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Founding Editors FREDERICK SEITZ DAVID TURNBULL SOLID STATE PHYSICS Advances in Research and Applications Editors HEN...

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Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors HENRY EHRENREICH

FRANS SPAEPEN

Division of Applied Sciences Harvard University Cambridge, Massachusetts

VOLUME 49

ACADEMIC PRESS San Diego New York London Sydney Tokyo

Boston Toronto

This book is printed on acid-free paper.

@

Copyright 0 1995 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc.

A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road. London NWl 7DX

International Standard Serial Number: 0081-1947 International Standard Book Number: 0-12-607749-5 PRINTED IN THE UNITED STATES OF AMERICA 95 96 9 7 9 8 99 0 0 E B 9 8 7 6 5

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3 2

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Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

DAVID K. FERRY (2831, Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287 RICHARD H. FRIEND(l), Cnvendish Laboratory, Cambridge CB3 OHE, United Kingdom NEIL C . GREENHAM (11, Cavendish Laboratory, Cambridge CB3 OHE, United Kingdom

HAROLD L. GRUBIN (2831, ScientiJic Research Associates, Glastonbury, Connecticut 06033 R. L. GUNSHOR (2051, School of Electrical Ensneering, Purdue Uniuersity, West Lafayette, Indiana 47907 P. M. Hut (1511, Department of Physics, The Chinese University of Hong Kong, Shatin, New Tem'tories, Hong Kong

NEIL F. JOHNSON (1511, Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, England, United Kingdom A. V. NURMIKKO (205), Division of Engineering and Department of Physics, Brown University, Providence, Rhode Island 02912

vii

Preface This volume is concerned with recently developed physics and materials for semiconductor and photonic devices. It discusses visible light emitters and lasers that use polymers and 11-VI compounds, photonic gap materials, and the modeling of electronic transport in semiconductor devices. Visible light emitters are becoming increasingly important for displays as light emitting diodes (LEDs) become ever more attractive. Photonic band gap materials are important for a variety of optical applications because they contain frequency ranges for which light propagation cannot occur. These materials consist of periodic dielectric structures, just as electronic band gap materials involve periodic arrays of atoms or mesoscopic layers. Sophisticated and more realistic semiconductor device modeling is becoming crucial as the number of transistors on a single chip continues to double about every three years. The first chapter by Greenham and Friend presents an extended overview, highlighted by nearly four hundred references, of an area that is engaging an increasing number of researchers. The optoelectronic properties of conjugated polymers are far less well understood than their inorganic counterparts, and therefore are of intrinsic physical interest. Since many conjugated organic materials exhibit strong photoluminescence, they are attractive candidates for electroluminescent devices. Moreover, their colors, which can be controlled by altering the chemical composition and structure, extend into the blue range of the spectrum. Inorganic blue LEDs, such as the 11-VI materials discussed by Nurmikko and Gunshor, have materials problems that remain to be solved. So do organic photoconductors; however, conjugated polymer LEDs already exhibit sufficiently high brightness, efficiency, and low operating voltages, that they may well be competitive in a large commercial display market over the long term. The physically and device oriented discussion uses the well understood ideas developed for the conventional inorganic semiconductor devices as a departure point for exploring the very different materials problems, electronic structure, elementary excitations, and transport properties of these materials. The chapter should therefore be readily accessible to those largely unacquainted with polymer science. The chapter by Hui and Johnson concerning photonic band-gap materials also has a significant pedagogical component. Their discussion of the electromagnetic field equations in periodic dielectric media with and without localized impurities is based on the crystal representation, k * p, and tight-binding approaches familiar from solid state physics. Their treatment is lucid, insightful, and complete. The two principal applications ...

Vlll

PREFACE

ix

of photonic band structure materials proposed thus far are the suppression of electron-hole recombination in semiconductor devices which would increase their efficiency, and the construction of single mode cavities utilizing the localized photon modes. As the authors point out, the possible applications, and hence the required level of fabrication precision, depend strongly on the frequency range and hence the operating wavelength of the required device. While millimeter wave detectors can be built at reasonable cost because of their macroscopic periodicity, optical devices will require fabrication techniques that are accurate on the scale of microns. Thus, the interface with semiconductor device applications still awaits further definition. The contribution by Nurmikko and Gunshor presents a sweeping account of both the fundamental physics and the device-related scientific aspects of 11-VI semiconductor visible light emitters. It explores the wide-ranging spectrum of fascinating fundamental problems typical of 11-VI heterostructures, which results in large part from their strongly ionic character. Excitons in these materials are relatively stable and without doubt play an important role in light emission. Doping and electrical contacts present special problems. Defect formation energies are relatively small and promote device degradation. The importance of these phenomena is qualitatively different from that encountered in 111-V light emitters. Furthermore, the underlying physics is only partially explored. The technological aspects of the field are developing very rapidly. This “snapshot” of the present technical state is nonetheless of great value, because of its scholarly account of the modern physics of these semiconductors. Furthermore, these same physical ideas may well be of importance in understanding GaN, another blue light emitter that has been the focus of recent attention, and also polymer LEDs. The last, extensively referenced and didactic chapter by Ferry and Grubin describes quantum transport theory and its application to semiconductor devices in much needed detail. Quantum effects become quite significant when, as is now anticipated, transistor gate lengths become comparable to 0.1 mm. As the authors point out, these effects appear in many forms, for example, in the modification of statistical thermodynamics describing device operation, the introduction of new length scales, the occurrence of ballistic transport, and in quantum interference and fluctuations. The quantitative description of these phenomena involves sophisticated applications of nonequilibrium statistical mechanics. The authors provide information relevant to all parts of this field related to semiconductor device physics. These include all the commonly used formalisms due to Wigner, Kubo, and Keldysh, respectively, and the utilization of

X

PREFACE

density matrix and Green’s function approaches. Applications are made to a number of specific structures that frequently occur in semiconductor devices, for example, the resonant tunneling diode. The selection of new device-related physics presented in this volume will without doubt interest many practitioners and newcomers who wish to broaden their perspective. HENRY EHRENREICH FRANS SPAEPEN

SOLID STATE PHYSICS. VOL. 49

Semiconductor Device Physics of Conjugated Polymers NEILc. GREENHAM AND RICHARD H. FRIEND Cavendish Laboratory. Cambridge CB3 OHE. United Kingdom

1. Introduction

I1 .

111.

IV .

V.

VI .

VII . VIII .

.......................................

1. Semiconducting and Conducting Polymers . . . . . . . . . . . . . . . . . . . 2. Organic Semiconductor Devices . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Synthesis of Semiconducting Polymers . . . . . . . . . . . . . . . . . . . . . 4. Doped Conducting Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Molecular Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Physical Properties of Conjugated Polymers . . . . . . . . . . . . . . . . . . Optical and Electronic Properties of Conjugated Polymers . . . . . . . . . . . . 7. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Electronic Structure-Ground State . . . . . . . . . . . . . . . . . . . . . . 9. Electronic Structure-Excited States . . . . . . . . . . . . . . . . . . . . . . 10. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Device Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Charge Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Carrier Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. ExcitonDecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field-Effect Diodes and Transistors 14. The Field-Effect Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Polyacetylene Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Polythiophene and Polfi p-phenylenevinylene) Devices . . . . . . . . . . . 17. Thiophene Oligomer Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 18. OtherDevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light-Emitting Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Conjugated Polymer Electroluminescence . . . . . . . . . . . . . . . . . . . 21 . Control of Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Electrical Characteristics of Single-Layer Devices . . . . . . . . . . . . . . 23. Multilayer Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Probing of Excited States in LEDs . . . . . . . . . . . . . . . . . . . . . . . 25. Optical Properties of LEDs . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. Novel Device Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Technological Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photoconductive and Photovoltaic Devices . . . . . . . . . . . . . . . . . . . . . 28. Molecular Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Polymeric Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 2 7 12 12 25 27 29 32 32 35 39 51 63 64 73 80 84 84 91 103 111 113 115 115 117 118 124 130 134 137 142 143 144 144 145 149

Copyright 0 1995 by Academic Press. Inc . All rights of reproduction in any form reserved

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2

NEIL C. GREENHAM AND RICHARD H. FRIEND

1. Introduction

This article is concerned with the new semiconductor device physics that can be realized using thin films of semiconducting polymers as the active layers, in devices ranging from field-effect transistors to electroluminescent diodes and photoconductive devices. Semiconducting properties in organic materials are not new, but the availability of film-forming semiconducting polymers is very recent. There is a high level of interest in the potential that these structures might have for use in large-area thin-film electronics, because of the prospect for low-cost fabrication of devices with very acceptable performance. In addition, the investigation of the properties of these devices has provided a very fruitful method for studying the basic semiconductor physics of these materials. It is this aspect with which we are concerned here. The subject that we are discussing has generated a huge amount of literature, and we cannot attempt to mention here all that has been done, nor even all the important work that has been done. We have attempted to provide an outline of the background to this field, and have discussed in greater detail those aspects most directly relevant to the device physics. We have drawn our examples extensively from work that has come from Cambridge. 1. SEMICONDUCTING AND CONDUCTING POLYMERS Semiconducting or conducting properties associated with organic materials usually derive from the presence of extended rr orbitals formed in carbon-containing compounds that show sp2 + p , hybridization. The best known example is that of graphite in which the geometry of the extended “honeycomb” sheets of carbons is determined by the sp2 a bonds within the plane, and in which the p z orbitals overlap to form extended rr orbitals above and below the plane. The electronic structure is largely two dimensional, with weak coupling through van der Waals interactions between planes, and the semimetallic properties are due to the extended rr orbitals that form energy bands with substantial dispersion. Modification to the filling of the n- valence and rr* conduction bands can be done by forming charge-transfer complexes with donor or acceptor species, which are accommodated as intercalants between the graphite sheets.’ Moving from extended sheets to smaller fragments of extended rr orbitals gives a very wide range of molecular semiconductors, among which anthracene (see Fig. 1) provided the protypical example for the extensive ‘M.S. Dresselhaus and G. Dresselhaus, M

u . Phys. 30, 139 (1981).

SEMICONDUCTOR DEVICE PHYSICS

3

FIG.1. Structure of anthracene.

work carried out in the 1960s and 1 9 7 0 ~ .Electronic ~ transport over macroscopic distances requires that there be contact between adjacent molecules, which is possible in materials of this type because there is direct overlap between the rr electrons on neighboring sites. The energy gap between filled rr and empty rr* states for anthracene is 3.9 eV, very much higher than that for materials with more extended rr orbitals, and the scope for introducing electronic charges is restricted by the high ionization potential and low electron affinity for this high-gap semiconductor. However, charge generation through photoexcitation has been extensively investigated and molecular materials of this type have been developed as photoconductors that are now used in the major of xerographic copiers and printer^.^ The recent interest in organic semiconductors dates from the discovery that molecular semiconductors in which the molecular units are arranged in stacks and which can be formed with nonintegral occupancy of the rr states could show very high electronic conductivities. Compounds such as tetrathiafulvalene tetracyanoquinodimethane (TTF-TCNQ) were found in the early 1970s to show metallic conductivities above 100 S cm-', along with the other characteristics of metals such as Drude optical refle~tivity.~,' The material forms as segregated stacks of TTF and of TCNQ, with good intermolecular interactions along the stacks. The metallic behavior results from charge transfer from TTF stack to TCNQ stack of about 0.59 electrons per formula unit, giving partially filled bands on both stacks, which both contribute to the electrical conductivity. The charge-transfer salts have provided an extremely rich area of investigation of low-dimensional physics, with a range of competing low-temperature ground states including Peierls insulator, spin-density wave, metal, and superconductor. Superconductivity, first reported in 1980 in the hexafluorophosphate salt of a derivative of TTF, TMTSF (see Fig. 21, with a critical temperature of

'

M. Pope and C. E. Swenberg, Electronic Processes in Organic Crystals, Clarendon Press, Oxford (1982). 'P. M. Borsenberger and D. S. Weiss, Organic Photoreceptorsfor Imaging Systems, Marcel Dekker, New York, (1993). 4D. JCrome, Science 252, 1509 (1991). 'J. M. Williams, A. J . Schultz, U. Geiser, K. D. Carlson, A. M. Kini, H. H. Wang, W. K. Kwok, M. H. Whangbo, and J. E. Schirber, Science 252, 1501 (1991).

4


ds>=<sB S

H 3 C Xs H3C

Se

TTF

S

K s y c H 3 Se

TMTSF

CH3

N TCNQ

N &a“

NQC

+N

FIG.2. Structures of some donors and acceptors used to form charge-transfer salts. ITF and TMTSF are donors and TCNQ and DDQ are acceptors.

around 1 K, has now been found in a number of these materials, and critical temperatures of above 12 K are reported.4~~ Though these molecular conductors are of great interest, the scope for using them has always been hampered by the fact that the crystals are always fragile and often small. For this reason, the report in 1977 from the University of Pennsylvania of high conductivity in charge-transfer complexes formed with a polymer, polyacetylene, which exhibited extended T conjugation along the polymer chain, provoked considerable excitement.6 Polymers that are used commercially are all materials that can be possessed, often very conveniently, and the immediate interest was that these polymers could be used as electrical conductors. Another aspect that attracted interest was the very simplicity of the polymer structure. Polyenes are indeed the simplest conjugated structures, and they are found in

k.K. Chiang, C. R. Fincher Jr., H. Shirakawa, E. J. Louis, S. C. Gau, and A. G . MacDiarmid, Phys. Reu. Lett. 39, 1098 (1977).


5

cis-transoidal, or "cis" polyacetylene

trans-transoidal, or "trans" polyacetylene

FIG.3. The cis and trans isomers of polyacetylene.

several biological materials. A widespread structural type consists of an alternating copolymer of acetylene and methylacetylene, that is, -4-CH = CH-CH = C(CH3)-)-,,; thus, the polyene where n = 5 is a functional part of the visual pigment rhodopsin, and that with n = 10 is found in the yellow pigment p-carotene. Polyacetylene, -(-CH-)-,, , is the simplest polymer of this type, and interest by Lennard-Jones in the rr electron structure dates back more than 50 years7 As the length of the conjugated sequence is increased, the energy gap between the filled rr and empty rr* states falls, though in the long chain limit the gap remains finite, for reasons discussed in III.8.a, first appreciated by Longuet-Higgins? and takes a value of about 1.5 eV. Polyacetylene has served as the prototypical conjugated polymer; the simplicity of its structure has allowed theoretical modeling, and recently there has been a ready supply of polymer in a form suitable for a range of experiments. The polymer can exist in several isomeric forms and the trans-transoidal isomer, usually referred to as "trans-polyacetylene," which is the thermodynamically stable isomer at room temperature, is shown in Fig. 3. The obvious route to polyacetylene is addition polymerization of acetylene, and the work that demonstrated high electrical conductivity in polyacetylene was performed with material synthesized at the interface between a solution of a Ziegler-Natta initiator and gaseous acetylene.' Polyacetylene produced by this technique is obtained in morphologies varying from an isotropic fibrilla mat of low density through to a highly . ~ that these could be anisotropic dense material." Chiang et ~ 1 showed chemically doped with electron acceptors such as AsF, and iodine, or ' 5 . F. Lennard-Jones, Proc. Roy. Soc. London A158, 280 (1937). 'H. C. Longuet-Higgins and L. Salem, Proc. Roy. Soc. London A521, 172 (1959). 9T. Ito, H. Shirakawa, and S. Ikeda, J. Polym. Sci. Polym. Chem. Ed. 12, 11 (1974). "'G. Wegner, Angew. Chem. Int. Ed. Engl. 20, 361 (1981).

6


electron donors such as sodium, to give values of conductivity up to 1000 S cm-' at room temperature. This value of conductivity is high, and just about in the range of metallic behavior. This was the first demonstration of metallic behavior within the intramolecular rr electron system along the polymer chain, and the significance of these results was quickly picked up by many other groups worldwide. The behavior of the conductivity with both acceptor and donor dopants is suggestive of the conventional substitutional p and n doping of a semiconductor. The appropriate sign for the thermopower is indeed observed." Doping, of course, is not substitutional; the dopant species reside alongside the polymer chain and there is a charge-transfer reaction akin to the intercalation chemistry known for graphite.' Very high concentrations of dopants can be taken up by the polymer, and the high, metallic values of conductivity are usually measured on samples with molar fractions of dopants in the range 1 to 20% per carbon atom. These high values of dopant concentration cause large structural changes, and though there is some tendency to form ordered "intercalation" structures,'*' l 3 there is usually a high level of disorder in the doped phase. This must limit the value of the metallic conductivity, indeed the conductivity values, though metallic at room temperature, fall at lower temperatures, in contrast to the behavior expected from a metal. This is generally attributed to disorder, both at the microscopic level and also at the macroscopic scale, in the contact between crystallites or fibrills in the sample. This article is concerned primarily with the properties of conjugated polymers, that is, polymers which show extended rr bonding along the polymer chain, though we make reference where appropriate to complementary work using molecular materials. Because the development of the field has been so much controlled by the availability of processible materials, we have set out in Section I1 a brief survey of methods for synthesis and processing of those polymers that form the basis for many of the studies of device properties currently performed. Whereas conventional polymers are readily processed in solution or in the melt and can be cheaply manipulated into desirable forms, this is not in general possible for conjugated polymers. The delocalized rr electron system makes the molecular chains rigid, with resultant high melting points and low solubilities. The dilemma of a potentially attractive material that "Y. W. Park, A. Denenstien, C. K. Chiang. A. J. Heeger, and A. G . MacDiarmid, Solid State Commun. 29, 747 (1979). 12 R. H. Baughman, N. S. Murphy, G . G . Miller, and L. W. Shacklette, J . Chem. Phys. 79, 1065 (1983). 13 M. Winokur, Y.B. Moon, A. J . Heeger, J . Barker, D. C. Bott, and H. Shirakawa, Phys. Reu. Lett. 58, 2329 (1987).


7

cannot be processed is not new in materials science in general or polymer science in particular. Two well-established lines of attack on such problems involve either modifying the molecular structure so as to retain the property of interest while rendering the material processible, or carrying out the processing stages with a more tractable precursor, which can be converted subsequently to the desired material. Both of these approaches have been successfully applied to the synthesis and processing of most classes of conjugated polymers, and the methods adopted are summarized in Section I1 for the major structural classes. The semiconductor physics of these polymers has become a major area of interest, and it is this that forms the subject of this article. There has been a considerable theoretical and experimental effort directed toward understanding the way in which charges can be stabilized on the chains, and the nature of the transport processes that can then occur. In contrast to three-dimensionally bonded inorganic semiconductors, these materials behave as “molecular materials,” and there is a considerable reorganization of the local T electron bonding in the vicinity of extra charges added to the chains. This results in self-localization of the added charge, to form, in general, polarons, though for the particular symmetry of the trans isomer of polyacetylene, these take the form of bond-alternation defects, or solitons. The theoretical models developed to describe these processes are discussed in Section 111, along with a brief description of some of the salient experimental results that characterize the electronic structure of conjugated polymers. 2. ORGANIC SEMICONDUCTOR DEVICES The effectiveness of molecular semiconductors as active materials is demonstrated well by their successful use as the photoconductive elements in xerography. One of the potential advantages is that since no strong chemical bonds need to be broken at interfaces or surfaces of molecular materials, there need not be any surface or impurity states that lie within the semiconductor gap. It has indeed been shown that relatively simple fabrication techniques can be used very successfully to build up device structures as a series of layers. We review in Section IV the current understanding of the processes of charge injection, transport, and recombination, and of exciton decay, which is relevant to the understanding of the devices of interest here. The availability of film-forming semiconducting polymers in the late 1970s resulted in attempts to fabricate a range of semiconductor devices, principally two-terminal diodes formed as sandwich structures with metal-

8


lic electrodes to either side of a film of polymer. The first report on polyacetylene formed by the Shirakawa route, of in siru polymerization of acetylene gas onto the bottom electrode, revealed that Schottky barriers could be formed against metals with appropriate work functions (here the polyacetylene was functioning as a p-type semicond~ctor).'~ Efforts were also made to observe a field effect in a three-terminal device.I5 However, these early experiments were constrained by the poor processibility of the polymers then available, and in spite of considerable efforts to study photovoltaic and photoconductive devices,I6 progress was relatively unspectacular. Nevertheless, in siru polymerization by electrochemical oxidation to form polythiophene was used by Koezuka and co-workers" to produce working metal-insulator-semiconductor (MIS) field-effect transistors (FETs) that functioned as enhancement-mode devices. The arrival of solution-processible polymers in the mid-1980s allowed a more rapid development of this field, with the demonstration of devices with desirable characteristics. We illustrate this here with reference to the Schottky-barrier diode, which has been made using a range of solutionprocessed polymers.I8-*' Figure 4 shows a Schottky-barrier diode structure formed using polyacetylene prepared by the Durham precursor route" (see Section 11.3.b), which is p-type as prepared. The bottom gold electrode provides an ohmic contact to the polyacetylene formed above it, and the top contact is formed by thermal evaporation of a low work-function metal to give the Schottky barrier. The current versus voltage characteristics of this device are shown in Fig. 5, from which it can be seen that the rectification ratio is reasonable, approaching 5 X lo5 at biases of f 1.5 V. The variation of current density, J , with bias voltage, V is usually parameterized as (2.1)

I4P. M. Grant, T. Tani, W. D. Gill, M. Kroubni, and T. C. Clarke, J . Appl. Phys. 52, 869 (1981). IS E. Ebisawa, T. Kurokawa, and S. Nara, J . Appl. Phys. 54, 3255 (1983). 16 J. Kanicki, in Handbook of Conducting Polymers (T. J. Skotheim, ed.), Vol. 1, p. 644, Marcel Dekker, New York (1986). "H. Koezuka, A. Tsumara, and T. Ando, Synfh. Met. 18, 699 (1987). 18 J. H. Burroughes, C. A. Jones, and R. H. Friend, Nature 355, 137 (1988). 1Y J . H. Burroughes and R. H. Friend, in Conjugated Polymers (J. L. BrCdas and R. Silbey, eds.), p. 555, Kluwer, Dordrecht (1991). '"H. Tomozawa, D. Braun, S. Phillips, and A. J. Heeger, Synfh. M e f . 22, 63 (1987). 21 H. Tomozawa, D. Braun, S. D. Philips, R. Worland, A. J. Heeger, and H. Kroemer, Synfh. Met. 28, C687 (1989).

9

SEMICONDUCTOR DEVICE PHYSICS gpld

,plyacetylene

catjode

ajode

aluminium

FIG. 4. Schottky-barrier diode structure formed with Durham-route polyacetylene. The device is fabricated on a flat glass substrate, with a thin (20-nm) layer of gold as the ohmic contact, onto which the polyacetylene layer is prepared via the Durham precursor. The top metal contact, which is applied by thermal evaporation, is chosen to give the Schottky barrier (aluminum, chromium, or indium). [J. H. Burroughes and R. H. Friend, in Conjugated Polymers (J. L. Brtdas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of KIuwer Academic Publishers.]

where q is the electronic charge, k , is Boltzmann’s constant, and T is temperature. The constant n is termed the ideality factor and is larger than 1 for devices that suffer from defect states or interfacial insulating layers. For the device shown in Fig. 5 the value of n is around 1.3 at low bias. Direct evidence that this type of device functions in the conventional manner, with the width of the depletion region controlled by the applied bias V , is obtained from the variation of device capacitance per unit area, C / A , with V , which for an abrupt junction (uniform concentration of

-

10-l2

1

-2

.

-1

1

1

0

1

.

2

bias voltage M

FIG.5. Modulus of current versus applied bias voltage for a Schottky barrier diode, of the type shown in Fig. 4,formed with an aluminum contact to provide the Schottky barrier and a gold contact to provide the ohmic junction. The device area is 0.64 cm2 and the thickness of the polyacetylene film is 500 nm. [J. H. Burroughes, C. A. Jones, and R. H. Friend, Nature 335, 137 (1988).1

10

NEIL C. GREENHAM AND RICHARD H. FRIEND 3 16

x 10

-1

C2

0

‘

-2

1

’

-1

’

0

1

.

1

1

.

2

bias voltage (V)

FIG.6. l / C z versus bias voltage for a polyacetylene Schottky diode of the type shown in Fig. 4. formed with an aluminum top Schottky barrier. The sample area is 0.63 cm’. [J. H. Burroughes, C. A. Jones, and R. H. Friend, Nature 335, 137 (1988).]

acceptors, N,), is given by (2.2) where Vd, is the “built-in” voltage.22 Results for these polyacetylene diodes are shown in Fig. 6, in which it is seen that there is a linear slope of 1/C2 versus bias in the reverse bias regime. Values for the acceptor concentration, N,, as determined by the gradient of 1/C2 versus V , are - ~ this p ~ l y m e r . ” ~ ’ ~ found in the range of 1016 to 2 X 10” ~ r n for Field-effect transistors using polyacetylene and also soluble deriva~ ~ Section II.3.e) were shown to have reprotives of p ~ l y t h i o p h e n e(see ducible properties, albeit with very low field-effect mobilities in the range l o p 5 to cm2 Vp‘ s ’. The fabrication of devices of this type is illustrated in Fig. 7. Note that this is a buried gate device in which the silicon substrate is n-doped so that it functions also as the gate, and that the insulator layer is formed by thermal oxidation of the silicon. Source and drain contacts are formed by lithographic patterning of the layer of electrode material formed on top of the silicon dioxide layer, and the semiconductor polymer layer is applied as the last process step. FET devices were shown to be particularly effective experimental vehicles for “9

22S. M.Sze, Physics of Semiconductor Deuices, Wiley, New York (1981). 23 A. Assadi, C. Svensson, M. Willander, and 0. Inganas, Appl. Phys. Lett. 53, 195 (1988).


11

FIG 7. Schematic diagram for a polyacetylene MISFET structure as used by Burroughes

et ul Dimensions shown are to scale, except the channel width (20 pm) and length (1.5 m).

Note that in this example, the source and drain contacts are of poly-silicon; gold is more commonly used since its high work-function provides relatively good matching for hole injection for a wide range of polymers. [J. H. Burroughes, C. A. Jones, and R. H. Friend, Nature 335, 137 (1988).]

studying the nature of the electronic excitations in conjugated polymers, as we discuss in Section V. Very much higher field-effect mobilities have now been reported, using either well-ordered o l i g o m e r ~ ~or~ well-ordered -~~ p~lymer,~' and are comparable to those found in amorphous silicon. There is at present a high level of interest in the development of these devices to form arrays over large areas. Thin-film electroluminescence (EL) using conjugated polymers has provided the other major area of device-related activity. Diodes formed with molecular semiconductors have long been known to exhibit electroluminescence when conditions are arranged so that electrons and holes are injected at opposite electrodes, as was first discovered for a n t h r a ~ e n e . ~ ~ - ~ ' The discovery that conjugated polymers could act as both transport and emissive layers, reported in 1990,3' has generated a very high level of interest. Device fabrication can be very straightforward, with a layer (or F. Gamier, G. Horowitz, X. H . Peng, and D. Fichou, Adu. Mafer. 2, 92 (1990). "F. Gamier, A. Yassar, R. Hajlaoui, G. Horowitz, F. Deloffre, B. Dervet, S. Ries, and P. Alnot, J . Am. Chem. SOC.115, 8716 (1993). "F. Gamier, R. Hajlaoui, A. Yassar, and P. Srivastava, Science 265, 1684 (1994). "H, Fuchigami, A. Tsumura, and H. Koezuka, Appl. Phys. Lett. 63, 1372 (1993). 2x M. Pope, H. P. Kallmann, and P. Magnate, J . Chem. Phys. 38, 2042 (1963). 24

"W.

Helfrich and W. G. Schneider, Phys. Reu. Lett. 14, 229 (1965). W. Helfrich and W. G. Schneider, J . Chem. Phys. 44, 2902 (1966). 31 J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend,

P. L. Burn, and A. B. Holrnes, Nufure 347, 539 (1990).

12

NEIL C. GREENHAM AND RICHARD H. FRIEND poiy(pphenyleneviny1ene)

aluminium, magnesium External Circuit

FIG.8. Structure of an electroluminescent diode based on a conjugated polymer, PPV. The polymer film is formed on a glass substrate coated with indium-tin oxide, and the top electrode is then formed by thermal evaporation.

layers) of polymer sandwiched between two electrodes, one of which is transparent. This is illustrated in Fig. 8 for the case of the first E L diodes made with poly(p-phenylenevinylene), PPV.31 The polymer was prepared as a thin film, of thickness of order 100 nm, by spin-coating a “precursor” polymer from solution, using a standard photoresist spin-coater, and subsequently converting the “precursor” polymer to the semiconducting PPV by heating. This polymer layer was formed on a glass substrate coated with indium-tin oxide (IT01 and the other electrode then formed by thermal evaporation of the selected metal, as shown in Fig. 8. Rapid progress has since been made, and EL diodes with a wide range of emission colors and with quantum efficiencies (photons/electron) of several percent are now reported. The development of this area of device physics is discussed in Section VI. Interest in photoconductive and photovoltaic devices based on organic semiconductors has a long history, and the arrival of the film-forming polymers has only recently made an impact in this area. We review some of this work in Section VII.

II. Materials

3. SYNTHESIS OF SEMICONDUCTING POLYMERS

a. Introduction The chemical structures of some common conjugated polymers are shown in Table I. The optical and electronic properties arising from their conjugated nature are explained in Section 111; this section is concerned with the chemical synthesis of conjugated polymers.

13

SEMICONDUCTOR DEVICE PHYSICS TABLE1. SOMECOMMON CONJUGATED POLYMERS ~~

POLYMER

PA

NAME CHEMICAL

trans-polyacetylene

PDA

polydiacetylene

PPP

poly( p-phenylene)

PPV

pol$ p-phenylenevinylene)

FORMULA7T-?TTT* ENERGY GAP" (ev)

en

1.5

1.7

on

3.0 2.5

OR

n

2.2

PT

2.0

polythiophene R

P3AT

2.0

poly(3-alkylthiophene) R

PTV

3.1

polypyrrole H

PAni

polyaniline

"The n-r* energy gap is taken as the energy at the maximum slope, da/dE, of the spectrum of the optical absorption ( a ) .

14


Maximum overlap of p , orbitals occurs when a conjugated polymer adopts a planar configuration. Conjugated polymers therefore tend to be rigid, rod-like molecules, and insoluble in common organic solvents. Polyacetylene, poly(p-phenylenevinylene), poly(p-phenylene), and polythiophene all fall into this category. Direct chemical synthesis is therefore difficult because the product will tend to precipitate out from solution while its molecular weight is still small. This insoluble solid product is then difficult to form into the thin films needed both for optical experiments and for electronic devices. A number of methods have been used to overcome this problem, as described later. Electrochemical polymerization onto a conducting electrode has been used extensively for producing films of intrinsically conducting polymers.32 These films tend to be highly doped by the inclusion of counter-ions from the electrolyte solution, and their microstructure is often not fully dense. Films prepared by this method have potential applications in sensors (e.g., polypyrrole in gas sensors33), but they are not ideal for semiconducting electronic applications such as light-emitting diodes (LEDs) and transistors, though electrochemically prepared poly(3-methylthiophene) has been used successfully as the active semiconductor in MISFETs. 17*34 A more convenient strategy is to prepare a soluble “precursor” polymer, which can be processed from solution to form thin films. These films are then converted to the final polymer by a further processing step such as heating. The Durham route to p~lyacetylene~’ and the Wessling route to PPV36’37are both examples of this method. The films formed through the precursor route are insoluble, and therefore unlikely to be affected by subsequent processing steps. It is sometimes convenient to produce polymers that are soluble in their final form. A common way to achieve this is to substitute flexible alkyl or alkoxy chains as side-groups on the polymer backbone. The poly(2,5-dialkoxyppheny1enevinylene)s and the poly(3-alkylthiophene)~are examples of this class of material, and are both soluble in organic solvents such as chloroform. 32

A. F. Diaz and J. Bargon, in Handbook of Conducting Polymers (T. A. Skotheim, ed.), Vol. 1, p. 81, Marcel Dekker, New York (1986). 33 J. M. Slater, E. J. Watt, N. J. Freeman, I. P. May, and D. J. Weir, Analysr 117, 1265 (1992). 34 D. M. Taylor, H. L. Gomes, A. E. Underhill, S. Edge, and P. I. Clemenson, J . Phys. D: Appl. Phys. 24, 2032 (1991). 35J. H. Edwards and W. J. Feast, Polymer 21, 595 (1980). 36 R. A. Wessling and R. G. Zimmerman, U.S. Patent 3401152 (1968). 37 R. A. Wessling and R. G. Zimmerman, U.S. Patent 3406677 (1972).


A

B

15

C

FIG.9. Durham precursor route to polyacetylene.

Real polymers, especially in the solid state, are far from the idealized one-dimensional chains represented by the chemical formulas in Table I. In general, a polymer sample will have a range of molecular weights, and will also contain various defects. These defects may be either chemical, such as the inclusion of saturated units or chain branches, or conformational, where the conjugation is broken by geometrical distortions of the chain due to its local environment. It is useful to define the “conjugation length” of a polymer as the average number of repeat units between conjugation-breaking defects. The electronic properties of a conjugated polymer are highly dependent on the nature and quantity of such defects, which, in turn, are strongly affected by the synthetic route and subsequent processing. Other defects that are likely to affect the physical properties include impurities incorporated within the polymer film, such as metal ions or low molecular weight compounds from the chemical synthesis. b. The Durham Precursor Route to Polyacetylene In attempts to circumvent the processing problems associated with polyacetylene produced by the direct route, the solution-processible precursor methods developed in Durham have proved particularly The steps for this are summarized in Fig. 9, and we refer to the materials produced by this technique as Durham polyacetylene. Monomer A is available in a one-step high-yield process from commercially available reagents; ring opening metathesis polymerization (ROMP) operates exclusively on the cyclobutene unit to give the polymer B, which is soluble and can be characterized by solution-phase methods. The stereochemistry of the backbone vinylene units in B depends on the nature of the ROMP initiator. In most reported work the reactive and fairly nondiscriminating WCl,/Me,Sn-derived initiator system has been used, which gives atactic 38W.J. Feast and R. H. Friend, J . Mater. Sci. 25, 3796 (1990).

16


polymers with random distributions of cis and trans vinylenes; polymer B is therefore expected to have a disordered microstructure and this accounts for its ready solubility. During the last few years the synthesis, characterization, and use of well-defined metathesis initiators has made spectacular progress and, for example, the initiators M(CH-tBu)(NAr)(O-t-Bu), (where M = W or Mo) give samples of B in which the backbone vinylenes are t r a n . ~ ? ~ . ~ ~ Because B is soluble it can be processed into a variety of forms by standard techniques. Thermal elimination of hexafluoroorthoxylene from samples of the processed precursor polymer B is symmetry allowed and gives rise to C. All the newly formed double bonds in C have cis geometry, and at or above room temperature they isomerize to trans polyacetylene, as indicated in Fig. 9. The process from precursor B to Durham polyacetylene involves the following steps: a chemical elimination of hexafluoroxylene; the diffusion of the hexafluoroxylene through the sample prior to its evaporation; the cis-to-trans isomerization of the vinylenes; and the crystallization of the polyacetylene chains. These processes overlap in time and space during the formation of the polyacetylene and together they determine the detailed order and structure of the final polyacetylene sample; thus the protocol adopted (i.e., heating rate, temperature, time, pressure, etc.) for the B to C to polyacetylene sequence determines the nature of the sample and hence its properties. A particularly useful aspect of this approach is the ability to orient samples by stretching prior to or during conversion to p ~ l y a c e t y l e n e . ~ ~ , ~ ~ c. Precursor Routes to Po&(p-phenylenevinylene), PPV The sulfonium polyelectrolyte precursor route to PPV was developed by groups at the University of Massachusetts (USA)36,37and the Sumitomo Chemical Company (Japan).43 In this route, shown in Fig. 10, an alkylsulfonium precursor polymer is formed by the base-induced polymerization of a p-xylylenebis(a1kylsufonium) salt, usually in aqueous solution. The final polymer is formed by the elimination of the alkylsulfonium leaving group. Early work used dimethylsulfonium precursors, however it has since been found that using a cycloalkylene leaving group, such as tetrahydrothiophene (THT), gives polymers with higher molecular weights and also "K. Knoll and R. R. Schrock, J . Am. Chem. SOC.111, 7989 (1989). J. H. F. Martens, K. Pichler, E. A. Marseglia, R. H. Friend, H. Cramail, D. Parker, and W. J. Feast, Polymer 35, 403 (1994). 41 G. Leising, Polym. Commun. 25, 201 (1984). 42 M. M. Sokolowski, E. A. Marseglia, and R. H. Friend, Polymer 27, 1714 (1986). 43M. Kanbe and M. Okawara, J . Polym. Sci.: A-l 6, 1058 (1968). 40


17

FIG.10. The sulfonium precursor route to poly(p-phenylenevinylene), PPV.

allows more efficient elimination of the leaving group, thus increasing the conjugation length of the final material.44 The polymerisation reaction to form the THT precursor has been found to proceed more effectively if a mixture of water and methanol is used as the solvent.45 The PPV used for much of the work in Cambridge was synthesized according to the scheme shown in Fig. ll.45Monomer 2 was synthesised by the reaction of a , a’-dichloro-p-xylene (1) with THT in methanol. Monomer 2 in methanol/water solution was polymerized by the slow addition of aqueous sodium hydroxide (0.9 M equivalent) at 0°C. After stirring for 1 hr, the reaction mixture was neutralized with hydrochloric acid and dialyzed against water for three days to remove low-molecular-weight impurities. The solvent was removed from the resulting solution and precursor polymer 3 was redissolved in methanol. After processing to form thin films, the precursor was converted to the final polymer 4 by thermal elimination of hydrochloric acid and THT. This reaction was typically performed under dynamic vacuum ( < mbar) at 220°C for 12 hr, and is frequently referred to as the “conversion” step. At room temperature in methanol solution, the THT leaving group of polymer 3 is slowly substituted by methoxy groups. To suppress this reaction it is necessary to store the precursor solution at low temperatures ( - 20°C). The methoxy groups formed by this reaction are not eliminated by thermal treatment, and therefore give a partially conjugated final polymer. An alternative route to PPV is, however, possible using this rea~tion.~B ‘ ,y~heating ~ 3 at 50°C in methanol, a fully substituted polymer (5) is f0rmed.4~ This “methoxy leaving group” polymer is soluble in 44 R. W. Lenz, C.-C. Han, J. Stenger-Smith, and F. E. Karasz, J . Polym. Sci. A . 26, 3241 (1988). 45 P. L. Burn, D. D. C. Bradley, R. H. Friend, D. A. Halliday, A. B. Holmes, R. W. Jackson, and A. Kraft, J . Chem. SOC.Perkin Trans. 1, 3225 (1992). 46 T. Momii, S. Tokito, T. Tsutsui, and S. Saito, Chem. Lett. 7, 1201 (1988). 47 S. Tokito, T. Momii, H. Murata, T. Tsutsui, and S. Saito, Polymer 31, 1137 (1990).

18


6 *

MeOH

1

50'C

v

NaOH MeOH-H20 O'C

\

5

FIG.11. The THT precursor route to polyfp-phenylenevinylene), PPV. [P. L. Burn, D. D. C. Bradley, R. H. Friend, D. A. Halliday, A. B. Holmes, R. W. Jackson, and A. Kraft, J . Chem. SOC. Perkin Trans. 1, 3225 (19921.1

chloroform. By heating 5 at 220°C in an atmosphere of HCI in flowing argon, full elimination of the methoxy groups can be achieved, to form PPV (4). This route is, unfortunately, not attractive for the fabrication of a range of devices including LEDs, because the HCI gas released at high temperatures is reactive and attacks, for example, IT0 electrodes. d. Synthesis of Soluble PPV Derivatives Polymers that are soluble in their final form avoid the need for a thermal treatment of the thin film after deposition, and allow purification of the final polymer in solution. This may be an advantage for device applications. In addition, a much wider range of synthetic routes is available once the need for a precursor is removed. The standard way of making soluble PPV-based polymers is to substitute long flexible side-groups onto the phenyl ring in order to increase the entropic contribution to the free energy in solution (Fig. 12). Most workers have used 2,5-dialkoxy substituents.


19

FIG.12. 2.5-substituted PPV.

The synthesis of insoluble substituted PPVs with short alkyl or alkoxy groups ( R i , R , =CH,O-, C,H,O-, CH,-), has been reported by several groups using the sulfonium precursor The first soluble PPV derivative was poly(2,5-dihexyloxy-p-phenylenevinylene),also synthesized by the precursor route5' [Fig. 13(a)l. To make the sulfonium precursor soluble, it was necessary to form a partially conjugated precursor polymer by adding excess base to the polymerization reaction.,' An alternative precursor route to soluble poly(2,5-dialkoxy-p-phenylenevinylene)uses a

2 eq xylene K08u

RO

?\b

q-$q RO

/&

an OR

RO

FIG.13. Synthetic routes to poly(2,5-dialkoxyphenylenevinylene).(a) THT precursor route. (b) Chlorine leaving-group precursor route. (c) Direct route. 48

S. Antoun, D. R. Gagnon, F. E. Karasz, and R. W. Lenz, Polym. Reprints 27, 116 (1986). I. Murase, T. Ohnishi, T. Noguchi, and M. Hirooka, Synth. Met. 17, 639 (1987). S. Antoun, F. E. Karasz, and R. W. Lenz, J . Polym. Sci. A 26, 1809 (1988). (I S. H. Askari, S. D. Rughooputh, F. Wudl, and A. J. Heeger, Polym. Reprints 30, 157 (1989). "S. H. Askari, S. D. Rughooputh, and F. Wudl, Synth. Met. 29, El29 (1989). 4Y

i(1

20


Q7J 0

/ FIG.14. Poly(2-methoxy-5-(2'-ethyl-hexyloxy)-p-phenylenevinylene~ (MEH-PPV).

chlorine leaving group precursor, which is itself soluble, and may be eliminated by heating to 100"C53 [Fig. 13(b)]. Under different reaction conditions, this reaction may proceed directly to the final polymer, amounting to a dehalogenation reactions4 [Fig. 13(c)l. Asymmetrically substituted PPVs have been found to have improved solubility in a range of solvents. Figure 14 shows the structure of poly(2methoxy-5-(2'-ethyl-hexyloxy)-p-phenylenevinylene)(MEH-PPV)>5 which has a branched alkoxy chain. This polymer has been extensively investigated, both for its photophysical proper tie^^^ and for use in L E D s . ~ ~ Phenyl substituents have also been used to increase solubility, and poly(phenyLphenyleneviny1ene) (Fig. 15) has been extensively

L

'n

FIG.15. Polyiphenyl-phenylenevinylene). "W. J. Swatos and B. Gordon, Polym. Preprints 31, 505 (1990). 54 T. Nakano, S. Doi, T. Noguchi, T. Ohnishi, and Y.Iyechika, European Patent 91301416.3 (1991). 55 F. Wudl, P.-M. Allemeand, G. Srdanov, Z . Ni, and D. McBranch, in Materials for Nonlinear Optics: Chemical Perspectives (S. R. Marder, J. E. Sohn, and G. D. Stucky, eds.), ACS Symp. Ser., Val. 455,p. 683,ACS, Washington, D.C. (1991). Sh T. W. Hagler, K. Pakbaz, K. F. Voss, and A. J. Heeger, Phys. Reu. E 44, 8652 (1991). 57 D. Braun and A. J. Heeger, Appl. Phys. Lett. 58, 1982 (1991). SR M.Gailberger and H. Bassler, Phys. Reu. E 44,8643 (1991). 59H. Vestweber, J . Oberski, A. Greiner, W. Heitz, R. F. Mahrt, and H. Bassler, Adu. Mater. Opt. Electron. 2, 197 (1993).

21

SEMICONDUCTOR DEVICE PHYSICS H13C60

CN

FIG. 16. Synthesis of CN-PPV. [N. C. Greenham, S. C. Moratti, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, Nature 365 628 (1993).]

Polymers of this type are synthesized by palladium-catalyzed coupling of dihalogenoarenes and ethylene.'" Substitution at the vinylic carbons can also be useful for tuning the electronic properties of a PPV-based polymer for particular applications. For LED applications, the substitution of cyano groups at the vinylic carbons next to alternate phenyl rings can be used to increase the electron affinity of the polymer. The synthesis of the soluble cyano-substituted derivative of dihexyloxy-PPV, which we refer to elsewhere in this article as CN-PPV, is shown in Fig. 16." The method used is a Knoevenagel condensation polymerization, which was earlier used by a number of authors to produce insoluble cyano-substituted PPVS.'*,'~ The reaction conditions for the synthesis of CN-PPV must be chosen with great care in order to avoid Michael addition across the double bond. Figure 17 shows the structure of an asymmetrically substituted CN-PPV derivative, which

FIG.17. An asyrnetrically substituted CN-PPV derivative. [D. R. Baigent, N. C. Greenham, J. Griiner, R. N. Marks, R. H. Friend, S. C. Moratti, and A. B. Holrnes, Synth. Met. 67, 3 (1994).1 60

H. Martelock, A. Greiner, and W. Heitz, Makromol. Chem. 192, 967 (1991). N. C. Greenham, S. C. Moratti, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, Nature 365, 628 (1993). 62 R. W. Lenz and C. E. Handlovitis, J . OR. Chem. 25, 813 (1960). "HH.-H Horhold, Z . Chem. 12, 41 (1972). hi

22


b R

R

FIG. 18. Synthetic routes to pol$3-alkythiophene). (a) Grignard reaction. (b) Oxidative coupling.

has a slightly larger energy gap and is soluble in a wider range of solvents.64 e. The Synthesis of Poly(3-alkylthiopheneh Polythiophene and its derivatives have been widely studied for their conducting properties in the doped state. Unsubstituted polythiophene is insoluble, and has traditionally been synthesized electro~hemically.~~ Poly(3-alkylthiophene)~with long alkyl chains are found to be soluble and fusible.66 For soluble derivatives, the problems associated with electrical p~lymerization~~ can be overcome using a chemical synthesis. Two commonly used chemical routes are Grignard coupling of 2,5-dihalo-3-alkylthiophene using a nickel (11) chloride catalyst 68 [Fig. 18(a)], and oxidative coupling of 3-alkyl thiophenes using an iron (111) chloride catalyst 69*70 [Fig. 18(b)]. Removal of the catalyst is performed by Soxhlet extraction; however, note that complete removal of all metallic impurities is extremely difficult and that the poly(3-alkylthiophene)~ are always doped to some extent. Figure 19 shows the structure of an alkoxy-substituted polythio64

D. R. Baigent, N. C. Greenharn, J. Griiner, R. N. Marks, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synth. Met. 67, 3 (1994). 65G.Tourillon and F. Garnier, J . Electroanul. Chern. 135, 173 (1982). 66

M. Sato, S. Tanaka, and K. Kaeriyarna, J . Chern. Soc., Chern. Commun. 873 (1986). B. Krische and M. Zagorska, Synth. Met. 28, C263 (1989). 68 M. Kobayashi, J. Chen, T.-C. Chung, F. Moraes, A. J. Heeger, and F. Wudl, Synth. Met. 9, 77 (1984). 69 R. Sugirnoto, S. Takeda, H. B. Gu, and Y. Yoshino, Chern. Express 1, 635 (1986). '"M. Leclerc, F. M. Diaz, and G. Wegner, Mukromol. Chern. 190, 3105 (1989). 67

23


r

1

I

OC12H25

FIG.19. Structure of poly( P'-dodecyloxy-a,a',-a,a"terthienyl), (poly DOT,).

phene derivative, known as polyDOT3, which has been used in MISFET devices (see Section V.16.b). Depending on the relative positions of the alkyl substituents, alkythiophene units may couple in head-to-head, head-to-tail, or tail-to-tail configurations (Fig. 20) The standard syntheses all give some proportion of head-to-head linkages. Recently, it has been possible to synthesize regioregular poly(3-alkylthiophene)~ with almost entirely head-to-tail linka g e s . ' ' ~ In ~ ~the solid state, these regioregular polymers are better ordered than their regiorandom counterparts and show significantly different optical and electronic properties. f. Poly(p-phenylene)s

Poly( para-phenylene), PPP, is another simple conjugated polymer and, like polyacetylene, it is insoluble and infusible. Attempts to make PPP have an even longer history than polyacetylene and were initially motivated by reasons related to its expected high thermal stability and to curiosity concerning the mechanical behavior of a true rigid-rod polymer, rather than any interest in its optical or electrical properties. Because the product is so intractable, early direct syntheses via oxidative coupling of benzene (either chemically or electrochemically) or via catalyzed Grignard coupling of paradibromobenzene ran into severe problems of product characterization and processing. Early samples of PPP were, like polyacetylene, obtained as intractable powders; however, despite its unattractive processing

*b *& R

R

H:H

H:T

-ps' iQR T:T

FIG.20. Head-to-head (H:H), head-to-tail (H:T), and tail-to-tail (T:T) linkages in P3AT. 71

R. D. McCullough and R. D. Lowe, J . Chern. Soc., Chern. Comrnun. 70 (1992). "T. A. Chen and R. D. Rieke, J . Am. Chern. SOC.114, 10,087 (1992).

24


A

tot-*

IB

H3COCd

PPP

OCOCH,

C

FIG.21. Precursor route to polyi p-phenylene).

properties (only powder compaction was feasible), PPP proved to be thermally and oxidatively stable and worth the attention of synthetic chemists. Effective processing via a precursor polymer was first achieved by an elegant route, summarized in Fig. 21, established by ICI worker^.^^.^^ The starting material for this route is obtained via a microbiological oxidation of benzene. The cyclohexadienediol A is esterified to give monomer B, which undergoes free radical initiated polymerization to give a processible precursor polymer C. This polymer is soluble and can be solution processed; heating results in the elimination of methanol and carbon dioxide and the formation of PPP. The one minor defect in this route is that the radical polymerization step does not give an exclusively 1,4 addition polymer and so the material is structurally inhomogeneous. Improvements in control of this aspect of the synthesis, however, have been reported,75 and it remains the best route to polyphenylene currently available. Improved processibility of polyphenylenes by attachment of side-chains has been investigated, with increased interest since the appreciation that this polymer is capable of blue luminescence. PPP with substitution at the 2 and 5 positions with alkyl chains was reported to be solution p r o c e ~ s i b l e . ~ ~ Soluble copolymers containing substituted and nonsubstituted PPP units have also been r e p ~ r t e d . ~More ~ , ~ recent ~ work has been with ladder73D.G. H. Ballard, A. Courtis, I. M. Shirley, and S. C. Taylor, J . Chem. SOC., Chem. Commun. 954 (1983). 74 D. G. H. Ballard, A. Courtis. I. M. Shirley, and S. C. Taylor, Macromolecules 21, 294

(1988). 7sD. L. Gin, V. P. Conticello, and R. H. Grubbs, J . Am. Chem. SOC. 116, 10,507 (1994). 7h M. Rehahn, A.-D. Schliiter, G. Wegner, and W. J. Feast, Polymer 30, 1054 (1989). 77 G. Leising, G. Grem, G. Leditzky, and U. Scherf, Proc. SPIE 1910, 70 (1993). 78 W.-X. Jing, A. Kraft, S. C. Moratti, J. Griiner, F. Cacialli, P. J. Hamer, A. B. Holmes, and R. H. Friend, Synth. Mer. 67, 161 (1994).


b

R1 H

25

H R1

FIG. 22. (a) Ladder-type polyphenylene. (b) Ladder-type polyphenylene with short p phenblene spacer units. [J. Huber, K. Miillen, J. Salbeck, H. Schenk, U. Scherf, T. Stehlin, and R. Stern. Acta Polym. 45, 244 (199411.

structure polyphenylenes of the type shown in Fig. 22(a).79 These polymers give efficient blue luminescence in solution, but the emission in the solid state is at lower energies due to interchain This problem has been overcome by the introduction of short p-phenylene spacer units [see Fig. 22(b)], which disrupt the interchain ordering.83 4. DOPEDCONDUCTING POLYMERS Although the interest in this article is directed primarily at polymers that are semiconducting, it is nevertheless important to mention the development of materials that are of greatest interest in their doped state, where they are formed as charge-transfer complexes. The two materials that show the greatest scope for low-cost fabrication with adequate levels of electrical conductivity and with proven stability against dedoping or other mechanisms for damage are the polypyrroles and the polyanilines. Polypyrroles are prepared by oxidation of pyrrole either by chemical or electrochemical means, and are generally formed in situ since they are not easily processed further. Oxidation at a level of around 0.4 electrons per pyrrole unit is achieved during polymerization, and this material shows a 79

U. Scherf and K. Miillen, Makromol. Chem. Rapid Commun. 12, 489 (1991). G. Grem and G. Leising, Synrh. Met. 55-57, 4105 (1993). XI J. Huber, K. Miillen, J. Salbeck, H. Schenk, U. Scherf, T. Stehlin, and R. Stern, Acta Polym. 45, 244 (1994). "J. F. Griiner, H. F. Wittmann, P. J. Hamer, R. H. Friend, J. Huber, U. Scherf, K. Miillen, S. C. Moratti, and A. B. Holmes, Synth. Met. 67, 181 (1994). to G. Grem, C. Paar, J. Stampfl, G. Leising, J. Huber, and U. Scherf, Chem. Muter. 7, 2 no

(1995).

26


conductivity of around 10 S cm-’, which is adequate for a number of applications. A considerable range of counter-ions can be used, and efforts have been made to select those that give the greatest stability to the charge-transfer complex formed with the polymer. A number of applications have been found for this material, including its use for through-hole plating of double-sided printed circuit boards,84 its use as the electrolyte in high-performance electrolytic capacitors,8’ and its use to provide electromagnetic screening when impregnated in a range of textiles.86 Polyaniline is an old material, which was revisited following the discovery of high levels of electrical conductivity in doped polyacetylene. It shows a more complicated range of phases than the polymers mentioned earlier because, in addition to supporting a range of oxidation states, it can also support different degrees of protonation at the nitrogen, and at least four phases are established, as indicated in Fig. 23. Of these, the oxidized and protonated form (emeraldine, named on account of its green color) is electrically conducting, and considerable efforts have been made to find ways to achieve a material that is resistant to deprotonation and is also processible from ~ o l u t i o n . ~ ~ - ~ ~ Polyaniline is synthesized by oxidation of aniline, and this can be achieved either chemically or electrochemically without great difficulty. It is not readily soluble in convenient solvents, through N-methylpyrrolidinone is a poor solvent for the emeraldine base form, and if this solvent is used the polymer needs subsequent treatment with acid to regain electrical conductivity. Alternatively, the conductive emeraldine phase can be processed from solution in strong acids such as sulfuric acid,” or can be processed from solution in meta-cresol if a suitable sulfonic acid is used to provide the counter-ion (e.g., camphor sulfonic acid).” The latter materi-

84

W. Metzger, J. Hupe, and W. Kronenberg, Plat. Surf. Finish. 77, 28 (1990). Y. Kudoh, M. Fukuyama, T. Kojima, N. Nanai, and S. Yoshimura, in Intrinsically Conducting Polymers: An Engineering Technology (M. Aldissi, ed.), NATO AS1 Series E, Vol. 246, p. 191, Kluwer, Dordrecht (1993). X6H.H. Kuhn, in Infrinsically Conducting Polymers: An Emerging Technology (M. Aldissi, ed.), NATO AS1 Series E. Vol. 246, p. 25, Kluwer, Dordrecht (1993). 87 A. G. MacDiarmid and A. J. Epstein, in Science and Applications of Conducting Polymers (W. R . Salaneck, D. T. Clark, and E. J. Samuelsen, eds.), p. 117, Adam Hilger, Bristol (1991). 88 A. J. Epstein, in Conjugated Polymers (J. L. BrCdas and R. Silbey, eds.), p. 211, Kluwer, Dordrecht (1991). nv A. J. Epstein, J. Joo, C. Y. Wu, A. Benatar, C. F. Faisst, J. Zegarski, and A. G . MacDiarmid, in Intrinsically Conducting Polymers: An Emerging Technology (M. Aldissi, ed.), NATO AS1 Series E, Vol. 246, p. 165, Kluwer, Dordrecht (1993). YO A. Andreatta, A. J. Heeger, and P. Smith, Polym. Commun. 31, 275 (1990). 91 Y.Cao, G. M. Treacy, P. Smith, and A. J. Heeger, Appl. Phys. Letf. 60,2711 (1992). 8s


27

leuco emeraldine

emeraldine base

pernigraniline

A'

A'

emeraldine salt

FIG. 23. Structures of polyaniline at different stages of oxidation and protonation. The conducting form is the emeraldine salt. The anions, X-,are selected to achieve stability or processibility in solution.

als can show high conductivities (above 100 S cm-l) and might be used to provide optically transparent conducting layers. This material has also been found to be particularly useful as a high work-function electrode in a range of semiconductor devices, as we discuss later.

5. MOLECULAR SEMICONDUCTORS EL devices using molecular organic materials were first demonstrated in the 1960s, but suffered from problems of high drive voltages and the need to use highly reactive electrodes. Thin-film devices, using Langmuir-Blodgett filmsg2or sublimed molecular films,93gave reduced drive voltages, but 92 G. G. Roberts, M. McGinnity, W. A. Barlow, and P. S. Vincett, Solid State Commun. 32, 683 (1979). 93 P. S. Vincett, W. A. Barlow, R. A. Ham, and G. G. Roberts, Thin Solid Films 94, 171 (1982).

28


C

FIG.24. Structures of some molecular semiconductors that have been used in thin-film EL devices. (a) Alq, is used as an electron-transport and emissive layer. (b) TPD is used as a hole-transport layer. (c) PBD is used as an electron-transport layer.

still remained inefficient. Interest in molecular organic materials was revived by Tang and Van who demonstrated efficient electroluminescence in two-layer sublimed molecular film devices comprising a holetransporting layer of an aromatic diamine and an emissive layer of 8-hydroxyquinoline aluminum (Alq,). Since the work of Tang and Van Slyke, a large number of other molecular materials have been used as the chargetransporting or emissive layer in LEDs. The structures of Alq, and TPD (a commonly used hole-transporting materialg5) are shown in Fig. 24, along with that of 2-(4-biphenylyl)-5-(4-tert-butylphenyl)-1,3,4-oxadiazole (PBD), an oxidiazole derivative that has frequently been used as an electrontransporting layer in organic L E D s . ~Compared ~ to conjugated polymers, the synthesis and purification of molecular semiconducting materials is relatively straightforward, and many useful materials are available commercially. In the context of conjugated polymer devices, molecular materials are of interest for a number of reasons. First, they are frequently used in combination with conjugated polymers, particularly as electron-transport ing materials. Second, the device physics is often very similar in polymer and molecular devices. Third, molecular materials provide well-defined systems that are useful in understanding the fundamental physics of 94

C. W. Tang and S. A. Van Slyke, Appl. Phys. Lett. 51, 913 (1987). M. Abkowitz and D. M. Pai, Phil. Mug. B . 53, 103 (1986). 96C.Adachi, S. Tokito, T. Tsutsui, and S. Saito, Jpn. J . Appf. Phys. 27, L713 (1988). 95

29


R FIG. 2.5. Structure of hexamers of polythiophene, a-sexithiophene (R a,w-di(hexy1)sexithiophene (R = C,H,,).

=

H), and

conjugated polymers. Of particular interest here are oligomers, that is, molecular materials comprising a well-defined number of polymer repeat units. Sexithiophene (Fig. 25),97 an oligomer of polythiophene, has recently attracted much interest both as a model system for understanding the photophysics of conjugated materials, and as the active layer in MISFET tran~istors.~'Gamier and co-workers have shown that mobilities as high as 7 x 10-' cm2 V-' s-l can be attained in substituted sexithiophene, giving performance approaching that of amorphous ~ilicon.'~ Amorphous sublimed films have a tendency to recrystallize after deposition, especially under the conditions experienced in device operation. Conjugated polymers, which are morphologically stable, overcome these problems, and also offer the advantage of being easy to deposit over large areas using solution-processing methods. 6. PHYSICAL PROPERTIES OF CONJUGATED POLYMERS

In the solid state, the physical properties of a polymer can be greatly affected by the packing of the polymer chains relative to one another. X-ray and electron diffraction have both been used to study the structure of conjugated polymers such as Durham-route p ~ l y a c e t y l e n e loo ~ ' ~and ~~~ PPV'o'.'02 on a molecular scale. Although the rigidity conferred on the polymer chains by the extended conjugation should encourage a high degree of clystallinity, the methods of preparation of the polymers often prevent ordering on a large scale. The solution-processed polymers of interest here generally show relatively little order. For example, spin-coated films of Durham polyacetylene show some local ordering, but the coherence length of the small crystal97J. Kagan and S. K. Aroda, J . 0%.Chem. 48, 4317 (1983). 98 G. Horowitz, X. Z. Peng, D. Fichou, and F. Gamier, Solid State Commun. 72, 381 (1989). YY C. S. Brown, M. E. Vickers, P. J. S. Foot, N. C. Billingham, and P. D. Calvert, Polymer 27, 1719 (1986). ""'H. Kdhlert and G. Leising, Mol. Crysr. Liq. Cryst. 117, 1 (198.5). ,Ill M. A. Masse, D. C. Martin, E. L. Thomas, F. E. Karasz, and J. H. Petermann, J . Muter. Sci. 25, 311 (1990). '"*J. H. F. Martens, E. A. Marseglia, D. D. C. Bradley, R.H. Friend, P. L. Burn, and A. B. Holmes, Synth. Met. 55-57, 434 (1993.; ibid. 449.

30


lites is no more than 3 to 4 nm." PPV prepared via the precursor route is more ordered, and the crystalline order extends over microcrystallites having typical dimensions of 5 nm.'"' To obtain detailed information on the structure, it is necessary to use aligned polymer films, which can sometimes be obtained by stretching during the conversion process for the precursor routes to polyacetylene and to PPV. It has been found that the degree of crystallinity can be increased considerably in the Durham polyacetylene by stretch-orienting the polymer during conversion, and this affects all its electronic properties, lowering the T-T* gap as the chain extension is improved!', lo", In PPV the degree of crystallinity, the alignment of microcrystallites, and the amount of disorder within crystallites are all strongly dependent on the type of precursor used, the processing conditions, and the sample geometry. In thin films, the degree of chain alignment within the plane of the film is critical in determining the anisotropy of the optical constants. A n example where both the optical and microstructural properties of PPV are affected by the choice of precursor route was demonstrated by Halliday et ~ 1 . ' "In~ this work, rigid conjugated segments were introduced into the precursor polymer before spin-coating, leading to enhanced structure in the optical absorption and emission spectra after conversion. Electron diffraction measurements on unstretched spin-coated films indicated significantly improved intrachain ordering, with the intrachain coherence length increased to 9 nm, compared to 4 nm for PPV prepared as by the method discussed in Section II.3.c. Other conjugated polymers also show microcrystalline structure. The structure of various PPV derivatives has been studied by Martens,"" and the poly(3-alkylthiophene)~have been investigated by several group^.'"^-'^^ As the fraction of alkyl chain increases in substituted polymers, the packing of these side-chains has an increasingly important effect on the overall structure. The structure of the poly(3-alkylthiophene)~is therefore strongly dependently on the regioregularity of the polymer.Io8 "I3J. H. F. Martens, K. Pichler, E. A. Marseglia, R. H. Friend, H. Cramail, and W. J. Feast, Synth. Met. 55-57, 443 (1993). 104 D. A. Halliddy, P. L. Burn, D. D. C. Bradley, R. H. Friend, 0. M. Gelsen, A. B. Holmes, A.Kraft, J. H. F. Martens, and K. Pichler, Adu. Mater. 5, 40 (1993). 105 M. Sato, T. Shimizu, and A. Yamauchi, Synth. Met. 41, 551 (1991). '"'K. Tashiro, K. Ono, Y. Minagawa, M. Kobayashi, T. Kawai, and K. Yoshino, J . Polym. Sci. B: Polym. Phys. 29, 1223 (1991). 107 T. J. Prosa, M. J. Winokur, J. Moulton, P. Smith, and A. J. Heeger, Macromolecules 25, 4364 (1992). IOU R. D.McCullough, S. Tristram-Nagle, S. P. Williams, R. D. Lowe, and M. Jayaraman, J . Am. Chem. Soc. 115, 4910 (1993).


31

The larger scale morphology of a polymer film is also important for device applications, with the ideal being a uniform, fully dense, pinhole-free film. With sufficient care and cleanliness in preparation, most films used for devices are found to be uniform under an optical microscope, both in transmission and fluorescence. Transmission electron micrographs of precursor-route PPV spin-coated onto NaCl substrates show structure on the 100-nm scale.Io2 The exact appearance again depends strongly on the sample preparation, but no samples were free of voids and defects. Less is known about the morphology of the PPV derivatives, but it is clearly important to bear in mind the film morphology when discussing device and interface characteristics. The mechanical properties of conjugated polymers have not been studied in great detail, but Smith and co-workers have demonstrated that the high tensile strengths expected for chain-extended polymers can be demonstrated when suitable control is exercised in the fabrication of fibers."'"-'" The thermal properties of conjugated polymers are highly dependent on the presence and nature of the side-groups. At one extreme, the precursor-route PPV shows little evidence for structural changes up to high temperatures (300 to 400"C), though some increase in amplitude of ring-rotation modes is seen."2 In contrast, the dialkoxy-substituted PPVs and the poly(3-alkylthiophene)~ are soft, and have low glass transition temperatures and relatively low melting temperatures; for example, 190°C for poly(3-hexylthiophene) and 80°C for poly(3-d0decylthiophene)."~ The stability of conjugated polymers to atmospheric attack from water and oxygen, in the dark and under illumination, varies very considerably among the range of polymers studied. Stability of the undoped polymers is generally higher than that of the doped polymers. Polyacetylene is particularly susceptible to oxidative degradation, and is not generally considered to be a stable material.Il4 In contrast, the polythiophenes and the PPVs are relatively stable in the dark, though they do show photo-oxidation, as has recently been reported for PPV."s IIIV

Y. Cao, P. Smith, and A. J . Heeger, Polymer 32, 1210 (1991). A. J . Heeger and P. Smith, in Conjugated Polymers (J. L. BrCdas and R. Silbey, eds.), p.

11(1

141, Kluwer, Dordrecht (1991). 111 S. Tokito, P. Smith, and A. J. Heeger, Polymer 32, 464 (1991). "'G. Mao, J. E. Fischer, F. E. Karasz, and M. J . Winokur, J . Chem. Phys. 98, 712 (1993). 113 G. Gustafsson, 0. Inganls, W. R. Salaneck, J. Laakso, M. Loponen, T. Taka, J.-E. Osterholm, H. Stubb, and T. Hjertberg, in Conjugated Polymers (J. L. BrCdas and R. Silbey, eds.), p. 315, Kluwer, Dordrecht (1991). 114 N. C. Billingham, P. D. Calvert, P. J. S. Foot, and F. Moharnmad, Polym. Degrudut. Sfubil. 19, 323 (1987). 115 F. Papadimitrakopoulos, K. Konstadininis, T. M. Miller, R. Opila, E. A. Chandross, and M. E. Calvin, Chern. Muter. 6, 1563 (1994).

32


111. Optical and Electronic Properties of Conjugated Polymers

7. INTRODUCTION The electronic structure of conjugated molecules and polymers can be conveniently described in terms of u bonding formed by overlap of sp2 hybrid orbitals, and rr bonding formed by overlap of p z orbitals on adjacent carbons. This description then allows a useful parameterization of the electronic properties in which the contribution of the u electrons provide the force constant for the carbon-carbon bonds, and in which the IT electrons are described using Huckel (tight-binding) methods. This approach has proved to be particularly important for describing the coupling of the lattice to the electronic excitations in the rr electrons caused by photoexcitation from 7r to IT* or by charge injection. However, the input parameters in such models need to be chosen empirically, and the use of more sophisticated quantum chemical calculations has been important in gaining an accurate description of the electronic structure. We briefly review both approaches here. The description of the electronic structure of conjugated molecules such as anthracene dates back several decades.2,8,''6 The overlap of p, orbitals on adjacent sites creates 7r molecular orbitals, and with one electron per site the lower lying half of these orbitals is occupied (IT, bonding orbitals) and the upper half is unoccupied ( r r * orbitals). Since the occupancy of the rr orbitals contributes to the bonding energy, carbon-carbon bond lengths are dependent on the electronic state. Therefore, promotion of electrons from IT to rr* states results in a change in equilibrium geometry of the molecule, and this is seen experimentally through the coupling of electronic and vibrational transitions. The absorption and emission from an organic molecule is represented in Fig. 26. In this figure, the energies of the ground and excited states are shown as a function of the configuration coordinate of the system, which represents the positions of the atomic nuclei. In general, the minimum of the excited state potential curve is at a different configuration coordinate to that of the ground state. Because electronic transitions take place on a much faster timescale than nuclear motions (less than lo-'' s compared to s), nuclei do not have time to change their configuraapproximately tion during an electronic transition." Electronic transitions are therefore

'

'I6L. Salem, 7he Molecular Orbital 7heoty of Conjugated Systems, Benjamin, New York (1966). "k.Keanvell and F. Wilkinson, in Transfer and Storage of Energy by Molecules ( G . M. Burnett and A. M. North, eds.) Vol. 1, Wiley, New York (1969).


33

FIG.26. Absorption and emission processes between the So and S, electronic states of an organic molecule. [A. Kearwell and F. Wilkinson, in Transfer and Storage of Energv by Molecules (G. M. Burnett and A. M. North, eds.), Vol. I, Wiley, New York (196911.

represented by vertical lines on Fig. 26 and are known as vertical, or Franck-Condon, transitions. The vibrational probability functions corresponding to the various vibrational excited states are also shown in Fig. 26. The probability of an electronic transition between two states depends on the overlap between the vibrational wavefunctions of the initial and final states. The vibrational part of the transition probability between vibra2 tional states I x,,)and I x n ) is given by the Franck-Condon factor ISmnl where Smn =

( XmI

xn)

(7.1)

The transition from the first vibrationally excited state of one electronic level to the vibrational ground state of another electronic level, for example, is known as a 1-0 transition.

34


A molecule in a vibrationally excited state will undergo radiationless relaxation to the lowest vibrational state. Experimentally this occurs very quickly, on timescales of approximately 10- l 3 s. Emission therefore occurs from the lowest vibrational state of the electronic excited state, and the absorption and emission spectra show a mirror symmetry, with the emission occurring at lower energy. The amount of oscillator strength in each of the vibrational peaks is highly dependent on the change in equilibrium configuration coordinate between ground and excited state, AQ;, that is, the amount of structural relaxation that occurs in the excited state. The amount of relaxation is conveniently described by the Huang-Rhys parameter, Siwhere

(7.2) w, is the phonon frequency, and M is the reduced mass for the mode Q,. The Huang-Rhys parameter gives the energy of relaxation in units of the phonon energy, hw;. Moving from molecules to polymers raises some interesting issues, since it is possible to describe the electronic structure within the framework of the molecular orbital description mentioned earlier, or by using the formalism of band theory as applied to a one-dimensional periodic chain. The crossover between the description of localized molecular orbitals to the fully delocalized orbitals implicit in band theory has been a source of great debate. It is now appreciated that the effect of the electron-lattice and electron-electron interactions, as evident in the optical properties of conjugated molecules (see Fig. 26), is to cause localization of excited electronic states on the polymer chain. These are variously described as solitons, polarons, bipolarons, or excitons, depending on the symmetry of the polymer chain and charge on the excitation, as we discuss in more detail in Section 111.9. The application of models for infinite isolated chains to measurements made on materials in which the polymer chains comprise relatively short straight conjugated sections, separated by conformational or chemical defects, requires some caution, and we discuss later how interchain interactions and disorder may modify these isolated-chain descriptions. A model for the electronic properties of an infinite one-dimensional chain of the polymer trans-polyacetylene was developed by Su, Schrieffer, and Heeger."'. This model and its refinements aimed at modeling polymers such as PPV are described in Sections 111.8 and 111.9. lit(

W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Left. 42, 1698 (1979). W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Phys Reu. R 22, 2099 (1980); erratum, 28, 1 I38 (1 983). 11')


35

FIG.27. The two alternative senses of bond alternation for a trans-polyacetylene chain.

8. ELECTRONIC STRUCTURE-GROUND STATE

a. Tight-Binding Models The simplest conjugated polymer, and therefore the easiest to model theoretically, is trans-polyacetylene. Because an infinite chain of transpolyacetylene has two equivalent structures with the same energy (Fig. 271, it is known as a degenerate ground state polymer. The band structure of trans-polyacetylene has been modeled by several groups'20, 12'; the most widely used model was developed by Su, Schrieffer and Heeger (SSH),"s,"9 and involves a tight-binding calculation for a polymer chain with cyclic boundary conditions, neglecting electron-electron interactions. The polymer chain is represented as a series of carbon atoms, with the u bonding represented as an effective spring constant, K , between neighboring atoms. Electron-phonon interactions are considered by introducing a nearest neighbor overlap integral into the Hamiltonian, which is expanded linearly about the equilibrium separation of neighboring atoms, t = to + a ( u , - u,, ,), where u, is the displacement of site n, and the u electrons are parameterized through the force constant for the bonds, K. This ' ~ be expressed by the SSH (Su, Schrieffer, model, due to Su et ~ 1 . , " ~ "can Heeger) Hamiltonian shown here:

(8.1)

The first term gives the kinetic energy of the rr electrons (c, and c,' are creation and annihilation operators at site n), the second term gives the kinetic energy of the lattice ( p , is the momentum of the atom at site n), and the third term gives the cr bond strain energy. The rr electron system of the trans isomer of polyacetylene can be considered as an example of a 120

M. J. Rice, Phys. Lett. 71A 152 (1979). A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, Rev. Mod. Phys. 60, 781 (1988).

121

36


- rla

- r12a

rl2a

xl a

-

r12a 0 d2a FIG.28. (a) Band structure for trans-polyacetylene before and after dimerization, in the extended zone scheme. (b) Band structure in the reduced zone scheme after dimerization, showing the energy gap, 2A. Occupied states are shown as thick lines; unoccupied states as thin lines.

one-dimensional, half-filled electron band, and the bond dimerization that is found for the ground state of the chain can thus be considered as the result of a Peierls distortion of the regular chain.8p'22This opens up an energy gap in the dispersion curve at k = 7r/2a, where a is the unit cell length before dimerization, as shown in Fig. 28(a). The band structure may then be represented in the reduced zone scheme as shown in Fig. 28(b). Within the Peierls electron-phonon coupling model for the chain dimerization and n - ~ * band gap, the gap (2AJ is determined by the dimen-

122

R. Peierls, Quantum Theory of Solids. Oxford University Press, UK (1955).

37


sionless electron-phonon coupling constant, A, through the relation’23 A.

=

8t,exp[-(l

+ 1/2A)],

(8.2)

where A is related to the parameters in the SSH Hamiltonian through A=-

2a2 r t ,K

’

(8.3)

Taking experimental values of 1.7 eV for the gap and 12 eV for the bandwidth, the value of A is required to be of order 0.2, rather larger than can be attributed to electron-phonon coupling alone,’23 and it is widely held that the enhanced bond dimerization is due to the effects of the on-site Coulomb repulsion energy, U,. This is shown, for example, in the work of Baeriswyl and Maki,’24 in which, for U, small ( < 4 t ) , bond dimerization is enhanced. It is possible to choose realistic values of A ( = 0.1) and U 0 / 4 t ( = 0.6-0.7) and to obtain a satisfactory value for the energy gap. In spite of the clear evidence for the importance of Coulomb interactions and their role in determining the properties of conjugated molecules, the SSH model and its continuum version125are widely used, particularly for the modeling of polyacetylene in its excited states with the various parameters (bandwidth, electron-phonon coupling, etc.) chosen empirically. The trans isomer of polyacetylene is a special case by virtue of its symmetry with respect to the sense of bond alternation. Most other polymers, including those listed in Table I, have a nondegenerate ground state, that is, a preferred sense of bond alternation. A modification to the SSH Hamiltonian that models this was introduced by Brazovskii and Kirova,126and introduces an “intrinsic” contribution to the gap parameters, Ai,which is added to the Peierls contribution that arises by the mechanism discussed earlier. For the polymers of interest, most of which are formed by the coupling of aromatic or heteroaromatic units, the r - r * gap is usually very much larger than that for trans-polyacetylene (see Table I) so that the contribution from the “intrinsic” gap is larger than that from the Peierls mecha123

D. Baeriswyl, D. K. Campbell, and S . Mazumdar, in Conjugated ConductingPolymers (H. l e s s , ed.), Vol. 102, p. 7, Springer-Verlag, Berlin (1992). 124D.Baeriswyl and K. Maki, Phys. Reu. B 31, 6633 (1985). 12’H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Reu. B 21, 2388 (1980). %. A. Brazovskii and N. N. Kirova, JEPT Lett. 33, 4 (1981).

38


nism. In this situation it is more realistic to begin with the molecular orbitals of the appropriate monomer units (e.g., benzene to form PPP). The polymer electronic structure is then built up by introducing nearest neighbor interactions between the monomer units. In PPP, for example, this interaction causes the broadening of the occupied n and unoccupied A orbitals to form rr and rr* bands in the polymer. As the intermolecular interaction is increased, the rr and rr* bands become broader, and the energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is lowered. b. Quantum Chemical Calculations The usefulness of the empirical Hiickel models discussed in Section III.8a derives from their correspondence to the results obtained from more sophisticated calculations. A wide variety of computational techniques has been used, ranging from ab initio calculations to highly parameterized semiempirical methods. '27 Calculations for polymers rapidly become computationally expensive unless the technique is carefully chosen. A method frequently used for conjugated polymers is the valence effective Hamiltonian (VEH) method, following optimization of the unit-cell geometry using the Hartree-Fock semiempirical AM1 (Austen Model 1) technique. The details of these techniques are described elsewhere.I2* These calculations provide many of the basic parameters that define the electronic structure relevant to the operation of semiconductor devices, including the energy gap between the rr and rr* states, the widths of the rr and rr* bands, and the energies of these bands with respect to the vacuum level. To obtain good fits to experimental data for conjugated polymers, the theoretical ionization potentials predicted by the VEH method must be reduced to allow for the polarization energy in the solid state and other effects not included in the model. Despite this shortcoming, VEH calculations remain a useful technique for predicting the effect of a change in chemical structure on the energy gap and ionization potential of a conjugated polymer. For example, ionization potentials and energy gaps are important for the description of the operation of LEDs, for which PPV is often used. The VEH method predicts values of ionization potential and energy gap of 5.05 and 2.32 eV, respectively, for PPV in the solid state.'*' The majority of calculations have been performed on isolated chains, and therefore give no information about the character of interchain 127

J. L. Brtdas, AdLl. Mater. 7, 263 (1995). J. L. Brtdas, in Handbook of Conducting Polymers (T. A. Skotheim, ed.), Vol. 2, p. 859, Marcel Dekker, New York (1986). 12Y J. L. Brtdas and A. J. Heeger, Chem. Phys. Left. 217, 507 (1993). I2X

39


interactions. However, the metallic levels of electrical conductivity observed in the doped polymers show that interchain charge transport can be easy, and it is the case that all the polymers that can be used to transport charge (as required for the device applications of interest in the present article) have structures that allow some overlap of the rr orbitals on adjacent chains. Direct calculations of these interactions require full structural information on the three-dimensional packing of the polymer chains, and this is not generally known. However, there are three-dimensional local density functional calculations for both polyacetylene and PPV,13' and these indicate that the transverse bandwidths are of order 0.4 eV for both materials.

"'

9. ELECTRONIC STRUCTURE-EXCITED STATES

The description of the rr electrons in conjugated polymers that we outlined earlier is one in which there is strong coupling between the chain geometry and the occupancy of the rr states. As is the case for small conjugated molecules (see Section 111.7), we expect a strong coupling between electronic and vibrational excitations, driven by the different equilibrium geometries in the ground and excited states. The special feature that needs to be considered for the polymers is the length over which these excited states are extended along the chain. The general finding from both theory and experiment is that these states are localized, and the form of this localization depends on the symmetry of the polymer chain. a. SSH Model

The fundamental excitations of the Peierls-distorted chain with a half-filled band are known to be phase kinks, or solitons, in the pattern of the bond alternation. This was shown for polyacetylene"8-'20 to take the form of the bond alternation defects shown in Fig. 29. An important insight into the nature of these excitations from the work of Su et ~ 1 . " ' ~is~that ' ~ the bond alternation defect is not localized at a single carbon site, as indicated for convenience in the schematic representation in Fig. 29, but is spread over some 10 to 15 carbon sites. This delocalization is crucial to the energetics of the stabilization of the soliton and is clearly demonstrated experimentally. For this situation, it is possible to use a continuum model I311 131

1'. Vogl and D. K. Campbell, Phys. RaI. Lett. 62, 2012 (1989). P. Gommes da Costa, R. G. Dandrea, and E. M. Conwell, Phys. Rer. B . 47, 1800 (1993).


40

FIG.29. Schematic representation of the formation of a pair of solitons on a polyacetylene chain. (a) shows the undistorted chain and (b) shows the associated band scheme. The interband T-T* optical transition is shown as a dotted arrow. If two electrons are introduced onto the chain, initially into the conduction band as shown in (b), the chain relaxes to the form shown in (d), with a reversed sense of bond alternation in the center of the chain separated by two solitons. Associated with the two solitons are nonbonding T states in the gap, created from one (doubly occupied) valence and one (empty) conduction band state. These two states are doubly occupied, as shown in (c), and each carries a negative charge. New optical transitions from the soliton level to the conduction band are indicated with a dotted arrow; the oscillator strength for the interband transition is weakened through the loss of the band states. A complementary picture holds for positive charge, which is accommodated as unoccupied soliton levels.

for the polyacetylene chain, within which various simple analytic results are found. Thus, the gap parameter A varies through the soliton as A(x)

=

A, tanh(x/&)

(9.1)

with the soliton half-width 5 = ( h u , / A , ) a where u , is the Fermi velocity, ~ extra lattice energy associated with the formation of or ( 2 t , / A , ) ~ . ' ~The the soliton is (2/7r)A, and the dynamical mass M, of the state described by Eq. (9.1) is given by (9.2) where 0, is an averaged bare phonon freq~ency.'~'The term A, and hence & and M, depend very sensitively on the value of A [through Eq. 132

B. Horovitz, Solid State Commun. 41, 729 (1982).


41

(8.211. The soliton has associated with it an energy level which within one-electron theory lies at mid-gap and is of p , nonbonding character. This is as shown in Fig. 29, where, for the injection of negative charge, this state is doubly occupied. The corresponding positively charged state is unoccupied. Within the SSH model and its continuum version, storage of added charge on the mid-gap state created with the soliton is energetically favored over storage in the band states; the energy of creation of the soliton is 2A/7r, but the availability of the mid-gap state allows storage of the charge in a state of energy A lower than the appropriate band edge state. Thus the energy of stabilization of the soliton over charge storage in the band edges is A(l - 2/77); this is about 0.3 eV taking the experimentally measured value of A = 0.8 eV. We note here that this has direct consequences for the operation of devices such as the MIS structure; charges injected onto the chains are stored in soliton states, and the threshold for injection into these states is at energies of kA(1 - 2 / n ) relative to the gap center, rather than at the conduction and valence band edges, at + A . We return to this in more detail in Section V.15. The formation of solitons through the addition of charge to the polymer chain modifies the optical properties, with new optical transitions allowed between the mid-gap soliton levels and the band edges, as indicated for the negative soliton to conduction band transition in Fig. 29. The dipole matrix element for this transition is large, and is enhanced above that for the T - V * interband transition by a factor of 7r2 [ / u , where u is the average carbon-carbon spacing projected along the chain.'33*134 The effect of Coulomb interactions, characterized through the on-site Coulomb repulsion, Uo, is to modify the energies of the electronic transitions between band states and the localized soliton state^.'^'-'^' Transitions involving charged solitons are reduced in energy, and modeling the effects of U as a perturbation in the one-electron (9.3)

133

N. Suzuki, M. Ozaki, S. Etemad, A. J. Heeger, and A. J. MacDiarmid, Phys. Reu. Lett. 45, 1209 (1980); erratum, 45, 1463 (1980). 134 S. Kivelson, T.-K. Lee, Y. R. Lin-Liu, I. Peschel, and L. Yu, Phys. Reu. B 25, 4173 (1982). 13sZ. Vardeny and J. Tauc, Phys. Reu. Left. 54, 1844 (1985). 136 Z . Vardeny and J . Tauc, Phys. Reu. Lett. 55, 1510 (1986). 137 D. Baeriswyl, D. K. Campbell, and S. Mazumdar, Phys. Rev. Lett. 56, 1509 (1986).

42


A value of aU0/35 of about 0.4 eV is consistent with the values of A E obtained from photoinduced absorption m e a ~ u r e m e n t s . 'Associated ~~ with the formation of the mid-gap soliton absorption band is the loss of 7~ and T* band states and the bleaching of the T-T* interband transitions. In addition to the electronic excitations of the soliton, there are also new vibrational modes associated with translation of the soliton (T modes) and with shape or amplitude modes of the soliton (A modes). The T modes are associated with the coupling of those vibrational modes of the (CHI, unit cell that modulate the bond dimerization amplitude to the translation of the soliton along the hai in.'^'*'^^ These modes appear as Raman-active modes for the dimerized chain, with frequencies determined by the phonon response function,

(9.4) where A,/A are the relative coupling parameters for the modes that couple to the bond dimerization ( n = 1 to 3 for polyacetylene) and w," are the bare phonon frequencies for these modes. D,(w) determines the frequencies of the Raman modes of the dimerized chain, through D,(w) = - 1/(1 - 2A) where A is the effective electron-phonon coupling constant. It determines the frequencies of the T modes of the soliton, through D o ( @ )= - l / ( l - a > where a is a parameter that characterizes the pinning of the soliton due to disorder, etc., and, finally, it determines the frequencies of the Raman-active A modes of the soliton, through D o ( o ) = - 1/(1 - 2KA), where K is calculated to be 0.723.'30-'4' b. Polymers with a Nondegenerate Ground State In polymers with a nondegenerate ground state, such as PPV, the two alternative senses of bond alternation do not have equivalent energies (Fig. 30). The soliton is clearly not a stable excitation in these materials, since the high-energy form can only exist over a finite length of chain. The charged excitations of a nondegenerate ground state polymer are termed polurons or bipolarons and represent localized charges on the polymer chain, with an accompanying local rearrangement of bond alternation, as shown in Fig. 31. These states may be considered as being equivalent to a 138E.Ehrenfreund, Z. Vardeny, 0. Brafman, and B. Horovitz, Phys. Re[>.B 36, 1.53.5 ( 1 987).

C. Hicks and G. A. Blaisdell, Phys. Re[).B 32, 919 (lY85). A. Terai, Y. Ono, and Y. Wada, J. Phys. SOC. Jpn. 55, 2889 (1986). I4lA. Terai, Y. Ono, and Y.Wada, Synth. Met. 28, D3.53 (1989).

'"J.

1411


43

benzenoid

4 quinoid

-

-

-

FIG.30. Benzenoid and quinoid forms of PPV. The quinoid form is of higher energy, and is therefore not stable over large lengths of polymer chain.

confined soliton pair, and in this model, the two nonbonding mid-gap soliton states form bonding and antibonding combinations, thus producing two gap states symmetrically displaced about mid-gap ( k o,)as shown in Fig. 32. These levels can be occupied by 0, 1, 2, 3, or 4 electrons, giving a positive bipolaron (bp2+1, positive polaron (p'), polaron exciton, negative polaron (p-), or negative bipolaron (bp2- 1, respectively. The presence of polarons or bipolarons gives rise to new optical absorptions below the energy gap, as shown in Fig. 33. In the continuum models of Brazovskii ~ ~ FBC model), the preferred sense of and Kirova'26 and Fesser et ~ 1 . l(the bond alternation is introduced through an extrinsic contribution to the gap parameter, A, as

A

=

A,,

+ A,,

(9.5)

where A,, is the contribution to the gap due to the Peierls mechanism, and A , is the extrinsic contribution. It is useful to define a confinement parameter, y , as (9.6)

Y

FIG.31. Negative charge on a PPV chain, represented as a region of quinoid character. I4*K. Fesser, A. R. Bishop, and D. K. Campbell, Phys. Reu. B 27, 4804 (1983).

44


FIG.32. Polaron, bipolaron, and singlet exciton energy levels in a nondegenerate ground state polymer (energy gap 2A). Luminescence from the singlet exciton is also shown.

where A is the effective electron-phonon coupling constant, such that A

=

A,exp(y).

(9.7)

The extent of the polaron along a polymer chain and the position in the gap of the localized energy levels are determined in these one-electron models by the size of y. The case of near degeneracy allows the polarons to be relatively extended with electronic levels near the center of the gap, whereas strong breaking of the degeneracy keeps them much more compact, with levels near the band edges. Hence, the degree of nondegeneracy can be conveniently parameterized by an experimentally accessible ratio o o / A , where 2 0 , is the separation between the intragap polaron levels and 2A is the energy gap. For the case of those excitations for which the occupations of the upper and lower intragap states are equal (the polaron exciton and bipolaron), 0, is related to y through 142 -sin-'( 0 0

T)

(9.8)

1-

P+

-

b p2+

FIG.33. Sub-gap optical transitions due to the presence of a positive polaron and a

positive bipolaron.


45

For the case of the singly charged polaron, the gap states are more closely tied to the band edges, with a limiting separation of + A / 6 about the center of the gap for y = 0. The greater degree of relaxation of the bipolaron over the polaron stabilizes the coalescence of two like-charged polarons to form a bipolaron. The addition of Coulomb interactions to the problem introduces several new aspects to the behavior. It is widely observed that where there are optical transitions associated with the two band to gap transitions shown for the bipolaron in Fig. 33, the two transitions are of approximately equal intensity. This, and their asymmetric displacement about mid-gap is inconsistent with the predictions of the noninteracting electron continuum 14* Calculations of the effects of inclusion of on-site and intersite electron-electron interaction terms in the model H a m i l t ~ n i a n ' ~ ~show -~~' that weak on-site interactions do not alter the predictions of the noninteracting model much for the relative transition intensities, and only when the strong coupling limit is approached can the theories obtain intensity ratios consistent with experiment. Weak interactions can, however, explain the asymmetric displacement of the levels. Within a perturbation theory treatment,'43 the two bipolaron transitions are found at energies: (9.9)

where Ueff = aU0/31 with U, the bare Coulomb repulsion for two electrons on one p r orbital, and l / a the length in units of carbon-carbon bonds for the bipolaron. An estimate of the interaction strength, U,,, can thus be obtained from the observed difference between the gap energy and the sum of the two bipolaron transition energies. Though this model has been used often, there are problems in making quantitative comparison with experiment. A number of factors may be important; Choi and Rice146have used a more realistic model for the form of the 7~ bonding, and have been able to get a more satisfactory description of the relative strengths of the optical transitions between the band edges and the gap states (see Fig. 33) than that predicted by the FBC model. c. Extensions of the Simple Models The quantum chemical techniques that have been used to describe the ground state electronic structure have also been used with considerable 143

D. K. Campbell, D. Baeriswyl, and S. Mazumdar, Synth. Met. 17, 197 (1987). U. Sum, K. Fesser, and H. Biittner, Solid State Commun. 61, 607 (1987). I4'U. Sum, K. Fesser, and H. Buttner, Phys. Rev. B 38, 6166 (1988). I4'H.-Y. Choi and M. J. Rice, Phys. Rev. B 44,10521 (1991). 144

46


success to model the electronic structure and chain geometry of charged excited states.128,147-'5" In general, these results have provided confirmation that the simple parameterizations used in the Huckel methods are useful. However, in several areas the Huckel models are seen to fail. Many of these problems are due to the neglect of exchange and correlation effects caused by electron-electron interactions. In particular, the simple model predicts that the energy of photons emitted by luminescence should be equal to the spacing between the two bipolaron levels. Experimentally, the luminescence is found to occur at significantly higher energy than the bipolaron level spacing as determined from photoinduced absorption. 15' According to the SSH model, the first excited state of a conjugated polymer is coupled radiatively to the ground state only through a one-photon transition, implying that all the photoexcited states should decay radiatively. Nonradiative decay, which occurs to some extent in all conjugated polymers, requires the breaking of charge-conjugation symmetry. This can be induced by impurities or by other defects in the solid state, and can also be achieved by the introduction of symmetry-breaking terms such as electron-electron interactions into the Hamiltonian. Some conjugated polymers, including trans-polyacetylene, exhibit no luminescence, implying solely nonradiative decay. It has been suggested that this is due to the lowest energy excited state having the wrong parity to couple radiatively to the ground state,'52contrary to the predictions of the SSH model. Several attempts have been made to include electron-electron interactions144, 153,154 and other symmetry-breaking effect^'^^.'^^ into the SSH framework; however, the importance of these corrections remains a matter of controversy. The effects of interchain interactions on the stabilization of polarons in conjugated polymers have been considered in the three-dimensional density functional calculations performed on polyacetylene"" and PPV.'3'.156 These calculations indicate that the polaronic effects as modeled in the 147

J. L. BrCdas, R. R. Chance, and R. Silbey, Phys. Reu. B 26, 5843 (1982). J. L. BrCdas, B. ThCmans, J. G. Fripiat, J. M. AndrC, and R. R. Chance, Phys. Rec. B 29, 6761 (1984). 149 J. L. BrCdas and G. B. Street, Acc. Chem. Res. 18, 309 (1985). ISII P. Broms, M. Fahlman, K. Z. Xing, W. R. Salaneck, P. Dannetun, J. Cornil, D. A. dos Santos, J. L. BrCdas, S. Moratti, A. B. Holmes, and R. H. Friend, Synfh Met. 67, 93 (1994). I5 I R. H. Friend, D. D. C. Bradley, and P. D. Townsend, J . Phys. D 20, 1367 (1987). 152 Z. G. Soos, S. Etemad, D. S. Galvio, and S. Ramasesha, Chem. Phys. Len. 194, 341 (1992). "'M. Gabrowski, D. Home, and J. R. Schrieffer, Phys. Re[,. B 31, 7850 (1985). 154 M. Sasai and H. Fukutome, Prog. Theor. Phys. 73, 1 (1985). '"P. L. Danielson and R. C. Ball, J . Physique 46, 1611 (1985). Is6H. A. Mizes and E. M. Conwell, Phys. Rev. Len. 70, 1505 (1993). 14n


47

isolated chain models may be considerably less strong when interchain interactions are taken into account, and the same concern has been raised by emir^.'^', '51 d. Excitons Though the description of the excited states of the conjugated polymer chain has been couched in the framework of a noninteracting electron model, with coupling to the lattice, we have already indicated that electron-electron interactions play a very important role in description of the electronic structure, for both ground and excited states. The effects of electron-electron interactions are to a considerable extent taken into account in the description of the ground state by the empirical parameterization of these Huckel models. However, this breaks down in the description of the excited states, particularly for the neutral excited states, which are usually excitonic in nature. Excitonic effects are well known in many solid-state systems, including ionic crystals, 111-V semiconductors, and molecular organic crystals.', 1599 These effects become particularly important when the exciton binding energy is large, as predicted for PPV in the presence of strong electron-electron interactions. l6 ' A n exciton can be considered as a bound electron-hole pair, and may be classified as a Frenkel exciton if the electron-hole pair is located on one molecular unit, or as a Mott-Wannier exciton if it extends over many molecular units. The intermediate case, where the exciton extends over a few adjacent molecular units, is termed a charge-transfer exciton. The description of the absorption and emission from a conjugated molecule, as illustrated in Fig. 26, is inherently that of a Frenkel exciton. Because the molecular orbitals are necessarily confined to a small region (the size of the molecule), the density of the excited state wavefunction on the carbon-carbon bonds is high, so that there can be significant rearrangement of the molecular geometry in this excited state. The strong coupling of the electronic and vibrational excitations that is illustrated in Fig. 26 therefore follows. '"D. Emin, Phys. Rev. B 33, 3973 (1986). isx D. Emin, in Handbook of Conducting Polymers (T. J. Skotheirn, ed.), Vol. 2, p. 915, Marcel Dekker, New York (1986). 159 J. T. Devreese and F. Peeters, eds., Polarons and Excitons in Polar Semiconductors and Ionic Crystah, NATO AS1 Series B, Vol. 108, Plenum Press, New York (1984). IhO D. C. Reynolds and T. C. Collins, Excitons. Their Properties and Uses, Academic Press, San Diego, (1981). I6 I Z. Shuai, J . L. Br&das,and W. P. Su, Chern. Phys. Lett. 228, 301 (1994).

48


When molecular units are brought together to form a solid, it is necessary to consider whether the exciton can extend over many molecular units. The Mott-Wannier exciton may be considered a hydrogen-like bound state of an electron and a hole, producing a series of energy levels given by

G E = Ec - n2

f o r n = 1 , 2 , 3 ,...,

(9.10)

where EG is the energy required to form a “free” electron and hole, and G is the binding energy of the exciton, given as (9.11)

where p is the reduced effective mass of electron and hole (1/p = l/m, + l/mh), E, is the relative permittivity, rnp is the proton mass, and R, is the Rydberg constant (13.6 eV). This gives the exciton radius r as r=

4rrE”Qfi*

w2

n2.

(9.12)

The condition for the Mott-Wannier theory to be applicable is that the exciton radius be considerably larger than the intermolecular spacing. This condition is met when the effective mass is low and the relative permittivity is large, as is the case for the group IV and 111-V semiconductors. Molecular crystals formed with molecules such as anthracene have high effective masses (since intermolecular contacts are not large) and relatively low relative permittivities. They are therefore not described by Mott-Wannier theory, the excitons being substantially confined to individual molecular units. The case of the conjugated polymers is interesting in that the effective mass for motion along the polymer chain is relatively low. The Mott-Wannier treatment has been applied to PPV, which is treated as an anisotropic semiconductor with anisotropic effective masses ( p,,= 0.0421 m e , p I = 2.66m,) and anisotropic relative permittivities ( E , , = 8, = 3), by Gommes da Costa and Conwe11.162The results of this calculation give an exciton with an anisotropic ellipsoid, of extension about 2 nm along the 162

P. Gommes da Costa and E. M. Conwell, Phys. Reu. B 48, 1993 (1993).


49

chain and 0.4 nm transverse to the chain, and binding energy [equal to C in Eq. (9.101 of about 0.4 eV. This value of binding energy and degree of localization gives a strongly “molecular” character to the exciton which is usually assumed to be confined to a single chain. Therefore, electron-lattice interactions are also strong, so that absorption and emission spectra show vibronic coupling. The inclusion of electron-lattice interactions is made using the Fesser, Bishop and Campbell one-electron description of polar on^,^^* and the neutral excited state (commonly termed the polaron exciton) is as shown in Fig. 32. However, serious problems arise when trying to obtain quantitative agreement with this model.’51 A more complete description of the exciton must therefore take into account both the Coulomb and the electron-lattice interactions, and there have been several computations that take both into account. One consequence of the electron-electron interaction is that singlet and triplet excitons are no longer of the same energy, nor of the same size. The expectation that the triplet exciton is considerably more localized than the singlet exciton (as a result of the reversed sign of the exchange interaction) is confirmed in calculations for PPV and its oligomers by Shuai et ~ 1 . ’ ~ ’ and by Beljonne et ~ 1 . Beljonne l ~ ~ et al. use a semiempirical intermediate neglect of differential overlap (INDO) Hamiltonian with configuration interaction techniques to show that the triplet exciton in PPV is stabilized by 0.65 eV with respect to the singlet exciton, and is localized over not much more than a single polymer repeat unit, whereas the singlet is considerably more extended. They are also able to calculate higher excited states, including a higher lying triplet which is measured experimentally in photoinduced absorption (see Section 111.10.b).The arrangement of ground and excited state energies for PPV is shown in Fig. 34. We have limited discussion so far to the description of low-lying excitations. It is now established that the modeling of higher energy excitations is only realistic in models that include electron-electron interactions. Thus, the UV absorption spectrum of PPV can be described using configuration interaction methods’64 or by Pariser-Parr-Pople methods.’65 Chandross et al.I6’ consider that strong Coulomb interactions are necessary to describe these properties, and they propose a binding energy for the singlet exciton as high as 0.9 eV.

163

D. Beljonne, Z. Shuai, R. H. Friend, and J. L. BrCdas, 1.Chem. Phys. 102, 2042 (1994). J. Cornil, D. Beljonne, R. H. Friend, and J. L. BrCdas, Chem. Phys. Left.223, 82 (1994). I65 M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, 2. V. Vardeny, E. W. Kwock, and T. M. Miller, Phys. Rev. B 50, 14702 (1994). 164

50

;:'...:

NEIL C. GREENHAM AND RICHARD H. FRIEND singlet

triplet

absorption

intersystem crossing absorption

1.4 eV

T

luminescence

SO FIG.34. Electronic transitions in PPV, showing both singlet and triplet states. Values for S"-S' and T'-T* are as determined experimentally.25 has been estimated by Beljonne et al. to be 0.65 eV. [D. Beljonne, Z. Shuai, R. H. Friend, and J . L. BrCdas, J . Chern. Phys. 102, 2042 (1994).]

We have indicated that the ability of the exciton to select its extension along the polymer chain allows it to minimize its energy without recourse to extension to neighboring chains. This does seem to be appropriate for those polymers that have smaller r r r * energy gaps, but polymers with gaps in the blue or UV have poor intrachain delocalization and can show changes in optical properties as they are brought together in the solid. Interchain interactions are therefore important for these materials, and lead to the formation of more extended excited states, which would be described as charge-transfer excitons within the framework of the molecular semiconductors.2 If the intermolecular contacts are optimized via a geometrical change following excitation, such excitons are described as excimers (where the exciton extends over identical molecular units) or exciplexes (where the exciton extends over two or more different molecular units). Excitons are mobile within the solid, and their motion, either coherent or diffusive, plays a very important part in the photophysics of conjugated polymers. The theory of Frenkel excitons in molecular organic crystals is traditionally developed by the tight-binding method,' using a basis set of wavefunctions each with the excited state localized on a different molecule. Introduction of nearest neighbor interactions between molecules produces an exciton band, in which the exciton is a delocalized Bloch state of the crystal. If there is significant electron-phonon interaction, the exciton


51

rapidly loses its coherence, and it is more appropriate to think of it as a localized state that moves by hopping between sites. In practice, disorder in site energies plays a very important part in causing this loss of coherence. Hopping between sites can still occur, either by tunneling (requiring overlap of the wavefunctions of the initial and final states) or by the resonant process known as Forster transfer. Forster transfer is mediated by a near-field dipole-dipole interaction, and does not require the overlap of initial and final wavefunctions. The original theory by Forster'66 has since been applied and extended by several authors to study exciton diffusion in organic systems.', 167, 168 Note that the Forster process operates most effectively when the exciton has a strong dipole matrix element to the ground state, and contributes very significantly to the diffusion of singlet excitons in polymers such as PPV. In contrast, this process is ineffective for triplet excitons, which can therefore only hop by the tunneling mechanism. 10. EXPERIMENTAL RESULTS

This section reviews some of the experimental observations of the optical properties of conjugated polymers. Polyacetylene is a special case because it has a degenerate ground state and therefore has soliton-like excitations. From an experimental point of view, the absence of strong luminescence restricts the information that can be obtained from optical measurements. We concentrate, therefore, on PPV, since it is representative of the larger class of polymers with nondegenerate ground states and because it has been widely studied. A compilation of measured spectra is shown in Fig. 35 for PPV with good intrachain ordering.'69 a. Optical Absorption and Emission Typical optical absorption and emission spectra for PPV are shown in Fig. 36. As expected, the emission occurs at lower energy than the absorption, and is assigned to the radiative decay of singlet excitons. In contrast to the symmetric spectra shown in Fig. 26, the absorption here is much broader than the emission, and shows less well-pronounced vibrational structure. ""T. Forster, Ann. Phys. VI 2, 55 (1948). I67 R. S. b o x , 77zeory of Excifons, Academic Press, New York (1963). I68 V. M. Kenkre and R. S. b o x , Phys. Rev. B 9, 5279 (1974). IhY K. Pichler, D. A. Halliday, D. D. C. Bradley, P. L. Burn, R. H. Friend, and A. B. Hnlmes, J . Phys.: Condens. Matter 5, 7155 (1993).

52

NEIL C. GREENHAM AND RICHARD H. FRIEND C

.-

EL.-

d04e+ U

m.32 C.

3 EL.' u 2e .c Q

Qga Egx

Aid k UU+"

'5%

+g

gb d U

0.5

I

I

I

I

1

I

1.5

2

2.5

3

Energy I eV

10-l~

3.5

FIG. 35. Absorption, photoinduced absorption, electroabsorption, and photoconductivity spectra for PPV with good intrachain ordering. [D. A. Halliday, P. L. Burn, D. D. C. Bradley, R. H. Friend, 0. M. Gelsen, A. B. Holmes, A. Kraft, J. H. F. Martens, and K. Pichler, Adu. Muter. 5, 40 (1993).]

This can be explained in terms of the range of conjugation lengths present in PPV, with the shorter conjugated segments having electronic transitions shifted to higher energies due to confinement of the excited states. Absorption samples the whole range of conjugation lengths, and is therefore broadened, whereas the excitons responsible for emission are able to diffuse between conjugated segments during their lifetime, and will move preferentially onto the low-energy, better ordered segments where they decay. Exciton diffusion is well known in molecular organic systems,' and accounts for the red shift observed in emission from conjugated polymers

5

Energy I eV

FIG.36. Absorption (line) and emission (circles) spectra of PPV at 300 K.


53

x W

C

.-'E v)

W

8

/-

i Localisation ;threshold

Excitation Energy FIG.37. Site-selective fluorescence results showing the 0-0 emission peak energy versus excitation energy. [After U. Rauscher, H. Bassler, D. D. C. Bradley, and M. Hennecke, Phys. Reis. B 42, 9830 (1990).1

on short timescales after photoe~citation.'~"17* Fu rther evidence for the importance of this process in conjugated polymers is provided by the site-selective fluorescence experiments of Rauscher et In this experiment, a tunable, narrow-linewidth laser is used to excite the polymer. The emission spectrum is then measured as the excitation wavelength is moved through the absorption spectrum. The energy of the 0-0 emission peak is shown schematically in Fig. 37 as a function of the excitation energy. For excitation above a certain energy, known as the localization threshold, the emission energy is independent of the excitation, whereas below this value there is a linear relationship between the emission and excitation energies. This result is explained in terms of diffusion of excitons within a Gaussian density of states. Excitons with energies above the localization threshold can hop to conjugated segments with lower energies, until they reach an energy where the hopping rate is less than the decay rate, giving emission at this energy. Excitons excited below the localization threshold have little chance to hop to lower energy segments, and therefore decay quasiresoargue that the difference in energy between absorpnantly. Heun et 170 I. D. W. Samuel, B. Crystall, G. Rumbles, P. L. Burn, A. B. Holmes, and R. H. Friend, Synth. Met. 54, 281 (1993). 171 U. Lemmer, R. F. Mahrt, Y. Wada, A. Greiner, H. Bassler, and E. 0. Gobel, Appl. P h y ~ Lett. . 62, 2827 (1993). I7'R. Kersting, U. Lemmer, R. F. Mahrt, K. Leo, H. Kurz, H. Bassler, and E. 0. Gobel, Phys. Reu. Lett. 70, 3820 (1993). 173U. Rauscher, H. Bassler, D. D. C. Bradley, and M. Hennecke, Phys. Reu. B 42, 9830 ( 1990). 174 S. Heun, R. F. Mahrt, A. Greiner, U. Lemmer, H. Bassler, D. A. Halliday, D. D. C. Bradley, P. L. Burn, and A. B. Holmes, J . Phys.: Condens. Matter 5, 247 (1993).

54


tion and emission below the localization threshold (0.01 eV in standard PPV) can be accounted for by coupling of the 0-0 transition to low-energy vibrational modes of the conjugated segments. The shape of absorption and emission spectra in conjugated polymers has been found to depend strongly on the amount of disorder in the sample. An example of this is described by Halliday et. where PPV prepared with a significantly improved intrachain ordering showed a sharpening of vibronic structure in absorption and emission, with a shift of oscillator strength into the 0-0 transition. Hagler et ~ 1 have . studied ~ ~ the absorption and emission spectra of MEH-PPV blended in a polyethylene gel. Tensile drawing of the blends produces a similar sharpening of the vibronic structure and shift of oscillator strength. A Franck-Condon analysis has been used to fit these data to a one-dimensional SSH model, broadened by disorder. These authors argue that the site-selective fluorescence data of Rauscher et al. can be attributed to the existence of a “mobility edge” similar to that seen in amorphous inorganic semiconductors, where excitation below this edge produces localized, immobile states, as opposed to the mobile band states produced by excitation above the edge. Electroabsorption (EA) spectroscopy, where the modulation of the absorption spectrum by an applied electric field is studied, has been used extensively to examine the electronic structure of organic materials. The EA spectra of PPV (see Fig. 35) and MEH-PPV can be approximated at low energies by the first derivative of the absorption spectrum.’75 By comparison with the first derivative EA lineshape in crystalline polydiacetylene, where the absorption is definitely excitonic in origin, this has traditionally been interpreted as a Stark shift of the excitonic absorption. Recently, however, Hagler et al.’76have argued that the same first derivative lineshape can be obtained from a band model using perturbation theory. b. Photoinduced Absoiption Photoinduced absorption is used to study the presence of new absorptions below the energy gap after photoexcitation. This technique has been used extensively for trans-p~lyacetylene,’~~ and we show in Fig. 38 results obtained on polymer prepared as an unoriented thin film by the Durham precursor route. Note that this form of polyacetylene is relatively disordered and that the optical transitions are seen at higher energies than in 175

0. M. Gelsen, Ph.D. Thesis, University of Cambridge (1993). T. W. Hagler, K. Pakbaz, and A. J. Heeger, Phys. Rev. B 49, 10,968 (1994). 177 J. Orenstein, Z. Vardeny, G. L. Baker, G. Eagle, and S. Etemad, Phys. Reu. B 30, 786 (1984). 176

55

SEMICONDUCTOR DEVICE PHYSICS I

I

33

x 10

- LT

2-

T

0

loo0

2000 3ooo 4000 probe energy (cm9

\.,-,.

-2

0.6

10

1.4 18 Probe Energy (eV)

::,..:.+

.At--.

2.2

2

(b) FIG.38. Photoinduced optical absorption spectrum for unoriented Durham polyacetylene, (a) measured at 77 K with excitation at 488 nm [R. H. Friend, H. E. Schaffer, A. J. Heeger, and D. C . Bott, J . Phys. C 20,6013 (1987)], and (b) measured at 20 K with excitation at 457.9 nm. [P. D. Townsend and R. H. Friend, Phys. ReLl. B 40, 3112 (1989). IOP Publishing Limi tcd]

better ordered material. The near-IR and visible spectra show bleaching of the interband T ~ T *transitions, peaking near 2.1 eV, and induced absorption bands at 0.5.5 and 1.6 eV. The band at 0.5.5 eV is due to the formation of long-lived separated charges, which are stored as charged solitons, and the absorption then arises as the optical transition between the mid-gap

56


soliton level and the band edges, as indicated in Fig. 29. Note that since this is a transition involving an initially charged soliton level, the transitions are reduced in energy from mid-gap by the Coulomb term shown in Eq. (9.3). The absorption band at 1.6 eV is less easily assigned, and is generally considered to be extrinsically stabilized. The induced absorption spectrum in the IR shows again the “mid-gap’’ absorption due to electronic excitation of the charged soliton, and shows in addition three vibrational feature: a broad band at 625 cm-’, a weak feature at 1287 cm-’, and a strong feature at 1373 cm-’. These three absorptions arise from the translation of the charged soliton along the chain; the band at 625 cm-’ is expected at zero frequency in the absence of disorder (the Goldstone mode) and the other two arise from coupling of the translation to the Raman-active modes of the chain that modulate the dimerization amplitude, as given by Eq. (9.4). The relative strengths of these vibrational and electronic features provide a means for measuring the effective mass of the charged soliton, via Eq. (9.2), and this is considered to be not far different from the electron mass. In a nondegenerate ground state polymer such as PPV, the photoinduced absorption can be due to polaron or bipolaron transitions, or excitations of triplet excitons as shown in Figs. 33 and 34. For PPV, photoinduced absorptions have been seen at 0.6 and 1.6 eV and have been assigned to bipolaron transitions.” 178 Similar sub-gap absorptions are also seen in chemically doped PPV.’79 For better ordered PPV, however, these transitions are not observed under photoexcitation, indicating that bipolarons may be stabilized by defects or disorder.’69 A further photoinduced absorption is seen in PPV at approximately 1.4 eV, as shown in Fig. 35, and is assigned to a transition between the triplet exciton and the excited state of the triplet exciton (Fig. 34). Triplet excitons are long-lived excited states in PPV (with a lifetime of around 1 ms at 10 K), since the radiative transition to the ground state is dipole forbidden. Since the transition from the ground state to triplet state is forbidden by symmetry, triplets are not generated by direct photoexcitation in PPV. Any triplets observed after photoexcitation must therefore be generated by intersystem crossing from the singlet state. The mechanism of this intersystem crossing is not well understood, but may arise from the process of charge separation and subsequent recombination. The existence

176

N. F. Colaneri, D. D. C. Bradley, R. H. Friend, P. L. Burn, A. B. Holmes, and C. W. Sprangler, Phys. Rev. B 42, 11,670 (1990). 17’D. D. C. Bradley, G . P. Evans, and R. H. Friend, Synth. Met. 17, 651 (1987).


57

of triplet excitons in PPV has been confirmed by optically detected magnetic resonance experiments,'x0 and the assignment of the 1.4 eV photoinduced absorption feature to a triplet-triplet transition has been confirmed by absorption-detected magnetic resonance.'" In EL devices, excitons are formed by the recombination of nongeminate electrons and holes, and can therefore form as triplets or singlets. Triplet excitons therefore play an important part in the operation of E L devices. Simple spin statistics arguments suggest that triplet excitons are three times more likely to form than are singlets, though this ratio is altered in more sophisticated models'82; evidence for formation of triplet excitons in LEDs is presented in Section VI.24. There is some debate about the assignment of the induced absorption bands at 0.6 and 1.6 eV in PPV (and for similar absorption bands in other polymers). It is established from their vibrational IR activity that they are associated with charged excited states, and they have been usually assigned to doubly charged bipolarons because similar features in the optical absorption are seen in chemically doped material for which it is known that the charged states have low spin. However, singly charged polarons are expected at low dopant concentrations, by the law of mass action, and although the magnetic signature for polarons is observed, the optical transitions are not easily established. Doping studies on model oligomers, such as the sexithiophene shown in Fig. 25, show in fact that the two absorption features are a property of the singly charged molecule,183~'84 ~ 8 suggested that the and F u r u k a ~ a and , ~ ~Bassler ~ ~ ~ ~et ~~ 1 . ~ ~ have charged photogenerated excited states in PPV may also be singly charged. In this case, the absence of the magnetic response may arise from interchain pairing.189 Optical measurements of injected charges in FET structures provide further information on this issue and are discussed in Section V.

ixo

L. S. Swanson, P. A. Lane, J. Shinar, and F. Wudl, Phys. Reu. B 44, 10,617 (1991). X. Wei, B. C. Hess, Z . V. Vardeny, and F. Wudl, Phys. Rev. Lett. 68, 666 (1992). 1x2 E. L. Frankevich, A. A. Lymarev, I. Sokolik, F. E. Karasz, S. Blumentengel, R. H. Bau hman, and H. H. Hohold, Phys, Reu. B 46, 9320 (1992). If3D. Fichou, G. Horowitz, and F. Gamier, Synth. Met. 39, 125 (1990). 1x4 D. Fichou, G. Horowitz, B. Xu, and F. Gamier, Synth. Met. 39, 243 (1990). Ix5A.Sakamoto, Y. Furukawa, and M. Tasumi, J . Phys. Chem. 98, 4635 (1994). in6 Y . Furukawa, Synth. Met. 69, 629 (1995). 1H7 J. M. Oberski, A. Greiner, and H. Bassler, Chem. Phys. Lett. 184, 391 (1991). IXX M. Deussen and H. Bassler, Chem. Phys. 164, 247 (1992). in9 M. G. Harrison, R. H. Friend, F. Gamier, and A. Yassar, Synth. Met. 67, 215 (1994). I8 I

58


c. Dark Conductivity and Photoconductwity We are not concerned in this chapter with the processes of electrical conduction in doped conjugated polymers; low carrier concentrations are of greater relevance to the understanding of device operation. Most of the conjugated polymers that have been studied in detail have low ionization potentials, and are therefore doped more readily by electron acceptors than by donors to give p-type material. Most of these polymers are found to be lightly p-doped (e.g., as assessed by the sign of the thermopower) as synthesized, without deliberate doping, and both residual catalyst and included oxygen are considered capable of producing this level of doping (typically < 1017 charges cm-'). Lower energy gap polymers are often more heavily doped than higher gap materials. Unoriented spin-coated films of Durham polyacetylene19nhave a room-temperature conductivity of around lo-' S cm-' and a concentration of mobile carriers as assessed from capacitance-voltage measurements of Schottky diodes (see Section 2) of between 10l6 and 1017 ~ m - The ~ . temperature dependence of the dark conductivity, shown in Fig. 39, exhibits an activation energy of about 300K 250K 200K I '

.

l5OK

FIG. 39. Dark conductivity, photoconductivity, and photocarrier mobility for unoriented Durham polyacetylene. Plotted against reciprocal temperature are A, the dark current I d ; B, the photocurrent, I,,; C, I,, minus its value at 20 K; and D, the photocarrier mobility. [P. D. Townsend and R. H. Friend, Phys. Reu. B 40, 3112 (1989).]


59

0.4 eV at room temperature, but it is not clear how this activation energy is attributed to the concentration of mobile carriers or to their mobilities. Photoconductivity measurements are therefore of considerable use, and for the case of polyacetylene it is possible to measure the temperature dependence of the concentration of the charge carriers from the temperature dependence of the photoinduced absorption band associated with charged carriers.’” The measured photoconductivity shows a temperature-independent “fast” component and a temperature-dependent component that is due to the long-lived photogenerated charged solitons. The results of this analysis are shown in Fig. 39; the photocarrier mobility is obtained from the photocurrent, Zp, and the concentration of photocarriers at temperature T, N T ) , from the relation A a / a = N(T)u where A a / a is the fractional change in absorption at 0.55 eV and (T is the optical cross section per carrier. Using a value of u of 8 X lo-’‘ cm-2 obtained from measurements on chemically doped samples, Townsend and Friend’” obtained a value for the room-temperature mobility of 5 X cm2 V - ’ s - ’ . Figure 39 demonstrates that the mobility of the photocarriers above 200K shows an activated behavior similar to that of the dark carriers, though with a slightly reduced activation energy of 0.31 eV. The similarity of the mobilities of the dark and photoexcited carriers indicates a common mechanism for both, and Townsend and Friend consider that the appropriate process for the rate limiting step is that of “bipolaron” hopping of bound (or weakly bound) soliton-antisoliton pairs between adjacent chains. In this model, motion of one charge to an adjacent chain creates two singly charged “polarons” at a cost in energy of wh,and the second charge can then follow the first so as to transfer the pair of charges between chains. For three-dimensional hopping, the mobility, p, obtained in this model using the Einstein diffusion relation is

(10.1) where a is the distance between hops, W, is the hop energy, and w is the attempt-to-hop frequency. If wh is identified as the experimentally measured mobility for the photocarriers and w is taken to be the average of the frequencies of the optic phonons that modulate the dimerisation IYO

P. D. Townsend and R. H. Friend, Phys. Keu. B 40, 3112 (1989).

60


amplitude (3.9 x l O I 3 Hz), then a value for a of 3 nm is obtained. This is considerably larger than the interchain separation (0.42 nm), and can be taken as a measure of the average distance traveled between interchain hops. This value is close to the dimension of the crystallites in the unoriented Durham material.” PPV shows a relatively low c ~ n d u c t i v i t y , ~which ~ ~ ~ ”is~dependent on the amount of exposure to oxygen and can be lower than lo-” S cm-’. It shows a strong photoconductivity, and the action spectrum is shown in Fig. 35. Note that the onset of the photocurrent occurs at the onset of optical ~ ~ used this to argue that the fundaabsorption. Lee and c o - w o r k e r ~ ’have mental optical absorption in PPV produces carriers that have negligible excitonic binding energy. An alternative explanation has been proposed, based on the dissociation of singlet excitons.”~’’4 In this model, optical absorption creates intrachain singlet excitons, that is, with electron and hole on one conjugation length. A finite proportion of these excitons dissociates to form interchain excitons, that is, with the electron and hole on different conjugation lengths. These interchain electron-hole pairs can then diffuse apart under the electric field until they form free carriers. This process is frequently modeled by a theory credited to On~ager.’’~ More recently, an improved theory has been developed by Noolandi and Hong,lY6which takes into account the rapid relaxation of singlet excitons to the vibrational ground state. Photoconductivity data for poly(Zpheny1p-phenylenevinylene) have been shown to give a good fit to Onsager theory,sx with an activation energy of 0.165 eV corresponding to the coulombic binding energy of an interchain exciton with radius 5 nm. Photoconductity in the crystalline polydiacetylenes was extensively measured in the 1970s,lg7and it was found that absorption into the well-defined exciton peak near 1.8 eV produced little photocurrent. However a photocurrent was seen with an onset of around 2.5 eV, and this has been interpreted as indicating that the excitons are strongly bound (binding energy 0.7 eV) so that the strong photocurrent response is only achieved at

I9’S. Tokito, T. Tsutsui, R. Tanaka, and S. Saito, Jpn. J . Appl. Phys. 25, L680 (1986). IY2S.Tokito, T. Tsutsui, S. Saito, and R. Tanaka, Polym. Commun. 27, 333 (1986). 193 C. H. Lee, G . Yu, and A. J. Heeger, Phys. Rev. B 47, 15543 (1993). 194 R. N. Marks, J. J. M. Halls, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, J . Phys.: Condens. Matter 6, 1379 (1994). 195 L. Onsager, Phys. Rev. 54, 554 (1938). Iy6J. Noolandi and K. M. Hong, J . Chem. Phys. 70, 3230 (1979). Iy7D. Bloor and R. R. Chance, eds., Polydiacetylenes. Synthesis, Structure and Electronic Properties, NATO AS1 Series E. Vol. 102, Martinus Nijhoff, Dordrecht (1985).


61

excitation energies sufficient to generate “free” carriers. This contrasts with the results for PPV and many related polymers, but the explanation for this difference in behavior may lie in the very rapid nonradiative decay of photogenerated excitons in the polydiacetylenes ( < 2ps), which therefore do not have time to d i s ~ o c i a t e . ’ ~ ~ . ’ ~ ~ In practice, the response of conjugated polymer photoconductive devices is often limited by recombination of photogenerated charge carriers before they reach the electrodes, so that the photocurrent peaks before the peak of optical ab~orption.”~Under high illumination, space-charge effects also become important in limiting the current. Due to the difficulties of separating the effects of photogeneration, recombination, transport, and intrinsic charges, it is extremely difficult to obtain reliable information about mobilities from photoconductivity experiments with PPV and its derivatives. We discuss mobilities in PPV further in Section IV.12. d. Ionizution Potentials Because all of the semiconductor devices we consider here involve contacts between electrode metals and the organic layers, or between two organic layers, information about the ionization potential (the energy between the HOMO level and the vacuum level) and the electron affinity (the energy between the LUMO level and the vacuum level) is important to obtain. Although values can be predicted from quantum chemical models, it is not straightforward to obtain independent experimental measurements of the ionization potential and electron affinity of conjugated polymers. The VEH results for PPV give an ionization potential of 5.02 eV and an electron affinity of 2.76 eV.’99For dimethoxy-PPV, the ionization potential and electron affinity are both reduced, to 4.69 and 2.68 eV, re~pectively.’~~ Note that VEH calculations do not account for the effects of packing bulky side-groups in the solid state, and that extending the alkyl side-chain in dimethoxy-PPV, for example, is not expected to produce a significant change in the VEH values for the energy gap and ionization potential. Ultraviolet photoemission spectroscopy (UPS) is a useful method to obtain information about the density of occupied states in a conjugated polymer.200The UPS spectrum is highly sensitive to surface contamination, and the experiment must therefore be performed in ultrahigh vacuum with carefully prepared samples. The UPS spectrum of PPV has been measured 19 h

A. Kohler, H. F. Wittmann, R. H. Friend, M. S. Kahn, and J. Lewis, Synth. Met. 67, 245 (lYY4). I99 F. Meyers, A. J. Heeger, and J. L. BrCdas, J . Chern. Phys. 97, 2750 (1992). 2011 W. R. Salaneck, Rep. Prog. Phys. 54, 1215 (1991).

62


by Sato et ~ 1 . and ~ ~ has ' been found to give reasonable agreement with the positions of peaks in the density of valence states predicted by the VEH method. It is difficult to obtain absolute values of the ionization potential from UPS measurements, since it requires the position of the Fermi level in the polymer to be estimated. The ionization potential of 6.2 0.1 eV quoted by Sato el al. for PPV is considerably higher than the value expected from the analysis of the heights of barriers to hole injection in polymer LEDs (see Section VI.22). Electrochemical techniques have also been used to determine reduction and oxidation potentials for conjugated polymers. For these measurements, a thin film of conjugated polymer is coated onto a conducting electrode, and immersed in an electrolyte solution, forming one half of an electrochemical cell. A standard reference electrode is used to complete the cell. In cyclic voltammetry,2"2 the cell current is measured as the voltage across the cell is ramped positive and negative in a cyclic fashion. The potentials required to achieve oxidation or reduction are measured relative to the reference electrode. The difference between oxidation and reduction potentials gives a measure of the difference between HOMO and LUMO energies. It is difficult to extract absolute values of ionization potentials and electron affinities from these data, nevertheless this technique is useful to determine differences in these parameters between different polymers. In practice, the situation is complicated by a number of factors. In an electrochemical oxidation, for example, the negative counter-ion from the electrolyte solution must diffuse into the polymer to maintain electrical neutrality. The presence of this counter-ion can significantly alter the structure of the polymer, and hence alter the electronic properties. Since the onset of oxidation, for example, is not sharp, it is often not clear which point on the oxidation curve should be equated to the ionization potential. For most conjugated polymers, at least one of oxidation or reduction is irreversible, which further complicates the problem. The oxidation and reduction potentials for a series of PPV derivatives are reported by Helbig and HOrh~ld.*"~ Despite the difficulties mentioned here, these data give reasonable agreement with the measured optical gaps, and with the results of quantum chemical calculations.

'"IN. Sato, M. Logdlund, R. Lazzaroni, W. R. Salaneck, J.-L. BrCdas, D . D . C. Bradley, R. H. Friend, and K. E. Ziemelis, Chem. Phys. 160, 299 (1992). 2"2P.H. Rieger, Electrochemistry, Chapman & Hall, New York (1994). 2"3M. Helbig and H.-H. Horhold, Mukromol. Chem. 194, 1607 (1993).


63

Other methods have also been used to estimate the ionization potential of PPV, including the Kelvin vibrating capacitor method.204This gives a value of 4.59 eV,*"' considerably lower than the values obtained by the methods outlined earlier. Ideally, it would be possible to use the positions of the HOMO and LUMO levels as measured experimentally or predicted theoretically to model the energy barriers at the electrodes in a polymer LED or transistor. As demonstrated earlier, however, the values obtained by different techniques for PPV show a wide spread. Since small changes in barrier height can greatly affect the charge injection (see Section IV.ll), it is impossible to draw detailed conclusions about barrier heights on the basis of the available information. Even the work functions of various electrode materials are not well known and depend strongly on the measurement technique used. It is, however, possible to draw more general conclusions about the barrier heights in LEDs; for example, one can confidently predict that the barrier to hole injection from I T 0 into PPV is much smaller than the barrier to electron injection from aluminum. In practice, although independent theoretical calculations or experimental measurements of energy levels can establish relative trends between polymers, it is often easier to infer information about the barrier heights from the performance of polymer devices, particularly if different polymers and electrode materials are compared.

IV. Device Processes

In this section we consider some of the models that have been used to describe the processes of charge carrier injection and transport, recombination, and exciton decay in semiconductors. We first present some simple models for injection at an ideal contact between a metal and a conventional semiconductor. These models provide a starting point for understanding injection into conjugated polymers. Since mobilities in conjugated polymers are very low, space charge effects are likely to be important in determining the electrical characteristics of polymer devices. It is therefore important to distinguish between injection effects and bulk effects such as

2114

N. W. Ashcroft and N. D. Mermin, Solid Stare Physics, Holt, Reinhard and Winston,

New York (1976). 201

G. Leising, personal comrnuncation (1994).

64


the build-up of space charge. The difficulty in modeling polymer devices such as LEDs and photodiodes is the fact that so many of the important properties of conjugated polymers are poorly known at present. Among these are electron and hole mobilities, ionization potentials, electron affinities, trap concentrations and distributions, doping levels, interfacial barriers. and recombination mechanisms. 11. CHARGEINJECTION

Charge injection from metallic contacts into solids, and in particular into 206-208 To understand the semiconductors, has been extensively injection process, it is first necessary to examine the detailed nature of the interface and to identify any barriers formed. The current flowing in a device is frequently limited by these barriers at the contacts rather than by the properties of the bulk of the solid. The two principal mechanisms for injection of charge carriers in the presence of a barrier are thermionic emission and quantum mechanical tunneling. These processes are described in Sections IV.ll.b and IV.ll.c, respectively. a. Metal-Semiconductor and Metal-Insulator Contacts When a metal and another solid are brought into intimate contact, their Fermi levels must be equal at the interface in order to achieve thermodynamic equilibrium. Since the work function of the metal and the Fermi energy of the solid (as measured from the vacuum level) are not necessarily equal, in general this requires a transfer of charge between the metal and the solid. The theory of contacts between doped semiconductors and metals was developed in 1938 by MottZoyand Schottky.210The situation for a contact between a metal and a lightly-doped n-type semiconductor is shown in Fig. 40. The results quoted here are for the injection of electrons into an n-type semiconductor. Similar results can be obtained for holes and p-type semiconductors. In the case shown, where the work function of the metal (&) is larger than the Fermi energy of the semiconductor (&I, negative charge from the semiconductor conduction band is transferred to the 20hH.K. Hensich, Semiconductor Contacts: An Approach to Ideas and Models, Clarendon Press, Oxford (1989). 207 E. H. Rhoderick and R. H. Williams, Meral-Semiconductor Contacts, Clarendon Press, Oxford (1988). 208 K. C. Kao and W. Hwang, Electrical Transport in Solids, Pergamon Press, Oxford (1981). 204 N. F. Mott, Proc. Cambr. Phil. SOC.34, 568 (1938). 2111 W. Schottky, Naturwiss. 26, 843 (1938).


65

vacuum level

m

metal semiconductor FIG.40. Blocking contact between a metal and an n-type semiconductor.

metal contact, leaving a positive space charge in the semiconductor due to the ionized dopants. The width of this depletion region is approximated by

w=

J2'4m NDq2

'

(11.1)

where ND is the dopant concentration, X is the electron affinity of the semiconductor, and E , ~is its permittivity.22 W term represents the distance over which the bands are bent. The barrier to electron injection formed at this interface is known as a Schottky barrier. The height of this barrier is given simply by 4,,, - X , the difference between the work function of the metal and the electron affinity of the semiconductor. This type of contact is known as a blocking contact. For an applied bias, V , across the junction (with the metal negative) the depletion width becomes (11.2) If the Fermi energy of the semiconductor is larger than the work function of the metal then electrons must flow from the metal into the semiconductor, as shown in Fig. 41. Since there are now free electrons in the semiconductor conduction band at the contact, the current flow at low biases is no longer limited by the barrier to injection, and the contact is known as ohmic. Under bias, as shown in Fig. 42, electrons are released from the point where d V / h = 0 with zero kinetic energy. This point is known as the virtual cathode. As the applied bias is increased, this point moves toward the contact until they coincide, at which point the current becomes limited by the barrier at the electrode.211 Similar analyses have *"G. T. Wright, Solid State Electron. 2, 165 (1961).

66


metal semiconductor FIG.41. Ohmic' contact between a metal and an n-type semiconductor.

been performed for the case of a trap-free intrinsic semiconductor (or insulator). The exact solution is complicated, but the characteristic length over which band-bending occurs is the ambipolar intrinsic Debye length A,, given by A,

c,kT =

(11.3)

q2(n,+pol '

qv=o

FIG.42. An ohmic contact under applied bias, showing the change in W with bias voltage. [K. C. Kao and W. Hwang, Electrical Transport in Solids, Pergamon Press, Oxford (1981).]


67

where no and p o are the intrinsic concentrations of electrons and holes, The length over which band-bending occurs is clearly much greater for intrinsic materials than for extrinsic semiconductors. In the case of an insulator or intrinsic semiconductor with traps, it is the density of traps that determines the band-bending.'12 For a density N, of shallow traps at an energy E, below the conduction band, the width of the depletion region is given approximately by

(11.4)

To model the operation of a polymer EL device, it is clearly important to determine the thickness W of the accumulation or depletion regions relative to the thickness of the device. If W is much greater than the thickness of the device, then in equilibrium the semiconductor cannot support sufficient charge to cause significant band-bending, and the bands are effectively flat. Conjugated polymers used for LEDs are prepared with doping levels as low as possible, and are sometimes referred to as intrinsic semiconductors. The intrinsic ambipolar Debye length for PPV is many orders of magnitude longer than a typical device thickness, and the trap-free intrinsic case would therefore correspond to bands that are effectively flat. The exact nature and quantity of trapping sites in PPV are not well known, although trapping of electrons, for example, by oxygen impurities, is frequently put forward as an explanation for the fact that electron mobilities in conjugated polymers are much smaller than hole mobilities. For PPV, however, even assuming one trapping site per repeat unit and taking a trap depth of 0.4 eV (a large value for a shallow trap) gives a value of of approximately 250 pm, which is far in excess of typical device thicknesses. Although the syntheses of conjugated polymers for devices are designed to minimize the impurity concentration, most conjugated polymers are lightly p-doped. For the poly(3-alkylthiophene)~(P3ATs), for example, this doping is thought to arise largely from residual amounts of catalyst (in the form of FeCl,), which is difficult to remove ~ o m p l e t e l y . ~ "The ~ ~ ' capaci~ tance-voltage characteristics of MIS devices using P3AT as the semicon"'.I.G. Simmons, J . Phys. Chem. Solids 32, 1987 (1971). ?Ii

K. E. Ziemelis, A. T. Hussain, D. D. C. Bradley, R. H. Friend, J. Riihe, and G. Wegner, Phys. Rec. Left. 66, 2231 (1091). ?I 4 G. Gustafsson, 0. Inganas, J. 0. Nilsson, and B. Liedberg, Synth. Met. 26, 297 (1988).

68


ductor can be successfully modeled by treating the polymer as a p-type semiconductor. The concentration of acceptor sites varies from 1015 cm-3 to more than 1019 depending on the synthesis and purification ~ ~ e d The . ~synthesis ~ ~ of , PPV ~ ~does ~ not , involve ~ ~ the ~ use of a catalyst, but the material is still believed to be lightly p-doped. The acceptor sites in this case could be associated with residual precursor material, or with oxygen impurities. PPV samples prepared in Cambridge have been found to have a low dark conductivity, implying a low doping level which is S cm-’ difficult to measure directly. Using a quoted conductivity of cm2 V - ‘ sS1 (see Sec. even taking a mobility as low as or IV.12.a) gives a dopant concentration of less than 1014 ~ m - Karg ~ . et however, obtain a doping level of 10l6 to l O I 7 cm-3 in their material by analysis of the response of LED structures. Among substituted PPV derivatives, clear p-type behavior has been observed in dimethoxy-PPV, with a doping concentration of order l O I 7 cm-3,219 but longer chain soluble dialkoxy-PPVs have much lower doping levels.220All these values would be expected to depend strongly on the chemical synthesis, conversion conditions, and sample handling. Note that while a material may be considered “pure” in the organic chemistry sense (i.e., less than 0.1% impurity), this still allows for significant levels of doping that can greatly affect the semiconducting properties. Even for dopant concentrations as low as 1013 ~ m - the ~ , Fermi level for PPV is only about 0.4 eV above the valence band. This implies that the intrinsic model is not appropriate to determine whether a metal/PPV contact is ohmic or blocking. For a barrier height of 0.5 eV, for example, dopant concentrations of 1013 and 10” cmP3 correspond to accumulation layer thicknesses of approximately 2 wm and 7 nm, respectively. Because these two figures represent values considerably smaller and considerably larger than a typical device thickness, the appropriate model for describing device operation will clearly be highly dependent on doping concentration. For this reason, the injection characteristics of a number of barrier models are considered in the next sections. 215

D. Braun, G. Gustafsson, D. McBranch, and A. J. Heeger, J . Appl. Phys. 72, 564 (1992). ‘lhN. C. Greenham, A. R. Brown, D. D. C. Bradley, and R. H. Friend, Synth. Met. 57,4134 (1993). 217R.N. Marks, Ph.D. Thesis, University of Cambridge (1993). 218 S. Karg, W. Riess, M. Meier, and M. Schwoerer, Mol. Cryst. Liq.Cryst. Sci. Tech. A 334, 619 (1993). 21 9 M. G. Harrison, K. E. Ziemelis, R. H. Friend, P. L. Burn, and A. B. Holmes, Synrh. Met. 55, 218 (1993). 220 K. Pichler, personal communication (1994).


69

b. Themionic Emission For thermionic emission over a triangular barrier of height (bb from a metal into a high-mobility semiconductor, the current density is given by J =

4rrqk2m* h3

T 2 exp(-(b,/kT)

= A T T 2 exp(-(bb/kT),

(11.5)

where AT is known as the effective Richardson constant and m* is the electron effective mass.’’ The field dependence is given by the image-force lowering of the barrier height, as illustrated in Fig. 43. When an electron is placed at a distance x from a metal surface, a positive charge is induced on the surface of the metal. The effect of this charge is the same as that of an ‘‘image charge” placed at a distance x behind the surface. The potential due to the force between these charges is given by (11.6)

When the effect of the applied field, F , is superposed on the image force, a field-dependent lowering of the barrier is obtained, given by (11.7) For a Schottky barrier under a bias I/,formula (11.7) becomes

A4

=

[

q6(qv -

4blND

2(8rrl2E:

(11.8)

FIG. 43. Image-force lowering of barrier height. Under an applied field F , the barrier is lowered by an amount A+.

70


In a semiconductor with a low mobility, it is necessary to take into account the diffusion of carriers within the barrier region back towards the contact. The analysis is complicated, and depends on the type of barrier assumed. A full treatment is given by Kao and Hwang.*”*The effective Richardson constant AT is now replaced by AT, which depends on both field F , temperature T , and mobility p. In the high-field limit, the case for an intrinsic semiconductor gives

For a Schottky barrier in the low-mobility limit, and also at high field, the relationship between AT and AT is given by

(11.10)

where x , is the position of the maximum barrier height (taking into account image-force lowering), as shown in Fig. 44. Under nonequilibrium conditions such as the injection of charge carriers, the carrier populations no longer follow the usual Boltzmann relations. It is useful to replace the Fermi energy by the quasi-Fermi energies for electrons and holes so that the modified Boltzmann relations still apply. In Eq. (11.101, I(/ represents the energy difference between the quasi-Fermi energy for electrons and the conduction band.

FIG.44. A Schottky barrier between a metal and an n-type semiconductor, including image-force effects and showing the quasi-Fermi energy for electrons.


71

c. Field Emission At sufficiently low temperatures, or for large barriers at high fields, emission due to quantum mechanical tunneling through the barrier, known as field emission, can become important.20RThe rate of tunneling through a triangular barrier is given at low temperatures by (11.11)

where a=

4rr(2m*

)I”

h

(11.12)

Tunneling currents are frequently analyzed using a Fowler-Nordheim plot, where log(J/F2) is plotted against 1 / F to give a straight line, the gradient of which can be used to extract the barrier height. For a Schottky barrier, the tunneling rate is given by

where (11.14)

The essential characteristic of field emission is that the current is insensitive to temperature, but strongly dependent on the applied field. Since the tunneling rate depends strongly on the width of the barrier, thermal excitation can significantly increase the tunneling current. The exact expressions for thermally assisted tunneling are complicated, but for a Schottky barrier the current has the form

J

= J,

exp(qV/nkT),

(11.15)

where J, is essentially a constant for low doping levels, and n = 1 at high temperatures.”

72


d. Nonideal Contacts Keep in mind that the contacts described so far in this section are ideal contacts, where the metal is assumed to be in perfect contact with a semiconductor whose structure at the surface is identical to the bulk structure. For a real contact, the possibility of defect states at the interface must be considered. If the charge contained in these states is sufficient to cause the band-bending necessary to equalize the Fermi levels of metal and semiconductor, then the barrier to injection will be determined solely by the surface states, independent of the metal work function.22 Another possibility is the existence of an interfacial barrier, which can radically alter the injection characteristics. In a PPV LED, such a barrier may be caused by the reaction of the I T 0 electrode with the HCI gas produced during the PPV conversion. Interfacial layers may also be caused by chemical bonding or doping occurring at the metal-polymer contact. Salaneck2"0 has made extensive studies of the interactions between polymer surfaces and submonolayers of various metals deposited by in situ evaporation. Both X-ray and ultraviolent photoemission spectroscopy (XPS and UPS) have been used to study the chemical interactions between polymer and metal. For the case of the alkali metals, such as sodium, the interaction with PPV is to form fully ionized sodium cations, with n-doping of the PPV, which stores charge in two "bipolaron" bands within the n-r* gap.221 The cyano-substituted " contrast, the reacPPVs (Fig. 16) are similarly n-doped by ~ o d i u m . ' ~In tion between aluminum and PPV is covalent, with formation of a bond between the aluminum and the vinylic carbons, causing the loss of conjugation along the chain.222These interactions are also modeled in quantum chemical calculations.222The case of the alkaline earth metals is particularly interesting since the comparison of the work function with the electron affinity of PPV puts these at the threshold for the charge-transfer reaction. Brown et al? consider that there should be a barrier for electron transfer from metallic calcium to pristine PPV, and Parker 224 also considers there to be a barrier present at the negative electrode in ITO/MEH-PPV/Ca devices. However, the UPS and XPS data for Ca 22 I M. Fahlman, D. Beljonne, M. Liigdlund, A. B. Holmes, R. H. Friend, J . L. BrCdas, and W. R. Salaneck, Chem. Phys. Left. 214,327 (1993). 222P. Dannetun, M. Liigdlund, M. Fahlman, M. Boman, S. Stafstrom, W. R. Salaneck, R. Lazzaroni, C. Fredriksson, J . L. BrCdas, S. Graham, R. H. Friend, A. B. Holmes, R. Zamboni, and C. Taliani, Synfh. Met. 55-57, 212 (1993). 223A.R. Brown, D. D. C. Bradley, J. H. Burroughes, R. H. Friend, N. C. Greenham, P. L. Burn, A. B. Holmes, and A. Kraft, Appl. Phys. Left. 61, 2793 (1992). 2241. D. Parker, J . Appl. Phys. 75, 1656 (1993).


73

formed on poly(2,5-diheptyl-p-phenylenevinylene) show clear evidence that Ca is present as the dication, and that the surface layer of the polymer is reductively doped.225This illustrates that the energetics of the chemical doping reaction of the conjugated polymer involve both the energy of the transferred electron and also the energy terms associated with the accommodation of the calcium ion in the polymer. We note that diffusion coefficients for polyvalent ions in conjugated polymers are very low, and it is considered that the penetration of the calcium will not be beyond the first few nanometers, so that calcium can provide a relatively stable electrode for LEDs when protected from atmospheric degradation.226 It is important to note that the UPS and XPS studies that Salaneck has reported are performed under UHV conditions, and with polymer surfaces prepared so as to have low levels of contamination. In contrast, most of the device work that is in the literature is made under less carefully controlled conditions, and it is very likely that significant amounts of oxygen are have made XPS studies under such conditions present. Gao et and find evidence for the formation of a calcium oxide interfacial layer. All the injection models presented here were developed for conventional three-dimensionally bonded semiconductors. In the conjugated polymers, structural relaxation and polaron formation may considerably complicate the situation. Few attempts have been made to incorporate these effects into models of conjugated polymer-metal interfaces, and this should be borne in mind when comparing experimental results with the simple models described earlier. ~

1

.

~

~

~

9

~

~

~

12. CARRIER TRANSPORT

a. Single-Cam'er Currents In a trap-free semiconductor, the transport of electrons is described by the mobility, p o ,which represents the average drift speed of an electron per unit applied field. The drift current is given by (12.1)

where nr is the number density of free carriers. '"P. Dannetun, M. Fahlman, C . Fauquet, K. Kaerijama, Y. Sonoda, R. Lazarroni, J. L. Bredas, and W. R. Salaneck, Synfh. M e f . 67, 133 (1994). '16F. Cacialli, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synth. Met. 67, 157 (1994). '*'Y. Gao, K. T. Park, and B. R. Hsieh, J . Chern. Phys. 97,6991 (1992). '"Y. Gao, K. T. Park, and B. R. Hsieh, J . Appl. Phys. 73,7894 (1993).

74


In a semiconductor with significant trapped charge that does not contribute to the transport, the total number density of charge, n , is made up of free and trapped parts n f and n , , such that n

=

nf

+ n,.

(12.2)

It is useful to define an effective mobility, p e , where nl’

pe

=

(12.3)

and JIL = n p , F .

(12.4)

In three-dimensionally bonded inorganic semiconductors, room-temperature mobilities are high, typically lo2 to l o 4 cm2 V - ’ s - ’ , and are limited by scattering from impurities, phonons, etc. In molecular crystals, however, the mobility is limited by hopping of the charge between conjugated units, and mobilities are typically in the range of to 10’ cm2 V - ’ s - ’ . ~For the conjugated polymers, hole mobilities have been found to be typically less than l o p 4 cm2 V - ’ s-I, as discussed for the case of photocarriers in Durham-route polyacetylene in Section III.lO.c, and as reported for other p ~ l y r n e r s . ~ ~Thi ” ~ s’ ~low mobility is attributed to larger hopping distances, increased disorder, and also to the effect of traps. Electron mobilities have proved difficult to measure by time-of-flight techniques, and are estimated This to be at least two orders of magnitude lower than hole difference is frequently attributed to the trapping of electrons at defect sites due to impurities such as o ~ y g e n . ~ ’ ~ . ~ ~ ~ The theory of the field and temperature dependences of hopping mobilities in disordered materials has been developed largely for charge-transporting molecules dispersed in an inert polymer matrix. Two theories have been put forward, one based on the assumption that structural relaxation in the excited state is important (the “polaron” and the other based on the assumption that disorder is the dominant effect.234 The 22yT.Takiguchi, D. H. Park, H. Ueno, K. Yoshino, and R. Sugimoto, Synth. Met. 17, 657 (1987). 230 J. Obrzut, M. J. Obrzut, and F. E. Karasz, Synfh. Met. 29, El03 (1989). 23’N.T. Binh, L. Q. Minh, and H. Bassler, Synth. Met. 58, 39 (1993). 232B.R. Hsieh, H. Antoniadis, M. A. Abkowitz, and M. Stolka, Polym. Prepn‘nts 33, 414 (1992). 233L.B. Schein, Phil. Mug. B 65, 795 (1992). 234H.Bissler, Phil. Mag. B 50, 347 (1984).

75


temperature and field dependences of the mobility predicted by the two models are different. The polaron model gives In p a - ( T , , / T ) 2at constant field

(12.5)

In p a F-’ sinh(aF) at constant temperature

(12.6)

and whereas the disorder model gives In p a

-

( T , / T ) 2at constant field

(12.7)

and In p a F’/*at constant temperature.

(12.8)

In practice, neither theory can adequately explain all the available data, and it is difficult to measure over a sufficient range of temperatures to distinguish the T - and T-’ temperature dependences without causing a morphological change that affects the results. The disorder model is, however, particularly attractive since it is based on the same hopping arguments used to describe the motion of excitons in a disordered material, as described in Section 1II.lO.a. In some materials it has been found experimentally, that above a certain field mobilities begin to decrease. This has been explained in the disorder theory by introducing off-diagonal disorder, that is, disorder due to the distribution of distances between molecules, in addition to the diagonal disorder due to the different interaction of each molecule with its local e n v i r ~ n m e n t . ~In~ ’the presence of off-diagonal disorder, at low fields there are enhanced routes for charge transport involving some short hops not in the direction of the applied field. As the field is increased, these routes are suppressed due to the increased energy required to hop against the field, and the mobility is decreased. Transport in conjugated polymers in highly dispersive, meaning that the broadening of the leading edge of the transient photoconductivity signal measured in time-of-flight experiments is larger than that predicted by the Einstein relation

’

qD

=

kTp,

(12.9)

where D is the diffusion coefficient for charge carriers. Dispersive transport is a general feature of disordered systems,236although in PPV the 21s 23h

P. M. Borsenberger, L. Pautrneier, and H. Bassler, J . Chem. Phys. 94, 5447 (1991). R. Richert, L. Pautmeier, and H. Bassler, Phys. Reu. Left. 63, 547 (1989).

76

NEIL C. GREENHAM A N D RICHARD H. FRIEND

effect of trapping is also likely to be important. As well as making it more difficult to obtain reliable data from time-of-flight experiments,232dispersive transport greatly complicates the modeling of the current-voltage and transient characteristics of polymer LEDs, for example. When a single-carrier type is injected into a semiconductor, provided that the contact is able to supply sufficient current, the total current will be .~~~ limited by the build-up of space charge within the s e m i c o n d ~ c t o r The results presented next are derived for electrons, but identical results apply for holes. For a trap-free material, neglecting diffusion currents, and when the equilibrium charge density is negligible in comparison to the injected charge density, the space-charge-limited (SCL) current is given by (12 .lo) where V is the applied voltage and d is the sample thickness. At low voltages, where the equilibrium charge density, n,, is larger than the injected charge density, the ohmic current (12.11) is predominant over the SCL current. For a sample with traps at a single level, the SCL current is given by replacing p with pe [see Eq. (12.311: (12.12) Expressions for more complicated trap distributions are given by Lampert and Mark,237for example. If the injected charge density is high enough that the quasi-Fermi energy moves above the trapping level, then the traps become completely filled. In this case, known as the trap-filled limit (TFL), the density of charge in traps becomes negligible compared to the injected charge, and the current reverts to the trap-free value. If the traps are sufficiently deep, as the voltage is increased the current can change directly from the ohmic regime to the TFL regime, as shown in Fig. 45. 237M.A. Lampert and P. Mark, Curreni Injection in Solids, Academic Press, New York (1970).


77

In V

FIG.45. Current density ( J ) versus ( V )for single-carrier injection, showing the transition from ohmic to space-charge-limited conditions.

The SCL current represents the maximum single-carrier current that can be passed under any circumstances. In the conjugated polymers, where mobilities are low, space charge effects are likely to be important. In a polymer LED, it will be important to establish whether the current is limited largely by the injection process, by space charge effects, o r by a combination of the two. b. Electron -Hole Capture- Two-Cam'er Currents In an E L device, carriers of opposite sign are injected from either side of the semiconductor. In calculating space charge effects, the total charge from both electrons and holes must now be considered, and the situation becomes considerably more complicated than the single-carrier case. The capture of electrons and holes to form bound, neutral excited states (excitons in the case of conjugated polymers) is clearly of great importance, both because it is a prerequisite for the emission of light in a polymer LED, and because the removal of mobile charged carriers can have a profound effect on the current-voltage characteristics of a twocarrier device. A simple capture argument states that a bound excited state will form when an electron and a hole approach within a distance r,, such that the coulombic binding energy is greater than the thermal energy, kT, giving

r,

=

~

q2 4m, kT

(12.13) '

78

NEIL C . GREENHAM AND RICHARD H. FRIEND

For E , = 4, a value of r, = 14 nm is obtained at 300 K. This argument is clearly an oversimplification, because it neglects both the effect of the external electric field and the anisotropy of the conjugated polymer. For electrons and holes, number densities n and p , approaching each other with average relative speed u, it is useful to define a recombination cross section, a,, such that the generation rate for excitons per unit volume, G,, is given by G,

=

npua,.

(12.14)

For conventional semiconductors, I I is usually taken to be the average relative thermal velocity, and is therefore independent of the applied field. In a conjugated polymer, where the transport occurs largely by hopping and the mean free path of a charge carrier is small, this picture may not be appropriate. In the limit where thermally induced hopping is very much less likely than field-induced hopping, the relevant speed is the relative drift velocity. Writing the electron and hole drift speeds in terms of their mobilities pe and ph gives (12.15) According to the simple capture model presented here a, = rr:, but Eq. (12.14) can be used to define a capture cross section for any capture mechanism. When an electron and a hole are generated by absorption of a photon, their subsequent capture is referred to as recombination. This term is also used to describe electron-hole capture in a two-carrier device, despite the fact that in this case the electrons and holes are injected from opposite electrodes. Where both electron and hole currents are injection limited, and space charge effects are negligible, the two-carrier problem can be treated essentially as a superposition of two single-carrier currents. The quantity and distribution of exciton formation can then be determined using Eq. (12.14). If electron-hole capture is not complete, however, the hole current at the electron-injecting contact can significantly affect the barrier to electron injection, for example. Also, electron-hole capture occurring close to one of the contacts can alter the injection rate.206 Where the two-carrier current is limited by space charge effects, the current depends strongly on the effectiveness of recombination. In the trap-free case where diffusion currents are neglected and the contacts are


79

FIG. 46. Electron ( n ) and hole ( p ) distributions for two-carrier injection. (a) Recombination cross section, a,, large. (b) Recombination cross section, u,,small. [L. M. Rosenberg and M. A. Lampert, J . Appl. Phys. 41, 508 (1970).]

assumed to act as infinite reservoirs of charge, an exact analytical solution has been found by Parmenter and R ~ p p e I The . ~ ~current ~ is given by

J =

9 -6 s

pe

V2 f f 21

(12.16)

where peffdepends on the electron and hole mobilities and the capture cross section. It should be noted that this solution and its subsequent developments assume that the recombination is given by npua,, where u is independent of mobility and field. As explained earlier, this may not be appropriate for the conjugated polymers. Two limits of the Parmenter-Ruppel solution with p, = 2ph are shown in Fig. 46.239In the limit of no recombination (ua; + 01, the negative and positive space charges cancel, giving overall neutrality. In this case there is no limit to the current due to space charge, and currents can be very large. In the limit of large recombination cross section, there is no region of overlap of electron and hole space charges, and the current is given by the sum of the individual electron and hole SCL currents (peff= p, + &). The position "'R. H. Paramenter and W. Ruppel, J . Appl. Phys. 30, 1548 (1959). 13'L. M. Rosenberg and M. A. Lampert, J . Appl. Phys. 41, 508 (1970).

80


of the recombination zone depends on the relative magnitudes of p, and p h . If, for example, pe e p h , then recombination occurs close to the electron-injecting electrode and the current is essentially equal to an SCL hole current. For intermediate values of u q , recombination takes place in an extended zone, the width of which decreases with increasing oar. Approximate solutions for this case have been found using the regional approximation method, as described by Lampert and Mark.237In this approach, the problem is solved for four regions, two adjoining the electrodes where the current is assumed to be due to a single carrier type, and two in the center of the sample where recombination takes place. Lampert and Mark also describe solutions for the case with traps, and make attempts to include diffusion corrections. The approximate analytical solutions for all but the simplest cases become increasingly complex, and numerical methods are now likely to provide a more fruitful approach.

13. EXCITON DECAY LED devices operate by double charge injection and subsequent electron-hole capture to give excitons. Once an exciton has been created by electron-hole capture, the probability of obtaining light output is determined by competition between radiative and nonradiative decay channels. If radiative and nonradiative decay occur by monomolecular processes with lifetimes T~ and T,,,, respectively, the rate of decay of a population n of excitons is given by (13.1)

with the solution

The efficiency of radiative decay is given by

’

=

photons produced excitons generated

T

= -

T~

(13.3)


81

In an EL device, excitons are produced by the capture of nongeminate electrons and holes. Simple spin statistics arguments predict that the ratio of production of triplet to singlet excitons is 3:1, since the triplet state is threefold degenerate. Since the radiative transition from the triplet exciton to the ground state is dipole forbidden, triplet excitons have long radiative lifetimes. In the presence of faster nonradiative decay channels, they will thus tend to decay nonradiatively. In photolurninescence, all excitons are initially generated in the singlet state, and the amount of intersystem crossing between singlet and triplet states is usually small. It is therefore argued that the efficiency of radiative decay for excitons generated in EL is a factor of 4 lower than the photoluminescence efficiency. Various processes leading to deviations from this factor of 4 have been pr0posed.2~’ If the absorption and emission spectra are mirror images of one another, the radiative lifetime can be related to the absorption coefficient, a( v), by the StricMer-Berg relationshipz4’ (13.4) where

nref is the refractive index, Nv is the number of absorbing species per unit volume, and L(v) is the emission spectrum as a function of photon frequency, v . The radiative lifetime (in the absence of any reflecting interfaces) is thus an intrinsic property of the material, whereas the nonradiative lifetime may be influenced by extrinsic factors such as the presence of quenching sites. In PPV, the presence of impurities has been found to enhance nonradiative decay, and thus to reduce PL effi~iency.’~’ For this reason, high levels of chemical purity have been found to be essential for producing efficient LEDs. Metal-polymer interfaces are also found to enhance nonradiative decay of singlet e x ~ i t o n s . ’ ~ ~ C. Greenham, Ph.D. Thesis, University of Cambridge (1995). S. J. Strickler and R. A. Berg, J. Chem. Phys. 37, 814 (1962). 242 M. Yan, L. J. Rothberg, F. Papadirnitrakopoulos, M. E. Galvin, and T. M. Miller, Phys. Reci. Lett. 73,144 (1994). 243 G. Cnossen, K. E. Drabe, and D. A. Wiersrna, J . Chem. Phys. 98, 5276 (1993). 240N. 24 I

82


Intrinsic factors are also important in determining rates of decay of excitons. The ordering of 'Ag and 2B, states has already been mentioned as a reason why some conjugated polymers exhibit no luminescence (see Section 111.9.~).Conformational effects, both in solution and in the solid state, are also likely to influence nonradiative decay. Gettinger et ~ 1 . ' ~ ~ have found a correlation between the intrinsic persistence length and the PL quantum efficiency for a series of dialkoxy-PPVs in solution. The intrinsic persistence length is a measure of the stiffness of the polymer chain in solution, and Gettinger et al. show that increasing the stiffness of the chain decreases the amount of nonradiative decay. In the solid state, various strategies have been used to reduce the amount of nonradiative decay, and thus to improve PL efficiencies. These include increasing the length of alkyl (or alkoxy) side-chains, introducing nonconjugated units into the polymer backbone, and dispersing the conjugated polymer as a blend in an inert matrix. It is argued that all these methods reduce the probability of excitons diffusing to a quenching site. Interchain processes such as excimer formation can also lead to nonradiative decay, especially for materials with a large energy gap.2 Due to the difficulties involved in direct measurements of PL efficiencies, various estimates of quantum efficiency have been made using indirect time-resolved method^."'^^^^ The total lifetime for a polymer in the solid state can be measured directly by observing the fast decay of the PL. Unfortunately, however, it is not possible to extract both radiative and nonradiative lifetimes independently from a single-exponential overall decay. Measurement of PL efficiency in solution is possible by comparison with a standard solution, and simultaneous measurement of the total lifetime allows the radiative lifetime to be determined. One strategy to estimate solid state PL efficiencies is to use the radiative lifetime of the polymer or a model compound in solution to estimate the value in the solid state. The radiative lifetime, however, depends on both the absorption spectrum and the refractive index, and great care must be taken to allow for the change in refractive index between solution and solid. Solid state PL efficiencies obtained by this method must therefore be treated with considerable caution. In practice, photoluminescence decay for PPV films shows a behavior that cannot be fitted to a single-exponential decay. Two- and three-exponential decays have been used to fit these data, and the fast and slow 244 C. L. Gettinger, A. J . Heeger, J. M. Drake, and D. J. Pine, J. Chem. Phys. 101, 1673 (1994). 24 5 I. D. W. Samuel, B. Crystal, G. Rumbles, P. L. Burn, A. B. Holmes, and R. H. Friend, Chem. Phys. Lett. 213, 472 (1993).

83


TABLE 11. EFFICIENCIES FOR VARIOUS CONJUGAI'ED POLYMERS. AT THE SPECIFIED EXCITATION WAVELENGTHS POLYMER PPV MEH-PPV CN-PPV MEH-CN-PPV"

PL EFFICIENCY 0.27 0.10-0.15 0.35-0.46 0.48

ERROR

EXCITATION (nm)

f 0.02

458 488 488 488

* 0.02 f0.01 * 0.02

"Denotes the asymmetrically substituted CN-PPV derivative shown in Fig. 17.

components have been attributed to nonradiative and radiative decay, respectively.170,171,245 There is, however, no generally accepted physical model leading to this behavior. Typical lifetimes reported for the fast component of decay in PPV prepared by conventional routes are in the range of 70 to 250 ps at room t e m p e r a t ~ r e . ' ~ "Y,an ~ ~et~ , ~ ~ ~have recently demonstrated that oxygen impurities, probably in the form of carbonyl groups, act as effective quenching sites in PPV. The rate of nonradiative decay is therefore highly sensitive to the amount of residual oxygen remaining after conversion, and also to any subsequent photooxidation of the sample. For films of PPV converted under hydrogen, and believed to have a very low density of oxygen defects, the PL decay is fitted well by a single exponential with a time constant of approximately 1.2 ns.242 This figure seems a reasonable estimate of the radiative lifetime, and is comparable to the radiative lifetimes of PPV oligomers in the solid state. A nonradiative lifetime of 250 ps therefore corresponds to a PL quantum efficiency of 17%. Yan et ul.242 have found that their data for PPV containing quenching sites can be fitted well to a stretched exponential decay derived from a one-dimensional diffusion model for excitons. In this model, the diffusion of excitons to quenching sites is limited by one-dimensional diffusion along the chains to sites where interchain hopping can readily take place. Recently, direct measurements of PL efficiencies in conjugated polymers have been reported, using an integrating sphere to collect the emitted In Table 11, we show the measured PL efficiencies for some of light.246-24R the polymers used in Cambridge. We note that the value for PPV is 24 h D. Braun, E. G . J. Staring, R. C . J. E. Demandt, G. L. J. Rikken, Y. A. R. R. Kessener, and A. H. J. Venhuizen, Synth. Met. 66, 75 (1994). 24 7 E. G. J. Staring, R. C. J. E. Demandt, D. Braun, G. L. J. Rikken, Y. A. R. R. Kessener, T. H. J. Venhuiozen, H. Wynberg, W. Tenhoeve, and K. J. Spoelstra, Ad:. Mater. 6, 934 (1994). 24X N. C. Greenham, 1. D. W. Samuel, G . R. Hayes, R. T. Phillips, Y. A. R. R. Kessener, S. C. Moratti, A. B. Holmes, and R. H. Friend, Chem. Phys. Left. 241, 89 (1995).

84


FIG.47. The MIS device fabricated with n-silicon (MI, silicon dioxide (I), and polyacetylene (S) with a top, ohmic contact layer of gold. Typical thicknesses for the polyacetylene layer are in the range of 10 to 100 nm. [J. H. Burroughes and R. H. Friend, in Conjuguted Polymers (J. L. BrCdas and R. Silbey; eds.), Kluwer Academic, Dordrecht, (1991). Reprinted by permission of Kluwer Academic Publishers.]

consistent with the values estimated above from time-resolved measurements. On the basis of picosecond stimulated-emission measurements, ~ ~ recently proposed that only about 10% of the species Yan et ~ 1 . ’have produced by photoexcitation in PPV are intrachain singlet excitons, and that the rest are interchain excitons, which cannot decay radiatively. The measured values of 0.27 for the PL efficiency in PPV, however, show that at least 27% of absorbed photons produce singlet excitons, and on the basis of these measurements combined with time-resolved measurements of the PL decay, we believe that intrachain singlet excitons are produced with an efficiency close to The high values of PL efficiency for substituted PPVs shown in Table I1 and reported by other groups are encouraging for the development of polymer LEDs with high quantum efficiencies.2463 247 V. Field-Effect Diodes and Transistors

14. THE FIELD-EFFECT DEVICE a. The Field Effect in Conjugated Polymers The addition of an insulating layer between the semiconductor and one of the metal layers of the diode structure shown in Fig. 4 results in the MIS structure, which is widely used in a range of silicon semiconductor devices. Figure 47 shows an example of this device as fabricated on a silicon wafer substrate, with polyacetylene as the semiconductor and silicon dioxide as 249

M. Yan, L. Rothberg, B. R. Hsieh, and R. R. Alfano, Phys. Rev. E . 49, 9419 (1994).

85


FEC v >>o

v>o

a) V O

A

unoccupied

..._.._....._. ._......_. _.._.._.. A

0.64A

- - - - - - - - - - _ _ _ - _ __ __ ___ _ - - -

FIG.50. MIS band scheme appropriate for p-type polyacetylene. (a) Energy level scheme present for charge accumulation. As discussed in the text, the threshold for hole injection occurs when E, is A ( 1 - 2 , ’ ~ ) = 0.36A above E(,,and the charge injected is accommodated in the mid-gap states associated with the solitons created to store this charge. (b) Scheme that describes the onset of strong inversion. There is a charge depletion region, in which the soliton states associated with the extrinsic carriers are absent, and inversion occurs as E, is brought down to A ( 1 - 2 , ’ ~ ) above E,. The negative charges in the inversion layer are stored in the mid-gap soliton levels created to accommodate this charge (these are now doubly occupied). [J. H. Burroughes and R. H. Friend, in Conjuguted Polymers (J. L. BrCdas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of Kluwer Academic Publishers.]

94


bias voltage (V) FIG.51. The differential capacitance, aQ/av, and aln(T)/aV, where T is the optical transmission at 0.8 eV, versus bias voltage for an n-Si/SiO,/polyacetylene MIS device (SiO, thickness, 200 nm; polyacetylene thickness, 60 nm). Both quantities are measured with an ac modulation of 0.25 V at 500 Hz. [J. H. Burroughes and R. H. Friend, in Conjugafed Polymers (J. L. Bredas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of KIuwer Academic Publishers.]

layer; analysis here of the measured value gives the same value for the polyacetylene layer thickness as that measured directly (60 k 5 nm). Analysis of the variation of capacitance in the region of bias for which the width of the depletion layer is varying, through Eq. (14.21, gives values of N, = 8.5 X 10l6 cm-3 and = 1.6 V. Studies of the frequency dependence of the complex impedance provide information about the carrier mobilities for motion of the mobile holes from the gold top contact through to the polymer-insulator interface, and very low values are found, of order cm2 V-' s - ' . These mobility values are considerably lower than the field-effect mobilities, which are measured under similar conditions (see Section V.15.c), and are taken to indicate that the polymer chains lie preferentially in the plane of the film. The use of silicon dioxide as the insulator layer provides robust devices since the dielectric strength of the silicon dioxide is so high. However, it is of interest to use a wider range of insulators, including polymeric insulators, to demonstrate the feasibility of producing all-organic devices, and MIS diodes using Durham-route polyacetylene have been manufactured using polyimide with polyacetylene formed on top, and with poly(methy1methacrylate) formed on top of a polyacetylene layer. There is evidence that in both types of structure the surface layer of the polyacetylene at the interface with the insulator is better ordered than when formed at the interface with silicon dioxide."- l 9

5

95


01 0.4

0.6

0.8

1-0

1!

Energy (eV) FIG.52. Fractional change in the optical transmission, T , for the MIS devices shown in Fig. 51, obtained between gate voltages of 0 and V , with V = - 10, -20, -30, and -40 V. Modulation frequency is 330 Hz. [J. H. Burroughes and R. H. Friend, in ConjugatedPolymers (J. L. BrCdas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of Kluwer Academic Publishers.]

MIS devices of the type shown in Fig. 47 can be arranged to show partial optical transparency below the silicon band gap (1.1 eV), and can therefore be used to measure changes in the optical transmission associated with the presence of injected charges. The MIS structures operate in both charge accumulation and depletion, and there is particular interest in the charge accumulation layer in that the charges injected are present without any associated dopant. Extensive experiments have been carried out on the electro-optical properties of these MIS structures, covering both the electronic excitations of the solitons at mid-gap and also the I R and Raman vibrational excitations of the soliton. Figure 52 show the spectrum for the MIS structure previously characterized for its electrical properties in Fig. 51. An absorption band, peaking near 0.8 eV, is seen to grow with increasing negative gate voltage. The form of this absorption band is as expected for the charged soliton to band edge transition, though the energy of the transition measured here is significantly raised with respect to the value of 0.55 eV found for photoinduced charges (Fig. 38) and for the extrinsic charges measured in a similar manner for Schottky diodes.18.19 This is attributed to a higher level of disorder in the surface layer of the polyacetylene at the interface with the S O , , which locally raises the polyacetylene band gap. One of the important advantages of using the MIS device for optical measurements is that it allows quantitative measurements of the optical response, and we can gain very direct information about the way in which

96


charge is stored in the device. Figure 51 shows the differential capacitance, C = d Q / d V , as a function of voltage for the device discussed in the previous paragraph. It also shows the variation with bias voltage of d ln(T)/dT, where T is the optical transmission at the peak of the mid-gap absorption band (0.8 eV). Comparison between d Q / d V and d ln(T)/dT shows that the optical response sets in at the same voltage at which the capacitance data show the onset of formation of the accumulation layer, and for negative gate voltages we see a constant ratio between the two quantities. This ratio gives the optical cross section per injected charge, u ,through (15.2) A value for u at the peak of the mid-gap absorption band of 1.2 X lo-'' cm2 is found, and this is consistent with values obtained from chemical doping experiment^.'^^ It is important to note that the value of u is constant throughout the region of accumulation, and that it reaches this value at the onset of accumulation. It provides, therefore, very clear evidence that almost all the charge that is injected into the structure, which is measured by the differential capacitance, is stored in states that give rise to the new optical behavior, which is characteristic of the formation of a charged soliton-like excitation on the polymer chain. There is no evidence for the presence of trap states, which if present would have reduced the value of u ,particularly at the onset of accumulation. The size of the charge-induced optical response can be quite large for these MIS structures. For the device shown in Fig. 52, the peak value in the fractional change in absorption is some 0.7%, and the limit, set by Eq. (14.4) and the dielectric strength of the insulator, is about 2.4%. Besides the electronic transitions associated with the soliton state, there is extra IR activity due to new vibrational modes that couple to motion of the charged soliton along the polymer chain. These are seen in both doping2s8 and photoexcitation experiments259 (Fig. 38) in unoriented Durham polyacetylene. Figure 53 shows the results from an FTIR experiment on an MIS structure, showing the difference in transmission for the MIS device biased between 0 and -50 V with respect to the gate. Three peaks are observable; an absorption feature above 4000 cm-' (this is the low-energy side of the electronic absorption, as in Fig. 521, and two sharp *'*R. H. Friend, D. D. C. Bradley, P. D. Townsend, and D. C. Bott, Synth. Met. 17, 267 (1987). 25 9 R. H. Friend, H. E. Schaffer, A. J. Heeger, and D. C. Bott, J. Phys. C 20, 6013 (1987).


1000

1700

2400

3100

97

3800 4’

Wavenumber (cm-1)

FIG.53. Voltage-modulated transmission for a MIS diode, [T(O) - T ( - 5 0 V)]/T(O) in the spectral range 1000-5000 cm-’. [J. H. Burroughes and R. H. Friend, in Conjugated Polymers (J. L. BrCdas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of Kluwer Academic Publishers.]

vibrational features at 1379 and 1281 cm-I. These are identified as the two higher frequency translation modes of the soliton, and can be compared with the photoinduced absorption data shown in Fig. 38. The value of 1379 cm-l lies between that measured for photoinduced 1373 cmand for dopant-induced charges,2581410 cm- I , and provides information about the chain conformation and degree of conjugation. The intensity of this mode in relation to that of the electronic absorption band is much lower than that found in photoinduced absorption measurements. The oscillator strength scales inversely with effective mass of the charge carriers, and Burroughes and Friend estimate that the mass for the charges present in the accumulation layer is a factor of 3.6 higher than that of the photoexcited charge carriers in oriented Durham polyacetylene, consistent with the higher degree of disorder for the surface layer of the polyacetylene where the field-effect charges are confined. Measurements below 1000 cm-’ were not possible, so that the lowest frequency, “pinned” translational mode was not observed. New Raman-active modes associated with the width modulation of the soliton are also expected. These have not been seen in doping or photoexcitation experiments because of the inherent experimental difficulties, but the modulation experiment that can be performed with the MIS structure is well suited for this measurement. Experiments were carried out using the MISFET structure of the type shown in Fig. 7, but with a very thin polyacetylene layer (10 nm) to ensure that the whole depth of the polymer

’,

98


film was accessible for Raman scattering.260Raman scattering off the top surface of the structure was measured, and spectra were obtained from subtraction of a “background” scan from a “signal” scan. Here the “signal” was the measured scattering from a device with the gate at - 30 V with respect to the source and drain, and the “background” was the same scattering signal with the gate set to 0 V. Under these conditions the charge density in the accumulation layer is equivalent to 2.2 X charges/carbon atom for a 100 8, film. The spectral region from 900 to 1700 cm-’ was studied at a range of laser wavelengths from 476.5 to 584.0 nm. Two different classes of feature are observed in the difference spectrum of accumulation layers in the Durham polyacetylene shown in Fig. 54. The main class of feature is negative and arises from the bleaching of the resonance-enhanced A, modes that dominate the normal Raman spectrum of trans-polyacetylene. The two main modes are near 1100 and 1500 cm-’ with the exact frequency a function of the excitation energy. The bleaching of these two main modes dominates the difference spectrum. The dispersion of the bleaching is the same as that of the normal spectrum, but the bleaching fraction (the ratio of the integrated bleaching signal to the normal signal) changes with excitation energy, being 7 X at at 514 nm, and 1.6 X at 476 nm, 6.9 X at 488 nm, 7.7 x 584 nm. There are also small positive features to the low energy side of each of the bleaching features. These are much less intense, do not disperse with excitation energy, and occur at 1065 and 1455 cm-I. ~ ” the positive features in the difference Lawrence el ~ 1 . ~attribute spectrum to the width-modulating modes of the soliton. The presence of a defect on the polyacetylene chain is also expected to bleach the normal Raman spectrum. Terai et calculate that this bleaching will extend over a range of 30 carbon atoms-that is a 30% bleaching for a 1% soliton density. This large bleaching effect corresponds to the enhancement in the band-to-band optical absorption coefficient as calculated by Kivelson et The soliton density was calculated simply from the field across the insulating layer to be 2.2 X charges/carbon atom. Using the calculated enhancement factor of 30, this should give a bleaching of 0.007, which was indeed observed in the blue, but it was below that observed in the red. In summary, the studies of electrical and optical properties of polyacetylene MIS devices formed using thermally grown silicon dioxide as insulator ’‘OR. A. Lawrence, J. H. Burroughes, and R. H. Friend, in Electronic Properties of Conjugated Polymers IZI (H. Kuzmany. M. Mehring, and S. Roth, eds.), Springer Series on Solid State Sciences, Vol. 91, p. 127, Springer-Verlag,Berlin (1989).

99


620

0

Differcnce specurn

Normal spectrum

-10 0 C

1200

iioo

1000

0

W avenumbcr 54. Difference spectra due to charge accumulation layer in a polyacetylene MISFET structure. The solid line is the difference signal; the dashed line is the normal Raman spectrum. The excitation wavelength is 584 nm. [R. A. Lawrence, J. H. Burroughes and R. H. Friend, in Electronic Properties of Conjugated Polymers I11 (H. Kuzmany, M. Mehring, and S. Roth, eds.), Springer Series on Solid State Sciences, Vol. 91, p. 127, Springer-Verlag, Berlin (1989). Copyright 0 1989 Springer-Verlag] FIG.

provide very clear evidence for the formation of a charge accumulation layer in the polyacetylene, with the full set of signatures of charged soliton formation, including the electronic mid-gap transition, and the IR-active and the Raman-active vibrational modes. The spectroscopic information provides detailed information about the electronic structure of the region of the polyacetylene in which the accumulated charge is stored, which is expected to be within 10 nm of the interface with the insulator. Thermally grown silicon dioxide seems to produce a higher level of disorder at the interface than in the bulk, as indicated by the high energy of the mid-gap absorption (0.8 eV> and the high soliton effective mass. Other measurements made using polyimide and polytmethylmethacrylate) as insulators show that the mid-gap absorption is found at 0.55 eV, which is the same energy as that found in the photoinduced absorption measurements (Fig. 381, and which therefore indicates that the surface order is similar to that

100


a

0 100

v

c

C

5c .$

-2104

-4

10.4

-20

-1 5

-10

-5

Source Drain Voltage (V)

0

vSs

FIG.55. I,, versus V,, at several values of for a polyacetylene FET made using an improved Durham-route polymer with a biphenyl end group. The device structure is similar to that shown in Fig. 7, but with gold source and drain contacts, separation of 5 pm,channel width of 90 cm, and a silicon dioxide thickness of 20 nm. Note that the contribution to Idr due to the bulk resistance of the film has been subtracted. [C. P. Jarrett, R. H. Friend, G. Widawski, and W. J. Feast (unpublished observations)].

of the bulk for these polymer insulators. Such sensitivity to the nature of the interfacial order is probably also the case for the field-effect mobility. c. FETs The MIS field-effect transistor, or MISFET, operates by modulation of the conductance of the semiconductor in the depletion, accumulation, or inversion modes, with control through the potential on the gate. Polyacetylene FETs have been studied with two types of source-drain contacts. Gold, which has a high work function, should give ohmic contact to the charge accumulation layer, and is an obvious choice in this role. Burroughes et al.'' have also used poly n-silicon, capped with aluminum, with the anticipation that the lower work function for these materials should give better charge injection to the inversion layers. Figure 55 shows some recent measurements of the channel current, Ids, against the drain voltage, Vds,for various negative gate voltages, %,, for the FETs fabricated with gold source and drain contacts, and which use Durham-route polyacetylene synthesized using Schrock catalysts and with a para-biphenyl chain end.*6' It can be seen that the source-drain current, Ids, is controlled by the gate-source voltage, %,, rising strongly as the polycetylene layer is brought into accumulation for negative values of 5,. 26 1

C. P. Jarrett, R. H. Friend, G. Widawski, and W. J. Feast (unpublished observations).

101

SEMICONDUCTOR DEVICE PHYSICS 0.03 0.03 0.02 0.02 0.01

0.01

0.00 -20

&,

-1 5

-5

-10

vg.sat

0

(v)

FIG.56. where Ids is the saturation current (measured at Vds = - 20 V), versus gate cm2 voltage for the device shown in Fig. 55. A field-effect mobility for this device of V-’ s - is obtained from the gradient of the response at larger values of gate voltage (shown as a solid line). [C. P. Jarrett, R. H. Friend, G. Widawski, and W. J. Feast (unpublished observations)].

’

The field-effect mobility can be obtained from the variation of the saturaversus tion current with gate voltage, using Eq. (14.7). The plot of gate voltage is shown in Fig. 56. Values for the field-effect mobility obtained from this form of polyacetylene are in the range 1 to 4 X cm2 V - ’ s - ’ , This is considerably higher than the values reported earlier l 9 for FETs with gold source-drain contacts and standard Durham polyacetylene, which are typically in the range to cm2 V-’ s-’. the properties of MISFETs with poly ~ ~ Burroughes et ~ 1 . ’ investigated n-silicon source-drain contacts to promote the formation of the charge inversion layer for positive gate voltages. The structure of these devices is as shown in Fig. 7. The polyacetylene layer for these devices was kept thin, typically to no more than 20 nm; this allows full depletion at the appropriate bias, and thus gives very good “on” to “off’ g, ratios. Figure 57 shows the variation of Z, with Vgsfor fixed V,, = 10 V. Note that I d s has a minimum at Vgs= +10 V, corresponding to full depletion across the polyacetylene layer. For voltages above this, the conductivity rises rapidly, and is identified with the formation of a charge inversion layer at the polyacetylene-SiO, interface. For voltages lower than + 10 V, I d , rises very sharply over one or two decades. This sharp rise is most likely due to the decrease in width of the depletion layer, so that the “knee” in the curve, seen at around + 5 V, corresponds to the flat-band voltage at which

JIds

‘‘9

102

NEIL C . GREENHAM AND RICHARD H. FRIEND

FIG. 57. MISFET structure with poly n-silicon source and drain contacts, as shown in Fig. 7, with a Durham polyacetylene layer thickness of 20 nm. Shown here is the variation of Ids with 5. for Vds = + 10 V. [J. H. Burroughes and R. H. Friend, in Conjuguted Polymers (J. L. Brtdas and R. Silbey, eds.), Kluwer, Dordrecht (1991). Reprinted by permission of Kluwer Academic Publishers.]

the measured conductivity is just that of the bulk polymer. Further reduction in the gate voltage produces a further increase in Ids, which is associated with the formation of a charge accumulation layer, even though the poly n-silicon source and drain might have been expected to give blocking contacts to the majority carriers in the accumulation layer. The change in conductivity for these structures is very high, with a ratio of lo5 between I/gs = + 10 and -40 V. The evidence that this type of structure shows for operation in the inversion mode is interesting because it has in general proved very difficult to demonstrate. Burroughes et al.” point out that the polarity of the gate voltage to achieve inversion is such as to attract the dopant ions toward the insulator layer (positive gate voltage and negative ions for the case here of p-type doping). If sufficient ions are able to move in response to the applied field, then this would prevent formation of the inversion layer, and Burroughes et al. observe that inversion was only obtained for devices made with very thin films of polymer, for which the total density of dopant ions per unit area was insufficient to prevent inversion. The properties of these FETs are in accord with the properties of the MIS diodes presented in Section V.15.b, with clean switching between depletion and accumulation. However, the field-effect mobilities are not high, and behave in many ways analagously to the mobility of the photocarriers also measured on these spin-coated films of polyacetylene. Mea-


103

surements of the temperature dependence of the field-effect currents have been reported,'8. l 9 and show weakly thermally activated properties, with activation energies near room temperature of around 0.1 eV. Improvement in field-effect mobility can be achieved by producing better ordered polymers, and there is direct evidence from the optical spectroscopy of the accumulation layer charge in the MIS diodes that the surface layer of the polymer formed against the insulator layer can be more disordered than the bulk of the polymer. At the time of writing, the very recent results obtained with polyacetylene, which has better stereoregularity, shown in Fig. 55 and 56, are very promising, showing an improvement in mobility of as much as a factor of 1000 over earlier results, and it is considered that this improvement is associated with better ordered polymer chains. 16. POLYTHIOPHENE AND POLY(P-PHENYLENEVINYLENE) DEVICES a. MIS Diodes Polyacetylene is distinct from most other conjugated polymers in supporting soliton-like excitations that have associated electron states that are nonbonding and therefore lie at mid-gap. Other polymers, which are represented here by the various derivatives of polythiophene and of PPV, support polaron-like excited states, which can be either singly or double charged, as indicated in Fig. 32. Charge introduced in such polymers can therefore be distributed between both types of excitation, and there have been considerable efforts to use the electro-optical measurements of the charges injected into MIS diodes to determine how the charge is stored. The density of charges present as the accumulation layer in polymer LEDs is always low, i.e., very much less than the number density of polymer repeat units [see Eq. (14.4)]. We can expect therefore at room temperature that the equilibrium between bipolarons, BP2+,and polarons, P'. BP2

-

2Pt,

(16.1)

is heavily weighted toward the dissociated polarons by the higher configurational entropy of the polarons, even if there is a contribution to the internal energy that favors the associated b i p ~ l a r o n . It ~'~ might therefore be expected that the only response seen would be that due to the polarons. However, the position that has emerged is considerably more complicated than might have been predicted on this basis, and it seems that the presence of disorder plays an important role in the energetics of some of the excitations seen.

104


FIG. 58. Structure of an MIS diode with poly(2,5-dimethoxy-p-phenylenevinylene), PDMeOPV, used for visible and near-IR spectroscopy. [M. G. Harrison, K. E. Ziemelis, R. H. Friend, P. L. Burn, and A. B. Holmes, Synth. Met. 55, 218 (19931.1

MIS devices that demonstrate optical transmission over a wider spectral range than the devices built up on silicon substrates have been developed, using silicon dioxide deposited onto semitransparent metal layers on glass substrates to provide insulator and gate, as illustrated in Fig. 58. Ziemelis et d 2 l 3 reported the field-induced absorption spectra due to injected charges in poly(3-hexylthiophene), (see Fig. 181, and found that the dominant contribution to the induced absorption spectrum associated with injected charges (after correction for electromodulation of the n- T* absorption edge) showed two features that peaked at 1.8 and at 0.5 eV, with associated bleaching of the n-rr* interband transitions at 2.5 eV (the peak of the r r - ~ * absorption for these samples), as shown in Fig. 59. The induced absorption at 1.8 eV was associated with the dipole-allowed transition between the two gap states of the polaron (Fig. 32) and the lower transition to the transition between the band edge and gap states. Note that the energies of these transitions are not affected by the Coulomb ~ that potential within the pertubation theory due to Baeriswyl et ~ 1 . , ' *so they provide a direct measure of the positions of the one-electron states in the gap. The similarity of the absorption at 1.8 eV with the position of the peak in the singlet photoluminescence, at 1.7 eV for this polymer, is reassuring, because this indicates that the polaronic binding energies for these excitations are similar, as is required if the capture of polarons of opposite charge can produce the exciton responsible for the electroluminescence that is observed in this and other conjugated polymers (see Section VI). There is also some evidence for the presence of induced absorption at energies similar to those seen in chemical doping and


105

a

-5

1.0

2.0

1.5

3 .O

2.5

Energy (eV)

(a) 5

"0

I

a5

1.0

Energy (ev) (b) FIG.59. Voltage-modulated optical absorption spectra for poly(3-hexylthiophene), showing the fractional change in optical transmission, - A T / T , as a function of photon energy. (a) Electromodulation of the absorption band (data on RH scale) and the absorption produced by charge injection (data on LH scale) above 1 eV. (b) Absorption produced by charge injection in the IR. [K. E. Ziemelis, A. T. Hussain, D. D. C. Bradley, R. H. Friend, J. Riihe, and G. Wegner, Phys. Reo. Lett. 66,2231 (1991).]

photoinduced absorption experiments, at 1.2 eV, and Ziemelis et ~ 2 1 . ~ ' ~ consider that this might be associated with some defect-stabilized bipolarons. The study of the poly( p-phenyleneviny1ene)s has revealed somewhat different results.219The dimethoxy derivative of PPV, poly(2,5-dimethoxyp-phenylenevinylene), PDMeOPV (see Fig. 131, was found to be convenient for study since it is sufficiently p-doped as prepared to be suitable for

106


FIG.60. Voltage-modulated transmission spectrum of a PDMeOPV MIS device in accumulation. ( + 2-V ac modulation at 367 Hz about - 10-V dc is applied to the gate contact.) The features at 0" are attributed to bipolarons (0.71 and 1.65 eV) and polarons (2.0 eV) and that at + 131" ( - 2.27 eV) to electroabsorption. The feature at 15", around 1.2 eV, may be extrinsic. [M. G. Harrison, K. E. Ziemelis, R. H. Friend, P. L. Burn, and A. B. Holmes, Synth. Met. 55, 218 (19931.1

+

use in these devices without further doping. Several charged excitations are present, and these can be separated by analysis of the phase and frequency dependence of the optical response with respect to the ac modulation of the gate voltage. Figure 60 shows the optical spectra obtained when the device was modulated about the bias voltage at which accumulation is achieved. The features attributed to charged excitations all occur at the same phase (indicated as 0" in Fig. 60). The high-energy feature is attributed to electromodulation of the interband transition, and occurs at a different phase ( 131" in accumulation, + 166" at the onset of depletion). A third distinct component is observed around 1.2 eV. In general, features in the absorption spectrum that are attributed to charge injection have a bias dependence which follows the differential capacitance, as was also observed for the polyacetylene MIS diodes (see Fig. 51). cm2 for the transition at This gives an optical cross section of 2 x 0.7 eV. The peaks at 0.71 and at 1.65 eV are present in both the bulk polymer (in depletion) and in accumulation. Their energy levels agree well with those attributed to the long-lived bipolarons observed in photoinduced absorption262and electrochemical doping experiments.263Harrison et al. suggest that the 2.0-eV peak may be due to a polaronic contribution,

+

"*H. S. Woo, S. C. Graham, D. A. Halliday, D. D. C. Bradley, R. H. Friend, P. L. Burn, and A. B. Holmes, Phys. Reu. B 46, 7379 (1991). 263J. B. Schlenoff, J. Obrzut, and F. E. Karasz, Phys. Reu. B 40, 11822 (1989).


107

which appears to be more stable at the interface than in the bulk. Though this assignment to bipolarons and polarons is consistent with the description of charge storage in the doped polymers in bipolarons, this picture may need to be revised if the "bipolaron" absorptions at 0.71 and 1.65 eV are reassigned to singly charged polarons, as suggested r e ~ e n t l y . " ~ -The '~~ measurements on the hexamer of thiophene, discussed in Section V. 17, provide further indication of the role of extrinsic factors in the stabilization of polaron-like excitations. One further optical experiment that has been performed with MIS devices of this type is the measurement of the modulation of the photoluminescence from the conjugated polymer layer as a function of the applied gate ~ o l t a g e . ~It' i~s found , ~ ~ ~that the luminescence is quenched when an accumulation layer is present, and this is attributed to the role of charged polarons or bipolarons in quenching the singlet excitons. From analysis of the size of the quenching, a value of 5 nm can be set as the larger of the exciton diffusion range or the spatial extent of the charge accumulation layer away from the insulator layer into the bulk of the polymer. b. FETs A considerable number of studies have been made of the electrical characteristics of polythiophene-based FETs, using electrochemically prepared polymers'7s34 and solution-processed poly(3-alkylthiop h e n e ) ~ . ~ ~ .In ~ "general, ~ ~ ' ~ devices show properties similar to those of the polyacetylene FETs discussed in Sec. V.15, with relatively poor fieldeffect mobilities. These mobilities are considerably higher for FETs that use poly(3-alkylthiophene)~with short alkyl chains, up to cm2 V-' s - I for poly(3-methylthi0phene),~~than with the longer side-chains, for which mobilities can be lower than cm2 V - ' s-'. Efforts have been made to improve mobilities by fabricating devices with stretch-aligned p0lymer,2~~ and using the regioregular but they have not 2b4

265

P. Dyreklev, 0. Inganas, J. Paloheimo, and H. Stubb, J . Appl. Phys. 71, 2816 (1992). J. Paloheimo, P. Kuivalainen, H. Stubb, E. Vuorimaa, and P. Yli-Lahti, Appl. Phys. Lett.

56, 1157 (1990). 2bh

J. Paloheimo, E. Punkka, H. Stubb, and P. Kuivalainen, in Lower Dimensional Systems

and Moleculur Electronics (R. M. Metzger et al., eds.), p. 635, Plenum, New York (1991). 267 J . Paloheimo, H. Stubb, P. Yli-Lahti, and P. Kuivalainen, Synfh. Met. 41-43, 563 (1991). 268

J. Paloheimo, H. Stubb, P. Yli-Lahti, P. Dyrekelev, and 0. Inganas, Thin Solid Films

210-211, 283 (1992). "'2. Xie, M. S. A. Abdou, X. Lu, M. J. Deen, and S. Holdcroft, Can. J . Phys. 70, 1171

(1992). 270 P. Dyreklev, G. Gustafsson, 0. Inganas, and H. Stubb, Solid State Commun. 82, 317 (1992). 27 I K. Pichler, R. H. Friend, K. A. Murray, A. B. Holmes, and S. C. Moratti, Mol. C y f . Liy. C y t 256, 671 (1994).

108


succeeded in raising mobilities to significantly higher values. Efforts have also been made to reduce the source-drain separation to submicron dimensions, using either lithographic techniques272or by using a discontinuous metal film on the insulator layer,273but these have not resulted in improved mobilities. There have been fewer studies of the poly(phenylenevinylene)s, probably because these polymers have been less readily available. The poly(2,5-dimethoxy-p-phenylenevinylene) studied in the MIS diodes25" (see Section cm2 V-' s-'. V.16.21, shows a low field-effect mobility of 1.5 X Polfip-phenylenevinylene), PPV, is not directly usable as prepared because it shows too high a dark resistivity, but Pichler et ~ 2 1 . ' ~ have ~ succeeded in raising the conductivity by doping with iodine, both by exposure of the polymer film to iodine vapor and by implantation with iodine ions, and have reported that FET behavior can be observed, though with very low mobilities of up to cm2 V-' s - l . In contrast to these very unpromising results, Fuchigami et aLZ7 have achieved the highest reported value of a polymer field-effect mobility using the methoxyeliminating precursor route (shown in Fig. 11 for PPV) to polfithiophenevinylene), PTV (see Table I). This polymer has a relatively low n-r* gap (1.8 eV) and can show relatively good order as judged from the vibronic absorption band.275The FET devices used in this structure in the ~ - r * work were fabricated with deposition of the source and drain electrodes on top of the polymer film, which was kept thin (60 to 80 nm), and field-effect mobilities of 0.22 cm2 V-' s-' were measured from the analysis of the saturation current regime. The mobility was found to be highly dependent on the degree of conversion of the polymer from its precursor, and it is considered that high mobilities require very extended conjugation and a well-ordered structure. We have mentioned that the majority of studies of the field effect have been made on polymers that are not intentionally doped, and that the problem with most doping protocols is that the dopants used are able to diffuse within the polymer matrix. However, promising results have now been obtained using molecular doping species such as DDQ, as mentioned in Section V.14.a.254-256DDQ is considered to form a doubly charged anion in the solid solutions obtained with poly( p '-dodecyloxy-a, a',-a',a" terthienyl), poly DOT,. It is possible to control the concentration 272S.Franssila, J. Paloheirno, and P. Kuivalainen, Electron. Leu. 29, 713 (1993). 273J. Paloheimo, H. Stubb, and L. Gronberg, Synrh. Met. 55-57, 4198 (1993). 274 K. Pichler, C. P. Jarrett, R. H. Friend, B. Ratier, and A. Moliton, J . Appl. Phys. 77, 3523 (1 995). 275A.J. Brassett, N. F. Colaneri, D. D. C. Bradley, R. A. Lawrence, R. H. Friend, H. Murata, S. Tokito, T. Tsutsui, and S. Saito, Phys. Rev. E 41, 10,586 (1990).

109


10-9 1019

I

1020

. . . . ....

I

1021

. . . . ...

Acceptor Density ( ~ m - ~ )

1022

FIG.61. Conductivity in polyDOT, doped with DDQ as a function of acceptor density. The solid circles represent data points where the acceptor density is calculated from the dopant concentration. The open circles represent acceptor densities as obtained from MIS diode measurements. The solid line shows a variation of the conductivity, u ,with acceptor density, No, of the form (T a N,’ with y = 3.5 [C. P. Jarrett, A. R.Brown, R. H. Friend, and D. M. de Leeuw, J . Appl. Phys. 77, 6289 (1995).1

of carriers from l O I 9 to 5 x 10” cm-3 and therefore to control the bulk conductivity of this system over many orders of magnitude (lop8to 10- S cm-’) through control of the concentration of the DDQ, and this has allowed the study of the field-effect mobility in materials spanning this range of bulk c o n d u c t i ~ i t y The . ~ ~interest ~ ~ ~ ~ ~in these studies is that the mobility of the carriers introduced through doping rises very quickly with the dopant concentration. At low concentrations of extrinsic charge carriers, mobilities are low because the conduction process requires hopping from one localized state to the next. As the concentration of carriers increases, there is increasing overlap between adjacent states, so that transport switches to a “metallic” o r band-like process. This higher conductivity is clearly required to explain the high values of dc conductivity in heavily doped polymers276 where the conductivity can reach values of several tens of thousand S cm-I. Field-effect mobilities have been measured over this wide range of bulk conductivities2”, 256; for the more heavily-doped polymers this requires some care to measure a small field effect on top of the substantial bulk current. The rapid increase in carrier mobility with doping is seen in Fig. 61, which shows the variation of the bulk conductivity of the polyDOT, doped with DDQ as a function of bulk carrier density. The variation of the

’

216 T. Schimmel, D. Glaser, M. Schwoerer, and H. Naarmann, in “Conjugated Polymers” (J. L. BrCdas and R. Silbey, eds.), p. 49, Kluwer Academic Publishers, Dordrecht (1991).

110

NEIL C. GREENHAM AND RICHARD H. FRIEND 10’

1o 9 109

108 107

l o 6 105 104 1 0 3 lo-* 10”

loo

Conductivity (Scm-’) FIG.62. Dependence of the field-effect mobility in polyDOT, doped with DDQ on the bulk conductivity. The solid circles represent the data taken by progressive thermal dedoping of a highly doped device; the open circles represent data from devices made by varying the initial doping concentration with no thermal dedoping. The square points represent mobility values calculated from the conductivity divided by the elementary charge and the acceptor density. These data points indicate that the average mobility of bulk charge carriers is about two orders of magnitude smaller than the average mobility of accumulated charge carriers. [C. P. Jarrett, A. R. Brown, R. H. Friend, and D. M. de Leeuw, J . Appl. Phys. 77, 6289 (1 9951.1

conductivity, u ,with acceptor density, No, is of the form u a N,‘ with y = 3.5. Note that the concentration of bulk carriers is determined here both from the concentration of DDQ present and also from the space charge density determined from the capacitance versus bias voltage dependence of MIS diodes made alongside the FETs; the agreement between the two methods provides confirmation that all the DDQ present acts as dopant. Figure 62 shows the variation of the field-effect mobility with bulk conductivity, and also the carrier mobility for the bulk carriers. Both field-effect mobility and bulk carrier mobility increase rapidly with bulk conductivity, fitted in Fig. 62 as a power-law of the form p a u 8 with S = 0.7. Note that the mobility of the field-effect charges is a factor of 100 larger than that of the bulk charges, and Jarrett el ~ 1 . ~suggest ’ ~ that this indicates that the majority of the carriers produced by doping are pinned to the dopants by the Coulomb potential. These results provide insight into the processes that control carrier mobilities, and Brown et UL.*~’ have shown that this form of dependence of the carrier mobility on bulk conductivity is a universal law for a wide range of “disordered” molecular or polymeric semiconductor field-effect devices. The actual values of mobility measured are still relatively low, and new strategies are required


111

if FETs with useful switching ratios and useful carrier mobilities are to be produced. A n approach to improving the switching ratio of the FET is to introduce blocking contacts at the source and drain electrodes, as has been reported by Kaezuka et al.I7 An interesting alternative is to use materials that are very much better ordered, as has been successfully demonstrated by Garnier and coworkers using the hexamers of thiophene, as discussed in Section V. 17. 17. THIOPHENE OLIGOMER DEVICES

The use of molecular semiconductors in field-effect devices has been using a range of phthalocyanines. Of more advanced by Madru et a1.277*278 direct interest to the work with conjugated polymers is that recently carried out using oligomers of thiophene. The structure of the sexithiophenes is shown in Fig. 25. These oligomers show many of the properties of the polymer, and, for example, sexithiophene shows an onset of r r r * absorption at a little above 2 eV. The particular interest in these materials is that they can be processed into highly crystalline thin films by vapor d e p ~ s i t i o n . ~Garnier ~ ~ ” ~ et ~ al. ~ ~were able to show that these well-ordered sublimed films could exhibit high field-effect m ~ b i l i t i e s , and ~ ~ . there ~ ~ has been a high level of interest in this material.2R”The high mobilities are attributed to good intermolecular charge transport in these well-ordered films, which can be arranged to grow so that the direction of optimum intermolecular contact is within the plane of the film.25 Control of the structure of the film can be made through the selection of the molecular structure, and Garnier et al. have produced their best results using a ,w-di(hexyl)sexithiophene26 (see Fig. 25); they report field-effect mobilicm2 V-l s-I. They have also succeeded in ties of typically 7 X producing FETs with this level of field-effect mobility by printing electrodes on either side of a polymer thin film which was used as the insulator layer (polyethylene terephthalate of thickness 1.5 pm).26 Harrison et ~ 1 . ’have ~ ~ carried out a range of measurements on the optical spectroscopy of the field-induced charges in MIS diodes made with sexithiophene, using structures similar to those shown in Fig. 58. It was 217 R. Madru, G. Guillaud, M. A. Sadoun, M. Maitrot, J. J. AndrC, C. Clarisse, M. L. Contellec, and J. Simon, Chem. Phys. Left. 142, 103 (1987). 278 R. Madru, G . Guillaud, M. A. Sadoun, M. Maitrot, J. J. AndrC, J . Simon, and R. Even, Chrm. Phys. Lett. 145, 343 (1988). 219 X. Peng, G . Horowitz, D. Fichou, and F. Garnier, Appl. Phys. Lett. 57, 2013 (1990). ?HI1 H. Akimichi, K. Waragai, S. Hotta, H. Kano, and H. Sakaki, Appl. Phys. Left. 58, 1500 (1991).

112

NEIL C. GREENHAM AND RICHARD H. FRIEND 0.5

1

0.5

I

1 .o

I

1.5

I

2.0

2.5

Energy (eV) FIG. 63. Optical spectrum of field-induced charge for a sexithiophene thin-film MIS device. The measurements were made at room temperature with a 2-V peak-to-peak modulation voltage at 367 Hz, superimposed on 0-V dc, applied to the gate electrode. Features characteristics of the bulk material (low charge density) are probed under these conditions. Note that the data are presented as detected at two phases (solid and dotted lines), in quadrature to one another, with respect to the gate voltage modulation; the phases are selected to give maximum separation of different features in the spectrum. [M. G. Harrison, R. H. Friend, F. Garnier, and A. Yassar, Synth. Met. 67,215 (19941.1

found that the optical spectra associated with the injected charges were considerably more complicated than those shown for the conjugated polymers, and Harrison et al. were able to identify three distinct charged excitations, which were observed to varying degrees under conditions where the bulk space charge or the surface accumulation charge was modulated. Figure 63 shows the optical spectrum of field-induced charges for a thin-film MIS device with modulation of the gate voltage about zero bias, under which conditions the charged states probed are largely those in the bulk, at the edge of the depletion regime. A range of new absorptions is seen. Some of these new absorptions can be identified in relation to 184 The absorptions at those seen for chemically doped sexithi~phene.’~~? 0.83 and at 1.70 eV are associated with singly charged cations (polarons). When the same device is modulated about a negative bias (-4 V) as shown in Fig. 64, so that the accumulation regime is studied, an additional feature at 1.34 eV is observed (at a different phase than the 0.83- and 1.70-eV features), and this is assigned to the absorption due to the dication (bipolaron). This is expected at high concentrations of charge, as achieved under accumulation. The third charged state shows transitions at 0.55, 1.01, and at 2.05 eV and is seen when the diodes are driven in depletion, particularly when the film has been annealed (some of these features are

113

SEMICONDUCTOR DEVICE PHYSICS 0.5 0.0

t

g

*

s

-0.5

-1.0

-1.5

-2.0 -2.5 0.5

1.o

1.5

Energy (eV)

2.0

2.5

FIG.64. Optical spectrum of field-induced charge for a sexithiophene thin-film MIS device operating in accumulation. The measurements were made at room temperature with a 2-V peak-to-peak modulation voltage at 127 Hz, superimposed on -4-V dc gate bias. The accumulation layer is probed under these conditions. [M. G. Harrison, R. H. Friend, F. Garnier, and A. Yassar, Synrh. Mer. 67,215 (19941.1

seen in Fig. 63). Harrison et al. propose that this state is formed as an intermolecular 7r dimer, which is formed in well-ordered regions of the film. The assignment of the various optical transitions to these different excited states is illustrated in Figs. 65 and 66. Strong transitions are only observed in the oligomers between adjacent states; this is required to get a nonvanishing dipole matrix element. Note that the optical transitions of the singly and double charged oligomers are different than those observed in the polymers; the twin absorption bands are observed for the singly charged state for the oligomer (Fig. 65) and for the doubly charged state for the polymer (Fig. 33).28' 18. OTHERDEVICES The modulation of optical properties caused by charge injection into the field-effect devices can be used to provide an electro-optical modulator, although this is slow because of the low carrier mobilities. A larger depth of modulation can be achieved by arranging for the light signal to be passed parallel to the active layer of the device, and this has been reported "lJ.

Cornil and J. L. BrCdas, Adv. Muter, 7, 295 (1995).

114

NEIL C. GREENHAM AND RICHARD H. FRIEND LUMO+I

I UM0+2

................. ......

LUMO+l . T p y M O

&HOMO

Neutral 6T

Radical Monocation 6T"

Dication 6T2+

FIG. 65. Proposed energy-level diagram for sexithiophene, showing highest occupied molecular orbitals (HOMOS), singly occupied orbitals (SOMOs), and lowest unoccupied molecular orbitals (LUMOs). The evolution of the electronic energy levels is shown from the neutral species, first to those of the radical monocation (6T" ) at low charge densities, then to the spinless dication, (6T"). [M. G. Harrison, R. H. Friend, F. Garnier and A. Yassar, Synrh. Met. 67, 215 (1994).]


Cation-Cation n-d imer (6T)TL22' 2


FIG.66. Proposed energy-level diagram for sexithiophene showing the electronic energy levels of a spinless r dimer (6T'+),, as a result of splitting of the energy levels of the constituent monocations in close proximity. [M. G. Harrision, R. H. Friend, F. Garnier, and A. Yassar, Synth. Mer. 67, 215 (1994).]


115

for a device in which the light is waveguided to pass between the source, drain, and gate electrodes of a FET ~ t r u c t u r e . ~ ~ ” A very different type of three-terminal device has been reported by Yang and Heeger.282 They have built a device in which a “grid” of solution-processed conducting polyaniline (doped with camphor sulfonic acid), in the form of a network of polyaniline fibrils, is sandwiched between layers of MEH-PPV (Fig. 141, which are contacted to hole-injecting electrodes. The potential on the grid is used to control the flow of holes through the spaces between polyaniline fibrils, and the device therefore is considered to function as a solid state triode. A gain of up to 5 has been reported. VI. Light-Emitting Diodes 19. BACKGROUND

Elecroluminescence (EL), the process of light emission from a material after injection of electrical charge carriers, has been the subject of interest for several decades. This area has many applications, for example in telecommunications and information display. Light-emitting diodes (LED$ using p-n junctions of inorganic semiconductors are an established techn0logy,2*~however they are difficult to form over large areas, and efficient blue emission has only recently been achieved.284-286 Most microelectronic circuits are based on silicon, which does not have the direct band gap required for light emission, so it is not possible to fabricate light-emitting components directly on the chip. Attempts have been made to overcome this problem by treating the silicon surface to form a porous microstructure.**’ Electroluminescent devices have been fabricated from this “porous s i l i ~ o n , ”but, ~ ~ despite ~ . ~ ~ intensive ~ work, efficiencies remain low and stable devices are difficult to fabri~ate.’~” 28 2

Y. Yang and A. J. Heeger, Nature 372, 344 (1994). E. W. Williams and R. Hall, Luminescence and the Light Emitting Diode, Pergarnon Press, Oxford, 1978. 284 K. Ohkdwa, A. Uneno, and T. Mitsuyu, Jppn. J . Appl. Phys. 30, 3873 (1991). 28 5 H. Jeon, J . Ding, A. V. Nurrnikko, W. Xie, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Lett. 60, 892 (1992). 2X h T. Nakayarna, Y . Itoh, and A. Kakuta, Appl. Phys. Lett. 63, 594 (1993). lx7L.T. Canharn, Appl. Phys. Lett. 57, 1046 (1990). 28RN.Koshida and H. Koyarna, Appl. Phys. Lett. 60, 347 (1992). 28 Y F. Namavar, H. P. Maruska, and N. M. Kalkhoran, Appl. Phys. Lett. 60, 2514 (1992). 290 S. S. Iyer, L. T. Canharn, and R. P. Collins, eds., Light Emissionfrom Silicon, Mat. Res. Soc. Symp. Proc., Vol. 256, Materials Research Society, Pittsburgh, (1992). 2x3

116


Inorganic phosphors have for some time been used to fabricate large-area electroluminescent display^.^^'-^^^ In a typical display the active material is separated from the electrodes by dielectric layers, and high fields are used to accelerate charges across these layers and into the material.294This type of display has found some applications, but is not widely used, partly due to the high driving voltages required and also due to problems of degradation under operation. Many conjugated organic materials exhibit strong photoluminescence, and are therefore attractive as possible materials for electroluminescence. The first report of electroluminescence in such a material was by Pope, Kallmann, and Magnate in 1963,28 using single crystals of anthracene. Subsequently, many other organic molecular crystals were also found to electrol~minesce.~ Alth ~ ~ough - ~ ~ ~some good quantum efficiencies were achieved, driving voltages typically in excess of 100 V were required due to the thickness of the crystals. In addition, many of these devices required reactive electrodes containing sodium to give efficient electron injection, making them unsuitable for practical applications. . used ~ ~ Driving voltages were reduced significantly by Roberts et ~ 1 who Langmuir-Blodgett techniques to produce thin films of an anthracene derivative. Further progress was made by Vincett et who deposited amorphous thin films of anthracene by vacuum sublimation. This method allows uniform thin films to be deposited over large areas, and is now the standard method of production for molecular organic E L devices. Interest in the field was renewed in 1987 when Tang and Van Slyke reported the use of additional hole-transporting layers to achieve good quantum effic i e n c i e ~ .Adachi ~ ~ . ~ ~et~ have extended this work by introducing a third, electron-transporting layer and have reported high quantum efficiencies and high brightnesses with low drive voltages and a range of emission ~ o l o r s . The ~ ~se~ devices - ~ ~ ~have attracted a great deal of commercial interest, and are now being investigated by a number of industrial ~

291

1

.

~

~

3

~

~

~

J. W. Allen, J. Lumin. 31-32, 665 (1984). 292G.0. Muller and R. Mach, J. Lumin. 40-41,92 (1988). 2y3H.Kobayashi, Proc. SPIE 1910, 15 (1993). 294R.Mach and G. 0. Mueller, Proc. SPIE 1910, 48 (1993). 295 G. Vaubel, Phys. Stat. Sol. 35, K67 (1969). 2Y6 J. G. Basturo and Z. Burshtein, Mol. Crysl. Liq. Cryst. 31, 211 (1975). 297Z.Burshtein, Mol. Cryst. Liq. Cyst. 60, 207 (1980). 29 8 C. W. Tang, S. A. Van Slyke, and C. H. Chen, J. Appl. Phys. 65, 3610 (1989). 299 C. Adachi, S. Tokito, T. Tsutsui, and S . Saito, Jpn. J . Appl. Phys. 27, L269 (1988). 3"0C.Adachi, T. Tsutsui, and S. Saito, Appl. Phys. Lett. 56, 799 (1990). 301S.Saito, T. Tsutsui, M. Era, N. Takdda, C. Adachi, Y. Hamada, and T. Wakimoto, Proc. SPIE 1910, 212 (1993).


117

companies. Charge-transporting layers have also proved useful for improving conjugated polymer EL devices, as discussed in Section VI.23. 20. CONJUGATED POLYMER ELECTROLUMINESCENCE Conjugated polymers exhibit a number of attractive properties for electroluminescence. As described in Section 1.2, thin films can be formed over large areas by spin-coating from solution. These films can be highly fluorescent, and different polymers give emission colors spanning the whole of the visible spectrum. Molecular organic EL devices often suffer from recrystallization of the amorphous film during operation or storage, leading to rapid degradation of the device performance. Conjugated polymers, which have no tendency to recrystallize, avoid this problem, and offer the prospect of producing more stable devices. Electroluminescence in the conjugated polymer poly(ppheny1enevinylene) (PPV) was first observed in Cambridge in 1989.31 In the course of investigating its dielectric properties, a thin film of PPV was sandwiched between aluminum electrodes, one of which was coated with an interfacial layer of aluminum oxide, and yellow-green EL was seen through one of the electrodes when a voltage was applied. These early devices were inefficient, and suffered from problems of nonuniformity and unreliability. Improvements were made using indium-tin oxide for hole injection, and internal quantum efficiencies of 0.01% have been achieved, still using aluminum for electron injection.”’ A schematic energy-level diagram for a PPV LED is shown in Fig. 67. As described in Section 1.2, polymer LEDs operate by the injection of electrons and holes from negative and positive electrodes, respectively. Electrons and holes capture one another within the polymer film and form either singlet or triplet excitons. Of these, the singlet excitons may decay radiatively, giving out light which is observed through one of the electrodes, which must be semitransparent. The internal quantum efficiency q n t defined , as the ratio of the number of photons produced within the device to the number of electrons flowing in the external circuit, is given by (20.1) where y is the ratio of the number of exciton formation events within the device to the number of electrons flowing in the external circuit, r,, is the N. C. Greenham, R. H. Friend, A. R. Brown, D. D. C. Bradley, K. Pichler, P. L. Burn, A. Kraft, and A. B. Holmes, Froc. S H E 1910, 84 (1993). 3112

118


IP

IT0

PPV

FIG.67. Schematic energy-level diagram for an ITO/PPV/Al LED, showing the ionization potential (IP) and electron affinity (EA) of PPV, the work functions of I T 0 and Al (@,To and @*,Ir and the barriers to injection of electrons and holes ( A E , and A E h ) .

fraction of excitons formed as singlets, and q is the efficiency of radiative decay of these singlet excitons. To achieve efficient luminescence, it is therefore necessary to have good balancing of electron and hole currents, efficient capture of electrons and holes within the emissive layer, and efficient radiative decay of singlet excitons. Braun et a1.579303demonstrated that improved efficiencies could be obtained in MEH-PPV devices by using calcium for electron injection, indicating that electron injection is the limiting factor in determining device efficiency. A similar improvement is found in PPV devices, and efficiencies of up to 0.1% can be obtained from single-layer PPV devices with calcium.223 As shown in Fig. 68, the electroluminescence spectrum of a PPV LED is very similar to the PL spectrum for PPV, indicating that radiative decay of the same singlet exciton is responsible for the emission in both cases.3o4 21.

CONTROL OF COLOR

One of the attractive features of polymer EL is that the color of emission can be controlled by altering the chemical structure of the polymer. PPV gives emission in the yellow-green; the emission color can be moved toward the red by the substitution of electron-donating groups such as alkoxy chains at the 2- and 5- positions on the phenyl ring.57 Substituents 303

D. Braun, A. J. Heeger, and H. Kroemer, J . Electron. Muter. 20, 945 (1991). 3"4D. D. C . Bradley, A. R. Brown, P. L. Burn, J. H. Burroughes, R. H. Friend, A. B. Holmes, K. D. Mackay, and R. N. Marks, Synth. Met. 43, 3135 (1991).

119

SEMICONDUCTOR DEVICE PHYSICS 1

I

I

2.8

3.3

I' 1.8

2.3

Energy (eV)

3.8

FIG.68. El spectrum of a PPV LED, compared with the PL and absorption spectra of a PPV film. [A. R. Brown, D. D. C. Bradley. J. H. Burroughes, R. H. Friend, N. C. Greenham, P. L. Burn, A. B. Holrnes, and A. Kraft, Appl. Phys. Lett. 61, 2793 (19921.1

can also cause changes in energy gap through steric, rather than electronic, effects by disrupting the conjugation along the chain. The substitution of bulky cholestanoxy groups, for example, has been used to obtain green emission in soluble polymers.305 Other polymer systems, not based on PPV, can also be used for EL; the poly(alkylthiophenes), for example, conveniently give emission in the red region of the spectrum.215~216,306,307 A very large number of polymers have now been investigated for their EL properties, and we do not intend to review all the reported polymers here. The emission wavelengths and EL efficiencies of a number of conjugated polymers are shown in Table 111, but this list is not intended to be exhaustive. Polymer LEDs emitting in the blue part of the spectrum are a particularly attractive target for research, because blue inorganic LEDs have only recently become available. One strategy to obtain blue-shifted emission is to introduce nonconjugated units into a PPV backbone in order to reduce the average conjugation length. Using a random copolymer of PPV and the methoxy leaving-group precursor to dimethoxy-PPV, Burn et ~ 1 . ~ " ' have obtained blue-green emission. Due partly to the problem of exciton 305

C. Zhang, S. Hoger, K. Pakbaz, F. Wudl, and A. J. Heeger, J . Electron. Muter. 22, 413 (1993). 306 Y. Ohmori, M. Uchida, K. Muro, and K. Yoshino, Jpn. J. Appl. Phys. 30, L1938 (1991). 307 Y. Ohmori, M. Uchida, K. Muro, and K. Yoshino, Solid State Commun. 80, 605 (1991). 108 P. L. Burn, A. B. Holrnes, A. Kraft, D. D. C. Bradley, A. R. Brown, R. H. Friend, and R. W. Gymer, Nature 356, 4 1 (1992).

120


migration, however, the blue-shift compared to standard PPV is not large. The introduction of nonconjugated units is also found to give an increase in the PL efficiency.24692473308 This improvement is usually attributed to the suppression of the diffusion of singlet excitons to quenching sites where they can decay nonradiatively. Partially eliminated PPV precursors have also been used to obtain blue-shifted e m i s s i ~ n , " ~ ~as" ~have blends of polytp-phenylphenylenevinylene) in poly(9-vinylcarbazole).31' It is relatively easy to find molecular materials, for example, oligomers of PPV, which show blue emission. These materials tend to lack the morphological stability of the conjugated polymers, and therefore present some problems for device stability. Several groups have used dispersions of molecular ThiS materials in inert polymer matrices to inhibit re~rystallization.~'~-~'~ work led to the idea of attaching the molecular emitter directly to an inert polymer, either as a pendent side-group316 or incorporated into the polymer m a i n - ~ h a i n . ~These ' ~ devices typically require high operating voltages and are not stable over long periods of time, possibly due to the large fraction of nonactive material in the device making injection and transport difficult. An alternative strategy to obtain blue emission is the use of completely different conjugated polymer systems. Blue EL has been reported in polytp-phenylene) (PPP),318 p~lfialkylfluorene),~~~ fluorinated polyquinoline,320and PPP-based ladder copolymer^.^'^^^^ One problem that is found for the larger gap polymers is that although the luminescence from isolated polymer chains (in solution or in solid solution) may be blue, solid films often show red-shifted emission. This is observed for the PPP-ladder polymers, for which the dominant emission band is in the yellow part of 309

C. Zhang, D. Braun, and A. J . Heeger, J . Appl. Phys. 73, 5177 (1993). P. L. Burn, A. B. Holmes, A. Kraft, D. D. C. Bradley, A. R. Brown, and R. H. Friend, J . Chem. SOC.,Chem. Commun. 32 (1992). 31 1 C . Zhang, H. von Seggern, K. Pakbaz, B. Kraabel, H.-W. Schmidt, and A. J. Heeger, Synth. Met. 62, 35 (1994). 31 2 A. R. Brown, Ph.D. Thesis, University of Cambiridge (1992). 313 H. Vestweber, R. Sander. A. Greiner, W. Heitz, R. F. Mahrt, and H. Bassler, Synth. Met. 64, 141 (1994). 314 J. a d o , K. Hongawa, K. Okuyama, and K. Nagai, Appl. Phys. Lett. 64, 815 (1994). 315 J. Kido, M. Kohda, K. Okuyama, K. Nagai, and Y . Okamoto, Proc SPIE 1910,31 (1993). ?'I6D. R. Baigent, R. H. Friend, J. K. Lee, and R. R. Schrock, Synfh. Met. 71, 2171 (1995). 31 7 I. Sokolik, Z. Yang, F. E. Karasz, and D. C. Morton, J . Appl. Phys. 74, 3584 (1993). 3in G. Grem, G. Leditzky, B. Ullrich, and G. Leising, Adu. Mater. 4, 36 (1992). 314 Y. Ohmori, M. Uchida, K. Muro, and K. Yoshino, Jpn. J . Appl. Phys. 30, L1941 (1991). 320 I. D. Parker, Q. Pei, and M. Marrocco, Appl. Phys. Lett. 65, 1272 (1994). 321J. Griiner, P. J. Hamer, R. H. Friend, J.-J. Huber, U. Scherf, and A. B. Holmes, Adv. Mater. 6, 748 (1994). 310

T-LE 111. SUMMARY OF CONJUGATED POLYMERS USEDFOR ELECTROLUMINESCENCE POLYMER PPV MEH-PPV RO-PPV

PPPV Cyano-substituted dihexyloxy-PPV Cyano-substituted PPV/PTV copolymer P3AT

Interrupted PPV copolymer Partially converted PPV PPP PPP copolymers Polfialkylfluorene)

EMISSION NEGATIVE TRANSPORTYOQUANTUM (nm) ELECTRODE LAYERS EFFICIENCY

PEAK

565

Al Ca

~

-

PBD

0.001-0.01 0.1

1.o

-

Ca In or Mg

5 90

Ca

-

570 560 560 495 695

Ca Al Al Ca Al

-

720

Al

690

Mg/In Ca or In Ca

-

0.05 0.2

Ca

-

> 0.1

Ca Al

-

Blue-green 590

-

530 550 465 420 400 470

~

PPV PPV -

-

Ca PBD Ca PPV Ca,In, or Al Mg/In -

a -c

d d

1.o

605 580

~

REF.

NOTES

?

0.4

0.3 0.001 0.01 0.16 0.2 4 0.2 ?

< 0.0025

0.07-0.21

0.75

0.01 1 0.5 ?

e

R = CsHl,C14H29 R , = CH, R2 = CIOH,, R = cholestanoxy Blended with PS Blended with PVK

f

g

h i, j k 1

PL peaks at

m

R = C13H2,CnH4s R = C,H,, R = C,H,, R = CIZH2S PPV-based RO-PPV-based R-PPV-based

n

840 nm

0

P 9 g r

q>s 1

U

RO-substituted R-substitued

u

w X

(CONTINUES)

TABLE 111. (CONTINUED) POLYMER

L

h) h)

EMISSION NEGATIVE TRANSPORT % QUANTUM (nm) ELECTRODE LAYERS EFFICIENCY

PEAK

Fluorinated polyquinoline

450

Ca

Polydiphenylquinoxaline Ladder polymers and derivatives

490 600

Mg/Ag CaorAl Ca Ca Ca

Blue oligomers dispersed in polymer matrix (a selection) Oligomers in main chain Oligomers as side-chains Regioregular-substituted polythiophenes PolyphthalidylineaIylene Poly(phenylacety1ene)

450 475 475 450 465 450 450-560 480 600

Al

Mg/Ag

Al

Ca In CuorCr Ca

0.003 PVK&PBD 4 Various ? ? PPV 0.6 0.9 0.2 ? Alq3 Various -

?

-

0.3

-

?

-

0.01 0.1-0.5

NOTES

REF. Y z u , aa

Copolymer

ab ac

ad

i

ae, af

ag

ah ai

ai

ak

“J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burn, and A. B. Holmes, Nature 347,539 (1990). bS. Karg, W. Riess, V. Dyakonov, and M. Schwoerer, Synth. Met. 54, 427 (1993). ‘W. RieB, S. Karg, V. Dyakonov, M. Meier, and M. Schwoerer, J. Lurnin. 60-61, 906 (1994). dA. R. Brown, D. D. C. Bradley, J. H. Burroughes, R. H. Friend, N. C. Greenham, P. L. Burn, A. B. Holmes, and A. Waft, Appl. Phys. Lett. 61, 2793 (1992). eD. Braun and A. J. Heeger, Appl. Phys. Lett. 58, 1982 (1991). fS. Doi, M. Kuwabara, T. Noguchi, and T. Ohnishi, Synth. Met. 55-57, 4174 (1993). gD.Braun, E. G. J. Staring, R. C. J. E. Demandt, G. L. J. Rikken, Y. A. R. R. Kessener, and A. H. J. Venhuizen, Synth. Met. 66, 75 (1994). hC. Zhang, S. Hoger, K. Pakbaz, F. Wudl, and A. J. Heeger, J. Electron. Muter. 22,413 (1993). ‘H. Vestweber, J. Oberski, A. Greiner, W. Heitz, R. F. Mahrt, and H. Bassler, A h . Muter, Opt. Electron. 2, 197 (1993).

'H. Vestweber, R. Sander, A. Greiner, W. Heitz, R. F. Mahrt, and H. Bassler, Synth. Met. 64, 141 (1994). kc.Zhang, H. von Seggern, K. Paibaz, B. Kraabel, H.-W. Schmidt, and A. J. Heeger, Synth. Met. 62 35 (1994). 'N. C.Greenham, S. C. Moratti, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, Nature 365, 628 (1993). "'D. R. Baigent, P. J. Hamer, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synth. Met. 71, 2175 (1995). "Y.Ohmori, M. Uchida, K. Muro, and K. Yoshino, Jpn. J. Appl. Phys. 30,L1938 (1991). OD. Braun, G. Gustafsson, D. McBranch, and A. J. Heeger, J. Appl. Phys. 72,564(1992). PN. C.Greenham, A. R.Brown, D. D. C. Bradley, and R.H. Friend, Synth. Met. 57,4134 (1993). qP. L. Bum, A.B.Holmes, A. Kraft, D. D. C. Bradley, A. R.Brown, R. H. Friend, and R.W. Gymer, Nature 356,

47,(1992).

'E. G. J. Staring, R. C. J. E. Demandt, D. Braun, G. L. J. Rikken, Y. A. R. R.Kessener, T. H. J. Venhuiozen, H. Wynberg, W. Tenhoeve, and K. J. Spoelstra, A h . Muter. 8h6,934 (1994). "C.Zhang, D. Braun, and A. J. Heeger, J . Appl. Phys. 73,5177 (1993). 'G. Grem, G. Leditzky, B. Ullrich, and G. Leising, Adu. Muter. 4,36 (1992). "G. Leising, G. Grem, G. Leditzlq, and U. Scherf, Proc. SPIE 1910,70 (1993). "W.-X. Jing, A.Kraft, S. C. Moratti, J. Griiner, F. Cacialli, P. J. Hamer, A. B. Holmes and R.H. Friend, Synth. Met.

67,161 (1994).

"G. Grem and G. Leising, Synth. Met. 55-57, 4105 (1993).

"Y.Ohmori, M. Uchida, K. Muro, and K. Yoshino, Jpn. J . Appl. Phys. 30,L1941 (1991). yI. D.Parker, Q. Pei, and M. Marrocco, Appl. Phys. Lett. 65, 1272 (1994). IT. Yamanoto, T.Inoue, and T. Kanbara, Jpn. J. Appl. Phys. 33, L250 (1994). ""J. Huber, K. Miillen, J. Salbeck, H. Schenk, U. Scherf, T. Stehlin, and R. Stem, Actu Polym. 45,244 (1994). ab J. F. Griiner, H. F. Wittmann, P. J. Hamer, R.H. Friend, J. Huber, U. Scherf, K. Miillen, S. C. Moratti, and A. B.

Holmes, Synth. Met. 67, 181 (1994). "'J. Griiner, P. J. Hamer, R. H. Friend, J.-J. Huber, U. Scherf, and A. B. Holmes, Ado. Muter. 6,748 (1994). O d A R. Brown, Ph.D. Thesis, University of Cambridge (1992). "'J. Kido, K. Hongawa, K. Okuyama, and K. Nagai, Appl. Phys. Lett. 64, 815 (1994). OrJ. Kido, M. Kohda, K. Okuyama, K. Nagai, and Y. Okamoto, Proc. SPIE 1910,31 (1993). Sokolik, Z. Yang, F. E. Karasz, and D. C. Morton, J. Appl. Phys. 74,3584 (1993). oh D. R. Baigent, R.H. Friend, J. K. Lee, and R. R.Schrock, Synth. Met. 71, 2171 (1995). (liR. E.Gill, G. G. Malliaras, J. Wilderman, and G. Hadziioannou, Ado. Muter. 6, 132 (1994). "'I. L. Valeeva, A. N.Lachinov, V. A. Antipin, and M. G. Zolotukhin, JETP 78, 83 (1994). ak L.S. Swanson, F. Lu,J. Shinar, Y.W. Ding, and T. J. Barton, Proc. SPIE 1910, 101 (1993).

124


the and it has been established that this is due to formation of aggregates in the solid film, which support excitons that extend over more than the single chain.323The tendency to form aggregates can be controlled by introduction of disorder, as in the ladder-PPP copolymers shown in Fig. 22, which can produce good blue emission.”*321Soluble copolymers of PPP and alkyl- or alkoxy-substituted PPP have also been A major problem in obtaining EL from used to produce blue EL.77,78,80 materials with a high energy gap is that the barrier to electron injection, hole injection, or both are inevitably larger than in materials with lower energy gaps. This makes the problem of balancing electron and hole injection even more difficult, and tends to lead to devices with high driving voltages and low efficiencies. CHARACTERISTICS 22. ELECTRICAL

OF

SINGLE-LAYER DEVICES

Modeling of the electrical characteristics of polymer LEDs is a challenging task, and at present there is no consensus as to the most appropriate model to apply. As discussed in Section IV, the energy levels, mobilities, diffusivities, doping levels, and the detailed structure of interfaces in polymer LEDs are poorly known at present. It is therefore difficult to distinguish between the effect of the bulk and the effect of the interfaces on the electrical behavior. It is also likely that many of the important parameters vary considerably between different polymers, and even between the different samples of nominally the same polymer. The current density-voltage (J-V) and luminance-voltage (L-V) characteristics of a typical ITO/PPV/Ca device are shown in Fig. 69. The luminance is approximately proportional to the current, indicating that the quantum efficiency is approximately constant over a wide range of currents. The voltage required to produce a given current density (J) has been found to be proportional to the thickness of the device, implying that the current density is simply a function of the average electric field across the d e ~ i c e . ~The ” J-V characteristics show diode-like behavior, with rectification ratios usually in excess of lo3 between forward and reverse bias. As discussed in Section III.lO.d, it is difficult to make accurate predictions about the barriers to electron and hole injection ( A E , and AE,,, respectively, in Fig. 67). Reasonable assumptions about the polymer ionization potential and electrode work functions make it clear that the barrier to injection of electrons from aluminum must be significantly larger than the 322 J. Griiner, H. F. Wittmann, P. J. Hamer, R. H. Friend, J. Huber, U. Scherf, K. Miillen, S . C. Moratti, and A. B. Holmes, Synth. Met. 67, 181 (1994). 323A.Kohler, J. Griiner, R. H. Friend, U. Scherf, and K. Miillen Chern. Phy. Lett. (in press).


4.0

I

I

I

I

125

I

Voltage (V) FIG.69. Current density (squares) and luminance (circles, arbitrary units) versus voltage for an ITO/PPV/Ca LED.

barrier to injection of holes from IT0.312The majority of the current is therefore expected to be due to holes. Electroluminescence, however, requires the simultaneous injection of electrons, and the quantum efficiency will therefore depend strongly on the barrier to electron injection. The improvement in efficiency in changing the electron-injecting electrode from aluminum to calcium is consistent with the reduction in barrier height for electron injection as the work function of the electrode is decreased. The J-V characteristics for PPV LEDs have been studied in detail by Marks et al.2'7,324 for various electrode materials and temperatures. These results show a power-law dependence at low voltages, which is attributed to a space-charge-limited current with significant trapping. The effective mobilities obtained by fitting these data are between lop9 and lo-' cm2 v-1 s - I , indicating that the injection may be filamentous, or that the current is due to injection of carriers with genuinely low mobility. At higher voltages, Marks found that the current was predominantly an interface-limited hole current. The injection mechanism proved difficult to fit with simple models, and it was tentatively proposed that the current was limited by thermionic emission through an interfacial barrier. The importance of interfacial barriers has also been identified by Vestweber et who concentrate particularly on oxide layers at the interface between polytphenylphenylenevinylene) and Al. As discussed in Section IV.l l.d, the existence of an oxide layer at a PPV/Ca interface has been demonstrated 324

R. N. Marks, D. D. C. Bradley, R. W. Jackson, P. L. Burn, and A. B. Holmes, Synth. Met. 55-57, 4128 (1992).

126


by Gao et a1.228using XPS techniques, but the presence of this layer is sensitive to sample preparation conditions. Riel3 et al.32s have recently demonstrated that improved efficiency can be obtained in ITO/PPV/Al devices by intentionally depositing a thin layer of A1203between the PPV and Al. The characteristics of MEH-PPV LEDs have been studied in detail by Parker.224 He chose electrode materials to give currents that are either almost entirely due to holes or almost entirely due to electrons. At high fields, he obtained good fits to the Fowler-Nordheim theory for tunneling through a triangular barrier, with barrier heights that are in reasonable agreement with the values predicted by the electrode work functions. The model used to fit these data assumes that the bands are flat, which requires that there be negligible space charge within the device, and that the doping concentration be low enough for the width of any Schottky barriers formed to be significantly greater than the thickness of the device. The weak temperature dependence of the current suggests that tunneling is the predominant injection mechanism. For devices with two injecting electrodes, the current is found to be significantly greater than the sum of the currents in the equivalent single-carrier devices. This indicates that space charge effects must be significant in limiting the single-carrier currents, which contradicts the assumptions made in the model used to fit the data. Although the barrier heights extracted from the Fowler-Nordheim plots are reasonable, the observed currents are several orders of magnitude lower than those predicted by Eq. (11.111, indicating further problems with the simple tunneling model. Riel3 and coworkers at the University of Bayreuth have proposed a Schottky barrier model for the operation of ITO/PPV/Al LEDs.218,326-32y They argue that the current is predominantly carried by holes, and that this hole current is limited by the Schottky barrier formed at the PPV/AI interface, rather than by any barrier at the ITO/PPV interface. They model the current -voltage characteristics using the equation for thermionic emission across a Schottky barrier from a semiconductor into a metal

I

= I,[exp(

g) -

11.

(22.1)

325W.RieR, M. Meier, S. Karg, E. Buchwald, M. Colle, J. Gmeiner. M. Greczmiel, P. Posch, and P. Strohriegl, Oral Presentation, CMMP Conference, Warwick, UK (1994). 326 S. Karg, W. Riess, V. Dyakonov, and M. Schwoerer, Synth. Met. 54,421 (1993). 321 S . Karg, W. Reiss, M. Meier, and M. Schwoerer, Synfh. Met. 55-57,4186 (1993). 32 8 J. Gmeiner, S. Karg, W. RieR, P. Strohriegl, and M. Schwoerer, Actu Polymer. 44, 201

(1993). 329 W. Riel3, S. Karg, V. Dyakonov, M. Meier, and M. Schwoerer, J . Lumin. 60-61, 906 (1994).


127

This formula gives good fits to the data only between 1 and 2 V, and values of n between 1.8 and 2.4 are required, compared to n = 1 for the ideal case. The voltage, V , in Eq. (22.1) represents the voltage across the Schottky barrier. In conventional doped inorganic semiconductors, the voltage dropped across the remainder of the sample is usually small, which simplifies the analysis of device performance. In conjugated polymer devices, however, the resistivity of the polymer is sufficiently high that the voltage dropped across the bulk of the device is no longer negligible. Riel3 et al. attempt to allow for this by writing the voltage, V , across the junction as V = I: - IR,

(22.2)

where is the total voltage across the device, and R represents the bulk resistance.329This allows them to fit their data to higher voltages, and they argue that the fit to this model supports the mechanism of thermionic emission over a Schottky barrier. It is, however, not clear that the bulk resistance in these materials can be modeled well by a constant resistance R. At higher currents and voltages, it is possible that the current becomes limited by space charge, by a barrier at the I T 0 electrode, or by an interfacial oxide layer. Note that the doping levels estimated for these devices are in the range 1016 to 10" ~ m - considerably ~ , higher than the values estimated by Marks et al. for their devices. Impedance analysis, where the complex impedance is measured as a function of frequency, has been applied to give additional information about PPV LEDs. The small ac voltage required to measure the impedance is superimposed on a dc bias, which can be altered to simulate different driving conditions for the device. Real and imaginary parts of the impedance are plotted with frequency at the implicit variable to give curves that typically comprise a series of semicircles. The difficulties involved in interpreting these data have been discussed by Marks,217who has shown that the impedance data for PPV devices can be fitted to more than one equivalent circuit of resistors and capacitors. One such circuit is the series-connected circuit shown in Fig. 70. In the interpretation favored by RieB et al. for devices with aluminum electrodes,329 the two resistor-capacitor pairs represent the junction (the Schottky barrier) and the bulk of the device, respectively. The bulk resistance is relatively insensitive to bias, whereas the junction resistance decreases rapidly with increasing bias. ~ ' used internal photoemission spectroscopy to study Rikken et ~ 1 . ~have 330 G. L. J. A. Rikken, D. Braun, E. G. J. Staring, and R. Demandt, Appl. Phys. Lett. 65, 219 (1994).

128


C1

c2

FIG. 70. A possible equivalent circuit for complex impedance data from a polymer LED.

the interface between dialkoy-PPV and calcium. In this experiment, infrared light of various energies is incident on an LED structure, causing promotion of electrons over the injection barrier. The additional current is measured using phase-sensitive techniques as a function of the applied voltage. A simple model involving image-free lowering of a Schottky barrier (see Section IV.11.b) gives a barrier height of 0.62 eV, considerably larger than that expected from the work function of calcium and the electron affinity of dialkoxy-PPV, and Rikken therefore proposes a model where the region of polymer next to the electrode is doped with calcium ions, leading to a modified barrier, in accord with the results of Dannetun et a[. (see Section 1 ~ . 1 i . d ) . ~ ~ ~ The position of the region where electron-hole capture occurs within the device is also not yet well established. If the electron current is small compared to the hole current, then the capture zone is expected to extend from the negative electrode into the device. The range of electrons within the device depends on the mobilities of electrons and holes, and on the capture cross section. Using two-layer PPV/MEH-PPV devices, Brown et al. have shown that the range of electrons in PPV can be as large as 200 nm.3319332 Vestweber et al. have analyzed the performance of PPPV LEDs using a model where the recombination rate is independent of the drift velocity of the carriers within the device.313They attribute the low efficiency in PPPV devices to a large fraction of the injected electrons passing through the device without capture. They see a significant increase in E L efficiency on blending the PPPV with polystyrene, consistent with a smaller hole mobility, leading to increased density of holes, and a higher probability of electron capture. This model, however, neglects the influence of the space charge due to holes on the hole current itself. A similar effect to that proposed by Vestweber et al. might also explain the increase in efficiency seen on blending PPV derivatives with the electron-transporting molecule PBD (see Section II.5).333 331

A. R. Brown, N. C. Greenham, J. H. Burroughes, D. D. C. Bradley, R. H. Friend, P. L. Burn, A. Kraft, and A. B. Holmes, Chem. Phys. Lett. 200, 46 (1992). 332N.C. Greenham, R. H. Friend, A. R. Brown, J. H. Burroughes, D. D. C. Bradley, P. L. Burn, A. Kraft, and A. B. Holmes, Roc. SPIE 1910, 111 (1993). 333 C. Zhang, S. Hoger, K. Pakbaz, F. Wudl, and A. J. Heeger, J . Electron. Ma/er. 23, 453 (1994).


129

Transient measurements, where the EL and current are studied as a function of time after a rapid change in the driving voltage, have been used by several groups to study transport in organic LEDs.59,217,313,334-340 In the simplest model, the time delay t , between voltage turn-on and the appearance of the first EL is related to the sum of the drift mobilities for electrons ( p,) and holes ( p h )by (22.3) where d is the device thickness and F is the electric field. Since the electron and hole mobilities are seldom well matched, this approach usually gives the mobility of the more mobile carriers, typically the holes. Hole mobilities of lo-’ cm2 V-’ s - ’ have been obtained from poly(3dodecylthiophene) LEDs using this method.337 In practice, the transient response of polymer LEDs is complicated by dispersive transport, space charge effects, and by the influence of trapped charges and extrinsic carriers present in the device in the “off’ state. The turn-on of EL typically has a fast component, lasting up to about 10 ps, frequently accompanied by a slow component lasting up to about 1 ms. Trapped charges in the device make it difficult to distinguish a true transit time, since they remove the necessity for the transport of charges through the entire thickness of the device before the first EL is observed. By allowing the device to stay in the “off’ state for long periods before ~ ’ been able to obtain clear applying the voltage pulse, Karg et ~ 1 . ~have transit times from PPV LEDs, which allow them to estimate a hole mobility that is strongly field dependent, ranging from to 3 X cm2 V-’ s-l . The dependence on the time in the “off” state of the initial EL turn-on has also been investigated by Vestweber et d 3 I 3 for devices 334 C. Hosokawa, H. Takailin, H. Higashi, and T. Kusumoto, Appl. Phys. Lett. 60, 1220 (1992). 3.15 C. Hosokawa, H. Tokailin, H. Higashi, and T. Kusurnoto, Appl. Phys. Lett. 63, 1322 (1993). 33 h D. Braun, D. Moses, C. Zhang, and A. J. Heeger, Appl. Phys. Lett. 61, 3092 (1992). 337Y.Ohmori, C. Morihima, M. Uchida, and K. Yoshino, Jpn. 1. Appl. Phys. 31, L.568 (1992). 338S.Karg, V. Dyakonov, M. Meier, W. RieR, and G. Paasch, Synth. Met. 67, 16.5 (1994). 33Y N. C. Greenham, R. H. Friend, D. D. C. Bradley, S . C. Moratti, and A. B. Holmes, Oral Presentation, MRS Spring Meeting, Symposium L on Electroluminescent Polymers, San Francisco (1994). 340 P. Delannoy, G. Horowitz, H. Bouchriha, F. Deloffre, J.-L. Fave, F. Gamier, R. Hajlaoui, M. Heyman, F. Kouki, J.-L. Mongre, P. Valat, V. Wintgens, and A. Yassar, Synth. M e f . 67, 197 (1994).

130


based on PPPV. Marks’” has extracted higher values for the hole mobility cm’ V-’ s by fitting the fast component of the turn-on of around in PPV devices to a model for the arrival of the diffusively broadened leading edge of a charge distribution at an electrode, but it is not clear how trapped charges affect this analysis. The fast turn-on times (100 ns) reported by Braun et uf.336 in MEH-PPV LEDs are likely to be a consequence of recombination of injected charges with charges trapped in the device. The slow component of the E L turn-on observed in many devices is difficult to interpret. Ohmori et ~ 7 1 attribute . ~ ~ ~ the slow rise in efficiency in poly(3-octadecylthiophene) LEDs to an enhancement of the PL efficiency through joule heating. However, the slow features observed in other polymers exist over a wide range of device powers, and are therefore unlikely to be thermal in origin. The transport of low-mobility carriers, probably electrons, is likely to be important here, but without a better understanding of the operation of the devices it is difficult to produce a model to fit the transient data. Vestweber et af.3’3propose that in LEDs formed with blends of tris(4-methoxysti1bene)amine in polycarbonate, the initial low efficiency is caused by holes passing through the device without recombination, and that the EL increases with time over a period of tens of microseconds as more electrons are injected, thus increasing the probability of recombination. 23. MULTILAYER DEVICES The use of low-work-function metals such as calcium for electron injection is not attractive for potential applications, because these metals are highly reactive and require careful encapsulation. An alternative strategy, first demonstrated by Tang and Van S l ~ k with e ~ ~molecular organic LEDs, is the use of two-layer structures. With PPV, electron injection is more difficult than hole injection, and it is therefore helpful to use an additional electron-transporting layer between the PPV and the negative electrode. ~ the electron-transporting material PBD (see Section Brown et ~ f . * *used 11.9, which had earlier been used in the form of sublimed films by Adachi et al. in molecular organic L E D s . ~To ~ alleviate problems of recrystallization, the PBD was dispersed in an inert matrix of polymethylmethacrylate (PMMA). This PBD/PMMA blend was deposited by spin-coating from chloroform solution. A schematic energy-level diagram for this device is shown in Fig. 71. The inclusion of the electron-transporting layer in a device with a calcium electrode gave an internal quantum efficiency of 1%, an improvement


I

131

Neaative

Electron-transporting layer FIG.71. Schematic energy-level diagram for a device incorporating an electron-transporting layer, ignoring the effects of charges within the device. A E , denotes the barrier to hole injection from the emissive layer into the electron-transporting layer.

of a factor of 10 on the value without the additional layer.223Similar improvements were seen with magnesium and aluminum electrode^.^"' The reasons for the improvement in efficiency are as follows. The electrontransporting layer has a large ionization potential, and hence there is a large barrier to injection of holes from the PPV into the PBD. The hole current passing through the device is reduced due to the presence of this barrier. The positive space charge built up within the PPV layer results in a larger field at the negative electrode, thus improving the balance of charge injection. Also, the region of luminescence is moved away from the metallic negative electrode, causing a reduction in nonradiative decay of singlet excitons. There may also be a further improvement in the efficiency of radiative decay of singlet excitons due to a decrease in the radiative lifetime as the luminescence is moved away from the electrode (see Section VI.25). Electron-transporting layers have now been used with a number of other 320, 341 In this context, conjugated polymers to improve EL efficiencie~.~~. precursor route polymers such as PPV have the advantage of being insoluble in the solvent used to deposit the electron-transporting layer. With soluble emissive layers, it is necessary to choose the solvent system for the electron transporting layer carefully, in order to avoid washing away the emissive layer. Blends of PBD in polymer matrices tend to suffer from rapid degradation, and there are recent reports of polymers contain34 1

A. J. Heeger, I. D. Parker, F. Klavetter, C. Zhang, Y. Yang, Q.-B. Pei, and N. Colaneri, Oral Presentation, MRS Spring Meeting, Symposium L on Electroluminescent Polymers, San Francisco (1994).

132

NEIL C. GREENHAM AND RICHARD H. FRIEND LUMO

PPV

..

... eV

LUMO

CN-PPV

- Ca -Al

HOMO

FIG.72. Approximate energy levels for PPV, CN-PPV, and various electrode materials.

ing a range of high electron affinity groups within the main chain of the polymer, which can show improved stability.342 An alternative approach to the problem of electron injection is to synthesize emissive polymers which themselves have an increased electron affinity. This technique was demonstrated by Greenham et a1.,6' using the polymer CN-PPV described in Section II.3.d. This polymer is found, by electrochemical technique^^"^,^^^ and by quantum chemical calculation^'^^ to have an electron affinity 0.9 eV greater than that of PPV, and an ionization potential 0.6 eV greater than that of PPV. Single-layer devices made with CN-PPV and calcium or aluminum negative electrodes showed red emission, with internal quantum efficiencies in the range of 0.1 to 0.2%. Although the efficiencies of these devices were not high, the fact that changing the negative electrode from calcium to aluminum had very little effect indicated that the barrier to electron injection was no longer the critical factor in determining the efficiency. A large increase in efficiency and a decrease in driving field were achieved using two-layer devices with the structure ITO/PPV/CNPPV/Al. These devices gave red EL with a spectrum similar to the PL spectrum of CN-PPV, indicating that the EL was originating only in the CN-PPV layer. Internal quantum efficiencies of 4% were found, with calcium or aluminum electrodes. These figures are high for a polymer LED, particularly using aluminum as the negative electrode, and show that the balance of carrier injection is good. The relative positions of the energy levels for PPV, CN-PPV, and various electrode materials are shown in Fig. 72. This figure also shows the barriers to the injection of holes from PPV into CN-PPV, and to the injection of electrons from CN-PPV into PPV. For the reasons discussed in Sections 1II.lO.d and IV. it is difficult to 34 2 M. Strukelj, F. Papadimitrakopoulos, T. M. Miller, and L. J . Rothberg, Science 267, 1969 (1995). 343S.C. Moratti, R. Cervini, A. B. Holmes, D. R. Baigent, R. H. Friend, N. C. Greenham, J. Gruner, and P. J. Hamer, Synth. Met. 71,2117 (1995).


133

CN-PPV

FIG.73. Energy-level diagram for an ITO/PPV/CN-PPV/AI showing the Fermi energy and the HOMO and LUMO levels.

device under zero bias,

draw an accurate energy-level diagram for a two-layer device. An attempt at such a diagram for an ITO/PPV/CN-PPV/AI device is shown in Fig. 73. For the sake of simplicity, this diagram assumes that the width of any depletion or accumulation regions is significantly less than the thickness of the polymer layers; this may not be true in practice. The PPV acts here as a hole-transporting layer, as has been demonstrated in a number of other polymer The fact that no EL is observed from the PPV suggests that all the injected electrons recombine in the CN-PPV, and that none is injected over the large barrier from the CN-PPV into the PPV. Space charge within the PPV layer redistributes the electric field in order to maintain identical hole currents at the ITO/PPV and PPV/CN-PPV interfaces. A similar performance with calcium or aluminum electrodes indicates that the electron current is determined by space charge effects rather than by any barrier at the negative electrode. As in single-layer LEDs, the observation that the light output is approximately linear with current is difficult to explain using simple models. We propose that in these devices the electron current would be small in the absence of injected holes, and that the injection of holes causes an increase in the electron current due to the cancellation of space charge, thus providing a plausible mechanism for the balancing of electron and hole currents. A great deal more work is necessary, however, before a quantitative model for the operation of these two-layer devices can be established. The synthetic route to CN-PPV shown in Fig. 16 lends itself to considerable variation, and the asymmetrically substituted polymer shown in Fig. 344 34 5

S. Doi, M. Kuwabara, T. Noguchi, and T. Ohnishi, Synth. Met. 55-57,4174 (1993). H. Antoniadis, B. R. Hsieh, M. A. Abkowitz, and M. Stolka, Appl. Phys. Lett. 62, 3167

(1 993).

134


I N

10-2

103

E

2

102

10’

m -

‘E

10-8

0

1

2

3

Voltage (V)

4

5

6

FIG. 74. Current density and luminance versus voltage for an ITO/PPV/asymetrically substituted CN-PPV/Al device. [D. R. Baigent, N. C. Greenham, J. Griiner, R. N. Marks, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synfh. Met. 67, 3 (19941.1

17, for example, gives similar or slightly improved performance to CN-PPV in two-layer devices, with slightly blue-shifted emission. This polymer is soluble in toluene, from which it is easy to spin-coat very thin uniform films, allowing devices with low operating voltages to be fabricated. The characteristics of one of these devices are shown in Fig. 74, which shows that luminances in excess of 100 cd m-’ can be achieved at a voltage of 5 V.64Polymers LEDs emitting in the near infrared have also been reported using cyano-substituted materials.346 Extension of the emission range of cyano-substituted polymers into the blue using this kind of two-layer structure will require the development of suitable insoluble, high-energygap, hole-transporting layers. 24.

PROBING OF

EXCITED STATES

IN

LEDS

The amount of information about the operation of polymer LEDs that can be gained from their electrical and emission characteristics is limited. Therefore, it is useful to develop noninvasive techniques to study their operation in detail. The techniques described in this section are induced absorption and magnetic resonance measurements. As described in Section III.lO.b, induced absorption has been used extensively to study the excited states produced by photoexcitation in ~ ’ used induced absorption to conjugated polymers. Brown et ~ 1 . ~have 34 6

D. R. Baigent, P. J. Hamer, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synfh. Met.

71, 2175 (1995). 34 7 A. R. Brown, K. Pichler, N. C. Greenham, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, Chern. Phys. Lett. 210, 61 (1993).

135


Energy / eV b

-6

3

4

2

2

1 2

. c:

%l

4

Ln

0

!z y1

0

0

0

'20.5

0.7

0.9

1.1

1.3

Energy (eV)

1.5

1.7

3

'

-1

Fiti. 75. Induced absorptions in an ITO/PPV/Ca device at 20 K. (a) Due to triplet excitons. (b) Due to charged excited states. The two signals are recorded using two-phase detection, and separated as described for Fig. 63. [A. R. Brown, K. Pichler, N. C. Greenham, D. D. C. Bradley, R. H. Friend, and A. B. Holrnes, Chrm. Phys. Lett. 210, 61 (1993).1

study the long-lived excited states formed in PPV LEDs under forward bias. In this experiment, a sub-gap probe beam is incident on the LED, passes through the polymer film, and is reflected back by the negative electrode. Modulation in the probe beam is detected using phase-sensitive techniques as the device current is modulated. Two components can be distinguished in the induced absorption: a strong feature at around 1.4 eV, and two weaker features at 0.6 and 1.6 eV, as shown in Fig. 75. By comparison of its shape, lifetime, and temperature dependence with the photoinduced absorption in this material, the feature at 1.4 eV is identified as being due to triplet excitons, thus confirming the presence of triplet excitons in polymer LEDs. Brown et al. have estimated the rate of generation of triplets from the size of the photoinduced absorption, and find it to be broadly consistent with the 3:l ratio of triplet-to-singlet production expected from spin-statistics arguments for nongeminate recombination of electrons and holes, although the uncertainties in this

136


estimate are large. The features at 0.6 and 1.6 eV are coincident with the photoinduced absorption features seen in poorly ordered PPV"' and are attributed to charged excited states, probably bipolarons. In the PPV used for these devices, however, no features are seen at these energies in photoinduced absorption. This difference can be explained by the conditions in an EL device, where there is significant net charge density throughout the device, which are very different from the case of photoexcitation, where the photogenerated charge carriers are geminate in origin. From the size of the 0.6- and 1.6-eV-induced absorption signals, and making reasonable assumptions about carrier mobilities and absorption cross sections, Brown et al. have concluded that the species responsible for these features is not the same species responsible for charge transport in the devices. Optically detected magnetic resonance (ODMR) has been extensively used to study recombination and luminescence processes in crystalline and it has been used for detailed amorphous s e m i c ~ n d u c t o r s .In ~ ~addition, ~ investigations of the triplet state in organic molecules.349In photoluminescence-detected magnetic resonance (PLDMR), the absorption of microwaves by a sample in a magnetic field is detected by its effect on the photoluminescence as the magnetic field is swept through resonance. Resonance occurs when the microwave photon energy, hv, is equal to the energy spacing between the magnetic sublevels of the species under investigation in the applied magnetic field B . This occurs when hv

=

pBgBAm,,

(24.1)

where pB is the Bohr magneton, g is the Land6 g factor, and Ams is the change in magnetic quantum number. Many conjugated polymers have now been investigated by PLDMR.'80s350, 351 For PPV, three characteristics signals have been observed in the PLDMR spectrum using X-band microwaves.laOA broad PL-enhancing feature (approximately 100-G wide) is observed at Ams = 1, accompanied by a signal of width approximately 25 G corresponding to a transition with Ams = 2, known as a half-field signal. The presence of this half-field signal confirms the assignment of these features to the presence of triplet excitons. From the lineshape of the triplet PLDMR, the spatial extent of the triplet wavefunction can be determined. The results indicate that the triplet is localized to an area not much larger than one benzene ring, in accord with the results of quantum chemical calculation^.'^^ The 348B.C. Cavenett, A h . Phys. 30, 475 (1981). 349N.M. Atherton, Electron Spin Resonance, Wiley, New York (1973). ""L. S. Swanson, J. Shinar, and K. Yoshino, Phys. Reu. Lett. 65, 1140 (1990). 351J. Shinar and L. S. Swanson, Synth. Met. 49-50, 621 (1992).


137

third PLDMR signal observed in PPV is a narrow PL-enhancing feature at full-field. This signal is attributed to the Am, = 1 transitions of distant polaron pairs formed by the dissociation of singlet excitons. These polaron pairs are believed to be coulombically bound, but to have a sufficient separation to give a negligible exchange energy. The exact mechanism by which altering the populations of the magnetic sublevels of triplet excitons and distant polaron pairs affects the PL is not yet well established, and is discussed e l ~ e w h e r e . ~ ~ ~ , ~ ~ ~ Similar techniques can be used to study the excited states in E L devices. In an electroluminescence-detected magnetic resonance (ELDMR) experiment, the E L device is placed in an identical environment to that used for PLDMR, except that the excitation is provided electrically rather than optically, and changes in the EL and conductivity are measured. ELDMR using ITO/PPV/Ca experiments were performed by Swanson et LEDs. The half-field triplet features observed in these devices for ELDMR and conductivity-detected magnetic resonance (CDMR) are shown in Fig. 76, along with the PLDMR for comparison. The ELDMR and CDMR signals provide clear evidence for the importance of triplet formation in PPV LEDs. The shapes of the ELDMR and CDMR signals are different from the PDLMR signal, and apparently arise from the presence of two distinct components. Swanson et al. tentatively attribute these two components to two different types of triplet exciton, one of which may be associated with one of the interfaces in the LED. A narrow full-field feature is seen in both ELDMR and CDMR, and is similar in shape to the narrow “polaron” feature seen in the PLDMR, but is quenching rather than enhancing in sign. This feature is attributed to the resonant enhancement of the production of excited states with S = 0 (probably bipolarons) from pairs of like-charged polarons. Enhanced bipolaron production will lead to a reduction in the number of mobile charge carriers, and may also cause direct quenching of singlet excitons, thus accounting for the reduction in both current and light output on resonance. As in induced absorption experiments, the differences in signal between photoexcitation and electrical excitation can be explained by the different charge densities present in the two cases.

25.

OPTICAL

PROPERTIES OF LEDS

Organic LEDs are complicated optical systems comprising several layers of different refractive indices. The angular distribution of the light emitted 15 2

J. Shinar el al. (to be published). 353L.S. Swanson, J. Shinar, A. R. Brown, D. D. C. Bradley, R. H. Friend, P. L. Burn, A. Kraft, and A. B. Holmes, Phys. Reu. B 46, 15072 (1992).

138


Magnetic Field (kG)

FIG.76. PLDMR, ELDMR and CDMR signals at half-field for an ITO/PPV/Ca LED at 10 K. [L. S. Swanson, J. Shinar, A. R. Brown, D. D. C. Bradley, R. H. Friend, P. L. Burn, A. Kraft, and A. B. Holmes, Phys. Reu. B 46, 15072 (19921.1

from the LED is determined by refraction and reflection at the various interfaces in the device, and by the location and orientation of emitting dipoles within the device. Since the layer thicknesses are typically of the same order as the wavelength of the emitted light, optical interference between light taking different optical paths through the device can also be important in determining the light output. It is important to understand the origin of the angular distribution in order to relate the light output per unit solid angle in the forward direction, Lo (which is easy to measure experimentally, and determines the brightness of a display device) to the


139

total amount of light produced within the emissive layer, Ftotal (which is related to the internal quantum efficiency). The simple case for an LED containing isotropically oriented emitting dipoles, neglecting interference effects, has been set out by Greenham et ~ 2 1 . ~The ' ~ relationship between L,, and Ftotal in this case is

Ftotal = 27rn2L0.

(25.1)

Whereas in molecular organic LEDs the emissive dipoles are likely to be isotropically oriented, in conjugated polymers there is likely to be a preferential orientation parallel to the plane of the substrate. Evidence for this preferential orientation is provided by anisotropy in the refractive and from polarized absorption spectroscopy.356 In the extreme case of perfect alignment within the plane of the film, the emission is weighted more toward the forward direction, and Eq. (25.1) becomes (25.2) In both cases, the angular distribution of emission outside the device is approximately Lambertian, that is, L ( e ) = L , COS(B).

(25.3)

Since the calculation of internal quantum efficiency from Lo depends on a number of assumptions about refractive indices and orientation of emitting dipoles, quoted values should be treated with great caution. A large fraction of the emitted light undergoes total internal reflection within the device, and may be coupled out at the edges of the device, scattered in the forward direction, or absorbed within the device. Integrating sphere measurements can be used to measure the total amount of light escaping from the device in all directions and, thus, to set a lower limit for the internal quantum efficiency. The importance of absorption within the device is not yet clear. The effect of dielectric and metal interfaces on the emission from molecules has been studied by many authors, and particularly by Drex' hage.357The first calculations on organic LEDs were by Saito et ~ l . ? ~and 154

N. C . Greenham, R. H. Friend, and D. D. C. Bradley, Adri. Muter. 6, 491 (1994). R. W. Gymer, R. H. Friend, H. Ahmed, P. L. Bum, A. M. Kraft, and A. B. Holmes, Synfh. Met. 55-57, 3683 (1993). j5'D. McBranch, I. H. Campbell, D. L. Smith, and J. P. Ferraris, Appl. Phys. Lett. 66, 1175 155

(1995). 157

K. H. Drexhage, in Progress in Optics (E. Wolf, ed.), Vol. 12, p. 163, North-Holland

(1974).

140


these demonstrate how the angular distribution of emitted light can be affected by interference effects in LED structures. The most important interface to consider in a polymer LED is the boundary between the polymer and the metallic negative electrode, which is usually highly reflecting. More sophisticated calculations are currently under way, taking into account all the interfaces in an LED based on a model for oscillating dipoles in a multilayer system due to Lukosz and Kunz359.36” and Crawford,361which has been applied to inorganic thin-film E L devices by Poelman et al.362 The presence of dielectric interfaces affects not only the angular distribution of the emitted light from a molecule, but also the radiative lifetime. For a molecule oscillating parallel to a perfect mirror at a distance x , the lifetime can be shown to be r =

70

1 - $z-’ sin z -

;Z-~COSZ

+ $-’sin

z’

(25.4)

where z = 47~nx/A,, T~ is the lifetime in the absence of any interfaces, A, is the wavelength of emission in free space, and n is the refractive index of the medi~m.3’~ This phenomenon has been investigated experimentally by many authors, including, most recently, Cnossen et ~ 1 . ’and ~ ~ Tsutsui et ~ 1 . ’ ~Full ~ quantum-mechanical treatments of this problem by M o r a w i t ~ ’and ~ ~ by Chance et al.365give good agreement to the predictions of the classical theory used by Dre~hage.’~’Because the efficiency for radiative decay of a singlet exciton is determined by a competition between radiative and nonradiative decay according to Eq. (13.31, altering the local environment will cause a change in the PL efficiency. For a dipole oscillating parallel to and very close to a perfect reflector, for example, the radiated power will be very small, leading to a large increase in the radiative lifetime and decrease in the PL efficiency. Very close to an interface (within an exciton diffusion range of 5 to 10 nm), nonradiative decay can be increased due to exciton quenching, independent of any 358S.E. Burns, N. C. Greenham, and R. H. Friend, Synth. Mer. (in press). 35yW.Lukosz and R. E. Kunz, J. Opt. SOC. Am. 67, 1607 (1977). 360 W. Lukosz and R. E. Kunz, J. Opt. SOC. Am. 67, 1615 (1977). 36 I 0. A. Crawford, J. Chem. Phys. 89, 6017 (1988). 362 D. Poelman, R. L. Van Meirhaeghe, W. H. LaflSre, and F. Cardon, J . Phys. D 25, 1010 (1992). 363 T. Tsutsui, C. Adachi, S. Saito, M. Watanabe, and M. Koishi, Chem. Phys. Lett. 182, 143 (1991). 364 H. Morawitz, Phys. Rev. 187, 1792 (1969). 3h5R.R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 62, 2245 (1975).


141

change in the radiative rate. In an EL device, the change in efficiency of radiative decay of excitons due to optical interference effects depends strongly on the location and orientation of the emitting dipoles within the device, which are not well known. Preliminary calculations suggest that interference effects cause significant reductions in efficiency (possible as much as a factor of 2) in typical polymer L E D s . ~ ~ ’ Several groups have designed “microcavity” organic LEDs in order deliberately to tune the angular distribution and spectrum of the emission. Using two highly reflecting electrodes, directed emission from the edges of organic LEDs has been observed.36h By making one of the electrodes partially transmitting, etalon-like structures can be achieved. This strategy ~ ’ in has been applied in molecular organic LEDs by Takada et ~ 1 . ~and . authors ~ ~ see ~ a narrowing conjugated polymers by Wittmann et ~ 1 These of the emission spectrum around a wavelength A corresponding to a resonant mode of the cavity, given by 2ndcos 0

=

mA

form

=

1,2, ...,

(25.5)

where d is the cavity thickness, n is the refractive index, and 0 is the emission angle in the cavity. The use of multilayer dielectric stacks instead of semitransparent metals allows more careful control of the characteristics of the microcavity, as has been demonstrated in photoluminescence by Dodabalapur et al.”’ and by Ochese et A n I T 0 electrode can be incorporated into the dielectric stack as the top layer, thus allowing microcavity LEDs to be f a b r i ~ a t e d . ~ ~Tsutsui l - ~ ~ ~et ~ 1 . ~ ’ ’used a europium-complex emitter with an intrinsically narrow emission spectrum, and hence obtained sharply directed emission, the angle of which was controlled by altering the optical characteristics of the microcavity. Control of the emission spectrum by design of the optical properties of polymer LEDs is likely to be important in obtaining the colors required for display devices. M. Hiramoto, J. Tani, and M. Yokayama, Appl. Phys. Lett. 62,666 (1993). N. Takada, T. Tsutsui, and S. Saito, Appl. Phys. Lett. 63,2032 (1993). 3hX H. F. Wittmann, J. Griiner, R. H. Friend, G. W. C. Spencer, S. C . Moratti, and A. B. Holmes, Adc. Muter. 7, 541 (1995). 365 A. Dodabalapur, L. J. Rothberg, T. M. Miller, and E. W. Kwock, Appl. Phys. Lett. 64, 2486 (1994). 3711 A. Ochse, U. Lemmer, M. Deussen, J. Feldmann, A. Greiner, R. F. Mahrt, H. Bassler, and E. 0. Giibel, Mol. Cryst. Liq. Cryst. 256, 335 (1994). j7’T. Tsutsui, N. Takada, S. Saito, and E. Ogino, Appl. Phys. Lett. 65, 1868 (1994). 372U.Lemmer, R. Hennig, W. Guss, A. Ochse, J. Pommerehne, R. Sander, A. Greiner, R. F. Mahrt, H. Bassler, J. Feldmann, and E. 0. Gobel, Appl. Phys. Left. 66, 1301 (1995). 373A. Dodabalapur, L. J. Rothberg, and T. M. Miller, Electron. Lett. 30, 1000 (1994). 374 A. Dodapalapur, L. J. Rothberg, and T. M. Miller, Appl. Phys. Lett. 65, 2308 (1994). 3hh 367

142


26. NOVELDEVICE STRUCTURES

The processing advantages of conjugated polymers and their desirable physical properties allow a number of novel and potentially useful device structures to be realized. Yang et ul.375have shown that the conducting emeraldine salt form of the conjugated polymer polyaniline (see Section 11.4) can be used as the hole-injecting electrode for polymer LEDs. Thin films of this polymer, formed by spin-coating from meta-cresol, are sufficiently transparent over most of the visible range to be acceptable for use with red and green LEDs. In addition to providing improved efficiency and lower operating voltage due to having a higher ionization potential than ITO, they can also be deposited on flexible transparent substrates such as polyethyleneterephthalate (PET). Using this strategy, Gustafsson et ~ 1 . ~ ~ ' have fabricated flexible polymer LEDs, which may be useful in applications where rigid or flat substrates are inconvenient. Further reductions in driving voltage have been made by using networks of polyaniline with high surface areas as hole-injecting electrode^.'^^ These networks are made by depositing blands of polyaniline and polyester resin, and then selectively removing the polyester resin to leave a rough, porous network of polyaniline, which enhances the local field in the polymer LED. A common problem in the design of microelectronic devices is the interfacing of light-emitting devices to silicon-based driving circuitry. The use of conjugated polymers may alleviate this problem by allowing lightemitting diodes to be fabricated on silicon substrates, which themselves contain the driving circuitry. Parker and Kim378have demonstrated the use of doped silicon substrates as injecting electrodes, although it is difficult to remove completely the native oxide layer from the silicon surface, leading to high driving voltages. The use of silicon, which is opaque to visible radiation, as a substrate requires the use of a semitransparent top electrode. Parker and Kim used thin, semitransparent layers of calcium or gold in this role. An alternative approach using two-layer PPV/CN-PPV devices was demonstrated by Baigent et al.,379 who evaporated a film of aluminum onto a doped silicon substrate to act as an electron-injecting electrode. A layer of CN-PPV followed by a layer of PPV precursor was 37s

Y . Yang and A. J. Heeger, Appl. Phys. Lett. 64, 1245 (1994). 376G.Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, Nature 357, 477 (1992). 377Y.Yang, E. Westenveele, C. Zhang, P. Smith, and A. J. Heeger, J . Appl. Phys. 77, 694 (1995). 37n I. D. Parker and H. Kim, Appl. Phys. Lett. 64, 1774 (1994). 37yD.R. Baigent, R. N. Marks, N. C. Greenham, and R. H. Friend, Appl. Phys. Lett. 65, 2636 (1994).


143

FIG. 77. Polymer LED on a silicon substrate. [D. R. Baigent, R. N. Marks, N. C. Greenham, and R. H. Friend, AppL Phys. Lett. 65, 2636 (1994).1

then deposited by spin-coating. Thermal conversion of the PPV precursor was found not to damage the underlying CN-PPV layer. A positive electrode of IT0 was then deposited by rf sputtering, to give the device structure shown in Fig. 77. These devices gave similar performance to ‘‘conventional’’ ITO/PPV/CN-PPV/AI devices, and could be operated in atmospheric conditions without encapsulation. 27. TECHNOLOGICAL ISSUES

Conjugated polymer LEDs already show sufficiently high brightness and efficiencies and sufficiently low driving voltages to be attractive for a number of commercial applications. The major issue remaining to be resolved before widespread applications are found is the question of device lifetime and degradation. This is a major topic of research for industrial and academic groups interested in developing technological applications for polymer LEDs, but very little work has yet been published in this area. Oxygen and moisture are known to be particularly damaging to organic LEDs, both through reactions with the electrodes and through oxidation of the emissive material. The quenching of the luminescence in PPV by photo-oxidation has been clearly demonstrated by Yan et aZ.242Careful encapsulation is therefore likely to be necessary in practical devices.380 Other degradation mechanisms that may be important include delamination at the polymer/electrode interface, chemical reactions at the elec380 P. E. Burrows, V. Bulovic, S. R. Forrest, L. S. Sapochak, D. M. McCarty, and M. E. Thompson, Appl. Phys. Lett. 65, 2922 (1994).

144


trodes, the migration of mobile ions within the device, and possible damage to the polymer layers by the passage of charge carriers. Degradation is likely to be more rapid in the softer polymers with low glass transition temperatures. 1 have . reported ~ ~ ~lifetimes in excess of 1000 hr for Cacialli et ~ ITO/PPV/Ca devices operated in a vacuum of mbar at current densities between 300 and 800 mA cm12. These results are encouraging, and suggest that lifetimes of more than 10,000 hr, which are required for many applications, may be achieved with further refinements to polymer synthesis, processing, and device structures in the future. VII. Photoconductiveand Photovoltaic Devices

28. MOLECULARDEVICES

A considerable amount of effort has been invested in the search for thin-film solar cells using a wide range of semiconductors, both organic and inorganic. Solar energy conversion is only economic under special circumstances, and only devices that use inorganic semiconductors have been developed. However, applications as photodetectors, particularly for applications that require large active areas such as medical imaging, may provide a different end use in which the criteria for performance are very different, and there is an increasing level of interest in this area.3x1 The first problem that has to be considered with the use of organic, molecular semiconductors is that the excited states produced by photon absorption are usually excitons that have relatively high binding energies and do not dissociate to give electrons and holes. Exciton ionization in the bulk is therefore not a promising method to follow. However, interfaces between molecular semiconductors or with electrodes can provide the correct energetics to allow charge separation, and one of the more promising routes to follow is to use a two-layer cell, similar in some ways to the two-layer LED structures of the type illustrated in Fig. 73.3x2This approach has been followed by several groups, and Tang3x3 has obtained some of the best results, using a cell formed on glass coated with indium-tin oxide, on which layers of a copper phthalocyanine (hole-collecting semiconductor) and a perylene tetracarboxylic derivative are formed by subli'"U. Schiebel, N. Conrads, N. Jung, M. Weibrecht, H. Wieczorek, T. Zaengel, M. J. Powell, I. D. French, and C. Glasse, Proc. SPIE 2163, 129 (1994). 382D.Wohrle and D. Meissner, Adu. Muter. 3, 129 (1991). 3x3C.W. Tang, Appf. Phys. Left. 48, 183 (1986).


145

mation, capped with an evaporated silver electrode. Excitons that diffuse to the interface between these two semiconductors ionize, and electrons and holes are then collected. Cells of this type can show power conversion efficiencies of 1% and, if sufficiently thin, good fill factors (0.65). Two difficulties are commonly encountered. First, carrier mobilities are low, so that power loss is likely to be problematic at high levels of illumination. Second, although the absorption coefficients for organic semiconductors are very high ( > l o 5 cm-'), the absorption depth is usually greater than the diffusion range of the excitons created by the absorption process. Thus, only a fraction of the excitons generated is able to find the interface between the two semiconductors at which ionization can occur. A strategy to improve performance is to use high-purity, highly crystalline molecular semiconductors, which can show greater diffusion ranges for e x ~ i t o n s A . ~different ~~ approach is to arrange a structure in which there is a very large surface area, so that all absorbing regions lie close to an interface at which ionization can occur. This approach was who used a sintered electrode of TiO, demonstrated by Gratzel et u21.38s*386 onto which they surface-absorbed a layer of an organic dye (ruthenium bipyridinium complex). Absorption in the ruthenium complex results in electron transfer into the TiO, and the circuit is completed via a redox couple of iodine/iodide to a back electrode to give a photoelectrochemical cell with a reported energy conversion efficiency of 7 to 12%.385 DEVICES 29. POLYMERIC The attractive feature in the use of polymers lies in the scope for convenient processing of films over large area, which may be structurally stable and resistant to recrystallization. There have been a number of studies of the photoconductive and photovoltaic response of a wide range of conjugated polymers, some of which were mentioned earlier, for example, photoconduction in polyacetylene (Fig. 39) and in pold p-phenylenevinylene) (Fig. 35). Photovoltaic efficiencies were found to be low, with low open-circuit voltages and low overall efficiencies. Initial efforts were with p ~ l y a c e t y l e n e ' and ~ * ~some ~ ~ of the p ~ l y t h i o p h e n e s . ~ ~ ~ 3x4

N. Karl, A. Bauer, J. Holzapfel, J. Marktanner, M. Mobius, and F. Stolzel, Mof. C y s t . Liq. C y s f . A . 252, 243 (1994). 3x5 B. O'Regan and M. Gratzel, Nature 353, 737 (1991). 386 M. Gratzel and K. Kalyanasundararn, Cur.Sci. 66, 706 (1994). 3M7 B. R. Weinberger, M. Akhtar, and S. C. Gau, Synth. Met. 4, 187 (1984). 388 S. Glenis, G . Tourillon, and F. Gamier, Thin Solid Films 139, 221 (1986).

146


- 1 21 - 2 - 1

1

0

2 3 Voltage (V)

1

4

5

FIG.78. I-V curves for an ITO/PPV/Mg diode, of the type shown in Fig. 8 with PPV layer of thickness 120 nm, illuminated by 457.9-nm radiation through the IT0 electrode. [R. N. Marks, J . J. M. Halls, D. D. C. Bradley, R. H. Friend, and A. B. Holmes, J . Phys.: Condens. Mutter 6, 1379 (1994). IOP Publishing Limited]

Mechanisms for photogeneration of carriers in conjugated polymers are not well understood at present, not least because there are widely differing views on the size of the binding energy of the photogenerated excitons, as discussed in Section 1II.lO.c. The recent interest in the properties of electroluminscent diodes, discussed in detail in Section VI, has regenerated interest in the photoconductive and photovoltaic responses of devices of this type, and Karg et ul.326,327were the first group to report the photovoltaic response of devices that also functioned as LEDs. Results for diodes fabricated as shown in Fig. 8 have been reported by several groups.194, 326,327,389 Figure 78 shows current versus voltage characteristics for a diode formed with PPV between IT0 and Mg, and demonstrates an open-circuit voltage that rises to 1.2 V at high illumination intensities (approximately equal to the work-function difference between the two electrodes), and a short-circuit current that gives a quantum efficiency (electrons collected per incident photon) of up to 1%. At high forward and reverse biases, however, the quantum efficiency can reach very high values; l ~ ~ an efficiency of up to 500% under forward bias Marks et ~ 1 . report (indicating that there are multiple transits of carriers across the device within the recombination lifetime), and several groups have reported efficiencies above 50% under reverse bias, particularly for devices that 389

H. Antoniadis, B. R. Hsieh, M. A. Abkowitz, S. A. Jenekhe, and M. Stolka, Synfh. Met. 62, 265 (1994).

147

SEMICONDUCTOR DEVICE PHYSICS 1.2

2.0

PPV

2.5

3.0

3.5

Energy (eV)

Fic. 79. Comparison between the spectral response of an ITO/PPV/AI diode of the type shown in Fig. 8 when illuminated through the I T 0 contact (bias at 0 and 1.5 V) and the semitransparent Al contact (bias at 0 V). [R. N. Marks, J. J. M. Halls, D. D. C . Bradley, R. H. Friend, and A. B. Holmes, J . Phys.: Condens. Matter 6, 1379 (1994). IOP Publishing Limited]

make use of Ca and IT0 electrode^.'^^^"^"^' These high efficiencies under bias (typically in the range up to 10 V> have generated some interest in the potential use of this type of structure as a large-area p h o t ~ d e t e c t o r . ~ ~ ~ Information about the nature of the mechanism for carrier photogeneration can be obtained from the spectral dependence of the photovoltaic response, and this is shown in Fig. 79 for a PPV diode with Al and I T 0 electrodes, both of which were transparent or semitransparent. Under conditions of reverse internal field, obtained here when the diode is held at 0 V (short circuit), the internal field within the polymer layer will sweep electrons toward the Mg or Al electrode, and holes to the IT0 electrode. When the diode is biased at a voltage greater than the open-circuit voltage, however, then the charges are pulled in the opposite direction. The spectral responses indicate that these diodes can collect the photogenerated charges only when the electrons are generated close to the electrode to which they travel. Thus, a strong photoresponse for excitation in the T-T* absorption band is seen when the device is illuminated from the Al electrode at 0 V and from the IT0 at 1.5-V bias. In contrast, illumination at 0-V bias from the I T 0 electrode only gives a strong response for excitation at the edge of the T-T* absorption band, at the wavelength that optimizes the fraction of absorbed photons close to the Al electrode. 390

G . Yu, C. Zhang, and A. J. Heeger, Appl. Phys. Lett. 64, 1540 (1994). X. Wei, M. Maikh, Z. V. Vardney, Y. Yang, and D. Moses, Phys. Rev. B 49, 17480 (1994). 392 G. Yu, K. Pakbaz, and A. J. Heeger, Appl. Phys. Lett. 64, 1 (1994). 7Y1

148


Marks et ~ 1 . consider ’ ~ ~ that photogeneration occurs in the bulk, by field-assisted dissociation of the singlet exciton, and that electrons tend to become trapped and recombine with holes, rather than traveling the full . ~ width of the polymer layer. Recent work by Antoniadis et ~ 1 indicates that carrier generation is assisted by extrinsic factors, such as the presence of oxygen. An application to the case of the polymers of the two-layer heterojunction structure that has been used successfully with the molecular semicond u c t o r ~ has ~ * ~recently been reported. The fullerene C,,, provides a useful high electron affinity semiconductor, and it is found that blends of this with a range of soluble PPV and polythiophene derivatives show very efficient charge separation following p h o t o e ~ c i t a t i o n . ~Two-layer ~~-~~~ diodes of c6, and MEH-PPV were reported initially to show poor effi~ i e n c i e s , ~but ~ ’ photovoltaic quantum efficiencies (short circuit) above 5% have now been obtained for PPV/C6, di0des.3~~ The principle demonstrated by O’Regan and Gratze13” of using a large effective area to ensure that all photons are absorbed close to an interface between electronaccepting and hole-accepting semiconductors has recently been applied to . of ~ MEH-PPV ~ ~ and CN-PPV (Fig. 16) the polymers by Halls et ~ 1 Blends are expected to form interpenetrating networks when cast from solution and correctly annealed, and films formed in this way give high photoconductive responses when formed between Al and I T 0 electrodes. This summary of activity on photoconductive and photovoltaic responses of the conjugated polymers is intended to provide a brief description of those processes that are most relevant to the device structures that form the area of principal interest in this article, particularly the LEDs. There is increasing activity in this area, and the prospect for large-area photodetectors provides a realistic goal for eventual applications.

393H.Antoniadis, L. J . Rothberg, F. Papadimitrakopoulos, M. Yan, M. E. Galvin, and M. A. Abkowitz, Phys. Reu. B 50, 14911 (1994). 394N.S. Sariciftci, L. Smilowitz, A . J. Heeger, and F. Wudl, Synth. Met. 59, 333 (1993). 395 N. S. Sariciftci, L. Smilowitz, A. J. Heeger, and F. Wudl, Science 258, 1474 (1992). 396B.Kraabel, C. H. Lee, D. McBranch, D. Moses, and N. S. Sariciftci, Chem. Phys. Lett. 213, 389 (1993). 397N.S . Saricifitci, D. Braun, C. Zhang, V. I. Srdanov, A. J. Heeger, G. Stucky, and F. Wudl, Appl. Phys. Left. 62, 585 (1993). 39x J. J. M. Halls, K. Pichler, R. H. Friend, S. C. Moratti, and A. B. Holmes, Synth. Met. (in press). 399 J. J. M. Halls, C. A. Walsh, N. C. Greenham, E. A. Marseglia, R. H. Friend, S. C. Moratti, and A . B. Holmes, Nature 376, 498 (1995).

~

~


149

VIII. Conclusions

Conjugated polymers have provided a rich area of research for several areas of science, from synthetic chemistry to materials science and semiconductor physics. We have been concerned here with the electronic properties of conjugated polymers. The study of the characteristics of a wide range of semiconductor devices has allowed detailed characterization of the electronic phenomena that these materials demonstrate. We have made mention of the synthesis and processing of those polymers that have been found to be useful in this application, and have discussed some of the issues that relate to device fabrication. The progress made with device fabrication and measurement has demonstrated that polymers can be employed in devices with useful electrical or electro-optical performance, most particularly with the LEDs. The combination of the potential for convenient processing of polymers to form large-area thin films, together with good device characteristics, has generated a high level of interest in possible applications. ACKNOWLEDGMENTS This chapter has drawn on the work of many people in the Cambridge group, both in the Department of Physics and in the Department of Chemistry, and those in the several groups elsewhere with which we have worked. We are grateful to all those whose work and ideas have influenced the contents of this article, and particularly to those whose work we have cited. We thank Dr. R. N. Marks for his helpful comments on this article during its preparation. One of us (NCG) thanks Clare College, Cambridge, for support.

SOLID STATE PHYSICS. VOL. 49

Photonic Band-Gap Materials P. M. Hur Department o f Physics, The Chinese Uniuersity of Hong Kong, Shatin, New Territories, Hong Kong

NEILF. JOHNSON Department of Physics, Clarendon Laboratory, Uniuersity of Oxford, Oxford OX1 3PU, England, United Kingdom

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experimental Systems and Results . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Photonic Band Theory: Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 5. Theoretical Calculations and Results . . . . . . . . . . . . . . . . . . . . . . . . IV. Conclusions and Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 157 157 165 170 170 186 199

1. Introduction

1. SURVEY

The properties of wave propagation in a periodic system have always been of great interest to physicists. A well-studied example in solid state physics is that of an electron in a periodic potential in either one dimension (lD), two dimensions (2D), or three dimensions (3D).'-3 Such a periodic potential arises in naturally occurring crystalline solids (3D) and in artificially made heterostructures (1D and 2D). The eigenstates of such an electron are known as Bloch states. As a result of Bragg scattering at the Brillouin zone boundary, gaps of disallowed energies appear in the energy spectrum

'

N. W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College, Philadelphia (1976). 'J. Callaway, Quantum Theory of the Solid State, Chaps. 4 and 5, Academic Press, New York (1974). 'J. Callaway, in Solid State Physics (F. Seitz and D. Turnbull, eds.), Vol. 7, p. 100, Academic, New York (1958).

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Copyright 0 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

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P. M. HUI AND NEIL F. JOHNSON

of the electron.' The continuous energy-wavevector dispersion relation characteristic of free space is therefore modified; this energy spectrum is referred to as the electronic band structure of the system. The quantitative details of a given electronic band structure depend on the specific form of the periodicity being considered. As a result, different crystalline solids have different electronic band structures.', 2 * 4 An analogous situation can arise for electromagnetic waves propagating through a medium with a periodic variation in the dielectric constant. These systems, which are generally known as photonic band-gap (PBG) materials or photonic crystals, can be designed with a periodic variation in the dielectric constant in either one, two, or all three spatial A typical structure consists of a regular array of objects (e.g., spheres) of one dielectric material embedded in a second material with a different dielectric constant. Due to the periodic structure of the array, there may be ranges of frequencies at which no allowed modes exist for electromagnetic wave propagation in a given direction. These ranges of frequencies are termed photonic band gaps in analogy with the electronic band gaps. The dispersion relation (i.e., the frequency-wavevector relation) for electromagnetic wave propagation in PBG materials is referred to as the photonic band structure. The photonic band structure of a given PBG material depends on the crystal structure, the lattice constant, the shape of the embedded dielectric object, the dielectric constants of the constituent materials, and the filling fraction, which is the percentage of total crystal volume occupied by any one of the materials. The existence of full photonic band gaps in PBG materials has important potential applications in the area of quantum optics and laser technology.' The idea of combining two different materials in order to create a new structure with novel properties has been repeatedly employed in solid state physics in recent years. In semiconductor superlattices? for example, layers of semiconductors are put together to create an additional periodicity in one direction. This procedure is known as band-gap engineering since the resulting structure can have a band gap that is quite different from those of the constituent semiconductors. Multilayered systems of magnetic and nonmagnetic materials have been shown to exhibit enhanced 4E. 0. Kane, in Narrow Gap Semiconductors (W. Zawadski, ed.), p. 19, Lecture Notes in Physics, Vol. 33, Springer-Verlag, New York (1981). %ee, for example, the articles published in the special issue of J . Opt. SOC. A m . B. 10, No. 2 (1993). bE. Yablonovitch, J . Phys.: Condens. Matter, 5, 2443 (1993). 'P. St. J-Russell, Phys. World, p. 37, Aug. 1992. 'P. R. Villeneuve and M. Piche, Prog. Quant. Electr. 18, 153 (1994). 9D. L. Smith and C. Mailhiot, Reu. Mod. Phys. 62, 173 (1990).

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magnetoresistance, the so-called "giant magnetoresistance" phenomenon." Giant magnetoresistance has also been observed in granular magnetic composites in which small, magnetic particles are randomly dispersed in a nonmagnetic host medium. The latter systems are examples of the larger class of materials collectively known as macroscopically inhomogeneous media, the physics of which has recently been reviewed by Bergman and Stroud." The underlying interest in such systems concerns the effective response (e.g., electrical, thermal, dielectric, optical, or mechanical) when two or more kinds of materials with different macroscopic properties are put together in either an ordered or disordered way. In this respect PBG materials are another example of such composite systems, with the relevant property being the effective optical response. The idea of a system with a photonic band gap is not new. There is a history of work on 1D waveguides and on periodic 1D structures made from slabs of dielectric materials, yielding photonic gaps for electromagnetic waves propagating in the direction normal to the slabs.12 The PBG materials are therefore the logical extension to higher dimensions of the well-known 1D waveguides and attenuators already in use. Surprisingly however, the possibility of fabricating such a PBG system in higher and John.I4 The dimensions was only raised in 1987 by Yablon~vitch'~ important advantage of such higher dimensional PBG materials over their ID counterparts is the possibility of a full photonic band gap throughout the entire Brillouin zone, thereby preventing propagation of electromagnetic waves in all directions as opposed to just along one particular direction. In the five years that followed this initial suggestion, many groups became attracted to PBG material research as a result of advances in fabrication techniques and the increasing awareness of the potential importance of PBG 'OR. E. Camley and R. L. Stamps, J. Phys.: Condens. Matter 5, 3727 (1993). I'D. J. Bergman and D. Stroud, in Solid State Physics (H. Ehrenreich and D. Turnbull, eds.), Vol. 46, p. 147, Academic Press, New York (1992). 12 P. Yeh, Optical Waues in Layered Media, John Wiley & Sons, New York (1991). "E. Yablonovitch, Phys. Reu. Lett. 58, 2059 (1987). I4 S. John, Phys. Reu. Lett. 58, 2486 (1987). "S. L. McCall, P. M. Platzman, R. Dialichaouch, D. Smith, and S. Schultz, Phys. Reu. Lett. 67,2017 (1991). 16 E. N. Economou and A. Zdetsis, Phys. Reu. B 40, 1334 (1989). I'E. Yablonovitch and T. J. Gmitter, Phys. Reu. Lett. 63, 1950 (1989). in E. Yablonovitch and T. J. Gmitter, J. Opt. SOC. Am. 7, 1792 (1990). 19 K. M. Leung and Y. F. Liu, Phys. Reu. B 41, 10,188 (1990). 'OK. M. Leung and Y. F. Liu, Phys. Reu. Lett. 65,2646 (1990). 21K.M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).

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One of the principal motivations driving the PBG field has been to design a PBG material (2D or 3D) that has a full photonic band gap in the optical frequency range and yet which is both lossless and easy to fabricate. Despite numerous attempts, the successful achievement of this goal has yet to be reported. Several of the proposed technological applications for PBG materials rely on a complete photonic band gap throughout the entire Brillouin zone. Nevertheless, other applications have been suggested that are less dependent on such a full gap. Therefore, there is merit in considering the general class of PBG materials, whether or not they actually show a full photonic band gap for all directions of propagation. As is customary in this field, therefore, we will use the general term PBG material irrespective of whether or not a full photonic band gap actually exists throughout the Brillouin zone in the particular sample under consideration. We shall also adopt the terminology photonic band structure even though the band structure arises from a strictly classical treatment of the problem, that is, from solving Maxwell's equations. The aim of this article is twofold. First, we discuss recent experimental and theoretical progress in understanding the properties of PBG materials. Second, we attempt to provide a rigorous formalism describing electromagnetic properties in PBG materials, akin to that currently available in the literature for electronic properties in crystalline solids.', 40 The complica*'R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. B 44, 13,772 (1991); 10,961 (1991). 23J. Pendry and A. M. MacKinnon, Phys. Reu. Lett. 69, 2772 (1992). 24 K. M. Ha, C. T. Chan, and C. M. Soukoulis, Phys. Reu. Lett. 66, 393 (1990). "S. Satpathy, Z. Zhang, and M. R. Salehpour, Phys. Reu. Lett. 64, 1239 (1990); 65, 2478(E). 26 Z. Zhang and S. Satpathy, Phys. Reu. Lett. 65, 2650 (1990). "M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, Opt. Commun. 80, 199 (1991). 28M. Plihal and A. A. Maradudin, Phys. Reu. E 44, 8565 (1991). 29 N. Stefanou, V. Karathanos, and A. Modinos, J . Phys.: Condens. Matter 4, 7389 (1992). ' O S . Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Reu. E 46 10,650 (1992). "E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Bromrner, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991). 32 R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992). 33 E. Yablonovitch, T. J. Gmitter, and K. M. b u n g , Phys. Reu. Lett. 67, 2295 (1991). 34H.S. Soziier, J. W. Haus, and R. Inguva, Phys. Reu. E 45, 13,962 (1992). 3s C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991). 36 G. Henderson, T. K. Gaylord, and E. N. Glytsis, Phys. Rev. B 45, 8404 (1992). 37P. R. Villeneuve and M. Piche, J . Opt. SOC. Am. A 8, 1296 (1991). 38 G. X. Qian, Phys. Reu. B 44, 11,482 (1991). 39P. R. Villeneuve and M. Piche, Phys. Reu. E 46, 4969 (1992); 4973 (1992). 40 E. 1. Blount, in Solid State Physics (F. Seitz and D. Turnbull, eds.), Val. 13, p. 305, Academic, New York (1962).


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tion for PBG crystals compared to electronic crystals is that Maxwell’s equations describing the propagation of electromagnetic waves are vectorial in nature, while the Schroedinger equation describing electronic waves is scalar. It is this feature that can give rise to nontrivial generalizations of the well-known properties for electrons in solids. The sections in this article on experimental fabrication techniques (Section 11.2), potential applications (Section 11.31, and theoretical calculations of photonic band structures (Section 111.5) are intended for general readership. The formalism section (Section 111.4) is more advanced, but is designed to be self-contained and can therefore be read independently. Because PBG materials are generally artificially made structures, there are an infinite number of possible PBG materials depending on the crystal structure, the dimensionality, the lattice parameter, the filling fraction, and the dielectric constants of the constituent materials. The search for the PBG material with the largest photonic band gap is therefore far from easy, given the range of possible parameters. Initially it was thought that an fcc (face-centered-cubic) structure should have a large photonic band gap, because of its nearly spherical Brillouin zone.41 This provoked many theoretical and experimental investigations of PBG materials with an fcc crystalline form. Section 11.2 examines the various sample constructions and experimental arrangements that have been employed so far in the search for a photonic gap in 2D and 3D PBG materials. We also review the experimental results to date. The technical aspects will only be discussed in sufficient detail so as to highlight the number of new experimental techniques that have had to be developed in the PBG field. In this way, we emphasize the extent to which investigation of the photonic band gap differs from that of the electronic band gap. For detailed technical discussions of the experimental apparatus, we refer the reader to the original sources. The proposed applications for PBG materials have been both numerous and diverse. They are all, however, based on the idea of the modification of the free-space photon dispersion relation from w = ck, where k is the wavevector, to the more general form o = w ( k ) . These applications employ either a perfectly periodic PBG material (crystal) or consider a PBG material containing “defects.” In perfect PBG crystals, the introduction of gaps in the photon dispersion prevents propagation of photons at those frequencies, thereby allowing applications of PBG materials in integrated optics as filters and polarizers. The frequency gap also provides a means of tailoring the optoelectronic properties of a semiconductor device placed within the PBG material. In particular, the electron-hole recombination

4’

E. Yablonovitch, J . Opr. Soc. Am. B 10, 283 (1993).

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process, which is a major form of loss of efficiency in many devices, can be suppressed if the energies of any photons that would be emitted in free space fall in the gap of the PBG material.41Introducing a “defect” into the periodic PBG material can introduce a finite number of localized defect modes, just as in a doped semiconductor system. Such discrete levels lying in the gap imply that such “doped” PBG materials could provide useful single-mode cavity systems operating in the technologically important optical frequency range. Section 11.3 discusses the various applications that have been suggested for PBG materials. Again, technical details are not given in detail because our goal is to emphasize the general mode and diversity of the proposed applications. A major distinction between the properties of electronic and photonic band structures, as mentioned earlier, results from the vector nature of electromagnetic waves as opposed to the scalar nature of electronic waves. There are additional fundamental differences. Charge and the Pauli exclusion principle do not enter into the discussion of PBG materials as photons are chargeless, spin-one entities. Also photon-photon interactions are negligible, in contrast to the important role of electron-electron interactions in solids. A description of photonic band structures therefore has an advantage over that of electronic band structures in that a singleparticle description is exact; that is, solving Maxwell’s equations for a single electromagnetic wave can yield exact results. This is in sharp contrast to the electronic case for which many-electron interactions can qualitatively affect the properties predicted by the single-electron Shroedinger equation. This allows the presentation in Section 111.4 of an exact formalism describing photonic band structures in PBG crystals. The various representations are discussed for the photonic crystal. The corresponding equations are given both in the absence and presence of defects. Similarities and differences between the photonic and electronic problems are emphasized. In practice, photonic band structures have been calculated theoretically for various PBG crystals in 2D and 3D using extensions of traditional, numerical electronic band structure techniques. In Section 111.5, we discuss the various theoretical approaches to date. In addition we demonstrate the extent to which predictions from different theoretical approaches are consistent with the experimental results discussed in Section 11.2. Given that there are an infinite number of possible PBG crystals, it is neither practical nor instructive to provide a complete catalog of all the theoretical band structures that have been reported. We choose rather to focus on a few representative results that have been widely cited in the literature and that can be compared directly to experiment. In so doing, we will attempt to highlight trends that have emerged with respect to the


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photonic properties as the PBG material parameters are varied. The most popular calculational technique employed to date has been a numerical plane-wave expansion (see, for example, Ref. 20). Accurate calculations of photonic band structures must account for the vector nature of the electromagnetic waves. Such vector calculations have so far been numerically intensive and have necessitated powerful computers. Given the wide choice of PBG material parameters, it would be useful for the design and fabrication of PBG materials to have some approximate analytic expressions for the photonic band gap as a function of filling factor, dielectric contrast, lattice parameter, and lattice structure. This is particularly important given the rather unrelated structures in which full photon band gaps have been observed. Because the main frequency region of interest lies in the vicinity of a photonic band gap, it is reasonable to search for alternative theoretical schemes that allow more insight into the nature of the dispersion relation in the vicinity of band maxima and minima. Such a scheme is well known in conventional electronic theory and is the so-called k .p It is shown in Section 111.5 that a photonic k * p approach can be successfully used to calculate the dispersion relation near the photonic band gap. Another method that has proved useful for understanding electronic properties in solids is the tight-binding method.’ Section 111.5 also illustrates that a tight-binding method can be employed successfully in PBG materials. We conclude in Section IV with a discussion of possible future directions for research into PBG materials. II. Experimental Overview

2. EXPERIMENTAL SYSTEMS AND RESULTS

In principle, the construction of a photonic band-gap material is straightfoward. One needs to introduce a periodic variation in the dielectric constant of a material in one, two, or three spatial dimensions, thereby producing a lD, 2D, or 3D periodic dielectric material. In practice, there are many technical complexities and subleties in the fabrication and measurement of PBG materials; these have become apparent through experimental trial and error.41 One-dimensional PBG materials operating in the microwave regime significantly predate the “invention” or 2D or 3D PBG material^.^' Such 1D systems could be built with relatively low precision because of the relatively long microwave wavelengths. In addition, since the periodicity only had to be introduced along one direction, samples could be made fairly easily using slabs of dielectric materials. The challenge of fabricating

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a PBG material with a full photonic band gap at optical frequencies introduces two significant complications. First, the periodicity must be introduced in two or three dimensions because it is only in 2D or 3D PBG materials that full photonic band gaps can exist. Second, the requirement of a photonic band gap in the optical frequency spectrum requires precision machining of dielectric periodicity on the optical length scale. Indeed the optical regime remains the hardest, but arguably the most useful, frontier in the fabrication of PBG materials with a full band gap. So far, the successful fabrication of a PBG sample with a full band gap in the visible spectrum has yet to be reported because of these difficulties at such small length scales. For example, PBG materials fabricated by hole-drilling, as discussed later, must have holes that are uniform on this length scale in order to obtain a true gap in a perfect crystal. To date, investigations of PBG material properties in both 2D and 3D have therefore been mostly limited to the microwave regime, where precision in machining is less important. The search for a full photonic band gap has prompted interest in the fabrication and investigation of both 2D and 3D PBG structures. Typically investigations in the microwave regime have been carried out on 2D and 3D samples about 20 periods long and a few centimeters in length.5 In general, 2D microwave samples can be made by arranging long dielectric rods in a regular lattice or by cutting holes into a dielectric slab; 3D samples are fabricated either by arranging dielectric objects such as spheres in a regular lattice or by creating voids in a dielectric b10ck.~The advantage of 2D structures over 3D is that they are generally easier to fabricate. An important distinction can be made between the nature of experiment needed to determine the photonic band gap in a given sample of a PBG material, and that needed to determine the electronic band gap in conventional solids. In semiconducting and insulating samples, the electronic band gap can be measured reasonably easily using photon absorption. This is because the band gap separates valence bands full of electrons from conduction bands that are essentially empty. A photon will then be absorbed if it has sufficient energy to take an electron from one of the full valence bands up to the empty conduction band.2 The absorption of photons for energies below the band gap is therefore essentially zero. As the band gap energy is reached, the absorption rises sharply, particularly in low-dimensional systems. The band gap is then obtained from this threshold photon energy. The photonic band gap in PBG materials cannot be measured in the same way because, unlike the electronic system, the band gap does not separate a full valence band of carriers from an empty conduction band. The photonic band gap is simply a window of frequencies in which propagation through the crystal cannot occur. Some ingenious


159

methods have therefore had to be developed to determine the possible presence of a photonic band gap throughout the Brillouin zone. We now look more closely at the various techniques that have been employed to fabricate a given PBG material sample and measure its photonic band structure in the microwave frequency range. Such microwave experiments have been carried out by a number of groups; the different setups are all based on determining the phase and amplitude of the field transmitted through a PBG material sample for various orientations. The refractive index ratio between the two PBG material constituents in typical microwave experiments has usually been chosen to be similar to that between a common semiconductor such as GaAs and air ( - 3.6:l). In this way, any dispersion relations obtained in the microwave regime can then simply be scaled down in size in order to gain insight into the corresponding photonic band structure in the optical regime.41 One experimental setup for investigating photonic band structures in a given PBG material in the microwave regime involves an anechoic chamber, as used by Yablonovitch and Gmitter.” The arrangement consists of a long chamber with walls that absorb microwaves, which provide essentially plane waves incident on the PBG material sample. This pump-probe setup ” allowed investigation of electromagnetic wave propagation through PBG samples over the frequency range 1 to 20 GHz. To determine the photonic gap, the frequency range was swept with the radiation incident along a given crystal direction, and the signal transmission was measured. To obtain the corresponding wavevector k in the PBG material, Yablonovitch and Gmitter” exploited the fact that the k component parallel to the surface was conserved and that the frequencies at the onset of the gap correspond to wavevectors at the Brilloin zone edge. By rotating the crystal and using both polarizations of incident radiation separately, they were able to map out parts of the photonic band structure. A drawback of this method is that they could only investigate the band structure in the vicinity of the gap; in addition, they could not investigate the bands along the T-X and T-L directions because these directions do not lie in the surface of the Brillouin zone. Using this setup, Yablonovitch and Gmitter ”, built and investigated 3D fcc structures consisting of 8000 dielectric spheres of 6-mm radius. The spheres were made of Al,O, and were supported in the lattice by a thermal-compression-molded dielectric 3, and the foam refractive index foam. The sphere refractive index was 1. A true photonic band gap was not observed, however, at any was filling fraction. They then tried replacing the dielectric spheres with spherical airholes in a dielectric background. This was achieved by drilling holes in opposite faces of dielectric slabs with a spherical drill bit and then stacking these

’’

-

N

160


slabs on top of each other so that the hemispheres faced each other; this produced the desired fcc array of spherical voids inside a dielectric block. However, no photonic band gap was observed. They then tried increasing the size of the airholes to such an extent that the voids were closer than closed-packed; the voids therefore overlapped and allowed continuous holes to pass through the structure. Initially, they thought they had observed a full photonic band gap centered at 15 GHz and forbidden for both p01arizations.I~ However, it turned out that their finite-size PBG sample, which was only 12 unit cells wide, had not shown up band crossings in the dispersion relation near the W and U symmetry points. These crossings later emerged from theoretical calculations and are discussed further in Section 111.5. The finite size of the sample had the effect of limiting the wavevector resolution and restricting the dynamic range in transmission. The major experimental breakthrough yielding the first observation of a photonic band gap in the microwave regime came when Yablonovitch and co-workers found that these symmetry-induced degeneracies in the fcc structure could be lifted by building the PBG material from nonspherical atoms.33 Using a mask with a triangular array of holes, they drilled three sets of holes at various oblique angles to obtain an fcc structure of nonspherical, elongated, air-filled atom^.^^,^' The resulting structure was a fully 3D fcc crystal. For a dielectric refractive index of 3.6 they observed a full gap corresponding to 19% of the mid-gap frequency in a structure made from 22% dielectric. They also measured the mid-gap attenuation to be 10 dB per unit cell; this indicates that the 3D PBG crystal need not be many layers thick in order to expel the photon field at that frequency. The significant feature of the hole-drilling procedure of Yablonovitch et is that the holes did not need to be individually drilled. Furthermore, similar structures were also grown by reactive ion e t ~ h i n g , ~which ’ left oval holes in the material. It was found33 that the forbidden gap width increased slightly in such structures. This opened the possibility that the nonspherical “atoms,” which were observed to lift the degeneracies and open a complete photonic gap, might be fabricated in the optical frequency regime using reactive ion etching. Meade et used a similar experimental pump-probe setup to investigate 2D PBG materials. Their samples consisted of air columns drilled into a dielectric material of refractive index 3.6, forming a triangular lattice array. They found a gap for both polarizations between 13 and 16 GHz for a sample of air columns with both diameter and lattice constant of almost 1 cm. These two groups ~ o l l a b o r a t e don ~ ~the investigation of defect modes in a 3D PBG sample with a single defect. Their sample consisted of the fcc structure with nonspherical airholes in a dielectric material with a refrac-


161

tive index of 3.6. The hole diameter and lattice constant were about 0.5 and 1 cm, respectively. Defects were introduced into the PBG crystal in one of two ways: by adding one dielectric sphere (donor) in a hole of the lattice or by breaking one of the connections between the airholes (acceptor). Transmission through the defect modes was achieved by replacing the plane wave (single k) incident on the sample with a spherical point source. In this way, waves were incident on the sample with a range of possible k component's along the normal to the surface. This allowed generation of a wide range of possible defect modes at a given frequency. The results are shown in Fig. 1. For the sample with a broken connection [Fig. l(b)l the appearance of a single mode in the forbidden gap is quite striking when compared to the strong attenuation in the perfect crystal at this frequency [Fig. l(a)]. Because of its relatively equal frequency separation from the two band edges, the defect mode in Fig. l(b) is labeled as a deep acceptor.

1

5J

10-1

,

-

C

10-2-

z5 10-3-

a

10"

11

12

13

14

15

Frequency in GHz

16

17

FIG.1. Experimental results for attenuation of microwave radiation passing through (a) a perfect 3D PBG fcc crystal of thickness 8 to 10 atomic layers. Crystal consists of air atoms cut from a background dielectric; the forbidden gap is indicated. (b) The same PBG crystal with a single acceptor defect made by cutting a slice through the dielectric material in one unit cell; the acceptor yields modes near the mid-gap. (c) The same PBG crystal with a single donor defect consisting of a dielectric sphere; the donor yields modes near the band edge. [Taken from Fig. 3 of E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 67, 3380 (1991).]

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For the sample with an extra dielectric sphere in the middle of an airhole, multiple modes were observed; these modes appeared close to the upper frequency edge of the gap since the defect represented a relatively weak perturbation to the system. These modes were hence labeled as shallow donor modes. A theoretical discussion of PBG defects and the analogies with the more familiar case of doped semiconductors is given in Section III.4.c. A second technique for measuring PBG photonic dispersion is the waveguide method used by McCall et ul.1s,42to look at 2D photonic band structures, both with and without single defects. For frequencies in the range 6 to 20 GHz, they used a parallel-plate metallic waveguide with plate separation of 1 cm and plate dimensions of 46 X 51 cm. At each end of the chamber, which contained the PBG sample, 8- to 12-GHz waveguide fittings allowed injection and detection of microwaves. The waveguide system contained a single TEM mode and was essentially 2D since the electric field for such frequencies was nearly constant along the transverse direction. Their PBG samples were arranged in square or triangular lattices consisting of 1-cm-long low-loss dielectric rods. The rods (about 900) were placed in the form of a finite lattice in a precision-drilled foam template. The rods were positioned in such a way that the electric field was polarized parallel to the rods (TM: transverse magnetic mode) and the radiation propagated in the plane perpendicular to the rods. They were able to sweep the microwave frequency and make measurements of the power transmitted through the sample. Removing a rod introduced a defect (vacancy) into the lattice. Their 2D result.^'^,^^ are similar to the 3D data of Meade et d3*They managed to map out the position-dependent fields in the resulting standing-wave structure (i.e., defect modes) by coupling them to a noninvasive probe through a lattice of small holes in the upper plate. They found experimentally large attenuation consistent with the presence of a photonic band gap in a various directions. They also found localized defect modes with a decay length of the order of a lattice constant.42 A third technique of measuring PBG material properties is the so-called COMITS (coherent microwave transient spectroscopy) technique emThis technique involves the generation of ployed by Robertson et ~

42

1

.

~

~

3

~

~

D. R. Smith, R. Dalichaouch. N, Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, 1. Opf.SOC.Am. B 10, 314 (1993); D. R. Smith, S. Schultz, S. L. McCall, and P. M. Platzmann, J . Mod. Opt. 41, 395 (1994). 43 W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 68, 2023 (1992). 44 W. M. Robertson, G. Aqavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, J . Opt. SOC. Am. B 10, 322 (1993).


163

picosecond electromagnetic transients via optoelectronically pulsed antennas. The technique permits investigation of the phase of the transmitted signal and hence mapping out of the photonic dispersion relations. Measurements can be made over a broad frequency range (15 to 140 GHz) and the technique is sensitive to polarization. It has been used by Robertson to investigate both bulk and surface state dispersion in PBG samples. The setup has a coplanar stripline antenna transmitter and receiver placed either side of a sample in air. A dc bias is used to generate a short optical pulse ( 2 ps), which triggers a short current pulse in the transmitter; this current pulse travels down the stripline and then radiates into the air. The radiation pulse is focused toward the sample using a hemispherical lens. The transmitted pulse is then picked up by the transmitter on the other side of the sample. The Fourier transform of the time-dependent signal at the receiver gives the transmitted amplitude as a function of frequency. The ratio of the Fourier transformed signals both with and without the sample gives the phase information. Robertson et a1.43,44investigated two 2D PBG materials made, respectively, of long, 0.37-mm radius, ceramic rods and long, 0.75-mm radius cylindrical holes in a 3.6 refractive index background dielectric. Samples with holes were investigated for both square and triangular lattices. Figure 2 shows an example of their results.43 Although the finite time window of only 200 ps limited the spectral resolution of the signal, some evidence of gaps could still be seen. The agreement between these experimental results and theory is discussed in Section 111.5. Robertson et al.4s also investigated nonradiative surface modes, exponentially decaying away from the surface, at the boundary surface of terminated PBG materials. Their setup used a phase-matching prism to couple microwave radiation to the surface mode; this enabled coupling of the radiation to the surface modes, which, by themselves, could not radiate into the crystal because they lay in the band gap, nor could they radiate into air because of the requirement of energy and wavevector conservation. For a 2D PBG sample of dielectric rods terminated by hemispherical rods in a square array, Robertson et al. found surface modes; however, no such modes were observed for samples of arrays terminated by full cylindrical rods. Finally we mention a recently introduced technique for fabricating 3D PBG materials due to Ozbay ef al.46 They employed the ordered stacking of micromachined (110) silicon wafers in order to build a periodic struc-

-

45 W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Opt. Lett. 18, 528 (1993). 4h E. Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).

164


0.0

4.2

8.4

12.6

WAVEVECTOR ( c m l )

16.8

FIG. 2. Comparison of experimental and theoretical (plane-wave expansion) photonic band structure for a 2D PBG crystal consisting of a square lattice of alumina-ceramic cylindrical rods with e = 8.9. Cylinder radius is 0 . 2 ~where a is the crystal lattice constant. Propagation of electromagnetic waves is along the (10) direction. Electric-field polarization is parallel to the rod axis in (a) and perpendicular in (b). Solid dots are experimental COMITS (coherent microwave transient spectrum) results; dashed lines are theoretical (plane-wave expansion) results. [Taken from Fig. 3 of W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 68, 2023 (1992).]

ture that had been predicted t h e ~ r e t i c a l l yto ~ ~form a full 3D band gap. They demonstrated a full band gap in this structure centered at 100 GHz with a window of about 40 GHz. Their fabrication technique employed the anisotropic etching of silicon by aqueous potassium hydroxide, which etches the (110) planes of silicon very rapidly, leaving the (111) planes relatively untouched. Hence using (110koriented silicon, they could etch arrays of parallel rods into wafers, and then stack these to make the desired structure. The silicon wafers were each 3 in. in diameter and 390 p m thick. High-resistivity wafers were chosen to minimize absorption losses in the silicon. These authors suggested46 that such a technique could also be used to construct PBG materials in the optical frequency regime. 41

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid. State Commun. 89, 413 (1994).


165

We note that a combination of electron-beam lithography and reactive ion-beam etching has recently been used to fabricate a 2D periodic structure in GaAs/GaAlAs with features on the 50-nm scale.48

3. APPLICATIONS Many applications have been proposed for photonic band structure materials, based on the modification of the free-space photon dispersion relation. The two main applications that have most caught the interest of workers in the field are the suppression of the electron-hole recombination process in semiconductor devices, thereby removing a major form of loss of efficiency, and the possibility of constructing single-mode cavities via localized photon modes in PBG materials with defects. In this section we look at these two applications in more detail, before discussing briefly other suggested uses for PBG materials that have appeared in the literature. a. Suppression of Electron-Hole Recombination One of the most interesting applications of a full photonic gap in PBG materials is to alter the radiative recombination rate of electrons and holes in a semiconductor. The importance of this application lies in the fact that such processes often lead to energy loss, noise, and lowering of the characteristic operating speed in semiconductor devices. In heterojunction bipolar transistors, for example, there are certain regions of the transistor current -voltage characteristic where the gain is determined by such electron-hole recombination processes. In solar cells, recombination limits the maximum output voltage. Yablonovitch has suggested an application in the field of laser diodes4’ If a correlated flow of electrons were to be used to drive a laser diode, there would be a constant flux of output photons; this phenomenon is known as photon-number-state squeezing and yields a low bit-error rate for optical communication. In practice, laser diode efficiency is limited by random spontaneous emission in all directions. Employing a PBG material with a full band gap eliminates unwanted electromagnetic modes that would otherwise increase the bit-error rate; the photon squeezing effect would therefore be enhanced. The simplest way of seeing the effect of a PBG material on such electron-hole recombination is from Fermi’s golden The rate of downward transition between filled and empty levels, such as in an atom, is 4xP. L. Gourley, J. R. Wendt, G. A. Vawter, T. M. Brennan, and B. E. Hammons, Appl. Phys. Leti. 64, 687 (1994). 4Y E. A. Hinds, Advances in Atomic, Molecular and Optical Physics, Vol. 28, p. 237, Academic, New York (1991).

166


proportional to I(flVli)l%(E);here ( f l V l i ) is the matrix element of the interaction V coupling the atom to the field, between initial and final The interaction V in the Coulomb gauge is states i and f of the sy~tern.~’ essentially just A p where A is the vector potential and p is the momentum operator; p ( E ) is the density of final states per unit energy. Altering the dispersion relation from its free-space form ( w = c k ) affects the density of photon states per unit frequency and hence p ( E ) . In free space there are modes at every frequency (i.e. the spectrum is continuous), and hence the probability of finding a system such as an atom or semiconductor in a given excited state (exciton) will decay exponentially with time. If the only modes present are evanescent due to the presence of a PBG, then the atom becomes “dressed” by a photonic cloud, and the resulting electronic properties of the atom differ from those in free space. The only decay channels now available to the electron-hole pair (exciton) are nonradiative, such as phonon emission or higher order photon processes. b. Single-Mode Cavities In atomic physics there has been a recent surge of interest in what is called cavity quantum electrodynamics?’ The basic idea is that the boundary conditions imposed by a finite cavity can modify the photon spectrum in such a way that only discrete mode frequencies are available, just like a particle-in-a-box for an electron. The radiation properties of an atom in a cavity depend on the availability of such modes at the frequency of recombination In practice, cavities with no losses have been difficult to build. In particular, metal cavities, which work well at microwave frequencies, become too lossy at optical frequencies. In contrast, PBG materials with full band gaps have an advantage; since they are made of low-loss dielectrics, they can in principle work well at optical frequencies. Optical devices such as lasers often require a single well-defined photon mode to operate. Such defect modes can be introduced into a PBG material by creating a defect in an otherwise perfect PBG crystal. According to the specific defect design, a defect mode can be pulled out of the continuum to form a localized state in the gap; the electromagnetic fields for this defect mode are localized around the defect and decay exponentially with increasing distance from the defect. The decay distance can be chosen to be of the order of the lattice spacing by placing the defect mode near the center of the gap. The dependence of the defect mode frequency on the characteristics of the defect is further discussed in Sections III.4.c and 111.5.

.’”

‘“P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).


167

The defects in PBGs can be thought of as higher dimensional versions of a quarter-wavelength slab at the center of a Fabry-Perot etalon. The only losses in such defect modes lying in a true photonic gap are due to absorption of the dielectric material itself. The Q factor measures the extent of such losses and the coupling of the mode itself to the surroundings. It is given by the number of optical cycles needed for the radiation field energy in the cavity to decay by a factor e-'. The Q factor of the defect modes in doped PBGs materials can be made very large by choosing low-loss dielectrics. At the same time, the frequency range can, in principle, be made large enough by finding a combination of dielectric materials yielding a sufficiently large photonic band gap. It has been suggested that reliable and thresholdless light-emitting diodes (LEDs) could be made from such high-Q single-mode PBG microcavitie~.~' Just as with conventional LEDs, a voltage can be applied to a PBG material. Typical current commercial laser diodes can have several hundred longitudinal modes within the spectral bandwidth of the gain medium, yielding a low quantum efficiency of much less than 1% (the quantum efficiency is given by the ratio of output to input powers). Single-mode operation in such a situation results from mode competition and gain saturation. Although Fabry-Perot microresonators have the advantage that there is only one high-Q longitudinal mode, the open-sided cavity couples the atoms to free-space modes; the resulting quantum efficiency is still less than 1%. The potential advantages of high-Q PBG single-mode LEDs are discussed more fully in Refs. 41 and 6. c. Other Applications There have been many other suggestions for devices based on PBG materials. The possible applications, and hence the level of fabrication precision required, depend very much on the frequency range and hence operating wavelength of the required device. Millimeter detectors can therefore be built fairly cheaply because of their relatively macroscopic periodicity (millimeters). On the contrary, optical devices will need fabrication techniques that are accurate on the scale of microns. As alluded to earlier, PBG materials can provide a new electromagnetic environment for atomic and mesoscopic physics. Mossberg and Lewenstein5' have shown that a PBG material's full band gap can drastically alter the properties of single-atom resonance fluorescence. For example, when the radiative emission on one or both of the strong-field resonance fluorescence sidebands is suppressed using a photonic band gap, the atom 51

T. W. Mossberg and M. Lewenstein, J . Opt. Soc. Am. B 10, 340 (1993).

168


behaves like a perfect classical dipole and is the source of coherent monochromatic radiation that is insensitive to fluctuations in the drivingfield intensity. Photon antibunching, a fundamental aspect of atomic fluorescence in free space, does not occur in the PBG frequency range. Kurizki et al.52 considered the effect on the physics of an atom of a localized mode arising from a defect in a PBG material; they show the existence of new quantum-electrodynamic effects in resonant field-atom interaction, owing to the spatially modulated standing-wave character of the field. In particular, they use a two-level atom interacting with the quantized field of these defects to show the existence of oscillatory patterns of the atomic population inversion, fluorescence spectra, and nonclassical field states. Dowling and BowdenS3 considered theoretically the behavior of two similar dipoles, with almost the same frequency, that radiate in frequency near a photonic band edge such that one of the dipoles was in the gap, while the other was in the band. They showed that although the dipole in the gap could not radiate directly, its properties could be probed via beats in the output power of the dipole lying in the band. De Martini et af.54have investigated both theoretically and experimentally spontaneous and stimulated emission in the thresholdless microlaser, which is based on a PBG material environment. Chu and Ho55 studied the spontaneous emission from electron-hole pairs in cylindrical dielectric waveguides and the spontaneous emission factor of microcavity ring lasers. They conclude that microlasers based on strongly guided single-mode dielectric waveguides are promising devices for achieving high . ~ experimen~ efficiencies and low lasing thresholds. Erdogan et ~ 1 studied tally and theoretically the enhancement and inhibition of radiation in ~~ theocylindrically symmetric, periodic structures. Bullock et u I . showed retically the possibility of using 2D PBG materials as 2D Bragg reflector mirrors for semiconductor laser-mode control. Dowling et have noted that the zero photon group velocity at the band edge of a 1D PBG corresponds to a very long optical path length; in the presence of an active 52G.Kurizki, B. Sherman, and A. Kadyshevitch, J . Opt. Soc. Am. B 10, 346 (1993); B. Sherman, G. Kurizki, and A. Kadyshevitch, Phys. Reu. Lett. 69, 1927 (1992); A. G. Kofman, G . Kurizki, and B. Sherman, J . Mod. Opt. 41, 353 (1994). 53J. P. Dowling and C. M. Bowden, J . Opt. SOC. Am. B 10, 353 (1993). 54 F. De Martini, M. Marrocco, P. Mataloni, and D. Murra, J . Opt. SOC. Am B 10, 360 (1993). 55D. Y.Chu and S. T. Ho, J . Opt. SOC.Am. B 10, 381 (1993). 56T. Erdogan, K. G. Sullivan, and D. G. Hall, J . Opt. SOC. Am. B 10, 391 (1993). 57 D. L. Bullock, C. C. Shih, and R. S. Margulies, J . Opt. SOC.Am. B 10, 399 (1993). 58 J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, J . Appl. Phys. 75, 1896 (1994). See also J. P. Dowling and C. M. Bowden, J . Mod. Opt. 41, 345 (1994); M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J . Appl. Phys. 76, 2023 (1994).


169

medium this increase in path length could yield a large gain, suggesting possible application in vertical-cavity surface-emitting lasers. Control of signal propagation in millimeter and microwave integrated circuits is a further application of PBG materials. The photonic gap is chosen so as to reduce the power lost by emission into the dielectric investisubstrate, and hence enhance the emission into air. Brown et gated the radiation properties of a planar bow-tie-shaped antenna based on a PBG material. They concluded that highly efficient planar antennas could be made on PBG material regions fabricated in semiconductor substrates such as GaAs. A related application is that of shielding in Y junctions in integrated optical circuits, in order to reduce leakage of radiation. Delay lines have also been suggested. The possibility of constructing highly anisotropic 3D photonic band structures has provided the motivation for investigations of highly directional radiation for imaging. Nonlinear effects in dielectric materials could be enhanced by PBG materials. By suppressing one-photon events, higher order multiphoton effects, ordinarily swamped by the one-photon effects, might be observable. In particular, two-photon lasers could be built operating at a frequency w / 2 , where w lies in the photonic band gap. These lasers have been predicted to have properties as diverse as frequency tuning, emission of squeezed states, and short-pulse generation.' Disordered PBG materials can also modify the atomic radiative dynamics. In particular, a small amount of disorder can induce localized modes in the gap; the experimental results discussed in Section 11.2 for a single defect represent the extreme limit of such localization. The analogous behavior for electrons is that of Anderson localization. For electrons, however, the presence of electron-phonon scattering and electron-electron interactions hinder experimental observation of the effect. For photons in a disordered PBG, the optical version should be observable, as first pointed out and developed subsequently by J ~ h n . ' ~ This , ~ ~ is? particularly ~' likely because photon-photon interaction and photon-phonon effects are negligible, and careful engineering should be able to control the exact disorder introduced into the PBG. Experimental work has included studies at optical frequencies in 3D random arrangements of scatterers. In a 2D disordered array of rods, localized states have been observed.62As disorder is introduced into a perfect PBG crystal, photonic band gaps will be replaced by pseudo-gaps with a small, finite density of states. Like their E. R. Brown, C. D. Parker, and E. Yablonovitch, J . Opt. Soc. Am. B 10, 404 (1993). S. John, Physics Today, p. 32 (May 1991); Physica B 175, 87 (1991). hl S. John and J. Wang, Phys. Rev. Letr. 64, 2418 (1990); Phys. Rev. B 43, 12,772 (1991). h2 R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature 354, 53 (1991). 59

60

170


perfect crystal counterparts, disordered PBG materials can also therefore reduce the rate of spontaneous emission in semiconductors at optical frequencies. Genack and Garcia63 have suggested a new class of compact filters, transducers, and switches based on the characteristics of localized waves in nonabsorbing media. These characteristics include narrow resonances, giant transmission fluctuations, strong spatial correlation, and extreme sensitivity to variations in the dielectric function of the medium. More recently, the localization of light in 1D Fibonacci dielectric multilayers has been in~estigated.~~

111. Theoretical Background

4. PHOTONIC BANDTHEORY: FORMALISM In this section we provide a vector formalism for photonic properties in PBG materials that parallels the well-known scalar formalism for electronic properties in s0lids.2,~'In particular, we develop a vector representation theory employing various basis sets whose scalar analogs have proved useful in electronic systems. To date, most theoretical calculations of photonic properties in PBG materials have employed only a plane-wave basis (i.e., plane-wave representation). However, given the success of other representations in electronic systems,' it is worth investigating their photonic, vector analogs with the hope that these will yield additional insight into PBG properties. In particular we develop the crystal momentum representation (CMR), the crystal coordinate representation (CCR), and the effective mass representation (EMR). For a perfect PBG crystal, these representations are shown to lead to the vector analog of the well-known k p theory for electronic systems; the electronic k * p theory has proved to be an invaluable tool for describing the dispersion relation of electrons near an electronic band gap in semiconductor^.^ In addition, a tight-binding description of photonic band structures is obtained. For a PBG crystal plus defect, the representations lead to a vector analog of the electronic effectiveness-mass equation for the case when the defect represents a slowly varying perturbation. When the defect represents a well-localized perturbation, a generalization of the electronic Koster-Slater impurity model2 is obtained. 63

A. 2. Genack and N. Garcia. J . Opt. SOC. Am. B 10, 408 (1993).

W. Gellerman, M. Kohmoto, B. Sutherland, and P. C. Taylor, Phys. Reu. Lett. 72, 633 (1994). 64

171


a. Wave Equations For electromagnetic waves propagating in a medium characterized by a spatially dependent dielectric constant E(r) and magnetic permeability p(r), the electric and magnetic fields satisfy the wave equations. 1 V x -V p(r)

X

1 d2 E(r, t ) + - ,.E(r)E(r, c 2 dt

t)

=

0

(4.1)

and 1 V x -V E (r)

1 d2 c2 d P

x H(r, t ) + - - (r)H(r, t ) = 0.

(4.2)

We are considering, for simplicity, systems where E and p are scalar functions of position. The extension to anisotropic systems, with a tensorial dielectric constant and magnetic permeability, is straightforward. Consider a monochromatic wave of frequency w with e-'"' time dependence; the wave equations become 1 V x -V p(r)

x E(r)

-

w2

7E(r)E(r) C

=

0

(4.3)

=

0.

(4.4)

and V

X

1 -V E (r)

X

w2

H(r) - yCp ( r ) H ( r )

These equations are of identical form and one can be obtained from the other by interchanging the roles of (E, E ) and (H, p). Solving either the E equation [Eq. (4.3)l or the H equation [Eq. (4.411 exactly will lead to the same physical results. The E equation and the H equation are similar in form to the envelopefunction equation for electrons in semiconductor heterostructures within the effective mass approximation.2 In the H equation, E(r) plays the role of a spatially dependent effective mass, u$(r)/c2 plays the role of kinetic energy, and hence p(r) is similar to a spatially dependent potential. In the E equation, p(r) plays the role of effective mass, w2E(r)/c2 plays the role of kinetic energy, and hence E(r) is now analogous to a spatially dependent potential. Further details on the analogy between the photonic and elecThere are, tronic problems have been discussed by Henderson et however, fundamental differences. Photonic problems deal with vector

172


fields. Charge and the Pauli exclusion principle do not enter into the problem because photons are chargeless, spin-one entities. In addition, the photon-photon interaction is not important, in contrast to the situation in solids where Coulomb interactions can be significant. In PBG materials, the dielectric constant is periodic and obeys

where R is any lattice vector. The PBG materials considered to date have been nonmagnetic, hence p = 1 everywhere. The E and H equations now become V

X

V

X

w2

E(r) - x E ( r ) E ( r ) C

=

0

(4.6)

=

0.

(4.7)

and 1 V x -V 4r)

X

w2

H(r) - x H ( r ) C

Equations (4.6) and (4.7) form the starting point for many of the reported photonic band structure calculations in PBG materials. We note that it is also possible" to start with an equivalent equation, similar to Eqs. (4.6) and (4.71, for the displacement field D(r). As is discussed in Section 111.5, some of the earlier 3D PBG calculations used a scalar wave equation of the form.

The scalar wave equation has the advantage of being computationally easier to solve. However, it can lead to qualitatively incorrect results for 3D photonic band structures. In contrast, for lower spatial dimensions, a scalar wave equation can be exact. In particular, for 2D PBG crystals, the vector wave equations for the TE (transverse electric) and TM (transverse magnetic) polarizations can be exactly reduced to equivalent scalar wave eq~ations.''-~~ In this section, we will keep the formalism as general as possible by dealing directly with the exact vector wave equations. 65

T. K. Gaylord, G. N. Henderson, and E. N. Glytsis, J . Opt. SOC. Am. B 10, 333 (1993). N. F. Johnson, P. M. Hui, and K. H. Luk, Solid Stare Commun. 90, 229 (1994). 67P.M. Hui, W. M. Lee, and N. F. Johnson, Solid State Commun. 91, 65 (1994). 66


173

b. Representation Theory Bloch Functions The eigenfunctions of Eqs. (4.6) and (4.7) satisfy Floquet's therorem and are Bloch functions of the form

(4.10) where n is a band index and k lies within the first Brillouin zone (BZ). The functions u;(k, r) and uy(k, r) are periodic in r with the same periodicity as &), and C! is the volume of the crystal. The corresponding eigenvalues are oik,yielding the band structure or the dispersion relation. The Bloch functions form the basis set for the crystal momentum representation (CMR). The orthogonality relations are

and

where the integral is over the whole volume a. Note the presence of the weight function E(r) in the orthogonality relation for the eigenfunctions of the E equation. The orthogonality relation for the H Bloch functions should similarly be weighted by ,u(r), which has been taken to be unity throughout the system. Plane- Wave Expansion Most photonic band structure calculations are based on the plane-wave expansion (PWE) method. Since u(k,r) is periodic, it can be expanded in terms of plane waves with wave vectors G being reciprocal lattice vectors. Thus, E(r) and H(r) can be expanded as

(4.13) Hk(r) = ~ H k ( G ) e ' ( k + G ) ' r . G

(4.14)

174


The periodic dielectric constant E(r) can be expanded in a similar fashion as (4.15) The Fourier coefficients E(G) can be chosen to be real and expressed as (4.16) where the integration is over the volume u, of a unit cell. In some systems, such as isolated spheres of one dielectric in another, E ( G )can be calculated analytically; in other systems it has to be calculated numerically.' Substituting Eqs. (4.13) and (4.14) into the original E and H wave equations yields matrix equations for the Fourier coefficients E,(G) and H

and v(G - G')(k

+ G ) x H,(G')

w2

=

0, (4.18)

where v(G) are the Fourier coefficients of the inverse of the dielectric constant, that is, of the function v(r) = l/E(r). For each k, Eqs. (4.17) and (4.18) each represents an infinite-dimensional matrix equation. Diagonalizing the matrix gives infinitely many w,(k), each of which is labeled by a band index n. Repeating the calculation for different wave vectors gives the band structure. In principle, solving either Eq. (4.17) or (4.18) gives identical results. However, the infinte-dimensional matrix problem must be truncated in practice. After truncation, it is not generally the case that the E and H equations give identical results, even when the same finite number of plane waves is used. Equation (4.18) is a standard eigenvalue problem and is often used for photonic band calculations. Two methods have been used to obtain the Fourier coefficients v(G - G I . They can be obtained by Fourier transforming the function l / E ( r ) directly or by inverting the matrix E(G - G') obtained by Fourier transforming the function 4r). It has been found that the latter approach leads to a faster conver-


175

gence of the eigenvalues than direct transformation of l/E(r), and that the eigenvalues obtained are close to those calculated from the E equation when the same finite number or plane waves is used.8 Implementation of PWE methods in practice is discussed further in Sec. 111.5. Equations (4.17) and (4.18) are the general matrix equations for PBG materials. The size of the H equation matrix, for instance, can be reduced by noting that V . H = 0. Equation (4.14) implies that H,(G) can be regarded therefore as having only two components, each of which is orthogonal to k + G. Thus, H,(G) can be expressed as

where Zk, (G) is a unit vector and {k + G, ;,,(GI, zk2(G))forms a triad. Similar reduction in the size of the E equation matrix results from noting that V . V X E = 0. These equations may be further reduced to simpler forms in some special cases, for example, in 2D systems in which an ordered array of parallel rods of one dielectric is embedded in a second dielectric. In this case, the size of the matrix can be reduced by decoupling the fields into two orthogonal polarizations. Of particular interest in such a 2D PBG material is the TM polarization where the magnetic field is transverse to the rods (and E is parallel to the rods) and the TE polarization where the electric field is transverse to the rods (and H is parallel to the rods). These two situations are discussed further in Sec. 111.5. Wannier Functions Whereas the Bloch functions give a k-space description, a real space description can be achieved by defining Wannier functions. Suppose that the Bloch functions Hnk(r) of the H equation are known; then the Wannier functions can be defined in terms of the set of Bloch functions within a band n as (4.20)

where the sum is over k s within the first BZ and N is the number of unit cells. Because there are N allowed k values in the first BZ, Eq. (4.20) simply defines N Wannier functions associated with the nth band by forming suitable linear combinations of N Bloch functions within the band. These functions are localized around the lattice point labeled by R.

176


The orthogonality relation is

where the integration is over the crystal volume R. The Wannier functions form a complete set for all n and R. They form the basis set for the crystal coordinate representation (CCR), which is useful in describing the effects of impurities. For ordered PBG materials, the Wannier functions can be used to formulate a tight-binding description of the band structure. Similar constructions can be carried out for the Bloch functions of the E equation, leading to Wannier functions E,(r - R), which are localized around the site R. The orthogonality relation for E,(r - R) also carries the weight function 4 r ) as in Eq. (4.11). Kohn-Luttinger Functions and k .p formalism The Kohn-Luttinger functions@ xnk(r)of the H equation can be defined in an analogous way to those in the electronic problems as

(4.22) where k, is some fked wavevector within the BZ. The Kohn-Luttinger functions satisfy the orthogonality relation (4.23) Kohn-Luttinger functions can be similarly constructed for the E equation. To set up a k p formulation for photonic bands, we can start with either the H equation or the E equation. We will work with the H equation. We expand H(r) in Eq. (4.7) in terms of Kohn-Luttinger functions

-

(4.24) where the sum is, in principle, over all bands. The expression of eigenfunctions in terms of the Kohn-Luttinger functions is usually referred to as the efectiue mass representation (EMR). A set of equations for the expansion coefficients anj(k) can be obtained by substituting Eq. (4.24) into the wave 6M

J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).

177


equation [Eq. (4.711 and using the orthogonality relation Eq. (4.23). The resulting equation is

[2( 1

I

wj)[) - W i k ) 6 / j - p/,(k)]a.,(k) =

(4.25)

O,

where

Here s = k - ko is the deviation from the wavevector k o . Two types of terms are involved: The first few terms are linear in s = Isl; the last two terms are quadratic in s. The s . p term is analogous to the k p term in the electronic problem in that p(rj, involves integrals over a unit cell of volume u, of two u's and a differential operator,

-

Owing to the vector nature of the problem, p(,j, is more complicated than the momentum matrix element for the electronic problem, which is proportional to /@Vuj dr, where the u's are scalar functions in the electronic case. The simplifying feature is that the whole integral can be treated as a parameter as in the electronic problem, and can either be obtained by fitting to band structures calculated using numerical methods, o,' by fitting to available experimental data. The a@element of the matrix Q(rj,is given by

where [u], is the a component of u. In Eq. (4.261, the matrix and

6

q(/j) = TrG(/j)*

GT is the transpose of (4.29)

178


The k * p formalism is exact up to this point. Solving for the photonic band structure mik in Eq. (4.25) amounts to finding the eigenvalues of a matrix H with elements Hlj given by

In principle, the band indices 1 and j run over all bands and the matrix is of infinite dimension. In practice, the matrix will be truncated to sizes that can be handled either analytically or numerically without performing heavy computations. As is shown in Section 111.5, analytically solvable two- or three-band models are often sufficient to describe the dispersion of the bands accurately if one is only interested in the dispersion around k, . It is an advantage of the k . p method that, after truncation, only a few parameters are needed to characterize the band structure in the region of interest around the photonic band gap. The dispersion relation of a particular band n around k, can be obtained using perturbation theory. In this case, k = k, and thus s is small. Keeping terms to second order in s, we have

(4.31)

where

Motivated by the form of the dispersion relation in a uniform medium w 2 = c 2 k 2 /(€PI, we define 1

[GF].,

=

1 a2(m,2k) 2c2 aka ak, '

(4.33)

a quantity related to the reciprocal of the refractive index. For nonmagnetic materials, we have p = 1 everywhere, and Eq. (4.33) defines a reciprocal effective dielectric tensor for the material. Using Eq. (4.31) for

179


w : ~ ,we have

which is analogous to the f-sum rule for the reciprocal effective mass tensor in the electronic problem.* A similar k . p approach can be formulate starting with the E equation. Results can be easily obtained from those of the H equation by noting the identical forms of the original wave equations [Eqs. (4.3) and (4.411. We note that this photon k * p formulation may be used for studying “dielectric superlattices” consisting of alternating layers of different periodic dielectric crystals and “dielectric heterostructures” in a way that is analogous to the k * p envelope-function approach employed in treating semiconductor h e t e r o ~ t r u c t u r e-. ~”~ c. Defects As in electronic problems, the introduction of defects and impurities in PBG materials leads to defect states (modes) in the gap. These states behave as donor or acceptor states depending on the characteristics of the defect. As mentioned in Section 11.3, such defect modes are useful because many optical devices require some well-defined photon modes to operate. Numerical calculations of such modes are further discussed in Section 111.5. The disorder of a dielectric structure can be described by a positiondependent dielectric function E(r). This function can be written E(r)

=

E(r)[l

+ V(r)l,

(4.35)

where 4 r ) is periodic in real space and E(r)V(r) describes the deviation from the periodic d r ) at any point r. The detailed mechanisms of the disorder is not important here; it could be positional disorder, a missing or additional dielectric object, shape or size irregularities, or a periodic structure consisting of two or more kinds of dielectric embedded in a host (i.e., a PBG alloy). Substituting Eq. (4.35) into the H equation yields V

X

1

-V E(r)

X H(r)

+ V X U(r>V X H(r)

w2

-

TH(r) C

=

0, (4.36)

N. F. Johnson, H. Ehrenreich, P. M. Hui, and P. M. Young, Phys. Reu. E 41,3655 (1990). N. F. Johnson, P. M. Hui, and H. Ehrenreich, Phys. Rev. Lett. 61, 1993 (1988). ” N. F. Johnson, Ph.D. Thesis, Harvard University (1989). 69

711

180


where (4.37) Without the second term, this is the wave equation for an ordered PBG material from which the Bloch functions, Wannier functions, and Kohn-Luttinger functions can be constructed (see Section III.4.b). A general method to solve the problem is to expand H(r) in terms of some complete set of basis functions, such as Bloch functions, Kohn-Luttinger functions, or Wannier functions leading to the CMR, EMR, and CCR representations, respectively. Crystal Momentum Representation of Bloch functions as

We expand H(r) in Eq. (4.36) in terms

(4.38)

where A , , =,,(nklH> are the expansion coefficients and thus the wave function in the CMR. Substituting into Eq. (4.36) leads to

(4.39) which is the CMR equation for a defect in a PBG material. Effective Mass Representation Luttinger functions as

We expand H(r) in terms of the Kohn-

(4.40) where the C,, are the expansion coefficients and hence the wave function in the EMR. Substituting Eq. (4.40) into Eq. (4.36) gives the EMR

181


equation for a defect in a PBG material:

where Pnn,is given by Eq. (4.26). In the ordered case, the last term in Eq. (4.41) vanishes and the equation reduces to the k . p formalism of Sec. 111. 4.b. Cystal Coordinate Representation We expand H(r) in terms of Wannier functions as

(4.42) where BnR=,(nRIH) are the expansion coefficients and hence the wave functions in the CCR. Substituting in Eq. (4.36) gives the CCR equation for a defect in a PBG material:

+ c [/H:(r n' R'

-

R) . V

X

U(r)V

X

1

H,,(r - R') dr BntR,= 0, (4.43)

where we have defined the function (4.44) Without the disorder term (U = 0), this CCR formulation in Eq. (4.43) amounts to a tight-binding approach based on the Wannier functions centered around each lattice point. The photonic dispersion relation can hence be regarded as resulting from overlaps of neighboring Wannier functions. In principle, the hopping integrals can be calculated from first

182


principles. An empirical approach would be to fit the dispersion relation obtained by numerical methods, such as PWE, to the tight-binding form

where a i , y i , and Si are the tight-binding parameters for the ith band. The parameter a describes the position of the band, and y and 6 are the nearest neighbor (nn) and next nearest neighbor (nnn) overlap integrals, respectively. Empirical formulas for these parameters as a function of the system variables would provide a valuable tool for gaining insight into photonic band structures over a wide range of PBG materials, without going through extensive computations. Such an approach is similar to the empirical tight-binding method developed in solid state physi~s.’~In Sec. 111.5, we will show that such an approach can yield highly accurate photonic band structures. For the case of a slowly varying defect perturbation, the CCR equation [Eq. (4.43)] can be used to formulate an envelope-function approach that is analogous to the effective mass approximation treatment of shallow impurities in a semiconductor. Consider Eq. (4.43); for a slowly varying perturbation U (r) (i.e., shallow impurity), Bloch states in the vicinity of the extremum of band n’ will become mixed just as in the electronic problem. Expanding the dispersion relation around the extremum at k , in terms of the x, y, and z components of the wavevector k yields

where A,,, are general coefficients. Using standard summation relations for plane waves2 it follows from Eq. (4.44) that

, where w;,(k’ -+ -iVRr + k,) denotes the dispersion relation o ; , ~from Eq. (4.46) with k’ replaced by -iVR, + k , . The vector operator VR, corresponds to differentiation with respect to the components of R . The 12

W. A. Harrison, Electronic Structure and the Properties of Solids, W. H. Freeman and Company, San Francisco (1980).

183


CCR equation can now be rewritten as

+

nR

[

/H:,(r - R’) * V

X

U(r)V

X

I

H,(r - R) dr B,,

=

0, (4.48)

and is still exact. The integral in the last term can be rewritten as73 M,,,z(R’,R) = / [ V x H,,(r - R’)]* U(r)[V x H,(r - R)] dr. (4.49)

We now assume that U(r) varies slowly over a unit cell and over the spatial extent of the Wannier function. Since V X H,(r - R) is expected to be localized around the site R, following the behavior of H,(r - R), the overlap between the curl of Wannier functions centered around different sites will be small. The perturbation U(r) can be taken to be constant over the region of space for which the integral is appreciable. Assuming that the coupling between different bands is small, the approximation M,,,(R’,R) = 8,,t8RR,AntU(R) can be made, where d,,is an integral defined as in Eq. (4.49) but with U(r) set to unity, n = n’, and R‘ = R. The one-band equation for a slowly varying perturbation is therefore given by

[

$wa(k’

4

-iVRt

+ k,) + U(R’)d,, -

B,,,,

=

0.

(4.50)

The analogy with the effective mass equation for semiconductors becomes clear when we replace R’ by the continuous variable r and we rename the “envelope function” B,,,, as Fnr(r).2,71 The effects of the periodic 4 ) are embedded in the dispersion relation w : ~ ,which appears in the first term. As a simple example, consider the low-frequency limit where the photonic dispersion relation of the ordered system around k, = 0 can be written as

73

K. M. Leung, J . Opt. SOC.Am. B 10,303 (1993).

184


where l/Eeff is the inverse of the effective dielectric constant. In this case, Eq. (4.50) becomes (4.52) Note that for systems without cubic isotropy the effective dielectric function Eefl will be a tensor, analogous to the effective mass tensor for electrons. Next consider the technologically interesting case where the dispersion relation of the ordered system has a finite-frequency photonic band gap at k, = 0, and the dispersion relation in the vicinity of the gap can be written as w,(k) = o,(O)+&k2.

(4.53)

Squaring w,(k) and retaining terms up to order k 2 , Eq. (4.50) now becomes

We note that this envelope-function formulation can also be obtained starting from the equivalent CMR and EMR equations, respectively. A convenient way to study the impurity problem is by means of Green’s functions. A Green’s function can be defined for the nth band c2

GAo)(R’- R, o) = - C N

&k.(R‘-R) W’

(4.55)

-

The CCR equation can then be rewritten in terms of Green’s functions as

X

[IHz,(r - R’) V *

X

U(r)V

X

H,(r

-

1

R) dr BnR = 0, (4.56)

which is still exact. The Green’s function method will be particularly useful when coupled with the empirical tight-binding approach because G(O) can then usually be expressed in terms of complete elliptical integrals if only 74G.F. Koster and J. C. Slater, Phys. Reu. 95, 1167 (1954).


185

nearest neighbor couplings are important, or obtained numerically using the tight-binding form of wnk when next nearest neighbor couplings have to be taken into account. This method amounts to taking the second term in Eq. (4.36) as the perturbation term and then constructing the total Green’s function G of the perturbed problem using the unperturbed Green’s function G‘”’. The poles of G provide information regarding the energy of the defect modes. As an illustration, we assume that the photonic bands can be treated independently. Equation (4.56) now takes the simpler form

with M given by Eq. (4.49). Consider a single impurity problem in which a region of one of the dielectrics E , in the unit cell, say, at the origin, is replaced by a third dielectric with dielectric constant e C .In a 3D (2D) system with spheres (cylinders) of one dielectric positioned within another, this can be visualized as being the replacement of the sphere (cylinder) centered at the origin by another dielectric sphere (cylinder) of the same = U within the volume of the size. In this case, U(r) = - & / ( E , E , ) replaced region, and is zero elsewhere; here 6~ = E , - E , is the difference in the dielectric constants of the impurity and the original material. Since the Wannier functions are localized around lattice sites, the integral in Eq. (4.57) can be approximated as M,,,,(R’,R ) = ~ 5 S R ~ ,U ~ jn ” ,where j,,is an integral defined in the same way as An but with the integration only carried out over the region of the replaced material. Note that 2, has implicit w dependence through Maxwell’s equations. The defect modes can then be obtained by solving the equation (4.58)

and identifying roots having frequencies outside the continuum of the band. This is reminiscent of the Koster-Slater treatment for a localized impurity in the electronic Finally we note two interesting extensions of the defect problem in PBG materials. The first concerns a single defect (impurity) with nonlinear optical properties situated in a PBG crystal. It has recently been found for the electronic case75.76that nonlinear impurities can lead to results concerning the existence of a bound state that are qualitatively different from ”M. 1. Molina and G. P. Tsironis, Phys. Reu. B 47, 15,330 (1993), Phys. Reu. Lett. 73, 363 (1994). “K. M. Ng, Y. Y. Yiu, and P. M. Hui, Solid State Comrnun. 95, 801 (1995).

186


the usual linear problem. Similarly in PBG materials, the presence of a nonlinear defect could play an important role in determining whether or not a truly localized photon mode exists in the photonic band gap. Such a defect may, for example, consist of a sphere made of a frequency-dependent dielectric. This would provide an additional degree of freedom to tune the position of localized modes. The above Green’s function formulation can be generalized to study such nonlinear impurities. The second situation concerns a PBG crystal containing many impurities. This problem becomes one that is similar to the alloy problem in solid state physics.2.77 Standard techniques, such as the average t-matrix approximation and the coherent potential a p p r o ~ i m a t i o ncan ~ ~ be applied to study this problem using the tight-binding picture78 or k . p formalism79 as a starting point.

5. THEORETICAL CALCULATIONS AND RESULTS We now turn to a discussion of the various theoretical approaches that have been employed in practice to calculate photonic properties in PBG materials. These methods are basically generalizations of techniques originally developed for electronic calculations. We also discuss the extent to which the theoretical predictions are in agreement with the results of experiments discussed in Section 11.2. As mentioned in Section 1.1, the fact that there are an infinite number of possible PBG crystals means that we cannot provide a complete catalog of all possible photonic band structures. Instead we choose to focus on a few representative theoretical calculations and compare them with the corresponding experimental results, while emphasizing trends that have emerged concerning photonic properties as a function of PBG material parameters. Many theoretical papers have attempted to solve either the exact vector wave equation (Section III.4.a) or a simplified scalar wave equation. These calculations are essentially numerical and computationally intensive. For both vector and scalar equations, the most prevalent approach to obtaining solutions has been based on a plane-wave expansion of the position-dependent fields as discussed in Section III.4.b. This approach will in principle provide exact solutions of both the vector and scalar wave equations if an infinite set of plane waves is considered; the resulting matrix to be diagonalized is infinite dimensional [Eqs. (4.17) and (4.1811. In practice, a 77 H. Ehrenreich and L. M. Schwartz, in Solid Stute Physics (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), Vol. 31, p. 149, Academic, New York (1976). 78 E. N. Economou, Green’s Functions in Quantum Physics, Springer Series in Solid Stute Sciences, Vol. 7, Springer-Verlag, Berlin (1979). 79 E. Siggia, Phys. Reu. B 10, 5147 (1974).


187

finite set of plane waves must be employed. Given a large enough computer and sufficient computing time, such a calculation would indeed yield numerical answers which converge to the exact result. For the case of the scalar wave equation, the similarity to the scalar Schroedinger equation meant that plane-wave algorithms and numerical techniques developed in standard electronic band theory calculations could be used directly. For the vector wave equation, the numerical routines had to be modified to account for the vector nature of the fields. In addition, attention had to be paid to the errors introduced as a result of the plane-wave truncation procedure. The size and origin of these truncation errors, and hence the resulting accuracy of the calculations, depend on the specific truncation procedure employed. Such errors can be more significant than in the electronic case because of possible discontinuities in, for example, the electric field at a boundary between two dielectrics; a large number of plane waves may be needed for an accurate representation of the field and convergence may be slow. Similar difficulties arise in describing the step-like discontinuity in the dielectric constant at the boundary. This general difficulty of representing a discontinuous function by a finite series of plane waves is a manifestation of Gibb's phenomenon in Fourier transform theory. Gaussian functions and high-order supergaussian functions that provide good convergence have been The results of the numerical calculation after truncation may also depend on whether one is solving for B(r), D(r), E(r), or H(r). A detailed review of PBG photonic band structure properties based on a plane-wave analysis is given by Villeneuve and Piche.8 Initially plane-wave expansions were used to solve the simpler scalar wave equation for 3D structures, as opposed to the full vector equations. The hope was that the results would be qualitatively, and perhaps quantitatively, similar to the results of a full vector theory. Examples of such a numerical calculation include Refs. 16, 19, 25, and 80. It soon became clear, however, that the scalar wave approximation" was not reliable for 3D PBG crystals." In particular, the details of the photonic band gap predicted by the scalar wave theory in 3D were often quantitatively different from those observed experimentally, for example, for the fcc structure. In fact, the scalar wave calculations predicted that many structures should be capable of exhibiting full photonic gaps in 3D,30whereas experimentally the opposite was found to happen. xu

S. John and R. Rangarajan, Phys. Reu. B 38, 10101 (1989). Born and E. Wolf, Principles of Opfics,Pergamon, Oxford (1965). 82 As noted in Section III.4.a. the vector wave eqtiation for TE and T M polarizations in a 2D PBG can be reduced t o scalar form. " M.


188

Leung and Liu2' and Zhang and Satpathy26 carried out a numerical plane-wave solution for the full vector wave equations in a 3D PBG crystal. It was thought that an fcc structure, with its nearly spherical Brillouin zone, would be the most promising candidate for a full photonic band gap. They therefore chose to examine theoretically a 3D fcc structure of spherical airholes in a background dielectric. These calculations showed degeneracies that were symmetry induced and therefore prevented a photonic band gap. Ho et aL2' turned to the 3D diamond structure, which broke the near-spherical symmetry. They found that a photonic band gap could form in this structure. In particular, for diamond structures comprising either airholes in a dielectric (refractive index 3.6) or dielectric spheres in air, they predicted full photonic band gaps of around 20% of the mid-gap frequency. Their results are summarized in Figs. 3 and 4. Ho et al. chose to consider the wave equation in H(r) rather than E(r) or D(r). The motivation was the fact that H(r) is continuous at an interface between media with different dielectric constants, whereas E(r) and D(r) are both discontinuous. Despite the discontinuity in the first derivative of H(r), the H(r) expansion is therefore expected to have a faster convergence. They then expanded in a basis of just transverse plane waves, rather than expanding in a complete basis, solving for the normal modes, and then projecting the longitudinal solutions. Subsequently, it was realized that any choice of nonspherical atoms should lift these symmetry-induced degeneracies mentioned earlier, yield0.7 0.6

0.5

z

0.4

W

3

a

LL

0.3

0.2 0.11

0.0

*

1 1 U

L

/J/ r

1 1 X

W

K

Wavevector

FIG. 3. Theoretical photonic band structure, calculated numerically using a plane-wave expansion, for a 3D diamond structure of dielectric spheres of refractive index 3.6 in an air background. A filling fraction of 34% for the dielectric implies the spheres are just touching each other. A full photonic band gap appears between the second and third bands. Frequency is in units of c/a where a is the lattice constant. [Taken from Fig. 2 of K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Reu. Lerr. 65, 3152 (19901.1

189


-

-o- air spheres

0 0.25

0.0

0.4

0.2

--c--

0

dielectric spheres

1

1.0

0.8

0.6

filling ratio dielectric spheres

air spheres

2

3

. ....c--*

4

5

6

-...

0’

7

refractive index ratio FIG. 4. Ratio of full photonic band gap width to the mid-gap frequency in a 3D PBG diamond structure, shown as a function of (a) the filling fraction and (b) refractive index ratio. Results were obtained numerically using a plane-wave expansion. The two cases of airholes in a dielectric and dielectric spheres in air are shown. The dashed line in (b) corresponds to air spheres occupying 81% of total volume; the solid line corresponds to dielectric spheres with a filling fraction of 34%. These filling fractions are chosen so as to maximize the gap. As in Fig. 3, the dielectric refractive index is 3.6 and the gap appears between the second and third bands. [Taken from Fig. 3 of K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).]

ing a full 3D photonic band gap. Yablonovith et ~ 1 then . fabricated ~ ~ such a structure with nonspherical airholes (see Section 11.2) and found experimentally a full 3D photonic band gap in agreement with Ho et d ’ s vector ’ predicted, based on plane-wave calculations, theory. Chan et ~ 1 . ~ have that the diamond structure with a refractive index contrast as low as 1.87

190


to 1 will still open up a full photonic gap. In addition, the simple cubic structure was found to produce a gap.”*R3This finding is interesting since such simple-cubic systems are likely to be easier to fabricate in 3D than those with complex unit cells. Soziier and Hauss3 reported theoretical results for both spheres and scaffold structures in 3D. As mentioned earlier, it had been thought that a spherical Brillouin zone would favor a large photonic band gap. Soziier and Haus showed, however, that the bcc (body-centered-cubic) structure, whose Brillouin zone is rounder than simple cubic, had surprisingly small gaps. By solving the vector equations in 2D using the plane-wave expansion, it was shown theoretically 32.39,R4 that the 2D hcp (hexagonal-close-packed) lattice of long parallel rods should yield a full photonic gap in the perpendicular plane. Also structures such as arrays of air rods with square cross section in a background dielectric have been predicted to give 2D photonic band gaps.39 Maradudin and McGurnR5have studied the effect of finite rod lengths on 2D photonic properties. Using the plane-wave numerical routines of Meade et al., Robertson et u1.43v44obtained good agreement between the theoretical dispersion relation and the experimental results of the COMITS measurements for 2D square and ’triangular lattices of dielectric rods. An example of this agreement between theory and experiment is shown in Fig. 2. In addition, both theory and experiment found that transmission via certain modes was forbidden by symmetry. ~ undertook a study of two complementary 2D PBG Meade et ~ 1 . ’ then structures in order to try to understand what features determine whether a given PBG material will have a large or small photonic band gap. The advantage in carrying out this theoretical investigation in a 2D, as opposed to 3D, PBG material is that the eigenmodes of the 2D system are either TM with electric field parallel to the rods, or TE with the electric field perpendicular to the rods. The polarizations in a 3D system are much harder to visualize. The separation of these modes in 2D allows independent investigation of the factors underlying the presence of a gap for each polarization. Their numerical results for the 2D dispersion relations are shown in Fig. 5. Note that the sample considered in Fig. 5(a) corresponds to the COMITS sample, which was featured in Fig. 2. Meade et ~ 1 . ’ ~ concluded that it is concentrations of dielectric material that are important for developing a gap for the TM polarization, while connectivity within the 83H. S. Soziier and J. W. Haus, J . Opt. SOC. Am. B 10, 296 (1993). 84 J. M. Gerard, A. Izrael, J . Y. Marzin, R. Padjen, and F. R. Ladan, Solid State Electron. 37, 1341 (1994). n5 A. A. Maradudin and A. R. McGurn, J . Opt. SOC. A m . B 10, 307 (1993). 86 R. D. Meade, A. M. Rappe, K. D. Brornrner, and J. D. JWdnnOpOUlOS, J . Opt. SOC.Am. B 10, 328 (1993).


191

0.8 0.7

0.6 0.s 0.4

0.3 0.2 0.1

0

0.6

0.5 0.4

0.3 0.2

0.1

0

Fic;. 5. Theoretical photonic band structure cdkulated numerically using a plane-wave expansion, for two 2D photonic (PBG) crystals. In (a) the PBG crystal is the same as that in Fig. 2. In (b) the PBG consists of a square array of square holes (side length 0 . 8 4 ~ ~in) a dielectric with E = 8.9. Cross sections of the crystals are shown in the insets. Solid lines represent TM modes [fields along ( E , , H,, H,,)]; dashed lines represent TE modes [fields along ( H , , E x , E , )I. Brillouin zones are shown as insets. [Taken from Fig. 1 of R. D. Meade, A. M. Rappe, K. D. Brommer, and J . D. Joannopoulos, J . Opt. Soc. Am. B 10, 328 (1993).]

192


plane is important for band gaps in TE polarization. At low dielectric contrast (e.g., 13 for GaAs) the structure that optimizes both of these, and therefore produces a gap for both polarizations, is a triangular lattice of air columns; at a higher index contrast the optimal structure is a square lattice of air columns. More recently, Ho et have predicted, using the numerical plane-wave expansion, a novel 3D PBG material with a full photonic band gap. The 3D PBG crystal is made by stacking layers of dielectric rods that repeat every four layers. The rods can be circular, elliptical or rectangular in shape. A sample with this crystal structure was then built by Ozbay et u I . , ~ ~ as discussed in Section 11.2. The sample showed a mid-gap attenuation in good agreement with the theoretical values. Chan et al." have predicted, using the same plane-wave expansion as Ho et af.,a new class of structures with rhombohedra1 symmetry that possess large photonic gaps. They point out that most of the 3D structures known to date to have full gaps fall into this broad class of so-called A7 structures." Chan et note that the local connectivity, that is, the number of rods per joint, of the structure is very important in determining the existence of a full gap; this finding is ' Chan et al. consistent with the earlier conclusions of Meade et ~ 1 . ~Indeed find that structures composed of a high-dielectric component forming a percolating network with the lowest coordination number seem to favor a full gap. Their analysis suggests that diamond, with its near-spherical Brillouin zone and low connectivity, should still be the preferred structure for PBG materials. Theoretical calculations of photonic dispersion relations have also been carried out recently for PBG materials where one of the constituent materials is either or superconducting." In addition the plane-wave expansion method has been use to calculate the effects of anisotropy in a photonic crystal with spheres made from an anisotropic dielectric?' Other anisotropic PBG materials studied have included exotic cholestric blue phases.y3 Enhanced dispersion forces in ordered colloidal suspensions have been shown theoretically to alter the phase diagram for hard-sphere transitions in such system^.'^

87

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 50, 1988 (1994). "J. Slater, Quantum Theory of Molecules and Solids, vol. 2, McGraw-Hill, New York (1965); L. M. Falicov and S. Colin, Phys. Rev. 137, A871 (1965). 89 A. R. McGurn and A. A. Maradudin, Phys. Rev. B 48, 17,576 (1993). 90 R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. B 49, 10,914 (1994). "M. W. Lee, P. M. Hui, and D. Stroud, Phys. Rev. B 51, 8634 (1995). y21. H. H. Zabel and D. Stroud, Phys. Reu. B 48, 5004 (1993). 93R.M. Horneigh and S. Shtrikman, Phys. Rev. E 47, 2067 (1993). 94 C. L. Adler and N. M. Lawandy, Phys. Rev. Lett. 66,2617 (1991).


193

For PBG materials with defects, ideally one would like to calculate the properties of a single defect in an otherwise perfect (infinite) PBG material. Such an approach is computationally prohibitive using a plane-wave expansion. Instead the plane-wave expansion has been used to describe defects placed in a finite unit cell of PBG material using the supercell m e t h ~ d . ~The ~ , ~supercell ' method places a single defect in a single cell, which is then repeated. Meade et aLZ2used 8-atom conventional cells of they decided to deal with the H(r) field diamond. Following Ho et because of its continuous nature at dielectric boundaries. They expanded the H(r) field in a basis of plane waves up to a finite frequency; they included about 130 plane waves per polarization per primitive unit cell. Based on a test case calculation for a 1D system of periodic dielectric slabs for which an exact solution was available, they estimated the frequency error of their numerical routine to be within 5%. However, they found that the convergence in frequencies was rather slow compared to electronic structure calculations; they deduced that this slow convergence was related to the discontinuity in the first derivative of H(r) at the dielectric boundaries. They sampled frequencies at 48 k points in the irreducible Brillouin zone of the 8-atom unit cell in order to calculate the density of states of the PBG energy spectrum and then coarse-grained the frequencies. The coarse graining of the Brillouin zone led to some difficulty in reproducing the typical density of states at low frequencies. Meade et a1.22 chose to study defects in the diamond structure studied by Ho et al." with air spheres in a background dielectric, because of the prediction of a large photonic band gap. They considered two types of impurities22: dielectric spheres in air, located in the bond-center site, and air spheres in the dielectric, located at the hexagonal site. The air sphere acted like a repulsive potential in that it attempted to repel the field lines from it; the air sphere therefore pushes a state out of the valence band. The dielectric sphere attracted field lines and hence acted as a negative potential. It therefore pulls a state down from the conduction band. Subsequently, Meade et managed to improve on the accuracy of these theoretical calculations by drastically reducing the computational time required in their plane-wave calculations; the convergence rate of the calculation was subsequently increased. This improvement was achieved by exploiting properties of particular matrix operations involved in the planewave numerical routines. Typically the improved method, which was still supercell-based, employed 750,000 plane waves for each eigenmode; it was estimatedy5 that the resulting frequencies converged to better than 0.5%. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alerhand, Phys. Reu. B 48,8434 (1993). 95

194


2 u. 0.450

’ 1

’

2

’ ’ ‘ 3 4 5 Defect Volume

’ 6

’ 7

-

8

FIG.6. Comparison of experimental and theoretical (plane-wave expansion) defect mode frequencies for the 3D fcc photonic crystal considered in Fig. 1. The defect volume is normalized by A/n’ where n is the refractive index and A is the mid-gap vacuum wavelength. Experimental values (obtained from sample considered in Fig. 1) are shown as solid circles. “Old Theory” from E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 67, 3380 (1991), is shown as a dotted line. “New Theory” with improved plane-wave convergence is from R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alerhand, Phys. Reu. B 48, 8434 (1993). Solid lines are nondegenerate modes while dotted-dashed lines are doubly degenerate. Modes on the left result from the air impurity (acceptor); modes on the right result from the dielectric defect (donor). [Taken from Fig. 4 of R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alcrhand, Phys. R ~ L B, . 48, 8434 (1993).]

Figure 6 compares the original calculational method of Meade et a1.22 (labeled “Old Theory”) with the improved scheme” (labeled “New Theory”) for defect modes in the 3D fcc structure of Yablonovitch et aL3’; this structure was discussed in Section 11.2 and is featured in Fig. 1. Also shown in Fig. 6 are the experimental results3’ for this same 3D structure. The agreement between theory and experiment is very good for the air defect (acceptor), but less so for the dielectric sphere defect (donor). In particular, only one donor mode was identified experimentally while three were found theoretically. Meade et aLY5noted that the other two modes had not been investigated experimentally due to the difficulty in performing the experiment. The reason proposed by Meade et al. for the fact that the calculated frequencies were higher than the experimental ones (see Fig. 6) was that the dielectric defect had to be positioned manually in the crystal and held in place; the air defect, on the other hand, could be accurately and directly placed in the PBG crystal by machining. A couple of alternative approaches have been suggested to the planewave expansion method. One of these is by Pendry and MacKinnon.23 Their approach closely reflects the nature of experimental PBG samples, which all consist of a finite number of lattice periods. Pendry and MacKinnon employ a finite-element method for calculating the transmission


195

coefficient through finite numbers of PBG slabs. Their method, while still numerical, avoids having to store large matrices; it also has some other significant features. First, conventional plane-wave calculations calculate w at a given real k-value; unfortunately, this technique proves difficult for complex k, which are of interest for localized modes. Pendry’s method is a photon equivalent of the on-shell scattering method employed in lowenergy electron diffraction theory. The method allows calculation for both real and complex k(w). Second, for systems with a single defect, we mentioned earlier that the plane-wave calculations can converge slowly, and can therefore be extremely time consuming. The on-shell method however expands the field over a surface as opposed to a volume. This calculation is far more efficient. Third, the method of Pendry allows a straightforward application to metallic systems at microwave frequencies. In this limit, E(r) can be complex with large imaginary values, hampering conventional plane-wave methods. The finite-element method divides space into a set of small cells with coupling between adjacent cells. The stability of the method depends crucially on the choice of lattice and coupling constants. There is also a subtle problem related to the finite-element procedure. The longitudinal solutions of Maxwell’s equations can be regarded as zero frequency, dispersionless modes; finite-element models in which the longitudinal modes are only approximately zero will therefore confuse such modes with the low-frequency transverse modes of interest. Hence Pendry and MacKinnon2’ placed a criterion on their model such that it produces a set of zero-frequency, dispersionless modes, which converge to the longitudinal modes in the limit of small unit cell size. The computer time saved by their method can be great. In particular for a dielectric structure containing L x L x L cells, the dimensions of the transfer matrix are only 4L2 instead of the 2 L3 expected otherwise; this leads to a speed increase of a factor of 100 when L = Figure 7 compares the calculations of Pendry and M a ~ K i n n o nto~ ~the COMITS measurements of Robertson et al. (see Fig. 2) for a 2D array of dielectric cylinders. They divided the unit cell of the system into a 10 X 10 X 1 mesh. For each cell an average was taken over the dielectric constant within that cell. The transfer matrix was then found by multiplying together the matrices for each of the 10 slices. The transmission coefficient was calculated by stacking together slabs of one cell thickness using the multiple scattering formula familiar in the theory of low-energy d i f f r a ~ t i o n .The ~ ~ agreement in Fig. 7 is very good, and lies within the experimental error of 5 GHz. Sigalas ef aZ.96used this transfer96 M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, Phys. Reu. B 48, 14,121 (1993); M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, Phys. Reu. B 49,

11,080 (1994).

196

P. M. HUI AND NEIL F. JOHNSON (a)

E

perpendicular to rods theory

-experimeni

tc+f+

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C

3

.-C

‘0

.-9

iw

E

B

b

(b)

100 GHz E parallel to rods

100

t c c theory ~ exuerimenl

-

Ecd

0.0

1.o

FIG.7. Lefi: Comparison of experimental and theoretical photonic band structures for the 2D PBG crystal of dielectric cylinders described in Fig. 2. Solid lines are theoretical [finite-element method of J. Pendry and A. M. MacKinnon, Phys. Reu. Lett. 69, 2772 (1992)l; solid dots are experimental (COMITS data from Fig. 2). Right: Power transmitted through a seven-row array of dielectric cylinders. Dotted curve shows instrument response in absence of cylinders. In each case, E polarization is perpendicular to rods in (a), and parallel to rods in (b). [Taken from Fig. 1 of J. Pendry and A. M. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992).]

matrix technique to investigate 2D ordered and disordered PBG materials, also obtaining good agreement with experiment. A second theoretical approach, which is not directly plane-wave based, is that of L e ~ n gwho ~ ~ presented a calculation of defect modes in PBG crystals using a Green’s function technique within a basis of vector Wannier functions (see Section III.4.b). In addition, Leung and Q u and ~ M ~ o~ r ~ zhave ~ ~ considered a photonic multiple-scattering theory analogous to the well-known KKR (Korringa-Kohn-Rostoker) theory for electrons. This method was found 97K. M. h u n g and Y.Qui, Phys. Reu. B 48, 7767 (1993). 9nA.Moroz, Phys. Reu. B 51, 2068 (1995).

197


to give very good agreement with the plane-wave expansion method for a 2D PBG crystal.y7 These numerical solutions provide a well-defined prescription for obtaining electromagnetic properties in PBG materials. However, it would also be invaluable in aiding in the design of PBG materials to have some approximate analytic expressions for the photonic band gap as a function of the PBG material parameters. Since the main frequency region of interest lies in the vicinity of a gap, we might look for alternative theoretical schemes that allow more insight into the nature of the dispersion relation near photonic band maxima and minima. A well-known scheme in conventional electronic theory is the k * p method." Exploiting the analogy with electronic systems, we have developed''? y9. loo this k p approach to calculate the photonic dispersion relation near the photonic band gap in both the scalar wave approximation and the full vector theory (see Section III.4.b for the general vector formalism). As a demonstration of the ease of implementation and accuracy of the k . p method described in Section III.4.b, we now compare the k p results with the numerical plane-wave expansion method discussed earlier. We consider the 2D PBG sample employed in Figs. 2, %a), and 7 consisting of a square array of dielectric cylinders. Both k and k, have only x and y components, and s in Eq. (4.26) takes on the form ( s x , sy ,0) (see Section III.4.b). Consider the TM modes in which the u for the electric field takes on the form u = [O, 0, u ( x , y ) ] . The zz element is the only pcssible nonvanizhing_matrix element of the matrix Thus, both the s Q * s and the (Q + Q T ) k, terms in Eq. (4.26) vanish. In Eq. (4.27) the vector p(!,, reduces to the form ( p , , p y , 0) and is given by p(,,) = (2i/n,)/uT Vu, dr. The dotted lines in Fig. 8 show the plane-wave band structure for low-lying TM bands near high-symmetry points in the Brillouin zone. [These are the same TM modes as shown in Fig. 5(a).l The eigenstates (i.e., the u functions) at the high-symmetry points ( X and I') obtained from the plane-wave method were then used to calculate the matrix elements PI, in Eq. (4.30). The band structures at X ( l 3 can easily be described within our k . p formalism using a two- (three)-band model. Equation (4.30) now has the form of a 2 X 2 (3 X 3) matrix at X ( T ) and can be analytically diagonalized. The resulting k p band structures are shown by solid lines in Fig. 8, out to 10% of the separation along the line between the high symmetry points. The agreement is remarkably good. In Fig. 8(a) the slight upward curvature of the plane-wave bands from X -+ M can now be explained within the k p formalism by the 673

-

-

6.

-

Y'J

N. F. Johnson and P. M. Hui, Phys. Reu. B 48, 10,118 (1993). N. F. Johnson and P. M. Hui, J . Phys.: Condens. Mutter 5, L355 (1993).

loll

198


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(0,O) FIG. 8. Theoretical TM band structure calculated using k . p theory (solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, Xa), and 7. Analytic diagnalization of truncated k . p matrix is described in text. The X point is (n/a,O), the M point is ( n / a , n / a ) , and r is (0,O). Frequency w is in units of c / a , where c is the speed of light, and wavevector k is in units of l / a . (a) Lowest two bands near the X point. (b) Bands 2, 3, and 4 near the r point.


199

presence or the q(,,,,, terms. The small differences between the results of the two methods are consistent with the neglect of the higher bands in the k . p matrix. The higher bands, if included, would have the effect of “repelling” (i.e., pushing down) those bands shown in Fig. 8, yielding even better agreement. Using similar reasoning, the T E modes can also be found. These are shown in Fig. 9, where they are again compared to the plane-wave method band structure [i.e., to the TE modes in Fig. %a)]. Again the agreement is very good. We emphasize that, for the purpose of comparison, we calculated the k . p input parameters using the numerical plane-wave method eigenstates in order to demonstrate how accurate even a few-band k p model can be. The usefulness of the k . p method in practice is that these parameters can be obtained by fitting either to published band structures calculated using numerical techniques or to experimental data. Finally we note the application to photonic problems of another wellknown approach from electronic band structure theory, the tight-binding method or ETBM (empirical tight binding m e t h ~ d ) . ’ ~ This method has been applied successfully to a variety of semiconductor systems to describe their electronic band structures. We have developed a similar scheme for photonic band structures“” (also see Section 111.4.~).Within this approach, we have demonstrated that a good fit to the 2D photonic band structure can be achieved using just a few tight-binding parameters. Figure 10 shows the lowest five TM-mode bands for the same 2D PBG sample as was considered in Figs. 2, %a), and 7. The solid lines are tight-binding bands, whereas the dotted lines are plane-wave method bands. The parameters used in Fig. 10 are obtained simply by fitting to the plane-wave method band structure at the symmetry points r, X , and M in the BZ. Improved values can be obtained by more careful fits to more points within the BZ. For the lowest band, it is sufficient to include only the nearest neighbor term in order to obtain a good fit. For the higher bands, a reasonable fit can be obtained by taking into account the next nearest neighbor term. An ETBM theory can then be developed by studying the dependence of the tight-binding parameters on the system parameters.’”’ IV. Conclusions and Future Direction

We have presented a review of recent experimental and theoretical results concerning the properties of photonic band-gap materials. Both 2D and 3D PBG systems have been discussed, with and without defects. Most “” W. M. Lee, M. Phil. Thesis, The Chinese University of Hong Kong (1994); W. M. Lee, P. M . Hui, and N. F. Johnson (unpublished).

200

P. M. HUI AND NEIL F. JOHNSON oa/2nc

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-

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M

(a) oa/21tc 0.725 1

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1

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FIG. 9. Theoretical TE band structure calculated using k . p theory (solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, 5(a), 7, and 8. Analytic diagonalization of truncated k . p matrix is described in text. The X point is (r/a, O), the M point is ( r / a , r / a ) , and r is (0,O). Frequency w is in units of 2nc/a, where c is the speed of light, and wavevector k is in units of r / a . (a) Lowest two bands near the X point. (b) Bands 2, 3, 4, and 5 near the r point. (c) Lowest four bands near M point.

20 1


0.9

0.0

0.7 0.6

0.5

0.3

0.2 0.1

0

r

X

M

r

FIG.10. Theoretical TM band structure calculated using a next nearest-neighbor empirical tight-binding scheme (ETBM-solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, 5(a), 7, 8, and 9. The X point is ( T / u , O ) , the M point is (?r/a, ?r/a), and r is (0,O).Frequency w is in units of 2?rc/u, where c is the speed of light.

reported theoretical work has relied on numerical solution of the vector wave equation using a plane-wave expansion. The agreement between recent numerical plane-wave results and available experimental data is generally very good in both 2D and 3D systems. We have, in addition, provided a self-contained discussion of the representation theory for PBG materials. Formal results have been obtained that represent the vector-wave analogs of the well-known expressions for electronic systems. In particular,

202


we have demonstrated that theoretical approaches such as k p theory and the tight-binding method can easily be implemented in PBG materials. These two approaches, which have already provided valuable insight into electronic properties in solids, yield results in PBG materials that are in quantitative agreement with the numerical plane-wave results. The outstanding experimental challenge in the fabrication of PBG materials is to make a 2D or 3D crystal with a full photonic band gap throughout the Brillouin zone in the optical frequency range. From the technological viewpoint, such a structure should have the additional property that it is easy to fabricate. On the more fundamental side, the ability to modify atomic and semiconductor radiation properties via a modification of the radiation channels opens many possibilities for investigating electron-photon effects in such systems. PBG materials may therefore prove very useful in the field of cavity quantum electrodynamics. We have also raised the possibility of introducing nonlinear defects into PBG crystals. The numerical plane-wave routines for calculating PBG properties have become very sophisticated in the last few years. However, one of the main theoretical challenges that has eluded researchers is to provide a simple, preferably analytic, recipe for the approximate value of the photonic band gap as a function of the PBG parameters. Given that the number of possible PBG crystals is infinite, such a result would be invaluable to experimentalists searching for the PBG crystal with the optimum gap at a given frequency. We believe that photonic k p theory, which focuses on the frequency region around the photonic band gap, and the empirical tight-binding method could prove useful in this search. Our discussion throughout this article has focused on “bulk” PBG crystals. Given the analogies that exist between photonic and electronic band structures, one can raise the possibility of following the semiconductor (e.g., GaAs-GaAlAs) heterostructure field by making PBG heterostructures. For example, PBG (quantum) wells, wires, dots, and superlattices are all possible, given the existence of two PBG materials with full photonic band gaps G.e, the analogs of GaAs and GaAlAs, respectively). In the same way as the envelope-function formalism based on bulk k . p theory provides the most useful description for the electronic the formalism of Section 111.4 can similarly be generalized to these new PBG systems. Just as for semiconductor heterostructures,69- 7 1 the extra flexibility in layer widths, etc., implies that such PBG heterostructures should exhibit a wide range of photonic band-gap behavior.


203

ACKNOWLEDGMENTS This work was supported in part by grants from the British Council under the UK-HK Joint Research Scheme 1993-94 and 1994-95. PMH acknowledges additional support from the Chinese University of Hong Kong under a Strategic Research Program and a Direct Grant for Research 1994-95, and useful discussions with S. Y . Liu during the course of this work. NFJ acknowledges additional support from the Nuffield Foundation, and the hospitality of thc Universidad de Los Andes (Colombia) where part of this work was prepared. W e thank R. D. Meade for useful discussions and W. M. Lee and K. H. Luk for fruitful collaborations.

SOLID STATE PHYSICS, VOL. 4Y

Physics and Device Science in II - VI Semiconductor Visible Light Emitters A. V. NURMIKKO Dillision of Engineering and Deparmenl of Physics, Brown University, Proiidence, Rhode Island

R. L. GUNSHOR School of Electrical Engineering, Purdue Unic,ersity, West Lafayetle, Indiana

1. Overview of Wide Band-Gap 11-VI Semiconductors and Blue-Green Light Emitters . . . . . . . . .. . . . ... . . . . . . . .. 11. Electronic States in Wide Band-Gap 11-VI Quantum Wells. . . . ..... I . Electronic Confinement and Band Offsets: General Comments 2. ZnCdSe/ZnSe Quantum Wells: A Case Study of Two-Dimensional Excitons . . . . . . . . ..... . . . . ...... 3. Tellurium Isoelectronic Centers in ZnSe and Exciton-Phonon Interaction 4. Toward Quantum Wires and Dots . . ... ... . 111. DopingandTransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Overview and Current Status . . . . . .. ..... ....... 6. p-Type Doping of ZnSe and Related Compounds .... ... 7. Development of Low-Resistance Contacts to p-ZnSe ....... IV. Blue-Green LEDs and Diode Lasers . . . . . . . .... . . ... 8. Light-Emitting Diodes. . . . . . . . . .. . . ..... 9. Diode Lasers . . . . . . . . . . . . . . . . .. . . . . . . . ..... 10. Degradation and Defects . . . . . . . . . .. . . . . . . . 11. Surface-Emitting Lasers: First Steps . . . ... . . . ..... V. Physics of Gain and Stimulated Emission in ZnSe-Based Quantum Well Lasers 12. Overview Comments. . , . . . . . . . .... . . . .. ... 13. Excitonic Processes in Optically Pumped Lasers at Low Temperature. . . . 14. Gain Spectroscopy of Blue-Green Diode Lasers . . . . . . . . . . . .. VI. Summa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

..

.. .

..

... . . . .. . . ....... ... ... . .. .. . . .. . . ... . . . . . .. . . . ... . .. .. . . . . . ..... .. . ... ... . . . .... . .... ..... .. . .. .. . .. . . . . . . . .. .. . . .... . . ... ... .... . . . ..

205 210 210 217 225 229 233 233 235 24 1 244 245 250 259 264 267 268 269 277 28 1

I. Overview of Wide Band-Gap II-VI Semiconductorsand Blue-Green Light Emitters

The current spotlight on wide bandgap semiconductors in general, and the 11-VI compounds in particular, has been firmly switched on by recent 205 Copyright 0 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

206

A. V. NURMIKKO AND R. L. GUNSHOR

progress in the blue-green light-emitting devices (LEDs) and diode lasers. At this writing (late 1994), technological possibilities ranging from highdensity optical storage to high-brightness displays are emerging on the horizon. Two recent milestones, in particular, are of note in this connection: (1) the demonstration of brief room-temperature continuous-wave operation of ZnSe-based quantum well lasers around the 500-nm wavelength’ and (2) the commercial introduction of high-brightness blue and green LEDs in GaN-based heterostructures. The attention to both the 11-VI and 111-V wide band-gap semiconductors is, in many respects, both a belated and renewed one, coming on the heels of a long research period that produced a substantial body of basic research with very little concrete signs of encouragement toward applications-until recently. In this article we do not review the evolving story of the 111-V nitrides, except for occasional comparisons and contrasts concerning the basic physical properties of the 11-VI and 111-V semiconductors. The history of wide band-gap 11-VI semiconductors reaches back to at least the early part of this century, with the use of ZnO in a rectifying point contact diode. In a later “real” application, ZnS has long been used as a host material in phosphors were doping by transition metal elements produces high cathodoluminescence efficiencies. Apart from cathode-ray tubes, such phosphors are being exploited today in flat-panel display screens, based on high field electroluminescence phenomena. With the emergence of compound semiconductors in general, a surge in basic semiconductor physics of the 11-VIs occurred in the 1960s, when interband optical spectroscopy of ZnSe, ZnTe, ZnS, CdSe, CdTe, and related mixed crystal compounds near their band-gap photon energies was instrumental in the study and discovery of the new exciton and polariton phenomena.’ On the other hand, although the device prospects for compact, directly electrically excited light emitters across the visible spectrum were triggered by the demonstration of the GaAs diode laser in 1960, they remained frustrated for three decades in the wide-gap 11-VIs due to the poor electrical and electronic properties of the bulk materials. Even if a large body of important basic work was accumulated during this period2z3 the lack of progress towards optoelectronic applications caused many (especially industrial) research laboratories to curtail work in the field, ‘Deleted in proof. M. Aven and J. S. Prener, Physics and Chemisrry of 11-VI Compounds, North-Holland, Amsterdam (1967). ’Proceedings of the International Conference on 11-VI Compound Semiconductors, from 1967 to 1985.

207

VISIBLE LIGHT EMITTERS

which retreated by some measure into relative academic obscurity. In contrast, basic research with narrow-gap II-VIs, notable Hg -,Cd,Te, was closely coupled with parallel by infrared detector applications, notably in the important 10-pm wavelength region. From the vantage point of today, key events were set into motion in the early 1980s, when a few research groups launched research expeditions into the wide-gap II-VIs with the new nonequilibrium epitaxial growth techniques, notably by molecular beam epitaxy (MBE), changing the materials character of the field in a fundamental way? In this article, we do not discuss the epitaxial growth techniques but refer the reader to, for example, Ref. 4 as well as the recent proceedings of the International Conference on II-VI Compounds?,6 The two main consequences of the epitaxial approach to II-VI semiconductor synthesis were (1) improvement of impurity and defect control, which eventually led to the realization of useful ambipolar doping and p-n junctions, while also allowing the study and identification of the underlying microscopic reasons as to the perennial doping difficulties, and (2) the realization of flexible heterostructures such as quantum wells (QW) and superlattices. The new heterostructures, in particular, quickly opened a new domain of optical physics so that the electronic states of a number of wide-gap II-VI QW and superlattice systems were characterized in detail by optical spectroscopy by the late 1980s, with ZnSe, ZnTe, and CdTe as the primary starting compounds. Within the constraints of lattice mismatch strain, specific type I quantum wells were identified as choices for quasi-2D systems that would offer significant advantages for radiative recombination, partly due to the enhanced electron-hole overlap that influenced the interband transition rate (excitons), and partly due to the efficient carrier collection, which reduced the detrimental effects of nonradiative traps in the bulklike portions of the heterostructures. These benefits were demonstrated, for example, by Jeon et al.’ in the room-temperature optical pumping of a (Zn, Cd)Se/ZnSe multiple quantum well (MQW) laser.

,

For example, R. Gunshor, L. Kolozdiejski, and A. Nurmikko, “Molecular Beam Epitaxy of 11-VI Semiconductor Microstructures”, in Semiconductors and Semimetals (T. Pearsall ed.), Vol. 33, pp. 337-409, Academic Press, San Diego (1990). ’Proceedings of the 5th International Conference on Il-VI Compounds, Tamano, Japan, J . Cryst. Growth, Vol. 117, Nos. 1-4, North Holland, Amsterdam, (1992). hProceedingsof the 6th International Conference on II-VI Compounds, Newport, RI, J . Cvst. Growth Vol. 138, Nos. 1-4, North Holland, Amsterdam, (1992). ’H. Jeon, J. Ding, A. V. Nurmikko, H. Luo, N. Samarth, J. Furdyna, W. Bonner, and R. Nahory, Appl. Phys. Lett. 57, 2413 (1990).

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The key advance in the control of electrical properties occurred in 1990 when Park et aL8 and Ohkawa et aL9 independently showed how p-type doping of ZnSe could be accomplished with free hole concentrations reaching to the order of 10” cm-3 by using nitrogen as the dopant element within the MBE growth and thereby paving the road for useful p-n junction and related “vertical injection” heterostructures. This achievement overcame a principal hurdle that decades of work on bulk materials had simply been unable to surmount. For example, while ZnSe could be doped n-type to provide a reasonable electron density (say, in excess of 10’’ ~ m - ~ p-type ), doping was apparently “forbidden.” In the case of ZnTe, it had been the n-type doping that was practically impossible while p-doping was “allowed.” (Interestingly, the n-type doping of ZnTe remains a research challenge even today.) When combining the p-n junction and the new QW heterostructures, Haase et al.’” demonstrated blue-green diode laser operation from a (Zn, Cd)Se/ZnSe/Zn(S, Se) device in 1991, followed by the authors’ To date, at least a dozen research groups worldwide have succeeded in showing some form of blue-green diode laser operation under pulsed electrical injection, based on heterostructures, which are the major focus of this article. Threshold current densities at or below 1 kA/cm2 at room temperatures have been realized, with threshold currents reaching down to the milliampere range for short, ridge-type waveguide structures. The reader should be aware of the fact that both the device physics and engineering of these new laser structures is the subject of intense current research efforts; hence, the examples of device performance summarized later in this article will most likely require substantial revision on a timescale of a year. At this point, cw operation at room temperature been achieved at bias conditions slightly = 6 V and below Ith= 10 mA. Currently, the device research is below focusing on the solving the rapid device degradation seen under the cw conditions, a circumstance that now poses the major technical challenge for the 11-VI community from the standpoint of commercial viability of blue-green lasers and LEDs. At this writing, the initial indications are that the origin for the device degradation is dominated by specific classes of

vh

‘R. M. Park, M. B. Troffer, and C. M. Rouleau, J. M. De Puydt, and M. A. Haase, Appl. Phys. Lett. 57, 2127 (1990). ’K. Ohkawa, T. Karasawa, and 0. Yamazaki, Jpn J . Appl. Phys. 30, L152 (1991). ’OM. Haase, J. Qui, J. DePuydt, and H. Cheng. Appl. Phys. Lett. 59, 1272 (1991). “ H . Jean, J. Ding, W. Patterson, A. V. Nurmikko, W. Xie, D. C. Grillo, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Lett. 59, 3619 (1991). 12 H. Jean, J. Ding, A. V. Nurmikko, W. Xie, D. C. Grillo, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Lett. 60, 2045 (1992).


209

extrinsic, growth-related crystal defects. We include a description of research work in this problem area also in this article. Having duly emphasized the importance of the wide band-gap 11-VI semiconductors to optoelectronic device work, we hasten to underline that the new heterostructures and short-wavelength lasers offer a rich opportunity for basic and device physics in a class of semiconductors that have their own fascinating idiosyncrosies. The overall physical properties of ZnSe and its alloys, for example, are dictated by the fact that in terms of the constitution of their chemical bonds these “semi-insulators” lie considerably on the polar side of a materials chart when compared for example, with, GaAs. As a consequence, electron-hole pair interaction (excitonic effects) are very strong especially in the quasi-2D case of a quantum well, and the coupling of the electronic states with the lattice (e.g., via the Frohlich interaction) is likewise very potent. Mechanical properties are also influenced by the rather large ratio of the polar to the covalent component in the chemical bond of a ZnSe crystal. We illustrate the impact of the polar character on both optical and transport properties by a range examples in the following sections. Of course, such physical characteristics must also figure into the design and fabrication of optoelectronic devices, including eventual issues of their manufacture and production. This article is organized as follows. In Section 11, we introduce the range of compounds from which present light-emitting structures are being fabricated, by focusing on the description and characterization of the electronic states that are central to the quantum well and heterojunction physics of ZnSe-based heterostructures. Optical properties are emphasized in this section. In Section I11 we concentrate on doping and electron-hole transport of the wide band-gap 11-VI semiconductors, both in thin films and in heterostructures (vertical transport). In Section IV we show examples of current device characteristics and performance, both for LEDs and diode lasers. We again caution the reader that the device field is advancing sufficiently rapidly so that further developments are very likely in a time that it takes a volume such as this to go to press. In section V, we examine the physics of light emission by 11-VI heterostructures in the context of spontaneous and stimulated emission by Coulomb correlated quasi-2D electron-hole pairs. Concluding remarks are made in Section VI regarding the opportunities for future research and development for brining the new blue-green emitters into the realm of practical devices. The examples used in this article to illustrate the present status of the field provide, from necessity, a limited and somewhat subjective sampling of an active field. In attempting some continuity of the key themes within a finite amount of space, we have drawn fairly heavily on work in the authors’ laboratories and have occasionally had to omit references to

210


numerous important contributions by colleagues to whom we apologize for such omissions. The field is still expanding and many new contributors are having an important and fresh impact on both the basic and device science of 11-VI semiconductors. II. Electronic States in Wide Band-Gap Il-VI Quantum Wells

1. ELECTRONIC CONFINEMENT AND BANDOFFSETS: GENERAL COMMENTS In this section we consider the electronic states in wide band-gap 11-VI heterostructures, primarily from the standpoint of their impact on the optical properties and design of blue-green light emitters. As shown later, the ability to fabricate such structures with a good deal of versatility in the control of both electrical and optical properties has been a crucial element in the early success of the demonstration diode lasers, and has played a dominant role in the LEDs as well. Different manifestations of “band structure engineering” are found in the design of the optically active region as well as the contact region, to cite two prime examples. Hence, quantitative determination of the electronic states near the conduction and valence band extrema are a key for utilizing the benefits offered, for example, by MBE. In this article we do not review or list the state of present knowledge of all 11-VI heterostructures that have been studied to date. Rather, we focus on those systems that appear as most important to the present blue-green diode lasers and LEDs. An excellent and comprehensive review of band offsets in semiconductors has appeared in a previous volume of this series.” The influence of strain in a specific case of the ZnSe/Zn(S,Se) quantum wells has been detailed in Ref. 14, which introduces the general methodology for accounting lattice mismatch strain in band offset calculations. In case of the 11-VI heterostructures, reason. able accuracy has been reached in the (mainly) experimental determination of band offsets for a number of key combinations of binary, ternary, and quaternary materials. Nonetheless, the problems of lattice mismatch strain and composition control lead to uncertainties in these results so that the values quoted here are subject to a margin of error of several tens of meV ( = kT at room temperature). In any contemporary semiconductor laser, the chief design criterion is to combine electronic and optical confinement by enveloping a quantum well 13

E. T. Yu, J. 0. McCaldin, and T. C. McGill, in Solid State Physics (H. Ehrenreich and D. Turnbull, eds.), Vol. 46, pp. 2-146, Academic Press, San Diego (1992). I4C. G. Van de Walle, Phys. Rev. B 39, 1871 (1989).


211

or a superlattice with an optical waveguide embedded within a multilayer heterostructure. This composite is generically called the separate confinement heterostructure (SCH). Optical confinement is not an important requirement for an LED per se; however, for devices in which radiative recombination involves mobile electron-hole pairs (as opposed to electronic excitations localized at impurities), electronic confinement is also a key feature for the incoherent II-VI light emitters. Elementary considerations for useful electronic confinement (in a type I QW) require for the conduction and valence band offsets that A E c , A E l , s k T , EFc,E F I ,at a given operating temperature and injection level (with E,, and E,, corresponding the electron and hole quasi-Fermi levels, respectively). As a case example of an important contemporary multilayer structure, Fig. l(a) shows a schematic band structure for a layered arrangement that has been successfully implemented in recent room-temperature cw diode laser devices, based on the combination of a (Zn, Cd)Se QW, Zn(S, Se) electronic barrier and optical waveguide layers, and ZnMgSSe outer optical waveguide cladding layers. Because the (Zn, Cd)Se quantum well is under compressive biaxial strain, the heavy and light hole degeneracy is removed due to the uniaxial component, as indicated in the inset of the figure. The effective mass in the growth direction ( m , I ) is heavy-hole-like, while the ) its dispersion resembles that of the light hole near in-plane mass ( m h I ,and k 2: 0 (see inset of Fig. 1). Consequently, the in-plane density of states is reduced from the bulk value for ZnCdSe, and aids in reducing the threshold inversion density required for gain in the conventional degenerate electron-hole plasma model. In that sense the ZnCdSe/ZnSe is quite analogous to the InGaAs/GaAs QW system. One distinction is that precise effective mass values for the p-type alloys in our case are not accurately known. Indirectly, however, magneto-optical spectroscopy of the so-called 1s and 2s exciton states in the ZnCdSe/ZnSe QW has indicated the following in-plane effective mass values: me = 0.17m0 and mhl,2: 0.2m0 (see later discussion and Ref. 38). We review in this section some of the basic quantum well physics that provides both a diagnostic for testing the strength of electronic confinement in such structures and indicates some of the special features of II-VI semiconductor heterostructures, when compared, for example, with GaAs-based heterostructures. A measure of the degree of overall electronic confinement for this important heterostructure is obtained from Fig. l(b) from the optical transmission spectrum at T = 10 K, which shows the excitonic band gaps for the three constituent semiconductors for a sample composed of a 75-A-thick active Zn,-,Cd,Se QW layer with Cd concentration of approximately x 2: 0.20. (Optical properties of II-VI het-

212


(a)

% : z

ZnSSe

E,

/ \

ZnMgSS i

T=l OK

a u . m

am -

a . ZnCdSe QW

1

FIG. 1. (a) Schematic of the conduction and valence bands in the SCH diode laser heterostructure composed of a ZnCdSe QW, ZnSSe optical/electrical confinement layer, and the ZnMgSSe outer cladding. (b) Absorbance of a typical laser laser structure (in direction of growth) at T = 10 K, showing the relationship between the (excitonic) energy band gaps of the constituent materials. The low energy oscillations are due to Fabry-Perot interference effects.

erostructures are discussed in more detail in the next section.) The sulfur concentration in the ternary ZnS,Se, was approximately y = 0.07 and y’ = 0.12 in the Zn,-,Mg,S,~Se,-, quanternary layer in which the Mg concentration was z = 0.09. The QW layer was under a biaxial strain of approximately 1.4%, whereas the ternary and quanternary layers were lattice matched to the GaAs buffer in this pseudomorphic structure. Apart from influencing the electronic energies and “intrinsic” band offsets, lattice mismatch strains place severe practical constraints on the design of wide-gap 11-VI quantum well and SCH devices. A severe penalty is imposed if either portions of or the entire heterostructure exceeds the so-called critical thickness limits for strain relaxation, that is, a condition is

-,

VISIBLE LIGHT EMI'ITERS

213

reached during growth that leads to the formation of misfit disclocations.'' The threading dislocations associated with the misfit dislocations severely tax the active parts of a light emitter, forming a sort of an electronic graveyard for electron-hole pairs due to enhanced nonradiative recombination at these extended defects. As described elsewhere in this article, the initial presence of dislocations (although not necessarily associated with lattice misfit) in as-grown material, in general, has been clearly identified as a primary source of device degradation in the blue-green diode lasers under the high current density typical in such devices. As a practical matter, the design of SCH configurations such as that in Fig. 1 requires at least one quaternary alloy, given the present material spectrum of wide-gap 11-VI semiconductors that have been found compatible with epitaxial growth techniques. A guide map for designing light-emitting 11-VI heterostructures is illustrated in Fig. 2, in which the room-temperature band gap is graphed against the lattice constant for a number of other cubic semiconductors for reference. The band gaps of the noncubic GaN and InN are also included for comparison because these semiconductors have emerged as another strong class of technological candidates for short-wavelength light emitters. The shaded area, spanned by ZnSe, ZnS, MgS, and MgSe, defines the binary perimeter for the synthesis of the ternary Zn(S, Se) and quaternary (Zn, MgXS, Se), which figure centrally in the currently developing SCH lasers. In particular, the alkaline earth cation Mg (column IIa), which was introduced by researchers from Sony Corporation relatively recently, has expanded the electronic structure design of pseudomorphic blue-green light emitters significantly.16 Optical investigation of the electronic confinement in the CdTe/(Cd,Mg)Te QW has also been reported." Figure 2 shows clearly that, unlike the case with the ubiquitous GaAs/(Ga, ADAS heterojunction, lattice mismatch is an omnipresent design constraint for the 11-VI materials, among with ZnSe can be viewed as forming the "hub" for the blue-green light emitters. Beyond the actual 11-VI heterostructure, the choice of the substrate involves critical additional considerations that are related to the density of defects in the as-grown materials for the diode lasers and LEDs. To date, most of the light-emitting structures have been grown on GaAs substrates, which presents a 0.25% lattice mismatch to ZnSe, but they can be fully lattice matched to Zn(S, Se) and (Zn, MgXS, Se) "Strained Luyer Superluttices, Semiconductors and Semimeials (T. Pearsall, ed.), Vols. 32 and 33, Academic Press, San Diego (1990); see also L. B. Freund, J . Appl. Phys. 68, 2073 (1 990). '"H. Okuyama, F. Hiei, and K. Akimoto, Jpn. J . Appl. Phys. 31, L340 (1992). "6. Kuhn-Heinrich, W. Ossau, H. Heinke, F. Fischer, T. Litz, A. Waag, and G. Landwehr, Appl. Phys. Lett. 63, 2932 (1993).

214


>Mgse

5.4

5.6

MnSe

+ GaN

MnTe

-

ZnTe AlSb Inp

n

II

I

II

II II

5.8 6.0 Lattice Constant

6.2

(A,

6.4

InN

CdTe

6.6

FIG.2. Energy band gap versus lattice constant for common 11-VI, 111-V, IV, and IV-VI semiconductors in the cubic phase. Solid lines indicate some of the ternary combinations and dashed lines suggest near lattice matching onto specific substrates. Note that there is some uncertainty in the values for MgSe and MgS in the cubic phase. Band gaps for the wide-gap 111-V (noncubic) GaN and InN are also shown.

of particular compositions. Evidence has shown that the inclusion of a GaAs buffer layer is also very important; even so there are key questions that are being researched today in terms of the interfacial defects and connected issues of interface chemistry at the heterovalent ZnSe/GaAs interface. Expanded efforts are presently under way to employ the (In, Ga)P and other phosphide-based compounds as a buffer layer'' for the 11-VI epitaxy, and results based on ZnSe hemoepitaxy are just beginning to appear.'' In principle, such homoepitaxy offers obvious major advantages, provided that the quality of the ZnSe bulk material is suitable and can be made sufficiently conductive. Considerable progress is expected in the blue-green light emitters grown homoepitaxially. As a final comment about MBE in general, we note that complicating the epitaxy of the 11-VI IR

L. Kolodziejski, K. Lu, E. Ho, C. A. Coronado, P. A. Fisher, and G. S. Petrich, J . Cryst. Growth 138, 1 (1994). 19 G. Cantwell, W. C. Harsch, H. L. Cotal, B. G. Markey, S. W. McKeever, and J. E. Thomas, J . Appl. Phys. 71, 2931 (1992).


215

ternaries and quaternaries is the fact that none of the elements has a unity sticking coefficient on the growth surface, in strong contrast with most 111-V semiconductor cases. This makes the composition control, for example, of the quaternary ZnMgSSe quite challenging from the standpoint of reproducibility. Within the past decade, a substantial number of 11-VI quantum wells and other heterostructures have been synthesized and studied in wide-gap semiconductors. Much of this work predeeded the demonstration of the diode laser in 1991 and formed an important testing ground for epitaxial growth techniques while generating valuable information about the electronic confinement effects. The heterostructures included such combinations as the ZnSe/Zn(S, Se),20 ZnSe/ZnS?l CdTe/ZnTe?2 CdSe/ZnTe (with CdSe in cubic phase),23 CdS/(Zn,Cd)S,24 and CdS/ZnSZ5 type I structures, the ZnTe/ZnSe type I1 structure,2h and so on. As already indicated, we will not review this important literature here but rather refer to the article by Yu et a1.13 in this series, which compiles and analyzes the band offsets for various 11-VI (and 111-V) heterojunctions, including a thorough discussion of the contemporary physical models and their shortcomings to the problem. In addition, the recent proceedings of the International Conference on 11-VI Semiconductors give the reader ample opportunity to explore the recent developments in 11-VI heterostructures.'.' Our primary focus in this section is on the optical physics of the (Zn, Cd)Se/ZnSe and related QWs, in preparation for the subsequent sections that feature the blue-green light emitters. Broadly stated, it has been generally found by experiment that there is a tendency of the 11-VI heterostructures to follow the common anion "rule" in terms of apportioning the conduction and valance band offsets. Hence, for example, the CdTe/(Cd, Mn)Te QW was found to have an intrinsically 2"S. Fugita, Y. Matsuda, and A. Sasaki, Appl. Phys. Left.47, 597 (1985); K. Shahzad, D. J. Olego, and C. G. Van de Walk, Phys. Rev. B 38, 1417 (1988); K. Nakanishi, I. Suemune, H. Masato, Y. Kuroda, and M. Yamanishi, Jpn. J . Appl. Phys. 29, L2420 (1990). 2'T. Yokogawa, M. Ogura, and T. Kajiwara, Appl. Phys. Left.49, 1702 (1986); Y. Kawakani and T. Taguchi, J . Vuc. Sci. Technol. B7, 789 (1989). 22 R. Miles, G. Wu, M. Johnson, T. McGill, J.-P. Faurie, and S. Sivananthan, Appl. Phys. Letf. 48, 1383 (1986); Y. Hefetz, D. Lee, A. V. Nurmikko, S. Sivananthan, X. Chu, and J.-P. Faurie, Phys. Re(,.B 34, 4423 (1986). "N. Samarth, H. Luo, J. K. Furdyna, S. B. Qadri, Y. R. Lee, A. K. Ramdas, and N. Otsuka, Appl. Phys. Left. 54, 2680 (1990); M. C. Phillips, E. T. Yu, Y. Rajakarunanayake, D. A. Collins, and T. C. McGill, J . Crysf.Growth 111, 820 (1991). 24T.Karasawa, K. Ohkawa, and T. Mitsuyu, J . Appl. Phy.9. 69, 3226 (1991). 25 G. Brunthaler, M. Lang, A Forstner, C. Giftke, D. Schikora, S. Ferreira, H. Sitter, and K. Lischka, J . Crysl. Growth 138, 538 (1994), and references therein. "H. Fujiyasu and K. Michizuki, J . Appl. Phys. 57, 2960 (1985).

216


small (near zero) A whereas the ZnSe/Zn(S, Se) system offered an impractically small A Ec.28The overall band-gap difference in the highly strained binary CdTe/MnTe system is as large as 1.6 eV; as a consequence, wavelength shifts induced by electronic confinement were seen in low-temperature photoluminescence from the infrared through the visible region into blue in ultrathin QWS.~'These examples with a score of other experimental observations seem to imply that elementary considerations of the energy positions for the conduction band minimum and valence band maximum at the Brillouin zone center (dominated by the s electron orbitals of the cation and the p orbitals of the cation) suffice to estimate the band offsets, at least to zeroth order, if strain is properly accounted for. Qualitatively, in a (Zn, Cd)Se/Zn(S, Se) QW the inclusion of Cd in the ternary for the well layer induces the confinement in the conduction band; sulfur (in the barrier layer) performs this function for the valence band states. In such a description, based entirely on bulk properties, the role of interfaces is totally ignored. Once the MBE growth of (Zn,Cd)Se was shown to be feasible,3o experimental evidence was quickly obtained that clearly showed the potential usefulness of the ZnCdSe/ZnSSe heterostructure for light-emitting purposes. The experimental evidence was acquired by employing (Zn, Mn)Se barrier layers, so that Zeeman spectroscopy at the principal exciton transitions could be performed to measure directly the penetration of the electron-hole wavefunctions into the barrier.31 This approach uses the properties of the diluted magnetic semiconductors (DMS) that are well known to exhibit very large Zeeman effects at low lattice temperatures due to the short range, Heisenberg-type exchange interaction. This feature has been exploited recently to study, for example, magnetic field induced type I to type I1 conversion in DMS quantum wells at low lattice t e m ~ e r a t u r e . ~ We ' , ~ ~consider the fundamen21

S.-K. Chang, A. V. Nurmikko, J.-W.Wu, L. A. Kolodziejski, and R. L. Gunshor, Phys. Reu. B 37, 1191 (1988). 28 K. Mohammed, D. J. Olego, P. Newbury, D. A. Cammack, R. Dalby, and H. Corneliassen, Appl. Phys. Lerf. 50, 1820 (1989). 29 S. M. Durbin, J. Han, 0. Sungki, M. Kobayashi, D. R. Menke, R. L. Gunshor, Q. Fu, N. Pelekanos, A. V. Nurmikko, D. Li, J. Gonsalves, and N. Otsuka, Appl. Phys. Leff. 55, 2087 (1989); N. Pelekanos, Q. Fu, J. Ding, W. Walecki, A. V. Nurmikko, S. Durbin, M. Kobayashi, and R. L. Gunshor, Phys. Reu. B 41, 9966 (1990). SO N. Samarth, H. Luo, J. K. Furdyna, S. B. Qadri, Y. R. Lee, A. K. Ramdas, and N. Otsuka, Appl. Phys. Len. 54, 2680 (1990). "W. Walecki, A. V. Nurmikko, N. Samarth, H. Luo, J. K. Furdyna, and N. Otsuka, Appl. Phys. Leff.57, 466 (1990). 32 X. C. Liu, W. C. Chou, A. Petrou, J. Warnock, B. T. Jonker, G. A. Priz, and J. J. Krebs, Phys. Reu. Letf. 63, 2280 (1989). 33N.Dai, H. Luo, F. C. Zhang, N. Samarth, M. Dobrowolska, and J. K. Furdyna, Phys. Reu. Left. 63, 2280 (1989).


217

tal optical processes of the ZnSe-based 11-VI heterostructures in the next section. Relatively few first principles electronic structure calculations have appeared in the literature for the prediction of the expected band offsets or comparison with experiment for the wide band-gap 11-VI compounds. Reference 13 reviews some of the theoretical approaches and their correlation with experiment for selected 11-VI heterointerfaces. Recently, Nakayama has employed the pseudopotential approach in the local density a p p r o ~ i m a t i o n for ~ ~ evaluating the band offsets for a range of 11-VI compounds, including the role of Mg, as well as other alkaline earth elements in Zn(S, Se).35 2. ZnCdSe/ZnSe QUANTUM WELLS:A CASESTUDYOF TWO-DIMENSIONAL EXCITONS

The majority of experimental information on the electronic structure of 11-VI quantum wells and superlattices has been retrieved by optical spectroscopy at the fundamental absorption edge. As mentioned in the introduction, exciton physics influences the lowest interband resonances profoundly in the bulk,' and even more so in the quantum wells and related type I confining structures. (Many standard optical experiments are performed at near normal incidence to the QW or superlattice layer plane so that polariton effects are not of prime importance due to the breakup of wavevector conservation). The exciton effective mass parameters are well who established for bulk ZnSe; we cite here the work by Holscher et employed high-resolution two-photon spectroscopy to obtain a value for the exciton binding energy E2D= 17 meV. For comparison, the exciton binding energy for GaAs is approximately Ex = 4 meV; for CdTe, Ex = 9 meV; ZnTe, Ex = 11 meV; and CdS, Ex = 30meV. In a quantum well, the quasi-2D electronic confinement leads to a strong enhancement of the electron-hole Coulomb interaction. For the GaAs/AlGaAs case, increases in the binding energy from a value of Ex = 4 meV in the bulk to approximately 10 meV in a QW have been frequently reported. The oscillator strength also increases, by factor of 8 in the fully 2D theoretical limit. More specifically, in this hypothetical limit, the envelope part of the electron-hole wavefunction can be written

34

T. Nakayama, Jpn. J . Appl. Phys. 32, L725 (1993). 3sT. Nakayama, Jpn. J . Appl. Phys. 33, L211 (1994). 3hH,Holscher, A. Nothe, and C. Uichlein, Phys. Reu. B 31, 2379 (1985).

218

A. V. NURMIKKO A N D R. L. GUNSHOR

where the effective Bohr radius is now one-half of the 3D exciton radius u B . The absorption coefficient in turn is proportional to (2.2)

where the interband energy relative to the band gap Eg is En = Eg R/(n - 1/21’, with R given by the 2D exciton Rydberg R = 4E,3D.The linewidth broadening is phenomenologically introduced through the parameter 8. The most completely characterized II-VI system in terms of its exciton properties so far is the (Zn,Cd)Se/ZnSe QW and we discuss its optical properties here as a case study. The band gap of Zn,Cd, -,Se as a function of Cd composition has been studied by several groups; we quote here results by Wu et uL3’ who fit their optical data on epitaxial layers to the expression

E g ( x ) = xE,(CdSe)

+ (1 - x)E,(ZnSe)

-

b,,,x(l

-

x),

where the room-temperature band gaps for ZnSe and (cubic) CdSe of 2.72 and 1.67 eV, respectively, were adopted, together with a bowing parameter of 0.26. Direct experimental evidence for the enhanced electron-hole Coulomb . ~ interaction was found from absorption spectroscopy by Ding et ~ 1 and , ~ ~ which a number of confirming observations Pelekanos et ~ l . following have been made by several groups. As an example Fig. 3 shows the absorption spectrum from T = 2 K to T = 400 K of a structure containing six Zn,,,Cd,,,Se QWs of L , = 30 = A thickness [Fig. 3(b)]. The spectra are compared with a bulk case in the Fig. 3b, obtained from a sample housing six 200-A-wide QWs in the same ZnCeSe/ZnSe material system.38 (The exciton Bohr diameter in ZnSe is approximately 90 A.) The effect of confinement is immediately demonstrated in that the n = 1 HH exciton resonance remains distinct far beyond room temperature in the quasi-2D case (narrow well sample). Moreover, the peak value for the absorption coefficient in the QW reaches a value of 1.8 X lo5 cm-I, in spite of the larger role of inhomogeneous broadening by alloy potential fluctuations in 37 Yi. H. Wu, K. Ichino, Y. Kawakami, S. Fuijita, and S. Fujita, Jpn. J . Appl. Phys. 31, 1737 (1992). 3X J. Ding, N. Pelekanos, A. V. Nurmikko, H. Luo, N. Samarth, J. Furdyna, Appl. Phys. Lett. 57, 2885 (1990). 3’) N. T. Pelekanos, J. Ding, M. Hagerott, A. V. Nurmikko, H. Luo, N. Sarnarth, and J. Furdyna, Phys. Rev. B 45, 6037 (1992).

~

219


L= ,

a

200 8.

b

L=30 A

L

13 L

-m Y-

+ W 0

u

T-15OK

C

.-0

i

+I

a L 0

m n

d:

2.400

2.600

2.800

3.000 2.400

2.600

2.800

3.000

P h o t o n E n e r g y (eV) Photon Energy (eV) FIG.3. Comparison of exciton absorption in the (Zn, Cd)Se/ZnSe QWs between (a) 3D, ( L w = 200 A) and (b) 2D ( L , = 30 A) cases, as a function of temperature. The absorption edge is dominated by the n = 1 HH exciton. [From Ref. 38.1

-

the narrower QW case. For a Wannier exciton, the broadening is propor2, x is the Cd concentration, u tional to A E [x(l - x ) ] 1 / 2 ( u / u g ) 3 /where is the lattice constant, and a B is the exciton Bohr radius. A rough fit to the measured transition energies is obtained for the band offsets of roughly AE,. = 140 meV and AE,, = 60 meV, the latter including a dominant contribution by the nearly 1% lattice mismatch strain. More recently, Cingolani and co-workers4" have applied photocurrent spectroscopy on ZnCdSe/ZnSe QW p-i-n heterostructures to resolve several excitonic transitions and the continuum edge; these data were used to obtain band offsets and an assignment for the exciton binding energy in general agreement with values obtained by other investigations. They also studied the dependence of the exciton binding energy on the Cd concentration and well width (i.e., confinement) in the same QW system and show how the binding energy is reduced to a value of about E x = 30 meV for the concentration x = 0.10.41Figure 4 shows the dependence of the experimentally determined values of Ex on QW width for two Cd compositions -x = 0.11 (circles) and x = 0.23 (triangles)-together with results obtained from a variational calculation for a uniform dielectric constant of 40 R. Cingolani, M. Didio, M. Lomascolo, R. Rinaldi, P. Prete, L. Vasanelli, L. Vanzetti, F. Bassani, A. Bonanni, L. Sorba, and A. Franciosi, Phys. Reu. B 50, 12179 (1994). 41 R. Cingolani, L. Calcagnile, A. Franciosi, L. Sorba, and L. Vancetti, in Blue-Green Lasers (R. Gunshor and A. Nurmikko, eds.), Proc. SPIE 2346, 112 (1994).

220


->,

-,-

I

" "

I

" "

I

'A

E 35

Y

a

30

W

C

25 P,

.-C

.-m

20 0

5

10

15

20

Well Width [nrn] FIG.4. Heavy hole exciton binding energy for (Zn, Cd)Se/ZnSe QW for two compositions ( x = 0.11, circles; and x = 0.23, triangles) as a function of well thickness. Solid lines represent calculations for c(ZnSe); dashed lines represent E extrapolated from dZnSe) and dCdSe). [From Ref. 41.1

ZnSe ( E = 9.2; dashed lines) and for a dielectric constant obtained from linear interpolation of ZnSe and CdSe (solid lines). . analyzed ~ ~ the strain contributions to the offsets Earlier, Wu et ~ 1 had in the ZnCdSe/ZnSe system in some detail, including an assignment for . ~ ~ rigorous and the valence band effective masses. Young et ~ 1 performed detailed calculations of the exciton absorption spectra for this strained system by superlattice formalism, which has led to the elucidation of the HH and LH exciton states, band offsets, and to very good agreement with the experimental spectra of Fig. 3. (Reference 42 contains a compilation of relevant physical parameters for ZnSe and CdSe.) These authors also analyzed the exciton linewidths and reached results in close contact with experiment. Perhaps the most problematic remaining issue in comparing experiment with theory is the uncertainty concerning the actual strain and its spatial distribution, which may depend on the growth history of a particular ZnCdSe/ZnSe structure. In particular, the experimental reports to date have focused on multiple QW configurations in which the lattice mismatch strain is at least partially relaxed by the formation of misfit dislocations. Nonetheless, the consensus obtained from these investigations from a device standpoint indicates that the valence band offset is modest at best, implying considerable thermal excitation of holes out of the quantum well at room temperature so that an efficient light emitter based on the ZnCdSe/ZnSe appears impractical. As indicated in Fig. 1, the confinement that is necessary for a diode laser is achieved by increasing the valence band offset with the use of ZnSSe barrier layers (and the 42P. M. Young, E. Runge, M. Ziegler, and H. Ehrenreich, Phys. Reu. 49, 7424 (1993).

221


2550 n

5

E

2540

v

x

F W c w

2530 2520

0

c a 0

5

10

15

20

25

Magnetic field (Tesla) FIG.5. Energy of the 1s and 2s n = 1 exciton states as a function of magnetic field. The solid line is a theoretical fit. The inset shows a relevant portion of the absorption spectrum at B = 23.5 T. [From Ref. 39.1

ZnMgSSe outer barrier layers to define a “reservoir” of electron-hole pairs within the ternary). Although careful analysis of the optical data for the ZnCdSe/ZnSe QW gives, in principle, a means to extract a value for the 2D exciton binding energy, better insight is obtained if the exciton states are studied under a magnetic field. For example, as shown in Fig. 5, both the 1s and 2 s exciton states associated with the lowest n = 1 H H exciton state can be identified and tracked in such a magneto-optical experiment (circles) and compared with theory (solid line).39 This assignment enables the determination of the quasi-2D exciton binding energy from the difference between the zero-field 1s and 2s exciton energies. For 3D excitons IE,, = 0.75E2D,where as the 2D excitons IE,, - E,,I = 0.88EiD.From the data of Fig. 4, the binding energy of the quasi-2D exciton for the Zn,,,Cd,,,,Se/ZnSe QW is Ex = 41 meV, a value larger than either the longitudinal optical (LO) phonon energy ( h w L o = 31 meV in bulk ZnSe) or kT at room temperature. We also wish to note important recent work in which the magnetoexciton states in the ZnCdSe/ZnSe system have been ~ the influence of the HH-LH mixing detailed by Puls et ~ 1 In . this~ work has been clearly demonstrated, apart from the measurement of the exciton binding energy (the latter is in broad agreement with the results from other groups). The reduced HH exciton in-plane mass of p,,= O.lm, is in 43

J. Puls, V. V. Rossin, and F. Henneberger, Muter. Sci. Forum 182 / 184, 743 (1995).

222


FIG. 6. Schematic illustration of the major scattering processes contributing to the LO phonon-defined lifetime of the lowest lying exciton for Ex < fi wLo and Ex > fi wLo, respectively. The shaded area represents the continuum states.

good agreement with the Luttinger parameters for ZnSe (apart from a slight increase due to the finite Cd content). The increase in the binding energy of the 2D exciton in a ZnCdSe/ZnSe QW has, in addition to an enhanced oscillator strength, another important consequence. In particular, for a sufficiently large Cd concentration ( x > 0.2), Ex exceeds the energy for the LO phonon nuLo. Such a situation cannot be reached, for example, in the GaAs/AlGaAs QWs. In bulk ZnSe, the Frohlich interaction provides a potent scattering channel both for interband and intraband processes (in comparison with GaAs, this interaction is approximately seven times larger for the electrons alone). However, for the case Ex > h w L O ,the dissociation rate of excitons into electron-hole pair continuum states by LO phonon scattering is energetically inhibited (or strongly reduced). This fact, schematically illustrated in Fig. 6, makes it possible to observe experimentally the distinct n = 1 HH exciton absorption peak seen for up to T = 400 K in the narrow QW of Fig. 3, in spite of the large Frohlich interaction typical of wide-gap 11-VI semiconductors. The contribution to the absorption linewidth by lifetime broadening, appropriate to the 1s exciton ground state by one LO phonon scattering, can be expressed within first-order perturbation theory as r(T) = r j n h = rLo[exp(fiw,,/k,T) - I]-', where r i n h is the linewidth of inhomogeneous origin, and rLo represents the strength of the exciton-LO phonon coupling. As an example of the hierarchy with increasing polarity for bulk GAAs, ZnTe, and ZnSe are 10, of the material, the values for rLo 30, and (at least) 60 meV, respectively. From the temperature-dependent


223

linewidth deduced in the data of Fig. 3, a reduction by about a factor of 3 in the exciton-LO phonon scattering rate was deduced (this scattering promotes excitons to their excited states). The low-temperature linewidth = 7 meV in the of the n = 1 HH exciton absorption is approximately rinh data, consistent with broadening based on alloy potential fluctuations for the particular composition. For weaker confinement, Miyajima et have recently reported that the binding energy for the exciton in the ZnSe/ZnMgSSe QW remains below that the of the LO phonon so that a large broadening with temperature of the exciton linewidth is observed. Attempts to reduce the inhomogeneous linewidth of the ZnCdSe QW while retaining good electronic confinement have led to the synthesis of a ZnSe/CdSe “pseudoalloy” in the form of a short period superlattice. Luo et ~ 1 first . ~achieved ~ the growth of these structures, which have been subsequently analyzed optically by several groups. However, although some reduction in the low-temperature linewidth is observed, a certain amount of interface and compositional disorder remains.46 The limits of inhomogeneous linewidth due to compositional disorder in ZnCdSe QWs can be significantly reduced in heterostructures where electron-hole confinement takes place within a binary ZnSe well. Figure 7 shows the transmission spectrum at T = 10 K of a ZnSe/ZnSSe/ZnMgSe pseudomorphic heterostructure in which the thickness of the single ZnSe QW is approximately 75 A.47 The n = 1 H H and LH exciton states are both clearly discernible, and display a linewidth of approximately 0.6 meV, presumably limited by QW well thickness fluctuations. The corresponding peak absorption coefficient for the H H exciton exceeds 10‘ cm-’. For a sulfur concentration of y = 0.07, chosen for reasons of lattice matching, the QW confinement is relatively weak, especially in the conduction band, so that the luminescence efficiency decreases (and absorption linewidth increases) rapidly for increasing temperature ( T > 50 K). We also mention who interpreted their measurements in the work of Inoue et ZnSe/ZnSSe superlattices by means of Brewster-angle reflection spectroscopy in terms of monolayer-scale terrace structure. 44 T. Miyajirna, F. P. Logue, J. F. Donegan, J. Hegarty, H. Okuyama, A. Ishibashi, and Y. Mori, Appl. Phys. Lett. 66, 180 (1995). 45 H. Luyo, S. W. Short, S. H. Xin, A. Yin, M. Dobrowolska, and J . K. Furdyna, Proc. 22nd Ini. Conf.Physics of Semiconduciors, Vancouver (1994). World Scientific, Singapore. 46 H. Zajicek, P. Juza, E. Abrarnof, 0. Pankratov, H. Sitter, M. Helm, G. Brunthaler, W. Faschinger, and K. Lischka, Appl. Phys. Leti. 62, 717 (1993); Z. Zhu, H. Yoshihara, K. Takebayashi, and T. Yao, Appl. Phys. Lett. 63, 1678 (1993). 41 P. Kelkar, V. Kozlov, P. Kelkar, A. V. Nurrnikko, D. C. Grillo, J. Han, M. Ringle, and R. L. Gunshor (submitted for publication). 4n K. Inoue, T. Kuroda, K. Yoshida, and 1. Suernune, Appl. Phys. Lett. 65, 2830 (1994).

224

A. V. NURMIKKO AND R. L. GUNSHOR 1 .o

0.9 0.8

-

T= 10 K

0.7 0.6

-

0.5 0.4 0'8.!!8

2.79

'

2.80

'

2.81

2.82

2.03

'

2.84

Photon Energy (eV)

FIG. 7. Transmission spectrum of a single OW ZnSe heterostructure at T displaying the n = 1 HH and LH exciton states. [From Ref 47.1

=

10 K,

In addition to obtaining adequate electronic confinement, an important issue with the present status of 11-VI semiconductors concerns the control of point defects that contribute to nonradiative recombination processes. This point is critical for the efficiency of LEDs and impacts on the threshold current density of a diode laser. Perhaps more important, point defects are presently understood to play a major role in the process of degradation and failure of the lasers. Although it is not clear at this writing what the dominant point defects are (presumably vacancies), some progress has been made empirically in their control in epitaxial growth. Temperature dependence of the QW photoluminescence efficiency is usually a direct qualitative indicator of the concentration of points defects. The luminescence spectra from the ZnCdSe QW in an SCH sample (such as Fig. 1) from T = 77 K to room temperature is shown in Fig. 8 where the composition and the thicknesses of the layers have the following typical v a l u e s ~ ~ Z n ~ , ~ ~75C A; d ~ZnS0,07Se0,9,, . ~ ~ S e ~ 2ooo A; Zn0,,,Mg0,,,S0,,7Se0,9,, 8000 A.49 Note that the spectrally integrated luminescence efficiency decreases only by a factor of 4 to 5 over this temperature range. In contrast with a more typical drop by about two orders of magnitude, this modest decrease illustrates how the point defect density can be controlled through careful adjustments of the epitaxial growth conditions. We consider the physics of radiative recombination phenomena, especially under high injection conditions, in more detail in Section V, including real-time measurements of the relevant recombination rates. Finally, in concluding this section, we note that spectroscopy of 11-VI QWs in the presence of applied electric fields has yielded useful additional 4Y

J. Han, R. L. Gunshor, and A. V. Nurmikko, Phys. Stat. Sol. B 187, 285, (1995).

225

VISIBLE LIGHT EMI'lTERS

2.44

2.46

2.48

2.50

2.52

2.54

2.56

2.58

Photon Energy (eV) FIG. 8. Temperature dependence of the QW luminescence from a ZnCdSe/ZnSSe/ ZnMgSSe SCH sample from T = 77 to 300 K [From Ref. 49.1

insight regarding electronic confinement. For example, Wang and cow o r k e r ~ ~have ~ used the quantum confined Stark effect in the ZnCdSe/ZnSe for demonstration of electro-optic modulators in the bluegreen. Since this article is devoted to light emitters, we will not cover this area of potential device importance any further. 3. TELLURIUM ISOELECTRONIC CENTERS IN ZnSe EXCITON-PHONON INTERACTION

AND

In addition to the specific case of the Frohlich interaction discussed earlier, the wide band-gap II-VIs are in general expected to show a more pronounced coupling of electronic and lattice excitations than, for example, GaAs due to the larger polar component in the bond for ZnSe. Such differences influence the issues of doping in a profound way (see Section III), but can also be probed with nonequilibrium electronic excitations (electron-hole pairs). As a particular illustration of the manifestation of the strong coupling of electronic excitations to the lattice, as well as the usefulness of such a process to an LED, the case of isoelectronic centers of Te is quite illustrative. In particular, the study of the ZnSe:Te system by SO S. Y. Wang, Y. Kawakami, J. Simpson, H. Stewart, K. A. Prior, and B. C. Cavenett, Appl. Phys. Lett. 62, 1715; see also 63, 857 (1993).

226


optical spectroscopy yields insight into the physics of exciton self-trapping in a deformable lattice, a phenomenon usually associated with an ionic crystal. The theoretical treatment of self-trapping was largely developed by Toyozowa and c o - w o r k e r ~ ,aimed ~ ~ at ionic crystals such as AgCl. Beyond the optical properties, the observation of strong exciton-phonon coupling in ZnSe:Te has a certain phenomenological kinship and connection to the physics of p-type doping in ZnSe as well. When considering material systems with relatively large electron-phonon interactions, such as a silver halide retain, the elementary excitation of the system is often described as a “phonon-dressed” electron with a renormalized effective mass. In the case of electron-optical phonon interaction in the ionic crystals, the strong electrostatic interaction results in a phonon-dressed electron of “polaron” with a renormalized mass that increases continuously with increasing interaction strength. On the other hand, for the case of the deformation potential interaction between electrons and acoustical phonons, the renormalized mass abruptly increases by several orders of magnitude as the interaction strength reaches a critical value.” The origin of these differences can be traced to the ranges of the respective forces involved. The electron-optical phonon interaction is Coulombic (Ir - r’l-’) at large distances, which implies that there will always be a bound state and therefore no abrupt transition in the nature of the state occurs. In contrast, the deformation potential interaction between electrons and acoustical phonons is short ranged, [ 6 ( r - r’)], essentially vanishing everywhere except where the lattice is distorted. For this case the binding potential must exceed the kinetic energy of the localized particle (the kinetic energy being due to localization as required by the uncertainty principle) if a bound state is to occur. Therefore, the system will undergo an abrupt transition from a free state to a trapped state when this energy condition is met. Figure 9(a) shows schematically, in a configurational coordinate diagram, how, due to the strong exciton-phonon interaction, the system can be stabilized by energy E L , as the lattice relaxes to the new equilibrium position indicated by the lattice distortion Q, = c.” Note in this figure the introduction of the parameter D,which is the fluctuation of the site energy due to lattice vibrations. The other extreme, namely, the free exciton in a rigid lattice which describes the system in the absence of lattice distortion and allowing for site transfer, would be described by the potential represented in Fig. 9(c), where B is the half the free exciton bandwidth. In an intermediate regime, shown in Fig. 9(b), the system may be parameterized by the 51 Y. Toyozawa, in Relaxations of Elementary Excitations, (R. Kubo and E. Hanamura, eds.), pp. 3-18, Springer-Verlag, New York (1980).

227


(a)

(b)

iC)

Fio. 9. Configurational coordinate model for (a) localized exciton, (b) exciton near the free to self-trapped transition, and (c) excitonic band in a rigid lattice. [From Ref. 51.1

parameter g, = E , R / B , which describes the kinetic energy of the exciton system. Specifically, the exciton can become localized by the lattice distortion it induces if EL, > B , and will remain delocalized for E L , < B. Therefore, as the critical condition of g, = 1 is reached, the system undergoes an abrupt transition from a free ( F ) to a self-trapped state (S). This abrupt transition in the nature of the exciton is strongly reflected in the optical emission of the system. Although we expect a relatively sharp resonant emission for the system in the F state, the emission for the S state will be Stokes shifted by 2ELR - B and will be broadband. Although it is true that the 11-VI materials do not generally exhibit the form of “intrinsic” self-trapping that has been observed in silver halides, such behavior has been seen in specific 11-VI materials, including ZnSe and CdSS2in which a small concentration of tellurium has been introduced as an isoelectronic center. In this case, one has the situation where the exciton-phonon interaction may not be large enough for self-trapping to take place, nor is the bare Te impurity potential strong enough to bind the exciton in the rigid lattice, but the cooperative effect of these two shortrange interactions is sufficient to bind the exciton with strong lattice relaxation effects. Such a case has been theoretically modeled51 to show that an self-trapped state can be the lowest energy state provided that the (short-range) isoelectronic impurity potential is sufficiently strong. This effect is therefore called “extrinsic” self-trapping and is the physical phenomenon responsible for the large spectral shifts seen in the emission 42

A. Reznitsky, S. Permogorov, S. Verbing, A. Naumov, Yu. Korostelin, N. Novozhilov, and S. Prokoviev, Solid Sfate Cornrn. 52, 12 (1984); J. D. Cuthbert and D. G. Thomas, J . Appl. Phys. 39, 1573 (1968).

228


FIG.10. (a) Schematic of conduction and valence band potentials for doping of ZnSe QW by ZnTe ultrathin layers. (b) Photoluminescence from (top) an undoped ZnSe/(Zn, Mn)Se QW at the n = 1 lowest exciton, (middle) same QW except for additional Te planar isoelectronic doping, and (bottom) the absorption edge of the middle sample. [From Ref 53.1

of the tellurium-doped 11-VI materials. Figure 10(a) shows a schematic in which the ultrathin ZnTe induces a deepening for the hole potential. Figure 10(b) shows the photoluminescence spectrum at T = 2 K a heterostructure in which the Te isolectronic centers have been introduced by planar doping during the MBE growth within a ZnSe QW. For comparison, the top panel of Fig. 1Mb) shows the photoluminescence from an undoped QW for the n = 1 lowest exciton, and the bottom panel displays the absorption edge of the middle The absorption edge for both the undoped and &doped samples exhibits the onset of interband exciton absorption as a sharp ( < 5-meV) resonance. Although the emission is also narrow in the undoped QW, it is significantly broadened and Stokes shifted on the scale in excess of 100 meV for the isoelectronically doped case. The structure in the emission of the middle panel of Fig. 10(b) is related to the formation of two dominant groups of small Te clusters as 53

L. A. Kolodziejski, R. L. Gunshor, Q. Fu, D. Lee, A. V. Nurmikko, J. M. Gonsalves, and N. Otsuka, Appl. Phys. Lett. 52, 1080 (1988); Q. Fu, D. Lee, A. V. Nurmikko, R. L. Gunshor, and L. A. Kolodziejski, Phys. Reu. E 39, 3173 (1989).


229

a consequence of finite interdiffusion of the monolayer Te “sheet” into the adjacent region (on a monolayer scale). In our interpretation, small Te clusters initially provide a locally (atomic scale) attractive potential, chiefly for the holes. Upon capture, the interaction with the lattice leads to configurational coordinate changes so that the hole’s self-energy is effectively renormalized as much as 300 meV, depending on the details of the Te cluster. Magneto-optical spectroscopy has further verified how the exciton selftrapping in such ZnSe:Te heterostructures occurs chiefly as a result of the hole state becoming strongly localized. As one consequence, the radiative lifetime of the recombining exciton increases by approximately two orders of magnitude (from about 100 ps for the free exciton at T = 10 K to more With increasing Te content (from one to four monolayers of than 10 ZnTe), the Stokes shifts increases to approximately 0.5 eV, thereby providing, in principle, a means of constructing light-emitting structures from the blue to the yellow within the ZnSe:Te system. These possibilities are discussed in more detail in Section IV.

4. TOWARDQUANTUM

WIRES AND

DOTS

Beyond quantum wells, there are ongoing efforts to fabricate nanostructures in wide band-gap 11-VI semiconductors, where size-dependent effects due to optical and electronic confinement may provide additional flexibility for blue-green light emitters. One specific goal is the realization of a quantum wire (and/or a quantum dot) laser. Given the enhancement by 2D confinement on the exciton properties in quantum wells, there is also ample motivation to pursue the fabrication of such lower dimensional structures from a basic point of view. The standard attempts to fabricate quantum wires and quantum dots generally rely on lateral patterning to complement the usual layered growth. Relatively broad “wires” (widths > 1 pm) are now routinely fabricated from ZnSe-based multilayer heterostructures for index guide laser devices, as described in Section IV. The methods of fabrication are based on both dry and wet chemical etching techniques, following appropriate sequence of lithographic steps. In addition to these etching methods, a impurity-induced disordering technique has been introduced recently as an alternative for optical and possible electronic confinement in the lateral direction.55 The fabrication chal54 D. Lee, Q. Fu, A. V. Nurrnikko, R. L. Gunshor, and L. A. Kolodziejski, Superlattices Microstruct. 5, 345 (1989). 55 T. Yokogawa, P. D. Floyd, M. M. Hashemi, J. L. M e n , H. Luo, and J. Furdyna, Appl. Phys. Lett. 62, 3488 (1993).

230


lenges, however, increase rapidly because lateral dimensions below 100 mm are being sought. There is only limited progress so far in the wide-gap 11-VI semiconductors, due to the detrimental role played by the surface damage generated in the fabrication process. The Glasgow group of Sotomayor Torress6 have investigated the use of electron-beam lithography combined with various dry etching agents to prepare wires on ZnTe- and ZnSe-based heterostructures. Ding et aLS7 have also employed such methods to study systematically the optical properties of quantum well wire (QWW) geometries, specifically through the measurement of the photoluminescence (PL) efficiency as a function of wire widths as small as 500 A. The quantum wells are based on the (Zn,Cd)Se/Zn(S,Se) system, which forms the active region in most of the recent blue-green diode laser work. Measurement of the PL efficiency as a function of temperature shows that the surfaces created in the reactive ion (dry) etching process play a detrimental role in providing a large density of nonradiative centers, which substantially reduces the luminescence efficiency for narrow QWW samples. Scanning electron microscope (SEM) images of typical fabricated wires are shown in Fig. 11, with the wire widths ranging from approximately L , = 3000 A to 500 A?’ The slightly overcut (triangular) profile is a result of the edge of the titanium mask being partially removed in the dry etching process. This conclusion is drawn from the SEM comparison between the titanium mask width before and after the reactive ion etching (RIE) process. The measured PL from the QWW structures originated from the n = 1 heavy hole exciton transition. Study of the PL efficiency as a function of both wire width and temperature shows that the surfaces generated by the dry etching process play a dominant role as sources of nonradiative recombinations centers. Figure 12 shows the relative PL efficiencies of the wire samples with L , = 1300 8, (solid circles), L , = 3400 A (triangles), and the original unpatterned QW sample (squares), respectively, at different temperatures. The lines are guides for the eye and the relative PL efficiency for each sample is normalized to its value at T = 10K. One can see that PL efficiency from the QWW sample decreases much faster with temperature than that of the QW sample, which is indicative of the existence of nonradiative recombination mechanisms in addition to those for normal QW structures. Therefore, surface passivation techniques for such wide band-gap 11-VI nanostructures are imperative 56

M. A. Foad, A. P. Smart, M. Watt, C. M. Sotornayor Torres, and C. D. W. Wilkinson, Electron. Lett. 27, 73 (1991); M. A. Foad, A. P. Smart, M. Watt, C. M. Sotomayor Torres, C. D . W. Wilkinson, W. Kuhn, H. P. Wagncr, S. Bauer E. Leiderer, and W. Gebhardt, Semicond. Sci. Technol. A6 115 (1991). 57 J . Ding and A. V. Nurmikko, Appl. Phys. Left.63, 2254 (1993).

231


FIG. 11. Scanning electron micrographs of QWW structures fabricated by electron-beam lithography and RIE processes. The pitch size is 6000 A. The approximate wire width is indicated. [From Ref. 57.1

before their design and application for blue-green light emitters can be pursued. An alternative method to obtain lower dimensional structures is by some means of lateral modulation in the epitaxial growth process itself. Recently, self-organizing growth of quantum dots, for example, has been demonstrated in highly strained 111-V semiconductors. In 11-VI semicon-

100-

90 80

c

*

...

~

70

60 50 40

30 20

' ~

~

~

-

10 -

0

r

. .

. .+ A

I

~ ~ O O A Q W W 3400AQWW QW

*

I

*

*' I*

Temperature (K) FIG. 12. Relative photoluminescence efficiencies (solid circles), defined as the spectrally integrated PL intensity multiplied by the ratio of the pitch size and the wire width at T = 10 K for wires of different widths. The efficiency of the quantum well sample before the fabrication processes is normalized to be 100%. The solid curve is a fit of the data. [From Ref. 57.1

232


700

19

600

Wavelength (nm) 500

1

.

I

* '

I

I

400

I

I

,

I

,

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Energy (eV)

FIG.13. Absorption spectra at T = 10 K for CdSe quantum dots embedded in optically clear polyvinyl butyral films. [From Ref. 60.1

ductors, Magnea58 has reported on submonolayer features of ZnTe grown in a CdTe matrix to produce quantum dots that show evidence of exciton localization. Epitaxy with lateral features on a nanometer scale, whether through coherent self-assembly or by growth on patterned substrates, has much potential in terms of both basic and applied research in the 11-IVs and will undoubtedly grow as a research area in the near future. A very different recent approach toward the fabrication of quantum dots in 11-VI semiconductors has been through the chemical synthesis of ~ been nanocrystallites in either liquid or solid media. Norris et u L . ~ have particularly successful in demonstrating the preparation of CdSe quantum dots by a technique that involves the pyrolysis of organometallic reagents by injection into a hot coordinating solvent. Figure 13 shows the absorption spectra of such monodisperse dots imbedded in polyvinyl film at T = 10 K, demonstrating clearly the effects of electronic confinement for a wide range of particle diameters D (in Angstroms). Elsewhere, the inclusion of fluorescent centers (Mn ions) in CdS quantum dots has been "N. Magnea, in Proc. 6ih International Conference on II-VI Compounds, Newport, RI, J . Crysf. Growth 138, 550 (1994). 59D. J . Norris, A. Sacra, C. B. Murray, and M. G. Bawendi, Phys. Reu. Leif. 72, 2612 (1994).


233

recently studied with a view toward application in new forms of light-emitting material.60

111. Doping and Transport

In this section we treat the very broad and still not completely understood subject of doping in the wide-gap 11-VIs by focusing selectively on those issues and accomplishments that are most relevant to present blue-green LEDs. Our aim is not to provide a comprehensive technical treatment but rather to give the reader a physical flavor of the “critical exponent” among the physical properties in these semiconductors and their heterostructures. The subject is still a developing one and the reader is encouraged to gain more insight of current progress, for example, from the Proceedings of the International Conference on 11-VI Semiconductors?.

5. OVERVIEW AND CURRENT STATUS The term semi-insulators would long have been more applicable to the 11-VI compounds, which confounded researchers by being apparently impossible to dope in an ambipolar sense. Apart from intrinsic reasons for such difficulties, bulk materials also suffered from an omnipresence of extrinsic defects that obstructed progress.* Differing contemporary theoretical viewpoints exist regarding the intrinsic microscopic description for ambipolar doping of wide-gap 11-IVs. Extrinsic problems with impurities in bulk materials, which led to models of sometimes complex defect chemistry, have by now been substantially overcome by epitaxial growth techniques. Thus the gap between experiment and theory has considerably narrowed and a reasonable physical understanding has emerged to guide experiment in the choice of specific dopants for device applications. In ZnSe, which is our main focus here, n-type doping with a shallow donor such as C1 leads readily to electron concentrations of 10’’ cm-3 and beyond MBE,h’,62as illustrated in Fig. 14 from Zhu et who employed planar doping to reach very high electron densities. Gas source MBE was 60

R. N. Bhargava, D. Callagher, X. Hong, and A. Nurmikko, Phys. Reu. Lett. 72, 416 (1994). hl K. Ohkawa, T. Mitsuyu, and 0. Yamazaki, J . Cyst. Growth 86, 329 (1987). h2 K. Ohkawa, T. Mitsuyu, and 0. Yamazaki, J . Appl. Phys. 62, 3216 (1987). h3Z.Zhu, H. Mori, and T. Yao, Appl. Phys. Lett. 61, 2811 (1992).

234


0

-0000

-

..LIILyyI.14.Y..

10’3 o A o

.

planar

F:

E

0

G

TEMPERATURE (K)

FIG.14. Temperature-dependent resistivity and electron concentration for heavily n-doped films of ZnSe by planar (open symbols) and uniform (closed symbols) methods. [From Ref. 63.1

also shown to produce high n-type doping levels.64 On the other hand, p-type doping with column V elements other than nitrogen (such as As and P) has been strikingly unsuccessful (except for lithium, which becomes highly mobile in the lattice under thermal electrical excitation, however). For ZnTe, heavy p-type doping is readily accomplished with nitrogen,65 while meaningful n-type doping is yet to be achieved. More generally, although very qualitatively, the relatively high degree of polarity associated with the chemical bonds in ZnSe and related compounds plays a direct role in making doping a much more significant challenge than in, for example, Si or GaAs. Pedagogically, it is useful to examine the relative ratio of the covalent and polar components to the total bond energy, which is shown in Table I for GaAs, ZnSe, and GaN (after the definition and calculation of these “hybrid” bond energies by Harrison66. The table also shows the vast differences in epitaxial growth temperatures of the three compounds. MP. A. Fisher, E. Ho, J. L. House, G. S. Petrich, L. A. Kolodziejski, M. S. Brandt, and N. M. Johnson, Mat. Res. Soc. Syrnp. Proc. 340, 451 (1994). 65 J. Han, T. Stavrinides, M. Kobayashi, M. Hagerott, and A. V. Nurmikko, Appl. Phys. Lett. 62, 840 (1993). hh W. A. Harrison, in Electronic Structure nnd the Propeaiies of Solids, Chap. 7, Freeman & Co., San Francisco (1980).

235

VISIBLE LIGHT EMITTERS TABLEI. COMPARISON OF VALUES Tgrowth

(“C)

GaAs ZnSe GaN

650 300 800

HYBRID COVALENT ENERGY (eV)

5.55 5.55 8.85

HYBRID POLARENERGY (eV)

1.88 3.80 3.92

Although not a unique definition of the bond polarity, an observation can nonetheless be immediately made that ZnSe has the largest ratio of the polar to covalent energy of the three compounds (although GaN has the largest polar energy in absolute terms). This aspect has an influence on many of the physical properties of wide-gap 11-VIs, ranging from electrical to optical, and to mechanical. In the context of doping, there is already a hint from the comparison of values in Table I that the expected coupling of electronic states to the lattice is likely be a factor influencing the energetics of formation of donor and acceptor impurities in ZnSe, that is, increasing the likelihood of the formation of deep (electrically inactive) levels. Next we outline some of the present issues associated with p-type doping in ZnSe and its alloys Z d S , Se) and (Zn, Mg)(S, Se), that is those compounds that form the present nucleus for the blue-green light emitters. We address the important question of the practical limits to free hole densities that can be expected with these materials by MBE growth. The free hole densities figure dominantly in determining the shaping of vertical and lateral transport in the heterostructures, including interface resistances and contact impedances. Of course, in a high current density environment such as a laser device Joule losses present a major problem of heat dissipation and contribute to device degradation and device failure. As part of our discussion we review recent transport measurements on epitaxial layers that have shed considerable light on the nature of acceptor states for the case of nitrogen doping. Following this we turn to vertical transport and review the techniques of forming low-resistance contacts to the p-type materials. 6.

TYPE DOPING OF ZnSe AND RELATED COMPOUNDS

As mentioned in the introduction to this article a major advance in the doping of ZnSe p-type material occurred in 1990 when Park et af.8 and, independently, Ohkawa et aL9 showed how net acceptor concentrations slightly in excess of 10” cm-3 and resistivities on the order of 1 Qcm

236


could be achieved by using nitrogen as the substitutional acceptor. The nitrogen was derived from an rf plasma source, and this method of doping is now the norm in many laboratories worldwide. The achievement of p-doping made the fabrication of useful p-n junctions a reality and led rapidly to the first demonstration of the blue-green diode lasers, a subject that is described in the following section. Although doping with nitrogen in the context of MBE has been successful to the extent of obtaining useful resistivities in p-type ZnSe, note that attempts at p-doping by MOCVD and gas source MBE approaches have been frustrated by the detrimental effects of compensating hydrogen, which is generally present in the dopant source^.^^-^^ In the MBE approach, the introduction of the substitutional nitrogen is implemented by in situ incorporations of a nitrogen rf plasma cell within the growth chamber. There have been differing viewpoints about the dynamics of the process of nitrogen incorporation at the growth surface at a microscopic level, mainly involved with whether the adsorbed species is in an atomic or excited molecular state.70 We do not review this controversy here, although it is important to note that the use of a plasma-based source (whether rf or electron cyclotron resonance) inherently produces a copious range of nitrogen in excited states and dissociation products, some of which may contribute little or have an adverse effect. A more ideal source of atomic nitrogen both for the MBE and (especially) for the MOCVD growth awaits further development. Such a source would have implications beyond the doping of ZnSe, because the growth rates of the group I11 nitrides would presumably be impacted by increased atomic species of nitrogen. The physical pictures describing restricted doping as currently experienced with MBE growth have traditionally been divided into issues of compensation due to defect generation7' or processes that govern the solid solubility of the substitutional dopant.'* One challenge to theory73 has 67

J. A. Wolk, J. W. Ager, K. J . Duxstad, E. E. Haller, N. R. Taskar, D. R. Dorman, and D . J. Olego, Appl. Phys. Lett. 63,2756 (1993). 68 A. Kamata, H. Mitsuhashi, and H. Fujita, Appl. Phys. Lett. 63,3353 (1993). 69 E. Ho, P. A. Fisher, J. L. House, G. S . Petrich, L. A. Kolodziejski, J. Walker, and N. M. Johnson, Appl. Phys. Lett. 66,1062 (1995). 70 For example, T. Uenoyama, T. Nakao, and M. Suzuki, in Proc. 6th International Conference on 11-VI Compounds, Newport, RI, J . Cryst. Growth 138,301 (1994). 7'D. J. Chadi, Phys. Rev. Lett. 72, 534 (1994). 72 D. B. Laks, C. G. Van de Walle, G. F. Neumark, and S. T . Pantelides, Phys. Reu. Lett. 66, 648 (1991); D. B. Laks, C. G. Van de Walle, G. F. Neumark, P. E. Blochl, and S. T. Pantelides, Phys. Rev. B 45, 10,965 (1992). 73 Y. Marfaing, in Proc. 6th International Conference on 11-VI Compounds, Newport, RI, J . Cryst. Growth 138,305 (1994).

237

VISIBLE LIGHT EMITTERS l 0 ' 8 p

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FIG. 15. Temperature dependence of the free hole concentration in pZnSe:N, p Z d S , Se):N, and (Zn, MgXS, Se):N epitaxial layers, obtained from Hall measurements. [From Ref. 74.1

been to justify the choice of nitrogen as the p-type dopant in ZnSe and its ternary and quaternary alloys. The importance of lattice relaxation has been emphasized by Chadi7' who has modeled deep level formation in p-ZnSe in analog to the DX center problem in (Ga, AI)As by first principle total energy calculations. The predictions are that unsuccessful acceptor candidates such as P or As distort the tetrahedral bonding arrangement sufficiently strongly (including bond breaking) to result in the formation of deep levels. For p-ZnSe, calculated trends among the different column V (substitutional) acceptors suggest that the lightest element, nitrogen, is least likely to form a deep level, localized state. Experimentally, a series of observations has been made recently in Hall experiments on p-ZnSe:N, p-Zn(S, Se):N, and (Zn, Mg)(S, Se):N epitaxial layers, under nominally the same nitrogen doping conditions. These observations provide a glimpse into how the DX-like lattice relaxation effects may develop as the band gap and the degree of disorder increase.74 Figure 15 displays the hole concentration as a function of inverse temperature for these epilayers. Two observations are immediate: (1) The hole concentration decreases distinctly as the band gap increases, while ( 2 ) the acceptor activation energy deepens from about 90 meV to nearly 180 meV. In fact, closer examination of the Hall data for the quaternary shows a second slope, indicative of a second acceptor state, hence suggesting deviations from a simple hydrogenic behavior. 74 J. Han, in Proc. 6th International Conference on 11-VI Compounds, Newport, RI, J . Cryst. Growth 138, 464 (1994).


238

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Fic. 16. Temperature dependence of the free hole concentration in p-(Zn, Mg)(S, Se) in the dark and under illumination, showing the persistent photoconductivity effect. The inset shows a configurational coordinate diagram for the AX center. [From Ref. 75.1

The nature of the deeper acceptor state in p-ZnMgSSe is elucidated in Fig. 16, which shows the photoactivation of the free hole states at lower lattice temperatures. This persistent photoconductivity effect was observed in p-(Zn, Mg)(S, Se) epitaxial layers in recent measurements by Han et al.” The resistivity of the epitaxial layer decreases by several orders of magnitude upon illumination with light at wavelength of approximately 900 nm (1.4 eV) or shorter. From corresponding Hall data, the free hole concentration is similarly modulated by light. The transport experiments support the proposition that this acceptor state is configurationally lattice relaxed. The inset to Fig. 16 is drawn in the configurational coordinate representation, which implies the coexistence of free hole and metastable “hydrogenic” acceptor states with a lattice relaxed AX state at a deeper energy. Two potential energy barriers exist in this schematic, separating the lattice relaxed state from the free and shallow metastable acceptor states, respectively. The situation is rather similar to the DX center in (Ga,AI)As except that the holes are at issue; in fact, a DX-like behavior 7s J. Han, M. Ringle, Y. Fan, R. L. Gunshor, and A. V. Nurmikko, Appl. Phys. Lett. 65, 3230 (1994).


239

had not previously been observed for acceptors. These observations show that although there is no evidence for AX centers in ZnSe:N, the challenges to p-type doping increase substantially with increasing Mg and S concentrations in ZnMgSSe, as one attempts to design light emitters for deep blue wavelengths. The occurrence of lattice relaxation is, of course, not unexpected in the relatively polar wide-gap 11-VI semiconductors. As already shown in the previous section, Te isoelectronic centers were shown to exhibit trends toward extrinsic self-trapping for photoholes both in bulk material as well as in quantum wells.53 Furthermore, persistent photoconductivity effects have been recently observed in other n-type 11-VI semiconductors such as the DX centers in CdTe and (Cd, Mr~lTe.~' Of special relevance to the discussion of contacts to p-ZnSe discussed in the next section is the observation that the nitrogen plasma cell, so effective for the p-doping ZnSe, can also be used to implement even higher p-doping levels in ZnTe. It was found that when the ZnTe epilayers were grown under the same MBE growth conditions, and with the same nitrogen plasma operating conditions as for p-ZnSe, free hole concentra~ , easily obtained. In choosing between tions exceeding 1 X 1019 ~ m - were the self-compensation picture and the arguments for solubility, the compensation picture does not appear to provide the limiting factor for the growth conditions reported here-transport measurements conducted by the authors revealed a compensation ratio in ZnSe:N epilayers of 10% or less. Some insight into the factors limiting p-doping levels in ZnSe resulted from combining SIMS measurements, shown in Fig. 17, with temperaturedependent Hall data for a comparison between ZnSe:N and ZnTe:N for samples grown under the same condition^.^^ SIMS indicated a nitrogen ~ , is surprisingly consistent with density in ZnSe of 1.2 X 10l8 ~ m - which the nitrogen acceptor concentration of N, = 0.5 X 1017 cm-3 derived from temperature-dependent Hall effect measurements on this sample. It is therefore concluded that, at the doping levels and particular growth conditions reported here, close to 100% of the nitrogen atoms incorporated in ZnSe are substitutional at the Se lattice sites and serve as active acceptors. It is worth mentioning that, while large nitrogen concentrations (in the 10'' cm-3 range) have been reported, the net acceptor concentrations were still below l o t x ~ m - The ~ . implication is that in those experi76 N. G. Semaltianos, G. Karczewski, T. Wojtowicz, and J. K. Furdyna, Phys. Reu. 47, 12540 (1993). 77 Y. Fan, J. Han, L. He, R. L. Gunshor, M. Brandt, J . Walker, N. Johnson, and A. V. Nurmikko, A&. Phys. Lett. 65, 1001 (1994).

240


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FIG. 17. Depth profiles of nitrogen concentration for heavily doped ZnSe and ZnTe epitaxial layers grown under similar conditions, from SIMS measurements. [From Ref. 77.1

ments either the excess nitrogen was not incorporated substitutionally or there was a significant degree of compensation from donor-like defects. In the case of ZnTe the nitrogen density was found to be much higher, for example 8 X lo1' cmP3,than that obtained for ZnSe, with a free hole concentration of 7 X lo'* cmP3. Again, as in the case of ZnSe:N grown under the same conditions, 100% of the nitrogen atoms incorporated into ZnTe are at the Te lattice sites and serve as acceptors. One is led to the conclusion that, at least for the MBE growth conditions employed in the authors' laboratory, the p-doping level in ZnSe:N is limited by the solubility of the nitrogen. Of course, it is not clear to what extent the existing solubility theory, which was based on assumptions of thermodynamic equilibrium, is applicable to the kinetics of MBE growth. Concluding this brief review of the status of p-type doping in ZnSe and its wider band-gap alloys, one might summarize the current situation as follows: On the one hand, the practical levels of doping that can be routinely achieved in MBE growth appear adequate for LED and diode laser exploration in the green, as shown in the next section. At a microscopic level, however, there is still considerable uncertainty, for example, when considering the incorporation of nitrogen dopants on substitutional and relaxed sites, especially in the alloys of ZnSe. In the widest band-gap alloys the maximum levels of acceptor atom incorporation yield only a fraction as useful free holes. This leads to device resistances that are still too high, especially for the diode lasers. With increasing band gap up to and beyond 3 eV in the ZnSSe ternary and ZnMgSSe quaternary materials


24 1

it seems that the effective levels of doping exhibit precipitous decreases at the point when the existence of a deep, lattice relaxed state becomes likely. In that event, multilayer designs that employ band structure engineering ideas need to be called on. Furthermore, exploration of other wide-gap ternaries and quaternaries, in which Mg is substituted for by other column IIa elements (including ZnS and CdS as base compounds), has barely begun. 7. DEVELOPMENT OF LOW-RESISTANCE CONTACTS TO p-ZnSe

In this section we focus on the problem of electrical (ohmic) contacts, since much recent work has been expended in this key area. Undoubtedly, much more information about vertical transport will be forthcoming as a result of ongoing research efforts. One common denominator that underlies the present difficulties in general is the still rather low level of p-type doping in the ZnSe-based wider-gap alloys. Attempts to obtain a low-resistance contact between a metal and a wide band-gap semiconductor ( p - and n-type) present a fundamental dilemma of electrically bridging two vastly dissimilar materials in terms of their electronic band structure. The problem scales, roughly, in its degree of difficulty with the band gap of the semiconductor. In the case of ZnSe and its alloys, the relative ease of achieving heavy n-type doping, together with the position of the surface Fermi level, presents a moderately low barrier for electron injection with many common metals (such as indium). Furthermore, in the typical diode laser structures, the electron injection occurs from n-GaAs into the n-type ZnSe-based material so that the problem is transformed to a question about the microscopics at the heterovalent heterointerface. Some potential energy discontinuity is present, perhaps up to 300 meV for GaAs/ZnSe (and a few hundred meV for GaAs/ZnMgSSe), and, together with interface traps, contributes to a finite heterojunction impedance (hence a voltage drop).74 Although at present this is not the major hurdle, issues about the influence of the microstructure at the heter~interface~' (i.e., interface states) remain open and are the subject of ongoing research. The main challenges exist at the p-ZnSe/metal contact, where two contemporary approaches are being followed: (1) the implementation of a graded p-ZnTe/ZnSe heterostructure to facilitate an ohmic ont tact^^.'' and (2) the deposition of a narrow 78

C . T. Walker, J . M. DePuydt, M. A. Haase, J. Qui, and H. Cheng, Physica B 185, (1993). Y. Fan, J. Han, L. He, J. Saraie, R. L. Gunshor, M. Hagerott, H. Jeon, and A. V. Nurmikko, Appl. Phys. Left. 61, 3160 (1992). '"F. Hiei, M. Ikeda, M. Ozawa, T. Miyajima, A. Ishibashi, and K. Akimoto, Electron. Lett. 29, 878 (1993). 7')

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band-gap semimetallic layer of p-HgSe.8’ In this review we highlight approach (1) from the authors’ laboratories while noting that at this writing it is undetermined whether any of these contact schemes is sufficiently robust and offers low enough contact resistance to withstand the requirements of a practical (cw) blue-green diode laser device. This particular contact scheme has also been demonstrated at Sony Laboratories, where the interpretation, however, was based on a different physical concept of resonant hole tunneling.*” As already noted, the approach that has been found effective for the p-doping of ZnSe and its alloys by nitrogen can be used to obtain a very high degree of p-doping within the MBE growth of ZnTe.65 Free hole concentrations approaching 10” cm-3 have been readily achieved. For example, palladium, platinum, gold, and their multilayers form a low-resistance contact to such highly doped p-ZnTe epilayers,** so that one is naturally led to consider the use of ZnTe as an intermediate “electrical buffer” for contracting to p-ZnSe. However, the large valence band offset (AE[, = 1 eV) between ZnTe and ZnSe layers forms a barrier to the hole injection at a p-ZnTe/p-ZnSe interface (ZnTe/ZnSe is a type I1 QW). The consequence of this is shown schematically in Fig. 18(c). A possible solution for removing the energy barrier in the valence band (in reality a potential energy spike due to the electrostatic charge redistribution) is to introduce a graded band-gap p-ZnSe,_,Te, layer in which the Te concentration varies from y = 0 to 1 across such a contact layer. Due to the practical difficulty of controlling the Te concentration in an MBE-grown Zn(Se, Te) alloy (selenium and tellurium compete for surface incorporation), a “pseudograded” p-type, strained ZnTe/ZnSe ultrathin layer structure (20 8, per cell) was designed and grown, with the ZnTe and ZnSe layer thicknesses in each cell varying to approximate a graded band-gap material.79 Although the overall “superlattice” thickness ( = 340 8,) exceeds substantially the critical thickness for the 7% lattice mismatch between ZnSe and ZnTe, electron microscopy shows that the dislocations in the contact layer do not propagate into the adjacent quaternary in a SCH-SQW diode laser structure. Results of conductivity measurements of p-ZnSe epilayers for three different contacting schemes consisting of (1) Au/pZnTe/p-Zn(Se, Te)/ p-ZnSe, housing the graded pseudoalloy, (2) direct Au/p-ZnSe, and (3) an Au/p-ZnTe/p-ZnSe heterostructure are compared in Fig. 18. As seen in Fig. Ma), the graded contact appears to be quite ohmic, showing a straight ni 82

J. Ren, Y. Lanzari, J . W. Cook, Jr., and J. F. Schetzina, J . Electr. Mat. 22, 973 (1993). M. Ozawa, F. Hiei, A. Ishibashi, and K. Akimoto, Electron. Lett. 29, 503 (1993).

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FIG.18. Current-voltage characteristics at room temperature for contact arrangements on p-ZnSe epilayer whose schematics are shown in the inset: (a) Au/Zn(Te,Se) graded gap layer/p-ZnSe; (b) Au/p-ZnSe, and (c) Au/p-ZnTe/p-ZnSe. [After Ref. 79.1

line through the origin. The inset shows that the I-V characteristics maintain the same slope even at a few millivolts from the origin. Figure 18(b) shows the characteristics of the contact formed by gold deposited onto an as-grown ZnSe:N epilayer. The I-V characteristic corresponds to two back-to-back Schottky diodes; the observed turn on voltage is the reverse-bias breakdown voltage. Increasing the doping level is expected to reduce the turn-on voltage of the contact. Figure 18(c) shows the I-V characteristic when p-ZnTe is used to inject holes into the ZnSe epilayer in the absence of the graded region. The specific contact resistance for a structure such as (i) but with Au replaced by Pd was determined by transmission line measurements to be as low as 3 x R cm2, which can be considered acceptable with LEDs but is somewhat high for laser diode devices. One might well expect an improvement in performance of this contact scheme as the growth is modified to more closely approach a continuous alloy grading and as p-doping levels in the selenide layers are increased. We close this section by noting that many other topics related to transport in II-VI heterostructures are potentially important to the future of the blue-green light emitters. Because the overwhelming majority of devices are presently fabricated on GaAs substrates and buffer layers, the issue of vertical transport across the (n-type) III-V/II-VI heterointerface is of direct relevance. It is currently assumed that this interface adds to the

244


voltage burden of an LED or a diode laser by less than 1 V due to the small conduction band offset; however, only preliminary studies of this transport have been conducted.78 Vertical transport across the 11-VI heterointerfaces in the light emitters is not yet completely understood and needs to be examined closely, for example, from the viewpoint of leakage currents in the QW diode lasers. In addition, lateral transport in the heterostructures is another subject that will have a major impact on future device geometries such as the surface-emitting lasers. Finally, and perhaps most important from a technological point of view, the question of p-type doping in MOCVD and MOMBE growth of ZnSe and its alloys still appears unanswered. In spite of reports of some success with nitrogenbased doping and demonstration of LED action,83 a blue-green diode laser grown by MOCVD or by other forms of gas source epitaxy awaits realization. IV. Blue-Green LEDs and Diode Lasers

This section is devoted to the device concepts and engineering of 11-VI blue-green light emitters, which at this writing (late 1994) are under basic research and development in a number of laboratories worldwide. It is very important for the reader to recognize that from a technological perspective, these devices are still in their infancy, following the the first demonstrations of the diode laser at a cryogenic temperature in 1991.'0-'' Today, both LEDs and diode lasers are subject to a wide spectrum of vigorous studies that range from their basic physics of operation to device design and engineering. For example, because the first demonstration of the cw diode laser operation are barely a year ~ l d , ~ the ~ ~technologically * ~ v ~ ~ crucial issues of device degradation and reliability have only now become feasible and relevant subjects for systematic laboratoy study. On the other hand, given the pace of progress and the technical accomplishments of the past three years, there is considerable optimism about further advances that can be expected with the blue-green light emitters on their journey 83T.Yasuda, I. Mitsuishi, and H. Kukomoto, Appl. Phys. Lett. 52, 57 (1988); Sz. Fujita, T. Asano, K. Maehara, T. Tojyo, and Sg. Fujita, in Proc. 6th International Conference on II-Vl Compounds, Newport, RI, J . Cyst. Growth 138, 737 (1994). 84 a N. Nakayama, S. Itoh, K. Nakano, H. Okuyama, M. Ozawa, A. Ishibashi, M. Ikeda, and Y. Mori, Electron. Lett. 29, 1488 (1993). X4 b N. Nakayama, S. Itoh, K. Nakano, H. Okuyama, M. Ozawa, A. Ishibashi, M. Ikeda, and Y. Mori, Elecron. Lett. 29, 2194 (1993). 85 A. Salokatve, H. Jeon, J. Ding, M. Hovinen, A. Nurmikko, D. C. Grillo, J. Han, H. Li, R. L. Gunshor, C. Hua, and N. Otsuka, Electron. Lett. 29, 2192 (1993).

VlSlBLE LIGHT EMITTERS

245

toward future optoelectronic applications such as optical storage and multicolor displays. Current laser devices are still subject to very rapid degradation and failure (by the standards of a mature technology), and continued improvement in underlying material quality and the ability to control defects, both electronic and mechanical, are examples of key issues in this context. From another viewpoint, the ability to approach material challenges by designing heterostructures for tailored physical properties, coupled with the very attractive optical properties inherent to 11-VI semiconductors, offers much space for innovative device physics and engineering for future light emitters. In this section, we describe the principal technical approaches that have been implemented in the design of the blue-green LEDs and diode lasers, including a status report on recent progress with these heterostructures. 8. LIGHT-EMITTING DIODES

The conventional approach to visible LEDs is through the use of p-n junctions fabricated from the bulk or configured as wide heterostructures. Until recently, the only commercially available blue LEDs were based on Sic, with electrical-to-optical efficiencies on the order of The devices were fabricated from bulk Sic, which, as an indirect gap material, is unsuitable for a diode laser. Due to the indirect nature and the presence of a variety of crystal defects and polytypes, the emission from the S i c LEDs is spectrally somewhat broad; however, the cost of commercially available components is quite low. Within the past couple of years, epitaxially grown GaN has shown promise as a LED, and bright (In, Ga)N/GaN p-n junction emitters at output powers reaching the milliwatt range and external efficiencies of about are now being produced in Japan.8h In the GaN blue emitters, the dominant radiative transition is centered around A = 450 nm and involves Si, Mg and/or Zn impurities within the active region of the heterostructure. The role of the impurities is in part to provide efficient capture of electron-hole pairs and subsequent radiative recombination at localized, relatively long-lived ( - 500 ns) fluorescent electronic centers. Such optical emission process are not compatible with the physical requirements for a diode laser; furthermore, limitations in the substrates of choice have complicated the fabrication of optical resonator structures as well. However, considerable research re86

S. Nakamura, T. Mukai, and M. Senoh, Appl. Phys. Lett. 64, 1687 (1994).

246


sources are now being invested at both materials and device development level, and good progress is being made with AIN, GaN, InN, and their heterostructures so that the prospects of a diode laser demonstration are improving. In many ways, the wide-gap nitrides represent a class of semiconductors that is both complementary to the 11-VI compounds yet share similarities in terms of the challenges presented by their physical properties (e.g., p-type doping) in the development of light emitters at short visible wavelengths. The main purpose of this section is to examine the physics of spontaneous emission in the current 11-VI LEDs to show the reader that there are good prospects for these emitters from a fundamental point of view. Many of the current issues that are connected to device reliability and lifetime share common elements with the diode lasers; we defer this discussion to Section IV.9. In contrast to the current wide heterostructure designs in nitride-based LEDs, or the bulk-like configuration in S i c devices, it appears that a QW configuration is fundamentally beneficial for an LED in the wide-gap 11-VI compounds. This conclusion is based in part on the efficient electron-hole pair collection by a QW in a properly designed heterostructure, as well as on the strong electron-hole pair Coulomb pairing effects, which can enhance the radiative transitions in ZnSe-based quantum wells even at room temperature. In an LED, the key requirement for the active region is to provide a radiative recombination rate that is competitive with the nonradiative processes due to impurities, defects, etc., in the active region of the device. Selected impurities or defects may, on occasion, serve as useful centers for localizing electronic excitations for a subsequent bound transition, however, and we return to the example of isoelectronic centers for green 11-VI LEDs below. As already discussed in Section 11, in a narrow ZnSe-based QW ( L , = 50 A), the quasi-2D exciton binding energy can exceed the LO phonon energy so that the dissociation of excitons into free electron-hole pairs in inelastic ionizing collisions with the phonons is much reduced. Instead, such scattering leads to a population distribution among exciton excited states, which can feed the exciton ground state at wavevector K = 0 for efficient radiative recombination, provided that significant nonradiative processes due to defects, etc., are not present. The blue-green LEDs that have so far been demonstrated in ZnSe-based heterostructure, in which the interband transitions, unaided by impurity states, provide the means of radiative transitions, are typically configured as p-n heterojunctions with ZnCdSe QW as the active layer material and ZnSSe and/or ZnMgSSe cladding layers to provide electronic confinement. Since these structures are also very similar to those employed in the diode lasers described later, we refer the reader to Section IV.9 for more details. The current-voltage and light output versus current are shown in

VISIBLE LIGHT EMI'ITERS

247

Currant @A) FIG.19. Output power from a blue room-temperature (Zn, Cd)Se/Zn(S, Se) MQW LED as a function of input current. The inset shows the I-V characteristics. [From Ref. 87.1

Fig. 19 for a ZnCdSe/ZnSSe QW LED that was grown on GaAs substrate and buffer layers; the peak external efficiency reached in this type of device was approximately l%.87 A schematic of the device structure, including the p-ZnTeSe graded band-gap contact and a transparent indium-tin oxide contact, can be found in Ref. 87 for the nonoptimized device whose room temperature emission peaks at A = 494 nm. Figure 20 shows a photograph of a seven-segment test display device in which the optical emission from the ZnCdSe QWs is retrieved through a transparent top electrode material.87 Considerable progress has been made recently in improving the efficiency and lifetime of such LED structures. Researchers at Sony Laboratories have reported efficiencies up to several percent 88 and comparable results have been obtained at North Carolina State University x9 where the homoepitaxial growth on transparent ZnSe substrates gives a significant extra advantage for future device design and power output. At current densities well below 100 A/cm2, lifetimes into the range of thousands of hours have been reached. Figure 21 shows the emission spectra of the ZnCdSe/ZnSSe QW LED of Fig. 19 at T = 77 and 300 K, and compares these with the absorption spectrum of the unexcited material in the vicinity of the lowest exciton 87 M. Hagerott, H. Jeon, J. Ding. A. V. Nurmikko, W. Xie, D. C. Grillo, M. Kobayashi, and R. L. Gunshor, Appl. Phys. Leu. 60, 2825 (1992). xn A. Ishihashi, private communication (1994). *')D. B. Eason, Z. Yu, C. Boney, J. Ren, L. Churchill, J. W. Cook, and J. S . Schetzina, in Proc. 6th International Conference on II-VI Compounds. Newport, RI, J . C y s t . Growth 138, 703 (1994).

248


ENERGY (eV)

ENERGY (eV)

FIG. 21. Comparison of emission and absorption spectra near the n = 1 HH exciton resonance of a (Zn, Cd)Se/Zn(S, Se) OW LED at T = 77 and 300 K. [After Ref. 87.1

resonance ( n = 1 HH exciton).8' The spectral coincidence alone strongly suggests that the recombination involves excitons. Additional evidence can be obtained from radiative lifetime studies, an example of which is the temperature-dependent lifetime shown in Fig. 22 for the SCH structure whose optical absorption was shown in Fig. 1. These time-resolved photoluminescence measurements indicate that the lifetime increases from approximately 180 ps at T = 77 K to nearly 900 ps at room temperature." Estimates for the free electron-hole pair recombination process (in the effective mass approximation, and without Coulomb interaction) suggest an underlying radiative decay (oscillator strength) that is as much as one order of magnitude weaker than observed. In these experiments one must ensure sufficiently high levels of photoexciation so that nonradiative recombination centers related, for example, to impurities and other point defects are saturated. In fact, from a practical point of view, the present situation in the 11-VI LEDs is still such that under the typically low level of injection that characterizes an LED, the radiative efficiency of the ZnSe-based QWs, while strikingly high at cryogenic temperatures (approaching 100% internal conversion), typically drops by about one order of magnitude by room temperature even in the best material. This decrease is due to the presence of the miscellaneous point and other structural defects in the material that capture electron-hole pairs prior to their reaching the 9oJ. Ding, M. Hagerott, P. Kelkar, A. V. Nurmikko, D. C. Grillo, Li He, J. Han, and R. L. Gunshor, in Proc. 6th International Conference on 11-M Compounds, Newport, RI, J . Cryst. Growth 138, 301 (1994).

FIG.20. Numeric display device operating ath=494 nm, based on the (Zn,Cd)Se/Zn(S,Se) QW LED. [From Ref. 87.1

FIG. 23. Photographic images of the LED emission for devices with IML (green) and 2ML (yellow/green) of ZnTe imbedded within the (Zn.Cd)Se/ZnSe QW segment ("&doped") of the p-n junction, respectively. [From Ref. 91 .]

FIG.30. Electroluminescence (EL) images from a gain guided ZnCdSe QW SCH diode laser at T=300 K . For reference, (a) is a phase contrast micrograph showing a paired defect (arrow) visible within the laser stripe through transparent contact layer. (b) EL image shortly after laser turn-on at J=900A/cm2. (c) after 4 min, and (d) after 10 min. [From Ref. 110a.1

FIG.42. Photographic illustration of the output from a blue ZnSe QW diode laser, operating at a wavelength of h=463 nm

249

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0

FIG.22. Photoluminescence decay time of the SCH QW structure of Fig. 1, obtained from time-resolved spectroscopy at the spectrally integrated n = 1 QW transition. [After Ref. 90.1

QW ground electronic states. Further material improvements are now being witnessed that have in part been responsible for the advances (e.g., the Sony Laboratories and NCSU studies mentioned earlier). At the same time, the application of the graded band-gap contacting technique gives the LEDs a relatively low turn-on voltage of 2.5 to 3 V, so that once the room-temperature radiative efficiency improves further, the blue LEDs based on 11-VI QWs should have good prospects for reaching the commercial marketplace. The strong coupling between electronic and lattice excitations in the wide-gap 11-VI semiconductors can also be exploited to extend the wavelength range of the high-efficiency QW emitters in LED applications. The idea is based on the role of Te as an isoelectronic center in ZnSe where trapping of excitons was shown to strongly redshift the emission wavelengths in Section 11.3. The practical implementation of the idea for QW light emitters is conveniently realized by imbedding monolayers of ZnTe within the ZnCdSe-based QWs, as discussed earlier in Section 11. Such p-n junction heterostructures have provided green and yellow-green emission?l thereby spanning the range from about 490 to 560 nm when complementing structures without isoelectronic doping. (We comment later on the extension of emission deeper into the blue in connection with recent developments in diode lasers.) Figure 23 shows the emission through the top transparent electrodes of the two Te planar doped structures (one and two monolayers of ZnTe within the QW, respectively), which should be compared with the emission from the undoped QW structure in Fig. 20. Eason et al.92 have used the Te isoelectronic centers recently to fabricate ')I

M. Hagerott, J. Ding, H. Jeon, A. V. Nurmikko, Y . Fan, L. He, J. Han, Saraie, R. L. Gunshor, G. C. Hua, and N. Otsuka, Appl. Phys. Lett. 62, 2108 (1993). <J2 D. B. Eason, W. C. Hughes, J. Ren. K. Bowers, 2. Yu, J. W. Cook, Jr., and J. F. Schetzina, Electron. Lett. 30, 1178 (1994).

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very high efficiency ( 5%) green LEDs in the ZnSe-based heterostructures, which also feature a means of low-resistance contacting that employs the narrow-gap p-HgSe. When Mn ions are added into the QWs as isoelectronic luminescent centers, their d-electron states can also effectively capture free electron-hole pairs and release the energy as emission whose wavelength is influenced by the crystal field arrangement. This “doping,” which is widely used in ZnS:Mn high-field ac electroluminescent flat-plant displays, offers in principle an additional means of producing yellow as well as red emissionsg3 from the ZnSe-based p-n junction heterostructures. A real possibility therefore exists for multicolor LEDs over the entire visible range from the ZnSe-based heterostructures, which are compatible with multilayer growth within a single epitaxial chamber. Furthermore, due to the high density of Te isoelectronic centers, which can be embedded in the active QW region, it may be possible to achieve green-yellow diode laser operation from such structures as well. Another potentially important approach to green LEDs has been recently proposed and reported in a design where a n-CdSe/p-ZnTe heterojunction, separated by a graded Cd, -xMg,Se injecting region, allows low-voltage injection of electrons into the heavily doped p-ZnTe for radiative r e c ~ m b i n a t i o n .This ~ ~ scheme, which circumvents the difficulty of realizing a ZnTe p-n junction, is an example of the versatility that is offered by band structure engineering techniques, which are very likely to be increasingly widely implemented in the wide band-gap 11-VI semiconductors. Such approaches may be especially relevant to LEDs, which lack many of the additional constraints and structural limitations imposed in diode lasers. 9. DIODELASERS A historical schematic that broadly summarizes the evolution of various blue-green diode laser structures from 1991 to date is sketched in Fig. 24. Much of this evolution [from Fig. 24(a) to (d)] has been shaped by concurrent developments in 11-VI epitaxy, with the aim toward implementation of the pseudomorphic SCH-QW configuration, as introduced in the beginning of this article (Fig. 1). The advantages of the joint electronic/optical confinement in an SCH structure are, of course, very 93 A. V. Nurmikko, Q. Fu, D. Lee, R. L.Gunshor, and L. A. Kolodziejski, Proc. 19th Int. Conf. Physics of Semiconducfors, Warsaw, p. 1523, (1988). 94 M. C. Phillips, M. W. Wang, J. F. Swenberg, J . 0. McCaldin, and T. C. McGill, Appl. Phys. Lett. 61, (1992).

VISIBLE LIGHT EMI’ITERS

25 1

Flc. 24. Schematic of blue-green diode laser structures in rough chronological order of their development since 1991, from (a) to (d).

well known in 111-VI semiconductor lasers, but limitations of lattice matching constraints prevented its realization in the ZnSe-based emitters until the introduction of the (Zn, Mg)(S, Se) quaternary material at Sony L a b ~ r a t o r i e s .Other ~~ pseudomorphic configurations such as the gradedindex design, which have been hampered by strain-induced defects in 11-VI lasers, are only now becoming possible. A high misfit dislocation density (beyond lo7 per cm2) was present in the structures that were first demonstrated as diode lasers in 1991 [Fig. 24(a)]. On the other hand, configurations such as the pseudomorphic structure in Fig. 24(b) possessed a poor optical confinement factor (the optical intensity overlap with the confined electronic states of the QW, llop2 and lower). In either case, it was difficult to reach room-temperature operation, although with reflective end facet coatings, pulsed threshold current densities of about 1 kA/cm2 were ~ b t a i n e d . ’ ~Even ? ~ ~ at low duty cycles ( however, such devices operated only a few minutes. On the other hand, cw opera-

-

95

H. Okuyama, K. Nakano, T. Miydjirna, and K. Akimoto, Jpn. J . Appl. Phys. 30, L1620 (1991). Vh H. Jeon, M. Hagcrott, J. Ding, A. V. Nurmikko, D. C. Grillo, W. Xie, M. Kobayashi, and R. L. Gunshor, Opt. Lerr. 18, 125 (1993).

252


0

500

I[ [mAl

1000

1500

FIG.25. Output characteristics of a gain guided ZnCdSe/ZnSSe/ZnMgSSe SCH-SQW 7 = 50 ns). [From Ref. 98.1 diode laser, operated under low cycle pulsed injection (5 X

tion at T = 77 K was relatively easily accomplished, producing output powers on the order of a milliwatt. The structure of Fig. 24(c) was introduced by the group at Sony Laboratories in 1992 to show wavelength shortening effects by the choice of the wide band-gap ternary (Zn, Mg)Se as a barrier material for a ZnSe QW;’ but the operation was still limited to cryogenic temperatures, probably due to both inadequate optical and electronic confinement (recall the common anion argument for valence band offsets). Indicating marked improvement and the direct impact of the SCH configuration, Fig. 25 shows the output characteristics obtained by the group at Philips in 1993 from the ZnCdSe/ZnSSe/ZnMgSSe SCH-SQW diode laser [Fig. 24(d)l, operated under low duty cycle, very short pulsed injection (T = 50 ns) but reaching well beyond room t e m p e r a t ~ r e In . ~ ~these gain guided lasers, with the Cd concentration typically x = 0.15 to 0.20, sulfur concentration y = 0.07, and the Mg concentration x‘ = 0.10 to 0.20, the room-temperature current injection density was measured as approximately 500 A/cm2 for devices of about 1 mm in length, an excellent figure of merit. Differential quantum efficiency of about 17% was also obtained. From the measured values for the index of refraction for ZnMgSSe, a typical value for the electronic/optical confinement factor is about r = 0.03 for a quantum well thickness of L , = 75 A and 2000 A for the ZnSSe optical guiding layer. Very similar diode laser structures were fabricated in the authors’ laboratories, incorporating the graded gap ZnSeTe contact layer,99 and by groups at 3M and Sony.

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97

H. Okuyama, T. Miyajima, Y.Morinaga, F. Hiei, M. Ozawa, and K. Akimoto, Electron. Lett. 28, 1798 (1992). 98J. M. Gaines, R. R. Drenton, K. W. Haberern, T. Marshall, P. Mensz, and J. Petruzzelo, Appl. Phys. Lett. 62, 2562 (1993). Y9 D. C . Grillo, Y. Fan. J. Han, H. Li, R. L.Gunshor, M. Hagerott, H. Jeon, A. Salokatve, G. Hua, and N. Otsuka, Appl. Phys. Lett. 63, 2723 (1993).


253

Both single and multiple quantum well structures have been fabricated; the relative insensitivity of the threshold current versus the number of QWs may be an indicator of the role of leakage currents in these heterostructures. At present, given the relative uncertainty about the values for band offsets, band bending at the heterointerfaces, and the details of overall vertical transport, the issue of leakage clearly calls for more systematic experimental and theoretical studies. The commonly chosen composition of the SCH-QW structure, especially that of the ZnCdSe QW as the active material, places the room-temperature laser wavelength into the green range of A = 510 to 530 nm. In recent exploratory work on ZnCdSe/Zn, - rt Mg,,S,,Se, -yt/Zn, -x.Mgx.Sy.Se, -y,r and ZnSe/Zn, -x,Mg,tSy,Se,-yt/Zn, -x.Mgxt,Sy. Se, - y , , heterostructures at Sony Laboratories and the authors' group, the emission wavelengths under pulsed excitation at room temperature have been shortened from about 480""' to 470'"' to 460 nm'"' by judiciously increasing the Mg concentration in the barrier layers. Figure 26 shows the spectrum from a blue ZnSe QW laser both below and above the lasing threshold (Jth= 2 kA/cm2 per QW).1"2A portion of the spectrum from an InGaN/GaN LED8h is included for comparison. Two particular problems await at the moment for a solution in attempts to reach into the cw regime of operation with these devices: (1) the rather rapid drop of the free hole concentration in p-ZnMgSSe with Mg concentration (typically into the range of < 1016 cm-3 for Eg = 3.0eV) and (2) the appearance of a compositional modulation attributed to phase separation in the quanternary along a specific crystallographic direction. Cross-sectional TEM revealed the presence of one-dimensional quasi-periodic compositional modulation along the [110] direction in the ZnMgSSe outer cladding layers of the laser structure.'"' The reduction in the level of p-doping can be understood from the discussion in Section 11.3, whereas the effects of compositional modulation of the alloy are not yet clear on the diode laser performance. By and large, these devices are presently test structures designed for feasibility studies, whose evaluation will pave the way for shortening the emission wavelengths deeper into the blue at lower thresh101l H. Okuyarna, S. Itoh, E. Kato, M. Ozawa, N. Nakayarna, K. Nakamo, M. Ikeda, A. Ishibashi, and Y. Mori, Electron. Left. 30, 415 (1994). I0 I T. Ohata, S. Itoh, N. Nakayama, S. Matsurnoto, K. Nakano, M. Ozawa, H. Okuyarna, S. Torniya, M. Ikeda. and A. Ishibashi, in Proc. Inf. Workshop on ZnSe-Based Blue-Green Laser Structures, Wurzburg, p. 13 (1994); see also H . Okuyama, E. Kato, S. Itoh, N. Nakayarna, T. Ohata, and A. Ishibashi, Appl. Phys. Lett. (1995). 1112 D. C. Grillo, J . Han, M. Ringle, G. Hua, R. L. Gunshor, P. Kelkar, V. Kozlov, H. Jeon, and A. V. Nurrnikko, Electr. Letf. 30, 2131 (1995).

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440

,

,

450

.

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460

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GaN LED I

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288

DAVID K. FERRY AND HAROLD L. GRUBIN

problems in transport has been the subject of considerable discussion.'* In general, the semiclassical approach assumes that the scattering processes are perturbations distinct from those of the driving fields, that the scattering occurs instantaneously in both space and time, and that potential and density gradients are slow on the scale of the de Broglie wavelength of the carrier^.'^ In future ultra-submicron semiconductor devices, all of these assumptions can be expected to be violated. Some work has already appeared concerning the interaction of the driving fields and the scattering processes, an effect known as the intracollisional field effect (ICFEX3' - 3 4 The problem of the rapid spatial variation of the potential is of course what leads to the quantization effects in the first place,35 and the multiple interactions this causes is the major problem to be addressed in this review. The transition between semiclassical dynamics and quantum dynamics is one that remains in question in basic quantum theory,36 but quantum transport has been discussed for some time. One aspect of this is that the basic equations are Markovian in nature, but under the conditions in which a one-electron distribution function is used, these can become non-Markovian in nature due to memory effects introduced by the scattering. Under strong fields and scattering, a new nonperturbative basis of electron states, rather than a simple perturbation of the Boltzmann equation, needs to be used.37 To be sure, this problem-the steady state of the far-from equilibrium system under high fields-is not new, and appears equally as well in the semiclassical transport problem. The first to suggest that this new dissipative steady state was different was L a n d a ~ e r . ~It' was pursued extensively by the Brussels but the major point here is that the transition from semiclassical to quantum dynamics is also not a simple perturbative process. It is these major differences that create much of the problem in trying to develop quantum-mechanical treatments of the transport for strongly nonequilibrium systems such as occur in semiconductor devices. 29 G. V. Chester, R e p . Prog. Phys. 26, 411 (1963). 3''J. R. Barker, J . Phys. C 6, 2773 (1973). 31 K. K. Thornber, Sol. State Electron, 21, 259 (1978). 32D.K. Ferry and J. R. Barker, J . Phys. Chem. Sol. 41, 1083 (1980). 33D.Lowe, J. Phys. C 18, L209 (1985). 34P.Lipavslj, F. S. Khan, F. Abdolsalami, and J . W. Wilkins, Phys. Rec. E 43,4885 (1991). 35 See, e.g., J . R. Barker, in Granular Nanoelectronics (D. K. Ferry, J . R. Barker, and C. Jacoboni, eds.), p. 1, Plenum Press, New York (1990). 36 See, e.g., the discussion of C. George, I. Prigogine, and L. Rosenfeld, Nature 240, 25 (19721, and W. H. Zurek, Phys. Today, p. 36 (Oct. 1991). 37P. Lipavslj, V. Spitka, and B. Velickf, Acta Phys. Polon. A73, 319 (1988). "See the discussion in R. Landauer, Phys. Today 31, 23 (Nov. 1978). 39See,e.g., B. Misra, Proc. Natl. Acad. Sci. USA 75, 1627 (1978).

QUANTUM TRANSPORT IN SEMICONDUCTOR DEVICES

289

2. THEDIFFERENCE FROM BOLTZMA”TRANSPORT

The basic transport equation for studying carrier behavior in semiclassical models of semiconductor devices has been the Boltzmann transport equation:

where p = m v is the momentum, and r is the position. Here, it is assumed that the momentum is related to the energy of the particles by a welldefined single-electron band structure; for example, the spectral density is defined by A ( E , p) = 2 d ( E - p 2 / 2 m ) (the spectral density is related to the dispersion relation between energy and momentum; integration over the momentum produces the density of states). Moreover, it is also assumed that the effect of the potential arises solely from the value of the first derivative, the field F in Eq. (2.1). Finally, it is assumed that the distribution function f(p, t ) varies slowly on the temporal scale of the relaxation processes, in that it is the local distribution at time t that appears in Eq. (2.1) and not some retarded value of the distribution (which one would assume would be the distribution at the time the appropriate free path began). There are then two approaches to solving this equation to obtain transport coefficients: 1. It is assumed that the variation of the distribution from the equilibrium Maxwell-Boltzmann one (nondegenerate statistics are assumed) is small, and the value of f in the derivatives is’ replaced by the equilibrium value. This leads to what is usually referred to as the relaxation-time approximation.

2. For complicated, anisotropic scattering processes, or for high-field transport, the above approximation fails, and one must actually solve for the distribution function. This, in fact, is the major problem in hot carrier transport.

In the case of quantum transport, each of the preceding assumptions fails. In particular, the spectral density is no longer a simple delta function, and one must find its form in the interacting system of many electrons with scattering by impurities, phonons, and other electrons. At low temperatures, and near equilibrium, the spectral function is usually found to be a Lorentzian, in which broadening exists around the value specified for the energy-momentum relation of the semiclassical model. In addition, the

290


potential leads to nonlocal behavior, in which the last term on the left-hand side of Eq. (2.1) includes an entire hierachy of derivatives, such as originally introduced by Wigner.40 Finally, the collisions are no longer localized in space and time, so that the collision integral on the right-hand side of Eq. (2.1) becomes a non-Markovian retardation integral. This leads to a modified hierarchy of solution for the quantum distribution function: 1. The spectral function must first be determined in the interacting system. 2. For near-equilibrium systems, or at low temperatures, it may be assumed that the quantum distribution is given by small deviations from the equilibrium Fermi-Dirac distribution. 3. For complicated, anisotropic scattering processes, or for high-field and strongly nonequilibrium transport, the above approximation fails, and one must actually solve for the distribution function.

To be sure, in some approaches this sequence is finessed by using single-time functions, such as the density matrix and the Wigner distribution function, which essentially integrate out the spectral function, but retain the full spatially nonlocal nature of the potential interactions that lead to the hierarchy of derivatives appearing in the transport equation that replaces Eq. (2.1). We will illustrate this further later. Nevertheless, the transport problem, as in the semiclassical case, remains a balance between the driving forces, primarily the potential, and the relaxation forces represented in the collision integraL4' In the quantum-mechanical case, there has been an argument for some time over whether or not the application of an electric field to a crystal would destroy the bulk band structure and create a Stark ladder of discrete states.42 In fact, it is known that this does not occur in bulk crystals, where the use of the electric field creates a Franz-Keldysh shift of the bands, which is quite useful in modulated electroreflectance to study the band structure.43 Some Stark ladder effects are seen in well-correlated superlattice structures under optical illumination, but, in general, the effects are washed out in bulk materials by the scattering process found there.44 40

E. Wigner, Phys. Rev. 40, 749 (1932). 415.G . Kirkwood, J . Chem. Phys. 14, 180 (1946). 42 See, e.g., P. N. Argyres and S. Sfiat, Phys. Lett. A 146, 231 (1990), and references contained therein. 43See, e.g., the discussion in D. E. Aspnes, Phys. Re[!. 147, 554 (1966). 44 A review is found in M. A. Stroscio, in Introduction to Semiconductor Technology: GaAs and Related Compounds. (C. T. Wang, ed.), p. 551, John Wiley, New York (1990).


29 1

f

FIG.1. The Wigner distribution showing the density variation adjacent to an infinitely high potential in the space x < 0. The oscillations at higher momentum value arise from interference of the waves peaking closer to the interface.

a. Statistical Thermodynamics and Quantum Potentials As we have discussed, the potential in quantum systems creates actions that are nonlocal to the actual potential, that is, they can occur some distance from the potential. Let us consider how this nonlocality arises. Consider a simple potential energy barrier (for which the density is shown in Fig. 1): V ( x ) = e K u ( - x ) , where u ( x ) is the Heavyside step function. We assume that a nonzero density exists in the region x > 0, and the question is how the density varies near the barrier, a quite typical problem in introductory quantum mechanics and in devices. Here, however, the problem refers in general to a statistical mixed state, rather than to a single quantum state. In classical mechanics (in the absence of any self-consistent Poisson equation solutions to find a new, self-consistent potential), the density varies as exp( - p V ) , where p is the inverse electron temperature (= l / k , T ) , and for this case is uniform and constant up to the barrier, dropping abruptly to zero in the half-space x < 0. In quantum mechanics, however, the wave function is continuous, and any member of the statistical ensemble must be small at the interface (vanishingly small for the case V, + m). This then leads to a different behavior on the part of the density. In Fig. 1, we show the Wigner distribution function (which, for the moment, can be thought of as the quantum-statistical mechanical analog of the classical phase-space distribution) for this situation. The parameters here are appropriate to bulk GaAs, with n = 2 X 10” cm-3.4s We note that, far from the barrier, the distribution approaches the classical Maxwellian form, but near the barrier, the distribution differs 45

A. M. Krirnan, N. C. Kluksdahl, and D. K. Ferry, Phys. Reu. B 36, 5953 (1987).

292


greatly from the uniform classical case. The repulsion of density from the barrier is required by the continuity of the wavefunction at the barrier, but the first peak in the wavefunction away from the barrier occurs close to the barrier for high momentum states. This leads to much of the complication evident in the figure, and to a momentum-dependent positional correction to the density away from the potential barrier. The density peak away from the barrier is governed by physics similar to the peak in density away from the semiconductor-oxide interface in a MOSFET, and assures that net charge neutrality is maintained (which means that Poisson's equation is included in the solution). The deviation in the density occurs over several thermal de Bro lie wavelengths (evaluated with the thermal momentum) A, = h2/3mk,T. This suggests that nonlocal deviations from classical results can be expected to occur in most semiconductor devices over a range of 20 to 40 nm even at room temperature! It is clear that the density no longer varies simply as exp( - p V), and that modifications to the statistical mechanics need to be made. The development of quantum corrections to statistical thermodynamics, especially in equilibrium, has a rich and relatively old history. Unfortunately, there is no consensus as to the form of the correction to this simple exponential behavior. It we could find such a correction, it could be utilized in the semiclassical hydrodynamic equations developed from moments of more basic transport equations such as Eq. (2.1). One of the original efforts to obtain quantum corrections to classical distribution functions was done by Wigner, who introduced a quantum distribution f ~ n c t i o n . ~In' this regard, it can be considered as an attempt to find an additional term that can be added to the classical potential to produce the desired results. The Wigner potential has been put in the form46

F

(2.2) This represents a quantum correction to the mean kinetic energy of a distribution of particles. Bohm4' also introduced an effective potential, in his discussions. For a distribution of particles, in the single-electron approximation, the Bohm potential represents a nonelectrostatic force, acting on a particle distribution whose value is determined by the form of 46 47

G. J . Iafrate, H. L. Grubin, and D. K. Ferry, J . Physique (Colloq., C-10) 42, 307 (1981). D. Bohm, Phys. Rev. 85, 166 (1952); Phys. Reu. 85, 180 (1952).


293

the particle distribution. In a sense, this potential is determined through an interaction of the particle with itself quantum mechanically. The Bohm potential is given by

These two differ numerically only in a minor way, even though their conceptual origins are quite different. Feynman and Hibbs4' suggested a variational approach by which the classical potential would be weighted by a Gaussian spreading function. Later work by Feynman and K l e i r ~ e r textended ~~ this to the development of a general variational form for the effective potential, in which a nonlocal smoothing function is applied to the actual potential, and new terms arise to represent quantum diffusion. A new version, based on a Green's function solution of the effective Bloch equation for the density matrix in the nonlocal potential has been developed, but untried in actual device simulation^.^" This is discussed further later. It must be emphasized that the variation of the wavefunctions, or the consequent quantum distribution functions, away from confining barriers leads to quantization within a small system. This quantization is the overriding property of small systems, such as quantum wires and quantum b ~ x e s . ~In' some cases, the narrow minibands that result from this quantization have been suggested as a method cutting down on phonon scattering, by ensuring that the width of a miniband is small compared to the optical phonon energy, while the spacing of the minibands is larger than this energy.52 Most devices, however, are (erroneously) thought to be unconstrained in the direction to/from the contacts (or the reservoirs, as they will often be called below), so that these effects are not likely to be observed in most realistic devices. In fact, determining the contact effects in these "open" systems will be quite complicated, a point to which we return later. 48 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Sec. 11.1-2, McGraw-Hill, New York, (1965). 49 R. P. Feynman and H. Kleinert, Phys. Reo. A 34, 5080 (1986); H. Kleinert, Phys. Lett. B 181, 324 (1986). '"D. K. Ferry and J. R. Zhou, Phys. Reu. B 48, 7944 (1993). 51 See, e.g., U. Riissler, in Quantum Coherence in Mesoscopic Systems (B.Kramer, ed.), p. 45, Plenum Press, New York, (1991). s2H. Sakaki, Jpn. J . Appl. Phys. 19, L735 (1980); Inst. Phys. Conf. Ser. 63, 251 (1982); J . Vac. Sci. Technol. 19, 148 (1981).

294


b. Phase Interference While current devices have gate lengths in the 0.25- to 0.7-pm range, at least for production devices (the shorter ones are GaAs microwave devices), future devices will reach far smaller sizes. It is conceivable that the gate length will then be comparable to those in which quantum effects are studied. The relevant quantity for discussion of quantum interference effects is the phase change of the carrier as it moves through the semiconductor device. Interference between differing electron waves, or differing electrons on their individual trajectories, can occur over distances on the order of the coherence length of the carrier wave, and this latter distance is generally taken to be of the order of the inelastic mean free path, or phase breaking length. Ballistic, and therefore coherent and unscattered, transport has been observed through the base region of a GaAs/AlGaAs hot electron transistor.s3 From this, it is estimated that the inelastic mean free path for electrons in GaAs may be as much as 0.12 p m at room temperature. The inelastic mean free path is of the order of (and usually equal to) the energy relaxation length 1, = UT,, where T~ is the energy relaxation time and u is a characteristic velocity. This tells us that even in Si the electron inelastic mean free path may be 50 to 100 nm. Thus, the inelastic mean free path can be quite long, and can be comparable to the gate length in these devices. Since the phase remains coherent over the range of the correlation function of the electrons (in space or time), there can be interference effects in the overall conductance of the device. The small device will then reflect the intimate details of the impurity distribution in the particular device, and macroscopic variations can then arise from one device to another, an effect well known in mesoscopic devices, where it leads to nonlinearities and fluctuation^.'^ The basic concepts were expressed rather early by Landauer,” in which the conductance through a region with localized scatters was expressed by very sample specific properties, known as the Landauer formula 2e2 G=-T. h

(2.4)

53These are reviewed by M. Heiblum, in High Speed Electronics (B. Kallback and H. Beneking, eds.), Springer-Verlag, Berlin (1986), and J. R. Hayes, A. J. F. Levi, A. C. Gossard, and J. H. English, in High Speed Electronics (B. Kallback and H. Beneking, eds.), SpringerVerlag, Berlin (1986). 54R.Landauer, in Nonlinearity in Condensed Matter (A. R. Bishop, D. K. Campbell, P. Kumar, and S. E. Trullinger, eds.), p. 2, Springer-Verlag, Berlin (1987). 55 R. Landauer, IBM J . Res. Develop. I, 223 (1957).


295

Here, the formula is expressed for one dimension, but it can be expanded to more dimensions by interpreting T as the total transmission of all modes (electrons) through the region of interest. In this formula, the potentials, used to calculate the conductance, are determined at the reservoirs (or contacts). While the original formula was obtained for noninteracting electrons, recent work has shown that a similar, but more complicated, form can be obtained in the interacting electron case.56 The most usual study of the sample specific variations of the conductance with gate bias, applied bias, or magnetic field, all of which provide fluctuations in the local potential in the inhomogeneous sample (and all samples are inhomogeneous in this phase coherent regime), has dealt with universal conductance fluctuation^.'^ However, it is also possible to have a net coherent backscattering from the impurities, without losing the phase coherence, and this leads to the concept of weak localization, a form of increased resistance due to the interactions,’5s57One additional effect that has been suggested, but not studied well, is the fact that the random impurities cause a significant deviation in the current density from the uniform (average) value, especially where the cross section of a single scatterer exceeds that of its equivalent volume of background semiconductor. Then, any single scatterer is likely to affect a greater current, due to the detour of current lines away from other scatterers. This can lead to a greater effect of each scatterer and, hence, a larger contribution to the resistance of the device.58 This effect reaches its peak in the presence of a magnetic field, which can be coupled through the two phase coherent paths, and which leads to the Aharonov-Bohm effect.59 The effect is most commonly studied in metal loops, coupled to a pair of reservoir^.^^-^^ However, anytime two mutually uncorrelated quantum channels are connected at the reservoirs, one must expect that there will be flux-sensitive fluctuations of the conduc56

i7

Y. Meir and N. S. Wingreen, Phys. Reu. Left. 68, 2512 (1992). S. Namba, C. Hamaguchi, and T. Ando, eds., Science and TechnologV of Mesoscopic

Structures, Springer-Verlag, Tokyo (1992). 5X R. Landauer and J. W. F. Woo, fhys. Reti. B 5, 1189 (1972). 59 Y. Aharonov and D. Bohm, fhys. ReLi. 115, 485 (1959). 61) D. Yu. Sharvin and Yu. V. Sharvin, JETf Left. 34, 272 (1981). hl B. Pannetier, J. Chaussy, R. Rammal, and P. Gandit, fhys. Reu. Lett. 53, 718 (1984). h2R.A. Webb, S. Washburn, C. Umbach, and R. A. Laibowitz, fhys. Reu. Lett. 54, 2696 (1985); S. Washburn, C. P. Umback, R. B. Laibowitz, and R. A. Webb, Phys. Rev. E 32,4789 (1985). hi V. Chandrasekhar, M. J. Rooks, S. Wind, and D. E. Prober, fhys. Reu. Lett. 55, 1610 (1985). “M. Buttiker, Ann. New York Acad. Sci. 480, 194 (1985).

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t a n ~ e . ~ ~One - ~ ' can ask the question as to whether the presence of inelastic scattering in one or both arms of the loop will cause these oscillations to be damped, and the answer is generally positive. The study of this damping effect led to the general development of the multichannel version of Eq. (2.417' Nevertheless, the presence of many channels of transport through the active gate region can be expected to lead to relatively large fluctuations in the overall conductance, if the conditions are properly attained. Each of these effects is likely to begin to impact devices, as the size is reduced, even at room temperature. The most likely is universal conductance fluctuations, especially in the turn-on characteristics of the device, where the conductance is low and the impurities are being charged/discharged. Consider a small device, perhaps with a gate length and width of 0.05 x 0.1 pm, respectively. If the number of carriers in the inversions channel is 2 X lo'* cm-', there are only 100 electrons under the gate. If there is a fluctuation of a single impurity (between ionized and neutral), one might expect a change of conductance of order 1% in the thermally averaged classical regime. The change can be much larger if the carriers are phase coherent. The phase coherence7' and the charging of such single impurities has been detected at low temperature in Si M O S F E T S . ~It~ is clearly established now that the effect can be much larger than one would expect, and this largeness is due to the quantum interference caused by the change in trajectories of individual electrons. The conductance change in the phase interference process can be of the order of Eq. (2.41, which is about 40 $3. If our device were to exhibit outstanding conductance of 1000 mS/mm (gate width), the absolute conductance would only be 100 pS, so that the fluctuation could be of the order of 40% of the absolute conductance! This is a very significant fluctuation, and arises from the lack "C. P. Umbach, S. Washburn, R. B. Laibowitz, and R. A. Webb, Phys. Rev. B 30, 4048 (1984). hh J. C. Licini, D. J . Bishop, M. A. Kastner, and J . Melngailis, Phys. Reu. Left. 55, 2987 (1985). "R. G. Wheeler, K. K. Choi, A. Goel, R. Wisnieff, and D. E. Prober, Phys. Reu. Left. 49, 1674 (1982). 68 S. B. Kaplan and A. Harstein, Phys. Reu. Lett. 56, 2403 (1986). 69P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Reu. B 35, 1039 (1987). 'OM. Biittiker, Physica B 175, 199 (1991); Phys. Reu. Left. 68, 843 (1992). 71 M. Biittiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985); M. Biittiker, Phys. Rev. Lett. 57, 1761 (1986); M. Biittiker, Phys. Reu. B 32, 1846 (1986). 12 W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D. Jackel, D. M. Tennant, and A. D. Stone, Phys. Reu. Lett. 56, 2865 (1986). 73W.J. Skocpol, in Physics and Fabrication of Microsttuctures and Microdevices (M. J. Kelly, ed.), p. 255, Springer-Verlag, London (1986).


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of ensemble averaging of these effects in the phase coherent transport through the device. It may well be a limitation in the performance of such devices. In fact, our 0.1-pm gate width device is quite nearly a device formed on a quantum wire, since the width is also comparable to the inelastic mean free path of the carriers. Thus, as the gate potential is varied, one may well expect to see (even at room temperature) conductance fluctuations arising from the effects just discussed, well as other quantum interference effects in the device. These conductance fluctuations appear as noise, but are not temporal variations-they arise in dc measurements and are quite repeatable. Such fluctuations, and their effects on device performance and behavior, can only be modeled with full quantum-mechanical transport, and electrostatic, models.

3. OPENSYSTEMS AND CONTACTS The implications of Eq. (2.4) are that the conductance of a localized tunneling (or scattering) barrier can be calculated from its transmissive If there are reservoirs on the left- and behavior between two reserv0i1-s.~~ right-hand sides of the transmissive region, which may be considered for the present as the deuice, it may be assumed that the system is in steady-state thermodynamic equilibrium deep within the reservoirs. The device resistance is then composed of the active region and the contacts, which connect the latter to the reservoirs. In a quantum-mechanical sense, this is represented by the incident and outgoing wavefunctions, their occupation probabilities, the multidimensional transport through the system, and the included inelastic processes. Nevertheless, it is still possible, in principle, to calculate the transmissivity between the incident wavefunctions and output wavefunctions in the exit reservoir. Even if the transmission through the active region is totally elastic (the ballistic transport of the next paragraph), dissipation and ultimately irreversibility occur through relaxation in the exit reservoir and the contact region adjacent to it. While physics normally considers closed systems, it is the macroscopic open system with its contacts and reservoirs that are important to the consideration of devices.75 Indeed, simulations of electron wave packet transport through quantum wires are sensitive to the details of the treatment of the 14

R. Landauer, in Analogies in Optics and Micro Electronics (W. van Haeringen and D. Lenstra, eds.), p. 243, Kluwer Academic Publ., Amsterdam (1990). 75 R. Landauer, Z. Phys. B 68, 217 (1987).

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RESERVOIR,

RESERVOIR,

FERMI OR

FERMI OR

BOLTZMANN

BOLTZMANN

FIG.2. A schematic depiction of the embedding of a device in its surroundings: boundaries, contacts, etc.

r e ~ e r v o i r s a, ~result ~ that is also known for studies of weak localization and universal conductance fluctuations in mesoscopic devices.77 This becomes more important in devices, because the problems of devices intrinsically involve open systems. As illustrated in Fig. 2, carriers within the device are interacting with a reservoir at each end. The electrons or holes in these reservoirs have been characterized as either satisfying a Boltzmann or Fermi distribution, in equilibrium, or as a displaced distribution in a nonequilibrium state of bias. Such a characterization implies that a local equilibrium exists in the “contact” and is fraught with all of the uncertainties associated with this characterization. It will be quite necessary to discover just how the characterization of the reservoirs, or contacts, actually impacts the simulations of the devices. We will see in the simulations in the following sections that the boundaries and contacts play a very essential role, a role that is well known in the semiclassical world. a. Ballistic Transport The idea of a quantum trajectory, which resembles the classical phase space trajectory, dates to the early ideas of de Broglie and his pilot waves. It was resurrected by B ~ h r nto~explain ~ considerable detail of his wave approach. Nevertheless, it has been difficult to incorporate these trajectory ideas within quantum mechanics, because the probabilistic interpretation of the wavefunction tends toward the lack of a well-defined single trajectory for the wave packet. This is reflected in the summation over probabilistically weighted trajectories in path integrals.48 Nevertheless, there is a consistent interpretation of quantum mechanics using trajectories as its basis.78 This becomes important when we want to talk about ballistic transport of carriers from one contact reservoir to another. If the distance between the two reservoirs is less than the elastic mean free path, then 76

L. F. Register, U. Ravaioli, and K. Hess, J . Appf. Phys. 69, 7153 (1991).

77V.Chandrasekhar, D. E. Prober, and P. Santhanam, Phys. Reu. Lett. 61, 2253 (1988); V.

Chandrasekhar, P. Santhanam, and D. E. Prober, Phys. Reu. B 44 11203 (1991). 78 R. B. Griffiths, Phys. Reu. Left. 70, 2201 (1993).


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carriers injected into the active region from one reservoir will drift under the applied fields to the other reservoir. It is not convenient to think about this motion in any other manner than as the transport of the centroid of the carrier wave packet along a semiclassical trajectory. Indeed, ballistic transport theory may be set up by choosing the appropriate fields to accelerate the carriers through the device; the problem becomes completely nontrivial if the fields are computed in a fully self-consistent manner.35 The principles behind the Landauer equation [Eq. (2.4)l are not dependent on this view, because one may define the channels by various modes of a waveguide, but it is often convenient to think of the Landauer equation in this fashion. We will see this later in connection with transport in high magnetic fields. In general, the nature of ballistic transport goes beyond structures whose lengths are less than the elastic mean free path. Rather, the important length over which the transport is essentially ballistic is the inelastic mean free path, or phase breaking length. That is, the important length is that over which the transport remains coherent. This was demonstrated by Buttiker,” who showed that transport through two series-connected phase-coherent regions produced the normal additivity of resistances only if an inelastic scattering process occurred between the two regions. In the absence of this inelastic process, coherent addition of the two regions resulted. Surprisingly, if the overall transmission probability were low, then the addition of weak inelastic processes actually lowered the overall resistance, but the classical additivity of resistances was recovered as the degree of inelastic scattering was increased. It is thought that this initial lowering results from resonant transmission through the scatterer, and this resonance produced higher transmission, because this effect did not occur for transmission probabilities of the order of 0.5. The study of ballistic transport in quantum waveguide structures, in which a coherent structure supporting only a few occupied channels (a channel is one mode of transverse quantization) was placed between two reservoirs, has been pursued by a number of authors. One reason for this is that the modification of Eq. (2.4) for the case of multiple modes has been somewhat controversial. To be sure, the overall conductivity is related to the overall transmission matrix of the multimode structure.80 Yet the number of probes (side-arms are often added as voltage contacts, while the reservoirs serve as current contacts) affects the resultant formulas in many cases, and the result of reversing the magnetic field in a multiprobe measurement must yield the proper symmetries, consistent ”M,Buttiker, Phys. Re(,. B 33, 3020 (1986).

XI1

D. S. Fisher and P. A. Lee, Phys. Reis. B 23, 685 (1981).

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with the Onsager relations." Indeed, if we have a four-terminal structure, with terminals 1 and 2 being used to provide the source and sink of current, and terminal 3 and 4 being used to measure the voltage, then Eq. (2.4) can be generalized to

and T = T,,= T12. Clearly, if the last fraction is ignored, then Eq. (2.4) is recovered. There are caveats to these equations as well. The "ballistic leads," for example, the regions between the reservoir/contacts and the active region of measurement, must be connected in a nonreflective manner with the reservoirs, and the electrochemical potentials must be measured deep in the reservoirs to assure thermalization of the carriers. If this geometry is not respected, then deviations can occur in measurements, and even in some theories, and this is important to considerations of potentials.82 Consequently, the study of the existence of the proper transmission formula and resonances in the transmission under certain conditions ~ ~addition, considerable continues to be a topic of some d i s c u s ~ i o n .In effort is now being expended on the study of the high-frequency forms of the conductance of these ballistic structure^,^^ as well as the effect that bends in the ballistic structure play in the overall conductance.8s One of the most interesting confirmations of the Landauer equation is the experimental observations of quantized conductance through a constriction that could be varied by the gate voltage. Indeed, steps in the conductance were found to be in exact agreement with Eq. (2.4), because the measurements were essentially two-terminal measurement^.^^^'^ However, the most extensive study of the multiterminal version [Eq. (3.111 has ni M. Biittiker, IBM J. Res. Develop. 32, 317 (1988); in Electronic Properties of Multilayers and Low-Dimensional Semiconductor Structures (J. M. Chamberlain et al., eds.), p. 51, Plenum Press, New York (1990). M. Landauer, Physica Scripta T42, 110 (1992). "P. J. Price, Sol. State Commun. 84, 119 (1992); Appl. Phys. Lett. 62, 289 (1993). n4 M. Biittiker and H. Thomas, Inst. Phys. Conf. Series 127, 19 (1992); M. Biittiker, A. Pr&tre,and H. Thomas, Phys. Rev. Leu. 70, 4114 (1993). 85 H. U. Baranger, in Computational Electronics (K. Hess, J. P. Leburton, and U. Ravaioli, eds.), p. 201, Kluwer, Norwell, MA (1991). n6B.J. van Wees, H, van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhouven, D. van der Marel, and C. T. Foxon, Phys. Reu. Lett. 60, 848 (1988). ni D. A. Wharam, U. Ekenberg, M. Pepper, D. G. Hasko, H. Ahmed, J. E. F. Frost, D. A. Ritchie, D. C. Peacock, and G. A. C. Jones, Phys. Reu. E 39,6283 (1989).


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been in the quantum Hall effect." In the presence of a high magnetic field, scattering of the carriers is suppressed as the ballistic trajectories are folded into Landau orbits, in which the essentially one-dimensional transport along the orbit hinders the scattering proce~s.'~Only those trajectories that reflect from the lateral boundaries move from one contact to another; these edge-located carriers are essentially what are now called edge In a great many studies, which are not the central point of this review, the multiterminal version [Eq. (3.111 has now been verified. It should also be pointed out that, although Eqs. (2.4) and (3.1) are basically obtained for noninteracting electrons, the results may not be significantly affected by the presence of carrier-carrier interaction^.^^*'^ Although nearly all of the discussed ballistic electron studies have been carried out at low temperature, the basic nature of ballistic transport carries through from the semiclassical regime and supports the basic trajectory nature of the transport of carriers, even in the quantum regime. This is likely to be an important consequence for the small semiconductor devices, in which we envision a need to include detailed quantum transport models. The ballistic transport provides one limit of the transport process and must be reflected in accurate models. b. Role of the Boundaries and Contacts

One of the important consequences from Eq. (3.1) is that the actual resistance, or conductance, that is measured is dependent on the details of the probes that are connected to the conducting channel. That is, it specifically depends on whether there are two or four probes, or the details about how well the probes absorb or reflect incoming ballistic trajectories. In general, there are a large number of possible measurement probe geometries, and each delivers different possible overall conductance^.^^ In short, one way of looking at this problem is that there is a variability in the possible voltages measured, for a given current, and it is important to know just where in space the voltage measurements are made. A slightly differxx M. Biittiker, Phys. Reii. Lett. 62, 229 (1989); in Nanostructure Physics and Fabrication (M. A. Reed and W. P. Kirk, eds.), p. 319, Academic Press, New York (1989); in Nanostruclured Systems (M. A. Reed, ed.), p. 191, Academic Press, New York (1992). XY M. Biittiker, Phys. Reu. B 38, 9375 (1988). YO P. Streda, J. Kucera, and A. H. MacDonald, Phys. Rev. Lett. 59, 1973 (1987). 91 T. Martin and S. Feng, Phys. Reu. Lett. 64,1971 (1990). 92 A. M. Chang, Sol. State Commun. 74, 871 (1990). 'JS F. Sols, in Phonons in Semiconductor Nanostruclures (J. P. Leburton, J. Pascual, and C. M. Sotomayor-Torres, eds.), Kluwer Nonuell, MA (1992). 94 L. W. Molenkamp and M. J . M. de Jong, Phys. Reu. B 49, 5038 (1994). "R. Landauer, J . Phys. Cond. Matter 1, 8099 (1989).

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ent view of this is that any particular (and certainly small) device is actually embedded within its environment. The boundary effects play an essential role in determining the physical properties of semiconductor quantum wires.96 The performance of the device is not usually separable from its environment, and the environment can in fact completely determine the performance of the device." Specific studies, with regard, for example, to resonant-tunneling diodes clearly show this is the case,97 a point that we will continue to see in the simulations discussed in later sections. The role of the interaction between the contacts (or probes or reservoirs, as the case may be) and the device can be significant. Indeed, it is actually possible for scatterers near the contact to induce oscillations in the electrochemical p~tential,~'which further complicates the contact potential drop. The importance of the contacts and probes is best exemplified in the case of the quantum Hall Since the edge states are the penultimate ballistic (and nondissipative) channel, the entire conductance and voltage distribution depends on the details of the current and potential probes. The nature of this interaction can in fact be studied by varying the confinement potential to study the transition from local (classical) to nonlocal, ballistic transport.'" The idea of environment must be extended beyond just the concepts of probes and contacts. Indeed, it is the entire environment of a particular device that can lead to changes of device behavior. The presence of continuous devices opens the door to transfer between such devices, which has been especially studied in coupled quantum wires.ln2Another important scattering process, in addition to internal scatterers within the device, is interaction with the interface modes of the lattice.ln3 In fact, in many cases the scattering from remote and interface modes may be more likely than scattering within the active device region, simply because of the fraction of phase space sampled by any particle wavefunction inside the device may be smaller than that part outside. This is also true for the 96

G. Y. Hu and R. F. O'Connell, J . Phys. Cond. Matter 4, 9623 (1992). S. Ho and K. Yamaguchi, Semicond. Sci. Technol. 7 , B430 (1992). 98 M. Biittiker, Phys. Reu. B 40, 3409 (1989). 99 M. Biittiker, Ann. New York Acad. Sci. 581, 176 (1990); Festkorperprobleme 30, 41 (1990). 1Ol1 M. Biittiker, Phys. Reu. B 38, 13297 (1988). 1111 G. Miiller, D. Weiss, K. von Klitzing, K. Ploog, H. Nickel, W. Schlapp, and R. Losch, Phys. Reu. B 46, 4336 (1992). 102 M. Macucci, U. Ravaioli, and T. Kerkhoven, Superlutt. Microsfmc. 12, 509 (1992); U. Ravaioli, T. Kerkhoven, M. Raschke, and A. T. Galick, Superlutt. Microstmc. 11, 343 (1992). 103 M. A. Stroscio, G. J. Iafrate, K. W. Kim, M. A. Littlejohn, A. Bhatt, and M. Dutta, in Integrated Optics and Optoelectronics (K.-K. Wong and M. Razeghi, eds.), Crit. Reviews of Opt. Science and Technology, vol. CR45, p. 341 (1993). Y7


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surface, which may have extensive influence on the nature of the wave packets.ln4 We will examine the specific formulation that leads to the detailing of the device-environment interaction in a great deal more depth later. 4. SOMEPOTENTIALLY IMPORTANT QUANTUM DEVICES

It has already been pointed out above that the normal Si MOSFET actually incorporates quantization within the channel, in the direction normal to the interface between the oxide and the Si inversion layer. Such quantization also appears in the GaAs/AlGaAs HEMT in a similar fashion. Carriers are created by dopants placed in the AlGaAs, and these carriers transfer into the GaAs to create an accumulation layer on the GaAs side of the heterojunction interface. In both cases, the inversion/accumulation layers are created in a self-consistent potential with the actual size (thickness) of the layer being larger than the classical width due to the wavefunction of the carriers. This leads to a number of observable quantum effects in these devices, but most occur at low temperatures. Moreover, much of these effects is only of second order in the transport properties. Nevertheless, full understanding of the ultrasmall device will require a more advanced quantum transport treatment. As was already mentioned, one of the most obvious quantum effects that can occur in the transport for ultrashort gate lengths is tunneling through the gate depletion region, and study of this effect is impossible in a classical treatment. Tunneling is a fully quantum-mechanical process in which a carrier penetrates into and traverses a barrier region, where the amplitude of the barrier exceeds the kinetic energy of the carrier. It first became of interest in semiconductors in highly-doped p - n junctions, where the conduction band on the n-type side lay below the valence band edge on the p-type side (so-called degenerately doped junction^).'^^ The theory of such interband tunneling, which can also occur in semiconductors under very high electric fields (where it is often referred to as Zener tunneling), has been worked out over many decades, and has been reviewed extensively.'""'08 Even with a long history of work, there remain questions about the details 1114

M. C. Chen, W. Porod, and D. J. Kirkner, J . Appl. Phys. 75,2.54.5 (1994). Io5L. Esaki, Phys. Reu. 109, 603 (19.58). Ill6 See. e.g., E. 0. Kanc, J . Phys. Chem. Sol. 12, 181 (19.59), and the next two references. 1117 P. N. Argyrcs, Phys. Rec. 126, 1386 (1962). Ill8 P. J. Price in Handbook of Semiconductors, Vol. 1 (revised) (P. T. Landsberg, ed.), North-Holland, Amsterdam (1992).

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of real tunneling processes in the presence of dissipative mechanism^,'^', ' l o and the tunneling time, the physical time required for a carrier to move from one side of the tunneling barrier to the other, remains quite controversial." Tunneling is also the basis of the scanning tunneling microscope,'12 a new method of studying surfaces with atomic resolution. Nevertheless, the key device of interest at present is the resonant-tunneling diode.

'

a. The Resonant- Tunneling Diode The idea of using two tunneling barriers within a single conduction (or valence) band is a relatively old idea. However, the concept of using band-gap engineering with semiconductor heterojunctions to create realistic barriers is only a few decades old,'I3 and such structures have clearly shown negative differential conductivity as soon as they were made.'14 The basic idea is shown in Fig. 3. The barriers are formed by thin layers of AlGaAs, and the well and boundary layers are formed from GaAs. A quasi-bound state forms in the well layer. With no applied bias, the tunneling through the structure is greatly reduced, because the tunneling length must extend over the entire width of the three barrier and well layers. On the other hand, when an applied bias is present, the anode layer and well layer are pulled to lower energies. When the quantum well level becomes degenerate with the occupied conduction band states in the cathode layer [Fig. 3(b)], current begins to flow through the structure, since there is now a resonant state available to the electrons, and the tunneling distance is now just that of the first barrier. If the two barriers were equal when the quantum level aligns with the cathode-filled states, the transmission coefficient would rise to unity. When further bias is applied, so that the quantum level in the well drops below the conduction band edge of the cathode, current no longer flows. Thus, current flows only for a finite range of bias, and negative resistance is obtained on the high voltage side of this region.Il5 We return to this later, with a more detailed description in each 109

A. 0. Caldeira and A. J. Leggett, Phys. Reu. Lett. 46, 211 (1981). "OH. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985). Ill See, e.g., the review by M. Jonson, in Quantum Transport in Semiconductors (D. K. Ferry and C. Jacoboni, eds.), Plenum Press, New York (1992); recent studies have continued to be published, for example, A. P. Jauho and M. Jonson, J . Phys. Cond. Matter 1, 9027 (1989); J. A. St$vneng and A. P. Jauho, Phys. Rev. B 47, 446 (1993). '''B. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57 (1982). 113 L. Esaki and R. Tsu, IBM J . Res. Develop. 14, 61 (1970); R. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562 (1973). 'I4L. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24, 593 (1974). 115 D. K. Ferry, in Physics of Quantum Electronic Devices (F. Capasso, ed.), p. 77, SpringerVerlag, Berlin (1990).


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FIG.3. The potential distribution for a double-barrier resonant-tunneling diode. (a) The potential is shown in equilibrium. The full states (shaded) on either sides of the barriers are coupled only by tunneling through the entire structure. (b) Under bias, however, the resonant level is aligned with the states in the emitter, allowing them to tunnel through with a probability approaching unity.

of the following sections, and include a discussion of nonresonant tunneling as well. More recently, the resonant-tunneling diode has found applications in microwave circuits for amplification and oscillation. Throughout the development of the resonant-tunneling diode, there have been a number of controversies. The first was whether the electrons tunneled completely through the structure coherently (in one step) or sequentially (in two step^)."^ In the end, it turns out that the actual calculation of the current seems to be independent of this, but the only certainly sequential processes involve scattering of the carriers, especially when the scattering is The second controversy, and perhaps more meaningful, was over the role of trapped charge in the quantum well. For increasing bias, the quantum well has to be emptied as the current shuts off. When "'T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, and D. 0. Peck, Appl. Phys. Lett. 43, 588 (1983). "'See, e.g., M. Buttiker, IBM J . Res. Develop. 32, 63 (1988). i i nA. P. Jauho, Phys. Rev. B 41, 12327 (1990). 119 J. A. St$vneng, E. H. Hauge, P. Lipavslj, and V. SpiEka, Phys. Reu. E 44,13595 (1991). "OM. Sumetskii, Phys. Lett. A 153, 149 (1991). ''I M. Wagner, unpublished results.

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the bias is again reduced, the well must be charged to begin the current. This suggests that there should be some hysterysis in the current-voltage curve.122To study this effect, one needs self-consistent calculations for the current-voltage curve. Many of these have been carried out and are discussed in the later sections of this paper. Nevertheless, some more straightforward approaches have also appeared; more straightforward only in the sense that they try not to get involved in the detailed calculations of 124 quantum transport.'23% The resonant-tunneling diode is the "fruit fly" for quantum transport studies, since the classical description cannot provide any insight into the process-the tunneling process is the key ingredient. Thus, we will see it treated again and again in our discussions here. In addition to the current, charge within the well, and the overall device characteristics, interest has focused recently on simple evaluations of the high-frequency conductivity and the n ~ i s e . ' ~When ~ * ' ~the ~ resonant-tunneling diode is also laterally ~ o n f i n e d , ' ~ ' -one ' ~ ~ gets a quantum dot (the laterally confined quantum well) with quite a complicated level structure within the dot. b. Quantum Dots

Lateral patterning to create an isolated region in which electrons (or holes) can be localized has led to considerable effort in the study of quantum dots and the electronic structure in these dot^.^'^'^^ Usually, as in solids, the energy structure is calculated for the one-electron states, but consideration of the carrier spin'30, and multiparticle states has also occurred.'32 The quantum dot is usually thought of as a localized region IZ2F.W. Sheard and G . W. Toornbs, Appl. Phys. Left. 52, 1228 (1988). 121 H. P. Joosten, H. J. M. F. Noteborn, K. Kaski, and D. Lenstra, J . Appl. Phys. 70, 3141 (1991). 124 T. Fiig and A. P. Jauho, Appl. Phys. Left. 59,2245 (1991); Su$. Sci. 267, 392 (1992). lZ5C.Jacoboni and P. J. Price, Sol. State Commun. 75, 193 (1990); in Resonant Tunnelingin Semiconductors (L. L. Chang ef al., eds.), p. 351, Plenum Press, New York (1991). 126 J. H. Davies, P. Hyldgaard, S. Hershfield, and .I. W. Wilkins, Phys. Reu. B 46, 9620 (1992). 127 M. A. Reed, J. N. Randall, R. J. Agganval, R. J. Matyi, T. M. Moore, and A. E. Westel, Phys. Reo. Lett. 60, 535 (1988). 128 H. Mizuta, C. Goodings, M. Wagner, and S. Ho, J . Phys. Cond. Marrer 4, 8783 (1992). 129 M. P. Stopa, Surf. Sci. 367, 286 (1992); M. P. Stopa and Y. Tokura, in Science and Technology of Mesoscopic Sfructures (S. Namba, C. Harnaguchi, and T. Ando, eds.), p. 297, Springer-Verlag, Tokyo (1992); M. Stopa, Phys. Reo. B 48, 8340 (1993). 130 A. S. Sachrajda, R. P. Taylor, C. Dharrna-Wardana, P. Zawadski, J. A. Adarns, and P. T. Coleridge, Phys. Rev. B 47, 6811 (1993). 131 M. Wagner, U. Merkt, and A. V. Chaplik, Phys. Rev. B 45, 1951 (1992). "'U. Merkt, J. Huser, and M. Wagner, Phys. Reu. B 43, 7320 (1991).


307

Fic;. 4. The gate electrodes deplete the quasi-2D electron gas under them to create a quantum dot region in the enclosed area. The size of this area can be adjusted by depleting additional states with the “tuning” electrode. This adjusts the resonances and transmission through the structure.

defined by gate potentials, as shown in Fig. 4, but it can actually be a waveguide resonator attached to an electron (or hole) ~ a v e g u i d e . ’The ~~ quantum dot is an interesting mesoscopic structure in its own right; it is a mini-Aharonov-Bohm ring when edge states cycle through the ~ t r u c t u r e , ’ ~ ~ and it can occur, and be studied, in a variety of manners, not the least of which is by STM probing.’35 We now are beginning to see studies of various “interactions” in the dot to study its properties specifically; for example, recently Feng et created an additional gate-controlled region in the center of the dot to act as an “impurity.” Obviously, the central quantum well of a resonant-tunneling diode also plays the role of a quantum dot. In fact, many aspects of quantum dots coupled to waveguides, or other probing regions, play much the same role as the tunneling coupling. However, the quantum dots can also be coupled capacitively, which is a classical interaction. Quantum dot effects begin to occur when the capacitors coupling the central region to its environment begin to be sufficiently small that the change in energy of the capacitor, when one electron transits it, is larger than the thermal energy or any bias energy. This is the so-called Coulomb blockade regime, and leads to the I33 See, e.g., Z. Shao, W. Porod, and C. S . Lent, in Proc. 3rd Intern. Workshop Cornp. Elecfron.,(S. M. Goodnick, ed.), p. 219, Portland, Oregon (1994); W. Porod, Z. Shao, and C. S . Lent, Phys. Reis. B 48, 8495 (1993). 134 R. P. Taylor, A. S. Sachrajda, P. Zawadski, P. T. Coleridge, and J. A. Adams, Phys. Reu. Left. 69, 1989 ( 1992). ‘jSM.Sumetskii, J . Phys. Cond. Matter 3 , 2651 (1991). 13h Y. Feng, A. S. Sachrajda, R. P. Taylor, J. A. Adams, M. Davies, P. Zawadski, P. T. Coleridge, D. Landheer, P. A. Marshall, and R. Barber, Appl. Phys. Lett. 63, 1666 (1993).

308


field of single-electron tunneling (SET).137One advantage of the use of capacitively coupled dots is the ability to modulate the barriers and produce various device-like effects through this gate m ~ d u l a t i o n . ' ~ * - ' ~ ~ Others have reversed this to use the oscillating barriers to study tunneling through these barrier^.'^^,'^^ Noise has also been suggested as a mechanism to study the tunneling properties of the barriers them~e1ves.l~~ For our purposes here, the quantum dot is another interesting variant of the resonant-tunneling diode, and we will see several approaches to treat the detailed transport through these devices. It should be pointed out, though, that arrays of quantum dots form an interesting lateral surface superlattice, which leads to a number of other interesting new physical effects, particularly in the magnet~transport.'~~ Random arrays of the SET devices are a major candidate for future logic applications in the ultrasmall regime,'37x'45 It should also be pointed out that multiple quantum dots illustrate SET effects even in Si MOS structure^,'^^ a result that is expected from semiclassical modeling.'47 II. The Quantum Equations

Although there are different formulations of quantum mechanics, nearly all approaches that lead to modeling of semiconductor devices derive from 137

See, e.g., D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids (B. L. Altshuler, P. A. Lee, and R. A. Webb, eds.), p. 173, North-Holland, Amsterdam (1991); K. K. Likharev, in Granular Nanoelectronics (D. K. Ferry, J. R. Barker, and C. Jacoboni, eds.), Plenum Press, New York (1992). 138 L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. Mooij, H. Pothier, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Reu. Lett. 64, 2691 (1990). 139H.Pothier, P. Lafarge, P. F. Orfila, C. Urbina, D. Esteve, and M. H. Devoret, Physica B 169, 573 (1991); Europhys. Lett. 64, 2691 (1992). 140 M. P. Stopa, Surf. Sci. 263, 433 (1992); J. P. Bird, K. Ishibashi, Y. Aoyagi, and T. Sugano, Sol.-State Electron., 37, 709 (1994). I4'M. Sumetskii, Phys. Rev. B 46, 4702 (1992). I4*J. D. White and M. Wagner, Phys. Reo B 48, 2799 (1993). 143J. W. Wilkins, S. Hershfield, J. H. Davies, P. Hyldgaard, and C. J. Stanton, Phys. Scripta T42, 115 (1992); S. Hershfield, J. H. Davies, P. Hyldgaard, C. J. Stanton, and J. W. Wilkins, Phys. Reu. B 47, 1967 (1993). 144 D. K. Ferry, Prog. Quantum Electron. 16, 251 (1993). 145K.Nakazato, T. J. Thornton, J. White, and H. Ahmed, Appl. Phys. Letr. 61, 3145 (1992); K. Nakazato, R. J. Blaikie, J. R. A. Cleaver and H. Ahmed, Electron. Lett. 29, 384 (1993). 146 H. Matsuoka, T. Ichiguchi, T. Yoshimura, and E. Takeda, IEDM Conf. Proc., p. 781, IEEE Press, New York (1992). 14'K. Yano and D. K. Ferry, Superlatt. Microstruc. 11, 61 (1992); Phys. Reu. B 46, 3865 (1992).


309

the Schrodinger equation, (11.1) in one dimension (here taken to be x ) so that 1I' = 9 ( x , t ) . Note that we have taken a particular form for the momentum, in that it is assumed that the particle energy is quadratic in the momentum. In devices, modeling usually proceeds from another formulation, which arises from Eq. (11.1) as is shown later. The form of Eq. (11.1) is dissipationless, since the potential is normally the applied or built-in electrostatic potential. Although other forms are usually used, work has been done to apply the Schrodinger directly in simulations. For this, it must usually be assumed that the length of the region being simulated is considerably smaller than any characteristic dissipation length. Various characteristic lengths are important in the quantum mechanical description of tran~port.'~'Of most interest are the elastic mean free path, which describes a characteristic scattering length for elastic processes that do not break the phase coherence or relax the energy, and the inelastic mean free path, which describes processes that do break the phase coherence. Some processes can break phase coherence, but do not relax the energy, although almost all energy relaxing processes break the phase coherence. Processes that can break phase coherence without relaxing the energy can arise from elastic processes that are sufficiently strong that they introduce localization. T h o u l e ~ s 'suggested ~~ that one should relate the coherence length to the inelastic mean free time as L,=

fi,

(11.2)

where D is the carrier diffusion constant (it is assumed that the transport is diffusive, which implies that it is not ballistic or that there is considerable elastic scattering occurring within this length). Generally, it is still true that there is not a particularly good theoretical basis for calculating L , as yet.14* Indeed, most estimates for its value are taken from experimental studies of the material in a particular device configurati~n.'~ The basic concepts of transport in mesoscopic systems in the presence of localized scatters can be traced in Landauer." It is now recognized that slowly varying elastic potentials can lead to localization, and hence phase breaking in the ~ y s t e r n . ' ~ More ~ ~ ' ~ importantly, ' there is a wealth of work 148

B. Kramer and J. MaSek, Z . Physik B 76,457 (1989). 14')D.J . Thouless, Phys. Reu. Lett. 39, 1167 (1977). "('H. Cruz and S. Das Sarma, J . Physique 1 3 , 1515 (1993). R. Bertoncini, unpublished results.

'"

3 10


now that clearly shows that the onset of inelastic scattering will suppress many of the quantum effects that are of interest; for example, quantum interference effect^."^ There are many techniques to simulate this, even 154 Indeed, the role of scattering by large with the Schrodinger energy exchange processes, such as optical phonons, has clearly been demonstrated in studies of the double-barrier resonant-tunneling diode (DBRTD).1]8-120.155- 157 The treatment of transport with the Schrodinger equation has followed several approaches. In one case, the scattering matrix formulation utilized by Buttiker and Thomasg4 has been used to study simple waveguides in which elastic scatterers have been imbedded.'58 Here, a new concept has been introduced, and that is that the wave nature of the electron can be ~ used to treat the transport of the electron as a guided-wave p r ~ b l e m , 'just as in the case of microwave waveguides. The approach here uses Eq. (2.4) with the total transmission defined as (11.3) n,m

where Tn, is the transmission from mode m of the input to mode n of the output contact. To develop this, it is usually assumed that the waveguide is created in an otherwise quasi-2D electron gas. The experimental waveguide itself can be defined either by physically creating a waveguide region by reactive-ion etching or by defining it electrostatically with lateral gates.13 Then it is possible to write Schrodinger's equation in two dimensions as (time independent, however)

with

152 F. W. J. Hekking, Yu. V. Nazarov, and G. Schon, Europhys. Lett. 14,489 (1991); 20, 255 (1992). 153 F. Sols, in Nanostmctures and Mesoscopic Systems (M. A. Reed, ed.), p. 417, Academic Press, New York (1992); in Phonons in Semiconductor Nanostmctuwes (J. P. Leburton, J. Pascaul, and C. M. Sotomayor-Torres, eds.), Kluwer, Nonvell, MA (1992). 154L.F. Register and K. Hess, Phys. Rev. B 49, 1900 (1994). 155 N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins, Phys. Reu. Lett. 61, 1396 (1988). 156W.Cai, T. F. Zheng, P. Hu, B. Yudanin, and M. Lax, P h y ~ Rev. . Lett. 63, 418 (1989). "'H. A. Fertig, S. He, and S. Das Sarma, Phys. Rev. B 41, 3596 (1990). IS'S. Datta, M. Cahay, and M. McLennan, Phys. Reu. B 36, 5655 (1987); M. Cahay, M. McLennan, and S. Datta, Phys. Reu. B 37, 10125 (1988); H. R. Frohne, M. J. McLennan, and S. Datta, J . Appl. Phys. 66, 2699 (1989).


311

and the first term on the right-hand side is the confinement potential defining the lateral extent of the waveguide, whereas the last term is any applied potential describing bias or impurities, etc. The general solution of the wavefunction in any small region (these regions are then connected159) over which the lateral confinement potential is constant (which means that the waveguide has uniform properties) is given by (11.6) where in general, (11.7)

for hard-wall confinement of the waveguide (in hard-wall cases, it is usually assumed the wavefunction vanishes at the confinement wall). Other approaches sometimes use soft walls with quadratic potentials in which the lateral modes are described by harmonic oscillator wavefunctions. The longitudinal modes are described, in general, by a combination of forward and backward waves, as +,,,CX>

=

a,,eYt,* + b,,e-"r1",

(11.8)

where y,, is the propagation constant. If the mode is a propagating mode, then = ik,, and describes the wave nature of the mode. If, on the other hand, the mode is evanescent, then y,, is a real quantity describing the decay of the mode. It is very important to note that proper inclusion of the evanescent modes is vely important in studying waveguide discontinuities by this method, just as it is in microwave waveguides. At the interface between two regions, in each of which the mode properties are uniform, the total wavefunction and its derivative are matched across the interface. (Note that if there is an applied potential at the interface site, the normal derivative is discontinuous by an amount determined by this potential.I6" This approach has been used to study the role of rounded corners at crossing waveguides,161 waveguide stubs,162and the effects of impurities in the stub r e g i ~ n , " ~ as * ' well ~ ~ as other configurations mentioned later. "'M. Macucci and K. Hess, Phys. Rev. B 46, 15357 (1992). IhO Y. Takagaki and D. K. Ferry, Phys. Reu. B 45, 8506 (1992). Ihl Y. Takagaki and D. K. Ferry, P h y . Reu. B 44, 8399 (1991). 162 F. Sols, M. Macucci, U. Ravaioli, and K. Hess, in Nanosfmcture Physics and Fabrication (M. A. Reed, ed.), p. 157, Academic Press, New York (1989). 161 A. Weisshaar, J. Lary, S. S. Goodnick, and V. K. Tripathi, Appl. Phys. L e f f .55, 2114 (1992); Proc. SPIE 1284, 45 (1990). I64 Y. Takagaki and D. K. Ferry, Phys. Reu. B 45, 6715 (1992).

312


One particularly interesting application is the study of a waveguide with a double constriction. That is, two narrow waveguides are separated by a wide region, and contacted with wider reservoirs on the ends, as shown in Fig. 5(a).'65v'66This structure is the waveguide equivalent of the DBRTD described previously. Modes are allowed in the wide central region, which are below cutoff in the constricted regions, and this allows for tunneling into the central region if the constrictions are sufficiently short, and consequently a negative-differential conductance can be obtained in the structure. In Fig. 5(b), the overall transmission probability is shown for such a structure as the energy of the incident wave is varied. This may be used in a conventional tunneling calculation to determine the current -voltage characteristics, and the resonance peak in the figure allows for the existence of the negative-differential conductance in these

characteristic^.'^^

Note that the waveguide mode-matching technique is fully compatible with formulation of scattering matrices,'67 although one normally thinks of the matching of the wavefunction and its derivatives in terms of a transfer matrix approach. The waveguide approach has great versatility, as long as the active regions are easily defined in terms of waveguide sections. It has been applied to tunneling between different waveguides,'68,lh9 to the onset of localization arising from rough waveguide boundarie~,'~'to bends,17' and to resonances from multiple bends in the waveguide.I7* In the latter, results comparable to lattice Green's function approachess5 have been obtained. One problem is that self-consistent solutions to the waveguide mode propagation have not been obtained, although the role of carrier-carrier scattering between various sub-band modes has been stud165

A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Tripathi, IEEE Electron Dev. Lett. 12, 2 (1991). 166 Y. Takagaki and D. K. Ferry, Phys. Reu. B 45, 13494 (1992). 107 A. Weisshaar, J . Lary, S. M. Goodnick, and V. K. Tripathi, J . Appl. Phys. 70, 355 (1991). 168 M. Macucci, U. Ravaioli, and T. Kerkhoven, Superlutt. Microstnrc. 12, 509 (1992). 169 Y. Takagaki and D. K. Ferry, Phys. ReLl. B 45, 12152 (1992); J . Appl. Phys. 72, 5001 (1992); J. Phys. Cond. Matter 5, 1975 (1993). 171) Y. Takagaki and D. K. Ferry, J . Phys. Cond. Matter 4, 10421 (1992); Phys. Rev. B 46, 15218 (1992). '"F. Sols and M. Macucci, Phys. Reci B 41, 11887 (1990). I?? J. C. Wu, M. N. Wybourne, W. Yindeepol, A. Weisshaar, and S. M. Goodnick, Appl. Phys. Lett. 59, 102 (1991); J. C. Wu, M. N. Wybourne, A. Weisshaar, and S. M. Goodnick, Proc. SPIE 1676,40 (1992); W. Yindeepol, A. Chin, A. Weisshaar, S. M. Goodnick, J. C. Wu, and M. N. Wybourne, in Nanosttuctures and Mesoscopic Systems, p. 139, Academic Press, New York (1992); J. C. Wu, M. N. Wybourne, A. Weisshaar, and S. M. Goodnick, J . Appl. Phys. 74, 4590 (1993).


n

P

v

10

313

-

Y

C

piL

3

5-

0 O r

0

.

5

10

15

20

Bias Voltage (mV)

(b)

FIG.5. A quantum waveguide equivalent of the DBRTD and quantum dot. (a) The wide region of the waveguide provides mode propagation at a lower energy than the thinner waveguides. This provides the equivalent of the resonant energy in the DBRTD and the dot levels in the quantum dot. (b) Computation of the current transmission through the structure leads to a negative-differential conductance just as in the DBRTD. In this calculation, W, = W, = 25 nm, W, = 45 nm, and W , = W, = 500 nm. The length of the center section is 25 nm and that of the two barrier sections is 20 nm. In addition, a depletion in from the edge is assumed under bias, with 6, = 0, 6 , = 0.5, 6, = 1.0, and 6, = 1.5 nm. The Fermi energy is 5 mV, and the temperature is assumed to be 0 K. [After Goodnick et al., IEEE Electron Deu. Lett. 12, 2 (1991), with permission; 0 1991 IEEE.]

314


ied,'73,'74 and the general many-body problem has been f o r m ~ l a t e d . ' ~ ~ Also, a generalized density-functional theory has been proposed that could be used to incorporate many-body effects in the waveguide theory.I7' In general, however, the modeling approaches discussed in the remainder of this review are better suited for incorporation of many-body effects and self-consistency with the Poisson equation for the potential.

5. THEDENSITY MATRIXAND ITS BRETHREN In general, one can solve Eq. (11.1) by assuming an expansion of the wave-function in terms of a set of static basis functions that satisfies the time-independent equation as

in which E, is the energy level corresponding to the particular basis function. Then, the total wave function can be written as

iE,t

T).

(5.2)

If we now multiply each side of this equation with h ( r ) and integrate over the position, we can evaluate the coefficient in terms of the total wavefunction at any arbitrary time, which we here take to be to. Then, Eq. (5.2) can be rewritten as

This may be rewritten as

173

G. Y. Hu and R. F. OConnell, Phys. Rev. B 42, 1290 (1990). "'I. K. Marmorkos and S. Das Sarma, Phys. Reu. B 48, 1544 (1993); S. Das Sarma, in Topics in Condensed Matter Physics, Nova Science, New York (1993). I7'B. Y.-K. Hu and S. Das Sarma, Phys. Reu. Lett. 68, 1750 (1992); Phys. R w . B 48, 5469 (1993). 17h M. G. Ancona, Superlatt. Microstmc. 7 , 119 (1990); in Proc. Computational Electronics, Univ. Illinois Press, Urbana, IL (1992).


315

Here, K(r, f ; r’, to) is our propagator kernel, or Green’s function. The kernel in Eq. (5.4)describes the general propagation of any initial wave function at time t o to any arbitrary time t (which is normally > t,, but not necessarily so). There are a number of methods of evaluating it, either by differential equations (which we pursue here), or by integral equations known as path integral^.^','^^ In general, the form shown here is for a system described fully by a well-developed set of basis functions, which are characteristic of the entire problem. For example, it is often the case that the time is taken to be imaginary, in which the substitution ( t - t,) + - & p , where p = l / k , T is the inverse temperature, and the resulting form of the kernel is that of a system in thermal eq~ilibrium.”~ In this case, we talk about Eq. (5.4) representing a simple mixture of pure states. The usual case is that the exponential is separated into the two temporal parts, and then each time-varying basis function is expanded in an arbitrary (but different) set of wavefunctions, so that we have a mixed system, and we write the kernel as

The equal-time version of this is termed the density matrix

There are many different (in detail) definitions of the density matrix. It can be defined just by the coefficients in the expansion, so that it is a c-number matri~.’~’It also appears as the thermal equilibrium formed defined earlier (for the time-independent form)

The last form of Eq. (5.6) defines it in terms of field operators, in which creation and annihilation operators replace the expansion coefficients, and these operators excite or deexcite each of the “modes” of the basis set. In any of these definitions, it is important to recall that the density matrix is the equal-time version of the Green’s function. 177

W. P6tz and J . Zhang, Phys. Reu. B 45, 11496 (1992). ”‘M. G. Ancona and G. J. Iafrate, Phys. Reo. B 39, 9536 (1989). 17V U. Fano, R ~ L Mod. !. Phys. 29, 74 (1957).

316


a. The Liouuille and Bloch Equations In general, the density matrix is best characterized (for the present argument) in terms of the field operator form, which is the last part of Eq. (5.6). The temporal equation of motion for the density matrix can then be developed using Eq. (11.1) as . dP h-dt

=

[ H, p ] ,

(5.8)

or'''

which is termed the Liouuille equation. Sometimes, a higher order operator algebra is used, since the Hamiltonian H is an operator in the Hilbert space of the density matrix (defined by some basis set of functions). In this case, Eq. (5.8) can be written

where fi is a commutator-generating superoperator.I8', 182 There is no problem in incorporating a dissipative term in the Hamiltionian and treating it by perturbation theory. In fact, this is quite viable method of treating irreversible transport, as has been discussed repeatedly.'83-'85 On the other hand, if we accept the general view of the density matrix represented by Eq. (5.71, then it is natural to introduce the time as an imaginary quantity t + - i h p , and -

180

dp = @

[ "( -

2rn

d' dr2

-

-$)+

1

V(r) - V(r') p(r,r'), (5.11)

See, e.g., F. Rossi, R. Brunetti, and C. Jacoboni, in Hot Cum'ers in Semiconductor Nanostrctures, p. 153, Academic Press, New York (1992). 181 R. Zwanzig, J. Chem. Phys. 33, 1338 (1960). 182 H. Mori, frog. Theor. Phys. 34, 399 (1966). 183 W. Kohn and J. M. Luttinger, Phys. Reo. 108, 590 (1957). 1x4 J . L. Siege1 and P. N. Argyres, Phys. Reu. 178, 1016 (1969); P. N. Argyres, in Lectures in Theoretical Physics. Vol VIII-A, p. 183, Univ. Colorado Press, Bolder (1966). lXsP.N. Argyres and J. L. Sigel, Phys. Reu. Lett. 31, 1397 (1973).


317

FIG.6. The density matrix, in equilibrium, for a DBRTD. Charge in the quantum well is indicated by the small peak centered in the gap.

which is normally termed the Bloch equation in the symmetrized space of the density matrix. In a sense, this form is a quasi-steady-state, or quasiequilibrium, form in which the time variation is either nonexistent or sufficiently slow as to not be important in the form of statistical density matrix. There exist mathematical proofs that a unique monotonic solution of this equation exists for the density matrix.186Before leaving this subject, it is also important to note that there exists an adjoint equation to Eq. (5.1 1) which arises from the anticommuntator relationship as178 -dP =

ap

[ c(2 + y )+ -

2m

dr2

V(r)

+ V(r')

p(r,r'). (5.12)

arr2

This will be used later in a discussion of the connection to semiclassical behavior. In many respects, it should be noted that the density matrix is quite often found to be self-adjoint. In Fig. 6, we show a plot of the density matrix for a DBRTD in the absence of bias. The main diagonal, for which x = x' in this one-dimensional model, represents the density variation through the device. The off-diagonal parts represent the spatial correlation that exists in the system. 1Xb A. Amold, P. A. Markowich, and N. Mauser, in Muthemaiical Models and Methods in Applied Sciences, Vol. 1, p. 83, World Sci. Press, Singapore (1991).

318


b. Wigner Functions and Green's Functions A problem with the density matrix in many semiconductor problems arises because it is defined in position space, with the important quantum interference effects occurring between two separated points in space. Many problems submit best to a phase space representation (such as the electron-phonon interaction) and are difficult with the density matrix representation. Even so, it is a function of six variables, plus of course the time (or the temperature). In many cases, it would be convenient to describe things in terms of a phase space function, whose six variables arise from a single position vector and a momentum vector. Although this is not the normal case in quantum mechanics, it certainly can be arranged.40,lX7 To see how this is achieved, we rewrite Eq. (5.9) in terms of a new set of coordinates, the center-of-mass and difference coordinates as R

=

1 -(r 2

+ r'),

s

=

(r

-

Then, the Liouville equation can be rewritten as i h z dt

=

[ iz) + + I) V (R

d2

-

-

(5.13)

r').

V(R -

I)]

p ( R , S, t ) . (5.14)

If we now introduce the phase-space Wigner d i s t r i b u t i ~ n in , ~ three ~ spatial dimensions, 1 f w ( R , p , t )= - / d 3 s (hI3

p ( R , ~ , t ) e ~ ~ ' ~ / ' , (5.15)

which is often called the Weyl t r a n s f ~ r r n , ' ~then ~ - ~the ~ ~ Liouville equation can be written as dfw

1

d

ih

(5.16) '"G. A. Baker, Phys. Reu. 109, 2198 (1958). ""J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949). IHY T. B. Smith, J . Phys. A 11, 2179 (1978). lYoA.Janussis, A. Streklas, and K. Vlachos, Physica 107A,587 (1981). '''A. Royer, Phys. Rev. A 43, 44 (1991).


319

in the absence of any dissipative processes. This can be rewritten in a more useful form as

1

-

h3 /d3PW(R,

P)f,(R, p

+ P, t ) = 0,

(5.17)

where W(R,P)

=

/d’qsin ( y ) [ V ( R

+ ;j

-

V(R -

:j].

(5.18)

The use of the Wigner function is particularly important in scattering problems,’92and it clearly shows the transition to the semiclassical world.’” Reviewing the preceding approach, it may be simply stated that the Wigner function is the Fourier transform, in the difference coordinate, of the density matrix. These are two equivalent methods of looking at a problem, one (the density matrix) is entirely in a position space, while the second is in phase space and often has the classical behavior as a limit. In Fig. 7, the Wigner function for a DBRTD, in the absence of bias, is shown. This should be compared to the density matrix version in Fig. 6. Clearly, the Wigner function shows much of the behavior of Fig. 1, in which a nonclassical behavior is encountered near the potential barriers. In general, the Wigner function described by Eq. (5.15) is not positive definite. This is a consequence of the uncertainty relationship between position and momentum. If Eq. (5.15) is integrated over all momentum, then the square magnitude of the wavefunction results, and this is a positive definite quantity, being related to the density. By the same token, integrating Eq. (5.15) over all position provides the square magnitude of the momentum density, which is also a positive definite quantity. It has been proved that the Wigner function provides a smooth continuous solution to the equation of motion [Eq. (S.17)].’94 Because of its close relation to the semiclassical Boltzmann equation, it may also be shown that it provides a robust solution to the coupled Liouville and Poisson 192

E. A. Rernler, Ann. fhys. 95, 455 (1975).

Iy3J. R. Barker and S. Murray, fhys. Lett. 93A, 271 (1983). 194

A. Arnold, P. A. Markowich, and N. Mauser, Technical Report 142 Purdue Univ. (1991); A. Arnold and P. A. Markowich, in Applied and Industrial Mathematics (R.Spigler, ed.), p. 301, Kluwer Nonvell, MA (1991).

320


FIG.7. The Wigner distribution function, in equilibrium, for a DBRTD. Again, charge in the quantum well is indicated by the spread in the small peak centered in the gap.

eq~ations,’’~ and therefore is quite usable for device modeling. More recently, Arnold has shown that the solution of Eq. (5.17) remains a “physical Wigner function” in the sense that it is a mixed quantum state consisting of a combination of pure states with nonnegative distribution weights, for all times r > O.I9‘ In fact, however, the Wigner function shown in Fig. 7 is positive definite, and this is a general result for the equilibrium “ground state.””’ In both the density matrix and the Wigner function, only a single time variable appears in the problem, as it was assumed that the two wave functions, or field operators, in Eq. (5.6) were to be evaluated at equal times. In essence, these two approaches build in correlations in space but do not consider that there may be correlations in the time domain. This approach does not have to be taken, and this leads to the concept of the use of Green’s functions themselves to describe the behavior of quantum systems.’’’ In general, one separates the kernel in the wavefunction’s integral expression for the propagator into forward and reverse times in order to have different functions for retarded (forward in time) and advanced (backward in time, in the simplest interpretation) behavior. We do this by introducing the retarded Green’s function as (for fermions) Gr(r,r’; t , t ‘ )

1Y5

=

-iO(t

-

t’)(K(r, r’; t , t ’ ) )

A. Arnold, P. Degond, P. A. Markowich, and H. Steinruck, Appl. Math. Len. 2, 187 (1989); A. Arnold and P. A. Markowich, Bolletino U.M.I. 4-B, 449 (1990). 196 A. Arnold, private communication. 19’ P. Carruthers and F. Zachariesen, Rev. Mod. Phys. 20, 245 (1983). IYxD.N. Zubarev, SOL).Phys. Uspekhi 3, 320 (1960).


321

where the angle brackets have been added to symbolize an ensemble average, which is also the summation over the proper basis states, and the curly braces denote the anticommutator product for fermions. O n the other hand, the advanced Green’s function is given by

and one can then write the kernel itself as ( K ( r , r ’ ; t , t ’ ) ) = i[Gr(r,r’;t,f’)- Ga(r,r’;t,tf)].

(5.21)

Finding the Green’s functions from the Schrodinger equation or from the Liouville equation is not difficult for simple Hamiltonians, as for any quantum-mechanical problem. Proceeding for complicated Hamiltonians, such as in the case of many-body interactions or electron-phonon interactions, is not as simple and a perturbation approach is usually used. However, this approach is not without its problems, in that the perturbation series is difficult to evaluate and may not converge. Generating the perturbation series usually relies on the S-matrix expansion of the unitary operator 199 (5.22) where V ( t ) is the perturbing potential interaction in the interaction representation. In nearly all cases, it is necessary to expand any perturbation series in terms of wavefunctions, and Green’s functions, in the absence of the perturbation, which means at t -+ --oo. In the equilibrium situation, we can take the opposite limit as well, for the upper limit of the integral in Eq. (5.221, because it is assumed that the system is in absolute equilibrium at the point t -+ as well. In the nonequilibrium situation, which is the normal case in nearly all active semiconductor devices, the latter limit is just not allowed. Then one must seek a better approach, and this has been given by the real-time (nonequilibrium) Green’s functions developed by Keldysh2”’ and Kadanoff and Baym.’”l I”

R. A. Craig, J . Math. Phys. 9, 605 (1968). zoo L. V. Keldysh, Sou. Phys. JETP 20, 1018 (1965).

20 I L. P. Kadanoff and G . Baym, Quantum Statistical Mechanics, Benajamin, New York (1962).

322


FIG.8. The path of integration for real-time Green’s functions that are out of equilibrium. The tail extending downward connects to the thermal equilibrium Green’s functions, where appropriate.

To avoid the need to proceed to t + a in the perturbation series, a new time path for the real time functions was suggested by Blandin et af.2”2 (There may well have been others, but this seems to be the work that put it in proper context.) This new contour is shown in Fig. 8, where the contour evolves from the equilibrium (thermal) Green’s function at t,, - ihp to a real-time function at to. The contour then extends in the forward time direction to max(t, r’), hence returning in the anti-time-ordered direction to In many cases, one lets to -+ - a if we are not interested in the initial transients of the system. The handling of the Green’s function, when both wavefunctions are on either the upper or lower branch is straightforward. On the other hand, when these two functions are on different branches, two new functions, the correlation functions, must be defined.*04 These are the “less-than’’ function

which has the opposite sign for bosons, and the “greater-than” function

In general, these four Green’s functions are all that are needed to handle the complete nonequilibrium problem (in relatively lowest order, as is discussed later), but it is often found that two other Green’s functions are useful. These are the time-ordered and anti-time-ordered Green’s func’‘’A. Blandin, A. Nourtier, and D. W. Hone, J . Physique 37, 369 (1976). 20 3 M. Wagner, Phys. Reu. B 44, 6104 (1991); 45, 11595 (1992). 2‘145.Rammer and H. Smith, Reu. Mod. Phys. 58, 323 (1986).


323

tions, in which the ordering is in the positive time progression around the contour of Fig. 8. These two are

and

There are obviously relationships between these six Green’s functions, and these can be expressed as

For systems that have been driven out of equilibrium, the ensemble average brackets, indicated in the definitions of the Green’s functions, no longer signify thermodynamic averaging or averaging over the ground state (at T = O), because the latter quantities are ill defined. Instead, the bracket indicates that some average needs to be taken over the available states of the nonequilibrium system, in which these states are weighted by the nonequilibrium distribution. The equation of motion for the Green’s functions is basically derivable from the Liouville equation. The development of this equation for the various Green’s functions will be put off until Section V. Here, we note that there are many methods of collapsing the Green’s functions into single-time functions, which lead to a variety of transport equations.205Let us consider how to arrive at the Wigner function from the Green’s function. We note that the definition of the density matrix, which led to the Wigner function, is basically quite similar to that of C < . This is the proper association, because the latter function relates to the nonequilibrium distribution f u n c t i ~ n . Thus, ~ ~ ’ ~we ~ ~introduce ~ the center-of-mass

“”H. Schoeller, Ann. Phys., 229, 320 (1994).

324

DAVlD K. FERRY AND HAROLD L. GRUBIN

and difference coordinates [Eq. (5.13)], and equivalent ones for time (with T the average time and T the difference time). Then it is clear that fw(R,p,T)

=

lim / d 3 ~ G < ( R , ~ , T , ~ ) e - i P ’ s / (5.29) *,

7’0

or

(5.30) where the difference time has been Fourier transformed into a frequency in the last expression. It is clear from this last expression that the difference coordinates, introduced into the Green’s function as in the density matrix, are Fourier transformed, and that the Wigner function is obtained from the Green’s function by averaging out the frequency (or energy) dependence. Thus, the crucial kinetic variable in the Wigner function is the momentum, and not the energy, although the two are certainly related through a dispersion relation. c. Reduced Density Matrices and Projection Operators In each of the descriptions introduced here-the density matrix, the Wigner distribution, and the real-time Green’s functions-it has more or less been assumed that one is dealing only with the electron system. Indeed, it has mainly been assumed that one is dealing with single noninteracting electrons so that an equation for the equivalent one-electron distribution function is adequate. Before proceeding, it is of interest to consider how a quasi-kinetic picture can be obtained for the equivalent one-electron density matrix (or other approach) from a general system in which the electrons and the lattice all contribute to the density matrix, which can be a many-electron (and many atom) function. In general, the system is described by the Hamiltonian

where the terms on the right-hand side represent the electronic motion, the external fields (in the scalar potential gauge), the lattice motion, and the electron-phonon interaction, respectively. The lattice can include the Coulomb interaction between impurity atoms and the electrons. The Hamiltonian H,, includes all of the appropriate many-body terms and energy shifts appropriate to the full electron many-body problem, the


325

details of which are not treated here. The field term represents the driving fields through a simple form H, = - e F . r. The total density matrix p is defined over the entire system: electrons, lattice, and interaction."' If p is to be represented in terms of a complete set of eigenstates for the electron and lattice systems separately, the general wavefunction will be a product of the individual wavefunction basis sets. The total trace operation, which appears as the representation of the ensemble average for a matrix form of the density operator, can be separated into a succession of separate trace operations Tr,> and Tr,, which represent the partial traces, or partial ensemble averages, over the lattice and electron components, respectively. This allows us to define the electronic density matrix pr = T r L ( p ) . It is probably worth noting at this point that it is not at all obvious that this decomposition of the density matrix into clearly definable electron and lattice contributions will hold except in the steady-state case."' In essence, the objections are equivalent to those that limit the use of the effective mass approximation to relatively long time scales for the interaction. We will explore this more in later sections. The approximations above are immediately invoked by introducing Eq. (5.31) into the Liouville equation and then summing over the lattice degrees of freedom as described. This gives (5.32) Clearly, the first term on the right-hand side is the electronic motion within whatever effective mass approximation may be suitable. The second term is the electron-lattice interaction. The trace over the lattice coordinates is equivalent to the summation over the phonon wavevectors in the electron-phonon interaction. At this point, it is important to project out the desired part of the electron density matrix, which is usually the one-electron equivalent density matrix. This will be done by the use of projection operators."' To begin Eq. (5.32) is Laplace transformed with s taken to be the Laplace transform variable conjugate to the time. Then (5.33)

+

where He = H, H,, and the superoperator notation for the commutators has been used. We now obtain the one-electron density matrix

326


through the projection operator P through182, PPe.

Pel =

(5.34)

P2=P, Q=1-P.

This particular projection operator commutes with the trace over the lattice variables, so that we may define a scattering operator as

With these definitions, Eq. (5.33) can now be written asL2 (5.36)

It is clear at this point that one only needs products of various projections of the resolvent operator (the fraction term above), and the projection operator identity 1

1

S - H

s - QHQ

_-iP+Q +Q

1

,,,Q,

QfiP

)

1 s

-

d FfiP -

,

1 s - QHQ

(5.37)

s - QHQ

where

d = PHQ

1 *

*QfiP.

s - QHQ

(5.38)

The latter term connects a “diagonal” element to an off-diagonal element, and then reconnects them by the conjugate operation, so it clearly relates to a second-order interaction of the one-electron density matrix through the electron-lattice interaction. This leads us to rewrite Eq. (5.35) as

(5.39)


327

The temporal equation is obtained again by retransforming the density matrix. This results in the quantum kinetic equation dPt!I -=

dt

-

1 t.i -PH,@p,,(t) - T / Z ( t h h o

-

t’)p,,(t’) dt’.

(5.40)

Clearly, the first term on the right-hand side gives rise to the spatial variations of the one-electron density matrix that appear in Eq. (5.9) including the potential terms. The last term on the right-hand side is the scattering term, which has been ignored in our discussions. This term incorporates retardation of the scattering, which by itself is a great difference from the Boltzmann equation. On the other hand, if the scattering term 5 varies slowly, then the convolution integral can be separated, and the Boltzmann transport equation reasserts itself as the semiclassical limit. Indeed, to lowest order, the scattering integral, the last term on the right-hand side of Eq. (5.401, is readily shown to give the Fermi golden rule for perturbation theory. The essence of the argument here is that the electron-phonon interaction, the electron-impurity interaction, and even the electron-electron interaction can all be treated by perturbation theory, so that the one-electron (or one-hole) treatments developed in the preceeding paragraphs are all the equilibrium, or zeroorder, formulations to begin with, and the higher order interactions will work to perturb these solutions. 6. THEKUBOFORMULA AND LANGEVIN EQUATIONS

Although a kinetic theory, such as that of the preceding paragraphs, is usually used as the basis of transport theory, there are alternative approaches tied rather directly to the use of correlation functions, either within a linear response formalism or with a Langevin equation formalism. The latter has existed for quite some time, but the formal development of the linear response formalism is relatively recent, following the work of Kubo.206Because this approach is often used, we review it in this section and then relate it to the various moment equations that can be obtained from the kinetic transport equations. a. The Kubo Formula and Correlation Functions In the Kubo approach, we want to find the response of the coupled electron-phonon system to a time-dependent perturbation by calculating 206

R. Kubo, J . Phys. Soc. Jpn. 12, 570 (1957); Can. J. Phys. 34, 1274 (1957).

328


to lowest order the change in the density matrix. As earlier, the Hamiltonian is written in the form of Eq. (5.31). The difference here is that it is the electric field term HF that is taken to be the perturbing potential. The quantity that is of interest is the current response to this electric field; the field is the forcing function and the current is the response to that force. To show this in detail, the field (assumed to be an ac field for the moment) is written in the vector potential gauge, so that the perturbing Hamiltonian term may be written as

where j(r) is the paramagnetic part of the symmetrized total current operator. Now, one can either transform everything into the interaction representation, keeping just the lowest order terms in the exponential expansion [Eq. (5.2211 for linear response, or expand the density matrix in the Liouville equation [Eq. (5.8)].The latter approach is used, although the end result is the same. The density matrix is expanded as

where po describes the system prior to the application of the perturbation and is thus a system in which the electrons and the lattice are in equilibrium with each other, and with any internal electric fields that may result from inhomogeneous distributions of dopants within the semiconductor device. The linearized Liouville equation is then given by

(6.4)

Here, H = H , + HL + H e L . The linear perturbation in the density matrix can then be obtained to be 6p(t) =

i -e-iHr/h h

dt'[ po,HF(t')leiH'/h.

(6.5)

In this last equation, it has been assumed that the perturbing Hamiltonian has its own time variation (which is the case for the field in the vector potential gauge), but vanishes for negative times. The exponentials in the


329

unperturbed Hamiltonian and the time variation of the perturbation will result in the current operators winding up as the interaction representation, although we did not begin with it. Note that the equilibrium density matrix produces no current, so that

where the cyclic properties of the trace have been used, the displacement current has been ignored, and j(r, t ’ ) = e - r H ~ / j(r)erHr/*. h

(6.7)

It is possible to uncouple the retardation that appears in Eq. (6.6) by taking the Fourier transform of this equation and recognizing that the electric field is related to the vector potential by F(r, w ) = ioA(R, w ) . This leads, for a homogeneous electric field, to the result for the conductivity (6.8) which is homogenous only for size scales large compared to the inelastic mean free path.207The term in the ensemble average (the trace operation) is a retarded two-particle Green’s function. It has an imaginary part that must be canceled by the displacement current at equilibrium, because the conductance is a real quantity at zero frequency. There have been many discussions through the years about the accuracy and applicability of the Kubo formula of Eq. (6.8),208but it has now been fairly well established that it is accurate and gives results that agree with other treatment. Kubo himself used the approach to give one of the first detailed quantum treatments of high magnetic field galvanomagnetic One important consequence of the discussion here is that the current response and the conductivity are now defined in terms of the fluctuations of the current itself. In essence, this is just the correlation function (or Green’s function) that appears in the integral, and is a verification of the importance of the fluctuation-dissipation theorem. The current that flows 207 See e.g., B. Kramer and J. Masek, in Quantum Fluctuations in Mesoscopic and Macroscopic Systems (H. A. Cerdeira, F. G. Mpez, and U. Weiss, eds.), p. 3, World Sci. Press, Sin apore (1991); B. Kramer, J. Masck, V. SpiEka, and B. Veliclj, Su$. Sci. 229, 316 (1990). “‘See, e.g., P. N. Argyres and D. G. Resendes, J . Phys. Cond. Matter 1, 7001 (1989). 209 R. Kubo, in Solid State Physics, (F. Seitz and D. Turnbull, eds.), Vol. 17, p. 269, Academic Press, New York (1969.

330


(the conductivity) is determined directly by the dissipation arising from the presence of the scattering processes and the electric field. The correlation function describes these fluctuations, and the Kubo formula is no more than a direct statement of this important theorem. In fact, it is a powerful technique with which to calculate the noise properties of mesoscopic (as well as macroscopic) conductors.210Since the correlation function involves the fluctuations due to scattering processes, it directly incorporates these interactions within the calculation of the two-particle Green’s function. Indeed, the role of carrier-carrier scattering in weak localization2” and of impurity scattering in universal conductance fluctuations utilizes this app r ~ a c h , ~ even ’ , ~ ~in inhomogeneous field situations such as these. Even in the so-called ballistic limit, (61) can be extended to study the ac response of quantum wires effectively.’l2 The Kubo formula of Eq. (6.8) is valid at finite temperatures and for transverse as well as longitudinal fields. At low frequencies, however, there is a problem related to the coefficients in front of the integral. This is related to the fact that the vector potential diverges at zero frequency is approached. To get around this problem, the derivation can be modified by the use of an approach outlined by Mahan.2’3 For this approach, the commutator given earlier is rewritten using the identity (6.9) which is easily proven using the properties of thermal Green’s functions, with p = l/k,T. With this identity, the conductance of Eq. (6.8) can be rewritten as

Although this result no longer contains the frequency, it still contains a nonlocal integration over the position variables and still assumes that the electric field is uniform throughout the active region. Note also that the 210 See, e.g., L. Reggiani and T. Kuhn, in Granular Nanoelectronics (D. K. Ferry, ed.), p. 287, Plenum Press, New York (1991); T. Kuhn and L. Reggiani, I1 Nuouo Cim. 14D, 509 (1992); T. Kuhn, L. Reggiani, and L. Varani, Superlatt. Microstmc. 11, 205 (1992); Sernicond. Sci. Technol. 7 , 8495 (1992). 21 I See, e.g., B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Systems (A. L. Efros and M. Pollak, eds.), p. 1, North Holland, Amsterdam (1985). 212 B. Veliclj, J. MaSek, and B. Kramer, Phys. Lett. A 140, 447 (1989). 213 G. D. Mahan, Many-Particle Physics, Plenum Press, New York (1981).


33 1

expectation value is now a proper correlation function and no longer contains a commutator product. At this point, it is important to point out that we have made a significant shift in theoretical emphasis. Until this section, the entire set of derivations and considerations has focused on the streaming terms of the kinetic equations, and not on the relaxation and/or scattering terms. With the present discussion of the Kubo formula, the opposite has now occurred, in that here we are focusing on the relaxation/scattering terms that give rise to the fluctuations and the streaming terms have been buried in reaching the Kubo formula [Eq. (6.1011. To be sure, the two approaches are not separable, and one must still evaluate the two-particle correlation functions that are the heart of the Kubo formula, and details of the relaxation effects of these two-particle Green's functions will expose the overall response of the system.214This concentration on the dissipative processes, rather than the details of the streaming terms, will also be present in the next paragraph. b. Retarded Langevin Equations Another alternative to transport theory depends on the (retarded) Langevin equation and the Onsager relations. In this approach, the time-rate of change of a dynamic variable, such as the velocity, is related to a dissipative function, or to a series of forcing terms through dissipative coefficients.2's The general form of these equations is (6.11) where a = { a , }is the set of all observables, suitably ensemble averaged to produce the c-number observable, and the F, are the thermodynamic forces.21hThe memory functions K , , are time-dependent correlation functions, just as in the Kubo formula. Indeed, in some sense, Eq. (6.11) is a generalized form of the Kubo formula. The general approach to finding the equations that make up the set of Eq. (6.11) is to expand the density matrix in linear response, as done in Eq. (6.3) and then use this to compute ensemble averages of the dynamic variables, which is just the procedure used for the Kubo formula. Note that the ordinary Langevin equation (absent the convolution integral in time) may be obtained if the thermodyW. A. Schwalm and M. K. Schwalm, Phys. Rev. B 45, 1770 (1992). "'L. Onsager, Phys. Rev. 37, 405 (1931); 38, 2265 (1931). "'R. Zwanzig, Phys. Rec. 124, 983 (1961); R. Zwanzig, K. S. J. Nordholm, and W. C. Mitchell, Phys. Re(,. A 5, 2680 (1972). 'I4

332


namic forces vary slowly on the scale of the relaxation forces, and the memory functions vary sufficiently rapidly (and decay sufficiently rapidly) that the integral over time is not sensitive to the final time; for example, the time integrals must be convergent. An advantage of using the Langevin equation approach is the equal treatment of the frictional (dissipative) forces (which give a nonvanishing result in the memory function integration) and the fluctuating forces, which average to zero in any time i n t e g r a t i ~ n . ~Moreover, ’~ the generalized Langevin equation is a reduced description of the system, which can profit from the use of many approaches developed for statistical physics. Although it can be argued that this approach does not now need to worry about the density matrix, the latter is inherent in the treatment, and one still must worry about the details of the evaluation of the memory functions, which in keeping with the Kubo formula are two-particle correlation functions. Nevertheless, it is possible to take the generalized Langevin equation beyond linear response to treat, for example, localization of the carriers”’ and high-electric-field t r a n s p ~ r t . ~ ~Because *~’~-~~~ one must inherently calculate the memory function as a many-particle interaction, it is easy to extend it to a many-body i n t e r a ~ t i o n . ~ ~ ~ , ~ ~ ~ During the past few years, an alternative approach has cropped up to generate these Langevin equations. This approach is given many names, but the most useful is force balance. We can understand this approach from the simple fact that, in analogy to the Liouville equation, the time variation of the expectation value of an operator, which itself is not an explicit function of time, is given by (6.12) 21 7

G. W. Ford, J. T. Lewis, and R. F. O’Connell, fhys. Rev. A 36, 1466 (1987); R. F. O’Connell and G. Y. Hu, in Granular Nanoelectronics (D. K. Ferry, ed.), p. 313, Plenum Press, New York (1991). 218 G. Y. Hu and R. F. O’Connell, J . fhys. Cond. Matter 2, 5335 (1990). 21 9 D. K. Ferry, in Physics of Nonlinear Transporr in Semiconductors (D. K. Ferry, J. R. Barker, and C. Jacoboni, eds.), p. 577, Plenum Press, New York (1980); J. Zimmerman, P. Lu li, and D. K. Ferry, Sol. State Electron. 26, 233 (1983). “‘J. J. Niez and D. K. Ferry, fhys. Reu. B 28, 889 (1983). ’”G. Y. Hu and R. F. O’Connell, fhysica A 153, 114 (1988). ”*G. Y. Hu and R. F. O’Connell, fhysica A 149, 1 (1988); Phys. Reu. B 38, 1721 (1988); Phys. Reu. B 39, 12717 (1989). 223 J. Devreese, R. Evrard, and E. Kartheuser, fhys. Reu. B 12, 3353 (1975); E. Kartheuser, R. Evrard, and J. Devreese, fhys. Reu. B 19, 546 (1979). 224 G. Y. Hu and R. F. O’Connell, fhys. Reu B 36, 5798 (1987); J . fhys. C 21,4325 (1988); Phys. Reu. B 40, 3600 (1989); J . fhys. Cond. Matter 2, 9381 (1990); fhys. Reu. B 44, 3140 (1991).


333

At first sight, these results appear to be the same as those of the retarded Langevin equation given earlier, and in some cases this is the proper interpretation. However, the manner in which these force balance equations has been used to calculate the resisticiity, not the conductiuity. Recall that it is the field (or the potentials) that is applied to the system, and it is the current that is the response of the system, so that the proper Kubo formula, or Langevin equation, deals with the conductivity or the momentum, respectively, for the response. Much of the literature on the force balance arises from Lei and his colleagues,225although Peeters and Devreese226have also used this approach to compare the force balance with the Feynman path integral approach. In principle, the evaluation of the resistivity is conceptually thought to be easier since, for weak scattering, one only has to keep the low-order terms. However, this approach has been criticized rather heavily, and there is an indication that the correct evaluation of the resistivity (and corresponding coefficients for other equations than the current) still requires an infinite summation over terms, even for weak scattering.**' While the former authors feel that they have answered (successfully) such criticism, the approach remains controversial in many aspects, particularly because many approximations made in the earlier formulations reduced the scattering integrals to the Fermi golden rule and to Boltzmann transport.22R One problem with this approach is that Eq. (6.12) requires some sort of ensemble average to be performed, and the question is just which ensemble is to be used. Obviously, it is not the equilibrium ensemble, since there is no current in this ensemble. Thus, the non-equilibrium distribution, or density matrix, must be computed as part of the problem, and this part of the problem is often ignored. In particular, the nature of the distribution can affect the quantum corrections to the streaming terms in the quantum kinetic equations. On the other hand, more recent calculations explicitly using a quasi-equilibrium density matrix (as well be discussed in the next section), yield results that are in keeping with other appro ache^.^^^.^^^ With the quasi-equilibrium density matrix approach, the results seem to be equivalent representations of the Retarded Langevin Equation and the Kubo formula. '25X. L. Lei and C. S. Ting. Phys. Reu. B 32, 1112 (1985); L. Y. Chen, C. S. Ting, and N. J. M. Horing, Sol. State Commun. 73, 437 (1990); Phys. Reu. B 42, 1129 (1990). 220 F. M. Peeters and J . Devreese, Phys. Rev. B 23, 1936 (1981). '"P, N. Argyres, Phys. Reu. B 39, 2982 (1989). 22xA.-P.Jauho, unpublished. 224 W. Cai, P. Hu, T. F. Zheng, B. Yudanin, and M. Lax. Phys. Reu. B 40, 7671 (1989). 27U X. J. Lu, N. J. M. Horing, and R. Enderlein, in Proc. 19th Int. Conf. on Phys. Semtconductors, Warsaw (1988); X. J. Lu and N. J. M. Horing, Phys. Reu. B 41, 2966 (1990).

334


These approaches have been applied to a variety of problems, in which estimates of the correlation functions have been made through explicit calculations, such as in the case of noise and high-electric-field transport.231- 2 3 5 Transport in quasi-two-dimensional systems and wires has also been c o n ~ i d e r e d , ~as~ has ~ - ~nonlinear ~~ transport with nonequilibrium ph~nons.*~" 7. BOLTZMANN-LIKE APPROACHES

The Boltzmann transport equation [Eq. (2.01 has been studied for a great many decades, and considerable insight into its functioning now exists. On the other hand, quantum transport has been studied for a considerably smaller amount of time, and in most cases such insight is nonexistent. For this reason, many theoretical approaches have been aimed at trying to establish Eq. (2.1) as a result of a limiting process from a more fundamental quantum basis, that is, one attempts to determine a quantum Boltzmann equation.241In general, these approaches do not simply reduce to the normal Boltzmann transport equation, since the proper manner in which to take the limit has never been defined well. Nevertheless, these derivations have been pursued for quite some time.242In general, however, it is necessary to formulate the problem with one of the more exact quantum transport equations, and then connect one of the quantum distributions to the semiclassical Boltzmann distribution through some sort of a n ~ a t z . * "This ~ - ~approach ~~ tends to work well when the major problem 23 1 G. Y. Hu and R. F. O'Connell, fhysica A 163,804(1990); fhys. Reii. B 41, 5586 (1990). 232J.J. Niez, K . 4 . Yi, and D. K. Ferry. fhys. Reu. B 28, 1988 (1983); J. J. Niez, Superluff. Microstmc. 2, 219 (1986). 23 3 F. Peeters and J. T. Devreese, in Functional Integration: Theory and Applicafions (J.-P. Antoine and E. Tirapequi, eds.), p. 303, Plenum Press, New York (1994). 234 X. L. Lei, J. Cai, and L. M. Xie, fhys. Reo. B 38, 1529 (1988). 235 X. L. Lei and N. J. M. Horing, in Physics of Hot Electron Transport, (C. S. Ting, ed.), p. 1 World Scientific Press, Singapore (1992); H. L. Cui, X. L. Lei and N. J. M. Horing, unpublished results. 23 h X. L. Lei, J. L. Birman, and C. S. Ting, J . Appl. fhys. 58, 2270 (1985). 231 J. Cai, Commun. Theor. Phys. 13, 249 (1989). 23x P. Warmenbol, F. M. Peeters, and J. T. Devreese, Phys. Reu. B 37, 4694 (1988); 39, 7821 (1989); Sol. State Elecfron.32, 1545 (1989). 23Y G. Y. Hu and R. F. O'Connell, J. fhys. Cond. Matter 3, 4633 (1991). 240 X. Lei and N. J. M. Horing, fhys. Rev. B 35, 6281 (1987); Sol. Stale Electron. 31, 531 (1988). 24 I See, e.g., G. D. Mahan, fhys. Reporfs 145, 231 (1987), and references therein. 242P.N. Argyres, J . fhys. Chem. Sol. 19, 66 (1961). 243A.P. Jauho, Phys. Reu. B 32, 2248 (1985).


335

is one of high scattering rates in an otherwise weak scattering process.244 In most cases, the results look just like the Boltzmann equation, although some novel approaches have been suggested.245 In most cases, the quantum transport may be cast into the form of the Boltzmann equation from, for example, real-time Green's functions. Then known forms of quantum animals such as the polarization246or polaron effect247 can be used in the transport equation as scattering terms. In other cases, the major quantum-mechanical effect is the dynamic change of the density of states, such as in Landau quantization, and this can be ~~~,~~~ incorporated within Eq. (2.1) by using a multi-band p i ~ t u r e ,although this is sometimes done through the force balance equation, which is subject to the concerns mentioned above. Other effective approaches use the real-time Green's functions to treat the quantum transport, and the limit is approached through taking the equal-time results, so that variations in densities are the main r e s ~ l t . ' ~ ' In ) ~ this ~ ~ ~latter case, the equation of interest is a moment of the kinetic transport equation, which relates it to the approach of the last section, although many higher order moments can be used as well.250 The major point to be made here is that the transition from quantum to classical (or semiclassical) transport must be approached carefully, and the proper route is not at all obvious. Consequently, some of the more obviously useful approaches are through moments of the transport equations themselves, because as there is often a more obvious connection between the quantum and classical worlds. This was the basis of the previous subsection, as well as the next. Nevertheless the transition often 24 4

See, e.g., L. Reggiani, Physcia Scripfa T23, 218 (1988). See, e.g., L. Reggiani, P. Poli, L. Rota, R. Bertoncini, and D. K. Ferry, Phys. Stat. Sol. ( b ) 168, K69 (1991); P. Poli, L. Rota, L. Reggiani, R. Bertoncini, and D. K. Ferry, Phys. Stat. Sol. ( h ) 176, 203 (1993). 246 B. Y.-K. Hu and S. Das Sarma, Phys. ReLl. B 44, 8319 (1991); Semicond. Sci. Technol. 7, B305 (1992). 24 7 J . T. Devreese and R. Evrard, ICTP Internal Report 74/93; Phys. Stat. Sol. ( b ) 78, 85 ( 1976). 24 K P. Warmenbol, F. M. Peeters, J. T. Devreese, G. E. Alberga, and R. G. van Welzenis, Phys. Reu. B 31, 5285 (1985); P. Warmenbol, F. M. Peeters, and J. T. Devreese, Sol. State Electron. 31, 771 (1988); Phys. Reu. B 38, 12679 (1988); P. Warmenbol, F. M. Peeters, X. Wu, and J. T. Devreese, Phys. Rev. B 40, 6258 (1989). 249 N. J. M. Horing, H. L. Cui, and X. L. Lei, Phys. Rev. B 35, 6438 (1987). 2suW.Cai, M. C. Marchetti, and M. Lax, Phys. Reti. B 37, 2636 (1988); T. F. Zheng, W. Cai, P. Hu, and M. Lax, Phys. Reit. B 40, 1271 (1989). "ID. B. I'ran Thoai and H. Haug, Phys. Stat. Sol. ( b ) 173, 159 (1992). 252 F. Rossi, T. Kuhn, J. Schilp, and E. Scholl, in Proc. 21st Int. Conf. Physics of Semiconductors, World Sci. Press, Beijing (1992); T. Kuhn and F. Rossi, Phys. Reu. Lett. 69, 977 ( 1992). 24 5

336

DAVID K. FERRY A N D HAROLD L. GRUBIN

incorporates effects that are not present in the classical world and make the limiting process difficult to achieve. 8. MOMENTEQUATIONS: THETRANSITION FROM

CLASSICAL TO QUANTUM

The discussion of the previous two sections, regarding how to achieve force balance (or other balance) equations in a method that does not utilize the rigor of the Kubo formula (or the equivalent retarded Langevin equation), suggests that one should approach this topic with a great deal of care. Foremost, it is important to know that the ensemble averages that must be computed in, for example, Eq. (6.11) depend on the details of the ensemble, or distribution function, itself, so that this crucial computation must still be carried out. In many situations, it is possible to create a quasi-equilibrium distribution function, which is parameterized in the observables. For each of these parameters, an equation of motion, describing its temporal evolution, must be formulated-this leads to the set of so-called balance equations. This approach is quite old, having been basically studied for more than half a century (not counting the work on the classical Boltzmann transport equation). The structure of this is that the evolution must be decoupled from the initial condition, usually by interparticle scattering, so that a quasi-equilibrium density matrix can be defined by its integral invariants.253Indeed, it may generally be said that the quasi-equilibrium density operator can be written as254

where Z is the partition function, f k is the integral invariant force, and Pk is the conjugate quantum-mechanical operator. Examples of these are the Fermi energy, the average momentum, and the inverse temperature p for forces, which are proportional to the conjugate force for the number, momentum, and energy operators, respectively. For each of these pairs, we must have a “balance” equation that describes the temporal evolution of the appropriate quantity.255In many cases, these balance equations offer a more convenient method of solving for transport properties, or for modeling devices, than the more detailed full solutions of the appropriate quantum-mechanical distribution. 25 3

N. N. Bogoliubov, Problemi dinam. teorii L’ stat. Fir. (Moscow, 1946). N. Zubarev, Nonequilibrium Statistical Mechanics, Consultant’s Bureau, New York (1974). 25 5 V. P. Kalashnikov, Physica 48, 93 (1970). 254 D.


337

Even with this approach, one is still faced with the manner in which the classical limit should be approached. Consider, for example, Eq. (5.141, which can be rewritten in the form (in the absence of scattering) dp -

dt

i--h d2p m dRds

+ "V(R + h

-

V(R

-

ij]p(R,s)

=

0. (8.2)

It would be nice to accept the classical limit of this equation as the Boltzmann equation (2.1). For this to be the case, however, several assumptions and limits must be introduced. One that is not too difficult to make is that the corresponding momentum is given by p = -ih(a/ds). Indeed, this is the result expected from the transformation into the Wigner formulation. The next term is more critical. To obtain the classical force term in Eq. (2.1), we need to expand the potential in a Taylor series, and then associate the difference coordinate with ( i / h ) s + - d/dp, which again is not unusual because it connects p and s as conjugate operators. The problem arises in the fact that we must also limit the potential to be of no higherpower than quadratic in the coordinates. This is just not the normal case, and we must face the fact that if the potential varies with higher powers than quadratic, the quantum transport equation will not reduce to the classical one! There will be extra terms that represent these higher order variations; indeed, these extra terms correspond to the WignerKirkwood expansion used in estimating the Wigner transport equation. But this behavior is just the situation that is expected from the density variation shown in Fig. 1 earlier. It is clear that we do not approach the classical limit by simply letting h -+ 0; the limiting process is more involved than this. In many respects, the behavior of the quantum distribution, and the density, must incorporate a nonlocal quantum potential, which produces the behavior of Fig. 1. We introduced this topic in the early parts of this As review, where we discussed the Wigner4" and B ~ h mpotentials. ~ ~ mentioned, other approaches have been based on variational appro ache^,^^ or other estimation^.^^' There has been a great deal of discussion in particular about the Bohm potential, because it tends to be linked closely to a different interpretation of quantum mechanics, which is somewhat controversial.2s7 Nevertheless, it has demonstrated the ability to predict quantum energy levels and is useful in many application^,^^' particularly in 256A.M. Kriman, J.-R. Zhou, and D. K. Ferry, Phys. Lett. A 138,8 (1989). 2s 7 See, e.g., P. R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press (1993). zs8D. Bohm and B. J. Hiley, Found. Phys. 11, 179 (1981); 11, 529 (1981); 12, 1001 (1982); 14, 255 (1984).

338


mesoscopic devices.259Here, however, we will take a different approach to find an equivalent quantum potential. To this end, we write the adjoint of the Bloch equation in the center-of-mass and difference coordinates as (again, in the absence of scattering processes)

where

Within the approximation that the logarithm of the density matrix can be approximated as linear in the potentials, it can be shown then that the density matrix may be found from Eq. (8.3) ass0

-p<WQ+ UQ)-

3 2

- In(

p)

ipd's +n

-

( 5 ) 2 } ,( 8 . 5 )

-

2P h

where the first term is found from the equation

-

h2 d2p 2m d s 2

+ W(R,s),

(8.6)

and W(R, s) is given by Eq. (8.4). The solution to this is given as5'

(8.7)

Thus, the actual potential, and an additional density-dependent function, are smoothed by the exponential Green's function. Here, A, is the thermal de Broglie wavelength introduced earlier. The exact actual rela259J. R. Barker, Semicond. Sci. Technol., 9, 911 (1994).


339

tion of Eq. (8.7) to either the Wigner or the Bohm potential is not currently known. However, it has now been shown that, in the limit of slowly varying potentials where only low-order terms in the potential expansion are treated, Eq. (8.7) reduces to simply

so that it is the Bohm potential that first modifies the statistical thermodynamics of the distribution function. a. The Hydrodynamic Moment Equations The density matrix has the usual characteristics that, in the limit s -+ 0, and with appropriate other limiting processes (e.g., in connection with h), it becomes the normal density n(R). The equation of motion for the density matrix, which should have some asymptotic connection with the Boltzmann equation for n(R), is obtained from the Liouville equation, and is given from Eq. (8.2) as (in the absence of dissipation) dP .fi -I--

dt

a%

m dRds

h

p(R,s)

=

0.

(8.9)

Here, the sinh function has been obtained in the same manner as the cosinh in the preceeding paragraph as a representation of the two displaced evaluations of the potential. We note that this last term leads only to odd orders of derivatives of the potential, as already noted, and the higher orders (higher than the first order) are clearly quantum correction terms because they do not appear in the Boltzmann equation. If we now take the limit of this equation as s + 0, we find

where we have used the operator definition of the momentum given earlier, asserting that an averaging process takes place as well. This is the well-known continuity equation. However, there are some problems of interpretation, since in the absence of dissipation the transport must be reversible. In a strict sense, the distribution is symmetric across the diagonal axis (s = O), so that the average momentum and the time derivative both actually vanish. The equation strictly has meaning when dissipation in the system (along with the proper driving forces) create asymme-

340


tries in the density matrix and drive it out of equilibrium. This does not mean that we cannot create an asymmetric distribution representing ballitic transport through a dissipation-free region. However, to do so requires special considerations in the contacts (hence the need for treatment of an open system) to maintain this special distribution f ~ n c t i o n . ’ ~ Then, to consider Eq. (8.10) as having meaning separate from the special situations of the contacts is improper, because the contacts provide the sources and sinks of nonequilibrium carriers that make up the asymmetric distribution. The dissipative terms, which are discussed later, then provide these sources and sinks on a local scale throughout the dissipative region where transport is being considered. Indeed, it is impossible to define a diffusion or drift current from Eq. (8.9) without the dissipative terms, because the currents due to these processes must result from a careful balance of driving forces and dissipative forces. Nevertheless, we can use Eq. (8.10) to define the appropriate moment of the distribution. Within this interpretation, and limitations, (8.11)

and the trace is a local evaluation that produces the classical (or semiclassical) density variations. In a similar fashion, the first-order moment equation can be developed by taking the derivative of Eq. (8.9) with respect to the difference variable. Then, on passing to the limit, we find d

(8.12)

The first term on the right-hand side is the normal divergence of the “momentum pressure” tensor in the classical limit. As is often done in the semiclassical case, we may approximate this by taking the pressure tensor as an isotropic scalar quantity, which leads to (8.13)

Using our derived quantum potential, this latter quantity becomes (E)=

3 u,,, + + -,P2 2p

2m

(8.14)


34 1

where

Thus, the effective quantum potential is the difference between the smoothed value and the unsmoothed value. This means that if the driving functions, the potential itself, and the density variations that respond to this potential are slowly varying functions on the scale of the thermal de Broglie wavelength, the effective quantum potential goes away and is not a factor. In this sense, we recover the classical forms for the balance equations in the limit that the spatial variations become slow on the scale of the thermal de Broglie wavelength. In some sense, the first corrections appear at a WKB level of variation, as may be expected. In fact, in the limit in which the potential is slowly varying so that only the low orders of the potential expansion need be maintained, it is found that the effective potential reduces to the Wigner potential. Thus, while the statistical distribution is governed by the Bohm potential in this limit (discussed earlier), the transport behavior is governed by the Wigner potential. To obtain the energy equation, we need to take the second derivative of Eq. (8.9) with respect to the difference coordinate, and then pass to the limit of vanishing s. For this, however, we need to evaluate the third derivative of the density matrix, with respect to his difference coordinate. In keeping with previous approximations, this leads to (again, in the scalar approximation with no dissipation in the system) the energy equation (with no dissipation)

and the term in large square brackets may be evaluated as ih3

(8.17)

This particular form of the energy arises from the particularly simple manner in which the scalar approximation has been used to achieve the diagonal result. A more careful evaluation of the full tensor approach will yield a slightly different result, and some of the numerical factors can

342


change (for example, the last term in the parenthesis often appears as 5/2 rather than 3/2XZ6O We note in these developments that the effective quantum potential has contributions from both the spatial variation of the potential and from the density variation that is a response to that potential variation. This differs from earlier treatments, in that early WKB expansions easily obtained additional terms in a Taylor series expansion of the potential, with the assertion that this connected to the logarithm of the density through a simple classical partition function. This produced an effective quantum potential that is one-third of the Wigner potential [Eq. (2.2)1.403178.261 Actually, it is important to note that the form of the effective potential arises from the manner in which the second and third moments of the density matrix in the difference coordinate, as represented by these respective derivatives, are related to the potential and average momentum and Nevertheless, it is not possible to simply relate the higher order derivatives of the potential directly to the density matrix, because the actual effective potential arises from the self-consistent interactions between the density and the potential. b. Applications in Modeling Deuices The classical hydrodynamic equations have a long history in semiconductor device m ~ d e l i n g . The ~ ~ ~extension - ~ ~ ~ of these approaches to include the modifications of the statistical mechanics due to the quantum effects has allowed one to begin to model smaller devices without the need to use a fully quantum kinetic a p p r ~ a c h . ~ ~ . Here, ' ~ ~ , ~we ~ ' concentrate on the effect played by the effective quantum potential in the device. Grubin et al.26',267 and Gardner26" have used the quantum hydrodynamic equations, with the effective potential taking the value of 1/3 of the Wigner potential, to model a single tunneling barrier and a DBRTD. In particular, the former authors compared the results with exact calculations using the density matrix directly, and found good agreement between the two 260

C. L. Gardner, SLAM J. Appl. Math. 54, 409 (1994). H. L. Grubin, T. R. Govindan, J . P. Kreskovsky, and M. A. Stroscio, Sol. State Electron. 36, 1697 (1993). 2h2M.A. Stroscio, Superlatt. Microstruc. 2, 83 (1986). 263 See, e.g., the discussion in D. K. Ferry, Semiconductors, Macmillan, New York (1991). 264 H. Frohlich and V. V. Paranjape, Proc. Phys. Soc. London 869, 21 (1969). 265 K. BI$tekjaer, IEEE Trans. Electron Dev. 17, 38 (1970). 26 h R. Bosch and H. W. Thim, IEEE Trans. Electron Deu. 21, 16 (1974). 267H.L. Grubin and J. P. Kreskovsky, Sol. State Electron. 32, 1071 (1989). 268 J.-R. Zhou and D. K. Ferry, in Computational Electronics (K. Hess, J. P. Leburton, and U. Ravaioli, eds.), p. 63, Kluwer, Nonvell, MA (1991). 26 I


343

0.0

2

-

0

c

-.l

C 0

c

0

a

5

-.2

+ C

0

3

0

I

I

-1000

- 500

V

I

1

V

I -.3

3 -.4

0

500

1000

Distance (Angstroms) Along The Diagonal

FIG.9. The quantum potential for the hyperbolic tangent potential cited in the text. The solution area of the overall device is 200 nm.

approaches. For the transport, a Fokker-Planck form of dissipation was added to Eq. (8.9) in the form

2s

4Ds2

dp

(8.18)

In Fig. 9, the effective potential is shown in the region of the single barriers described by

v,(x)= 5[tanh( 2

7 ) y)], -

tanh(

(8.19)

where V,, = 0.5 V, a = 15 nm, and b = 1.315 nm. It can be seen from this figure that the effective potential has a peak that is almost 60% of the actual potential in a region on either side of the barrier. Using the Wigner potential itself, Zhou and Ferry269 have modeled MES-FETs in GaAs and Sic, and HEMTs in the AlGaAs/GaAs and strained-Si/GeSi heterojunction systems for ultra-submicron gate lengths ( L g < 0.1 pm). For these studies, the Wigner potential form for the 26 L) J.-R. Zhou and D. K. Ferry, IEEE Trans. Electron Deu. 39, 473 (1992); 39, 1793 (1992); 40, 421 (1993); Sol. State Electron. 36, 1289 (1993); SemiconductorSci. Technol., 9, 775 (1994).

344


effective potential was used, and a relaxation-time approximation was used for the dissipation, as

Separate relaxation times are used for momentum and energy, and the energy-dependent values for these are found from ensemble Monte Carlo calculations for homogeneous material. It should be remarked that po(R) is not the equilibrium Fermi-Dirac distribution, but is the actual spatially dependent diagonal density matrix found self-consistently with the built-in potentials from gates, doping variations, or barriers. For example, in the DBRTD, it is the density appropriate to either Fig. 6 for the density matrix or Fig. 7 for the Wigner distribution (which gives essentially the same hydrodynamic equations). Normally, for example, one expects Eq. (8.20) to vanish when continuity equation (8.10) (the lowest order moment equation) is computed. This is true, however, only as long as the local density remains appropriate to the unbiased self-consistent potential. In active devices, the density can deviate from this latter value when applied potentials exist, and the scattering decay term in the continuity equation corresponds to the diffusive restoring forces that work to reduce the charge fluctuations. In essence, these are spatial generation-recombination terms that correspond to the movement of charge under the scattering-induced diffusive processes. We return to this point later. In Fig. 10, the effective quantum potential is shown for the GaAs MESFET. Here, the potential is large at the interface between the active layer and the semi-insulating substrate, as well as around the gate deple-

Source

FIG.10. The quantum potential for a MESFET near pinchoff. The gate length is 24 nm, and the active layer is 39 nrn thick and doped to 10'' crK3.


f

8.0101’ 60 10”

345

r -

Y

.e

4.0 Id’

CI

iii 2.0 Id’ 0.0 ld

~

: 0

20

40

60

Distance from gate (om)

80

100

FIG. 11. The carrier density across the channel, under the gate, in a direction normal to the gate for a 24-nm GaAlAs/GaAs HEMT. The dashed curve is the case in which the quantum potential is set to zero.

tion layer. The quantum potential works to retard channel pinchoff, leading to a generally higher current through the device. In Fig. 11, the carrier density profile in the plane normal to the gate (and under the gate) is shown for a GaAlAs/GaAs HEMT. The quantization in the depleted GaAlAs layer leads to a reduction in the real-space transfer of carriers out of the active channel. On the other hand, the tendency of the quantum potential to retard pinchoff also works to reduce this real-space transfer. These two effects lead to a current enhancement of as much as 20% over the classical results (where the quantum potential is set to zero). It is clear that the quantum effects will be quite important in devices with gate lengths below 0.1 pm.27” On the other hand, it is clear that we have just begun to examine the quantum effects that can occur in such small devices. All of the preceding simulations have been either one- or twodimensional simulations. As discussed in the introduction, a small device with 0.05 x 0.1 p m gate (length times width) will have only of order 100 electrons in the active region. This suggests that the nonuniformities, and the individualness, of the impurities and electrons, will lead to significant quantum fluctuations.271 In fact, Monte Carlo estimates of the inelastic mean free path in Si at room temperature suggests that this quantity is of the order of 0.1 p m at low fields and 2 0.05 p m for fields up to lo5 V / C ~ . This ~ ’ ~suggests that effects such as universal conductance fluctuations may be observed at room temperature in such small devices. To fully 270J.-R.Zhou and D. K. Ferry, IEEE J . CZSI Design, in press. 27 I D. K. Ferry, Y. Takagaki, and J.-R. Zhou, Jpn. J . Appl. Phys., 33, 873 (1994). 27 2 H. Miyata, T. Yamada, and D. K. Ferry. Appl. Phys. Lett. 62, 2661 (1993).

346


investigate these effects, three-dimensional simulations are required in which the charge is a set of spatially localized impurity atoms and the corresponding electrons, rather than smoothed distributions. Or, more exact quantum-mechanical simulation approaches are required. 111. Modeling with the Density Matrix

In writing the density matrix as Eqs. (5.6) or (5.71, it is assumed to be a single-particle quantity. In fact, the density matrix starts life as a many-body quantity,273and it must be projected to a description as a single-particle density matrix. This process is the same as that followed in Sec. II.5.c. In essence, the projection operator that accomplishes this task integrates out the variables of all particles except for one, leaving a two-particle interaction that couples the one-particle density matrix to a two-particle density matrix.274Approximations to the latter particle-particle term lead to concepts such as Hartree and Hartree-Fock interactions and to higher order scattering processes that involve dissipation. 9. SOME CONSIDERATIONS ON THE DENSITY MATRIX

To use the density matrix to describe transport in quantum structures, we must find a set of governing equations that describes the behavior of particles in the appropriate structures, including the role of scattering, and the all important effects of the boundary conditions. In this sense, the situation is exactly the same as for the classical modeling problem, except here our equations have a quantum origin, as discussed earlier. A key issue in any modeling problem is the computation of the current, and with the density matrix this may well be the most difficult part of the problem. In this section, we attempt to illustrate how this is done by way of example, and use some particularly simple cases, such as Ohm’s law, before proceeding to more complex calculations for barriers. We will do this in somewhat more detail with the density matrix than with the other approaches to be discussed in subsequent sections, because the basic techniques do not differ in most cases, but are important to understand. In general, dissipative transport is treated through a perturbative treatment, as introduced in 273R.P. Feynman, Statistical Mechanics, A Set of Lectures, Benjamin/ Cummings, Reading, MA (1972). 274 N. N. Boboliubov, J. Phys. Sowjet. U. 10, 256 (1946); M. Born and H. S. Green, Proc. Roy. Soc. London A188, 10 (1946); J. G. Kirkwood, J . Chem. Phys. 14, 180 (1946); J. Yvon, Act. Sci. Ind. 542, 543 (1937).


347

the preceding section through the interaction representation. This approach has had a rich history.'s3s275-279 The density matrix approach has been used to compute transport in the presence of a uniform, high electric field in order to sense the changes arising from the quantum distribution function.28"-28s Monte Carlo techniques have been developed to study transport in this regime,283.286especially for studying transient transPort.287-289 It is possible to work with a density matrix that is described in the momentum representing; e.g., the density matrix is a function of two momentum variables rather than the two position variables in Eq. (5.9). Nevertheless, we describe only the case in the position representation, for which the equation of motion is given by Eq. (5.9). This latter equation is a partial differential equation permitting specification of conditions at boundaries as well as of an initial condition at some time to. For those situations where current flows and dissipation is an issue, such as in actual devices, Eq. (5.9) must be modified to include a term representing the decay of the density matrix due to scattering processes. We will spend considerable time with this last term. As we discussed earlier, the problems with devices involve open systems. If the boundaries are characterized by a local equilibrium, as is often the case, the form of the density matrix may be obtained from the semiclassical Fermi-Dirac distribution through the inverse transform of Eq. (5.15). To date, the description of transport in 27sH.Ehrenreich and M. H. Cohen, Phys. Reu. 115, 786 (1959). 27 h E. N. Adams and P. N. Argyres, Phys. Reu. 102, 605 (1956); 104, 900 (1956). 277P,N. Argyres, J . Phys. Chem. Sol. 4, 19 (1958); Phys. Rec. 109, 1115 (1958); J . Phys. Chem. Sol. 8, 124 (1959); Phys. Reu. 117, 315 (1960). 278N.J. Horing and P. N. Argyres, in Proc. Int. Conf. Phys. Semocond., Exeter, p. 58 Inst. Phys., London (1962). P. N. Argyres and L. M. Roth, J . Phys. Chem. Sol. 12, 89 (1959). 27yJ.B. Krieger and G. J. Iafrate, Phys. Reu. B 33, 5494 (19861, 35, 9644 (19871, 40, 6144 (198Y); G. J . Iafrate, J . B. Krieger, and Y. Li, in Electronic Properties of Mulrilayers and Low-Dimensional Semiconductor Systems (J. M. Chamberlain et al., eds.), p. 211, Plenum, New York (1990). 2X'IN.Sawaki and T. Nishinaga, J . Phys. C 10, 5003 (1977). 2n'J.R. Barker, Sol. Slate Electron. 21, 267 (1978). 2n 2 X. L. Lei and C. S. Ting, Phys. Reu. B 30, 4809 (1984); X. L. Lei, D. Y. Xing, M. Liu, C. S. Ting, and J . L. Birman, Phys. Reu. B 36, 9134 (1987). 2x 3 F. Rossi, R. Brunetti, and C. Jacoboni, in Granular Nanoelectronics (D. K. Ferry el al., eds.), Plenum Press, New York (1991); C. Jacoboni, Sernicond. Sci. Technol. B7, 6 (1992). 2X4P.N. Argyres, Phys. Lett. A 171, 373 (1992). 2x5 H. Schoeller, Ann. Phys., 229, 320 (1994). 2nh F. Rossi and C. Jacobini, Europhys. Lett. 18, 169 (1992). 2x7 R. Brunetti, C. Jacoboni, and F. Rossi, Phys. Rev. B 39, 10781 (1989); F. Rossi and C. Jacoboni, Sol. State Electron. 32, 1411 (1989). '*'H. Mizuta and C. J. Goodings, J . Phys. Cond. Matfer 3, 3739 (1991). ~ X Y F. Rossi and C. Jacoboni, Semicond. Sci. Technol. B7, 383 (1992).

348


devices has been confined to the cases where the particles are free in two directions (one-dimensional transport through the devices), which for specificity we take as the y and z directions. Hence, we can set y = y’ and z = z’, so that for parabolic bands,

where p = E / k , T , p F = E F / k B T ,h d = \ / h 2 / 2 m k B T (which differs from the thermal Debye length introduced earlier); all energies are measured from the conduction band edge; and N, = IY:)/(27r2Ai). In the limit that x + x’, the terms following the effective density of states become just the Fermi-Dirac integral F,,2( p F ) .There are two limiting cases that may be easily analyzed. In the high-temperature limit, where Boltzmann statistics apply, Eq. (9.1) reduces to the Gaussian

(9.2)

For a material such as GaAs, hd at room temperature is 4.7 nm and the ~ . a nominal density effective density of states is about 4.4 X 1017 ~ m - For of lo” cmP3,the Fermi level lies about 1.5k,T below the conduction band edge, which is borderline on the applicability of the nondegenerate form. The second case of interest is the low-temperature limit, the so-called quantum limit, at T = 0. Here, the Fermi level must lie in the conduction band for any reasonable carrier density, and Eq. (9.1) becomes

(9.3) where j , ( r ) is a spherical Bessel function. In the limit x x’, the second fraction becomes simply 3. One of the earliest applications in which Eq. (9.3) appeared was a discussion by BardeenZ9’ in which he showed that the -j

29 0

J. Bardeen, Phys. ReE. 49, 653 (1936).


349

electron density profile a distance x from an infinite potential barrier was given by

(9.4) and zero for x < 0. Oscillations in the density occur as a result of the spherical Bessel function in Eq. (9.3). These oscillations depend on the Fermi wavevector, which in turn is a function of the density. These oscillations do not occur in the nondenegerate limit of Eq. (9.2). A more exact calculation that interpolates between these two limiting cases can be expected to show a gradual damping of the oscillatory behavior as the temperature is raised. This is shown in Fig. 12, where the density matrix is plotted (for a homogeneous system) as a function of the off-diagonal, or correlation, distance. It may be seen that the oscillations are quite weak already at 77 K, and do not appear at 300 K. At higher densities, the oscillations will exist to higher temperatures.

1.5 I

-T=O.O K

5 0.5

r"

............ T=77 K

/---

I

-0.5

I

I

I

I

0

50

100

150

200

Correlation Distance (nrn)

FIG.12. The density matrix for a homogeneous system, in which the density is plotted as a function of the off-diagonal distance, or correlation distance, for GaAs with an electron density of lOI7 cm-'.

350


The preceding solutions provide some indication of what the density matrix coordinate representation profiles are for the cases most closely related to the standard classical equilibrium distribution functions. It may be anticipated, purely on physical grounds, that a problem examined using the classical distribution function in momentum space would yield the same physical results, with respect to the observables, as obtained in the coordinate representation. Consider, for example, the Boltzmann distribution exp[ - V ( x ) / k , T ] (recall that V is an energy-the potential energy). If we introduce a potential step of amplitude k,T ln(10), classical theory leads to the conclusion that the density is reduced by a factor of 10 (assuming of course that we are talking about allowed states in both regions). The same basic result is obtained for the density matrix, but if the temperature is low, so that the above oscillations are not damped, there is a more complex transition region between the two asymptotic potential values. This is shown in Fig. 13 for GaAs at T = 0 and a doping of 10” cm-3 in the region with the potential and an order of magnitude higher in the region in which the potential is V ( x ) = 0. A comparable situation with respect to potential energy and density occurs when, instead of solving a non-self-consistent equation with a barrier, we solve the self-consistent equation for the spatially varying density. This is shown in Fig. 14 for the same doping parameters as Fig. 13. The solution region extends for 100 nm on either side of the abrupt change in the doping; for example, the potential is created by the abrupt change in doping profile. Again, the solution is at T = 0. No attempt is made to develop the origin of the carriers, rather the regime is chosen because the detailed values of the density can be obtained independently. Here, the solution to the equation of motion of the density matrix is provided self-consistently by also solving the Poisson equation (9.5) There are a number of interesting features about Figs. 13 and 14. First, if we concentrate on the shape of the density matrix in the uniform field regions (away from the transition), the ripples in the density matrix indicate that the sharper Fermi edge structure plays a pronounced role as the temperature is lowered, as was pointed out earlier. In addition, the curvature of the density matrix at the main diagonal, but in the direction normal to the main diagonal, is more pronounced in the region with lower potential. This directly relates to the (average) kinetic energy of the

0.05

26 .................................................

-- 0.04

25 -

-

T.24

g

-

v F

._ Q

-

23-

22

- 0.03 T b -

-

- 0.02

5 5 -

- 0.01

2

............

-

..................................................

21

I

I

0

-PI

. I

0.00

I

b

FIG.13. (a) The charge density and potential are shown for a abrupt (non-self-consistent) potential step. The parameters are for GaAs at 0 K and a doping level of lo'* cmV3 in the region of low potential, and an order of magnitude lower in the region of high potential. (b) The density matrix for this situation.

352


0 05

26

25

-- 0.04

-

-

- Density -

-

003 9 0)

- 002

Potential Energy

c

m

- 0.01 22 -

21

-

-

I

F

5 -m

I

I

I

2

0.00

-001

FIG. 14. (a) The charge density and self-consistent potential for the same situation as in Fig. 13. (b) The density matrix.


353

density through2” E(r)n(r)

= rlim ‘+r

[

-(

fi2

-

8m

dr - s ) 2 p ( r , r f ) ] .

(9.6)

This should be compared with the same form in Eq. (8.13). As the temperature is raised, the curvature will be decreased, as the particles spread and the correlation extends further. Moreover, increased density results in increased curvature due to the increase of the average energy by the increase in the Fermi energy. a. Statistics of a Single Barrier Consideration of the situation with a finite barrier offers similar insights. For example, if the barrier is sufficiently wide (the characterization of sufficiently wide will be discussed later), we expect that the density within the interior of the barrier, far from the potential transitions, can be described by its classical values, again assuming that this region remains classically allowed. This case is shown in Fig. 15 for the same amplitude barrier as above; that is, the density is taken to be 10l8cm-3 in the region far from the barrier and an order of magnitude smaller (classically) within the barrier. However, when the width of the barrier is reduced, there is an increase in the density in the central region, which is also shown in Fig. 15(a). While the explanation of the variation of the density through the barrier region can be described in terms of the internal wave function reflections at the interfaces, along with the normal continuity of the wavefunctions and their derivatives, a more practical description of the density variation can be found using the quantum potential U, introduced in the previous section. By comparing the density matrix solutions with the approximate Boltzmann-like solutions using the quantum potential, it is found that a quite good description of the variation through the barrier can be obtained.291This is true for nondegenerate statistics for sure, but a comparable equivalence has not yet been shown to be valid for strongly degenerate Fermi statistics. Even so, the results found later give strong correlation to this interpretation. The quantum potential is one of the more interesting concepts that can be probed through simple solutions of the density matrix equation of motion. In this sense, the density matrix can be solved exactly for simple barrier problems, and the quantum potential can then be calculated from 291

H. L. Grubin, T. R. Govindan, B. J. Morrison, D. K. Ferry, and M. A. Stroscio,

Semicond. Sci. Technol. B7,360 (1992).

354


a

I22

100 nm barrier 10 nm barrier

I

I

I

1

0

50

100

150

7.00

Distance (nm)

FIG.15. (a) The density profiles for a wide and a narrow barrier. The doping variations are the same as in Figs. 13 and 14, which provide a barrier height of 42.7 meV. (b) The density matrix for the case of the wide barrier.


355

0.05

0.04 -

- Quantum Potenti: ............ Potential Energy

0.03 -

0.02 -

z

w

F I I

z1

e

0.00

-0.02

.......

i I

I

I

I

0

50

100

150

-0.03 -0.04

200

Distance (nrn)

FIG.16. The quantum potential (solid curve) and barrier potential (dotted curve) for the narrow barrier of Fig. 15.

the resulting density through the use of, for example, Eq. (2.2). For example, the quantum potential that results for the narrow barrier in Fig. 15 is shown in Fig. 16, along with the non-self-consistent potential barrier. Within the barrier, where the curvature of the density is positive, the quantum potential is negative, and the net result is that the effective potential energy seen by the electrons is less than V ( x ) .This results in a density that is larger than the classically expected value. Immediately outside the barrier region, where the density begins to increase (and has a negative curvature), the quantum potential is positive and the density is below its classical value. In fact, in the center of the barrier, the quantum potential for this example has reached a value that almost cancels the barrier potential We can extend this simple evaluation approach to the case of a single heterostructure barrier, such as occurs at the interface between GaAs and GaAlAs, which is modulation doped with the impurities residing in the latter material. In Fig. 17(a), the charge distribution, potential energy, and quantum potential are shown for a 200-nm region, at T = 300 K , in which the interface lies at the center of this region. The doping level is taken to be lo” cm-’ in the GaAs and 10l8 cm-3 for the GaAlAs (which is the

356 26

25

22

... .

Potential Energy i! ! ill

Quantum Potential :J I

- -0.15

j/

i

21

1

I

I

-0.20

FIG.17. (a) The charge density, self-consistent potential energy, and quantum potential for

a heterojunction between GaAIAs, and GaAs. The composition of the former material is

taken to give a 0.3-eV conduction band offset. (b) The density matrix for the charge in this heterojunction. The GaAlAs is the region x > 100 nm.


357

region 100 < x < 200 nm). In addition, the composition of the latter material is assumed to be such that a 0.3-eV barrier is created by the offset of the two conduction bands. It is easily seen that there is a reduction in the carrier density in the GaAlAs near the interface, with a resulting formation of an inversion layer in the GaAs adjacent to the interface. The peak in the density in the inversion layer is actually higher than the background doping of the GaAlAs. Note that the applied potential energy difference across the interface has been chosen to yield flat-band conditions, and is equal to the height of the barrier plus the built-in potential of the “junction.” For the situation where the charge depletion occurs in the GaAlAs at the heterobarrier interface, the amount of this depletion is such that the curvature of the potential energy within the vicinity of the barrier is approximately constant. As a consequence, when the height of the heterobarrier increases there is an increase in the width of the depletion zone on the GaAlAs side of the structure. Under flat-band conditions, where the net charge distribution is zero, there is a corresponding increase in the charge on the GaAs side. The quantum potential is negative on the GaAlAs side of the junction and tends to give a charge density that is actually larger than the classical value that would be expected. This also has the result of giving a slightly lower value on the opposite side of the junction than what would normally be expected. The small region of negative quantum potential on the GaAs side of the junction is a consequence of oscillations in the density that arise from the Fermi distribution. b . Multiple-Bam‘er Structures The simplest multiple-barrier structure is the double-barrier resonanttunneling diode, which was introduced previously as one of our prototypical quantum devices. The characteristic feature of the multiple-barrier structures is the existence of quasi-bound states between each pair of barriers (if there are more than two barriers). The density between the barriers, as well as within the barriers themselves, depends on the potential height of the barrier, the configuration (spatial variation) of the barrier, the doping levels, and the size of the regions between the barriers. The value of the quantum potential in the region between the barriers is approximately equal to the energy of the lowest quasi-bound state, relative to the bottom of the conduction band.292 It has been shown previously that, in the absence of current normal to the barriers, the total energy in this region is given by E = V(x) + UQ(x).257We show this by considering 292 H. L. Grubin, J. P. Kreskovslq, T. R. Govindam, and D. K. Ferry, Semicond. Sci. Technol., 9, 855 (1994).

358


a region in which two 5-nm barriers, 0.3 eV high, separated by 5 nm, are placed in the central part of the simulation region (200 nm long). The doping is taken to be 1Ol8 ~ m - except ~ , in a central 40-nm region in which it is reduced by two orders of magnitude. In Fig. 18, we plot the density distribution, the donor concentration, the quantum potential, and the self-consistent potential. It may be seen that the peak density in the center of the quantum well rises to a value that is about 40% of that in the heavily doped regions. The quantum potential, as expected from our discussion, is negative in the barriers and is positive in the quantum well. For this results of this figure, the quantum potential rises to approximately 84 meV, which is quite near the computed value of the quasi-bound state. (If the barriers are reduced to 200 meV, the quantum potential peak is reduced to about 70 meV.) The connection of the quantum potential to the quasi-bound state is an important feature of modeling with the density matrix, because it allows an easy evaluation of bound energy levels in complicated structures. A second calculation may be used to further examine the expected results. Here, the double-barrier structure from earlier is placed into a 40-nm quantum well, with the depth of the well a variable. As the depth of this well was increased, the quantum potential value in the central well remained independent of position, but increased slightly in value due to the extra confinement effect of the external (large) quantum well. When this larger quantum well was 150 meV deep, the quantum potential, and hence the quasi-bound level, increased to about 94 meV. For this condition, the various observables, and the quantum potential are shown in Fig. 19. It may be seen from this figure that the density between the two barriers has increased. Such an increase has at least two origins: (1) the increased density on either side of the barriers, due to the confining effect of the larger well, and (2) the lowering of the quasi-bound state relative to the Fermi level in the heavily doped regions. The quantum well itself is delineated in the figure by the subsidiary offset of the potential energy at the boundaries of the large quantum well. F r e n ~ l e yhas ~ ~ ~used the single-particle density matrix to study a double-barrier resonant-tunneling diode, and also looked at the case with more built-in barriers. The partial differential equation for the density matrix was solved using finite-difference techniques similar to those used to solve in conventional semiconductor devices. For the simulations, he treated the boundaries as ohmic contacts. He finds that the methods of solution are easier than the Wigner distribution function (discussed in the next section), but are more complicated to interpret, as can be inferred from the preceding discussion. He assumed 50-nm barriers and well, and 29 3

W. R. Frensley, J . Vac. Sci. Technol. B 3, 1261 (1985).


a 25

-

24

-.

* E r

.z 0

23

C

0 rn

- Density

-

............ Background 22

r; ,... . ..

21 0

50

359

.

100

150

200

Distance (nrn)

0.4

,

b

0.3

0.2 0.1

3

0.0

g

-0.1

$

w

-0.2 -0.3 -0.4

-0.5

I

0

I

1

I

50

100

150

200

Distance (nrn)

FIG. 18. A double-barrier structure with 5-nm barriers, 0.3 eV high, and a 5-nm well is solved by the density matrix technique. (a) The doping profile and the self-consistently determined density in the structure. (b) The self-consistently determined potential and the quantum potential for this structure. The peak of the quantum potential in the well defines the quasi-bound state energy, as discussed in the text.

360


a 25

24

21

20

b 0.4

0.3 0.2 0.1

& h

0.0

-0.2

-0.3 -0.4 -0.5

I

0

I

I

I

50

100

150

I

200

Distance (nm)

FIG.19. The double-barrier structure in Fig. 18 is immersed in a wider (40 nm wide) quantum well with depth of 0.15 eV. In (a), the doping and self-consistently determined density are shown, while in (b), the self-consistent potential and the quantum potential are plotted.


361

used a heterostructure offset of 0.39 eV. This work is the first step toward directly calculating the properties of real quantum devices, and points out the importance of such simulations to gain insight into the operation of the devices. Nevertheless, incorporation of the ohmic boundary conditions greatly complicated the numerical simulation algorithm, and the system was subject to the growth of numerically unstable modes. However, we point out later that within the Wigner formulation there exists a methodology to overcome his problem, and recent simulations indicate a method of overcoming this problem for the density matrix. G r o ~ h e v ~has ’ ~ also used the density matrix to simulate the Coulomb blockade (single-electron tunneling) regime of the resonant-tunneling structure. In this case, he used a three-dimensionally configured structure in an attempt to identify lateral modes and fine structure in the tunneling current. However, he reduced the problem to a pseudo-hopping formalism, which did not need to study carefully the spatial charge distribution self-consistently nor did it carefully examine the effects of the boundary conditions. AND 10. DISSIPATION

CURRENT

FLOW

One conclusion we can draw from the previous considerations is that, for both the self-consistent and non-self-consistent solutions of the potential, the solutions for the density and the potential sufficiently far from the interface are basically the same as that expected using the classical Boltzmann equation. When current flows through the structure, the semiclassical approach is usually pursued either by drift-diffusion or by hydrodynamic approaches, or through more extensive simulation of the Boltzmann equation through ensemble Monte Carlo procedures.“, Here, we want to begin to understand how current transport in quantum structures can be treated via the density matrix equations of motion. For cases where the ends of the “device” are heavily doped IZ’regions, boundary conditions on the numerical procedures are formulated to assure that the numbers of particles entering the “cathode” end of the device is equal to the number of particles leaving the “anode” end of the device (for electron flow). An alternative schema is to adopt procedures that incorporate sufficient dissipation in these boundary regions to thermalize the carriers to a near-equilibrium distribution. This latter schema should yield the same results for the charge and potential within the active portion of the device, but the major problem is how to deal with the dissipation 294A.Groshev. Phys. Reu. B 42, 5895 (1990).

362


processes within the simulations. To date, only very approximate schemes have been adopted, but it must be emphasized that some procedure for treating the dissipation must be adopted if transport in such devices is to be discussed sensibly. A low-order (Born approximation) perturbation treatment of phonon interaction in a homogeneous system has been treated with the density matrix by A r g y r e ~ .Similar ~ ~ ~ approaches have been used for h e t e r o s t r u ~ t u r e s A . ~ ~more ~ extensive renormalized phonon treatment has also been treated for homogeneous systems.297 We will follow a somewhat different approach here, in order to try to develop an effective approach for detailed device simulations. Following Caldeira and Leggett,’09q298we consider a system A interacting with a second system R (which is taken normally to be the reservoir) and described by the Hamiltonian H , = H HR H I , where the latter three terms describe system A , the reservoir and the interaction between the two. Clearly, we can follow the approach of Sec. II.S.c, and define the projection operator as being a trace over the reservoir variables, so that the reduced density matrix is described byI2

+

p&,r’, t )

=

+

Pp(r, r’,X,X’,t ) = Trx,xr{p(r,r’,X,X’,f ) } ,

(10.1)

where X,X‘ are the coordinates of the reservoir. In this approach, we do not need detailed information about the reservoirs, only information about their interactions with, and influence on, the electron system in the active device region. The method that is normally invoked is to develop a perturbation theory for the interaction and dissipation that follows from both internal dissipative processes and boundary effects (one such is surface-roughness scattering; another appears as changed boundary conditions, which will be heavily used in the next section on Wigner functions). The equation of motion for the density matrix [Eq. (5.9)] is rewritten to include a scattering contribution, which must be evaluated, which is the next task. In determining the form of the perturbation theory result, one is faced with a variety of approaches to take. One approach that encourages an intuitive prescription is simply to determine how standard Boltzmann scattering terms would look in the appropriate coordinate representation. While this will lose information regarding short-time events and build-up of the scattering process, much insight is gained into the form of scattering 295

P. N. Argyres, Phys. Reo. 132, 1527 (1963). W. Cai, M. C. Marchetti, and M. Lax, Phys. Rev. B 34, 8573 (1986), 35, 1369 (1987). 297S.Das Sarma, J. K. Jain, and R. Jalabert, Phys. Rev. B 41, 3561 (1990). 298A. 0. Caldeira and A. J. Leggett, Physica 121A, 587 (1983). 296


363

to low-order in perturbation theory. In the Boltzmann picture, when Fermi statistics can be safely ignored, the scattering rate is simply

(10.2) where the scattering processes are assumed to occur locally in space and W(r, k, k ) is the standard Fermi golden rule transition probability. Our approach then assumes that f(r, k) is replaced in the quantum treatment by a Wigner distribution and the inverse of the Weyl transform [Eq. (5.1511 is used as S

=

G/d’kfW(R,k)e-ik.s, 1

(10.3)

where the reduced coordinates [Eq. (5.1311 are used. After some simple manipulations, the scattering term of Eq. (10.2) becomes dp(R + ) ~

dt

I

= dlss

- ( - ) 21/ d 3 k

47r

(10.4) where p(R ,) = p(R + s / 2 , R - s/2). The structure of the scattering term in the coordinate representation may now be obtained from Eq (10.41, at least as the scattering is derived to lowest order in the Boltzmann scattering framework. For scattering that is mainly local in space, the difference coordinate s is small, and the second term in the square brackets can be expanded in a Taylor series, retaining only the leading nonvanishing term, for which

(10.5)

364


It may be recognized, from standard semiclassical treatment of scattering processes,263that the integration over k produces a momentum relaxation rate. Introducing Eq. (5.15), and integrating once by parts on the s coordinate, the dissipation rate can be rewritten as

We note that the integration over k incorporates the rapidly varying exponential, which in turn is approximately a delta function on the difference in the nonlocal variables, so that one really achieves an average value of the momentum relaxation rate, and (10.7) This term can now be added to the Liouville equation [Eq. (5.9)] on the right-hand side (after multiplying by in, of course). This form of the dissipative term has also been discussed by Dekker,299 as well as in the work cited earlier.298Density matrix algorithms for modeling devices have been reported that incorporate the dissipative contributions developed here.292 Because of numerical difficulties at higher bias levels, modifications to the scattering term were introduced that go beyond the approximations introduced in arriving at Eq. (10.7). We turn to these next. It is possible to rewrite the dissipative term by observing that the current density itself is a function of the nonlocal variable, through the fact that

m

ds

,

(10.8)

where we have used coordinate transformation (5.13). With this form of the current, the dissipation term can now be rewritten as - -

im --T(R)s

-j

S

-

:). 2

(10.9)

The current term can now be rewritten in terms of a velocity, under the assumption that this local velocity incorporates any diffusion effects directly, which is a local quantity of the average (center-of-mass) coordinate, 299

H. Dekker, Phys. Reii. A 16, 2126 (1977).


365

or j(r,r') = u(R)p(r,r'). Further it may be noted that the relaxation rate plays a nearly semi-classical role in near equilibrium, in that the potential drop between any two points can be described in terms of this dissipation and the velocity, much as a quasi-Fermi level is introduced, or

For small values of s, this latter equation is approximately - s * v(R)mr(R), and the dissipative term finally becomes

(10.12) so that the dissipative term has precisely the same form as the potential driving term, but with the potentials replaced by the (quasi-) Fermi levels. This has the result of creating a dissipative term on the right-hand side of Eq. (5.9) that is still real. The manipulations associated with the preceding discussion were predicated on finding a means of computing the current. In fact, a definition of the current was introduced in arriving at the final form of Eq. (10.12). In the computations that follow, an assumption is made that the carriers at the upstream boundary are in local equilibrium and that the distributions are either a displaced Maxwellian o r a displaced Fermi-Dirac distribution. This implies that, at the upstream boundary, the zero current quantum . s]. Since the density matrix is replaced by p(r, r') exp[(im/h)v(Rboundary) current is introduced as a boundary condition to the problem, a prescription is necessary to find its value. This leads to an auxiliary condition. To find the value of the current used in the Liouville equation for the density matrix, moments of the revised form of Eq. (5.9) are taken in the same sense as those of Sec. 11.8. Under time-independent conditions, the momentum balance equation (8.12) can be written as 2-

dE dR

dV

+ -p(R) dR

+ mT(R)u(R)p(R)

=

0,

(10.13)

where E is the total energy, kinetic plus potential, and not the energy per particle as indicated in Eq. (8.13). Under the assumption of current continuity, and the condition that the term u(R)p(R) is independent of

366


distance (Kirchoff s current law), and the condition that the local energies of the entering and exiting carriers are equal, current is then obtained from the condition

J =

-

E,(x) j;

EF(X’) [ mr( XI’ x” ) I -

(10.14)

where we have restricted the considerations to one spatial dimension. The simplest type of calculation to deal with is that of a free particle. For this case, with current introduced as a boundary condition through the preceding approach, the resulting density matrix is complex. The real part is symmetric in s, the distance from the principle diagonal. On the other hand, the imaginary part (which is necessary for the current) is antisymmetric about the principal diagonal. In Fig. 20, we plot the real and imaginary parts of the density matrix for a 200-nm region of GaAs, doped . bias of 10 meV has been applied across to a level of l O I 7 ~ m - A~ small this length (a field of 500 V/cm). It has been assumed that r corresponds to a mobility of 2580 cm2/Vs, or a scattering time of 0.1 ps. The mean carrier velocity is found to be about 1.3 X lo6 cm/s. Increasing the applied bias results in an increase in the velocity and an increase in the kinetic energy of the carriers, the latter of which manifests itself as increased curvature of the imaginary part of the density matrix in the correlation direction (normal to the principal diagonal). Let us now turn to a nonuniform sample in which the mobility varies with position in the structure. The material is again taken to be GaAs, with the parameters discussed in the previous paragraph for Fig. 20. Here, however, the scattering rate will be greatly increased over the central 2 nm of the structure. On the basis of the preceding discussion, this decrease in the scattering time will result in a sharp drop in the quasi-Fermi level over this region. The density cannot change as rapidly, and has a characteristic “Debye” length over which it changes. The density variation, the quasiFermi level, and the potential energy are shown in Fig. 21 for two cases corresponding to an increase of scattering rate by one and two orders of magnitude (actually the background scattering rate also is varied in the second). In Fig. 22, the opposite results, for a central region with lower scattering rate, are shown. For the case where the cladding region has a higher mobility (lower scattering rate), most of the potential drop is across the central, low mobility region (Fig. 21). Conversely, when the cladding region has a lower mobility than the central region, most of the potential drop is across the cladding region (Fig. 22). Of course, these results are clearly expected from classical considerations. What is not usually appreciated is that both cases lead to strong variations in the local, self-consistent density and in the resulting quasi-Fermi levels. It is in these regions, where


367

FIG. 20. The real (a) and imaginary (b) parts of the density matrix for a uniform GaAs structure in a constant electric field of 500 V/cm.

the density varies considerably from the normal doping levels, that we expect to see the largest quantum-induced changes in the results. We now turn to more device-like simulations. A single-barrier tunneling structure is considered. The basic structure is taken to be GaAs with nominal doping of 10l8 ~ m - A~ central . 30-nm region is assumed to be

368


a

I

- Potential (1.0;O.Olps)

0.000

-0.002

-g

9 -0.004

6

-0.006

-0.008 -0.010 -0.012

,

Femi (0.l;O.Olps)

0

I

I

I

50

100

150

200

Distance (nm)

b

1-

"I5

10~000

Density (1.O;o.olps)

1.10

1.ooo 6-

-

1.05

h

E

: .-E

1.00

0"

0.100

.-E

5

'C

r:

2

0.95

v)

0.010 0.90

-- Scattering (1.0:O.Olps) Scattering (0.1;O.Olps)

0.85

I

I

I

0.001

Distance (nm)

FIG.21. A central 5-nm region is assumed to have different scattering properties from the bulk. The two cases consider (1) a scattering time of 0.1 ps in the bulk and 0.01 ps in the central region, and (2) increase of the scattering time to 1 ps in the bulk (with 0.01 ps in the central region). In (a), the potential and quasi-Fermi levels are plotted, while in (b), the density is plotted for these two conditions.


369

a - Potential (0.l;l.Ops)

0.000

............ F e n i (0.l;l.Ops)

-0.002 -0.004 Y

x

e,

5

-0.006 -0.008

-0.010

-0.012

50

0

100

150

200

Distance (nm)

b 1.004

1 .o

1.003 1.002 o^

1.001

5r3

1.000

5

-a v)

E" P

F

'C

'2 i3 0.999

t:

%

(I)

0.998 0.997

0.1

0.996

0

50

100

150

200

Distance (nm) FIG.22. A central region of 50 nm is assumed to have a lower scattering rate (1 ps versus 0.1 ps in the bulk). In (a), the potential and quasi-Fermi level are plotted, while (b) illustrates the density variation.

370


unintentionally doped. A barrier with 0.3 eV height, and 10 nm width, is placed in the center of the lightly-doped region. The scattering time is assumed to be constant throughout the structure at a value of 0.1 ps. In Fig. 23, the potential and charge density are shown for a variety of applied bias levels, ranging from 0 to 0.4 eV. The bias is applied at the collector boundary (the zero of potential is maintained at the source end). A typical potential well forms on the source side of the barrier as the bias is increased (we will see similar behavior in the DBRTD of the next section). In all cases, the potential decreases essentially linearly across the barrier, which indicates that there is little charge accumulation (or depletion) in the barrier itself. On the other hand, there is significant charge accumulation on the source side of the barrier at the higher bias levels. In Fig. 24, the quasi-Fermi level (relative to the equilibrium Fermi level) is shown for the same bias levels. Most of the change in the quasi-Fermi level occurs within the barrier, and on the collector side of the barrier. It matches the applied bias well at the boundaries, as required for consistency. The current-voltage curve for this structure shows the expected exponential behavior, although the charge accumulation modifies slightly the pure exponential behavior of simple theory. The shape of this latter curve is also sensitive to the exact dimensions of the tunneling barrier, since this modifies the fraction of the potential that is dropped across the barrier. 11. FURTHER CONSIDERATIONS

ON THE

DENSITY MATRIX

a. An Alternative Approach to the Density Matrix An alternative approach to dealing with transport through use of the

density matrix has been proposed by Krieger and Iafrate.279These authors considered both uniform and nonuniform fields in an approach that allowed them to examine transport on a short timescale. For example, in a discussion of electrons interacting with impurities and a homogeneous time-dependent electric field in the momentum representation, they used first-order time-dependent perturbation theory to obtain

(11.1) which contains several important features. First, the basis functions used in obtaining this equation are the instantaneous eigenfunctions of the


371

Distance (nrn)

b 1.e+24

1.e+2?

.-

l.e+22

Z

i.e+21

-m

g

Y

0 fn

- 0.0 ev

1.e+20

-___

-0.2ev

............ -0.4 ev

l.e+19 100 150 200 Distance (nrn) FIG. 23. The potential (a) and density variation (b) across a single tunneling barrier, embedded in a central lightly doped region (the parameters are discussed in the text). The parameter for the various curves is the applied bias.

0

50

372

DAVID K. FERRY AND HAROLD L. GRUBIN 0.1

-2

0.0

Y

e SI W

I

h

-0.1

Fermi Energy = 0.04189ev

.E

.4 I -0.2

I

d

- 0.0 ev

____ -0.2ev

-0.3

........... -0.4ev

...............................................

-0.4 100

50

0

150

200

Distance (nm)

FIG.24. The variation of the quasi-Fermi level for the structure and biases of Fig. 23.

Hamiltonian:

H$

1

=

-2m (P

- eA)

- (p - eA)$ = E,$,

(11.2)

where

2m

,

k(K,t)=K--,

eA fi

(11.3)

and

E(K, t ) = E,(K, t>+ (KIH'IK).

(11.4)

Second, the off-diagonal elements of the energy in the last equation and the density matrix itself are expressed only in terms of the perturbed diagonal elements. Thus, initial conditions are not subject to the usual condition of an initial equilibrium state. The first-order perturbed density matrix in the momentum representation was used to obtain the time rate of change of the diagonal components themselves. The exponential term, which is field- and time-dependent, permits a discussion of short-time behavior as well as the long-time energy-conserving delta-function behav-


373

ior representation of the Fermi golden rule. The intracollisonal field effect enters at this point. These authors also obtained the Liouville equation in an accelerated Bloch representation, where it was demonstrated that the major result is to introduce a term in the perturbing matrix elements that connects states of the same general momentum K value, but with different band indices, thus leading to a contribution from interband tunneling. It is expected that, as the Wigner function formalism provides an initial form for the modification of the equation of motion for the density matrix, this approach will find application to the examination of transport on the short timescale. The situation with nonuniform fields is different, and was treated by these authors accordingly. They specifically examine solutions to the Liouville equation in the Wannier representation: a representation in terms of localized wavefunctions in real space. This representation bears close resemblance in form to the density matrix in the coordinate representation. The transformation to the Wannier representation is

where +,,(k) is the Fourier transform of the coordinate wave function $,(x). Here, (xlk) is the Wannier function satisfying the orthonormality conditions

(11.6)

The density matrix can then be transformed into the Wannier representation as

where (nl plm) is the density matrix in the Wannier representation (it should be pointed out this this would be true for any arbitrary set of coordinate representation wavefunctions, but here we have asserted that these wavefunctions (xlm) are the localized Wannier wavefunctions). Since the basis functions satisfy the Schrodinger equation, but with the

374

DAVID K. FERRY A N D HAROLD L. GRUBIN

proviso that the energy is now a matrix quantity, the Wannier functions satisfy n

-

-V2(x1n) 2m

=

C E ( m ,n ) ( x l m ) .

(11.8)

m

Then, the single-band equation of motion (with the shorthand notation p,, = (nl plm)) is

(11.9)

This latter equation was discussed by Krieger and Iafrate*” with a band index included. Because the density matrix in the coordinate representation with a finite lattice is indexed by the lattice points, it may be regarded as a pseudocoordinate representation. It is intriguing to compare the structure of Eq. (1 1.9) to the Liouville equation itself in the coordinate representation. First, we note that the energy operator in the coordinate representation is a second derivative and, in establishing its contribution, difference equations are implemented that generally involve at most nearest and second-nearest neighbors. The potential energy contributions in this representation do not involve a summation over the coordinates, but are local in the site. Thus, Eq. (11.9) differs from the Liouville equation in the treatment of the potential terms, just as arises in the Wigner representation. The density matrix in the Wannier representation, particularly for well-localized values of the momentum, will likely involve significant contributions from lattice sites well removed from that of interest. On the other hand, where the density is well localized, such as in a quantum well, the summation may well involve only a few nearest-neighbor terms. As usual, the choice of the best representation is one of convenience to the problem at hand. Rather than treat the perturbation of the density matrix in a specific representation, another approach3’” is to deal with perturbation about the equilibrium density operator itself. Here, we represent H,(t) as the unperturbed Hamiltonian, so that the field and corresponding perturbations are initialized at t = 0. We represent the perturbing operator as ”OH. L. Grubin and M. A. Stroscio, unpublished results.


375

V ( t ) ,in analogy with Eq. (5.22). Then, to second order in the expansion of the unitary perturbation p(t)

= p,(t)

+ p , ( t ) + p2(t),

(11.10)

where p o ( t > = tr,+G< ,

07.3)

and relations (5.27) and (5.28) have been used. Interchanging the rows of Eq. (V.2) leads to another commonly used form3’”

[

Gr

G3= 0

GK

Gal.

3s”A.I. Larkin and Yu. N. Ovchinnikov, Sou. Phys. JETP 41, 960 (1975).

(V.4)

406


We can now develop the equations of motion for the noninteracting forms of these Green’s functions; for example, the equations that the functions will satisfy in the absence of any applied potentials and perturbing interactions. For this, we assume that the individual field operators are based on wavefunctions that satisfy the basic Schrodinger equation (11.1). This leads to the two matrix equations

[

a

-ihz -

(V.6)

where I is the unit matrix (unity on the diagonal and zeroes off the diagonal). The “0” subscript has been added to indicate that this is the noninteracting form (there is no interparticle or particle-lattice interaction). In the next few sections, we show how this is expanded to include the electron-phonon and other interactions, and how it is applied to study some device structures. 16. HOMOGENEOUS, LOW-FIELD SYSTEMS

Transport, as has been stated earlier, arises as a balance between the driving forces and the dissipative forces. To achieve a description of transport with the Green’s functions, it is necessary to add some interaction terms to the Hamiltonian. The interaction term is treated as a perturbation in the usual case, along the lines of the S-matrix expansion of Eq. (5.22). These terms are usually expressable in terms of Feynman diagrams with the use of Wick’s theorem.213In the present context, where we want to treat dissipation through the electron-phonon interaction, this procedure works relatively well, and has been almost universally used. The assumption is that projecting the time axes back to the initial time allows the use of the pseudoequilibrium to justify the Wick’s theorem expansion. The various parts of the diagrams may be regrouped then into terms that represent the Green’s function itself and terms that represent the dissipative interaction with the lattice, which is referred to as the self-energy. The self-energy may also be expressed as a matrix Z just as is the Green’s function G itself. We will not treat the actual Feynman expansion here, because its form is available in numerous textbooks and review articles. Following this procedure, it is now possible to write the equation of motion


407

for the full Green’s function as (16.1)

[

a

1

- i h Y - H O W )- V(r’) G 2.t

=

h1

+ GX,

(16.2)

where, in general, the Green’s function matrix and the self-energy matrix are of the form of Eq. 07.4). Note that the reduced notation

involves an integration over the included internal variables, and the shorthand notation 1 = (rl, t l ) is used, so that the integration is over three spatial variables and a time variable. It is necessary to point out here that the total self-energy has been split into two parts: (1) a single-site part that arises from the external potential V(r), which does not require the integral Eq. (16.3) since it occurs at a single point in space, and (2) the nonlocal self-energy arising from the interaction of the electrons (or holes) with the phonons of the lattice or with other electrons (or holes). The normal Fermi golden rules assumes that the interaction takes place instantaneously and at a single point in space, which are assumptions that reduce the nonlocal self-energy to a single-site representation of the self-energy. This approach has been followed by some, who purport to follow more general quantum transport,351 but reduces the results to little more than the Boltzmann equation treatment.228 A somewhat similar approach has been used by Datta and coworkers,352in which the self-energy for the scattering process is represented in the general form.

(16.4)

where w is the Fourier-transform variable corresponding to the difference in time t - t’. A similar description is used for the retarded self-energy, and this has been applied to treat impurity scattering in mesoscopic 35 I

X. L. Lei and C. S. Ting, Phys. Reo. B 32, 1112 (1985). S. Datta, Phys. Rev. B 40, 5830 (1989); J . Phys.: Cond. Mutfer 2, 8023 (1990); Y. Lee, M. J. McLenna, and S. Datta, Phys R ~ LB’ . 43,14333 (1991). 152

408


devices. Here, however, we follow a more general treatment for the electron (or hole)-phonon interaction. a. Retarded Function In low fields, the general approach is to seek the quantum transport equivalent of the Boltzmann equation. The traditional Boltzmann equation is expressed in terms of the distribution function f(R, p, t ) . In previous sections, it was necessary to transform the density matrix or the Wigner distribution to achieve this quantity. Here, we want to describe the transport equation for low fields in a homogeneous system. This is, in essence, a linear approximation. In the process, we will introduce a phase-space distribution, along with the Wigner coordinates (the centerof-mass coordinates). The approach we follow is that of Mahan and ~ 0 ~ 0 r k e r ~and , 3 ~treat ~ ’ only ~ ~ ~the low electric field case. It has been extended to the case of both electric and magnetic field^.^'^^^^^ We begin by separating out the single entry in the matrix equation given earlier for the retarded Green’s function, which leads to the pair of equations

[

d

-in- dt’ =

-

1

H o b ’ ) - V(r’) Gr(r,r’)

S(r - r’)

+ /d4x” Gr(r,r”)Zr(r”,r’).

(16.5)

These are now two equations that specify the retarded Green’s function, and both must be satisfied. It is convenient at this point to introduce the change of variables [Eq. (5.1311 with the equivalent set for the time axes ( T is the average time and T is the difference time in this approach). 353G.D. Mahan and W. Hansch, J . Phys. F 13 , L47 (1983); W. Hansch and G. D. Mahan, Phys. Rev. B 28, 1902 (1983); G. D. Mahan, Phys. Rep&. 110, 321 (1984), and references therein. 354 L. Reggiani, P. Lugli, and A. P. Jauho, Phys. Rev. B 36, 6602 (1987). 35SG. D. Mahan, J . fhys. F 13, L257 (1983); 14, 941 (1984). 356 N. J. Morgenstern-Horing, H. L. Cui, and G. Fiorenza, Phys. Reu. A 34, 601 (1984).


409

Equations (16.5) now become, after adding and subtracting the two transformed equations

ih-

a

=

in-

dT

+ -V m

h2 + -V2

h2 d 2 2 m as2

+ -- + eF

1 8 ( ~ ) 8 ( s )+ -/d4x"(Z,G, 2

+ G,Z,),

1

. - + e F . s G,(s,R, T , T ) = /d4x"(ZrGr - GrZr). as

(16.6)

The functions inside the integrals on the right-hand sides of these two equations have not yet been transformed to the new coordinates (this is a complicated process we will deal with later). The two equations in Eq. (16.6) describe the relative motion about the center-of-mass motion and the latter motion itself. To proceed, it is now useful to Fourier transform the relative motion coordinates in order to achieve G,(k, R, T , R). Then the preceding equations become

=

/ ~ 3 s d ~ e i k ' s ~ i " " 7 / d 4 ~f r G,Z,). (Z,G,

(16.7)

The second of these equations has the same streaming terms on the left-hand side as the Boltzmann equation (2.1); however, the first equation has some problems, which arise from the streaming terms in the large parentheses (the first term). This term gives rise to a driving force that has an unusual position dependence in the otherwise homogeneous system. This leads to a size dependence that is not at all in keeping with the physics of the structure. On the other hand, if this is combined with the "frequency" it is apparent that these two terms together represent the

410


gauge variation of the energy and potential. This suggests a change of frequency (and spatial) variable as eF .R

a+- n

+ w,

eF d

v +v + h dw

(16.8)

With these changes, Eqs. (16.7) become

8m

=

/ d 3 s dr eik'"-"/d4x''(C,Gr

-

G,C,).

(16.9)

The large omega in the integral stills needs to be transformed for these equations. It is clear that this variation in the exponential in the transformed collision terms lead to phase interference events for distances small compared to the inelastic mean free path. The second of Eq. (16.9) contains a term that is linear in the field, but it is important to remember that the major factor leading to nonlinear transport is from the distribution function itself and not from nonlinear terms in the field. In fact this equation is exact to higher orders in the field. The scattering terms can be quite complicated, and some form of simplification is necessary to evaluate them to any great degree. Since the treatment in this subsection is supposedly for small electric fields in an otherwise homogeneous medium, it is possible to use a gradient expansion.201 To proceed, the functions are expanded about the center-of-mass coordinates, which assumes that both of the latter are much larger than the relative coordinates. This expansion limits the applicability of the results to systems in which the potentials are slowly varying in both space and time. This is hardly the case in mesoscopic semiconductor devices, but is a good approximation for linear transport in a homogeneous semiconductor. To

41 1


illustrate this procedure, only the first term in the collision integral on the right-hand side of Eq. (16.9) is considered. This integral is

First, the center-of-mass coordinates y integral is now a function of

=

x - xr

are introduced so that the

x+xf-y

2

Then, the coordinates are again transformed to s = x - R = (x + xf)/2. Finally, the variable change w = s - y is made, and Eq. (16.10) becomes XI,

W

- Gr

2

Y w,R- -,r,T2

The center-of-mass variables all appear with offsets, so that these may be separated by expanding the functions in a Taylor series, and the Fourier transforms then taken so that the scattering function now becomes

where the bracket operator is dGr {Cr l Gr } - - .d-C- ,L . -dGr +-.---.-

dR

dk

dk

dR

dCr

dGr

dR

dT

dCr dT

dGr

dfl‘

(16.13)

This latter term has a natural symmetry that is also found in the Poisson brackets of classical mechanics, and the shorthand notation of the brackets is often used for this purpose. The frequency derivatives are with respect to the unshifted frequency at this point, so that we must still correct for the change in frequency. This latter operation leads to

{Cr ) Gr } = -a.x-rdR

dGr

dk

dC. dGr 2.-+ -d.C-,- - .dGr dw dT d k dR

e F ( d C r dGr dEr dGr) + - -. - - - . -

h

dw

dk

dk

dw

dCr

dGr

dT

dw

(16.14) ’

412


Already the first term in the gradient expansion has produced a term that is linear in the electric field. If higher order terms are retained, they lead to higher order terms in the electric field. Since these terms have been neglected, the resulting expansion is valid only to linear terms in the field. With these results, it is now possible to write down a relatively closed, but still approximate, set of equations for the retarded Green's function. Combining the above results, these equations become

( ; + v k . v + e-F. n

[( "')

1--hdw d k + ( v k + ""-)?]}G, h d k dw (16.15)

where the bracket on the right-hand side now refers only to the first term on the right-hand side of Eq. (16.14). These equations simplify considerably for near equilibrium systems (which are the ones of interest) that are spatially homogeneous and in steady state, so that the derivatives with respect to R and T vanish. Then, the preceding equations can be written

ieF.

[(

d 1 - -J Z r ) n d w dk

"1

+ (vk + ' Z r ) - G, = 0. h d k dw

(16.16)

The first equation can be solved quite simply to yield a result that will also satisfy the second equation for sufficiently small electric fields F. The retarded Green's function is then finally found to be (retaining only terms linear in the field)

The retarded Green's function in this equation has no first-order terms in the electric field. Thus, its form is unchanged from that of the equilibrium function, although of course the self-energy may have undergone some changes. It may also be expected that the spectral density A(k, R, w , T ) =


413

- 2 Im{G,(k, R, w , T ) }will also have the equilibrium form. The equilibrium form of this function is a Lorentzian, which arises from normal collisionalbroadening. These important results will have a major effect in achieving a quantum Boltzmann equation later, after finding the “less-than’’ function. Before proceeding, however, note that if the quadratic terms in the field are retained in the retarded Green’s function, then the inverse transform into the difference coordinates will contain Airy functions. This observation provides a guide to a powerful approach to the high-field terms in the next section.

b. The “Less-Than’’Function The next step is to repeat the above procedure for the “less-than” function G < . It is this latter function that is related to the distribution function that we normally incorporate in the Boltzmann equation. As before, we will encounter two equations which arise from the two matrix equations, (16.1) and (16.2) We can write these two equations as

where the scattering functions are given by

In the absence of any collisions, the right-hand sides vanish, of course. The left-hand side of the second equation has the same form as the Boltzmann equation found earlier (except for the second term in the square brackets). The derivative with respect to the frequency suggests a constraint that can be placed on the connection between the classical and the quantum forms of the Boltzmann equation. However, if we properly connect the Wigner distribution function with the classical distribution function, then using Eq. (5.30) to form this function from the preceding equation will clearly illustrate that the frequency derivative will vanish from the classical limit. The gradient expansion can now be used on the terms I * in the same manner as previously. Our interest will remain focused on the homoge-

414


neous, steady-state situation in semiconductors, where derivatives with respect to R and T can be neglected. This also means that the gradient expansion terms can be evaluated at the center-of-mass coordinates with the relative coordinates appearing in their Fourier-transformed form. Then, Eq. (16.18) become

E )d kz+ (vk + 5f i d)k 4

e F . [ ( 1 - dw

dw ]

+2rG< G < (16.20)

where the following definitions have been introduced.

J(u)

dC'

du

dw

dk

= --

-

dC'

--.

vu

d k dw

(16.21)

We recognize the spectral density discussed earlier in these definitions. Similarly, r is related to the lifetime of the states involved. It is now possible to make a further simplification for the near-equilibrium situation. Since the field multiples the two bracketed terms, the Green's function within those terms may be replaced by the field-independent form since we are only seeking the linear response terms. In equilibrium, it is possible to separate the less-than function with the ansatz G < = iP( w)A(k, w ) , where P( w ) is the Fermi-Dirac distribution function. Each term multiplying the electric field contains either a factor of P ( w ) or the derivative of this quantity with respect to frequency, since both G < and C < are proportional to P( w ) . In the absence of the field, G < is just the equilibrium form, and the two collision terms 2 r G ' and AC' must balance each other (which is just another statement of detailed balance). Thus, it is the deviation of the distribution from the Fermi-Dirac form that leads to the transport. Thus, those terms involving the derivative of P ( w ) must be of importance, and indeed the other terms will drop out.


415

Thus, the second equation of Eq. (16.20) becomes

=

AC< - 2 r c < .

(16.22)

This result is the steady-state homogenous form of the quantum Boltzmann equation. The right-hand side demonstrates a gain-loss mechanism that is somewhat different from that normally seen, but it may easily be cast into the normal form. To linear terms in the field, this equation is exact and its derivation by Mahan and coworkers represents a major step forward in the use of the real-time Green's functions.353 Note that the coefficient of the second term in the square brackets [left-hand side of Eq. (16.22)] goes to zero at the peak of the spectral density. We can reasonably therefore neglect this second term, as will the velocity correction arising from the momentum derivative of S in the first term. This latter term is a correction to the velocity that arises from the correction to the single-particle energies arising from the real-part of the self-energy, just as the second term is a dispersive velocity correction arising from lifetime effects. The neglect of these terms really means that we are ignoring any corrections to the velocity structure of the singleparticle energy states due to the scattering processes, which is more or less akin to assuming that the self-energy shifts the bands in a rigid manner. Certainly this is appropriate to assume for the linear response case. Then, Eq. (16.22) can be rewritten (16.23) This may be rearranged to give ( 16.24)

At this point, it is necessary to introduce a form for the self-energy. It is easy to show that it may be expressed in terms of the Green's function itself and a phonon Green's function (we treat here only the case of phonon scattering). Thus, C < = iD < C < , but where the Green's function is evaluated at the shifted momentum and frequency according to the

416


phonon frequency and momentum. That is, we can write the self-energy in the form

where the term in the magnitude brackets is the matrix element and includes the phonon occupation factors, and the summation runs over emission and absorption processes. It has been assumed here that optical phonons, which are taken to be dispersionless, are the only phonons of interest. To proceed, we now will make the ansatz that Eq. (16.24) can be rewritten as

where we now have to evaluate the function A(k, w ) . If we use this definition in Eq. (16.251, we find

2' (k, w )

=

iP(w)T

-

d P ( w ) 2rr ieF . vkdw h

c*

where (note that a shorthand notation is used here, and that both A and A have the arguments listed for their product)

r=-c/-2T d3q IM, (k, q>I2A(kf q, h

*

(2rr)

w

f wq).

(16.28)

This now gives us back our initial assertion on the Green's function [Eq. (16.26)] if the scattering kernel satisfies the integral equation


417

The quantity A(k, w ) can be referred to as a vertex correction, in that the integral equation is solved to find a modified scattering strength. For example, if we had assumed that there was also carrier-carrier scattering that modified the Green’s functions, this would also appear in the scattering kernel A as a correction for the carrier response, which is one form of dynamic screening of the scattering process. It also has to be considered that if the phonon occupations are driven from their equilibrium forms, then the matrix element above is modified and a set of equations must be solved for the phonon Green’s functions. We note that the set of equations and the paths used to solve them is just as described in Sec. 1.2. We have first solved for the retarded Green’s function in order to find the spectral density. We then solved for the “less-than’’ Green’s function, which is related to the distribution function. The equation for this latter quantity was found to be very similar to the Boltzmann equation, at least in the linear response used here. A similar approach will be followed in nearly every case of quantum transport with the Green’s functions. The introduction of the two-time formalism has given us the additional need to find the spectral density. This was not necessary in the one-time formalism since there was an integration over the frequency (energy) domain. 17. HOMOGENEOUS, HIGH-FIELD SYSTEMS The development of a tractable quantum transport approach, through the use of the real-time Green’s functions, which incorporates both the collisional broadening of the spectral density and the intracollisional field effect, has proven difficult. (The latter is the interference between the driving terms in the field and the collisions that arise from the use of the nonequilibrium Green’s functions in the self-energy.) Moreover, this task is further complicated by the need to deal with the length and timescales relevant to modern mesoscopic semiconductor devices. The general approach has followed that of the linear response approach of the last section. Although the overall Green’s function approach is rigorous in principle, most applications have been limited by the introduction of the center-of-mass approach and the gradient expansion. These two processes, especially the latter, tend to limit the results to low fields. One of the earliest to go beyond this was the work of Jauho and W i l k i n ~ , ~in’ ~which high-field transport in a resonant-level impurity system of scattering was treated. In this work, the self-energies for electron-phonon as well as 35 7

A. P. Jauho and J. W. Wilkins, Phys. Reu. B 29, 1919 (1984).

418


impurity scattering were formulated, although in the end the gradient expansion was invoked in order to achieve a result for the kinetic equation. Nevertheless, the power of the Green’s function technique, when compared to other approaches, is still evident.358 To separate the “distribution function” part of the less-than Green’s function from the spectral density, one has usually had to invoke an ansatz, such as that leading to Eq. (16.22) in which G < = iP( w)A(k, w ) is assumed. The form of this ansatz has been the topic of some debate, primarily because there are several methods of approaching the separation. In fact, the ansatz is usually a method of recovering a single-time equivalent of the Boltzmann equation. The use of the gradient expansion is just one type of separation approach. A more rigorous approach, which is a zeroth-order approximation in the expansion of the correlation functions in terms of the collision duration time, has been developed by Lipavsk9 et a1.359,36”This form relates the Green’s function to the density matrix as

where

is the kinetic momentum operator. This approach was found to overcome a number of limitations and effects that arise from the normal ansatz used in the ~eparation.’~’ We will see later, however, that an approach using the Airy transforms brings out the separation automatically without having to make an unwarranted ansatz. Other approaches use expansions in the field362or models for the spectral density.”’ The other item mentioned earlier, which has been somewhat controversial, is the intracollisional field effect. In general approaches to quantum transport, a finite duration for the collision is found to occur, and the field can interact with the particle while it is undergoing its collision. Hence, the amount of energy and momentum gained or lost in the collision is modified by the effect of the field during the c ~ l l i s i o n . ~It ”was ~ ~ expected ~~ 358L.Reggiani, Ph~sica134B, 123 (1985). 359 P. Lipavslj, V. SpiCka, and B. Veliclj, Phys. Reu. B 34,3020 (1986). 3hU H. C. Tso and N. J. Morgenstern Horing, Phys. Reu. B 44, 1451 (1991). 3h’A. P. Jauho, J . Phys. F 13, L203 (1983). 362 N. J. Morgenstern Horing and H.-L. Cui, Phys. Reo. B 38, 10907 (1988). 363 P. Lugli, L. Reggiani, and C. Jacoboni, Superlatt. Microstruc. 2, 143 (1986); Physica Scripfa 38, 117 (1988).


419

that this could have a significant effect on the transport properties.’2,”2,3403364 In another approach, in which the collisions were treated as being completed, no intracollisional field effect was observed.365 Later work, however, showed that it would appear, but would be of such a size as to be negligible in normal completed collision^,^^ a result in keeping with numerical studies of the effect.366To date, there is no strong evidence that the intracollisional field effect is a significant process, with the possible exception of impact ionization in wideband-gap semiconductors, in which very high fields are coupled with relatively slow p r o c e ~ s e s . ~ ~ ~ . ~ ~ ~ The collision duration itself is another problem, and one that still has to be solved. There is not much work applicable to this, but one approach using the Green’s functions is due to Lipavslj et al.369These authors have also shown that different definitions of the instantaneous approximation for the self-energy will lead to different effects of the field on the collision, an effect that is negated if wavefunction, and therefore Green’s function, renormalization is properly included.370 The use of the center-of-mass transformations introduces some nonphysical variables into the description of transport. These, in fact, make the explicit assumption that the center-of-mass time T = ( t + t ’ ) / 2 has some inherent significance. This is not the case, and nothing makes the point clearer than the need to modify the ansatz used to separate the distribution function from the less-than function as shown in Eq. (17.1). Consider the velocity autocorrelation function, which is a function of t - t‘, where t’ is the initial time of this function. The center-of-mass time does not enter into any consideration of physical quantities. The same consideration is true for the Green’s functions of interest here. If a transformation is made to the center-of-mass coordinates, and homogeneity constraints (such as the assertion that the solutions are not functions of R) are applied to the space scale and timescales, this is fully equivalent to coarse-graining the Green’s functions over a timescale and space scale corresponding to the inelastic mean free path and inelastic mean free time. It is well known, particularly in mesoscopic systems, that the local 364 A. P. Jauho and L. Reggiani, Sol. State Electron. 31, 535 (1988); L. Regiani, P. Lugli, and A. P. Jauho, J . Appl. Phys. 64, 3072 (1988); A. P. Jauho, Sol. State Electron. 32, 1265 (1989). 365 F. S. Khan, J. H. Davies, and J. W. Wilkins, Phys. Re(*. B 36, 2578 (1987). 3hh D. K. Ferry, A. M. Krirnan, H. Hida, and S. Yarnaguchi, Phys. Reu. Lett. 67, 633 (1991). 367 D. K. Ferry, in The Physics and Technology of S O , (R. A. B. Devine, ed.), p. 365 Plenum, New York (1988). 3hX W. Quade, F. Rossi, and C. Jacoboni, Serniconduc. Sci. Techno/. 7, B502 (1992); W. Quade, F. Rossi, C. Jacoboni, and E. Scholl, Microelectron. Engr. 19, 287 (1992). “’P. Lipavsk9, F. S. Khan, A. Kalvovi, and J. W. Wilkins, Phys. Reu. B 43, 6650 (1991). 3711 P. Lipavslj, F. S. Khan, and J. W. Wilkins, Phys. Reu B 43, 6665 (1991).

420


current is highly nonuniform, even in the otherwise homogeneous systems, and that this leads to universal conductance fluctuations (dc fluctuations of the current with potential, rather than in time). This means that it will be impossible to utilize these center-of-mass formulations for such important applications as proper transient response and the calculation of fluctuation properties on a scale smaller than these coarse-graining times and lengths. Because many modern semiconductor devices have characteristic lengths smaller than, or comparable to, the inelastic mean free path (about 0.1 p m in Si at room temperature), these simplified approaches are doomed to failure in applications to these devices. In fact, a constant electric field breaks the symmetry of the system, so that momentum along the field is no longer a good quantum number, at least on the scale of the inelastic mean free path. If the field is treated in the scalar potential gauge, it is the spatial translational symmetry that is broken, and this was seen to be the case in the approach of the last section where it was necessary to remove an artificial dependence on position by a transformation of the frequency (energy) variable in Eq. (16.8). For this reason, many authors have placed the field in the vector potential, but this breaks the temporal translational symmetry, as evidenced from the results of Eq. (17.2). These facts have greatly complicated the search for high-field solutions for the Green’s functions. a. The Airy Function Retarded Green S Function For the preceding reasons, a different approach is suggested for the treatment of high-field transport with the Green’s functions if the proper space and time variations are to be retained. We remarked earlier, in connection with the retarded Green’s function, that retention of corrections due to the field introduced Airy functions in the inverse transformation to real space and time. Indeed, if the Schrodinger equation is solved in the presence of a uniform electric field, the basis functions that result are the Airy functions. The approach to be used here is to begin at the start by transforming the position variable along the electric field direction using the generalized Airy transform (nevertheless, it will be found that the treatment still is for the steady state). This will allow us to diagonalize the unperturbed Green’s function in the presence of the field alone, and to achieve a simpler form of the Dyson’s equation.37’ Nevertheless, it is 37’R.Bertoncini, A. M. Kriman, and D. K. Ferry, Phys. Reu. B 40, 3371 (1989); 41, 1390 (1990).

42 1


assumed that the system has translational symmetry in the coordinates normal to the field. The general Airy transform is defined by

where 6 is the two-dimensional position vector in the ( x , y ) plane and the field is assumed to be oriented along the z axis. For the Green’s function, of course, there are two Airy transformations on the two coordinates. In this transform space, a function that is diagonal in momentum k (assumed) and s variables is translationally invariant in the transverse plane but not in the z direction. The retarded Green’s function, in the presence of the field, still satisfies Eq. (16.5). These two equations must be Airy transformed. In the absence of the self-energy, the solutions are the equilibrium Green’s function with the field present, which may be shown to be 1

-uo(t fi

-

[: 1

t ’ ) exp - - E k , s ( t - t’) 6(s - S‘),

(17.4)

where ‘k,s

=

h2k2

+ eFs

(17.5)

in parabolic bands. This form has the distinct advantage that the Fourier transformation of the difference variable in time leads to (for a temporally homogeneous system) (17.6) which has the generic form expected for the retarded Green’s function. There is a problem in using this form of the retarded Green’s function because it assumes that the field was turned on in the infinite past (as indicated by the convergence factor T / ) , but this ignores the transient part of the integration path used for the real-time functions. However, this is consistent with the ignoring of the “tail” segment connecting to the thermal equilibrium. Nevertheless, the approach to be used here is only

422


for the steady state, and cannot handle the transient solution. The use of this equilibrium form allows us to write Eqs. (16.5) as integral equations, which are the Dyson’s equations. In the Airy transformed form, the first of these equations becomes

(17.7) where the diagonality of the equilibrium function has been assumed in the prefactor of the integral on the right-hand side. To proceed, it is now necessary to develop an approximation for the self-energy that appears in Eq. (17.7). The self-energy is given by the general Green’s function expansionzo4

The operator ordering in the last term (note that there is no convolution integral in this operator product) is such that it vanishes as the density goes to zero. As a result, for nondegenerate semiconductors, it is possible to ignore the second term. Moreover, in most semiconductors, the scattering is relatively weak and the self-energy can be calculated in the Born approximation-essentially the Green’s function in the first term is replaced by its equilibrium, noninteracting form, which is the free propagator. Since the retarded Green’s function is still solved self-consistently within the Dyson’s equation, collisional broadening will still appear in the final form, and the field’s presence in the free propagator introduces the high-field effects within the scattering process. The phonon Green’s function, for nonpolar optical scattering, can be expressed as (for equilibrium phonons) (17.9) and the matrix element includes the phonon occupation functions. The sum runs over the emission and absorption processes. With these two


423

approximations, the retarded self-energy can be written (where the carrier wavevector is split into its field-directed part and transverse part)

w

=

k

+ kzaz

-

q,

w’= k‘ + kia,

- q.

(17.10)

To be useful, the momentum dependence in the z direction will be transformed back to real space and then Airy transformed. In general, the matrix element for nonpolar optical phonons is not momentum dependent, so there is no complication in the matrix element from these transformations. Finally, the self-energy is found to be3”

(17.11)

where

Y =

eFs - h w

0 3 =

3(ehFl2 2m

*

(17.12)

The real and imaginary parts of the self-energy are plotted in Fig. 38 for two different values of the electric field (and for parameters appropriate to Si at room temperature). The oscillatory behavior of the real part of the self-energy is quite interesting and indicates that the interaction of the field and the scattering process is creating an equivalent quasi-2D behavior in the carrier gas, which is reinforced by looking at the imaginary part, which is related to the density of states. The steplike structure reinforces this interpretation. Because the self-energy is diagonal in the airy variable s, the integral in Dyson’s equation can be removed by assuming that the retarded function is also diagonal in this variable or by taking a delta-function dependence

424


~

U

-0.4‘

-5

1 .o

e

9

0.8

5

0

I 10

-

0.6 0.4

0.2

-

-

-r (b) FIG. 38. The real (a) and imaginary (b) parts of the retarded self-energy for the Airy function model in high electric fields. The reduced units are = ( t i 0 - ~ F s ) / @ ’ / ~ ,=@ (eF/L)’/’.

for the difference. This then leads to the final form for the retarded Green’s function

which is the equilibrium form, although the field is present in the transformed energy and in the self-energy. This result differs from that of the previous section. The spectral density is then twice the negative imaginary

425


part of the retarded Green’s function, and is

which, again, has the equilibrium form, but differs by the explicit incorporation of a field dependence. This is shown in Fig. 39, and the shift and distortion due to the field is evident. Note, however, that the normalization is maintained and a single-time function, which would integrate over this spectral density, does not exhibit any large effect from the field distortion, as discussed earlier. b. The Less- Than Function

With the preceding approximations, it is possible to achieve a good representation of the spectral function, which is correct (within the approximations) to all orders in the electric field. Although it is constrained to the case of weak scattering, it demonstrates both the intracollisional field effect and the collisional broadening effects. With this spectral function, it is now possible to calculate the “less-than’’ Green’s function, G < . In the

-0.05

0.00

0.05

-5 Flc. 39. The spectral function, A(k,sw) = -2Im{G,(k,s,w)}. The units and scales here are the same as in Fig. 38.

426


low-field case of the previous section, it was found that we had to solve an integral equation for a scattering function A. The latter represented aspects of the deviation from the Fermi-Dirac distribution function. Here, an integral equation will also result, but in this case it will lead to the quantum distribution function directly, without the need for an ansatz. However, this distribution function will not be the Wigner distribution function. Rather, the quantum distribution function will be a function of the energy alone, just as is found in the equilibrium case. To obtain the Wigner distribution function, we will need to couple the quantum distribution with the spectral density and integrate out the frequency (energy). Our starting point, as in the previous section, is the matrix equations, (16.1) and (16.2), which after introducing the Fourier transforms on the lateral position and the difference time, and the Airy transform on the dimension along the applied field, becomes

+c‘ ( k , s , sn, w)G,(k, s”,s’, w ) ] .

(17.15)

At this point, it has already been established that the retarded, and advanced by their connection, Green’s functions and self-energies are functions of a single Airy variable. This eliminates the integration through the use of a delta function on the right-hand side of the latter equations. We can then formulate the sum and differences of these two equations, which in the present case yield the same resulting equation. In essence this result means that these two equations are self-adjoint. The resulting equation is just

=

G,(k, S , w)G,(k, s’, w ) C < (k, S , S ’ ,

0).

(17.16)

We note that the retarded and advanced Green’s functions have imaginary parts sharply peaked around the diagonal; in fact, we have taken them to be diagonal. In the last line of Eq. (17.161, we can now assume that the


427

less-than self-energy is also diagonal without any great loss of generality. With this approximation, we can now rewrite Eq. (17.16) as

since the first line contains the spectral density in the numerator of the fraction, and the quantum distribution function is obtained, without any ansatz, to be (17.18) Although at this point it is not obvious that this function will be independent of the momentum, this will be found to be the case later, at least for nonpolar optical phonon scattering. The less-than self-energy function can be developed as easily for the case of the nonpolar optical phonon. As previously, it will be assumed that this function is diagonal in the Airy variable, so that we can write it (as in the previous section) as

(17.19) which depends, of course, on the less-than Green's function. Now, we note an important fact about this latter equation. The matrix element for the nonpolar optical phonon is independent of the phonon momentum vector, so that any change of variables in the integration over this momentum will not affect the final result for the self-energy. In fact, such a change of uariables will integrate out all of the momentum dependence on the right-hand side of the equation. This means that the less-than self-energy is a function only of the Airy variables s and the frequency (energy) w . A casual look Eq. (17.11) shows that the retarded self-energy likewise is a function only of these variables. Thus, the quantum distribution function can only be a

428 0.2

I

!I 4I

’

I

I

.

!

’

,,

I I 01 ,,

6

i

s -

E= 100kVIcm

4

0.1 -

.I

cc

I

E= POOkVIcm .......................... .......

,

S.C. _-_-__

I

.. 0.0 .~

-5.0

A

0.0

L

I

50

.

.

1

10.0

.

I

15.0

.

I

20.0

25.0

hmq

FIG.40. The distribution function (or more properly, the local density of particles) as a function of the electron energy. Parameters appropriate to Si are used. [After R. Bertoncini and A. P. Jauho, Phys. Reu. Len. 68, 2826 (19921, with permission.]

function of these two variables. With these thoughts, the integral equation for the quantum distribution function becomes

where (17.21)

is the effective density of states at the final energy of the scattering process. We note that the denominator of the prefactor plays the same role as r in the low-field equations. In Fig. 40, such a distribution function calculated for the same conditions as the earlier plots (Si at room temperature) is shown. It should be remarked that the integral equation for the distribution function has a certain characteristic form typical of that obtained from the Boltzmann equation as well. This integral equation can be retransformed into real space, and a Monte Carlo procedure developed for it.245 It is not obvious that the present formulation retains the desired property of gauge invariance. This topic has been considered in connection


429

with the Airy transform and the normal Keldysh approach.37z It has been shown that gauge invariance carries through to this approach in a relatively straightforward manner. Moreover, the resulting form for the Green’s functions in the Airy approach can be transformed directly back into the more normal formalism in a manner that maintains the gauge i n ~ a r i a n c e . ~ ~ ~ Finally, we note that the Airy Green’s functions can be connected directly to the Kubo formula for the current, and the resulting conductivity can be expressed as374

where the s variation has been removed from the kinetic energy and placed in the variable R = w - eFs/h. 18. FEMTOSECOND LASER EXCITATION The transport of photoexcited carriers is usually described with the Boltzmann transport equation, in which scattering processes, and the excitation process itself, are described via Fermi golden rule transition rates. However, today it is possible to excite a semiconductor with a laser pulse that is as short as 6 fs.375It is no longer a simple case of exciting an electron-hole pair with a photon from one of these lasers; now one must begin to deal with the correlation that exists between this electron and hole on the short timescale ( < 100 fs). Indeed, the polarization of the pair itself can now be In general, the semiclassimeasured with some degree of 372 R. Bertoncini and A. P. Jauho, Phys. Reu. B 44, 3655 (1991); Semiconduc. Sci. Technol. 7, B33 (1992). 373R.Bertoncini, Int. J . Mod. Phys. B 6, 3441 (1992). 374 R. Bertoncini and A. P. Jauho, Phys. Reo. Lett. 68, 2826 (1992). 37s R. L. Fork. C. H. Brito-Cruz, P. C. Becker, and C. V. Shank, Opt. Lett. 42, 112 (1979). 376 A. Mysyroviecz, D. H u h , A. Antonetti, A. Migus, W. T. Masselink and H. Morkoc, Phys. Reii. Lett. 56, 2758 (1986); D. Hutin, A. Antonetti, M. Joffre, A. Migus, A. Mysrovicz, N. Pey hambarian and H. M. Gibbs, Reu. Phys. Appl. 22, 1269 (1987). 3B?A. von Lehmen, J. E. Zucker, J. P. Heritage, and D. S. Chemla, Opt. Lett. 11,609 (1986). 37x N. Peyghambarian, S. W. Koch, M. Lindberg, B. Fluegel, and M. Joffre, Phys. Reu. Lett. 62, 1185 (1988). 37’) W. H. Knox, D. S. Chemla, D. A. B. Miller, J. B. Stark, and S. Schmitt-Rink, Phys. Reu. Lett. 62, 1189 (1988). 3x0 C. Hirliman, J . F. Morhange, M.A. Kaneisha, A. Chevy, and C. H. Brito-Cruz, Appl. Phys. Lett. 55, 2307 (1989).

430


cal description will break down on this timescale. First, an optical pulse creates carriers in quantum states that are definitely not semiclassical in that the initial carrier wavefunctions are superpositions of conduction and valence band states.381 As long as this phase coherence between the electron and hole states-the interband polarization-remains important, the carriers are not characterizable as plane waves with characteristic momenta from either the conduction or the valence band. Hence, on the short timescale, this polarization must be taken explicitly into account if the system is to be described with any sort of accuracy. The second major point is that the short timescale of the optical pulse gives rise to a very broad Fourier spectrum of photons impinging on the semiconductor. The broadening in excitation photon spectrum that comes from this short time interaction must be taken into account in a manner equivalent to the collisional broadening. The approach to treating rapid excitation of electron-hole pairs in semiconductors is a problem that is inherently a natural for the Green’s function technique, although this study is relatively recent in application. In fact, one will ultimately want to reduce the treatment to a single time variable, just as in the density matrix approach. However, the inherent correlations that must be treated make this a problem better treated by the Green’s functions where the temporal correlations are a natural result of the basis. In fact, the problem has been treated to some degree both by the density matrix382 as well as by the real-time Green’s function tech.iqUe.25l,383-385 0n the short timescales, the coherence introduced by the laser pulse cannot be neglected, so that the interband polarization must be treated as another kinetic variable. The resulting development leads to the optical Bloch equation.252~3s6-388 The presence of the correlation, which is represented by the interband polarization, is quite significant. The shock of the intense laser pulse fully and completely separates the electron-hole gas from the thermal equilibrium background. Although there has been some work on treating the development of the nonequilibrium system from the thermal distribution 3n 1

V. M. Galitsky and V. F. Yelesin, Resonant Interaction of Elecromagnetic Fields with Semiconductors, Energoatomizdat, Moscow (1986). 38 z R. Zimmermann, Phys. Stat. Sol. ( b ) 159, 317 (1990); J . Lum. 53, 187 (1992). 3H3H.Haug, Phys. Stat. Sol. ( b ) 173, 139 (1992); L. BBnyai, D. B. Tran Thoai, C . Remling, and H. Haug, Phys. Stat Sol. ( b ) 173, 149 (1992). 384 H. C. Tso and N. J. Morgenstern Horing, Phys. Rev. B 44, 8886 (1990). 385D. B. Tran Thoai and H. Haug, Phys. Reu. B 47, 3574 (1993); Z. Physik B 91, 199 ( 1993). 38 6 S. Schmitt-Rink and D. S. Chemla, Phys. Rev. Lett. 57, 2752 (1986). 3n 7 M. Lindberg and S . W . Koch, Phys. Reu. B 38, 3342 (1988). 3nRT. Kuhn and F . Rossi, Phys. Rev. B 46, 7496 (1992).


43 1

(along the tail of the contour of Fig. 8),389.39"this approach is not possible in the current context. This means that it is generally not possible to utilize Wick's theorem to discard the normal-ordered products in the Green's function expansion of Feynman diagrams. These products are part of the initial correlations that remain after the laser excitation, and must be treated by higher order diagrams in the Green's function e x p a n ~ i o n . ~ ~ ' , ~ ~ ~ In this section, we want to describe the manner in which the initial correlations and the carrier-carrier interactions fit into the development of the Green's function equations of motion for the two carrier types. To achieve the goal of writing the Green's functions as products of two types of carriers, we expand the definition of the Keldysh matrix given by, for example, Eq. (V.4). We consider that each of the field operators can be either an electron or a hole operator (an electron in the valence band). Then it is possible to write the equivalent of Eq. (V.4) as (18.1) which is a resulting 8 X 8 matrix, where c and u stand for conduction and valence band carriers, respectively. The end result will be eight equations for Green's functions for various types of particles and their interactions. Of course, there are certain symmetry properties among the various Green's functions. In the laser-excited situation, however, we are interested in the equivalent density matrices for the two carrier types and in the polarization (the cross terms in the subscripts of the preceding equation). This means that the final set of equations will have a single time variable, although we will develop these equations from the two-time formalism. but the Various approaches have appeared in the l i t e r a t ~ r e193839393 ,~~ approach we will follow is largely that of K ~ h n . ~ ~ ~ For the investigation of fast laser excitation of semiconductors, and the consequent coherent dynamics, we need to consider explicitly the coupling between the carriers and the light field and to include these terms in the system Hamiltonian. Here, we will take the equal time form of the 38 9

P. Danielewicz, Ann. Phys. 152, 239 (1984). P. A. Henning, Nucl. Phys. B 342, 345 (1990). '"A. G. Hall, J . Phys. A 8, 214 (1975). 392 S. G.Tikhodeev, Sou. Phys. Dokl. 27,492 (1982), 27,624 (1982); Yu. A. Kukharenko and S. G. Tikhodeev, Sou. Phys. JETP 56, 831 (1982). 393A.V. Kuznetsov, Phys. Reu. B 44, 8721 (19911, 44, 13381 (1991). 3')4 T. Kuhn, E. Binder, F. Rossi, A. Lohner, K. Rick, P. Leisching, A. Leitenstorfer, T. Elsaesser, and W. Stolz, to be published; J. Schilp, T. Kuhn, and G. Mahler, Semiconduc. Sci. Technol. B , 9, 439 (1994). 3YU

432


Hamiltonian (replacing the field operators by simple second-quantized creation and annihilation operators)

H,

=

CE;ck+c,

+ C E i d l d , + Cho,b,+b,

k

+

9

k

[M, * F'+'c:d', k

+ M* k

*

F'-)d - k c k

9

(18.2)

where c i , d i , bg ( C k , d , , b,) denote the creation (annihilat in) operators for the electrons, holes, and phonons, respectively, Eke = E, + h2k2/2m, and EL = h2k2/2m, are the electron and hole energies (measured from the valence band edge), M, is the dipole matrix element connecting the hole state with the electron state in optical absorption, and F ( * ) is the Fourier coefficient of the field at positive and negative frequency by

with oL the center frequency and F , ( t ) the optical pulse shape. The basic variables now used to describe the system response to the optical fields are the distributions of the electrons and holes

and the interband polarization

Note that these are equal-time Green's functions, and therefore are actually mixed density matrices. It will further be noted that these four functions are the four blocks of the super-Green's matrix [Eq. (18.111. The Hamiltonian of Eq. (18.2) does not have any interactions among the various excitations (electrons, holes, and polarizations), and in the absence of these, the equations of motion become

df; dt

dfhk

1

dt

lh

- = - = g : ( t ) = - [M,

F , ( t ) e - i " L ' p~ M:

F:(t)eimL'p,],

(18.6)


433

The main difference between the coherent generation rate gE(t) and the Fermi golden role of optical absorption is that in this approach the generation of carriers is a two-step process. The light wave first creates a polarization excitation in the semiconductor, and then this polarization is ionized into a separate electron and hole. The connection with the previous section is made by, for example, (18.8)

and this strange state is achieved by noting that d - k , creates a hole in the state -k, which in effect annihilates an electron in the valence band electron state k. This change in notation is a result of a “rotation” of valence electron states into hole quasi-particle states. (In reading the various literature on this topic, one should be aware of the vagaries of notation used. The notation here is that of Kuhn394 while the alternative notation in terms only of electron operators is used by, for example, Haug and coworkers.3R3)The rotation and renormalization in terms of quasi-particles is often referred to as a Bogoliubov transformation. Dissipative processes can be added to Eqs. (18.6) and (18.7) quite easily. The interaction with phonons is directly added to the equations for the electron and hole distribution functions through self-energy terms as described in the last section (the limitation to equal times simplifies the process immensely). In fact, in this formulation, the electron and hole losses to the phonons can almost be as easily treated by semiclassical scattering terms, and this is one approach that has been ~ ~ e d On . ~ the other hand, the inclusion of a collision broadening of the appropriate spectral function requires the retention of the proper Green’s functions expansions, and this has been pursued by Haug and among others. In Fig. 41, the energy distribution for the electrons is shown at different times for the Bloch equation model treated by Kuhn and coworkers, and is compared to semiclassical results (ignoring the polarization effects). The multiple peaks are phonon replicas of the initial excitation. No carrier-carrier interactions are included in the model. In Fig. 42, the average kinetic energy and the polarization are shown for comparison. A 100-fs pulse width is assumed in these calculations. Carrier-carrier interactions can be added to the process by including an additional term in the Hamiltonian, which has the usual form (again, written in a single time formulation)

(18.9)

~

~

.


.~

.~ i! i!

-

I!

-----

!;:J

........... -. -. -, .

!! !h!

. :,

1;

: :.:I

.. .. : ?

(0:

0 PS

0.5 ps 1.0 ps

1.5 ps

Energy (meV)

FIG.41. Energy distributions of electrons at different times during and after the pulse (a) for the full Block equation model and (b) for the semi-classical Boltzmann equation model. [After T. Kuhn, el al., Proc. NATO ARW on Coherent Optical Interactions in Semiconductors, Cambridge, 1993.1

where the three terms represent electron-electron, hole-hole, and electron-hole interactions. In addition, the interband exchange term has also been neglected (in which the electron and hole exchange momenta). Using the Heisenberg equations of motion for the various density matrices, we find the extra term for the time rate of change of the electron distribution function to be

f(c$+qdl(-qdk‘Ck)

-

(c:d;-qdk’ck-q)]

(18.10)

>

and analogous equations are obtained for the hole distribution function and the polarization. If Wick‘s theorem is used to factorize the products of (b)

6,

-lCRl ----- ClRl

‘L.

-.-.---

4

.2

d

s n 9 c

D

ga 0

------______ -0

(PSI FIG.42. The kinetic energies (a) and polarizations (b) as a function of time for a 100-fs laser pulse. [After T. Kuhn el al., Proc. NATO ARW on Coherent Optical Interactions in Semiconductors, Cambridge, 1993.1


435

the four operators into separate distribution functions and polarizations, then energy shifts are obtained corresponding to the Hartree and Hartree-Fock (exchange) contributions. The polarization leads to a renormalization of the internal field that excites the carriers. There are, however, second-order contributions, which are the higher order correlation f ~ n c t i o n s , ~3y2” ~These two-particle correlations, when treated in the lowest order, lead to terms (and the need for equations of motion for these terms)

In general, these lowest order terms now lead to the lifetimes of the electron hole, and polarization states.3y5However, it is not at all clear that the higher order terms can be ignored and, normally, the four-operator Green’s function must be determined from a self-consistent integral equation, often called the Bethe-Salpeter equation. In fact, these terms are also important for dynamic screening of the carrier interactions, and the basic electron-electron term leads to what is also known as the polarization bubble (polarization here of the electron gas separately). These higher order terms are also important for weak localization and universal conductance fluctuations at low temperature^.^^.'^ To date, few of these more complicated approaches have been utilized in the femtosecond excitation world, and this area remains one in which many fruitful advances are still to be made. In keeping with the general treatment of the scattering as semiclassical, it is also possible in numerical simulations of the femtosecond response to include the electron-electron interaction in real space as a molecular dynamics potential, and this has led to some understanding of the effect of the interacting gas on this ti mescal^.^'^ There have also been attempts to include the finite duration of the collision that would arise by retaining the full two-time Green’s function approach to the femtosecond response.3y7 19. THEGREEN-KUBOFORMULA

In Section 11.6, the Kubo formula was developed in linear response to the applied fields, as represented in the vector potential. It was noted there 34s

A. A. Abrikosov, L. Gorkov, and I. Ye. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics, 2nd. ed., Pergamon, Oxford (1965). 3Yh D. K. Ferry, A. M. Kriman, M. J. Kann, and R. P. Joshi, in Monte Carlo Simulations of Semiconductors and Semiconductor Deuices (K. Hess, ed.), Kluwer, Nonvell, MA (1991). 3Y7 D. K. Ferry, A. M. Kriman, H. Hida, and S. Yamaguchi, Phys. Rev. Left.67,633 (1991).

436


that the use of the Kubo formula was a significant change from the normal treatment of the dominant streaming terms of the Boltzmann equation, or of the equivalent quantum transport equation, to the relaxation and/or scattering terms. With the Kubo formula, one concentrates on the relaxation processes through the correlation functions that describe the transport. Here, we would like to close this loop once again and talk about how the real-time Green’s functions fit into the Kubo formula, with the combination termed the Green-Kubo formula. Now, if we note that the quantum-mechanical current is described by dq(r) dq*(r) .l=eh [ q * ( r ) y dr

(19.1)

2 im

then it is not too difficult to develop the Green’s function form of this quantity to use in the Kubo formula [Eq. (6.8)l. Using these ideas, it is possible to write the equivalent Kubo formula in terms of the Green’s function for the carrier system as

where we carry out the indicated integrations and limits, and A(r, t ) is the vector potential (19.3)

A h , t ) = - j ‘ d t F(r, t’).

Note here that the displacement current has been ignored. Now, this can be Fourier transformed to give the ac conductance

+ G < ( k 2+

i,

w r ) G r ( k 2-

5,w’ - w

11

,

(19.4)


437

which, after some further changes of variables becomes

XIG,(kl, w ‘ ) G < (k,

-

k,

O’ - W )

(19.5)

+ G < ( k l , w’)Gr(k, - k, W’ - w ) ] .

Note that this conductivity does not diverge as the frequency tends to zero, due to the integral over the shifted frequency. This form has certain advantages, but it has not been utilized very much with the real-time nonequilibrium Green’s functions. It is now useful to rearrange the terms into those more normally found in the equilibrium and low-temperature forms of the Green’s functions. For this, we make the ansatz G‘ (k, W )

=

if(o)A(k, W )

=

-f<w)[G,(k,

W )

-

G,(k, w ) ] . (19.6)

This ansatz was shown to be correct for the low-field Green’s function and for the Airy transformed Green’s functions in previous sections. It is also correct in equilibrium for zero-temperature and thermal Green’s functions, but it is not in keeping with the normal high-field ansatz used in the Kadanoff-Baym and Keldysh representation^.^^^ Nevertheless, we shall pursue this definition. Using this in Eq. (19.5) leads us to

The first two products of Green’s functions will cancel one another. This can be seen by changing the frequency variables as w“ = w’ - w , then using the fact that v(k, w ) = c*(k, - w ) , and G, = G,*. Thus, we are left just with the last term the curly braces. For low frequencies, the distribution function can be expanded about w ‘ , so that a(k. w )

=

e2h

l-

-I I

d3k,

do’ i-kl. .7T

X G,(k,, w’)G,(k, - k,

k

W’

-

af(w’) OJ)-

dw’

.

(19.8)

438


Finally, at low frequencies and for homogeneous material, we arrive at the form

In the case of low temperatures, the derivative of the distribution function is essentially a delta function at the Fermi energy. For this case, and with a change to the velocities in the arguments, one arrives at (19.10)

where wF = E F / h . The sum over the momentum counts the number of states that contribute to the conductivity and results in the density at the Fermi energy. In mesoscopic systems, where only a single transverse state may contribute, the Landauer formula, Eq. (2.4) (an additional factor of 2 arises from a summation over spin states), is recovered when it is recognized that JGr(k,,wF)I2 is the transmission for the mode. Even if there is no transverse variation, the integration over the longitudinal component of the wavevector will produce the difference in the Fermi energies at the two ends of the samples. The approach of Eq. (19.9) has been extensively utilized by the Purdue group to model mesoscopic systems with the equivalent Landauer formula for nonequilibrium Green's functions. For mesoscopic waveguides in the linear response regime, even with dissipation present, they have shown that this form can be extended to the use of a Wigner distribution, which then can be used to define a local electrochemical potential, but that sensible results are obtained when these potentials are defined in an average sense over regions the size of a de Broglie wavelength.398 They have also investigated the simulation of dissipative scatterers by the use of voltage probes, but find that the formulation is robust enough to introduce the scatters directly into the Green's function.39yThis latter has led to a general multiprobe formula400 and to ac In Fig. 43, the 398 M. J. McLennan, Y. Lee, and S. Datta, Phys. Reu. B 43, 13846 (1991); in Computalional Electronics (U. Ravaioli, ed.), Kluwer Norwell, MA (1991); Y. Lee, M. J. McLennan, G. Klimeck, R. K. Lake, and S. Datta, Superlafr. Microstruc. 11, 137 (1992). 3yyS. Datta and R. K. Lake, Phys. Rec. B 44, 6538 (1991). 40"S. Datta, Phys. Reu. B 45, 1347 (1992); Phys. Rev. B 46, 9493 (1992). 40 I S. Datta and M. P. Anantram, Phys. Reti. B 45, 13761 (1992).


439

FIG.43. The solutions for the chemical potential in a quantum wire with a stub waveguide attached. (a) The chemical potential, and (b) the current density as a function of position in the guide. Notice in particular the extensive fraction of the total potential drop which occurs at the contact regions. [After M. J. McLennan, Y.Lee, R.Lake, G. Neofotistos, and S . Datta, in Computational Electronics (U. Ravaioli, ed.), p. 247, Kluwer, Norwell, MA (19911, with permission.]

solutions for the electrochemical potential and current density are shown for a quantum waveguide with a stub sidearm. One important result of this calculation is the large potential drops at the cathode and anode contacts apparent in the structure. The role of surface scattering in mesoscopic quantum waveguides has also been investigated through the use of the Kubo formulation.402In this latter work, it is found that the presence of edge roughness can open gaps in the energy spectrum of the carriers for a one-dimensional wire, but this does not occur in higher dimensions. The Green’s function approach has also been utilized to study magnetotransport in these quasi-1D Usually, the Kubo formula and the Green’s functions developed earlier have been limited to the nearly (or exactly) homogeneous semiconductor. Jauho has extended this approach to the strongly inhomogeneous situation by the introduction of a new an sat^.^'^ Here, the Green’s function G < (k,,, Az, w , R ) is expanded as a product of the spectral density and a quasidistribution, the latter of which is composed of a mini-(Wigner) distribution for each transverse mode present at R and the wave functions 402

M. K. Schwalm and W. A. Schwalm, Phys. Reu. B 45, 6868 (1992). N. Mori, H. Momose, and C . Hamaguchi, Phys. Reu. B 45,4536 (1992). 404 M. Wagner, Phys. Rev. B 45, 11606 (1992). 4115 A. P. Jauho and 0. Ziep, Physicu Scriptu T25, 329 (1989). 401

440


propagating in the z direction as

")2 (19.11) Here, the inhomogeneous potential along a mesoscopic device may be treated nonperturbatively. Others have looked at other generalizations of Eq. (19.9),s6and at the time-dependent transport in nonlinear s t r ~ c t u r e s . ~ ' ~ Finally, there has been an effort to extend the general approach to obtain, for example, an energy balance equation along the line of the treatment of Sections 11.7 and II.8.407 Formula (19.10) has found significant usage at low temperatures to model quantum wires with what are known as real-space lattice Green's These were studied initially functions, or recursive Green's in connection with universal conductance fluctuations,8' and have since been used extensively to study general conductance in multiterminal mesoscopic d e v i ~ e s . ~ ~Some , ~ ~ formal ' - ~ ~ ~considerations on extending on arbitrary Hamiltonian result to more general Hamiltonians has also been presented.413It has also been extended to the consideration of Coulomb interactions among the electrons on the lattice and to selfconsistent fields due to gated potentials.41s The general approach to the site-representation evaluation of the Kubo formula begins with writing a new Hamiltonian for the total "energy" in 406

N. S. Wingreen, A. P. Jauho, Y . Meir, Phys. Reu. B 48, 8487 (1993). 4"7R. Lake and S. Datta, Phys. Reu. B 46, 4757 (1992). 408 G. Czycholl and B. Kramer, Sol. State Commun. 32, 945 (1979). 409 D. J. Thouless and S. Kirkpatrick, J. Phys. C 14, 235 (1981). 410 B. Kramer and J. MaSek, 1. Phys. C 21, L1147 (1988); J. MaSek, P. Lipavsky, and B. Kramer, J . Phys. Cond. Matter 1, 6395 (1989); J. MaSek and B. Kramer, Z . Phys. B 75, 37 (1989); B. Kramer and J. MaSek, in Science and Engineering of One- and Zero-Dimensional Semiconductors (S. P. Beaumont and C. M. Sotomajor-Torres, eds.), p. 155, Plenum, New York (1990). 4"H. Baranger, A. D. Stone, and D. P. DiVicenzo, Phys. Reo. B 37, 6521 (1988). 4'2S. He and S. Das Sarma, Sol. State Electron. 32, 1695 (1989); Phys. Rev. B 40, 3379 (1989); Phys. Reu. B 48, 4629 (1993); S. Das Sarma and S. He, Int. J. Mod Phys. B 7, 3375 (1993). 413W.A. Schwalm and M. K. Schwalm, Phys. Reu. B 37, 9524 (1988). 414 W. Hausler, B. Kramer, and J. MaSek, Z. Phys. B 85, 435 (1991). 415 Q. Li and D. K. Ferry, Superlatf. Microstmc. 11, 245 (1992).


44 1

the system. This may be done as

f

= { G ; ' ( j ) + G R ( j )= { G ; ' ( j >+

(19.18) The Green's functions are propagated separately from the left and the right directions, and then the two versions are added to give the total Green's function. Once this is done, the conductivity can be computed as a function of the energy E. Variations of this general approach have been used by Sols et ~ 2 1 . ~to' ~ study a stub-tuned waveguide "transistor." In Fig. 44, the general structure of the device and the transmission coefficient for a range of parameters are shown. The approach has also been used to study the resonant-tunneling diode in the presence of electron-phonon coupling."y9417Figure 45 shows the general structure of the device and the tunneling coefficient as a function of energy in the presence of the dissipative interaction. The density in the quantum well is illustrated in Fig. 46 as a function of the transmission coefficient, and the build-up time is plotted as a function of the well width in Fig. 47. 20. THERESONANT-TUNNELING DIODE

Section 11.4 discussed the variety of devices that have become the test vehicles for examining the role of quantum transport. Foremost among these is the resonant-tunneling diode. A variation of this, the quantum dot, in which a resonant section of an electron waveguide is coupled to two other (input/output) waveguides, was also discussed. It turns out that these are essentially the only devices in which any extensive study of quantum transport with the Green's functions has been made. One ap41 h

F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. L e f f .54, 350 (1989); 1.Appl. Phys. 66, 3892 (1989); M. Macucci, V. Pevzner, L. F. Register, and K. Hess, in New Concepts for Low-Dimensional Electronic Systems (G. Bauer, H. Heinrich, and F. Kuchar, eds.), Springer-Verlag, Heidelberg (1993). 4'7J. A. St$vneng and E. H. Hauge, Phys. Reu. B 44, 13582 (1991).


443

PVQ

r

TVD

FIG.44. (a) Schematic view of the three-terminal structure used in the calculations of the waveguide transmission. (b) The squared modulus of the transmission coefficient for the first mode, as a function of the effective length of the stub L* and of the electron energy E ( L , = L , = 10 nm). [After F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. Lett. 54, 350 (1989), with permission.]

proach in fact uses a single-particle Green’s function and the poles of this latter function to identify the resonant states within the resonant-tunneling Using the Kubo formula, it is conceptually easy to determine the current through the resonant-tunneling diode, although the introduction of a self-consistent potential distribution complicates this task. A quasi41R

W. R. Frensley, Superlatt. Microstruc. 11, 341 (1992).

444


a

0

200

100

300

E (meV)

b

FIG. 45. The transmission as a function of energy (a) and the barrier structure (b) for a double-barrier AlGaAs/GaAs structure. The transmission corresponds to 5-nm barriers (40% Al concentration) and a 6-nm well. [After J. A. Stbvneng, E. H. Hauge, P. Lipavsky, and V. SpiEka, Phys. Reu. B 44, 13595 (19911, with permission.]

analytic theory to determine the self-consistent population of the resonant state was introduced by Lu and H ~ r i n g Here, . ~ ~ ~it was found that there could be a suppression of the tunneling current, due to the exclusion principle, when the resonant state was significantly occupied. On the other have demonstrated that the semiclassical rate hand, Davies et d4*’ equation picture holds both in the coherent limit, in which transport through the double barrier is described in a simple wave mechanics picture, and in the limit where scattering in the resonant state makes a classical “sequential tunneling” picture more appropriate. In the case where the electrons are incident with an energy greater than that of the resonant state, elastic scattering can “focus” the electrons into final states whose motion perpendicular to the barrier planes closely match the energy of the resonant state.421This implies that the lateral momentum contribu4’9X. J. Lu and N. J. M. Horing, Phys. Stat. Sol. ( b ) 174, 27 (1992).

420 J. H. Davies, S. Hershfield, P. Hyldgaard, and J. W. Wilkins, Phys. Reu. B 47, 4603 (1993). 42 I H. A. Fertig and S. Das Sarma, Phys. Reu. B 40, 7410 (1989).


445

- exact

FIG.46. The time evolution of the particle density in the quantum well of a double barrier. Here, a hopping model is used to simulate the transport. The long and short dashed curves are the result of averaging over an energy window around the resonant energy level of varying width, where the range is defined by the reduce wavevector r, = 1.2 X 10-3(27r/a), and a is the lattice constant. [After J. A. Stovneng and E. H. Hauge, Phys. Reu. B 44, 13582 (19911, with permission.]

tions are randomized in such a manner that the longitudinal momentum closely matches the energy of the resonant level. The role of inelastic scattering in the resonant-tunneling diode has been examined by a variety of authors within the Green's function formalism.The earliest studies focused on the tunneling of carriers that were assisted by an infrared photon beam, with the result that the photon-induced resonant

0

1

2

3

4

5

6

7

0

9

Ax/a FIG.47. The build-up time for charge in the quantum well as a function of the initial wave packet width A x . The amplitude of the barrier is taken to be four times the hopping energy between sites [After J. A. St4vneng and E. H. Hauge, Phys. Reu. B 44, 13582 (1990, with permission.]

446


1.55

.775

29.4

0.00

-

Energy (eV)

.:

*OS5

FIG.48. The Green's function for a double-barrier tunneling diode with inelastic scattering present. The plot illustrates the "current" density as a function of both position and energy. There is a clear transition of energy within the barrier indicating an inelastic tunneling process. [After R. Lake and S. Datta, in NASCODE VZI (J. J. H. Miller, ed.), Front-Range Publ., Boulder, CO (1991).]

tunneling could be switched and controlled by the change of the applied bias.422Later work showed that the role of the inelastic processes depended on the transparency of the barriers, particularly in thicker barriers where the charge in the well would be larger.423In Fig. 48, the density in the resonant-tunneling diode is shown as a plot of the distribution function versus both energy and position. Here, it is clear that a phonon emission process is under way during the tunneling of the particles through the structure. The possible role of multiple scattering (higher order interactions) has also been studied recently.424 The tunneling time for carriers through the double-barrier structure has been of some interest in connection with the debate over the tunneling time for particles moving through a barrier. One approach is to calculate the ohmic conductance through the structure as a function of the fre422W.Cai. T. F. Zheng, P. Hu, M. Lax, K. Shum, and R. R. Alfano, Phys. Reu. Lett. 65,104 (1990); W. Cai. T. F. Zheng, P. Hu, and M. Lax, Mod. Phys. Left. B 5, 173 (1991). 423G.Neofotistos, R. Lake, and S. Datta, Phys. Reu. B 43, 2442 (1991); R. Lake and S. Datta, Phys. Reu. B 45, 6670 (1992); R. Lake, G. Klimeck, and S. Datta, Phys. Reu. B 47, 6427 (1993); R. Lake, G. Klimeck, M. P. Anantram, and S. Datta, Phys. Reu. B 48, 15132 (1993). 424P.Hyldgaard, S. Hershfield, J. H. Davies and J. W. Wilkins, Ann. Phys. 236, 1 (1994).


447

quency of the bias potential.425 From these studies, it is found that the basic ac conductance rolls off at a frequency corresponding to the width of the resonance level A E, divided by Planck’s constant, or (20.1)

However, these latter authors find that there is a peak in the conductance if the barriers are unequal, which in fact is the usual case. Other studies indicates that lateral confinement also affects the ac conductance,426as does the presence of charge in the where the absence of charge in the well basically slows the transient switching due to the need to charge the well and its intermediate state. A more complete theory is one in which it is necessary to discuss the differences between transits through the well, charging of the well, and tunneling processes.428 From this latter, it is demonstrated that there are two characteristic timescales of importance-that of the steady-state current characteristic time (which appears in the conductance) and that of the charge in the well Q,. These two define the dwell time and the transit time, where the latter is defied by Q W / j d c .The transit time is an intrinsically static quantity, whereas the response time (similar to the roll-off frequency discussed above) is important for ac properties. The response time is found to be dominated by the escape rates out of the quantum well for the stored charge in the well. Carrier-carrier scattering does not affect these conclusions to any great extent.429 As pointed out in the Introduction to this article, the quantum dot is a mesoscopic equivalent of the resonant-tunneling diode, except that it is usually studied in the few-electron regime. The mesoscopic quantum dot has been studied in analogy to that of the diode. Here, in an approach that treats the charging of the dot region through a Hubbard-like term in the Hamiltonian, it is found that charging of the dot region strongly affects the overall transmission.430Of course, the few-electron quantum dot is subject to the normal single-electron charging with conductance peaks as electrons charge and discharge the dot r e g i ~ n . ~ ” ,The ~ ’ ~ overall behavior has 425

C. Jacoboni and P. J. Price, Frontiers in Condemed Matter Theory, Ann. New York Acad. Sci. 581, 253 (1990); in Resonant Tunneling in Semiconductors (L. L. Chang, ed.), p. 351, Plenum, New York (1991). 42hW.Cai, P. Hu, and M. Lax. Phys. Reu. B 44, 3336 (1991). 427W.Cai and M. Lax, Phys. Reu. B 47, 4096 (1993). 42X E. Runge and H. Ehrenreich, Phys. Rev. B 45, 9145 (1992); Ann. Phys. 219, 55 (1992). 424 T. Brandes, D. Weinrnann, and B. Krarner, Europhys. Lett. 22, 51 (1993). 430 A. Groshev, T. Ivanov, and V. Valtchinov, Phys. Reu. Lett. 66, 1082 (1991). 43 I J. MaSek and B. Krarner, PTB-Mitreilungen 101, 327 (1991). 432Y.Meir, N. Swingreen, and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991).

448


recently been reviewed433and is not dealt with to any great extent here. The use of Green’s functions has, however, identified the importance of carrier-carrier interactions,434 and the nonlinearities introduced by the change in the The use of the real-time Green’s functions has mainly been limited in practice to nearly homogeneous high-field structures and to linear response in inhomogeneous structures. Nevertheless, it has the capability to provide the most in-depth look at the actual physics of nanodevices, and its use is still open to considerable study: ACKNOWLEDGMENTS The authors are indebted to a great number of people who have contributed to this effort and provided permission to use their own figures and results. Our own work, along with our students and collaborators, has been supported by ONR, ARO, and AFOSR. Unpublished algorithm development and results on the density matrix were developed along with T. R. Govindam (SRA, Inc.), who also contributed to the understanding of the formulation of dissipation in the density matrix. Major discussions over the years with A.-P. Jauho (Technical University of Denmark), W. R. Frensley (University of Texas at Dallas), S. Goodnick (Oregon State University), U. Ravaioli and K. Hess (University of Illinois), J. R. Barker (Glasgow University), A. M. Kriman (SUNY Buffalo), M. Buttiker (ETH Zurich), R. Landauer (IBM Research), R. Bertoncini (CRS4 Cagliari, Italy), and L. Reggiani and C. Jacoboni (University of Modena), plus a multitude of students, have contributed greatly to our level of understanding of this field, and the authors are properly cognizant of the debt owed to these people. 433

G.-L. Ingold and Yu. V. Nazarov, in Single C h a g e Tunneling (H. Grabert and M. H. Deverot, eds.), p. 21, Plenum, New York (1991). 434 W. Pfaff, D. Weinmann, W. Hausler, B. Kramer, and U. Weiss, unpublished results; Wa. Hausler and B. Kramer, Phys. Rev. B , 47, 16353 (1993); T. Brandes. W. Hausler, K. Jauregui, B. Kramer, and D. Weinmann, Physica B 189, 16 (1993). 435H.Scholler, unpublished results; C . Bruder and H. Scholler, Phys. Rev. Lerr., 72, 1076 (1994).

Author Index

Numbers in parentheses are footnote reference numbers and indicate that an author’s work is referred to although the name is not cited in the text. Arjavalingan, G., 162, 163, 164, 190(43, 44) Armstrong, J. P., 169 Arnold, A., 317, 319, 320, 385, 386, 391 Aroda, S. K., 29 Aronov, A. G., 330 Asano, T., 244 Ashcroft, N. W., 63, 151, 152(1), 157(1) Askari, S. H., 19 Aspnes, D. E., 289 Assadi, A,, 10, 88, 107(23) Atherton, N. M., 136 Aven, M., 216, 233(2) Averin, D. V., 308, 399(137)

A

Abdolsalami, F., 288, 419(34) Abdou, M. S. A., 107 Abkowitz, M. A,, 74, 76(232), 133, 146, 148 Abkowitz, M., 28 Abramof, E., 223 Abrikosov, A. A., 435 Adachi, C., 28, 116, 130(96), 140 Adams, E. N., 347 Adams, J. A., 306, 307 Adler, C. L., 192 Ager, J. W., 236 Aggarwal, R. J., 306 Aharonov, Y., 295 Ahmed, H., 88, 108(250), 115(250), 139, 308 Akhtar, M., 145 Akimichi, H., 111 Akimoto, K., 213, 242, 251, 252 Alberga, G. E., 335 Alerhand, 0. L., 193, 194 Alfano, R. R., 84, 446 Allemeand, P.-M., 20 Allen, J. W., 116 Alnot, P., 11, 29(25), 111(25) Altshuler, B. L., 286, 295(15), 309(15), 330, 435(15) Anantram, M. P., 438, 446 Ancona, M. G., 314, 315, 317(178) Anderegg, V. F., 308 Anderson, S. F., 285 Ando, T., 8, 14(17), 107(17), 111(17), 287, 295, 392(19) Andrt, J. J., 111 Andrt, J. M., 46 Andreatta, A,, 26 Antipin, V. A,, 123 Antonetti, A., 429 Antoniadis, H., 74, 76(232), 133, 146, 148 Antoun, S., 19 Aoyagi, Y., 308 Arends, J. H., 287 Argyres, P. N., 289, 303, 316, 326(185), 329, 333, 334, 347, 362

B

Baccarani, G., 284 Baeriswyl, D., 37, 41, 420371, 45, 104(123) Baigent, D. R., 22, 120, 123, 124, 132, 134(64), 142, 143 Baker, C. A., 318 Baker, G. L., 54, 96(177) Ball, R. C., 46 Ballard, D. G. H., 24 BBnyai, L., 430, 433(383) Baranger, H. U., 300,312(85), 440(85) Baranger, H., 440 Barber, R., 307 Bardeen, J., 348 Bargon, J., 14 Barker, J. R., 285, 288,319,326(30), 332(32), 338,347,361(11), 380(30,35), 400, 418(30), 419(32, 3401, 448 Barker, J., 6 Barker, R., 285, 288(12), 302(12), 326(12), 4 19(12) Barlow, W. A,, 27, 116(92, 93) Barton, T. J., 123 Bassani, F., 219, 272 Bassler, H., 20, 53, 57, 60(58), 74,75,82(171), 83(171), 107(187, 1881, 120, 122, 125(313), 129(59, 3131, 130(313), 141 Bassous, E., 284 449

AUTHOR INDEX Bastiaans, M. J., 380 Basturo, J. G., 116 Baude, J. F., 256 Bauer, A,, 145 Bauer, S., 230 Baughman, R. H., 6, 57 Bawendi, M. G., 232 Baym, G., 321, 323(201), 410(201) Becker, P. C., 429 Beenakker, C. W. J., 300 Behr, T., 272, 274(119) Belicky, B., 330 Beljonne, D., 49, 50, 72, 136(163) Benatar, A., 26 Berg, R. A,, 81 Bernstein, G., 285 Bertoncini, R., 309, 335, 420, 423(371), 428, 429, 448 Bhargava, R. N., 233 Bhatt, A,, 302 Billingham, N. C., 29, 30(99), 31, 60(99) Binder, E., 431, 433(394) Binh, N. T., 74 Binnig, B., 304 Bird, J. P., 308 Birman, J. L., 334, 347 Biship, A. R., 43, 44(142), 45(142), 49(142) Bishop, D. J., 296 Biswas, R., 163, 164, 192(46, 47) Blaikie, R. J., 308 Blaisdell, G. A,, 42 Blandin, A,, 322 Blochl, P. E., 236 Bloemer, M. J., 168 Blood, P., 278, 279(125) Bloor, D., 60 Blatekjaer, K., 342 Blount, E. I., 154, 170(40) Blumentengel, S., 57 Boboliubov, N. N., 346 Bogel, E. O., 53, 82(171), 83(171) Bohrn, D., 292, 298(47), 337 Bolmes, A. B., 17 Boman, M., 72 Bonanni, A., 219, 272 Boney, C., 247 Bonner, W., 207 Born, M., 187, 346 Borsenberger, P. M., 3, 75 Bosch, R., 342 Bott, D. C., 6, 55, 96, 97(258)

Bouchriha, H., 129 Bowden, C. M., 168 Bowers, K., 249 Bradley, D. D. C., 11, 12(31), 17, 18, 21, 29, 30,31(102), 46,49(151), 51, 52,53, 54(104), 56, 60, 61(194), 62, 67, 68(213), 72, 88, 96, 97(258), 103(213), 104(213), 105, 106, 107(213), 108, 117, 118, 119, 120, 122, 123, 125, 128, 129, 130(223), 131(223,302), 132(61), 134, 135, 136(178), 137, 138, 139, 146, 147, 148(1Y4), Brafman, O., 42 Brandes, T., 447 Brandt, M. S., 234 Brandt, M., 239, 240(77) Brassett, A. J., 108 Braun, D., 8, 20, 68, 83, 84(246, 247). 118, 119(215), 120, 122, 123, 127, 129, 130(336), 148 Brazovskii, S. A., 37, 45(126) Brtdas, J. L., 9, 38, 46, 41, 49, 50, 61, 62, 72, 73, 85, 113, 128(225), 132(129), 136(163) Brennan, R. M., 165 Brenner, K. H., 380 Brito-Cruz, C. H., 429 Brommer, K. D., 154, 160(31, 32), 161, 162, 163, 164, 190, 191, 192(86), 193, 194 Broms, P., 46, 72(150) Brown, A. R., 11, 12(31), 68, 72, 88, 109, 110, 117, 118, 119, 120, 122, 123, 124(312), 128, 130(223), 131(223, 302), 134, 135, 137, 138 Brown, C. S., 29, 3(x99), 60(99) Brown, E. R., 169 Bruder, C., 448 Brunetti, R., 316, 347, 361(180) Brunthaler, G., 215, 223 Buchwald, E., 126 Bullock, D. L., 168 Bulovic, V., 143 Buot, F. A., 386, 387 Burn, P. L., 11, 12(31), 17, 18, 29, 30,31(102), 51,52, 53, 54(104), 56,68, 72,82,83(170, 2451, 104, 105(219), 106, 117, 118, 119, 120, 122, 123, 125, 128, 130(223), 131(223, 302), 136(178), 137, 138, 139 Burnett, G. M., 33 Burns, S. E., 140

AUTHOR INDEX Burroughes, J. H., 8, 9, 10, 11, 12(31), 72, 85, 87(18, 19), 88(18), 92(18, 191, 93, 94, 95, 97, 98, 99, 100(18), lOl(18, 19), 102, 103(18, 19). 117(31), 118, 119, 122, 128, 130(223), 131(223) Burrows, P. E., 143 Burshtein, Z., 116 Biittiker, M., 295, 296, 299, 300, 301, 302, 310(84), 448 Biittner, H., 45, 46(144, 145) C Cacialli, F., 24, 73, 123, 124(78), 144(226) Cahay, M., 310 Cai, J., 334 Cai, W., 310, 333, 335, 362, 446, 447 Calcagnile, L., 219, 220(41), 272 Caldeira, A. O., 304, 362, 364(298) Callagher, D., 233 Callaway, J., 151. 152(2), 154(2), 157(2), 158(2), 170(2), 171(2), 179(2), 184(2), 185(2), 186(2) Calvert, P. D., 29, 30(99), 31, 6M99) Cammack, D. A,, 216 Campbell, D. K., 37, 39, 41, 42(130, 1371, 43, 44(142), 45, 46(130), 49(142), 104(123) Campbell, I. H., 139 Canham, L. T., 115 Cantwell, G., 214 Cao, Y., 26, 31, 142 Carcenac, F., 285 Cardon, F., 140 Carrier, G. F., 385 Carroll, C. W., 380 Carruthers, P., 320, 382(197) Cavenett, B. C., 136, 225, 270 Cerdeira, H. A,, 286, 435(14) Cervini, R., 132 Chadi, D. J., 236 Chan, C. T., 153, 154, 164, 187(30), 188, 189, 192, 193(21), 195 Chance, R. R., 46, 60, 140 Chandrasekhar, V., 295, 298 Chandross, E. A., 31 Chandross, M., 49 Chang, A. M., 301 Chang, L. L., 304 Chang, S.-K., 216 Chaplik, A. V., 306

451

Chaussy, J., 295 Chemla, D. S., 429, 430 Chen, J., 22 Chen, M., 303 Chen, T. A,, 23 Cheng, H., 208, 241, 244(10, 781, 251(78), 256, 259, 260 Chester, G. V., 288 Chevy, A,, 429 Chiang, C. K., 4, 5(6), 6 Chien, W., 376 Choi, H.-Y., 45 Choi, K. K., 296 Chou, W. C., 216 Chu, D. Y., 168 Chu, X., 215 Chung, T.-C., 22 Churchill, L., 247 Cingolani, R., 219, 220(41), 272 Clarisse, C., 111 Clarke, T. C., 8 Cleaver, J. R. A., 308 Clemenson, P. I., 14, 107(34) Cnossen, G., 81, 140(243) Cohen, M. H., 347 Colaneri, N. F., 56, 108, 136(178) Colaneri, N., 131, 142 Coleridge, P. T., 306, 307 Colle, M., 126 Collins, D. A,, 215 Collins, R. P., 115 Collins, T. C., 47 Commes da Costa, P., 39, 42(131), 46(131) Conner, M., 380 Conrads, N., 144 Contellec, M. L., 111 Conticello, V. P., 24 Conwell, E. M., 39, 42(131), 46, 48 Cook, J. W., 247 Cook, J. W., Jr., 242, 249 Corneliassen, H., 216 Cornil, J., 46, 49, 72(150), 113 Coronado, C. A,, 214 Cotal, H. L., 220 Courtis, A,, 24 Craig, R. A,, 321 Cramail, H., 16, 30 Crawford, 0. A,, 140 Cruz, H., 309 Crystal, B., 82, 83(245) Crystall, B., 53, 83(170)

452

AUTHOR INDEX

Cui, H. L., 335, 408, 418 Czycholl, G., 440 D

Dai, N., 216 Dalby, R., 216 Dalichaouch, R., 162, 169 Dandrea, R. G., 39, 42030, 46(131) Danielewicz, P., 431 Danielson, P. L., 46 Dannetun, P., 46, 72, 73, 128(225) Das Sarma, S., 309, 310, 314, 335, 362, 440, 444

Datta, S., 154, 187(30), 192, 310, 407, 438, 439, 440, 446

Davies, J. H., 306, 308, 419, 444, 446 Davies, M., 307 de Jong, M. J. M., 301 de Leeuw, D. M., 88, 109, 110 De Martini, F., 168 Deen, M. J., 107 Degond, P., 320 Dekker, H., 364 Delannoy, P., 129 Deloffre, F., 11, 29(25), 111(25), 129 Demandt, R. C. J. E., 83,84(246,247), 12M246, 2471, 122, 123, 127

Denenstien, A., 6 Dennard, R. H., 284 DePuydt, J., 208, 235(8), 241, 244(10, 781, 251, 256,259, 260

Dervet, B., 11, 29(25), 111(25) Deussen, M., 57, 107(188), 141 Devoret, M. H., 308 Devreese, J. T., 47, 334, 335 Devreese, J., 332, 333 Dharma-Wardana, C., 306 Dialichaouch, R., 153, 162(15) Diaz, A. F., 14 Diaz, F. M., 22 Didio, M., 219 Diessel, A., 272, 274(119) Ding, J., 115, 207, 208, 216, 218, 219(38),

221(39), 230, 231(57), 244, 247, 248, 249, 251,255(85), 256(85), 260, 264,269,270, 271(114, 1161, 273(116), 275(116), 278 Ding, Y. W., 123 DiVicenzo, D. P., 440 Dobrowolska, M., 216, 223

Dodabalapur, A., 141 Doi, S., 20, 122, 133 Donegan, J. F., 223 Dorman, D. R., 236 dos Santos, D. A., 46, 72(150) Dowling, J. P., 168 Drabe, K. E., 81, 14M243) Drake, J. M., 82 Drenton, R. R., 252 Dresselhaus, G., 2, 6(1) Dresselhaus, M. S., 2, 6(1) Drexhage, K. H., 139 Drury, R., 287 Durbin, S. M., 216 Dutta, M., 302 D w t a d , K. J., 236 Dyakonov, V., 122, 126, 129, 146(326) Dyrekelev, P., 107 Dyreklev, P., 107 Dzyaloshinskii, I. Ye., 435

E Eagle, G., 54, 96(177) Eason, D. B., 247, 249 Eastman, L. F., 285 Ebisawa, E., 8 Economou, E. N., 153, 186, 187(16), 195 Edge, S., 14, 107(34) Edwall, D. D., 376 Edwards, J. H., 14, 15(35) Ehrenfreund, E., 42, 42(138) Ehrenreich, H., 179, 186, 202(69, 701, 220, 276, 277(122), 347, 447

Elasesser, T., 431, 433(394) Ell, C., 402 Emin, D., 47 Enderlein, R., 333 English, J. H., 294 Engquist, B., 390 Epstein, A. J., 26 Era, M., 116 Erdogan, T., 168 Esaki, L., 303, 304 Esteve, D., 308 Etemad, S., 41, 42(133), 46, 54, 96077) Etiene, B., 285 Evans, G . P., 56 Even, R., 111 Evrard, R., 332, 335

453

AUTHOR INDEX F

Fahlman, M., 46, 72, 73, 128(225) Faisst, C . F., 26 Falicov, L. M., 192 Fan, F., 264 Fan, Y., 239, 240(77), 241, 242(79), 243(79), 249,252, 256, 257(104), 258, 260, 261(108), 263(108), 265(104), 277(104) Fano, U., 315 Faschinger, W., 222 Fauquet, C., 73, 128(225) Faurie, J.-P., 215 Fave, J.-L., 1 Feast, W. J., 14, 15, 16, 24, 30, 100, 101 Feldmann, J., 141 Feng, S., 301 Feng, Y., 307 Ferraris, J . P., 139 Ferreira, S., 215 Ferry, D. K., 285, 287, 288, 291, 292, 293, 302(12), 304, 308, 311, 312, 326(12), 332, 334, 335, 337, 338(50), 342, 343, 345, 353, 357, 361(11), 364(263,292), 378(46), 380, 382(45), 383, 384(45), 38.5, 386(320), 387, 388, 390, 392(328), 396, 400, 419, 420, 423(371), 428(245), 435, 440 Fertig, H. A,, 310, 444 Fesser, K., 43,44(142), 45,46(144, 149, 49(142) Feynman, R. P., 293,298(48), 315(48), 337(49), 346, 380 Fichou, D., 11, 29, 57, 111, 112(183, 184) Fiegna, C., 285 Filig, T., 306 Fincher, C. R., Jr., 4, S(6) Fiorenza, G., 408 Fischer, F., 213 Fischer, J. E., 31 Fischetti, M. V., 286 Fisher, D. S., 299, 330(80), 440(80) Fisher, P. A., 214, 234, 236 Flattk, M. E., 276, 277(122) Floyd, P. D., 229 Fluegel, B., 429 Foad, M. A,, 230 Foot, P. J. S., 29, 30(99), 31, 60(99) Ford, G. W., 332 Fork, R. L., 429 Forrest, S. R., 143 Forster, T., 51

Forstner, A,, 215 Fowler, A. B., 287, 392(19) Foxon, C. T., 300 Franciosi, A,, 219, 220(41), 272 Frankevich, E. L., 57 Franssila, S., 108 Fredriksson, C., 72 Freeman, N. J., 14 Freeman, W. H., 182, 199(72) French, I. D., 144 Frensley, W. R., 358, 385, 389, 392, 396, 400, 401, 443,448 Freund, L. B., 213 Friend, R. H., 8, 9, 10, 11, 12(31), 15, 16, 17, 18, 21, 22, 24, 25, 29, 30, 31(102), 46,49, 50, 51, 52, 53, 54(104), 55, 56, 57, 58, 59, 60, 61, 62, 67, 68, 72, 73, 82, 83, 84(248), 85, 87(18, 191, 88, 92(18, 19), 93, 94, 95, 96, 97(258), 98,99, 100, 101, 102, 103(18, 19, 213), 104, 10.5, 106, 107, 108, 109, 110, 111(189), 112, 113, 114, 115(250), 117, 118, 119, 120, 122, 123, 124(78,321), 128, 129, 130(223), 131(223, 302), 132, 134, 135, 136(163, 178), 137, 138, 139, 140, 141, 142, 143, 144(226), 146, 147, 148 Fripiat, J . G., 46 Frohlich, H., 342 Frohne, H. R., 310 Fu, Q., 216, 228, 219, 239(53), 250 Fu, Y., 88 Fuchigami, H., 11, 88(27), 108(27) Fugita, S., 215 Fuijita, S., 218, 220(37) Fujita, H., 236 Fujita, S., 218, 220(37) Fujita, Sg., 244 Fujiyasu, H., 215 Fukutome, H., 46 Fukuyama, H., 296,330(69) Fukuyama, M., 26 Furdyna, J. K., 215, 216,223, 239 Furdyna, J., 207, 218, 219(38), 221(39), 229, 269, 270, 271(114) Furukawa, Y., 57, 107(18.5, 186) G

Gabrowski, M., 46 Gaensslen, F. H., 284

454

AUTHOR INDEX

Gagnon, D. R., 19 Gailherger, M., 20, 60(58) Gaines, J. M., 252 Galhraith, I., 270 Galick, A. T., 302 Galitsky, V. M., 430 Galvlo, D. S., 46 Calvin, M. E., 31, 81, 83(242), 143(242), 148 Gandit, P., 295 Ganin, E., 285 Gao, Y., 73, 126(228) Garcia, N., 170 Gardner, C. L., 342 Gamier, F., 11, 22, 29, 57, 111, 112, 113, 114, 129, 145 Gau, S. C., 4, 5(6), 145 Gaylord, T. K., 154, 171(36), 172 Gehhardt, W., 230 Gellerman, W., 170 Gelsen, 0. M., 30, 52, 54 Genack, A. Z . , 170 George, C., 288 Gerard, J. M., 190 Gerber, Ch., 304 Gerrligs, L. J., 308 Gettinger, C. L., 82 Giftke, C., 215 Gill, R. E., 123 Gill, W. D., 8 Gin, D. L., 24 Glaser, D., 109 Glasse, C., 144 Glenis, S., 145 Click, A. T., 287 Glytsis, E. N., 154, 171(36), 172 Gmeiner, J., 126 Gmitter, T. J., 153, 154, 159(17, 18), 160(17, 31, 33), 161, 189(33), 193(31), 194(31) Gobel, E. O., 53, 141 Goel, A., 296 Gogoliuhov, N. N., 336 Colin, S., 192 Comes, H. L., 14, 107(34) Gommes da Costa, P., 48 Gonsalves, J. M., 228, 239(53) Gonsalves, J., 216 Goodhue, W. D., 305 Goodings, C. J., 347 Goodings, C., 306 Goodnick, S. M., 312 Goodnick, S. S., 311

Goodnick, S., 448 Gordon, B., 20 Gorkov, L., 435 Gossard, A. C., 294 Gourley, P. L., 165 Govindam, T. R., 357,364(292), 378,342,353 Grahert, H., 304 Gradley, D. D. C., 68, 119(216) Graham, S. C., 106 Graham, S., 72 Grant, P. M., 8 Gratzel, M., 145 Greczmiel, M., 126 Green, H. S., 346 Greenham, N. C., 21, 22, 68, 72, 81, 83, 84(248), 117, 118(223), 119, 122, 123, 128, 129, 130(223), 131(223, 3021, 132, 134, 135, 137(240), 139, 140, 141(240), 142, 143, 148 Greiner, A., 20, 21, 53, 57, 82(171), 83(171), 107(187), 120, 122, 125(313), 129(59, 3131, 130(313), 141 Grem, G., 25, 120, 123, 124(77, 801, 131(77) Griffiths, K., 278, 279(125) Griffiths, R. B., 298 Grillo, D. C., 208, 223, 224(47), 244,247,248, 249(90), 251,252,253,254(102), 255(85), 256(85), 260,261(108), 262(108), 264, 266, 276, 278 Grillo, D., 256, 257(104), 258(104), 265(104), 277( 104) Gronberg, L., 108 Groshev, A., 361, 447 Grubhs, R. H., 24 Gruhin, H. L., 292, 342, 353, 357, 364(292), 374, 378, 380(46), 394 Griiner, J. F., 25, 123 Gruner, J., 132 Gruner, J., 22,24, 120, 123, 124(78, 3211, 134, 141 Gu, H. B., 22 Guha, S., 256, 259, 260 Guillaud, G., 111 Guinea LBpez, F., 286, 435(14) Gunshor, R. L., 115,208,216,223,224, 225(49), 228, 229, 238,239, 240(77), 241, 242(79), 243(79), 244, 247, 248, 249, 250, 251, 252, 253, 254(102), 2SS(85), 256, 257(104), 258(104), 260,262(108), 263(108), 264,265(104), 266,276, 277(104), 278

AUTHOR INDEX Guss, W., 141 Gustafsson, G., 31, 67, 68, 88, 107, 119(215), 123, 142 Gutowski, J., 272, 274(119) Gyrner, R. W., 88, 108(250), 115(250), 119, 12(x308), 123, 139 H

Haase, M. A,, 208,235(8), 242,244(78), 251(78), 256, 259, 260 Haase, M., 208, 244(10) Haberern, K. W., 252 Haddad, G. I., 400 Hadziioannou, G., 123 Hagedorn, M. S., 256 Hagerott, M., 218,221(39), 234,241,242(79), 243(79), 247, 248, 249, 251, 252, 270, 271(116), 273(116), 275(116), 278 Hagler, T. W., 20, 54 Hajlaoui, R., 11, 29(25), 111(25, 26), 129 Haken, H., 380 Hakki, B. W., 256, 257(105) Hall, A. G., 431, 435(391) Hall, D. G., 168 Hall, R., 115 Hailer, E. E., 236 Halliday, D. A., 17, 18, 30,51,52,53, 54(104), 56(169), 106 Halls, J. J. M., 60, 61(194), 146, 147, 148 Hamada, Y., 116 Hamaguchi, C . , 287, 295, 439 Hamer, P. J., 24, 25, 120, 123, 124(78, 321), 132, 134 Harnrnons, B. E., 165 Han, 238 Han, C.-C., 17 Han, J., 216, 223, 224, 225(49), 234, 237, 239, 240(77), 241, 242(79), 243(79), 244, 248, 249, 252, 253, 254(102), 255(85), 256, 257(104), 258(104), 260,261(108), 263(108), 264,265(104), 266,276, 277(104), 278, 285 Handlovitis, C. E., 21 Hangleiter, A,, 280, 281(127) Hann, R. A,, 27, 116(93) Hinsch, W., 408, 415(353) Harder, C . , 278, 279(125) Hark, V., 280, 281(127) Harris, J. S., Jr., 376

455

Harrison, M. G., 57, 68, 89, 104, 105(219), 106, 111(189), 112, 113, 114 Harrison, W. A., 182, 199(72), 234 Harrison, M. G., 88, 108(250), 115(250) Harsch, W. C., 214 Harstein, A,, 296 Hashemi, M. M., 229 Hatano, T., 404 Haterott, M., 269, 271(114) Haug, H., 272,402,335,430,431(251), 433(383) Hauge, E. H., 305, 310(119), 442, 444, 445 Haugen, G. M., 260 Hauksson, I., 270 Haus, J. W., 154, 187(34), 190 Hausler, W., 440, 448 Havinga, E. E., 88, 109(255), llO(255) Hayes, G. R., 83, 84(248) Hayes, J. R., 294 He, L., 239, 240(77), 241, 242(79), 243(79), 249, 253,254(102), 256, 257(104), 258(104), 265(104), 277(104) He, Li, 248, 249(90) He, S., 310, 440 Heeger, A. J., 6, 8, 19, 20, 22, 26, 30, 31, 34, 35, 38, 39(118, 1191, 41, 42(133), 54, 55, 60, 61, 68, 82, 96, 115, 118, 119, 120, 122, 123, 128, 129, 130(336), 131, 132(129), 142, 147, 148 Heenecke, M., 53 Hefetz, Y., 215 Hegarty, J., 223 Heinke, H., 213 Heitz, W., 20, 21, 120, 122, 125(313), 129(59, 313), 130(313) Hekking, F. W. J., 310 Helbig, M., 62, 132(203) Helfrich, W., 11 Helm, M., 223 Henderson, G. N., 172 Henderson, G., 154, 171(36) Henneberger, F., 221, 272 Hennecke, M., 53 Hennig, R., 141 Henning, P. A., 431 Henry, C . H., 278, 279(124), 280(124) Hensich, H. K., 64, 67(206), 78(206) Hershfield, S., 306, 308, 444, 446 Hess, B. C . , 57 Hess, K., 298, 310, 311, 442, 443, 448 Heun, S., 53

456

AUTHOR INDEX

Heyman, M., 129 Hibbs, A. R., 293, 298(48), 315(48) Hicks, J. C., 42, 42(139) Hida, H., 419, 435 Hiei, F., 213, 242, 252 Higashi, H., 129 Hiley, J. M., 337 Hinds, E. A., 165, 166(49) Hiramoto, M., 141 Hirliman, C., 429 Hirooka, M., 19 Hjertberg, T., 31 Ho, E., 214, 234, 236 Ho, K. M., 153, 154, 163, 164, 187(30), 188, 189, 192, 193(21), 195 Ho, S. T., 168 Ho, S., 302, 306 Hofler, G. E., 256, 259 Hoger, S., 119, 122 Hoger, S., 128 Hohold, H. H., 57 Holdcroft, S., 107 Holland, P. R., 337, 357(257) Holmes, A. B., 11, 12(31), 18, 21, 22, 24, 25, 29, 30, 31(102), 46, 51, 52, 53, 54(104), 56,60,61(194), 68,72,73,82,83,84(248), 104, 105(219), 106, 107, 117, 118, 119, 120, 122, 123, 124(78, 3211, 125, 128, 129, 130(223), 131(223, 302), 132, 134, 135,136(178), 137, 138, 139,141, 144(226), 146, 147, 148 Holscher, H., 217 Holweg, P. A. M., 308 Holzapfel, J., 145 Home, D., 46 Hommel, D., 272, 274(119) Hone, D. W., 322 Hong, K. M., 60 Hong, X., 233 Hongawa, K., 120, 123 Hopfel, R. A,, 402 Hopfield, J. J., 206, 217(1) Horhold, H.-H., 21, 62, 132(203) Horing, N. J. M., 333, 334, 335, 402, 444 Horing, N. J., 347 Horneigh, R. M., 192 Hornreich, R. M., 192 Horovitz, B., 40, 42 Horowitz, G., 11, 29, 57, 111, 112(183, 184) 129 Hosokawa, C., 129

Hotta, S., 111 House, J. L., 234, 236 Hovinen, M., 244,255(85), 256(85), 260, 262(108), 263(108), 264 Howard, R. E., 296 Hsieh, B. R., 73, 74, 76(232), 84, 126(228), 133, 146 Hu, B. Y.-K., 314, 335 Hu, G. Y., 302, 314, 332, 334 Hu, P., 310, 333, 446, 447 Hua, C., 250, 255(85), 256035) Hua, G. C., 249,253,254(102), 260,261(108), 263(108) Hua, G., 252 Huber, J., 25, 120, 123, 124(81) Huber, J.-J., 120, 123, 124(321) Hughes, W. C., 249 Hui, P. M., 172, 179, 185, 192, 197, 199, 202(69, 70) H u h , D., 429 Hupe, J., 26 Huser, J., 306 Hussain, A. T., 67, 68(213), 88, 103(213), 104(213), 105, 107(213) Hwang, W., 64, 66, 70(208), 71(208) Hybertson, M., 276, 280(123) Hyldgaard, P., 306, 308, 444, 446

I Iafrate, G. J., 292,302,315,317(178), 342(46), 347,370(279), 374(279), 378(46), 380(46) Ichiguchi, T., 308 Ichino, K., 218, 220(37) Ikeda, M., 242, 244, 253, 254(84a) Ikeda, S., 5 Imry, Y., 296 Inganas, O., 10, 31, 67, 88, 107 Ingold, G.-L., 448 Inguva, R., 154, 187(34), 190(34) Inoue, K., 223 Inoue, T., 123 Isawa, Y., 404 Ishibashi, A., 223, 242,244,247, 253, 254(84a), 264, 265(111) Ishibashi, K., 308 Ishihara, T., 269,270,271(114, 116), 273(116), 275(116) Ito, T., 5

457

AUTHOR INDEX Itoh, S., 244, 253, 254(84a) Itoh, Y., 115 Ivanov, T., 447 Iwai, H., 285 Iyechika, Y., 20 Iyer, S. S., 115 Izrael, A,, 190

Joo, J., 26 Joosten, H. P., 306 Joshi, R. P., 435 Jung, N., 144 Juza, P., 223

K J Jackel, L. D., 296 Jackson, R. W., 17, 18, 125 Jacobini, C., 285, 306, 316, 347, 361(11, 180), 418, 419, 447,448 Jacobs, J. P., 278, 279(125) Jacobsen, K. W., 310 Jain, J. K., 362 Jalabert, R., 362 Janussis, A., 318 Jarrett, C. P., 88, 100, 101, 108, 109, 110 Jauho, A. P., 304, 305, 306, 310(118), 333, 334, 407(228), 408, 417, 419, 418, 428, 429, 439, 440, 448 Jeglinski, S., 49 Jenekhe, S. A., 146 Jensen, K. L., 386, 387 Jeon, H., 115, 207, 208, 241, 242(79), 243(79), 244,247,248(87), 249,252,255(85), 256(85), 266, 269, 270, 271(114, 116), 273(116), 273116) JCrome, D., 3, 4(4) Jin, Y., 285 Jing, W.-X., 24, 123, 124(78) Joannopoulos, J. D., 154, 160(31, 32), 161, 162, 162(32), 163, 164, 190, 190(32), 191, 192(86), 193(22), 193, 194 Jobst, B., 272, 274(119) Joen, H., 251 Joffre, M., 429 John, S., 153, 169, 187 Johnson, M., 215 Johnson, N. F., 172, 179, 197, 199, 202(69, 70, 71) Johnson, N. M., 234, 236 Johnson, N., 239, 240(77) Jones, C. A,, 8, 9, 10, 11, 87(18, 19), 88(18), 92(18, 19), 94(18, 19),95(18, 191, 100(18), lOl(18, 19), 102(19), 103(18, 19) Jonker, B. T., 216 Jonson, M., 304

Kadanoff, L. P., 321, 323(201), 410(201) Kadyshevitch, A., 168 Kaerijama, K., 22, 73, 128(225) Kagan, J., 29 Kahbn, F. S., 419 Kahlert, K., 29, 30(100) Kahn, M. S., 61 Kajiwara, T., 215 Kakuta, A., 115 Kalashnikov, V. P., 336 Kalkhoran, N. M., 115 Kallmann, H. P., 11, 116(28) Kalvovh, A., 419 Kalyanasundaram, K., 145 Kamata, A., 236 Kanbara, T., 123 Kanbe, M., 16 Kane, E. O., 152, 157(4), 170(4), 303 Kaneisha, M. A., 429 Kanicki, J., 8, 145(16) Kann, M. J., 435 Kano, H., 111 Kao, K. C., 64, 66, 70(208), 71(208) Kaplan, S. B., 296 Karasawa, T., 208, 215, 235(9) Karasz, F. E., 17, 19, 29, 30(101), 31, 57, 74, 106, 120, 123 Karathanos, V., 154 Karczewski, G., 239 Karg, S., 68, 122, 126, 129, 146(326, 327) Karl, N., 145 Kartheuser, E., 332 Kaski, K., 306 Kastner, M. A., 296 Kato, E., 253 Kawai, T., 30 Kawakami, Y., 218, 220(37), 225, 270 Kawakani, Y., 215 Kearwell, A., 32 Keanvell, G. M., 33 Keldysh, L. V., 321, 403200)

45 8

AUTHOR INDEX

Kelkar, P., 223, 224(47), 248, 249(90), 266, 276, 278 Keller, F., 272 Kenkre, V. M., 51 Kerkhoven, T., 287, 302, 312 Kern, D. P., 285 Kersting, R., 53 Kesler, M. P., 278, 279(125) Kessener, Y. A. R. R., 83, 84(246, 247, 248), 120(246, 2471, 122, 123 Khan, F. S., 288, 419 Kido, J., 120, 123 Kikharev, K. K., 308, 399(137) Kim, H., 142 Kim, K. W., 302 Kirkner, D. J., 303 Kirkpatrick, S., 440 Kirkwood, J. G., 289,346 Kirova, N. N., 37, 45(126) Kivelon, S., 35 Kivelson, S., 41, 42(134) Klavetter, F., 131, 142 Kleinert, H., 293, 337(49) Klimeck, G., 438, 446 Klingshirn, C., 272 Kluksdahl, N. C., 380, 385, 386(320), 387, 388, 390, 392(328), 396, 398 Kluksdahl, N. V., 291,382(45), 383(45), 384(45) Knight, P. L., 166 Knoll, K., 16 Knox, R. S., 51 Knox, W. H., 429 Kobayashi, E., 287 Kobayashi, H., 116 Kobayashi, M., 30, 115,208,218,234,244(11), 247, 248(87), 251 Koch, S. W., 429, 430 Koezuka, H., 8, 11, 14(17), 88(27), 107(17), 108(27), 111(17) Kofman, A. G . , 168 Kohda, M., 120, 123 Kohler, A., 61 Kohmoto, M., 170 Kohn, W., 176,316 Koishi, M., 140 Kojima, T., 26 Kolodziejski, L. A., 226, 228, 229, 234, 236, 239(53), 250 Kolodziejski, L., 207, 214 Konoda, Y., 73, 128(225)

Konstadininis, K., 31 Koremer, H., 8 Koshida, N., 115 Koster, G. F., 184, 185(74) Kouki, F., 129 Kouwenhouven, L. P., 300 Koyama, H., 115 Kozlov, V., 223,224(47), 256,257(104), 258(104), 265(104), 266, 276, 277(104) Kraabel, B., 120, 122, 148 Kraft, A. M., 139 Kraft, A., 17, 18, 24, 30, 52,54(104), 72, 117, 118(223), 119, 120, 122, 123, 124(78), 128, 130(223), 131(223, 302), 137, 138 Kramer, B., 309, 329,330, 440, 447, 448 Krebs, J. J., 216 Kreskovsky, J. P., 342, 357, 364(292) Krieger, J. B., 347, 370(279), 374(279) Kriman, A. M., 291,337,382(45), 383, 384(45). 385,386(320), 388, 392(328), 396, 420, 423(371), 435, 448 Kriman, A., 285 Krische, B., 22 Kroemer, H., 118 Kroll, N., 162 Kromer, H., 376 Kronenberg, W., 26 Kroubni, M., 8 Kubo, R., 327, 329 Kucera, J., 301 Kucharska, A. I., 278, 279(125) Kudoh, Y., 26 Kuhn, H. H., 26 Kuhn, T., 330, 335, 430, 431, 433(252, 388, 394), 434 Kuhn, W., 230 Kuhn-Heinrich, B., 213 Kuivalainen, P., 107, 108 Kukharenko, Yu. A., 431, 435(392) Kukomoto, H., 244 Kumar, V., 380 Kunz, R. E., 140 Kuo, L. H., 260 Kurizki, G., 168 Kuroda, T., 223 Kuroda, Y., 215 Kurokawa, T., 8 Kurz, H., 53 Kusumoto, T., 129 Kuwabara, M., 122, 133 Kuzmany, H., 99

AUTHOR INDEX

Kuznetsov, A. V., 431 Kwock, E. W., 141

L Laakso, J., 31 LaBlanc, A. R., 284 Lachinov, A. N.. 123 Ladan, F. R., 190 Lafarge, P., 308 Laflkre, W. H., 140 Laibowitz, R. B., 295, 296 Lake, R. K., 438, 439 Lake, R., 440, 446 Laks, D. B., 236 Lampert, M. A,, 76, 79, 80(237) Landauer, M., 300, 394(95) Landauer, R., 288,294,295,296,297,309(55), 310(74), 448 Landheer, D., 307 Landwehr, G . , 213 Lane, P. A., 57, 136(180) Lang. M., 215 Lanzari, Y., 242 Larkin, A. I., 405 Lary, J., 311, 312 Lasher, G . , 281 Launois, H., 285 Laux, S. E., 286, 287 Lawandy, N. M., 191, 192 Lawrence, R. A., 98, 99, 108 Lax, M., 310, 333, 335, 362, 380, 446, 447 Lazarroni, R., 73, 128(225) Lazzaroni, R., 62, 72 Le, H., 260 Leclerc, M., 22 Leditzky, G., 25, 120, 123, 124(77), 131(77) Lee, C. H., 60, 148 Lee, D., 215, 228, 229, 239(53), 250 Lee, J. K., 120, 123 Lee, M. W., 192 Lee, P. A., 286, 295(15), 296, 299, 309(15), 330(69, 80). 435(15), 440(80), 447 Lee, T.-K., 41, 42(134) Lee, W. M., 199, 203 Lee, Y. R., 215, 216 Lee, Y., 407, 438, 439 Leggett, A. J., 304, 362, 364(298) Lei, X. L., 287, 333, 334, 335, 347, 407 Leiderer, E., 230

459

Leisching, P., 431, 433(394) Leising, G . , 16, 2.5, 29, 30(100), 63, 120, 123, 124(77, 801, 131(77) Leitenstorfer, A., 431, 433(394) Lemmer, U., 53, 82(171), 83(171), 141 Lennard-Jones, J. F., 5 Lenstra, D., 306 Lent, C. S., 307 Lenz, R. W., 17, 19, 21 Leo, K., 53 Leung, K. M., 153, 154, 157(20), 160(33), 183, 184(73), 187(19), 188(20), 189(33), 196, 197(97) Levi, A. J. F., 294 Levinson, I. B., 381, 400(314) Lewenstein, M., 167 Lewis, J. T., 332 Lewis, J., 61 Li, D., 216 Li, H., 244, 252, 255(85), 256(85), 264, 276, 278 Li, Q., 440 Li, Y., 347, 370(279), 380 Licini, J. C., 296 Liedberg, B., 67 Lin-Liu, Y. R., 37, 4(x125), 41, 42(134) Lindberg, M., 429, 430 Lipavsky, P., 440, 444 Lipavslj, P., 288, 305, 310(119), 418, 419, 437(359), 442(119) Lischka, K., 215, 223 Littlejohn, M. A,, 302 Littlewood, P. B., 276, 280(123) Litz, T., 213 Liu, M., 347 Liu, X. C., 216 Liu, Y. F., 153, 157(20), 187(19), 188(20) Logan, R. A., 278, 279(124), 280(124) Lagdlund, M., 62, 72 Logue, F. P., 223 Lohner, A,, 431, 433(394) Lomascolo, M., 219, 272 Longuet-Higgins, H. C., 5 , 32(8), 36(8) Loponen, M., 31 Losch, R., 302 Louis, E. J., 4, 3 6 ) Louisell, W. H., 380 Lous, E. J., 88 Lovergine, N., 272 Lowe, D., 288 Lowe, R. D., 23, 30

460

AUTHOR INDEX

Lowisch, M., 272 Lu, F., 123 Lu, K., 214 Lu, X. J., 333, 444 Lu, X., 107 Lugli, P., 332, 408, 418, 419 Luk, K. H., 172, 197(66, 671, 203 Lukosz, W., 140 Luo, H., 207, 215, 216, 218, 219(38), 221(39), 229, 269, 271(114) Luttinger, J. M., 176, 316 Luyo, H., 223 Lymarev, A. A., 57 M

MacDiarmid, A. G., 4, 5(6), 6, 26 MacDiarmid, A. J., 41, 42(133) MacDonald, A. H., 301 Mach, R., 116 Mackay, K. D., 118 Mackay, K., 11, 12(31), 117(31), 122 MacKinnon, A. M., 154, 194(23), 195(23), 196 Macucci, M., 302, 311, 312, 442, 443 Madru, R., 111 Maehara, K., 244 Magnate, P., 11, 116(28) Magnea, N., 232 Mahan, G. D., 330, 334, 405(213), 406(213), 408,415(353) Mahler, G., 431, 433(394) Mahrt, R. F., 20, 53, 82(171), 83(171), 120, 122,125(313), 129(59,313), 130(313), 141 Maikh, M., 147 Mailhiot, C., 152 Mailly, D., 285 Mains, R. K., 400 Maitrot, M., 111 Majda, A., 390 Maki, K., 37, 40(125) Malliaras, G. G., 123 Mandel, L., 380 Mankiewich, P. M., 296 Mao, G., 31 Maradudin, A. A., 154, 190, 192 Marchetti, M. C., 335, 362 Marfaing, Y., 237 Margulies, R. S., 168 Mark, P., 76, 80(237) Markey, B. G., 214

Markowich, P. A., 317, 319, 320 Marks, R. N., 11, 12(31), 22, 60, 61(194), 68, 72(217), 74(217), 117(31), 118, 122, 125, 127(217), 129(217), 134(64), 142, 143, 146, 147, 148(194) Marktanner, J., 145 Marmorkos, I. K., 314 Marrocco, M., 120, 123, 131(320) Marseglia, E. A., 16, 29, 30, 31(102) Marshall, P. A,, 307 Marshall, T., 252 Martelock, H., 21 Martens, J. H. F., 16, 29, 30, 31(102), 52, 54(104) Martin, D. C., 29, 30(101) Martin, T., 301 Maruska, H. P., 115 Marzin, J. Y., 190 Masato, H., 215 MaSek, J., 309, 329, 330, 440, 448 Masse, M. A,, 29, 30(101) Mataloni, P., 168 Matsuda, Y., 215 Matsumoto, S., 253 Matsuoka, H., 308 Matsuoka, T., 287 Matyi, R. J., 306 Mauser, N., 317, 319 May, I. P., 14 Mazumdar, S., 37,41, 42(137), 45,49, 104(123) Mazuta, H., 306 McBranch, D., 20,68, 119(215), 123, 139, 148 McCaldin, J. O., 210, 250 McCall, S. L., 153, 162, 169 McCarty, D. M., 143 McCullough, R. D., 23, 30 McGill, T. C., 210, 215, 250 McGill, T., 215 McGinnity, M., 27, 116(92) McGurn, A. R., 190, 192 McKeever, S. W., 214 McLennan, M., 310 McLennan, M. J., 310,407,438,439 Meade, R. D., 154, 160(31,32), 161, 162, 163, 164, 190, 191, 192(86), 193, 194, 203 Mehring, M., 99 Meier, M., 68, 122, 126, 129, 146(327) Meir, Y., 295, 440, 447 Meissner, D., 144 Melngailis, J., 296

461

AUTHOR INDEX

Menke, D. R., 216 Mensz, P., 252 Merkt, U., 306 Mcrmin, N. D., 63, 151, 152(1), 157(1) Merritt, F. R., 278, 279(124), 28N124) Merz, J . L., 229 Metzger, W., 26 Meyers, F., 61 Michel, E., 163, 164(46), 192(46) Michizuki, K., 215 Michler, P., 280, 281(127) Migus, A,, 420 Miles, R. E., 287 Miles, R., 215 Miller, D. A. B., 429 Miller, D. R., 387 Miller, G. G., 6 Miller, T. M., 31, 49, 81, 83(242), 132, 141, 143(242) Milonnia, P. W., 166 Minagawa, Y., 30 Minh, L. Q., 74 Misra, B., 288 Mitchell, W. C . , 331 Mitsuhashi, H.. 236 Mitsuishi, I., 244 Mitsuyu, T., 115, 215, 233 Miyajima, T., 223, 242, 251, 252 Miyata, H., 345 Miyoshi, T., 391, 398, 402, 404 Mizes, H. A,, 46 Mizuta, H., 347 Mobius, M., 145 Modinos, A,, 154 Mohammad, F., 31 Mohammed, K., 216 Molenkamp, L. W., 301 Molina, M. I., 185 Moliton, A,, 108 Momii, T., 17, 19(47) Momose, H., 439 Mongre, J.-L., 129 Mooij, J., 308 Moon, Y. B., 6 Moore, T. M., 306 Moraes, F., 22 Moratti, S. C., 21, 22, 24, 25, 73, 83, 84(248), 107, 120, 122, 123, 124(78), 129, 132, 134, 141, 144(226), 148 Moratti, S., 46, 72(150) Morawitz, H., 140

Morgenstern Horing, N. J., 418, 430 Morgenstern-Horing, N. J., 408 Morhange, J . F., 429 Mori, H., 233, 244(63), 316, 326(182) Mori, N., 439 Mori, Y., 223, 244, 253, 254(84a) Morihima, C., 129, 13N337) Moringa, Y., 252 Moritz, A., 280, 281(127) Moroz, A., 196 Morrison, B. J., 353 Morrocco, M., 168 Morton, D. C., 120, 123 Moses, D., 129, 130(336), 147, 148 Mossberg, T. W., 167 Mott, N. F., 64 Moulton, J., 30 Moyal, J. E., 318, 380088) Mriman, A. M., 419 Mueller, G. O., 116 Mukai, T., 245, 253036) Mullen, K., 25, 120, 123, 124(81) Muller, G. O., 116 Muller, G., 302 Murase, I., 19 Murata, H., 17, 19(47), 108 Murmikko, A. V., 115 Muro, K., 119, 120, 123 Murphy, N. S., 6 Murra, D., 168 Murray, C. B., 232 Murray, K. A., 107 Murray, S., 319 Mysrovicz, A., 429 Mysyroviecz, A,, 425

N Naarmann, H., 109 Nagai, K., 120, 123 Nahory, R., 207 Nakamo, K., 253 Nakamura, S., 245, 253(86) Nakanishi, K. 215 Nakano, K., 244, 251, 253, 254(84a) Nakano, T., 20 Nakao, T., 236 Nakayama, N., 244, 253, 254(84a) Nakayama, T., 115, 217 Nakazato, K., 308

462

AUTHOR INDEX

Namavar, F., 115 Namba, S., 295 Nanai, N., 26 Nara, S., 8 Nazarov, Yu. V., 310, 448 Neikirk, D. P., 387 Neofotistos, G., 439, 446 Neukirch, U., 272, 274(119) Neumark, G. F., 236 Newbury, P., 216 Newman, E., 285 Ng, K. M., 185 Ni, Z., 20 Nickel, H., 302 Nier, F., 386 Niez, J. J., 332, 334 Nilsson, J. O., 67 Nishinaga, T., 347 Noguchi, T., 19, 20, 122, 133 Noolandi, J., 60 Nordholm, K. S. J., 331 Norris, D. J., 232 North, A. M., 33 Noteborn, H. J. M. F., 306 Nothe, A,, 217 Nourtier, A,, 322 Nurmikko, A. V., 207, 208, 215, 216, 218, 219(38), 218, 221(39), 223, 224, 225(49), 228, 229, 230, 231(57), 233, 234, 238, 239, 240(77), 241, 242(79), 243(79), 244, 244(11), 247, 248, 249, 250, 251, 255(85), 256,256(85), 257(104), 258(104), 260, 261(108), 263(108), 264, 265(104), 266, 269,270,271(114, 116), 273(116), 275(116), 276, 277(104), 278 0

O’Connell, R. F., 302, 314, 332, 334 O’Regan, B., 145 Oberski, J. M., 57, 107(187) Oberski, J., 20, 122, 129(59) Obrzut, J., 74, 106 Obrzut, M. J., 74 Ochse, A,, 141 Ogawa, M., 391,398, 402, 404 Ogino, E., 141 Ogura, M., 215 Ohata, T., 253 Ohguro, T., 285

Ohkawa, K., 208, 215, 233,235(9) Ohmori, Y., 119, 120, 123, 129, 130(337) Ohnishi, T., 19, 20, 122, 133 Ojeda-Castaneda, J., 380 Okamoto, Y.,123 Okawara, M., 16 Okhawa, K., 115 Okuyama, H., 213, 223, 244, 251, 252, 253, 265(484a) Okuyama, K., 120, 123 Olego, D. J., 215, 216, 218 Ono, K., 30 Ono, M., 285 Ono, Y., 42, 42(140, 1411, 98(141) Onsager, L., 331 Onsger, L., 60 Opila, R., 31 Orenstein, J., 54, 96(177) Orfila, P. F., 308 Osman, M. A., 380, 390(312) Ossau, W., 213 Osterholm, J.-E., 31 Otsuka, N., 215, 216, 228, 239(53), 244, 249, 252,253, 254(102), 255(85), 256(85), 260, 261(108), 263(108) Ovchinnikov, Yu. N., 405 Ozaki, M., 41, 42(133) Ozawa, M., 242, 244, 252, 253, 254(84a) Ozbay, E., 163, 164(46), 192(46)

P Paar, C., 25 Paasch, G., 129 Padjen, R., 190 Pai, D. M., 28 Pakbaz, K., 20, 54, 119, 120, 122, 128, 147 Paloheimo, J., 107, 108 Pankratov, O., 223 Pannetier, B., 295 Pantelides, S. T., 236 Paoli, T. L., 250, 251(105) Papadimitrakopoulos, F., 31, 81,83(242), 132, 143(242), 148 Paramenter, R. H., 79 Paranjape, V. V., 342 Park, D. H., 74 Park, K. T., 73, 126(228) Park, R. M., 208, 235(8) Park, Y.W., 6

463

AUTHOR INDEX Parker, C. D., 169, 305 Parker, D., 16 Parker, 1. D., 72, 88, 108(250), 115(250), 120, 123, 126(224), 131, 142 Patil, M. B., 287 Patterson, W., 208, 244(11) Pautmeier, L., 75 Pearson, C. E., 385 Peck, D. O., 305 Peeters, F. M., 333, 335 Peeters, F., 47, 334 Pei, Q., 120, 123, 131(320) Pei, Q.-B., 131 Peierls, R., 36 Pelekanos, N. T., 218, 221(39) Pelekanos, N., 216, 218, 219(38) Pendry, J., 154, 194(23), 195(23), 196 Peng, X. H., 1 I , 11l(24) Peng, X. Z . , 29, 111(98) Peng, X., 111 Permogorov, S., 227 Peschel, I., 41, 42(134) Petermann, J. H., 29, 30(101) Petrich, G. S., 214, 234, 236 Petrou, A., 216 Petruzzelo, J., 252 Pevzner, V., 442 Peyghambarian, N., 429 Pfaff, W., 448 Phil, M., 199 Phillips, M. C., 215, 250 Phillips, R. T., 83, 84(248) Phillips, S. D., 8 Phillips, S., 8 Piche, M., 152, 154, 169(8), 174(8), 175(8), 187(8), 190(39) Pichler, K., 16, 30, 51, 52, 54(104), 56(169), 68, 107, 108, 117, 131(302), 134, 135 Pine, D. J., 82 Pinhas, S., 296 Platzman, P. M., 153, 162, 169 Plihal, M., 154 Ploog, K., 302 Poelman, D., 140 Poli, P., 335, 428(245) Pommerehne, J., 141 Pomp, A., 88, 109(255), 1lO(255) Pope, M., 3, 1 I , 32(2), 47(2), 50(2), 51(2), 52(2), 82(2), 116(28) Porod, W., 303, 307 Posch, P., 126

Pothier, H., 308 Potz, W., 315, 380, 390(312) Powell, M. J., 144 Prener, J. S., 206, 233(2) Prete, P., 219, 272 PrStre, A,, 300, 310(84) Price, P. J., 287, 300, 303, 306, 447 Prigogine, I., 288 Prior, K. A,, 225, 270 Priz, G. A,, 216 Prober, D. E., 295, 296, 298 Prock, A., 140 Prosa, T. J., 30 Puls, J., 221, 272 Punkka, E., 107

Q Qadri, S. B., 215, 216 Qian, G. X., 154 Quade, W., 419 Qui, J., 208, 241, 244(10, 78), 251(78), 256, 259 Qui, Y . , 196, 197(97)

R Rajakarunanayake, Y.,215 Ramasesha, S., 46 Ramdas, A. K., 215, 216 Rammal, R., 295 Rammer, J., 322,323(204), 334(204), 405(204), 422(204) Randall, J. N., 306 Rangarajan, R., 187 Rappe, A. M., 154, 16(x31,32), 161, 162, 163, 164, 190, 191, 192(86), 193, 194 Raschke, M., 302 Ratier, B., 108 Rauscher, U., 53 Ravaioli, U., 287, 298, 302, 311, 312, 380, 390(312), 442, 443, 448 Reed, M. A,, 286, 306, 310(13) Reggiani, L., 330,335,408,418,419,428(245), 448 Register, L. F., 298, 310, 442 Rehahn, M., 24 Remler, E. A,, 319 Remling, C., 430, 433(383)

464

AUTHOR INDEX

Ren, J., 242, 247, 249 Resendes, D. G., 329 Reynolds, D. C., 47 Reznitsky, A,, 227 Rhoderick, E. H., 64 Rice, M. J., 35, 39(120), 45 Rice, T. M., 276, 280(123) Richert, R., 75 Rick, K., 431, 433(394) Rideout, V. L., 284 Rieb, W., 126, 129 Rieger, P. H., 62 Rieke, R. D., 23 Ries, S., 11, 29(25), 111(25) RieB, W., 122, 126 Riess, W., 68, 122, 126, 146(326, 327) Rikken, G. L. J. A,, 127 Rikken, G. L. J., 83, 84(246, 247), 120(246, 247), 122, 123 Rinaldi, R., 219, 272 Ringhofer, C., 385, 386(320), 387, 388, 390, 392(328), 396 Ringle, M. D., 260, 261(108), 263(108) Ringle, M., 223, 224(47), 238, 256, 257(104), 258(104), 260, 265(104), 266, 267(104) Rishton, S., 285 Risken, H., 380 Roberts, G. G., 27, 116(92, 93) Robertson, W. M., 162, 163, 164, 190(43, 44) Roghberg, L. J., 81, 83(242), 143(242) Rohrer, H., 304 Rooks, M. J., 295 Rosenberg, L. M., 79 Rosenfeld, L., 288 Rossi, F., 316, 335, 347, 361(180), 419, 430, 431, 433(252, 388, 394) Rossin, V. V., 221 Rossler, U., 293 Rota, L., 335, 428(245) Roth, L. M., 347 Roth, S., 99 Rothberg, L. J., 132, 141, 148 Rothberg, L., 84 Rouleau, C. M., 208, 235(8) Royer, A., 318 Rughooputh, S. D., 19 Riihe, J., 67, 68(213), 88, 103(213), 104(213), 105, 107(213) Rumbles, G., 53, 82, 83(170, 245) Runge, E., 220, 276, 277(122), 447

Ruppel, W., 79 Ryan, J., 285 S

Saad, Y., 287 Sachrajda, A. S., 306, 307 Sacra, A., 232 Sadoun, M. A., 111 Saggia, E., 186 Sai-Halasz, G. A., 285 Saito, M., 285 Saito, S., 17, 19(47), 28, 60, 108, 116, 130(96), 140, 141 Sakaki, H., 111, 293 Sakamoto, A,, 57, 107(185) Salamanca-Riba, L., 260 Salaneck, W. R., 31, 46, 61,62,72, 73, 128(225) Salbeck, J., 25, 120(81), 123, 124(81) Salehpour, M. R., 154, 187(25) Salem, L., 5, 32, 36(8) Salokatve, A,, 244,252, 255(85), 256, 257(104), 258(104), 260,265(104), 277( 104) Samarth, N., 207, 215,216,219, 221(39, 114) Samuel, I. D. W., 53, 82, 83, 84(248) Sander, R., 120, 122, 125(313), 129(313), 130(313), 141 Santhanam, P.,298 Sapochak, L. S., 143 Saraie, J., 241, 242(79), 243(79), 249 Sariciftci, N. S., 148 Sasai, M., 46 Sasaki, A,, 215 Sato, M., 22, 30 Sato, N., 62 Satpathy, S., 154, 172(26), 187(25), 188(26) Sawaki, N., 347 Scarlora, M., 168 Schaff, W. J., 285 Schaffer, H. E., 55, 96 Schein, L. B., 74 Schenk, H., 25, 120(81), 123, 124(81) Scherf, U., 25, 120, 123, 124(77, 81, 321). 13“7) Schetzina, J. F., 242, 249 Schetzina, J. S., 247 Schiebel, U., 144 Schikora, D., 215

AUTHOR INDEX Schilp, J., 335, 430(252), 433(252) Schimmel, T., 109 Schirber, J. E., 3, 4 6 ) Schlapp, W., 302 Schlenoff, J. B., 106 Schlip, J., 431, 433(394) Schliiter, A,-D.. 24 Schmidt, H.-W., 120, 122 Schmitt-Rink, S., 429, 430 Schneider, W. G., 11 Schoeller, H., 323, 347 SchoI1, E., 335, 419, 430(252), 433(252) Scholler, H., 448 Scholz, F., 280, 281(127) Schon, G., 310 Schottky, W., 64 Schrieffer, J. R., 34, 35, 39(118, 119), 46 Schrock, R. R., 16, 120, 123 Schroder, D. K., 377, 378(302) Schultz, S., 153, 162, 169 Schwalm, M. K., 331, 439, 440 Schwalm, W. A,, 331, 439, 440 Schwartz, L. M., 186 Schwoerer, M., 68, 109, 122, 126, 146(326, 327) Semaltianos, N. G., 245 Senoh, M., 245, 253(86) Sfiat, S., 289 Shacklette, L. W., 6 Shahzad, K., 215 Shambrook, A,, 154 Shank, C. V., 429 Shao, Z., 307 Sharvin, D. Yu., 295 Sharvin, Yu. V., 295 Shaw, M. P., 394 Sheard, F. W., 306, 396(122) Sheng, P., 154 Sherman, B., 168 Shih, C. C., 168 Shimizu, T., 30 Shinar, J., 57, 123, 136, 137, 138 Shirakawa, H., 4, 5, 6 Shirley, 1. M., 24 Shokin, Y. I., 385, 386(318) Short, S. W., 223 Shtrikman, S., 192 Shuai, Z., 49, 50, 136(163) Shum, K., 446 Siegel, J. L., 316 Sigalas, M., 163, 164, 192(46, 471, 195

465

Sigel, J. L., 316, 326(185) Silbey, R., 9, 46, 85, 140 Simmons, J . G., 67 Simon, J., 111 Simpson, J., 215, 270 Sitter, H., 215, 223 Sivananthan, S., 215 Skocpol, W. J., 296 Slater, J. C., 184, 18374) Slater, J. M., 14 Slater, J., 192 Smart, A. P., 230 Smith, D. L., 139, 152 Smith, D. R., 162 Smith, D., 153, 162(15) Smith, H., 322, 323(204), 334(204), 405(204), 422(204) Smith, P., 26, 30, 31, 142 Smith, T. B., 318 Smolowitz, L., 148 Snowden, C. M., 286, 287 Sokokik, I., 57, 120, 123 Sokolowski, M. M., 16, 29(42), 30(42) Sollner, T. C. L. G., 305 Solomon, P., 394 Sols, F., 301, 310, 311, 312, 442, 443 Sommers, C., 192 Soos, Z. G., 46 Sorba, L., 219, 220(41), 272 Sotomayor Torres, C. M., 230 Soukoulis, C. M., 153, 154, 164, 187(30), 188, 189, 192, 193(21), 195 Sozuer, H. S., 154, 187(34), 190 Spencer, G. W. C., 141 SpiEka, V.,288,305,310(119),329,418, 437(359), 442(119) Spoelstra, K. J., 83, 84(247), 120(247), 123 Sprangler, C. W., 56, 136(178) Srdanov, G., 20 Srdanov, V. I., 148 Srivastaqva, P., 11, 111(26) St. J-Russel, P., 152 Stafstrom, S., 72 Stampfl, J., 25 Stanton, C. J., 308 Staring, E. G. J., 83, 84(246, 2471, 120(246, 2471, 122, 123, 127 Stark, J. B., 429 Stavrinides, T., 234 Stefanou, N., 154 Stehlin, T., 25, 120(81), 123, 124(81)

466

AUTHOR INDEX

Steinruck, H., 320, 382 Stenger-Smith, J., 17 Stern, F., 281, 287, 392(19) Stern, R., 25, 120(81), 123, 124(81) Stewart, H., 225, 270 Stolka, M., 74, 76(232), 133, 146 Stolz, w., 431, 433(394) Stolzel, F., 145 Stone, A. D., 296, 330(69), 383, 440 Stopa, M. P., 306, 308 Stwneng, J. A., 304, 305, 310(119), 442, 444, 445 Stpcka, V., 444 Streda, P., 301 Street, G. B., 46 Streklas, A., 318 Strickler, S. J., 81 Strohriegl, P., 126 Stroscio, M. A., 289, 302, 342, 353, 374, 378 Stroud, D., 192 Strukelj, M., 132 Stubb, H., 31, 107, 108 Stucky, G., 148 Su, W. P., 47, 49(161) SU, W.-P., 34, 35, 39(118, 119) Suemune, I., 215, 223 Sugano, T., 308 Sugimoto, R., 22, 74 Sullivan, K. G., 168 Sum, U., 45, 46(144, 145) Sumetskii, M., 305, 307, 308, 310(120) Sundberg, M., 88 Sungki, 0..216 Sutherland, B., 170 Suzuki, M., 236 Suzuki, N., 41, 42(133) Svensson, C., 10, 88, 107(23) Swanson, L. S., 57, 123, 136, 137, 138 Swatos, W. J., 20 Swenberg, C. E., 3, 32(2), 47(2), 50(2), 51(2), 52(2), 82(2) Swenberg, J. F., 250 Swock, E. W., 49 Szafer, A., 383 Sze, S. M., 10, 64(22), 65(22), 69(22), 71(22), 72(22), 86(22), 91(22) T

Taguchi, T., 21.5 Taka, T., 31

Takada, N., 116, 141 Takagaki, Y., 311, 312, 345 Takailin, H., 129 Takayama, H., 37, 40(125) Takebayashi, K., 223 Takeda, E., 308 Takeda, S., 22 Takiguchi, T., 74 Taliani, C., 72 Tanaka, R., 60 Tanaka, S., 22 Tang, C. W., 28, 116, 130(94), 144, 148(383) Tani, J., 141 Tani, T., 8 Taniguchi, K., 287 Tanimoto, H., 287 Tannewald, P. E., 305 Tashiro, K., 30 Taskar, N. R., 236 Tasker, P. J., 285 Tasumi, M., 57, 107(185) Tauc, J., 41, 42(135, 136) Taylor, D. M., 14, 107(34) Taylor, P. C., 170 Taylor, R. P., 306, 307 Taylor, S. C., 24 Tenhoeve, W., 83, 84(247), 120(247), 123 Tennant, D. M., 296 Terai, A., 42, 98(141) Ththans, B., 46 Thim, H. W., 342 Thomas, D. G., 216, 217(1) Thomas, E. L., 29, 30(101) Thomas, H., 300, 310(84) Thomas, J. E., 214 Thompson, M. E., 143 Thornber, K. K., 288 Thornton, T. J., 308 , 440 Thouless, D. .I.309, Tiberio, R. C., 285 Tikhodeev, S. G., 431, 435(392) Ting, C. S., 333, 334, 347, 407 Tojyo, T., 244 Tokito, S., 17, 19(47), 28, 31, 60, 108, 116, 116(96), 130(96) Tokura, Y., 306 Tomiya, S., 253 Tomozawa, H., 8 Toombs, G. W., 306, 396(122) Tourillon, G., 22, 145

467

AUTHOR INDEX Townsend, P. D., 46, 49(151), 55, 56(151), 58, 59, 96, 97(258) Toyozawa, Y., 226, 22761) Tran Thoai, D. B., 335,430, 431(251), 433(3833 Treacy, G. M., 26, 142 Tripathi, V. K., 311, 312 Tristram-Nagle, S., 30 Troffer, M. B., 208, 235(8) Tsironis, G. P., 185 Tso, H. C., 402, 418, 430 Tsu, R., 304 Tsuchiya, H., 391, 398, 402, 403, 404 Tsurnara, A,, 8, 11, 14(17), 88(27), 107(17), 108(27), 111(17) Tsutsui, T., 17, 19(47), 28, 60, 108, 116, 130(96), 140, 141 Tuttle, G., 163, 164(46), 192(46)

U Uchida, M., 110, 120, 123, 129, 130(337) Ueno, H., 74 Uenoyama, T., 236 Uichlein, C., 217 Ullrich, B., 120, 123 Umbach, C . P., 295, 296 Umbach, C., 205 Underhill, A. E., 14, 107(34) Uneno, A,, 115 Urbina, C., 308 V

Valat, P., 129 Valeeva, I. L., 123 Valtchinov, V., 447 Van de Walle, C. G., 210, 215 van der Marel, D., 300 van Houten, H., 300 Van Meirhaeghe, R. L., 140 van Wees, B. J . , 300 van Welzenis, R. G., 335 Vancetti, L., 219, 220(41) Vande Walle, C . G., 236 Van Slyke, S. A,, 28, 116, 130(94) Vanzetti, L., 219, 272 Varani, L., 330 Vardeny, 2. V., 49, 57, 147 Vardeny, Z., 41, 42, 54, 96(177)

Vasanelli, L., 219 Vaubel, G., 116 Vawter, G. A,, 165 Velicu, B., 288, 329, 418, 437(359) Venhuiozen, T. H. J., 83, 84(247), 120(247), 123 Venhuizen, A. H. J., 83, 84(246), 120(246), 122 Verbing, S., 227 Veresegyhazy, R. K., 287 Vestweber, H., 20, 120, 122, 125(313), 129(59, 313), 130(313) Vickers, M. E., 29, 30(99), 60(99) Villeneuve, P. R., 152, 154, 169(8). 174(8), 175(8), 187(8), 190(39) Vincett, P. S., 27, 116(92, 93) Vlachos, K., 318 Vogl, P., 39, 42(130), 46(130) von Klitzing, K., 302 von Seggern, H., 120, 122 Voss, K. F., 20, 54(56) Vuorimaa, E., 107 W

Waag, A,, 213 Wada, Y., 42, 53, 82(171), 83(171), 98(141) Wagner, H. P., 230 Wagner, M., 305,306,308,322,405(203), 439 Wakirnoto, T., 116 Walecki, W., 216 Walker, C . T., 241, 244(78), 27(78) Walker, J., 236, 239, 240(77) Walsh, C . , 148 Wang, J., 169 Wang, M. W., 250 Wang, S. Y., 225 Waragai, K., 111 Warmenbol, P., 334, 335 Warnock, J., 216 Washburn, S., 295, 296 Watanabe, M., 140 Watt, E. J., 14 Watt, M., 230 Webb, R. A., 286, 295,296, 309(15), 435(15) Weckendrup, D., 272, 274(119) Wegner, G., 5, 22,24, 67, 68(213), 88, 103(213), 104(213), 105, 107(213) Wei, X., 49, 57, 147 Weibel, E., 304 Weibrecht, M., 144

AUTHOR INDEX

Weidlich, W., 380 Weinberger, B. R., 145 Weinmann, D., 447, 448 Weir, D. J., 14 Weiss, D. S., 3 Weiss, D., 302 Weiss, U., 286, 304, 435(14), 448 Weisshaar, A., 311, 312 Wendt, J. R., 165 Wessling, R. A,, 14, 16(36, 37) Westel, A. E., 306 Westenveele, E., 142 Whangbo, M. H., 3, 4(5) Wheeler, R. G., 296 White, J. D., 308 White, J., 308 Widawski, G., 100, 101 Wieczorek, H., 144 Wiersma, D. A,, 81, 140(243) Wigner, E., 289, 292(40), 318(40), 337(40), 342(40) Wilderman, J., 123 Wilkins, J. W., 288, 306, 308, 310, 419, 444, 446 Wilkinson, C. D. W., 230 Wilkinson, F., 32, 33 Willander, M., 10, 88, 107(23) Williams, E. W., 115 Williams, J. M., 3, 4(5) Williams, R. H., 64 Williams, S. P., 30 Williamson, J. G., 300 Wind, S., 295 Wingreen, N. S., 295, 310, 440, 447 Winokur, M. J., 30, 31 Winokur, M., 6 Wintgens, V., 129 Wisnieff, R., 296 Wittmann, H. F., 25, 61, 120, 123, 141 Wijhrle, D., 144 Wojtowicz, T., 239 Wolf, E. D., 285 Wolf, E., 187, 380 Wolk, J. A., 236 Woo, H. S., 106 Woo, J. W. F., 295 Wordeman, M. R., 284, 285 Worland, R., 8 Wright, G. T., 65 Wu, B. J., 256, 259, 260 Wu, C. Y., 26

Wu, G., 215 Wu, J. C., 312 WU,J.-W., 216 wu, x., 335 Wu. Yi. H., 218, 220(37) Wudl, F., 19, 20, 22, 119, 122, 128, 136(180), 148 Wybourne, M. N., 312 Wynberg, H., 83, 84(247), 120(247), 123

X Xhuai, Z., 47, 49(161) Xie, L. M., 334 Xie, W., 115, 208, 244(11), 247, 248(87), 251 Xin, S. H., 223 Xing, D. Y., 347 Xing, K. Z., 46, 72(150) Xu, B., 57, 112(184)

Y Yahlonovitch, E., 152, 153, 154, 155, 156(41), 157(41), 159(17, 18, 40, 160(17, 31, 33, 41), 161, 165(41), 167(6,41), 169, 189(33), 193(31), 194(31) Yakagaki, Y., 311 Yamada, T., 345 Yamaguchi, K., 302 Yamaguchi, S., 419, 435 Yamanishi, M., 215 Yamanoto, T., 123 Yamauchi, A,, 30 Yamazaki, O., 208, 233, 235(9) Yan, M., 81, 83(242), 84, 143(242), 148 Yang, Y., 115, 131, 142, 147 Yang, Z., 120, 123 Yano, K., 308 Yao, T., 223, 233, 234(63) Yassar, A., 11, 29(25), 57, 111(25, 26, 189), 112, 113, 114, 129 Yasuda, N., 287 Yasuda, T., 244 Yelesin, V. F., 430 Yi, K.-S., 334 Yin, A., 223 Yindeepol, W., 312 Yli-Lahti, P., 107 Yokayama, M., 141

AUTHOR INDEX Yokogawa, T., 215, 229 Yoshida, K., 223 Yoshihara, H., 223 Yoshimo, K., 119 Yoshimura, S., 26 Yoshimura, T., 308 Yoshino, K., 30, 74, 120, 123, 129, 130(337), 136 Yoshino, Y., 22 Yoshitomi, T., 285 Young, P. M., 179, 202(69), 220 Yu, E. T., 210, 215 Yu, G., 60, 147 Yu, H.-N., 284 Yu, L., 41, 42(134) Yu, Z., 247, 249 Yudanin, B., 310, 333 Yui, Y. Y., 185 Yvon, J., 346 Z

Zabel, L. H. H., 192 Zachariesen, F., 320, 382(197) Zaengel, T., 144 Zagorska, M., 22

469

Zajicek, H., 223 Zamboni, R., 72 Zawadski, P., 306, 307 Zdetsis, A,, 153, 187(16) Zegdrski, J., 26 Zhang, C., 119, 120,122, 123, 128,129, 130(336), 131, 142, 147, 148 Zhang, F. C., 216 Zhang, J., 315 Zhang, Z., 154, 172(26), 187(25), 188(26) Zheng, T. F., 310, 333, 335, 446 Zhou, J. R., 293, 338(50) Zhou, J.-R., 337, 342, 343, 345 Zhu, X., 276 Zhu, Z., 223, 233, 234(63) Zie, Z., 107 Ziegler, M., 220 Ziemelis, K. E., 62,67, 68,88, 103(213), 104, 105, 106, 107(213) Ziep, O., 439 Zimmerman, J., 332 Zimmerman, R. G., 14, 16(36, 37) Zimmermann, R., 430 Zolotukhin, M. G., 123 Zubarev, D. N., 320, 336 Zurek, W. H., 288 Zwanzig, R., 316, 325(181), 331

Subject Index

A

B

Absorption processes conjugated molecules Frenkel excitons and, 47 polarons, bipolarons and optical, 43 conjugated polymers, optical, 51-54 electronic band gap measurement using photon, 158 optically-pumped laser, 271 -272 organic molecules and electronic, 32-34 photoconductive and photovoltaic devices, 144-145 photoinduced conjugated polymers, 54-57 LEDs, 134-136 PPV and PT MIS diodes, 106-107 PPV and PT MIS diode optical, 104-107 thiophene oglimer field-effect devices, 112 Acceptors, see also doping in blue emitters, 245 charge-transfer salt, 4 nitrogen as substitutional, 235-236, 239-240 in Schottky-barrier diodes, 8-9 Accumulation region, see also Depletion region MIS diode, 89-90 polyacetylene MIS diodes, 98-99 width in EL devices, 67 Aharonov-Bohm effect, 295-296 Airy function retarded Green’s function, 420-425 Aligned polymer films, 30 Alkali metal-PPV interactions, 72 Ambipolar intrinsic Debye length, 66-66, 67 Amorphous sublimed films, 29 Anilines, polyaniline synthesis from, 26-27 Anisotropic semiconductors, 48-49 Anthracene, see also Conjugated molecules EL devices from thin films of, 116-117 as prototypical molecular semiconductor, 2-3 Anti-time-ordered Green’s function. 322-323

471

Balance equations, 336, see also Force balance equations Ballistic transport, 298-301 Band-gap engineering description, 152 tunneling barriers, 304 Band gaps, 158-159, see also Photonic band-gaps Band offsets anion rule and 11-VI heterostructures, 215-216 barriers created by conduction, 357 electronic structure calculations predicting, 217 lattice mismatch stain, accounting for, 210 Band structures Franz-Keldysh shift in bulk crystals, 290 versus periodicity, 151-152, see also Photonic band structure trans-polyacetylene (PA), 36 Barriers, amplitude, see also Double-barrier resonant-tunneling diodes; Schottky barriers multiple, structures of, 357-361 single dissipation and current flow calculations, 367-370 statistics of, 353-357 Biexcitons, optically-pumped lasers and, 272 (2-(4-Biphenylyl)-5(4-fert-butylphenyl)-l,3,4oxadiazole) PPV LED, improving efficiency of, 128 structure, 28 Bipolarons in conjugated polymers with nondegenerate ground states, 42-45 photoinduced absorption of PPV, 57 LEDs, excited states in, 137 PPV and PT MIS diodes, 103, 106 Bloch functions CMR representation expansion, 180 density matrix and, 316-318 photonic band-gap representations, 173 Wannier functions in terms of, 175

472

SUBJECT INDEX

Bloch states, 151-152 Blocking contacts definition, 65 PPV- and PT-based FETs, 111 Blue-green diode lasers, see also Blue-green light emitters confinement and band offset, 220-221 description, 250-259 excitonic processes in optically pumped, 269-217 gain spectroscopy, 277-281 low resistance contacts for cw,242 overview, 244-245 pulsed electrical injection, 208 surface-emitting, 264-267 Blue-green emission color, from LEDs, 119-124 Blue-green light emitters, see also Blue-green diode lasers degradation and defects, 259-264 doping and transport, 233-244 contacts to p-ZnSe, low-resistance, 241-244 overview and current status, 233-235 p-type of ZnSe and related compounds, 235-241 LEDs, description, 245-250 overview, 205-210 summary, 281-282 Blue-shifted emission color in LEDs, 119-124 Bohm potential particle interactions, 292-293 quantum energy level prediction, 337-338 Boltzmann transport equation description, 289-297 device quantization and, 286 dissipation and current flow calculations, 362-364 quantum derivations of, 334-336 statistical thermodynamics and quantum potentials, 291-293 versus quantum transport, 289-291 Bond alternations conjugated polymers nondegenerate ground states, 42-45 trans-polyacetylene chains, 37 polymer chain solitons and, 39-42 Boundaries, see also Contacts; Open systems density matrix equations of motion, 361 local equilibrium characterization, 347

numerical discretization and, 389-391 reservoir contacts and, 301-303 Boundary conditions partial differential equations for, in density matrices, 353 Wigner function modeling and, 382-384, 389-381 Brillouin zone Bragg scattering at boundary of, 151 full photonic band-gap, 153, 188, 202 Buried gate devices, 10, 11 C

Capacitance differential, 375-378 versus voltage in Schottky-barrier diodes, 8-9 Capacitive coupling, quantum dot effects and, 307-308 Capture, electron-hole blue LED, 245 PPV LEDs, 128 two-carrier currents, 77-80 Cavity quantum electrodynamics, 166, 202 CCR, see Crystal coordinate representation Charge carriers Boltzmann transport equation versus quantum transport of, 289-297 tield effect and, 85-86, 87-88 injected under nonequilibrium conditions, 70 photogeneration in conjugated polymers, 146-148 temperature-dependence of polyacetylene, 59 transport of, see also Quantum transport modeling single-carrier currents, 73-77 two-carrier currents, 77-80 Charge injection field emission, 71 of holes, polyacetylene devices, 92 in ITO/PPV/CA LEDs, 124-125 mechanisms for, in presence of barriers, 64 metal-semiconductor and metal-insulator contacts, 64-68 MIS diodes, 90 nonideal contacts, 72-73

SUBJECT INDEX into n-type ZnSe, 241 onto polymer chains with solitons, 41 thermionic emission, 69-70 Chargc stability on polymer chains, 7 Charge storage MISFETs, 41 polyacetylene devices PA, 87 soliton or mid-gap states, 91-92 Charge-transfer complexes donor and acceptor structures, 4 doped conducting polymers, 25 electrons from metallic calcium to PPV, 72 low-temperature ground states, 3-4 CMR, see Crystal momentum representation CN-PPV, see Dihexyloxy-poldp-phenylenevinylene) cyano-substituted derivative Coherent microwave transient spectroscopy, 162-163 Collisional-broadening equilibrium form of Green’s function, 413 homogenous, high-field systems, 417 Collisions, Green’s function and field effects on, 419 Color, controlling polymer EL emission, 118-124, 141 COMITS, see Coherent microwave transient spectroscopy Conductance of ballistic structures, 300 Green-Kubo formula and ac, 436-437 of localized tunneling/scattering barriers, 297-298 phase interference and, 294-297 universal fluctuations of, 419-420, 435 Conducting conjugated polymers description, 2-7 doping description, 25-27 films of intrinsically, 14 Conductivity, .set also Photoconductivity; superconductivity dark, 58-61 dissipation from scattering processes and electric field, 329-330 molecular semiconductor, 3-4 PBG materials, theoretical calculations for, 192 polyaniline, 26 PPV and PT-based FET, 108-110

473

Configuration coordinates of conjugated molecule energies, 32-34 Confinement, see Electronic confinement; Optical confinement Conjugated molecules, see also Conjugated polymers aromatic/heteroaromatic unit coupling, 37-38 Coulomb interactions, 37 A orbitals and ground states of, 38 electronic structure, 32-34 Frenkel excitons and, 47 SSH Hamiltonian, 36-37 Conjugated polymers, see also Conjugated molecules conducting description, 2-7 doping, 25-27 films of intrinsically, 14 device processes, 63-84 carrier transport, 73-80 charge injection, 64-73 exciton decay, 80-84 electroluminescence, used for, 121-123t field effect in, 84-89 materials doped conducting polymers, 25-27 molecular semiconductors and, 27-29 physical properties, 29-31 synthesis of semiconducting, 12- 15 optical and electronic properties, 32-63 photoconductive and photovoltaic devices, 144- 148 semiconducting chemical structures of common, 12-14 description, 2-7 organic, 7-12 synthesis of, 12-25 Conjugation-breaking defects, 15 Contacts, see also Boundaries; Electrodes; Open systems ballistic transport and, 298-301 boundaries and, 301-303, 378 metal-semiconductor and metalinsulator, 64-68 nonideal, 72-73 numerical discretization and, 389-391 open systems and, 297-298 p-ZnSe low-resistance, 241-244

474

SUBJECT INDEX

Continuum models FBC, 43-45 SSH modeling of polyacetylene in excited states, 37 solitons on polyacetelene chains, 39-41 Coulomb blockage regime definition, 307 density matrix simulation of, 361 Coulomb interactions bipolarons and optical transitions, 45 blue-green diode laser gain spectroscopy, 279-281 conjugated molecule properties, 37 dissipation in Wigner models and, 402 exciton description, 49 111-V versus 11-VI QW lasers, 268-269 quantum well quasi-2D electronic confinement, 217-218 UV absorption spectrum of PPV, 49 ZnSe-based QW lasers, 267-268 Crystal coordinate representation defect modes, photonic band-gap, 181-186 Green’s functions, 184-185 k . p theory vector analog, 170 Crystallinity of conjugated polymers, 29-31 Crystal momentum representation Bloch functions, 173 defect modes, photonic band-gap, 180 k . p theory vector analog, 170 Crystals electroluminescent, 116 Franz-Keldysh shift in bulk, 290 molecular conductor, fragility of, 4 PGB versus electronic, 154-155 photonic, see Photonic band-gap materials Current flow in density matrices, 361-370 Current versus bias voltage blue-green LED, 246-247 for Schottky-barrier diodes, 8, 9 D

Dark spot defects, 260-263 DBRTD, see Double-barrier resonanttunneling diodes DDQ, see 2,3-Dichloro-5,6-dicyano-1,4benzoquinone Defect modes 3D PBG materials, 160-162

losses in photonic gap, 167 photonic band-gap formalism, 181-186 single-mode cavities and, 166-167 Defects, see also Impurities bleaching Ramon spectrum for polyacetyline MIS diodes, 98 blue-green light emitters, 259-264, 281-282 bond alternation, polymer chain solitons and, 39-42 breaking of charge-conjugation symmetry and, 46 conjugated polymer, 15 device degradation, causing, 208-209 at metal-polymer interface, 72 PBG materials advantages of, 156 calculations, theoretical, 193-197 photonic band-gap formalism, 181-186 representing perturbations, 170 wide band-gap 11-VI semiconductor point, 224 Degradation, device blue-green light emitters, 259-264, 282 research on solving cw,208-209 A orbitals, conjugated molecule, 38 Density matrices, see also Quasi-equilibrium density matrices equal-time Green’s functions as mixed, 432 Green’s functions, 318-324 Langevin equation and, 332 Liouville and Bloch equations, 316-317 modeling using describing quantum transport using, 346-353 differential capacitance, 375-378 dissipation and current flow, 361-370 multiple-barrier structures, 357-361 overview, 346 statistics of single barriers, 353-357 uniform and nonuniform fields, 370-375 projection operators and reduced, 324-327 quantum distribution function and, 290 temperature and oscillatory behavior in, 349-353 wave functions and, 314-315 Wigner functions, 318-324, 378-404 Depletion region, see also Accumulation region in double-barrier resonant-tunneling diodes, 392-394

SUBJECT INDEX polyacetylene field-effect devices, 92-94 polyacetylene MISFETS, 101-102 width in n-type semiconductors, 65 in polymer EL devices, 67 Diagonal disorder, field mobilities and, 75 2,3-dichloro-5,6-dicyano1,4-benzoquinone FET doping, 88 PPV and PT-based FET doping, 108-110 Dielectric constant PBG material construction and, 157 periodic variations in, 152 in photonic band-gap wave equations, 171 in PWE method, 174 Dielectric interfaces, , LED, 140 Dielectric materials dielectric function expressing disordered, 179-180 nonlinear effect enhancement, 169 Differential capacitance, 375-378 Dihexyloxy-pol$ p-phenylenevinylene) cyano-substituted derivative in LEDs, 132-134 synthesis of, 21-22 Diluted magnetic semiconductors, 216 Diode lasers, see Blue-green diode lasers; ZnCdSe cw diode lasers Diodes, see also Laser diodes MIS, 89-90, see also Metal-insulatorsemiconductor devices resonant-tunneling, see Resonanttunneling diodes thin-film electroluminescence in, 11-12 N,N'-Diphenyl-N,N'(3-methylphenyl)-l,l'-

methynl-4,4'-diamine, structure, 28 Disorder model, 74-76 Dispersion relations calculations of photonic, 192 for EM wave propagation in PBG materials, 152 measuring PBG photonic, 162 photonic bands determining for specific, 178-179 minima and maxima, 157 Dissipation and current flow in density matrices, 36 1-370 femtosecond laser excitation and, 433 Wigner model and, 399-402 DMS, see Diluted magnetic semiconductors

475

Donors, see also Acceptors; Doping in blue emitters, 245 charge-transfer salt, 4 Doping, see also Impurities conjugated polymers conducting, 25-27 field effect devices, 87-88, 108-1 10 films of intrinsically conducting, 14 interfacial layer causation by, 72 ionization potentials and, 58 for LEDs, 67 organic semiconductors, 5-6 photoinduced absorption and, 57 poly(3-alkythiophene), 22 p-type, 67-68, 85-86, 87-88 density matrix and abrupt change in, 350-353 in multiple-barrier structures, 358 n-type with PPVs and sodium, 72 tellurium, 225-229 wide band-gap 11-VI semiconductor contacts to p-ZnSe, low-resistance, 241-244 overview and current status, 233-235 p-type of ZnSe and related compounds, 235-241 Double-barrier resonant-tunneling diodes, see also Resonant-tunneling diodes density matrix for Wigner distribution, 344 as multiple-barrier structure, 357-361 potential distribution for, 305f Wigner distribution and, 391-399 Wigner functions numerical discretization, 384-389 simulation of, 391-399 1D PBG materials, 157 1D waveguide, 153 2D PBG materials calculations, theoretical, 186-192 fabricating electron-beam lithrogrdphy, using, 164-165 for optical frequencies, 157 pump-probe investigations of, 160 2D ZnCdSe/ZnSe quantum wells, 217-225 3D PBG materials calculations, theoretical, 186-192 defect modes, 160-162 diamond-shaped, 188-190

476

SUBJECT INDEX

3D PBG materials (Continued) fabricating fcc, 160 for optical frequencies, 157 silicon wafers, using michromachined, 163- 165 full photonic band-gap, attempts to calculate for, 188-190 scalar wave approximation limitations, 187 Durham polyacetylene dark conductivity and photoconductivity, 58-59 MIS diodes of, 94 MISFETS of, 100-101 molecular scale studies of, 29-30 photoinduced absorption studies of, 54-57 Schottky-barrier diodes and, 9 synthesis, 15-16 Dyson equations, 420-425

E EA spectroscopy, see Electroabsorption spectroscopy E equation defect modes, photonic band-gap, 179-180 Kohn-Luttinger functions and, 176-179 photonic band-gap wave equations, 171-172 Effective mass representation defect modes, photonic band-gap, 180-181 Kohn-Luttinger functions and, 176-178 k p theory vector analog, 170 EHP, see Electron-hole plasma driven gain mechanism Elastic mean free path, 309 Electrical-optical efficiency of blue LEDs, 245 Electroabsorption spectroscopy, 54 Electrochemical polymerization, 14 Electrodes, see also Contacts energy barriers at polymer LED, 63 interface with polymers in FETs, 88-89 LEDs containing silicon injecting, 142 work functions dependency on measurement techniques, 63 versus Fermi energy of semiconductors, 64-65 polyaniline, 27

Electroluminescence conjugated polymers using, 121-123t Hiickel and SSH models compared, 46 turn-on rate for polymer LEDs, 129-130 Electroluminescence-detected magnetic resonance, 137 Electroluminescent devices, see also Bluegreen diode lasers; Blue-green light emitters; Light-emitting diodes accumulation and depletion region width, 67 conjugated polymer devices description, 117-118 diodes, 11-12 molecular semiconductors, 27-29 exciton formation in, 57, 81 two-carrier currents, 77-80 Electron-hole pairs excitons and, 47 in femtosecond laser excitation, 429-431 in inversion layers, 86 Electron-hole plasma driven gain mechanism, 268 Electronic confinement 11-IV QWs, spectroscopy of, 224-225 quantum well quasi-2D, 217-218 wide band-gap 11-VI heterostructures, 210-217 Electronic excitation-lattice coupling blue-green LED, 249 Te isolectronic centers, 225-229, 249 Electronic properties conjugated polymer and molecule description, 32-35 electronic structure, 35-39, 39-51 experimental results, 51-63 LEDs, 117-118 polyacetylene MISFETs, 102-103 PT and PPV-based FETS, 107-111 LEDs, single-layer, 124-130 regioregular versus regiorandom polymers, 23 Electronic states nonpertubative basis, 288 quantum well, 210-217 Electronic transitions conjugated molecule, 32-34 PPV excitation states in shorter conjugated segments, 52 singlet and triplet, 50

477

SUBJECT INDEX Electronic transport electron-hole capture, 77-80 materials used for, molecular, 28 T electron overlap in molecular semiconductors, 3 single-carrier currents, 73-77 Electron-phonon coupling Frenkcl excitons and, 50-51 optical, 226 trans-polyacetylene chains, 36-37 Electrons BIoch states, 151-152 inelastic mean free path of, 294 T , in anthraccne, 3 Electron-transporting layer, 130-132 Emeraldine phase, polyaniline synthesis, 26-27 Emission, simulated, ZnSe-based QW laser, 267-282 Emission color, LED, controlling, 118- 124 Emission processes conjugated molecules, Frenkel excitons and, 47 conjugated polymers, optical, 51-54 optical, blue emitter, 245 from organic molecules, 32-34 Emission spectrum LED, control of, 141 SCH/SQW devices, 256-257 Empirical tight binding method, photonicband structures, 199 EMR, see Effective mass representation Energy barriers, at polymer LED electrodes, 63 Energy gap, n and n* conjugated polymers, 13 of polyacetylene, 5 Epitaxy, see also Molecular beam epitaxy blue-green laser, 281-282 quantum dots, 231-232 quantum well wires, 231 Equations of motion current transport in quantum structures, 361-370 Heisenberg, for density matrices, 434 noninteracting forms of Green’s functions, 406 nonuniform fields and, 374 Poisson’s equation solution to, 350 Wigner functions numerical discretization, 384-389 and time-dependent, 381

Equilibrium form of Green’s function, collisional-broadening, 413 ETBM, see Empirical tight binding method Excim e rs nonradiative decay and, 82 polymers with gaps in blue or UV, 50 Exciplexes in polymers with gaps in blue or UV, 50 Excited states conjugated polymers and molecules excitons, 47-51 extensions of simple models, 45-47 in field effect devices, 86-87, 103 in LEDs, 134-137 nondegenerate ground states, 42-45 SSH model, 39-42 Exciton decay in conjugated polymers, 80-84 Exciton-phonon interaction, tellurium isoelectronic centers in, 225-229 Excitons blue-green LED, 247-248 conjugated polymer chains and, 47-51 conjugated polymers in EL devices, 57, 81 localization threshold and, 53-54 optically pumped lasers, 269-277 photoconductive and photovoltaic devices, 144-145 in polymer LEDs, 135-136 quasi-2D quantum wells, 217-218,267-268 self-trapping in deformable lattice, 226-229 singlet energy levels in nondegenerate ground state polymers, 44 PPV photoconductivity, 60 triplet in LEDs, 135-136 photoinduced absorption and PPV, 56-57 radiative lifetime, 81 ZnCdSe/ZnSe 2D QW, 218, 221-223 F

Face-centered-cubic structures 3D crystals, 160 investigations of PBG materials with, 155 FBC continuum model, 43-45 Fcc, see Face-centered-cubic structures

478

SUBJECT INDEX

Femtosecond laser excitation, 429-435 Femtosecond pump-probe spectroscopy, 271 Fermi-Dirac distribution density matrices and, 347-350 Wigner function modeling and, 383 Fermi energy levels charge transfer across metalsemiconductor/insulator contacts, 64-65 PPV dopant concentrations versus, 68 quasi-Fermi energies, 70, 366 Fermi’s golden rule, recombination in PBG materials and, 165-166 Field-effect devices, see also Metal-oxidesemiconductor field-effect transistors diodes, metal-insulator-semiconductor, 89-90 early attempts to develop, 8 MEH-PPV layers with polyaniline grid, 115 modulation of optical properties, 113-1 15 polyacetylene, 91-103 polymers, field effect in conjugated, 84-89 thiophene oligomer, 111-1 13 transistors, see also Field-effect devices; MIS (metal-insulatorsemiconductor) devices description, 90-91 polyacetylene and polythiophene derivatives in, description, 10-1 1 PPV and PT, 107-111 Field effects in conjugated polymers, 84-89 intracollisional, 288 Field emissions, charge injection and, 71 Field mobilities in disordered materials, 74-75 Films aligned polymer, 30 amorphous sublimed, 29 electron diffraction measurements on unstretched, 30 Langmuir-Blodgett, 27-28 large-scale morphology of polymer, 31 nonradiative decay and PPV, 82-83 polymeric photoconductive and photovoltaic devices, 145 sublimed molecular, 27-28 thin, see Thin films Finite-element method, 194-196 Fluctuation-dissipation theorem, 329-330

Fluorescence single-atom resonance, 167- 169 site selective, in conjugated polymers, 53-54 Fokker-Planck equation, 343 Force balance equations, 332-333 Formulas, conjugated polymer chemical, 13 Forster transfer process, 5 1 Fourier transforms, Wigner function and, 319 Fowler-Nordheim plots, 71 Franck-Condon transitions, 32-33 Franz-Keldysh shift in bulk crystals, 290 Frenkel excitons electron-hole pair location, 47 in molecular organic crystals, 50-51 Frequencies defect modes, EM field localization around, 166 optical fabricating 2D and 3D PBG materials for, 158 single-mode cavities and, 166-167 photon dispersion gaps in perfect PBG materials, 155-156 Fullerenes in photoconductive and photovoltaic devices, I48 G

GaAs, see Gallium arsenide Gain blue-green laser, 257 ZnSe-based QW laser, 267-282 excitonic processes, 269-277 optical resonances, 268 Gallium arsenide, components of total bond energy in, 234-235 Gallium arsenide lasers gain spectra of QW, 278 historical role of diode, 206 GaN covalent versus polar components of total bond energy, 234-235 LEDs from epitaxically-grown, 245 Gates lengths of MOSFET, 284-285 quantum effects, 294 quantum dots and potentials on, 306-307

479

SUBJECT INDEX Gaussian wave packets, DBRTD potential and, 387-389 Greater-than function, density matrices and, 322 Green-Kubo formula, 435-442, see also Kubo formula Green LEDs, attempts to fabricate, 250 Green’s functions density matrices and, 320-324 modeling using femtosecond laser excitation, 429-435 Green-Kuho formula, 435-442 high-field systems, homogenous, 417-429 low-field systems, homogenous, 406-417 overview, 404-406 resonant-tunneling diodes, 442-448 photonic band-gap impurities and, 184-186 quantum energy level prediction and exponential, 338-339 quantum transport and real-time, 335 retarded, 320-324 Grignard coupling route to poly(3alkythiophene), 22 Ground states of charge-transfer salts, 3-4 conjugated polymers and molecules extensions of simple models, 45-47 nondegenerate, 42-45 quantum chemical calculations, 38-39 tight-binding models, 35-38 vibrational. 33-34 H

Hakki-Paoli method, 257-258, 277 Hamil tonians differential overlap (INDO), 49 electron-phonon system response to time-dependent perturbations, 328-329 femtosecond laser excitation, 431-434 low-field systems, homogenous, 406 reduced density matrices, 324-326 SSH (Su, Schrieffer and Heeger), 35-38 valence effective, 38 Hartree-Frock semiempirical AM1 (Austen Model 1) technique, 38 Hcp, see Hexagonal-close-packed lattice HEMT, see High-electron mobility transistor

H equation defect modes, photonic band-gap, 179- 180 Kohn-Luttinger functions and, 176-179 photonic band-gap wave equations, 171- 172 Heterostructures, see also Wide band-gap 11-VI heterostructures photonic band-gap, 202 Hexagonal-close-packed lattice, 2D PBG, 190 High-electron mobility transistor quantization, 287 tunneling, 303-304 High-field systems, homogeneous Airy function retarded Green’s function, 420-425 description, 417-420 less-than function, 425-429 Hole concentration in p-ZnSe and p-ZnTe, 231, 239 Holes, see Capture, electron-hole HOMO levels, see Ionization potential, conjugated polymer Hiickel (tight-binding method) electronic structure and chain geometry of charged excited states, 46 T electron descriptions, 32 Hydrodynamic moment equations, 339-332 8-Hydroxyquinoline aluminum (Alq,) structure, 28

1

ICFE, see Intracollisional field effect Ideality factor for Schottky-barrier diodes, 8-9 Impurities, see also Defects; Doping charge-conjugation symmetry breaking and, 46 Green’s functions and photonic band-gap, 184-186 in PBG materials dielectric and air spheres, 193-194 nonlinear, 185-186 phase coherence and, 296 in poly(3-alkythiophene)s, 22 radiative decay in PPVs, 81 Incoherent 11-VI light emitters, 211 Indium-tin oxide electrodes, interfacial barriers in PPV LEDs, 72

480

SUBJECT INDEX

Inelastic mean free path ballistic transport over, 299 phase interference and, 294 Inelastic scattering in resonant-tunneling diodes, 445-446 Initiators, Durham polyacetylene and, 16 Insulator layer, MIS device description, 84-85 polyacetylene diodes, 94 Insulators, charge transfer across metal contacts and, 64-68 Interchain interactions polaron stabilization, 46-47 polymers with gaps in blue or UV, SO Interfacial barriers contaminated surfaces and oxygen, 73 effects on charge injection, 72 in PPV LEDs, 125-126 Internal photoemission spectroscopy, 127-128 Intracollisional field effect, 288 importance or lack thereof, 418-419 less-than function, 425 Intrinsic bistability in double-barrier resonant-tunneling diodes, 395 Intrinsic semiconductors band-bending versus trap density, 67 in LEDs, 67 trap-free, 65-66 Inversion layer MIS diode, 89-90 polyacetylene MISFETS, 101 p-type semiconductors, 85-86 Ionization potential, conjugated polymer doping and, 58 experimental results, 61-63 Isoelectronic centers of tellurium blue-green LEDs, 249-250 in ZnSe, 225-229 Isolated chains, models for infinite, 34 I T 0 electrodes, see Indium-tin oxide electrodes ITO/PPV light emitting diodes electrical characteristics, 124-125 multilayer, 132-133

K Keldysh matrix description, 405

Green’s functions as products of two carriers, 431 Kelvin vibrating capacitor method, 63 Kinetic transport functions Kubo formula and correlation functions, 327-331 relaxation/scattering versus streaming terms, 331 Knoevenagel condensation polymerization, 21 Kohn-Luttinger functions EMR representation expansion, 180-181 photonic band-gap theory, 176-179 Koster-Slater impurity model, 170 k * p formalism desgining PBG materials using, 197-199 dispersion relations near photonic bandgap, 157 Kohn-Luttinger functions and, 176-179 representations for vector analog of, 170 search for PBG crystal with optimum gap, 202 Kubo formula, see also Green-Kubo formula correlation functions and, 327-331 resonant-tunneling diodes and, 443-445

Landauer equation ballistic transport and, 299 conductance through regions with localized scatters, 294-295 quantized conductance experiments and, 300-301 Langevin equations, 327, 331-334 Langmuir-Blodgett films, 27-28 Large-area thin-film electronics, 2 Laser diodes photon-number-state squeezing, enhancing, 165 single-mode cavities and, 167 Lasers, see also Blue-green diode lasers femtosecond excitation of, 429-435 optically pumped excitonic processes, 269-277 room-temperature multiple-QW, 207, 269

SUBJECT lNDEX quantum well multiple, 21 1 separate confinement heterostructure, 21 1 ZnSe-based, 206 Lattice mismatch strain in band offset calculations, 210 design constraints, 212-213 ZnCdSe/ZnSe quantum wells, 220 Lattices reduced density matrices and, 324-325 self-trapping excitons in deformable, 226-227 Lax-Wendroff explicit time differencing method, 385-386 Less-than function density matrices and, 322 Green’s functions for modeling devices, 413-417, 425-429 Light-emitting diodes blue-green, see Blue-green light emitters color, control of, 118-124 q a n o group substitution, 21-22 device structures, novel, 142-143 doping levels of conjugated polymers for, 67 electroluminescence conjugated polymer, 117-118 generating in, 115-117 electron-hole capture in polymer, 77-80 energy barriers at electrodes in polymer, 63 excited states in, probing, 134-137 high-Q single-mode PBG microcavities and, 167 interfacial barriers in PPV, 72 molecular materials in, 28 multilayer, 130-134 optical properties, 137-142 SCL current in polymer, 77 single-layer, electrical characteristics of, 124-130 technological issues, 143-144 Liouville equation density matrices and, 316-317,318 differential capacitance, 376 dissipation and current flow calculations, 364-365 Lippman-Schwinger equation, 383-384 Localization threshold, conjugated polymer, 53

481

Low-field systems, homogeneous description, 406-408 less-than function, 413-417 retarded Green’s function, 408-413 Luminescence, 46, see also Electroluminescence LUMO levels, see Ionization potential, conjugated polymer M

Magnetic resonance electroluminescence-detected, 137 optically detected, 136 photoinduced absorption in PPV, 56-57 photoluminescence-detected, 136-137 Magnetoresistance, giant, 153 Materials conjugated polymer doped conducting, 25-27 molecular semiconductors, 27-29 physical properties, 29-31 synthesis of semiconducting, 12-25 electroluminescent, 116- 117, 117-1 18 photonic band-gap applications, 165-170 calculations and results, theoretical, 186- 199 conclusions and future directions, 199-202 experimental systems and results, 157-165 formalism, theoretical, 170-176 survey, 151-157 Matrix equations Keldysh function, 405 less-than function, Green’s, 413, 426 for PBG materials, 175 retarded Green’s function, 408 MBE, see Molecular beam epitaxy Mechanical properties, ZnSe crystal, 209 MEH-PPV, see Poly(2-methoxy-5(2’-ethylhexyloxy)-p-phenylenevinylene) Melting points, conjugated polymer, 6-7 Metal-insulator-semiconductor devices diodes, 89-90 polyacetylene, 92-100 PT and PPV. 103-107

482

SUBJECT INDEX

Metal-insulator-semiconductor devices (Continued) field effect transistors active layer materials, 29 early development of, 8 polyacetylene, 100- 103 polyDOT, in, 23 solitons and charge storage on polymer chains, 41 structure, 11, 84-85 of sexithiophene, 111- I12 Metallic conductivity, compounds with high, 3 Metallic Green’s functions, 442 Metal-oxide-semiconductor field-effect transistors, 284-285, 294 Metathesis initiators, 16 Microcavity organic LEDs, 141 Microcrystallites, conjugated polymer, 30 Microscopy, transmission electron, 260-263 MIS devices, see Metal-insulatorsemiconductor devices Mobility electron- hole polymer LED, 129 semiconductor, 73-75 field-effect PPV and PT-based FET, 107-108, 109- 111 thiophene oligomer devices, 111 photocarrier determining polyacetylene, 59 photoconductivity in PPV, 61 photovoltaic carrier, 145 polyacetylene MIS diode carrier, 94 Models, see also Quantum transport modeling electron-hole capture, 77-78 of hopping mobilities in disordered materials, 74-75 Koster-Slater impurity, 170 of polymer LED electrical characteristics, 124-130 Molecular beam epitaxy historical role of, 207 ZnCdSe/ZnS,Se heterostructures and, 216 ZnSe -type doping, 236 Molecular orbitals conjugated polymer electronic and vibrational excitations, 33, 47 localized versus delocalized, 34

Molecular organic LEDs multilayer, 130 optical properties, 139 oxygen and moisture effects, 143 Molecular organic materials electroluminescent blue emission from, 120 crystals, 116 Frenkel exciton theory, 50-51 thin-film EL in, 27-29 Molecular semiconductors extended T orbital fragments, 2-3 photoconductive and photovoltaic devices, 144-145 thiophene oligomer field-effect devices, 111-113 Moment equations applications in modeling devices, 342-346 for classical-quantum transition, 336-346 hydrodynamic moment equations, 339-332 Momentum variables, in density matrices versus position variables, 347-350 MOSFETs, see Metal-oxide-semiconductor field-effect transistors Mott-Wannier excitons electron-hole pair locations, 47 energy levels, 48 Multilayer light-emitting diodes description, 130-134 PPV/CN-PPV, 142-143 Multiple-barrier structures, 357-361 Multiple quantum well lasers, 207

N Nanostructures attempts to fabricate, 229-233 Green’s functions and study of, 448 Narrow-gap 11-IV semiconductors, 207 Nitrogen, ZnSe p-doping and, 235-236, 239-240 Nondegenerate ground states, conjugated polymer, 42-45 Nonideal contacts, 72-73 Nonlocality in quantum systems, 291-293 Nonluminescent dark defects, 259-263 Nonradiative decay exciton decay and, 80 Fermi’s golden rule, 166 inducing in conjugated polymers, 46

483

SUBJECT INDEX of photogenerated excitons in polydiacetylenes, 61 Nonradiative surface modes in PBG materials, 163 Normalization of Green’s functions, 404-405 N-type semiconductors, see also P-type semiconductors; Semiconductors chaTge injection into, 64-68 Schottky barrier between metal and, 70 Numerical discretization and solutions, 384-388 0

Off-diagonal diborder, field mobilities and, 75 Ohmic contacts definition, 6.5 depletion region width versus applied bias voltage, 66 p-ZnSe low resistance, 241-244 Oligomers field-effect devices of thiophene, 111-113 as molecular materials, 29 photoinduced absorption and, 57 Onsager relations ballistic transport and, 300 Langevin equation, and retarded, 331-334 Open systems, 297-298, see also Boundaries; Contacts Optical absorption in conjugated polymers, 51-54 Optical confinement of wide band-gap 11-VI heterostructures, 210-217 Optical emission, conjugated polymer, 51-54 Optically detected magnetic resonance, 136 Optically pumped lasers excitonic processes, 269-277 room-temperature multiple-QW, 207, 269 Optical properties conjugated polymer and molecule, 32-35 absorption and emission, 51-54 dark conductivity and photoconductivity, 58-61

experimental results ionization potentials, 61-63 photoinduced absorption, 54-57 polyacetylene MIS diodes, 95-96 PPV and PT MIS diodes, 104-107 field effect and, 87

field-effect devices MIS diodes, 90 thiophene oligomer, 111-1 13 LED, 137-141 PBG single-defect crystal with nonlinear, 185186 regioregular versus regiorandom polymers, 23 transitions Coulomb interactions and bipolarons, 45 solitons and, 40, 41-42 subgap, in presence of polarons and bipolarons, 44 Optical pumping of MQW lasers, 207, 269 Optical resonances in ZnSe-based QW lasers, 268-269 Optical spectroscopy, 217 Organic light-emitting devices, 137-139 Organic molecules, absorption and emission from, 32-34 Organic semiconductors, see also Conjugated polymers; Polymers; Semiconductors description, 7-12 high electronic conductivities of, 3 molecular, 27-29 Organic systems, exciton diffusion on, 51, 52 Oxidation in polyaniline synthesis, 26-27 Oxidation potentials, conjugated polymer, 62 P

PA, see Polyaniline: trans-Polyacetylene Pariser-Parr-Pople methods, 49 Partial differential equations boundary conditions in density matrices, 353 multiple barriers and single-particle density matrix, 3.58-361 Particles current and dissipation calculations with free, 366 projection of density matrices as single, 346 P3AT, see Poly(3-alkylthiophene) PBD, see (2-(4-Biphenylyl)-5(4-tertbutylphenyl)-1,3,4-oxadiazole) Peierls electron-phonon coupling model, 36-37 Periodic systems, wave propagation in, 151-152

484

SUBJECT INDEX

Perturbations CCR equation for slowly varying, 182-183 of density matrix diagonal elements, 372-373 dissipation and current flow, calculating, 362-365 of equilibrium density operator, 374-375 response of electron-phonon system to time-dependent, 327-331 Perturbation series, density matrices and, 321-322 Phase coherence, breaking, 309 Phase interference, quantum, 294-297 Phase kinks, see Solitons Phase-space Wigner distribution, 3 18-320 Phonon Green’s function for nonpolar optical scattering, 422-423 Phonon scattering, small system quantization and, 293 Phosphors, 206 Photocarrier mobility determining polyacetylene, 59 PPV, photoconductivity experiment results, 61 Photoconductive devices, molecular, 144- 145 Photoconductivity in conjugated polymers, 58-61 in p-doped ZnSe, 238-239 Photoconductors, molecular materials as, 3 Photocurrent spectrscopy, 219 Photoemission spectroscopy internal, 127-128 ultraviolet, 61-62, 72-73 X-ray, 72-73 Photoexcitation, charge generation through, 3 Photoinduced absorption conjugated polymers, 54-57 LEDs, 134-136 PPV and PT MIS diode, 106-107 Photoluminescence, 107, see also Electroluminescence Photoluminescence-detected magnetic resonance, 136-137 Photoluminescence efficiency nonradiative decay and, 81, 82 in PPV LEDs, 120 quantum well, 224 quantum well wire geometries, 230-231 Photonic band-gap materials applications, 165-170 for disordered, 169-170

proposed, 155 recombination, suppression of electron-hole, 165-166 single-mode cavities, 166-167 calculations and results, theoretical, 186-199 conclusions and future directions, 199-202 designing, k * p method for, 197-199 experimental systems and results, 157-165 fabrication techniques, microwave frequency range, 159-160 formalism, theoretical, 170-176 heterostructures, 202 muhidimensional, attempts to design, 154 PBG crystals defect with nonlinear optical properties, 185-186 versus electronic crystals, 154-155 pseudo-gaps in, 169-170 theoretical calculations for 2D and 3D, 156-157, 186-192 photonic properties, calculating, 186-199 survey, 151-157 Photonic band-gaps defect modes, 179-186 definition, 152 full 3D diamond-shaped structure, 188-190 2D hcp lattice structure, 190-192 experimental challenges, 202 fabricating 2D and 3D materials with, 157-158 strong-field resonance fluorescence sideband suppression, 167-168 Photonic band-gap theory calculations and results, 186-199 defects, 179-186 crystal coordinate representation, 181- 186 effective mass representation, 180-181 representation theory Bloch functions, 173 Kohn-Luttinger functions, 176-179 k p theory vector analog to, 170, 176-179, 202 plane-wave expansion, 173-175 Wannier functions, 175-176 wave equations, 171-172

SUBJECT INDEX Photonic band structure 2D and 3D PBG crystals, 156 definition, 152 versus electronic, 156 equations for calculating, in PBG materials, 172 ETBM method and, 199 Photonic crystals, see Photonic band-gap materials, PBG crystals Photonic dispersion, 162 Photonic multiple-scattering theory, 196- 197 Photon-number-state squeezing, 165 Photons in disordered PBG materials, 169 simple model prediction for energy emitted, 46 Photovoltaic devices, molecular, 144-145 Photoxidation, PPV, 31 Physical properties of conjugated polymers, 29 ZnSe and alloys, 209 T electron bonding in conjugated molecules and polymers, 32 extended, conjugated polymer, 6 T electrons, Huckel (tight-binding) method and, 32 T orbitals conjugated polymers, interchain charge transport, 39 semiconducting and conducting properties, 2 P T * energy gap conjugated molecules formed by aromatic/heteroaromatic unit coupling, 37-38 SSH Hamiltonian, 36-37 conjugated polymers, 13 lowering, for Durham polyacetylene, 30 polyacetylene, 5 quantum chemical calculations, 38 thiophene oligomer devices, 111 Planar antennas, PBG materials, 169 Plane-wave expansion method defects, calculating properties of PBG, 1935194,202 experimental results and numerical, 201 photonic band-gap representations, 173- 175 scalar and vector wave equation solutions, 186-187 Plane waves, infinite set of, 186-187

485

PLDMR, see Photoluminescence-detected magnetic resonance PL efficiency, see Photoluminescence efficiency Poisson’s equation device transport models and, 286 differential capacitance, 376-377 equation of motion solution of density matrix, 350 Polaron model of hopping mobilities in disordered materials, 74-75 Polarons, see also Solitons in conjugated polymers charge injection and, 73 FBC one-electron description of, 49 interchain interactions and stabilization of, 46-47 with nondegenerate ground states, 42-45 photoinduced absorption of PPV, 57 PPV and PT MIS diodes, 103, 106 Polyacetylene continuum modeling bond alternation defects and solitons, 39-41 of excited-state chains of, 37 difficulty of synthesizing, 14 discovery of high conductivity of, 4 Durham-route dark conductivity and photoconductivity, 58-59 photoinduced absorption studies of, 54-57 precursor, 15-16 Schotthybarrier diodes and, 9 experimental observations of optical properties, excitations, 51 photoconduction and photovoltaic efficiencies, 145 physical properties, 31 production of, 5-6 quantum chemical calculations, 39 structural types, 4-5 trans-Polyacetylene bond alternation, senses of, 35 charge storage and field effect, 87 formula and energy gap, 13 infinite one-dimensional chain model, 34 photoinduced absorption studies of, 54-56 tight-binding models, 35-37

486

SUBJECT INDEX

Polyacetylene field-effect devices charge storage in mid-gap states, 91-92 FETs, 100-103 MIS diodes, 92-100 Poly(alkathiophenes), 119 Poly(3-alkylthiophene) formula and energy gap, 13 H:H, H:T, T T linkages, 23 microcrystalline structure, 30 p-doping, 67-68 producing, 14 synthesis of, 22-23 Polyaniline electrode work functions, 27 field-effect devices with grid of, 115 in flexible LEDs, 142 formula and energy gap, 13 synthesis of conducting, 26-27 Polydiacetylene formula and energy gap, 13 photoconductivity in crystalline, 60-61 Poly(2,S-dialkoxy-p-phenylenevinylene) formula and energy gap, 13 producing, 14 in soluble PPV derivative synthesis, 18-20 Poly( P’-dodecyloxy-a a , -a’, cY”terthieny1) DDQ-doped FETs, 108-109 structure, 22-23 PolyDOT,, see Poly( P’-dodecyloxy-cY a , -a a” terthienyl) Polymeric photoconductive and photovoltaic devices, 145-148 Polymerization CN-PPV synthesis, 21 intrinsically conducting polymer films, 14 to produce MISFETs, 8 Polymer LEDs angular distribution of light emitted, 140 energy barriers at electrodes, 63 excitons, 135-136 flexible substrates, 142 hole mobility, 129 silicon substrates, 142-143 technological issues, 143-144 transient response, 129-130 Polymers, 23, see also Conjugated polymers I ,

Poly(2-methoxy-5(2’-ethyl-hexyloxy)-pphenylenevinylene) absorption and emission spectra, 54 LEDs, 118, 126

Poly(2-methoxy-5(2’-ethyl-hexyloxy)-pphenylenevinylene) (MEH-PPV), structure, 20 Poly(pheny1-phenylenevinylene), structure, 20 Poly(2-phenyl-p-phenylenevinylene),photoconductivity, 60 Poly( p-phenylene), blue EL from, 120- 124 Poly( p-phenylene) PPP formula and energy gap, 13 synthesis description, 23-25 difficulty of, 14 Poly(p-phenylenevinylene), see also PPV devices benzenoid and quinoid forms of, 43 conductivity of, 60 difficulty of synthesizing, 14 dispersive transport in, 75-76 excitons and, 48-49 formula and energy gap, 13 intrinsic ambipolar Debye length for, 67 ionization potentials, 61-63 mechanical properties, 31 microcrystalline structure, 30 molecular scale studies of, 29 nonideal contacts and, 72 nonradiative decay, 81, 82-83 optical absorption and emission, 51-52 photoconduction and photovoltaic efficiencies, 145 photoinduced absorption, 56-57 PL efficiency, 81-84 precursor-route, description, 16-18 quantum chemical calculations, 39 quenching of luminescence in, 143 synthesis p-doping and, 67-68 of soluble derivatives, 18-22 Polypyrrole (PPy) formula and energy gap, 13 synthesis, 25-26 Polythiophene (PT) difficulty of synthesizing, 14 FETs, 107-1 11 formula and energy gap, 13 MIS diodes, 103-107 sexithiophene oligomer, 29 synthesis of poly(3-alkylthiophene)s, 22-23 Poly(2,5-thiophenevinylene), formula and energy gap, 13

SUBJECT INDEX PPV, see Poly( p-phenylenevinylene); PPV devices PPV devices diodes carrier photogeneration in, 147-148 electroluminescent, 12 MIS, 103-107 FETs, 107- 111 LEDs, 117-118 electrical characteristics, 124- 130 emission color, 118-120 multilayer, 130-134 photoinduced absorption studies, 134- 136 PLDMR studies, 136-137 Precursor polymers, thin films from, 14 Precursor routes Durham, to polyacetylene, 15-16 to polfip-phenylene), 24 in soluble PPV derivative synthesis, 19-20 sulfonium polyelectrolyte, 16-17 in soluble PPV derivative synthesis, 19 Processibility, PPP improvements to, 24-25 Projection operators, 324-327, 346 Protonation, 26, 27 P-type semiconductors, 85-86, see also N type semiconductors; Semiconductors PWE method, see Plane-wave expansion method Pylenes, 4-5, see also Polyacetylene P, orbitals, conjugated polymer, 14, 32

Q Q factor, PBG material defect modes and, 167 Quantization in devices, 286-288 as overriding property of small systems, 293 Quantum chemistry of charged excited states, 45-46 of ground states, 45-46 Quantum devices Boltzmann transport, differences from description, 289-291 phase interference, 294-297 statistical thermodynamics and quantum potentials, 291-293

487

moment equations and modeling, 342-346 open systems and contacts ballistic transport, 298-301 boundaries and contacts, role of, 301-303 description, 297-298 overview, 283-286 potentially important overview, 303-304 resonant-tunneling diodes, 304-306 quantization in, 286-289 quantum dots, 306-308 Quantum dots description, 306-308 epitaxy, 231-232 fabrication of, 229-233 resonant-tunneling diodes and, 306, 447-448 Quantum dynamics, transition from classical to, 288 Quantum effects in semiconductors, 284-286, see also Quantum transport modeling Quantum efficiency photoconductive and photovoltaic device, 146- 147 photoluminescent, 81-84 Quantum equations Boltzmannn-like approaches, 334-336 density matrices and related functions, 314-327 Kubo formula and Langevin equations, 327-334 moment equations for classical-quantum transition, 336-346 overview, 308-314 Quantum kinetic equation, 327 Quantum potentials for GaAs MESFETs, 344-346 hydrodynamic moment equations and, 341-342 in multiple-barrier structures, 357-361 quantum distribution and nonlocal, 337 statistical thermodynamics and, 291-293 statistics of single barrier, 353-357 Quantum trajectories, ballistic transport and, 298-301 Quantum transport modeling density matrices, 346-378 alternative approaches, 370-375 differential capacitance, 375-378

488

SUBJECT INDEX

Quantum transport modeling (Continued) density matrices (Continued) dissipation and current flow, 361-370 multiple-barrier structures, 357-361 single barrier, statistics of, 353-357 devices Boltzmann transport, difference from, 291-297 open systems and contacts, 298-303 overview, 283-286 potentially important, 303-308 equations, quantum, 308-346 Boltzmann-like approaches, 334-336 density matrices and related, 314-327 Kubo formula and Langevin equations, 327-334 moment equations, classical versus quantum, 336-346 Green’s functions, 404-448 femtosecond laser excitation, 429-435 Green-Kubo formula, 435-442 high-field systems, homogenous, 417-429 low-field systems, homogenous, 406-41 7 resonant-tunneling diode, 442-448 open systems and contacts, 297-303 ballistic transport, 298-301 boundaries and contacts, 301-303 overview, 283-286 quantization in devices, 286-288 quantum devices, 303-308 quantum dots, 306-308 resonant-tunneling diodes, 304-306 Wigner distribution, 378-404 dissipation, role of, 399-404 equation-solving methods, 381-391 resonant-tunneling diode, doublebarrier, 391-399 Quantum well devices, see also Quantum dots; Quantum wires blue-green diode lasers, 253-254 blue LEDs, 246, 254 lasers developmental milestones, 206 gain and simulated emission of ZnSebased, 267-282 lattice mismatch strain, 212-213 Quantum wells DBRTD charge storage, 395-396 development of, 207 double-barrier structures in, 358

electronic states confinement and band offsets, electronic, 210-217 ZnCdSe/ZnSe 2D, 217-225 excitons quantum wires and dots, 229-233 tellurium isoelectronic centers and exciton-phonon interaction, 225-229 quantum dots and, 307 in resonant-tunneling diodes, 305-306 ZnCdSe/ZnSe 2D quantum wells, 217-225 Quantum wires fabrication of, 229-233 modeling using real-space lattice Green’s functions, 440 Quasi-equilibrium density matrices, 333 QW, see Quantum wells

R Radiative decay exciton decay and, 80 optical interference in LEDs and, 140-141 PPV LEDs, 118 SSH model and, 46 Raman-active modes in polyacetylene MIS devices, 97-98 Real-space lattice Green’s functions, 440 Recombination, electron-hole blue-green LEDs, 248-249 electronic confinement and, 211 suppressing in PBG materials, 155-156 in semiconductors, 165-166 two-carrier current devices, 78-80 Recursive Green’s functions, 440 Red shifts, conjugated polymer, 52-53 Reduction potentials, 62 Refractive indices, organic LED, 137-139 Regional approximation method, 80 Regiorandom polymers, 23 Regioregular polymers Poly(3-alkythiophene)s, 30 synthesis and advantages, 23 Relaxation of vibrationally excited conjugated molecules, 34 Relaxation-time approximation, 399-400

SUBJECT INDEX Reservoirs conductance of localized tunneling/scattering barriers, 297-298 projection operator and dissipative processes, 362 Resistivity, calculating force balance equations using, 333 Resonance fluorescence, 167-169 Resonant processes, Forster transfer and, 51 Resonant-tunneling diodes, see also Double-barrier resonant-tunneling diodes description, 304-306 modeling using Green’s functions, 442-448 quantum dots in, 307 Wigner distribution and double-barrier, 391-399 Ring opening metathesis polymerization initiator, 15 RO-PPV, see Dihexyloxy-poly(p-phenylenevinylene) cyano-substituted derivative

S Scalar wave equations 3D photonic band structures, 172 plane-wave expansion of positiondependent fields, 186-187 Scattering inelastic, in resonant-tunneling diodes, 305, 400-401 self-energy and, 407 transport in presence of localized, 309-310 Scattering terms Boltzmannn equations dissipation and current flow, 362-364 quantum effects in, 335 versus streaming in kinetic transport functions, 331 SCH, see Separate confinement heterostructure Schottky-barrier diodes, 8-10 Schottky barriers definition, 65 forming against metals, 8 ITO/PPV/Al LEDs, 126-127 thermionic emission and, 69, 70, 126, 127 tunneling rate through, 71

489

Schroedinger equation PGB versus electronic crystals and, 154-155 quantum mechanical semiconductor modeling and, 308-310 quantum waveguides, 310-314 SCH/SQW devices, 250-258 Se1f-en ergy Airy function retarded Green’s function, 422 for electron-phonon and impurity scattering, 417-418 Green’s function and, 406-407, 415-416, 427 Semiconducting polymers description, 2-7 quantum chemical calculations, 38-39 Semiconductors, see also Wide band-gap 11-VI semiconductors band-gap engineering, 152 charge injection, 86 charge transfer across metal contacts and, 64-68 chip density, implications of increasing, 283-286 inorganic, LEDs with p-n junctions of, 115 intrinsic band-bending versus trap density, 67 in LEDs, 67 trap-free, charge injection and, 65-66 molecular, 2-3, 27-29, 111-113 n-type, Schottky barrier between metal and, 70 quantum effects in, 284-286 quantum transport modeling, see Quantum transport modeling Semi-insulators, 233, see also Wide band-gap 11-VI semiconductors Separate confinement heterostructures, 211 Separate confinment heterostructure devices blue-green diode lasers, 250-251 lattice mismatch strain, 212-213 SET, see Single-electron tunneling Sexithiophene, 29, see also Oligomers D bonding in conjugated molecules and polymers, 32 Signal propagation, control of IC, 169 Silicon electroluminescence from, attempts to create, 115 LED substrates of, 142

490

SUBJECT INDEX

Silicon dioxide, 94 Simple models, 45-47 Simulated emission, ZnSe-based QW laser, 267-282 Single-atom resonance fluorescence, 167-169 Single-electron tunneling, 307-308, see also Tunneling; Tunneling time Single-layer light-emitting devices, 124-130 Single-mode cavities, 166-167 Site-selective fluorescences, 53-54 Solid conjugated polymers exciton extension over molecular units, 48 exciton motility in, 50 Solitons, see also Polarons charge storage in polyacetylene devices, 91-92 in conjugated polymers with nondegenerate ground states, 42 measuring mass of charged Durham polyacetylene, 56 in Peirels-distorted polymer chains, 39-42 in photoinduced absorption of Durham polyacetylene, 55-56 polyacetylene MIS devices, 97-98 Solubility conjugated polymers PPP, 24 PPV derivative synthesis and, 18-22 problems, 6-7 Solution-processed polymers, 29-30 Space charge injection effects versus, 63-64 limiting single-carrier current, 76 two-carrier currents, 77, 78, 79 Space-charge-limited (SCL) current, 76-77 Spectral function less-than Green’s function, 425 in quantum transport, 289-290 Spectroscopy blue-green diode laser gain, 277-281 coherent microwave transient, 162-163 femtosecond pump-probe, 271 11-VI QWs, electronic confinement and, 224-225 optical, 217 of polyacetylene MIS devices, 96-100 SSH (Su, Schrieffer and Heeger) model conjugated polymer electronic structure excited states, 39-42 ground state, 35-38 implication of radiative decay, 46 site-selective fluorescence, 54

Stacking faults, 263 Stark ladder effects, 290 Statistical thermodynamics, quantum potentials and, 291-293 Sublimed molecular films, 27-28 Substrates flexible, transparent LED, 142 for 11-IV heterostructures, choice of, 2 13-214 Sulfonium polyelectrolyte precusor route to PPV, 16-17 Supercell method, 193-194 Superconductivity in charge-transfer salts, 3-4 PBG materials, theoretical calculations for, 192 Superlattices development of, 207 electronic structure, 217-225 quantum dot arrays, 308 Stark ladder effects, 290 Surface-emitting lasers, 264-267 Synthesis of conjugated polymers conducting polyaniline, 26-27 polypyrrole (PPy), 25-26 PPV precursor routes, 16-18 semiconducting description, 12-15 Durham precursor route to polyacetylene, 15-16 poly(3-alkylthiophene)s, 22-23 polfi p-phenylenels, 23-25 soluble PPV derivatives, 18-22 T

Te, see Tellurium isoelectronic centers; Transverse electric polarization Tellurium isoelectronic centers blue-green LEDs, 249-250 in ZnSe and exciton-phonon interactions, 225-229 TEM, see Transmission electron microscopy Temperature ballistic electron studies and, 301 dark conductivity versus, 58-59 field emission and, 71

491

SUBJECT INDEX hopping mobilities in disordered materials, 74-7s low excitonic processes in optically pumped lasers, 269-277 quantum distribution function, 290 oscillatory behavior in matrices and, 349-350 p-ZnSe hole concentration and, 237, 239 QW photoluminescence efficiency and, 224 Tetrahydrothiophene precursor, 16-18 Tetrathiafulvalene tetracyanoquinodimethane compound, 3 Thermionic emission across Schottky barriers in PPV LEDs, 126, 127 in charge injection processes, 69-71 Thin-film electroluminescence (EL) of conjugated polymers, 11-12 of molecular organic materials, 27-29 Thin films EL devices from anthracene, 116-117 from PPV, 117-118 photoconductive and photovoltaic devices, 144-145 of polymers soluble in final form, 18 producing from conjugated polymers, 14 thiophene oligomer devices, 111 Thiophene oligomer field-effect devices, 111-1 13 Tight-binding models conjugated polymer, 35-38 Frenkel excitons in molecular organic crystals, 50 photonic-band structures, 199 Time-ordered Green’s function, 322-323 Time variables, Green’s functions, 404 TM, see Transverse magnetic polarization TMTSF, superconductivity of, 3-4 TPD, see N,N’-Diphenyl-N,N’[3-methylphenyl]-l,l ’-methynl-4,4’-diamine Transient behavior of double-barrier resonant-tunneling diodes, 396 Transient response, polymer LED, 129-130 Transistors, see also Field-effect devices, transistors; Metal-insulatorsemiconductor devices, field effect transistors bipolar, 165 high-electron mobility, 287, 303-301

Transmission electron microscocopy, 260-263 Transport, transition from quantum to classical, 335-336 Transverse electric polarization 2D PBG system eigenmodes, 190-192 k . p method and, 199, 200f wave versus scalar equations, 172 Transverse magnetic polarization 2D PBG system eigenmodes, 190-192 k * p method and, 197-199, 201f wave versus scalar equations, 172 Trap-free intrinsic semiconductors,’ insulators charge injection and, 65-66 space-charge-limited current, 76 Trapping sites band-bending versus number of, 67 PPV, 67 trap-filled limit (TFL), 76-77 Triplet excitons photoinduced absorption and PPV, 56-57 radiative lifetime, 81 Truncation errors, plane-wave, 187 TTF-TCNQ, see Tetrathiafulvalene tetracyanoquinodimethane compound Tunneling, see also Field emission Frenkel exciton, 51 in high-electron mobility transistors, 303-304 PPV LEDs, 126 single-electron, 307-308 Tunneling barriers description, 304 modeling single, 342-343 Tunneling time definition, 304 in double-barrier resonant-diodes, 398-399, 446-447 Two-carrier currents. 77-80

U Ultraviolet photoemission spectroscopy conjugated polymer ionization potentials, 61-62 polymer-metal interaction studies, 72-73 Universal conductance fluctuations femtosecond laser excitation, 435 in homogeneous systems, 419-420

492

SUBJECT INDEX

UPS, see Ultraviolet photoemission spectroscopy UV absorption spectrum, PPV, 49 V

Valence effective Hamiltonian method conjugated polymers and, 38 PPV ionization potential, 61 Vector wave equations 2D PBG crystals, 172 plane-wave expansion of positiondependent fields, 186-187 VEH, see Valence effective Hamiltonian method Vertex correction, 417 Vibrationally excited states in conjugated molecules, 33-34 PPV, 51-52 Virtual cathode point, 65

W Wannier functions CCR representation expansion, 181-183 density matrices and nonuniform fields, 373-374 photonic band-gap theory, 175-176 Wave equations for ordered PBG materials, 180 for photonic band-gap theory, 171-172 Waveguides ID, 153 Green-Kubo function and mesoscopic, 438-439 PBG photonic dispersion measurement using, 162 quantum, ballistic transport in, 299 Schroedinger equation and quantum, 310-314 Wigner function simulations of electronic, 402-404 Wave packets DBRTD potential and Gaussian, 387-389 quantum trajectories, 298 Wave propagation, electromagnetic in PBG materials, dispersion relation, 152 in periodic systems, 151-152

Weyl transform, see Phase-space Wigner distribution Wicke’s theorem femtosecond laser excitation, 434-435 low-field systems, homogenous, 406 Wide band-gap 11-VI heterostructures design guide, 213-214 development of flexible, 207 Wide band-gap 11-VI semiconductors diluted magnetic, 216 doping and transport, 233-244 history of, 206-210 low-resistance contact between metal and, 241-244 nanostructures, attempts to fabricate, 229-233 overview, 205-210 point defects, 224 quantum wells, see quantum wells self-trapping, 227 Wigner distribution functions density matrices and, 318-320, 323-324 dissipation, role of, 399-402 double-barrier resonant-tunneling diodes, 391-399 electron waveguide simulations, 402-404 Green-Kubo function and, 438 methods of solving boundary conditions for simulations, 389-381 initial state, 382-384 numerical discretization and solutions, 384-389 overview, 381 other devices, 402-404 quantum distribution function and, 290 in quantum modeling, advantages of, 378-381 statistical thermodynamics and quantum potentials, 291-292 Wigner potential, quantum device modeling and, 343-344 Work functions, electrode dependency on measurement techniques, 63 versus Fermi energy of semiconductors, 64-65 polyaniline, 27

SUBJECT INDEX

X XPS, see X-ray photoemission spectroscopy X-ray photoemission spectroscopy, 72-73 Z Zeeman effects, 216 ZnCdSe cw diode lasers external and wall-plug conversion efficiencies, 259 gain spectra, 278-279 ZnCdSe/ZnSe quantum wells 2D, 217-225

493

electronic states, 210-217 lattice mismatch strain, 220 ZnSe covalent versus polar components of total bond energy, 234-235 n-doping, electron concentrations, 233-234 p-doping description, 235-241 difficulties of, 234 low-resistance contacts to, 241 -244 tellurium isoelectronic centers blue-green LEDs, 249-250 and exciton-phonon interaction, 225-229

Cumulative Author Index, Volumes 1 - 49

A Abrikosov, A. A,: Supplement 12-Introduction to the Theory of Normal Metals Adler, David: Insulating and Metallic States in Transition Metal Oxides, 21, 1 Adrian, Frank J.: see Gourary, B. S. Akamatu, Hideo: see Inokuchi, H. Alexander, H., and Haasen, P.: Dislocations and Plastic Flow in the Diamond Structure, 22, 28 Allen, Phillip B., and MitroviC, Bozidar: Theory of Superconducting Tc, 37, 1 Amelinckx, S., and Dekeyser, W.: The Structure and Properties of Grain Boundaries, 8, 327 Amelinckx, S., Supplement 6-The Direct Observation of Dislocations Anderson, Philip, W.: Theory of Magnetic Exchange Interactions: Exchange in Insulators and Semiconductors, 14, 99 Appel, J.: Polarons, 21, 193 Ashcroft, N. W., and Stroud, D.: Theory of the Thermodynamics of Simple Liquid Metals, 33, 1 Axe, J. D., Moss, S. C., and Neumann, D. A.: Structure and Dynamics of Crystalline C,,,, 48, 149

B Bastard, G., Brum, J. A,, and Ferreira, R.: Electronic States in Semiconductor Heterostructures, 44,229 Becker, J. A,: Study of Surfaces by Using New Tools, 7, 379 Beenakker, C. W. J., and van Houten, H.: Quantum Transport in Semiconductor Nanostructures, 44, 1 Beer, Albert C.: Supplement 4Galvanomagnetic Effects in Semiconductors Bendow, Bernard: Multiphonon Infrared Absorption in the Highly Transparent Frequency Regime of Solids, 33, 249 Bergman, David J.: Physical Properties of Macroscopically Inhomogeneous Media, 46, 147

Bertram, H. Neal: Fundamental Magnetization Processes in Thin-Film Recording Media, 46, 271 Beyers, R., and Shaw, T. M.: The Structure of Y,Ba,Cu,O,., and Its Derivatives, 42, 135 Blatt, Frank J.: Theory of Mobility of Electrons in Solids, 4, 200 Blount, E. I.: Formalisms of Band Theory, 13, 306 Borelius, G.: Changes of State of Simple Solid and Liquid Metals, 6, 65 Borelius, G.: The Changes in Energy Content, Volume, and Resistivity with Temperatures in Simple Solids and Liquids, 15, 1 Bouligand, Y.: Liquid Crystals and Their Analogs in Biological Systems, in Supplement 14-Liquid Crystals, 259 Boyce, J. B.: see Hayes, T. M. Brill, R.: Determination of Electron Distribution in Crystals by Means of X-Rays, 20, 1 Brown, E.: Aspects of Group Theory in Electron Dynamics, 22, 313 Brown, Frederick C.: Ultraviolet Spectroscopy of Solids with the Use of Synchrotron Radiation, 29, 1 Brum, J. A,: see Bastard, G. Bube, Richard H.: Imperfection Ionization Energies in CdS-Type Materials by Photoelectronic Techniques, 11, 223 Bullet, D. W.: The Renaissance and Quantitative Development of the TightBinding Method, 35, 129 Bundy, F. P., and Strong, H. M.: Behavior of Metals at High Temperatures and Pressures, 13, 81 Busch, G., and Guntherodt, H.-J.: Electronic Properties of Liquid Metals and Alloys, 29, 235 Busch, G. A., and Kern, R.: Semiconducting Properties of Gray Tin, 11, 1 C

Callaway, Joseph: Electron Bands in Solids, 7, 100 495

496

CUMULATlVE AUTHOR INDEX, VOLUMES 1-49

Callaway, J., and March, N. H.: Density Functional Methods: Theory and Applications, 38, 136 Cardona, Manuel: Supplement 1I-Optical Modulation Spectroscopy of Solids Cargill, G. S., 111: Structure of Metallic Alloy Glasses, 30, 227 Carlsson, A. E.: Beyond Pair Potentials in Elemental Transition and Semiconductors, 43, 1 Charvolin, Jean, and Tardieu, Annette: Lyotropic Liquid Crystals: Structures and Molecular Motions, in Supplement 14-Liquid Crystals, 209 Chen, Chia-Chun: see Lieber, Charles, M. Chou, M. Y., see de Heer, Walt A. Clendenen, R. L.: see Drickamer, H. G. Cohen, Jerome B.: The Internal Structure of Guinier-Preston Zones in Alloys, 39, 131 Cohen, M. H., and Reif, F.: Quadrupole Effects in Nuclear Magnetic Resonance Studies of Solids, 5, 322 Cohen, Marvin L., and Heine, Volker: The Fitting of Pseudopotentials to Experimental Data and Their Subsequent Application, 24, 38 Cohen, Marvin L.: see de Heer, Walt A. Cohen, Marvin L.: see Joannopoulos, J . D. Compton, W. Dale, and Rabin, Herbert: F-Aggregate Centers in Alkali Halide Crystals, 16, 121 Conwell, Esther M.: Supplement 9-High Field Transport in Semiconductors Cooper, Bernard R.: Magnetic Properties of Rare Earth Metals, 21, 393 Corbett, J. W.: Supplement 7-Electron Radiation Damage in Semiconductors and Metals Corciovei, A,, Costache, G., and Vamanu, D.: Ferromagnetic Thin Films, 27, 237 Costache, G.: see Corciovei, A. Currat, R., and Janssen, T.: Excitations in Incommensurate Crystal Phases, 41, 202 D

Dalven, Richard: Electronic Structure of PbS, PbSe, and PbTe, 28, 179

Das, T. P., and Hahn, E. L.: Supplement 1-Nuclear Quadrupole Resonance Spectroscopy Davison, S. G., and Levine, J. D.: Surface States, 25, 1 Dederichs, P. H.: Dynamical Diffraction Theory by Optical Potential Methods, 27, 136 de Fontaine, D.: Configurational Thermodynamics of Solid Solutions, 34, 74 de Gennes, P. G.: Macromolecules and Liquid Crystals: Reflections on Certain Lines of Research, in Supplement 14-Liquid Crystals, 1 de Heer, Walt A,, Knight, W. D., Chou, M. Y.,and Cohen, Marvin L.: Electronic Shell Structure and Metal Clusters, 40, 94 de Jeu, W. H.: The Dielectric Permittivity of Liquid Crystals, in Supplement 14Liquid Crystals, 109 Dekeyser, W.: see Amelinch, S. Dekker, A. J.: Secondary Electron Emissions, 6, 251 de Launay, Jules: The Theory of Specific Heats and Lattice Vibrations, 2, 220 Deuling, H. J.: Elasticity of Nematic Liquid Crystals, in Supplement 14-Liquid Crystals, 77 Devreese, J. T.: see Peeters, F. M. de Wit, Roland: The Continuum Theory of Stationary Dislocations, 10, 249 Dexter, D. L.: Theory of the Optical Properties of Imperfections in Nonmetals, 6, 355 Dimmock, J. 0.:The Calculation of Electronic Energy Bands by the Augmented Plane Wave Method, 26, 104 Doran, Donald G., and Linde, Ronald K.: Shock Effects in Solids, 19, 230 Drickamer, H. G.: The Effects of High Pressure on the Electronic Structure of Solids, 17, 1 Drickamer, H. G., Lynch, R. W., Clendenen, R. L., and Perez-Albuerne, E. A,: X-Ray Diffraction Studies of the Lattice Parameters of Solids under Very High Pressure, 19, 135 Dubois-Violette, E., Durand, G., Guyon, E., Manneville, P., and Pieranski, P.: Instabilities in Nematic Liquid Crystals, 147

CUMULATIVE AUTHOR INDEX, VOLUMES 1-49 Duke, C. B.: Supplement 10-Tunneling in Solids Durand, G.: see Dubois-Violette, E. E Echenique, P. M., Flores, F., and Ritchie, R. H.: Dynamic Screening of Ions in Condcnscd Matter, 43, 230 Ehrenreich, H., and Schwartz, L. M.: The Electronic Structure of Alloys, 31, 150 Einspruch, Norman, G.: Ultrasonic Effects in Semiconductors, 17, 217 Eshelby, J . D.: The Continuum Theory of Lattice Defects, 3, 79 Evans, A. G., and Zok, F. W.: The Physics and Mechanics of Brittle Matrix Composites, 47, 177

F Fan, H. Y.: Valence Semiconductors Germanium and Silicone, 1, 283 Ferreira, R.: see Bastard, G. Ferry, D. K., and Crubin, H. L.: Modeling of Quantum Transport in Semiconductor Devices, 49, 283 Flores, F.: see Echenique, P. M. Frederikse, H. P. R.: see Kahn, A. H. Friend, R. H.: see Greenham, N. C. Fulde, Peter, Keller, Joachim, and Zwicknagl, Gertrud: Theory of Heavy Fermion Systems, 41. 1 G

Galt, J. K.: see Kittel, C. Geballe, Theodore, H.: see White, Robert

M.

Gilman, J. J., and Johnston, W. G.: Dislocations in Lithium Fluoride Crystals, 13, 148 Givens, M. Parker: Optical Properties of Metals, 6, 313 Glicksman, Maurice: Plasmas in Solids, 26, 275 Goldberg, I. B.: see Weger, M.

497

Comer, Robert: Chemisorption on Metals, 30, 94 Gourary, Barry S., and Adrian, Frank J.: Wave Functions for Electron-Excess Color Centers in Alkali Halide Crystals, 10, 128 Greenham, N. C., and Friend, R. H.: Semiconductor Device Physics of Conjugated Polymers, 49, 1 Grubin, H. L.: see Ferry, D. K. Gschneidner, Karl, A,, Jr.: Physical Properties and Interrelationships of Metallic and Semimetallic Elements, 16, 275 Guinier, AndrC: Heterogeneities in Solid Solutions, 9, 294 Gunshor, R. L.: see Nurmikko, A. V. Giintherodt, €I.-J.: see Busch, G. Guttman, Lester: Order-disorder Phenomena in Metals, 3, 146 Guyer, R. A,: The Physics of Quantum Crystals, 23, 413 Guyon, E.: see Dubois-Violette, E. H Haasen, P.: see Alexander, H. Hahn, E. L.: see Das, T. P. Halperin, B. I., and Rice, T. M.: The Excitonic State at the SemiconductorSemimetal Transition, 21, 116 Ham, Frank S.: The Quantum Defect Method, 1, 127 Hashitsume, Natsuki: see Kubo, R. Hass, K. C.: Electronics Structure of Copper-Oxide Superconductors, 42, 213 Haydock, Roger: The Recursive Solution of the Schrodinger Equation, 35, 216 Hayes, T. M., and Boyce, J. B.: Extended X-Ray Absorption Fine Structure Spectroscopy, 37, 173 Hebel, L. C., Jr.: Spin Temperature and Nuclear Relaxation in Solids, 15, 409 Hedin, Lars, and Lundqvist, Stig: Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids, 23, 2 Heeger, A. J.: Localized Moments and Nonmoments in Metals: The Kondo Effects, 23, 284

498

CUMULATIVE AUTHOR INDEX, VOLUMES 1-49

Heer, Ernst, and Novey, Theodore B.: The Interdependence of Solid State Physics and Angular Distribution of Nuclear Radiations, 9, 200 Heiland, G., Mollwo, E., and Stiickmann, F.: Electronic Processes in Zinc Oxide, 8, 193 Heine, Volker: see Cohen, M. L. Heine, Volker: The Pseudopotential Concept, 24, 1 Heine, Volker, and Weaire, D.: Pseudopotential Theory of Cohesion and Structure, 24, 250 Heine, Volker: Electronic Structure from the Point of View of the Local Atomic Environment, 35, 1 Hensel, J. C., Phillips, T. G.. and Thomas, G. A,: The Electron-Hole Liquid in Semiconductors: Experimental Aspects, 32, 88 Herzfeld, Charles M., and Meijer, Paul H. E.: Group Theory and Crystal Field Theory, 12, 2 Hideshima, T.: see Saito, N. Hirth, J. P.: see Pond, R. C. Huebener, R. P.: Thermoelectricity in Metals and Alloys, 27, 64 Hui, P. M., and Johnson, Neil F.: Photonic Band-Gap Materials, 49, 151 Huntington, H. B.: The Elastic Constants of Crystals, 7, 214 Hutchings, M. T.: Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields, 16, 227 I

Inokuchi, Hiroo, and Akamutu, Hideo: Electrical Conductivity of Organic Semiconductors, 12, 93 Ipatova, I. P.: see Maradudin, A. A. Isihara, A.: Electron Correlations in Two Dimensions, 42, 271 Iwayanagi, S.: see Saito, N.

J James, R. W.: The Dynamical Theory of X-Ray Diffraction, 15, 55

Jan, J.-P.: Galvanomagnetic and Thermomagnetic Effects in Metals, 5 , 3 Janssen, T.: see Currat, R. Jarrett, H. S.: Electron Spin Resonance Spectroscopy in Molecular Solids, 14, 215 Joannopoulos, J. D., and Cohen, Marvin, L.: Theory of Short-Range Order and Disorder in Tetrahedrally Bonded Semiconductors, 31, 71 Johnson, Neil F.: see Hui, P. M. Johnston, W. G.: see Gilman, J. J. Jmgensen, Kluxbull, Chr.: Chemical Bonding Inferred from Visible and Ultraviolet Absorption Spectra, 13, 376 Joshi, S. K., and Rajagopal, A. K.: Lattice Dynamics of Metals, 22, 160

K Kanzig, Werner: Ferroelectrics and Antiferroelectrics, 4, 5 Kahn, A. H., and Frederikse, H. P. R.: Oscillatory Behavior of Magnetic Susceptibility and Electronic Conductivity, 9, 257 Keller, Joachim: see Fulde, P. Keller, P., and Liebert, L.: Liquid-Crystal Synthesis for Physicists, in Supplement 14-Liquid Crystals, 19 Kelly, M. J.: Applications of the Recursion Method to the Electronic Structure from an Atomic Point of View, 25, 296 Kelton, K. F.: Crystal Nucleation in Liquids and Glasses, 45, 75 Kern, R.: see Busch, G. A. Keyes, Robert W.: The Effects of Elastic Deformation on the Electrical Conductivity of Semiconductors, 11, 149 Keyes, Robert W.: Electronic Effects in the Elastic Properties of Semiconductors, 20 37 Kittel, C., and Galt, J. K.: Ferromagnetic Domain Theory, 3, 439 Kittel, C.: Indirect Exchange Interactions in Metals, 22, 1 Klemens, P. G.: Thermal Conductivity and Lattice Vibrational Modes, 7, 1 Klick, Clifford C., and Schulman, James H.: Luminescence in Solids, 5, 97

CUMULATIVE AUTHOR INDEX, VOLUMES 1-49 Knight, W. D.: Electron Paramagnetism and Nuclear Magnetic Resonance in Metals, 2, 93 Knight, W. D.: see de Heer, Walt A. Knox, Robert S.: Bibliography of Atomic Wave Functions, 4, 413 Knox, R. S.: Supplement 5-Theory of Excitons Koehler, J . S.: see Seitz, F. Kohn, W.: Shallow Impurity States in Silicone and Germanium, 5, 258 Kondo, J.: Theory of Dilute Magnetic Alloys, 23, 184 Koster, G. F.: Space Groups and Their Representations, 5, 174 Kothari, L. S., and Singwi, K. S.: Interaction of Thermal Neutrons with Solids, 8, 110 Kriiger, F. A,, and Vink, H. J.: Relations between the Concentrations of Imperfections in Crystalline Solids, 3, 310 Kubo, Ryogo, Miyake, Satoru J., and Hashitsume, Natsuki: Quantum Theory of Galvanomagnetic Effect at Extremely Strong Magnetic Fields, 17, 270 Kwok, Phillip C. K.: Green’s Function Method in Lattice Dynamics, 20, 214 L Lagally, M. G.: see Webb, M. B. Lang, Norton D.: The Density-Functional Formalism and the Electronic Structure of Metal Surfaces, 28, 225 Laudise, R. A,, and Nielsen, J. W.: Hydrothermal Crystal Growth, 12, 149 Lax, Benjamin, and Mavroides, John G.: Cyclotron Resonance, 11, 261 Lazarus, David: Diffusion in Metals, 10, 71 Liebfried, G., and Ludwig, W.: Theory of Anharmonic Effects in Crystals, 12, 276 Levine, J. D.: see Davison, S. G. Levy, Peter M.: Giant Magnetoresistance in Magnetic Layered and Granular Materials, 47, 367 Lewis, H. W.: Wave Packets and Transport of Electrons in Metals, 7, 353 Lieber, Charles, M., and Chen, Chia-Chun: Preparation of Fullerenes and Fullerene-Based Materials, 48, 109, 349 Liebert, L.: see Keller, P.

499

Linde, Ronald K.: see Doran, D. G. Liu, S. H.: Fractals and Their Applications in Condensed Matter Physics, 39, 207 Lobb, C. J.: see Tinkham, M. Low, William: Supplement 2-Paramagnetic Resonance in Solids Low, W., and Offenbacher, E. L.: Electron Spin Resonance of Magnetic Ions in Complex Oxides, Review of ESR Results in Rutile, Perovskites, Spinel, and Garnet Structures, 17, 136 Ludwig, G. W., and Woodbury, H. H.: Electron Spin Resonance in Semiconductors, 13, 223 Ludwig, W.: see Leibfried, G. Lundqvist, Stig: see Hedin, L. Lynch, R. W.: see Drickamer, H. G. M McCaldin, James, 0.:see Yu, Edward T. McClure, Donald, S.: Electronic Spectra of Molecules and Ions in Crystals. Part 1. Molecular Crystals, 8, 1 McClure, Donald S.: Electronic Spectra of Molecules and Ions in Crystals. Part 11. Spectra of Ions in Crystals, 9, 400 McGill, Thomas C.: see Yu, Edward T. McGreevy, Robert L.: Experimental Studies of the Structure and Dynamics of Molten Alkali and Alkaline Earth Halides, 40, 247 MacKinnon, A,: see Miller, A. MacLaughlin, Douglas E.: Magnetic Resonance in the Superconducting State, 31, 1 McQueen, R. G., see Rice, M. H. Mahan, G. D.: Many-Body Effects on X-Ray Spectra of Metals, 29, 75 Manneville, P.: see Dubois-Violette, E. Maradudin, A. A., Montroll, E. W., Weiss, G. H., and Ipatova, I. P.: Supplement 3-Theory of Lattice Dynamics in the Harmonic Approximation Maradudin, A. A,: Theoretical and Experimental Aspects of the Effects of Point Defects and Disorder on the Vibrations of Crystals-1, 18, 274 Maradudin, A. A.: Theoretical and Experimental Aspects of the Effects of Point

500


Maradudin, A. A.: (Continued) Defects and Disorder on the Vibrations of Crystals-2, 19, 1 March, N. H.: see Callaway, J. Markham, Jordan J.: Supplement 8F-Centers in Alkali Halides Mavroides, John G.: see Lax, B. Meijer, Paul H. E.: see Herzfeld, C. M. Mendelssohn, K., and Rosenberg, H. M.: The Thermal Conductivity of Metals at Low Temperatures, 12, 223 Miller, A., MacKinnon, A,, and Weaire, D.: Beyond the Binaries-The Chalcopyrite and Related Semiconducting Compounds, 36, 119 Mitra, Shashanka, S.: Vibration Spectra of Solids, 13, 1 MitroviC, Bozidar: see Allen, P. B. Miyake, Satoru J.: see Kubo, R. Mollwo, E.: see Heiland, G. Montgomery, D. J.: Static Electrification of Solids, 9, 139 Montroll, E. W.: see Maradudin, A. A. Moss, S. C.: see Axe, J. D. Muto, Toshinosuke, and Takagi, Yutaka: The Theory of Order-Disorder Transitions in Alloys, 1, 194 N Nagamiya, Takeo: Helical Spin Ordering- 1 Theory of Helical Spin Configurations, 20, 306 Nelson, D. R., and Spaepen, Frans: Polytetrahedral Order in Condensed Matter, 42, 1

Neumann, D. A,: see Axe, J. D. Newman, R., and Tyler, W. W.: Photoconductivity in Germanium, 8, 50 Nichols, D. K., and van Lint, V. A. J.: Energy Loss and Range of Energetic Neutral Atoms in Solids, 18, 1 Nielsen, J. W.: see Laudise, R. A. Nilsson, P. 0.:Optical Properties of Metals and Alloys, 29, 139 Novey, Theodore B.: see Heer, E. Nurmikko, A. V., and Gunshor, R. L.: Physics and Device Science in 11-VI Semiconductor Visible Light Emitters, 49, 205

Nussbaum, Allen: Crystal Symmetry, Group Theory, and Band Structure Calculations, 18, 165

0

Offenbacher, E. L.: see Low, W. Okano, K.: see Saito, N.

P

Pake, G. E.: Nuclear Magnetic Resonance, 2, 1

Parker, R. L.: Crystal Grown Mechanisms: Energetics, Kinetics, and Transport, 25, 152 Peercy, P. S.: see Samara, G. A. Peeters, F. M., and Devreese, J. T.: Theory of Polaron Mobility, 38, 82 Perez-Albuerne, E. A.: see Drickamer, H. G . Peterson, N. L.: Diffusion in Metals, 22, 409 Pettifor, D. G.: A Quantum-Mechanical Critique of the Miedema Rules for Alloy Formation, 40, 43 Pfann, W. G.: Techniques of Zone Melting and Crystal Growing, 4, 424 Phillips, J. C.: The Fundamental Optical Spectra of Solids, 18, 55 Phillips, J. C.: Spectroscopic and Morphological Structure of Tetrahedral Oxide Glasses, 37, 93 Phillips, T. G.: see Hensel, J. C. Pickett, Warren, E.: Electrons and Photons in C60-Based Materials, 48, 225 Pieranski, P.: see Dubois-Violette, E. Pines, David: Electron Interaction in Metals, 1, 368 Piper, W. W., and Williams, F. E.: Electroluminescence, 6, 96 Platzman, P. M., and Wolff, P. A,: Supplement 13-Waves and Interactions in Solid State Plasmas Poirier, D. M., and Weaver, J. H.: Solid State Properties of Fullerenes and FullereneBased Materials, 48, 1 Pond, R. C., and Hirth, J. P.: Defects at Surfaces and Interfaces, 47, 287

CUMULATIVE AUTHOR INDEX, VOLUMES 1-49 R Rabin, Herbert: see Compton, W. D. Rajagopal, A. K.: see Joshi, S. K. Rasolt, M.: Continuous Symmetries and Broken Symmetries in Multivalley Semiconductors and Sernimetals, 43, 94 Reif, F.: see Cohen, M. H. Reitz, John R.: Methods of the OneElectron Theory of Solids, 1, 1 Ricc, M. H., McQueen, R. G., and Walsh, J. M.: Compression of Solids by Strong Shock Waves, 6, 1 Rice, T. M.: see Halperin, B. I. Rice, T. M.: The Electron-Hole Liquid in Semiconductors: Theoretical Aspects, 32, 1 Ritchie, R. H.: see Echenique, P. M. Roitburd, A. L.: Martensitic Transformation as a Typical Phase Transformation in Solids, 33, 317 Rosenberg, H. M.: see Mendelssohn, K. S Safran, S. A,: Stage Ordering in Intercalation Compounds, 40, 183 Saito, N., Okano, K., Iwayanagi, S., and Hideshima, T.: Molecular Motion in Solid State Polymers, 14, 344 Samara, G. A.: High-pressure Studies of Ionic Conductivity in Solids, 38, 1 Samara, G. A., and Peercy, P. S.: The Study of Soft-Mode Transitions at High Pressure, 36, 1 Scanlon, W. W.: Polar Semiconductors, 9, 83 Schafroth, M. R.: Theoretical Aspects of Superconductivity, 10, 295 Schnatterley, S. E.: Inelastic Electron Scatlering Spectroscopy, 34, 275 Schulman, James, H.: see Klick, C. C. Schwartz, L. M.: see Ehrenreich, H. Seitz, Frderick: see Wigner, E. P. Seitz, Frederick, and Koehler, J. S.: Displacement of Atoms during Irradiation, 2, 307 Sellmyer, D. J.: Electronic Structure of Metallic Compounds and Alloys: Experimental Aspects, 33, 83

501

Sham, L. J., and Ziman, J. M.: The Electron-Phonon Interface, 15, 223 Shaw, T. M.: see Beyers, R. Shull, G. C., and Wollan, E. 0.:Applications of Neutron Diffraction to Solid State Problems, 2, 138 Singh, Jai: The Dynamics of Excitons, 38, 295 Singwi, K. S.: see Kothari, L. S. Singwi, K. S., and Tosi, M. P.: Correlations in Electron Liquids, 36, 177 Slack, Glen A.: The Thermal Conductivity of Nonmetallic Crystals, 34, 1 Smith, Charles S.: Macroscopic Symmetry and Properties of Crystals, 34, 1 Sond, Ajay K.: Structural Ordering in Colloidal Suspensions, 45, 1 Spaepen, Frans: see Nelson, D. R. Spector, Harold N.: Interaction of Acoustic Waves and Conduction Electrons, 19, 291 Stern, Frank: Elementary Theory of the Optical Properties of Solids, 15, 300 Stockmann, F.: see Heiland, G. Strong, H. M.: see Bundy, F. P. Stroud, D.: see Ashcroft, N. W. Stroud, David: see Bergrnan, David J. Sturge, M. D.: The Jahn-Teller Effect in Solids, 20, 92 Swenson, C. A.: Physics at High Pressure, 11, 41

T Takagi, Yutaka: see Muto, T. Tardieu, Annette: see Charvolin, Jean Thomas, G. A,: see Hensel, J. C . Thomson, Robb: Physics of Fracture, 39, 1 Tinkham, M., and Lobb, C. J.: Physical Properties of the New Superconductors, 42 91 Tosi, Mario P.: Cohesion of Ionic Solids in the Born Modek, 16, 1 Tosi, M. P.: see Singwi, K. S. Turnbull, David: Phase Changes, 3, 226 Tyler. W. W.: see Newman, R.

502


V Vamanu, D.: see Corciovei, A. van Houten, H.: see Beenakker, C. W. J. van Lint, V. A. J.: see Nichols, D. K. Vink, H. J.: see Kroger, F. A. W Wallace, Duane C.: Thermoelastic Theory of Stressed Crystals and Higher-Order Elastic Constants, 25, 302 Wallace, Philip R.: Positron Annihilation in Solids and Liquids, 10, 1 Walsh, J. M.: see Rice, M. H. Weaire, D.: see Heine, V. Weaire, D.: see Miller, A. Weaire, D.: see Wooten, F. Weaver, J. H.: see Poirier, D. M. Webb, M. B., and Lagally, M. G.: Elastic Scattering of Low-Energy Electrons from Surfaces, 28, 302 Weger, M., and Goldberg, I. B.: Some Lattice and Electronic Properties of the B-Tungstens, 28, 1 Weiss, G. H.: see Maradudin, A. A. Weiss, H.: see Welker, H. Welker, H., and Weiss, H.: Group 111-Group V Compounds, 3, 1 Wells, A. F.: The Structures of Crystals, 7, 426 White, Robert M., and Geballe, Theodore, H.: Supplement 1s-Long Range Order in Solids Wigner, Eugene P., and Seitz, Frederick: Qualitative Analysis of the Cohesion in Metals, 1, 97 Williams, F. E.: see Piper, W. W. Wokaun, Alexander: Surface-Enhanced Electromagnetic Processes, 38, 224 Wolf, E. L.: Nonsuperconducting Electron Tunneling Spectroscopy, 30, 1 Wolf, H. C.: The Electronic Spectra of Aromatic Molecular Crystals, 9, 1

ISBN O-L2-607749-5

Wolff, P. A,: see Platzman, P. M. Wollan, E. 0.:see Shull, C. G. Woodbury, H. H.: see Ludwig, G. W. Woodruff, Truman 0.:The Orthogonalized Plane-Wave Method, 4, 367 Wooten, F., and Weaire, D.: Modeling Tetrahedrally Bonded Random Networks by Computer, 40, 1

Y Yafet, Y.: g Factors and Spin-Lattice Relaxation of Conduction Electrons, 14, 1 Yeomans, Julia: The Theory and Application of Axial king Models, 41, IS1 Yonezawa, Fumiko: Glass Transition and Relaxation of Disordered Structures, 45, I79 Yu, Edward T., McCaldin, James O., McGill, Thomas C.: Band Offsets in Semiconductor Heterojunctions, 46, 1

Z

Zak, J.: The &-Representation in the Dynamics of Electrons in Solids, 27, 1 Zhang, Zhe: see Lieber, Charles, M. Zheludev, I. S.: Ferroelectricity and Symmetry, 26,429 Zheludev, I. S.: Piezoelectricity in Textured Media, 298, 315 Zhu, Jian-Gang: see Bertram, H. Neal Ziman, J. M.: see Sham, L. J. Ziman, J. M.: The Calculation of Bloch Functions, 26, 1 Zok, F. W.: see Evans, A. G. Zunger, Alex: Electronic Structure of 3d Transition-Atom Impurities in Semiconductors, 39, 276 Zwicknagl, Gertrud: see Fulde, P.

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