Advances in Electronics and Electron Physics, Volume 38

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 38 CONTRIBUTORS TO THIS VOLUME Raymond Bowers K. Frank Jeffrey F...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 38

CONTRIBUTORS TO THIS VOLUME Raymond Bowers K. Frank Jeffrey Frey F. T. Hambrecht Hermann A. Haus Bruce D. McCombe Robert A. Puce1 M. E. Scharfe F. W. Schmidlin Hermann Statz Robert J. Wagner

Advances in

Electronics and Electron Physics EDITEDBY L. MARTON Smithsonian Institution, Washington, D .C . Assistant Editor CLAIRE MARTON

EDITORIAL BOARD E. R. Piore T. E. Allibone H. B. G . Casimir M. Ponte W. G. Dow A. Rose A. 0. C. Nier L. P. Smith F. K . Willenbrock

VOLUME 38

1975

ACADEMIC PRESS

New York San Francisco London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 0 1975, 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.

111 Fifth Avenue, New York,New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI

LIBRARY OF CONGRESS CATALOG CARDNUMBER:49-7504 ISBN 0-12-014538-3 PRINTED IN THE UNITED STATES O F AMERICA

CONTENTS

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

vii

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

ix

CONTRIBUTORS TO VOLUME 38 FOREWORD ..

.

Intraband Magneto-Optical Studies of Semiconductors in the Far Infrared II

BRUCED . MCCOMBE AND ROBERT J . WAGNER V . Bound Carrier Resonances . . . . . . . . . . . . . . . . . . VI . Interaction of Free and Bound Carriers with Collective Excitations . VII . Future Directions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

1 19 47 50

The Future Possibilities for Neural Control

F. T. HAMBRECHT AND K . FRANK I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . TI . Potential Applications under Investigation . . . . . . . . . . . . I11. Concepts and Techniques . . . . . . . . . . . . . . . . . . . . IV Future Possibilities . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

55

56 68

77 79

Charged Pigment Xerography

.

M . E SCHARFE AND F. W . SCHMIDLIN I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . General Discussion of the Xerographic System . . . . . . . . . . I11. Physical Discussion of the Photoreceptor Subsystem and Its Coupling to the Development System . . . . . . . . . . . . IV . Physical Basis for Development . . . . . . . . . . . . . . . . V . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . Y

. .

.

83 85 100 113 144 144

vi

CONTENTS

The Impact of Solid State Microwave Devices: A Preliminary Technology Assessment JEFFREY FREY AND RAYMOND BOWERS

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Solid State Microwave Sources . . . . . . . . . . . . . . . . . . I11. Microwave Integrated Circuits . . . . . . . . . . . . . . . . . . IV . Applications of Microwave Solid State Devices . . . . . . . . . . . V . Benefits and Problems . . . . . . . . . . . . . . . . . . . . . VI . Invasion of Privacy and Interception of Data Transmission . . . . . VI1. Conclusion; . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

148 153 169 169 178 191 191 191

Signal and Noise Properties of Gallium Arsenide Microwave Field-Effect Transistors ROBERT A . PUCEL.HERMANN A . HAUS.AND HERMANN STATZ

I . Introduction

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

195 204 224 228 244 252 Appendix I: Derivation of Gate Capacitance Expression . . . . . . 261 Appendix 11: Derivation of 1 . . . . . . . . . . . . . . . . 262 References . . . . . . . . . . . . . . . . . . . . . . . . . . 264

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Noise Figure . . . . . . . . . . . . . . . . . . . . . . . . . VI . Experimental Data . . . . . . . . . . . . . . . . . . . . . . .

11 The Intrinsic FET . . . . . . . . . . 111 The FET with Parasitic Resistances . .

ho:

AUTHORINDEX.

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

267

SUBJECT INDEX.

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

274

CONTRIBUTORS TO VOLUME 38 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

RAYMOND BOWERS, Program on Science, Technology, and Society and Department of Physics, Cornell University, Ithaca, New York (147)

K. FRANK, Laboratory of Neural Control, National Institute of Neurological Diseases and Stroke, National Jnstitutes of Health, Bethesda, Maryland (55)

JEFFREY FREY,Department of Electrical Engineering, Cornell University, Ithaca, New York (147) Laboratory of Neural Control, National Institute of F. T. HAMBRECHT, Neurological Diseases and Stroke, National Institutes of Health, Bethesda, Maryland (55) HERMANN A. HAUS,Electrical Engineering Department and the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts (195) BRUCED. MCCOMBE, Naval Research Laboratory, Washington, D.C. ( I ) ROBERTA. PUCEL,Research Division, Raytheon Company, Waltham, Massachusetts (1 95) M. E. SCHARFE, Xerox Corporation, Joseph C. Wilson Center for Technology, Rochester, New York (83) F. W. SCHMIDLIN, Xerox Corporation, Joseph C. Wilson Center for Technology, Rochester, New York (83) HERMANN STATZ,Research Division, Raytheon Company, Waltham, Massachusetts (195) ROBERT J. WAGNER, Naval Research Laboratory, Washington, D.C. ( I )

vii

This Page Intentionally Left Blank

FOREWORD Our preceding volume contained the first part of a review by B. D. McCombe and R. J. Wagner on “ Intraband Magneto-Optical Studies of Semiconductors in the Far Infrared.” The second part of that review, published here, deals with bound carrier resonances and with interactions of free and bound carriers with collective excitations. A few years ago, in our Volume 30, we had two interesting contributions about the electronic control of muscular action. This fascinating subject is again reviewed, here by F. T. Hambrecht and K. Frank under the title “The Future Possibilities for Neural Control.” The introductory sentence of their review is the best indication of its scope: “The thought of directly controlling certain aspects of the human nervous system or utilizing signals from the nervous system to directly control external devices is exciting to some and disturbing to other people.” In the last 10 years or so, we have all become so used to the widespread facilities for copying documents that we now take for granted the existence of the various devices used for this purpose. Perhaps the best known of all the processes used is the one called xerography, and its principles and technology form the subject of a review by M. E. Scharfe and F. W. Schmidlin entitled “ Charged Pigment Xerography.” The properties of a latent image formed by electrostatic charges and its “ development ” into a visible image are discussed at length. To describe the next review in this volume, by J. Frey and R. Bowers, I again turn to the authors’ words: “ Most technological developments have brought in the wake of their primary intended effect a series of unforeseen secondary effects, some adverse and some beneficial. It is characteristic of much technological development that those. concerned with the development of a new device are so preoccupied with the primary effect that they give inadequate attention to possible secondary consequences.” The review is entitled “The Impact of Solid State Microwave Devices: A Preliminary Technology Assessment,” and this title speaks for itself. Our last review is on “Signal and Noise Properties of Gallium Arsenide Microwave Field-Effect Transistors” by R. A. Pucel, H. A. Haus, and H. Statz. Of the many possible solid state microwave devices assessed in the previous review, the ones examined here have gained prominence due to their favorable properties. This review constitutes an in-depth study of the many parameters affecting the signal-to-noise ratio, offers a theory for the small-signal noise properties, and compares the theory with experiment. ix

X

FOREWORD

For the next few volumes of Advances in Electronics and Electron Physics the following subjects and authors are scheduled :

Interpretation of Electron Microscope Images of Defects in Crystals Energy Distribution of Electrons Emitted by a Thermionic Cathode Afterglow Phenomena in Rare Gas Plasmas Between 0" and 300' K Advances in Molecular Beam Masers Development of Charge Control Concept Semiconductor Microwave Power Devices. I and I1 Time Measurements on Radiation Detector Signals The Excitation and Ionization of Ions by Electron Impact Nonlinear Electron Acoustic Waves. I1 The Photovoltaic Effect In Siru Electron Microscopy of Thin Films Physics and Technologies of Polycrystalline Si in Semiconductor Devices Charged Particles as a Tool for Surface Research Electron Micrograph Analysis by Optical Transform Electron Beam Microanalysis Electron Polarization in Solids X-Ray Image Intensifiers Electron Bombardment Semiconductor Devices Thermistors High Power Electronic Devices Atomic Photoelectron Spectroscopy. I and I1 Electron Spectroscopy for Chemical Analysis Laboratory Isotope Separators and Their Application Recent Advances in Electron Beam Addressed Memories Nonvolatile Semiconductor Memory Devices Light Emitting Diodes, Methods and Applications. I and I1 Generation of Images by Means of TwoDimensional Spatial Electric Filters Mass Spectroscopy Multiphoton Processes High Injection in a Two-Dimensional Transistor SUPPLEMENTARY VOLUME:Charge Transfer Devices

M. J. Whelan W. Franzen and J. Porter J. F. Delpech, J. Boulmer, and J. Stevefelt D. C. Laink J. te Winkel S. Teszner and J. L. Teszner S. Cova John W. Hooper and R. K. Feeney R. G . Fowler Joseph J. Loferski A. Barna, P. B. Barna, J. P. Pbcza, and I. Pozsgai J. Kobayashi J. Vennik G . Donelli and L. Paoletti D. R. Beaman M. Campagna, D. T. Pierce, K. Sattler, and H. C. Siegmann J. Houston D. J. Bates G. H. Jonker G . Karady S. T. Manson D. Berenyi S. B. Karmohapatro J. Kelly J. F. Vervey H. F. Matare H. F. Harmuth

F. E. Saalfeld, J. J. Decorpo, and J. R. Wyatt J. Bakos W. L. Engl C. H. Sequin and M. F. Tompsett

FOREWORD

xi

We are fortunate to have acquired many good friends since Advances in Electronics and Electron Physics started. They have helped us with advice, with contributions and, last but not least, with the production of these volumes. In repeating our heartfelt thanks to all those who helped, we would like to renew our invitation for suggestions and contributions. L. MARTON CLAIRE MARTON


Intraband Magneto-Optical Studies of Semiconductors in the Far Infrared. II? BRUCE D. McCOMBE

AND

ROBERT J. WAGNER

Naval Research Laboratory, Washington. D.C. V. Bound Carrier Resonances ............................................................ A. Nearly Hydrogenic Centers .................................................................... B. Shallow Acceptor Levels in Semiconductors with Zone-Centered Degenerate Valence Bands ........................................................................ VI. Interaction of Free and Bound Carriers with Collective Excitations ..................... A. Frohlich Theory of the Electron-Polar (LO) Phonon Interaction ................... B. Resonant Electron-Phonon Coupling.. ..................................................... C. Resonant Electron-2NPO Phonon Coupling ............................................. D. Nonresonant Electron-Phonon Coupling E. Electron-Plasmon (Plasmaron) Interaction................................................ VII. Future Directions ..................................................................................... ............. ..... References ......................

1 1 16 19 20 23 36 39 43 47 50

V. BOUNDCARRIER RESONANCES A . Nearly Hydrogenic Centers 1. Theory of the Hvdrogenic Atom in a Magnetic Field

The electronic energy levels of isolated impurities in semiconductors can frequently be described quite accurately in the "effective mass approximation ( 1 1 ) . The case of an electron (hole) in the long range slowly varying potential due to isolated, randomly distributed impurity ions is treated in a manner similar to that used for the free electron in an applied magnetic field (see Section 11,C).For this situation, a Schrodinger-like equation is obtained for the envelope wave functions which describes the localization of the electron (hole) around the donor (acceptor) site. For the case of an isotropic conduction band with a minimum at k = 0, the energy levels of a singly charged donor impurity center; i.e., an atom whose nuclear charge differs by 1 from the atom normally occupying the particular location in question, are "

t Part I of this article (Sections I to I V ) appears in Volume 37 of Adrances in Electronics and Electron Physics. pp. 1 7 8 .

2

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

solutions of (neglecting spin), F,(x) = EF,(x).

Here the inverse effective mass m,/m* is obtained from Eq. (37a).The Coulomb potential at a distance I x I from the ion ( I x I B lattice constant) is given by

Ix I

(87) since the ion is immersed in a medium of background dielectric constant, E ~ In this approximation the effective mass is independent of energy. The solutions of Eq. (86) are the “effective” Rydberg series? -e2/%

where Ry* is the effective Rydberg or impurity ionization energy. Due to the small effective mass and large dielectric constant found in typical semiconductors, Ry* can be orders of magnitude smaller than the atomic hydrogen Rydberg (13.6 eV). In the presence of a uniform, externally applied magnetic field, Eq. (86) becomes (in the symmetric gauge),

(x’ + y’)

2

Pz + 2m* + wc2 L, ~

--

~~

~~

x F ( x ) = E F ( x ) , (89)

where spin has been neglected for simplicity. Solutions of this equation have been studied by a number of authors (127-133). It is useful to consider two limiting cases: (1) very small magnetic field, and (2) very large magnetic field. In the first case, the magnetic field may be taken as a perturbation, and one has the usual Zeeman effect in which the magnetic field causes small shifts and splittings of the unperturbed levels. In the latter case the reverse is true (the Coulomb term is small) and the energy levels are drastically different than those at low fields. A convenient parameter which characterizes the relative strength of the magnetic field is the ratio of free electron zero-point energy in the magnetic field to the effective Coulomb binding energy, y = +tro,/Ry*,

(90)

t We use n to denote the principal quantum number in the Rydberg series and the radial quantum number in the presence of a magnetic field. Where confusion might exist with the Landau quantum number, also denoted by n, the distinction will be clearly made.

.

INTRABAND MAGNETO-OPTICAL STUDIES. I1

3

or alternately

y = a6:',

(91)

where a: = &,ti2/m*e2 is the effective Bohr radius and 6 = eB/ch. In cylindrical polar coordinates the effective mass equation is

where p , z, cp are the usual cylindrical polar coordinates, energy is measured in units of Ry*, length in units of a t , and the strength of the magnetic field by the parameter y. In general Eq. (92) is not completely separable, and typically solutions have been obtained using variational approaches. Differences among the various theoretical treatments reside primarily in the choice of trial functions for the variational calculations. a. High jield limit. In the high field case (y >> 1 ) Yafet, Keyes, and Adams (127)have shown that in the limit y -, co,exact solutions of Eq. ( 9 2 ) can be written FNMd(p, z, q ) = @NM(P,q ) f N M d ( z ) .

(93)

The ONMare identical to the transverse part of the free carrier wavefunction - pt/2m* in Eq. (15) with N = n (the Landau quantum (solutions of Z,, number) and M = ml), and thefNM,(z)satisfy the one-dimensional equation

where VNM(z)is an effective, one-dimensional potential obtained by averaging the Coulomb potential in the transverse directions. The energy levels corresponding to the F N M i are

where I indicates the number of nodes of the wave function along the z-direction. The eNMd assume discrete negative values with magnitude small compared with y. Hence in this limit when the continuum states (Landau levels) are included, the energy levels consist of a set of discrete impurity levels located just below each Landau level. A schematic energy level diagram depicting a number of impurity states associated with the lowest two Landau levels is shown in Fig. 26.

4

BRUCE D. MCCOMBE AND ROBERT J . WAGNER

n=i /

/'

FIG. 26. Schematic diagram of the impurity levels associated with the two lowest 1. The Landau levels are shown as sections continuum Landau levels in the high field limit, ;' of parabolas while the impurity levels are shown as flat lines. The three predominant impurity transitions are indicated by the solid arrows, and cyclotron resonance is shown by the dashed arrow.

+

Solutions of Eq. (92) which correspond to continuum states also exist. These solutions have the same energy along z, E = h 2 k i / 2 m * , and the same density of states as free electron solutions; but due to the Coulomb interaction the wave functions are altered (129).For energies far above the continuum edge (bottom of Landau subband), the wave functions become free-electron-like (plane waves). Wallis and Bowlden (128)t and Hasegawa and Howard (130) have calculated the optical selection rules for impurity transitions in the high field limit. They find CRA circular polarization: S )I B AN

=

+ I or 0 ;

CRI circular polarization; S AN = O o r - 1 ;

E

(1

AM

= +1

A 1 even

AM

=

-1;

Aieven

/I B

B ; S IB AN = O ;

AM = O ;

A1odd.

t Wallis and Bowlden use a different notation from Yafet, Keyes, and Adams. and Hasegawa and Howard. Since the latter notation is the most common, it will be used in all subsequent discussion except to note that Wallis and Bowlden's (ImA)are related to the ( N M j . ) by N = I + 1/2(m + I m I ), m = M and 1 = 1.


5

Since parity and the z-component of orbital angular momentum commute with the Hamiltonian of Eq. (89) for arbitrary magnetic field, the AM and Ail selection rules are rigorously valid. However, the AN selection rule can be broken. Wallis and Bowlden calculated the absorption coefficient for a number of transitions and found that the predominant ground state to excited state transitions in the high field limit are (OOO) (OTO), (OOO) -+ (Ool),and (OOO) -+ (110). These authors also found strongly allowed transitions from impurity to continuum states. However, it has been shown by others [see, e.g., Hasegawa and Howard (130)] that these ionizing transitions are an artifact of the approximation used by Wallis and Bowlden. In fact the continuum transitions are weak and approach zero as B -+ co. In this limit the oscillator strength goes entirely into the (OOO) -+ (110) transition and all other transitions vanish as some inverse function of y (130). b. Low j e l d limit. For y -4 1 the magnetic field terms of Eq. (89) may be treated as a perturbation on the effective hydrogen atom energy levels, and one obtains the usual Zeeman effect. Namely, the low lying p-like hydrogenic states are split into three components labeled by the zcomponent of orbital angular momentum, m = 1,O. The quantum numbers describing this situation are n, I, m where n is the quantum number of the radial wave function, 1 is the total orbital momentum quantum number, and m is the z-component of the angular momentum. The electric dipole selection rules in this case for transitions from the ground state to excited states are CRA : -+

A1=+1,

Am=+1

A1 = -1,

Am

A1=+1,

Am=O

CRI :

E

(1

=

-1

B:

The connection between the low field and high field states has been the subject of some conjecture. The concept of “ nodal surface conservation (134) has been used to provide a relation (129). According to this principle the correspondence is :

”

I-

Irnl + A ;

n-l-l-+N-;(M+

[MI).

Boyle and Howard (135) have used the “ n o crossing” principle and the fact that parity and the z-component of orbital angular momentum are the only good quantum numbers for arbitrary field strengths to establish a somewhat different correspondence. They found that the only high field states corresponding to low field bound states are those with N = 0 or N > 0, N = M.

6


All other high field states, N > 0, N > M are metastable, since they are degenerate with continuum states to which they are connected by the Coulomb interaction. The relationship between low and high field states for both of these approaches is compared in Table VIII. Recently Baldereschi and Bassani (132) have made a concurrence by extrapolating and joining high and low field calculations. From their results they conclude that the Boyle and Howard relationship is correct. TABLE VIII CORRESPONDENCE BETWEEN Low FIELD( n h ) AND HIGHFIELD( N M I ) ENERGY LEVELS High Field Low Field (nW 1s

2p ( m = - 1 ) 2p ( m = 0) 2p(rn= + I ) 3p ( m = - 1 ) 3p ( m = 0) 3p(m= + I )

Elliott and Loudon (129) (NM4

Boyle and Howard ( 1 3 5 ) (NMI)

000 OTO 00 1 110

000 OTO 001 110 0 T2 003 112

1TO 101 210

The correspondence problem, the extension of the effective mass theory to include nonparabolicity and degenerate bands, the problem of central cell corrections, and the breakdown of the effective mass theory for deep impurities will be discussed in the following sections. 2. Experimental Results on Donor Impurities in Materials with Zone Center Conduction Band The foregoing theoretical discussion finds quantitative application in only a limited number of experimental situations. Many different impurity atoms have been introduced into a large number of semiconductor hosts with the result that there is a great deal of magneto-optical data. In some cases the results have been readily explained in the light of the preceding theoretical development. In other cases, the data are not well understood. The usual difficulties posed in this case are the need to use a realistic effective mass theory rather than a constant m* and the need for a more realistic Coulombic potential when the impurity atom is not well screened by the host lattice, i.e., central cell corrections. From this viewpoint, the simple theory provides both an explanation of some of the impurity magnetooptical results and a context for examining complicated data and introdu-


7

cing more sophisticated effective mass models and central cell corrections as required. The experimental discussion will first deal with relatively wellunderstood data for InSb and GaAs. These two materials, because of differences in effective mass and dielectric constant, yield information about hydrogenic energy levels over a wide range of y. a. Highfield limit (y % 1): ZnSb. While the first observation of impurity magneto-optical effects in InSb was made by Boyle and Brailsford (136), Kaplan (137) has carried out the most extensive study of the (000)-, (OTO), (000) --t (001), and (000) -+ (110) high field transitions using a Fourier transform spectrometer. These measurements were taken over a range of magnetic field (10-100 kG) such that y covered the range 7-70. The field dependence of the transition energy of these lines is shown in Fig. 27 along

FIG. 27. Field dependence of the energies of the three predominant impurity transitions. Experimental data are shown as the solid circles while the curves show the following theoretical results: Solid lines, Wallis and Bowlden (128); dotted line, Hasagawa and Howard (130); dashed line, Larsen (131). Note the change in energy scale for the (000) + (110) transition. [From Kaplan (237).]

with relevant theoretical calculations. Note that for the two low energy transitions, (000) -, (010) and (000)+ (OOl), the theories underestimate the transition energy over the entire range of magnetic field. For the high energy

8

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

transition, (OOO) -+ (1 lo), the theory, which assumes parabolic bands, ouerestimates the transition energy in such a way that the deviation between theory and experiment is an increasing function of magnetic field. In the latter case the discrepancy may be qualitatively understood as follows. With a slightly generalized form of Eq. (95) the transition energy may be written Ell0 - Eooo = El - Eo

+

l&OOO

I

-

IEllO

I,

where El and Eo are the energies of the n = 1 and n = 0 Landau levels, respectively. The predominant contribution to the deviation comes from the fact that the cyclotron resonance energy (El - E , ) is actually a sublinear function of B due to nonparabolicity (see Section IV,A). In addition the effective mass, m*, is also a function of energy due to nonparabolicity ;hence I caOO I and I c l 1 0 1, which are both proportional to m*, increase with the energy of their associated Landau levels. Since (110)lies higher in energy than (000) (by the cyclotron energy), I E~~~ I increases faster than I cOOOI and the difference, I E~~~ I - I c 1 I decreases with increasing magnetic field. This makes an additional contribution to the discrepancy. Larsen (131) has included these nonparabolic effects by adding the impurity potential to the effective Hamiltonian [Eq. (43)] and utilizing the Bowers and Yafet (BY) (38) approach to treat the strongly interacting bands in a magnetic field. However, a direct quantitative comparison with the experimental transitions is complicated by the fact that the energy cOOOmay be additionally modified by central cell effects which are not taken into account in the theory. The data for (OOO) + (010) and (000) -+ (001) in Fig. 27 suggest the presence of such central cell effects. As the magnetic field is increased, each of the impurity level wave functions is compressed nearer to the impurity center. Thus the screening by the host crystal (via cB) is reduced and the Coulombic binding energy is increased. This effect is greater for the ground state (000) (s-like) level than for the (OTO), (001), or (110) levels with their more extended wave functions. Qualitatively, the effect is to increase the observed transition energies over those calculated neglecting central cell corrections. If one assumes that central cell effects are small for the (010) level, then the effect on the (000) level should be directly apparent in the (000) -+ (010) transition. From physical arguments about the field dependence of the electronic charge density (resulting from the shrinkage of the electronic wave function transverse to the field) Kaplan has estimated that the central cell correction should increase linearly with field. While the experimental points departed from the theoretical field dependence of the (000) + (010) energy, the data were not sufficiently good to establish a functional dependence. In order to isolate a parameter independent of cOOO(and thus indepen-


9

dent of central cell corrections) for direct comparison with the nonparabolic theory of Larsen, Kaplan has measured the energy differences, (Ello - Eooo) - (El - E,) (the energy difference between “impurity shifted ” cyclotron resonance and free carrier cyclotron resonance) and EoTo- Eooo (from direct observation of the FIR transition shown in Fig. 27). The difference between these two energies yields cl10 independent of E ~ , , . In the parabolic approximation this difference is zero. Due to the nonparabolic dependence of m* on energy, I E~ I is greater than lcOTO I for a given magnetic field, and the difference increases with magnetic field. Kaplan found good agreement between the experimentally measured energy difference and that calculated by Larsen over the range 14 < y < 70. Thus it appears that the field dependent portion of the discrepancy between theory and experiment in Fig. 27 is adequately accounted for by proper consideration of conduction band nonparabolicity. Recent work by Demeshina et al. (138)indicates the possible observation of excited state transition(s), in particular, (010) + (011). Since these resonances involve initial and final states, both of which lie below the lowest Landau level, they occur in the region of lo00 pm, still a rather difficult region for experimental investigation. b. Low field limit (y < 1): GaAs. In GaAs, a heavier conduction band effective mass (m*/mo = 0.067) combined with a smaller dielectric constant (cB = 12.6)allow the exploration of a considerably different range of y from that in InSb. Here y x 1 at 65 kG. A number of interesting high resolution magneto-optical studies have been carried out for y < 1 on high quality epitaxial GaAs. The first measurements of donor impurity spectra were performed by Kaplan et al. (33a) and Stillman et al. (33b). Both groups observed the transitions (OOO) + (OTO), (000) + (001), and (OOO) + (110) in addition to higher energy transitions. From the field dependence and selection rules of these lines, it was clear that (OTO), (001), and (110) corresponded to 2p (m = - 1, 0, + l), respectively, as expected from both correspondence schemes. The splitting of the (OOO) -+(110) and (000) + (010) lines gave excellent agreement with previous effective mass measurements. While hydrogenic variational theory was used to fit the 1s + 2p (m = L- 1) transitions, theoretical results were not available to fit the other observed spectral features, i.e., the higher energy transitions which were assigned to 1s + 3p transitions. Narita and Miyao (139) have obtained more extensive photoconductivity data in both Faraday and Voigt geometries. In order to compare this data with the hydrogenic model, they performed a variational calculation utilizing a judicious choice of high field trial functions. The experimental results along with the variational calculation are shown in Fig. 28. From the

10

BRUCE D. MCCOMBE AND ROBERT J. WAGNER 1301

I

I

I

,

20

25

30

1

110 "OI

-

100

5

90

Y

80 2 W 3 70 O W K 6o LL

50

40 30 2o0

5

10

15

35

MAGNETIC F I E L D ( k G )

FIG.28. Experimental and theoretical transition energies as a function of magnetic field for GaAs. The theoretical lines are obtained from a variational calculation which utilizes high field trial functions. The experimental points are obtained as follows: Circles, Narita and Miyao (139); squares, Kaplan et ul. (33a); triangles, Stillman et al. (33b). [From Narita and Miyao

(1W.l close agreement between the experimental and calculated transition energies, Narita and Miyao were able to identify the transitions (000)--* (lTO), (0o0) -+ (112), and (000)-+ (210), as well as those previously identified. In addition, from a comparison of the peak intensities in the two geometries they found that the assignment of (000) + (112) to 1s + 3p ( m = 0) (33a, 336) was erroneous since it did not correspond to a Am = 0 transition. On the other hand, they were not able to achieve an unambiguous correspondence between the high and low field levels. (The correspondence between high and low field states is discussed further in connection with recent measurements in CdTe.) A rather surprising outcome of this work is that high field trial functions can be used to fit the observed transitions down to y z 0.1 with some success. The availability of extremely pure GaAs has led to the observation of a number of interesting effects which, although generally present in all impurity studies, tend to be obscured in poor and (or) high impurity concentration material. The importance of central cell corrections has been clearly established by the observation of a splitting of the Is + 2p ( m = + 1) transition by Fetterman et al. (140) utilizing a FIR laser. As in Kaplan's work on InSb, the central cell corrections are presumed to effect only the 1s state. In sub-

11


sequent high resolution studies utilizing Fourier transform spectroscopy such splittings have been observed in each of the 1s -+ 2p transitions (141). Some of these data are shown in Fig. 29. Each line of the triplet of 1s -+ 2p (rn = + 1) lines on this figure represents a slightly different 1s binding energy for a different donor type. Perturbation theory was employed by Fetterman et al. to compare the anticipated field dependence of the splitting with experiment (140). This work has now been refined sufficiently that a low level concentration of specific impurities, e.g. Sn, in GaAs can be identified (142, 143).

In related work Summers et al. (144) reported zero field measurements of the 1s 2p transition in GaAs doped with Ge, Si, Se, and S. Shifts in the position of the 1s 2p line were observed as a function of dopant. These shifts were attributed to central cell effects with positive central cell corrections of up to 2.5 cm- (for Ge). This value is about a factor of 2 larger than the largest shift reported by Fetterman et al. (140). The observed Is 2p transitions of Summers et al. (144) were more than 10 times broader than the lines observed by Fetterman et al. due to the relatively high impurity concentrations and possible banding of the 2p states. Thus the observed photoconductivity peaks do not necessarily reflect the true 1s + 2p transition energy. Stillman et al. (145) have utilized the splitting of the 2p (rn = 0, - 1) levels to assess the degree to which the simple hydrogenic effective mass theory (outlined at the beginning of this section) is valid. Using variational calculations of the two levels, they find that they are able to fit the energy difference from 20-55 kG to within k0.015 cm-'; thus effective mass theory is verified to within 0.15%. In spite of this accurate verification, the same authors (141) found large discrepancies between the Zeeman mass defined by -+

-+

-+

m:

=

heB/c[E(2p, rn =

+ 1) - E(2p, rn = - I)]

and the cyclotron mass, m:. According to the effective mass theory they should equal one another for arbitrary magnetic field. The Zeeman mass was found to be lower than the cyclotron mass by as much as 8% at low fields with the percentage of deviation, (rn: - rn:)/rnr x 100, a strongly decreasing function of magnetic field. Furthermore, the forbidden 1s -+ 2s transition was observed (Fig. 29). Stillman et al. suggest that these anomalies in the spectrum of the neutral shallow donor are a manifestation of the Stark effect caused by the electric fields of small numbers ( - lOI3 cmP3) of ionized donors and (or) acceptors. That is, they argue that the electric field couples the 2s and 2p (rn = 1)levels causing repulsion of the 2p levels from the 2s level. This appears as an apparent increase in the splitting AE[2p (m = k l)], i.e., a decrease in rn: at a given field. In addition, the 2s

12


FREQUENCY (an-')

FIG.29. Photoconductive response as a function of frequency for a high purity epitaxial sample of GaAs. ( T = 1.75"K, H = 10.400 kG.) The transition labeled Is --t 2s is forbidden in the absence of electric field perturbations. [From Stillman er al. (141).]

level takes on some 2p ( m = 0) character due to the perturbation, and as a result, the 1s + 2s transition becomes weakly allowed. Stillman et al. calculated the additional splitting due to the Stark electric fields using secondorder perturbation theory. A good fit to the experiment was obtained using a single adjustable parameter. A more extensive and compelling theoretical treatment of similar effects in studies of the line shape of the 1s + 2p ( m = - 1) transition has recently been published (146). c. Hydrogenic impurities in other materials. Although donor impurities in InSb and GaAs have yielded the most precise information concerning hydrogenic energy levels in a magnetic field, a number of other materials have revealed similar, although less detailed, features. High field donor impurity level transitions in epitaxial InAs have been reported by Litton et al. (49). In this work free carrier cyclotron resonance and " impurity shifted " cyclotron resonance (000 -+ 1 10)were observed over the magnetic field range 35-90 kG. (y x 4.5-10.5). From the measured mass a binding energy of 14.3 cm- ' was calculated with Eq. (88). It should be possible to explore the region of y x 1 with the material at about 8 kG; however, due to problems with sample and substrate transparency, Litton e f al. were unable to study the low y region. The 1s + 2p transitions have been observed in epitaxial specimens of InP by Chamberlain et al. (147). From the measured zero field 1s -+ 2p energy these authors obtained a donor binding energy of 61.8 cm-'. The 1s + 2p ( m = 0, 1) transitions in a magnetic field were also observed and an effective mass of 0.081mo was determined from the 2p ( m = + 1) - 2p (rn = - 1) separation, in good agreement with the cyclotron resonance measurements (50). In later work Stradling et al. (143) were able to resolve

13

INTRABAND MAGNETO-OPTICAL STUDIES. 11

structure in the 1s + 2p (rn = 0, & 1) transitions due to central cell effects on the ground state. There appeared to be two dominant donor species in these materials. Magneto-optical studies of hydrogenic centers in CdTe were first reported by Cohn et al. (248). Their interest focused on the bound electronLO phonon interaction as it appeared in the 1s -+ 2p (rn = 1) transitions. However, they did note that the CdTe impurity binding energy, Ry* = 115.5 cm- ', deduced from the 1s -,2p absorption, was in good agreement with that predicted from Eq. (88) for the effective Ry*. This is surprising since the heavy mass, 0.0963rnO,and small dielectric constant, 10, result in strong binding. Thus one would anticipate strong central cell corrections to the 1s level. Simmonds et al. (249), using higher purity materials, have observed as many as 6 donor species with shifts as large as 8 cm-'. Examples of these effects are shown in Fig. (30). Wagner and McCombe

*

al

55

60

65

0 (kG)

-.

FIG.30. Central cell splittings observed in two different bulk samples of CdTe. Each of the lines is due to the 1s 2p ( m = 1) transition associated with a different impurity species. The 78 pm line of a H,O vapor laser was used as a source for these high resolution experiments. [After Simmonds et al. (149).]

+

(250) have observed 1s -, 3p (rn = 0, & 1) transitions in CdTe. By comparison with the variational calculation of Narita and Miyao (139), they concluded that the 3p (rn = - 1, 0, + 1) levels correspond to (OT2), (003), (112), respectively. This supports the correspondence principle of Boyle and Howard (235) and the calculations of Baldereschi and Bassani (132). d. Excited state and other low energy transitions. Recently Chamberlain et al. ( 1 5 1 ) have compiled their results on low energy impurity transitions in various 11-VI and 111-V semiconductors known to have simple hydrogenic energy levels. These data are shown in Fig. 31. These transitions do not involve the ground state since the transition energies are less than the 1s -, 2p separation. These results, when compared to the calculated energy levels, suggest that the transitions observed are 2p (rn = - 1) -, 2s and possibly 2s -, 3p (rn = + 1).

14

BRUCE D. MCCOMBE A N D ROBERT J. W A G N E R

-

-

Other reported low energy impurity transitions that are not as yet understood are magneto-absorption studies on CdS (152) and ZnO (153).Both of these materials have a relatively large effective mass: 0.2m0 for CdS with B 1 b axis and 0.3mo for ZnO with B 11 c axis. Thus deeply bound centers should be anticipated with effective Rydbergs of 40 and 60 meV for CdS and ZnO, respectively. The data on CdS cannot be related to hydrogenic impurities but may be due to a shallow trap (252).However, for ZnO, two impurity transitions with zero field values of 5.9 and 7.2 meV are observed. Both lines split with magnetic field at the rate which would be expected for 1s -, n p ( m = 1). This is surprising since the hydrogenic model predicts a zero field 1s -,2p energy of 45 meV, nearly an order of magnitude larger than those observed. 3. Experimental Results on Donor Impurities in Materials with Multiple Equivalent Conduction Bands

A somewhat more complicated class of impurity magneto-optical studies is that of Group V donors in Ge or Si. Here the presence of a number of equivalent anisotropic conduction bands suggests that the hydrogenic levels


15

and optical spectra may be more complicated than for donors in materials with a zone-centered conduction band. Since high quality samples of Ge and Si have long been available, extensive studies of impurity spectra both with and without such external perturbations as uniaxial stress and magnetic field have been made. This has resulted in a relatively complete understanding of this class of impurities. The interested reader is referred to the review of Fisher and Ramdas ( 1 5 4 ) who have discussed the experimental studies of both donors (Group V) and acceptors (Group 111) in Ge and Si. While Fan and Fisher (155), Boyle (156), and Zwerdling et al. ( 1 5 7 ) first reported magneto-optical features of donor impurities in Si and Ge, the more recent work of Horii and Nisida (158)on As- and Sb-doped Ge will be considered here as an illustration. In order to appreciate these results, some of the differences between this case and the simpler donor cases previously discussed will be pointed out. As before, the localized impurity potential modifies the ground state wave function beyond that anticipated for a singly charged Coulomb potential. Here in addition to the usual central cell correction,? the short range character of the potential mixes large k-value terms from each of the four ( 1 11) conduction band valleys (for Ge). Alternately stated, there are nondiagonal matrix elements of the potential between wave functions representing the different valleys. This splits the ground state (s-state) into a singlet and triplet level at zero field. In the case of the excited states, the axial symmetry of the impurity Hamiltonian which results from anisotropic effective mass parameters splits the state m = 0 from the state in = 1 at zero field. With this background, the spectra of Horii and Nisida, Fig. 32, for Asdoped Ge become more intelligible. They point out that all transitions shown on the figure originate from the singlet s-state. Thus field-dependent transition splittings are caused by the final state splittings. (Such is not the case for Sb-doped Ge where both triplet and singlet s-states contribute as the initial states.) As the field is increased. the various np ( m = k 1) states split into four states, np ( m = ? I)*. B . With the magnetic field oriented along a ( 111) direction, two different groups of valleys develop: the A-valley being the valley along the field direction and the three B-valleys oriented at an angle with respect to the field direction. Since the impurity Hamiltonian includes the effective mass, the different cyclotron effective masses of the A and B valleys result in different impurity state wave functions and eigenvalues. Nisida and Horii (161 ) have performed variational calculations t Recently. some effort has been expended to modify the effective mass treatment of impurity states to account for central cell effects. For example. Pantelides and Sah ( 1 5 9 ) and Schechter (160) have used a pseudo-potential approach to calculate the binding energies. E,. ofdonors in Si where E , 5 40 meV.

16


FIG.32. A plot of the energy of the transition minima vs magnetic field for As-doped Ge. The initial state of each transition is the singlet s-state with the final state as indicated in the figure. The solid and dashed lines are drawn only as visual aids. [From Horii and Nisida (158).]

using both low field hydrogen-like and high field harmonic oscillator-like trial functions. By comparing the calculated results with experiment, they are able to make the level assignments shown in Fig. 32 and to indicate the field regions of applicability for the various trial functions. B. Shallow Acceptor Levels in Semiconductors with Zone-Centered Degenerate Valence Bands

In contrast to the relatively well-understood hydrogenic donor case discussed above, the magneto-optical properties of shallow acceptors remain a complicated and little-explored area. This is due primarily to the complexity of the band edge structure as has been discussed in Section IV,C,l. Ob-


17

viously, before attempting to include the effect of the magnetic field in the Hamiltonian, the theory for the zero field case must be developed. Schechter (162), Mendelson and James (163), and Lipari and Baldereschi (164) have each calculated the impurity levels in zero magnetic field starting with the Hamiltonian of Luttinger, Eq. (76), and including a Coulomb term:

Schechter and Mendelson and James then proceed to construct variational trial functions on the basis of the point group symmetry (full double tetrahedral) of the acceptor site. From this they obtain a set of energy levels, the trial functions of which form bases for r6,r,, or, T8representations of the point group. The wave functions are further differentiated by noting the parity (+ or -) of the total wave function. In the work of Mendelson and James, an index for the number of nodes of the radial function is included. Finally where more than one root of the secular equation exists, a quantum number is included for each root, ordered by increasing energy, i.e., (8 - Ol), (8 - 02), etc. The schematic diagram of Fig. 33 illustrates these levels as well as the “ alphabetical notation used by experimentalists for the allowed optical transitions. Generally speaking, experimental results (154, 165-167) have confirmed this effective mass approach for the positions of the excited states relative to each other and the continuum. However, this approach consistently underestimates the binding energy of the ground states. This is understood to be due to the overlap of the dominantly s-like ground state wave function with the central cell of atoms surrounding the impurity. Here the impurity is not screened by host atoms as it is for the more extended excited state wave functions. These central cell corrections to the ground state are typically much more important for acceptor impurities than for donor states associated with simple conduction bands. Lipari and Baldereschi (164)have also considered the zero magnetic field acceptor problem. Their approach, while giving comparable results to that of Schechter and Mendelson and James, introduces a unifying point of view which allows one to estimate acceptor energy level positions for any material for which the Luttinger valence band treatment is valid. They note that Eq. (96) resembles the equation which would apply to a spin 3/2 particle in a Coulomb potential. Using this analogy, they rewrite the Hamiltonian as a sum of two parts: ”

18


G

E

D

CC'

n'

FIG. 33. Schematic diagram of the acceptor ground state and p-like excited states in G e without (left side of drawing) and with (right side ofdrawing) magnetic field. [From Lin-Chung and Wallis (170).]

where the Ps and J s are second rank tensor operators. Lipari and Baldereschi point out that by comparing

6 = ( Y 3 - Y2)/Y*r (98b) the relative size of the spherical and cubic terms can be estimated. For example, p = 0.767 and 6 = 0.102 for Ge. Thus they neglect the effect of the cubic term in the initial solution. This term is included by perturbation they introduce theory. Utilizing the symmetry properties of Sspherica,, atomic-like eigenfunctions to use as variational trial functions. They compute the resultant energy levels for 0 I p 5 1. Although this approach has only been carried out for the zero field case, they suggest that the procedure may be useful when considering the effect of a magnetic field.


19

Bir, Butikov, and Pikus (168) Suzuki, Okazaki, and Hasegawa (169) and Lin-Chung and Wallis (170) have considered the effect of a perturbing magnetic field in the low field region, using the trial functions of Schechter or Mendelson and James. They estimate the splitting factors, g l land g, , of the zero field levels. The splittings are shown schematically in Fig. 33. In addition, Lin-Chung and Wallis calculate the optical selection rules which apply to the field split A, C, . . . transitions. At present, there are only a few experimental results which can be compared to theory. Soepangkat and Fisher (171 ) have studied the Zeeman spectrum of B- and Th-doped Ge. They observed the splittings of the transitions C, D, and G of Fig. 33. From these results, they were able to assess the g-values for the ground state and the 8-02 state. Kaplan (166) has reported work on Cd-, Zn-, and Ag-doped InSb. Although he observes a splitting for the C, D, E, and G lines, material quality was such that each of the line components was not well resolved. Thus, estimation of g-factors was not possible. Moore (172) has studied the magneto-optical properties of neutral double acceptors such as Be, Zn, etc. in Ge. He discusses his results assuming that the splittings he observed could be described by double acceptor wave functions constructed from the single acceptor results. Since small splittings were unresolved, a detailed comparison to theory was not possible. Although these references are not an exhaustive tabulation of experimental results, they indicate the dilemmas facing the experimentalist. O n one hand, the theoretical development is very complicated. O n the other, insufficiently pure materials broaden spectral features so that the various line components are not resolved. OF FREE AND BOUNDCARRIERS WITH VI. INTERACTION COLLECTIVE EXCITATIONS

The previous sections have been concerned with magneto-optical transitions among single particle (one electron or hole) electronic states. In this discussion the effects of the lattice ions which form the crystal are manifest only through the periodicity of the rigid lattice which gives rise to energy gaps, effective masses, etc., in the one-electron energy spectrum. Actually, the lattice is in constant motion and the electrons (holes) can interact with the quantized lattice vibrations (phonons) via several mechanisms: (1) polar or longitudinal optical (LO) phonon interaction, (2) deformation potential interaction, and ( 3 ) piezoelectric interaction. In addition, when a large number of electrons (holes) are present, as is frequently the case, these charged particles interact via their Coulomb fields, and there are additional modes of excitation (plasma modes or plasmons) corresponding to collective motion of the electron gas as a whole. These

20


modes bear a certain similarity to the LO phonon modes, and the single particle excitations can interact with the plasmons in a fashion somewhat analogous to the electron-LO phonon interaction (I 73).

A . Frohlich Theory of the Electron-Polar ( L O ) Phonon Interaction

The electron-LO interaction has received by far the most theoretical and experimental attention due to its widespread appearance (all compound semiconductors are polar in varying degrees). The importance of the polar phonon interaction has been demonstrated in transport measurements, Raman scattering, linewidths of cyclotron resonance, etc. The electron-LO phonon interaction is also of basic interest since it provides a model system for the case of a single particle interaction with a quantum field. For these reasons, and since other single particle-collective excitation interactions can be cast into a similar form, we give a brief description of the electron-polar (LO) phonon interaction in this section. Frohlich (1 7 4 ) provided the basic theoretical framework for subsequent work by describing the problem in Hamiltonian form x t o t

=

selec

+ sphonon + sinteraction .

(99)

To obtain an expression for this Hamiltonian, Frohlich used a model for an electron interacting with a polar lattice based on macroscopic dielectric theory (I 75). In this treatment the effect on the lattice of a single electron is described in terms of a polarization field (Plol)which is split into optical (Po,,) and infrared (PIR)contributions,

Here, Pop,corresponds to the electronic resonances of the dielectric function PI, corresponds to the lattice resonance at the transverse optical phonon frequency. If attention is restricted to longitudinal modes (for long wavelengths only the longitudinal modes carry a macroscopic polarization field), it is easily shown that

E(w),and

and

47T


21

where D is the electric displacement, and E , is the " high frequency" dielectric constant. Equation (102) holds since the lattice cannot respond to a sufficiently high frequency field, defined as a frequency well above the optical phonon frequencies but below the electronic resonant frequencies, i.e., below the fundamental energy gap of a semiconductor. For the electron-LO phonon interaction it is the lattice polarization, PI,, which is of interest, and from Eqs. (101) and (102)

At this point, two further assumptions are made: (1) the dielectric constant is independent of wave vector (valid for wavelengths long compared with the lattice spacing); and (2) the frequencies of interest lie below those of the optical oscillator (energy gap). Since D(x) is the externally applied displacement field, and P(x),, [Eq. (103)] is the effective longitudinal polarization field induced by the presence of an electron, the interaction energy density is given by -D(x) * P(x),, . With these results, the total Hamiltonian can be quantized and written in terms of creation and annihilation operators, b: and b,, for LO phonons of frequency wLo and wave vector q as

where R is the crystal volume, 1/B = (l/cm) - ( 1 / ~ ~ ) ,and the three terms correspond (in order) to the three terms of Eq. (99). It is convenient from a theoretical point of view to rewrite Eq. (104)in dimensionless units. For this purpose: (1) energy is expressed in units of ho,,; (2) length is expressed in ' ~ ~ ;(3) momentum is units of the "polaron radius" rP = ( h / 2 m * o 1 ~ ~ )and expressed in units of h/rp. In dimensionless units, the Hamiltonian as obtained by Frohlich is

where the dimensionless coupling constant a is given by

22


It should be pointed out that the interaction Hamiltonian, the third term of Eq. (104), is independent of the electron effective mass, and the mass appearing in the coupling constant, Eq. (106), comes about from writing the Hamiltonian in dimensionless form. In addition, note that the coupling constant reflects the ionic polarizability through the factor ( l/em) - ( l/cs). The polaron problem thus reduces to solutions of the Schrodinger equation with the Hamiltonian of Eq. (105). The type of approach that can be used depends on the size of the parameter a.For M < 1, perturbation theory is expected to be adequate, while for larger CI some form of variational technique has generally been used to obtain the energy spectrum. For M 6 1 it is easily shown from second-order perturbation theory that the energy is (again in dimensionless units) Epcrl=

-M

+ ~ ’ ( 1+ ~ / 6 ) .

(107)

This result is valid only for p < 1. The first term is a self-energy correction, i.e., the energy of the electron is reduced by an amount -cthoL,; and the second term represents a “dressing” of the effective mass, i.e.,

mP = m*/(l

-

~/6),

(108)

where m* is the unperturbed ‘‘ band mass. The presence of an external magnetic field can be taken into account by assuming that Eq. (105) is an adequate representation of this case when the momentum operator p is replaced by p - e A / c as in Section 11. (This has never been rigorously justified on theoretical grounds.) In the Landau gauge, and neglecting spin, ”

In terms of electronic creation and annihilation operators (see Section 11) the interaction Hamiltonian may be written ( I 76)

where


23

Here a:, a,, are creation and annihilation operators for electrons in Landau state v = n, k , , k,, respectively. Larsen (177) was the first to consider the correction to the Landau level spectrum resulting from the interaction Ham< 1. iltonian [Eq. ( 1 lo)] by using a perturbation approach valid for w, /aLo More recently Bajaj (178), using a slightly different approach, has obtained the Landau level spectrum for weak coupling and w,/wLo< 1.

where m* is the “band” effective mass and w, = eB/m,*c.The “corrected” cyclotron frequency for transitions between levels n and n + 1 is given by

For transitions originating at k , = 0 on the n = 0 Landau level, Eq. (112) yields the result obtained by Larsen (177). This describes an “effective nonparabolicity due to the electron-LO phonon (polaron) interaction with a very low field effective mass given by m&, = m,*/(l - 4 6 ) ; i.e., low field cyclotron resonance gives the polaron mass of Eq. (108). The “polaron nonparabolicity [second term of Eq. ( I 12)] results in a mass that increases with magnetic field (or frequency) as (3a/20)(wC/wLo)(n+ 1) for transitions at k , = 0. The polaron Landau levels at k , = 0 are shown schematically as the low field solid lines in Fig. 34 for n = 0, 1. For small coupling (a % l), the polaron increase in mass is very small and is extremely difficult to separate from the “ band nonparabolicity. However, in somewhat larger coupling semiconductors, this effect can be observed experimentally as discussed below. ”

”

”

B. Resonant Electron-Phonon Coupling 1. Theory of Resonant Electron-LO Phonon Coupling

An applied external magnetic field allows the tuning of individual electron (or hole) quantum states through a wide frequency range which can encompass the characteristic frequency of the phonons. Hence, a study of

24


q$$$.$$

Upper ranch

YO FIG. 34. Schematic diagram depicting the magnetic field dependence of the lowest two Landau levels (neglecting spin) as modified by the electron-LO phonon interaction (solid lines). The shaded area represents the continuum with threshhold given by the dashed line, E , . The two dashed lines which originate at the origin represent the unperturbed Landau levels.

magneto-optical transitions can reveal changes in the individual particle energy levels, transition linewidths, or transition probabilities (or a combination of all three) which are a result of the perturbation of the individual particle states due to the interaction. [A number of recent reviews discuss with varying emphasis resonant magneto-optical studies of electron-phonon interaction in polar semiconductors, see, e.g., Larsen et al. (I 79-182).] As shown initially by Johnson and Larsen (183, 183a) even a weak electron-phonon interaction can drastically affect optical transitions in semiconductors under condition of " resonance." Resonance, in this case, is achieved by magnetic-field tuning an appropriate pair of electronic levels until their energy separation is equal to (resonant with) the energy of the phonons of interest. In resonance, the upper state of the pair of levels is strongly perturbed, and this can be observed as a splitting into two branches (and broadening of the upper branch) of an optical transition which terminates on this level. A consideration of the case of free carriers in a parabolic band in the extreme quantum limit is illustrative. The interaction Hamiltonian is taken to be Eq. (1 10) and dispersion of the LO phonons is, as usual, neglected as is


the spin. The energy of an electron in the n of electron-phonon coupling is given by

=

25

1 Landau level in the absence

E(1, 0, k H ) = +ha, + (h2ki/2m*). Here the zero indicates the phonon ground state. The state (1,0, k H ) can be coupled to a state in the n = 0 Landau level with the emission of one LO phonon (0, 1, k H ) via the Hamiltonian Eq. (110). The energy of this unperturbed coupled state “containing” one LO phonon is (conservation of momentum requires k;, q H = k,)

+

where 4, is the component of phonon wave vector along the magnetic field. This expression neglects the offset to each of these Landau levels below oLo and the correction to the effective mass. The offset is schematically indicated in Fig. 34. For each value of k , , Eq. (114) forms a continuum with a threshold (where qH = k H ) E,

=

(ho,/2)

+ hoLo.

(115)

This threshold is plotted as a dashed line in Fig. 34. The density of states in the continuum is proportional to ( E - E,)-1’2. The n = I, no phonon Landau level crosses this threshold at ho,= ho,, - hzki/2m* and moves into the continuum at higher values of B. When the electron-LO phonon interaction is taken into account, these levels are expected to mix strongly near the region of crossover (resonance). With the continuum replaced by a single degenerate level at E , , the problem reduces to the often encountered quantum mechanical levelcrossing calculation. This was the approach used by White and Koonce (184), and their result is qualitatiuely correct. Namely, two branches are E + -, E , and E - -, (1, obtained, an upper, E + , and lower, E - : for small o,, 0, k,); and for large o,,E + --t (1,0, k,) and E - -, E, (as indicated schematically in Fig. 34). However, this calculation neglects the physical fact that the upper branch is degenerate with the continuum, and thus is not a true eigenstate of the system. Johnson and Larsen, who were the first to discuss theoretically the resonant electron-LO phonon interaction (183, 183a), treated the problem using Wigner-Brillouin (WB) perturbation theory. This calculation yielded a qualitatively different behavior for the upper branch near resonance. In the WB perturbation calculation, solutions for the upper branch terminate at a value of w, slightly below the crossover point. Solutions do not exist at lower The WB perturbation treatment breaks down for the upper values of 0,.

26


branch in the region of resonance due to the singularity in the continuum density of states at E , . It should also be pointed out that the WB approach is concerned only with the energies of the states and cannot give information about line shapes. On the other hand, this technique provides good results for the energies in the impurity case, where the singularity does not exist. The theoretical difficulties discussed above were overcome by Nakayama (285) who utilized a Green’s function approach to the problem. A previous Green’s function treatment (286), which was concerned only with the pole of the Green’s function, yielded results for the upper branch similar to the WB results with a slightly lower threshold. Nakayama showed that in spite of a singularity in the self-energy,

M ( -E

+ iq, k H )= A ( E , k R ) + iT(E,k,,),

q

+ 0,

(116)

FIG.35. A comparison of the Green’s function (heavy solid lines) and Wigner-Brillouin perturbation theory (dashed line for the upper branch) results for the magnetic field dependence of cyclotron resonance in the region of resonant interaction. The light solid line represents unperturbed cyclotron resonance. [After Nakayama (185).]

which is due to the singularity in the density of states of the continuum, the spectral weight function,

is well behaved for all values of E. Here A and r are the real and imaginary parts of the self-energy. The results for the peak in the spectral weight function above and below E , , E + and E - , respectively, are shown in Fig. 35 compared with the results of a WB perturbation calculation. In contrast to the WB result, E’ persists to very low values of magnetic field although with continually decreasing


27

intensity. Both treatments yield the same result for the energy of the lower the results for the upper branch coinbranch, and at high fields (0, % oLo) cide. In addition, both calculations predict a definite " offset '' in the cyclotron resonance transition energy going from w, < oLoto w, > wLo. The magnitude of this offset is slightly less than but approximately equal to aho,, and is weakly dependent on magnetic field (187). 2. Experimental Results It is convenient to categorize magneto-optical resonant coupling experiments according to the number of electronic " levels" participating in the process (181). In the simplest situation there are only two electronic levels involved. Thus, at resonance, the level separation, the phonon frequency, and the light frequency are all equal (in the absence of interaction). However, this can lead to experimental difficulties as discussed below. When possible, it is advantageous experimentally to use a three-level system. In this case, the energy separation between two excited electronic states is made equal to the phonon energy, while the initial state of the optical transition lies on a third level (ground state) and terminates on the upper level of the pair of excited states. Thus the energy of the photon absorbed can be considerably different from the phonon energy. The advantages of this situation will become apparent in the following discussion. a. Two-level experiments. In two-level experiments the optical transition between electronic levels may be overwhelmed by the optical activity of the phonons of interest, and thus may not be experimentally observable. This is the situation that occurs in studies of the resonant electron-LO phonon interaction in polar semiconductors. In addition, the theoretical situation is somewhat more complicated since, in this case, " vertex " contributions should be taken into account in addition to self-energy contributions in the Green's function formu1ation.t In spite of these difficulties, a number of useful two-level experiments have been carried out. The most detailed of these is the photoconductivity study of donor impurity transitions in InSb by Kaplan and Wallis (189). Their results are shown in Fig. 36. Although the experiment was performed ?For localized carriers only vertex contributions are present in two-level coupling, while in three-level coupling only self-energy effects are important. In fact, the results are of identical form for both contributions, and early calculations which treated only self-energy effects nonetheless gave correct results. The distinction can be more important for free carriers since self-energy and vertex contributions are ofdifferent form in this case. Here the relative contributions are strongly concentration dependent, and differences are most marked at high carrier concentrations. However, to date, measurements have been confined to the purest samples available, and thus these effects have not been experimentally revealed. For more details, see Kaplan and Ngai (181) and Economou et a/. (188).

28


I

o%oooo--A

I

I

30

l

l

l

l

35

l

l

l

l

l

l

l

I

40

l

l

45

l

l

l

l

I

50

MAGNETIC FIELD (kOe)

FIG. 36. A plot of “impurity cyclotron resonance’. energy vs magnetic field through the region of resonant interaction in InSb. The solid lines are the result of a Green’s function calculation (IYO), and the data points are taken from Kaplan and Wallis (189).

in regions of lattice absorption (at oT0) and reststrahlen reflection (between oT0 and wLo)a splitting at wLo is observed. This is made possible by the use of very thin samples and the extreme sensitivity of the photoconductivity technique. A number of other splittings are also observed at energies slightly above wLo. The electronic transition studied is the impurity transition (000) + (110) (Fig. 26). Two-level resonant splitting occurs when the (000) + (110) separation is equal to wLo [for coupling back to the (000) state]. In addition, three-level resonant splittings are observed when (000) + (110) is equal to wLo + (OlO), wLo + (020),etc. (for coupling to the various impurity excited states associated with the n = 0 Landau level). A theoretical fit to the data (190) is shown by the solid lines. A satisfactory fit is obtained with a coupling constant of a = 0.02. The origin of the splittings at A, B, C has not been conclusively identified (181). Resonant donor electron-LO phonon coupling has also been observed in CdTe (148), which is a much more polar material (a z 0.3-0.4). This

29


is shown in Fig. 37. In this experiment 1s -+ 2p (rn = f 1) Zeeman transitions were studied in magnetic fields sufficiently high that the 1s -+ 2p (rn = + 1) energy could be made to pass through the LO phonon energy. In addition, at the highest fields (-200 kG) the 2p (rn = 1) - 2p (m= - 1) energy can also be made to approach ha,, . Deviations of the experimental points from the calculated 1s -+ 2p (rn = + 1) transition energies based on band theory alone are apparent in the left-hand portion of the figure. The solid lines are obtained from a variational calculation for the hydrogenic atom in a magnetic field (131). In the right-hand side of the figure the same experimental points are shown compared with a WB perturbation calculation which takes into account resonant interaction between the 2p (rn = + 1) state and the 1s state + 1 LO phonon, and also the 2p (rn = - 1) state + 1 LO phonon. The value of the coupling constant used was a = 0.4, somewhat larger than that calculated from Eq. (106) (a = 0.28). Since there is a rather wide range of reported values of the high and low frequency dielectric constants in the literature (149, 1-50), it is not clear whether the discrepancy represents an inadequacy of the Frohlich model, an inadequacy of the variational calculation, or merely rather large inaccuracies in the

+

hw Y=$ 0 .I .2 .3 A .5 .6 .7 .8 ,

,

I

,

,

,

,

,

,

I

I

(IS-2P, M = - U 70 0

50

100

150

200

MAGNETIC FIELD (kG)

(a)

250

0

50

100

150

200

250

MAGNETIC FIELD (kG)

(b)

FIG.37. Experimental (solid circles) and theoretical (solid lines) magnetic field dependence of the donor electron Is --t 2p ( m = 1) transition frequencies in CdTe. (a) Theoretical calculation in the absence of electron-LO phonon interaction (a = 0). (b) Theoretical calculation for an electron-LO phonon interaction with a = 0.4. [From Cohn et al. (148).]

30


values of E, and E,. A discrepancy was also observed in a fit to cyclotron resonance data in CdTe. (See Section V1,D.) As pointed out above, for energies slightly greater than the resonant energy, the upper branch should approach an energy which is “offset” above the continuation of the low field w vs B curve for cyclotron resonance. This effect was initially observed and discussed by Dickey et al. (187) in experiments on InSb. Since the “offset” is rather small ( - 4 cm-’) compared to the transition energy ( > 200 cm- ’) and the experiments must be carried out close to the reststrahlen region, the experimental scatter was rather large, and a least squares fit to the low field data was used to demonstrate the effect. The results were consistent with a = 0.02. Following this first experiment, others have experimentally verified the presence of an offset in several materials (49,57,191).One of the clearest examples is the work of which is shown in Fig. 1 1 . From a Kinch and Buss ( 5 7 ) on Hg,,,,,C,,,,,Te careful fit of their low field data to the BY model these authors obtained a value of approximately 5 cm- for the offset which yielded a = 0.037.This is in excellent agreement with a calculated from Eq. (106) with experimental values of E, , c, and wLo for this particular alloy. This result may be fortuitous in view of the fact that Hg, -,Cd,Te is a mixed crystal system which exhibits two sets of LO and TO modes. In the present experiments coupling was observed only with the lower frequency LO mode. This point will be considered further with regard to the three-level experiments of McCombe (79). In addition to the splitting and offset effects just described, the resonant coupling also leads to an enhanced broadening of the upper branch near resonance. This broadening can be clearly observed even in two-level experiments. The effects of resonant coupling on the line widths of cyclotron resonance was initially considered by Harper (I 92), using perturbation theory. This theory predicts a discontinuous change in the width of the cyclotron resonance line which is proportional to c( and has a square root singularity (w - wLo)- This singularity reflects the density of final states in the continuum to which the excited electron can decay via the emission of one LO phonon. Experimental results for the linewidth in InSb are shown compared with the perturbation theory in Fig. 38. Reasonable agreement is obtained with a value of a between 0.02 and 0.03. However, it should be pointed out that experimental linewidths obtained from field swept experiments may be somewhat misleading due to the rapid bending of the upper By way of comparison the width branch toward a horizontal line at oLo. obtained from the spectral weight function [Eq. ( 1 17)] in the Green’s function treatment does not show a singularity at all, but rather an increasing width with decreasing frequency that saturates near w = wLo (185).In addition, the spectral weight function exhibits a very asymmetric shape im-

’”.

31


I

I

1.01

I

I

I

I

1.2

1.4

1.6

1.8

-0.02

20

30 40 50 MAGNETIC FiELD (kG)

60

FIG. 38. A comparison of experimental (open circles) and theoretical line widths as a function of magnetic field immediately above the resonant coupling region in InSb. Experimental points were obtained by subtracting a linear extrapolation of the low frequency line widths from the measured widths above wLo. The function 2Y(w,) is the line width obtained from the perturbation calculation discussed in the text. [From Summers et al. (191).]

mediately above wLo, a feature which has been confirmed by experiment (193). Thus, extremely careful analysis is required to obtain reliable values of CY from such studies. The two-level experiments described above, although demonstrating all of the qualitative aspects expected from resonant electron-LO phonon interaction, do not allow accurate determination of the coupling constant. Hence, they do not provide a stringent test of the Frohlich model and its generalization to include the presence of an applied magnetic field. b. Three-level experiments. Experimental investigations of three-level systems can provide a more stringent test of the theories since they are free from most of the complicating effects inherent in the two-level systems discussed above. The primary advantage of three-level experiments derives from the following fact. The electronic state to which the final state of the optical transition is coupled via the emission of one optical phonon is not the initial state of the optical transition. Thus, provided the coupled state lies far enough in energy above the initial state, the optical transition will occur in a spectral region free from strong lattice absorption, i.e., well above the reststrahlen frequency. However, in order to obtain a plot which shows a horizontal discontinuity at the energy of the optical phonon as in the two-level case, the energy difference between the coupled state and the initial state must be subtracted from the experimental transition energies. Unfortunately, there are only a few situations in which three-level experiments can be carried out. The first such experiment was the interband magneto-absorption study of Johnson and Larsen on InSb (183, 183a). These results established the presence of strong resonant effects, but due to the complexity of the valence

32

BRUCE D . MCCOMBE A N D ROBERT J. WAGNER

band states (Section IV) and further complications due to excitonic effects, a quantitative analysis has proved to be very difficult. In the first intraband three-level experiments Dickey and Larsen (194) made use of the localized electron analog of the combined resonance transition (Section IV,B) to provide a closer examination of resonant effects. In this three-level case, the EM absorption takes place between the (000; spin up) and (110; spin down) states, while the electron-LO phonon interaction connects the final state (110; spin down) with (000;spin down). These experiments revealed unexpected structure (three absorption peaks rather than two) in the resonant region. The double splitting was interpreted as resonant interaction with both LO and T O phonons with comparable coupling strengths. Subsequently, McCombe and Kaplan (I 95) reported combined resonance studies of both free and localized electrons. These results indicated that the additional splitting in the localized electron combined resonance was due to LO phonon coupling between the final state, (110; spin down) and (010; spin down) in analogy to the impurity cyclotron resonance studies described above (Fig. 36). Other measurements have also shown that the electron-TO phonon interaction must be weak (196). Only one splitting was observed in the free carrier measurements, and no evidence of T O phonon coupling was found. The free carrier results are shown in Fig. 39, compared with the Green’s function calculation of Nakayama (185). The experimental results are in good agreement with theory for c( = 0.02. To our knowledge these are the only detailed experimental results which have been obtained throughout the resonance region for free carriers. In related work McCombe (79) has reported the observation of resonant coupling in Hg, -,Cd,Te (x = 0.203). The Hg,-,Cd,Te mixed crystals all exhibit two-mode ” behavior throughout the composition range. In such crystals two sets of LO and TO modes are observed, one identifiable with Cd-Te and one with Hg-Te vibrations at arbitrary alloy compositions; the mode strengths are roughly proportional to the concentration of the constituents. Splittings were observed at each of the LO modes and the data were fit to a Green’s function calculation with a parameterized interaction Hamiltonian assumed to be given by “

where each of the terms has the form of Eq. (110) with “effective coupling constants u 1 and a 2 , respectively. The values a , (Cd-Te) = 0.02 f 0.005 and a2 (Hg-Te) = 0.0175 f 0.005 were obtained from a best fit to the data. These values must be viewed simply as parameters determined from the ”

INTRARAND MAGNETO-OPTICAL STUDIES. I1

33

B(kG) FIG. 39. Magnetic field dependence of the free electron spin-down cyclotron resonance energy, (I, - ) -+(O- -), through the resonant coupling region in InSb. Experimental points [from McCombe and Kaplan (195)] were obtained by subtracting the measured spin resonance energy, (0, - ) + (0, ), from the combined resonance energy, ( I . - ) -+ (0, + ). Solid lines are taken from the theoretical results of Nakayama (185) with CY = 0.02.

+

assumed model of the interaction, since it is not clear how the Frohlich model should be rigorously generalized to the mixed crystal case. However, such measurements should provide motivation for additional theoretical work on this problem. The experiments described thus far have been concerned with the coupling of electrons occupying states in s-like conduction bands to optical phonons. For this case there is no compelling experimental evidence for the coupling of these carriers to long wavelength (q = 0) TO phonons (196). This is consistent with symmetry arguments (197) which show that the matrix elements for carrier-1 nonpolar optical (NPO) phonon interaction, via the optical deformation potential, vanish in the long wavelength limit for s-like bands. The deformation potential is the only possibility for TO

34


phonons since there is no macroscopic polarization field associated with these modes. On the other hand, carriers in p-like bands can have a nonvanishing deformation potential interaction with q z 0 TO phonons. The distinction between s-like and p-like bands for this interaction has been clearly demonstrated in tunneling experiments (198). Due to the complicated nature of the free carrier states in the degenerate valence bands of zinc-blende and diamond structure semiconductors and the resulting multiplicity of observed transitions, (Section IV,C), magnetooptical studies of the free hole-phonon interaction in these materials are extremely difficult. On the other hand, localized acceptor states, although quite complicated as discussed in Section V,B, nonetheless form an inherently simpler system for experimental study. This is particularly true in the high field region at low temperature where a unique acceptor ground state is occupied (299). Acceptor transitions are also more advantageous for the study of carrier-TO phonon coupling since the magnitude of the effective deformation potential interaction increases with increasing localization, in contrast to the Frohlich interaction. This can be seen from a comparison of the interaction Hamiltonians in the two cases. The Frohlich Hamiltonian is given in Eq. (1 lo), and the Hamiltonian for the deformation potential coupling of holes to TO phonons in the degenerate valence bands of a zincblende semiconductor is given by (193)

Here

where i, is the unit polarization vector of the vibration, M A and MB are the masses of the atoms, po is the mass density of the crystal, do is the optical phonon deformation potential, Af(&,)is a 4 x 4 matrix, FC(x) is an envelope function for an acceptor in state v , j andj' run over the four degenerate band edge states, and a. is the lattice constant. In this case the entire q-dependence comes from the matrix element of Eq. (120) since the prefactor is independent of q. This is to be contrasted with the polar interaction, Eq. (1 lo), where the prefactor is proportional to 1 q 1- '. Thus, the cutoff on the q-summation [Eq. (119)] is determined roughly by the inverse of the carrier orbit dimension; for localized carriers


35

this is given by l/a;F; with a: an appropriate effective Bohr radius; for free carriers 1 = 9- = (&/eB)’/’ specifies the orbit radius. At typical magnetic fields of interest a;F can be much less than 1, and hence the effective interaction with localized carriers can be much stronger. To date, the only clear experimental evidence of hole-optical phonon coupling from magneto-optical investigations is the experiment of Kaplan ef al. (193) for localized acceptors in InSb. At the magnetic fields of interest in these experiments (20-60 kG), the acceptor transitions separate into two general categories; those associated with light hole levels, and those associated with “heavy” hole levels. The former states appear to be quite similar to the y % 1 donor electron states discussed in Section V,B, while the latter states, since the effective y is much less than 1, form a series which lies above the valence band continuum states. The transition studied by Kaplan er al. connects the acceptor ground state to an excited state associated with the (2, - 3/2) light hole Landau level [in the (n’, m,) notation of Section IV,C and Fig. 201. This transition was studied at magnetic fields such that the separation between the final state and an impurity excited state associated with the heavy mass continuum states was comparable to the optical phonon energies. The results are shown in Fig. 40. The optical transition appears split into three branches, corresponding to two nearly horizontal discontinuities at 31.5 and 33.0 meV. The coupled state of this three-level system was identified as a weakly field dependent excited acceptor state associated with the heavy mass continuum “

”

1

L100

MAGNETIC FIELD ( k O e )

FIG.40.Magnetic field dependence of acceptor transition energies in InSb. This transition connects the ground state to an acceptor excited state associated with the (2, -3/2) light hole Landau level. Solid lines are the result of the Green’s function calculation described in the text. [From Kaplan et ul. (193).]

36


states. This level is also observed as the final state of a strong, sharp transition which moves from 8.1 to 8.9 meV between 20 and 60 kG. It appears that this transition is related to the zero-field transition labeled “ C ” in Fig. 33. When the data is reduced by subtracting this energy from the observed transition energies of Fig. 40, the two discontinuities appear at 22.8 and 24.3 meV, in excellent agreement with the energies of zone center TO and LO phonons in InSb. In order to provide a quantitative comparison with theory, the authors made use of a model approximation to the actual acceptor wave function in Eq. (120), since rigorous wave functions are not available. The solid lines in Fig. 40 show the calculated results for the peak of the spectral weight function with Eqs. (110) and (119) taken to be the interaction Hamiltonians for LO and TO coupling, respectively. The Frohlich coupling constant was fixed at a = 0.02, and do was adjusted to obtain the best fit to both line position and shape. The near equality of the observed splittings for both TO and LO interactions derives from the fact that the coupled acceptor state is associated with heavy mass continuum states and is thus quite localized. From these results a value of do = 45 eV was obtained. This confirms the importance of TO phonon scattering in the analysis of hole mobilities in semiconductors (200). C . Resonant Electron-2NP0 Phonon Coupling

1. Background It has generally been assumed that the interaction of carriers with multiple phonons can be neglected in experiments such as those described above. These effects are of higher order and thus are expected to be weak compared to the one-phonon interaction. In addition, there was no experimental study or quantitative theoretical calculation providing evidence to the contrary. Recently, however, experimental and theoretical studies of resonant electron-optical phonon interaction have demonstrated that the effective electron-2 nonpolar optical (NPO) phonon interaction is readily observable (201). It is useful to examine the form of the interaction in some detail in order to point out certain important features which may not be otherwise apparent. The nonpolar interaction between carriers and phonons can be expanded in powers of the lattice displacement, u m p , Xint = -

c

Ump

mB

. V R U(X- R m p )


31

where m and fi specify the unit cell and the atom within the unit cell, respectively, R,, is a lattice vector, and U ( x - Rmp)is the potential at lattice site m, p. The first term gives rise to the usual one-phonon deformation potential interaction discussed in Section VI,B,2. The term bilinear in the phonon amplitude, A?:?;, corresponds to interactions in which two phonons are simultaneously created and/or destroyed. However, interactions involving two phonons can also occur via second order matrix elements of the first term, A?:?,,. Thus the matrix element (ME) for a transition or scattering from one electronic state to another via the emission and/or absorption of two phonons must be written schematically as

where v, v’, and v” refer to the initial, final, and intermediate states, respectively. Each state is specified by a set of quantum numbers for the phonons and a set for the electrons. Of course, energy must be conserved between initial and final states, and momentum must be conserved between all pairs of connected states. The possibility of interference between the two types of terms in Eq. (122) makes it difficult to draw any quantitative conclusions about the magnitude of the bilinear term, particularly since it has been shown (202) that there is a general tendency toward cancellation between the two terms. Ngai and Johnson (201) have presented arguments which indicate that the cancellation should be much less severe for the case of resonant coupling in InSb. These arguments were based, in part, on the fact that the I-NPO interaction for q z 0 T O phonons vanishes in s-like bands, as discussed above. For large q TO phonons these symmetry arguments are not valid. Thus, since the sum over intermediate states in Eq. (121) must include all points in the Brillouin zone, due to conservation of momentum there may be some contribution in the first term from large q phonons. It is difficult to assess the magnitude of this contribution ; hence the results discussed below must be viewed as yielding values of an effective 2 N P 0 phonon deformation potential, 9, and not the value of the deformation potential corresponding to A?::;, 9 2 N P O . 2. Experimental Results In the experiments of Ngai and Johnson two weakly allowed electronic transitions were studied in the spectral region of two optical phonon absorption. These transitions are: (1) LO phonon assisted cyclotron resonance, which will be discussed in more detail in Section VI,D, and (2) a transition which occurs close to the first harmonic of cyclotron resonance (- 204). The

38

BRUCE D. MCCOMBE AND ROBERT J. W A G N E R

latter transition is apparently a high field hydrogenic impurity transition originating on the (000) state and terminating on a state associated with the n = 2 Landau level, probably (210). (The nature of these “harmonic” transitions is discussed further in Section V1,E.) The experimental results are shown in Fig. 41. A zero field transmission spectrum is shown on the left where arrows indicate features which are identified with two-phonon absorption peaks. Two-phonon energies at various points in the Brillouin zone are indicated on the right. A number of splittings of the magneto-optical transitions are observed as shown in the central portion of the figure. Attention was focused on the splitting in the “harmonic” transition in the vicinity of 2T0, and 2T0,. The magnitude of this splitting is roughly 1 meV.t

+--

c-c--

‘ l 04

TRANSMISSION (ARB. UNITS)

H (kG)

FIG.41. Left, photon energy dependence of the lnSb sample transmission in zero magnetic field. Center, magnetic field dependence of the transition energies for LO phonon-assisted cyclotron resonance (solid circles) and the “first harmonic” of impurity cyclotron resonance (open circles). Right, two-phonon energies as obtained from one-phonon neutron scattering data. [From Ngai and Johnson (201).]

The interaction Hamiltonian for an electron interacting with 2 N P 0 phonons can be written in terms ofa parameter (9p112)9(181, 201), where B denotes the set of 2 N P 0 critical points in the Brillouin zone (e.g., X , - X ) , 9 is the effective 2 N P 0 deformation potential, and p9 is a number less than one, which is a measure of the relative volume contributing to the twophonon critical point s.This number is not easily obtained since its evaluation requires a knowledge of the lattice spectrum throughout the Brillouin zone. From a Green’s function calculation using the Hamiltonian just described, Ngai and Johnson determined the splitting to be proportional to

t It should be pointed out that independent measurements of direct cyclotron resonance in the region of the two-TO phonon energies have failed to reveal a splitting (203). Hence further experimental verification of the resonant 2 N P 0 coupling may be in order.


39

( 9 2 p ) z . A comparison with the data yielded a value ( 9 ~=~ 1.5’x ~ lo4) eV ~ (181). Since p? is less than one, and since there is certainly some cancellation

By way of to be expectd, this value is probably a lower limit for gZNPO. = 3.46 x lo5 comparison Lin-Chung and Ngai (204) have obtained 9,,,, eV from a simplified orthorgonalized plane wave calculation, a result compatible with experiment. It appears that the 2 N P 0 interaction may be important in Raman scattering, electrical transport measurements, free carrier absorption, and superconductivity (181, 205). D . Nonresonant Electron-Phonon Coupling

Unlike the case of resonant coupling described above, it is generally difficult to isolate the effects of electron-phonon interaction under nonresonant conditions. Since in most semiconductors the coupling is weak, i.e., CI < 1, the resulting small self-energy and mass renormalization are difficult to separate from band structure (e.g., nonparabolicity) effects. This is in contrast to the case of strong coupling materials such as the alkali, silver, and thallous halides (CI % 1.6-4) where a number of useful cyclotron resonance experiments have been carried out under nonresonant conditions (34, 182). Nonetheless, careful measurements on high quality weak coupling materials can yield information concerning the electron-LO phonon interaction as demonstrated by the case described below (CdTe). Another consequence of the electron-LO phonon interaction observed in the nonresonant region is the possibility of weak, second-order transitions involving the emission or absorption of an LO phonon. These transitions have been studied by a number of workers and are discussed below. Finally, a number of nonresonant experiments involving the coupling of electrons to acoustic phonons via the piezoelectric interaction are also discussed in this section. 1. Polaron Cyclotron Resonance: CdTe

As pointed out previously, the electron-LO phonon interaction gives rise to a self-energy shift and correction to the effective mass of the charge carrier which are proportional to the strength of the interaction, a. However, the mass obtained from low frequency cyclotron resonance measurements is the “dressed” polaron mass [Eq. (log)], and since accurate independent determinations of the band mass are not typically available, it is difficult to assess the effects of the interaction from such measurements. Fortunately, as indicated in Eq. (1 12), the interaction leads to a polaron nonparabolicity ” in addition to the usual band nonparabolicity. Although in small gap, weak coupling semiconductors this polaron contribution is small compared to the “

40


band nonparabolicity, a more favorable situation occurs in CdTe where E , = 1.6 eV and a x 0.3 - 0.4. Thus Waldman el al. (206) studied the magnetic field dependence of electron cyclotron resonance in CdTe at low temperatures and at frequencies well below wLo . The resonance was observed through the use of several FIR lasers which covered the spectral region between 30 cm-' and 84 cm- Accurate measurements of the magnetic field position of cyclotron resonance peak were made with an in situ NMR probe. The results, expressed in terms of an effective mass, defined by mcy= e B / o , c, are shown in Fig. 42. In order to fit the experimental data, a

'.

tt

0.106

0.104

P

O.Io2 0.100

--

t

I

v / a

= 0.3

t/EL-

0

20

40

60

80

100

12

MAGNETIC FIELD (kG)

FIG.42. Magnetic field dependence of cyclotron effective mass in CdTe. The solid circles are experimental points (with error bars indicated), and the solid lines are the results of the theoretical calculation described in the text for the indicated values of the coupling constant, c(. [From Larsen (179).]

more sophisticated variational calculation was used for the polaron nonparabolicity, and the band nonparabolicity was accounted for with a simplified BY model. The theoretical calculations are plotted as the solid lines in Fig. 42. It is clear that band nonparabolicity alone (a = 0) cannot account for the field dependence of the mass. The data is best fit with a = 0.4, in agreement with the resonant two-level donor impurity studies, but again in disagreement with a calculated from Eq. (106). As discussed in Section VI,C the reasons for this discrepancy are not clear.


41

2. LO Phonon Assisted Cyclotron Resonance Since the Frohlich interaction connects two Landau levels of arbitrary n, photon absorption can take place via second-order processes involving the emission or absorption of an LO phonon and the simultaneous elevation of an electron from Landau level n to n’. The theory of these LO phonon assisted cyclotron resonance (LOCR) transitions was initially presented by Bass and Levinson (207), and similar treatments have since been given by others (42,208).Briefly, the LOCR is a second-order process which involves [Eq. (lo)], and the electronboth the electron-radiation interaction, PR, [Eq. (104)l. From second-order perturbation LO phonon interaction, Xint, theory the transition rate may be written

where 0, i, f refer to initial, intermediate, and final states, respectively. Each state is specified by a set of electronic quantum numbers (n, k , ,and k H ) ,and a quantum number for the phonons. The theoretical results for free carriers show that there is resonant absorption for E’ I B at frequencies satisfying

where spin and nonparabolicity have been neglected, r = 1,2,3,. . . ,and the i-sign corresponds to emission and absorption, respectively. For E /I B there is no resonant absorption for simple, spherical bands (208). The LOCR transitions have been observed by a number of workers in several materials, both for free carrier (73, 208-211) and localized (42, 196) electronic states. For spherical conduction bands all of the qualitative features including polarization selection rules, position of the resonance, and order of magnitude of the intensity (208) have been verified by experiment. Saleh and Fan (210) have recently extended theory and experiment to encompass many-valley semiconductors. Additional features expected for many-valley semiconductors (e.g., resonant absorption for E /I B) were experimentally observed in n-PbTe. Although these experiments provide qualitative verification of calculated effects of the electron-phonon interaction, quantitative results are not expected from such measurements since the coupling constant appears only in the expression for the intensity of the transitions.

42


3. Piezoelectric Electron-Phonon Interaction: Piezopolaron

Charge carriers can be coupled to acoustic phonons via the electric fields associated with certain acoustic modes in piezoelectric materials. Such coupling was initially considered by Mahan and Hopfield (212) to provide a theoretical explanation for a discrepancy between effective masses derived from low temperature microwave cyclotron resonance (213, 214) and from high temperature or high frequency measurements [see Mahan and Hopfield (212) for references] in the strongly piezoelectric semiconductor CdS. The cyclotron resonance measurements gave rn,*(ll) = 0.157~1,and rn,*(l) = 0.177m0, while the other measurements yielded values of m* between 0.19~1, and 0.21m0. In other words, the low frequency measurements gave an average effective mass approximately 15% smaller than that obtained from high frequency or high temperature experiments. From a perturbation theoretical calculation of the self-energy correction due to the piezoelectric interaction, Mahan and Hopfield obtained a mass shift in the semiclassical limit (kB T % hw) that agreed with experiment in sign (negative) and order of magnitude. In this analysis the assumption is made that the high frequency (temperature) results give the “undressed band mass since the acoustic phonon frequencies are much lower. It should be emphasized that the mass shift in this case is opposite to that obtained from the usual LO phonon interaction, Eq. (108). The difference in sign of the shift is due to the fact that the major contribution to the piezopolaron self-energy comes from real acoustic phonons which are excited at finite temperature ; the usual polaron self-energy corrections, on the other hand, come from oirtual emission and absorption of LO phonons. Although the results of Mahan and Hopfield were in qualitative agreement with experiment, there are a number of possible objections to this approach. In particular, the semiclassical limit is not appropriate for the experiments since ho,> k , T . As a result, since this early work, a number of authors have considered the problem under different approximations and with differing results (215-21 7). These calculations are basically in agreement concerning the sign of the mass (or frequency) shift, but not the magnetic field and temperature dependence [see, e.g., Miyake (217)]. Unfortunately, FIR cyclotron resonance studies have contributed little toward the clarification of the problem. Temperature dependent studies (218-220) have been complicated by interference and Faraday rotation effects (221, 125) mentioned in Section IV. Thus it is not possible to draw reliable conclusions concerning the temperature dependence of the mass shift from these measurements. Likewise, low temperature measurements of the magnetic field dependence of “cyclotron” resonance in the range 20-70 cm-’ (152, 220) are complicated by the fact that a straight line fit to ”


43

the data extrapolates to % 7 cm-’ (rather than zero) at zero field. Thus, although the slope of the line through the data points gives a constant effective mass, m,* 0.19rn0,the individual resonance peaks, when converted to effective mass, yield values that increase from 0.15m0 to 0.18m0 over the region of observation. It has been suggested (152) that the resonance is actually associated with a shallow trap, and is thus not true cyclotron resonance. Again, due to this uncertainty it is not possible to ascertain reliably the magnetic field dependence of the mass. The diversity of experimental and theoretical results precludes the possibility of drawing conclusions about the importance of the piezoelectric electron-phonon interaction at this time. Additional careful experimentation, particularly in the low frequency region between 5 cm- and 20 cm- could prove to be extremely useful in this regard.

-

’

’,

E. Electron-Plasmon (Plasmaron) Interaction

A number of years ago Lundqvist (173) pointed out the similarity between the electron-LO phonon interaction and the interaction between single particle and collective excitations of an electron gas. In the random phase approximation the total Hamiltonian for the interacting electron gas can be written (173, 222) ZIOl =

+ .Yrrppl,

(125)

where

and

with

In these equations o,(q) is the frequency of the plasmon with wavevector q, b: and b, are creation and annihilation operators, respectively, for plasmons of wavevector q, and E ~ ( wis) the wavevector and frequency dependent dielectric function for the electron gas, which is assumed to be immersed in a . sum is over those plasmedium of background dielectric constant E ~ The mon modes which are stable against decay into an electron-hole pair, i.e., those modes which have I q I less than a critical value I qc I defined by the

44


intersection of the plasmon dispersion curve with the single particle electron-hole pair dispersion curve. In the lossless case in the long wavelength limit the dielectric function is &qaO(W)

&B[l - o;/02],

(129)

where o p= [4mVe2/m*~B]'/Z is the plasma frequency for q x 0. In this simple case X e _ phas l precisely the same form as the Frohlich Hamiltonian [Eq. (104)], with the Frohlich coupling constant [Eq. (106)] replaced by (222)

In the presence of a magnetic field this analogy is modified since, in contrast to LO phonons, the plasmons are strongly affected by the field. For q sufficiently large that cq > o the magnetoplasmon modes are given by (223) w:(e) = + o;) [+(o:+ - o;o;C O S ~01112, (131) where 8 is the angle between the direction of propagation and the magnetic field. The interaction Hamiltonian for the magnetic field case can be written (224, 225)

X e P p=l zI/:;,(q)[bL, vv'q

+ bq]u,?,a,,

(132)

where v denotes a single particle Landau state, and the prime indicates that the sum over q must be restricted to values such that the magnetoplasmon is a reasonably well-defined excitation. For o + magnetoplasmons

A similar expression with a, and w - interchanged holds for coupling to the o- magnetoplasmons. The implications of this interaction for magneto-optical studies were initially considered by Teitler et al. (222). It was determined that resonant coupling experiments must be ruled out since o,can be made to cross the magnetoplasmon frequency only for q parallel to the magnetic field. For this case the matrix element in Eq. (133) between two adjacent Landau states vanishes. On the other hand, these authors concluded that second-order transitions analogous to the LO phonon-assisted cyclotron resonance discussed in Section VI,D might be observable. McCombe et ul. (224) have reported the observation of such magnetoplasmon assisted magneto-optical transitions in InSb. An experimental


45

transmission trace is shown in Fig. 43. The data were obtained in the Faraday geometry with a FIR laser and circular polarizers that were approximately 90% efficient. This geometry was chosen since there is no “classical” coupling between the cyclotron motion and the longitudinal plasma motion (5). A prominent feature of the data is the absorption line (1) in the CRI polarization. The strong broad absorption in the CRA polarization with a wing extending into CRI results from over-absorbed cyclotron resonance in this thick sample. Due to the imperfect polarizer the CRI line (1) is weakly replicated in the nominally CRA polarization. Voigt magnetoplasma resonance measurements were also carried out on the sample; the frequency of

I

16 14

-

IZ 10 8 6 4 2 0 2 4 6 6 10 12 14 I6 18 20 2224 26 28 Nepallve +-. I F’oslllve

0 (kG)

FIG. 43. Magnetic field dependence of the transmission of 86.7 cm-’ laser radiation through n-type InSb for the two senses of circular polarization. The magnetic field position of free electron cyclotron resonance determined from a separate measurement on a thin sample of pure InSb is shown by the arrow labeled B , . The plasmaron “fundamental” and “harmonics” are indicated by the arrows labeled (l), (2),(3), respectively. [From McCombe et a/. (224).]

this resonance is w = [w: + w,5]”’. At lower laser frequencies the CRI line appeared at lower fields than the magnetoplasma resonance; as the operating frequency was increased the two lines merged together. In addition, at a given laser frequency the CRI line moved to lower fields as the carrier concentration (i.e., up)was increased. The CRI line was attributed to a second-order transition shown schematically in Fig. 44. The first step in the transition is virtual electron-hole pair creation at A accompanied by magnetoplasmon emission (momentum - q) via the interaction Hamiltonian of Eq. (132). In this step an electron in state I n = 0, k,, k,) makes a transition to a virtual intermediate state I n = 1, k , + q,, kH + q H ) at B. Momentum but not energy is conserved in the

46


TE

/

n=2

FIG. 44. Schematic diagram depicting the second-order processes responsible for the plasmaron “fundamental” (via intermediate state B) and first “harmonic” (via intermediate state C). The effective Fermi level at T = 0 is denoted by c F .Wavy lines indicate that part of the transition induced by the electron-radiation interaction. The size of the wavy lines is not intended to indicate the energy of the photon absorbed, which must be obtained from energy and momentum conservation as discussed in the text.

intermediate state. In the second step at B a photon is absorbed, and the qH). electron makes a transition to the final state I n = 0, k , q y , k , The second step also determines the polarization selection rule, CRI since An = - 1. The energy of the photon absorbed can be obtained from energy and momentum conservation

+

h o = ha+

+ [E(n = 0,

kH

f q H ) - E(n = 0, k H ) ] ,

+

(134)

where w + is the frequency of the appropriate magnetoplasmon propagating nearly perpendicular to the magnetic field [a+z [a: + wi]1’2 from Eq. (131)l; and the second term, the increase in kinetic energy of the excited electron, is of the order of the effective Fermi energy, E ~ at, T = 0. Thus, c c in the quantum limit, the CRI line should occur at higher since E ~ ( B ) 1/B2 energies (lower fields) than the magnetoplasma resonance but should approach the latter at higher fields, in qualitative agreement with experiment. A quantitative theoretical calculation of the absorption coefficient for this process gave generally good agreement with the experimental results for peak position and strength, as well as qualitative agreement for line shape (224). Finally, in addition to the “ fundamental” absorption process, harmonic transitions are also allowed in which the final electronic state lies on successively higher Landau levels. The “first harmonic” process is indicated in Fig. 44. The phonon energy for these transitions is given by hw = h[rw, + w + ] , where r = 1, 2, 3, .. . (225, 226). For w, $ w p these resonances occur at slightly lower fields than exact harmonics of cyclotron resonance. This process appears to provide an explanation for the observed “

”

47


“ harmonics”, (2) and (3) in Fig. 43, for Y = 1 and 2, respectively. A quantitatively theoretical calculation for the absorption constant is in quite reasonable agreement with the position and strength of the observed harmonics at low magnetic fields (226). O n the other hand, for very pure samples or high magnetic fields, donor impurity transitions become important. Miyake (227) has recently calculated the strength of impurity cyclotron resonance harmonics [(OOO) + (Nlil), N >, 21 for InSb in the high field limit. The calculated intensities appear to be in order of magnitude agreement with experimentally observed harmonics in high purity InSb (228). For intermediate fields and impurity concentrations it is not clear which of these mechanisms is operable or, indeed, whether or not both contribute to the observed lines. Further careful experimental studies as a function of temperature and carrier concentration are required to establish definitively the mechanism for the “ harmonic” transitions. Nonetheless, the electronplasmon interaction must be considered a viable explanation for the free carrier “fundamental” and “harmonics,” and thus it appears that this interaction may be important in other areas, e.g., free carrier absorption at low temperatures.? “

”

VII. FUTURE DIRECTIONS In this review a large variety of experiments on many different semiconducting materials have been described. From the preceding it is clear that FIR studies of semiconductors is an area of research that has reached some maturity and in which measurement techniques are rapidly becoming routine. Thus in the future one can anticipate even more widespread use of such techniques for a whole range of investigations, e.g., surface properties, manybody effects, defects, and nonequilibrium properties. We discuss in this section a number of specific examples of research problems which appear fruitful for the near future. Some of these are simply extensions of work discussed in the body of this review in which theoretical or experimental understanding is incomplete. On the other hand, we describe some interesting new problems which are being investigated with FIR magneto-optical techniques, and we also indicate some possible areas of future interest which have received little or no previous attention.

t In a recent paper, J. Blinowski and J. Mycielski [Phys. Lett. A 50, 88 (1974)], have questioned whether the E field can couple to the electron-(magneto)plasmon system. Although this criticism is valid for an unbounded plasma oscillation with an energy independent effective mass in a translationally invariant system, it has been shown that the presence of impurities and/or nonparabolicity allows the coupling to occur [J. J. Quinn, T. L. Reinecke, K. L. Ngai, B. D. McCombe, S. Teitler, and R. J. Wagner, Bull. Amer. Phys. SOC. [2] 20, 494 (1975)l.

48


As was apparent in the foregoing sections, cyclotron and related free carrier resonances have been extensively studied and have been extremely useful in elucidating band-edge features in semiconductors. This is true not only for small gap semiconductors with spherical conduction bands, but also for materials with degenerate (Ge) and anisotropic (Te) bands. As new materials become available FIR free carrier resonance experiments will certainly be useful in the determination of band parameters, particularly in heavy mass, low mobility materials. It is doubtful, however, that unusual or unexpected effects will be discovered in other zinc-blende or diamond structure semiconductors. On the other hand, there are a number of heavy mass materials in which band structure calculations are in a rather rudimentary state. For materials such as layer compounds, ferroelectrics, and alloy and magnetic semiconductors, FIR magneto-optical experiments may not only provide information for band structure calculations but may also reveal new effects due to interactions of the carriers with other excitations or perturbations. The level of understanding of simple hydrogenic impurities in semiconductors has reached a state of sophistication where magneto-optical techniques may be useful in analytic studies of impurity type and concentration. In contrast, acceptor states associated with degenerate valence bands have not been extensively investigated via magneto-optical techniques. This is due both to the scarcity of adequately pure materials and the theoretical complexity of the degenerate valence bands. While some low field Zeeman studies have been performed, the development of these low field states into high field limit states has not been investigated. In cases where the donor or acceptor binding energy is large the effective mass approximation becomes inadequate to describe the more tightly bound states. This difficulty, caused by the wave function localization around the central cell of the impurity, is apparent in the discrepancies between theory and experiment for ground state energies of specific impurities. Although the typical excitation frequencies for the deeply bound impurities occur in the middle or near infrared, magnetic field splittings of the ground state can be comparable to FIR photon energies. Thus it might be possible to study the ground state splittings and thereby provide a considerable aid to understanding the central cell effects. Although it has been widely studied because of its simplicity, the hydrogenic impurity center is only one of many defects which may occur in semiconductors. Other defects are known to exist and to have important technological consequences for both electrical and optical devices. To date, there have been very few attempts to study these defects via magneto-optical techniques. Even though the energy levels and optical selection rules for such centers are expected to be much more complicated than for simple


49

hydrogenic impurities, it seems likely that FIR magneto-optical experiments could be useful in the characterization of these defects. As has been indicated in Section VI both free and bound carrier resonances have been used to study interactions between the charged carriers and other excitations and/or perturbations in the solid. The principle example of such investigations is the extensive work on the electron-LO phonon interaction. Another illustration is the use of bound carrier resonance experiments to study the effect of the electric fields of ionized impurities on the neutral donors in GaAs. These examples suggest that magneto-optical resonances, once well-characterized from a single-particle viewpoint, may be generally useful for the study of interactions of the carriers with other microor macroscopic features of the sample. Additional experiments previously mentioned which typify this approach are: cyclotron resonance line shape studies to elucidate scattering processes at low temperatures; the studies of " harmonic " cyclotron resonance which have given evidence of electron2 N P 0 phonon interaction; attempts to observe the effects of electronpiezoelectric (acoustic) phonon interaction from cyclotron resonance measurements ; and finally, the observation of weakly allowed transitions in InSb which provide evidence for an observable electron-plasmon interaction. While each of these experiments represent promising new initiatives, in a number of cases further experimental and/or theoretical work is required. Furthermore, as new, high quality semiconductor materials become available, the importance of these interactions may be more clearly and conclusively demonstrated. Very recently, promising experimental work has been reported in two new areas. In the first of these FIR cyclotron resonance experiments have been used to study the two-dimensional electron gas induced in the semiconductor (Si) surface of a metal-oxide-semiconductor (MOS) field effect transistor (229,230).These experiments were concerned with both the electronic band structure and the effects of electron-electron interaction in the surface layer. These MOS devices provide a unique system for the study of manybody effects since the gate voltage influences the surface carrier density. This, in turn, can be related to a parameter which characterizes the importance of electron-electron interaction. A difference between the effective mass from these experiments and that obtained from Shubnikov-deHaas measurements was attributed to mass renormalization due to electron-electron interaction (230). In a second experiment microwave free electron cyclotron resonance has been used to monitor the decay of microscopic electron-hole droplets in Ge (231). These droplets are created when large numbers of free excitons produced by intense pulsed near infrared radiation condense into a dense metallic droplet of free electron-hole pairs. The droplet phenomenon, since

50


it provides evidence of a unique many-body ground state, has attracted a great deal of theoretical and experimental interest. While to date there have been no reports of FIR magneto-optical properties of the droplet itself, it is anticipated that significant contributions could be made in this area. In the droplet experiments and in the recent experiments of Otsuka et al. (232) biasing infrared radiation has been used to change the normal thermal equilibrium conditions of the sample. Thus the microwave radiation probes a system of free or bound carriers with generally altered, and sometimes very interesting, properties. A more common biasing technique used in most semiconductor devices is the application of pulsed or continuous electric fields. Thus it appears that characterization of optically or electrically biased systems with magneto-optical techniques could prove to be interesting from both a physical and a technological point of view. A number of the experiments described here and in the body of the review have depended for their success upon recent advances in FIR instrumentation. It seems likely that future advances may also open up new areas of fruitful research. Since a great deal of effort is being expended toward increasing power and/or frequency tunability, it seems clear that significant improvements will be made in FIR laser sources in the near future. Another area which seems ripe for development is in more sophisticated detection schemes such as those which have been so successful in the microwave region. If these microwave bridge techniques which utilize homodyne or heterodyne detection can be adapted for use in the FIR, a number of additional experimental studies would be possible with the increased sensitivity. The developments described above suggest that FIR magneto-optical studies will be an increasingly important probe of solid state phenomena in the future.

ACKNOWLEDGMENTS We have benefited from helpful discussions with S. Teitler, J. C. Hensel, R. Kaplan, R. Ranvaud, K. L. Ngai, and W. Dreybrodt during the course of this work. J. C. Hensel and R. Ranvaud kindly provided figures and manuscripts prior to publication. We would like to express our graditude to Mrs. L. Graham, Mrs. G. Garrett, Mrs. L. Blohm, Mrs. C. Hepler, and Miss I. Lajko for their continued efforts in typing various drafts of the manuscript. One of us (B.D.M.)would like to thank the Max-Planck-Institut fur Festkorperforschung, Stuttgart. for hospitality extended during a sabbatical year while part of the manuscript was prepared.

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INTRABAND MAGNETOGPTICAL STUDIES. I1

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W. S. Baer and R. N. Dexter, Phys. Rev. 135, A1388 (1964). K. Sawamoto, J . Phys. Soc. Jap. 18, 1224 (1963). D. M. Larsen, Phys. Rev. 142,428 (1966). M. Saitoh and A. Kawabata, J . Phys. Soc. Jap. 23, 1006 (1967). S. J. Miyake, Phys. Reu. 170, 726 (1968). K. J. Button, B. Lax, and D. R. Cohn, Phys. Rev. Lett. 24, 375 (1970). S. Narita, K. Nagasaka, and G . Kido, Proc. Int. Con$ Phys. Semicond., loth, Cambridge, Massachusetts, p. 158. USAEC, Div. Tech. Info., Oak Ridge, Tenn., 1970. K. J. Button, B. Lax, D. R. Cohn, and W . Dreybrodt, Proc. lnt. ConJ Phys. Semicond., loth, Cambridge, Massachusetts, p. 153. USAEC, Div. of Tech. Info., Oak Ridge, Tenn., 1970. G. Kido, K. Nagasaka, and S. Narita, J . Phys. SOC.Jap. 32, 1969 (1972). S. Teitler, B. D. McCombe, and R. J. Wagner, Proc. Int. Con$ Phys. Semicond., loth, Cambridge, Massachusetts, p. 177. USAEC, Div. Tech. Info., Oak Ridge, Tenn., 1970. N. V. Celli and D. Mermin, Ann. Phys. (New York) 30,249 (1964). B. D. McCombe, R. J. Wagner, S. Teitler, and J. J. Quinn, Phys. Rev. Lett. 28, 37 (1972). K. W. Chiu, K. L. Ngai, and J. J. Quinn, Solid State Commun. 10, 1251 (1972). K. L. Ngai, K. W. Chiu, and J. J. Quinn, Proc. lnt. ConJ Phys. Semicond., 1 I th, Warsaw, 1972 p . 335, PWN-Polish Sci. Publ., Warsaw, 1972. S. J. Miyake, J . Phys. SOC.Jap. 35, 551 (1973). J. C. Apel, T. 0. Poehler, and C. R. Westgate, Appl. Phys. Lett. 14, 161 (1969). G. Abstreiter, P. Kneschaurek, J. P. Kotthaus, and J. F. Koch, Phys. Rev. Lett. 32, 104 (1974). S. J. Allen, Jr., D. C. Tsui, and J. V. Dalton, Phys. Rev. Lett. 32, 107 (1974). J. C. Hensel, T. G. Phillips, and T. M. Rice, Phys. Rev. Lett. 30,227 (1973). E. Otsuka, T. Ohyama, and T. Sanada, Phys. Rev. Letr. 31, 157 (1973).


The Future Possibilities for Neural Control F. T. HAMBRECHT

AND

K. FRANK

Laboratory of Neural Control, National Institute of’Neurologica1 Diseases and Stroke, National Institutes of Health, Bethesda, Maryland

I. Introduction .......................................................................

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C . Respiratory Control .............................

E. Auditory Prosthesis F. Epilepsy Control.. ..

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111. Concepts and Techniques

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B. Supernormal Humans

I. INTRODUCTION The thought of directly controlling certain aspects of the human nervous system or utilizing signals from the nervous system to directly control external devices is exciting to some and disturbing to other people. It is true that the possibility exists of unscrupulous use of such techniques to the detriment of others. This does not mean that research in this field should be avoided but rather that attempts should be made to foresee the social consequences of such developments and to institute the proper safeguards. In this way, neural control is similar to pharmaceuticals that affect the nervous system. The potential benefits are large but accompanying these benefits are inherent risks. The future of neural control will depend largely on advances in three principal areas: (1) Development of techniques for transfer of information into the nervous system (inward information transfer). This involves the use 55

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of temporo-spatial patterns of stimuli to modify the electrical behavior of neurons in desirable ways. (2) Techniques for determining the temporospatial activity of selected neurons for use as control signals (outward information transfer). (3) Understanding of the basic control mechanisms operating within the nervous system. As advances are made in these areas they will be applied to alleviate neurological handicaps. For example, in patients with paralysis from spinal cord injuries neural control will be used to effectively bypass the injured neural tissue. Ideally, this would involve detecting motor signals in the brain for control of muscle stimulators. Simultaneously signals from artificial transducers or signals detected from neurons innervating intact touch, joint, and proprioceptive transducers would be fed back to the sensory areas of the brain. Such an application involves functional neuromuscular stimulation (FNS) and is discussed in more detail later. There are several examples of neural control which are already widely accepted in clinical medicine. The best known is the artificial cardiac pacemaker for patients with defects in their natural pacemakers or pacemaker conduction systems. The artificial pacemaker supplies impulses which trigger cardiac muscle contraction. This is a simple form of inward information transfer. An example of a system which uses outward information transfer is the electromyogram (EMG) controlled arm prosthesis (Wirta and Taylor, 1970). The EMG is an electrical signal that is recorded from muscles in the amputees stump and/or other muscles in the shoulder region and is used to control actuators in the prosthesis. Attempts are also being made to detect the force output and the joint angles of arm prostheses as feedback signals for more precise control (Mann and Reimers, 1970). The problem of intractable chronic pain is being treated in many medical centers by electrically stimulating portions of the spinal cord known as the dorsal columns (Shealy et al., 1967; Nashold and Friedman, 1972). The dorsal columns normally convey sensory information to the brain but are not felt to directly convey information about pain. The action of electrical stimulation of the dorsal columns may be related to the theory of pain proposed by Melzack and Wall (1965) and result in a masking or gateing effect similar to scratching the area around a mosquito bite. 11. POTENTIAL APPLICATIONSUNDER INVESTIGATION

A . Functional Neuromuscular Stimulation

As mentioned earlier, the use of electrical stimulation to activate or inhibit skeletal muscles in a purposeful manner is known as functional neuromuscular stimulation (FNS).The fact that an external source of electricity

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could cause muscle contractions was discovered over 180 years ago, and sporadic attempts have been made since that time to use electrical stimulation to treat paralyzed patients. If paralysis results from peripheral nerve injury and if the nerve does not regenerate, severe muscular atrophy or wasting occurs which cannot be reversed or prevented by electrical stimulation. In cases where regeneration does occur there is considerable controversy as to whether electrical stimulation has any effect on the rate or degree of functional return. Recently, significant advances have been made in the medical treatment of patients paralyzed from lesions of the central nervous system. Patients suffering from such afflictions as spinal cord injuries and cerebral palsy now have a longer life expectancy and consequently a greater need for rehabilitation. The atrophy that occurs in their paralyzed muscles closely resembles disuse atrophy and can be reversed with electrical stimulation. These factors in addition to recent progress in solid state electronics have renewed interest in FNS. Liberson et al. (1962) reported a method of correcting foot drop in hemiplegic patients. Paralysis of the muscle, tibialis anterior, in the lower leg results in a dragging of the foot during the swing phase of gait. Liberson placed electrodes on the skin over the peroneal nerve which innervates the tibialis anterior. A switch in the heel of the patient’s shoe closed during the swing phase of gait and activated a portable stimulator connected to the electrodes. More than 100 patients were treated in this manner and most obtained improvement in gait (Liberson, 1972). Although commercially marketed for several years, the system failed to achieve clinical acceptance. All of the apparatus was worn externally and had to be attached daily. Problems arose from broken wires, inaccurate placement of electrodes by the patients and psychological factors. Subsequently, other investigators have continued the development of peroneal nerve stimulation. Current systems have implanted stimulators and utilize radio transmission from the bee1 switch command signal (Vodovnik, 1971). Clinical evaluation is underway in several centers both in the United States and in Europe. In patients with spinal cord injuries, the degree of disability is dependent on the level of the lesion. If the spinal cord is interrupted in the lower back, the legs are paralyzed (paraplegic) while injury in the neck region can result in paralysis of both arms and legs (quadriplegic). Clearly, rehabilitation efforts will depend on both the location and the severity of the spinal cord injury. One of the most important functions which a quadriplegic loses is grasp. In many cases, movement of the upper arm is relatively unimpaired even though paralysis of the lower arm and hand is complete. In an attempt to restore grasp, Long and Masciarelli (1963) designed a splint for the hand that incorporated a means of stimulating the extensor digitorum, a muscle

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which extends the fingers, The splint braced the wrist and thumb in a fixed position. The index and middle fingers were held together in a hinged splint and brought into opposition with the thumb by a spring. By controlling the amplitude of stimulation to the extensor digitorum muscle the patient could overcome the spring force and separate the fingers from the thumb. The device was useful for some feeding activities, as well as grasping small items such as typing sticks and page turners, but was never extensively tested. Its contribution was in demonstrating problems in using FNS which were not readily apparent from the peroneal nerve stimulation studies. For example, operation of the hand splint often required sustained contractions of the stimulated muscle. This resulted in muscle fatigue with reduction in the force output. Also, spasticity and muscle spasms which often occur after paralysis interfered with proper operation of the device. The problem of muscle fatigue during FNS appears to be related to an unphysiological activation of muscle fibers. In nonparalyzed normal muscle each nerve impulse to a muscle fiber results in a muscle twitch. If the twitches occur rapidly enough (about 30-40 per second) or if the twitches of many muscle fibers occur asynchronously, the muscle contracts smoothly. An increase in force output is achieved principally by recruiting additional muscle fibers that contract asynchronously rather than increasing the activation frequency of individual fiber twitches. This reduces muscle fatigue by keeping the activation frequency of individual muscle fibers low. In the early studies of FNS, only one pair of electrodes was used for stimulation and muscle fibers were activated in synchrony. For a smooth contraction, the frequency of stimulation had to be high, resulting in rapid fatigue of individual fibers during a sustained contraction. To reduce fatigue during FNS, an attempt is being made to simulate the physiological situation. Multiple, spatially separated, electrodes are placed in a muscle and stimuli are applied sequentially through them. Most muscle fibers are activated by only one electrode at a frequency below that at which significant fatigue occurs. The whole muscle, however, acts as a mechanical low-pass filter and smooths the twitch contractions of the individual groups of stimulated muscle fibers. Smooth, fatigue resistant contractions have been demonstrated in a forearm muscle of man (flexor digitorum) with as few as three electrodes (Peckham, 1973). Ideally, more electrodes should be used to permit stronger contractions under more precise control, but the number of electrodes is limited by difficulties in placing electrodes in muscle as well as overlap in muscle fiber groups activated by individual electrodes. Another possible method of reducing fatigue is the conversion of fatigue sensitive muscle fibers to a fatigue resistant type. Most mammalian muscles contain a mixture of these muscle fiber types. Recent investigations have indicated that a relationship exists between the normal activation pattern of


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a muscle fiber and its fatigue properties (Burke et al., 1971). By exercising a muscle with a low frequency stimulation pattern, histochemical and physiological evidence compatible with conversion has been demonstrated (Mortimer, 1973). A limiting factor which is already affecting application of even simple FNS systems is a source of reliable proportional control signals. A subject must have the means of initiating and controlling the stimulation of paralyzed muscles. In addition this process should involve a minimum of conscious attention. Attempts to use EMG control from intact muscles have generally not been successful. The more severe the disability, the fewer control muscles there are available and the more functions an FNS system must perform. Also, signal-to-noise properties of EMG signals are marginal, especially when interfering signals are present from the stimulus current (Vodovnik and Rebersek, 1971). The lack of a functional relationship between a control muscle EMG and the desired control signal can be partially overcome by training but little evidence for this has been demonstrated (Radonjic and Long, 1970). Position transducers that detect the location of one part of the body with respect to another have been used for control signals. They are superior to EMG with respect to signal-to-noise characteristics but are even more restricted in the number of potential control sites. Also the subject may be required to maintain awkward, uncomfortable positions. Some examples of such control systems that have been tested are eye position, tongue position, head position and shoulder position. The possible use of electrical activity recorded directly from the brain as a source of control signals is discussed in Section III,A. Very little work has been done on sources and types of feedback signals necessary in FNS systems. In the case of peroneal nerve stimulation, the heel switch acts both as a command signal and as a feedback transducer that detects the removal of the heel from the floor. Experiments on implementing grasp have relied on visual feedback which is rather unsatisfactory. The visual system is needed for a multitude of other functions and should only be used for occasional checking of control system operation. Although the need for feedback information has been generally recognized, considerable disagreement exists as to what type is optimal. For example, a quadriplegic using FNS to grasp a small item such as a pencil needs the following information. First he must know when he has made contact with the pencil and the orientation of the pencil with respect to his fingers. As he lifts the pencil, he needs to squeeze hard enough to hold the pencil but without excessive pressure. At this point would force feedback or detection of slip be more appropriate? Functional use requires knowledge of joint angles and pressure of the tip of the pencil on the writing surface. Some of this information will

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be indirectly relayed to intact sensory receptors above the level of sensory interruption. How to supply the remaining information in the simplest possible manner has not been answered. A long-range possibility is to derive information from sensory receptors in the fingers, joints, and muscles. However, these receptors are extremely small and the signals they produce are rarely larger than a few millivolts. Not only will amplification be required but also isolation of the signals from interference due to the spread of the stimulus current. A more realistic approach at this time may be the use of miniature artificial transducers either attached to the skin or implanted. The choice of feedback signals will also depend on how they are to be utilized. If it is desired that the subject be made consciously aware of some aspect of a task he is performing, feedback information must be supplied to his sensory system. The quadriplegic generally has no sensation in the lower arm or hand. One solution may be to encode the feedback information for activation of tactile transducers attached to the skin above the level of sensory loss. Again such a technique is restricted by the limited number of sites with intact sensation and the ability of the subject to interpret such coded information. Collins (1970) and Bach-y-Rita (1972) have done extensive studies on the use of such tactile stimulators for sensory substitution systems and are quite optimistic about such plasticity in the sensory system. In the more distant future it may be possible to input such information directly to the sensory areas of the brain. Investigations related to this are discussed in Section III,D. It is hoped that a good deal of the feedback information needed will not require conscious attention. Tasks which are often repeated, involving essentially identical movements, could be preprogrammed in the control logic. Feedback information for these tasks could then be returned directly to the control logic. Paralyzed individuals often develop uncontrollable muscle spasms and spasticity. Spasticity in a muscle is characterized by increased muscle tone and an exaggeration of the reflexes. The exact cause of these conditions is not known and methods of controlling them are far from optimal. A spastic muscle or a muscle during spasm is difficult, if not impossible, to control. One theory postulates that lesions of the central nervous system result in a reduction of normal inhibition on neurons innervating muscles (motor neurons). This results in hyperactive response to intact excitatory inputs. Morris et ul. (1968) have attempted to control spasticity by electrically stimulating afferent nerves which supply inhibitory inputs to motor neurons. Although they were able to demonstrate functional reduction in spasticity in only 2 of 13 spastic subjects, this approach does show some promise. As the problems such as muscle fatigue, spasticity, sources of feedback, and control signals are solved, the sophistication of FNS systems will


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increase. This will involve stimulation of multiple muscles simultaneously for joint stabilization as well as motion. Eventually, such systems should be capable of restoring important motor functions such as grasp and ambulation.

B. Bladder EGacuation by Electrical Stimulation In many persons with lesions of the motor aspects of the central nervous system, the loss of bladder function results in chronic urinary tract infections often with fatal complications. The use of catheters or surgical drainage procedures does not solve the problem and often adds to psychological trauma by causing incontinence. A means of restoring bladder function with voluntary control of evacuation is needed. This should be accomplished without causing incontinence and without leaving residual urine in the bladder after evacuation. The normal physiological mechanism of bladder evacuation is not fully understood but basically it consists of two components. The muscular component of the body of the bladder known as the detrusor contracts, increasing the pressure in the bladder. Simultaneously, or shortly before, the sphincters (internal and external) open, allowing urine to discharge through the urethra. These functions are normally under control of the central nervous system. Since the paralyzed bladder will usually contract in response to electrical stimulation applied directly to its walls and the bladder is readily accessible surgically, investigators have attempted to implant electrodes on or in the detrusor muscle (Bors and Comarr, 1971; Bradley et al., 1963; Stenberg et al., 1967). These electrodes were connected to radio receivers implanted beneath the skin and were activated by radio transmitters, external to the body. Bradley and associates (1971) have reviewed some of the reasons why this technique has had only limited clinical success. They feel one of the most important problems that has not been solved is effective stimulus coupling to the smooth muscle cells of the detrusor. Other problems are pain and spread of the stimulus current to the sphincters causing partial or total occlusion of the urethra. To increase the effectiveness of detrusor contraction, multiple electrodes with large exposed surface areas have been implanted. This requires more complicated surgery, increased trauma to the bladder, more complicated hardware and greater energy demand from the stimulation electronics. Some attempts have been made to stimulate the pelvic nerves which innervate the detrusor. Studies in dogs indicate that functional bladder pressures can be achieved (Staubitz et al., 1966). However, fibrosis and inflammatory tissue around the electrodes with eventual response failures have been reported (Hald, 1969). Also, in humans the pelvic nerves are

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spread out diffusely in a plexus which makes their identification and electrode placement very difficult. Recently, Nashold et al. (1972) have demonstrated bladder evacuation by stimulating the lower spinal cord of paralyzed individuals. Electrodes were implanted in the sacral region of the spinal cord near the origin of the pelvic nerves. An advantage of this over previously discussed techniques is that the cell bodies of the pelvic nerves are relatively concentrated in this region. They require less energy to activate and a more uniform bladder contraction occurs during stimulation. Pain has not been a problem but only spinal cord injury patients with sensory loss have been studied. Contraction of the external sphincter, pelvic floor, and lower limb musculature as well as undesirable autonomic responses such as sweating and piloerection have occurred. These effects are felt to be caused by spread of stimulus current to nearby neurons in the spinal cord that control these functions. More studies are needed to determine the distribution of the pelvic neurons in the spinal cord, means of controlling stimulus current, and methods of inhibiting sphincter contraction. Even if these problems are solved, the procedure will still require a neurosurgical procedure to expose the spinal cord and will not be applicable if a lesion has occurred that destroys the pelvic nerves. C . Respiratory Control

When the spinal cord is interrupted in the upper cervical region of the neck, normal breathing is not possible because the pathway for respiratory control is interrupted. The two phrenic nerves which innervate the diaphragm leave the spinal cord at the level of third through fifth cervical vertebrae. Until recently, few persons with this type of injury lived and those who received immediate treatment had to be maintained continuously on an artificial respirator. With improvement in detection and treatment of spinal cord injury patients, the number of persons surviving such injuries has increased. If the phrenic nerves are not injured and if the lungs and diaphragm are normal, it is possible to electrically stimulate the phrenic nerves and free such patients from mechanical respirators. This technique is also applicable to patients with so-called “ hypoventilation of central origin” in which the central nervous system does not supply appropriate control signals for adequate ventilation. Glenn et al. (1970, 1972) have described the successful treatment of patients with this form of electrophrenic respiration (EPR). Electrodes were placed around one or both phrenic nerves and were connected to a receiver-stimulator implanted beneath the skin. Open loop control is achieved by a transmitter whose antenna is placed on the skin over the receiver.


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Continuous stimulation of a single phrenic nerve has resulted in fatigue of the diaphragm after 12-16 hours. The cause of this fatigue is not clear, but it is probably similar to the fatigue noted in functional neuromuscular stimulation. Stimulation with large electrodes around the phrenic nerve results in synchronous activation of nerve fibers at the pulse repetition rate (25 pps). This relatively high frequency causes fatigue of either the junction between the nerve and muscle (myoneural junction) or the muscle fibers themselves. Glenn and his colleagues have found that modulating the amplitude of their constant current stimulating pulses with a linear ramp is useful in reducing fatigue. The mechanism of this effect is probably the recruitment of additional muscle fibers as the amplitude of the pulses increases. However, the first fibers recruited are continuously subjected to high frequency stimulation during the inspiratory interval and are probably the fibers that are responsible for the fatigue. It may be worthwhile to use multiple electrodes and asynchronous low frequency stimulation (less than 8 pps) according to the method of Peckham (1973) to further reduce fatigue. Presently Glenn is stimulating the single phrenic nerve implants only at night and alternating the activation of the dual implants every 12 hours. Future EPR will most likely have more sophisticated control. When using the existing systems during the swallowing of food or fluid the patient must either turn off the transmitters or time the swallowing to occur during the expiratory phase. Otherwise aspiration into the trachea and lungs would occur. With appropriate feedback the device could be automatically deactivated during these functions. Normal speaking involves considerable fine control of diaphragm motion. Provision for patient override and control of the stimulator during speech would be worthwhile. Further in the future, blood oxygen, pH, and carbon dioxide sensors may be incorporated to permit true closed loop respiratory control. A provision for coughing would also be valuable so that the tracheobronchial tree could be cleared on demand.

D. Visual Prosthesis Electrical stimulation of a portion of the brain known as visual cortex results in sensations of light known as phosphenes. Foerster (1929) demonstrated a mapping of the visual field onto visual cortex and Krause and Schum (193 1) reported similar localized, well-defined phosphenes during electrical stimulation in a patient who had been blind for over 8 years. These experiments created interest in the possibility of a visual prosthesis for the blind. Such a prosthesis would consist of an array of light sensors, or a television camera, which after appropriate electronic processing, would modulate radio transmitters placed over the scalp. Radio

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receivers beneath the scalp would be connected to an array of electrodes for stimulation of the visual cortex. In 1967, Brindley and Lewin (1968) implanted an array of 80 electrodes on the surface of visual cortex in a blind nurse. The device was experimental and no provision was made for coupling to a light sensor. Although only a few electrodes could be addressed simultaneously, some significant results were obtained including: (1) Stimulation through many of the electrodes produced sensations of small, single spots of light (phosphenes) whose intensity could be modulated by a suitable variation of the stimulus. (2) Topographical mapping of the visual field onto visual cortex was confirmed. ( 3 ) Simultaneous stimulation of several points produced recognizable simple patterns. (4) During eye movements the phosphenes moved with the eyes but retained their spatial relationships relative to each other. ( 5 ) Phosphenes usually extinguished immediately following stimulation but after high level stimulation they persisted for up to 2 minutes. (6) The visual cortex can tolerate an electrode array in contact with it, as well as periodic short sessions of electrical stimulation, for at least 6 years. Encouraged by these results, Brindley et al. (1972) implanted a second electrode array on visual cortex of a man who had been blind for over 30 years. Results are similar to those obtained with the first patient except the phosphenes are larger and less well defined. One experiment involves the presentation of braille symbols by modulation of six phosphenes. Although the subject can recognize these phosphene generated symbols, he can read braille more rapidly by touch. These are preliminary results and may be altered by training and experience. Research supported by the National Institute of Neurological Diseases and Stroke (NINDS) on a visual prosthesis has attempted to determine the feasibility of continuous long-term electrical stimulation of the nervous system. It was decided not to support long-term implants in blind volunteers until feasibility studies had shown that such implants are safe and will withstand long-term stimulation. This work has been performed by investigators at the University of Rochester, Johns Hopkins University, Massachusetts General Hospital, Huntington Institute of Applied Medical Research, University of Florida, University of Utah, Tyco Laboratories, and the NINDS Laboratory of Neural Control. Potential biomaterials have been screened for toxicity by implantation on and in the brains of animals. Histological examination of surrounding meningeal and brain tissue is one method used for evaluating the effects of these materials. The electrochemical properties and the corrosion characteristics of potential electrode materials have been studied while passing current through them into a bath of simulated cerebrospinal fluid. The results of these tests indicate that suitable biomaterials are available for use in a visual


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prosthesis. Significant problems, however, remain to be solved. Normally blood vessels that supply the brain will not allow certain molecular species to pass through their walls, even though they readily penetrate vessels elsewhere in the body. This is known as the blood brain barrier and can be demonstrated by injecting certain dyes into the blood stream. During electrical stimulation with levels necessary for producing phosphenes in humans, these dyes escape from the vessels in the region of the stimulating electrode. The mechanism and significance of this finding are not known. During human studies, phosphenes often fade during continuous electrical stimulation while animal studies indicate a similar increase in threshold for detection of stimulation. This effect appears to be reversible if stimulation is periodically interrupted. Brindley’s first patient occasionally experienced phosphenes which continued after stimulation was stopped. This phenomenon, known as afterdischarge, has been studied in animals by recording the electrical activity of single nerve cells in visual cortex. It appears that the margin of safety between threshold for detection ofelectrical stimulation and production of afterdischarges is small and that stimulus ranges will have to be more tightly restricted than was originally anticipated. One of the most important questions that remains to be answered is the rate at which information can be transferred into visual cortex by electrical stimulation (Dobelle et al., 1974). This depends on many factors such as the stimulus modulation frequency, the number of electrodes that can be placed on cortex, as well as interaction of phosphenes. Human studies have indicated that it is difficult to resolve phosphenes produced by stimulation through electrodes closer than 2 mm. The feasibility of a prosthesis for the blind utilizing direct electrical stimulation of the surface of visual cortex has not as yet been demonstrated. If the technological problems can be solved, the usefulness of such a device will have to be determined by human blind volunteers. E. Auditory Prosthesis

An application of neural control which is quite similar in concept to a visual prosthesis is the development of an auditory prosthesis for the deaf. It has been known for some time that electrical stimulation of various portions of the nervous system concerned with auditory function produces sensations of sound. Before discussing the possible approaches to an auditory prosthesis, a basic review of the normal processing of auditory signals and types of deafness would be useful. Vibrations of the eardrum are transferred to the fluid in the cochlea of the inner ear by three small bones (malleus, stapes, incus) in a mechanical linkage. Disease processes involving this linkage

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produce what is known as conduction deafness. Hair cells in the cochlea detect vibration of the fluid. Auditory nerve fibers are in intimate contact with hair cells and the nerve activity (nerve impulse rate) is controlled by the hair cells. Deafness caused by destruction of hair cells and/or nerve fibers is called sensorineural deafness. The auditory nerve relays information to many centers in the brain and indirectly some of this information reaches a portion of the brain known as auditory cortex. Deafness resulting from disturbances in the brain is known as central deafness. The cochlea containing the hair cells and nerve fiber endings is essentially a coiled tube resembling a snail shell. The nerve fibers in the apical turn respond maximally to low acoustic frequencies and those in the basal turn to high frequencies. However, this sharp frequency dependence seen for acoustic stimuli is absent for electrical stimuli to which all nerve fibers tend to respond alike with only a small dependence on frequency (Kiang and Moxon, 1972). Since conduction deafness can usually be treated adequately with hearing aids or surgery and central deafness is rare, an auditory prosthesis using electrical stimulation would be most useful for cases of sensorineural deafness. The principal function of such a prosthesis would be for recognition of speech. Several locations in the auditory pathway are potential sites for electrical stimulation. Electrodes have been placed experimentally in the cochlea of deaf human subjects (Doyle et al., 1964; Michelson, 1971; House, 1973).This approach attempts to take advantage of the splaying out of nerve fiber endings along the cochlea for greater stimulus selectivity. It is limited by cochlear fluid shunting of stimulus current and traumatic damage produced by attempting to insert an electrode array without visualization into the coiled cochlea. Other investigators (Djourno and Eyries, 1957; Simmons, 1966) have placed electrodes directly into the auditory nerve. Since this nerve is a compact bundle of approximately 30,000 nerve fibers (Rasmussen, 1940), selective stimulation of individual or small groups of fibers is very difficult. It is also possible that significant trauma to the nerve occurs. Clearly, both of these approaches require viable auditory nerve fibers even though hair cells may be nonfunctional. Reliable data is not available on the number of patients with sensorineural deafness that retain viable auditory nerve fibers nor on the eventual fate of the nerve fibers after hair cell death. Auditory sensations reported by deaf subjects during electrical stimulation of the auditory nerve through a single monopolar or bipolar electrode have been described as buzzing, ringing, grating, and clicking sounds. With stimulus frequencies above 1000 Hz, the sound has been described as “noise” (Doyle et al., 1964), “total confusion” (Djourno and Eyries, 1957), and “high pitched steady sounds (Simmons, 1966). Most investigators ”


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agree that pitch discrimination is difficult above about 500 Hz when stimulating with a single electrode. This corresponds with the maximum impulse rate at which auditory nerve fibers can respond to electrical stimulation. Obviously, synchronous firing of many nerve fibers results in redundancy and would limit the rate at which information could be transferred into the nervous system. In normal hearing, nerve fibers do not discharge in unison or necessarily discharge in response to each stimulus but they d o tend to respond to a particular phase of the stimulus with preferred phases distributed over a 360” range (Kiang and Moxon, 1972). Presumably the brain is capable of performing a spatio-temporal integration of the inputs from many nerve fibers and this effectively increases the information transfer rate. A prosthetic system with multiple electrodes which is capable of activating small groups of auditory nerve fibers independently could conceivably mimic this effect. As Simmons (1966) points out, however, this will probably produce a “foreign language” and the ability of the mature brain to interpret this for useful communication is unknown. Another site that has been considered for an auditory prosthesis is temporal cortex. Penfield and Jasper (1954) noted that electrical stimulation of a small portion of the temporal lobe (transverse temporal gyri of Heschl) caused auditory sensations, usually a tone, a buzzing, or a knocking sound. Unfortunately this area is buried in the brain and is not easily accessible. Also it is not clear whether a tonotopic map exists in human cortex as has been suggested from animal experiments (Dobelle et al., 1973).

F. Epilepsy Control Recent advances in the medical treatment of epilepsy have greatly reduced the number of patients disabled by this disorder. However, a considerable number of individuals do not respond to, cannot be controlled by or experience toxic side effects to anti-epileptic medications. It is this group of patients who may be helped by neural control. Near the end of the 19th century, Lowenthal and Horsley (1897) and Sherrington (1897) noted that electrical stimulation of a portion of the cerebellum produced inhibition and release of rigidity in an animal preparation. Extensive investigation since that time has shown that the output of neurons of the cerebellum (Purkinje cells) exerts only inhibitory action upon their target neurons (Ito, 1970). It is also known that neural pathways exist between cerebellar neurons and cerebral neurons (Sasaki rt al., 1972). This information suggested to several investigators the possibility of electrically stimulating the cerebellum to prevent or interrupt epileptic seizures. Cooke and Snider (1955) reported that cerebellar stimulation could terminate experimentally induced cerebral seizures in the cat. Using a different

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FRANK

experimental seizure model, Dow et al. (1962) noted similar inhibition of epileptic activity in rats. However, in an attempt to repeat these findings in the cat, Reimer et al. (1967) noted that electrical stimulation of the cerebellum usually initiated or prolonged cerebral seizures. The reasons for this apparent inconsistency are not clear but there are several possibilities. There are many experimental animal models of epilepsy which resemble some of the human types (Purpura et al., 1972). The effects produced depend not only on the model selected but also on the animal species studied. The particular part of the cerebellum that is stimulated as well as the stimulus values selected are other variables. For example, Dow and Moruzzi (1958) described an inhibitory effect of cerebellar stimulation which occurs with stimulation in the range of 30-300 pps while reversal of response occurs in the range of 2-10 pps. Recently Gilman (1973) described some studies done in collaboration with Dr. I. S. Cooper in which several patients with medically intractable grand ma1 epilepsy and/or psychomotor epilepsy, received cerebellar stimulation for as long as 6 months. Electrodes were implanted on the surface of the cerebellar cortex and were connected to radio receiver-stimulators placed beneath the skin. Gilman and Cooper claim a dramatic decrease in the frequency and severity of attacks in these patients and are quite optimistic about the future of the technique. These are preliminary results, however, and more work is needed to determine the most effective stimulus sites and values. Also the long-term effects of cerebellar stimulation are not known. Since inhibitory as well as excitatory connections between nerve cells are present in the central nervous system and are necessary for its normal function, it is reasonable to search for a population of neurons that, when electrically excited, might have a net inhibitory action on epileptic neurons. Another possibility, that has received very little attention, might be to directly inhibit epileptic neurons in a reversible manner. Several techniques have been reported for inhibiting neurons such as localized cooling and localized pressure, but at the present time they are not developed sufficiently for long-term clinical application. 111. CONCEPTS AND TECHNIQUES

Consider once more the basic concept of neural control. In simple animal forms some perturbation of the environment is sensed by the animal through his afferent or sensory nervous system and this change in nervous activity or inward information is communicated through a more or less complex and poorly understood central nervous system to produce a change in the efferent or motor outward information. This produces a change in the


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environment which is generally appropriate for survival of the species. Basically, the same principles of organization apply to all animal forms-the higher, the more varied the sensitivity to environmental change and the more complex the motor response. A fault anywhere within the system may lead to an abnormal or inadequate response to a normal stimulus. Stimulation of the nervous system by artificial means (inward information transfer) may be used to augment the response to environmental changes. Similarly, activity of the nervous system may be detected in various ways (outward information transfer) and used to control devices that change the environment. A . Sources of Control Signals

Let us focus attention now on outward neural control, that is, control of the environment by information derived from activity of the nervous system. We have already described control of an artificial limb or paralyzed hand by electrical signals derived from muscle contractions in the Introduction. A potential source of control signals are the electrical transient signals detected with electrodes placed close to nerve fibers or nerve cells within the nervous system. Such signals may be classified according to the part of the nervous system from which they are derived. (1) Signals from peripheral motor nerves, to be useful, require that virtually the entire nervous system be intact. (2) Signals from the brain can be used when the spinal cord or peripheral motor nerves have been damaged. (3) Signals from the sensory nervous system may also be used ( e g , for feedback control) but do not include modification by central nervous system processing as do the others. The electrical signal from a nerve cell detected by an electrode nearby depends in amplitude on many factors including the distance from cell to electrode. Large electrodes which are about equally distant from a number of different nerve cells tend to average their individual signals and record what is called a gross potential. Examples are the electroencephalogram (EEG), from gross electrodes on the scalp, or the electrocortiogram (ECG) from gross electrodes on the surface of the brain. Microelectrodes, whose recording tips are small compared to the size of a nerve cell, record individual action potentials or “spikes” from the nearest cells. These potentials are much larger than the gross potentials averaged from more distant cells and can be used to monitor single cell activity. Gross electrodes are technically easier to make and implant at the present time but if the cells whose activity they record are associated with different functions (e.g. with contraction of antagonistic muscle groups) the signals will be confused and ambiguous. Nahvi et al. (1975) have considered

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the use of one or a number of gross electrodes as a source of control signals suggesting that pattern recognition techniques might be used to separate signals of different functional significance. At the present time, however, the usefulness of gross potentials as control signals appears to be limited and is greatest where nature has separated function anatomically. A promising technique for deriving control signals utilizes implanted microelectrodes in the motor cortex to detect spikes from individual nerve cells. The activity of some of these cells appears to be associated with specific movements. Neurons can be found whose irregular background firing rate changes about 50- 150 msec before a particular movement is initiated-as though these cells were in “line of command” over the neurons whose signals initiate muscle contraction directly. However, the nervous system does not appear to be organized so simply that the activity of each such cell is rigidly associated with contraction of a particular muscle. While some cells are quite dependable as predictors of specific movements others change their firing patterns in complex ways, changing their correlation with particular aspects of the movement. The experimental arrangement, which has been used by Evarts ( 1968) and others for basic research into the physiology of motor control and by Schmidt et al. (1975) for exploring potential control signals is described as follows: A monkey is trained to move a lever against an opposing force. If he performs the task within the experimental limitations imposed, a drop of fruit juice is delivered through a tube near his mouth. One or more microelectrodes implanted near cells in the motor cortical area normally associated with the required movement are used to record single cell activity. Such an arrangement demonstrates one form of neural control and serves to illustrate several fundamental problems. 1. Correlation of Cortical Cell Activity with Movement The variability in correlation of the activity of a single nerve cell with a particular movement may be due to some, as yet unknown, aspect of organization of the brain or it may be simply due to a variation in patterns of muscle contractions used to accomplish the same movement (Schmidt et al., 1974a). Cortical cells whose firing patterns correlate with a particular movement behave in various ways. Such a cell may increase or decrease its background firing frequency just before the beginning (on cell) or just before the end (off cell) of the movement. Or it may increase or decrease its firing rate before both beginning and end of the movement (on-off cell). These are called phasic cells. Rarely a (tonic) cell may have a firing frequency proportional to the amplitude or force of the movement lasting as long as the


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movement but with a phase lead of 50-150 msec. Such correlation of nerve cell firing with movement is relatively constant from one trial to another provided the pattern of movement does not vary. Humphrey et al. (1970) have shown that the combination of signals from several such cells can increase the accuracy of predicting some feature of a learned movement such as force or position. However, the firing pattern of these cells varies if the learned movement is associated with other movements so that the algorithm for combining the cells’ signals must be changed with changing movement patterns. Suppose that a control signal is desired to move an artificial hand so as to duplicate a monkey’s learned hand movement. If all other movements are absent or constant the problem is simplified. Signals from a variety of nerve cells can be combined to control the prosthesis. But different movement patterns are associated with different patterns of cell activity. Ideally, the artificial hand should move wherever the natural hand would have moved no matter what other movements are included. Thus a “ filter is needed to separate the subclass of cell firing patterns all of which are associated with the desired movement. Such a filter has not yet been devised but is not inconceivable. Apparently the spinal cord accomplishes just such a task of pattern recognition. ”

2. Voluntary Control of Cell Firing Fox and Rudell have demonstrated that a cat can learn to modify some details of the evoked electrical activity recorded from gross electrodes in the visual cortex, such as a reduction in the negative component occurring at a particular time following a light flash, but simultaneous control of several features in the recorded potential has not been shown (Fox and Rudell, 1968). A single electrode recording the activity of a single cell for control of a single movement component is a simple concept and quite within the limits of our present technology. But multiple channels which are (ideally) mutually independent are more difficult to obtain. An animal can learn to modify the firing pattern of almost any single cell (Olds, 1965). Fetz (Fetz and Baker, 1973) and Schmidt et al. (1974b) have arranged to reward a monkey for modifying the firing pattern of a cortical nerve cell rather than for the movement normally associated with it. The animal was even able to learn to inhibit the movement while still increasing the firing rate of the cell. In one experiment Fetz (Fetz and Baker, 1973) was able to show differential modification of the firing patterns of two cells recorded simultaneously. This is the limit of individual, independent, nerve cell control which has been experimentally demonstrated. Yet, the nervous system itself is capable of supplying a large array of generally independent parallel output signals as

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demonstrated by such complex motor behavior as playing the piano. Extracting such information from the patterns of brain cell firing is a fascinating challenge.

3. Built-In Organization The application for which a control signal is desired affects the choice of locations from which such a signal is derived. As an illustration consider the Moss prosthetic arm for amputees which has been previously described. The muscles whose EMGs are used to control the externally powered arm are all muscles which were normally involved in moving the natural arm. In general they are the muscles that position the upper arm and shoulder and provide reaction forces for movement of the lower arm against an opposing environment. Neural control of the normal limb is part of the “built-in organization” of the nervous system. This complex control of coordinated movements regulates the forces of contraction and the timing patterns of individual muscles. Thus the amputee’s problem of learning to control his artificial arm is simplified. He has merely to “will” the movement as before amputation in order to produce appropriate EMG signals from his remaining muscles. In a similar manner, it seems appropriate for more complex control problems to utilize cortical cells whose activity is naturally associated with the desired movement. 4. Learning

In addition to its elegant anatomical and physiological organization the nervous system has the property of adaptation or learning. Since one can learn to execute a new pattern of movements, one must certainly be able to learn to generate the new patterns of nerve cell firing which produce the movements. The patterns which can be learned may be limited, however, to normal movement control signals falling within the repertoire set by the system. That is, it may not be as easy to learn to control the firing of an arbitrary set of neurons in an arbitrary pattern. The choice of control signals and patterns of cell firing for operation of a particular device will likely utilize combinations of the built-in organization of the nervous system together with its power for adaptation. Those applications that mimic previously learned functions can lean heavily on built-in organization while the control of some new unfamiliar device will necessarily require additional learning. But this may be limited to learning to execute the correct sequence of rather familiar subpatterns of cell firing. Thus, we come back to the fundamental problem of recognizing a subpattern of cell firing in the face of a variety of other nervous activity.

FUTURE POSSlBILITIES FOR NEURAL CONTROL

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5. Types of Control Many attempts have been made to utilize a single on-off type of signal source for controlling an external device. Examples are the pillow switch for starting or stopping a tape recorder or the sequential radio control of a model airplane. In the latter, the control surfaces of the plane move through a circular sequence such as right, up, left, and down. The limitations of such a system as compared to multichannel, proportional control systems has been painfully demonstrated by numerous crashes. However, to the quadriplegic patient who has minimal residual movements even a single channel on-off switch can be a boon. It is fortunate that in the development of neural control there is a continuum of problems to solve from the simplest, almost trivial, control of a single on-off switch to systems as elegantly complex as the nervous system itself. The solutions are limited only by the rapidly changing technology available and the understanding of the basic mechanisms by which the nervous system operates. B. Techniques f o r Outward Information Transfer

Nerve cells transmit information by modification of the time sequences of their all-or-nothing action potentials. Each action potential or “spike is an electrochemical event accompanied by a variety of signs. There are changes in the concentrations of certain ions both inside and immediately outside the cell. There are undoubtedly small changes in temperature, changes in light scattering properties, sol-gel changes in the intracellular cytoplasm, and changes in electrical potentials across the cell membrane. All but the latter are either at or below the limit of detection for a single information signaling event in a single nerve cell. Thus the detection of the electrical spikes accompanying neural activity appears to be the only promising technique available for leading information directly from the nervous system. It should be mentioned that changes in cell membrane permeability to different ion species result in electrical current transients that generate magnetic fields but these fields are too small to be detected at any useful distance from a cell. Consequently, electrodes with appropriate electrical and mechanical properties must be introduced to detect electrical field changes. The development of such electrode techniques is one of the most important challenges of neural control. As previously mentioned, the size of the electrode determines whether it will record only the average of many localized potential changes or the transients from single cells which are larger in amplitude than the background average. The maximum signal-to-noise ratio (noise includes spikes from other cells) is obtained when the noninsulated portion of the electrode is inside a nerve cell. These spikes are on the order of 100 mV and last about ”

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1 msec. For the largest cells in the central nervous system the diameter of the intracellular electrode tip must be about 1 pm. With the best intracellular techniques in use today such a penetrated cell will last from a few seconds to a few hours before it “dies ” and becomes inactive. Larger electrodes between 10 and 100 pm positioned just outside a nerve cell can record extracellular spike potentials of a few microvolts to several millivolts from hours to months and possibly indefinitely. The signal-tonoise ratio is lower than with intracellular electrodes but is usually adequate for resolution of individual spikes. An implanted electrode constitutes a foreign body and generally produces a tissue reaction. The degree of reaction and damage to neighboring nerve cells depends in part upon the materials used to make the electrode and on the current passing from the electrode through the electrolytes surrounding it. The electrochemical reactions at the electrode surface are much more serious in the case of stimulating electrodes (Hambrecht, 1973) but even with recording electrodes some current must flow in order to detect a signal. Fortunately, amplifying techniques are available today which, combined with ideal electrode materials appear to permit indefinite recording if other requirements are satisfied (Salcman and Bak, 1973). Mechanically the brain in the rigid skull is more like a gel than a solid. Linear acceleration within normal limits produces negligible displacement of brain tissue relative to the skull. However, angular acceleration causes parts of the brain to lag behind the skull (Ommaya and Genarelli, 1974). If stiff electrodes penetrate the brain and are rigidly attached to the skull such relative movements will cause the electrodes to plow out an area of damage. Clearly, for long-term recording from single neurons, this form of damage must be prevented. It may be worthwhile to describe a number of schemes for introducing electrodes into the nervous system which have either been tried or at least considered in the hope that other possible techniques may be suggested to the reader. Here we will restrict consideration to electrodes intended to record the electrical activity of single nerve cells. An electrode which has shown some promise in early trials is called the “ thumbtack” or “map pin” electrode (Salcman and Bak, 1973). A 1-3 mm stiff tungsten or iridium wire 25-50 pm in diameter is electroetched to a tapering point 1-5 pm across and insulated except at the tip. Such a wire is stiff enough to penetrate the brain a short distance and its position is stabilized in relation to the brain surface by a round head where it is attached to a very flexible insulated lead. This lead connects the electrode to a connector fastened to the skull. A bio-adhesive such as cyanoacrylate is used to secure the head of the electrode to the brain surface after implantation and until


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normal healing processes form a scar around the head of the electrode. Such electrodes have been used to record single cell spike signals for several months. However, some movement of the nerve cells relative to the recording tip remains as the signals from individual cells appear to wax and wane over a period of several days. It is not yet possible to remain “connected” to a particular nerve cell for a long period of time. A large parallel array of such electrodes fastened to a common “ head where they are connected to a flexible cable of leads could, presumably, provide additional stabilization. (Such a “ bed of nails electrode array has not yet been tried.) There is considerable redundancy in the brain so that similar information is contained in the firing patterns of quite a number of single cells. Thus, it may be possible, in coupling to an external device, to shift control from one cell to another as their signals from an array of electrodes wax and wane. Another technique that has not yet been tried in the brain utilizes a cable of fine flexible insulated leads crimped into the end of a thin needle. This needle is just stiff enough to penetrate the brain either from side to side or along a semicircular path in and out of the surface. A short, stiff electrode similar to the tip of a “thumbtack electrode described above is welded to an appropriate spot on each wire of the cable. As the cable is pulled forward the electrodes follow in the same path but when the cable is pulled backward, slightly, each electrode moves away from the cable path out into “virgin” brain tissue which has not been damaged by the needle or cable of leads. Such a scheme has been used to record from individual fibers in muscle. Springy wire electrodes can be stiffened enough to permit penetration by confining them inside a hypodermic needle which is subsequently withdrawn (Marg et al., 1973). Alternately, it has been suggested that a fine wire electrode could be stiffened enough to permit penetration by coating it with some substance which would gradually dissolve after insertion. It should be clear from these suggestions that no satisfactory method has yet been found for chronic recording of single nerve cell signals without the danger of destroying nervous tissue. However, such a connection with the nervous system is absolutely essential to the ultimate success of outward information transfer. ”

”

”

C . Other Considerations In attempting to assess the future possibilities of neural control it might be well to list some of the problems which have to be overcome or at least further investigated.

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1. Location of Electrodes

Outward neural control is primarily concerned with the detection of neural signals while inward control involves the excitation or inhibition of neural activity. Both involve the appropriate placement of electrodes in close proximity to single nerve cells or groups of nerve cells which have appropriate functions. Much more needs to be known about the anatomy and the physiology of the brain if electrodes are to be optimally located. Because of the intermingling of cells of different function the nature of an individual cell can only be determined after the electrode has been implanted. Either the electrodes must be moved after insertion to find the appropriate cells or a very large number of electrodes must be implanted so that enough will be correctly placed by chance. The functional electrodes can then be selected for use. Fixation of electrodes and movement of cells or cell processes within the brain are problems which have already been mentioned.

2. Signal and Power Transfer Considerable effort has already been spent on problems of power transfer across the skin for electrical stimulation. Inductive coupling has been used with small numbers of electrodes but large arrays will require more elaborate multiplexing circuits to provide for distribution of stimuli in space and time as well as control of stimulus parameters. Separate signal and power channels are indicated and either light or ultrasonic pressure waves might be used to obviate the need for percutaneous connectors-probably inductive coupling for power and light or ultrasound for wide band width signal transfer (R. L. White, personal communication). Transmission of signals across the skin may require percutaneous connectors or new techniques for amplifying and multiplexing these signals before transmission across the skin.

3. Long-Term Efects

In the case of excitation or inhibition of neural activity it is necessary to study the long-term effects of modifying cellular activity. There may be problems of chemical exhaustion or fatigue at synapses and cells may be stimulated to alter their structure by abnormal forced activity. Possible problems of rebound " must also be considered, i.e., when cells are excited or inhibited artificially for a period of time they may show inhibition or excitation, respectively, when the artificial input is removed. "


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D. Lilly Hypothesis Just as pattern recognition is of vital importance to outward information transfer, inward neural control requires a knowledge of which patterns of cell firing are meaningful. The Lilly hypothesis, named after Dr. John C . Lilly, proposed that the state of the nervous system, its sensory input patterns, its conscious thoughts, its emotional state, its motor output is altogether determined by the pattern of activity of all its nerve cells. Thus, if each nerve cell in an individual could simultaneously be made to reproduce an earlier temporal firing pattern the individual would be forced to relive and reenact the earlier experience. Fortunately, it is not necessary to duplicate so precisely the activity patterns of the nervous system in order to effect a useful level of neural control. Sometimes quite unnatural stimulation of cells is interpreted as a familiar sensory experience as in the case of phosphene patterns produced by stimulation of the visual cortex. Much information about which input patterns are meaningful, can be derived from observation of naturally occurring nerve cell activity patterns and experimental stimulation of neurons.

IV. FUTUREPOSSIBILITIES A . Nervous System Regeneration

The marvelous biological property of regeneration and repair is limited in the central nervous system of mammals. No new nerve cells are formed after birth and, it has been estimated, that of the 10'' nerve cells in the human brain some lo5 are lost per day. Thus, after 30 years there has been a loss of about 10%. No one knows what role this loss of cells plays in the gradual changes which take place in the aging nervous system. Such irreversible damage is normal. When additional damage to nerve cells occurs through accident or disease there may be loss of normal function such as motor paralysis or sensory deficit. Some day it may be possible to overcome the mysterious barrier which blocks functional regeneration in the mammalian central nervous system. Until then, artificial means must be used whenever possible to help restore function in patients with neural deficits. B. Supernormal Humans

Already some elementary forms of neural control are practical as discussed in the section on current applications. But as the problems which have been described and others not yet foreseen are solved or bypassed, new

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applications of newfound basic knowledge will become apparent. At present inward control appears to have more applications than outward control using currently available technology. More distant is the development of outward control of prosthetic limbs and other mechanical devices. When the present muscular link is eliminated it is not unreasonable to anticipate a reduction in fatigue and possibly an increase in rate of outward information transfer. Combinations of inward and outward control may be expected such as functional neuromuscular stimulation under direct control of cells in the motor cortex or cerebellum, conscious control of bladder evacuation, or bypassing lesions of the nervous system which have interrupted communication pathways. For some time to come efforts will be focused on artificial means to alleviate neural deficits so that subnormal individuals will become more normal. However, there is no automatic limit to stop augmentation of function when the individual has reached a “normal” level. For example, a normal wrist which rotates through 180” may be replaced by an artificial wrist which rotates indefinitely over a wide range of torque and speed. Augmentation of normal strength limitations are already commonplace and await only the elimination of the muscular control link to achieve supernormal outward neural control. Communication is likely to become one of the more important applications of both inward and outward neural control. With about 10 independent parallel channels of outward information it should be possible to control a stenotype machine without intervention of muscular effort. Very slow single channel control of a typewriter has already been tried (B. Tork, personal communication). Direct connection of recording electrodes in the brain to a computer might provide some interesting possibilities for pattern recognition and complex direct outward control of external devices. Inward information transfer for communication could begin with as primitive a system as the Morse code. But hopefully combinations of stimuli could be developed which carry larger pieces of information such as words or even phrases. It is certainly fanciful but not altogether absurd to think that ultimately one brain might be able to communicate directly with another through the transformations provided by a computer. As one particular neural control application after another becomes possible one may expect various man-machine combinations to be developed with a high degree of specialization. To some extent the machine part may be interchangeable from one individual to another but more likely the circuitry implanted will fix the capabilities of that man-machine combination. Evolution in this direction may be expected to gradually increase the differences between individuals with the ultimate development of a class system where individuals differ physically, mentally, and emotionally from one


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another. Different classes may be developed to perform different functions within the society. In such a society the rights of the individual may be subjugated to the advantages of the society as a whole. Stimulation of the brain has already been shown to affect such emotional aspects as pain, fear, aggression, sex, pleasure, and feeding behavior. An animal may press a lever which stimulates his brain through an implanted electrode until he is exhausted, or stimulation in another location may be avoided as though it were painful. In the face of these facts it would be blind to suppose that neural control through stimulation of the nervous system will not be misused for purposes of domination and subjugation as well as for control of motivation. Nevertheless, like TNT and atomic energy, the impact of neural control on civilization is certain to be profound and hopefully will be of great benefit to mankind. REFERENCES P. Bach-y-Rita, “Brain Mechanisms in Sensory Substitution.” Academic Press, New York, 1972. E. Bors and A. E. Comarr. ‘‘Neurological Urology.” Univ. Park Press, Baltimore, Maryland, 1971. W. E. Bradley, S. N. Chou, and L. A. French, J . Neurosurg. 20, 953 (1963). W. E. Bradley, G. W. Timm, and S . N. Chou, Urol. Int. 26, 283 (1971). G. S. Brindley and W. S . Lewin, J . Physiol. (London) I%, 479 (1968). G. S. Brindley, P. E. K. Donaldson. M. A. Falconer, and D. N. Rushton, J . Physiol. (London) 225, 57 (1972). R. E. Burke, D. N. Levine, and F. E. Zajac, Science 174, 709 (1971). C. C. Collins, IEEE Trans. Man-Machine Syst. 11, 65 (1970). P. M. Cooke and R. S . Snider, Epilepsia [3] 4, 19 (1955). A. Djourno and C. Eyries, Presse M e d . 65, 1417 (1957). W. Dobelle, M. G. Mladejovsky, S. Stensaas and J. B. Smith, Ann. Otol. Rhinol, Laryngol. 82, 445 (1973). W. Dobelle, M. G . Mladejovsky and J. P. Girvin, Science, 183, 440 (1974). R. S. Dow and G . Moruzzi, “The Physiology and Pathology of the Cerebellum.” Univ. of Minnesota Press, Minneapolis, 1958. R. S. Dow, A. Fernandez-Guardiola, and E. Manni, Electroencephalogr. Clin. Neurophysiol. 14, 383 (1962). J. H. Doyle, J. B. Doyle, and F. M. Turnbull, Arch. Otolaryngol. 80, 388 (1964). E. V. Evarts, Electroencephalogr. Clin. Neurophysiol. 29, 83 (1968). E. E. Fetz and M. A. Baker, J . Neurophysiol. 36, 179 (1973). 0. Foerster, J . Psychol. Neurol. 39, 413 (1929). S. S. Fox and A. P. Rudell, Science 162, 1299 (1968). S. Gilman, I n “Neural Organization and its Relevance to Prosthetics,” p. 371, Symp. Specialists, Miami, 1973. W. W. L. Glenn, W. G. Holcomb, J. B. L. Gee, and R. Rath, Ann. Surg. 172, 755 (1970). W. W. L. Glenn, W. G . Holcomb, A. J. McLaughlin, J. M. OHare, J. F. Hogan, and R.Yasuda, N . Engl. J . Med. 286, 513 (1972). T. Hald, Dan. M e d . Bull. 16, 1 (1969).

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Charged Pigment Xerography M. E. SCHARFE

F. W. SCHMIDLIN

AND

............... .......... 83 ................................................ 85 A. Steps in the Xerographic Process .............................................................. 85

I. Introduction .................. 11. General Discussion of the

B. The Tone Reproduction Curve and Its Optimization

...................................................... ....................................... IV. Physical Basis for Development A. The Fundamental C o n n e d tic Image Development and Charge Neutralization.. .......................................................................... B. Description of the Driving Force ............................. C. Viscosity Controlled Development-Aerosol and Electr ....................... D. Adhesion Controlled Development ......... E. Cascade as a Mixed Inertia-Adhesion Controlled System .............................. V. Summary ................................................................................................. References ........

94 I00 100

104 113 113 116 I18 130 I39 144 144

I . INTRODUCTION Electrophotography began with its invention by Chester Carlson in 1938. The first electrophotographic system consisted of a photoconductive layer of fused sulfur, a developer medium of lycopodium powder, and a sheet of waxed paper (1-3). The images were produced in a four-step process. The photoconductor was first electrostatically charged or sensitized by rubbing it with a cloth. The sensitized layer was then exposed through a contact transparency of the desired image which produced a latent electrostatic image on the surface of the photoconductor. The latent electrostatic image was developed by pouring the lycopodium powder over the fused sulfur. The image was transferred to waxed paper by pressing the paper onto the developed image. This type of electrophotographic printing was later to be named xerography after the Greek words xeros and graphos, which together mean “dry writing” ( 4 ) . Electrophotography has been expanded considerably since Carlson’s invention and now includes a wide variety of electrophotographic processes. 83

84

M. E. SCHARFE A N D F. W. SCHMIDLIN

In 1961, an IRE subcommittee began a classification system in an effort to classify electrophotographic processes (5). In this classification, xerography came to mean any imaging system in which visible or ultraviolet electromagnetic radiation was employed to produce a latent image in the form of an electrostatic charge pattern. The charge pattern in turn was assumed to produce an electric field variation ( 6 E ) which acted upon electrostatically charged pigment particles to render the latent image visible (i.e., development). Each pigment particle carried some constant charge (Q) which served as the “handle” ( 6 )by which the electric field selectively pulled the pigment particle to the charge pattern. After 1961, a variety of new electrophotographic imaging systems were introduced which clearly belonged in a different class from xerography (7-10). In these systems, the electromagnetic radiation acted upon the pigment to produce a latent image in the form of a charge variation (SQ) directly in the pigment particles. Thus, the process of carrying a charged pigment developer to a latent electric field image was no longer required. In these systems, the latent image was developed by applying a constant electric field ( E )which sorted the pigment particles according to their charge. In order to clarify the fundamental difference between the abovementioned new imaging systems and xerography, a broad description of electrophotography was formulated by considering the total variation in electric force Q E over the image plane ( 6 ) .The total variation in QE is 6 ( Q E ) = Q SE

+ E SQ.

(1) The first term in this expression (Q 6 E ) is identified with xerography. Xerography has two subclasses : charged pigment xerography in which the charge Q resides on pigment particles; and noncharged pigment xerography in which the charge Q does not reside specifically on pigment particles [e.g., FROST (11)]. The second term in Eq. (1)is identified with the newer class of imaging systems where the latent image is a charge variation SQ in the pigment (7-1 0). Because of the wide variety of electrophotographic systems now in existence, it is not possible to discuss all of them in this short review article. Therefore, we have chosen to concentrate on the status of the physical understanding of charged pigment xerography. The paper is divided into three major sections. In the first section, we define and discuss the seven operational steps in the xerographic process, show how these steps combine to produce the tone reproduction curve, and give an example of how certain steps in the xerographic process can be optimized for a given xerographic system. In the second and third sections, we provide a detailed discussion of the physics that control the two most important steps in the xerographic process, the formation of the electrostatic image, and the development of the latent image into a real image.

85

CHARGED PIGMENT XEROGRAPHY

11. GENERAL DISCUSSION OF THE XEROGRAPHIC SYSTEM A . Steps in the Xerographic Process

The number of steps in the xerographic process depends on the type of photoreceptor to be used in the system. It takes seven steps to produce a single copy when the photoreceptor is to be reused in a multiple copy situation. It takes four steps in the case where the photoreceptor is a photoconductive paper which is used to produce only one copy. The basic xerographic steps for a multiple copy situation are shown in Fig. 1. These steps include (13, 1 4 ) : (a) Charging or sensitizing the photoreceptor. (b) Exposing the photoreceptor to form the latent electrostatic image. (c) Developing the electrostatic image. (d) Transferring the image to paper. (e) Fusing or fixing the image onto the paper. (f) Cleaning the extra toner from the photoreceptor. (g) Erasing the electrostatic image. \\

-/ /

Se FILM

>+-

PAPER

TONER CARRIER

-

(C)

,HEATER

PAPER

TON&

(el

(P)

FIG. 1. The seven basic steps in the xerographic process. (a) Sensitization, (b) exposure, (c) development, (d) image transfer, (e) fixing, (f) cleaning, (g) erase. From Tabak et al. ( 1 3 ) with permission.

86


If a photoconductor paper is used, steps (d), (f), and (g) are eliminated and the image is fused onto the paper. In the remainder of this section, we will discuss each step in the xerographic process to show how these steps are performed and how successive steps interact in producing a final image. We shall discuss only the phenomenological aspect of each step. The detailed physical discussion of these processes and their interrelationship appear in later sections. a. Sensitization. A xerographic photoreceptor is sensitized when it is uniformly charged to its maximum surface charge density. The most practical method of sensitization is to charge with a corona discharge. A corona discharge is a high voltage discharge obtained by applying a severalthousand volt bias between a conductive wire and the substrate of the photoreceptor plate (2, 15, 16). The extremely high fields near the conductive wire ionize the air producing ions which drift along the field lines between the wire and the photoreceptor and deposit on the free surface of the photoreceptor plate. The amount of charge supplied to the surface depends on the voltage on the corona wire, the spacing between the wire and the photoreceptor, and the length of time the ion current is allowed to flow. The magnitude of the surface potential produced by the corona charging is equal to o/C where t~ is the total charge per unit area supplied to the material and C is the capacitance per unit area. There are two types of corona charging devices. These are shown diagrammatically in Figs. 2 and 3. A “corotron” is shown in Fig. 2. This is

/SHIELD

PHOTOCONDUCTOR

FIG. 2. Schematic diagram of the elements in a corotron. Dimensions are in inches. From R. G . Vyverberg, in “Xerography and Related Processes” (J. H. Dessauer and H. E. Clark, eds.) by permission of Focal Press, London.

simply a corona wire which’is surrounded by a ground shield (15, 16). The ground shield serves two important purposes. It reduces variations in the charging current produced by irregularities in the wire diameter and it compensates for a nonuniform spacing between the corotron and the photoreceptor.


87

DIRECTION OF TRAVEL CORONA EMITTING WIRES

,-SCREEN

7 16

?I4

PHOTOCONDUCTOR

FIG. 3. Schematic diagram of the elements in a screen-controlled corona charging device (scorotron). Dimensions are in inches. From R. G . Vyverberg, in “Xerography and Related Processes” (J. H. Dessauer and H. E. Clark, eds.) by permission of Focal Press, London.

A “scorotron” is shown in Fig. 3. The scorotron is essentially a modified corotron with a control grid (15, 17). The control grid serves two purposes. It further improves the uniformity of the charging and it makes it virtually impossible to overcharge the photoreceptor. The control grid is usually biased at the desired photoreceptor surface potential. The control grid allows ions to flow to the photoreceptor surface until the surface pokntial is equal to the grid potential. At that potential, the ion current stops and the surface potential will remain equal to the grid potential. There are several other methods of charging photoreceptors. Most of these are not suitable for practical use. These methods include charging with ions generated by a radioactive source exposed to air ( 2 ) ,charging by rolling a biased rubberized roller over the photoreceptor surface (18),and induction charging by bringing a biased or charged electrode close to a grounded photoreceptor then ungrounding the photoreceptor and removing the bias electrode (2). In general, any process which can put a uniform charge on the surface of the photoreceptor can be used in a xerographic system. b. Formation of the latent image. The second step in the xerographic process is the formation of the latent electrostatic image. The latent electrostatic image is the electrical charge pattern on the surface of the photoreceptor which is produced by exposing the sensitized photoreceptor to the optical image. The formation of the electrostatic image is shown diagrammatically in Fig. 4. The light from the input image impinges on the surface of the photoreceptor. The incident photons are absorbed by the photoreceptor creating free hole-electron pairs in proportion to the intensity variation in the optical image. The hole-electron pairs are separated by the electric field and each carrier drifts toward the appropriate electrode. In this example, the holes drift toward the substrate and the electrons drift toward the illuminated surface. When the carriers reach the appropriate electrode, they recombine

88


8 + (electron)

4 (electron)

+(hole)

+ (hole )

4

+

___________ FIG. 4. Illustration of the internal photogeneration and charge transport in a photoreceptor.

with the surface charge. Since the number of photogenerated carriers varies with the incident light intensity, a latent electrostatic image patterned after the optical image appears on the surface of the photoreceptor. In discussing the development of latent electrostatic images, it is customary to talk of surface potentials rather than surface charge densities. This is done for two reasons. First, it is the surface potential, not surface charge density, that is measured experimentally. The surface charge density is determined from the potential through the expression (T = CV where C is the capacitance of the sample per unit area and I/ is the measured surface potential. Second, the electric fields which drive the development processes are usually proportional to the surface potential and the differences in surface potentials (contrast potentials). Exceptions to this are discussed in Section II1,A. One method of measuring the surface potential is shown in Fig. 5 (19). The photoreceptor is charged to its optimum potential by a corotron in one position and transported to a second position where the photoinduced discharge measurements are performed. The surface potential is monitored by a thin wire or Nesa glass probe which measures the magnitude of the potential induced on the probe. The surface potential is measured as a function of exposure by monitoring the surface potential while simultaneously exposing the material through a thin wire or transparent Nesa glass

HIGH VOLTAGE COROTRON

n

MONOCHROMATIC

m

OSCILLOSCOPE

r111:3+-

CHARGE POSITION

AMPLIFIER

MEASUREMENT POSITION

FIG. 5. Schematic diagram of the experimental apparatus used to measure the photoinduced discharge curve (PIDC). From Scharfe (19) with permission.


89

probe. The resulting surface potential versus exposure curve is defined as the photoinduced discharge curve (PIDC). This curve is used as the transfer function which relates the broad area optical input exposure to the surface potential and eventually to broad area development. c. Development. The latent electrostatic image can be developed by some very simple methods. One method is to pour a powder over the material as Carlson and Kornei did in producing the first xerographic prints ( I ) . In the pouring process, some of the powder particles acquire an electrostatic charge which is opposite to that of the latent image. These particles are attracted to the surface of the photoreceptor and adhere in numbers proportional to the strength of the latent image and produce a visible image. Another method is simply to blow smoke over the latent image (20). Some of the particles in the smoke are electrostatically charged and those particles of the correct polarity will be attracted to the latent image. In general, charged pigment particles of any kind can develop the latent image provided that they are sufficient in number and of the correct polarity. Several novel xerographic development systems have been invented to develop the latent image on a large scale practical basis. These development systems include cascade, magnetic brush, liquid ink, aerosol, electrophoretic, frost, fur brush, etc. With the exception of frost and liquid ink, these systems all use very fine charged pigment particles to develop the image. The differences between the charged pigment systems lie in the methods of transporting the pigment particles to the latent image. i. Cascade development. Cascade development is one of the most common methods of xerographic development. In this system, the developer material consists of “ toner” and “carrier beads (21,22).The toner is a fine pigment powder which is triboelectrically attached to a much larger carrier bead. This is shown diagrammatically in Fig. 6 . The carrier bead is primarily a transport vehicle which brings the toner to the electrostatic image. The development is accomplished by flowing or cascading ” the developer material over the photoreceptor plate. The agitation of the developer shakes off some of the toner which senses the electric field produced by the image and drifts to the surface of the photoreceptor. In addition, some of the toner is electrostatically pulled from the carrier beads when the beads come in ”

“

CARRFR PHOTOCONDUCTOR

\ BASE FIG.6. Schematic diagram of cascade development of electrostatic images (2).

90


close contact with the surface. This occurs when the electrostatic force between the photoreceptor and toner becomes larger than the adhesive force between the toner and carrier. A development electrode can be used in cascade development. The development electrode is a grounded or biased conducting plate which is placed close to the surface of the photoreceptor. In this case, the developer is cascaded between the development electrode and the surface of the photoreceptor. The development electrode enhances the electric fields in the development zone which in turn enhances the development process. ii. Magnetic brush development. The magnetic brush development system is shown diagrammatically in Fig. 7. The system consists of a magnet, a mass of iron beads or filings, and toner particles which are electrostatically attached to the iron beads or filings. The mass of iron filings are attracted to the magnet forming long chains which appear as bristles on a brush, hence the name magnetic brush (23-25). Development is accomplished by passing the latent electrostatic image beneath the magnetic brush. The electric field produced by the image electrostatically strips the toner from the iron beads and deposits it onto the surface of the photoreceptor.

/

FIG.7. Schematic diagram of magnetic brush development of electrostatic images. From R. M. Schaflert, “Electrophotography,”by permission of Focal Press, London.

The electric field in the magnetic brush developer can be larger than that in electroded cascade development. This is due to the greater bead density and the magnetic alignment of the beads, which produces a smaller effective spacing between the magnetic brush and the photoreceptor surface. We discuss the effects of development electrodes and development fields in Section 111. iii. Electrophoretic development. The electrostatic image can be developed with liquid developers (26-28). Liquid developers consist of a suspension of charged pigment particles in a dielectric liquid such as an insulating hydrocarbon. The development proceeds by covering the latent image with the developer fluid. The charged particles in the fluid sense the electric fields produced by the image and drift along these field lines to the surface of the photoreceptor.

91


iv. Fur brush development. A fur brush can be used in a manner similar to that of the magnetic brush development system. In this case, the toner is triboelectrically attracted to a cylindrical brush made from animal fur (20). The toner is continuously supplied to the fur brush which rotates and oscillates while in contact with photoreceptor. This system is difficult to control in practical systems, since humidity changes have a great effect on the triboelectric properties of the fur. v. Frost deuelopment. Frost development is a conceptually different xerographic process (11). In this process, the image is observed as a deformation in a resin layer and not by the density of pigment particles on the surface of the photoreceptor. One type of frost imaging geometry and process is shown in Fig. 8a. A transparent insulating resin film is placed over the surface of the photoreceptor. The resin-photoreceptor layer is then charged and simultaneously exposed to the input image. This produces large fields in the exposed region of the resin layer. The magnitude of these fields varies in proportion to the total amount of exposure. The resin layer is then softened by heat or solvent vapor. The softening allows the resin layer to deform in proportion to the field across the film. The developed image appears as a “frost ”-like image due to the light-scattering effect of the deformation in the resin layer. The image can be kept permanently or can be reused by heating the layer to relax the image. An alternative frost process is shown in Fig. 8b. CHARGE A N D EXPOSE SIMULTANEOUSLY W H I L E OVERCOATING IS SOFT

DARK

LIGHT

t+ttttttttttt+t

I I

I

I ‘\

\

(1) CHARGE

(2) EXPOSE TO

OPTICAL IMAGE

“TRANSPARENT“CHARG1NG DEVICE

( 3 ) CHARGE TO ZERO POTENTIAL (0)

(4)SOFTEN

(HEATOR VAPOR 1

(b)

FIG 8. Schematic diagram of the “frost” development process. (a) Simultaneous, (b) sequential. From Gundlach and Claus ( I I ) with permission.

92


d. Image transfer. The next step in producing a xerographic copy is to transfer the developed toner image from the photoreceptor to paper or material which will permanently retain the image. There are several methods of transferring images (29-31). Two of the more practical methods are electrical transfer and adhesive transfer. The image can be transferred electrically by placing paper over the image and charging the paper with a corotron of the same polarity used to charge or sensitize the photoreceptor. This is illustrated in Fig. 9. The charging PAPER

FIG.9. Illustration of electrostatic transfer. From Tabak et

a/.

( 1 3 ) with permission.

produces a strong electrostatic field in the space between the paper and the toner on the surface of the photoreceptor. This field lifts the toner from the photoreceptor and transfers it to the paper, producing an image on the paper. The paper is then removed from the photoreceptor and the image is ready to be fused or fixed permanently. The transfer can also be made with semiconductor rollers biased at very large potentials (32).In this case, the paper is fed between a biased roller and the photoreceptor. The roller is biased at about lo00 V above the substrate of the photoreceptor producing a large electric field in the nip between the roller and the photoreceptor. The pressure is adjusted so that the paper is in intimate contact with both the roller and the photoreceptor. The image is transferred by the bias field as the paper passes through the nip. The image on the photoreceptor can also be transferred by pressure sensitive adhesives coated onto the paper. In this process, the adhesive force between the paper and the toner exceeds the electrostatic force between the toner and the photoreceptor. The transfer is made by placing the paper firmly on the photoreceptor and removing it. As the paper is pulled away, the adhesive forces strip the toner from the photoreceptor transferring the image to the paper. Adhesive transfer is a very efficient process which transfers toner more uniformly than electrostatic methods. Adhesive transfer is especially useful in situations where a faithful, continuous tone reproduction is needed. The main problem with adhesive transfer occurs in transferring extremely dense


93

uniform images. The adhesive has to make contact with all the toner it is to transfer. If the toner layer is quite thick, some of the toner will not touch the adhesive and will not be transferred. e. Fusing or Jixing. The toner particles used in the development of the electrostatic image are usually made from resins which have low melting points. These resins are colored by blending them with colored pigments. The toner image is fixed to the paper by heating the paper until the toner begins to soften and flow. The toner particles first wet and coalesce, then wet and fuse to the paper (30, 33). The toner actually migrates a small distance into the paper and forms a permanent bond. The toner can also be fused by solvents and solvent vapors (34). In this case, the vapors chemically soften the resin toner particles until they bond together and to the paper. As the solvent evaporates, the toner rehardens forming the permanent image. Other methods of fixing include coating the image with a transparent lacquer or pressure fusing by forcing the toner into the paper at high pressures (32). One novel method employs toner particles with encapsulated ink. The encapsulated ink is released by crushing the toner particles at high pressures between rollers (35).The released ink stains the paper in the form of the image. For mechanistic discussions of fusing and fixing techniques refer to Lee (33). f: Cleaning. Some toner remains on the photoreceptor after the image has been transferred. This toner has to be removed before the next input document can be copied. The most common method of cleaning is to wipe the surface with a fur brush or oscillating blade. Once the brush or blade frees the toner from the surface, the free toner is removed from the system by vacuum suction. It is also possible to clean the photoreceptor by cascading a granular cleaner over the residual image (36). The granular cleaner attracts the residual toner in a manner similar to the triboelectric attraction of the toner to the carrier beads. This process is called “scavenging” (37) and can be quite effective in removing toner which is tightly bound to the photoreceptor surface. g . Erasure. The final step in the xerographic process is the erasure of the latent electrostatic image. This is accomplished by uniformly exposing the photoreceptor to an erase lamp. The erase exposure is adjusted to be just large enough to drive the potential to zero over the entire surface. It is important not to overexpose the photoreceptor too much since the internal dark decay of most xerographic photoreceptors increases with erase exposure. This can cause reproducibility problems if the photoreceptor is to be cycled frequently.

94

M. E. SCHARFE AND F. W. SCHMIDLIN

B. The Tone Reproduction Curve and I t s Optimization Each of the process steps described above affect the fidelity or character of the output image. Two of the process steps, however, clearly dominate the system performance and may be appropriately called the heart of the xerographic process. These are the latent image formation and development steps. In this section, we show how these two steps transform an image from the input exposure to the output copy, assuming all the other process steps have no effect on the image at all. To facilitate the discussion, it is customary to subdivide the latent image formation step into separate exposure and photoreceptor discharge processes even though they occur simultaneously in the same operational step. This separation makes it possible to define the three most important xerographic subsystems: exposure, photoreceptor, and development. Each xerographic subsystem can be described by a transfer function. The transfer function is the descriptor which shows how each subsystem influences the output image. In general a given transfer function has meaning only with respect to a particular input image. In most cases, images are predominantly composed of lines of different width (w) or broad (solid) areas. We deal with these cases analytically later, but for now it is sufficient to note that it is necessary to consider transfer functions for at least these two types of input images. In this section, we discuss the general nature of the transfer functions for each of the key subsystems and show how they may be adjusted to obtain optimal performance of the overall system. The manner in which the transfer functions combine to produce an overall input-output characteristic is conveniently diagramed by the four-quadrant plot shown in Fig. 10. Early discussions of such four-quadrant plots can be found in Bixby et al. (38)and Bickmore et al. (39, 40). We now discuss the transfer functions in each quadrant. 1. Transfer Functions for the K e y Subsystems

a. Exposure system. The exposure system transfers the input image into an input exposure on the surface of the photoreceptor. Several different transfer functions which describe this process for broad areas are shown in the lower right quadrant of Fig. 10. The horizontal axis from the origin out to the right is image input density and the vertical axis from the origin downward is log of the input exposure on the photoreceptor. The density of the input is defined by: I

= - 1%

( I ,/ I s ) ,

(2)


95

FIG. 10. Relationship between the three most important xerographic subsystems. The upper right quadrant shows the relationship between the density of the input image and the density of the output copy. The other quadrants represent possible transfer functions for the development system, photoreceptor system, and exposure system.

where I , is the reflected light intensity from the image and I , is the reflected Iight intensity from some standard white sheet of paper. Under this definition, the background density of the standard is zero regardless of the amount of reflected light. Input documents other than the standard have an average minimum density (background density) different from zero. The input exposure and the input density are related through the expression : log X

=

log X , - D,

(3)

where X is the local exposure on the photoreceptor corresponding to the input density D,and X , is the maximum exposure due to light reflected from the standard white sheet of paper. The input exposure produced by light reflected from the background density is defined as the background exposure XBg.It is evident that as X , is varied by varying the intensity of the exposure lamp, the exposure for a given input density varies accordingly.

96


The dashed lines in Fig. 10 refer to transfer functions with different values of Xm.

The transfer function for line images should be modified by the modulation transfer function (MTF) of the lens, etc. In order to focus attention on the electrical aspects of the system, we assume a lens with a flat MTF so that the same exposure transfer function can be used for both lines and broad areas. b. Photoreceptor system. The photoreceptor is characterized by the photoinduced discharge curve (PIDC). The PIDC is the transfer function which transforms a broad area input exposure into a surface potential on the photoreceptor. An example PIDC is shown in the lower left quadrant of Fig. 10. The horizontal axis is in units of potential and the vertical axis is in units of log of the input exposure. The surface potential produced by an input image of density DI depends on the maximum exposure, X , , the functional form or “shape” of the The choice of X , is critical in most xeroPIDC, and the local density (0,). graphic situations since different values of X , can produce significant differences in the efficiency of developing a given image. We will discuss the optimization of X , in the following section. For the present case, assume that X , has been chosen to just reduce the surface potential to zero for an input density of D = 0. The exposure transfer function for this value of X , is represented by the solid line in the lower right quadrant of Fig. 10. Under these conditions, the input exposure produced by an input density D, is X , as indicated by the arrows in the figure. E, in turn produces a potential V, on the surface of the photoreceptor. It is evident that if X , is changed (refer to dashed lines), the same input density would produce a significantly different surface potential. c. Development system. The development transfer functions are typically complex since they depend on the type of developer, the particular development process or system, and the amount of interaction between the developer and photoreceptor. We discuss the underlying physics for two special development systems in Section IV. For the present, we simply use linearized approximations for both lines and solid areas for some unspecified developer. A typical linearized solid area transfer function has the form: Ds = YO(
0.1 and is essentially a representation of the confluence of the various graphs of& in Fig. 12 for this current range.

ROBERT A. PUCEL ET AL.

218

Normollzed draln current Id/ I (0)

-2 P

c VI

.

0

c

-101

10'

2

4

6810'

2

4

6910°

Normalized drain current I d / I , (C )

FIG. 12. Transconductance and drain resistance as a function of normalized drain current for several values of the saturation index 5 with the drain voltage fixed. V,,/Wo0= 1, gm = (WKoo)f~, rd= (Wooi4)L.(a) Liu = 3, (b) Llu = 5, (c) Liu = 10.

2. Drain Resistance

id

The drain resistance rd is the ratio of the change in drain voltage to the differential change in drain current producing it when the gate voltage is fixed, that is id=

.

-dKd /dld IV.

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

219

The negative sign stems from the fact that I d is directed outward from the drain contact. By a differential process analogous to that used for gm , it has been shown by the authors (Statz et al., 1974) that rd can be expressed in the form where

Notice that the quantity in braces is the same as the denominator off,. In the limit L2 -+ 0, p = d , the expression (25) reduces to rd = (L/Zgo)x (1 - d ) - which was obtained by van der Ziel (1962) for the case of no velocity saturation. Graphs of the resistance function, or rather the product

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