Charge Transport in Low Dimensional Semiconductor Structures: The Maximum Entropy Approach [1 ed.] 3030359921, 9783030359928

Table of contents :
Preface
Contents
About the Authors
1 Band Structure and Boltzmann Equation
1.1 Crystal Structure
1.2 The Energy Band Structure
1.3 The Si Band Structure and the Semi-classical Picture
1.4 The Boltzmann Equation
1.5 H-Theorem and the Null Space of the Collision Operator
1.6 Quantum Confinement and Quasi 2DEG
1.7 Derivation of the Transport Equation Along the Longitudinal Direction
2 Maximum Entropy Principle
2.1 The Entropy
2.1.1 Properties of the Shannon–Jaynes Entropy
2.1.2 Shannon–Jaynes Entropy in the Continuous Case
2.2 Maximum Entropy Inference of a Distribution: The Discrete Case
2.3 Examples of Distribution Functions Deduced with MEP
2.3.1 Maxwell–Boltzmann Distribution
2.3.2 Fermi–Dirac and Bose–Einstein Distributions
2.4 Maximum Entropy Inference of a Distribution: The Continuous Case
3 Application of MEP to Charge Transport in Semiconductors
3.1 The Electron Transport Equation and the Maximum Entropy Principle
3.2 Further Considerations
3.3 Solvability of the MEP Problems in Semiconductors
3.3.1 Statement of the Main Result
3.3.2 A General Result by Csiszar
3.3.3 The Weight Functions
3.3.4 The Moment Cone
3.3.5 The Entropy Functional
3.3.6 The Lagrange Multipliers
3.3.7 Proof of the Main Theorem
4 Application of MEP to Silicon
4.1 Moment Equations and Closure Problem
4.2 Lagrange Multipliers
4.3 Constitutive Equations for the Fluxes
4.4 Closure Relations for the Production Terms
4.4.1 Acoustic Phonon Scattering
4.4.2 Non Polar Optical Phonon Scattering
4.4.3 Scattering with Impurities
4.5 Parabolic Band Approximation
4.6 Application to Bulk Silicon
4.7 The Energy Transport Limit Model
4.8 The Drift-Diffusion Limiting Model
4.8.1 Parabolic Band Case
4.8.2 Kane's Dispersion Relation
4.9 Formulation of the Model in the Framework of Linear Irreversible Thermodynamics
4.10 A Numerical Approach Based on Finite Elements
4.10.1 Simulation of a n+-n-n+ Silicon Diode
4.10.2 Simulation of a 2D Silicon MESFET
4.10.3 Simulation of a 2D Silicon MOSFET
5 Some Formal Properties of the Hydrodynamical Model
5.1 Hyperbolicity of the MEP Hydrodynamical Model
5.2 Nonlinear Asymptotic Stability of the Equilibrium State
5.2.1 Basic Equations and Formulation of the Problem
5.2.2 Formulation of the Auxiliary Problems
5.2.3 Asymptotic Stability of the Equilibrium State
5.2.4 Explicit Expressions the Production Terms
5.2.5 Estimates for J(0), J(2)
6 Quantum Corrections to the Semiclassical Models
6.1 Wigner Equation
6.2 Equilibrium Wigner Function
6.3 The Collision Operator
6.4 Quantum Corrections in the High Field Approximation
6.5 The Quantum Moment Equations
6.6 Entropy Balance Equation
6.7 Energy-Transport and Drift-Diffusion Limiting Models
7 Mathematical Models for the Double-Gate MOSFET
7.1 Semiclassical Model for DG-MOSFET
7.2 The Moment System and Its Closure by the MEP
7.3 Energy-Transport Model
7.4 Boundary Conditions and Initial Data
7.4.1 Boundary Conditions and Initial Data for the SP-Block
7.4.2 Boundary Conditions and Initial Data for the ET Block
8 Numerical Method and Simulations
8.1 Discretization of the Schrödinger–Poisson Equations
8.2 Discretization of the Energy-Transport Equations
8.3 Numerical Simulations
9 Application of MEP to Charge Transport in Graphene
9.1 Kinetic Description
9.2 Moment Equations
9.3 Closure Relations
9.4 Production Terms of Acoustic Phonon Scattering
9.5 Production Terms of Optical Phonon Scattering
9.5.1 Intraband Optical Phonon Scattering Production Term
9.5.2 Interband Optical Phonon Scattering Production Term
9.6 Production Terms of K-Phonon Scattering
9.6.1 Intraband K-Phonon Scattering Production Term
9.6.2 Interband K-Phonon Scattering Production Term
9.7 Numerical Results with Constant Lattice Temperature
9.8 Inclusion of the Crystal Heating
9.8.1 Carrier Moment Equations
9.8.2 The Phonon Moment System
9.9 The Closure Problem
9.10 Inversion of the Constraint Relations and Definition of the Temperature
9.10.1 Phonons
9.10.2 Temperature
9.11 Closure Relations
9.11.1 Electrons and Holes
9.11.2 Phonons
9.12 Hyperbolicity of the Model
9.13 Numerical Results with Variable Lattice Temperature
9.14 Conclusions
A 2DEG: Closure Relations for the Kane Case
B Closure Relations for the Parabolic Case
C Useful Computational Relations
D Crystal Vibrations and Phonons
E Simulation Codes
References
Index

Citation preview

Mathematics in Industry 31 The European Consortium for Mathematics in Industry

Vito Dario Camiola Giovanni Mascali Vittorio Romano

Charge Transport in Low Dimensional Semiconductor Structures The Maximum Entropy Approach

Mathematics in Industry The European Consortium for Mathematics in Industry Volume 31

Managing Editor Michael Günther, University of Wuppertal, Wuppertal, Germany Series Editors Luis L. Bonilla, University Carlos III Madrid, Escuela, Leganes, Spain Otmar Scherzer, University of Vienna, Vienna, Austria Wil Schilders, Eindhoven University of Technology, Eindhoven, The Netherlands

The ECMI subseries of the Mathematics in Industry series is a project of The European Consortium for Mathematics in Industry. Mathematics in Industry focuses on the research and educational aspects of mathematics used in industry and other business enterprises. Books for Mathematics in Industry are in the following categories: research monographs, problem-oriented multi-author collections, textbooks with a problem-oriented approach, conference proceedings. Relevance to the actual practical use of mathematics in industry is the distinguishing feature of the books in the Mathematics in Industry series.

More information about this subseries at http://www.springer.com/series/4651

Vito Dario Camiola • Giovanni Mascali • Vittorio Romano

Charge Transport in Low Dimensional Semiconductor Structures The Maximum Entropy Approach

Vito Dario Camiola Department of Mathematics and Computer Science University of Catania Catania, Italy

Giovanni Mascali Department of Mathematics and Computer Science University of Calabria Arcavacata di Rende, Italy

Vittorio Romano Department of Mathematics and Computer Science University of Catania Catania, Italy

ISSN 1612-3956 ISSN 2198-3283 (electronic) Mathematics in Industry The European Consortium for Mathematics in Industry ISBN 978-3-030-35992-8 ISBN 978-3-030-35993-5 (eBook) https://doi.org/10.1007/978-3-030-35993-5 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of prof. Angelo Marcello Anile

Preface

Nissuna umana investigazione si pò dimandare vera scienzia s’essa non passa per le matematiche dimostrazioni, e se tu dirai che le scienzie, che principiano e finiscono nella mente, abbiano verità, questo non si concede, ma si niega, per molte ragioni, e prima, che in tali discorsi mentali non accade esperienzia, sanza la quale nulla dà di sé certezza. Leonardo da Vinci Quelli che s’innamoran di pratica sanza scienzia son come ’l nocchier ch’entra in navilio senza timone o bussola, che mai ha certezza dove si vada. Leonardo da Vinci

In 1948 W. Shockley, W. Brattain and J. Bardeen invented the first transistor at the Bell laboratories and in 1958 Jack Kilby, an engineer at Texas Instruments, announced the creation of the first integrated circuit. Since then, progression in electronic engineering has meant miniaturization. By continuing to shrink the dimension of electronic devices, the effects of quantum confinement become more and more relevant. In particular, these effects need to be well understood in metal-oxide-semiconductor field-effect transistors (MOSFETs) and in the more recent double gate MOSFETs (DG-MOSFETs), which are the backbone of the modern integrated circuits. The DG-MOSFET, with its extra gate, is considered one of the most appropriate structures for minimizing, in nanometer devices, the short channel effects which cause a deterioration in the transistor performance—this thanks to the enhanced gate control (two gates) over the channel and to the reduced silicon layer thickness. In a DG-MOSFET the potential between the two gates and the oxide layers confines the electrons in the transversal direction, producing a quantum well whose length is comparable to the de Broglie wavelength. A similar effect is also present in hetero-structures like AlGa-Ga [22, 62, 63, 126]. Recent years have also witnessed great interest in 2D materials for their promising applications. The most investigated material is graphene, but of late the single-layer transition metal dichalcogenides (TMDCs), such as molybdenum vii

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disulfide and tungsten diselenide, and black phosphorus have received a certain amount of attention (see for example [204]). Graphene is considered a potential new material for future applications in nanoelectronic [20, 48, 55, 121, 151, 172] and optoelectronic devices [204] due to the fact that it has very good mechanical properties, is an excellent heat and electricity conductor, and also has noteworthy optical properties. It is a twodimensional allotrope of carbon which consists of carbon atoms tightly packed into a honeycomb hexagonal lattice. Thanks to this structure it has, as first approximation, a conical not curved band structure, so the effective mass of the electrons is zero and charge carriers exhibit a photon-like behavior. In a purely quantum approach, electric properties of nanoscale devices, such as current-voltage curves, can be computed via the non-equilibrium Green function [50, 200]. Another way to tackle the problem is in the framework of quantum kinetic theory via the Wigner function, which gives the macroscopic physical quantities of interest as expectation values [99, 136, 177]. Other approaches are based on the master equation [75]. However, in structures like DG-MOSFETs, the confining effect is in the direction transversal to the oxide, while in the other directions the electrons flow freely from the source to the drain and the electron transport can be treated semiclassically when the typical longitudinal length is of the order of a few tens of nanometers. In fact, in these conditions the electrons, as waves, achieve equilibrium along the confining direction in a time which is much shorter than the typical transport time, so that one can adopt a quasi-static description along the confining direction by a coupled Schrödinger–Poisson system which leads to a subband decomposition. The transport along the longitudinal direction can be described by a semiclassical Boltzmann equation for each subband. Numerical integration of the Boltzmann–Schrödinger–Poisson system has been performed with Monte Carlo methods or deterministic schemes for solving the transport part [4, 28, 79, 134, 167, 168, 196], but they are very expensive, from a computational point of view, for computer-aided design (CAD) purposes. This has prompted substitution of the Boltzmann equations with macroscopic models such as drift-diffusion or energy-transport ones [27, 64]. In this monograph, in order to describe the electron transport, models of an energy transport type will be adopted. They are deduced, under a suitable diffusion scaling, from a system of equations derived from the Boltzmann equations by using the moment method. The moment equations are closed by resorting to the maximum entropy principle (MEP). The results fit into the framework of extended thermodynamics [97, 153]. In the context of the Bayesian interpretation of the probability, Jaynes [94, 96] showed that it is possible to develop the MEP as an inference method to obtain the results achieved in the context of the statistical mechanics or, more generally, in all cases where the complete information, necessary for a detailed description, is lost or is not available. The term entropy was introduced for the first time in classical thermodynamics by Clausius in 1865. In 1948 Shannon [186] used the same term to indicate the amount of ignorance about a system in the context of information theory. The formal

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expression for this quantity is the same as that of the statistical Boltzmann entropy in the canonical ensemble and in 1957 Jaynes [94, 96] showed that it is possible to obtain all the expressions of statistical mechanics by maximizing it. An approach based on the MEP is also described for the treatment of charge transport in graphene. It leads to a hydrodynamical model which has been tested by a comparison with direct simulation Monte Carlo results. The model appears to be sound and sufficiently accurate for systematic use in CAD simulators for complex electron devices. The book is addressed to applied mathematicians, physicists, and electronic engineers. It is written for graduate or PhD readers but the initial chapter contains a modicum of semiconductor physics, which makes the book self-consistent and useful also for undergraduate students. The plan of the book is as follows: In Chap. 1 an overview of semiconductor band structure and of the Boltzmann equation is given, with emphasis on the case of confined transport. In Chap. 2 the MEP is presented in a general way, focusing attention on its meaning as a method of inference. Chapter 3 is devoted to the general features of application of MEP to bulk (three-dimensional) semiconductors with a special focus on the problem of the realizability. Detailed results for silicon are reported in Chap. 4, where in addition the numerics are discussed and applied to the most common semiconductor devices. Some formal properties are discussed in the subsequent chapter. In Chap. 6 a quantum model is introduced by using a Wigner approach which allows us to get, under a suitable scaling, explicit quantum corrections to the semiclassical model presented in Chap. 4. The remaining parts constitute the core of the book. The mathematical model for charge transport in DG-MOSFETs is formulated and a suitable numerical scheme is presented, along with simulations in cases of interest for the electrical engineering design. The book concludes with an application of MEP to graphene. In order to help those who want to use the models presented in this monograph, in the last Appendix detailed numerical codes are reported and commented upon for the simulation of a one-dimensional silicon diode. The interested reader can easily modify these codes to deal with more complex devices. The authors wish to acknowledge the stimulating and fruitful discussions with a number of colleagues. We would like to mention Nauofel Ben Abdallah, Giuseppe Alì, Luigi Barletti, Alexander Blokhin, Luis Bonilla, Carlo De Falco, Massimo Fischetti, Giovanni Frosali, Irene Gamba, Carlo Jacoboni, Ansgar Jüngel, Michael Junk, Antonino La Magna, Salvatore Fabio Liotta, Armando Majorana, Americo Marrocco, Omar Morandi, Orazio Muscato, Claudia Negulescu, Paola Pietra, Giovanni Russo, Ferdinand Schürrer, Jean Michel Sellier, and Francesco Vecil, and we apologize for any missing names. An expression of thanks also goes to the PhD students Marco Coco, Liliana Luca, and Giovanni Nastasi, who have read the first version of the book and helped the authors in improving the presentation of the topics.

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Preface

The book is dedicated to the memory of Prof. Angelo Marcello Anile, who was the scientific mentor of two of the authors (G.M. and V.R.) and also indirectly a guide for V.D.C. The approach to the matter treated in this monograph has been deeply influenced by his view of science. Catania, Italy Cosenza, Italy Catania, Italy September 2019

Vito Dario Camiola Giovanni Mascali Vittorio Romano

Contents

1

Band Structure and Boltzmann Equation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Crystal Structure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Energy Band Structure . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 The Si Band Structure and the Semi-classical Picture . . . . . . . . . . . . . . . 1.4 The Boltzmann Equation .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 H-Theorem and the Null Space of the Collision Operator .. . . . . . . . . . 1.6 Quantum Confinement and Quasi 2DEG . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Derivation of the Transport Equation Along the Longitudinal Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 5 8 14 18

2 Maximum Entropy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 The Entropy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Properties of the Shannon–Jaynes Entropy . . . . . . . . . . . . . . . . . 2.1.2 Shannon–Jaynes Entropy in the Continuous Case. . . . . . . . . . 2.2 Maximum Entropy Inference of a Distribution: The Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Examples of Distribution Functions Deduced with MEP .. . . . . . . . . . . 2.3.1 Maxwell–Boltzmann Distribution .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Fermi–Dirac and Bose–Einstein Distributions . . . . . . . . . . . . . 2.4 Maximum Entropy Inference of a Distribution: The Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

29 29 33 36

3 Application of MEP to Charge Transport in Semiconductors . . . . . . . . . . 3.1 The Electron Transport Equation and the Maximum Entropy Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Solvability of the MEP Problems in Semiconductors .. . . . . . . . . . . . . . . 3.3.1 Statement of the Main Result. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 A General Result by Csiszar. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.3 The Weight Functions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.4 The Moment Cone . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.5 The Entropy Functional.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47

23

37 40 40 42 44

47 52 53 55 58 59 61 62 xi

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3.3.6 3.3.7

The Lagrange Multipliers. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of the Main Theorem.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63 66

4 Application of MEP to Silicon . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Moment Equations and Closure Problem.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Constitutive Equations for the Fluxes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Closure Relations for the Production Terms. . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Acoustic Phonon Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Non Polar Optical Phonon Scattering .. .. . . . . . . . . . . . . . . . . . . . 4.4.3 Scattering with Impurities . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Parabolic Band Approximation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Application to Bulk Silicon . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 The Energy Transport Limit Model .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 The Drift-Diffusion Limiting Model.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.1 Parabolic Band Case . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.2 Kane’s Dispersion Relation. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.9 Formulation of the Model in the Framework of Linear Irreversible Thermodynamics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.10 A Numerical Approach Based on Finite Elements .. . . . . . . . . . . . . . . . . . 4.10.1 Simulation of a n+ − n − n+ Silicon Diode . . . . . . . . . . . . . . . 4.10.2 Simulation of a 2D Silicon MESFET . .. . . . . . . . . . . . . . . . . . . . 4.10.3 Simulation of a 2D Silicon MOSFET . . .. . . . . . . . . . . . . . . . . . . .

69 69 72 75 76 76 77 79 81 83 88 93 93 96 99 105 109 113 121

5 Some Formal Properties of the Hydrodynamical Model . . . . . . . . . . . . . . . . 5.1 Hyperbolicity of the MEP Hydrodynamical Model .. . . . . . . . . . . . . . . . . 5.2 Nonlinear Asymptotic Stability of the Equilibrium State. . . . . . . . . . . . 5.2.1 Basic Equations and Formulation of the Problem .. . . . . . . . . 5.2.2 Formulation of the Auxiliary Problems .. . . . . . . . . . . . . . . . . . . . 5.2.3 Asymptotic Stability of the Equilibrium State . . . . . . . . . . . . . . 5.2.4 Explicit Expressions the Production Terms .. . . . . . . . . . . . . . . . 5.2.5 Estimates for J (0) , J (2) . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

131 131 135 136 140 145 151 157

6 Quantum Corrections to the Semiclassical Models . .. . . . . . . . . . . . . . . . . . . . 6.1 Wigner Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Equilibrium Wigner Function .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 The Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Quantum Corrections in the High Field Approximation .. . . . . . . . . . . . 6.5 The Quantum Moment Equations.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Entropy Balance Equation.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Energy-Transport and Drift-Diffusion Limiting Models.. . . . . . . . . . . .

159 160 166 173 175 178 184 187

7 Mathematical Models for the Double-Gate MOSFET . . . . . . . . . . . . . . . . . . . 7.1 Semiclassical Model for DG-MOSFET . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 The Moment System and Its Closure by the MEP . . . . . . . . . . . . . . . . . . . 7.3 Energy-Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Boundary Conditions and Initial Data . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

191 192 194 201 204

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7.4.1 7.4.2

Boundary Conditions and Initial Data for the SP-Block . . . 204 Boundary Conditions and Initial Data for the ET Block . . . 209

8 Numerical Method and Simulations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Discretization of the Schrödinger–Poisson Equations . . . . . . . . . . . . . . . 8.2 Discretization of the Energy-Transport Equations . . . . . . . . . . . . . . . . . . . 8.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

211 212 214 217

9 Application of MEP to Charge Transport in Graphene . . . . . . . . . . . . . . . . . 9.1 Kinetic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Moment Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Closure Relations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Production Terms of Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . 9.5 Production Terms of Optical Phonon Scattering .. . . . . . . . . . . . . . . . . . . . 9.5.1 Intraband Optical Phonon Scattering Production Term . . . . 9.5.2 Interband Optical Phonon Scattering Production Term . . . . 9.6 Production Terms of K-Phonon Scattering .. . . . . .. . . . . . . . . . . . . . . . . . . . 9.6.1 Intraband K-Phonon Scattering Production Term .. . . . . . . . . 9.6.2 Interband K-Phonon Scattering Production Term .. . . . . . . . . 9.7 Numerical Results with Constant Lattice Temperature . . . . . . . . . . . . . . 9.8 Inclusion of the Crystal Heating . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8.1 Carrier Moment Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8.2 The Phonon Moment System .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.9 The Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.10 Inversion of the Constraint Relations and Definition of the Temperature .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.10.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.10.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.11 Closure Relations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.11.1 Electrons and Holes . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.11.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.12 Hyperbolicity of the Model . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.13 Numerical Results with Variable Lattice Temperature .. . . . . . . . . . . . . . 9.14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

229 230 239 241 249 250 251 254 256 257 258 260 266 269 269 270 272 272 273 274 274 276 279 280 280

A 2DEG: Closure Relations for the Kane Case . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 285 B Closure Relations for the Parabolic Case . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 289 C Useful Computational Relations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 D Crystal Vibrations and Phonons . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 295 E Simulation Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 303 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 325 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 335

About the Authors

Vito Dario Camiola received his Master’s degree in Theoretical Physics and his PhD in Applied Mathematics from the University of Catania. He then worked as a postdoctoral researcher at the Nanoscience Institute of CNR and Scuola Normale Superiore di Pisa. At present, he is a postdoctoral researcher and temporary professor of Analytical Mechanics at Universitá Politecnica delle Marche. His main research fields are theory of electron transport in low-dimensional devices and quantum thermodynamics. Giovanni Mascali received his Master’s degree in Theoretical Physics from the University of Catania in 1994, and his PhD in Mathematics at the Catania-MessinaPalermo consortium of universities in 2000. In 2002 he joined the University of Calabria as Assistant Professor. He is now Professor of Mathematical Physics at the Department of Mathematics and Computer Science of the University of Calabria. He was one of the editors of the 9th volume of Scientific Computing in Electrical Engineering, Series: Mathematics in Industry, and one of the directors of the international school MOMINE09 (Modeling and optimization in micro- and nano-electronics, Cetraro, Italy, September 2009). His research interests focus on extended thermodynamics, radiative fluid dynamics, and mathematical modeling of heat and charge transport in semiconductors and low-dimensional structures. He has published more than 60 papers in international journals, books, and conference proceedings. Vittorio Romano graduated in Mathematics in 1989 and obtained his PhD in Mathematics in 1994 at the University of Catania, Italy. He was Assistant Professor at the Polytechnic of Bari from 1996 to 2001, when he joined the University of Catania as Associate Professor. Since 2011 he has been Full Professor in Mathematical Physics. At present he is head of the Interdepartmental Centre of Mathematics for Technology A. M. Anile and a member of the Scientific Committee of the Italian National Group of Mathematical Physics and the Program Committee of the SCEE (Scientific Computing in Electrical Engineering) conferences. He has xv

xvi

About the Authors

been local coordinator of the node of the University of Catania for the projects EU-RTN “COMSON” and ENIAC JU “ERG”, director of the international schools MOMINE08 (Modeling and optimization in micro- and nano-electronics, Baia Samuele, Italy, June 2008) and MOMINE09 (Cetraro, Italy, September 2009), chair of the ICCT 2015 (International Conference on Transport Theory, Taormina, September 2015), and SCEE 2018 (Taormina, September 2018). His main current research interests include mathematical modeling and simulation of charge and phonon transport in semiconductors and low-dimensional structures, and he is the author of more than 100 papers, book chapters, and contributions in volumes and proceedings on these subjects.

Chapter 1

Band Structure and Boltzmann Equation

In this chapter a short overview will be given regarding the physics of semiconductors and some properties of the Boltzmann equation that is the starting point for describing the semiclassical transport of charge carriers. The reader is referred to [92, 99, 108] for more details.

1.1 Crystal Structure The structure of a solid is ideally given by a periodic repetition of a set of atoms, called basis, in the three dimensional space. This repetition gives rise to a three dimensional crystal lattice (see Fig. 1.1) whose translational periodicity is defined by three non coplanar vectors a1 , a2 , a3 such that the crystal remains identical to itself if translated by a generic vector T = n1 a1 + n2 a2 + n3 a3 ,

n1 , n2 , n3 ∈ Z.

The vectors a1 , a2 , a3 and the set L = {T = n1 a1 + n2 a2 + n3 a3 : n1 , n2 , n3 ∈ Z} ⊂ R3 are called primitive vectors and Bravais Lattice respectively. The primitive cell of L is a connected set D ⊂ R3 , whose volume equals that of the parallelepiped spanned by the basis vectors,1 i.e. vol(D) = |a1 · (a2 ∧ a3 )|. The cell L fills the whole space if translated by vectors belonging to L. In Fig. 1.2 a primitive cell in a bidimensional

1 The

symbol “∧” denotes the vector product.

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_1

1

2

1 Band Structure and Boltzmann Equation

Fig. 1.1 Example of a crystal lattice with a basis of one atom

Fig. 1.2 Primitive cell for a two dimensional lattice

case is depicted. In particular, the special primitive cell  3

D= T∈R :T=

3  n=1

  1 1 αn an , αn ∈ − , 2 2

is called the Wigner-Seitz cell. The reciprocal lattice L∗ of L is defined by   L∗ = G = n1 a∗1 + n2 a∗2 + n3 a∗3 : n1 , n2 , n3 ∈ Z , where the primitive vectors a∗1 , a∗2 , a∗3 ∈ R3 are the dual basis given by a∗1 = 2π

a2 ∧ a3 ; a1 · a2 ∧ a3

a∗2 = 2π

a3 ∧ a1 ; a1 · a2 ∧ a3

a∗3 = 2π

They satisfy the relations am · a∗n = 2πδmn with m, n = 1, 2, 3.

a1 ∧ a2 . a1 · a2 ∧ a3

1.2 The Energy Band Structure

3

According to the previous definitions, the volume of the reciprocal primitive cell is a∗1 · (a∗2 × a∗3 ) , while the Wigner-Seitz cell of the reciprocal lattice is  B = k ∈ R3 : k =

3  n=1



1 1 βn a∗n , βn ∈ − , 2 2

 (1.1)

and is called the first Brillouin zone.

1.2 The Energy Band Structure The energy structure of a semiconductor is strictly correlated with the periodicity of the crystal lattice. In the Bloch approximation, the dynamics of an external electron inside a semiconductor, without the presence of external fields, is described by the time-dependent Schrödinger equation [108] i h¯

∂ Ψ (r, t) = H Ψ (r, t) ∂t

with the Hamiltonian H =−

h¯ 2 Δ − qVL (r). 2m

The function Ψ is the electron wavefunction, Δ is the Laplacian operator, h¯ is the reduced Planck constant, −qVL (r) is the potential energy of the electron in the periodic field of the lattice having, therefore, the periodicity of the direct lattice. VL is due to the nuclei, to the core electrons and to the average interaction with all the other external electrons [92]. q represents the (positive) elementary charge. Et Assuming the ansatz Ψ (r, t) = ψ(r)e−i h¯ , the time independent or stationary Schrödinger equation is obtained H ψ(r) = Eψ(r)

(1.2)

with E ∈ R. The general form of the solutions of (1.2) is expressed by the following Bloch theorem [17, 108]. Theorem 1.2.1 Let VL be a periodic potential, i.e. VL (x + T) = VL (x) for all x ∈ R3 and T ∈ L (the Bravais lattice ). Then the eigenvalue problem for the Schrödinger operator H =−

h¯ 2 Δ − qVL (x), 2m

x ∈ R3 ,

4

1 Band Structure and Boltzmann Equation

can be reduced to an infinite set of eigenvalue problems for the Schrödinger equation on the primitive cell D of the lattice, indexed by k ∈ B (the Brillouin zone), H ψk = Eψk

in D,

ψk (x + T) = eik·T ψk (x),

x ∈ D, T ∈ L.

(1.3)

For each k ∈ B, there exists a sequence En (k), n ≥ 1, of eigenvalues with associated eigenfunctions ψn,k . The eigenvalues En (k) are real functions of k, periodic and symmetric on B. The spectrum of H is given by the union of the closed intervals {En (k) : k ∈ B} for n ≥ 1. The eigenfunctions of (1.3), called Bloch functions, can be written as ψn,k (x) = eik·x un,k (x),

x in D, k ∈ B

and they are plane waves modulated by the periodic function un,k . The vector k is termed (pseudo) wave-vector. Inserting the above expression in the Schrödinger equation, it is possible to see that un,k is solution of

 h¯ 2

h¯ 2 2 − Δun,k + 2ik · ∇un,k + k − qVL (x) un,k = En (k)un,k , 2m 2m where k = |k|, augmented with the periodic boundary conditions un,k (x + y) = un,k (x),

x ∈ R3 , y ∈ L.

The function En (k), due to the time reversibility of the Schrödinger equation, is even with respect to k and is called dispersion relation, while the set {En (k) : k ∈ B} is the n-th energy band. The union of the ranges of En over n ∈ N is not necessarily the whole real line R, i.e. there may exist energies E ∗ for which there is no number n ∈ N and no vector k ∈ B such that En (k) = E ∗ . An energy range where no electron state exists is called energy gap. It is important to remark that the quantity h¯ k is not in general the momentum of the electron, but reduces to it only in the free case (VL = 0). Because of its relation to the periodicity of the crystal, h¯ k is called crystal momentum[17]. The collection of the bands gives the band structure of the material. In the single electron approximation each state is an eigenstate corresponding to a ψn,k solution of the Schrödinger equation. At zero temperature electrons occupy the lowest energy states (taking into account also the spin degeneracy). The Fermi energy EF at zero temperature is the energy of the highest occupied energy state in the single electron scheme. Two cases may occur: • A certain set of bands is completely filled while the remaining ones are empty, that is the Fermi energy does not fall inside a band. In this case the difference between the energy of the highest occupied level and the energy of the lowest unoccupied level is known as band gap EG . If EG is of order of few eV’s or

1.3 The Si Band Structure and the Semi-classical Picture

5

greater the solid is an insulator while if EG is between 0.1 and about 2 eV the solid is a semiconductor. Of course the distinction between insulators and semiconductors is not sharp. The energy bands below EF are called valence bands because they determine the chemical bindings, those above EF are called conduction bands because they are responsible for the electrical conduction properties of the material. • Some bands are partially filled, that is the Fermi energy lies within the range of one or more bands. The electrons have infinite states which are available with an energy close to EF . In this case the solid is a conductor. Semiconductors, therefore, are characterized by a sizable energy gap between the valence and the conduction bands, the formers are not fully filled at the thermal equilibrium. In fact, upon thermal excitation, electrons from the valence band can jump to the conduction band, leaving behind vacancies in the valence band. These vacancies are in turn available free states, the electrons in the valence bands may move to. This generates a motion of vacancies which can be described in terms of positively charged particles that are named holes in the language of quasiparticles. Thus the transport of charge is achieved through both negatively charged (electrons) and positively charged (holes) carriers. The creation of electron-hole pairs by thermal excitation increases with the temperature. Usually a small percentage of foreign atoms (e.g. impurities) are present in the crystal. If the impurities have a number of valence electrons in the outer atomic shell greater than that of the atoms of the semiconductor, additional free electrons are provided to the conduction bands. Instead, if such a number is smaller, additional holes are created in the valence bands. The impurities of the first type are called donors, those of the second type are called acceptors. For example, in the case of Si, atoms of the group V, like P, As and Sb, are donors, while atoms of the group III, like B, Al, Ga and In, are acceptors. A semiconductor with a negligible presence of impurities is said intrinsic. Instead, we speak of extrinsic semiconductors of n-type or p-type if the impurities are prevalently donors or acceptors. The energy band structure of crystals depends on the explicit form of the potential VL . For the most common semiconductors the energy band structure has been obtained at the cost of intensive numerical calculations and also semiphenomenologically by quantum theory of solids, e.g. with the pseudo potentials [17, 108]. In the next section the energy band structure of Si will be examined more in detail in view of the applications in the subsequent chapters.

1.3 The Si Band Structure and the Semi-classical Picture The Si crystal has the diamond structure with cubic symmetry and therefore its first Brillouin zone has the shape of a truncated octahedron (Fig. 1.3). There are essentially three conduction bands and three valence bands (Fig. 1.4). The energy gap is about 1.1 eV. The lowest conduction band has six related minima

6

1 Band Structure and Boltzmann Equation

Fig. 1.3 Silicon first Brillouin zone

along the main crystallographic directions Δ at about 85% from the center of the first Brillouin zone, near the X points [108] (see Fig. 1.3). The valence band maximum occurs at k = 0, where two degenerate bands meet giving rise to the light holes and the heavy holes. The third valence band (the split-off band) has a maximum 0.044 eV below the maximum of the other two valence bands. However, in order to describe electron transport, for most applications, a simplified description of the energy bands is adopted. It is based on simple analytical approximations. In the sequel we will consider only the conduction bands and the dynamics of holes will be neglected. At normal operating device conditions, it is sufficient to take into account only the lowest conduction band, because the others are scarcely populated. Moreover, electrons are essentially located in the neighbourhoods of the local minima, the so-called valleys. As said, in the lowest conduction band Si has six valleys, termed as X-valleys because they are close to the X points. The energy-wave-vector relation in the neighborhood of an extremum can be approximated by a quadratic form. Assuming k0 as a point of minimum then ∇k E(k0 ) = 0 and the Taylor expansion reads  3 

 1  ∂2 E(k) = E(k0 ) + E (ki − ki0 ) kj − kj 0 + O(|k − k0 |3 ), 2 ∂ki ∂kj k=k0 i,j =1

where k = (k1 , k2 , k3 ) and O(|k|3 ) denote terms of order |k|3 .

1.3 The Si Band Structure and the Semi-classical Picture

  (0,0,0)

 direction

7

X  (1,0,0)

Conduction bands

energy gap

Forbidden region

Valence bands

Fig. 1.4 A schematic representation of the silicon band structure along some crystallographic directions (E(k) versus k in arbitrary units). There are reported the valleys of the the valence bands and the valley of the lowest conduction band near one of the X points. For reason of symmetry the band structure presents six X-valleys. Note the presence of the maximum at the Γ point in the valence bands. The energy bands for holes are obtained from those of valence electrons by reversing the sign

1



∂2 En ∂ki ∂kj



The terms 2 are the components of the effective inverse mass h k=k0   ¯  1 1 tensor . Since the tensor must be positive definite, in the diagonal form M M one can write

  k22 k32 k12 1 T k · + ∗+ ∗ . k= M m∗1 m2 m3 with m∗1 > 0, m∗2 > 0, m∗3 > 0 scalar effective masses that, in the isotropic approximation are assumed to be equal m∗1 = m∗2 = m∗3 = m∗ .

8

1 Band Structure and Boltzmann Equation

In the case of weak applied electric field, the parabolic band approximation is a good dispersion relation E(k) − E(k0 ) =

h¯ 2 (k − k0 )2 , 2m∗

(1.4)

where k − k0 is consistently assumed to vary in all R3 . If one considers higher applied fields, the electron energy is more appropriately described by the Kane dispersion relation (E(k) − E(k0 )) [1 + α (E(k) − E(k0 ))] =

h¯ 2 (k − k0 )2 , 2m∗

(1.5)

where α is the non-parabolicity parameter (for silicon α = 0.5 eV−1 in each Xvalley). The anisotropic version is (E(k) − E(k0 )) [1 + α (E(k) − E(k0 ))] 



2 2 2  k2 − k2,0 k3 − k3,0 h¯ 2 k1 − k1,0 = + + . 2 m∗1 m∗2 m∗3 The electron velocity v(k) in a generic band depends on the energy E(k) by the relation [17] v(k) =

1 ∇k E. h¯

Explicitly in the conduction band we get in the parabolic case v=

h¯ (k − k0 ) m∗

(1.6)

while for the Kane dispersion relation one has v=

h¯ (k − k0 ) . ∗ m [1 + 2α (E(k) − E(k0 ))]

(1.7)

1.4 The Boltzmann Equation The Boltzmann equation has been the first attempt to describe the evolution of a system of many particles in terms of a distribution function in the history of science [49] and it is the background for the development of the macroscopical semiclassical charge transport model we are going to present.

1.4 The Boltzmann Equation

9

Let us consider an electron gas of N particles inside a semiconductor of volume V . The occupation number f (r, k, t) furnishes the number of particles per state in r and in k, at time t. Therefore, the following normalization condition holds 2 (2π)3

  f (r, k, t) d 3 k d 3 r = N, B

 2 f (r, k, t) d 3 k = n(r, t) is the particle space density at time t. The (2π)3 B 2 term (2π) 3 is the state density and the factor 2 takes into account the spin degeneracy. The semiclassical Boltzmann equation for electrons in the conduction band is while

∂f q + v(k) · ∇r f − E · ∇k f = C[f ], h¯ ∂t

(1.8)

where v is the electron velocity, E the electric field and C[f ] is the collision term that describes the electron scatterings. The symbols ∇r and ∇k denote the gradient with respect to position and wave-vector, respectively. The scattering mechanism we will deal with is only the electron-phonon interaction. Electron-electron interaction, impurities and roughness effects will be neglected. At nonzero temperature, the crystal ions vibrate around the equilibrium positions represented by the points of the ideal Bravais lattice. Several normal modes can be excited in the lattice according to the number of atoms contained in a primitive cell. The modes with a dispersion relation like that of the acoustic waves form the acoustic branches. The modes that can interact with electromagnetic radiation and characterize the optical properties of the material form the optical branches [92]. The normal modes of vibration in a dielectric solid are described from a quantum point of view by means of quantized quasi-particles called phonons [17, 108]. At equilibrium, they are distributed according to the Bose–Einstein distribution  −1   h¯ ω NB = exp −1 , KB TL where TL is the lattice temperature, KB is the Boltzmann constant, and h¯ ω stands for the phonon energy, ω being the angular frequency. The interactions with the phonons produce a change in the energy and momentum of the electrons. Therefore if initially an electron has a wave-vector k, after the collision its wave-vector will be k . The exchange of energy can leave the electron in the same band (intra-band transition) or push it into another band (inter-band transition). As said, in the conduction bands the electrons are essentially located in the valleys. After a collision the electron can remain in the same valley (intra-valley scattering) or be pushed into another valley (inter-valley scattering).

10

1 Band Structure and Boltzmann Equation

The expression for the collision term is obtained by the Fermi’s golden rule [92, 126]. For a generic intravalley and intraband interaction, C[f ] can be written as [92] C[f ] =

1 (2π)3



  P (k , k)f (k )(1 − f (k)) − P (k, k )f (k)(1 − f (k )) d 3 k .

B

(1.9) Here P (k, k ) is the transition probability per unit time from the state k to the state k . The first term in (1.9) represents the gain, that is the number of electrons that are scattered to the state of wave-vector k. It is proportional to f (k ) and to 1 − f (k) on account of the Pauli exclusion principle. With analogous meaning, the second term is the loss. Note that the factor 2 is missing because scattering does not change the electron spin. Under the assumption that the electron gas is dilute, f 1, the collision operator can be linearized with respect to f and can be approximated as 

1 C[f ] = (2π)3



 P (k , k)f (k ) − P (k, k )f (k) d 3 k .

(1.10)

B

The Fermi–Dirac distributions  −1   E−μ +1 , feq = exp KB TL where μ is the chemical potential, represents equilibrium state, as will be better specified in Sect. 1.5. In the case of a dilute gas the previous expression becomes   E−μ feq ≈ exp − , KB TL

(1.11)

which is the Boltzmann distribution similar to that of classical gases. Substituting the Fermi–Dirac distribution or its approximation (1.11) in (1.10), one must have    P (k , k)feq (k ) − P (k, k )feq (k) d 3 k = 0. B

The detailed balance principle affirms that this integral is zero because the integrand is zero, therefore P (k , k)feq (k ) = P (k, k )feq (k) from which one obtains  P (k, k ) − E(kK )−E(k) B TL = e . P (k , k)

1.4 The Boltzmann Equation

11

The transition probabilities for the scattering with phonons, in the case of silicon, are given by the following expressions: P (ac) (k, k ) =

2πKB TL Ξd2 δ(E(k) − E(k )) hρv ¯ s2

P (no) (k, k ) = Zif

(1.12)

1 1 π(DT K)2 (NB + ∓ )δ(E(k ) − E(k) ∓ hω) ¯ ρω 2 2

(1.13)

respectively for acoustic and non polar optical phonons. The constant ρ is the density of the material, vs is the sound speed, Ξ and DT K are the acoustic and optical coupling constants respectively, Zif is the degeneracy of the final valley with respect to the initial one (for the values of these parameters see Appendix A) and δ is the Dirac distribution. The minus (−) sign refers to absorption phenomena and the plus (+) to emission ones. Each optical phonon transition is the sum of both an absorption and an emission contribution. In the first case the phonon energy is absorbed by an electron whose energy before the scattering, E(k), becomes E(k) + h¯ ω. In the second case a phonon is created at the expense of an electron whose energy becomes E(k) − hω. ¯ For the scattering with acoustic phonons, the elastic approximation is used. The Boltzmann equation is an integro-differential equation and to solve it is a very hard task, both analytically and numerically (see for example [52, 53, 56, 182] and references therein). In general one is not interested directly in the distribution function but in macroscopic quantities like average electron density, energy, velocity, etc. and macroscopic models are formulated to obtain the equations for such macroscopic quantities. For this purpose, let us consider a function μA (k), regular enough. We define the moment relative to μA (k) as MμA = MA (r, t) :=

2 (2π)3

 μA (k)f (r, k, t)d 3 k.

(1.14)

B

All the macroscopic (integrated with respect to the wave-vector k) quantities can be expressed as moments (or expectation values) corresponding to some suitable weight function. For example, the electron density is the moment relative to the weight function μA (k) = 1, the average energy is relative to E(k) and so on. By multiplying equation (1.8) by the function μA (k) and integrating over B, one finds the moment equation ∂MA + ∂t

 B

q 2μA (k) v(k) · ∇r f d 3 k − E · (2π)3 h¯

 B

2μA (k) ∇k f d 3 k = (2π)3

 B

2μA (k) C[f ]d 3 k. (2π)3

(1.15)

12

1 Band Structure and Boltzmann Equation

Since 





μA (k)∇k f d 3 k =

f ∇k μA (k)d 3 k

μA (k)f ndσ −

B

∂B

B

with n outward unit normal vector field on the boundary ∂B of the domain B and dσ surface element of ∂B, Eq. (1.15) becomes     ∂MA 2μ (k) 2μA (k) 2μ (k) q f A 3 v(k)d 3 k + E · f ∇k A 3 d 3 k − f ndσ + ∇r · 3 ∂t h¯ (2π) (2π) B B ∂B (2π)  2μA (k) = C[f ]d 3 k. (1.16) 3 B (2π)

The last term on the l.h.s. vanishes either when B is expanded to R3 (because in order to guarantee the integrability condition f must tend to zero sufficiently fast as k → ∞) or when B is compact and μA (k) is periodic and continuos on ∂B. This latter condition is a consequence of the periodicity of f on B and of the symmetry of B with respect to the origin. Note that the moment equations depend only on the independent variables r, t. This leads to a reduction of the numerical complexity. Various models employ different expressions of μA (k). For example, let us consider μA (k) = 1, 2 (2π )3 2 (2π )3

 

B

2 ∂ ∂f 3 d k= ∂t (2π )3 ∂t

 f d 3k = B

∇r f (k) · v(k)d 3 k = ∇r · B

2 (2π )3

∂n(r, t) ∂t  f (k)v(k)d 3 k = ∇r · (n(r, t)V(r, t)) B

where V is the macroscopic (average) velocity defined according to n(r, t) V(r, t) =

2 (2π)3

 f (k)v(k)d 3 k. B

In the unipolar case, charge must be conserved and therefore  C[f ]d 3 k = 0 B

as can be verified by a direct check of the collision term. Altogether, we get the continuity equation ∂n(r, t) + ∇ · [n(r, t)V(r, t)] = 0. ∂t

1.5 H-Theorem and the Null Space of the Collision Operator

13

If one introduces the electric current J = −qnV, the continuity equation gives (in the unipolar case) the charge conservation 1 ∂n(r, t) − ∇ · J = 0. ∂t q

(1.17)

Defining other moments, it is possible to obtain a system of equations for all the macroscopic quantities one is interested in. However, other complications emerge from these equations, since there are more unknowns than equations (the closure problem). For example, (1.17) is one equation with two unknowns: n and J. Phenomenological closure assumptions have been proposed in the literature, e.g. the celebrated drift-diffusion equation (Van Roosbroek [195], see [185] for a complete review), where the electric current is expressed as the sum of a Fick like term, proportional to the density gradient, and a drift contribution proportional to the electric field J = μn KB TL ∇n − q nμn ∇Φ ,       diff usion

(1.18)

drif t

where Φ is the electric potential. The quantity μn is called mobility and depends in the simplest version on the electric field E. There is a huge literature about fitting formulas for μn [185]. One of the most used is the Caughey–Thomas formula  μn = μ0 1 +



μ0 | E | vs

2 −1/2

where vs is the saturation velocity and μ0 is the low field mobility. Note that μn |E| → vs as |E| → +∞. It is important to remark that the drift-diffusion models assume that the electron temperature is constant and equal to the lattice temperature (isothermal flow) and therefore they are not suited for describing hot electron effects at high fields. In the next chapter a systematic strategy to tackle the closure problem will be explained.

14

1 Band Structure and Boltzmann Equation

1.5 H-Theorem and the Null Space of the Collision Operator The importance of H-theorems in kinetic theories cannot be overlooked. In fact, an H-theorem allows the introduction of the concept of entropy which is crucial for the analysis of the relaxation towards equilibrium. For these reasons the Htheorem plays such an important role in the kinetic theories of gases [49]. For the semiclassical semiconductor Boltzmann Transport Equation H-theorems were obtained in [169, 170] under the assumption that the transition probabilities are bounded functions. In [130–132] an H-theorem has been derived for the physical electron-phonon operator in the homogeneous case without electric field. The same problem has also been discussed in [117] in the parabolic case. Here we review the question in the case of an arbitrary form of the energy band and in the presence of an electric field, neglecting the electron-electron interaction and assuming the electron gas sufficiently dilute to neglect the degeneracy effects. A physical interpretation of the results is also suggested. As showed in the previous section, the transition probability from the state k to the state k can be written for a generic branch of phonons as [92]    P (k, k ) = G (k, k ) (NB + 1)δ(E  − E + hω) ¯ + NB δ(E − E − hω) ¯

(1.19)

where h¯ ω stands for the phonon energy, δ(x) is the Dirac distribution and G (k, k ) is the so-called overlap factor which depends on the band structure and the particular type of interaction [92] and enjoys the properties G (k, k ) = G (k , k)

and G (k, k ) ≥ 0.

Let us introduce the integer d ∈ {2, 3} in order to treat the cases of different dimension in a unified way. Assigned an arbitrary function μ(k) for which the following integrals exist, the chain of identities [130–132] 

 B

C [f ]μ(k)d d k =



=

B×B

 =

B×B

B×B



 P (k , k)f (k ) − P (k, k )f (k) μ(k)d d k d d k

 P (k, k )f (k) μ(k ) − μ(k) d d kd d k    G (k, k ) (NB + 1)δ(E  − E + hω) ¯ + NB δ(E − E − hω) ¯



×f (k) μ(k ) − μ(k) d d kd d k   

 = G (k, k )δ(E  − E − hω) (NB + 1)f (k ) − NB f (k) μ(k) − μ(k ) d d kd d k ¯ B×B

holds.

1.5 H-Theorem and the Null Space of the Collision Operator

15

Let us suppose that the energy E(k) is a smooth function of the modulus of dE(|k|) > 0 and denote by k, that is E = E(|k|). Let suppose, moreover, that d|k| dg(E) |k| = g(E) the inverse of E (of course g  (E) := > 0). For example, in the √ ∗ dE 2m E(1 + αE) case of Kane’s dispersion relation g(E) = and in the parabolic h¯ √ 2m∗ E band limit g(E) = . h¯ In terms of g(E) we can write d d k = |k|d−1 d|k|dΩd = (g(E))d−1 g  (E) d EdΩd , where dΩd is the solid angle element of the unit sphere in Rd . For a function μ(k), we will denote by μ(E, ˜ n) its expression in terms of E and k the unit vector n = , k = 0. By following [131], we set without loss of generality |k|   E f (k) = h(k) exp − KB TL and in analogy with the case of a classical gas we take μ(k) = KB log h(k). By using the definition of δ(x), one has  C [f ] log h(k)d d k

KB 

B

=

B×B

  G˜(k, E  , n )δ(E  − E − hω) (NB + 1)f˜(E  , n ) − NB f (k) ¯



d−1   × μ(k) − μ(E ˜  , n ) g(E  ) g (E ) d E  dΩd d d k     E      ˜ ˜ = G (k, E , n )δ(E − E − hω) (NB + 1)h(E , n ) exp − ¯ KB T L B×B   

d−1  

E −NB h(k) exp − g (E ) d E  dΩd d d k × μ(k) − μ(E ˜  , n ) g(E  ) KB T L      E   ˜ + hω, = KB G˜(k, E + hω, n ) exp − NB h(E ¯ n ) − h(k) ¯ KB T L B×B    d−1  d ˜ + hω, × log h(k) − log h(E g (E + hω) ¯ ¯ n ) (g(E + hω)) ¯ dΩd d k ≤ 0,

(1.20)

16

1 Band Structure and Boltzmann Equation

thanks to the elementary inequality (a − b) log

a ≥ 0 ∀a, b > 0. b

Therefore along the characteristics of Eq. (1.8)  −

log h(k) B

df d d d k=− dt dt

 

 log h(k) df

 dd k = −

C [f ] log h(k)d d k≥0.

B

B

This implies that H =

      2 E d k= 2 K log h(k) df d K f dd k f log f −f + B B KB TL (2π)d (2π)d B B

(1.21) can be considered as a Liapunov function for the Boltzmann equation (1.8). The first two terms are equal to the opposite of the entropy arising in the classical limit of a Fermi gas (see the next chapter), while the last term is due to the presence of the phonons. The function H represents the nonequilibrium counterpart of the equilibrium Helmholtz free energy, divided by the lattice temperature. It is well known in thermostatics that for a body kept at constant temperature and volume, the equilibrium states are minima for H . The balance equation for H is explicitly written as follows. If we set  η(f ) = KB f log f − f +

multiplying Eq. (1.8) by

E f KB TL

 .

2 2 η (f ) = ∂f η(f ) and integrating with respect (2π)d (2π)d

to k, one has    qE ∂ 2 2 2 d d · η(f )d k + ∇r · η(f )v d k − η (f )∇k f d d k d ∂t B (2π )d (2π ) (2π )d h ¯ B B  2 η (f )C [f ]d d k. = (2π )d B By taking into account the periodicity condition of f on the first Brillouin zone, the integral 

η (f ) B

∂ f dd k = ∂k j

 B

∂η(f ) d d k= ∂k j

 η(f )nj d d k ∂B

1.5 H-Theorem and the Null Space of the Collision Operator

17

vanishes and the balance equation for H assumes the usual form ∂H + ∇r · F = G , ∂t with F =

2 (2π)d

 η(f ) v d d k B

the free energy flux and 2 G = (2π)d



η (f )C [f ]d d k. B

The electric field contributes neither to the production nor to the flux. The existence of the Liapunov function H means that the collision term leads the system toward the equilibrium and constitutes, in the language of kinetic theory, a version of H-theorem for semiconductors. Strictly related to that is the problem of determining the null space of the collision operator which consists of finding the solutions of the equation C (f ) = 0. The resulting distribution functions represent the equilibrium solutions. Physically one expects that, asymptotically in time, the solution to a given initial value problem will tend to such a solution in the absence of an externally applied electric field. In [169, 170] it has been proved that the solutions of C (f ) = 0 are the Fermi–Dirac distributions under the assumption that the scattering probabilities are bounded functions. However this hypothesis is not satisfied by some scattering mechanism (like that with non-polar optical phonons in GaAs). The problem of determining the null space for the physical electron-phonon operator was tackled and solved in general in [131] where it is proved that the equilibrium solutions are not only the Fermi–Dirac distributions but form an infinite sequence of functions of the kind f (k) =

1 1 + h(k) exp (E(k)/kB TL )

(1.22)

where h(k) = h(E) = h(E + hω) is a periodic function of period hω/n, n ∈ ¯ ¯ N. This property implies a numerable set of collisional invariants and hence of conservation laws. However, when several scattering with phonons are considered and their energies are incommensurable, as it is in the real case, then the null space contains only the global thermal equilibrium Fermi–Dirac at the lattice temperature TL as it is to be expected on physical grounds.

18

1 Band Structure and Boltzmann Equation

1.6 Quantum Confinement and Quasi 2DEG In the previous sections the case of bulk electrons, i.e. without limitations about the possible positions, has been dealt with. In this section we treat the case of an ensemble of electrons confined along one dimension, called quasi two dimensional electron gas (2DEG). This situation arises when the length scale in one (the confined) direction is of the order of de Broglie wavelength of electrons, while in the non confined directions the length scale is much longer. In other words electrons are in a quantum regime in the confined direction and exhibit a semi-classical behaviour in the non confined ones. Let us suppose that electrons are quantized along the z direction and free to move in the orthogonal x–y plane. We assume that the domain is represented by Ω = [0, Lx ] × [0, Ly ] × [0, Lz ] ⊂ R3 . Here Lx , Ly , Lz > 0 are fixed. More general cases with a variable confining direction can be easily incorporated. Since the variation along the confinement direction is faster than that in the longitudinal directions, we introduce the small parameter σ defined as the ratio between the longitudinal typical length scale Ll and the transversal typical length scale Lt σ =

Ll Lt

which can be assumed to be a small quantity, σ 1. In order to solve the Schrödinger equation, it is customary to assume the following ansatz about the wave function 1 ψ(k, r) = ψ(kx , ky , kz , x, y, z) = √ ϕ(σ r|| , z)eik|| ·σ r|| A

in

Ω, (1.23)

with k|| = (kx , ky ) and r|| = (x, y) denoting the longitudinal (parallel to the x-y plane) components of the wave-vector k and the position vector r, respectively, and A symbolizing the area of the x-y cross-section (Fig. 1.5). The previous expression of ψ is inserted into the stationary Schrödinger equation in the effective mass approximation 

 h¯ 2 − ∗ Δ + EC (r) ψ = E ψ. 2m

(1.24)

Here EC is the conduction band minimum, EC = −q(VC + V ), where VC (z) is the confining potential and V (r|| , z) is the self-consistent electric potential. Under the assumption that the confining potential gives rise to a practically infinite barrier, as happens, for example, at the oxide/silicon interfaces, one can solve Eq. (1.24) by imposing the boundary conditions ψ(r|| , z) = 0

at z = 0

and z = Lz ,

1.6 Quantum Confinement and Quasi 2DEG

19

Fig. 1.5 Schematic representation of electron confinement in one dimension due to a potential barrier

and taking EC = −q V . If we introduce the slow variable r˜ || = σ r|| and insert (1.23) into (1.24), one has −

 h¯ 2 2  2 Δ σ ϕ(σ r , z) + 2ik · ∇ ϕ(σ r , z) − k ϕ(σ r , z) || || || || r ˜ r ˜ || || || 2m∗ −

h¯ 2 ∂ ϕ(σ r|| , z) − qV (r|| , z)ϕ(σ r|| , z) = Eϕ(σ r|| , z). 2m∗ ∂z2

In the limit σ → 0+ , one formally gets that the envelope function ϕ must solve the reduced Schrödinger equation 

 h¯ 2 d 2 − ∗ 2 − qV (r|| , z) ϕ(z) = εϕ(z), 2m dz

0 ≤ z ≤ Lz ,

(1.25)

with boundary conditions ϕ=0

at z = 0

and z = Lz . 

Lz

We require also the normalization condition

(1.26)

|ϕ|2 d z = 1. Observe that now

0

the solution depends only parametrically on r|| , that is ϕ = ϕ(r|| , z), and more in general also on time t if a non-steady solution is considered. Equations (1.25) and (1.26) constitute a one dimensional Sturm–Liouville problem. Therefore one finds a countable set of eigen-pairs (subbands) (ϕν (r|| , z), εν (r|| )), ν = 1, 2, . . .. Moreover the eigenvalues are simple and do not cross. The potential V is obtained from the Poisson equation ∇ · (d ∇V (r, t)) = −q(ND (r) − n(r, t)),

(1.27)

20

1 Band Structure and Boltzmann Equation

where d is the dielectric constant, ND (r) is the doping donor concentration, and n is the electron density given by n(r, t) =

+∞ 

ρν (r|| )|ϕν (r|| , z)|2 ,

ν=1

with ρν the (areal) density of electrons of the ν-th subband. Of course the Schrödinger and Poisson equations are coupled and must be solved simultaneously. The limit for getting (1.25) here has been performed only in a formal way. A rigorous justification can be found in [25] for a similar model. The above Schrödinger–Poisson model is able to describe only the ballistic case, because scattering is not included. In order to take into account this latter, several approaches are available. One can add in the Hamiltonian a term describing the electron-phonon interaction and solve the corresponding Schrödinger equation for the wave function in the electron-phonon space or use a Green density function method [61, 62]. However, for devices with longitudinal characteristic length of a few tens of nanometers, which is the case we are going to consider, the transport of electrons in the longitudinal direction is semi-classical within a good approximation. Electrons in each subband can be considered as different populations whose state is described by a semiclassical distribution function. Therefore the description of the electron transport along the longitudinal direction is included by adding to the Schrödinger–Poisson model the system of coupled Boltzmann equations for the distributions fν (x,y, kx ,ky , t) of electrons in each subband ∞  1 q ∂fν + ∇k|| Eν · ∇r|| fν − Eeff Cν,μ [fν , fμ ], ν · ∇k|| fν = h¯ h¯ ∂t μ=1

ν = 1, 2, . . . (1.28)

1 ∇r εν (r|| ). The justification of the previous equations is postponed q || to the next section. The density ρν is expressed in terms of fν by eff

where Eν

=

ρν (r|| , t) =

2 (2π)2

 B2

fν (r|| , k|| , t)d 2 k|| ,

B2 indicating the 2D Brillouin zone, which will be approximated with R2 consistently with the effective mass approximation. In each subband the electron energy is the sum of a transversal contribution εν and a longitudinal contribution ε|| Eν (r|| ) = εν (r|| ) +

 h¯ 2  2 kx + ky2 ≡ εν (r|| ) + ε|| , ∗ 2m

1.6 Quantum Confinement and Quasi 2DEG

21

where a parabolic band approximation has been used for the longitudinal contribution. Consequently, the longitudinal electron velocity is v|| =

1 h¯ k|| ∇k|| ε|| = ∗ . m h¯

For more details the interested reader is referred to [143]. It is important to stress that the Boltzmann equation in each subband has a drift term which is given by the gradient of the eigenenergy at variance with the transport equation in the bulk case, where electrons are driven by the gradient of the electrostatic potential. A rigorous derivation has been deduced in [25] by exploiting the Wigner transform. A further improvement of the subband transport model is obtained replacing the parabolic band approximation for the longitudinal component of the energy with the Kane dispersion relation in order to include the non-parabolicity effects at high fields ⎞ ⎛  1 ⎝ h¯ 2  2 Eν (r|| , k|| ) = εν (r|| ) + 1 + 4α ∗ kx + ky2 − 1⎠ ≡ εν (r|| ) + ε|| (k|| ), 2α 2m where α is the non-parabolicity parameter. Consequently, the longitudinal electron velocity is v|| =

h¯ k|| 1 . ∇k ε|| = ∗

h¯ || m 1 + 2αε||

(1.29)

In the limit case α → 0+ , it reduces to the parabolic band approximation. Regarding the collision operator for the transport of a 2DEG, also in this case the Fermi golden rule is the starting point, see for example [126]. In the non degenerate approximation, each term contributing to the collision operator has the general form 1 Cν,μ [fν , fμ ] = (2π)2



 B2

 Sμν (k|| , k|| ) fμ − Sνμ (k|| , k|| ) fν d 2 k|| .

When μ = ν we have intra-subband scatterings; when μ = ν we have inter-subband scatterings. The function Sμν (k|| , k|| ) is the transition rate from the longitudinal state with wave-vector k|| , belonging to the μ-th subband, to the longitudinal state with wave-vector k|| , belonging to the ν-th subband, and fμ ≡ fμ (r|| , k|| , t). The relevant 2D scattering mechanisms in Si are the acoustic phonon scattering, and the non-polar optical phonon one. Scattering with impurities will be not considered in this book, but it is relevant only at low temperature or low field [126]. For the acoustic phonon scattering in the elastic approximation, the transition rate is given by (ac) Sνμ (k|| , k|| ) = A(ac) Gνμ δ(Eμ (k|| ) − Eν (k|| )),

22

1 Band Structure and Boltzmann Equation

with A(ac) =

KB TL Ξd2

, TL the lattice temperature, which will be kept constant h¯ ρvS2 throughout the book, ρ the silicon density, Ξd the acoustic phonon deformation potential and vS the longitudinal sound speed. Their values are also The Gνμ ’s are the interaction integrals  Gνμ =

+∞ −∞

 |Iνμ (qz )| dqz , 2

Lz

Iνμ (qz ) =

ϕν (z)ϕμ (z)eiqz z dz,

0

with q denoting the 3D-phonon wave-vector, and the bar indicating complex conjugation. We note that Gνμ = Gμν holds. The interaction integrals couple the subbands in the scattering. Similarly, for the non-polar optical phonon scattering, one has   1 1 (no) δ(Eμ (k|| ) − Eν (k|| ) ∓ h¯ ω), (k|| , k|| ) = A(no) Gνμ Nq + ∓ Sνμ 2 2 where A(no) =

(Dt K)2 , 2ρω

Nq is the Bose–Einstein distribution of phonons, Dt K is the non-polar optical phonon deformation potential, and hω ¯ is the phonon energy. Their values are reported in Appendix A. An additional scattering due to the roughness of the interface oxide/semiconductor can be added. The interested reader is referred to [126]. The above quantum picture can present further specific features. For example in the Si–SiO2 interface of a Metal-Oxide-Semiconductor-Field-Effect-Transistor (MOSFET), a quantized inversion layer having a (100)-oriented surface has two sets of subbands, called ladders: one coming from the projection of the two valleys with longitudinal mass in the direction perpendicular to the interface, and the other one originating from the four valleys having transverse mass in the direction perpendicular to the interface. Here for the sake of simplicity only one ladder with a mean effective mass will be considered. However, our results can be easily generalized in order to include both ladders by taking into account also intervalley scatterings between subbands belonging to different ladders and solving a Schrödinger equation for each effective mass (longitudinal and transversal). If the longitudinal length scale is comparable with the de Broglie wavelength, a purely quantum approach must be employed also for the transport description. Electric properties of these nanoscale devices, like current-voltage curves, can be computed via the non-equilibrium Green function, [50, 90, 129, 200]. Another possibility to tackle the problem is within the framework of quantum kinetic theory via the Wigner function [99, 136, 177]. Other approaches are based on the master equation [75].

1.7 Derivation of the Transport Equation Along the Longitudinal Direction

23

1.7 Derivation of the Transport Equation Along the Longitudinal Direction In this section the transport equation along the longitudinal direction (1.28) is deduced. We look for a wave function Ψ (r, t) that can be written as sum of terms of the type ϕ(r|| , z, t) χ(r|| , t)

in

(1.30)

Ω,

where χ(r|| , t) is to be determined, while ϕ(r|| , z, t) satisfies the transversal Schrödinger equation 

 h¯ 2 ∂ 2 − ∗ 2 − qV (r|| , z) ϕ = εϕ, 2m ∂z

0 ≤ z ≤ Lz ,

(1.31)

with homogeneous boundary conditions ϕ = 0 at z = 0 and z = Lz , 

Lz

and the normalization condition

(1.32)

|ϕ|2 d z = 1. As said, the solutions depend

0

parametrically on (r|| , t). One has a countable set of solutions represented by the eigenfunction-eigenenergy couples (ϕn , εn )n∈N , with real εn and ϕn that can be chosen real-valued as will be supposed in the sequel. We can write the solution of the complete time-dependent Schrödinger equation i h¯

∂Ψ h¯ 2 (r|| , z, t) = − ∗ ΔΨ (r|| , z, t) − qV (r|| , z)Ψ (r|| , z, t) ∂t 2m

in the form Ψ (r|| , z, t) =



ϕn (r|| , z, t) χn (r|| , t)

n∈N

with the coefficients χn (r|| , t) to be determined. For each n ∈ N, the following equation must be satisfied    h¯ 2

∂ ∂ i h¯ χn ϕn + ϕn χn = − ∗ χn Δr|| ϕn + ϕn Δr|| χn + 2∇r|| ϕn · ∇r|| χn ∂t ∂t 2m +εn ϕn χn .

24

1 Band Structure and Boltzmann Equation

In order to get an evolution equation for χn , we project the previous equation over the subspace spanned by ϕn by applying the operator Πn defined as  Lz Πn g(r|| , z) = ϕn (r|| , z) g(r|| , z )ϕn (r|| , z ) d z 0

for any function g(r|| , z) regular enough. After observing that  Lz ∂ 1 ∂ Πn ϕn = ϕn ϕn2 d z = 0, ∂t 2 ∂t 0  Lz Πn ∇r|| ϕn = ϕn ϕn ∇r|| ϕn d z = 0,  Πn Δr|| ϕn = ϕn

0



Lz

ϕn Δr|| ϕn d z=ϕn

0



Lz

= −ϕn 0

0

Lz



 2 1 Δr|| ϕn2 − ∇r|| ϕn d z 2

∇r ϕn 2 d z, ||

we get

 Lz 2 ∂ h¯ 2 h¯ 2 ∇r ϕn d z χn . i h¯ χn = − ∗ Δr|| χn + εn + || ∂t 2m 2m∗ 0 Let us introduce the Wigner transform of χn   y||   y||  y|| ·p|| wnh¯ (r|| , p|| , t) = χn r|| + , t χn r|| − , t e−i h¯ d 2 y|| 2 2 R2y

(1.33)

∀ p|| ∈ R2 ,

||

where, in view of the future limit, an explicit dependence on the parameter h¯ has been indicated. Here p|| = h¯ k|| is the longitudinal crystal momentum. If we define ρn (r|| , s|| , t) = χn (r|| , t)χn (s|| , t), introduce the coordinate transformation y|| y|| r|| = x|| + , s|| = x|| − 2 2 and set  y|| y||  u(x|| , y|| , t) = ρn x|| + , x|| − , t , 2 2

1.7 Derivation of the Transport Equation Along the Longitudinal Direction

25

the Wigner transform of χn can be rewritten as wnh¯ (x|| , p|| , t) = F u(x|| , p|| , t), where F is the Fourier transform operator.2 The function u satisfies the following evolution equation (see for example [99, 136]) 

  ∂

i

∂u = ρ r|| , s|| , t = − Hr|| − Hs|| ρ r|| , s|| , t ∂t ∂t h¯

(1.34)

where Hr|| and Hs|| are the Hamiltonians acting on the variables r|| and s|| respectively. The Hamiltonian is given by Hr||



 Lz 2 h¯ 2 h¯ 2 ∇r ϕn d z . = − ∗ Δr|| + εn + || 2m 2m∗ 0

We observe that 

 1

Δr|| − Δs|| u = divy|| ∇x|| u . 2 Therefore, if we set Veff = εn +

h¯ 2 2m∗



Lz 0

∇r ϕn 2 d z, ||

the evolution equations for u is given by

 i 

i h¯ ∂u − ∗ divy|| ∇x|| u + δV x|| , y|| , t u = 0, ∂t m h¯ where

 δV x|| , y|| , t = Veff (r|| , t) − Veff (s|| , t).

any function g(x) belonging to the space L1 (Rd ) of the summable functions defined over Rd , with d integer, we define the Fourier transform as  i g(x)e− h¯ p·x dx ∀p ∈ R d . F g(p) =

2 For

Rd

Note that p has the dimension of a momentum if x has the dimension of a length. For any function h(p) ∈ L1 (Rd ), the inverse Fourier transform is given by  i 1 h(p)e h¯ p·x dp ∀x ∈ R d . F −1 h(x) = d (2π h¯ ) Rd

26

1 Band Structure and Boltzmann Equation

By Fourier transforming, one has



 i h¯ i ∂wnh¯ (x|| , p|| , t) − ∗ F divy|| ∇x|| u (x|| , p|| , t) + F (δV u) (x|| , p|| , t) = 0. h¯ ∂t m Since



 F divy|| ∇x|| u (x|| , p|| , t)  y|| ·p||

 i divy|| ∇x|| u e−i h¯ d 2 y|| = p|| · ∇x|| wnh¯ (x|| , p|| , t) = 2 h ¯ R and  F (δV u) (x|| , p|| , t) = =

1 (2π h¯ )2



R2

y|| ·p||

 δV x|| , y|| , t F −1 wnh¯ (x|| , y|| , t)e−i h¯ d 2 y|| 

R2 ×R2

y|| ·(p|| −p|| )

 h¯ δV x|| , y|| , t wnh¯ (x|| , p|| , t)ei d 2 y|| d 2 p||

which with the change of variable y|| = h¯ η can be rewritten as F (δV u) (x|| , p|| , t) =

1 (2π )2



 R2 ×R2

 x|| +

Veff

   h¯ h¯ η, t − Veff x|| − η, t 2 2



×wnh¯ (x|| , p|| , t) ei η·(p|| −p|| ) d 2 η d 2 p|| ,

one gets the quantum Vlasov equation ∂ h¯ p|| w + · ∇x|| wnh¯ + θh [Veff ]wnh¯ = 0, ∂t n m∗

(1.35)

where θh [Veff ] is the pseudo-differential operator which acts on wnh¯ as follows θh [Veff ]wnh¯ :=

i (2π)2

 R2η

 R2 

p||

     h¯ h¯ 1 Veff x|| + η, t − Veff x|| − η, t 2 2 h¯ 

×wnh¯ (x|| , p|| , t)e−iη·(p|| −p|| ) d 2 η d 2 p|| . We recall that a pseudo-differential operator is an operator whose Fourier transform acts as a multiplication operator on the Fourier transform of the function. In our case        Veff x|| + h2¯ η, t − Veff x|| − h2¯ η, t  F wnh¯ (x|| , η, t). F θh [Veff ]wnh¯ (x|| , η, t) = i h¯

1.7 Derivation of the Transport Equation Along the Longitudinal Direction

27

The multiplicator is called the symbol of the pseudo-differential operator. For the previous operator the symbol is     Veff x|| + h2¯ η, t − Veff x|| − h2¯ η, t

 δVeff h¯ (x|| , η, t) := i . h¯ Formally, it holds

 lim δVeff h¯ = i∇x|| εn · η.

h¯ →0+

Setting fn (x|| , p|| , t) = lim wnh¯ (x|| , p|| , t), h¯ →0+

as formal limit when h¯ → 0+ , we get the semiclassical Vlasov equation p|| ∂fn + ∗ · ∇x|| fn − ∇x|| εn · ∇p|| fn = 0. ∂t m If the previous equations is rewritten in terms of the wave-vector and if the collision term is added in order to take into account also the collisions with phonons, the semiclassical Boltzmann equation (1.28) 1 ∂fn 1 + ∇k|| En · ∇r|| fn − ∇x|| εn · ∇k|| fn = C[f ] h¯ h¯ ∂t is obtained.

Chapter 2

Maximum Entropy Principle

The description of a physical system requires the knowledge of some information, for example the prediction of the motion of a point particle in classical mechanics requires the knowledge of its initial position and momentum besides, of course, the system of forces acting upon it. In the case of a great quantity of particles, i.e. of the order of the Avogadro’s number (6.022 × 1023), the information for a detailed description of the motion of every particle are practically not available; therefore distribution functions and statistical methods are introduced in order to describe the behavior of complex systems. In the context of the Bayesian interpretation of the probability, Jaynes [94, 96] showed as it is possible to develop the Maximum Entropy Principle (hereafter MEP) as an inference method to obtain the results achieved in the context of the statistical mechanics or, more in general, in all cases where the complete information, necessary for a detailed description, are lost or not available.

2.1 The Entropy The term entropy was introduced for the first time in classical thermodynamics by Clausius in 1865 (for an overview see [116]). It was coined from the old greek ν (inside) and τρoπη (transformation) and was used to indicate a thermodynamical potential. Boltzmann, by his famous expression S = KB log W , linking the entropy S to a quantity W proportional to the number of microstates of the system, furnished a physical interpretation of the entropy and built a bridge between the macroscopic and the microscopic world. In 1948 Shannon [186] used the same term to indicate the amount of ignorance about a system in the context of information theory. The formal expression for this quantity is the same as that of the statistical Boltzmann entropy in the canonical

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_2

29

30

2 Maximum Entropy Principle

ensemble and in 1957 Jaynes [94, 96] showed that it is possible to obtain all the expressions of statistical mechanics by maximizing it. To better understand this last statement, it is important to underline that the Clausius-Boltzmann entropy is a property of the system, while from the Shannon– Jaynes point of view it is a property of the knowledge of the observer, so that the maximum state of entropy for a system means the state of maximum ignorance of the observer. Let us consider first the discrete case of a random variable X which takes the values in the set Ω = {x1 , x2 , · · · , xn } with a priori probability p1 , p2 , . . . , pn , satisfying, of course, the condition n 

pi = 1.

(2.1)

i=1

Let us suppose that we have only a partial knowledge about the probability distribution (p1 , p2 , . . . , pn ). Shannon proposed as measure of the uncertainty associated with this system the function S(p1 , p2 , · · · , pn ) determined to within a constant factor by the following conditions. Axiom 2.1.1 (Continuity) S(p1 , p2 , · · · , pn ) is a continuous function of pi , i = 1, 2, · · · , n. Axiom 2.1.2 (Equally Likely Cases) If all pi are equal, the quantity   1 1 1 , ,··· , S is a monotonically increasing function of its number of arguments n n n n. Axiom 2.1.3 (Composition Axiom)  If the values of X are     Law or Grouping grouped in m disjoint sets x11 , · · · , x1k1 , x21, · · · , x2k2 , · · · , xm1 , · · · , xmkm , the probabilities of these events are wj = pj 1 + pj 2 + · · · pj kj , j = 1, · · · , m. The grouping axiom requires 

 p11 p1k1 ,··· , + ··· w1 w1   pm1 pmkm . (2.2) ,··· , +wm S wm wm

S(p1 , p2 , · · · , pn ) = S(w1 , · · · , wm ) + w1 S

The first axiom is just a technical assumption. The second one is a formalization of the intuitive idea that there is more uncertainty in rolling a die than tossing a coin. The third axiom is the least intuitive. We justify it when the values of X are collected into two disjoint sets A = {x1 , · · · , xr } and B = {xr+1 , · · · , xn } by constructing a compound experiment: one of the two sets, A or B, is selected with

2.1 The Entropy

31

probability w1 and w2 respectively. If the set A is chosen then we select xi with the conditional probability  pi if i = 1, · · · , r P (X = xi |A) = w1 0 otherwise Similarly if the group B is chosen. Let Y be the result of the compound experiment. If xi ∈ A, P (Y = xi ) = P (A)P (Y = xi |A) = w1

pi = pi = P (X = xi ). w1

One has the same result if xi ∈ B. Therefore X and Y have the same distribution. The uncertainty before the experiment is S(p1 , p2 , · · · , pn ). When one reveals which group has been selected the amount of uncertainty S(w1, w2 ) is eliminated. pr p1 p2 , ,··· , If the group A or B is chosen, the remaining uncertainty is S w1 w1 w1   pn pr+1 pr+2 , ,··· , or S . w2 w2 w2 Thus, the average uncertainty, remaining after the group has been specified, is  w1 S

p1 p2 pr , ,··· , w1 w1 w1



 + w2 S

pr+1 pr+2 pn , ,··· , w2 w2 w2

 .

We expect that the uncertainty about the compound experiment minus that removed by selecting the group must equal the average uncertainty after the group is specified, that is  S(p1 , p2 , · · · , pn ) − S(w1 , w2 ) = w1 S

p1 p2 pr , ,··· , w1 w1 w1



 + w2 S

pr+1 pr+2 pn , ,··· , w2 w2 w2

which is Axiom 2.1.3 in the particular case of two subsets. The previous conditions determine the form of S(p1 , . . . , pn ) as1 S(p1 , . . . , pn ) = −C

n 

pi log pi ,





(2.3)

i=1

where the constant C > 0 depends on the unit to be used.2 The function S is called the information entropy. It can be considered as the expectation value of the random

1 log x

will be always intended as the natural logarithm. Shannon used log2 in his definition because he had in mind the two binary states (0 and 1) of the information so that entropy is measured in bits, but this detail is not important for the purposes of this book.

2 Indeed

32

2 Maximum Entropy Principle

variable which takes values −C log pi with probability pi . The expression of S is valid also if pi = 0 for some i, formally setting 0 log 0 := 0 which preserves also the continuity of S(p1 , . . . , pn ) since limx→0+ x log x = 0. In order to prove that (2.3) is the only function satisfying Axioms 2.1.1–2.1.3 we need the following lemma. Lemma 2.1.1 Let us define  1 1 ,··· , H (n) = S , n n    

n = 1, 2, · · ·

n

Then H (n) = C log n with C a positive constant. Proof Let us suppose that pi =

1 , i = 1, · · · , n, and collect them in k groups n

p1 , p2 , · · · , pr1 , pr1 +1 , · · · , pr1 +r2 , · · · , pn−rk +1 , · · · , pn .          r1

r2

rk

From the grouping axiom one has  S

1 1 ,··· , n n

 = H (n) = S

r

1

n

rk   ri + H (ri ). n n k

,··· ,

i=1

In the particular case r1 = r2 = · · · = rk = m, it follows H (mk) = H (k) +

k  m i=1

n

H (m) = H (k) + H (m).

(2.4)

We have H (1) = H (1)+H (1) which implies H (1) = S(1) = 0, expressing the fact that there is no uncertainty associated with an experiment with only one outcome. By induction it is straightforward to see that H (nk ) = k H (n) for all n, k

positive integers.

Of course H (n) = C log n solves the previous functional equation. It is possible to prove that it is the sole family of solutions [16].  Theorem 2.1.1 The only expression satisfying Axioms 2.1.1–2.1.3 is (2.3). Proof First let us consider a system Ω having " n states xi with associate probability " pi = ggi with gi positive integers. From ni=1 pi = 1 we have that g = ni=1 gi .

2.1 The Entropy

33

Let us suppose also to have n systems Bi , each of them with gi equally likely states bi1 , bi2 , · · · , bigi , depending on Ω in the following way. If Ω is in the state xi then the system Bi is chosen. By the previous lemma S(Bi ) = C log gi and therefore the mean " conditional entropy of the set B consisting of the subsystems Bi is S(B|Ω) = C ni pi log gi . If we now consider the compound system ΩB, the joint probability to have Ω in the state xi and B in a specific state of Bi is given by pi × g1i = g1 , that is the system ΩB has a uniform distribution with g states. This implies that S(ΩB) = C log g. By applying the grouping axiom one finds S(Ω) = S(ΩB) − S(B|Ω) = C log g − C

n 

pi log gi = −C

i=1

= −C

n 

n 

pi (log gi − log g)

i=1

pi log pi .

i=1

By the continuity axiom the results can be extended to the case with general values of the probability pi [16]. 

2.1.1 Properties of the Shannon–Jaynes Entropy According to the fact that entropy represents the amount of ignorance, it satisfies the following properties (or desiderata in Jaynes terms [96]): 1. Entropy attains a maximum for equal probabilities, i. e.   1 1 1 , ,..., S ≥ S(p1 , p2 , . . . , pn ). n n n This means that the observer’s ignorance is maximal when all the n outcomes have equal"likelihood. In fact, let us maximize S(p1 , p2 , . . . , pn ) under the constraint ni=1 pi = 1. By introducing the Lagrange multiplier λ to take into account the constraint, we have to find the unconstrained stationary points of the Lagrangian

n  pi . L(p1 , p2 , . . . , pn , λ) = S(p1 , p2 , . . . , pn ) − λ 1 − i=1

One gets 0=

∂L = −C (log pi + 1) + λ, ∂pi

0=

 ∂L = −1 + pi , ∂λ n

i=1

i = 1, 2, · · · , n,

34

2 Maximum Entropy Principle

wherefrom pi = eλ/C−1 for all i and from the constraints it follows that the uniform distribution pi = n1 is the only stationary distribution for L. To see that it represents a maximum, we have to prove that −

n 

pi log pi ≤ −

i=1

n  1 log n−1 = log n n i=1

that is 0 ≤ log n +

n 

pi log pi = log n

i=1

n  i=1

pi +

n 

pi log pi =

i=1

n 

pi log

i=1

pi . 1/n

From the elementary inequality x log x ≥ x − 1 valid for x ≥ 0 (the case x = 0 of course must be intended as limit), one has  i

 1 pi 1 pi pi = log ≥ pi log 1/n n 1/n 1/n n i

i



 pi − 1 = 0. 1/n

 The above proof also highlights that, by maximizing the Shannon entropy , one justifies Laplace’s principle of insufficient reason according to which if one has no information about the probability of the various outcomes then the uniform distribution must be used. In the original formulation this statement is considered in a negative sense, that is a reasonable choice if no further knowledge about the system is available. Instead the procedure based on MEP deduces and justifies it since the uniform distribution maximizes the ignorance on the system avoiding any spurious biasing assumption. 2. Entropy is unaffected by extra states of zero probability. S(p1 , p2 , . . . , pn−1 , 0) = S(p1 , p2 , . . . , pn−1 ). If there is no possibility that an event occurs, then the ignorance is no larger than it would have been if the observer had not included the event in the list of possible outcomes. 3. The entropy of a system is correlated to the entropy of its parts. This property needs a bit longer explanation. Let us consider another random variable Y taking its values in the set Γ = {y1 , y2 , · · · , ym }, m ∈ N, with probability distribution q(yl ) = ql . Let rkl = P (xk , yl ) be the joint probability for the two outcomes xk and yl and ckl = P (xk |yl ) =

rkl P (xk , yl ) = q(yl ) ql

2.1 The Entropy

35

" " the conditional probability. Of course nk=1 P (xk |yl ) = nk=1 ckl = 1, for each l = 1, 2, . . . , m. If we suppose that the two random variables X and Y describe some properties of two interacting systems Ω and Γ , the ignorance function related to the compound system is given by S(ΩΓ ) = S(r11 , r12 , . . . , r1m , r21 , . . . , rnm ) = S(c11 q1 , c12 q2 , . . . , c1m qm , c21 q1 , . . . , cnm qm ). The knowledge of one outcome yl connected with the system Γ modifies the ignorance regarding the system Ω as S(Ω|Y =yl ) := S(c1l , . . . , cnl ), where S(Ω|Y =yl ) is the conditional entropy S(Ω|Y =yl ) = −C

n 

ckl log ckl .

k=1

Therefore the combined ignorance decreases from S(ΩΓ ) to S(Ω|Y =yl ) . The expected ignorance with respect to all the outcomes of Γ is furnished by the average S(Ω|Γ ) Γ :=

m 

ql S(Ω|Y =yl )

l=1

and one is led to the following result S(ΩΓ ) = −C



⎡ ckl ql log(ckl ql ) = −C ⎣

k,l

=

 l

=





ql −C

 k





ckl ql log(ckl ) +

k,l

ckl log(ckl ) − C



⎤ ckl ql log(ql )⎦

k,l



ql log(ql )

l



ckl

k

   =1

ql S(Ω|Y =yl ) + S(Γ ) = S(Ω|Γ ) Γ + S(Γ ) ,

l

so that S(ΩΓ ) = S(Ω|Γ ) Γ + S(Γ ) .

(2.5)

36

2 Maximum Entropy Principle

In other words taking measures on the system Γ modifies the observer’s knowledge of the system Ω. Analogous results can be obtained if more than two systems are considered. If Ω and Γ are not interacting, the probabilities are independent so that S(Ω|Y =yl ) = S(Ω) and S(ΩΓ ) = S(Ω) + S(Γ ) . The ignorance regarding two uncorrelatd (independent) systems is additive, i. e. entropy is extensive, a well know property of the Clausius-Boltzmann entropy in classical thermodynamics.

2.1.2 Shannon–Jaynes Entropy in the Continuous Case Let us suppose now that X is a continuous random variable with (continuous) probability density p(x), x ∈ Rn . A natural extension of the definition of entropy in the continuous case is  S[p(x)] = −C p(x) log p(x)dx. (2.6) Rn

However, this expression is not invariant under a transformation of coordinates. To overcome this problem a measure m(x) is introduced in the definition of entropy 

 S[p(x)] = −C

Rn

p(x) log

 p(x) dx. m(x)

(2.7)

In fact let us consider a globally invertible transformation x = f (z), z ∈ Rn . The distribution and the measure change according to q(z) = p(z)|J |,

n(z) = m(z)|J |,

where J is the Jacobian of the transformation. Therefore       p(z)|J | q(z) dz = −C dz = S[q(z)], S[p(x)] = −C q(z) log q(z) log m(z)|J | n(z) Rn Rn so that the value of S remains unchanged. Entropy in this last expression is named relative entropy [202] and m(x) is considered, if properly normalized, a probability distribution. In particular Jaynes considers it as the prior distribution describing the complete ignorance of X [94], that is without any knowledge of constraints on the system.

2.2 Maximum Entropy Inference of a Distribution: The Discrete Case

37

  E(k) is chosen as measure density, E(k) If the Maxwellian distribution exp − KB T and k being the energy band and the wave-vector respectively, and identification C = KB is made, one gets  S[p(k)] = −KB

Rn

p(k) log

  ε p(k) 3 p(k) log p(k)d 3 k − d k = −KB p(k) d 3 k n n m(k) T R R

which represents the Helmholtz free energy divided by T with reverse sign. It is well known from thermodynamics that at equilibrium the Helmholtz free energy attains a minimum in a mechanically isolated system kept at constant temperature [91]. The discrete version of the relative entropy reads S[p] = −C

n  i=1

pi log

pi . mi

Another important difference between the continuous and discrete case is given by the fact that (2.6) is not even defined in sign and can be used only to calculate the change (increase or decrease) of the uncertainty. However, if (2.7) is limited from below, a suitable constant can be added to restore the positive sign since the additive constant plays no role in determining the MEP distribution.

2.2 Maximum Entropy Inference of a Distribution: The Discrete Case Let us consider a random variable X having n" possible outcomes x1 , x2 , . . . , xn . Let us suppose that some averages fr (x) = ni=1 pi fr (xi ), r = 1, 2, .., m, are known with fr : {x1 , x2 , . . . , xn } → R assigned functions. We want to estimate the corresponding unknown probabilities p(x1 ), p(x2 ), . . . , p(xn ) having at disposal only the knowledge of the previous mean values. Of course, the problem is illposed. From a statistical point of view we want to introduce an estimator of the probability distribution of the random variable X which is the least biased. Since the information entropy measures the degree of ignorance on the system, the estimator obtained by maximizing S should guarantee that no spurious information is added. This remark is also in agreement with the basic tenets of thermodynamics and statistical physics where the equilibrium distributions are just obtained requiring that the macrostate has the greatest number of associated microstates.

38

2 Maximum Entropy Principle

From the above considerations, one is led to the following maximum entropy principle (or method in a context no strictly related to physical phenomena): the least biased estimator of p(xi ), i = 1, 2, . . . , n, with respect to the only knowledge of the averages " fr (x), r = 1, 2, . . . , m, is given by the maximization of the entropy S = −C i pi log(pi /mi ) under the constraints n 

pi = 1,

(normalization condition),

i=1

fr (x) =

n 

r = 1, 2, . . . , m.

pi fr (xi ),

i=1

By introducing the Lagrange multipliers λr , r = 0, 1, 2, . . . ., m, the previous problem is equivalent to freely maximize

 n  n n m     pi  pi log − λ0 pi − 1 − λr pi fr (xi ) − fr (x) . S = −C mi i=1

i=1

r=1

i=1

It is simple to find out the solution "m

pi = mi e−λ0 −

r=1

λr fr (xi )

,

(2.8)

where the constant C has been included into the multipliers by mean of the λ0 λr → λ0 , → λr . Using the normalizing condition and transformation 1 + C C the other constraints, one finds pi =

mi − "m λr fr (xi ) r=1 e , Z

λ0 = log Z,

where Z=

n 

mi e −

"m r=1

λr fr (xi )

i=1

is the partition function. To show that (2.8) is indeed a maximum for S, we need the following algebraic lemma Lemma 2.2.1 If (p1 , p2 , · · · , pn ) and (q1 , q2 , · · · , qn ) are two probability vectors then the inequality −

n  i=1

holds.

 qi pi qi log ≤− qi log mi mi n

i=1

(2.9)

2.2 Maximum Entropy Inference of a Distribution: The Discrete Case

39

Proof Relation (2.9) is equivalent to n 

qi log

i=1

qi ≥ 0. pi

From the elementary inequality x log x ≥ x − 1 valid for x ≥ 0 (the case x = 0 of course must be intended as limit), one has n 

qi log

i=1

 qi  qi qi = pi log ≥ pi pi pi pi n

n

i=1

i=1



 qi − 1 = 0. pi 

Now we can prove the following property Theorem 2.2.1 The distribution pi = mi exp −λ0 −

m 

λr fr (xi )

r=1

is a maximum for S. Proof We observe that SIp := −C

n  i=1



m  pi pi log = C λ0 + λr < fr (x) > . mi r=1

Let (q1 , q2 , · · · , qn ) be an arbitrary probability vector satisfying the constraints. One has

n n m    qi pi −C qi log ≤ −C qi log = C λ0 + λr < fr (x) > = SIp . mi mi i=1

i=1

r=1

 Remark 2.2.1 If the random variable X assumes values in a countable set, additional technical hypotheses must be added in order to assure the convergence of the series present in the maximum entropy estimator. This usually gives restriction on the choice of the functions fr .

40

2 Maximum Entropy Principle

2.3 Examples of Distribution Functions Deduced with MEP Now, assuming X to be some microscopic quantity of a physical system, in principle it is possible to obtain all the distributions of statistical mechanics [94]. In the next subsections some basic examples are shown.

2.3.1 Maxwell–Boltzmann Distribution Let us consider a system of N particles at thermodynamical equilibrium"having a total energy E. Our purpose is to" determine the number of particles ni ( ni = N) occupying the energy level εi ( i εi ni = E) for each level i. Let Gi be the degeneracy of the i-th energy level, that is the number of quantum states available with energy εi . In the case of Maxwell–Boltzmann statistics, the particles are considered so distant to each other that exchange effects can be neglected, and distinguishable; in other terms, their main distance is much greater than the thermal de Broglie wavelength, so they obey with a good approximation to the laws of classical mechanics. Now let pi be the probability that a particle has energy εi or in other words that it is found to be in one of the Gi quantum states of energy level εi . Defining the average energy as ε = E/N, the constraints read 



pi = 1,

i

pi εi = ε.

i

We consider the relative entropy with mi = Gi S = −KB



pi log

i

pi , Gi

where the constant C has been set equal to the Boltzmann constant in analogy with statistical mechanics. If we maximize it under the above constraints, the MEP solution is Gi −λεi e , Z

pi =

(2.10)

with the partition function Z=



Gi e−λεi ,

i

where λ is the Lagrange multiplier relative to the energy constraint. As done above, the Boltzmann constant has been included into the multipliers.

2.3 Examples of Distribution Functions Deduced with MEP

41

If we introduce the absolute temperature T of the system by the classical thermodynamics relationship λ = 1/KB T [116, 153], (2.10) becomes the Maxwell– Boltzmann distribution. In order to highlight the features of the MEP approach, we show also the standard approach of getting the Maxwell–Boltzmann distribution. In statistical mechanics entropy is defined as [91] S = KB log W with W number of microstates associated to a given macrostate. For particles obeying the Maxwell–Boltzmann statistics, W can be calculated by observing that N! ways (recall that N particles can be partitioned among the energy levels in ' i ni ! " i ni = N and that the particles are distinguishable) and in turn the ni particles n can occupy the Gi slots of energy εi in Gi i ways. Therefore N! ( ni W = ' Gi i ni ! i

and, by using the Stirling formula log n! ≈ n log n − n (valid for large n), log W can be approximated as log W ≈ N log N −



ni log ni +

i



ni log Gi .

i

If one considers the thermodynamic limit, characterized by N → +∞ and ni /N remaining finite, S becomes



ni S = KB N log N − ni log Gi i  n  ni i = −KB N log /Gi . N N

= KB

 i

  ni ni log N − log Gi

i

By the law of large numbers, for N  1 one has

ni ≈ pi and N

S = NS Therefore maximizing the information entropy gives the same results of maximizing the statistical entropy but the first approach is based on fewer assumptions and is more systematic.

42

2 Maximum Entropy Principle

2.3.2 Fermi–Dirac and Bose–Einstein Distributions Regarding quantum statistical distributions, the discussion is a little more involved. Here the particles are identical and indistinguishable and, in the case of fermions, have to obey to Pauli’s exclusion principle: a single quantum state can be occupied by one particle at most. Let pij n be the probability that the j -th quantum state of the energy level εi contains n particles. The average number of particles in the energy level εi is Gi  

ni  =

pij n n.

j =1 n

The entropy we use is S = −C

Gi   i

pij n log pij n

j =1 n

and the constraints are Gi   i

Gi   i

pij n n = N,

j =1 n

pij n n εi = E.

j =1 n

The MEP solution reads e−(λ1 +λ2 εi )n pij n = " −(λ +λ ε )r , 1 2 i re where λ1 and λ2 are the Lagrange multipliers relative to the number and energy λ1 λ2 constraints, respectively, with the transformation → λ1 and → λ2 . C C Now one has to distinguish two cases: • Fermi–Dirac distribution. According to Pauli’s exclusion principle n = 0 or n = 1 and therefore "Gi "1 ni  =

j =1

"1

n=0 ne

n=0 e

−(λ1 +λ2 εi )n

−(λ1 +λ2 εi )n

=

Gi e−(λ1 +λ2 εi ) Gi = . 1 + eλ1 +λ2 εi 1 + e−(λ1 +λ2 εi )

2.3 Examples of Distribution Functions Deduced with MEP

43

The spin is included in the degeneracy factor Gi . • Bose–Einstein distribution. Here Pauli’s exclusion principle does not hold and n = 0, 1, . . .. In this case "Gi "+∞ j =1

ni  =

−(λ1 +λ2 εi )n n=0 ne . −(λ1 +λ2 εi )n n=0 e

"+∞

By using the formulae +∞  n=0

e−an =

+∞ 

1 , 1 − e−a

ne−an =

n=0

e−a , (1 − e−a )2

a > 0,

the above expression becomes < ni >=

Gi λ +λ e 1 2 εi

−1

,

provided λ1 + λ2 εi > 0. If we set λ1 = −

μ , KB T

λ2 =

1 , KB T

where μ is the chemical potential, we get the classical Fermi–Dirac or Bose– Einstein distribution in the discrete form  −1   < ni > εi − μ f (εi ) = ±1 = exp , Gi KB T where the upper sign refers to fermions and the lower sign to bosons. The ratio < ni > /Gi represents the i-th occupation number. Of course for fermions it is less or equal to one. Also for fermions and bosons the results of MEP are the same as those in statistical mechanics. In the case of fermions, the number of microstates is given by the product of simple combinations (we observe that it must be ni ≤ Gi ) ( Gi  , W = ni i

while in the case of bosons the number of microstates is given by the product of combinations with possible repetitions ( ni + Gi − 1 W = . ni i

44

2 Maximum Entropy Principle

Therefore by applying Stirling’s formula log W ≈



Gi log Gi −

i

log W ≈

 i

−



ni log ni −



(Gi − ni ) log (Gi − ni )

i

(ni + Gi − 1) log (ni + Gi − 1) −



(Gi − 1) log (Gi − 1)



(fermions),

ni log ni

(bosons).

By maximizing under the constraints 

ni = N,



i

ni εi = E,

i

one gets 1 ni . = λ +λ ε Gi e 1 2 i ±1 In the thermodynamic limit ni ≈< ni > and the two approaches, MEP and statistical mechanics, coincide. These examples make clear the use of MEP as inference method and it is remarkable to observe that the well known results in the context of the statistical mechanics are obtained under a fewer number of physical assumptions.

2.4 Maximum Entropy Inference of a Distribution: The Continuous Case In the continuous case we procede in analogy with the discrete one. Let us suppose to have a distribution function f : Rn → R+ 0 and let us consider a family of weight functions μA : Rn → R d A ,

A = 1, 2, · · · N,

with dA integers, such that the mean values  < μA >=

Rn

μA (x)f (x) d x,

A = 1, 2, · · · N,

there exist and are finite. We want to determine an estimator of f under the only knowledge of the values of the averages < μA >, A = 1, 2, · · · N.

2.4 Maximum Entropy Inference of a Distribution: The Continuous Case

45

Let us introduce the functional space Fμ constituted by the distribution functions g : Rn → R+ 0 such that the integrals  Rn

μA (x)g(x) d x,

A = 1, 2, · · · N

there exist and are finite. By following what done in the discrete case, we assume again the validity of MEP: We estimate f by maximizing the (relative) entropy  S[g] = −C

Rn

g(x) log

g(x) d x, m(x)

under the constraints  μA (x)g(x) d x =< μA >,

with g ∈ Fμ ,

A = 1, 2, · · · N.

Rn

In other words, the MEP estimator of f (x), which we name fMEP (x), is the solution of the optimization problem  fMEP (x) = max S, g∈Fμ

Rn

μA (x)g(x) d x =< μA >,

A = 1, 2, · · · N. (2.11)

By introducing the Lagrange multipliers λA to take into account the constraints (2.11)2, we have to determine the solutions of δS  = 0, where δS  is the first variation of the functional 

S [g] = S[g] +

N 

  λA < μA > −

Rn

A=1

 μA (x)g(x) d x .

One finds      N g(x) δS = − +1 + δg C log λA μA (x) d x = 0, m(x) Rn 



A=1

and therefore

N  λA fMEP (x) = m(x) exp − 1 + μA (x) C A=1



∀δg,

46

2 Maximum Entropy Principle

with the Lagrange multipliers that satisfy the system  < μA >=

Rn

μA (x)fMEP (x) d x,

A = 1, 2, · · · N.

In order that the problem (2.11) admits a solution, restrictions on the weight functions μA are usually necessary. For example, if we want to estimate a distribution 2 3 f : R → R+ 0 and we know only the expectation values < 1 >, < x >, < x >, in general the MEP estimator does not exist; in fact it should be of the form   fMEP (x) = exp λ0 + λ2 x 2 + λ3 x 3 which is not summable since limx→+∞ fMEP (x) = +∞ or limx→−∞ fMEP (x) = +∞, unless λ3 = 0. Only if one is able to prove analytically that λ3 = 0, the problem (2.11) has a solution. However, if one has to resort to a numerical procedure (which is the realistic situation) the problem is numerically ill posed in any case. The situation is even worse if the mean values are estimated by a statistical sample. It is unavoidable that < x 3 >= 0, due to the statistical uncertainty and finite size of the sample, and the summability issue arises again. A viable way to overcome the problem is to consider an even exponent in the mean value of the highest order power and impose that the sign of the relative multiplier is negative.

Chapter 3

Application of MEP to Charge Transport in Semiconductors

MEP can be used for solving the closure problem related to the moment systems associated to the electron transport equations. Here the case of 3D electron gas is considered. Lower dimensional electron gases will be treated in the next chapters. For applications to classical particles or other physical problems, the interested reader is referred to [71, 97, 153].

3.1 The Electron Transport Equation and the Maximum Entropy Principle As already seen in Chap. 1, the semiclassical Boltzmann equation for electrons in the conduction band is q ∂f (r, k, t) + v(k) · ∇r f (r, k, t) − E · ∇k f (r, k, t) = C[f ](r, k, t). h¯ ∂t

(3.1)

By multiplying equation (3.1) by the functions μA (k), A = 1, 2, · · · , N, and integrating over B, one finds the moment equations  ∂MA 2 f μA (k)v(k)d 3 k + ∇r · ∂t (2π)3 B     2 2 q 3 f ∇ μ (k)d k − μ (k)f ndσ = μA (k)C[f ]d 3 k, + E· k A A h¯ (2π)3 B (2π)3 B ∂B A = 1, 2, · · · , N, (3.2)

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_3

47

48

3 Application of MEP to Charge Transport in Semiconductors

where MA (r, t) =

2 (2π)3

 μA (k)f (r, k, t)d 3 k

(3.3)

B

represent the moments of the distribution f with respect to the weight functions μA (k), A = 1, 2, · · · , N. Of course, the previous relations make sense provided that the involved integrals there exist and are finite. The system of PDEs (3.2) is not a closed set of evolution equations for the moments MA because additional quantities appear like the fluxes and the production terms, that is we have to face with the closure problem. By applying MEP it is possible to get appropriate constitutive relations by expressing the fluxes and the production terms as functions of the basic moments MA , A = 1, 2, · · · , N. Firstly we have to find out the expression of the entropy in the continuum ni approximation. To this aim, recalling that fi = , we observe that for fermions in Gi the discrete case      S = KB Gi log Gi − ni log ni − (Gi − ni ) log (Gi − ni )

= KB

 

i

i

Gi log Gi −



i

Gi fi log(Gi fi ) −

i

= −KB





 Gi (1 − fi ) log Gi (1 − fi )

  Gi fi log fi + (1 − fi ) log (1 − fi ) .

i

In order to take the continuum limit, we resort to the standard formula (see for example [92])  i

2V h(ki )Gi →  (2π)3

 h(k)d 3 k

as the number of addends tends to infinity,

B

valid for any regular enough function h(k), V being the spatial volume (that of the crystal in the case of semiconductors). The factor 2 takes into account the spin degeneracy. The sum must be intended over all the discrete wave vectors ki inside the first Brillouin zone. In this way one gets the entropy density S := −

2 KB (2π)3

 B

  f log f + (1 − f ) log(1 − f ) d 3 k.

(3.4)

3.1 The Electron Transport Equation and the Maximum Entropy Principle

49

Let us suppose that a finite number of moments MA (r, t) =

2 (2π)3

 μA (k)f (r, k, t)d 3 k

(3.5)

B

of the distribution f with respect to some weight functions μA (k), A = 1, 2, · · · , N, be known. In order to include the equilibria, we will require that among the weight functions there are μ1 (k) = 1 and μ2 (k) = ε(k), whose moments represent the electron density and energy density. If we apply MEP with the entropy (3.4) under the constraints (3.5), by introducing the Lagrange multipliers λA , the problem to maximize S under the constrains (3.5) is equivalent to maximize 

S =S −



 λA

A

2 (2π)3





μA (k)f (r, k, t)d k − MA (r, t) . 3

B

Imposing δS  = 0 gives    1−f − λA μA δf d 3 k = 0, KB log f B and, since δf is arbitrary, it follows fME =

exp( K1B

"

1 A λA μA )

+1

.

Observe that the condition 0 ≤ fME ≤ 1 is satisfied, no matter the values of the Lagrange multipliers. If only the two weight functions μ1 = 1 and μ2 = ε are used, the continuous Fermi–Dirac distribution   ε(k)−μ −1 KB T 1+e μ 1 . is obtained with the identification, as in the discrete case, λ = − , λε = T T In thermodynamic equilibrium, T is equal to the lattice temperature TL . If further weight functions are considered, MEP gives an estimator of f in a non equilibrium state. In most cases the electron occupation number is rather low, that is 0 ≤ f 1 (non degenerate electron gas). By expanding the entropy density around f = 0 up to first order, it can be approximated as S =−

2 KB (2π)3

 (f log f − f ) . B

50

3 Application of MEP to Charge Transport in Semiconductors

After introducing the Lagrange multipliers λA , the problem to maximize S under the constrains (3.5) is equivalent to maximize S = S −



 λA

A

2 (2π)3



 μA (k)f (r, k, t)d 3 k − MA (r, t) B

and δS  = 0 gives

1  log f + λA μA δf = 0. KB A

Since this relation must hold for arbitrary δf , it follows fME = e

− K1

"

B

A λA μA

(3.6)

.

At equilibrium, with the identification of the Lagrange multiplier relative to the −λ −

ε(k)

energy considered above, one has fME = e 0 KB TL . In the parabolic band h¯ 2 k 2 approximation ε(k) = , evaluating λ0 from the relation 2m∗ 2 e−λ0 n= (2π)3

 R3

the Maxwell–Boltzmann distribution √ nπ 2π (mKB TL )

2 2

e

e 3/2

− 2mh∗¯ Kk

B TL

d 3 k,

2 2

− 2mh∗¯ Kk

B TL

is recovered. Returning to the general case, in order to complete the procedure, one has to express the multipliers λA ’s as functions of the moments MA by inverting the constrains (3.5) and arriving to the final form of the MEP estimator of f fME (r, k, t) = f˜ME (M1 (r, t), · · · , MN (r, t), k) = f˜ME (M(r, t), k), where M(r, t) = (M1 (r, t), · · · , MN (r, t)). Regarding the inversion of the relations moments—multipliers, the following property holds. " Lemma 3.1.1 Let us suppose that all the integrals exist and set χ = K1B A λA μA .  (χ) is definite in sign, the inversion is always achievable. Provided that fME

3.1 The Electron Transport Equation and the Maximum Entropy Principle

 Proof In fact the Jacobian matrix

∂MA ∂λB

51

 is definite in sign because

 ∂  2 δzA δzC = μA fME (χ)d 3 kδzA δzC ∂λC ∂λC B (2π)3 A,C

2   1 2  = f (χ) μA δzA d 3 k, (2π)3 KB B ME

 ∂MA A,C

A

with δzA arbitrary increments. In the degenerate case f (χ) =

1 eχ + 1

and f  (χ) = −

eχ < 0, (eχ + 1)2

∀χ.

In the non degenerate case f (χ) = e−χ

and f  (χ) = −e−χ < 0,

∀χ.

Therefore, if the moments with respect to the considered weight functions exist, the inversion is guaranteed both in the degenerate and non degenerate case.  Once the Lagrange multipliers have been expressed in terms of the moments, no matter if the degenerate or non degenerate case is dealt with, one finds the closure relations for the fluxes and the production terms by replacing the original set of moment equations (1.15) with ∂MA (r, t) 2 + ∇r ∂t (2π)3



f˜ME (M(r, t), k)μA (k)v(k)d 3 k B

 q 2 + E· f˜ME (M(r, t), k)∇k μA (k)d 3 k (2π)3 B h¯  2 = μA (k)C[f˜ME (M(r, t), k)]d 3 k, (2π)3 B A = 1, 2, · · · , N.

If we rewrite the previous system by using as field variables the Lagrange multipliers, the moment system reads   N N  2  1 ∂MA ∂λC  3 v(k)μ μ f (χ)d k ∇r λC + A C ME ∂λC ∂t KB (2π)3 B

C=1

C=1

= GA (λ1 , · · · , λN , E)

(3.7)

52

3 Application of MEP to Charge Transport in Semiconductors

where  q 2 GA (λ1 , · · · , λN , E) = − E · fME (λC (r, t), k)∇k μA (k)d 3 k h¯ (2π)3 B  2 + μA (k)C[fME (λC (r, t), k)]d 3 k. (2π)3 B By introducing the Jacobian matrices  A0 =

∂MA ∂λB



 ,

Ai =

1 2 KB (2π)3

 B

 vi (k)μA μB fME (χ)d 3 k

 ,

i = 1, 2, 3

and the vectors Λ = (λ1 , · · · , λN )T and G = (G1 , · · · , GN )T , the system (3.7) can be rewritten as ∂Λ  ∂Λ + Ai = G. ∂t ∂xi 3

A0

i=1

Since A0 is invertible and the matrices Ai are symmetric, (3.7) forms, at least in the case when the electric field is considered as an external field, a quasilinear hyperbolic system in the time direction. This implies that the Cauchy problem is well-posed thanks to the Fisher and Marsden theorem [77]. A study of the analytical properties of (3.7) can be found in [1].

3.2 Further Considerations MEP gives an approximation of the exact distribution function in terms of a finite number of moments and requires to solve a constrained optimization problem (see [111] for a numerical treatment of the nonlinear case). However, this latter does not always admit solution because of the lacking of integrability, as in the case of gas dynamics when one considers moments with respect to weight functions represented by polynomials in the microscopic velocity and the highest degree of the velocity is odd [102, 103, 117]. Since in the parabolic approximation for the energy band of electrons, the choice of the weight functions 1, v, E(k), vE(k) leads to a moment system similar to the Grad eight-moment one, the same drawback of the gas dynamics arises. However, in [105] it has been shown that when one employs the Kane model for the energy band, the corresponding maximum entropy models are symmetric hyperbolic systems with convex domains of definition and that the equilibria are interior points, guaranteeing the validity of expansions around equilibrium states. The previous issue will be presented in detail in the next section. Here we would like to mention that some attempts have been also made to use MEP in quantum

3.3 Solvability of the MEP Problems in Semiconductors

53

transport for closing the moment system arising from the Wigner equation, which represents the quantum analogue of the semiclassical Boltzmann equation. Already Jaynes [96] tried to devise a procedure based on the maximization of the von Neumann entropy of a quantum system. Recently this issue has been tackled in [66, 68], where quantum drift-diffusion and quantum energy-transport models have been derived by resorting to a quantum formulation of MEP. Further applications of such a procedure have appeared in the literature, e.g. for graphene [21] or carbon nanotubes [98]. The interested reader is refereed to [99, 100] for a comprehensive review. An alternative approach has been used in [177] where under a high field— collision balanced scaling, quantum corrections to the semiclassical case based on MEP have been obtained.

3.3 Solvability of the MEP Problems in Semiconductors From a mathematical point of view MEP gives an approximation of the exact distribution function in terms of a finite number of moments and leads to solving a constrained optimization problem. However this latter does not always admit solution as in the case of gas dynamics when one considers moments with respect to weight functions represented by polynomials in the microscopic velocity of degree higher than two [102–104, 117]. Since the parabolic approximation for the energy band of electrons leads to a moment system with the same type of weight functions, the same drawback of the gas dynamics arises. Here, by following Ref. [105], we show that such a problem is overcome when one employs the Kane model for the energy band. It is proved that the corresponding maximum entropy models are symmetric hyperbolic systems with convex domains of definition and that the equilibria are interior points, guaranteeing the validity of expansions around equilibrium states. We recall that when the approximation of Kane is used, one has h¯ 2 |k|2 E(k) = = ) ∗ 1 + 1 + 2 mα∗ h¯ 2 |k|2 m 1



h¯ 2 |k|2 1 1 , + − ∗ 2 2αm 2α 4α

k ∈ R3

where α is the non-parabolicity parameter, and the corresponding electron velocity is v(k) = )

h¯ k. ∗ m 2α 2 1+ m ¯ |k|2 ∗h 1

54

3 Application of MEP to Charge Transport in Semiconductors

All the macroscopic quantities of interest can be written as suitable moments of the distribution function f . To keep the analysis as general as possible, we introduce the weight functions ai : Rd → R and the corresponding moments ρi = f, ai  ,

i = 1, . . . , m

2 and k integration. Of course the case (2π)d of physical interest are represented by d = 1, 2, 3. Practically, all the macroscopic models are based on expectation values of vpolynomials and powers of the energy E respectively. For such a reason, we split the vector of weight functions a into two subgroups. The first m1 components of a are chosen as (P1 (v(k)), . . . , Pm1 (v(k))) where P1 , . . . , Pm1 are linearly independent polynomials with P1 (v) = 1, and the remaining m2 components give rise to energy moments (E(k)Q1 (v(k)), . . . , E(k)Qm2 (v(k))) where, again, Q1 , . . . , Qm2 are linearly independent polynomials and Q1 (v) = 1. The maximum entropy method can be reformulated as where ·, · denotes multiplication by

maximize H (f ) = −kB f, log f − 1 with f ≥ 0 and f, a = ρ.

(3.8)

As seen in the previous sections, for general ai , the formal solution of (3.8) is obtained with the method of Lagrange multipliers. We introduce the Lagrange functional L(f, λ) : = H (f ) − λ · (ρ − f, a) where λ is the vector of Lagrange multipliers. The necessary condition that all directional derivatives vanish in the maximum fλ leads to 0 = δL(fλ , λ) = (− log fλ + λ · a) δfλ so that fλ = exp(λ · a).

(3.9)

Finally, the Lagrange multipliers λ are chosen in such a way (if possible) that the moment constraints ρ = fλ , a are satisfied which gives rise to a function λ = λ(ρ). Depending on the choice of weight functions ai , it can happen that problem (3.8) is not always solvable, i.e. that there exist moment vectors ρ which cannot be written as a-moments of any exponential density fλ = exp(λ · a). Since the domain of definition U of fluxes and production terms in the balance equations is given by those moment vectors for which the solution of (3.8) exists, the non-solvability

3.3 Solvability of the MEP Problems in Semiconductors

55

implies that U does not coincide with the set of all a-moments (which is an open, convex cone). This structural deficiency of U is accompanied by the disadvantage that the equilibrium states are located on ∂U and that they are singular points of the flux functions. This has been demonstrated for maximum entropy moment systems which are based on polynomial weight functions [72, 102–104, 106]. Since the parabolic band approximation also leads to such polynomial weights, similar conclusions apply. Kane’s model is superior to the parabolic band approximation in the sense that the corresponding moment system has a desirable mathematical structure: it is a symmetric hyperbolic system with an open and convex domain of definition. The equilibria are interior points and the fluxes are regular at these states so that expansions around equilibria are reasonable in contrast to the parabolic band case. As already mentioned, the hyperbolicity is an immediate structural feature of the moment system and since equilibria are contained in U , they have to be interior points if the domain of definition is open. The smoothness of the fluxes follows from the inverse function theorem using the fact that λ → fλ , a is continuously differentiable with a positive definite Jacobian matrix fλ , a ⊗ a: in fact, for any vector 0 = ξ ∈ Rm , we have

2 , m m   * + ξi ai fλ , ai aj ξi ξj = fλ , i,j =1

i=1

which is strictly positive if the weight functions are, for example, continuous and linearly independent. Thus, what remains to be checked is that U is open and convex. We prove this fact by showing the solvability of (3.8) for all possible moment vectors ρ, or in other words, by showing that U coincides with the open convex cone of all a-moments.

3.3.1 Statement of the Main Result In order to state our main result, we ) first rewrite (3.8). For notational convenience, √ we measure E, k, v in units 1/(2α), m∗ /(2α h¯ 2 ), and 1/ 2αm∗ which leads to E(k) =

) 1 + |k|2 − 1,

v(k) = .

k 1 + |k|2

,

(3.10)

where the same notation, for the sake of sinmplicity, has been used for the scaled and unscaled variables.

56

3 Application of MEP to Charge Transport in Semiconductors

For small k, we see a similarity to the parabolic band approximation because E(k) ∼ |k|2 /2 and v(k) ∼ k. For large k, however, v(k) is bounded and E(k) grows only linearly due to the estimates |v(k)| < 1,

|k| − 1 ≤ E(k) ≤ 2|k| + 1.

(3.11)

Since we have chosen E and v and two sets {P1 , . . . , Pm1 }, {Q1 , . . . , Qm2 } of linearly independent polynomials with P1 = Q1 = 1, we write the weight functions as a = (P1 (v), . . . , Pm1 (v), EQ1 (v), . . . , EQm2 (v))T .

(3.12)

Since the assumption of a three dimensional k-space is not relevant for our argument, we assume k ∈ Rd . The moment set related to the weights ai is generated by the functions in F = {f ≥ 0 : f ≡ 0, |a|f ∈ L1 (Rd )}.

(3.13)

Here, f ≥ 0 and f ≡ 0 are to be understood in the measure theoretic sense, i.e. {x : f (x) > 0} should have positive Lebesgue measure. The corresponding moments are collected in M = {f, a : f ∈ F }.

(3.14)

Using this notation and the definition of the entropy functional H (f ) = − f, log f − 1 ,

(3.15)

we can restate (3.8) as maximize H (f ) subject to f ∈ F and f, a = ρ

(3.16)

Our main result is Theorem 3.3.1 The maximum entropy moment problem (3.16) is uniquely solvable for any ρ inside the open, convex cone M . The solution is an exponential density exp(λ · a) for some λ ∈ Rm depending on ρ. To prove Theorem 3.3.1, we are going to use a general result of Csiszar about the solvability of minimum relative entropy problems on sets of probability measures [60]. The connection between the two results is based on a few simple transformations. First, we observe that, up to normalization, every f ∈ F can be viewed as a probability density. The normalization f ∗ = f/ f, 1 is abbreviated by a ∗-superscript and its image of F is denoted F ∗ . Since we assume a1 = 1, the

3.3 Solvability of the MEP Problems in Semiconductors

57

moment vector of f ∗ has the structure * ∗ + f , a = (1, ρ2 /ρ1 , . . . , ρm /ρ1 )T ,

ρ = f, a ,

which gives rise to a normalization operation acting on vectors in Rm α ∗ = (α2 /α1 , . . . , αm /α1 )T ,

α ∈ Rm , α1 > 0.

Note that a ∗ = (a2 , . . . , am )T because a1 = 1 and thus f, a = ρ implies f ∗ , a ∗  = ρ ∗ . Apart from the transition to probability measures, we consider the functional of relative entropy. If P and R are probability measures on the Borel sets B on Rd , such that P has a density with respect to R, i.e.  P (A) =

pR dR,

A∈B

A

the relative entropy (or I-divergence) is defined as  I (P ||R) =

pR log pR dR.

As measure R, we are going to use 

g ∗ dk,

R(A) =

g(k) = exp(−E(k))

(3.17)

A

where g is integrable since E(k) grows linearly (see (3.11)). Then, if Pf ∗ is a probability having density f ∗ ∈ F ∗ with respect to the Lebesgue measure, it has density f ∗ /g ∗ with respect to R and  I (Pf ∗ ||R) =

f∗ f∗ log dR = g∗ g∗



f ∗ log

f∗ dk g∗

Using the definition (3.15) of H and log f ∗ = log f − log f, 1 ,

log g ∗ = −E − log g, 1 ,

we obtain the relation I (Pf ∗ ||R) = −

g, 1 f, E 1 H (f ) + 1 + log + . f, 1 f, 1 f, 1

(3.18)

58

3 Application of MEP to Charge Transport in Semiconductors

Since f, 1 and f, E are constant on the set of densities f ∈ F with f, a = ρ, we see that maximizing H subject to f, a = ρ is equivalent to minimize I (Pf ∗ ||R) + * subject to f ∗ ∈ F ∗ and f ∗ , a ∗ = ρ ∗

(3.19)

In summary, we have Proposition 3.3.1 Let ρ ∈ M . Then problem (3.16) has a unique solution f ∈ F if and only if (3.19) has a unique solution f ∗ ∈ F ∗ . The relation between f and f ∗ is given by f = ρ1 f ∗ . In particular, if f ∗ = c exp(ξ · a ∗ ) for some ξ ∈ Rm−1 and some c > 0, then f = exp(λ · a) with λ = (log(cρ1 ), ξ1 , . . . , ξm−1 )T . Csiszar’s result, which is presented in the next section, shows that (3.19) is even uniquely solvable with an exponential density if Pf ∗ is replaced by general probability measures P on Rd which have the correct moments. In connection with Proposition 3.3.1 this immediately yields Theorem 3.3.1.

3.3.2 A General Result by Csiszar Csiszar’s result [60] applies to general measurable spaces (X, H ) with weight functions a ∗ = (a2 , . . . , am ) being H -measurable. Note that ai∗ can be general measurable functions here. We only keep the previous notation to be consistent with our use of Csiszar’s theorem. By P we denote the set of probability measures on (X, H ) and for P , R ∈ P, we write P R if P has an R-density, i.e.  P (A) =

pR dR,

A∈H.

A

The entropy of P relative to R is defined as / I (P ||R) =

P R

pR log pR dR

+∞

P  R

.

To state Csiszar’s theorem, we introduce for any ρ ∗ ∈ Rm−1 the set of all probability measures with moments ρ ∗ ∗

∗

 1

E (ρ ) = {P ∈ P : a ∈ L (P ),

a ∗ dP = ρ ∗ },

 where L1 (P ) is the space of functions a such that

|a| dP there exists and is finite.

Since our aim is to find a minimizer of the relative entropy I (P ||R) over P ∈ E (ρ ∗ )

3.3 Solvability of the MEP Problems in Semiconductors

59

for some fixed R ∈ P, we restrict ourselves to those moment vectors ρ ∗ for which at least one corresponding P ∈ P has finite relative entropy, i.e. AR = {ρ ∗ ∈ Rm−1 : ∃P ∈ E (ρ ∗ ) such that I (P ||R) < ∞}. The last ingredient is the set TR which contains all coefficient vectors ξ ∈ Rm−1 for which the density exp(ξ · a ∗ ) is R-integrable TR = {ξ ∈ Rm−1 : exp(ξ · a ∗ ) ∈ L1 (R)}. Using this notation, Theorem 3.3 of [60] can be formulated as Theorem 3.3.2 Assume TR is open and let ρ ∗ ∈ int AR . Then the problem min I (P ||R)

P ∈E (ρ ∗ )

has a unique solution P ∈ P with P R and density pR = c exp(ξ · a ∗ ) for some vector ξ ∈ TR and some c > 0. Above for a set A, int A means the set of its interior points. In order to apply Theorem 3.3.2 to our particular choice of weight functions and moment vectors, we just have to check its assumptions. In the following sections, we show that for any ρ ∈ M there exists f ∈ F such that f, a = ρ and f log f ∈ L1 (Rd ). In view of (3.18) this implies that the set of all normalized moments + * M ∗ = { f ∗, a∗ : f ∗ ∈ F ∗} is contained in AR and since M ∗ is open (Corollary 3.3.1), we have M ∗ ⊂ intAR . The remaining condition that TR is open is shown in Proposition 3.3.6. While the condition ρ ∗ ∈ intAR follows from very basic properties of the weight functions (linear independence and analyticity), the condition that TR is open requires a detailed investigation of the growth behavior of a(k) for |k| → ∞. Finally, we want to remark that Theorem 3.3.2 is not helpful in the parabolic band approximation where E(k) = |k|2 /2 and the weights are simply polynomials in k where at least one of them grows faster than quadratically, say as |k|4 . While it is still possible to show ρ ∗ ∈ intAR , it is generally not true that TR is open, essentially because, as seen in Chap. 2, exp(−|k|2 /2+0·|k|4 ) ∈ L1 but exp(−|k|2 /2+|k|4 ) ∈ L1 for any  > 0 (see also [102, 103]).

3.3.3 The Weight Functions Before considering the moment set M in more detail, we collect a few basic properties of the weight functions. Our first result concerns the linear independence.

60

3 Application of MEP to Charge Transport in Semiconductors

Proposition 3.3.2 The components of the vector of weight functions a defined in (3.12) are linearly independent. " Proof Assuming the opposite, we could find γi ∈ R with |γi | > 0 such that " γi ai (k) = 0 for all k ∈ Rd . Setting P (v) =

m1 

Q(v) =

γi Pi (v),

i=1

m2 

γi+m1 Qi (v)

i=1

this implies P (v(k)) + E(k)Q(v(k)) = 0,

∀k ∈ Rd .

(3.20)

If we show " that (3.20) implies P = Q = 0, the linear independence of Pi and Qi leads to |γi | = 0, in contradiction to the assumption. To show P = Q = 0, we take any t ∈ Rd with |t| = 1 √ We note that v(st) = s(1 + s 2 )−1/2 t and (s) = E(st) = 1 + s 2 − 1 and set   s p √ = P (st ), 1 + s2

 q

s √ 1 + s2

 = Q(st),

s ∈ R.

Relation (3.20) implies 

s



p √ 1 + s2

 + (s)q

√



s 1 + s2

=0

∀s ∈ R.

Dividing by (s) and sending s to ±∞ yields q(±1) = 0 so that q(s) = (1 − ˜ Hence s 2 )q(s).     s 1 s p √ + (s) = 0, q ˜ √ 1 − s2 1 + s2 1 + s2

∀s ∈ R.

and by sending again s to ±∞, we find p(±1) = 0, i.e. p(s) = (1 − s 2 )p(s). ˜ Altogether, we get  p˜ √

s 1 + s2



 + (s)q˜

s

√ 1 + s2

 = 0,

∀s ∈ R.

Repeating the argument, we conclude that p = q = 0, since otherwise the degree of p and q would be larger than any fixed number. Since we picked t arbitrarily, we also find P = Q = 0 which concludes the proof.  While linear independence only implies that the zero set {k : β · a(k) = 0} of a linear combination β · a of the weight functions cannot be very big (the whole space

3.3 Solvability of the MEP Problems in Semiconductors

61

Rd ), we will need the stronger property that the zero set must be very small in the following sense. Definition 3.3.1 A set of measurable functions a1 , . . . , am on Rd has the pseudoHaar property if for any 0 = β ∈ Rm , the zero set of β · a has zero Lebesgue measure. Proposition 3.3.3 The components of the vector of weight functions a defined in (3.12) have the pseudo-Haar property. Proof We follow the argument presented in [119]: since E, v, Pi , Qi are analytic, also the weights ai are analytic and since the zero set of any non-zero analytic function on Rd has vanishing Lebesgue measure, the linear independence of the weight functions implies the pseudo-Haar property. 

3.3.4 The Moment Cone Since M = {f, a : f ∈ F } is the image of F under the linear mapping f → f, a, the obvious property of F being a convex cone carries over to M . Proposition 3.3.4 The moment set M is an open convex cone in Rm . Proof It remains to show that M is open which we do by using the same argument as in [119]. Assuming that ρ¯ = f, a is a boundary point of M , there exists 0 = ¯ · β ≥ 0 for all ρ ∈ M due to convexity. However, the β ∈ Rm such that (ρ − ρ) function h ∈ F defined by  h(k) =

3 2 f (k) 1 2 f (k)

β · a(k) < 0 β · a(k) ≥ 0

has a moment vector ρ = h, a which satisfies 1 (ρ − ρ) ¯ · β = h − f, β · a = − f, |β · a| < 0 2 which is strictly negative because |β · a| can vanish at most on a set of measure zero due to the pseudo-Haar property. Hence, the assumption that ρ¯ is a boundary point leads to a contradiction and M is therefore open.  We remark that the set M ∗ of normalized moments is obtained by intersecting the cone M and the hyperplane {1} × Rm−1 , i.e. M ∩ {1} × Rm−1 = {1} × M ∗ .

62

3 Application of MEP to Charge Transport in Semiconductors

Consequently, M ∗ is also convex because both M and {1}×Rm−1 are convex. Since M is open in Rm , the intersection is open in the relative topology of {1} × Rm−1 which is equivalent to the usual topology of Rm−1 (up to the bijection (1, α) → α between {1} × Rm−1 and Rm−1 ). Corollary 3.3.1 The set of normalized moments M ∗ is open and convex in Rm−1 .

3.3.5 The Entropy Functional A minimal requirement for the maximum entropy problem (3.16) to have a solution is the existence of at least one density f ∈ F with f, a = ρ which has a finite entropy. This question is considered here. Proposition 3.3.5 Let ρ ∈ M . Then there exists a function f ∈ F with f, a = ρ and f log f ∈ L1 (Rd ). In particular, f ∗ satisfies f ∗ , a ∗  = ρ ∗ and f ∗ log f ∗ ∈ L1 (Rd ). Proof Since M is open, there exists a hypercube with vertices η1 , . . . , η 2m in M which has ρ as its center point. By definition of M , each ηi is the moment vector of some fi ∈ F . For j ∈ N, we set  (j ) fi (k)

=

min{j, fi (k)}

|k| < j

0

|k| ≥ j

Using the dominated convergence theorem, it then follows that (j )

ηi

0 1 (j ) = fi , a −−−→ fi , a = ηi . j →∞

(j )

Hence, for j large enough, the vectors ηi will also be in M and ρ will be in their convex hull., i.e. there exist ωi ≥ 0 which add up to one, such that m

ρ=

2 

(j )

ωi ηi

i=1

(the argument is based on a simple application of the implicit function theorem—see " (j ) [104] for details). Setting f = i ωi fi , we thus have ρ = f, a with f being bounded and of compact support. Consequently, f log f ∈ L1 (Rd ). The result for f ∗ follows by normalization. 

3.3 Solvability of the MEP Problems in Semiconductors

63

3.3.6 The Lagrange Multipliers In [102–104] it is shown that the topology of the set Λ = {λ ∈ Rm : exp(λ · a) ∈ L1 (Rd )}

(3.21)

(the so called Lagrange multipliers) determines the solvability of the maximum entropy problem. Also in Csiszar’s theorem, it is decisive that the set TR = {ξ ∈ Rm−1 : exp(ξ · a ∗ ) ∈ L1 (R)}

(3.22)

is open in Rm−1 . Note that Λ and TR are closely related if we choose the probability measure R according to (3.17). In fact, if ξ ∈ TR then exp(ξ ·a ∗ ) exp(−E) ∈ L1 (Rd ) so that the vector H (ξ ) ∈ Rm defined by H (ξ )·a = ξ ·a ∗ −E is contained in Λ. Note that H (ξ ) = Bξ − b is an affine linear mapping, the vector b being the unit vector which picks out the component E of the weight vector, i.e. b · a = E, and B being the canonical embedding operator of Rm−1 into Rm , i.e. Bξ = (0, ξ1 , . . . , ξm−1 )T . Conversely, H (ξ ) ∈ Λ implies that ξ ∈ TR (by multiplying H (ξ ) = λ by B T and using that B T B is the identity on Rm−1 ). Altogether, we conclude that TR is the preimage of Λ and since H is continuous we find that TR is open in Rm−1 if Λ is open in Rm . In the following, we therefore restrict our considerations to a characterization of Λ. Introducing the polynomial vectors P = (P1 , . . . , Pm1 )T , Q = (Q1 , . . . , Qm2 )T and splitting λ ∈ Rm into λ1 ∈ Rm1 , λ2 ∈ Rm2 , we have by definition of the weight functions λ · a = λ1 · P (v) + λ2 · Q(v)E. In view of (3.11), the velocity v(k) is a bounded function of k and thus also P (v(k)) and Q(v(k)). Integrability of exp(λ · a) can therefore only be achieved if the factor λ2 · Q(v) in front of E is uniformly negative for large k since E grows linearly (see (3.11)). Observing that v(k) tends to the unit sphere for |k| → ∞, this leads to the integrability condition on λ2 λ2 · Q(t) < 0,

∀t ∈ Sd−1 ,

(3.23)

where Sd−1 is the unit sphere of Rd . With the following result, we make our considerations more precise (for ease of notation, we identify (λ1 , λ2 ) ∈ Rm1 ×Rm2   with λλ12 ∈ Rm1 +m2 ). Lemma 3.3.1 Let λ1 ∈ Rm1 and λ2 ∈ Rm2 such that (3.23) is satisfied. Then (λ1 , λ2 ) ∈ Λ.

64

3 Application of MEP to Charge Transport in Semiconductors

Proof Assume λ2 · Q(t) < 0 for all |t| = 1. Then there exists μ > 0 such that λ2 · Q(t) ≤ −μ for all |t| = 1. By continuity, we can find δ > 0 such that λ2 · Q(v) ≤ −μ/2,

1 − δ ≤ |v| ≤ 1

and we remark that, in view of (3.10), |v| ≥ 1 − δ

1−δ = D. |k| ≥ . 1 − (1 − δ)2

⇔

Since |v(k)| < 1 for all k ∈ Rd , we have |λ1 · a 1 (k)| ≤ |λ1 | max |P (v)| = C1 |v|≤1

and thus λ · a(k) ≤ C1 −

μ E(k), 2

|k| ≥ D.

On |k| ≤ D we can find uniform bounds so that λ · a(k) ≤ C2 , with a suitable C2 > 0. By taking into account that for |k| ≥ 2 ) |k| E(k) = |k| 1/|k|2 + 1 − 1 ≥ |k| − |k|/2 = , 2 we eventually get the estimate λ · a(k) ≤ C − ν|k|,

k ∈ Rd ,

|k| > δ¯

for some C, ν, δ¯ positive constants. This yields exp(λ · a(k)) ≤ exp(C − ν|k|), i.e. λ ∈ Λ.  The next result shows that (3.23) is also necessary for integrability. Lemma 3.3.2 Let λ1 ∈ Rm1 and λ2 ∈ Rm2 such that (3.23) is violated. Then (λ1 , λ2 ) ∈ Λ. Proof Our aim is to show that λ · a is bounded from below on a set Ωs¯ of infinite measure. If Qλ2 (v) = λ2 · Q(v) satisfies Qλ2 (t 0 ) ≥ 0 with |t 0 | = 1, we set Ωs¯ = {st 0 + δ : s ≥ s¯ , |δ| ≤ 1, δ · t 0 = 0} which is a cylinder of radius one around the direction t 0 . In particular, |Ωs¯ | = ∞ for all s¯ ≥ 0. An obvious parameterization of Ωs¯ is given by [¯s , ∞) × B where

3.3 Solvability of the MEP Problems in Semiconductors

65

B = {δ ∈ Rd : |δ| ≤ 1, δ · t 0 = 0}. Then k(s, δ) = st 0 + δ and s(k) = k · t 0 , δ(k) = k − (k · t 0 )t 0 . For |k| we have the estimate s 2 ≤ |k(s, δ)|2 ≤ s 2 + 1

(3.24)

s ≤ |k(s, δ)| ≤ 2s.

(3.25)

and in particular for s¯ ≥ 1,

Using the estimates |k|/2 ≤ E(k) ≤ 2(1 + |k|) for |k| ≥ 2 from the proof of Lemma 3.3.1, we conclude that s/2 ≤ E(k(s, δ)) ≤ 4(1 + s),

s ≥ 2, δ ∈ B.

(3.26)

Using (3.24) and the definition of v, we find 0 ≤ 1 − |v(k(s, δ))|2 ≤ 1 −

1 s2 =1− . 2 s +2 1 + s22

An elementary estimate shows 1 − 1/(1 + x) ≤ x for −1 < x ≤ 1 and hence 0 ≤ 1 − |v(k(s, δ))|2 ≤

2 , s2

s ≥ 2, δ ∈ B.

(3.27)

Similarly, we find s 0 ≤ 1 − v(k(s, δ)) · t 0 ≤ 1 − √ 2 + s2 √ and the elementary estimate 1 − 1/ 1 + x ≤ x for 0 ≤ x ≤ 1 yields 0 ≤ 1 − v(k(s, δ)) · t 0 ≤

2 , s2

s ≥ 2, δ ∈ B.

(3.28)

Using the relation |v − t 0 |2 = |v|2 − 1 + 2(1 − v · t 0 ) together with (3.28) and (3.27), we get √ 8 |v(k(s, δ)) − t 0 | ≤ , s

s ≥ 2, δ ∈ B.

(3.29)

In particular, for any  > 0, we can find s¯() > 2 such that |v(k) − t 0 | <  for all k ∈ Ωs¯() . Assuming first that Qλ2 (t 0 ) > 0, we can find  > 0 such that Qλ2 (v) ≥ 0

66

3 Application of MEP to Charge Transport in Semiconductors

for all |v − t 0 | < . Hence, Qλ2 (v(k))E(k) ≥ 0 for all k in the cylinder Ωs¯() and if we set C1 = max|v|≤1 |λ1 · P (v)|, one has exp(λ · a) ≥ exp(−C1 ),

∀k ∈ Ωs¯() ,

which implies λ ∈ Λ since Ωs¯() has infinite measure. In the case that t 0 is a root of Qλ2 , we can find  > 0 and C > 0 such that |Qλ2 (v)| ≤ C|v − t 0 |,

if |v − t 0 | < .

(3.30)

Hence, on Ωs¯() , by taking into account (3.26), (3.29), (3.30) and setting C1 = max|v|≤1 |λ1 · P (v)|, we have √ √ 8 ≥ −C1 − 8 8C = −K, λ · a(k(s, δ)) ≥ −C1 − 4(1 + s)C s

s ≥ s¯ () ≥ 2.

Again exp(λ · a) ≥ exp(−K) on the set Ωs¯() of infinite measure implies λ ∈ Λ.  Altogether, we have shown that Λ coincides with the set C = {(λ1 , λ2 ) : λ1 ∈ Rm1 , λ2 ∈ Rm2 , λ2 · Q(t) < 0 ∀|t| = 1}.

(3.31)

Proposition 3.3.6 The set Λ defined in (3.21) is a non-empty, open, convex cone in Rm . In particular, TR defined in (3.22) with R given by (3.17) is open in Rm−1 . Proof In view of our considerations above, it suffices to show that C defined in (m ) (3.31) is a non-empty, open, convex cone. Since Q1 = 1, we find (0, −t 1 2 ) ∈ (m2 ) C = ∅, where t 1 is the first unit vector in Rm2 . Also convexity follows easily from the definition. Finally, if (λ1 , λ2 ) ∈ C, we can find μ > 0 such that max|t |=1 λ2 · Q(t) ≤ −μ. Since Q(t) is bounded for |t| = 1, we also find δ > 0 such that |δ 2 · Q(t)| < μ/2 for all δ 2 ∈ Rm2 with |δ 2 | < δ. Hence (λ1 + δ 1 , λ2 + δ 2 ) ∈ C for all |δ 1 |, |δ 2 | < δ which shows that C is open. 

3.3.7 Proof of the Main Theorem In view of Proposition 3.3.1 it suffices to show that, for given ρ ∈ M , problem (3.19) has a unique solution of exponential type. Since ρ ∈ M implies ρ ∗ ∈ M ∗ , Corollary 3.3.1 and Proposition 3.3.5 in connection with (3.18) can be used to show that ρ ∗ ∈ intAR . Finally, TR is open according to Proposition 3.3.6 and Theorem 3.3.2 shows that (3.19) has a unique solution with R-density c exp(ξ · a ∗ ),

3.3 Solvability of the MEP Problems in Semiconductors

67

or equivalently, with Lebesgue density c exp(ξ · a ∗ )

exp(−E) = exp(λ · a) exp(−E), 1

where λ = (log(c/ exp(−E), 1), ξ1 , . . . , ξm−1 )T . This concludes the proof of Theorem 3.3.1.



Remark 3.3.1 From the proof of Theorem 3.3.1 one can see that the boundedness of |v(k)| is the important property. Therefore a similar result can be expected for more general dispersion relations which exhibit an effect of saturation for the modulus of v(k).

Chapter 4

Application of MEP to Silicon

In this chapter MEP is applied to close the moment equations for electrons in silicon semiconductors. In our model we consider the electrons distributed in the six Xvalleys assumed as equivalent. The approximation given by Kane will be used as dispersion relation. In order to illustrate how to practically implement the models presented in this book, and in particular in this chapter, in Appendix E an example of numerical code for the simulation of a silicon diode is given. The interested reader can easily modify this code to deal with more complex devices. The results of this chapter are based on references [7, 173, 174]. The interested reader can find applications to the optimal design of electron devices in [70, 190]. Models for holes, GaAs, GaN and SiC can be found in [3, 142, 147].

4.1 Moment Equations and Closure Problem Since the six valleys in the main crystallographic directions, close to the X-points, are considered as equivalent, it is sufficient to give the evolution equations only for a single valley. As seen in the previous chapters, one can get the balance equations for macroscopic quantities associated to the electron flow by multiplying Eq. (1.8) by a function μ(k) and integrating over B. In the model we propose μ equal to 1, h¯ k, E and E v. This represents the minimal set of weight functions which gives a model beyond the classical hydrodynamical ones. Of course, approaches with more moments can be investigated.

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_4

69

70

4 Application of MEP to Silicon

By considering such expressions for μ one has the following balance equations (repeated indices are considered as summed) ∂n ∂(nV i ) = 0, + ∂t ∂x i ∂(nP i ) ∂(nU ij ) + + nqE i = nCPi , ∂t ∂x j

(4.1) i, = 1, 2, 3

∂(nW ) ∂(nS j ) + + nqVj E j = nCW , ∂t ∂x j ∂(nS i ) ∂(nF ij ) i + + nqEj Gij = nCW , ∂t ∂x j

(4.2) (4.3)

i, = 1, 2, 3

(4.4)

where n=

2 (2π )3

/ R3

f d 3k

is the electron density,

1/ i 3 is the average electron velocity, 3 fv d k n R / 1 3 W = (2π2 )3 is the average electron energy, 3 E(k)f d k n R 1/ i 3 S i = (2π2 )3 is the energy flux, 3 f v E(k)d k n R

 1/ P i = (2π2 )3 is the average crystal momentum, 3 f hk ¯ i d 3 k=m∗ V i +2αS i n R 1/ i U ij = (2π2 )3 is the flow of crystal momentum, 3 f v hk ¯ j d3k n R 1/ 1 ∂ f (Evi )d 3 k, Gij = (2π2 )3 3 n R h¯ ∂kj 1/ i j 3 F ij = (2π2 )3 is the flux of energy flux, 3 f v v E(k)d k n R 1/ CPi = (2π2 )3 is the production of the crystal momentum, 3 C [f ]hk ¯ i d 3k n R / 1 3 CW = (2π2 )3 is the production of the energy, 3 C [f ]E(k)d k n R / i = 2 1 CW C [f ]v i E(k)d 3 k is the production of the energy flux. (2π )3 n R3

Vi =

2 (2π )3

The moment equations do not constitute a set of closed relations. We will use MEP for obtaining the needed constitutive relations [7, 173]. If we assume as fundamental variables n, V, W and S, which have a direct physical interpretation, the closure problem consists in expressing P i , U ij , F ij , i as functions of n, V i , Gij and the moments of the collision term CPi , CW and CW W and S i . It is important to stress that the role of the mean velocity V i here is radically different from that played in gas dynamics. In fact, for a simple gas the explicit dependence of fluxes on the velocity can be predicted by requiring galilean

4.1 Moment Equations and Closure Problem

71

invariance of the constitutive functions. Instead Eqs. (4.1)–(4.4) are not valid in an arbitrary galilean reference frame, but they hold only in a frame where the crystal is at rest (in the applications it may be considered as inertial and it is thus possible to neglect the inertial forces). Therefore V is the velocity relative to the crystal and the dependence on it in the constitutive functions cannot be removed by a galilean transformation. If n, V, W and S are assumed as fundamental variables, the vector of the Lagrange multipliers reads Λ = (λ, λ, λW , λW ) where the components of Λ represent the Lagrange multipliers relative to the density n, to the velocity V, to the energy W and to the energy flux S respectively. In order to simplify the notation we have absorbed the factor 1/KB into the definition of the Lagrangian multipliers and denoted them with the same symbol. In this case the maximum entropy distribution function reads    fME = exp − λ + λW E + λ · v + λW · vE .

(4.5)

Recalling that in the approximation of the Kane dispersion relation ,  v := |v| =

2E (1 + αE) m∗ (1 + 2αE)2

and for E  1 1 v∼ √ := v∞ , 2αm∗ in order to have integrability for fME we require that for E  1 0 < λW + λW · v ∼ λW + λW · l v∞ , l being an arbitrary vector of the unit sphere of R3 . Therefore, by taking into account that λW > 0, we get the following sufficient condition of realizability in agreement with the general results of the previous chapter v∞

| λW | −1 0 

∞

x ν−1 exp(−ax)dx =

0

1 Γ (ν), aν

with Γ (ν) the special Gamma function, that satisfies for positive integers p   √ 1 π Γ p+ = p (2p − 1)!!. 2 2 Concerning the Lagrange multipliers one has λ = − log 

λW =

3 , 2W

n 4 ∗ 3 πm W

3/2 ,

(4.46)

(4.47)

82

4 Application of MEP to Silicon

9m∗ 21m∗ V+ S, 4W 4W 2 9m∗ 27m∗ = V − S. 4W 2 20W 3

λ=− λW

(4.48) (4.49)

The distribution function given by the maximum entropy principle in this case reads    3 2  21m∗ n exp(−λW E) 9m∗ 27m∗ 9m∗ f ME(P ) ≈  S · v + E − V + S ·v V− 3/2 1 + 4W 4W 2 4W 2 20W 3 4 πm∗ W 3

(4.50) and the constitutive equations become 2 W δij , 3 10 2 W δij , = 9  1

= ∗ Uij + W δij . m

Uij(P ) = m∗ Fij(P ) ) G(P ij

(4.51) (4.52) (4.53)

Analytical expression can be also obtained for the production terms in the parabolic limit. For the acoustic phonon scattering one finds (ac) a11 (ac)

a12

(ac)

√ 3/2  2πKac ∗ 3/2 2 W (m ) , 3 h¯ 3 √ 5/2  2πKac ∗ 3/2 2 W = 32 (m ) , 3 h¯ 3 32 = 3

(ac) a12 , m∗ √ 7/2  2πm∗ Kac 2 W = 128 . 3 h¯ 3

a21 = (ac)

a22

(4.54) (4.55) (4.56) (4.57)

Concerning the non polar optical phonon scattering, if one introduces the modified Bessel function of second kind √  π(z/2)ν ∞  Kν =  exp(−z cosh t) sinh2ν t dt, z, ν > 0, (4.58) Γ ν + 12 0

4.6 Application to Bulk Silicon

83

after some algebra one obtains up to the first order the following expressions −1/2 √   2 2π(m∗ )3/2 (h¯ ωnp )2 2 1 1 ±ζ = Knp nB + ∓ W e × 3 2 2 h¯ 3     h¯ ωnp ∓ 2ζ − 1 [K2 (ζ ) ∓ K1 (ζ )] , (4.59) exp ± KB TL  −1/2 √   2π(m∗ )3/2 (h¯ ωnp )2 4 2 1 1 ±ζ W e × = Knp nB + ∓ 3 3 2 2 h¯ 3 

CW

(np)

a11

(np)

a12

(np)

a21

(np) a22

[K2 (ζ ) ∓ K1 (ζ )] , 4 √   2π(m∗ )3/2 (h¯ ωnp )2 4 2 1 1 ±ζ W = Knp nB + ∓ e × 3 3 2 2 h¯ 3

=

(4.60)

{3K2 (ζ ) + 2ζ [K1 (ζ ) ∓ K2 (ζ )]} ,

(4.61)

(np) a12 , m∗

(4.62)

 3/2 √   2πm∗ (h¯ ωnp )2 4 2 1 1 ±ζ W ∓ e × n = K + np B 3 3 2 2 h¯ 3      K2 (ζ ) 12 ∓ 9ζ + 4ζ 2 + K1 (ζ ) 3ζ ∓ 4ζ 2 ,

(4.63)

with ζ = 3h¯ ωnp /(4W ). In a similar way, one can get the expression of the contribution to the production terms relative to the scattering with impurities.

4.6 Application to Bulk Silicon The physical situation is represented by a silicon semiconductor with a uniform doping concentration which we assume sufficiently low so that the scatterings with impurities can be neglected (Fig. 4.1) (see also [8, 11, 14]). The values of the physical parameters are reported in Tables 4.1 and 4.2. On account of the symmetry with respect to translations, the solution does not depend on the spatial variables. The continuity equation gives n = constant and from the Poisson equation one finds that E is also constant. Therefore the remaining balance equations reduce to the following set of ODEs for the motion along the

84

4 Application of MEP to Silicon E contact

contact

Semiconductor

Vbias

Fig. 4.1 A simple device with uniform doping Table 4.1 Values of the physical parameters me m∗ TL ρ vs Ξd r rO 0

9.1095 × 10−28 g 0.32 me 300 K 2.33 g/cm3 9.18 × 105 cm/s 9 eV 11.7 3.9 8.85 × 10−18 C/V µm

Electron rest mass Effective electron mass Lattice temperature Density Longitudinal sound speed Acoustic-phonon deformation potential Si relative dielectric constant SiO2 relative dielectric constant Vacuum dieletric constant

Table 4.2 Coupling constants and phonon energies for the inelastic scatterings in silicon

A 1 2 3 4 5 6

Zf 1 1 4 4 1 4

h¯ ω (meV) 12 18.5 19.0 47.4 61.2 59.0

Dt K(108 eV/cm) 0.5 0.8 0.3 2.0 11 2.0

direction of the electric field   c qE 2αqEG  c11 d 12 V|| = − ∗ + + − 2αc21 V|| + − 2αc22 S|| , (4.64) ∗ ∗ ∗ dt m m m m d W = −qV|| · E + CW , (4.65) dt d S|| = −qE G + c21 V|| + c22 S|| , (4.66) dt

4.6 Application to Bulk Silicon

85

where V|| and S|| are the component of V and S along the electric field. As initial conditions for (4.64)–(4.66) we take V(0) = 0,

(4.67)

3 KB TL , 2 S(0) = 0.

W (0) =

(4.68) (4.69)

The stationary regime is reached in a few picoseconds. The solutions of (4.64)–(4.66) for several values of the electric field are reported in Figs. 4.2 (velocity), 4.3 (energy) and 4.4 (energy flux). The typical phenomena of overshoot and saturation velocity are both qualitatively and quantitatively well described (see [194, Fig. 3.22] for a comparison with the results obtained by MC simulations). For the sake of completeness also the non parabolic band case has been integrated, Figs. 4.5, 4.6, and 4.7. The differences, especially in the energy, confirm the opinion that the parabolic band is an oversimplification of the real band structure.

4.5 4

7 velocity (10 cm/s)

3.5 3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1 1.2 time (ps)

1.4

1.6

1.8

2

Fig. 4.2 Velocity (cm/s) versus time (ps) for |E| = 10, 30, 50, 70, 100, 120, and 150 kV/cm

86

4 Application of MEP to Silicon 0.7

0.6

energy (eV)

0.5

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1 1.2 time (ps)

1.4

1.6

1.8

2

Fig. 4.3 Energy (eV) versus time (ps) for the same values of the of the electric field as in Fig. 4.2

1.4

7

energy flux (10 eV cm/s)

1.2

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1 1.2 time (ps)

1.4

1.6

1.8

2

Fig. 4.4 Energy flux (eV cm/s) versus time (ps) for the same values of the electric field as in Fig. 4.2

4.6 Application to Bulk Silicon

87 Electric field 30 kV/cm 2

0.8

1.8

0.7

1.6

0.6

1.4

velocity (107 cm/s)

7

velocity (10 cm/s)

Electric field 10 kV/cm 0.9

0.5 0.4 0.3

1.2 1 0.8

0.2

0.6

0.1

0.4

0

0

0.5

1 time (ps)

1.5

0.2

2

0

Electric field 50 kV/cm

0.5

1 time (ps)

1.5

2

Electric field 70 kV/cm

3

4.5 4

2.5

3

7

velocity (10 cm/s)

2

7

velocity (10 cm/s)

3.5

1.5

1

2.5 2 1.5

0.5

0

1

0

0.5

1 time (ps)

1.5

2

0.5

0

0.5

1 time (ps)

1.5

2

Fig. 4.5 Velocity (cm/s) versus time (ps) in the parabolic band case (dashed lines) and for the Kane dispersion relation

88

4 Application of MEP to Silicon Electric field 10 kV/cm

0.065

Electric field 30 kV/cm

0.18 0.16

0.06

0.14 energy (eV)

energy (eV)

0.055 0.05 0.045

0.12 0.1 0.08 0.06

0.04 0.035

0.04 0

1 1.5 time (ps)

2

0.02

Electric field 50 kV/cm

0.35 0.3

0.6

0.25

0.5

0.2 0.15

0.05

0.1

0.5

1 1.5 time (ps)

2

1 1.5 time (ps)

2

Electric field 70 kV/cm

0.3 0.2

0

0.5

0.4

0.1

0

0

0.7

energy (eV)

energy (eV)

0.5

0

0

0.5

1

1.5

2

time (ps)

Fig. 4.6 Energy (eV) versus electric field (kV/cm) in the parabolic band case (dashed lines) and for the Kane dispersion relation

4.7 The Energy Transport Limit Model Other macroscopic models, simpler than the hydrodynamic models but more accurate than the drift-diffusion ones [136], are constituted by the energy transport (ET) models, which are based on the balance equations for density and energy. The ET

4.7 The Energy Transport Limit Model

89

Electric field 10 kV/cm

Electric field 30 kV/cm

0.07

0.25

0.06

energy flux (10 eV cm/s)

0.05 0.04

0.15

7

energy flux (107 eV cm/s)

0.2

0.03 0.02

0.1

0.05 0.01 0

0

0.5

1 time (ps)

1.5

0

2

0

Electric field 50 kV/cm

0.5

1 time (ps)

1.5

2

Electric field 70 kV/cm

0.7

1 0.9

0.6

energy flux (107 eV cm/s)

7

energy flux (10 eV cm/s)

0.8 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4 0.3 0.2

0.1 0

0.1 0

0.5

1 time (ps)

1.5

2

0

0

0.5

1 time (ps)

1.5

2

Fig. 4.7 Energy flux (eV cm/s) versus electric field (kV/cm) in the parabolic band case (dashed lines) and for the Kane dispersion relation

models currently employed in the device simulation, have been derived in [51, 128] in a semi heuristic way. A more formal derivation has been proposed starting on the spherical harmonic expansion (SHE) mode l (see [24, 26] and references therein) under the basic assumption that the dominant scattering mechanism is the electron-

90

4 Application of MEP to Silicon

electron one. However this scaling has a rather dubious physical validity since in the situations encountered in the applications the electron density is instead very low so that the degeneracy effects and consistently the electron-electron interactions are usually neglected in almost all the electron device simulations even if Monte Carlo techniques are employed [76]. Here we derive from the hydrodynamic model presented in the previous section an energy transport model as asymptotic limit under physically more appropriate scaling assumptions. As further limit of vanishing energy relaxation time a driftdiffusion model is recovered as well. In the last section the energy-transport formulation of the MEP model is cast in the framework of linear irreversible thermodynamics. We assume that the following scaling holds (for the sake of simplicity in the notation we continue to denote scaled and original variables with the same symbols)  t =O

1 δ2

 , xi = O

    1 1 , V = O (δ) , S = O (δ) , τW = O , (4.70) δ δ2

where τW is the energy relaxation time, defined as CW = −

W − W0 τW

with W0 energy at equilibrium. The symbol O(·) represents, as usual, a quantity of order (·). Relation (4.70)1 means a long time scaling, while (4.70)2 indicates a diffusion approximation. The assumptions (4.70)3 and (4.70)4 are consistent with the small anisotropy condition in the derivation of the fME , while the relation (4.70)5 means that the energy must relax to equilibrium slower than the velocity and energy flux. The following proposition gives the ET limiting model deduced from the hydrodynamic model (4.1)–(4.4). Theorem 4.7.1 Under the scaling assumption (4.70), for smooth solutions, the following compatibility conditions arise from system (4.1)–(4.4) ∂n ∂(nV i ) + = 0, ∂t ∂x i

(4.71)

∂(nW ) ∂(nS j ) + + nqVk E k = nCW , ∂t ∂x j

(4.72)

with ∂ ∂ ∂ log n + D12 (W ) i W + D13 (W ) i V , i ∂x ∂x ∂x ∂ ∂ ∂ S i = D21 (W ) i log n + D22 (W ) i W + D23 (W ) i V , ∂x ∂x ∂x

V i = D11 (W )

(4.73) (4.74)

4.7 The Energy Transport Limit Model

91

where V is the elecric potential. The elements of the diffusion matrix D = (Dij ) are given by D11 =

c22 U − c12 F , c11 c22 − c12 c21

D12 =

c22 U  − c12 F  , c11 c22 − c12 c21

D13 = −q

c22 − c12 G , c11 c22 − c12 c21 (4.75)

D21 =

c11 F − c21 U , c11 c22 − c12 c21

D22 =

F

U

c11 − c21 , c11 c22 − c12 c21

D23 = q

c21 − c11 G . c11 c22 − c12 c21 (4.76)

The prime denotes derivative with respect to W . Proof We shall procede in a formal way. The assumption (4.70)2 and the relation between the electric field and the electric potential imply E = O(δ). By including i the scaling (4.70) into the system (4.1)–(4.4) and observing that CPi and CW are of order δ, we have 

 ∂n ∂(nV i ) + = 0, ∂t ∂x i   ∂(nU ij ) ∂(nP i ) i i +δ + nqE − nC δ3 P = 0, ∂t ∂x j   ∂(nW ) ∂(nS j ) k + δ2 + nqV E − nC k W = 0, ∂t ∂x j   i ∂(nF ij ) 2 ∂(nS ) ij i +δ δ + nqEj G − nCW = 0, ∂t ∂x j

δ2

and by putting equal to zero the coefficients of the various powers of δ in the previous system, one gets the balance equations (4.71) and (4.72) of density and energy and moreover ∂ ∂ nV i = 0, nS i = 0, ∂t ∂t 1 ∂ nU = −qE i + c11 V i + c12 S i , n ∂x i 1 ∂ nF = −qE i G(0) + c21 V i + c22 S i . n ∂x i The last two relations allow us to express V and S as functions of n, W and V and give (4.73) and (4.74) by a simple computation.  Remark 4.7.1 In the stationary case the original model (4.1)–(4.4) and the limiting ET one (4.71) and (4.74) are equivalent, at least for smooth solutions

92

4 Application of MEP to Silicon

Remark 4.7.2 At variance with the above quoted references on the ET models [24, 26, 51, 128], we use W as variable and not the temperature. In fact the definition of a nonequibrium temperature in the context of non equilibrium thermodynamics is itself a controversial question. For the problem of electron transport in the parabolic band approximation, the analogy with the monatomic gas gives an indication about the definition of temperature. When the Kane dispersion relation is employed there is no such an analogy and the introduction of the concept of nonequilibrium temperature becomes very questionable, mainly for the physical interpretation when comparisons with measurements are made. From a mathematical point of view the two balance equations of the resulting ET model for the density and the energy constitute a parabolic system of PDEs. In fact in Fig. 4.8 we have plotted the eigenvalues of the matrix Dˆ =



D11 D12 D21 D22

 .

for values of W of interest in the applications. In the considered range of energies, the matrix Dˆ is negative definite. In the stationary case the two balance equations (4.71) and (4.72) supplemented by the Poisson equation form an elliptic system of PDEs.

0 −0.005

eigenvalues (eV/ps)

−0.01 −0.015 −0.02 −0.025 −0.03 −0.035 0

0.1

0.2

0.3 0.4 energy (eV)

0.5

0.6

0.7

Fig. 4.8 The eigenvalues of the matrix Dˆ versus the energy W (eV) in the parabolic case (dashed lines) and for the Kane dispersion relation (continuous lines)

4.8 The Drift-Diffusion Limiting Model

93

4.8 The Drift-Diffusion Limiting Model If we consider a short time scaling of the energy relaxation time, the formal limit for τW → 0 in (4.71)–(4.72) gives W = W0 and one has the unipolar drift-diffusion model ∂n + div(nV) = 0, ∂t J = nV = D11 (W0 )∇n + nD13 (W0 )∇V .

(4.77) (4.78)

By comparing (4.78) with the expression of J typical of the drift-diffusion model J = −Dn ∇n + μn0 n∇V , one can identify the diffusivity coefficient Dn and the low field mobility μn0 as Dn = −D11 (W0 ),

μn0 = D13 (W0 ).

(4.79)

In order to study the main physical properties of the limiting drift-diffusion model, we separately treat the case of parabolic band and Kane’s dispersion relation.

4.8.1 Parabolic Band Case In this subsection the energy band is assumed to be described by the parabolic approximation. For the low field mobility at room temperature of 300 K by a direct calculation the value μn0 = 1182 cm2 /s V has been found while, in order to get the high-field mobility, we have performed a numerical integration in the stationary homogenous case with a constant electric field along a fixed direction, by using the same physical parameters as in [173, 174] and including the scatterings of electrons with acoustic and non-polar optical phonons. In Fig. 4.9 the relation between the electric field and the velocity is shown. In particular for the saturation velocity the value vs = 1.1810×107 cm/s has been derived. Since in the considered case the motion is along the direction of the electric field, one gets the high-field mobility by the relation μ =| V | / | E | which has been plotted in Fig. 4.10. In the latter figure for comparison we have also reported the mobility given by the Caughey–Thomas formula [89, 185]  μn = μn0 1 +



μn0 | E | vs

2 −1/2

94

4 Application of MEP to Silicon 1.4

1.2

velocity (107 cm/s)

1

0.8

0.6

0.4

0.2

0

0

10

20

30 40 50 electric field (kV/cm)

60

70

80

Fig. 4.9 Velocity (107 cm/s) versus electric field (kV/cm) in the parabolic band approximation

1200

mobility (cm2 /s V)

1000

800

600

400

200

0

0

10

20

30 40 50 electric field (kV/cm)

60

70

80

Fig. 4.10 Mobility (cm2 /s V) versus electric field (kV/cm) in the parabolic band approximation. ‘asterisk’ the model obtained in this paper, ‘open circle’ the Caughey–Thomas formula

4.8 The Drift-Diffusion Limiting Model

95

1200

N =1014/cm3

mobility (cm2/ s V )

1000

+

800

600 N =1017/cm3 +

400

200

0

0

10

20

40 30 electric field (kV/ cm)

50

60

70

Fig. 4.11 Mobility (cm2 /s V) versus electric field (kV/cm) in the parabolic band approximation for different impurity concentrations

where the previous values of μn0 and vs are used for consistency. The two models of mobilities are very close for high fields, but there is a significant difference of about 20% for moderate fields. In order to take into account the influence of the impurities, in Fig. 4.11 the mobility for an impurity concentration of 1014 cm−3 and 1017 cm−3 is plotted versus the electric field. As can be seen at low concentrations of impurities the results are essentially the same as for bulk silicon, but at higher concentrations the mobility, at low field, is significantly lower. Now we pass to investigate the validity of the Einstein relation Dn = μn

KB T , q

(4.80)

which allows us to express the diffusion coefficient in term of the electron mobility. The Einstein relation (4.80) is justified in statistical mechanics only at thermal equilibrium and its use in the current drift-diffusion models has not a theoretical rationale. In the parabolic approximation the same equation of state for the monatomic gas holds. Therefore, if we neglect the quadratic terms in the velocity, W = 3/2KB T and (4.80) can be rewritten as Dn = μn

2W , 3q

(4.81)

96

4 Application of MEP to Silicon

which is verified in our model if and only if c22 (W )U (W ) − c12 (W )F (W ) =

2 W [c22 (W ) − c12 (W )G(W )] . 3

By using the expressions of U , F ,G one verifies that the previous relation is indeed, up to first order in the velocity, an identity for each W . Therefore in the parabolic band case up to first order in the velocity the Einstein relation is verified for any energy W and this in turn implies that (4.80) is valid for any electric field.

4.8.2 Kane’s Dispersion Relation Now we pass to consider the Kane dispersion relation case . By a numerical integration the relation between the velocity and electric field has been obtained and plotted in Fig. 4.12. In this case we find a saturation velocity vs = 1.0150 × 107 cm/sec and a low-field mobility μn0 = 1054 cm2 /s V. The corresponding high-field mobility is shown in Fig. 4.13 as function of the energy and in Fig. 4.14 versus the electric field. It is compared again with the Caughey- Thomas formula, by taking the same values of saturation velocity and low-field mobility. The qualitative behavior is as in the parabolic case.

1.4

1.2

velocity (107 cm/s)

1

0.8

0.6

0.4

0.2

0

0

10

20

30 40 electric field (kV/ cm)

50

Fig. 4.12 Velocity versus electric field in the Kane dispersion relation case

60

70

4.8 The Drift-Diffusion Limiting Model

97

1200

mobility (cm2 /s V)

1000

800

600

400

200

0

0

0.1

0.2

0.3 0.4 enegy (eV)

0.5

0.6

0.7

Fig. 4.13 Mobility versus energy (eV) in the Kane dispersion relation case

1200

mobility (cm2/s V)

1000

800

600

400

200

0

0

5

10 electric field (kV /cm)

Fig. 4.14 Mobility versus electric field in the Kane dispersion relation case

15

98

4 Application of MEP to Silicon 1100 1000

2 mobility (cm / s V )

900 N+=1014/cm3

800 700 600 500

17

3

N =10 /cm

400

+

300 200 100

0

10

20

30 40 electric field (kV/ cm)

50

60

70

Fig. 4.15 Mobility versus electric field in the Kane dispersion relation case for different impurities concentrations

The influence of impurity concentration on the mobility is analyzed by considering different impurity concentrations as shown in Fig. 4.15. Concerning the validity of the Einstein relation some theoretical problem arises. In the case of the Kane dispersion relation one cannot simply take the same equation of state as in the parabolic case. For the appropriate generalization of (4.80) out of equilibrium one has to express the nonequilibrium temperature T in term of the energy Lagrange multiplier as KB T =

1 , λW

which represents the most theoretically sound definition of T [97, 153, 155]. Therefore the general formulation of the Einstein relation looks D11 = −

D13 , qλW

(4.82)

which in our model reads c22 (W )U (W ) − c12 (W )F (W ) =

1 [c22 (W0 ) − c12 (W0 )G(W0 )] . (4.83) λW

4.9 Formulation of the Model in the Framework of Linear Irreversible. . .

99

At variance with the parabolic case the expressions of U, F, G and the coefficients cij are not known in an analytical explicit form. However it is still possible to show that also in the Kane case (4.83) is an identity. The proof is a trivial consequence of the following properties Theorem 4.8.1 U , F and G are related by U=

1 , λW

F =

1 G. λW

(4.84)

The previous relations can be proved by a simple integration by part. We can therefore affirm that up to first order in the velocity the Einstein relation is valid for any value of the electric field also in the Kane case.

4.9 Formulation of the Model in the Framework of Linear Irreversible Thermodynamics Let us start briefly recalling the basic formulation of linear irreversible thermodynamics. Let sn = sn (un , n) be the electron entropy density (where un is the electron energy density and n the electron density) and Tn the electron temperature (the lattice and the electron system are assumed to be at different temperature). The first principle of thermodynamics gives Tn dsn = dun − νn dn

(4.85)

where νn can be interpreted as the chemical potential. For the next considerations, it is convenient to introduce the electrochemical potential φˆ n by the following definition φˆn = −νn + qV

(4.86)

In standard irreversible thermodynamics [69] the entropy fluxes JLs of the lattice and of the electron system Jns are related to the particle and energy fluxes by the following relations: JLs =

JLu , TL

Jns =

Jnu − νn Jn , Tn

(4.87)

where JLu and Jnu are the lattice energy flux density and electron energy flux density nS, Jn = nV is the electron particle flux, TL and Tn are the lattice and electron gas temperature respectively. Now, according to the tenets of linear irreversible thermodynamics [69], the thermodynamic forces are linearly related to

100

4 Application of MEP to Silicon

the thermodynamic fluxes as follows: Jn =

L11 1 ∇ φˆ n + L12 ∇ , Tn Tn

(4.88)

Tn Jns =

L21 ˆ 1 ∇ φn + L22 ∇ , Tn Tn

(4.89)

where the matrix Lij is symmetric if, and only if, the Onsager reciprocity relations hold. In terms of electron energy flux Jnu one rewrites the above equations as L11 1 ∇ φˆ n + L12 ∇ , Tn Tn

 L L 1 11 21 + Jnu = νn ∇ φˆn + (νn L12 + L22 )∇ . Tn Tn Tn Jn =

(4.90) (4.91)

Now, for the electron system, described by a probability density function f , we recall that the entropy density in the Maxwellian limit of the Fermi–Dirac statistics is given by sn = −KB



2 (2π)3

(f log f − f )d 3 k,

(4.92)

B

where k is the electron wave-vector which belongs to the first Brillouin zone. This latter both in the parabolic and non-parabolic band approximation is extended to all R3 . By introducing the function η(f ) = −KB

2 (f log f − f ), (2π)3

one can write the electronic entropy sn as  sn =

η(f )d 3 k.

(4.93)

B

As shown, to the first order the maximum entropy distribution function is   (0) fME ≈ exp −λ − λW E (1 − χ) = fME (1 − χ) ,

(4.94)

where λ, λW are the Langrange multipliers relative to the density and energy and

 (0) fME = exp −λ − λW E . It easily follows that η(fME ) = KB

   2 W λ + λ 1 + − χ) f E + o(χ), (1 ME (2π)3

(4.95)

4.9 Formulation of the Model in the Framework of Linear Irreversible. . .

101

wherefrom dη = −KB

      2 (0) fME λ + EλW (1 − χ) + χ (dλ + EdλW ) + λ + EλW dχ . 3 (2π)

(4.96) Consequently, for the differential of the entropy density, dsn , we have  dηd 3 k = −KB λ ndλ − KB d(λλW )un − KB ZλW dλW ,

dsn =

(4.97)

B

where n is the electron particle density n=

2 (2π)3

 fME d 3 k = B

2 (2π)3

 B

(0) 3 fME d k,

(4.98)

un is the electron energy density un = nW =

2 (2π)3

 EfME d 3 k = B

2 (2π)3

 B

(0) 3 EfME d k,

and Z is the quantity 2 Z= (2π)3



2 E fME d k = (2π)3 B 2



3

B

(0)

E 2 fME d 3 k.

By taking the differential of relation (4.98), we have dn = −n dλ − un dλW .

(4.99)

Likewise, from the definition of un one gets dun = −un dλ − Z dλW .

(4.100)

Now, from (4.97), (4.99), and (4.100), one easily obtains dsn = KB λdn + KB λW dun

(4.101)

which has the same form as Eq. (4.85) provided the following identifications are made λW =

1 , KB Tn

λ=−

νn . KB Tn

(4.102)

102

4 Application of MEP to Silicon

Therefore, we have proved that the expression for the entropy differential as obtained from the maximum entropy ansatz can be written in the form of linear irreversible thermodynamics. un Now, it is easy to see that, in terms of W = , the energy per particle, we can n write Eq. (4.99) as dn = −dλ − W dλW . n

(4.103)

From the relation 4.19 we obtain λ = − log n − M(W ),

(4.104)

with M(W ) = − log

π 2 h¯ 3

, √ m∗ 2m∗ d0 (W )

and the substitution into the Eq. (4.102)2, yields νn = KB Tn log n + KB Tn M,

W dTn . dνn = KB Tn d log n + KB log n + KB M − Tn

(4.105) (4.106)

By substituting (4.105), (4.106) and (4.86) into Eqs. (4.90) and (4.91) one gets     qL11 L11 W L12 Jn = −L11 KB ∇ log n + ∇V − KB log n + KB M − + 2 ∇Tn , Tn Tn Tn Tn q (4.107) Jnu = −(νn L11 + L21 )KB ∇ log n + (νn L11 + L21 ) ∇V − Tn      νn L11 + L21 νn L12 + L22 W KB log n + KB M − + ∇Tn . Tn Tn Tn2

(4.108) By comparing these last two equations with the maximum entropy derived constitutive equations for the energy transport model (4.73) and (4.74) we have the following identification KB L11 qL11 , D13 = , n nTn     1 L11 W L12 dTn , =− KB log n + KB M − + 2 n Tn Tn Tn dW

D11 = −

(4.109)

D12

(4.110)

4.9 Formulation of the Model in the Framework of Linear Irreversible. . .

103

KB (νn L11 + L21 ) νn L11 + L21 , D23 = q , (4.111) n nTn     1 νn L11 + L21 W νn L12 + L22 dTn , =− KB log n + KB M − + n Tn Tn Tn2 dW

D21 = − D22

(4.112) which can be considered as an overdetermined linear system in the unknowns Lij . The compatibility conditions for such a system are D11 = −D13

KB Tn , q

(4.113)

D21 = −D23

KB Tn , q

(4.114)

The condition (4.113) represents the Einstein relation, while (4.114) is a sort of generalized Einstein relation linking, in the expression (4.74) for S, the diffusion coefficient with the drift one. By using (4.84) it is simple matter to prove that for the MEP model these conditions are indeed two identities. This assures that there exists a unique solution to (4.109)–(4.112). After some simple algebra, one gets L11 = −

nD11 , KB

(4.115)

nD12 nD11 + (νn − W ) , W KB dλ KB dW nD21 nD11 =− + νn , KB KB

L12 =

(4.116)

L21

(4.117)

L22 =

nD22 nD11 + νn (νn − W ) − νn L12 − L21 (νn − W ) , W KB dλ KB dW

where we have used the following results log n + M = νn λW ,

1 = λW . KB Tn

Note that the Onsager conditions are no longer valid outside equilibrium.

(4.118)

104

4 Application of MEP to Silicon

At this point we rewrite the density in terms of the so-called electron Fermi quasiνn level3 ϕn = − V [184] q

V + ϕn  n = Nc (Tn ) exp q , KB Tn

Nc = 2

2πKB m∗ Tn  3 2

h¯ 2

,

and we want to transform, in the stationary case, the system (4.71)–(4.71), (1.27) in the following form −div Jn = 0, −div JTn + n

W − W0 = 0, τW

div D = q(ND − n),     ϕn 1 + A12 ∇ − , Jn = A11 ∇ Tn Tn     ϕn 1 T Jn = A21 ∇ + A22∇ − , Tn Tn

(4.119) (4.120) (4.121) (4.122) (4.123)

where Jn is the electron current −qnV, D is the electric displacement vector and JTn = −Jun − V Jn = −nS + nqV V. The generation-recombination effects have been neglected because the unipolar case is considered. In order to get the sought transformation we need to find the analytical expression of the coefficients Aij as functions of the Dij ’s. Let us start by finding the coefficients A11 and A12 . First of all, we note that   Jn = −qnV = −qn D11 ∇ log n + D12 ∇W + D13 ∇V , (4.124)   1 ∇Tn = −∇ (4.125) Tn2 , Tn     V 1 ∇V =∇ (4.126) −V∇ , Tn Tn Tn        1 ϕn + V 1 V + φn 3 ∇ log n = ∇n = ∇ Nc (Tn ) exp q ∇Tn . = ∇ q + n n KB T n KB T n 2Tn

(4.127)

3 The

electron Fermi quasi-level is defined up to an additive constant. We assume that

 n NC exp q VK+ϕ is equal to the intrinsic concentration ni . B Tn

4.10 A Numerical Approach Based on Finite Elements

105

Now, comparing Eqs. (4.124)–(4.127) with Eq. (4.122) and by taking ino account the compatibility conditions (4.113) and (4.114), we get4 ⎛ A11 = −q 2

nD11 , KB

A12 = q 2

⎞

⎟ nD11 qn ⎜ ⎜D11 3 − D12 ⎟ . V − ⎝ W W KB KB 2λ dλ ⎠ dW

(4.128)

Likewise, we obtain A21 = q

2 nD11

KB

nD21 V −q , KB

A22

nD21 = KB

 qV −

3 2λW

 +

nD22 − A12 . dλW KB dW (4.129)

Note that the matrix A is not symmetric.

4.10 A Numerical Approach Based on Finite Elements The MEP energy-transport model given as in (4.119)–(4.123) has a similar structure of the family of models studied in [150]. They are characterized by a diffusion matrix A, which depends on parameters β, c, χ, γ , explicitly given by 

A11 A22

 KB Tn γ − ϕn + χ , A21 = A12 (4.130) = nμn eTn , A12 = nμn eTn β e 2     KB Tn 2 KB Tn γ − ϕn + χ + (β − c) = nμn eTn β (4.131) e e

By an appropriate choice of the parameters, the Stratton energy-transport model or the Degond energy-transport model can be easily recovered as well as some simplified hydrodynamic models, for example the spherical harmonic expansion (SHE) [26, 191]. In [15] the numerical scheme developed in [149] has been adapted in order to take into account the expression of the diffusion matrix for the MEP model. Numerical approach for the hyperbolic model can be found in [10, 13]. Let us recall some key features of such a numerical scheme. • Mixed finite element approximation (the classical Raviart–Thomas, RT0 , is used for space discretization, see [42, 137, 149] for more details).

the parabolic approximation 2λ3W = W . In the case of the Kane dispersion relation it can be used as a reasonable approximation.

4 In

106

4 Application of MEP to Silicon

• Operator-splitting techniques are used for solving saddle point problems arising from mixed finite elements formulation [84]. • Implicit scheme (backward Euler) is used for time discretization of the artificial transient problems generated by operator splitting techniques. • A block-relaxation technique, at each time step, is implemented in order to reduce as much as possible the size of the successive problems we have to solve, by keeping at the same time a large amount of the implicit character of the scheme. • Each non-linear problem coming from relaxation technique is solved via the Newton–Raphson method. Concerning the block relaxation technique, three main steps have to be considered • A step related to the Poisson equation for the computation of V k+1 and Dk+1 with the other unknowns, frozen at the last known values, i.e. ϕnk , Jkn , Tnk , JkTn . k+1 , Jk+1 are computed • A second step in which the variables ϕnk+1 , Jk+1 n , Tn Tn simultaneously. The reasons of such a procedure are explained in [149] and are essentially due to the strong coupling between the continuity equation and the energy balance one. Let us detail, now, the problem we have to solve during the second step of the relaxation process. Ω will represent the integration domain with boundary ΓN ∪ΓD , with ΓN and ΓD the parts of boundary where Newmann and Dirichlet conditions are imposed. • We have to find  k+1  ϕnk+1 U1 k+1 Tn : Ω −→ R2 = U = U2k+1 − T1n and  Jk+1 =

Jk+1 n JTn



 =

Jk+1 1 Jk+1 2



 2 : Ω −→ Rd ,

where d is the space dimension, such that5 : Sijk (x)

Ujk+1 − Ujk Δt

− divJk+1 + Fi (x, Uk+1 ) = 0, i

i = 1, 2,

Jk+1 = Aij (x, Uk+1 ) ∇Ujk+1 , i = 1, 2. i

5 We

recall that a summation must be intended with respect to repeated dummy indices.

(4.132)

4.10 A Numerical Approach Based on Finite Elements

107

with the boundary conditions: Uik+1 = Gi (x) on ΓD

and

Jk+1 · ν = 0 on ΓN , i = 1, 2. i

(4.133)

Sijk is a positive definite matrix which may be chosen (and adapted) by the user to create a false transient, and ν represents the unit external normal to Ω in ΓN , while the dot means the standard scalar product. Gi (x) is a given function. If other unknowns of the problem appear, for example in Fi and Aij , they are considered at the previous step k. The dual mixed variational formulation, which allows us the mixed finite element approach, needs the introduction of the following functional spaces H (div) = {ω | ω ∈ (L2 (Ω))2 , div(ω) ∈ L2 (Ω)}, V0 = {ω | ω ∈ H (div), ω · n = 0 on ΓN }, and is given by the following steps.   2

Let us find Uk+1 , Jk+1 ∈ L2 (Ω) × [V0 ]2 such that /

/

/ Ujk+1 −Ujk k V S (x) dx − Ω Vi div Jk+1 dx i i Ω / ij Δt + Ω Vi Fi (x, Uk+1 )dx = 0, ∀V ∈ [L2 (Ω)]2,

(4.134)

/ k+1 dx Bˆ ij (x, U k+1 ) Wi · Jk+1 j dx + Ω div Wi Ui / 2 − ΓD Gi (x) Wi · νdΓ = 0, ∀W ∈ [V0 ]

(4.135)

Ω

with ˆ B(x, Uk+1 ) = [A(x, Uk+1 )]−1 , and dΓD surface element on ΓD . The finite element formulation is obtained by replacing the spaces L2 (Ω) and V0 by finite dimensional subspaces generated over a triangulation Th of Ω and by using Raviart–Thomas RT0 elements, i.e. Lh = {vh ∈ L2 (Ω) | ∀K ∈ Th , vh |K = const}, 2    3 αK x V0h = ωh ∈ V0 | ∀K ∈ Th , ωh = + γK . βK y where αK , βK and γK are numerical coefficients [137]. Applying Newton– Raphson method to the discrete version of the system (4.134) and (4.135) and after the elimination of the primal variable U we obtain a sequence of linear problems

108

4 Application of MEP to Silicon

(denoted by the index l) for the dual variable Jl+1 only    Bˆ ij Wi · Jl+1 dx + Λpq div Wp + ∂Up (Bˆ ij ) Wi · Jlj div Jl+1 q dx j Ω Ω     l k k Ur − Ur = + Fq dx Λpq div Wp + ∂Up (Bˆ ij ) Wi · Jlj Sqr Δt Ω  − divWi Uil dx + Gi Wi · νdΓ = 0, ∀W ∈ [V0h ]2 .



Ω

ΓD

(4.136) Then the primal variable Ul+1 is recovered, element by element, by ⏐ ⏐ Uil+1 − Uil ⏐

K

= −Λij

 ⏐ ⏐ U l − Urk ⏐ , i = 1, 2, + Fj (U l ) − div Jl+1 Sjkr r j ⏐ Δt K (4.137)

where 



Λij (x, U ) = l



Sijk (x) Δt

−1 + ∂Uj (Fi (x, U )) l

.

(4.138)

In order to adapt the numerical scheme to the formulation of the diffusion matrix for the MEP model, we need first the inverse of the matrix A, Bˆ = A−1 , and then (due to the Newton method) the partial derivatives of the elements of B with respect ∂ Bˆ ij ϕn to the entropy variables (U1 = , U2 = − T1n ) i.e. , i, j, k = 1, 2. Tn ∂Uk In the models considered in [137], the Aij are given by very simple expressions, so it was relatively easy to find analytically the matrix Bˆ and its partial derivatives. In the present case the construction of the diffusion matrix A can be summarized as follows C −→ D −→ A which has to be completed by the step A −→ Bˆ = A−1 for algorithmic reasons. It is easy to recognize that it is extremely complicated to express analytically the ˆ However what we effectively need in the numerical process partial derivatives of B. is a way to evaluate a partial derivative of Bˆ ij for a given value of the unknown variables. This can be easily done by following the rules used in software devoted to automatic differentiation of (FORTRAN) codes like, for example, Odyssée [74]. For the MEP energy-transport model only in the parabolic case one can directly express cij as function of Tn . In the non-parabolic case the equation of state is

4.10 A Numerical Approach Based on Finite Elements

109

not explicitly known, i.e. an analytical relation between W and Tn is not available. Therefore, in the Kane case cij ,

CW ,

U,

F,

G,

λW

are considered as functions of the electron energy W . The derivatives of the previous quantities with respect to the entropic variables are then obtained by using the relation (4.102) between Tn and λW . In order to speed up the computation, cij , CW , U, F, G, λW have been evaluated in discrete points and approximated via cubic splines to recover C 1 functions.

4.10.1 Simulation of a n+ − n − n+ Silicon Diode As first problem the simulation of a ballistic n+ − n − n+ silicon diode is presented. The n+ regions are 0.1 μm long, while the channel has different lengths. Moreover several doping profiles, with abrupt junctions, are considered according to the Table 4.3. For comparison purposes, a direct Monte Carlo simulation has also been performed by employing the code (available on web) in [85]. The results for the MEP model along with the classical energy-transport models are collected in the Figs. 4.16, 4.17, 4.18, 4.19, 4.20, and 4.21. It is evident the superiority of the MEP model on the others in all the considered cases. For the nanoscale diode there is a loss of accuracy in the velocity but the MEP model has anyway a better performance. There are only slight differences between the Stratton model [191] and the variant proposed by Chen et al. [51]. The poorer accuracy is given by the SHE or Lyumkis et al model [24, 26] and the reduced hydrodynamic one [18]. Apart from the MEP model, the other ones give comparable values for the simulation of energy, but evident differences in the velocity that has a more pronounced peak near the second junction in the SHE and the reduced hydrodynamic models. The energy obtained with the MEP model is always in excellent agreement with the MC one, but that given by the other models is much higher. This is likely ascribed to the dubious relation between energy and temperature.

Table 4.3 Length of the channel, doping concentration and applied voltage in the test cases for the diode Test # 1 2 3

Channel length Lc (µm) 0.4 0.2 0.05

+ ND (×1017 cm−3 ) 5 10 10

ND (×1017 cm−3 ) 0.02 0.1 0.1

Vb (V) 2 1 0.6

110

4 Application of MEP to Silicon

x 107

2.5

Electron Velocity (cm/s) 2

1.5

1

0.5

0

micron −0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.16 Stationary solution for the electron velocity (cm/sec) in the diode with LC = 0.4 µm. The dots are the MC solution, the continuous line is the MEP model, the crosses and dashed line are the Stratton and Chen model respectively, the open circle with line is the SHE model, the dotted dashed line is the reduced hydrodynamic model 0.4

Electron Energy (eV) 0.35 0.3 0.25 0.2 0.15 0.1 0.05

micron 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.17 Stationary solution for the electron energy (eV) in the diode with LC = 0.4 µm. The notation is as in Fig. 4.16

4.10 A Numerical Approach Based on Finite Elements

2.5

111

x 107

Electron Velocity (cm/s) 2

1.5

1

0.5

0

micron −0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 4.18 Stationary solution for the electron velocity (cm/s) in the diode with LC = 0.2 µm. The notation is as in Fig. 4.16

0.35

Electron Energy (eV) 0.3

0.25

0.2

0.15

0.1

0.05

micron 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 4.19 Stationary solution for the electron energy (eV) in the diode with LC = 0.2 µm. The notation is as in Fig. 4.16

112

4 Application of MEP to Silicon

4

x 107

Electron Velocity (cm/s) 3.5 3 2.5 2 1.5 1 0.5 0

micron −0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 4.20 Stationary solution for the electron velocity (cm/s) in the diode with LC = 0.05 µm. The notation is as in Fig. 4.16

0.35

Electron Energy (eV) 0.3

0.25

0.2

0.15

0.1

0.05

micron 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 4.21 Stationary solution for the electron energy (eV) in the diode with LC = 0.05 µm. The notation is as in Fig. 4.16

4.10 A Numerical Approach Based on Finite Elements

113

4.10.2 Simulation of a 2D Silicon MESFET As second benchmark, the validity of the MEP energy-transport model and the efficiency of the numerical method have been checked by simulating a bidimensional Metal Semiconductor Field Effect Transistor (MESFET). The shape of the device is square as pictured in Fig. 4.22. The device has a 0.4 µm channel. The source and drain depths are 0.1 µm and the contact at the gate is 0.2 µm. The distance between the gate and the other two contacts is 0.1 µm. The lateral subdiffusion of the source and the drain region is about 0.05 µm. The following doping concentration is considered  3 × 1017cm−3 in the n+ regions ND (x) − NA (x) = in the n region 1017cm−3 with abrupt junctions. The reference frame has axes parallel to the edges of the device. The numerical domain representing the MESFET is Ω = [0, 0.6] × [0, 0.6]

0.6

source

0.55

n+

gate

drain n+

n

0.0

0.1

0.2

0.4

Fig. 4.22 Schematic representation of a bidimensional MESFET

0.5

0.6

114

4 Application of MEP to Silicon

where the unit length is the micron. The regions of high doping n+ are the subset [0, 0.1] × [0.55, 0.6] ∪ [0.5, 0.6] × [0.55, 0.6]. ΓD denotes the part of ∂Ω, the boundary of Ω, which represents the source, gate and drain ΓD = {(x, y) : y = 0.6, 0 ≤ x ≤ 0.1, 0.2 ≤ x ≤ 0.4, 0.5 ≤ x ≤ 0.6} . The other part of ∂Ω is labelled as ΓN . The boundary conditions are assigned as follows. We have ohmic contacts at source and drain: V = φint + Vapp ,

ϕn = −Vapp

(4.139)

At source Vapp = VS while at drain Vapp = VD . On the gate we have a Schottky contact V = φint + VB + Vapp ,

ϕn = −Vapp

(4.140)

where VB is the barrier potential modelling the Schottky contact [185] and Vapp = VG . On the source, drain and gate the built in potential φint is the solution of F (φint ) = q(n(φint ) − p(φint ) − ND + NA ) = 0 where the quasi-Fermi level is taken as ϕn = 0. The other boundary conditions are W = W0 ,

on ΓD ,

ν · ∇n = 0, ν · ∇W = 0, ν · ∇V = 0,

(4.141) on ΓN .

(4.142)

Here ∇ is the bidimensional gradient operator while ν is the unit outward normal vector to ∂Ω in the considered points. For the numerical simulation we start from the equilibrium state (Vapp = 0) and afterwords a desired bias point is reached via potential increments. First the path VS = 0,VG = 0 and VD = 0 to 1.6 V by steps of 0.2 V is followed and after, starting from the point VS = 0,VG = 0 and VD = 1.6, the gate potential VG is decreased from 0 to −5 V by steps of 0.2 V. When the gate potential is sufficiently low, no (“significant”) current flows from drain to source. As for the diode, the simulations with the other energy-transport models have been performed again. In Fig. 4.23 the comparison for the velocity is shown and in Fig. 4.24 that for the energy. Also in the simulation of the MESFET the MEP model presents a better accuracy. Note that close to the gate the depletion region has in the MC simulation practically no particles and this explains the results in the figures.

4.10 A Numerical Approach Based on Finite Elements

16

115

x 106

electron x−velocity (cm/s) 14 12 10 8 6 4 2 0

micron −2

0

2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x 107

electron x−velocity (cm/s) 1.5 1 0.5 0 −0.5 −1 −1.5

micron −2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.23 Stationary solution for the electron velocity (cm/s) in MESFET in the cross section y = 0.56 µm and y = 0.58 µm. The notation is as in Fig. 4.16

116

4 Application of MEP to Silicon 0.18 Electron Energy (eV) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 micron 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6

0.7

0.2 Electron Energy (eV) 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 micron 0

0

0.1

0.2

0.3

0.4

0.5

Fig. 4.24 Stationary solution for the electron energy (eV) in MESFET in the cross section y = 0.56 µm and y = 0.58 µm. The notation is as in Fig. 4.16

In Figs. 4.25 and 4.26 the characteristic curves have been analyzed and again the MEP model gives the results closer to the MC ones. The 2D results of the simulation with the MEP model are shown in Figs. 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, and 4.33.

4.10 A Numerical Approach Based on Finite Elements

117

4 Current (Ampere/cm) 3.5 3 2.5 2 1.5 1 0.5 Volt 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 4.25 Characteristic curve in MESFET. IS in Ampere/cm as function of VD at VG = −0.8 V. The notation is as in Fig. 4.16

4.5 current (Ampere/cm)

4 3.5 3 2.5 2 1.5 1 0.5

Volt 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 4.26 Characteristic curve in MESFET. IS in Ampere/cm as function of VD at VG = 0 V. The notation is as in Fig. 4.16

118

4 Application of MEP to Silicon Electron Density

17

x 10

3 2.5

1/cm3

2 1.5 1 0.5

0 2

0 7

4 6

5

4

3

2

1

8

0

cm

−5

x 10

−5

x 10

6

cm

Fig. 4.27 Stationary solution for the electron density (cm−3 ) in MESFET Electron Energy

0.22 0.2 0.18 0.16

eV

0.14 0.12 0.1 0.08 0.06 0

0.04 0.02 7

2 4 6

5

4

3

2

−5

1

0

8

x 10

cm

Fig. 4.28 Stationary solution for the electron energy (eV) in MESFET

−5

x 10

6 cm

4.10 A Numerical Approach Based on Finite Elements

119

Electron Velocity X Component 7

x 10 3

2

cm/s

1

0

−1

−2 0 0.5

−3 7

6

5

4

−4

3

2

1

0

x 10

1 cm

−5

x 10

cm

Fig. 4.29 Stationary solution for the x-component of the velocity (cm/s) in MESFET Electron Velocity Y Component

7

x 10 12 10 8

cm/s

6 4 2 0

0 2

−2 7

4 6

5

4

3

2

−5

x 10

−5

x 10

6 1

0

8 cm

cm

Fig. 4.30 Stationary solution for the y-component of the velocity (cm/s) in MESFET

120

4 Application of MEP to Silicon Electric Field X Component 5

x 10

4 3 2

Volt/cm

1 0 −1 −2 −3 −4

0 0.5

−5 7

6

5

4

3

2

1

0

−4

x 10

1 cm

−5

x 10

cm

Fig. 4.31 Stationary solution for the x-component of the electric field (V/cm) in MESFET Electric Field Y Component

5

x 10

7 6 5

Volt/cm

4 3 2 1 0

0 −1 7

2 4 6

5

4

3

2

−5

x 10

−5

x 10

6 1

0

8

cm

cm

Fig. 4.32 Stationary solution for the y-component of the electric field (V/cm) in MESFET

4.10 A Numerical Approach Based on Finite Elements

121

Electrostatic Potential

1

0.5

Volt

0

−0.5

−1 0 2 −1.5 7

4 6

5

−5

6 4

3

2

1

0

−5

x 10

x 10

8 cm

cm

Fig. 4.33 Stationary solution for the electric potential (V) in MESFET

4.10.3 Simulation of a 2D Silicon MOSFET In this section the performance of the MEP energy-transport model and the efficiency of the numerical method is studied by simulating a bidimensional Metal Oxide Semiconductor Field Effect Transistor (MOSFET). The shape of the device is pictured in Fig. 4.34. The device has a 0.2 µm channel. The source and drain depths are 0.1 µm and the contact at the gate is 0.15 µm. The distance between the gate and the other two contacts is 0.025 µm. The lateral subdiffusion of the source and the drain region is about 0.05 µm. The gate oxide is 0.15 µm long and 6 nm thick. The doping concentration is 2 ND (x) − NA (x) =

1017 cm−3 in the n+ regions −1014 cm−3 in the p region

with abrupt junctions. The contacts on the source, drain and gate along with the base contact on the bulk are assumed to be Ohmic. Homogeneous Neumann conditions are assumed on the remaining part of the boundary.

122

4 Application of MEP to Silicon

source 0.4

gate SiO2

n+

n+

0.4

p

0.0

0.4

Fig. 4.34 Schematic representation of a bidimensional MOSFET

If the axes of reference frame are chosen parallel to the edges of the device, the silicon part of the MOSFET is represented by the numerical domain [0, 0.4] × [0, 0.4] and at the top of the silicon part the silicon oxide domain is [0.125, 0.275] × [0.4, 0.406] where the length is the micron. The regions of high-doping n+ are the subset [0, 0.1] × [0.35, 0.4] ∪ [0.3, 0.4] × [0.35, 0.4]. A grid of 4644 elements has been used (see Fig. 4.35): 3344 in the bulk zone, 343 in the n+ source zone, 357 in the n+ drain zone and 600 in the oxide zone.

4.10 A Numerical Approach Based on Finite Elements

123

Fig. 4.35 Mesh used for the computation

In order to reach the desired bias, VD = 1.0 VS = 0 and VG = 0.5, we first compute the equilibrium state and then the previous point is reached by continuation on the applied potential. First, we go to Vd = 1.0 by steps of 0.1 V and after we go to Vg = 0.5 by two steps of 0.25 V. In order to estimate how fine the mesh should be, the source current for different meshes have been computed. The case of the parabolic approximation has been considered because the various coefficients can be evaluated analytically. The results are summarized in Table 4.4. There are not relevant differences among the results with meshes 3, 4 and 5. Therefore we use in the simulations the mesh 3 of the table because less time expensive. In the last column of Table 4.4 the results for the Kane case are also reported. As in the previous sections, we have compared the results obtained with the MEP and the other energy-transport models. In Figs. 4.38, 4.39, and 4.40 cross sections of the numerical solutions relative to the x and y component of velocity and energy are plotted and it is again clear the superiority of the MEP models. Also the characteristic curves, reported in Figs. 4.36 and 4.37, show that the MEP results are very close to the MC ones. The standard energy-transport models overstimate the drain current with an error of about 25%, while the error for the MEP model is always less than about 13%. The 2D solution for the MEP model is plotted in Figs. 4.38, 4.39, 4.40, 4.41, 4.42, 4.43, 4.44, and 4.45.

124

4 Application of MEP to Silicon

Table 4.4 Comparison of the computed source current for several meshes Computed source current Is Applied voltage Vd 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0

Vs 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Vg 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.25 0.50

For the MOSFET Parabolic band Meshes and currents (A/cm) Mesh 1 Mesh 2 Mesh 3 1198 el. 2521 el. 4644 el. 0.12245 0.12526 0.12722 0.17631 0.18050 0.18278 0.21275 0.21792 0.22038 0.24416 0.25017 0.25282 0.27392 0.28074 0.28358 0.30315 0.31079 0.31382 0.33224 0.34071 0.34393 0.36139 0.37068 0.37412 0.39071 0.40083 0.40450 0.42022 0.43120 0.43509 1.0192 1.0406 1.0522 1.7228 1.7555 1.7772

Mesh 4 6094 el. 0.12748 0.18307 0.22067 0.25311 0.28387 0.31410 0.34420 0.37439 0.40476 0.43534 1.0535 1.7798

Mesh 5 9796 el. 0.12740 0.18273 0.22015 0.25244 0.28307 0.31317 0.34314 0.37320 0.40345 0.43393 1.0540 1.7835

Kane Mesh 3 4644 el. 0.1478 0.2020 0.2354 0.2647 0.2926 0.3198 0.3465 0.3729 0.3992 0.4253 0.9782 1.5610

The total computing time was less than 6 30 for mesh 1 and about 18 for mesh 2 on a laptop computer. The various presented figures are relative to mesh 3 0.45 Current (Ampere/cm) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Volt 0

0

0.5

1

1.5

Fig. 4.36 Characteristic curve for the MOSFET. The drain current ID versus the drain-source applied voltage VD at VG = 0. The notation is as in Fig. 4.16

4.10 A Numerical Approach Based on Finite Elements

125

2 Current (Ampere/cm) 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Volt 0 −2

−1.5

−1

−0.5

0

0.5

1

Fig. 4.37 Characteristic curve for the MOSFET. The drain current ID versus the gate applied voltage VG at VS = 0. The notation is as in Fig. 4.16

20

x 106

electron x−velocity (cm/sec)

15

10

5

0

micron −5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 4.38 Stationary solution for the x-component of the velocity (cm/s) in MOSFET in the cross section y=0.375 µm. The notation is as in Fig. 4.16

126

4 Application of MEP to Silicon

15

x 106

electron y−velocity (cm/sec)

10

5

0

micron −5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 4.39 Stationary solution for the y-component of the velocity (cm/s) in MOSFET in the cross section y=0.375 µm. The notation is as in Fig. 4.16 0.35 electron energy (eV)

0.3

0.25

0.2

0.15

0.1

0.05 micron

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 4.40 Stationary solution for the energy (eV) in MOSFET in the cross section y = 0.375 µm. The notation is as in Fig. 4.16

4.10 A Numerical Approach Based on Finite Elements

127

electron density 17

x 10 8 7 6

1/cm

3

5 4 3 2 1 0 −1 2 1 0 −5

−1

x 10

−2 −3

cm

2

1

0

5

4

3

−5

x 10

cm

Fig. 4.41 Stationary solution for the electron density in MOSFET (cm−3 ) electron energy

0.35 0.3 0.25

eV

0.2 0.15 0.1 −3 0.05

−2 −1

0 4.5

−5

0 4

3.5

3

2.5

2

1.5

1

0.5

0

2

−5

x 10

x 10

1

cm

Fig. 4.42 Stationary solution for the electron energy in MOSFET (eV)

cm

128

4 Application of MEP to Silicon electrostatic potential

1 0.8 0.6

Volt

0.4 0.2 0

−0.2 −0.4 −0.6 −0.8 2 1 0 −1

−5

x 10

−2 −3 cm

2

1

0

5

4

3

−5

x 10 cm

Fig. 4.43 Stationary solution for the electrostatic potential in MOSFET (V) x−component electric field

5

x 10

0.5 0 −0.5

Volt/cm

−1 −1.5 −2 −2.5 −3 −3

−3.5 −4 4.5

−2 −1 4

0 3.5

3

2.5

2

1.5

−5

x 10

−5

x 10

1 1

0.5

0

2

cm

cm

Fig. 4.44 Stationary solution for the x-component of the electric field in MOSFET (V/cm)

4.10 A Numerical Approach Based on Finite Elements

129

y−component electric field

5

x 10

0.5 0

Volt/cm

−0.5 −1 −1.5 −2 −2.5 −3

−3 −3.5 4.5

−2 −1 4

−5

0 3.5

3

2.5

2

1.5

−5

x 10

x 10

1 1

0.5

0

2

cm

cm

Fig. 4.45 Stationary solution for the y-component of the electric field in MOSFET (V/cm)

Chapter 5

Some Formal Properties of the Hydrodynamical Model

In this chapter we investigate the formal properties of the hydrodynamical model for semiconductors based on MEP. We will prove that it forms a hyperbolic system in the physically relevant region of the space of the dependent variables. Such a property is a consequence of the general theory developed in Chap. 3 when applied to the complete non linear model but since we have performed an expansion of the original non linear MEP distribution function, the hyperbolicity have to be checked. In this chapter the basic stability features of the MEP model will be also studied by analyzing the typical 1-D problem represented by the n+ −n−n+ ballistic diode. When the applied voltage is negligible the system is expected to tend to the global thermodynamical equilibrium where the charge is at rest with the same temperature as the crystal. By following [36] it is proved that for the model under consideration the equilibrium solution is asymptotically stable in the parabolic band case under certain restrictions on the doping profile. The reader not interested in analytical questions can skip this chapter without compromising the comprehension of the rest of the book.

5.1 Hyperbolicity of the MEP Hydrodynamical Model Let us consider the quasilinear system of PDE’s  ∂ ∂ (0) F (U) + F (i) (U) = B(U), ∂t ∂xi 3

(5.1)

i=1

with m 3 U = U(x, t) : D × R+ 0 → R , D open subset of R ,

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_5

131

132

5 Some Formal Properties of the Hydrodynamical Model

and F (β) , B : Ω → Rm ,

β = 0, 1, 2, 3,

sufficiently smooth functions and U(D) ⊂ Ω ⊂ Rm . If we consider a smooth solution, we can introduce the Jacobian matrices A(β) = ∇U F (β) ,

β = 0, 1, 2, 3.

We recall that the system (5.1) is said to be hyperbolic in the t-direction if   det A(0)(U) = 0 and the eigenvalue problem det

3 

ni A (U) − λA (i)

(0)

(U) = 0

(5.2)

i=1

has real eigenvalues and the eigenvectors span Rm for all unit vectors n = (n1 , n2 , n3 )T . In the case of the system (4.1)–(4.4) we have (indeed it is computationally more convenient to substitute the equation for Pi with a linear combination of the Eqs. (4.2) and (4.4) in order to have an equation for nVi ) ⎛

⎞ ⎛ ⎛ ⎞ ⎞ n 1 Vi ∗ ⎜V⎟ ⎟ ⎜ ∗ ⎟ ⎜ ⎟ , F (0) = n ⎜ m V ⎟ , F (i) = n ⎜ (U − 2αm F )ei ⎟ , i = 1, . . . , 3, U=⎜ ⎝W ⎠ ⎠ ⎝ W ⎠ ⎝ Si F ei S S where ei is the i-th column of the 3 × 3 identity matrix I3 , and the Jacobian matrices are given by ⎛

A(0)

1 01×3 0 ⎜ m∗ V m∗ nI3 03×1 =⎜ ⎝ W 01×3 n S 03×3 03×1 ⎛

A(n) =

3  i=1

ni A(i)

⎞ 01×3 03×3 ⎟ ⎟, 01×3 ⎠ nI3

⎞ 0 01×3 n·V nnT ⎜ (U − 2αm∗ F )n 03×3 n(U  − 2αm∗ F  )n 03×3 ⎟ ⎟. =⎜ ⎝ n·S 01×3 0 nnT ⎠ Fn 03×3 nF  n 03×3

5.1 Hyperbolicity of the MEP Hydrodynamical Model

133

Here the prime denotes partial derivation with respect to W and 01×3 is the vector (0, 0, 0).   Let us introduce the region Ωˆ = U ∈ R8 : n > 0, W > 0 and the functions g1 (W ) = (U + m∗ F  − W U  + 2αm∗ (W F  − F ))2 − 4m∗ (U F  − U  F ), (5.3) . (5.4) g2 (W ) = U + m∗ F  − W U  + 2αm∗ (W F  − F ) − g1 (W ), g3 (W ) = (U − 2αm∗ F )F  − (U  − 2αm∗ F  )F.

(5.5)

In [174] the following algebraic lemma has been proved Lemma 5.1.1 If the inequalities g1 (W ) > 0,

g2 (W ) > 0,

g3 (W ) > 0

(5.6)

are satisfied, in the region Ωˆ the system (4.1)–(4.4) is hyperbolic and the eigenvalues are given by (5.7) λ1,2,3,4 = 0, √ 2 λ±± = ± U + m∗ F  − W U  + 2αm∗ (W F  − F )± 2  1/2 31/2 ∗   ∗  2 ∗   (U + m F − W U + 2αm (W F − F )) − 4m (U F − U F ) . (5.8) Proof For U ∈ Ωˆ one has det(A(0)) = (m∗ )3 n7 > 0. In order to check the second condition of hyperbolicity, after some algebra one finds that the eigenvalues are given by λ1,2,3,4 = 0, (5.9) √ 2 U + m∗ F  − W U  + 2αm∗ (W F  − F )± λ±± = ± 2  1/2 31/2 ∗   ∗  2 ∗   (U + m F − W U + 2αm (W F − F )) − 4m (U F − U F ) . (5.10) If the inequalities (5.6)1 and (5.6)2 are satisfied, the eigenvalues λ±± are real, distinct and different from zero. Then the hyperbolicity is guaranteed if the dimension of the kernel of A(n) is four.

134

5 Some Formal Properties of the Hydrodynamical Model

Since n · n = 1, at least one of the components nj is different from zero. Let us suppose that n1 = 0 and consider the submatrix of A(n) ⎛

n1 V 1 ⎜ n1 (U − 2αm∗ F ) Aˆ = ⎜ ⎝ nk S k n1 F

n1 n 0 0 n1 n(U  − 2αm∗ F  ) 0 0 0 n1 nF 

⎞ 0 0 ⎟ ⎟. n1 n ⎠ 0

The determinant of Aˆ is given by n61 n3 g3 (W ). If either n2 or n3 is different from zero we get the previous result with n1 substituted by either n2 or n3 . Since the rank of A(n) cannot be greater than four, under the condition g3 (W ) > 0 the eigenspace associated to λ = 0 has therefore four ˆ We remark that in the one dimensional case the independent eigenvectors in Ω. system becomes strictly hyperbolic with eigenvalues λ±± .  Now, let us check the conditions (5.6). In the parabolic band limit one has g1 (W ) =

160 2 W , 81

g2 (W ) =

√ 4 (5 − 10)W, 9

g3 (W ) =

20 2 W 27

ˆ The eigenvalues are and the conditions (5.6) are trivially satisfied in Ω. λ1,2,3,4 = 0

and λ±±

) √ = ± (10 ± 2 10)W .

(5.11)

In the case of the Kane dispersion relation, we have numerically evaluated the functions g1 (W ), g2 (W ) and g3 (W ) for the range of values of W typically encountered in the electron devices. Figure 5.1 shows that the relations (5.6) are satified also in the non parabolic case. Therefore we can conclude that at least for the values of W of practical interest the system (4.1)–(4.4) is hyperbolic. Remark 5.1.1 The above results are an important step in the study of the analytical properties of the model. It is well known that the deduction of hydrodynamiclike models from the transport equation can lead to balance equations presenting undesirable features like the appearance of instabilities, e.g. the Burnett or superBurnett equations in gas dynamics [38, 86]. In most cases the loss of stability of the equilibrium states can be ascribed to the loss of hyperbolicity at certain wavelengths in the Fourier space. Recently some general procedures have been devised to restore hyperbolicity for hydrodynamic models beyond Navier-Stokes-Fourier equations [59] and macroscopic models deduced from the Boltzmann equation with a collision operator in the relaxation time approximation [107].

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

135

0.14

g1 (W) (eV2)

0.12 0.1 0.08 0.06 0.04 0.02 0 0

0.05

0.1

0.15

0.2

0.25 W(eV)

0.3

0.35

0.4

0.45

0

0.05

0.1

0.15

0.2

0.25 W(eV)

0.3

0.35

0.4

0.45

0

0.05

0.1

0.15

0.2

0.25 W(eV)

0.3

0.35

0.4

0.45

0.4 0.35 g2(W) (eV)

0.3 0.25 0.2 0.15 0.1 0.05 0

0.4 0.35

g3(W) (eV2)

0.3 0.25 0.2 0.15 0.1 0.05 0

Fig. 5.1 The functions g1 (W ), g2 (W ) and g3 (W ) versus the energy W (eV) in the parabolic case (dashed line) and for the Kane dispersion relation (continuous line)

5.2 Nonlinear Asymptotic Stability of the Equilibrium State In view of the last remark, in this section the stability properties of the MEP model are investigated for the typical 1-D problem of the n+ − n − n+ ballistic diode, that has been intensively studied by employing different numerical methods and models,

136

5 Some Formal Properties of the Hydrodynamical Model

e.g. [9, 12, 83, 174, 179, 180]. Roughly speaking, physically, the situation is given by a semiconductor divided into three parts: two regions of high doping (the n+ regions) with a region of low doping (the n region) in between. The dynamics of charge carriers depends on the applied potential (the bias voltage). When the applied voltage is negligible the system is expected to tend to the global thermodynamical equilibrium where the charges are at rest with the same temperature of the crystal. We prove that for the model under consideration the equilibrium solution is asymptotically stable in the parabolic band case under certain restrictions on the doping profile. A similar question has been previously tackled in [33, 35, 37] for a simplified version of the model, obtained by linearizing with respect to the momentum and energy-flux. Here the full nonlinear model is considered according to [36]. Usually, the stability is investigated by an appropriate Liapunov function. In principle MEP gives an expression for the entropy which can be used as Liapunov function. However, in order to get explicit constitutive functions, an expansion in term of a small anisotropy parameter has been introduced in and for the resulting system an entropy function is not known and the analysis must be performed by using a specific approach. For a study of global existence of the initial boundary value problem on the whole space for the models based on MEP the interested reader can see [1]. Other hydrodynamical models have been investigated for example in [2, 120, 122, 123, 127, 135].

5.2.1 Basic Equations and Formulation of the Problem Let us introduce the dimensionless variables1 R= qi =

Si m∗ C30

n , N+ ,

ϕ=

ui =

Vi , C0

Φe , m∗ C20

E= τ=

W , m∗ C20

C0 t , L

x˜ i =

xi . L

Here R, E, ϕ, ui , q i , i = 1, 2, 3 are new dependent variables, τ , x˜ i , i = 1, 2, 3 are i i + new independent variables (further we will write again ) x instead of x˜ ), N is the

B T0 is a sort of sound speed, doping density N in the n+ region (see [7]), C0 = Km∗ T0 is the lattice temperature, KB is the Boltzmann constant and L is the width of the n+ − n − n+ channel.

1 Only in this section the symbol E will be used for the adimensional average energy. In the rest of the book, this symbol indicates the energy bands.

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

137

The evolution equations (4.1)–(4.4) in the dimensionless divergence form read ∂(Rui ) ∂R = 0, + ∂τ ∂x i ∂(Rui ) ∂( 23 RE) ∂ϕ + = R i + R8 CiP , i = 1, 2, 3, ∂τ ∂x i ∂x

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (5.12)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 10 2 ⎪ i ⎪ ∂(Rq ) ∂( 9 RE ) 5 ∂ϕ ⎪ i ⎭ 8 = + R C , i = 1, 2, 3, + RE W ∂τ ∂x i 3 ∂x i ∂ϕ ∂(RE) ∂(Rq i ) + = Rui i + R8 CW , i ∂τ ∂x ∂x

ε2 #ϕ = R − ρ,

(5.13)

where ρ= 8 CiP = CPi

1 , β

N , N+

ε2 =

,

8 CW = CW

L m∗ C20

β= L

m∗ C30

,

e2 L2 N + , m∗ C20 i 8 CiW = CW

L m∗ C40

.

For 1-D problems the system of balance equations (5.12) and (5.13) becomes Uτ + BUx = F (Q, U ),

(5.12 )

ε2 ϕxx = R − ρ.

(5.13 )

Here ⎛

⎞ R ⎜ J ⎟ ⎟ U =⎜ ⎝RE ⎠ , Rq

⎞ 1 0 0 ⎜ 0 23 0⎟ ⎟, B=⎜ ⎝ 0 0 1⎠ 20 2 − 10 9 E 0 9 E 0 ⎛

0 0 0

⎞ 0 ⎜ RQ + R8 C1P ⎟ ⎟, F =⎜ ⎝ J Q + R8 CW ⎠ 5 81 3 REQ + R CW ⎛

J = Ru, Q = ϕx ,    R8 C1P c˜11 c˜12 J = (see Sect. 5.2.4), R8 C1W c˜21 c˜22 Rq   2 E−1 . ˆ (see Sect. 5.2.4), P = R R8 CW = cP 3 



138

5 Some Formal Properties of the Hydrodynamical Model

According to the previous section the eigenvalues of the matrix B are

1/2 ⎫ √ ⎪ ⎪ 10 + 2 10 ⎪ E =± ,⎪ ⎪ ⎪ ⎬ 9

λ1,2



λ3,4

1/2 ⎪ √ ⎪ ⎪ 10 − 2 10 ⎪ =± ,⎪ E ⎪ ⎭ 9

(5.14)

and therefore the hyperbolicity condition amounts to E > 0. The system (5.12 ) can be rewritten in the form Rτ + Jx = 0, Jτ + Rx + Px = RQ + cˆ11 J + c˜12 Θ, 3 Pτ + Jx + Θx = J Q + cP ˆ , 2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ P 2 ⎪ Θτ + (P + )x = P Q + cˆ21 J + cˆ22 Θ,⎭ 5 R

(5.12 )

where Θ = Rq − 52 J , cˆ11 = c˜11 + 52 c˜12 , cˆ21 = 25 c˜21 − c˜11 + 52 cˆ22 , cˆ22 = 25 c˜22 − c˜12 . The scaled doping density ρ = ρ(x) must be considered as a known function defined on [0, 1]. The coefficients c, ˆ c˜11 , c˜12 , c˜21 , c˜22 must be considered as functions of E (the explicit expression of these coefficients can be found in [173] and it is reported in the Sect. 5.2.4). We assume that the function (ρ(x) − 1) is sufficiently smooth and finite and 1 ≥ ρ(x) ≥ δ > 0, x ∈ [0, 1]. By taking into account the expressions of the eigenvalues (5.14), we assign two boundary conditions at x = 0 and two boundary conditions at x = 1 for equations (5.12 ) along with two further conditions for (5.13 ). According with [29, 83], the following boundary conditions are imposed R(τ, 0) = R(τ, 1) = 1,



P (τ, 0) = P (τ, 1) = 0, ϕ(τ, 0) = A,

ϕ(τ, 1) = A + B,

(5.15) (5.16)

where A and B are constants. Without loss of generality, we assume that A = 0 and B > 0 which represent the bias across the diode. Of course, also the initial data at τ = 0 must be assigned. The first relation of (5.15) is the condition of charge neutrality at the contacts while the second one expresses the fact that in a ohmic contact thermal equilibrium is assumed.

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

139

Following [29], we give an equivalent formulation of the mixed problem (5.12 ), (5.13 ), (5.15), and (5.16) by considering the system (5.12 ) coupled with the relation 1 ε Qτ =

J (τ, s) ds − J (τ, x) = l[J ]

2

(5.17)

0

instead of the Poisson equation (5.13 ). Equation (5.13 ), rewritten in the form ε2 Qx = R − ρ,

(5.18)

will be treated as an additional stationary law which, in particular, the initial data have to satisfy. From the boundary conditions (5.16) it follows that the relation 1 Q(τ, s) ds = B

(5.19)

0

is fulfilled. Therefore the initial data must also satisfy this relation. The electric potential ϕ = ϕ(τ, x) is found from the evident equality x ϕ(τ, x) =

Q(τ, s) ds.

(5.20)

0

Thus, instead of the mixed problem (5.12 ), (5.13 ), (5.15), and (5.16) one can consider the problem (5.12 ), (5.17), and (5.15), with additional conditions (5.18) and (5.19), which actually are requirements on the initial data. It is easy to show that these two formulations are equivalent, at least for smooth solutions. Problem (5.12 ), (5.13 ), (5.15), and (5.16) has for B = 0 the global thermodynamical equilibrium ⎫ J (τ, x) = J= = 0, i.e. u(τ, x) = uˆ = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ = = P (τ, x) = P = 0, i.e. E(τ, x) = E = ,⎪ ⎪ ⎪ 2 ⎬ = = 0, i.e. q(τ, x) = qˆ = 0, ⎪ Θ(τ, x) = Θ ⎪ ⎪ ⎪ ⎪ ˆ ⎪ = = eϕ(x) ⎪ R(τ, x) = R(x) , ⎪ ⎪ ⎪ ⎭ ϕ(τ, x) = ϕ(x), ˆ

(5.21)

140

5 Some Formal Properties of the Hydrodynamical Model

as stationary solution, with ϕ(x) ˆ satisfying the Poisson equation =− ρ ε2 ϕˆ  = R

(5.22)

ϕ(0) ˆ = ϕ(1) ˆ = 0.

(5.23)

and the boundary conditions

It is obvious that if ε can be considered as a small parameter, the solution to boundary value problem (5.22) and (5.23) can be approximated as ϕ(x) ˆ = ln ρ(x) + O(ε).

(5.24)

In the sequel the function ϕ(x) ˆ is assumed to be sufficiently smooth and finite. From the physical considerations, one expects what follows. Remark 5.2.1 Let B = 0. The solution to (5.12 ), (5.13 ), (5.15) and (5.16) is expected to tend to the equilibrium state as τ → ∞, i. e., J (τ, x) → 0, P (τ, x) → 0, Θ(τ, x) → 0, = R(τ, x) → R(x), ϕ(τ, x) → ϕ(x). ˆ Below this fact will be proven under some restrictions on the doping density, the initial data and coefficients c, ˆ c˜11 , c˜12 , c˜21 , c˜22 , β. A similar fact has been proven in [35] for a simpler mixed problem than (5.12 ), (5.13 ), (5.15), and (5.16). The problem (5.12 ), (5.13 ), (5.15), and (5.16) has been also studied in [34] in the linear case.

5.2.2 Formulation of the Auxiliary Problems In this section it is given a different formulation of the mixed problem (5.12 ), (5.17), and (5.15) with additional requirements (5.18) and (5.19) in the case B = 0. First let us introduce a potential H = H (τ, x) (do not mix up with the electric potential ϕ) such that J = Hτ R = −Hx

 .

(5.25)

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

141

The first condition in (5.12 ) is fulfilled automatically while the boundary conditions (5.15) can be rewritten as Hx (τ, 0) = Hx (τ, 1) = −1



P (τ, 0) = P (τ, 1) = 0

.

In a view of (5.25), Eq. (5.17) becomes >

? ε2 Q(τ, x) − l[H ] = 0 τ

that is ε2 Q = A0 (x) + l[H ], where 1 H (τ, s) ds − H (τ, x).

l[H ] = 0

From the additional relation (5.18), it follows that A0 (x) = −ρ(x), i.e. x A0 (x) = C −

ρ(s) ds. 0

Here C is a constant. By taking into account (5.19), one has 1 C=

(1 − s)ρ(s) ds 0

and x A0 (x) = −

1 ρ(s) ds +

0

(1 − s)ρ(s) ds. 0

(5.15 )

142

5 Some Formal Properties of the Hydrodynamical Model

For the sake of convenience, we introduce a new independent variable U (τ, x) instead of H x H (τ, x) = U (τ, x) −

= ds. R(s)

(5.26)

0

Considering Eq. (5.22) as an ordinary differential equation in the unknown function ϕ(x) ˆ with boundary conditions (5.23), one obtains 1 ϕˆ = ϕ(x) ˆ =β

 = − ρ(s) ds, G(x, s) R(s)

0

where G(x, s) is the Green function, G(x, s) =

 s(x − 1), if

0 < s ≤ x,

x(s − 1), if

x < s < 1.

Differentiating the expression for ϕ(x) ˆ with respect to x, one has 

x

ϕˆ (x) = β

 = − ρ(s) ds − β R(s)

0

1

 = − ρ(s) ds. (1 − s) R(s)

(5.27)

0

In view of (5.26) and (5.27), one has Q(τ, x) = β (l[H ] + A0 (x)) = βl[U ] + ϕˆ  (x).

(5.28)

By using (5.26), from (5.25) we find J = Uτ ,



= − R = Ux , L=R

.

(5.25 )

Moreover, the second equation in (5.12 ) can be rewritten as   = Jτ − Lx − cˆ11 J + Px − Rβl[U ] − c˜12 Θ + βl[U ] + ϕˆ  L = 0,

(5.29)

and (5.15 ) takes the form L(τ, 0) = L(τ, 1) = P (τ, 0) = P (τ, 1) = 0.

(5.15 )

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

143

Differentiating (5.29) by x, we have Lτ τ − Lxx + Pxx + F = 0.

(5.30)

Here F = t1 Lτ + t2 Pτ + χ1 L + χ2 P + F0 ,  σ  = − f1 Px − c˜12 f0 J + σ R = f1 , F0 = f0 − f1 Lx − βL2 − βl[U ]R R R R 3 t1 = c˜12 − cˆ11 , t2 = c˜12 , 2

 = − ρ , χ2 = −cˆc˜12 , χ1 = β 2 R f0 = βl[U ] + ϕˆ  , σ =

P , R

 cˆ11 =

Note that

  f1 = cˆ11 J + c˜12 Θ,

d cˆ11 (E) dσ

and so on.

2 E = σ + 1. 3

Differentiating cross-wise the two last equations in (5.12 ), one eliminates Θ from the left-hand sides and comes to the relation ˜ xx + Lxx + G = 0, aP ˜ τ τ − bP where a˜ =

3 , 2aˆ

b˜ =

1 + 52 (1 + 2σ ) , aˆ

5 aˆ = 1 − σ 2 , 2

G = t3 Lτ + t4 Pτ + χ3 L + χ4 P + G0 , 3 2 1 5 = 2 2 G0 = −5Rf2 + R σ − g0 − F0 , aˆ 2    1 5

cˆ21 − cˆ22 − t1 , t3 = aˆ 2   15 1 cˆ + cˆ22 + t2 , t4 = − aˆ 4   5 1 1 χ3 = − χ1 , χ4 = cˆ cˆ22 + c˜12 , aˆ aˆ 2 f2 =



 1

= − Lx , Px − σ R R

(5.31)

144

5 Some Formal Properties of the Hydrodynamical Model

 = ] − Lf0 + cˆ11 J + c˜12 Θ − Px + Lx + βJ l[J ] g0 = f0 Rβl[U  

   P 5  f0 Px + P ϕˆ  − βL + cˆ22 Jf0 + cˆ21 + cˆ (Pτ + σ Lτ ) − J + cˆ22 Θ f2 . R 2 Remark 5.2.2 While deriving (5.30) and (5.31), Θx has been expressed by using the third equation in (5.12 ). Remark 5.2.3 Underlined aggregates turn into zero in the case of uniform doping, = ≡ 1, ϕ(x) i.e. when ρ(x) ≡ 1, 0 ≤ x ≤ 1 (in this case R(x) ˆ ≡ 0, 0 ≤ x ≤ 1). We rewrite Eqs. (5.30), (5.31) in the following way. Summing up the Eq. (5.30) and the Eq. (5.31) multiplied by 1˜ , one gets b

Lτ τ +

a˜ b˜ − 1 Pτ τ − Lxx + F + b˜ b˜

1 G = 0, b˜

(5.30 )

summing up (5.30) and (5.31), we have ˜ τ τ − (b˜ − 1)Pxx + F + G = 0. Lτ τ + aP

(5.31 )

Unifying (5.30 ), (5.31 ), we obtain the system ALτ τ − BLxx + T Lτ + XL + Λ = 0.

(5.32)

Here

  ˜ b˜ b−1 L 1 0 a˜ , B= , L= , A = a˜ P 1 a˜ 0 b˜ − 1       T1 T2 T1 T0 0 T80 T = = ST + CT , = + T3 T4 T0 T4 −T80 0       80 X1 X2 X1 X0 0 X X= = + 80 0 = SX + CX, X3 X4 X0 X4 −X ˜ 1 + t3 ˜ 2 + t4 bt bt , T2 = , T3 = t1 + t3 , T4 = t2 + t4 , a˜ a˜ T2 + T3 T2 − T3 T0 = , T80 = , 2 2 ˜ 1 + χ3 ˜ 2 + χ4 bχ bχ , X2 = , X3 = χ1 + χ3 , X4 = χ2 + χ4 , X1 = a˜ a˜

T1 =

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

145

X2 + X3 80 = X2 − X3 , , X X0 = 2 2

Λ=

˜ 0 +G0 bF a˜ F 0 + G0

.

The boundary conditions (5.15 ) take the form L(τ, 0) = L(τ, 1) = 0.

(5.15 )

Finally, differentiating (5.32) by τ , we come to the system ADτ τ − BDxx + T Dτ + XD + K = 0,

(5.33)

where D = Lτ , K = Λτ + Aτ Dτ − Bτ Lxx + Tτ D + Xτ L, (see (5.29), (5.32) and the last equation in (5.12 )) = Jτ = Lx + cˆ11 J − Px + Rβl[U ] + c˜12 Θ − f0 L, Lxx = B −1 (ADτ + T D + XL + Λ) , ? 5> P Q + cˆ21 J + cˆ22 Θ − Px − (Rσ 2 )x . Θτ = 2 The boundary conditions for (5.33) follow from (5.15 ). Remark 5.2.4 We note that in a small neighborhood of the equilibrium state (5.21) inequalities aˆ > 0, b˜ > 0, b˜ − 1 = 25aˆ (1 + σ )2 > 0 hold. So, we have A, B > 0 (positive definite) in a small neighborhood of the equilibrium state.

5.2.3 Asymptotic Stability of the Equilibrium State Now we start to construct the global a priori estimate which will enable us to get the asymptotic stability of the equilibrium state. By using the obvious relations2 (we recall that A, B are symmetric matrices) 2 (Dτ , ADτ τ ) = (Dτ , ADτ )τ − (Dτ , Aτ Dτ ) , 2 (Dτ , BDxx ) = 2 (Dτ , BDx )x − (Dx , BDx )τ − 2 (Dτ , Bx Dx ) + (Dx , Bτ Dx ) ,

2 (·, ·)

indicates the standard scalar product on Rn .

146

5 Some Formal Properties of the Hydrodynamical Model

(D, ADτ τ ) = (D, ADτ )τ − (Dτ , ADτ ) − (D, Aτ Dτ ) , (D, BDxx ) = (D, BDx )x − (Dx , BDx ) − (D, Bx Dx ) and by multiplying (5.33) by 2Dτ , one obtains {(Dτ , ADτ ) + (Dx , BDx ) + (D, SXD)}τ − 2(Dτ , BDx )x + 2 {(Dτ , ST Dτ ) + (Dτ , CXD)} + 2 (Dτ , K) − (Dτ , Aτ Dτ ) − (D, (SX)τ D) + 2 (Dτ , Bx Dx ) − (Dx , Bτ Dx ) = 0.

(5.34)

Instead by multiplying the same system by 2D, one comes to the expression {2 (D, ADτ ) + (D, ST D)}τ − 2(D, BDx )x + 2 {− (Dτ , ADτ ) + (Dx , BDx ) + (D, SXD) + (D, CT Dτ )} + 2 (D, K) − (D, (ST )τ D) − 2 (D, Aτ Dτ ) + 2 (D, Bx Dx ) = 0.

(5.35)

Arguing in the same way, we multiply (5.32) first by 2Lτ and then by 2L, getting {(D, AD) + (Lx , BLx ) + (L, SXL)}τ − 2(D, BLx )x + 2 {(D, ST D) + (D, CXL)} + 2 (D, Λ) − (D, Aτ D) − (L, (SX)τ L) + 2 (D, Bx Lx ) − (Lx , Bτ Lx ) = 0.

(5.36)

{2 (L, AD) + (L, ST L)}τ − 2(L, BLx )x + 2 {− (D, AD) + (Lx , BLx ) + (L, SXL) + (L, CT D)} + 2 (L, Λ) − (L, (ST )τ L) − 2 (L, Aτ D) + 2 (L, Bx Lx ) = 0.

(5.37)

Equation (5.29) and the last two equations in (5.12 ) together yield the evident identity 2 3 2 2 3 2 2 2 − 2(J L)x + 2(J P )x + 2(P Θ)x J +L + Θ + P 5 2 τ > ? = + 2 −cˆ11J 2 − c˜12 J Θ − cˆ22 Θ 2 − cˆ21 J Θ − cP ˆ 2 − Rβl[U ]J + 2 (J L − P J − ΘP ) f0 + 2Θ (Px σ + Pf2 ) = 0.

(5.38)

If one multiplies (5.34)–(5.38) by positive arbitrary constants α1 , α2 , α3 , α4 , α5 , sum up them and integrates with respect to x from 0 to 1 by taking into account the

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

147

boundary conditions (5.15 ), one comes to d (0) J + J (1) = Π. dτ

(5.39)

Here J (0) =

 3 1 2 2 2 Θ + J2 (Y, A0 Y ) + α1 (Dx , BDx ) + α3 (Lx , BLx ) + α5 dx, 5 0

⎞ Dτ ⎟ ⎜ Y = ⎝ D ⎠, L ⎛

⎛

α1 A ⎜ A 0 = ⎝ α2 A 0

R = α3 A + α2 ST + α1 SX, R1 = α5 A1 + α4 ST + α3 SX,

J (1) = 2

α2 A R α4 A

⎞ 0 ⎟ α4 A ⎠ , R1

  3 A1 = diag 1, , 2

1 2 (Y, A1 Y ) + α2 (Dx ,BDx ) + α4 (Lx , BLx ) 0

 3 = + α5 −cˆ11 J 2 − μ12 J Θ − cˆ22 Θ 2 − Rβl[U ]J dx,

μ12 = c˜12 + cˆ21 , ⎞ ⎛ α1 ST − α2 A α1 CX 0 ⎟ ⎜ A1 = ⎝ α2 CT α2 SX + α3 ST − α4 A α3 CX ⎠. α4 SX + α5 diag(0, −c) ˆ 0 α4 CT

The aggregate Π, although is not given here, can be easily written down. Remark 5.2.5 From 1 l[U ] =

1 U (τ, s) ds − U (τ, x) =

0

0

⎞ ⎞ ⎛ s ⎛  1  s ⎝ Uz (τ, z) dz ⎠ ds = ⎝ L(τ, z) dz ⎠ ds, x

0

it follows that 1

⎛ 1 ⎞ 12  |L| ds ≤ ⎝ L2 ds ⎠ .

0

0

|l[U ]| ≤

x

148

5 Some Formal Properties of the Hydrodynamical Model

Then 1 ⎛ 1 ⎞  1  1 1 2 Rβl(U = L2x dx ⎠ . )J dx ≤ 2β |l||J | dx ≤ β ⎝εˆ J 2 dx + 2ˆε 0

0

0

0

While deriving the last inequality, we have used the Cauchy inequality with εˆ > 0 and the Poincare inequality (see [148]) 1

1 L ds ≤ 2

1 L2x dx.

2

0

0

From now on we assume that ρ(x) ≡ 1 (see Sect. 5.2.2, Remark 5.2.3), i.e. = R(x) ≡ 1. We shall use argumentations similar to those employed for problems of this type in [30, 31]. Let us suppose that the problem (5.12 ), (5.17), and (5.15) has the smooth (classical) local solution on a sufficiently small interval [0, τ∗ ]. We define the constant 2 3 M∗ = max max $L(τ )$C[0,1] , max $Lx (τ )$C[0,1], max $Lτ (τ )$C[0,1] . τ ∈[0,τ∗ ]

τ ∈[0,τ∗ ]

τ ∈[0,τ∗ ]

At the equilibrium state the quadratic form under the integral sign in J (0) and (see below) are positive definite as shown in Sect. 5.2.5. By continuity these quadratic forms are still positive in a small neighborhood of the equilibrium state. Let us suppose that the value of M∗ is sufficiently small that the quadratic forms under the integral sign in J (0) and J (2) to be positive definite in [0, τ∗ ]. The equality (5.39) can be rewritten as follows J (2)

d (0) J + J (2) ≤ |Π| , dτ

(5.39 )

where

J

(2)

1 2 =2 (Y, A1 Y ) + α2 (Dx , BDx ) + α4 (Lx , BLx )+ 0

 3  β εˆ β 2 2 2 2 J − Lx + α5 −cˆ11 J − μ12 J Θ − cˆ22 Θ − dx. 2 4ˆε

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

149

Since J (0) and J (2) are positive quadratic form with constant coefficients, there exists a constant M1 > 0 (which is finally determined through the constant M∗ ) such that J (2) ≥ M1 J (0).

(5.40)

In view of (5.40), the inequality (5.39 ) can be rewritten as follows:  3 d (0) 2 J + M1 J (0) ≤ M2 J (0) , dτ

(5.41)

with M2 > 0 a constant which is determined by M∗ . The right-hand side in (5.41) is obtained by estimating |Π| (see [30, 35]) with the use of some standard embedding relations (see, for example, [32, 113, 187]) as3 (see inequality (5.39 )) 8b $Lxx (τ )$L (0,1) $L(τ )$C[0,1], $Lx (τ )$C[0,1] ≤ Mb $L(τ )$W 2 (0,1) ≤ M 2 2



8b M3 J (0)(τ ) ≤M

1 2

,

 1 2 $J (τ )$C[0,1], $Jx (τ )$C[0,1] ≤ Mb $J (τ )$W 2 (0,1) ≤ Mb M4 J (0) (τ ) . 2

8b are constants of embedding, while M3 , M4 > 0 are constants Above Mb , M determined by M∗ . Lemma 5.2.1 Let us consider the differential inequality z ≤ −M1 z + M2 z 2 , 3

(5.42)

with M1 and M2 positive constants. The solutions z(τ ) satisfying the initial condition 0 < z(0) < (M1 /M2 )2 enjoy the property z(τ ) ≤  1−

z(0)

−M τ 2 e 1 ,

M2 √ M1 z(0)

τ > 0.

(5.43)

Proof Since z(0) > 0 we can introduce the change of dependent variable y = √ 1/ z. By dividing the differential inequality by z3/2 , one gets y  ≥ M21 y − M22 , which is a linear differential inequality.

is the Sobolev space of the real functions on [0, 1] that are square-integrable and admit first and second generalized derivatives which are also square-integrable.

3 W 2 (0, 1) 2

150

5 Some Formal Properties of the Hydrodynamical Model

From e−M1 /2τ y  ≥ e−M1 /2τ M21 y − e−M1 /2τ M22 , it is straightforward to prove that the solutions with y(0) > M2 /M1 satisfy e−M1 /2τ [y(τ ) − M2 /M1 ] ≥ y(0) − M2 /M1 , wherefrom e−M1 /2τ y(τ ) ≥ y(0) − M2 /M1 , which expressed in terms of z(τ ) gives the relation (5.43). 3

If z ≥ 0, then F (z) = −M1 z + M2 z 2 is negative at 0 < z < Consequently, if the initial data are sufficiently small and  J

(0)

(0)
0.

(5.44)

Remark 5.2.6 The global estimate (5.44) has been derived provided that ρ(x) ≡ 1, = 0 ≤ x ≤ 1, i.e. R(x) ≡ 1, ϕ(x) ˆ ≡ 0, 0 ≤ x ≤ 1. Clearly, this estimate remains valid if the doping density ρ(x) slightly differs from the uniform density ρ(x) ≡ 1, 0 ≤ x ≤ 1. Existence of the estimate (5.44) means that4 J (τ, x), Θ(τ, x) ∈ W22 (0, 1), ˚ 1 (0, 1), L(τ, x) ∈ W22 (0, 1) ∩ W 2 ˚ 1 (0, 1) for all τ ≥ 0. ϕ(τ, x) ∈ W24 (0, 1) ∩ W 2 Clearly, J (τ, x), P (τ, x), Θ(τ, x) → 0, = R(τ, x) → R(x) in C 1 [0, 1], if ϕ(τ, x) → ϕ(x) ˆ 4W ˚ 1 (0, 1) 2

in C 3 [0, 1], if

τ → ∞; τ → ∞.

is the closure, with respect to the norm of W21 (0, 1), of the set of the real functions in [0, 1], compactly supported and having derivatives of any order.

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

151

Therefore we can state the following theorem Theorem 5.2.1 The equilibrium state is asymptotically stable by Liapunov for the mixed problem (5.12 ), (5.13 ), (5.15), and (5.16) for sufficiently small initial data and smooth perturbed solutions.

5.2.4 Explicit Expressions the Production Terms In this and the next sections some details about the expressions appearing in the estimates presented above are collected. The reader who is not interested in the mathematical details can skip these two sections. In the parabolic band approximation, the production term of the energy balance equation can be written as 8 CW = CW

6  . L = μˆ A ζA K1 (ζA ) Δ(ζA ), 3 ∗ m C0 A=1

where 1

μˆ =

(m∗ )2 (h¯ ωnp ) 2 L(Dt K)2 3

3

h¯ 2 (KB T0 ) 2 π 2 ρ0

z,

h h¯ = 2π , h is the Plank constant, hω ¯ np is the optical phonon energy, Dt K is the deformation potential for optical phonons, ρ0 is the mass density of the material, K1 (ζ ) is the modified Bessel function of second kind of order one,

Δ(ζ ) = nB eζ − (1 + nB )e−ζ ,

nB =

1 , eξ − 1

ζ =

3ξ , 4E

ξ=

hω ¯ np , KB T0

z is the number of final equivalent valleys in the intervalley scattering. Since Δ(ζ ) = nB e−ζ

⎞ ⎛ 1    ˆ ˆ e2ζ − e2ζ = −2nB eζ ⎝ e2λ(ζ−ζ ) dλ⎠ (ζˆ − ζ ) 0

⎛

= −2nB ζ eζ ⎝

⎞

1 e 0

2λ(ζˆ −ζ )

P dλ⎠ , R

152

5 Some Formal Properties of the Hydrodynamical Model

where ζˆ =

ξ 2

= = 3 ), we have the expression for the coefficient c: (E = E ˆ 2 R8 CW = cP ˆ ,

cˆ = −2

6 

⎛ 1 ⎞  ˆ μˆ A nB(A) ζA eζA K1 (ζA ) ⎝ e2λ(ζA −ζA ) dλ⎠ . 3 2

A=1

0

Concerning 8 C1P , 8 C1W , we have

8 C1P

⎛

8 C1W

=⎝

·

⎞ ⎡⎛

L

0

0

L m∗ C04

m∗ C02

⎠ ⎣⎝

(ac) a21

9m∗ C0 4W 2 27m∗ − 20W 3 0

∗

− 21m W 9m∗ 4W 5

⎛ ⎞⎤ (np) (np) a11 a12 ⎠+⎝ ⎠⎦ (ac) (np) (np) a22 a21 a22

(ac) (ac) a11 a12

⎞

u

0 m∗ C30

q

=



c˜11 c˜12 u c˜21 c˜22

8+ = νˆ ac A

6  A=0

q

=A νˆnp(A) A

u , q

where (ac) a11

√ 3  2 32 2π ∗ 32 2 W = Kac (m ) , 3 3 h¯ 3

(ac)

a21 = Kac =

(ac) a12 , m∗

(ac)

a22 =

(ac) (ac) a12 = 2W a11 ,

8 (ac) W a21 , 3

KB T0 Ξd2 , 4π 2 h¯ ρ0 vs2

Ξd is the deformation potential of acoustic phonons, vs is the sound speed,  − 6  2 4 2 W = 3 3 1

(np) a11

A=1

×

√ 3 2π(m∗ ) 2 (h¯ ωnp(A) )2 h¯ 3

+ (nB(A) + 1)e

−ζA

2 zA Knp(A) nB(A) eζA (K2 (ζA ) − K1 (ζA )) +

3 (K2 (ζA ) + K1 (ζA )) ,

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

  6  2 4 2 W = 3 3 1

(np) a12

A=1

√ 3 2π(m∗ ) 2 (h¯ ωnp(A) )2 h¯ 3

153

2  zA Knp(A) nB(A) eζA 3K2 (ζA )+

 + 2ζA (K1 (ζA ) − K2 (ζA )) + (nB(A) + 1)e−ζA [3K2 (ζA ) + 2ζA (K1 (ζA ) 3 + K2 (ζA ))] , (np)

(np)

=

a21

(np) a22

a12 , m∗

1  3 √ 6  2 2π(m∗ ) 2 (h¯ ωnp(A) )2 4 2 W = zA Knp(A) · 3 3 h¯ 3 A=1 2      · nB(A) eζA K2 (ζA ) 12 − 9ζA + 4ζA2 + K1 (ζA ) 3ζA − 4ζA2 +

+(nB(A) + 1)e

−ζA

   K2 (ζA ) 12 + 9ζA + 4ζA2

+ K1 (ζA ) 3ζA + 4ζA2 Knp =

3 ,

h¯ (Dt K)2 , 8π 2 ρ0 (h¯ ωnp )

is the modified Bessel function of second kind of order two, √  3 2 2 8L 2π(m∗ )2 Kac , νˆ ac = 3 h¯ 3 √ 2π(h¯ ωnp )2 Knp L ) z, νˆ np = 2 3 4 C h ¯ 0 3 ⎞ ⎛

−22E − 35 aˆ 11 aˆ 12 1 ⎟ ⎜ = 8 A = √ A, A = ⎝ . ⎠, A = E aˆ 21 aˆ 22 18 2 −40E − 5 E

K2 (ζ )

Here aˆ 11 =

aˆ 12 =

Δ (ζ ) [4ζ K1 (ζ ) − 22K2 (ζ )] + Δ(ζ ) [28K1 (ζ ) − 4ζ K2 (ζ )]

, 3 E2     12 −Δ (ζ ) 35 K2 (ζ ) + 12 5 ζ K1 (ζ ) + Δ(ζ ) 5 ζ K2 (ζ ) − 3K1 (ζ ) 5

E2

,

154

5 Some Formal Properties of the Hydrodynamical Model

   2 1 16 2 100 aˆ 21 = E − 2 Δ (ζ ) ζ − 40 K2 (ζ ) − ζ K1 (ζ ) + 3 3  3 76 16 +Δ(ζ ) ζ K2 (ζ )− ζ 2 K1 (ζ ) , 3 3 2     3 8 18 16 2 + ζ K2 (ζ ) + aˆ 22 = E − 2 Δ (ζ ) ζ K1 (ζ ) − 5 5 5 3 16 + Δ(ζ ) ζ [K2 (ζ ) + ζ K1 (ζ )] , 5 Δ (ζ ) = nB eζ + (1 + nB )e−ζ . Note that

Δ(ζˆ ) = 0,

Δ (ζˆ ) =

ˆ

2eζ

e2ζˆ − 1

.

The constants appearing in the above relations are reported in Tables 5.1 and 5.2 [174, 175]. Table 5.1 Values of the physical parameters N+ T0 L β

Doping density in the n+ region Lattice temperature Width of the n+ − n − n+ channel 2 2 + β = e L∗ N2

5 × 1023 m−3 300 K 6 × 10−7 m and 3 × 10−7 m 1.07494 × 104 and 2.68734 × 103

Ξd Kac νˆ ac

Deformation potential

1.44171 × 10−18 J 1.05274 × 10−35 Jm3 /s 4.99755 and 2.49878

m C0

Table 5.2 Coupling constants and phonon energies A z h¯ ωnp (J) Dt K (J/m) ξ nB Knp (Jm3 /s) μˆ νˆ np −21 −10 1 1 1.922 × 10 8.010 × 10 0.464 1.692 1.913 × 10−37 0.037 5.506×10−3 0.019 2.753×10−3 −21 −9 −37 2 1 2.964 × 10 1.282 × 10 0.716 0.956 3.176 × 10 0.119 0.022 0.059 0.011 3 4 3.044 × 10−21 4.806 × 10−10 0.735 0.921 4.349 × 10−38 0.068 0.013 0.034 6.277×10−3 −21 −9 −37 4 4 7.593 × 10 3.204 × 10 1.834 0.190 7.748 × 10 4.747 1.392 2.374 0.696 5 1 9.804 × 10−21 1.762 × 10−8 2.368 0.103 1.815 × 10−35 40.794 13.591 20.397 6.796 6 4 9.451 × 10−21 3.204 × 10−9 2.283 0.114 6.225 × 10−37 5.296 1.733 2.648 0.866

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

155

Here we discuss the conditions which make the quadratic forms under the integral signs in J (0), J (2) positive definite. Recall that entries of the matrices A, B, T and X are taken at the equilibrium state (see (5.21)). First one notes that the matrices A, B at the equilibrium state are positive definite (see Remark 5.2.4); in fact they read A=

7 3

1

1

 ,

3 2

B=

5 3

0

0

 .

5 2

Preliminarly we write down the conditions which provide ST , SX > 0. Since (see Sect. 5.2.2) 5 ST = SX =

5 1 3 (μ12 − cˆ11 − cˆ22 ) 4 (μ12 − 2cˆ22 ) − 3 cˆ 5 1 −cˆ − 15 4 (μ12 − 2cˆ22 ) − 3 cˆ 4 cˆ22 5 3β 5 ˆ cˆ22 6 c(

− c˜12 )

5 ˆ cˆ22 − c˜12 ) 6 c( 5 2 cˆcˆ22

,

,

these matrices are positive definite if χ = μ12 − cˆ11 − cˆ22 > 0,     5 15 1 2 5 (μ12 − 2cˆ22 ) − cˆ > 0, d = (cˆ11 + cˆ22 − μ12 ) cˆ + cˆ22 − 3 4 4 3   1 25 β cˆcˆ22 − cˆ2 (cˆ22 − c˜12 )2 > 0. t= 6 6 (5.45) From the equation det (ST − λA) = 0,

det (SX − λA) = 0

we find for the smaller eigenvalues

kT = λ− =

lT −

)

lT2 − 85 d

2

,

kX = λ− =

lX −

)

2 − 8t lX 5

2

where 5 2 lT = −cˆ11 − cˆ22 − c, ˆ 2 3

5 2 lX = β + cˆcˆ22 + cˆc˜12 . 3 3

,

156

5 Some Formal Properties of the Hydrodynamical Model

Then ST ≥ kT A,

SX ≥ kX A.

(5.46)

For the next considerations we set k=

T80 , 8 X0

! = cˆ11 cˆ22 −

(5.47) μ212 . 4

(5.48)

with 5 1 T80 = (c˜12 − cˆ21 ) − c, ˆ 4 3 80 = 5 c( X ˆ cˆ22 − c˜12 ) (see Sect. 5.2.2). 6 It follows that CT = kCX. Table 5.3 contains values of c, ˆ cˆ11 , c˜12 , cˆ21 , and cˆ22 at the equilibrium state as well as χ, d, t, kT , kX , k, !. It is evident that the conditions so that ST and SX are positive definite at equilibrium are satisfied.

Table 5.3 Adimensional parameters = cˆ = c( ˆ E) = + 5 c˜12 (E) = cˆ11 = c˜11 (E) 2 = c˜12 = c˜12 (E) = − c˜11 (E) = + 5 cˆ22 (E) = cˆ21 = 25 c˜21 (E) 2 2 = − c˜12 (E) = cˆ22 = c˜22 (E) 5

χ d t kT kX k !

L = 6 × 10−7 m −21.108 −343.950 −11.610 −133.520 −20.363 219.182 2.036 × 104 1.923 × 107 20.994 703.963 1.036 1.738 × 103

L = 3 × 10−7 m −10.554 −171.975 −5.805 −66.760 −10.181 109.591 5.090 × 103 1.202 × 106 10.497 175.991 2.071 434.528

5.2 Nonlinear Asymptotic Stability of the Equilibrium State

157

5.2.5 Estimates for J (0) , J (2) Now J (0) and J (2) are analyzed. Let us begin with J (0). By taking into account (5.46), we have (see Sect. 5.2.3)

J

(0)

 3 1 2 2 2 Θ + J2 = (Y, A0 Y ) + α1 (Dx , BDx ) + α3 (Lx , BLx ) + α5 dx 5 0

1 2

 (Y, A2 Y ) + α1 (Dx , BDx ) + α3 (Lx , BLx ) + α5

≥

2 2 Θ + J2 5

3 dx,

0

where ⎞ ⎞ ⎛ α1 A α2 A 0 α1 α2 0 A2 = ⎝α2 A nA α4 A⎠ = ⎝α2 n α4 ⎠ ⊗ A, 0 α4 n1 0 α4 A n1 A ⎛

n = α3 + α2 kT + α1 kX , n1 = α5 k1 + α4 kT + α3 kX , 4 2 is such that A1 ≥ k1 A, k1 = 1 − 5 ⊗ is the Kronecker product. Since A > 0, the matrix A2 > 0 if ⎛ ⎞ α1 α2 0 ⎝α2 n α4 ⎠ > 0, 0 α4 n1 i.e. if α1 n − α22 > 0, α1 nn1 − n1 α22 − α1 α42 > 0.

(5.49)

With the choice α1 = kα2 ,

α3 = kα4 ,

(5.50)

where k is determined from (5.47) (see Table 5.1), the inequalities (5.49) are obviously fulfilled.

158

5 Some Formal Properties of the Hydrodynamical Model

Now we pass to analyze the aggregate J (2)

J

(2)

1 > (Y, A1 Y ) + α2 (Dx , BDx ) + α4 (Lx , BLx )+ =2 0

+ α5

 ?  β εˆ β −cˆ11 J 2 − μ12 J Θ − cˆ22 Θ 2 − dx. J 2 − L2x 2 4ˆε

Note that (Y, A1 Y ) = (Y, SA1 Y ), where SA1 =

A1 + A∗1 . 2

Moreover, by choice of (5.50), the matrix SA1 has the form

 SA1 = block diag α1 ST − α2 A, α2 SX + α3 ST − α4 A, α4 SX + α5 diag(0, −c) ˆ 

(5.51) ≥ diag α2 (kkT − 1), α2 kX + α4 (kkT − 1), α4 kX ⊗ A > 0. We can take α5 > 0 and εˆ > 0 sufficiently small. For example, α5 = εˆ 2 , where εˆ is so small that α4 (Lx , BLx ) − α5

β 2 L > 0. 4ˆε x

Remind that ! > 0 (see Table 5.1), this assures, by using (5.51), that the quadratic form under the integral sign in J (2) is positive definite. We have obtained that the quadratic forms under the integral signs in J (0), J (2) are positive definite at equilibrium. If the constant M∗ is small, this statement remains valid in a neighborhood of the equilibrium state.

Chapter 6

Quantum Corrections to the Semiclassical Models

Based on the considerations of the previous chapters, a natural way to get a quantum macroscopic model is to use MEP in a quantum framework to close the moment system arising from the Wigner equation. The general guidelines can be found in [94, 202]. Recently this approach has been followed in [66] (see also [68] and references therein). However one has to deal with operatorial equations which are very complex to solve numerically. Moreover drastic simplifications of the collision terms are introduced in order to make the problem tractable. Here another strategy is adopted to close the moment system arising from the Wigner equation. The Wigner function and the relative transport equation are expanded in powers of h¯ 2 . The zero order part of the collision operator is supposed to be the same as the semiclassical one, while the first order contribution is supposed to act only on the h¯ 2 correction of the Wigner function and is modeled in a relaxation form. Therefore at the zero order the Wigner function is given by the solution of the semiclassical Boltzmann equation, while the h¯ 2 order correction is obtained with a Chapman–Enskog expansion in the high field scaling. Note that a similar Chapman–Enskog expansion has been used in a semiclassical context in [169] and recently in [39, 65] for a rigorous derivation of a quantum driftdiffusion model in the high-field case. Also in [82] a quantum hydrodynamic model has been obtained via a Chapman–Enskog expansion but using a BGK collision operator and assuming a different scaling: the energy relaxation time longer than the momentum relaxation time, and a small influence on the dynamics by the barrier potential with respect to the self-consistent field. What follows is based on references [177, 178].

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_6

159

160

6 Quantum Corrections to the Semiclassical Models

6.1 Wigner Equation The typical physical situation we want to describe is the case when the main contribution to the charge transport is semiclassical while the quantum effects enter as small perturbations. For example, this is reasonable for devices such as MOSFETs of characteristic length of about ten nanometers under the effect of strong electric fields. The main assumption is that there is a balance between the h¯ 2 drift and collision terms. This can be motivated by observing that in the semiclassical scatterings the collision frequencies (scattering rates) increase as the energy raises up. So, since we expect that quantum effects are relevant at high fields, there should be high energies with consequently high collision frequencies. Similarly to the case of charge transport in confined structures already seen in the first chapter, the starting point for the derivation of the quantum corrections to the semiclassical model is the single particle Wigner–Poisson system which represents the quantum analog of the semiclassical Boltzmann–Poisson system.1 For any r and s ∈ R3 , let us introduce the single-particle density matrix ρ(r, s, t) which is related to the wave function ψ by ρ(r, s, t) = ψ(r, t) ψ(s, t)

(6.1)

ρ(r, r, t) = n(r, t)

(6.2)

and satisfies the relation

with n(r, t) the (average) electron probability density. The time evolution of the density matrix is described by the von Neumann equation (see for example [99, chapter 10]) i h¯

∂ ρ(r, s, t) = (Hr − Hs ) ρ(r, s, t) ∂t

(6.3)

where Hr and Hs represent the Hamiltonians acting with respect to the r and s variables respectively. If E (p) is the energy band in terms of the crystal momentum p = h¯ k, the Hamiltonian reads H (r, p) = E (p) − qV (r, t)

(6.4)

with V (r, t) an external potential, here assumed to be real. Moreover, we assume that E (p) is a even function of the modulus of p.

1 For

the sake of simplicity only the case of a single valley in the conduction band is considered.

6.1 Wigner Equation

161

On account of the quantum mechanics correspondence principle p → −i h¯ ∇r , the von Neumann equation reads i h¯

∂ ρ(r, s, t) = (E (−i h∇ ¯ r ) − E (−i h∇ ¯ s )) ρ(r, s, t) − (qV (r, t) − qV (s, t)) ρ(r, s, t). ∂t

(6.5) We have to specify the meaning of the operator E (−i h¯ ∇r ) ρ(r, s, t). Given a function g ∈ L1 (R3 ), let us denote by F [g](η) its Fourier transform  F [g](η) =

R3v

g(v) e−iv·η d v,

and let us denote by F −1 the inverse Fourier transform F −1 [h(η)] =

1 (2 π)3

 R3η

h(η) eiv·η d η.

We define the operator E (−i h¯ ∇r ) ρ(r, s, t) as a multiplication operator in the Fourier transform space, mapped back in the r-space E (−i h∇ ¯ r ) ρ(r, s, t) :=

1 (2π h) ¯ 3

 R3η ×R3r



E (p) ρ(r , s, t)e−ip·(r −r)/h¯ dp d r . (6.6)

Note that p is the conjugate momentum to r . In order to derive a transport equation, analogously to what done in Chap. 1, let us introduce the single electron Wigner quasi-distribution w(x, p, t), depending on the position x, momentum p = hk ¯ and time t,     y y y  y  w(x, p, t) = F ρ x + , x − , t (x, p, t) = ρ x + , x − , t e−ip·y/h¯ d y. 2 2 2 2 R3

(6.7) If we set u(x, y, t) := ρ(x + y2 , x − y2 , t), then w(x, p, t) = F [u](x, p, t) and, of course,  1 −1 u = F [w] = w(x, p, t)eip·y/h¯ d p. (2π h¯ )3 R3 Now let us denote with p the conjugate momentum associate to y. Under the change of coordinates y r=x+ , 2

y s=x− , 2

162

6 Quantum Corrections to the Semiclassical Models

after observing that ∇r = 12 ∇x + ∇y and ∇s = 12 ∇x − ∇y , one has that the symbols corresponding to E (−i h¯ ∇r ) and E (−i h¯ ∇s ) become   1 E p+ η 2

  1 and E p − η , 2

where the fact that E is an even function has been used. η is the conjugate momentum to x. Fourier transforming Eq. (6.3) gives i h¯

     ∂ 1 1 F [u](x, p, t) = F E p+ η −E p− η u(x, y, t) ∂t 2 2     y  y  −q V x + , t −V x − , t u(x, y, t) (x, p, t). 2 2

One has       1 1 F E p+ η −E p− η u(x, y, t) (x, p, t) 2 2       1 1 1 = η − E p − η × F E p + (2π h¯ )3 2 2 R3η ×R3x   ×u(x , y, t) eiη·(x−x )/h¯ d η d x (x, p, t)       1 1 1 η − E p − η = E p + (2π h¯ )3 R3y ×R3η ×R3 2 2 x



=

1 (2π h¯ )3

 R3η ×R3x

× eiη·(x−x )/h¯ u(x , y, t)e−ip·y/h¯ d y d η d x      1 1  eiη·(x−x )/h¯ w(x , p, t) d η d x E p+ η −E p− η 2 2

and     y  y  V x + , t − V x − , t u(x, y, t) (x, p, t) 2 2       y y  ip ·y/h¯  d p (x, p, t) V x + , t − V x − , t w(x, p , t)e 2 2 F

1 = F (2π h¯ )3 =

1 (2π h) ¯ 3

 

R3p

   y  y   V x + , t − V x − , t w(x, p , t)ei(p −p)·y/h¯ d p d y. 3 3 2 2 R  ×Ry p

6.1 Wigner Equation

163

From the previous relations it follows that the Wigner function satisfies the collisionless Wigner–Poisson system ∂w(x, p, t) + S[E]w(x, p, t) − q Θ[V ]w(x, p, t) = 0, ∂t

(6.8)

div (d ∇V ) = −q(ND − NA − n).

(6.9)

which is the quantum counterpart of the Vlasov–Poisson system. S[E] and Θ[V ] represent the pseudo-differential operators       h¯ h¯ i S[E]w(x, p, t) = ν, t − E p − ν, t E p + 2 2 h¯ (2π)3 R3 ×R3ν x



×w(x , p, t) e−i(x −x)·ν d x d ν       h¯ h¯ i V x + η, t − V x − η, t Θ[V ]w(x, p, t) = 2 2 h¯ (2π)3 R3  ×R3η p 

×w(x, p , t) ei(p −p)·η d p d η. Remark 6.1.1 In the effective mass approximation, S[E]w(x, p, t) coincides with the semiclassical expression v · ∇x w(x, p, t). In fact, if E (p) =

p2 then 2m∗     h¯ h¯ hp ¯ ·ν E p+ ν −E p− ν = 2 2 m∗

and therefore2 i S[E]w(x, p, t) = (2π)3

 R3x ×R3ν

p·ν  w(x , p, t) e−i(x −x)·ν d x d ν m∗

 1 p  = · ∇x w(x , p, t) e−i(x −x)·ν d x d ν 3 ∗ 3 (2π) m Rx ×R3ν  p p w(x , p, t)δ(x − x) d x = ∗ · ∇x w(x, p, t). = ∗ · ∇x m m R 3 x

 2 Recall

that 1 (2 π)3

 R3

e−iη·p d p = δ(η).

164

6 Quantum Corrections to the Semiclassical Models

Remark 6.1.2 In the limit3 as h¯ → 0+ the term Θ[V ]w(x, p, t) tends to −∇x V (x) · ∇p w(x, p, t). In fact, we have      h¯ h¯ 1 V x + η, t − V x − η, t = ∇x V (x) · η + O(h¯ ). h¯ 2 2 and therefore lim Θ[V ]w(x, p, t) =

h¯ →0+

=−

i (2π)3





R3p ×R3η

w(x, p , t) ∇x V (x) · η ei(p −p)·η d p d η

1 ∇x V (x) · ∇p (2π)3





R3p ×R3η

w(x, p , t) ei(p −p)·η d p d η = −∇x V (x) · ∇p w(x, p, t).

 In general w(x, p, t) is real valued but not positive [136]. However it is possible to calculate the macroscopic quantities of interest as expectation values (moments) of w(x, p, t) in the same way as the semiclassical case. For example, the average density is given by4 n(x, t) =

1 (2π h¯ )3

 R3

w(x, p, t) d p.

(6.10)

In fact, one has 1 (2π h) ¯ 3

 R3

w(x, p, t) d p =

1 (2π h) ¯ 3



 R3

w(x, p, t) eip·y/h¯ d p

= [u(x, y, t)]y=0 y=0

= ρ(x, x, t) = n(x, t).

Similarly, for the average linear momentum density the relation n(x, t)V(x, t) =

1 (2π h¯ )3

 R3

v w(x, p, t) d p.

holds.

3 Of

course h¯ must be intended in a scaled form. For a possible scaling see [99]. spin will be not included. Otherwise a factor 2 must be added.

4 The

(6.11)

6.1 Wigner Equation

165

Firstly we prove (6.11) in the parabolic band case. From the time-dependent Schrödinger equation i h¯

∂ ψ(x, t) = H ψ(x, t) ∂t

with the Hamiltonian H =−

h¯ 2 Δ − qV (x), 2m∗

where Δ is the Laplacian operator and V (x) is the potential (assumed real valued), one gets the continuity equation ∂n + ∇ · (nV) = 0 ∂t

(6.12)

provided that one defines the average momentum density as nV = −

 i h¯

¯ ψ∇ψ − ψ∇ ψ¯ . ∗ 2m

(6.13)

The previous expression can be rewritten as nV = −

 i h¯  ∇y u(x, y, t) y=0 m∗

(6.14)

and therefore, under the standard hypotheses for the existence of the Fourier transform of the derivative,    i h¯ 1 ip·y/h¯ nV = − ∗ w(x, p, t) e dp ∇y m (2π h¯ )3 R3 y=0    p 1 1 = w(x, p, t) eip·y/h¯ d p = v w(x, p, t) d p. 3 ∗ (2π h¯ ) (2π h¯ )3 R3 R3 m y=0 Indeed, what we have proved for the density and velocity holds for the average of any operator A whose symbol a(p) is assigned. In order to prove that, let us evaluate the spatial average di A in the coordinate representation  < A >=< ψ|A |ψ >=

R3 ×R3

ψ(r) < r|A |s > ψ(s) d r d s.

By the variable transformation y r=x− , 2

s=x+

y 2

166

6 Quantum Corrections to the Semiclassical Models

the previous relation reads 

  y y y y < x − |A | x + > ψ x + d xd y ψ x− 2 2 2 2 R3 ×R3

 1 y y = w(x, p, t)eip·y/h¯ < x − |A | x + > d x d y d p (2π h¯ )3 R3 ×R3 ×R3 2 2    y 1 y ip·y/h¯ |A | x + > d y d x d p. = w(x, p, t) e < x − (2π h¯ )3 R3 ×R3 2 2 R3 The expression  R3

 =

R3

eip·y/h¯ < x −

y y |A | x + > d y 2 2

y y |A | x − > d y := a(x, p). 2 2

e−ip·y/h¯ < x +

(6.15)

is the so-called Wigner–Weyl transform of the operator A and for any fixed x is its symbol. The Wigner–Weyl transform is a way, not the unique, to deal with operators in the phase-space in analogy with the classical case. Therefore, the density of the observable associated to the operator A is 1 (2π h¯ )3

 R3

w(x, p, t)a(x, p) d p.

(6.16)

Other relevant moments of w are the average energy and energy-flux densities  1 E(p) w(x, p, t) d p, (2π h¯ )3 R3  1 n(x, t)S(x, t) = E(p) v w(x, p, t) d p. (2π h¯ )3 R3

n(x, t)W (x, t) =

(6.17) (6.18)

6.2 Equilibrium Wigner Function Let us denote with ρˆ the density matrix operator. It is related to ρ by the relation  (ρφ)(x, ˆ t) =

R3

ρ(x, y, t)φ(y) d y

for any suitable test function φ. In other words, ρ(x, y, t) is the kernel of ρ. ˆ

6.2 Equilibrium Wigner Function

167

ρˆ solves the operatorial Liouville von-Neumann equation i h¯

∂ ρˆ = [H, ρ], ˆ ∂t

where [H, ρ] ˆ = H ρˆ − ρH ˆ is the commutator. In a steady state, and in particular at ∂ equilibrium, ρˆ = 0 and therefore [H, ρ] ˆ = 0, that is H commutes with ρ. ˆ ∂t The equilibrium density matrix can be obtained by maximizing the quantum entropy [95] which, according to von Neumann [197], is given by SQ = −KB tr(ρˆ ln ρ), ˆ

(6.19)

where tr is the trace operator. If we consider the electrons moving in a thermal bath of phonons at the temperature T , the equilibrium density matrix has to satisfy the constraints trρˆ = 1,

< H >= tr(ρH ˆ ).

(6.20)

Since at equilibrium H commutes with ρ, ˆ there exists an orthonormal basis such that both H and ρˆ have a diagonal representation (for the sake of simplicity we assume that the spectrum is discrete) H =



Ei |ψi >< ψi |,

ρˆ =

i



ρi |ψi >< ψi |.

i

Maximizing the entropy (6.19) under the constraints (6.20) means to maximize the objective function −



kB ρi ln ρi + α 1 −



i

ρi

+β −

i



ρi Ei ,

i

with α and β Lagrange multipliers. One gets, after an obvious renormalization of the Lagrange multipliers, ρi = exp (−β(Ei − ΦF ))) where ΦF is the Fermi level and β =

1 , TL being the lattice temperature. KB TL

168

6 Quantum Corrections to the Semiclassical Models

Therefore, the equilibrium density matrix operator is given by5 ρˆeq = exp(−β(H − ΦF )),

(6.21)

We will denote by ρeq (r, s, β) the density matrix at equilibrium. The dependence on β has been explicitly included. Expanding exp(−βH ) it is possible to get an approximation of ρeq (r, s, β) at the several orders in h¯ . An alternative approach is based on the Bloch equation as follows. By deriving with respect to β, one has  ∂ ρˆeq 1

= −H ρˆeq = − H ρˆeq + ρˆeq H ∂β 2 where the commutation relation between H and ρˆeq has been used. For any suitable test function φ, we have  R3

∂ρeq (r, s, β) 1 φ(s) d s = − ∂β 2



 R3

 Hr ρeq (r, s, β)φ(s) + ρeq (r, s, β)Hs φ(s) d s.

From general considerations in quantum mechanics, we require that H must be self-adjoint 

 R3

ρeq (r, s, β)Hs φ(s) d s =

R3

Hs ρeq (r, s, β)φ(s) d s

and therefore from the previous relations we get the Bloch equation  ∂ρeq (r, s, β) 1 = − Hr ρeq (r, s, β) + Hs ρeq (r, s, β) . ∂β 2

(6.22)

Since at equilibrium ΦF is constant, from the properties of the exponential operator, it is enough consider first the case ΦF = 0 and then mutiply the result by exp (βΦF ).

5 We

recall that if A is a Hermitian operator and f a function regular enough, f (A) is defined as follows. In a basis where A has a diagonal representation  ai |ψi >< ψi |, A= i

with ai eigenvalues of A, we set f (A) =

 i

provided f (ai ) makes sense.

f (ai )|ψi >< ψi |,

6.2 Equilibrium Wigner Function

169

Let us consider a Hamiltonian of the general form (6.4). After the change of variables x+

h¯ η, 2

x−

h¯ η 2

and by Fourier transforming, the Bloch equation reads ∂weq (x, p, β) 1 =− ∂β 2



1 (2π)3



     h¯ h¯ E p+ ν +E p− ν 2 2 R3 ×R3ν x



−

q (2π)3

×weq (x , p, β) e−i(x −x)·ν d x d ν 

     h¯ h¯  V x + η + V x − η × weq (x, p , β) ei(p −p)·η d p d η , 3 3 2 2 R  ×Rη



p

(6.23) where weq (x, p, β) is the equilibrium Wigner function . Since for β = 0 we must have ρˆeq = 1 when ΦF = 0, it follows  R3

ρeq (r, s, 0)φ(s) d s = φ(r)

which implies ρeq (r, s, 0) = δ(r − s), wherefrom weq (x, p, 0) = 1.

(6.24)

Equation (6.23) augmented with (6.24) allows us to determine weq . In view of the application in the next sections, we look for solutions of the form weq (x, p, β) = w(0) (x, p, β) + h¯ 2 w(1) (x, p, β) + o(h¯ 2 ). By taking into account that     h¯ h¯ 1 ∂ 2 E(p) νi νj h¯ 2 + o(h¯ 2 ), E p + ν + E p − ν = 2E (p) + 2 2 4 ∂pi ∂pj     h¯ h¯ 1 ∂ 2 V (x) ηi ηj h¯ 2 + o(h¯ 2 ), V x + η + V x − η = 2V (x) + 2 2 4 ∂xi ∂xj

170

6 Quantum Corrections to the Semiclassical Models

up to the first order in h¯ 2 the Bloch equation reads ∂weq (x, p, β) h¯ 2 ∂ 2 E(p) ∂ 2 weq (x, p, β) = −E(p)weq (x, p, β) + ∂β 8 ∂pi ∂pj ∂xi ∂xj +qV (x)weq (x, p, β) −

q h¯ 2 ∂ 2 V (x) ∂ 2 weq (x, p, β) . 8 ∂xi ∂xj ∂pi ∂pj

(6.25)

At the zero-th order one has ∂w(0) (x, p, β) = −E(p)w(0) (x, p, β) + qV (x)w(0) (x, p, β) ∂β whose solution with w(0) (x, p, 0) = 1 is w(0) (x, p, β) = exp [−βE(p) + qβV (x)] . At the first order in h¯ 2 one has ∂w(1) (x, p, β) 1 ∂ 2 E(p) ∂ 2 w(0) (x, p, β) = −E(p)w(1) (x, p, β) + ∂β 8 ∂pi ∂pj ∂xi ∂xj +qV (x)w(1) (x, p, β) −

q ∂ 2 V (x) ∂ 2 w(0) (x, p, β) . 8 ∂xi ∂xj ∂pi ∂pj

(6.26)

We solve the last equation by looking for solutions of the form w(1) (x, p, β) = g(x, p, β)w(0) (x, p, β) with the function g satisfying the equation ∂g(x, p, β) ∂β =

1 ∂ 2 E(p) ∂ 2 w (0) (x, p, β) ∂ 2 V (x) ∂ 2 w (0) (x, p, β) q − ∂xi ∂xj ∂pi ∂pj 8w (0) (x, p, β) ∂pi ∂pj 8w (0) (x, p, β) ∂xi ∂xj

(6.27) and the initial condition g(x, p, 0) = 0. One finds g(x, p, β) =

  qβ 2 ∂ 2 E(p) ∂ 2 V (x) β 3 2 ∂ 2 E(p) ∂V (x) ∂V (x) ∂ 2 V (x) q + −q vi vj , 8 ∂pi ∂pj ∂xi ∂xj 24 ∂pi ∂pj ∂xi ∂xj ∂xi ∂xj

(6.28) (we recall that v = ∇p E(p) is the electron velocity).

6.2 Equilibrium Wigner Function

171

Altogether, by incuding also the factor exp (βΦF ), we get the equilibrium Wigner function 

qβ 2 h¯ 2 ∂ 2 E(p) ∂ 2 V (x) 8 ∂pi ∂pj ∂xi ∂xj   β 3 h¯ 2 2 ∂ 2 E(p) ∂V (x) ∂V (x) ∂ 2 V (x) q + o(h¯ 2 ). + −q vi vj 24 ∂pi ∂pj ∂xi ∂xj ∂xi ∂xj

weq (x, p, β) = exp [−(βE(p) − ΦF ) + qβV (x)] 1 +

(6.29) In particular for a parabolic band (6.29) becomes   βp2 qβ 2 h¯ 2 weq (x, p, β) = exp − ∗ + βΦF + qβV (x) 1 + ΔV (x) 2m 8m∗   q 2 β 3 h¯ 2 m∗ ∂ 2 V (x) 2 + |∇V (x)| − + o(h¯ 2 ). vi vj 24m∗ q ∂xi ∂xj which is the same result as that obtained by Wigner in his original paper [201]. It is convenient to express the equilibrium Wigner function in terms of the equilibrium local density n(x, t) =

1 (2π h¯ )3

 R3

weq (x, p, β) d p.

After some algebras one gets  

qβ 2 ∂ 2 V (x) q 2 β 3 ∂V (x) ∂V (x) n(x, t) exp (−βE(p)) 2 1 + h¯ weq (x, p, β) = + A0 (β) 8 ∂xi ∂xj 24 ∂xi ∂xj

   Bij (β) ∂ 2 E(p) Aij (β) qβ 3 ∂ 2 V (x) − vi vj − (6.30) − + o(h¯ 2 ), ∂pi ∂pj A0 (β) 24 ∂xi ∂xj A0 (β)

where 1 A0 (β) = (2π h¯ )3 Bij (β) =

1 (2π h¯ )3

 

R3

R3

e

−βE

d p,

1 Aij (β) = (2π h¯ )3

e−βE vi vj d p.

 R3

e−βE

∂ 2E d p, ∂pi ∂pj

172

6 Quantum Corrections to the Semiclassical Models

In the case of the Kane dispersion relation v=

p , m∗ (1 + 2αE)

  ∂ 2 E(p) 1 2α δij − ∗ = ∗ pi pj , ∂pi ∂pj m (1 + 2αE) m (1 + 2αE)2 √ √  4π m∗ 2m∗ +∞ −βE . 4π m∗ 2m∗ A0 (β) = e E(1 + αE)(1 + 2αE) d E := d0 (β), (2π h) (2π h) ¯ 3 ¯ 3 0   √  +∞ . 4π m∗ 2m∗ 4α [E(1 + 2αE)]3/2 −βE Aij (β) = d E, δij e E(1 + αE) − (2π h) 3(1 + 2αE)2 ¯ 3 0 8π √ ∗ Bij (β) = 2m δij 3(2π h) ¯ 3



+∞

0

e−βE

4α [E(1 + 2αE)]3/2 d E, 3(1 + 2αE)2

obtaining (2π h¯ )3 n(x, t) exp (−βE(p)) √ 4πm∗ 2m∗ d0 (β) 2  2 2  qβ ∂ V (x) q 2 β 3 ∂V (x) ∂V (x) 2 + × 1 + h¯ 8 ∂xi ∂xj 24 ∂xi ∂xj

weq (x, p, β) =



2αpi pj δij δij − − ∗ m∗ (1 + 2αE) (m∗ )2 (1 + 2αE)3 m d0 (β)

 +∞ . 4α [E(1 + 2αE)]3/2 −βE e E(1 + αE) − dE 3(1 + 2αE)2 0

  +∞ 3/2 2δij qβ 3 ∂ 2 V (x) −βE 4α [E(1 + 2αE)] vi vj − − e d E, 24 ∂xi ∂xj 3m∗ d0 (β) 0 3(1 + 2αE)2

+o(h¯ 2 ). (6.31) It is worthwhile to stress that in the homogeneous case the h¯ 2 correction vanishes in the parabolic band limit and the equilibrium Wigner function reduces to the semiclassical Maxwellian weq (x, p, β) =

β 3/2(2π h¯ )3 n(x, t) exp (−βE) . √ √ 4πm∗ 2m∗ π

6.3 The Collision Operator

173

Instead, when the Kane dispersion relation is used, even in the homogeneous case, a non trivial contribution h¯ 2 remains in the presence of a constant electric field E = −∇V 2 q 2β 3 (2π h¯ )3 n(x, t) exp (−βE(p)) Ei Ej √ 1 + h¯ 2 24 4πm∗ 2m∗ d0 (β)

weq (x, p, β) = 

δij ∗ m (1 + 2αE) 

+∞

× 0

−



e−βE

2αpi pj ∗ 2 (m ) (1 + 2αE)3

−

δij ∗ m d0 (β)

. 4α [E(1 + 2αE)]3/2 E(1 + αE) − 3(1 + 2αE)2

 dE

.

This implies that the quantum correction affects all the transport parameters even in homogeneous case in the presence of an electric field when more realistic energy bands are used. Besides that of von Neumann, other expressions of the quantum entropy have been also adopted. The interested reader is refereed to [23].

6.3 The Collision Operator If we want to tackle the problem of charge transport, an additional term C [w] must be added to (6.8) representing the Wigner transform of the contribution to the Hamiltonian arising from the electron-phonon interactions. This leads to the collisional Wigner equation ∂w(x, p, t) + S[E]w(x, p, t) − q Θ[V ]w(x, p, t) = C [w] ∂t

(6.32)

which is a sort of quantum Boltzmann equation. A detailed treatment of C [w] can be found in [19, 78, 92], but its complexity makes it not suitable for applications in the simulations of electron devices. In [177] a perturbation of the semiclassical collision term has been used, which is useful for the formulation of macroscopic models. To simplify the notation, in the remaining part of the chapter we rescale the Wigner function as follows 1 w(x, p, t) → w(x, p, t). (2π h¯ )3

174

6 Quantum Corrections to the Semiclassical Models

We suppose that the expansion w = w(0) + h¯ 2 w(1) + O(h¯ 4 )

(6.33)

holds. By proceeding in a formal way, as h¯ → 0 the Wigner equation gives the semiclassical Boltzmann equation q ∂w(0) + v · ∇x w(0) + ∗ ∇x V · ∇v w(0) = C (0) [w(0) ]. ∂t m

(6.34)

Therefore we identify w(0) (x, p, t) with the semiclassical distribution f (x, p, t). At first order in h¯ 2 one finds ∂w(1) ∂ 3E ∂ 3 w(0) 1 +v · ∇x w(1) − ∂t 24 ∂pi ∂pj ∂pk ∂xi ∂xj ∂xk +q∇x V · ∇p w(1) −

∂3 V ∂ 3 w(0) q = C (1) 24 ∂xi ∂xj ∂xk ∂ pi ∂pj ∂pk

(6.35)

with C (1) to be modeled. As a general guideline C [w] should drive the system towards the equilibrium. Let us consider electrons in a thermal bath at the lattice temperature TL = 1/kB β. We make the following assumption Assumption 1   (1) C [w] = C (0) [w(0) ] + h¯ 2 C (1) [w(1) ] = CC [w(0) ] − h¯ 2 ν w(1) − weq + O(h¯ 4 ) (6.36)

with

CC [w(0) ]

the semiclassical collision operator

and ν > 0 quantum collision frequency. Remark 6.3.1 At variance with other approaches, only collision term has a relaxation form. This assures that semiclassical scattering of electrons with phonons and w(0) > 0 and therefore the semiclassical expression of sense.

the h¯ 2 correction to the as h¯ → 0 one gets the impurities. We note that the collision term makes

The value of the quantum collision frequency ν is a fitting parameter that can be determined by comparing the results with the experimental data. We require that C [w] conserves the electron density Assumption 2  R3

C [w] d p = 0.

(6.37)

6.4 Quantum Corrections in the High Field Approximation

175

Property 6.3.1 The collision operator C [w] of the form (6.36) satisfies up to terms O(h¯ 2 ) the following properties: 1. Ker (C [w]) is given by the quantum Maxwellian (0) (1) weq = weq + h¯ 2 weq , (0) the classical Maxwellian. with weq

2.  −kB

R3

C (0) [w(0)] ln

w(0) d p = −kB exp(−βE)



 R3

 ln w(0) + βE C (0) d p ≥ 0,

3.   1 (1) − C (1)[w(1) ] w(1) − weq ≥0 2 Moreover the equality holds if and only if w is the quantum Maxwellian, defined above. Properties 1 and 3 are straightforward. Property 2 is based on the proof in [130–132] valid in the classical case.

6.4 Quantum Corrections in the High Field Approximation In the case of high electric fields and parabolic energy bands E(p) =

1 p2 = m∗ v 2 , ∗ 2m 2

it is possible to get an explicit approximation for w(1) by a suitable Chapman– Enskog expansion. Let us introduce the dimensionless variables x˜ =

x , l0

t˜ =

t , t0

v˜ =

v , v0

with l0 , t0 and v0 = l0 /t0 typical length, time and velocity. Let lV be the characteristic length of variation of the electrical potential V . For high fields we expect that lV is small. Therefore we assume

 V = V lV x˜

176

6 Quantum Corrections to the Semiclassical Models

which implies ∇x V (x) =

1 ∇x˜ V (x). lV

Since m∗ v20 represents a characteristic energy, we set qV = m∗ v20 V˜ . Let us denote by w∗ a typical value of w(1) and set w(1) = w∗ w˜ (1) . We want to have the terms containing the derivative of the electric potential of the same order in the Wigner equation. This is accomplished if we rescale w(0) as w(0) = w¯ w˜ (0) with v0 l 3 l0 w¯ = (m∗ )2 V w∗ . lV t0 Let 1/tC be the characteristic collision frequency. We rescale it according to ν˜ = tC ν. Equation (6.35) can be rewritten as6   3 V˜ 3w (0) ∂ ∂ w˜ (1) ∂ ˜ l 1 0 + v˜ · ∇x˜ w˜ (1) + ∇x˜ V˜ · ∇v˜ w˜ (1) − lV 24 ∂ x˜i ∂ x˜j ∂ x˜k ∂ v˜ i ∂ v˜ j ∂ v˜ k ∂ t˜  t0  (1) . = − ν˜ w˜ (1) − w˜ eq tC Let us introduce the characteristic length associated with the quantum correction of the collision term (a kind of mean free path in a semiclassical context) lC = v0 tC We assume that the quantum effects occur in the high field and collision dominated regime, where drift and collision mechanisms have the same characteristic length. Therefore we set formally lC =1 lV and observe that in the high frequency regime the Knudsen number α=

lC l0

is a small parameter. Substituting in the previous equation, one gets α

∂ 3 V˜ ∂ w˜ (1) ∂ 3 w˜ (0) 1 = + α v˜ · ∇x˜ w˜ (1) + ∇x˜ V˜ · ∇v˜ w˜ (1) − 24 ∂ x˜i ∂ x˜ j ∂ x˜k ∂ v˜ i ∂ v˜ j ∂ v˜ k ∂ t˜   (1) . −˜ν w˜ (1) − w˜ eq

6 Summation

over repeated indices is understood.

6.4 Quantum Corrections in the High Field Approximation

177

The zero order in α gives   1 ∂ 3 w˜ (0) ∂ 3 V˜ (1) . (6.38) ∇x˜ V˜ · ∇v˜ w˜ (1) − = −˜ν w˜ (1) − w˜ eq 24 ∂ x˜i ∂ x˜j ∂ x˜k ∂ v˜ i ∂ v˜ j ∂ v˜ k Coming back to the original (dimensional) variables, by Fourier transforming with respect to the variable v one has7  F

−1

w(1) (x, v, t) =

 iq ∂3 V − ηi ηj ηk F w(0) (η) 24m∗3 ∂ xi ∂ xj ∂ xk ν + miq∗ η · ∇x V 3 (1) +νF weq (η) (x, v, t). 1

(6.39) Here η has the dimension of the inverse of a velocity. Approximating w(0) with fME , we obtain w(x, v, t) ≈ fME (x, v, t) + h¯ 2 w(1) (x, v, t).

(6.40)

For example, in the 8-moment case the following explicit approximation for the Wigner function is obtained (see the results of Sect. 4.5) w(x, v, t) ≈ ∗v2 2   exp(− 3m 9m∗ 21m∗ 4W ) +E V(0)  3/2 n 1 − − 4W 4W 2 4 ∗ 3 πm W    3 9m∗ 27m∗ (0) + −E S ·v 4W 2 20W 3   ∂3 V 1 iq 2 −1 ηi ηj ηk F w(0) (η) +h¯ F − iq ∗3 24m ∂ x ∂ x ∂ x i j k ν + m∗ η · ∇x V ? (1) +νF weq (η) (x, v, t)

(6.41)

7 With obvious meaning of the symbol, we are now expressing w as function of the velocity v instead of p.

178

6 Quantum Corrections to the Semiclassical Models

which can be used for evaluating the unknown quantities in the moment system, associated with the Wigner equation . V(0) and S(0) represent the average velocity and the average energy flux at the zeroth order in h¯ 2 .

6.5 The Quantum Moment Equations In analogy with the semiclassical case, multiplying (6.29) by suitable weight functions φ, depending in the physical relevant cases on the velocity v, and integrating over the velocity, one has the balance equation for the macroscopic quantities of interest    ∂ q w φ(v) d v + ∇x · φ(v)v · w d v + ∗ φ(v) Θ[V ]w d v ∂t R3 m R3 R3  φ(v) C [w] d v. (6.42) = R3

In the 8-moment model the basic variables are the macroscopic density, velocity, energy and energy-flux, that are the moments relative to the weight functions 1, m∗ v, 12 m∗ v2 , 12 m∗ v2 v. By evaluating (6.42) for φ = 1, under the assumption that the necessary moments ∂ 3 w(0) of w(1) (x, v, t) and with respect to v exist, one has ∂ vi ∂vj ∂vk     q q q (0) 2 Θ[V ]w d v = ∇ V · ∇ w d v + h ∇ V · ∇v w(1) d v ¯ x v x m∗ R 3 m∗ m∗ R3 R3   ∂3 V ∂ 3 w(0) q − d v = 0, 24(m∗ )3 ∂ xi ∂ xj ∂ xk R3 ∂ vi ∂vj ∂vk obtaining the continuity equation ∂(nVi ) ∂ n+ = 0. ∂t ∂x i

(6.43)

In order to get the other moment equations we observe that from (6.38) it follows   q ∂ 3 w(0) q ∂3 V (1) ∇ V · φ(v) ∇ w d v − φ(v) dv x v m∗ 24(m∗ )3 ∂ xi ∂xj ∂xk R3 ∂ vi ∂vj ∂vk R3    (1) dv=0 φ(v) w(1) − weq +ν R3

(6.44) for each weight function φ(v) such that the integrals exist.

6.5 The Quantum Moment Equations

179

By taking into account (6.44), multiplying Eq. (6.29), expanded at the first order in h¯ 2 , by the weight functions m∗ v, 12 m∗ v2 , 12 m∗ v2 v, and after integration one finds the balance equations for momentum, energy and energy-flux   ∂(nUij ) ∂ (0) (0) V (0) (nm∗ Vi ) + + n q E = nP , V , S W , (6.45) i i i i ∂t ∂x j ∂(nSj ) ∂ (0) (nW ) + + nqVk E k = nCW (W (0) ), (6.46) ∂t ∂x j   ∂(nFij ) 5 q ∂ (0) (0) (0) S (0) (nSi ) + n W . (6.47) + E W = nP , V , S i i i i ∂t ∂x j 3 m∗ (0)

(0)

Here Vi , W (0) and Pi are the zero order components of the average velocity, energy and energy-flux. The components of the flux of momentum and the flux of energy-flux are defined as Uij =

1 n(x, t)

Fij =

1 n(x, t)

 

R3

R3

m∗ vi vj w(x, v, t) d v,

(6.48)

1 ∗ m vi vj v2 w(x, v, t) d v. 2

(6.49)

The production terms are defined as  n PiV

= 

nPW =  n PiS =

m∗ vi C [w] d v,

(6.50)

R3

1 ∗ 2 m v C [w] d v, 2

(6.51)

R3

1 ∗ 2 m v vi C [w] d v 2

(6.52)

R3

Remark 6.5.1 The quantum corrections affect only the free streaming part, while the drift and production terms appear only at the zero order.     Therefore P W (W (0) ), PiV W (0) , Vi(0) , Si(0) and PiS W (0) , Vi(0) , Si(0) are the same as in the semiclassical case. The system (6.43), (6.45)–(6.47) is not closed for the presence of the unknown quantities Uij , Fij , PiV , P W and PiS . We solve the closure problem with the approximation (6.40), assuming a collision dominated high field regime for the quantum effects.

180

6 Quantum Corrections to the Semiclassical Models

In order to evaluate the unknown quantities present in the moment equations, the following formal lemmas are useful Lemma 6.5.1 

  ∂ ∂ vi1 · · · vik w(1) d v = i k ··· F w(1) (η) ∂ηi1 ∂ηik R3 η=0

(6.53)

Lemma 6.5.2 (1) F weq (0) =

∂ ∂ ∂ ∂ (1) (1) F weq (0) = F weq (0) = 0 ∂ηik ∂ηi ∂ηj ∂ηk

∂ ∂ h¯ 2 β q ∂ 2 V (1) F weq (0) = n(x, t) ∂ηi ∂ηj 12m∗2 ∂xi ∂xj   ∂ 2V ∂ ∂ h¯ 2 q (1) Δη F weq (0) = −n(x, t) + 5 ΔV δ ij ∂ηi ∂ηj ∂xi ∂xj 12m∗3

(6.54) (6.55) (6.56)

The proofs follow by a simple computation. With the aid of these lemmas we get the following closure relations Property 6.5.1 Ji = n Vi = n Vi(0) + O(h¯ 4 ), W = W (0) −

h¯ 2 β q Δ V + O(h¯ 4 ), 24m∗

(6.57) (6.58)

h¯ 2 β q ∂ 2 V 2 (0) + O(h¯ 4 ), (6.59) W δij − 3 12m∗ ∂xi ∂xj   h¯ 2 β 2 q 2 h¯ 2 q ∂ ∂V ∂2V ∂ V (0) Si = Si − + ΔV − Δ V + O(h¯ 4 ), 2 ∂xi ∂xr ∂ xr ∂ xi 24m∗2 ν 8m∗2 ν ∂ xi

Uij =

(6.60) Fij =

h¯ 2 β q 3 ∂ V ∂ 2 V ∂V 10 (W (0) )2 − ∗ 9m 3m∗3 ν 2 ∂ x(i ∂xj ) ∂xr ∂xr   2 h¯ 2 q 2 h¯ 2 β q 3 ∂V ∂V ∂ 3V ∂V 2 ∂ V − ∗3 2 − Δ V + |∇ V | 4m ν ∂xi ∂xj ∂xr ∂xr 12m∗3 ν 2 ∂ xi ∂ xj ∂xi ∂xj   2 2 2 2 h¯ q h¯ q ∂ΔV ∂ V ∂ V − ∗3 2 − ΔV δij + 5 ∗2 ∂xi ∂xj 4m ν ∂x(i ∂xj ) 24m   2 3 h¯ q ∂ΔV ∂ V − ∗2 Vj ) + Vk + O(h¯ 4 ). (6.61) 4m ν ∂x(i ∂x(i ∂xj ∂xk)

6.5 The Quantum Moment Equations

181

In the previous relations parentheses indicate symmetrization, e.g.  1

Aij k + Aikj , 2  1

= Aij k + Aikj + Aj ik + Aj ki + Akij + Akj i . 3!

Ai(j k) = A(ij k)

Proof We prove only the relation for Si . The other ones can be obtained in a similar way. First we observe that up to first order terms in h¯ 2 Si ≈

1 n(x, t)

 R3

  2  1 ∗ ¯ (0) h m∗ vi v2 w(1) d v. m vi v2 w(0) + h¯ 2 w(1) d v = Si + 2 2 n R3

From Lemma 6.5.1 one has    1 ∗ i ∂ m vi v2 w(1) (x, v, t) dv = − Δη F w(1) (η) (x, t) 2 ∂ηi R3 2 η=0 and a lengthy calculation gives ∂ηi Δη F w

(1)

(η) = −4

ν

q 2 ∂V ∂V (m∗ )2 ∂xi ∂xr 3 + i mq∗ η · ∇x V

 −i

∂ 3V ∂ q ∗ 3 24(m ) ∂xa ∂xb ∂xc ∂ηr

   ∂ (0) (1) ηa ηb ηc F w (η) + ν F weq (η) − 2

∂ηr ν +i

 × −i

iq ∂V m∗ ∂xr 2 q m∗ η · ∇x V

 ∂ 3V ∂ ∂  q (0) η η η F w (η) a b c 24(m∗ )3 ∂xa ∂xb ∂xc ∂ηi ∂ηr  iq ∂V ∂ ∂ q2 m∗ ∂xi (1) 2 +ν F weq (η) + 6 2 |∇V |

4 ∂ηr ∂ηi m ν + i q∗ η · ∇x V

 × −i

m

∂ 3V

q (1) ηa ηb ηc F w(0) (η) + ν F weq (η) 24(m∗ )3 ∂xa ∂xb ∂xc

q2 |∇V |2 m2 3 + i mq∗ η · ∇x V

−2

ν  × −i



  q ∂ ∂ 3V ∂  (0) (1) ηa ηb ηc F w (η) + ν F weq (η) 24(m∗ )3 ∂xa ∂xb ∂xc ∂ηi ∂ηi

182

6 Quantum Corrections to the Semiclassical Models

−

ν

+

iq ∂V m∗ ∂xi 2 + i mq∗ η · ∇x V

1 q ν + i m∗ η · ∇x V

 −i

  ∂ 3V q (0) η Δ η η F w (η) η a b c 24(m∗)3 ∂xa ∂xb ∂xc  (1) (η) +ν Δη F weq

 −i

  q ∂ 3V ∂ (0) Δ η η F w (η) η η a b c 24(m∗ )3 ∂xa ∂xb ∂xc ∂ηi  ∂ (1) +ν Δη F weq (η) ∂ηi

The previous relation evaluated at η = 0 reduces to 2iq ∂V ∂ ∂ iq ∂V (1) (1) F weq (η) − Δη F weq (η) + ν m∗ ∂xr ∂ηr ∂ηi ν m∗ ∂xi   ∂ 3V ∂ 1 q (0) F w (η) Δη (ηa ηb ηc ) . + −i ν 24(m∗ )3 ∂xa ∂xb ∂xc ∂ηi η=0 −

Since 

 ∂ Δη ηp ηj ηk ∂ηi

 η=0



= 2 δip δj k + δij δpk + δik δpj ,

by taking into account Lemma 6.5.2, relation (6.60) follows.



h2

Remark 6.5.2 The zero order in ¯ is the same as that obtained in [7, 173]. Remark 6.5.3 Since 

 R3

w

(1)

dv=

R3

(1) weq dv=0

we have  R3

C [w] d v = 0

that is the assumption (6.37) is fulfilled. Remark 6.5.4 The density n is not expanded in powers of h¯ 2 . However if one wants to make a comparison with other models present in the literature, the following formal expansion can be introduced by using the parametrization of weq in terms of the quasi Fermi potential Φ n(x, t) = n(x, t)(0) + h¯ 2 n(x, t)(1) + O(h¯ 4 ),

6.5 The Quantum Moment Equations

183

where n(x, t)(0) = exp (−β E + βq (V + Φ))   βq h¯ 2 β 2 q 2 n(x, t)(1) = n(x, t)(0) |∇V | ΔV + . 12m∗ 2 As already mentioned, other hydrodynamical models have been obtained by using a Chapman–Enskog type expansion. In particular in [82] a model based on the standard balance equations of density, energy and momentum has been investigated. The collision term has been modeled requiring that the productions of momentum and energy have a relaxation form and the scaling has been performed to be appropriate in situations where momentum relaxes to equilibrium much faster than energy, the influence of the barrier is small compared with the effect of the external potential, and there is a dominance of the collisions. It has been found that the quantum corrections to the energy, pressure tensor and energy flux are, as in our case, related to the second order moments of the h¯ 2 part of the equilibrium distribution, which has been determined by solving the Bloch equation with the aid of a smoothed quantum potential [81]. This latter is more suited for dealing with the discontinuities arising from barrier potentials than relationships including high order derivatives of V . However the semiclassical limit of the above model leads to the Baccarani-Wordeman hydrodynamical model which suffers some serious drawbacks, e.g. the violation of Onsager’s reciprocity condition and the presence of a unjustified Fourier law for the heat flux [6, 189]. Remark 6.5.5 In the limit of high frequency ν → ∞ one has the simplified model Ji = n Vi = n Vi(0) + O(h¯ 4 ), W = W (0) − Uij =

h¯ 2 β q Δ V + O(h¯ 4 ) 24m∗

2 (0) h¯ 2 β q ∂ 2 V W δij − + O(h¯ 4 ) 3 12m∗ ∂xi ∂xj (0)

Si = Si Fij =

(6.62)

+ O(h¯ 4 ),

h¯ 2 q 10 (0) 2 (W ) − 9m∗ 24m∗2

(6.63) (6.64) (6.65)

  ∂ 2V ΔV δij + 5 + O(h¯ 4 ). ∂xi ∂xj

(6.66)

From Eq. (6.39) one sees that in the limit ν → ∞, w(1) reduces to the (1) quantum correction of the equilibrium Wigner function weq . The resulting quantum corrections to the tensor Uij are the same as those obtained in [80] by using a shifted Wigner function, but with the semiclassical contribution which contains also a heat flux, not added ad hoc.

184

6 Quantum Corrections to the Semiclassical Models

Remark 6.5.6 Under the assumption (see [80]) that the equilibrium relation ∇V = (q β)−1 ∇ ln n + O(h¯ 2 )

(6.67)

is valid, one can recover a formulation of the quantum corrections of density gradient type. The density gradient version is more familiar because the quantum corrections take a form similar to the Bohm potential arising in the Madelung model of quantum fluids in the zero temperature limit. Moreover some numerical experiments [80] lead to consider it more robust than the original formulation in terms of the derivatives of the electric potential. In the limit ν → ∞ the closure relations in the density gradient form explicitly read (0)

Ji = n Vi = n Vi

(6.69)

2 (0) h¯ 2 ∂ 2 ln n W δij − + O(h¯ 4 ), 3 12m∗ ∂xi ∂xj

(6.70)

Si = Si(0) + O(h¯ 4 ), Fij =

(6.68)

h2

¯ Δ ln n + O(h¯ 4 ), 24m∗

W = W (0) − Uij =

+ O(h¯ 4 ),

10 h¯ 2 (0) 2 n(W ) δ − ij 9m∗ 24m∗2 β

(6.71)   ∂ 2 ln n . Δ ln n δij + 5 ∂xi ∂xj

(6.72)

6.6 Entropy Balance Equation Now we investigate the properties of the entropy of the proposed model based on the approximation w(x, v, t) ≈ fME (x, v, t)+h¯ 2 w(1) (x, v, t). It is shown that a suitable entropy functional satisfies a balance equation with a negative entropy production. Property 6.6.1 The following additional balance equation holds along the solution of the system (6.34) and (6.35)   ∂W (1) 2 + ∇x · Σ + q∇x V · −βm∗ e−β E v(w˜ (1) − w˜ eq ) dv ∂t R3    √ βπ 3/2 β 4 m∗ q ∂ 2 V  +n 2 vvr vs w(1) d v m∗ 24 ∂xr ∂xs R3    ∂ 3 w(0) ∂ 3V q (1) w˜ (1) − w˜ eq d v = P ≤ 0, − ∗ 3 24(m ) ∂xk ∂xr ∂xs R3 ∂vk ∂vr ∂vs

(6.73)

6.6 Entropy Balance Equation

185

where  W = kB +

 R3

h¯ 2 2

 R3

 Σ = kB

   w(0) ln w(0) − 1 + β Ew(0) d v

R3

 2 (1) e−β E w˜ (1) − w˜ eq d v, (6.74)

  v w(0) ln w(0) + β Ew(0) d v − 2kB nV(0)

  2 h¯ 2 (1) + ve−β E w˜ (1) − w˜ eq d v, (6.75) 2 R3       h¯ 2 (1) (1)  (1) (0) (0) (1) kB ln w + β E C + C [w ] w˜ − w˜ eq P= d v, 2 R3 (6.76) (1)

with w˜ (1) and w˜ eq defined through the relations w(1) = e−β E w˜ (1) ,

(1) (1) weq = e−β E w˜ eq .

  Proof First we multiply Eq. (6.34) by kB ln w(0) + β E ≡ h(x, v, t) and integrate with respect to v. Observing that  R3

h(x, v, t)

∂ ∂w(0) d v = kB ∂t ∂t



 R3

   w(0) ln w(0) − 1 + β E w(0) d v,  R3

 = ∇x ·

 R3

v h(x, v, t) w

R3

 h(x, v, t) ∇v w(0) d v = kB

dv−



w(0) v · ∇x h(x, v, t) d v

R3

 R3

v · ∇x w(0) d v

  v h(x, v, t) w(0) − kB w(0) dv,

   ∇v w(0) ln w(0) − 1 + β Ew(0) d v = 0, 

R3

R3

v h(x, v, t) w(0) d v − kB

= ∇x · 

(0)



= ∇x ·

R3

v h(x, v, t) · ∇x w(0) d v

and taking into account the property (6.3.1), one finds ∂W (0) + ∇x · Σ (0) = P (0) ≤ 0, ∂t

(6.77)

186

6 Quantum Corrections to the Semiclassical Models

where  W (0) = kB



R3

 Σ (0) = kB

R3

 P (0) =

R3

   w(0) ln w(0) − 1 + β Ew(0) d v,

  v w(0) ln w(0) + β Ew(0) d v − 2kB nV(0) ,

  kB ln w(0) + β E C (0) d v. (1)

Now we multiply Eq.(6.35) by (w˜ (1) − w˜ eq ) and integrate with respect to v. We observe that  R3

(1) (w˜ (1) − w˜ eq )

 = =

R3

(w˜

∂ 1 ∂t 2

(1)

(1) − w˜ eq )

 R3

∂w(1) dv ∂t ∂ (1) (w(1) − weq )d v + ∂t

(1)

R3

(1) (w˜ (1) − w˜ eq )

∂weq dv ∂t

(1) 2 e−β E (w˜ (1) − w˜ eq ) dv

since 

 R3



w(1) d v =

 R3

(1) weq d v = 0,

 vr vs w(1) d v =

R3

R3

(1) vr vs weq d v.

Similarly, since 

 R3

(1) vi weq d v = 0 and

(1) vi vr vs weq d v = 0,

R3

one has 

 R3

(1) (w˜ (1) − w˜ eq )v · ∇x w(1) d v = ∇x ·



√ βπ − 2 m∗

3/2

R3

1 −β E (1) (1) 2 (w˜ − w˜ eq ) dv ve 2

  β 3q ∂ 2V ∇x n vvr vs w(1) d v 3 24 ∂xr ∂xs R

and  R3

(1) (w˜ (1) − w˜ eq )∇v w(1) d v = −βm∗

 R3

(1) 2 e−β E v(w˜ (1) − w˜ eq ) dv

  √ βπ 3/2 β 4 m∗ q ∂ 2 V  +n 2 vvr vs w(1) d v, m∗ 24 ∂xr ∂xs R3

6.7 Energy-Transport and Drift-Diffusion Limiting Models

187

obtaining ∂ 1 ∂t 2



(1) 2 e−β E (w˜ (1) − w˜ eq ) d v + ∇x ·

R3



√ βπ − 2 m∗   ∗ −βm

3/2

R3

 R3

1 −β E (1) (1) 2 ve (w˜ − w˜ eq ) dv 2

  ∂ 2V vvr vs w(1) d v + q∇x V · ∇x n 24 ∂xr ∂xs R3

β 3q

(1) 2 e−β E v(w˜ (1) − w˜ eq ) dv

   √ βπ 3/2 β 4 m∗ q ∂ 2 V  (1) +n 2 vvr vs w d v m∗ 24 ∂xr ∂xs R3    ∂ 3 w(0) ∂ 3V q (1) (1) w ˜ − w ˜ dv − eq 24(m∗ )3 ∂xk ∂xr ∂xs R3 ∂vk ∂vr ∂vs  (1) 2 = −ν e−β E (w˜ (1) − w˜ eq ) d v ≤ 0. R3

(6.78)

The proof is completed by multiplying Eq. (6.78) by h¯ 2 and summing with Eq. (6.77).  Remark 6.6.1 W is an entropy in a mathematical sense, that is as a Liapunov function. Indeed W (0) is the semiclassical physical enthalpy with reverse sign and W (1) plays the role of a quantum correction at the order h¯ 2 . Remark 6.6.2 Let us consider the spatially homogeneous case with V constant. Then (6.6.1) gives d W (w) = P(w) ≤ 0 dt with P = 0 only if w = weq , which implies that W (w) tends to W (weq ).

6.7 Energy-Transport and Drift-Diffusion Limiting Models As already seen in the semiclassical case, under a suitable scaling, it is possible to get an energy model from the hydrodynamic system of equations and then in the isothermal limit a drift-diffusion model. First we rewrite the energy production term in a relaxation form PW = −

W (0) − kB TL τW

188

6 Quantum Corrections to the Semiclassical Models

with τW energy relaxation time, and as in [174] we assume that the following scaling  t =O

1 δ2



 ,

τW = O

1 δ2

 ,

xi = O

  1 , δ

V = O (δ),

S = O (δ) (6.79)

holds, with δ a formal parameter. The first condition is a long time scaling, that is almost stationary regime. The second one means that the energy relaxation time must be sufficiently long with respect to the typical time of the transient. Equation (6.79)3 is the typical diffusion scaling, while (6.79)4 and (6.79)5 are consistent with the expansion made to get the closure relations. Under the conditions (6.79), equating to zero at the various order in δ the terms appearing in the balance equations one gets the following compatibility conditions Property 6.7.1 Under the diffusion scaling one gets (formally) the energytransport model ∂n ∂(nV i ) + = 0, ∂t ∂x i

(6.80)

∂(nW ) ∂(nS j ) + + neVk E k = nPW , ∂t ∂x j E = −∇x V ,

(6.82)

ΔV = −e(ND − NA − n),

(6.83)

(6.81)

with the closure relations Vi =

2    3 Uik ∂n Fik ∂n 1 ∂Uik ∂V ∂Fik 5q (0) ∂V c22 + −q + − W − c12 n ∂xk ∂xk ∂xi n ∂xk ∂xk 3m∗ ∂xi Δ˜

(6.84)

   3 2 Fik ∂n Uik ∂n 1 ∂Fik 5q ∂Uik ∂V (0) ∂V − c Si = + − W + − q , c11 21 n ∂xk ∂xk 3m∗ ∂xi n ∂xk ∂xk ∂xi Δ˜

(6.85) where ˜ (0) ) = c11 c22 − c12 c21 , Δ(W supplemented by the relations of the property 6.5.1. The proof is as in [174].

6.7 Energy-Transport and Drift-Diffusion Limiting Models

189

If one introduce the equation of state W (0) =

3 kB T 2

(6.86)

the previous energy-transport model can be written by using the electron density and temperature T , besides the electrical potential, as variables. However it is crucial to remark that (6.86) is valid only in the parabolic band case in analogy with the monatomic gas dynamics and it is not justified in the non parabolic case, e.g. with the Kane dispersion relation. In this latter case it is more appropriate to retain the energy W as fundamental variable. Remark 6.7.1 As τW → 0 one has (formally) the drift-diffusion model ∂n ∂(nV i ) + = 0, ∂t ∂x i E = −∇x V ,

(6.88)

ΔV = −e(ND − NA − n),

(6.89)

(6.87)

with Vi given by (6.84) evaluated for W (0) = 32 kB TL . In the limit ν → ∞ one has 2 c22 kB TL −

 5 2 c (k T ) 12 B L ˜ B TL ) 2 m∗ Δ(k   ∂n ∂V 5 × − qn c22 − c12 kB TL ∗ ∂xi 2m ∂xi      h¯ 2 βq h¯ 2 q ∂ 2V ∂ ∂ 2V − c22 − nc12 n ΔV δik + 5 , ∗ ∗ 2 ∂xk 12m ∂xi ∂xk 24(m ) ∂xi ∂xk 1

Ji = nVi =

which can be rewritten in the usual drift-diffusion form J = nV = −Dn ∇n + n μn0 ∇V + ∇ · Q where Dn = −

1 ˜ B TL ) Δ(k

 c22 kB TL −

5 c12 (kB TL )2 2 m∗

is the diffusion coefficient, μn0

  5 =− c − c12 ∗ kB TL ˜ B TL ) 22 2m Δ(k q



190

6 Quantum Corrections to the Semiclassical Models

is the low-field mobility and Q has components Qij = −



1

n c22

˜ B TL ) Δ(k

h¯ 2 βq ∂ 2 V h¯ 2 q − n c 12 12 m∗ ∂xi ∂xk 24(m∗ )2

 Δn δik + 5

∂2n ∂xi ∂xk

 .

The semiclassical part satisfies the Einstein relation Dn = μn0 kB TL /q. If (6.67) is valid, the quantum correction to the current is given by the divergence of the tensor Qik = −

1 ˜ B TL ) Δ(k

 n c22

h¯ 2 ∂ 2 ln n h¯ 2 kB TL − n c 12 12 m∗ ∂xi ∂xk 24(m∗ )2

  ∂ 2 ln n Δ ln n δik + 5 , ∂xi ∂xk

which can be rewritten as Qik = n μn0

h¯ 2 h¯ 2 kB TL ∂ 2 ln n 1 + Δ ln n δik . n c 12 ˜ B TL ) 12 m∗ q ∂xi ∂xk 24(m∗ )2 Δ(k

Chapter 7

Mathematical Models for the Double-Gate MOSFET

In this chapter the ideas presented in the previous sections will be employed to get a mathematical model for the simulation of a Double Gate MOSFET (hereafter DGMOSFET). This model is based on the Schrödinger–Poisson system coupled to a set of energy-transport equations, one for each subband, arising from the moment systems associated to the Eq. (1.28) with closure relations obtained by MEP. The DG-MOSFET is devised as one of the most promising devices to overcome short channel effects that are particularly relevant at nanoscale dimensions [164]. A typical schematic representation is depicted in Fig. 7.1. Assuming the device of infinite length along the y axis, we can limit ourselves to simulate a 2D section. Due to the symmetries and dimensions of the device, the transport is, within a good approximation, one dimensional and along the longitudinal direction with respect to the two oxide layers, while the electrons are quantized in the transversal direction like in Fig. 1.3. We assume that the oxide gives rise to an infinitely deep potential barrier at the oxide/silicon interfaces; in fact realistic values of the potential barrier are higher than 3 V and it is very unlikely to find electrons with such an energy in the device under consideration for the typical values of the applied electric fields. Six equivalent valleys are considered with a single effective mass m∗ = 0.32me , me being the free electron mass. A possible generalization could include both longitudinal and transverse masses. What follows is based on references [143, 145]. A mathematical model for a mixture of 2D and 3D electron gases can be found in [5, 43, 47] and for silicon wires in [159].

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_7

191

192

7 Mathematical Models for the Double-Gate MOSFET

Fig. 7.1 Simulated double gate MOSFET. Along the y axis the device is considered as infinite

7.1 Semiclassical Model for DG-MOSFET The device is represented by the domain [0, Lx ]×[0, Lz +2tox ]. The semiconductor occupies the subset D = [0, Lx ] × [tox , Lz + tox ], the remaining part represents the oxide. tox is the width of each oxide layer, while Lx and Lz are the lengths along the x and z directions. According to Chap. 1, the semiclassical description of the DG-MOSFET is given by the equations we summarize below. • The equation for the envelope function ϕ(z) 

 h¯ 2 d 2 − ∗ 2 − qV ϕ(z) = εϕ(z), 2m dz

tox ≤ z ≤ Lz + tox ,

(7.1)

with the normalization 

Lz +tox

|ϕ(z)|2 d z = 1.

tox

• The Poisson equation for the electrostatic potential V overall the device, semiconductor and oxide, ∇ · (d ∇V ) = −q(ND (r) − n),

(7.2)

7.1 Semiclassical Model for DG-MOSFET

193

where n is the electron density n(r, t) =

+∞ 

ρν (x, y, t)|ϕν (z, t)|2 .

ν=1

• The Boltzmann equations for the electron distributions fν (x,y, kx ,ky , t) in each subband ∞  ∂fν q eff 1 Cνμ [fν , fμ ], ν = 1, 2, . . . + ∇k|| Eν · ∇r|| fν − Eν · ∇k|| fν = ∂t h¯ h¯ μ=1 eff

where Eν

=

(7.3)

1 ∇r εν (r|| ). q ||

ρν is expressed in terms of fν by 2 ρν = (2π)2

 B2

fν (r|| , k|| , t)d 2 k|| ,

with B2 indicating the 2D Brillouin zone, which will be approximated with R2 consistently with the effective mass approximation. εν (r|| ), ν = 1, 2, · · · , are the eigenvalues, solutions of (7.1). The above system of course must be augmented with suitable boundary conditions. In each subband the energy is the sum of a transversal contribution εν and a longitudinal (kinetic) contribution ε|| . The Kane dispersion relation will be used and therefore the energy band reads ⎞ ⎛  1 ⎝ h¯ 2  2 1 + 4α ∗ kx + ky2 − 1⎠ ≡ εν (r|| ) + ε|| (k|| ), Eν (r|| , k|| ) = εν (r|| ) + 2α 2m where α is the non-parabolicity parameter. Consequently the longitudinal electron velocity is v|| =

h¯ k|| 1 . ∇k ε|| = ∗

h¯ || m 1 + 2αε||

(7.4)

The parabolic band approximation (α → 0+ ) will be also considered later. In such a case Eν = ε ν +

 h¯ 2  2 2 k + k y ≡ εν + ε|| , 2m∗ x

194

7 Mathematical Models for the Double-Gate MOSFET

and the longitudinal electron velocity reads v|| =

1 h¯ k|| ∇k|| ε|| = ∗ . h¯ m

The relevant 2D scattering mechanisms, we will include, are acoustic phonon scattering (treated as elastic), and non-polar phonon scattering. Their expressions are given in Chap. 1.

7.2 The Moment System and Its Closure by the MEP The system (7.1), (7.2), and (7.3) furnishes a complete mathematical model for the simulation of the DG-MOSFET of Fig. 7.1. Attempts to directly solve it can be found, for example, in [28, 79, 168, 196] where numerical schemes based on finite differences have been employed. In [196] comparisons with Monte Carlo simulations have been also performed. However the direct numerical approach consisting of solving the Schrödinger–Poisson–Boltzmann system is a daunting computational task and requires very long computing times. This has prompted the development of simpler macroscopic models for CAD purposes. These models can be obtained as moment equations of the Boltzmann transport equations under suitable closure relations. The moment of the ν-th subband distribution with respect to a weight function a(k|| ) reads Ma,ν =

2 (2π)2

 B2

a(k|| )fν (r|| , k|| , t)d 2 k|| .

In particular we take as basic moments  2 fν (r|| , k|| , t)d 2 k|| , (2π)2 B2  1 2 v|| fν (r|| , k|| , t)d 2 k|| , the longitudinal mean velocity Vν = (2π)2 ρν B2  2 1 ε|| fν (r|| , k|| , t)d 2 k|| , the longitudinal mean energy Wν = (2π)2 ρν B2  2 1 ε|| v|| fν (r|| , k|| , t)d 2 k|| . the longitudinal mean energy-flux Sν = (2π)2 ρν B2

the areal density ρν =

(7.5) (7.6) (7.7) (7.8)

The corresponding moment system reads  ∂ρν Cρ,νμ , + ∇r|| · (ρν Vν ) = ρν ∂t μ

(7.9)

7.2 The Moment System and Its Closure by the MEP

195

 ∂(ρν Vν ) (0) + ∇r|| · (ρν F(0) CV,νμ , ν ) + ρν Gν ∇r|| εν = ρν ∂t μ

(7.10)

 ∂(ρν Wν ) + ∇r|| · (ρν Sν ) + ρν ∇r|| εν · Vν = ρν CW,νμ , ∂t μ

(7.11)

 ∂(ρν Sν ) (1) + ∇r|| · (ρν F(1) CS,νμ , ν ) + ρν Gν ∇r|| εν = ρν ∂t μ

(7.12)

where

 1 2 1 (7.13) = v|| ⊗ v|| fν (r|| , k|| , t) d 2 k|| , (2π)2 ρν B2 ε|| ⎛ ⎞ 1

 (0) v ∇ 1 2 Gν ⎜ h¯ k|| || ⎟ 2 (7.14) ⎝1 (1) =

 ⎠ fν (r|| , k|| , t) d k|| , 2 ρ (2π) Gν ν B2 ∇k|| ε|| v|| h¯

   1 2 Cρ,νμ 1  , k ) f  − S (k , k ) f d 2 k|| d 2 k|| , (k = S μν νμ ν || || μ || || (2π)2 ρν B2 ε|| CW,νμ (0)

Fν (1) Fν



CV,νμ CS,ν,μ



=

1 2 2 (2π) ρν B2



v|| ε|| v||

(7.15) 

 Sμν (k|| , k|| ) fμ − Sνμ (k|| , k|| ) fν d 2 k|| d 2 k|| .

(7.16) The above-written moment system is not closed because there are more unknowns than equations. Therefore constitutive relations in terms of the fundamental variables are needed for the extra-fluxes (7.13), (7.14) and the productions terms (7.15), (7.16). By following the guidelines of Chap. 2, MEP will be employed to accomplish such a program. Actually, in a semiconductor electrons interact with phonons, which describe the thermal vibrations of the ions placed at the points of the crystal lattice. However, if one considers the phonon gas as a thermal bath, one has to maximize only the electron component of the entropy. Moreover, the electron gas will be assumed to be sufficiently dilute, so that degeneracy effects can be neglected. Therefore in each subband the expression of the entropy obtained as semiclassical limit of that arising from the Fermi statistics will be taken. However, the total entropy must also consider the contribution from the quantized part.

196

7 Mathematical Models for the Double-Gate MOSFET

We define the entropy density of the system as  +∞ 2kB  2 S [f1 , f2 , · · · ] = − |ϕν (r|| , z, t)| (fν log fν − fν ) d 2 k|| ,    B2 (2π)2 ν=1   quantum effect  semiclassical contribution (7.17)

Remark 7.2.1 The proposed expression of the entropy has been introduced for the first time in [143, 145]. It combines quantum effects and semiclassical transport along the longitudinal direction, weighting the contribution of each subband with the square modulus of the ϕν (z, t)’s arising from the Schrödinger–Poisson block. Of course, the same expression can be adopted also for confinement along two directions, e.g. in quantum wires. According to MEP, the fν ’s are estimated with the distributions fνMEP ’s that ∀(r|| , z) ∈ D, t > 0 solve the problem: max S [f1 , f2 , · · · ] f1 (r|| , ·, t) ∈ F2D f2 (r|| , ·, t) ∈ F2D ··· under the constraints MaA ,ν

2 = (2π)2

(7.18)

 B2

aA (k|| )fν (r|| , k|| , t)d 2 k|| , (7.19)

where MaA ,ν are the basic moments (7.5)–(7.8) and F2D is the space of the summable functions with respect to k|| such that the moments MaA ,ν there exist for any A = 1, 2, . . . , Nν and ν = 1, 2 . . . , Nν being the number of moments used in the ν-th subband.   With the above choice of the functions aA (k|| ) = (1, v|| , ε|| , ε|| v|| ), the resulting maximum entropy distribution functions read (the factor kB has been 2 included into the multipliers as well as the factor (2π) 2)   

fνMEP = exp − λν + λV,ν · v|| + λW,ν + λS,ν · v|| ε|| . In order to complete the procedure one has to insert the fνMEP ’s into the constraints (7.19) and express the Lagrange multipliers as functions of the basic moments ρν , Vν , Wν , Sν . In general it is not possible to assure that such a procedure can be accomplished. The solvability of the MEP problem depends on the band structure and on the choice of the moments. In the case of the Kane dispersion relation the solvability of the MEP problem is guaranteed by the properties proved in Chap. 3.

7.2 The Moment System and Its Closure by the MEP

197

Expressing the Lagrange multipliers in terms of the fundamental moments, except for few particular situations, cannot be accomplished in an analytical way and requires a numerical inversion, which is not practical for numerical simulations of electron devices, since it must be performed at each time or iteration step (see [112] for the case of silicon). To overcome such a difficulty, the same approach as in [7, 140, 173] is used: we assume a small anisotropy of the distribution functions and expand them up to first order with respect to the vectorial Lagrange multipliers λV,ν , λS,ν around zero 

  fνMEP ≈ exp −λν − λW,ν ε|| 1 − λV,ν · v|| + λS,ν · v|| ε|| .

(7.20)

By inserting (7.20) into the constraints (7.5) and (7.6), one gets the following expressions for the densities and the energies1 ρ= ρW =

m∗ h2

π¯

m∗ π h2 ¯



+∞ 0





exp −λ − λW ε|| 1 + 2αε|| dε|| dθ,



+∞

exp (−λ) 0



 ε|| exp −λW ε|| 1 + 2αε|| dε|| ,

wherefrom λW =

1 − 2αW +

.

(1 − 2αW )2 + 16αW , 2W



π h¯ 2 λ = − ln ρ g(W ) , (7.21) m∗

with  g(W ) =

1 2α + λW (λW )2

−1 .

Remark 7.2.2 In the parabolic band limit (α = 0) one has λW =

1 W

in according to the equipartition theorem because the system has two degrees of freedom. Remark 7.2.3 According to non equilibrium thermodynamics [97, 153], at equilibrium each energy Lagrange multiplier is related to the temperature of the system by λW =

1 Here

1 , k B TL

and whenever possible we omit the subband index for simplicity.

198

7 Mathematical Models for the Double-Gate MOSFET

where TL is the lattice temperature. Out of equilibrium, the previous relation is a definition of the electron temperature, which, in general, is different from that of the lattice. In this chapter the lattice temperature will be kept constant assuming the phonons as a thermal bath. For the inclusion of heating effects of the crystal lattice see [138, 139, 157, 158, 180, 181]. By substituting (7.20) into the remaining constraints for the velocities and the energy-fluxes and inverting, one has λV = b11 (W ) V + b12 (W ) S,

λS = b21 (W ) V + b22 (W ) S,

(7.22)

where bij (W ) =

 (−1)i+j −1 m∗  γ(5−i−j ) (W, 0) + α γ(6−i−j ) (W, 0) , b(W ) g(W )

b(W ) = [γ1 (W, 0) + α γ2 (W, 0)] [γ3 (W, 0) + α γ4 (W, 0)] − [γ2 (W, 0) + α γ3 (W, 0)]2 .

Note the symmetry of the coefficients, b12 = b21 , which reminds us of the Onsager reciprocity conditions [154]. The γ ’s are reported in Appendix C. Once one has the explicit expressions of the fνMEP ’s, the needed constitutive relations are obtained by inserting into the extra-variables (7.13)–(7.16) the fνMEP ’s (1) (0) (1) as estimators of the fν ’s. The obtained closure relations for F(0) ν , Fν , G ν , G ν , Cρ,νμ , CW,νμ , CV,νμ CS,νμ are reported in Appendix A. The parabolic band model recovered from the Kane model in the limit α → 0+ is reported in Appendix B. Remark 7.2.4 Note that trying to get directly a MEP model in the parabolic approximation for the energy bands is not possible for the lacking of integrability. Since the MEP distributions have been expanded, one cannot guarantee that the resulting balance equations remain still a quasilinear hyperbolic system. Regarding that the following property holds. Property 7.2.1 The moment system of the subbands augmented with the MEP closure relations forms a quasilinear hyperbolic system in the time direction in the physically relevant range of Wν . Proof Since the differential part of each subband is decoupled in the moment system, we can limit our analysis to the study of a single subband. The subband energies εν are assumed as known functions. Let us consider the quasilinear system of PDEs  ∂ ∂ (0) F (U) + F (i) (U) = B(U, x, t), ∂t ∂x i 2

i=1

(7.23)

7.2 The Moment System and Its Closure by the MEP

199

with U(x, t) vector field belonging to a connected open set Ω ⊆ Rm , ∀ t > 0 and ∀ x = (x1 , x2 ) belonging to a domain D ⊆ R2 . Let F (β) : Ω → Rm ,

β = 0, 1, 2

be functions smooth enough. Defining the Jacobian matrices A (β) = ∇U F (β) ,

β = 0, 1, 2,



the system (7.23) is said to be hyperbolic in the t-direction if det A (0) (U) = 0 and the eigenvalue problem det

2 

ni A

(i)

(U) − μA

(0)

(U) = 0

(7.24)

i=1

has real eigenvalues and the eigenvectors span Rm for all the unit vectors n = (n1 , n2 ) of R2 . In the case under consideration, by omitting the subband index and taking into account the expressions reported in Appendix A, we have ⎞ ρ ⎜V ⎟ ⎜ 1⎟ ⎜ ⎟ ⎜V ⎟ U = ⎜ 2⎟, ⎜W ⎟ ⎜ ⎟ ⎝ S1 ⎠ S2 ⎛

⎞ 1 ⎜V ⎟ ⎜ 1⎟ ⎜ ⎟ ⎜V ⎟ = ρ ⎜ 2⎟, ⎜W ⎟ ⎜ ⎟ ⎝ S1 ⎠ S2 ⎛

F (0)

⎞ V1 ⎜ F (0) ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 ⎟ =ρ⎜ ⎟, ⎜ S1 ⎟ ⎜ (1) ⎟ ⎝F ⎠ 0 ⎛

F (1)

⎞ V2 ⎜ 0 ⎟ ⎟ ⎜ ⎜ (0) ⎟ ⎜F ⎟ =ρ⎜ ⎟, ⎜ S2 ⎟ ⎟ ⎜ ⎝ 0 ⎠ F (1) ⎛

F (2)

and the Jacobian matrices are given by ⎛

1 ⎜ ⎜V1 ⎜ ⎜V2 A (0) =⎜ ⎜W ⎜ ⎜ ⎝S1 S2

0 ρ 0 0 0 0

0 0 ρ 0 0 0

0 0 0 ρ 0 0

0 0 0 0 ρ 0

⎞ ⎛ 0 n·V ⎟ ⎜ 0⎟ ⎜n1 F (0) ⎟ ⎜ 2 (0)  ⎜ 0⎟ (i) =⎜n2 F ⎟, A (n) = n A i ⎟ ⎜ 0⎟ ⎜ n·S i=1 ⎟ ⎜ ⎝n1 F (1) 0⎠ ρ n2 F (1)

n1 ρ 0 0 0 0 0

The prime denotes partial derivation with respect to W . The equation   det A (n) − μA (0) = 0

n2 ρ 0 0 0 0 0

0 n1 ρ(F (0) ) n2 ρ(F (0) ) 0 n1 ρ(F (1) ) n2 ρ(F (1) )

0 0 0 n1 ρ 0 0

⎞ 0 ⎟ 0⎟ ⎟ 0⎟ ⎟. n2 ρ⎟ ⎟ ⎟ 0⎠ 0

200

7 Mathematical Models for the Double-Gate MOSFET

gives the eigenvalues μ1,2 = 0,

(7.25)

μ3,4,5,6

(7.26)

with multiplicity 2  . a(W ) ± a(W )2 − 4b(W ) =± 2

where a(W ) = F (0) + (F (1) ) − W (F (0) ) ,

b(W ) = F (0) (F (1)) − (F (0) ) F (1) .

In Fig. 7.2 the eigenvalues μ3,4,5,6 are plotted in the range 0–1 eV which covers at room temperature of the crystal the physical relevant range of the longitudinal mean energy W . The four eigenvalues μ3,4,5,6 are real and distinct. Therefore each of them has a corresponding eigenspace of dimension one. Concerning the eigenvalue μ = 0, we observe that whatever n we take, the first and fourth rows of A (n) are linearly independent; in fact ρ > 0, n1 and n2 cannot be both zero, moreover  det

F (0) ρ(F (0) ) F (1) ρ(F (1) )

 = ρ b(W )

and, since the eigenvalues μ3,4,5,6 are real and different from zero, it must be b(W ) > 0. Fig. 7.2 The eigenvalues μ3,4,5,6 (108 cm/s) versus the mean longitudinal energy W ranging from 0 to 1 eV

7.3 Energy-Transport Model

201

The second and third rows are proportional and similarly the last two rows. Moreover the second and third rows are linearly independent from the last to rows. It is also evident that the first and fourth rows are linearly independent from the other ones. Therefore the rank of A (n) is four. This implies that the eigenspace associated to μ = 0 has dimension two and completes the proof of the hyperbolicity of the system (7.23). ' & In the one dimensional case one has only the eigenvalues μ3,4,5,6 and similar considerations lead again to the hyperbolicity of the balance equations. In the approximation of longitudinal parabolic energy of the subbands (α = 0), we explicitly have μ3,4,5,6 = ±

4

2±

√  W 2 m∗

which are real and distinct provided W > 0.

7.3 Energy-Transport Model From the moment system closed with the MEP procedure, it is possible to deduce a reduced system of balance laws for the longitudinal density and energy density in each subband, named energy-trasport (ET) equations. These kinds of models are widely used in semiconductor simulations and very efficient numerical schemes have been devised for them. Moreover since the energy-transport equations are of parabolic nature, they enjoy better regularity properties. Under a suitable diffusion scaling, we show how to get an energy-transport approximation of the hyperbolic model obtained in the previous section. In each subband the production terms of the velocities and the energy fluxes can be written in a compact form as (see Appendix A)      CV,ν c11,ν (Wν ) c12,ν (Wν ) Vν = . CS,ν Sν c21,ν (Wν ) c22,ν (Wν ) The subbands are coupled in the productions of the velocities and the energy-fluxes only through the differences of the subband bottom energies. As in [141, 144, 174] we introduce the diffusion scaling  t=O

1 δ2



  1 , r|| = O δ

(7.27)

202

7 Mathematical Models for the Double-Gate MOSFET

and, coherently with the hypothesis of small anisotropy of the distribution functions, we assume V = O (δ) , S = O (δ) as well. Moreover we assume that    Cρ,νμ = O δ 2 , Cρ,ν (Wν ) =

CW,ν (Wν ) =

μ

(7.28)



  CW,νμ = O δ 2 . (7.29)

μ

The physical meaning of these latter relations is as follows. It is sufficient to discuss the first relation, the other is similar. If we introduce the interband relaxation times τνμ , we can write Cρ,νμ = −

ρν − ρμ . τνμ



Therefore (7.291) amounts to require τνμ = O δ −2 , that is τνμ much longer than the typical length scale of the device (long time approximation). Inserting the previous scaling into the moment equations (7.9)–(7.12) one gets     ∂ρ    ν 2 + ∇r|| · (ρν Vν ) = O δ 2 ρν O δ Cρ,νμ , ∂t μ     ∂(ρ V )  ν ν (0) = O (δ) ρν ) + ρ G ∇ ε CV,νμ , + O (δ) ∇r|| · (ρν F(0) O δ3 ν r ν ν ν || ∂t μ     ∂(ρ W )    ν ν 2 O δ CW,νμ , + ∇r|| · (ρν Sν ) + ρν ∇r|| εν · Vν = O δ 2 ρν ∂t μ    ∂(ρ S )   ν ν (1) + O (δ) ∇r|| · (ρν F(1) CS,νμ , O δ3 ν ) + ρν Gν ∇r|| εν = O (δ) ρν ∂t μ

and formally equating to zero the several coefficients of the powers of δ the following energy transport model is deduced ∂ρν + ∇r|| · (ρν Vν ) = ρν Cρ,ν (Wν ), ∂t ∂(ρν Wν ) + ∇r|| ·(ρνSν ) + ρν ∇r|| εν · Vν = ρν CW,ν (Wν ), ∂t

(7.30) (7.31)

7.3 Energy-Transport Model

203

where the index ν runs over the considered subbands and Vν = D11,ν (Wν )∇r|| log ρν + D12,ν (Wν )∇r|| Wν − D13,ν (Wν )∇r|| εν , Sν = D21,ν (Wν )∇r|| log ρν + D22,ν (Wν )∇r|| Wν − D23,ν (Wν )∇r|| εν . The coefficients Dij,ν are given by D11,ν =

c22,ν Fν(0) − c12,ν Fν(1) , cν

D13,ν =

(0) c12,ν G(1) ν − c22,ν Gν , m∗ cν (1)

D21,ν =

c11,ν Fν

(0)

D23,ν =

D12,ν =

c22,ν (Fν(0) ) − c12,ν (Fν(1)) , cν

D22,ν =

c11,ν (Fν ) − c21,ν (Fν ) , cν

(0)

− c21,ν Fν , cν

(1)

(0)

(1)

c21,ν Gν − c11,ν Gν , m∗ cν

with cν = c11,ν c22,ν − c12,ν c21,ν . The system (7.30) and (7.31) coupled with the Schrödinger–Poisson block represents the Schrödinger–Poisson-energy transport (SPET) model. We will use it for the simulations of DG-MOSFETs. Remark 7.3.1 In the stationary case the original hyperbolic system and the corresponding energy-transport model are equivalent at least for smooth solutions in the sense that the constitutive relations (7.30) and (7.31) are exact in the steady case. In order to classify the ET equations, let us rewrite the system (7.30) and (7.31) as ⎛ ⎞ ⎞ ⎛

⎞ ⎛  0 D11,ν − Wν D12,ν ∇r|| ρν + D12,ν ∇r|| (ρν Wν ) ∂ ⎝ ρν ⎠ ⎠ + r.t. = ⎝ ⎠ + divr|| ⎝

 ∂t 0 D21,ν − Wν D22,ν ∇r|| ρν + D22,ν ∇r|| (ρν Wν ) ρν Wν

where r.t. stands for the remaining lower order derivative terms. We would like to show that the diffusion matrix ⎛ ⎞ D11,ν − Wν D12,ν D12,ν ⎠ (7.32) Dˆ ν = ⎝ D21,ν − Wν D22,ν D22,ν in the relevant physical cases is negative definite, that is ξ · Dˆ ν ξ < 0, ∀ξ ∈ R2 , ξ = (0, 0)T . The elements of Dˆ ν indeed depend on the bottom of the subbands εν and envelope functions ϕν (z), ν = 1, 2, 3, . . . that can be evaluated only numerically in a DG-MOSFET. However, if we consider the case of an infinite

204

7 Mathematical Models for the Double-Gate MOSFET

potential barrier, one has the explicit formulas εν =

ν 2 π 2 h¯ 2 2L2z m∗

 ,

ϕν (z) =

2 νπ sin (z − tox ), Lz Lz

  z ∈ tox , tox + Lz ,

ν = 1, 2, . . .

By evaluating the eigenvalues of Dˆ ν with the previous expressions of the bottom energy of the first three subbands, one finds the results plotted in Fig. 7.3. No additional subbands are considered because as will be shown in the last chapter dedicated to the simulations, the inclusion of more than three subbands is practically irrelevant for the investigated physical cases. We find that for the relevant range of energy Wν there are two distinct and real negative eigenvalues in each subband. Therefore at least in the case of an infinite potential barrier, employing the first three subbands, the ET model is represented by two parabolic equations for each subband coupled to the Schrödinger–Poisson system. The reader interested in analytical questions about general ET models is referred to [99, 163]. A recent result of existence for the semiclassical ET MEP model has been obtained in [30].

7.4 Boundary Conditions and Initial Data The device depicted in Fig. 7.1 is considered infinite in the y direction and, therefore, we can assume a traslational symmetry of the solution with respect to such direction. This implies that the solution does not depend on y. With an obvious meaning, we will substitute the dependence on r|| with that on x by retaining, for the sake of simplifying the notation, the same symbols of the variables. Regarding the boundary conditions and the initial data, we discuss separately the Schrödinger–Poisson (SP) block and the energy-transport (ET) equations.

7.4.1 Boundary Conditions and Initial Data for the SP-Block Dirichlet boundary conditions are taken at the gates, that is V = Vug at the upper gate and V = Vlg at the lower gate, with Vug and Vlg prescribed voltages. Homogeneous Neumann boundary conditions are assumed at the oxide external boundaries ∂V = 0, ∂ν with ν unit outward normal (see Fig. 7.4).

eigenvalues of the diffusion matrix of the subband 3

eigenvalues of the diffusion matrix of the subband 2

eigenvalues of the diffusion matrix of the subband 1

7.4 Boundary Conditions and Initial Data

205

0 −0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014

0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 longitudinal mean energy W(eV)

0.8

0.9

0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 longitudinal mean energy W(eV)

0.8

0.9

0

0.1

0.2

0.8

0.9

0 −0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014

0 −0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014

0.3

0.4

0.5

0.6

0.7

longitudinal mean energy W(eV)

Fig. 7.3 Eigenvalues of the matrix Dˆ ν for ν = 1, 2, 3 versus the longitudinal mean energy

206

7 Mathematical Models for the Double-Gate MOSFET

Fig. 7.4 Schematic representation of the boundary conditions

More involved are the boundary conditions at source and drain contacts. In the semiclassical case Dirichlet conditions are imposed, but they are no longer valid in a quantum context due to the confinement. In [171] Neumann conditions are used in order to allow the electrostatic potential to float and guarantee the charge neutrality. In [28] a different approach has been proposed. First a one dimensional Schrödinger–Poisson problem is solved at the boundaries representing the drain/source contacts and its solution is used as boundary condition for the 2D problem at thermal equilibrium. Then the boundary conditions at source and drain for arbitrary bias voltage are fixed adding Vb to the result obtained at equilibrium. Here we employ a different approach using both the previous ideas. We first look for the thermal equilibrium solution, which is obtained setting VS = VD = Vug = Vlg = 0. In order to get the boundary condition as in [28] we proceed solving first the SP equation at x=0. For symmetry reasons, the same result holds at the contact at x = Lx . Since the potential is defined up to an additive constant, we impose V (0, tox ) = 0. Moreover we require that the normal derivative is zero ∂V (0, tox ) at the oxide /silicon interface = 0. On account of the symmetry of the ∂z device, the same conditions hold at z = tox + Lz , that is V (0, tox + Lz ) = 0 and ∂V (0, tox + Lz ) = 0. The following system is therefore solved ∂z ⎧  2 d2 ⎪ h ¯ ⎪ ⎪ − − qV ϕ(z) = εϕ(z), tox ≤ z ≤ tox + Lz , ⎨ 2m∗ dz2 (7.33)   ⎪ ∂V (0, z) ∂ ⎪ ⎪ d = q(n(0, z) − ND ), 0 ≤ z ≤ Lz + 2 tox , ⎩ ∂z ∂z with the density given by the relationship / tox +Lz n(0, z) =

tox

∞ ND (0, η) d η 

Z (0)

ν=1



(eq) εν (0) exp − |ϕν (0, z)|2 k B TL

(7.34)

7.4 Boundary Conditions and Initial Data

207

where Z is the partition function Z (x) =

∞  ν=1



(eq)

εν (x) exp − k B TL

(eq)

and εν (x) is the equilibrium ν-th bottom energy. The formula for n(0, z) is based on the following considerations. If we introduce the Fermi level εF , which at equilibrium is constant, the equilibrium distribution can be written as fν(eq) (x, k|| , t)

(eq)

= exp −

εν

(eq)

(x) + ε|| (k|| ) − εF k B TL

.

As a consequence  2 = f (eq) (x, k|| , t)d 2 k|| (2π)2 R2 ν (eq)

(eq)   εν (x) − εF m∗ 2 k . = exp − T + 2α(k T ) B L B L k B TL π h¯ 2 ρν(eq) (x, t)

By imposing the neutrality condition at the source contact, one has 

tox +Lz

 ND (0, η) d η =

tox

tox +Lz

n(0, η) d η tox

that is 

tox +Lz

ND (0, η) d η =

tox

=

 ρν(eq)

ν=1

+∞  m∗  ν=1

+∞ 

π h¯ 2



tox +Lz

|ϕν (0, η)|2 d η

tox



kB TL + 2α(kB TL )2 exp −

(eq)

εν

(eq)

(0) − εF k B TL

.

It follows exp

(eq) εF

k B TL



tox +Lz

ND (0, η) d η =

m∗ π h¯ 2



tox

kB TL + 2α(kB TL )2

  (eq) εν (0) exp − ν=1 kB TL

 "+∞

208

7 Mathematical Models for the Double-Gate MOSFET

and therefore from n(0, z) =

+∞ 

ρν(eq) (0)|ϕν (0, z)|2

ν=1

one has the relationship (7.34). Once the solution of the system (7.33) is obtained, we pass to determine the equilibrium solution for the whole device with a procedure which makes use of the solution of Eq. (7.33) for specifying the boundary data at x = 0 and x = Lx as follows (the details are given only for x = 0, since the same considerations hold for x = Lx ). The boundary part representing the source contact (Fig. 7.5) ? > Γ = (x, z) ∈ R2 : x = 0, 0 ≤ z ≤ Lz + 2 tox is divided into two disjoint parts (Fig. 7.5) Γ1 and Γ2 , where > ? Γ1 = (x, z) ∈ R2 : x = 0, tox + δ ≤ z ≤ Lz + tox − δ and > ? Γ2 = (x, z) ∈ R2 : x = 0, 0 ≤ z < tox + δ ∪ > ? (x, z) ∈ R2 : x = 0, Lz + tox − δ < z ≤ Lz + 2tox ,

Fig. 7.5 Boundary condition at source

7.4 Boundary Conditions and Initial Data

209

with δ a positive constant less than Lz /2 and tox . We take homogeneous Neumann ∂V = 0 along Γ2 and V (0, z) = V (0, z)1d along Γ1 , with V (0, z)1d conditions ∂ν the potential arising from the solution of the system (7.33). The natural boundary conditions, with homogeneous Neumann conditions on the oxide and Dirichlet conditions on the contact, are recovered setting δ = 0. However, when δ = 0, due to the loss of regularity where the boundary conditions change type, we have a boundary layer. In order to regularize the solution, we have extended the Neumann part also inside the contact. With the above boundary conditions, the thermodynamical equilibrium without transport is obtained by solving the SP block with the density given by  n(x, z) =

tox +Lz

tox

ND (0, η) d η  ∞ Z (0)

ν=1

(eq) εν (x) |ϕν (x, z)|2 . exp − k B TL

(7.35)

At the end in order to include the transport, we couple the SP block with the ET one, taking Dirichlet boundary conditions for the electrostatic potential V (0, z) = V (0, z)(eq) ,

V (Lx , z) = V (0, z)(eq) + Vb ,

(7.36)

where V (0, z)(eq) is the equilibrium potential obtained solving the SP block alone and Vb the applied bias voltage.

7.4.2 Boundary Conditions and Initial Data for the ET Block At variance with the standard approach in the literature, we do not impose Dirichlet conditions on the areal density and on the longitudinal mean energy at source and drain, but ∂ ∂ ρν (x, t) = Wν (x, t) = 0, ∂x ∂x

x = 0, Lx ,

t > 0,

ν = 1, 2, . . .

which are more flexible from a numerical point of view. Fixing Dirichlet boundary conditions for the energy at source and drain, although common in the literature, leads to an inconsistency with Monte Carlo simulations in the semiclassical case [176]. By using Boltzmann statistics, the following initial data e−εν / kB TL ρν (x, 0) = " −ε / k T N D (x), μ B L μ e

Wν (x, 0) = kB TL ,

ν = 1, 2, . . .

210

7 Mathematical Models for the Double-Gate MOSFET

are taken, where  N D (x) =

Lz +tox

ND (x, z) d z tox

is the integrated doping with respect to the transversal coordinate.

Chapter 8

Numerical Method and Simulations

The aim is to simulate the DG-MOSFET of Fig. 7.1 with the model presented in Chap. 7 consisting of the Schrödinger–Poisson block (1.25) and (1.27) coupled to the energy-transport equations (7.30) and (7.31). For the sake of clarity, the complete model is summarized below   h¯ 2 d 2 − ∗ 2 − qV (x, z, t) ϕ(x, z, t) = εϕ(x, z, t), tox ≤ z ≤ Lz + tox , 2m dz (8.1)     ∂ ∂ ∂V (x, z, t) ∂V (x, z, t) d + d = −q(ND (x, z) − n(x, z, t)), ∂x ∂x ∂z ∂z (8.2) ∂ρν (x, t) ∂ + (ρν (x, t)Vν (x, t)) = ρν (x, t)Cρ,ν (Wν (x, t)), (8.3) ∂t ∂x ∂ ∂εν (x, t) ∂(ρν (x, t)Wν (x, t)) + (ρν (x, t)Sν (x, t)) + ρν (x, t) Vν (x, t) = ∂t ∂x ∂x ρν (x, t)CW,ν (Wν (x, t)), (8.4) where the index ν runs over the subbands and n(x, z, t) =

+∞ 

ρν (x, t)|ϕν (x, z, t)|2 ,

ν=1

with ϕν and εν eigenfunctions and eigenvalues of (8.1) satisfying the normalization  Lz +tox |ϕν (x, z, t)|2 d z = 1. The variables Vν and Sν are the comcondition tox

© Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_8

211

212

8 Numerical Method and Simulations

ponents along the x direction of the longitudinal mean velocity and of the mean energy-flux The previous equations are augmented with the boundary conditions discussed in Sect. 7.4. The numerical method we propose, is based on references [45, 46] and advances in time by an explicit discretization of (8.3) and (8.4) with the energy subband bottoms εν frozen at the previous time step. This allows to split the numerical scheme into two parts: the discretization of the Schrödinger–Poisson (SP) block and the discretization of the energy-transport (ET) equations. The discretization of the Schrödinger–Poisson (SP) block will be performed by using the Gummel method for guaranteeing a faster convergence of the iterations, while the energy-transport equations will be numerically solved by a suitable adaptation of the Scharfetter-Gummel scheme. Along the x direction let us introduce the grid points xi , i = ia , . . . , ib , with xi+1 − xi = h = constant, and the middle points xi±1/2 = xi ± h/2. Along the z direction let us introduce the grid points zj , j = ja , . . . , jb , with zj +1 − zj = k = constant. Moreover let us take a uniform time step Δt and set unij := u(xi , zj , n Δt), for a generic function u. It is straightforward to extend the scheme to a non uniform grid and time step.

8.1 Discretization of the Schrödinger–Poisson Equations The discretization of the Schrödinger–Poisson equations has been performed following standard approaches. Let suppose that the time is fixed at the level n, which will be omitted for the sake of clarity in the notation. In order to couple equation (8.1) with the Poisson equation (8.2), an iterative scheme based on the Gummel (l) method has been adopted. For each subband let us denote with ϕνij the value of ϕνij at the l-iterate. The Schrödinger equation is discretized in each slice x = xi with central differencing (l)

(l)

(l)

h¯ 2 ϕi,j +1 − 2ϕi,j + ϕi,j −1 (l) (l) − ∗ − qVi,j ϕi,j = εi ϕi,j , 2m k2 ϕi,ja = 0,

ϕi,jb = 0,

j = ja + 1, · · · , jb − 1, (8.5)

(8.6) (l)

where zja = tox and zjb = tox + Lz . Obtained ϕνi,j from the previous relations, an approximation of n(xi , zj , nΔt) is reconstructed as n(l) i,j =

 ν

ρνi (xi )|ϕν(l) |2 , i,j

(8.7)

8.1 Discretization of the Schrödinger–Poisson Equations

213

and this latter is inserted into the discretization of the modified Poisson equation 

    di+1,j + di,j  (l+1) di,j + di−1,j (l+1) (l+1) (l+1) − Vi−1,j Vi+1,j − Vi,j Vi,j − 2 2    1 di,j +1 + di,j (l+1) (l+1) Vi,j +1 − Vi,j + 2 k 2  di,j + di,j −1  (l+1) (l+1) Vi,j − Vi,j − + Q(l) −1 i,j 2 1 h2

=

(l) q 2 ni,j 

kB TL

(l+1)

Vi,j

 (l) − Vi,j , i = ia + 1, · · · , ib − 1,

j = ja + 1, · · · , jb − 1, (8.8)

plus B.C. where  (l) Qi,j

=

q(NDi,j − n(l) i,j ), if i = ia · · · ib , 0, otherwise,

j = ja · · · jb , (silicon body), (oxide layers),

and the dielectric constant is a piecewise constant function 2 d =

Si in silicon, ox in the oxide.

The r.h.s. of the previous equation gives the nonlinear coupling of the Gummel (l) method. As Vi,j converges to the exact solution, the r.h.s. vanishes and one has the solution of the original Poisson equation. A simple coupling with zero r.h.s. is possible but the convergence is very slow, and therefore not practical for realistic simulations. Instead the Gummel method assures a rather fast convergence. In the literature other approaches have been tried, e.g. the Newton–Raphson one (the interested reader can see [28]). The iteration is continued until the stopping criterion (l+1)

||ni,j

(l)

− ni,j ||∞

||n(l+1) i,j ||∞

(l+1)

< tollerance,

||Vi,j

(l)

− Vi,j ||∞

(l+1) ||Vi,j ||∞

< tollerance

is fulfilled. In the numerical simulation we set the tollerance equal to 10−6 and take (0) as initial guess ϕi,j the result of the previous time step. In particular for the first time (0)

step we take as initial guess the solution of (8.5) with Vi,j = 0. At each iteration (8.5) leads to ib − ia + 1 independent eigenvalues problems, that are solved with the subroutine DSTEVX of the LAPACK library. The linear system arising from (8.8), after ordering the nodes by columns, is solved with the subroutine DGESV of the LAPACK library, without preconditioner.

214

8 Numerical Method and Simulations

Remark As discussed in Sect. 7.4, when we integrate the SP block to get the thermodynamical equilibrium, homogeneous Newman boundary conditions are imposed on Γ2 and Dirichlet conditions on Γ1 in order to regularize the solution. This allows the solution to float, but can create a spurious difference of potential between the gate and the edges of the source and drain contacts. To avoid such a drawback, we need to iterate the solution as follows. At each iterate, the difference of potential between the edges of the contact (source or drain) and the gate, ΔV = V (0, zc ) − Vg , zc = tox , Lz + tox , is evaluated and then we impose in the 1D SP block V (0, zc ) = −ΔV . After few iterations ΔV becomes negligible. As stopping criterium |ΔV | < 10−5 has been adopted. Of course, such further iterations are not necessary for non the equilibrium solutions.

8.2 Discretization of the Energy-Transport Equations The key point for the formulation of the numerical scheme is the following proposition which can be proved by a direct calculation. Property 8.2.1 The following relations G(0)ν = λνW F (0)ν ,

G(1)ν = λνW F (1)ν ,

ν = 1, 2, . . .

(8.9)

hold. Proof We prove the first relation. The other one can be obtained similarly. By taking into account the relations in Appendix A, from the definition of F (0)ν and integrating by parts, by omitting the subband index, one has λW

 +∞  +∞ m∗ (0) y + αy 2 −λW y y + αy 2 −λW y 1 − 2α e F = e dy = dy g(W ) 1 + 2αy λW (1 + 2αy)2 0 0 =

m∗ G(0) . g(W )

' & Thanks to the above proposition in each subband the current density J = ρV and the energy-flux density Z = ρ S can be rewritten as (the subband index is omitted) J = J(1) − J(2),

Z = Z(1) − Z(2) ,

(8.10)

where each term is in a drift-diffusion form c22 c c12 = c

J(1) = J(2)



 ∇r|| (ρF (0) ) + ρλW F (0) ∇r|| ε ,   ∇r|| (ρF (1) ) + ρλW F (1) ∇r|| ε ,

(8.11) (8.12)

8.2 Discretization of the Energy-Transport Equations

c11 c c21 = c

Z(1) = Z(2)

215



 ∇r|| (ρF (1) ) + ρλW F (1) ∇r|| ε ,   ∇r|| (ρF (0) ) + ρλW F (0) ∇r|| ε ,

(8.13) (8.14)

with c = c11 c22 − c12 c21 . cij (r) The drift-diffusion form is evident if one identifies F , r = 0, 1, as c generalized mobilities and qλW as the inverse of a sort of thermal potential. This fact is even more evident in the parabolic case, where λW = 1/W , which at thermal equilibrium reads λW = 1/kB TL . The previous consideration allows us to formulate a numerical method for the transport part by a suitable extension of that proposed in [176] for the semiclassical MEP energy-transport model. We discretize the balance equations (8.3) and (8.4) in the one dimensional case as (J )ni+1/2 − (J )ni−1/2 ρin+1 − ρin + − ρin Cρni + O(h2 , Δt) = 0, Δt h

(8.15)

n n − εi−1 − (ρ W )ni Zi+1/2 − Zi−1/2 Ji+1/2 + Ji−1/2 εi+1 (ρ W )n+1 i + + Δt h 2 2h n

n

n

n

n − ρin CW + O(h2 , Δt) = 0. i

(8.16)

In order to evaluate the components of the currents in the middle points, let us consider the cells Ii+1/2 = [xi , xi+1 ], and expand J (r) , r = 0, 1, in Taylor’s series in Ii+1/2 (hereafter the variables with no temporal index are evaluated at the time step tn = nΔt)

∂J (r) J (r)(x) = (J (r))i+1/2 + (x − xi+1/2 ) ∂x

+ o(h). i+1/2

Moreover, we introduce UT = q λ1W , which, as said, plays a role analogous to the thermal potential in the drift-diffusion model (see [176] for more details), and indicate by U T its piecewise constant approximation U T =  1 1 1 + in the cell Ii+1/2 . Then we define the local mobilities 2 λW (Wi+1 ) λW (Wi ) c22 ρF (0) , c c11 ρF (1) , = c

c12 ρF (1) , c c21 ρF (0) , = c

g11 =

g12 =

(8.17)

g21

g22

(8.18)

216

8 Numerical Method and Simulations

where cpq is a piecewise constant approximation of cpq in the cell Ii+1/2 , p, q = 1, c (W )+c W 2, given by c = pq i pq ( i+1 ) , and c = c c − c c . As in [67], we define pq

11 22

2

12 21

also the local Slotboom variables 

skr = exp ε/q U T gkr

k, r = 1, 2.

Since in each cell J

(1)

Z (1)

 c22 c22 ρF (0) ∂ε (0) + ρF , c cq U T ∂x   ∂ c11 c11 ρF (1) ∂ε ≈ ρF (1) + , ∂x c cq U T ∂x

∂ ≈ ∂x



J

(2)

Z (2)

 c12 c12 ρF (1) ∂ε (1) + ρF , c cq U T ∂x   ∂ c21 c21 ρF (0) ∂ε ≈ ρF (0) + , ∂x c cq U T ∂x

∂ ≈ ∂x



the local Slotboom variables satisfy

 ∂s1r (x)  exp ε/q U T J (r)(x) = ∂x ⎧ ⎫

⎬ (r)

 ⎨ (r) ∂J exp ε/q U T (J )i+1/2 + (x − xi+1/2 ) + o(h) . (8.19) ⎩ ⎭ ∂x i+1/2

At each time step ε is approximated in Ii+1/2 by a piece-wise linear function ε(x, nΔt) = εin +

 x − xi n εi+1 − εin . xi+1 − xi

Integrating (8.19) over Ii+1/2 , one has  (s1r )i+1 − (s1r )i =

exp xi

(r)

= Ji+1/2

h q UT n − εn εi+1 i



xi+1

ε(x) q UT

 (r)

Ji+1/2 dx + O(h2 )

     εi+1 εi exp − exp + O(h2 ), q UT q UT

which, with some algebra, gives (g1r )i+1 − (g1r )i h (g1r )i+1 + (g1r )i , r = 1, 2 +σi+1/2 h

(J (r) )i+1/2 = σi+1/2 coth σi+1/2

where σi+1/2 =

εi+1 −εi . 2 q UT

(8.20)

8.3 Numerical Simulations

217

With the same procedure the following discrete expressions for the two parts of the energy flux are obtained (g2r )i+1 − (g2r )i h (g2r )i+1 + (g2r )i , r = 1, 2. +σi+1/2 h

(Z (r) )i+1/2 = σi+1/2 coth σi+1/2

(8.21)

The spatial error in formulas (8.20) and (8.21) is O(h2 ).

8.3 Numerical Simulations The numerical experiments indicate that it is sufficient to take into account only the first three subbands, at least for the examined cases. The steady state is reached after about 5 ps. By using as units picoseconds for time, microns for length and electronvolts for energy, it is not necessary to adimensionalize the variables. The physical parameters are reported in Tables 4.1 and 4.2 of Appendix B. In order to fix the number of grid points, some preliminary numerical simulations of the thermal equilibrium, which represents also the initial data for the other simulations, suggest to take about 30–40 nodes in the transversal direction. So we take 37 points along the z-axis, but we consider along the longitudinal direction grids with 16, 32 and 64 cells (17, 33 and 65 grid points respectively), in the case VD = 0.5 V and Vlg = Vug = 0 V, with VD voltage applied at the drain with respect to that at the source, and Vlg and Vug voltages respectively applied at the lower and the upper gate. The results relative to the areal density and the longitudinal average energy are plotted for each subband in Fig. 8.1. At variance with the simplified cases considered in [145], the behaviour of the error is not uniform due to the coupling with the Schrödinger–Poisson block. In the drain one has a degradation of the convergence rate cR which is worse in the third subband than in the first subband. If we denote by u(k) the generic variable of the solution with k cells, at each grid point of the coarser grid, cR can be estimated as cRi

u(16)i − u(32)i , = log2 u(32)i − u(64)i

i = 1, 2, . . . , 17,

(8.22)

while a global estimate is given by the average 1  cRi . 17 17

cR =

(8.23)

i=1

In Table 8.1 the mean convergence rates are reported. The longitudinal energies present a converge of higher order than the areal densities. On the base of the

8 Numerical Method and Simulations 6 energy (eV) subband 1

areal density (cm−2) subband 1

218

5 4 3 2 1 0

0

5

10 15 20 25 30 35 40 nanometer energy (eV) subband 2

areal density subband 2

2.5 2 1.5 1 0.5 0

0

5

10 15 20 25 30 35 40 nanometer

energy (eV) subband 3

areal density subband 3

0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04

0

5

10 15 20 25 30 35 40 nanometer

0

5

10 15 20 25 30 35 40 nanometer

0

5

10 15 20 25 30 35 40 nanometer

0.22

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04

0

5

10 15 20 25 30 35 40 nanometer

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06

Fig. 8.1 Comparison of the results for the areal density and the average longitudinal energy in the case VD = 0.5 V and Vlg = Vug = 0 V by using 17 (dotted lines), 33 (asterisk), 65 (open circle) grid points

previous considerations in the following simulations a grid with 65 × 37 grid points is used. As first case we consider a symmetric situation: VD = 0.5 V and Vlg = Vug = −3 V. In Figs. 8.2 and 8.3 we plot the steady state density and the potential. The solution does not present any spurious oscillation or boundary layer and reflects the symmetry of the problem. It is evident that the boundary conditions at source and drain are completely different from the semiclassical ones that are simply n = ND .

8.3 Numerical Simulations

219

Table 8.1 Mean convergence rate by comparing meshes with 17, 33, 65 grid points in the case VD = 0.5 V and Vlg = Vug = 0 V ρ1 1.1066

cR

ρ2 0.7701

ρ3 0.7302

W1 1.4020

W2 1.4849

W3 1.4430

1.4 1.2 1.4

1

n [1020 cm-3]

1.2

0.8

1

0.6

0.8 0.6 0.4

0.4

0.2

0

0.2

0 10 8 0

6 5

10

15

4 20

x [nm]

25

30

z [nm]

2 35

40 0

U[eV]

Fig. 8.2 Stationary density in the case VD = 0.5 V and Vlg = Vug = −3 V

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 10 8 0

6 5

10

15 x[nm]

4 20

25

30

z[nm]

2 35

40 0

Fig. 8.3 Stationary electrostatic potential energy in the case VD = 0.5 V and Vlg = Vug = −3 V

220

8 Numerical Method and Simulations

1.4 1.2 1.4

1

n [1020 cm-3]

1.2

0.8

1

0.6

0.8 0.6 0.4

0.4

0.2

0

0.2

0 10 8 0

6 5

10

15

4 20

x[nm]

25

30

z[nm]

2 35

40 0

U[eV]

Fig. 8.4 Stationary density in the case VD = 0.5 V and Vlg = −3 V, Vug = 3 V

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 10 8 0

6 5

10

15 x[nm]

4 20

25

30

z[nm]

2 35

40 0

Fig. 8.5 Stationary electrostatic potential energy in the case VD = 0.5 V and Vlg = −3 V, Vug = 3V

In Fig. 8.61 the first three subband bottoms are shown. We have a good qualitative agreement with the other numerical simulations known in the literature [28, 79, 196]. As second case we take VD = 0.5 V, Vlg = −3 V and Vug = 3 V. In Figs. 8.4 and 8.5 we plot the density and the potential respectively, while in Fig. 8.62 the first

8.3 Numerical Simulations

221

0.4 subband 3

subband bottom energy (eV)

0.3 0.2 0.1

subband 2 0 subband 1 −0.1 −0.2 −0.3 −0.4 0

5

10

15

20 25 nanometer

30

35

40

35

40

0.4 subband 3

subband bottom energy (eV)

0.3 0.2 0.1 0

subband 2

−0.1 subband 1

−0.2 −0.3 −0.4 0

5

10

15

20 25 nanometer

30

Fig. 8.6 First three subbands at the steady state in the case VD = 0.5 V and Vlg = Vug = −3 V (up), VD = 0.5 V and Vlg = −3 V, Vug = 3 V (down)

222

8 Numerical Method and Simulations

three subband bottoms are shown. One can note the depletion region beneath the upper gate. From Fig. 8.7 it is evident a very accurate current conservation, proving the robustness of the numerical method. In the second case the current is reduced by one half due to the gate voltage in agreement with the behaviour of the density. The areal density, average velocity and energy measured from the first subband bottom and the current in the first three subbands are shown in Figs. 8.8, 8.9, and 8.10. The areal density is not symmetric between source and drain within each subband but the total areal density is so. The drift (mean) velocity has been evaluated according again to the formula " ν ν ρ V V = "ν μ . μρ Similarly the global longitudinal mean energy has been evaluated taking as reference value the bottom of the first subband according to the formula " W =

ν

ρ ν (W ν + εν − ε1 ) " μ . μρ

The maximum drift velocity in the channel is one and half times the saturation velocity when Vlg = Vug = −3 V, while it is about two times the saturation velocity when Vlg = −3 V and Vug = 3 V. Moreover in the first case the velocity in the first subband is lower than that in the first subband in the second case. Instead the velocity in the second and third subbands is higher in the first case, but with a resulting lower total longitudinal current. The energy has an evidently different value between source and drain as happens in the semiclassical case. The use of Dirichlet conditions for the energy at the contacts misses such an effect. At last the characteristic curves have been calculated and shown in Fig. 8.11 by fixing Vlg = −3 V and varying Vug from − 3 to 3 V. With increasing Vug the average longitudinal current increases as consequence of the controlling effect of the gate voltage on the electric characteristics of the device. The comparison with the parabolic band case regarding the current is reported in Table 8.2. For moderate values of VD the currents differ of about 10%, but at higher fields (≈0.35 V) the difference becomes of about 25%, clearly showing the influence of the band structure and the fact, already known in the semiclassical case, that the currents are lower when the non-parabolicity effects are included.

8.3 Numerical Simulations

223

70

longitudinal current density (A/cm)

60 total 50 subband 1

40 30 subband 2 20 10 0 −10 0

subband 3

5

10

15

20 nanometer

25

30

35

40

35

40

35

longitudinal current density (A/cm)

30 25

total

20

subband 1

15 10 subband 2 5 0 subband 3 −5 0

5

10

15

20 nanometer

25

30

Fig. 8.7 Average areal current in the first three subbands and global areal current in the case VD = 0.5 V and Vlg = Vug = −3 V (up), VD = 0.5 V and Vlg = −3 V, Vug = 3 V (down)

224

8 Numerical Method and Simulations 8 7

total areal density

areal density (10

13

−2

cm )

6 5 subband 1 4 subband 2 3 2

subband 3

1 0

0

5

10

15

20 nanometer

25

30

35

40

30

35

40

8 7

total areal density

areal density (10

13

−2

cm )

6 subband 1

5 4

subband 2

3

subband 3

2 1 0

0

5

10

15

20 nanometer

25

Fig. 8.8 Areal density in the first three subbands in the case VD = 0.5 V and Vlg = Vug = −3 V (up), VD = 0.5 V and Vlg = −3 V, Vug = 3 V (down)

8.3 Numerical Simulations

225

2 subband 1 total average

velocity (107 cm/sec)

1.5

1

0.5 subband 1

subband 2

0

−0.5 0

5

10

15

20 25 nanometer

30

35

40

2.5 total average

velocity (107 cm/sec)

2

subband 1

1.5

1

subband 2

0.5

0 subband 3 −0.5 0

5

10

15

20 25 nanometer

30

35

40

Fig. 8.9 Average velocity in the first three subbands and global mean velocity in the case VD = 0.5 V and Vlg = Vug = −3 V (up), VD = 0.5 V and Vlg = −3 V, Vug = 3 V (down)

226

8 Numerical Method and Simulations 0.5 0.45

average energy (eV)

0.4

subband 3

0.35 0.3 subband 2

0.25 0.2 0.15

subband 1 0.1

average 0

5

10

15

20 25 nanometer

30

35

40

0.4

0.35

average energy (eV)

subband 3 0.3

0.25

0.2

0.15

average

subband 2

0.1

0.05

subband 1

0

5

10

15

20 25 nanometer

30

35

40

Fig. 8.10 Average total energy measured from the bottom of the first subband W ν + εν − ε1 in the first three subbands and global mean energy in the case VD = 0.5 V and Vlg = Vug = −3 V (up), VD = 0.5 V and Vlg = −3 V, Vug = 3 V (down)

8.3 Numerical Simulations

227

70

longitudinal current A/cm

60

50

40

30 Vg

20

10 0.05

0.1

0.15

0.2

0.25 0.3 V (V) D

0.35

u

0.4

0.45

0.5

Fig. 8.11 Longitudinal mean current (A/cm) versus the source- drain voltage VD with Vlg = −3 V and Vug ranging from −3 to +3 V according to the arrow

Table 8.2 Total longitudinal currents (A/cm) versus VD (V), for Vlg = Vug = 0 V in the parabolic and non-parabolic case VD (V) longitudinal current (A/cm): parabolic case longitudinal current (A/cm): non-parabolic case

0.15 23.065 21.026

0.2 25.256 23.289

0.25 27.436 24.597

0.3 29.113 25.391

0.35 32.790 25.939

Chapter 9

Application of MEP to Charge Transport in Graphene

The last years have witnessed a great interest in 2D-materials for their promising applications. The most investigated one is graphene, but lately also the single-layer transition metal dichalcogenides (TMDCs), like molybdenum disulphite, tungsten diselenite, and black phosphorus have received a certain attention (see for example [204]). Graphene is considered as a potential new semiconductor material for future applications in nano-electronic [20, 40, 48, 55, 121, 151, 172] and optoelectronic devices [204], due to the fact that it has very good mechanical properties, is an excellent heat and electricity conductor, and also has noteworthy optical properties. It is a two-dimensional allotrope of carbon which consists of carbon atoms tightly packed into a honeycomb hexagonal lattice (see Fig. 9.1), due to their sp2 hybridization. Thanks to this structure it has, as first approximation, a conical not curved band structure, so the effective mass of the electrons is zero and charge carriers exhibit a photon-like behavior [48]. Among the graphene-based structures promising for applications in nano electronics and spintronics [183], the graphene nano-ribbons (narrow strips obtained by cutting a piece from a sheet or by removing hydrogen atoms from graphane [73, 188]) are of particular importance [88]. A kinetic approach has been used in [121] for simulating the semiclassical charge transport by adopting a finite difference scheme. Monte Carlo simulations have been performed in [57, 182]. However, the computational complexity of directly solving the transport equations [54–58] makes it desirable to have less complex models, like drift-diffusion, energy-transport and hydrodynamical models, more suitable for CAD purposes. Here a hydrodynamical model is formulated for charge transport in monolayer suspended graphene based on the maximum entropy principle, similarly to that already seen in the previous chapters. The charge carriers are divided into electrons and holes in order to overcome some integrability problems related to the existence of the expectation values of interest and to have a more symmetric description between the valence and the conduction band. The evolution equations © Springer Nature Switzerland AG 2020 V. D. Camiola et al., Charge Transport in Low Dimensional Semiconductor Structures, Mathematics in Industry 31, https://doi.org/10.1007/978-3-030-35993-5_9

229

230

9 Application of MEP to Charge Transport in Graphene

for macroscopic variables like density, energy, velocity and energy-flux are obtained by taking the moments of the transport equations. The constitutive relations needed to have a closed system of balance laws are deduced by resorting to MEP. The maximization problem, MEP leads to, is proved to be globally solvable in the physically relevant region of the field variables. In the same region the evolution equations are shown to form a hyperbolic system of conservation laws. All the main scattering mechanics are included: acoustic phonon, optical phonon and K-phonon interactions. Degeneracy is taken into account as well. Other macroscopic models are also available in the literature. For example, in [101, 203] a spinorial Wigner function has been employed to get drift-diffusion like models for charge transport in graphene, including spin transport as well. Quantum effects have been introduced with a Wigner formalism in [150, 152]. What follows is based on reference [43, 146]. For further developments the interested reader is refereed to [124, 125].

9.1 Kinetic Description The atoms in graphene form two interpenetrating triangular Bravais sub-lattices and each unit cell contains two atoms belonging to different sub-lattices [48] (see Fig. 9.1). The first Brillouin zone is hexagonal as the Bravais lattice (Fig. 9.2). Each atom has three nearest neighbor (nn) atoms belonging to the other sublattice. The nn vectors, that is the vectors drawn from the considered atom to the nn ones, are for a sublattice (see Fig. 9.1) δ1 =

√ √ a a (1, 3), δ 2 = (1, − 3), δ 3 = a(−1, 0), 2 2

a being the distance between nn carbon atoms (a  Å). Similar vectors can be introduced for the other sublattice but the results are the same up to a phase factor. There are six σ bonds per unit cell whose orbitals yield six bands, three of them (called σ ) are below the Fermi energy and the other three are above (the σ ∗ bands). Moreover, each primitive cell contributes two 2pz -orbitals that participate in π bonding. The pz electrons can be treated independently from the other valence electrons, since the overlap between the corresponding orbitals and the s or px and py orbitals is strictly zero by symmetry. Therefore, within the tight binding approximation, the energy dispersion relations of the π bands can be calculated by solving the eigenvalue problem for the approximate1 effective tight binding

1 It

does not take into account the next nearest neighbor hopping correction.

9.1 Kinetic Description

231

Fig. 9.1 Direct graphene lattice with the two sub-lattices in different colors. The rhombus is the primitive cell. Note that it contains two atoms. a1 and a2 are the primitive vectors. a is the distance between two nearest neighbor atoms. Also the vectors connecting the nearest neighbor atoms are drawn for each sublattice

Fig. 9.2 First Brillouin zone in graphene. b1 and b2 are the primitive vectors. K and K are the Dirac points

232

9 Application of MEP to Charge Transport in Graphene

Hamiltonian (see [198])  H (k) = tO

0 γk∗ γk 0

 ,

with tO the nn C–C tight binding overlap energy and γk := eik·δ1 + eik·δ2 + eik·δ3 the sum of the nn phase factors. One finds for the energy bands  E ± = ±tO |γk | = ±tO 3 + 4 cos

√ √ 3a a 3 kx cos ky + 2 cos a 3ky , 2 2

where the + refers to conduction (π ∗ ) band, and the − to the valence (π) band. The points where the bands touch each other are called Dirac points. These points, where one has E ± = 0, coincide with the vertices of the first Brillouin zone     2π 2π 2π 2π  , √ ,− √ K= , K = 3a 3 3a 3a 3 3a (the other vertices have obvious coordinates) and are divided into two nonequivalent sets of three points. These two sets are named K and K -points. In the proximity of the Dirac points, for |k|a 0 the length of all the device, the numerical domain. System (E.1) and (E.2) has to be supplemented with the following boundary

E Simulation Codes

305

conditions at the contacts: n(0, t) = nD (0), n(L, t) = nD (L),

∂W ∂W (0, t) = (L, t) = 0, φ(0) = 0, φ(L) = Vb . ∂x ∂x

(E.6) Note that at variance with the standard approaches, we impose a Neumann condition on the energy instead to fix equal to the thermal value associated to the room temperature. As seen in Chap. 4, this conditions is in better agreement with the MC simulations. The expressions of U , F , CW , cij are reported in Chap. 4. They require a numerical evaluation of some integrals. In order to improve the efficiency of the simulation code, numerical tables have been created for U , F , CW , cij and the values of interest during the run are obtained by interpolation with splines. The tables of the functions U , F , CW , cij are saved in the files fluxes_silicon.kan and produc_silicon.kan which are called by the main program. The readers can create these file data with the codes at the end of this Appendix. The main program is constituted by the MATLAB function etMEP_1d. This function has the following arguments: 1. 2. 3. 4. 5. 6. 7. 8.

the final time (in ps); the bias voltage (in V); the coefficient in the CFL condition (a pure number); the length of the device (in µ); the doping in the n+ region normalized to 1012 cm−3 ; the doping in the n region normalized to 1012 cm−3 ; position of the first junction (in µ); position of the second junction (in µ).

The grid points are fixed at the lines 72 and 73. The first grid point has been set equal to 2 because a ghost point is needed at each boundary for properly taking into account the boundary conditions. The default number of grid points is 97 but the user can change it by modifying the parameter jb. At line 76 the time step is fixed according to a parabolic CFL conditions. At lines 85–87 the doping profile is regularized by using a hyperbolic tangent fitting. The smearing can be tuned through the parameter s at line 81. The initial conditions are set at lines 106–107: the initial density is equal to the doping concentration, the initial energy density is taken equal to the equilibrium 3kB TL times the doping, with TL the room temperature (300 K). Indeed, energy 2 when the Kane dispersion relation is used, the equilibrium energy is slightly different but from a practical point of view the previous choice works well. The files fluxes_silicon.kan and produc_silicon.kan are uploaded in lines 111– 112. The default length of rows is 155. If the user wants to use his own file, he must change the value of the variable n2f lux representing the length of the previous files, fixed at line 115.

306

E Simulation Codes

The program advances in time evaluating at the current time the electrostatic potential, and then the solution at the next time step is obtained by using such potential. The potential is evaluated in the function Poisson1. For the numerical scheme we adopt the strategy delineated in [176], and based on the substitution of the Poisson equation with a parabolic equation which for large time has the solution of the Poisson equation as stationary solution. This approach may appear too sophisticated for a simple 1D problem, and, indeed, a simple discretization with central differences is sufficiently accurate and faster. We adopt the more involved scheme for a didactic purpose: the aim is to make the reader acquainted with the scheme which is very useful in more involved situations like the simulation of MOSFETs or DG-MOSFETs. The Poisson equation is therefore replaced with the following one φt −

∂ ∂x

  ∂φ d = q(ND − n). ∂x

(E.7)

If we introduce a fictitious time step Δtˆ and set φir = φ(xi , rΔtˆ), (E.7) can be discretized in an explicit way as  φir+1 = φir + d Δtˆ

 1

φi+1,j − 2φi,j + φi−1,j + q(NDi − ni ) 2 h

 (E.8)

with the notable advantage to take easily into account the different types of boundary conditions. The price to pay is that at each time step, we need to reach the stationary state of (E.7) by using a time step satisfying the CFL condition, usual for parabolic equations, Δtˆ ≤

h2 2

with h the spatial step. However the computational effort is comparable with that required by direct methods. In fact, after a very short initial transient, the convergence is reached in few iterations. As stopping criterion we require that the difference of two iterates in the L∞ -norm is less than an assigned tolerance. The user can modify the tolerance at line 320. The time evolution is based on an exponential fitting like that employed in the Scharfetter-Gummel scheme for the drift-diffusion model of semiconductors. The current density J and the energy-flux density H are rewritten as J = J (1) − J (2) ,

H = H (1) − H (2)

(E.9)

E Simulation Codes

307

and then each term is put into a drift-diffusion form J (1) =

c22 D



 ∂ ∂ (nU ) − qn φ , ∂x ∂x

  F ∂ c12 ∂ (nF ) − qn φ , D ∂x U ∂x





H (1) =

J (2) =

F ∂ c11 ∂ (nF ) − qn φ , D ∂x U ∂x

H (2) =

c12 D



(E.10)  ∂ ∂ (nU ) − qn φ . ∂x ∂x (E.11)

We introduce the grid points xi , with xi+1 − xi = h = constant, and the middle points xi±1/2 = xi ± h/2. A uniform time step Δt is used. Let us denote with uli the numerical approximation of u(xi , l Δt). The balance equations are discretized as follows nl+1 − nli Ji+1/2 − Ji−1/2 i + = 0, Δt h

(E.12)

(n W )l+1 − (n W )li Hi+1/2,j − Hi−1/2,j Ji+1/2 + Ji−1/2 φi+1 − φi−1 i + −q Δt h 2 2h (E.13) − ni (CW )i = 0. The variables without temporal index must be considered evaluated at time level l. First, we introduce UT = U (W )/q, which plays the role of a thermal potential (Wi+1 ) and indicate by U T its piecewise constant approximation U T = U (Wi )+U in 2q the cell [xi , xi+1 ]. Then we introduce the local mobilities g11 = −

c22 D

nU,

g12 = −

c12 D

nF,

g21 = −

c11 D

nF,

g22 = −

c12 D

nU,

where cpq is a piecewise constant approximation of cpq , p, q = 1, 2, given by  Wi +Wi+1 in the cell [xi , xi+1 ], and the local Slotboom variables cpq = cpq 2

 skr = exp −φ/U T gkr

k, r = 1, 2.

The currents are approximated as follows (g1r )i+1 − (g1r )i (g1r )i+1 + (g1r )i + zi+1/2 , h h (g2r )i+1 − (g2r )i = −zi+1/2 coth zi+1/2 h (g2r )i+1 + (g2r )i , r = 1, 2, +zi+1/2 h

(J (r))i+1/2 = −zi+1/2 coth zi+1/2 (H (r))i+1/2

308

E Simulation Codes

−φi where zi+1/2, = φi+1 . 2U T The function production provides the necessary values of U , F , CW , cij by linear interpolation. If the current value of W is lower than W (n1f lux) or exceeds W (n2f lux), a linear extrapolation is performed. Note that when the variable z (the variable aax in the code) is very small, in order to avoid numerical instabilities, the original expressions for the currents are substituted with a Taylor expansion (lines 185–191).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Numerical simulation of the MEP energy-transport % model for semiconductors by using the Scharfetter-Gummel scheme. % The Poisson equation is solved with a false transient % method. % % A 1D silicon diode is simulated. % % References: % % V. Romano, 2D numerical simulation of the MEP energy-transport model % with a finite difference scheme, J. Comp. Physics 221 439-468 (2007). % % The following files are required: % % fluxes_silicon.kan % produc_silicon.kan % % which contain the fluxes and production terms. % These files are created by the function Silcon_fluxes_production.m % The values of the first and last nodes, in the energy grid, % must be given to the parameters % n1flux, n2flux % % Units: length in micron % time in ps % energy eV % potential in Volt % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

35 36 37

function UU = etMEP_1d(tmax,Vbias,CFL,xmax,Np,Nm,g1,g2)

38 39 40 41 42 43 44 45 46 47 48

% % % % % % % % % %

arguments: tmax = final time of simulation (ps) Vbias = bias voltage (V) xmax = length of the device (micron) CFL = Courant condition: ratio dx/dt (for Vbias = 1 V, set CFL = 0.025) Np = doping in the n+ region normalized to 10^(12) cm ^(-3) Nm = doping in the n region normalized to 10^(12) cm ^(-3) g1 = position of the first junction g2 = position of the second junction

49 50 51 52 53 54

global kbt0sue epsilon mbar

E Simulation Codes 55

%

309

physical parameters

56 57 58 59 60 61 62

% % % % % %

epsilon = dielectric constant in Si/elementary charge in 1/(V micron) elettrone = electron charge (C) tauw= energy relaxation time (ps) mstar= effective electron mass in Si (kg) mbar is the coefficient to change units in the mobilities

63 64 65 66 67 68 69

kbt0sue=0.0259; elettrone=0.1602d-18; epsilon=11.7*8.85d-18/elettrone; mstar=0.32*9.11d-31; mbar=elettrone*1.d-12/mstar;

70 71

%

parameters for the simulations

72 73 74

ja=2; %first node jb=ja+96; %last node

75 76 77

dx=xmax/(jb-ja); dt=dx^2/CFL;

78 79

% normalization of the density in 1/micron^3

80 81 82

s= 1d-3; %spread of the doping around the junction

83 84

dc = (Np-Nm)/2;

85 86 87 88

for j=ja:jb conc(j) = Np-dc*(tanh(((j-ja)*dx-g1)/s)-tanh(((j-ja)*dx-g2)/s)); end

89 90

nn=ja:jb;

91 92

xx=(nn-ja)*dx; %vector of abscissas

93 94

figure

95 96 97 98

plot(xx,conc(ja:jb)) %plot of the doping xlabel('position (micron)') ylabel('doping concentration (10^{17} cm^-3')

99 100 101 102 103

% Initial conditions: % density (variable v(:,1)) equal to doping % energy density (variable (v(:,2)) equal to the equilibrium value

104 105

t=0 %initial time

106 107 108

v(ja:jb,1)=conc(ja:jb); v(ja:jb,2)=1.5d0*conc(ja:jb)*kbt0sue;

109 110

%

load of the production terms

111 112 113

load fluxes_silicon.kan load produc_silicon.kan

114 115 116

n1flux=1; %first energy node in the files fluxes_silicon, produc_silicon.kac n2flux=155; %last energy node in the files fluxes_silicon, produc_silicon.kac

117 118 119

pr=produc_silicon; fx=fluxes_silicon;

120 121

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

310 122 123 124 125 126 127

E Simulation Codes

ww(n1flux:n2flux) = pr(:,1); %values of W prod(n1flux:n2flux,1)= pr(:,2); %c11 (ps) prod(n1flux:n2flux,2)= pr(:,3); %c12 (1/ps) prod(n1flux:n2flux,3)= pr(:,4); %c21 (1/ps/eV) prod(n1flux:n2flux,4)= pr(:,5); %c22 (eV/ps) prod(n1flux:n2flux,5)= pr(:,6); %CW (eV/ps)

128 129

% load of fluxes

130 131 132

flux(n1flux:n2flux,1) = fx(:,2); % U(eV) flux(n1flux:n2flux,2) = fx(:,3); % mstar*F(eV^2)

133 134

phi=Poisson1(v,ja,jb,dx,conc,Vbias);

135 136 137 138

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % evolution %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

139 140

while t j + 1

165 166

aax = (phi(j)-phi(j-1))/(Tpot(j)+Tpot(j-1));

167 168

for p=1:2

169 170 171 172

if abs(aax) > 1.0d-4 cur1 = -aax/tanh(aax)*(mu1(j,p)-mu1(j-1,p))/dx+ ... aax*(mu1(j,p)+mu1(j-1,p))/dx;

173 174 175

cur2 = -aax/tanh(aax)*(mu2(j,p)-mu2(j-1,p))/dx+ ... aax*(mu2(j,p)+mu2(j-1,p))/dx;

176 177

cur(j,p,1)=cur1-cur2;

178 179

else

180 181

aax=2.0d0*aax;

182 183 184 185 186

cur1 = -0.5d0*(exp(aax)+1)* ... (1.0d0 - aax/2.0d0 + aax^2/12.0d0 - aax^4/720.0d0)*... (mu1(j,p)-mu1(j-1,p))/dx+2.0d0*aax*(mu1(j,p)+mu1(j-1,p))/dx;

187 188

cur2 = -0.5d0*(exp(aax)+1)* ...

E Simulation Codes

311 (1.0d0 - aax/2.0d0 + aax^2/12.0d0 - aax^4/720.0d0)* ... (mu2(j,p)-mu2(j-1,p))/dx+ 2.0d0*aax*(mu2(j,p)+mu2(j-1,p))/dx;

189 190 191

cur(j,p)=cur1-cur2;

192 193

end

194 195

end

196

end

197 198 199 200

for j=ja+1:jb-1

201 202

nold = v(j,1);

203 204

v(j,1)=v(j,1)-dt/dx*(cur(j+1,1)-cur(j,1));

205 206

v(j,2)=v(j,2)-dt/dx*(cur(j+1,2)-cur(j,2)) ... + dt/dx*0.5d0*(cur(j,1)+cur(j+1,1))*... 0.5d0*(phi(j+1)-phi(j-1))+dt*v(j,1)*Cw(j);

207 208 209 210

end

211 212 213 214

% boundary conditions at source:

215 216

v(ja,1)=Np; v(ja,2)= v(ja+1,2);

217 218

%Dirichlet condition for the density %Homogenous Neuman condition for the energy

219

% boundary conditions at drain:

220 221

v(jb,1)=Np; v(jb,2)=v(jb-1,2);

222 223

%Dirichlet condition for the density %Homogenous Neuman condition for the energy

224

phi=Poisson1(v,ja,jb,dx,conc,Vbias);

225 226

t=t+dt;

227 228 229

end

230 231 232

cur(ja,1)=cur(ja+1,1); cur(jb,1)=cur(jb-1,1);

233 234

UU=[v(ja:jb,:) cur(ja:jb,:) phi(ja:jb)'];

235 236

%the oputput is saved in the file et_output1.dat

237 238

save 'et_MEP_output.dat' UU -ascii

239 240 241 242 243 244

%warning: the variable in v are defined in the grid points while % the currents are defined in the middle points % The output file contains: % density (10^{17} cm^{-3}), energy density (eV * 10^{17} cm^{-3}, % current/elettrone, energy current, potential (V)

245 246 247 248

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % end evolution %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

249 250 251 252

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

253 254 255

%plot of the density

312 256 257 258 259

E Simulation Codes

figure plot(xx,v(ja:jb,1)) xlabel('position (micron)') ylabel('density (10^{17} cm^{-3}')

260 261

%plot of the energy

262 263 264 265 266

figure plot(xx,v(ja:jb,2)./v(ja:jb,1)) xlabel('position (micron)') ylabel('energy (eV)')

267 268

%plot of the velocity

269 270 271 272 273

figure plot(xx(1:length(xx)-1) + dx/2,cur(ja:jb-1,1)./v(ja:jb-1,1)*10.0d0) xlabel('position (micron)') ylabel('velocity (10^7 cm/sec)')

274 275 276

%plot of the current %current. Note that it is defined in the middle points

277 278 279 280 281

figure plot(xx(1:length(xx)-1) + dx/2,cur(ja:jb-1,1)*1.0d1*elettrone*1.0d19) xlabel('position (micron)') ylabel('current density (A/cm^{-2})')

282 283 284

%plot of the electric potential

285 286 287 288 289

figure plot(xx,phi(ja:jb)) xlabel('position (micron)') ylabel('Potential (V)')

290 291

%plot of the electric field

292 293 294 295 296

electric = - (phi(ja+1:jb) - phi(ja:jb-1))/dx; %electric field in V/micron plot(xx(1:length(xx)-1) + dx/2,electric*1.0d1) xlabel('position (micron)') ylabel('Elecric field (kV/cm)')

297 298 299

end

300 301 302 303

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % auxiliary functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

304 305

function phi=Poisson1(v,ja,jb,dx,conc,Vbias)

306 307 308 309 310

% % %

Solution of the Poisson equation in the one dimensional case by using an iterative scheme: a parabolic equation is solved whose solution tends asymptotically to the solution of the Poisson equation

311 312 313

global kbt0sue epsilon mbar

314 315 316

factor=0.9d0; deltat=factor*0.5d0*(dx^2);

317 318 319

nmaxiterate=10000; tolerance=1.0d-5;

320 321 322

% potential at source

E Simulation Codes 323

313

phi(ja) = 0.0d0;

324 325 326

% potential at drain phi(jb) = Vbias;

327 328 329

ermax=1; niterate=1;

330 331

while (ermax >tolerance)&(niterate = n1flux) & (ncount= ww(n2flux)

509

j1=n2flux;

510 511 512

else

513

j1=max(find(ww 0

322

z=t.^k.*(1+alpha.*t).^2.*(1+2.*alpha.*t).*... exp(-lambdaW.*t);

323 324 325 326

end

327 328

if p