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 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

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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

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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.

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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-

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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-

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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-

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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.

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

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 )

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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.

INTRABAND MAGNETO-OPTICAL STUDIES. 11

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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.

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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 ”

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. 11

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

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

“ 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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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

INTRABAND MAGNETO-OPTICAL STUDIES. I1

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

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

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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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).

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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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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.

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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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F. T. Hambrecht, I n “Neural Organization and Its Relevance to Prosthetics,” p. 281, Symp. Specialists, Miami, 1973. W. House, I n “Neural Organization and its Relevance to Prosthetics,” p. 273, Symp. Specialists, Miami, 1973. D. R. Humphrey, E. M. Schmidt, and W. D. Thompson, Science 170, 758 (1970). M. Ito, Int. J . Neurol. 7, 162 (1970). N. Y. S. Kiang and E. C. Moxon, Ann. Otol. 81,714 (1972). F. Krause and H. Schum, (1931). I n “Neue Deutsche Chirurgie” (H. Kuttner, ed.), Vol. 49a, p. 482. Enke, Stuttgart. W. T. Liberson, I n “Functional Neuromuscular Stimulation, Report of a Workshop,” p. 147. Nat. Acad. Sci., Washington, D.C., 1972. W. T. Liberson, H. J. Holmquest, D. Scot, and M. Dow, Proc. Int. Congr. Phys. Med., 1960 p. 705 (1962). C. Long and V. D. Masciarelli, Arch. Phys. Med. Rehabil. 44, 499 (1963). M. Lowenthal and V. Horsley, Proc. R o y . Soc. 61, 20 (1897). R. W. Mann and S. D. Reimers, l E E E Trans. Man-Machine Syst. 11, 110 (1970). E. Marg, W. W. Alberts, and B. Feinstein, In “Neurophysiology Studied in Man” (G. G . Somjen, ed.), p. 14. Excerpta Med. Found., Amsterdam (1973). R. Melzack and P. D. Wall, Science 150, 971 (1965). R. P. Michelson, Ann. Otol. 80, 9 14 (1971). J. M. Morris, S. F. Pollack, and H. A. Hinchliffe, ‘’ Elicitation of Periods of Inhibition in Human Muscle by Stimulation of Cutaneous Nerves,” Tech. Rep. No. 59. Biomechanics Lab., University of California, Berkeley, 1968. J. T. Mortimer, In “Neural Organization and its Relevance to Prosthetics,” p. 131, Symp. Specialists, Miami, 1973. M. J. Nahvi, C. D. Woody, R. Ungar, and A. R. Sharafat, Electroencephalogr. Clin. Neurophysiol. 38, 191 (1975). B. S. Nashold and H. Friedman, J . Nrurosurg. 36. 590 (1972). B. S. Nashold, H. Friedman. J. F. Glenn, J. H. Grimes, W. F. Barry, and R. Avery, Arch. Surg. (Chicago) 104, 195 (1972). J. Olds, lnt. Congr. Physiol. Sci. [Proc.], X r d , 1965 p. 372 (1965). A. K. Ommaya and T. Genarelli, Brain 97. 633 (1974). P. H. Peckham, I n “Neural Organization and its Relevance to Prosthetics,” p. 131. Symp. Specialists, Miami, 1973. W. Penfield and H. Jasper, “Epilepsy and the Functional Anatomy of the Human Brain.” Churchill, London, 1954. D. P. Purpura, J. K. Penry, D. Tower. D. M. Woodbury, and R. Walter, “Experimental Models of Epilepsy.” Raven Press, New York. 1972. D. Radonjic and C. Long, Proc. I n t . Symp. Ext. Contr. Hum.Extremeties, 3rd, 1969 p. 59 (1970). A. T. Rasmussen, Laryngoscope 50, 67 (1940). G . R. Reimer, R. J. Grimm, and R. S. Dow, Electroencephalogr. Clin.Neurophysiol. 23, 456 (1967). M. Salcman and M. Bak, I E E E Trans. Bio-Med. Eng. 20, 253 (1973). K. Sasaki, S. Kawaguchi, Y. Matsuda, and N. Mizuno. E v p . Brain Res. 16, 75 (1972). E. M. Schmidt, R. G . Jost, and K. K. Davis, Brain Res. 73, 540 (1974a). E. M. Schmidt, M. J. Bak, and J. D. Mclntosh, Fed. Amer. Soc. Exp. B i d . Abstr. (1974b). E. M. Schmidt, R. G . Jost, and K. K. Davis, Brain Res. 83, 213 (1975). C. N. Shealy, J. T. Mortimer, and J. B. Reswick, Anesth. Analg. (Cleveland) 47, 489 (1967). C. S. Sherrington, Proc. Roy. Soc. 61, 243 (1897). F. B. Simmons, Arch. Otolaryngol. 84, 24 (1966).

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Charged Pigment Xerography M. E. SCHARFE

F. W. SCHMIDLIN

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

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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

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

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.

CHARGED PIGMENT XEROGRAPHY

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

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

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.

CHARGED PIGMENT XEROGRAPHY

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

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

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

CHARGED PIGMENT XEROGRAPHY

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

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

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

CHARGED PIGMENT XEROGRAPHY

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)

CHARGED PIGMENT XEROGRAPHY

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

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

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(<

-

Vb),

(4)

where D, is the output density, yo is a constant, Vp is the magnitude of the surface potential, and Vb is the bias on the development electrode. Equation (4) is assumed to hold for V, 2 Vb, and D, = 0 for V, < V,. For the case shown in Fig. 10, Vb = 0. In the next section, we discuss methods of choosing an optimal value for Vb.

CHARGED PIGMENT XEROGRAPHY

97

A linearized line development function has the typical form :

where D, is the density of the line, yw is another constant, and V,, is the background potential produced by the background exposure. D,, like D, (discussed in the next section), is determined by the background voltage adjacent to the lines ( V,,) and the image potential produced by the density of the line. For the line development function shown in Fig. 10, it is assumed that V,, = V, = 0. Thus, the only difference between the line and broad area development functions for this example is in the slope. The line development function can also be written in the form: DL

=

(7,

+ ? O H AK

-

yO(Vb

-

VBg),

(6)

where AV, is defined as the contrast potential (V, - V,,). This expression shows that line development is proportional to contrast potential, assuming V, and V,, remain constant. We will use this form of the line development transfer function in the next section where we discuss the optimization of the process. The final input-output transfer functions for both lines and solid areas are shown in the upper right quadrant of Fig. 10. The solid area inputoutput transfer function is defined as the tone reproduction curve. These curves show that lines of a given input density develop more strongly than large solid areas of the same input density. In a following section, we show that the difference in developability is largely due to the difference in development fields associated with both lines and solid areas.

2. Optimal Choice of Operating Parameters for the Xerographic Process The input-output characteristic for any given xerographic system is sensitive to two adjustable operating parameters: the maximum exposure X , and bias voltage Vb. The selection of these parameters depends upon the image characteristic of primary interest. For example, if one is interested only in line copy (typewritten images), it may be advantageous to select the exposure to enhance the contrast density of some low input density lines. On the other hand, if one is interested in continuous tone reproduction (broad area) it may be advantageous to adjust the exposure to provide the maximum output density range or latitude. We illustrate the above with two examples. Suppose first that one is interested in developing the largest possible output density range for the photoreceptor system shown in Fig. 10. To achieve this, one must choose a maximum exposure (X,) which is just large enough to reduce the surface

98

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

potential of the photoreceptor to zero. This exposure allows the entire PIDC to be used as the photoreceptor transfer function. The exposure transfer function for this value of X , is given by the solid line in the lower right quadrant. To avoid losing any of the total available density range in the development system, the bias voltage ( Vb) must be set equal to zero. Negative bias voltages must be avoided since they generally produce a high output background. As a second example of the optimization process, we assume that it is most important to maximize the output density of low density input lines (D = 0.3) relative to a low background density (D 0.03 or less). The output density of these lines is maximized by choosing the background exposure (XBg)which produces the largest possible contrast potential for a 0.3 input density [refer to Eq. (6)]. The optimal background exposure is obtained from the contrast potential curves shown in Fig. 11. In this figure, we have redrawn the PIDC

-

/

'0.3'

'1.0 '0.5

LOG EXPOSURE

FIG. 11. Photoinduced discharge curve (PIDC) and associated contrast potentials for the PIDC shown in Fig. 10.

shown in Fig. 10 and have added a number of contrast potential curves. The contrast potential curves are a family of curves which transform differences in input exposure (differences in input density) into differences in surface potentials (contrast potentials). Each curve in this family describes the contrast potential as a function of background exposure for a different input density. The contrast potential curves shown are for the 0.3, 0.5, and 1.0 input density differences. These are only examples. There is a contrast potential curve for every possible input density difference.

99

CHARGED PIGMENT XEROGRAPHY

The contrast potential curves are calculated directly from the PIDC and do not require a special measurement. The contrast potential at a given background exposure is determined by subtracting the magnitude of the surface potential at that exposure from the magnitude of the surface potential produced by the exposure corresponding to the chosen input density. The input density exposure is given by Eq. (3). The entire contrast potential curve is obtained by repeating this calculation for all possible background exposures. For example, the contrast potential curve for the 1.0 input density difference is obtained by subtracting the surface potentials which are separated by one order of magnitude in exposure and plotting the difference at the higher exposure. The background exposures which produce the maximum contrast potential and therefore the maximum output density differences are determined directly from Fig. 11. For example, the contrast potential for a 0.3 input density difference peaks when the background exposure is X o . 3 . Background exposures above or below X o . 3 produce less contrast potential and consequently less development. Similarly, the contrast potential for a 1.0 input density difference peaks at a background exposure X,,,, etc. It is evident that each input density has a unique optimum background exposure.

\

11.6

LINES

A

E

S

1.4

REPRODUCT'ION CURVES

DEVELOPMENT

J. 0.4 0.6 9.8 1.0 1.2

\

VOLTS

\

'

t

v0.3

,

,-. . * '

INPUT DENSITY

/

?

PHOTORECEPTOR

EXPOSURE

FIG.12. Optimal relationship between the xerographic subsystems for copying 0.3 density lines.

100

M. E. SCHARFE AND F. W. SCHMIDLIN

X 0 , 3 is chosen as the background exposure in this example since it is important to maximize the output density of 0.3 density lines. The background potential associated with the background exposure is Vo.3 as indicated in Fig. 11. The electrode bias V, is chosen to be equal to Vo.3 or slightly higher. This assures a white output background density ( D = 0) for an input density equal to the background ( D 0.03). The maximum exposure X , is obtained from Eq. (3) by substituting D = 0.03 for the background density and X = X0,3for corresponding background exposure. The optimized system is shown in Fig. 12. The exposure system transfer function is given by Eq. (3) with X, approximately equal to Xo.3. The development system transfer functions are given by Eqs. (4) and (6) with both VBg and V, equal to V0,3.The resulting tone reproduction curves provide the largest possible output density for low density ( D 0.3) lines.

-

-

DISCUSSION OF THE PHOTORECEPTOR SUBSYSTEM 111. PHYSICAL AND ITSCOUPLING TO THE DEVELOPMENT SYSTEM A . Calculation of Electric Field in the Development Zone The role of the photoreceptor in any xerographic system is to transform an optical input image into an image field variation ( 6 E ) above the photoreceptor. The field variation in turn interacts with the development system in producing a developed image on the photoreceptor. The purpose of this section is to discuss the field variations above the photoreceptor for a twolayer dielectric slab model of the photoreceptor and development system and to show how the geometry and dielectric constants of both systems modulate these fields. The dielectric model of the photoreceptor and development system is shown in Fig. 13. The photoreceptor is represented by a region of thickness s and dielectric constant E, . The development system (developer media) is represented by a region of thickness d and dielectric constant q,. The photoreceptor is supported by a conducting substrate and the developer media is contained by a conducting development electrode which is biased at a voltage of V, relative to the substrate. The coordinate, z, represents the spatial distance from the surface of the photoreceptor to some point in the developer media. In the discussion to follow, we will formulate the relation between the two-dimensional variations in surface potential produced by exposure to an image and the electric field above the photoreceptor which drives the development process.

C H A R G E D PIGMENT XEROGRAPHY

2

d,a* t +t+

S, as

101

DEVELOPER MEDIA

,U(x)=uotukc o s K x

+

t +tt

t

t t + t

PHOTORECEPTOR

/,-/,: 7-, 7-:727,:7-SUBSTRATE

FIG. 13. Dielectric model of photoreceptor and development system. From Ing ( 4 3 ) and Scharf ( 4 2 ) with permission.

We first consider the electric field in the development zone produced by a simple periodic surface charge density in one dimension : C(X)

=

Co

+

d k COS

kx,

(7 )

where cro and g k are constants and k is the spatial radial frequency ( k = 27c/il where 1 is the wavelength). Poisson's equation for this charge distribution can be solved to obtain E"(Z) = (oO/&s)fo + h / E s ) f k cos kx -fo(v,/s)

(8)

where En is the normal component of the electric field above the plate at the bias voltage V,, and

is identical tofk as k --t 0. It can be shown, after a little algebraic manipulation, that Eq. (8) is identical to the one given by Schaffert (41). In the preceding section, the photoreceptor transfer function was described in terms of the broad area PIDC. The voltage in this case is the free (open circuit) surface potential without any development electrode present. In the absence of space charge throughout the bulk of the photoreceptor, which is assumed in this analysis, the open circuit surface voltage is related to the surface charge density by

fo

y

= rJ,S/&,

*

(10)

102

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

By solving Poisson’s equation for the open circuit condition (or taking it can be shown that Eq. (10) can also be used the limit of Eq. (8) as d + a), for spatially varying surface potentials and surface charge densities as long as ks < 1. For slowly varying images (ks < l), Eq. (10) can be used to transform the surface charge densities in Eq. (8) into PIDC surface potentials. Making the substitutions (42, 43)

for the peak image potential and

for the background potential. Equation (8) becomes

It should be emphasized that a simple periodic intensity pattern would not produce a simple periodic charge (or surface potential) pattern on the photoreceptor due to the nonlinear PIDC. Therefore small amplitude exposures or some means of exposure compensation must be employed to experimentally generate purely sinusoidal charge patterns. It will be shown in Section IV that the developed density variations can be obtained from Eq. (12) by simply multiplying through by a constant. This is valid provided that the development process does not appreciably neutralize the image and that the minimum electric field does not reverse direction over the surface of the photoreceptor. This field condition is defined as the “small signal” case. When the minimum electric field does reverse, a different formulation relating field to development applies. This condition is defined as the “large signal” case. Examples of these cases are shown in Fig. 16. To obtain an analytic approximation for the electric fields associated with rectangular line images, one must consider a Fourier expansion of a single line of width w. It can be shown that a good approximation for the magnitude of the electric field at the center of the line can be obtained by neglecting all Fourier components except the fundamental (1 = 2w). The higher components can be viewed as building the rectangular line shape. This approximation is especially good for practical xerographic systems since the edges of the rectangular shape are generally softened by the nonconstant MTF of the exposure system lens.

CHARGED PIGMENT XEROGRAPHY

103

Using Eq. (12) as an approximate expression for the electric field, the peak electric field for rectangular lines (at x = 0) can be expressed in the form

This field is proportional to the output density at the center of the line for subneutralized development. It should be noted that this expression bears a close relationship to the development characteristic given by Eq. ( 5 ) . Equation (13) describes the relationship between the output of the photoreceptor transfer function (PIDC) and the electric field which drives the development process. The principal factors in this coupling are f k and fo which Schmidlin (6) called the “coupling coefficients” between the surface charge density associated with the latent image and the developer. The photoreceptor thickness enters into Eq. (13) simply because the surface charge has been expressed in terms of the surface potential. The development fields for an infinitely large solid area are obtained In this limit, the perpendicular field from Eq. (13) by letting k -,0 (1-+ a). becomes : This expression shows that the development fields for large solid areas are proportional to the difference between the image potential and the bias on the development electrode. It also shows that fo is the proportionality constant which describes how the dielectric constants and thicknesses of the photoreceptor and development system combine to modulate solid area development. The development fields for narrow lines are dominated primarily by the fk term in Eq. (13).fk consists of two parts. One part cash k(d - Z) cosh d

describes how the electric field varies as a function of distance above the photoreceptor. The second part

shows how the electric field varies as a function of the dielectric constants and relative thicknesses of the photoreceptor and development system. f k is greater than fo over a wide range of line frequencies for systems where the development zone is thicker than the photoreceptor (d > s). This

104

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

accounts for the more efficient line development observed in many xerographic systems. In these systems,f, is equal tofo at k = 0 (solid areas), increases to a maximum as k increases (line widths become smaller), and finally decreases exponentially in k as k + co. For systems where d < s,f k does not increase to a maximum but continually decreases for increasing k. The absolute magnitude offk in all systems depends strongly on the values of E,, , E, , z, d, etc. By fixing the values of z and k and varying s, it can be seen that f k increases monotonically with s until it saturates when ks 7c. This implies that typewritten lines, as well as broad areas, develop more efficiently with increasing photoreceptor thickness for a given charge density. Practical photoreceptors are limited in thickness because of trapping limitations which severely impair the discharge of the material. We can draw several conclusions about the photoreceptor-development system interaction based on the preceding equations and discussions. These are: (a) Always charge a photoreceptor as high as possible to maximize the development fields for both lines and solids. (b) Lines produce larger development fields than solids over a wide frequency range when the development zone is thicker than the photoreceptor (d > s). (c) Line and solid area development fields can be enhanced by increasing the photoreceptor thickness. (d) Solid area development fields increase with decreasing thickness of the development zone.

-

B. Physical Basis f o r Photoreceptor Performance

The fundamental xerographic photoreceptor characteristic is the photoinduced discharge curve (PIDC). In order to produce a useful PIDC, a photoreceptor material must exhibit three basic photoconductive properties. It must be able to convert the input exposure into free hole-electron pairs. It must allow the free holes and electrons to drift through the bulk of the material without any trapping, and it must hold the latent image charge for a sufficient time to develop the image. In this section, we discuss these properties and show how each property is important in producing the PIDC. 1. Physical Processes Producing the PIDC

Schmidlin has described the xerographic latent image formation process in terms of the fundamental charge generation and displacement processes (6). In this formulation, the generation rate per unit volume is given by g(x, t ) = r/clF,e-""(l - R ), (15)

CHARGED PIGMENT XEROGRAPHY

105

where F o is the flux of incident photons per unit area; 1 - R is the probability an incident photon is not reflected; CI is the absorption coefficient, or probability per unit distance that the photon is absorbed; and q is the probability that an absorbed photon is converted into a mobil charge pair. The expected displacement of a positive charge carrier (hole), assumed to be moving toward the substrate, is S+(s) = 1+(1 - e - ( s - x ) / r + )

(16) where 1;’ is the probability per unit displacement along the field that a hole becomes effectively stopped by a “deep trap.” In xerography, a deep trap is defined as one whose mean release time is comparable to or longer than the process or development time. The corresponding expected displacement of an electron, which moves in the opposite direction of the hole, is &(x)

=

1-(1 - e - x / f - )

(17) where 11’ is the probability per unit displacement along the field that an electron becomes effectively stopped by a deep trap. The reciprocal capture probabilities, or attenuation lengths (l+ and 1- ), reduce to the familiar Schubwegs (,u+ z + E ) and (,u- z - E ) in materials in which the random thermal speeds of both carriers are large compared to the drift speeds ( p + E and ,u- E ) . ,u+ and ,u- are the drift mobilities of the hole and electron, respectively, and z + and z- are their respective deep trapping lifetimes. The instantaneous broad area voltage discharge rate is obtained from Eqs. (15) and (16)by combining the generation rate with the total displacement. This gives:

This expression is valid as long as the time spent in displacement is short compared to the duration of the generation process. The PIDC is obtained from Eq. (18) by integrating over time and plotting V as a function of exposure. For an alternative discussion of the processes controlling the PIDC, refer to the work by Mort and Chen (44). 2. Generation Limited Discharge

To examine the limitations produced by the photogeneration process alone, consider the case in which the combined expected displacement (6, + 6 _ ) occurs in times which are short compared to the duration of the

106

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

generation process and is equal to the thickness of the photoreceptor. Under these conditions, Eq. (18) reduces to

for M % s-' (strongly absorbed light). To integrate this equation, it is necessary to know how the fundamental conversion efficiency, r], depends on electric field ( E ) ,photon wavelength (A),and light intensity (F,). A field dependence in r] was first identified in selenium by Pai and Ing (45), and Tabak and Warter (46). It has since been identified in many other commercially important photoreceptors. A variety of models have been proposed to explain the origin of the field dependence. These models include the Poole-Frenkel (45, 46), Warter dimensional (47), and the Onsager formulation (48, 49). Materials which exhibit flux variations in r] have not been identified experimentally in practical xerographic photoreceptors. However, we include these effects in this formulation to show how flux variations would affect the PIDC should they occur. For the purpose of integrating Eq. (19), it is convenient to represent the field and intensity dependence of r] by a power law. This is clearly an approximation and is applicable over a limited range of the dependent variables. In practical cases, it may be necessary to join several regions of different field dependence together in order to obtain an accurate PIDC. The integral of Eq. (19) with r] independent of light intensity and its field dependence approximated by

is "(1 1 - P,

-

(~)'-'')

=

+ 4 s(1 - R)voF,t -

ES

or

This solution is valid for P, # 1. For P, = 1, the solution is a simple exponential. The PIDC is obtained from Eq. (21) by plotting V as a function of exposure ( F , t). The photoinduced discharge curves obtained from Eq. (21) are shown in Fig. 14 (50). These curves show how the field dependence of the generation efficiency modulates the shape of the PIDC. It is evident that as the power of

107

CHARGED PIGMENT XEROGRAPHY

0

2

4

6

8 10 12 EXPOSURE (a 1

R

" -1.50

-1.00

-0.50

0

14

16

I -

-0.50

18 20

--I 1.00

LOG EXPOSURE ( b)

PIDCs for generation limited photoreceptors. P , is the power of the field dependence of the photogeneration process. The potential coordinate is normalized to Vo = 1. Theexposureaxis is normalized to (q/e,)s( 1 - R ) q o .From Chen and Mort (50) with permission. FIG. 14.

the field dependence increases the discharge becomes less steep producing PIDCs with broader exposure range (extended range) but with less contrast potential. An example of a PIDC for a commercially important material which is only generation limited is shown in Fig. 15 (50). This is the PIDC for amorphous Se. The value of P, for this material is 0.5 at fields above lo4 V/cm and is independent of light intensity. Charge displacement, or transport limitations, in amorphous Se are small due to a long deep trapping lifetime and a moderately large drift mobility. It is evident from Eqs. (15) and (19) that ( F o t ) can be treated as the independent variable as long as r] is independent of F o . Generalization to the case where r] is a function of both F o and E independently has been treated by Chen and Mort (50). The results of these calculations show that, if such effects occur, the discharge becomes less steep as the power of the flux dependence decreases. This is similar to changes produced by increasing the power of the field dependence of r] (refer to Fig. 14).

108

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

-

>

160 -

-

-

-

0

0

~ 4

8

~ ~ 12 TIME (sec)

16

~

20

~

”

FIG.15. PIDC for amorphous Se ( P , = 4). E , = 2.4 x lo5 V/cm. The shape of the PIDC is controlled almost entirely by the field dependent photogeneration process. From Chen and Mort (50) with permission.

3. Transport Limited Discharge

There are two types of charge transport limitations, range and transit time. Range limitations occur when an appreciable number of the photogenerated carriers become trapped in the bulk before they drift completely through the material. Trapping is a particularly severe limitation since it reduces the contrast potential, produces large residual potentials, and virtually eliminates a photoreceptor from use in cyclic electrophotographic systems. Transit time limitations, on the other hand, occur when the photogenerated carriers drift through the bulk of the material in times long compared to the time it takes to expose and develop the image. In these cases, the functional form or shape of the PIDC depends on the time of measurement. Transit time limitations also limit the contrast potential and produce “residual” potentials but the limitation becomes less severe as the measurement or development time is increased. a. Range limited discharge. To examine the consequence of a range limitation, we consider the case of a PIDC produced by strongly absorbed light, c1 + 1; Under this exposure, all the carriers are generated near x = 0, and Eq. (18) reduces to

’.

This expression is valid only near t = 0 where the electric field in the photoreceptor can be considered uniform. Equation (22) has considerable historical interest since it was used in early attempts to determine the hole range in amorphous selenium (51). It was subsequently shown by Tabak and Warter that this analysis did not

’

’

CHARGED PIGMENT XEROGRAPHY

109

apply to amorphous selenium since the large residual voltage predicted by Eq. (22) was not observed. By carefully extending transient photocurrent measurements to very low fields, Tabak and Warter eventually found the true range limitation in this material (46). According to Eq. (22), a large fraction of the carriers become trapped within the photoreceptor as soon as the field drops to the point where p r E becomes comparable to s. Knowing pz, one can estimate the residual potential from the field at which p z E s. As an example, consider a 60 pm sample of amorphous Se charged to 900 V and exposed to a highly absorbed light sec. flash. The deep trapping lifetime in amorphous Se is typically 60 x The drift mobility is 0.13 cm2/V-sec. Substituting for p, z, and s, and solving for E, we find that the residual field is 7.7 x 10' V/cm (13).This corresponds to a residual potential which is less than 4.6 V. A residual this small is an insignificant perturbation on the PIDC. In general, it can be concluded that pzE must be much larger than the photoreceptor thickness to be of commercial value xerographically (47). Photoreceptors with p z E comparable to the thickness at high fields are of less interest. AS'S, is an example of such a photoreceptor. As2& exhibits range limitations at fields of the order of 5 x lo5 V/cm for samples 17 x cm thick (52).This corresponds to a residual of 850 V across the sample which is unacceptable for xerographic use. b. Transit time limited discharge. To examine the effect of slowly moving carriers on the PIDC, it is necessary to use an alternative formulation for the instantaneous voltage discharge rate. This formulation is equivalent to that of Batra et al. (53, 5 4 ) and is applicable as long as deep trapping can be neglected. In this formulation, it is assumed that the input exposure is a highly absorbed light flash with a flash duration which is much shorter than the transit time of the fastest photogenerated carriers. Under these conditions, the number of carriers in motion remains constant until they leave the photoreceptor at the substrate. The fundamental equations which relate the electric field to carrier density and motion are as follows:

-

J c ( x ,t ) = en(x, t)p(E)E(x,t),

(23)

110

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

Equation (23) is the conduction equation. Equation (24) is the continuity equation. Equation (25) is Poisson’s equation. Equation (26) is Kirchoffs law which states that the external current flow is zero for an electrostatically charged photoreceptor. In these equations, E ( x , t ) is the electric field and n(x, t ) is the carrier density. Integration of Eq. (26) across the photoreceptor gives dV

-=

dt According to Eqs. (23) and (25)

--I

l S

E

J,dx.

O

J , = pE(dE/dx). (28 ) With this equation and Eq. (26), it can be shown that the electric field acting on any carrier moving along with the conduction current does not change with distance or time; i.e., any carrier moving with the velocity p E continues to move with the same constant velocity until it completely crosses the sample. This is true whether p depends on the electric field or not. Equation (27) reduces to an elementary integral by substituting for J , from Eq. (28). If we assume that the mobility is field dependent and can be approximated by the power law, p = p,, EP,the integration of Eq. (27) gives:

where VO E, = Eo = -, S

for t

and q N , = CI, Eo is the total charge per unit area generated by the flash. Physically, E , is the field acting on the first carrier to transit the sample, EL is the field acting on the last carrier to transit the sample, and E, is the field acting on any carrier in between. E, for a given carrier is just the field required to make the carrier transit the sample in time t (moving all the while at the constant velocity of = p(Ef)E,. Since E, decreases from Eo down to EL it is clear from Eq. (29) that the discharge stops after a time equal to the transit time of the last carrier. This means that after a normal exposure one must wait for an additional time equal to the transit of the last generated carrier in order for the PIDC to become stabilized, i.e., to become independent of measurement time. The field (or surface potential) corresponding to the last generated car-

CHARGED PIGMENT XEROGRAPHY

111

rier is given by Eq. (19). In other words, the discharge appears generation limited after the last carrier has drifted completely through the material. The point we wish to emphasize is that the generation and displacement events occur sequentially in time and are independent of each other for strongly absorbed light. Consequently, carriers in motion cannot perturb the generation of subsequent carriers even though they remain in transit and thereby contribute to the surface potential. This means that the previous discussions of the PIDC and the procedures for selecting the optimal exposure, etc., are valid only when the time between exposure and development is equal to or greater than the transit time of the last generated carrier. It is clear from Eq. (29) that the part of the PIDC requiring the longest time to stabilize is the low surface potential or high exposure region. In the case where the input exposure is large enough to completely discharge the photoreceptor, the PIDC in principle takes an infinite amount of time to stabilize since there is no field available to push the last carrier across the sample. However, an exposure which leaves a finite surface potential always leaves some field to push the last carrier across the sample. Materials with very low field dependent drift mobilities require a moderately long time to stabilize the PIDC. As an example, consider a 60 pm sample of amorphous As,Se, charged to 900 V and exposed to light which is highly absorbed near the illuminated surface and is intense enough to completely discharge the material. The drift mobility in amorphous As,Se, depends on the electric field with the functional form (19, 55, 56) P = PoE/Eo (30) where p o / E o is typically 2 x 10- l o cm3/Vz-secfor samples this thick. With this mobility, the carriers photogenerated at 900 V surface potential take 1.33 x sec to drift through the material (T, = Eo L/p,,E2). On the other hand, those carriers generated at 73 V take 0.2 sec to drift through the bulk of the material. This implies that even if there is no trapping the PIDC will have an appreciable “residual” after 0.2 sec for any exposure which is large enough to discharge the surface below 73 V. The surface potential as a function of time for any flash exposure is obtained by integrating Eq. (29). This gives 1 t V ( t )= v, 1 ( 1 - (1 - c I 0 ) P + * } 0 < t < T,(E,) P+2 W O ) 1

1

~

-+

112

M. E. SCHARFE AND F. W. SCHMIDLIN

This expression can be used to calculate the “residual” surface potential in the unstable region after a given exposure. As an example, we again consider the above As,Se, sample initially charged to 900 V. We know that after 0.2 sec any exposure which is large enough to discharge the sample to below 73 V would leave a residual which is larger than the ultimate stable value. If the exposure is very large (a,, = l), the surface potential in the full exposure region after 0.2 sec is

This “residual” is nearly as high as the 73 V, above which the PIDC is stable. This example shows that there is little advantage in trying to use exposures in excess of that needed to obtain a stable PIDC at the desired development time.

4. Material and Contact Requirements for Xerographic Photoreceptors We have seen that a good xerographic photoreceptor must be relatively free of bulk traps and have a sufficiently high mobility in order to permit the formation of the latent image in the available process time. We will now discuss the additional dark electrical requirements which the photoreceptor must satisfy in order to preserve the latent image long enough for development. These have been discussed in detail in another article (57).We simply list the key results here: (a) The free surface of the photoreceptor must contain surface traps of sufficient depth and density to hold the corona deposited charge. The traps must be deep enough so that their mean release time is longer than the process or development time. (b) The substrate must form a blocking contact to injection of the carrier opposite in polarity to the corona charge. This is necessary only for bipolar photoreceptors such as selenium. If the opposite polarity carrier is not mobile, as in As2Se,, then the substrate contact is less important. Even with a blocking contact, the injection current Ii must satisfy (Ii7devG E , Eo), where z~~~is the development time. (c) The total number of free carriers thermally generated throughout the bulk of the photoreceptor must be small compared to the number on the surface; i.e., egthtdevS

&sE0

9

where &,Eo is the surface charge density o,, g t h is the bulk thermal generation rate and z~~~ is the time from sensitization to completion of the development.

CHARGED PIGMENT XEROGRAPHY

113

Although it is not a mandatory charging requirement, it is important to note that the full photoreceptor thickness cannot be effective in coupling the latent image fields to development unless qn, s < a, = E, Eo .

In this expression, n, is the bulk free carrier concentration for the dominant charge carrier. If qn, s is larger than asand if a, is as large as the material can sustain without electrical breakdown, the photoreceptor will charge with a depletion layer in the bulk. The depth of the depletion layer is given by xd = a,/qn,

and

xd < s.

Under these conditions the analysis of the previous sections must be adjusted to include the nonuniform fields associated with the noncapacitive charging of the material.

IV. PHYSICAL BASISFOR DEVELOPMENT Thourson has recently reviewed the variety of ways in which a latent electrostatic image can be developed into a real image (58). Included in this review are methods of development which do not involve charged pigment particles, such as frost (11) and liquid ink (12), as well as all the currently known methods of charged pigment development such as aerosol (59, 60), electrophoretic (26, 27), touchdown (61-64), magnetic brush (23-25), and cascade (21, 22). In this review, we focus on the underlying physics of the charged pigment development systems. We have divided this topic into five subsections. The first subsection is a discussion of the relationship between electrostatic image development and charge neutralization, i.e., it relates optical density to the effective charge per unit area delivered to the imaging surface. The second subsection is a brief description of the electric fields which are the driving force for the development process. The third subsection is a discussion of viscosity controlled development. This includes both aerosol and electrophoretic development. The fourth subsection is a discussion of adhesion controlled development. This includes touchdown and magnetic brush development. The final section is a discussion of cascade development which is a mixed inertia-adhesion controlled development process. A . The Fundamental Connection Between Electrostatic Image Development and Charge Neutralization

The basis of development in CPX (charge pigment xerography) is controlled coverage of white area with black pigment particles using the electrostatic charge on the pigment as a handle by which the latent image field

114

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

pulls the pigment to the appropriate location. The first consideration in a quantitative description of development is to determine the relationship between optical density or area covered by black pigment, and the effective charge per unit area delivered to the imaging surface. The relationship between optical density and area covered by pigment has been studied extensively in the fields of graphic arts and printing. In general, the relation between density and area covered is difficult to derive theoretically since one needs to know, in a statistical sense, the ultimate destiny of a photon incident on a surface bearing a particular distribution of pigment. This is a difficult problem requiring a numerical Monte Carlo analysis. We circumvent this difficult problem by making use of an empirical relation provided by Yule and Nielsen (65):

D

= -b

log (1 -fa( 1 - 10-DJh))

(32)

where D is output print density, D, is the solid area relection density of the pigment (or toner),f, is the fractional area covered by the pigment, and b is a dimensionless number of order 2 selected to fit measured D V S . ~ , . This empirical relation was formulated for halftone patterns with variable dot size, however a similar expression is expected for a variable number of dots of fixed size. The important features of this expression are: (1) It gives an empirical measure for the enhanced photon capture cross section of small toner due to light scattering in the material supporting the toner. This as can be seen by expanding the logarenhancement factor is b(l ithm for small f a . ( 2 ) It goes to the correct maximum density of D, in the limit of full coverage (fh = 1). The dependence offn on number of deposited toner per unit area (n), each covering a projected geometrical area ( u ) is simply nu (33) providing that the areas of the individual toner do not overlap. If toner fell at random, it can be shown (66) that the fractional area covered should be f = 1 - ,-no. (34) fa

=

At low coverage, however both coverage assumptions reduce to D

=

Bna,

for nu

<1

(35)

with

p

(log,, e)b(l - 10-D.$/b).

(36)

f or This linear relation is a good approximation to Eq. (32) for nu I DI 0.3. For nu 2 f significant departure from linearity occurs if all toner covers new area, but linearity will be sustained to much larger nu if toner

CHARGED PIGMENT XEROGRAPHY

115

deposits randomly. Applicability of the Yule-Nielsen relation to xerographic toner has been verified experimentally by J. Knapp (private communication). Experiments typically show linearity between D and n up to D 1. For transmission density, linearity persists to very high values as expected. In charged pigment electrophotography, the fundamental parameters that characterize a toner are the area it covers and its charge. The area together with the intrinsic optical density of each particle determines the total optical density. The charge controls the strength of the electric force which pulls the toner to the latent electrostatic image. Once attracted to the image, the charged toner partially neutralizes the image, thereby reducing the ability of the image to attract additional toner. These are the only two ways in which a toner can interact and affect a latent image. Knowing the number of toner per unit area deposited on a surface, the charge collected per unit area is

-

c, = nQ (37) where Q is the average charge per toner and n is the number of toner per unit area. n can be obtained experimentally from optical density measurements and knowledge of the average projected area per toner. c,can be measured directly by measuring the changes in surface potential on a photoreceptor or any other dielectric layer on which the toner deposits. The change in surface potential is given by

A K = ~ , “ S / E S ) + (rt/41 (38) where S / E is the electrical thickness of the receiver sheet, r, is the average toner radius, and E , is the dielectric constant of the toner. Very often r, << s and can be neglected. For simplicity, we deal only with this case in these discussions. Other slight corrections in Eq. (38) may also be necessary in a more careful analysis. For example, toner can stack on top of each other and thereby not contribute to the density. Toner can also transfer charge to the photoreceptor and subsequently leave (this process is called scavenging). In addition, charge displacements can occur inside the photoreceptor producing changes in the latent image (dark decay). Separation and identification of these processes is important for a detailed understanding of the image neutralization mechanisms and for establishing the magnitude of the charge on the toner before it arrives at the photoconductor. The effects of stacking, scavenging, and dark decay have been identified and quantified for a magnetic brush developer by Schein (67). Actually, many of the corrections compensate each other, and an even smaller error is made than one might first imagine providing the surface potential measurement is treated as the “effective toner charge.” By definition, the “effective

116

M. E. SCHARFE AND F. W. SCHMIDLIN

toner charge quantifies the degree of solid area image neutralization and with negligible photoreceptor dark decay and scavenging it gives the correct charge on the arriving toner. The quantity, dD/da,,is defined as the “development gain.” It is always a constant for the system at low and intermediate densities which makes it a convenient parameter for characterizing a developer. The physical significance of the “gain” can be seen by noting the maximum density contrast possible for a given latent image. The maximum density contrast occurs when enough toner has been deposited to completely neutralize the charge contrast in a latent image. Thus ”

6D,,,

(dD/da,)60 (39) where 60 is the charge contrast in a latent image. We alternatively refer to the developer gain by the parameter, D,, (“density per charge”). D,, can be expressed in terms of the charge per particle using Eqs. (35) and (37):

-

D,,

=

= dD/du, = Pa/Q.

(40)

Since P is typically 1 (0.8 is used throughout), D,, is seen to be basically the projected area of a toner per unit charge carried by it. It is also closely related to the reciprocal of the surface charge density on the toner (the connection factor being 4 for spherical toner). These formulations reduce the problem of understanding CPX development to one of accounting for the accumulated toner charge, O, .

B. Description of the Driving Force A quantitative description of xerographic development must include a description of the latent image which drives the development process. It must also account for changes in the latent image as development progresses. These changes occur because of the charge carried to the photoreceptor surface by the toner. We refer to this phenomenon as image neutralization,” and the associated impact on the image as feedback.” Once the surface charge density is known, the problem of computing the electric field pattern is a straightforward electrostatics problem. Since electrostatic formulations are linear, Fourier transforms can be used to reduce a general latent image into a superposition of simple periodic images. Unfortunately the development dynamics are almost never linear. Consequently, much of the motive for using a Fourier transform disappears. We take the point of view, however, that the development of a simple periodic image is still important to study. None of the significant forces controlling development depend on the structure of the image so an understanding of the development of periodic images should be of value in clarifying the relevant “

“

CHARGED PIGMENT XEROGRAPHY

117

physics involved in developing all images. In fact, it will be seen that a study of the effects of development on the amplitude and wavelength of the periodic image field can be used to confirm or deny any anticipated development dependence on the geometry of the image. In the following discussions, we deal with a combined uniform and periodic image of the form

EZ = Eo + Ek COS k x where Eo

= (oO/&s).fo -

(W)fO

Ek = (‘k/Es)f.

The surface charge density amplitudes (0, , ok)and coupling coefficients ( f o , f k ) were discussed in Section II1,A. A schematic diagram of the field patterns for the following three cases are shown in Fig. 16. a. Small signal case. This occurs when

and I E , I < E , . This is defined as the small signal ” case because the field lines are in the same direction over the entire photoreceptor surface, as depicted in Fig. 16a. b. Large signal case. This case corresponds to the same wavelength condition (ks < 1, kd $ l), but with I E k I > E,. This is defined as the “large signal case because the field lines reverse direction at different areas over the surface of the photoreceptor, as depicted in Fig. 16b. This condition is typical for most line copy situations. It is a natural consequence off, > f o for typical development system geometries. c. The broad area case. This case corresponds to a periodic wavelength which is long compared to the distance to the counter electrode “

”

ks 4 1

and

kd

< 1.

This condition is characterized by field lines which are nearly parallel everywhere, as illustrated in Fig. 16c for I E , I < E,. The “straightness” of the field lines persists for both large and small signals but the direction of the field is reversed in the low charge areas under large signal conditions. This case describes “broad area development fields for most common development systems. These three cases describe the complete set of electric field circumstances considered in the sections to follow. ”

118

M. E. SCHARFE AND F. W. SCHMIDLIN

I

I

PHOTORECEPTOR

I

I

PHOTORECEPTOR

d

3 I I I11111111ll1111I I I I Y I( C )

-

L

A

FIG. 16. (a) Electric field pattern for the “small signal” case. (b) Electric field pattern for the “large signal” case. (c) Electric field pattern for large solid areas; N and h are computer generated field plots provided by L. Marks.

C. Viscosity Controlled Development-Aerosol

and Electrophoretic

1. Toner Motion and the Controlling Forces

Knowing that Q 6E is the available driving force for development in CPX, one would naturally look for a means of placing the charged pigment within range of the image field with minimal disturbance or restraining forces. This situation clearly prevails in aerosol (powder cloud) and electrophoretic development where the charged pigment is simply suspended in a fluid medium. The term aerosol is used in conjunction with a gaseous

119

CHARGED PIGMENT XEROGRAPHY

suspension and the term “electrophoretic is used in conjunction with a liquid suspension. In both cases, the equation of motion for a toner suspended in a fluid is ”

m(f

-

9)

+ 6 n ~ ~ ,-( iv,)

=

QE

(41)

where m is the mass of the toner (or pigmented particle); r is toner acceleration; g is gravitational acceleration; q is the coefficient of viscosity; r is toner velocity; Q is net toner charge; and E is the local electric field. We first consider the special case of a perfectly quiescent fluid where only static forces prevail. The only static forces available are the gravitational and electrostatic forces. Equating these we find that E =

(m/Q)s

(42)

is the minimum field required to oppose gravity. The effect of gravity has been recently observed by Muntz, Welkowsky, and Morsel1 (651) by using very weakly charged toner (Q/m = 1.5 pC/gm) in a very low field ( E = 65 V/cm). Although these authors attributed a sensitometric significance to the gravitational force, we believe the effect can always be made smaller via a change in system orientation or a bias. It is appropriate to note that with one additional exception to be discussed later (cf. p. 139).the above case is the only occurrence in electrophotography in which toner mass directly plays any role in development. Typically, gravitational forces are at least an order of magnitude smaller than the electrostatic force and can be neglected. In addition, all but the fluctuations can, in principle, be biased out, thereby minimizing the importance of gravitational forces. With gravity neglected, Eq. (41) is easily solved for the case of a uniform electric field and fluid velocity, r

=

(v,

+ pE)[t + t( 1 - e-”‘)I

(431

where p = Q/6nyr, is the toner mobility, and t = m/6n~r,is the relaxation time for the damping of inertial forces. This result shows that toner will pE. This is not simply follow the “electroviscous flow lines” defined by v, true if v, + pE changes by a significant fraction of itself over a distance of the order of I v, + pE I t ; i.e., if d(v, + pE)t/dx 1 . v, + pE usually does not vary this fast in electrophotography. The fact that toner follows the resultant v, + pE instead of the field lines alone shows that fluid motion can significantly influence a developed image. In fact, there are many manifestations of fluid motion in aerosol and electrophoretic development. Some particularly interesting, though complex and yet unquantified cases, have been discussed by Bickmore (39).

-

+

120

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

The effect of fluid motion in high quality, high gain aerosol systems can be minimized by dispersing the toner via turbulence far from the field variations of a latent image. Consider, for example, the schematic of a commercial system shown in Fig. 17. The field variations associated with an image fill the space between the photoreceptor and grid. In general some field variations will extend all the way to the grid since a general image will have Fourier components with wavelengths comparable to the gridphotoconductor spacing. Thus one would expect that it should be necessary for all turbulent motion associated with the toner injection to damp out before the toner passed through the grid. Indeed, this would be necessary if one hoped to achieve faithful (proportionate) development of an arbitrary image. Fortunately, useful development requirements are much less restrictive as will become evident in the discussions to follow. 2. Idealized Viscosity Controlled Development Lewis and Stark (69) discussed the initial stages of development in the aerosol system shown in Fig. 17. They described the development as if the image were immersed in a homogeneous sea of charge compensated toner.

-b

30cm

43.5cm lOcm

1.

SELENIUM SURFACE OPEN WIRE GRID

ASPIRATION-TYPE POWDER CLOUD GENERATOR

FIG. 17. Schematic diagram of a commercial aerosol development system. From Lewis and Stark (69) with permission.

A physical argument in support of this assumption goes as follows: “Turbulent” macroscopic motion of the air carries toner through the grid and essentially “deposits” it on field lines in a random fashion. The moment of “deposit” corresponds to the moment that I u, I becomes smaller than IpE 1 . If the turbulent motion damps out far above the image (i.e., where 6E -, 0), and statistically delivers equal amounts of toner to each elemental area, then the time average density should tend toward a uniform value. Given a region between the image and grid (counter electrode) where the toner density is continuously sustained at a constant value, it immediately follows that the toner accumulation rate per unit area at the photoreceptor is just the normal component of the toner flux:

dnldt

=

nbvz = nb@, ,

(44)

CHARGED PIGMENT XEROGRAPHY

121

where nb is the bulk density of toner and p is the mobility. Multiplication by the toner charge gives do,Jdt = Q , n , p E ,

=

C ,E, ,

(45 1

where o,is the accumulated toner charge per unit area and C , is the “toner conductivity.” A constant value of nb is required for these expressions to be valid. This is assured for all time only in the absence of space charge (V * E = 0), as was shown by Lewis and Stark (69) and Onsager (49). For sufficiently short development times, feedback due to image neutralization can be neglected. Under this condition, Eq. (44) indicates that toner deposits in proportion to the normal component of the electric field at the surface of the photoreceptor. Experimental confirmation of this result for aerosol development is described by Lewis and Stark (69). They point out, however, that regions on the surface of the photoreceptor bridged by field lines which return to the photoreceptor rather than lead up into the toner supply must be excluded. This exclusion effect is substantiated by the good agreement between the computed and measured edge deletion for a weakly developed step image. The manner in which the exclusion region changes as the developed portion of the image progresses toward neutralization is not yet available in the literature. To solve Eq. (45)for extended time intervals with both feedback and the exclusion effect properly taken into account is a rather tedious task requiring numerical analysis. Unfortunately, example solutions of this problem are not yet available, and the degree of nonlinearity introduced by the exclusion effect is difficult to assess. We suspect the effect is small; however, it continues to be a source of substantial controversy. In the case of small signals, there is no excluded region and Eq. (45) becomes truly linear. In this case, it is easy to incorporate the feedback effect due to neutralization of the surface charge. Since the deposited toner charge is proportional to field at the surface of the photoreceptor, the Fourier transform of o, contains the same Fourier components as E , . The total normal electric field therefore becomes

where the Fourier components of CT,, namely o,, and o,,k are dependent on time. The final solution of Eq. (45)now becomes o,(X)

where

= CJt,

0

+ 0,. COS kx k

(47)

122

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

and

Substitution of (ao,/Q)forf, into Eq. (32) gives the developed density. If the final density is low enough (unity or less), the linearized version of Eq. (32) can be used, and the density can be obtained from Eq. (47) by simply multiplying by the developer gain, D,, . Thus D ( x ) = Do

+ Dk COS kx,

where Do

=

[

Dpcol.o= D p c y (i - exp - f o c,t

and

The basic requirements for the validity of this result (first obtained by R. B. Lewis, private communication) are a small signal and a constant value for the bulk toner density nb . Neither of these conditions are strictly valid in practical viscosity controlled developers, and the consequences due to departure from the idealized conditions are discussed later. Despite the idealized conditions underlying Eqs. (50)-(52), a number of important features of CPX development are shown in the result and several useful conclusions can be drawn. i. For sufficiently brief development times, the exponentials can be expanded, keeping only the first term. Thus Dk

=

Dpc Bk(fk/Es)ct

=

Dpc Ek

ct

t.

(53)

This means that feedback due to neutralization becomes negligible and development becomes proportional to the electric field. It also shows that D,, C, t is the common factor which transforms the normal electric field into developed density. These same factors transform Eq. (12) (discussed in Section III,A) into the developed density for sinusoidal charge distributions. Multiplying through by D,, C, t, the density becomes:

CHARGED PIGMENT XEROGRAPHY

123

This is the developed density expressed as a function of the broad area surface potentials (PIDC). Similarly, y o and yw in Eqs. (4)and (5) reduce to the values given in Eq. (55) when the first Fourier component of a rectangular line is used to represent the density at the center of the line (x = 0). Equations (54) and (55) must be restricted, however, to small signals, otherwise the density would try to assume negative values which is clearly not possible. The development gamma for a given spatial frequency is defined as

The development gamma for broad areas is yo which is obtained from Eq. ( 5 6 ) by letting k + 0. It is not possible to formulate a rigorous expression for the development gamma for the large signal case because of the nonlinearity in the dynamics as mentioned earlier. Under highly subneutralized development conditions, however, the maximum developed density is still given by Eq. (56)providing that the toner density, or conductivity (C,),remains constant. In this case, the development gamma for the condition where VBg = V, is given by

This result is identical to that obtained from Eq. ( 6 )when k is identified with the predominant Fourier component of a line of width w (i.e., Yk + yw). Note that the large signal gamma [Eq. (57)] is less than or equal to the small signal gamma [Eq. (56)]. It should be emphasized that the preceding results apply to the idealized case only where the bulk density of toner (n,) is constant. The proportionality between developed density and field is preserved, however, in practical systems where n, is no longer constant. In these cases, the proportionality constant, y k , may be different. ii. For sufficiently extended development, the exponential terms vanish and the transformation from image surface charge density ( b k ) to developed density ( D k )is completely given by the developer gain, D,, . iii. The maximum developed density variations are limited by the development gain for any particular developer and higher gain is required to achieve the same density under subneutralized development conditions. Thus the development gain must always satisfy Dpc

2

Dk,max/'k

5

where D k , max is the maximum required contrast density. iv. The coupling coefficient f k , which contains the dependence of development on system geometry (s and d ) and dielectric constants (es and

124

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

E,,) as well as image structure (A), is a rate controlling factor. In the present viscosity controlled case, the quantity

llTk = f k C t / & s (58) defines a development time constant. But in general, fk must enter in a similar manner for all development systems because the ultimate density corresponding to complete neutralization of the image must be independent Of f k *

v. Because of (iv), it follows that the appearance 0f.h in the development characteristic implies at least some degree of incomplete neutralization. As first pointed out by R. B. Lewis (private communication), it also follows from (iv), that short wavelength image variations neutralize faster than long wavelength variations. However, the specific quantitative dependence on wavelength expressed by Eq. ( 5 8 ) cannot be expected to hold in general. vi. The image development rate as t -+ 0, is dDk/dt

=

(Dpc/zk)Ok

’

(59)

This suggests defining a gain-rate product,” “

GRP

= D,,/zk

(60)

as a figure of merit for the developer. It contains all the parameters associated with the developer. Since it includes the coupling coefficient fk, the GRP also depends upon photoreceptor thickness. This reflects the fact that the coupling should be considered a part of the development system. 3. Status of Practical Viscosity Controlled Development Systems

a. Physical basis for the direrence between ideal and practical systems. Practical viscosity controlled systems differ from the idealized case analyzed in the preceding section. The primary reason for these differences can be attributed to the fact that the developer as a whole must start out nearly charge neutral, at least before any field is applied. Consequently the charge on the pigment must be compensated in some way. In aerosol systems the compensating charge resides on other toner. In electrophoretic systems much of the compensating charge (or all of it) can exist in the form of ions. It will be shown later that the principal manifestation of the presence of counter charge in both systems is in the kinetics of the development process. A secondary reason why practical systems differ from the idealized one is that the concentration of pigment particles (n,,) does not always remain

CHARGED PIGMENT XEROGRAPHY

125

constant in time. This applies primarily to aerosol developers. It is also possible that the charge on the pigment particles may not be stable. This is more likely to occur in a liquid suspended toner, though it has not yet been identified as a serious problem in the electrophoretic systems discussed herein. b. Comparison of the physical parameters characterizing the ideal and practical systems. Before discussing a number of the peculiarities associated with each of the viscosity controlled systems, it is appropriate to first list the characteristic parameters for several practical systems and compare them with the theoretical characteristics computed using the formulas derived for the idealized case. The input data required to compute the theoretical gain and rate constant are the toner diameter d,, the toner charge Q, and toner concentration. These data for several practical systems are listed in Table I (66, 68-71). Mean toner size and charge values determined via electron micrographs in combination with collected charge or mass are considered the most reliable. We selected values determined by this method whenever possible. [The method of determination was not specified in Van Engeland et al. (71).] Using the viscosity values and pigment concentration per unit volume indicated in the table, we computed the theoretical mobility ( p ) and toner conductivity (C,)in each case. Experimental values for the toner mobility are not known. However, experimental values for the total conductivity (C = C, + C i , toner conductivity plus ionic conductivity) of electrophoretic inks were measured and the values obtained by the respective authors are listed in the table. In each case the experimental values are two to three orders of magnitude smaller than the theoretical values for the toner conductivity alone. We discuss some of the proposed reasons for this later. Using the size, charge, and theoretical toner conductivity in Table I, we computed the theoretical electrophotographic characteristics listed in Table 11. To compute the broad area rate constants, the values assumed for the broad area coupling coefficientf, are listed in the table. In each case [except the first one by Dahlquist and Brodie (66)]f, was computed using the dielectric constants, photoreceptor thickness, and developer thickness of the experimental system investigated. In the case of Dahlquist and Brodie, development was done through a slit which made the geometry difficult to analyze. The experimental development gain was determined from the slope of density versus surface charge density collected, ADlAcr,. A value of 0.8 was assumed for p in the theoretical gain. It is significant that the experimental and theoretical development gains are in good agreement. Of course, this agreement may be simply due to the fact that charge collection measurements only provide the effective charge per particle (Qerf)and that all this charge does not have to be carried to the

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

CHARGED PIGMENT XEROGRAPHY

127

TABLE I1 ELECTROPHOTOGRAPHIC CHARACTERISTICS OF VISCOSITYCONTROLLED DEVELOPERS 1

DPC

(m’/C) Electrophoretic Th. 1.0 x ~ x p 1.0 3 Th. 7.5 x lo3 ~ x p5. - 7.5 x 103 Th. 1.1 x lo3 ~ x p 1.1 . x 103 Aerosol Th. 7.5 x lo4 ~ x p4.2 . x 104 Th. 1.9 x lo5

(sec-’)

373 0.3 2 0.3 15

0.4 1.8 x 10-4

fil

10-

’

GRP (m’/C-sec)

Comment and reference

3.72 x 105 Dahlquist and Brodie (66)

10102.5 x lo-’ 2.5 x

’

2 x 10-3

1.4 x lo4 Stark and Menchel (70) 1.6 x lo4

VanEngeland et al. (71) 14

Lewis and Stark (69) Muntz et a!. (68)

collecting surface on a toner. The implications of this with respect to the development kinetics is discussed later. A general rule, however, that must be followed in considering models in which Qeff differs from the true toner charge Q, is that Qeff must be proportional to Q and scale with the projected toner area. Otherwise, the established agreement between the theoretical and experimental development gains will become upset. The above is a rather severe scaling law and should prove useful in ruling out many models from consideration. It is noteworthy that the gain of all electrophoretic developers is of the order of lo3 m2/C. This is evidently a consequence of its practical application for picture reproduction, where it is important to come as close as possible to unit slope in the tone reproduction curve without enhancement of edges. This, in turn, requires development to neutralization, and surface charge densities on typical xerographic photoreceptors are of the order of 10-3 C/m2. Sensitivity of the tone reproduction curve on toner charge was pointed out by Van Engeland et al. (71). We emphasize the fact that, in any system where development is carried to completion, the development gain alone must be matched to the surface charge density in a latent image to obtain any specified tone reproduction curve, independent of the system geometry (photoreceptor and developer thicknesses, s and d ) . Geometry only affects the rate constant. Sensitivity in picture reproduction is unimportant. Hence the relatively low gain in electrophoretic developers has been found acceptable. The relatively high gain achieved with powder cloud development accounts for its selection as the developer in electroradiography. The gain in

128

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

powder cloud is at least an order of magnitude higher than for any other xerographic developer. To complete the comparison of development gain for various CPX developers it is of interest to note typical gain values for cascade and magnetic brush. These gains are about lo4 and 3 x lo3 m2/C, respectively, assuming representative tribo values of 7 and 20 pC/gm and D, = 10 pm. It can be shown that the relatively low gain of these systems is related to the fact that the electrostatic driving force must overpower adhesion. Turning to a comparison of the theoretical and experimental rate constants, we find that the experimental rate constants are much lower than the theoretical. Stark and Menchel ( 7 0 ) have pointed out that the experimental rate constant of an electrophoretic developer could be accounted for approximately if the toner conductivity were replaced by the total ink conductivity, C, + Ci. This idea was subsequently adopted and supported experimentally by Van Engeland et al. (71). Earlier attempts (72, 7 3 ) to formulate a quantitative description of the development kinetics included the effect of ion motion, but did not include the dependence on dielectric constants, geometry, and wavelength as incorporated in the coupling coefficient fk . The proper dependence on these parameters together with the effect of ion motion was incorporated by Schaffert for the broad area limit (41). It is easy to show that Schaffert’s expression which describes the collective effect of the dielectric constants and geometry is identical to our fo . This, of course, is expected since the relevant analysis involves only electrostatic considerations. Our solution for the idealized case is more general in that it also contains the wavelength dependence as needed to describe the development of a periodic image. The solution for a periodic image now provides a vantage point from which it can be argued that replacement of C, by C, + Ci is not a generally justifiable procedure. First note that fk is typically much larger thanf, and that fk is of the order of unity in the high frequency limit of ks 9 1. As fk approaches unity, zk approaches the dielectric relaxation time of the ink (te= E/C).Stark and Menchel ( 7 0 )argued that the smallness of z, compared to zo justified the assumption of space charge neutrality in the ink. We now see that the development of high spatial frequencies (or narrow lines and edges) should develop at the same rate. Qualitatively, this first appears to be in agreement with the so-called “edge enhancement commonly observed in CPX. However, closer quantitative scrutiny of edge development normally reveals a fall-off in the developed density at the very edge of a charge step, due to the limited availability of toner along the field lines. (Field lines rapidly decrease in length at edges or high frequencies.) As far as we know electrophoretic development is no exception (though it should be reduced in relation to other forms because of the high toner density). The simple conductivity model cannot account for such a behavior. ”

CHARGED PIGMENT XEROGRAPHY

129

Perhaps more direct weaknesses of the conductivity model are that it cannot explain the observed departures from the simple exponential decay law, nor can it reasonably explain the magnitude of the difference between the experimental total conductivity and the theoretical toner conductivity. Dahlquist and Brodie attempted to explain the failure of the exponential decay law in terms of toner diffusion near the photoreceptor surface. It is difficult to reconcile this with the fact that drift tends to sustain a uniform density right up to the accumulation point. If important, diffusion would also produce a high background since it is not controlled by the latent image. Stark and Menchel (70) estimated the rate of toner diffusion through a screen-type counter electrode and concluded that it was negligible. They suggested that the decaying time constant may be due to toner depletion. But for solid area development this seems unlikely since the total toner contained between the photoreceptor and grid far exceeds that required for D,,, . Thus toner deletion should be manifest only at high spatial frequencies. It can easily be shown that the ratio of C/C, would be consistent with an effective toner charge which is C/C, times larger than the actual (unshielded) toner charge. But this would mean that the actual toner charge would be of the order of lo-’ of the charge on a single electron in two out of the three cases listed in Table 11. To us this seems unreasonable. We conclude from the above that there are many problems remaining for a clear understanding of the kinetics of electrophoretic development. However, the key concepts required to understand the final development characteristics are on sound physical grounds. The understanding of aerosol development is on a similar footing. The physical basis for the final development characteristics is again sound but much remains with respect to understanding the kinetics. The central problem here is that the toner density is not constant with time, as required in the simple idealized case. Lewis and Stark (69) have shown that the toner density falls rapidly within the duration of single “puff.” The time constant for the decay is somewhat longer than that expected for the average transit time, assuming that the drifting toner are not clustered. In practice, it is typical procedure to wait many transit times between dispensing puffs and apply 10-20 puffs to complete the development of a single image. This means that the amount of toner delivered to the image is actually limited by the amount supplied rather than being continuously present as in the idealized model. It can be shown, however, that development under supply limited conditions produces characteristics identical to the idealized characteristics, except for broad areas (kd < 1) and providing C, t is replaced by E, A K I s E , , where

AK

= rst.

S/E,

is a measure of the average quantity of toner deposited on the photoreceptor

130

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

per unit area. In the supply limited model the rate constant becomes meaningless since the rate of development is now simply controlled by the rate at which toner is dispensed into the system. Of course, the waiting time between dispensing events must somehow depend on the rate at which turbulence associated with the dispersing event damps out, but there is no apparent reason for the toner collection rate to be simply linear with t , n b , E o , and p. This means the often observed linear dependence of collection efficiency on bias field is left without an explanation. D . Adhesion Controlled Development

When toner is delivered to an image by way of some vehicle such as a belt or bead, the force holding the toner to the vehicle must be overcome before toner can be removed from the vehicle. This force is called the adhesion force. These forces have been investigated experimentally by Donald ( 7 4 ) using a centrifuge. It was shown that the total adhesive force could be described in terms of the net charge Q on the toner:

F,

=

F,

+ k*Q2.

(61) The first term ( F , ) is independent of the charge while the second term is proportional to QZ,as one would expect for the electrostatic image force. The graph of F , is shown in Fig. 18. Also shown is the negative of the electrostatic force Q E , where E is the total field including any bias. In order to remove toner from the delivery vehicle by an electrostatic force alone (no direct contact with the photoreceptor), it is essential that the magnitude of

P-

FIG.18. Relationship between the adhesive force which holds toner particles of charge Q to some delivery vehicle and the electrostatic force ( - Q E ) which tends to pull the toner from the delivery vehicle. All toner with charge between Q 1and Q 2 will be pulled from the delivery vehicle in this example.

CHARGED PIGMENT XEROGRAPHY

131

Q E exceed Fa. In this example, only toner possessing a charge between Q1 and Q 2 can be removed from the delivery vehicle at the indicated field, assuming F, and k* are constant. F, depends on the contact area and materials involved, while k* depends on the toner size (k* cc r;’). For a particular value of F , and toner size, it can be shown from the foregoing force balance that a threshold field E,, = 2(k*F,)”’ exists for an optimal charge of Qx = ( F n / k * ) l i ’ . Larger or smaller charge than this requires a larger threshold field. Inserting the representative value of N for F , and 5 pm for r , , one finds a threshold field of 4 x lo4 V/cm and Q, = 5 x C. These values are typical of the field and toner charge required to achieve development in any CPX developer involving a delivery vehicle. It is important to emphasize that F , determines both the threshold field and optimal charge. It is also significant that smaller toner require a larger field, assuming that F , ccr, (as one would expect on a contact area basis for spherical particles) and k*ccr;

’,

1 . Idealized Touchdown Development The simplest idealized development system which involves only the adhesive force is shown in Fig. 19 (75, 76). This is a sectional view of a flat delivery electrode with a number of toner stuck to some small area called a “picture element.” The latter is selected as the smallest area A j ( E ) over which the local electric field lies between E and E + AE. The quantity we want to compute is the fractional area of the photoreceptor which will become covered with toner when the field reaches some specified value. To obtain this area, consider a very large broad area A , composed of many picture elements. Each picture element is covered with a set of toner selected at random from a population defined by a distribution function which varies with respect to the adhesive force. Within any picture element where the field is E, we sum the projected area of all toner for which

-PICTURE ELEMENT

-7 FIG. 19.

Idealized touchdown development system. From Schmidlin (75) with permission.

132

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

the adhesive force is less than QE. These toner will transfer to the photoreceptor and cover a fractional area given by

where the sum over i is such that FOi< Qi E. Since Qi and Fai are characteristic toner quantities, it is appropriate to define an “adhesive field,” X, by

x

FaiIQi .

(63)

The transfer criterion then becomes x < E . Both E and x can be treated as continuous stochastic variables. Then f a can be written in the continuum limit, as

where we have assumed that all toner are of the same polarity. In this equation, n is the number of toner per unit area available to transfer; g ( X ) dX is the probability that the adhesive field lies between x and x + dX; a ( x )is the projected area per toner of the subset of toner which transfers when E is between x and x d x ; andf(E) dE is the number of picture elements for which the field lies between E and E + dE. In general, a ( x )can be written as the area of a toner times another probability function. We circumvent this complexity by assuming (a) is a constant equal to the average geometrical projected area. When (a) is factored out of the above integral, the remaining double integral is recognized as a convolution. This suggests defining

+

whose density function becomes

h(E*) =

5

m

f,(E)q(E

- E*)

dE

0

The advantage of defining E* as a stochastic variable is that it naturally introduces a way to include statistical variations in all contributors to the electric field. The total electric field is

+

E = Eb En E, where Eb is the applied bias field; En is the noise field due to fluctuations in the dark decay, and E, is the field produced by the image. Thus the total variance in E* is O:(E*) = a:(&)

+ a:(&,) + o,Z(E,) + o:(X).

(67)

CHARGED PIGMENT XEROGRAPHY

133

In Schmidlin (75) the magnitude of these quantities was estimated for a typical xerographic photoreceptor and all of the field variances were shown to be small compared to a,’(x). Thus, one can think of h(E*) as having essentially the same shape as g(x) which we assume to be Gaussian with a constant variance &). The effect of the electric field is to shift the mean value of E*. In the absence of any electric field E* = -X. The effect of the bias field E b is to make E* more negative (reverse bias) or less negative (forward bias) as desired. The number of toner experiencing a force large enough to transfer them to the photoreceptor is determined by the area under h(E*) which extends beyond the origin. The fractional area covered by these toner is given by

f, = nu

1

W

h(E*) dE*.

0

(68)

The above integral is indicated by the shaded area in Fig. 20. The case shown corresponds to an initial bias such that E* = i?b - is strongly

Eb

-E

0

FIG.20. Relationship between the density function of E* and the image and bias fields ( E , , E J . The shaded area represents those toner which will transfer to the photoreceptor from the donor vehicle.

negative for the background exposure (D = 0). The effect of exposure is indicated by EI shifting E* back toward the origin. As the exposure decreases, more of E* becomes positive. Eventually the complete density function h(E*) is shifted in the positive direction until it completely crosses the origin. At this point all the (correct sign) toner presented to the image will have transferred to the photoreceptor. In the previous analysis of this system (75, 76), a uniform monolayer of toner was assumed to be present on the delivery electrode (i.e., nu = 1). It was also assumed that the electrode could be placed arbitrarily close to the photoreceptor without the toner actually touching the photoreceptor. This

134

M. E. SCHARFE A N D F. W. S C H M I D L I N

has three consequences: ( 1 ) lateral motion is constrained so that toner transfers directly under the point where it is presented to the image; (2) the electric field, effecting the transfer, E,,is the normal component of the electric field at z = d - r , ; and (3) development is locally complete upon transfer of a single toner-meaning feedback due to image neutralization is negligible. It is also implicit that one is dealing with an ordered stacking of the toner during transfer with no chance of overlapping the projected areas of the toner. Generalization of the formalism to random stacking is achieved by writing = 1 - exp [-nuP,(E*, o,(E*))] (69) where

S, h(E*) dE* a3

P,(E*, 0") =

(70)

is the transfer probability of each toner presented to the image. It is assumed that P, is independent of the number of toner presented to the image. Substitution of,f, into Eq. (32) results in the development characteristics shown in Fig. 21. To obtain a universally useful characteristic, we use E*/o,(E*) as the independent variable. Curve (1) corresponds to ordered stacking of a single compact monolayer (nu = 1 ) with the Yule-Nielsen parameter b equal to unity. Curve (2) corresponds to the random deposition of three monolayers (nu = 3) with b = 2. The failure to reach the full reflection density assumed to be D, = 2, simply means that the uncovered

t D

2.0-

-

1.6-

-

1.2 -

-

0.0-

-

0.4-

-

0

3

2

1

0

1

2

3

4

E,+&X a,(E*)

FIG. 21. Development characteristics for idealized touchdown development. Curve 1 corresponds to an ordered monolayer (na = 1). Curve 2 corresponds to random deposition of three monolayers (na = 3 ) . Curve 3 corresponds to the random deposition of five monolayers (nu = 5).

135

CHARGED PIGMENT XEROGRAPHY

areas (holes) which remain after delivering three monolayers still have a significant effect on the saturation density. If the number of monolayers is increased to five, Curve (3) is obtained (again with b = 2), and the maximum saturation density is very nearly reached. In all three of the above cases, feedback due to image neutralization is neglected. This is an excellent assumption almost independent of the toner charge for Curve (1) since only one closely placed monolayer of toner is required to achieve D,,, . In the other two cases of random stacking, a larger electrode spacing (d) is clearly required, and neutralization feedback could be important if the monolayers are presented sequentially. In these calculations, we consider only the case where the toner charge is small enough to neglect feedback. It was shown in Schmidlin ( 7 6 ) that this is an important case from the standpoint of ultimate sensitivity and is reasonably attainable in practice. The primary result of this analysis is that the complete development characteristic is predominantly governed by the adhesive force distribution function g(x). In addition, all the key characteristics such as the sensitivity and gamma are predominantly controlled by the variance in x, a,2(~).To emphasize this, note from Fig. 21 that a bias background of

E,

=

x - 2o,(E*)

is required to reduce the background (Dmin)density to 0.01 for case (1). In the other two cases only a slightly larger bias is required. Then, in all cases a field change via exposure, IT,,of very nearly one a, is required to increase the density above Dminby 0.10 density units. The reciprocal of the exposure required to do this is, by definition, the photographic sensitivity. Consequently the sensitivity of an adhesion controlled system is almost entirely controlled by o,(x). To show that the development gamma is also determined by g,(x), we turn to the intermediate density region (0.5-1.5 density units). Note from Fig. 21 that an extended constant slope region appears over this density range, which is also the maximum slope. It can be shown by direct computation that nap, 1 in this region even though no analytical simplification of the equations is possible. For the extended linear region, we obtain directly from Fig. 21 that

-

and dD = 0.95 d(E*/a,(E*)) ~ _ _ _ _ _ _ _

for nii

=

5.

136

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

Thus we conclude that the slope of D vs. (E*/oV) is relatively insensitive to nlj and is of order unity for the typical number of toner monolayers required to achieve the maximum density of D, . Transforming the slope to a density vs. electric field curve, we obtain

This shows that the gamma, with respect to the electric field, of an adhesion controlled system is overwhelmingly determined by the variance in alone. The slope with respect to the normalized variable can be treated as a slight correction factor, dependent only on the amount of toner delivered to the image, nlj. Typically, na will be close to 5 in order to achieve the saturation density, but, in any case it is a determinable quantity for any particular development system. The constant factor which transforms field variations into density variations for subneutralized development is therefore just a; '(x),to a good approximation. For completely neutralized development, image density is related to the surface charge density through the developer gain D,, , just as it is for viscosity controlled development. The intervening region of partially neutralized development is more complex, requiring feedback via the development probability functions Pt(E*).We ignore this complexity in this review. Apart from the low density region, which one must consider carefully in selecting the bias to suppress background, a linearized approximation to the development characteristic shown in Fig. 21 for nli = 5 is a good one providing that the slope is matched in the central region. As a result, the appropriate development gammas for periodic contributions to the peak field at the center of a line are given by

x

where the change in field with respect to the surface potential is obtained from Eq. (12). The solid area gamma is obtained from Eq. (74) by letting k + 0. y k and y o also correspond to yw and y o in Eqs. (4) and ( 5 ) when those equations refer to linearized approximations of adhesion limited development. Before turning to a discussion of practical adhesion controlled systems, we indicate how the analysis changes when the toner is delivered close enough to the photoreceptor to make physical contact. In this case, one must replace the average adhesive field by the difference between the tonerd ) and the toner-photoreceptor adhesive delivery vehicle adhesive field field ,), i.e.,

(x,,

(xt,

x

+

xt.d

- xt,p;

CHARGED PIGMENT XEROGRAPHY

137

and the variance in y, is replaced by the sum of the variances

4(x)

+

4(xt.d)

+

&t,

,).

To distinguish practical systems in which toner makes physical contact to the photoreceptor from those in which the toner does not make contact with the photoreceptor (until after transfer), the two are sometimes called “contact touchdown” and “almost touchdown.”

2. Status of Practical Adhesion Controlled Systems (Touchdown, Magnetic Brush) Several practical development systems similar in character to the preceding idealized case have been suggested (61-64) or studied. Most of the configurations considered involve a number of monolayers loaded onto a belt or roller and presented to the image in a single contact event. These systems are somewhat more complicated than the above in that cohesive forces between the toner also exist, causing possible transfer in clusters or en masse. The central advantages of a touchdown system are its potentially high development rate (limited only by mechanical considerations) and flat spatial frequency response (by making d s). The disadvantage of contact touchdown is its notoriously high background. This is presumably due to the inability to identify and control the factors contributing to the variance in the adhesive field. Such control is clearly far more important in “contact touchdown” than in “almost touchdown.” Development characteristics of practical touchdown systems are not abundant in the literature. One recently reported characteristic (77) for a system called “Impression Development” is shown in Fig. 22. In this system, a uniform layer of toner is loaded onto a roller with a specially designed toning system. The toner is then corona charged to obtain a more uniform charge than could be achieved by triboelectric effects alone. Development is accomplished by pressing the pretoned developer roll against the photoreceptor in a single nonsliding contact event. It is significant that the development characteristic of the impression system is very similar to the idealized system (five monolayer case) shown in Fig. 21. The principal difference is that the “impression system” appears to have a sharper bend in the low density region. It is possible that this may be due to transfer en masse or in clusters. Magnetic brush development is one of the most commonly used methods of development in commercial copy machines. Although it is sometimes considered similar to cascade development, the only real similarities between the two are: (1) both use beaded carrier vehicles for the toner, and (2)

-

‘I

”

138

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

FIG. 22. Copy reflection density versus contrast voltage for an “impression development system. From Chang and Wilbur (77) with permission.

”

the triboelectric relation between the toner and beads is selected to obtain a preferred charge on the toner. The basic nature of the forces involved in holding the toner onto the beads (adhesion) or pulling it off (electrostatic) are no different than the ones already considered. The only new force appearing in the system is the magnetic force, but since there is generally no magnetic material* in the toner, the magnetic force serves only to form brush-like filaments and transport the developer as a whole. A scale drawing of a single brush filament is shown in Fig. 23 (58). Thourson used this figure to show that the brush developer appears like a closed-spaced electroded system with good solid area capability. It is evident from the figure that a filament must behave like a tiny touchdown system with some toner making physical contact with the photoreceptor but with much more toner making “almost” contact, i.e., with space between the toner and photoreceptor. The central difficulty with analyzing the system quantitatively is finding a convenient way to handle the awkward geometry.

* Even if there were magnetic material in the toner, its effect could presumably be treated as an adhesive force.

CHARGED PlGMENT XEROGRAPHY

139

TONER

PARTICLE

EL ECTRl C FLUX LINES 0

/ +

FIG. 23. Schematic diagram of a single magnetic brush filament. From Thourson with permission.

Clearly, a statistical approach must be used as a natural extension of the procedure used in analyzing the idealized adhesion controlled development system.

E. Cascade as a Mixed Inertia-Adhesion Controlled System Cascade (21, 22) is fundamentally different from all other CPX developers in that gravitational energy transforms into random kinetic energy of the beads while the developer is cascaded over an image. The random part of the bead motion results in collisions between beads or with the walls (photoreceptor and counter electrode) of the development chute, as indicated in Fig. 24. In a collision, the toner riding on a bead experiences an inertial force which tends to keep the toner moving in the same direction as the bead is moving just prior to a collision. For example, suppose the velocity of a bead normal to the wall is U just prior to a collision with the wall; and, after the collision it recedes from the wall with a velocity which is somewhat less than U , say pU. If the time duration over which this velocity change occurs is denoted by z, , the average inertial force experienced by the toner during this time is

Fi = m ( l

+ p)(U/z,).

(75)

140

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

FIG.24. Schematic diagram of the possible toner release events in cascade development bead-wall collisions, bead-bead collisions, and electric field stripping.

This force is directed normal to the wall, although in bead-bead collision it - Ufina,). The magnitude of Fi is along the change in bead velocity (Uinitia, for a typical bead velocity of 0.2 m/sec, a collision time of sec, and a toner 10 pm in diameter is of the order of 2 x lo-' N. This is of the same order of the short range adhesive force. Hence the inertial force is an important contribution in the release of toner from beads. During the time a bead changes its direction in a collision, four forces must be considered in determining whether the toner can move away from the surface of a bead. These are related through the expression: Fi cos

4

+ Q E 2 F , + k*QZ,

(76) where C#I is the angle between Fi and the normal to the surface of the bead. Since F , is of the same order of magnitude as the short range (F,) and long range (k*Q2) parts of the adhesive force, it isevident that the electrostatic force can be very small and a toner can still separate from the surface. The collision force equation contains an important size dependence. With Qar:, k*ar; ', F , art and Fia$, it is evident that Fi quickly loses its

141

CHARGED PIGMENT XEROGRAPHY

power of assistance as r, decreases. Therefore, toner smaller than 5 pm in diameter receive little help from the inertial force in being released from a bead. With Fi small, the remaining equation becomes identical to the purely adhesion controlled case discussed earlier. Consequently, without inertial assist, large fields would be required to remove the toner, and the smaller the toner, the larger the critical field. In general, the development fields for 10 pm toner are not large enough to strip toner unless the beads happen to come very close to the edge of a line (cf. illustration in Fig. 24). Another important consequence of the inertial force is that k*Q’ can now be less than F, and still provide adequate development. This is because the inertia force helps remove toner from the carrier beads providing, of course, that the toner particles are large enough. It can be shown that this process gives potentially higher development gain (smaller Q/r;) for cascade development as compared to any of the purely adhesion controlled systems. Donald and Watson (78) point out that after the toner breaks away from the bead surface, it must retain sufficient kinetic energy to achieve an escape trajectory. In other words, the inertial energy must be large enough to overpower the long range image force contribution to the adhesive force. The latter persists much longer than the lops sec during which the bead velocity is reversed. An escape energy criterion is obtained by integrating the adhesive force acting on the toner from the point of contact with the bead to a point where the adhesive force is balanced by the electrostatic force. This is contingent on the electrostatic force being in the proper direction to pull the toner away from the bead. The effect of the electrostatic force is to lower the energy binding the toner to the bead as is shown in Fig. 25. An approximate escape energy criterion can be written as

im(l

+ p)’U’

cos’

4 2 k*Q’rl;(E,

Q),

where 4 is again the angle between the direction of the inertial force (change in bead velocity) and the normal to the bead surface (x in Fig. 25); and

.fi(E,Q ) = [I

-

(E/k*Q)”’12

is the fractional lowering of the energy barrier by the normal component of the local electric field E. This is only an approximate f 1 , since the field actually changes with time due to the changing field enhancement at the surface of a metal bead as it approaches another bead or a wall with a high dielectric constant. This effect, as well as the dependence of the escape energy criterion on the angle 4, were both neglected by Donald and Watson. It can be shown that neglect of the time dependence is justifiable for toner sufficiently far from the impact point. Toner residing on the side of the bead opposite to the impact point (4 > 4 2 ) cannot escape since the inertial force there pushes them into the bead more strongly. The dependence of the

142

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

PE.=O

REDUCTION IN BARRIER HEIGHT

I

/ COULOMB BARRIER U = QE, / ( r t x )

a w

A P P I IFn

IER RADIUS ( r )

t CARRIER BEAD SURFACE

FIG.25. Potential energy diagram of toner particle on a carrier bead. The external electric field reduces the binding energy which holds the toner to the bead. From Donald and Watson (78) with permission.

release criterion on angle from the impact point is extremely important since it rules out a number of previously proposed models of cascade development. All previous models of cascade development have considered only beadwall impacts as the means by which the inertial force assists toner release from the beads. Sullivan and Thourson (79) described a mechanism in which toner was assumed to release on bead-wall collisions with the photoreceptor on a discharged area just next to the image (cf. Fig. 24). Such a mechanism is not tenable, however, when one considers the relative direction of the inertial and electrostatic forces. In such an area, the electrostatic force on the contact side of the bead opposes the inertial force and thereby keeps the toner on the bead. The few toner which make physical contact with the photoreceptor may still transfer, however, because of the adhesive bond made between the toner and photoreceptor. This is undoubtedly the principal cause of background. But the overwhelming majority of the toner on a bead which do not make physical contact with the photoreceptor are either pushed into the bead by inertia (all those on the hemisphere facing the developer) or are pushed back onto the bead by the local electrostatic force. This is true even if inertia does succeed in lifting toner off the surface of the bead initially. The same argument also rules out bead-wall impacts with the counter electrode as a significant release mechanism.

CHARGED PIGMENT XEROGRAPHY

143

The preceding considerations leave only bead-photoreceptor impacts directly on the image and bead-bead collisions throughout the bulk of the developer as significant toner-release mechanisms. For line copy, it can be shown that bead-bead collisions are the dominant mechanism. Experimental support of this exists in a study of periodic image development by Stark and Menchel (80). The location of the first developed toner for a partially developed large signal case is shown in Fig. 26. If one examines where the

PHOTORECEPTOR

FIG.26. Schematic diagram showing location of toner in a cascade developed periodic image.

toner lies and follows the electrostatic force lines backward, it is clear that the toner can only come from the space above the point where the field lines arch back onto the photoreceptor. Thus development appears very much as one would expect of a powder cloud system. This result is consistent with the view that toner is released by bead-bead collisions throughout the bulk. Since the bead radius is about the size of the arch, toner cannot be released by bead-bead collisions except above the arch. The total thickness of the development zone is about three times the thickness shown, as indicated by the broken lines, with field lines being nearly straight the rest of the way. The conspicuous lack of toner in the region subtended by the arch shows that significant toner was not released by impacts directly on the photoreceptor-even in areas where the electrostatic force is directed into the photoreceptor. The close correspondence between the distribution of toner on the image and the separation between field lines returning to the photoreceptor and leading into the bulk of the developer shows that the predominant source of toner is bead-bead collisions. A more quantitative description of how toner is released from the beads and attracted by the latent image in a cascade system is imminent.

144

M. E. SCHARFE A N D F. W. S C H M I D L I N

V. SUMMARY In this article, we have described and discussed one subclass of xerography-Charged Pigment Xerography. The complete xerographic process consists of seven operational steps. Each of these steps affects the fidelity or character of the output image. Two of these, however, clearly dominate the system performance and are appropriately called the heart of the xerographic process. These are the latent image formation and development steps. The latent image is directly related to 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. The PIDC is governed by two fundamental processes-the field dependent photogeneration of free carriers and the subsequent charge transport through the bulk of the material. The latent image can be developed by several different types of development systems. These include viscosity controlled, adhesion controlled, and a mixed inertia-adhesion controlled development system. The theoretical development characteristics of the idealized viscosity and adhesion controlled systems are in good agreement with their practical counterparts (aerosol, electrophoretic, and touchdown). The kinetics (or rate of development) in the viscosity controlled case is less well understood. In the case of practical adhesion controlled systems (magnetic brush) and the mixed inertia-adhesion controlled system (cascade), it appears that the relevant underlying physics has been identified but a complete quantitative explanation of the development characteristics is presently unavailable or unpublished.

REFERENCES 1 . C. Carlson, in “Xerography and Related Processes” (J. Dessauer and H. Clark, eds.), Chapter 1. Focal Press, London, 1965. 2. J. H. Dessauer, G . R. Mott, and H. Bogdonoff, Phorogr. Eng. 6, 250 (1955). 3. R. Hammer, Fortune 66, 155, 1962. 4. R. M. Schaffert and C. D. Oughton, J . Opt. Soc. Amer. 38, 991-998 (1948). 5. IRE Standards on Electrostatographic Devices, Proc. I R E 49, 619 (1961). 6. F. Schmidlin, IEEE Trans. Electron Deoices 19, 448 (1972). 7. R. W. Gundlach, Japanese Patent 422,242 (1936). 8. V. Tulagin and L. M. Carreira, US.Patent 3,384,565 (1968);V. Tulagin, J . Opt. SOC.Amer. 59, 328-331 (1969). 9. R. Luebbe, M. Maltz, G . Reinis, and W. A. VanDorn. “Electrophotography,” l n t . Con$ Electrophotogr, SPSE, 2nd (D. White, ed.), to be published. 10. W. L. Goffe, Photogr. Sci. Eng. 15, 304 (1971); A. L. Pundsack, Photogr. Sci. Eng. 18, 642 (1974);S. Tutihasi, Photogr. Sci. Eng. 18, 394 (1974).

CHARGED PIGMENT XEROGRAPHY 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 2 7. 28. 29. 30. 31. 32. 33. 34. 35. 36. 3 7. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

145

R. W. Gundlach and C. J. Claus, Photogr. Sci. Eng. 7, 14 (1963). R. W. Gundlach, US. Patent 3,084,043 (1963). M. D. Tabak, S. W. Ing, and M. E. Scharfe, I E E E Trans. Electron Devices 20, 132 (1973). R. M. Schaffert, “Electrophotography,” Chapter 2. Focal Press, London, 1965. R. G. Vyverberg, in “Xerography and Related Processes ” (J. Dessauer and H. Clark, eds.), Chapter 7. Focal Press, London, 1965. C. Carlson, US. Patent 2,588.699 (1952). L. E. Walkup, US. Patent 2,777,957 (1957). V. E. Straughan and E. F. Mayer, Proc. Nut. Electron. Con$ 13, 959 (1958). M. E. Scharfe, Phys. Reu. 2, 5025 (1970). R. W. Gundlach, in “Xerography and Related Processes ” (J. Dessauer and H. Clark, eds.), Chapter 9. Focal Press, London, 1965. E. Wise, U S . Patent 2,618,552 (1952). L. E. Walkup, US.Patent 2,618,551 (1952). C. J. Young and H. G. Greg, R C A Reir. 15, 471 (1957). E. Giaimo, US. Patent 2,786,440 (1957). C. J. Young, U.S. Patent 2,786,441 (1957). K. A. Metcalfe, J . Sci. Instrum. 32, 74 (1955). C. J. Claus and E. F. Mayer, in “Xerography and Related Processes ” (J. Dessauer and H. Clark, eds.), Chapter 12. Focal Press, London, 1965. R. M. Schaffert. “ Electrophotography,” Chapter 5, Part 11. Focal Press, London, 1965. R. M. Schaffert, US. Patent 2,576,047 (1951). P. G. Andrus and F. W. Hudson, in “Xprography and Related Processes” (J. Dessauer and H. Clark, eds.), Chapter 14. Focal Press, London, 1965. R. M. Schaffert, “Electrophotography,” Chapter 2, Part I. Focal Press, London, 1965. W. D. Bolton and W. E. Goety, Pllotogr. Eng. 7, 137 (1956). L. H. Lee, “Electrophotography,” Int. Con$ Elecrrophotogr., SPSE, 2nd (D. White, ed.) to be published. C. R. Mayo, US. Patent 2,684,301 (1954). National Cash Register Co., German Patent 1,079,081 (1960). H. E. Copley, U S . Patent 2,484,782 (1949). 0. G. Hauser and R. S. Menchel, presented at Ann. Symp. S P S E , Washington D.C., 1968. W. E. Bixby, P. G. Andrus, and L. E. Walkup. Phorogr. Eng. 5, 195 (1954). J. T. Bickmore, in “Xerography and Related Processes ” (J. Dessauer and H. Clark, eds.), Part 111, p. 285. Focal Press, London, 1965. J. T. Bickmore, R. E. Hayford, and H. E. Clark. Pkorogr. Sci. Eng. 3, 210 (1959). R. M. Schaffert, “Electrophotography,” Part 111, p. 285. Focal Press, London. 1965. M. E. Scharfe, “Electrophotography,” Int. Con$ Electrophotoyr, S P S E . 2nd (D. White, ed.) to be published. S. W. Ing has made similar substitutions (private communication). J. Mort and I. Chen, in “Applied Solid State Science” (R. Wolfe, ed.). Academic Press, New York (to be published). D. M. Pai and S. W. Ing, Phq’s. Rev. 173, 729 (1968). M. D. Tabak and P. J. Warter, Jr., Phys. Rev. 173, 899 (1968). P. J. Warter, Jr., Proc. lnt. Con/; Photocond, 3rd, 1900, p. 311 (1971). D. M. Pai and R. C. Enck, Phys. Rev. (to be published). L. Onsager, J . Chem. P h p 2, 599 (1934). 1. Chen and J. Mort, J . Appl. Phys. 43, 1164, (1972). H. T. Li and P. J. Regensburger, J . Appl. P h j x 34, 1730 (1963).

146 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79, 80.

M. E. SCHARFE AND F. W. SCHMIDLIN

S. W. Ing, Jr., J. H. Neyhart, and F. Schmidlin, J. Appl. Phys. 42, 696 (1971). I. P. Batra, K. K. Kanazawa, B. H. Schectman, and H. Seki, J. Appl. Phys. 42, 1124 (1971). I. P. Batra, K. K. Kanazawa, and H. Seki, J . Appl. Phys. 41, 3416 (1970). D. M. Pai and M. E . Scharfe, J. Non-Cryst. Solids 8-10, 752 (1972). J . M. Marshall and A. E. Owen, Phil. M a g . [8] 24, 1281 (1971). F . W. Schmidlin, 26th Annu. Con$ SPSE, Rochester, 1973 (Abstr. No. 54, p. 68). T. L. Thourson, IEEE Trans. Electron Deuices 19, 495 (1972). J . T. Bickmore, M. Levy, and J. Hall, Photogr. Sci. Eng. 4, 37 (1960). E . Inoue, H. Kohado, K. Kosei, and H. Saito, Denshi Shashin 7 , 71 (1967). H . G . Greig, U S . Patent 2,811,465 (1957). C . R. Mayo, U.S.Patent 3,245,823 (1966). C. R. Mayo, U.S. Patent 2,895,847 (1959). R. W. Willmott. U.S. Patent 3,232,190 (1966). J . A. C . Yule and W. J. Nielsen, Proc., Tech. Ass. Graphic Arts 3, 65 (1951). J . A. Dahlquist and I. Brodie, J. Appl. Phys. 39, 3020 (1969). L. B. Schein, “Electrophotography,” Int. Con$ Electrophotogr., S P S E , 2nd (D. White, ed.) to be published. E. P. Muntz, D. D. Welkowsky, and D. D. Morsel], “Electrophotography,” Int. Conf Electrophorogr., SPSE, 2nd (D. White, ed.) to be published. R. B. Lewis and H. M. Stark, in “Current Problems in Electrophotography” (W. F. Berg and K. Hauffe, eds.), p. 322. de Gruyter, Berlin, 1972. H. M. Stark and R. Menchel, J . Appl. Phys. 41, 2905 (1970). J . VanEngeland, W. Verlinden, and J. Marien, “Electrophotography,” Int. Conf: Electrophotogr., SPSE, 2nd (D. White, ed.) to be published. I. V. Anfilov and V. M. Fridkin, Zh. Nauch. Prikl. Fotogr. Kinematogr. 5, 367 (1960). T. Kurita, Denshi Shashim 3, 26 (1961). D. K. Donald, J. Appl. Phys. 40, 3013 (1969). F . W. Schmidlin, 24th Annu. Con$ SPSE, Chicago, 1971 (Abstr. No. 11, p. 19). F. W. Schmidlin, 25th Annu. Con$ SPSE, San Francisco, 1972 (Abstr. No. 60, p. 112). L. S. Chang and C. U. Wilbur, “Electrophotography,” lnt. Con$ Electrophotogr., SPSE, 2nd (D. White, ed.) to be published. D. K. Donald and P. K. Watson, IEEE Trans. Electron Devices 19, 458 (1972). W. A. Sullivan and T. L. Thourson, Photogr. Sci. Eng. 11, 115 (1967). H . Stark and R. Menchel, to be published.

The Impact of Solid State Microwave Devices: A Preliminary Technology Assessment"? JEFFREY FREY Department qf Electrical Engineering AND

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

I. Introduction . .... ........ . .. . ., .. _.. ...... . , . .. ., ... .., . .. ... . . . . . . ..... . . , ... A. Features of the Microwave Portion of the Spectrum .................................... B. Microwave Applications . ........, . , ... . , .. ..., . ., . . .. ... . . . . ..... .. . .. .. . . . . . ., ..., , , , ..., , , , ., C. Microwave Device Development ...... ................... D. Microwave Growth Stimuli .................................................................... E. Nature of the Problem ................................................................. 11. Solid State Microwave Sources .. .... . . . . . . . ...... .... . ....... . . ..... . . . ............. . ..... . . ...... A. The Transferred Electron, or Gunn Oscillator ............................................ B. Avalanching Junction Devices _. C. Transistors ........................................................................... sistor Technology ... . , .... ... . . . . ..... .., , , ..., ., , ....., , ....., .....

148 149 150 150 151 152 153 153 158 162 165 ............................ 169 169 A. Communications.. . ... . .. . . ...... ... . . . . .. . ... .. . . . . ... . ... .. . . ..... . . . . . .... .. . .... 169 B. Control Systems ....................................................................... 173 177 V. Benefits and Problems . .. . ................... 178 A. Impact o n the Use of the Electromagnetic Spectrum ................................... 179 B. Minimization of Spectral Congestion .... . . .. . ....... .. . . ..... .. . .. .... . ....... .. ... ... ..... 183 C. Biophysical Hazards of Microwave Radiation . .... . . . ... ... . . . .... .. . .. . .. 186 VI. Invasion of Privacy and Interception of Data Transmission . . . . ...... . ..... 191 VII. Conclusions ..... .. . . . . .. .. . . .. . .. _ _.... . . .. . .. .. . .. .. . .. .... .., , , , , ....., , , ...., ., , , ., , ., , ., . _.., , ..... 191 References ....... . . . . .. ................. .. . ..... ..... .. . .. ... ..... . .. . ..... . . . .... .. , .. ..., . .. ... 191

* Supported by funds provided to the Cornell University Program on Science, Technology, and Society by the National Science Foundation, the General Electric Foundation, and the Sloan Foundation. t Abbreviated descriptions of this work have appeared in Scientijc American 226, 13 (1972) and in IEEE Specrrum 9, 41 (1972). 147

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JEFFREY FREY A N D R A Y M O N D BOWERS

I. INTRODUCTION 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 possibility of anticipating secondary consequences that may result from the introduction of new technologies has received much attention in recent years and has led to the concept of “Technology Assessment” (National Academy of Engineering, 1969; National Academy of Sciences, 1969; Brooks and Bowers, 1970). Technology assessment not only anticipates secondary technological effects, but also attempts to foresee some of the potential social impact of a new technology. While it is generally agreed that the ability to anticipate the consequences of technological change is a desirable goal, skeptics doubt that we have the methodology, the imagination, or the understanding to achieve successful or worthwhile analysis in view of the very great changes that can be brought about by new technology. An example that might support this view is the following: in 1948 there were 100,000 television sets, the next year there were a million, a decade later there were 50 million. Faced with this kind of extraordinary growth and consequent impact, one can understand the view that analyzing potential societal impacts of devices such as television is beyond our present capacities, no matter how desirable. It is indeed possible to find many examples, especially in the field of biology, of technological changes that could have such a profound effect on society that it would be difficult to know where to begin the process of analysis. The possibility of increasing longevity and predetermination of sex of children are two examples. However, the fact that one can specify problems of such large magnitude should not deter us from trying to do a better job of anticipating consequences of technological development in areas where the social consequences are less profound, where the new technology is almost at our doorstep and where it does not seem difficult to foresee some possible secondary and tertiary consequences. Experience gained in analyzing these simpler situations may help in future attempts to deal with more important areas. The recently developed, potentially very cheap solid state sources of low-power microwave energy comprise a new technology which seems a suitable subject for technology assessment. The physical characteristics of the new devices are now fairly well understood, and it is possible to foresee at this stage a number of technological options that will become available.

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The devices should result in a major extension in the application of microwaves in anevolutionary sense: the new devices will enable us to do things we already know how to do, but much more simply and cheaply. Thus past experience with microwaves will be useful in projecting future uses. In this respect, we are faced with a simpler problem than one would encounter with a device, such as the telegraph, which was revolutionary in its time. The telegraph, which provided instantaneous communication between very distant points, made it possible to perform many functions that could not be carried out previously in any fashion. In this article a preliminary attempt at a technology assessment of solid state microwave devices, we shall attempt to foresee the range of new microwave applications suggested by these devices. The analysis of secondary consequences is here largely restricted to the problem of the regulation of microwave devices to ensure proper use of the electromagnetic spectrum, but the assessment will also touch briefly on the issues of potential health dangers, invasion of privacy, and effect of this technology on other technologies. Our purpose is not only to identify possible adverse effects, but also to suggest areas of investigation that will increase the potential for benefit that can come from the application of these new devices.

A . Features of the Microwave Portion of the Spectrum The term “microwave ” describes electromagnetic radiation spanning a particular portion of the spectrum of all radiation: the microwave band is generally taken to mean the frequency range lo9 to something over 10’ Hz; the wavelengths in free space range from roughly 30 cm to 0.3 cm. The location of the microwave band within the spectrum as a whole is illustrated in Fig. 1.

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JEFFREY FREY AND RAYMOND BOWERS

Microwave radiation possesses two features that explain its importance in the technology of signal transmission. First, microwave radiation covers a very wide portion of the frequency spectrum and the large bandwidth thus available allows microwave beams to carry more information than can beams at lower frequencies. Second, their small wavelength allows microwave beams to be focused and spatially directed with relative ease. While the frequency range above 10" Hz is even more advantageous from these points of view, as yet no devices to generate and control energy in this frequency range are available that match the combination of output power, cost, and reliability of contemporary devices in the microwave band. In addition, signals at the higher frequencies are much more subject to attenuation due to moisture in the atmosphere than at microwave frequencies; this attenuation will present a serious obstacle to their use for transmission in the atmosphere.

B. Microwave Applications The bandwidth and directionality properties of microwave radiation have determined the range of past applications. The first important application of microwaves was in radar, developed just before and during the 1939-1945 war; radar depends upon the highly directional properties of microwave radiation and the ease with which it is reflected from solid objects. The second major application of microwave systems was in communications, first making possible the relaying of network TV programs between cities in 1948; in this case the wide-band property of microwaves was as important as beam directionality. Microwave applications thus fall into the two general categories of control and communications.

C . Microwave Device Development A qualitative idea of the effects of technological development on microwave solid state devices and systems in the last few years is given in Fig. 2. The cw power available from such devices has increased almost five orders of magnitude, from sub-milliwatt levels available in the early sixties to the 10-W level of 1973. In 1962 the only way to generate even milliwatts of microwave energy using semiconductor devices was by harmonic multiplication of a low frequency signal generated by a transistor oscillator; in 1973 the IMPATT (Impact Avalanche Transit Time) diode could directly generate almost 10 W. Simultaneously, techniques were developed to replace microwave waveguide circuits with thin metal lines, delineated in the proper pattern by photographic techniques on ceramic, plastic, or other insulating substrates. Such microwave integrated circuits can lend themselves relatively

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easily to low-cost mass production. The overall cost of microwave systems to perform simple functions has also been reduced through the use of automated circuit analysis and synthesis procedures. Finally, a great reduction in the cost of the electronics required to process and analyze microwave signals has resulted from the application of large scale integrated circuits. The trend of development of microwave systems since their first application has been from very expensive systems, purchased by the military and the communications industry, to somewhat less expensive ones, purchased by smaller functional units such as airlines. Recent technological developments, however, indicate that a large proliferation in the numbers of microwave systems can be expected, with microwave system ownership reaching the individual level.*

D. Microwave Growth Stimuli In addition to developments in microwave technology, developments in computer technology and increases in computer use will also stimulate the growth of microwave systems. Transmission of information between remote computer terminals and a central computer-as between a branch bank and

* Our concern here is with relatively low-power microwave sources. We are not considering the use of microwaves for heating, since it does not seem likely at this time that recent developments in the solid state field are likely to have much impact on the generation of high average microwave power levels. Hence, we are excluding (for example) microwave cooking ovens from most of our discussion.

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JEFFREY FREY A N D RAYMOND BOWERS

its central office-can be performed using telephone lines; however, a private microwave system, with the capability for transmitting a thousand times as much information in the same time as the telephone system, could be more efficient in many applications. Furthermore, when computers themselves are interconnected for simultaneous computation, very fast, i.e., wideband, data links are required. The computer-oriented uses of microwave systems can be expected to increase regardless of any new developments in microwave technology, but the microwave developments described here will facilitate the use of computers, partially by facilitating remote access to them. E . Nature of the Problem

Devices with prices within reach of many householders will soon be on the market and it is likely that within a decade these devices will be used in the home, and in the private auto and boat; they may even be incorporated into toys. Commercial organizations may also use them on a massive scale for transmission of information. The proliferation of microwave devices could conceivably match that of television sets. In order to assess the potential growth and consequences that may result from that growth, a number of questions must be considered : (a) What are likely performance and cost levels for these devices? (b) What are the estimated costs of complete microwave systems? (c) What are likely uses for the new devices? (d) What will be the effect of microwave device technology on other technologies ? (e) What impact will inexpensive microwave systems have on the use of the electromagnetic spectrum, considered as a consumable or saturatable resource ? (f) What special problems will there be in reducing interference among systems ? (g) If segments of the frequency spectrum are too congested to allow for uncontrolled proliferation of new systems, what are the principles that should determine allocation and priority within the frequency spectrum? (h) Are there potential health hazards resulting from increased likelihood of exposure to microwave radiation? (i) Will the new devices present special problems with respect to privacy and the protection of confidential information? These questions represent only a fraction of those that should be considered in a comprehensive technology assessment. Several of them deal with the use of an important and limited resource, the electromagnetic spectrum. The list does not include more important nontechnological issues such as the impact of the new devices on social processes. Certainly the introduction of devices such as the telephone and the automobile have influenced where

IMPACT OF SOLID STATE MICROWAVE DEVICES

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and how people live. We are not suggesting that the introduction of new, cheap microwave sources will have comparable social effects; nevertheless, it would be nearsighted to assume that these new sources will not have substantial social impact eventually, since they will profoundly affect both communication and transportation networks. This article reports our attempts to take the first steps toward the assessment of microwave solid state technology. We hope that these preliminary efforts will encourage social scientists to collaborate with technologists in order to investigate these potential social implications. Such work by several groups, extending over several years, will be required before we can have a reliable picture of the potential impact of this new technology.

11. SOLIDSTATEMICROWAVE SOURCES With the exception of the transistor, it is little more than a decade since the theoretical ideas leading to the microwave solid state generators we are discussing here were published and early experiments demonstrated their enormous potential (Read, 1958; Ridley and Watkins, 1961; Hilsum, 1962). In the years following the first demonstration of high frequency oscillations in gallium arsenide by Gunn in 1963, and of the Read and IMPATT diodes by Tager in the USSR in 1960 and by Lee et al. (1965) and Johnston et al. (1965), an armory of devices has been developed that, in certain ranges, can now compete in output power and efficiency with the older and wellestablished electron tube technologies. An additional, an almost bewildering array of microwave-oscillatory phenomena can be found in the technical journals, many of which might lead to important devices in the future. However, it is already clear that transistors and the microwave transferred electron (Gunn and Limited Space-Charge Accumulation, LSA) oscillators, IMPATT diodes, and TRAPATT (Trapped Plasma Avalanche Triggered Transit) diodes, have achieved a high level of development and will be of major importance in the development of this new phase of microwave technology. In this section we shall describe the operation of the most common of today’s microwave semiconductor devices and delineate their inherent performance limits. A. The Transfrrred Electron, or Gunn Oscillator (Carroll, 1970) 1. Means of Operation

This “bulk negative differential conductance device depends on the fact that increasing the electric field in certain materials above a particular “critical” value results in a decreased current through the material. The origin of ”

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JEFFREY FREY A N D RAYMOND BOWERS

this effect is as follows: in certain materials, such as GaAs and InP, it is possible to excite (or “ transfer ”) electrons, using moderate electric fields, from their normal high mobility states, which carry large currents, into higher energy states that have much lower mobility and consequently transport less current. When small electric fields are applied, the electrons increase their velocity (and hence the current they carry) as the electric field is increased. In this range the conductivity is normal or “positive.” However, as the electric field is further increased, electrons are excited (or transferred) into higher energy states of lower mobility. When this condition is reached, increasing electric fields result in decreasing current and the effective differential conductivity becomes “ negative.” The existence of a negative conductivity can lead to oscillations. The transferred electron device (TED) is usually operated under conditions that lead to the formation of domains of space charge that traverse the sample in fractions of a microsecond; it is the motion of these domains that is responsible for the oscillatory microwave currents.

2. Limitations on Performance The efficiency of a transferred electron oscillator (TEO) is dependent upon the ratio of the highest electron drift velocity achievable in the material to the lowest velocity, since the current is proportional to carrier velocity. As in all signal generators, large efficiency values require large current and voltage oscillations. Other materials, such as InP, may exhibit higher peakto-valley-current ratios than does GaAs, and are being investigated for use in high efficiency TEDs. The TEO is a bulk negative differential resistance device that, in principle, should oscillate in any circuit which has a positive ac resistance smaller than the negative resistance and that allows voltage and current waveforms to develop in the proper phase, and with the proper shape, to generate energy. However, the TEO is rarely operated in the bulk negative resistance mode, because localized field nonuniformities in real devices usually lead to “buildup” of electrons and the formation of localized dipole charge layers called domains. These domains are stable-i.e., do not decay-because of the negative differential mobility characteristics of the material and the values of electric fields in and near the domain. In most TEOs, therefore, the generation of microwave energy is due to the generation of localized domains, which then pass through the device to be collected at the contacts out of phase with the voltage pulse across the device that originally produced them. Most TEOs are, thus, transit-time devices; since the domain moves with the high-field velocity of electrons in GaAs ( lo7 cm/sec), and the passage

IMPACT OF SOLID STATE MICROWAVE DEVICES

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time of a domain across the device should be of the order of an rf period, the length of a TEO operating purely in the domain mode must be roughly L

-

vel/freq

=

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For microwave TEOs, devices of the order of 1-10 pm in length are required. Current materials technology, both liquid phase and vapor phase epitaxial growth of GaAs materials (Knight el al., 1965; Kang and Greene, 1967), is capable of producing layers of such thickness and of a high enough quality for TEOs. 3. Modes of Transferred Electron Oscillators

TEOs can operate in various modes-the "Gunn" mode or the limited space charge accumulation (LSA) mode-or in a mode that is a hybrid of these. Most actual device operation is probably in the hybrid mode, but a brief discussion of the extreme modes will be useful. a. Gurin mode. In this mode, a stable domain forms at one end of the device and moves to the other end. The condition for such operation is that the domain growth time be less than the domain transit time. Using the basic constants of the material, it can be shown (Copeland, 1967a) that this requires

N , 1 > 10' 2/cm2, where N , is the donor density in the n-type GaAs and 1 is the length of drift region. b. L S A mode. If a GaAs device is placed in a resonant circuit tuned so that its oscillation period is much less than the time required for a domain to form, formation of a complete domain will be inhibited during the cycle. However, during the brief period when the highest field in the device is above threshold (i.e., the value at which a negative conductance appears) a domain will begin to form. In order for this partially formed domain to be allowed to dissipate, the field in the domain region must fall to a value below threshold relatively quickly. To allow domain dissipation, the time the field is below threshold must be greater than the dielectric relaxation time for the domain. These considerations result in the requirement that (Copeland, 1967b) 2 x lo4 < N d / f > 2 x lo5, where,fis the operating frequency. The charge trapped in the domain during the high field (high voltage) portion of the cycle is not available as terminal current, so that current falls during the high voltage part of the cycle. Thus, the device appears as an ac negative resistance between its terminals. The use of this bulk negative resistance in the LSA mode allows the generation of

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JEFFREY FREY A N D RAYMOND BOWERS

relatively high amounts of power, since almost all of the volume of the device can effectively be put to use. However, LSA devices and circuits must be carefully designed to prevent formation of domains due to material imperfections or improper circuit tuning, resulting in localized fields large enough to cause avalanching. Circuit tuning must also be adjusted to suppress domain-mode oscillations at the lower frequencies at which the microwave circuit or bias circuit might be resonant. c. Hybrid Modes. If a field high enough for domains to form is applied to a TEO it will oscillate even though operated in a circuit resonant at a frequency other than that determined by the domain transit time. If the period of the resonance is longer than the transit time, no domain will exist in the device for part of the rf cycle; if the period is shorter than the transit time, the domain will be “quenched” before it makes a complete transit. In either case rf current is generated which is out of phase with the applied rf voltage, so that the device appears to be a negative resistance. Since a TEO will oscillate if its circuit is tuned in either direction from the basic carrier transit-time frequency, this device can be used in very wide-band applications. For example, a single TEO has been used in an electronically tuned oscillator covering the band 7.9-20 GHz (Green and Melick, 1973), and very wide-band transferred electron amplifiers have also been reported.

4. TOE Limits The best performance currently obtained from cw Gunn effect devices is shown in Fig. 3. The experimental points shown were obtained in the laboratory for devices with conventional heat sinks. With a typical input power of about 17 W, about 2.5 W of output power can be obtained (almost 15%

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efficiency). Considerably higher output power can be expected from similar devices with better heat sinking possibly involving water cooling, multiplemesa construction, or diamond heat sinks. The performance achieved by LSA diodes and potential limits are summarized in Fig. 4; the maximum peak power for an LSA device is plotted as a function of operating frequency. The optimized theoretical diode represented in this figure is assumed to be fabricated with a doping gradient to minimize effects of nonuniform heating and to be mounted on a diamond heat sink. The LSA diode must be operated in a pulsed mode due to the high power densities involved; the best duty cycle yet achieved for these diodes is 0.001. Thus, the best average power outputs attainable can be obtained approximately by multiplying the average power levels of Fig. 4 by 0.001. 5. T E O Applications

The domain-type TEO can provide moderate amounts of cw power at moderate efficiency levels. In addition, it is relatively simple to fabricate and build a TEO circuit given GaAs epitaxial material of sufficiently high quality. Current GaAs material costs are high and the resultant devices are

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JEFFREY FREY A N D RAYMOND BOWERS

expensive despite the greater simplicity of fabrication and circuit design. However, as material costs are lowered, the cw TEO can be expected to be used in low-power radars and in applications where its wide-band properties are important. In addition, the cw TEO is a relatively low-noise device that may find an application as a low level amplifying stage in communications apparatus. The LSA oscillator, producing large peak power with relatively high efficiency and low duty cycle, has a number of military radar applications. Great care must be exercised in the fabrication of LSA devices and in associated circuit design; need for such care indicates that the cost of LSA devices may remain too high for widespread public use. B. Avalanching Junction Devices

A number of important microwave devices generate microwave energy by exploiting the time delays inherent in producing avalanche breakdown in a junction and the large numbers of mobile carriers produced in the avalanche. The original device of this type was proposed by Read in 1958. The first Read-type structure was made in the United States by Lee et al., but the simultaneous discovery that a much simpler avalanching p-n junction could also generate microwaves led to concentration of effort on the latter device, which came to be called the IMPATT diode. In 1967, an IMPATT diode was unexpectedly found to produce a large amount of microwave power, at a lower frequency but with a higher efficiency than would have been expected of the IMPATT mode; this phenomenon led to the discovery and analysis of the TRAPATT mode (Deloach and Scharfetter, 1970). Recent work, aimed at improved performance of avalanche-type diodes, has led to a re-examination of the complex Read structure and to successful attempts to fabricate it. While it is common to treat IMPATT and TRAPATT diodes as completely distinct devices, we will treat them here as merely two extreme modes of operation of the same device. Similarly, other modes, such as the “relaxing avalanche mode” (RAM) (Zappert and Lee, 1972) are intermediate states in a continuum of modes of operation of simple p-n junctions. 1. IMPATT and Read Diodes In these devices, an intense radio frequency field across a reverse-biased semiconductor p-n junction results, during part of the cycle, in the creation of a large concentration of mobile charge carriers in the vicinity of the junction due to impact ionization caused by the accelerated electrons. If this cloud of charge is allowed to drift from its site of creation to a collector electrode, the current pulse associated with this flow of charge will be out of

IMPACT OF SOLID STATE MICROWAVE DEVICES

159

phase with the applied field. The phase difference will be determined by the time required for the avalanche to create the cloud of charge plus the time required for the cloud to travel to the collector electrode. The former time lapse depends on material properties and is not easily adjustable, but the latter time lapse can be set during fabrication to any desired value. Consequently, it is possible to arrange for the current collected to be one-half cycle out of phase with the applied field. Under such conditions, the device acts as a negative resistance and can oscillate. The IMPATT diode is similar in principle to the Read diode, but, to simplify fabrication, the avalanching and drift regions are combined in a single “compromise ” region. In the IMPATT, the charge cloud is much wider than in the Read diode and different parts of it can have quite different transit times. Hence, less of the collected current will be in the exact out-ofphase relationship to the applied voltage to produce output power, and as a result the IMPATT diode is much less efficient than the Read diode. Doping profiles, field shapes during operation, and output waveforms are illustrated for Read, IMPATT, and TRAPATT diodes in Fig. 5.

1

q 0

>

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D I STA N C E (0)

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JEFFREY FREY A N D RAYMOND BOWERS

2. T R A P A T T Mode The IMPATT and Read structures are so designed that the density of charge carriers in the avalanche-created plasma does not greatly affect the electric field in the drift region. In the TRAPATT diode, however, the doping in the drift region is relatively small, so that a given number of carriers caused by the avalanche can be a significant fraction of the background impurity density. In this case, the magnitude of the electric field in the drift region can be reduced by the presence of the approximately neutral plasma of electrons and holes created by the avalanche. Since the voltage across the diode is proportional to the integral of the electric field within the drift region, a reduction of field will give rise to a reduction of voltage to a value that can be much below its dc bias level. This reduction in voltage is much greater in the TRAPATT diode than in the IMPATT diode, and the large voltage swings (peak-to-peak oscillations) that can arise in the former device can result in the generation of much higher power levels at much higher efficiency values than in the IMPATT diode. Further, the TRAPATT diode must operate at a lower frequency than an IMPATT of equal drift length, because the field in the drift region is reduced to such a low value by the plasma that the carrier drift velocity in the drift region is significantly reduced. 3. Avalanche Device Performance Limits

The highest level of development of cw avalanche devices is today found in GaAs diodes which have approximately the Read-proposed profile and which utilize sophisticated heat-sinking techniques. GaAs devices have consistently provided better power and efficiency than Si devices. The reasons for this advantage are not clear, although they may be related to the ionization rates in GaAs, and to the higher electron mobility in that material. Experimental results with GaAs and Si IMPATT diodes and Si double-drift IMPATT diodes are shown in Fig. 6, with a theoretical prediction of expected performance of GaAs Read-type structures and that of the best silicon diodes. Theoretical calculations of the expected power in the pulsed TRAPATT mode (Scharfetter, 1970) and selected experimental results are shown in Fig. 7. 4. Avalanche Device Applications

Cw avalanche devices can generate higher power at any given frequency than TEOs; the AM noise in avalanche devices, however, is much greater than that in TEOs. Hence, avalanche devices will be useful as final-stage

IMPACT OF SOLID STATE MICROWAVE DEVICES

161

'Or A

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devices in communications transmitters, well into the millimeter-wave frequency range. TRAPATT diodes will be useful in mobile radar systems, producing large values of peak output power with high efficiency. These devices, however, are not useful above about 12 GHz because of the difficulty in designing the circuits required to support the complex waveforms required for their proper operation at these frequencies.

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JEFFREY FREY A N D R A Y M O N D BOWERS

C . Transistors

It is interesting that transistors have not, until recently, been prominent in most considerations of microwave solid state devices. This apparent neglect is a result of the technological difficulty involved in producing microwave transistors. The devices considered above, TEOs and avalanche devices, perform in the microwave frequency range due to an intrinsic property of the material (electron drift velocity) and the possibility of fabricating devices with appropriate drift lengths to cause oscillation. The frequency of operation of transistors, however, does not depend on intrinsic properties of the material, except in limiting cases. Transistors operate by modulation of a current flowing between two electrodes (emitter to collector, or source to drain) by means of a third electrode. This modulation is achieved by varying current into the base electrode of a bipolar transistor or by varying the voltage on the gate of a field effect transistor. Transistor performance is usually limited by materials parameters such as maximum carrier velocity, device parameters such as base or channel length, or parasitic circuit elements. Recently great efforts have been made to increase the maximum permissible frequency of transistor operation by reduction in dimensions; only recently has this frequency been moved well into the microwave spectrum. The implications of the above discussions are the following: as fabrication technology improves, transistor operating frequencies will rise ; since operating frequency below the limit is not determined by device or materials parameters, the transistor will be used in wide-band applications; finally, the third-terminal modulation capability of the transistor will result in its widespread use as an amplifier. 1. Limits on Transistor Performance

A number of treatments of transistor performance limits have been given (Johnson, 1965; DeLoach, 1967) and it is appropriate to update those here. Considering only transit time in the base of a bipolar transistor, or in the channel of a field effect transistor, the transit-time cutoff frequencyfT of a transistor is defined as .fT =

1/2w

where T is the average time for a charge carrier moving at the average carrier velocity to traverse the effective base width L,, . The latter will differ from the geometrical base width due to formation of depletion regions under bias. The maximum permissible value of field in the device, which determines the maximum rated voltage V,, , is the breakdown field Em of the semiconductor.

IMPACT OF SOLID STATE MICROWAVE DEVICES

163

Utilizing these parameters, one can formulate the optimum value product (VmfT 2 2 x 10” V/sec for Si and GaAs.) If X, is the parasitic capacitive reactance across the output terminals of the device, its maximum power output P , will be related to V, by roughly,

Consequently the maximum power limit:

can be formulated. This limit is plotted in Fig. 8, along with experimental data obtained between 1965 and 1973. The interesting point about the curves of Fig. 8 is that some experimental points lie above the “theoretical limits” curve, casting doubt on the experimenter’s accuracy or on the basis of the ‘‘limits’’ themselves. Indeed, a

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FIG. 8. Values of ( P , X , ) 1 ’ 2products achieved by laboratory transistors, and theoretical limit curves due to Johnson” and Frey. The latter limit curves are drawn for uniform doping in the base region; the former for uniform electric field. F4 is a silicon field-effect transistor; F5 is a gallium arsenide field-effect transistor.

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JEFFREY FREY A N D RAYMOND BOWERS

re-examination of the fundamental limits indicates that the “saturated velocity for GaAs may not be the proper value to use in the above formulas for two reasons: (1) electric fields along the path between electrodes might not reach values high enough for velocity saturation, and (2) the carrier and lattice dynamics may be such that, even though the fields are very high, electrons pass through the device in too short a time for their energy to equilibrate to a value consistent with a velocity of l o 7 cm/sec. (Ruch, 1972). A third possible effect, the influence of the negative differential mobility region in GaAs, has been shown to be of negligible importance in most GaAs transistors (Dawson and Frey, 1973). It now seems likely that the second effect may be of considerable importance, at least in devices made of materials (such as GaAs and InP) in which carrier scattering is largely by excitation of optical phonons. Theoretical calculations of the problem, currently underway in several laboratories, indicate that the proper carrier velocity to use in transistor calculations may be considerably higher than the peak values formerly considered. ”

2. Bipolar us. Field Effect Transistors

Figure 8 shows that the highest P , X , products have been achieved by GaAs FETs. That these FETs are superior to bipolar devices is at least partly because the FET is a horizontally oriented surface device, while the bipolar transistor is a vertically oriented “bulk device. That is, the current in an FET flows in a direction parallel to the surface of the semiconductor, while that in the bipolar transistor travels normal to the surface. The channel width in an FET is thus determined by surface processing of the device; the base width in a bipolar transistor is determined by control of the depths which impurities reach inside the semiconductor. The properties of the FET channel are thus easier to control during fabrication than those of the bipolar base. Two properties of GaAs, relative to those of Si, lead to superior performance of GaAs devices. First, the low-field mobility of electrons in GaAs is more than double that in Si, and second, it is possible to dope GaAs substrates so that they acquire a very high resistivity ( p > lo7 R-cm). The second property can lead to a great reduction of parasitic elements, compared to levels that can be achieved in Si devices. The first property applies to GaAs bipolar devices as well as to FETs, but the difficulty of introducing impurities into GaAs has prevented much progress with the former device. In fact, the GaAs FET is an exceedingly simple device to make, given a GaAs crystal with a high resistivity substrate and an epitaxial ”

IMPACT OF SOLID STATE MICROWAVE DEVICES

165

layer of appropriate doping and thickness. To make such a device one need only put two metal ohmic contacts on the surface, for source and drain, and place a Schottky barrier contact between them for the gate. No operations that affect the bulk of the device are necessary. 3. Transistor Applications On the basis of the transistor fundamental limits given in Fig. 8, one can predict that transistors will eventually be able to produce cw powers of the order of 5-10 W in X band (8-12 GHz) and 2&30 W in C band (4-8 GHz). Furthermore, transistors exhibit much lower noise than any junction device that depends upon impact ionization : bipolar transistors are currently available with noise figures of 2 dB at 2 GHz and 3.3 dB at 4 GHz (HewlettPackard HP35870) as compared to 10 dB at 10 GHz for TEOs. GaAs FETs are expected to exhibit lower noise than silicon bipolar transistors; a GaAs FET is already on the market with a noise figure of 4 dB at 8 GHz (Fairchild FMT900). As techniques of epitaxial-layer growth and fine line delineation improve and FET gain increases in the near future, one can expect noise figures below 4 dB to extend well in Ku band. Consequently, one can expect transistors to be useful in communications systems, both in low-level, low-noise amplifying stages and in high-level final-transmitter stages. The power levels achievable with transistors may never equal those achievable with IMPATT diodes above X band, but the simplicity of transistor use in amplifying systems, where they do not require circulators, will ensure their widespread use. D. Developments in Transistor Technology

The desirability of increasing the output power and efficiency and of optimizing such other parameters of microwave semiconductor devices as noise figure and bandwidth when used as amplifiers has led to advances in semiconductor processing technology that would not have been necessary for lower frequency devices. In particular, the technologies of liquid- and vapor-phase epitaxial growth of gallium arsenide, and of Schottky barrier contact deposition on both GaAs and Si, have been driven forward by the demands of makers of TEOs, IMPATT and Read-IMPATT diodes, and field-effect transistors. Similarly, as transistors find more application in microwave systems, the demands on their performance and cost levels will naturally increase, and fabrication technologies, particularly as applied to mass production, will improve. In addition, the search for new materials that may lead to devices with enhanced performance will accelerate.

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JEFFREY FREY A N D R A Y M O N D BOWERS

1. Materials Research

Indium phosphide, a 111-V compound semiconductor material, offers some hope for yielding devices with improved performance in the near future. Figure 9, a curve of the velocity-field dependence for Si, GaAs, and InP, indicates why: a higher peak electron velocity (along with a larger negative differential conductivity) is available in this material than in GaAs. This higher peak velocity can lead to higher cutoff frequencies in InP fieldeffect transistors ; the larger negative differential conductivity can lead to higher power output and efficiency from transferred electron devices. As yet the technology of InP epitaxial materials growth is crude, no satisfactory metal has been found to form a Schottky barrier on the material, and semiinsulating InP substrates are not available, but the attractive velocity-field characteristic of InP has led many experimenters to begin research on this material. Beyond InP, one can foresee a large number of possible pseudo-binary semiconductor materials, such as GaAlAs, which might prove advantageous in microwave applications. Certainly, materials experimentation along these lines will increase. 2. Microwave Transistor Improvements Of all the microwave semiconductor devices discussed above, transistors have been least developed with respect to their potential performance. This situation is largely due to the technological complexity of microwave 3 L

c

0 0 W

, ln

z u

- 2

c>u

s W

>

LL I-

E l n

I

I

FIELD

I

(KVICM)

FIG. 9. Equilibrium velocity-field profiles for silicon, gallium arsenide, and indium phosphide.

TABLE I FIELD-EFFECT TRANSISTORS-TECHWOLOGICAL FACTORS

Parameter Channel doping

Direction of change UP Down

Affected performance parameter Ym

UP Down

Ym

Channel length

UP

Lower field in channel for given Ve: hence gm through higher p gm through lower transit time

Size

UP Down

Doping levels t 10'' cm-- 3

Gate breakdown voltage; Lower pinch-off voltage so lower field in channel (higher velocity in GaAs)

Epitaxial thickness

Down

Technology

Lower pinch-off voltage

Power output Bandwidth;,f, (due to parasitic capacitances)

Layers < 1 p to be repeatably grown; good interface with substrate (buffer layer) to reduce noise

Electron-beam resist exposure for L < 1 p

TABLE I1 BIPOLAR TRANSISTORS-TECHNOLOGICAL FACTORS Parameter

Direction of change

Affected performance parameter

Technology

Negligible tradeofis

Base Width W

Down

.fT: hfe; %I)

Emitter resistance re

Down

fT(TF)hfe(DE); F(hfc)

Emitter depth

Down

Base field shaping

Optimized

hf,,

(4

Diffusion control; Arsenic emitter; ion implantation Higher emitter doping (As emitter; ion implantation); new geometries; selective etching Diffusion control (ion implantation; As emitter) 4

Diffusion or implantation control

f T ( T B ) ; hfe

4U

Significant Tradeoffs

Base doping

Collector doping

Size

Down

fT(TB

UP

due to reduced C,, fT; hf,, Due to reduction in E-B depletion length

Down

fT(r,) Due to reduced C, bandwidth

) Due to reduced depletion width;fT(r,)

UP

fT(5

UP

Power output (through periphery/area ratio) fT; Bandwidth due to all capacitances Power Reliability

Down Operating temperature

hfe;fT(TE)

UP Down

If epi layer is so thin that rc r 0, effect of increased C , can be minimized by circuit design As doping is raised collector width must be reduced to reduce re; good epi growth Geometric design utilized to maximize peripher y/area

New metals required for contacts; new semiconductors (e.g., Sic)

8 E

m

E

IMPACT OF SOLID STATE MICROWAVE DEVICES

169

transistors, which must be made with extremely small geometries. O n the other hand, small geometries lead to high power densities in power devices, and means have to be found to reconcile these conflicting requirements. The trends of technological development that can be expected for microwave transistors are summarized in Tables I and I1 (for bipolar and field-effect transistors, respectively). It will be seen from these tables that all of the latest processing technologies, such as ion implantation doping and electron-beam pattern definition, will eventually be brought to bear on the microwave transistor problem. Indeed, when these devices reach their full potential, they may occupy a predominant position among all microwave semiconductor devices, due to the relative ease of their fabrication (a surface-oriented, not bulk, geometry for FETs), their low noise output (no impact ionization), and the simplicity of design of amplifying circuits (three terminals are available).

INTEGRATEDCIRCUITS 111. MICROWAVE Great reductions in the cost and size of microwave systems, in part made possible by the semiconductor devices discussed above, will also be facilitated by the use of microwave integrated circuits. Such circuits will be made by high precision photographic and thin-film techniques which are easier and much cheaper to adapt to mass production than the machining techniques required for waveguide and coaxial microwave circuits. For example, a typical microwave receiver, in the past, had to be assembled from precisely machined metal waveguide parts and consequently occupied considerable space and was expensive. Such construction was acceptable when microwave systems were assembled in small quantities for specialized uses, but the clumsiness and expense of sophisticated waveguide circuits would, if they were necessary, certainly hinder the popularization of microwave applications. Waveguides are replaced by strips of gold a few thousandths of an inch thick and less than 0.050 in. wide, placed in the desired pattern on a substrate of glass, ceramic, sapphire, or other materials by techniques that can be as simple as silk screening. A microwave integrated circuit receiver need occupy no more than a few cubic inches, and photolithographic delineation of patterns followed by etching can be much cheaper than the precision machining of fine parts.

Iv. APPLICATIONS OF MICROWAVE SOLID STATE DEVICES A . Communications

Probably the most widespread civilian use of microwaves now is the field of communications. Current and proposed long-distance communications links [AT&T, Western Union, Microwave Communications, Inc. (MCI)

170

JEFFREY FREY A N D RAYMOND BOWERS

Datran], operate in, or are proposed for, the 3.7-4.2 GHz and 5.925-6.425 GHz bands; as these bands become saturated, new links will appear in the 11.7- 12.2 GHz band. As lower frequency bands become occupied, microwave solid state devices will become dominant in terrestrial communications systems. This prediction arises from the following considerations. Microwaves do not bend with the curvature of the earth, so that repeaters, which receive, amplify, and re-transmit the signal, must be used at regular intervals between terminals of long links. The spacing between repeaters in the lower two (4 and 6 GHz) microwave bands is determined by the curvature of the earth, features of the terrain, and acceptable antenna heights. Above 10 GHz, however, atmospheric attenuation becomes important, so that the repeaters have to be more closely spaced. In the lower two bands, 30 miles is a normal spacing between repeaters. Since the cost of the microwave electronics (exclusive of antennas) in a typical repeater is less than 10% of the total cost of the repeater (including tower construction, etc.) (Cosgrove and Chipp, 1968) and expensive equipment must be provided at each terminal to switch incoming and outgoing calls to the proper circuits, the fraction of the total system cost attributable to microwave components for a system operating below 10 GHz is very small. Consequently, there is little cost benefit to be gained by using solid state devices below 10 GHz. As the 4 and 6 GHz bands become saturated, however, use of high frequency bands becomes necessary; here the cost situation is significantly different. The main argument for the use of solid state sources in trunk application is summarized in Fig. 10; this figure gives the repeater spacing required for a fade margin of 40 dB in an environment of 10.2 cm of rain per hour over the entire path length, as a function of frequency (Tillotson, 1969). Above 10 GHz, attenuation of microwaves due to water vapor in the atmosphere increases markedly, reducing the required repeater spacing from 30 miles to below 3 miles. The order-of-magnitude increases in the number of repeaters required for a long system at millimeter wave frequencies should be accompanied by at least the order-of-magnitude reduction in cost and increase in reliability of solid state sources relative to thermionic sources. This is certainly the claim of some source manufacturers. Indeed, the Bell System is basing its plans for wideband, millimeter-wave trunks on the availability of microwave solid state sources and microwave integrated circuits (Welber, 1970). In addition to being used in systems for communication over long distances, millimeter-wave systems have been proposed for local distribution of signals carried by long-distance lower frequency trunks. For example, MCI, which offers a private wideband communications service over its own intercity microwave network, has petitioned the Federal Communications

171

IMPACT OF SOLID STATE MICROWAVE DEVICES 30-

25

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15-

5 a

&

l0-

5 -

1

0

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4

1

1

8

1

1

12

1

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16

1

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1

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24

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28

Commission (FCC) to set aside the 38.6-40 GHz band for such local distribution. Solid state sources will surely be used in such local distribution systems because of lower cost and improved reliability. Not all of the expected future expansion of wideband communications capacity can be above ground, because of the lack of adequate spectrum space. A number of foreign post and telegraph organizations, as well as the Bell System in the United States, are already installing prototype wideband millimeter-wave waveguide systems below ground. The Bell System approach utilizes a bandwidth of 70 GHz, from 40-1 10 GHz, composed of 120 individual channels. Two-level phase shift keying at a 274 Mbit rate is anticipated for the system. Repeaters, spaced about 20 miles apart, will contain 120 IMPATT diodes with power outputs ranging from about 100 mW at 40 GHz, to about 25 mW at 110 GHz (Welber, 1970). Because of cost and reliability factors, a nationwide network of such trunks would not be possible without IMPATT diodes or without other semiconductor millimeter-wave devices for local oscillators and lower frequency devices as amplifiers. Application of low-power microwave transistors in the associated logic circuits is also likely. The use of microwaves for dedicated short-distance communications links will probably increase in the future. Such links may be specialized: between remote computer terminals and a central computer; between a CATV head end and the distribution center (a band between 12.7 and 12.95 GHz has been set aside by the FCC for this purpose); between a main plant and a remote location (a firm in Syracuse, New York is using such a

172

JEFFREY FREY A N D RAYMOND BOWERS

system for data communications during the day and surveillance at night), etc. Microwave semiconductor devices will be utilized in communications systems in which synchronous earth satellites are used for the relay of wideband signals over long distances, supplanting or replacing terrestrial systems. As the price of microwave systems comes down, broadcast to smaller receiving units becomes feasible. Satellites are currently used to relay network TV programs from continent to continent, but the technology will shortly be extended to the relay of programs directly to homes. First tests of such systems will be conducted using NASA's ATS-F satellite, scheduled for launching in 1974. During the first year of its operation, this satellite will be positioned over the western hemisphere, and used to broadcast educational programs to 114 remote towns in the Rocky Mountains, Alaska, and Appalachia. Subsequently, the satellite will be positioned over India and used to relay programs to some five thousand village receivers in that country. The cost of the ground terminals for these systems will be under twenty-five hundred dollars, about one-hundredth of the cost of the terminals now in use, and will make extensive use of simplified semiconductor circuits. The large expansion foreseen for satellite-to-small-unit broadcast is exemplified by the spectrum allocations that were made for this service at the 1971 World Administrative Radio Conference in Geneva. At that time, the bands 0.62-0.79, 22.5-23.0, 41 .O-43.0, and 84.0-86.0 GHz were allocated for such purposes. The anticipated large expansion in the service (and its political potentialities) have already been recognized by the USSR, which recently introduced at the United Nations a draft treaty calling for strict international regulation of such transmission. Additional experiments scheduled for ATS-F illustrate other applications in which microwave solid state devices may be of importance. A prototype satellite air traffic control system will provide continuous communications among, and navigational data for, ships, aircraft, and ground control stations. Such a system could lead to a large increase in the capacity of air traffic lanes through higher precision position location than is now possible. ATS-F will also be used to relay data from a satellite orbiting at a lower altitude to a ground station the lower satellite could not reach directly. Finally, ATS-F will be used to perform experiments on the propagation of millimeter-wave signals (13 and 18 GHz up-links and 20 and 30 GHz down-links) to and from satellites, in various weather conditions. As satellite-to-earth transmission (down-link) frequencies above 12 GHz come into use, solid state sources will be used in satellites themselves. Indeed, a proposal has been made to use 120 low-power narrow band Gunn oscillator microwave sources to replace a single traveling-wave tube* in a * Fairchild Camera and Instrument Corp., private communication.

IMPACT OF SOLID STATE MICROWAVE DEVICES

173

communications satellite. With this approach, power per unit bandwidth for the solid state source can be made about equal to that of the electron tube. Congestion of the U H F bands may necessitate the construction of microwave systems for omni-directional land-based broadcast TV transmission (Feldman et al., 1969). Such a system might be more efficient for broadcast use over small areas than direct satellite transmission, and would provide a large area of application for microwave solid state devices as receiver local oscillators. Microwave solid state sources can be expected to have an effect on the future development of land mobile radio services. These services, formerly confined to the 450-470 MHz band, have been growing so rapidly-at the rate of almost 300,000 transmitters per year for the last 4 years-that in many areas it is impossible to add new transmitters. In response to this congestion, the FCC has truncated the U H F TV band, allowing mobile transmitters to operate in a region around 890 MHz, and is experimenting with relaxing the block allocation system to allow land mobile use of spectrum where it is not otherwise required for U H F TV. The new developments in solid state sources have the potential for a major direct improvement in land mobile communications systems, possibly making automobile telephones very common. There simply may not be enough room in the U H F bands, now available or proposed, for mobile communications to increase the number of automobile phones to a substantial percentage of the 10' automobiles in use in this country. Microwave or millimeter wave bands are clearly needed. The directionality of radiation at microwave wavelengths would, however, necessitate a large number of local transmitting/receiving terminals, with the phone signal being passed along from terminal to terminal to follow the motion of the car. A system for locating the car (one which probably also involves microwaves) would also be required. Once again the cheapness and reliability of solid state devices might vastly increase this application of microwaves. B. Control Systems

In the field of guidance and control-radar, radio location, process control, etc.-the availability of microwave solid state devices is, again, not expected to result in many wholly new applications but, rather, to greatly increase the number of systems in use. 1. Radars for Collision Avoidance

Simple solid state radar systems have already appeared for use in light aircraft and boats: a unit called AWARE (Aircraft Warning Avoidance Radar Equipment) utilizes a solid state source to project range gates around

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JEFFREY FREY AND RAYMOND BOWERS

the aircraft; a radar echo from another aircraft within the gate initiates an audible alarm, and causes a simple display to indicate the approximate position of that aircraft. The system is intended to warn a pilot only of the proximity of another aircraft, not its location. A handheld, suitcase-size FM/cw doppler radar intended for use on boats is now available for under seven hundred dollars.* Intended for such purposes as finding harbor inlets and avoiding collisions in dense fog, the “display” of this unit is an audible tone. The frequency of the tone increases as the distance to the reflecting object decreases. Radars for use in automobiles have attracted a great deal of interest among microwave firms, since they represent a potential market of lo7 units per year. For automobile use, stringent conditions of low cost and high reliability must be met, and, in fact, units have yet to be developed that meet these requirements and can cost under fifty dollars to manufacture. The environment in which such units must work reliably is the following: temperature range, 40°F + 175°F (250°F if under hood); vibration, 5 G vibration amplitude, 10-55 Hz; shock, 30 G for 10 msec. Automobile manufacturers such as General Motors and Toyota, and systems manufacturers such as Bendix and RCA, have been working on various aspects of this problem, as has the Department of Transportation’s Transportation Systems Center in Cambridge. Frequencies from X band up to 36 GHz are being investigated, with one of the most interesting systems being developed by RCA. The RCA unit requires installation of a passive nonlinear reflecting element, such as a varactor diode, at the rear of each vehicle. The radar receiver then is made responsive only to the second harmonic of the transmitted signal, and is thus unaffected by spurious signals caused by trees or guard-rails that may appear within the transmitted beam when the car is turning a curve. Limited function auto radars may also be developed for use in indicating clear freeway lanes for passing obstacles when reversing, and for indicating speed by using doppler reflections from the road surface. 2. Radars for Position Sensing and Coritrol

High power pulsed sources-TRAPATT and LSA diodes, and eventually transistors-may be used in air traffic control (ATC) radar beacons, in distance measuring equipment (DME) transmitters, and in transponders in ships and airplanes. These devices are used for both identification and location. Currently, the national ATC and DME systems work around 1090 MHz; ATC units produce 1 kW peak transmitter power with a 1% duty cycle and DME units produce 500 W at 0.1% duty cycle, with 30 dB

*

”

WhistlerTMRadar,” manufactured by Kimball Products Co., Sudbury, Massachusetts.

IMPACT OF SOLID STATE MICROWAVE DEVICES

175

gain as amplifiers. These performance levels are only slightly beyond those currently achievable with LSA, TRAPATT, and transistor devices. Transponders effectively increase the sensitivity of location radars; when they receive a radar signal, these units reply, implicitly indicating location and explicitly identifying themselves. The Transportation Systems Center of the United States Department of Transportation (private communication) is investigating the use of microwave radars to sense both the speed and location of trains approaching grade crossings, and of microwave links to transmit this information to the crossing barrier. There are almost two hundred thousand unprotected railway crossings in the United States, and each year more than one thousand people die in grade crossing accidents. The Department of Transportation sees a need for roughly one thousand sensor and telemetry systems per year. Microwaves can also be used for various types of process control, especially in systems where microwave radiation is sensitive to the properties of a finished product. For example, the thickness of a latex backing applied to carpets can be monitored using microwaves. Simple radars are now in use in a midwest brewery to indicate when beer bottles are filled, at a southern hospital to automatically open operating-room doors, and at an amusement park in Florida, to tell plastic crocodiles when to roar. Vehicle location will be facilitated using microwaves. Buses, service trucks, police cars, or cars equipped with mobile telephones could be equipped with coded low-power transmitters, sensed by simple receivers on buildings or utility poles, that in turn relay location information to base by wire. Alternatively, vehicles may be equipped with passive identifiers which respond to microwave interrogation in a unique way, perhaps by reflecting a harmonic of the transmitted frequency.

3. Intrusion and Theft Alarms Radar methods that are used to detect objects in undesirable places-as, for example, a stationary car in front of a moving one-can naturally be used without modification as intrusion alarms, detecting unwanted animate objects in places where none are supposed to be. In fact, the requirements on such alarms are much less stringent than on collision avoidance radars, and a number of simple cw doppler systems are already commercially available, one for under three hundred dollars. The FCC has already set aside a special “mini” spectrum slot at 10.525 GHz for such systems in the United States, and several thousand are already in use in the United Kingdom. A related application is in shoplifting detection. In one system, transmitters operating at 915 MHz and receivers operating at 1830 MHz are positioned at a store exit. If the passive junction-diode multipliers that are

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JEFFREY FREY AND RAYMOND BOWERS

embedded in “inventory control” tags on display items are taken through the system, they re-radiate the 1830 MHz harmonic of the 915 MHz signal to trigger a shoplifting alarm. 4. Telemetry An automated meter-reading system that uses a microwave transmitterreceiver and magnetic recording units, mounted in a slow moving van, to interrogate small transponders on each house to obtain and record meter readings, has been tested.* Some argue, however, that broadcast meter reading is a wasteful use of the spectrum and claim that more rational alternatives have been put forward by cable TV companies and the common carriers. Biotelemetry is also possible, since changes in physiologically significant parameters of the circulatory and respiratory systems can be detected by simple microwave transmission or reflection measurements at low-power levels (Schwan and Vogelhut, 1968). It is conceivable that this use of microwaves, or even of implantable microwave monitors, will become an accepted medical technique.

5. Microwave T o y s Cheap microwave sources might also be incorporated into toys. The technology exists to provide children with a primitive radar to watch cars going by, and it takes little imagination to invent a new generation of military games that have a primitive radar component. This is a potentially troublesome area because of the implications of uncontrollable use of cheap systems with poor frequency stability and undirected beams, which might interfere with other, more serious, microwave systems. 6 . Future Technologies

Finally, many other areas of technology now in the laboratory demonstration phase may utilize microwave semiconductor devices. These uses are still further away from practical realization than those that have been discussed above. A number of future possibilities, with years considered likely before their realization, has been forecast by the Electronic Industries Association (1972). Those events relevant to the growth of microwave applications with approximate year of introduction, include: individual portable two-way communications devices carried by most Americans (1990);

*

I’

Purdax” System, Sangamo Electric Corp., Springfield, Illinois.

177

IMPACT OF SOLID STATE MICROWAVE DEVICES

computer-controlled network stock transfer system (1978); integrated financial services, with automated transfer of funds (1978); TV systems used for instruction in 10% of all schools (1977); TV networks linking campuses (1980); use of radar prosthetic devices, e.g., radar for the blind (1985).

C. Summary-Applications

of Microwave Solid State Devices

Table I11 summarizes the applications mentioned, applications which may range from the ridiculous (crocodile “ears”) to the profound (satelliteto-home TV). All are in some way important to somebody and together indicate the very many and widespread applications of microwave solid state sources. An idea of the scale of commercial applications of microwave devices is given in Fig. 11, which shows the quantity of a system likely to be sold at a certain acceptable price. [This chart was derived as follows: current figures for total markets (e.g., number of automobiles, number of boats under sixteen feet in length, etc.) were obtained. Then, the price for common electronic systems at which a significant penetration of these markets can be expected was determined.]

lo7

AUTO RADARS SATELLITE-TO HOME TV TRUCK RADARS AUTO TELEPHONES

-

ul

c

2

I06

SMALL BOATS

3

LIGHT AIRCRAFT

LL

0 LL

lo5

LARGE BOATS

w m

COMMERCIAL 8 CORPORATE AIRCRAFT COMMON CARRIER MILITARY

10

I02

lo3 lo4 PRICE PER U N I T

lo5 (DOLLARS)

lo6

10’

FIG. 1 1 . A rough estimate of the expected numbers of microwave systems in use as a function of system price.

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JEFFREY FREY A N D RAYMOND BOWERS

In some cases, social considerations modify the perception of “ acceptability” and “need” as a function of price. For example, a Department of Transportation requirement for collision avoidance radars on every car would have the effect of drastically modifying the market for such a device. Thus, great uncertainty exists in predictions of the number of systems that may come into use as a function of price. TABLE I11 APPLICATIONSOF MICROWAVE SYSTEMS

I. Common carrier communications A. mm Waveguide B. mm Wave trunks C. Local distribution (12 GHz) 11. Specialized common carrier communications A. Local distribution (12 GHz) B. mm Wave trunks 111. Dedicated communications system A. Industrial B. CATV C. Educational D. Hospitals IV. Satellite communications A. LOs in ground stations B. Satellite TWT replacements V. Radar A. Aircraft 1. Collision avoidance 2. DME 3. ATC

V. Radar (cont.) B. Automobiles and trucks 1. Collision avoidance a. Front b. Backup 2. Passive restraint devices 3. Speedometers 4. Passing (pullout) clearance C. Miscellaneous 1. Railway grade crossings 2. Intrusion alarms 3. Process control (e.g., bottle fill level) 4. Door opening 5. Toys V1. Miscellaneous A. Process control B. Anti-theft C. Remote identification D. Bio-diagnostics

V. BENEFITSAND PROBLEMS

The benefits that could result from the development of new microwave sources are manifold. Domestic and international communications networks could be improved by opening up the frequency range above 10 GHz, thus relieving congestion at lower frequencies. The new sources also could provide a potentially economic means of communication for places where “wired systems are unavailable or impractical. Microwave systems have the potential to improve the technologies associated with transportation (including traffic control and reduction of injury by collision), reduction of damage by fire, crime detection and prevention, and health care. ”

IMPACT OF SOLID STATE MICROWAVE DEVICES

179

These benefits, however, will be accompanied by many problems which need attention soon if we are to maximize the benefits and minimize the problems. Most of the projected problems ultimately result from the possibility of a massive proliferation of microwave devices, resulting in heavy demands on part of the electromagnetic spectrum and the likelihood of mutual interference. Also, if widespread use comes to pass, we shall have to devote much more attention to possible health hazards that might result from exposure to microwave beams. We have past experience and institutional arrangements for controlling microwave systems (largely military and industrial) whose numbers do not exceed a few tens of thousands and whose cost is anywhere from lo4 to lo7 dollars per system. But if unit prices fall to the range of a thousand dollars and there is widespread use in light aircraft and large private boats, the numbers could mount to hundreds of thousands or perhaps a million, as shown in Fig. 11. We are surely unprepared to deal with the numbers of units that will be in use when the unit price of microwave systems falls below one hundred dollars and the systems are widely used in automobiles and trucks and the number of units rises to 10 or 100 million. A. Impuct on the Use of the Electroinugnetic Spectrum

An understanding of the FCC’s system of block spectrum allocation is essential to an understanding of the spectrum conservation problem. In this system, each specific spectral use (as foreseen when the system was first set up) is allocated a block of the spectrum. Figure 12 shows the allocation of the spectrum between 3 and 10 GHz, with bands reserved for government use (with some sharing by amateurs) cross-hatched. Blocks are allocated for the use of radar, in radio location, navigation, altimeter, and meteorological applications; and for communications, in the form of common carrier links, private industrial links, TV relays, etc. Some of the newer applications of microwaves discussed in Section IV (such as telemetry) do not fit into the fixed block classifications, and have been located by the FCC in the general purpose ‘‘ industrial, scientific, and medical” band from 890-942 MHz. In all of the bands mentioned, and in all of the uses summarized,* microwave solid state devices will be applied, with a resulting drop in systems costs compared to the previous generation of devices. In a study published in 1965, the Joint Technical Advisory Committee of the Institute of Electrical and Electronics Engineers noted that “the spectrum space above 10 Gc/s is unique at this point in history, in that there are * W i t h the probable exception of weather radar, which requires very large peak pulse power.

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JEFFREY FREY A N D RAYMOND BOWERS

FIG. 12. Microwave portion of the frequency spectrum, between 3 and 10 GHz, as divided into blocks for specific uses by the FCC. Cross-hatched blocks are reserved for government use (e.g., military, NASA) with some amateur sharing. Partially cross-hatched blocks are shared by government and civilian users.

relatively few services implanted in the band.. . . ” Today, however, that situation is changing: new common carrier land transmitters are assigned to the band 11.7-12.2 GHz; community antenna relays are at 12.7-12.95 GHz; and most satellite-to-earth links may well be above 12 GHz. These services, with a potential for great growth, are being forced into the band above 10 GHz by shortage of spectrum space, not because it is technologically advantageous. While the spectrum in this country is allocated for specific uses up to 90 GHz-and recent proposals would extend block allocations to 300 GHz-not all of this spectrum is in use. One of the problems in dealing with spectrum use is that “virtually none of the statistics explicitly required in the examination of the spectrum usage is available in the existing data base maintained by the FCC ” (Joint Technical Advisory Committee, 1968a), so that information about use must be inferred from a variety of other sources, such as number of applications, number of stations authorized, etc. From the data that are available, however, one can conclude that the spectrum below 10 GHz is filling rapidly to the point where future growth of microwave services may be affected. A measure of the growth of microwave common carrier transmission systems, for example, is shown in Fig. 13. Before 1969 the predominant microwave common carriers were AT & T and Western Union. In August of that year, however, the FCC decided that in the future other carriers would be allowed to compete with those already established to provide wideband

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microwave links, for data communications, or for other purposes. Since that decision, the FCC has received some two thousand applications for new microwave stations for these uses. At some point, the spectrum below 10 GHz, as currently allocated, will become completely saturated at all locations; and it would have become so even had solid state microwave devices not come on the scene. Spectrum congestion is a local problem; congestion is already severe in some areas, and in some cases particular bands are saturated. Congested areas include such diverse places as New York City, Houston, Fargo (North Dakota), and Venice (Louisiana). Figure 14 illustrates the fact that the nature of the locality determines the type of congestion ;the commercial and entertainment center, New York, is afflicted with complete saturation in the 3.7-4.2 GHz common carrier band, and severe congestion in the other two common carrier bands below 12 GHz, while the safety and special service bands are nearly full in the offshore oil drilling region of Venice (Communications and Systems, Inc., 1970). The FCC’s block allocations system does not allow the transfer of common-carrier spectrum to safety and special services, or vice versa. Of course some flexibility is allowed in interpretation of spectrum “saturation” because microwave beams can be aimed so that saturation can occur in one small area, at one frequency, but may not exist in another location nearby at the same frequency. Congestion does occur,

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1.9 GHZ INDUSTRIAL

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however, since microwave beams tend to converge, at mountain passes, in city centers, etc. Due to congestion of this part of the spectrum, there is going to be widespread use of the spectrum above 10 GHz when economic and reliable systems are available. Indeed, if the very large number of systems that are implicit in the potential application of microwave techniques are to be accommodated, the only logical place for these services is in the frequency range above 10 GHz. Since solid state devices already span a range of frequencies up to 100 GHz, it is tempting to assume that the new increase by a factor of 10 in the microwave frequency range available should accommodate all expected uses without difficulty. Such an optimistic assumption may not be justified. Congestion is almost inevitable given the block system of allocation illustrated in Fig. 12 and the administrative system set up to manage it. Specific FCC bureaus are charged with looking after their own blocks only. The Common Carrier Bureau, therefore, has little control over parts of the spectrum outside the common carrier bands. Further, the FCC does not regulate, and therefore does not monitor or compile data on, a substantial collection of bands amounting to almost half of the spectrum between 3 and 10 GHz; this spectrum is reserved for government (military, NASA, etc.) use, and is managed by the Office of Telecommunications Policy in the executive branch. As a result of the division of management among bureaus within the FCC and among separate offices of the government, it is possible for a carrier seeking a station license to be denied one on the grounds of spectrum congestion in the bands allocated to the service for which a license

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is sought-even though a nearby spectrum, in a band under control of another agency, is unused. While the rigid application of the block allocation system has worked well in the past, a new or modified approach may soon be necessary.

B. Minimization of Spectral Congestion Four activities might facilitate preparation for the proliferation of microwave systems for communications, guidance, and other uses: (1) Calculations of estimated use of spectrum should be made. (2) An adequate data base for such calculations, and for correlating such calculations with the real situation, should be established and made accessible. (3) Consideration should be given to new frequency assignment principles and procedures. (4) Technological means for conserving spectrum should be explored. It would be desirable to attempt model calculations for communities of varying population densities equipped with all of the foreseeable microwave systems of the future, including fixed and mobile communications systems, car radars, etc. The bandwidth requirements of each application and the narrow beaming possible at high microwave frequencies should be considered in order to assess the magnitude of the expected congestion. In the absence of such calculations, prudence based on past experience would surely lead us to assume that congestion is likely to arise, and in the future spectrum allocation must take into account the potential for such congestion. One of the requirements for minimizing spectrum congestion is complete information on spectrum use: a computerized data base containing information on location, frequency, radiated power, and power contour for each transmitter, both operating and proposed. Only those organizations that are financially involved in the problems, such as the Western Electric Company and MCI, now compile this information; their data usually covers only the common carrier bands, and is not normally regarded by the possessors as public information. A more comprehensive data system might well be developed and maintained by an appropriate government agency. It should be noted that lack of an adequate data base must result in application of excessive safety margins on spectrum overlap, so that the spectrum may not be optimally utilized. The principle of block allocation of frequencies in the microwave spectrum should be reconsidered. In addition, performance requirements, differing for different uses in different parts of the spectrum and applicable to both transmitting and receiving units, must be established. Simple block

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allocation is excessively rigid, as was seen in the cases of New York City and Venice, Louisiana, and flexibility should be expressly built into the system. Account should be taken of beam directionality and polarization, and multiple use of the same frequency might well be allowed, even in the same area. Some standard unit of utilization of spectrum space, such as the PODAF (power density exceeding a specified level over an area within an assigned frequency band) (Joint Technical Advisory Committee, 1968b) might be used in allocating spectrum, rather than just field intensity and contour. Special bands analogous to the Citizens Band in the lower frequency region might be set aside for very low power uses, which will be hard to regulate, with certain general performance criteria enforced for these systems. In order to conserve spectrum, more engineering attention should be given to antenna design. Maximum antenna beamwidths allowed by the FCC range from 12” at frequencies below 6 GHz to 3” at frequencies above 10 GHz. Beamwidth and radiated power of a transmitter are interrelated; the smaller the beamwidth, the larger the power density impinging upon a receiver, for a given radiated power. Interference could be reduced were beamwidths reduced, and techniques for the design of inexpensive narrowbeam antennas, perhaps fiber glass paraboloids or molded dielectric units, may not have kept up with advances in the development of microwave sources. There is no standard at all for receiving antennas, so that it is possible for a user with a poor receiving antenna to block the assignment of spectrum space to another user on the basis of “harmful interference,” because the original user’s antenna has poor side or back lobe selectivity. A receiving antenna standard might be combined with a receiver selectivity standard to minimize adjacent-channel, as well as spatial, overlap. Active antennas, employing some amplification on the receiving end, might be developed in order to allow use of lower transmitter powers. Above 10 GHz, frequencies for specific applications should be chosen, wherever possible, to take advantage of natural attenuation by the atmosphere at certain frequencies. Optimum modulation techniques, in which a minimum amount of bandwidth is used to transmit a maximum amount of information, should be required. The determination of such optimum modulation techniques is, however, a subtle problem. Model calculations can be of use in the determination of optimum bandwidth for a communication system as well as merely in predicting expected interference. One such calculation has already been done (Ruthroff and Tillotson, 1969). The model involves a closely spaced group of transmitters, half of which operate on each of two permitted frequencies. Considering effects of antenna discrimination, dense packing of transmitters, and modulation methods, one result of the calculation is shown in the curve of Fig. 15, which gives a measure of the communication capacity

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of the system as a function of the ratio of spectrum width B to highest modulating frequency f,. The curve is plotted for an antenna response approaching the theoretical limit in directionality, but the results are similar in nature for real antennas. The conclusion in this case is that, if maximum information flow through a dense network is the criterion of efficiency of spectrum use, minimum spectrum ‘‘use’’ is the best solution to the spectrum crowding problem. Additional factors, such as regulatory politics and the future expansion of common carrier transmission into microwave or optical waveguide, are also important in selecting modulation methods. For example, the Bell System plans to transmit video telephone signals over its digital long-haul network ; the analog-to-digital conversion expands the bandwidth required for the transmission. Whether this is the most efficient solution to this transmission problem depends on a complex mixture of technical and nontechnical factors, discussion of which is beyond the scope of this review. Finally, there are some who feel that in order to conserve spectrum the use of microwaves in toys and in systems for which wired transmission can also do the job should be prohibited. The argument is that, since the electromagnetic spectrum in free space is a saturatable resource, it should not be used for trivial purposes or when alternative wired systems are available. It seems, however, to be wholly impractical ab initio to prevent the development of such devices, and indeed such prevention involves a restraint on the use of the spectrum that impinges on the rights of some developers. It seems much more realistic to assume that such systems will be developed and to allocate them frequency ranges well separated from more sensitive and vital functions, such as common carrier communications, data transmission, and safety and special services, thereby preventing serious or even disastrous

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interference. Of course, the time might well come when the frequency spectrum is so heavily congested that an embargo becomes necessary on all new systems transmitting microwaves through free space, if the function could be performed via cables or wave guides.

C. Biophysical Hazards of Microwave Radiation Since it is reasonable to expect that large numbers of microwave systems will be in the hands of private individuals and therefore will be only loosely supervised, the need for exploring the biological effects of microwave radiation is urgent. Standards that were established at a time when microwave systems were fairly uncommon and when the average person was unlikely to be irradiated by a microwave beam may be inadequate when microwave beams are emitted from many automobiles, traffic signals, and utility poles. A measure of the magnitude of the problem can be obtained by considering automobiles with radar. A collision-avoidance radar on an automobile might have an average power output of 50 mW; if this power was transmitted within a beam angle of 2", the power density at a distance of 5 m from the vehicle would be more than 100 pW/cm2 (neglecting near-field antenna effects). It is unlikely that anyone would be irradiated by a single beam for any appreciable length of time, but he could be exposed to beams from many vehicles. There is no doubt that microwave radiation can be harmful to living organisms, but there is considerable controversy over the levels of irradiation required to produce significant effects, over the permanence of the effects, and over the physiological events that cause them.

1. Factors Afecting Microwave Absorption

Whether or not microwave radiation has a significant effect on a human is determined both by the magnitude of the radiation incident upon the subject and by the percentage of that radiation that is absorbed. The absorption by an object depends on the wavelength of the radiation relative to the size of the object, the object's dielectric properties, and localized diffraction and refraction effects related to the shape of the object. When the wavelength is small compared to the irradiated object, frequency affects absorption because the dielectric constant of tissue varies with frequency. The impedance presented to the radiation and the number of wavelengths within each type of material also vary with frequency. The frequency variation of the dielectric constant and conductivity of skin, fat, and muscle is shown in Fig. 16(Schwan and Li, 1956a,b). Using the data of this figure the attenua-

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SKIN MUSCLE FAT

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FIG. 16. Relative dielectric constant, and conductivity, of human tissue.

tion of microwaves in human tissue is calculated to be about 3 Np/cm at 10 GHz, and 1 Np/cm at 3 GHz for skin, and 0.5 Np/cm at 10 GHz, 0.2 Np/cm at 3 GHz for fat (England, 1950). The expected absorption by a person, allowing for varying amounts of muscle, fat, skin, and clothing can be calculated approximately using a one-dimensional model constituting layers of imperfect dielectrics with the properties of Fig. 16. Because there can be large variations in all of the important parameters mentioned, the only general statement about absorption one can safely make is the following: clothing, and the spaces between it and the skin, can affect the absorption of microwave radiation by introducing more impedance-discontinuity planes between the radiation and the irradiated object, and thus enhancing reflection. Clothing can reduce absorption to 50% or less of the incident radiation (Marha et al., 1971).Of course, several important parts of the body often are not covered by clothing.

2. Thermal Effects of Microwave Radiation If the absorbed microwave energy is transformed into kinetic energy of the molecules of the absorbing tissue, as opposed to, say, potential energy represented by an altered internal electric or chemical balance, then the effect of the radiation is thermal. If the capacity of the body to remove the heat generated by localized radiation is insufficient, a temperature rise in the area of the radiation occurs; otherwise, a generalized temperature rise results.

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Two parts of the body are particularly sensitive to local temperature increase: the lens of the eye, which has very small heat transfer capability through blood circulation with the rest of the body, and the testes, which must be maintained at a temperature lower than normal body temperature in order to avoid degeneration. Cw radiation at 100 mW/cm2 at 2.54 GHz for 1 hr is sufficient to produce cataracts in rabbits (Carpenter et al., 1960); but cataracts can also be produced by pulsed microwaves with average power levels lower than those required to produce cataracts by cw radiation (Michaelson, 1972). On the basis of experiments with dogs, a power level of 5 mW/cm2 has been suggested as a threshold for testicular effects in man (Mumford, 1961). A case is in the literature of a radar technician who performed maintenance on an operating high-power unit and was rendered infertile (Rosenthal and Beering, 1968). If the rate of exchange of heat from the body as a whole to its environment is insufficient, or if total irradiation at a high level occurs, the temperature of the irradiated subject can rise to lethal levels. Many animals have been killed in pursuit of radiation tolerance data; a typical result is that 28 mW/cm2 at 24 GHz can kill a rat in a little over 2 hr. (Mumford, 1961). Admittedly, rats and other fur-bearing animals transfer heat to their environment less efficiently than man, but the microwave absorption by such creatures is also affected by their fur. Since neither modifying phenomenon has been adequately quantified, the use of the data for rats to project standards for man may be an unreliable procedure. In summary, the major noticeable physiological effects of microwaves at cw power densities of the order of 1-10 mW/cm2 and greater are thermal, with the greatest danger being to the eyes and testes. The effect of pulsed radiation, particularly on the eye, is not so clear, and it seems advisable to limit the exposure of one’s eyes to such radiation, even if the average power density is less than the figure given above.

3. Athermal Effects of Microwave Radiation

Athermal effects are here classified as those not directly attributable to heating. The potential hazard of athermal effects to humans is much less certain than that of thermal effects, and substantial controversy has arisen over the experimental data on athermal effects, and its interpretation. In nonhuman laboratory specimens, pronounced effects, which may well be athermal, can be caused by microwaves at relatively low power levels. The metabolic activity of embryonic chick hearts has been shown to be disturbed by 75 mW/cm2 at 24 GHz (since the cardiac rate did not change in these experiments, the effect was felt to be athermal) (Paff et al., 1962);the

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development of insect pupae can be adversely affected by very low power microwaves at 10 GHz (Carpenter and Livstone, 1971); radiation at very low levels is so selectively “lethal” to plants and seeds, even with relatively short exposure times, that it has been suggested as a method of weed control (Davis et al., 1971); rats have been shown to be able to detect the presence of 2.45 GHz radiation at doses much lower than those required to produce any measurable (to 0.05’C) temperature rise (King et al., 1971). In humans, some subjects can “ hear” pulse-modulated radiation at average power densities below 2 mW/cm2 (Frey, 1962).Soviet publications contain evidence of disturbed heart rhythm, hypertension, decreased sensitivity of sense organs, changes in blood pressure, and changes in the brain’s electrical rhythm (Roth, 1968).These findings, however, have been presented with insufficient data to allow the experiments to be definitively reproduced, and the few attempts to reproduce them have failed. It is not clear, finally, whether all of the observed athermal effects on humans are reversible, whether there is an irradiation threshold for reversibility, or, indeed, whether some of the effects observed are not beneficial rather than hazardous. Various mechanisms to explain athermal effects have been proposed, but none have yet been verified (Westin, 1968). Most of the mechanisms incorporate the fact that many of the molecules and other subelements of living systems are both dipolar and not very massive, and can thus easily interact with an applied electromagnetic field, to have their physical condition (location, orientation) changed. Changes in internal potentials, and hence chemical reactivity, might ensue from this localized change. Recently, an alternative hypothesis has been formulated to explain atherma1 effects (Lebovitz, 1972). The inner ear, in which a stagnant fluid pool exists, may be subject to an abnormal thermal load even at low microwave power levels. The inner ear is an extremely sensitive system, and the thermal load might be transformed into a “significant neuronal input.” Some connection may yet be made between this ‘‘ neuronal input’’ and observable symptoms. In sum, various symptomatic effects on man in microwave radiation at densities much less than 10 mW/cm2 have been reported. Explanations for these effects are still hypothetical. Possible mechanisms for both thermal and athermal microwave effects on living organisms are summarized in Fig. 17, in which the overlap between thermal and athermal effects is demonstrated: large amplitude molecular vibration clearly represents a thermal effect that would be reflected in heating, but low-level molecular vibration, while it might raise the temperature on a microscopic scale-and even this is disputed-(Lebovitz, 1972) might also result in local changes in the electrical, and hence chemical, behavior of the molecule.

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4. Clrrrent Safety S1a)idurds The recommended maximum permissible average radiation level in the United States was set over a decade ago at 10 mW/cm2 for an 8-hr exposure period, or 25 mW/cm2 for 10 min out of each hour over an 8-hr period. The extensive Tri-Service research program that recommended this standard dealt largely with high radiation levels, and sought out thermal effects observable immediately, or very shortly after irradiation (Michaelson, 1971). Indeed, the United States standard is of the order of the amount of heat the human body can transfer to its environment under normal circumstances (Michaelson, 1972). Consequently, some workers feel that the TriService research is “largely irrelevant” (Frey, 1971) to the subject of low-power microwave radiation hazards. Taking the possibility of such hazards into account the USSR has set maximum exposure limits far below the American levels: the Soviet limits are 10 pW/cm2 for 6 hr daily, 100 pW/cm2 for 2 hr daily or 1 mW/cm2 for 15 min daily. 5. Conclusions on Biophysical Hazards

It should be clear from the discussion above that the effects of microwave radiation on biological systems are still poorly understood. Plainly it is necessary to do much more research in this area (emphasizing low-power

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effects) and to reexamine safety standards before microwave devices proliferate. The work should be concerned with both human and nonhuman biological systems. If this research is not done, public controversy will surely develop once the devices proliferate, just as controversy has arisen over low-level radiation emitted from nuclear reactors. In the case of microwaves it is still possible to investigate the low-level effects before massive deployment of systems. OF PRIVACY AND INTERCEPTION OF VI. INVASION DATATRANSMISSION

Many people are concerned with the possibility of new developments in electronics being used for the invasion of privacy. Also, as more information is transmitted through microwave beams, industrial organizations and banks may also be concerned about the possibility of interception of transmitted data. The problem is complex and one must consider not only the communication technique used but also the interplay between communications systems and computer data banks. The new microwave solid state sources do not appear to the authors of this review to present a special new problem in the privacy area in the sense of adding a new dimension to the privacy problem. In some respects, the new microwave systems seem to have some advantages over telephone lines with respect to the maintenance of privacy. First of all, one must gain access to the beam, whose position may not be physically apparent. Second, moderately secure transmission seems somewhat easier to achieve in the microwave range than in the standard telephone system. One technique uses double frequency transmission : the first frequency carries the coded message and the second frequency transmits the code. Anybody trying to intercept the information will have to find both frequencies, and the signal which transmits the code can occupy an exceedingly narrow band. Such techniques can also be used on telephone wires, but we believe they can be accomplished more simply with microwave links. Obviously, it is almost impossible to prevent somebody who is really determined to do so from intercepting information. Our concern has been with how to make interception difficult enough to discourage frequent or casual interception. If large amounts of information, whether thought to be sensitive or not, are to be transmitted by microwave systems, it should be normal practice for them to be coded at least in some simple fashion so that access to this information would require the possession of simple " key." Since such coding is not hard to do, its implementation as standard procedure would seem prudent.

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Our conclusions with respect to privacy are tentative, but the issue must be brought before the technical community in order to stimulate public discussion. The ethics of information interception is not at issue here; but the citizen should be informed of what techniques are possible so that he can decide for himself what steps to take if he wishes to ensure privacy.

VII. CONCLUSIONS We have discussed some of the prospects, benefits, and problems that may be associated with the development of cheap solid state microwave sources. Our discussion has been limited to problems associated with use of the spectrum, potential hazards to health, and the issue of privacy. We have not attempted to assess the impact on social processes; they are likely to be substantial in those areas where wired communication systems do not exist (e.g., the Indian and Alaskan projects). Firm conclusions cannot be presented at this stage; areas where more detailed analysis is necessary have been identified. The technical community should devote some of its resources at meetings and in its publications to the continued discussion of these problems, inviting contributions from social scientists as well as from physical scientists and technologists. This is part of the public responsibility of the research and development community. ACKNOWLEDGMENTS The authors are indebted to many colleagues, in industry and in universities, for suggestions and critical comments. In particular, we are grateful to Professors L. F. Eastman, G. C. Dalman, and C. A. Lee of Cornell University for their reviews; and to Professor H. J. Carlin for his help in initiating this project. Harvie Branscomb was an invaluable research assistant.

REFERENCES H. Brooks and R. Bowers, Sci. Amer. 222, 13 (1970). R. L. Carpenter and E. M. Livstone, I E E E Trans. Microwaue Theory Tech. 19, 173 (1971). R. L. Carpenter, D. K. Biddle, and C. A. van Ummerson, I R E Trans. Med. Elecfron. 7, 152 (1960). J. E. Carroll, “ H o t Electron Microwave Generators,’’ Chapter 3. Arnold, London, 1970. J. A. Copeland, I E E E Trans. Electron. Deoices 14. 55 (1967a). J. A. Copeland, J . Appl. Pkys. 38, 3096 (1967b). T. Cosgrove and R. D. Chipp, I E E E Trans. Commun. Techno/. 16, 513 (1968). Communications and Systems, Inc., “Frequency Assignment Techniques for Microwave Systems” (prepared for Federal Communications Commission) (Contract No. RC-10090). Communications and Systems, Inc., Greenbelt, Maryland, 1970. F. S. Davis, J. R. Wayland, and M. G . Merkle, Science 173, 535 (1971). R. H. Dawson and J . Frey. 9th Workskop Compound Semicotid. Devices. 1973 (1973). B. C. De Loach, Jr., Aduan. Microwaves 2, 43 (1967).

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B. C. De Loach, Jr. and D. L. Scharfetter, I E E E Trans. Electron. Devices 17, 9 (1970). Electronic Industries Association, “Electronics 1985.” Electronic Industries Association, Washington, D.C.. 1972. T. S. England, Nature (London) 166, 480 (1950). J. Feldmann, B. Rehfeld, G. J. Rosseler, and K. H. Sakowski, I E E E Trans. Commun. Technol. 17, 475 (1969). A. H. Frey, J . Appl. Physiol. 17, 689 (1962). A. H. Frey, I E E E Trans. Microwave Theory Tech. 19, 153 (1971). W. R. Green and D. R. Melick, Proc. Bien. Cornell Elec. Eny. ConJ, 4th p. 279 (1973). J. B. Gunn, Solid State Electron. 1, 88 (1963). C. Hilsum, Proc. I R E 50, 185 (1962). E. 0. Johnson, R C A Rev. 26, 163 (1965). R. L. Johnston, B. C. De Loach, Jr., and B. G. Cohen, Bell Syst. Tech. J . 44, 369 (1965). Joint Technical Advisory Committee, .‘Spectrum Engineering-The Key to Progress,” Suppl. 3. IEEE, New York, 1968a. Joint Technical Advisory Committee, “Spectrum Engineering-The Key to Progress,’’ Suppl. 8. IEEE, New York, 1968b. Joint Technical Advisory Committee of the Institute of Electrical and Electronics Engineers (JTAC), “Radio Spectrum Utilization.” IEEE, New York, 1965. C. S. Kang and P. E. Greene, Appl. Phys. Lett. 11, 171 (1967). N. W. King, D. R. Justesen, and R. L. Clarke, Science 172, 398 (1971). J. R. Knight, D. Effer, and P. R. Evans, Solid State Electron. 8, 178 (1965). R. M. Lebovitz, “The Sensitivity of Portions of the Human Central Nervous System to ‘Sale’ Levels of Microwave Radiation,” Rep. R-983-RC. Rand Corporation, Santa Monica, California, 1972. C. A. Lee, R. L. Batdorf, W. Wiegmann, and G. Kaminsky, Int. Solid State Circuits Con& Dig. of Tech. Papers, 1965 p. 28 (1965). K. Marha, J. Musil, and H. Tuha, “Electromagnetic Fields and the Life Environment,” p. 26. San Francisco Press, San Francisco, California, 1971. S. M. Michaelson, I E E E Trans. Microwave Theory Tech. 19, 131 (1971). S. M. Michaelson, Proc. I E E E 60, 389 (1972). W. W. Mumford. Proc. IRE 49, 427 (1961). National Academy of Engineering, “A Study of Technology Assessment,” Report of the Committee on Public Engineering Policy. Committee on Science and Astronautics, US. House of Representatives, Washington, D.C., 1969. National Academy of Sciences, *‘Technology: Processes of Assessment and Choice,” Report. Committee of Science and Astronautics, US. House of Representatives, Washington, D.C., 1969. G. H. Paff, W. B. Deichman, and R. J. Boucek, Anat. Rec. 142, 264 (1962). W. T. Read, Jr., Bell Syst. Tech. J . 37, 401 (1958). B. K. Ridley and T. B. Watkins, Proc. Phys. Soc., London 78, 293 (1961). D. S. Rosenthal and S. C. Beering, J . Amer. Med. Ass. 205, 105 (1968). E. M. Roth, ed.. “Compendium of Human Responses to the Aerospace Environment,” Vol. 1, NASA Rep. NASA-CR-1205(1).NASA, Washington, D.C., 1968. J. G. Ruch, I E E E Truns. ElectranDeoic*es 19, 652 (1972). C. L. Ruthroff and L. C. Tillotson, Bell Syst. Tech. J . 49, 1727 (1969). D. L. Scharfetter, Bell Syst. Tech. J . 49, 799 (1970). H. P. Schwan and K. Li, Proc. IRE 44, 1572 (1956a). H. P. Schwan and K. Li, IRE Trans. Med. Electron. 4, 45 (1956b).

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H. P. Schwan and P. 0. Vogelhut, in “Microwave Power Engineering” (E. C. Okress, ed.), Vol. 2, p. 235. Academic Press, New York, 1968. L. C. Tillotson, Bell Syst. Tech. J. 48, 1563 (1969). I. Welber, I E E E Int. Cono. Dig. p. 136 (1970). J. B. Westin, J . Occup. Med. 10, 134 (1968). F. G. Zappert and C. A. Lee, Electron. Lett. 8, 245 (1972).

Signal and Noise Properties of Gallium Arsenide Microwave Field-Effect Transistors ROBERT A. PUCEL,* HERMANN A. HAUS,?' HERMANN STATZ*

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................................................. B. Noise Figure Derivation ..........

I. INTRODUCTION

High frequency gallium arsenide field-effect transistors (GaAs FETs) have demonstrated remarkably low noise figures and high power gains at microwave frequencies. Consequently they are excellent candidates for lownoise amplifiers and receivers for communications and radar applications. For example, single stage GaAs FET amplifiers have exhibited in the laboratory noise figures below 4 dB and gains in excess of 10 dB at 10 GHz (Liechti et al., 1972; Baechtold et al., 1973).

* Research Division, Raytheon Company, Waltham, Massachusetts. Electrical Engineering Department and the Research LabOrdtOry of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts. f'

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The basic principle of operation of the field-effect transistor was first described by Shockley (1952). He proposed the device as a new semiconductor amplifier based on majority carrier flow, rather than on bipolar flow as for the conventional transistor. As conceived by Shockley the FET is in essence a semiconductive device containing a current path, whose conductance is modulated by the application of an electric field transverse to the direction of current conduction. Figure 1 illustrates the model of the junction field-effect transistor p -type gote

Ohmic

/+c;:: Ohmic source contoct

\

p-type gote

FIG. 1. Shockley’s model of the junction field-effect transistor.

(JFET) proposed by Shockley. Shown is a slab of n-type semiconductor with an ohmic contact on either end, and two p-type junction contacts on opposite sides. When a positive potential V,, is applied between the two ohmic contacts as shown, electrons flow from the left contact, called the source electrode, to the right contact, the drain electrode. If in addition a bias voltage V,, , negative with respect to the source, is applied to the remaining control or gate electrodes (assumed connected together), the p-n junctions become reverse biased. The resulting transverse field depletes carriers from the vicinity of the junction, forming a depletion or space-charge region. As a result, the cross section of the current path, called the channel, becomes

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constricted. Since the extent of the depletion region can be controlled by the gate bias, the drain current will be modulated by the bias voltage. In principle, the modulation of the current path requires negligible power because the junction is reverse biased; therefore, the FET is an active device, capable of power gain. The conductance of the channel under the gate can also be controlled by varying the concentration of carriers, rather than the cross section. This is the principle of the metal-oxide-semiconductor FET (MOSFET). In this device, the gate junction is replaced by a metallized oxide layer over the semiconductor. MOSFETs so far have not proved suitable for microwave applications and will not be considered here. Junction FETs may use either a p-n junction for the gate electrode, as assumed by Shockley, or a Schottky barrier junction as proposed by Mead (1966). Of these two types, the Schottky barrier field-effect transistor (SBFET) has exhibited superior microwave properties, especially higher isolation between drain and gate electrodes because of the reduced electrostatic coupling. At present all GaAs FETs are Schottky barrier devices. Although silicon can also be used for microwave SBFETs, because of the higher electron mobility and saturated velocity of GaAs, GaAs FETs can operate at higher microwave frequencies. We shall treat exclusively GaAs SBFETs in our analysis. With changes in some material parameters, the treatment will hold also for silicon devices. The analysis, in principle, also applies to p-n junction structures. Microwave SBFETs are fabricated by the planar process on a thin conducting GaAs epitaxial layer which has been deposited on a semi-insulating GaAs substrate. The Schottky barrier gate electrode is formed by deposition of a metal, such as gold, on the epitaxial layer. The operation of a JFET is reminiscent of a vacuum tube wherein the source, gate, and drain electrodes are analogous to the cathode, grid, and plate electrodes, respectively. Indeed, the drain current-voltage characteristics of a JFET based on Shockley’s model and illustrated in Fig. 2 look very much like the pentode characteristics of a vacuum tube. For operation below the “knee” of the curves, the depletion region of the gate does not extend across the channel. The region above the knee, in the current “saturation” regime, corresponds to “pinch off” of the channel, that is when the two depletion regions in Fig. 1 have merged into one. Although in principle no carriers are present in the depleted channel, carrier flow is assumed to exist in an infinitesimal sheet. Basic to Shockley’s treatment of the FET is his assumption that the field distribution between the gates can be treated as a superposition of two one-dimensional fields, the longitudinal field, corresponding to charge flow between the ohmic contacts, and the transverse field, corresponding to

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Source - drain bias ( V )

FIG.2. Drain-current-voltage characteristic of Shockley’s FET model. The gate and drain bias potentials are expressed in terms of the pinch-off voltage Woo.

charge depletion in the channel. This allows a full analytic treatment of the FET. An assumption underlying this field decomposition is that the channel cross section must vary ‘‘ slowly” as one moves from the source to the drain, the so-called gradual channel approximation (GCA). The implication of this approximation is that the channel gate length L must be approximately three or more times the channel thickness dimension a. Shockley applied his gradual channel approximation only for bias operating conditions below the knee of the drain current characteristic, that is below the pinch-off point, and assumed that beyond pinch off, the current saturated. He recognized the inadequacy of the GCA beyond saturation because of the strong two-dimensional field distribution near the drain produced by the free charges on that electrode. The strong longitudinal field component generated by these charges violates a basic assumption of the gradual channel approximation. Before we embark on our exposition, it is appropriate to discuss the various modifications of Shockley’s original theory that have been proposed to account for the current saturation characteristics of the FET. Since an excellent review of this subject has been given in a previous volume of this series by Yang (1972), our discussion shall be brief. Shockley, in extending his FET analysis beyond pinch off, later divided the channel into two regions: the gradual region from the source electrode to the pinch-off point, and the depleted region from the pinch-off point to the drain electrode. In the depleted region, it was assumed that the electric field and electrostatic potential were determined by the ionized impurity charges

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in the depletion region and the free charges on the drain electrode. The joining of the electrostatic potentials for the two regions was postulated to occur at a predetermined position in the channel by a graphical procedure. Later, a more elaborate analytic matching technique was used by Prim and Shockley (1953). Recently, Wu and Sah (1967) presented a refined matching procedure, in which both potential and electrostatic fields were matched simultaneously. The boundary between the two regions was determined by the matching conditions. Since this boundary position was a function of operating biases, particularly the drain voltage, they showed that current saturation no longer occurred, as in Shockley’s model, and that a finite drain resistance could be accounted for by the motion of this boundary with drain voltage. This mechanism for explaining finite drain resistances had been proposed earlier by Reddi and Sah (1965) and by Hofstein and Warfield (1965) for MOS FETs, but had not been developed in detail. Dacey and Ross (1955)were the first to suggest that current saturation in FETs might be attributable to the decrease in mobility at high electric fields. To test this hypothesis, they introduced a field-dependent mobility in Shockley’s gradual channel approximation using Shockley’s theory (1951) and Ryder and Shockley’s (1951) experimental data for germanium. Trofimenkoff (1965) and Tarney (1966) extended Dacey and Ross’ treatment by using a better analytic approximation for the experimental mobility data. Zuleeg (1967), carrying this approximation still further, made an analysis based on a three-piece linear approximation of the velocity-field characteristic, which included velocity saturation beyond a certain field. He measured the saturation current for silicon devices as a function of temperature and found the same temperature dependence as that of the limiting drift velocity (1965). This was the first experimental evidence relating current saturation in FETs to velocity-limited flow. Velocity saturation in GaAs FETs appears to have been invoked first by Winteler and Steinemann (1967) to explain current saturation in their experimental devices. Their data suggested a limiting velocity of 10’ cm/sec for electrons. The significance of field-dependent mobility in JFETs with small channel length : thickness ratios was treated in detail first by Hauser (1967) who calculated the shape of the junction depletion region near the ends of the gate using a potential distribution characteristic of semicylindrical electrodes. A significant feature of this model, which differs from previous models, is the absence of complete channel pinch off in the current saturation regime even with no velocity saturation! Inclusion of a limiting carrier velocity greatly decreased the saturation current. The importance of velocity saturation was found to depend on a single parameter, namely E, L/Woo,where E, is a critical field denoting the onset of velocity saturation, L is the gate length,

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and Woothe pinch-off voltage. Hauser found that effects of velocity saturation became important when this parameter satisfied the condition EsL/Wo, < 1/3. Hauser’s treatment, which was based on the idealized Shockley configuration, Fig. 1, was later extended by Mo and Yanai (1970) to short-gate devices having a more practical geometry. Supporting experimental data were presented. Chiu and Ghosh (1971) also treat the shortgate geometry, and, in addition, postulate carrier accumulation in the velocity-saturated region but offer no experimental justification for this assumption. Hofstein and Warfield (1965), however, had shown earlier that the electric fields produced by mobile charges in the velocity-saturated region could be neglected in comparison to the fields originating from free charges on the drain electrode. The first attempt to analyze velocity saturation effects in GaAs was made by Turner and Wilson (1969). To account for a velocity-saturated electron flow, they postulated a finite channel opening at the drain end of the gate at the onset of drain current saturation. These workers still retained the gradual channel approximation of Shockley throughout the entire channel (with a constant mobility) but imposed the condition that the onset of velocity saturation always began precisely at the drain end of the gate; that is, the critical field for velocity saturation remained “pinned” at this point in the channel. Saturation current levels corresponding to different gate bias voltages could be accommodated by a widening or narrowing of the channel opening at the drain end. Lehovec and Zuleeg (1970) modified this model by replacing the constant mobility with the approximate field-dependent expression proposed earlier by Trofimenkoff (1965). However, although their model assumed complete velocity saturation at the end of the gate, an infinite channel field was necessary at this point at the onset of current saturation, because of the analytic form of the assumed velocity-field expression. About this same time Grebene and Ghandhi (1969) proposed a twosection model of the FET, based on a two-piece linear approximation of the velocity-field characteristic. As in the Turner-Wilson model the mobility was assumed constant below a critical field E, ,and the velocity was assumed constant above this field as shown in Fig. 3. In operation above pinch off the FET was divided into two sections as indicated in Fig. 4. In region I near the source the mobility was assumed constant and Shockley’s gradual channel approximation was applicable. In the contiguous region near the drain, region 11, the carrier velocity was assumed saturated. In this second region, a conductive channel of finite opening was postulated to account for current continuity and the finite carrier velocity as in the Turner-Wilson model. Unlike the Turner-Wilson model, however, the plane corresponding to the onset of velocity saturation was not “ pinned ” at the drain end of the gate.

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20 1

Electric field E

FIG. 3. Two-piece linear approximation of the velocity-field characteristic of a semiconductor exhibiting velocity saturation.

Rather, it was allowed to move into the channel with bias voltage variations as the field distribution demanded, its position being determined by the location at which the longitudinal channel electric field equaled the critical value E, . Because of the automatic adjustment of this plane, the model not only applies to operating conditions below the knee of the I-I/ characteristic, but also above. Lehovec and Zuleeg (1970) also had suggested this modification of the Turner-Wilson model, but at the outset had assumed that the length of the velocity-saturated region was small compared with the epitaxial channel thickness. Unfortunately, as we shall demonstrate later, this condition is rarely, if ever, fulfilled in microwave devices biased into the current-saturated regime. The small-signal properties of junction FETs were first analyzed semiquantitatively by Shockley in his original paper. His analysis, which was applied to germanium and silicon devices, was restricted to the case of constant mobility. Dacey and Ross (1955) extended Shockley's treatment to a field-dependent mobility, more representative of actual devices and coinpared the predicted performance with Shockley's case. The frequency and power handling capabilities of FETs also were considered. G

S

D

FIG.4. Two-section model of the FET used by Grebene and Ghandhi (1969) based on the velocity-field characteristic of Fig. 3.

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The first detailed quantitative treatment of the small-signal parameters was made by van der Ziel and Ero (1964). Returning to the constant mobility case of Shockley, they considered the region under the gate as an active, nonuniform transmission line. By obtaining an approximate solution of the resulting nonlinear wave equation in a power series in frequency, accurate quantitative expressions for the intrinsic two-port small-signal parameters were derived which were applicable up to moderately high frequencies. Hauser (1965) also obtained approximate solutions to the transmission line model using an iterative technique. His results, which were in agreement with van der Ziel and Ero, also encompassed the more complicated dualgate structure. Somewhat later Geurst (1965) and Paul (1967) obtained an exact solution of the van der Ziel-Ero wave equation in terms of parabolic cylindrical (Weber) functions which extended the applicability of the solutions to still higher frequencies. However, because of the complexity of these functions, a power series expansion in frequency was necessary. All of the small-signal treatments discussed so far are only valid up to the point at which the channel becomes pinched-off, that is below the “knee of the I-V characteristic. Unfortunately, FETs are usually operated in the current saturation mode. Therefore, the treatments above postulate that the small-signal parameters evaluated at the pinch-off point (just at the knee) can be applied beyond pinch off with negligible error. This extension appears to be a reasonably good approximation for the input (gate-source) and forward transfer (transconductance) parameters; however, it fails completely to predict a finite output resistance, since the drain current is assumed to remain fixed beyond the knee. As pointed out earlier, this deficiency in the model was removed about this time by others who invoked channel length modulation as the cause of finite drain resistance. The first treatment of GaAs FETs to include the effects of velocity saturation on the small-signal parameters was made by Turner and Wilson (1969) who attempted to explain the microwave performance of short-gate GaAs FETs using their model described earlier. Based on this same model, the small-signal analysis was developed in a somewhat different way by Wolf (1970) who also derived a small-signal equivalent circuit. Hower and Bechtel (1973), using a modification of Hauser’s iterative technique, developed the Turner-Wilson small-signal model in more detail. Taking parasitic contact resistances into account, they compared the predicted dc characteristics and small-signal parameters with experimental results and appeared to obtain reasonable agreement. Turner and Wilson’s small-signal model, like van der Ziel’s, is valid below the knee of the I-V characteristic only, and must be extrapolated beyond into the saturation regime ; therefore it also predicts an infinite drain resistance. ”

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The first comprehensive treatment of noise in FETs was made by van der Ziel in a series of classic papers (1962, 1963a). These analyses, which were based on the small-signal analysis of Shockley’s device published later (van der Ziel and Ero, 1964), identified the main source of noise in FETs as thermal noise in the channel. Using a Green’s function analysis for the thermal voltage fluctuation distribution along the channel, van der Ziel derived expressions for the drain current and induced gate current noise and also for the correlation between them. A constant mobility was assumed throughout the channel. Shoji (1966) extended van der Ziel’s analysis to MOS devices. Klaassen (1967), using Geurst’s (1965) wave analysis for the high frequency parameters of the FET, and van der Ziel’s treatment for the thermal noise, extended van der Ziel’s results to higher frequencies. He also showed that, unlike van der Ziel’s result, the correlation coefficient was complex, rather than imaginary. He attributed this to the influence of the high frequency gate-channel coupling on the noise of the drain current, which van der Ziel neglected. However, it appears that this correction is important only beyond the useful frequency of operation of the FET. Baechtold, in a series of papers (1971, 1972) was the first to include velocity saturation effects on the noise performance of microwave FETs. Using van der Ziel’s analytic treatment of the thermal noise, and the TurnerWilson model, Baechtold derived noise coefficients as a function of the parameter E, L/Woo mentioned earlier. These coefficients also were presented in convenient graphical form. Baechtold also included a fielddependent noise temperature based on his experimental data, to account for the “hot electrons. His analysis, unfortunately, has the same limitations as the Turner-Wilson model, upon which it is based, in that it supposes a velocity saturated region of vanishing extent. Thus it neglects any noise that may be generated by carriers traveling at their limiting velocities. The assumption that velocity saturation only occurs at the drain edge of the gate, the basis for the Turner-Wilson model, cannot be justified for operating conditions well above the knee of the I-V characteristic for microwave devices. For example, for a typical microwave device, L 1-2 x l o p 4 cm, and for a drain bias in the current saturated regime V,, 7 2 V, the average longitudinal channel field is 10 kV/cm, well above the critical field E, 3 kV/cm at which velocity saturation effects first become important. In the analysis to be developed here, we abandon the hypothesis of Turner and Wilson, and instead adopt the two-section model of Grebene and Ghandhi. We show by a correct application of this model that velocity saturation can manifest itself over most of the channel length for GaAs microwave devices having gate lengths of the order of 1 pm. Our analysis will show that this two-section model not only yields a ”

-

-

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good representation of the dc and small-signal characteristics of the GaAs FET, but the noise properties as well. The treatment of the noise for the region below velocity saturation is based on the analyses of van der Ziel, suitably modified to account for the variable boundary position between the two regions. For this region we also adopt Baechtold’s modification for the electron temperature. The noise analysis for the second, or velocitysaturated region, however, is new. It turns out that the noise generated in this region can be very important in microwave devices, and indeed, may dominate the contribution of the unsaturated (van der Ziel) region. Because the correlation between drain and gate noise is also higher, strong noise cancellation occurs, so that the minimum noise figure can still be attractively low and in accordance with experiment. Our noise treatment of the saturated velocity region in some ways follows a suggestion by Shockley et d.(1966),according to which the high field diffusion constant implies the formation of dipole layers which drift through the high field region toward the drain contact. The high field diffusion constant for GaAs has been measured by Ruch and Kino (1968) and evaluated by Fawcett and Rees (1969). Both the measurements and the calculations show a peak in the parallel diffusion constant near the saturation field, followed by a rapid drop to much lower values. The measurement of the diffusion constant is complicated by trapping effects. We obtain better agreement with noise figure measurements if we use in our expressions values which are consistent with the calculations of Fawcett and Rees. In our analysis, we neglect variations of the diffusion constant with electric field. We turn now to our analysis of the dc, small-signal, and noise characteristics of GaAs microwave FETs. 11. THEINTRINSIC FET A . Introduction

A practical microwave GaAs FET is usually fabricated by deposition or diffusion of source, gate, and drain contacts on the surface of an appropriately doped thin epitaxial n-type layer. This layer, in turn, is grown on a semi-insulating wafer by either a vapor or liquid epitaxial technique. The resistivity of the substrate is of the order of loM R-cm. On the other hand, the epitaxial layer has a resistivity of about R-cm, and a thickness ranging from 0.1 to 0.4 ,urn. Thus the substrate for all practical purposes can be considered an insulator insofar as conduction processes are concerned. A perspective sketch of a practical FET is shown in Fig. 5. This figure, in addition to illustrating the geometrical configuration, also serves to introduce appropriate nomenclature for later reference. Note that all three con-

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tacts are in the plane of the epitaxial surface because of the planar technology used. Also observe that a small but finite spacing must separate the gate contact from the source and drain electrodes. For microwave transistors, this spacing is limited to a lower value of approximately 1 pm, dictated by photolithographic restrictions, but is usually several times this value. The gate length L, which is one of the crucial dimensions in the FET, is typically of the order of 1-4 pm. Because of the finite separation of the contacts, small parasitic resistances are introduced between the edges of the gate electrode and the source and drain contacts. Added to these resistances are the interfacial resistances under the source and drain electrodes. These parasitic resistances, although important, do not affect the intrinsic operation of the transistor, and can be ignored in a study of the intrinsic characteristics. They affect strongly the noise performance of the FET. For analytic convenience, we shall consider the idealized model of the FET shown in Fig. 6a. Later we shall show how the analysis of this model can be modified to include the effects of the parasitic resistances introduced by the source and drain contacts. The structure of Fig. 6a can be considered equivalent to one-half of the symmetric FET shown in Fig. 6b which was the configuration conceived by Shockley (1952) and analyzed by van der Ziel (1962). The symmetry plane plays the role of the nonconducting substrate. This structure is a good approximation for the conduction processes of the planar structure of Fig. 5 because of the negligible substrate conductivity; however, it is not an appropriate representation for the capacitance between the source and drain contacts of the planar structure because of the nonzero permittivity of the semi-insulating substrate. For computation of this parasitic capacitance the actual planar geometry will be used. We shall use the symmetric FET of Fig. 6b for our analysis. It will be mentioned, where necessary, what numerical factors will convert the results to the asymmetric model. Figure 6b defines the pertinent parameters based on the notation of van der Ziel (1962). To avoid unnecessary confusion, and

FIG.5 . A perspective of a practical FET structure fabricated by the planar technology.

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FIG. 6. Idealized model of the FET used in the small-signal and noise analysis. (a) Asymmetric model with semi-insulating substrate. (b) “Equivalent” symmetric model. The symmetry plane plays the role of the semi-insulating substrate.

to facilitate comparison with van der Ziel’s results, we shall follow his convention and assume positive carriers, that is a p-type conducting layer. However, since all GaAs transistors are made of n-type material, we shall use in our numerical examples saturated velocities, noise temperatures, mobilities, etc., characteristic of electrons. Van der Ziel’s analysis, as mentioned earlier, is not valid in the current saturation regime, that is when the channel at the drain end is pinched off. The present analysis, in addition to including the presaturation regime, is also valid beyond pinch off;however, we redefine the onset of pinch off to mean the condition at which the maximum longitudinal electric field in the channel first reaches the critical value E , , rather than when full channel depletion occurs, since for a finite E , full channel depletion cannot take place. Our assumption of a saturation field above which carriers travel at constant velocity, as illustrated in Fig. 3, disregards a region of negative mobility in the actual velocity-field characteristic. Although this negative mobility is responsible for domain formation and rf oscillations in Gunn diodes, only limited evidence of such instabilities in GaAs FETs has been demonstrated

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(Winteler and Steinemann, 1967; Zuleeg, 1968, 1969) and only for doping levels at least an order of magnitude lower than that used for practical FETs. Indeed the computer solutions of Himsworth (1972) point to the possibility of negative resistance effects for lightly doped epi-layers. The apparent minor role played by the negative resistance region in practical short-gate FETs suggests that rf instabilities due to this region, if they exist, occur at frequencies far above the normal frequency regime of microwave FETs, or alternatively, that domain formation is inhibited by the two-dimensional character of the field configuration within the FET (Copeland, 1968) or by the space charge of the carriers at the high injection levels present (Watson, 1969).

B. The dc Conditions We derive first the dc characteristics of the FET and obtain from these, by a perturbation analysis, relevant small-signal or ac parameters. Let x = 0 and x = L represent, respectively, the source and drain reference planes of the channel. The channel of the transistor is separated into two longitudinal sections, Fig. 4, corresponding to the two piecewise linear segments of the assumed velocity-field characteristic of Fig. 3. In region I, x < L,, Ohm's law is assumed to hold, p = po. In region 11, L 1 < x I L, velocity saturation applies, u = us. The boundary between these two regions, x = L1, is specified by the condition E = E,. The piecewise linear approximation of the velocity-field characteristic necessarily involves compromises in the choice of values used for E, ,po ,and 0,. We have selected po = 4500 cm2/V-secand E, = 2.9 kV/cm which yield a value for us = po E, = 1.3 x lo7 cm/sec. The choice for po is indeed typical of doping levels used in practical FETs, i.e., 1016-1017cm-3. The value for the saturation field E, is somewhat lower than the range 3-4 kV/cm observed and calculated for these doping levels (Ruch and Kino, 1968). Its choice was based on the requirement that the resultant velocity would not exceed the computed range by too great a margin (Ruch and Fawcett, 1970). Despite the compromises, the dc, small-signal, and noise parameters based on them agree extremely well with experiment as will be demonstrated later. Let the reference or source potential be at ground, and the gate and drain electrode potentials referred to the source be denoted by V,, and V,, ,respectively, as illustrated in Fig. 7a.* Also, let the potential of the channel, at a distance x from the source be V ( x ) and its value at the pinch-off point

* In the idealized model being considered here (no contact resistances) one may assume that the gate and drain channel potentials are equal to the applied bias potentials, i.e. V,, = V,,, Ed

=

'dd'

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x = L , be denoted by Vp, where V, = V(L,). It shall be necessary in our analysis to utilize the channel potential referred to the gate electrode. We denote this potential at point x, by W ( x ) .(See Fig. 7.) Thus

W ( x )=

Kg + 4 - V ( x ) ,

(1)

where 4 is the “built-in” potential of the gate junction. This barrier potential is typically of the order of 0.8-0.9 V for GaAs Schottky barrier junctions using metals such as gold. The values of W ( x )at the source end of the channel, W,, at the drain end, Wd , and at the joining point, W, ,are given by

W, = Kg + 4,

Kg + 4 - v,, Wd = K g + 4 - K d . W, =

(24 (2b)

(2c) Figure 7c illustrates schematically the various potential differences described by these equations. For a specified source-drain current I d (with positive carriers), the E-field is directed from left to right as indicated in Fig. 7a and the channel potential decreases monotonically to the source-drain bias potential E d as one moves toward the drain. Therefore, the depletion region between gate and source widens as the drain electrode is approached. The expressions to be developed later become simplified if we introduce the reduced “potentials”

where Woois the gate-to-channel potential required to deplete the channel of carriers given by Woo = (qN/2KEo)U2.

(4)

Here q = 1.6 x C , N is the doping density in the channel in atoms/cm3, K = 12.5 is the dielectric constant of GaAs, and t o = 8.854 x F/cm is the permittivity of free space. The dimension u is the half ~

FIG. 7. Cross-sectional diagram of the idealized FET showing various geometrical dimensions and potentials used in the analysis. (a) Bias voltages, channel dimensions, and reduced potentials. In the idealized model shown, parasitic resistances are absent, so that EE= VEBand V,, = Vdd. (b) Longitudinal electric field distribution along symmetry plane of channel. (c) Potential distribution along symmetry plane. Also shown is gate bias potential and gate-to-channel potential.

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thickness of the semiconductor between the gate electrodes. Note that Woo represents the “ pinch-off voltage ” for the Shockley model. The channel-to-gate potential W ( x ) is obtained by integration of the one-dimensional Poisson equation in the y-direction in the depletion region, for a volume charge density N . This treatment of the depletion layer potential is justified on the basis that the potential changes along the channel are gradual or, equivalently, that the longitudinal field in the channel is negligible compared to the transverse field in the depletion region (Shockley, 1952).This condition is fulfilled if the channel thickness is much smaller than the total gate length. Since there is no surface charge density at the boundaries of the conducting channel, i.e., the edges of the depletion region, the transverse electric field vanishes there. With this boundary condition, we obtain for W ( x )the expression W ( x )= Woo[1 - b(x)/aI2,

(51

where 2b(x) is the “height” or opening of the channel at point x (Fig. 7a). The drain current is given in terms of W ( x )by Ohm’s law, I , = 2b(x)Zo(dW/?x), (6) where 8 W / a x = E,(x) is the longitudinal channel field, Z is the width of the gate electrode, Fig. 5, and o = q p o N is the conductivity of the epitaxial layer. Here po is the low-field mobility and N is the carrier density. Integrating Eq. ( 6 ) from x = 0 to x = L,, using the reduced potential w ( x ) , one obtains

where j-,(S,

p)

=

p 2 - s 2 - 3 ( p 3 - s3).

(8)

Here go = 2ao

(9) is twice the sheet conductivity of the epitaxial layer of the asymmetric transistor, and L1 the, as yet, undetermined length of region I . The form of Eq. (7) was first derived by van der Ziel(l962). (For future reference, whenever go appears in an expression, this expression also applies to the asymmetric transistor provided that the factor of 2 in go is dropped.) We may determine L , by utilizing the current continuity between regions

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I and 11. In region I1 the carriers are assumed to travel at their limiting velocity u, where us = POES as indicated in Fig. 3. Thus in region I1 Id

7

= gOZEs(l - P ) =

Is(l - P )

(10) (114

(W

where (12) is the maximum possible drain current that could exist if the channel were fully undepleted and the carriers traveled at their saturated velocity. We shall call I , the saturation current. It will be used as a convenient normalizing factor for the drain current in the ensuing analysis. Equating Eqs. (7) and (ll), we obtain the relation E, L1 /Woo= f l /( 1 - p ) or, equivalently, Is

L1 where

= 90ZES

= L[fl(S? P ) / W - P)1

(13)

<

= E,L/Woo (14) is a dimensionless potential parameter which we label the saturation index. The product E, L is the voltage drop along the channel which would exist if the longitudinal channel field were constant and equal to Es . A comparison of 5 with the normalized source-drain potential V,, /Wooexpresses in a quantitative way the importance of velocity saturation. The smaller is 5, the greater is the role of velocity saturation. For microwave transistors, 5 is of the order of 0.05-0.4O, with the lower values corresponding to the upper end of the microwave range (X band). Once the reduced potentials s and p are known, the length of region I is specified and the current I , is determined. Conversely, given the gate bias or (reduced) potential s, and the drain current I d ,p is determined from Eq. (11) and hence L , is specified from Eq. (13). Assuming that the channel current is given, one may calculate the channel source-to-drain potential drop by integrating the longitudinal electric field along the channel from x = 0 to x = L. This integration is done in two parts, from x = 0 to x = L1, and from x = L1 to x = L, since the electric field is due to different sources in either region. Integration of the electric field in region I yields the potential drop

v, = -(Wp - W,) = -Wo0(p2 - s2)

(154 (15b)

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as is obvious from an application of Eqs. (1) and (2). In region I1 the longitudinal electric field is determined entirely by free charges on the drain electrode (if carrier accumulation is neglected) since the carriers travel at their saturation velocity. The potential produced by the free-charges satisfies Laplace’s equation. The most general solution of this equation valid for L1 5 x I L which vanishes at the boundary plane x = L , , and at the gate electrodes y = + a is of the form n=+m

2a

n= 0

where n is an integer. This form was first suggested by Shockley (1952) and later proposed for the saturated velocity case by Grebene and Ghandhi (1969). Following Grebene, it is sufficient for our purposes to approximate Eq. (16) by the lowest space harmonic because of the rapidly varying exponentials. Since the longitudinal electric field at x = L I is assumed to be continuous at the junctions of regions I and I1 and equal to E , , we obtain for the simplified Laplacian potential q x , y)

%

-

2a E, cos nY- sinh----4 x - LlJ . 71 2a 2a ~-

To this we must add the x-independent Poisson potential produced by the ionized impurities of the depletion region. A particular solution of Poisson’s equation is obviously the constant potential - Wo,(p2- sz) within the carb, = a ( l - p ) , and a potential within the depletion rier stream I y I I region b, I I y I I a, increasing parabolically toward the gate. The longitudinal channel drop in region I1 can be obtained from Eq. (17) by letting x = L. Adding this drop to the drop in region I, we obtain for the source-drain potential

This equation, in conjunction with Eq. (13), forms a pair which allows one to eliminate the unknown L1 and to solve for p given the applied gate and drain potentials V,, and V,, . Equation (1 la) then yields I d . These highly nonlinear equations in L1and p are most conveniently solved by computer. 1 are valid. AlterOnly the solutions satisfying the conditions 0 < s < p I natively, one may specify & and I d [which fixes p through Eq. (11)] and solve for s. As we shall show below, it is often convenient to proceed this way since pertinent small-signal and noise parameters can be conveniently displayed as functions of the normalized drain current I d / I , .

SIGNAL AND NOISE PROPERTIES OF

GaAs FETS

213

It is illuminating to display as functions of the normalized drain current, solution sets for the reduced potentials s and p or equivalently the normalized channel openings at the source and drain ends of the gate, (1 - s) and (1 - p ) , respectively. Figure 8 is a graph of (1 - s) and (1 - p ) as functions of Id/Is for various values of the ratio L/a and 5, typical of microwave transistors. In the usual operating range of current, I d / l s is of the order of 0.5 or less. Notice that although the channel opening on either end of the gate is a very strong function of the drain current, or equivalently of the gate bias, the percentage variation of the channel width from source to drain is of the order of 10-20% at most, except at very small currents. In other words, the channel boundary is nearly parallel to the symmetry plane, so that the

Source -end

10-2

2

4

6

8

lo-'

2

4

6

8 10'

Normalized drain current I d /I,

FIG. 8. Dependence of channel opening on normalized drain current for a specified source-drain voltage and saturation parameter 5. V,,/W,, = 1, 5 = 0.1.

approximation s x p is usually good at high currents, especially for short gate lengths, i.e., L/a 5 5. The values for s and p are insensitive to the value of the drain potential. On the other hand, the length of the velocity-saturated region L2 is a strong function of y d , though not of the ratio Id / I , . Figure 9 illustrates the dependence of L2 on V,, for a typical value of z d / Z s and 5. Note that the length of region I1 increases with V,, . For dimensions typical of microwave devices,

214

ROBERT A. PUCEL ET AL.

for example L = 1 pm, a = 0.3 pm, region I1 encompasses approximately 96% of the region under the channel for K d = Woo.The strong penetration of the velocity-saturated regime into the gate region, as depicted by Fig. 9 also has been demonstrated numerically by Himsworth (1972) with his twodimensional computer analysis of the FET. Although we have considered one current ratio, the results depicted in Fig. 9 are typical of other drain current ratios also. It is interesting to replot the data of Fig. 9 against the ratio L / a for Kd/Woo = 1, but with the length of region I1 normalized to the epitaxial thickness a. This is done in Fig. 10. Notice that L , / a is insensitive to the value of L / a and the saturation index l. Indeed, one may conclude that the extent of the saturated velocity regime is approximately three times the epitaxial thickness for drain biases of the order of the pinch-off voltage. C. Small-Signal Parameters

The small-signal equivalent circuit of the FET, valid up to moderately high frequencies, is shown in Fig. 11. In Fig. l l a a perspective sketch indicates the equivalent circuit parameters and the approximate region of the FET responsible for each element. Figure 1l b is the corresponding circuit representation. For the present discussion, we shall focus attention on the

Normalized source -to- drain voltage V,,/W,,

FIG.9. Relative length of velocity-saturated region as a function of the normalized drain voltage and ratio L/a. Id/13= 0.5, ( = 0.1.

SIGNAL AND NOISE PROPERTIES OF

215

GaAs FETs

fP Jk

O

2

4

6

8

k 10

Gate length -to-channel thickness ratio L / a

FIG.10. Length of velocity-saturated region L , relative to epitaxial film thickness a as a function of the geometric ratio L/a and saturation index t. The drain voltage and drain current are held fixed. V,,/Woo = 1, l d / I s= 0.5.

intrinsic portion of the FET, the section enclosed by dashed lines, and assume that the parasitic resistances R,, R d , , and R, are absent. The gain mechanism is embodied in the transconductance g,. The output or drain resistance is represented by the resistor rd . The depletion layer capacitance under the gate electrode is denoted by the source-gate capacitance C,, ,and its charging resistance in the channel by R , . The parasitic capacitances C,, and c , d represent, respectively, the fringing capacitances between the drain electrode and the gate and source contacts. In the analysis to follow we shall derive expressions for these various circuit elements in terms of the geometric and material parameters and the operating bias conditions.

1. Transconductance g, The transconductance and drain resistance are evaluated by a perturbation of the dc characteristic resulting from small changes in the applied gate and drain potentials. The transconductance is defined as the ratio of the

216

ROBERT A. PUCEL ET AL. Source

Gate

Drain

Epitaxial layer

Substrate

-

r - - - - - - - - -

1

FIG.11. Small-signal equivalent circuit of FET. (a) Perspective sketch indicating geometric region responsible for each equivalent circuit element. (b) Circuit schematic based on (a).

small change in drain current produced by a small change in gate voltage when the source-drain voltage V,, is fixed, that is

The gate potential, characterized by s, causes not only a change in the pinch-off potential p but also of the length L1.Differentiating Eqs. (13) and (18), using the relation L , L2 = L, one may solve for dp/ds, and obtain

+

SIGNAL A N D NOISE PROPERTIES OF

GaAs FETs

217

the expression g m = (Is/WOo)&(s, P, t),

where & is the nondimensional function given by

(21) In the limit L z + 0 and p = d, gm = (gOZ/L)(d- s), which is van der Ziel and Ero’s result (1964). The normalizing conductance factor is independent of the doping level. To obtain a measure of the magnitude of this factor, let Z = 5 x lo-’ cm and a = 2 x lo-’ cm-typical microwave design values. With IC = 12.5 and us lo7 cm/sec, we obtain I , /Woo x 100 mmho. The dependence of gm on the applied gate and drain potentials is embodied in the function &. We have computed fg as a function of the normalized drain current for several values of the saturation index and geometric ratio L/a typical of microwave designs. These are shown by the solid lines in Figs. 12a-c for the case V,,/Woo = 1. The transconductance function is insensitive to the ratio K d / w o o when this ratio is of the order of 0.5 or greater, as would be the case under normal operating conditions. Observe that although gm increases rapidly for large values of drain current, in the practical range corresponding to a reverse-biased gate junction, i.e., Id/ I , x 0.5 or less,& is only a mild function of current and the ratio L/a, and is of the order of 0.40.8. It is also insensitive to the saturation index for the range shown; however, one may show that when velocity saturation becomes unimportant, that is, when 5 + co,gmincreases significantly. Thus, for short-gate (microwave) GaAs FETs the most effective way to increase gm is to increase the gate width Z, since the conductance factor Is/Woovaries linearly with this dimension. The expression for gmsimplifies considerably for a short-gate device, that is when the approximations s x p , L , % 0 hold. In this case

-

I

1

gm x s woo 2P -

1 -

Id/])*

The second expression is a remarkably good approximation when l d / I , > 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 <(a/L)f,,are shown in dotted lines in Figs. 12a-c. Note that at low drain currents rd is somewhat more sensitive to Id than is gm. Not only is rd sensitive to drain current (gate bias), but it also varies strongly with drain voltage. Curiously enough, the variation with drain voltage at constant drain current is nearly linear over a wide range of drain voltages as Fig. 13 shows. In practical

._ c

e

80

U

60

z

L/a=5

40 L/a = 3

20 0

--0.2

0.4

0.6

0.8

1 .o

Normalized source-to- drain voltage Vsd /Woo

FIG.13. Normalized drain resistance as a function of normalized source-drain bias and the geometric ratio L/a. The curves are drawn for a fixed drain current. l d / l s= 0.5, rd

=

(&3/4)”t.

220

ROBERT A. PUCEL ET AL.

devices, the drain resistance is usually lower than the value computed and is a rather arbitrary function of the operating potentials because of leakage paths on the surface or through the substrate (Reiser, 1970). For short gate lengths and not too low drain current, the expression for rd simplifies to

which is independent of the gate length! Note that the linear dependence of rd on source-drain potential is evident now. Also observe the linear dependence on drain current.

3. Gate-Source Capacitance C,, The gate-source capacitance C,, is the rate of change of the free charge on the gate electrode with respect to the gate bias voltage when the drain potential is held fixed. Actually this ratio represents the sum of the sourcegate and drain-gate capacitances, but for practical microwave devices the drain-gate capacitance contribution is relatively small as experimental evidence indicates (Liechti et a/., 1972; Baechtold, 1972). Thus we define

The gate charge Q, is the integral over the gate electrode of' the field component normal to the electrode. In region I, this field corresponds to the space-charge potential W ( x ) ,Eq. (5). However, in region I1 an additional component of field is introduced by the Laplacian potential @(x, y ) , Eq. (17). It can be shown (Appendix I) that the resultant charge on both gates is given by the expression

where

f&, p ) = S ( p 3

- s3) - + ( p 4 -

s4).

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

22 1

The first term in Eq. (29) represents the contribution of region I. Performing the differentiation indicated by (28), we show in Appendix I that

c,, = 2KEOZf,(%P? 9 ) wheref,

=fCl

+fC2

(31)

+ 1.56 with

x

[ f;9 2--

P cosh (rrL2/2a)

In this expressionf,, andf,, represent the contributions of regions I and 11, respectively. The numerical additive term takes into account the fringing capacitance at either edge of the gate electrode as computed by Wasserstrom and McKenna (1970). For short gate lengths of the order of 1 pm, this fringing contribution can account for as much as 25% of the total capacitance. Equation (31) also applies to the asymmetric FET if the factor of 2 is dropped. Typical values of C, for a microwave design fall in the range 0.1-1.0 pF. The capacitance functionf, or ratherf, /(L/a) is graphically displayed in Figs. 14a-c as a function of normalized drain current and representative parameter values. Notice that the capacitance function is weakly dependent on the saturation index. For short gate lengths and gate biases not too near the pinch-off value Woo,the expression for C, becomes particularly simple. For this case, using p = s, L1 x 0, only the first term off,, remains and one obtains

=

2Ki,z[fi(

j+

1 a 1 - ld/ls

where the first term corresponds to a " parallel-plate " capacitor of thickness up. Note that C, is approximately linearly dependent on gate length, as one might expect. The simplified expression (33b) is an excellent approximation of all three sets of curves displayed in Fig. 14.

222

ROBERT A. PUCEL ET AL. 10' 8

I

I

,

E=0.2

2t

10-'

2

4

6 8 10"

2

4

6 8 10'

10'

I

10.'

, 2

6 810-'

4

(a)

-e,

4.

+

2 -

0 C U

0

U 0

I 6 810'

(b)

6 -

-"

4

Normalized droin current I d /IS

Normalized drain current I d /I,

"

2

L/O.lO

=0.05

30.2

-1

E.0.l

4t Norrnolized drain current 'Id/Is (C)

FIG.14. Normalized source-gate capacitance as a function ofnormalized drain current for several values of the saturation index 5. Kd/Wnn = 1. (a) L/u = 3, (b) L/u = 5, (c) L/u = 10.

4. Drain-Gate and Source-Drain Capacitances Cd, and

c,d

The drain-gate capacitance Cd, and source-drain capacitance C s d should be considered parasitic elements of the FET since to first order they are not intrinsic to the gain mechanism of the FET, but rather are fringing capacitances between electrodes. To evaluate them properly, it is necessary to revert to the actual planar geometry, Fig. 5, rather than the idealized symmetric transistor, Fig. 6b, used so far. The proximity of the contacts in the idealized model would yield an overestimate of these capacitances, particularly for the drain-gate capacitance.

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

22 3

A good estimate of these capacitances can be obtained by considering the electrostatic coupling between two parallel conductors on a surface of a semi-infinite dielectric, in this case the GaAs chip, since the interelectrode spacings are small compared to the chip (substrate) thickness. Figures 15a and b are the models to be used in our analysis of Cd, and Csd.

(0)

(b)

FIG. 15. Perspective sketches defining geometrical configuration and dimensions used in deriving expressions for parasitic capacitances (a) Model for drain-gate capacitance, (b) Model for source-drain capacitance

Smythe (1968) has developed an expression for interelectrode capacitances between parallel strips immersed in an infinite dielectric medium. By properly modifying his expressions to account for the air dielectric above the electrodes, one obtains an approximate expression for the capacitance applicable to both c d g and C,d of the form

where K ( k ) is the complete elliptic integral of the first kind. Equation (34), of course, applies only to the asymmetric FET. The argument k for each capacitance is given below

where L d , and Lsd are, respectively, the interelectrode spacings between the drain contact and the gate and source electrodes. In these expressions, it has been assumed that the length of the drain electrode Ld is large compared to the length of the gate electrode Ld 9 L, a good approximation, and that the length of the drain and source electrodes (assumed equal, Ls = Ld) are large compared to their spacing, Ld 9 L,d, which is also true for practical devices. Since expression (34) does not involve the FET mechanism. obviously Cd, and Csd are independent of operating potentials. For Csd, generally speaking, this is consistent with experiment; however, a weak dependence on

224

ROBERT

A. PUCEL

ET

AL.

bias conditions is observed for c d , . Experimentally, good agreement is obtained with the theoretical value of C s d , even though the shielding action of the gate electrode has been ignored. Calculated values of c d , appear to be somewhat higher than the experimental values; therefore, the analytic value for c d g should be considered an upper bound. Typical values of c d , and C s d for a 500 pm wide gate are 0.03 pF and 0.06 pF, respectively. Figures 16a and b illustrate the dependence of c d g and C s d on electrode dimensions and spacings. Note that c d , does not depend strongly on gate length and is a rather mild function of interelectrode spacing. The sourcedrain capacitance is nearly independent of interelectrode spacing for the practical range plotted, and is a weak function of electrode size.

111. THEFET WITH PARASITIC RESISTANCES One may modify the preceding dc and small-signal analyses to include the ever-present parasitic contact and bulk resistances between the edges of the gate electrode and the source and drain contacts. These are denoted, respectively, as R, and Rdrand are indicated in the schematic of the planar FET, Fig. 11. It is assumed that these resistances are the same for both the signal and the dc response. Let us consider first the effect on the dc characteristics. Since the drain current passes through both resistances, it will have the effect of dropping part of the externally applied drain-source bias, here denoted by the symbol T/dd, to the internal level & by an amount I d ( & + &). Thus we have the relation K d = b d - zd(Rf + Rdr). (36) In addition the potential drop across R, serves to “back-bias’’ the gate electrode so that we have the relationship K g = &g + I d Rf (37) where V,, is the externally applied gate-source bias. If one inserts these two expressions into Eqs. (18) and (2a), respectively, then it is possible to derive the terminal I-I/ dc characteristics of the FET once the values ofR, and Rdr are known. It is not easy to model these resistances analytically, and even if this were possible, the variability of interfacial contact resistances due to process variations would negate the usefulness of the model. Rather, it is more fruitful to obtain these resistances experimentally, by measuring the slope of the ld-I/dd characteristic near the origin as a function of gate bias and correct for the channel resistance contribution. A method for doing this has been described recently by Hower and Bechtel (1973). Typically, values of R, and R,, fall in the range from 5-20 R, depending on the gate width. Usually R d , is greater

SIGNAL AND NOISE PROPERTIES OF

E

1.4

L.2.0prn

u

k

a

I

225

GaAs FETs

1.3-

L=1.5prn

1.2-

L=l.Opm

N

3 f

U .-

3

.-C

1.1-

c

L

a a a

L=0.5t~rn

1.0-

0.9-

c .-

u

!0.8a

c

0.7C

A

r

1.o

0

I

2 .o

3.O

4.0

5.0

Drain-gate separation Ldg (prn) (a1

I

10

20

30

40

50

60

I

70 80 90 100

Source and drain length L, ( p m )

(b)

FIG. 16. Theoretical dependence of parasitic capacitances on electrode dimensions and spacings. (a) Drain-gate capacitance C,, , (b) Source-drain capacitance C,, .

226

ROBERT A. PUCEL ET AL. 110.

I

I

UNIT 70632'H

100 -

80

-

60 -

-2 v

-4 v 0

2

1

Drain voltage

Vdd

3

4

(VI

FIG. 17. Comparison between the calculated and measured drain current-voltage characteristic for an X-band GaAs FET. The dashed line represents the locus of operating bias voltages at the onset of velocity saturation. To the right of the line, velocity saturation becomes ; 0.34 p n ; Z = SO0 pm; L = 1.0 pm; R , = 6.5 Q; important. N , = 6.5 x 10l6 ~ m - ~u = R,, = 11.3 +f(V,,) Q. (--) Theory, ( 0 )experiment.

than Rf because of the wider interelectrode spacing between the gate and drain electrodes. Figure 17 demonstrates the excellent agreement obtained between calculated and measured dc characteristics for an X-band device having a 1.0 pm gate length. The parasitic resistances were determined by Hower's method, but an heuristically determined voltage dependence of Rdr was necessary as indicated in the legend of Fig. 17. The locus L2 = 0 represents the operating bias values corresponding to the onset of velocity saturation at the drain end of the channel. It is evident that in the current saturation regime, vdd > 2 V, velocity saturation occurs for all values of the applied gate and drain biases. One usually operates an FET in this region. Note the nonzero slope (finite drain resistance) predicted for the region above the knee of the characteristics. The small-signal performance is affected by R, as a negative feedback, analogous to the degeneration produced by an unbypassed cathode resistance of a vacuum tube. This feedback reduces the internal gmto a terminal value g: given by gf

=

grn/(l

+ SrnRf).

(38)

SIGNAL AND NOISE PROPERTIES OF

227

GaAs FETs

This feedback is most pronounced at low gate bias voltages, that is, when g, is high. For a well-designed microwave device, g z is within 20-30% of the internal value. The parasitic drain contact resistance can be neglected, insofar as smallsignal performance is concerned, since it is always small compared with the drain output resistance. An additional parasitic resistance, which though unimportant for dc considerations, must be included in the small-signal equivalent circuit, is the resistance of the thin gate-metallization, represented by R, in Fig. 11. This resistance, which may exhibit the skin effect, taken together with R, has a pronounced effect not only on the input resistance at high frequencies but perhaps more importantly, on the noise figure, since it is a source of thermal noise. Indeed R, and R, are the major sources of external noise in a GaAs microwave transistor, as we shall show later. Figure 18 is a comparison of the calculated and measured source-gate capacitance and transconductance for the device whose dc characteristics are displayed in Fig. 17. As can be seen, the theoretical model agrees very well with the data points.

32

I

I

I

UNIT 70632-H

LL

24 -

n

v

E

20

._ B

-

0

n

1,

16-

-0.4

8

! 6"

-

._ P

- 0.3

€ c

I 0

(3

40.2 -1

-2 Gate-source bias voltage V,

-3

-4

(vl

FIG. 18. Comparison between the theoretical and experimental values of the source-gate capacitance C,, and terminal transconductance g.: The applied drain bias Vdd is held fixed (Vdd = 3.0 V ) and the gate bias V,, is varied. N , = 6.5 x 10l6 cm-'; a = 0.34 pm; Z = 500 pm; L = 1.0 pm; R , = 6.5 a. (-) Theory, ( 0 )experiment.

228

ROBERT A. PUCEL ET AL.

IV. NOISE A. Introduction 1. The Intrinsic Noise Sources Noise in a microwave GaAs FET is produced both by sources intrinsic to the device and by thermal sources associated with the parasitic resistances. The intrinsic noise arises from two mechanisms. The first of these is the thermal or Johnson noise produced in the “ohmic” section of the channel, that is, region 1. This source has been analyzed for the JFET by van der Ziel (1962, 1963a) and for the MOSFET by Shoji (1966). The second mechanism is the diffusion noise in the velocity-saturated section of the channel, region 11. This noise mechanism, which has not been included in FET analyses before, can be the dominant one for short-gate devices as we shall show. Before we proceed with our noise analysis, it may be appropriate at this point to digress briefly to consider, in somewhat general terms, the distinction between the noise generated in an ohmic conductor and the noise produced in a conductor supporting velocity limited carrier flow. This will serve to establish the physical basis for the noise analysis. The constitutive law of conduction relating the one-dimensional current densityj to the electric field E and the density gradient anlax is given by j = oE

+ qD(dn/dx) + j ,

(39) where j , represents the noise generating mechanisms. The noise generating mechanisms at different spatial positions are uncorrelated. As shown by Shockley et al. (1966) and by van der Ziel and van Vliet (1968), the spectrum ofj, is given by ~,(X);(X’) =

4q2(Dn/A)Af6(x - x’)

(40)

where A is the cross-sectional area of the conductor, in our case the channel. This is a general result and applies even if the diffusion coefficient is a function of the electric field. The delta function appearing in this expression must be interpreted to be zero, for x # x’, and equal to l / A x when x = x’, where A x is the length of the incremental sections into which the conduction region has been partitioned for the purpose of analysis. (In the limit A x will be made to approach the differential length dx.) The source becomes very large as A x is made small. This is a natural consequence of the fact that statistical fluctuations in a system increase with the decrease of system size. In order to be consistent with the assumptions made in deriving Eq. (#), one must choose Ax to be

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

229

nonzero but small compared with the macroscopic dimensions of the system. Instead of re-deriving (40) here, we shall show that it is physically plausible in that it leads to the familiar Nyquist or Johnson noise formula for a conductor obeying Ohm’s law. We will then show how it can be used to explain the noise generating mechanism in the strongly non-ohmic case of carrier velocity saturation. For an ohmic conductor, the diffusion coefficient is independent of the applied field and is related to the low-field mobility p o by the Einstein relationship D = Do = ( k T / q ) p o ,hence the identity qZDn/kT = o

(41)

holds. This relation transforms the general law (40) into the more familiar special case j,(x);(x’) = 4(0/A)kTAfd(x - x’).

(42) We shall show now that this expression is consistent with the well-known Nyquist formula for a conductor of resistance R at thermal equilibrium. For this purpose one must restrict the analysis to processes of sufficiently high conductivity and low concentration gradient so that the diffusion term in Eq. (39) can be neglected. If the conductor is opencircuited, that is i f j = 0, then at low frequencies E = -j, 10, where j , constitutes a set of mutually uncorrelated, randomly activated spatial impulse functions extending from x to x + Ax, of “content” j , A x = (4akTAflA)’i2(Ax)’i2 (43) assigned to the frequency increment A5 The open-circuit mean square voltage is obtained as the mean square superposition of the voltages EAx produced by each of these sources. Thus

1 AX = 4RkTAf lu21 = I E A x II ~ = - 4kTAf OA 1

(44)

where R = L / o A is the resistance and L = Ax is the length of the conductor. This is the well-known Nyquist formula. In his theory of FET noise, van der Ziel used the Nyquist law with T assumed independent of position. When the carriers are driven by an E-field, their random velocities may increase above the values corresponding to the lattice temperature and hence their temperature T is greater than the lattice temperature. To the extent that the carriers have an (electric-field produced) average velocity small compared with their random velocities, their statistical properties may still be assumed to be those of thermal equilibrium but at the elevated carrier temperature. For this reason one may introduce into the

230

ROBERT A. PUCEL ET AL.

constitutive law a Gaussian noise source whose spectrum is derived on the basis of equilibrium considerations, the rise in carrier temperature being taken into account by treating T as a function of the local E-field. The implications of a field-dependent carrier temperature were considered briefly by Klaassen (1971) and by van der Ziel (1971a,b) for FETs in which velocity saturation was assumed absent. The first detailed quantitative treatment incorporating hot electron temperatures in a noise analysis was made by Baechtold (1971, 1972) for Si and GaAs devices. However, Baechtold did not include a second velocity-saturated region in his model. The treatment of the noise in region I1 is completely different from the above. Here the carriers drift at their saturated velocity u s , and Ohm’s law no longer holds. Yet, interestingly enough, the noise sources may still be obtained from quasi-equilibrium considerations as we shall argue below. The basis for our discussion, however, is no longer the specialized form, Eq. (42), but the general equation (40). When the electric field has increased the carrier energy so that an appreciable number of carriers have energies comparable to the optical phonon energy, an increase in the electric field will cause energy transfer from the carriers to the lattice without a change in the carrier drift velocity. This, of course, is the phenomenon of velocity saturation. Even under this condition the rapid randomization of the direction of carrier velocity is sufficient to maintain a roughly spherical velocity distribution in velocity space. To the extent that a roughly spherically symmetric velocity distribution (in a frame of reference moving at the carrier drift velocity) implies a random walk process, the noise sourcej,, described by Eq. (40),associated with the velocity-saturated carrier flow, can be attributed to charge displacements produced by the random motion of the carriers. This, of course, is the process responsible for diffusion. Hence, the carriers still experience diffusion, even though their response to a change in the electric field has been eliminated for the most part. According to Eq. (40) the current spectrum is white. Hence, at any position x within the spatial increment Ax, the microscopic currents in(x)= jn(x)A occur in short bursts, uncorrelated from one instant of time to the next. A current burst within the spatial interval x, x + Ax, leads to a charge displacement from x to x + Ax, resulting in an electric dipole layer with one charge sheet at x, and equal and opposite charge sheet at x + Ax. When diffusion is dominant and ohmic conduction absent in a reference frame moving with the average velocity u s , the constitutive law can be used but with CT = 0. This indicates that a displacement of charge, with an attendant gradient of carrier density, produces a diffusion current component. Since a dipole layer is highly singular, one might expect that the associated diffusion current qD(dn/dx)would in time greatly alter the carrier distribu-

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

23 1

tion in the dipole layer. This is indeed the case; however, in a low frequency noise analysis only the lowest spatial Fourier components of the drifting dipole layer density contribute to the noise. These Fourier components may be shown to experience negligible diffusion during the transit time within the drift region. For this reason one may compute the induced gate noise on the basis of dipole layers drifting undisturbed at the saturated velocity zi, provided one restricts the analysis to “low frequency” noise. This is the foundation for the authors’ treatment of the noise originating in region 11. The interpretation of high field noise in FETs as diffusion noise has been suggested by van der Ziel (1971b) also, but he did not consider the case of velocity saturation. It is convenient for noise figure calculations to represent the internal noise sources of the intrinsic FET by noise generators suitably connected to the external terminals of the intrinsic FET (now considered noiseless). One may do this for any linear two-port (Rothe and Dahlke, 1956).A particularly convenient representation which relates closely to the physical noise generating process in the FET is a pair of noise current generators, one connected across the input port (gate-source) and one connected across the output port (drain-source). These are labeled as i, and i d , respectively, in Fig. 19a. The

FIG. 19. Representation of a noisy FET by terminal noise sources. (a) Intrinsic F E T with two noise sources. The current source i, represents the induced gate noise, the output current source I,, the drain circuit noise. (b) The intrinsic FET with parasitic resistances. The voltage generator em represents the thermal noise of the gate contact resistance R , , the voltage generator e , , the thermal noise of the source contact resistance R , .

232

ROBERT A. PUCEL ET AL.

direction of the current generators follows the convention adopted by van der Ziel. The representation in terms of external current generators is useful because the output generator can be identified with the short-circuit channel noise generated in the source-drain path, while the input generator can be related to the noise current induced in the gate circuit by the charge fluctuations in the drain current. Because of the coupling between the drain noise and the gate noise, the current generators are partially correlated. 2. The Extrinsic Noise Sources

A practical FET always contains additional noise sources external to the intrinsic FET which are associated with the parasitic resistive paths in series with the terminals. These sources generate Johnson noise and can be expressed in terms of the resistance values. For a microwave FET there are two principal sources of parasitic noise. These are illustrated in Fig. 19b. The first, represented by the voltage generator em arises from the thermal noise produced in the gate metallization resistance R, . Usually this contribution is not negligible for microwave FETs because the resistance of the thin films used for the gate fingers cannot be neglected. The second source, labeled e, is associated with the gate-source path resistance Rf external to the channel. (See Fig. 11.) The drain contact resistance R,, (Fig. 11) also generates thermal noise, but this source can be neglected in comparison with the other sources after amplification. When the FET is connected into a circuit, additional thermal noise is generated by the lossy circuit elements. These must be taken into account when one computes the noise figure of an FET amplifier; however, we do not associate these noise sources with the FET. 3. A Synopsis of the Noise Analysis

It will be our principal objective in the present section to analyze the various internal noise mechanisms contributing to the current generators and to obtain analytic expressions for this noise. Specifically,we shall obtain analytic representations for the mean square values I i,” 1, I id’ I where the overbar indicates a statistical average, and for the correlation coefficientj C defined by the relation ~~

~

~~

1 I I id” I ) ‘ I 2

j C = ig*id/( ig

(45)

where the asterisk denotes the complex conjugate and j = ( - l)”*. Our analysis is restricted to frequencies below which transit-time effects in the channel play a major role (a reasonable assumption for present microwave FETs) ; therefore, the transconductance gm and the spectrum of

SIGNAL AND NOISE PROPERTIES OF ~

233

GaAs FETs

I ii I are independent of frequency. Since i, is produced by capacitive coupling of the gate circuit to the noise s o u r - in the drain c i r c u i t , m will vary as frequency squared, i.e. as wz,and i*i as jw, where w = 2nfdenotes g P the radian frequency. Therefore, the correlatlon coefficient is imaginary and independent of frequency. The analysis of the internal noise will be somewhat complex, and to make it easier to understand, it will be useful to outline briefly the method of presentation. In the first part, Section IV,B, we will derive an expression for the open-circuit noise voltage I ui I developed across the source-drain path, and by the simple circuit transformation I ii I ri = I u; I we shall convert this voltage to the equivalent noise current generator. (It is convenient to proceed from the open-circuit voltage, because all incremental noise voltage fluctuations distributed alongthe channel are in series and can be added.) The determination of I ui I will begin with a calculation of the thermal noise generated in region I. In this analysis we shall draw heavily on the earlier work of van der Ziel (1962, 1963a). However, we will add two new features to van der Ziel's treatment to apply it to the GaAs FET. First, we include the field dependence of the noise temperature of the (hot) electrons, as suggested by Baechtold (1972).Second, we take into account the enhancement of this thermal noise as it propagates through region 11. Let the voltage fluctuation developed across region I be called u l . Then by its modulation of the position of the joining plane x = L1,u1 is amplified as it appears at the far end of the velocity-saturated region. Let the resultant open-circuit noise voltage be denoted by u d l . Proceeding to region 11, we investigate the noise mechanism under velocity-saturated conditions. Unlike region I, the noise here is attributed to spontaneous generation of dipole layers, rather than Johnson noise. We integrate this distributed noise over region I1 and obtain a mean square voltage contribution, 1 uiz I. Since this noise source is not correlated with the Johnson noise produced by region I, the total mean square drain voltage is the sum of I uil I and I & I . Thus the short-circuit drain current noise is ~

~

~

I id" I

=

( 1 ui1 I +

~ 1 4 2

I )hi.

_

(46)

In the next step in our analysis, Section IV,C, we derive expressions for the induced gate noise produced by the elementary noise voltage fluctuations in the channel. The approach used here is to assume short-circuited drain conditions, as van der Ziel does, and to interpret the resulting fluctuations of the drain current id as charge fluctuations in the channel. These charge fluctuations produce opposite charge fluctuations on the gate electrode which can be expressed in terms of the short-circuited gate current i, . We begin by calculating the induced gate charge produced by Johnson noise generated in region I. The elementary noise voltage fluctuations in this

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ROBERT A. PUCEL ET AL.

region induce charge fluctuations on the gate electrode via two coupling mechanisms, one local, involving region I only, the other remote, involving region 11. In the first coupling the noise fluctuations cause, locally, a channel width fluctuation, or equivalently a depletion layer charge fluctuation. The latter induces an equal but opposite charge fluctuation on the segment of the gate electrode in region I adjacent to the channel height fluctuation. This is the mechanism invoked by van der Ziel(1963a) in his treatment of gate noise in FETs with no velocity saturation. The indirect coupling to region I1 occurs somewhat differently, although channel height modulation is also involved. The component of the short-circuit drain current fluctuation produced by the noise sources in region I, namely i d l = o d l /rd, requires that the entire (uniform) channel height in region I1 fluctuate or “breathe” in synchronism with this current fluctuation because of the saturation of the carrier velocity. This channel height modulation, like that in region I, also requires a compensating fluctuating charge on the gate electrode segment in region 11. The two charge fluctuation components are fully correlated. If we define the direct component of induced gate charge, q1 and the indirect component q 1 2 , then the total charge qgl induced on the gate by fluctuations in region I is

+ q12

(47) We next determine the induced charge fluctuations produced by the diffusion noise of region 11. Since a dipole layer carries no net charge, it cannot couple a charge directly onto the elementary gate segment in region I1 adjacent to the dipole, unlike the direct charge induction that takes place in region I. However, indirectly the dipole can couple a charge over the entire gate length, via the short-circuit drain current fluctuations id2 = Ud2/rd produced by the dipole induced voltage drop vd2 generated in region 11. The indirect coupling is by the same channel height “breathing” mechanism discussed earlier for q 1 2 . Let qg2 denote the total induced gate charge produced by the dipoles. Since this charge fluctuation is uncorrelated with qgl, the mean square gate charge fluctuation is given by qg1 =

~~~

2 Iqg

I

411

= Iqi1

I+

Id2 ~

The short-circuit mean square gate noise above by the relation

*

I ii 1

I.

(481

can be obtained from the

~~~

(49) liil = ( igl2 I + l i i 2 I where I I = w2 I qgZl I and I I = w 2 m . Finally, in Section IV,D, using the results outlined above, we calculate the correhtion coefficientjC, Eq. (45). We now proceed to the calculation of the various noise components.

iil

ii2

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

235

B. Drain Circuit Noise

1. Open-Circuit Drain Voltage Fluctuation Produced by Sources in Region I The mean square noise voltage developed across region I by thermal fluctuations can be taken from the treatment of van der Ziel(l962). The rms voltage fluctuation at the end of region I produced by an infinitesimal section of channel of length Ax at position x is proportional to the incremental channel resistance at that point, corrected for the field-dependent electron temperature TJx). The result expressed in terms of the reduced channel potential W ( X ) is shown in Appendix I1 to be

where we have taken the limit Ax + dx. For T, we employ the expression proposed by Baechtold (1972) based on his experimental results,

T,/To = 1

+ c~(E/E,)~

( 5 la)

where T, is the effective noise temperature, To is the zero-field temperature, assumed to be 300"K, 6 is an empirical constant, E is the electric field, and E, is the saturation field. In terms of the reduced potentials,

where for E ( x ) we have used the relation

obtained from Eqs. (5), (6), and (11). Next we evaluate the open-circuit voltage fluctuation at the end of region I1 attributable to Au,. Because the drain fluctuation current is zero under the assumed open-circuit drain conditions, the width of the channel in region I1 is fixed, as determined by the direct current I, and so is the channel voltage W, at the pinch-off point. It follows that the only parameter that can vary is the position of the pinch-off point L,, that is, the length of region I . The field at x = L , by definition is equal to the saturation value E,; therefore, the change in length A L l , required to absorb the noise voltage fluctuation Aul is given by ALl = A u , / E , .

(53)

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ROBERT A. PUCEL ET AL.

The source-to-drain fluctuation Audl resulting from this position modulation can be obtained from Eq. (18) as

do,,

=

dEd/dLl

Is.

P

A L , = Aul cosh ( n L 2 / 2 a )

(54)

where we have used the relation L1 + L2 = L. The hyperbolic factor represents the “enhancement” of the Johnson noise by region 11. The mean square noise voltage I uil I is obtained by integration of JAv;, I over region I, 0 Ix I L , or, equivalently, s I W(X)I p . One obtains (Statz et al., 1974)

The quantities Po and P, are defined as Po

=

( j - 1 ) - “(p’ - s 2 ) - 4(p3 - s3)

+ f ( p 4 - s”]

(56)

and pa = 2 6 ( f l ) - ’ ( i - p ) 3

1 - P . I-s]

(57)

Equation (55) also holds for the asymmetric transistor provided the value of g o appropriate for the one-sided device is used. The portion of I uil I containing Po is identical to Eq. (13) of van der Ziel’s paper (1962). The new term proportional to Pa represents the hot electron contribution.

2. Open-Circuit Drain Voltage Fluctuation Produced by Sources in Region I1 We are dealing here with a region in which carriers are drifting at their saturated velocity. As stated earlier, the noise for this condition should properly be interpreted as diffusion noise. The authors have shown elsewhere (Statz et al., 1974) that this diffusion noise may be represented as a uniform distribution of spontaneously generated dipole layers of strength qAxo/A created at the rate r = (2DnA)/Axo,where A = 2(1 - p)aZ represents the cross-sectional area of the channel and n = N the density of carriers. As explained earlier, we may assume that these dipoles drift unchanged from their point of generation at x = x o , where L , < x o < L, to the drain electrode x = L. Since to the authors’ knowledge the analytic treatment of diffusion noise in the velocity-saturated drift region of an FET is new, a somewhat detailed analysis will be presented here.

SIGNAL AND NOISE PROPERTIES OF

237

GaAs FETs

The potential field associated with a dipole layer of strength q A x o / A situated at the plane x = x o in region I1 can be expanded in a twodimensional space harmonic series, each term of which decays exponentially in magnitude in the x-direction on either side of the plane of the dipole layer. Because of the rapid rate of decay of the higher order terms, the dipole layer potential Y ( x , y) can be approximated very well by the first term of this series, namely

where the ( + ) sign applies for x < x o , and the ( - ) sign for x > 0. This dipole potential, by itself, will upset the potential and field at the joining plane of regions I and I1 which were established by the Laplacian field O(x, y) to satisfy the boundary conditions there. Therefore a compensating potential will be induced on the open-circuited drain electrode to restore the boundary conditions. Taking into account the drifting of the dipole layer toward the drain, one may show that the compensating (timedependent) potential disturbance induced on the drain electrode by the dipole has the form

which is valid for 0 I t - to I (I. - xo)/u,. The spectral density of this pulse has frequency components extending up to and beyond the range corresponding to the inverse transit time of region 11, namely L2 / u s . Since we are not interested in frequencies this high, we need concern ourselves only with the spectral density at low frequencies, where the density is flat and corresponds to “white ” noise. Equation (59) represents the induced voltage of one of the many dipoles generated at the plane x = x o . Since the dipoles are generated at random times at this plane, they are completely uncorrelated. To obtain the mean square value of Audz, we must first evaluate the spectral density of one pulse (at low frequencies, w + 0) and multiply this by the rate of dipole generation r. Doing so one may show(Statz et al., 1974) that IAU;,

I

=

32a2 q2DNAf sin2 n4u: (ao)’b,Z

(2)([E exp

( L - xO)] -

1)

2

dxo

(60) in the differential limit. We obtain the total mean square voltage produced

238

ROBERT A. PUCEL ET AL.

by dipoles throughout region I1 by integrating (60) over the range L1 I xo IL. There results

-

4 exp-71L2 2a

+3+

a

For the asymmetric transistor, the numerical factor of 16 must be replaced by 64, provided I d is then interpreted as the channel current for the onesided transistor. Notice that I ui2 1, unlike I v,”, 1, depends explicitly on the drain current, increasing with it and is proportional to the high field diffusion coefficient. We shall see the implications of this later. Since the noise voltage contributions of the two regions are uncorrelated, their mean square values add. Thus the equivalent short-circuit current generator id across the drain-source terminals Fig. 19a consists of the components

C . Gate Circuit Noise

1. Short-circuit Gate Current Fluctuations Produced by Sources in Region I We have shown in Section IV,B that there are noise voltage fluctuations along the channel. Since the gate is capacitively coupled to the channel, there will be noise charges induced on the gate. These charges are time dependent; therefore there will be noise currents in the gate circuit. Van der Ziel has evaluated these noise currents for the nonsaturated transistor by setting the noise voltage equal to zero at the end of the ohmic region. Because of the additional velocity-saturated region the boundary conditions at the end of region I are different in the present case. When we evaluate the induced gate current, we have to add effects due to the “breathing” of the channel in region 11. Finally we have to allow for a field-dependent noise temperature. When proper account is taken of the modified boundary conditions at the end of region I, van der Ziel’s expression for the induced gate charge

SIGNAL AND NOISE PROPERTIES OF

239

GaAs FETs

Aq, produced by an elementary thermal voltage fluctuation at a point xo in region I becomes (Statz et ul., 1974) A411

=

+ yw(xO)]Audl

(2aqNZLl / r d I d ) [ - k

(63)

where Audl is the open circuit drain voltage fluctuation corresponding to the elementary voltage fluctuation at x o . (Observe that the factor Audl /rd is simply the short-circuit drain current Aidl resulting from the elementary fluctuation.) The parameters k and y are given by

k

=

( f i ) - ’ [ - 4 ( p 3 - s3)

+ +(p4

- s4)

+ (s’

- + s 3 ) ( p - s)]

+ yp (64)

y

=

woo E,L,

1 + 2p(l - p ) -11

-

-}

1 cash (71/2~)L2 ~

The parameter y represents the modification of van der Ziel’s boundary condition at the end of region I. When velocity saturation is absent, i.e., when L2 = 0 and p = d, y becomes unity and the expression for Aq, reverts to van der Ziel’s result (1963a). We remind the reader that in our case AqI1 represents the induced charge appearing on that portion of the gate electrode adjacent to region I. To this we must add an additional component Aq12 induced on the remaining segment of the gate electrode. As we mentioned earlier, this contribution originates from the “breathing” of the channel height in region I1 in response to the current fluctuation ALidl. Since the velocity of the carriers is saturated, this current fluctuation demands that the depletion layer give up (or retrieve) additional carriers with a charge per unit length of Aid, / u s .The induced charge on the gate, therefore, is of opposite sign, thus

,

where we have used the relation I d = 2aqNZv,(l - p ) . This charge is fully correlated with Aq,, since it arises from the same elementary disturbance. Adding the two induced charges, we obtain

240

ROBERT A. PUCEL ET AL.

where Aq, is the total induced gate charge caused by an elementary noise voltage fluctuation in region I. The modified parameter k is given by k =k

+ (L2/L1)(1 - p).

(681

The mean square value of this charge can be evaluated after we express Av,, as a function of position. This voltage is proportional to the thermal voltage fluctuation A W, at xo, as shown by van der Ziel(l962) and also in Appendix 11, modified by the enhancement factor cosh (nL2/2a). We find

The elementary voltage fluctuation has a mean square value expressible in terms of the channel resistance of an incremental section Axo as

where T, is the elevated carrier temperature at xo. Proceeding to the differential limit and using the identities E(xo) dxo = dW = 2WO0w dw and I , = 2ab(x0)ZE(x0),this becomes

Taking the mean square value of (67), using (69) and (70b), and integrating over the range s I w I p, one obtains I qf I . The mean square value of the corresponding induced short-circuit gate current 1 itl I is obtained by multiplication by w 2 ; thus

where Ro = (fl)-31(kf)2(p2 - s2)

4

- -k(k 3

+ YNP3 - s3)

+ j1 [(k')2 + 4ky + yz](p4 - s4) - 4 (ky + y2)(p' - s5) + Y32 (p" - s")), -

R, = 6(1 - ~ ) ~ ( f , ) - ~

p

-s

+ In- 1 - s

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

24 1

For an asymmetric transistor the numerical factor of 64 should be changed to 16 and the go appropriate for a one-sided device must be used. In the limit L1 L, the portion of I i2 I proportional to R o is identical 8? to the expression obtained by van der Ziel (1963a); furthermore, the term proportional to Rd represents the hot electron contribution obtained by Baechtold (1972). --f

2. Short-circuit Gate Current Fluctuations Produced by Sources in Region I1 The fluctuating charges induced on the gate by the noise sources in region I1 stem from the channel height modulation in regions I and I1 produced by the fluctuating channel current generated by the dipoles under short-circuit drain conditions. The charge induced by the dipoles on the gate segment adjacent to region I, which we denote as qzl is given by L1

qZ1= -2qNZ

Ab(x) dx

J0

(744

With the identity db/dld = - Udw/dld,and the relation Idx

= gOzwOO

.fl(s,

w)

obtained from Eq. (7), one may express the integral in terms of the reduced . over the range s I wI p , we obtain potential, ~ ( x )Integrating

q21

=

-

2uqNZLlk(y = 0 ) .

(75)

142 Id

The charge induced on the gate in region I1 is simply q 2 2 = -(L2/us)id2. (76) Combining q Z 1 and q Z 2 ,the total induced gate charge produced by the dipoles is

q2 = q21

_

+ q22

- 2uqNZL, k ' ( y

= 0) OdZ

(77)

'

I d rd

Taking the mean square value and multiplying by short-circuit gate current I it2 1,

L1k'(y = 0 )

d,we obtain the sin2 -

242

ROBERT A. PUCEL ET AL.

This equation also applies to the asymmetric transistor provided the factor 16 is changed to 64 and the Id and rd apply to the one-sided transistor.

D . The Correlation Coeficient Some correlation must exist between the short-circuit gate and drain noise currents since the elementary noise voltages in the channel are responsible for both. Using i, = i g l + ig2 and id = i d , + id2 in the expression for the correlation coefficient, Eq. (45) and recognizing that the pairs ( i g 1 ,i d 2 ) and (ig2, i d , ) are uncorrelated because they arise from uncorrelated sources, one may conveniently express the correlation coefficient jC as the sum of two terms, j C , and j C , ,corresponding to regions I and 11, respectively, where

with

,

Here j C , and j C 2 2 , respectively, are the "self" correlation coefficients applicable to regions I and 11. Except for the different boundary conditions and the inclusion of a field-dependent temperature, the derivation of C 1 coincides with the treatment of van der Ziel (1963a) for the nonsaturating transistor. It can be shown (Statz et al., 1974) that

,

where

so = ( j 1 ) - 2 { k " ( p 2 - s2)

- 4(p3 - s3)

+ y[ - 5 ( p 3

+

f(p4

- s3)

- s")]

+ (p" - s3) - 3(p5 - s')]}

(82)

and

For no electron heating, S, = 0. If in addition there is no velocity saturation so that y = 0 and p = d , then C , , coincides with van der Ziel's result. Turning to C 2 2 ,an inspection of Eq. (77) reveals that q2 is related to ud2, hence i d 2 , by a spatially independent factor. Thus id2 and ig2 are fully correlated, ii2id2 = j ( 1 ii2 1 1 ii2 )'I2. Hence C 2 , = 1. ~~

I

SIGNAL AND NOSE PROPERTIES OF

243

GaAs FETs

E. Noise Coeficients We shall now define a set of dimensionless noise coefficients based on the preceding analysis which are convenient for noise figure calculations. Let

and

+

R1 R2 (85b) where PI, P , and R , , R , are of the same form as P and R, respectively, but with the subscripts (1) and (2) attached to id and i, . Thus the subscripts on the P and R coefficients refer to regions I and 11. In terms of these coefficients the algebraic factor in the correlation coefficient takes the form =

Using (20) for g m , ( 2 5 ) for Yd, and (31) for Csprthe components of the P and R coefficients of the asymmetric transistor are

i2(erpTL’

-

To obtain f 3 we have used the relation Do diffusion constant.

4exp--71L2 2a

U

=

+3+ a

(88) (kTO/q)pofor the low-field

244

ROBERT A. PUCEL ET AL.

The coefficients P and R are strong functions cf the drain current because of the dependence of the dipole noise contribution on this current. Indeed at high drain currents, Id/Is> 0.1, corresponding to gate bias voltages near zero, the dipole contribution is the dominant one for L/a ratios even as high as 10. This point is illustrated in Fig. 20a where we have plotted the fraction of the total drain and gate noise contributed by region 11, that is the ratios P , / P and R2 / R , respectively. Note that for I d /I, > 0.2, over 75 % of the noise is contributed by region I1 for the long gate device, L/a = 10. This is raised to 95% for the short gate design. The predominance of dipole noise is illustrated even more vividly by the correlation coefficient. In Fig. 20b we have plotted the contribution of region I1 to the total correlation coefficient, that is, the fractional contribution of the second term j C , in Eq. (79). Note also that the total correlation coefficient, which is also graphed, is near unity. In the next section we shall make use of the P, R, and C functions in the derivation of the noise figure of the FET amplifier.

V. NOISE FIGURE A . Introduction The validity of our noise theory can be tested only by its application to a practical device for which the necessary design parameter values are known, and for which the parasitic resistances have been measured. It is not necessary to include all of the equivalent circuit parameters of the FET since some have a small effect on the noise figure. For example, for simplicity we shall neglect the (small) feedback drain-gate capacitance c d , as well as the source-drain capacitance Csd,Fig. 11. The small perturbation of the noise figure produced by these capacitances can be added later if necessary (Klaassen and Prins, 1969; Dahlke, 1955; Hartmann and Strutt, 1973). We may also neglect the small effect of the output drain resistance rd because the product gmrd<< 1 (van der Ziel, 1962). Finally, any small source-lead inductance attributable, for example, to wire bonds, will not be considered. This inductance, whose effect on the noise figure also may be added later, can actually improve (lower) the noise figure slightly (Anastassiou and Strutt, 1974). The equivalent circuit of the FET to be used in our noise figure analysis is shown in Fig. 21. This was derived from the circuit of Fig. 11 by setting Cdg, Csd, and g d = r; to zero. Notice that no noise generator is shown with the intrinsic depletion layer charging resistance Ri . The noise associated with Ri is imbedded in the gate noise generator ig. This has been demonstrated theoretically by van der Ziel and Ero (1964) and later verified

c

8

B

2-

g

0

e

I

Normalized drain current

Id/Is

(b)

FIG. 20. Fraction of the total intrinsic noise of the GaAs FET attributable to the diffusion noise of Region 11. Shown are gate and drain noise and the correlation coefficient as a function of the normalized drain current for several values of the geometric ratio L/a. V,,/W,, = 1, 5 = 0.1. (a) Gate and drain noise, (b) correlation coefficient.

246

ROBERT A. PUCEL ET AL.

iZS

I I I I

FIG.21. Equivalent circuit of FET used in noise analysis.

experimentally by Bruncke and van der Ziel(l966). The trans-admittance y , representing the current generator of the intrinsic transistor can be approximated by the form y , = gme-jor where oz is a phase shift produced by the transit time T of the electrons in the channel (Wolf, 1970; Liechti et al., 1972). Since we have neglected transit time effects in our small-signal and noise analysis, to be consistent we shall also neglect them here. The intrinsic transistor in its simplified form can be represented by two short-circuit Y parameters, namely Y, the input admittance, and Y,, the input-to-output transfer admittance. In terms of the equivalent circuit parameters these can be written as

,

Y11

= .bCsg/(l

Y21

= gm/('

+ jwCsgRi)

(89a)

+bCsgRi)

where, in the expression for Y2,, we have set y ,

(89b) =

g,.

B. Noise Figure Derivation The configuration shown in Fig. 21 with the source-terminal common to input and output is often called the grounded-source or common-source connection. Although we shall derive the expression for the noise figure for this circuit, our results will also apply with negligible error to the commongate and common-drain configurations (van der Ziel, 1969; Kasser, 1972).

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

247

The noise figure F can be expressed in the form F = l +

I imO + ifo + ig0 + i d 0 l2 I is0 l2 ~~

where imO,ifo, i g 0 , i d o , and iso are the noise current components in the short-circuited drain-source path produced by the noise generators em,el, i, , i d , and e, , respectively. The noise generators representing the extrinsic thermal sources are given by their mean square values, I e i 1 = 4kT0R, AJ 1 e: I = 4kT0R, AJ and I e: I = 4kT0 R, Af; where R, denotes the real part of Z , and To = 300°K. By a straightforward circuit analysis Eq. (90) becomes ~

~

R,+R,+

~

Z

I ,i,” I ~ 4kToAf

’

-u,I~=

1 + YllZ, ~ - +

~ 4kToAf

where Z, = Z, + R, + R, = R, + R, + R f + jX,,and “ R e ” denotes the “real part of.” The first two contributions, embodied in R, and R, , represent the thermal noise of the gate and source parasitic resistances. The third and fourth contributions refer to the induced gate current and the drain current noise of the intrinsic transistor and the last term the correlation between these two currents, modified by the extrinsic elements Z,, R,, and R f . It is convenient at this point to represent the intrinsic current sources i, and id in terms of noise conductances genand gdn, respectively, where ~

ggn = ~

I i,” I /4kToAf = ( W 2 C g / g m ) R I id‘ I /4kToAf = g, P.

(924

(92b) Then by a suitable reorganization of terms the expression for F can be put in the useful form gdn =

12)

F = 1 + (l/Rs)(rn + g n I Zs + Zc (93) where rn and gn are the so-called noise resistance and noise conductance, respectively, and Z , is the correlation impedance (Rothe and Dahlke, 1956). The expressions for Y,, g,,, and Z, can be written in the form

248

ROBERT A. PUCEL ET AL.

T=0

T=O

I

I I

0

intrinsic FET

I I

In terms of r , , g,, and Z , , all the noise properties of the FET with parasitics are embodied in a very simple noisy network, shown in Fig. 22 which precedes the FET (now considered noise-free). Thus r, represents a thermal voltage generator at the standard temperature To,g, a shunt thermal current generator at the same temperature, and 2, an impedance at absolute zero. The noise figure of this combined network is the same as that of the original noisy FET. The parasitic resistances R, and Rf appear as a series combination in both the equivalent noise resistance r , and the correlation impedance Z , , but not in the equivalent noise conductance 9,. One may show that the noise contributions of the parasitics enter in r , , whereas the resistive properties of R, and Rf are relevant in Z , . In other words, at Kelvin temperatures other than the standard temperature, T # To = 300"K, thesum R, + Rf in r, should be multiplied by the ratio T/To. If we insert the expressions for gen and gd, in terms of the noise functions R and P, Eq. (92), the expressions for r , , g, , and Z , take on a particularly simple form, r,

=

(R,

+ Rf) + K ,

Z,

=

(R,

+ Rf)+

KC

(95c) y, 1 where K , , K,, and K , are functions of P, R, and C. We shall consider these -

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

249

K functions as our fundamental noise coefficients. They are given by the rather complicated forms, K,

=

P[(1 - C ( R / P ) ” 2 ) 2+ (1

K,

=

1 - C(R/P)’12 (1 - C ( R / P ) ” 2 ) 2+ ( 1 - C’)R/P

K,

=

R ( l - C’) ( 1 - C(R/P)”’)’ + (1 - C 2 ) R / P*

-

C2)R/P]

(964

These coefficients are plotted as a function of normalized drain current in Figs. 23a,b,c for three different L/a ratios and saturation indices 5, typical of practical devices. The coefficient values for other design parameter values can be obtained by interpolation. These coefficients are also displayed as functions of the ratio L/a in Fig. 24. To compute these graphs, we have used 6 = 1.2 in Eq. (51a) to fit Baechtold’s experimental T,(E) data (1972) with our choice of E, = 2.9 kV/cm. In addition we have chosen D = 35 cm2/sec for the high field diffusion coefficient. This choice was dictated by the best fit between computed and measured noise figures, and will be discussed more fully later. Notice that of the three coefficients, K , shows the strongest increase with drain current, reflecting the strong current dependence of the drain noise function P. The least dependence on drain current is exhibited by K , , which also is mildly dependent on Lla. Observe that whereas K , and K , increase strongly with increasing Lla, K , decreases. The curves illustrated in Figs. 23 and 24 apply for the drain bias condition K d j w O o = 1. There is a small drain voltage dependence of these K coefficients. This is illustrated in Fig. 25. Observe, though, that in the range 0.5 < Kd/woo < 1, which encompasses most practical bias conditions, the variation is small. The variation in the optimum noise figure, as we shall show later, is even less. It is to be noted that for short gate lengths, when region I1 encompasses most of the channel and dipole noise dominates so that C z 1, the noise coefficients simplify drastically. In particular, K , approaches zero.

C . Minimum Noise Figure

The first stage of a low-noise amplifier chain is usually designed to have a minimum noise figure. The noise figure is optimized by proper choice of the source impedance 2, = R, + j X , . This optimization can be achieved by a suitable impedance matching network between the signal source and the input (gate-source) terminals of the FET. As a function of the source

ROBERT A. PUCEL ET AL.

250 101,

I

2-

I , ? :I : r

I

h

10'8-

6-

162

10-2

2

4

6

8 0.'

2

6 8 10'

4

2

4

6

8 10

'

2

4

6 8 10'

Normalized drain current Id11,

Normalized drain current Id/Is

(bl

lo1 101,

2

4

6 8 10.'

2

Normalized drain current

4

6 8 10'

Id/Is

(C)

FIG.23. Noise coefficients as a function of normalized drain current for various values of the saturation index 5. V,,/W,, = 1. (a) L/a = 3, (b) L/a = 5, (c) L/a = 10.

SIGNAL AND NOISE PROPERTIES OF

25 1

GaAs FETs

I

1

1

I

I

I

I

I

i

3

FIG, 24. Noise coefficients as a function of the ratio of the gate length to the epitaxial layer thickness ld/ls= 0.5, V,,/W,, = 1, = 0.1.

<

impedance, the noise figure expression Eq. (93) has its minimum value when the reactive component of 2, is equal in magnitude and opposite in sign to the imaginary part of the correlation impedance Z , = R, jX,, and the real part of Z, has its optimum value R,, o p t . Using Eq. (93), it is easy to show that the components of the optimum source impedance are given by

+

~ s , o p t= xs,opt

=

[K + - x c

(~n/gn)l”~

.

(97b)

The minimum value of F can be expressed as Fmin =

1 + 2gn(Rc

(97a)

+ 4,opt)

or in decibels as Fmin(dB) = 10 log,, Fmin.

252

ROBERT 6.01

I

A.

I

I

PUCEL ET

AL.

I

I

I

1

I

L /a =5

5.0 -

”

Y

7 0

y:

01

Y

4.03.0-

YI

c

C

.-

-

; 2.00 0

u

;.

1.0-

2

0

I

I

1

1

0.2

0.4

0.6

0.8

Normalized source -to-drain voltage

I

1 .o

qd/Woo

FIG. 25. Noise coefficients as a function of the normalized drain voltage for the case L/a = 5. The drain current is held fixed. I, / I , = 0.5, 5 = 0.1.

It shall be helpful for later discussions to expand the rather complicated appearing frequency dependence of Fminin a power series in w. By straightforward algebra, we obtain Fmin =

1 + 2(uCsg/gm){Kg[Kr + gm(Rm + Rf)13”2

+ 2(wCsg/gm)2[Kggm(Rm + R, + KcRi)] + ....

(99) The appearance of gm and the parasitic resistances in the linear frequency term emphasizes the dominant role played by these parameters in the optimum noise performance of an FET. Notice that R, appears in the quadratic term only. The ratio wCSg/gm appearing in this expansion can be expressed asff, , wheref, = gm/2nC,, is the gain-bandwidth product of the FET. It is also the frequency at which the current gain of the FET drops to unity. FETs usually are not operated at frequencies abovef, because of the severe degradation in power gain asf, is exceeded.

VI. EXPERIMENTAL DATA A . Introduction

We have applied our small-signal and noise theory to a practical n-type GaAs FET whose published design values stated by Brehm (1973) and Brehm and Vendelin (1974) are as follows: L = 2 pm, a = 0.2 pm,

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

253

Z = 2.85 x 10- cm, Nd = 1017 cm- 3, and Woo= 2.9 V. The values of the parasitic resistances are R, = 15 R and R, = 0.8 R. This design closely parallels geometries studied at various laboratories including our own at the Research Division of the Raytheon Company. The noise coefficientsand small-signal parameters were calculated by us by first solving for the internal parameters s and p as a function of the bias voltages. A computer program based on the equations of Sections I1 and 111 was written for this purpose. As a by-product we also obtained the dc solutions for the I-I/ characteristics. As a spot check of our dc and smallsignal theory, we compute for a zero gate bias and a 3 V drain bias, a = 28.3 mA* and a terminal conductance saturation drain current ldss g t = 19 mmho, both values within 5% of Brehm and Vendelin’s experimental data (1974). We also have made similar calculations for our own device designs, and obtained equally good agreement with experiment. It might be of interest to point out that for this device design our computer results show that for drain-source bias values corresponding to current saturation, i.e., above the “knee” of the Id-Vdd curves, the length of the saturated velocity region is of the order of 30% of the gate length L. (For shorter gate lengths, of the order of 1 pm, this fraction can be as high as 90%.)The computed channel opening for the bias conditions stated above is almost uniform along the entire length of the gate, s M 0.68, p z 0.76. Thus, the longitudinal voltage drop along the channel in region I is negligible compared to the corresponding drop in region 11. The noise figure calculation requires a value for the gate charging resistance R,. Since there is at present no analytic formula for R, for the tworegion channel model, we have assumed that the R,C,, product is approximately proportional to the transit time 7 through the channel, where

There are sound theoretical grounds for this assumption, since in the constant mobility case, i.e., L , = L, Ri C,, is proportional to 7 (Hower et al. 1969, 1971). In this case 7 is strongly dependent on bias conditions of course. When velocity saturation is present, the transit time calculated from (100) is very nearly equal to the constant value L/u,, as if region I1 extended over the entire gate length! Using the values of Ri and C,, derived from the experimental data of Brehm and Vendelin (1974), we set Ri C,, = 4 x 10- sec. In passing we note that this product seems to scale with gate length, so that for a gate * I,,, the saturation drain current at zero gate bias, should not be confused with I , , the normalizing saturation current used earlier.

254

ROBERT A. PUCEL ET AL.

-

length L = 0.9 pm, this product is 1.8 x sec (Liechti et al., 1972). In any case, a precise value for Ri is not important, since it appears in the second-order term of the power series expansion for F,,,, Eq. (99). B. Experimental Results As a preliminary to our noise figure presentation, we illustrate in Fig. 26 the calculated dependence of the noise conductances genand gdnon the gate bias or more precisely on the normalized drain current Id/Idss(see footnote

vd d = 3 . 0 V f =

4.0 GHz

I L

C

..-0

al

8 C

.c

-0 ?!

z

10-1

I

0

0

I

I

I

1

0.2

0.4

0.6

0.8

10-1 1.0

Normalized drain current I d /Idss

FIG. 26. Equivalent gate noise conductance ygn, drain noise conductance gdn, and correlation coefficient magnitude as a function of drain current for a specified value of the high field diffusion constant D. The design parameters for the FET are: L = 2.0 {tm, u = 0.2 pm, 2 = 285 pm,and N = 10" cm13.

SIGNAL AND NOISE PROPERTIES OF

255

GaAs FETs

p. 253) for a particular value of the high field diffusion constant, D. For other values of D our computations show that these conductances are nearly proportional to D except at the low end of the drain current scale. This illustrates the predominance of the high field dipole noise of region 11. This is further emphasized by the near unity correlation coefficient displayed in this graph. Figure 27 illustrates the dependence of Fminon the normalized drain current for a particular value of drain bias and several values of the high field diffusion constant. Also shown are the experimental results for the two devices reported by Brehm and Vendelin (1974). It is evident that the choice

I

0

I

0.2

I

I 0.4

I

I 0.6

Normalized drain current

I

I

0.8

I

I I.o

I

I

1.2

I,, /Idas

FIG.27. Comparison between theoretical and experimental noise figures for a microwave GaAs FET as a function of drain current. The solid curve labeled D = 35 cm2/sec is the best fit to the experimental data. Shown also are theoretical curves for values of the high field diffusion coefficient above and below the value giving the best fit. The rapid increase of Fmi,with drain current is caused by the diffusion noise of the velocity-saturated region. This is made evident by the curve labeled D = 0 which corresponds to no diffusion noise. The upturn in noise figure to the left of the minimum is caused by the rapid decrease ofg, at low currents. The curve labeled R , + R , = 0 represents the intrinsic noise of the FET. +, 0 , Experiment (Brehm and - - -, theory. (Experimental data, Brehm and Vendelin, 1974, courtesy of Vendelin, 1974);-, Microwaves.)

256

ROBERT A. PUCEL ET AL.

D = 35 cm2/sec provides the best fit to the experimental data. This value is substantially lower than the low-field number Do = k T o p o / q = 110 cm2/sec, and the high-field values measured by Ruch and Kino (1968). However, it is in good agreement with the experimental high-field values reported recently by Castelain et al. (1974), and is in general agreement with the high-field levels computed by Fawcett and Rees (1969). Since the circuit losses were not stated by Brehm and Vendelin, we could not correct the measured noise figure for them. We may speculate that these losses could increase the noise figure by approximately 0.25-0.5 dB, so that the choice of D might have to be revised downward. Obviously more experimental data are necessary to resolve this uncertainty. Notice that our theoretical results exhibit all of the features of the experimental data, in particular the rapid rise of Fmi,at high drain currents, a property not shared by other noise theories. This rapid rise in Fminat high drain currents and the strong dependence on the diffusion constant is a measure of the strong influence of the diffusion noise of region 11. This dependence is embodied primarily in the noise coefficient K,. To further demonstrate the dependence on D, we also have plotted in Fig. 27 the curve for D = 0, that is, for no dipole noise. Note that Fminno longer increases with drain current, but actually decreases ! The degradation of the noise figure to the left of the minimum, on the other hand, is a consequence of the sharp drop-off of g, at extremely low currents, as is evident from Eq. (99). Because of this, the width of the trough in the Fmincurve can be quite broad if the decrease in g, extends over an extended current region. This would occur, for example, if the doping profile at the substrate interface were not sharp but tapered off slowly as the interface is approached. We also show in Fig. 27 the value of Fminin the absence of parasitics (R, = R, = 0). The low level of Fminderives from the strong cancellation of noise due to correlation, as predicted by the expression for Fminwhich for this case simplifies to Fmin= 1

+ ( ~ C O C ~ ~ / ~ , ) [ -P RC( I)] + ... 2

112

(101)

where the factor (1 - C’) is very small. The effectiveness of the noise cancellation produced by correlation is diminished drastically when the parasitic elements are introduced. It is obvious then that parasitic resistances should be kept to a minimum, not only because they introduce noise of their own, but also because they indirectly cause an increase in the contributions of the noise sources associated with the intrinsic device. The importance of maintaining low parasitic resistances is illustrated further in Fig. 28. This is a graph of the predicted minimum noise figure as a function of parasitic resistance. Note the considerable degradation in noise figure as (R, + R,) exceeds the design value.

SIGNAL AND NOISE PROPERTIES OF

I

0

10

1

GaAs FETs

I

20 30 Parasitic resistance R~ +R,

257

I

40

50

(R )

FIG.28. Theoretical noise figure as a function of parasitic resistance for the device design considered in the text. L = 2 pm, a = 0.2 pm, N , = l O I 7 cm-3, Z = 285 pm. Vdd = 3.0 V, f = 4.0 GHz.

When the device is biased well above the knee of the I-I/ characteristic, that is, in the current saturation regime, the minimum noise figure is a mild function of the drain bias voltage. This is illustrated by the experimental results of Brehm (1973) and Brehm and Vendelin (1974) shown in Fig. 29. Note the excellent agreement between their data and our theoretical results. It is informative to predict the minimum noise figure for frequencies other than 4 GHz. Figure 30 is a graph of Fminfor the frequency range extending from 1 to 10 GHz. This graph was computed with the exact expression for Fmin,Eq. (98). Note the nearly linear dependence on frequency. Figure 31 shows how the noise figure varies with gate length. Note the significant reduction in Fminfor a 1 pm gate. The predicted noise figure for the shorter gate lengths are substantially lower than the results reported in the literature (Liechti et al., 1972) for 10 GHz operation. This disagreement may indicate some limitations in the validity of our analysis for these short gate lengths, since the gradual channel approximation in region I may no longer be valid. Only more experimental data can resolve this question. It is possible, however, that the experimental noise figures reported to date for X band are higher than predicted because of other reasons. First, circuit losses neglected in our computations become increasingly important at the higher microwave frequencies and may possibly explain some of the discrepancy. Second, processing techniques for the shorter gate lengths are

258

ROBERT A. PUCEL ET AL.

-

3.0 -

-

-m -

0

0 0

4

U

?!

2.0

-

-

-

-

-

-

P

r

0

? Z

1.0

0

1

I 1.0

I

I 2.o

I

I 3.0

I

I 4.0

1

5.0

Source-drain bias (volts)

FIG. 29. Comparison of the theoretical and experimental noise figure as a function of source-drain potential for a fixed drain current.J= 4.0 GHz, I , = 10 mA. ( 0 )Experiment theory. (Experimental data, Brehm and Vendelin, 1974, (Brehm and Vendelin, 1974), (--) courtesy of Microwaves.)

not as well developed as for the longer gate lengths, so that the practical device may not be represented accurately enough by the simple equivalent circuit used in our analysis. Third, trap noise originating in the epitaxial layer, or more likely at the interface between the epitaxial layer and the substrate, may be significant. There is some experimental evidence that trap-related noise may be important at microwave frequencies. For example, experiments performed by the authors have demonstrated that sources at the epitaxial substrate interface, presumably traps, can generate drain current noise whose spectrum extends from below the video band (30 MHz) up through C band. Luxton (1973) of the Plessey laboratories reports that the growth of a high resistivity epitaxial buffer layer between the substrate and the channel layer improves the noise figure of microwave frequencies. This may indicate that the buffer layer reduces the effectiveness of short lifetime interface traps. Noise originating from short lifetime traps may also explain the higher than theoretical noise figure observed at the low end of the microwave band

SIGNAL AND NOISE PROPERTIES OF 5.0

GaAs FETs

I

I

1

I

I

I

1

1

259

4.0 -

m D

I

t&

3.0-

: P

c

.-I

2.0-

C

f

E

1.0-

i

.O

FIG.30. Theoretical minimum noise figure as a function of frequency for the device design considered in the text. L = 2.0 pm3 a = 0.2 pm, Z = 285 pn, N , = 10l7cm-3, v,, = 3.0 V.

I

4.01

:

t

I

f=lOGHz

I 0

I

0.5

/

I

1.0

I

1.5

0

Gate length L ( p m )

FIG.31. Predicted minimum noise figure as a function of gate length for the device design , = 3.0 V. considered in the text. a = 0.2 pm. Z = 285 Am, N , = l O I 7 C I I - ~ Vdd

260

ROBERT A. PUCEL ET AL.

extending from 0.5 to 3 GHz. The observed noise figure in this band does not continue to decrease with frequency at the nearly linear rate exhibited by the graph in Fig. 30; rather it decreases at a much slower rate (Baechtold et al., 1973; Luxton, 1974). The linear frequency variation at small w originates from the quadratic frequency dependence of the gate noise I I or ggn. [One may show this by expressing R in Eq. (101) in terms of the gate noise generator using Eq. (92a).] Now, single-level traps introduce a noise component in the drain current varying with frequency as

it

-

K/(l + w 2 T 2 ) (102) (van der Ziel, 1963b; Halladay and Bruncke, 1963; Copeland, 1971),where K is a constant of the device and the traps. The induced gate noise, consequently, will have a component of the form

-

K'w2/(1

+w

T )

(Jordan and Jordan, 1965). At frequencies substantially higher than the inverse trap lifetime, w % T - ', this gate-current component will become frequency independent. Thus, presumably, in the low microwave band the total gate noise current can be approximated by the expression

I it 1

__

-

A

+ Bm2

(104) where the constant term is produced by traps. Thus Fmin would decrease at a slower than linear frequency rate when the second term became comparable to the first. There is obviously a need for further research on the role of traps in microwave FETs, but it appears hopeful that improved technology for the growth of epitaxial layers, including high resistivity buffer layers, will reduce the noise contributed by traps. C . Summary and Conclusion

We have shown that the saturated velocity region plays a significant role in determining the small-signal and noise properties of GaAs microwave FETs. The theory presented here differs in several important respects from previous theories. First, the extent of the velocity-saturated regime is not restricted to the very edge of the channel as Turner and Wilson (1969) and Baechtold (1972) have assumed, but is allowed to extend into the channel a significant distance from the drain end, depending on the bias conditions. A consequence of this is that a finite drain resistance is predicted. Second, because of the nonzero extent of the saturated velocity region, a new noise generating mechanism is introduced.

SIGNAL AND NOISE PROPERTIES OF

26 1

GaAs FETs

We have shown for microwave designs that the noise produced in the velocity-saturated region increases with the dc level of drain current, and explains the experimental dependence of the noise figure on this current. At the higher drain currents, this noise predominates. We have presented an extensive series of curves for the small-signal and noise parameters. These are given in a form suitable for the device engineer to predict the performance of a particular design. While we have demonstrated good agreement between our theory and some limited experimental results, it is still too early to establish the limits of validity of this theory until more extensive experimental data becomes available for a wider range of design parameters. APPENDIX I : DERIVATION OF GATE CAPACITANCE EXPRESSION We derive the expression for the field-component normal to the surface of the gate electrode, i.e., E,(x, a), then integrate it over the surface to obtain the total gate charge, Q , . The gate capacitance is the derivative of Q, with respect to gate bias. The y component of field consists of two parts, E,,(x, a) in region I, and EYz(x,a) in region 11. In region I the field is simply that produced by the ionized impurities of the depletion region. Thus E,l(X,

4 = (2Woo/a)[l- b(x)/aI.

(105)

In region I1 there is an additional component produced by the Laplacian potential Eq. (17), so that

+

Ey2(x,a ) = (2Woo/a)p E, sinh [ n ( x - L l ) / 2 a ] where p = 1 - b,/a. Integrating this field over both gate electrode areas, one obtains the total gate charge Q,,

which after some manipulation, becomes

where f z ( s , p ) is given by Eq. (30). Performing the differentiation C,, = dQ, / d E , , holding V,, constant and using the identity (108a)

262

A. PUCEL

ROBERT

ET AL.

from Eqs. (19b) and (20), and the relation (108b) obtained by differentiation of (18), we obtain for Cgg

c,, = 2 K E O m l ( S P1 , 5 ) + L ~ ( S , p , t;) + 1.561

(109)

wheref,.(s, p , t) andf,,(s, p , 5 ) are given by Eq. (32), and the constant term has been added to account for fringing. Assuming that the feedback capacitance Cd, is small compared to C,, ,as experimental data shows, we may write (110) so that (109) can be considered an approximation for C, . This is the expression (31) given in the text. =

csg

- cdg

APPENDIX11: DERIVATION OF I Au:

I

We derive in this appendix the open-circuit noise voltage Au,, and its mean square value I Au: I , developed across region I by an elementary thermal noise voltage AWxoat some point x o in region I. The noise disturbance will cause a perturbation in the channel height 2b and in the channel field dW/dx such that the total current I , is constant. Let 6W be the disturbance in the channel potential W at some x > x o ; then

which leads to the relation d(6W) dW b+-6b=O dx dx

or d(6W) 1dW - -~ -6b dx b dx for the perturbation in the field. The channel width perturbation 6b can be eliminated by subjecting the relation 2b

=

2 4 1 - (W/Woo)”2]

(114)

a (WooW)”* 6 W . 2

(115)

to a perturbation. This yields 6b

= -

~

SIGNAL AND NOISE PROPERTIES OF

GaAs FETs

263

Substitution of (115) into (113) gives 1 d(6W) - a 1 dW 6 W dx 2b (Woo W)”’ dx ’ ~

Using (114) for b, and W

~

=

wzWoo,we may cast (116) into the form

1 d(6W) - dw 6 W dx 1-w’

~~

We integrate (117) from xo to L,, using the definitions 6W(xo) = AW,,, 6W(Ll) = AD,. Thus

Taking the mean square value, IAV?

I

1-(

=

1 - w(xo) 1-P

m.

The mean square value of the elementary noise voltage can be expressed in terms of the channel resistance of an elementary section of length dxo at xo . Thus

I AWf, I

4kT(xo) Af (dx0/2ob(xoP)

(120) where T,(xo) is the elevated electron temperature at xo . Using the identities =

E(x0) dxo = d W = 2 Woow dw Id

=

2ab(xo)ZE(x0)

(121) (122)

in (120) we obtain

Inserting this expression into (119), we finally get Eq. (50) of the text,

where xo = x is any point in region I.

ACKNOWLEDGMENT The authors take pleasure in expressing their appreciation to various colleagues, especially Dr. Charles F. Krumm and Mr. Daniel Masse, who supplied the necessary experimental data and on whom the various theories were tested; to Mrs. Laura Spiniello and Mrs. Janet Newell

264

ROBERT A. PUCEL ET AL.

of the Computer Group who performed the extensive calculations; to members of the Publications Department for their excellent graphical and photographical services; and to the staff of the Research Division library. Last, but not least, a special note of gratitude is due Mrs. Donna Schilling for her invaluable assistance in the preparation of the manuscript.

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AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. Born, M., 20(175), 52 Bors, E., 61, 79 Boucek, R. J., 188, 193 Bowers, R., 148, 192 Bowlden, H. J., 2(128), 4, 7, 50 Boyle, W. S., 5 , 6, 7, 13, 15, 51 Bradley, C. C., 11(143), 12(143), 13(149, 151), 14(151), 29(149), 40(206), 42(151), 51, 52 Bradley, W. E., 61, 79 Brailsford, A. D., 7, 5f Brandt, R. C., 11(145), 51 Brehm, G. E., 252, 253, 255, 257, 258, 264 Brindley, G. S., 64, 79 Brodie, I., 114(66), 125(66), 126, 127, 146 Brooks, H., 148, 192 Bruncke, W. C . , 246,260, 264 Bunts, R. C., 61, 81 Burke, R. E., 59, 79 Burnett, W. H., 61, 81 Butikov, E. I., 19, 51 Button, K . J., 14(152, 153), 15(157), 42(218, 220), 43(152), 5 1 , 5 3

A Abstreiter, G., 49(229), 53 Adams, E. N., 2(127), 3, 50 Alberts, W. W., 75, 80 Allen, S. J., 49(230), 53 Anastassiou, A., 244, 264 Andrus, P. G . ,92(30), 93(30), 94(38), 145 Anfilov, I. V., 128(72), 146 Apel, J. C., 47(228), 53 Avery, R., 62, 80

B Bach-y-Rita, P., 60, 79 Baechtold, W., 195, 203, 220, 230, 233, 235,241, 260,264 Baer, W. S . , 42(213), 53 Bajaj, K. K., 23, 52 Bak, M. J., 71, 74,80 Baker, M. A,, 71, 79 Baldereschi, A., 2( 1321, 6, 13, 17, 51 Barry, W. F., 62, 80 Bass, F. G., 41,52 Bassani, F., 2(132), 6, 13, 51 Batdorf, R. L., 153, 193 Batra, I. P., 109, 146 Bechtel, G. N., 202, 224, 264 Beering, S. C . , 188, 193 Bickmore, J. T., 94, 113(59), 119, 145, 146 Biddle, D. K., 188, 192 Bir, G. L., 19, 51 Birch, J. R., 13(149), 29(149), 51 Bittmann, C. A., 253,264 Bixby, W. E., 94, 145 Bogdonoff, H., 83(2), 86(2), 87(2), 89(2), I44 Bolton, W. D., 92(32), 93(32), 145

C

Cairns, B. R., 253, 264 Carlson, C., 83(1), 86(16), 89(1), 144, 145 Carpenter, R. L., 188, 189, 192 Carreira, L. M., 84(8), 144 Carroll, J. E., 153, 192 Castelain, R., 256, 264 Celli, N. V., 44(223), 53 Chamberlain, J . M., 12, 13, 14,42(151), 51 Chang, L. S . , 137(77), 138, 146 Chen, I., 105, 106(50), 107, 145 261

268

AUTHOR INDEX

Cheng, S . F., 61,81 Chipp, R. D., 170, 192 Chiu, K. W., 42(225). 44(225), 46(225,226), 47(226), 53 Chiu, T. L., 200,264 Chou, S. N., 61, 79 Clark, H. E., 94(40), 145 Clarke, R. la., 189, 193 Claus, C. J., 84(1 I), 90(27), 91(1 I), 113(11, 27), 145 Cohen, B. G., 153, 193 Cohen, J., 195, 220,246,254, 257, 264 Cohn, D. R., 13, 14(152, 153). 28(148), 29,40(206), 42(218, 220), 43(152), 51, 52,53 Collins, C. C., 60, 79 Cornarr, A. E., 61, 79 Constant, E., 256, 264 Cooke, P. M., 67, 79 Copeland, J. A., 155, 192, 204, 207, 228, 260,264,265 Copley, H. E., 93(36), 145 Cosgrove, T., 170, 192

Dow, M., 57,80 Dow, R. S., 68, 79,80 Doyle, J. B., 66, 79 Doyle, J. H., 66, 79 Dreybrodt, W., 14(152),42(220), 43(152), 51,53

E Eaves, L., I1(143), 12(143), 51 Economou, E. N., 27,52 Effer, D., 155, 193 Ehrenreich, H., 33(197), 52 Elliott, R. J., 2( I29),4( 1291, 5( I29), 6, 50 Enck, R. C., 41(208), 52, 106(48), 145 England, T. S., 187, 193 Ergun, H . B., 12(147), 51 Ero, J. W., 202, 203, 217, 244, 265 Evans, P. R., 155, 193 Evarts, E. V., 70, 79 Eyries, C., 66, 79

F Fairman, R. D., 253,264 Falconer, M. A., 64, 79 Fan, H.Y.,15, 41(208, 210), 51, 52 Dacey, G. C., 199,201, 264 Fawcett, W., 204, 207, 256, 264, 265 Daetwyler, K., 195, 260, 264 Feinstein, B., 75, 80 Dahlke, W., 231, 244,247,248,264,265 Dahlquist, J. A., 114(66), 125(66), 126, 127, Feldmann, J., 173, 193 Fernandez-Guardiola, A., 68, 79 146 Fetterman, H. R., 10, 11(142), 51 Dalton, J. V., 49(230), 53 Fetz, E. E., 71, 79 Davis, F. S., 189, 192 Fisher, P., 15, 17(154), 19, 51 Davis, K. K., 70, 80 Foerster, 0, 63, 79 Dawson, R. H., 164,192 Forster, T., 195, 260, 264 Deichrnan, W. B., 188, 193 Fox, S. S., 71, 79 DeLoach, B. C., Jr., 153, 158, 162, 192, French, L. A., 61, 79 193 Frey, A. H., 189, 190, 193 Demeshina, A. I., 9, 17(165), 51 Frey, J., 164, 192 Dennis, R. B., 30(191), 31(191), 52 Fridkin, V. M., 128(72), 146 Dessauer, J. H., 83(2), 86(2), 87(2), 89(2), Friedrnan, H., 56, 62, 80 144 Frohlich, H., 20,52 Dexter, R. N., 42(213), 53 Dickey, D. N., 27(187), 30, 32, 52 Dingle, R.. 11(144), 51 G Djourno, A., 66, 79 Dobelle, W.,65, 67, 79 Gee,B. L., 62, 79 Donald, D. K., 130, 141, 142, 146 Gehring, K.A., 12(147), 51 Donaldson, P. E. K., 64,79

D

269

AUTHOR INDEX

Genarelli, T., 74, 80 Geurst, J. A., 202, 203, 264 Ghandhi, S. K.,200,201,212,264 Ghosh, H. N., 200,264 Giaimo, E., 90(24), 113(24), 145 Gillen, H. W., 61, 81 Oilman, S., 68, 79 Girvin, J. P., 65, 79 Glenn, J. F., 62, 80 Glenn, W. W. L., 62, 79 Goety, W. E., 92(32), 93(32), 145 Coffe, W. L., 84(10), 144 Gowen, F., 195,220, 246,254,257, 264 Greatbatch, W., 61, 81 Grebene, A. B., 200,201,212,264 Green, W. R., 156, 193 Greene, P. E., 155, 193 Greg, H. G., 90(23), I13(23), 145 Greig, H. G., 113(61), 146 Grimes, J. H., 62, 80 Grimm, R. J., 68,80 Gundlach, R. W., 84(7, 1 I), 89(20), 91(11,20), 113(11, 12). 144,145 Gunn, J. B., 153, 193

H Hald, T., 61, 79 Hall, J., 113(59), 146 Halladay, H. E., 260, 264 Hambrecht, F. T., 74,80 Hammer, R., 83(3), 144 Harper, P. G., 24(182), 30(191), 31(191), 39( I82), 52 Hartmann, K., 244,264 Hasegawa, H., 2(130), 4, 5 , 7, 19,50, 51 Haus, H. A., 219, 236,237,239,242, 265 Hauser, J. R., 199, 202, 264 Hauser, 0. G., 93(37), 145 Hayford, R. E., 94(40), 145 Helbig, R., 14(153), 51 Hensel, J. C., 49(231), 53 Henvis, B. W., 31(193), 34(193), 35(193), 52 Herring, C., 37(202), 52 Hill, D. E., 11(144), 51 Hilsum, C., 153, 193 Himsworth, B., 207, 214,264 Hinchliffe, H. A., 60,80

Hodby, J. W., 24(182), 39(182), 52 Hofstein, S. R., 199, 200, 264 Hogan, J. F., 62, 79 Holcomb, W. G., 62, 79 Holmquest, H. J., 57,80 Holmquist, B., 61, 81 Hooper, W. W., 253,264 Hopfield, J. J., 42.52 Horii, K., 15, 16, 51 Horsley, V., 67, 80 Hoult, R. A., I1(143), 12(143), 51, House, W., 66,80 Howard, R. E., 2(130), 4, 5 , 6, 7, 13,50, 51

Hower, P. L., 202,224, 253, 264 Huang, K., 20(175), 52 Hudson, F. W., 92(30), 93(30), 145 Humphrey, D. R., 71,80

I Ing,S. W., 85(13),92(13), 101, 102(43), 106, 109(13, 52), 145, 146 Tnoue, E., 113(60), 146 Ipatova, I. P.,28(190), 52 Ito, M., 67,80 Ivanov-Omskii, V. I., 41(211), 52

J James, H. M., 17, 51 James, R. P., 204,228, 265 Jasper, H., 67, 80 Johnson, E. J., 24, 25,27(187), 30(187), 31, 36(201), 37, 38, 52 Johnson, E. O., 162, 193 Johnston, R. L., 153,193 Jordan, A. G., 260,264 Jordan, N. A., 260,264 Jost, R. G., 70,80 Justesen, D. R., 189, 193

K Kasser, R. D., 246,264 Kaminsky, G., 153, 193 Kanazawa, K. K., 109(53, 54), 146

270

AUTHOR INDEX

Kang, C. S., 155,193 Kaplan, R., 7, 17, (166), 19, 24(181), 27(181), 28(181, 190), 31(193), 32, 33, 34(193, 199), 35, 38(181,203), 39(181), 51,52 Kawabata, A., 42(216), 53 Kawaguchi, S., 67,80 Keyes, R. W., 2(127), 3, 50 Kiang, N. Y. S., 66, 67,80 Kido, G., 42(219, 221), 53 King, N. W., 189, 193 Kino, G . S., 204, 207, 256, 265 Kirkman, R. F., 17(167), 51 Klaassen, F. M., 203, 230, 244,264 Kleiner, W. H., 5(134), 51 Kneschaurek, P., 49(229), 53 Knight, J. R., 155, 193 Koch, J. F., 49(229), 53 Kohado, H., I I3(60), 146 Kolomiets, B. T., 41(21 I), 5 2 Koonce, C. S., 25.52 Korn, D. M., 12(146), 51 Korovin, L. I., 26(186), 52 Kosei, K., 113(60), 146 Kotthaus, J. P., 49(229), 53 Krause, F., 63, 80 Kurita, T., 128(73), 146

L Larsen, D. M., 2(131), 7,8, 10(140), ll(140, 141, 145), 12(141, 146), 13(148), 23, 24, 25, 27(187), 28(148), 29(131, 148), 30(187), 31, 32,40(206), 42(215), 50, 51, 52, 53 Lax, B., 13(148), 14(152, 153), 15(157), 28(148), 29(148), 40(206), 42(218, 220), 43(152), 51,52,53 Le Borgne, A., 256,264 Lebovitz, R. M., 189, 193 Lee, C. A., 153. 158, 193, 194 Lee, L. H., 93(33), 145 Lehovec, K., 200,201, 264 Lehrer, W., 253, 264 Levine, D. N., 59, 79 Levinson, 1. B., 41,52 Levy, M., 113(59), 146 Lewin, W. S., 64, 79 Lewis, R. B., 120, 121, 125(69), 126, 127, 129, I46

Li, H. T., 109(51), 145 Li, K., 186, 193 Liberson, W. T., 57, 80 Liechti, C. A,, 195, 220,246, 254,257,264 Lin-Chung, P. J., 18, 19, 39, 51, 52 Lipari, N. O., 17, 51 Livstone, E. M., 189, 192 Long, C., 57, 59, 80 Loudon, R., 2(129), 4(129), 5(129), 6, 50 Lowenthal, M., 67, 80 Luebbe, R., 84(9), 144 Lundgvist, B. I., 20(173), 43, 52 Luxton, H. E. G . , 258, 260,264

M McCombe, B. D., 13, 29( I SO), 32( l96), 33(196), 41(196, 209), 43(222), 44(222, 224), 45,46(224), 51, 52, 53 Mclntosh, J. D., 71, 80 McKenna, J., 221,265 McLaughlin, A. J., 62, 79 Mahan, G.D., 42,52 Maltz, M., 84(9), 144 Mann, R. W., 56,80 Manni, E., 68, 79 Maradudin, A. A., 28(190), 52 Marg, E., 75, 80 Marha, K., 187, 193 Marien, J., 125(71), 127(71), 128(71), 146 Marshall, J. M., 1 I1(56), 146 Masciarelli, V. D., 57, 80 Matsuda, Y., 67, 80 Mayer, E. F., 87(18), 90(27), I13(27), 145 Mayo, C. R., 93(34), 113(62,63), 145, 146 Mead, C. A., 197, 264 Melick, D. R., 156, 193 Melzack, R., 56, 80 Menchel, R. S., 93(37), 125(70), 126, 127, 128, 129, 143,145,146 Mendelson, K.S., 17.51 Merkle, M. G., 189, 192 Mermin, D., 44(223), 53 Metcalfe, K. A., 90(26), 113(26), 145 Michaelson, S. M., 188, 190, 193 Michelson, R. P., 66,80 Miura, N., 11(143), 12(143), 51 Miyake, S. J., 42(217), 47(227), 53 Miyao, M., 9, 10, 13,51

271

AUTHOR INDEX

Mizuno, N., 67,80 Mladejovsky, M. G., 65, 67, 79 Mo, D. L., 200, 264 Mohr, T. O., 195,260,264 Mollwo, E., 14(153), 51 Moore, W. J., 19, 52 Morris, J. M.,60, 80 Morsell, D. D., 119, 125(68), 126(68), 127, 146 Mort, .I., 105, 106(50), 107, 145 Mortimer, J. T., 56, 59,80 Moruzzi, G., 68, 79 Mott, G. R.,83(2), 86(2), 87(2), 89(2), 144 Moxon, E. C., 66,67,80 Mumford, W. W., 188,193 Muntz, E. P., 119, 125(68), 126, 127, 146 Murzin, V. N., 9(138), 17(165), 51 M u d , J., 187, 193

N Nagasaka, K., 42(219, 221), 53 Nahvi, M. J., 68, 80 Nakayama, M., 26, 30(185), 32(185), 33, 52 Narita, S., 9, 10, 13,42(219, 221), 51, 53 Nashold, B. S . , 56, 62, 80 Neyhart, J. H.,109(52), 146 Ngai, K. L., 22(176), 24(181), 27(181, 188), 28(181), 31(193), 34(193), 35(193), 36(201), 37, 38(181), 39(181,205), 42(225), 44(225), 46(225, 226), 47(226), 52,53 Nielsen, W. J., I14(65), 146 Nisida, Y., 15, 16, 51

0

O’Hare, J. M., 62, 79 Ohyama, T., 50(232), 53 Okazaki, M., 19, 51 Olds, J., 71, 80 Ommaya, A. K., 74,80 Onsager, L., 106, 121, 145 Otsuka, E., 50, 53 Oughton, C . D., 83(4), 144 Overhauser, A. W., 33(197), 52 Owen, A. E., 1 I l(56), 146

P Paff, G. H., 188, 193 Pai, D. M., 106(48), 111(55), 145, 146 Pantelides, S., 15, 51 Parker, C. D., 11(142), 51 Paul, R., 202, 264 Pavlov, S. T., 26(186), 52 Peckham, P. H., 58, 63,80 Penfield, W., 67, 80 Penry, J. K., 68, 80 Perichon, R., 256, 264 Phillips, T. G., 49(231), 53 Pikus, G. E., 19, 51 Poehler, T. O., 47(228), 53 Pollack, S . F., 60, 80 Praddaude, H. C., 2( I33), 51 Prim, R. C., 199,264 Prins, J., 244, 264 Prinz, G . A., 32(196), 33(196), 41(196, 209), 52 Pucel, R. A., 219, 236, 237,239,242,265 Pundsack, A. L., 84(10), 144 Purpura, D. P., 68,80

Q Quinn, J. J., 42(225), 44(224, 225), 45(224), 46(224, 225, 2261, 47(226), 53

R Radonjic, D., 59, 80 Ramdas, A. K., 15, 17(154), 51 Rasmussen, A. T., 66,80 Rath, R., 62, 79 Read, W. T . , Jr., 153, 158, 193 Rebersek, S., 59, 81 Reddi, V. G. K., 199,264 Rees, H. D., 204, 256, 264 Regensburger, P. J., 109(51), 145 Rehfeld, B., 173, 193 Reimer, G. R., 68,80 Reimers, S . D., 56,80 Reinis, G., 84(9), 144 Reiser, M., 220, 265 Reswick, J. B., 56,80 Rice, T. M., 49(231), 53

272

AUTHOR INDEX

Ridley, B. K., 153, 193 Rosseler, G. J., 173, 193 Rosenthal, D.S., 188, 193 Ross, J. M., 199,201,264 Roth, E. M., 189, 193 Rothe, H., 23 I , 247, 248, 265 Ruch, 1. G., 164, 193,204,207,256, 265 Rudell, A. P., 71, 79 Rushton, D. N., 64, 79 Ruthroff, C. L., 184, 193 Ruvalds, J., 27( I88), 52 Ryder, E. J., 199,265

S Sah, C. T., 15,51, 199, 264,265 Saito, H., 113(60), 146 Saitoh, M.,42(216), 53 Sakowski, K. H., 173,193 Salcman, M., 74,80 Saleh, A. S., 42(208, 210), 52 Sanada, T., 50(232), 53 Sasaki, K., 67, 80 Sawamoto, K., 42(214), 53 Schaffert, R. M., 83(4), 85(14), 90(28), 92(23, 31), 101, 128(41), 144, 145 Scharfe, M.E., 85(13), 88(19), 92(13), 101, 102(42), 109(13), 1 I l(19, 5 9 , 145, 146 Scharfetter, D. L., 158, 160, 193 Schechter, D., 15, 17,51 Schectman, B. H., 109(53), 146 Schein, L. B., 115,146 Schmidlin, F., 84(6), 103, 109(52), 112(57), 131(75,76), 133(76), 135, 144, 146 Schmidt, E. M., 70, 71,80 Schum, H., 63,80 Schwan, H. P., 176, 186,193,194 Scot, D., 57,80 Seki, H., 109(53, 54), 146 Sharafat, A. R., 68,80 Shealy, C. N., 56,80 Sheregii, E. M., 41(21 I), 52 Sherrington, C. S.. 67,80 Shockley, W., 196, 199, 204, 205, 210, 212, 228,264,265 Shoji, M., 203, 228, 265 Simmonds, P. E., 13(149), 14(149), 29(149), 42,SI

Simmons, F. B., 66,67,80 Smith, J. B., 67, 79 Smith, S. D., 30(191), 31(191), 52 Smythe, W. R., 223, 265 Snider, R. S., 67, 79 Soepangkat, H. P., 19,5I Stark, H. M., 120, 121, 125(69, 70), 126, 127, 128, 129, 143, 146 Statz, H., 219, 236, 237, 239, 242, 265 Staubitz, W. J., 61, 81 Steinemann, A., 199, 207, 265 Stenberg, C. C., 61,81 Stensaas, S., 67, 79 Stillman, G. E., 10(140), 1 l(140, 141, 142, 145), 12(141), 51 Stradling, R. A., I1(143), 12(147), 13(149, 151), 14(151), 17(167), 24(182), 29(149), 39(182), 42(151), 51,52 Straughan, V. E., 87(18), 145 Strutt, M. J. O., 244, 264 Sullivan, W. A., 142, 146 Summers, C. J., 11, 30 (1911, 31, 51, 52 Suzuki, K., 19, 51

T Tabak, M. D., 85(13),92, 106, 109(13), 145 Tannenwald, P. E., 10(140), 11(140), 40(206), 51, 52 Tarney, K.,199,265 Taylor, D. R., 56,81 Teitler, S., 43(222), 44(222, 224), 45(224), 46(224), 53 Thompson, W. D., 71,80 Thourson, T. L., 113(58), 138(58), 139, 142, 146 Tillotson, L. C., 170, 184, 193, 194 Timm, G. W., 61, 79 Tower, D., 68,80 Tremere, D. A., 253, 264 Trofimenkoff, F. N., 199,200,265 Tsui, D. C., 34(198), 49(230), 52, 53 Tuha, H., 187,193 Tulagin, V., 84(8), 144 Turnbull, F. M., 66, 79 Turner. J. A.. 200. 202,260,265 TutihLi, S., 84(10), 144

273

AUTHOR INDEX

U Umarov, L. M., 9(138), 17(165), 51 Ungar, R., 68,80

V van der Ziel, A., 202,203,205 210, 217, 219, 228, 230, 231, 234, 235, 236, 239,240,241,242,244,246,260,264, 265 Van Dorn, W. A., 84(9), 144 Van Engeland, J., 125(71), 127, 128, 146 van Ummerson, C. A., 188,192 van Vliet, K. M., 228, 265 Vendelin, G. D., 252,253,255,257,258, 264 Verlinden, W., 125(71), 127(71), 128(71), 146 Vodovnik, L., 57, 59,81 Vogelhut, P. O., 176, 194 von Ortenberg, M., 14(153), 51 Vyverberg, R. G., 86(15), 87(15), 145

W Wagner, R. J., 13,29(150), 32(196), 33(196), 41(196, 209), 43(222), 44(222, 224), 45(224), 46(224), 51, 52, 53 Waldman, J., 10(140), Il(140, 142),40, 51,52 Walkup, L. E., 87(17), 94(38), 145 Wall, P. D., 56, 80 Wallis, R. F., 2(128), 4, 7, 18, 19, 28(190), 50, 51, 52 Walter, R., 68, 80 Walter, W., 195,260, 264 Warfield, G., 199, 200,264 Warter, P. J. 106, 109, 145 Wasserstrom, E., 221, 265 Watkins, T. B., 153,193 Watson, H. A., 207,265

Watson, P. K., 141, 142, 146 Wayland, J. R., 189, 192 Welber, I., 170, 171, 194 Welkowsky, D. D., 119, 125(68), 126(68), 127,146 Westgate, C. R., 47(228), 53 Westin, J. B., 189, 194 Wherret, B. S.,30(191), 31(191), 52 White, R. M., 25, 52 Wiegmann, W., 153, 193 Wilbur, C. U., 137(77), 138, 146 Wiley, J. D., 36(200), 52 Willmott, R. W., 113(64), 146 Wilson, B. L. H., 200, 202. 260, 265 Winteler, H. R., 199,207,265 Wirta, R. W., 56, 81 Wise, E., 89(21, 22,) 113(21, 22), 139(21,22), 145 Wolf, P., 195, 202, 246, 260, 264, 265 Wolfe, C. M., ll(141, 142, 145), 12(141), 51

Woodbury, D. M., 68,80 Woody, C. D., 68,80 Wu, S. Y.,199,265

Y Yafet, Y., 2(127), 3, 50 Yanai, H., 200,264 Yang, E. S., 198,265 Yasuda, R., 62, 79 Young, C. J., 90(23, 25), 113(23, 25), 145 Yule, J. A. C . , 114(65), 146

z Zajac, F. E., 59, 79 Zappert, F. G., 158,194 Zuleeg, R., 199, 200, 201, 207, 264, 265 Zurlo, P. 61,81 Zwerdling, S., 15, 51

SUBJECT INDEX

A

C

Cable TV, 17 1 Cadmium sulfide impurities in, 14 as piezoelectric semiconductor, 42 Cadmium telluride cyclotron resonance data in, 39 hydrogenic impurities in, 13 other impurities in, 10 resonant donor electron-LO phonon coupling in, 27 Cascade development, in xerography, 89,

Adhesion controlled development, in charged pigment xerography, 130-139 idealized touchdown development and, 131-137

Afterdischarge, phosphenes in, 65 Aircraft Warning Avoidance Radar Equipment, 173 American Telephone and Telegraph Co., 180

Athermal effects, of microwave radiation, 188-190

139-144

ATS-F satellite, 172 Auditory prosthesis, 65-67 Automobiles, radar for, 174, 186 Avalanche devices, 158-161 see also IMPATT; TRAPATT AWARE system, 173

Cell firing, voluntary control in, 71-72 Cell membrane, information transfer across, 73-75 Cerebellar stimulation, seizure termination through, 68 Charged pigment zerography, 83-144 see also Xerographic process; Xerography adhesion controlled development in,

B

130-139

cascade development in, 139-144 charge neutralization in, 113-1 16 defined, 84 driving force in, 116-118 electrostatic image development vs. charge neutralization in, 113-1 16 feedback in, 135 impression development in, 138 magnetic brush developer in, 115, 137-

Bell System, microwave communication in, 17 1

Biomaterials, brain implantation of, 64 Biotelemetry, microwave devices in, 176 Bipolar transistors, technological factors in, 168 see also Transistor(s) Bladder evacuation, by electrical stimulation, 61-62 Blind prosthesis for, 65 radar for, 177 Body movement, cortical cell activity in, 70 Bound carrier resonances, 1-19 nearly hydrogenic centers in, 1-16 Brain biomaterial implantation in, 64 pain in, 79

139

physical basis for development in, 1 13143

subneutralized development in, 136 toner in, 115 Charge neutralization, in charged pigment xerography, 1 13-1 16 Chronic pain, spinal cord stimulation in, 56

Cochlea, deafness and, 65

274

275

SUBJECT INDEX

Collective excitation electron-plasmon interaction in, 4 3 4 7 Frohlich theory of electron-phonon (LO) interaction, 20-23 resonant electron-phonon coupling in, 23-36 resonant electron-2NPO phonon coupling in, 36-39 Common Carrier Bureau, 182 Communications, microwave solid state devices in, 169-173, 178 Communications and Systems, Inc., 181 Conduction deafness, 66 Control signals, in neural control, 69-73 Control systems, microwave solid state systems and, 173-177 Corona charging devices, in xerography, 86 Corotron, in xerography, 86 Cortex, tonotopic map in, 67 Cortical cell activity, movement and, 70 CPX, see Charged pigment xerography Cutoff frequency, of transistor, 162 Cyclotron effective mass, magnetic field dependence of in CdTe, 40

D Data transmission, interception of, 19 1-192 Deafness conduction, 66 sensorineural, 66 Development system, in xerographic process, 89, 100-144 Development zone, in xerographic process, 100- 104 Diamond structure sem;conductors, free carrier states in valence bonds of, 34 Diaphragm, fatigue of, 63 Distance measuring equipment (DME) transmitters, 174 Donor impurities in excited state and other low-energy transistors, 13 high-field limit in, 7-9 low-field limit in, 9-12 in materials with zone center conduction band, 6-14 multiple equivalent conductor bands and, 1417 Dorsal columns, electrical stimulation of, 56

E Ear, electrical stimulation of, 65-67 Effective mass approximation, for semiconductor impurities, 1 Electrocortiogram (EGG), 69 Electrode(s) “bed of nails,” 75 cell reaction to, 73 location of, 75 types of, 74 Electroencephalogram, 69 Electromagnetic field, see microwave radiation Electromagnetic spectrum block allocation of, 183-186 congestion in, 181-184 impact of microwave solid state devices in, 179-183 interference in, 185 modulation techniques in, 184-185 Electromyogram (EMG) controlled arm prosthesis, 56, 72 Electron-LO phonon interaction, 20-23, 40 collective excitation and, 44 Frolich theory of, 20-23 self-energy shift and, 40 Electron-2NPO phonon interaction, 50 Electron-plasmon interaction, 43-47 Electrophoretic development, in xerography, 90, 124130 Electrophotography classification in, 84 development of, 83-84 Electroradiography, powder cloud development in, 127 EMG, see Electromyogram Epilepsy control, 67-68 F

Federal Communications Commission, 170-171, 175, 182 Feedback in functional neuromuscular stimulation, 56-61 in neural control, 76 in xerographic process, 116 FET, see Field effect transistor: Gallium arsenide field effect transistor

276

SUBJECT INDEX

Field effect transistor see also Gallium arsenide field effect transistor vs. bipolar transistor, 164 current saturation vs. velocity-limited flow in, 199-200 dc characteristics of, 207 idealized model of, 205-207, 209 intrinsic, 204224 junction, see Junction field effect transistor noise in, 203, 228 operating principle of, 196 with parasitic resistances, 224228 pinch-off voltage of, 210 small-signal equivalent circuit of, 214-215 small-signal parameters of, 2 14-224 symmetry plane of, 205 technological factors in, 167 two-section model of, 20 I FIR cyclotron resonance studies and experiments, 4041, 50 FIR laser sources, improvements in, SO FNS, see Functional neuromuscular stimulation Foot drop, in hemolytic patients, 57 Free and bound carriers, interaction of with collective excitations, 19-47 Free. electron spin-down cyclotron resonance energy, magnetic field dependence of, 33 Frolich coupling constant, 45 Frolich interaction, 20-23, 34 see also Electron-LO phonon interaction photon absorption in, 40 Frost development, in xerography, 91 Functional neuromuscular stimulation (FNS), 56-61 fatigue during, 58

G Gallium aluminum arsenide, in transistors, 166 Gallium arsenide in field effect transistors, 164 impurities in, 10-12 Gallium arsenide field effect transistor

see also Field effect transistor; Intrinsic

field effect transistor channel opening in, 213 channel-to-gate potential in, 210 correlation coefficient for, 242 drain circuit noise in, 235-238 drain-gate and source-drain capacitances in, 222-224 drain resistance in, 218-220 experimental noise data for, 252-26 I extrinsic noise sources in, 232 fabrication of, 208-209 gate capacitance expression for, 26 1-262 gate circuit noise in, 238-242 gate-source capacitance in, 220-222 instabilities in, 21 1 intrinsic noise sources in, 228-232 noise in, 228-244 noise analysis in, 232-234 noise coefficients for, 243-244 noise figure for, 244-252 open-circuit drain voltage fluctuation in, 235-238 open circuit noise voltage for, 262-263 short-circuit gate current fluctuations in, 238-242 signal and noise properties of, 195-263 small-signal characteristics of, 204, 214-224, transconductance in, 215-218 Turner-Wilson model of, 201-204 velocity saturation in, 201-204, 21 1 Gate capacitance expression, for GaAs FETs, 261-262 General Motors Corp., 174 Gradual channel approximation (GCA), 198 Grand ma1 epilepsy, cerebellar stimulation in, 68 Green’s function, in resonant electron-LO phonon coupling, 26, 39

H Hemiplegic patients, foot drop in, 57 Hole-electron pairs, in xerography, 87 Human body, heat sensitivity in, 188 Hydrogenic atom donor impurities and, 6-14

277

SUBJECT INDEX

high-field case for, 3-5 low-field limit for, 5-6 magneto-optical techniques in study of, 49 in magnetic field, 1-6 Hypoventilation of central origin, 62

I Image neutralization, in xerographic process, 135 IMPATT (impact avalanche transit time) diode, 152, 155, 158-161, 168, 174 Impression development, in charged pigment xerography, 138 Impurity cyclotron resonance energy, vs. magnetic field, 29 Impurity ionization energy, 2 Impurity transitions, low-energy, 13-14 Indium antimonide cyclotron linewidths in, 31 cyclotron resonance in, 30 donor impurity transitions in, 28 impurities in, 12 magnetoplasmon assisted magnetooptical transitions in, 44 photon energy dependence of, 39 Indium phosphide hydrogenic impurities in, 12 in transistors, 168-1 69 Information transfer, outward, 73-75 Institute of Electrical Electronics Engineers, 179 lnterband and intraband magneto-absorp tion studies, 32 lntraband magneto-optical studies bound carrier resonances and, 1-19 interaction of free and bound carriers with collective excitations in, 20-47 Intrinsic field effect transistors, 204-224 see also Field effect transistor Intrusion alarms, microwave devices in, 175

J

Johnson noise, in field effect transistors, 233-234

Joint Technical Advisory Committee, 184 Junction field effect transistor (JFET), 196-197 field-dependent mobility in, 199 noise sources in, 228 operation of, 197 small-signal properties of, 202

K Kirchhoffs law, for photoreceptors, 110

L

Landau level, impurity level and, 3 Landau level spectrum, 23 Landau quantum number, 3 Learning, nervous system and, 72 Level separation, in magneto-optical resonant coupling experiments, 27 Lilly hypothesis, in neural control, 77 LO phonon assisted cyclotron resonance (LOCR), 39,41 LO phonon interaction, 20-23, 36 Low- and high-field energy levels, hydrogenic atom and, 3-6 LSA diode, 174 Lycopodium powder, in xerography, 83

M

Magnetic brush development, in xerography, 90, 116 Magnetic field Coulomb binding energy and, 2 hydrogenic atom in, 1-6 impurity cyclotron resonance energy and, 29 Zeeman effect in, 2 Magneto absorption studies, inter- and intraband three-level, 32 Magneto-optical resonant coupling experiments, levels in, 27 MCI Corp., 186 Metal-oxide-semiconductor field-effect transistor (MOSFET), 197 noise in, 203, 228

278

SUBJECT INDEX

Microwave applications, 150, 169-1 78 see also Microwave solid state devices Microwave band, defined, 149-150 Microwave integrated circuits, 169 Microwave radiation athermal effects of, 188-1 89 biophysical hazards of, 186-191 invasion of privacy and data interception through, 191-192 physiological changes in, 188-1 89 safety standards in, 190 thermal effects of, 187-1 88 Microwave solid state devices applications of, 169-178 for automobile radar, 174, 186 benefits and problems of, 178-192 in communications, 169-173 in control systems, 173-177 development of, 150-151 future technologies in, 176 health hazards of, 152, 186-191 impact of on electromagnetic spectrum, 179-183 for intrusion on theft alarms, 175 performance and cost levels of, 150-1 52 in telemetry, 176 in toys, 176, 185 Microwave sources, solid state, 153-169 Microwave technology, growth of, 151, 176 Modulation methods, in electromagnetic spectrum, 184-1 86 MOS (metal-oxide-semiconductor) field effect transistor, 49 Moss prosthetic arm, 72 Motor control, physiology of, 70 Movement, cortical cell activity in, 70-71 Multiple equivalent conduction bands, donor impurities and, 14-17 Muscle fatigue, in functional neuromuscular stimulation, 58 Muscle fibers, fatigue resistant, 58 Myoneural junction, 63

Nerve cell electrical signals from, 69-70 information transmission through, 73-75 Nervous system artificial stimulation of, 69 electrodes in, 73-75 learning and, 72 regeneration of, 77 Neural control concepts and techniques in, 68-77 control signal sources in, 69-73 electrodes in, 75 epilepsy control in, 67-68 feedback in, 76 future possibilities in, 55-79 Lilly hypothesis in, 77 long-term effects in, 76 nervous system regeneration in, 77 signal and power transfer in, 76 supernormal humans and, 77-78 visual prosthesis and, 63-65 Neuromuscular stimulation, functional, 56-61 Noise drain circuit, 235-238 in GaAs field effect transistors, 235-238, 252-26 1 gate circuit, 238-242 white, 237 Noise coefficients, for GaAs field effect transistor, 242-244 Noise figure, 244-252 Noncharged pigment xerography, 84 see also Charged pigment xerography

0

Office of Telecommunications Policy, 182 Outward information transfer, techniques for, 73-75 P

N National Aeronautics and Space Administration, 172, 182 National Institute of Neurological Diseases and Stroke, 64-65

Pain brain stimulation and, 79 spinal cord stimulation and, 56 Parasitic resistances, in field effect transistor, 224-227 Phosphenes, in visual prosthesis, 64-65

279

SUBJECT INDEX

Photoinduced discharge curve (PIDC), in xerography, 88, 96-97, 101-104 physical production of, 104-105 stabilizing of, 1 1 1 transport limited discharge and, 108-1 12 Photoreceptor, 87, 92-100 in cascade development, 144 dielectric model of, 100-101 Kirchhoff’s law and, 110 material and contact requirements for, 1 1 1-1 13 photoinduced discharge curve and, 104112 physical basis for performance of, 104I13 Photoreceptor subsystem, coupling of to development system, 100-113 Photoreceptor transfer function, 98 Phrenic nerves, respiratory control and, 62-63 Piezoelectric electron-phonon interaction, 4243 Piezopolaron, 4 2 4 3 Plasmaron, absorption process in, 4 6 4 7 p-n junction, 201 PODAF allocation system, in electromagnetic spectrum, 184 Polaron, “nonparabolicity” of, 39 see also Electron-LO phonon interaction Polaron cyclotron resonance, for cadmium telluride, 39-40 Polaron Landau levels, 23 Privacy, invasion of by microwave devices, 191-192 Prosthesis for amputees, 72 auditory, 65-67 visual, 63-65

Q Quadriplegic patients feedback in, 59-60 grasp in, 57

R Radar for automobiles, 174, 186 for blind persons, 177

for collision avoidance, 173-174 Raman scattering, 20, 39 RCA Corp., 174 Relaxing avalanche mode (RAM), 158 Resonances, bound carrier, see Bound carrier resonances Resonant electron-LO phonon coupling, 23-36 cyclotron resonance linewidths in, 30 Green’s function approach to, 26 three-level experiments in, 31 Resonant electron-2NPO phonon coupling, 36-39 background of, 36-37 experimental results in, 37-39 Respiratory control, phrenic nerves in, 62-63 Rydberg ionization energy, 2 S

SBFET (Schottky barrier field effect transistor), 197 Schottky barrier gate electrode, 197 Scorotron, in xerography, 87 Semiconductors donor impurities and zone center conduction band in, 6-14 FIR studies of, 48 hydrogenic impurity center in, 49 impurity level of, 3 - 4 intraband magneto-optical studies of, 1-50 low- and high-field energy levels for, 3-6 low-energy impurity transitions in, 14 magneto-optical properties of shallow acceptor levels in, 16-19 piezoelectric, 42 Rydberg or impurity ionization energy for, 2 Sensorineural deafness, 66 Solid state devices see also Microwave solid state devices impact of, 147-194 microwave region and, 169-178 Spasticity, muscle tone and, 60-61 Spinal cord electrical stimulation of, 56 respiratory control and, 62-63

SUBJECT INDEX

280

Spinal cord injuries, rehabilitation in, 57-58 Supernormal humans, neural control in, 77-79 Surface charge density, in xerography, 88

U UHF bands, microwave systems in, 173 United Nations, 172

V

T Technological changes, social effects of, 148-150 Telemetry, microwave solid state devices in, 176 TEO, see Transferred electron oscillator Theft alarms, microwave devices in, 175 “Thumbtack” electrode, 74 Toner, in charged pigment xerography, 115 Tone reproduction curve, in xerographic process, 94-99 Tonotopic map, in human cortex, 67 TO phonon coupling, 34-36 Touchdown development, in charged pigment xerography, 13 1-138 Toyota automobile, radar for, 174 Toys, microwave solid state devices in, 176, 185 Transferred electron oscillator (TEO), 153-154, 165 applications of, 157 limits of, 156-157 modes of, 155-156 Transistor(s) see also Gallium arsenide field effect transistor applications of, 165 bipolar vs. field effect, 164, 168 performance limits for, 162-164 in solid state devices, 162-165 transit-time cutoff frequency of, 162 Transistor technology, materials research in, 167-168 Transportation Department, U.S., 175 Transportation Systems Center, 174,175 TRAPATT (trapped plasma avalanche triggered transit) diodes, 153, 158-161, 174 TV networks, microwave devices in, 177 2 N P 0 phonon coupling, 36-39

Velocity saturation, in GaAs FETs, 201 204 Visual cortex, electrical stimulation of, 63 Visual prosthesis, 63-65

W Western Electric Co., 183 Western Union Co., I80 Wigner-Brillouin perturbation theory, 25 World Administrative Radio Conference, 172

X Xerographic process see also Charged pigment xerography; Xerography adhesion controlled development in, 130-139 adhesive transfer in, 92 development in, 89,9697, 100-104, 113-144 edge enhancement in 128 electrostatic image in, 87-88 electroviscous flow lines in, 119 erasure in, 93 exposure system in, 94-96 feedback in, 116 frost development in, 91 latent image in, 1 1 W 1 7 ideal and practical development systems in, 124-130 image transfer in, 92 input-output characteristics of, 97-99 operating parameters in, 97-99 operational steps in, 85-93 photoinduced discharge curve in, 96-98

28 1

SUBJECT INDEX

photoreceptor in, 94-101 photoreceptor materials and contact requirements for, 112-1 13 photoreceptor subsystem in, 100-1 13 powder cloud development in, 127 sensitization in, 86-87 tone reproduction curve in, 94-100 touchdown development in, 13 1-138 viscosity controlled development in, 118-130

Xerography, 83 see also Charged pigment xerography; Xerographic process development in, 113-144 electrostatic image development vs. charge neutralization in, 113-1 16 magnetic brush development in, 90

A 5 B 6

c 7 D B € 9 F O

G I

H 2 1 3 1 4

physical basis for development in, 113-144

scorotron in, 87 surface charge potential in, 88

Z Zeeman effect, magnetic field and, 2 Zeeman mass, vs. cyclotron mass, in hydrogenic atom donor impurities, 11 Zinc blende semiconductors, free carrier states and valence bands of, 34 Zone center conduction band, donor impurities and, 6-14 Zone centered degenerate valence bands, 16-19

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