Topics in High Field Transport in Semiconductors

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Cover Illustration: First Brillouin zone in GaAs showing electron distribution

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constant electric field strength of 400 kV/cm. Each simulated electron is represented by a single blue dot. by Michael

Weber.

TOPICS IN HIGH FIELD TRANSPORT IN SEMICONDUCTORS

SELECTED TOPICS IN ELECTRONICS AND SYSTEMS Editor-in-Chief: M. S. Shur

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Selected Topics in Electronics and Systems - Vol. 22

TOPICS IN HIGH FIELD TRANSPORT IN SEMICONDUCTORS

Editors

Kevin F Brennan Georgia Institute of Technology, USA

P Paul Ruden University of Minnesota, USA

V f e World Scientific wb

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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TOPICS IN HIGH FIELD TRANSPORT IN SEMICONDUCTORS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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FOREWORD

The dawn of the third millennium coincides with what has often been referred to as the information age. The rapid exchange of information in various formats has become one of the most important activities of contemporary society. Shannon's early recognition that information in its most basic form is reduced to a series of bits has led to a vast infrastructure devoted to the rapid and efficient transfer of information in bit form. The technical developments that make this infrastructure possible result from a blending of computing and telecommunications. Basic to these industries is the semiconductor hardware that provides the essential tools for information processing, transfer, and display. The remarkable advances in computing technology stem mainly from the continued miniaturization of semiconductor based integrated circuits. Each new generation of integrated circuits provides higher performance at lower cost, thus enabling the rapid growth in computing capability witnessed over the last decades. Modern computers can process vast amounts of information in relatively short periods of time. Concurrent developments in telecommunications, mainly in fiber optic and wireless telephony, have enabled the rapid transfer of digital information. The expansive growth in the internet in recent years is but one example. Much of the improvement in telecommunications has arisen from advances in optoelectronic and high frequency amplifier semiconductor devices. Finally, advances have also occurred in the manner in which information is displayed. New technologies, such as flat panel displays, have been developed to aid the user in the interpretation and understanding of the transferred information. As semiconductor devices are scaled to smaller and smaller dimensions, denser, faster, and cheaper integrated circuits are produced. This development fuels the explosive growth in computing and information processing capability. In telecommunications, device limits are pushed to support faster bit rates and to operate at higher frequencies and power. In both instances, semiconductor devices are required to operate at relatively high electric field strengths, posing new challenges and problems for device design. The small dimensions and relatively high electric field strengths in modern devices call into question some of the standard assumptions employed in semiconductor device simulators. For very high power and high frequency performance, new semiconductor materials with larger energy band gaps than silicon or gallium arsenide must be employed to avoid device breakdown. Finally, the enormous range of applications of semiconductor technology calls for new, unconventional semiconductor materials that may be amenable to low-temperature, i.e. low-cost, device processing.

vi

Foreword

It is the purpose of this special issue of the International Journal of High Speed Electronics and Systems to examine some of the charge carrier transport issues encountered in modern semiconductor devices and novel materials. The special issue has also been reprinted as a book. Theoretical approaches to understand and model the relevant phenomena are described in eight papers written by leading experts and pioneers in this field. The first four papers of this volume are devoted to the theoretical issues that arise in devices with very small spatial dimensions. The papers by Ferry et al and Jacoboni et al. focus mainly on issues related to quantum transport, which is expected to dominate charge carrier transport in devices with dimensions approaching the de Broglie wavelength of the electron. As discussed in the first paper by Ferry et al, "there is no present approach to the quantum regime that has proved effective." The first paper provides a discussion of the problems and some of the techniques that have been advanced to address quantum transport in ultra-small devices. The second paper by Jacoboni et al. elaborates on some of the techniques introduced in Ferry et al. The paper by Jacoboni et al. examines the use of Wigner functions in quantum transport. In their paper, these authors review the state-of-the-art in this field and show how Monte Carlo simulation can be used in the context of Wigner functions. The third and fourth papers by Reynolds et al. and by Register and Fisher, focus principally on high field effects in semiconductors and their theoretical treatment. In the chapter by Reynolds et al, the influence of local inhomogeneities on the electric field dependent properties of Bloch states is examined for single-band dynamics. The article by Register and Fisher examines how collision broadening affects high field transport dynamics particularly the impact ionization threshold energy and large potential barriers. The next three papers by Ridley and Zakhleniuk, Reigrotzki et al, and Bellotti et al, examine transport issues in wide band gap semiconductors used in high power, high frequency device applications. These materials are better able than the more conventional, but relatively narrow energy band gap, silicon and gallium arsenide to support the very high field strengths encountered in many of the applications envisioned. Ridley and Zakhleniuk investigate transport in a two-dimensional electron gas induced by polarization charges at the heterointerface between layers of different Ill-nitride compounds. In their paper, these authors discuss the various scattering mechanisms and their effect on the transport dynamics of the electrons in these heterostructures. The sixth paper by Reigrotzki et al. examines impact ionization and high field effects in wide band gap semiconductors. This paper discusses how high field quantum corrections, such as the intra-collisional field effect, affect the impact ionization process and its calculation. The seventh paper by Bellotti et al. discusses the complexities encountered in simulating the wide band gap semiconductors, which often crystallize with relatively low symmetry lattices. These authors present a discussion of the key ingredients to transport calculations for wide band gap semiconductors, such as the band structure, the scattering mechanisms, and multiple band transport issues.

Foreword

vii

The special issue and book concludes with a paper by Campbell and Smith on electrical transport in organic semiconductors. Organic semiconductor devices are of growing interest for light emitting diodes, of potentially great importance in future displays, they are candidates for low-cost integrated circuits, and for semiconductor circuits on flexible substrates. This paper examines transport issues in materials where hopping conduction, as opposed to band conduction, is dominant. In summary, the special issue of the International Journal of High Speed Electronics and Systems and the reprinted book provide an overview of the transport physics that arises in very small devices and in novel semiconductor materials. The papers included in this volume are mainly theoretical in scope. The volume provides a comprehensive overview that we hope will be beneficial to newcomers as well as to engineers and researchers already engaged in this exciting field. We thank Dr. Michael Shur for inviting us to edit this volume. As editors, we are indebted to all the contributors. Their expertise, dedication, and promptness are greatly valued. We also appreciate the help extended by World Scientific Publishing in the production of this volume. Kevin F. Brennan Professor, School of ECE Georgia Tech Atlanta, GA 30332-0250, USA P. Paul Ruden Professor, Department of ECE University of Minnesota Minneapolis, MN 55455, USA

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Dr. Kevin F. Brennan, Professor, Georgia Tech, Atlanta, GA Kevin F. Brennan is a Professor of Electrical and Computer Engineering at the Georgia Institute of Technology in Atlanta, GA where he has been since 1984. His research interests lie in the general area of semiconductors and microelectronics, with a particular emphasis on the physics and device application of emerging semiconductor materials for future high power, high frequency and photonic detection applications. His research experience includes the design, evaluation and optimization of infrared, optical and ultraviolet photonic detectors and emitters. Dr. Brennan holds a BS in Physics from the Massachusetts Institute of Technology, Cambridge, MA, a MS in Physics and Ph.D. in Electrical Engineering from the University of Illinois at Urbana-Champaign, IL. He has published numerous papers in scientific peer reviewed journals, been granted 5 U.S. patents, and is the author of the book, The Physics of Semiconductors with Applications to Optoelectronic Devices, Cambridge Univ. Press, 1999. Dr. Brennan is a member of the IEEE, American Physical Society, the Optical Society of America, and Sigma Xi.

Dr. P. Paul Ruden, Professor, University of Minnesota, Minneapolis, MN P. Paul Ruden worked on semiconductor superlattices at the Max-Planck-Institute for Solid State Research in Stuttgart, West Germany from 1979 to 1983. He received his Ph.D. in Physics from the University of Stuttgart in 1982. From 1983 to 1985, he was with the Electronics Science and Technology Division, Naval Research Laboratory, Washington, DC, and, concurrently, with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC. In 1985, he joined the Corporate Research Laboratory, Honeywell Inc., Bloomington, MN, where he led research efforts in the areas of semiconductor photodetectors and III-V heterostructure field effect transistor technology. Since 1989, he has been on the faculty of the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, where he is currently Professor of Electrical Engineering. His research focuses on modeling of semiconductor materials and devices. He has published more than 100 papers in technical journals in these fields. Dr. Ruden is a senior member of the IEEE and also holds memberships in the American Physical Society and the Materials Research Society.

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CONTENTS

Foreword K. F. Brennan and P. P. Ruden

v

Quantum Transport in Semiconductor Devices D. K. Ferry, D. Vasileska, and H. L. Grubin

1

Quantum Transport and its Simulation with the Wigner-Function Approach C. Jacoboni, R. Brunetti, P. Bordone, and A. Bertoni

25

Bloch Dynamics in Spatially Local Inhomogeneous Electric Fields J. P. Reynolds, G. J. Iafrate, and J. He

63

Collision Broadening through Sequences of Scattering Events: Theory, Consequences and Modeling within Semiclassical Monte Carlo L. F. Register and B. Fisher Transport in a Polarization-Induced 2D Electron Gas B. K. Ridley and N. A. Zakhleniuk Impact Ionization and High Field Effects in Wide Band Gap Semiconductors M. Reigrotzki, J. R. Madureira, A. Kuligk, N. Fitzer, R. Redmer, S. M. Goodnick, and M. Diir

93

117

149

Simulation of Carrier Transport in Wide Band Gap Semiconductors E. Bellotti, M. Farahmand, M. Goano, E. Ghillino, C. Garetto, G. Ghione, H.-E. Nilsson, K. F. Brennan, and P. P. Ruden

163

Electrical Transport in Organic Semiconductors /. H. Campbell and D. L. Smith

223

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International Journal of High Speed Electronics and Systems, Vol. 11, No. 2 (2001) 363-385 © World Scientific Publishing Company

QUANTUM TRANSPORT IN SEMICONDUCTOR DEVICES D. K. FERRY and D. VASILESKA Department of Electrical Engineering and Center for Solid State Electronics Research Arizona State University, Tempe, AZ 85287-5706 H. L. GRUBIN SRA, Inc.P. O. Box 1058, Glastonbury, CT 06033 It is clear that continued scaling of semiconductor devices will bring us to a regime in which device gate lengths are less than 50 nm within another decade. Pushing to dimensional sizes such as this will probe the transition from classical to quantum transport, and there is no present approach to the quantum regime that has proved effective. Contrary to the classical case in which electrons are negligibly small, the finite extent of the momentum space available to the electron sets size limitations on the minimum wave packet—this is of the order of a few nanometers. While quantum transport formalism has been applied to a variety of problems, in most cases it has not been overly successful. In this paper, we discuss the problems and some of the approximate approaches which will ease the above-mentioned transition.

1.

Introduction

Whether we are talking about the density of integrated circuits, or a single highfrequency device, the important critical dimension (the gate length) continues to decrease each year. For example, 0.18 jam is the state-of-the-art process technology in VLSI, but even smaller dimensions are expected in the near future. Microwave devices have been fabricated with gate lengths as small as 20 nm,1"3 while MOSFETs have been fabricated in the 10-30 nm range.4'5 In these small devices, it is expected that quantum effects will begin to play a significant role, both in confinement of the carriers and in direct sourcedrain tunneling.5,6 As a consequence, it may be expected that future efforts in modeling and simulation of these devices will require the use of quantum transport approaches. Up to now, device modeling and simulation has predominantly been based upon solutions of the Boltzmann transport equation (BTE). This is an ideal semi-classical theory, which has the twin virtues of both conceptual and mathematical simplicity, and it often works far better than anyone could reasonably expect. Quantum transport theory enjoys no such status!7 It is neither conceptually nor mathematically simple, and it often reduces to the semi-classical BTE result based upon the approximations introduced to solve the problem.7 Yet, it is still necessary to understand the quantum approach, both for the ab initio validation of the BTT and for treatment of those intrinsically quantum l

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D. K. Ferry, D. Vasileska & H. L.

Grubin

effects that will occur in small device structures. In spite of the difficulties, quantum theory has been used to effectively model structures such as the resonant-tunneling diode.8"" It is important to note that the appearance of quantum effects in devices has been known for some time. Some of the earliest experimental studies of quantum transport at low temperatures have been in the Si MOS device,12 which ultimately led to a nobel prize. At the same time, there have been a great many investigations of quantum transport, primarily in nearly homogeneous systems, mostly metallic in nature. There also have been many attempts to introduce quantum transport theory to device modeling.14 In the next section, the differences between the BTE and quantum transport will be outlined, and some misconceptions about these differences will be addressed as well. We then turn to the generic non-equilibrium Green's function, and show how it reduces to the BTT, and what the assumptions in this reduction mean for devices. Following this, we introduce the reduced representations of the density matrix and the Wigner distribution function. The latter will be further developed for approximate simulations in the treatment of devices in subsequent sections.

2.

Quantum vs. Classical: Where does the Difference Arise?

In general, the BTT is one in which processes are viewed on a coarse-grained time scale / » Tcoii, where the latter is the collision duration time. In fact, it is much more limiting in that the time scale, over which changes in the distribution function occur, is required to be longer than any of the relevant relaxation times for momentum, energy, etc. In addition, each collision is treated as an irreversible process that is completed prior to the next one and is local in both space and time—e.g., there is no time duration to the scattering process and it occurs at a single point. The collisions are also assumed to be independent of any driving fields. It is often assumed that any deviation from these assumptions requires a quantum transport approach, but this is not correct. In fact, the first assumption listed above, concerning the long-time limit in which the BTE is valid, is a property of the Boltzmann transport equation itself, rather than of classical transport. This lies in the fact that the collision integral in the BTE is Markovian, and neglects the convolution that is important at short times. While such convolution is certainly important in quantum transport, and is emphasized there, general treatments of this can be expressed in classical transport.15 Such non-Markovian behavior is intrinsic in simulations using the ensemble Monte Carlo technique, and several studies have used the correlation functions of the carriers for estimating various relaxation times and distances.17'18 As a result, the factors that separate classical transport from a quantum treatment are the locality of the scattering and the non-interference of driving and relaxation forces. These two effects are not separable and are closely connected. An additional factor, only alluded to in the previous paragraph, is the independence of each of the scattering events. In fact, this may be the most important

2

Quantum

Transport in Semiconductor

Devices

365

difference, as it is the one first violated in quantum treatments, even in the near equilibrium situations. In classical, and semi-classical, treatments of transport, such as BTT or ensemble Monte Carlo approaches, the independence of the scattering events means that the energy continues to be a well-defined function of the carrier momentum. That is, if we define the carrier energy as E = r\co, and the momentum is defined through a function £k(k), then semi-classical treatments assume that the spectral density function (the relation between co and k) is given by A(k,co) = S(r\o)-Ek).

(2.1)

In the quantum mechanical parlance, we say that the energy is defined on the shell determined by £ k . The most common breakdown occurs in heavily-doped systems, where the electron (or hole) density is high. The interactions among the electrons, the many-body interactions, lead immediately to a renormalization of the energy bands (referred to as band-gap narrowing in the electron device world) and dressing of each particle leads to a change in the effective mass. The spectral density A(k,a) is no longer defined by the delta function of (2.1), but is a generalized distribution. As a result, the energy is no longer easily connected to the function £k. Again, in the quantum mechanical parlance, we begin to have off-shell contributions to each of the important interactions. In near equilibrium situations, transport is handled through Green's functions, either the zero-temperature Green's functions used for low temperatures, or the thermal Green's functions used for thermal-equilibrium systems.19'20 Perhaps the most important goal of these treatments is the determination of the spectral density and the modified density-of-states function that results from it. The non-locality of the scattering processes is a little more complicated, in that it arises due to the fact that the collision actually requires a non-zero amount of time to complete. This is different from the non-locality of quantum mechanics itself—a point to which we return later. Here, the non-locality means that the scattering process has an interaction volume and a collision duration time, both of which violate the assumptions basic to BTT. These also allow for multiple collisions to occur simultaneously, but this is not a quantum effect, and can occur in Coulomb scattering within a BTT picture. In fact, the inclusion of carrier-carrier, and carrier-impurity scattering through a real-space Coulomb interaction can be accomplished within the ensemble Monte Carlo approach, and easily incorporates this multiple scattering effect.'8'2'"3 However, the inclusion of the collision duration time, and the non-locality of the scattering, is a fully quantum mechanical effect, although attempts have been made to include them in some ensemble Monte Carlo techniques.24 Why this becomes important can be understood by consideration of Fig. 1, where we show an initial and final state of a phonon absorption process in the presence of a very inhomogeneous conduction band edge—just the type of conduction band edge that one may expect to find in a typical ultra-submicron device. Here, a particle approaches a potential barrier and absorbs a phonon near the barrier. Classically, the transition indicated in Fig. 1 is forbidden. First, it is non-local in space. Secondly, absorbing a phonon leads to an increase in the momentum wave vector, but

3

366 D. K. Ferry, D. Vasileska & H. L. Grubin

here the final state has a lower value of momentum wave vector due to the rapid increase in the potential energy. The horizontal lines are meant to be the spatial extent of the initial and final wave packets (which are both moving). In quantum mechanics, the energy that is conserved is the total energy, not just the kinetic energy. In Fig. 1, it is the intra-collisional field effect15 which rapidly changes the momenta of the two states with position (time). A ballistic electron approaching the top of a 0.3 eV barrier is traveling at a velocity that approaches 108 cm/s. The collision duration is a few femtoseconds,26'27 but the carrier will still cover 3-5 nm (which may be the entire barrier thickness) during the collision.

*E

V(x)

H - *

FIG. 1. Conceptual picture of an electron absorbing a phonon near a potential barrier.

It can be seen from the above discussion that the introduction of a collision duration time creates a great deal of difficulty in the fully quantum transport treatment. In some sense, this is readily incorporated in most cases through a proper development of a self energy, whose real part describes the many-body renormalization of the energy bands and whose imaginary part describes the detailed scattering dynamics that are present. Nevertheless, the interference between the scattering processes and the local fields and potentials is an important effect in quantum transport theory. Although difficult to properly treat, the neglect of such effects gives results that are little better than BTT itself. Before continuing, it is worthwhile to point out that many people think of quantum transport as arising from size quantization in the system. Working just in this mode, one can find the eigenstates of the system and compute transfer matrices for use in the Landauer formula, provided that the carriers are non-interacting and the system is relatively stationary. Determining the eigenfunctions in an interacting system can usually be done only when there are a dozen or fewer electrons, as a general rule.28 As a result, this approach is usually confined to very low temperatures and non-interacting systems. But, in fact, this type of system can be handled, if the spatial variation is slow, by semi-classical BTT through the incorporation of scattering rates between the various

4

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quantum states (just as intervalley scattering is handled in a multi-valley band structure). It is the rapid spatially varying system, in which it is difficult to think of stationary quantum eigenstates, that creates the real need for quantum transport theory. It is in these systems (which describe most ultra-submicron semiconductor devices) that the above principles need to be carefully incorporated into the transport, and this requires quantum transport theory.

3.

Non-Equilibrium Green's Functions

Let us consider a Hamiltonian, which for the moment contains only the kinetic energy and Coulomb interaction terms, in second-quantized notation29

H = jdxi//+(x) -

fi2V2^ 2m

+

(3.1)

+

+ jdx jdx V (x)y/ (x')V(x - x V(x'){/(x) where the carats denote operators. The field operators satisfy the general commutator relationships