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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Transistors: Types, Materials and Applications : Types, Materials and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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ELECTRICAL ENGINEERING DEVELOPMENTS

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TRANSISTORS: TYPES, MATERIALS AND APPLICATIONS

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ELECTRICAL ENGINEERING DEVELOPMENTS

TRANSISTORS: TYPES, MATERIALS AND APPLICATIONS

BENJAMIN M. FITZGERALD

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

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Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Transistors : types, materials, and applications / editors, Benjamin M. Fitzgerald. p. cm. Includes bibliographical references and index. ISBN 978-1-61728-074-0 (e-book) 1. Transistors. I. Fitzgerald, Benjamin M. TK7871.9.T7334 2010 621.3815'28--dc22 2010014117

Published by Nova Science Publishers, Inc. New York

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CONTENTS

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Preface

vii

Chapter 1

Carbon Nanomaterial Transistors and Circuits Scott Chilstedt, Chen Dong and Deming Chen

Chapter 2

Electronic Properties and Self Consistent Simulations of Carbon Nanotubes in Transistor Technology Mutlu Avci and Serhan Yamacli

35

Chapter 3

Nanowire Field-Effect Transistors Guozhen Shen and Di Chen

73

Chapter 4

Operating Characteristics of Mosfets in Chaotic Oscillators R. Trejo-Guerra, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, C. Cruz-Hernández and C. Sánchez-López

97

Chapter 5

On the Variational Inequalities Approach to Study Electrical Circuits with Transistors K. Addi and D. Goeleven

119

Chapter 6

Photocurrent Study of the Transport Mechanism in Molecular Self-Assembling Field Effect Transistors V. Andrei Pakoulev, Dmitry Zaslavsky and Vladimir Burtman

133

Chapter 7

Organic Field-Effect Transistors: Tetrathiafulvalene Derivatives as Highly Promising Organic Semiconductors M. Mas-Torrent, P. Hadley, S.T. Bromley, J. Veciana and C. Rovira

175

Index

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191

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PREFACE Transistors play a central role in many electronic circuits, where they usually function as either a switch or an amplifier. This book reviews research in the field of transistors including a new class of transistors whose channels are made from semiconducting carbon nanomaterials; the evolution of these designs and the highlights of the work that has driven their development. Also discussed, herein, are the electronic properties and self-consistent simulations of carbon nanotubes in transistor technology; the future developments in the nanowire field-effect transistor research area; the implementation of chaotic oscillators by using transistors designed with CMOS integrated circuit technology and others. Throughout much of the semiconductor industry’s history, the digital transistor maintained a familiar metal-oxide-semiconductor (MOS) structure using a polycrystalline silicon gate, a silicon dioxide insulator, and a single crystal silicon channel. However, continued technology scaling brings a number of new design issues, and alternative structures are becoming advantageous. Recently, high-K dielectrics and metal gates have seen widespread adoption to combat rising leakage power consumption. To keep scaling in the future, alternatives to the conventional silicon channel must also be considered. In Chapter 1, the authors describe a new class of transistors whose channels are made from semiconducting carbon nanomaterials. These nanomaterials come in two forms: carbon nanotubes (CNTs), and graphene nanoribbons (GNRs). The research community has given specific attention to these two carbon allotropes because of their outstanding electrical properties, including high mobilities at room temperature, high current densities, and micronscale mean free paths. There are many possible transistor designs involving CNTs and GNRs, and each offers a unique set of benefits. They also face a number of challenges. This chapter covers the evolution of these designs, and highlights the works that have driven their development. In order to be useful to the semiconductor industry, transistors must be connected together to form higher order circuits. Therefore, state of the art carbon nanomaterial modeling techniques are reviewed, and their application towards nanoscale VLSI circuit evaluation is discussed. The authors also introduce logic gates and small scale circuit structures that use these nanomaterial transistors. Throughout the chapter, an emphasis is placed on identifying the opportunities and challenges involved in the adoption of carbon nanomaterial transistors. Chapter 2 describes the properties of metallic and semiconducting carbon nanotubes and their self-consistent atomic simulations. In the future VLSI technology, semiconductor carbon

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viii

Benjamin M. Fitzgerald

nanotubes are expected to be used as the channel regions of the carbon nanotube field effect transistors (CNTFETs) while the metallic carbon nanotubes are to be used as the interconnects between VLSI circuit components. The advantage of the utilization of the carbon nanotubes stems from the ballistic electronic transport. Current densities up to 109A/cm2 can be carried through the carbon nanotubes when the self–heating and electromigration is much less compared to conventional bulk materials. This chapter starts with reviewing the properties of both metallic and semiconducting carbon nanotubes. The classification of carbon nanotubes according to their indices is given. Boundary conditions, electronic wavefunctions and dispersion relationships of both metallic and semiconductiong carbon nanotubes are derived. In addition to these physical properties, Brillouin zone reciprocal lattice definitions of metallic and semiconducting nanotubes are shown which are necessary for complete understanding of carbon nanotubes in CNTFET operation. The chapter continues with the presentation of the ab initio (first principles) self consistent simulation method of nano scaled devices. As the ab initio solution method, density functional theory (DFT) is reviewed together with its governing equations and assumptions. The non–equilibrium Green function (NEGF) method is also defined which is used for the simulation of carbon nanotubes in non–equilibrium state. The ab initio simulations of various metallic and semiconducting carbon nanotubes are given using ATK® software which implements DFT and NEGF. The properties important from the electronics engineering point of view such as bandgap, conductivity, current – voltage variations, mean free path of electrons and energy band diagrams are obtained from the ab initio simulations and plotted in three dimensional graphics to emphasize these properties and their variation by the change of the carbon nanotube type. The chapter ends with the review of the obtained electronic properties of the carbon nanotubes and their comparison with the conventional silicon technology. The projections on the usage of carbon nanotubes in digital, low frequency analog and high frequency analog circuits are given. One-dimensional (1-D) semiconductor nanostructures have attract great attention in recent years due to their special structures and potential applications in many areas, such as electronics, optoelectronics, sensors, mechanics, etc. Configuring 1-D semiconductor nanostructures as nanoscale field-effect transistors has opened a new and exciting research area named nanowire electronics. In Chapter 3, the authors review variety of nanowire building blocks that have been available to researchers till now and discuss a range of nanoscale devices that could find exciting applications in the future. In the first part, the authors give a brief introduction to nanowire electronics, and then discuss the typical forms of 1-D nanostructures. In the third part, the application of nanowire FETs as high-performance nanoscale sensors, light-emitting diodes, nanolasers, nanowire solar cells, and transparent electronics will be introduced in detail. The chapter will end up with some perspectives and outlook on the future developments in nanowire field-effect transistors research area. Integrated circuit design covers a wide range of signal processing applications which are implemented by linear and nonlinear analog or digital circuits. Actually, in the nonlinear analog domain, chaotic oscillators play an important role because they have the potential to be used to implement secure communication systems. In this manner, Chapter 4 introduces guidelines for designing chaotic oscillators using Metal-Oxide-Semiconductor Field Effect Transistors (MOSFETs). The authors show the operating characteristics of the MOSFET

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Preface

ix

when it is used to realize linear operations such as: amplification, integration, addition and subtraction; and nonlinear operations such as: saturation and in the design of negative resistances. Theoretical results are provided on how to tune the circuit element values of the chaotic oscillator by numerical simulation, and the authors provide guidelines to design a chaotic oscillator by MOSFETs and by using the behavioral simulation in aware of the circuit parasitics. They show how the combination of applying system and transistor level of design leads us to establish a systematic procedure for designing transistor-based chaotic oscillators. Furthermore, to highlight the usefulness of the design, the authors introduce the design of a secure communication system by applying Hamiltonian forms and observer approach for the synchronization part. The main object of Chapter 5 is to discuss the use of variational inequalities models to study electrical circuits involving devices like diodes and transistors. Chapter 6 focuses on the intrinsic charge transport in organic field-effect transistors (OFETs) based on self-assembled monolayers (SAMs) and on the nature of transport in organic systems, in which surface and bulk properties are undistinguishable due to scale of consistent materials. Recently developed SAM-OFETs are characterized by photovoltaic measurements. The dynamics of charge transport are determined and used to clarify a transport mechanism. Taken together, these SAM devices provide a unique tool to study the fundamentals of polaronic transport on organic surfaces and to discuss the SAM OFET performance. An outline is presented of the outstanding problems that are now becoming experimentally reachable owing to the development of SAM-OFETs. Vapor phase molecular self-assembly of 1,4,5,8-Naphthalene-tetracarboxylic diphenylimide (NTCDI) having a rich -stacking charge delivery system is used to enhance the performance of molecular fieldeffect devices. Charge mobility in SAM-OFET could achieve values of more than 30 cm2 V1 -1 s . The dynamics of charge transport in NTCDI-derived SAM-OFETs were probed using time-resolved measurements in an NTCDI-derived photovoltaic cell device. Time-resolved photovoltaic studies allow us to separate the charge annihilation kinetics in the conductive NTCDI channel from the overall charge kinetic in a SAM-OFET device. It has been demonstrated that tuning of the type of conductivity in NTCDI SAM-OFET devices is possible by changing Si substrate doping. In addition, the possibility of measuring transport in highly ordered SAM structures shines light on the polaron charge transfer in organic materials. The authors’ study proposes that a cation-radical exchange (redox) mechanism is the major transport mechanism in SAM nanodevices. The role and contribution of the transport through delocalized states of redox active surface molecular aggregates of NTCDI are exposed and investigated in this chapter. The processing characteristics of organic semiconductors make them potentially useful for electronic applications where low-cost, large area coverage, and structural flexibility are required. Contrary to amorphous silicon, which is widely used in solar cells and flat screen displays, organic materials offer the benefits that they can be deposited on plastic substrates at low temperature by employing solution-based printing techniques, as explained in Chapter 7. These deposition techniques would, therefore, reduce the manufacturing costs dramatically. The challenge now lies on finding organic semiconductors which are processable, stable and, at the same time, exhibit high enough mobilities ( >0.1 cm2/Vs) and ON/OFF current ratios (>106) to be used for applications in modern microelectronics.

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In: Transistors: Types, Materials and Applications Editor: Benjamin M. Fitzgerald, pp. 1-34

ISBN: 978-1-61668-908-7 © 2010 Nova Science Publishers, Inc.

Chapter 1

CARBON NANOMATERIAL TRANSISTORS AND CIRCUITS Scott Chilstedt, Chen Dong and Deming Chen Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign

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Abstract Throughout much of the semiconductor industry’s history, the digital transistor maintained a familiar metal-oxide-semiconductor (MOS) structure using a polycrystalline silicon gate, a silicon dioxide insulator, and a single crystal silicon channel. However, continued technology scaling brings a number of new design issues, and alternative structures are becoming advantageous. Recently, high-K dielectrics and metal gates have seen widespread adoption to combat rising leakage power consumption. To keep scaling in the future, alternatives to the conventional silicon channel must also be considered. In this chapter, we describe a new class of transistors whose channels are made from semiconducting carbon nanomaterials. These nanomaterials come in two forms: carbon nanotubes (CNTs), and graphene nanoribbons (GNRs). The research community has given specific attention to these two carbon allotropes because of their outstanding electrical properties, including high mobilities at room temperature, high current densities, and micronscale mean free paths. There are many possible transistor designs involving CNTs and GNRs, and each offers a unique set of benefits. They also face a number of challenges. This chapter covers the evolution of these designs, and highlights the works that have driven their development. In order to be useful to the semiconductor industry, transistors must be connected together to form higher order circuits. Therefore, state of the art carbon nanomaterial modeling techniques are reviewed, and their application towards nanoscale VLSI circuit evaluation is discussed. We also introduce logic gates and small scale circuit structures that use these nanomaterial transistors. Throughout the chapter, an emphasis is placed on identifying the opportunities and challenges involved in the adoption of carbon nanomaterial transistors.

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Introduction To combat the increasing power consumption at recent technology nodes, silicon dioxide insulators have been replaced by high-k dielectrics, and gates have been changed from polycrystalline silicon to metal. As feature sizes continue to scale, quantum effects will necessitate further modification to the traditional transistor structure. At the high integration densities allowed by nanometer feature sizes and 3D fabrication techniques, the most critical design consideration becomes power density. This means that the minimization of power consumption will be critical not just to save energy, but also to reduce the amount of heat generated by billions of closely packed transistors. In effect, future designs will be power limited. The reduction of leakage power due to gate electrostatics will continue to be critical, and will motivate the exploration of alternative structures such as tri-gate transistors and vertical cavity transistors. However, large power savings can also be achieved by increasing the conductance of the channel. A popular technique to increase conductance in silicon transistors is the use of compressive and expansive strain on the crystalline lattice. This allows hole mobility enhancements of 2-4 times higher on a (100) wafer [1]. However, strained silicon can only go so far, and for continued mobility increases alternative channel materials will eventually need to be considered. Two near term candidates are germanium, and III-V compounds such as InGaAs, InAs, and InSb. Both offer higher mobilities than silicon and, now that high-k dielectrics are being used, no longer have the disadvantage of lacking a stable native oxide. These materials offer respectable increases in mobility but, even when combined with generous strain engineering, will soon hit their own limits of power density. In addition, silicon, germanium, and III-V compounds are likely to become less effective when scaled to atomic dimensions. A more promising long-term solution is the transition to carbon nanomaterial channels. Carbon is a group 14 element that resides above silicon in the periodic table. Like silicon and germanium, carbon has four electrons in its valence shell. It is most commonly found as an amorphous non-metal. However, when carbon atoms are arranged in crystalline structures composed of hexagonal benzene-like rings, they form carbon nanomaterials that offer exceptional electrical properties. For the replacement of silicon in future transistor channels, the two most promising of these allotropes are carbon nanotubes and graphene. In their semiconducting forms, these nanomaterials exhibit room temperature mobilities over ten times greater than silicon. This translates to devices with significant improvements in performance and power savings, allowing higher integration at the same power density. In addition, they can be scaled to smaller feature sizes than silicon while maintaining their electrical properties. It is for these reasons that the Emerging Research Devices and Emerging Research Materials working groups of the ITRS (International Technology Roadmap for Semiconductors) selected carbon-based nanoelectronics as their recommended “Beyond CMOS” technology [2]. Despite the recent discovery of carbon nanomaterials, there has already been a significant research effort toward the creation and characterization of carbon-based field effect devices (FETs). This work can generally be divided into two categories: those focusing on carbon nanotube FETs (CNFETs), and those focusing on graphene nanoribbon FETs (GNRFETs). CNFETs are field effect devices that use carbon nanotubes (CNTs) for their channels. The

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CNTs are commonly assumed to be in parallel arrays, but can also be made using networks of random connections. Most works focus on single-walled carbon nanotubes because they offer better performance and controllability than multi-walled nanotubes. The nanotubes in the channel are assumed to be semiconducting, as a metallic CNT would create an electrical short between the source and drain, greatly reducing the transistor ION/IOFF ratio. These semiconducting nanotubes can be either doped n-type and p-type to form MOSFETs, or contacted with specific metals and electrically biased to form Schottky-barrier FETS (SBFETs). As in a modern silicon FET, a metal gate is used to apply an electric field to the device to control its state. CNFETs have been studied longer than GNRFETs, and have the advantage of more mature fabrication and modeling techniques. The GNRFET is a field effect device that is similar to a CNFET, but instead of using carbon nanotubes as the channel material, it uses thin ribbons of graphene called graphene nanoribbons. These devices are typically made from monolayer or bilayer graphene for the best performance, but trilayer or many-layer graphene may also be used. The semiconducting behavior of a graphene-based channel is dependent on the narrowness and edge states of the GNRs. For increased drive strength, many thin ribbons are used in parallel instead of one wide ribbon. The primary advantage of GNRFETs over CNFETs is the two-dimensional structure of graphene. Since graphene is created in large homogeneous sheets, it can be patterned using standard lithographic techniques [3]. This makes it easier to work with than nanotubes, which require a bottom-up method of fabrication in which the nanotubes must be aligned and placed during growth or in a subsequent processing step. Graphene can also offer enhanced electrostatics for gate electrodes, especially when a back gate and a top gate are both used. CNFETs and GNRFETs show incredible promise for use in future integrated circuits. Their main advantage is that they offer improved performance at the nanoscale dimensions where silicon-based transistors become increasingly difficult to work with. Early digital applications are likely to involve a hybrid of traditional silicon CMOS logic for higher level interfaces and carbon-based transistors to implement dense small-feature-size logic. This would allow the performance and power benefits of carbon nanomaterial FETs to be leveraged at critical dimensions, while maintaining the advantages of the mature CMOS process for non-critical sections. In this chapter, we present the latest research in carbon-nanomaterial transistors. The chapter is organized as follows. In Section I, we begin with a discussion of the intrinsic properties of carbon nanomaterials that make them desirable for future electronic applications. Then we discuss the various CNFET (Section II) and GNRFET (Section III) designs that have been developed to date. Section IV reviews the high level modeling techniques used to characterize such devices, and Section V describes how these devices can be assembled into higher order structures to create carbon-based circuits and logic. Challenges and Opportunities are presented in Section VI, and Section VII concludes the chapter.

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I. Carbon Nanomaterials Carbon nanomaterials have received significant interest and investment from the research community due to their unique electrical and physical characteristics. This section details the structure and these devices, and their desirable physical and electrical properties.

A. Atomic Composition

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Carbon nanomaterials are composed primarily of benzene-like hexagonal rings of carbon atoms. Each edge of the hexagon is a single or double carbon-carbon bond with a bond length of roughly 0.14 nm. These sp2-bonded carbon rings connect together to form a number of carbon allotropes that exhibit different properties. For example, when the rings are connected in a soccer ball like arrangement of 60 carbon atoms, they form a molecule known as a buckminsterfullerene. When the rings are placed on a single plane in a repeating honeycomb pattern, they form a crystalline structure known as monolayer graphene. This hexagonal graphene pattern also exists in rolled up cylindrical tubes, a family of allotropes known as carbon nanotubes. If single sheets of graphene are stacked on top of one another, they form bilayer graphene, trilayer graphene, many-layer graphene, and eventually graphite (10+ layers). Since stacked graphene relies on molecular attraction between layers instead of interlayer chemical bonds, the layers can be sheared from one another, giving graphite its useful properties for pencil lead. The relationships between these allotropes can be seen in Figure 1 [4]. For their potential usefulness in transistor channels, our focus will be on the two semiconducting allotropes: carbon nanotubes and graphene.

Figure 1. The relationship between graphene, buckminsterfullerenes, carbon nanotubes, and graphite [4], © 2007 Macmillan Publishers Ltd, reprinted with permission.

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(a)

5

(b)

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Figure 2. Multi-wall carbon nanotubes discovered in 1991 (a) Schematic (b) TEM image [5], © 2002 Elsevier, reprinted with permission.

Carbon nanotubes can be categorized into two groups: single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) (Figure 2). A SWCNT is a hollow cylinder with a diameter of roughly one to four nanometers, and can be thought of as a rolled up sheet of single-layer graphene. A MWCNT is composed of a number of different diameter SWCNTs nested inside one another, and can be thought of as a rolled up sheet of multi-layer graphene. MWCNTs have dimensions greater than SWCNTs and are typically from four to several tens of nanometers in diameter. Carbon nanotubes vary in length, and can be made up to 1 mm long. Since they have diameters of less than 10 nm, this yields exceptionally high aspect ratios, meaning carbon nanotubes can be thought of as a onedimensional material. Due to cylinder symmetry, there are a discrete number of ways a graphene sheet can be rolled to form a SWCNT. To characterize each direction, two atoms in the graphene sheet are chosen, one of which serves as the origin. The sheet is rolled until the two atoms coincide. The vector pointing from the first atom to the second is called the chiral vector, and its length is equal to the circumference of the nanotube. The direction of the nanotube axis is perpendicular to the chiral vector. The properties of a given SWCNT can be determined by its chiral vector (n, m) or in other words, the direction that the graphene sheet has been rolled. A SWCNT with a chiral vector (n, m) indicates that during rolling, the carbon atom at the origin is superimposed with the carbon atom at the lattice location (n, m). Figure 3(a) illustrates possible chiral vectors [6]. Depending on the rolling method, three different types of SWCNT can be synthesized: armchair nanotubes with m = n, zigzag nanotubes with m = 0, and chiral nanotubes with n ≠ m ≠ 0 (Figure 3(b)). MWCNTs are not characterized in this way because they are composed of nanotubes with varying chirality. Carbon nanotubes will be either metallic or semiconducting depending on their chiral vector. This will be described in Section II.

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(a)

(b) Figure 3. (a) Chiral vectors of SWCNTs (b) SWCNT types: armchair, zigzag, chiral [6], © 2007 Springer Science+Business Media, reprinted with permission.

Like carbon nanotubes, graphene can exist in a number of forms. Monolayer graphene is made from a sheet of carbon exactly one atom thick, making it a pure two-dimensional crystal. Before graphene was first isolated in 2004, two-dimensional crystals were assumed not to exist at room temperatures [4]. Since graphene does not wrap around and connect back to itself like a carbon nanotube, its edges are free to bond with other atoms. Because unbonded edges are unstable, the edges are usually passivated by absorbents such as hydrogen. Other possible edge passivations include oxygen, hydroxyl groups, carboxyl groups, and ammonia [7]. Bulk planar graphene can be patterned by lithography to define narrow strips known as graphene nanoribbons (GNRs). Such ribbons can also be created through other techniques, such as chemical synthesis [8] and the ‘unzipping’ of carbon nanotubes [9]. The narrower the nanoribbon, the greater the impact of its edge structure on its properties. The crystallographic

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orientation of the edges is especially important. Figure 4 demonstrates two possible edge state orientations, known as armchair and zigzag, and common width designations for each [10]. In Figure 4(a), an armchair-GNR is shown with a size of N = 10, where N is the width in number of carbon atoms. In (b), a zigzag GNR is shown with a size of N = 5, where N is the width in number of zigzag chains.

Figure 4. Graphene nanoribbon classification: (a) armchair, (b) zigzag [10] © 1996 American Physical Society, reprinted with permission.

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When multiple layers of graphene are stacked together, the layers can be aligned with different orientations, as shown by the patterns in Figure 5 [11]. This creates a number of possible stacking structures. Simple AA stacking is electrically unfavorable, so alternating orientations such as AB are more commonly found. It is also possible to stack layers off-axis to create rotational defects, which has the effect of canceling some of the interlayer interactions found in AB stacking, resulting in properties similar to monolayer graphene [12].

Figure 5. Common graphene stacking orientations [11], © 2006 American Physical Society, reprinted with permission.

B. Physical Properties The regular crystalline structure of both graphene and carbon nanotubes gives them robust physical properties. For instance, the breaking strength of graphene has been measured as over 200 times greater than that of steel, making it the strongest material ever tested [13]. Due to the strength of the bonds in the graphenic lattice, carbon nanomaterials are stable at room temperature, and remain so even when scaled to atomic dimensions.

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The strength and stability of carbon nanomaterials make them attractive for use in nanoelectronic devices that require stiffness, lightness, and robustness, such as NEMS (nanoelectrical mechanical systems). They have also opened new opportunities in flexible electronics, since carbon-based devices are more flexible than rigid silicon devices. Even traditional electronics can see a benefit in the form of increased reliability against environmental stresses and shocks. Carbon nanotubes exhibit unusually high thermal conductivity, comparable to that of pure diamond and nearly an order of magnitude higher than copper. This allows them to dissipate the heat generated by electrical switching and device leakage more effectively, making them useful as thermal conduction paths in high-density circuits. To this end, research is being done to investigate the viability of incorporating carbon-based thermal vias and channels for thermal management in future integrated circuits. This application is especially important given the anticipated shift to thermally-limited three-dimensional integrated circuits.

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C. Electrical Properties While the physical properties of carbon nanomaterials are desirable, the real opportunity lies in their electrical properties. Advances in silicon technology will continue for the foreseeable future, but a highly scaled silicon MOSFET will face formidable problems in terms of reduced drive current and increased short-channel effects such as drain-induced barrier lowering (DIBL). Carbon nanomaterials, on the other hand, have electrical properties that allow them to overcome these challenges and achieve strong performance at sub-10 nm dimensions. The high quality of the crystal lattice in carbon nanomaterials gives them a recordbreaking mean free path up to microns in length, which results in the near-ballistic transport of charge carriers. More importantly, this mean free path is achieved at room temperature, allowing for very high mobilities. Under ideal conditions, the room temperature electron mobility can reach about 100,000 cm2/ V·s in carbon nanotubes, and about 200,000 cm2/ V·s in graphene, making them significantly more attractive than silicon at 1,400 cm2/ V·s, and comparable to undoped InSb at 77,000 cm2/ V·s [14]. However, the ideal conditions for such high mobilities require the samples to be suspended in air. When carbon nanomaterials are placed on a substrate, the vibrations of the substrate are transferred to the carbon lattice, causing remote interfacial phonon scattering [14]. Still, even with such effects, room temperature mobilities over ten times greater than silicon are achievable. Furthermore, unlike III-V compounds, such mobilities remain high in both electrically and chemically doped devices with doping concentrations up to 1012 cm-2 [4]. These mobilities result in high carrier velocities, low conductance, and the possibility of terahertz-speed devices. In addition to their high current densities, carbon nanomaterials are more robust to short channel effects. Structures such as double-gated graphene and gate-all-around SWCNTs offer nearly ideal control of channel electrostatics, minimizing effects such as DIBL. With longer mean free paths, larger mobilities, and better electrostatics, carbon nanomaterial channels will consume less power and dissipate less heat than their silicon counterparts. Much like the switch from bipolar transistors to lower-power MOSFETs, switching from silicon-based transistors to carbon nanomaterial-based transistors would allow a greater number of devices to be integrated under a given power density requirement.

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II. Carbon Nanotube FETs (CNFETs) A. General As described in Section I, single walled carbon nanotubes can be described by a chiral vector (m, n). The conductivity of a SWCNT can be determined by its chiral vector. If the CNT has a chiral vector such that m – n = 3x, where x is an integer, it demonstrates metallic behavior. Metallic SWCNTs have high electron mobility and robustness, and can carry a current density ~1000 larger than Cu, making them attractive for uses in nanoscale interconnect and nanoelectromechanical systems. If a CNT has a chiral vector where m – n ≠ 3x, it behaves as a semiconductor, with a band gap that typically varies between ~0.5 eV and ~2 eV [6]. The conductance of a semiconducting nanotube is strongly dependent on the applied electric field. Due to its crystalline structure and nanoscale dimensions, a semiconducting nanotube demonstrates ballistic electronic conduction and insensitivity to electromigration. These advantages make SWCNT transistors a promising candidate for use in future nanoelectronic systems. In the past decade, many studies have been published regarding the architecture, fabrication, and testing of carbon nanotube field effect transistors (CNFETs). In this chapter, we review the most representative of these developments.

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B. Transistor Types The first reports of operational room temperature CNFETs came in 1998 from groups at IBM [15] and Delft University of Technology (TUDelft) [16]. The structures of the two CNFET devices are shown in Figure 6. These designs have similar architectures: a single nanotube (either single-walled or multi-walled) behaves as the channel region and rests on metal source/drain electrodes. The design from [15] has a 140 nm thick SiO2 dielectric on top of a silicon back gate and 30 nm thick Au electrodes defined by e-beam lithography, whereas the design from [16] uses Pt electrodes with a 300 nm thick SiO2 insulator. Au and Pt contacts are selected because their work functions are close to the carbon nanotube work function of 4.5 eV.

(a)

(b)

Figure 6. Schematic cross section of Si back gated CNFET (a) with Au S/D contacts [15], © 1998 American Institute of Physics (b) and Pt S/D contacts [16], © 1998 Macmillan Publishers Ltd, reprinted with permission.

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(a)

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(b) Figure 7. I-VG transfer characteristics of the devices in Figure 6 (a) IBM [15] , © 1998 American Institute of Physics (b) TUDelft [16], © 1998 Macmillan Publishers Ltd, reprinted with permission.

The I-VG characteristic of the CNFET developed in [15] is shown in Figure 7(a) for different source-drain voltages. As the gate voltage is swept from +6 V to -4 V, the sourcedrain current increases strongly, indicating that the device operates as a FET. The increase of current at negative gate voltages is identical to that of a p-channel MOSFET and is evidence that holes are the majority carriers. The currents in Figure 7(a) start to saturate as VG < 0V. This saturated current value corresponds to a resistance of approximately 1.1MΩ, which is mainly due to the metal-to-CNT contacts. A total conductance difference of five orders of magnitude is observed. Figure 7(b) presents the I-VBias measurement from device in [16]. The non-linear I-VBias characteristic indicates a dependency of the source-drain current on gate voltage. As the gate voltage switches to a negative value, the CNFET turns on and linear I-V curves can be observed. The saturated current corresponds to a resistance of 1 MΩ, which matches the value from [15]. Like in [15], conductance through this CNFET varies by over six orders of magnitude. These pioneering works demonstrate the use of the CNFET as a switch for future integrated circuit design. However, it is difficult to integrate devices based on the layouts in Figure 6 because they use the silicon substrate as a back gate. This means that unless the same

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gate voltage is applied to all of the devices on a substrate, a substrate isolation technique such as shallow trench isolation (STI) must be used. In 2001, the group from TUDelft enhanced their previous CNFET design by using aluminum local gates to control individual transistors [17]. This design uses a narrow Al gate insulated by a thin native Al2O3. The Al gate is defined by e-beam lithography on silicon oxide, and the gate insulator is grown by exposing the Al gate to air. Single-wall nanotubes are then deposited onto the wafer on top of the predefined gates. Finally, Au source and drain contacts are created by e-beam lithography. The resulting I-VSD characteristic shows that this new CNFET works as an enhancement-mode p-type device [17]. In order to build logic functions, it is necessary to have both n-type and p-type transistors available. However, all of the aforementioned CNFETs are p-type. Results in [18] demonstrate that n-type CNFETs can be obtained by annealing of a p-type device or through doping. Figure 8(a) illustrates the transformation of a p-type device annealed in a vacuum at a temperature 700K for 10 min. The transistor is turned on when positive gate voltages have been applied and turned off when the gate voltage is negative. The inset in Figure 8(a) compares the IV characteristic of the device before and after annealing. It clearly shows the transition from a negative threshold voltage into a positive threshold voltage. The transformed n-type device has an ION/IOFF ratio of over three orders of magnitude. In addition to annealing, doping is shown to be effective in creating n-type devices. In [18], potassium is selected as the electron donor to shift the Fermi level to the conduction band. Figure 8(b) shows that the IV characteristic of an n-type device are similar to those of the annealed device.

Figure 8. IV characteristics of n-type CNFETs created by (a) annealing in vacuum (b) doping with potassium [18], © 2001 American Chemical Society, reprinted with permission.

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A major improvement was made in 2002 through the creation of an MOS structure, with a gate electrode on top of the nanotube channel and a thin SiO2 dielectric . The structure of this top gate design resembles common silicon MOSFET designs and is shown in Figure 9(a). Top gated designs offer several important improvements over back-gated devices. For one, back-gated devices use a relatively thick ~100 nm oxidation layer, which requires a high gate voltage to switch the device on, whereas top-gated CNFETs can have a thin gate dielectric of ~15-20 nm, allowing for lower voltage operation. Secondly, top gating dramatically reduces the gate source/drain overlap capacitance, which is critical to high-frequency operation. In addition, with back-gated devices, the carbon nanotubes are exposed to air. Such exposure can cause the n-type devices obtained from previously mentioned techniques to revert to ptype. In top-gated devices, the carbon nanotubes are encapsulated in gate oxide, avoiding this electrostatic instability problem and helping to improve reliability. A top-gated CNFET design was presented in [19]. In this design, the CNT is fabricated on top of a heavily doped single-crystal silicon wafer coated with 120 nm of thermal SiO2 (Figure 9(a)). Next, titanium source/drain electrodes are patterned by e-beam lithography with a spacing of ~200 nm. A thin layer of gate oxide is then deposited, and finally the titanium gate electrodes are patterned by e-beam lithography. Figure 9(b) compares this top-gated device with a back gate design. The IV characteristics of the two structures have the same shape, but the operating voltages of the top-gated device are much lower (~ -0.1V to -0.5V over threshold) than the operating voltages in the bottom-gated design (~ -3.5V to -15.5V over threshold).

(a)

(b)

Figure 9. Topgated CNFET (a) device structure and (b) IV comparison to back gate designs [19], © 2002 American Institute of Physics, reprinted with permission.

High-k dielectrics allow high gate capacitance without requiring nanometer-thin layers of gate oxide, and are widely used in modern process technologies. High-k dielectrics can boost MOSFET performance by enhancing current injection into the channel, and at the same time reduce gate tunneling to minimize leakage power. Integration of thin films of ZrO2 high-k dielectrics into CNFETs is demonstrated in [20] (Figure 10(a)). First, molybdenum source/drain electrodes with dimension of 50 nm are patterned on a silicon wafer with 500 nm thick SiO2. SWCNTs are then grown by CVD to bridge the predefined source/drain. The 8 nm ZrO2 gate insulation layer is then deposited by atomic layer deposition (ALD). Finally, ebeam lithography and lift-off are used to create a 60 nm Ti top gate.

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(a)

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(b)

(c) Figure 10. (a) Cross section of a high-k CNFET with ALD-ZrO2 as the gate dielectrics, (b) I-Vgs characteristic, and (c) I-Vds characteristic [20], © 2002 Macmillan Publishers Ltd, reprinted with permission.

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(a)

(b)

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Figure 11. (a) Cross section of a CNFET with a channel of multiple parallel nanotubes and (b) dense arrays of SWCNTs [21], © 2007 Macmillan Publishers Ltd, reprinted with permission.

By using a high-k gate insulator, the normalized transconductance and carrier mobility of a CNFET were reported to reach 3,000 Sm–1 and 3,000 cm2V–1s–1 respectively. Compared to a 60 nm Intel MOSFET with transconductance 800 Sm–1, a high-k CNFET offers over a threefold improvement. High-k CNFET devices have been reported to have excellent subthreshold swing as well. Based on the plot in Figure 10(b), a slope of 70 mV per decade can be measured, which is very close to theoretical limit of 60 mV/decade. The measured I-V characteristics (Figure 10(c)) show clearly defined linear and saturation regions in a PMOS device. Single nanotube devices reveal great performance improvements over existing solutions. However, the integration of single nanotubes into existing integrated circuits is still a great challenge. Due to limited fabrication control of nanotube properties, a single CNT device is susceptible to large performance fluctuations. One feasible solution is to use densely packed, horizontal arrays of non-overlapping SWCNTs in the channel, as shown in Figure 11(a) [21]. This creates parallel conducting paths that can provide larger current than a lone CNT. Having multiple carbon nanotubes in the channel also statistically averages the device-todevice variation and offers increased reliability against a single tube failure. There are two major challenges in fabricating a multiple carbon nanotube device, both of which have been addressed in [21]. The first challenge is the creation of large scale, high density, perfectly aligned nanotube arrays. These can be achieved by using photolithographically defined parallel patterns on a quartz surface and growing carbon nanotubes with CVD along these predefined patterns. Using this technique, nanotube arrays can be successfully fabricated with average diameters of ~1 nm and lengths greater than 300 μm, with 99.9% alignment Figure 11(b). The nanotubes can then be transferred to the desired substrate, such as silicon or even a flexible plastic. The second challenge is that intrinsically, one third of the fabricated carbon nanotubes are metallic. These nanotubes are always conducting, which deteriorates the transistor ION/IOFF ratio. Metallic nanotubes can be removed by techniques such as electrical breakdown [22]. After the electrical breakdown process, ION/IOFF ratios can be improved by four orders of magnitude [21]. Multiple nanotube channel CNFETs allow for special post-process tuning of the device characteristics that is not possible with single-tube channels. This is because after fabrication, each of the nanotubes in the multiple nanotube channel has its own electrical properties. Therefore, there will be a distribution of threshold voltage across the nanotubes within a

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single FET. Results in [23] demonstrate how to use these properties to adjust device threshold voltage and ION/IOFF ratios. Post-processing techniques such as the electrical breakdown method mentioned previously, which removes metallic nanotubes within the channel region, can also be applied to pruning nanotubes with unwanted threshold voltages. Figure 12(a) illustrates two experiments where different ratios of high VT nanotubes are removed from the tail of the distribution. As more nanotubes are removed, the device current density decreases as there are fewer conducting paths. Measurement data from Figure 12(a) show that the overall VT was changed from an initial value 3.6 V, to 3.1 V, and then to 2.4 V. At the same time, ION decreased from 70 μA, to 60 μA, and then to 47 μA. Another interesting result comes from the variance of VT before and after shifting. For the devices that were tested, the initial VT had a Gaussian distribution with a mean of 2.4 V and a standard deviation of 0.43 V. After adjustment, the average VT was lowered to 1.8 V and the standard deviation was reduced to 0.29 V (Figure 12(b)). This experiment indicates that the VT shifting technique is not only effective for optimizing device performance, but also for unifying device characteristics. A similar process was used to remove high leakage nanotubes to improve device ION/IOFF ratios with favorable results, as shown in Figure 12(c).

(a)

(b)

(c)

Figure 12. (a) Selective VT removal. (b) Variation of VT before and after VT shifting, and (c) ION/IOFF ratio shifting [23], © 2009 IEEE, reprinted with permission.

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III. Graphene Nanoribbon FETs (GNRFETs)

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The intrinsic physical and electrical properties of graphene make it desirable for applications ranging from biosensors to flexible electronics to solar cell electrodes. One of the most exciting applications is the use of graphene channels in future high performance transistors. To that end, physicists and materials scientists have been characterizing and experimenting with graphene to understand how it could be used to make such devices a reality. Few-layer graphene was initially discovered at the University of Manchester in 2004 [24]. The first graphene transistor-like device was created shortly thereafter, a simple test structure constructed to measure graphene’s field effect behavior. The major challenge with graphene is that in its native state as a large sheet, it behaves like a zero-band gap semiconductor, or semi-metal, meaning that it conducts electrons freely. This is desirable if graphene is to be used for interconnect, and of course many researchers are exploring this possibility. However, in order to be effective in transistors, graphene must be made semiconducting and demonstrate a high ION/IOFF ratio. To obtain the necessary off state, a band gap needs to be introduced.

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(b)

Figure 13. (a) GNR test structures of varying width and crystallographic orientation (b) The effect of GNR width on conductance at 300K (squares) and 1.6K (triangles) [25] , © 2007 American Physical Society, reprinted with permission.

One way to open a band gap is to pattern graphene into a narrow ribbon to laterally confine the charge carriers in a quasi-one-dimensional system somewhat analogous to a CNT. This idea was experimentally demonstrated in [25], where researchers at Columbia University used e-beam lithography to define two dozen graphene nanoribbons with widths ranging from 10 to 100 nm and varying crystallographic orientations. These GNRs were contacted to form the test structures in Figure 13(a). In this figure, the nanoribbons are the thin lines between contacts. Conductance of the GNRs was measured at both 300 K and 1.6 K. The measurements show that for a given crystallographic direction, the energy gap depends

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strongly on the width of the GNR, as shown in Figure 13(b). As the ribbons are made smaller, less conductance is observed, which indicates a stronger semiconducting behavior. A similar experiment in [26] comes to a similar conclusion. In both cases, researchers also note an apparent dependence of the electrical behavior on the edge states of the nanoribbons. One of the first works to demonstrate sub-10 nm width GNRFETs is [27]. The authors are able to achieve such dimensions because instead of patterning GNRs from a planar sheet with e-beam lithography, they started with GNRs that had been chemically derived at smaller dimensions using the process described in [8]. In this process, exfoliated graphene is dispersed into a chemical solution by sonication, creating very small fragments. The solution is then applied to a substrate and dried. The resulting GNRs are identified with atomic force microscopy. These GNRs ranged from monolayer to trilayer, and were deposited on a SiO2 dielectric over a highly doped silicon back gate, and contacted with Pd source/drain electrodes. Devices were created using the bilayer GNRs, including both wide (10-60 nm) and small (< 10 nm) GNRFETs. When tested, the large GNRs demonstrated metallic behavior due to vanishingly small band gaps, while the sub-10 nm GNRFETs were found to be semiconducting. Compared to the earlier works on GNRs of 20 nm width, the semiconducting GNRFETs show 105 higher ION/IOFF ratio at room temperature, ~20 times higher current density (at Vds = 1V), and ~100 times higher transconductance per μm. This is due to larger band gaps, higher GNR quality, thinner gate oxide, and shorter GNR channels [27]. The importance of GNR edge states was predicted by first-principles physics calculations [10][28]. As described in Section I, the edges of graphene can either be zigzag or armchair, depending on the orientation of the graphene lattice along the edge. As in carbon nanotubes, this chirality was predicted to play a role in the semiconductivity of the sample. A recent experiment used scanning tunneling microscopy to verify this prediction, and confirms that the crystallographic orientation of the edges significantly influences the electronic properties of nanometer-sized graphene [29]. By measuring the band gap of graphene samples and noting their predominant edge chirality, the authors observe that predominantly zigzag edges are metallic, while predominantly armchair edges are semiconducting [29]. For GNR transistors, all-semiconducting armchair edges are the most desirable. However, in the ribbons produced so far, the edges are not always atomically smooth and often contain a mixture of segment types, as seen in Figure 14. In these cases, the semiconducting properties weaken, and the band gap becomes dependent on the ratio of armchair segments to zigzag segments [29]. One of the advantages of graphene is that its two-dimensional structure allows for strong electrostatic control of the channel. As with CNTs, a number of gate configurations are possible. Back gate designs are the simplest to create and are useful for the testing and characterization of individual transistor properties. Top gate designs are better suited to creation of large-scale integration. Ideal gate electrostatics come from double gate designs, where input signals are applied to both a top and back gate simultaneously, allowing higher drive current and reducing off-state leakage current. Top and bottom gate can also be driven independently in dual gate designs, where the back gate bias is used to electrically dope the device into n-type or p-type behavior, and the top gate is used as a logical input. In this way, multiple devices could share the same backgated substrate to minimize the need for substrate isolation through trenches or implants. Recently, a dual gate design was used to test the electrostatic band gap control in bilayer graphene (Figure 15) [30]. Measurements of the device for varying bottom gate voltages

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demonstrated a gate-controlled, continuously tunable band gap of up to 250 meV. This means that graphene can be effectively doped without uncontrolled chemical doping, and that bilayer graphene can potentially be used in novel nanophotonic devices for infrared light generation, amplification, and detection [30].

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Figure 14. GNR with different dominant edge chiralities (top: Armchair GNR with 0.38eV band gap, middle: Zigzag GNR with 0.14eV band gap, bottom: Zigzag GNR with 0.12eV band gap) [29], © 2009 Macmillan Publishers Ltd, reprinted with permission.

There are two ways in which the source and drain of a graphene nanoribbon transistor can be contacted. One is with direct metal contacts in order to form Schottky contacts for a Schottky-barrier FET (SBFET). The other is with heavily doped source and drain extensions to create MOSFET-like behavior [31]. Schematics of these designs are shown in Figure 16. In [31], models of ideal devices were compared, and MOSFETs were shown to offer significantly better device characteristics than SBFETs in almost every way. Ideal MOSFET devices have a larger maximum on-off ratio, 50% larger on-state current, larger transconductance, better saturation behavior, a 30% higher cutoff frequency, and 20% faster switching speed. In addition, calculations demonstrate that MOSFETs are more robust than SBFETs against the influences of defects or impurities [31]. The drawback is that MOSFETS are more difficult to fabricate, show more variation due to doping, and occupy a larger area than SBFETs. To construct a MOSFET, the source and drain must be doped. Under ambient conditions, the edges of GNRs readily absorb hydrogen, oxygen, hydroxyl groups, and carboxylic groups from the air, resulting in p-doping. This is why most manufactured GNRFETs are slightly ptype. In [7], GNRFET devices were made to be n-type by high-power electrical annealing (eannealing). First, the native p-type absorbents were annealed out in a vacuum, and then NH3 (ammonia) was introduced and annealed into the GNR. The resulting electron-rich GNRs were shown to have similar mobilities to pristine p-type GNRs, and all tests indicated that the nitrogen bonds existed on the edges. These results suggest that edge doping represents a new and straightforward approach to graphene doping, something not possible in edge-free carbon nanotubes [7].

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Figure 15. Dual-gated bilayer GNRFET (a) device (b) schematic [30], © 2009 Macmillan Publishers Ltd, reprinted with permission.

Figure 16. (a) SBFET with metal contacts (b) MOSFET with doped source and drain extensions.

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In addition to logic applications, GNRFETs are well suited for individual ultrahighfrequency analog transistors [32]. Top-gated graphene transistors of various gate lengths have been fabricated with peak cutoff frequencies up to 26 GHz for a 150 nm gate [33]. Results also indicate that if the high mobility of graphene can be preserved during the device fabrication process, a cutoff frequency approaching terahertz may be achieved for graphene FETs with a gate length of 50 nm, [32][33].

Figure 17. All-graphene devices. Metal semiconductor junctions based on (a) GNR edge chirality, (b) n-type and p-type GNR doping, and (c) varying GNR widths, and (d) an example GNRFET based on chirality changes [34], © 2007 American Chemical Society, reprinted with permission.

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Figure 18. Effect of different bends on GNR edge chirality.

Another interesting possibility is the creation of all-graphene transistors by patterning a large piece of graphene to contain both semiconducting channels and metal contacts. To date, such devices have not been made, but a number of groups have studied the idea. In [34], firstprinciples transport calculations show that all-GNR field effect transistors can achieve high performance levels similar to those made from single-walled carbon nanotubes. These designs could be based on pn junctions formed by three different methods as shown in Figure 17: a change in GNR chirality (a), a change from a p-doped GNR to an n-doped GNR (b), or a change in GNR width (c). Using these junctions, transistors could be created, such as the one shown in Figure 17(d). This transistor is based on the observation that certain connection angles change the chirality of the edges, forming a metal-semiconductor junction. The chirality effects of GNR connection angles are shown in Figure 18.

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A. CNFET Modeling CNFETs are generally modeled using one of two approaches: the non-equilibrium Green’s function (NEGF) approach [35], or a simple modeling approach based on the assumption of ballistic transport. Many physical aspects of the problem are captured in the NEGF approach, but it involves a great deal of numerically intensive calculations. These calculations make it difficult to explore the design space and nearly impossible to integrate in a commercial circuit simulator such as SPICE. On the other hand, the simple technique of assuming ballistic transport allows for high speed equations that match reasonably well to measured performance. Since they directly relate to the underlying device physics, such models can be intuitive understood and allow for straightforward design space explorations.

1. SPICE Compatible MOSFET Models To maximize ease of use, models should be compatible with SPICE, the industrialstandard circuit simulator. The most comprehensive and well known SPICE compatible CNFET model was created by Stanford University and presented in [36] and [37]. The Stanford University CNFET model is a ballistic model that covers MOSFET-like structures, and is implemented in three levels, as shown in Figure 19. Level 1 models near-

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ballistic transport in the intrinsic channel region under the metal gate. This level does not include any parasitic capacitance and resistance. The equivalent circuit for the intrinsic channel region including the trans-capacitance network is shown in Figure 20. Like traditional silicon MOSFET SPICE models, the core part of the equivalent circuit is the voltage controlled current sources. The three voltage controlled current sources represent the thermionic current contributed by the semiconducting sub-bands (Isemi), the current contributed by the metallic sub-bands (Imetal), and the leakage current (Ibtbt) caused by bandto-band tunneling. Note that Imetal is equivalently modeled as a voltage-dependent conductance. As described in [36], Isemi can be expressed as:

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Figure 19. Relationship between the three levels of the Stanford CNFET model [36], © 2007 IEEE, reprinted with permission.

Figure 20. Equivalent circuit for the intrinsic CNT channel [36], © 2007 IEEE, reprinted with permission.

4e2 I semi (Vch, DS ,Vch ,GS ) ≈ h

( E −ΔΦ )/ kT ⎡ kT ⎛ 1 + e m ,0 B Tm ⋅ ⎢Vch, DS + ln ⎜⎜ ∑ ( Em ,0 −ΔΦ B + eVch , DS )/ kT e km ⎝ 1+ e ⎣⎢ M

m =1

⎞⎤ ⎟⎟ ⎥ ⎠ ⎦⎥

Where k is the Boltzmann constant, T is the temperature in degrees Kelvin, Vch , DS , Vch ,GS denote the Fermi level near the source/drain region, ΔΦ B is the channel surface

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potential change, Tm represents transmission probabilities related to scattering and Em,l is the carrier energy of the (m,l) quantum state. For a semiconducting channel region where metallic carbon nanotubes are completely removed by electrical burning, Imetal = 0. However, if there are metallic nanotubes in the channel, the channel current will be dominated by Imetal and the device will have a degraded ION/IOFF ratio. The level 1 model also contains transcapacitances between the G, S, D, and B terminals to capture the AC response of the CNFET device. Detailed information can be found in [36]. The level 2 model [37] is an extension of the level 1 model that considers device nonidealities such as elastic scattering within the channel region, resistance and capacitance of the doped source/drain regions, and Schottky barriers formed by the metal contacts. In this model, circuit elements are added around the ideal CNFET level 1 model. The potential drop caused by elastic scattering is modeled, as well as parasitic resistances and capacitances of the heavily doped source/drain regions, and Schottky barrier resistances to account for the metalnanotube contacts. The level 3 model [37] is an extension of the level 2 model for a channel region containing an array of multiple nanotubes. The N nanotubes in the channel can be categorized into two groups: the two carbon nanotubes on the edges and N – 2 nanotubes in the middle. The nanotubes in these groups are connected in parallel for increased drive strength and reliability. All of the CNTs within the same group are treated identically and the groups each consider charge-screening effects.

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2. SBFET Models A drawback of the SPICE compatible model is that it only works for MOSFET devices. These devices assume source and drain regions that include heavily doped CNTs. Such devices are desirable because they suppress ambipolar conduction, which minimizes leakage current and allows for thinner gate oxides and higher power supply voltages. In addition, because the Schottky barriers between the source and channel are reduced, such transistors will be capable of delivering greater on current [38]. However, at nanoscale dimensions, such doping might be impractical, and an SBFET design might prove more effective. A ballistic transport-based SBFET model is presented by Stanford University in [39]. This model builds on a previously published ballistic channel model and incorporates effects due to Schottky barriers at the source/drain contacts, band-toband tunneling, and ambipolar conduction.

B. GNRFET Modeling Since GNRFETs and CNFETs share similarities, it should follow that GNRFET models can leverage the developed CNFET models for physics based phenomena. However, due to their edges, GNRs display a number of effects that are not present in CNTs, including edge scattering, edge bond relaxation, and the third nearest neighbor interaction [40]. Unlike with CNTs, which have had the benefit of over a decade of study, there are no analytical compact models of GNRFETs available for direct use in SPICE simulations. In the analytical CNFET models based on the Landauer approach, a simple tight binding calculation with first nearest

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neighbor interactions gives accurate results for carrier transport [41]. However, when applied to GNRFETs, this method fails to predict a correct band gap when compared to ab initio physical models based on tight binding. To correct this, a simple analytical model has been developed in [41] to describe the effects of edge bond relaxation, the third nearest neighbor interaction, and edge scattering. However, this approach still involves numerical computation for calculating the charge density integral and solving the self-consistent electrostatics, and is not scalable to large circuits [42]. While such approaches may eventually lead to compact circuit compatible analytical models, many of the present works still employ a NEGF approach to accurately model GNRFET behavior in both MOSFETs and SBFETs. In this approach, the numerical simulation for GNRFETs is achieved by using a NEGF-based equation to simulate quantum transport and solving it self-consistently with a coupled three dimensional Poisson equation for the treatment of electrostatics [40][43]. This approach is illustrated by Figure 21 (adapted from [44]).

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Figure 21. NEGF formalism for a genetic transistor.

Figure 22. NEGF self-consistent approach for solving devices at equilibrium.

The first step is to derive the Hamiltonian matrix [H] for the isolated channel. Then the self-energy matricies Σ1, Σ2, and ΣS are computed to describe how the channel couples to the source contact, drain contact, and the dissipative processes respectively [44]. After these have been computed, the equilibrium transport is solved by computing the Green’s function along with the self-consistent electrostatic potential U [44]. The result of this calculation is then used to recalculate the electrostatics U using the Poisson equation. This cycle, shown in Figure 22, repeats until the values converge [35]. Note that this solution is numerically intensive and expensive to compute. Part of the reason is that this real space representation computes all modes (subbands) together. The simulation time can be reduced significantly if only a few modes are considered at a time [40]. This technique is called the mode space approach, and it is used to decouple the two

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dimensional GNR lattice into several one dimensional lattices by a basis transform along the GNR width [40].

V. Logic Gates and Circuit Structures In order to be useful to the semiconductor industry, transistors must be connected together to form higher order circuits. Because carbon nanomaterials exhibit unique characteristics, alternate logic families to static CMOS can be considered. In this section, we introduce how CNFET and GNRFET devices can be assembled into higher order structures to allow carbon-based circuits and logic.

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A. CNFET Logic Structures As mentioned in Section II, CNFETs have demonstrated promise as future electronic devices. Therefore, the research community is making a significant effort to integrate basic CNFET devices into complex logic circuits. The first CNFET logic gates were demonstrated in [17]. A range of digital logic operations were demonstrated, including an inverter, a NOR gate, an SRAM cell, and an AC ring oscillator consisting of one, two, and three-transistor circuits. These gates were implemented using resistor-transistor logic, in which the p-type CNFETs were connected to large off-chip resistors. The ring oscillator was implemented by connecting three inverters in series, and achieved an operating frequency of 5Hz. This low frequency was due to the gigaohm resistance and 100pF parasitic capacitance of the wires connecting to the off-chip bias resistors. The performance of CNFET ring oscillators was enhanced in a following work by [45]. In this work, SWCNT arrays were synthesized by chemical vapor deposition (CVD) on substrates pre-patterned with catalyst, and local gating was obtained by using tungsten metal back gates. More importantly, local doping was applied to convert p-type CNFETs into ntype, allowing complementary CNFET logic to be created for the first time. The resulting three stage ring oscillator had a measured frequency of 220 Hz, much higher than previously achieved with resistor-transistor logic. This speed can be further optimized by reducing the resistance of electrode to carbon nanotube contacts and the parasitic capacitance of the interconnect. Recently, a multi-stage top gated complementary CNFET ring oscillator was built on a single 18-mm-long SWCNT [46]. This ring oscillator consists of 12 individual CNFETs: six p-type FETs with Pd metal gates and six n-type FETs with Al metal gates. Five inverter stages were used for oscillation and another inverter was used for reading the signal. A frequency response of 52 MHz was measured. This frequency was still limited by the parasitics rather than by the intrinsic nanotube speed. In both of the CNT oscillator designs, a single SWCNT rope was used for the transistor channels. For large scale integration, multiple CNTs need to be used. This requires the creation of nanotubes on the wafer-scale, and precise directional control to allow parallel arrays for minimum density logic. Existing works demonstrate that CNTs grown on quartz offer significantly better alignment than those grown on silicon [21][47]. However, if the nanotubes are grown on quartz, they must then be moved to the desired substrate for use. In

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[48], researchers from Stanford University presented techniques for addressing this problem. First, they demonstrated wafer-scale growth of SWCNTs on a quartz substrate. Then, they detail the wafer-scale transfer of the CNT arrays on to a silicon substrate using a process based on thermal release tape. Although the CNTs are aligned to the crystal orientation of the single crystal quartz substrate they are grown on, a very small fraction of CNTs may be misaligned during growth or become misaligned during the transfer process. This misalignment can cause shorts between transistor source and drain (Figure 23(a)), reducing the on/off ratio of the gate and possibly causing incorrect gate logic.

(a)

(b)

(c)

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Figure 23. (a) Misaligned CNT, (b-c) misalign immune techniques [49], © 2008 IEEE, reprinted with permission.

(a)

(b)

(c)

Figure 24. (a) CNFETs with CNT random network channel regions (b) Upper: etched channel trenches, Lower: trench current density (c) A CNFET row decoder on a polyimide substrate [50], © 2008 Macmillan Publishers Ltd, reprinted with permission.

A misalignment immune logic design strategy is presented in [48] and [49]. This technique allows CNT circuit to be created on imperfect CNT arrays, and is illustrated in Figure 23. Three possible gate structures for a NAND gate are shown. Design (a) has the two pull-up gates at different horizontal levels to create a compact layout in the horizontal

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direction. Design (b) and (c) on the other hand have the two pull-up network gates at the same horizontal level, and separates them by either an undoped region (b) or an etched region (c). Ideally, all CNTs should be grown perfectly aligned and pass underneath the gates, however as shown in (a), misaligned CNTs can short source and drain, creating incorrect logic functions. Design (b) and (c) are immune to this problem because the section of misaligned CNT that passes through the gates is either undoped or etched away, which means it will not be conducting. This design technique reveals a possible new research direction for computeraided design tools. CNFET circuit designs could be made misalignment immune by automatically inserting undoped/etching regions where misalignment defects could cause faults. Instead of using arrays of perfect aligned CNTs, another design approach is to use low cost random networks of CNTs to form transistors, as shown in Figure 24 [50]. In such transistors, CNTs are synthesized randomly on a wafer and have varying diameters and mixed types. The transistor active regions are then defined by etching. During the etching step, trenches are etched along the channel in the direction of transport, as shown in Figure 24. These trenches break up long metallic CNTs or purely metallic pathways between the source and drain to minimize direct path connections between the two electrodes. Random-network CNFETs are low cost because they avoid difficult fabrication steps such as the growth of perfectly aligned CNT arrays and CNT transfer. However, there are limitations on the scalability of such devices because shorter channel lengths will have a larger probability of direct paths between source and drain. Random network CNFETs were used to build circuits in [50]. One design was the four-bit row decoder containing 88 transistors, which was fabricated on a polyimide substrate (Figure 24(c)). The random CNT networks were grown using CVD and then transfer printed onto the polyimide. Au source and drain electrodes are used to contact the channel region. Since all of the components in the device can operate under physical strain, including the random CNT network, this circuit is an example of flexible electronics impossible using traditional silicon. The fabricated device presents excellent electrical properties, with high mobilities of 80 cm2V-1s-1, subthreshold slopes of 140mVdec-1, on/off ratios of 105 and switching speeds in the kilohertz range.

B. GNRFET Logic Structures With the prevalence of GNRFET transistors, logic devices are a logical next step. Some early works have already been done into such structures. The first inverter based on integrating two graphene transistors of opposite types was presented in [51]. The two GNRFETs were produced on a single flake of a monolayer graphene as shown in Figure 25. The left-hand transistor in Figure 25(a) was electrically annealed to obtain an n-type FET while the right-hand transistor contained a pristine p-type FET. The transistors were backgated by a highly doped silicon substrate insulated with a layer of SiO2. The voltage transfer characteristics of the fabricated inverter are shown in Figure 26 and exhibit clear voltage inversion. The inverting behavior of the shaded region was demonstrated up to frequencies of 10 kHz for a 3.3V swing power supply [51]. A disadvantage of this design is that input and output logic voltage levels are not the same, so inverters could not be directly cascaded. In addition, when the device is operated in the shaded region of Figure 26, both transistors are on and there is static power dissipation.

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Figure 25. Complementary GNRFET inverter (a) design (b) SEM image (c) schematic [51], © 2007 American Institute of Physics, reprinted with permission.

Figure 26. Voltage transfer curve of the GNRFET Inverter (solid line) [51], © 2007 American Institute of Physics, reprinted with permission.

Another graphene logic family is presented in [52]. This design takes advantage of the observations that GNRFETs do not exhibit drain current saturation effects, and instead behave as simple voltage controlled resistors whose resistance depends only on the applied gate voltage. This means that small changes in the gate input voltage can be detected by measuring the resulting SD resistance. Taking two input signals and using the average of them to drive the gate voltage allows the state of the two input signals to be determined by the corresponding output resistance of the transistor. By using the small signal response of the drain voltage in response to the varying gate voltages, logical output signals were derived for XOR, NAND, OR, and NOT functions using a two terminal GNRFET [52]. This allowed the

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realization of four different logic gates with only a single graphene transistor. The main drawback of this design is that the gates would be very susceptible to noise and variations, as the logical values depend on fractional changes to the gate input voltage. Another problem is that the gate is always conducing, constantly consuming static power. This power dissipation could be reduced by using a graphene transistor with a high resistance, but this would decrease the speed of the gate [52].

C. Circuit Structures

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Of course, single gates are not very useful in isolation. In order to be effective in integrated circuits, a large number of devices must be connected together to form higher order structures. Initially, carbon-nanomaterial devices could be treated as a direct replacement to their silicon counterparts, so that existing circuit designs, architectures, and CAD tools could be leveraged. However, the differences between carbon nanomaterials and silicon mean that such an approach might not be optimal. If the unique properties of carbon nanomaterials are to be fully exploited, new circuits, architectures, and CAD tools must be developed.

Figure 27. Ambipolar CNFET gate (a) Schematic (b) NOR behavior (c) NAND behavior [53], © 2009 American Chemical Society, reprinted with permission.

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Figure 28.(a) Array of field effect transistors fabricated on monolayer graphene (b) Example FET [3], © 2009 American Chemical Society, reprinted with permission.

One possible circuit technique exploits the ambipolar conduction found in carbon nanomaterials. As mentioned previously, this ambipolar conduction allows devices to switch between p-type and n-type through electrical doping. When a negative gate bias is applied to a CNT, the conduction and valence bands shift up to generate hole carriers. With a positive gate bias, the conduction and valence bands shift down to generate electron carriers. In [53], ambipolar CNFETs were assembled to form complimentary logic gates that could be switched between NOR and NAND behavior simply by exchanging VDD and GND. This gate is shown in Figure 27, and demonstrates that ambipolar behaviors can be leveraged to create flexible circuit designs or complex gates in reduced area. Despite the large number of nanomaterial-based devices that have been proposed, there are relatively few that have specifically focused on carbon-nanomaterial based architectures. In one such work [54], a CNT-based FPGA architecture was proposed. This architecture uses arrays of multiple-tube top-gated CNFETs to form the memory decoder in FPGA lookup tables, and uses metallic CNTs to build nanotube-based memories. To account for variations in nanoscale fabrication such as the number of CNTs in a FET channel, a variation-aware CAD flow was developed [54]. While works like this represent a good start, much more research is needed to determine the best circuit designs, architectures, and CAD techniques to fully exploit carbon nanomaterial-based systems.

VI. Challenges and Opportunities Carbon nanomaterials offer many advantages for integrated electronics, but a number of hurdles must be overcome before CNFETs and GNRFETs can be used in large-scale circuits. The first of such challenges is the construction of the nanomaterials themselves. The major issue in CNFET development is the lack of a controlled growth process to determine placement, alignment, chirality, conductivity, diameter, and number of walls. As a result, metallic and semi-conducting nanotubes are mixed together after synthesis. Although postprocessing techniques such as electrical breakdown can help, a fully controlled synthesis recipe needs to be developed to create low cost, high quality semiconducting nanotubes. Even if the type of carbon nanotubes can be predetermined, techniques need to be developed to control the location and alignment of the CNTs on the target wafer. Other challenge that have

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yet to be solved are the hybrid fabrication of carbon nanomaterials and CMOS components and the interfacing of small nanomaterial transistors with larger devices. Similarly, the major challenge in the development of graphene has been the lack of a reasonable process for growing graphene epitaxially on a suitable substrate with predictable edge states and number of layers. Most of the known large-scale fabrication techniques yield areas that contain a varying mix of monolayer, bilayer, trilayer, and many-layer graphene. However, recent developments in the use of growth on copper foils and films have produced promising results. In [3], a technique is demonstrated for producing 1 cm single layer graphene (SLG) on copper films, and patterning GNRFETs on this layer (Figure 28). This allows direct fabrication of uniform transistor arrays using known thin film technology, without the need for delicate transfer processes. Furthermore, the devices demonstrate a low failure rate (0 and a decrease in conductance for Vg 0, 0 ≤ k 2 ≤ 4, 0 ≤ k3 < 2

βγ k 1 α .

(23)

A simplified schematic of this unidirectional coupled system using CCIIs is shown in Figure 9. Here, the coupling gain k1 is given by Rk . The other k i values are zero. Experimental results are shown in the time and the phase plane Figure 10.

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k1 =

1 Rk + 2 R X

Figure 9. Unidirectionally coupled Chua's oscillators.

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(a)

.

(b) Figure 10. Experimental synchronization in (a) time x1, x2; and (b) phase x1, x2

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Several forms have been proposed to encrypt information into a continuos chaotic signal. Some of the principal approaches are [45, 46]: •

•

•

Additive chaotic masking: the information signal is simply added to the chaotic carrier in the transmitter and later, in the receptor the chaotic carrier is reconstructed to be subtracted to the received signal, thus recuperating the original information. Chaos Shift Keying (CSK) several autonomous chaotic systems are switched according to the message which is to be transmitted, they have different parameters. Then, the synchronization will be presented in one of the matched receivers allowing the message to be decoded. Chaotic modulation: in this case the transmitter is an autonomous chaotic system controlled by the message signal. The receiver must synchronize and later apply an inverse operation to its state and the transmitted to recover the message. A general modulation/demodulation approach is seen in [47].

Other possible approaches include [48]: • •

Multiplicative chaotic mixing. Parametric modulation (indirect encoding)

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Experimental message encryption under the Hamiltonian forms synchronization procedure already explained can be seen in [49, 50] using chaos shift keying and in [34] using additive chaotic masking. Although some of the encryption techniques have relatively low secure levels, we stress that the usage of continuous chaos for practical applications is still in its beginning. The mentioned applications have to cope with noisy conditions and uncertain parameter values; however, more robust synchronization approaches such as H-infinity [51], adaptive methods [41, 42, 52], modulation methods [47], as well as correlation based approaches like those seen in discrete chaotic systems [53, 54, 55] could help in overcoming these problems. As a result, the realization of these approaches by using transistors, are open problems in integrated circuit design. In addition, the field of applied chaos is still relatively young. Much is to be done before we see the high quality application of chaos respect to other nonlinear techniques. At this point we can see that potential applications of continuous chaos are in the fields of secure communications and encryption [49, 56, 57, 58], uncorrelated noise generators [59, 60], radar metrics [61] and signal processing [62].

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Conclusion In the generation of linear dynamical systems, many circuit topologies can arise, as the variety of active cells is wide. Some of these cells will present easy coupling characteristics with the rest of the circuit, while others will cost extra stages. Automatic optimization procedures can take place in this respect. In the other hand, the construction of electronic nonlinear dynamical systems can be made simply for those which use PWL functions as nonlinear elements. This is the case of some of the most known multiscroll dynamical systems. The operation conditions for the MOSFETs in a PWL cell were exposed for some circuit topologies, the reader is encouraged to study a particular case in the required detail to capture useful information and develop a satisfactory behavior model and circuit realization. For instance we point out that the explained models of the active cells which can be implemented with CMOS integrated circuit technology were obtained in frequency ranges where it was observed that their terminal resistances were constant. The circuit poles where located far from the chosen operation frequencies. However, the changing of transistor state in the saturated cell can introduce hysteretic effects, because the parasitic capacitance Cgs has slightly changes when the MOSFET goes from saturation to triode and virtually disappear in the cutoff region. To implement saturated step-like functions, notice that active slopes must be as high as possible and breakpoints must be quickly reached as well. In this case, the required active slope gain accuracy loses significance respect to the need of rapid saturation. Thus, the nonlinear function requirements (PWL in this work) will also point the approximation objectives. In this chapter we show how to design a fully CMOS compatible chaotic oscillator. Recommendations have been given to avoid some problems related to the system’s most dominant characteristic: the exponential divergence of perturbed trajectories. Since it is chaotic, special care has to be paid in the signal distribution, extra coupling has to be made and parameters are preferred to be related to an internal reference amount instead of an external process-independent value.

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Acknowledgments We acknowledge the support of CONACyT under the project numbers 48396-Y, J49593Y, P50051-Y and for the sabbatical leave of the second author at University of California Riverside during 2009-2010. This work has been partially supported by Promep México, under the projects UATLX-PTC-088, UATLX-CA-197 and by Consejeria de Innovacion, Ciencia y Empresa, Junta de Andalucia, Spain, under the project TIC-2532. The fifth author thanks the support of the JAE-Doc program of CSIC, co-funded by FSE.

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References [1] Frost, G. (2008). TechTalk Serving the MIT Community (Official Massachusetts Institute of Technology newspaper, 52(24). http:// web. mit. edu/ newsoffice/ 2008/ techtalk5224.pdf [2] González, O. A., Han, G., Pineda de Gyvez J. & Sánchez, S. E. (2000). Lorenz-Based Chaotic Cryptosystem: A Monolithic Implementation. IEEE Trans Circ Sys Fund Theor Appl, vol 47(8), 1243-1247. [3] Fujiwara, T., Horio, Y. & Aihara, K. (2003). An Integrated Multi-Scroll Circuit with Floating-Gate Mosfets. International Symposium on Circuits and Systems (ISCAS’03), 180-183. [4] Rodríguez-Vázquez, A. & Delgado-Restituto, M. (1993). CMOS Design of Chaotic Oscillators Using State Variables: A Monolithic Chua's Circuit. IEEE Trans Circ Sys II, vol 40(10), 596-613. [5] Elwakil, A. S., Salama, K. N. & Kennedy, M. P. (2000). A System for Chaos Generation and its Implementation in Monolithic Form. International Symposium on Circuits and Systems (ISCAS’00), 217-220. [6] Tsukutani, T., Higashimura, M., Kinugasa, Y., Sumi, Y. & Fukui, Y. (2003). A realization of multiple circuit transfer functions using OTA-C integrator loop structure. IEICE Trans Fund Electron Comm Comput Sci, vol E86A(2), 509-512. [7] Tlelo-Cuautle, E. & Duarte-Villaseñor, M. A. (2008). Evolutionary electronics: automatic synthesis of analog circuits by GAs, In Success in Evolutionary Computation, Series: Studies in Computational Intelligence; Editor Yang, A., Shan, Y. Bui, & L. T. Springer-Verlag, Vol. 92, 165-188. [8] Tlelo-Cuautle, E., Duarte-Villaseñor, M. A. & Guerra-Gómez, I. (2008). Automatic synthesis of VFs and VMs by applying genetic algorithms, In Circuits, Systems and Signal Processing, Birkhäuser: Boston, MA, vol. 27(3), 391-403. [9] Laker, K. R. & Sanasen, W. M. C. (1994). Integrated Circuits and Systems; Mc Graw Hill Series in Electrical and Computer Engineering, Mc Graw Hill: New York, NY. [10] Gray, P. R., Hurst, P. J., Lewis, S. H. & Meyer, R. G. (2001). Analysis and Design of Analog Integrated Circuits, John Wiley & Sons, INC: New York, NY. [11] Tlelo-Cuautle, E., Sánchez-López, C., Martínez-Romero, E. & Tan, S. X. D. (2010). Symbolic analysis of analog circuits containing voltage mirrors and current mirrors. Analog Int Circ Signal Proc. DOI: 10.1007/s10470-010-9455-y

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[12] Tan, S. X. D. & Shi, C. J. R. (2004). Efficient Approximation of Symbolic Expressions for Analog Behavioral Modeling and Analysis. IEEE Trans Comput Aided Des Integrated Circ Sys, vol 23(6), 907-918. [13] Fakhfakh, M., Tlelo-Cuautle, E. & Fernández, F. V. (2010). Design of Analog Circuits through Symbolic Analysis, Bentham Sciences Publishers Ltd. [14] Fernández, F. V., Rodríguez-Vázquez, A., Huertas, J, L. & Gielen, G. (1997). Symbolic Analysis Techniques: Applications to Analog Design Automation, Wiley-IEEE PRESS. [15] Tlelo-Cuautle, E., Sánchez-López, C. & Moro-Frías, D (2010). Symbolic analysis of (MO)(I)CCI(II)(III)-based analog circuits. Int J Circ Theor Appl. DOI: 10.1002/cta.582 [16] Tlelo-Cuautle, E., Guerra-Gómez, I., Reyes-García, C. A. & Duarte-Villaseñor, M. A. (2010). Synthesis of Analog Circuits by Genetic Algorithms and their Optimization by Particle Swarm Optimization, In Intelligent Systems for Automated Learning and Adaptation: Emerging Trends and Applications, Editor Chiong, R., IGI Global: Hershey, PA, 173-192. [17] Tlelo-Cuautle, E., Moro-Frías, D., Sánchez-López, C. & Duarte-Villaseñor, M. A. (2008). Synthesis of CCII-s by superimposing VFs and CFs through genetic operations, IEICE Electronics Express, vol 5(11), 411-417. [18] Tlelo-Cuautle, E., Moro-Frías, D. & Fakhfakh, M. (2008). Systematic design of CCI(II)(III)s by combining UGCs, 3rd International Design and Test Workshop, 2008 (IEEE IDT 2008), 350-353. [19] Biolek, D., Senani, R., Biolkova, V. & Kolka, Z. (2008). Active elements for analog signal processing: Classification, review, and new proposals. Radioengineering 17(4), 15-32. [20] Lü, J. & Chen, G. (2006). Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications. Int J Bifurcat Chaos Appl Sci Eng., vol 16(4), 775-858. [21] Muñoz-Pacheco, J. M. & Tlelo-Cuautle, E. (2010). Electronic design automation of multi-scroll chaos generators, Bentham Sciences Publishers Ltd. [22] Serrano-Gotarredona, T. & Linares-Barranco, B. (2002). Current-mode fullyprogrammable piece-wise-linear block for neuro-fuzzy applications. Electron Lett, vol 38(20), 1165-1166. [23] Bhat, M. S., Rekha, S. & Jamadagni, H. S. (2006). Extrinsic Analog Synthesis using Piecewise Linear Current-Mode Circuits. Proceedings of the 19th International Conference on VLSI Design Held Jointly with 5th international Conference on Embedded Systems Design (VLSID’06), 51-56. [24] Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering; Westview. [25] Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos; Springer-Verlag: New York, NY. [26] Hoppensteadt, F. C. (2000). Analysis and Simulation of Chaotic Systems; Applied mathematic sciences 94; Springer Verlag; New York, NY, Vol. 94. [27] Khalil, H. K. (2002). Nonlinear Systems, Macmillan Publishing Company: New York, NY. [28] Kennedy, M. P. (1993). Three Steps to Chaos Part I: Evolution. IEEE Trans Circ Sys Fund Theor Appl, vol 40(10), 640-656. [29] Kennedy, M. P. (1993). Three Steps to Chaos Part II: A Chua’s Circuit Primer. IEEE Trans Circ Sys Fund Theor Appl, vol 40(10), 657-674.

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[30] Silva, C. P. (1993). Shil’nikov’s Theorem, A Tutorial. IEEE Trans Circ Sys Fund Theor Appl, vol 40(10), 675-682. [31] Madan, R. N. (1993). Chua's circuit: a paradigm for chaos; World Scientific Publishing Company. [32] Tlelo-Cuautle, E., Gaona-Hernández, A. & García-Delgado, J. (2006). Implementation of a chaotic oscillator by designing Chua's diode with CMOS CFOAs. Analog Int Circ Signal Proc, vol 48(2), 159-162. [33] Sánchez-López, C., Trejo-Guerra, R. & Tlelo-Cuautle, E. (2008). Simulation of Chua's Chaotic Oscillator Using Unity-Gain Cells. 7th International Caribbean Conference on Devices, Circuits and Systems, 2008 (ICCDCS 2008), 1-4. [34] Trejo-Guerra, R., Tlelo-Cuautle, E., Cruz-Hernández, C. & Sánchez-López, C. (2009). Chaotic communication system using Chua’s oscillators realized with CCII+s. Int J Bifurcat Chaos, Appl Sci Eng., vol 19(12), 4217-4226. DOI: 10.1142/S0218127409025304 [35] Tlelo-Cuautle, E. & Muñoz-Pacheco, J. M. (2008). Automatic simulation of 1D and 2D chaotic oscillators. J Phys Conf Ser, vol 96(1), 1-8. [36] Tlelo-Cuautle, E. & Muñoz-Pacheco, J. M. (2007). Numerical simulation of Chua’s circuit oriented to circuit synthesis. Int J Nonlin Sci Numer Simulat, vol 8(2), 249-256. [37] Sánchez-López, C., Trejo-Guerra, R., Muñoz-Pacheco, J. M. & Tlelo-Cuautle, E. (2010). N-scroll chaotic attractors from saturated functions employing CCII+s. Nonlinear Dynamics. DOI: 10.1007/s11071-009-9652-3 [38] Muñoz-Pacheco, J. M. & Tlelo-Cuautle, E. (2009). Automatic synthesis of 2D-n-scrolls chaotic systems by behavioral modeling. J Appl Res Tech, vol 7(1), 5-14. [39] Muñoz-Pacheco, J. M., Tlelo-Cuautle, E. & Carbajal-Gómez, V. H. (2009). A CADTool for the Design of n-Scrolls Chaotic Systems from Behavioral Modeling. Second International Workshop on Nonlinear Dynamics and Synchronization (INDS'09), 198202. [40] Pikovsky, A., Rosenblum M. & Kurts, J. (2001). Synchronization A universal concept in nonlinear sciences; Canmbridge Nonlinear Science Series 12; Cambridge University Press: New York, NY. [41] Chen, G. (1999). Controlling Chaos and Bifuractions on Engineering Systems; CRC Press: Boca Raton, FL. [42] Wu, Y., Zhou, X., Chen, J. & Hui, B. (2009). Chaos synchronization of a new 3D chaotic system. Chaos, Solitons and Fractals, vol 42, 1812-1819. [43] Pecora, L. M. & Carroll, T. L. (1990). Synchronization in chaotic systems. Phys. Rev. Lett., vol 64, 821-824. [44] Sira-Ramírez, H. & Cruz-Hernández, C. (2001). Synchronization of Chaotic Systems: A Generalized Hamiltonian Systems Approach. Int J Bifurcat Chaos Appl Sci Eng, vol 11(5), 1381-1395. [45] Dachselt, F. & Schwarz, W. (2001). Chaos and Cryptography, IEEE Trans Circ Sys Fund Theor Appl, vol 48(12), 1498-1509. [46] Stavroulakis, P. (2006). Chaos Applications in Telecommunications; CRC Press. [47] Wade, P. T., Oppenheim, A. V. & Rosales, R. R. (2001). Generalized Frequency Modulation, IEEE Trans Circ Sys Fund Theor Appl, vol 48(12), 1405-1412. [48] Silva, C. P. & Young, A. M. (2002). Chaos, fractals, and wavelets in communications & signal processing, IEEE CAS Section & PIMS Seminar.

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[49] Cruz-Hernández, C., López-Mancilla, D., García-Gradilla, V., Serrano-Guerrrero, H. & Núñez-Pérez, R. (2005). Experimental Realization of Binary Signals Transmission Using Chaos. J Circ Sys Comput, vol 14(3), 453-458. [50] Trejo-Guerra, R., Tlelo-Cuautle, E., Cruz-Hernández, C., Sánchez-López, C. & Fakhfakh, M. (2008). Current Conveyor Realization of Synchronized Chua's Circuits for Binary Communications. Design and Technology of Integrated Systems in Nanoscale Era, (IEEE DTIS’08), 1-4. [51] Suykens, J. A. K., Paul, F. C., Vandewalle, J. & Chua, L. O. (1997). Robust Nonlinear Hoo Synchronization of Chaotic Lur’e Systems. IEEE Trans Circ Sys Fund Theor Appl, vol 44(10), 891-904. [52] Leung, H. & Lam, J. (1997). Design of Demodulator for the Chaotic Modulation Communication System. IEEE Trans Circ Sys Fund Theor Appl, vol 44(1), 262-267. [53] Sushchik, M., Tsimring, L. S. & Volkovskii, A. R. (2000). Performance analysis of correlation-based communication schemes utilizing chaos. IEEE Trans Circ Sys Fund Theor Appl, vol 47(12), 1684-1691. [54] Tam, W. M., Lau, F. C. M. & Tse, C. K. (2006). Generalized correlation-delay-shiftkeying scheme for noncoherent chaos-based communication systems. IEEE Trans Circ Sys Fund Theor Appl, vol 53(3), 712-721. [55] Kaddoum, G., Charge, P. & Roviras, D. (2009). A Generalized Methodology for BitError-Rate Prediction in Correlation-Based Communication Schemes Using Chaos. IEEE Comm Lett, vol 13(8), 567-569. [56] Gámez-Guzmán, L., Cruz-Hernández, C., López-Gutierrez R. M. & García-Guerrero, E. (2009). Synchronization of Chua’s Circuits with Multi-Scroll Attractors: Application to Communication. Comm Nonlinear Sci Numer Simulat, vol 14, 2765-2775. [57] Aguilar-Bustos, A. Y., Cruz-Hernández, C., López-Gutiérrez, R. M., Tlelo-Cuautle, E. & Posadas-Castillo, C. (2010). Hyperchaotic Encryption for Secure E-Mail Communication, In Emergent Web Intelligence; Advanced Information Retrieval, Series: Advanced Information and Knowledge Processing. Chbeir, R., Badr, Y., Abraham, A., Hassanien, & A. E. (Eds.). AI&KP book series of Springer Verlag XVI. DOI: 10.1007/978-1-84996-074-8 [58] Muñoz-Pacheco, J. M., Tlelo-Cuautle, E., Trejo-Guerra, R. & Cruz-Hernández, C. (2008). Synchronization of n-Scrolls chaotic systems synthesized from high-level behavioral modeling. 7th International Caribbean Conference on Devices, Circuits and Systems, 2008 (IEEE ICCDCS’08), 1-5. [59] Demirkol, A. S., Özoğuz, S., Tavas, V. & Kilinc, S. (2008). CMOS Realization of a Double-Scroll Chaotic Circuit and its Application to Random Bit Number Generation. International Symposium on Circuits and Systems (ISCAS’08), 2374-2377. [60] Yalҫin, M. E., Suykens, J. A. K. & Vandewalle, J. (2004). True Random Bit Generation From a Double-Scroll Attractor. IEEE Trans Circ Sys, vol 51(7), 1395-1404. [61] Liu, Z., Zhu, X., Hu, W. & Jiang, F. (2007). Principles of Chaotic Signal Radar. Int J Bifurcat Chaos Appl Sci Eng., vol 17(5), 1735-1739. [62] Dedieu, H. & Ogorzalek, M. (2000). Chaos-Based Signal Processing. Int J Bifurcat Chaos Appl Sci Eng, vol 10(4), 737-748.s

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In: Transistors: Types, Materials and Applications ISBN 978-1-61668-908-7 c 2010 Nova Science Publishers, Inc. Editor: Benjamin M. Fitzgerald, pp. 119-132

Chapter 5

O N THE VARIATIONAL I NEQUALITIES A PPROACH TO S TUDY E LECTRICAL C IRCUITS WITH T RANSISTORS K. Addi and D. Goeleven AIM, Universit de La R´eunion

Abstract The main object of this paper is to discuss the use of variational inequalities models to study electrical circuits involving devices like diodes and transistors.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Keywords: Variational inequalities, Mathematical modelling, Diodes, Transistors.

1.

Introduction

Various problems arising in the Engineering Sciences can be formulated as variational inequalities of the form: Find u ∈ Rn such that hM u + q, v − ui + Φ(v) − Φ(u) ≥ 0, ∀v ∈ Rn ,

(1)

where M ∈ Rn×n is a real matrix, q ∈ Rn is a vector and Φ : Rn → R ∪ {+∞} is a proper convex and lower semicontinuous function. Here, for x, y ∈ Rn , the notation phx, yi = Pn n hx, xi to i=1 xi yi is used to denote the euclidean scalar product on R and kxk = denote the corresponding norm. Problem (1) is called ”variational inequality of the second kind” or ”mixed variational inequality” (see e.g. [12], [14], [17] and [23]). The development of variational inequalities methods to study circuits in electronics is a recent research topic ([1]-[11], [15] and [21]) that appears particularly promising. Our aim in this paper is to discuss a mathematical modelling methodology which is of particular interest to study electrical circuits involving devices like transistors.

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120

2.

K. Addi and D. Goeleven

Set-valued Ampere-Volt Characteristics

Electrical devices like diodes are described in terms of Ampere-Volt characteristics (i, V ) that is a graph expressing the difference of potential V across the device as a function of current i through the device. i

-

+

V Figure 1. Electrical Device.

The schematic symbol of a circuit element is given in Figure 1. The conventional current flow i will be depicted on the conductor in the direction of the arrow and the potential V := VA − VB across the device will be denoted alongside the device. Here VA (resp. VB ) is the potential of point A (resp. B). Experimental measures as well as empirical and physical models lead to a variety of monotone graphs that may present Let us suppose here that we may write: (∀i ∈ R) : V ∈ F(i) for some set-valued function F : R ⇉ R. The domain D(F) of F is defined by D(F) = {x ∈ R : F(x) 6= ∅}. We assume that F is maximal monotone, i.e.

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(z1 − z2 )(x1 − x2 ) ≥ 0, ∀ x1 , x2 ∈ D(F), z1 ∈ F(x1 ), z2 ∈ F(x2 ) and the graph G(F) of F, i.e. G(F) := {(x, y) ∈ R × R : x ∈ D(F), y ∈ F(x)} is not properly included in any other monotone subset of R × R. A classical result in convex analysis (see e.g. Proposition 1.3.15 in [13]) ensures that there exists a proper, convex and lower semicontinuous function ϕ : R → R ∪ {+∞} such that (∀i ∈ R) : F(i) = ∂ϕ(i). (2) Here ∂ϕ(i) denotes the convex subdifferential ∂ϕ(i) (see e.g. [18], [24]) of ϕ at i. Let us here recall that for a proper, convex and lower semicontinuous function Φ : Rn → R ∪ {+∞}, the convex subdifferential of Φ at x ∈ Rn is defined by: ∂Φ(x) = {w ∈ Rn : Φ(v) − Φ(x) ≥ hw, v − xi, ∀v ∈ Rn }. The set ∂Φ(x) describes the differential properties of Φ by means of the supporting hyperplanes to the epigraph of Φ at (x, Φ(x)). Moreover, a well known result in convex analysis (see e.g. [18], [24]) ensures that: z ∈ ∂Φ(x) ⇐⇒ x ∈ ∂Φ∗ (z) ⇐⇒ Φ(x) + Φ∗ (z) = hx, zi Transistors: Types, Materials and Applications : Types, Materials and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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where Φ∗ denotes the Fenchel transform of Φ, i.e. the proper, convex and lower semicontinuous function defined by (∀z ∈ R) : Φ∗ (z) = sup {hx, zi − Φ(x)}, x∈D(Φ)

with D(Φ) := {x ∈

Rn

: Φ(x) < +∞} denoting the domain of Φ.

The function ϕ in (2) is called the electrical superpotential (determined up to an additive constant) of the device. Roughly speaking, the electrical superpotential ϕ appears as a ”primitive” of F in the sense that the ”derivative” (in the generalized sense) of ϕ recovers the set-valued function F (see [3]). Remark 2.1. The concept of superpotential has been introduced in convex mechanics by Moreau [20] and developed in nonconvex mechanics by Panagiotopoulos [23]. We have V ∈ F(i) ⇐⇒ V ∈ ∂ϕ(i) ⇐⇒ i ∈ ∂ϕ∗ (V ) ⇐⇒ ϕ(i) + ϕ∗ (V ) = iV.

2.1.

Diode Models

Example 2.1. (Ideal diode model) The diode is a device that constitutes a rectifier which permits the easy flow of charges in one direction but restrains the flow in the opposite direction. Diodes are used in power electronics applications like rectifier circuits, switching inverter and converter circuits. Figure 2 illustrates the ampere-volt characteristic of an ideal diode. i

-

+

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

V

Figure 2. Ideal diode model.

This is a model in which the diode is a simple switch. If V < 0 then i = 0 and the diode is blocking. If i > 0 then V = 0 and the diode is conducting. We first see that the ideal diode is described by the complementarity relation V ≤ 0, i ≥ 0, V i = 0 which can be written equivalently as min{−V, i} = 0. The electrical superpotential of the ideal diode is  0 if x ≥ 0 (∀x ∈ R) : ϕD (x) = ΨR+ (x) := . +∞ if x < 0

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K. Addi and D. Goeleven

We have indeed

  R− if x = 0 0 if x > 0 (∀x ∈ R) : ∂ϕD (x) := ,  ∅ if x < 0

so that the complementarity relation V ≤ 0, i ≥ 0, V i = 0 can be written equivalently as V ∈ ∂ϕD (i). Example 2.2. (Practical diode model) Figure 3 illustrates the ampere-volt characteristic of a practical diode model. i

-

+

V

V (Volts) 1 V1 i (mA)

-100

V2

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Figure 3. Practical diode model.

There is a voltage point, called the knee voltage V1 , at which the diode begins to conduct and a maximum reverse voltage, called the peak reverse voltage V2 , that will not force the diode to conduct. When this voltage is exceeded, the depletion may breakdown and allow the diode to conduct in the reverse direction. Note that usually | V2 |>>| V1 | and the model is locally ideal. The electrical superpotential of the practical diode is  V1 x if x ≥ 0 . (∀x ∈ R) : ϕP D (x) = V2 x if x < 0 Then

  V2 if x < 0 [V2 , V1 ] if x = 0 , (x ∈ R) (∀x ∈ R) : ∂ϕP D (x) =  V1 if x > 0

recovers the ampere-volt characteristic (i, V ).

2.2.

Transistor Models

A junction transistor is a semiconductor triode capable of producing amplification. A P-N-P (resp. N-P-N) transistor consists of a silicon (or germanium) crystal in which a layer Transistors: Types, Materials and Applications : Types, Materials and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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123

PNP TRANSISTOR

IE E Emitter

I

C C

°

° Collector

I

B

°B Base

EBERS-MOLL MODEL α I IC I

°E

αN I E I

E

C

°

N

P

°

°

N

P

°C

°

I'

I

IB

VE

VC

°

Figure 4. Transistor P-N-P. NPN TRANSISTOR I E Emitter

I

E

C C ° Collector

°

I

B

°B Base

EBERS-MOLL MODEL α IC I I

°E

αN I E I

E

C

°

P

N

°

°

I

N

P

°

°C

I'

V

I

V E

B

C

°

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 5. Transistor N-P-N.

of N-type silicon (resp. P-type) is sandwiched between two layers of P-type silicon (resp. N-type). The three portions of transistor are known as emitter, base and collector. The behavior of a transistor can be described by means of the Ebers-Moll model (see e.g. [19]) involving two diodes placed back to back and two dependent current-controlled sources αI IC and αN IE shunting the diodes. Here αN ∈ [0, 1[ is known as the current gain in normal operation and αI ∈ [0, 1[ is known as the inverted common-base gain current. Throughout this paper, we will use the notations and conventions of Figures 4 and 5. Example 2.3. (Ideal transistor model) Let us here assume that the two diodes of EbersMoll are ideal. That means that each diode acts as a simple switch. If VE < 0 (resp. VC < 0) then I = 0 (resp. I ′ = 0) and the diode is blocking. If I > 0 (resp. I ′ > 0) then VE = 0 (resp. VC = 0) and the diode is conducting. We may then write VE ≤ 0, I ≥ 0, VE I = 0 and VC ≤ 0, I ′ ≥ 0, VC I ′ = 0. That is also: 

−VE −VC



≥ 0,



I I′



    −VE I ≥ 0, h , i=0 −VC I′

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(3)

124

K. Addi and D. Goeleven

or equivalently

Moreover:





VE VC

I I′





=

∈





1 αN

∂ψR+ (I) ∂ψR+ (I ′ ) αI 1





.

IE IC

(4) 

(5)

and: IB = −(IE + IC ).

(6)

The relations in (4)-(6) constitutes a handy mathematical model for the transistor. Example 2.4. (General transistor model) A general mathematical model of a transistor can be given provided that there exist proper convex and lower semicontinuous functions ϕE , ϕC such that the ampere-volt characteristics of the two diodes of Ebers-Moll model can be formulated as: VE ∈ ∂ϕE (I), VC ∈ ∂ϕC (I ′ ). The function ϕE is called the emitter electrical superpotential of the transistor while the function ϕC is named the collector electrical superpotential. The mathematical model of the transistor reads:     ∂ϕE (I) VE , (7) ∈ ∂ϕC (I ′ ) VC 

I I′



=



1 αN

αI 1



IE IC



(8)

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

and IB = −(IE + IC ).

(9)

The different models of diodes that have been discussed in the previous Section can be here used so as to define corresponding models of transistors.

3.

Mathematical Modelling Approach

In this paper, we illustrate our mathematical modelling approach with a rectifierstabilizer circuit (Figure 6).

The rectifier-stabilizer circuit involves four diodes D1 , D2 , D3 and D4 , a Zener diode Dz , a N-P-N transistor T , two resistors R1 and R2 and two capacitors C1 and C2 . This circuit is supplied by the signal input u. We first follow a classical compartmental approach to split it into two blocks: the ”rectifier” circuit depicted in Figure 7 and the ”stabilizer” one presented in Figure 8.

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125

T

D4

D1

R₁

u

C2

C1

R₂

DZ

D2

D3

Figure 6. Rectifier-Stabilizer circuit.

V1

D4

i1

u

V4

D1

i4 R

V2

V

C1

D3

V3

D2

i3

i2

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 7. Rectifier circuit.

VC

VE

R₁

C2 R₁

Figure 8. Stabilizer circuit.

We suppose that all diodes of the rectifier block are ideal. We denote by Vi the voltage of diode Di (1 ≤ i ≤ 4), V the voltage of the capacitor and use the other notation indicated Transistors: Types, Materials and Applications : Types, Materials and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

126

K. Addi and D. Goeleven

on Figure 7. Kirchhoff’s laws yield the system:  i1 + i4 = VR + C1 dV  dt    = V + V3  −V4 i3 = i4 + i1 − i2    −V = V + V3 − u 1   −V2 = −V3 + u.

We have

 −V4    −V1 −V2    −V3

∈ ∈ ∈ ∈

−∂ϕD (i4 ) −∂ϕD (i1 ) −∂ϕD (i2 ) −∂ϕD (i3 )

Moreover V3 ∈ ∂ϕD (i3 ) if and only if i3 ∈ ∂ϕ∗D (V3 ). Setting (∀ V ∈ R) : θD (V ) = ϕ∗D (−V ) we get (∀ V ∈ R) : ∂θD (V ) = −∂ϕ∗D (−V ) Note that here θD (V ) = ΨIR− (−V ) = ΨIR+ (V ) = ϕD (V ). Therefore V3 ∈ ∂ϕD (i3 ) ⇔ i3 ∈ −∂ϕD (−V3 ). Setting

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(∀x ∈ R4 ) : Φ(x) = ΨR+ (x1 ) + ΨR+ (x2 ) + ΨR+ (x3 ) + ΨR+ (x4 ) = Ψ(IR+ )4 It results that the dynamical behaviour of the circuit in Figure 7 is descibed by the system:   B i4 z }| {  −V3  −1 1 1 dV  = V + 0 0  (10)  i1  , dt RC1 C1 C1 i2 y

C

N

yL

z }| { z z }| z }| {  }| {   −V4 1 0 −1 0 0 i4  i3   0   1 0 1 −1   −V3        −V1  =  1  V +  0 −1 0 0   i1 −V2 0 0 1 0 0 i2

F

z }| { 0   0  +    −1  u 1 {

(11)

and y ∈ −∂Φ(yL ).

(12)

At equilibrium, the dynamical circuit in Figure 7 reduces to the circuit in Figure 9 and the stationary solutions of (10)-(12) satisfy the problem:  −1  RC1 V + ByL = 0 (13)  4 hN yL + CV + F u, v − yL i + Φ(v) − Φ(yL ) ≥ 0, ∀ v ∈ R .

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127

D4

V1

V4

D1

i1

i4

u

R

V

D3

V2

D2

i2

V3

i3

Figure 9. Rectifier circuit. From the first equation of (13) one deduces that V = RC1 ByL , so that y = (N + + F u and our problem reduces to the variational inequality VI(M, Φ, Fu):

1 a CB)yL

yL ∈ R4 : hM yL + F u, v − yL i + Φ(v) − Φ(yL ) ≥ 0, ∀ v ∈ R4 . with

(14)

 R −1 R 0  1 0 1 −1   M := N + RC1 CB =   R −1 R 0  . 0 1 0 0 

Let us now consider the stabilizer block as in figure 10.

iE

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iC iR

VC

VE

R₁

R₂

iz

V

Dz

Vo Vz

Figure 10. Stabilizer circuit. We denote by VE , VC and Vz the voltages of the transistor and the Zener diode respectively as indicated on Figure 10. Note that we omit the capacitor C2 , thanks to the equilibrium, and use the other notation indicated on Figure 10. Kirchhoff’s laws yield the system:  = V + −R1 (iz + ie + ic )  −Vz −Vz − VC = V  VE − VC = V − R2 ie .

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K. Addi and D. Goeleven

The N-P-N transistor behavior is described by means of the Ebers-Moll model as given in the previous section while the ideal Zener diode behavior is depicted in Figure 11.

i

-

+

V

s

Figure 11. An ideal Zener Diode.

Setting Vze = Vz − Vs Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

we see that the ideal Zener diode is then described by the complementarity relation Vze ≥ 0, iz ≥ 0, Vze iz = 0. We have     −R1 −R1 −1 0 0 Vze  −1 0 1   −VE  =  0 0 0 −R2 0 −1 1 −VC and



iE iC



1 = 1 − αI αN



1 −αN

    − R1 iz V + Vs 0   ie  +  V + Vs  . 0 ic V −αI 1



I I′



and thus       −1 0 0 Vze −R1 K R1 (αN − 1) R1 (αI − 1) iz  −1 0 1   −VE  = 1   I  0 0 0 K 0 −1 1 −VC I′ 0 −R2 R2 αI   V + Vs +  V + Vs  , V

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129

where K = 1 − αI αN . Then Θ

w

z

}| { z  }| { z }| { R1 (1 − αN ) R1 (1 − αI ) Vze R1 K iz  −VE  = 1  R1 K R1 (1 − αN ) + R2 R1 (1 − αI ) − αI R2   I  K R1 K R1 (1 − αN ) R1 (1 − αI ) −VC I′ z 

q

z 

}| { V + Vs +  V + Vs  . V

We have also:

  Vze −VE  −VC

Setting

∈ −∂ΨIR+ (iz ), ∈ −∂ΨIR+ (I), ∈ −∂ΨIR+ (I ′ ).

(∀x ∈ IR+ ) : Ξ(x) = ΨIR3+ (x) = ΨIR+ (x1 ) + ΨIR+ (x2 ) + ΨIR+ (x3 ), we obtain the variational inequality model VI(Θ, Ξ, q): z ∈ R3 : hΘz + q, v − zi + Ξ(v) − Ξ(z) ≥ 0, ∀v ∈ R3 .

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4.

(15)

Conclusion

The study of electrical circuits with diodes and transistors may benefit a lot from the variational inequalities modelling approach. Such mathematical models can indeed be qualitatively and rigorously studied by means of powerful mathematical tools from convex and nonlinear analysis and solved numerically by means of appropriate numerical methods. For example, the variational inequalities modelling the rectifier-stabilizer circuit of the previous section has been studied in this way in [6] where existence, uniqueness and numerical simulations have been obtained. More precisely, let u : IR+ → IR be a given supplied voltage. Using the results proved in [6], we may assert that for each t ∈ IR+ , the rectifier output signal V (t) is uniquely defined by (∀t ∈ IR+ ) : V (t) = R(i1 (t) + i4 (t)) (16) where for each t ∈ IR+ , i1 (t) and i4 (t) are computed as solutions inequality VI(M, Φ, Fu(t)) in (14). Setting   V (t) + Vs (∀t ∈ IR+ ) : q(t) =  V (t) + Vs  V (t)

of the

variational

with V (t) defined in (16), we may also use the results proved in [6], to assert that for each t ∈ IR+ , the stabilizer output signal Vo (t) is uniquely defined by (∀t ∈ IR+ ) : Vo (t) =

R2 (I(t) − αI I ′ (t)) K

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130

K. Addi and D. Goeleven

where I(t) and I ′ (t) are computed as solutions of the variational inequality VI(Θ, Ξ, q(t)) in (15). 15 Vi V 10

voltage

5

0

−5

−10

−15

0

0.01

0.02

0.03

0.04

0.05 time

0.06

0.07

0.08

0.09

0.1

Figure 12. Rectifier circuit. 15 input voltage u in/out V output voltage Vo

5

voltage

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

10

0

−5

−10

−15

0

0.01

0.02

0.03

0.04

0.05 time

0.06

0.07

0.08

0.09

0.1

Figure 13. Stabilizer circuit. More general existence and uniqueness theorems applicable to the study of a large class of models in electronics have been proved in [4] and [5]. The Lyapunov stability of equilib-

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131

rium points of electrical circuits with diodes is also a question that can been studied through variational inequalities as in [8].

References [1] Acary V., Brogliato B., and Goeleven D., Higher Order Moreau’s Sweeping Process: Mathematical Formulation and Numerical Simulation, Mathematical Programming, 113, 133-217, 2008. [2] Addi K., On Mathematical Analysis of a Non-regular Electronic Circuit, Pac. J. Optim., 5, 3-16, 2009. [3] Addi K., Adly S., Brogliato B., and Goeleven D., A method Using the Approach of Moreau and Panagiotopoulos for the Mathematical Formulation of Non-regular Circuits in Electronics, Nonlinear Analysis, Series C : Hybrid Systems and Applications, 1, 30-43, 2007. [4] Addi K., Brogliato B., and Goeleven D., A Qualitative Mathematical Analysis of a Class of Linear Variational Inequalities via Semi-Complementarity Problems, To appear in Mathematical Programming, 2009. [5] Addi K. and Goeleven D., A Spectral Condition for Variational Inequalities: Theory, Methods and Applications in Electronics, in New Research on Nonlinear Analysis, Nova Publishers, 2008. [6] Addi K. and Goeleven D., Variational inequality formulation of a rectifier-stabilizer circuit, AIM Research Report 2009, Submitted for publication.

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[7] Brogliato B., Some Perspectives on the Analysis and Control of Complementarity Systems, IEEE Transactions on Automatic Control, 48, 918-935, 2003. [8] Brogliato B., and Goeleven D., The Krakovskii-LaSalle Invariance Principle for a Class of Unilateral Dynamical Systems, Mathematics of Control, Signals and Systems, 17, 57-76, 2005. [9] Camlibel M.K., Heemels W.P.M.H. and Schumacher H., On Linear Passive Complementarity Systems, Eur. J. Control, 8, 220-237, 2002. [10] Camlibel M.K. and Schumacher J.M., Existence and uniqueness of solutions for a class of piecewise linear dynamical systems, Linear Algebra and its Applications, 351-352, 147-184, 2002. [11] Denoyelle P., and Acary V., The Non-Smooth Approach Applied to Simulating Integrated Circuits and Power Electronics Evolution of Electronic Circuit Simulators Towards Fast-SPICE Performance, INRIA Research Report, ISNN 0249-0803, RhneAlpes, France, 2006. [12] Facchinei F., and Pang J.-S., Finite Dimensional Variational Inequalities and Complementarity Problems, Springer Verlag, Berlin, 2003. [13] Goeleven D., Motreanu D., Dumont Y. and Rochdi M., Variational and Hemivariational Inequalities – Theory, Methods and Applications. Vol.1: Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Boston, 2003. Transistors: Types, Materials and Applications : Types, Materials and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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[14] Goeleven D. and Motreanu D., Variational and Hemivariational Inequalities – Theory, Methods and Applications. Vol.2: Unilateral Problems and Unilateral Mechanics, Kluwer Academic Publishers, Boston, 2003. [15] Goeleven D., An existence and uniqueness result for a linear mixed variational inequality arising in electrical circuits with transistors, JOTA, 138, 397-406, 2008. [16] Isac G., Complementarity Problems, Lecture Notes in Mathematics 1528, Springer Verlag, Heidelberg, 1993. [17] Konnov I.V., and Volotskaya E.O., Mixed Variational Inequalities and Economic Equilibrium Problems, Journal of Applied Mathematics, 6, 289-314, 2002. [18] Hiriart-Urruty J.B. and Lemarechal C., Fundamentals of Convex Analysis, Springer Grundlehren Text Editions, Heidelberg, 2001. [19] Millman J., and Halkias C.C., Integrated Electronics, McGraw-Hill Kogakusha, LTD, Sydney, 1985. [20] Moreau J.J., La Notion du Surpotentiel et les Liaisons Unilat´erales on Elastostatique, Acad. Sci. Paris, 167A, 954-957, 1968. [21] Murakami Y., A Method for the Formulation and Solution of Circuits Composed of Switches and Linear RLC networks, IEEE Transactions on Circuits and Systems-I, 49, 315-327, 2002. [22] Murty K.G., Linear Complementarity, Linear and Nonlinear Programming, Available at http://www-personal.engin.umich.edu/∼ murty/book/LCPbook/, 1997. [23] Panagiotopoulos P.D., Inequality Problems in Mechanics and Applications, Convex and Nonconvex Energy Functions, Birkha¨user, Basel, 1985.

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[24] Rockafellar R.T., Convex Analysis, Princeton Univ. Press, Princeton, 1970. Peer reviewers: • Prof. Aleksandar Rodic Mihajlo Pupin Institute, Robotics department University of Belgrade 11060 Belgrade, Zvezdara, Volgina 15 SERBIA [email protected] • Prof. Samir Adly DMI-XLIM, University of Limoges 87060 Limoges FRANCE [email protected]

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Chapter 6

PHOTOCURRENT STUDY OF THE TRANSPORT MECHANISM IN MOLECULAR SELF-ASSEMBLING FIELD EFFECT TRANSISTORS V. Andrei Pakouleva, Dmitry Zaslavskyb and Vladimir Burtmanc,* a

Department of Chemistry, University of Wisconsin-Madison, 1101 University Ave., Madison, WI, USA b School of Chemical Sciences, University of Illinois at Urbana-Champaign, 106 Noyes Lab, Urbana, USA c University of Utah, Department of Geology and Geophysics Salt Lake City, Utah, USA

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Abstract This chapter focuses on the intrinsic charge transport in organic field-effect transistors (OFETs) based on self-assembled monolayers (SAMs) and on the nature of transport in organic systems, in which surface and bulk properties are undistinguishable due to scale of consistent materials. Recently developed SAM-OFETs are characterized by photovoltaic measurements. The dynamics of charge transport are determined and used to clarify a transport mechanism. Taken together, these SAM devices provide a unique tool to study the fundamentals of polaronic transport on organic surfaces and to discuss the SAM OFET performance. An outline is presented of the outstanding problems that are now becoming experimentally reachable owing to the development of SAM-OFETs. Vapor phase molecular self-assembly of 1,4,5,8-Naphthalene-tetracarboxylic diphenylimide (NTCDI) having a rich -stacking charge delivery system is used to enhance the performance of molecular fieldeffect devices. Charge mobility in SAM-OFET could achieve values of more than 30 cm2 V1 -1 s . The dynamics of charge transport in NTCDI-derived SAM-OFETs were probed using time-resolved measurements in an NTCDI-derived photovoltaic cell device. Time-resolved photovoltaic studies allow us to separate the charge annihilation kinetics in the conductive NTCDI channel from the overall charge kinetic in a SAM-OFET device. It has been demonstrated that tuning of the type of conductivity in NTCDI SAM-OFET devices is possible by changing Si substrate doping. In addition, the possibility of measuring transport in highly ordered SAM structures shines light on the polaron charge transfer in organic *

E-mail address: [email protected].

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V. Andrei Pakoulev, Dmitry Zaslavsky and Vladimir Burtman materials. Our study proposes that a cation-radical exchange (redox) mechanism is the major transport mechanism in SAM nanodevices. The role and contribution of the transport through delocalized states of redox active surface molecular aggregates of NTCDI are exposed and investigated in this chapter.

Introduction

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1. OFETs: Achievements vs. Problems The invention of an organic donor-acceptor structural analog of a silicon p-n junction in 1974 [1] marked the beginning of the intensive research in the field of molecular electronic devices. As a part of the semiconductor research area, this field has traditionally been closely related to applications, and is unique in the synergy between device development and fundamental science. The emergence of new approaches for molecular engineering is a vital requirement for progress in molecular electronics and should eventually lead to molecular devices with pre-designed properties that may compete with silicon technologies, especially due to the small size involved. In the last 10 years a number of different experimental strategies have been used to fabricate organic field effect transistors (OFETs) and to study electron transport in these devices. Despite the very impressive results in device fabrication and initial insight into device transport, the new field also brought many open questions [2, 3]. One of the most complete analyses of this field could be found in a review by Frisbie et al. [4]. In the current chapter we will focus on kinetic studies in OFET structures with a predefined epitaxial order and on the development of adequate analytical tools to analyze charge transport in these structures. A lack of basic understanding of structure-property relationships in OFET materials (and in organic solids in general) results in a knowledge gap that impedes rational efforts to design organic semiconductors with further enhancements in electrical performance [4]. One of the major obstacles for achieving efficient OFET is the low charge mobility in most organic materials. This is an acute problem, since carrier mobilities in disordered organic semiconductor films are several orders of magnitude smaller than in inorganic semiconductors. In contrast, carrier mobility in organics could be dramatically enhanced in ordered structures, such as spontaneously ordered regio-regular polythiophene [RR-P3HT] films. However, the inherent problem of such ordered systems is the interchain coupling. The charge-transfer probability between charged polarons and bipolarons belonging to neighboring chains [5], along with high electron-phonon coupling and the Peierls transition, leads to charge trapping [6]. The strong coupling could result in low mobility and even induce pseudo-gaps [7]. In addition, disordered thin films of small organic-molecule-based electronic devices have considerable charge quenching due to carrier trapping at defects, and charge recombination at grain boundaries and interfaces that result in low electron mobility. The scope of questions about SAM-OFET systems provides a host of other unanswered issues, inherent in traditional OFET technology, including the following: What is the role of orderdisorder interplay in search for good mobilities? In particular, how is crystalline or partially crystalline ordering involved in the transport mechanism in organic solids? What is an impact of the thin film technology on device properties [4]? What are the structural and electronic factors that facilitate or impede transport in organic semiconductor films? Are there particular molecular structures and crystal packing motifs that are especially favorable for transport?

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What is the role of film quality, defects and grain boundaries? Most of these unanswered questions are a direct consequence of the lack of basic knowledge in transport mechanisms in organic semiconductors. The other challenge in OFET fabrication is preparation of an efficient n-type conductive channel. OFETs based on n-channel (also commonly referred to as “n-type”) semiconductors conduct electrons and achieve their high conductivity “on” state with positive gate voltages, whereas most OFETs (e.g., pentacene OFETs) are p-channel devices (i.e., they conduct holes) and turn on with negative gate voltages. The motivation for seeking good n-channel OFETs is that they enable complementary circuit design. Electrical engineers are proficient at developing low-power complementary circuits that utilize both positive and negative gate voltages to turn transistors on and off. Consequently, the development of good n-channel OFETs with performance comparable to that of pentacene OFETs is a major goal for organic electronics. Relatively fewer n-type materials have also been reported, including C60 and its derivatives [8, 9], oliothiophene and its derivatives [10], copper hexadecafluorophthalocyanine [10], naphthalene diimides, and perylene diimides [11, 12, 13, 14]. However, some organic n-channel semiconductor materials cannot operate in air due to electron trapping by oxygen and carrier injection issues [15]. The approaches to achieve n-channel materials with air stability and high mobility have been reported by the incorporation with strong electronwithdrawing groups, such as -CN and –F [16, 17, 18]. Currently, the highest mobility of ntype organic semiconductor is obtained from a thiazole oligomer derivative (1.83 cm2 V-1 s-1) measured in a vacuum environment [19] or from a perylene diimide derivative with a mobility of 0.64 cm2 V-1 s-1 measured in air.

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2. Analytical Tools to Study OFET Device Operation Lack of structure-property understanding is partly due to a shortage of adequate analytical tools to study OFET device operation. For example, at the current stage of development, mobility is still the most important characteristic in OFETs. Does an OFET device have a special structure, special organic-metal interface or a peculiar charge transport mechanism? Is it just a mobility in organics, or should the whole OFET device structure be optimized? The situation is complicated by the strong dependence of the current through the molecular junction on the nature of the chemical bond to the electrodes [20]. Chemically bounded to the semiconductor, molecules are likely to affect the energy and density of surface states and, therefore, the semiconductor band bending [21]. Furthermore, the presence of a dipolar layer at the semiconductor/metal interface can affect the barrier for electron transport inside the semiconductor [22, 23]. It is clear that OFET experimental results should be analyzed in terms of multiple mechanisms, and it is crucial to establish whether this is justified or if, in doing so, new mechanisms are ignored. It would be also very helpful to study independently charge injection into- and charge transport in the organic channel. However, the methods for observation and evaluation of such kinetics in organic devices at the component level have yet to be developed. Current state of the art employs varieties of Kelvin probe-derived techniques that are widely used to determine electron work function and bend banding in organic-inorganic heterostructures. UV photoemission and inverse photoemission spectroscopies (UPS, IPES) are used to determine the position of the highest occupied and lowest unoccupied molecular

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orbitals (HOMO and LUMO, respectively). X-ray photoelectron spectroscopy (XPS) measurements served to estimate band bending. Comparison of these techniques with measured I-V characteristics enables one to determine mechanism(s), which control charge transfer in molecular diodes. These methods are static, since probing interfaces in time-scale compared to interface charge transfer is very challenging. Fast spectroscopy was applied to clarify dynamic of charge transfer in organic polymers, polymer-fullerenes and hybrid singlewall-carbon-nanotube strucrures and even in hybrid structures like dye-sensitized photovoltaic cells. Nevertheless, direct observation of complete dynamic of charge transfer in the interface is still far from the scope of these methods. A relatively simple and sensitive photocurrent method to probe transport mechanisms in organic ordered system in time resolved regime is introduced in this chapter.

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3. Self-Assembly and SAM OFET Devices The way to achieve epitaxial control in organic devices is not trivial too. The transport mechanism in low-dimensional organic structures is intimately related to the ordering and dimensionality of the underlying electronic system, which may transform with the molecular packing [24]. Therefore, a systematic research in organic semiconducting materials and structures with controlled order is a pre-requirement for efficient OFET fabrication. This fine structural tuning is a very challenging goal since tunable organic epitaxial structures have not been studied at all, in contrast to inorganic semiconductors. We have developed such approach [25]. We explored ordered self assembled monolayer (SAM) structures of small molecules with the potential of high charge mobilities and low intermolecular interaction. SAMs [26, 27] are ordered assemblies of functional molecules, which are formed spontaneously on an appropriate surface and are typically used to modify surface properties of materials. The tendency of certain types of molecules to spontaneously form assemblies arises from their characteristic, amphiphilic chemical structure. Self-assembling molecules are typically composed of two groups: a head group which has high affinity for an appropriate surface and a tail group with a high affinity for similar tail groups of other, neighbouring molecules. When such molecules are brought in contact with the appropriate surface, the head groups of the molecules will physisorb or form a chemical bond to the surface (chemisorption). As the density of a SAM increases, the tail groups of the molecules on the surface will come closer and start to interact, typically giving rise to the order within the selfassembled monolayer. Often the molecules also contain a functional end group which allows further physical or chemical functionalization. Both the self-assembling and self-ordering properties of SAMs, combined with the presence of this functional end group and with the SAMs stability (due to the surface bond) are the keys to a wide range of possibilities and applications such as surface engineering and surface modification for controlling adhesion, corrosion, lubrication, (bio)chemical sensing and mimicking, etc. By choosing precursors with suitable physicochemical properties, it is possible to exert a fine control on the formation processes in order to obtain complex architectures. Clearly, when it comes to designing a synthesis strategy for a material the most important tool is the knowledge of the chemistry of the building units. SAM is a process that is easily influenced by external parameters. This can make synthesis more problematic due to the many free parameters that require control. On the

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other hand, this has the exciting advantage that a large variety of shapes and functions on many length scales can be obtained [1].

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4. NTCDI Thick Film and SAM OFETs NTCDI-derived compounds were already tried in organic devices, including thin and thick film OFETs [29, 30, 31, 32]. The highest mobilities of solution-processed n-type semiconductors were achieved from quaterthiophene [33] (0.2 cm2 V-1 s-1) measured in a vacuum and from naphthalene diimide (NTCDI) with a long-chain fluorinated alkyl group (0.01 cm2 V-1 s-1) solution-processed in air by Katz et al. The highest n-channel mobilities of these devices, which were based on NTCDI derivatives were calculated to be about 1.2 × 10-3 cm2 V-1 s-1 [34]. Direct fabrication of ordered organic structures as SAM and there incorporation in molecular device is still a challenging task, and only few reports are available. Several known and new SAM deposition and characterization techniques were explored and optimized to obtain the required, qualitative SAMs for use in organic thin-film transistors for a first time in ref. [35]. Various SAMs were then applied to different interfaces in SAM-OFETs, which allowed a better control of the morphology and properties of the organic semiconducting material. This resulted in a significant improvement of the electrical performance of the organic transistors and even enabled a new way to pattern the organic semiconductor. Alternatively, using a technique called „molecular layer epitaxy‟ [36, 37], multilayer structures with 2D -stacking and semiconducting properties have been obtained. This structures have been applied in organic light emission dioded (SAM-OLEDs), and in SAM-OFETs fabrication [38, 39]. A room-temperature electron mobility as high as 90 cm2 V-1s-1 was reported using MLE [40]. The MLE technology is a vapor-phase oriented technique that allows the buildup of organic heterostructures via epitaxial growth of subsequent layers by interlayer covalent bonding. Vapor-phase self-assembly protocol for device fabrication introduced below is a modification of MLE approach used for new generation of SAM-OFETs. The same approach was used for a fabrication of SAM photovoltaic devises (SAM-PVC) to study photophysics of charge transfer phenomena in NTCDI channel and to model charge transport in NTCDI SAM-OFETs.

5. Transport in NTCDI SAM OFETs This chapter is focused on self-assembled monolayres of small organic molecules with extended -electron system, NTCDI, containing two benzene rings, four carboxyl groups and two anhydrides. These two anhydrides are used to connect NTCDI molecules to substrate by chemical imide bonds (out-of-plane bonding), while two adjacent NTCDI molecules held together in a solid monolyares by van der Waals forces (in-plane bonding). Weak van der Waals bonding in in-plane direction and weak intermolecular overlap of electronic orbitals lead to the narrow electronic bands (a typical bandwidth W~0.1 eV is two orders of magnitude smaller than that in silicon) and the low mobility of charge carriers ( ~1–10 cm2 V-1 s-1at room temperature). Anisotropy of the transfer integrals between adjacent molecules reflects the low symmetry of the molecular packing. It is believed that the most adequate description of charge transport in these organic semiconductors is based on the concept of

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polarons, molecular states with a correlation between the electronic and lattice degrees of freedom at the scale of the order of the lattice constant [41, 42, 43, 44]. After several decades of intensive research, our basic understanding of charge transport in small-molecule organic semiconductors remains limited [45]. The complexity of transport phenomena in these systems is caused by the polaronic nature of charge carriers and the strong interaction of the polarons with defects. OFETs are aimed to be used at room temperature. Therefore, the high-temperature polaronic transport should be developed. At room temperature, which is typically comparable to or even higher than the characteristic phonon energies, lattice vibrations might become sufficiently strong to destroy the translational symmetry of the lattice. In this regime the fluctuation amplitude of the transfer integral becomes of the same order of magnitude as its average value [46]. The band description breaks down, and a crossover from the transport in delocalized states to the incoherent hopping between localized states is predicted with temperature increase [47, 48, 49, 50, 51, 52, 53]. While time-of-flight experiments demonstrated that a possibility of the intrinsic (not limited by static disorder) charge transport can be realized in the bulk of these crystals [54], the mechanism of the charge transport on the surface of organic semiconductors is less studied [55, 56, 57]. It is clear that in SAM-OFET devices field-induced charges should propagate along the interface between an SAM and a gate dielectric, which are true surface conditions. The differences in transport mechanism for surface and bulk media include: (i) the density of carriers, which in field-effect experiments can exceed that in bulk measurements by many orders of magnitude, approaching the regime when the intercharge distance becomes comparable with the size of polarons [58] (ii) motion of charge carriers in the field-induced conduction channel may be affected by the polarization of the gate dielectric [59], and (iii) molecular packing on the surface can also be different from that in the bulk. This chapter will expand the study of correlation between dynamic characteristics (time-of-flight measurements) and charge mobility in bulk crystals to the case of SAM system. Exploration of the polaronic transport on organic surfaces is crucial for a better understanding of fundamental processes that determine operation and ultimate performance of organic electronic devices. Fundamental research has been hampered by the lack of a proper tool for exploring the polaronic transport on surfaces of organic semiconductors. The most common organic electronic device whose operation relies on surface transport is the organic thin-film transistor. Over the past two decades a large effort in the development of OFETs has resulted in improvement of the characteristics of these devices [60], so that currently the best organic OFETs outperform the widely used amorphous silicon transistors. However, even in the best OFETs, charge transport is still dominated by the presence of structural defects and chemical impurities.

6. The Scope of This Chapter In this chapter we present a brief overview of the experimental results obtained with NTCDI SAM-OFETs over the last eight years. Because we focus on the physics of electronic processes in these devices, many surface chemistry oriented issues will not be discussed here; we refer the reader to a bible of self-assembling by A. Ulman for details on SAM fabrication and characterization. This much-cited, definitive text has become the standard resource on the SAM topic worldwide. In the Materials and Methods section we briefly describe the SAM

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growth, OFET and organic photovoltaic cells (PVC) fabrication techniques and measurement techniques. The Results section focuses on the observation of intrinsic polaronic transport in SAM devices. Electronic mechanisms of charge transport are discussed in the Discussion section. The Conclusion section outlines several basic issues that are now starting to be experimentally reachable due to the development of SAM-OFETs and for which theoretical work is still to be done.

Materials and Methods Several variations of SAM devices were studied for the same 1,4,5,8-Naphthalenetetracarboxylic diphenylimide (NTCDI) SAM system to evaluate a transport mechanism in SAM-OFETs. These SAM devices are: (i) SAM-OFETs and (ii) hybrid self-assembled photovoltaic cells (SAM-PVC). SAM-PVC devices were fabricated on p-and n-Si substrate.

1. Preparation of Organic Films

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Step-A SAM on SiO2 and glass for SAM-OFET conductive channels and SAM-PVCs (Figure 1A). Thick, impermeable to light wafers of n-type (500 cm, Virginia Semiconductors) Si(100) having 20 Å of SiO2 coating and glass slides were cleaned and functionalized with amino groups (step a), Figure 1A as described in ref. [61]. In experiment, which was aimed to invert the conductivity type of NTCDI channel, a p-type Si(100) having 20 Å of SiO2 (100 cm, Virginia Semiconductors) was used in addition to the n-type silicon wafers. NTCDA (1,4,5,8naphthalene tetracarboxylic anhydride) evaporated at 110°C and reacted with the NH2functionalized surface for 45 min in a Bell Jarr chamber at 10-5 Torr (step b). The product of assembly lacks the amino group preventing formation of the second layer. The substrate was kept on a heated sample holder (180°C) preventing physadsorption of the precursor. Vacuum deposition modified over 60% of the surface amino groups [62]. A layer of 4-aminophenylthiol was added within 20 min (step c). This reaction tops the surface with SH-groups reactive toward metals [63]. The chip was gently rinsed with 2-propanol and heated at 80°C for 1 h. A silver contact was placed on top of the films by placing a drop of colloidal silver in acetonitrile and allowing the solvent to evaporate; the area or the contact between the silver and the coated silicone was ~3 mm2.

Step-B SAM on Au electrodes for SAM-OFET source and drain (Figure 1B). First, a 3 mM solution of 3-chloro-1-propanethiol (Cl-Pr-SH) in ethanol was spin-coated on the OFET substrate to form AuS bonds (i in Figure 1B). The substrates were held under Ar flow for 20 min after spin-coating and after that the substrates were rinsed with isopropyl alcohol and heated at 100°C for 1 h. Next, a 5 mM solution of 1,5-Diaminonaphtalene (DAN) in ethanol was spin coated on the surface followed by annealing at 100 °C for 1 h to form imine bonds with template layer (ii in Figure 1B). The excess DAN was removed by an ethanol rinse and

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annealed at 100°C for 1 h. A self-assembly route described above for oxidize surface and depicted at Figure 1A was completed SAM-OFET device fabrication. Different self-assembly of template layers on OFET conductive channel, following step a in Figure 1A, and electrodes, following step ii in Figure 1B insure formation of unified NTCDI layer across SAM-OFET surface and device integrity.

Figure 1. Fabrication of SAM devices. (A) The three-step assembly of monomolecular films of NTCDI on oxides Si and Si3N4 substrates (conductive channel at SAM-OFET and at SAM-PVC). (B) The twostep assembly of amino-coated template layer on Au electrodes used only for SAM-OFET source and drain. SAM of NTCDI was continued from stage b at panel A, following assembly of aminated template layer on Au.

In addition, few samples were prepared for SAM-PVC devices. Thick (~300 nm) films of NTCDA and C6-NTCDI were prepared by physadsorption of NTCDA on the cold functionalized surface of silicon. Sparse NTCDI films were obtained from thick films of

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NTCDA by heat desorption of the excess of NTCDA followed by step c. Step B was not employed in SAM-PVC devices.

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2. Characterization of the NTCDI Monolayer A step-by-step buildup of the layers was characterized by contact angle (CA) changes, IR spectroscopy, XPS, and UV-vis spectroscopy. The results agree well with previous characterization of NTCDI SAM monolayers and are briefly summarized below. After step a, CA changed from 17° to 45°, and the appearance of the 3200 cm-1 alkylamine IR peak was observed. The N (1s) XPS core level spectra showed a predominantly 398.8 eV (85%) peak on the amine surface because of the nonprotonated NH2-group [64]. The minor N1(s) component at 400.6 eV is attributed to protonated NH3+ (15%). The UV-vis spectrum of the film grown on the glass slide did not reveal any peaks. After step b, the CA changed from 45° to 92° and the 1655 cm-1 IR peak of the imide bond formation was detected. In accord with previous studies [65], formation of imides on the surface results in disappearance of the majority of the original peaks at 400.6 eV (29%) and 399.8 eV and formation of a new single broad N(1s) peak at 399.6 eV (67%) because of formation of imido groups with a possible presence of amido groups [66] When step b was performed on a glass slide, two peaks appeared in the UV spectrum: 360 and 390 nm; OD ~ 0.004 and 0.006, respectively (Figure 3). We also observed an appearance of a new broad peak in the greenish-orange region (OD = 0.001) that characterized a formation of an in-plane ordered organic heterostructure in SAM structure. After step c, the CA changed from 92° to 60° and a new peak (236 eV) associated with HS groups was observed in the XPS spectrum. No significant changes were found in the UV-vis and the IR spectra after step c. Variable angle spectroscopic ellipsometery (VASE, Woollam Co.) was used to verify the monolayer growth as it was reported for NTCDI SAM structure. The variable angle spectroscopic ellipsometer measured spectra with 5-nm intervals in the range 300-1700 nm. The structural model for fitting ex-situ ellipsometry data uses the data of three different incident angles: 65°, 70°, and 75°. Measured and fitted ellipsometric data for a structure containing Si/SiO2, siloxane matrix, NTCDI-benzene thiol exhibit molecular c-axis interplanar spacing are 20 Å for SiO2 native oxide, 3.5 Å for the siloxane matrix, 7.0 Å for NTCDI, and 6.9 Å for the benzene thiol layer.

3. SAM-OFET Electrode Configuration Substrates for OFETs and tunneling devices were fabricated using reactive ion etchgrown silicon nitride (100 nm) as the insulating layer on highly doped n-type silicon, which also functioned as the gate electrode. Gold source and drain interdigitated contacts were photolithographically defined on the silicon nitride to give total channel width W1=6000 μm (width is 400 μm for 15 meander elements) and channel lengths L1=20 μm. Figure 1A shows a top view of the interdigitated finger array. The substrates were cleaned by immersing for 1 h into a solution containing H2O/H2O2/NH3 (5:1:1) while sonicating. After that, they were washed with deionized water (17.8MΩcm) and acetone, subsequently, and heated in an oven for 30 min at 100 °C. Following fabrication step A and step B, which are describe above,

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indium contacts were attached to drain and source Au pads (Figure 2). Silver paste was used to connect to bottom gate electrode.

4. Determination of the Absorption and External Quantum Efficiencies in Closed-Circuit SAM-PVCs The spectra of dense and sparse monolayers of NTCDI grown on glass slides were recorded in a Shimadzu spectrophotometer. Comparison of the films grown on Si and glass is justified by the surface titration, which shows that the density of amino groups on glass and SiO2 coating of Si(100) is approximately the same [62] (2-3 per 100 Å2). The EQE was calculated as a ratio of the number of electrons passed through a 500 resistor to the number of photons in the light flow. The output of a Xenon lamp/monochromator assembly (8-nm bandwidth) standardized to the Oriel calibrated light source was used to illuminate the photoelement. The EQE was also measured at 532 nm by referring to nonsaturating output of CW YAG laser calibrated with a photocalorimeter.

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5. Time-Resolved Photovoltaic Measurements in Closed-Circuit SAM-PVCs These were performed with a picosecond laser system consisting of a Ti: Sapphire femtosecond laser, stretcher/compressor/amplifier of picosecond laser pulse (“Titan”, Quantronix), and optical parametric amplifier of superflourescense (“TOPAS”, Qunantronix/ Light Conversion). The output pulse parameters were: wavelength 532 nm, energy 10 J, pulse duration 1 ps, and repetition rate 1 kHz. To avoid saturation effects, the pulse energy was attenuated to a 0.1 J level with a beam diameter of 1 mm. The photovoltage transients were recorded by Tektronix TDS 3025 digital oscilloscope bypassed with a 7.6 k resistor (Figure 3).

Figure 2. FET electrode configuration. Optical image of electrode meander before deposition.

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Figure 3. Experimental setup for a time-resolved photovoltage measurement in closed-circuit SAMPVCs.

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6. Capacitance Coupling (Open-Circiut) Measurements in SAM-PVC To learn the charge separation we studied the formation and decay of the surface photovoltage by an auxiliary transparent electrode, which serves as a capacitive probe that picks the transient surface photovoltage. This method has been used in electronics and biophysics but is novel for the surface redox chemistry. The probe both provides the direction of the charge transfer and resolves its dynamics. Its application is independent of the optical properties of the semiconductor. Laser initiation of the reaction permits quantitative measurements of the reactions with the rate constants ~108-103 s-1. The back electron transfer is poly-disperse with the components ranging from ~10 microseconds and up. Measurement scheme was the same as at Figure 3, but instead of upper electrode a semi-transparent indium tin oxide electrode was used to achieve a capacitor coupling. More details for this measurement setup and principles of capacitive probe operation are shown at Discussion section 2.2, Figure 11 and Figure 12.

Results 1. NTCDI SAM OFET Devices In order to have high mobility and low charge trapping in organic semiconductors, we have focused on the class of organic materials with herring-bone crystallographic packing. Herring-bone packing could prevent interchain quenching and, at the same time, to have high electron and hole mobilities. The aromatic polycyclic structures, which are based on small molecules, and have a natural herring-bone pattern packing in the crystals [67] are promising

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candidates to that end. For example, one of them with the highest mobility in crystalline form ( = 3∙10-3 cm2 V-1 s-1), naphthalene tetracarboxylic dianhydride (NTCDA) has such herringbone pattern (Figure 4) [68]. Moreover, it was shown that NTCDA conductance in crystalline could be enhanced by two orders of magnitude by increasing charge concentration through doping [69]. Our research showed that SAM of NTCDI molecules results in ordered structures, which have even higher electron mobility.

(A)

(B)

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Figure 4. A) Natural herring-bone pattern packing of NTCDA molecules. B) Scheme of orthogonal NTCDI intermolecular plane configuration.

Figure 5. Resulting ID-UD curves at different gate biases.

Figure 5 shows the I-V curves of a SAM-OFET device. The device exhibits saturation regime at high enough drain voltages. VD of 0.5 V was required in order to obtain the drain current. The nature of this barrier will be clarified at discussion section following photocurrent studies in NTCDI hetero-structures. Detailed discussion of the ways to decrease

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metal-organic injection barrier in SAM-OFET is beyond the scope of our contribution. Our preliminary experiments indicate that doping of template layer by molecules carrying positive and negative charge could essentially affect this barrier. In particular, using of 0.01 M solution of tri-phenyl amines decreases this barrier to 0.1V without effecting essentially mobility in SAM-OFET device. 20

-ID ( A)

15

VD = 2V

10 5 0 0

1

2

3

4

UG(V) Figure 6. Transconductance measurements of SAM-OFET device showing n-type behavior at VD = 2V

The mobility of this n-type device was calculated at the saturation region using the nex equation:

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e

tOX

where COX

ID L 1 1 VG W VD COX

is capacity of SiO2 layer, tOX is thickness of SiO2 layer,

(1)

S

he static

0

S

permittivity of the material and

0

is dielectric constant, L and W are length and width of

conductive NTCDI channel, VD is drain bias, under which mobility is calculated,

I D and

VG are drain current and gate bias range in which mobility is calculated. Here COX

2 10 6 4 8.86 10

ratio I D VG

6 10 2

3

14

cm , L and W are 20 m and 6000 m, V is 2V (Figure6) and D F / cm

A . n-type mobility (μ ) of 31.5 cm2 V-1 s-1 is obtained by applying eq. 1 e V

to the data in figure 4 and a high on/off ratio of up to 109 is obtained. We note that such a high mobility was achieved only in 10% of fabricated devices. Totally 20 SAM-OFET devices were tested. 20% of tested devices were shortened and about 35% of devices were damaged during measurements. Roughly 35% of devices have a lower mobilities or exhibit properties of gated tunneling diodes. These tunneling diodes demonstrated typical FlowlerNordhaim tunneling features [70]. Here we will discuss transport properties only of the best devices. The device reproducibility aspect will not be discussed in this chapter. This is a rather high mobility, considering the thickness of the organic semiconducting layer. The

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transconductance of device is shown in Figure 6 and demonstrates the turn on voltage of the device in an n-type mode on the same device. In this device a VD = 2V drain voltage was required to overcome the blocking contacts, consequently the channel currents were measured at much lower drain voltages. Mobility in p-channel was 102 lower compared to n-channel. Thus, the possibility to have high mobility in SAM-OFET molecular systems is demonstrated. These studies, along with photophysics transport studies of SAM structures, demonstrate that molecules with natural herring-bone pattern packing might be favorable for OFET applications. This peculate will be discussed in detail in Discussion section.

2. NTCDI SAM PVC Devices

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To achieve hybrid SAM-PVC we self-assemble the NTCDA molecules, which are the same molecules used in conductive channel of SAM-OFET, on n-type silicon substrate. The self-assembly strategy followed step-A synthetic protocol (Figure 1A). Two kind of measurement set up were used to evaluate transport in NTCDI-based SAM-PVC. The first measurements scheme relies on closed and second on open electrical circuit measurements. In first set up the Ag upper electrode was used to collect photocurrects (section 2.1, Discussion). Semi-transparent ITO electrode was used to probe electric field created by photogenerated charges in second set-up (section 2.2, Discussion). Combination of these measurements should (i) provide a kinetic of charge transfer in 2D NTCDI molecular system and (ii) prove that this charge dynamic solely attributed to transport properties of 2D NTCDI molecular system, and not to capacitor discharge and interface charge transfer at SAM-PVC. The last section ((section 2.3) of SAM-PVC sub-chapter demonstrates possibility to change the type of conductivity in 2D NTCDI molecular system by changing of substrate type from traditional n-type Si to p-type Si substrate.

2.1. Measurements of In-Plane Currents in SAM-PVC: Closed-Circuit Electrical Measurements We topped the film with a small-area silver electrode and found that the resulting Ag/NTCDI monolayer/Si sandwich (Figure 7) was sensitive to light. In the dark, the current-voltage (I-V) curve goes through the origin (Figure 8) and straightens at the slopes that correspond to ~30 and ~10 k , respectively, as the positive or negative voltage increases. Illumination with continuous monochromatic light changes this characteristic dramatically. Now, the I-V curve also goes through the bottom right quadrant where the current through the element opposes the external bias. This plainly manifests that light generates electromotive force, which induces a negative charge on silicon (photocathode) and a positive charge on silver (photoanode). The maximal photovoltage at the saturating light power was as high as 280 mV. In the absence of the film or only with the amino layer present, the maximal photovoltage was below 0.1 mV and could not be characterized accurately. The mutual arrangement of the cell and the incident light, the external quantum efficiency (EQE) of 0.4-0.7 (Figure 9), which cannot be accounted for by the absorption efficiency of the film, the dissimilarity of the EQE, and the absorption efficiency of the film and resemblance of

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EQE to the spectrum of silicon [71] suggest that the light is productively absorbed by the semiconductor (Figure 7).

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Figure 7. Operation of the SAM-PVC. Incident light creates holes (h) in the valence band of silicon. Upon ejection of electrons (reaction 1) from the immobilized molecules into the holes, cation radicals are formed. Cation radicals rapidly exchange with the neutral molecules (2); the positive charge travels through the film, resulting in the in-plain current. Rereduction of cation radicals by silver (3) completes the photovoltaic element. Reactions 1-3 generate electromotive force. The diode (1) reflects rapid photoejection of electrons from the film and slow back-reaction. The external electric connections used in the voltammetric experiments (Figure 2) include a power source, an ammeter, and a voltmeter.

Figure 8. Electrooptical properties of the Si/NTCDI/Ag heterostructure. All experiments have been carried out at room temperature. Constant monochromatic 400-nm light illuminated an ~10-mm spot centered ~5 mm away from the silver electrode. When the Ag electrode is highly electronegative, the element behaves as a photoamplifier and light noticeably increases the asymmetry of the I-V curve. Marked “Fitted area” will be used in discussion section to model transport mechanism in SAM NTCDI channel.

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Figure 9. Effect of the surface density of NTCDI on the film spectra and light-harvesting efficiency. Dense film A assembled on glass has a prominent aggregation band in the greenish-orange region. The external quantum efficiency of the corresponding Si cell illuminated by constant nonsaturating light is represented by spectrum C. Sparse film B has no aggregation band in the spectrum. The external quantum efficiency of the corresponding cell was too low to measure. Insert show the measurement setup under continuous illumination.

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A

d = 3 mm

B

d

d = 7 mm d = 10 mm

Figure 10. Scanning time-resolved photovoltage probing. A) Measurement setup. Distance d is between illumination point and harvesting upper Ag electrode. The transients were induced by pulses centered at different distances apart from the silver electrode. B) Kinetic of photoresponse at different d. The inset shows the dependency of the T1/2 of the voltage rise on the distance l between the bright spot and the silver photoanode. The solid line approximates the points by a parabola T1/2=l2/D with the parameter D~106 cm2/s

This contrasts with light harvesting in the dye-sensitized photovoltaic cells [72, 73] or the hybrid nanorod-polymer cells [74] which require special junctions with enormous contact

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areas between the materials to enhance their absorption efficiency. The geometry of the cell also suggests that light harvesting incorporates a longitudinal spatial energy transfer. In contrast to natural photosynthesis [75] the conserved energy does not migrate in the form of excitons. The absence of long range light harvesting, when the uncoated side of the chip was illuminated, establishes importance of the film and rules out energy transfer within the silicon bulk. The process starts with a redox reaction separating charges between silicon and film (reaction 1, Figure 8). Because of the spatial separation of the cathode and the anode chemistry, this reaction can be identified straight from the polarity of the cell, which shows that electrons from the NTCDI molecules are ejected into silicon, that is, NTCDI molecules oxidize. Charge separation is supported by the asymmetry of multiplication of the dark current with respect to the bias direction (Figure 8), which cannot be accounted for by the influence of nonpolar excited states. Indeed, regardless of the amplification mechanism, they would not be able to discriminate the bias directions. On the contrary, charge separation explains this asymmetry easily. Since the positive charges in the NTCDI film “reflect” in silicon “mirror”, they are accompanied by their countercharged electrostatic images and migrate as trans-surface dipoles. As these dipoles approach the Ag/film/Si junction, they are either attracted into the contact area by parallel or repelled by the antiparallel external field. In agreement with photooxidation mechanism of energy conservation, light harvesting was observed in the thick films of N,N‟-dihexyl-naphthalene tetracarboxylic diimide (C6-NTCDI) physadsorbed on the silicon surface, but not in the thick films of NTCDA, because NTCDA is much harder to oxidize. This photooxidation incorporates several events. Absorption of light results in elevation of an electron to the conductance band leaving a vacancy in the valence band. This vacancy can be filled either by back recombination of the electron from the conductance band (unproductive decay) or by transfer of an electron from the film into silicon (reaction 1, Figure 8) since the affinity of this vacancy for an electron is apparently high enough to oxidize a molecule of NTCDI. As a result, similarly to the photochemistry of dyesensitized cells [76, 77], an extra electron remains in the conductance band of silicon, while the oxidized molecules of NTCDI become cation radicals (Figure 7, inset). These radicals combine essential structural motifs of the oxidized forms of two ubiquitous redox cofactors: NAD+ and semiquinone. These features ease oxidation of neutral molecules. The energy of light is conserved in two forms: the electrostatic energy of transsurface dipoles and the free energy of reduction of cation radicals by silver (reaction 3, Figure 8). The equilibrium of this reaction is shifted toward neutralization of cation radicals, and we could not attain electric currents in the films assembled on the surface of glass because measurements of such currents even over microgaps require very sensitive equipment [78]. To study migration of cation radicals to the photoanode (reaction 2), we used short light pulses focused at different distances from the anode to provide both time and spatial resolution. The photovoltage transients consist of voltage rise and decay (Figure 10). The decay is exponential with ~ 40 s at the given external load. This component represents the discharge of the cell because of both the backflow and the functional current through the load. The voltage rise originates from accumulation of charges at the anode and its kinetics depends on the distance between the illuminated spot and silver (Figure 10). To avoid a model bias, we described the kinetics of the voltage rise by its half-time T1/2 (Figure 10, inset), which reflects the travel time of the cation radicals. Simple estimate shows that corresponding speed of charge exchange exceeds 104 m/s. Despite some spatial

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uncertainty due to the finite beam size and a limited number of data points, one can see that the travel time appears to be proportional to the second power of the distance between the illuminated area and the silver electrode rather than to its first power. Since the area (distance quadrate) covered by a random-walking particle is proportional to its travel time, this dependency (Figure 7, reaction 2) is consistent with the random walk fashion of migration of cation radicals within the film during successive self-exchange reactions between the neutral molecules and the cation radicals. It is driven by the spatial gradients of their electrochemical potential, which are present because the anode reduces the cation radicals. Migration of cation radicals requires proximity of the molecules forming the monolayer and acting as redox centers. Such proximity manifests itself as a broad absorbance band in the green-orange region (Figure 9), attributed to formation of closed packed aggregate channel [62]. This migration, that is, time-resolved growth of the photovoltage, was not observed in the sparse monolayers of NTCDI, in the thick films of NTCDA, or without organic film. Only minor signals were observed in these control devices especially with the laser beam incident at the edge of the Ag contact. When the center of the beam was moved away from the edge, these signals disappeared rather than develop slower. The slight decrease of the total charge collected by the anode is probably due to partial recombination of trans-surface dipoles. The charges formed in the remote-illuminated area have to travel within the film over “gigantic” distances at finite speed. The vast majority of studies of lateral charge transfer in monomolecular structures were limited to the self-assembling and Langmuir-Blodgett monolayers [79, 80, 81]. Some of these studies have revealed that electrochemical activity of the assembling molecules is a requirement for charge transfer within these monolayers [82, 83]. In analogy to that we believe that the efficient delivery of positive charges to the photoanode becomes possible because of the macroscopic connectivity of the network of NTCDI molecules acting as redox centers. This connectivity permits rapid transfer of large net charge through an extremely narrow cross section and originates from assembling NTCDI molecules at the positions predetermined by arrangement of the NH2-groups within the dense interlinked siloxane network [27].

2.2. Measurement of Out-of-Plane Charge Recombination in SAM-PVC: Open-Circuit Electrical Measurements When redox active molecules are covalently tethered to the surface of the n-doped silicon, light induces their oxidation by the semiconductor (section 2.1). One particular problem to be solved by the new method originates from our abovementioned results. In principle, the energy can be conserved for further spatial transfer in two ways. First, it can be converted into the energy of the excited state as in the photosynthetic light harvesting antennas. Second, similarly to the dye-sensitized cells, the energy can be conserved as the free energy of charge back recombination (Figures 11 and 12). In the second case, the actual case as we show, the role of the organic molecules in these SAM-PVCs is different from the role of the dye in the dye sensitized cells. The NTCDI molecules do not absorb light but act exclusively as redox couplers and form cation-radicals upon ejection of an electron into the light-induced hole in the valence band of silicon. However, distinguishing between the two possible mechanisms is challenging because adoption of the traditional methods used for the dyes-sensitized cells would make the endeavor prohibitively difficult. Switching to the IR

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region where silicon is transparent is demanding in terms of the equipment and still cannot help the fact that, the monolayers contain too few molecules for time-resolved spectroscopy and only a small fraction of these molecules actually forms radicals. These molecules react too fast to make single molecule techniques applicable even in principle. Finally, an attempt to produce model nanoparticles would require an adequate adaptation of the existing modification technology and there is now guarantee for the adequacy of the model system studied in the solution compared to the actual element functions in the solid state. Experimental results of the previous section show that the diffusion time of about 1 s or less at the given distances is apparently much shorter than the life time of the cation radicals in SAM-PVC. The similar back recombination in the dye-sensitized TiO2 takes milliseconds [84]. Direct measurements in the closed-circuit configuration do not allow answering the question about back charge recombination, because in this configuration a long-time part of the charge transfer kinetics depends on the input impedance of the measurement system and internal impedance of the monomolecular devise, which in turn depends on many other external parameters. Therefore, the purpose of our study became two-fold. The first goal was to find a universal and highly sensitive method to study interfacial charge transfer. The second goal was to prove that illumination of the NTCDI-modified surface of silicon results in formation of cation-radicals at the junction and thus to test the potential of the method. In order to match these goals we have opted to the surface photovoltage technique, which is sensitive to the net charge displacement [85, 86], the very basic property of redox reactions. Since the spatially organized charge transfer inevitably causes electric displacement perpendicular to the junction (Figure 11), we have expected this elegant non-spectroscopic approach to be extremely useful in the studies of the surface redox chemistry. Here we show that the charge displacement perpendicular to the surface is sufficiently strong for direct measurements. To pick this electrostatic signature we utilize an auxiliary transparent electrode that act as a capacitive probe (Figure 12). This probe proposed by Bergmann in 1932 selectively picks the ac voltage [87]. Previously the transparent auxiliary electrode has been used to study the photovoltage generated by the p-n junctions. This nondestructive no contact method proved to be convenient for the quality control of these junctions before and without equipping them with electrodes [87, 88] or for measurement of the lifetimes of the minority charge carriers [89, 90]. In those measurements the ac photovoltage has been generated by a chopped light beam and its amplitude has been measured. While repetitive laser represents a limit case of the chopped beam, application of short pulses permits time-resolved measurements. Interestingly, a related approach has been utilized in the studies of the photosynthetic reaction centers in the essentially 3D membrane suspension samples [91, 92]. The method measures the transient voltage drop between the plates of the capacitor formed by the semiconductor chip and a probe ITO electrode transparent to the visible light. The photovoltaic response of this device contains two components. The first one is characteristic of capacitor between doped Si and ITO electrodes and the second one is a response from organic heterojunction itself. The relaxation time constant k for organic heterojunction is independent on device area and identical to the constant of charge annihilation on the single molecule or cluster. That is in opposite to a capacitor response, which is area dependent.

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Figure 11. Light indiced charge transfer at the silicon-NTCI(A) interface. The relative arrangement of the energy levels of silicon and NTCI(A) is necessitated by the observed polarity of the photo-induced charge transfer and by the fact that the light is absorbed by silicon. The structure of the putative cation radical is presented on the left. Vector D stands for the induced electric displacement.

Figure 12. The putative surface electrochemistry and its electrostatic signature. The silicon substrate forms the capacitor plane 1. The neutral NTCDI(A) molecules and the putative cation-radical formed upon illumination form the imaginative plane 2. This organic layer is separated from the tin-doped indium oxide electrode (plane 3) by a 25 micrometer insulating film. The capacitor planes 1 and 3 are connected respectively to the ground and signal input of the oscilloscope (V), which has a standard input resistance of 1 MOhm.

2.2.1. Theoretical Background for the Capacitive Probe Measurements To proceed further, we need to develop a theory of the capacitive probe response to distinguish between the surface charge transfer dynamics and the instrumental response. Let us denote the silicon surface, the organic film and the probe electrode as planes 1, 2 and 3 respectively (see Figure 12). Charge separation between the organic film and the silicon substrate will cause formation of an electric field between the planes 1 and 2. Under uniform

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illumination the concentration of the surface dipoles is the same throughout the film plane. The film becomes a surface of equal potential and can be considered as a plate of a complex 3-plate capacitor, which in turn is formed by two serial capacitors: C21 (organic film/Si) and C23 (organic film/ITO electrode). The voltage jump U21 will cause an electric current through the outer circuit formed by the probe capacitor C23, cable and internal resistance R of the oscilloscope. The capacities of the organic film and probe electrode were estimated to be C21 ~ 1 F and C23 ~ 100 pF, respectively. Direct measurement of the cable capacity demonstrated C = 360 pF. Input capacity of the oscilloscope CS = 13 pF is much smaller than the cable capacity, and in the following up calculations CS was included into C. Observed signal (voltage US(t) on the oscilloscope) reflects the dynamics related to the charge exchange between capacitors and to the finite lifetime of the dipoles created in organic film and semiconductor surface. Charge annihilation in the film will cause a backward current I21(t) = kQ21(t) , where Q21(t) is the charge accumulated in the capacitor C21, and k is the rate of the transient charge annigilation. As it was mentioned above, the constant k is defined by electrochemical properties of the sample, and does not depend on the area of illumination. To analyze the system behavior we consider three time scales: 1) charging the organic film buy the picosecond laser pulse; 2) fast (nanosecond scale) charge redistribution between capacitors C21, C23, and C; 3) discharge of the system (microsecond to millisecond time scale). We assume the C21 charging event is much faster than any discharge or charge exchange process, because it is defined by the laser pulse duration ~1 ps. With this 0

assumption, the initial voltage U21 is defined by the charge Q21 generated during the light

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action: U 21 (0)

0 Q21 / C21 . The initial charges of the probe capacitor C23 and the cable are

supposed to be negligible: Q23(0) = 0, QC(0) = 0. Fast voltage jump U21 causes charge exchange in the system. Now the cable and probe electrode are being charged by the capacitor C21. At this step we take into account finite impedance of the cable Z = 50 Ω. The initial voltage jump is compensated by the voltage drop UC = IZ , where I is the electric current through the cable. At this time scale we can neglect the current through the oscilloscope, because its input impedance is R = 1 MΩ, so that RC ~ 400 s, and RC >> ZC ~ 20 ns. At this relatively fast time scale of the charge exchange we also can neglect the C21 discharge and consider U21 to be constant. (We expect the typical relaxation time of transient dipoles in the organic film to be in the range of 10-100 s, while time constants ZC23 and ZC are in the nanosecond range). Finally, we can describe the second step of the charge exchange by the next system of Kirchhoff‟s equations according to the equivalent electric circuit shown on Figure 13:

U 21 U 23 U C U S I Q23 QC Taking into account Ohm‟s low UC = IZ, noting that U 21

US

(2) 0 Q21 / C21 , U 23

Q23 / C23 ,

QC / C , and denoting for the simplicity: k23 = 1/ZC23 and kC = 1/ZC, we can rewrite the

system (1) in the next form:

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k23Q23 QC Q23

kC QC 0

QC

0 Q21 C21Z

(3)

Using a Laplace transform with initial conditions Q23(0) = 0, QC(0) = 0, we can convert the system of differential equations (3) to the system of linear equations, the latter being presented in the matrix form as follows:

k23 s

s kC s

Q23

0 Q21 / sC21Z

QC

0

(4)

Here, Q23 ( s) and QC ( s ) are Laplace transforms of Q23 (t ) and QC (t ) , respectively. The solution for the system (4) is

1 Q23

s kC

QC

1 s kC

k23

s kC s ( s kC k23 )

k23

k23 s ( s kC k23 )

0 Q21 sC21Z

(5)

0

It can be written in the simplified form:

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Q23

QC

0 Q21 1 C21Z s( s kC

(6)

k23 )

Applying an inverse Laplace transform to (5) we get:

Q23 (t ) QC (t )

0 Q21 (1 e C21Z (kC k23 )

( kC k23 ) t

)

(7)

Figure 13. Equivalent circuit for the capacitive probe shown on Figure 12 in the nanosecond time scale.

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Equation (7) leads to the quite obvious conclusion. Because the charge exchange on the nanosecond time scale is controlled by the same current I

Q23 (kC

capacitor are charged equally, and at the time scale t

QC , the cable and probe

k23 ) 10 ns the charge and

voltage distribution are defined just by the capacities of the appropriate elements: 0 Q23

U

0 Q21 CC23 C21 (C C23 ) 0 Q21 C , U C0 C21 (C C23 )

QC0

0 23

(8)

0 Q21 C23 C21 (C C23 )

These values may be taken as initial conditions to evaluate the system dynamics on the microsecond time scale. The equivalent electric circuit for the measurement system at this time scale is shown on Figure 14. Let us consider some additional approximations to simplify the final analysis. After the fast restoration of the balance in the system, we can neglect the voltage drop in the cable. The discharge of the cable is defined now by the input impedance R of the oscilloscope, while the cable impedance Z = 50 Ω is negligible compared to R = 1 MΩ. Another assumption is based on the fact that the faster time constant observed in experiment is in the range of 10 – 20 s, which is shorter by, at least, the order of magnitude than any time constant related to the electrical discharge through R. This fact allows us to conclude, that the discharge of the organic film is mainly caused by the electrochemical annihilation, and like in the previous case we can neglect the current to the outer circuit, so that I 21

Q21

kQ21 . With these

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assumptions we can write the system of Kirchhoff‟s equations for equivalent electric circuit presented on Figure 14 as:

U 21 U 23 U S I 21 I3

Q21 IC

IR

(9)

kQ21 0

Figure 14. Equivalent circuit for the capacitive probe in microsecond time scale.

In the last equation, which represents the currents in the node d, we have taken into account that in the shown configuration the current IC is discharging C, while I3 is charging the probe capacitor C23, so IC

QC , I3

Q23 , and IC

U S / R QC / RC . The voltages

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U23 = Q23 /C23, US = QC /C, U21 = Q21 /C21. Let as also keep notation K21 = 1/RC21, KC = 1/RC and K23 = 1/RC23. (These constants are much lager than k23 = 1/ZC23 and kC = 1/ZC). Now the system (9) may be rewritten in the form:

0

K 21Q21 K 23Q23 KC QC 0

Q21 kQ21 Q23 QC

(10)

KC QC

0

Applying a Laplace transform with the initial conditions (8), we convert (10) to the system of linear equations presented in the matrix form as:

`

K 21

K 23

KC

Q21

0

s k

0

0

Q23

0 Q21

0

s

s KC

QC

0

(11)

The solution of the system (11) is:

Q21 Q23 QC

0 K 23 K C s ( K 23 K C ) 1 ( s k )( s K C ) K 21 ( s K C ) D( s) s(s k ) K 21s

0 KC (s k ) K 23 ( s k )

0 Q210 , (12) 0

where D(s) is a discriminant of the matrix presented in (11):

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D( s)

s 2 ( K 23 ( K 23

K C ) s (kK 23 kK C

K C )( s k )( s

k23 K C ) kK 23 K C (13)

K 23 KC ) K 23 KC

The solution for QC (s) can be found in form:

QC Here we denote K

0 s K 21Q21 D( s )

0 K 21Q21 s ( K 23 KC ) ( s k )( s K )

K23 KC /( K23

(14)

KC ) 1/(C23 C ) R . Applying the inverse

Laplace transform to (13), we can find QC(t) and the final signal US (t) = QC(t)/C:

U S (t )

( K 23

0 K 21Q21 (ke k t KC )(k K )C

K e

K

t

)

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(15)

Photocurrent Study of the Transport Mechanism… The last expression can be modified to the simplified form if we denote 1/ K 0 K 21Q21 ( K 23 KC )C

157 and

C23 0 Q21 U0 : (C23 C )C21 U S (t )

U0 (k e k t k 1

e t/ )

(16)

The expression (16) shows that a monodisperse charge recombination gives rise to a biexponential signal. Let us consider rapid recombination of the dipoles: k 1 . Now, it is more convenient to discuss the response (16) if it is presented in the form:

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U S (t ) U 0 e k t

U0 k

1

(e k t

e t/ )

(17)

The first additive term in the equation (17) reflects the kinetics of the surface charge displacements. The second term is the deviation of the measured signal from the charge transfer dynamics. It is important that the transient crosses the US = 0 level before recovering the base line. Such behavior is an intrinsic property of the RC chain and reflects the simple fact that the probe charges and then discharges during the measurements, while the oscilloscope measures the current through its input resistance. So, the second term in equation (17) reflects the input resistance of the oscilloscope and the capacity of the sample/probe capacitor and therefore depends on the size of the sample, length of the cable and other instrumental parameters. However, in the discussed case, k 1 , the amplitude of the instrumental part of the response is considerably smaller than the amplitude of the signal related to charge dynamics in the sample. Therefore, the best condition for the measurements 1/(C23 C ) R , which may be controlled by increasing the input impedance, is k 1/ capacity of the probe C23 (i.e., by increasing the size of the surface, or decreasing the thickness of the spacer) or the cable capacity C (i.e., by increasing the length of the cable). In our experiments the instrumental time constant was in the range 0.2–2 ms. 1 , the rapidly formed voltage U0 will dissipate as a result of charge When k redistribution between the plates 1 and 3 through the oscilloscope. Now, the faster component R(C21 C ) k . At longer time scale the will reflect the instrumental time constant system dynamics will reflect the dynamics of charge annihilation. Because the amplitude of the first term in Equation (16) becomes small compared to instrumental term, this configuration is not very useful for measurements. However, even in this configuration it is possible to estimate charge annihilation rate k. 2.2.2. Experimental Results The functionalized with amino groups silicon chips were prepared and left as is or topped with the either sparse or dense layers of NTCI(A) as described in step-A materials and method section. The electrometric measurements have been performed using set up shown schematically on Figure 3, but with uncoupled upper electrode. This electrode was a 200 nm

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50 Induced voltage, mV

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thick ITO (indium tin-oxide) film deposited on a glass plate and separated from the NTCDI(A) film on the silicon surface by a 0.025 mm insulating spacer (Figure 12). The input impedance of the Tectronics TDS 3032 oscilloscope was set to 1 MΩ. In the experiments with an external 1:10 voltage splitter the apparent input resistance was 10 MΩ. In these experiments a 1/10 of the actual voltage has been applied to the input channel of the oscilloscope. The excitation light pulses at 800 nm were generated by the Ti/sapphire laser system. The pulses had 1 ps duration, 10 μJ energy (attenuated with the set of neutral density filters), and 1 kHz or 200 Hz repetition rate. Note that NTCI(A) does not absorb at 800 nm in contrast to 532 nm utilized in the experiments of previous section. The signals were recorded by averaging of 512 oscilloscope traces. The maximum sampling rate was 2.5 GS/second. A typical photovoltaic signal from the sparse film of the NTCI(A) molecules is shown in the Figure 15. It is complex and includes ~20 ns grouing up part, heterogeneous decay below V = 0 and recovery to the base line. The polarity of the signal corresponds to charging the NTCI(A) film positively. This observation perfectly agrees with our previous assumption (section 2.1) that silicon photo-oxidizes the neutral NTCI(A) molecules, which form cationradicals upon ejection of electrons into the semiconductor. Minor electric responses are observed even in the absence of the NTCI(A) coating. Bare silicon surfaces are often contaminated with electrochemically active impurities. Such “as is” surfaces give highly irreproducible electric responses, which strongly decrease after rinsing the surface with acetonytrile (the data are not shown). This artifact is almost completely gone after NH2-functionalization of the silicon surface (Figure 15). The remaining signal could originate from certain polarization of the silicon waffle due to introduction of the gradient of the minority charge carriers near the surface [72]. Another possibility is the light-induced change of the built-in potential [93]. Regardless of its nature, this artifact is minor. No signal is observed when the surface of silicon is protected by a thick (>200 nm) insulating layer of SiO2. The insulating oxide layer prevents both the interfacial redox chemistry and band bending.

NTCDI(A)

40

O

O

O

O

N

O

30 20 200nm SiO2

10 0

-Si-(CH2)3-NH2

-10 0

20

40

1000

2000

Time, microseconds

Figure 15. Light-induced voltage transients. Samples were used as capacitor faces separated from the ITO probe electrode by a 25 μm insulating spacer. The voltage transients represent polarization and depolarization of the Si/film junctions: Si/NTCDI(A) (solid line), Si/SiO2 (dotted line), Si surface derivatized with amino groups (dashed line).

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While the initial charge separation is too fast to be resolved electrometrically, the heterogeneous recombination is time-resolved clearly. The fast and intermediate phases are independent of the input resistance and have a time constant of ~ 10 μsec and ~100 μsec, respectively. The contribution of the slower component into the signal varies among the samples and is about 1/3 of the total amplitude of the signal presented in Figure 13 (spare film of NTCDI(A)). The third, slowest negative component, demonstrates a clear proportionality to the input resistance and stretches 10-fold upon addition of a 9 MΩ additional resistor. Therefore, the baseline recovery likely represents the instrumental function as outlined in equations 16 and 17. This recovery component is difficult to measure because of its small amplitude. It is important to note that in the dense NTCI(A) films the contribution of the slow (~100 μsec) component of the photovoltage decay does not exceed 10% [94] (Figure 14). This observation well agrees with our previous conclusion that in the dense films the molecules rapidly exchange the oxidizing equivalent with their neighbors (section 2.1). Therefore, the positive charges formed in the “slow” molecules drain back to the silicon substrate via the two-step process: (i) exchange with the “rapid” molecules within connected redox network and (ii) the following discharge of these “rapid” molecules. Overall this process is apparently faster than the direct discharge of the “slow” molecules. It is also important to note that these “slow” NTCI(A) molecules can be pre-charged in the discontinuous networks with the intense constant light [95]. A strong continuous illumination decreases the total amplitude of the signal, mainly at the expense of the slow component while the fast component remains essentially unchanged. The heterogeneity of the recombination most probably reflects the micro-heterogeneity introduced by the silicon cleaning procedures and/or the microscopically uneven thickness of the siloxane layer. Similar kinetic heterogeneity has been observed in PTCDI-doped SAMs grown on ITO by single molecule fluorescence analysis [96] and has been attributed to the differences in the local environment. In contrast to the photosynthetic light-harvesting antennas, the energy of light is conserved by the SAM-PVC via interfacial electron transfer. The silicon/film junction forms an electrochemical capacitor charged by the light. This capacitor stores both the electrostatic energy and the free energy of the back recombination [96]. It is important to be able to measure and control the discharge characteristics of this capacitor in order to utilize it as an energy source. The electric field formed within this capacitor is picked by the auxiliary electrode and can be measured directly. The time resolution limits of capacitive probing are defined instrumentally. A typical 2Gsample/second oscilloscope is convenient to measure charge transfers that are not much faster than ~ 108 s-1. The reactions, which are slower than the instrumental time-constant, are also inconvenient to measure. Therefore, the ideal dynamic range for the capacitive measurements is between 103-108 s-1. Capacitive probing has lower time-resolution than the pump-probe spectroscopy but is free of certain limitations of the latter. First, it is independent on the optical properties of the semiconductor. Second, it works with flat surfaces which contain much less molecules than suspensions of nano-particles [97]. Indeed, separation of only 5.5x109 charges per cm2 across ~ 20 Å will induce a strong enough signal with amplitude of ~ 1 mV (for ε ≈ 2). This number of molecules dissolved in 1 cm3 corresponds to a less than 10 pM concentration which is far below the sensitivity limit of any spectroscopic method except for fluorescent techniques

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[96]. However, the electrometric approach works with non fluorescent molecules and provides a nanosecond time-resolution. The technique also provides the direction of the charge transfer. Measurement of interface charge transfer response from ultra-fast spectroscopy is complicated in general case. The interpretation of transfer kinetics is even more challenging. For example, in well-studies Gratzell photovoltaic cells [98], the origin of two charge transfer kinetics in femtosecond range [99] is not clear. Luckily, for SAM-OFET such delicate measurements of interface charge transfer are not required, since the kinetics of charge transport in conductive NTCDI channel is in microsecond range [119]. Nevertheless, in this scale several components of kinetics (~10 μsec and ~100 μsec) have been observed. We tested few control sample devices to discriminate direct charge injection kinetics. We note that thick NTCDIA films measured with capacitor probe showed mostly long type kinetics. This likely can be attributed to trapping of the charge the molecules, which do not have a contact with Si substrate. Exact physics of long-component response in thick NTCDI film still have to be proved unambiguously. However, it is clear that short-component of decay function (~10 sec) characterize interfacial back charge transport from NTCDI to the substrate. This component was the main one in pure SAM-devices. Therefore, we can apply developed model to determine kinetics of charge recombination and compare with kinetic of charge transport.

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2.3. Reversing Conductivity Channel in SAM-PVC (Closed-Circuit Measurements) The preparation of efficient n-type conductive channel in addition to p-type conductive channel is another challenge in OFET fabrication. Moreover, it is highly desirable that n- and p-channel OFET devices were fabricated by technologically comparable methods. Mainstream research direction in OFET is attempting to develop suitable organic materials for n-type conductive OFETs. Contrary to this mainstream studies, we are trying to develop switching of type of conductive channel based entirely on type of substrate, which is p- or ntype Silicon. The idea behind this study is the fact that „static‟ charge transfer self-assembled monolayer should be greatly affected by type of substrate due to proximity effect. Thus different type of „Shotkey barrier‟-like structure is conceivable. In turn, it could change the type of conductive channel in SAM material. To verify this assumption we perfomed a photophysical study in SAM-PVC devices. The silicon chips functionalized with amino groups were prepared and topped with the dense layers of NTCI(A) on n-type and p-type silicon substrates in the same manner as described in step-A section of materials and methods. Here, p-type silicon substrates (Si(100), 100 cm, Virginia Semiconductors) were tested in addition to silicon wafers of n-type (Si(100), 500 cm, Virginia Semiconductors). Both wafers had 20 Å of SiO2 coating. SAM-PVC devices were tested by time-resolved photovoltaic measurements in closed-circuit SAM-PVCs as described in section 2.1. Devices were illuminated at the distance 10 mm from collective electrodes. Result of these measurements is shown in Figure 15. The change of the response poliarity clearly indicates the conductivity type in NTCDI conductive channels switches and depends on conductivity type of the substrate. Difference in response kinetics is related to difference in heterostructures, doping level of Si and possible SAM density. Direct SAM-OFET testing

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of this idea is more complicated. Such test will require development of metal electrodes and surface chemistry to establish chemical bond between metal and organics, which would have requested charge injection ability. Nevertheless, the described experiment could be taken as proof of concept of such approach. Complementary circuit design should be very straightforward once this approach will be developed for SAM-OFET. The simplicity of this approach will be due to the possibility to use the same material (as NTCDI SAM matrix) for different types of conductivity.

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Discussion: Polaron Transport Mechanics This chapter is focused on study of a correlation between dynamic properties and charge mobility in NTCDI SAM systems. This research is connected to similar studies in bulk crystals of N. Karl group [100]. These publications reported a direct time of flight measurements of charge carriers in bulk crystals of organic aromatic compound. It is remarkable that ten years ago, in 1998 N. Karl bring question “Fast electronic transport in organic molecular solids?” in the title of their paper [101]. It has been demonstrated unambiguously that charge transport in the bulk crystals is intrinsic. Therefore, these researches bring a hope for generation of scientists, who developed the field of molecular electronics. In this spirit, in the beginning of this section, the question “Fast electronic transport in SAMs?” will be discussed. We have to note a difference between our experiments and the research of N. Karl group [101]. The time of flight experiments were carried out in thin organic crystals (0.2 to 1 mm thick) by applying electric field across the sample and registering the transient electric current created by illumination of the surface of the crystal by a short laser pulse. Karl et al were focused on charge mobility measurements, but simple estimate shows that the speed of charges in the sample achieved 103 m/s (at 300 K) to 5∙104 m/s (at 40 K) with applied electric field 20-40 kV/cm. In our experiments we did not apply longitude (in-plane) or interface (out-of-plane) biases to avoid bias polarization phenomena in SAM. In addition, our photovoltaic study was aimed to distinguish between charge transport and charge recombination processes. Nevertheless, we have sown that in SAM the speed of charge diffusion can be as high as 104 m/s at room temperature (Figure 10). One could expect that this speed may be even higher if the bias is applied along the surface. Even more surprising is the capability to initiate an electron or hole conductivity in SAM-NTCDI (Figure 16) by changing type of doping of the Si substrate as it was observed in perylene crystal by applying different polarity of the electric field. Due to technological reason, we could not self-assemble perelene molecules on Si (sublimation temperature of perylene tetracarboxyl dianhydride is 200 C higher than of NTCDI). However, naphalene and perylene belong to the same class of disc-shaped aromatic compounds with extended -electron system. Therefore, a certain analogy of transport properties is anticipated, if similar packing is achieved. It is possible to conclude that, despite of the difference between photocurrent measurements in organic crystal and SAM film, the transport phenomena are very similar and velocities of charge waves in crystal and SAM film are roughly at the same range. Moreover the speed of charges in SAM film exceeds the same rate in organic crystal. This observation is an opposite of what could be expected from transport in 3D vs. 2D systems [101]. This discrepancy could be explained by role of Si substrate in NTCDI SAM/ Si heterostruture, as discussed below. Due to barrier structure of this interface, out-of-plane charge transfer from

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NTCDI to Si could sustain in-plane current. The contribution of this mechanism is absent in crystal.

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Figure 16. Charge recombination in connective and sparse films.

Figure 17. (A) Photoresponse of NTCDI SAM-PVC placed on n-doped and p-doped silicon wafer; (B) possible scheme of molecular level, which explain switching conductivity channel in SAM-NTCDI.

Experimental results clearly indicate on the possibility of polaron transport in NTCDISAM 2D channel, which is located along the substrate plane. Such possibility was theoretically predicted by Horovitz [102]. He suggested that polarons dominate the photocurrent I due to a novel electric field–assisted tunneling route for which ln I ~ E−2/3. The negative area of our experimental SAM-PVC I-V (Figure 8) taken under illumination and dark conditions is presented at Figure 15 for in-plane currents. The Horovitz model (ln I ~ −E−2/3) is used to fit our experimental data. We take these results as an evidence of possible polaron transport mechanism in SAMNTCDI conductive channel. Moreover, an estimate of front velocity of electronic wave in SAM-NTCDI system, (Figure 10) result in the diffusion parameter D~104 cm2/s, which is close to speed of the polaron transport. In addition, the photovoltaic studies make possible to discuss a possible type of polaron system in SAM matrix. This polaron transport is likely realized in SAM-PVC device through random walk mechanism. Excited redox state likely employs percolation-defined random walk [103], which is exemplified in Figure 18.

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Figure 18. Photoresponse of NTCDI SAM-PVC, as “marked area” at Figure 7 and fit ln I ~ E−2/3 over experimental data for (A) photocurrent and (B) dark current.

The set of redox reactions occurring in NTCDI channel (Figure 18) includes three steps: 1. Si* + NTCDI (Si-) + (NTCDI+): photo-oxidation (18) 2. (NTCDI+) + NTCDI NTCDI + (NTCDI+): dark self-exchange 3. (NTCDI+) + Ag NTCDI + Ag+: dark harvesting.

Results of photocurrent measurement in SAM-PVC could rationalize a high mobility measurement in SAM-OFETs. Indeed, assuming initiation of the same redox transport in SAM-OFET, a polaron level mobility might be expected. Assuming polaron nature of transport in SAM-OFET, we could expect to increase the charge mobility, which currently is achieved to be as high as 31.5 cm2 V-1 s-1. It is a few orders of magnitude higher than in the bulk organic crystals. This mobility level could bring performance of SAM-OFET on the level of amorphous silicon FETs. This conclusion could be confirmed by photovoltaic response in control devices (defined at materials and method section), which were illuminated

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by light at the vicinity of contact upon different scope loads: 50 Ohm and 7.6 kOhm (Figure 19). A different scope load in this case define the time-scale and sensitivity of measurement.

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Figure 19. Variety of possible cation-radicals in NTCDI molecules on molecular level. (A) Formation of anion-radical state. (B) Formation of cation-radical state. This redox state could be exchanged extremely fast with neighboring neutral NTCDI molecule and transport a redox state to harvesting electrode thought self-exchange reactions. This transfer of redox state is illustration of polaron transport in SAM-NTCDI on nano-scale level.

Figure 20. Possible cation-radicals in NTCDI molecules at macroscopic scale. (A) 3D cartoon of random walk transport of redox state from illuminated area to harvesting Ag electrode. (B) Top view of redox transport at macroscopic scale. Note that this system could potentially handle a large current density due to possible multi-channel transport in organic 2D system. Only one conductive channel is shown here for simplicity.

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In addition to SAM film, a thick (~300 nm) films of NTCDA and C6-NTCDI were prepared by physisorption of NTCDA on the cold functionalized surface of silicon. This sample has been marked above as physisorbed film. Sparse NTCDI films were obtained from thick films of NTCDA by heat desorption of the excess of NTCDA followed by step c. This sample has been marked above as incomplete SAM. Response of Si-Ag point contact was included for comparison. Only the sample with complete SAM demonstrates a long-range charge delivery (~ 1 cm illumination from the electrode). The maximum voltage in SAM films exceeds that in incomplete SAM and physisorbed films by 2–3 orders of magnitude. These results clearly demonstrate that any incompleteness and disorder in SAM conductive channel affects redox self-exchange reactions on microscopic level. The time-scale of the SAM device response also demonstrates an ability of complete SAM structure to deliver charges from remote areas, in contrast to other tested devices. It is also notable that only SAM-device and incomplete-SAM-devices give rise to photo-signal at finest time scaleinput resistance of the oscilloscope. The fact that Si-Ag device has a resolution wit 50 higher efficiency than the device based on NTCDI-C6 physisorbed film is due to shielding of incoming light by a thick film. Transport properties of NTDCI SAM OFETs and PVCs clearly point on polaron nature of the transport. The way to prove this hypothesis is to find the way to block cation-radical exchange in the transport channel and observe change in transport properties in these devices. Experiment which addresses this possibility is shown at Figure 20. It proves the importance of redox capability to polaron transport on the molecular level. PV response of ~150 mV was observable in NTCDI-C6 film under illumination 3 mm away from Ag harvesting electrode, while no PV effect was observed in the case of NTCDA film. NTCDA molecules could not be oxidized at normal conditions. Thus, any un-reacted physisorbed molecular cluster will serve as an isolator for polaron transport („redox blockade‟). It is also known that OFET based on NTCDA molecules shows mobility of 10-4 cm2 V-1 s-1 [104], while NTCDI-based OFET routinely demonstrates mobilities, which are higher by factor 103 [105]. Thus, the experiment on Figure 20 demonstrates evidence of polaron transport on the molecular level, in addition to importance of the macroscopic order. It is also notable that OFET based on NTCDA molecules [105] shows mobility of 10-4 cm2 V-1s-1, while NTCDI-based OFET [105] routinely demonstrates mobilities, which are by 103 factor higher. Therefore “redox blockade” experimental (Figure 20) and literature data prove the polaron transport in the NTCDI-SAM channel. Thus, any unreacted physisorbed molecular cluster will serve as an isolator and trap for polaron transport. Polaron conductive mechanism could explain why at the current stage it is so difficult to achieve high mobility in SAM-OFET. Both macroscopic and microscopic orderings are important for polaron transport and this impose strict conditions on quality of SAM conductive channel. Taking into account short range of dipole-dipole interaction, any defect could trap redox state. In addition to that, trapped redox state will form stable charged trap, which will prevent transport not only through the defect point, but also in the vicinity of this defect. Trapped redox state should affect the efficiency of charge transfer more than existence of non-reactive molecular cluster, which not carrying any charge. As the number of such trap exceeds percolation limit, no polaron transport will exist in such a system. Thus, in some analogy to biological systems, a redox active path in SAM matrix and good packing are the pre-condition for an efficient charge transport in the SAM systems.

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Based on this assumption, we suggest that switching of conductivity type is possible in conductive channel. It is due to the fact that (i) various forms of cation-radical are possible and (ii) proximity of interface with inorganic semiconductor. Formed anion- and cation radicals as shown at Figure 17 correspond to p- and n-type conductive channel in OFET. Our SAM-PVC measurements indicate that changing doping of substrate could indeed change the current sign and, thus, the type of conductive channel in SAM-PVC. Although this assumption should be confirmed in SAM-OFET system, our observation could result in alternative way to achieve efficient OFETs with tunable type of conductive channel. Moreover, one could conceive the cascade FET devices, which would have variable mobility in conductive channel according to selective doping of substrate.

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Figure 21. PV response in control devices with illumination at the vicinity of the contact upon different scope loads: 50 Ω and 7.6 kΩ.

Figure 22. Light harvesting in thick films: molecular redox. Only film composed from C6-NTCDI-C6 molecule, which contains nitrogen atom (red color on the left panel), was active in long-range harvesting. NTCDA molecule with unsubstantiated oxygen (blue color on the right panel) could not sustain long-range any redox transport. We took this fact as an evidence of polaron transport in latent channel in SAM-PVC and SAM-OFET devices.

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EVAC

LUMO CB EF

0.4-0.7 eV

1.1 eV HOMO

3.2 eV

1.6-1.8 eV

VB

Si

NTCDI ML

Ag

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Figure 23. Energy diagram of heterostructure of 2D NTCDI transport channel according to SAM-PVC studies.

Finally, we could summarize results of our SAM-PVC and reported studies [106, 107] at energy diagram, which shows positions of the energy levels at the heterostructure of p-type 2D NTCDI transport channel, at Figure 21. Left part of this diagram shows conductive (CB) and valence (VB) bands in silicon substrate and slope of the bands, as they approach to the interface with SAM NTCDI. Middle panel demonstrate HOMO-LUMO (HOMO = highest occupied molecular orbital, LUMO = lowest occupied molecular orbital) gap and COB (COB = continuum of band, [108]) of NTCDI monolayer. The right part of diagram shows position of Fermi level EF of Ag electrode. The Si/NTCDI interface of SAM-PVC (left part of Figure 21) models the interface of conductive channel on silicon in SAM-OFET device. In this interface the out-of plane photooxidation of NTSDI takes place. Structure of NTCDI SAM is modeled at the middle part of diagram. This part is responsible for the charge transport in SAM-OFET NTCDI 2D system. Set of thin lines in the vicinity of EF shows continuous of band (COB) of 2D NTCDI system, which is also called sometime “the-states-in-the gap”. Transport through COB is in-plane dark self-exchange step in equation 18. Note that SAM-PVC device does not have direct access to HOMO-LUMO levels, neither SAM-OFET. Of course, molecular orbitals change essentially their distribution as the redox undergoes through molecule, but charges do not have to overcome energy barriers, which involve molecular LUMO orbitals. Boundaries of molecular stacks are the only places where tunneling is governing transport mechanism [109, 110, 111]. The right panel of Figure 21 shows the Fermi level of the electrode inside the COB. The possibility of such position of EF was proven at ref. [113]. This part of SAM-PVC system corresponds to out-of-plane dark harvesting (equation 18) and should model charge injection into NTCDI channel from Au electrodes in SAM-OFETs. Figure 21 represent the “static” model which describe SAM heterostructure in neutral manifold. While the polarons are propagating along the in-plane COB channel, the out-of-plane electronic “pump” (reaction 1 at Figure 7) could provide an electrical motion force for the charge-delivery. This

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contribution could explain the higher charge speed in SAM layer than in organic crystal, as discussed in first paragraph of Discussion section. Therefore, the charge kinetic studies described in section 2.2 demonstrate that the out-ofplane charge injection can be discriminated. This discrimination is possible through illumination of the SAM-PVC device in different positions. Thus, variable kinetic of in-plane charge transport could be separated from the constant value of charge injection. The whole balance of device kinetics can now be visualized and used for optimization of the SAMOFET design. Our studies underline the importance of careful comparison of the electrical and optical data to discuss major transport mechanism in SAM system. Section 2.3 of this chapter demonstrates that a photocurrent study was the only way to split and estimate performance of different OFET component, such as conductive channel and electrodes. Using optic and electronic studies we were able to evaluate the major transport mechanism in SAMOFET. A complex interplay of the molecular properties and solid-state properties was highlighted by SAM-PVC studies in NTCDI conductive channel. There are questions about the optimal structure of conductive channel in the OFET, which include determination of what kind of crystal packing (e.g., herringbone versus stacking) is most favorable for the charge transport and whether intermolecular electronic cloud overlap can be enhanced by tuning the crystal packing (e.g., the degree of intermolecular ring slip) or by increasing the strength of intermolecular bonding [112,113]. Our results indicate that high charge mobility could be achieved in a herring-bone packing molecules, for example NTCDI molecules, which have one of the highest mobility for organic molecules. Significantly, closing the structure-property knowledge gap on lowdimensional phase transitions in organic solids is required component to get insight into structure-property relationship of SAM-OFETs and other organic devices with epitaxial order. We also want to note that SAM-OFET was used without any protection layer and demonstrated good device stability, which is perhaps one of the most crucial issues in OFET. SAM-OFETs were studied in an operational regime for two weeks without any protection from the ambient. Two-week check of SAM-OFET does not show any decrease of the semiconductor sensitivity due to O2 and H2O. The I-V characteristics do not show a hysteresis as a function of device operating parameters. This result agrees with previous results on OFET performance based on functionalized NTCDI system, such as NTCDI-CH2C6H4CF3 [4]. These structures offer the best combination of high mobility and air stability.

Conclusion 1. Estimating Mobility in NTCDI SAM-OFETs Photocurrent investigation of transport mechanism in 2D NTCDI-based SAM-OFET demonstrates that a moderately high mobility could be achieved in this device. Mobility which we achieve in NTCDI SAM OFET is roughly 10 times larger than the value reported for NTCDI thick film OFETs. We believe that improvement in mobility has been achieved due to the better packing in OFET conductive channel and better condition for redox exchange transport, as is evident from structure-photoresponse study depicted in Figure 19. Experiment with cation-radical blockade in photovoltaic device (Figure 19) and essentially

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lower mobility in NTCDA thick film OFET device impose the polaron nature of charge transport in NTCDI based thick film OFETs. The possible ways to improve performance of 2D NTCDI-based SAM OFET are (1) proper molecular engineering of a metal-SAM charge injecting interface, (2) better contact between SAM layer on electrodes and contacts, (3) usage of more efficient redox transport mechanism, (4) usage of smaller gap between electrode to limit possible number of defects and (5) optimizaton of macro- and micro-scope structure of organic conductive channel to insure the best connectivity of aromatic rings. However, it is unlikely that charge mobility in 2D SAM OFET devices will exceed electron mobility in amorphous silicon, which is ~ 1000 cm2 V-1 s-1. This limitation arises from proposed polaron nature of charge transport in studied organic 2D system. The better charge mobility in SAM OFET could be achieved using (i) different electrode and device configuration in organic compounds or (ii) materials, which demonstrate higher intrinsic mobilities. The first solution could be exemplified by self-assembly of benzene electrons from molecule between two Pt electrodes with following hybridization of 3 aromatic core and electrons of electrodes [114]. The result is conductivity, which exceed G0 in tunneling regime in this system. The celebrating example of second approach for achieving higher mobility is graphine material. Recently physicists have demonstrated that graphene, a single-atom-thick sheet of graphite, has a greater ability to conduct electricity than any other known material at room temperature ( = 2 105 cm2 V-1 s-1) [115]. In fact, electrons can travel up to 100 times faster in graphine than silicon, making it a likely candidate for semiconductor material replacement in devices like computer chips and sensors. Graphene also has a resistivity of 1.0 Ohm-cm, which is just 35% less than copper. That figure would also make graphene the lowest resistivity material at room temperature. A first field-effect experiments with graphine demonstrated formation of gap state at low temperatures [116], which is extremely promising for a SAM-OFET application. These two systems, graphene and planar benzene congiguration utilize transport mechanisms, which are different from one that we found in NTCDI-based SAM-OFET system.

2. Alternative Pathways for Development of SAM-OFETs A possible avenue for developing 2D NTCDI-based SAM-OFET systems for electronic devices is an approach comprising “intramolecular circuits”, suggested by C. Joachim at 2002 [117]. Here, instead of pursuing highest performance in three terminal FET devices, the new logic devices based on intermolecular circuits in organic 2D system are introduced. The circuit is assumed to be connected by many electrodes. Development of this concept should result in a different design of a logic or intramolecular gate. Another path, which might be explored in 2D NTCDI-based systems, is spin transport in 2D aggregates. It was noted that weak spin-orbit and hyperfine interaction in organic solids is favorable for spintronic applications. This research could also lead to new logic devices, such as spin-OFETs. The specific feature of these two examples is that high mobilities are not pre-required and special properties of organic solids in 2D are utilized to achieve new electronic devices, which are currently not available. It has been suggested that 2D NTCDI-based systems could be used as an artificial model of charge transfer in a biological membrane [119]. In developing this idea, a new type of biosensor could be developed. A new type of biosensor could utilize in real time latent skin current in SAM–OFET and SAM-PVC as a sensor for probing bio-

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recognition events on interface of SAM. This assumption is based on the robustness of NTCDI-derived structures, pronounced skin-current in SAM-OFET and SAM-PVC devices and chemical capability to continue self-assembly.

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3. Photocurrent Study as Tool to Elucidate a Transport Mechanism in 2D NTCDI-Based SAM-OFET We demonstrated in this chapter the development of adequate analytical tools, and combine electric and opto-electronic studies that are crucial for future progress beyond the current stage of SAM-OFETs. In the current state of the art, a variety of Kelvin probe-derived techniques are widely used to determine electron work function and bend banding in organicinorganic heterostructures. UV and inverse photoemission spectroscopy (UPS, IPES) are used to determine the position of the highest occupied and lowest unoccupied molecular orbitals (HOMO, LUMO). XPS measurements served to estimate band bending. Comparison of these techniques with measured I-V characteristics enables one to determine the mechanism(s) that control charge transfer in molecular diodes. These methods are static, since probing interfaces in time-scale compared to interface charge transfer is very challenging. Ultra-fast spectroscopy was applied to clarify dynamic of charge transfer in organics. Nevertheless, direct observation of the complete dynamics of charge transfer in the interface is still beyond the scope of these methods. We are suggesting using a photocurrent method in a timeresolved regime as a new tool, supplementary to direct electrical ID-UD measurements to probe hybrid molecular interfaces. The possibility of studying charge transfer kinetics described in section 2.2 demonstrates that the charge out-of-plane injection and in-plane transport in an organic channel pathway can be discriminated. Thus, a new photocurrent approach is an important analytical tool for the molecular engineering of SAM OFETs. It should be possible to evaluate and compare the rate of charge injection and charge transport in these hybrid devices to achieve optimal device performance. The application of photophysics studies of SAM devices makes possible the analysis of transport properties in 2D organic systems and should provide future insight into charge transfer in low-dimensional objects.

Acknowledgments We thank A. A. Gewirth for the valid suggestions and helpful discussions, A. Bezryadin (Physics Department, UIUC) for the loan of the electrometer and supportive discussions, R. Gennis (Chemistry Department, UIUC) for the use of UV–Vis instrumentation, R. Hash (MRL, UIUC) for XPS measurements, R. Strange (MRL, UIUC) for reflection spectroscopy, and M. Sardela (MRL, UIUC) for XRD measurements. We benefited from useful discussions with L. Woichek, V. Vardeny, A. Rogachev (all of the Physics Department, UU), and J. Wright (Chemistry Department, WMU). We thank M. Zudov (School of Physics and Astronomy, UM) for his review and critical comments of our manuscript.

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Review This chapter was reviewed by Professor Michael Zudov, School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN, 55455; email zudov@ physics.umn.edu.

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Chapter 7

ORGANIC FIELD-EFFECT TRANSISTORS: TETRATHIAFULVALENE DERIVATIVES AS HIGHLY PROMISING ORGANIC SEMICONDUCTORS M. Mas-Torrenta, P. Hadleyb, S.T. Bromleyc, J. Vecianaa and C. Roviraa

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a

Institut de Ciència de Materials de Barcelona, Campus UAB, 08193 Bellaterra, Spain. b Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands c Departament de Química Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain

The processing characteristics of organic semiconductors make them potentially useful for electronic applications where low-cost, large area coverage, and structural flexibility are required.[1] Contrary to amorphous silicon, which is widely used in solar cells and flat screen displays, organic materials offer the benefits that they can be deposited on plastic substrates at low temperature by employing solution-based printing techniques. These deposition techniques would, therefore, reduce the manufacturing costs dramatically. The challenge now lies on finding organic semiconductors which are processable, stable and, at the same time, exhibit high enough mobilities (μ>0.1 cm2/Vs) and ON/OFF current ratios (>106) to be used for applications in modern microelectronics. Transistors play a central role in many electronic circuits, where they usually function as either a switch or an amplifier. A field-effect transistor can be described as a three terminal device in which the current through a semiconductor linked to two terminals (namely source and drain) is controlled at the third terminal (called gate) by a voltage. A typical organic fieldeffect transistor (OFET) configuration is shown schematically in Figure 1. The voltage applied to the gate (VG) induces an electric field through the dielectric and causes the formation of an accumulation layer of charges at the interface of the semiconductor deposited above. Then, by applying a source-drain voltage (VSD) it is possible to measure current between the source and the drain (ISD).

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M. Mas-Torrent, P. Hadley, S.T. Bromley et al. conducting channel source

semiconductor semiconductor

drain ISD

insulator

VSD

gate VG

Figure 1. Schematic drawing of an OFET. This device is biased the way a p-type semiconducting OFET would be. The source is grounded and a negative voltage is applied to the drain and the gate to induce holes in the semiconducting channel

Depending on how strong the interactions between the molecules of the semiconductor are, the electron wavefunctions that correspond to these energy levels can either be localized or extended. Strong molecular interactions lead to extended electron states with the allowed energy levels arranged into bands of allowed energies separated by gaps of forbidden energies. Weak molecular interactions lead to localized states and hopping conduction. Band conductors, with extended electron states, can be good conductors or good insulators depending on the location of the Fermi energy. Electron states much more than kBT above the Fermi energy are unoccupied. Here kB is Boltzmann's constant and T is the absolute temperature. States much less than kBT below the Fermi energy are occupied and states within kBT of the Fermi energy are partially occupied. The Fermi energy, EF, is defined implicitly by the relationships, ∞

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n=

∫ g ( E ) f ( E )dE and

−∞

1

f (E) = e

E − EF k BT

+1

Here E is the energy, f(E) is the Fermi function, n is the electron density, and g(E) is the energy density of states per unit volume. If the electron density and distribution of energy levels are known, the Fermi energy can be found by iterating these equations. A material with partially filled bands is a metal and a material with the Fermi energy in a forbidden band gap is an insulator if the difference in energy from the Fermi energy to the allowed bands is much greater than the energy of thermal fluctuations kBT. Semiconductors that are easier to oxidize than to reduce are typically p-type semiconductors where the Fermi energy is in the bandgap but within kBT above the top of the valence band. In this case, current will flow predominantly due to the motion of positively charged holes. Semiconductors that are easier to reduce than to oxidize are typically n-type semiconductors where the Fermi energy is within kBT beneath the bottom of the conduction band. In this case, current will flow predominantly due to the motion of negatively charged electrons. Figure 2 shows a schematic energy diagram for all these types of materials. The Fermi energy can be shifted either chemically or electrostatically. For instance, adding a dopant that increases the number of electron states below the Fermi energy will decrease the Fermi energy. Oxidation shifts the Fermi energy down and reduction shifts the

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Fermi energy up. In a transistor, a voltage applied to the gate will shift the Fermi energy electrostatically and can thereby modulate the conductivity of the channel. An applied gate voltage can make an originally conducting channel an insulator (depletion mode) or an originally insulating channel conducting (enhancement mode). A gate voltage can even convert an n-type semiconductor into a p-type semiconductor (inversion). Whenever two materials come into contact, a transfer of charge takes place to compensate for the difference in the work functions. Electrons can lower their energy by moving to a material with a larger work function. They will do this until the electric field at the interface increases enough to hold them back and the Fermi energy is the same in both materials. Such electric fields typically exist at the source and drain contacts of a transistor. These fields locally shift the energy of the electron states with respect to the Fermi energy. This phenomenon is known as band bending. Consequently, depending on how the bands are bent, the semiconducting region near a contact can be a better conductor or a worse conductor than the rest of the channel. Energy diagrams for two metal/semiconductor contacts are illustrated in Figure 3. In Figure 3a, the Fermi energy is in the middle of the allowed energy band of the metal on the left and it is just above the HOMO states on the right. This means that the semiconductor is p-type. The electron energy states in the semiconductor are bent down at the contacts indicating that the work function of the metal (i.e. the minimum energy required to remove an electron from the Fermi level to the vacuum level) is lower than the work function of the semiconductor. The states near the contacts are further away from the Fermi level making this region less conducting than the middle of the channel. This region of poor conductivity is known as a Schottky barrier. Figure 3b shows the same p-type semiconductor in contact with a high work-function metal. This bends the bands up in the semiconductor forming good ohmic contact to the p-semiconductor. Similarly a low workfunction metal makes ohmic contact to an n-type semiconductor while a high work-function metal forms a Schottky barrier at an interface with an n-type semiconductor.

Figure 2. The conductivity of a material depends strongly on the position of the Fermi energy, EF. The gray shaded areas indicate energies where electron states exist. A forbidden energy gap Eg is indicated for all five cases. (a) A metal with a small work function. (b) An n-type semiconductor. (c) An insulator. (d) A p-type semiconductor. (e) A metal with a large work function.

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semiconductor

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Figure 3. Band bending at interfaces due to differences in work functions. The gray regions indicate the energies at which electron states exist. Just the valence band of the semiconductor is drawn for simplicity. (a) A contact formed between a metal with a low work function and a p-type semiconductor. A poorly conducting Schottky barrier appears at the contact. (b) A contact between a high workfunction metal and a p-type semiconductor. This results in ohmic contact.

Figure 4. The MOSFET model. Here ISD is the current from source to drain, VSD is the voltage from source to drain, VG is the gate voltage, VT is the threshold voltage, L is the length of the transistor from source to drain in the direction that the current flows, W is the width of the transistor, C is the capacitance between the gate and the channel, and μ is the mobility. The MOSFET equations are plotted on the right for a transistor where WCμ/L = 3.3 × 10-8 Ω-1 and the threshold voltage is VT = 30 V. The thick gray line separates the linear regime from the saturation regime.

The current that flows through a transistor depends on all these shifts caused by bringing different materials in contact as well as the shifts caused by applying gate and source voltages. Fortunately there is a relatively simple model that describes the current. The standard model for a Metal Oxide Semiconductor Field Effect Transistor (MOSFET) was originally constructed to describe inorganic semiconductors such as silicon but is commonly applied to OFETs. The equations for the MOSFET model are given in Figure 4. There are two regimes in this model, the linear regime, where paradoxically the current is described by a parabola, and the saturation regime where the source-drain current is independent of the source-drain voltage. An important parameter in this model is the mobility. This parameter

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can differ by orders of magnitude depending on the materials used for the semiconducting channel. Poorly conducting organic semiconductors have mobilities of μ = 10-4 cm2/Vs or lower, good quality spin-on organic materials have mobilities of μ ~ 10-2 cm2/Vs and well ordered organic materials have mobilities in the range μ ~ 1-10 cm2/Vs. Amorphous silicon has a mobility of μ ~ 1 cm2/Vs, and crystalline silicon has mobilities on the order of μ ~ 103 cm2/Vs. Exceptionally clean systems of inorganic semiconductors can have mobilities of 107 cm2/Vs. There are two main families of organic semiconductors (Figure 5) that can be used as a component in OFETs: i) conjugated polymers (e.g. polyphenylene, polythiophene, poyphenylenevinylene) and ii) small conjugated molecules with low molecular weight (e.g. pentacene, oligothiophene, phthalocyanine). Small conjugated molecules form regular crystalline structures where the conductivity is determined by the overlapping of the π−orbitals. The intermolecular bonds in molecular crystals are weaker than the interatomic bonds in inorganic semiconductors like silicon, so thermal fluctuations result in larger displacements in molecular crystals. These displacements disrupt the crystal order and result in lower mobilities than are observed in inorganic semiconductors. However, crystals of small molecules typically form more ordered films and have higher OFET mobilities compared to polymers.

Pentacene n

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Polyphenylene

S

S S

n

S S

S

Sexithiophene

Polyphenylenevinylene

S

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Figure 5. Molecular formula of organic semiconductors: a) conjugated polymers, b) low-molecular weight materials.

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Polymers are chains with good conductivity along the chains and weaker conductivity between the chains. Interchain electron transport takes place primarily due to π−orbital overlap between the chains. Sometimes the chains fold into ordered crystalline structures over short length scales but usually they have a microstructure that resembles cooked spaghetti and, thus, the conductivity is limited by disorder. While polymers are mostly soluble in organic solvents, devices based on small molecules are typically prepared by evaporation of the organic material due to their low solubility. The performance of OFETs in the last 20 years has improved enormously.[1] Nowadays, mobilities of the same order as amorphous silicon (0.1-1 cm2/Vs) are achieved in the best OFETs perfomance. Thiophene and, especially, acene derivatives are considered to be the benchmark in OFETs and most of the best mobilities have been found in these families of compounds. A mobility of 1.5 cm2/Vs has been reported for pentacene,[2] and recently a mobility of 15 cm2/Vs was reported for rubrene.[3] Some works have focused on directly functionalizing pentacene in order to tune its electronic properties[4] and enhance the intermolecular overlap.[5] Comparative studies of OFETs based on oligothiophenes regarding the deposition method[6] and their molecular structure have also been carried out.[7][8] All such studies are of great interest in order to gain a better understanding of the requirements that the organic semiconductors should accomplish to achieve high OFET performance. However, OFET performance depend on a large number of parameters and, therefore, it is essential to understand their role in the device operation and to have control over all of them in order to ensure reproducibility and to be able to establish comparative studies between the different devices. Below some of the key experimental variables for the OFETs fabrication are illustrated.

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1. Device Configuration The design of the device configuration will influence on the measured electronic properties. For instance, the choice of metal electrodes will affect the contact resistance. As mentioned before, it is important to choose the electrode metal according to the nature of the organic semiconductor to have efficient charge injection. That is, for an n-type material it will be more suitable to have a metal with a lower work function, whereas for p-type material a metal with high work function would be desired. The dielectric material will influence the electric field created along it, the current leakage through the gate insulator and the quality of the interface between the organic semiconductor and the dielectric. In addition, the characteristics of the dielectric can be modified with surface treatments. For example, growing a self-assembled monolayer of an organic molecule on silica will change the hydrophilic nature of the oxide to hydrophobic. It has been demonstrated that these treatments often have strong effects on the film structure and on the resulting electrical characteristics.[9] Likewise, the source and drain electrodes can be evaporated on the top of the organic material (top contact configuration) or on the dielectric before depositing the organic semiconductor (bottom contact configuration). Usually, the electrical contact with the organic semiconductor is better in a top contact configuration. However, the evaporation of the metal might damage the organic material. This top-architecture does not permit the patterning of the electrodes through conventional lithography and, thus, the contacts have to be deposited through a shadow mask, which does not allow channel lengths shorter that tens of

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micrometers.[10] On the other hand, the protruding electrodes in the bottom-contact configurations might cause inhomogeneities during the film formation.

2. Integration of the Organic Material in the Device The organic semiconductor can be deposited on the device employing solution based techniques like drop-casting or spin-coating. The use of such techniques is the most promising route to produce low manufacturing costs and to fabricate devices of large-area coverage. In addition, by combining them with stamping or printing techniques,[11] it is possible to pattern organic semiconductors eliminating the use of lithography. However, since often organic semiconductors are not very soluble, an alternative deposition method is by sublimation of the organic material in a variety of vacuum depositions systems. Parameters such as pressure and substrate temperature determine the morphology and quality of the resulting films. Typically, the use of vapor phases allows for the obtaining of higher purity materials and, hence, the higher OFET mobilities have been achieved in this way.

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3. Organic Semiconductor Obviously, the choice of the organic semiconductor will mainly determine the device performance. Here, it will be important to take into account the intermolecular interactions that the molecules exhibit as the stronger the electronic coupling between neighboring molecules is, the higher mobilities will be achieved. In this way, it will be thus crucial to have materials with very high molecular order to avoid disruption of the electron conduction path. Typically, solution processed polymers form complex microstructures, where microcrystalline domains are embedded in an amorphous matrix. The disordered matrix limits the charge transport resulting in low field-effect mobilities. For this reason, the most studied polymer for OFETs is poly(3-hexylthiophene) (P3HT), which has given the highest OFET mobility found for an organic polymer OFET, 0.1 cm2/Vs.[12]2 This high mobility is related to structural order in the polymer film induced by the regioregular head-to-tail coupling of the hexyl side chains (Figure 6).[13] Additionally, when p-xylene or cyclohexanone are used as solvents, P3HT forms fibers in the solution, which can then be deposited on the substrate (Figure 7).[14] Hole mobilities in the range of 0.02- 0.06 cm2/Vs have been reported very recently for single nanofibers and films of aligned fibers.[14]b{-[15] The formation of similar fibers has been observed in other conjugated polymers and is believed to be governed by the π-stacking of the conjugated chains.[16] The importance of the molecular ordering for fabricating OFETs with high performance is not restricted to polymer based transistors. For instance, Dimitrakopoulos and Mascaro demonstrated that films of pentacene evaporated at different temperatures show a variety of crystallinity which correlates with the resulting OFET performance.[17]

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Figure 6. Schematic lamellar type ordering in P3HT films. Reprinted with permission from reference 13.

Figure 7. AFM image of a device prepared by drop casting a solution of P3HT in p-xylene whilst applying an ac voltage (2 V, 102 kHz) on the right electrode of the image. This resulted in the formation of aligned nanofibers. Reprinted with permission from reference [14b].

The Langmuir-Blodgett technique, which consists in transferring on a substrate amphiphilic molecules ordered on the water surface-air interface, offers a unique approach of preparing well-ordered thin films. The application of this technique for the preparation of OFETs has already been demonstrated.[18][19] The fundamental material characteristics of organic semiconductors are, however, most clearly measured in single-crystals owing to their higher molecular ordering and to the fact that they do not present grain boundaries that limit the mobility.[20] The highest OFET mobility reported to date has been observed in rubrene crystals (15 cm2/Vs).[20] a All these organic crystals are typically grown from the vapor phase. Additional parameters such as the purity or stability of the material and processability determine its potential for applications. For this reason, some attempts have focused on synthesizing soluble pentacene and oligothiophenes precursors which, after heating, are converted to their parents compounds,[21] since these latter show high OFET performances but very often have to be processed using vacuum-techniques due to their low solubility in common organic solvents.

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Overall, to progress in this field towards device applications, it is very important to search for new molecules that are able to act as organic semiconductors, and also to develop easy methods to grow single crystals or films. Along these lines, we recently reported the preparation OFETs using single-crystals of tetrathiafulvalene (TTF) derivatives as the organic semiconductor.[22] These crystals were grown from solution by drop casting, a very simple method. Moreover, responding to the current needs of gaining a better understanding of the relationship between crystal structure and field-effect mobility, a systematic study of the dependence of the device performance on the crystal structure was carried out. This comparative study allowed for the focused investigation of the influence of the intermolecular interactions on the electronic transport properties.

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OFETs Based on Tetrathiafulvalene (TTF) Derivatives Since the discovery of the first organic metal tetrathiafulvalene-7,7’,8,8’tetracyanoquinodimethane (TTF-TCNQ) over thirty years ago,[23] tetrathiafulvalene (TTF) and its derivatives have been successfully used as building blocks for charge transfer salts giving rise to a multitude of organic conductors and superconductors. The crystallisation of TTF derivatives is governed by the π-π stacking, which permits, together with the S···S interactions, an intermolecular electronic transfer responsible for their transport properties. TTF derivatives are generally soluble in various solvents, are easily chemically modified, and are good electron donors. TTFs have already been extensively studied for the preparation of a wide range of molecular materials.[24] Considering thus all the above, TTF derivatives also promise to be good candidate molecules for the preparation of OFETs, due to the possibilities of synthesising tailored derivatives which can be easily processed, either in vacuum or from solution.[25] With the aim of establishing a correlation between crystal structure and charge carrier mobility, we recently prepared OFETs based on neutral TTF crystals grown from solution.[22] We studied eight different TTF derivatives, which are shown in Figure 8, namely bis(ethylenedithio)-tetrathiafulvalene (BEDT-TTF), (ethylenethio)-(ethylenedithio)tetrathiafulvalene (ETEDT-TTF), bis(ethylenethio)-tetrathiafulvalene (BET-TTF), (ethylenethio)-(thiodimethylene)-tetrathiafulvalene (ETTDM-TTF), dithiophene-tetrathiafulvalene (DT-TTF), (thiophene)(thiodimethylene)-tetrathiafulvalene (TTDM-TTF) and (ethylenethio)(thiophene)-tetrathiafulvalene (ETT-TTF). Taking, thus, into account the crystal packing in which these molecules are arranged, this family of compounds can be classified into three groups. BEDT-TTF and ETEDT-TTF are from the first group (G1). Their supramolecular organisation consists of dimers sustained by hydrogen bonds, forming chains along the a axis due to lateral S···S interactions. The chains are arranged perpendicularly to each other avoiding therefore the formation of stacks. In the second crystal structure group (G2), BETTTF and ETTDM-TTF crystallise forming chains of quasi planar molecules along the a axis interacting side-by-side. These chains stack into layers giving rise to a bidimenssional electronic structure. Finally, the molecules from group 3 (G3), DT-TTF, TTDM-TTF, and ETT-TTF, crystallise forming uniform stacks of almost planar molecules along the b axis. The interplanar distance between molecules of one stack is very short (3.56-3.66 Å).

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Figure 8. Molecular structures of the studied TTF derivatives for OFETs. The molecules are classified into three groups according to their crystal packing.

The OFET devices were prepared by drop casting a solution of the TTF derivative onto the gate insulator and the prefabricated source and drain gold electrodes. The solution was allowed to evaporate at room temperature which resulted in the formation of long regular crystals, some of which connected two of the microfabricated electrodes (Figure 9). The electrical characteristics of the devices were studied by measuring the source-drain current versus the applied source-drain voltage for different gate voltages applied to the silicon substrate. Figure 10 shows the electrical measurements performed on a DT-TTF crystal. All the crystals exhibit a p-type behavior where the conductivity increases as a more negative gate is applied. We measured 67 different single-crystal OFETs using the TTF derivatives mentioned above. A clear correlation between crystal structure and device performance was found as a notable improvement of the charge carrier mobility was observed on going from group 1 to group 3. This trend suggests therefore that the crystal packing of group 3 is the most suitable for this family of materials for the preparation of OFETs. The highest mobility found was of 1.4 cm2/Vs for a DT-TTF crystal. This is the highest mobility found for a solution processed material, which makes this material very interesting for potential applications. Au electrodes TTF crystal

Si 100 μm

SiO2

Figure 9. Left: Device configuration used. Right: Optical microscope image of a DT-TTF single crystal lying on the gold electrodes. Reprinted with permission from reference [22] a. Copyright 2004 American Chemical Society.

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Figure 10. Source-drain current versus source-drain voltage at different gate voltages for a DT-TTF single crystal OFET. Reprinted with permission from reference [22b]. Copyright 2004 American Chemical Society.

At room temperature the charge mobility of semiconducting organic materials is often determined by a hopping transport process, which can be depicted as an electron or hole transfer reaction in which an electron or hole is transferred from one molecule to the neighboring one. Two major parameters determine self-exchange rates and, thus, the charge mobility:[26] (i) the electronic coupling between adjacent molecules, which needs to be maximized, and (ii) the reorganization energy (λreorg), which needs to be small for efficient charge transport. The reorganization energy of self-exchange in a hole-hopping material is defined as the sum of the geometrical relaxation energies of one molecule upon going from the neutral-state geometry to the charged-state geometry and the neighboring molecule upon going through the inverse process [ A(0) + A’(+) → A(+) + A’(0)]. These two portions of λreorg are typically nearly identical.[27] Previous theoretical works have attempted to explain the high mobility pentacene OFETs in terms of its low λreorg.[25a]-[28] We performed density functional (DF) calculations to calculate λreorg for all the studied TTF derivatives. All reported calculations were performed at a 6-31G(d,p)/B3LYP[29] level of theory using Gaussian 98.[30] In the neutral state, TTF derivatives are most energetically stable when in a distorted boat conformation, but adopt planar conformation for the +1 charged state.[31] First we calculated λreorg employing the stable boat conformation for each neutral isolated molecule and then, again, employing the closest local minimum energy structure for the neutral molecule which best approximated the conformation found in their crystal structure. For Group 2 and Group 3 crystal structures, the constituent TTF derivatives are found to be almost perfectly planar. Thus, in our calculations, to represent this packing-induce planarity, isolated neutral molecules were optimized to local energetic minimum planar conformations. For the Group 1 crystal packing, the TTF derivatives are not planar but adopt boat-like conformations. For BEDT-TTF we employed the lowest energy boat conformation, and, in the case of ETEDT-TTF, a minimum energy structure very close to the lowest energy boat conformation was found to be the best isolated molecular representation. For each calculation the lowest energy planar +1 molecular conformation was used. These results are summarized in Table 1. Comparing the resulting λreorg of all these molecules, we find that λreorg values calculated while always employing the lowest energy neutral molecule boat conformations do

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not follow any obvious pattern relating to the corresponding crystal structure. However, when employing the isolated neutral conformations, approximating the crystal packing-constrained molecular geometries, a trend of λreorg values emerges in line with that of the experimentally measured mobilities i.e. molecules with better mobilities tend to have lower λreorg values. The main reason for the difference in these two sets of calculations comes from the use of planar neutral molecules to represent the respective planar TTF derivates in the crystals of Group 2 and Group 3, which results in a notable drop of λreorg. This decrease in the λreorg values can be simply explained by the fact that the geometries of the neutral and charged planar molecules become closer, which is not the case for molecules of Group 1. The calculated differences in isolated molecular λreorg values, however, do not appear to fully explain the relatively large measured differences found in the mobilities, suggesting that intermolecular interactions should also be taken into account. As mentioned above, the other parameter that determines mobilities, in addition to the reorganization energy, is the intermolecular electronic coupling, often estimated via the calculation of transfer integrals.[32] In Table 1 we thus further report transfer integrals for the TTF-derivative crystals displaying the highest mobility (and highest symmetry) in each crystal structure group. The single value, tmax, quoted for each Group is the maximum transfer integral for that crystal, based on a systematic search of all nearest neighbor dimeric possibilities for the strongest electronic coupling. Remarkably, the transfer integral values also follow the trend found for the experimentally measured mobilities i.e. the crystals with higher mobilites also have higher maximum transfer integrals. For DT-TTF, tmax is found along the packing crystal axis b, which also corresponds to the long crystal direction and, thus, the device channel direction. However, for BET-TTF, whereas the long crystal axis is known to correspond to the crystallographic b+c direction, tmax is found between the lateral S···S contacts present in the ab plane. Finally, for BEDT-TTF the interdimer tmax is also found to be between the lateral S·S along the a axis. The extremely encouraging trends, in both λreorg and tmax values, are thus far based on calculations employing isolated molecules/dimers. Such studies can further benefit by taking into account a fuller representation of the true crystal environment e.g. the boat conformations in the crystals of Group 1 are stabilized by intermolecular hydrogen bonding. Previously we demonstrated that an additional drop in the λreorg value of DT-TTF is observed if one considers crystal-embedded molecules, rather than using solely their isolated planar molecular representations.[3] Following this line of reasoning, we hope in the future to further investigate the role of intermolecular interactions on TTF derivative crystal transport properties. All these results are of great importance for the future design of new TTF derivatives with promising OFET performance. Following therefore this studied correlation, we prepared OFETs with the organic semiconductor dibenzo-TTF (DB-TTF), which crystallises similarly to the molecules in group 3 and has values of transfer integral (0.037 eV) and reorganisation energy (0.248 eV) close to the ones found for DT-TTF. Interestingly, the devices based on this material also displayed very high mobilities of the order of 0.1-1 cm2/Vs.[34] Other recent works have also prepared OFETs by evaporation of some TTF derivatives.[35] The best device performance employing vacuum techniques has been achieved with the dinaphto-tetrathiafulvelene analogue, which exhibited a maxim mobility of

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0.42 cm2/Vs, whereas the DB-TTF shows in this kind of device a maximum mobility of 0.06 cm2/Vs,[34]dwhich is one order of magnitude lower than the one found for the single crystal. Table 1. Mean values of charge carrier mobilities (μm) and their standard deviations (in parenthesis), maximum transfer integrals of the symmetric derivatives (tmax), and isolated molecular reorganization energies of the TTF derivatives using the minimum energy neutral and +1 conformations (λreorg) and using the neutral local minimum energy conformation best approximating the crystal structure conformation (λreorg (c))

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TTF Derivative ETEDT-TTF G1 BEDT-TTF ETTDM-TTF G2 BET-TTF ETT-TTF G3 TTDM-TTF DT-TTF

μM, Cm2/Vs 1.4·10-4 (9.8·10-5) 1.2·10-3 (1.7·10-3) 1.8·10-3 (9.0·10-4) 1.5·10-2 (1.7·10-2) 6.2·10-2 (3.7·10-2) 1.5·10-1 (8.0·10-2) 2.5·10-1 (4.0·10-1)

λReorg, Ev 0.506 0.550 0.432 0.326 0.438 0.512 0.574

λReorg (C), Ev 0.488 0.550 0.314 0.246 0.238 0.258 0.238

Tmax, Ev 0.019 0.021 0.034

In summary, organic field-effect transistors are very promising for potential applications in large-area and low-cost electronics. However, further research is required to gain a better understanding and control over the parameters that influence the device performance. In addition, since most of the organic semiconductors are p conductors, searching for high performance and stable organic materials which are n conductors and which are ambipolar (conduction by holes and electrons) is also necessary to progress in the field. We also described the influence of the crystal structure on the performance of singlecrystal OFETs of TTF derivatives. A trend in mobilities through the different crystal structures was observed which was further strongly corroborated by calculations of both the molecular reorganization energies and the maximum intermolecular transfer integrals. The early obtained results with OFETs with TTF derivatives already point out the high potential of these materials, which can be easily processed either in vacuum or from solution.

References [1] Dimitrakopoulos, C. D. & Malenfant, P. R. L. (2002). Adv. Mater. 2002, 14, 99. b) Y. Sun, Y. Liu, D. Zhu, J. Mater. Chem. 2005, 15, 53. c) Katz, H. E., Chemistry of Materials, 16, 4748. [2] Nelson, S. F., Lin, Y. Y., Gundlach, D. J. & Jackson, T. N. (1998). Appl. Phys. Lett. 72, 1854-1856. [3] Sundar, V. C., Zaumseil, J., Podzorov, V., Menard, E., Willett, R. L., Someya, T., Gershenson, M. E. & Rogers, J. A. (2004). Science, 303, 1644-1646 [4] Meng, H. M., Bendikov, M., Mitchell, G., Helgeson, R., Wudl, F., Bao, Z., Siegrist, T., Kloc, C. & Chen, C. H. (2003). Adv. Mater. 15, 1090-1093. [5] Anthony, J. E., Brooks, J. S., Eaton, D. L. & Parkin, S. R. (2002). J. Am. Chem. Soc. 2001, 123, 9482-9483. b) J. E. Anthony, D. L. Eaton, S. R. Parkin, Org. Lett, 4, 1, 15-18.

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[6] Yanagi, H., Araki, Y., Ohara, T., Hotta, S., Ichikawa, M. & Taniguchi, Y. (2003). Adv. Funct. Mater, 13, 767-773. [7] Halik, M., Klauk, H., Zschieschang, U., Schmid, G., Ponomarenko, S., Kirchmeyer, S. & Weber, W. (2003). Adv. Mater, 15, 917-922. [8] Videlot, C., Ackermann, J., Blanchard, P., Raimundo, J. M., Frère, P., Allain, M., de Bettignies, R., Levillain, E. & Roncali, J. (2003). Adv. Mater. 2003, 15, 306-310. [9] Gundlach, D. J., Nichols, J. A., Zhou, L. & Jackson, T. N. (2001). Appl. Phys. Lett. 2002, 80, 2925. b) I. Kymissis, C. D. Dimitrakopoulos, S. Purushothaman, IEEE Trans. Electron Devices, 48, 1060. [10] Horowitz, G. (2004). J. Mater. Res., 19, 1946. [11] Katz, H. E. (2004). Chem. Mater. 2004, 16 (23), 4748. b) M. M. Ling, Z. Bao, Z. Chem. Mater, 16 (23), 4824. [12] Sirringhaus, P. J., Brown, R. H., Friend, M. M., Nielsen, K., Bechgaard, B. M. W., Langeveld-Voss, A. J. H., Spiering, R. A. J., Janssen, E. W., Meijer, P. & Herwig, D. M. (1999). de Leeuw, Nature, 401, 685. [13] Mena-Osteritz, E., Meyer, A., Langeveld-Voss, B. M. W., Janssen, R. A. J., Meijer, E. W. & Bäuerle, P. (2000). Angew. Chem. Int. Ed, 39, 2680 [14] Ihn, K. J., Moulton, J. & Smith, P. (2004). J. Polym. Sci. Part B: Polym. Phys. 1993, 31, 735. b) M. Mas-Torrent, D. Den Boer, M. Durkut, P. Hadley, A. P. H. J. Schenning,. Nanotechnology, 15, S265. [15] Merlo, J. A. & Frisbie, C. D. (2003). J. Phys. Chem. B 2004, 108, 19169; J. A. Merlo, C. D. Frisbie, J. Polym. Sci. Part E: Polym. Phys., 41, 2674. [16] Ph. Leclère, E., Hennebicq, A., Calderone, P., Brocorens, A. C., Grimsdale, K., Müllen, J. L. & Brédas, R. (2003). Lazzaroni, Prog. Polym. Sci., 28, 55. [17] Dimitrakopoulos, C. D., Mascaro, D. J. IBM J. Res. & Develop, 2001, 45, 11. [18] Paloheimo, J., Kulvalainen, P., Stubb, H., Vuorimaa, E., Yli-Lahti, P. Appl. Phys. Lett. 1990, 56, 1157. [19] Matsui, J., Yoshida, S., Mikayama, T., Auki, A. & Miyashita, T. Langmuir, 2005, 21, 5343. [20] Xu, G., Yu, W., Xu, Y., Xu, G., Cui, D., Zhang, Y. & Liu, D. Zhu, Langmuir, 2005, 21, 5391. [21] Podzorov, V., Sysoev, S. E., Loginova, E., Pudalov, V. M. & Gershenson, M. E. (2003). Appl. Phys. Lett, 2003, 83, 3504. b) Sundar, V. C., Zaumseil, J., Podzorov, V., Menard, E., Willett, R. L., Someya, T., Gershenson, M. E. & Rogers, J. A. (2004). Science, 303, 1644. De Boer, R. W. I., Klapwijk, T. M. & Morpurgo, A. F. (2003). Appl. Phys. Lett, 83, 4345. Ichikawa, M., Yanagi, H., Shimizu, Y., Hotta, S., Suganuma, N., Koyama, T. & Taniguchi, Y. (2002). Adv. Mater, 14, 1272. [22] Herwig, P. T. & Müllen, K. (1999). Adv. Mater. 1999, 11, 480. (b) Gelink, G. H., A. Huitema, H. E., van Veenendaal, E., Cantatore, E., Schrijnemakers, L., van der Putten, J. B. P., Geuns, T. C. T., Beenhakkers, M., Giesbers, J. B., Huisman, B. H., Meijer, E. J., Mena Benito, E., Touwslager, F. J., Marsman, A. W., van Rens, B. J. E. & de Leeuw, D. M. (2002). Nature Materials 2004, 3, 106.A. Afzali, C. D. Dimitrakopoulos, T. L. Breen, J. Am. Chem. Soc., 124, 8812. [23] Mas-Torrent, M., Durkut, M., Hadley, P., Ribas, X. & Rovira, C. (2004). J. Am. Chem. Soc. 126, 984. b) Mas-Torrent, M., Hadley, P., Bromley, S. T., Ribas, X., Tarrés, J., Mas, M., Molins, E., Veciana, J. & Rovira, C. (2004). J. Am. Chem. Soc., 126, 8546.

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[24] Ferraris, J., Cowan, D. O., Walatka, V. V. & Perlstein, J. H. (1973). J. Am. Chem. Soc., 95, 948. [25] See review papers in the issue Chem. Rev., 2004, 104. [26] M. Mas-Torrent, C. Rovira, J. Mater. Chem. in press. [27] R. A. Marcus, Rev. Mod. Phys. 1993, 65, 599. b) N. E. Gruhn, D. A. da Silva Filho, T. G. Bill, M. Malagoli, V. Coropceanu, A. Kahn, J.-L. Brédas,. J. Am. Chem. Soc. 2002, 124, 7918. [28] M. Malagoli, J.-L. Brédas, Chem. Phys. Lett. 2000, 327, 13. [29] H. M. Meng, M. Bendikov, G. Mitchell, R. Helgeson, F. Wudl, Z. Bao, T. Siegrist, C. Kloc, C.-H Chen, Adv. Mater. 2003, 15, 1090. b) J. Cornil, J. Ph. Calbert, J.-L Brédas, J. Am. Chem. Soc. 2001, 123, 1250. [30] D: Becke, J. Phys. Chem. 1993, 98, 5648 [31] M. J. Frisch, et. al., Gaussian 98, Revision A.9, Gaussian, Inc.: Pittsburgh PA 1998. [32] Demiralp, W. A. Goddard III, J. Phys. Chem. A 1997, 101, 8128. [33] J. L. Brédas, J. P. Calbert, D. A. da Silvo Filho, J. Cornil, J. Proc. Natl. Acad. Sci. 2002, 99, 5804. [34] S. T. Bromley, M. Mas-Torrent, P. Hadley, C. Rovira, J. Am. Chem. Soc. 2004, 126, 6544. M. Mas-Torrent, P. Hadley, S. T. Bromley, N. Crivillers, J. Veciana, C. Rovira, Appl. Phys. Lett. 2005, 86, 012110. Iizuka, M., Shiratori, Y., Kuniyoshi, S., Kudo, K. & Tanaka, K. (1998). Appl. Surf. Sci. 130-132, 914. b) Noda, B., Katsuhara, M., Aoyagi, I., Mori, T. & Taguchi, T. (2005). Chem. Lett., 34, 392. c) Katsuhara, M., Aoyagi, I., Nakajima, H., Mori, T., Kambayashi, T., Ofuji, M., Takanishi, Y., Ishikawa, K., Takezoe, H. & Hosono, H. (2005). Synthetic Metals, 149, 219. d) Naraso, J. I., Nishida, S., Ando, J., Yamaguchi, K., Itaka, H., Koinuma, H., Tada, S., Tokito, & Yamashita, Y. (2005). J. Am. Chem. Soc, 127,10142

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INDEX

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A abstraction, 98, 106 acetone, 141 acetonitrile, 139 adaptation, 151 adhesion, 136 adjustment, 15, 109 AFM, 182 aggregates, ix, 134, 169 aggregation, 148 alcohol, 139 alternatives, vii, 1, 67 aluminum, 11, 79 aluminum oxide, 79 amines, 145 ammonia, 6, 18 ammonium, 77 amplitude, 105, 138, 151, 157, 159 annealing, 11, 18, 139 annihilation, ix, 133, 151, 153, 155, 157 aromatic compounds, 161 aromatic rings, 169 arsenic, 75 assumptions, viii, 35, 155 asymmetry, 147, 149 atomic force, 17 atoms, 4, 5, 6, 37 authors, 17 automation, 116 availability, 98 averaging, 110, 158

B background, 91 bandgap, viii, 16, 17, 18, 23, 32, 36, 41, 51, 63, 176 bandwidth, 99, 137, 142 barriers, 167

behavior, 3, 9, 16, 17, 18, 23, 26, 28, 29, 38, 41, 65, 75, 98, 105, 109, 114, 123, 128, 145, 153, 157, 184 Belgium, 172 bending, 135, 136, 158, 170, 177, 178 benzene, 137, 141, 169 bias, 17, 24, 29, 53, 55, 86, 87, 91, 103, 104, 108, 145, 146, 149, 161 binding, 22, 23, 37 biological systems, 165 biosensors, 16, 84, 85 blocks, viii, 66, 73, 124 Boltzmann constant, 21 bonding, 137, 168, 186 bonds, 7, 18, 137, 139, 179 breakdown, 122 building blocks, 74, 82, 89, 183 bulk materials, viii, 35 burning, 22

C candidates, 2, 30, 38, 144 carbon, vii, viii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 17, 18, 20, 22, 24, 28, 29, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 42, 45, 48, 49, 50, 51, 53, 67, 79, 85 carbon atoms, 2, 4, 7, 36, 41 carbon nanotubes, vii, viii, 2, 3, 4, 5, 6, 7, 8, 12, 14, 17, 20, 22, 29, 30, 31, 32, 35, 36, 38, 39, 40, 41, 42, 45, 48, 49, 53, 67, 79, 85 carboxylic groups, 18 Caribbean, 117, 118 carrier, 8, 14, 22, 23, 75, 78, 79, 113, 134, 135, 183, 184, 187 cartoon, 164 casting, 182, 183, 184 catalyst, 24, 83 cation, 147, 149, 150, 151, 152, 158, 166 cell, ix, 16, 24, 37, 51, 88, 89, 99, 100, 103, 104, 105, 109, 114, 133, 146, 148, 149 channels, vii, 1, 2, 4, 8, 14, 16, 17, 20, 24, 30, 62, 76, 139, 160 chaos, 98, 105, 106, 114, 116, 117, 118

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Index

chaotic behavior, 98, 102, 106, 107 chaotic masking, 113, 114 charge density, 23, 48, 59 charge trapping, 134, 143 chemical bonds, 4 chemical reactions, 76 chemical vapor deposition, 24 chemisorption, 136 China, 73 chirality, 5, 17, 19, 20, 29, 30, 55, 67 City, 133 classification, viii, 7, 35, 36 cleaning, 159 communication, 98, 117, 118 communication systems, 118 community, vii, 1, 4, 24, 98 compatibility, 84 complementarity, 121, 122, 128 complexity, 102, 138 components, viii, 26, 30, 35, 36, 55, 89, 143, 151, 160 composition, 74, 82, 87, 88, 91 compounds, 2, 8, 77, 137, 180, 182, 183 computation, 23, 63 computer technology, 49 computing, 23, 73 concentration, 77, 144, 153, 159 conductance, 2, 8, 9, 10, 16, 17, 21, 61, 64, 75, 84, 85, 144, 149 conduction, 8, 9, 11, 22, 29, 54, 85, 138, 176, 181, 187 conductivity, viii, ix, 8, 9, 29, 36, 133, 135, 139, 146, 160, 161, 162, 166, 169, 177, 179, 180, 184 conductor, 36, 54, 120, 177 conductors, 176, 183, 187 configuration, 100, 142, 144, 151, 155, 157, 169, 175, 180, 184 confinement, 42, 74, 80 connectivity, 150, 169 conservation, 149 construction, 29, 98, 114 consumption, vii, 1, 2, 54, 66, 108 control, 3, 8, 11, 14, 17, 24, 29, 91, 108, 110, 136, 137, 150, 159, 160, 163, 166, 170, 180, 187 conversion, 48, 88 copper, 8, 30, 135, 169 correlation, 52, 114, 138, 161, 183, 184, 186 correlation function, 52 corrosion, 136 costs, ix, 73, 175, 181 couples, 23 coupling, 45, 99, 108, 109, 110, 112, 114, 134, 143, 181, 185, 186 covalent bond, 137 covalent bonding, 137 crystal structure, 183, 184, 185, 186, 187 crystalline, 2, 4, 7, 9, 134, 144, 179, 180 crystallinity, 83, 181 crystallisation, 76, 183

crystals, 6, 79, 138, 143, 161, 163, 179, 182, 183, 184, 186 current ratio, ix, 55, 56, 175 CVD, 12, 14, 24, 26, 78, 80, 82, 86 cyclohexanone, 181

D decay, 86, 143, 149, 158, 159, 160 decomposition, 83 decomposition temperature, 83 defects, 7, 18, 26, 30, 134, 135, 138, 169 definition, 109 degradation, 36, 102 delivery, ix, 133, 150, 165 density, viii, 2, 8, 9, 14, 15, 17, 24, 25, 35, 43, 45, 46, 48, 49, 50, 51, 52, 57, 58, 59, 63, 86, 135, 136, 138, 142, 148, 158, 160, 164, 176, 185 density functional theory, viii, 48 depolarization, 158 deposition, ix, 12, 75, 76, 81, 82, 83, 90, 91, 137, 139, 142, 175, 180, 181 derivatives, 135, 137, 180, 183, 184, 185, 186, 187 desorption, 141, 165 destruction, 105 detection, 18, 84, 85, 86, 89 deviation, 157 DFT, viii, 35, 36, 48, 50 dielectric constant, 48, 75, 145 dielectrics, vii, 1, 2, 12, 13, 32 differential equations, 154 diffusion, 151, 161, 162 diffusion time, 151 dimensionality, 74, 136 diodes, ix, 66, 86, 119, 120, 123, 124, 125, 129, 131, 136, 145, 170 direct measure, 151 direct observation, 136, 170 discharges, 157 discrimination, 168 disorder, 134, 138, 165, 180 dispersion, viii, 35, 37, 41, 42 displacement, 151, 152 distilled water, 81 distribution, 14, 15, 58, 59, 76, 108, 114, 155, 167, 176 distribution function, 58, 59 divergence, 105, 114 donors, 183 dopants, 75 doping, ix, 8, 11, 18, 19, 22, 24, 29, 30, 54, 55, 74, 75, 84, 91, 133, 144, 145, 160, 161, 166 drawing, 176 duration, 142, 153, 158 dynamical systems, 98, 100, 105, 114, 131

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E Education, 35, 92 electric current, 149, 153, 161 electric field, 3, 9, 146, 152, 159, 161, 162, 175, 177, 180 electrical breakdown, 14, 15, 29, 32 electrical conductivity, 84 electrical properties, 2, 4, 8, 14, 16, 26, 30, 74 electricity, 169 electrochemical deposition, 76 electrochemistry, 152 electrodeposition, 83 electrodes, 3, 9, 12, 16, 17, 26, 31, 52, 54, 58, 74, 91, 135, 139, 140, 151, 160, 161, 167, 168, 169, 180, 181, 184 electroluminescence, 87 electrolyte, 55 electromagnetic, 48 electromigration, viii, 9, 35 electron, 8, 9, 11, 29, 36, 38, 42, 43, 44, 46, 47, 48, 49, 51, 54, 57, 58, 59, 63, 74, 79, 134, 135, 137, 143, 144, 149, 150, 159, 161, 169, 170, 176, 177, 178, 180, 181, 183, 185 electron beam lithography, 74 electron state, 176, 177, 178 electronic circuits, vii, 175 electronic structure, 183 electron-phonon coupling, 134 electrons, viii, 2, 16, 36, 38, 42, 43, 44, 45, 46, 47, 48, 49, 55, 58, 62, 63, 135, 142, 147, 149, 158, 169, 176, 187 email, 171 emission, 87, 88, 137 encoding, 113 encryption, 109, 114 energy, viii, 2, 16, 22, 36, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 57, 60, 88, 111, 135, 142, 149, 150, 152, 158, 159, 167, 176, 177, 185, 186, 187 energy density, 176 energy transfer, 149 environment, 84, 135, 159, 186 epitaxial growth, 137 equilibrium, 23, 46, 47, 102, 105, 110, 111, 126, 127, 149 estimating, 109 etching, 26 ethanol, 85, 139 ethylene, 76 ethylene glycol, 76 Euro, 69, 70, 71 evaporation, 77, 180, 186 evolution, vii, 1, 102 excitation, 86, 87, 158 experimental condition, 87 exposure, 12, 84, 85

193

F fabrication, 2, 3, 9, 14, 19, 26, 29, 30, 82, 88, 89, 134, 135, 136, 137, 138, 139, 140, 141, 160, 180 failure, 14, 30 family, 4, 27, 183, 184 feedback, 99, 102 Fermi level, 11, 21, 167, 177 fibers, 181 film formation, 181 films, 30, 134, 139, 140, 142, 149, 150, 159, 160, 162, 165, 166, 179, 181, 182, 183 filters, 158 fine tuning, 87 flexibility, ix, 175 flight, 161 fluctuations, 14, 105, 176, 179 fluorescence, 159 focusing, 2, 91 foils, 30 fragments, 17 France, 131 free energy, 149, 150, 159 freedom, 138 functionalization, 85, 136

G gallium, 86 gases, 79, 81, 84, 85 generation, 18, 102, 109, 114, 137, 161 germanium, 2, 36, 122 Germany, 31 goals, 151 gold, 184 grain boundaries, 134, 135, 182 graph, 120 graphene sheet, 5, 30, 38 graphite, 4, 169 groups, 5, 6, 9, 20, 22, 36, 84, 135, 136, 137, 139, 141, 142, 157, 158, 160, 183, 184 growth, 3, 25, 26, 29, 30, 33, 74, 75, 77, 78, 79, 81, 82, 83, 91, 139, 141, 150 growth mechanism, 75, 91 growth rate, 75 guidelines, viii, ix, 97

H Hamiltonian, ix, 23, 48, 97, 98, 110, 114, 117 harvesting, 148, 149, 150, 163, 164, 165, 166, 167 heat, 2, 8, 141, 165 heating, 54, 182 hemoglobin, 85 heterogeneity, 159 host, 74, 134

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HRTEM, 78, 80, 82, 83 hybrid, 3, 30, 136, 139, 146, 148, 170 hybridization, 169 hydrogen, 6, 18, 183, 186 hydrogen bonds, 183 hydrothermal process, 81 hydroxyl, 6, 18 hydroxyl groups, 6, 18 hyperfine interaction, 169 hypothesis, 165 hysteresis, 168

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I ideal, 8, 18, 22, 80, 121, 122, 123, 125, 128, 159 identity, 110 illumination, 84, 86, 88, 148, 151, 152, 153, 159, 161, 162, 165, 166, 168 image, 5, 27, 41, 76, 77, 78, 79, 81, 82, 86, 87, 91, 142, 182, 184 images, 77, 78, 80, 81, 82, 83, 87, 88, 89, 91, 149 implementation, vii, 36, 98, 99, 106, 107 impurities, 18, 75, 138, 158 indices, viii, 35, 38, 39, 40, 41, 53 indium, 77, 87, 90, 142, 143, 152, 158 industry, vii, 1, 24, 30, 73 inequality, 106, 119, 127, 129, 131, 132 initiation, 143, 163 insight, 134, 168, 170 instability, 12, 105 insulation, 12 insulators, 2, 176 integrated circuits, 3, 8, 14, 28 integration, ix, 2, 14, 17, 24, 50, 52, 59, 92, 97, 98, 99, 107, 108 integrity, 108, 140 interaction, 22, 23, 109, 136, 165 interactions, 7, 23, 48, 53, 84, 176, 183 interface, 36, 82, 84, 135, 136, 138, 146, 152, 160, 161, 166, 167, 169, 170, 175, 177, 180, 182 intermolecular interactions, 181, 186 inversion, 26, 177 investment, 4 IR spectra, 141 irradiation, 79 isolation, 11, 17, 28 Israel, 172 I-V curves, 10, 84, 85, 88, 89, 144

K kinetic studies, 134, 168 kinetics, ix, 133, 135, 149, 151, 157, 160, 168, 170

L laser ablation, 78, 80 lasers, 87, 88 lattices, 24 laws, 126, 127 leakage, vii, 1, 2, 8, 12, 15, 17, 21, 22, 54, 180 lifetime, 153 ligand, 76 light beam, 151 light-emitting diodes, viii, 73, 74, 84, 86 limitation, 169 line, 27, 66, 91, 148, 157, 158, 178, 186 lithography, 6, 9, 11, 12, 16, 17, 180, 181 low temperatures, 169 LTD, 132 Lyapunov function, 110 lying, 184

M management, 8 manufacturing, ix, 175, 181 matrix, 23, 89, 90, 111, 112, 119, 141, 154, 156, 161, 162, 165, 181 measurement, 10, 67, 139, 143, 146, 148, 151, 153, 155, 163, 164 measures, 120, 151, 157 media, 138 melt, 77 melting, 79 membranes, 76 memory, 29, 55 metal organic chemical vapor deposition, 75 metal oxides, 80, 84 metal-oxide-semiconductor, vii, 1, 33 metals, 3, 74, 81, 84, 139 microelectronics, ix, 175 micrometer, 152 microscope, 85, 184 microscopy, 17, 87 microstructure, 180 microstructures, 181 migration, 149, 150 miniaturization, 36, 73 minority, 151, 158 mixing, 113 mobility, ix, 2, 8, 9, 14, 19, 55, 75, 79, 91, 133, 134, 135, 137, 138, 143, 144, 145, 146, 161, 163, 165, 166, 168, 169, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187 model, 20, 21, 22, 23, 37, 43, 57, 59, 60, 61, 62, 63, 65, 74, 114, 121, 122, 123, 124, 128, 129, 137, 141, 147, 149, 151, 160, 162, 167, 169, 178 model system, 74, 151 modeling, vii, 1, 3, 20, 30, 32, 55, 117, 118

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Index models, ix, 18, 20, 21, 22, 23, 30, 48, 56, 63, 65, 78, 114, 119, 120, 124, 129, 130, 167 molecular structure, 134, 180 molecular weight, 179 molecules, 84, 85, 135, 136, 137, 143, 144, 145, 146, 147, 149, 150, 151, 152, 158, 159, 160, 161, 164, 165, 168, 176, 179, 180, 181, 182, 183, 184, 185, 186 molybdenum, 12 momentum, 43 monolayer, 3, 4, 7, 17, 26, 29, 30, 136, 141, 146, 150, 160, 167, 180 Monte Carlo method, 56 morphology, 91, 137, 181 motion, 36, 43, 54, 138, 167, 176 motivation, 135 multiples, 107 multiplication, 149

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N NAD, 149 nanobelts, 74, 77, 80, 81, 82, 84 nanodevices, ix, 134 nanoelectronics, 2 nanofibers, 181, 182 nanomaterials, vii, 1, 2, 3, 4, 7, 8, 24, 28, 29, 30, 82, 84 nanometer, 2 nanometers, 5, 79 nanoparticles, 77, 151 nanophotonics, 86 nanoribbons, vii, 1, 3, 6, 16, 17, 31, 32 nanorods, 74, 77, 78 nanostructures, viii, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92 nanosystems, 83 nanotube, viii, 2, 5, 6, 9, 12, 14, 24, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 45, 48, 49, 50, 51, 53, 67, 82 nanowires, 42, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 94 naphthalene, 135, 137, 144 neglect, 153, 155 Netherlands, 175 network, 21, 25, 26, 84, 150, 159 neural network, 61 neural networks, 61 next generation, 89 nitrogen, 18, 166 nodes, 2 noise, 28, 55, 114 nonlinear systems, 98, 105

O objectives, 114

195

observations, 27 one dimension, 24 operator, 48, 49, 99 Operators, 102 optical properties, 159 optimization, 54, 102, 114, 168 optoelectronics, viii, 73, 74, 86, 94 order, vii, 1, 3, 8, 11, 16, 18, 24, 28, 36, 43, 48, 53, 54, 56, 106, 134, 136, 138, 143, 144, 151, 155, 159, 165, 168, 179, 180, 181, 186, 187 ordinary differential equations, 98 organic compounds, 169 organic polymers, 136 organic solvents, 180, 182 orientation, 7, 16, 17, 25 oscillation, 24, 109 oxidation, 12, 76, 149, 150 oxide thickness, 30, 55 oxides, 22, 81, 140 oxygen, 6, 18, 135, 166

P parameter, 64, 105, 107, 108, 114, 148, 162, 178, 186 parameters, 49, 50, 52, 60, 61, 62, 63, 74, 78, 91, 105, 106, 107, 108, 109, 110, 113, 114, 136, 142, 151, 157, 168, 180, 182, 185, 187 parents, 182 particles, 38, 49, 77 pathways, 26 percolation, 165 permit, 180 permittivity, 145 perylene, 135, 161 PET, 74 pH, 89 phase transitions, 168 photoemission, 135, 170 photolithography, 74 photoluminescence, 86, 88 photonics, 87 photons, 38, 86, 142 photooxidation, 149, 167 photoresponse, 148 photosynthesis, 149 photovoltaic cells, 136, 139, 148, 160 physical properties, viii, 7, 8, 30, 35, 36, 50, 74 physicochemical properties, 136 physics, 17, 20, 22, 74, 138, 160, 171 Poincaré, 105 Poisson equation, 23 polarity, 149, 152, 158, 161 polarization, 85, 138, 158, 161 polyimide, 25, 26 polymer, 76, 181 polymers, 74, 179, 180, 181 poor, 177

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Index

potassium, 11 power, vii, 1, 2, 3, 8, 12, 22, 26, 28, 54, 66, 73, 86, 88, 108, 121, 146, 147, 150 prediction, 17 pressure, 79, 82, 181 probability, 26, 43, 57, 58, 134 probability density function, 57 probability distribution, 43 probe, 136, 143, 146, 151, 152, 153, 154, 155, 157, 158, 160, 170 production, 55 program, 115 proportionality, 159 prostate, 85 protocol, 137, 146 pruning, 15 pulse, 142, 153, 161 purification, 77 purity, 78, 181, 182 PVC, 139, 146

Q quality control, 151 quantization, 41, 79 quantum dot, 32 quantum dots, 32 quantum mechanics, 53 quantum state, 22 quartz, 14, 24, 25, 78

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R radar, 114 radiation, 76 radio, 66 radius, 75 random walk, 150, 162, 164 range, viii, 24, 26, 39, 43, 73, 74, 79, 97, 102, 105, 108, 136, 141, 145, 149, 153, 155, 157, 159, 160, 161, 165, 179, 181, 183 reaction center, 151 reaction temperature, 76, 77, 78 reading, 24 reality, 16 reason, 23, 36, 161, 181, 182, 186 reasoning, 186 recall, 120 recognition, 170 recombination, 134, 149, 150, 151, 157, 159, 160, 161, 162 recombination processes, 161 recovery, 158, 159 rectification, 87 red shift, 87 redistribution, 153, 157 reflection, 43, 170

region, 9, 15, 21, 22, 26, 36, 45, 46, 47, 48, 53, 54, 55, 56, 59, 90, 91, 103, 105, 107, 114, 141, 145, 148, 150, 151, 177 relationship, 4, 37, 48, 55, 62, 63, 64, 74, 91, 168, 183 relaxation, 22, 23, 151, 153, 185 reliability, 8, 12, 14, 22 resistance, 10, 21, 22, 24, 27, 28, 36, 38, 53, 54, 56, 63, 100, 103, 104, 108, 152, 153, 157, 158, 159, 165, 180 resolution, 73, 86, 149, 165 response time, 86 rings, 2, 4, 137 robustness, 8, 9, 170 rolling, 5, 38, 41 room temperature, vii, 1, 2, 6, 7, 8, 9, 17, 137, 138, 147, 161, 169, 184, 185 Royal Society, 81

S salts, 183 sampling, 50, 52, 158 sapphire, 158 saturation, ix, 14, 18, 27, 53, 56, 62, 97, 102, 103, 104, 105, 114, 142, 144, 145, 178 savings, 2 scaling, vii, 1, 31 scattering, 8, 22, 23, 43, 54, 55, 56, 62, 63, 64 schema, 88, 89 Schottky barriers, 22, 59 search, 134, 183, 186 searching, 187 second generation, 100 secure communication, viii, ix, 97, 98, 109 self-assembly, ix, 133, 137, 140, 146, 169, 170 self-ordering, 136 semiconductor, vii, viii, 1, 9, 16, 19, 24, 35, 36, 41, 50, 53, 67, 73, 74, 79, 122, 134, 135, 137, 143, 147, 150, 151, 153, 158, 159, 166, 168, 169, 175, 176, 177, 178, 180, 181, 183, 186 semiconductors, ix, 74, 81, 134, 135, 136, 137, 138, 143, 175, 176, 178, 179, 180, 181, 182, 183, 187 sensing, 84, 85, 87, 136 sensitivity, 84, 85, 86, 107, 159, 164, 168 sensors, viii, 73, 74, 84, 85, 86, 88, 169 separation, 143, 149, 152, 159 shape, 12, 31, 41, 109 shortage, 135 Si3N4, 81, 82, 140 signals, 17, 27, 99, 103, 110, 150, 158 silane, 83 silica, 180 silicon, vii, viii, ix, 1, 2, 3, 8, 9, 10, 11, 12, 14, 17, 21, 24, 25, 26, 28, 30, 36, 73, 74, 82, 84, 88, 89, 122, 123, 134, 137, 138, 139, 140, 141, 146, 147, 149, 150, 151, 152, 157, 158, 159, 160, 162, 163, 165, 167, 169, 175, 178, 179, 180, 184

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Index silver, 139, 146, 147, 148, 149, 150 simulation, viii, ix, 23, 33, 35, 49, 50, 52, 56, 60, 61, 65, 67, 97, 117 single crystals, 79, 183 single walled carbon nanotubes, 9 SiO2, 9, 12, 17, 26, 31, 54, 75, 83, 139, 141, 142, 145, 158, 160, 184 skin, 169 smoothness, 30 soccer, 4 software, viii, 36, 63, 109 SOI, 55, 69 solar cells, viii, ix, 73, 74, 84, 88, 89, 175 solid state, 151 solubility, 180, 182 solvents, 181, 183 space, 20, 23, 33, 41, 51, 105 Spain, 115, 175 spectroscopy, 136, 141, 151, 159, 160, 170 spectrum, 43, 48, 52, 53, 65, 66, 87, 91, 105, 141, 147, 148 speed, 18, 20, 24, 28, 56, 149, 150, 161, 162, 168 spin, 139, 169 stability, 8, 89, 105, 110, 111, 112, 130, 135, 136, 168, 182 standard deviation, 15, 187 steel, 7 storage, 91 strain, 2, 26 strategies, 73, 75, 134 strength, 3, 7, 8, 22, 109, 168 stress, 114 strong interaction, 138 structural defects, 138 substrates, ix, 24, 79, 89, 91, 139, 140, 141, 160, 175 subtraction, ix, 97, 98, 110 Sun, 31, 93, 95, 187 supply, 22, 26, 103 surface chemistry, 138, 161 surface modification, 136 surface properties, 136 surface treatment, 180 surfactant, 76, 79, 81 suspensions, 159 switching, 8, 18, 26, 38, 55, 56, 91, 121, 160, 162, 166 symmetry, 5, 37, 75, 102, 108, 137, 138, 186 synchronization, ix, 97, 98, 109, 110, 113, 114, 117 synthesis, 6, 29, 30, 76, 83, 91, 115, 117, 136 system analysis, 106

T temperature, ix, 8, 11, 21, 50, 52, 57, 79, 81, 82, 84, 138, 161, 175, 176, 181 terminals, 22, 100, 175 thermal treatment, 83 thin films, 12, 134, 182

197

threshold, 11, 12, 14, 15, 55, 64, 88, 91, 178 time resolution, 159 tin, 143 tin oxide, 143 titanium, 12 topology, 100, 101, 104, 108 TPA, 87, 88 tracking, 100 trajectory, 105 transformation, 11 transistor, vii, ix, 1, 2, 3, 4, 11, 14, 17, 18, 20, 23, 24, 25, 26, 27, 28, 30, 31, 33, 74, 91, 97, 98, 104, 114, 122, 123, 124, 127, 128, 138, 175, 177, 178 transition, 2, 11, 79, 134 transition metal, 79 transmission, 22, 43, 45, 48, 52, 53, 58, 59, 79, 91 transmission electron microscopy, 5, 77, 79, 80, 83 transparency, 91 transport, viii, ix, 8, 20, 21, 23, 26, 33, 35, 36, 38, 51, 52, 56, 58, 74, 91, 133, 134, 135, 136, 137, 138, 139, 145, 146, 147, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 180, 181, 183, 185, 186 tungsten, 24 tunneling, 12, 17, 21, 22, 55, 141, 145, 162, 167, 169 Turkey, 35

U uncertainty, 150 uniform, 30, 76, 77, 80, 81, 82, 152, 183 unique features, 85 UV irradiation, 83, 86 UV spectrum, 141

V vacuum, 11, 18, 84, 135, 137, 177, 181, 183, 186, 187 valence, 2, 29, 147, 149, 150, 167, 176, 178 vanadium, 79 vapor, 74, 75, 77, 79, 81, 82, 181, 182 vapor-liquid-solid, 75, 88 vapor-solid (VS), 75 variables, 47, 82, 98, 102, 105, 180 variance, 15 vector, 5, 9, 38, 39, 41, 44, 110, 111, 119 velocity, 38, 43, 62, 162 versatility, 99 VLS, 75, 83 VSD, 175, 176, 178, 185

W wave number, 42 wave vector, 38, 41, 43, 58

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198

Index XRD, 170

weak interaction, 109 web, 115 wires, 24 working groups, 2

Z X

zinc, 77 ZnO, 77, 78, 81, 83, 84, 85, 86, 87, 88, 89 ZnO nanorods, 77, 78

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X-ray photoelectron spectroscopy (XPS), 136, 141, 170

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