, qap ∫ - A∂x [k (T )∂xT ]. (27.28)

The factor A = pdt is the cross-sectional area, where d and t ª 0.34 nm are the diameter and the thickness of the CNT, respectively. The lattice thermal conductivity k(T) is given by k(T) = k0T0/T due to the Umklapp phonon-phonon scattering [30]. The roomtemperature thermal conductivity has been estimated to be between 15 and 60 W/(cm K) for individual CNTs [31]. The second term in Eq. (27.27) determines the heat removal through the substrate, which is modeled as a contact resistance [32] qsub ∫ g0 (T - T0 ). (27.29) The parameter g0 denotes the thermal coupling between the CNT and the substrate which has been experimentally estimated between 0.05 and 0.20 W/(Km) for SiO2 [13, 14]. The lattice temperature boundary conditions between CNT and leads are set by the presence of a contact resistance

k∂xT |x=± L/2 =

T ( ± L / 2) - T0 (27.30) th

with th ~ 107 K/W for m-CNTs [33].

27.2.4 Heat Flow Diagram

The coupled nonlinear integrodifferential equations for the carrier and lattice temperatures [Eqs. (27.5) and (27.27)] require a selfconsistent solution. The interplay between carrier and phonon heat transfer [Eq. (27.24)] is depicted in Fig. 27.3, where both forward and backward carrier populations gain energy from the electric field. In the high-field regime, OP population builds up (hot phonon effects) due to the enhanced scattering by phonon emission but because of their small group velocity, heat removal by OPs is not significant. Consequently, either they transfer their energy back to the carrier population in reabsorption processes or they decay into APs,

Model

thereby heating the lattice. Energy can be removed from the system through the leads (by both carriers and APs) or to the substrate (by APs), as illustrated in Fig. 27.3b. A key factor that characterizes heat transfer is the ratio t aep /t app which determines the fraction of heat transferred from the lattice [34, 35]. On the one hand if t aep /t app  1, most of the energy gained by the OP population is transferred to the APs by instantaneous phonon decay and carried by the lattice. On the other hand, if t aep /t app  1, the energy of an OP emitted by carrier scattering is more likely to return to the carrier population by reabsorption rather than decay into APs. Hence, most of the energy remains within the carrier and OP populations. However,

Figure 27.3  (a) Flowchart of the heat transfer. The power given to the system by the external field is transferred to the forward and backward carrier populations. These populations exchange heat by emission/absorption processes with OP populations. In addition OPs decay into APs, transferring heat to the lattice. (b) Heat removal from the nanotube: by carriers (black arrows) through the leads and by APs (white arrows) through the leads and to the insulating substrate if present.

363

364

High-Field Electrothermal Transport in Metallic Carbon Nanotubes

since OPs cannot carry significant amount of heat (because of their small group velocity), heat is removed by the electron populations through the leads. In this case, the presence of a substrate would make no difference—considering that t aep or t app are not affected by the presence of the substrate—because only a small fraction of heat is transferred to the lattice.

27.3 Results

27.3.1 Low-Field Regime As heating of carrier or lattice population is negligible in the lowfield regime (|T± (x) – T0|,|T(x) – T0| ≪ T0), the collision integrals for OP scattering have analytic expressions (see Appendix). Combining Eq. (27.4) with Eqs. (27.15) and (27.A1), we obtain the Matthiessen rule for the low-field resistivity r(T ) =



1 Ê 1 1 ˆ Á eff + eff ˜ , (27.31) gcG0 Ë l AP lOP ¯

eff eff where l AP and lOP are the effective AP and OP mean-free path given by eff l AP =



and

eff lOP

ÏÔ =Ì ÓÔ

Âu t

2

a

ep F a

T0 l AP,0 T

(27.32) -1

¸Ô . (27.33) 2˝ sinh[hwa /(2kBT )] ˛Ô hwa /(2kBT )

As shown in Fig. 27.4, the analytical expression in Eq. (27.31) accurately describes the temperature dependence of the lowfield resistivity obtained experimentally in m-CNT [36]. The set of parameters lAP,0 ª 800 nm, tZB ª 85 fs, t OP1 ª 250 fs, hwZB = 0.16 eV, hw OP1 = 0.20 eV, and T0 = 300 K are consistent with experimental values found in the literature [8, 9].

Results

Figure 27.4 Temperature dependence of the low-field resistivity: theory (solid line) and experiment (symbols) from Purewal et al. [36].

In Fig. 27.5 we plot the forward carrier (solid line) and lattice (dashed line) temperature profiles corresponding to the low-field regime (I = 0.01 mA) for 0.3- and 3-mm-long m-CNTs. These profiles are determined for both SS and FSCNT configurations, for which the thermal coupling to the substrate are g0 ª 1.5 ¥ 10–3 W (cm K) and 0, respectively. In the case of symmetric boundary conditions for carrier temperatures, i.e., T+(–L/2) = T–(L/2), and lattice temperature, i.e., T(–L/2) = T(L/2), the carrier temperature profiles satisfy T+(x) = T–(–x) due to the electron-hole symmetry, thereby reducing the number of differential equations. In this regime the electric field along the channel remains constant. For short CNTs the profiles are similar regardless of the presence/absence of substrate with the heating in the carrier population being more significant than in the lattice. The forward carrier temperature increases along the tube and the maximum is reached at the drain as a consequence of the long mean-free path (compared to the CNT length) [37]. In the case of long m-CNTs, despite of substantial inelastic scattering that heats the AP population, the carrier temperature increase is larger than that of the lattice. In particular, for FSCNT lattice heating is comparable to that of the carriers while in the SSCNT the lattice temperature is flattened by the efficient heat removal through the substrate and remains closer to the substrate temperature. The maximum lattice temperature is located exactly at the m-CNT midlength while the forward carrier temperature peak is shifted toward the right lead.

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High-Field Electrothermal Transport in Metallic Carbon Nanotubes

Figure 27.5 Forward carrier (solid line) and lattice (dashed line) temperature difference to the contact temperature T0 corresponding for I = 0.01 mA (lowfield regime): (a) 0.3-mm-long SSCNT, (b) 0.3-mm-long FSCNT, (c) 3-mm-long SSCNT, and (d) 3-mm-long FSCNT.

27.3.2 High-Field Regime In Fig. 27.6 we display the electrical characteristics (solid lines) corresponding to the SS and FS configurations of 0.3- and 3-mmlong CNTs that are in excellent agreement with experiments [9, 10]. The parameters used in the calculations are k(300 K) = 20 W/(cm K), g0 = 1.5 ¥ 10–3 W/(cm K), and ZA ª 5 eV. The relaxation times have been used as fitting parameters with values of tZB ~ 85 fs, t OP1 = t OP2 ~ 200 fs, and t app = 1 ps, which lie within the range estimated by experiments [9, 10] and theory [19]. Nonlinear electrical characteristics are obtained for all CNTs. In the high-bias regime, the CNTs in FS configuration exhibit NDR induced by the inefficient heat removal that increases scattering. However, while in 0.3-mm-long FS and SS CNTs the current levels approach the 20 mA limit attributed to the onset of OP emission [7], in the 3-mm-long FSCNT the current level is substantially reduced (but still exhibiting NDR features). By contrast in the 3-mm-long SSCNT, the current increases monotonically in the bias range shown and no saturation is observed due to the fact that the electric field is about an order of magnitude smaller than in the 0.3-mm-long SSCNT for the same voltages.

Results

Figure 27.6 IV characteristics of 0.3- and 3-mm-long CNTs in both SS and FS configurations. Symbols correspond to experimental data from Javey et al. [9] (*) and Pop et al. [10] ().

The corresponding lattice and forward carrier temperature profiles are shown in Fig. 27.7. For short (0.3-mm-long) m-CNTs, the temperature profiles in both FS and SS configurations have a parabolic shape, as shown in Figs. 27.7a and b. Unlike the low-field regime (Fig. 27.5), the temperature maxima for forward carriers in the high-field regime occur at the center of the m-CNT in both cases, in agreement with breakdown experiments in m-CNTs [38, 39]. We note that for the same channel length, the values of the carrier and lattice temperature maxima are lower in the SSCNT than in the FSCNT as heat is removed by the substrate. In SSCNTs, the reduced temperature lowers scattering rates and consequently higher current levels are achieved compared to their FS counterparts. By contrast, in 3-mm-long SSCNTs the efficient heat removal through the substrate flattens the temperature profiles as depicted in Fig. 27.7c, as observed recently [18]. Because of the electron-hole symmetry [T+(x) = T–(–x)] forward and backward carrier temperatures are equal to one another in the channel but are different in the regions close to the contacts. However due to the effective heat transfer to the substrate, carrier populations are not locally in thermal equilibrium with the lattice. For the 3-mm-long FSCNT (Fig. 27.7d), carrier and lattice temperature profiles have a parabolic shape and markedly higher values due to limited heat removal, which enhances carrier scattering and phonon decays. The current level reduction as well as the near thermal equilibrium between carrier and phonon

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High-Field Electrothermal Transport in Metallic Carbon Nanotubes

populations indicate that transport approaches the diffusive regime limit [40].

Figure 27.7 Lattice (left) and carrier (right) temperature profiles as a function of the bias voltage Vds for the IV characteristics shown in Fig. 27.6. From top to bottom: 0.3-mm-long SSCNT, 0.3-mm-long FSCNT, 3-mm-long SSCNT, and 3-mmlong FSCNT.

27.3.3 Heat Flow In Fig. 27.8 we simultaneously display the local heat production given by Joule’s law [Eq. (27.24)] and the fraction of the local heat carried by electrons [ qel in Eq. (27.25)], APs [ qap in Eq. (27.28)],

Figure 27.8 Two-dimensional (2D) heat flow profiles as a function of Vds and distance for 0.3- and 3-mm-long CNTs, for the IV characteristics and temperature profiles depicted in Figs. 27.6 and 27.7, respectively. From top to bottom: 0.3-mm-long SSCNT, 0.3-mm-long FSCNT, 3-mmlong SSCNT, and 3-mm-long FSCNT. First column corresponds to the Joule’s heat production (q tot ) Second, third, and fourth column display the fraction of heat carried by the electrons (qel ) , acoustic phonons (qap) , and substrate (qsub) , respectively. The profiles are normalized to the maximum power density across the channel at Vds = 1 V.

Results 369

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High-Field Electrothermal Transport in Metallic Carbon Nanotubes

and substrate [ qsub in Eq. (27.29)] for the IV characteristics shown in Fig. 27.6. For illustrative purposes, the profiles have been normalized to the maximum power per unit length across the channel for Vds = 1 V. Since the current remains constant along the channel the q profile is proportional to the electric field profile and presents a peak at the middle of the m-CNT. Despite significant carrier heating (right column of Fig. 27.7), the fraction of heat carried by electrons (second column in Fig. 27.8) is small ( 1 in the expression of the voltage-dependent resistance. Hence under high bias the nonuniform thermal profiles in FSCNTs, depicted in Fig. 27.7d, induce local electric field inhomogeneities along the CNT shown in Fig. 27.8d [45].

27.3.5  Electrical Power vs Length

Electrical breakdown by oxidation in CNTs has been estimated to occur at around 1000 K [46, 47]. In Fig. 27.10 we display the electrical powers P(1000 K) (solid line) and P+(1000 K) (dashed line) as a function of the CNT length for various values of the thermal coupling to the substrate g0. The parameters P(1000 K) and P+(1000 K) correspond to the electrical power (P = IV) dissipated along the CNT when the maximum lattice temperature and maximum carrier temperature reach 1000 K, respectively. For the FSCNT (g0 = 0), both electrical powers are determined from the self-consistent solution of the BTE and heat equation. For SS-CNTs (g0 > 0), we compute the power as P = I(IRc + FL) by assuming that the electric field is homogeneous along the conductor (vanishing thermal gradients) and solving for the values of electric field and currents as described in Section 27.3.3. In the former case, i.e., g0 = 0 (□), P(1000 K), and P+(1000 K) are approximately equal (both curves are indistinguishable when L ≳ 1 mm) and scale inversely proportional to the length. In contrast the dissipated powers in SSCNTs increase linearly with length for sufficiently long m-CNTs, when the voltage drop at the contacts is smaller than that along the channel. Similar trends have been reported experimentally in the electrical breakdown of multi-walled CNTs in both FS and SS configurations [39]. We point out that at a fixed length P+(1000 K) and P(1000 K) increase with the substrate coupling. Additionally, the difference P(1000 K) – P+(1000 K) widens by the enhancement of g0 indicating more prominent hot-electron effects as carrier populations reach the 1000 K temperature at lower power values than the lattice.

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Figure 27.10 Electrical powers, P(1000 K) and P+(1000 K), dissipated in the CNT as a function of length when the maximum lattice (solid lines) and carrier (dashed lines) temperatures reach 1000 K, respectively. The symbols indicate the different values of the thermal coupling to the substrate.

27.4 Conclusions A model for electrothermal transport in 1D metallic systems in the high-temperature (incoherent) regime has been presented. We use the BTE formalism to describe the interdependent e-p dynamics in high field. By accounting for the emergence of carrier and phonon populations out of equilibrium we quantify the heat production and dissipation by each of these populations depending on the experimental configuration. The phonon decay rate is the bottleneck that regulates heat exchange between carriers and lattice. Our model shows remarkable agreement with high-field transport experiments and offers a qualitative interpretation of breakdown experiments in multiwalled CNTs. We attribute the emergence of NDR observed in suspended CNTs to the inhomogeneous electric field and selfheating in the high-field regime.

Acknowledgments

One of the authors (M. K.) acknowledges the support of the Department of Physics and the Department of Electrical and Computer Engineering at the University of Illinois at UrbanaChampaign.

References

Appendix: A  nalytical Expressions for the Collision Integrals For carriers in different branches in thermal equilibrium (i.e., T+ = T– º Tel) and when hot phonon effects are neglected (i.e., t ep  t app or low current level), analytical expressions for the zeroth and first moments of the collision integral [Eq. (27.6)] and the OP decay into APs [Eq. (27.26)] can be obtained:

F0a, ± = ±

gc2kBTel

hpuFt aep

{[Neq + 1][G(W + ) - G(W - )]

+ Neq [G( -W - ) - G( -W + )]}, F1a, ± = -

gc2 hwa kBTel 2hpuFt aep

(27.A1)

{[Neq + 1][G(W + ) + G(W - )]

- Neq [G( -W - ) + G( -W + )]}, qlat,a =

gc2 wa kBTel 2puFt aep

{[Neq + 1][(W + ) + (W - )]

- Neq [( -W - ) + ( -W + )]},

(27.A2)

(27.A3)

hwa ± ( m + - m - ) x and W ± = . exp( x ) - 1 kBTel The expressions are valid in the low-bias regime, where heating in the carrier population and lattice can be neglected. In addition, the omission of hot phonon effects is valid approximation for the long FSCNTs because the minor thermal imbalance between OP and AP (lattice) populations. where we define ( x ) =

References

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8. J.-Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. Ustunel, S. Braig, T. Arias, P. Brouwer, and P. McEuen, Nano Lett. 4, 517 (2004). 9. A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, and H. Dai, Phys. Rev. Lett. 92, 106804 (2004).

10. E. Pop, D. Mann, J. Cao, Q. Wang, K. Goodson, and H. Dai, Phys. Rev. Lett. 95, 155505 (2005).

11. P. Sundqvist, F. J. Garcia-Vidal, F. Flores, M. Moreno-Moreno, C. GomezNavarro, J. S. Bunch, and J. Gomez-Herrero, Nano Lett. 7, 2568 (2007). 12. M. Lazzeri and F. Mauri, Phys. Rev. B 73, 165419 (2006).

13. H. Maune, H.-Y. Chiu, and M. Bockrath, Appl. Phys. Lett. 89, 013109 (2006). 14. E. Pop, D. A. Mann, K. E. Goodson, and H. Dai, J. Appl. Phys. 101, 093710 (2007).

15. I.-K. Hsu, R. Kumar, A. Bushmaker, S. B. Cronin, M. T. Pettes, L. Shi, T. Brintlinger, M. S. Fuhrer, and J. Cumings, Appl. Phys. Lett. 92, 063119 (2008). 16. C. T. Avedisian, R. E. Cavicchi, P. M. McEuen, X. Zhou, W. S. Hurst, and J. T. Hodges, Appl. Phys. Lett. 93, 252108 (2008).

17. V. V. Deshpande, S. Hsieh, A. W. Bushmaker, M. Bockrath, and S. B. Cronin, Phys. Rev. Lett. 102, 105501 (2009). 18. L. Shi, J. Zhou, P. Kim, A. Bachtold, A. Majumdar, and P. L. McEuen, J. Appl. Phys. 105, 104306 (2009).

19. M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 95, 236802 (2005). 20. M. Oron-Carl and R. Krupke, Phys. Rev. Lett. 100, 127401 (2008).

21. G. Pennington, S. J. Kilpatrick, and A. E. Wickenden, Appl. Phys. Lett. 93, 093110 (2008). 22. D. Song, F. Wang, G. Dukovic, M. Zheng, E. D. Semke, L. E. Brus, and T. F. Heinz, Phys. Rev. Lett. 100, 225503 (2008).

23. M. A. Kuroda, A. Cangellaris, and J.-P. Leburton, Phys. Rev. Lett. 95, 266803 (2005).

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45. For a fixed bias Vds, the local heat dissipation q (x) µ F(x) as the current I remains constant along the channel [Eq. (27.24)].

46. M. Radosavljevic, J. Lefebvre, and A. T. Johnson, Phys. Rev. B 64, 241307(R) (2001). 47. F. Cataldo, Fullerenes, Nanotubes, Carbon Nanostruct. 10, 293 (2002).

Chapter 28

Atomic Vacancy Defects in the Electronic Properties of Semi-metallic Carbon Nanotubes

Hui Zeng,a,b,c Jun Zhao,a Huifang Hu,c and Jean-Pierre Leburtonb,d

aCollege of Physical Science and Technology, Yangtze University, Jingzhou, Hubei 434023, China bBeckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA cCollege of Material Science and Engineering, Hunan University, Changsha, Hunan 410082, China dDepartment of Electrical and Computer Engineering and Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We investigate the electronic properties of semimetallic (12,0) carbon nanotubes in the presence of a variety of monovacancy, divacancy, and hexavacancy defects, by using first principle density Reprinted from J. Appl. Phys., 109, 083716-1–083716-6, 2011. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2011 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Atomic Vacancy Defects in the Electronic Properties of Semi-metallic CNTs

functional theory combined with nonequilibrium Green’s function technique. We show that defect states related to the vacancies hybridize with the extended states of the nanotubes to modify the band edge, and change the energy gap. As a consequence, the nanotube conductance is not a monotonic function of the defect size and geometry. Paradoxically, tetravacancy and hexavacancy nanotubes have higher conductance than divacancy nanotubes, which is due to the presence of midgap states originating from the defect, thereby enhancing the conductance.

28.1 Introduction

The unique electronic, mechanical, and transport properties of single walled carbon nanotubes [1] (SWNTs) make them promising candidates for the ultimate miniaturization of electronic functions at the molecular level [2–4] Extensive experimental [1] and theoretical effects [5] have been deployed to unveil novel transport properties [6]. In particular, the ability to modify the nanotube properties through deformation, doping [7], or the creation of single- and multiple-atom vacancies has attracted considerable attention [6]. In this context it has been shown that electron [8] and ion irradiation [9] can be used to artificially remove individual carbon atoms from SWNTs. Recently, atomically resolved scanning tunneling microscopy (STM) has clearly shown that small holes can be created in metallic multiwall carbon nanotubes with diameter of about 10 nm [10]. While the presence of carbon vacancies governs both the electronic [11] and mechanical properties [12] of nanotubes, they can be used to control the operation as chemical sensors [13]. Nanotube with vacancies also can be used as catalysts for thermal dissociation of water [14], whereas reconstruction of vacancies due to dangling bond saturation can provide active sites for atomic adsorption [15]. The structure and transformation energy of atomic vacancies in nanotubes, and especially, the existence of vacancy clusters are of fundamental importance for understanding the formation as well as the emergence and conditions for fractures in nanotubes [16]. Although some work has been devoted to carbon nanotube with monovacancy [17] and divacancies, by contrast electronic properties of nanotubes with clusters of vacancies have received little attention

Model

[18]. Of special interest is the influence of atomic vacancy defects on the small bandgap opened by the curvature of metallic (m,0) CNTs. Duo to their chirality, (m,m) CNTs do not have this effect [6]. In this chapter, we investigate by ab initio simulation, the combined influences of curvature effect and presence of vacancy defects of various sizes and geometries on the electronic properties of (12,0) semimetallic CNTs. Structure stability is assessed based on the length of the new bonds formed during reconstruction around the defects, and the nature of the defect states are studied by identifying and analyzing the band structure. Our results show that the current in nanostructures with six atoms removed and displaying a highly symmetric pattern is larger than SWNT with divacancy defects. While the point defect divacancy is the most stable because of its smallest transformation energy in contrast to the hexavacancy with largest transformation energy, that does not preclude that hexavacancies are energetically unfavorable in multiwalled nanotube. Higher-order defects can still be formed by removing a group of atoms at once with high energy impacts or chemical etching [16, 19].

28.2 Model

Our study focuses on the zigzag (12,0) SWNT with 10 Å diameter, using a supercell with length L = 42.6 Å (10 unit cells). We use 30 Å distance to the directions perpendicular to the tube axis, which is sufficient to prevent the tube-tube interaction [24]. The optimization structure calculations are performed in the framework of density functional theory, using a basis set of numerical pseudoatomic orbital as implemented in the SIESTA code [20, 21]. Standard norm-conserving Troullier–Martins [22] pseudopotentials orbitals are used to calculate the ion-electron interaction. The exchange correction energy is calculated within the generalized gradient approximation, as parameterized by Perdew et al. [23]. We adopt a double-z polarized basis set with an energy cutoff (for real space mesh points) of 200 Ry for structural relaxations. To test the accuracy of the structure of defected tube after reconstruction, we compare our relaxed structures with published works and found good agreement [18, 24].

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Atomic Vacancy Defects in the Electronic Properties of Semi-metallic CNTs

The I–V characteristics of the nanotube device are obtained for a two-probe device geometry where the central region contains the vacancy defects and both leads consist each of one supercell pristine tube (see Fig. 28.1). The electrochemical potentials for left and right electrode are described by mL(0) and mR(0), provided that selfconsistent calculations of two electrodes are completed. Once a bias voltage Vb is applied, the difference in chemical potentials between the two probes is equivalent to the bias voltage: Vb = mR(Vb) – mL(Vb). Thus, the current is evaluated by means of the Laudauer formula 2e h

Ú

mR (Vb )

T (E , Vb ) dE , (28.1)



I(Vb ) =



T(E, Vb) = 4Tr[Im(SLGrSRGa),

m L (Vb )

where the transmission coefficient T as a function of the electron energy E is obtained by using self-consistent real space, nonequilibrium Green’s function formalism and the density functional theory [25, 26]

(28.2)

where SL (SR) represents the self-energies of the left (right) electrode and Gr (Ga) is the retarded (advanced) Green’s function. It is calculated from the relation

GCr = [ESC - HC - S Lr - S Rr ] = (GCa ) , (28.3)

where SC and HC are the single-electron Hamiltonian matrix and overlap matrix. The conductance G(E) is evaluated at the Fermi level of the device such that G(E) = G0T(E), with G0 = (2e2)/h denoting the quantum conductance.

Figure 28.1 Schematic structure of the two-probe transport model, two semi-infinite left (L) and right (R) electrodes are comprised by one unit cell and extend to z = ± • with periodic boundary condition. The direction of transport is denoted by z.

Results and Discussions

28.3 Results and Discussions In Fig. 28.2, our structure optimization calculations show different configurations of carbon vacancies, i.e., monovacancies (1Va), divacancies (2Va), tetravacancies (4Va, 4Vb), and hexavacancies (6Va, 6Vb) obtained by spontaneously reconstructing stable or metastable structures. The charge density for each configuration is also shown in Fig. 28.2 with a color code for the charge distribution. It is seen that the pristine tube as well as the 2Va, 6Va, and 6Vb structures has a perfectly symmetric charge distribution. Specifically, it indicates that the charge density near the defect in all configurations is not very affected by the vacancy. More importantly, we find the charge density around the defects in 1Va, 4Va, and 6Vb configurations is much larger than that in divacancy configurations. However, the charge density around the defects in 4Va is very small.

Figure 28.2 Charge density of all structural configurations. (a) Ball-andstick model for (12,0) SWNT. (b), (c), (d), (e), and (f) are charge densities of (12,0) SWNT with no-vacancy, monovacancy, divacancy, tetra-vacancy, and hexavacancy, respectively. The defect areas are highlighted by the elliptical solid lines. Blue represents high density; red low density.

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Atomic Vacancy Defects in the Electronic Properties of Semi-metallic CNTs

Table 28.1 lists the calculated transformation energy of all configurations [27]. In the case of monovacancies (1Va), the reconstruction around the defect forms a dangling bond (DB) as well as a pentagon, and leaves a so called 5–1 DB defect, as obtained from our optimization procedure which is in agreement with existing data [28, 29]. Some bond lengths associated with the defect area is depicted in the Fig. 28.3, as can be seen that the C–C bond lengths of the pentagon stretch after relaxation. In particular, the newly formed C–C bond length is increased to 1.53 Å, and the dangling bond carbon forms the newly C–C bond with length approximately 1.39 Å. More importantly, the carbon atom that has the dangling bond moves 0.2 Å along the radial direction while the pentagon is moved toward the axis of the SWNT. Thus, this dangling bond is more likely to form a bridge for connecting two tubes [15] or to provide an active site for atomistic adsorption [30]. The 1Va configuration that only contains a 5–1 DB defect is one of the most stable structures [28]. In the case of divacancies (2Va), four uncoordinated carbon atoms around the missing carbon defects have rebonded together to stabilize the nanostructure, producing two newly formed C–C bonds with length approximately equal to 1.48 Å, thereby a so-called 5–8–5 defect [pentagon-octagon-pentagon (Fig. 28.2)] defects are formed. Note that the shrinkage of the tube diameter caused by the presence of the defects is about 0.67 Å. The 5–8–5 defects in divacancies are the most favorable structure in small diameter tubes [31]. For the tetravacancy cluster, the 4Va configuration contains two pentagonoctagon connected by the quadrangle, while the 4Vb configuration contains an decagon in connection with two pentagon pairs on both left and right sides. Among four missing atom nanostructures, these two configurations are the most stable nanostructures for their comparatively small transformation energy. The separated 5–8 defects in the 4Va configuration resembles the 5–8–5 defects in the 2Va configuration, while the reduction of the tube diameter cause by the two separated 5–8 defects in the 4Va is much smaller than that in the 2Va nanostructure. These differences can be ascribed to largescale spatial distribution of the defects for sharing stress in the 4Va nanostructure. In contrast, the local defect in the 4Vb configuration remarkably affects its circumference, namely, the diameter propagate about 0.56 Å by the pentagon while shortens 0.9 Å by the decagon. The C–C bond lengths of the four pentagons

Results and Discussions

in this nanostructure seems to experience less change, as shown in the Fig. 28.3d. Hence, the effect of diameter propagation extends 14 Å along the tube axis, which substantially decrease the electronic transport due to its atomic reconstruction. It can be seen clearly that the different optimized nanostructures between the 4Va and 4Vb configurations mainly originated from the spatial scale of the defects, and which eventually leading to distinct different electronic and transport properties. For the hexavacancy cluster, the 6Va configuration contains two unsaturated atoms compared with 6Vb with six unsaturated atoms. On the other hand, the 6Va structure manifests as a big hole with 14 number ring connected to two adjacent tetragons, while the length of the whole tube shortens 0.1 Å after relaxation. The diameter of the structure near the defect drastically shrinks which originates from the large hole. As shown in the Figs. 28.3e and f, it is found that the C–C bond length of the

Figure 28.3 Optimized atomic structure of all nanostructure configurations. Ball-and-stick model for (12,0) (a) monovacancy, (b) divacancy, (c) 4Va, (d) 4Vb, (e) 6Va, (f) 6Vb configurations. The defect areas are highlighted by yellow, and some C–C bond lengths are depicted by the blue numeral (in Å).

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Atomic Vacancy Defects in the Electronic Properties of Semi-metallic CNTs

tetragon is remarkably increased to 1.68 Å in the 6Va nanostructure, while the bond length suffers negligible change after relaxation in the 6Vb nanostructure. As for the comparison between unrelaxed and relaxed structures, we find the area of defects become smaller after optimization, but the length of the simulated sample do not change at all. Therefore, it can be seen that except for the cases of 4Vb and 6Va configurations, the existence of six missing atoms only affects the local nanotube electronic structure. Table 28.1 Transformation energy Configurations

Transformation energy

1Va

5.85 eV

4Vb

8.17 eV

2Va 4Va 6Va

6Vb

3.69 eV 6.95 eV

21.31 eV 18.97 eV

In the Fig. 28.4, the energy band structures of the 6 defective (12,0) tubes are displayed with the pristine (12,0) band structure for comparison. The dotted line in the figure indicates the Fermi level. In the pristine (12,0) tube, the highest occupied (lowest unoccupied) band is labeled a (a ¢); both are doubly degenerate, and cross at the G-point, which contributes to two quantum transmission channels. In the 1Va structure, the defect lifts the degeneracy of the two doubled degenerate a and a ¢ electron bands at the G-point, which are now labeled b- and b ¢-bands. Concomitantly, the defect gives rise to a new band labeled g, the lowest unoccupied band that crosses the b-band at the G-point [32]. This g-band results from quasibound unsaturated a orbitals [17], and is partially occupied near the G-point at the expense of the b¢-bands. In the 2Va nanotube, the band structure resembles the 1Va structure with a large splitting of the b-bands at the G-point. Here, the g-band has disappeared, and the d-band is very close and below the Fermi level with a very small dispersion that flattens near the X-point; this flat d-state, which is also predicted by Berber et al. [24] is partially occupied at the expense of the b¢bands near the G-point. The existence of vacancy clusters in the 4Va and 4Vb nanostructures generate a defect state above and below its

Results and Discussions

Fermi level, respectively. In the 4Va nanotube, the b ¢- and a ¢-bands upward shift with a large splitting between them. And the b-bands also moves upward located near the fermi level. Unexpectedly, it is found that the g-band induced by the defects locates 0.65 eV far from the G-point. In the 4Vb configuration, the defects induced g-band, however, it emerges below the Fermi level with a flat tail. In addition to the presence of defect states, the b¢ upward moving is also observed. The striking feature for this configuration is the upward moving for both the a- and a ¢-bands leading to the a-band cross the Fermi level. In the 6Va configuration, the splitting between the two b¢-bands increases over the whole Brillouin zone; both the g-band and the d state reappear as in the 1Va CNTs. Here, the b-bands below the Fermi level are quasidegenerate. The 6Vb configuration consists of a large hole in the nanotube, which produces a new defect state labeled d¢, very close to its parent d-state, above the Fermi level.

Figure 28.4 Band structures of (12,0) SWNTs with various vacancies. The band structure of the pristine (12,0) CNT is given for comparison.

Figure 28.5 shows the local density of state (LDOS) results for specific atoms associated with the defects shown in the Fig. 28.3. In the monovacancy configuration, the two LDOS peaks near the Fermi level derive from the unsaturated atom, thus the unsaturated could provide active site for adsorption. While the highest peak located below the Fermi level derives from the newly formed C–C bond after relaxation. In the divacancy configuration, the small LDOS peak that is positioned slightly above the Fermi level originates from the carbon atoms that forming the octagon. And the formation of the octagon also gives rise to the highest peak below the Fermi level. In the 4Va configuration, it is noted that the two octagons lead to

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Atomic Vacancy Defects in the Electronic Properties of Semi-metallic CNTs

quasibound state resonances near the Fermi level. The highest peak above the Fermi level arises from the appearance of the tetragon. Conversely, the highest peak above the Fermi level arises from the appearance of the two pentagons. It can be seen that comparable large LDOS associated with quasibound states will lead to remarkable reduction of conductance. In the 4Vb nanostructure, the LDOS peaks are mainly originated from the four atoms that connect the adjacent two pentagons in the two pentagon pairs. Moreover, the variation of resonant energies corresponding to the peaks indicates that comparable large LDOS associated with quasibound states will lead to remarkable reduction of conductance in this configuration. In the

Figure 28.5 Local density of states (LDOS) of (12,0) defective tubes for (a) 1Va, (b) 2Va, (c) 4Va, (d) 4Vb, (e) 6Va, (f) 6Vb configurations as a function of resonance energy for specific atoms that are highlighted by yellow in Fig. 28.3.

Results and Discussions

6Va nanostructure, the two unsaturated atoms not only give rise to the two peaks that slightly above and below the Fermi level, but also the striking peak at about 1.25 eV. In the 6Vb nanostructure, it is found that all LDOS peaks are originated from the six unsaturated atoms, thus the distribution of vacancy in this configuration yield more presence of striking peaks at certain resonance energy. It can be seen that the presence of unsaturated atoms related with the LDOS peak near the Fermi level is favorable for electronic transport. In addition, the LDOS results also indicate that the variation of diameter induced by the defect could substantially alter the SWNT’s electronic structure. Figure 28.6 shows the density of states (DOS) on a finer scale around the gap for each vacancy configuration with the (12,0) pristine nanotube. Due to the SWNT curvature effect, the pristine semimetallic (12,0) tube has a small DOS-gap of about 0.07 eV [33], not shown on Fig. 28.4 because of the larger scale. This small gap is symmetric with respect to the Fermi level, and the DOS exhibits two 1D singularities at the edge of a (a¢) bands. The split in the two 1D peaks is due to the lifting of degeneracy of the a (a ¢) bands by the curvature effect. In the 1Va configuration the presence of the 5–1DB defect results in a loss of symmetry of the DOS around the Fermi level, and gives rise to two extremely high DOS peaks around the gap. However, the DOS-gap is smaller compared with the gap in the pristine tube, which is paradoxical given the presence of the defect. We attribute this bandgap reduction to the presence of the g-state that hybridizes with the b- and b ¢-bands. Similar effects have been recently predicted in graphene, where the presence of midgap states due to vacancy defects enhances the metallic character of the material [35]. In the divacancy configurations, the 1D DOS singularities of the pristine structure around the gap broaden into shoulders. In the 2Va structures, there is a DOS peak located at –0.027 eV, which are caused by the presence of the 5–8–5 defects. The symmetric 5–8–5 defects in the divacancy structures substantially lower the DOS that is still symmetric around the Fermi level with a gap of 0.066 eV. In the 4va configuration, it is found that the symmetric DOS gap is lowered to 0.03eV by the introduction of the 5–8–4–8–5 defects, which is in consist with previous predication. Here, our results confirm that some particular defects maybe favorable for transport

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under certain conditions. In the meantime, the defects substantially decrease the 1D singularities below the Fermi level, and give rise to a small DOS peak at 0.008 eV below the Fermi level. The DOS results for the 4Vb configuration are far more complex. The symmetric DOS gap is about 0.03 eV and a small peak below the Fermi level is also observed. However, the 1D DOS singularities near the Fermi level almost vanished, especially for the results above the Fermi level. The emergence of a high DOS peak at 0.038 eV below the Fermi level is due to the presence of defects.

Figure 28.6 Density of states (DOS) of (12,0) defective tubes as a function of energy, DOS of the pristine (12,0) CNT is given for comparison.

In the case of the 6Va structure, the gap is reduced to 0.042 eV, and is symmetric around the Fermi level, except for a large DOS peak at 0.018 eV and a smaller peak at –0.013 eV. Such a big defect

Results and Discussions

induces a sizable modification of the 6Va configuration around the radial direction of the tube, which strongly affects and substantially reduces the DOS near the Fermi level. In the 6Vb configuration, the gap is reduced to 0.058 eV, whereas the dip in the DOS at 0.034 eV originates from resonant scattering by quasibound states [34]. Note that the presence of the two 1D DOS singularities in the valence band is practically unaffected by the defect, but lightly more split than in the pristine nanotube. Figure 28.7 displays the I–V characteristics of the (12,0) structures over a voltage range of 6 mV, which corresponds to an electric field range of 15 KV/cm (1 meV is equal to 2.5 KV/cm). All the I–V characteristics are linear with different conductances (inset). As expected, the conductance of the pristine nanotube is the highest, closely followed by the 1Va conductance, which indicates that in addition to the reduced bandgap (see Fig. 28.4) the 5–1 DB defect does not remarkably scatter electrons. On the opposite, the lower conductance of the 2Va nanostructure is mainly due to the strong electron scattering arising from the 5–8–5 defects. Unexpectedly, the 4Va and 4Vb exhibit different I–V behaviors, that is, the conductance of 4Va is even larger than that of the 2Va. It is due to the particular geometry of the defects, which reduce the DOS gap and the 1D singularity maintains. In contrast, the electron scattering arises from the curvature effects in 4Vb is more pronounced than 4Va due to the distinct shrinking takes place by the presence of pentagon (shown by Fig. 28.3).

Figure 28.7 I–V characteristics and conductance (inset) of (12,0) defective tubes; the results of pristine (12,0) CNT are given for comparison. The color in the I–V curve and conductance (inset) denotes the same structure.

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More importantly, we found that the superior conductance of the 4Va configuration and hexavacancy configurations compared with the divacancy structures. This phenomenon is attributed to the geometry of the defect with highly symmetric patterns (Fig. 28.3), which affects the band and DOS as the larger number of missing atoms reduces the curvature effects. In addition, the 6Va nanostructure has a gap as small as in the 1Va CNT (Fig. 28.4), which boosts the conductance to the level comparable to the 2Va CNTs, although four more atoms are moved compared with the situation in divacancy configuration. In the 6Vb structure the effect of the larger gap is offset by the presence of the d-d ¢ bands (Fig. 28.4) that enhances conductance, especially in the valence band. Therefore, the conductance is not a monotonic function of the defect size and geometry but depends on its spatial symmetry and specific atomic distribution. Specifically, local shrinking induced by the defects remarkably lowers the nanotube conductance.

28.4 Conclusion

In conclusion, we have shown the conductance variation due to the presence of atomic vacancies in carbon nanotubes is not monotonous function of the number of missing C-atoms, but of the reconstruction around the defect and its spatial symmetry. Therefore, the ability to tailor the atomic structure of carbon nanotubes provides new ways to control their transport properties.

Acknowledgments

The authors thank Professor A. Bezryadin, Dr. J. W. Wei, and Dr. T. Markussen for helpful discussion. We also thank Marcelo A. Kuroda for technical assistance in the MAC OS X Turing cluster. This work is financially supported by Natural Science Foundation of China (grant nos. 11047176 and 90923014) and the Research Foundation of Education Bureau of Hubei Province of China (grant no. B20101303).

References

References 1. S. Iijima and T. Ichihashi, Nature (London) 263, 603 (1993).

2. P. G. Collins, A. Zettl, H. Bando, A. Thess, and R. E. Smalley, Science 278, 100 (1997). 3. S. J. Tans and C. Dekker, Nature (London) 404, 834 (2000).

4. C. Zhou, J. Kong, E. Yenilmez, and H. Dai, Science 290, 1552 (2000).

5. J. Han, M. P. Anantram, R. L. Jaffe, J. Kong, and H. Dai, Phys. Rev. B 57, 14983 (1998). 6. J. C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. 79, 677 (2007).

7. M. K. Ashraf, N. A. Bruque, R. R. Pandey, P. G. Collins, and R. K. Lake, Phys. Rev. B 79, 115428 (2009). 8. J. A. Rodriguez-Manzo and F. Banhart, Nano Lett. 9, 2285 (2009).

9. P. M. Ajayan, V. Ravikumar, and J.-C. Charlier, Phys. Rev. Lett. 81, 1437 (1998).

10. T. Aref, M. Remeika, and A. Bezryadin, J. Appl. Phys. 104, 024312 (2008). 11. C. Gomez-Navarro, P. J. De Pablo, J. Gomez-Herrero, B. Biel, F. J. GarciaVidal, A. Rubio, and F. Flores, Nat. Mater. 4, 534 (2005).

12. M. Sammalkorpi, A. Krasheninnikov, A. Kuronen, K. Nordlund, and K. Kaski, Phys. Rev. B 70, 245416 (2004). 13. J. A. Robinson, E. S. Snow, S. C. Badescu, T. L. Reinecke, and F. K. Perkins, Nano Lett. 6, 1747 (2006). 14. M. K. Kostov, E. E. Santiso, A. M. George, K. E. Gubbins, and M. B. Nardelli, Phys. Rev. Lett. 95, 136105 (2005).

15. M. Terrones, H. Terrones, F. Banhart, J.-C. Charlier, and P. M. Ajayan, Science 288, 1226 (2000). 16. C. Jin, K. Suenaga, and S. Iijima, Nano Lett. 8, 1127 (2008).

17. H. L. Choi, J. Ihm, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 84, 2917 (2000). 18. J. Kotakoski, A. V. Krasheninnikov, and K. Nordlund, Phys. Rev. B 74, 245420 (2006).

19. A. Zobelli, A. Gloter, C. P. Ewels, and C. Colliex, Phys. Rev. B 77, 045410 (2008). 20. P. Ordejón, E. Artacho, and L. M. Soler, Phys. Rev. B 53, R10441 (1996).

21. J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys: Condens. Matter 14, 2745 (2002).

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22. N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1993).

23. J.-P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 24. S. Berber and A. Oshiyama, Phys. Rev. B 77, 165405 (2008).

25. J. Taylor, H. Guo, and J. Wang, Phys. Rev. B 63, 245407 (2001).

26. M. Brandbyge, J. L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys. Rev. B 65, 165401 (2002).

27. The transformation energy Et is defined as Et = Ed – (Nd/Np) Ep, where Ed is the total energy of the defective tube, Ep is the total energy of the pristine tube, Nd and Np are the number of the atoms in the defective tube and pristine tube, respectively. 28. A. J. Lu and B. C. Pan, Phys. Rev. Lett. 92, 105504 (2004).

29. J. Rossato, R. J. Baierle, A. Fazzio, and R. Mota, Nano Lett. 5, 197 (2005). 30. B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, Science 315, 77 (2007).

31. R. G. Amorim, A. Fazzio, A. Antônio, F. D. Novaes, and A. J. R. Silva, Nano Lett. 7, 2459 (2007).

32. We simulated the band structure for the SWNT configurations with different number of cells (CNT lengths) and found no qualitative change among them. The only difference are that the δ state becomes flatten with increasing CNT length.

33. M. Ouyang, J. L. Huang, C. L. Cheung, and C. M. Lieber, Science 292, 702 (2000). 34. Y.-W. Son, M. L. Cohen, and S. G. Louie, Nano Lett. 7, 3518 (2007). 35. S. H. M. Jafri et al., J. Phys. D 43, 045404 (2010).

Chapter 29

Chirality Effects in Atomic Vacancy– Limited Transport in Metallic Carbon Nanotubes

Hui Zeng,a,b,d Huifang Hu,a and Jean-Pierre Leburtonb,c

aCollege of Material Science and Engineering, College of Physics and Microelectronic Science, Hunan University, Changsha 410082, China bBeckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA cDepartment of Electrical and Computer Engineering, Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA dCurrent address: College of Physical Science and Technology, Yangtze University, Jingzhou 434023, China [email protected]

We use first principles density functional theory combined with nonequilibrium Green’s function technique to investigate the electronic and transport properties of metallic armchair and zigzag carbon nanotubes (CNTs) with different kinds of multivacancy defects. While the existence of a small band gap in pristine zigzag Reprinted from ACS Nano, 4(1), 292–296, 2010.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2010 American Chemical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Chirality Effects in Atomic Vacancy–Limited Transport in Metallic CNTs

(12,0) CNTs lowers its conductance compared to pristine armchair (7,7) CNTs, transport properties in the presence of multi (hexa)vacancy are superior in the former nanostructure, that is more sensitive to defect size and topology than the latter. In addition, in the zigzag structures hexavacancy nanotubes have higher conductance than divancancy nanotubes, which is due to the presence of midgap states that reduce the transmission gap and enhance the conductance.

29.1 Introduction

Single walled carbon nanotubes (SWNTs) have attracted continued interest since their discovery in 1993, due to their outstanding physical properties [1, 2]. In this context, extensive experimental efforts [1] have been developed to investigate the kinetics of atomicscale defect formation in the nanostructure [3, 4], since defects play a decisive influence on their electronic, transport, and mechanical properties [5]. High-resolution transmission electron microscopy (TEM) studies have shown clear evidence of the presence of point defects such as monovacancy [6, 7], interstitial-vacancy defects [8], and pentagon-heptagon defects [9] in carbon nanotubes (CNTs) and graphene. More recently, Florian reported the possibility of artificially creating individual vacancies in carbon nanostructures by using an electron beam of 1 Å diameter [10]. Although the presence of vacancies deteriorates the genuine properties of nanotubes, in term they may have beneficial effects [11], as for instance, a dangling bond can provide active sites for atomic adsorption [12], which can be used as catalysts for thermal dissociation of water [13]. While considerable theoretical works have been devoted to the understanding of the electronic properties of carbon nanotubes with monovacancies [14] and divacancies [15], transport properties in the presence of clusters of vacancies have received little attention [16]. Such studies are timely since small holes of a few atoms feature sizes that have recently been created in metallic carbon nanotubes with diameter of about 10 nm [17], as clearly evidenced by scanning tunneling microscopy. It was also shown that higherorder defects can be formed by removing a group of atoms with high energy impacts [7] or chemical etching [18]. Among the important

Model

fundamental and technological issues related to the formation of vacancy clusters are the influences of defect size and their spatial symmetry as well as the CNT chirality on the transport properties. In this chapter, we investigate the electronic and transport properties of zigzag and armchair CNTs in the presence of mono-, di-, and hexavacancy defects. For this purpose, we focus on (12,0) and (7,7) nanostructures as both are metallic with different chiralities, where the former (zigzag) is known to develop a small band gap of the order of –50 to 70 meV (semimetallic) [19], whereas the latter (armchair) is gapless around the Fermi level [20]. We use density functional theory to assess the stability of each structure with atomic vacancies subjected to the formation of new bonds during reconstruction estimated from energetic and structural considerations [21]. Quantum transport properties in the stable nanostructures are then computed by means of the nonequilibrium Green’s function technique [22]. Our most striking result is the different conductance behavior between zigzag and armchair nanotubes with atomic vacancy sizes. Indeed, while conductance in the (7,7) CNT is monotonously decreasing with increasing defect size, the conductance in the hexavacancy (12,0) CNT is larger than the divacancy nanostructures. We attribute this different behavior to the presence of the gap in the zigzag CNT. In both cases, different hexavacancy configurations have different influences on the conductance.

29.2 Model

We simulate two metallic (12,0) and (7,7) single wall CNTs (SWNTs) with 10 Å diameter by using a supercell of length L = 42 Å. We perform optimization calculations using the density functional theory, as implemented in the SIESTA code [23]. The standard normconserving Troullier–Martins [24] pseudopotential orbitals are used to calculate the ion-electron interaction. We use the numerical double-z plus polarization (DZP) basis set, and plane cutoff energy is chosen as 200 Ry. The exchange correction term is calculated within the generalized gradient approximation [25]. To test the accuracy of the structure of defected tube after reconstruction, we compare our relaxed structures with published works and found good agreement [15, 16].

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To compute the transport properties of the defective CNTs, we consider a two-probe geometry system [26] constructed in such a way that the central region consists of an optimized supercell (contains 10 unit cells for (12,0) nanotube and 17 unit cells for (7,7) nanotube) containing the defects, which is surrounded by two leads made of one unit pristine supercell on each side. Our band structure calculations for both structures are shown to be consistent with previous works [27, 28] and ensure that our results are reliable. More details about the NEGF formalism can be found in Ref. [22]. The current is calculated by means of the Laudauer formula:

I=

2e h

Ú

mmax

mmin

dE( f l - f r )T (E , Vb )



(29.1)

where the transmission coefficient T as a function of the electron energy E is given by T(E, Vb) = 4Tr[Im(SlGRSrGA)]

(29.2)

Sl (Sr) represents the self-energies of the left (right) electrode, GR (GA) is retard (advanced) Green’s function, fl (fr) is the corresponding electron distribution function of the electron eigenstates of the left (right) electrode. Furthermore, mmin = min (m + Vb, m)[mmax = max (m + Vb, m)] denotes the minimum (maximum) electrochemical potential m of the electrodes. The calculations are performed at T = 300 K.

29.3 Results and Discussion

Figure 29.1 shows different stable or metastable configurations of carbon vacancies, that is, mono- (1Va), di- (2Va), and hexavacancy (6Va, 6Vb) in armchair and zigzag SWNTs obtained after structural optimization by SIESTA. The optimization process is performed by spontaneously reconstructing the CNT around the defects; Table 29.1 lists the transformation energy for all configurations [29]. Generally, the reconstruction around the monovacancy (1Va) defect gives rise to a dangling bond (DB) and leaves a so-called 5–1DB defect [30, 31]. This dangling bond can provide an active site for atomistic adsorption. The 5–1DB defect is one of the most stable structures. Note that the orientations of the 5–1DB defect in (7,7)

Results and Discussion

Figure 29.1 Vacancy configurations of all structural models. (a and b) Balland-stick model for (7,7) and (12,0) SWNT with monovacancy, divacancy, and hexavacancy, respectively. The newly formed C–C bonds during the reconstruction are highlighted by yellow, and the atoms at the far side are omitted for clarity.

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and (12, 0) SWNTs are different. In this framework, we define the defect orientation as the angle made by the axis of defect (indicate by the red arrowed in Fig. 29.1) with the horizontal direction. Hence, the 5–1DB defect in the (7,7) SWNT is titled at 38° with respect to the axis, while the orientation of the 5–1DB defect in the (12,0) SWNT is parallel to the tube axis, which is a consequence of the CNT chirality. In the situation of divacancies (2Va), the optimization process shows that four uncoordinated carbon nanotubes around the missing carbon atoms have bonded together, forming a pentagon–octagon– pentagon (5–8–5) defect. We also observe that the orientation of the 5–8–5 defect in the (7,7) SWNT is titled by 32° with respect to the axis while it remains parallel to the axis in (12,0) SWNT. It is worth mentioning that the 5–8–5 defect is the most stable defect because of its lowest transformation energy (see Table 29.1) [4]. For the hexavacancy clusters, our simulation shows that the (7,7) nanotubes have the lowest formation energies. For this armchair nanostructures, the hexavacancy configurations do not contain any unsaturated atoms, unlike zigzag (12,0) CNTs, for which the 6Va configuration exhibit two unsaturated atoms compared with six unsaturated carbon atoms in the 6Vb configuration. Except for the 6Vb defect in the armchair (7,7) nanotube, all configurations manifest as a large hole in the tube. Specifically, in the (12,0) nanotube, the 6Va configuration contains a symmetric tetra-decagon (14-bond ring) and the 6Vb configuration manifests as a missing hexagon that results in six unsaturated carbon atoms. The transformation energy of the hexavacancy configurations are the largest, as shown in Table 29.1, suggesting that the large holes may split into smaller size vacancies through atomic reconstruction. The lowest transformation energy for the hexa-vacancies occurs for the 6Vb configuration in the (7,7) nanotube, made of two 5–7 defects separated by a hexagon. This configuration is associated with atomic bond shrinking around the defects area, where bonds perpendicular to the nanotube axis are more affected due the CNT curvature. Overall, it can be seen that all defects only affect the local structure of the SWNT, whereas the diameter of the structure near the defect shrinks accordingly.

Results and Discussion

Table 29.1 Transformation energy Ef Configuration

(7,7) SWNT

(12,0) SWNT

1Va

6.41 eV

5.85 eV

6Vb

6.97 eV

18.97 eV

2Va

6Va

3.57 eV

9.41 eV

3.69 eV

21.31 eV

The electronic band structure of the two types of nanotubes with various defect configurations and their pristine analogues are displayed in Fig. 29.2. In the band structure of the pristine (7,7) nanotube, the highest occupied band a and the lowest unoccupied band a ¢ cross at the Fermi level, away from the G-point. In the armchair 1Va configuration, the titled 5–1DB defect creates a statelabeled g that appears above the Fermi level, and results from quasibound unsaturated s-orbitals [14]. This g -state hybridizes with the a, a ¢-bands to evolve into the b, b ¢-bands that anticross at the Fermi level close to the G-point. The g-band is now the lowest unoccupied band [32] which anticross with the b-band opening the direct band gap of about 0.05 eV. In the 2Va configuration, the g-state has moved closer to the Fermi level. The reconstruction process of the tilted divacancy has introduced twisting-type deformations and opened a band gap of about 0.11 eV at the G-point, which is consistent with previous work [15]. Except for the anticrossing of the low b-band close to the X-point (ca. –0.6 eV), the band structure away from the Fermi level does not experience much change; therefore, the large scale deformation (5–8–5 defects) dominates the electronic structure of the tilted di-vacancies nanostructure. In the 6Va configuration, the split b, b ¢-bands move away from the Fermi level significantly, while the g-state approaches the Fermi level, especially close to the X-point. This roll-off at large k vectors effectively reduces the band gap and makes the current in this configuration a little larger than in the 6Vb case (see Fig. 29.3). The band gap of the 6Vb configuration is further increased to 0.325 eV, as b, b ¢- and g-bands move apart. Overall, it can be seen that the band gap in the various vacancy configurations of the (7,7) SWNTs is monotonously increasing with the increasing numbers of atomic vacancy (or the size of the vacancy cluster).

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Figure 29.2 Band structure of (a) (7,7) and (b) (12,0) SWNT with various vacancies. The band structure of the pristine nanotube is also given for comparison.

Figure 29.3 I–V curve and conductance (inset map) of (a) (7,7) and (b) (12,0) defective SWNT, the result of pristine is given for comparison. The same color in I–V curve and in conductance (inset map) denotes the same structure.

In the pristine (12,0) nanotube, the a, a ¢-bands converge at the G-point; both are doubly degenerate and contribute to two quantum transmission channels. Finer scale simulation however reveals a small

Results and Discussion

band gap due to the curvature effect [19]. In the presence of vacancy defects, the doubled degenerate a, a ¢-bands split at the G-point and become the modified b, b ¢-bands. In the 1Va structure, the defect introduces a g-state that results from quasi-bound unsaturated s-orbitals [14], as in the (7,7) SWNT. It crosses the b-band at the G-point, and is partially occupied around the symmetry point at the expense of the b ¢-bands. In the 2Va nanotube, the band structure resembles the 1Va structure with a larger splitting of the b-bands at the G-point. Here, there is a new state called d, very close and below the Fermi level with a very small dispersion that flattens near the X-point, also predicted by Berber et al. [15]. The existence of a large hole in the 6Va configuration opens the band gap by moving the b, b ¢-bands apart, and the d-state is now very flat and above the Fermi level, while the g-band reappears. The band structure resembles the one in the 6Vb configuration of the (12,0) CNTs but with d-state. The 6Vb vacancy that consists of a large hole in the nanotube produces a new defect state labeled d ¢, very close to its parent d-state, above the Fermi level. Figure 29.3 shows the I–V characteristics of the two sets, that is, (7,7) and (12,0), of carbon nanotubes, with the corresponding conductances for each vacancy size and configuration in insets. The voltage range corresponds to an electric field range of 15 KV/cm, for which all I–V characteristics are linear with the applied voltage. For both chiralities, the conductance of the pristine nanotube is the highest, closely followed by the 1Va conductance, indicating that the 5–1DB defect does not remarkably scatter electrons. Also, as expected the conductances of the pristine armchair (7,7) nanostructures are higher than the pristine zigzag (12,0) CNTs because of the absence of transmission gap in the former. The conductance of the divacancy (7,7) nanotube is the second highest due to the (5–8–5) defects that gives rise to significant electron backscattering, and translate into a decrease of the transmission coefficient larger than the 1Va CNT. The conductance continues to decrease in the 6Va (7,7) nanostructures, because backscattering increases compared with the 2Va case. Finally, the 6Vb (7,7) configuration has the lowest conductance because of the smaller transmission coefficient at the Fermi level compared with that of the 6Va CNT. Generally speaking, the (7,7) nanotube conductance is a monotonic function of the defect size and geometry.

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In the (12,0) CNTs, the drop in the conductance of the divacancy nanostructure is slightly deeper than in the 2Va (7,7) CNTs, which we attribute to the presence of the localized d-state close to the Fermi level in the former. Unexpectedly, the conductance of both hexavacancy (12,0) configurations is equal or superior to that of the divacancy (12,0) nanotube. Indeed, both defects have a highly symmetric pattern that reduces the curvature effect compared with the divacancy structure, and thereby reduces the transmission gap, which in the 6Va configuration boosts the conductance to a level comparable to the 2Va CNTs. In the 6Vb (12,0) structure this effect is enhanced by the presence of the b, b ¢-bands, which also enhances the transmission around the gap. This defect is spatially more symmetric than the 6Va vacancy, which reduces backscattering compared to the latter. Moreover, the topology of the defect is directly responsible for the higher conductance of the 6Vb (12,0) CNT compared to its analogue in the armchair (7,7) CNT, as the band gap in the former is even smaller than in the latter structure. The curvature effect can be clearly seen in comparing the distortion produced by the 6Vb defect on both the armchair and zigzag structures in Fig. 29.1. The anomalous conductance variation in the (12,0) nanotubes shows that the transport in a defective CNT is not a direct function of the number of missing atoms but of the chirality and defect pattern in the nanostructure.

29.4 Conclusion

In this chapter, we have shown unexpected conductance variations between armchair and zigzag metallic carbon nanotubes in the presence of multivacancy defects. Specifically, conductance in zigzag CNTs is not a monotonic function of the number of missing C-atoms compared with the conductance of armchair metallic CNTs. In the former, the conductance depends on the reconstruction around the defect and its spatial symmetry. Our results suggest the ability to tailor the atomic structure of carbon nanotubes and provide new ways to control their transport properties.

References

Acknowledgment The authors thank Prof. A. Bezryadin, T. Markussen, Dr. J. W. Wei, and Ph.D. Y. Xu for fruitful discussion. We also acknowledge technical assistance from T. Markussen and M. A. Kuroda. Extensive calculations are performed in the MAC OS X Turing cluster. This work is supported by National Basic Research Program grant no. 2006CB921605. Hui Zeng would like to thank the China Scholarship Council and the Beckman Institute for supporting his study in the University of Illinois at Urbana-Champaign as a joint Ph.D. candidate.

References

1. Iijima, S., Ichihashi, T. Nature 1993, 263, 603–605.

2. Saito, R., Dresselhaus, G., Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1998. 3. Telling, R. H., Ewels, C. P., El-Barbary, A. A., Heggie, M. I. Nat. Mater. 2004, 2, 333–337. 4. Amorim, R. G., Fazzio, A., Antonelli, A., Novaes, F. D., da Silva, A. J. R. Nano Lett. 2007, 7, 2459–2462.

5. Ajayan, P. M., Ravikumar, V., Charlier, J. C. Phys. Rev. Lett. 1998, 81, 1437–1440.

6. Hashimoto, A., Suenaga, K., Gloter, A., Urita, K., Iijima, S. Nature 2004, 43, 870–873. 7. Jin, C., Suenaga, K., Iijima, S. Nano Lett. 2008, 8, 1127–1130.

8. Urita, K., Suenaga, K., Sugai, T., Shinohara, H., Iijima, S. Phys. Rev. Lett. 2005, 94, 105502. 9. Suenaga, K., Wakabayashi, H., Koshino, M., Sato, Y., Urita, K., Iijima, S. Nat. Nanotechnol. 2007, 2, 358–360. 10. Rodriguez-Manzo, J. A., Banhart, F. Nano Lett. 2009, 9, 2285–2289. 11. Krasheninnikov, A. V., Banhart, F. Nat. Mater. 2007, 6, 723733.

12. Terrones, M., Terrones, H., Banhart, F., Charlier, J.-C., Ajayan, P. M. Science 2000, 288, 1226–1229.

13. Kostov, M. K., Santiso, E. E., George, A. M., Gubbins, K. E., Nardelli, M. B. Phys. Rev. Lett. 2005, 95, 136105.

14. Choi, H. J., Ihm, J., Louie, S. G., Cohen, M. L. Phys. Rev. Lett. 2000, 84, 2917–2920.

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15. Berber, S., Oshiyama, A. Phys. Rev. B 2008, 77, 165405.

16. Kotakoski, J., Krasheninnikov, A. V., Nordlund, K. Phys. Rev. B 2006, 74, 245420. 17. Aref, T., Remeika, M., Bezryadin, A. J. Appl. Phys. 2008, 104, 024312.

18. Zobelli, A., Gloter, A., Ewels, C. P., Colliex, C. Phys. Rev. B 2008, 77, 045410. 19. Ouyang, M., Huang, J.-L., Cheung, C. L., Lieber, C. M. Science 2001, 292, 702–705. 20. Kane, C. L., Mele, E. J. Phys. Rev. Lett. 1997, 78, 1932–1935.

21. Jones, R. O., Gunnarsson, O. Rev. Mod. Phys. 1989, 61, 689746.

22. Datta, S. Quantum Transport: Atom to Transistor; Cambridge University Press: New York, 2005.

23. (a) Ordejon, P., Artacho, E., Soler, J. M. Phys. Rev. B 1996, 53, R10441– R10444. (b) Soler, J. M., Artacho, E., Gale, J. D., Garcia, A., Junquera, J., Ordejon, P., Sanchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745– 2779. 24. Troullier, N., Martins, J. L. Phys. Rev. B 1993, 43, 1993–2006.

25. Perdew, J. P., Burke, K., Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865– 3868. 26. (a) Taylor, J., Guo, H., Wang, J. Phys. Rev. B 2001, 63, 245407. (b) Brandbyge, M., Mozos, J.-L., Ordejon, P., Taylor, J., Stokbro, K. Phys. Rev. B 2003, 65, 165401. 27. Yamamoto, T., Nakazawa, Y., Watanabe, K. New J. Phys. 2007, 9, 245.

28. Rocha, A. R., Padilha, J. E., Fazzio, A., da Silva, A. J. R. Phys. Rev. B 2008, 77, 153406. 29. The transformation energy Ef is defined as Ef = Ed–Nd/Np ¥ Ep, where Ed is the total energy of the defective tube, Ep is the total energy of the pristine tube, and Nd and Np are the number of the atoms in the defective tube and pristine tube, respectively. 30. Lu, A. J., Pan, B. C. Phys. Rev. Lett. 2004, 92, 105504.

31. Rossato, J., Baierle, R. J., Fazzio, A., Mota, R. Nano Lett. 2005, 5, 197– 200.

32. We simulated the band structure for the SWNT configurations with different number of cells (CNT lengths) and found no qualitative change among them. The only difference is that the S-state becomes flattened with increasing CNT length.

Chapter 30

Vacancy Cluster–Limited Electronic Transport in Metallic Carbon Nanotubes

Hui Zeng,a,b Jean-Pierre Leburton,b,c Huifang Hu,d and Jianwei Weie aCollege

of Physical Science and Technology, Yangtze University, Jingzhou, Hubei, 434023, China bBeckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA cDepartment of Electrical and Computer Engineering, Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA dCollege of Physics and Mircoelectronic Science, Hunan University, Changsha, Hunan, 410082, China eCollege of Mathematics and Physics, Chongqing University of Technology, Chongqing 400054, China [email protected]

We investigate the electronic properties of metallic (7,7) carbon nanotubes (CNTs) in the presence of a variety of tetra- and hexaReprinted from Solid State Commun., 151, 9–12, 2011. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2010 Elsevier Ltd. Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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vacancy defects, by using the first principles density functional theory (DFT) combined with the non-equilibrium Green’s function technique. From the view point of energetic stability large vacancies tend to split into pentagon and heptagon (5–7) defects. However, this does not preclude the presence of “holes’’ in the carbon nanotube by the nanoelectronic lithography technique. We show that the states linked to large vacancies hybridize with the extended states of the nanotubes to modify their band structure. As a consequence, the hole-like defects in the CNT lead to more prominent electronic transport compared to the situation in the defective CNT consisting of pentagon–heptagon pair defects. Our study suggests the possibility to improve the electronic properties of a defective carbon nanotube via morphological modifications induced by irradiation techniques.

30.1 Introduction

The unique transport properties of single walled carbon nanotubes (SWCNT) have attracted extensive interest in nanoscale electronics since their discovery in 1993 [1, 2]. High-resolution transmission electron microscopy (HRTEM) is widely used as a powerful tool to explore the atomic-scale structures of carbon nanotubes (CNTs) [3]. In addition, the presence of point defects such as monovacancy [4, 5], interstitial vacancy defects [6], and pentagon-heptagon defects [7] have been clearly shown by HRTEM. More recently, Florian et al. demonstrated the possibility of artificially creating individual vacancies in carbon nanotubes by using an electron beam of 1 Å diameter [8], indicating that precisely engineering and shaping carbon nanotubes is possible. The presence of vacancies in CNTs may offer extraordinary mechanics and transport properties [9], as for instance, a dangling bond can provide active sites for atomic adsorption [10], vacancies can reduce the band gap of semiconducting CNTs [11]. Indeed, significant theoretical literatures have concentrated on the electronic properties of CNTs with monovacancies [12, 13], divacancies [14, 15] and a few numbers of vacancies [16]. However, the transport properties in the presence of vacancy clusters have received much less attention [17], although vacancy clusters can lead to the understanding of vacancy formation processes. It is timely

Model

to address the issue because clear evidence has been shown that vacancy clusters of a few atoms can be created in metallic carbon nanotubes with diameter of about 10 nm [18]. More importantly, experimental results revealed that the formation of high-order defects in nanostructures can be precisely tailored by high energy impacts [5] or chemical etching [19]. Interesting results related to the reduction in the energy gap are associated with the anomalous conductance increase in defective CNTs with vacancies [11]. We present a theoretical study of the electronic and transport properties of armchair CNTs with tetra- and hexa-vacancy defects. Through comparing the results of transformation energies and band structure of both metastable and the most stable configurations, we found that large vacancies are energetically unfavorable because they tend to splits into pentagon and heptagon (5–7) defects. However, this does not preclude the presence of “holes” in the carbon nanotube by nanoelectronic lithography technique. The formation of the hole-like defects is more favorable for electronic transport when compared with the 5–7 pair defects case, because the states linked to large vacancies hybridize with the extended states of the nanotubes to modify its band structure. Our study suggests the possibility to improve the electronic properties of defective carbon nanotube via morphological modifications induced by irradiation techniques.

30.2 Model

In order to obtain accurately the electronic structures of defective CNT configurations, we employ density functional theory (DFT) implemented in the SIESTA code [20, 21] to perform the optimization of metallic (7,7) SWCNT with 17 unit cells in length [17]. We used standard norm-conserving Troullier–Martins [22] pseudopotential orbitals to calculate the ion–electron interaction. The numerical bias set is chosen as double-z plus polarization (DZP), and sufficient kinetic cutoff energy is chosen to ensure the convergence of total energy for the whole system. The exchange and correction term is calculated within the generalized gradient approximation [23]. The relaxed structures are found to be in consistence with published work, ensuring the accuracy of all configurations of defective CNT after atomic reconstruction [9, 16].

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Transport properties of defective CNTs are calculated within the two-probe system [24, 25] based on the framework of the nonequilibrium Green’s function (NEGF) [26] technique. The system is composed of three sections, namely, two semi-infinite electrodes and a central scattering region consisting of an optimized supercell containing the defects. We follow the procedure described by Ref. [13] to build the two-probe system structure, and the calculation is performed under room temperature. More details concerning the NEGF formalism can be found in the Ref. [26].

30.3 Results and Discussions

Figure 30.1 depicts the stable or metastable configurations of defective nanostructures after structural optimization by using periodic DFT calculation, that is, tetra- (4Va, 4Vb) and hexa-vacancy (6Va, 6Vb). The stability of each configuration with atomic vacancies is subjected to the formation of new bonds around the defects during reconstruction. The transformation energies of all configurations are listed in Table 30.1 for comparison.a In general, the curvature effect in the carbon nanotube allows shortening the bonds and locally decreasing the diameter of SWCNT, which is opposite to the situation in graphene because the stress can be accommodated on the two-dimensional plane [14, 27]. As in the cases of tetravacancy, the 4Va configuration in the absence of any unsaturated atoms is the most favorable nanostructure, which contains two 5–7 pairs connected with each other by the twisted hexagon. In contrast, the formation of a hole (5–5–5–9 defects) in the 4Vb configuration gives rise to larger transformation energy. The difference between the two transformation energies originates from the fact that the bonds perpendicular to the nanotube axis are more affected due to the CNT curvature, thus the former with the maximum number of perpendicular bonds is the energetically favored nanostructure. On the contrary, 5–5–5–9 defects in the graphene is the preferred structure rather than the 5–7–7–5 defects in that the stress can be shared on the two-dimensional plane in graphene. Similar to the results of tetravacancy, the 6Va configuration with the 5–7–7–5

transformation energy Et is defined as Et = Ed – Nd ¥ Ep, where Ed is the total Np energy of the defective tube, Ep is the total energy of the pristine tube, Nd and Np are the number of the atoms in the defective tube and pristine tube respectively. aThe

Results and Discussions

defects is the most energetically favorable nanostructure among hexavacancies in the armchair CNTs. Unlike the 6Va configuration, the defects in the 6Vb configuration manifests as a large hole in the tube, which is made of four pentagons symmetrically connected with a decagon in the center. The transformation energy, as shown in the Table 30.1, implies that the large hole may split into smaller size vacancies through atomic reconstruction. It can be seen that the vacancy clusters here only affect the local structure of the SWCNT, while the diameter of the structure near the defects shrinks accordingly.

Figure 30.1 Schematic structures of defective carbon nanotube after atomic reconstruction. The vacancy defects are highlighted by blue atoms. Table 30.1 Transformation energy Configurations

Transformation energy (eV)

4Va

6.56

6Vb

9.41

4Vb 6Va

8.45 6.97

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Vacancy Cluster–Limited Electronic Transport in Metallic Carbon Nanotubes

Figure 30.2 Electronic band structures of defective SWCNT with various vacancy clusters. The result of the pristine is also shown for comparing.

The electronic band structure of SWCNTs with various defect configurations is shown in Fig. 30.2. In the band structure of the pristine nanotube, the highest occupied band a and the lowest unoccupied band a′ cross at the Fermi level, away from the G-point. In the 4Va configuration, the 5–7–7–5 defect creates a defect state labeled g that is located above the Fermi level. The g-band is the lowest unoccupied band at present, and it hybridizes with the a- and a ′-bands to evolve into the b- and b ′-bands that anticross at the Fermi level. As a result, the defect state opens a direct band gap of about 0.25 eV at the G-point, while the indirect gap of this configuration is about 0.22 eV. The band structure away from the Fermi level does not experience much change, except for the onset of new states near the Fermi level. Correspondingly, the electronic structure is dominated by a large scale deformation (5–7–7–5 defects). In the 4Vb configuration, it is interesting to notice that its Fermi level is shifted downward in comparison with the pristine CNT. Therefore, the main features of the 4Vb band structure are: (1) the lifted b-band crosses the Fermi level; (2) no distinct defect state near the Fermi level is observed. This downward shift of the Fermi level makes the conductance of this configuration much larger than that in the 4Va case. Except for the downward movement of the Fermi level, the electronic band of this configuration does not experience much change. The 6Va band structure is similar to that of the 4Va, while the gap in the former is further increased to 0.325 eV due to the split b, b ′-bands and the g-bands move away from the Fermi level significantly. In the 6Vb configuration, the g-state approaches

Results and Discussions

the Fermi level with a long dispersionless tail, especially close to the X-point. This roll-off at large k vectors effectively reduces the direct band gap to about 0.246 eV and indirect gap to about 0.115 eV. Generally, band structure results demonstrate that the vacancy cluster can effectively reduce band gap in contrast to the comparable large band gap induced by small pieces of pentagon and heptagon (5–7) defects. Figure 30.3 shows the I–V characteristics of defective SWCNTs are linear with the applied voltage for all configurations. As expected, the conductance of the pristine is the highest, closely followed by the conductance of the 4Vb, suggesting that the 5–5–5–9 defect does not remarkably scatter electrons. In contrast with the 4Vb, the 4Va conductance significantly decreases due to the strong electron backscattering induced by the 5–7 pair defects. The conductance continues to slightly decrease in the 6Va configuration, because backscattering associated with removed atoms increases when compared with the 4Va case. There can be seen that the formation of 5–7 pair defects are unfavorable for electronic transport. In addition, the tiny difference in conductance between the 4Va and 6Va configurations indicates that such 5–7 pair defects seem to uniformly affect the electronic and transport properties; correspondingly, removing more atoms leads to larger energy gap and lower conductance. Considering smaller size vacancies (pentagon and heptagon) are expected to create energetically favorable defects with lowest transformation energy [16], we find our results are in good agreement with the experimental data observed by HRTEM [9], that is, the conductance monotonously decreasing with missing atoms. Unlike the effects of 5–5–5–9 defects on the transport of 4Vb configuration, the formation of the hole-like (5–5–5–5–10) defects in the 6Vb configuration does not improve its conductance significantly when compared with that of the 6Va configuration. We attribute these different behaviors to the different band gaps modified by the defects (see Fig. 30.2). Both defects have highly symmetric patterns that reduce backscattering compared with the nanostructures consisting of 5–7 pair defects. As is well known, the transport properties of defective carbon nanotube depend on their special topology of the defect. This distortion effect can be clearly seen by comparing between the defects in the a type (5–7–7–5

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defects) and b type (5–5–5–9 or 5–5–5–5–10 defects) in Fig. 30.1, where the band gap in the former is larger than that in the latter. Here, we conclude that the hole-like defects in the carbon nanotubes can effectively improve the electronic transport when compared with the situation in the defective CNT consisting of smaller defects, although in certain conditions such improvement is limited.

Figure 30.3 I–V curves and conductances (Inset map) of (7,7) defective tubes; the results of pristine (7,7) CNT are given for comparison. The color in the I–V curve and conductance (inset map) denotes the same structure.

30.4 Conclusion We use first principles calculation combined with the nonequilibrium Green’s function to investigate the electronic and transport properties of armchair carbon nanotubes with vacancy cluster (tetra-vacancy and hexa-vacancy). Molecular dynamics calculation reveals that large vacancies tend to split into pentagon and heptagon (5–7) defects. However, this does not preclude the presence of “holes” in the carbon nanotube by nanoelectronic lithography technique. The hole-like defects in the carbon nanotubes is favorable for electronic transport, as the states associated with large vacancies can hybridize with the extended states to modify

References

their band structure. Consequently, the presence of the hole-like defects in the CNT enhances its conductance compared to that of the defective CNT with the same missing atoms but containing smaller defects. Our results suggest the possibility to improve the electronic properties of defective carbon nanotube via morphological modifications induced by irradiation techniques.

Acknowledgments

We thank Prof. A. Bezryadin for providing some crucial experimental results and Dr. T. Markussen for fruitful discussion. We gratefully acknowledge technical assistance from Dr. M. A. Kuroda and T. Markussen for performing the calculation in the Mac OS X Turing cluster. This work was supported by Science Foundation funded project under grant no. 801080010111 from Yangtze University and grant no. 2008EDJ01 from Chongqing University of Technology, and also from Natural Science Foundation of China under grant no. 10947161.

References

1. S. Iijima, T. Ichihashi, Nature 263 (1993) 603.

2. R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998.

3. Ado Jorio, Gene Dresselhaus, Mildred S. Dresselhaus, Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer-Verlag Press, Berlin, 2008.

4. A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, S. Iijima, Nature 43 (2004) 870. 5. C. Jin, K. Suenaga, S. Iijima, Nano Lett. 8 (2008) 1127.

6. K. Urita, K. Suenaga, T. Sugai, H. Shinohara, S. Iijima, Phys. Rev. Lett. 94 (2005) 105502. 7. K. Suenaga, H. Wakabayashi, M. Koshino, Y. Sato, K. Urita, S. Iijima, Nat. Nanotechnol. 2 (2007) 358. 8. J.A. Rodriguez-Manzo, F. Banhart, Nano Lett. 9 (2009) 2285.

9. A.V. Krasheninnikov, F. Banhart, Nature Mater. 6 (2007) 723.

10. M. Terrones, H. Terrones, F. Banhart, J.-C. Charlier, P.M. Ajayan, Science 288 (2000) 1226.

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11. Yuchen Ma, P.O. Lehtinen, A.S. Foster, R.M. Nieminen, New J. Phys. 6 (2004) 68.

12. H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 84 (2000) 2917.

13. A.R. Rocha, J.E. Padilha, A. Fazzio, A.J.R. da Silva, Phys. Rev. B 77 (2008) 153406. 14. Rodrigo G. Amorim, A. Fazzio, Alex Antonelli, Frederico D. Novaes, AntonioJ. R. da Silva, Nano Lett. 7 (2007) 2459. 15. S. Berber, A. Oshiyama, Phys. Rev. B 77 (2008) 165405.

16. J. Kotakoski, A.V. Krasheninnikov, K. Nordlund, Phys. Rev. B 74 (2006) 245420. 17. Hui Zeng, Huifang Hu, J.-P. Leburton, ACS Nano 4 (2010) 292.

18. T. Aref, M. Remeika, A. Bezryadin, J. Appl. Phys. 104 (2008) 024312.

19. A. Zobelli, A. Gloter, C.P. Ewels, C. Colliex, Phys. Rev. B 77 (2008) 045410. 20. P. Ordejón, E. Artacho, J.M. Soler, Phys. Rev. B 53 (1996) R10441.

21. J.M. Soler, E. Artcho, J.D. Gale, A. Garia, J. Junquera, P. Ordejón, D. Sánchez-Portal, J. Phys.: Condens. Matter. 14 (2002) 2745. 22. N. Toullier, J.L. Martins, Phys. Rev. B 43 (1993) 1993.

23. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.

24. Jeremy Taylor, Hong Guo, Jian Wang, Phys. Rev. B 63 (2001) 245407.

25. M. Brandbyge, J.L. Mozos, P. Ordejón, J. Taylor, K. Stokbro, Phys. Rev. B 65 (2002) 165401.

26. S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, New York, 1995. 27. J.M. Carlsson, M. Scheffler, Phys. Rev. Lett. 96 (2006) 046806.

Chapter 31

Vacancy-Induced Intramolecular Junctions and Quantum Transport in Metallic Carbon Nanotubes

Hui Zeng,a,* Jun Zhao,a,* Jean-Pierre Leburton,b and Jianwei Weic aSchool

of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou, Hubei 434023, China bBeckman Institute for Advanced Science and Technology, Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA cCollege of Optoelectronic Information, Chongqing University of Technology, Chongqing 400054, China [email protected]

First-principles calculations are carried out to study atomic reconstruction and the electronic structures of metallic armchair and zigzag single-walled carbon nanotubes (SWCNTs) in the presence *H.Z.

and J.Z. contributed equally to this work.

Reprinted from J. Phys. Chem. C, 118(40), 22984–22990, 2014. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2014 American Chemical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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of hexa-vacancy defects. The introduction of hexa-vacancy defects can effectively give rise to the formation of intramolecular junctions in both armchair-and zigzag-type SWCNTs. Two kinds of vacancy distribution, six atoms removed along the zigzag chain and a pristine hexagonal lattice, lead to asymmetric and symmetric intramolecular junctions with respect to the axis of the SWCNT after reconstruction, respectively. It is found that the (7,7) symmetric intramolecular junction exhibits more favorable electronic transport compared to the corresponding asymmetric intramolecular junction, whereas the (12,0) symmetric exhibits smaller current in contrast to the asymmetric configuration. This unexpected behavior is attributed to a competition between a widening of the transmission windows that opens additional transmission channels and transport quenching due to orbital misalignment induced by the applied bias.

31.1 Introduction

Owing to their unique electronic and transport properties, SWCNTs are promising candidates for the ultimate miniaturization of nanoelectronic integrated circuits [1, 2]. These quasi-onedimensional (1D) nanomaterials exhibit either metallic or semiconducting properties depending on their diameter and chirality [3]. For this reason, they have stimulated intensive research in a large number of scientific areas [4], which along with their synthesis and characterization, include technological developments with their applications in novel nanoscale devices. Moreover, it has been demonstrated that two SWCNT segments of different diameters and chiralities can be seamlessly joined together to form intramolecular metal–metal, semiconductor–metal, and semiconductor– semiconductor junctions around local defects in the nanostructures [5, 6]. These remarkable 1D nanostructures anticipate exciting new phenomena with potential applications in nanoelectronics [7, 8], such as intramolecular junctions with rectifying properties [9] and negative differential resistance behaviors [10]. Consequently, SWCNT intramolecular junctions could be functional building blocks in carbon-based nanoelectronic circuits [11].

Introduction

Experimentally, the use of energetic particles irradiation is a powerful technique to create point defects such as pentagon or heptagon in two-dimensional graphene and 1D carbon nanotube (CNT) materials. Recently, Rodriguez-Manzo et al. have demonstrated the possibility of introducing vacancies in CNTs with high precision by controlling the electron beam [12]. Although it was suspected that the presence of vacancy defects introducing lattice disorder deteriorates the performance of carbon-based devices, experimental observations have shown that the various defects could, on the contrary, have beneficial influences in practical applications. This is due to the ability of electron and ion irradiation technology to tailor the atomic structure of CNTs. For instance, it has been shown that the formation of intramolecular junction originates from the reconstruction of vacancy clusters through a series of structural transformations [13]. In this context, energetic particle irradiation can be utilized to artificially generate intramolecular junctions in SWCNTs by engineering their atomic structure for controlling their transport properties [14, 15]. On the theoretical side, first-principles calculations have concluded that the formation of hole-like defects is more favorable for electronic transport compared to the pentagon– heptagon (5–7) pair defects [16], whereas two 5–7 defects separated by a perfect hexagon in hexa-vacancy defected CNT is the most stable configuration in armchair CNTs [17, 18]. Because the formation of intramolecular junctions is related to the presence of defects in SWCNTs [2], it is crucial to understand the nature of the vacancy defects and their influences on the SWCNTs’ electronic and transport properties. In this chapter, we conduct a density functional theory (DFT) study on the electronic and transport properties of intramolecular junctions formed in defective zigzag and armchair SWCNTs by introducing hexa-vacancy defects. We focus on the (7,7) and (12,0) nanostructures of approximately 10 Å in diameter, because both are metallic with different chiralities, where the latter (zigzag) is known to develop a small band gap, in contrast to the former (armchair) that has no gap around the Fermi level [19]. Structure stability is evaluated by using first-principles calculation, and the four lowestenergy structures or metastable configurations are obtained after atomistic reconstruction. In both armchair and zigzag SWCNTs, the formation of tilted 5–7–6–7–5 defects makes the tube twist, and

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thus asymmetric intramolecular junctions are produced. In contrast, a symmetric intramolecular junction made up of 4–8–8–4 defects is shown to be stable in armchair tubes, whereas in zigzag tubes, the symmetric junction is made up of 5–7–7–5 defects. It is found that the (7,7) symmetric intramolecular junction exhibits better electronic transport than the corresponding twisted nanostructure, whereas the (12,0) symmetric intramolecular junction exhibits smaller current compared to its corresponding asymmetric configuration. To understand this unexpected result, we investigate quantum transport through these intramolecular junctions and provide a detailed analysis below.

31.2 Computational Methods and Models

The atomically reconstructed process for defective CNTs is implemented by using density functional theory in the framework of the SIESTA code [20, 21]. The interaction between valence electrons and the atomic core is computed within the standard normconserving Troullier–Martins pseudopotential [22]. The numerical double-z plus polarization (DZP) parametrization is chosen as basis set, and the kinetic cutoff energy is set to 200 Ry. The generalized gradient approximation (GGA) in the form of Perdew–Burke– Ernzerhof is used for the exchange-correlation functional [23]. All nanostructure geometries converged until forces acting on all atoms dropped below 0.01 eV/Å. The calculations are performed at electronic temperature T = 300 K. The defective nanostructures were fully relaxed in terms of 17 unit cells for (7,7) nanotube (associated with 476 atoms) and 10 unit cells for (12,0) nanotube (associated with 480 atoms), respectively. Hence, we used a vacuum space of 30 Å as the distance of directions perpendicular to the tube axis, which is sufficient to avoid the tube–tube interaction. According to the previous reports carried out by dynamic simulations and ab initio total energy calculations, the generation of hexa-vacancy along the zigzag chain and one pristine hexagonal lattice would result in the formation of asymmetric and symmetric intramolecular junctions after reconstruction [17, 18]. In order to obtain the intramolecular junctions with different symmetry, we initially remove six atoms along the zigzag chain and one hexagonal

Computational Methods and Models

lattice in the atomic network of both armchair and zigzag pristine CNTs. Therefore, we consider four defective nanostructures, two asymmetric ((7,7) 6Va and (12,0) 6Va) and two symmetric ((7,7) 6Vb and (12,0) 6Vb) configurations according to structural symmetry, as shown in Fig. 31.1. With the corresponding transformation energies of all nanostructures listed in Table 31.1 , the asymmetric intramolecular junctions (6Va) consisting of the 5–7–6–7–5

Figure 31.1 Schematic representation of various intramolecular junction nanostructures after atomic reconstruction. The (7,7) armchair (top panel) and the (12,0) zigzag (bottom panel) asymmetric intramolecular junctions consisting of 5–7–6–7–5 defects (left column of (a) and (c)) are identified as (7,7) 6Va and (12,0) 6Va configurations. The symmetric intramolecular junction consisting of 4–8–8–4 defects (right column of (b)) in armchair tubes and 5–7– 7–5 defects (right column of (d)) in zigzag tubes are identified as (7,7) 6Vb and (12,0) 6Vb configurations, respectively. Top, side, and front sectional views of the optimized nanostructures illustrate the changes in the nanotube lattice. The defect areas are highlighted by yellow color, and the blue bars in figures (b) and (d) correspond to the newly formed (6,6) and (11,0) SWCNT segments, respectively. The atomic lattice of the removed atoms (close-up image and denoted by green color) in the pristine CNT is shown in the hollow area of the front sectional view of the tube.

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defect are present in both armchair and zigzag tubes, as shown in Fig. 31.1a,c. (Note: The transformation energy Et is defined as Et = Ed – Nd/Np ¥ Ep, where Ed is the total energy of the defective tube, Ep is the total energy of the pristine tube, Nd and Np are the number of the atoms in the defective tube and pristine tube, respectively.) In contrast, the symmetric intramolecular junctions (6Vb) consist of the 4–8–8–4 defects in the armchair and the 5–7–7–5 defects in the zigzag SWCNTs. Meanwhile, the symmetric intramolecular junctions in both armchair and zigzag SWCNTs shown in Fig. 31.1b,d are found to be energetically favorable (see Table 31.1). Specifically, removing a hexagonal lattice from pristine CNT could result in the formation of symmetric intramolecular junctions in the nanostructures, which is in good agreement with the dynamic analysis reported previously [13]. To validate our structures of the defective SWCNTs after reconstruction, we have compared our reconstructed nanostructures with previous works in the literature and found excellent agreement [13, 17]. Table 31.1 Transformation energy Configurations

Transformation energy (eV)

(7,7) 6Va

6.97

(12,0) 6Vb

5.42

(7,7) 6Vb

(12,0) 6Va

10.69 14.61

After performing geometry optimization, accurate electronic band structures were calculated. By using a different number of cells (12, 14 unit cells for the (7,7) tube and 6, 8 unit cells for the (12,0) tube), we found no qualitative change among them except for the g-state becoming flat with increasing CNT length, ensuring the reliability of the electronic structure calculation. In order to simulate the electronic transport properties of the defective SWCNTs, a two-probe computational approach is implemented [24–26]. The theoretical model is built in such a way that the central region consists of an optimized supercell with the hexa-vacancy defects, which is surrounded by two semi-infinite leads made up of one pristine primitive cell on each side. The 3 ¥ 3 ¥ 60 Monkhorst–Pack k-point sampling is employed for these two leads during quantum

Results and Discussion

transport calculation. The conductance as a function of electron energy and the I–V characteristics are calculated in the framework of Laudauer formalism:

È T (E ,Vb ) = 4Tr ÍIm Ê Î Ë I=

2e h

Ú

mmax

mmin

ÂG ÂG l

R

r

Aˆ˘

¯ ˙˚

(31.1)

dE( f1 - fr )T (E ,Vb ) (31.2)

where S1 (Sr) represents the self-energies of the left (right) electrode, GR (GA) is retarded (advanced) Green’s function, fl (fr) is the corresponding electron distribution function of the electron eigenstates of the left (right) electrode. Furthermore, mmin = min(m + Vb, m)(mmax = max(m + Vb, m)) denotes the minimum (maximum) electrochemical potential m of the electrodes. Moreover, the detailed methodology and practical implementation of the nonequilibrium Green’s function approach (NEGF) is available in Ref. [26].

31.3 Results and Discussion

As for the (7,7) 6Va intramolecular junction shown in Fig. 31.1a, the 5–7–6–7–5 defect makes the tube twist around the defect area, which locally increases the diameter of the SWCNT at the location of the two pentagons. One notices that the orientations of the 5–7–6– 7–5 defect in (7,7) and (12,0) SWCNTs are different. In this context, we define the defect orientation as the angle made by the axis of the defect (indicated by the angle j and f in Fig. 31.1a,c, respectively) with the horizontal direction. Hence, the 5–7–6–7–5 defect in the (7,7) SWCNT is tilted at j = 22° with respect to the horizontal axis, while the orientation of such defect in the (12,0) SWCNT is f = 73°, as a result of the differences in SWCNTs chirality. In addition, this configuration is the most energetically favorable nanostructure among hexa-vacancies with the armchair chirality [17]. As shown in Fig. 31.1b, the defective area comprising the symmetric 4–8– 8–4 defects (parallel to the axial direction), named (7,7) 6Vb, is predicted to be another metastable configuration compared with previous reports [18, 27]. Unlike the 6Va configuration, the defects in this intramolecular junction manifest a remarkable indentation in the tube, which is clearly evidenced by the front view. The

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symmetric tetragons lead to spherical bulge along radial direction, while the bond that connected two adjacent octagons causes prominent shrinkage at this location, and eventually gives rise to the formation of a (6,6) segment, which is visualized by the side view in Fig. 31.1b. The presence of this symmetric intramolecular junction is accompanied by large-scale deformation around the vacancy defects. Hence, the 4–8–8–4 defects are not only subject to comparable stress, but also result in the largest transformation energy among hexa-vacancy configurations, which is consistent with previous work [17]. The (12,0) 6Va intramolecular junction configuration, containing the laterally oriented 5–7–6–7–5 defects, is found to be a metastable configuration [28]. In this asymmetric configuration, the major changes caused by the lateral hexa-vacancy manifest also as a twisting-type deformation after atomic rearrangement, which resembles that of the (7,7) 6Va configuration. The intramolecular junction formed by the 5–7–7–5 defects (parallel to the axial direction) is the minimum-energy configuration among hexavacancy defective nanostructures in zigzag SWCNT [13], as opposed to the case of (7,7) nanostructures. The conjugation of the 5–7 pairs constitutes the architecture of the (11,0) CNT, leading to remarkable shrinkage at the defective area and the formation of the metal–semiconductor–metal (M–S–M) junction. The opposite side of the defective area of this symmetric intramolecular junction configuration, however, undergoes insignificant changes. The structural formation of this intramolecular junction agrees well with previous DFT results for semiconductor–metal–semiconductor (S–M–S) junction in the (16,0) SWCNT [13], validating the reliability of our simulated results. The electronic band structures of the defective SWCNTs are shown in Fig. 31.2. The pristine (7,7) tube is of strongly metallic type because of its gapless electronic band (Fig. 31.2a) [15, 19]. In the (7,7) 6Va configuration, it is noted that the lowest unoccupied band above the Fermi level, marked as g-band, derives from a defect state due to the presence of the 5–7–6–7–5 configuration. Moreover, the bands near the Fermi level are significantly shifted with respect to the Fermi level, which opens a band gap of 0.33 eV. Conversely, the (7,7) 6Vb configuration remains metallic owing to the crossing of two defect states, that is, the g-band and the d-band located

Results and Discussion

above and below the Fermi level, respectively. The emergence of the symmetric intramolecular junction in this nanostructure not only gives rise to a band shift but also lifts the band degeneracy. In general, the large-scale deformation (4–8–8–4 defects) dominates the electronic structure of this symmetric configuration.

Figure 31.2 Band structures of the armchair (top panel) and the zigzag (bottom panel) CNT consisting of hexa-vacancy defects: (a) (7,7) pristine CNT; (b) and (c) (7,7) 6Va and (7,7) 6Vb configurations shown in Fig. 31.1a,b, respectively; (d) (12,0) pristine CNT; (e) and (f) (12,0) 6Va and (12,0) 6Vb configurations shown in Fig. 31.1c,d, respectively. The Fermi level is indicated by the dotted line (red); the coupling misalignments due to band alterations are denoted by the gray shadows, which correspond to the suppression of quantum conductance.

In the pristine (12,0) nanotube, one should point out that the aand the a¢-bands are doubly degenerate (Fig. 31.2d). Owing to the curvature CNT effect [29], they do not merge at the G-point, which

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produces a tiny band gap of approximately 0.08 eV. In the (12,0) 6Va configuration, the a- and a¢-bands evolve into the modified b- and b¢-bands, respectively. Both of them are pronouncedly shifted away from the Fermi level, generating a direct band gap of 0.38 eV and an indirect band gap of 0.28 eV. In the (12,0) 6Vb configuration, the g-state created by the defects is the lowest unoccupied band with dispersionless tail extending to the X-point. The occurrence of the defect state effectively reduces the direct band gap to about 0.06 eV by anticrossing with the b-band. Compared with the modifications on electronic bands arising from the various hexa-vacancy defects, it is concluded that the tilted 5–7–6–7–5 defects yield more pronounced variations than symmetric intramolecular junction configuration. Hence, these modifications correspond to various effects on the CNT conductance, which are analyzed below. The conductance spectra of the defective SWCNTs are shown in Fig. 31.3. The quantized conductances of the pristine SWCNTs, including both armchair and zigzag types, manifest ballistic transport in the low-dimensional system, and the sharp dip located at the Fermi level in the (12,0) pristine tube derives from curvature effects [8]. For the defective tubes, the conductances are expected to dramatically reduce due to the defect-induced symmetry breaking in the reconstructed nanostructures [30]. For the (7,7) 6Va configuration with the presence of the 5–7–6–7–5 defect (Fig. 31.3a), the conductance reduction at the Fermi level originates from coupling mismatch between its highest occupied subband and lowest unoccupied subband. The variations in conductance are featured by a first plateau at about 1G0 around the Fermi level, indicating roughly 50% suppression of the double available electron transmission channels [31]. This can be explained by its modified electronic bands as its s band is lifted and its s* band is downward shifted, as shown in Fig. 31.2b, which results in transmission channel breaking. In addition, a smooth dip located at about –0.88 eV is the signature of resonant scattering by quasibound state induced by the defect. On the opposite, the 4–8–8–4 defects in the (7,7) 6Vb configuration give rise to less modifications on the conductance at the Fermi level than that in the (7,7) 6Va configuration, which finds their origin in the band structure. As in the (7,7) 6Va configuration, the most noticeable features of the conductance changes in the (7,7) 6Vb nanostructure are concentrated around the Fermi level. Hence,

Results and Discussion

the conductance behavior in the conduction band is analogous to the former configuration, whereas its conductance behavior below EF resembles that of the pristine CNT, due to the preservation of band overlap. In addition, a conductance dip at 0.56 eV below EF can be attributed to a transmission resonance through a quasibound state. As shown in Fig. 31.3b, the 5–7–6–7–5 defects in the (12,0) 6Va nanostructure open a 0.24 eV conductance gap, resulting in full suppression of the transmission channel at the Fermi level. Except for the conductance gap, there are no appreciable changes in conductance around EF compared to that of the pristine. For the (12,0) 6Vb configuration, the conductance spectrum around EF is similar to the pristine SWCNT, despite the quantitative difference on the first plateau. Noticeably, the resonant backscattering associated with the quasibound states is created below the Fermi level, indicating that one of the transmission channels is suppressed.

Figure 31.3 Conductance vs electron energy for the (a) (7,7) and (b) (12,0) SWCNTs consisting of hexa-vacancy defects. The Fermi level is set to be zero. The quasibond states, as denoted by the arrows, are associated with their respective band structures, shown in the insets.

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Vacancy-Induced Intramolecular Junctions and Quantum Transport

The I–V curves of the defective SWCNTs are shown in Fig. 31.4. Clearly, atomic vacancy defects in the nanostructures induce nonlinear I–V characteristics compared to the pristine situation. It is interesting to notice that Fig. 31.4 displays chirality-dependent differences in the I–V characteristics. The (7,7) 6Vb configuration exhibits more favorable electronic transport compared with the twisted nanostructure of the (7,7) 6Va configuration, whereas the analogous (12,0) 6Vb nanostructure manifests unexpected smaller current in contrast to the case of (12,0) 6Va configuration. Generally, the distinct I–V behaviors of the two chiral tubes rely on their band structure and transmission spectra at equilibrium. Paradoxically, as shown in Fig. 31.2e,f, band structure simulations imply that the symmetric intramolecular junction exhibits smaller band gap than that of the asymmetric intramolecular junction, which is consistent with the conductance spectra shown in Fig. 31.3. However, the differences in the conductance spectra of the two configurations at EF are found, as illustrated in the inset of Fig. 31.4. Indeed, the conductance at EF is mainly dependent on whether p electrons can overlap with and hop to p« orbitals. Consequently, the conductance decrease shown in Fig. 31.4a is attributed to misaligned coupling between the p and p« induced by external bias, which is consistent with the I–V curves.

Figure 31.4 I–V characteristics of the (a) armchair- (left column) and (b) zigzagdefective SWCNTs (right column), with the corresponding pristine results. The conductance at EF as a function of applied bias is shown in the insets. The colors in the I–V curves identify the nanostructures.

Results and Discussion

In order to understand in detail the influences of chirality and symmetry of the defects on the I–V behaviors of defective SWCNTs, the transmission spectra as a function of electron energy and bias are plotted in Fig. 31.5. The (7,7) pristine nanostructure is featured by considerable transmission coefficients under various biases. The presence of the 5–7–6–7–5 defect in the (7,7) 6Va nanostructure gives rise to a large region of transmission suppression around the Fermi level, as displayed in Fig. 31.5b. This means that only few electrons can travel from one end of the CNT to the other end, provided that the bias voltage does not exceed the energy range bordered by the bright line (Fig. 31.5b). There also exists a region of moderate transport quenching caused by the 4–8–8–4 defect in the (7,7) 6Vb nanostructure. However, for the transmission displayed in Fig. 31.5c, it is observed that the quenching region does not cross the Fermi level and this region is remarkably narrowed. As a result, electron transport in the (7,7) 6Vb nanostructure is much more prominent than that in the (7,7) 6Va nanostructure, which is also consistent with the I–V characteristics displayed in Fig. 31.4. For the (12,0) pristine nanostructure at zero bias, a sharp transmission gap is found at the Fermi level, which under bias, shifts away from it. Such transmission gap at the Fermi level evolves into two discrete transmission quenching regions located symmetrically with respect to the Fermi level and moving apart with increasing bias. Comparison with Fig. 31.5e and f, d shows significant differences in the transmission coefficients between the pristine CNT and the defective nanostructure in the energy window considered. It should be pointed out that the pristine nanostructure has fairly large transmission coefficient within the critical energy window, and thus, considerable current flowing through the system can be achieved once a small bias is applied, as shown in Fig. 31.4b. In sharp contrast, for the (12,0) 6Va configuration, not only is the transmission quenching region broadened but the transmission coefficient within the critical energy window becomes smaller. Therefore, the current of the (12,0) 6Va configuration is much smaller than that of the pristine CNT under the same bias. As shown in Fig. 31.5f, the transmission coefficient of the (12,0) 6Vb configuration is characterized by a large quenching window spreading over a continuous region indicated by blue color. Furthermore, the transport quenching region is found to broaden with the applied bias. By comparing Fig. 31.5e with

429

Figure 31.5 Color plot of the transmission spectrum as a function of electron energy E (horizontal direction) and applied bias (vertical direction). The color scale on the right indicates the conductance values. The top panel and bottom panel correspond to the armchair- and the zigzag-defected CNTs, respectively: (a), (b), (c) are for the pristine, the 6Va, and the 6Vb configurations of the (7,7) nanostructure; (d), (e), (f) are for the pristine, the 6Va, and the 6Vb configurations of the (12,0) nanostructure. The transmission suppression region referred to the critical energy window is indicated by white solid line.

430 Vacancy-Induced Intramolecular Junctions and Quantum Transport

Conclusions

Fig. 31.5f, we find a difference in the size of the transport quenching area; that is, the suppression of electron transmission in Fig. 31.5f is much more pronounced than that in Fig. 31.5e under the same bias. Therefore, the current of the (12,0) 6Va nanostructure is superior to that of the (12,0) 6Vb nanostructure, which can be understood in terms of these two effects: (1) electron transport channels are available in the former; (2) the states far from the Fermi level provide considerable contributions to the current caused by the widening of the transmission windows in the former with increasing bias. The transmission spectra under different bias thus are virtually responsible for the differences of the I–V curves.

31.4 Conclusions

In summary, we have performed first-principles calculations to investigate spontaneous reconstructions in defective (7,7) and (12,0) SWCNTs. Two types of defect distributions, asymmetric and symmetric intramolecular junction configurations, are studied in both armchair and zigzag SWCNTs after atomic rearrangements. The stability of all defective configurations is assessed in terms of transformation energy. The I–V characteristics of symmetric configurations in the armchair and zigzag SWCNTs are found to exhibit different behaviors compared with those of the corresponding asymmetric nanostructures, although the asymmetric and symmetric defects induced similar band-structure modifications on the two chiral SWCNTs, respectively. This unexpected behavior is attributed to a competition between a widening of the transmission windows that opens new transmission channels and current suppression due to orbital mismatch under applied bias, which is confirmed not only by the variation of the conductance amplitude at the Fermi level but also by the whole transmission spectra as a function of the electron energy for different applied biases. Our calculation provides insight into the formation of intramolecular junctions in SWCNTs and their influences on electron transport in defective SWCNT materials, which are relevant for practical applications in SWCNT-based nanoelectronics.

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Acknowledgments We gratefully acknowledge Prof. K.-L. Yao for fruitful discussions and Dr. Z.-Q. Fan, J.-W. Liang for technical assistance on performing transport properties and relax calculation. This work is financially supported by Natural Science Foundation of China (grant nos. 11304022, 11347010, 11404037, and 11204391), the Research Foundation of Education Bureau of Hubei Province of China (grant nos. Q20131208 and XD2014069), and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (grant no. KJ130831).

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Chapter 32

On the Sensing Mechanism in Carbon Nanotube Chemiresistors

Amin Salehi-Khojin,a Fatemeh Khalili-Araghi,b,c Marcelo A. Kuroda,d Kevin Y. Lin,a Jean-Pierre Leburton,b,c,e and Richard I. Masela

aDepartment of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, USA bBeckman Institute, University of Illinois at Urbana-Champaign, USA cDepartment of Physics, University of Illinois at Urbana-Champaign, USA dDepartment of Computer Science, University of Illinois at Urbana-Champaign, USA eDepartment of Electrical and Computer Engineering, and University of Illinois at Urbana-Champaign, USA [email protected]

There has been recent controversy whether the response seen in carbon nanotube (CNT) chemiresistors is associated with a change in the resistance of the individual nanotubes or changes in the resistance of the junctions. In this study, we carry out a network Reprinted from ACS Nano, 5(1), 153–158, 2011.

Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2011 American Chemical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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analysis to understand the relative contributions of the nanotubes and the junctions to the change in resistance of the nanotube network. We find that the dominant mode of detection in nanotube networks changes according to the conductance level (defect level) in the nanotubes. In networks with perfect nanotubes, changes in the junctions between adjacent nanotubes and junctions between the contacts and the CNTs can cause a detectable change in the resistance of the nanotube networks, while adsorption on the nanotubes has a smaller effect. In contrast, in networks with highly defective nanotubes, the changes in the resistance of the individual nanotubes cause a detectable change in the overall resistance of a chemiresistor network, while changes in the junctions have smaller effects. The combinational effect is also observed for the case in between. The results show that the sensing mechanism of a nanotube network can change according to the defect levels of the nanotubes, which may explain the apparently contradictory results in the literature.

32.1 Introduction

Interpenetrating networks of metallic-semiconducting carbon nanotubes (CNTs) have been increasingly used as one of the key electronic materials for new classes of chemiresistors [1–9]. They consist of arrays of nanotubes between two gold contacts and produce a highly sensitive response compared to other solid-state gas chemiresistors. At this point, there is some controversy whether the response arises from the modulation of nanotube themselves, modulation of the junctions between gold and the nanotubes, or modulation of the junctions between two adjacent nanotubes. For example, Bradley et al. [10] showed that NH3 mainly interacts with carbon nanotubes themselves. In contrast, Peng et al. [11] suggested that the modulation of nanotube–metal electrode junctions influence the response to NH3. In another experiment, Liu et al. [12] observed that both nanotube channels and nanotube–gold junctions play a role in the detection process of NH3. The objective of our work was to do calculations to see if we can understand why different sets of careful experiments give different results. In particular we were interested in determining whether changes in the properties of the nanotubes could change

Introduction

the dominant sensing mechanism. Recall that Gomez-Navarro et al. [13] found that the resistance of a nanotube changes by 3 orders of magnitude as defects form on the nanotube surface. We were interested if such changes were sufficient to switch the dominant mode of sensing. Our results show that the dominant sensing mechanism is highly dependent on the resistance of the nanotube. In particular, we show that in networks consisting of highly conductive (perfect) nanotubes the chemiresistor response is determined by the junctions between adjacent metal nanotubes and the junctions between the nanotubes and the gold. However, in networks with low conductive (heavily defective) nanotubes, the chemiresistor response is determined by modulations in the resistance of the nanotubes themselves. The combinational effect is also observed for the case in between. This conclusion arises from a detailed systematic analysis of the network electric transport analysis considering both metallic and semiconducting CNTs and corresponding homogeneous– heterogeneous junctions. We did such an analysis and asked the question, “How do changes in the resistance of the metal nanotubes, the semiconducting nanotubes, and the junctions between them influence the response of a nanotube network?” Our numerical technique is to vary the resistance of metallic and semiconducting nanotubes and corresponding homogeneous–heterogeneous junctions one at a time within the experimental range and determine the overall change of network conductance. We also considered a similar resistance for junctions between gold electrodes and nanotubes and junctions between two adjacent metallic nanotubes. In detail, each CNT was modeled as a stick of length L that is randomly positioned on a 2D surface. One end of the tube is positioned randomly on the surface and its other end is determined after picking an arbitrary orientation with a uniform probability distribution between –180 and 180 degrees. Coordinates of the CNT junctions (nodes) are determined, and a connectivity matrix (that identifies pairs of nodes that are connected to each other with a finite resistance) is defined to represent the CNT network. In the produced network, the resistance arises because of two contributions: the resistance of the individual CNT and the resistance of junctions. On the basis of Gomez-Navarro et al.’s work [13] we choose three different scenarios in our simulation for CNTs: (i) perfect tube, (ii)

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slightly defective nanotubes, and (iii) heavily defective nanotubes. The resistance of each tube was calculated using the expression in Ref. [14] with a prefactor of f (1 £ f £ 1000) [13] for the defects part as follows:

R= f

R0 È L ˘ Í1 + ˙ (32.1) 2 Î leff ˚

in terms of the quantum resistance R0 = h/(2e2) (where e is the electron charge and h is Planck’s constant) and the effective carrier mean free path leff. In the case of metallic CNTs, the effective mean free path used in the computation was on the order of 600 nm after accounting for the contribution of both acoustic and optical phonons [14]. In the case of intrinsic semiconducting nanotubes, their low carrier loading prevents the electrical conduction in the absence of doping, and their effective resistance becomes more than 4 orders of magnitude larger than their metallic counterparts. The junction resistances were assigned to 15.38R0 for metal–metal junctions, 33.3R0 for semiconducting–semiconducting junctions, and 100 times higher than that of metal–metal for metal–semiconducting junctions [15]. Applying Kirchhoff’s laws to the resulting network of resistors, the overall conductance of the network was calculated. The simulations corresponding to a certain network loading were repeated between 300 and 1000 times (depending on the CNT loading), and average values for the conductance of the network over these repetitions are reported.

32.2 Results and Discussion

Figure 32.1 displays the nanotube networks’ conductance obtained from simulation for the nanotube loadings per unit area up to 6 mm–2. This is equivalent to 4 mg/L nanotube in solution, which will be discussed later. We define P as the probability of finding at least one conducting path in the network. The inset of this figure shows the percolation probability of the chemiresistors, P, as a function of the CNT loading. The dependence of network conductivity (s) on CNT loading obtained from simulation is in accordance with the standard percolation theory described by s µ (N – Nc)a, where N is the volume loading of the nanotube solution, Nc is the critical volume

Results and Discussion

loading of the CNT corresponding to the percolation threshold (Nc = 1/p(4.236/Ltube)2, and a is a critical fitting exponent. The best fit of s µ (N – Nc)a to the theoretical curve shown in Fig. 32.1 results in a = 1.92, which is close to the theoretical prediction of a = 1.94 [16]. Here we point out that the simulations performed in this study provide detailed information about nanotube networks, such as distribution of the current among metallic and semiconducting nanotubes, which cannot be accessed from experiment and analytical percolation equations.

Figure 32.1 (a) Experimental and simulated conductance vs nanotube loading. The average conductance for the ensemble of networks fits very well to experimental results using a conversion factor of 0.66, the ratio of mass per unit volume of nanotube solution (mg/L) over nanotube loading in unit area (mm–2).

We also did experiments to verify the predictions of the computations. CNT networks shown in Fig. 32.2 were fabricated using liftoff photolithography as described in our previous papers [17, 18] and the conductance shown in Fig. 32.1 was measured. The average conductance measured for the network shows very good agreement with the simulations using a conversion factor of 0.66, the ratio of mass per unit volume of nanotube solution (mg/L) over nanotube loading in unit area (mm–2). Figure 32.3 shows the change in overall conductance of a network with equivalent loading of 1 mg/L by separately varying the resistance of the metallic nanotubes, the resistance of the semiconducting nanotubes, the resistance of the junctions between

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adjacent metal nanotubes, the resistance of junctions between adjacent semiconducting nanotubes, and the resistance of the junctions between adjacent metallic and semiconducting nanotubes. We did the analysis for three different conductance levels of nanotubes: (a) perfect nanotubes (f = 1 in Eq. 32.1), (b) heavily defective nanotubes (f = 500), (c) slightly defective nanotubes (f = 70). For perfect nanotubes (Fig. 32.3a), we found that if the resistance of the junctions between adjacent metallic nanotubes changes as one might expect if gas adsorbs, there is a large change in the conductance of the network. In contrast variations in the resistance of metallic nanotubes, semiconducting nanotubes, and junctions between two adjacent semiconducting nanotubes and between the adjacent metallic–semiconducting nanotubes have little effect.

Figure 32.2 (a) SEM images of fabricated nanotube chemiresistors. The gap zone and electrodes are labeled G and E in the SEM images, respectively. (b) Display of a typical Raman spectrum of our nanotube chemiresistors (1 mg/L) that include nanotube peaks, which consist of a sp3-like disorder band (D) around 1340 cm–1 and a sp2-like tangential band (G) around 1590 cm–1. The Raman spectra were obtained using an Ar+ laser operating at 514.5 nm (2.41 eV), a spot size of 50 mm diameter, and 3 mW of power on the sample.

Results and Discussion

By comparison when the nanotubes are highly defective, changes in the resistance of the defective nanotubes have a significant effect on the overall conductance of the network. However, the variations in the resistance of all kinds of junctions have negligible effects. We also see the combinational effects for the case of slightly defective nanotubes, as shown in Fig. 32.3c. These results clearly indicate that different conductance levels of nanotubes yield different sensing mechanisms in a network. Physically, pristine nanotubes have a very low resistance. In such a case, the resistance of the nanotubes is low compared to the resistance of the junctions between adjacent nanotubes, so large percentage changes in the conductance of the nanotubes (i.e., changes larger than one would expect for gas adsorption) do not produce a significant change in the conduction of the network. The semiconducting nanotubes also have little effect because they have such high resistance that there is little or no current through them. Thus, only changes in the resistance of the junctions, either between adjacent nanotubes or between the nanotubes and the gold, have a significant effect of the resistance on the network. Defective nanotubes show the opposite effect. Defects can vary the resistivity of the nanotubes by 3 orders of magnitude [13]. In that case the resistance of the nanotubes is large compared to the resistance of the junctions. Hence, the changes in the resistance of the nanotubes have an important effect on the overall conduction of a network. The analysis above considered only Shottkey–Richardson conduction in the nanotubes (i.e., conduction through the conduction band), but at higher voltages electron transport also occurs via Poole–Frenkel conduction if there are defects in the nanotube. If electrons are transported via the Poole–Frenkel mechanism [17, 19, 20], the electron hopping through nanotube defects can inject accumulated charges at the defect sites to the conduction band of the nanotubes, which in turn changes the conductance of the chemiresistor upon gas adsorption. That causes an enhancement in the sensitivity of the nanotubes because the analyte concentration is higher on defects than on pristine regions of the nanotubes. Details of such a mechanism have been explained in Ref. [17].

441

Figure 32.3 Variation of the nanotube network conductance as a function of the change in the resistance of metallic nanotubes, the change in the resistance of semiconducting nanotubes, and the change in the resistance of junctions between two adjacent nanotubes for the case of (a) perfect nanotube, f = 1; (b) highly defective nanotubes, f = 500; and (c) slightly defective nanotubes, f = 70.

442 On the Sensing Mechanism in Carbon Nanotube Chemiresistors

Results and Discussion

To determine the efficient current required for achieving pure electron hopping, we considered the experimental condition where we applied a constant external current and obtained the histogram of currents passing through nanotube segments. Figure 32.4a shows the histogram results presented in the form of the fraction of nanotubes that carry a specific current at each nanotube loading, while Fig. 32.4b shows the sensitivity of the chemiresistors to adsorption of gas. The details of the experimental section have been explained in Refs. [17, 18]. In this calculation, we assumed that our network was made of highly defective nanotubes since the Raman spectra shown in Fig. 32.2b show a large density of defects. It is useful to compare parts a and b of Fig. 32.4. We observe a similar trend for both sensitivity and fraction of nanotubes vs nanotube loading for current equal to or less than 100 nA. We also observe that there are no nanotube segments at higher loadings that carry currents higher than 100 nA. The results suggest that 100 nA is an upper limit for the minimum current required for an efficient electron-hopping process. Figure 32.4 also indicates that the fraction of nanotubes that carry a 100 nA current decreases with the increase in the network loading. This result suggests a similar trend in the electron-hopping distance (or range) vs nanotube loading. If this is the case, thus, one should also expect a similar trend in the charge fluctuation and consequently in the 1/f noise level. To check this hypothesis, we measured the 1/f noise level of our chemiresistors at three different applied currents for different nanotube loadings. Results are displayed in Fig. 32.5. We attribute the increased 1/f noise level at lower nanotube loadings to higher charge fluctuations in these networks due to higher extent of electron-hopping processes. A diminished trend of 1/f noise with network loading is also consistent with the reduction in the electron-hopping distance (or range), as shown in Fig. 32.4. The results indicate that the electron-hopping mechanism in Poole– Frenkel conduction well describes the sensitivity and noise level in highly defective nanotube chemiresistors.

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On the Sensing Mechanism in Carbon Nanotube Chemiresistors

Figure 32.4 (a) Length fraction of nanotubes that carries specific current vs nanotube loading. 100 nA is the maximum current that the length fraction of all networks is able to carry. Lower currents are not efficient, and higher currents exist only in lower nanotube loadings. (b) Typical normalized response to 10 ppm DMMP gas molecules for nanotube chemiresistors with loadings of 0.75 to 8 mg/L. The inset shows the response to ammonia molecules.

To summarize, we performed network simulation to differentiate the sensing mechanism in nanotube chemiresistors. We found that the change in the resistance of nanotubes can modify the dominant detection mechanism in nanotube chemiresistors. Our results showed that networks with highly defective nanotubes

Methods

were influenced only by change in the resistance of the nanotube themselves, while networks with pristine nanotubes were modulated only by modulation of the resistance of junctions. Clearly, in the latter case, among junctions between a gold electrode and nanotubes, and junctions between adjacent metallic nanotubes, the one with lower conductance dominates the overall conductance of nanotube networks. Our results explain how seemingly identical studies done carefully on different nanotubes and with different fabrication techniques can reach different conclusions on the dominant sensing mechanism in nanotube networks, as reported by Bradley et al. [10], Peng et al. [11], and Liu et al. [12].

Figure 32.5 Measured noise density for nanotube chemiresistors with nanotube loadings ranging from 0.75 to 8 mg/L at three applied electric currents. The maximum noise level is at 1 mg/L. The noise level decreases with the network loading and then stays fairly constant. A similar trend was observed in Figs. 32.4a and 32.4b.

32.3 Methods 32.3.1 Fabrication and Design of the Chemiresistor Nanotube chemiresistors were fabricated using standard lift-off photolithography. A silicon substrate with a thermal oxide layer (500/0.6 mm Si/SiO2) was patterned with chrome and gold (17/100 nm Cr/Au) for source and drain electrodes separated by a 6 mm gap. 3-Aminopropyltriethoxysilane (APTES, 1%) (Gelest, Inc.) was

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utilized as a supportive coating to enhance the interaction between the SWNT film and the silicon substrate.

32.3.2 SWNT Preparation and Deposition on Silicon Substrate

A highly concentrated SWNT suspension (400 mg/L) was made from 10 mg of SWNT powder (Unidym, High Purity HIPCO) and 1% (w/v) sodium dodecyl sulfate (SDS) in water. Multiple sets of 10 min, low-powered ultrasonication (at 40% power and 90% frequency), 1 h stirring, and 3 h centrifugation (at 2800g or 4100 rpm) were performed to homogenize and uniformly disperse the suspension. The highly concentrated, homogeneous SWNT suspension was then diluted to a selected concentration (mg/L) in a 25 mL solution before being vacuum filtered with mixed cellulose ester (MCE) membranes (Millipore, 0.22 mm pore size). After the SWNT was successfully deposited onto the membranes, the wet MCE-SWNT membrane was dried for at least 2 h under 381 mmHg gauge pressure before multiple rinsings with approximately 80 mL of purified and deionized water (Millipore, Milli-Q water). Multiple rinsing was intended to completely remove the SDS residue from the MCE-SWNT membrane. Finally, a stamp technique was used to transfer homogeneous, randomly aligned CNT films to the APTEStreated silicon surface.

Acknowledgment

This work was supported by grants from Cbana Laboratories and Dioxide Materials. The authors would like to thank Dr. Behnam for his useful comments on nanotube network design.

References

1. Kong, J., Franklin, N. R., Zhou, C., Chapline, M. G., Peng, S., Cho, K., Dai, H. Nanotube Molecular Wires as Chemical Sensors. Science 2000, 287, 622–625.

2. Lee, C. Y., Sharma, R., Radadia, A. D., Masel, R. I., Strano, M. S. On-Chip Micro Gas Chromatograph Enabled by a Noncovalently Functionalized Single-Walled Carbon Nanotube Sensor Array13. Angew. Chem. 2008, 120, 5096–5099.

References

3. Novak, J. P., Snow, E. S., Houser, E. J., Park, D., Stepnowski, J. L., McGill, R. A. Nerve Agent Detection Using Networks of Single-Walled Carbon Nanotubes. Appl. Phys. Lett. 2003, 83, 4026–4028.

4. Peng, G., Tisch, U., Haick, H. Detection of Nonpolar Molecules by Means of Carrier Scattering in Random Networks of Carbon Nanotubes: Toward Diagnosis of Diseases via Breath Samples. Nano Lett. 2009, 9, 1362–1368. 5. Robinson, J. A., Snow, E. S., Badescu, S. C., Reinecke, T. L., Perkins, F. K. Role of Defects in Single-Walled Carbon Nanotube Chemical Sensors. Nano Lett. 2006, 6, 1747–1751. 6. Robinson, J. A., Snow, E. S., Perkins, F. K. Improved Chemical Detection Using Single-Walled Carbon Nanotube Network Capacitors. Sens. Actuators, A 2007, A135, 309–314.

7. Snow, E. S., Perkins, F. K., Houser, E. J., Badescu, S. C., Reinecke, T. L. Chemical Detection with a Single-Walled Carbon Nanotube Capacitor. Science 2005, 307, 1942–1945.

8. Vichchulada, P., Zhang, Q., Lay, M. D. Recent Progress in Chemical Detection with Single-Walled Carbon Nanotube Networks. Analyst (Cambridge, U. K.) 2007, 132, 719–723. 9. Zribi, A., Knobloch, A., Rao, R. CO2 Detection Using Carbon Nanotube Networks and Micromachined Resonant Transducers. Appl. Phys. Lett. 2005, 86, 203112/1-203112/3.

10. Bradley, K., Gabriel, J. C. P., Star, A., Gruner, G. Short-Channel Effects in Contact-Passivated Nanotube Chemical Sensors. Appl. Phys. Lett. 2003, 83, 3821–3823. 11. Peng, N., Zhang, Q., Chow, C. L., Tan, O. K., Marzari, N. Sensing Mechanisms for Carbon Nanotube Based NH3 Gas Detection. Nano Lett. 2009, 9, 1626–1630. 12. Liu, X., Luo, Z., Han, S., Tang, T., Zhang, D., Zhou, C. Band Engineering of Carbon Nanotube Field-Effect Transistors via Selected Area Chemical Gating. Appl. Phys. Lett. 2005, 86, 1–3.

13. Gomez-Navarro, C., Pablo, P. J. D., Gomez-Herrero, J., Biel, B., GarciaVidal, F. J., Rubio, A., Flores, F. Tuning the Conductance of Single-Walled Carbon Nanotubes by Ion Irradiation in the Anderson Localization Regime. Nat. Mater. 2005, 4, 534–539. 14. Kuroda, M. A., Leburton, J. P. High-Field Electrothermal Transport in Metallic Carbon Nanotubes. Phys. Rev. B 2009, 80, 165417.

15. Fuhrer, M. S., Nygard, J., Shih, L., Forero, M., Yoon, Y.-G., Mazzoni, M. S., Choi, H. J., Ihm, J., Louie, S. G., Zettl, A., et al. Crossed Nanotube Junctions. Science 2000, 288, 494–497.

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16. Stauffer, G. Introduction to Percolation Theory; Taylor & Francis: London, 1985. 17. Salehi-Khojin, A., Field, C. R., Yeom, J., Shannon, M. A., Masel, R. I. Sensitivity of Nanotube Chemical Sensors at the Onset of Poole-Frenkel Conduction. Appl. Phys. Lett. 2010, 96, 163110–163113.

18. Salehi-Khojin, A., Lin, K. Y., Field, C. R., Masel, R. I. Nonthermal Current Stimulated Desorption of Gases from Carbon Nanotubes. Science 2010, 329, 1327–1330. 19. Jombert, A. S., Coleman, K. S., Wood, D., Petty, M. C., Zeze, D. A. PooleFrenkel Conduction in Single Wall Carbon Nanotube Composite Films Built Up by Electrostatic Layer-by-Layer Deposition. J. Appl. Phys. 2008, 104, 094503/1–094503/7. 20. Suehiro, J., Imakiire, H., Hidaka, S.-i., Ding, W., Zhou, G., Imasaka, K., Hara, M. Schottky-Type Response of Carbon Nanotube NO2 Gas Sensor Fabricated onto Aluminum Electrodes by Dielectrophoresis. Sens. Actuators, B 2006, B114, 943–949.

Chapter 33

Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons

Hui Zeng,a,b Jean-Pierre Leburton,b,c,d Yang Xu,e and Jianwei Weif aCollege

of Physical Science and Technology, Yangtze University, Jingzhou, Hubei 434023, China bBeckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA cDepartment of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA dDepartment of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA eDepartment of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, China fCollege of Mathematics and Physics, Chongqing University of Technology, Chongqing 400054, China [email protected]

Reprinted from Nanoscale Res. Lett., 6, 254, 2011. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2011 Zeng et al.; licensee Springer Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons

The electronic transport of zigzag-edged graphene nanoribbon (ZGNR) with local Stone-Wales (SW) defects is systematically investigated by first principles calculations. While both symmetric and asymmetric SW defects give rise to complete electron backscattering region, the well-defined parity of the wave functions in symmetric SW defects configuration is preserved. Its signs are changed for the highest-occupied electronic states, leading to the absence of the first conducting plateau. The wave function of asymmetric SW configuration is very similar to that of the pristine GNR, except for the defective regions. Unexpectedly, calculations predict that the asymmetric SW defects are more favorable to electronic transport than the symmetric defects configuration. These distinct transport behaviors are caused by the different couplings between the conducting subbands influenced by wave function alterations around the charge neutrality point.

33.1 Introduction

As a truly two-dimensional nanostructure, graphene has attracted considerable interest, mainly because of its peculiar electronic and transport properties described by a massless Dirac equation [1, 2]. As such, it is regarded as one of the most promising materials since its discovery [3–5] because charge carriers exhibit giant intrinsic mobility and long mean-free path at room temperature [6, 7], suggesting broad range of applications in nanoelectronics [8–11]. Several experimental [4, 8, 12, 13] and theoretical [2, 14, 15] studies are presently devoted to the electronic, transport, and optical properties [16] of graphene. By opening an energy gap between valence and conduction band s, narrow graphene nanoribbons (GNR) are predicted to have a major impact on transport properties [17, 18]. Most importantly, GNR-based nanodevices are expected to behave as molecular devices with electronic properties similar to those of carbon nanotubes (CNTs) [19, 20], as for instance, Biel et al. [21] reported a route to overcome current limitations of graphenebased devices through the fabrication of chemically doped GNR with boron impurities. The investigation of transport properties of GNRs by various experimental methods such as vacancies generation [22],

Model and Methods

topological defects [23], adsorption [24], doping [25], chemical functionalization [26–28], and molecular junctions [29] have been reported. Meanwhile, defective GNR with chemically reconstructed edge profiles also have been experimentally evidenced [30] and have recently received much attention [31, 32]. In particular, StoneWales (SW) defects, as one type of topological defects, are created by 90° rotation of any C–C bond in the hexagonal network [33], as shown by Hashimoto et al. [34]. More recently, Meyer et al. [35] have investigated the formation and annealing of SW defects in graphene membranes and found that the existence of SW defects is energetically more favorable than in CNTs or fullerenes. Therefore, the influences of SW defects on electronic transport of GNRs is crucial for the understanding of the physical properties of this novel material and for its potential applications in nanoelectronics. In this brief communication, we investigate the influence of SW defects on the electronic transport of zigzag-edged graphene nanoribbons (ZGNRs). It is found that the electronic structures and transport properties of ZGNRs with SW defects can very distinctively depend on the symmetry of SW defects. The transformation energies obtained for symmetric SW defects and asymmetric SW defects are 5.95 and 3.34 eV, respectively, and both kinds of defects give rise to quasi-bound impurity states. Our transport calculations predict different conductance behavior between symmetric and asymmetric SW defects; asymmetric SW defects are more favorable for electronic transport, while the conductance is substantially decreased in the symmetric defects configuration. These distinct transport behaviors result from the different coupling between the conducting subbands influenced by the wave function symmetry around the charge neutrality point (CNP).

33.2 Model and Methods

The optimization calculations are done by using the density functional theory utilized in the framework of SIESTA code [36, 37]. We adopt the standard norm-conserving Toullier-Martins [38] pseudopotentials orbital to calculate the ion-electron interaction. The numerical double-z polarized is used for basis set and the plane cutoff energy is chosen as 200 Ry. The generalized

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Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons

gradient approximation [39] proposed by Perdew and Burke and Ernzerhof was employed to calculate exchange correction term. All nanostructure geometries were converged until no forces acting on all atoms exceeded 0.01 eV/Å.

Figure 33.1 Schematics of the molecular device considered in the calculation of ZGNR. The whole device is composed of scattering region and two electrodes containing the corresponding pristine ZGNRs. The SW defects are highlighted by yellow atoms (a) ZGNR with symmetric SW defects, some C-C bond lengths and angles are shown by (c); (b) ZGNR with asymmetric SW defects, while some C-C bond lengths and angles are shown by (d).

The electronic transport properties of the nanoribbon device have been performed by using non-equilibrium Green’s function (NEGF) methodology [40, 41]. In order to self-consistently calculate the electrical properties of nanodevices, we construct the two-probe device geometry where the central region contains the SW defects and both leads consist each of the two supercell pristine ZGNR, as shown in Fig. 33.1. The equilibrium conductance G is obtained from the Landauer formula such that G = G0T (E), where G0 is the

Results and Discussions

2e2 . The transmission h coefficient T as a function of the electron energy E is given by

quantum conductance with relationship G0 =

T (E) = 4Tr [Im (SlGR SrGA)]

(33.1)



GR = (e – Hs – Sl – Sr)–1 = (GA)†

(33.2)

where Sl (Sr) represents the self-energies of the left (right) electrode,

GR (GA) is retard (advanced) Green’s function. It is calculated from the relation:

where Hs is the Hamiltonian of the system. More details about the NEGF formalism can be found in Ref. [42]. In this study, we consider symmetric and asymmetric SW defects contained in 6-ZGNRs, where 6 denotes the number of zigzag chains (dimers) across the ribbon width [18]. Taking into account screening effects between electrodes and central molecules, we use 10-unit cell’s length as scattering regions, and 2 units as electrodes to perform transport calculation. The electron temperature in the calculation is set to be 300 K.

33.3 Results and Discussions

In Fig. 33.1, we show the geometry of defective ZGNR after relaxation. After introducing symmetric SW defects, the GNR shrinks along the width axis, by 0.526 Å, and correspondingly, the nearest four H atoms move toward the central region by 0.21 Å. As a result, the bond angles of the edge near the SW defects are reduced from 120 to 116°, as shown in Fig. 33.1c, d. In contrast to the shrinking along the width axis, the SW defects stretch from 4.88 to 5.38 Å along the length axis direction. No distinct change for the H-C bond length at the edge is observed. Thus, the effect of symmetric SW defects on the geometry modification is limited to the defective area, with mirror reflection around their axis. However, the presence of asymmetric SW defects to the geometry modification is far more complex. They twist the whole structure by shifting the left side upward, while the right side is downward shifted. Hence, the mirror symmetry is broken because of the asymmetric SW. The transformation energies for symmetric and asymmetric SW defects are 5.95 and 3.34 eV, respectively. These results imply that the asymmetric SW defects are energetically more favorable than the symmetric SW defects.

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Defect Symmetry Influence on Electronic Transport of Zigzag Nanoribbons

Wave functions of electronic states at the Gamma point of the highest-occupied electronic states (HOES) and the lowestunoccupied electronic states (LUES) are depicted in Fig. 33.2. As expected, the wave functions of the pristine even-index ZGNR at the Gamma-point associated to the HOES and LUES exhibit the welldefined parity with respect to the mirror plane, and their eigenstates in the case of symmetric SW defects, the HOES and LUES, keep the same parity because of the potential induced by the symmetric defects [25, 43]. Note that, although the wave functions of both the pristine and symmetric SW defects have well-defined parity, the sign of their wave functions, especially for the electronic states below the CNP, are precisely opposite. For the asymmetric SW defects, the welldefined parity of the wave functions is not preserved. Moreover, the wave function symmetry in this configuration is broken leading to substantial electron backscattering below and above the CNP.

Figure 33.2 Wave functions at the gamma point of defective ZGNRs. Wave functions at the gamma point associated with the LUES above the CNP (top panel) and the HOES below the CNP (bottom panel) for ZGNR with no defects (a, d), symmetric SW defects (b, e) asymmetric SW defects (c, f). Blue and red colors correspond to the opposite signs of the wave function.

The central issue of this study is to investigate the influence of SW defects in the ZGNRs on their electronic and transport behavior. ZGNRs are known to present very peculiar electronic structure, that is, strong edge effects at low energies originated from the wave functions localized along the GNR edges [44]. Spin-unpolarized calculations reveal that all ZGNRs are metallic with the presence of sharply localized edge states at the CNP [25, 43, 44], while ab initio

Results and Discussions

calculation with spin effect taken into consideration found that a small band gap opens up [18]. The electronic band structures of defective nanoribbons and the corresponding pristine GNRs are shown for comparison. In the case of pristine GNR, zone-folded effects give rise to nondegenerated bands for a- and b-spin states, and the corresponding spin bands shift upward and downward with respect to the CNP, respectively. It also leads to gapless electronic structure as well as 3G0 conductance in the vicinity of CNP (see Fig. 33.3). Meanwhile, zone-folded effects create more subbands near the CNP, namely, four a-spin subbands around 0.4 eV and four

Figure 33.3 Electronic band structures of defective ZGNRs. (a) for the pristine, (b) for the symmetric SW defects and (c) for the asymmetric SW defects. The solid red (dotted blue) line denotes the a-spin (b-spin) bands. The dashed black line indicates the CNP, and solid circles indicate defect states.

b-spin subbands around 0.4 eV. The presence of symmetric SW defects substantially split the electronic bands, especially for the b-spin bands above the CNP, resulting from the bands anticrossing at G or p point. More importantly, the symmetric defects open a band gap of about 0.12 eV for a-spin bands and 0.09 eV for b-spin bands, which is attributed to the mismatch coupling between its LUES and HOES wave functions due to the presence of defects. It is interesting to note that a defect state deriving from the a-spin subband is located at about 1.15 eV above the CNP producing a localized state, where complete backscattering is obtained (see red dashed line in Fig. 33.3). Thus, these changes in the band structures arising from introducing symmetric SW defects are unfavorable to electronic transport. In contrast to the extensive split produced by

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the symmetric SW defects, the electronic structure modification due to the asymmetric SW defects is slight. Except for some bands splitting that could be unfavorable to electron transport, the band structure away from the CNP does not experience much change. Similar to the emergence of defect states induced by the symmetric SW configuration, two defect states are observed in the asymmetric SW configurations; one defect state arising from the a-spin subband locates at about 0.6 2eV above the CNP, and the other one from the b-spin is –1.20 eV below the CNP. Both defects give rise to localized states that lead to conductance gaps (see, dotted line in Fig. 33.3). Overall, the band structure results reveal that the SW defect states near the CNP lead to complete electron backscattering region, where the location depends on the spatial symmetry of the defects. The electronic transport results are displayed in Fig. 33.3. The states induced by H atoms at the edge produce a conductance peak in the vicinity of CNP in the pristine ZGNR. In this study, our results show a good agreement with previous studies [43–45]. The first conductance plateau corresponding to the occupied and unoccupied states is G0. In the case of symmetric SW defects in the ZGNR, the conductance in the vicinity of CNP is decreased as a result of the four H atoms shrinking. The conductance with symmetric defects remarkably decreases below the CNP, manifesting monotonous reduction of conductance with increasing electron energy. We attribute this effect to the antisymmetry (opposite sign at every position) of wave functions, with respect to the pristine GNR in the wave functions (see Fig. 33.4e) that block the electronic transport. On the other hand, the orientation of about 50% of all wave functions corresponding to LUES is reversed, which gives rise to a conducting plateau (about 0.5 G0) that ranged from 0.04 to 0.8 eV above the CNP. More importantly, strong electron backscattering induced by the coupling between all states are expanded to lead to full suppression of the conduction channel at particular resonance energies. Accordingly, a smooth conductance valley around 1.12 eV corresponding to complete electron backscattering is observed. Concerning the transport properties of asymmetric SW configuration, we find that the absence of conductance peak at the CNP is due to the breaking of edge states. In addition, localized states in the vicinity of CNP lead to reduced conductance. The main feature of the first conducting plateau below the CNP is preserved

Results and Discussions

except for the smooth conductance valley located at about –1.2 eV. This illustrates the obvious different transport behaviors between the symmetric and asymmetric SW defects. We indeed found that such different transport behaviors result from different coupled electronic states supported by the wave function results. The HOES and LUES wave functions of asymmetric SW defect configuration are very similar to that of the pristine GNR except for the defective area. Therefore, the first conducting plateau near the CNP is preserved for the asymmetric configuration. Naturally, the asymmetric SW defects are responsible for the two conductance valleys, namely, a smooth valley at –1.2 eV and a sharp valley at 1.48 eV. The large reduction of conductance at these areas induced by the asymmetric SW defects corresponds to complete electron backscattering region. which is different from the situation in CNTs, where SW defects induce suppression of only half of the conductance channels [46]. However, the impact of the two conductance valleys on the ZGNRs is limited because they are far away from the CNP. The transport properties of asymmetric SW configuration are predicted to be comparable with that of the pristine GNR in spite of non-preservation of the geometry and wave function symmetry for the former. We note that similar results have been obtained under spin-dependent calculation by Ren et al. [47] very recently. Overall, the electronic transport calculations predict that it is more likely to be observed for asymmetric SW defects in the ZGNR, since these defects are more favorable for electronic transport in contrast to the substantially transport degradation in the symmetric defects configuration.

Figure 33.4 Conductance of defective ZGNRs as a function of electron energy. The black solid line, red dashed line, and blue dotted line denote the results of pristine, symmetric SW, and asymmetric SW defects.

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33.4 Conclusion In summary, we investigate the influence of local structural defects on the electronic transport of ZGNR using first principles calculations. The transformation energies reveal that the asymmetric SW defects is energetically more favorable than the symmetric SW defects. Both defects give rise to complete electron backscattering region that depends on the spatial symmetry of the defects. Our transport calculations predict that the asymmetric SW defects are more favorable for electronic transport in contrast to the substantially decreased in the symmetric defects configuration. We attribute these distinct transport behaviors to the different coupling between the conducting subbands influenced by the wave function modification around the CNP.

Abbreviations

CNP: charge neutrality point; GNR: graphene nanoribbons; HOES: highest-occupied electronic states; LUES: lowest-unoccupied electronic states; SW: Stone-Wales; ZGNR: zigzag-edged graphene nanoribbon.

Acknowledgments

The authors gratefully thank Prof. K.-L. Yao and Dr. M. A. Kuroda for their technical assistance with performing ab initio transport properties and the relax calculation in the Mac OS X Turing cluster. This study is financially supported by the Scientific Research Foundation of Yangtze University (grant no. 801080010111) and the Chongqing University of Technology (grant no. 2008EDJ01), and the Natural Science Foundation of China under grant no. 11047176.

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13. Lee C, Wei X, Kysar JW, Hone J: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 2008, 321:385– 388. 14. Fujita M, Wakabayashi K, Nakada K, Kusakabe K: Peculiar localized state at zigzag graphite edge. J Phys Soc Jpn 1996, 65:1920–1923.

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Chapter 34

Controllable Tuning of the Electronic Transport in Pre-designed Graphene Nanoribbon Hui Zeng,a,c,* Jun Zhao,a,* Jianwei Wei,b Dahai Xu,a and J.-P. Leburtonc

aCollege of Physical Science and Technology, Yangtze University, Nanhuan Road 1, Jingzhou, Hubei 434023, China bCollege of Optoelectronic Information, Chongqing University of Technology, Chongqing 400054, China cBeckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected]

We make use of ab initio density functional theory calculation to explore the electronic and transport properties of zigzag-edged graphene nanoribbon (ZGNR) with peculiar designed electronic transport channels by tailoring the atomic configuration of the nanostructure. Tailoring the atomic structure has significant influences on the electronic transport of the defective nanostructure, and eventually the metal-semiconducting transition are identified with the increasing number of missing atoms. Our results *These

authors contributed equally to the work. Reprinted from Curr. Appl. Phys., 12(6), 1611–1614, 2012. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2012 Elsevier B.V. Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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demonstrate that pre-designed graphene nanoribbon by selective tailoring with high precision is expected to be served as the basic component for nanoelectronic device.

34.1 Introduction

Graphene, a perfect two-dimensional material consisting of a single atomic plane of carbon atoms [1], has attracted considerable amounts of both theoretical and experimental studies owing to its novel electronic properties [2], such as ballistic transport [3] and quantum Hall effects [4]. Its excellent transport properties, such as high conductivity and the high mobility [3], motivate intense interest to apply it in the realization of nanoscale electronics since experimental synthesis of stable single graphene layer at room temperature was achieved [5–7]. Graphene nanostructure opens a new prospective for the minimization of electronic device since it can be easily tailored via conventional lithographic techniques [8]. Specifically, graphene nanoribbons (GNRs) are composed of very narrow atomic strips [9], which can be viewed as quasi-onedimensional nanomaterials assembled by graphene component. The electronic structure of graphene nanoribbon directly depends on their width as well as the shapes of the edges (armchair or zigzag) [10]. As one of the most important shapes of the edges, zigzagedged GNRs are more intriguing in that they present localized spinpolarized states character [11]. More recently, Munoz-Rojas et al. proposed that a finite zigzag graphene ribbon bridging two metallic graphene electrodes featuring 100 percentage magnetoresistance [12]. There are still many attempts to access component of molecular device by the thinner 1D carbon chain [13–15]. To make the graphene-based nanodevice to be realized, the crucial issue is controlling the electronic current through pre-designed graphene. Various defects in the carbon-based nanomaterials have been demonstrated to be able to effectively tune the electronic structure and transport properties of the nanosystem [16, 17]. Jin et al. introduced a new approach to obtain quite stable freestanding carbon chains connected with two graphene flakes by employing energetic electron irradiation inside a transmission electron microscope (TEM) [18]. It provides an powerful route to

Model

produce both a pure carbon constriction of the thinnest GNRs and a channel contacted by carbon atoms as leads. Furthermore, they have performed the simulation by density functional theory and the calculation results show that zigzag-edged GNRs are energetically favorable for connecting the chain. Meanwhile, Andrey Chuvilin et al. have reported analogous experimental results, hence one channel and two channels connection with two carbon atomic leads [19]. In this chapter, we explore the electronic and transport properties of pre-designed GNRs chain with zigzag edges. Using the framework of density functional theory calculation, we find that transport properties of the pre-designed GNRs are mainly governed by the symmetry of the geometry. It is possible to design the graphenebased nanodevice components by tailoring the atomic structure, especially the symmetry of the pattern, and the shape of the GNRs edge. We believe that modification of the transport properties by means of tailoring pre-designed structure making GNR as promising candidate for the next generation nanoscale device.

34.2 Model

We make use of the framework of ab intio density functional theory [20], as implemented in the SIESTA code [21], to relax the atomic geometry structure and obtain the electronic structures. Following previous work [22], the length of ZGNR width are classified by the number (N) of the zigzag chains. we refer to a ZGNR with N zigzag dimers contained in its units cell as an N-ZGNR. In this work, we present the theoretical investigation of electronic transport properties of 8-ZGNR containing 12 unit cells. The standard normconserving Troullier–Martins [23] pseudopotential orbitals are used to calculate the ion–electron interaction. The exchange correction term is calculated within the generalized gradient approximation [24]. The transport properties are calculated by the nonequilibrium Green’s function method [25, 26]. We use numerical double-z polarized basis set and the plane cutoff energy is chosen as 200 Ry. The relax process are done until the force less than 0.01 eV/Å, and the calculations are performed at electronic temperature T = 300 K. According to the nonequilibrium Green’s function method [26], the system is divided into three parts, namely, left and right leads as

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Figure 34.1 Schematic geometry and charge density of all structural configurations. Color blue represents high density; the red low density. The top figure is the pristine results with symmetric pattern (a); the middle is the symmetric nanostructure results with symmetric pattern (b); the bottom is the asymmetric nanostructure initiated by removing more atoms from the model of symmetric configuration.

Results and Discussions

well as central parts. The two leads are semi-infinite with periodic boundary conduction applied, and they are seamless connected with the central part. More detail about the nonequilibrium Green’s function method can be found in the Ref. [27]. We have built three defective structural configurations, including the pristine case, the symmetric configuration, and the asymmetric configuration formed by removing more atoms (as shown in the Fig. 34.1). We perform the calculations on the electronic and transport properties of various configurations in comparison with the pristine results.

34.3 Results and Discussions

The spin-polarized density of states (DOS) are displayed in the Fig. 34.2. As for the pristine case, the edge state gives rise to two extremely high peaks near the Fermi level, which are distributed below the Fermi level for a-spin state and b-spin state, respectively. This phenomenon is induced by the two spin states shifting in the opposite direction with respect to the Fermi level resulting from the edge state. It is noted that the sharp peak induced by the b-spin state is larger than that of the a-spin state. Previous study also report similar results based on the spin-polarization calculations [28], and it unveiled that perfect ZGNRs are semimetallic in antiferromagnetic configuration, with a band gap opening as a result of the presence of edge states [22]. Hence, the splitting between the aand b-spin states in the Fermi level suggests that the pristine ZGNRs are antiferromagnetic [29]. The resonant states near the Fermi level found in the pristine lead to prominent influence on the electronic transport. Moreover, the substantial DOS peaks located around ±1.2 eV are associated with the enhancement of quantum conductance, as shown in the Fig. 34.3. For the symmetric nanostructure, the edge state induced sharp peak is remarkably decreased and moves toward the Fermi level because edge states are broken by tailoring its atomic structure, such modification in the electric structure has crucial influence on the transport properties around the Fermi level, which is evidenced by the conductance result displayed in the Fig. 34.3. In particular, the DOS peak at about –1.5 eV is much higher than that of the pristine, indicating more electrons are localized here as a result of charge transfer from the edge state denoted by

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Figure 34.2 The spin-polarized density of states (DOS) for (a) the pristine, (b) symmetric configuration, and (c) asymmetric configuration. The a-spin DOS (top panel) and b-spin DOS (bottom panel) are shown by solid line (red color) and dotted line (blue color), respectively. The Fermi level is denoted by the zero.

Figure 34.3 Conductances of the defective ZGNR after tailoring structural atoms. (a) the pristine, (b) symmetric configuration, and (c) asymmetric configuration. The majority spin conductance and minority spin conductance are shown by solid line (red color) and dotted line (blue color), respectively. The Fermi level is denoted by the dotted black line.

Results and Discussions

the striking peak. The a-spin state and b-spin state that far from the Fermi level are shown to be symmetric. Specifically, there emerges two smooth DOS dips approximately located at ±1.34 eV in both the a- and b-spin states, indicating the presence of resonant backscattering, whose features are mainly determined by the symmetry and the atomic nanostructure. Obviously, the DOS result far from the Fermi level of this configuration resembles the pristine, leading to the similar transport behavior shown in the Fig. 34.4. As for the asymmetric configuration, it is found that the electronic structure of this configuration is very similar to the situation in the symmetric configuration although there are numerical differences between them. The main changes taking place in the electronic structure is that the smooth DOS gap at 0.39 eV in the a-spin state is substantially increased, and correspondingly similar result is also observed at about –0.39 eV in the b-spin state. We attribute the distinct results to the enhancement of resonant energies arising from the different coupling due to the tailored nanostructure.

Figure 34.4 I–V characteristics of the defective ZGNR after tailoring structural atoms. (a) the pristine, (b) symmetric configuration, and (c) asymmetric configuration.

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The spin-dependent conductance of the defective GNRs are shown in the Fig. 34.3. For the pristine GNR, both the majority spin and minority spin contribute 1G0 conductance at the Fermi level, which is consistent with previous study [30]. It can be seen clearly that the different contributions are derived from the majority spin and minority spin when the magnetic configuration with parallel spin orientations are taken into account, which can be explained from the staggered sublattice potentials on the hexagonal lattice due to edge magnetization. In general, the symmetric geometry of the pristine GNR gives rise to symmetric conductance with respect to the Fermi level. The quantum conductance is also associated with the van Hove singularity in DOS results shown in the Fig. 34.2. In the case of the symmetric configuration, the conductance in the vicinity of Fermi level is remarkably decreased to about 0.3G0 for each spin-dependent channel. The major feature of the first plateau is maintained, which is extended to ±1.12 eV around the Fermi level. However, the conductance is smaller than 0.5G0 due to the breaking of electron coupling near the Fermi level. Moreover, two conductance peaks (±0.2 eV) are found to be located below and above the Fermi level, which is in accordance with electronic band calculations. And two sharp peaks located ±1.36 eV are originated from the localized electrons at this region. The conductance peaks of this configuration are symmetric with respect to the Fermi level. Overall, the conductance decrease after tailoring in this configuration. As for the asymmetric configuration, the conductance in the Fermi level is virtually absent. Note that the first plateau extend to the energy window –1.12 eV, 1.12 eV, which is consistent with the plateau range of the pristine GNRs. In spite the first plateau is preserved, the value of the conductance is reduced to almost zero, thus the conductance gap exhibits near the Fermi level except for two inconspicuous peaks located at ±0.2 eV. It is noted that there exist two small peaks located at ±1.5 eV, which are attributed to the movement of the two sharp peaks located ±1.36 eV in the smooth configuration that away from the Fermi level. Although the tailoring in this configuration leads to the asymmetric nano-structure, the conductance is symmetric with respect to the Fermi level in general. Therefore, the conductance result confirms that the electronic transport is fundamentally suppressed arising from the tailoring of more atoms in the hexagonal lattices, and eventually lead to metal-semiconducting transition.

Conclusion

Figure 34.4 shows the I–V curves of the defective GNRs, containing the pristine GNR result for comparison. The I–V curve of the pristine ZGNR manifests oscillate characteristics at the range of –1.4 V to 1.4 V because of the increase of the transmission gap in the case of even-ZGNR, and the current is determined by the competition between the bias and the transmission gap [31, 32]. Beyond this range, the current starts to increase tempestuously. The symmetric I–V curve with respect to the Vbias = 0 arise from its symmetric nanostructure. The oscillate characteristics of the I–V curve also present in the symmetric pattern configuration, within the same range –1.4 V to 1.4 V, which is in consistence with the results of the pristine. However, the oscillation intensity has substantially decreased. The main differences in I–V curve between the pristine and the symmetric configurations is that the electronic transport has significantly deteriorated as a result of the atomic geometry of the smooth configuration. The asymmetric configuration associates with more atoms removed; the oscillate characteristics of its I–V curve is almost evanescent, but it has a long plateau ranged from –1.2 V to 1.2 V originated from geometry change. Compared with the results of the pristine, the I–V curve illustrates that the tailoring atoms could effectively lead to the metal-semiconducting (M–S) transition. The I–V curve of this configuration, associated with asymmetric nanostructure, seems to be symmetric with respect to the bias. As a matter of fact, the current is asymmetric but the differences are not distinct. The pronounced asymmetric I–V characteristics are expected be present provided that more atoms are tailored in a comparative large nanosystem. Our calculation results of the pre-designed GNRs indicate that it is possible to obtain particular electronic and transport properties through tailoring to utilize GNRs as diode or other nanodevice.

34.4 Conclusion

In summary, we have investigated the electronic and transport properties of zigzag-edged GNRs with pre-designed transmission channels. The electronic band gap increase in the defective GNRs after tailoring atoms from the pristine GNR. The I–V curve of symmetrical atomic structure present symmetrical pattern with

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respect to the Vbias = 0. However, the asymmetrical I–V curve patterns due to the symmetry break of the atomic geometry indicate that this configuration can be used as diode or other nanodevice.

Acknowledgment

The authors acknowledge Dr M. Kuroda, Z-Q. Fan, Prof. K-L. Yao for technical assistance on performing ab initio transport properties and the relax calculation in the Mac OS X Turing cluster. This work is supported by the National Science Foundation of China grant nos. 11047176 and 10947161 as well as the Research Foundation of Education Bureau of Hubei Province of China under grant nos. Q20111305, B20101303, and T201204.

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21. J.M. Soler, E. Artacho, J.D. Gale, A. Garcia, J. Junquera, P. Ordejon, D. Sanchez-Portal, J. Phys. Condens. Matter 14 (2002) 2745. 22. Y.-W. Son, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 97 (2006) 216803. 23. N. Troullier, Jose Luriaas Martins, Phys. Rev. B 43 (1991) 1993.

24. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. 25. J. Taylor, H. Guo, J. Wang, Phys. Rev. B 63 (2001) 245407.

26. M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, K. Stokbro, Phys. Rev. B 65 (2002) 165401.

27. S. Datta, Quantum Transport: Atom to Transistor, Cambridge University Press, New York, 2005. 28. S.S. Yu, W.T. Zheng, Q.B. Wen, Q. Jiang, Carbon 46 (2008) 537.

29. L. Yang, C.-H. Park, Y.-W. Son, M.L. Cohen, Steven G. Louie, Phys. Rev. Lett. 99 (18) (2007) 186801.

30. S.M.-M. Dubois, Z. Zanolli, X. Declerck, J.-C. Charlier, Eur. Phys. J. B 72 (2009) 1–24.

31. H. Zeng, J.-P. Leburton, Y. Xu, J.-W. Wei, Nanoscale Res. Lett. 6 (2011) 254.

32. Z. Li, H. Qian, J. Wu, B.-L. Gu, W. Duan, Phys. Rev. Lett. 100 (2008) 206802.

473

Chapter 35

Quantum Conduction through Double-Bend Electron Waveguide Structures

T. Kawamura and J.-P. Leburton

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

Quantum transport properties through a variety of electron waveguide structures that have a double-bend geometry are investigated using the recursive Greens function technique. The conductance is calculated as a function of the chemical potential using the two-probe, multichannel Landauer–Büttiker formula. The results for the right-angle double-bend structure are in agreement with previous calculations based on mode-matching techniques. For multiple double-bend structures in series, the existence of an energy gap between the first and second subband threshold energies where Reprinted from J. Appl. Phys., 73(7), 3577–3579, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 American Institute of Physics Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Quantum Conduction through Double-Bend Electron Waveguide Structures

the conductance is suppressed is shown. The effects of disorder and thermal broadening on the conductance are also investigated. Advances in crystal growth and electron-beam lithography within the last few years have made it possible to probe the quantum transport properties of semiconductor nanostructures in the quantum ballistic regime. This regime is realized when the typical dimensions of the system are less than the elastic mean free path l and the phase coherence length lf. The elastic mean free path, which is limited by impurity scattering from remote ionized donors in the barrier, and unintentional residual impurities in the channel, can be greater than 1 mm in ultrapure GaAs/AlxGa1–xAs heterojunctions. At low temperatures, the scattering by phonons is significantly suppressed, and the phase coherence length can become relatively large lf > 10 mm, so that the electrons are not scattered by impurities or phonons. In this way the electrons maintain their phase coherence as they ballistically traverse the device. Neglecting all imperfections, an idealized sample becomes an electron waveguide where the quantum transport properties are solely determined by the geometry of the conductor and the wavelike nature of the electrons. In contrast to conventional transistors, which operate in the classical diffusive regime, the quantum ballistic transport regime may lead to the development of novel electronic devices based on quantum interference effects [1, 2]. This motivates the study of geometrical effects on the transmission of electrons through such structures. In this communication we investigate the quantum conductance properties of electron waveguides which contain the double-bend geometry as illustrated in Fig. 35.1a. An analysis of the transmission through such a double-bend structure was first carried out by Weisshaar et al. [3] using a mode-matching technique. Experimental work on the low-temperature conductance of the double-bend waveguide has been carried out by Wu et al. [4]. We focus mainly on the quantum conduction through the various conductors when only the first transverse mode in the leads is propagating. Our starting point is to calculate the transmission amplitudes across the double-bend structure by solving the relevant two-dimensional Schrödinger equation. The confining boundaries that define the geometry of the conductor are assumed to be hard walls. In the regions denoted I, II, and III in Fig. 35.1a the potential

Quantum Conduction through Double-Bend Electron Waveguide Structures

is taken to be uniformly zero. The electron motion in the strongly confined epitaxial direction is taken to be completely frozen out. The relevant amplitudes can be numerically calculated by the recursive Green’s function (RGF) method. This technique has been extensively used to analyze the quantum transport properties of mesoscopic systems. We comment only briefly on this technique since it is described in detail in the literature [1, 5–7].

Figure 35.1 (a) Double-bend waveguide structure with lead width W, lateral length L, and horizontal length D. (b) Slanted double-bend waveguide structure with a uniform lateral width W. (c) Two double-bend waveguides in series, (d) Three double-bend waveguides in series.

In the RGF method the continuum Schrödinger equation for the structure is spatially discretized. The resulting tight-binding Hamiltonian (TBH), is then exactly solved. The transmission amplitudes across the sample are obtained by calculating the relevant Green’s functions across the structure. The RGF technique proceeds to build up the Green’s function matrices from one perfect semi-infinite lead (for which the Green’s functions can be derived analytically [8]) to the other, by attaching one column of the discretized lattice in the sample at a time. The necessary recursion relations are obtained by taking the appropriate matrix elements of Dyson’s equation G = G0 + G0UG, which relates the Green’s function G0 for the system without coupling to the isolated column, to the total Green’s function G with the coupling U. The conductance

477

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Quantum Conduction through Double-Bend Electron Waveguide Structures

can then be evaluated by the two-probe multichannel Landauer– Büttiker formula, G = (2e2/h)Tr(tt@), where t is the flux-normalized transmission matrix. By formal quantum scattering theory the transmission matrix t is related to the total Green’s function matrix across the sample [5, 9]. For ideal waveguides built up of rectangular sections it is efficient to carry out the recursion in the mode representation since the propagators for finite sections with uniform lateral widths can be evaluated analytically [1]. However, for systems with disorder it is advantageous to carry out the recursion across the scattering region in the site representation. This allows one to avoid evaluating the eigenvalues and eigenmodes for each column in the sample.

Figure 35.2 Conductance G in units of 2e2/h for double-bend geometry (W = 20a, L = 80a, and D = W) as a function of the Fermi level. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

Using the RGF technique, we calculate the quantum conductance of various structures that contain the double-bend geometry as illustrated in Figs. 35.1a–d. In Fig. 35.2 we show the conductance of a right-angle double-bend structure (see Fig. 35.1a), as a function of the Fermi level EF. We set W = 20a, L = 80a, and D = W. The first two threshold energies for propagation in the ideal leads are E1 = 0.0246 and E2 = 0.0979, in units of the hopping matrix element V = h2/2m*a2 (where a is the lattice constant between neighboring sites). For a = 25 and 50 Å, the lead widths are W = 500 and 1000 Å, and the magnitudes of the hopping matrix element are V = 91.0 and 22.75 meV, respectively, for m* = 0.067m0. We see that there are well-defined transmission peaks below the second threshold

Quantum Conduction through Double-Bend Electron Waveguide Structures

energy associated with the two bends separated by a length L. The resonances become more numerous and sharper as L is increased, as first noted by Weisshaar et al. [4]. Resonant transmission due to internal reflections in the structure, and the existence of bound and quasibound states has been theoretically investigated in other structures [10–14]. Figure 35.3 shows the conductance of the slanted double-bend waveguide illustrated in Fig. 35.1b for W = 20a, L = 40a, and D = W. The conductance is quantized in units of 2e2/h, as in a quantum point contact. In contrast to a constriction, the slanted chicane waveguide has a uniform lateral width along the y direction. Note that the conductance steps occur at energies that correspond to twice the threshold energies in the ideal leads. This scaling of the energies at which the steps occur arises from the collimation of the electrons along the slanted chicane so that the lateral confinement length is effectively reduced by a scale factor 2. The threshold energies scale as the square of the inverse lateral width which leads to the doubling of the threshold energies for the slanted chicane. We observe similar transmission resonance at the onset of each quantized step, as in standard quantum point contacts [10, 11].

Figure 35.3 Quantized conductance G in units of 2e2/h for the slanted doublebend structure with a uniform lateral width (W = 20a, L = 40a, and D = W). The arrows indicate the location of twice the magnitude of the threshold energies E2, E3, and E4 for propagation in the ideal leads. The arrow for 2E1 = 0.0492 is not shown. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

In Fig. 35.4 we show the conductance due to multiple doublebend electron waveguides in series (see Figs. 35.1c and d). For two

479

480

Quantum Conduction through Double-Bend Electron Waveguide Structures

double-bend structures in series with W = 20a, L = 40a, and D = W for each double bend, we see that we get a region of finite transmission which is well separated from a “noisy” section above an energy EF = 0.07. The destructive interference due to internal reflections in the multiple double-bend structure creates a broadened valley for energies 0.05–0.07. For W = 1000 Å, this corresponds to an energy gap of ~0.45 meV and a temperature ~5 K. The broadening of the gap is enhanced by the addition of a third double-bend structure in series (not shown). An oscillatory resonance structure in the conductance is present for energies up to 0.05, where the number of peaks is correlated with the number of double-bend structures in series. Disorder due to impurities is modeled by adding a random value d to each site energy ϵij + d in the TBH within an energy amplitude |d| £ D/2 [12, 15]. For D = 0.1 in units of V the conductance of the two double-bend electron waveguides in series is given by the dotted line in Fig. 35.4. The disorder parameter corresponds to D~2.3 meV for W = 1000 Å. We see that the effect of disorder is to slightly shift the positions of the resonance peaks and decrease their amplitude below unity. As the disorder parameter D is increased, the resonances are destroyed, and the conductance is no longer a characteristic of the waveguide geometry, and becomes dominated by the particular impurity configuration in the conductor.

Figure 35.4 Conductance G in units of 2e2/h for multiple double-bend structures. Two double-bend waveguides in series are shown where W = 20a, L = 40a, and D = W, for each double bend. The conductance for an ideal waveguide is given by the solid line. For the disordered case where a random value d is added to each site energy ϵij + d, where |d| £ D/2, the conductance for D = 0.1 is given by the dotted line. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

Quantum Conduction through Double-Bend Electron Waveguide Structures

The finite-temperature conductance is easily calculated by convoluting the zero-temperature result with the derivative of the Fermi distribution [10, 11]. In Fig. 35.5 we show how the conductance profile for the multiple double-bend waveguide changes with temperature. We set the temperature to three values kBT = 0.001, 0.004, and 0.008 in units of V. For a lateral width W = 500 Å, the corresponding temperatures are T = 1.06, 4.22, and 8.45 K. For a wider width W = 1000 Å, the corresponding temperatures are T = 0.26, 1.04, and 2.11 K. As the temperature is increased the transmission in the valley region rises, and the peak amplitude decreases. Note that at higher temperatures the doublepeak structure can no longer be resolved.

Figure 35.5 Finite-temperature conductance G in units of 2e2/h vs the Fermi level for two double-bend structures in series. For each double bend W = 20a, L = 40a, and D = W: kBT = 0.001 (solid line); kBT = 0.004 (dashed line); and kBT = 0.008 (dotted line). The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

The resonance transmission centered at EF ~ 0.4 is still fairly well resolved for a temperature T = 8.45 K for W = 500 Å, and T = 2.11 K for W = 1000 Å. For even narrower waveguides the relevant thermal energies are increased. However, in such samples the effects of inelastic phonon scattering will become important, which are not included in the present model calculations. In summary, using the RGF method we have calculated the quantum conductance of a variety of electron waveguides that contain the double bend as a geometric element. The slanted double-bend structure effectively acts as a constriction with subband energies

481

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Quantum Conduction through Double-Bend Electron Waveguide Structures

which correspond to twice the threshold energy for propagation in the leads. The multiple double-bend structures develop an energy gap where the transmission is suppressed. Resonance features that are characteristic of the geometry of the sample are not destroyed by low levels of disorder. As the number of double-bend structures in series is increased the gap region becomes better defined. The main transmission peak below the gap is quite robust against thermal broadening and shows an oscillatory structure correlated with the number of double-bend waveguides in series.

Acknowledgments

We would like to thank M. Macucci, Song He, and Rodolfo A. Jalabert for valuable discussions on the numerical techniques. This work is supported by the United States Army Research Office grant no. DAAL03-91-G-0052 and the Joint Service Electronics Program.

References

1. F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. Lett. 54, 350 (1989); J. Appl. Phys. 66, 3892 (1989). 2. S. Datta, Superlattices Microstruct. 6, 83 (1989).

3. A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Tripathi, Appl. Phys. Lett. 55, 2114 (1989).

4. J. C. Wu, M. N. Wybourne, W. Yindeepol, A. Weisshaar, and S. M. Goodnick, Appl. Phys. Lett. 59, 102 (1991); J. C. Wu, M. N. Wybourne, A. Weisshaar, and S. M. Goodnick, SPIE Proc. 1676, 200 (1992); W. Yindeepol, A. Chin, A. Weisshaar, S. M. Goodnick, J. C. Wu, and M. N. Wybourne, in Proceedings of the International Symposium on Nanostructures and Mesoscopic Systems, edited by M. A. Reed and W. P. Kirk (Academic, New York, 1992), p. 139.

5. P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981).

6. D. J. Thouless and S. Kirkpatrick, J. Phys. C 14, 235 (1981); A. MacKinnon, Z. Phys. B 59, 385 (1985).

7. H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, Phys. Rev. B 44, 10637 (1991).

8. E. N. Economou, in Green’s Functions in Quantum Physics, Springer Series in Solid State Sciences, Vol. 7, edited by M. Cardona, P. Fulde, and H. J. Queisser (Springer, Berlin, 1983), p. 79.

References

9. A. D. Stone, and A. Szafer, IBM J. Res. Develop. 32, 384 (1988). 10. G. Kirczenow, Phys. Rev. B 39, 10452 (1989).

11. A. Szafer and A. D. Stone, Phys. Rev. Lett. 62, 300 (1989). 12. H. U. Baranger, Phys. Rev. B 42, 11479 (1990).

13. R. L. Schult, D. G. Ravenhall, and H. W. Wyld, Phys. Rev. B 39, 5476 (1989). 14. E. Tekman and S. Ciraci, Phys. Rev. B 40, 8559 (1989); K.-F. Berggren and Z.-L. Ji, ibid. 43, 4760 (1991). 15. S. He and S. D. Sarma, Phys. Rev. B 40, 3379 (1989).

483

Chapter 36

Quantum Ballistic Transport through a Double-Bend Waveguide Structure: Effects of Disorder

T. Kawamura and J.-P. Leburton

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA [email protected]

We investigate the quantum transport properties through a variety of double-bend electron-waveguide structures using the recursive Green’s-function technique. The conductance is calculated as a function of the chemical potential using the two-probe multichannel Landauer–Büttiker formula. Detailed numerical calculations are presented to study the effects of waveguide geometry, impurity scattering, interface roughness, and finite temperature on the quantum conduction. We find that the roundness of the corners washes out the resonance structure by increasing the conductance Reprinted from Phys. Rev. B, 48(12), 8857–8865, 1993. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1993 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

in the valley regions. Impurity scattering and interface roughness slightly shift the peak positions and decrease their amplitudes. Thermal averaging of the conductance leads to a broadening of the resonance peaks, and a corresponding decrease of the peak amplitudes.

36.1 Introduction

Recent advances in microfabrication techniques have made possible the routine fabrication of nanostructures whose behavior at low enough temperatures is dominated by electron phase-coherence effects. A variety of interesting physical phenomenon have been observed in mesoscopic systems which show the importance of quantum coherence effects, such as weak localization, universal conductance fluctuations, Aharonov-Bohm oscillations, and resonant tunneling [1]. If the typical dimensions of a conductor are smaller than the elastic and inelastic mean free paths, the electrons are not scattered by impurities or phonons, and they maintain their phase coherence as they ballistically traverse the device. An idealized sample becomes an electron waveguide where the quantum transport properties are solely determined by the geometry of the conductor and the wavelike nature of the electrons. A remarkable manifestation of the successful achievement of quantum ballistic transport through a semiconductor nanostructure is the observation of quantized conductance steps through a narrow construction, as the number of one-dimensional (1D) channels is systematically varied by the application of an external gate voltage [2]. In contrast to conventional transistors which operate in the classical diffusive regime, the quantum ballistic transport regime may lead to the development of electronic devices based on quantum interference effects [3, 4]. This motivates the study of geometrical effects on the transmission of electrons through such structures. The main problem we investigate is the quantum conductance properties of electron waveguides which contain the double-bend (DB) structure illustrated in Fig. 36.1a, which in its simplest form can be described as a series of two laterally offset leads. We focus mainly on quantum conduction through the various conductors in the lowenergy regime, when only the first transverse mode in the leads

Numerical Method

(regions I and III) is propagating. An analysis of the transmission through such a DB structure was first carried out by Weisshaar et al. [5] using a mode-matching technique. Experimental work on the low-temperature conductance of a DB electron waveguide has been carried out by Wu and co-workers [6–8]. The transmission probabilities has also been studied by mapping the transfer matrix of the structure in the single-mode approximation into a 1D squarewell problem [9, 10]. In this chapter we use the recursive Green’s function (RGF) method to evaluate the quantum transport properties of such electron waveguides [11]. This technique allows us to take into account the effects of disorder rather easily in comparison to mode-matching methods. In particular, we investigate the effects of impurities, interface roughness, and roundness of the corners on the conductance profile as a function of the chemical potential. The numerical method used is briefly outlined in the following section.

Figure 36.1 (a) DB waveguide with lead width W, lateral length L, and longitudinal length D. (b) Slanted DB waveguide structure with a uniform lateral width W and length D. (c) Two DB waveguides in a series, (d) Three DB waveguides in a series.

36.2 Numerical Method Our starting point is to calculate the transmission amplitudes across the DB electron-waveguide structure by solving the two-dimensional Schrödinger equation. The model Hamiltonian is given by

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

H=-



h2 È ∂2 ∂2 ˘ Í 2 + 2 ˙ + V ( x , y ), (36.1) 2m * ÍÎ ∂x ∂y ˙˚

where M* is the effective mass of the electron. The confining boundaries which define the geometry of the conductor are assumed to be hard walls so that the potential V(x, y) identically vanishes in regions I, II, and III in Fig. 36.1a, otherwise V(x, y) = •. The electron motion in the strongly confined epitaxial direction (the z direction) is taken to be completely frozen out. The relevant amplitudes can be numerically calculated by the mode matching or the RGF method. Both of these methods have been extensively used to analyze the quantum transport properties of mesoscopic systems. In the mode-matching scheme the continuum problem is approximately solved by a matrix truncation [12, 13], whereas in the RGF method the corresponding discretized problem is solved exactly [14–17]. Each region of the conductor is represented by a square grid with lattice constant a, so that the x and y coordinates in Eq. (36.1) only take on the discrete values xi = ia and yj = ja (where i, j = 0, ±1, ±2, . . .). For a ≪ lF, where lF denotes the Fermi wavelength, the discretized system accurately simulates the transport properties of the corresponding continuum problem. The discretized Schrödinger equation is equivalent to the familiar tight-binding Hamiltonian HTB =



Âe

i,j | i ,

i,j

+V

j Ò·i , j | + V

Â(| i , j Ò·1 + j | + H.c.) i,j

Â(| i , j Ò·i , j + 1 | + H.c.). i,j

(36.2)

Here H.c. denotes the Hermitian conjugate, ϵi,j is the site energy at the ith column and jth row, and V = –h2/2m*a2 is the hopping matrix element between neighboring sites separated by the lattice constant a. Consider the electron-waveguide geometry in Fig. 36.2. Take the number of lateral sites J to be uniform throughout the sample. The system consists of two perfect semi-infinite leads on the rightand left-hand sides of a finite scattering region which extends from column 1 to N. The scattering region defines the conductor whose quantum transport properties we wish to investigate. In general, the scattering region defines an area with nonuniform site energies and lateral widths. The right lead ends at column N + 1 and extends

Numerical Method

to positive infinity, whereas the left lead ends at column i = 0 and extends to negative infinity. From formal quantum-mechanical scattering theory [15, 16, 18] it can be shown that the retarded Green’s function between column i = 0 and itself is related to the reflection amplitude, and that the Green’s function between columns i = N + 1 and i = 0 is related to the transmission amplitude, The RGF method provides us with a numerically stable way of building up the relevant Green’s functions [19].

Figure 36.2 The sample from columns i = 1 to i = N, between two perfect semi-infinite leads. The presence of asterisks indicate nonuniformity in the site energies within the scattering region. Hard wall boundary conditions along j = 0 and j = J +1.

The first step is to evaluate the Green’s function of the perfect semi-infinite right lead. The wave functions yk,l and corresponding eigenvalues Ek,l of the semi-infinite right lead are specified by two labels. The label k is continuous and denotes the longitudinal wave vector, whereas l is a discrete label that denotes the lateral subband mode number. By definition, the retarded Green’s function of the right lead is given by

G r ( x i' , y j' ; x i , yi ) =

 k,l

* y k,l ( x i' , yj' )y k,l ( x i , yj )

E - E k,l + ig

, (36.3)

where g is a positive infinitesimal number (g > 0). For our purposes we only need to evaluate the Green’s function for the semi-infinite right lead at column N +1. Now we slightly change our notation to emphasize the fact that the Green’s function defined in Eq. (36.3) defines a Green’s-function matrix. The Green’s function of the right lead between column N + 1 and itself, is expressed as Gj'jr (N + 1, N + 1). Since the lateral indices j and j′ range from 1–J we obviously have a square matrix of dimension J ¥ J.

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

Our ultimate goal is to evaluate the total Green’s function associated with the Hamiltonian given by Eq. (36.2). As mentioned before, the relevant Green’s-function matricies we must evaluate are Gj¢j(0,0), and Gj¢j(N + 1,0), since these Green’s functions are directly related to the reflection and transmission amplitudes, respectively [15, 16, 18], of the perfect semi-infinite right lead. The next step in the RGF method is to attach a single column of sites at i = N to the perfect semi-infinite right lead which terminates at column i = N + 1. The relevant total Green’s-function G can be calculated in terms of Gr, and the Green’s function of the isolated column at i =N, gN. This is accomplished by deriving a set of closed recursive relations among the Green’s functions with the aid of Dyson’s equation G = G0 + G0UNG, where UN contains all the coupling terms between the right lead which terminates at i = N + 1 and the isolated column at i = N and G0 = Gr + gN. By attaching one column in the scattering region at a time, we are able to generate the following sequences of Green’s functions: Gr(N + 1, N + 1 ) Æ G(N, N) Æ . . . Æ G(1,1) and Gr(N + 1, N + 1) Æ G(N + 1, N) Æ . . . Æ G(N + 1, 1). In the final recursion step we attach a perfect semi-infinite left lead which terminates at i = 0, and not just a single isolated column to obtain Gj¢j(0, 0), and Gj¢j(N + 1, 0). Thus far, the Green’s functions have been expressed in the site representation. In particular, the indices j and j′ in the Green’s function Gj¢j(0,0) refer to the lateral site coordinates on column i = 0. We can also choose to work in the mode representation where the Green’s function is denoted as G l'l (0, 0), and the indices l and l′ refer to the mode number on the perfect column at i = 0. By carrying out a similarity transformation G = S G †, we obtain the corresponding Green’s-function matrix in the site representation, where the matrix S is a J ¥ J matrix whose columns are made up of the lateral eigenfunctions of the perfect leads. Since the transformation is unitary, we have that S†S = SS† = 1. The RGF method outlined above is easily generalized to systems with nonuniform lateral widths. The only essential difference is that the relevant matricies are of rectangular dimensions. From a numerical point of view, for perfect systems composed of rectangular sections it is efficient to carry out the recursion in the mode representation as done by Sols et al. [3]. In this case, one can analytically solve for the retarded propagators for each finite

Results and Discussion

rectangular section with uniform lateral width. Mode-matching methods are also ideally suited for analyzing such simplified geometries without any disorder. However, for systems with position-dependent disorder it is advantageous to carry out the recursion process in the site representation [20]. The conductance G can then be evaluated by the two-probe multichannel Landauer–Büttiker formula, J



2e2 2e2 G= Tr(tt  )= | t nm|2 , (36.4) h h n,m=1

Â

where n and m refer to the lateral modes in the perfect leads, and t is the flux-normalized transmission amplitude that is related to the Green’s function across the scattering region in the mode representation. Current conservation requires that the unitarity condition is satisfied, S(|tnm|2 + |rnm|2) = 1, where rn,m is the fluxnormalized reflection amplitude. This sum rule serves as a check of the numerical calculations. In all our results the unitarity condition was satisfied within a tolerance of 1 ¥ 10–4. Some results for perfect waveguides without disorder have been checked with modematching techniques and were found to be in excellent agreement with the RGF results in the low-energy regime. At high energies near the band edge where the energy dispersion is nonparabolic, one expects a discrepancy between the continuum and discrete models.

36.3 Results and Discussion 36.3.1 Ideal Structures

Using the RGF technique, we calculate the quantum conductance of various structures which contain the DB geometry as illustrated in Figs. 36.1a–d. The following parameters define the structure: (1) W the lateral width of the leads (regions I and III), (2) L the lateral width of the midsection (region II), and (3) D the longitudinal length of the midsection. In some cases, instead of semi-infinite leads, we have finite leads of length d as shown in Fig. 36.3a, which are attached to wide semi-infinite regions that model the 2D source and drain contacts to the waveguides. For all the DB structures, we set W = D = 20a and for structures with finite leads we set d = 10a. The width

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

of the source and drain regions were typically set to 80a. Unless otherwise specified, we set the temperature T = 0 in all cases. For a given lattice constant a = A ¥ 10–10 m, the hopping matrix element is given by

|V | =

3.81 ¥ 103

(m */m0 )A2

meV, (36.5)

where m* is the effective mass of the electron, and A is the lattice constant in units of angstroms. We set m* = 0.067m0, the value in GaAs.

Figure 36.3 (a) DB waveguide structure with lead width W, lateral length L, horizontal length D, and lead length d attached to wide 2D regions which model the source and drain contacts. The parameter R defines the roundness of the corner to which it points. All sites at a radial R or greater define a hard wall boundary. Note that R ≥ W. (b) Single-stub T structure, and symmetric doublestub cross structure.

In Fig. 36.4 we show the conductance for two ideal DB waveguide structures with different lateral widths L = 60a and 80a. For semiinfinite leads, without the wide source and drain regions, the conductance is given by the dotted lines. In units of |V| [where V is given in Eq. (36.2)], the first two subband energies in the leads are given by El = 0.025 and E2 = 0.098, respectively. For energies below E1 there are no propagating states and the conductance is identically zero. Right above E1 the conductance becomes finite rather abruptly. As L is increased, there is a corresponding increase in the number of resonances where the conductance reaches it’s peak value of unity. This is expected on simple qualitative grounds since a larger

Results and Discussion

midsection, or “box” of dimensions D ¥ L supports more bound states through which resonant transmission can occur [21]. The number of resonance peaks that are experimentally observed roughly correspond to the number predicted by a simple counting method of the modes in a closed cavity with hard wall boundary conditions [7]. Since there is no disorder in the structures, the conductance profile is determined by the geometry of the samples. The transmission features through such ideal conductors at zero temperature arise from the scattering of electron waves by the boundaries defining the geometric shape of the conductor.

Figure 36.4 The conductance G in units of 2e2/h for the DB waveguide as a function of electron energy. For semi-infinite leads of width W = 20a the conductance is given by the dotted line. The conductance for finite leads of length d = 10a, attached to a wide 2D region, is given by the solid lines, (a) L = 60a and (b) L = 80a. In both cases, D = W. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

Propagating below the first threshold energy E1, via evanescent states, becomes possible with the addition of wide regions as shown in Fig. 36.3a, which simulate the 2D source and drain contacts. The

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

conductance is shown by the solid lines in Fig. 36.4 for L = 60a and L = 80a. Although the high-energy portion of the transmission remains relatively unchanged, we clearly see some prominent peaks right below the first subband threshold energy. These resonances are due to the tunneling through quasibound states in the double-bend structure. Locally, at each L-shaped right-angle bend, a quasibound state can form. As a result, one expects a double-bend structure to possess two quasibound states corresponding to a symmetric and antisymmetric superposition of the local bound states at each bend. The existence of such quasi-bound states can be illustrated quite directly by mapping the problem on to an effective one-dimensional scattering problem through two square wells [9, 10]. Note that for the shorter length L (Fig. 36.4a), one can just begin to resolve the two peaks corresponding to the symmetric and antisymmetric states. For larger L (Fig. 36.4b), the symmetric-antisymmetric energy gap decreases, and the peak separation is insignificant. However, if the lead lengths d are increased, the elastic broadening of the resonant peaks becomes much narrower, and the splitting of the peaks can be resolved even for the structure with larger L (not shown). The T structure and the symmetric cross [22, 23] structures illustrated in Fig. 36.3b also support quasibound states below the first threshold energy [24]. In Fig. 36.5 we show the conductance of both structures with and without the wide source and drain contact regions. The quantum transport properties of the T structure and its possible use as a quantum modulated transistor has been theoretically investigated in detail by Sols et al. [3]. Note that the symmetric cross structure also supports quasibound states above the first threshold E1. Our results for the T and symmetric cross structures attached to wide contacts on either side are in excellent agreement with the mode-matching results of Berggren and Ji [24]. In real electron waveguides where the boundaries are defined via electrostatic confinement from metal gates, the assumption of sharp geometric features is inaccurate. A certain amount of roundness of the square corners is inevitable. The careful consideration of such geometrical features has led to an understanding of various magnetoresistance anomalies in narrow Hall bar junctions [1, 25]. In Fig. 36.6 we show the effects of a rounded corner on the conductance profile in a double-bend structure. Various degrees of roundness can be simulated by changing the hard wall radius

Results and Discussion

Figure 36.5 The conductance G in units of 2e2/h for (a) the T structure and (b) the cross structure, as a function of electron energy. For semi-infinite leads of width W = 20a the conductance is given by the dotted line. The conductance for finite leads of length d = 10a, attached to a wide 2D region, is given by the solid lines. For both structures the lateral width of the midsection of both structures is given by L = 50a, and its length D = W. The dashed lines are offset by 2e2/h. The energy is given in units of the hoping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

parameter R as illustrated in Fig. 36.3a. The boundary of a rounded corner is defined by the circle of radius R and its intersection with the adjacent walls of the corner itself. The minimum value of R is Rmin = W which defines the most extreme extent of roundness within our model. The maximum value of R is Rmax = 2 W which defines the case of perfect, sharp corners. All sites whose radial distance are greater than R become part of the hard wall boundary. It is clearly evident in Fig. 36.6 that a rounded corner increases the transmission between the resonant peaks. For R = W = 20a (see Fig. 36.6b) the resonance peaks are barely visible and the transmission through the sample is featureless and practically of unit value above the first threshold energy [7, 10, 26]. The quantum transport essentially

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

becomes adiabatic, significantly reducing any back-scattering which leads to perfect transmission [27]. In real systems the roundness is not so extreme, and the resonance peaks at unity are distinguishable for a lesser degree of roundness R = 22a (see Fig. 36.6a).

Figure 36.6 The conductance G in units of 2e2/h vs electron energy for the DB structure, for different degrees of roundness at the corners: (a) R = 22a and (b) R = 20a, where R defines the radius of the rounded boundaries at the corners (W = 20a and L = 80a). The dotted line gives the conductance for sharp corners. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

Figure 36.7 shows the conductance of the slanted DB waveguide illustrated in Fig. 36.1b. We set W = D = 20a and L = 40a. The conductance is quantized in units of 2e2/h, as in a quantum point contact [2]. In contrast to a constriction, the slanted DB waveguide has a uniform lateral width along the y direction. Note that the conductance steps occur at energies which correspond to twice the threshold energies in the ideal leads. This scaling of the energies at which the steps occur arises from the collimation of the electrons along the slanted DB so that the lateral confinement length is

Results and Discussion

effectively reduced by a scale factor 2. The threshold energies scale as the square of the inverse lateral width which leads to the doubling of the threshold energies for the standard DB. At the onset of each quantized step, we observe similar transmission resonances as seen in previous simulations of standard quantum point contacts [12, 17].

Figure 36.7 Quantized conductance G in units of 2e2/h vs electron energy for the slanted DB waveguide with uniform lateral width (W = 20a, L = 40a, and D = W). The arrows indicate the location of twice the magnitude of the threshold energies E2, E3, and E4 for propagation in the ideal leads. The arrow for 2 ¥ E1 = 0.05 is not shown. The energy is given in units of the hopping matrix element V = h2/2m*a2 where a is the lattice constant of the discretization.

The finite-temperature conductance is easily calculated by taking the convolution of the zero-temperature result G(E, T = 0) with the derivative of the Fermi distribution f(E),

G( m ,T ) =

Ú

∂f ˘ Í- ∂E ˙ G(E ,T = 0)dE . (36.6) Î ˚

•È

0

In Fig. 36.8 we show the effects of thermal averaging on the conductance of a DB electron-waveguide structure. The conductance as a function of the chemical potential is shown for four different temperatures kBT = 0, 0.001|V|, 0.004|V|, and 0.008|V|, where |V| is given by Eq. (36.5). For a = 25 Å the dimensions of the DB midsection are D = 50 nm, and W = 200 nm. Using parameter values appropriate for GaAs, the hopping matrix element |V| = 91 meV and the corresponding temperatures of the four curves in Fig. 36.8 are T = 0, 1.05, 4.22, and 8.44 K, respectively. At finite temperatures the zerotemperature result is averaged over an energy interval of ~4kBT

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

[12, 17], and the resonance peaks become broadened and lower in amplitude. For sufficiently high temperatures, all resonance features that are characteristic of the geometric shape of the waveguide are washed out. Note that the curves are qualitatively similar to the experimental results of Wu and co-workers [6, 7]. In their data, one can clearly see multiple resonance peaks as a function of the gate voltage, and a noticeable dip (antiresonance) in the conductance as the Fermi level is about to cross the second threshold energy for propagation.

Figure 36.8 Finite-temperature conductance G(m, T) in units of 2e2/h vs chemical potential for the DB structure (W = 20a, L = 80a, and D = W), for different temperatures: kBT = 0.0 (solid line), kBT = 0.001 (dashed-dotted line), kBT = 0.004 (dotted line), and kBT = 0.008 (dashed line). The lines are, respectively, offset by 1.5, 1.0, and 0.5 in units of 2e2/h. The chemical potential and thermal energies are given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

36.3.2 Systems with Disorder In this section we study the effects of disorder on the conductance of a single DB waveguide structure (with sharp corners) that is connected to a wide source and contact region as shown in Fig. 36.3a. A simple way to introduce disorder is to set a single site energy in the sample to a higher value, which simulates the presence of a repulsive impurity. Figure 36.9 shows the conductance for the case in which we place a single repulsive impurity in the narrow lead region. At a given impurity site we increase the site energy by 1.0|V |. This corresponds to introducing a short-range d-function–

Results and Discussion

type impurity. It is clear that the conductance profile is dependent on the exact location of the single impurity. As the impurity is moved across the lead, we see that although the general resonance structure is maintained, the peak heights are lowered from unity in varying amounts. Some peak amplitudes remain essentially unaltered at unity, whereas other peak amplitudes are lowered by as much as 0.2 in units of 2e2/h. Comparing Fig. 36.9 with the conductance of the ideal sample shown in Fig. 36.4b we see that the impurity breaks the degeneracy of the quasibound states below the first threshold energy, and it becomes possible to see a signature of an additional resonance peak at low energy. It follows that in a real experiment, the transport properties of a sample changes once it has been thermally cycled since the impurities in the system can migrate and change the potential profile in the waveguide.

Figure 36.9 The conductance G in units of 2e2/h vs electron energy for the DB structure attached to a wide 2D region (W = 20a, L = 80a, and D = 10a), for different locations of a single short-ranged repulsive impurity. The strength of the impurity is 1.0 and is located in one of the finite leads. The conductance of the ideal structure without the impurity is given by the dotted lines. In (a) xi = 2a, yi = 10a, (b) xi = 5a, yi = 10a. The energy is given in units of the hopping matrix element V = h2/2m *a2, where a is the lattice constant of the discretization.

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Figure 36.10 The conductance G in units of 2e2/h vs electron energy for the DB structure attached to a wide 2D region (W = 20a, L = 80a, and D = 10a), for different degrees of disorder. A random value d is added to each site energy eij + d, where |d| £ D/2: (a) D = 0.1 and (b) D = 0.5. The conductance of the ideal structure without any impurities is given by the dotted lines. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

Disorder throughout the DB waveguide structure (except in the contact regions) is introduced by adding a random value d to each site energy eij + d in Eq. (36.4), within an energy amplitude |d| £ D/2. Figure 36.10 shows the effect of increasing the disorder parameter D on the conductance [28, 29]. For low degrees of disorder (see Fig. 36.10a) we see that the general features of the transmission resonances which are characteristic of the sample geometry are preserved. Note that the peak amplitudes are, in general, reduced from unity, and that the peak positions become slightly shifted. Since the plots are for specific configurations of impurities there is no broadening of the peaks, which would result if we were to take an ensemble average over many configurations. As the degree of

Results and Discussion

disorder is increased (see Fig. 36.10b) the conductance features are no longer dominated by the scattering from the sample boundaries, and instead strongly dependent on the particular impurity configuration which exists in the conductor. The effects of interface roughness are simulated by breaking up the double-bend structure into sections, typically of length 3a–4a along the x direction. Within each section we randomly decide whether to shift or not to shift the upper and lower boundaries. Let the probability for keeping the boundary at its original position, the upper and lower boundaries, be 1–2p, then there is a probability 2p that a given boundary will be shifted by ±2a and ±4a. By increasing the probability for a nonzero deviation, we can model a system which is increasingly disordered due to interface roughness [30, 31]. Essentially, we arrive at a system in which the lateral dimension varies randomly along the sample. This produces a positiondependent threshold energy configuration within each section of the DB structure. In constricted regions the electron will have a lower longitudinal energy, while the opposite case holds for a wider region. As a result, a random change in the lateral width appears as a random change in potential. Figure 36.11 shows the effect of various degrees of interface roughness on the conductance a DB structure. The effects of interface roughness are qualitatively similar to that of impurities distributed throughout the sample. The recent work of Wu et al. [7] shows that it is experimentally possible to probe the quantum transport regime where the scattering of electrons is dominated by the geometric boundaries of a DB. However, the experimental peak amplitudes were significantly lower than those found theoretically based on a mode-matching analysis of ideal systems. Although the degree of the reduction is underestimated, our numerical calculations do indeed show that, in general, a reduction of the peak amplitudes due to disorder is to be expected. This discrepancy can probably be reduced by a more sophisticated treatment of the disorder potential that takes into account the long-range nature of the potential from the remote ionized impurities [32].

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

Figure 36.11 The conductance G in units of 2e2/h vs electron energy for the DB structure attached to a wide 2D region (W = 20a, L = 80a, and D = 10a), for different degrees of boundary roughness. For a given section each boundary is randomly displaced by 0, ±2, or ±4 in units of a. Larger values of the parameter p correspond to an increasing roughness of the boundary, (a) p = 0.2 and (b) p = 0.4. The conductance for the case of sharp corners is given by the dotted lines. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

36.3.3 Multiple Double-Bend Structures In Fig. 36.12 we show the conductance due to multiple DB electron waveguides illustrated in Figs. 36.1c and d. In the multiple DB case we see that we get a region of finite transmission which is well separated from a “noisy” section above an energy of EF = 0.07|V|. The destructive interference due to internal reflections in the multiple DB structures creates a broadened valley for energies of 0.05|V|–0.07|V|. The effect of disorder is shown in Fig. 36.12a for D = 0.1|V|. By attaching a wide 2D contact region we are able to see two additional peaks (see Fig. 36.12b) due to resonant tunneling

Results and Discussion

Figure 36.12 Conductance G in units of 2e2/h for the multiple DB structures, as a function of electron energy: (a) Two DB waveguides in a series with semiinfinite leads with width W = 20a, lateral width L = 40a for the middle section, and D = W. For D = 0.1 the conductance is given by the dotted line, (b) Same structure with finite leads of width d = 10a attached to a wide 2D region, (c) Three double-bend structures in a series with semi-infinite leads. The energy is given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

through quasibound states. For W = 1000 Å, the energy gap is ~0.45 meV, and the temperature is T ~ 5 K. In Fig. 36.13 we show the finitetemperature conductance for two DB waveguides in a series, attached

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Quantum Ballistic Transport through a Double-Bend Waveguide Structure

to wide contact regions. The valley region is quite robust against an increase in temperature and may be experimentally observable. Note that the first broadened peak in the multiple bend structure for kBT = 0.004 is much narrower and well denned than the peak in the single DB structure. The broadening of the gap is enhanced by the addition of another DB structure as seen in Fig. 36.12c. An oscillatory resonance structure in the conductance is present for energies up to 0.05|V|, where the number of peaks is correlated with the number of DB waveguides in a series.

Figure 36.13 Finite-temperature conductance G(m, T) in units of a 2e2/h vs chemical potential for two DB structures in a series attached to a wide 2D region (W = 20a, L = 40a, D = W, and d = 10a): kBT (solid line), kBT = 0.001 (dasheddotted line), kBT = 0.004 (dashed line), and kBT = 0.008 (dotted line). The lines are, respectively, offset by 1.5, 1.0, and 0.5 in units of 2e2/h. The chemical potential and thermal energies are given in units of the hopping matrix element V = h2/2m*a2, where a is the lattice constant of the discretization.

36.4 Conclusions In summary, using the RGF method we have calculated the quantum conductance of a variety of electron waveguides which contain DB as a geometric element. An ideal DB shows a resonance structure which depends on the length L, where a longer DB waveguide supports more resonance peaks with unit amplitude. The resonance structure is washed out for increasing degrees of roundness of the sharp corners, and higher temperatures. The roundness does not lead to

References

a decrease in the peak amplitudes in contrast to thermal averaging. The conductance profile is very sensitive to the exact distribution of impurities in the system. For small degrees of disorder the peak amplitudes are diminished, but are still characteristic of the geometry of the sample. Increasing the disorder further destroys all traces of the resonance structure. The effects of interface roughness on the conductance are qualitatively similar to that of impurities. The DB structure in a series have an energy region of low transmission which becomes better defined with the addition of a third DB waveguide. The lowest peak is rather robust with an increase in temperature and may be experimentally observed.

Acknowledgments

We would like to thank M. Macucci, Song He, and Rodolfo A. Jalabert for valuable discussions on the numerical techniques. This work was supported by the United States Army Research Office grant no. DAAL03-91-G-0052.

References

1. See C. W. J. Beenakker and H. van Houten, Solid State Phys. 44, 1 (1991) for an excellent review of the quantum transport field in semiconductor nanostructures. 2. B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988); D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ajmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. C 21, L209 (1988).

3. F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. Lett. 54, 350 (1989); J. Appl. Phys. 66, 3892 (1989). 4. S. Datta, Superlatt. Microstruct. 6, 83 (1989).

5. A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Tripathi, Appl. Phys. Lett. 55, 2114 (1989). 6. J. C. Wu, M. N. Wybourne, W. Yindeepol, A. Weisshaar, and S. M. Goodnick, Appl. Phys. Lett. 59, 102 (1991).

7. J. C. Wu, M. N. Wybourne, A. Weisshaar, and S. M. Goodnick, SPIE Proc. 1676, 200 (1992).

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8. W. Yindeepol, A. Chin, A. Weisshaar, S. M. Goodnick, J. C. Wu, and M. N. Wybourne, in Proceedings of the International Symposium on Nanostructures and Mesoscopic Systems, edited by M. A. Reed and W. P. Kirk (Academic, New York, 1992), p. 139. 9. H. Wu, D. W. L. Sprung, and J. Martorell, Phys. Rev. B 45, 11960 (1992). 10. J. Goldstone and R. L. Jaffe, Phys. Rev. B 45, 14 100 (1992).

11. T. Kawamura and J. P. Leburton, J. Appl. Phys. 73, 3577 (1993). 12. George Kirczenow, Phys. Rev. B 39, 10452 (1989).

13. R. L. Schult, D. G. Ravenhall, and H. W. Wyld, Phys. Rev. B 39, 5476 (1989). 14. D. J. Thouless and S. Kirkpatrick, J. Phys. C 14, 235 (1981); A. MacKinnon, Z. Phys. B 59, 385 (1985). 15. P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981).

16. A. Douglas Stone and Aaron Szafer, IBM J. Res. Develop. 32, 384 (1988). 17. Aaron Szafer and A. Douglas Stone, Phys. Rev. Lett. 62, 300 (1989). 18. H. U. Baranger and A. D. Stone, Phys. Rev. B 40, 8169 (1989).

19. For a review of Green’s-functions techniques, see E. N. Economou, in Green’s Functions in Quantum Physics, edited by M. Cardona, P. Fulde, and H. J. Queisser, Springer Series in Solid State Sciences Vol. 7 (Springer, Berlin, 1983). 20. However, a combination of mode-matching and transfer-matrix techniques can handle systems with varying widths and disorder. See E. Tekman and S. Ciraci, Phys. Rev. B 43, 7145 (1991); Y. Takagaki and D. K. Ferry, ibid. 46, 15218 (1992). 21. The cavity resonances below the second threshold energy are roughly given by E/|V| = npa/L)2 + (pa/D)2, where n = 1, 2, 3, . . . . Note that such an analysis is meant only to give an indication of the number of peaks that are expected, and is not meant to give the exact number or energies where the resonances occur. 22. A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Tripathi, IEEE Electron Dev. Lett. ED-12, 2 (1991); Y. Takagaki and D. K. Ferry, Phys. Rev. B 45, 13494 (1992). 23. T. Itoh, N. Sano, and A. Yoshii, Phys. Rev. B 45, 14131 (1992).

24. Karl-Fredrik Berggren and Zhen-Li Ji, Superlatt. Microstruct. 8, 59 (1990); Phys. Rev. B 43, 4760 (1991). 25. Harold U. Baranger, David P. DiVincenzo, Rodolfo A. Jalabert, and A. Douglas Stone, Phys. Rev. B 44, 10 637 (1991).

References

26. F. Sols and M. Macucci, Phys. Rev. B 414, 11887 (1990); C. S. Lent, Appl. Phys. Lett. 56, 2554 (1990). 27. A. Yacoby and Y. Imry, Phys. Rev. B 41, 5341 (1990).

28. Song He and S. Das Sarma, Phys. Rev. B 40, 3379 (1989).

29. D. van der Marel and E. G. Haanappel, Phys. Rev. B 39, 7811 (1989); C. S. Chu and R. S. Sorbello, ibid. 40, 5941 (1989); Y. Takagaki and D. K. Ferry, ibid. 45, 6715 (1992); Zhen-Li Ji, Semicond. Sci. Technol. 7, 198 (1992). 30. T. Ando, Phys. Rev. B 42, 5626 (1990).

31. E. Tekman and S. Ciraci, Phys. Rev. B 40, 8559 (1989).

32. J. A. Nixon and J. H. Davies, Phys. Rev. B 41, 7929 (1990); J. A. Nixon, J. H. Davies, and H. U. Baranger, ibid. 43, 12638 (1991).

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Chapter 37

Quantum Transport through OneDimensional Double-Quantum-Well Systems

T. Kawamura,a H. A. Fertig,b and J.-P. Leburtonc aBeckman

Institute for Advanced Science and Technology, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA bDepartment of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA cBeckman Institute for Advanced Science and Technology, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA [email protected]

We investigate the lateral tunneling properties of electrons through a one-dimensional double-quantum-well system using the recursive Green’s-function technique. The conductance exhibits a number of interesting quantum-interference effects, including a strong resonance at the onset of conductance, a “beating” effect due to competing characteristic times for the system, and a finite Reprinted from Phys. Rev. B, 49(7), 5105–5108, 1994. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 1994 The American Physical Society Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Quantum Transport through One-Dimensional Double-Quantum-Well Systems

conductance at large Fermi energies, where the densities of states for the individual wells is small. It is shown that the last property may be exploited to use this system as a quantum-interference transistor. Advances in crystal growth and electron-beam lithography have made it routinely possible to probe the quantum-transport properties of semiconductor nanostructures of a reduced dimensionality. The quantum-ballistic-transport regime in these small systems is realized when the dimensions of the system are less than the elastic mean free path l, and the phase coherence length lf. Neglecting all imperfections it follows that the quantum-transport properties are solely determined by the geometry of the system, and the wavelike nature of the electrons. In this chapter we investigate the transport properties of a one-dimensional (1D), double-quantum-well (DQW) tunneling structure in the quantum ballistic regime. Our study is motivated by the successful fabrication by Eisenstein, Pfeiffer, and West [1] of a GaAs/AxGal–x As-based 2D electron system in a DQW, in which two closely spaced 2D electron systems can be independently contacted. Such systems make it experimentally possible to study the mutual Coulomb drag between parallel 2D electron gases [2], and the tunneling conductance between two coupled, purely 2D systems [3]. There are also recent studies of lateral tunneling between 2D and 1D systems [4]. Although the geometry of the 1D DQW is quite simple, we find that the quantum conductance of the device possesses a number of remarkable properties. When the conduction-band minima in the two wells are aligned they are in balance, and we find that the conductance G is essentially unity in units of 2e2/h, as a function of the Fermi energy EF. The conductance profile changes drastically when the wells are set out of balance. In this case G has a rather sharp peak right at a subband threshold energy, and there is a “beating” effect which can be interpreted as arising from the interplay of two inherent time scales of the device. For increasing energies the peak heights of G approach a constant magnitude which is greatly suppressed from unity. We find that this behavior at higher energies is rather robust with respect to disorder and temperature. This property thus might be exploited to fabricated a quantum modulated transistor.

Quantum Transport through One-Dimensional Double-Quantum-Well Systems

Figure 37.1 Geometry of the one-dimensional double-quantum-well system. The length of the scattering region between the broken bonds is denoted by d. The lattice spacing of the tight-binding Hamiltonian is denoted by a. Perfect semi-infinite leads are attached to the left and right sides of the system. The layers are coupled by an interlayer hopping matrix element t, which is much smaller than the intralayer one V. The small arrows indicate where V identically vanishes. The bar graph on the left shows a possible weight distribution of an incident electron wave as it travels down along the wire with a longitudinal wave vector k.

The system under study is illustrated in Fig. 37.1. A rectangular midsection of length d, composed of two coupled (1D) wires is attached to semi-infinite left and right leads. The perfect left lead only contacts the bottom wire (layer 2), and the perfect right lead only contacts the top wire (layer 1). The Schrödinger equation for the system is solved by spatially discretizing the system with lattice constant a in each layer. The resulting tight-binding Hamiltonian is then exactly solved by using the recursive Green’s-function (RGF) technique [5–8]. The discretization introduces an energy scale into the problem which is given by V, the hopping matrix element between neighboring intralayer sites. It is given by V = h2/2m*a2, where m* is the effective mass of the electron (m* = 0.067m0 in GaAs). For a = 5 Å, one finds V = 2.27 eV; in what follows, all our energies will be given in units of V. An electron in a given layer can hop to an adjacent lattice site on the other layer with amplitude t, the tunneling matrix element. The site energies of layer 1 and layer 2 are given by ϵi1 and ϵi2, respectively, where i denotes the longitudinal lattice site index along the wires (the x axis). In the perfect semi-infinite left and right leads the site energies of the layers are fixed at some constant value ϵi1 = ϵ1, and ϵi2 = ϵ2. The effects of disorder due to short-ranged

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Quantum Transport through One-Dimensional Double-Quantum-Well Systems

impurities can be simulated by adding a random energy dis to each site in the midsection of length d, ϵis = ϵs + dis (where s = 1 or 2 is the layer index). The degree of disorder is given by the parameter D, where |dis| £ D/2. At the edge of the scattering region the effects of a gate which allow independent contact to each layer are modeled by locally setting the intralayer hopping matrix element to zero (see Fig. 37.1), creating a broken bond [9]. It can be shown from formal quantum-mechanical scattering theory that the transmission amplitude across the sample is related to the retarded Green’s function of the structure between columns in the perfect left and right leads [5]. The RGF technique provides us with a stable numerical procedure to build up the relevant Green’sfunction matrices from one perfect semi-infinite lead (for which the Green’s functions can be derived analytically in the absence of a magnetic field) to the other, by attaching one column of the discretized lattice in the sample at a time. The necessary recursion relations are obtained by taking the appropriate matrix elements of Dyson’s equation. The conductance can then be evaluated by the two-probe multichannel Landauer-Büttiker formula G = (e2/h)(tt†), where t is the flux-normalized transmission matrix. All of our RGF calculations obey unitarity to within a tolerance of 1 ¥ 10–4. Using the RGF technique, we calculate the quantum conductance of an ideal 1D DQW system with broken bonds. In Figs. 37.2a–c we show the conductance as a function of the electron energy for three consecutive energy intervals when the wells are in balance. The parameter values are a = 5 Å, d = 1015 Å, ϵ1 = ϵ2 = 2.0, and t = 0.01. (We remind the reader that the energy scale V = 2.27 eV.) In the perfect leads the subband energies are given by E1 = 1.99 and E2 = 2.01. For energies between E1 and E2 the conductance profile suggests that the midsection of length d behaves like a resonant box. For low energies the resonance peaks due to quasibound states in the scattering region are very sharp and become broadened as the Fermi energy EF Æ E2. For energies above E2 the conductance shows a complicated behavior and eventually settles down to a series of resonances and antiresonances. Note that the resonance peaks never exceed unity in units of 2e2/h.

Quantum Transport through One-Dimensional Double-Quantum-Well Systems

Figure 37.2 Conductance in units of 2e2/h as a function of the chemical potential. The energy is measured in units of the hopping matrix element V = h2/2m*a2 = 2.21 eV, where a = 5 Å is the lattice spacing of the sites in the tight-binding Hamiltonian. In (a)–(c) the length d = 1015 Å. The site energies of layer 1 and layer 2 are in balance ϵ1 = ϵ2 = 2.0, and the interlayer tunneling matrix element t = 0.01.

In Figs. 37.3a–c we show what happens to the conductance when the site energies of the quantum wells are out of balance, ϵ1 = 2.0, and ϵ2 = 1.0, with all other sample parameters identical to those in Fig. 37.2. In this case the subband energies in the perfect leads are given by E1 = 1.0 and E2 = 2.0. Nonzero conductance results when the energy exceeds the second subband energy E2. Compared to the balanced case, we see that the conductance peak values are greatly

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suppressed. The maximum conductance occurs just above E2, and then decreases abruptly to values below 0.002. As a check of our RGF results we have calculated the conductance for the ideal systems without disorder using mode-matching techniques [10]. We find that the continuum and discrete models are in excellent agreement as long as EF is not too far from the subband threshold energy, when nonparabolicity in the tight-binding dispersion relation becomes a dominant factor. One of the surprising features of our results is the nearly perfect transmission that one finds over a large range of Fermi energies for the balanced well case (Fig. 37.2). This is a coherent multiplescattering phenomenon, and is related to the large spatial overlap of the electron states in the two wells when they are in balance. Clearly, when the wells are out of balance (Fig. 37.3), wave functions of the individual wells at the same energies will not match nearly so well, since the effective Fermi wavelength for each well differs. It is interesting to note that the range of Fermi energies over which this perfect transmission occurs decreases with either decreasing t or d; thus, if one has a system with a very small tunneling probability, as might happen if the wells have too large a separation, the effect can always be made to occur by choosing a large enough sample length d. We will argue below that, by controlling the bias potential between wells, one can in principle use this effect to construct a transistor. Finally, we remark that there is a surprising broad suppression of the resonances just above the edge at which finite conductivity first sets in (Fig. 37.2b). At present we have no simple physical explanation for this phenomenon. We can also interpret the “beating” effect apparent in Fig. 37.3 in terms of two competing time scales for the system. In particular, if we imagine injecting a wave packet from the left in well 2 in Fig. 37.1, the wave packet will oscillate between the wells with a period given by tT = h/DE (DE = E2 – E1), and will move down the channel roughly at the group velocity ug = h(kl + k2)/4m*. If the time to reach the end of the channel, d/ug, is a half-integral multiple of tT, then the wave packet will lie primarily in the upper well (2) as it reaches the broken bond on the right. Most of the weight of the wave packet in this case is in the well with an unbroken bond, and the wave packet to a large extent is able to pass out of the overlapping region of length d. One thus expects a conductance maximum at such

Quantum Transport through One-Dimensional Double-Quantum-Well Systems

energies, and we find that the positions of the peaks in the beats may be predicted approximately from this simple consideration.

Figure 37.3 Conductance in units of 2e2/h as a function of the chemical potential. The energy is measured in units of the hopping matrix element V = h2/2m*a2 = 2.27 eV, where a = 5 Å is the lattice spacing of the sites in the tightbinding Hamiltonian. In (a)–(c) the length d = 1015 Å. The site energies of layer 1 and layer 2 are unbalanced ϵ1 = 2.0, and ϵ2 = 1.0, respectively. The interlayer tunneling matrix element t = 0.01.

Many of the results for the unbalanced well case may be understood from a first-order perturbation theory analysis. We start with two semi-infinite quantum wires, one in the region x < d, the other in x > 0, which to zeroth order are disconnected. The wave functions of this system are trivially given by

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yl(x) =



h2 ks2

2 / L sink1(d – x),

y2(x) =

2 / L sink2x,

where /2m* = E – ϵs (s = 1 and 2), and we have taken the wires to have finite length L. We add a term to the Hamiltonian that connects the wires over a finite interval 0 < x < d,

U = t[|1Ò ·2| + |2Ò ·1|]Q(d – x)Q(x).

The conductance of the system (in the limit L Æ • ) then takes the form

G = (2pe2L2/h)g1(EF)g2(EF)|Trs|2, (37.1)

where gs is the 1D density of states, r is a state in the left well, and s a state in the right well, both at the Fermi energy [11]. The T matrix is formally given by [12] Trs = ·r|U|sÒ + ·r|UGU|sÒ,

where G is the usual retarded Green’s function, G = 1/(EF – H + id). The Hamiltonian H = H0 + U, where H0 is simply the kinetic energy operator, and d > 0. To lowest order in t, one may show that

Trs = –(4h2t/2m*DϵL)[k2sinkld – kl sink2d], (37.2) where Dϵ = |ϵ2 – ϵ1|. Most of the properties of the conductance in Fig. 37.3 follow from Eqs. (37.1) and (37.2). Clearly, the fast oscillations arise from the sine terms in the T matrix, and are essentially due to the overlap of the wave functions in the individual wells. We can also understand the sharp peak at the band edge in the following way. Using the usual form for a 1D density of states, g(E) = (1/ph) m */ 2E , we find for E2 > E1 and EF ª E2,

G ª(e2/h)(2t2/Dϵ2)[EF(EF – Dϵ)]1/2(2m*d2/h2),

which, due to its proportionality to d2, which may be quite large, accounts for the sharp peak in the conductance at the band edge. We note that very similar peaks have been observed in 1D to 2D lateral tunneling experiments [4]. Similarly, we can see that the maximum peak heights in Fig. 37.3 at larger energies, which do not appear to fall off in magnitude with increasing energy, may be understood from this approximation. It is at first surprising that these peaks do

Quantum Transport through One-Dimensional Double-Quantum-Well Systems

not fall off, since the density of states for the wells each falls off as 1 / EF ; however, the T matrix turns out to increase with energy as

E F at the maxima of the sines in Eq. (37.2). This leads to a constant magnitude at high energy for the peak heights. In general, a plot of G in this approximation gives quite good agreement with Fig. 37.3, so long as one does not go so far out in energy that the differences between our tight-binding model and the continuum approximation discussed here become important.

Figure 37.4 Multichannel conductance in units of 2e2/h as a function of the chemical potential. The energy is measured in units of the hopping matrix element V = h2/2m*a2, where a = 5 Å is the lattice spacing of the sites in the tight-binding Hamiltonian. The length d = 1015 Å, and the width of each layer is 30 Å for five lateral sites along the y direction. In (a) ϵ1 = ϵ2 = 2.0, and in (b) ϵ1 = 2.0 and ϵ2 = 1.0. Energies here are measured in units of V = 2.27 eV.

Figure 37.4 illustrates the conductance for a parallel wire system with a finite width, such that there are five subbands per wire. One can see that the form of the conductance for each subband is essentially additive. In particular, the resonance effect for balanced wells is preserved, and shows an essentially steplike structure, and the beating effect for unbalanced wells contains the frequencies

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of all the propagating subbands. Thus, we expect that finite wire widths do not qualitatively change the phenomena we have found in the single-subband wires case, so long as the temperature scale of the system is lower than the single wire subband spacings. Finally, we note that the suppression of the tunneling conductance at high Fermi energies when the wells are unbalanced may in principle be used to construct a three-terminal transistor. While several quantum-interference effects have been suggested for constructing ultrasmall transistors [7, 13], the necessity of having very low temperatures and ultrapure samples have made fabrication of such devices quite difficult. However, we find that the suppression described here is quite robust with respect to both disorder and temperature. The results for the thermally averaged conductances with disorder in the energy interval EF = 2.4 – 4.0 are given in Fig. 37.5. Results are given for D = 0.3, and kBT = 0.01 (T = 263 K). When the wells are in balance we see that the conductance is larger than in the unbalanced case by two orders of magnitude. By attaching a side gate next to one of the wells, one can control the balance, and hence tunneling conductance, of the wells with a terminal separate from the source and drain. This result points to the potential usefulness of this structure in fabricating a quantum modulated transistor.

Figure 37.5 Finite temperature conductance as a function of the chemical potential. d = 1015 Å, ϵ1 = 2.0, t = 0.01, and D = 0.3. kBT = 0.01. Balanced case ϵ2 = 2.0. In the inset we show the unbalanced case for ϵ2 = 1.0.

In summary, using the RGF method we have calculated the quantum conductance of a 1D DQW system when the wells are in

References

and out of balance. Thermal broadening and random short-range impurities have been taken into account. The resonance features are essentially due to the wave-function overlap of each layer within the scattering region of length d between the broken bonds. A large modulation in the conductance occurs when the wells are unbalanced, which may be a potentially useful property in the realization of quantum devices.

Acknowledgments

We would like to thank Song He, and Rodolfo A. Jalabert and M. Macucci for valuable discussions on the numerical techniques. This work was supported by United States Army Research Office grant no. DAAL03-91-G-0052, and NSF grant no. DMR 92-02255.

References

1. J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. 57, 2324 (1990). 2. T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 66, 1216 (1991). 3. J. P. Eisenstein, T. J. Gramila, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 44, 6511 (1991). 4. C. C. Eugster and J. A. del Alamo, Phys. Rev. Lett. 67, 3586 (1991).

5. P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981); A. D. Stone and A. Szafer, IBM J. Res. Dev. 32, 384 (1988); H. U. Baranger and A. D. Stone, Phys. Rev. B 12, 8169 (1989). 6. D. J. Thouless and S. Kirkpatrick, J. Phys. C 14, 235 (1981); A. MacKinnon, Z. Phys. B 59, 385 (1985); H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, Phys. Rev. B 44, 10637 (1991).

7. F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. Lett. 54, 350 (1989); J. Appl. Phys. 66, 3892 (1989).

8. T. Kawamura and J. P. Leburton, Phys. Rev. B 48, 8857 (1993).

9. The calculations outlined here are easily generalized to the case where the 1D wires are given a finite lateral extent along the y axis. For N lateral sites per layer, the Green’s-functions matrices are of dimension 2N.

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10. G. Kirczenow, Phys. Rev. B 39, 10452 (1989); A. Szafer and A. Douglas Stone, Phys. Rev. Lett. 62, 300 (1989).

11. The system considered here for the T-matrix analysis is slightly different from the one illustrated in Fig. 37.1. Here we take the semiinfinite leads to consist of a single layer. For our parameter values we have checked by mode-matching methods that the conductance profiles essentially remain unaltered as long as EF is above both subband threshold values. 12. Quantum Mechanics, 2nd ed., edited by E. Merzbacher (Wiley, New York, 1970), pp. 492–499. 13. S. Datta, Superlatt. Microstruct. 6, 83 (1989).

Chapter 38

Cascaded Spintronic Logic with Low-Dimensional Carbon

Joseph S. Friedman,a,b Anuj Girdhar,c,d Ryan M. Gelfand,a,e Gokhan Memik,a Hooman Mohseni,a Allen Taflove,a Bruce W. Wessels,a,f Jean-Pierre Leburton,c,d,g and Alan V. Sahakiana,h aDepartment

of Electrical Engineering & Computer Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA bDepartment of Electrical & Computer Engineering, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, Texas 75080, USA cDepartment of Physics, University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, Illinois 61801, USA dBeckman Institute for Advanced Science & Technology, University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, Illinois 61801, USA eCREOL, The College of Optics and Photonics, University of Central Florida, 4304 Scorpius Street, Orlando, Florida 32816, USA fDepartment of Materials Science & Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA gDepartment of Electrical & Computer Engineering, University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, Illinois 61801, USA hDepartment of Biomedical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA [email protected]

Reprinted from Nat. Commun., 8, 15635, 2017. Physical Models for Quantum Wires, Nanotubes, and Nanoribbons Edited by Jean-Pierre Leburton Text Copyright © 2017, The Author(s) Layout Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-91-6 (Hardcover), 978-1-003-21937-8 (eBook) www.jennystanford.com

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Cascaded Spintronic Logic with Low-Dimensional Carbon

Remarkable breakthroughs have established the functionality of graphene and carbon nanotube transistors as replacements to silicon in conventional computing structures, and numerous spintronic logic gates have been presented. However, an efficient cascaded logic structure that exploits electron spin has not yet been demonstrated. In this chapter, we introduce and analyse a cascaded spintronic computing system composed solely of low-dimensional carbon materials. We propose a spintronic switch based on the recent discovery of negative magnetoresis tance in graphene nanoribbons, and demonstrate its feasibility through tight-binding calculations of the band structure. Covalently connected carbon nanotubes create magnetic fields through graphene nanoribbons, cascading logic gates through incoherent spintronic switching. The exceptional material properties of carbon materials permit Terahertz operation and two orders of magnitude decrease in power-delay product compared to cutting-edge microprocessors. We hope to inspire the fabrication of these cascaded logic circuits to stimulate a transformative generation of energy-efficient computing.

38.1 Introduction

Manipulation of the spin-degree of freedom for spintronic computing requires the invention of unconventional logic families to harness the unique mechanisms of spintronic switching devices [1–14]. Cascading, one device directly driving another device, has been well known as a major challenge and fundamental requirement of a logic family since von Neumann’s [15] 1945 proposal for a storedprogram electronic computer. If the input and output signals are not of the same type and magnitude, it is difficult to connect devices without an additional device for translation. This extra device consumes power, time and area, and severely degrades the utility of the logic family. Here we present an alternative paradigm for computing: allcarbon spin logic. This cascaded logic family creatively applies recent nanotechnological advances to efficiently achieve high-performance computing using only low-dimensional carbon materials [16–24]. A spintronic switching device is proposed utilizing the negative magnetoresistance of graphene nanoribbon (GNR) transistors

Results

[25–29] and partially unzipped carbon nanotubes (CNTs) [30, 31], unzipped [30, 32–35] from metallic CNT interconnect. These carbon gates can be cascaded directly; no additional intermediate devices are required between logic gates. The physical parameters necessary for proper switching operation are evaluated through mean-field tight-binding calculations of the band structure to enable an analysis of computational efficiency and to provide guidance for an experimental proof of concept. The results demonstrate the potential for compact all-carbon spin logic circuits with Terahertz operating speeds and two orders of magnitude improvement in power-delay product, thus motivating further investigation of the proposed device and computational structure.

38.2 Results

38.2.1 Device Structure and Physical Operation The active switching element is a zigzag GNR field-effect transistor with a constant gate voltage and two CNT control wires, as illustrated in Fig. 38.1. The gate voltage is held constant, and the GNR conductivity is therefore modulated solely by the magnetic fields generated by the CNTs. These magnetic fields can flip the orientation of the strong on-site magnetization at the GNR edges, which display local antiferromagnetic (AFM) ordering due to Hubbard interactions [36] (see Section 38.4). As shown in Fig. 38.2, the magnetization at each edge is controlled by its neighbouring CNT, with magnetization decaying towards the centre of the GNR. In the absence of an external magnetic field or with edge magnetizations of opposite polarities, the GNR exhibits global AFM ordering in the ground state. Significantly, GNRs with edge magnetizations of the same polarity exhibit global ferromagnetic (FM) ordering in the ground state. Mean-field tight-binding calculations show that the GNR global magnetic ordering determines the band structure, and therefore the conductivity. The Zeeman interaction can switch the magnetic ground state, causing spin-dependent band splitting. There are conduction modes in the FM state for all energies, but no conduction modes in the AFM state for Fermi energy EF within the AFM state bandgap. By tuning EF into the AFM bandgap through control of the

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Cascaded Spintronic Logic with Low-Dimensional Carbon

Figure 38.1 All-carbon spin logic gate. Magnetoresistive GNR unzipped from carbon nanotube and controlled by two parallel CNTs on an insulating material above a metallic gate. As all voltages are held constant, all currents are unidirectional. The magnitudes and relative directions of the input CNT control currents ICTRL determine the magnetic fields B and GNR edge magnetization, and thus the magnitude of the output current IGNR.

Figure 38.2 Graphene nanoribbon edge magnetization. (a) On-site magnetization profile of a zigzag graphene edge. The magnetic field created by an adjacent CNT current causes strong on-site magnetization at the GNR edge. The colour of each circle represents the spin species, while the radius corresponds to the magnitude of the magnetization. (b) The on-site magnetization of each site in a unit cell as a function of distance from the edge. (c) Graphene nanoribbon edge magnetization in the absence and presence of an externally applied magnetic field. In the absence of a magnetic field, the GNR exhibits global AFM ordering with edges of opposite polarities. The application of a magnetic field aligns the edge polarities, achieving global FM ordering.

Results

gate voltage, the application of magnetic fields at the GNR edges causes a colossal change in conductivity, switching the GNR from the resistive AFM state to a conductive FM state. Importantly, if EF is outside the AFM bandgap, conduction modes are always available, and switching of the magnetic ordering does not cause a change in conductivity. This is one possible explanation for the lack of magnetoresistance observed by Bai et al. [28] when applying an inplane magnetic field to a GNR. It can be further noted that given the proximity between the CNTs and GNR, the attractive van der Waals and repulsive Casimir forces may alter the electronic wavefunctions and energy dispersion. However, these effects do not change the nature of the highly conductive CNT transport, nor the electronelectron repulsion among lattice sites. As a result, the spontaneous AFM ordering and edge magnetization are not sufficiently affected to alter the GNR magnetoresistance.

38.2.2 Edge Effects and Operation Temperature

The spin-dependent band splitting is strongest with pristine zigzag edges [37] as achieved in Ref. [35], enabling a spin-polarized current [38] and closing the energy gap for the GNR in the FM state as shown in Fig. 38.3. Local edge defects in quasi-pristine GNRs cause local perturbation in the magnetic ordering around these defects [37]. However, the magnetic state is quickly regained within two unit cells (3 nm, increasing the magnitude of the critical magnetic field for large defect density. As the magnetic ordering originates from the GNR edges, it is not affected significantly by defects present away from the edges. In the case of very rough edges, a lack of sufficient contiguous zigzag portions to compensate for the presence of armchair edges may result in large switching fields. Smooth GNRs with long contiguous stretches of pristine zigzag edges have been experimentally demonstrated [35]. As defects affect the magnetization on the order of 1 nm around the defect location [37], this abundance of zigzag edges of 5 nm or larger elicits strong magnetic ordering. Sufficient contiguous zigzag edges between defects thus enable a persistence of the magnetic order.

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Cascaded Spintronic Logic with Low-Dimensional Carbon

Figure 38.3 Magnetoresistive behaviour of GNR controlled by adjacent CNTs. (a,b) Band diagrams for the AFM and FM global ordering of a 12-atomwide zigzag GNR with zero current in the CNTs and Hubbard parameter U = 2.7 eV, as in Eq. (38.1) of Section 38.4. In the global AFM state (a), there is a large gap between the valence and conduction bands, within which lies the Fermi energy, EF. Therefore, there are no available conduction modes, and the conductance is zero. In the FM state (b), there is no bandgap and there is at least one conduction mode at all energies. (c,d) The magnetic instability energy in meV for zigzag GNRs with widths of (c) 20 nm and (d) 35 nm. The blue region designates a positive instability energy (the insulating AFM state), while the red region indicates negative instability energies (the conductive FM state). In the narrower GNR transistor, the axes of the CNTs are 10 nm from the GNR edge, while the wider GNR has CNTs placed 1 nm away. The critical switching current, which depends on U, is denoted with a dashed line. (e) The transmission function T(E) of the AFM state defines the number of available conduction modes as well as the probability for an electron to travel across the device. Thus, for EF values within the bandgap, the GNR conductance switches when the global ordering switches between the FM and AFM states. (f) A typical switching event, where the GNR conductance increases by G0 when the CNT current overcomes the critical switching current IC.

Yazyev [39] indicated a GNR Curie temperature near 10 K, below which the spin correlation length grows exponentially. At temperatures around 70 K, correlation lengths are on the order of 10 nm, presenting a limitation for device operation. The correlation length approaches 1 nm at room temperature, making observation of the magnetization difficult in disordered systems. Therefore,

Results

low temperatures are desirable to minimize the required magnetic field and to ensure the manifestation of this effect in large samples. This concern may have been resolved, with magnetic order recently demonstrated in zigzag GNRs at room temperature [40].

38.2.3 Switching Behaviour

We performed simulations of the proposed all-carbon spintronic switching device to determine the system and material parameters required to ensure feasibility. The magnetic instability energy is dependent on the GNR width (Supplementary Note 1), and determines the edge magnetic field required to switch the global ground state from AFM to FM ordering. As shown in Fig. 38.3c,d, the CNT current sufficient to overcome the magnetic instability energy is strongly affected by the proximity of the CNT control wires to the GNR edges. The current requirement can be tuned through control of the Hubbard U parameter. The required current ranges from exceptionally small magnitudes to significant fractions of an Ampere, and can be minimized with a wide GNR positioned close to the CNT control wires. As the GNR width is increased, the magnetic instability energy decreases as nearly the inverse square of the width [25]. For many U values and GNR/CNT geometries, the 20 mA that can be passed through a single-walled CNT is sufficient to maintain the required switching current [41–43]. When the GNR switches from the AFM to the FM state, there is a massive change in conductance, as shown in Fig. 38.3f. The magnitude of the current through the GNR functions as the binary gate output, with binary 1 representing the large current of the conductive FM state and binary 0 representing the resistive AFM state. The GNR current flows through the CNT from which it was unzipped, and this binary CNT current is the input to cascaded GNR gates. It should be noted that unlike other spintronic logic proposals, logic gates can be cascaded directly through the carbon materials without requiring intermediate control or amplification circuitry.

38.2.4 Logic Gates and System Integration

The various combinations of input magnitudes and directions permit the computation of the logical OR and XOR operations. When there

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Cascaded Spintronic Logic with Low-Dimensional Carbon

is no difference in magnetization between the edges of the GNR, the GNR is in the resistive AFM state and outputs a binary 0. Application of current through the CNTs can cause the GNR to switch into the FM state and output a binary 1. The OR logic function of Table 38.1 is computed by CNT currents oriented in opposite directions that create aligned on-site magnetization at the GNR edges. This OR gate thus enables a highly conductive FM state in the presence of current in at least one input CNT. In the XOR logic function of Table 38.2, the input currents are oriented in the same direction. Therefore, large currents flowing through both CNTs cause AFM ordering in the XOR gate, resulting in a small output current. This GNR switching device provides the functionality necessary for general-purpose computing, as the OR and XOR gates form a sufficient basis set to generate all binary functions. Table 38.1 GNR OR gate truth table for input CNT control currents in opposite directions IA

IB

State

Io

0

0

AFM

0

FM

1

0 1 1

1 0 1

FM FM

1 1

Table 38.2 GNR XOR gate truth table for input CNT control currents in the same direction IA 0

0 1 1

IB

State

1

FM

0

0 1

Io

AFM

0

AFM

0

FM

1 1

Nanofabrication trends suggest potential techniques for efficiently constructing cascaded all-carbon spin logic integrated circuits scaled up to perform complex computing tasks. Parallel and perpendicular CNTs can be laid out on an insulating surface [44] above a metallic material used as a constant universal gate voltage for the entire circuit. As shown in Fig. 38.4, a complex

Results

Figure 38.4 All-carbon spin logic one-bit full adder. (a) The physical structure of a spintronic one-bit full adder with magnetoresistive GNR FETs (yellow) partially unzipped from CNTs (green), some of which are insulated (brown) to prevent electrical connection. The all-carbon circuit is placed on an insulator above a metallic gate with constant voltage VG. Binary CNT input currents A and B control the state of the unzipped GNR labelled XOR1, which outputs a current with binary magnitude A≈B. The output of XOR1 flows through a CNT that functions as an input to XOR2 and XOR3 before reaching the wired-OR gate OR2, which merges currents to compute CIN -(A≈B). This current controls XOR4 and terminates at V_. The other currents operate similarly, computing the one-bit addition function with output current signals S and COUT. (b) In the symbolic circuit diagram shown here with conventional symbols, the output of XOR1 is used as an input to OR2 and XOR3 along with CIN. The full adder S output is computed as S = CIN - ≈ (A≈B). OR2 outputs CIN -(A≈B), which is used along with S as an input to XOR4 to compute (CIN-(A≈B))≈(CIN≈(A≈B)). This output of XOR4 is equivalent to (A.CIN)-(B.CIN). OR3 takes this signal as an input along with the output of XOR2, which is equal to A.B, to compute COUT = (A.B)-(A.CIN)-(B.CIN). As the wired-OR gates simply sum the currents and have no significant delay, the total propagation time is that of three XOR gates, determined by the XOR1-XOR3-XOR4 worst-case path.

circuit composed of the logic gates of Fig. 38.1 can be created through selective CNT unzipping to form GNRs [24, 30, 32–35]. Electrical connectivity between overlapping CNTs [34, 45, 46] can be determined by the placement of an insulating material. The only external connections are to the supply voltage and user input/ output ports (for example, keyboard, monitor and so on), possibly with vertical covalent contacts of the type described by Tour [47].

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Cascaded Spintronic Logic with Low-Dimensional Carbon

All computing functionality is performed by the carbon materials alone, without the aid of external circuitry. As in other large-scale integrated circuits’ fabrication imprecision (for example, misaligned CNTs, imperfect CNT junctions, edge defects and so on) can be tolerated provided that the GNR logic gates function properly and the electrical connectivity between CNTs is correct. Though the possibility of miniaturization is an important figure of merit for conventional computing structures, the atomic dimensions of CNTs and GNRs make the concept of down-scaling irrelevant for all-carbon spin logic. Cascaded all-carbon spin logic gates can be connected by routing the GNR output currents through the CNT control inputs of other GNR gates. Four XOR gates and three wired-OR gates are cascaded in Fig. 38.4 to realize a full adder, an essential computational function traditionally performed with 28 CMOS transistors (Supplementary Fig. 1 and Supplementary Table 1). The supply voltage nodes V+ and V_ are held constant, thereby causing the polarities of all current paths to be constant. Binary switching results from changes in input current magnitudes due to changes in the output currents of other gates, without amplification, conversion or control circuitry. The output currents flow through the inputs of other logic and memory elements such as parity gates and toggle latches (Supplementary Figs. 2 and 3, and Supplementary Table 2). These circuits provide traditional logic functionality with far fewer devices, enabling compact spintronic computing systems.

38.3 Discussion

While these circuits may be implemented with other materials exhibiting high conductivity and negative magnetoresistance, the exceptional properties of CNTs and GNRs make these structures ideal candidates for use in this logic family. Current is the state variable in all-carbon spin logic, enabling exceptionally fast computation with switching delay determined by electromagnetic wave propagation. This is in stark contrast to conventional computing systems in which voltage is the state variable, leading to CMOS switching and RLC interconnect delays limited by charge transfer and accumulation.

Discussion

As described by Fig. 38.5, the GNR conductivity switches far faster than the signal can propagate through the CNTs [48]. The all-carbon spin logic switching time td = tmag + tgnr + tprop is the summation of the times required for a CNT current to switch a magnetic field in a neighbouring GNR (tmag), the GNR magnetoresistance to switch in response to a magnetic field (tgnr) and the electric field to propagate through the CNT to switch the current (tprop). The propagation time tprop is significantly larger than tmag and tgnr, and therefore determines td. The electromagnetic wave propagation speed in a CNT 1 is vf = = 800 kms–1, where LK = 400 pH nm–1 is the kinetic LK CQ inductance and CQ = 0.4 aF nm–1 is the quantum capacitance [48]. For 400 nm CNT interconnect length lcnt, the worst-case logic gate l switching time is td = tprop = cnt = 500 fs. High-performance circuits vf operating with clock frequencies of 2 THz can therefore be realized. The average power dissipation per logic gate is P = VsupplyIaverage = I +I Vsupply on off = 10 mW, given a differential supply voltage Vsupply = 2 V+ – V_ = 1V and an on-state current Ion = IC ª 20 mA sufficiently small to flow through a single-walled CNT. The power-delay product for each gate, a metric of computing efficiency, can be determined for all-carbon spin logic as PDP = Ptd ª 5 ¥ 10–18 J. This is approximately 100 times more energy-efficient than 22 nm CMOS.

Figure 38.5 Analysis of switching and propagation delay. Following a switch in the current through CNT0 at a time t = 0, the magnetic field at the edge of GNR1 switches at t = tmag, the resistance of GNR1 switches at t = tmag + tgnr and the current through CNT1 switches at t = tmag + tgnr + tprop. This marks the end of one complete switching and propagation cycle, and is immediately followed by the switching of GNR2.

Furthermore, the power dissipation of all-carbon spin logic is nearly independent of frequency, whereas conventional CMOS

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Cascaded Spintronic Logic with Low-Dimensional Carbon

circuits dissipate increasing power as clock frequency is increased. This two orders of magnitude improvement outweighs the power costs of low-temperature operation and leaves significant room for second-order parasitic effects. This direct comparison can be made due to the absence of additional circuitry between logic gates, in contrast to other spintronic logic proposals. In addition, each GNR switching device in the all-carbon spin logic family performs the functionality of between four and twelve CMOS transistors. The onebit full adder of Fig. 38.4, for example, has only four active GNR gates and a propagation time of 3td, yielding a PDP of 6 ¥ 10–17 J. All-carbon spin logic permits the development of cascaded spintronic logic circuits composed solely of low-dimensional carbon materials without intermediate circuits between gates, resulting in compact circuits with reduced area that are far more efficient than CMOS. Though a complete all-carbon spin logic system is several years away from realization, currently available technology permits experimental proof of the concept as shown in Fig. 38.6. By exploiting the exotic behaviour of GNRs and CNTs, all-carbon spin logic enables a spintronic paradigm for the next generation of high-performance computing.

Figure 38.6 Proposed proof-of-concept experiment. By measuring the change in IOUT in response to a change in IIN, the central component of all-carbon spin logic can be demonstrated. The carbon nanotube CNTOUT can be partially unzipped such that a portion forms a GNR. A second CNT, CNTIN, is then placed nearly parallel to CNTOUT. A constant voltage should be applied across CNTOUT. It is not necessary to achieve the dimensions described in this work; rather, to make the experiment more facile, CNTIN must merely be close enough to the GNR to cause a measurable response in IOUT when IIN is varied. Furthermore, as shown in the figure, CNTIN need not be as long as CNTOUT, thereby preventing the CNTs from making contact even if the CNTs are not perfectly parallel.

Methods

38.4 Methods 38.4.1 Hubbard Tight-Binding Hamiltonian The tight-binding Hamiltonian for a zigzag-edged GNR is =H



 ÊË t c

 ij is c js

ijs

+ h.c .ˆ + U ¯

Â(n

 iØ Ò + n iØ ·n i≠ Ò - ·n i≠ Ò·n iØ Ò ) i≠ ·n

i

Â

- gs mBmS BiZ (n i≠ - n iØ ), (38.1) i

where tij is the transfer integral between orbitals localized at sites  i and j of the GNR lattice, and c is is the annihilation (creation) operator for an electron of spin s at site i. Interactions up to the third nearest neighbour are considered, and the values for all transfer and overlap integrals are taken from set D of Hancock et al. [37]. In the second term, which represents the Coulomb interaction between electrons, U is the repulsive Hubbard parameter and n is is the on-site occupation of an electron with spin s. In the mean-field approximation, the expectation of the on-site occupation is used to reduce the complexity of the Hamiltonian, which can be solved with iterative methods [49, 50]. Finally, the third term represents the Zeeman interaction, where gS is the electron Landé g-factor, mB is the Bohr magneton, mS is the z-component of electron spin and BiZ is the z-component of the magnetic field at site i. The non-homogeneous magnetic field is generated by the Biot–Savart law and permeates everywhere in space, thereby affecting every atom in the GNR (not only the edges). The magnitudes of the magnetic fields in this study are small enough to neglect phase changes in the transfer integrals due to the magnetic field.

38.4.2 Diagonalization and the Secular Equation

By taking advantage of the translational symmetry of a GNR, the kth component of the Hamiltonian for spin s can be written as

 s (k ) = H  0s + V s e ik ◊a + V s e - ik ◊a (38.2) H

 0s represents the interactions within one unit cell, and V s where H is the interaction between one cell and the next (previous) unit cell

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at a displacement of a (–a). Each component of the single-particle states can be calculated by solving the secular equation

 s c sm = e m S s c sm . (38.3) H

m

Here c s is the eigenstate of spin s corresponding to the energy em and S s is the overlap matrix, which is the identity matrix in the investigated parameter set. The energies esk corresponding to N k states c s where N is the number of sites in the unit cell, define the band structure of the GNR. At the GNR edges, we assume hydrogen passivation sp2 dangling bonds, which is the standard treatment of edges in simulations of transport through GNRs. As there is no dangling bond, there is no reconstruction other than a slight modification of the bond angle between H–C and C–C bonds. This has a negligible effect on the magnetization of the edges. In our calculations, we use 30,000 k points in the Brillouin zone.

38.4.3 Mean-Field Approximation

At zero temperature, the N lowest states are populated, and the occupation of the ith site in the unit cell is given by

·n is Ò =

esk < e F

 S kj

k k*  is c js ijs c

(38.4)

This value is used as an input to the Hamiltonian for the next iteration, and this process is repeated until the maximum change in occupation at any site is