Advances in Electronics and Electron Physics, Volume 37

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 37 CONTRIBUTORS TO THISVOLUME K. M. Adams E. F. A. Deprettere Br...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 37

CONTRIBUTORS TO THISVOLUME K. M. Adams E. F. A. Deprettere Bruce D. McCombe Kohzoh Masuda Susumu Namba J. 0. Voorman Robert J. Wagner P. K. Weimer

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 M. Ponte H. B. G. Casimir W. G. Dow A. Rose L. P. Smith A. 0. C. Nier F. K. Willenbrock

VOLUME 37

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 O F THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION I N WRITING FROM T H E 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 N W l

LIBRARY OF CONGRESS CATALOG CARD NUMBER:49-7504 ISBN 0-1 2-014537 -5 PRINTED I N T H E UNITED STATES OF AMERICA

CONTENTS CONTRIBUTORS TO VOLUME37 .

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vii

FOREWORD.

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ix

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Intraband Magneto-Optical Studies of Semiconductors in the Far Infrared. I I. I1. I11. IV .

BRUCED. MCCOMBE AND Introduction . . . . . . . Theoretical Background . . . . Experimental Techniques . . . Free Carrier Resonances . . . References . . . . . . .

ROBERT J . WAGNER . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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i 2 17 28 75

The Gyrator in Electronic Systems K . M . ADAMS. E . F. A . DEPRETTERE, A N D J . 0. VOORMAN I . Introduction . . . . . . . . . . I1 . Reciprocity in Physical Systems . . . . I11. The Gyrator as Network Element . . . . IV . Filters . . . . . . . . . . . . V. Principles of Realization of the Gyrator . . VI . Basic Electronic Design . . . . . . . VII . Basic Gyrator Measurements . . . . . VIII . Trends in Gyrator Design and Applications . IX . Conclusion . . . . . . . . . . References . . . . . . . . . .

. . . . . . . . . .

. . . . . . 80 . . . . . . 80 . . . . . . 86 . . . . . . 94 . . . . . . 109 . . . . . . 128 . . . . . . 145 . . . . . . 170 . . . . . . 176 . . . . . . 177

Image Sensors for Solid State Cameras P. K . WEIMER I. I1 . 111. IV . V. VI . VII . VIII . IX . X. XI . XI1 .

Introduction . . . . . . . . . . . . . . Photoelements for Self-scanned Sensors . . . . . . Principles of Multiplexed Scanning in Image Sensors . . Early XY Image Sensors . . . . . . . . . . Multiplexed Photodiode Arrays . . . . . . . . Principles of Scanning by Charge Transfer . . . . . Charge-Transfer Sensors Employing Bucket Brigade Registers Characteristics of Charge-Coupled Devices (CCDs) . . . Experimental Charge-Coupled Image Sensors . . . . Performance Limitations of Charge-Coupled Sensors . . Charge-Transfer Sensors as Analog Signal Processors . . Self-scanned Sensors for Color Cameras . . . . . .

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182 183 187 193 198 202 208 213 225 . 236 . 247 . 251

vi

CONTENTS

XI11 . Peripheral Circuits for Solid State Sensors . . . . . . . . 253 XIV . Conclusions . . . . . . . . . . . . . . . . . 257 References . . . . . . . . . . . . . . . . . 259 Ion Implantation in Semiconductors

SUSUMU NAMBA A N D KOHZOHMASLDA I. I1. I11. I V. V. VI .

Introduction . . . . . . . . Concentration Profiles of Implanted Ions Enhanced Diffusion . . . . . . Annealing and Electrical Properties . Measurement Technique . . . . Devices . . . . . . . . . References . . . . . . . .

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and Defects . . . .

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264 271

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2x9 299 . 310 . 325 . 328

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ALTHOR

INDEX .

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331

SC RJtCT

INDEX

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339

CONTRIBUTORS TO VOLUME 37 Numbers in parentheses indicate the pages on which the authors’ contributions begin

K. M. ADAMS(79), Department of Electrical Engineering, Delft University of Technology, Delft, Netherlands (79), Department of Electrical Engineering, Delft E. F. A. DEPRETTERE University of Technology, Delft, Netherlands (11 Naval Research Laboratory, Washington, D.C. BRUCED. MCCOMBE

KOHZOHMASUDA(263), Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan SUSUMUNAMBA(263), Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan J. 0.VOORMAN(79), Philips Research Laboratories, N. V. Philips’ Gloeilampenfabrieken, Eindhoven, Netherlands

ROBERTJ. WAGNER(1), Naval Research Laboratory, Washington, D.C. P. K. WEIMER (181), RCA Laboratories, Princeton, New Jersey

vii

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FOREWORD

Our present volume starts with 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.” In a much broader survey of farinfrared radiation, Professor L. C. Robinson* confines magneto-optical effects in semiconductors necessarily to barely three pages. In their review, McCombe and Wagner offer an in-depth treatment of this important subject leading to a better knowledge of intraband transitions. A potentially very important device is discussed by K. M. Adams, E. F. A. Deprettere, and J. 0. Voorman in their review of “The Gyrator in Electronics Systems.” They point out that the electronic gyrator, as it now exists, is limited in industrial applications for several reasons. Nevertheless the subject is important enough for a thorough treatment and the reviewers’ presentation will, no doubt, stimulate thinking toward improved solutions. An entirely new trend in image pick-up and reproduction is the subject of P. K. Weimer’s review. Under the title “Image Sensors for Solid State Cameras,” he discusses the new devices more from a research viewpoint “with much greater emphasis on principles of operation, than on details of construction.” The five-year-old discovery of charge coupling has produced a tremendous growth of new devices, which shall be covered soon in a separate monograph presented as a supplement to this series. Weimer’s review is a detailed treatment of one aspect of this growing field. The technology of “doping” semiconductors has progressed considerably since the early days. A very interesting technique, permitting rather precise introduction of impurities, is discussed by S. Namba and K. Masuda in their article on “Ion Implantation in Semiconductors.” The method, which started as a laboratory operation, developed into an important industrial one, allowing large-scale production of semiconductor devices. Many important and interesting reviews are scheduled to appear in future volumes. The list on the next page indicates the contents of the volumes to come.

* L. C. Robinson, “Physical Principles of Far-Infrared Radiation,” Meth. Exp. Phys. (L. Marton and C. Marton, eds.), Vol. 10. Academic Press, New York, 1973. ix

x

FOREWORD

lntraband Magneto-Optical Studies of Semiconductors in the Far Infrared. 11 The Future Possibilities for Neural Control Charged Pigment Xerography The Impact o f Solid Statc Microwave Devices: A Preliminary Technology Assessment Signal and Noise Parameters of Microwave FET Advances in Molecular Beam Lasers Interpretation of Electron Microscope linagcs of Defects in Crystals Development of Charge Control Concept Energy Distribution of Electrons Emitted by a Thermionic Cathode Time Measurements on Radiation Detector Signals The Excitation and Ionization of Ions by Electron Impact Nonlinear Electron Acoustic Waves. 11 The Photovoltaic Effect Semiconductor Micro8ar.e Power Devices. 1 and IT Experimental Studies of Acoustic Waves in Plasmas Auger Electron Spectroscopy 11) S i r i r Electron Microscopy of Thin Films Afterglow Phenomena in Rare Gas Plasmas Between 0 and 300 K Physics and Tcchnologies of Polycrystaliine Si i n Semiconductor Devices Charged Particles as a Tool for Surface Research Electron Micrograph Analysis by Optical Transforms Electron Beam Microanalysis Electron Polarization in Solids X - R a j Imagc Intensifiers Eiectron Bombardment Semiconductor De\ ices Thermistors High Power Electronic Devices Atomic Photoelectron Spectroscopy Electron Spectroscopy for Chemical Analysis Laboratory Isotope Separators and Their Applications Recent Advances in Electron Bcam Addressed Memories Supplementary Volume: Charge Transfer De\ ices

B. D. McCombe and R. J. Wagner K . Pi-ank and F. T. Hanibrccht F. W. Schmidlin and M. E. Scharfe J . Frey and R. Bowers R. A. Haus, R. Pucel, and H. Statz D. C. Lame M. J. Whelan J. te Winkel W. Franzen and J . Porter S. Cova John W. Hoopcr and R. K. Fecney

R. G. Fowler Joseph J. Loferski S. Teszner and J. L. Teszner J. L. Hirschfield and J. M. Buzzi N. C. MacDonald and P. W. Palmberg A. Barna. P. B. Barna. J . P. Pbcia, and I. Pozsgai J. F. Delpech J. Kobayashi J . Vennik G. Donelli and L. Paoletti D. R. Beaman M. Canipagna, D. T. Pierce. K. Sattler, and H. C. Siegmann J. Houston D. .I.Bates G. H. Jonker G. Karady S. T. Manson D. Berenyi S. B. Karmohapatro J. Kelly C. H. Sequin and M. F. Tompsett

Throughout the years we have enjoyed the wholehearted cooperation of many friends. O u r warmest thanks go to them for the help they gave us. We would like to invite, as in the past. comments on the published volumes and suggestions for future ones. L. MAKTON CLAIRE MARTON

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 37

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

AND

ROBERT J. WAGNER

N a d Research Luhorutory, Washingrun, D.C.

I. Introduction.. ........................................................................................... 11. Theoretical Background.. ............................................................................ A. Optical Properties ....................... ............... B. Quantum Theory of Free Electrons in a Magnetic Field .............................. C. Electron in a Periodic Pote : The Effective Mass Approximation ............... 111. Experimental Techniques ..... ................................................................ A. Sources and Spectrometers ..................................................................... B. Methods of Detection ........................................................................... IV. Free Carrier Resonances ........................................................................... A. Electron Cyclotron Resonance ..... ..................... ................ B. Spin-Flip Resonances ........................................................................... C. Hole Cyclotron Resonance ..................................................................... References.. ..............................................................................................

1 2 3 6 11

17 17 25

28 28 42 55 15

I. INTRODUCTION The experimental and theoretical determination of the electronic structure of semiconductors has been of considerable interest for a number of years. Optical and magneto-optical studies have proved extremely valuable in such investigations. The optical experiments divide naturally into two categories according to whether the transitions involve quantum states in only a single energy band, intraband transitions, or states in two bands, interband transitions. Due to the nature of the energy bands, effective masses, etc. and available magnetic fields, this natural division also carries over to the spectral region in which the two types of transitions are observed; interband transitions occur in the near infrared through the ultraviolet while intraband transitions occur typically in the far infrared (FIR) or microwave region. Although interband optical and magneto-optical measurements have provided a great deal of useful information (I, Z), intraband magneto-optical studies provide more detailed and precise information concerning effective masses and g-factors of both electrons and holes, impurity I

2

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

states, and interactions among the single particle states and collective excitations such as phonons and plasmons. In recent years significant advances in FIR instrumentation have made possible the detailed investigation of a wealth of intraband phenomena with an attendant increase in information and understanding of the quantum electron (or hole) states and their interactions. This rapid evolution and expansion in knowledge indicates the need for an up-to-date in-depth review of this field.* On the one hand, it is important to understand what has been accomplished and what remains poorly understood; on the other hand, the rapid proliferation of experimental data has led to the publication of some misleading and/or erroneous results. In this article we present a review of the recent developments in intraband magneto-optics of semiconductors with emphasis on the above points. In order to achieve these aims it is necessary to ignore for the most part other useful experimental approaches and also similar studies of other material types, e.g. metals and insulators. In addition, we have made a number of subjective judgments concerning the material to be included and the method of presentation. For example, classical magnetoplasma effects are not discussed since these studies have been extensively reviewed by Palik and Furdyna (5). Finally, although the title delineates a rather precise spectral region (the FIR is usually defined to be between 50 and 1000 pm), in the interest of clarity and cohesiveness we have overstepped these spectral boundaries in several instances. The rationale for these judgments will be apparent in the discussion of experimental results. This review will be divided into two parts. The first part, presented here, treats both theoretical and experimental investigations of free carrier magneto-optical transitions in semiconductors. In addition, a brief section on experimental techniques is included in this part. Part 11, to be published in the next issue of this series, will discuss bound carrier magneto-optical studies and the interactions of both bound and free carriers with collective excitations, e.g. phonons and plasmons.

BACKGROUND 11. THEORETICAL In order to obtain information about the electronic states of semiconductors, one must be able to relate the macroscopic information provided by experiments, e.g. the sample transmission or reflection versus magnetic field, to the microscopic states and optical transition probabilities of the charged particles under investigation. In this section we present a brief description of * F o r a review of early, primarily microwave, intraband studies, the reader is relerred to Lax and Mavroides ( 3 ) . A recent review of magneto-optical experiments in general by Mavroides ( 4 ) has some information concerning the subject of this present work.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

3

the macroscopic optical properties of solids, proceeding from Maxwell’s equations, and we then show how these are related to the microscopic energy levels, wavefunctions, and transition probabilities. We also present a quantum mechanical discussion of the motion of a nonrelativistic electron in the presence of a uniform external magnetic field. This discussion is initially confined to free electrons, and then the necessary generalization to include the effects of the periodic array of ions in a crystalline solid is sketched. The nature of the energy levels and the selection rules for electromagnetic (EM) transitions (cyclotron resonance and spin resonance) are discussed in some detail. This provides a sound basis for the examination of the various experiments and the extensions of theory necessary to include the effects of impurities, more complicated energy bands, and the interactions of electrons (or holes) with other elementary excitations (e.g. phonons and plasmons) in semiconduct ors. A . Optical Properties

For a continuous isotropic medium characterized by a real conductivity, and a real dielectric constant, c R ,the following wave equation is obtained from Maxwell’s equations in unrationalized Gaussian units :

(T,

Here E‘ is the electric field vector of the EM radiation, c is the velocity of light in free space, and the magnetic permeability has been set equal to unity since only nonmagnetic or weakly magnetic materials are considered. In an anisotropic medium or in the presence of a magnetic field the tensor character of (T or cR serves to couple the different vector field components. However, for the purposes of exposition and in the interest of simpler notation it is sufficient to consider the isotropic case where (T and cR may be viewed as scalars. The equations are easily generalized when the tensor character must be made manifest. For propagation in the z-direction a solution of Eq. (1) for transverse waves is given by E’ = Eb exp i(mt - K z ) where

Here K is the complex propagation wavevector, o is the angular frequency, and u is the phase velocity of the wave in the medium.

4

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

A complex refractive index may be defined by ti

- iti' = cKJcu,

(3)

where K2

-

2 1 ~= ~ '471010.

/('2 -F ~ :

It is frequently desirable to describe the optical properties in terms of a complex dielectric function, E = eR + iE, defined by I-: = (ti

-

(4)

iK')2

with

c,

=

(5)

21t-h-'.

With these relationships the electric field of the wave may be written

E' = Eb exp ( - W K ' Z / C ) exp iw(t - t i z / c ) ; hence the amplitude of the wave is attenuated exponentially with an attenuation constant o t i ' / c . Experimentally the quantity of interest is the attenuation of power or intensity rather than amplitude. The ratio of the intensity, 1, at a distance z to the initial intensity 1, is I/l,

=

exp ( - 2 w t i ' z / c )

=

exp ( - q,z).

(6)

The absorption coefficient, M , , has the significance that the intensity falls to lie of its initial value in a distance l/xn. The single surface power reflectivity, R, between the medium and vacuum for normal incidence may be written in terms of the real and imaginary parts of the index of refraction

A=

(ti - 1 ) 2

(ti

+

1)2-+

+ tif2 KZ

'

(7)

and the transmission through a thickness d of the medium assuming ti' <

ti

1s

The various relationships for the macroscopic optical properties ( R , T', cq, , ti, etc.) must now be related to the microscopic properties of the system

under investigation. The total Hamiltonian of a system consisting of charged

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

5

particles plus radiation can be written as

1 (9)

where A‘ and cp’ are the vector and scalar potential of the radiation field respectively, e is the magnitude of the electronic charge, and the momentum operator p = -ihV. The electric and magnetic fields of the EM wave are given by E’ = - V p ’ - (l/’c)(ZA’/dt) and B’ = V x A‘ in terms of these potentials. In this treatment the particles are treated quantum mechanically and the EM field semiclassically. The radiation field could also be quantized and written in terms of creation and annihilation operators for photons. The result for the transition rate between unperturbed levels of the system of charged particles is the same in both cases. If a gauge is chosen such that V - A ’ = 0 and

p‘ = 0,

the interaction term between the EM field and the particles is found to be given by

The transition probability per unit time from state j’ to state j of the unperturbed particle Hamiltonian is obtained in second-order perturbation theory as

where

q;: = d X y v x i ” t Y y= ( j

I .wintI ,?),

Y j is the wavefunction of the unperturbed state, and dx

= dx d y dz. If A‘ is taken to be a simple harmonic field of amplitude Ah and polarization vector the transition rate is given in the dipole approximation by

p,

w . . ,= jJ

-

2ne2At I [t . p].., J J 12 qcl, h2 &c2

(0. J - cl,.,)]. J

The power absorption coefficient, c t 0 , for this transition is simply the power absorbed per unit volume divided by the energy flux (power/area) in the wave. Hence to obtain a o , w j j , is multiplied by the photon energy per transition and divided by the volume times the time-averaged Poynting

6

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

vector. With the occupation factors for initial and final states and the correction for stimulated emission included, a, is given by

Here j i is the probability that state ,j is occupied, Q is the volume, and the sum goes over all initial and final states of the system which satisfy energy conservation. The b-function in Eq. (12a) implies statesj andj' that are infinitely sharp. In practice the energy states are broadened to some degree. This may be taken into account by replacing the &function with a normalized broadening function usually of Lorentzian form, i.e.

( 12b) where ;represents a phenomenological broadening constant.

B. Quantum Theory of'Free Electrons in a Magnetic Field 1. Eizeryy Leuels

We consider a single electron of mass mo confined in a cubic box of sides Lo and volume R = L:. (The intrinsic electron spin is neglected for the present for simplicity. It is easily included at a later point.) The electron is subjected to a uniform dc magnetic field of induction B in the presence of an EM disturbance of frequency to. The vector and scalar potentials of the harmonic EM disturbance are A' and cp' as in Section II,A, and the same gauge is chosen. The dc magnetic induction is taken parallel to the z-axis with vector potential *

A

=

( -+BJ,~ B x0). ,

Under these conditions the Hamiltonian for a single electron is given by

F

=

2mo j p + ' A C +eAr)2. C

"This gauge (the symmetric gauge) IS particularly useful in probiding a clear picture of the ph)sics iinderlying the energy levels and the section rules for E M transitions [see. e.g.. Dicke and Wittke (6) and Johnson and Lippmann (7)]. It is also convenient for discussing the hydrogenic impurity states in a magnetic field to be considered in a later section. Another. more commonly used gauge, is the Landau gauge ( 8 ) A = ( - BJ,0. 0).

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

7

where the fact that the electronic charge is negative has been used. Neglecting terms of order A”, we may write

X o = - - 1( p 2 + 2m0

2e

v

+ eA/c)

C

and

In Eq. (14b) =X =

(l/mO)(p

P/mo .

Explicitly making use of the symmetric gauge, we have No =

1 ~

2m0

(p:

+ p?) +

(2+ y’) +

eB --L, 2mo c

+

Pt 2m0

___

where L, = ( x p y - y p x ) = z-component of the total orbital angular momentum. The first two terms of Eq. (15) represent the Hamiltonian for a twodimensional simple harmonic oscillator. It is easily shown that p,, YFsH0, and L, are three independent mutually commuting operators ; hence eigenfunctions may be chosen which are simultaneous eigenfunctions of all three. The energy can thus be separated into three parts: (1) the energy associated with motion in the z-direction, which is unchanged by the presence of the magnetic field; ( 2 ) the energy associated with motion in the x- and ydirections, which is the sum of energies of two linear simple harmonic oscillators of frequency 0,,/2 = eB/2mo c, where o,,is the cyclotron frequency; and (3) the energy associated with the operator L, , the angular momentum about the magnetic field. The wavefunctions may be written schematically as*

*The symbol k,, is taken to be the electron wavevector along the magnetic field, i.e. the eigenvalue of p,/h in this case.

8

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

where the three quantum numbers are defined by the eigenvalue equations

Thus the total energy eigenvalue is given by

where 17,

20

and

M,2

-in,

2 -x.

From parity considerations it may be shown that no and in, are either both even or both odd integers. Hence the more commonly used “Landau” quantum number, 11, may be defined by no

+ i n 1 = 2n 2 0,

and the wavefunctions specified equivalently by familiar form for the energy levels is obtained,

(21)

1 iz, in, , k H ) . Thus the more

Equation (22) for the energy levels is found directly if one uses the Landau gauge for A; however, the physics of the EM transitions among these levels is much more clearly seen in the symmetric gauge. A useful energy level diagram which shows the relationship among the three quantum numbers n, n o , and m, is given in Fig. 1 for kH = const. Since for any value of M there are a large number of combinations of no and m, which satisfy Eq. (21) (an infinite number for nz, -+ m) each energy state specified by n and kH in Eq. (22)is highly degenerate, as is indicated in Fig. 1. Thus far the intrinsic electron spin, s, has been omitted from the discussion. The effect of including spin is easily found by adding a term

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

-

n= 5 -

-

-

/

/

0

3

2

I

-,

/ -

/

9

4

5

p = n-ml= %-n

FIG. I . Schematic energy level diagram showing the relationship among the various quantum numbers described in the text. Typical cyclotron resonance transitions are indicated by the bold vertical arrows. The parity of the state is odd when both no and 111, are odd, and even when both no and m, are even. Each cyclotron resonance transition satisfies the selection rule for parity (An = odd) and for the z-component of angular momentum (Am, = + I).

to X o[Eq. (15)]. Since s commutes with all other operators of the Hamiltonian, the total wavefunction may be written as a simple product wavefunction

k , m,> = I m, k H ) I m,>, where Im,) is a two component spinor. The spin operators are 2 x 2 matrices which may be written in terms of the Pauli spin matrices as

I fi, m,

I

3

3

s

=

(h/2)o

with 0 -i

0 1

1 0

The energy eigenvalues of spin are Emv =

eh - Bm, m0 C

=

msgpBB,

where pB = I eh/2mo c I is the Bohr magneton, y is the electron g-value (= 2), and m, = k 1/2 is the spin quantum number. (The spin quantum number m, = +1/2 for the z-component of spin parallel to B, and ms = - 1/2 for the z-component of spin antiparallel to B.) Hence the total

10

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

electron energy is

The energy levels including spin are thus described by two sets of parabolas as a function of k,, one for spin parallel and one for spin antiparallel to the magnetic field. For a given n, the energy displacement between the two sets is gpBB, the spin energy, while for a given s, the energy displacement between adjacent levels within a set is ~ c J J the ~ ~ cyclotron . energy. 2. Optical Transitioi? urid Selrctioii Rules ci. Electric-dipole iiiteraction. The EM selection rules are most easily obtained from a consideration of the *' raising" and "lowering" operators for Landau levels, at and u, defined by

u' with

d =

=

[ P x + iP,]/h(Za)' l ; a

=

[P,

-

iP,]/h(2a)'/2

(28)

eB/hc. These operators have the following commutation rules

[a, u + ] = (aa'

-

a+a) =

I ;[n , u']

=

hco,at

;[ X

, a] = hcoc, a.

It is easily established that

I I?, 111, , k , , m,) a 1 11, 117, , k, , m,)

ut

= (I7 = 17'

+1

y I It

+ 1, + 1, k,

I 17 - 1, I T Z ~

112,

-

1, k , , tn,),

, n1J, (29)

that is, operation on an eigenfunction by at(.) yields a new eigenfunction with both n and i i i l increased (decreased) by unity. This corresponds to a vertical transition between adjacent levels on Fig. 1. Since in the presence of an external magnetic field the radiation interaction term is proportional to P (rather than p) [Eq. (14b)], the transition probability is proportional to matrix elements of P. For a circularly polarized wave propagating parallel to the z-axis the relevant quantities are P' = P, & iP,. . The operators P' can be rewritten in terms of at and u in order to obtain the selection rules from Eq. (29). For left circular polarization" (P' ) the selection rules are:

A I I=

+ I,

A I I I= ,

+ 1,

AUI,\= 0,

(30)

* In the circular polarization convention used here left circular polarization corresponds t o counterclockwise rotation of thc electric field vector of the wave 111 time when the wave is vie\\ed head on by an observer. i.e. with the magnetic field vector pointing or the observer [see. e.g.. Klein (9): this is a useful reference for general macroscopic optical properties]. To avoid an? possihle confusion over conventions, howcvcr, it is simpler to refer to cyclotron resonance active ( C R A ) and cyclotron resonance inactive (CRI) senses of circular polarization for electron,. This terminology will be used throughout the remainder of the article. For conipariwn. in

11

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

and for right circular polarization ( P - )

0. (31) Since the linear momentum of the FIR photon is typically small, conservation of momentum yields AkH z 0. These selection rules have a simple physical interpretation in terms of conservation of parity and z-component of angular momentum (in,). h. Muynetic-dipole interaction. The interaction of the spin angular momentum with the EM radiation is taken into account by adding a term An

= - 1,

Ant, =

-

1,

Am,

=

to the particle-radiation interaction Hamiltonian, Eq. (14b), where ;7 is a unit vector along B' and B, = I B 1. The optical magnetic induction, B , is polarized perpendicular to E' (and A') for linear polarizations; the sense of circular polarization is the same for B' and E'. This term can be treated in the same way as the equivalent term for electric-dipole interaction, and the transition probability is proportional to the square of matrix elements of fi . s. For circularly polarized radiation the quantities of interest are .s* = s, is,. It is easily shown that .s' are raising and lowering operators for the zcomponent of spin. Hence, the selection rules for CRA circular polarization are

A I ~=, 0, and for CRI circular polarization An

=

0,

A11

=

0,

AIR,= 0,

Ams = - 1,

(33)

+ 1.

(34)

Am5 =

This result again derives physically from conservation of the zcomponent of angular momentum. From conservation of linear momentum we have the additional condition AkH z 0. C. Electron in a Periodic Potential: The E f f c t i v e Mass Approximation 1. Eliergy Letlels

In the absence of spin and any external perturbations the solutions of the Schrodinger equation for an electron placed in the periodic potential, V(x), of a crystal lattice are given in terms of Bloch functions [see, e.g., Kittel (lo)] by yk(x) = eik'ylfk(x), (35) the above convention left circular polarization is CRA and right circular polarization is CRI for electrons.

12

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

where uk(x) is a function with the periodicity of the lattice (cell periodic function) and k is the wavevector (hk is the crystal momentum). Substitution of the Bloch functions for band r into the Schrodinger equation yields an equation for the cell periodic functions of the form

x, =

h2k2 p2 + - + V(x), 2m0 2m,

and

For small k around a band extremum, taken to be k = 0 for simplicity, Zk. may be treated as a perturbation. The energy is given to order k2 as

and

with R, the volume of a unit cell, u,, the band edge (k = 0) cell periodic functions, and E,(O) the energy of band r at k = 0. Here, (2m0/h2)@ is the reciprocal effective mass tensor for band r, and 6,, is the Kronecker delta. In the presence of an external magnetic field and the crystal potential, a more realistic Hamiltonian for a single electron including spin and the spinorbit interaction is given by X =

P2

2m, ~

+ V(X) + 4mih c2 - V V ( X ) x P + ~~

(r

e m, c

B

S.

(38)

It is easily shown that with spin-orbit interaction the velocity operator becomes P

h

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

13

Generally this correction to P is small and it will be neglected in what follows. The inclusion of an external magnetic field presents some difficulty because the unperturbed wavefunctions (Bloch functions) extend throughout the crystal. Thus the effects on the Bloch functions cannot be considered small, even though on an atomic scale of energies the magnetic field energy is certainly small. Luttinger and Kohn (LK) (11) solved this problem by recognizing that if the electrons of interest occupy a small region of k-space a b u t some k where the energy is an extremum (taken to be k = 0 for simplicity) a perturbation treatment in powers of the wavevector about k = 0 could still be used to approximate the Hamiltonian. A suitable complete orthonormal set of basis functions (LK functions) in this case is (39) Hence the solutions of the Sclirodinger equation %'" = E", where .Pis the Hamiltonian of Eq. (38), may be expanded in the LK functions with expansion coefficients Ar(k) x r , k,

in,(.)

=

etk ' x L 1 r0. . in,(.)-

here the spin index has been included in the band index, r . An eigenvalue equation for Ar(k) is found in the form

i

1

dk' dxeik'X.#'r,,eik"XAr(k')

=

EAr(k),

(41)

1st

Brill. w n e

where the Hamiltonian matrix (12, 13) is given by

Here, h& = P^ = p" + eA,/c, the 1, are the direction cosines of the magnetic field, the difference between Paand pa is neglected in the matrix element, and repeated Greek indices are summed over. With the introduction of the Fourier transform of Ar(k), Fr(x)=

1

eik'"Ar(k)dk,

1 s t B.z.

and the assumption that both the vector potential and F,(x) are slowly varying over a unit cell, an infinite set of coupled equations in real space can

14

B R U C F D. MCCOMRF 4 N D RORFRT J. WAGNFR

be obtained

In this equation the F,(x) are the real space “envelope” functions. The total wavefunction is given by Y

c F,.(x)u,,o(x).

=

(44)

).’

The problem now is to remove the off-diagonal terms in the band index and obtain an “effective” Hamiltonian for a single band as was done in the zero field case. Luttinger and Kohn utilized a series of unitary transformations to accomplish this, and thereby obtained a transformed effective Hamiltonian and transformed wavefunctions. To second order they found an effective mass equation in real space for a simple, nondegenerate (except for spin) band given by [D:,!P, P, -tpB Bib,o;.]F,.(x)= ( E , - E,(O))F,(x),

(45)

where the total wavefunction for band r is given in lowest order by

(46)

Y,.@)2 F,(X)&*(X).

The “effective.’ Hamiltonian for a single band can be rewritten as .Y?

eff

=

Dg’{P, , Pal + D;O[P, , PSI

+ /lB B?,,cJ;,,

(47)

where the band index has been dropped to simplify notation. The terms have been separated into symmetric (S) and antisymmetric (A) parts to take account of the fact that the P,’s do not commute in general. This lack of commutativity can lead to some important differences in the energy levels and selection rules when compared to the free electron case as we shall see below. In Eq. (47) ; P I , P,)

and Dc”

+ P, P, .

= P, P,

=

i(D:B

PA’= i(DZ,!

+ Dff), -

D!,?).

I n order to compare directly with the free electron discussion, we specialize to the case of a cubic semiconductor having inversion symmetry and take the Cartesian axes along the cube axes with the magnetic field in the :-direction. For this case (14) Dgfl = 0:: 6,, ,

DY

= -Dpz

A.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

15

and we have

It can also be shown that

where

The expectation values of [L,],.,for the two different spin states of band r (i.e. m, = 112 and m, = - 1/2) are equal in magnitude and opposite in sign. Hence [L,Jr,contributes to the effective spin magnetic moment of band r ; thus we may write

+

Y l eff =

D;:p, p ,

+ Y*VB

(Jz

B,

(48)

where g*

=

2(1

+ [L,],,).

(49)

The effective y-factor, g*, can be either positive or negative since [I,,],, can be large in magnitude and positive or negative. Thus the noncommutativity of the P,’s in a magnetic field may lead to an additional contribution to the spin magnetic moment and an effective y-factor which, in some cases, can be extremely large. This additional contribution vanishes in the absence of spin-orbit interaction. The effective Hamiltonian, Eq. (48), is identical to the free electron Hamiltonian with the free mass and g-factor replaced by the effective mass and effective g-factor. Hence the energy levels and wavefunctions can be found by simply taking the free electron energy levels and wavefunctions and making the appropriate substitutions. Although the simple effective mass Hamiltonian for a single band is not adequate to provide a quantitative description of the electronic states in all semiconductors, it does give a good qualitative picture for nondegenerate bands in most materials. This approach also yields quite good results for the problem of hydrogenic impurities discussed in Section V,A, and, with a minor extension, it gives a reasonable picture of the magnetic field states for the degenerate valence bands in diamond and zinc blende semiconductors. In addition, the basic theoretical development [through Eq. (43)] provides the basis for more extensive and detailed calculations which are sometimes necessary for a quantitative description. (See Sections IV,A,1 and IV,C,l.)

16

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

2. Matrix Elements The matrix element for electric dipole transitions between states 'Piand Y f is given in lowest order by

In the effective mass approximation with the zero-order wavefunctions of Eq. (39) this reduces to

c%

[Z.

PIif

.j.

dXFT(X)Fi(X)

crystal

unit cell

crystal

where

[ Z f PI,/

1 dxu$,(x)[?. p]u,,(x).

=

szo

";I1

cell

The first term of Eq. (50) is referred to as the interband term and the second is referred to as the intraband term. In the simple effective mass case under consideration, this separation is well defined ; thus we are concerned only with the second term. In the absence of spin-orbit interaction, the envelope functions Fi may be written in a form identical to the free electron functions of Section I1,B. Hence the selection rules are given by the free electron selection rules. Eqs. (30) and (31). A similar treatment may be carried through for magnetic-dipole interaction and the selection rules are again given by Eqs. (33) and (34). For a typical case of electrons in a crystal where there is nonzero spinorbit interaction, the situation is more complicated. In this case the antisymmetric term in the effective Hamiltonian can mix the spin and orbital parts of the wavefunctions as noted in the previous section. It follows that, even for a '' simple " band, electricdipole transitions involving a change in spin quantum number (and conversely magnetic-dipole transitions involving a change in Landau quantum number), can become weakly allowed. Such transitions are discussed in Section IV,B. In the case of several strongly interacting bands the lowest order

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

17

wavefunctions for a given band are, in general, composed of a mixture of products of band edge functions and envelope functions from all the interacting bands (15). In this case, a division into “ interband and “ intraband parts is no longer as useful, since both types of terms can contribute to the experimentally observed intraband transition (16). ”

”

111. EXPERIMENTAL TECHNIQUES Since much of the recent progress in semiconductor magneto-optical studies in the FIR has been a direct result of advances in experimental techniques, we include here a brief description of FIR methods. The intent of the present section is not to provide a detailed description of all the FIR instrumentation and equipment presently in use in the laboratory, but rather to give a short overview, comparing a number of the commonly used systems and pointing out their advantages and disadvantages. This allows the ensuing sections which are concerned with experimental studies of the physics of various magneto-optical effects to be relatively free of comments on experimental technique. For the reader interested in a more detailed description a number of excellent articles exist [see, e.g., Robinson (17), Houghton and Smith (It?), Chantry (19),and Moller and Rothschild (20)].In part A, the advantages and disadvantages of various FIR sources and spectrometers are discussed, and in part B, a number of devices and schemes for detecting FIR radiation are described and compared.

A . Sources and Spectrometers

Although FIR magnetospectroscopy has a long history ( 2 1 ) it has only recently become a popular experimental technique. This has been primarily due to the lack of radiant power available from classical, hot bodies in this spectral region. For example, an “ideal” blackbody of area 1 cm2 at a temperature of 2000°K provides less than 200 pW in the total spectrum of wavelengths longer than 50 pm with f l 3 optics. In contrast, FIR lasers now provide more power than this in many of the large number ( z 300) of lines a-Jailable in this region. Nonetheless, classical sources are still useful since they radiate continuously throughout the infrared, whereas a continuously tunable laser over any significant range in the FIR has not yet been achieved * *At present, a number of approaches to continuously tunable FIR laser radiation are under active laboratory investigation (22a-e); however, as of this writing a practical source for spectroscopy is not available.

18

BRUCE D. MCCOMBE AND ROBERT J . WAGNER

1. Grating Spectrometers

Historically, the first technique used in FIR magneto-optical studies of semiconductors was a classical source in conjunction with a monochromator. At its simplest, the monochromator consists of a prism or grating which disperses the radiation spatially with a slit (or slits) selecting a suitable small portion of the spectrum for the intended experiment. The obvious limitation of this procedure is that increased wavelength selectivity (resolution) brings with it a concomitant decrease in the amount of radiation in the selected wavelength increment. In practice, beyond 50 pm wavelength, only gratings are used since no adequate transparent dispersive materials exist. The use of gratings involves an additional complication since the radiation dispersed at a selected angle consists not only of the wavelength desired, but also higher orders (shorter wavelength submultiples A/2, A/3, etc.). Classical blackbodies such as a hot S i c rod or “globar” produce these higher order wavelengths with much greater intensity than the fundamental wavelength. A somewhat more useful source in the FIR is the “high pressure” (1-2 atm) Hg-arc. This source radiates much less than an equivalent black body in the near infrared since, in this region, the intensity is due to a continuum radiation resulting from electron-neutral atom collisions in the plasma. At longer wavelengths the plasma becomes self-absorbing, and the radiation spectrum approaches that of a blackbody at the plasma temperature (= 6000°K). In spite of this intensity reduction at higher frequencies, low (frequency) pass filters are still required to eliminate the unwanted higher orders. Although there are rather sophisticated filters in use (these range from transmission filters, such as capacitive metal meshes and crystal powder suspensions, to reflection filters, such as gratings and inductive metal meshes (23)) they are still somewhat lossy and thus reduce the power available. Consequently, monochromators are designed to collect a large solid angle of the radiant flux by using smallf-number optics. This necessitates the use of very large gratings and mirrors; and as a result, such spectrometers are large and bulky. Generally, these spectrometers are flushed carefully with dry air or evacuated in order to eliminate atmospheric absorption bands which occur throughout much of the FIR. In the latter case, the spectrometer is generally enclosed in a large cylindrical tank to withstand the resulting pressure difference. A grating spectrometer system designed for use in semiconductor, magneto-optical studies is shown in Fig. 2. Although many experiments can be performed by sweeping magnetic field with a fixed wavelength radiation incident on the specimen, it is frequently desirable to sweep the wavelength at fixed magnetic field. The latter is usually more difficult because the radiation intensity varies due to

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

19

FIG.2. A sophisticated FIR grating spectrometer system designed by D. Dickey. Here B is the Hg-arc source, G is a 7.5 x 8 in. grating. and the detector is located either at W (transmission experiment) or W' (reflection experiment). [From Johnson and Dickey (42).]

wavelength dependence of source intensity, filter response, detector response, and nionochromator throughput, in addition to possible atmospheric absorption bands. Thus it is necessary to remove the background variation by some means. This is usually accomplished by recording a spectrum with magnetic field on, and then a spectrum with field off, and dividing the two. Such a procedure has the additional advantage of removing field independent variations in the transmission of the sample itself. However, this advantage is obtained at the expense of doubling the noise and the requirements for system stability. In spite of the difficulties mentioned above it is possible with grating spectrometers to achieve a resolution of about 0.1 cmbetween 30 and 60 cm-' (24). 2. Fourier Transform Spectrometers

The ready availability of high speed digital computers has brought the technique of Fourier Transform Spectroscopy (FTS) into widespread use*

* For

an extensive and up-to-date general reference to this subject see Bell (25).

20

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

Due to the development of Fast Fourier Transform computation methods (26) and “mini” computers with sufficient speed and storage capability, this technique has evolved to the stage where it can be utilized even in laboratories where a large digital computation facility is not available. The FT spectrometer includes an interferometer which can be of the Michelson or Lamellar type, associated source, detector, signal processing electronics, and some means of calculating the Fourier transform of the interferogram. The interferometer itself consists of collimating optics, some means of dividing the source beam (a beamsplitter in the case of the Michelson interferometer, and large interleaved gratings for the Lamellar interferometer), and a drive unit to vary the path difference between the two beams. At present, a number of interferometers and complete Fourier Transform spectrometers are available commercially. A commercial Michelson interferometer is shown schematically in Fig. 3. The basic operating principle of these systems can be stated simply as follows: the interference pattern of a two beam interferometer obtained as a function of the path difference, x, between the two beams is the Fourier transform of the spectrum of the light passing through the interferometer. For a monochromatic source the interference pattern is a simple cosine function, and the intensity reaching the detector may be written as “

”

9(X)

= S[ 1

+ cos (2713X)],

(52) where 3 is the frequency of the monochromatic source in inverse centimeters, S = 0/27cc. If there is some spectral distribution of radiation from the source, S(il), then by simple extension of Eq. (52),the intensity at the detector will be given by Y(,)

=

1

1

d?S(G)[l + cos

(27LGX)]

‘ 0 s^

= +Y(O)

+ ’(.0

dSS(3) cos (271ijx),

(53)

where Y ( 0 ) is the intensity for zero path difference. Since Y ( x )is an even function of x, one may use the Fourier cosine transform to obtain the spectrum of the radiation reaching the detector: S(C) = 4

I’

1

‘ 0

L/S[;jY(X)

- +
(276x).

(54)

This is the basic equation of Fourier transform spectroscopy (FTS). In practice, of course, infinite path difference cannot be obtained, and the interferogram must be terminated at some maximum value of path difference, x,, . This sharp cutoff can produce artificial, and undesirable, sidebands on

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

21

M

d

-

d

wc 0

L

a 0

L

d

m

i;l

22

BRUCE D. MCCOMBE A N D ROBERT J . WAGNER

narrow line structure in the spectrum; hence the sharp cutoff is generally removed in the computed spectrum by a technique called apodization,” i.e. multiplying the interferogram by a function which smoothly goes to zero. This removes the sidebands at the cost of reduced resolution. (The resolution is approximately given by A? z 1/xmax.) In addition, the sampling interval cannot be made infinitesimal, and data must be taken at discrete values of path difference such that N Ax = x,, ,where Ax is the sampling interval and N‘ is the number of samples. From information theory, it can be shown that Ax must be chosen such that Ax 5 1/2tmaX where is the maximum frequency of interest. With these limitations the desired spectrum is easily obtained by digital techniques from Eq. (54) after converting the integral to an appropriate sum. The primary disadvantage of this technique is the requirement of fairly complicated computer processing of the data as obtained from the instrument (the interferogram). This means that there is usually a significant time delay before the data can be seen in the desired, interpretable form (the spectrum). However, through the use of small “mini” computers, this processing can now be done essentially on line in the laboratory and the spectrum obtained almost immediately. In any case, this disadvantage is more than offset by two advantages which are obtained with FTS. The first of these is the increased throughput of the F T spectrometer as compared to a grating spectrometer. Since the grating monochromator uses slits, it is essentially a one-dimensional instrument. The interferometer, on the other hand, is a two-dimensional instrument and thus possesses an inherent advantage in terms of throughput. In addition, the filtering requirements are somewhat less restrictive for the FT spectrometer; this also increases the amount of power available. The second advantage of FTS is the so-called multiplex or Fellgett advantage (27).The grating spectrometer is a sequential instrument which “sees” only one spectral element at a time at the detector. On the other hand, the FT spectrometer simultaneously sees all spectral elements throughout the measurement with a single detector (hence the term multiplex), and the spectrum is unraveled later by taking the Fourier transform of the observed interferogram. The advantage is obtained from the fact that the integrated signal is proportional to t, the time available for the measurement, whereas the noise, being random, is proportipnal to t ’ l 2 . In the sequential system with In spectral elements there is a time Tpiz available for each element, where T is the total time for the measurement; hence the signal to noise is proportional to (T/m)’/’. In the simultaneous system with the same total measurement time, there is a tlme T available for every spectral element and the signal to noise is proportional to T’ ‘. Thus for a constant measurement time there is an improvement in signal to noise for the FTS “

“

”

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

23

system which is proportional to wz1I2. Alternatively, for a constant signal to noise the number of spectral elements (resolution) can be improved, or the time required to take the data can be reduced in the FT spectrometer compared with the grating spectrometer. The actual improvement in performance from the Fellgett and throughput advantages depends to some extent on a number of external parameters such as detector response and size, spectral region to be investigated, source, particular experiment, etc. However, it is safe to say that FTS is more advantageous for high resolution studies of large spectral regions in the FIR. The best interferometer resolution obtained in solid state studies appears to be 0.035 cm-' (28). The question of the best technique for specific experiments is often a matter of dispute and the choice depends to a large extent on the background and prejudices of the experimenter. Because of the nature of the technique, experiments must be done at a fixed magnetic field. As in the case with the grating spectrometer, source intensity, detector sensitivity, etc. may vary with wavelength, and again this makes it necessary to obtain background data (field off) with which to normalize the sample data. Although wavelength-dependent spectroscopy is generally less convenient than magnetic-field-dependent spectroscopy, one important advantage should be mentioned. Some magneto-optical phenomena have only a very weak field dependence; in such cases, studies in which the wavelength is varied give unambiguous results. In contrast, studies in which the field is varied can overlook or distort important structure.

3. F I R Laser Sources The discovery of FIR laser action in a large'number of gases" has provided the newest practical device for magneto-optic studies. Hundreds of intense, monochromatic laser lines constitute ideal sources for magnetic field dependent studies. The use of the laser in magneto-optics is probably the simplest of the experimental techniques described here. Essentially, the laser replaces the classical source-monochromator. While it is not appropriate to fully describe the various FIR lasers here, some of their characteristics which are relevant to magnetospectroscopy will be enumerated. Laser action takes place in a large Fabry-Perot cavity, typically 3-4 in. diameter and 6-15 ft long. The flowing low pressure gas of 0.1-2 mm Hg of HCN, H,O, D20, CH30H, CH3CN, etc. is contained in a glass or metal * For a tabulation of lines reported prior to 1971, see Pressley (2%). For more recently reported lines, see Chang and McCee (2Yb),Fetterman et ul. ( ~ Y c )Dyubko , et al. (294, Wagner ef al. (ZYe), Plant et ul. (29f), and Hodges er a/. (299).

24

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

tube of 3-4 in. diameter with length equal to the Fabry-Perot cavity length. Population inversion occurs either through a high voltage discharge in the tube or by optical excitation with a near-infrared (CO,) laser. Both the discharge technique and the optically-pumped technique can be either continuous or pulsed. Each of the four possible variations has its advantages and disadvantages, depending on the needs of the experimenter for intensity, stability, and number of lasing lines. For example, while pulsed lasers typically produce many more lines than continuous lasers, they suffer from

SUPERCONDUCTING

INTEGRATOR

RATIO

€23 RECORDER

FIG.4. Schematic diagram of a pulsed FIR laser spectrometer used for magneto-optical studies of solids. Wavy lines indicate laser radiation. Boxcar” integration and electronic ratiometry are used in this system to reduce pulse-to-pulse instabilities and drift. [From Wagner and Prinz (30).] ‘I

pulse-to-pulse instabilities which are a source of noise. This problem can be overcome by utilizing averaging and ratio techniques. Such a pulsed F I R laser system used for magnetospectroscopy is shown schematically in Fig. 4. Practically, the resolution which can be obtained from this laser used in conjunction with a high homogeneity superconducting magnet is better than 0.01 cm- (30). FIR lasers are ideal for investigating selected spectral features which cannot be resolved through the use of the interferometer or for studying

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

25

magnetic field dependent absorption lines in the presence of strong background absorption or reflection, i s . in essentially opaque samples. Thus the FIR laser system and the FT spectrometer approach can be viewed as complementary techniques in the study of magneto-optical phenomena.

4. Light Pipes While focusing optical components can be used to direct the FIR radiation from source to sample to detector, more frequently light pipe techniques are utilized. It has been shown (31) that the transmission of FIR radiation through metal (brass, copper or stainless steel) tubes can be accomplished with moderate loss. Light pipes are easily purged or evacuated; thus energy loss due to strong atmospheric absorption is avoided. In addition, they are particularly convenient for use in cryogenic dewars and for magnetic field experiments where space is limited. On the other hand, the difference in reflectivity for E parallel and E perpendicular to the light pipe wall can lead to strong polarizing effects. These effects must be carefully considered when interpreting certain experimental results.

B. Methods of Detection * Frequently, the selection of the proper detector is as important to the success of an experiment as the selection of the radiation source/ spectrometer. The invention of sensitive new detectors during the last decade has contributed to the rapid increase in FIR magneto-optics. 1. Rooni Temperature Detectors

Prior to 1960, there existed only two reliable FIR detectors; the Golay cell and the optical thermocouple. In essence, the Golay cell depends on the expansion of gas in a pneumatic chamber when the gas is heated by infrared radiation. This flexes a mirror in the cell wall, and the mirror motion is converted to an electrical signal by a separate visible optical system. The primary advantage of this device lies in the fact that the gas cell absorbs radiation uniformly throughout the far infrared; hence it is a very “flat” detector. O n the other hand, the flexible mirror is quite fragile, which makes this device less reliable than the optical thermocouple. The optical thermocouple operates through the heating action of the infrared radiation incident on the thermocouple junction. The temperature *For a collection of reprints concerning all of the F I R detectors discussed in Sections III,B,I and 111,8,2 including performance characteristics see Arams (32).

26

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

developed at the junction is proportional to the radiation absorbed. This detector, although comparable in sensitivity to the Golay cell for wavelengths shorter than 40 pm does not absorb well in the FIR. As a result it becomes very insensitive beyond 100 pm. Recently, a great deal of development work has centered on room temperature pyroelectric detectors. In these devices, the incident radiation causes a change in detector temperature which in turn causes a spontaneous electric polarization of the pyroelectric material. The time rate of change of the polarization induces a current in an external circuit, which can be amplified, etc. These detectors, while generally less sensitive than the previously discussed devices, can have a much faster time response. Hence they are finding increased application as laser radiation detectors. These detectors, although convenient since they operate at room temperature, have been replaced in most magneto-optical studies of semi-conductors by cryogenic detectors. This has come about for a number of reasons. First, these new detectors are much more sensitive. Second, a number of the cryogenic detectors are much faster. Two of the thermal detectors mentioned above have very long response times (zz 100 msec). As a result, they cannot be used conveniently with pulsed lasers or with fast chopping rates. High frequency chopping is desirable to eliminate the problem of llf noise. (It should be noted that whenever the radiation source is continuous in time, the radiation is turned on and off repetitively with a mechanical “chopper.” This allows the use of synchronous detection techniques, which are desirable to achieve the highest possible signal-to-noise.) Finally, since the sample under study is almost always at cryogenic temperatures, it is reasonably convenient to immerse the detector in the same liquid bath. 2. Cryogenic Detectors

The most frequently used cryogenic detectors are semiconductor bolometers (either Ge or Si), impurity photoconductors (Ge doped with Ga, B, or Zn; or GaAs), and the free electron bolometer (InSb). The bolometer operation depends on the fact that the resistance of the element, either Ge or Si, is a strong function of temperature at the operating temperature. This is achieved by careful doping of the material. The element is then isolated b y some suitable thermal resistance, and absorption of radiation causes a corresponding change in temperature, which is detectable as a change in resistance. It is usually designed to have a rather long response time but has the advantage that its response is very flat in the FIR. This makes it ideal for use with the interferometer. Most of the Ge bolometers which have been used to date have required operation at 1.5”K or lower to

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

27

obtain peak sensitivity. However, both Ge and Si can be used at 4.2"K with some reduction in sensitivity. For peak sensitivity, fastest response, and convenience (operation at 4.2"K), the impurity photoconductive detector or the free electron bolometer (Kinch-Rollin detector with no magnetic field; Putley detector with a magnetic field) are most frequently used. These detectors, however, have a distinct disadvantage; a strongly varying spectral dependence of sensitivity. An understanding of the physical mechanisms which make them useful as detectors will also point out this shortcoming. As discussed in Section V of this paper, donor or acceptor impurities can be introduced into Ge or GaAs. The ground state of these impurities lies 5.0-10.0 meV from the continuum (conduction or valence band). The absorption of an infrared photon by a bound electron (or hole) excites the carrier to the conduction (or valence) band where it is mobile and increases the sample conductivity. However, if the energy of the infrared photon is less than the impurity binding energy, the radiation is not absorbed and no photoconductivity is observed. As a result these detectors typically have sharp low frequency edges in the infrared where the response drops rapidly toward zero. At frequencies lower than the edge they are very insensitive. Recent studies have shown that it is possible to achieve some useful response beyond the photoconductive edge ; however, this is typically accompanied by a degradation of the peak response. The operation of the free electron bolometer depends on the fact that free electrons in InSb have a mobility which rises as T3I2,where T is the electron temperature. Free electrons absorb radiation in the FIR at a rate proportional to A2 ; consequently, at very long wavelengths sufficient radiation is absorbed to heat the electrons and increase their mobility. This is detected as a change in the resistance of the detector element. Because of the A2 dependence of the absorption, the response is very weak at wavelengths shorter than 130 pm, but rises rapidly to a broad maximum around 500 pm. Both the photoconductive detector and the free electron bolometer are very fast detectors, suitable for use with pulsed lasers. In the magnetic-fieldswept experiment, the variation in spectral response is of little importance provided the feature of interest falls within the proper spectral range. 3. Other Detection Techniques Although some variation of the detection schemes described above is generally used, in some cases unique properties of the sample allow it to function as its own detector. For example, a strong absorption occurs at an energy equal to the 1s to 2p hydrogen-like impurity transition in zero magnetic field in GaAs. Although an electron in the 2p state is not mobile, it may

28

BRUCE D. MCCOMBE A K D ROBERT J. WAGNER

be thermally excited into the conduction band and thus provide a photoconductive signal. This photoresponse persists as one applies a magnetic field, albeit at energies reflecting the 1s + 2p+ 1, Is + 2p,, and 1s -,2p- separations. As a result, one can map out the hydrogenic spectrum of the GaAs impurities simply by studying the photoconductivity of the sample (33a. 33h). Since this is a null technique (i.e. there is photoresponse only at the transitions of interest) it can be much more sensitive than transmission studies. A related technique has been used in studies of cyclotron resonance of alkali halides (34).In this case the high resistance of these materials makes it impossible to measure conductivity as such. However, electrodes can be placed on the sample; and when charges are generated in the bulk, the application of an electric field causes these charges to be swept to the electrodes. Here they are sensed with a charge-sensitive amplifier. Self-detection schemes such as those described above, although not universally applicable, can provide an approach to studies of phenomena that are inaccessible via conventional transmission or reflection methods.

IV. FREE CARRIER RESONANCES A . Electron Cyclotron Resonance I . Bobvers and Yafet Model

Far infrared magneto-optical techniques have been particularly successful when applied to the study of zone-centered electrons in zinc blende semiconductors. In many of these materials, the application of modest magnetic fields results in cyclotron resonance conditions in the FIR due to the light mass electrons. Although sophisticated theories have been developed to account for interband magneto-optical results (35, 3 6 ) the experiments discussed here can generally be understood in the light of simpler theoretical models. We will use a model for the conduction (and valence) band suggested by Kane (37) for the zero field case and suitably extended for the presence of a magnetic field by Bowers and Yafet (BY) (38). The relevant energy bands for this case in zero magnetic field are shown in Fig. 5. The top of the valence band (r,)is twofold degenerate (fourfold with spin included) and consists of light and heavy hole bands. The other valence band of interest (r,)is split off to lower energy an amount A by the spin-orbit interaction. The s-like conduction band (r,)and the light hole valence band interact strongly away from k = 0 via the k p interaction. For small E , this

-

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

29

gives rise to large curvatures and correspondingly small effective masses for these two bands. The BY model treats interactions among these four bands (eight with spin) exactly, neglecting interactions with all other bands and neglecting the (generally small) interactions which arise from the inversion asymmetry of the zinc blende structure. This approximation should be good when the energy separations among levels of this set of bands are much smaller than the separation between this set and all other interacting bands. The basis

I

-

k

FIG.5. Schematic energy-momentum relationship at the Brillouin zone center for a typical zinc blende or diamond structure semiconductor. Here the effects of inversion asymmetry are neglected. The group theory designations are those which apply to the zinc blende case.

functions are chosen as the complete set of Luttinger-Kohn functions [Eq. (39)]. With the Hamiltonian of Eq. (38), the equation in real space for the envelope functions, F,(x), is given by Eq. (43). This infinite set ofcoupled equations is now restricted to the eight bands of interest. By choosing as a basis set of cell periodic functions those which diagonalize the zero field Hamiltonian including the spin-orbit interaction, one can obtain an explicit set of equations for the envelope functions. The basis functions are given in Table I. The states of total angular momentum J, and its projection along the z-direction, m, , are also shown (x,y , z are chosen along the cubic axes of the crystal). Periodic functions which transform like atomic s and p functions, under the symmetry operations of the Tf group at r are represented by S and X , Y , 2, respectively. The symbols T and 1 denote the projection of the electron spin with respect to the z-direction. With this set of basis functions, the eigenvalue equations can be written in matrix form (38) as

30

BRUCE

PE0 -E 0 0

0 0 0

D. MCCOMBE (+)1'2fEZ

(3)1'2PE-

0 -E 0 0 0 0

AND ROBERT J. WAGNER

(+)'"P E , 0 0 -E 0

0 Pk + 0 0 0 -E'

0

0

0

0

(+)"2PE,

(3)' 2P I;,

(f)'"P K , - (:)1;2Pk-

0 0 0

0 -A

-

0

E'

(+)'i*P!iZ 0 0 0 0 0 -A-E

=

I

(55) In Eq. (55) = (Lxk i&)/(2)'I2,P is Kane's interband momentum matrix element, P = (-ih/rn,)((S I pz 12)); E' = E - h2e/2rno, and the matrix elements of the free spin term CJ$ have been neglected. TABLE I CELLPERIODIC FUNCTIONS FOR

THE

Function

BOWERSAI\D YAFET MODEL' J.

inJ

Energy

" T h e characterization of the functions by the total angular momentum quantum numbers J . i n J is indicated.

Since all the Fi in Eq. (55) except F , and F , can be eliminated, the following two uncoupled equations are obtained: [ ( E , - E)E(E + A ) + P 2 C ( E + 2A/3) + P 2 h 2 A d /3 ]F , ( = ~ )0, [ ( E , - E ) E ( E A ) P 2 e ( E 2A/3) - P2h2Arr/3]F,(x)= 0.

+ +

+

(56) Here the small diagonal terms h 2 c / 2 m ocontaining the inverse free electron mass have been neglected. They can be included in the final results for the effective mass and g-factor. The functions F , and F , are orthogonal and solutions to Eq. (56) are obtained as harmonic oscillator functions. In the Landau gauge, E,

*Frequently, the momentum matrix element, P, is replaced by E , which is defined as 2m, P2/h'.

=

0.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

A

=

31

( - B y , 0,O)

+

Fl, r= exp (ik,x ik,z)@.,[(y - y,,)~ll/~], (57) where O,, is a normalized harmonic oscillator function and y o = k&. From Eq. (55) it is easily seen that the other F , are also described by harmonic oscillator functions. Thus the total wavefunctions for a given band are characterized by quantum numbers k,, n, and a “spin” projection along the magnetic field, & . The two different orthogonal solutions are labeled - and for the cases F , # 0, F , = F , = 0 and F 2 # 0, F , = F , = 0, respectively. The energy eigenvalue equations for the two orthogonal solutions for the conduction, light hole and split-off hole bands are obtained from Eq. (56) and (57) as

+

E“. +(En.&

-

Eg)(En,i-

+A)

- P2[4(2n

+ 1) + ki][E,, i + 2A/3]

P20A/3 = 0. ( 5 8 )

The heavy hole band which is flat and spin degenerate in this approximation has the eigenvalue E = 0. A useful approximate solution for the conduction band energy levels is obtained for En,+ < E, + 2A/3, where energy is measured from the bottom of the conduction band in zero magnetic field. This approximation is suitable for FIR studies of a number of 111-V and 11-VI semiconductors and has the advantage of yielding a simple expression for the energy levels. The result for the conduction band energy levels is

where o,= eB/m*c, and, neglecting the free electron contribution to the effective mass and g-factor, 1

m*

-

4P2 A + 3E,/2 3E$z2 A + E , ’

and S* =

mo m*A

A

+ 3E9/2’

The cyclotron resonance selection rules discussed in Sections II,B and II,C [Eqs. (30) and (SO)] hold for this nonparabolic model. Here the “spin” state is denoted by (+ ) or (-). However, due to the nonparabolicity, the

32

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

+

t-

w/o spin

w/ spin

FIG.6. Schematic energy level diagram depicting the Bowers and Yafet solution of Eq. (59) for the conduction band at k , = 0. Solid arrows-cyclotron resonance transitions for the two spin states of the lowest Landau level; dashed arrows-spin resonance transitions for the lowest two Landau levels; dot-dashed arrow-combined resonance transition.

separation between adjacent Landau levels of a given spin becomes progressively smaller as the Landau quantum number is increased. These features are illustrated in Fig. 6.

2. Experimental Studies o f Line Positions The initial infrared experiments on cyclotron resonance of electrons in 111-V semiconductors were performed with monochromators and globars or mercury arc lamps as sources of radiation. Here the wavelength limit was about 100 pm. Thus it is not surprising that most early experiments were performed at laboratories with high magnetic fields which were required to bring o,into the range of the monochromator. While these measurements tended to be crude by present standards of instrumentation, they yielded reasonably good values for the band edge effective masses. Furthermore, these experiments indicated the general applicability of the Bowers and Yafet model described above. As an example of the experimental results, some of the data and analysis on InSb will be presented. A brief discussion of experimental results for other materials will be given as an indication of the extent to which FIR cyclotron resonance methods have been used in electron effective mass determinations. a. InSb. The initial observations of infrared cyclotron resonance of electrons in InSb were made by Burstein et a/. (39). More extensive measurements by Palik er d . (40) and Lax et d.(41) followed somewhat different experimental approaches. In the case of Lax and co-workers, extremely high pulsed magnetic fields permitted the use of short wavelength radiation of 10-22 pm. While these measurements were performed at room temperature, they clearly showed the variation of effective mass with magnetic field predicted by the Bowers and Yafet model. Since Lax et al. observed cyclotron resonance only at fields 2 50 kG, it was somewhat difficult to deduce a

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

33

band edge effective mass. O n the other hand, a detailed fit of the theory to the high field results was carried out. The experiments of Palik and co-workers utilized somewhat different instrumentation. Facing a limitation of continuous magnetic fields with a maximum of 70 kG, they required longer wavelength radiation. Out to about 50 pm, radiation from a prism (CsI) or grating monochromator was used. From 50 to 170pm, a series of alkali halide reststrahlen reflection filters were used to isolate narrow ( + 2 c m - l ) bands of radiation. This approach was taken to provide a larger amount of FIR radiation. The cyclotron resonance line in this experiment was significantly narrowed by cooling the sample to 25”K, thus increasing z and w, 5.5 This resulted in a more accurate measurement of line position and, hence, effective mass. The most detailed study of cyclotron resonance in InSb is the work of Johnson and Dickey (42). These workers used a high resolution grating monochromator, a sensitive cryogenic bolometer, and lightly doped, high mobility InSb to study both cyclotron resonance transitions, 0, + 1, and 0, - -+ 1, - (see Fig. 6 ) , as well as one of the spin-flip transitions discussed in the next section. from 4 to 40 kG. A compilation of these results is presented in Fig. 7. The qualitative effect of the k p interaction is apparent, i.e. the plot of E versus B is not linear and the cyclotron resonance transitions from the ( + ) spin state occur at higher energy than those for the ( - ) spin state at the same magnetic field. Johnson and Dickey analyzed these results using a calculation of the four energy levels (0, + ; 0, - ; 1, + ; and 1, - ) from a generalized Bowers and Yafet model. The generalization was to relax the assumption that m* and g* are only affected by the interaction of the conduction band with the p-like valence bands and to include interactions with all other remote bands. These remote band terms were treated as a small perturbation in the calculation and provided corrections to the BY expressions for m* and g*. The calculated cyclotron resonance energies were fit to the experimental data of Fig. (7). The results of the best fit are shown by the solid lines. In obtaining the best fit only data below 18 meV were used due to polaron effects at higher energies (see Section V1,B). From these results JD obtained a band edge effective mass of (0.0139 k 0.0001) m,. While they also obtained a value of E,, care must be exercised in comparing this value with values obtained from the basic BY model previously described. The introduction of FIR lasers has led to a large number of papers reporting studies of cyclotron resonance of electrons in InSb. Some of these studies are primarily concerned with impurity effects and will be discussed in

+

+

-

*Equation (12b) may be used to describe a broadened cyclotron m o n a n c e line by replacing the transition frequency Q,- 01, , by coC.

34

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

the hydrogenic impurity section. Of the studies of free electron cyclotron resonance, several experiments have been performed which yield additional information about carrier properties. Kobayashi and Otsuka ( 4 3 ) have studied cyclotron resonance as a function of bias electric field applied to the

MAGNETIC FIELD ( k G )

FIG. 7. Variation of electron cyclotron resonance transition energies with magnetic field for InSb. The highest energy data points are due to impurity levels to be discussed in Section V.A: the intermediate energy points correspond to 0, + + 1, + ; and finally the lowest energy points correspond to 0. - + 1, - (see Fig. 6). [From Johnson and Dickey ( 4 2 ) . ]

sample. They noted changes of line intensity of the various cyclotron transitions, 0, + + 1, + ; 0, 1, - ; 1, + + 2, +, as the electric field is increased. From this they were able to deduce the effective electron temperature as a function of electric field. Cyclotron resonance has been used by Murotani and Nisida ( 4 4 ) to obtain information about nonequilibrium electron populations and lifetimes. In this work, the electron population was presumed to be shifted from impurity states into free carrier Landau levels by the high intensity laser radiation. This shifts the intensity of the photoconductive or photoHall response at free electron cyclotron resonance relative to the intensity at --f

INTRABANII MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

35

impurity cyclotron resonance. Utilizing these results, Murotani and Nisida deduced qualitative information about the electron concentration and mobility compared to the low intensity (no carrier heating) laser data. They then constructed rate equations to attempt to understand the concentrations and lifetimes that exist for the various free electron and impurity states. They concluded that the intensity of the impurity absorption drops with increasing radiation flux because the carrier population has been shifted to the p2 = 0, Landau level. The experiments discussed above do not provide any basically new information about the properties of InSb. However, they point toward a potentially fruitful application of cyclotron resonance techniques, namely the study of nonequilibrium properties of free and bound carriers. b. InAs, InP, GaAs, and their alloys. As a result of its small energy gap, effects of the k p interaction are strongly revealed in InSb as discussed above. All other 111-V compounds with the same zone center band structure have larger energy gaps, E,; thus the BY model is not expected to provide as accurate a description of the conduction band for these materials. In spite of this a number of cyclotron resonance studies of 111-V semiconductors have been interpreted using the BY model, and it generally appears to provide an adequate description. Cyclotron resonance of electrons in InAs was first studied by Keyes et al. (45), Palik et al. (46), and Palik and Wallis (47). Later, with the aid of continuous magnetic fields to 150 kG, Palik and Stevenson (48) observed both spin-up and spin-down cyclotron resonance and cyclotron resonance involving higher Landau levels. Some of this data is shown in Fig. 8. The calculated line positions from the BY model are indicated by the vertical lines. Most recently, Litton et al. (49) have confirmed the earlier effective mass measurements with m* = (0.0230 k 0.003) m, at 15°K. Cyclotron resonance of electrons in InP has been studied by Palik et al. (46)and Palik and Wallis ( 4 7 ) . More recently Chamberlain et al. (50) using a laser spectrometer have obtained a value of rn* = (0.0803 0.0003) rn, at

-

9°K.

The first infrared cyclotron resonance study of GaAs was performed by Palik et a/. (51). Since this experiment, which utilized bulk material, considerable success has been achieved in the preparation of high purity epitaxial films. As an example, both spin-up and spin-down cyclotron resonance in GaAs have been resolved by Fetterman et al. (52) as shown in Fig. 9. The percentage energy (magnetic field) separation of these two lines is only 0.25 % at 60 kG. These workers found a band edge effective mass of (0.0665 k 0.0001)rn0at 4.5"K. Similar results have been obtained by Chamberlain et ul. (50). In both these experiments the lines are so sharp that laser techniques were required to avoid instrumental broadening.

-

N O

.:-

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

37

W

m

z

0

a v)

W

Lz

W

I + V 3

0 Z

0 V

0

c

0

I 5

MAGNETIC F I E L D

FIG.9. Photoconductive response as a function of magnetic field through the cyclotron resonance region in GaAs. The two lines are associated with 0, + + 1, + and 0, - + 1. cyclotron resonance transitions. [From Fetterman er nl. (52).]

Recently interest has arisen in effective mass measurements on pseudobinary alloy systems with end members InAs, GaAs, InSb, and GaSb. This interest derives in part from the theoretical prediction that the inevitable disorder of Ga and In atoms in GaJn, -,As alloys, for example, will result in disorder-induced valence-conduction band mixing and observable departures of the effective mass from that predicted from k * p theory. While the first measurements of effective mass in these alloys did not use magnetooptical techniques, this seems to be the ideal approach to the problem. In the first magneto-optical measurement, Fetterman et al. (52) studied cyclotron resonance for Ga,In,-,As (0.846 < x < 1) samples. They calculated the anticipated mass values presuming a knowledge of E , , A, and nz* for the end members InAs and GaAs. An empirical relation obtained from other work was utilized to estimate E , for the alloys, and A was assumed to vary linearly across the alloy system. With this input the effective masses were calculated using the BY model and compared with the experimental results. While their calculated mass values involve a poor choice of E,(InAs) and apparent computational errors, it appears that there is a real disagreement between theory and experiment. Berolo ef al. ( 5 3 )have suggested that compositional disorder results in an increase in effective mass over that anticipated from k * p theory, in qualitative agreement with the experiments. It would be interesting to extend these measurements throughout this alloy system as well as to other systems as a more stringent test of the predictions.

38

BRUCE D. MCCOMHE AND ROBERT J. WAGNER

c. CdTe and Cd, H g , -, Te. A number of cyclotron resonance studies have been performed on the IILVI alloy system Cd,Hg, -,Te. While CdTe has the InSb-type zone center band structure (see Fig. 5 ) with an energy gap of about 1.6eV, the effect of alloying with HgTe is to reduce the (conduction)-I-, (valence band) separation. As these bands approach with increased alloying, a very strong k p interaction occurs. For T = 4 K and x = 0.16, T6 and Ts become degenerate at k = 0 with extreme band repulsion of the conduction and light hole bands. For x < 0.16, the r6 band is

-

l- IG. 10. Schematic energy-momentum relationship at the Brillouin zone center for the

.. inverted band structure. Here inversion asymmetry effects are neglected; the group theory ”

designations are those which apply to the zinc blende case.

lower than the Ts bands at k = 0. The repulsive k * p interaction causes the T, light valence band to invert to a TS light conduction band. On the other hand, the r6 conduction band now becomes a light valence band. The heavy mass T, valence band is not affected by the k p interaction with the rb band, and hence the Ts valence and conduction bands remain degenerate at k = 0. This new arrangement of bands, first suggested for a-Sn (54),is shown in Fig. 10. While electron cyclotron resonance experiments for HgTe could be discussed in this section, the degeneracy of the Ts bands requires a more complicated theoretical framework for proper analysis. This framework is discussed in the hole cyclotron resonance section. Hence the experimental studies of HgTe will also be discussed there. Infrared electron cyclotron resonance in CdTe has been observed by Waldman et al. (55). In this material, a strong electron-phonon coupling complicates the analysis of the data. However, the band edge value, m* = 0.0965 nz, at 4.2”K, is obtained after accounting for the coupling. This

-

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

39

is in good agreement with microwave cyclotron resonance results. Additional studies of this material will be discussed in Section VI. The first observations of cyclotron resonance in the Cd,Hg, ._,Te alloys appears to be due to Strauss et al. 56). In a magneto-reflection experiment in Cd,Hg, -,Te (x = 0.17 and 0.14) these authors observed both interband and intraband resonances. Using the Bowers and Yafet model they obtained a band edge effective mass of 4.0 x 10-4m,. Due to the range of photon energies used (40-60 meV) this experiment did not reveal the striking nonlinear behavior of E p h o t o n versus B (w,) that should result from the close approach of conduction and light mass valence bands. More recent FIR data by Kinch and Buss ( 5 7 ) shown in Fig. 1 1 demonstrate this behavior

FIG. 11. Variation of cyclotron resonance (0. + + 1. + ) frequency with magnetic field for Cd, ,,,Hy,,,,,Te. The dashed line is calculated from the BY model. [From Kinch and Buss (571.1

very clearly. The effective mass obtained by fitting to the BY model is m* = 4.66 x 10-3rno.These data also show an “offset” due to the electronphonon interaction which is discussed in Section VI,B. For the above materials a listing of the most reliable band edge effective masses obtained from FIR cyclotron resonance experiments is given in Table 11. d. Other materials. In some cases, the use of far infrared lasers for observing electron (or hole) cyclotron resonance provides an unique advantage over microwave sources. For a given material, higher magnetic fields are required to obtain resonance in the far infrared than in the microwave region. At these higher magnetic fields, it is more likely that w,z > 1 and hence that a clear resonance absorption can be distinguished. With the availability of far infrared lasers and high magnetic fields, one can contemplate studying low mobility materials with relatively short scattering lifetimes.

40

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

TABLE I1

EFFECTIVF MASES FROM FIR CYCL O T R O ~RESOZANCI. Mnteii‘il

T ( K)

InSb

10

0.0139 f 0.0001

InAs

15

0.0230 t_ 0.0003

Johnson and Dickey (42) Litton a at.

I nP

9

0.0803 f 0.0003

Chamberlain

CaAs

4.2

0.0665 f 0.0001

CdTe

4.2

0.0965 i 0.0005

Fetterman er ti/. ( 5 2 ) Waldman Y r d.( 5 5 )

rn*lm,

Reference

(49) (’1

(50)

Few experiments of this type have been reported, but the work of Button

er 01. (58) on SnO, is a good example of the utility of the technique. While previous estimates of electron mass in SnO, varied from 0.1 m, to 0.4 i n o , these workers, using a far infrared laser and high magnetic field solenoids, found a single anisotropic electron band with m* (Bllc) = (0.234 f 0.002) m, and m* (B I e ) = 0.299 4 0.002) m, . The SnO, sample was of quite high quality with a mobility at 97°K of z 9000 cm2/V-sec. In fact, the experiment could have been done using material of much lower mobility. This experiment suggests that useful characterization measurements may be performed on material previously classed as “unsuitable for cyclotron resonance studies. Obviously many large gap, heavy mass semiconductors fall into this category. However in these materials, an additional dilemma frequently arises. In order to obtain the longest scattering time 5, the experimental temperature is frequently less than 100’K. A t these temperatures, large gap semiconductors have very few free carriers. Thus, special detection and (or) carrier generation techniques are required to observe the resonance. In this regard, the microwave work of Hodby ( 3 4 ) on electron cyclotron resonance in the alkali halides is particularly suggestive. Although this optical pumping/photocapacitance approach may not be directly applicable to the FIR, adaptations of this technique seem to be possible. ”

3. Lirie Shape Studies In addition to measurements of peak positions to determine transition energies, the advent of high resolution methods in FIR spectroscopy has made possible detailed studies of cyclotron resonance line shapes. Since the

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

41

linewidth depends sensitively on the scattering mechanism (or mechanisms), such measurements provide an opportunity to obtain selective information concerning the detailed nature of these mechanisms. Thus far, however, very little experimental linewidth data has been obtained in the FIR. In addition there are only a few theoretical calculations [see, e.g., Kawabata (59), and Shin et al. (60)] that have considered the most important scattering mechanism for pure semiconductors (ionized impurity scattering) under the experimental conditions appropriate to FIR cyclotron resonance. A recent calculation by Shin et al. (60) has emphasized the importance of screening of the ionized impurities. For a screened Coulomb scattering potential these authors find that when the magnetic length, 1 = 1/ p = (hc/eB)1/2,is somewhat less than the screening length, a, of the ionized impurities, the cyclotron resonance linewidth in the quantum limit (all electrons in the lowest Landau level) should increase with increasing magnetic field. This is in disagreement with earlier results of Kawabata (59) obtained for an unscreened Coulomb potential. However, the latter calculation is for nonadiabatic (inter-Landau level) scattering; whereas the results of Shin et al. (60) are valid when “adiabatic” (intra-Landau level) scattering predominates. The question of the relative importance of the two types of scattering as a function of magnetic field is an important one. Shin et al. have shown that for l/a < 1 (high magnetic field) “adiabatic” scattering should predominate. The case for l/a B 1 is still not settled theoretically, but *‘ nonadiabatic scattering should certainly be more important in this regime. Experimental results bearing on these calculations are rather limited at this time. The experiments of Ape1 et al. (61) showed a linewidth that narrowed as the magnetic field was increased from 4 to 20 kG. However, this data may be misleading for several reasons: (1) The sample was not of the highest quality, as evidenced by the inagrritude of the linewidths; (2) it was several absorption lengths thick which made precise linewidth measurements difficult due to the exponential nature of the absorption; (3) only three useful data points were obtained; (4) the lowest frequency data were complicated by an impurity absorption line that overlapped the free carrier cyclotron resonance. More recently, Kaplan et a ] . (62) have obtained cyclotron resonance data on thin samples of higher quality InSb. These data converted to scattering times T = ~ / A W ~where ,~, is the equivalent angular frequency h~rlfwidth at half-maximum absorption, is shown in Fig. 12. These data are compared with the calculation of Shin et al. and Kawabata discussed above. The experimental data show a peak in scattering time near l/a = 0.5; i.e., for fields below this maximum in z, the linewidth decreases with increasing field, while at fields above the maximum the width increases with increasing field. “

”

”

42

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

This appears to demonstrate the critical importance of the parameter Ela in determining the field dependence of the linewidth. However, recent measurements on InP (50)show a decrease in linewidth as a function of field over a region where l/a is apparently less than 1, in contrast to the above results.

1008 06 0 5

0.:

04

0.80.7 -

-c m

0

0.60.50.4-

0.3-

/

0

0

5

10

MAGNETIC

15

20

25

FIELD (kOe)

F I G . 12. Cyclotron resonance linewidth expressed as a relaxation time (as defined in the text) as a function of magnetic field for InSb. The solid lines represent the calculations of Kawabata (59) (NONADIABATIC) and Shin er al. (60)(ADIABATIC). An error in the l/a scale and in the ADIABATIC theoretical curve due to neglect of the background dielectric constant has been corrected. [After Kaplan et al. (62).]

It thus appears that additional work, both experimental and theoretical, is required to explain these puzzling results before the full potential of the FIR techniques in elucidating scattering mechanisms can be realized. B. Spin- Flip Resonances

As mentioned in Section I1 the spin-orbit interaction can have a large effect on the energy levels and allowed transitions in a magnetic field. In addition to modifications of the g-factor due to interband coupling of orbital angular momentum into the spin states, the simple selection rules for EM transitions are changed. In general, transitions which change both the '' spin " and Landau quantum numbers become weakly allowed. These tran-

I N T R A B A N D MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. 1

43

sitions can be excited by either the electric or magnetic components of the incident radiation. The electric-dipole interaction, P,,, = eEd1, is typically much larger than the magnetic dipole interaction, &?,,,.d. = g*ehBb/rno c. (For n-InSb in a field of 50 k G I x c . d . l2 = 4 x 10’ I i%fm.d. 12. ) Consequently, in most practical situations it is necessary to consider only transitions excited by electricdipole interaction. Thus in the remainder of this section only electric-dipole spin-flip transitions are discussed, keeping in mind that magnetic-dipole transitions, although allowed, are typically orders of magnitude weaker. The precise nature of the coupling which allows electric-dipole spin-flip transitions via the spin-orbit interaction depends upon band structure details of the particular material considered. For the conduction band of zinc blende semiconductors it has been shown theoretically that electric-dipole spin-flip transitions can arise from two possible sources: (1) the lack of inversion symmetry (63); and (2) a small fundamental energy gap (64,65). For valence bands of diamond and zinc blende semiconductors the application of uniaxial stress removes the degeneracy of the upper valence bands and greatly reduces the complicated mixing of the magnetic energy states. In this case electric-dipole spin-flip transitions can also be identified (66). Additional theoretical calculations for other materials include those for the conduction band of Ge and Si (67), Bi (68), and a-Sn type semiconductors having the inverted zone center band structure (69). While experimental studies of spin-flip resonance have been somewhat limited due to the weak nature of the transitions, in several instances the mechanisms causing the transitions have been unambiguously identified. In addition, studies of these resonances can yield precise and useful band parameters, particularly effective g-factors, and hence provide information complementary to that obtained from the usual cyclotron resonance measurements. “

”

1. Conduction Band of Zinc Blende Structure Semiconductors

Since the most extensive and detailed experimental data exist for the conduction band of zinc blende semiconductors, it is useful to discuss briefly the wavefunctions and matrix elements for this situation. Experimental studies on InSb (70) have shown that the small gap mechanism (64,65) dominates the inversion-asymmetry mechanism (63) for this small gap semiconductor. Transitions due to the latter mechanism have never been conclusively identified experimentally (71) ; hence inversion-asymmetry effects are not included in the following discussion. The conduction band wavefunctions derived from the BY model (see Section IV,A) have been given explicitly by Zawadzki (72):

44

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

45

As a result of the admixture of valence band states, the wavefunction for a given Landau quantum number n and “spin” is composed of both spin-up (I) and spin-down (1)states as well as harmonic oscillator states of quantum numbers n and n _+ 1. Thus it is clear that matrix elements of P (electric-dipole transitions) are now allowed in which the “spin state changes (from + to - ”), or in which both the “spin” and orbital states change (by -t 1). As mentioned previously, the matrix element may be expressed as a sum of “interband” and “intraband” terms analogous to those of Eq. (50). However, due to the strong admixture of cell periodic and envelope functions in Eqs. (62) and ( 6 3 ) the predominant contribution to the matrix element for the intraband spin-flip transitions comes from the so-called “interband” terms (72), i.e. terms like (S I ? . p I X f iY). The first type of transition (pure spin-flip) arises from the first term of Y + and the fifth term of Y - (and vice versa), and the second type (both spin and orbital states change by f 1) is due to the first term of Y + and the sixth term of Y (and vice versa). The pure spin-flip resonance is excited in this case by the electric component of the radiation field. This transition, termed electric-dipole excited electron spin resonance (EDE-ESR), absorbs radiation in the CRI polarization at the energy En, - En,+ = hvs(n). The momentum matrix element (for E,,, 4 Eg)is given by ”

“

”

“

~

~

The combined resonance involves a change in both spin and Landau quantum numbers and occurs for the radiation electric field polarized parallel to the dc magnetic field (E’JjB)at an energy E,,+l,- - En.+ = hv,,,(n). The matrix element is

These are the results obtained by Sheka (65) who used a canonical transformation to eliminate interband terms and obtained an effective electron velocity operator. As is evident from Eqs. ( 6 6 ) and (67) the spin-flip matrix elements are quite similar in form. They differ in their dependence on electron wave vector; the EDE-ESR matrix element depends on k, (the wave vector parallel to the magnetic field) while the combined resonance matrix element is

46

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

proportional to d1’2 x B”’. On the other hand, the prefactors are identical in both cases; hence both matrix elements depend strongly on the inverse energy gap. Since the absorption coefficient is proportional to the matrix element squared, the possibility of observing such transitions decreases extremely rapidly with increasing energy gap. In practice, electric-dipole spin-flip transitions of this type have been observed only in quite narrow gap semiconductors. Indium antimonide ( E , = 0.235 eV) is the largest gap material in which these spin-flip resonances have been positively identified. The initial observation of combined resonance was made by McCombe et al. (73). In these experiments the transmission of n-type InSb was studied as a function of magnetic field using a conventional infrared monochromator, linear polarizers, and high magnetic field provided by Bitter-type solenoids. These high field measurements took advantage of the magnetic field dependence of the matrix element [Eq. (67)]. From these studies, the E’/JBselection rule was clearly established. Subsequently, detailed comparison of the calculated and observed magnetic field, carrier concentration, polarization, and orientation dependence of this transition (70) demonstrated conclusively that the small energy gap mechanism discussed above, and not the inversion-asymmetry mechanism, was responsible for the observed resonance in InSb. The EDE-ESR was not observed in these studies. The latter resonance was first reported by Bell (74) who utilized microwave techniques; however, no comparison with theory was presented in this work. McCombe et al. (75)have observed the EDE-ESR in the FIR using a pulsed gas laser spectrometer. The material investigated was the alloy semiconductor Cd,Hg, -,Te (x = 0.193). At this composition Cd,Hg, _,Te has the ‘‘ normal” zinc blende zone center band structure depicted in Fig. 5 with an energy gap of less than 60 meV. Figure 13 shows the results of a polarization study at two laser wavelengths. Again the polarization selection rules discussed above are clearly obeyed. From measurements of EDE-ESR, combined resonance and cyclotron resonance accurate band parameters for materials such as Cd,Hg, -,Te may be obtained. McCombe and Wagner (76) have studied these transitions in an alloy with x = 0.193 and fit their energy versus magnetic field dependence to that calculated from the BY model [Eq. (%)I. The results are shown in Fig. 14 with the indicated values of the parameters used for the calculated curves. These band parameters are in quite reasonable agreement with recent interband measurements (77);however it should be emphasized once again that care must be exercised when comparing band parameters obtained from two different models. It is worth mentioning here that intraband magneto-optical studies utilizing both cyclotron resonance and spin-flip resonances can provide a more accurate set of band parameters for a given

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

47

model, since both the separation between Landau levels and the separation between “spin” states are measured. Subsequent to the EDE-ESR measurements in Cd,Hg, -,Te this resonance was observed in InSb (78). Due to its larger energy gap the total

FIG. 13. Magnetic field dependence of the transmission of circularly polarized radiation through a sample o f Cd,Hg, _,Te ( Y = 0.193). The laser wavelengths are indicated in the figure. [From McCombe (79).]

20c

I

1oc C

FIG. 14. Transition frequencies versus magnetic field for several intraband transitions in Cd,Hg,_,Te (x = 0.193) with n = 3 x l o L 5c r K 3 and T = 4.5”K: combined resonance (0); cyclotron resonance ( 0 ) ;and EDE-ESR (A). The solid lines are derived from the Bowers and Yafet model with E , = 0.058 eV and E , = 19.1 eV. [From McCombe and Wagner (76).]

intensity is more than 200 times weaker in InSb for the same carrier density. The observed lines are also much narrower in TnSb, and the widths are strongly dependent on carrier concentration and temperature (76). At present, the theoretical explanation for the linewidths is in an unsatisfactory

48

BRUCE D. MCCOMBE AND ROBERT J . WAGNER

state. The observed lines in InSb are more than a factor of ten narrower than would be expected simply from broadening due to the nonparabolicity combined with the kH dependence of the transition matrix element [Eq. (66)] in the absence of scattering (79). Recent theories (80-82) which can be interpreted physically in terms of dynamical or "motional " narrowing, and which have been used to explain the linewidths of spontaneous spin-flip Raman scattering, give qualitatively reasonable results for the EDE-ESR linewidths in some cases. However, they do not correctly describe the observed temperature dependence in higher concentration samples. It is clear that further experimental and theoretical work is required to attain a quantitative understanding of the line shapes.

I " " " " '

00

Fic,. 15. Electron ~/-\alucsversus magnetic field for InSb. Experimental points werc obtained a t 4.3 K with Bi'[1 1 I]. The calculation of Pidgeon PI t i / . (83) (solid line) is for B [ l 1 I]. while that of Johnson and Dickey ( 4 2 ) (dashed line) is for BIl[1101. [From McCombe and Wagner (78).]

Nonetheless, in the quantum limit at low temperature the position of the EDE-ESR line yields a precise determination of the energy separation between the two spin states of the lowest Landau level and hence the y-value of this level. Such precise measurements can be used to test the quality of energy band models and parameters obtained for these models by other experimental methods. As an example of this the y-values for the lowest Landau level in InSb obtained from EDE-ESR measurements (78) are compared with values calculated from fits to two different band models in Fig. 15. Both models include effects of remote band interactions. Here the y-value is defined by y(n) = hvs(r7)/pBB. In the PMB model parameters were

49

I N T R A B A N D MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. 1

obtained from interband measurements, while in the J D model parameters were obtained from intraband measurements. The experimental points deviate from both calculated curves. The deviation from the Johnson and Dickey calculation is not quite as large as it appears, due to an anisotropy in the g-factor. On the other hand, the disagreement with the interband results (solid curve) is greater than 8% between 30 and 80 kG. This is well outside the combined experimental errors, and it is of the same order as the discrepancy between the calculated zero field g-factor and that measured by standard microwave ESR (84) (indicated by the triangle in Fig. 1s).Similar discrepancies between inter- and intraband measurements have been noted in the band edge electron and hole effective masses of several small gap semiconductors (85). At present, these discrepancies are not well understood.

2. Valence Bands of Zinc Blende and Diamond Structure Semiconductors under Uniaxial Stress

As mentioned above, the r-point valence bands of diamond structure or zinc blende structure materials under uniaxial stress also provide a system in which electric-dipole spin-flip resonance can be identified. For zero stress, the valence bands are degenerate in zero field (see Fig. 5). In the presence of a magnetic field the resulting energy levels may be grouped into four sets (see the following Section IV,C,l), which can be labeled for large quantum numbers as ‘‘ light holes ( + ) a and b or heavy holes ( - ) a and b, according to the separation between neighboring levels in a set. If the effects of the inversion-asymmetry of the zinc blende lattice are neglected, this discussion applies equally well for Ge or InSb. As shown in the following section, under certain approximations the wavefunctions for the degenerate Ts(J = 3/2) valence band (15,66) can be written: ”

“

Y : ( u ) = aF(n)@n-lI3/2, - 3 / 2 )

‘Y’(b)

=

b: (n)mn-

”

+ L Z ; ( ~ ) @ , , +1312, ~

-

1/2),

I 3/2, 1/2) + b: (.)a,,+ 1 3/2, - 3/2).

(68)

(69)

The correspondence between the band edge functions in the angular momentum notation and the notation used in the BY model is given in Table I. The energy levels are shown schematically in Fig. 20. The various a’s and b’s are approximately the same size for small n; hence, no unique “spin” state can be identified for a given level (i.e. rn, is not a good quantum number). In general, transitions between the two sets (a set s b set) as well as transitions between different states of a given set are allowed.

50

BRUCE D. MCCOMBE AND ROBERT 3. WAGNER

In zero field the presence of uniaxial stress lowers the cubic symmetry and removes the degeneracy of the f 3 / 2 and k t/2 valence bands. The bands are split into two pairs of Kramer doublets, m, = k 1/2 and ~n= , 3/2, which are separated by a small “energy gap” proportional to the magnitude of the stress. Under compressive stress the splitting is such that the M , = ~ f 1/2 states lie at higher energy than the m, = f 3 / 2 states, i.e. the i ~ i , = , &fstates (upper band) remain populated at large values of stress

v

c.b.

7-

n=-1

n= ____-----1-

2-

3-

1-

I

2-

4-

3-

2%

4-

‘&’ A

m, = -112

.112 (n-1)

n’= (n.1)

312

a-set

n.1

-

2-

3-

5.0.

b-set

n = -1

1-

2-

4-

m

- 412

(n-1)

- 312 (n.1)

t - 1 ~ 16. . Schematic energy-momentum relationship for the relevant bands of a zinc blende (neplecting inversion asymmetry) or diamond structure semiconductor in the presence o f uniaxial compressive stress (left). The first feu energy levels at k, = 0 in the presence of uniaxial stress and an applied magnetic field (right).

(see Fig. 16). For sufficiently large stress the strong admixture of wavefunctions is removed, and with the magnetic field and stress both along the (001) axis the wavefunctions for the upper band can be written

YY,(b)=d,,~,,-,(3/2,-3/2)+~,,+,/3/2,1/2),

M =

1,2.3. (71)

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

51

Here d,, is the ratio of the energy of the nth level to the stress splitting of the valence bands. For n = - 1 and 0, d, = 0 and for n 2 1, d, 6 1 in the above approximation. Thus Yn-( a )is predominantly composed of effective “ spin projection m, = - 1/2 with a small admixture of m, = + 312; and ’Y,-(b) is predominantly m, = 1/2 with a small admixture of in, = -312. Clearly for this situation m, is a “good” quantum number, and “spin”-flip transitions can be meaningfully discussed. The corresponding energies for these states at k , = 0 referred to the top of the upper band in the presence of stress are

”

+

Ed. f

= fio,(iz’

* +glPlg B,

+ 4)

(72)

where n’ = 0, 1, 2, . . . (see Fig. 16) ho, = ( y l - y2)z2wco,and y1 = 2 ~Here . w,, is the free electron cyclotron frequency, and yl, y 2 , and K are the Luttinger effective mass and effective g-value parameters discussed in the next section. Similar expressions are obtained for the tn, = & 312 levels. In order to calculate the momentum matrix elements for the “spin”-flip transitions in such degenerate bands, it is first necessary to determine the momentum operator appropriate to this situation. From the general expression pop = (m,/fi) d#/dk the proper momentum operator for this specific case has been determined (66,86). In the limit of large stress the combined resonance matrix element (An’ = + 1, AntJ = - 1) for E / \ B(86) is given by <(n’

+ I ) ~ m,, = -4 1 popI (n’)b,m,

=

++) cc (n’ + i ) i ’ 2 ~ 3 ’ 2 / 2 6, , (73)

where 28, is the stress splitting of the valence bands. This expression may be compared with Eq. (67) for the conduction band case. Note the much stronger magnetic field dependence of Eq. (73). The combined resonance originating on the 12‘ = 0 level is indicated on the right-hand side of Fig. 16 by the dashed arrow. It should also be noted here, by way of comparison, that the above treatment neglects k p interactions with the conduction band of the type which produces the spin-flip resonances described previously for InSb and Cd,Hg, -,Te. Transitions of similar origin are expected to be present in the valence bands for zero stress (6Y)* However, they are typically much weaker than the combined resonance of Eq. (73) since the strength of the former is roughly proportional to 1/E: while that of the latter is proportional to 118:. For InSb the maximum value of 28, attainable is about 60 meV while E, is about 235 meV. Combined resonance in the stress split valence bands was first observed

-

*This calculation was done for the “inverted” r-Sn band structure. It should provide an adequate description only for large Landau quantum numbers, since the r8 (light hole)-r, (heavy hole) band mixing is neglected by using the BY model.

52

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

by Hensel (86) using a microwave spectrometer as part of an extensive study of the valence bands of Ge under uniaxial stress. This work is discussed in more detail in the next section. Recently, Ranvaud (87) has extended these measurements to higher fields using a FIR HCN laser. Combined resonance as well as cyclotron resonance was observed in both Ge and InSb. Due to the strong dependence of the combined resonance matrix element on magnetic field [Eq. (73)], this transition is more easily studied in the FIR. Transmission data for Ge with E’IJBare shown in Fig. 17 for various values of

goL (K) ?O

0

MAGNET,?’

FIELD

FIG. 17. Transmission of 337 pni laser radiation through p-type Ge iis it function of magnetic field for several values of uniaxial stress. The combined resonance transition is labeled 3. The zero transmission point for each of the curves is indicated to the left. Values of stress are in units of 10’ dyn cm2. E 11 B 1, stress 11 1 I]. [After Ranvaud (H7).]

uniaxial stress along the [ 11 11 axis. Note that the combined resonance intensity decreases with increasing stress as predicted above. From the difference between the energies of the lowest combined resonance transition (0, 1/2) + (1, - l/2) and the adjacent cyclotron resonance transition (0, - 1/2) + (1, - 1/2) [levels are labeled by (n’, m,,)] one obtains the y-factor of the mJ = & $ states. This is given by g1 = 21c in the above approximation which neglects the Luttinger q parameter. As pointed out in the following section this parameter is related to the anisotropy of the g-factor. For example for stress I/ B 11 [ 11 11, when the effects of q are included, the g-factor is given by 9 , = 21i + (13/2)q. From a careful study of the small anisotropy of the above transitions Hensel and Suzuki (88) were able to

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

53

obtain values for both of these parameters in Ge for the first time. Their results are: K = 3.41 0.03, and q = 0.06 i- 0.01, which corresponds to a maximum anisotropy (between (111) and (001)) of about 5%.

3. Bismuth Apparently, the initial experimental observation of electric-dipole spin-flip resonances in the FIR was made by Burgiel and Hebel (BH) in Bi (89). Bismuth is semimetallic and crystallizes in a rhombohedra1 structure consisting of two interpenetrating face-centered cubic lattices which have been slightly distorted by elongation along a (1 11) axis. There are two atoms per unit cell, and pure Bi has three equivalent anisotropic conduction band minima at the L-points of the Brillouin zone and an overlapping valence band maximum at the T-points. The conduction band minima are separated from the valence band maxima at the L point by a direct energy gap of approximately 12 meV (90). The magnetic energy levels are shown in Fig. 18 for a magnetic field of 50kG directed along the threefold (trigonal) axis. For this orientation the spin-splitting is approximately half the Landau level separation. Also note that the (+) spin state lies at a higher energy than the ( - ) state in contrast to the InSb case discussed earlier. In the experiments of BH the transmission of 125--200pm thick Bi samples was measured as a function of frequency for a number of fixed values of magnetic field using a Fourier transform spectrometer. Experimental data are shown in Fig, 19 for the magnetic field oriented parallel to the threefold axis and light propagation parallel to the field (longitudinal or Faraday geometry). A rather complicated spectrum consisting of a number of transmission minima is observed. Cyclotron resonance occurs at approximately 50 cm-I for this field, and the general decrease in transmission is due to the wing” of this relatively very strong absorption line. The weak absorption lines were attributed by BH to various pure spin (Ams = k 1, An = 0) and combined (Ams = T 1, An = 1 ) transitions. Possible transitions from hole levels were ruled out from a comparison with the known spacing of the hole levels derived from studies of Shubnikovde Haas oscillations (91). The presence of several spin and combination frequencies was ascribed to the large nonparabolicity which makes the energy of a given transition a decreasing function of the Landau quantum number of the initial state. A number of possible initial states are populated at these magnetic fields as can be seen in Fig. 18. Burgiel and Hebel did not attempt to provide theoretical justification for the assignment of the various transitions. The matrix elements and selection rules have been calculated by Wolff “

54

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

(O,+)

?oo l -200 -

-I&

0

108

kZ (CM-0

FIG. 18. Energy-momentum relationship for Bi at the L-point of the Brillouin zone in the presence of a magnetic fieid along the threefold axis. [Calculated from Smith PI ui (91).] €,denotes the level to which the electrons fill the conduction band (the Fermi level). [From Burgiel and Hebel (89).]

B = 35.7kG

[L 2

16

18

20

22

24

2h

28

3O

FREQUENCY (cm-' )

FK. 19. Far infrared transmission of bismuth with B parallel to the trigonal axis. Arrows indicate the resonances discussed in the text. [From Burgiel and Hebel (KY).]

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICO&DUCTORS. I

55

(68) from the two-band model for Bi (92). This calculation yielded an allowed pure spin-flip transition (Ams = k 1, An = 0 ) for the two senses of circular polarization. The combined resonance is also allowed for E‘((B between states (n, + ) and ( i t 1, -); however these pairs of states are degenerate in the two-band model (hence the transition would not be experimentally observable). On the other hand, for B along the trigonal axis in Bi (the experimental configuration of BH) the two-band model breaks down, i.e. the spin splitting is only about halfthe Landau level separation (Fig. 18). Hence it is clearly not profitable to attempt a comparison with theory in this case. In the absence of polarization studies and a proper theory for ihe experimental configuration, it is not possible to precisely identify the various transitions and hence to obtain quantitative band structure information from these experiments.

+

C. Hole Cyclotron Resonance The progress of cyclotron resonance studies of holes has been considerably hampered by the complexity of the valence band structure of many semiconductors. While FIR studies should be productive of detailed information concerning band parameters for light and heavy hole valence band levels, most of the useful information about these materials has been obtained using microwave techniques. To a certain extent, this reflects the recent development of FIR techniques. O n the other hand it may indicate some potential limitations of the present infrared approach by comparison to established microwave methods. In this section, the Luttinger and Kohn (LK) formalism (Section 11) as applied by Luttinger to the case of the degenerate valence bands of Ge is sketched, and the success and limitations of the theory are indicated. The recent work of Hensel and Suzuki (HS) (88, 93-95) who have produced a prodigious experimental and theoretical study of microwave cyclotron resonance in Ge is then presented. It is in the light of this work that the few reported FIR studies of degenerate hole bands are discussed. Cyclotron resonance measurements of the conduction band of HgTe are also described in this section. Since HgTe has the “inverted band structure, conduction band studies are intimately related to valence band studies in semiconductors with the “normal band ordering. Finally, cyclotron resonance of holes in tellurium is reviewed. In this case, for free carriers, very complete and useful FIR experiments have been performed. ”

”

56

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

1. Degenerate Bands in Zinc Blende and Diamond Structure Semiconductors The valence bands of diamond and zinc blende semiconductors at the Brillouin zone center are constructed from a set of sixfold degenerate p-like states (three Kramers doublets). In the presence of spin-orbit interaction, the states are split into an upper fourfold set and a lower twofold set. The formal effective mass theory for the case of degenerate bands was first worked out by LK ( I I). Luttinger subsequently calculated energy eigenvalues for valence band Landau levels using the LK approach of coupled equations involving harmonic oscillator envelope functions (1.5). In this set of coupled equations the interaction of a given band in the set with all other farremoved bands is included to second order. The starting equation (15) is

2 [D$P"Pp + pg B)L,o,]F,,(~)= EFr(x),

(74)

?-'

where r, r' run over the degenerate set of bands. The effective mass matrix D$! includes interaction momentum matrix elements p",,, with other bands, r", outside the set r' as follows:

where E , is the energy of the degenerate band and pR,, is defined by Eq. (37b). Again, one may identify symmetric and antisymmetric effective mass matrices in a fashion similar to the single band case, with the antisymmetric part reflecting the hole g-factor. When the spin-orbit interaction is included, the 6 x 6 matrix, D:? (including spin), can be reduced to a 4 x 4 component (for the upper degenerate p-states) and a 2 x 2 component (for the lower spin-orbit split p-state). Using group theoretical arguments Luttinger has given the general form of D for the upper fourfold bands of a diamond structure semiconductor

where c.p. denotes cyclic permutations, {a, b ) = i ( a b + ba) indicates the symmetrized product, and the five Luttinger parameters y l , y 2 , y 3 , K , and q are related to sums of interband momentum matrix elements of the form of

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

57

Eq. (7.5). (They are generally treated as adjustable parameters to be determined by a fit to experiment.) The parameters yl,y 2 . and y 3 are related to the effective masses of the holes, while K and q determine the effective g-factors as discussed previously. In Eq. (76) the J’s are the 4 x 4 angular momentum matrices for spin 3/2 and I is a unit matrix. Although the 4 x 4 matrix equation [Eq. (76)] i s not easily solved in general, an approximate solution yielding analytic expressions for the eigenvalues may be obtained by making two rather restrictive assumptions; namely, y 2 = y 3 = 7 and q = 0. While the latter assumption was thought to be reasonably good (the g-factor anisotropy is only about 5 % as measured by HS), the former is known to be a drastic one. These parameters reflect the degree to which the magnetic energy level structure varies with the orientation of B relative to the crystal axes. For j12 = -y3 all directions are equivalent (the “spherical” assumption) and the energy levels are the same for all directions of B. Early cyclotron resonance experiments in Ge indicated that the energy surfaces were not spherical ; however. this approximation reveals some noteworthy general features which justify its presentation. With k , set equal to zero;” the 4 x 4 matrix decouples into two 2 x 2 matrices given in Table 111. The operators at, u in this table are creation and annihilation operators for harmonic oscillator functions, and utu is the harmonic oscillator “number” operator, i.e. utuQn = nQn. The upper block is TABLE 111

THELU TTINGER D MATRIXIN

THE SPHERLCAL

APPROXIMATION“

* The assumption that k, = 0 implies that when comparing calculated energy level separations with experimental transition energies, the observed transitions occur at k , = 0. The case k , # 0 was originally investigated by Wallis and Bowlden (96), and HS (88, 94, 9 5 ) have demonstrated the importance of nonzero k , for a quantitative understanding of the cyclotron resonance spectra.

58

BRUCE D. MCCOMBE AND ROBERT J . WAGNER

denoted “ a ” and the lower block .‘b.” The eigenvalues and eigenfunctions are E(a), $(a) and E(b), $(b) for the a- and b-sets, respectively. The two 2 x 2 matrices of Table I11 are reduced to numerical matrices, which may be solved directly, by the following choice of wavefunctions

where the On’s are harmonic oscillator functions and a,, u 2 , h,, h2 are the eigenvector components of the numerical matrices associated with the band edge functions having m, = + 312, - 1/2, + 112, and - 312, respectively. The choice of n is arbitrary to within a constant [the choice here is that of Pidgeon and Brown (35)].The energy eigenvalues in units of heB/m, c for the a-set are:

E-,(a)

-

=

-

y)

-

+ti,

The eigenvalues for the h-set are:

Thus, the energy levels fall into four “ladders,” u-set ( + or - ) and h-set ( + or -). In the limit of large quantum number, n, the + levels correspond to light holes and ‘’ - levels to heavy holes (according to their splitting in a magnetic field). The total wavefunctions for these four ladders are given in Eqs. (68) and (69) and the energy levels are shown schematically in Fig. 20. Although the levels / I = -1, 0 in the u-set have a separation more characteristic of the ) ? I , , = +3/2 levels, they are shown grouped (by the dotted lines) with the n i , , = - 1/2 levels. The reason for this grouping is easily seen. For / I = - 1, 0 the coefficient a: = 0 and Y ? ,(u) = @, 13/2, - 1/2) [see Eq. (68)]; hence the band edge component of the wavefunction is strictly m, = - 112 in this approximation. The grouping of levels in the b-set is straightforward. Note that the energy separation between the n = 0 and n = - 1 levels in ”

”

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

59

the a-set and b-set are particularly simple combinations of parameters :

E;(U) - E?,(u)

=

71

7, + 7. -

E , f ( b ) - E T , ( b ) = y1 (80) Transitions between these pairs of levels are called fundamental transitions and are indicated by the solid arrows in Fig. 20. The form of the transition energy of these fundamental transitions is unchanged even for y 2 # ;’3 and q 0 provided B is in the ( 1 11) or (001) direction. In these cases, 7 must be set equal to y 3 or y 2 , respectively. “

”

+

FIG.20. Schematic energy-momentum relationship for the relevant energy bands of zinc blende (neglecting inversion asymmetry) or diamond structure semiconductors neai- k = 0 (left). The first few energy levels for the valence bands in the presence of an applied magnetic field at k, = 0 (right). The dotted lines along with the niJ label indicate the grouping of states according to the dominant hand edge contribution. The Landau quantum number, n’, for each l e d is related t o its labeled harmonic oscillator quantum number, i~, as specified i n the figure.

A most important feature of these energy levels is the fact that for large ii’ both light and heavy hole levels are uniformly spaced, whereas for small n’ the energy separation between adjacent levels is strikingly nonuniform. This results from the strong admixture of harmonic oscillator states near the point of degeneracy. The distinction between what has been termed “classical” and “quantum” cyclotron resonance derives from this feature. Experimentally, the distinction arises from the difference in distribution of carriers among the quantum levels for different temperature regimes. For

60

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

k, T + h a , the holes are distributed according to Boltzmann statistics over a large number of Landau levels. Thus there are many overlapping cyclotron resonance “contributions” (n’ + n’ + 1) for the light and heavy holes, each of which has a different energy (or magnetic field) for small n’. The transition energy approaches a constant for large n’. The resulting spectrum consists of two “classical” cyclotron resonance lines ; one for light holes and one for heavy holes. On the other hand, for k , T 5 ho,, holes are distributed only in small n’ Landau levels. This results in a small number of cyclotron resonance “contributions,” each of which has a different frequency (or field); these may be resolved into separate lines experimentally. Evidence of the many constituent lines (Luttinger effects) was first observed by Fletcher et al. (97), and this indicated the need for a theory of the degenerate bands as described above. Attempts to analyze the low temperature cyclotron resonance spectra of holes in Ge, however, have met with little quantitative success until recently, primarily due to the neglect of kH-effects. The continued experimental and theoretical work of HS (88, 93-95) has succeeded in unraveling the complex cyclotron resonance spectra of Ge.” Theoretically, they extended Luttinger’s effective mass Hamiltonian to include the r7split-off valence band in the basic manifold, and took into account the effect of uniaxial stress in a general way. Furthermore, they made no assumptions about the relative sizes of the Luttinger parameters, about the k,-dependent contribution to the spectra, or about the direction of the magnetic field. Utilizing symmetry arguments, they developed a scheme for classifying eigenstates and electric-dipole selection rules. Rather than the relatively simple wave functions of the form of Eqs. (68) and (69) which occurred in the Luttinger treatment, HS assumed wavefunctions of the general form x)

yflh.

kff

= n,=O

+3;2

1

m,,=-3 2

amj(nU)$fla.

flh.

kr,

I J , “J>,

(81)

where I J , m J ) are the LK cell periodic functions for the p3,2 and p1 states and a,,(n,) are numerical coefficients. The symmetric gauge is used to specify the Landau functions $I,, n h , k H ; hence the wavefunctions are described by the two “harmonic oscillator” quantum numbers 17, and nb ( I T , is equivalent to the Landau quantum number, II, and nb is equivalent to I I - i n1 of Section 11). By substituting the wavefunction [Eq. (Sl)] into the k f f = EYI,,. k H , a secular matrix for the effective mass equation &Tflh. um,,(na)is generated. This infinite matrix is solved numerically by truncat111

* Fujiyasu r f a!. (Y8) have also reported microwave studies of cyclotron resonance of holes Ge under uniaxial stress.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

61

ing to an 80 x 80 matrix. An arbitrary selection of values for the y 2 , y 3 , q, K , and strain (deformation potential) parameters allows the matrix to be diagonalized, energy levels determined, and the anticipated spectra computed. However, with this approach it would be impossible to match theory uniquely with experiment. The large number of lines, their potential k,-dependence, and broadening effects would make it impossible to determine the constants unambiguously. In order to surmount part of the problem HS made use of the fact, previously noted, that the application of uniaxial compressive stress results in a considerable simplification of the degenerate bands and thus of the experimental results. Figure 21 is illustrative of this simplification. First, the light hole transitions below 1 kOe disappear as the stress is increased to 1900 kg/cm2; second, the large number of heavy hole transitions in the region 2-8 kOe are “reduced ” to a series of about 5 lines near 8 kOe; finally, and most importantly, the line (0, - 1/2) + (1, - 1/2) moves only slightly as stress is increased from 490 to 1900 kg/cm2. This is one of thefundamental transitions. These developments can be qualitatively understood from a consideration of the hole wavefunctions and energy levels as described previously. The compressive stress splits the k = 0 valence band degeneracy. This has the effect of reducing the hole population and cyclotron resonance strength for the lower band. In addition, the stress reduces the admixture of band edge wavefunctions. With magnetic field applied, four series of levels develop; two associated with the m, = f 1/2 states and two with the m, = & 312 states as shown in Fig. 16. In the high stress limit these ladders become equally spaced, and the cyclotron resonance series approach ellipsoidal low and high mass values. The lowest transitions, (0, -3/2) --t (1, -3/2) and (0, - 1/2) (1, - 1/2), are virtually independent of stress and simply related to the parameters y l , y 2 , and y 3 . These relationships are shown in Table IV for two high symmetry directions. The reason for the stress independence of these lines in lowest order is easily seen by examining the zero stress wavefunctions in the spherical approximation. For example, as mentioned above, both \ r l l ( u )and ‘ Y i ( a ) are proportional only to /3/2, - 1/2) and have no contribution from 13/2, +3/2). Hence, since the predominant effect of the stress is to “decouple” m, = - 312 from m, = + 1/2 and m, = 3/2 from m, = -- 112 states, the stress has no effect on the 0 1 transition in this approximation. By studying the stress dependence of the many cyclotron resonance lines which were observed, HS were able to identify the two fundamental transitions. They found that these lines were extremely sharp and, as a result, they were able to make the following very precise determinations of the Luttinger .--f

+

.--f

62

BRUCE D. MCCOMBE AND ROBERT J . WAGNER

m o

i

d

un

d

0

0

I

I

MAGF(ETK: FIELD IN OERSTEDS

FIG. 21. Valence band cyclotron resonance spectra in Ge showing the effects of iiniaxial siress. The data were obtained at 1.2' K and 52.9 GHz with B~~[111]1! stress. Hole resonances at

zero stress are designated by their effective mass values. For finite stress the transitions are labeled by the (17~. +ni,,) notation abbreviated to n'+ (all transitions shown are r n , = i 1,2). [From Hensel and Suzuki (95).]

parameters: =

13.38 f 0.02;

;'2

=

4.24 +_ 0.03;

;a3

=

5.69 & 0.02.

Hensel and Suzuki also noted that while the transition (0, -312) + (1, - 3 i 2 ) was stress independent to the limits of their experimental resolution, the transition (0, - 1/2) + (1, - li2) showed an extremely weak ( 1 '!()), but clearly observable. stress-dependence. From this dependence they were able to determine values for the valence band spin-dependent uniaxial deformation potentials.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

63

The spin-flip transitions which occur for E’IIB allowed a determination of the K and q parameters as discussed in the previous section. Using all of the Luttinger parameters determined in this manner HS proceeded to calculate the stress-free cyclotron resonance spectrum of holes in Ge on the basis of

TABLE IV EFFECTIVE MASS TENSORCOMPOKENTS FOR LOWESTTRANSITIONS I N EACH LADDERIN HIGH STRESSLIMIT".^ Stress

rn, =

1’2

m,

=

THE THE

i312

The parallel axes of the tensors are along the stress direction. After Hensel and Suzuki (Y5).

the theoretical model previously described. Assuming each component line to have Lorentzian shape with z = 2.7 x 10- l o sec, they calculated the integrated intensity of each line as a function of crystal momentum in the direction of the field, k,, and were thus able to synthesize the zero stress spectra. From this calculation they found that the dominant contribution to many of the cyclotron resonance lines did not come from transitions at k, = 0. As a result the lines were broadened and actually shifted from their k, = 0 positions. In view of this it is clear why early attempts to understand the spectra with the k , = 0 assumption were unsuccessful. In earlier FIR work, Button rt al. observed “Luttinger” effects in the hole cyclotron resonance spectra in InSb (99) and Ge (100) using an HCN laser. However, these experiments did not make use of uniaxial stress to clarify the data and lines were identified only on the basis of the spherical Luttinger model with k, = 0. At this writing a FIR magneto-optical study

64

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

comparable to the microwave work of HS has not been published. A detailed comparison of the techniques and experimental conditions for the two types of experiments is illuminating in this regard. Ranvaud's recent uniaxial stress measurements on hole cyclotron resonance in Ge and InSb using a FIR HCN laser (87) allow a direct comparison of the two methods. In the microwave work, the high purity sample is maintained at a temperature of 1.2"K while holes (and electrons) are optically excited with near infrared radiation. A high Q cavity is used in conjunction with a microwave bridge to increase sensitivity. Adequate cyclotron resonance signals can be detected with hole densities of only 108-10'ocm-3 in a thin layer only a few microns thick. In the FIR, a single pass transmission method is generally '

TABLE V COMPARISON OF THFRMAL A N I I PHOTONE \ ~ K G I EFSO R THE MICROWAVEA N D FIR LASER EXPFRIMENTS

DISCUSSED ic THE TEXT

Microwave

0.2 19

0.103 ( T = 1.2"K)

2.126

Laser

3.618

1.124 (T = 20'K)

2.133

used. This requires a significantly larger number of carriers in order to obtain an observable cyclotron resonance signal. Thus Ranvaud utilized samples 2-3 mm thick which were intentionally doped with 10'4-1016 acceptor impurities per cubic centimeter. Free carriers were thermally excited at temperatures 2 20°K. The two techniques are compared in Table V. Note that the condition ho > k , T is equivalently satisfied in both, even though the FIR frequency is 17 times the microwave frequency. In addition the necessity of using rather highly doped material to obtain sufficient carriers for the FIR experiments serves to degrade the carrier scattering time, 5, and to broaden the cyclotron resonance lines. In fact, where HS observed a dozen lines, Ranvaud (87) was able to resolve only four. O n the other hand, for materials such as InSb, where the inherent purity is significantly worse than that of Ge, the advantages of the microwave technique are lost and the necessary condition for sharp resonances, w c s >> 1, is better satisfied in the FIR. For the case of Ge Ranvaud was able to achieve a good fit to his uniaxial stress data using Luttinger parameters that agree well with those of HS.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

65

However in InSb Ranvaud found a qualitatively unexplained stress dependence for one of the fundamental transitions. In the absence of a suitable explanation for the stress dependence of this crucial transition, it is difficult to obtain a reliable set of Luttinger parameters from these measurements. As in the case of electron cyclotron resonance, FIR laser sources can be extremely useful for the measurement of classical hole cyclotron resonance when material quality is poor, i.e. when w,z < 1 in the microwave region. With sufficiently high magnetic fields wCoccurs in the FIR and the condition w, z > 1 can often be achieved. This is the approach recently used by Bradley et al. (101) in their measurement of the “classical” light and heavy hole masses in Gap, a typical zinc blende structure semiconductor. Using pulsed magnetic fields up to 200 kG these authors were able to resolve the cyclotron resonances for carriers with effective masses of 0.16 mo and 0.54 m, . This application o f F I R laser techniques promises to be extremely fruitful in the characterization of semiconductors with heavy mass valence bands; to date very few such materials have been investigated in any detail. “

”

2. 1nl;erted Band Ordering: HgTe

As discussed previously HgTe and alloys of Cd,Hg, -,Te with x 5 0.16 possess the “inverted” zone center band structure (Fig. 10). Due to the relatively small energy gap ( E , = 0.3 eV) and the degeneracy of the T, conduction and valence bands, a more general treatment than either the BY model or the Luttinger model for degenerate bands is necessary to properly account for both the nonparabolicity of the conduction band at higher energies and the “Luttinger” effects (similar to those just discussed for Ge) at lower energies. Groves et al. (16) have provided such a generalization based on the earlier work of Pidgeon and Brown (35) for InSb. In this treatment states at k = 0 are divided into two groups. One group (A) consists of the strongly interacting (closely bunched) states of interest, Ts,T8,and r, ; the other group (B) includes all the other states. Interactions of the k * p type between a given state in A and the states in B are removed to second order in k by the LK approach. The resulting renormalized interaction among the states in A gives an 8 x 8 interaction Hamiltonian which is diagonalized exactly by computer for k , = 0. Groves et al. have used this approach to fit their interband magnetoreflection data for both r6+ T8 and T8-+ Ts transitions in HgTe and to obtain an appropriate set of band parameters. Leung and Liu (102) have generalized the theoretical approach of Groves et al. (16) to include the effects of k, # 0. Experimental studies of interband (r, valence + T, conduction) and intraband (T, conduction) magnetoabsorption in HgTe in the spectral region between 2 mm and 100 pm have been reported by Tuchendler et al. “

”

66

BRUCE D . MCCOMBE A N D ROBERT J. WAGNER

(103). In addition transitions which did not extrapolate to zero energy at zero magnetic field were observed. The latter lines were attributed to transitions involving “resonant acceptor impurity states which are degenerate with the r; band continuum in zero field. Both carcinotron and FIR laser sources were used in these experiments. Examples of some of these data are shown in Fig. 22. Experiments were performed in both the Faraday (longitudinal) and Voigt (transverse) geometries. In the Voigt geometry, due to the ”

FARADAY CONFIGURATION

c

c

2’

e w

a

CR

I

0

I

10

20

1

30

I

I

40

50

B(kG) FIG. 22. Transmission versus magnetic field for a thin sample ( 5 10 pm) of HgTe. The trace at 775 pm was obtained with a carcinotron source while the other two traces were obtained with HCN (311 pm) and H,O (1 18.6 pm) laser sources. The line labeled CR is cyclotron resonance and that labeled S is “spin” resonance. [After Tuchendler er a / . (103).]

significant number of free carriers, the cyclotron resonance at lower frequencies was found to be “plasma-shifted” to lower fields as compared to the position of the cyclotron resonance in the Faraday geometry. At higher photon energies the positions of these lines coincided. The position of the plasma-shifted cyclotron resonance is usually given by = 47~Ne~/&,m* is the plasma frequency, N is where wpcr= [ m i + the carrier concentration, and E~ is generally taken to be the low frequency dielectric constant, independent of frequency. However, in a zero gap semiconductor such as HgTe the ri --+ ri interband transitions contribute a frequency dependent term to the dielectric function which can extend to zero frequency. Only by taking this contribution into account were these authors able to obtain agreement between the calculated and observed positions of the plasma-shifted cyclotron resonance.

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

67

Tuchendler et a/. also identified an electric-dipole “spin resonance for ”

E IB (labeled “ S ” in Fig. 22) and a “combined” resonance for E’IIB (not

shown). The spherical Luttinger model was used to provide a theoretical framework for the interpretation of these low energy transitions. As discussed in Section IV,B “spin” is not a good quantum number in these degenerate bands for the lowest levels; thus spin-flip transitions are not well defined. The “spin” transition is presumably a transition from state (1, - 1/2) in the a-set to state (1, -3/2) in the b-set. Again states are specified by ( H ’ , mJ).By reference to the discussion of Section IV,B it is easily seen that the total wavefunctions for these states are Y t ( a ) = (Dl I3/2.

-

1/2)

and

Y’,+(b) = (D, I3/2, -3/2),

respectively. It is not clear, however, why a transition matrix element between these states should be non-zero, since both initial and final states have odd parity. It is possible that inversion asymmetry (which has been neglected) may allow this transition by relaxing the parity selection rule. From a fit to the various observed transitions Tuchendler et a/. have obtained a set of Luttinger parameters which adequately describe their results as well as the interband rz + rC, results of Guldner et a/. (104). In the latter work low energy transitions ( h ~ - oE, < ~ 30~meV) ~ were ~ ~fit using ~ the spherical Luttinger model, while higher energy transitions were fit with the BY model. There is some justification for the use of the spherical approximation, since from the work of Groves et al. the anisotropy is not large (yz - y3)/yz = 0.12. However, the parameters obtained from these experiments, which are mutually consistent, are in some disagreement with the results of Groves et aI. except for the low temperature energy gap. The parameters are compared in Table VI. As can be seen from the table the largest discrepancy is in y1 (35%). The reasons for the discrepancy are not clear. Certainly the theoretical approach of Groves et a/. is more rigorous than the “marriage” of two models used by Guldner et a/. and Tuchendler et a/.; on the other hand, the experimental work of the latter authors is more extensive and detailed (especially since it includes intraband measurements for the r: band). It is possible that k, effects which were found to be very important in Ge by Hensel and Suzuki could account for this discrepancy since such effects were neglected in the interpretation of the experimental data by both groups. However, Guldner et a/. have calculated the k,-dependence of the r energy levels for high symmetry directions. These effects appear to be unimportant for the Ts band levels, and by intrainterband transitions as well as the implication for the P6+ band transitions. In view ofthis it seems that additional work is required for a complete understanding of the degenerate valence-conduction bands of HgTe.

68

BRUCE D. MCCOMBE A N D ROBERT J . WAGNER

TABLE VI

BAND PARAMETERS FOR HgTe"

0.3025 0.3025

- 12.8

- 10.5

8.4 -9.9

~

- 16.98

- 11.29

(103. 104) (16, 105)

" Luttinger parameters are calculated from the higher band parameters of Groves et a/. (16)as follows: 7, = j,2

=

~, , -

=

with E ,

=

?!b.

-

= - 16.98

;.;.b. .,hb.

-

=

~

&b

Ep/3E, Ep/6E, EP/6E, - E,/6E,

10.49 9.32 - 11.29 ~

= -

=

18.1 eV.

3. Tellurium n. Theoretical background. Recently, a large number of FIR magnetooptic experiments have been performed on tellurium. This is in part due to its relatively unique and interesting energy band structure, and in part due to the availability of high quality single crystal material. The crystal structure consists of atoms arranged in spiral chains with three atoms per spiral and spirals situated in a hexagonal arrangement. The Brillouin zone is a hexagonal prism with the crystal c-axis perpendicular to the prism face. In contrast to the diamond or zinc blende zonecentered valence bands, the valence band maxima of tellurium are located at the H-points, corners of hexagonal faces of the Brillouin zone. H u h (106) has suggested a description of the valence bands using a k * p perturbation approach. When the effects of spin-orbit interaction are included, two possible energy band schemes can occur as shown in Fig. 23. These two cases simply reflect different choices of the parameters in the energy-momentum relations (hole energies are measured positive downward with the energy zero taken to be the position of the valence bands at H in the absence of spin):

E E

= 3 , 4=

+ Bok: [ S : k ; + 4Af]1'2 - A 2 . Ak2L + B OkZ, f [S\kt + a2k:]' + A2 , Akt

(82)

where k: = k: + k : . In Fig: 23 and Eqs. (82) we have used a slightly modified version of the notation of Doi et al. (107) who have presented a rigorous calculation using the k p formulation. A comparison between these parameters and those used by other workers is given in Table VII. In Eqs. (82) the z-direction is

-

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

69

,

taken parallel to the c-axis, and El, E,, and E3, correspond to the H,, H , , and H, states, respectively. The parameters A , B o , S1, and S , represent momentum matrix elements between the valence bands and higher bands, and A 1 and A2 are proportional to the spin-orbit splittings of the valence bands. As is apparent from Eqs. (82) the curvature of H, in the z-direction

H

kz

+kZ

(a 1

(b)

FIG. 23. Schematic E versus k , relationship showing the valence band structure of tellurium in the neighborhood of the H-point of the Brillouin zone (this point is also designated M by some workers (108, 1 2 I ) ) with spin-orbit interaction included. In (a) ( I S : / B , I > 4 1 A , I ) the uppermost valence band exhibits the “camelback” structure. I n (b) ( I S : / B , I < 4 I A 1 I ) it has a single maximum at the H-point. The H, and H, bands are nondegenerate as are the H, bands (except at k , = 0) since their Kramers conjugate states occur at the nonequivalent point H’ due to the lack of inversion symmetry of the tellurium structure.

depends critically on the size of I S:/Bo I relative to 4 I A l I . With I S:/Bo I > 4 1 A 1 1 , the “camelback” of Fig. 23a results. If I S:/Bo 1 .< 4 1 A1 1 , a single maximum at the H-point occurs as in Fig. 23b. In the case of Fig. 23a, 1 kzO1, the magnitude of kZ at the maximum, is [Sf - 16Bg A:]”*/2 1 Bo S1 1 and the energy difference E(k,o) - E ( 0 ) 5 6

=

S:( 1

-

4Bo I A 1 I /S:)/4

I Bo I.

TABLE VII A

NOTATION FOR TELLURIUM BAND PARA METERS^

COMPARISON OF THE

Doi er a/. (107)

Betbeder-Matibet and H u h ( I 1I )

Weiler (108)

In order to avoid confusion with the magnetic induction, B, has been used instead of B in Doi er al. (107).

70

BRUCE D. MCCOMBE AND ROBERT J. WAGNER

Unlike the rather dramatic changes in shape of the H, band, the shape of

H, and H, remain relatively insensitive to changes in the parameters. Thus since the H, band lies highest and is easily populated with free holes, it provides an ideal situation for intraband magneto-optical studies to determine the band parameters. Rather than outline the detailed calculations of the upper valence band structure of tellurium in the presence of a magnetic field (108-210), we present here a simplified description of the upper (H4) band which is due to Betbeder-Matibet and H u h (111). This simple approach provides a good deal of physical insight into the energy levels and selection rules, while it avoids the mathematical complexities of the more rigorous treatments. Two cases can be distinguished: (a) BJIc,and (b) B Ic. For case (a) the energy in the z-direction is not modified by the presence of the magnetic field. Since the energy in the transverse direction is a quadratic function of k , the transverse motion goes over directly to the Landau quantization as in Section 11, i.e. Ak: + ( a l/;?)ho,. , where ha),, = -2AeB/h. In this somewhat simplified model the separation between adjacent Landau levels is a constant independent of either 11 or k , . However, when higher order terms in the E-k relationship are taken into account, this no longer holds true. The effects of including the higher order terms will be discussed later. For case (b) (B Ic) one must distinguish between the two possibilities shown in Fig. 23. The choice of parameters leading to Fig. 23b would not yield any qualitatively different effects in the presence of an applied magnetic field. O n the other hand the “camelback of Fig. 23a demands special attention. Betbeder-Matibet and H u h (111) have shown that for an electron in the “camelback” band with Bilk, one can obtain an effective onedimensional Schrodinger equation by making the following substitutions in the expression for E , [Eq. (82)]:

+

”

k,

---f

(AIB,)’ 4(eB/H)112X, k,

+

(B,/A)”4(eB/”)’ 2(d/dX)

and E

+ (AB,)’

2(eB/h)c

+ ,412;.

The resulting one-dimensional equation, which yields a good approximation to the energy levels. is d2Y -

dX2

+ V ( X ) Y = EY,

with V ( X )= x 2 - (A;

+ 4(2X2)’/2 + A,

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICOKDUCTORS. 1

7I

and

A,

=

2(AB,)- ' / 2 ( e B / h ) -'/,Al,

A,

=

( A l l , ) - "'(eB/h)- 1/2A2,

t2 = (Bo/A)"2(eB/h)-'(S,/2Bo). It should be noted that V ( X ) remains unchanged under inversion; X + - X ; hence parity is a good quantum number. For small values of B, V ( X )appears as two separated, harmonic oscillator-like potential wells with minima at

X

=

iX,,

where

X,

=

(Bo/A)'~4(~B/h)-''2k,o.

In the low field limit the iiiterwell barrier is 8 = ( A & - '/'(eB/h)- '6. In this case there is only a slight coupling between potentials. As a result, the two sets of Landau levels are nearly degenerate with E,,t

=

E n = ( n + +)(heB/m.c)

(84)

and the wavefunction can be written Y,,

= 2- 1'2[Qn(X

-

X,) i @),(X + X,)],

(85)

where Qn is a harmonic oscillator function. As B is increased, the separation 2 X , and barrier height 8 of the two wells decreases. This results in increased coupling of the two sets of Landau levels and a splitting of the previously degenerate n' states into n+ and y1-. It is clear that if the camelback condition prevails in tellurium, it will be apparent in the splitting of cyclotron resonance transitions at high fields. Electric dipole selection rules require n' + ( n + 1)-, i.e. allowed transitions are those between states of odd parity with respect to X . The inclusion of higher order terms in k, as done in the more rigorous treatments (108, 110) relaxes this rule somewhat. b. Esperimental results. The availability of high quality p-type tellurium has resulted in very detailed investigations of the valence band structure described above. From early low magnetic field sub-millimeter work (I12), it was concluded that the band was anisotropic with m* (Bilc) = 0.11 m, and m* (B Ic) = 0.26 mo . However, oscillatory magnetoresistance (Shubnikovde Haas) measurements suggested the presence of the camelback condition (113, 114) as did later microwave cyclotron resonance (115). Couder (116), using carcinotron sources from 405-1080 pm (740-278 GHz) and fields to 55 kG, first observed the splitting which is expected from the "camelback" condition. As shown in Fig. 24 for B Ic the primary cyclotron absorption F, observed at 278 and 308 GHz splits into F, and F, at 353 GHz. In addition to these free carrier resonances other lines S,, S, ,and I observed by Couder were attributed to transitions involving impurity states.

72

BRUCE D. MCCOMBE A K D ROBERT J . WAGNER

Following Couder's early work, a number of authors have employed submillimeter lasers (I 17,118) and FIR nionochromators ( I 19) to study the higher energy/magnetic field region of the E versus B plot. The data obtained prior to 1971 have been compiled by Nakao et al. (110) and these data are shown in Fig. 25. The calculation of Nakao et a/. for the transition

I

20

1

BkG)

55

FIG.24. Experimental traces o f transmitted power versus magnetic lield for Te with B i c at the indicated frequencies. The primary low frequency absorption line, F,, splits into two lines, F F z , at higher frequencies. Additional lines. S S,, and I, were attributed t o impurity state transitions. [After Couder (116).]

,,

,,

CY CLOIRON RESONANCE ( H i c)

MAGNETIC FIELD (kOe)

FIC;.25. Transition energies of cyclotron resonance in Te for the case of H I c'. The solid 0,x . and 0 and dashed curves are the calculations of Nakao et a/. (110).The symbols 0, represent the experimental data of Yoshizaki and Tanaka (119) Dreybrodt era[.(118), Couder (116), and Radoff and Dexter (115), respectively. The solid lines correspond to allowed transitions and the dashed lines correspond to forbidden transitions. [From Nakao et irl. (110).]

INTRABAND MAGKETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

73

energies is indicated by the solid (allowed) and dashed (forbidden) lines. Their choice of parameters yielded an energy of 2.54 meV for the height of the camelback ” maximum above the H-point energy. The forbidden transitions O + -+ I + and 0- + 1- have also been observed. These transitions were first identified by Dreybrodt et al. (118). When warping terms of the form k,(k: - 3k:) are included in the k p Hamiltonian, these forbidden transitions become allowed for B I/ bisectrix (B parallel to the hexagonal face of the Brillouin zone and perpendicular to the side of the prism); they remain unallowed for B 1 binary axis (108) (B parallel to the hexagonal face and parallel to the prism side). Experimental results have shown these forbidden transitions in both orientations. The reason for their appearance with B /I binary is not yet understood. Recent work by Couder et al. (120) has clarified the region of 1-2 meV and 10-40 kG. These measurements have confirmed the assignments shown in Fig. 25. Couder et al. also explored the dependence of cyclotron resonance on the angle between B and the c-axis for frequencies both above and below the splitting region. They were able to describe all aspects of their results satisfactorily with the simple camelback model. In the light of the above discussion, it is somewhat surprising that the B/lc orientation seems to have been a greater source of interpretational difficulty than B Ic. Although Couder observed a single absorption line as might be anticipated for low concentration samples, Yoshizaki and Tanaka (119) have reported a much more complicated spectrum for their samples with p z 10’4-1016cm-3. They argued that, although some of the observed spectral features were due to impurity transitions, two of the lines were free hole cyclotron resonance lines, one arising from transitions at the H-point and one from transitions at the valence band maxima, k k,, . It should be reiterated that for the simplified model described above there is no dependence of ho,,Il on k,; thus with this model there could be only one cyclotron resonance line irrespective of where the transitions originate. In the more sophisticated models (108, 110) higher order terms (k: and k t k : ) enter, and these terms give rise to a dependence of hw,, lI on k , as well as on n. At low energies (e.g., the work of Couder et a/.) these terms are unimportant. Bangert and Dreybrodt (121) have calculated the absorption coefficient for cyclotron resonance for the B/lc geometry. Using the model and band parameters of Nakao et al. (110), they found that if a weak broadening mechanism exists, the density of states singularity at the H-point is washed out, and the dominant contribution to the absorption coefficient comes from transitions near the peaks of the “camelback” ( k k , , ) rather than the H-point. This conclusion is in agreement with calculations by von Ortenberg et al. (122). On the basis of these calculations and the other experimental results, it must be concluded that the additional line observed “

-

“

”

74

BRUCE D. MCCOMBE A N D ROBERT J. WAGNER

by Yoshizaki and Tanaka was not an H-point cyclotron resonance transition. As discussed below, in less pure samples interference” effects can give rise to additional structure near cyclotron resonance, and this may be the explanation for the additional line. Bangert and Dreybrodt also noted that a significant line shift should occur on heating from 4.2” to 77°K. von Ortenberg et al. (122) have studied the line position for both BI/c and B Ic as a function of temperature up to 120°K. In order to avoid complications due to interference and (or) impurity effects, they were careful to use the highest purity samples available ( p “v 4 x cmp3). They attempted to fit the experimental results with three different models (108, 110, 123) each based on Doi’s zero field k * p formulation. They found that even qualitative agreement required the introduction of a temperature-dependent broadening factor. von Ortenberg et al. concluded that the model of Weiler (108) provided the best fit to their data. Although all experimental workers have reported some evidence of impurity lines, the results are frequently inconsistent with each other as well as with shallow acceptor results in other materials. Probably the most reliable information has been obtained by Couder et at. (120) using material with p z 1013 ~ m - From ~ . the field dependence of three impurity lines, they concluded that an impurity bound state exists with ionization energy of 1.24 meV. As in the case of shallow acceptors in most Groups IV or IIILV semiconductors, they find the intensity of the inpurity lines weakens with increased temperature. Similarly, Dreybrodt et ul. (118) have studied the temperature dependence of a number of lines, both cyclotron resonance transitions and impurity transitions. They found a dramatic difference in the temperature dependences of these two types of lines and made use of this to identify the impurity transitions. However, the impurity binding energy deduced from these measurements was 0.4 meV. In contrast to the above work, Yoshizaki and Tanaka (119,124). using more heavily doped material, found that the lines that they attributed to impurities increased in intensity with increasing temperature. Yoshizaki and Tanaka (119) and von Ortenberg et al. (122) both found that increased impurity concentration increased the intensity of the subsidiary (impurity) structure relative to the cyclotron resonance lines. It seems more likely that this is indicative of the presence of “interference” effects similar to those described by Cronburg and Lax (125), and not representative of impurity effects in the higher concentration samples. “Interference effects can arise when the index of refraction, K , of the material under investigation varies with respect to photon energy or magnetic field in the neighborhood of a strong absorption line. This change of index can manifest itself in at least two distinctly different ways. On the one hand, the effective sample “

”

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

75

thickness, ~ dcan , change with an attendant constructive or destructive interference when A(icd) is some multiple of a quarter wavelength. However, this effect requires a sample with parallel faces, a condition usually purposely avoided in the experiments. O n the other hand, “Faraday” rotation of the light polarization by the sample (which results from the difference between K ~ and ~ K, , - ~ ~can ) cause marked effects when light pipe techniques are used. If the radiation is asymmetrically distributed across the cross section of the light pipe, it is possible for the walls of the pipe between the sample and detector to act as an analyzer of Faraday-rotated light. This can also give rise to oscillations in the apparent sample transmission which resemble weaker absorption lines on the shoulder of the main line. It is clear that further work is required to clarify the situation concerning acceptor impurity transitions in Te. Some information about the conduction band at the H-point has been obtained from interband magneto-optical measurements. For this work, Doi et al. (126) have performed a k p calculation for the conduction band. In fitting their data, they obtain m* (B//c)= 0.05 m, and m” (B i c ) = 0.145 i n , . In the only apparent observation of electron cyclotron resonance Button et al. (117) found m* (B Ic ) = 0.135 m, at 200”K, in fair agreement with the interband results.

-

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. Wc would like to express our gratitude to Mrs. L. Graham, Mrs. G. Garrett, Mrs. L. Blohm, Mrs. C. Hepler, and Miss I. Lajko for their efforts in typing various drafts of the manuscript. One of us (B.D.M.) would like to thank the Max-Planck-lnstitut fur Festkorperforschung, Stuttgart for hospitality extended during a sabbatical year while part of the manuscript was prepared.

REFERENCES I. 2. 3. 4. 5.

6. 7. 8. 9.

M. Cardona, Solid State Pkxs., Suppl. 11 (1969). R. L. Aggarwal, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer. eds.), Vol. 9, p. 151. Academic Press, New York, 1972. B. Lax and J. G. Mavroides, Solid Slate Phys. 11, 261 (1960). J. G. Mavroides, in “Optical Properties of Solids” (F. Abelks, ed.). p. 351. North-Holland Publ., Amsterdam, 1972. E. D. Palik and J. K. Furdqna, Rep. P m p . Phys. 33, 1193 (1970). R. H. Dicke and J. P. Wittke, “Introduction to Quantum Mechanics.” p. 260. AddisonWesley, Reading, Massachusetts, 1960. M. H. Johnson and B. A. Lippmann, Pl7j.s. Rev. 76, 828 (1949). L. D. Landau, 2. Pkys. 64, 629 (1930). M. V. Klein, “Optics,” p. 487. Wiley, New York, 1970.

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BRUCE D. MCCOMBE AND ROBERT J. WAGNER

10. C. Kittel, “Quantum Theory of Solids,” p. 179. Wiley, New York, 1963. 11. J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). 12. T. Kjeldaas and W. Kohn, Phys. Rec. 105, 806 (1957). I?. Y. Yafet, Sofir! State P/IJ.\. 14, 15 (1963). 14. C. Kittel. “Quantum Theory or Solids.” p. 279. Wiley. New York, 1963. 15. J. M. Luttinger, Phys. REG.102, 1030 (1956). 16. S. H. Groves. R. N. Brown, and C. R. Pidgeon. Phys. Rec-. 161. 779 (1967). 17. L. C. Robinson, “Physical Principles of Far-Infrared Radiation,” Meth. Exp. Phys. (L. Marton and C. Marton, eds.), Vol. 10. Academic Press, New York, 1973. IX. J. Houghton and S. D. Smith, ”Infrared Physics.” Oxford Univ. Press (Clarendon), London and New York, 1966. 19. G. W. Chantry. *‘ Submillimeter Spectroscopy.’’ Academic Press, New York, 197 I. 20. K. D. Moller and W. G. Rothschild, “Far-Infrared Spectroscopy.” Wiley, New York, 1971. -71. E. D. Pdlik, “A Far-Infrared Bibliography,” Appendix to Moller and Rothschild, Ref. 20. -7Ztr. P. P. Sorokin, J. J . Wynne, and J. R. Lankard, A p p l . Ph!.s. Lett. 22, 342 (1973). 22h. V. T. Nguyen and T. J . Bridges, Pkys. Rer. Lett. 29. 359 (1972). -72c. R. L. Aggarwal, B. Lax, and G. Favrot, Appl. Phys. Lerr. 22, 329 (1973). 22d. D. H. Auston, A. M. Glass, and P. LeFur. Appl. Phys. Leu. 23, 47 (1973). 22e. E. Gornik, Phys. Rec. Lett. 29, 595 (1972). -73. K. D. Moller and W. G. Rothschild, “Far-Infrared Spectroscopy,” pp. 19-41. Wiley, New York, 1971. 24. F. Kneubiihl, Appl. Opt. 8, 505 (1969). 25. R. J. Bell, “Introductory Fourier Transform Spectroscopy.” Academic Press, New York. 1972. 26. J. W. Cooley and J. W. Tukey, Murh. Compurcrt. 19, 297 (1965). 27. P. Fellgett, J . P k ~ xRadium 19, 187 (1958). -78. S. D. Secombe and D. N. Korn. Solid State Comtnun. 11, 1539 (1972). 2%. R. J . Pressley, ed., “Handbook of Lasers.” pp. 334-349. Chem. Rubber Publ. Co., Cleveland, Ohio, 1971. -7Yb. T. Y. Chang and J . D. McGee, Appl. Phys. Lett. 19, 103 (1971). Z9c. H. R. Fetterman, H. R. Schlossberg, and J. Waldman. Opt. Cornmun. 6 , 156 (1972). 3 t f . S. F. Dyubko, V. A. Svich, and L. L. Fesenko, SOL..Phys.--JETP Lett. 16, 418 (1972). ?Ye. R. J. Wagner, A. J. Zelano, and L. H. Ngai, Opt. Cotnrnun. 8, 46 (1973). 29f: T. K . Plant, P. D. Coleman, and T. A. DeTemple, I E E E J . Quantum Electrot?. 9. 962 (1973). 2Y~q. D. T. Hodges, R. D. Reed, and D. H. Barker, fEEE J . Q ~ r u ~ t ~ iElrcfroti. ini 9. 1159 (1973). 30. R. J . Wagner and G. A. Prinz, Appl. Opt. 10, 2060 (1971). 31. R. C. Ohlmann, P. L. Richards, and M. Tinkham, J . Opt. Soc. Amer. 48, 531 (1958). 32. F. R. Arams, ed., “ Infrared to Millimeter Wavelength Detectors.” Artech House, Ded ham, Massachusetts, 1973. 33tr. R. Kaplan, M. A. Kinch, and W. C . Scott, Solid Store Conmun. 7, 883 (1969). 33h. G. E. Stillman, C. M. Wolfe, and J. 0. Dimmock, Solid S f a t e Cornrnun. 7, 921 (1969). 34. J. W. Hodby. iri ‘‘ Polarons in Ionic Crystals and Polar Semiconductors” (J. T. Devreese, ed.). p. 389. North-Holland Publ.. Amsterdam, 1972. 35. C. R. Pidgeon and R . N. Brown, Phys. Rec. 146, 575 (1966). 36. R . L. Bell and K. T. Rogers, Phys. Rer. 152, 746 (1966). 37. E. 0. Kane, J . Phys. Chern. Solids 1, 249 (1957). 38. R. Bowers and Y. Yafet, Phys. Rev. 115. 1165 (1959). ZY. E. Burstein, G. S. Picus. and H. A. Gebbie, Phys. Rev. 103, 825 (1956). 40. E. D. Pahk, G. S. Picus, S. Teitler, and R. F. Wallis, Phys. Rev. 122, 475 (1961). 41. B. Lax, J. G. Mavroides, H. J. Zeiger. and R. J . Keyes. Ph!..s. Ret:. 122, 31 (1961).

INTRABAND MAGNETO-OPTICAL STUDIES OF SEMICONDUCTORS. I

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 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. 81. 82. 83. 84. 85. 86.

77

E. J. Johnson and D. H. Dickey, Phps. Rec. B I, 2676 (1970). K. L. I. Kobayashi and E. Otsuka, Solid State Cownnun. 11, 815 (1972). T. Murotani and Y. Nisida, J . Pkys. Soc. J a p . 32, 986 (1972). R. J. Keyes, S. Zwerdling, S. Foner, H. H. Kolm, and B. Lax, Phys. Rev. 104, 1804 (1956). E. D. Palik, G. S. Picus, S. Teitler, and R. F. Wallis, Phys. Rec. 122, 475 (1961). E. D. Palik and R. F. Wallis, Phys. Rev. 123, 131 (1961). E. D. Palik and J. R. Stevenson, Pbys. Rec. 130, 1344 (1963). C. W. Litton, R. B. Dennis, and S. D. Smith, J . Phjis. C 2, 2146 (1969). J. M. Chamberlain. P. E. Simmonds, R. A. Stradling. and C. C. Bradley, Proc. Int. Conc P/ij.s. Setnicond., l l t b , Warsaw, 1972, p. 1016. PWN-Polish Sci. Publ., Warsaw, 1972. E. D. Palik, J. R. Stevenson, and R. F. Wallis, Pkys. Rev. 124, 701 (1961). H. Fetterman, J. Waldrnan, and C. M. Woife, Solid State Cornrnun. 11, 375 (1972). 0. Berolo, J. C. Woolley, and J. A. Van Vechten, Phys. Rev. B 8, 3794 (1973). S. H. Groves and W. Paul. Phys. Rev. Lerr. 11. 194 (1963). J. Waldman, D. M. Larsen, P. E. Tannenwald, C. C . Bradley, D. R. Cohn, and B. Lax, Pl7I.s. Rev. Left. 23, 1033 (1969). A. J. Strauss, T. C. Harman, J . G. Mavroides, D. H. Dickey, and M. S. Dresselhaus, Proc. I n t . Conj: PI7ys. Sernicond.. Exeter, Eng. p. 703. Inst. Phys., Phys. Soc., London, 1962. M. A. Kinch and D. D. Buss, .I. Phys. Chern. Solids 32, Suppl. 1; 461 (1971). K. J. Button, C. G. Fonstad, and W. Dreybrodt, Phys. Rev. B 4, 4539 (1970). A. Kawabata, J . Phys. SOC.Jap. 23, 999 (1967). E. E. H. Shin, P. N. Argyres, and B. Lax, Phj,.s. Rev. Left. 28, 1634 (1972); Phys. Rru. B 7, 3572 (1973). J. R. Apel, T. 0. Poehler. C. R. Westgate, and R. A. Joseph, Phys. Rea. B 4,436 (1971). R. Kaplan, B. D. McCornbe, and R. J. Wagner, Solid State Cornn~un.12, 967 (1973). E. I. Rashba and V. I . Sheka, Soc. Pby.s.-Solitf Sttrte 3. 1257. 1357 (1961). Y. Yafet, Solid Stare Pl71.s. 14, 41 (1963). V. 1. Sheka, Sou. Phys.-Solid State 6, 2470 (1965). G. E. Gurgenishvilli, Soc. Php.-Solid Srate 5, 1510 (1964). I. I. Boiko, SOC.Phps-Solid State 4, 1558 (1963). P. Wolff, J . Phys. Clzern. Solids 25, 1057 (1964). D. Kacman and W. Zawadzki. Pbys. Status Solitli 47, 629 (1971). B. D. McCombe, Phys. Rec. 181, 1206 (1969). A. S. Saleh and H. Y. Fan, Phys. Rev. 182, 790 (1969). W. Zawadzki, in “New Developments in Semiconductors” (P. R. Wallace, R. Harris, and M. J. Zuckermann. eds.), p. 441. NoordhoK, Leyden, 1972. B. D. McCombe, S. G. Bishop, and R. Kaplan, Phys. Rev. Lett. 18. 748 (1967). R. L. Bell, Phys. Reu. Lett. 9, 52 (1962). B. D. McCombe, R. J. Wagner, and G. A. Prinz. Phys. Rev. Lett. 25. 87 (1970). B. D. McCombe and R. J. Wagner, Proc. I n r . Conj. Pliys. Sernimnd.. I l t h , Wtrrsuw, 1972, p. 321. PWN-Polish Sci. Publ., Warsaw, 1972. S. H. Groves, T. C. Harman, and C. R. Pidgeon, Solid Stare Cornrnun. 9, 451 (1971). 3.D. McCombe and R. 1. Wagner, Phyx Rec. B 4, 1285 (1971). B. D. McCombe, Proc. I n f . Con$ Sernicond.-Seiiiimetal Transiriori, *Vice, 1973. S. R. J. Brueck and F. A. Blum, Phys. Rec. Lett. 28, 1458 (1972). R. W. Davies, Pbys. Rev. B 7, 3731 (1973). Y. C. S. Auyang (S. Y. Yuen), Ph.D. Thesis, Mass. Inst. of Technol., Cambridge, Massachusetts, 1971. Also S. Y. Yuen. P. A. WolH: and B. Lax. Pb1.s. Rer. B 9, .1394 (1974). C. R. Pidgeon, D. L. Mitchell, and R. N. Brown, Phys. Rea. 154, 737 (1967). R. A. Isaacson, Phys. Rer. 169. 312 (1968). S. H. Groves, J . Phys. Chetn. Solids 32, Suppl. I. 447 (1971). J. C . Hensel. Phys. Rev. Lett. 21, 983 (1968).

78 X7. XK.

X9. 90. Y1.

97. Yj. Y4. 95. Y6. Y7.

98. YY. 100. 101.

B R U C E D. MCCOMBE A N D ROBERT J. W A G N E R

R. Ranvatid. P1i.D. Thesis. Brown Unit.. Providence, Rhode Island, 1973. J . C. Hensel and K. Suzuki. Proc. f r i r . Cord. Phys. Semicond., 10tl7, Cu!tihr.it/yr, Mussuchusetts. p. 541. USAEC. DIV.of Tech. Info.. Oak Ridge. Tenn., 1970. J. C. Burgiel and L. C. Hebel, Phys. Rev. A 140, 925 (1965). E. J. Tichovolsky and J. G . Mavroides, So1id Stute Cornrizzrrl. 7. 927 (1969). G . E. Smith. G . A. Baraff. and J. M. Rowell, Plzys. Rer. 4 135. 1118 (1964). M. H. Cohen and E. 1. Blount, Phil. May. 5, 115 (1960). J. C. Hensel and K . Suzuki. N u t . Birr.. S f a n d . ( U S ) . Spec. Pirhl. 323, 431 (1971). K . Suzuki and J . C. Hensel, Phys. Rer. B 9, 4184 (1974). J. C. Hensel and K. Suzuki, PIrj..s. Rer. B 9, 4219 (1974). R. F. Wallis and H. J. Bowlden, Phjx Rev. 118, 456 (1960). R. C. Fletcher, W. A. Yager, and F. R. Merritt, Phys. Rec. 100, 747 (1955). H. Fujiyasu, K. Murase, and E. Otsuka. J . Phys. Soc. J a p . 29, 685 (1970). K. J . Button, B. Lax. and C. C. Bradley, Phys. Rei,. Lett. 21, 350 (1968). K. J. Button. in “Optical Properties of Solids” (E. D. Haidemenakis, ed.), p. 253. Gordon & Breach. New York. 1970. C. C. Bradley. P. E. Sinimonds, J. R. Stockton, and R. A. Stradling, Solid Srufe Cotiirtitr?z.

12, 413 (1973). 102. W. Leung and L. Liu, Phya. Rer. B 8. 381 I (1973). 103. J . Tuchendler, M. Grynberg. Y. Couder, H. Thome. and R. LeToullec. Ph!..\. Rr~i..B 8, 3884 (1973). 104. Y. Guldner, C. Rigaux. M. Grynberg, and A. Mycielski. Phj,s. Rec. B 8, 3875 (1973). 1 0 5 . C. R. Pidgeon and S. H. Groves. iri ”11-VI Semiconducting Compounds; 1967 International Conference” (D. G . Thomas. ed.). p. 1080. Benjamin. New York. 1967. 106. M. Hulin, J . Phys. Chem. Solids 27, 441 (1966). 107. T. Doi, K. Nakao. and H. Kamimura, J . P h y . SOL..Jtrp. 28, 36 (1970). 108. M. H. Weiler. Solid Sture Cor~nzurz.8, 1017 (1970). 109. M. S. Bresler and D. V. Mashovets. Phys. Sfatus Solidi 39. 421 (1970). 11(J. K. Nakao, T. Doi, and H. Kamimura. J . Phys. Soc. J u p . 30. 1400 (1971). 1 1 1 0. Betbeder-Matibet and M. Hulin. P/iy.\. Sicitu.< Solid; 36. 573 (1969). 112. J. C. Picard and D. L. Carter, J . PI1j.s. Soc. Jup. 21, Suppl., 202 (1966). 113. E. Braun and G. Landwehr, J . P h n . Soc. Jup. 21, Suppl., 380 (1966). 114 C . Guthmann and J. M. Thuillier. Soiitl State Corm7irn. 6. 835 (1968). 115. P. L. Radoff and R. N. Dexter. Pkrs. Srrriir.s Solid; 35. 261 (1969). 116 Y. Couder, Phy,$. Rec. Leu. 22, X90 (1969). 117. K. J. Button, G . Landwehr, C. C. Bradley, R. Grossc, and B. Lax. Phys. Rec. Lerr. 23. 14 ( 1969). 118 W. Dreybrodt, K. J. Button, and B. Lax, Solid Srure ~ O ~ J l i J i8,i ~ 1021 ~ . (1970). 1 1Y. R. Yoshizaki and S. Tanaka. .I. Phjx SOP.Jap. 30, 1389 (1971). 1-70. Y. Couder, M. H u h , and H. Thome. Phys. Reo. B 7, 4373 (1973). 121. E. Bangert and W. Dreybrodt, Solid Srare Corn~nurz.10, 623 (1972).

122.

M. von Ortenberg, K. J. Button. G. Landwehr, and D. Fischer, Phjx Rc’c. B 6. 2100 (1972).

123

W. Dreybrodt. M. H. Weiler. K. J . Button. R. Lax. and G. Landwehr. P r o c I u / . C’orif P / J ~ . .Sernicomi.. s. IOtk, Cumbrirlye. !2.luz.\cicliir.sert\, p. 347. USAEC. Div. of Tech. Info.. Oak Ridge. Tenn.. 1970. 1-74. R . Yoshizaki and S. Tanaka, Solid Srure Curnnnrri. 8. 17x9 (1970). 125. T. L. Cronburg and B. Lax, Plzr.s. Lett. A 37, 135 (1971). 126. T Doi. H. Kamimura, H . Shinno. R. Yoshizaki, and S. Tanaka. Prot.. 1 i i r . C O ~ JPhi.\. /. Se!nicoiid., I 1 f l i . Warsuw, 1972. p. 761. PWN-Polish Sci. Publ., Warsaw. 1972.

The Gyrator in Electronic Systems K . M . ADAMS

AND

E . F. A . DEPRETTERE

Depirrtnient of’ Elecrrictrl Engginerring Dert Unioersity of’Technology. Del$. Netherlands

.

AND

J . 0. VOORMAN Philips Research Laboratories. N .V . Philips’ Gloeilanipvilfubrieken. Eindhouen. Netherlands

I . Introduction .......... ....... .......... I1 . Reciprocity in Physic s ...... .......... A. Some Historical Background ..................................... B. Nonreciprocity ................................................................................. C . The Origin of Antireciprocity .............................................................. D . Further Generalizations ..................................................................... ......... 111. The Gyrator as Network Element ......... A . Historical Introduction .................................... B. Basic Properties of the Gyrator ............................................................ IV . Filters ................................................................................................... A . Introduction ............ B. Basic Configuration ........................... C . The Scattering Matrix ........................................................................ D . Ladder Filters .................................................................................... E . Resonance in Filters ........................................................................... F. Sensitivity ....................................................................................... ..................................................... G . The Gyrator in Filters .............. V . Principles of Realization of the Gyrator ........................ A . Physical Effects ...... ................................. B. Active Circuits .................................................. C . Ideal Active Network Elements .......................................... D. The Nullor ....................................................................................... E. Gyrators Constructed from Nullors and Resistors .................................... VI . Basic Electronic Design ........................................................................... A . Realization Based on Two Resistors in the Signal Path .............................. B. Realizations Based o n Four Resistors in the Signal Path ........................... C . Noise ............................................................................................. VI1 . Basic Gyrator Measurements ..................................................................... A. Introduction .................................................................................... B. The Gyrator as n-Port ........................................................................ C . The Gyrator in Its Applications ............................................................ 79

80 80 80 82 83 85 86 86 88 94 94 94 96 99 101 107 109 109 109 111

113 113 116 128 129 141 143 145 145 146 155

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VIII. Trends in Gyrator Design and Applications ................................................ A. Introduction ..................................... ......................................... B. Gyrators for Application in Consumer Pro s ...................................... C. Gyrators for Professional Use .............................................................. IX. Conclusion.. ..................................................................... References ............................................................................................

170 170 170

171 177

I. INTRODUCTION Recent developments in electronic technology combined with increasing insight into the theoretical basis of physical systems have led to a growing interest in the gyrator and its applications. On the one hand, theoretical results which have been known for more than twenty years are beginning to be applied in earnest to real systems. O n the other hand, the new technology which has made these applications possible has stimulated a reexamination of the theory and has led to new and interesting results, which in their turn can stimulate further developments in technology itself. In this survey article we report and comment on some of these developments. Our approach will be based on an attempt to extract unifying principles from the mass of data now available rather than to give an encyclopedic account of the state of the art. By so doing we cannot fail to adopt a somewhat subjective point of view, approving of some developments, disapproving of others, and being noncommittal on some questions which we d o not yet feel capable of resolving but which certainly deserve attention. We hope that by stating our premises and reasoning clearly, interest in this fascinating field will be further stimulated. Our survey will touch on historical matters, go into various theoretical questions, as well as be concerned with thoroughly practical matters such as system specification and actual hardware. The following sections are so arranged that they can be read largely independently of one another. We would emphasize, however, that for the reader who wishes to be able to apply the existing technology intelligently to his own particular system, a thorough grounding in the theory is essential. 11. RECIPROCITYI N PHYSICAL SYSTEMS

A . Some Historical Background

During the eighteenth and nineteenth centuries, mechanical systems were studied in great detail and in various degrees of generality. It was noticed that in nearly all these systems a reciprocity relation held which

THE GYRATOR IN ELECTRONIC SYSTEMS

81

manifested itself in certain symmetry relations in the coefficients of the constitutive equations of the system, and which ultimately depended on Newton’s third law of motion: the action of a particle A on a particle B is equal and is oppositely directed to the reaction of particle B on particle A.” This reciprocity relationship can be expressed in various forms such as: Let an infinitesimal external force F , acting on a system at a location P, give rise to an infinitesimal displacement x2 at a location P, when no other external force acts on the system. If now the same force F , acts externally on the system at P2 and no other external force is present, then the displacement occurring at P, is equal to x 2 . When it is possible to derive the forcedisplacement relationships from a potential energy function, then the reciprocity follows automatically as a consequence of the commutativity of the operations of partial differentiation with respect to different coordinates. Another example from classical mechanics is the reciprocity between impulsive forces and velocities, which follows from the existence of a kinetic energy function. A corresponding electrical analogue is the reciprocity relation between currents and flux linkages in a linear network of magnetically coupled coils, which follows from the existence of a magnetic energy function. As a result of relations of this type many people erroneously came to regard reciprocity as essentially connected to the concept of energy. From this point it was not difficult to infer (also erroneously) an apparently essential connection between passivity and reciprocity. This idea that passivity and reciprocity are somehow related has played an important role in the evolution of the concepts relating to reciprocity between force and velocity, but has also led to considerable harm. Thus when one comes to consider the reciprocity between voltages and currents in an electrical network, a new difficulty arises. There is no a priori reason for expecting the existence of a function which plays the same role as the energy functions already mentioned. It is true that in the case of a particular type of linear system the situation could be saved by the introduction of a dissipation function (Rayleigh, 1894). However, this step really involved begging the question, since the dissipation function could only be deduced knowing that the system (expressed in network theoretical terms) consisted of linear resistances and ideal transformers; from network theory it is known that such a system is reciprocal since it consists of reciprocal elements. In the case of nonlinear networks, for which either a content or a cocontent function exists (Millar, 1951), we find a similar reciprocity relation between the deviations of the voltages and the currents from a particular distribution of voltages and currents in the network. But again the existence of the content function can only be a priori guaranteed for a network of ideal transformers and one-port resistors with characteristics representing a “

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single-valued mapping of the currents on the voltages or of the voltages on the currents. Another development which has taken place is based on completely different considerations resulting from the study of reversibility in thermodynamic systems. Since 1931 attempts have been made to deduce symmetry relations on the basis of microscopic reversibility (de Groot and Mazur, 1962) in which the reversibility of time in the microscopic equations of a thermodynamic system is supposed to hold. However, in order to make the theory work, terms depending on a magnetic field strength have to be reversed when the time is reversed, and there are grave difficulties in deciding which variables should be regarded as thermodynamic forces and which as affinities. Further, the question of whether time should have a preferred direction of flow or be reversible and why this should be so, as well as the relation to the concepts of order and disorder (Prigogine et al., 1972), has not been satisfactorily resolved. The whole matter has been trenchantly criticized (Truesdell, 1969) as being devoid of a logical structure based on a set of sound and physically acceptable axioms. Even so it is not impossible that new studies of thermodynamics will shed light on this whole question. B. Nonreciprocit)) The physical evidence available to date indicates that there is no a priori reason for expecting any form of symmetry in the equations of general physical systems. If a symmetry does exist, then it is a result of the form of the constitutive equations of the system under investigation and not of any general physical principle. Thus the reciprocity principle as commonly understood is not a principle at all but a property of certain special classes of systems. As we have already seen in the previous section, there is a close relation between reciprocity and energy when certain other very restrictive conditions apply. There are two very simple physical situations where reciprocity does not apply. The first occurs in a mechanical context and concerns the relation between the velocities and the corresponding torques arising in the motion of a gyrostat about mutually orthogonal axes perpendicular to the spin axis. Here, as has been known since the eighteenth century, the matrix expressing this relation for an idealized gyrostat is antisymmetric (Pars, 1965), instead of symmetric as in the case of a mechanical system consisting for example of particles, rods, and dashpots, and which possesses the reciprocity property. In spite of the reality of such gyroscopic systems, many theorists persisted in ignoring their existence in their single-minded search for symmetry, or else quietly excluded gyroscopic terms from the equations of motion as an inconvenient detail.

THE GYRATOR IN ELECTRONIC SYSTEMS

83

The second example concerns the Lorentz force experienced by a charged particle moving in a uniform magnetic field. Here again the relation between the components of the forces and velocities along orthogonal axes in a plane normal to the direction of the magnetic field is given by an antisymmetric matrix.

C. T h e Origin of Antireciprocity If one has been brought up on the premise that symmetry and reversibility should always be present in some form in general physical systems, then it is rather startling to be confronted by an antireciprocal system. The natural reaction is to try to preserve the symmetry or reciprocity at all costs. In the more general form of the reciprocity relation (Rayleigh, 1894), one considers two different “states of the system, corresponding to two different sets of variables satisfying the system equations. Further one distinguishes various types of variables from one another, such as forces from displacements, velocities, and accelerations, or voltages and flux linkages from charges and currents. Denoting the states by A and B and the two types of external system variables by the vectors u and i respectively, the reciprocity relation becomes ”

Now in the case of the gyrostat we can satisfy this relation by taking for state B any member of the set of possible states A transformed by time reversal. (Here u and i represent the torque and velocity vectors respectively.) Then the spin direction is automatically reversed in state B as compared to state A. We can also regard state B as being a member of the set of possible states A but now referring to a d g t r e n t gyrostat, namely with the spin direction reversed, but with otherwise identical characteristics. The important distinction between the t w o approaches lies in the difference between what one regards as the state. In the first case the state refers to all relevant parameters, both internal and externally observable, and thus includes the spin vector of the gyrostat, which, however, is assumed constant. In the second case, only the “external” variables which enter explicitly in (1) are considered. Similar relations hold for a charged particle moving in a magnetic field, provided we reverse the direction of the magnetic field in state B. Of course one can argue that the magnetic field itself is caused by current flow, either macroscopic or microscopic, and if the time is reversed the current flow and thus the magnetic field will be reversed.

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It is tempting to speculate as to what is the basic cause of the antireciprocity in systems of this type.* Two important facts emerge which are evidently closely connected with antireciprocity and with the skew-symmetry of the system matrix. First of all we note that the basic equations in both systems include a vector product which is anticommuting. Thus for the gyrostat we have the equation of motion T

= Cnr

+ Ar

x r,

(2)

where T is the torque vector, r is a unit vector along the spin axis, A and C are principal moments of inertia, and n is the spin velocity. If the spin velocity is sufficiently large, we can neglect the last term and write

T = Cni.

(3)

The projection of the total angular velocity vector on the plane normal to the spin axis is Q=rxr. (4) The total power absorbed by the gyrostat as a result of the action of the external torques is T * Q= 0, (51 by virtue of ( 3 ) and (4). This is an example of what is known as a ??oneneryic system (Birkhoff, 1927). We also note from (3) and (4), that the component of angular velocity which is proportional to a given component of torque is perpendicular to that torque component. These two properties are sufficient to guarantee antireciprocity of the system and lead automatically to an antisymmetric system matrix. In the same way, we have for the charged particle, moving in a uniform magnetic field and in the absence of an electric field, the equation of motion:

F

= ev x

B,

(6)

where F is the force, e the charge, v the velocity, and B the magnetic flux density. Then the power absorbed by the particle as a result of the force is

F - v = 0. (7) Again the system is nonenergic and the force components are perpendicular to the velocity components to which they are proportional. We thus can see the intimate connection between antireciprocity, nonen* Antireciprocity is a special case of nonreciprocity. I f the normal reciprocity relation can be restored by a simple change of sign in some of the describing equations, then we refer to the system as antireciprocal.

THE GYRATOR IN ELECTRONIC SYSTEMS

85

ergicness, and the properties of three-dimensional Euclidean space, as exhibited by the properties of the vector product. D . Further Generalizations The concepts embodied in Eq. (1) can be applied to far more general systems than those so far mentioned. For example, if one considers linear time-invariant systems in which all the system variables are functions of time and of the form A iexp ( p t ) , with A iindependent of time and p a constant for all the system variables, then the reciprocity relation applies to a large class of such systems (Rayleigh, 1894). An example is a linear n-port network consisting of resistors, capacitors, inductors, and transformers. When the reciprocity concept is applied to three-dimensional field problems, the summation in Eq. (1) is replaced by a spatial integral and a relation between a volume integral and an integral over the bounding surface results. Such relations form the basis of the solution of integral equations by means of Green’s functions and are applicable to a wide class of problems (de Hoop, 1966). Other developments have been concerned with the application of relations like Eq. (1) to two specially related systems which are termed interreciprocal or adjoint (Penfield et d., 1970). The resulting equations lead to useful results for deducing general properties, for network computations, or for the numerical solution of integral equations. All of these developments, however, are special cases of the properties of linear operators and their adjoints in the theory of abstract spaces (Dunford and Schwartz, 1958). For example, in the theory of Hilbert spaces one has the relation

where T denotes a linear operator mapping a Hilbert space $ into itself, T* is the corresponding adjoint operator, x and y are elements of 9, and the symbol (, ) denotes the inner product in 5. Because of the importance of concepts like power and energy in physical systems, the Hilbert space concept with its inner product is especially relevant. Depending on the definition of the inner product that is chosen, a wide variety of reciprocity results of which (1) is the prototype can be obtained. These can refer to variables in the time domain, frequency domain, or spatial domains. The operator T and the space 9 form the abstract representation of the system equations; corresponding to the adjoint operator one can either construct a real system or postulate a fictitious one. As a result of (8) various terms can be made to disappear in the describing equations. It is precisely

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the desirability for the disappearance of such terms and the resulting simplications that have led many physicists to attach such importance to symmetry. Symmetry, however, is only one way to get rid of unwanted terms and corresponds to self-adjoint operators. The really important point to note, however, is that there is no reason to expect the existence of some universal physical principle that guarantees self-adjointness and that the correct use of the adjoint operator will usually allow one to achieve what is desired without having to impose the symmetry condition.

111. THEGYRATOR AS NETWORK ELEMENT A . Historical Introduction In view of Section lI,C, there is no fundamental objection to postulating the existence of an electrical two-port device described by the equations

where u1 and il are the input voltage and current and u2 and i, are the output voltage and current. Such a device is called the ideal gyrutor and is represented by the symbol shown in Fig. 1. The gyrator was introduced as a

FIG. 1. The ideal gyrator.

postulated electrical network element (Tellegen, 1948a) representing the simplest linear, passive nonreciprocal system, before it was clear whether such an element could be a reasonable approximation to some physically real, purely electrical system. I t is true that at about this time, the idea of an antireciprocal passive device was becoming quite clear to several people. Thus an antireciprocal two-port, electromechanical system and its cascade connection with a reciprocal magnetomechanical two-port was considered. The resulting two-port is an antireciprocal electrical two-port (Jefferson, 1945; McMillan. 1946). Indeed gyrator devices based on this principle were built and marketed in the 1960’s, but have not enjoyed much popularity. The ideal gyrostat was

THE GYRATOR IN ELECTRONIC SYSTEMS

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also proposed as a mechanical element (Bloch, 1944), and several of its important properties when used in conjunction with other mechanical elements were deduced. Bloch came very close to formulating the electrical gyrator concept, especially in his considerations of the analysis of mechanical networks. However, it was Tellegen who first clearly recognized the fundamental importance of the gyrator as an element, deduced its most important properties, and showed how to employ it in the synthesis of electrical networks (Tellegen, 1948b, 1949; Tellegen and Klauss, 1950, 1951). As a result, electrical network theory and especially network synthesis gained enormously in breadth. In fact as a recent text (Belevitch, 1968) shows, electrical network theory today without the gyrator would be inconceivable. Tellegen, however, not only developed the theory of the gyrator, but also considered possible realizations. This work led him to consider a hypothetical medium in which the electrical polarization can be influenced by the magnetic field strength and the magnetic polarization can be influenced by the electric field strength (Tellegen, 1948a). Such a medium was later found to exist physically (Astrov, 1960, 1961). He also considered the gyromagnetic effect in ferromagnetic materials as a basis for a possible realization. Further calculations by Polder (1949) and experimental work by Hogan led to the first practical gyrator device, operating at microwave frequencies (Hogan, 1952). In spite of all these and many recent developments, the gyrator has not been readily accepted by the electrical engineering and physics communities. Even among network theorists, with a few notable exceptions, there was considerable hesitation in accepting the element and doing theoretical work with it until about 1960. And today one can still find teachers of electrical engineering who prefer their students to remain in complete ignorance of the gyrator concept. We suggest that a major cause of such phenomena is the widespread and deeply-rooted misconception of the true nature of reciprocity. Amongst the designers and users of electronic apparatus, the erroneous notion that a nonreciprocal system cannot be passive is still persistent. It has arisen because all the devices these people came in contact with were either passive and reciprocal or active and nonreciprocal even though by a suitable combination of such devices, passive and nonreciprocal systems could be constructed, at least in principle. Matters have not been helped by references to the reciprocity theorem.” When formulated with any degree of rigor, this “theorem ” either reduces to the tautologous statement that reciprocal systems are reciprocal or to the deduction that a certain specific system constructed from components with certain specified constitutive equations possesses the reciprocity property. The authors of too many books and “

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

papers give the impression that they are appealing to some very general physical principle when they cite the reciprocity theorem or reciprocity principle.” “

”

“

B. Basic Properties of’the Gyrator 1. Siyri Concentions

Figure 2 shows the symbol for the gyrator with various reference voltagepolarity and current-direction signs. These signs and their interpretation as stated here now have fairly general usage. The interpretation is that a quantity such as u , ( t ) is representated by a positive number if the potential of terminal 1 is higher than the potential of terminal 1’ at the instant of time t.

FIG.

2. S i p conventions

Similarly a quantity such as il(t) is represented by a positive number if the current of port 1 flows in the direction indicated by the arrow at time f. In the opposite case the quantity is represented by a negative number. The functions u ( . ) and i(. ) are then the sets of these numbers ordered on the continuous set {t). We are free to choose these reference polarities and directions as we please. Once chosen, however, these references determine the signs to be attached to the algebraic symbols in the describing equations.

2. Noneneryicness The gyrator is an example of what is known as a norieneryic sjxstern (Birkhoff, 1927). That is to say, the instantaneous power absorbed by the element is always zero. Thus from the defining equations (91 we have u l ( t ) i l ( t )+ u 2 ( t ) i 2 ( t= ) 0, The ideal transformer, with defining equations

Vr.

(10)

THE GYRATOR IN ELECTRONIC SYSTEMS

89

in conjunction with the reference polarities and current directions of Fig. 3, is also nonenergic. The only other nonenergic, linear, time-invariant twoports are two-port combinations of the nonenergic one-ports : the short circuit and the open branch. The proof of this result is a simple exercise in linear algebra.

FIG.3. The ideal transformer with its sign convention in relation to thedefining equations: u I = n u 2 , i2 = - n i l .

The concept of nonenergicness has important uses in much more general contexts (Adams, 1974), but for our present purposes, it is not necessary to go into these details. It is clearly a special case of passivity.

3. Isolation and Power Transmission Consider the two-port networks shown in Fig. 4. If the gyrator is defined by Eq. (9), then the two-port resistance matrix of two-port a is

With the sign conventions of Fig. 2, this means that the power absorbed by port 1 is u,i, = Ri: 2 O

(13) if R > 0. On the other hand, the power absorbed by port 2 can be either positive or negative, depending on the ratio i l / i 2 . Since the resistor is passive, this means that power can be transmitted only in one direction through the two-port. Thus port 2 can deliver power to its surroundings but port 1 will always absorb power. In the same way we find for two-port b that the two-port conductance matrix is

[

G -2G

G O1

(G

=

1/R)

and that the same conclusions regarding power transfer apply. It is as a

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K. M. ADAMS. E. F. A. DEPRETTERE AND J.

0. VOORMAN

result of this property that the arrow associated with the gyrational resistance R in Fig. 2 is employed. This arrow can now be interpreted in two ways. Either we can agree that R is always positive, in which case the arrow denotes the actual direction of power flow in the gyrator-resistor two-port and Eq. (9) applies with R always positive, or we can regard the arrow as a reference direction which corresponds to the direction of power flow when R is positive, or is opposite to this direction when R is negative. If we then agree to introduce the reference voltage polarities and current directions of

=====P

FIG.4. The isolator and power flow

the ports in such a way that the arrow is directed between the terminals associated with plus signs of the two ports, and further choose the current references such that ui is positive when the port absorbs power, then the equations of the gyrator are given by (9). The use of the arrow alone thus enables us to write down the equations with the correct signs and to adopt a consistent reference for voltage polarities and current directions. If for any reason we wish to change any of these references then the corresponding equations can be obtained by replacing the appropriate voltages or currents by their negatives.

THE GYRATOR IN ELECTRONIC SYSTEMS

91

The gyrator-resistor combination discussed here is known as an isolator and has important applications in microwave technology (Beljers, 1956). 4. Antireciprocity If, as in Section II,C, we consider two states denoted by A and B, then we find that the defining equations yield

This relation can be obtained from Eq. (1) by converting a single minus sign to a plus sign, and thus corresponds to an antireciprocal system. When the impedance or the admittance matrix of an n-port exists, then the matrix is skew-symmetric if the n-port is antireciprocal. Antireciprocity is the simplest form of nonreciprocity. No other nonreciprocal element than the gyrator is required to characterize linear, time-invariant, nonreciprocal n-ports. It is one of the basic results of network synthesis that any passive n-port, which is characterized by linear ordinary differential equations with constant coefficients in the port voltages and currents, can be synthetized from the elements: resistor, capacitor. inductor, ideal transformer, and gyrator (Oono and Yasu-ura, 1954; Belevitch, 1968). As we shall see in the following sections, the ideal transformer and inductor can be deleted from this list. 5. Ideal Transformers A cascade connection of two gyrators is equivalent to an ideal transformer (Fig. 5). Conversely, any ideal transformer can be synthetized by a network of two gyrators with suitably chosen gyrational resistances.

hd t FIG.5. Ideal transformer synrhetized from two gyrators

6. Iininittance Inversion

Perhaps the most important property for engineering applications is the inversion of an impedance or admittance, as shown in Fig. 6. In particular, a capacitance with impedance llpC is converted by a gyrator into an impedance pR2C, corresponding to a self-inductance L = R 2 C . Especially this

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K . M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

property has led to increased interest in applying the gyrator concept to integrated circuits. More generally, any one-port network N is equivalent to a one-port formed by the cascade connection of a gyrator and a network N’, which has the structure of the dual network of N (if it exists) and with element values proportional to those of the dual network. Another generalization is the two-port equivalence shown in Fig. 7.

-Jj-=j

JY=;;.

FIG.6. Demonstrating the immittance inversion property

7. Iiiterclzange

of’ Variables and Duality

A generalization of the foregoing applies to n-ports. Let IZ gyrators, with gyrational resistors of 1 Q, be connected to the ports of an n-port network N as shown in Fig. 8. There results an 17-portnetwork N’, which is described by 1R

3-c 3-c

n-port

IQ

FIG. 8. Interchange of port variables

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THE GYRATOR IN ELECTRONIC SYSTEMS

equations obtained from those of N by interchanging the port voltages and the corresponding port currents. If the equations are expressed in terms of other electrical quantities, then these quantities are interchanged according to their relation to voltages and currents. Thus, for example, charges and fluxes are interchanged. Then network N ’ is equivalent in the n-port sense to the dual of N (when it exists). Thus we can use the gyrator as a dualizing converter in design or analysis. A simple application of this principle is shown in Fig. 9.

FIG.9. A special case of dualizing.

When the gyrator can be used freely, the distinction between current and voltage as ‘‘ through and “across variables disappears, since the topological structures inherent in these concepts and their duals are freely interchangeable. This is one of the reasons for the difficulties in attempts to lay the foundations of nonequilibrium thermodynamics in the conventional manner (de Groot and Mazur, 1962). ”

”

8. Reactioe Power Although the gyrator is nonenergic, this does not mean that the complex power absorbed by the gyrator is zero, but only that the real part of the complex power is zero. The imaginary part, or the reactive power, is given by Qc,

=

f Im ( U ,IT +

U 2IT) = Im ( R I , IT).

(16)

This is not surprising when we consider the relation of reactive power to stored energy, Thus the reactive power absorbed by an inductor is QL = -$ Im (jwLII*) = &wLI Z I z

=

20W, > 0

(w

> 0),

(17)

where W, is the mean stored magnetic energy. For a capacitor we find that Q c = -‘C 2 a (U12 = -2wW, > 0

(0>

O),

(18)

where We is the mean stored electric energy. Since reactive power is conserved in networks (Tellegen, 1952; Penfield et al., 1970), if an inductor is simulated by a gyrator terminated in a capacitor, the difference of the reactive powers absorbed by the capacitor and the simulated inductor must be equal to the reactive power absorbed by the

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

gyrator. It is readily verified from Eqs. (16)-( 18) that QL

= QC

+QG~ 7

when account is taken of the sign convention for currents and voltages and of the interconnection of the elements. This property of storing reactive power is in sharp contrast to the properties of the ideal transformer, for which both the absorbed active and reactive powers are zero. Once the gyrator has been introduced, the distinction between total stored electric energy and total stored magnetic energy of a system disappears. The total stored energy, electric and magnetic, is still relevant but the difference between the two types of energy disappears. In the Lagrangian formulation of networks, this disappearance is compensated by the appearance of linear terms in the Lagrangian function to account for the presence of gyrators (Adams, 1968).

IV. FILTERS A . Introduction

A filter is an apparatus that, when presented with a mixture of certain “things,” allows some of these “things” to pass through it in a specified manner and prevents the passage of the other “things.” In electrical engineering the “things” are usually signals which can be regarded as consisting of various components, some of which are desirable and are to be retained as far as is possible, and others which are undesirable and are to be rejected. Historically, the first filters consisted of linear, time-invariant networks, and were designed for separating the signal components in various frequency bands from one another. Today, the word filter has come to mean almost any signal-processing apparatus or algorithm or program that is remotely connected with the idea of separation of desirable from undesirable quantities. In many of these generalizations from the original simple filter networks, concepts such as reciprocity and antireciprocity apply. The gyrator concept then automatically appears and plays an important role (see, e.g., Fettweis, 1971). In this section we shall concentrate on the classical network filter from which these other more recent developments have evolved. B. Basic Configuration

The classical filter-network configuration consists of a signal source which is capable of delivering a certain maximum quantity of signal power, a passive transmission two-port network which is the actual filter, and a load

95

THE GYRATOR IN ELECTRONIC SYSTEMS

where the filtered signal is further processed and its accompanying electrical power is finally dissipated into thermal power (Fig. 10). The signal (energy) source can be represented by an ideal voltage source in series with an impedance (Thevenin circuit), or equivalently, an ideal current source in parallel with an admittance (Norton circuit). We shall restrict our discussion to the simplest but very important practical case in which both the source and the load impedances are pure resistances. We shall further restrict ourselves to the case where the signal source strength varies sinusoidally with time.

n a ,

Source

-

Load

Source

Filter I

Load

I I

FIG. LO. Basic filter configuration

Then one of the very important parameters for judging the performance of the filter is the ratio of the power received by the load to the maximum available power of the energy source as a function of frequency (Geffe, 1963; Saal, 1963; Skwirzynski, 1965).* The quality of such a filter is then often judged by such parameters as the closeness with which this ratio approaches unity in certain frequency bands (the passbands), the smallness of the ratio in certain other bands (the stopbands) and the narrowness of the frequency intervals in the remaining portions of the spectrum (the transition bands). These terms are illustrated in Fig. 11. T w o other points should be noted at this stage. The first is that intuitively we should expect the most efficient transmission in the passband, when the filter proper is lossless. At these frequencies the filter is ideally transparent to signals. Secondly, filtering is often connected with such matters as improving the signal-to-noise ratio. This is a power ratio and the noise power of a source is a maximum available noise power, so that, provided the filter itself does not generate any noise, the transmission parameter shown in Fig. I I is most appropriate.

* A filter theory based on this power ratio as primary characteristic is often referred to as the insertion-loss theory. Unfortunately, not all authors mean the same thing by this term, which is the reason why we prefer not to use it in this article.

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K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

FIG. 1 I . A typical filter characteristic. o is the ratio of the power dissipated in the load to the maximum available power of the source, A and E are the stop bands, C is a passband. and B and D are transition bands.

C. The Scattering Matri.u When analyzing and synthetizing transmission ti-ports, the scattering matrix S is frequently the most suitable tool to use. Amongst its advantages are the fact that it always exists if the n-port is passive, in contrast to most of the other network matrices (Carlin rf id., 19591, and that it is directly related to power transfer and thus to the specification of filter characteristics (Belevitch, 1968). The n-port equations are written in the form

b

=

Sa,

(20)

Here U and 1 are the (complex) port voltages and currents while R is a diagonal matrix with positive diagonal elements, and R'12 is also a diagonal matrix with positive diagonal elements which are equal to the square root of the corresponding elements of R. Then it is clear that

The network relations between the various quantities are illustrated in Fig. 12. The elements of a and b are called wave variables; a is the incident and b is the reflected wave vector. In ordinary (lumped-element) networks this terminology is rat her artificial, but in transmission lines and waveguides these quantities correspond to actual waves. The elements of R are called the

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THE GYRATOR IN ELECTRONIC SYSTEMS

normalizing resistances or preferably the reference resistances of the ports. They are real and positive. The diagonal elements S j j of S are called the reflection factors ; the other elements are the transmission factors. If the n-port is lossless, and the signals are sinusoidal, then S is unitary:

STS*

=

(251

1, ,

where 1, denotes the unit matrix of order n. In the more general case of exponential signals, S is para-unitary, i.e. STS* =

1,

(26)

where S, denotes S( - p ) . R.

h,

R2

FIG. 12. Scattering variables of a two-port. u1 = 4R;' ' D o , - Ri ' 1 , .

=

$R;' ' ( U ,

+ R , I,).

=

1. The Canonical Forin In the important special case of a lossless two-port constructed of a finite number of real, lumped, passive elements, S can be written in the form (Belevitch, 1968)

where g, h, f are polynomials in the complex frequency p ; f , , h , denote = & 1. In the case of sinusoidal signals, p = j w and h, = h*. From the defining equations, we see that the zeros of g are the natural frequencies of the network of Fig. 12 when the source intensities are zero, i.e. the original two-port network with the ports terminated in the reference resistances. This terminated network is evidently passive, so that the zeros of g must lie in the left-half of the complex plane, i.e. 9 is a Hurwitz polynomial. The para-unitary condition can now be expressed in the particularly simple form (Belevitch, 1968)

f(-p), h( - p ) , and

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

If now we choose U,, = 0 in Fig. 12, then a, = 0 and we have the basic filter configuration of Fig. 10. Then

U,

=

R;”(U, -I-b,)

=

R:”b,.

(29)

In the case of sinusoidal signals, the (average) power absorbed by the load is

5 I U2 I2/R2 = 4I b, 1.’

(30)

The maximum available (average) power from the source is

I uoi I2/8R, = 4I 12, a1

(31)

so that the fraction of the available power transmitted is

by (27). The fraction of the available power which is not transmitted is “reflected” back into the source. This is (33) by (25) and (28). A filter characteristic is often specified by the so-called characteristic function ) I defined by

I? * = j ,

(34)

which exhibits explicitly the zeros of the reflection and of the transmission factors. As a result of (28) and the fact that y is a Hurwitz polynomial, g is determinable once 1 $ I is known and the procedures of network synthesis then guarantee that a realization can be found (Belevitch, 1968; Youla, 1971).

2. The Circulator The gyrator as a lossless two-port has a particularly simple form of S-matrix if we choose the reference resistances equal to the gyrational resistance. We find

where q

=

+ I . From the gyrator we can construct a certain three-port in

THE GYRATOR IN ELECTRONIC SYSTEMS

99

either of the ways shown in Fig. 13. This three-port has an S-matrix

where y = 1 for network (a) and y = - 1 for network (b). In both cases the three-port is known as a circulator and has important uses especially in microwave engineering (Bosma, 1964). R

36 30

d

3

b

(cl

FIG. 13. The circulator: (a) the admittance matrix exists, (b) the impedance matrix exists, (c) symbol for the circulator.

D. Ladder Filters Most high-performance filters are realized in the form of an LC ladder network if this is at all possible. It has been found through long years of experience that ladder filters are relatively easy to build and to adjust so as to conform to the filter specification. The theoretical basis for this effect was

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

not fully realized until quite recently (Orchard, 1966). It was brought into focus through the difficulties arising from attempts to construct high performance active RC filters. There are two reasons for the favorable properties of the LC ladder network (see Fig. 14). First, the zeros of transmission are separately determined by the resonance frequencies of the arms of the ladder. Thus at such a transmission zero

FIG. 14. A ladder network. According to popular usage the arms labeled with Z ‘ s are termed the “series” arms and the arms labeled with Y’s are termed the “shunt“ arms.

(or attenuation pole) either one of the “series” arms of the ladder has an infinite reactance or one of the “shunt” arms has a zero reactance. The transmission zeros can thus be accurately adjusted by tuning the arms of the ladder. Since the frequency of the source can be accurately set, it is a simple matter to tune the appropriate ladder arm by adjusting the capacitor or inductor so as to produce a minimum in the output. In practice, because of the particular construction of ferrite pot cores, this tuning is easily carried out by a simple adjustment of the air gap of the core. Thus the zeros offare then accurately determined and we find that the stopband characteristic conforms very closely to the specification. The adjustment of the transmission zero(s) closest to the passband is especially important, since these zeros substantially determine the transition region(s). The second reason is that in the passband the first-order sensitivity of 1 S2 1’ to element-value variations at a zero of S1 is zero. It is very easy to see how this arises as a consequence of the losslessness property. Thus if the element values vary for any reason from their nominal values, the network still remains lossless (we suppose for the moment that the filter is constructed of ideal inductors and capacitors) so that Eq. (25) is still valid. In particular.

,

Is11

l2 +

IS21

l2 = 1

(37)

and a first-order variation yields

,

,

If 1 S , I = 0, then 6( I S,, I ) = 0, since 1 S z 1I and I S, I cannot simultaneously be zero. Further since S,, I 5 1, 16 I S , 1 I 5 1 and is thus finite.

I

THE GYRATOR IN ELECTRONIC SYSTEMS

101

Thus in the neighborhood of a reflection zero, the transmission characteristic varies little with variations in the element values. Furthermore, in the passband, the reflection factor will not normally differ much from zero if the filter is well designed, so that even at a maximum value of the modulus of the reflection factor, the variation in 1 S,, I will be small. Experience shows that the most sensitive region is near the passband edge, and is closely related to the resonance effect (Section IV,E). It is important to note that (37) and (38) remain valid if the terminating resistors deviate from their nominal values, so that the conclusions regarding the sensitivity also apply here. In practice one finds that very high quality filters can be built with components with 1 tolerance and that for many purposes a tolerance of up to 5 % is good enough. The ladder filter thus en-joysa remarkable property which is not known to hold for any other configuration except those which simulate the ladder exactly or which in some sense combine the properties of losslessness and the ability to set the attenuation poles very accurately (see, e.g., Bruton, 1973). If the reactive components have parasitic losses, then the performance and the sensitivity of the filter deteriorate (Neirynck and Thiran, 1967). It is thus important to reduce the losses as much as possible; the sharper the discrimination between the pass and stop bands, the higher needs to be the quality factor of the components. As a result of these considerations it would seem very attractive to consider replacing the inductances in a ladder filter by a gyrator-capacitor combination. Particularly since it is now possible to simulate inductances in this manner with extremely high and relatively stable Q's in the range 100010,000, this technique would seem to have much to recommend it. There are, however, some grave difficulties in practice which we shall discuss in the following sections. However, the same difficulties appear in many other proposals for filter circuits. E . Resonance in Filters

As already mentioned in Section IV,C, the natural frequencies of the free oscillations of the filter are the zeros of the polynomial g, i.e. the poles of S1 S,, , S, , Szl.If such a network is excited at a frequency which is close to one of the natural frequencies, resonance will occur and large voltages and currents will appear in the various components. Using Eq. (28), we can readily see qualitatively in what region this effect will be most pronounced. Thus for sinusoidal signals ( p = ,jw), Eq. (28) becomes

,

+ Ifl'=

yo.

(39) At a point on the j-axis of the complex plane near a zero of g, I g I is small 1912>

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K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

and thus I h I and I f 1 are small. Thus this point on thej-axis must be close to zeros of h and$ The only possibility which allows this to occur is for the resonance frequency to lie near the edge of the passband or in the transition band, between a reflection zero and a transmission zero (see Fig. 15). It

1

O

IS2,l-

IS,l-

1

x

+4

i

0

1

IW

Oh xf

t t

-a

FIG. 15. Relation between poles and zeros and the filter characteristic

follows that the narrower the transition band, other parameters being equal, the more pronounced will be this effect. We can give a simple quantitative relation for this effect (Dicke, 1948; Kishi and Nakazawa, 1963; Carlin, 1967; Kishi and Kida, 1967, 1970) based on Tellegen’s theorem (Tellegen, 1952; Penfield ef al., 1970), which is valid for any lossless two-port terminated in the reference resistances. We employ Tellegen’s theorem in the form

1 (U,* A l x + 1: AU,) = 0, a

where a is taken over all branches of a complete network, ( U , } and [I,) refer to a possible network distribution of voltages and currents obeying Kirchhoff’s laws, and (AU,) and (AI,} refer to a variation of voltages and currents about that distribution, also obeying Kirchhoff’s laws. We consider next the situation that port 1 of the lossless network is excited by an energy source while port 2 is terminated in the reference resistance. This results in a certain { U,, la}distribution. For the variational state we allow all the reactances to undergo a small variation but keep the other elements constant. From (40) we find after some calculation, correct to the first order of small quantities,

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THE GYRATOR IN ELECTRONIC SYSTEMS

where 8, = -arg (S,) (i,j = I, 2), T is taken over all reactances, P is the available input power, and suffix 1 refers to the fact that port 1 is excited by the source. If the variations of U , and I , result from a variation of the frequency, then we obtain from (41)

where W is the average total stored energy, or in derivative form,

In the same way, if port 2 is excited by a source and port 1 is terminated in its reference resistance, we have

From (43) and (44) we obtain, using (25),

where 5 is referred to as the total transmission time (group delay) from port 1 to port 2 and back to port 1. But

0 1 2+ 8,,

=

-arg (S12S21) = -arg

(F),

so that T =

d 2 d o arg (y) ~

We can factorize g as Y =

I<

iI (P - PA = 5

n

( p = J'LU).

(47)

( j w - .io, + xs),

(48)

5

where K is a constant and the zeros p, of g are -a, becomes

+ jw,.

Then (47)

From this expression we see that T is large for those points on the imaginary axis which are close to zeros of g, i.e. when a, 4 1 and 1 w - w, I 4 1. Thus T is large when the network is in resonance or close to resonance.

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K. M. ADAMS, E. F. A. DEPRETTERE A N D 3. 0. VOORMAN

If one of the zeros of g dominates in the frequency range of interest, then

where Q is the quality factor of the dominant zero of g, viz.

and - a O

+ jtno is the dominant zero of 61 I

C

FIG. 16. A typical low-order bandpass filter characteristic. showing the attenuation 70 log,, (I,’I S,, I ) and the group delay i = -(t//tfru) arg (.S2,).

We can also obtain another energy relation from (40), viz. for the case that only one reactance component varies. Then, denoting the element value of this component by xk,we obtain instead of (43)

THE GYRATOR IN ELECTRONIC SYSTEMS

105

where W, is the average energy stored in the component. Corresponding to (45) and (47), we obtain

This last result shows that the energy stored by component k is greatest when the argument of g is most sensitive to variations in the component. Various other formulas for the energy functions and their relation to sensitivity and to the scattering parameters can be given (Kishi and Kida, 1967, 1970) but they are of not much help in obtaining a good estimate of the necessary dynamic range of the components. For example, in gyratorless networks, or in networks in which the gyrator is used only to simulate an inductance, we can distinguish between the magnetic and the electric energies. However, near resonance these energies are nearly equal and the extra complication of the resulting expressions does not contribute significantly to further insight. Further, as (43) and (44) show, the difference between ( W / P ) ,and ( W / P ) ,is not great if the reflection factor is small. Thus

where i refers to either of the differential operators we have used, with appropriate interpretation of the left-hand side of the equation (E: = 0 or 1). If all the zeros of the reflection and transmission factors lie on the imaginary axis, as is the case for the important class of filters with Chebyshev characteristics, then the difference between the two energy-power ratios is zero. If now all the components of one type, and especially the inductances, have the same energy-handling capacity, then the most favorable configurations will be those in which the stored energy is equally distributed over the components. This unfortunately is seldom approximated in practice. As (53) shows, the optimum obtainable results when, at resonance, arg ( 9 ) varies equally as nearly as possible with respect to each reactive component value. In practice it will be necessary to calculate this variation, or some equivalent parameter, for the specific configuration being considered. We can, however, remark that in general it is preferable in this connection to choose a network in which the natural frequencies depend on all the components. This is in sharp contrast to the case of cascaded second-order systems containing isolating amplifiers, where the total energy at resonance is largely concentrated in two components. The limited energy-handling capacity of the reactive components presents the designer of sharply selective filters with various problems. If he chooses to use an LCR network, he will need to examine the deterioration of the quality factor of the coils with increasing amplitude, due to the nonlinear

I06

K. M.

ADAMS, E. F.

A. DEPRETTERE A N D J. 0. VOORMAN

characteristic of the magnetic material. It turns out that the filter characteristic near the band edge deteriorates rapidly with decreasing Q. If electronic components are used in conjunction with energy storing elements, the input signal of the filter must be kept sufficiently low to prevent overloading or saturation of the electronic components. In such a case the signal level may not be sufficiently above the noise level. One must remember that in any practical situation noise will be generated by the terminating resistors and any electronic circuitry that is present. If gyrators, and especially electronic gyrators are used, the designer will be confronted with the fundamental limitation of signal-to-noise ratio, selectivity of the filter, and dynamic range of the gyrators (which in turn is limited by the maximum allowable dissipation in the electronic circuitry). At the present state of electronic technology, it is not possible to build filters in this way with the most rigorous performance characteristics which modern communication systems demand and at the same time to stay within the power limits." Nevertheless there are many situations in telecommunications and electronic engineering where the highest performance is not called for or where the allowable power dissipation in the electronic circuitry need not be extremely low. Two examples of what is possible in this respect are given by Sheahan and Orchard (1967)and Orchard and Sheahan (1970):

Filter degree

Bandwidth

16 16

16-20 kHz 104-108 k H z

~

Max. output S,/N ratio

Maximum power consumption

YO dB 86 dB

85 mW 960 mW

~~

~~

If both filters had the same degree of absolute selectivity (equal transition width measured in Hz and equal minimum attenuation in the stopbands) we should expect the second filter to require 5 to 6 times the power consumption of the first filter in order to achieve the same signal-to-noise ratio, since with equal absolute bandwidths the Q of the zeros of g and thus the resonance effect is proportional to the upper band edge frequency. However, the electronic gyrators generate noise internally in the filter, which is transmitted to the output of the filter, with a power level at the resonance frequency, which is proportional to Q 2 . Since this noise is in general much greater than the noise produced by the signal source, we see that the signal-to-noise ratio per watt of supplied power should be proportional to 1/Q2. The above * Very recent work shows that with adaptively controlled power supplies it is now possible to satisfy some of the most stringent specifications. See Section VIII.

THE GYRATOR IN ELECTRONIC SYSTEMS

107

figures are indeed in close agreement with this expectation, namely 85 960

10‘8.6 9 . 0 )

2:

1 28 ’

which is very close to the square of the ratio of the upper band edge frequencies. We shall go into the whole question of noise generation in a later section.

F . Sensitivity We have seen in Section IV,D that the modulus of the power transmission factor is largely insensitive to variations in the component values. In fact, using Tellegen’s theorem it is possible to give a precise relation between the sensitivity of this transmission factor with respect to an element variation Axk and the stored energy in the element (Kishi and Kida, 1967):

This expression holds for any reciprocal lossless two-port with resistive terminations and subject to variations in the reactive elements. In the important special case that all zeros of S, and S,, lie on the imaginary axis, the right-hand side of (55), as a result of the remark following (54) reduces to

,

P’ so that

by (53). This sensitivity at the band edge can be quite high, depending on the magnitude of the characteristic function, but is further in accordance with the resonance effect. However, in the case of ladder networks, if the zeros of transmission are accurately adjusted, a deviation in the value of an inductance in a branch responsible for a transmission zero is accompanied by a corresponding deviation in the capacitance of that branch. We can see the consequences of this adjustment for the sensitivity as follows. Kishi’s and Kida’s derivation depends on an application of Tellegen’s

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

theorem in the form

C (AUrZ, - U , AZ,)

=

0.

I

The contribution of a branch, consisting of an impedance Z,, to the lefthand side of (58) is AZ,I,?. Using this expression, the relation between energy and the derivative of IS,, I is established as in (55). If now z k consists of an inductance and capacitance in series, both of which vary, but with invariant resonance frequency, then

where coo is the resonance frequency of the branch. Thus in Kishi and Kida’s derivation, when L, alone varies, we now have to replace AL, by ALk(l - o$02). The result is that the energy function wk is replaced by

w,1 1 - Lc);/w2 1 .

Similarly, when in Kishi and Kida’s derivation, C, alone varies, we have to replace AC, by AC,(oj2/wc - I), so that the energy function w k is now replaced by W, 1 0 2 / w i - 1 I . In the case of a parallel resonant circuit, L and C are interchanged in these relations. We thus see that the sensitivity of 1 S,, I with respect to variations in a reactive component of a branch responsible for a transmission zero is considerably less for a ladder with accurately adjusted transmission zeros than for a general lossless network in which this possibility is not readily available (see also the recent work of Temes and Orchard, 1973). Not all filters have their reflection and transmission zeros on the imaginary axis. In such cases the sensitivity relations are more complicated and the energy distribution at resonance is in general less uniform. Further, in the case of complex transmission zeros, the ladder network is not available. Other things being equal, the sensitivity of the power transmission of such filters is greater than for the case of the simple ladder with Chebyshev characteristics. Also for all lossless filters, the sensitivity of the group delay is high in the region of sharp selectivity, as can readily be seen by differentiating (52) or (53) with respect to LL). There is little that can be done to ameliorate this effect. If an amplitude or phase equalizer is required, the sensitivity and resonance effect can be quite tolerable, since in general the zeros of g do not lie close to the imaginary axis. Furthermore many networks of this type are not terminated lossless two-ports so that the preceding theory does not apply without modification. The necessary formulas can in any specific case be readily developed by the methods of Kishi and Kida (1967, 1970).

THE GYRATOR IN ELECTRONIC SYSTEMS

109

G. The Gyrator in Filters

The consequences of the foregoing discussion are that any reciprocal passive network, which by experience or analysis has been shown to be an acceptable filter network, can be considered as being fit for replacing the inductances by gyrator-capacitor combinations. If, however, sharp selectivity is required, then the necessary dynamic range of the components will certainly need to be checked. If the network contains transformers, then their simulation by gyrators and capacitors can be considered but the sensitivity will need to be investigated by a separate analysis. Further, any parasitic effects associated with the gyrator realization will need to be taken carefully into account. It may often be better to use an equivalent network with negative capacitors or inductors, thereby eliminating the transformers. This has been successfully done by Orchard and Sheahan (1970). There is, however, another more recent development which certainly deserves more attention than it has received to date. That is to design a truly nonreciprocal filter from first principles and to implement it with gyrators, capacitors, and resistors. There are various synthesis procedures available (see, e.g., Oono and Yasu-uru, 1954; Fettweis, 1969, 1970, 1971). Fettweis’ method in particular can provide interesting networks. Generally speaking such realizations demand a better realization of the gyrator than is needed for simple inductance simulation. Nevertheless, the technology has now progressed so far that some of these nonreciprocal realizations can be taken seriously.

V. PRINCIPLES OF REALIZATIONOF

THE

GYRATOR

A . Physical Efects

In this section we shall briefly discuss various physical effects which could form the basis of a practical gyrator deoice, as opposed to the theoretical rlenzent discussed in the preceding section. Some of these effects have already been mentioned in Section 11. We begin by listing the physical effects which correspond to linear passive systems. 1. Gyroscopic Forces

In a purely mechanical context, a well constructed gyrostat can be a good approximation to the ideal gyrostat, which is the mechanical analogue of the gyrator. Such systems lie outside the scope of this survey, which is devoted to electronic systems. Gyroscopic forces, however, are important in

110

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M. ADAMS, E.

F.

A.

DEPRETTERE A N D J. 0. VOORMAN

the gyromagnetic effect. This effect really involves two effects, viz. the gyroscopic relation between torques and angular velocities and the Lorentz force. The effect is responsible for the nonreciprocal Faraday rotation of the plane of polarization of electromagnetic waves passing through a magnetic medium subject to a constant bias magnetic field. At microwave frequencies and in the uhf band, practical gyrator-like devices based on these effects have been built and successfully operated (Hogan, 1952; Bosma, 1964). The design and application of such devices, however, have been extensively described in the microwave literature and will thus not be discussed further in this survey. There are also various other effects in the solid state involving the coupling between mechanical and magnetic moments, which, however, have not resulted in the development of a significant gyrator-like device.

2. Lorentz Force The Lorentz force plays a role in the gyromagnetic effect (Beljers and Snoek, 1950). It is also responsible for the Hall effect, which has been employed in the design of gyrator-like devices (Arlt, 1960). However, any device of this type is accompanied by high resistive losses so that it is a poor approximation to the ideal gyrator. Although the ratio of the gyrational resistance to loss resistance is proportional to the magnetic field strength at low intensities of the magnetic field, at higher values the nonlinear magnetoresistance effect comes into play and effectively prevents a gyrator of high quality from being realized. One could consider reducing the unwanted resistance by employing negative resistances, but then the realization would not be entirely based on passive system behavior, which is the subject of this section. Other working devices which effectively employ the Lorentz force are not known. 3. Electromechanical, Mechanico-magnetic Transducers

If an electrostatic microphone is acoustically coupled to an electrodynamic loudspeaker, the relation between the voltages and currents of the electrical terminals of the microphone and loudspeaker is nonreciprocal. In order to understand how this comes about, we consider an idealized situation in which the current in the electrical port of the loudspeaker is proportional to the mechanical force exerted on the acoustical medium. In the microphone the mechanical force at the acoustical medium interface is proportional to the voltage over the electrical port. If the whole system is lossless and incapable of storing energy, i.e. nonenergic, then it is readily shown using the results of Sections II,C and III,B,2 that the voltages and

THE GYRAFOR IN ELECTRONIC SYSTEMS

111

currents of the two ports must satisfy the gyrator equations. In fact the loudspeaker voltage and the microphone current are proportional to the particle velocity in the acoustical medium. The difficulty with devices of this type is to prevent the occurrence of various resonances in the system. Some energy storage is inevitable and the problem is to ensure that in the case of sinusoidal excitation the ratio of the product of stored energy and frequency to transmitted power is small in the frequency range of interest. Losses can also be a problem, but with careful design of the acoustical medium (e.g. a solid-state material with much higher acoustical impedance than that of the surrounding air), the losses can be largely limited to the electrical transducers. Devices of this type have been produced (Silverman, 1962, 1963; Silverman et al., 1961) with a relative bandwidth of about 10% but have not enjoyed much popularity. It is possible that with the present flourishing interest in mechanical surface-wave devices for signal processing in electronic systems, a high quality gyrator device in microform and based on this principle would be feasible, although realization of a suitable magnetomechanical transducer would not be easy. 4. Tellegen Medium Tellegen (1948a) in his original article on the gyrator proposed a medium with constitutive equations of the form

D=&.E+y*H,

B = y* . E +

*

H,

where Y* is the tensor adjoint to y. The coefficients y are responsible for the nonreciprocity of such a medium and the quality of any gyrator-like device constructed with such a medium as working material will depend on the ratios Although media with constitutive equations of this form are known, e.g. Cr,O, (Astrov, 1960; O’Dell, 1966), the desired effect is too small for forming the basis of a practical gyrator.

B. Active Circuits

The foregoing exhausts the known basic physical effects which correspond to passive systems and which could conceivably be used for practical gyrator realization. We consider next the possibilities offered by active

112

K. M. ADAMS, E. F. A . DEPRETTERE A N D J. 0. VOORMAN

systems. Before going into a detailed discussion of these matters, it is as well to recall what is meant by active and passive systems. According to popular usage, a resistor or a capacitor is regarded as a passive component, whereas a transistor is regarded as an active component because it is capable of power amplification. However, from the point of view of network theory, this is not an entirely satisfactory division of properties. Passivity is an observable property of a system. In order to make an observation, the system must have ports which form the interface with the environment, and in particular with the measuring instruments. We define an n-port network to be passive if it is incapable of delivering more energy to its surroundings through its ports than it has already absorbed from the surroundings through its ports during its previous history. Thus

1

I

.--r

u(.)”i(.) LIZ 2 0;

Qt, Qu(.), i ( . ) E N,

(61)

where u( . ) and i ( . ) are vectors of any simultaneously allowable port voltage and current functions, belonging to the class N which is determined by the network. An active ti-port is then an n-port which violates this condition for some u(. ) and i( . ) belonging to N. Viewed in this way, a transistor, which is a two-port, is passive. The transistor does not in itself produce electrical energy. It is a nonlinear system that converts electrical power in one form (the supply) into another distinctly recognizable form (output signals). This distinction between the supply and the signal power is possible because in the normal mode of amplifier operation, the characteristics are such that the signals correspond to the deviation of the voltage and current distribution in the transistor from some specified distribution, which is due to the power supply in the absence of any signals. This is the well-known situation of small”-signal operation. Amplification of signals can occur and it then makes sense to talk about a locally active system or network, because the source of energy supply and the conversion process are being ignored in the further discussion. The word local serves to remind us that we are only considering the situation about a fixed operating point on the characteristics. O n the other hand, the transistor is also used in situations where there is no separate supply, e.g. in certain pulse and logic circuits, in which case there is no possibility of amplification of signal power. Thus, whether a system or device is active or passive depends on the point of view of the observer; to be specific, on how the ports and their accompanying variables are chosen, as well as the mode of operation (i.e., the class N to which the port variables may belong), rather than on some absolute intrinsic property of the device itself. “

THE GYRATOR IN ELECTRONIC SYSTEMS

113

These considerations must be borne in mind when we come to consider the realization of electronic devices, which in some sense approximate the ideal gyrator. Any such device will be a two-port as far as the signals are concerned and may or may not be passive. If it is not passive, we shall have to be very careful about what we regard as an acceptable approximation, since an active system can be unstable and thus display a very different behavior from the passive system it is intended to approximate. On the other hand, we may choose to construct the device by interconnecting active devices together in such a way that the resulting two-port is passive. What happens is that the interconnections constrain the mode of operation of the active component devices in such a way that any power produced by some of the components is absorbed by the others and is not delivered to the ports of the device that we wish to construct. In the preliminary stages of design we can work with active components. We must not forget, however, that at some later stage the power supply and power conversion process will have to be taken into account and translated into hardware, and that particularly as a result of the power conversion, various unpleasant parasitic effects can occur.

C. Ideal Active Network Elements The number of electronic circuits that have been proposed for the realization of gyrators is so vast that we would be unable to compare them and discuss their design without having available some general and simplifying principles. The first simplification we can adopt is to introduce a set of basic active elements from which, in conjunction with the passive elements, resistor, and capacitor, all other active networks can be constructed. There are various ideal active elements in common use amongst theoreticians, such as ideal voltage and current sources (which are one-ports), transactors (ideal controlled sources, which are two-ports), the negative resistance, negative impedance converter, negative impedance inverter, and the nullor. This last element is especially interesting since it is the only active element from which all the other active elements can be constructed in conjunction with the passive elements, and since it is also a very natural first idealization of the basic building block of modern integrated circuits, viz. the transistor. Accordingly, we shall develop all our discussion in terms of the nullor. D. The Nullor The nullor concept was introduced into network theory by Tellegen (1956) under the name of ideal amplifier. The same concept was independently treated by Keen (1959) who called it the unitor. Finally the present,

114

K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

generally accepted name of nullor was given by Carlin (1964).The use of this concept in a theoretically responsible fashion was in the early stages not very widespread. More recently, however, the whole matter has been thoroughly discussed (Tellegen, 1966) and the correct use of the nullor concept is now fairly generally understood. The nullor is a two-port described by the equations u , = 0, i , =

0,

where for convenience port 1 is called the null-port and port 2 the )lor-port. The voltage and current of port 2 are arbitrary, which seems a rather strange and generally unsatisfactory state of affairs. However, the nullor only has network significance when it is used in conjunction with some other two-port network, in such a way that all the voltages and currents throughout the complete network are uniquely determinate when the initial state of energy distribution is specified. If, for example, the ports of the nullor are connected in parallel to the ports of another two-port, which may or may not contain certain sources, and with scattering matrix S, then the voltage and current distribution in the complete network is determinate if and only if S , , # 0. That is to say, there must be some feedback path from the nor-port to the null-port via the auxiliary network. This may seem a rather restrictive condition, but is really no stranger than the restrictions placed on the use of voltage sources, viz. voltage sources may not be connected together to form a closed loop if the algebraic sum of their source intensities is nonzero. The nullor does not possess a scattering matrix. However, we can take as the auxiliary network a second nullor connected with its null-port parallel to the nor-port of the first nullor and vice versa. Then all voltages and currents are determinate, viz. zero, but the result is not very interesting. For convenience in drawing circuits and also in analyzing them, we can split the nullor into two one-ports known as the nullator (null-port) and the norator (nor-port). These elements were introduced by Carlin and Youla (1961) and have been the source of some confusion. Provided, however, we recognize the nullator and norator always as a pair and as a shorthand for

null - p o r t

nor - p o r t

nullator

norator

FIG.17. The nullor and its equivalent nullator-norator form

THE GYRATOR IN ELECTRONIC SYSTEMS

115

the nullor, the theory will be in order (Tellegen, 1966). On no account can we consider networks in which the number of nullators is not equal to the number of norators. The advantage of using the norator and nullator is that various alternative realizations, involving different ways of combining the norators and nullators into nullors, can readily be seen from the same circuit diagram. As already mentioned, the nullor is a natural idealization of the transistor. Thus in Fig. 18, if we imagine the current amplification factor p of a

FIG. 18. The nullor derived as idealization of a transistor (three-terminal form) or of two transistors (galvanically separated two-port form).

transistor to tend to infinity, we obtain the condition i, = 0. If we also allow the differential emitter-base resistance to tend to zero, we obtain the condition u1 = 0. The collector-emitter voltage and collector current are then determined by the external circuitry. In fact modern integrated circuit techniques enable us to fabricate an artificial transistor which very closely approximates the nullor condition. As Fig. 18 shows, we can either approximate the three-terminal nullor or the two-port nullor, with galvanic isolation between the ports, by using one or two (artificial) transistors (Voorman and Biesheuvel, 1972). We can also regard the nullor as an idealization of the operational amplifier. In such an amplifier the gain is very high. Hence if the output voltage or current remains reasonably limited, both the input voltage and current will be very small. Again, the nullor equations seem a very reasonable idealization. Furthermore, the operational amplifier is always used in

116

K. M. ADAMS, E. F. A. DEPRETTERE AND J. 0. VOORMAN

conjunction with some kind of feedback network, which is in complete agreement with what the theory demands of the nullor. It matters little what the value of the gain A is, provided it is high. However, the frequency dependence of this factor, as well as other parameters of the amplifier, are of course important for the stability of the complete network in which the amplifier is embedded, and in such considerations, because A is not infinite,

FIG.19. The nullor derived as idealization of rhe operational amplifier

its argument as a function of the frequency is important. Nevertheless, in the preliminary design we can neglect all such considerations and check at a later stage of the actual hardware design whether the complete system is stable or not. E . Gyr-utors Constructed fiom Nullors and Resistors

It is evident that we cannot construct a gyrator from nullors alone, since the nullor equations and Kirchhoff's laws alone cannot lead to a relation connecting voltages with currents. In order to obtain the gyrator equations as the eliminant of the complete set of network equations, it is necessary to add some element that relates voltages to currents and the resistor does this in the simplest possible way. The next question that arises is: how many nullors and resistors d o we need? It has been proved that at least two resistors and at least two nullors are needed for the realization of a gyrator." Further, it is known that a gyrator can be realized using two resistors and three nullors or by using two nullors and four resistors. It has also been 'Recent work by the authors, submitted for publication.

THE GYRATOR IN ELECTRONIC SYSTEMS

117

conjectured that it is impossible to realize a gyrator from two resistors and fewer than three nullors or from two nullors and fewer than four resistors. A complete proof of this conjecture has recently been obtained.* 1. Networks with Two Resistors We shall now examine the circuits containing only two resistors. a. Parallel type. Consider first the equation u1 = - R i , . (62) Evidently a two-port network can satisfy this relation if we ensure that port current 2 flows through the resistor and that the voltage thus developed over the resistor is made equal to the voltage of port 1. In order to effect this equality of voltages without influencing the currents, we need to use the nullator. If galvanic isolation between the ports is required, then two norators will be needed. If we use two norators, then we must also employ two nullators, and we thus obtain the subnetwork of Fig. 20a. In the same way we obtain the subnetwork of Fig. 20b, which realizes the equation

u 2 = Ri,,

and finally we obtain the complete realization of the gyrator by connecting these subnetworks in parallel as shown in Fig. 20c. There are various interesting points to note about this realization. First, the galvanic isolation between ports and resistors is complete. If we are prepared to sacrifice this isolation, we can short-circuit a norator and a nullator, thereby rendering one nullor superfluous. We can thus devise three-nullor networks of four different types (Fig. 21), depending on whether (i) the ports have a common terminal (Fig. 21a), (ii) both resistors are incident to a terminal (Fig. 21b), (iii) one resistor is incident to terminals of different ports (Fig. 21c), or (iv) the two resistors are incident to different terminals of the same port (Fig. 21d). Secondly, a large number of gyrator circuits have appeared in the literature based on the antiparallel connection of two voltage-controlled current sources. Such a structure is an alternative way of regarding the networks of Figs. 20 and 21, since the network of Fig. 20a is the nullor-resistor equivalent of the voltage-controlled current source. If galvanic isolation can be sacrificed, one nullor in each such equivalent network can be replaced by short circuits. However, an extra nullor is still required for the 180" phase difference of the transconductances, or if one prefers it, the antiparallel connection. The final result is that one of the networks of Fig. 21 must result * Recent work

by the authors, submitted for publication.

118

K. M. ADAMS, E. F. A. DEPRETTERE AND J. 0. VOORMAN

0

1

FIG. 20. Realization of the gyrator equations: separately (a) and (b). of the combined gyrator equations; (c). by two resistors and four nullors. This is the basic parallel-connection form.

if each controlled source is replaced by its nullor-resistor equivalent and if the ports are not completely galvanically isolated. The third point to note is that since two physically distinct resistors are necessary, which in practice can never be exactly equal, we in fact synthetize a two-port with admittance matrix

which corresponds to a passive two-port, only if G I = G , . Otherwise the two-port is active. Thus we see that even in this early stage of design, where we assume that the nullors are ideal, a fundamental difficulty arises. This means that the behavior of any such two-port will have to be analyzed in

THE GYRATOR IN ELECTRONIC SYSTEMS

119

FIG.21. Three-nuilor, two-resistor realizations. There are just four distinct types.

detail in the context of the application for which it is intended before one can say whether it is an acceptable approximation to the ideal gyrator or not. However, there is one simple situation in which one of the primary properties of the ideal gyrator also applies to the two-port under discussion. Namely, if port 2 is terminated in an admittance Y, an impedance at port 1 can be observed satisfying

Z = R,R,Y.

(65)

That is, the impedance-inversion property of the gyrator applies and the resulting one-port is passive. Use of this type of circuit in conjunction with a capacitance for the simulation of an inductance is thus feasible (Orchard, 1970). As we shall see later, this basic circuit can be implemented electronically with excellent results.

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K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

The fourth property to note is that the ports and the resistors of the circuit of Fig. 20 can be interchanged without the properties or the configuration changing. In the practical implementation, such an interchange does have an effect due to the imperfections of the components used to realize the nullors. We note in passing that if one port and one resistor are interchanged, a positive impedance converter results.

7 (a)

2

1

2'

1'

(b)

FIG.22. Realization based on the series connection.

As has already been noted, the basic configuration can be regarded as the parallel connection of two voltage-controlled current sources. One may ask whether it is possible to realize a gyrator by the series connection of two current-controlled voltage sources. The answer is that it is possible but that apparently six nullors are required. The current-controlled voltage source with complete galvanic isolation between the ports is shown in Fig. 22a. The series connection of two such controlled sources is shown in Fig. 22b. It is not possible to reduce the number of nullors in a controlled source without losing galvanic isolation of the ports, and galvanic isolation is required for at least one of the controlled-source circuits for the series connection to be valid. However, in the complete network two of the nullors are superfluous.

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121

Thus in Fig. 22b, the nullators A , and A , serve only to constrain one terminal of R , to have the same potential as node 1. The norators B , , B, serve only to constrain the current in R , to be equal to the current through terminal 1’. We can thus eliminate A , and B , as shown in Fig. 23. Similar considerations apply to the nullators C , and C,

FIG.23. Derivation of the basic form from the series connection

and the norators D , and D , . The result of these operations shown in Fig. 23 is that the basic configuration of Fig. 20 is obtained. Thus the fifth point to note is that the approach via the impedance matrix does not offer anything new except the use of two superfluous elements. The sixth point to note is that in the configuration of Fig. 20, each port current flows through two norators and a resistor, which are separate circuits without common branches. The voltages developed across the resistors are brought over to the ports by distinct pairs of nullators. In the various three-nullor circuits of Fig. 21, this separation of signal paths is maintained; the only difference is that one of the signal paths requires only one norator or only one nullator. This property is of considerable importance when the electronic realization of the nullors is being considered. Various other forms o f three-nullor two-resistor configurations have been proposed which realize the defining equations directly (Daniels, 1969) but which involve signal paths containing more than the minimum number of norators and with branches common to both signal paths. From the point of view of practical realization such circuits, as a result of imperfections in the electronic circuitry, are much more likely to be adversely affected by undesirable and uncontrollable parasitic effects than the configuration of

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J . 0. VOORMAN

Fig. 20 and its derivatives in Fig. 21. Some of these less interesting circuits, which can all be derived from Fig. 21 by changing a norator connection, are shown in Fig. 24.

a

0

n

0

v

0

L '

0

A

0

4

b. Hybrid type. Another way to realize the gyrator is to use a mixed form of equations whereby the voltage of one port is expressed in terms of the voltage of the other port and the currents, or whereby a relation between the two port currents and a voltage is used. Thus we can write it1 =

-Ri,,

u2

u1

=

+ R(i, + i2),

(66)

(67)

THE GYRATOR I N ELECTRONIC SYSTEMS

123

or

i,

=

-Gu,,

il = i,

+ G(ul + u,).

These equations can be realized by controlled sources. Since, however, any gyrator realization based on nullors requires at least two distinct resistors, a direct realization of Eqs. (66) and (67) really results in realizing the equations

u1 = - R , i 2 , u,

R2(i1+ i,)

+ u1 = R,il + ( R , - Rl)i,.

(70) Since the resistances are never exactly equal, we obtain an impedance matrix with a nonzero diagonal element. If port 1 of such a two-port is terminated by a capacitor, then we observe at port 2 an impedance given by =

Z

=

R, - Rl

+ pR,R,C,

(71) which represents the series combination of an inductor and a resistor, which may be positive or negative. This is an undesirable parasitic effect which is not present in the realization of Section V,E,l,a. If port 2 is terminated by a capacitor, then an impedance equivalent to an inductance in parallel with a resistance (negative or positive) is seen at the other port. In order to get an idea of the magnitude of this parasitic effect, let us choose two nominally equal resistors with 1% tolerance. In the worst case then

If we now terminate both ports of the gyrator-like device in equal capacitors, we obtain a resonant circuit with resonance frequency given by

The Q of such a tuned circuit at resonance is

This means that with arbitrarily chosen resistances within the 17; tolerance range, the Q can have any value in the ranges (50, a )and ( - 50, - a). For this reason, realizations of a gyrator based on Eqs. (66)-(70) are less interesting than those based on Fig. 20, and will not be discussed further in this survey. For completeness, however, we give some of these realizations in Fig. 25 (Mitra, 1969).

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

.

FIG. 25. Some hybrid realization forms. [The right-hand circuit satisfies Eqs. (68) and (69) if the polarity of i i 2 is reversed.]

2. Networks w i t h FOLWResistors One of the first electronic circuits for the realization of a gyrator was proposed by Klein (1952). This circuit involved three ideal pentodes and one resistor. When reduced to its fundamental form, this results in a network consisting of three nullors and four resistors, which thus contains one more than the minimum number of nullors. A similar circuit, but with floating ports, was given by Sharpe (1957). This circuit contains four ideal pentodes and this is equivalent to a network of four nullors and four resistors. Since this early work, large numbers of gyrator circuits based on four resistors have been proposed. Here we restrict the discussion to those circuits containing the minimum number of nullors. Nonminimal circuits would appear to have little, if any advantage to offer above minimal circuits. The use of extra nullors means in general that in the electronic realization more supply power will be needed; also the use of extra nullors and extra resistors in the main signal paths of the circuit will usually result in extra noise superimposed on the signals. We now consider the basic type of minimal circuit based on four resistors. In view of the discussion of Section V,E,l,b, any realization in which the admittance matrix contains diagonal terms depending on the difference of two resistances will not lead, in conjunction with two capacitors, to a resonant circuit with a high and stable Q. Accordingly, any such realizations should be avoided. Two of these unsuitable circuits are shown in Fig. 26. It has been shown* that when the diagonal terms of the admittance matrix are always zero independently of the resistance values, and the resistors are suitably numbered, then one transfer admittance must be equal to * G I , *Recent work by the authors, t o bc submitted for publication.

THE GYRATOR IN ELECTRONIC SYSTEMS

125

FIG.26. Two two-nullor. four-resistor realizations with nonzero diagonal terms in the Z-matrix. For network (a), u , = -R,i,, u 2 = R , i , + ( R , R , Rb/R,)i,. For network (b) i d 1 = R , i,, 11, = - R , i, + ( R 2 - R , R h / R a ) i 2 . -

while the other is equal to G, GJG, . The synthesis of the first port equation

i,

=

-Gli12,

(75)

does not present a new situation in respect to the discussion of Section V,E,1. The second port equation, however,

strongly suggests that some form of current or voltage conversion, given by the ratio G3/G4, must take place. Such a voltage or current ratio, involving the values of the conductances in this simple way, cannot be generated by a network of positive resistances. Accordingly, we need one nullor to effect this operation on the voltages and currents. That leaves one nullor and two resistors available to establish a relation between currents and voltages. Since a gyrator cannot be synthetized from a nullor and two resistors alone, the relations which the subnetwork of one nullor and two resistors determine must be of the form i, = EG,Zf, ,

(77)

i,

(78)

=

EG~LI,,

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

where 6 = 1. These equations define a two-port known as a negative impedance inverter. It can be realized as shown in Fig. 27, whereby E can be either + 1 or - 1 according to the choice of reference polarity and current direction of one of the ports.

FIG.27. Negative-impedance inverter

It is evident that with such a simple circuit of two resistors and one nullor, complete galvanic separation of the ports is impossible. Furthermore, this network is the only configuration of these components which can correspond to Eqs. (77) and (78), as can readily be verified by investigating all the possible interconnections of two resistors, a nullator, and a norator. The subnetwork responsible for the current or for the voltage inversion must be capable of inducing a change of sign in one of the equations (77) and (78). Such a network is known as a negative impedance converter and can be

+

F u r

JG=+

u4

+"4

u2

FIG.28. Negative-impedance converter: (a) voltage inversion. (b) current inversion

realized either according to Fig. 28a (negative voltage inverter), thus corresponding to the equations

i,

= - i,

,

THE GYRATOR IN ELECTRONIC SYSTEMS

127

or according to Fig. 28b (negative current inverter)

A cascade connection of the network of Fig. 27 and of one of the networks of Fig. 28 in any of the four ways (Antoniou, 1969) shown in Fig. 29 results in a realization of the gyrator when

(d)

(C)

FIG.29. Gyrator realizations resulting from a cascade connection of the networks of Figs. 27 and 28.

B (a)

FIG.30. Gyrator realization resulting from Fig. 27 and a negative resistance.

Although there are eight different ways of combining these subnetworks in cascade, only four of them are distinct. There are, however, two other distinct possibilities, obtained by replacing one of the resistors in Fig. 27 by a negative resistor, which in turn can be effected by a combination of a negative impedance converter and a positive resistor. Although there appear to be many ways of doing this, the two networks of Fig. 30 are the only distinct possibilities (Antoniou, 1969). One can also realize the negative resistor by a combination of a positive resistor and a negative impedance inverter, but this does not lead to any new circuits.

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J . 0 . VOORMAN

The networks of Figs. 29d and 30b are particularly interesting from a practical point of view, since both norators are incident to a terminal node. In the operational-amplifier implementation of these circuits, this terminal can be connected to the earth or to a special current source of the power supply. Both networks are nullor-resistor equivalents of operational amplifier circuits which have been successfully employed in practice. Figure 29d shows a network due to Antoniou (1969) and further developed by Trimmel and Heinlein (1971). Figure 30b is a simple variant of the wellknown circuit due to Riordan (1967), which has been frequently applied to filter realizations (Orchard and Sheahan, 1970). 3. Comparison q f the Two-Resistor and Four-Resistor- Cor$gurution.s The question now arises, which of the various nullor-resistor networks is likely to lead to the best final realization? We cannot answer this question properly until we have considered the electronic design of the nullors and the accompanying supply circuits. This in turn will depend on the application and involves questions such as earthing. However, there are two advantages which the two-resistor configurations have over the four-resistor configurations. If an inductance is to be simulated, the accuracy will depend on the tolerance of the resistance values. Thus with 1% components, the maximum deviation in the inductance value will be 2% for the two-resistor configuration and 4% for the four-resistor configuration (assuming ideal nullors). Also, as we shall see later, the noise generated by the resistors and observable at the ports is in the optimum case twice as great for the fourresistor as for the two-resistor configuration. However, the total noise includes that generated by the electronic circuitry, and it is possible that, depending on the electronic realization, a twonullor four-resistor circuit will have a lower total noise power than a four- or three-nullor, two-resistor circuit with the same basic electronic realization of the nullor and supply circuits. Only the configuration of Fig. 20 has complete galvanic separation of the ports and is thus suitable for the realization of fully floating gyrator circuits. However, the question of floating ports is also intimately connected with the supply circuitry, and some of the other configurations with their proper power supply circuits can provide satisfactory realizations of gyrator circuits with semifloating ports.

VI. BASICELECTRONIC DESIGN In this section we shall consider the electronic realization of the nullors and their associated supply circuits. We can, however, discuss these matters separately and in general terms only to a limited extent, since the perfor-

THE GYRATOR IN ELECTRONIC SYSTEMS

129

mance of the complete circuit depends on the interaction of the various parts in a rather complicated way and involves such matters as input impedances, frequency response, stability, dissipation, noise, intermodulation distortion, offset voltages, and ‘‘ latch-up (multiple stable states). This interaction will vary from circuit to circuit so that it is necessary to consider all of these effects in the context of a specific design. ”

A . Realizations Based on Two Resistors in the Signal Path In order to get an idea of the problems involved let us first consider a very simple circuit (Blom and Voorman, 1971) based on Fig. 21a and shown in Fig. 31. Here, we approximate the nullor by a single transistor. In order to supply the transistors with direct current and at the same time to preserve the low short-circuit input admittances at the ports, current sources are needed. Simple circuits for realizing these current sources are shown in Fig.32. Because the circuit is directly coupled, we need both P N P and N P N transistors for both the signal processing and the supply current sources. We can judge the performance of the circuit in the first instance by determining the quality factor of a tuned circuit formed by terminating both ports with ideal capacitors. The transfer resistances of the two-port are primarily determined by R , in series with the differential resistances of the emitter-to-base diodes of the transistors T, and T, and by R , in series with

FIG.31. Simple gyrator circuit based on Fig. 21.

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

+

M

-

FIG.32. Basic current-source circuits.

the emitter-to-base diode of T, . Neglecting the equivalent parallel conductance of the current source, we find that the small-signal input admittance of port 1 when port 2 is short-circuited is given to a first approximation by

where b1 is the current gain (base to collector) of transistor T,, rl = Bl/(l + p,), and R; is the resistance of the series connection of R , and the emitter-base diodes of T, and T 3 . Similarly we find for port 2,

where R; is the resistance of R , in series with the emitter-base diode of T, . If we now terminate the ports with ideal capacitors, the conductances given by (84) and (85) will be in parallel with these capacitors and thus cause losses in the tuned circuit. From the impedance inversion property of the gyrator we

c1

pj?)

FIG 33 One-port equivalent ciicuit of a gyrator terminated by capacrtots w i t h loss conductances

obtain the equivalent circuit of Fig. 33. We find that the resonance frequency is given, with an error of O( 1/02),by

THE GYRATOR IN ELECTRONIC SYSTEMS

131

and the quality factor is given by

If the transistors T I and T, are equal, so that fil = p, ,Q will be a maximum when the two time constants C , R', and C , R; are equal. Then

Thus with a P of say 300, a Q of 150 should be obtainable with this circuit, and this is confirmed in practice at low frequencies. An important practical point should be noted in this respect. Any signal current flowing through the power supply will result in a reduced Q, owing to the internal resistance of the supply. Thus, instead of connecting the terminating capacitors to terminals 1, 1' and 2, 2', they should be connected between 1 and 2' (negative of the supply), and 2 and the positive of the supply. In this way, the capacitors and the transistors, together with their emitter resistors which deliver the port currents, form two loops which do not contain the power supply as branch. As the frequency increases, phase shifts in the transistors and parasitic capacitances in the wiring cause the Q to increase with frequency and eventually to become negative, so that instability results. We shall go into this matter later. At this stage, however, we want to consider ways of improving the Q at low frequencies and to reduce the dependence of the gyrational resistances on the diode resistances. Also we want to keep the dissipation down to a minimum. The first two improvements amount to obtaining a better approximation of the nullor. In order to reduce the dissipation, the bias currents need to be as low as possible. But to obtain accurate gyrational resistors we need high quality linear resistors for R , and R , . The emitterbase diode resistances need to be low compared with these linear resistors, even at low bias currents. Thus from this point of view we also need a better approximation to the nullor. 1. Improved Nullov Design

A simple way to improve the nullor approximation is to use the super$ configuration of Fig. 34. In Fig. 34 most of the signal current is carried by T,. The difference between i, and i, of T, is reduced by the addition of (I/@,I j 2 ) i c 2 , due to the action of T,. The whole circuit is equivalent to a transistor with

P re

PI&

f

11,

= re,lR2 = rzz ,

(89)

(90)

132

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

12

1% PN P

NPN

Fic;. 34. Suprr-/i transistors

where r, is the emitter-to-base differential resistance. In this circuit, rel 2 r e * ,since T, is fed by a supply current which is 1/& of that of T, . We can reduce r e l by giving T, a separate supply, e.g. via a resistor between the collector of T, and the emitter of T, . If we use this combination in the circuit of Fig. 31, a much higher Q than @,will result; however, unless special care is taken, the parallel conductance of the current sources can no longer be neglected, and at low frequencies the Q will normally be lower than $PI P 2 . As a result of (90), provided T, is given a special supply, the transfer function of the two-port can be made practically independent of the transistor parameters at low frequencies, without having to choose impractically large values for R , and R, . The circuits of Fig. 34 can be quite satisfactory if we have available high-quality P N P and N P N transistors. If, however, the gyrator is to be implemented as an integrated circuit, in which the lateral PNP transistors have a much lower than the vertical NPN transistors, then another approach is needed. One solution is shown in Fig. 35a (Voorman and Biesheuvel, 1972). Here an artificial P N P transistor is made using two NPN

a,

PNP

FIG. 35. Nullor circuits: (a) and (b) high-quality P N P and N P N artificial transistorb: (c) an artificial transistor with freedom of choice of dc collector-emitter voltage. but lower precision.

THE GYRATOR IN ELECTRONIC SYSTEMS

133

and one P N P transistor. The signal current that flows internally from terminal e to terminal c is delivered by the N P N transistor T I . Any significant voltage across b,e is detected by T, , converted to a current, which in turn is transmitted by T, without amplification, amplified by T I and finally delivered to e so as to reduce ube (provided port ec is passively loaded). We find that the parameters of this artificial transistor are given by

P = PIP,% = P i re

1 N

~

9

re2,

(91) (92)

PN

where the suffices 1,2,3 refer to transistors 1,2,3, and it is supposed that the N P N transistors have nearly equal parameters, indicated by suffix N. A related circuit, which as it stands is not suitable for integration, is shown in Fig. 35c. The emitter signal current of TI is fed into T4 and transferred to the collector of T, with transfer ratio a4. This results in a loss in performance in the nor-port of the nullor [ic - i, is now O(l/P) instead of O( 1/P2)], but it has the advantage that the collector-emitter dc voltage of the artificial transistor can be arbitrarily chosen within fairly wide limits. As a result, it is not necessary to have separate NPN and P N P artificial transistors, and with this arrangement we can realize a gyrator circuit in which all the terminals are at zero dc potential (Deprettere, 1971). The voltagedividing chain for supplying the bases of T, and T4 would not normally consist of linear resistors. The details of the supply circuits for the transistors, however, form a separate subject which will be discussed in a later section. In fact, the supply circuits for the current sources and for the bases of the transistors are responsible for roughly half of the dissipation. It is possible to reduce this dissipation if one is prepared to dispense with the stipulation that all terminals of the ports be at zero dc potential. In that case it is necessary to employ both NPN and P N P artificial transistors. An example of an N P N artificial transistor which has been successfully integrated is shown in Fig. 35b (Voorman and Biesheuvel, 1972). In this circuit, the differential-amplifier pair T, and T, have a very high (signal) impedance in their emitter circuits, owing to T , and its associated current source. Any voltage between b and e is converted into currents by T, and T, , which are amplified by T, and the Darlington combination of T, and T, . The result is that a current is delivered by T, to e, which tends to reduce u b e , provided port ec is passively loaded. O n analysis we find that the small-signal parameters of this artificial transistor are given by

B

= P4P5/%

P i PP

+

P3(P4Bs

+ %Ps/%)(l + P Z l U 3 ) ?

2

where it is assumed that the N P N transistors have equal p’s.

(93) (94)

134

K. M. ADAMS, E. F. A . DEPRETTERE A N D J. 0. VOORMAN

To calculate the equivalent r e , we first note that the base current of T, is small compared with the base current of T1, and is thus small compared with the sum of the collector currents of T, and T,. Thus I , , = P 4 I C 5Then . most of the signal voltage ube appears across re5 N PI P 4 r e 2 .The signal current i,, = ( P 4 8 3 8 2 ) - l i e z , so that

A more exact analysis shows that (95) is quite a good approximation provided P4 = Pp is not too small. Note that the two-stage Darlington combination is required to obtain a small r e , whereas only one of T, or T, would be necessary for a sufficiently high 8.

FIG. 36. Basic gyrator circuit with floating ports.

Using these circuits, high quality gyrator circuits can be built in which the transfer resistances are equal to R , and R , respectively, with an error of less than 0.1%. Further, the influence of the emitter-base diodes on the signal processing is so small that the nonlinear distortion is negligible within the normal working range of the transistors. The simple circuit which we have taken as prototype is based on Fig. 21a. In the same way a prototype based on Fig. 20 is readily obtained and has similar properties, except that it has a true two-port structure with galvanic separation of the ports. Also, in order to supply the signalprocessing transistors with equal bias currents, we need three current sources. If, however, both ports are to float above ground, we need four current sources, and if there is to be no dc voltage across the ports, we must avoid feeding the transistors via the gyrational resistors. The result is that we then need eight current sources in the supply (Fig. 36). The nullor approximations can then be further improved in the manner already discussed.

135

THE GYRATOR IN ELECTRONIC SYSTEMS

2. Stability There are two types of phenomena which determine the stability of a gyrator circuit whose ports are terminated by capacitors. The first concerns the stability of the clusters of transistors used to realize the nullors. Owing to the high degree of feedback in these circuits, internal high-frequency oscillations can be expected. In practice it turns out that such oscillations do occur in the artificial PNP transistor but can be suppressed by the addition of a reversed biased diode between the collector and emitter of T 3 , acting as a capacitance, and a conducting diode in the base lead of T3 (cf. Fig. 35a). High-frequency oscillations are in general highly nonlinear and accurate analysis is very difficult. In general, linear analysis of a simplified circuit, combined with experimental work, is the best approach to prevent the occurrence of such oscillations. The other type of stability concerns the complete circuit and is a lowfrequency phenomenon. Referring to Fig. 31 we find that the admittance matrix of the gyrator circuit with capacitors C , and C , in parallel with the ports is given by

+

We suppose now that each a is not a constant but is of the form i/(l p z ) , where is constant, p is the complex frequency, and z is a time delay. For low frequencies, this is a good approximation to the actual frequency dependence of a. The corresponding P is then of the form

P = B/( 1 + p T ) ,

T

=

PZ/~.

(97)

In order to keep the further calculation within reasonable bounds, we suppose that a1 = a2 = ag = a, R; = R; = R , C , = C2 = C. Further,

a/R’ a pc + -PR -,

N

2( 1 - pz)/R’,

=

q 1 + pT) pc + j?R’(l + pz) ~~

1 OR‘ ’

NpC+-

(98) ~

(99)

136

K. M. ADAMS, E. F. A . DEPRETTERE A N D J. 0. VOORMAN

since in general CR' 9 T / p 1 2. Thus we obtain as an approximate characteristic equation of the network, det Y = p2C2

1

+ 2p

where we have taken /? $ 1. The roots of this equation will lie in the left half of the complex plane provided 2

< CR'/B.

(101)

Thus from (86) and (88), the circuit will be stable provided the resonance frequency of the simulated tuned circuit thus formed satisfies 2w,2

<

1 QO

,

where Qo is the quality factor at resonance that would result if x and @were frequency independent. Thus for Qo = 100, we have w02 < 1/200, which means that the phase shift in a is less than 0.3", or the resonance frequency of the tuned circuit has to be less than about O.OOSf,, where f, is the cutoff frequency of the transistor, for stability to apply. For Qo = 1000, these figures are 0.03 and 0.0005 f, respectively. Under these circumstances, it is readily verified that the terms neglected in deriving (100) are negligible in the frequency range 0 < w < wO, so that (102) does indeed give a good estimate of the frequency at which the tuned circuit becomes unstable. If the x's of the transistors are not equal, then 2 2 has to be replaced by the sum of the time constants of the a parameters to which the transconductances are proportional. The relation (102) gives us a means to compare the frequency responses of gyrator circuits. The relevant figure of merit is 1/22, or equivalently G Q O , where 0 is the maximum (angular) resonance frequency for which the tuned circuit remains stable without special compensation measures. Next, we have to consider whether in the case where coo satisfies (102) the circuit is actually stable. In view of the approximations made, the foregoing analysis merely tells us that there is no natural frequency, corresponding to an unstable mode, in the neighborhood of the resonance frequency of the tuned circuit. At high frequencies, our approximation to the characteristic equation is not valid. The simplest way to deal with this question is to write the characteristic equation in the form F(p) 1=0 (103)

+

and to apply the Nyquist criterion to F ( j u ) , which corresponds to the '' open-loop'' transfer function in feedback control systems. From (96) we

THE GYRATOR I N ELECTRONIC SYSTEMS

137

can choose

where Y, and Y2 are the short-circuit input admittances of the gyrator circuit. The locus F(jw) lies in the first and fourth quadrants when co is small and passes close to the critical point (- 1 , O ) as o passes through oo. However, in the neighborhood of w o , 1 F ( j o ) I < 1, as is readily verified from the foregoing analysis. As o further increases, I F(jw) 1 rapidly decreases to a small value. Hence the locus of F ( j o ) does not encircle the critical point.

( 0 ) (b) FIG. 37. 'Uyquist diagrams of simple gyrator-capacitor circuits: (a) Fig. 31. subject to

inequality (101): ( b ) Fig. 31 with inequality (101) violated.

Moreover, provided the internal electronics circuitry has been designed to be separately stable, F ( p ) contains no poles in the right half-plane. Thus, according to the Nyquist criterion, the complete circuit with capacitive terminations will be stable. Note that the condition that F ( p ) has no poles in the right half-plane is very important. It is not very surprising to find that in a badly designed gyrator circuit with slovenly layout, either the nullor circuits oscillate internally or the gyrator circuit oscillates when one port is terminated in a capacitance and the other port is open. Under such conditions, further oscillations when both ports are terminated in capacitances can be expected. O n the other hand, if the whole circuit is designed according to the principles discussed here, if each nullor circuit and its power supply is separately

138

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

tested for stability and if necessary is corrected, and if clean layout with a minimum of parasitics is employed, then no stability problem for tuned circuits with the resonance frequency satisfying (102) should be experienced. It is also readily verified that the analysis remains substantially unchanged when we realize the nullors with the artificial transistors already discussed. In the neighborhood of wo it is permissible to take the equivalent x of the artificial transistor as associated with a single time constant, since coo is many decades below the cutoff frequency of the transistors. At high frequencies I F(jw) I retains its general qualitative character of decreasing monotonically to a very small quantity. At this point it is interesting to recall the remark made in Section V concerning the interchange of ports and gyrational resistances. If the nullors are ideal, then a gyrator circuit with the same properties results. In order to consider what happens in the nonideal case, we take the Y-matrix (96) and interchange R‘ and l/pC. This is not quite accurate since the emitter-base diode resistances should be taken in series with the ports rather than with the gyrational resistors. However, in the case of artificial transistors, the diode resistance is very small and can be neglected without disturbing the essential qualitative character. It is then no longer sensible to distinguish between R and R’. We obtain as the characteristic equation

This equation is not satisfied by any real o in the low-frequency region, provided coo = ( R ,R2 C1C2)-’” is much smaller than l / ~Since . for very low frequencies, cx and b can be taken as real constants, and then the lefthand side of (105) is clearly a Hurwitz polynomial, the root loci as functions of coo remain in the left half-plane in the low-frequency region. The circuit would appear to be a better configuration than Fig. 36. However, the highfrequency behavior is unfortunately unmanageable. We can see globally from Eq. (105) the kind of difficulty to be expected. Assuming for simplicity equal a’s, Fs, C‘s, R’s, and a single time constant T, we find that (M/P)’ + cx2 has a zero given by p z N - l/p + j . In the neighborhood of this zero, the coefficient of p2 in Eq. (105) is small, so that a root of Eq. (105) in this neighborhood, and thus with large modulus, can be expected. Whether such a root lies in the left or right half-planes depends on arg (cx2/lp2 + cx’) and on arg (alp). The single time constant model is not sufficiently accurate in the neighborhood of p~ = j to give a good estimate of the values of these argucx2 lies ments. We can, however, conclude that the closer the zero of (alp)’ to the imaginary axis, i.e. the larger is, the greater the chance that Eq. (105) will have a zero in the right half-plane in this neighborhood, and thus the greater the chance of instability. Experimental work confirms that this type

p

+

THE GYRATOR IN ELECTRONIC SYSTEMS

139

of circuit is invariably unstable when any reasonably high Q at low frequencies is realized. We note that if the capacitors are replaced by inductors, then in (96) the R terms are replaced by pL and the pC terms by R. A characteristic equation similar to (96) results and the stability properties are similar to those of the circuit of Fig. 36. Such a circuit, however, is not known to have any useful applications.

3. Current Sources The current sources which supply the nullor circuits with dc power must present a high impedance to the nullor terminals. This is a question of ensuring a sufficiently low Early effect, or a high value of p combined with a moderate resistance value of the resistor in the emitter supply (Fig. 32). This can be achieved sufficiently well with the N P N transistors. In the case of P N P transistors it will often be necessary to improve the performance, e.g. by using the super$ form (Fig. 34). A consequence of this realization is that at moderate frequencies, internal phase shifts cause the P N P current source to appear as a negative impedance with a numerically large value. In a gyrator-C resonant circuit this effect leads to Q-enhancement and results in instability at a lower frequency than otherwise would be the case. The design of good current sources is very strongly dependent on the techniques one has available for implementing integrated circuits. A second important point to be taken into account in the design is the question of equality of the currents delivered by the sources in a circuit such as that of Fig. 36. The sources connected to the positive supply can be made equal by employing transistors and emitter resistors which are as nearly equal as possible and by connecting their bases to the same potential. Similarly the equality of the sources connected to the negative supply can be established. In order to equalize the currents, we employ a control circuit such as is shown in Fig. 38. In this circuit, the current is determined by the series connection of R,, T, (working as a diode), and R. This fixes i, and i,. If i, > i, , then i, - i, is amplified by T, and causes an extra current to flow in T, (working as a diode) and R,. The potential of the bases of T, and T, rises, so i, increases, thus reducing il - i, . A third point to note is that in every nullor circuit, the bases of the transistors must be supplied with current. There is always at least one base which must be supplied externally to the nullor circuitry. If no provision is made for this in the current sources, then an equal current will flow in the nullor ports, i.e. offset current is generated. This effect can be reduced to a very low level by ensuring that il and i 3 differ by an amount equal to the total external leakage (base) currents of the nullor circuit. We must also

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

include the base currents of T, and T, in this total when several current sources are controlled by the same control network. If we require i, - i, = ni,/P, then we can effect this by increasing the voltage across R 2 and the emitter current ofT2 by ni,/P,but keeping i2 unchanged. This can be effected by the circuit of Fig. 39, whereby the super$ combination is used both for

,+= 11

/$\ 12

Nullor

u-

FIG. 38. Current-source control circuit

FIG. 39. Base-current compensation circuit

base-current compensation and for impedance enhancement of the current source. All of these design principles have been successfully implemented (Voorman and Biesheuvel, 1972). 4. ‘‘ Latch-up ’’

In electronic circuitry one is frequently confronted with the situation where a circuit “locks” in some parasitic equilibrium state and fails to work according to its specifications. What happens is that some or all of the transistors do not receive their correct bias currents so that they are either

THE GYRATOR IN ELECTRONIC SYSTEMS

141

cut off or bottomed. This situation is known as “latch-up.” The reason for the occurrence of this situation is that the complete set of nonlinear equations which describes the dc behavior of the complete circuit have more than one solution, any one of which represents an attainable (and sometimes stable) current and voltage distribution in the network. Unfortunately, the theoretical studies on this subject are not yet sufficiently well developed to aid the designer in correctly predicting the occurrence of this phenomenon and in adopting corrective measures. The only alternative is a laborious analysis of each circuit on the basis that a transistor either conducts normally, or bottoms, or is cut off. If more than one stable state is detected, then extra diodes can sometimes be introduced, which conduct heavily when an undesirable state is obtained but which do not conduct in the normal situation and thus do not influence the normal working of the circuit. The augmented circuit then has to be checked that only one stable state is possible. B. Realization Based on Four Resistors in the Signal Path

As already mentioned in Section V, there are two configurations of two nullors and four resistors which can serve as the bases for practical realizations. They lead to a well-known circuit due to Riordan (1967) and a more recent circuit due to Trimmel and Heinlein (1971) (Fig. 40). All of these configurations have the limitation that the ports are not galvanically separated. Until now, such circuits have only been used for simulating inductances and not as gyrator two-ports. Also to date, the nullors in these circuits have been realized only by operational amplifiers. It would seem worthwhile investigating the possibilities of using the artificial transistors discussed in Section VT,A in this type of configuration; to our knowledge this has not been done. The circuits of this type have been limited by the requirement that the operational amplifiers have a grounded output and that one port be grounded (Antoniou, 1969; Orchard and Sheahan, 1970). However, the more recent circuit of Trimmel and Heinlein (1971) is free of these restrictions since the amplifiers are fed through current sources. Although the complete performance of this type of circuit is dependent on the particular form of electronic circuitry used to realize the operational amplifiers, there are certain global conclusions which apply. As these circuits have been analyzed in some detail in the literature mentioned above, we shall only quote the results and comment on them. First, there are several ways of combining nullators and norators to form nullors, and secondly the polarity of the gain of each operational amplifier which approximates a nullor can be chosen in two ways. Only a few of the resulting circuits are stable (Antoniou, 1969).

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

In the Riordan circuit, voltage-controlled voltage sources have been employed. Provided the voltage gain is greater than 1000, the input and output impedances of the amplifiers are not important and the selfinductance simulated is approximately L = R , R3R4(C/R2)(1 4/A1) where A , is the gain of the first amplifier at low frequencies. The maximum Q attainable at low frequencies is approximately half the gain of amplifier 2.

+

1

Ri

1'0

f (b)

FIG.40. Two-nullor realizations: ( a ) Riordan circuit; (b) Trimmel and Heinlein circuit.

At higher frequencies, the same tendency for Q to become negative as in the case of the four-nullor, two-resistor configuration occurs. The time constant of amplifier 1 is largely responsible for this effect. Although it is possible to compensate this effect by adding suitable capacitances, the result is difficult to control properly, since it depends on the difference between the time constant of amplifier 1 divided by its gain, and a time constant determined by passive components. The circuit is subject to conditional stability and latch-up unless countermeasures are taken in the electronic design. The circuit of Trimmel and Heinlein employs voltage controlled current sources with high gain as.operationa1 amplifiers. The self-inductance simulated is approximately L = R , R 3 R,(C/R,)(I + 6 ! y I R ) , where y 1 is the transconductance of amplifier 1 and R is the mean value of the resistances when they are nearly equal. The Q at low frequencies is proportional to g 2 ,

THE GYRATOR IN ELECTRONIC SYSTEMS

143

and inversely proportional to the difference of two resistance ratios. With equal resistors and gains, the Q becomes negative when the time constant is taken into account. Compensation is possible as in the case of all gyrator circuits (Miiller, 1971; van Looij and Adams, 1968) but only in the form of subtracting two nearly equal quantities, which are subject to tolerances or different temperature coefficients. In filter applications, a negative or variable Q is acceptable, provided it is numerically large. The terminating resistors of the filter provide the necessary damping of any inherent low-frequency oscillations.

C. Noise The principal sources of noise in electronic gyrator circuits are the resistors and the current sources required for supplying the nullors. The nullors themselves, if they are well designed, contribute little to the total noise

(a)

(b)

FIG.41. Noise source representations of the nullor and its associated resistor: (a) thermal and shot noise; (b) flicker (llf) noise. (un, u,) = 2kT(2Rbb+ r,)df, ( i n , in) = 4(1/bre)kTdJ (i,,, in,) = 4kTdJR.

with the exception of the ljfnoise. In the ideal case, since the norator voltage and current are determined in the first instance by the external circuitry and not by the nullor circuitry, it is clear that any internally generated noise does not appear at the nor port. At the null port, there is a base-emitter diode which carries a bias current I,. This results in a shot-noise voltage in series with the port given by (unS, unS) = 2kTr, dJ where re is the diode resistance kT/qZ,. Since in the case of the artificial transistors, I, is very low, this shot-noise voltage can be rather high. There is also a thermal-noise contribution to the total noise voltage, (u,, u,) = 4kTRbbdJ due to the base resistance. A noise-current source across the emitter-base diode can usually be neglected (Fig. 41a; Blom and Voorman, 1971).There remains the llfnoise, which also produces a voltage in series with the null port (Fig. 41b; van der Ziel, 1970). Provided we choose the gyration resistors sufficiently large, the

144

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

total noise voltage in series with the null port will be primarily the thermal noise of the gyration resistor and the llf’noise. The result is that in the two-resistor four- (or three-) nullor realization, each resistor produces a noise current ( i n , in) = 4kTGdf which, as a result of the nullors, appears in parallel with the gyrator ports, together with the llf noise which contributes ( i s [ , i s l ) ( r e / R ) 2to the port current. The current sources produce similar effects, which appear as currents between the gyrator terminals and the ground (Fig. 42).

I

FIG.42. Noise-current sources due to the supply-current sources.

Since the thermal noise current of these sources is inversely proportional to Re + r e , where re is the emitter-base diode resistance and Re is the external emitter resistance, it is advisable not to make Re too small. On the other hand, a large Re is wasteful of supply power. The ratio of this noise current to that produced by the gyrational resistance is denoted by the noise factor F,. A reasonable practical value of F , for discrete circuits is 4 or 5 (Blom and Voorman, 1971), but F , can be ten times higher in the case of integrated circuits. In the case of the four-resistor, two-nullor realization similar conclusions apply, except that the contribution of the resistors to the noise current at the ports is twice as large in the case of equal resistors (Orchard and Sheahan, 1970). However, depending on the electronic realization one can save on the number of current sources required. When filters are constructed from capacitors and inductors simulated by gyrator circuits and capacitors, the spot noise power observed at the output of the terminated filter, due to the thermal noise of the gyrational resistances, is intimately connected with the average “magnetic energy” stored in the circuit at a given frequency, and is thus dependent only on the scattering matrix of the filter and not on the particular LC realization adopted. This noise power is given approximately by (Voorman and Blom, 1971):

THE GYRATOR IN ELECTRONIC SYSTEMS

145

where we employ the notation of Section IV,E. The noise is most pronounced near the band edge where -5 = d02,‘dw is a maximum (cf. Section IV,E). In the case of a reciprocal two-port, z = 2.2 [cf. Eq. (45)] and the maximum spot noise is approximately k T d f o z . This occurs at the same frequency where the resonance effect and the sensitivity are highest. From these results we can determine a lower limit to the minimum necessary dynamic range for a given signal-to-noise ratio in the output. This in turn determines the necessary power supply. It is important to note that if the dynamic range is fixed, the permissible input signal level is proportional to l / o z while the output noise is proportional to oz. The S/N ratio at the output is thus proportional to l/(wz)2. VII. BASICGYRATOR MEASUREMENTS A. Introduction

Any theoretical considerations and design philosophy must be supported by measurements on actual circuits. In this section we consider the basic description of a gyrator circuit as an object for measurement. Although in theory a gyrator is a two-port, an electronic gyrator circuit is in practice a

f FIG.43. The gyrator as a grounded four-port

grounded four-port (Fig. 43). In the ideal case the short-circuit admittance matrix Y is Y=

0 0

0 0

+G -G

-G +G

We can either measure the elements of this admittance matrix or of the scattering matrix directly, or we can measure some properties of the gyrator in one or more specific applications, e.g. with two capacitors as resonant circuit. Then it depends on the designer’s philosophy whether we must measure all parasitic effects in detail or only their order of magnitude. The circuit can

146

K. M. ADAMS. E. F. A. DEPRETTERE A N D 3. 0. VOORMAN

be designed such that for its main application all parasitic effects (or almost all of them) are so small that they can be neglected. In this case they may be strongly nonlinear or temperature dependent but because they are so small this is of no importance. For these parasitic effects, only rough measurements, preferably in the context of the main application(s), are necessary. On the other hand, if the parasitic effects are not so small and they must be taken into account or compensated, accurate measurements are required. Quantities which can be measured are accuracy, behavior at higher frequencies, intermodulation, signal-handling capability, noise, dc offset, etc. B. The Gyrator as n-Port Before considering measuring techniques, we must have an adequate description of the gyrator circuit as the object to be measured. We can consider the gyrator most accurately as a grounded four-port and further, depending on the circumstances, as a three-port, two-port, or one-port when one port is terminated by a n impedance. 1. The Gyrator as a Grounded Four-Port

The short-circuit admittance matrix of the ideal gyrator is in this case

0 0

0 0

Y=

+G -G

-G +G

The impedance matrix does not exist and the scattering matrix is not very suitable for describing the grounded four-port. Thus the scattering matrix normalized to G is

S

=+

I -

4

1

2

-2 2

2

1 4

-:I1 1

This is not a very suitable form to take as the basis from which deviations in ideal behavior can be measured. The best description here is the admittance matrix and it seems to be natural t o measure its elements directly. The admittance matrix Y can be written as the sum of its Hermitian and skew Hermitian part

Y

= f(Y

+ Y*') + +(Y

where T denotes the transpose and

-

Y*'),

(108)

* the complex conjugate of the matrix.

147

THE GYRATOR IN ELECTRONIC SYSTEMS

The admittance matrix of a lossless network is skew Hermitian. The Hermitian part gives a measure of the losses or of the activity of the network. The grounded four-port description is the most complete description of the physical gyrator. The elements of the four-port admittance matrix are not always different in value. Suppose we have a symmetrical voltage-controlled current source

(a)

( b)

FIG.44. Two identical symmetrical stages connected antiparallel, forming a gyrator.

(Fig. 44a) with a grounded four-port admittance matrix Y. Then, owing to the symmetry we have '21

= '34

7

3 '1

= '24

9

'41

'=

y14

9

y32

y33 =

= '23

y,,,

> y42

= y13

= '12,

y43

3

Y44 = Yll.

(109)

A second identical voltage-controlled current source connected to the former one as indicated in Fig. 44b has the following grounded four-port admittance matrix: y' =

'33

'31

'34

'32

'13

yll

y14

'12

For the complete circuit, a well-known gyrator type, we must add the admittance matrices Y and Y and we obtain a matrix with four different element values only. The structure of this grounded four-port gyrator admittance matrix is

where yl

yll

+ '22

>

y 2 = y]2

+

'24

9

3'

= '13

+

'21,

y4

= '14

+

y23

.

(112) If terminal 4 is grounded, then in the resulting semifloating gyrator the four different elements Yl, Y, , Y3, Y4 are still present.

148

K . M.

ADAMS,

E. F. A. DEPRETTERE A N D J. 0. VOORMAN

2. Grounded and Semzjloatiny Gyrators The description of the gyrator as a grounded two-port is applicable if both ports of the gyrator are grounded. Then. in fact. only a part of the fourport admittance matrix is used. For an ideal gyrator we have

The two-port scattering matrix normalized to the gyration conductance G,

0

-1

s = [+l

013

also gives a good description here.

I

FIG.45. Inductance simulation with a semItloating gyrator and a capacitor

Note that for the grounded gyrator the impedance matrix exists too. For the ideal gyrator we have

where R = 1/G is the gyration resistance. A semifloating gyrator (one port grounded, the other floating) can best be described by a grounded three-port admittance matrix. Note that we use each time a grounded-port matrix description because its parameters can be measured easily and accurately, as will be shown in Section VII,B,3. With floating ports this is much more difficult. Once the matrix elements of the grounded-network matrix have been measured, all properties of the circuit can be calculated. For example, if the grounded three-port matrix of a semifloating gyrator (Fig. 45) is

Y=

[I

-G+y21 +GJ'y y31

+G

+

4'12

-G

+

2)13

Y22

y23]

Y32

Y33

9

(116)

where generally / y k , 6 G, we can calculate the behavior of a floating inductor simulated between terminals 2 and 3 when port 1 is terminated by a

149

THE GYRATOR IN ELECTRONIC SYSTEMS

capacitor C. With I ,

=

-pCU1, we obtain

(1 17)

Because the simulation of an inductance between terminals 2 and 3 must be considered, it is better to introduce as new variables the port voltage Up = U2 - U , and the mean port current I , = (I2 - 13)/2.In addition, we can take for the third and fourth variable the common-mode voltage U,, = ( U , + U 3 ) / 2and the common-mode current I , , = I, l3(the current to the ground, see Fig. 46). In terms of these variables we get

+

I,,

=

[-

(h+ Y 3 & ! L + A PC + Y 1 1

+ 4’22 + y23 + y,, + J’33

1 u,,

(a)

FIG. 46. (a) The port voltage Up = U , - U , and the common-mode voltage U , , = $ ( U , + U 3 ) , where U , and li, are the terminal voltages. (b) The mean port current I , = + ( I 2 - I , ) and the common-mode current I , , = I, I , .

+

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

The coefficient of Up is the inductance term expressed in terms of the measured elements of the grounded three-port admittance matrix. We also see that the inductance value and particularly its quality factor depend on the common-mode signal voltage.

3. Admittance Measurements All input admittances of the short-circuit admittance matrix can be measured accurately with the standard bridge methods (e.g., Stout, 1960). For the transfer-admittance measurements, bridge methods can also be used. For example, the transformerless double bridge of Fig. 47 can be used.

FIG. 47. A transformerless double bridge circuit for accurate transfer-admittance measurements.

The conditions for balance give

Often y , can be taken to be zero. The usual rules for accurate bridge measurements apply here also. The substitution method can also be applied. For higher frequencies it can be very advantageous to use a Wagner earth connection. In fact, point E (Fig. 47) must be connected to earth and automatically the Wagner earth connection is implemented. 4. Measurement of Scattering Purmneters

In the microwave region the scattering parameters, which are related directly to forward and backward traveling waves, can be measured by comparing amplitudes and phases of the waves. In the low-frequency region we are obliged to use voltages and currents. Consequently, we shall translate the scattering parameters into quotients of voltages or currents.

151

THE GYRATOR IN ELECTRONIC SYSTEMS

First, let us consider a reflection factor Sll.We have: J K a , + ( U , R , I , ) , f i b , = f ( U , - R , I , ) . Hence

=

+

FIG.48. A reflection coefficient S , , expressed as a quotient of two voltages: S ,

=

U,/U3

FIG. 49. A transmission coefficient S , , expressed as a quotient of two voltages: S, = 2JRT*2 ( U 2 / U , ) .

,

In this way a direct measurement of the reflection coefficient is possible. It is also possible to measure the input impedance Zi = Ul/Il, followed by a calculation of the reflection factor from - R, s,, =-.Zi Zi + R ,

Measurement of a transmission factor Szl is even more simple (Fig. 49). With K a , = i ( U , + R , I , ) = i U o , &b2 = * ( U , - R , I , ) = U , , we obtain

Again a quotient of two voltages is to be measured, which can be done for example with a bridge method.

152

K. M. ADAMS, E. F. A . DEPRETTERE A N D J. 0. VOORMAN

5. Offset An electronic gyrator can d o its work properly only if it is biased. The bias voltages and currents should be balanced as well as possible, and in such a way that they are not observable at the ports as components of the port voltages and currents. In practice, perfect balance is not achieved. Unbalance and different dc levels are manifestations of offset. Small offset currents and voltages cannot be distinguished from signal currents and voltages. If is is the signal current vector and iff, is the offset current vector then the vector i of the total currents is i = i, + i,,,,, and similarly for the voltages u = U, + uoff,.Thus

i

=

Gu

-

Gsrr. + iff. ,

(124)

where G is the port conductance matrix. Small offsets, which fall within the signal-handling capacity, cannot be distinguished from signals. We can transform these voltage offsets u,&. into current offsets Cff, by

but generally not vice versa (when G - ' does not exist). Symbolically, we can always write (also if ~ ~ u o fexceeds f . ~ ~ the signal handling capacity) - GUoff.

Ioff. = i f f .

>

giving, with (124), i

=

Gu

+ Ioff,.

(126)

These equivalent offset currents I,,,,, determine the offset behavior of the circuit completely. Measurement of the offset current vector Ioff.can be carried out as shown in Fig. 50. The ports are adjusted to proper dc levels (ul. 2 , u 3 . 4) and the i.

A

FIG.50. Measurement of the offset current vector I,,, .

153

THE GYRATOR IN ELECTRONIC SYSTEMS

currents i , , i,, i 3 , i, are measured. If the gyrator is ideal, we have from (126)

0 -G +G

0

+G

-G

fG

u3.4

-G

u3.4

(127) As an example we shall calculate the offset of a parallel resonant circuit (Fig. 51), using the offset current vector and assuming that the conductance

FIG. 51. Offset calculation from the offset current vector Iocf,.

matrix G is the matrix of an ideal gyrator. We have i, u2 = u4 = 0, and consequently from (126)

=

i3

=

0,

u.3 = - R I l . f r . ,

' 4 = '3off.

+ '4oTf.

'

In a filter with cascaded gyrator sections which are dc coupled, the offset of one gyrator can decrease the signal handling capability of the others. 6. Noise For a linear noisy n-port, the noise can be completely described by y1 noise sources at the ports (Bosma, 1967). These sources can be wave sources but also voltage or current sources. For a floating port of a gyrator, dc sources supply the bias currents to the floating parts of the gyrator circuit. Noise currents are superimposed on the direct currents. Hence, it is natural to describe the noise behavior by a set of (noise) current sources (Fig. 52). The noise currents can be characterized by their mean squared values per hertz: G,, = (in,, i n k ) and their correlation can be characterized by the functions G,, = (in,, i,,), as observed through an ideal narrow-band filter and expressed in A2/Hz. All functions can be put together in a symmetric

154

K. M. ADAMS, E. F. A. DEPRETTERE AND J . 0. VOORMAN

noise matrix G, such that G,, = G,, = ( i n k , in,). Thus G can be formed by taking the mean value of each element of in iz. Using the Schmidt orthogonalization procedure (Gantmacher, 1959),we can transform in by the relation in = Aj, to an equivalent noise current vector j,, whereby A is so chosen that j, f = l , , where 1, is the fourth-order

A

FIG. 52. Complete noise description of the gyrator.

unit matrix. This transformation of noise currents results in a set of independent currents which can be treated separately. The matrix elements Gkl can be measured with standard methods using a correlator (Bittel and Storm, 1972). Let us consider a specific example (Fig. 53) derived from physical con-

FIG. 53. Noise model for a symmetrical electronic gyrator: inc is the noise current from the gyration conductances G, i n is the noise current from the electronic circuitry, F , are the noise in<,),k = 1, 2,3, 4; ( i n c ~ ink) . = 0. factors. (in<;, in,,) = 4kTG (A2/Hz); (i,,, i,J = Fk(inc,,

siderations (Blom and Voorman, 1971). The noise current sources (in,) over the ports represent the thermal noise of the gyration conductances: ( i n , , iq) =

4kTG

(A2/Hz).

The current sources in to ground represent the noise of the electronic circuitry and we can write, where F is a noise factor, (in

in>

= F(inc

>

in,>.

(129)

Usually this noise is partly llpnoise, shot-noise, and thermal noise, and the noise factor F behaves as a function of frequency (Voorman and Biesheuvel,

THE GYRATOR IN ELECTRONIC SYSTEMS

155

1972), e.g., as

F

=

+ 50/f

50

( f i n kHz).

For this example, Fig. 53 gives a representation where all noise currents are uncorrelated. Its noise matrix (see Fig. 52) can be directly calculated. It is l+F -1

0 0 l + F -1 -1 l + F

-1 1 + F 0 0

(130)

C. The Gyrator in Its Applications In Section VII,B the gyrator has been considered as an n-port. Its representation and measurements have been discussed without paying much attention to applications of the device. In this subsection we shall discuss those measurements which are of special interest for the gyrator in its applications, e.g. as an isolator, circulator, transformer, or as an inductance simulator in filters. Performance criteria are measured for these particular applications.

1. Isolator. Circulator, and lransformer Because the gyrator is antireciprocal (Penfield et al., 1970),it can be used most satisfactorily to realize an isolator. The arrow of the gyrator indicates the direction of transmission (Fig. 54a) or isolation (Fig. 54b). The source is matched to the two-port. An ideal isolator is best described by its scattering matrix S: =

[;:]

The accuracy of the isolator can best be estimated by measuring its scattering matrix (Section Vll,B,4). Let us consider the noise factor F of an isolator, in the same way as one considers the noise factor of an amplifier (see Fig. 55). For the gyrator we take the noise model of Fig. 53. Hence (Fig. 5 5 ) the mean square values per hertz are < i n , , inl) = < i n z ,inz)

=

(I

+ FG)4kTG,

(132)

where F , is the noise factor of the electronic circuitry of the gyrator. The thermal noise voltages per hertz of the conductances G are given by
=

< u n l ,unz)

=

4kTIG.

(133)

156

K. M. ADAMS, E. F. A. DEPRETTERE AND J. 0. VOORMAN

At port 1, the available signal power from the source U o is P,, = U:G/8 and the available noise power density is pn1= kT. At port 2, the available signal power is P,, = P,, and the available noise power density is p,,2= (3 + 2F~)kT.-Note that the noise voltage un2gives no contribution at port 2

G

i.

zGul

( b)

FIG.54. An isolator: (a) the arrow indicates the direction of transmission; (b) the load is isolated from the source.

FIG.55. The noise of an isolator.

and that we have to take the load resistance as noiseless. Finally, we obtain for the noise factor F of the isolator, ( 134)

With the help o f a wave analyzer, F and F , can be determined as functions of the frequency.

157

T H E GYRATOR I N ELECTRONIC SYSTEMS

The gyrator can also be used as a circulator (Fig. 56a) provided that all three ports are terminated in the conductance G (equal to the gyration conductance). 7'he scattering matrix of this three-port circulator is ideally

s=

[+;

0

0

+;

s].

-1

(135)

(Note that this matrix differs from the one mentioned in Section IV,C,2 owing to the different sign conventions for port 3 adopted here.) The dual of

G e u1

G t U -:

--

u iG

0

(a)

u+2--+G a

( b)

FIG.56. The gyrator as three-port circulator in dual configurations (a) and (b).

this circuit (Belevitch, 1968) in which the reference direction and polarity of port 2 are reversed as shown in Fig. 56b and which has all its ports grounded, has the scattering matrix 0

s = [ +;

0

+':

+1

01

(136)

in the ideal case. Two cascaded gyrators behave as a transformer, with a transformation ratio G1/G2. Here, either a chain or a transfer matrix (scattering variables) is the most appropriate. 2. Gyrator Resonant Circuit

If we terminate a gyrator on both sides in a capacitor we obtain a parallel resonant circuit (Fig. 57). This resonant circuit is an extremely good test circuit to estimate the gyrator behavior in selective circuits. In particular, the quality factor Q, the stability of the resonance frequency f o , the noise, and

Q'

=_

FIG.57. A symmetrical gyrator parallel resonant circuit.

158

K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

the intermodulation of the circuit give such a good insight into the finer points of the gyrator performance that it is worthwhile to pay a great deal of attention to this circuit. A gyrator with gyration resistance R and with one port terminated by a capacitor C simulates at the other port an inductance L = R2C (Fig. 57). Together with a second capacitor C, we obtain a symmetrical parallel resonant circuit with resonance frequency coo = 1/&C = l/RC. If the gyrator and the capacitors are completely lossless, the quality factor of the resonant circuit is infinite. a. The resonancefrequency and quality factor. Let us consider a nonideal gyrator with admittance matrix

terminated by two capacitive admittances Yl and Y2 (Fig. 58a). The nonzero

(a)

(b)

FIG. 58. In a nonideal gyrator, parallel resonant circuit, the input admittances of the gyrator can be regarded as shunts.

input admittances can be thought of as shunting the admittances Yl and Y2 (Fig. 58b). Generally, the (parasitic) input capacitances combine additively with the terminating capacitances, and the input conductances increase the tan 6 of the terminating capacitances. We define Yl = Y, + Y,, and r; = Y, + Y Z 2 . The impedance U j l of this nonideal parallel resonant circuit is

The following calculations can be made for quite different values of capacitances and conductances. However, for ease of understanding the essential phenomena, we shall only consider an almost symmetrical parallel resonant circuit. We write

THE GYRATOR IN ELECTRONIC SYSTEMS

159

where c, c , , c2 < C and g, g,, g2 < G. The negative values of tan 6 for the transfer admittances Y, and Y2 represent the internal delay in the gyrator circuitry. This representation is valid for low frequencies up to those frequencies where the device is no longer usable. In terms of these variables, the impedance U/Z of the resonant circuit becomes (to a very good approximation)

Introducing the resonance frequency coo = G/C and the quality factor Q by

we obtain the formula

I

I U/Z

An illustration of

1+Q

- + - .I: KO

as a function of frequency is given in Fig. 59. The

t

FIG 59 The impedance of the resonant circuit as a function of frequency, demonstrating relations between resonance frequency mug, quality factor Q. and various bandwidths. w1 = w O [- 1,2Q + (1 + 1 4Q')' '1, = w 0 [1/2Q + ( I + I 4Q2)' '1. B , d B = w0/Q, B , dB =

J3dQ.B l a d e

=

3u0lQ.

maximum value of the impedance occurs at the resonance frequency coo. Measurement of the 3 dB bandwidth gives the quality factor: Q = wo/B, d B . As the curve is steepest at the 3 dB points, this measurement of the quality

160

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

factor is more accurate than those which use the 6 dB or 10 dB bandwidths. The quality factor can also be measured by the Q-meter method or, if the Q is high, by determining the decay time of the free oscillations when the circuit is excited by a step function. The actual choice of method will depend on the equipment available, in particular, the short-term stability of the signal generator, the accuracy of the counter, or the range of the impedance measuring equipment or Q-meter. Particularly with high Q measurements, care must be paid to parasitics introduced by the measuring setup. As an example, let us consider the quality factor Q as a function of the resonance frequency coo. This coo can be changed by changing either the gyration conductance or the terminating capacitances. In the following we shall assume that yl, g 2 , c,, c 2 are independent of the values of G and C. In practice, this is not always the case. We can write

where the first expression gives Q = Q(wo) for a constant gyration conductance and the second expression gives Q = Q(wo) for constant terminating capacitances (see Fig. 60). At low frequencies we obtain for constant G a constant Q, and for constant C a constant bandwidth B = coo/Q.

FIG.60. The quality factor Q as a function or the resonance frequency: (a). G constant (Q constant for low frequencies) and (b). C constant ( B = coo/Q constant for low frequencies).

Finally, we note that the low-frequency behavior can be extended by several decades if small resistors in series with the capacitors are used, in such a way that the sum of the time constants is equal to the total delay time of the gyrator circuitry (van Looij and Adams, 1968). b. Impulse response and response to a suddenly switched sine wave. The impedance of a parallel resonant circuit, formed by a gyrator with its ports

161

THE GYRATOR IN ELECTRONIC SYSTEMS

terminated by equal capacitors C in parallel with equal conductances g, is given by

The voltage response of such a network to a current impulse i(t) = h ( t ) is u&t) =

1 H ( t ) exp ( - g t / c ) cos w o t , C

(144)

-

where H ( t ) is the Heaviside unit function, and oo= G/C. Since it is possible to obtain very high Q's with gyrator circuits, even at very low frequencies, the experimenter can be misled into believing that something is radically wrong with his circuit, when in fact all that is happening is that any induced transient persists for a long time. The effect can be used to measure the Q (Fig. 61). t

FIG.61. Impulse response of a parallel resonant circuit. From the exponential decay, the quality factor can be determined.

+

If now a sinusoidal current of the form i(t) = H ( t ) l sin (ot $) is suddenly applied to the circuit, supposed to be quiescent at t = 0, where o N coo, then two effects occur. First, free oscillations are induced in the circuit and secondly, a sinusoidal response is superimposed on these oscillations. If the Q is very high (e.g. > lOOO), the free oscillations persist for so long that one may be misled into believing that some nonlinear distortion is taking place, when in fact all that is happening is a perfectly normal linear process, with a long drawn out transient. The complete response is

- y exp ( -dot)sin (wot

+ G2)],

(145)

162

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

$z = arc tan

q

[

=

-

~

wo/w

1 - 6 tan

-

4

tan

(1 - 2 6 ( ~sin

4

41

-

arc tan [w(6 + E ) / o ~ ] ,

+ cos 4) sin +)'/2.

(147) (148)

I f Q S l a n d I w - w g l $w,,then

where

$;

=

-arc tan e,

$;

=

6 - arc tan E.

(150)

At a time t = n/lw - wo1 after the current is initially applied, the two sinusoidal terms reinforce, and if 60 is small compared with - coo [ / n ,the voltage will be approximately twice as large as it would have been if only the steady-state term were taken into account. As a result, the dynamic range of the circuitry may be inadvertently exceeded with, as a result, nonlinear distortion. We can also write (149) as

10

1

+ 21 (1 - e e 8 m t ) s i n ( o t + ~ - a r c t a n ~. ) (151) The voltage response thus consists of a slowly decaying sinusoidal carrier with low frequency amplitude modulation, and a slowly growing sinusoid. Because of the unusually high Q that is obtainable in gyrator circuits, this phenomenon can be expected to be noticeable in many situations when the signal conditions are changed. One must wait until the transient has died out before making any precision measurements. c. Noise. Again let us consider a gyrator-capacitor parallel resonant circuit. For a fairly high quality factor the noise is concentrated around the resonance frequency. If we measure this noise, we obtain in fact the spot noise of the circuit. This measurement gives an easy way to determine the noise factors F , of the gyrator, in the noise representation of Fig. 53 (Blom and Voorman, 1971), as a function of frequency. The noise voltage measured over a parallel resonant LC circuit is (u,, , 1 1 , ) ~ = kT/C. More generally, it is readily shown by a scattering matrix

THE GYRATOR IN ELECTRONIC SYSTEMS

163

formulation that the noise voltage across a resistor in a passive network, where only thermal noise sources at the same temperature are present, is also given by the same expression, where C = limp+m(l/p)Y(p) and Y(p) is the admittance of the one-port network formed by removing the resistor. We consider now the noise sources in the electronic realization of the gyrator. Their mean squared values per hertz are

+ (1 + F,)G] = 4kT[g2 + (1 + F3)G]

at port 1

(i,,, , i,,,)

at port 2,

and

where g1 and g2 are the conductances in parallel with the ports. We first take a special case, viz. where (1 + F , ) / g , = (1 + F 3 ) / g 2 . In this case we consider the system as a lossless, noiseless two-port terminated by two conductances g1 and g2 with a mean squared noise current per hertz given by in*,,>= 4kTg1[1

and (in*,, i:3)

:=

4kTg,[I

+ (1 + Fl)G/gJ

+ (1 + F,)G/g,].

FIG.62. The noise of a passively realized gyrator-capacitor, parallel resonant circuit can be regarded as due to the thermal noise of the loss conductances g1 and gz . 3

C; 2

4

FIG.63. The noise of an electronic gyrator-capacitor resonant circuit.

Therefore in accordance with the foregoing results, the noise voltage appearing at port 1 over the conductance g1 is given by

If (1 + F , ) / g , # (1 + F 3 ) / g 2 , then a more detailed calculation shows that we have to add a correction term given by

We can now translate these parameters into directly measurable quantities.

164

K. M. ADAMS, E. F. A . DEPRETTERE A N D J. 0. VOORMAN

we find that the exact expression for the noise voltage is given by

For a high Q circuit, the noise voltage is closely approximated as

which is the same result as obtained by neglecting the correction term (153) and neglecting g1 y 2 compared with G 2 . Measurement of the quality factor Q and (u,, L,u , , ~ for ) ~ different values of C,/C, gives us the noise factors F , and F, . This measurement is easy to perform because the noise voltage is high for high quality factors. With different terminals grounded, we measure in the same way F , and F,. The noise factors show the liif-noise of the circuit for low frequencies and a white noise for higher frequencies, e.g.; F , = 50/f'+ 40, where f is the frequency in kHz. Once we know the noise factor we can calculate the performance in different applications as for example in filters (Voorman and Blom, 1971). d. Intermodulation. Let us consider the case where our gyrator is not completely linear. In a first approximation the gyrator equations become

i,

=

i2

=

C ( U+ ~ (ELI: -C(u,

+ bu;), + UU: + 6 ~ ; ) .

( 156) Because each of these equations is realized by a separate voltage-controlled current source (VCCS), generally, we have no cross terms such as c u l u , . In a telephony channel filter, any such terms are responsible for intermodulation before filtering, so that they result in interference from adjacent channels. In practice, however, these terms can be made very small. We assume

;T;TlTT;-k2 -

C

FIG. 64. A slightly nonlinear gyrator gives intermodulation, which is calculated.

in

a remnant circuit frequency-dependent

165

THE GYRATOR I N ELECTRONIC SYSTEMS

that both of the equations show the same nonlinear behavior. Observe that for symmetrical VCCS’s we have a = 0. From this fact we may expect that a gyrator with symmetrical VCCS’s gives a lower intermodulation. Only this case (a = 0) will be considered in the following. Further, we have as equations for the resonant circuit (Fig. 64), i

--

il

=

C

--i2 = C

du 1 ~

dt

du2

dt ~

+ gu,, + gu2 ,

(157)

where the conductances g represent the losses of the circuit. Finally, combining these equations, we obtain

To find an approximation for the solution of these equations for an input current i ( t ) = I sin ot,we observe that the nonlinear terms Gbu: and Gbu: are relatively small. Therefore we shall use the solution of the linear equations (b = 0) substituted in the nonlinear terms. With the right-hand sides of the equations known, we can solve the new set of linear equations. This first iteration will give us a good insight into the influence of the nonlinearities. The solution of the linear equations (b = 0) is (see Section VII,C,2,b) (or - arctan 6

where w o = G/C, E

= Q(w;wl

0: =

+

o ~ i ( l g2/G2), Q

=

-

arctan E ) ,

(159)

Go,/2gwo, 6 = 01/2Qw, and

- w , / w ) . Further,

We substitute the solutions u l l ( t ) and u2,(t) in the nonlinear terms and use the relations 4 sin3 x

=

3 sin .Y

4 C O , S ~x

=

3 cos x

and

-

sin 3x

+ cos 3

~ .

166

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

The solutions of the equations with these terms, which can be added to u 1l(t) and u z l ( t ) ,can be found in analogy to u l , ( t ) and u z l ( t ) . We first obtain extra terms with frequency o,viz.

x {cos ( ~ - tarctan 6 - arctan E )

+ E sin (or - arctan 6 - arctan

x {sin (at - arctan E )

- E

E))

cos (ot- arctan E ) ] .

(162)

When the quality factor is high (6 4 l), the first term of the last expression can be neglected with respect to the second. The remaining expressions have in-band amplitudes proportional to (lQR)3Qand they give voltages Q times as high as would be expected from the expression bu3.If w approximates wl, they may become higher in amplitude than the original signals u1 l(t) and ~ ~ ~ In ( tmeasurements, ) . it seems as if the quality factor of the circuit increases with increasing signal level. In practical gyrators, this effect can be observed very well. However, the voltages u1 and u2 still remain approximately 90 deg out of phase, although both of them shift remarkably in the passband (change of E ) with respect to the input signal. Finally, it will be seen that the intermodulation terms show a much higher selectivity than the respectively. original terms, viz. by a factor (1 + c2)- and (1 + E ' ) In the same way, we find for the terms with frequency 3 0 the following approximate expressions (6 6 1):

x cos ( 3 ~ -t 3 arctan E - arctan c 3 ) ,

x sin ( 3 m - 3 arctan E - arctan E

~ ) ,

THE GYRATOR IN ELECTRONIC SYSTEMS

167

where c3 = Q{(3w/01) - (wl/3w)). Let us compare the amplitudes of the intermodulation terms with frequency o and those with frequency 3w. For both, the maximum occurs for approximately w = w1 and we see that the terms with frequency 3 0 are about 24 x Q smaller than the intermodulation terms with frequency w. For this reason these terms are hardly ever observed in a resonant circuit even if a strong intermodulation is present. In practice we observe the following. If we increase the amplitude of the input signal, the amplitude of the signal in the resonant circuit also increases but, due to the intermodulation, not completely proportionally. At a certain level, the output signal no longer increases in amplitude, while distortion cannot be observed. Only for extremely high input signals can higher harmonics be observed. When we use a gyrator-capacitor resonant circuit in an oscillator we shall not see any harmonic distortion from the resonant circuit but only from the extra circuitry employed as a limiter.

3. Filters

If gyrators are used in filters we can simply take the well-known lossless LC filters and replace each inductor by a gyrator and a capacitor. If a coil is floating and the gyrators are not, we can still make this replacement at the cost of one or more gyrators (Leich and van Bastelaer, 1968). Generally, these filters are lossless and terminated by dissipative elements such as resistors (Fig. 65). lossless

r----------_------------

I I

I

UO

-L - - _ _ - _ _ _ _ _ _ _ _ _ - - _ _ _ _ _ _ 1-

c

FIG.65. An LC-filter generally consists of a lossless two-port terminated by resistors.

Measurement of the insertion loss of a filter can be carried out in various ways but the intermodulation and noise measurements have to be clarified somewhat further. a. Intermodulation. In a gyrator capacitor filter the voltages and currents to be handled by the gyrators are not equal to the input voltage (current) of the filter. They may be higher. Due to the terminating resistors, the effective quality factors of the gyrator resonant circuits are relatively low.

168

K . M. ADAMS, E. F. A. DEPRETTERE A N D 3. 0. VOORMAN

Due to intermodulation of the gyrators, harmonics can be generated. These harmonics are filtered more or less before they reach the output of the filter. In Fig. 66 the distortion of a seventh-order elliptic low-pass filter is given as a function of frequency for a constant input voltage. The cutoff frequency

I

Distortion (dB)

-301

-40

FIG.66. The distortion of the output signal of a seventh-order elliptic low-pass filter as a function of the frequency ( f , is the cutoff frequency of the filter: 3415 Hz). The filter is C071059 (Saal. 1963).

Distortion (dB) f = 1 kHz

FIG.67. The distortion at 1 kHz as a function of the input signal level in the filter. L'% corresponds to an output of 270 mV.

.f, (3.415 kHz) is indicated. Values below -75 dB could not be measured very well. Figure67 shows the distortion for a frequency of 1 kHz as a function of the level of the input signal for the same filter. Measurements of this type determine, together with the criteria for maximum allowable distortion, the maximum admissible signal level for a filter in

THE GYRATOR IN ELECTRONC SYSTEMS

169

its application. Both of these curves were obtained for a filter constructed from the gyrator described by Voorman and Biesheuvel (1972). b. Noise. Once the maximum admissible signal level is determined, we wish to know the noise of the filter. An LC filter is noisy because of the noise of the terminating resistors. In a filter with capacitors and electronic gyrators we have more noise. The noise of the gyration resistors and of the electronic circuitry will find its way more or less filtered to the output. This noise can be calculated on the basis of a noise model for a gyrator (Voorman and Blom, 1971) and can also be measured. Generally, the power handling capacity of a filter is defined as the ratio between the power of a sine-wave at maximum admissible amplitude (maximum allowable distortion) and the total noise power, both at the output of the filter. To obtain the total noise power we have to integrate the spot noise (e.g. in pV2/Hz) over the relevant frequency band. For bandpass filters, the procedure described above will present no difficulties. However, for a low-pass filter, we obtain a noise spectrum as given in Fig. 68 (curve a). The llfnoise causes the spectrum to be proporSpot noise (,uV*/Hz)

0.I6

0

1

2

3

4

Fiti. 68. The spot noise of a seventh-order elliptic low-pass filter (curve a) and the same noise psophonietrically weighted (curve b).

tional to l/f’for,fJ 0 and the spectrum cannot be integrated. Before integration some limitation is necessary, e.g. a restriction in the passband for low frequencies can solve this problem. In telephony for voice frequency channels the noise is often weighted according to the loudness contour of the human ear, i.e. the noise is psophometrically weighted (Fig. 68, curve b) (Com. Consult. Int. Telephon., 1956) before it is integrated. The signal handling capacity of the filter is then calculated. Note that in general the conditions for maximum signal are defined for a signal of 800 Hz where the psophometer gives no attenuation.

170

K. M.

ADAMS, E.

F. A. DEPRETTERE AND J. 0. VOORMAN

VIII. TRENDS IN GYRATOR DESIGNAND APPLICATIONS A. Introduction

The electronic gyrator as it now exists can give a satisfactory performance in the laboratory but is not commonly used in industrial applications, for several reasons. Its circuitry is complex, the power consumption is high for practical values of the signal-to-noise ratio, and the behavior for higher frequencies is rather poor. Moreover, as an integrated circuit, the gyrator requires many terminals, i.e. contacts which are potential sources of trouble. This is in contrast to the coil which has only two terminals. It is today’s challenge to find solutions to these problems. In the following we shall discuss some of these problems. Also some solutions will be given and some advances in gyrator design and applications will be indicated. Given the present state of the art, we are fairly optimistic about future industrial applications of the electronic gyrator for consumer products as well as for professional use. B. Gyrators for Application in Consumer Products For consumer products such as radio’s, T.V.’s, and tape-recorders, filter specifications are generally not so severe as for products for professional use in for example telephony and telegraphy. This means that for consumer products, RC active filters can in general fulfill the requirements (all-pass, low-pass, high-pass filters) for the lower frequency range. Here they are cheaper and smaller than the gyrator-C and LC filters. For higher frequencies coils are so small, satisfactory, and cheap that the commonly used LC filters d o not seem to have electronic competitors at all.

FIG. 69. Sallen and Key’s second-order KC active filter section for the simulation of a resonance circuit.

Problems arise only in the low-frequency range when resonant circuits are required with a quality factor of let us say more than 20. This will be illustrated with an example (see Fig. 69)-the well-known circuit due to Sallen and Key (1955), consisting of an amplifier with high input and low output impedance and voltage amplification K together with two resistors

171

THE GYRATOR IN ELECTRONIC SYSTEMS

and two capacitors. The transfer function is

2 for the giving for the resonance frequency wo = 1/(R1R2C 1 C 2 )1 / and quality factor Q

In the neighborhood of an infinite quality factor, the denominator is almost zero, which is obtained by subtraction. Consequently, here the quality factor is very sensitive with respect to element deviations. To be more precise, we can write (assuming R , v R,, C , N C,, K 'v 3, Q 9 I), AQ

N

Q

Q

ARl

AR,

AC, AC2 2--+2-+3-, Cl c 2

K

(167)

and we see that with element deviations of 1%, not even a stable quality factor of 20 can be expected. Only resonant circuits consisting of lossless elements: L,C, gyrator, etc. d o not have this severe limitation. It is their nature to have low losses and hence to have high quality factors (100-1000). Hence, for consumer products in the lower frequency range, simple gyrators can solve these selectivity problems (e.g. for pilot, signaling, and carrier filters) for which RC active filters are not accurate enough. Because of the costs only simple gyrator circuits with a low number of terminals can be used. In this direction much work has still to be done.

C. Gyrators f o r Professional Use This division into gyrators for consumer articles and for professional use is not so strict as it seems to be from the foregoing, but for professional use power consumption and accuracy play a more important role than complexity, provided that the reliability is not affected. As a higher complexity is not forbidden here we have more possibilities, some of which will be discussed. 1. Class B Gyrators?

To reduce the power consumption of operational amplifiers, which were originally designed for class A operation (Cherry and Hooper, 1968), they were redesigned for class B operation. The same could be done and in fact is also done (Orchard and Sheahan, 1970) for the nullors used in gyrators.

172

K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

However, there are two important drawbacks. In the first place the intermodulation of the signals in class B operation is higher than in class A. In the second place the phase shift during the takeover of the complementary output circuits is relatively high and can deteriorate the gyrator behavior. At least the losses of the gyrator will be influenced, as a phase shift is nothing but an indication for the existence of losses in the gyrator. It seems to be better to remain in class A operation and to adapt the supply current to the amplitude of the signal in the gyrator. A gyrator where the supply current adapts itself to the signal amplitude will be called an “adaptive gyrator.” Until now the gyrator had to have supply currents so high in value that the maximum signal that could occur could be handled without distortion, but also for small signals this high supply current was present. That we gain enormously with adaptive gyrators will be shown by an example. For a low-pass telephony filter with 3 gyrators and 10 capacitors (Voorman and Biesheuvel, 1972) a signal-to-noise ratio of 90 dB was obtained with a total power consumption of 30 mW. If we assume that the mean signal level is 20 dB below the maximum level, the mean supply currents for the adaptive gyrators are 10 times lower, giving already a mean power consumption of 3 mW instead of 30 mW. Moreover, only if the signal frequency is in the neighborhood of the resonance frequency of the gyrator circuit is this value of the supply current required. Hence, in practice, the power consumption will actually be much lower than the 3 mW mentioned. Finally, with this low power consumption the noise of the electronic circuitry is also lower and we have a remarkable gain in S/N ratio at low signal levels. These three points indicate that it is worthwhile to consider adaptive gyrators even if they are somewhat more complex.

2. Thr Aduptiae Gjmtor-

An adaptive gyrator is a gyrator with the feature that its supply current adapts itself to the signal amplitude. Note that this gyrator is always operating in class A. One way to obtain a measure for the signal amplitude is to use a peak detector. However, a peak detector requires an extra capacitor which cannot be integrated. In fact we have to use two peak detectors for the two signal current loops as we can see in Fig. 70. One loop of the gyrator with ports “ I ” and “ 2 ” is formed by the capacitor C I , the collector-emitter path of transistor TI, the conductance G I , and transistor T; and the other by C , , T, G , and T i . In each loop we have to measure the signal amplitude and to adapt the supply current. A much simpler and better method can be applied in filters where the

.

173

THE GYRATOR IN ELECTRONIC SYSTEMS

:c2

FIG.70. Main signal diagram of an electronic gyrator-capacitor parallel resonant circuit

gyrators form resonant circuits. To show this, let us consider a symmetrical parallel resonant gyrator-capacitor circuit (Fig. 70 with G, = G2 and C, = C 2 ) .If near resonance the current in G I is a sin wt, then the current in G, is practically equal to a cos mot. The currents in the two loops differ 90” in phase and have approximately the same amplitude. Squaring the currents gives u2 sin’ w f and a2 cos2 cot, respectively, and addition gives u2. In this way we find the amplitude which can be used to directly control the supply current. Let us consider this control mechanism somewhat more closely. If we have a supply current Z ( t ) and the signal currents in the loops are il(t) and i2(t),the gyrator circuit behaves properly if under all circumstances I il(t) I < Z(t) and I i 2 ( t )I < I ( t ) . With two squaring devices we form two currents equal to i:/Z and i$/Z. Addition gives (if + i i ) / I and in the control loop we make I = (1 + &)(i: + i i ) / I or Z = (1 +‘ &)1’2(i: + i$)”’ with E > 0. In this way a supply current is formed which always fulfills the conditions Z ( t ) > I il(t) 1 and Z ( t ) > I iz(t)I. Moreover, this control mechanism works instantaneously and can be integrated completely. If the resonant circuit is damped, it has a low quality factor and there can be complex signals at the gyrator ports, which means that also the supply current can vary rapidly. If we have a high quality factor, this is not permissible, as a little intermodulation of the supply current with the signal current can influence the quality factor strongly. To study this we shall use a narrowband approximation. Let il(t) = u ( t )sin [coot

+ 4(t)],

(168)

where coo = G/C is the resonance frequency of the circuit and u ( t )and 4(t) are relatively slowly varying functions. With iz(t) = -RCdi,(t)/dt, we obtain for the controlled supply current.

+ w1o d t cos’ (coot + 4) + ... I@

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K. M. ADAMS. E. F. A. DEPRETTERE A N D J. 0. VOORMAN

If the quality factor is high, both a ( t ) and +(t)are slowly varying functions and the supply current consists of a slowly varying term on which are superimposed small rapidly varying terms.

3. Conjunctors? Generally the gyration resistors of a gyrator cannot be integrated for reasons of accuracy. This means that the I.C. for a fully floating gyrator requires at least 11 terminals (4 port terminals, 4 terminals to insert the gyration resistors, plus and minus, and a terminal to connect a resistor for dc current adjustment). For an adaptive gyrator the last terminal can be omitted but the remaining high number of terminals (a coil has only two) results in high expense and is a potential source of unreliability. Suppose we integrate the gyrator-resistors R , and R 2 . If they were meant to be 10 kR, in practice resistors of 8 kR to 12 kR could result, depending on the diffusion depth and concentration. Their absolute value is not correct but the relative accuracy of R , and R , can be made very high, depending on the circuit integration process. An electronic multiplication of R , as well as R 2 by the quotient ReIR,, where R, is also a monolithic resistor and Re is an accurate external resistor, gives as overall resistances R, R J R , and ReRJR,, which have substantially the same accuracy as R e . In this way, we can relate all monolithic resistors to a single external resistor R e . With this single external resistor we can tune not only the gyrator but also a complete filter. Moreover, this saves us 3 I.C. terminals. What is left is to show how the multiplication of a resistance by a quotient of two resistances can be implemented electronically. First, let us make two currents of a given ratio. To this end we allow a current I , to flow in a resistor R , (Fig. 71a), and in the nor-port of the nullor we find a current

Rm

Re

Rm

-

I

FIG. 71. To multiply the monolythic resistor R by the quotient R J R , , we first make a current ratio IJI,,, equal to R J R , (a) and multiply the current Ir by it (b).

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175

I, such that I J I , = R,/R,. Multiplication by a resistance ratio is reduced to multiplication by a current ratio. As multiplication is a nonlinear operation it cannot be performed by a (linear) resistor-nullor network alone. Therefore, we shall introduce ideal diodes defined by the equation U = (kT/q)In ( I / I o ) ,where l ois the saturation current which is assumed to be equal for all diodes. Note that a transistor can be modeled by a nullor and diodes, depending on the degree of idealization. Addition or subtraction of the diode voltages corresponds to multiplication or division of their currents. In Fig. 71b a loop of diode voltages is formed, giving the relation I,,, = IeIr/lm.The VCCS gives I , = Ui,/R. Hence,

giving the correct transfer admittance. In an analogous way we can control all monolithic resistors with R e . The circuit we have designed for reduction of the number of I.C. terminals can also be seen as a controlled gyrator, which is an essential ingredient of the conjunctor, a nonlinear three-port introduced by Duinker (1962). 4. Future Applications

It is felt not only that gyrator design as described in the previous sections can and will be improved but also that integration into telecommunication systems has to be studied thoroughly. It is not only that we have to use the same supply voltages and impedance levels but the gyrators must also fit into the system philosophy. The application of gyrators is more than merely replacing coils by gyrators and capacitors. It is the use of all extra gyrator features to obtain simpler and better solutions. Let us mention one of these new features. With a symmetrical parallel resonant gyrator-capacitor circuit, an oscillator can be made by applying a feedback which keeps the quality factor infinite. If the gyrator is a controllable one, as described in the former section, we can vary the gyration resistance R, = R,(t). Now, we have the equations du2 , dt

u1 = R,(t)C--

u2

= -R,(t)C-

du dt

Put u = u , +.ju2 and the equations can be combined. The solution is: u1 = a cos 4(t) and u2 = a sin 4 ( t ) , where a is a constant and d4/dt =

l/R,(t)C. Hence, the instantaneous frequency d$/dt of the oscillator can be controlled by R,(t) without any transients. This is an ideal FM/FSK oscilla-

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K. M. ADAMS, E. F. A. DEPRETTERE A N D J. 0. VOORMAN

tor, and is an example of a new device which is obtained by the introduction of the gyrator. The central part of all these devices-the adaptive gyrator or conjunctor, the ideal FM/FSK oscillator-has been monolithically integrated for experimental purposes (about 200 transistors) in the Philips Research Laboratories. For each application a breadboard control circuit (about 40 transistors) is added. The adaptation as well as the controllability (e.g. a factor of over 25 in R, with a deviation from linearity of about 2 % at the extremities) show very promising results. The FM/FSK oscillator is surprisingly good. All of these results will be presented in the near future. In this field of signal processing with gyrators, conjunctors, and probably also traditors (Duinker, 1959; Deprettere, 1973) it is felt that many new developments can be expected, which will be of great practical importance to future systems for professional use.

IX. CONCLUSION In this survey we have emphasized some important principles that are relevant to the design and application of gyrators to electronics and communications. A great deal of material dealing with the detailed electronic design of the complete circuit has had to be excluded for lack of space. We have not gone into a detailed comparison with RC-active circuits. There are, however, two recent developments which deserve a short comment. A development of the “leap-frog’’ circuit (Szentirmai, 1973) is a filter with the same favorable sensitivity properties as the terminated LC ladder. However, for a given degree of the filter, the circuit requires more nullors and supply circuitry than the corresponding gyrator-C filter. It also contains many more resistors, so that both the dissipation and the noise for a given dynamic range can be expected to be considerably worse than in the case of the g y r a t o r 4 filter. Another development is a second-order circuit due to Soderstrand and Mitra (1973). The authors claim zero sensitivity, but to achieve this requires an amplification equal to 16Q2, presumably without phase shift. The spread of element values can be impractical. If, for example, a Q of 500 is required (which is easily attainable with gyrator circuits), then one requires a resistance of 2 GR in parallel with a capacitance of 35 pF. If the 2 GR resistor is replaced by an open circuit, then the low sensitivity of the circuit vanishes. We can conclude that the gyrator as a concept and a device is fully established. To exploit the possibilities fully, a new theory will be needed. Further development of the synthesis theory of nonreciprocal networks, linear time-variable networks, nonlinear networks, and their application to communications is called for. Experience has shown that the technology of

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177

integrated circuits has a tremendous potential to meet the demands of current and future system specifications arising from theoretical considerations.

REFERENCES Adams, K. M. (1968). I n ”Network and Switching Theory” (G. Biorci, ed.), pp. 361-381, Academic Press, New York. Adams, K. M. (1974). I n “Physical Structure in Systems Theory” (F. J. Evans and J. J. van Dixhoorn, eds.). Academic Press, New York. In press. Antoniou, A. (1969). Proc. I E E E 116, 1838. Ark, G. (1960). Solid-State Electron. 1, 75. Astrov, D. N. (1960). Zh. Eksp. Teor. Fiz. 38, 984. Astrov, D. N. (1961). Zh. Eksp. T e o r . Fiz. 40,1035. Belevitch, V. (1968). “Classical Network Theory.” Holden-Day, San Francisco, California. Beljers, H. G. (1956). Philips Tech. Tijdschr. 18, 126. Beljers. H. G., and Snoek, J. L. (1950). Philips Tech. Rec. 11, 313. Birkhoff. G . D. (1927). “Dynamical System ‘ Anier. M a t h Soc.. Providence. Rhode Island. Bittel. H., and Storm, L. (1972). Rauschen Springer-Verlag. Berlin and New York. Bloch, A. (1944). Phil. Mag. 35, 315. Blom. D., and Voorman, J . 0. (1971). Philips Res. Rep. 26, 103. Bosma, H. (1964). I E E E Trans. b‘icrowaiv Theory Tech. 12, 61. Bosma, H. (1967). Philips Res. Rep., Suppl. 10. Bruton, J. J. (1973). Radio Electron. Eng. 43, 325. Carlin, H. J. (1964). I E E E Trans. Circuit Theory 11. 67. Carlin, H. J. (1967). Proc. I E E E 55, 482. Carlin, H. J., and Youla, D. C. (1961). Proc. IRE 49, 907. Carlin, H. J., Youla, D. C . , and Castriota, L. J. (1959). I E E E Trans. Circuit Theory6. 103, 317. Com. Consult. Int. Telephon. (1956). ‘‘ Recommandations de Principe et Mesures Relatives a la Qualite de Transmission Appareils Telephoniques,” C.C.I.T., Vol. 4, p. 122. Union Int. Telecommun., Geneva. Cherry, E. M., and Hooper, D. E. (1968). “Amplifying Devices and Low-Pass Amplifier Design.” Wiley, New York. Daniels, R. W. (1969). I E E E Trcins. Circuir Theory 16, 543. de Groot. S. R., and Mazur. P. (1962). “ Non-Equilibrium Thermodynamics.” North-Holland Publ., Amsterdam. de Hoop, A. T. (1966). A p p i . Sci Res. 16, 39. Deprettere, E. (1971). I E E E I n t . Symp. Network Theory, London, Dig. Tech. Pap. p. 65. Deprettere. E. (1973). I n t . J . Electron. 34, 237. Dicke, R. H. (1948). I n “Principles of Microwave Circuits” (C. C. Montgomery, R. H. Dicke, and C . M. Purceil, eds.). pp 130 -161. McGraw-Hill, New York. Duinker, S. (1959). Philips Res. Rep. 14, 29. Duinker, S. (1962). Philips Res. Rep. 17, 1. Dunford, N., and Schwartz, J . T. (1958). Linear Operators,” Part I. Wiley (Interscience), New York. Fettacis, A. (1969). .4rch. Elekrisch. Uehertr. 23, 605. Fettweis, A. (1970). I E E E Ti.trri.5.Circirit T/?eorj,17. 86. Fettweis, A. (1971). Arch. Ele!,rron. U e h w t r . 25. 79. Gantmacher, F. R. (1959). “ T h e Theory of Matrices.” Chelsea, Bronx, New York. Geffe. P. R. (1963). “Simplified Filter Design.” Ridder, New York. Hogan. C. L. (1952). Bell S j s t . Tech. J . 31, I. ‘I

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Jefferson, H. (1945). Phil. Mag. 36, 223. Keen, A. W. (1959). Proc. I R E 49, 1148. Kishi, G., and Kida, K. (1967). I E E E Trans. Circuit Theory 14, 380. Kishi, G., and Kida, K. (1970). I E E E Trans. Circuit Theory 17, 632. Kishi, G., and Nakazawa, K. (1963). IEEE Trans. Circuit Theory 10, 67. Klein, W. (1952). Arch. Elektrisch. Uebertr. 6, 205. Leich, H., and van Bastelaer, P. (1968). Rev. M B L E 11, 31. McMillan, E. M. (1946). J . Acoust. Soc. Amer. 18, 344. Millar, W. (1951). Phil. Mag. 42, 1150. Mitra, S. K. (1969). “Analysis and Synthesis of Linear Active Networks.” Wiley, New York. Miilier, J. H. W. (1971). I E E E f n t . S j ~ n pNetwork . Theory, London, Dig. Tech. Pap. p. 59. Neirynck, J.. and Thiran, J. P. (1967). Elecrron. Lett. 3, 74. O’Dell, T. H. (1966). I E E E Student J . p. 15. Oono, Y., and Yasu-ura, K. (1954). Mem. Fac. Eng., Kyushu Unit;. 14, 125. Orchard, H. J. (1966). Electron. Lett. 2, 224. Orchard, H. J. (1970). In “Active Filters, Lumped, Distributed, Integrated, Digital and Parametric” (L. P. Huelsman, ed.), pp. 90-127. McGraw-Hill, New York. Orchard, H. J., and Sheahan, D. F. (1970). I E E E J . Solid-Stute Circuits 5, 108. Pars, L. (1965). “A Treatise on Analytical Dynamics.” Heinemann, London. Penfield, P., Spence, R., and Duinker, S. (1970). “Tellegen’s Theorem and Electrical Networks.” MIT Press, Cambridge, Massachusetts. Polder, D. (1949). Phil. Mag. 40,99. Prigogine, I., Nicolis, G., and Babloyantz, A. (1972). Phys. Today 25, 23. Rayleigh, Baron (Strutt, J. W.) (1894). “The Theory of Sound.” Macmillan, London. Riordan, R. H. S. (1967). Electron. Lett. 3, 50. Saal, R. (1963). “Der Entwurf von Filtern mit Hilfe des Kataloges normierter Tiefpasse.” Telefunken GmbH, Backnang, Wuertt. Sallen. R. P., and Key, E. L. (1955). I R E Trans. Circuit Theor!.2. 78. Sharpe, G. E. (1957). I R E Trans. Circuit Theory 4, 321. Sheahan, D. F., and Orchard, H. J. (1967). Electron. Lett. 3, 40. Silverman, J. H. (1962). I R E Trans. Component Parts 9, 81. Silverman, J. H. (1963). Electronics 36, Feb. 22, 56. Silverman, J. H., Schoeffler, J. D., and Curan. D. R. (1961). Proc. Nat. Electron. Conf: 17, 521. Skwirzynski, J. K. (1965). Design Theory and Data for Electrical Filters.” Van NostrandReinhold, Princeton. New Jersey. Soderstrand, A. M., and Mitra, S. K. (1973). f E E E Trans. Circuit Theory 20, 441. Stout, M. B. (1960). Basic Electrical Measurements.” Prentice-Hall, Englewood Cliffs, New Jersey. Szentirmai, G. (1973). Bell Syst. Tech. J . 52, 527. Tellegen, B. D. H. (1948a). Philips Res. Rep. 3, 81. Tellegen, B. D. H. (1948b). Philips Res. Rep. 3, 321. Tellegen, B. D. H. (1949). Pliilips Res. Rep. 4, 31, 366. Tellegen, B. D. H. (1952). Philips Res. Rep. 7, 259. Tellegen, B. D. H. (1956). Rend. Semin. Mar. Fis. Milano 25, 134. Tellegen, B. D. H. (1966). I E E E Trans. Circuit Theory 13, 466. Tellegen, B. D. H., and Klauss, E. (1950). Philips Res. Rep. 5, 81. Tellegen, B. D. H., and Klauss, E. (1951). Philips R e x Rep. 6, 86. Temes, G. C., and Orchard, H. J. (1973). f E E E Trans. Circuit Theor), 20, 567. Trimmel, H. R., and Heinlein, W. E. (1971). I E E E Int. Symp. Network Theory, London, Dig. Tech. Pap. p. 61. “

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Truesdell. C. (1969). “Rational Thermodynamics.” McCraw-Hill, New York. van der Ziel, A. (1970). “Noise: Sources, Characterization, Measurement.” Prentice-Hall, Englewood Cliffs, New Jersey. van Looij, H. T., and Adams, K. M. (1968). Elecrron. Lett. 4, 431. Voorman, H. O., and Biesheuvel, A. (1972). I E E E J . Solid-state Circuits 7, 469. Voorman, J. O., and Blom, D. (1971). Philips Res. Rep. 26, 114. Youla, D. (1971). PFOC.IEEE 59, 760.

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Image Sensors for Solid State Cameras P. K . WEIMER R C A Luborururies. Princeton. Nett J e m )

I . Introduction .......................................................................................... I1 . Photoelements for Self-scanned Sensors ...................................................... 111. Principles of Multiplexed Scanning in Image Sensors .................................... A . Single-Line Sensors ........................................................................... B. Multiplexed Scanning of X Y Sensors ...................................................... IV. Early XY Image Sensors ................................................. A . Photoconductive Arrays ..................................................................... B. Phototransistor Arrays .................................................................. V . Multiplexed Photodiode Sensors ............. ..... ..... A . Single-Line Photodiode Sensors .......... B. X Y Addressed Photodiode Sensors ......................................................... .................................. V1 . Principles of Scanning by Charge Transfer ........ A . Single-Line Charge-Transfer Sensors ............................... B. Two-Dimensional Charge-Transfer Sensors ........ .................................... VII . Charge-Transfer Sensors Employing Bucket Brigade Registers ........................ A. .4 32 x 44 Element Bucket Brigade Camera ............................................. B. Experimental Tests of C harge-Transfer Readout of an MOS-Photodiode Array VIII . Characteristics of Charge-Coupled Devices (CCDs) ....................................... A . General Description of CCDs ............................................................... B. Transfer Losses in CCD’s ..................................................................... C . Noise Characteristics of C C D s ............................................................ IX . Experimental Charge-Coupled Image Sensors ................... A . Single-Line CCD Sensors ........................................ B. Two-Dimensional Area-Type Charge-Coupled Sensors .............................. X . Performance Limitations of Charge-Coupled Sensors .................................... A . Resolution ........... .......................................................... ;

vity .......................................................................... XI . Charge-Transfer Sensors as Analog Signal Processors .................................... A . Charge-Transfer Delay Lines ............................................................... B . Video Signal Processing within the Camera by Recycling of Signals through the Sensor Itself ....................................................................................... XI1. Self-scanned Sensors for Color Cameras ...................................................... XI11. Peripheral Circuits for Solid State Sensors ................................................... A . Input Circuit Design ........................................................................... B. Output Circuit Design ........................................................................ XIV . Conclusions ..................................................................... References .............................................................................................

181

182 183 187 188 190 193 194 197 198 198 199 202 202 205 208 209 211 213 213 216 220

229 236 231 239 247 247 249 251 253 253 254 259

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P. K. WEIMER

I. INTRODUCTION More than twenty-five years have passed since the invention of the transistor, but television camera and display equipment still depend almost entirely upon special beam-scanned tubes for picking up and reproducing the image. Even though television tubes have now reached a high level of performance the well-known advantages of solid state devices in reliability, compactness, and cost would be welcome in both the camera and receiver. In spite of major advances in solid state technology, it has proved to be extremely difficult to design solid state scanning systems which could match the elegant simplicity of the electron beam. Progress is being made, however, and an increasing number of self-scanned sensors have appeared on the market in the last few years. Single-line sensors are already finding application in page readers, in character recognition equipment, and in satellite cameras. These devices perform a unique function not always suitable for tubes, and are far easier to build than self-scanned area sensors. More recently, two-dimensional area sensors having limited resolution have also appeared on the market. These devices are intended for surveillance or identification and control purposes. They will be useful in applications requiring a very compact camera, with extreme durability and low power consumption. Although costs are still high and general performance falls short of camera tubes, the present rate of development of solid state sensors is very rapid. Many research laboratories, with support from the governmental agencies are now actively involved in work on image sensors (1). Industrial laboratories which have participated in this development over the past ten years include Bell Laboratories, General Electric Co., Fairchild Camera and Instrument Corporation, RCA Corporation, and Westinghouse. A continuing program on sensors has also been carried out at Stanford University where the specific objective has been to develop a reader for the blind (2). From the research point of view, work on image sensors has been both challenging and technically rewarding. T o achieve resolution comparable to camera tubes, sensors will require several hundred thousand picture elements. The development of a suitable scanning technique must be accompanied by a feasible fabrication technology. At RCA Laboratories, work o n sensors was started before silicon integrated circuits showed promise for devices of such complexity. Evaporated thin-film circuits (3),employing an early type of thin-film MOS transistor known as the TFT, were used to scan experimental sensors having up to 5 12 x 512 photoconductive elements. However, by 1970, the silicon technology had been refined and was so widely established that it clearly represented the more powerful technology for image sensors. The first solid state sensors utilized intersecting XY address strips (3)

IMAGE SENSORS FOR SOLID STATE CAMERAS

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connected to each picture element for scanning. By application of progressive scanning pulses to the address strips, the response at each picture element was measured in sequence yielding a time-varying video output signal. Although valid in principle, the XY address system was subject to difficulties in uniformity and in signal-to-noise ratio. In 1969 and 1970, two discoveries were made which permitted an entirely new approach to solid state scanning. These were the bucket brigade register by Sangster and Teer ( 4 ) and the charge-coupled device by Boyle and Smith (5).These registers, which are easily fabricated in silicon, permit the transfer of analog signals through many stages with very small losses. By means of such registers, the sensor can be scanned by transferring the picture charges in sequence from the array to an amplifier located at the edge of the sensor. Significant improvement in signal-to-noise ratio and in uniformity over earlier XY systems become possible. Charge-coupled devices (or CCDs), because of their lower transfer losses and higher packing density. now appear to be superior to bucket brigades for most applications. Their greater promise for semiconductor memories and signal processing devices has also contributed to the more extensive research effort which they have received. A 128 x 106 element chargecoupled image sensor was incorporated into a camera at Bell Laboratories (6) in 1972. Still larger CCD arrays are being investigated elsewhere. Cameras employing CCD sensors having up to ten thousand elements are now available on the market. Although the impact of the CCD development on solid state sensors has been truly revolutionary it cannot be concluded that charge-coupled sensors will necessarily be best for all types of applications. Advances continue to be made in both CCD and X Y addressed sensors. Many problems remain, however, and their solution will affect performance and relative cost. It is clear that an entire television camera, including sensor and all drive circuits, will eventually be formed on one or two chips of silicon. If this can be done at low cost, many new applications for television should soon follow. The present review discusses solid state sensors from the research point of view, with much greater emphasis on principles of operation than on details of construction. New types of sensors are described and projected performance is compared with existing camera tubes. The extension of solid state scanning techniques to other types of imaging devices and signal processors is considered. FOR SELF-SCANNED SENSORS 11. PHOTOELEMENTS

The function of an image sensor is to generate a time-varying video signal corresponding to the spatial variations in the incident optical image. One of the earliest lessons learned in the development of camera tubes (7)

184

P. K. WEIMER

was that high sensitivity operation would require the sensor to integrate the total light flux falling on each element throughout the whole scanning period. How this was accomplished in the vidicon ( 8 )is illustrated in Fig. 1. The vidicon was the first camera tube with a solid state target, and it provided the starting point for the earliest self-scanned image sensors.

PHOTOCONDUCTIVE rTARGET

VIDEO + SIGNAL OUT

ELECTRON G U N 7

V (a)

COMMUTATOR SWITCH N 250,000 FOR TV

-

FIG.1 . A cross-sectional drawing of a vidicon photoconductive camera tube ( a ) and the equivalent circuit of its tarset and scanning beam (b).

The continuous photoconductive layer in a vidicon target can be represented as an array of isolated photoconductive elements each shunted by a capacitor. Light integration is attained by choice of a high resistivity photoconductor whose RC time constant exceeds the scanning period of the electron beam. Light flux is integrated in the form of charges collected on each element at the surface of the photoconductor. The elemental charges are neutralized in sequence by the scanning beam, causing a video signal to appear in the lead connected to the transparent electrode on which the photoconductor is deposited. Only a limited number of high resistivity photoconductive materials have been used successfully in the vidicon. These include Sb,S, , PbO, CdSe, and amorphous selenium. A major advance in image sensing occurred with the development of the silicon diode array target ( 9 )for the vidicon. Although the volume resistivity

IMAGE SENSORS FOR SOLID STATE CAMERAS

185

of undoped silicon is too low to serve for charge integration, the depleted region under a reverse-biased photodiode element can be made to meet the requirement adequately for most purposes. As shown in Fig. 2, one form of silicon vidicon target consists of an array of approximately lo6 photodiode elements diffused into a thin wafer of silicon which is illuminated from the side opposite the diodes. The beam contacts the beam landing pads which connect to the diodes and serve to shield the insulating oxide regions between the diodes from the beam. The video signal is derived from the silicon substrate which serves the same function as the transparent conducting coating in the photoconductive vidicons. The silicon vidicon offers the advantages of dependable, long life even when exposed to direct sunlight,

FIG.2. Cross section of a typical silicon photodiode target for a vidicon camera tube

and its target has increased responsivity for visible and near infrared illumination. Its infrared response is useful for low-light-level surveillance applications but has to be filtered o u t when proper color rendition is required. Its complete freedom from photoconductive lag was also an advantage over the early evaporated targets. The development of silicon processing techniques for the vidicon target proved to be highly useful in the subsequent use of silicon in self-scanned sensors. An equally important development in silicon technology which enhanced the attractiveness of silicon for image sensors was the MOS transistor (10). The high ratio of on-to-off conductance of these insulated-gate field effect transistors has allowed them to replace the electron beam as the commutator in self-scanned charge storage sensors. Their simplicity of fabrication and their superb capability for large scale integrated circuits appears ideal for image sensors. Figure 3 shows how an MOS transistor can be

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P. K . WEIMER

combined with a photodiode to serve as a charge-integrating sensor element ( 2 2 ) . The diffused photodiode functions also as the source of the discharge transistor. Its potential variations throughout the scanning period are the same as the charge-discharge cycle of the scanned surface of the vidicon. The photogenerated holes which are collected at the photodiode while the transistor is biased off are discharged through the external circuit at the instant of scan.

i

SCAN PULSE

STRUCTURE

O% T! CURRENT

EOUIVALENT CIRCUIT

1

I\ OPERATING CYCLE

FIG.3 . The structure, equivalent circuit, and operating cycle of an MOS-photodiode imagc sensor element.

The recent development of charge-transfer scanning ( 4 , 5 ) has further advanced the capability of silicon as a detector. The CCD sensors have utilized the fact that an MOS capacitor, produced by covering a silicon semiconductor with an insulating layer and a metal gate, can also serve as an integrating photoelement similar to a diffused photodiode. The photosensitive region is formed by biasing the metal gate so as to deplete the semiconductor surface. Illumination of this region causes the minority carriers to collect at the semiconductor-insulator interface. The accumulated charges can be removed laterally by charge transfer (6) to an adjacent electrode or by charge injection (22, 12a, 13) back into the body of the silicon. The MOScapacitor photoelements should preferably be located on a thin slab of silicon and illuminated through the silicon to avoid light absorption in the conducting gate. However, fairly good sensitivity has been obtained in CCD sensors with top illumination by depending entirely on light which enters the silicon around the edges of opaque metal gates. Transparent polycrystalline silicon

IMAGE SENSORS FOR SOLID STATE CAMERAS

187

gates are also feasible but optical interference effects tend to introduce maxima and minima in the spectral response curves. For all of the above reasons, silicon is currently theoverwhelming favorite as the detector for image sensors operating over a wavelength range from 400 up to 1000 nm. It is not the ideal sensor, however, even for visible light. A wider bandgap material such as PbO, CdSe, or Sb,S, could have lower dark current at room temperature and would not require filtering to remove the unwanted infrared sensitivity. For far-infrared applications a narrower bandgap material than silicon would be preferred. Since the prospects are somewhat remote for achieving electronic properties in these materials equal to single-crystal silicon, their use in self-scanned structures will probably require a hybrid structure. That is, the detector would be used in combination with silicon scanning circuits. However, the low impedance of most infrared detectors makes them difficult to combine with solid state scanning. Fortunately, the number of infrared photons in many scenes of interest is so great that passive imaging is feasible without requiring full photon integration in the image sensor. It will become apparent in the discussion of sensor signal-to-noise ratio in Section X that a detector element which yields more than one carrier per incident photon could be superior in image sensor applications to a silicon photodiode whose quantum yield is always less than unity. Examples of such high gain detectors are the photoconductors CdS, CdSe, and PbS and silicon avalanche diodes. The uniformity of gain from one element to the next would have to be excellent for such detectors to be useful in imaging devices. High gain CdS-CdSe has been used successfully in the thin-film arrays (14) described in Section IV,A. In these devices the internal gain mechanism in the photonconductor (up to lo6 electrons per photon) was used to obtain photon integration without requiring charge storage. The effect of the light was to produce a cumulative increase in the conductivity of the sensor element. Under ideal conditions, this type of “excitation storage” could approach the efficiency of charge storage without seriously compromising the response time of the device. 111. PRINCIPLES OF MULTIPLEXED SCANNING IN IMAGE SENSORS Multiplexed scanning of solid state arrays is directly analogous to electron beam scanning in camera tubes. In each case the picture charge at the element is discharged through a local switch into a common electrode connected to the output amplifier. Prior to the development of charge-transfer scanning all experimental sensors operated in this manner. Multiplexed scanning will be discussed first in terms of single-line sensors, and then for two-dimensional XY arrays.

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A . Single-Line Sensors Figure 4a illustrates multiplexed scanning of a single-line sensor (I 1 ) in which each element consists of a photodiode in series with an MOS transistor as in Fig. 3. The scan generator is a parallel-output digital shift register with each stage connected to the gate of the elemental transistor. As the negative pulse progresses down the register each transistor is turned on in sequence, discharging the accumulated picture charge into the output bus and resetting the elemental capacitor for the next integration period. The resulting video signal current in the output bus is fed into an amplifier preferably located on the same silicon chip with the sensor and multiplexer.

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The elemental capacitor shunting each diode should be large compared to the stray capacitance shunting the transistor. The elemental capacitor may consist largely of the depletion layer capacitance between photodiode and the substrate or it may be formed of the oxide capacitance to an overlying electrode. As explained in the preceding section, the diffused photodiode

may actually be eliminated completely and the photogenerated carriers collected at the silicon insulator interface. In this case the transistor behaves like a single-stage charge-coupled device. Each picture element in a multiplexed system must contain a light detector and a switch. The switch could be a diode as well as a transistor. In the

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example illustrated in Fig. 4b the photodiode itself serves alternately as a reverse-biased photodetector during charge integration and as a forwardbiased discharge switch at the instant of scan (13). Sensors employing the discharge of the photogenerated carriers by injection into the substrate will be discussed in Section V,B,2. In case the RC conditions for charge storage are not met in the sensor it is still possible to operate using multiplexed scanning. Although sensitivity would be reduced without charge storage, this mode of operation might be necessary for sensor elements having very low impedance. A serious disadvantage of multiplexed scanning is that the video signals from successive elements are combined on an extended output bus whose total capacitance to ground may exceed the capacitance of each element by several orders of magnitude. The equivalent amplifier noise current would be increased by the square root of this factor beyond its value if it were connected to a single element. The low light threshold of the device as determined by noise would accordingly be raised by the same square root factor. The noise becomes even worse in a two-dimensional array having the added capacitance of XY address strips. A less fundamental disadvantage of multiplexed scanning is the tendency for spurious signals to be introduced into the output signal by variations in size of successive scan pulses. The fact that these pulses tend to be picked up capacitatively in the output bus through the multiplexer switches would not be particularly serious if the pulses were uniform from element to element. However. successive pulses and successive multiplexer switches are not identical so that low frequency variations are introduced which cannot be filtered out. The pictures produced by most of the early sensors contained vertical striations arising from this effect in the horizontal multiplexer. Such striations can be minimized by use of signal integration, double sampling, and sample-and-hold techniques in the output amplifier (see Section XII1,B). However, as the scanning frequencies are increased to accommodate larger arrays it becomes steadily more difficult to suppress the nonuniform switching transients picked u p during the rise and fall of successive pulses. The scan generator for the multiplexer can assume a variety of forms. The most common is a digital shift register or ring counter located adjacent to a single row of sensors or at the edge of the two-dimensional array. Such generators require from three to eight MOS transistors per picture element. The rate of progression of the scan pulse is accurately determined by the clock voltages. Figure 5 shows the circuit for a typical MOS scan generator. If a large two-dimensional array is being scanned, an additional driver stage may be required at each output to drive all the gates along a given row or column. Other types of analog scan generators have also been used. These include bucket brigade registers, which require only a single transistor per

190

P. K. WEIMER

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stage (see Section VII), and resistive-gated generators (15, 15a) in which the outputs are activated in sequence by a linear voltage ramp. Another type of scan generator utilizes an acoustic surface wave (15b) to contact successive elements. This type of system has the disadvantage that the scanning velocity is determined by the material properties of the medium and cannot be adjusted to match the preferred size of array.

B. Multiplexed Scanning of XY Sensors Although a single-line sensor duplicated N times would serve as a twodimensional array this approach is not usually suitable. Two-dimensional sensors which are to be scanned by multiplexing employ two perpendicular

19 1

IMAGE SENSORS FOR SOLID STATE CAMERAS

sets of X and Y address strips which connect to each element in the array. Figure 6 shows the elemental circuit for some of the XY sensors which have been built. Each element includes the basic components of a photodetector plus one or more switches connected to the address strips. These include: (a) A thin-film CdS photoconductive array (13)which has been scanned in sizes up to 512 x 512 elements. The switching diode was obtained by use of dissimilar contacts to the photoconductor. This array was operated by excitation storage rather than by charge storage (see Section IV,A). (b) A silicon phototransistor array (16) which was scanned in sizes up to 400 x 500

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FIG. 6. Sensor elements which have been used in XY image sensor arrays: (a) Photoconductor-diode array. (b) Phototransistor array (isolated collector rows). (c) Phototransistor array with an MOS switch. (d) Photodiode array with two MOS switches in series. ( e ) Photodiode array with an MOS amplifier at each element. ( f ) Photodiode array with a single MOS switch. (g) Photodiode array with cascaded MOS switches. (h) Photodiode array which operates by charge injection in the substrate.

elements. Charge storage occurred in the capacitance shunting the collectorbase junction while the emitter-base diode served as the switch (see Section IV,B). (c) A silicon phototransistor (17) array having a common substrate collector and an MOS switch at each element (100 x 100). (d) A silicon photodiode array (18) with common cathodes in the substrate and two MOS switches coupling the photodiodes to a common video output bus. (e)A proposed silicon photodiode array ( 1 9 ) similar to (d) but with an additional amplifying transistor T, at each element. ( f ) A silicon photodiode array (20) having a single MOS transistor at each element. This simple structure can be used with charge-transfer readout. (g) A silicon photodiode array (21) similar to a two-transistor random access memory array (50 x 50 sensor commercially available 1973). (h) A silicon photodiode array which operates by charge injection (12,13) into the substrate. Although most of the

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P. K. WEIMER

photodiode structures listed above were built with a diffused photodiode at each element, an equivalent MOS-capacitor array could be constructed for each as described in the preceding section. This step has already been taken with the charge-injection sensors described by Michon and Burke (12). (See Section V,B,2.) XY arrays are normally operated in conjunction with external multiplexers and scan generators located on the periphery of the array. The physical separation of the photodetectors and the scanning circuits is a potential advantage in permitting hybrid technologies for sensing and scanning. Three common systems of multiplexed scanning of XY arrays are illustrated in Fig. 7. The derivation of signals from the column buses as in (b)

FIG. 7. Three systems of multiplexed scanning of a n X Y image sensor. (a) Video signal from the rows. (b) Video signal from the columns. (c) Video signal from a common electrode such as substrate.

has the advantage that the signals from a whole row can be transferred in parallel to the output multiplexer, with a whole line time available for addressing each element of that row. In a sensor having no storage capacitors at each element, method (b) will give an increase of the total signal over method (a) by the ratio of the line time to the element time. The column bus capacitors required for line storage operation can be added externally, or the internal crossover capacitance of the address strips can be utilized. In sensors having a common substrate or a common video output bus to each element (such as Figs. 6c, 6d, 6e, 6f, 6g, and 6h) no external multiplexer is required. However, the total capacitance of the output electrode will be considerably higher than that of the output bus of the external multiplexer. Since the output electrode capacitance increases directly with the total number of elements the use of a common output electrode will give a degraded signal-to-noise ratio in large high resolution sensors. In small area sensors a common output bus can be used satisfactorily with sufficient illumination.

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193

With an external multiplexer the total capacitance shunting the input of the video amplifier is composed of the capacitance of the multiplexer output bus plus the capacitance of one column bus (or row bus) plus the capacitance of the element being scanned. Each bus contains many (off) switches whose gate capacitance adds to the capacitance of the bus itself. It is apparent that even with multiplexed scanning the XY array will suffer a degraded signal-to-noise ratio compared to the signal-to-noise ratio of a single element connected directly to the amplifier. In general, the rms noise contributed by this capacitance will vary directly as the square root of the total capacitance. The noise increase could be as large as 1&30 times for a 500 x 500 element array. Since the multiplex method of scanning of XY arrays yields a poor signal-to-noise ratio at low light levels it would be desirable to provide some means of increasing the signal level prior to scanning. The addition of an MOS amplifier stage at each element (such as shown in Fig. 6e) would improve signal-to-noise ratio but introduces other problems of complexity and nonuniformity. The use of an image intensifier stage prior to the array would be another method of increasing the final signal-to-noise ratio. In addition to problems of random noise, the multiplexed readout of the early XY arrays was subject to fixed pattern noise introduced by spatial variations in the address strips, in the multiplexer switches, and in the scan pulses received from the scan generators. The horizontal scan generator was more critical than the vertical because of its higher clock rate. Variations which fall within the video passband cannot be removed by filtering. They are repeated from line to line and appear as fixed vertical striations in the transmitted picture. In spite of their poor reputation, XY addressed arrays offer a degree of versatility and simplicity of construction not found in some of the more recent COD sensors. Recent advances in signal processing and integrated circuit technology would undoubtedly yield improved results over that obtained several years ago with this approach. The application of chargetransfer techniques to the scanning of XY photodiode arrays (Section VI,B,2) could provide an improved sensitivity which would be adequate for most applications. However, the capacitance of the address strips will always introduce noise which will limit the performance at very low light levels. IV. EARLYXY IMAGE SENSORS During the 1960s, the three principal approaches to solid state sensors employed photoconductive, phototransistor, or photodiode sensor elements. The first two systems appeared particularly attractive at that time because the internal gain in the element itself yielded higher signal levels than could

194

P. K . WEIMER

be easily obtained with photodiodes. Recent advances in MOS technology and in small signal processing have favored the photodiode systems which will be discussed in the next section. The photoconductor and phototransistor devices are of more than historical interest, however, since they illustrate scanning systems which may yet profit from future advances in technology. A . Photoconductive Arrays The thin-film photoconductive approach to image sensors (14) was started in the early 1960s, before the silicon integrated circuit technology had become established. This project therefore represented the development +V

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195

IMAGE SENSORS FOR SOLID STATE CAMERAS

T 3 cm 1

FIG.9. Photograph of the 256 x 256 element integrated thin-film image sensor deposited on two glass substrates mounted on a printed circuit board.

of a new thin-film technology (3)as well as a study of image sensors. As early as 1965, 180-stage integrated shift registers (22) employing thin-film transistors (TFT's) were produced for scanning of arrays. Figures 8 and 9 show the circuit and integrated thin-film embodiment of a 256 x 256 element photoconductor-diode sensor (13) which was fabricated at RCA Laboratories in 1968. The sensitive area of the sensor was one-half inch square with

196

P. K. WEIMER

the picture elements spaced on 50p centers. Each element of the array consisted of an evaporated CdS-CdSe photoconductor in series with a Schottky diode as illustrated in Fig. 6a. The diode action was obtained by use of one blocking contact (Te) and one ohmic contact (In) to the photoconductor element. The entire array including sensor and thin-film scanning decoders were deposited by evaporation onto two glass substrates which were interconnected by thin-film conductors. The scanning decoders employed TFTs, diodes, and resistors interconnected so that the 256 sequential scan pulses could be generated by two external 16-stage shift registers. The output signal was derived from an external multiplexer connected to the column buses so that a large signal increase due to line storage was obtained. The entire 256 x 256 array was scanned in 1/60 sec using a 4.8 MHz horizontal clock rate. The resolution obtained was consistent with the number of picture elements but the transmitted picture contained vertical striations arising from nonuniformities in the scan generators and defective elements in the sensor or decoder. Figure 10 shows pictures transmitted by a 256 x 256 photoconductive sensor. The uniformity was

FIG. 10. Pictures transmitted by the 256 x 256 element thin-film sensor shown i n k-igs. X and 9.

comparable to that obtained with other XY sensors of this period, but much inferior to recent CCD sensors. Pictures were transmitted at normal room illumination but the photoconductive lag was objectionable with moving scenes, particularly at lower illumination levels. Photoconductive arrays with integrated scanning decoders deposited on the same glass substrate were produced in sizes up to 512 x 512 elements (23). Integrated thin-film shift registers having up to 264 driver stages were also produced using complementary cadmium selenide and tellurium TFT's. The thin-film approach for image sensors was eventually discontinued in favor of the silicon technology using charge-transfer scanning.

197

IMAGE SENSORS FOR SOLID STATE CAMERAS

B. Phototransistor Arrays The combination of charge integration and signal gain obtainable from phototransistors (24) made this approach a favorite of several laboratories during the 1960's. Bipolar transistor technology was somewhat more advanced than MOS technology at that time so that large arrays were attempted. A 400 x 500 phototransistor array shown in Fig. 11 was fabricated

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by Westinghouse and incorporated into a camera (25). The equivalent circuit for each element of the array is shown in Fig. 6b. The collector rows were formed in the silicon substrate by an isolation diffusion process so that each collector was separated from all other collectors by back-to-back p-n junctions. Individual base and emitter regions were diffused into each collector at regular intervals, and the emitters along each column were connected to common column buses. The row and column address strips were each terminated in bonding pads, through which connections were made to external scanning circuits. Other types of phototransistor arrays were also built in which all elements had a common substrate collector. One method of accomplishing this was by use of a series MOS transistor (17) at each element, as shown in Fig. 6c. Another method was by driving the bases capacitively through the row bus as discussed in Weimer et al. (26). Although the phototransistor arrays have proved to be useful in relatively small sizes, such as a reading aid for the blind (2),they d o not presently appear promising for television applications. The major disadvantage was

198

P. K. WEIMER

the random variations in transistor gain and low level threshold caused by emitter offset. Transistor gains could vary as much as 2 to 1, far exceeding the nonuniformities observed in photodiode and photoconductor arrays. In addition, the sensitivity improvement expected from phototransistors was not obtained in practice. The nonlinear characteristic of the emitter base junction resulted in a threshold at low light levels. Advances in the ability to read out small signals from photodiode and CCD arrays has lessened the interest in sensor elements such as phototransistors and photoconductors which provide gain at each element.

V. MULTIPLEXED PHOTODIODE ARRAYS A . Single-Line Photodiode Sensors

The earliest photodiode arrays were single-line sensors in which each element consisted of an MOS transistor in series with the photodiode (11, 17). These were operated in the charge storage mode as discussed in Section 11. The elemental capacitance could be made sufficiently large to give ample stored signal in spite of the unity gain of the photoelement. Single-line sensors having several hundred elements in a row spaced on 2.5 mil centers were scanned by means of externally connected shift registers in 1966. Basically the same system is used in the single-line photodiode arrays which are now available on the market (27). The principal difference is that the digital scan generator is incorporated on the same silicon chip, and the total number of elements and the number of elements per unit length has been increased. Recently a photodiode sensor containing 1024 elements formed on a silicon chip more than an inch long has been announced (28). Its maximum video rate is 40 MHz and the sensor can be used to scan a 8.5 x 11 in. page in a fraction of a second. Applications for single-line sensors include pattern recognition equipment, in-process measurement or inspection of assembly lines, and mapping of the earth’s surface from a moving satellite (29). Motion of a single-line camera relative to the scene permits a two-dimensional picture to be reproduced. For high definition pictures a sequence of silicon chips has been joined to obtain several thousand photodiodes in a row. A critical requirement in the use of linear arrays for picture reproduction is that of uniformity of background from one element to the next. For example, any variation in dark current from adjacent photodiodes will produce streaks in the picture. Such a fixed pattern background can be subtracted out with subsequent signal processing circuits, if necessary, but this added complication would not ordinarily be justified.

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The presently available photodiode line sensors will soon be forced to compete with CCD line sensors. The characteristics in which they may have difficulty in matching CCD sensors are in resolution, sensitivity, signal-tonoise ratio, uniformity, and fixed pattern noise. The photodiode arrays currently available will operate at higher video frequencies than CCD sensors but this is probably only a temporary advantage. It is too early to predict a cost advantage for either. However, the photodiode sensor can be produced with standard silicon gate technology, which could offer a cost advantage for smaller arrays.

B. X Y

Addimsed Photodiode Sensors

1. MOS-Photodiodr Sensors

Photodiode area-type sensors were slow in being developed, probably because of the rather small picture signals which could be stored on the small elemental capacitors. A possible method for increasing the signal level is to include a voltage-sampling MOS current amplifier at each element (19). (See Fig. 6e.) This structure is too complex to be practical in high resolution arrays. However, 10 x 10 element photodiode arrays (19) with scan generators integrated on the same silicon chip were built in 1968. Larger photodiode arrays (50 x 50) having two MOS transistors at each element as shown in Fig. 6d were built in Japan (18). At the present writing, seif-scanned photodiode arrays in sizes up to 50 x 50 and 64 x 64 are available in solid state cameras. In each case the vertical and horizontal scan generators are included on the same chip with the array. The elemental circuit for the 50 x 50 array (28) is illustrated in Fig. 6g. Although a common video output bus is used which connects to all elements on the chip, the sensitivity is relatively high. An exposure of 2 x fc-sec of tungsten illumination (2870°K) is sufficient to cause saturation. At 400 frames per second dynamic ranges better than 100 to 1 have been measured. 2. ‘‘ Chal-ye-1njection”Sensor.s

In all of the experimental structures described so far the photoelectrons accumulated on the photodiode were returned to the substrate through one or more MOS switches, just as in the linear arrays. Concurrently with this work another type of photodiode array ( 1 3 ) was investigated in which the picture charge was removed by injection of the charge directly into the silicon substrate. As shown in Fig. 6h, each photodiode was coupled capacitively to the row and column address strips intersecting at this point.

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P. K. WEIMER

Figure12 illustrates the potential excursions of the photodiode during one frame period as the scan pulses are applied to the two address strips. It will be noted that both vertical and horizontal scan pulses have such a polarity as to forward-bias the diode, but that the diode is not actually discharged into the substrate until the coincidence of both pulses occurs at the instant of scanning. Between scans the photodiode remains reverse-biased so that it can collect minority carriers produced by the light. The accumulated charge is returned to the substrate by the coincidence of scan pulses. The video signal can be derived from the substrate or from the address strips.

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FIG.12. Operating characteristics for an early type of charge injection sensor [Weimer ef o/. ( 1 3 ) ] . Line (3) shows the potential variations of the photodiode during the chargeedischarge cycle. Line (6) shows an improved method of suppressing scanning transients used by Michon and Burke (12).

In the early work on charge-injection sensors two problems were noted: ( 1 ) The failure of the injected carriers (30) to recombine within the duration of the scan pulse could prevent the diode from being completely discharged in one scan. This caused signals to be carried over into subsequent frames giving lag. (2) The video signals tended to be masked by switching transients from the horizontal scan generator. A modified version of this sensor in which the diodes were replaced by a grounded base transistor was found (31) to be useful in reducing the switching transients in the output signal. Operation of arrays employing charge injection has recently been

IMAGE SENSORS FOR SOLID STATE CAMERAS

20 1

reported ( 1 2 . 1 2 ~ indicating ) that good performance can be achieved with this mode of operation. A 32 x 32 element photodiode sensor having a specially designed amplifier connected to the substrate was found to give excellent suppression of the capacitively-coupled switching transients. The signal to the substrate was first integrated to recover the initial scan pulses and then voltage sampled to obtain the video signal from the net injected charge. The transmitted pictures appeared to be entirely free of low frequency striations arising from variations in the scan pulses from either the horizontal or vertical scan generator. Line (6) of Fig. 12 illustrates the operation of the improved circuit. A significant difference in construction of the 32 x 32 element charge injection sensor from earlier sensors of this type is shown in Fig. 13. In the

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(b) FIG. 13. A comparison of two forms of charge injection sensors. (a) Photodiode-capacitor sensor elements [Weimer et ul. ( 1 3 ) ] .(b) MOS capacitor sensor elements [Michon and Burke (141.

lo1

earlier designs most of the minority carriers released by light absorbed in the depletion region of the silicon were collected on the diffused diode prior to injection into the substrate. In the recent charge injection sensors most of the carriers are collected under the gate electrodes at the silicon-silicon dioxide interface just as in a CCD sensor. The small diode connecting the two gate regions provides easy coupling between the two gate regions so that minority carriers can flow back and forth between adjacent gates without being injected into the substrate until both gates are driven into accumulation. The coupling diode could be omitted entirely provided the pair of gates at each element were sufficiently close spaced to permit charge transfer to occur. An improved 100 x 100 element charge injection sensor has recently been described ( 1 2 ~ )A. thin epitaxial layer of ti-type silicon on top of a p-type substrate has been found to give higher efficiency of injection of holes as compared to the homogeneous n-type substrates shown in Fig. 13. The time lag resulting from failure of injected carriers to recombine within the period of the scan pulse is thereby reduced.

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P. K. WEIMER

Although the derivation of video signal from the substrate permits a simplification of the scan generators and peripheral circuits the relatively large capacitance of the output to ground will degrade the signal-to-noise ratio at low light levels. Since the total capacitance increases directly as the number of elements this source of random noise will become increasingly objectionable for higher resolution sensors. A substantial reduction in output capacitance of an XY array can be obtained by taking the signal from each row or column separately, using an external multiplexer as shown in Fig. 7a or 7b.

VI. PRINCIPLES OF SCANNING BY CHARGE TRANSFER A . Single-Line Churge- Transfer. Sensors

An entirely new system of solid state scanning has resulted from the development of integrated charge-transfer registers. The construction of an MOS bucket brigade (32) and a charge-coupled register (5) is illustrated in

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(C)

FIG. 14. Structural cornparisoil of a bucket brigade MOS photodiode sensor and ii three-phase charge-coupled sensor. (a) Circuit for MOS bucket brigade line sensor. (b) MOS bucket brigade line sensor cross section. (c) Charge-coupled line sensor cross section.

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203

Fig.14. These devices differ from the earlier digital shift registers in that an utzalog signal introduced at the input will be transferred through the register as a spatial sequence of modulated charge packets. The application of clock voltages to the overlying electrodes causes all charge packets to be transferred along the surface of the semiconductor from one location to the next with relatively small losses per transfer. Although the clock waveforms may have two, three, or four phases, at any instant all packets will be located under the same phase electrode of each group. The present discussion will examine the various ways that charge-transfer registers can be used for scanning of arrays. The detailed operation of CCDs and bucket brigades will be considered later. Clearly, a spatially-varying charge pattern which has been introduced serially into a charge-transfer register can be reconverted into a time-varying (although delayed) signal by simply collecting the charges emerging from the end of the register. Similarly. if the spatial charge pattern is introduced optically to all elements of the register, operation of the register will yield a video signal equivalent to that obtained earlier by multiplexed scanning. The charge-transfer method of scanning offers two potential advantages over multiplexed scanning: 1. Signal-to-noise ratio. The input capacitance of an output amplifier placed on the same chip with the register need be no larger than the capacitance of a single element in the register. The signal-to-noise degradation previously introduced by the large capacitance of the long video bus of the multiplexer is eliminated. The signal-to-noise ratio of a scanned device can now approach that of a single element provided the noise introduced by transfer can be kept small. 2. Uniformity. Since the single discharge switch at the end of the register serves for the entire array the variations previously introduced by multiple switches and nonidentical scan pulses are eliminated. Successive elements are discharged by the recurring pulses from the clock drive, thus removing the need for digital scan generators. In general, all clock signals have at least twice the frequency of the video passband so that they can be readily removed from the video signal by low pass filters. Obviously, these advantages cannot be realized in practical devices unless the transfer losses and the noise introduced by transfer is very low. In the construction of charge-transfer sensors two methods may be used for introducing the photogenerated charge pattern into the register: 1. The register may be illuminated directly while the clocks are stopped for a light integration period (Figs. 14 and 15a). Minority carriers released within the silicon by the light will then be collected at the surface under those electrodes whose voltage forms the deepest potential well. The video

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P. K. WEIMER

signal will be generated at the output electrode during the scanning period when the clocks are started again. 2. Separate photocells are provided for accumulation of the photocarriers during the integration period (Fig. 15b). The registers are shielded from the light so that they need not be stopped except momentarily when the charges are transferred from the photocells. The use of two parallel registers, as shown in Fig. 15b, reduces the total number of transfers and allows the sensor elements to be more closely spaced.

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The illuminated-register sensor, although simple to build, imposes certain limitations on the relative proportion of the time allotted for scanning and for light integration. Unless the optical image is masked off during the scan period the light falling on the register during the scanning period will superimpose a smeared signal over the charge pattern which had developed during the integration period. Although some types of pictures can be reproduced fairly well with a scanning period about equal to the integration time, the scanning period normally should be no more than a few percent of the total time to avoid significant degradation of the image. Sensors with separate photocells offer an additional advantage over illuminated-register sensors in that hybrid technologies can be used for the photodetectors and the scanning circuits. This may be of particular importance for infrared pickup at wavelengths beyond that normally suitable for silicon (33).

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205

B. Two-Dimensional Charge- Transfer Sensors 1. Sensors with Internal Registers for Each Row or Coluinn

The simplicity of charge-transfer registers of the bucket brigade or CCD type has made it feasible to build two-dimensional sensors using a register for each row or each column of the sensor. With a suitable scanning organization it is possible to transfer the charge from every element to a lowcapacitance output amplifier located on the same chip without the charge having to pass through any stage whose capacitance is larger than that of a picture element. Figure 16 shows schematically two scanning systems which

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FIG. 16. Two systems of scanning two-dimensional charge-transfer sensors having illuminated registers. (a) Horizontal transfer along successive rows into a continuously running output register. (b) Vertical frame transfer along columns in parallel through a temporary store into the output register.

have been used for arrays in which the registers in the image area are illuminated directly by the light from the scene. In the horizontal transfer system (20),shown in 16a, the clock voltages for each row are cut off most of the time to allow the charge pattern to accumulate for approximately 1/60 sec. When a given line is to be scanned its gates are activated by the horizontal clocks causing all charges to be transferred into the output register. The output register, which operates continuously at the horizontal clock

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P. K. WEIMER

rate, transfers the charges to the output amplifier located in the upper right-hand corner of the device. A vertical scan generator which can be either a digital or analog shift register provides the necessary pulses for gating the horizontal clock supply to the proper line. The horizontal transfer system was demonstrated with the 32 x 44 element bucket brigade sensor described in Section VII. The same system could be used with CCD registers and with sensors having interleaved photocells. Since only one line time is used for moving the charge pattern out of the illuminated area the resulting image smear is very small. The vertical frame-transfer sensor (6) shown in Fig. 16b requires a separate silicon area for frame storage when operated with continuous illumination of the sensor area. The clock voltages in the sensor area are stopped during the normal field-scanning period to allow charge integration to occur. The resulting charge pattern is then transferred in parallel to the storage area during the vertical retrace period. The stored charge is then scanned during the following field period by transferring the charges in parallel a line at a time into the horizontal output register. Simultaneously during this field a new picture is being integrated in the sensor area. No scan generators or clock supply gates are required within the array, but the clock voltages applied to the storage area need to operate at two different rates. The horizontal output register runs continuously except during the horizontal retrace period when a new row of charges is transferred in parallel from the vertical registers to the output register. Although the vertical frame transfer format requires additional space on the silicon chip for the storage area, a compensating feature is that the charge in the sensor area can be integrated under different electrodes in successive fields (34) to provide an increase in effective vertical resolution. When one set of sensor gates is held on during the integration period while the other gates are off, the charges collect under the “ o n ” gate. By keeping different gates on during successive fields the effective center of the picture element is displaced, giving increased resolution when successive fields are viewed with interlaced scan. The resolution capability of a single field is unaffected by this process since it can be no greater than the total number of complete stages in the sensor register. Both the horizontal and vertical transfer systems could be built with two, three, or four-phase CCD registers, but the vertical system is somewhat easier topologically to lay out when multiphase clock registers are required. The frame transfer system was used in the three-phase 128 x 106 (interlaced) C C D array built by Bell Laboratories. (See Section IX,B.) The vertical and horizontal transfer systems can also be built with interleaved photocells and nonilluminated registers. The use of separate photocells is particularly advantageous with the vertical transfer system ( 3 5 )

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207

because the separate storage area can be eliminated as shown in Fig. 17a. Integration of charge by light incident on the photoelements occurs simultaneously with the readout of the charges from the unilluminated register. The newly developed charge pattern from the photocells is then transferred to the empty registers during the vertical retrace period. As before, the vertical registers shift the entire pattern down one line during each horizontal retrace period, at which time the horizontal output register is loaded with the line to be scanned.

FIG. 17. Two systems of scanning two-dimensional sensor arrays with nonilluminated charge-transfer registers. (a) Shielded registers interleaved with sensor elements. (b) External charge-transfer registers connected to XY address strips.

The use of nonilluminated registers in the vertical transfer format permits the total number of transfers and the total number of clock supplies to be reduced to the same as for the horizontal system of Fig. 16a. An additional feature of the vertical transfer system with interleaved photocells is that an input register can be used for introduction of bias charge and for recycling of signals back into the array for signal processing within the sensor itself (36).Although an input register can be included in the vertical frame transfer system of Fig. 16b, it cannot be used during the normal scanning period since the sensor clocks are stopped at this time.

2. Charge-Transfer Readout o f X Y Sensors In applications not requiring the lowest possible scanning noise a hybrid type of sensor, shown in Fig. 17b, could be useful. This system (20) combines the simplicity and versatility of an XY array with the improved uniformity and noise reduction of charge-transfer scanning in the horizontal direction. As described in Section VII,B, this method of scanning has been tested with

208

P. K. WEIMER

a silicon MOS-photodiode array but it would also be advantageous when nonsilicon detectors are to be used with silicon scanning. The construction of an array with infrared detectors may be simpler in XY form than with interleaved silicon registers. The noise level in the signal from an XY array cannot be less than the fluctuation noise associated with the capacitance of each vertical address strip in the array. Reference to Eq. (2) in the discussion of CCD noise sources in Section VII1,C indicates that a single strip capacitance of 10 pF would yield a noise fluctuation of approximately 1200 electrons. The resulting threshold sensitivity of a silicon XY sensor would be comparable to that of a silicon vidicon which is considered a relatively sensitive tube. In applications where the shot noise in the signal exceeds 1200 electrons this mode of scanning would be comparable in performance to a charge-coupled sensor with internal registers. For very low signal levels, however, where the shot noise in the signal is very much less than this, a CCD sensor having a much lower noise background would be superior.

VII. CHARGE-TRANSFER SENSORS EMPLOYING BUCKETBRIGADE REGISTERS The bucket brigade analog delay line described by Sangster and Teer ( 4 ) in 1969, and by Sangster (32) in 1970, provided a basis for building experimental charge-transfer sensors. A 16 x 16 element bucket brigade sensor (37,38)was built in 1970, and this was followed by a 32 x 44 element sensor (20) which was incorporated into a miniature camera (39). Although the bucket brigade register may be replaced by charge-coupled devices (40,41) for image sensors having internal registers, it offers certain features which may be useful in future devices. A structural comparison of the integrated MOS bucket brigade with the three-phase charge-coupled device was shown in Fig. 14. The principal difference was the presence of the diffused islands which permitted the bucket brigade to be constructed as a series of MOS transistors with all gates slightly offset in the direction of charge transfer. The fact that the bucket brigade device could be represented by such a simple circuit tended to obscure its significant advance over earlier forms of analog shift registers ( 4 2 ) .The MOS transistors do not function as simple switches connecting the capacitors but as active source-followers which cut themselves off when the charge packet has been transferred to the next capacitor. The loss of charge per transfer (43)can be relatively small ( - 10- per transfer) but not usually as small as can be achieved with CCD’s. The diffused islands tend to limit ultimate packing density in sensor arrays employing internal registers but are advantageous in applications where larger dimensions are required.

IMAGE SENSORS FOR SOLID STATE CAMERAS

209 0 OUT

FIG 18 Circuit didgram for a 32 x 44 element bucket brigade sensor The photodiodes formed by sources and drains of transistois are not shown in the diagram

A . A 32 x 44 Element Bucket Brigade Camera Figures 18 and 19 show the circuit and a photomicrograph of a 32 x 44 element bucket brigade sensor. The scanning organization for the sensor was the horizontal line-by-line system illustrated in Fig. 16a of the preceding VERTICAL SCAN GENERATOR -TRANSMISSION GATES OUTPUT AMPLI FIER-,

,'

4

OUTPUT REGISTER'

X 44 ELEMENT BUCKET BRIGADE PHOTOSENSITIYE ARRAY FIG. 19. Photomicrograph of the 32 x 44 element bucket brigade sensor. The elements

were spaced on 3 mil (0.0076cm) centers.

2 10

P. K. WEIMER

section. Each row consisted of a 44-stage bucket brigade register having 88 diffused sources and drains which also served as photodiodes. Alternate diodes were connected capacitively to the overlying metal gates which were activated by the horizontal clock voltages when a given line was to be scanned. During most of the 1/60 sec interval between scans the clock drive for each line was stopped to allow a picture charge pattern to accumulate on the photodiodes. When the charges on a given line were to be transferred to the output register the horizontal clock drive for that line was gated on by the vertical scan generator located on the left side of the drawing. The continuously running output register transported the charge up the right side of the array to the MOS output amplifier located on the same chip. The signal-to-noise ratio could be extremely good because the capacitance at the input to the amplifier was not appreciably larger than that of an individual picture element. The increasing delay introduced by the output register for the lines nearer the bottom of the array was compensated by gradually advancing the phase of the vertical scan pulse which controlled the gating of the horizontal clocks to the rows. For this purpose the vertical clock frequency in the camera was made slightly larger than the normal line frequency of the output signal. The problem of image smear expected from continued illumination of the internal registers during transfer was not visible in uniformly illuminated scenes. Such smearing should be even less significant with sensors having a full 500 lines since the ratio of smeared signal to integrated picture signal would be decreased to 1 part in 500. A lighting situation under which the illuminated registers could yield objectionable streaks occurred when local areas were illuminated at a light level which would overfill the picture element capacitors by one or more orders of magnitude. The excess charges at the illuminated element would then spill over into adjacent elements along the same row giving a bright streak instead of an enlarged round spot as in the silicon vidicon. The picture uniformity was excellent compared to early X Y sensors in spite of a sprinkling of dark current spots such as noted in early silicon vidicons. One feature of the bucket brigade sensor which would be useful for other scanning applications is the bucket brigade scan generator used for vertical scanning. An MOS bucket brigade circuit operated in the manner shown in Fig.20a will serve as a digital scan generator (38) having only one transistor per stage. A typical MOS shift register as shown in Fig. 5 would require six transistors for the same function. Operation in the scan generator mode requires a constant high level input current interrupted only by the “start pulse. The spurious background pulses are of no consequence for vertical scanning but would require some smoothing if used to drive a horizontal output multiplexer. An alternative scan generator mode is shown in Fig. 20b. ”

21 1

IMAGE SENSORS FOR SOLID STATE CAMERAS BUCKET BRIGADE SCAN GENERATOR

I

5 psldiv DOUBLE CLOCK

1

I

PMOS

5ps/div SINGLE CLOCK

FIG.20. Measured voltage waveforms from a P-MOS bucket brigade delay line operated in two modes as a digital scan generator.

B. Experiinental Tests of Cliaiye- Transfer- Readout of at? MOS-Photodiode Array

Bucket brigade registers are more versatile than CCD registers in applications requiring large center-to-center spacing of successive stages. In charge-transfer readout of XY arrays each vertical bus of the array can serve as one stage of a short register coupling the individual elements to the horizontal output register. Figure 21 shows a circuit for scanning an MOSphotodiode array by charge transfer, using the system outlined in Fig. 17b. This circuit was tested with an 8 x 8 element array constructed entirely of discrete components (20, 44). The accumulated charges for all elements along a given row were transferred in parallel via the columns to the row of

2 12

P. K. WEIMER

-V VIDEO OUT

FIG. 21. Circuit used MOS-photodiode array.

for

demonstrating

charge-transfer

readout

of

an

XY

peripheral capacitors C, . Although the high capacitance of the column buses will slow this transfer, a whole line time is available for completing the transfer. If desired, the transconductance of the transistors T, can be made extra large to minimize any carryover of signal into the next line. A transfer loss as large as lo-’ would be tolerable for the vertical transfer stages since so few stages are involved. It is noted that conservation of charge requires that the signal voltage swing on the high capacitance columns will always be less than on the elements and on the C, capacitors. A bucket brigade stage of the “tetrode” type ( 4 5 ) would be particularly suitable in coping with the capacitance of the columns. The charges held temporarily on the CTcapacitors are transferred to the output register during the horizontal retrace period. The output register could be a bucket brigade as shown, or it could be a CCD register. The single-gate MOS sensor illustrated in Fig. 21 is relatively simple to construct with or without diffused photodiodes at each element. However, the above method of parallel readout of an entire row in one line time could be applied with advantage to other types of XY arrays such as charge injection sensors (12). In the latter case this mode of operation would extend the time for recombination of injected carriers from one element time to one line time.

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213

VIII. CHARACTERISTICS OF CHARGE-COUPLED DEVICES (CCDs) A . General Description of CCD's

The original proposal of Boyle and Smith ( 5 ) was to store minority carriers in potential wells at the surface of a semiconductor and to transport these charges along the surface by moving the potential wells. In its simplest form a charge-coupled register can be made by depositing a series of closely spaced metal electrodes on top of a thin oxide layer ( - 1500 8, thick) covering a uniformly doped semiconductor. In operation, the metal electrodes are driven by clock voltages but are reverse biased so that the entire surface region of the semiconductor remains depleted of majority carriers. Although the applied voltage is large enough to invert the surface the insulating properties of the depletion region are sufficiently high that several seconds would be required for thermally generated minority carriers reaching the surface to reestablish equilibrium. In CCD operation, minority carriers are introduced into the semiconductor surface by electrical or optical means and transferred along the surface as an analog signal. A condition of quasi-stable equilibrium is established at the surface so that any contribution of thermally generated minority carriers from the body of the semiconductor is constantly swept toward the output electrode along with the signal carriers. The charge-transfer process can be readily explained in terms of the three-phase register illustrated in Fig. 22. The semiconductor in this case is

(CI

FIG.22. Operation of a three-phase charge-coupled shift register. (a) Cross section of the structure. (b) Surface potential profile for d 1 = - V , d, = 0, and d3 = 0, forming a potential well under the phase-1 electrode. (c) Transfer of charge from phase-1 electrode to phase-2 electrode illustrated at times shown in (d). (d) Voltage waveforms of the applied clock voltages.

214

P. I(. WElMER

n-type and the minority carriers to be moved along the surface are holes. The clock voltages shown at the right of the figure are applied to each of the three sets of electrodes. At the time t = t , all holes are initially located in the potential energy wells under the +1 electrodes which at this moment are at the most negative voltage relative to the grounded substrate. The crosshatched region represents the holes whose presence increases the surface potential energy and causes the well to become partially filled like a fluid in a container. At the time t , the 6, electrodes have also reached the maximum negative voltage creating a second well into which the holes under 41 begin to move. The transfer of holes to 42is aided at time t , by the rising voltage on which removes the well under 41. The barrier under electrode d 3 , which is the least negative electrode at this time, prevents any charges from moving back to the left. The transfer is complete at time t, and the original charge is now located entirely under the 4, electrode. Continued application of the clock voltages causes all charge packets to be transferred toward the right with relatively small losses from their original values. The achievement of sufficiently low transfer losses is a major problem which will be discussed in the next section. Charge-coupled registers are constructed so that the carriers are confined to a channel whose width is usually of the order of 10-100 pm. The confining barriers along the sides of the channel can be produced by any of the following methods: (1) Strips of thick field oxide ( - 12 pm thick) between the gate and substrate. (2) Diffused PI+ or p + channel stops of the same type as the substrate. (3) Conducting shields with a fixed bias located between the gate electrodes and the substrate, but insulated from the substrate. Efficient transfer of charge from one well to the next can occur in the three-phase register only if the electrodes are very closely spaced (approximately two microns or less). Too large a spacing permits a barrier to form between electrodes, and the exposed oxide is subject to surface charges which may cause instabilities. Another form of register shown in Fig. 23 employs overlapped aluminum and polycrystalline silicon gates (46,47.470). This structure can be operated with two or four phases. The overlapped construction protects the gaps from ambient effects and the close spacing provides a high uniform field between electrodes. An advantage of this structure is that it can be fabricated with small cell dimensions using standard silicon gate technology without requiring as precise control of the metal spacing as in the earlier three-phase metal gate structures. A two-phase CCD register would be preferred over three-phase structures because of the reduced number of drive voltages and the possibility of having simpler construction. The latter advantage is not always achieved in practice. In order to provide directionality of charge flow in a two-phase

IMAGE SENSORS FOR SOLID STATE CAMERAS

215

s- t

D- I INUM

7 CHANNEL WIDTH = 5 5

m11S

,I,

I

"5'

FIG.23. Cross-sectional view and labeled photograph of a 128-stage two-phase silicon gate CCD.

structure an asymmetry must be incorporated into each potential well which may actually increase the complexity of fabrication. Barriers to prevent back-flow can be obtained in one of the following ways: (1) The use of a thicker channel oxide under the rear portion of each gate (48). (2) The use of two gates for each phase with a dc offset voltage under the rear gate (46). (3) The use of an ion-implanted barrier in the silicon under the back part of each gate (49). (4) The use of two levels of fixed charge in the channel oxide over each gate (SO). The silicon gate structure shown in Fig. 23 can be operated with twophase clocks in either of two modes: (1) The silicon gate and aluminum gate for each phase can be tied directly together while depending on the thicker oxide under the aluminum to provide the barrier to prevent back-flow of charge. This mode of operation is more suitable for devices made on low resistivity substrates. (2) Adjacent silicon and aluminum gates can be driven with the same two-phase clocks but with an offset voltage applied to the aluminum gates to enhance their barrier action. Alternatively, all four gates can be driven independently with four-phase clocks.

216

P. K. WEIMER

A disadvantage of the silicon gate technology for large area devices is that the lateral resistance of the polycrystalline silicon gates (- 50 I2 per square) may lead to an RC delay of clock voltages along the register (47). Another form of sealed-channel CCD has been described consisting of four overlapping aluminum clock lines (51) which are separated by anodized aluminum. The high conductivity of the aluminum should allow better operation at high frequencies, but the opacity of the aluminum gates would be objectionable in image sensor applications if the register were to be illuminated from the gate side. Three-phase CCD sensor arrays with overlapping silicon gates have been described recently ( 5 1 ~ ) . Continuous operation of a CCD register requires means for introducing and removing the minority carriers which transport the signal. In image sensors the initial charge pattern can be provided by direct illumination of the depletion region during a period when the clocks are stopped. In delay lines the carriers are introduced electrically by means of a reverse-biased input diode such as S-1 as shown in Fig. 23. In either case the carriers are removed by means of a second more strongly biased diode D-1 which acts as a drain. The entire C C D register thus resembles an MOS transistor with a multiplicity of gates between source and drain. The added gates near the source and drain assist in controlling the input and output signals. The auxiliary output transistor (S-2, D-2) located on the same silicon chip has its gate connected to a diffused diode touching the channel region near the end of the register. This diode fluctuates in potential in proportion to the size of the charge packet passing along the channel. The signal amplification produced by this voltage-sampling transistor permits smaller charges to be detected with less interference from clock transients and output circuit noise than would be obtained from the drain D-1.

B. Trunsjer Losses in C C D s The most serious losses in CCD’s are caused by failure to transfer the entire signal packet from one stage to the next during a single clock cycle. A short signal pulse would accordingly be attenuated and shifted in phase as the residual charges are transferred during later clock cycles. A longer input pulse could reach full amplitude but with poor frequency response and excessive delay. In image sensors where thousands of transfers must occur at 10 MHz clock if the fractional loss per transfer should not exceed normal TV resolution is to be attained. Surface channel devices of the type discussed so far have difficulty in achieving such low losses at this clock frequency. Buried-channel C C D s which are capable of lower losses at high frequencies will be discussed separately below.

-

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217

1. Surjuce-Channel CCD’s

Carriers can fail to transfer to the next electrode within the proper clock cycle for two reasons: they may become trapped in fast interface states, or they may be limited by the dynamics of free charge motion. Three mechanisms control the free charge transfer (52):self-induced drift, thermal diffusion, and fringing field drift. Self-induced drift caused by mutual repulsion of charge is effective only for larger charge densities at the beginning of transfer and has little effect on the final transfer efficiency obtained. Thermal diffusion (53) results in an exponential decay of charge under the electrode with a time constant 12

where L is the center-to-center electrode spacing in microns and D is the diffusion constant. Equation (1) indicates the importance of keeping the electrode dimensions small and the diffusion constant and mobility high for operation at high clock rates. The transfer process can be speeded up by the fringing field ( 5 2 )between electrodes which has a major component in the direction of charge propagation. Its magnitude at the silicon-silicon dioxide interface increases with increasing oxide thickness and gate voltage, and decreases with increasing gate length and doping density. Carnes and Kosonocky (54)have calculated that a p-channel CCD should have losses no more than lop4 at a clock frequency of 10 MHz with a gate length of 7 pm and a substrate doping of 10’ cm- This assumes negligible trapping of carriers. In surface-channel devices the trapping of charge carriers in fast interface states ( 5 5 ) in the channel can cause additional transfer losses. These states can fill very rapidly at a rate determined by the number of free carriers, but their rate of emptying depends only on the energy level of the trapping state. Charges not released within the same clock cycle in which they were captured will be released later resulting in transfer loss and poor frequency response. It has been found that interface state losses can be minimized by introducing a constant background signal or “fat-zero” into the register. The effect of the background charge is to keep the slowest states filled (i.e. those states farthest from the band edge) so that they do not have to be filled and emptied by the signal charge. The background charge can be introduced electrically at the input diode of the register or optically by means of a uniform bias light in the case of an image sensor. Although the background signal causes some inconvenience and reduces the useful dynamic range of the register, it usually does not need to be larger than 10-30% of the full well

’.

218

P. K. WEIMER

16‘

I

I

I

I

0

,&tA //’

/

s

12-

//

I

0

OZ n. Y v)

v)

0 A

-3< 10 =

P

t;

!+to to = 0.85 nsec -

/

Tr I

U e

-

LL

16‘

bL pe - 7t I o

I

105103

I 10‘

I

I 105

106

r = 64nsec

CCD6-5-8

I 10’

108

capacity. Figure 24 shows the measured loss per transfer (47)as a function of clock frequency for a 64-stage two-phase silicon gate register of the type shown in Fig. 23. This register was fabricated on a 1.0 0-cm n-type substrate having (100) orientation. Branch A of the curve was measured without “fatzero current and provided data from which the fast interface state density was estimated. [Nss = 2.9 x 10” (cm’ - eV)-l.] Branch B of the curve showed the reduction in transfer loss observed when a 50% “fat-zero” current was added. The dotted curve on the right represents the calculated transfer loss for free charge transfer for 0.4 mil long gates assuming that self-induced drift dominates transfer for the first 99% of the charge with a characteristic time to = 0.85 nsec, and thermal diffusion dominates thereafter with a time constant of 64 nsec appropriate for L = 0.4 mil. The fringe field has little effect on transfer losses in this case because of the low resistivity of the substrate. Although the background charge is effective in reducing the effect of trapping in the regions directly under the transfer electrodes, a full charge packet spreads out further under the edges of the electrodes (56).The interface states over this additional area can also trap signal charge which will be reemitted when the charge moves on. The edge effect thus contributes to transfer losses regardless of background charge. A more sophisticated ”

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219

method of avoiding trapping in interface states is to replace the surface channel by a buried channel as described in the next section. The above discussion on transfer losses has not included the gain or loss of minority carriers from the channel to the substrate which might arise from dark current leakage in the depletion layer or from charge pumping” caused by improper clock voltages. The effect of dark current will be considered in the discussion of fixed pattern noise in C C D sensors in Section VIII,C,2. “

2. Buried-Channel CCDs Experience with the surface-channel CCD’s described above has shown that transfer losses caused by charge trapping in interface states can not be completely eliminated by the introduction of background “ fat-zero current. As will be discussed in the next section the presence of such states will also introduce noise into the transfer process as well as transfer loss. A modified CCD structure has been proposed in which the charges do not flow along the surface but are confined to a channel which lies beneath the surface (57). The number of traps in the bulk silicon are so much less than at the surface that a background current should be unnecessary to maintain low transfer losses. More efficient transfer at high frequencies can be obtained with buried channels because the fringing fields can be made higher and the carriers are at a greater distance from the electrodes. The larger mobility in the bulk silicon will also enhance the speed. The construction of the buried-channel CCD differs from an equivalent surface-channel device in that a thin layer of silicon having opposite conductivity type to that of the bulk is formed immediately under the insulator layer. This layer can be produced by ion implanation at the surface of a homogeneous silicon substrate or by use of an epitaxial surface layer or both. Figure 25 compares the energy level diagrams for a buried n-channel CCD with a surface n-channel device. The n-type surface layer is biased sufficiently positively with respect to the p-type substrate (by means of the n + output diode) that all electrons associated with the n doping are swept out. The clock gate electrodes are then biased so that a potential minimum is formed within the n-type layer rather than at the semiconductor surface. The charges are transferred along the buried channel by application of two, three, or four-phase clock voltages to the gate electrodes. However, the detailed effects of the gaps between electrodes and the barriers introduced by stepped oxide gates are somewhat different in the buried channel device. If too large a charge is introduced into a buried-channel register the channel becomes filled to the point that the carriers can interact with the traps at the surface, degrading the performance (58). For a larger storage ”

220

P. K. WEIMER

I

I~NS

SURFACE-CHANNEL

[ELECTRONS

------i P-TYPE SILICON

CTRONS

!I%

P-TYPE SILICON

(bl FIG. 25. Energy level diagrams [or two types of CCD registers. (a) Surface-channel CCD. (b) Buried-channel CCD.

capability it would be desirable to have a thin gate oxide and a buried channel which lies close to the surface. On the other hand, a deep channel is preferred for higher speed operation. Esser ( 5 9 ) has shown that a graded doping concentration of the surface layer, with highest concentration at the surface, will permit a large storage capacitance with high speed operation. In this case the main part of the charge packet which is closer to the surface is transferred by self-induced fields while the last fraction which determines the effective loss is driven further into the layer where much higher drift fields exist. per transfer at more than 100 MHz Transfer losses as low as 7 x have been reported for the so-called “peristaltic form of buried channel CCD described by Esser (59). In this device the surface layer consisted of a homogeneously doped epitaxial n-type layer of 4.5 pm thickness. Background “fat-zero’’ current had no influence on the losses. ”

C. Noise Clzaructeristics of’CCD’s I. Stntisticul Noise In applications such as image sensors, the noise properties of the chargecoupled registers are crucial in determining device performance. Fortunately, the noise introduced by charge transfer is small enough that many

TABLE I NOISE SOURCES I N

Category Electrical input circuit

Source of noise

CCD’S“

N

(derived)

Charge fluctiations set into first well

N,,= 400(C,,)1’2

N, = 40 for C,,

Transfer losses

N,,= [2zN,(N,

Trapping losses

m,,= [1.4kTN,, N,A,]“Z

= 0.01 (electrical “fat -zero ”)

Charge-coupled register Transfer processes

Storage processes

Output circuit

+ NrZ)]l’Z

External amplifier

m,,> 200 for i.N, = 0.2 N , = 950 for 2000 transfers (N,, = 10’o/cm2/eV)

aw

N, = 56 for N ,

= 3120 where I , = 10 nA/cmZ

Shot noise in thermally generated dark current Shot noise in op tically-generated background signal

N (typical values)

N, = NkL. where N,,

N, = 316 for N,, = l o 5 =

1/10 full well

(optical ‘‘fat-zero ”)

m, = 1300 for C,, N, = 130 for C,,

=

10 p F pF

= 0.1

C,, = capacitance of each potential well in picofarads; E = fractional transfer loss per gate; N , = number of gates or transfers; N , = number of signal carriers per charge packet; NFz= number of background carriers (“fat-zero”) per charge packet; A , = area of a single gate: p = number of phases; Nd = number of carriers per packet from thermally generated dark current. The values of N, given here apply for one scanning period ( T = 1/60 sec).

N

L 2

222

P. K. WEIMER

transfers can be tolerated. Very small signal levels are feasible provided the input and output circuits are carefully designed. In the following discussion, the various CCD noise sources will be considered along the lines outlined by Carnes and Kosonocky (60),but quantitative conclusions are tentative, subject to revision as more detailed studies become available. Experimental results reported to date (96) are in fairly good agreement with the predictions. As shown in Table I, noise generated in a CCD register can arise in three separate areas: (1) the input circuit, (2) the register itself, and (3) the output circuit. It is convenient to represent the signal in terms of the total number of carriers (N,) per charge packet and the magnitude of each noise source N, as the equivalent rms fluctuation in N , . For an example, a typical full well can hold lo6 carriers and the shot noise associated with this signal would be N,”’ or lo3.The other noise components can be added as the square root of the sum of the squares to obtain the resultant signal-to-noise ratio in the output circuit. An MOS-gated input circuit such as illustrated in Fig. 23 can be used for introducing a “ fat-zero” background or an electrical signal to be delayed. The inherent uncertainty in setting the charge into the first well is shown in Carnes and Kosonocky (60) to be given by

-

where C,, is the capacitance of the well in picofarads. As shown in Section XIII, the input circuit must be operated properly to achieve a background noise level as low as that given by relation (2). Unless such precautions are taken the background noise introduced electrically could equal or exceed the shot noise value = N;L2. The noise sources within the register can be separated into two groups: transfer process noise and storage process noise. Thornber and Tompsett (61) have shown that the correlation resulting when charges lost by one packet are gained by the succeeding packet results in a suppression of the low frequency portion of noise generated during the transfer process. The frequency spectrum for transfer noise is thus shifted toward higher frequen. cies regardless of whether losses are incurred by trapping or by free charge dynamics. As shown by Table I, interface trapping (55) represents the major source of transfer noise for a surface channel device. However, the suppressed low frequency content of this noise may tend to reduce its visibility in television applications.

m,

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223

No such correlation effect occurs in the storage process noise introduced into the register by dark current leakage or by illumination. The shot noise associated with these charges will have a flat energy spectrum over the useful bandwidth of the device from zero frequency up to the Nyquist limit,f,/2. The equivalent rms fluctuation in carriers resulting from dark current leakage in the register is given by

where 1, is the dark current leakage per unit area ( - 10 nA/cm2), A , the area of each gate ( - lop6cm), p the number of phases (3), and T is the total integration time of a charge packet (1/60 sec). For a simple delay line, T represents the total transit time of a charge packet through the register, but in image sensor applications, where the clocks are stopped to allow photocarriers to be accumulated at each picture element, T comprises the entire frame time. Assuming a uniform dark current of 10 nA/cm2 and other conditions as stated following Eq. (3), the total charge accumulated in each packet in 1/60 sec is N , = 3125 electrons with a shot noise of I?, = 56. (For line storage operation with T = 63.5 p e c , N , and N, would be 9 and 3 electrons, respectively.) The dark current leakage is obviously a function of temperature and the semiconductor processing. When the register is illuminated the optically generated signal N , will also be accumulated during the period T and have the shot noise N,”’ associated with it. In addition to their rms noise components both N , and N , may contain even larger “fixednoise” variations which may dominate the performance of the device (see next section). Two methods commonly used for extracting the signal from a chargecoupled register are: (1) connect the input of an external amplifier directly to the output diode, or (2) connect the external amplifier to an on-chip MOS transistor whose gate is driven by a floating diffusion located near the end of the register as in Fig. 23. In each of the above modes the potential of the output diode or floating diffusion will be reset once each clock period. Thus, no resistor need be connected to the diode, and the output noise will be determined by the error in resetting the floating diode. The reset noise is given by the same expression used for calculating the input noise N, = 400 C;i2. However, the capacitance is very much greater in case (1) above when the capacitance of the lead connecting to an external amplifier must be reset. The resulting value of N, = 1300 is comparable to the amplifier noise limitation in a conventional vidicon camera. In case (2) the

224

P. K. WEIMER

much smaller capacitance of the on-chip MOS gate connected to the floating diffusion yields an equivalent noise which is 0.1 that of a vidicon amplifier. Output circuits having still lower equivalent noise are currently being investigated (62). The relative importance of the various noise sources is determined by the level of signal which is to be transferred. As long as the rms sum of all noise sources is small compared to the shot noise associated with the signal these sources are obviously of small consequence. As will be seen later this situation is actually true for CCD sensors operating under rather dim illumination such as a living room at night. However, under lower light levels such as a landscape illuminated by a full moon the factors listed above could limit the performance. For operation at still lower illumination such as by starlight all of the noise sources listed will have to be reduced. (See Section X on low-light level performance of CCD’s.) The use of buried-channel registers is expected to provide a major reduction in interface state noise and in transfer noise. The dark current noise may be the most difficult to suppress, since cooling of the silicon will be required. Even more serious than the dark current noise is the fixed pattern signal associated with the spatial variations in dark current, discussed in the next section.

2. Fixed Pattern Noise Dark current which varies from one element to the next can produce a nonuniform background of charge whose mean deviation from the average far exceeds the rms noise fluctuations in dark current at each element. This factor can have a major effect in determining the useful sensitivity of an image sensor. It can also affect the operation of series-parallel-series delay lines (see Section XII) where the charge packets are held in parallel registers for relatively long periods. Whenever the signal is passed through parallel channels and then recombined, repetitive patterns will appear in the output signal if the parallel channels are not identical. Such fixed pattern noise is not a problem in a simple serial delay line since every charge packet is equally exposed to the same series of wells. Other types of fixed pattern noise in image sensors can arise from variations in sensitivity from one element to the next and from geometric irregularities in clock lines or in channel stop diffusions. The pickup of clock voltages in the output signal can also mask the useful signal. Fortunately, the clock supply has twice the frequency of the upper limit of the signal passband so that the signals can be separated by filtering. Low frequency interference within the video passband is less likely to occur than with the early forms of scanning by means of digital multiplexers.

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225

IX. EXPERIMENTAL CHARGECOUPLED IMAGESENSORS A . Single-Line CCD Sensors

1. Line Sensors with Illuminuted Registers The operation of a CCD register as a line sensor was first reported by Tompsett et al. (63) of Bell Telephone Laboratories in August of 1970. A simple optical image was projected on an eight-bit three-phase register while the clock voltages were stopped so that a charge pattern could form. The clocks were then started again and the accumulated pattern was transferred along the register to an output diode where the charge packets could be collected in sequence. The output signals were used to produce a twodimensional picture by mechanically moving the optical pattern across the row of sensors for successive lines. Since 1970, single-line illuminated-register sensors having up to 500 elements have been produced in various laboratories. These have been made with both three-phase and two-phase registers, and they normally include an input diode for introduction of background current or electrical signals to be delayed. (See,for example, Fig. 23.) Most single-line silicon CCD registers can be operated as a line sensor provided the clock voltages are stopped for a suitable integration period to allow a charge pattern to form. The two-phase silicon gate registers were also operated as line sensors, although light absorption in the silicon gate reduced the sensitivity relative to that obtained with three-phase aluminum gate registers having gaps between the electrodes.

2. Line Sensors with Noniltuminated Registers In most applications of line sensors it is not convenient to have to interrupt the scanning in order to expose the sensor. All problems of image smear resulting from simultaneous exposure and scanning can be eliminated by employing a separate row of photocells with the transfer register shielded from the light. The longest single-chip CCD line sensor of this type reported to date is a 1500 element CCD page reader (64). This device was constructed as illustrated in Fig. 15b using a single row of 1500 photocells coupled to two 750-stage shielded CCD registers located one on each side of the sensor row. With this arrangement exposure of the photocells and readout of the signals could be carried on simultaneously. MOS gates, not shown in Fig. 15b, were located between each sensor element and its adjacent register stage. All gates were normally kept closed,

226

P. K. WEIMER

FIG. 26. Photomicrograph of a buried-channel charge-coupled single-line image sensor having 500 photoelements [Kim and Dyck (65), courtesy of Fairchild Camera and Instrument Corp.].

IMAGE SENSORS FOR SOLID STATE CAMERAS

227

but were opened simultaneously when a new line of information was to be transferred to the registers. The signals from the parallel registers were recombined in the output so that all elements were read in proper sequence. Advantages of the parallel register format were that the total number of transfers and the transfer frequencies were reduced by a factor of two. Also, the sensor elements could be spaced more closely than would be possible if a single output register were used. The transfer losses of the two surface channel registers were sufficientlylow ( < 5 x per gate) that good resolution was achieved over the entire sensor. A 500-element buried-channel line sensor having a useful dynamic range of over lo00 to 1 has recently been reported (65).The scanning organization of this device, shown in Fig. 26, is similar to the 1500-element sensor described above, with two 250-stage three-phase registers located along the sides of the sensor row. All gates were of doped polysilicon with the undoped polysilicon between the gates forming a resistive sheet which minimized charging of the oxide. A final layer of aluminum was deposited over the registers as a light shield to prevent image smearing during transfer. The optically generated carriers in the sensor area were collected on the silicon surface under the central polysilicon photogate electrode. This area was subdivided by means of channel-stop diffusions into 500 separate elements with a center-to-center spacing of 1.2 mils. A short three-phase output register was used to recombine the signals from the two main registers and to transfer the signal to an on-chip gated-charge MOS amplifier similar to that illustrated in Fig. 23, The dynamic range of this sensor was enhanced by the use of buriedchannel registers, which avoid the interface state noise of surface-channel devices and require no " fat-zero" background current. Figure 27 shows three pictures transmitted by the sensor with an integration time of 500 psec and an output clock frequency of 1 MHz (0.5 MHz for the main registers). The illumination level for the top picture (30 fc of tungsten light at 2800°K) approached 90% of saturation in the bright areas. Pictures b and c were reduced 100 x and 1000 x , respectively. Picture b showed no degradation but picture c showed random noise which arose in the output circuit. The large dynamic range obtained with the 500 x 1 buried-channel sensor played a significant role in suggesting the potential capabilities of charge-transfer sensors for low-light-level television. However, it was recognized that the requirements of uniformity in dark current would be far more severe in an area sensor than in a line sensor. For example, the relatively short light integration period in the above tests (500 psec) reduced the dark current contribution in the output signal by a factor of 33 times compared to what it would have been if the usual 1/60 sec integration time had been used.

228

P. K. WEIMER

FIG. 27. Pictures taken with a 500-element linear imagmg device at three d~fferent illumination levels. The highlight signal level in the region around the boy‘s shoulder in (a) is approximately 90s; of saturation. The light levels in (b) and (c) are reduced by 100 x and 1 0 0 0 ~from (a). The operating frequency was 1 MHz [Kim and Dyck (65). courtesy of Fairchild Camera and Instrument Corp.].

IMAGE SENSORS FOR SOLID STATE CAMERAS

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B. Two-Dimensional Area- Type Charge-Coupled Sensors 1. Sensors with Illuminated Registers

The feasibility of imaging with two-dimensional charge-coupled sensors was demonstrated by Bell Telephone Laboratories (66) in 1971. Figure 28 shows the general layout for a three-phase surface-channel sensor (6) having 64 x 106 imaging cells. This sensor is of the vertical frame transfer type having illuminated registers in the sensor area, and a separate storage area for handling the signal during readout. The 106 vertical registers in the sensor portion each had 64 stages and the storage area was of equal size. The horizontal output register contained 106 stages, with an output diode for collecting the signal. In operation, the charge pattern which has accumulated in the illuminated sensor area during the first field period is transferred to the storage area during the following vertical blanking interval. The pattern continues to advance toward the output register during the next field period and each row of charges is transferred in parallel to the output register during horizontal blanking. The video signal appears at the output diode as the charges are transferred in succession along the fast output register. Meanwhile, during the second field period, a new image pattern has formed in the sensor area so that continuous output signals are obtained. The input register shown in Fig. 28 does not play a useful role in this mode of operation but was included so that the device could also be used as an analog delay line (see Section XII). A total of nine different clock voltages are required for driving the three sets of gates in the sensor, in the storage area, and in the output registers. In addition, other separately addressable gates were provided to allow independent operation of various portions of the device. The 64 x 106 sensor was fabricated on 20-40 Q-cm p-type silicon. An n-type phosphorus diffusion provided input and output diodes as well as the cross-unders required for driving the three-phase registers. A p-type boron diffusion served as channel stops to define the edges of the 106 vertical transfer channels and the two horizontal channels. After removing the masking oxides used for the diffusions, a layer of 1300-1400 8, of dry HCl gate oxide was thermally grown. Typical values for oxide charge and interface state density were 5 x 10" cm-' and 1 x 10'ocm-2, respectively. The coplanar gates were formed by depositing a single layer of tungsten, 1500 8, thick, which was subdivided into gates by etching. All gate electrodes were 9 pm wide in the direction of charge transfer, and were separated by 2 pm gaps, giving a total cell length of 33 pm. The active area of the device was 4 x 5mm.

230

P. K. WEIMER

INPUT GATE

-

l [.

T

. ... I

1

T

1

- .

1 . I 1 7 . 1

p2 OP3

0

!.. 1... 1. 1. 7.: .:I.: I: 1.: I: INPUT

REGISTER DIODE

CHANNEL BOUNDARY DIFFUSION IMAGING AREA

BOTTOM GATE

-

OUTPUT REGISTER PI

0

p20

ELECTRODES

. ...I.. ..I ..I.. ,1..:

r

'r

:.-. :.. r 1

:

1 - 1 1 - 1 1 1

1

' C O U T P U T GATE

P3O

FIG.28. A schematic diagram of the three-phase charge-coupled image sensor built by Bell Laboratories [Sequin et a/. (6)]. Frame transfer sensor with illuminated registers in sensor area.

IMAGE SENSORS FOR SOLID STATE CAMERAS

23 1

The sensor was mounted in the camera so that the optical image entered the silicon through the gaps in the metal electrodes. Figure 29a shows a picture transmitted by the sensor operating in the frame transfer mode just described. The element readout rate was 1 MHz and the field rate was 120 frames/sec. In another mode of operation the image was projected on the entire 106 x 128 element array for 1/30 sec, and was then read out by shifting ail rows downward a line at a time unfil they had all reached the output register. A recognizable picture was obtained, as shown in Fig. 29b, even when the integration period was equal to the readout period but the vertical image smear caused by illumination during readout was objectionable.

FIG.29. Television pictures transmitted by the three-phase charge-coupled image sensor shown in Fig. 28. (a) Operation in the frame transfer mode with 64 x 106 elements, (b) Operation with image projected on entire array of 128 x 106 elements with integration time equal to the readout time [Sequin et al. (6)].

The question arose as to how a vertical frame transfer sensor could conveniently provide a vertically interlaced signal such as required for most television systems. Sequin (34) has described a method of operation which not only achieves interlace but effectively doubles the number of vertical lines in a given sensor. In the normal frame transfer mode described above, the photocarriers were collected under the same phase electrode in each frame. The gate which was held most positive during the integration period would have a deeper potential well than its neighbors and would therefore collect the photoelectrons generated in its neighborhood. For interlaced operation, it was sufficient to collect the carriers under different electrodes in successive fields: e.g. under the phase 1 electrodes in the odd fields and the phase 2 + 3 electrodes in the even fields. Although the vertical resolution in either field could not exceed the 64 stages in the sensor registers, the effective

232

P. K. WEIMER

midpoint of each element would be shifted one-half an element up or down on successive fields. The total number of sampling points in the vertical direction was accordingly increased by a factor of two when considered over two fields. This improvement was clearly visible in the transmitted picture when viewed on an interlaced monitor. Figure 30 shows a comparison 'of interlaced and noninterlaced scanning on a small 45 x 60 element (noninterlaced) frame transfer sensor made at RCA Laboratories (36). The interlaced mode of operation can cause the spots resulting from high dark current in the sensor to be more conspicuous since different sets of defects will be visible in each field and they will flicker at a 30 Hz rate.

FIG.30. Operation of a 45 x 60 element three-phase CCD sensor under two conditions of operation. (a) Normal 1/60 sec integration time with interlace to give 90 x 60 elements. (b) Short integration time without interlace. (Dark current spots were more conspicuous in (a) because the integration time was much longer and because interlace exposes more defects.)

It should be noted that each element of a frame transfer sensor operating in the interlaced mode will have only half the integration time of an interlaced sensor whose vertical elements do not overlap in successive fields. The interlaced frame transfer sensor is analogous to a camera tube whose beam width is twice the center-to-center spacing of the scanning lines. Although the integration time in each case is only 1/60 sec instead of 1/30 sec, no charge is lost. The vertical resolution, however, of the interlaced frame transfer sensor appears to be fully equivalent to that of a noninterlaced sensor having twice as many scanning lines (67). The sensitivity and resolution of CCD sensors will be discussed in more detail in Section X. Although the 128 x 106 element (interlaced) sensor had far fewer elements than the early XY addressed sensors its sensitivity and overall uniformity was considerably better than the earlier XY sensors.

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233

Several investigations have been started at various laboratories to develop CCD sensors having full TV resolution. In spite of the extra silicon area required for storage, the frame transfer sensor with illuminated registers is a good candidate for extension to larger sizes. The simplicity of construction of the three-phase register permits element sizes to be as small as one square mil. However, a serious problem in fabrication is to avoid shorts or connecting bridges between electrodes spaced 2-2.5 pm apart. A 128 x 160 element sensor (256 x 160 elements interlaced) was built (68) at RCA Laboratories having aluminum electrodes with 2.5 pm spacing and a total gap length of approximately 12 ft. Very high quality masks are required to produce sensors of this size free of defects. Nevertheless, many experimental devices of this type were built and operated. An even larger experimental sensor (69) of the same type has been produced in the Electro-Optics Department of the RCA Electronic Components Division. The television pictures shown in Fig. 31 were taken with the

FIG.31. Television pictures transmitted by a 256 x 320 element C C D sensor made at the RCA Electronic Components Division [Rodgers (69)].

solid state camera, shown in Fig. 32. The sensor used in this camera (Fig. 33) contained 256 x 320 elements (512 x 320 interlaced). Such a sensor requires a silicon chip approximately 500 x 750 mils and has an active area of 250,000 square mils. This sensor has an important advantage over earlier CCD sensors in that it can be operated at standard TV scan rates and the picture displayed on a regular television monitor. The problem of avoiding blemishes or spots in the output signal can be expected to become more severe as larger area devices are built. Too many defects on the average will reduce the yield and increase the costs beyond what the market will bear. For this reason it is most important that the optimum design of sensor require as few steps in fabrication as possible. Although the frame transfer sensors described above are relatively simple in

234

P. K . WEIMER

FIG.32. Solid state camera incorporating the 256 x 320 element CCD sensor

construction, it may ultimately become necessary to increase their complexity in order to control the overload characteristic known as blooming.” This effect occurs when a brightly lit object in the scene produces excess charge in the sensor which spreads to adjacent elements along the register producing objectionable streaks in the transmitted picture. Blooming can be controlled (71, 72) by introducing diffused diode buses between the registers to draw off the excess charge. If such blooming buses prove to be necessary they can be added most readily to the relatively simple structure of the illuminated register sensors. Most of the large experimental frame transfer sensors made to date have been of the three-phase surface-channel type with the light entering the silicon through the gaps between the electrodes. Two-phase registers require simpler clock voltages and fewer crossovers in the sensor. The sealedchannel silicon-gate registers described in Section VIII could be used in illuminated register sensors, but a fraction of the light in the blue end of the spectrum may be absorbed in the silicon gates, if illuminated from the gate side. Illumination from the substrate side will require thinning of the silicon as in the silicon vidicon. Illuminated register sensors using buried channels have not been reported to date. “

IMAGE SENSORS FOR SOLID STATE CAMERAS

235

FIG.33. A comparison of the size of the developmental 256 x 320 element CCD sensor with a 213 in. silicon vidicon tube which is mounted in its focus and deflection coil assembly.

2. Sensors with Nonilluminated Registers and Interleaved Photocells A charge-transfer sensor having separate photocells and its registers shielded from light avoids all image smearing caused by illumination during transfer. A vertical transfer sensor of the type shown in Fig. 17a has been fabricated ( 7 3 ) in a 100 x 100 element size by the Fairchild Camera and Instrument Corporation. This sensor, which is currently being incorporated in cameras for special applications such as surveillance, represents the first area-type CCD sensor to reach the commercial market. The transport registers are of the two-phase buried-channel type with two levels of polysilicon gates. An overlying layer of aluminum forms electrical connections and shields the registers from light. An advantage of the nonilluminated register sensor is that the need for a separate storage area is eliminated. However, 30-50% of the light incident from the gate side is lost, and the elemental structure is more complex than required for sensors with illuminated registers. Figure 34 shows the potentials under the electrodes of the elemental cell (35)of the 100 x 100 sensor during (1) charge integration and scanning and (2) transfer of charge from the photocell to the register. During integration, electrons are collected in

236

P. K. WEIMER

the potential well under the positively charged transparent photogate electrode. When charges are to be transferred to the registers during the vertical blanking interval the photogate is made much less positive than the adjacent register gate. The scanning gates must be designed to provide a barrier

yrrL

VERTICAL S C A I 6ATE

INTEGRATION OF CHARGE

TRANSFER TO REGISTER

34. Operation of the 100 x 100 element area sensor showing potentials during integration and transfer of charge from photoelements to the registers [Amelio (35),courtesy of Fairchild Camera and instrument Corp.]. Yiti.

which prevents charge from spilling back into the sensor element while the register is running. Interlacing capability is achieved by shifting the evennumbered sensor elements into phase 1 electrodes in field one and the oddnumbered sensor elements into the phase 2 electrodes in field two. The resolution for the 100 x 100 sensor was reported (73) to be 75 lines in both the vertical and horizontal direction. A differential gated charge amplifier was included on the 100 x 100 sensor chip. Dynamic ranges of > 1000 to 1 have been measured at an output frequency of 500 kHz. Random noise set the limit of detection at approximately 400 signal electrons.

X. PERFORMANCE LIMITATIONS OF CHARGE-COUPLED SENSORS Sensitivity and resolution are the two most basic qualities of sensor performance. Other characteristics such as uniformity, the number of picture elements, freedom from spurious signals, reliability, and ease of manufacture are gradually being solved by advances in technology. However, it is the fundamental limitations on sensitivity and resolution which will determine whether CCD sensors will be able to satisfy the more demanding applications now being served by camera tubes. Since representative data on perfor-

IMAGE SENSORS FOR SOLID STATE CAMERAS

237

mance of experimental sensors is not readily available, the present discussion will be based largely on analytical predictions of ultimate limitations.

A . Resolution

A solid state sensor capable of meeting normal broadcast standards should have approximately 5 12 x 680 elements (interlaced) for balanced resolution. Although sensors this large will considerably exceed the size of existing CCD sensors their fabrication appears to be within the foreseeable capabilities of silicon technology. However, single-chip sensors which could match the limiting resolution of special camera tubes such as the return beam vidicon (67) (-4000 lines) are well beyond the present state of the art. In practice, it was soon apparent that the observed resolution in a CCD sensor could fall far below the maximum resolution set by the number of elements. Sampling effects, lateral diffusion of charge within the silicon, nonuniform dark current, clock transients, and illumination during transfer can all contribute to a loss in resolution. The charge loss per transfer, however, introduces the most fundamental limitation on the maximum resolution which can be achieved in a charge-coupled sensor. Transfer losses are normally expressed in terms of the nE product, where n is the total number of transfers and I is the fractional inefficiency (or loss) per transfer. The inefficiency product should be low for both the vertical and horizontal registers in order to maintain the expected spatial resolution. The effect of transfer losses on resolution has been calculated (74, 75), and the results are plotted in Fig. 35. The ordinate shows the degradation in the modulation transfer function (MTF) for various values of nE as a function of the normalized spatial frequency. The curves are terminated at a spatial frequency equal to one-half fo , the geometrical element frequency, since according to Nyquist’s sampling theorem, frequencies greater than this cannot be resolved. (Expressed another way, the useful video passband extends up to one-half the clock sampling frequency.) The block diagram at the right side of Fig. 35 shows the spreading and delay of a single charge packet for different values of the transfer inefficiency product. It is noted that the center of the degraded pulse is delayed independently of signal frequency by a number of stages approximately equal to the nE product. The MTF curves apply for a charge pattern which originates either optically at each element or which is introduced electrically at the input of the register. With optical input, of course, the n& product will become progressively larger for signals originating farther from the output terminal of the register. If the value of nE is larger than a few tenths in any portion of the sensor the picture detail in these areas will be smeared out and delayed either vertically or horizontally.

238

P. K. WEIMER >

0

z

%

w

DEGRADATION OF A SINGLE PACKEl

0

nE = 2.0

. 1 1 1 . -

01

0.2

0.3

0.4

0.5

INPUT FREQUENCY RELATIVE TO ELEMENT FREQUENCY f o

FIG. 35. Degradation in Modulation Transfer Efficiency (MTF) for various values of ~ i as c a function of the normalized spatial frequency. The block diagram at the right shows the delay and spreading of a single packet for different values of the inefficiency product (75), courtesy of Bell Laboratories.]

iw;.

[Tompsett

The ne product in the 64 x 106 element sensor made at Bell Laboratories (Section IX) was measured to be approximately 0.3 for both the vertical and horizontal registers. This value of ns produces a degradation in vertical and horizontal M T F of up to 55% in the most distant corner. The maximum drop in MTF which can be tolerated will depend upon the application and the requirements of the user. If we assume a maximum tolerable ne product of 0.1, we find that the loss per transfer in a 512 x 680 element interlaced three-phase frame transfer sensor should not exceed E = 2.8 x The maximum number of transfers in this example was 3576. A two-phase interlaced sensor with interleaved photoelements could tolerate larger transfer losses per transfer because fewer transfers would be required for the same number of elements. Table I1 shows the maximum tolerable loss per transfer calculated for

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239

TABLE I1

MAXIMUM TOLERABLE TRANSFER INEFFICIENCY E FOR VARIOUS SIZESAND TYPES OF CHARGE-TRANSFER SENSORS No elements

Type of sensor‘

45 x 60 (90 x 60 interlaced)

A

450

2.2 x 10-4

6.6 x

100 x 100

B

400

2.5

7.5 x 10-4

64 x 106 (128 x 106 interlaced)

A

702

1.4 x

128 x 160 (256 x 160 interlaced)

A

1248

8 x 10-5

2.4 x 10-4

256 x 320 (512 x 320 interlaced)

A

2496

4 x 10-5

1.2 x 10-4

256 x 680 (512 x 680 interlaced)

A B, C

3576 1872

2.8 x 5.3 x 10-5

8.4 x 10-5 1.6 x 10-4

4500 x 6000 (interlaced)

A B, C

31500 16500

3.2 x lo-‘ 6.1 x

9.5 x 10-6 1.8 x 10-5

max n

-

- _ _

for nE = 0 1 (MTF = 82%)

10-4

&

-__ for ne = 0 3 (MTF = 55%)

4.2 x 10-4

A-Three-phase illuminated register, vertical frame transfer. B-Twophase interleaved photoelements, vertical or horizontal transfer. C-Two-phase, illuminated register, horizontal transfer.

two assumed values of ne and for several different sizes and types of CCD sensors.

B. Low Light Sensitivity In a classic paper entitled “Television Pickup Tubes and the Problem of Vision,” Rose (76) has shown that the resolution of an ideal sensor at threshold illumination is limited by the fluctuations in the photons impinging on the light-sensitive area. Actual devices usually fall short of ideal performance

240

P. K. WEIMER

because of additional noise introduced by the scanning process and because of a low yield of photoelectrons in the primary photoprocess. After fifty years of research and development, camera tubes can now be produced in which scanning noise is negligible. This result has been accomplished in such combination tubes as the I-SIT or the I-isocon by the technique of providing electron image intensification (77) prior to scanning. Improvement of the scanning process alone (as in the isocon without intensification) has not been sufficient to reach photon-limited performance at all light levels. A nonintensified solid state image sensor such as the CCD sensors described above has no gain mechanism prior to scanning. Its only chance for ideal performance lies in attaining substantially noise-free scanning. Although the noise and spurious signals introduced by charge-transfer scanning are fairly low, they can still be large enough to limit the actual low-light performance. A partially compensating advantage of a silicon sensor over the photoemissive cathodes used in the most sensitive tubes is the higher responsivity of silicon and its extended spectral response into the near infrared. Table 111 (seep. 244) compares the measured responsivity of an S-20 multialkali photocathode (such as used in the I-SIT)with that ofa filtered and unfiltered silicon vidicon tube using a standard white light tungsten source at 2856°K. (Responsivity measured with such a standard source is normally expressed in mA/W-2856”K and should not be confused with the unit of responsivity at a single wavelength, A/W. The unit (W-2856 K ) is used to designate the total radiated power in watts, integrated over all wavelengths from a tungsten filament lamp operated at a color temperature of 2856°K. It can be converted directly into lumens by the relationship 1 W-2856°K = 20 lm.) In some of the analysis which follows, it will be convenient to express the measured responsivity of sensors in terms of an average quantum yield, 8, derived from the calculated responsivity of an “ideal” detector having a 100% quantum efficiency over the normal silicon range of 400-1 100 nm. The responsivity of such a detector (78) is taken to be 238 mA/W-2856”K. The measured responsivity of silicon devices usually fall far below this value because of the failure of quanta to be absorbed or because of other losses in the sensor. The Rose analysis (76), which gives the resolution of an ideal sensor as a function of illumination, has been used for many years for evaluating the performance of camera tubes at low light levels ( 7 ) . Similar calculations for C C D sensors have been carried out by Carnes and Kosonocky ( 7 9 ) and more recently by Campana (80). Although the method of analysis for each was essentially the same as the earlier model, the conclusions in the second paper regarding the tolerable C C D noise to match the sensitivity of the I-SIT tube were more stringent. Most of the differences lie in the assumptions regarding responsivity, contrast, threshold signal-to-noise ratio. and ”

IMAGE SENSORS FOR SOLID STATE CAMERAS

24 1

integration time. The derivation is simple and will be reproduced here using radiometric units as in the paper by Campana (80). The number of carriers accumulated at each element during the integration period is given by

N = HSA, t 4 '

(4)

where H is the image irradiance (Watts/m2-2856"K),S the effective responsivity A/W-2856"K), A , the geometric area of each element (m'), t the integration period (sec), and q is the electronic charge (coulombs). The responsivity S includes all light losses due to structure and internal reflection as well as the detector quantum yield 8. Defining the contrast in the charge image as C = A N / N , , the number of signal charges per element will be

ANs = CHSA,t/q,

(5)

while the shot noise associated with N , is given by

Combining N i l 2 with the effective rms noise N arising from all other CCD noise sources (60) (see Section VIII) yields the total rms noise per picture element :

At very low light levels where the observed resolution will be limited by noise the geometric element size (L, = must be replaced by a somewhat larger observable picture element whose side is L. The criterion for calculating the limiting resolution versus irradiance curve is that the ratio of the number of picture charges per observable picture element to the total rms fluctuations per observable picture element must exceed a minimum observable signal-to-noise ratio, k . That is,

6)

A N , . (L2/L2) - k. N,(L/Lif Relation (8) can be expressed in terms of resolution (in TV lines per picture height or line pairs per mm):

242

P. K. WEIMER

where R is the observable limiting resolution and R, is the geometrical resolution. Substituting (5) and (7) in (9) gives:

R

R k

CHSA,t/q (HSA,t/q + N2)1/2

= 3 .-

~

~

Expression (10) has been used to calculate limiting resolution versus irradiance for various assumed values of S, k, t, and fi. The results are plotted in Fig. 36. The irradiance level H , at which R = Rg is obtained by

-aE E

a W

a (0

a:

2

I,,,,,

I

I , , , , , ,

I

I,,,,,,

I

,,

I

I,,,

, , I1

I,,,

FIG. 36. Predicted resolution of silicon image sensors as a function of irradiance of tungsten light for various values of scanning noise and responsivity S . R was calculated from Eq. (10) assuming R , = 20 line pairs per mm (sensor with 1 mil pitch), C = 1, r = 0.2 sec. k = 1. 8, the average quantum yield for all incident photons between 400 and 1100 nm, is related to responsivity S as shown in Table 111: ( @ = 1 corresponds to S = 0.238 A,W-2856 K : 6 = 0.37 corresponds to S = 0.087 AiW-2856"K; 6 = 0.019 corresponds to S = 0.0048 Am-2856-K.) Curves (1)-(3) are for "ideal" sensors having zero scanning noise N = 0. Curves (4)-(8) are calculated for actual sensors with increasing scanning noise I < %< lo4. The dotted curves (9) and (10) are the measured resolution for the I-SIT and SIT camera tubes. (The I-SIT tube meahurcd had S = 0.0048 A W - B W K . ) representa the total scanning noise for the assumed integration period I = 0.2 sec.

m

m

solving (10) for H : H,

=

-

4

[k2

~-

2SA, t C 2

+ k ( k 2 + 4C2N2)"2].

(11)

Relation (11) is useful in assessing the effect of varying the values of the parameters used in calculating Eq. (10).

IMAGE SENSORS FOR SOLID STATE CAMERAS

243

The maximum geometrical resolution R, is assumed in each case to be 20 line pairs per millimeter corresponding to a CCD element spacing of 1.0 mil center-to-center. ( A , = 6.45 x lo-'' m'.) All effects of modulation transfer function and the Kell factor o n resolution are ignored in the present discussion. The contrast C has been taken to be unity in each curve. It is noted from (11) that decreasing C would shift the curves to the right in proportion to C2 if N is zero but in proportion to C if N is large. Relation (11) also shows that the value assumed for k, the minimum detectable signal-to-noise ratio, plays a sensitive role in predicting the irradiance threshold at whch resolution begins to deteriorate. Carnes and Kosonocky (79) used k = 5 as recommended by Rose for detecting small isolated objects. Campana (80) used k = 1, based on experimental tests of detecting bar patterns with an I-SIT camera system. Such a low value of k is reasonable here since the eye is able to sense the cooperative effect of the large number of picture elements that go into making the bar pattern, even though the signal-to-noise ratio of a single observable element is unity. A value of k = 1 was therefore taken for all curves plotted in Fig. 35. The curves would be shifted laterally as the square of k when N = 0 but only as the first power of k when N is large. Curves (l), (2), and (3) in Fig, 36 represent "ideal" sensors in which no noise is introduced in the scanning process, i.e. N = 0. The total noise in this case is shot noise in the signal current. In curve (1) S is assumed to be 238 mA/W-2856"K, corresponding to a theoretically perfect silicon detector having 0 = 1 over the wavelength range 400-1100 nm. In curves (2) and (4)-(8), S = 87 mA/W-2856"K, or 0 = 0.37, the same as an unfiltered silicon vidicon. (See Table 111.) Although this value is approximately three times larger than measured to date in top-illuminated CCD sensors, this value would be a reasonable expectation in a thinned CCD sensor illuminated from the substrate side (81). In all curves the integration time t has been assumed to be 0.2 sec corresponding to the integration time of the eye of the viewer rather than the frame time of the scanning system. It is well-known that the eye will integrate signal and noise in a television picture giving a visual improvement comparable to that of a camera photographing the television screen with an exposure time of 0.2 sec. The exact value of both S and t will have only a first power effect on the irradiance required. Curves (4b(8)show the effect of increasing the total rms scanning noise from 1 to 10,000 electrons per integration period. These curves are applicable for other types of silicon self-scanned image sensors, including the XY addressed photodiode arrays discussed in Section VI. The dotted curves in Fig. 36 are for camera tubes having a modified S-20 photocathode. Curve (3) was calculated for an ideal tube having S = 4.8 mA/W-2856"K, while curves (9) and (10) are the measured values of resolution versus irradiance for an

TABLE 111

RELATIONSHIP BETWEEN R E s P o w s i v r r Y S

AN11

AVERAGE QUANTUM YIELD 0

FOR FOUR TYPES OF

DETECTOR

ELEMENTS

Sensor

Responsivity (pA/lm)

Responsivity (mA/W-2856'K)

Average quantum yield 6 (400-1 100 nm 2856'K) ~~~~

Trialkali photocathode (S-20)

I60

Silicon vidicon with 1R absorbing filter

910

Silicon vidicon unfiltered Type V response

4350

Ideal silicon detector with loo"/, response over wavelengths (400-1 100 n m )

1 1.900

3.2

0.013

Quantum yield at peak raponsivity. 6' ~~

0.20 (at 420 n m )

18.2

0.076

87

0.37

0.83 (at 500 nm)

238

I .o

a 7:

245

IMAGE SENSORS FOR SOLID STATE CAMERAS

I-SIT tube (77a)and a SIT tube (77b).(The SIT tube contains a silicon vidicon target on which high energy electrons from an S-20 photocathode are imaged, producing an electron gain of several thousand prior to integration and scanning.) In the I-SIT tube, the photocathode of a SIT tube is coupled by means of fiber optics to the screen of a separate image intensifier tube, which also uses an S-20 photocathode. The horizontal scale in Fig. 36 shows the equivalent illuminance on the sensor in lumens per square foot (foot candles) using the conversion factor 1 lm/ft2 = 0.5 W-2856”K/mZ. The highlight illuminance on the sensor is approximately one-tenth that on the scene. A scene illuminance of 10- lm/sq ft is typical of starlight illumination on an overcast night. Comparison of the predicted CCD performance with ideal sensors and with the existing SIT and I-SIT tubes show that the scanning noise N must be kept to less than 10 electrons per element (80)per integration period if the CCD is to match the I-SIT under the assumed test conditions. Although a somewhat larger value of N might be tolerated if comparisons were made under conditions of lower contrast and higher signal-to-noise ratio [as in Carnes and Kosonocky (79)] it is clear that N should be kept as small as possible. Since the numerical values of N,, given in Table I were calculated for a single scanning period of 1/60 sec, those values should be increased by a factor of (0.2/0.016)1’2for proper comparison with the curves of Fig. 36 which assume an integration time of 0.2 sec. Although the various noise contributions arise in different ways, they will each be increased in proportion to the square root of the total number of scans in one integration period. The modified values of N,, arc listed below, and the prospects for reducing each are summarized as follows: 1. Shot noise in the electrically introduced “fat-zero’’ current (N= 138). This source of noise can be eliminated by use of registers which do not require a “fat-zero’’ background current (e.g. buried channel registers). 2. Transfer losses (N> 690). Various authors are in disagreement as to the existence and magnitude of this source. According to Carnes and Kosonocky (60) this type of noise would decrease as the transfer loss is reduced. 3. Trapping losses == 3290). This noise source would be most serious in surface-channel devices. It should be substantially smaller with buriedchannel registers. 4. Shot noise in the dark current (N= 194). The thermally induced dark current noise can be reduced by cooling the sensor. (Ndrops by $for each 10°C drop in temperature.) 5. Shot noise in optically generated background current = 1095) for N,, = 0.1 full well.) This source is an alternative to (1). Elimination of the need for “fat-zero’’ current would remove this source.

(m

(m

246

P. K. WEIMER

6. Output amplifier on chip (I7= 416). A method of reducing this noise now being explored is to voltage-sample successive output nodes to produce a distributed parallel-channel output amplifier. Floating-gate amplifiers (62) used singly or as distributed amplifiers may also reduce this noise factor significantly. The total rms noise introduced by the scanning process is obtained by taking the square root of the sum of the squares of each component. The total value of for all of the above components (assuming all were present) would be 3575 for 0.2 sec integration period. This method of summing does not take into account the fact that different noise sources may have a different frequency spectrum and hence may differ in their relative visibility. As stated in Section VIII, fixed pattern noise in practical devices may actually exceed the fluctuation noise discussed above, and can restrict the range of useful operation to higher light levels than predicted in Fig. 36. A major source of fixed pattern noise in experimental sensors is caused by local variations in dark current which will produce a mottled background in the transmitted picture. In this case, the fixed noise background will vary directly as N , and not as the square root. Although improvements in silicon processing would be expected to reduce this problem, operation at very low light levels would require a reduction in dark current variations by several orders of magnitude below that tolerable in a silicon vidicon. Cooling of the device to reduce dark current noise should reduce fixed-pattern noise as well. Spurious signals which remain fixed can also be removed by subsequent signal processing of the video signal. A method of background subtraction which is particularly appropriate for C C D s is to recycle the fixed pattern signal through the sensor itself (36). (See next section.) Another type of fixed pattern noise may arise from variations in clock voltage. The complete removal of all clock voltages from the signal is particularly difficult at very low signal levels. A major research effort is currently being carried out in a number of laboratories ( I ) to develop a simple low cost C C D sensor suitable for use at very low light levels. If it should prove too difficult to reduce sufficiently either the noise associated with scanning or the fixed pattern noise in the sensor, a possible solution would be to introduce image intensification prior to scanning as has been done with tubes. Many combinations of image intensification with C C D arrays are conceptually possible. While vacuum tube intensifiers are quite highly developed, the resulting combination would risk loss of the compact, low cost advantages anticipated for CCDs. Solid state intensifiers (82) combined with C C D sensors might be attractive but they present even more formidable problems in resolution and uniformity. For normal scene illuminations ranging from that of a full moon up to bright sunlight a back-illuminated C C D sensor with a scanning noise back-

IMAGE SENSORS FOR SOLID STATE CAMERAS

247

ground as large as 3000 electrons could still yield a good quality picture. Its operating sensitivity would be comparable to that of a silicon vidicon which is normally considered a sensitive tube, If its cost can be kept low a CCD sensor with an adequate number of elements would be a very attractive device indeed.

XI. CHARGE-TRANSFER SENSORS AS ANALOGSIGNALPROCESSORS The earliest video signal processors requiring delay utilized beamscanned storage tubes (83) operating similarly to camera tubes. More recently, storage tubes have been supplemented by acoustic delay lines, magnetic tape, and video disks. The development of charge-transfer scanning has reemphasized the connection between image sensors and analog signal processing. The sensor itself can be used as a highly versatile delay line, or it can be used simultaneously as a sensor and as a delay line for processing its own signal. Other types of analog signal processors such as transversal filters (84), correlators (85), and time division analog multiplexers (86) can be produced by modified forms of registers. The present section will consider only such video processing that can be carried out by a charge-transfer sensor having an input register. The equally important application of such devices for digital memories is beyond the scope of this paper. A . Charge-Transfer Delay Lines

Significant advantages of a charge-transfer register over an acoustic delay line are: (1) The delay time is electronically variable. (2) Continuous video delay of up to 1/30sec or more will be feasible. (3) The device is compact and easily integrated with other components. Delay time is calculated by dividing the total number of storage elements by the clock frequency. The time axis of the signal can be modified continuously over a wide range by varying the frequency. Sangster (84) has discussed the use of a bucket brigade register for correcting for variations in tape speed of an audio or video tape. Television pictures were shifted and stretched horizontally by changing the clocking frequency of a bipolar bucket brigade from 9 MHz read-in to 3 MHz readout. Video signals were passed through integrated bipolar registers having a total of up to 864 half-stages with negligible deterioration. (Charge amplifiers were included within each integrated register.) Continuous delay of a normal resolution television picture for 1/30 sec in a single serial register would require several hundred thousand transfers. Reference to Fig. 35 shows that the transfer inefficiency E per stage should not exceed lop6per stage to keep the total degradation in MTF to less than 50%. A charge-transfer register known as the series-parallel-series type (87),

248

P. K. WEIMER

n CI

n

5

c, ; FIG.37. Transfer sequence and clock voltages for a series-parallel-series delay line.

illustrated in Fig. 37, would require only 1100 transfers for the same storage capacity. In this case a value of E = 0.3 x l o p 4 would suffice for the same loss in MTF. Tompsett and Zimany (88) have operated the 106 x 128 element CCD sensor as a series-parallel-series delay line. The results are shown in Fig, 38. The right half of the picture was undelayed while the left half of USE OF A 106 x128 ELEMENT CCD TO DELAY HALF A PICTUREPHONE@FIELD

I

DELAYED

DIRECT

I

I DELAYED

DIRECT

FIG. 38. Signals delayed by transmission through a 106 x 128 element three-phase CCD array operating as a series-parallel-series delay line. The left half of each display was delayed one field period (16.7 mhec in the Picturephone system) while the right half was undelayed [Tompsett and Zimany (88),courtesy of Bell Laboratories].

IMAGE SENSORS FOR SOLID STATE CAMERAS

249

the picture was delayed by 16.7 nisec, one field period of the system being used. Only a slight difference is noted in the resolution of the two pictures. Another type of parallel-channel delay line has been proposed having multiplexed gates ( 4 ) rather than clock-driven gates. The multiplexed gate devices allow greater packing density of charges and require still fewer transfers for the same total number of storage elements (89). Parallelchannel delay lines d o not possess the advantage of inherently good uniformity found in a single-channel serial register. In the latter a high dark current spot in one element would add equally to the background signal of a11 charges passing this point. In a parallel-channel register a single spot in one element would be much more noticeable since its effect would appear only in those charge packets which passed through that particular channel. However, the visibility of a localized high dark current spot in a seriesparallel-series delay line could still be less conspicuous than in an image sensor where each current source is reproduced in the transmitted picture as a spot and not averaged over a whole column.

B. Video Signal Processing within the Camera by Recycling of Signals through the Sensor Itself Three signal processing operations which could utilize delay of the video signal for one or more fields are: (1) Multiple readout of the same charge pattern without discharging it. (2) Subtraction of a fixed background signal generated within the sensor. (3) Detection of a moving object in the presence of a stationary scene background. Frame delay in a charge-transfer camera can be carried out in either of two ways: (1) Include within the camera a second charge-transfer sensor which is operated in parallel with the first sensor as a series-parallel-series delay line. (2) Recycle the video signal through the sensor itself (36, 36a) so that it can serve for both sensing and delay. Method (2) is simpler and would be more effective for background subtraction since no new spurious signals are introduced by the delay device. Although an input register could be used with any type of sensor it is particularly convenient with the vertical transfer sensors shown in Figs. 16b and 17a. Figure 39 illustrates the timing sequences for video recycling in a sensor having interleaved photocells and nonilluminated registers. In multiple readout the signal is fed back in either polarity to the input at the same time it is being read out. At the end of one frame each elemental charge (or a constant minus the charge) will again be found at the same site in the vertical register. The horizontal blanking interval allows several additional horizontal clock cycles to be added to each line, if necessary to compensate for any delay in the feedback loop. The process can be repeated for as many fields as required

250

P. K. WEIMER

4

1 -SIGNAL

OUT

FIG. 39. Operation of a charge-transfer sensor with video recycling for multiple nondestructive readout, background subtraction, and as a motion detector. The sensor has nonilluminated registers and interleaved photocells.

until the MTF has become degraded by the nt: product becoming too large. The recycling of signals can also be used with short linear registers to measure E more accurately (90). In background subtraction the fixed pattern signal to be eliminated is normally caused by local variations in dark current. In the subtraction system illustrated in Fig. 39 the optical image is masked off by means of a shutter every other frame in order to allow the background signal to be regenerated. Figure 40 shows how the delayed and inverted background signal is added to the total signal in the next field. This process requires good linearity and dynamic range throughout the entire system.

I

' / I * I1

OR LINE

I1

U'

WHERE I,, ~I,,=I,,,etC. I,, AND I, ARE CONSTANT

FIG. 40. Signal level components for fixed pattern background subtraction in a sensor having nonilluminated registers shown in Fig. 39.

IMAGE SENSORS FOR SOLID STATE CAMERAS

25 1

In spite of the loss of signal by shuttering on alternate fields the subtraction process should permit operation at lower light levels in sensors whose threshold is limited by a fixed pattern background rather than by statistical noise. The requirement for cooling of the sensor to remove the background pattern could be mitigated by subtraction. Statistical noise is of course not removed by the subtraction process but is increased by a factor of the square root of two. A further reduction in signal-to-noise ratio results from the shuttering of every other field. Background removal by recycling is also possible with a vertical-transfer illuminated-register sensor but the signal loss due to shuttering would be three times instead of twice as in the above case. Various methods can be used to reduce the shuttering loss. Motion detection is obtained by recycling and subtracting alternate fields to remove the stationary portions of the image. The difference signal obtained in alternate fields identifies objects in motion. A continuous difference signal could be produced by use of a separate delay register with an external adder. Although the use of recycled signals for each of the above functions has been demonstrated (36),a complete evaluation of their effectiveness has not yet been made. Video signals can also be recycled through the output register alone in order to repeat a line or to apply vertical aperture correction,

XII. SELF-SCANNED SENSORS FOR COLOR CAMERAS Color cameras most widely used in broadcasting (91) employ a separate tube for each color channel, and have special optics for splitting the image into three components. Single-tube cameras having striped color filters in the image plane depend on structure in the target (92)or on various encoding systems (93) to produce a standard NTSC type of color signal. For special applications, single-tube field sequential cameras (94)have been constructed with rotating filter changers in front of the tube. It is apparent that each of these cameras could in principle be converted to solid state by substitution of an equivalent sensor for each camera tube. The relative advantage of such a conversion would depend on the system. For best results the color camera should be completely redesigned to take advantage of the particular properties of image sensors. The present discussion will be limited to a brief assessment of sensor characteristics as they relate to color pickup. 1. Compact design of a sensor. This feature would be particularly valuable in a three-sensor camera. An experimental version of such a camera (95) has already been built using the Bell 106 x 128 element sensor. Figure 41 shows the reduction in size of a typical three-tube camera if the tubes were

252

P. K. WEIMER

replaced with sensors. However, a three-sensor camera would still require special optics and would be more bulky than a single-sensor camera with elemental color filters.

FIG.41. Reduction in size of a three-tube color camera if self-scanned image sensors were substituted for the three camera tubes and focusing coils.

2. Accuracy of scan. This feature would reduce the registry problem in a multiple sensor camera to one of optics alone. In a single-sensor color camera the constant elemental frequency would permit color encoding by phase and amplitude modulation of a suitable reference subcarrier without requiring an accompanying index signal. The feasible systems for efficient color extraction from a single-sensor camera exceed that of a single-tube camera where variations in scanning speed and beam focus limit the choice of encoding system which can be used. However, transfer losses in the sensor must not be sufficiently high to allow color mixing to occur. 3. Low-light-level capability of a sensor. The higher signal-to-noise ratio obtainable in a solid state sensor is highly advantageous for color camerasparticularly for single-sensor cameras in which elemental filters are used. Such filters are more wasteful of light than the color reflective optics used in a multiple sensor camera. Although threshold scene illumination is not required for color transmission the sensors should be designed for high responsivity and low noise to approach the color discrimination of the eye under similar illumination. The responsivity of silicon is excellent for red

IMAGE SENSORS FOR SOLID STATE CAMERAS

253

and green light but may prove to be somewhat deficient for blue light if the sensor is illuminated through polysilicon gates. The infrared response of silicon requires this component of the light to be filtered out for proper color balance. Care has to be taken in sensor design to avoid sensitivity loss at particular wavelengths due to interference effects within the sensor structure itself. 4. Other operating characteristics. Additional characteristics of sensors valuable for color pickup include mechanical and electrical stability, low power consumption, instant turn-on, and freedom from time lag in moving scenes. Clearly, the development of a suitable color sensor will expand the applications of color television. A simple, reliable camera used in conjunction with low cost recording equipment could replace home movies as a new and more versatile form of electronic photography.

XIII. PERIPHERAL CIRCUITS FOR SOLIDSTATESENSORS In addition to the sensor a solid state camera should include inputoutput circuits, clock drives, timing generators, power supplies, and means for interfacing the signal with associated equipment. For maximum reduction in cost and size of camera the peripheral circuits should be integrated, either on the same silicon chip with the sensor or on a small number of associated chips. The equivalent number of components required is less than for the sensor itself and their functional design can be based on standard design rules. Only the input-output circuits will be discussed here since they are the most intimately connected with the operation of the sensor itself.

A . Input Circuit Design

An electrical input circuit can be used with some types of sensors for introducing either a constant background current (Section Vlll,B,l) or a modulated signal in case the sensor structure is to be used for analog delay (Section XI). Although background current can also be injected optically into a sensor, the bias light will yield shot noise fluctuation ( N = NA’’) which would limit performance at very low light levels. As indicated in Section VTII, an electrical input consisting of a diode and several control gates (such as shown in Fig. 23) is capable of a lower noise level, given by N, = 4 0 0 6 . The reduced noise level will be achieved, however, only if the circuit is operated so that the charge set into the first capacitor (Cpf)has reached complete equilibrium with the input signal during each clock cycle.

254

P. K. WEIMER

If, instead, charge were leaked into the first capacitor via a partially pinchedoff gate the rms fluctuations in the final charge would contain full shot noise. Noise fluctuations several times larger than shot noise have even been noted under certain conditions (96). The circuit requirements for a low noise input have been recognized independently by different workers (96, 97, 98) who have arrived at similar methods of operation. In a typical system ( 9 6 ) the first potential well is initially overfilled and then the excess charge is allowed to return to the source. The final input charge is established by the height of the potential barrier under the first gate. This system has the added advantage that the charge introduced is a linear function of the input voltage. B. Output Circuit Design The design of the circuit coupling the sensor to the video amplifier plays an important role in determining sensitivity and freedom from spurious signals. The major requirements on the output circuit are that it provide amplification with negligible introduction of noise, while suppressing all switching transients and fixed pattern noise arising from the horizontal clocks. Since the design considerations for an amplifier to be used with a charge-transfer sensor are somewhat different than for an XY-addressed sensor these cases will be discussed separately.

1. Output Circuits for Charge-Trumfer Sensors The small signals generated by a charge-transfer sensor at low illumination require very low noise levels in the output circuit to avoid further signal degradation. Spurious signals from the horizontal clocks are not a major problem with charge-transfer sensors since these signals are confined to frequencies of more than twice the video passband and can be removed by filtering. However, care must be taken in design of the clock supplies to obtain drive signals having the proper waveforms and containing minimum voltage fluctuations. The output amplifier should be located as close as possible to the output register to reduce clock pickup and to avoid unnecessary capacitive loading on the output lead. A common low-noise output circuit for charge-transfer devices has employed an MOS transistor with its gate connected directly to a diffused diode located in the channel of the output register. This arrangement was illustrated previously in Fig. 23 and is represented in Fig. 42a by the dotted lines. In operation, the potential of the floating diode must be reset once during each clock cycle to a fixed dark level by the transistor T, whose gate is connected to the clock. Noise fluctuations arising from the error in setting

255

IMAGE SENSORS FOR SOLID STATE CAMERAS

CLAMP

VIDEO OUT (0)

(b)

(C)

FIG. 42. Three types of low-noise output circuits suitable for use with charge-transfer sensors at low light levels. (a) Gated-diode output with correlated double-sampling circuit [White er a/. (US)]. (b) Distributed charge amplifiers connected to successive nodes of the output register [Weimer er a[. ( I O I ) ] . (c) Floating-gate amplifier which may also be used as a distributed amplifier [Wen and Salsbury ( 6 2 ) ] .

the dark level of the floating diode are estimated to be N, = 4 0 0 G . White et nl. (99) has found that this source of noise can be reduced by means of a correlated double-sampling process shown schematically by the solid lines in Fig. 42a. The video output from the sample-and-hold processor is obtained by taking the difference between the previously clamped reset level and the same reset level plus the signal increment introduced by the charge packet from the output register. The correlated double-sampling process removes switching transients as well as subtracting out the dark signal component which contains the reset noise. A related double sampling technique (100) was developed earlier for suppression of switching transients in multiplexed scanning of XY arrays. This system will be described in the next section. A different approach to the reduction of reset noise and clock transients makes use of the fact that the voltage change associated with a given charge packet can be sampled at successive nodes (62, 101) in the output register. A distributed amplifier of this type showing four parallel MOS amplifiers at successive nodes of a bucket brigade register (101)is illustrated in Fig. 42b. Auxiliary registers are used to compensate for the time spread as a given charge packet moves down the output register. The total amplifier noise would be expected to decrease in proportion to the square root of the number of parallel stages. Two MOS amplifiers connected to successive nodes of an output bucket brigade register (20) (e.g. see Fig. 21) have been

256

P. K. WEIMER

used with various sizes of two-dimensional charge-coupled sensors (36, 68, 69). The double-output system provides more effective elimination of the clock pulses from the signal as well as a modest improvement in signal-tonoise ratio. Another type of output amplifier shown in Fig. 42c makes use of one or more floating gates (62) in the output register in place of a floating diffusion for driving the MOS amplifier. The floating gate does not require resetting on each clock cycle, and has been proposed for use in distributed amplifiers (62). However, in the single-gate version reported to date the stage following the floating gate amplifier used a gated charge integrator which was reset at clock frequency. Calculations indicated that for a bandwidth of 1 MHz 50- 100 electrons could be detected at room temperature using a single stage floating gate amplifier. Still lower noise levels should be possible in a distributed amplifier of this type. 2. Output Circuits for X Y-Addressed Sensors

In the three common methods of extracting signal from an XY array (see Fig. 7) the output lead is already shunted by a capacitance to ground which may be several orders of magnitude larger than the elemental capacitance of a charge-transfer register. The performance at low light levels will therefore be limited by the fluctuations in the number of carriers required for charging this capacitance (N,= 4 0 0 a ) unless the output circuit can be designed to minimize this source of noise. An equally troublesome source of fixed pattern noise arises from the spatial variations in the multiplex switches and in the successive scan pulses. The resulting signal variations cannot be removed by low pass filters since their frequency falls within the video passband. Two general types of circuits which have been used for suppressing switching noise in multiplexed scanning systems are illustrated in Fig. 43. These include (a) a double sampling technique similar to that discussed in the preceding section, and (b) a differential amplifier scheme. In each case the fixed pattern noise is suppressed by being subtracted from itself. The double-sampled system having an integrating amplifier illustrated in Fig. 43a was described by Brugler (100) in 1968. The integration process provides a means of discriminating between the picture signal (which is dc) and the capacitive feed-through signal from the scan generator (which is ac). Integration of the output signal over the period of the scan pulse cancels out the unwanted capacitive signals but leaves the dc picture charge which is then detected by the second sample. The double sampled charge integration scheme was used by Plummer and Meindl (102)and later by Michon and Burke (12) for scanning XY photodiode arrays (see Fig. 12). Although the

IMAGE SENSORS FOR SOLID STATE CAMERAS

+”

257

P

+V

(b)

(0)

FIG.43. Two types of output circuits which have been used for reduction of switching transients in XY sensors. (a) Double-sampled integrating amplifier (Brugler (loo)]. (b) Differential amplifier with dual input [Weimer er a / . ( 1 3 ) ] .

charge integration technique should be as effective as the sample-and-hold technique described in the previous section for removal of switching transients it would probably not be as effective as that system for suppressing the fluctuation noise associated with the high capacitance of the output bus. All double-sampling schemes become progressively more difficult to implement as the scanning frequencies are increased. The application of a differential amplifier for suppression of switching transients in a single-line photodiode sensor is illustrated in Fig. 43b. This system (103) was found to be useful in scanning the 256 x 256 element photoconductive array shown in Fig. 8. When each elemental capacitor is discharged by the closing of the multiplexer switch the signal currents flowing in the two output leads V, and V, are of opposite polarity. However, the switching transients induced in each lead by the scan generator are approximately equal and will have the same phase. It is therefore possible to subtract one output signal Srom the other to obtain a larger output signal with smaller switching transients than were present in either output alone. The differential amplifier system does not require the auxiliary channel to contain picture signal to be effective. Various other structures have been used to provide a separate transient signal which could then be subtracted from the total picture signal plus transient signal contained in the other channel. As the clock frequencies are increased the achievement of a satisfactory cancellation of transient signals becomes increasingly difficult. XIV. CONCLUSIONS The replacement of beam-scanned camera tubes by solid state sensors, if it occurs, will be a most impressive achievement, surpassed only by the successful development of a solid state display panel suitable for television.

258

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The sophisticated objectives of television pickup have required more than fifty years of continuous research and development on camera tubes. The resulting tubes, in specialized forms, have approached ideal performance in sensitivity, resolution, and freedom from spurious signals and they are fairly low in cost when picture defects can be tolerated. However, camera tubes are often bulky, power consuming, critical to operate, short-lived, or unresponsive to the desired wavelengths, and they are far too costly if high performance is demanded. Recent advances in implementation of the powerful new concept of scanning by charge transfer now indicate the technical feasibility of building solid state sensors which approach the performance of camera tubes in most respects within five or ten years. Whether this can be done at sufficiently low cost to undersell the comparable tube will depend upon the size of the market and on the continuation of an extensive research and development program on sensors. In specialized applications, where the desirable features of solid state devices are required, a higher cost for image sensors may be tolerable for some time to come. Solid state sensors will be capable of more compact cameras, lower power consumption, a more rugged construction, higher responsivity, and longer life than ordinarily obtained with tubes. However, these features may be accompanied by other disadvantages such as a requirement for cooling, or the spreading of the signal around brightly illuminated objects. Fortunately, the fabrication of solid state sensors is based upon a silicon integrated circuit technology in which the cost per component on the chip has decreased at a fantastic rate within recent years. However, high resolution sensors will require active areas of silicon at least ten times larger than the largest present-day LST circuits. The prospects for cost reduction in sensors are fairly good but further advances in technology will be required. Solid state sensors have already found their first market applications in single-line sensors and limited-resolution area arrays. If low cost arrays can be built having a sufficient number of elements to be compatible with standard display equipment, many new industrial and consumer applications for sensors will unfold. The use of sensors with a low-cost TV recording system could provide a form of electronic photography which could rival home movie cameras. Economical and trouble-free operation appears to be the key to the expanded use of sensors in the home, business, school, and industry. Clearly, solid state sensors could coexist with camera tubes for many years with each serving its own market. ACKNOWLEDGMENTS The author wishes to thank G. F. Anielio. J . E. Carnes. R. H . Dyck, W. F. Kosonocky. R. L. Rodgers, 111. C. H. Sequin. G. Strull. M . F. Tompsetl. and their co-authors for permission to reprint figures which first appeared i n their papers.

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259

REFERENCES 1. D. F. Barbe and S. B. Campana, Electro-Opt, Conf;. Suss. Low Light Level Trlec. Syst., New York. 1973. 2. J. S. Brugler, J. D. Meindl, J. I>. Plummer. P. J. Salsbury, and W. T. Young, I E E E J . Solid-State Circuits 4, 304 (1960). 3. P. K. Weimer, H. Borkan. G. Sadasiv, L. Meray-Horvath, and F. V. Shallcross, Proc. I E E E 52, 1479 (1964). 4. F. L. J. Sangster and K. Teer. I E E E J . Solid-Srote Circuit5 4, 131 (1969). 5. W. S. Boyle and G. E. Smith, Be“ Syst. Tech. J . 49, 587 (1970). 6. C. H. Sequin, D. A. Sealer. W. J. Bertram. Jr.. M. F. Tompsett. R. R. Buckley. T. A. Shankoff. and W. J. McNamara, I E E E Trans. Electron Derices 20. 244 (1973). 7 . P. K . Weimer, Adcan. Elecrrorl. Ekectrori Piiys. 13, 387 (1960). 8. P. K. Weimer, S. V. Forgue. and R. R. Goodrich, Electronics 23, 70 (1950). 9. M. H. Crowell. T. M. Buck. E. F. Labuda. J. V. Dalton, and E. J. Walsh, Bell Syst. Tech. J . 46, 491 (1967). 10. S. R. Hofstein and F. P. Heiman, Proc. I E E E 51, I190 (1963). 11. G. P. Weckler, I E E E J . Solid-Stare Circuits 2, 65 (1967). 12. G. J. Michon and H. K. Burke, I E E E I n t . Solid-State Circuits Coi& Dig. Tech. Pap.? Philotlrlpliiu p. I38 (1973). 12u. G. J. hlichon and H. K. Burke. I E E E Sulid-Sttr/e Circuits Conf:, Diy. Tech. Pup., Pliiladelpliia. p. 26 (1974). 1 3 , P. K. Weimer, W. S. Pike, G. Sadasiv, F. V. Shallcross, and L. Meray-Horvath, I E E E Spectrum 6, 52 (1969). 14. P. K. Weimer, G. Sadasiv. J. E. Meyer, Jr.. L. Meray-Horvath. and W. S. Pike, Proc. I E E E 55, 1591 (1967). 15. J. W. Horton, R. V. Mazza, and H. Dym. P m . I E E E 52, 1513 (1964). 150. M. V. Whelan and L. A. Daverveld, I E E E I n r . Electron Devices Meet., I E D M Tech. Dig., Wtrshiiiyron, D.C., p. 416 (1973). l 5 h . 1. Kaufman and J. W. Foltz, Pvoc. I E E E 57, 2081 (1969). 16. G. Strull, W. F. List, E. L. Erwin, and D. Farnsworth, Apyl. Opr. 11, 1032 (1972). 1 7 . R. H. Dyck and G . P. Weckler, I E E E Trans. Elecrron Deuices 15, 196 (1968). 18. T. Ando et a/., J . Inst. Telec. h g . J a p . pp. 33-46 (1972). 19. P. J. W. Noble, I E E E Trans. Electron Devices 15, 202 (1968). 20. M. G. Kovac, W. S. Pike. F. V. Shallcross. and P. K. Weimer. Electronics 45, 72 (1972). 2 I . E. H. Snow, I E E E In/ercon Tech. Pap., N e w Y o l k Sess. 37, Pap. 37-2 (1973). 22. P. K. Weimer, G. Sadasiv. L. Meray-Horvath. and W. S. Homa, Proc. I E E E 54,354 (1966). 23. P. K . Weimer, R C A Rer. 32, 251 (1971). 24. R. A. Anders, D. E. Callahan, W. F. List, D. H. McCann, and M. A. Schuster, I E E E Trai7s. Electron Devices 15, 191, 1968. 25. M. A. Schuster and G. Strull, I E E E T r m y . Ekectrou Deriws 13, 906 (1966). 26. P. K. Weimer, F. V. Shallcross. and V. L. Frantz, I E E E J . Solid-Sttrte Circuits 6. 135 (1971). 27. C . P. Weckler, I E E E Intercon Tech. Pap. New f’ork. Sess. 1 , Paper 1;2 (1973). 28. Reticon Corporation, Mountain View, California. 29. See “Advanced Scanners and Imaging Systems for Earth Observations,” Ch. 4. N A S A Spec. Puhl. SP-335 (1973). 30. G. Sadasiv, Solitl-State Sensor Symp., Cur$ Re(,.. Minneupolis, Miriri. p. 13 (1970) (IEEE Catalog No. 70C25-SENSOR). 31. E. Arnold, M. H. Crowell. R. D. Geyer, and D. P. Mathur. I E E E Trirns. Electron Dei,ices 18. 1003 (1971).

260

P. K. WEIMER

3-7. F. L. J. Sangster, I E E E Irzt. Solid-Strife Circuits Conf:. Dig. Twh. Pap., 13. 74 (1970). 33. A. J. Steckl and T. Koehler, C C D A p p l . Con$ Proc., Nur.. Electron. Lab., Smi D i q o , CaliJ

TD-274, p. 247 (1973). C. H. Skquin, I E E E Truns. Electron Decices 20, 535 (1973). G. Amelio, I E E E intercon Tech. Pup., New York Pap. Ii3 (1973). P. K. Weimer, W. S. Pike, M . G. Kovac, and F. V. Shallcross. I E E E I n r . Solid-Srute Circuits Cot$, Dig. Tech. Pup., Philudelphiu p. 132 (1973): also in Data Comniiiii. Design 2, No. 3. 21 (1973). 360 P. K. Weimer. W. S. Pike, F. V. Shallcross, and M. G. Kovac, RCA R r r . 35 No. 3, 341 (1974). 3 7. M. G. Kovac, P. K. Weimer, F. V. Shallcross, and W. S. Pike, f E E E Int. E k f r o i i Detices Meer., Absri., Wnshington, D.C. p. 106 (1970). 38. P. K. Weimer, M. G. Kovac, F. V. Shallcross, and W. S. Pike, I E E E Trtrris. Electron Devices 18, 996 (1971). 3Y. W. S. Pike, M. G. Kovac, F. V. Shallcross, and P. K. Weimer, RCA Reo. 33. 483 (1972). 40. G. F. Amelio, W. J. Bertram, Jr., and M. F. Tompsett, I E E E Trans. Electron Devices 18, 986 (1971). 41. M. F. Tompsett. G. F. Amelio, W. J. Bertram, Jr., R. R. Buckley, W. J. McNamara. J. C . Mikkelsen, Jr., and D. A. Sealer, I E E E Trans. Elecfron Devices 18, 992 (1971). 42. W. J. Hannan, J. F. Schanne, and D. J. Woywood, I E E E Trans. Mil.Elecfroii. 9. 246 (1965). 43. C. N. Berglund and H. J . Boll, I E E E Trans. Electron Devices 19, 852 (1972). 44. M. G. Kovac, F. V. Shallcross, W. S. Pike, and P. K. Weimer, I E E E Electroii Derice.5 Meer., Abbrr., p. 74 (1971). 45. L. Boonstra and F. L. J. Sangster, Elecrroriics 45, 64 (1972). 46. W. F. Kosonocky and J. E. Carnes, I E E E I t i r . Solid-State Circuit5 Con$. Dig. Tech. PUIJ., p. 162 (1971). 4 7. W. F. Kosonocky and J. E. Carnes, R C A Rev. 34, 164 (1973). 4 7tr W. F. Kosonocky and J. E. Carnes. I E E E Inr. Elecrron Devices Meet., I E D M Tech. Dig., Washington, D.C. p. 123 (1973). 48. W. S. Boyle and G. E. Smith, I E E E Specrrurn 8, 18 (1971). 49. R. H. Krambeck. R. H. Walden, and K. R. Pickar, Bell Sysr. Tech. J . 51. 1849 (1972). 50. W. F. Kosonocky, private communication (1973). 51. D. R. Collins. S. R. Shortes, W. R. McMahon, R. C. Bracken. and T. C. Penn. J . Elecrrochern. Soc. 120, 521 (1973). 5 1( I C. H. SCquin, D. A. Sealer, W. J . Bertram, R. R. Buckley. F. J . Morris, T. A. ShankolT. and M. F. Tompsett, I E E E Solid-Srufe Circuits Corif: Dig. Tech. Pup., Philutfelphiu. p. 24 (1974). 52. J. E. Carnes, W. F. Kosonocky, and E. G. Ramberg, I E E E Truns. Electron Derices 19. 798 ( 1 972). .. i?C. K . Kim and M. Lendinger. .I. , ~ / J / I / . f'lij,\. 42, 3586 (1971). 54. J. E. Carnes and W. F. Kosonocky, R C A Eng. 18, 7 8 (1973). 5 5 . J. E. Carnes and W. F. Kosonocky, A p p l . Phys. Lett. 20, 261 (1972). 56. M. F. Tompsett, I E E E Trans. Electroi7 Devices 20, 45 (1973). 5 7 . R. H. Walden, R. H. Krambeck, R. J . Strain, J. McKenna, N. L. Schryer, and G. E. Smith, Bell Syst. Tech. J . 51, 1635 (1972). 58. K. C. Gunsagar. C. K. Kim, and J . D. Phillips, I E E E Int. Electron Deaices Meet.. I E D M Tech. Dig., Washiiiyton, D.C. p. 21 (1973). 5Y. L. J. M. Esser, CCD Appl. Con/; Proc.. NuL.. Electron. Lab., Sari Diego, Cul$ TD-274. p. 269 (19731. 60. J . E. Carnes and W. F. Kosonocky, RCA Rev. 33, 327 (1972). 34. 35. 36.

IMAGE SENSORS FOR SOLID STATE CAMERAS

26 1

61. J. K. Thornber and M. F. Tompsett, I E E E Trans. Electron Decicrs 20, 456 (1973). 62. D. D. Wen and P. J. Salsbury, I E E E Solid-State Circuits Conf, Dig. Tech. Pap., Philadelphia p. 154 (1973). 63. M. F. Tompsett, G. F. Amelio, and G. E. Smith, Appl. Phys. Lett. 17, 111 (1970). 64. M. F. Tompsett, D. A. Sealer, C. H. Sequin, and T. A. Shankoff, I E E E Intercon Tech. Pap., New York Sess. 1, Pap. 1/4 (1973). 65, C. K. Kim and R. H. Dyck, l’roc. I E E E 61, 1146 (1973). 66. W. J. Bertram, Jr., D. A. Sealer, C. H. Sequin, M. F. Tompsett, and R. R. Buckley, I E E E Intercon Dig. p. 292 (1972). 67. 0. H . Schade, Sr., R C A Rec. 31, 60 (1970). 68. M. G. Kovac, F. V. Shallcross, W. S. Pike, and P. K. Weimer, C C D Appl. Co$ Proc., Nac. Electron. Lab., Sun Diego, Calif TD-274, p. 37 (1973). 69. R. L. Rodgers, 111, I E E E 1tire.rcon Tech. Pap.. Sess. 2, Pap. 2,;3 (1974). 70. S. B. Campana, Electro-Opt. Syst. Design June. p. 22 (1971). 71. C. H. Sequin, Bell S ! s . Tech. J . 51, 1923 (1972). 72. W. F. Kosonocky. J. E. Carnes, M . G. Kovac. P. Levine. F. V. Shallcross. and R. L. Rodgers. 111. RC-1 Ref.. 35. No. 1 . 3 (1974). 73. L. Walsh and R. H. Dyck, C C D Appl. ConJ Proc., NUV. Elrcrron. Lab., Sun Diego. CuliJ TD-274, p. 21 (1973). 74. W. B. Joyce and W. J. Bertram, Bell S j r t . Tech. J . SO, No. 6, 1741 (1971). 75. M. F. Tompsett, J . Vac. Sci. Technol. 9, 1166 (1972). 76. A. Rose. Aduan. Electron. 1, 131 (1948). 77. R. W. Engstrom and G. A. Robinson, Electro-Opt. Syst. Design ST-4693 (1971). 77a. 4849 I-SIT Camera Tube. R<:A Electronic Components Data Sheet printed 7/73. 77h. 4804 SIT Camera Tube, RCA Electronic Components Data Sheet printed 7/73. 78. R. W. Engstrom, private communication (1973). 79. J. E. Carnes and W. F. Kosonocky, R C A Rec. 33, 607 (1972). 80. S. B. Campana, CCD Appl. Conj: Proc., Nar. Electron. Lub., Sail Diego, Culij: TD-274, p. 235 (1973). XI. S. R . Shortes, W. W. Chan, W. C. Rhines, J. B. Barton, and D. R. Collins, I E E E Irit. Elecrruri Decices Meet., IEDM Tech. Dig., Washington, D.C. p. 415 (1973). 82. F. H. Nicoll, in “ Photoelectronic Materials and Devices ” (S. Larach, ed.), Ch. 8, p. 313. Van Nostrand-Reinhold, Princeton, New Jersey, 1965. 83. B. Kazan and M. Knoll, “Electronic Image Storage.” Academic Press, New York. 1968. 84. F. L. J. Sangster, Philips Tech. Rev. 31, 97 (1970). 85. J. J. Tieman, R. D. Baerstsch, and W. E. Engler, CCD Appl. Cot$ Proc., Nur. Elrctron. Lab.. Sun Dieyo, Culij: TD-274, p. 103 (1973). 86. T. F. Cheek, Jr., A. F. Tasch. Jr., J. B. Barton, S. P. Emmons, and J. E. Schroeder, CCD Appl. Con$ Proc., Nar. Electron. Lub., Suit Diego, Calif: TD-274, p. 127 (1973). 87. P. K. Weimer, U S . Pat. No. 3,763.480 (1973). 88. M. F. Tompsett and E. J . Zimany, Jr., I E E E J . Solid-State Circuits 8, 151 (1973). 89. D. D. Collins, J. B. Barton, D. D. Buss, A. R. Kmetz, and J. E. Schroeder, I E E E In/. Solid-State Circuits Con$. Dig. Tech. Pap., Pliiludelphiu p. 136 (1973). 90. P. Levine, I E E E J . Solid-Stare Circuits 8, 104 (1973). 91. H. Breimer, W. Holm, and S. L. Tan, Philips Tech. Rep. 28, No. 11, 336 (1967). 92. P. K. Weimer, S. Gray, C. W .Beadle, H . Borkan, S. A. Ochs, and H. C. Thompson, I R E Truiis. Electron Devices 48, 147 (1960). 93. D. H. Pritchard, R C A Rec. 34, 217 (1973). 94. M. H. Mesner, R C A Eng. 19. 30 (1973). 95. M. F. Tompsett, W. J. Bertram, D. A. Sealer, and C. H. Sequin, Electronics 46, 162 (1973). 96. J. E. Carnes, W. F. Kosonocky, and P. A. Levine, R C A Reo. 34, No. 4, p. 553 (1973).

262 47. YX.

99. 100

I(J1. 102.

102.

P. K . WEIMER

M. F. Tompsett. CCD A p p l . Cot!/. Proc.. Wac.. Elecrrm. Lob.. S i r 1 Dirgo. Cu/I/:.TD-274. p. 147 (1973). S. P. Enimons and D. D. Buss. paper presented at Solid State Dcvice Research Conference, June 26-28, 1973, Denver. Colorado (Abstr). I E E E Trans. Elecrror7 D ~ l - i c r . ~ 20. 1172 (1973). M. H. White, D. R. Lampe. I. A. Mack, and F. C. Blalia. I E E E Solid-Sttrrr Circuits CouJ:,Dig. Tech. Pay., p. 134, (1973). J. S. Brugler. Tech. Rep. 4824-1, Solid State Electron. Lab., Stanford University. Stanford, California, May 1968. P. K. Weimer. M. G. Kovac, W. S. Pike. and F. V. Shallcross, Final Rep. Contr. NOO14-71-C-0415. Office of Naval Research. September 1972. J . D. Pluminer and J. D. Meindl, J . Solid Srute Circuits 7. I 1 1 (1972). P. K. Weiiner. G. Sadasiv, and W. S. Pike. Tech. Rep. AFAL-TR-68-82. Contr F336 15-67-C-1550. Wright Patterson Air Force Base, Ohio. April 1968.

Ion Implantation in Semiconductors SUSUMU NAMBA

AND

KOHZOH MASUDA

Facrrlty q j Engineeriri
I . Introduction ............................................................................................. A. Historical Review ................................................................................. B. Concept of Ion Implantation .................................................................. I1 . Concentration Profiles of Implanted Ions and Defects ....................................... A . Introduction ...................................................................................... B. Amorphous Target ............................................................................. C. Single Crystal Target ........................................................................... D. Channeled Particle .............................................................................. 111. Enhanced Diffusion .................................................................................... A . Introduction ....................................................................................... B. Diffusion Assisted by Vacancies Produced by High Energy Radiation ............ C . Interstitial Diffusion Mechanism for Enhanced Diffusion .............................. D . Radiation Enhanced Diffusion ............................................................... IV . Annealing and Electrical Properties ............................................................... A . Introduction ....................................................................................... B. Silicon ................................................................................................ C . GaAs ................................................................................................ V . Measurement Technique ........................................................................... A . Introduction ....................................................................................... B. Channeling Effect Technique .................................................................. C. Activation Analysis .............................................................................. D. ESR Method ...................................................................................... VI . Devices .................................................................................................. A. Introduction ...................................................................................... B. Gate-Masked Ion Implanted MOSFET ................................................... C . Threshold Voltage Control ..................................................................... D. Bipolar Transistor ................................................................................. E . IMPATT .......................................................................................... References ................................................................................................

263

264 264 267 271 271 272 285 288 289 289 291 295 297 299 299 300 305 3 10 310 311 323 324 325 325 325 326 326 327 328

264

SUSUMU NAMBA A N D KOHZOH MASUDA

I. INTRODUCTION A . Historical Review

Ion implantation has a long history in the field of fundamental physics but a rather short one in the development of technology. When the accelerated ions bombard a solid target, most of the energetic ions penetrate into the target and each ion stops at a certain point. This is the phenomenon of ‘‘ ion implantation” and is treated as a collision cascade of incident particles and primary knocked-on atoms of the target material. The processes of the energy transfer between incident atoms and target atoms depend upon the velocities of the incident ions, the masses of the incident ion and target atoms, and the angle between the direction of the scattered atoms and that of the incident beam. Rutherford developed the atomic model from the analysis of a-particle scattering experiments in 1911. In 1912 Stark predicated that the ions penetrate the crystal at a certain crystal direction deeper than other directions, because of the “open channels” which exist at this direction. This was a great discovery concerning the interactions between the beam and the crystal. But at that time, most scientists concentrated their attentions on the X-ray diffraction phenomena found by Laue rather than to channeling phenomena. Only Bragg considered the relation between them. But the development of the physics in this field was not successful because of the insufficient state of the art of ion beam technology at that time. Excellent fundamental studies were carried out on the interactions between energetic ions and solids, but few applications were made except in nuclear physics experiments. Recently, etching, machining, and sputtering by ion beam and ion implantation have been developed. Also stimulated research on the method of generation of ions, the formation of ion beams, and the ion beam radiation effects in solids have been performed. Fundamental knowledge of ion implantation was known relatively early, but project research for industrial needs started only in 1962. The reasons why this new technique was developed extensively are based upon two points. One is that the technical development of beam handling was well enough established to perform ion implantation in an industrial environment. The other is the recent requirement for the production of precise semiconductor devices. The two conditions coincided with each other in time. The research on radiation effects in solids and on thermal annealing, research on high energy and high current beams, and the discovery of channeling phenomena provide the technical backgrounds for ion implantation. But, on the other hand, a great motive force in accelerating the development

ION IMPLANTATION IN SEMICONDUCTORS

265

has been the recognition by scientists of the necessity for the appearance of a new technique-ion implantation-which will break the technical barrier arising from the imprecision of the thermal diffusion method. The first attempt to apply ion implantation to the controlling electrical properties in semiconductor devices was carried out by Oh1 of Bell Telephone Laboratories and his results were reported in 1952. Electrical properties of point contact rectifiers were improved by 20 30 keV He' ion bombardment on silicon single crystal heated at 300" 400°C, and a solar battery which had high sensitivity in the shortwave region was made by fabrication of the junction very near to the surface by H + bombardment on p + silicon. Patents relating to implantation were obtained by Watanabe and Nishizawa in 1950 and patents on the formation of p-n junctions were obtained by Schockley in 1954. Cussins in the United Kingdom experimented with the bombardment of ions from H + to Sb' accelerated with 5 90 keV. In 1955, he concluded that the effects of the bombardments on Ge are caused only by defects at the germanium surface which act as acceptors and are not caused by implanted ions. It was clear that p-n junctions can be made by ion bombardment but it was not clear whether the implanted ions or defects produced in the crystals by ion bombardment affect the electrical properties. This uncertainty of the origin of the p-n junction caused some delay in the development of the ion implantation technique even though much progress was made around 1955. In 1961 Lindhard in Denmark calculated the energy loss process when the ion penetrates amorphous materials. He obtained a theoretical equation in good agreement with experimental values. Around 1960, Davies in Canada carried out precise measurements of the depth distribution of implanted ions by isotope implantation. He counted the residual activity of the sample with successive etchings of the implanted surface. Davies' experiments on isotope implantation justified the existence of ions in the sample and the depth distributions of the implanted ions were found to coincide with Lindhard's calculation. In addition, Davies found some extra distributions of ions which were located deeper than Lindhards profile calculation. This was experimental proof of the channeling phenomena. Robinson in the United States did computer simulation on the energy loss of 1 10 keV C u ions in fcc and bcc diamond structure crystals and showed that the ion ranges are strongly dependent upon the crystal direction. The calculations are shown in Fig. 1. Nelson and Thomson in the United Kingdom did experiments on the penetration of 75 keV H f into 3000 A gold single crystal thin films and

-

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266

SUSlJMU NAMBA AND KOHZOH MASUDA 2000

I

lI

l

1

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100

50

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3

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100

200 300 PENETRATION

400

500

600

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5-kev Cu Atoms Slowing Down in Cu, Born-Mayer Potential; Static Lattice; Initial Directions as Given on Curves.

FIG.1. Directional effects of penetration of incident particles

found that the H f ions penetrate easily in the (1 10) direction. Those experiments, together with theoretical considerations, are proof of the existence of open channels. This occurred about 50 years after Stark's predictions of open channels. Applications of ion implantation to semiconductor devices were made about 1962 and the electrical properties of p-n junctions produced by ion implantation were reported in 1963. Great advantages in MOS fabrication technology were found in the United States where some of the barriers to the thermal diffusion technique were broken down. The directions of research on ion implantation can be seen by a consideration of the series of International Conferences. Sessions on ion implantation had been included in the conferences on atomic collisions in solids which were held in 1965, 1967, and 1969. The International Conference on the Application of Ion Beams to Semiconductor Technology was also held at Grenoble, France in 1967. The first International Conference on Ion Implantation in Semiconductors was held in the United States in 1970. Many fundamental studies on the interactions between the ion beam and atoms in the solids, and the possibilities of applications for new devices were introduced. The second International Conference (Garmisch-Partenkirchen)

ION IMPLANTATION IN SEMICONDUCTORS

267

and the European Conference were held in 1971. The U.S.-Japan Seminar was held in the summer of 1971 at Kyoto, Japan. Enhanced diffusion for the method of long range profile performance, low temperature annealing phenomena such as injection annealing and low temperature recovery of crystallinity from amorphous states, and backscattering measurement techniques for determining the atom locations in the lattice and lattice imperfections were presented. New methods of fabrication of devices were also presented. The third International Conference was held at the IBM Watson Research Center in the United States in 1972. Subjects of ion implantation were expanded to include metal. insulator, and all other materials rather than only semiconductors. Ion implantation is a superior technique for producing new materials which are very difficult or Impossible to produce by chemical methods. B. Concept of

loii

Implantation

1. Fundamentals a. Impurities. Generally speaking, the existence of many kinds of crystal structures give us models of the interaction forces between atoms or ions. Physical properties of the bulk materials are those of a perfect crystal. Small imperfections in nearly perfect crystals are very helpful in providing a clearer understanding of solid states physics. This is a sort of perturbation method. Light irradiation and a variation of the sample temperature were used to produce a change in the number of electrons and holes with photons and phonons. In this case the major configurations of the atoms are not changed. On the other hand, ion implantation is the introduction of foreign atoms or lattice irregularities in the solid. An important feature of the introduction of foreign atoms by ion implantation is that this method is applicable to all kinds of materials. Even though the applicability of ion implantation to all types of materials is obvious, the technique for the production of intense, stable, monoenergetic ion beams is difficult. One of the useful features of ion implantation in solid state physics research is the very high purity of the implanted ions. The ion implantation machine was originated from the isotope separator, so that generally the purity of the ions is extremely good; however in cases of the same masses, for example *'Si and (14N)2,separation of "Si and (I4N), by a magnetic field is impossible. Thus extra care should be taken in these cases. Depth profiles of the implanted ions can be calculated reasonably well by the Lindhard theory in the case of amorphous material. This is a very important result for understanding the elemental collision processes and

268

S U S U M U NAMBA A N D KOHZOH MASUDA

also for the application of ion implantation to the field of electronic device fabrication. The results obtained using the Lindhard theory show us that a more abrupt junction can be obtained by ion implantation than the conventional thermal diffusion method. Another method for making an abrupt concentration difference is epitaxial growth. However the uniformity of the impurity concentration within the epitaxial layer in a lateral direction is worse than that of the implanted layer. The depth profile of implanted ions in the case of an amorphous target is close to a Gaussian distribution. The Lindhard theory can tell us the exact range and the straggling of the distribution of implanted atoms in the vicinity of the range. At a depth less than the range, and in the tail deeper than the range, a decrease and an increase of the concentration of implanted atoms is observed, respectively. The backscattering effect of the incident particles and enhanced migration effects should be considered in these cases. These effects are subjects to be resolved in the future. Even below room temperature the migration of interstitial Si atoms was observed when they are bombarded by energetic particles. Many unresolved problems still remain at above room temperature experimental conditions. Fundamental studies are needed to understand the electronic and atomic behavior in the heat treatment process for annealing and age effects of devices. Techniques to determine the migration behavior and location of impurity vacancies and interstitial atoms are backscattering and ESR and NMR measurements. Some of the work on gold migration into silicon done by Mayer et al. by the backscattering method is one of the revolutionary attempts to understand the mechanism of abnormal diffusion. Watkins found the interstitial migration of Si at low temperatures by the measurement of the ESR signal of Al. Apart from bombardment techniques, which can move the atom during the measurement, possible methods for detecting the existence of impurity atoms and their location consist of NMR and ESR measurements. Quadrupole effects in the NMR spectra of metals implanted with I2B and 14N ions show that the implanted ions remain at the interstitial site until, after a certain lifetime, they change substitutional sites (I). In this case, it is suggested that the migration of the atoms is different in the cases of ion implanted species and thermally diffused species. ESR spectra of Sb implanted silicon are shown in Fig. 2a,b. Two kinds of isotopes of 121Sband 123Sbexist in natural antimony. The abundances are 57.25':/, and 42.75%. The number of components of the hyperfine splitting of the ESR signal of nuclear spin I is 21 + 1. (1)

269

ION IMPLANTATION IN SEMICONDUCTORS

Isotopes '"Sb and 12%b are separated when they are implanted. The ESR spectrum of the '"Sb implanted sample shows six equally separated components of the signal. This is a clear proof of the existence of "'Sb in the silicon substrate and also proof of the nonexistence of lZ3Sb and other impurities. A similar proof can be given in the case of lZ3Sbimplantation (2,3). 121 Sb(1.92

1. W9.,798592

123Sb(1=7/ 2 ) . Y=19.801502

GHz

GHz

7000 Oe 7100 Oe MAGNETIC flELD

FIG.2. ESR spectra of implanted '"Sb

7200 (a) and

oe

4

(b).

Normally the atoms occupy the most stable site in the crystal. The lattice point is one of the most stable places in the crystal for an atom. Other sites are considered to be unstable compared to the normal lattice site. A quantitative explanation for this difference is not yet available. But the results of ion backscattering, quadrupole shift in NMR, and radiation activation analysis strongly suggest that the ratio of the nonsubstitutional impurity to substitutional impurity in the case of implantation is larger than in the case of thermal diffusion. b. Vacancies. In the early stages of research on radiation effects, X-rays and ?-rays were used for the purpose of creating a uniform distribution of vacancies throughout the bulk of the sample. In the case of research on radiation effects, the mechanisms of the displacement of the atoms and the production and migration of vacancies and chemical reactions introduced by high energy radiation were studied. Many results in the field of semiconductors were summarized and reviewed at the Santa Fe Conference on Radiation Effects in Semiconductors ( 4 ) .In the case of fundamental research on radiation effects, the formation of isolated impurities without impurity-impurity interaction is

270

SLJSUMU NAMBA A N D KOHZOH MASUDA

necessary to obtain a clear understanding of the impurity state. But in the case of the application of ion implantation, especially in the semiconductor industry, we need considerably higher concentrations and thus observe important interactions between impurities. Even for fundamental research in the theory of impurities, we cannot ignore this occurrence of strong interactions. The problem is the same in the case of a vacancy. The efficiency of production of a foreign environment such as the introduction of impurity atoms or vacancies or their clusters is much greater in the case of ion implantation than in X-ray or ;'-ray irradiation. This high efficiency of production of vacancies is very useful from both fundamental and technological points of view. We have some methods involving ion implantation for device fabrication but few methods to obtain a clear understanding of the fundamental physics. Multivacancies rather than single vacancies are normally produced by ion implantation, because of the high flux density and high total dose of the implanted ions. One of the mechanisms in the production of multivacancies is a coagulation of single vacancies; in this case a strong ion beam intensity dependence is expected as is often observed in experiments. Another mechanism is production by successive collisions which makes divacancy formation relatively easy. Systematic studies have not yet been performed. The electronic structure of the isolated vacancies can be determined by the ESR method used for F centers and V centers in alkali halides and donor centers in Si, studied by Kanzig (5, 6) and Feher (7) respectively. The identification of many defects was made by Watkins, Corbett, Vook, and others." Resolution of the mechanisms of production and migration are the main subjects of research in the field of vacancies produced by ion implantation. Furthermore, a determination of the electronic structure should be undertaken. An incident ion creates a disordered region near its track, and overlapping of the disordered regions seems to produce the amorphous phase. Therefore, at extremely high total doses, amorphous phases are produced. The structure of this amorphous phase may be different from that in the fused silica or glass. X-ray Laue patterns show no crystalline state, ESR shows an isotropic signal which suggests the noncrystalline state, and backscattering yields a random spectrum which is typical evidence of an amorphous state. It would be interesting to know the electronic structure of the amorphous state. Carrier mobilities in the amorphous phase are low enough for them to be used as insulating layers. Isolation layers can be made by the

* They named such vacancies by using the first letter of their laboratories such as P. B. G, a i d S for Purdue University, Bell LdboratOrieS. General Electric Co., and Sandia Laboratories.

ION IMPLAhTATION IN SEMICONDUCTORS

27 1

production of an amorphous layer in an integrated circuit by ion implantation. The annealing temperature of the damaged region is often assumed to be proportional to the degree of damage. This is true until the degree of damage is less than in the amorphou's phase. When the degree of damage increases and the amorphous phase is completely formed, the annealing temperature for the electrical properties decreases to 550°C in silicon. The annealing behavior of the amorphous phase is very different from that of the only heavily damaged layer.

2. Technologies With the early applications of ion implantation, controllability, reproducibility, uniformity of impuritj doping, and cost of production were assumed to be superior to the conventional diffusion method. In fact, every item except cost is really more favorable than for the conventional thermal diffusion method. Many devices such as MOS, JFET, voltage variable capacitors, IMPATT, transistors, and IC are produced by the ion implantation technique. The most important features of ion implantation in these cases are controllability and uniformity of concentration of the implanted atoms. The application of ion implantation to group 111-V compounds such as GaAs is more complicated because of the differences of the diffusion coefficients of Ga and As and their vacancies in GaAs. The production costs of the ion implantation and thermal diffusion methods are considered to be similar, but exact comparisons are extremely difficult to obtain. Dry method. The performance of the dry method in the process of semiconductor fabrication is one of the ultimate objects of engineers. With the establishment of a significant ion implantation technology and the requirement of submicron technology, it is to be hoped that a combination of computer controlled processes with dry technologies such as molecular beam evaporation, electron beam fabrication, ion implantation and ion beam machining will be used in the near future over a considerably wide field in the semiconductor industry instead of the conventional wet method such as chemical treatments for development and etching. 11. CONCENTRATION PROFILES OF IMPLANTED IONSAND DEFECTS A . Introduction

Many calculations and experimental studies were carried out to obtain precise concentration profiles of implanted ions. Knowledge of the concentration profiles of ions and lattice defects is quite useful for device fabrication

272

SUSUMU NAMBA AND KOHZOH MASUDA

and for physical considerations of atomic collisions. Lindhard, Scharf, and S c h i ~ t (referred t to subsequently as LSS) calculated a concentration profile of both implanted ions and lattice defects. This calculation is the most acceptable computation of ion ranges. Since the trajectory of a scattered atom is easily changed by the shape of the atomic potential for a collision, it is difficult to obtain exact values for a series of scatterings. Many calculations on details of the concentration profile were made by Gibbons, Brice, Winterbon, and Furukawa. Experimental determinations were carried out by Davies, Gibbons, Dearnaley, and Namba. In the case of implantation into amorphous materials, projected ranges and range straggling may be calculated from LSS theory. Some assumptions concerning the shape of the potential field and approximations in polynomials and moments are necessary to perform the numerical calculations. The calculated curves under these conditions exhibit a nearly gaussian distribution. Some deviation from gaussian behavior was found but usually the skewness of the distribution is not large. This will be described in a later section. In the case of single crystals, enhancement of the penetration was observed. Radiation enhanced diffusion is assumed to result from the increase of radiation induced vacancies, the occurrence of the secondary channeling,' and interstitial migration. An analysis of the mechanisms of enhancement will give us new insight into the migration of atoms in crystals. This is also quite a useful technique for device fabrication. B. Amorphous Target The energy transfer between an ion and a target atom is treated as a series of single binary collisions. In this treatment an isotropic distribution of scattering centers is assumed; therefore an amorphous material is suitable for the target. Even a single crystal has atom rows but when the beams bombard a crystal in directions not parallel to any atom row, this single crystal is supposed to be effectively equivalent to an amorphous state in the scattering process. If the concentration profile is in agreement with the calculation of LSS, the technique of ion implantation is expected to result in a steeper abrupt junction than occurred in the conventional thermal diffusion method which is governed by the well-known complementary error function. In the case of light ion bombardment, the Thomas-Fermi assumptions in LSS theory are less justifiable but even in Li or He implantation the agreement between the calculation and experiment is satisfactory. An example of the agreement between experiment and LSS theory is shown in *A detailed explanation of the secondary channeling is described in Section 1I.C.

ION IMPLANTATION IN SEMICONDUCTORS

273

Fig. 3 (8).This sample was previously bombarded by Ne+ ions to obtain an amorphous phase. Profiles of 32P implantation are shown with the calculated values of LSS. 105 r

5

104

5 m W Y

z

z 0

103

Id

I

I

I

0.1 02 0.3 0.4 05 DEPTH, MICRONS OF SILICON (AMORPHOUS )

FIG. 3. Comparison between theoretical calculation and experimental value of concentration profile. 0 , Amorphous layer ("Ne), 40 keV, 5 x 101Z/cm2, 32P, room temperature; x , amorphous layer (*ONe), 120 keV. 5 x 10'Z/crnZ,32P, room temperature. Calculated from Lindhard er a/. (9).

1. LSS Theory

The calculation of the total range of implanted particles had been performed by Lindhard et al. (9, 20). The average energy loss per unit path length is dE

_dR _

=N.S=N('doT,

where A', S, do,and T are the number of scattering centers, the cross section for energy transfer, the differential cross section, and energy transfer, respectively.

274

S U S U M U NAMBA A N D KOHZOH MASUDA

R ( E ) is obtained as follows:

or ” 1 ’ dE‘/[S”(E‘)+ Se(E’)], N-0 1

R ( E )=

where S,(E’) and S,(E’) are nuclear stopping power and electronic stopping power, respectively. Then the average square fluctuation in range (AR)2 is given by

(4) and

(AE)’

=

NQ’(E) dR,

In the high energy region, the electronic stopping power increases with decreasing particle velocity and has a maximum for a certain velocity. But in the low energy region, the electronic stopping power is proportional to the particle velocity, c. The electronic stopping power in the latter region is

1’

< r1 = ro . z?’3,

(6)

where

i”, 2 z;C’,

go =

e”f2.

2 2 3

=

z: + zy, 3

In many cases of ion implantation with heavy ions and low velocities, the differential scattering cross section is given by

with the potential V ( r ) = ( Z ,z2e’a;-

1/.wA),

ION IMPLANTATION IN SEMICONDUCTORS

where

-

cis

T I T,

=

275

u = 0.8853~0z - ‘ I 3 ,

:‘E = 4hf,h’f2/(h’f1

+ h’f,)’E,

b (collision diameter) = 2 2 , Z 2e 2 / M , u2, and

M , = M,M,/(M,

+ M2).

In the case of s = 2, S, is approximated by a constant standard stopping cross section Sf, that is

Beside the power law potential calculation, screened potentials U ( r ) are used,

U ( r ) = ( Z ,Z 2 d 2 / r ) . qo(r/a),

(10)

where q, is the Fermi function a

=

a, . 0.8853(Z:’3 + Z22/3)-1’2.

(11)

Using a screened coulomb potential, one can obtain dimensionless parameters for range and energy:

p

=

Ml RNM24m2 ( M i + 6;)”

and

Then a universal differential cross section is obtained as nt

tla = nu2 -;

2t3

f(t‘”), 2

where

6 is the deflection in the center of gravity system. The functionf(t’/2) is shown in Fig. 4a.

276

S U S U M U NAMBA A N D KOHZOH M A S U D A

The nuclear stopping cross section is obtained from the equation

The function (d&/dp), is shown in Fig. 4b. The stopping power for s = 2, i.e. S:, and the electronic stopping power Se are also indicated in Fig. 4b.

0.6 GO.h n

$0.2 0

I bl

1

2

3

4

&’I2

FIG. 4. (a ) f ( r ’ ’) for T ’, Thomas-Fermi and Rutherford scattering potential. (b) Electronic and nuclear stopping power for I : ’ *.

For the total stopping cross section,

where

The projected range R,, which is a projection of the range on the direction of the incident beam, is observed experimentally. Vector range R,, which is the shortest distance between the incident point and the point where the ion is at rest, and R I which is a projection of R , onto the plane perpendicular to the incident beam are related by __

Rf=G+G,

(16)

as shown in Fig. 5. Fits between calculated and experimental values are shown in Fig. 6.

277

ION IMPLANTATION IN SEMICONDUCTORS

Good agreement between calculated and experimental values is obtained at higher incident energy. A small disagreement at lower energy is assumed to be caused by the channeling effect of ions in the polycrystalline Al. But also

FIG. 5. Schematic representation for R,, R,, and R,. Crosses, indicate the scattering centers.

P

30 2.0 1.o

.5

.2

1

.05 C

nq ."-

.002

,005

.01

02

.05

1

5

.2

L.

2.

lo

FIG.6 . The ranges for pure nuclear stopping are given by the solid curve, denoted as Th-F on the figure. A curve for an electronic stopping power k of 0.4 is also shown. Dotted line, p = 3.06c, is derived from the constant stopping power. 0 H 0 are experimental points of implantation of Na, A, Xe. and Cs into an A1 plate. respectively.

some enhanced diffusion may occur in this case as has been often observed in recent experiments. The moments are given by (R"-'(E))

=

N

1do,,,

E - T, L

where i means ith electron. and 1=N

(. do,,,, / R ( E ) - R ( E - T, 1 Tei -

i

.i))l,

(17)

278

SUSUMU NAMBA AND KOHZOH MASUDA

where

R(E) = (R(E)). Higher order terms in Eq. (18) are given by

2. Esamples of Numei~iccdCalcu1atioir.s Gibbons calculated the projected ranges and range straggling on the basis of LSS theory (12). He assumed a gaussian distribution, so that the following relation is justified:

N(R,) = 0 . 4 N d / a , , (20) where N d is the total number of implanted atoms per square centimeter. Then the concentration N ( x ) is expressed as 0.4Nd

exp [-(x ARP Brice calculated the projected ranges and stragglings for many kinds of ions into silicon and germanium on the basis of LSS theory (12). He calculated both the first and the second order terms in powers of the transferred energy T . Differences between the values of the first order approximation and the second order approximation are small for the range calculations but considerable differences occur between the values of the two approximations in the calculation of range straggling. Those differences are shown in Figs. 7

N(x) =

FIG. 7. Comparison of different approximations iii projected range soluiions foi- ' 'In. "Si. and I l B ions incident on amorphous silicon. - - - - - - Approximation (2) second order: approximation ( 1 ) first order.

~

~

TABLE 1 SOMER ~ S U L T SOF BRICE'SCALCULATION" BORON-11 S I L I C O N

E=400

E=300

EP

RP

DRP

DR*

EP

400 380 360 340 320 300 2 80 260 240 220 200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 305 617 9 36 1264 1601 1947 2305 2673 3056 3453 3866 4296 4747 5221 5721 6250 6810 7399 7983 8229 8248 8266 8283 8298 8311 8322 8330 8336 8339 8342

0 15 31 47 65 84 105 127 152 179 2 10 244 283 329 382 446 524 623 751 922 1027 1038 1048 1058 1068 1078 1087 1094 1101 1105 1110

0 6 19 37 61 90 125 166 2 14 269 333 406 492 591 708 846 1014 1220 1482 1817 2006 2024 2042 2059 2075 2089 2130 2114 2123 2129 2134

300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

Rp

0 35 1 713 1086 1471 1872 2288 2723 3178 3657 4161 4695 5260 5854 6444 6692 6711 6730 6746 6762 6775 6786 6794 6800 6803 6806

DRP

DR*

0 21 44 69 96 126 160 198 243 296 359 4 36 5 36 667 84 3 951 962 973 983 994 1004 1013 1021 1028 1032 1037

0 10 31 61 102 152 214 290 381 490 624 788 994 1259 1604 1801 1820 1839 1856 1873 1889 1903 1915 1924 1930 19 36

1 Z

h)

co TABLE I

0

C~oi7//mivd

BORON-11 SILICON

E=200

EslOO

EP

RP

DRP

DR*

200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2

0 428 872 1336 1824 2339 2883 3460 4066 4668 4920 4940 4959 4976 4991 5005 5016 5025 5031 5034 5037

0 34 73 117 169 2 30 306 404 5 36 718 832 84 3 855 866 877 888 89 8 907 914 919 924

0 19 61 214 211 327 478 676 941 1298 1506 1527 1546 1566 1584 1601 1616 1629 1640 1647 1653

1 0

EP

RP

DRP

DR*

100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 568 1203 1850 2493 2762 2783 2803 2822 2838 2852 2864 2874 2880 2883 2887

0 76 172 298 481 603 616 629 642 655 667 679 689 698 704 710

0 56 194 426 784 1013 10 36 1058 1080 1101 1121 1140 1155 1168 1177 1186

3

z

0

ARSENIC-75

SILICON

E=400

E=300

EP

RP

DRP

DR*

EP

RP

DRF

DR*

400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 203 360 503 636 762 883 999 1111 1220 1325 1427 1525 1621 1713 1803 1889 1973 2054 2134 2174 2178 2183 2187 2191 2196 2200 2205 2210 2215 2220

0 80 148 208 263 313 359 402 442 478 511 541 568 592 612 628 641 653 662 669 672 672 673 673 673 674 674 674 675 675 675

0 8 23 43 65 91 119 149 182 215 251 288 326 366 406 447 489 533 578 625 651 653 656 659 662 665 668 671 675 678 681

300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 204 355 486 608 724 833 938 1038 1135 1227 1316 1402 1486 1568 1610 1614 1618 1623 1627 1632 1637 1641 1646 1651 1656

0 89 157 216 267 313 35 3 389 420 447 469 487 499 510 518 521 522 522 522 523 523 523 524 524 524 525

0 10 29 53 80 110 142 177 214 252 291 332 375 419 467 494 497 499 502 505 508 512 515 519 522 526

b

TABLF I

N cio

COIIII‘IIIIL~~

ARSENIC-75

N

SILICON

E=200

E-100

EF-

RP

DRP

DR*

EP

RP

DRP

DR*

200 180 160 140 120 100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 19 7 332 450 557 657 752 842 929 1015 1058 1063 1068 1072 1077 1082 1087 1092 1097 1102 1107

0 96 161 212 255 289 316 336 349 359 36 3 36 3 363 364 364 364 365 365 365 366 366

0 14 38 67 100 136 174 214 257 305 332 335 338 341 344 348 351 355 358 362 366

100 80 60 40 20 10 9 8 7 6 5 4 3 2 1 0

0 176 286 382 475 523 528 533 5 38 543 548 553 559 564 570 5 76

0 95 144 175 190 196 196 19 7 197 197 198 198 198 199 199 200

0 22 55 93 138 165 168 171 174 178 181 185 188 192 196 200

“ E : incident energy 0 1 the particle: E P : the energy while i o n is DRP: straggling in projected range: DR”: lateral spread.

“it1

the m i d s t ” of

;I

collision: RP: projected t-ange:

283

ION IMPLANTATION IN SEMICONDUCTORS

and 8 for the cases of B, Si, and In, As shown in Fig. 8, broader range stragglings are obtained for the second order approximation. This gives us a better fit between the calculation and experiment than the first order approximation. The differences between the second order approximation and the third order approximation are less than one percent. Some of the calculated results are listed in Table I. A similar calculation was performed by Winterbon (13).

A R , (E,E')

(A)

FIG.8. Comparison of different approximations in the solutions for A R , ; '"In, *'Si, and * ' B incident on amorphous silicon. - - - - - - Approximation (2) second order; approximation ( I ) first order. -

3. Deciution from the Gaussian Distribution When the nuclear interaction potential is expressed in the form V ( r )= j,,r-'

higher moments are expressed as

where

In this case, the calculated concentration profile shows a significant deviation from the gaussian distribution, as shown in Fig. 9. In this figure a new

284

SUSUMU NAMBA A N D KOHZOH MASL'DA

R

CAI

FIG.9. Comparison of profiles of Furukawa's calculation (solid line) and the gaussian from LSS (dashed line).

analysis for the calculation is also shown. The energy distribution of an ion is calculated at collision steps in the substrate material to determine the range distribution, instead of solving the integral equations for the moments of the range which is necessary in the theory of LSS. The new calculations and the calculations of LSS with higher moments give almost the same results ( 2 4 ) .

4. Lateral Spread of the Distribution Lateral spreads of the implanted ions were calculated by Furukawa on the .basis of LSS theory and the assumption of a gaussian distribution. He found that the actual distribution of implanted ions through a slit is given by a complementary error function at the window edge as shown in Fig. 10 (15).

&A 1 1 1 1 I;, I 1 J 1 1 1 1 1 i i 1 h////, C O L L I M A T E D I O N BEAM

;I

I

SURFACE

OF

TARGET

I

Iar

w z

Y '

0 U

,

a

-a DISTANCE

FIG. 10. Actual distribution of implanted ions through a slic

ION IMPLANTATION IN SEMICONDUCTORS

285

The calculated values were confirmed experimentally by Akasaka and Horie in order to obtain a n exact design formula for an integrated circuit. Traces of stain etched figures show some disagreement between the calculated and experimental values as shown in Fig. 11 (16). Instead of the radio isotope

tI-

50

100 150 200 ENERGY ( k e V )

250

FIG.11. An example of the lateral spread. Comparison of the experimental and theoretical values. B + : 1 x 10'5/cm2. Sub. 0.1 (Q-cm).

method, the stain etching method was used to determine the experimental value. Therefore some of the effects of radiation damage centers create some ambiguity in the determination of the concentration profile as mentioned earlier. C. Single Crystal Target

In contrast to the case of the amorphous state, profiles of implanted ions have an extra tail beyond the peak of the distribution even when the direction of the incident beam is not parallel to the crystal axis. This distribution with the extra tail is called a non-gaussian distribution. There are three mechanisms which contribute to this tail; however, the relative amounts of contributions from each as a function of the various experimental parameters is not yet clearly understood. They are: (i) The enhancement of the diffusion coefficient by the effect of radiation induced vacancies. If radiation induced vacancies act like thermally produced vacancies, and if the thermal diffusion constant is proportional to the density of vacancies, then the diffusion constant increases with the number of radiation induced vacancies. This assumption is quite reasonable when the high energy ion produces many vacancies in the sample. Some of the experimental data support this mechanism. But some questions still

286

SUSUMU NAMBA A N D KOHZOH MASUDA

remain. What are the types of radiation induced vacancies? What are the lifetimes and mean free paths of those vacancies? These are chosen as parameters which are fitted to the experimental values without any definite determination. (ii) Channeling effect. A linear array of atoms in the single crystal makes a pipe of continuous high potential energy. This pipe produces a significant enhancement of effective scattering cross section for the incident ions. Thus the ions hardly penetrate within this array of atoms but they may penetrate between the arrays of atoms (open channel). The penetration depth of the incident atoms is quite large along this open channel. We can see major open channels in the (1 lo), ( 11l), and (loo} directions of a Si target. Other than those three open channels, many channels at low index direction also exist in the same crystal. If the incident ions are directed along a low index direction the penetration depth of the ions is very much deeper than if the bombardment directions are different. This is normally called channeling. The stopping power for ions which penetrate in the direction of an open channel is much smaller than for ions penetrating in some other direction. Thus even ions not injected along an open channel, after suffering a collision with a target atom may change their trajectory parallel to the direction of the open channel from the nonchanneling direction* Secondary channeling creates an extra tail for the ion profiles. (iii) Interstitial migration. The activation energy in the case of the migration of an interstitial atom may be smaller than that in the case of vacancy assisted diffusion. In this case interstitial migration will be observed. This phenomena might be an interstitial diffusion. Paths for the secondary channeling and the interstitial diffusion are quite similar in any case, but the temperature dependences are different for the two cases. Thus we can distinguish between those two mechanisms by varying the temperature. In the latter case, the extra tail of the profile shows an exponential decrease when the following assumptions are fulfilled. The density of the interstitial atoms at the beginning of the extra tail is constant during the whole processes of interstitial migration. Interstitial atoms are trapped by some trapping centers at the end of the migration. Those trapping centers are lattice imperfections. They might be vacancies, vacancy clusters, impurity atoms, or dislocations. If those lattice imperfections are uniformly distributed, the profiles of the deposited interstitial atoms will be exponential. Many of the experimental results are quite consistent with those considerations, but models for the lattice imperfections and the real path of the interstitial migration are still not clearly understood. We can say that the profiles in the case of amorphous samples calculated by LSS theory are quite acceptable, but in the case of single crystals, the situation for profile calculations is not * Henceforth this phenomenon is called secondary channeling

287

ION IMPLANTATION IN SEMICONDUCTORS

as satisfactory. Activation analysis, activation by a neutron or proton beam, and the backscattering method will be helpful in understanding the real atom distributions. The electrical properties should be known from the distribution of the atoms, but all implanted atoms are not always electrically activated so that the measurement of the profiles by the electrical method is often different from the profiles of atom distribution. Dearnaley (I 7) found tails beyond the gaussian distribution. He changed the conditions of acceleration voltages, the angles between the incident ion beams and crystal orientation, the total dose of implanted ions, and the temperature of the sample when it is implanted. As shown in Fig. 12 the tail of the profile is strongly dependent upon the ordering of the atom arrays.

10

'0

02

04

06

08

10

DEPTH (MICRONS

OF

12

14

I6

18

"0

02

06

04

08

1.0

DEPTH (MICRONS OF SILICON1

SILICON1

(b)

(01

t

o3

2102

z

2

0 U

10

'0 DEPTH (MICRONS OF SILICON1 (c I

02

06

0.4

DEPTH (MICRONS

0.8

1.0

OF SlLlCONl

(dl

FIG. 12. Exponential tail at the end of the profile. (a) x ( 1 lo), 12 keV, 5 x 10'2/cm2, 32P, room temperature; 0 (IIO), 40 keV, < 1.2 x 10' 3icm2, 32P, room temperature; ( 1 lo), 110 keV, 5 x 10'2/cm2, 32P, room temperature. (b) 0 (110), 40 keV, 1.2 x 10'3/cm2, "P, room temperature; 0 ( I lo). 40 keV, 8.9 x 1 0 1 3 / ~ m 2 ,3 1 P+ 32P, room temperature; x (110). 40 keV, 7.25 x 10'",cm2, 3 1 P+ ."P, room temperature. (c) 0 0 t o (110), 40 keV, 5 1.2 x 1013/cm2, 32P, room temperature; 0 2- t o ( 1 lo), 40 keV, 5 x 10IZ/crnZ,32P, room ( 1 lo), temperature; x 8" t o (110). 40 keV, 5 x 10'Z/cmZ, 32P, room temperature. (d) 40 keV, 5 x 10'2'cm2. 32P.400.C; 0 (110). 40 keV. 1.2 x IOL3/cm2.32P. room temperature.

288

S U S U M U NAMBA A N D KOHZOH MASUDA

D . Channeled Particle Extremely deep penetration of the implanted atoms by channeling is expected when the incident beam bombards a target in a direction parallel to an atom row. Experimental evidence of such deep penetration had been observed by Davies and Dearnaley in the case of implantation in silicon (18, 19). Even in the case of implantation with the beam parallel to the crystalline axis, a small percentage of the beam changes its direction by scattering by the nonuniform potential along the atom rows. Therefore, after such scattering a few percent of the parallel components of the beam become randomized. The ions missing from the aligned components have ranges varying from the value of the random component to that of the channeled component. As a result of the mixing of both random and channeled components typical profiles for the channeled beam appear as shown in Fig. 13. When the angle between the trajectory of the incident particle and the axis of the atom row is smaller than a critical angle $c the incident particle on the average can move almost parallel to the atom row even when the angle of incidence is not zero. In this case, when the incident particle hits the atom row it is reflected by the continuum potential of the atom row. On the other hand, when the incident angle ) I relative to an atom row is larger than the critical angle the incident particle passes through the barrier of the continuum potential similar to that of a nonchanneled trajectory. Ion concentration

Depth into crystat

FIG. 13. The depth distribution of implanted atoms in a slngle crystal under conditions 5uch that the beam is aligned with a major crystallographic axis. The shaded portion shows the distribution of perfectly channeled" ions which penetrate nearly t o the maximum channeling range R,,,. The dashed curves indicate the type ofdistributions that might be obtained under typical implantation conditions in silicon and gallium arsenide. I'

ION IMPLANTATION IN SEMICONDUCTORS

4.0-

' Silicon

2D -

289

'Sb
,

/ + /'

'9

-

*'-P(IIo>

This is for a well channeled particle. There is another effect for secondary channeled particles which enter a group of channeled particles from the group of randomly directed particles. Secondary channeled particles produce some contribution to enhanced diffusion which will be described in Section 111. 111. ENHANCED DIFFUSION

A . Introduction

The approximate shape of the depth distribution of implanted atoms is gaussian. LSS suggested some deviation of the depth distribution from a single gaussian distribution. In the case of the existence of a large deviation

290

SUSUMU NAMBA AND KOHZOH M A S U D A

from the gaussian distribution the calculation of the higher moments is important for the determination of the shape. Furukawa calculated the third moment for the heavy ion implantation and obtained a small deviation from a gaussian distribution with the assumption of a particular potential. Since the experimental data show some deviation from a gaussian distribution, this calculation of the higher moments gives a better fit between calculated and experimental values. Gibbons made similar comments on the possibility of explaining the skew of the distribution based on the third moment of the straggling of the range. But the accuracy of the activation analysis, backscattering analysis, and the analysis by nuclear reaction is not good enough to check the agreement between the calculated and experimental values when the deviation is small. Recently a measurement of the backscattered atoms from the implanted surface was made to detect the amount of gaussian distribution which is outside the sample surface. This method may possibly give detailed information of the skew of the straggling of the distribution (21). All experimental results of the profiles on amorphous solids are in good agreement with the theoretical calculation by LSS except for the small skewness of the profile. This minor disagreement between theory and experiment due to the small skewness will probably be solved by a calculation of the higher moment of the profile. The basic assumption of the theoretical calculation is that the length of the target in the direction of the ion beam is infinite. Strictly speaking, this assumption is not true. Backscattered ions across the surface of the target escape without scattering. A detailed comparison between the theoretical calculation and the experimental value in the backscattered ions outside the surface of the target is very difficult because of the existence of the assumption of abrupt discontinuity which is unfortunately extremely difficult to include. In contrast to the amorphous target, considerable enhancement of the penetration beyond the maximum peak is observed in the case of a single crystal target. A tail of the profile beyond the maximum of the peak is normally observed in the profile of the implanted ions in a single crystal. This tail shows a deeper penetration in a single crystal than in an amorphous target. The profiles of ions implanted in the channeling direction show a longer range than those calculated for an amorphous target. Therefore the reason for the appearance of the longer range is often assumed to be a channeling effect. However the enhancement of the penetration is observed to be independent of the direction of the incident beam relative to the crystal orientation. In other words, the enhancement of the penetration is observed even in the case of implantation in a nonchanneled direction. Thus the reason for the enhancement of the penetration is very complicated.

ION IMPLASTATION IN SEMICONDUCTORS

29 1

Even in the case of a well channeled beam a difference of the tail beyond the peak of the channeled particles is found. This phenomenon suggests the existence of some mechanisms other than channeling, even if they are very small. Profiles of this case are shown in Fig. 15 (17).

DEPTH ( p n l

FIG. 15. Different tails of the profiles of channeled particles.

The mechanisms to be considered for the enhancement of the profile in the tail are diffusion assisted by vacancies produced by the high energy radiation, diffusion of the interstitial foreign atoms which have a lower activation energy than substitutional foreign atoms, and secondary channeling. The enhanced diffusion is observed in the case of hot implantation. But it is also observed during the thermal annealing after implantation. B. Diffusion Assisted by Vacancies Produced by High Energy Radiation

The diffusion of atoms in the crystal depends upon the number of vacancies in the crystal and the activation energy of the migration of the atoms. The diffusion constant is written by Di = a2WoN(V'), (26) where a is the distance between neighboring atoms, W, the probability of jumping from one site to the nearest neighbor site, and N ( V ) is the number of vacancies at the nearest neighbor site. The distance between neighboring atoms is automatically determined when the sample is chosen as Si or GaAs, etc. The probability of jumping from one site to the nearest neighbor site is strongly dependent upon the potential barrier height between the two sites. This is normally approximated by the Boltzmann formula and expressed as

292

S U S U M U NAMBA A N D KOHZOH MASUDA

where E l is an activation energy for hopping. The hopping probability depends upon the temperature T, of the sample. V, is the frequency for the hopping. Hopping of the atom to the neighboring substitutional site does not occur when the neighboring sites are occupied by either host or foreign atoms. The probability of finding an empty nearest neighbor site is equal to that of finding a vacancy at the same site. This probability is proportional to the density of the vacancies. It is understandable that the diffusion constant of the foreign atoms depends upon the activation energy for the hopping, the temperature of the sample, and the density of vacancies because the distance between the neighboring atoms is a constant. Ions penetrating into the target produce many damage centers which are vacancies, host atom interstitials, and some lattice irregularities. Among these damage centers vacancies are most useful for the enhanced diffusion of foreign atoms. The fundamental ideas for the calculation of the number of displaced atoms had been given by Seitz and Kohler (22). The fundamental formula to calculate the total number of displaced atoms N , is

where T is the number of displaced atoms produced for each primary displacement, n the number of primary displacements per incident particle, and Y is the total number of incident particles. The assumption that the threshold energy E , is isotropic is often used for the calculation of N , . The number of primary atoms displaced by an incident particle in traversing a distance dx through a target is given by

where no is the number of atoms per unit volume and

E is the energy of the ion and

where M I , Z, are the mass and charge of the incident particle, M , , Z , the mass and charge of the target particle, uo the Bohr radius of the hydrogen atom, and R , is the Rydberg energy for hydrogen. The number of displaced atoms produced by subsequent displacements by a primary knock-on atom is very complicated to calculate. Calculations of each v(E)have been given by Chadderton (23). There are other difficulties such as the estimation of E d . The value for E ,

29 3

ION IMPLANTATION IN SEMICONDUCTORS

was assumed to be 25 eV by Seitz and Kohler. This value might depend on the material and also the crystal axis and the temperature. Experimental values of Ed are shown in Table 11. TABLE I1 SOMEEXPERIMENTAL VALUESOF Ed

Material Copper Copper Copper Silver Gold Nickel Iron Germanium ("-type) Germanium Silicon Graphite a

Property measured

Threshold energy (eV)

Temperature

Electrical resistivity Electrical resistivity Electrical resistivity Electrical resistivity Electrical resistivity Electrical resistivity Electrical resistivity Electrical conductivity

25 22" 19 28" 40" 34.5 24" 31

78 10 4.2 4.2 4.2 4.2 4.2 78

Minority carrier lifetime Minority carrier lifetime Electrical resistivity

14.5 12.9 25

(W

300 300 300

Denotes effective threshold displacement energies

Changes in the physical properties of materials are only observed when an energy greater than Ed is transferred to the atom. The electrical resistivity, Hall constant, minority carrier lifetime, ESR, and backscattering measurements can be used to detect the threshold energy for the occurrence of defect production. The number of displaced atoms Nd is calculated using Eqs. (28) and (29). We can assume that the number of vacancies is equal to or proportional to the number of displaced atoms. Then the amount of enhancement of the diffusion by the vacancies is estimated from N,. The number of the Frenkel pair n at thermal equilibrium is calculated from the condition to minimize the free energy as follows : n =(NN~)I/~~-~/Z~TS

(30)

where N, N,are the number of atoms at the lattice site and the number of interstitial sites and w is the formation energy of the vacancy interstitial pair. The diffusion constant is determined by the larger value of either the density of the vacancies at thermal equilibrium or the density of the vacancies produced by bombardment. In the latter case enhancement of the diffusion constant is expected.

294

S U S U M U NAMBA A N D KOHZOH MASUDA

The change in the diffusion constant below about 1000°Cfor a dose rate of 7.2 x 10'Z/cmZ is shown in Fig. 16 (24). The diffusion constant is determined by the density of the vacancies in the thermal equilibrium state over 1000°C and by the density of the vacancies produced by the bombardment below 1000°C.In the latter case, the diffusion constant is independent of temperature when the dose rate of the implantation is constant. If the dose rate is increased, the diffusion constant is increased because of the increase in the density of vacancies during the bombardment. TEMPERATURE

800

1

600 500

\

lo""' 06

a

0.6

' 1.b

' l.i '

(lOoO/T'K)

FIG. 16. Diffusion constant for various temperature of 0 1.2 x 10'2/cm2.sec; A 7.2 x 10'2/cm'. sec; normal diffusion.

hot

implantation.

The substrate temperature is effective in changing the diffusion constant in some cases. When the enhanced diffusion is controlled by the number of vacancies produced by bombardment, the vacancy yield and the lifetime of the vacancy are very important for determining the enhanced diffusion constant. Even at room temperature some of the vacancies produced by bombardment vanish by the thermal annealing effect. At an elevated temperature more intense annealing of the vacancies is expected. Much experimental evidence shows a considerable difference between hot and room temperature implantation. In the case of hot implantation, increases of W, and N ( v ) occur simultaneously so that enhanced diffusion is observed. But in the case of room temperature implantation, only enhancement of N ( u ) will be observed.

295

ION IMPLANTATION IN SEMICONDUCTORS

But because of the short lifetime of the vacancies, only part of the total number of radiation induced vacancies is effective. Therefore little enhancement of the diffusion is observed in the case of room temperature implantation. The diffusion coefficient in the region between room temperature and 800°C was measured. The relation between the implanted ions N ( x , To)and the diffusion coefficient D is expressed as

-

5 ' 20

[erfc(4DT0 ~

x2 - erfc20,~)

+

-~

where To is the duration of ion implantation, I , is the dose rate per unit area, and o,,is the standard deviation of the projected range. Equivalent diffusion coefficients calculated from Eq. (31) are shown in Fig. 16. In the higher temperature region the diffusion coefficient is controlled by the thermal diffusion equation. But in the temperature range between 800" and 500°C the effective diffusion coefficients are the same. In this region the diffusion coefficient is fixed by the number of radiation induced vacancies. When the flux density of the incident ion beam is increased the effective diffusion coefficient is increased because of the increase in the density of the radiation induced vacancies. At around room temperature this effect of the enhancement of diffusion coefficients is no longer observed. A certain energy is necessary to release the impurity atoms from the original site to the other site where the atom is able to migrate with the assistance of the adjacent vacancy. This energy is greater than the thermal energy of the lattice at about 300"K, so that the diffusion coefficient will drop towards the value of the normal thermal diffusion coefficients. C. Interstitial Difftision Mechanism for Enhanced Diffusion

Another mechanism for the enhanced diffusion with hot implantation is that of interstitial diffusion. The following evidence points to this conclusion. 1. Exponential Tail

Profiles for these cases show the exponential tail after the major gaussian peak. This exponential tail is calculated as follows.

296

SUSUMU NAMBA AND KOHZOH MASUDA

The one-dimensional diffusion equation can be written:

where K , is a unimolecular rate constant for trapping and D is the diffusion constant for the interstitial motion. Setting a N ( x ) / ? t = 0, we see that the equilibrium distribution has an exponential form (25)

N,,(x)

=

N,,(O) exp [ - (K,/D)"' . x],

(33)

so that the exponential tail can be explained by the interstitial migration with uniformly distributed trapping centers. 2. Foreign Aton? Locution in the Lattice Measurements of the lattice location show that most of the atoms are located neither at the exact substitutional site nor at the exact interstitial sites in the region of extra tail (26) in the case of Ga and In implantation in silicon. If the impurity atoms diffuse by exchange between their sites and adjacent vacancies, the final positions of the atoms should be at the substitutional sites. But in the case of another mechanism such as interstitial migration, the final sites of the impurity atoms will be some irregular positions because a trapped interstitial atom can easily shift its position due to the stress field around the trap or the irregularity of the surrounding potential field. The results of the He beam channeling experiments show that the enhanced tail is not caused by vacancy assisted diffusion but by interstitial migration. 3. Dependence on Eiierqj, oj the Iiiczileizt Particle

The ranges of the channeled particle are proportional to the square root of the incident energy. As shown In Fig. 17 the enhanced tail is not dependent upon acceleration voltage. In this case, channeling is not the major mechanism for producing enhanced diffusion. 4. Teniperatuw Dependence of the Projles

The enhanced tail increased at 500'C implantation more than that at room temperature implantation. This tendency suggests that the enhancement is a thermally assisted process such as diffusion or migration of impurity atoms and strongly rejects the possibility of channeling ( 2 4 ) .

ION IMPLANTATION IN SEMICONDUCTORS

297

FIG. 17. Energy dependence of ion profiles of implanted As in Si at room temperature. 0 35 kV, 5 x 10i3/cm2; A 45 kV, 5 x 10'3/cm2: A 60 kV, 2.5 x 10'J/cm2; 0 130 kV, 2 1 0 ~ ~ 1 ~ ~ 2 .

D. Radiation Enlzanced DijJusion The radiation enhanced diffusion (referred to as R.E.D.) means the bombardment of light ions in order to change the profile of impurity ions drastically. This is an excellent technique for device fabrication. R.E.D. was tried by several groups. One of the examples is shown in the following. Silicon slices containing a thermally diffused junction are irradiated at temperatures between 600' and 1200°C with protons of 0.2 1.0 MeV energy. The penetration depth of the protons is less than the original junction depth. In the portion of the sample immediately beneath the irradiated region, a significant increase in junction depth occurs, as can be seen in Fig. 18 (27). In this case, it is suggested that the vacancies produced by proton beam irradiation migrate into the deeper region which make the diffusion coefficient of impurities extremely high. The behavior of impurities and vacancies is similar to that in hot implantation. Glotin discovered the enhanced diffusion of phosphorus and boron in

-

298

SUSUMU NAMBA AND KOHZOH M A S U D A

6~ n - T Y P E Si

IRRADIATED REGION

7$-

MOVEMENT OF

FIG. 18. Schematic representation of the movement of a p-!I junction following proton irradiation at elevated temperature.

silicon by the measurements of radiotracer technique (28). First, he implanted 32Pat room temperature to a dose of 1014ions/cm2; he then increased the substrate temperature up to 700°C and implanted again the stable isotope 31Pto a dose of 3 x lOI5/cm2. He found that this second implantation caused an increase in the depth of the first implantation. The effective diffusion coefficient in this case was 2.7 x cm2/sec. Since the distribution profile in this case was gaussian the vacancy assisted diffusion mentioned previously will be a good explanation of this enhancement. Abe performed the experiment on a Sb enhanced diffusion in silicon layer using a 100 keV proton beam. The effective enhanced diffusion coefficient in the region of the epitaxial growth region was 1 x 10- cm’/sec. Profiles in this epitaxial layer are shown in Fig. 19. In the profile of impurities which were redistributed by enhanced diffusion there was not a single gaussian so he suggested that there was a nonuniform distribution of diffusion coefficients (29).

’

DEPTH (pm)

(r

=

FIG. 19. Shift of the boundary of the epitaxial layer. 0.33 p . 40 min; -- u = 0.5 p, 40 min.

---

u

=

0.25 p, 40 min;

--

299

ION IMPLANTATION IN SEMICONDUCTORS

In A1 implantation in silicon at room temperature an enhanced diffusion of A1 has been observed after annealing. Figure 20 shows the effects of surface layer removal on the enhanced diffusion of Al. Prior to annealling at 800°C for 20 min, steps were formed in the as-implanted wafer by means of anodic oxidation and HF stripping. The removal of the first layer of 160 8, does not cause any change in junction depth, but the second layer removal (total 320 A) caused a remarkable AMORPHOUS REGION \ (-16OA)

,\

(Ad (

JUNCTION n-Si

- 6 0 07 A -1200A

i

L FIG.20. Decrease in junction depth depend on removal of surface layers.

A)

decrease in junction depth. The third layer removal (total 480 also caused a considerable decrease, but the region deeper than 480 8, does not contribute to a detectable decrease in junction depth. From this it can be seen that the region which does contribute to the enhanced diffusion is not the amorphous layer but the isolated disordered region (30).

IV. ANNEALINGAND ELECTRICAL PROPERTIES A . Introduction In the case of implantation, normally annealing is necessary to obtain good electrical properties. Thermal annealing changes nonsubstitutional atoms into substitutional atoms which are effective in obtaining an electrically active center (31).A large fraction of the penetrating atoms just at the end of their path in the solid occupy nonsubstitutional positions because there is more space than for substitutional positions. The annealing temperature at which the conversion from nonsubstitutional to substitutional sites occurs is relatively lower than that for the occurrence of normal thermal diffusion. But in some cases thermal treatment for annealing is not effective in obtaining good electrical properties even in the case of high temperature annealing. In this case hot implantation is expected to be effective. Imperfections produced during ion implantation are also annealed by

300

SUSUMU NAMBA A N D KOHZOH MASUDA

heat treatment. The production efficiencies of the defects in both hot implantation and room temperature implantation are different. This is probably due to the interaction between the defects because of the high total dose of the implanted ions. Backscattering, ESR, and electron microscopy are useful methods for measuring the physical properties of the defects. Radiation induced defects decrease the electrical carrier mobility because of the action of trapping centers and decrease the efficiency of luminescence. B. Silicon The concentration of charge carriers is obtained by the Hall effect measurements in order to determine the effect of thermal annealing on Sb implanted Si. An abrupt increase in the number of charge carriers is observed at around 600°C as shown in Fig. 21a (20). Substitutional Sb can be a donor + Substitutional ,,,

0.8 R.T. Implant

r

Hot substrap 1rnphnt:Sb

0.4

0.4

;

;charge ; Carriers

C

200

LOO

600 800

Anneal Temperature

('c)

(a)

FIG. 21. (a ) Annealing effect of Sb implanted Si (room temperature implantation). (b) Annealing effect of Sb implanted Si (hot implantation).

center. But if the donor center is compensated, this donor center can not act as donor. It is very important to know what percentage of atoms are in the substitutional sites. Channeling experiments give the exact answer to this question. The ratio of the substitutional components to the total number of implanted atoms is measured in one sample. The growth of the substitutional component is exactly the same as for the generation of charge carriers in room temperature implanted Sb in Si, as mentioned earlier. In the case of hot implantation at 35OoC,as shown in Fig. 21b, the substitutional component constitutes almost 90% of all implanted atoms even at the low annealing temperature of 400°C. However the density of charge carriers is very low even though a high substitutional component is obtained. After annealing at 800 C, agreement between substitutional components and the density of the charge carrier is obtained. Between the annealing temperatures of 400" and

ION IMPLANTATION IN SEMICONDUCTORS

30 1

80O0C, some defects which compensate the donor carriers are generated from the substitutional Sb atoms. There exist two mechanisms which contribute to thermal annealing. One is the conversion from the nonsubstitutional site to the substitutional site. The other is the recovery of crystal defects. Defects produced by implantation are much more complicated than in the case of low dose irradiation of neutron beams or electron beams, which have already been reviewed in other books (32). The efficiency of production of defects by ion beams is much greater than by fast neutron beams so that the range ofdensities of the defects is widely spread. In the low density range, each defect is isolated from the others. The interaction between defects increases with increasing defect density and finally the amorphous phase is produced at the heaviest implantation dose. Thus we must study the behavior of the isolated defect, compound defects, and the amorphous phase in the solids. Figure 22 illustrates W

2 + V

a 1.0 -\ LL

400K

V

\\

W

\ \

W

0.1

“SATURATION”-!

-

8 8

x\

\

b 0

P” 200KeV 112HR 5W’C ANNEAL

10’~

10”

10”

1d6

DOSE-NO IONS crr-’

FIG22 Fraction of dose electrically active agalnst implantation at 1 W K ( x), 200 K(O), 300 K(O), 350 K(O), and 400’K(A).

the relations between doping efficiency and total dose for each implantation temperature. When the 3 ’ P ions were implanted in Si with 200 keV and post annealing at 55OoC, doping efficiency is dependent upon implantation temperature and total dose. The doping efficiency is almost 100% below the dose of 10’3/cn12which means that the number ofcarriers is almost equal to that of implanted ions. On increasing the dose, the doping efficiency first decreases gradually then increases gradually after a minimum point and again reaches 100%.The dose at which the doping efficiency reaches 100%is called the critical dose which is equal to that at which the amorphous phase occurs. At an extremely high dose, the concentration of the implanted ions

302

SUSUMU NAMBA A N D KOHZOH MASUDA

exceeds the solubility limit of the ions. Therefore the doping efficiency decreases in this region as shown by the dashed line indicated by “saturation” in Fig. 22 (33). The critical dose increases in the high temperature region because of the annealing effect of defects produced during the implantation. The doping efficiency does not reach 100% at 400°C implantation because of the large annealing effect during implantation. It is reasonably well understood that the easy recovery of the carriers in the low dose region is due to the fact that the defects can be annealed at low temperature. There are no clear explanations of why the carrier recovery is easier when the amorphous layer is formed compared to when there are only isolated amorphous islands. But the appearance of the amorphous layer has an important role in the annealing behavior of the implanted layer. This amorphous phase gives an isotropic ESR signal of g = 2.006. Therefore it can be one of the criteria of existence of the amorphous phase (34,35).

1200-

\

K‘SPECTRP RANDOM

1000-

ORIENTATION

-

~ ~ ~ . C , A N N E A L ’,

60

70

80

90

100

CHANNEL NUMBER

FIG. 23. Backscattering spectra for successive annealing of P implanted Si

The amorphous layer may recover by epitaxial recrystallization on the silicon crystal of the wafer. Figure 23 is an example which shows the recrystallization process of the amorphous layer produced by implantation of P at energy 40 keV with a dose of 1.3 x 10’5/cm2 by means of the backscattering method. The defect peak just after implantation is equal to that of the random spectrum, which means that the implanted layer has become amorphous (36). Little change in the spectra by annealing below 500°C means no recovery

ION IMPLANTATION IN SEMICONDUCTORS

303

of the crystallinity. O n annealing over 550°C, the defect peak decreases rapidly and also shifts toward the surface. This fact means that the recovery of the crystallinity starts from the boundary of the single crystal of the wafer and the amorphous layer. The spectra after annealing at 655°C are almost equal to that of the unimplanted silicon which means almost perfect recovery of the crystallinity. When the implantation was done at 280 keV 31P with 3 x 1014 atoms/cm2 at room temperature with post annealing at 600°C of 30 min, beyond the peak of the implanted region the transition region was formed which is very highly resistive as shown in Fig. 24 (37). There are heavily damaged states in the transition region. Thus the decrease in the number of charge carriers depends upon the heavily damaged state. In the case of B implantation reverse annealing effects are observed. In order to clarify our understanding of the effect of a heavily damaged phase, a

19

10

h

TE U v

H

E+z

,d8

w

V

Ba W

>

B 9

0 3 0

I

I

I

I' I

1

I

6 O

102 L 1o2

0 o

,w 0.2

0.4

I

0.6 DEPTH ( prn)

a8

I

FIG.24. Carrier concentration profile of P implantation

304

SUSUMU NAMBA A N D KOHZOH MASUDA

high density beam of neon was used to create extra damage in a boron preimplanted layer. Preimplantation of B was done with 10" atoms/cm2 at an acceleration voltage of 50 keV. Post bombardments of Ne were done with and 1014atoms/cm2 and zero. The results are shown in Fig. 25. 1

I

I

1

I

1

FIG.25. Neon dose dependence of annealing of boron implanted Si. B: SO keV . 10" cm'. Ne: 125 keV. R(Ne)/R(B)= 1.2. 0 1 0 ' h ~ c m zA ; 10'5/cmZ; 0 10"!cm2; 0 bithout Ne. Anneal time 10 min.

At the relatively low level of implantation damage of 10" atoms/cm2 reverse annealing was observed. The degree of the reverse annealing decreases with an increasing dose of Ne. Finally no reverse annealing effect was observed at a dose of 1OI6 atoms/cm2 which is a region of formation of a continuous amorphous phase (38). The dose dependence of the degree of reverse annealing was also done at 1.5 x lo", 7 x 1014,and 1 x 1014 atoms/cm2 of B at a substrate temperature of 80"K, as shown in Fig. 26. We see no reverse annealing effect at the

u Anneal Ternp.?C)

FIG.26. Dose dependence of annealing of boron implanted silicon for 50 keV at 80 K.

ION IMPLANTATION IN SEMICONDUCTORS

/

6 3 Ole

305

I 1000

Anneal temp. ("c)

FIG.27. The change of a fraction of the substitutional component of the implanted atoms at various annealing temperatures.

high dose of 1.5 x 10" atoms/cm2, but reverse annealing was observed at the dose of 1 x 1014 atoms/cm2. Reverse annealing was also observed in the behavior of the substitutional B concentration as shown in Fig. 27. The reverse annealing effect changes not only the carrier concentration but also the atom locations in the lattice. C. GaAs

Research on ion implantation in GaAs was delayed relative to Si because of the complexity of compound materials. In the case of Si, even though reverse annealing or interstitial migration occurs which creates some complexity in annealing phenomena, there is no stoichiometry problem. On the

306

S U S U M U NAMBA A N D KOHZOH M A S U D A

other hand, in GaAs, shifts from chemical stoichiometry easily happen when annealing at a high temperature because either Ga or As migrates faster, depending upon the environmental condition. Furthermore, hot implantation is often considered to be a better method of obtaining good electrical characteristics. When the substrate is hot, the difference between the speeds of migration of Ga and As in GaAs produces some complexity in the compositions and also electrical characteristics. Deviation from chemical stoichiometry also happens in the fabrication of p-n junctions. Mayer et al. (39) implanted Zn and Te into n and p type GaAs to get p and n type GaAs respectively. But a p-n junction was not obtained by implantation with 20 kV at 400°C and post annealing at 600°C. This conclusion was drawn from measurements of the V-I characteristics of the diode. The hole mobility of 50 cm2/V sec for Zn implantation was obtained, but the electron mobility was not obtained in the case of Te implantation because the diode characteristics were not abrupt. Hunsperger et ul. (40)also obtained a very broad junction width which is more likely a p-i-n junction rather than a p-n junction with 70 kV implantation at 400°C and annealing at 500°C. The depth distribution of implanted ions expected from this p-i-n structure is much deeper than that expected from LSS theory. The widths of the i region were 0.18 pm and 2.7 pm for the substrate dopant density of 1.8 x 10l8 atoms/cm3 and 1 x 10l6atoms/cm3, respectively. He showed that the changes in the mobilities and carrier concentrations depend upon the post-annealing temperature so as to stress the importance of the post-annealing effects. Roughan implanted Zn into GaAs at room temperature with 80 k V and annealed at 650°C. In this implantation, he found that it is possible to make a p-n junction at a dose of 10l6atoms/cm2 and a p-i-M junction was obtained at a dose of 1015atoms/cm3. This fact suggests that the interactions between implanted atoms and defects produced during implantation are important factors in determining whether a p-n junction is formed or not. Defects produced around the junction may suppress the migration of each component. Foyt et al. (41) reported that they obtained an abrupt junction by implantation with 70 kV at 500°C and post annealing at 800°C. Hunsperger and Marsh (42) tried to check in detail how the change of the i structure depends on the temperature of annealing. The width of the i layer increases with increasing temperature for implantation with 20 kV at 400°C in both cases of Cd and Zn implantation. But after properly prepared post annealing the i layer almost completely vanished. The dependence of the i layer on the annealing temperature is shown in Fig. 28. Under these circumstances the migration of As from the junction region was assumed, but some direct proof of the escape of the As is needed. Itoh

ION IMPLANTATION IN SEMICONDUCTORS

307

did a preimplantation of As which was expected to escape from the junction region. After this preimplantation of As, the same amount of Cd was implanted into the GaAs with 20 kV at 500°C and post annealing at 650°C. He obtained a p-n junction as was expected. Measurements were done by the C-V method and V-I characteristics. Without preimplantation of As, it was not possible to obtain a p-n junction. We can conclude that there are two successful methods. One is to suppress the escape of As by having the sample at low temperature. Another is to compensate for the As escape by preim-

:5 -

w

z 40 Y

?

32-

a

-

I -

O t

500

600 ANNEAL

700

800 TEMPERTURE ('C)

I 900

FIG. 28. Dependence of the i layer thickness on anneal temperature. - - - - - Substrate impurity concentration was 1 x 10'0/cm3. Implant conditions: dose 8 x 1015/cm2, 20 k V Cd' ions, substrate at room temperature, 10 rnin anneal period. -~~ Substrate 5 x 1015/cm3. Implant conditions: dose 1 x 1016/cm2, impurity concentration was 20 k V Zn' ions, substrate at room temperature, 10 min anneal period.

-

-

plantation. One of the advantages of the ion implantation over the thermal diffusion method was that it can be performed at relatively low temperature. This requirement for low temperature processing is more important in the case of GaAs than in the case of Si. But if one performs an ion implantation at low temperature, such as at room temperature, a high temperature annealing is necessary to obtain good electrical properties and this high temperature annealing produces many unexpected problems. A typically hot implantation with a substrate temperature of about 400°C is proposed to obtain good electrical properties. What is the operative mechanism of hot implantation in this case? Channeling experiments were carried out by Gamo et al. to clarify this situation. They observed that if the Te atoms were implanted at low temperatures, such as at room temperature, even after considerable high temperature annealing the lattice location of Te is not a substitutional site. This fact

308

SUSUMU NAMBA A N D KOHZOH MASUDA

was proved by the implantation of Te with 70 kV at room temperature and 500T with post annealing at 800°C and 900°C (43). The small fraction of substitutional components which appear at 400°C annealing is shown in Fig. 29. But even at an annealing temperature of

ANNEAL TEMPERATURE ( O C )

FIG.29. Dependence of the Tc amni location in the lattice on annealing temperature.

800 C , this fraction of the components is around 50% and there is no tenatoms/cm2 dency to increase this for both implantation doses of 1 x and 2 x 1016 atoms/cm2. O n the other hand, the fraction of the substitutional components of implanted atoms is 90% for hot implantation at 550-C. This result suggests that the thermal energy and the kinetic energy are necessary to place the Te ions at the substitutional sites at the end of the collision cascade. During post annealing the implanted atom has only thermal energy. Thus when the sample is set at the same temperature for both post annealing and hot implantation the conditions of the surroundings are quite different. This difference gives the advantage of lattice location for the case of hot implantation of GaAs. Moreover, the charge state of the defect during ion implantation is different from that during post annealing, causing the migration energies for the different charge state defects to be considerably different. For example, in the case of silicon the neutral vacancy has an activation energy of 0.33 eV whereas the negative vacancy has an activation energy of 0.18 eV. Group V atoms easily occupy substitutional sites even at room temperature implantation and post annealing. Implantations of P into GaAs were done successfully to obtain p-rz junctions (44-47). Fabrication of an insulating layer on the conductive GaAs wafer is also performed by proton beam irradiation. In this case a minimum carrier density of 10" carriers/cm3 (48) was obtained. This value is small enough to make an isolation pattern in an integrated circuit of GaAs. The optimum dose to obtain the most resistive layer was also found.

309

ION IMPLANTATION IN SEMICONDUCTORS

>

.

nonchanneled

Y

I

I

T

0

- lam

Ol

87'k implant

.t c

.,

v,

300Ok anneal

-

-800

296'k implant

-600

-

-400

unimplanted

-200 I

I

I

Energy ( keV

1

FIG.30. Shift of the damaged peak of ( I 1 I ) direction spectrum

As mentioned earlier, defects in GaAs easily migrate and some of them can migrate even at room temperature. Vook measured the backscattering of GaAs and found abnormal enhanced diffusion during the low temperature implantation of oxygen at over 275°K (49). The proof of this enhanced diffusion is the shift of the damaged peak of the 11 1) direction as shown in Fig. 30. A similar enhanced diffusion of defects is reported by Namba et a/. (50). They performed an experiment on photoluminescence intensity as a function of implanted dose and found a considerable decrease in the photoluminescence intensity which depended on the migration of the defects. For the abnormal enhanced diffusion, Stein suggested the charge state change mechanism for defects during both implantation and post annealing ( 5 2 ) . The pattern of the migration of defects in the implanted sample is not very well understood. One of the reasons for the complexity in the mechanism of defect migration is the complexity of the structure of the defect itself. As already mentioned the density of the defects is very large in the case of ion implantation. Overlap of defects creates multivacancies and stress between

<

3 10

SUSUMU NAMBA A N D KOHZOH MASUDA

neighboring defects produces a deformed multivacancy. Even in the very low dose region, very heavily damaged parts are still produced at the end of the collision cascade. Those circumstances are completely different from the case of electron or neutron irradiation where more uniformly distributed isolated defects are expected at a low dose. Even though the models of the defects are rather complicated some stages of the annealing are reported. The characteristic band edge luminescence in GaAs is completely quenched after ion implantation. Borders reported that this luminescence can be observed at 230°C annealing (52). Harries et al. found the complete recovery of this luminescence at 600°C annealing. Harries et al. carried out the channeling experiment to detect the annealing stages for the recovery of crystal order. They found two stages by using a 450 keV proton beam for the channeling experiments. The most heavily damaged region is located at about the depth calculated from the projected range. This heavily damaged region recovers at 300°C. A moderately damaged region remains until 650°C annealing. A Kikuchi pattern was observed at 450°C (53). This observation shows that the crystalline structure recovers to some extent at this temperature. Arnold and Whan (54) observed the expansion of the implanted layer of GaAs. Crystal defects almost completely anneal out at a temperature of 650°C. This is deduced from the channeling measurement (55). We have another stage for an increase in the carrier density at a temperature above 650°C. Thus a recovery process might still exist at a temperature higher than 650°C. Moreover, during the annealing Ca or As diffuses independently to the surface. SiOz or Si,N, films are used to avoid this diffusion. The mechanism for the protective ability of these films against Ga or As is still unknown. V. MEASUREMENT TECHNIQUE

A . Introduction

It is very important to be able to obtain precise data on the depth distribution of the implanted atoms for considerations of atomic processes in both the collision cascade and the fabrication technique of electronic devices. The present techniques for obtaining the depth distribution of the implanted atoms are activation analysis following the successive stripping of the layers by etching and backscattering analysis of He or H beams. Recently the He ions produced by the n-He nuclear reaction of loB were used to determine the depth profile of the B implantation. This method is a very useful technique in the case where the mass of the dopant ion is lower than

ION IMPLANTATION IN SEMICONDUCTORS

311

that of the host material, whereas the backscattering method of injected ions is useful in the case where the mass of the dopant ion is greater than that of the host material. Activation analysis has a higher accuracy for depth resolution than backscattering for a normal region of acceleration such as in the neighborhood of 2 MeV. Backscattering is a nondestructive method for the measurement of the depth distribution, and is superior to activation analysis in this respect. The electrical activity of the implanted ions strongly depends upon the lattice location of the implanted atoms. A channeling study of the high energy beam can tell us the lattice location of the implanted atoms by changing the direction of the beams relative to the crystal axis. Either the high energy particles ejected from the impurity atoms by nuclear reaction or the characteristic X-rays ejected from the impurity atoms can also be used. The relation between the capacitance and voltage are useful to determine the depth profile of the majority carriers in the sample. The relation between the majority carrier concentration and the capacitance of the depletion layer is

x = &A ~

C’

(35)

where e is the unit charge, E the dielectric constant, A the area of the junction, C the capacitance, and V the applied voltage. The Copeland method is used for the measurement of AC/AV. The mobility of the carriers is measured by the Hall effect and the Van der Pauw method. The microscopic electronic structure of the implanted atoms and of the vacancies is often determined by electron spin resonance (ESR). This method is very useful for understanding the electronic structure of the defect center when it is a paramagnetic center. The special techniques in the field of ion implantation will be described in the following sections. B. Channeling Effect Technique 1. Backscattering Phenomena

When the beams of high energy particles bombard the implanted target some of the particles are scattered backward on collision with a target atom which might be either a substrate atom or an implanted atom. The detected

3 12

S U S U M U NAMBA A N D KOHZOH M A S U D A

energy of a backscattered particle is given by

. xlcos 8

I

1

S*(E) dll K ,

-

1' .

0

S(E)d12,

(36)

xicos B z

where

S ( E ) is the stopping power along l,, S * ( E ) the stopping power along I,. M , the mass of the incident particle, M , the mass of the target material, 0, the angle between the incident particle beams and backscattered beams, x the depth at which the target atom existed, and 8,, e2,I,, I,, are the incident and scattered angles and incident and scattered path lengths of the particles, respectively. When the incident particle suffers a collision at the surface. the energy of the backscattered particle is Eobq(0).An energy loss for this phenomenon is only caused by the momentum transferred by Rutherford scattering. When the incident particle makes a collision at depth x the energy of the backscattered particle is Eobs(x). The energy loss in this case comes from both Rutherford scattering and electronic stopping in the target medium. The stopping power for an incident particle is given by S ( E ) and that for a backscattered particle is given by S*(E). The relation between S ( E ) and S * ( E ) suggested by Mayer et al. as S*(E) 0.7S(E) can be used for most purposes. Figure 31 shows the relation between the scattering yield and the energy of the backscattered particles scattered by the silicon and the implanted atoms which have larger masses than that of substrate material. If the implanted atoms are distributed in the bulk uniformly the spectrum of the backscattered particle is shown by the dot-dash line. Normally the acceleration voltages for this purpose are from 0.5 MeV to 3 MeV and the particles used are H or He, so the range of the implanted atom is negligibly small compared with the range of the high energy particle. Hence the energy spectrum of the backscattered particles from the implanted atoms sometimes looks like that of particles scattered from the surface only. When the thickness of the implanted layer exceeds the resolution depth we can determine the depth distribution of the implanted atoms by this method. The detector resolution of 15 keV is equivalent to a depth resolution of about 300 8, for 1 MeV He in Si. (To get more accurate information an activation analysis is desirable.) One example of the spectrum of compound elements is shown in Fig. 32 (56). The sample is a GaAs wafer coated with SO, film. The yield-energy spectrum of backscattered 1 MeV He has a sharp edge for "Ga and 75As. The difference between the spectra of "Ga and "As is very difficult to

-

ION IMPLANTATION IN SEMICONDUCTORS

313

SILICON

ENERGY

OF

SCATTERED HELIUM

FIG. 31. Energy relation of the backscattered particle. The lower part of the figure shows the resulting energy spectrum 01' scattered particles.

detect. Therefore, energy EI corresponds to the energy of the backscattered particle from either Ga or As. When the surface of the GaAs crystal was coated with the SiO, the spectrum edge of the GaAs crystal was shifted toward the lower energy side EY by an amount which depends upon the electronic stopping power of the SiO, film. Thus finding a signal at E: is proof of the existence of Ga or As at the surface of the SiO, layer. The spectrum tail from E: to EY is proof of the existence of G a or As in the SiOz film. The spectra of Si and 0 appear at much lower energies and they will not produce any disturbances in the spectrum of GaAs. The spectra of GaAs coated with a SiO, film under heat treatment for 0, 4, and 8 min are shown in this figure. After 4 min heat treatment at 750°C some yield appeared between the energies E ; and EY. Therefore it is clear that some of the Ga from the GaAs was diffused into the SO, film. The energy of the peak E: is identified as a Ga peak from measurements at higher energy (2.5 MeV) (57). After another 4 min both an increase of the diffused G a in the SiOz film and a considerable accumulation of Ga at the surface of the SiO, film is observed. SiOz and Si,N, films are used to inhibit the decomposition of compound semiconductors during heat treatment. But the fact that

3 14

S U S U M U NAMBA A N D KOHZOH MASUDA

% G a b Si02 STRUCTURE

1

400

s

v,

c

W

Z

3

0

1 7 Si t

GO

-----.\

t

300

0,

F

ENERGY (MeV)

E?

E’

(3

= 20c E W

t-

3 Y

V

2 loo

0 ENERGY ( M e V )

FIG. 32. Energy spectrum of the backscattered particle from GaAs coated by SiO,

outdiffusion of G a from GaAs through the SiO, film occurs indicates that a SiO, film is not a sufficiently protective material for GaAs. This is a good example of the investigation of diffusion through thin films. Backscattering gives us information on the composition and on the profiles of the density distributions. 2. Locution of’ Atoms in a Lattice

A schematic representation of the backscattering experiments is shown in Fig. 33. The intensities of the backscattered particles are greatly affected by changes of the angle between the incident beam and the crystal axis. An example of a Si crystal doped with Au and Sb is shown in Fig. 34. When the beam direction is tilted through the (110) axis the yield of the backscattered particle shows a very deep dip just in the (1 10) direction as shown in Fig. 34. This dip is caused by channeling along the Si atom rows. When the impurity atoms are located at the same sites as the host crystal, this row of impurity atoms is the same as that of the host atom. Therefore in such a case, the dips in the yields of the backscattered particle for host and

315

ION IMPLANTATION IN SEMICONDUCTORS I ENERGY ANALYSIS

1 1 TARGET

T

COLLIMATION

T

FIG.33. Schematic representation of the backscattering experiment.

impurity atoms are exactly the same. Differences in the curves of the yields of the dips suggest the existence of some deviation of the impurity atom sites from the normal host atom sites. When the crystal is irregular, the yield of the backscattered particle shows a constant value independent of the beam direction. In this case the direction of this beam is called a random direction.

FIG.34. Angular dependence of backscattered particles of Au, Sb implanted Si.

316

SUSUMU NAMBA A N D KOHZOH M A S U D A

Au atoms are distributed at positions different from the normal lattice sites of Si and there is no particular rule for the separations between host atom and impurity atom. Therefore yields of Au in Si show a constant value independent of the change of tilt angle. Alternatively, if the Au occupy tetrahedral interstitial sites yields show some dips and some peaks centered at the (1 10) direction because of the regularity of the Au atom positions in the crystal. A similar yield of spectra for both host atoms and impurity atoms for a particular axis of the crystal means that the impurity atoms are in the same atomic rows as the host atoms for this particular axis of the crystal. Thus Sb are aligned in the (1 10) direction in the case of Fig. 34. In the case of the channeled particles the row of atoms makes a pipe of a high potential field which does not permit any channeled particle to enter. The potential of this pipe can be assumed to be the average value of the potentials of the atomic row in the channeled direction. When the He beams bombard the crystal in the aligned direction they are traveling along this pipe of the potential field. The cross section of this pipe is assumed to be ny&, where ymin is the closest distance of approach between the channeled particle and an aligned row. The cross section of the pipe is the cross section for collision so that the total cross section Z, for the collision is given by

C,

=

nNd;$i,,

(37)

where N is the density of the atoms and (1 is the lattice spacing along the row. Incident particles are scattered with a total cross section Ct . Therefore the yield of the aligned beams has a minimum peak at the position equivalent to the surface. This minimum scattering yield is called xminwhich is C, . In some approximations y$n is replaced by the mean square vibrational amplitude P:. The atomic configuration in a (1 10) plane is shown in Fig. 35. Substitu-

@

0 X

LATTICE SITES INTERSTITIAL HOLES SPLIT INTERSTITIAL

FIG. 35 Various location of atoms

111

the (100) plane

ION IMPLANTATION IN SEMICONDUCTORS

317

tional sites, tetrahedral interstitial sites, and random sites (which are neither substitutional nor interstitial sites) are distinguishable by measurement of the channeling effect in (1 lo), (1 1l), or (100) and random directions. When the He beams bombard the crystal in the (1 11) direction impurities at the tetrahedral sites sit inside the potential pipe. Therefore those impurities are not scattering centers. On the other hand, the impurities at random sites are outside of this pipe so that they are scattering centers for the incident particles. The yield in the (1 11) direction gives the number of impurities at random sites in the sample. The yield in the (1 10) direction gives the number of impurities at random sites and tetrahedral interstitial sites. And the yield in a random direction which is not parallel to any of the atom rows gives the number of impurities at random sites, tetrahedral interstitial sites, and substitutional sites. The latter number is detectable by measurements of the channeling effects in three directions which are random, (IIO), and ( I l l ) . In general the statistical error is very large at the low count level of the yield. Great care is necessary to perform the subtraction between two numbers which may have considerable statistical errors. Information obtained from measurements of the channeling effect is summarized in Fig. 36. TWO-DIMENSIONAL MODEL

DIRECTIONAL

YES

EFFECT

?

YES

FIG.36. Information from the channeling effect measurements.

318

S U S U M U NAMBA A N D KOHZOH MASUDA

Figure 37 shows examples of the spectra of TI implanted Si with 1 MeV He beams, on the right-hand side of the figure. The spectra of Si targets are shown on the left. By comparing the areas under the T1 peaks with Fig. 36 one can conclude that approximately 40% of the TI atoms are

CHANNEL

NUMBER

FIG.37. Energy spectra of helium ions backscattered from a thallium-implanted cr)ctal.

located on substitutional sites and another 40”/0 on the tetrahedral interstitial sites in Si. The area of the peak for the direction (111) (20”/,) corresponds to the number of atoms located at random sites within an error of about 5 % depending on xmin(20). 3. Flu\- Peaking A simple theory for the determination of the atom location in the crystal lattice is easy to obtain but for more detail, especially in quantitative measurements, the flux peaking effect creates some difficulties iii the estimation of the number of atoms. The spectrum of silicon implanted with 200 keV Yb ions has a prominent peak over the random level at the center of the Si dip for the (110) direction (58,59). This peak value over the random level can not be understood by the simple theory described in Section V,B,2. This anomalous peak is explained by considering a nonuniform spatial distribution of the flux density of the He beam in the crystal.

ION IMPLANTATION IN SEMICONDUCTORS

3 19

The computer simulation of channeling gives the results on the variation with depth of the spatial distribution of channeled ions. One example of the distribution for 1.0 MeV He ions is shown in Fig. 38. Between 200 and 300 b;

a

20 16

12

8 4

4 2

1

400

1

1

1

1

1

800

1

,

1

,

1

1

1200

DEPTH(%)

FIG.38. Computed variation of the maximum flux for 1 MeV helium ions in the (100) channel of copper. Curve a-no multiple scattering; curve b-O"C, inelastic niultiple scattering and a beam collimation of i0.06,; curve c-as (b) but with a beam collimation of k0.23'.

flux peaking of about 4 is expected from the calculation under the assumptions of inelastic multiple scattering and a beam collimation of f0.23'. This flux density is calculated for the position at the center of the channel. Therefore, when the interstitial atom is located at the center of the channel such as in the case of the ( 1 10) direction in Fig. 35, the scattered beam from an interstitial atom is enhanced more than that.from a substitutional atom or a random atom. This is the reason why the scattering yield from the interstitial atoms exceeds the random level. The assumption of the potential pipe of the row ofatoms is based on the average continuum potential method formulated by Lindhard. According to this theory, when the positions of the impurity atoms are exactly the same as that of the host atoms then the width (full width at half maximum) of the dips for impurities and host atoms should be the same. The potential is averaged out uniformly along the row so that this potential makes a pipe of

320

SUSUMU NAMBA AND KOHZOH MASUDA

the potential barriers. When the inpurity atoms are located outside the average potential (pipe) the width of the dip for the impurities is different from that for the bulk material. If one finds the same width for both impurity and bulk atoms one can deduce that the sites of the impurities are at substitutional sites. Thus it is always necessary to perform angle scanning to determine the lattice location. Some calculations of the spatial distribution of the flux peaking effect in the plane transverse to the incident beam are shown in Fig. 39. The expected

FIG. 39. One example of computed equiflux contours in the (100) transverse plane of copper for perfect alignment.

value for the flux peaking for well aligned beams is about 7 times larger than the average value. These calculations are the results averaged over a depth of 1080 The channeling effect also occurs for the space between planes. Planar channeling may be treated in a similar manner. Channeling also occurs for compound semiconductors such as GaAs and Gap, and other crystals such as alkali halides.

A.

4. Dgfect

The channeling effect is very sensitive to slightly displaced atoms. Thus channeling is very useful for detecting lattice irregularities. Many lattice irregularities are created by the radiation damage produced during ion implantation. The depth distribution of the damaged region is measured by the channeling effect, as shown in Fig. 40 for the case when the depth resolution is better than the straggling distribution of the damage (60). These results can be obtained by using a magnetic energy analyzer which has better energy resolution than an SSD detector. In the case of amorphous formation, the amorphous yield is equal to the

ION IMPLANTATION I N SEMICONDUCTORS

32 1

100KEV Sb OAMAGE I N Si

- 800 $ 700-

-

3 0

-

u)

9 600-

1,

0

4

2-

500-

-

a

-

..

100-

-

-

SPECTROMETER FLUX ( G A U S S )

FIG. 40. Spectrum of the backscattered particle from damaged region. Intensity of spectrometer flux corresponds with the depth. Curve a-nondamaged crystal; curve b-with damaged region; curve c-random spectrum. (This is equivalent to the complete damaged crystal.)

random level so that the yield in the amorphous region is saturated for the dose of ion implantation. In other words the saturation of the yield for the dose coincides with the formation of amorphous regions.

5. Nuclear Reaction Method a. Thermal neutron beam method. Backscattering of He ions is very useful for many impurities suitable for semiconductor dopants. But unfortunately as described earlier such is not the case for B. The backscattered yield from an impurity atom which has a lighter mass than the host atoms appears over the yield from host atoms. The peak from the B impurities is very weak compared with that of bulk Si because of the very intense yield from bulk Si atoms. Therefore it is very difficult to get information from the analysis of the B peak. Recently another method to detect the B profile has been reported (61). The experimental setup is shown in Fig. 41. One example of this method was carried out using the neutron beam of a reactor with B doped Si. A thermal

322

S U S U M U NAMBA A N D KOHZOH MASUDA

neutron beam of 2.3 x 10' neutrons/cm2 . sec was used to obtain the following nuclear reaction :

+ n + 4He(1471 keV) + 7Li(839 keV), l o B + 12 -+ 4He(1777 keV) + 7Li(1014 keV). loB

(38)

This nuclear reaction has an extremely large thermal cross section (about 3837 barns), and what is more important, this cross section is about LO7 times that for almost all other stable nuclei (including I'B). This produces a good signal-to-noise ratio.

EMRAhcE WNWW 02"AI

I

k

EXIT WlNWW .OYAL

FIG.41. Experimcntal setup of backscattered helium from activated B by thermal neutron.

The attenuation coefficient of the thermal neutrons is negligibly small so that the energy loss of the He atom from the loB to the target (equivalent to the surface) is necessary to determine the depth of the implanted log.Four peaks of the spectrum are observed as described by Eq. (38). The width of the peak of He (1471 keV) is used to calculate the straggling of the implanted loB by using the deconvolution method. h. p-x Reaction. In the case of B implantation, the nuclear reaction of I l B (p,x)*Be is also used to determine the depth distribution of B. In this

ION IMPLAXTATION IN SEMICONDUCTORS

323

case the proton beam (450 keV) from the accelerator for ion implantation is useful. In both the nuclear reactions B (n,s)Li, B(p,a)Be the He atom emitted from the B atom is in the blocking condition. Thus the lattice location of B is also detectable by the angular scanning method (62). C. Activatio~Analysis The intensity of the radiation from the irradiated impurity atoms is normally expressed as A =

b --

fJ

3.7

N Id- , [l - exp (-0.693TD/T,)]. 1034

(39)

where A is the activation in curies, G~ the reaction cross section in barns, Ni the number of reactive nuclei, d, the flux of a flow of particles in neutrons/ cm2 . sec, the half-life of the nucleus produced by the reaction, and T i is the duration of the irradiation. The original elements, production elements, half-life, and produced activity in the case of neutron activation are listed in Table 111. B, N, and C are TABLE 111 ACTIVATION BY THERMAL NEUTRONBOMBARDMEN? Atom

Production

Half-life ~~

2XAI

32P "Gd

'%a "As 1 1 4mIn 1 I6m1"

lz2Sb ""Sb 206T1 3IO B ~

2.27 m 14.3 d 21.4 m 14.2 h 36.5 h 49.0 d 54.0 m 2.75 d 60.0 d 4.19 m 5.0 d

Activity ~

1.6 x lo-' 4.1 x lo-"' 5.4 x 1 0 - 7 4.8 x 10-8 8.5 x 10-9 1.5 x lo-* 5.5 10-5 3.1 x 10-8 5.7 x 1 0 - ' O 5.2 x lo-' 5.6 x IO-"

a l O I 5 atoms are irradiated by 2.8 x l O I 3 thermal neutrons/cm2 . sec.

not suitable elements for neutron activation because of their short lifetimes and small cross sections. The sensitivity of this method for As, for example, is loi3 atoms total depending upon the condition of elimination of the background.

324

S U S U M U NAMBA A N D KOHZOH MASUDA

The depth distributions of the implanted atoms are obtained by successive etching following anodic oxidization. Thermal annealing is often necessary to achieve anodic oxidization when the specific resistivity of the sample is high ( 2 a few R-cm). This trouble is almost negligible when the specific resistivity is low I( 0.1 Q-cm). The thickness of the Si removed by this method is obtained by a measurement of the p r a y intensity. In the case of a GaAs substrate, GaAs is also activated by thermal neutrons. Therefore, chemical separation of the impurity is necessary to detect the profile of the impurity. Isotope implantation into GaAs is more convenient than the thermal neutron activation method.

D. E S R Method The channeling technique is the only direct measurement for detecting the slight displacement of the atoms from the lattice points. This is not so sensitive for vacancy detection. One vacancy, divacancy, or other multivacancies are normally trapping centers for the electrons or holes. When those centers trap the electron or hole and remain spin (1/2) they are detectable by electron spin resonance (ESR). In this case, the electron detected by ESR interacts with neighboring nuclei and other unpaired spins. The electronic structure of the center will be very well understood by analyzing those interactions (63). Profiles of defect centers are also well examined by this method (64, 65). The magnetic field across the sample chamber produces the level splitting for the ESR. The separation AE of two levels of the unpaired spin are shown to be

AE = gPHo. Therefore, the microwave of the frequency 11

=

(40)

AE/h

is absorbable by those levels. Here g is the spectroscopic splitting factor, p the Bohr magneton, H o the static field, and h is Plank's constant. When the spin has surrounding nuclei, it interacts with them through the magnetic dipole interaction between the electron spin and nuclear spin. In such cases, one can observe the hyperfine structure of the ESR spectrum as shown in Fig. 2 where the separations caused by the nuclear magnetic dipole moments are shown. Moreover in the case of high densities of spins interactions between spins predominate. The exchange effects between the spins arising from the overlapping of the wavefunctions, hopping between many sites, and reorientation between equivalent potential sites are expected. This is very often observable in the case of ion implantation.

ION IMPLANTATION IN SEMICONDUCTORS

325

The intensity I A of the absorption by N , spins is expressed as

where T, is the temperature and S the spin. Therefore, if T,, g, and S are almost the same IA is proportional to N , almost exactly. This is very useful for the comparison of the absolute number of centers which corresponds to that of defect centers if they are all paramagnetic centers. In this case there is no difficulty in assuming the oscillator strength, which is often the problem in the spectroscopic study of defects. VI. DEVICES A . Introduction

Ion implantation is now a very popular method for preparing certain semiconductor devices. At an earlier stage of device fabrication by ion implantation, a gate-masked ion implanted MOSFET was the most exciting application of this technique together with semiconductor nuclear particle detectors which have excellent position linearity. Other applications have been in IMPATT diodes, hyperabrupt capacitancevoltage diodes, and high value resistors. Recently the control of the threshold voltages for COS/MOS and the fabrication of high speed microwave bipolar transistors have become extremely interesting. Applications of GaAs in the fabrication of optical IC, modulators, and couplers are also exciting aims in the future. The production of a rail for moving bubbles in magnetic memory materials shows considerable success. Among those, some successful results in silicon devices will be explained in the following. B. Gate-Masked Ion Implanted MOSFET

The only defect of MOSFET was its low speed compared to bipolar devices. This defect was overcome by the gate-masked ion implantation method. The principle of the gate-masked ion implanted MOSFET is illustrated briefly in Fig. 42. A low parasitic capacitance structure is easily made by using ion implantation with a gate-masked system. Except for the part indicated C the device is fabricated by the normal diffusion technique. An overlap tolerance of about 2 pm between the gate and drain is necessary to fabricate this structure by a conventional diffusion method. But exact adjustment to minimize the overlap between gate and drain is very easily performed by ion implantation because no penetration of ions is possible behind the mask (66).

326

S U S U M U NAMBA A N D KOHZOH MASUDA EVOLUTION CF THE ION-IMPLANTED MOSFEl

FIG 42. Gate-masked implanted MOS. (a) Diffused passivated: (b) ion-implanted. not pasivated; (c) ion-implanted and diffused, not passivated; (d) ion-implanted and diffused, passivated.

C. Threshold Voltage Control It is difficult to control the threshold voltage of a MOS transistor. Small amounts of implanted ions affect the shift of threshold voltage. Both p channel and n channel MOS can be controlled by a small amount of implantation. The relation between the dose of the implanted ions and the shift of

e' ion dose ( Id' mi2)

FIG.43. Relation between threshold voltage and implanted dose.

the threshold voltage is linear in the low-dose region as is shown in Fig. 43. The error is generally a few percent but it depends on the thickness of the oxide film (67). D. Bipolar Transistor

A high value of the product of gain and bandwidth, low base resistance, and small collector capacity are required simultaneously in a microwave

327

ION IMPLANTATION IN SEMICONDUCTORS

transistor. For these purposes, an extremely thin base width just under the emitter and an abrupt impurity profile of emitter are necessary. 5.6 GHz for silicon, 11 GHz for germ'anium, and 15 GHz for gallium arsenide were obtained for.f, which is the highest frequency at which the transistor can give a gain 2 6 dB and a noise figure i 6 dB. The structure of this transistor is shown in Fig. 44 (68). Emitter

Emitter electrode Base electrode

/

/

/

\

n Epitaxial layer

J-$

L

Oxide

/ k 'P-diffused

P'guard Implanted t n s e

layer

base

FIG.44. Structure of high frequency transistor.

Recently the double peaked boron profile is used to get a precise control of device parameters over a wide range.

"

1.2pm THICK DOPlNG-10~7cm-~

328

SUSUMU NAMBA AND KOHZOH MASUDA

After the growth of an epitaxial layer of 1.4 p on to the n + Si substrate, B ions of 100 keV and 200 keV acceleration voltage are implanted. A p + layer of 0.15 p is formed by thermal diffusion after ion implantation. This treatment is effective for thermal annealing of the implanted region. The capabilities of this IMPATT diode are an efficiency of 14% and maximum power of 1 W at 50 GHz. One example of the profile of the dopants is shown in Fig.45(69). Two peaks appeared in the B profile in accord with double implantation with different acceleration voltages.

REFERENCES 1 . T. Minamisono, K. Matuda, A. Mizobuchi, and K. Sugimoto, J . Phys. Soc. Jap. 30, 311 (1971). 2. K. L. Brower and J. A. Borders, A p p l . Phys. Lett. 16, 169 (1970). 3. J. A. Borders and K. L. Brower, Radiat. E f j 6, 135 (1970). 4. F. L. Vook, “Radiation Effects in Semiconductors.” Plenum. New York, 196X. 5. W. Kanzig, Phys. Rev. 99, 1890 (1955). 6. W. Kanzig and T. 0. Woodruff, J . Phy.5. Chern. Solids 9, 70 (1959). 7 . G. Feher, Phys. Rec. 114, 1219 (1959). 8. G . Dearnaley. U . K . At. Energy Auth., Res. Group, Rep. AERE-R 6559 (1970).

9. J. Lindhard, M. Scharff, and H. E. Schiott, Kgl. Dan. Vidensk. Selsk., Mut.-Fys. Medd. 33, No. 14 (1963). 10. H. E. Shiott, Kgl. Dun. Vidensk. Selsk., Mat.-Fys. Medd. 35, No. 9 (1966). 11. W. S. Johnson and J. F. Gibbons, “Projected Range Statistics in Semiconductors.” Stanford Univ. Bookstore. Stanford. California. 1969. 12. D. K. Brice, Res. Rep. Sandiu Lab. SC-RR-71-0599 (1971). 13. K. B. Winterbon, Radiat. Eff: 13, 215 (1972). 14. S. Furukawa and H. Ishiwara. J . Appi. Phys. 43, 1268 (1972). l i . S. Furukawa, Proc. U.S.-Japcni Sernin. I o n Inip[unt. Semicond.. K y o t o p. 73 (197 I ). 16. Y. Akasaka, K. Horie, and S. Kawazu, Appl. Phys. Letf. 21, 128 (1972). 17. G. Dearnaley, U . K . At. Energy Auth., Res. Group Rep. AERE-R 6559 (1970). In. G. Dearnaley. J. H . Freeman, G. A. Card. and M. A. Wilkins. Car7. J . Phvs. 46.587 (1968). 19. J. A. Davies, G. L. Ball. F. Brown. and B. Domeij, Can. J . Phys. 42, 1070 (1964). 20. J . W . Mayer, L. Eriksson, and J. A. Davies, “ I o n implantation in Semiconductors: Silicon and Germanium.” Academic Press, New York, 1970. -71. J. A. Davies, personal communication (1972). 22. F. Seitz and J. S. Kohler, Solid Stute Phys. 2, 305 (1956). 23. L. T. Chadderton, “Radiation Damage in Crystals,” p. 29. Methuen, London and Wiley, New York, 1964. 24. K. Gamo, K. Masuda, S. Namba, S. Ishihdra, and 1. Kimura, A p p i . Phys. Lrrr. 17. 391 (1970). 25. J. A. Davies and P. Jespersgard, Can. J . Phys. 44, 1631 (1966). 2 6 K . Masuda. K. Gamo, A. Imada. and S. Namba, Proc. U.S.-Japtrt7 Seniin. lor1 Iniplunt. Srn~rconc/.,Kyoto p. 145 (1971). 27. J. C. Pfister, “ Radiation Damage in Semiconductors.” Dunod, Paris, 1965. 28. P. M. Glotin, Can. J . Phys. 46, 705 (1968). 2Y. T. Abe, Y. Ohmura. M. Konaka. S. Mimura. K. Kanazawa. and H. Ohtsubo. Pro<. L .S-Japan Seniin. IOU Implunt. Setnicoiitl., Kyoto p. 57 (197 I).

ION IMPLANTATION IN SEMICONDUCTORS

329

30. T. Itoh and I. Ohdomari. Proc. US.-Japan Seniin. Iotz Implant. Sernicond., Kyoto p. 49 (1971). 31. W. Schockley, “Electrons and Holes in Semiconductors.” Van Nostrand, New York, 1950. 32. F. Seitz and J. S. Kohler, Solid State Phys. 2, 305 (1956). 33. F. F. Morehead, B. L. Crowder, and R. S. Title, J . Appl. Phys. 43, 1112 (1972). 34. B. L. Crowder, R. S. Title, M. H. Brodsky, and G. D. Pettit, Appl. Phys. Lett. 16. 205 (1970). 3.5. K. Murakami, K. Masuda, K. Gamo, and S. Namba, J a p . J . A p p l . Phys. 12, 1307 (1973). 36. J. W. Mayer, L. Eriksson, S. T. Picraux and J. A. Davies, Can. J . Phys. 46, 663 (1968). 37. B. Crowder, Proc. U S . - J a p a n Semin. lon Implanr. Senzicotzd., Kyoto p. 63 (1971). 38. N. Yoshihiro, Proc. US.-Japun Semin. loti lmplanr. Semicond., Kyoto p. 33 (1971). 39. J. W. Mayer, 0. J. Marsh, R. Mankarious, and R. Bower, J . Appl. Phys. 38, 1975 (1967). 40. R. G . Hunsperger, 0. J. Marsh, and C. A. Mead, Appl. Phys. Lett. 13, 295 (1968). 41. A. G . Foyt, J. P. Donnelly, and W. T. Lindley, A p p l . Phys. L e t f . 14, 372 (1969). 42. R. G. Hunsperger and 0. J. Marsh, Met. Trans. I, 603 (1970). 43. K. Gamo, M. Takai, K. Masuda, and S. Namba, Proc. Cotif Solid State Decices, 4th. T o k y o p. 130 (1972). 44. P. E. Roughan and K. E. Manchester, J . Elecrrochem. Soc., Solid Stare Sci. 116, 278 (1969). 45. A. G . Foyt, J. P. Donnelly, and W. T. Lindley, A p p l . P/?Js. Lett. 14, 372 (1969). 46. J. D. Sansbury and J. F. Gibbons, A p p l . Pkys. Lett. 14, 31 1 (1969). 47. V. M. Zelevinskaya, G . A. Kachurin, N . B. Pridachiu, and L. S. Smirnov, Soc. Ph1.s.Semicond. 4, 258 (1970). 48. A. G. Foyt, W. T. Lindley, C. M . Wolf, and J. P. Donnelly, Solid State Electron. 12, 209 (1969). 49. F. L. Vook and S. T. Picraux, Proc. Int. Conf: Ion Implant. Semicond., 2nd, GurmiscliPurtenkirchen p. 141 (1971). 5 0 . S. Namba, K. Masuda, K. Gamo, and K. Aoki, Proc. U S . - J a p a n Seniin. Ion Inip/trtit. Setnicond, Kyoto p . 163 (1971). 5 1 . H. J. Stein, Rudiat. Eff: 9, 195 (1971). 52. J. A. Borders, A p p l . Phjs. Lett. 18, 16 (1971). 53. E. D. Wolf and R. G. Hunsperger, Appl. Phys. Lett. 16, 526 (1971). S4. G. W. Arnold and R. E. Whan, Radiar. EjJ 7, 109 (1971). 55. J. S. Harris and F. H. Eisen, Radiat. Elf 7. 123 (1971). S6. J. W. Mayer, I. V. Mitchell, and M. A. Nicolet, Proc. lnt. Conf Ion Implant. Semicond., ?rid, Garriiisch-Partenkircheri p. 274 (1971). .i7. J. Gyulai, J. W. Mayer, and I. V. Mitchell, Appl. Phys. Lett. 17, 332 (1970). 58. J. U . Andersen, 0. Andreasen, J. A. Davies, and E. U g g e r h ~ j Radiat. , Eff 7, 25 (1971). SY. F. H. Eisen, Proc. Int. Cot$ Ion Itnplurit. Setnicoiid., 2nd Garmisch-Partenkirchen p. 287 (1971). 60. E. B6gh, P. Hdgied, and I. Stensgaard, Rudinr. 15ff7, 115 (1971). 61. J . F. Ziegler, G. W. Cole, and J. E. E. Baglin, I B M Res. RC 3759 (1972). 62. J. U . Andersen, W. M. Gibson. and E. Uggerhnj, Proc. Int. Conf Appl. Ion Beanis Semicotiti. Technoi., Grenohie p. 153 (1967). 63. K . L. Brower, Radiar. Elf 8, 213 (1971). 64. K. L. Brower, F. L. Vook, and J. A. Borders, A p p l . Phys. Lett. 16, 108 (1970). 65. H. Aritome, T. Ikegami, T. Nishimura, K. Masuda, and S. Namba, Proc. Conf: Solid State Devices, 4th, T o k y o p. 136 (1972). 66. R. W. Bower, H. G. Dill, K. G. Aubuchon, and S. A. Thompson, I E E E Trans. Elecrron Devices 15, No. 10, 757 (1968).

3 30

S U S U M U NAMBA A N D KOHZOH MASUDA

T. Tokuyama. T. Warahisako, and I. Yoshida. Pvoc. L'.S.-Japtr!i Setnirl. 10~7 I r u p l m i / . SwiiC O ! l t / . , KJYJfU p. I13 (1971). 68. K. Fujinuma, T. Sakamoto, T. Ahe, K. Sato, and Y. Ohmura, Proc. Cot!/. Solid Srtrte D e r i c ~ ~1ss, t . TokJ.0p. 71 (1969). 6Y. H . G. Dill, R. M. Finnila, A. M. Lenpp. and T. N. Toonihs. Solid Srute T r c h ~ d p. . 27 (Dec. 1972).

67.

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. Belevitch, V., 91, 96, 97, 98, 157, 177 Beljers, H. G., 91, 110, 177 Bell, R . L., 19, 28(36), 46, 76, 77 Berglund, C . N., 208(43), 260 Berolo, 0..37, 77 Bertram, W. J., Jr., 183(6), 186(6), 206(6), 208(40,41), 216(51a), 229(6, 66), 230(6), 231(6), 237(74), 251(95), 259.260. 261 Betbeder-Matibet. 0..69( 1 I I), 70, 78 Biesheuvel, A., 1 15, 132, 133, 140, 154, 155, 169, 172, 179 Birkhoff, G. D., 84, 88, 177 Bishop, S. G., 46(73), 77 Bittel, H., 154, 177 Blaha, F. C., 255(99), 262 Block, A,, 87, 177 Blom, D., 129, 143, 144, 154, 162, 169, 179 Blount, E. I., 55(92), 78 Blum, F. A,, 48(80), 77 Bogh, E., 320(60), 329 Boiko, I . I., 43(67), 77 Boll, H. J., 208(43), 260 Boonstra, L., 212(45), 260 Borders, J. A., 269(2, 3). 310(52), 324(64), 328,329 Borkan, H., 182(3), 251(92), 259. 26/ Bosma, H., 99, 110, 153. 177 Bower, R..306(39), 325(66), 329 Bowers, R.,28, 29(38), 76 Bowlder, H. J., 57, 78 Boyle, W. S . , 183, 186(5), 213, 215(48), 259. 260 Bracken, R. C., 216(51), 260 Bradley, C. C., 35(50), 38(55), 40(50, 5 5 ) . 42(50), 63(99), 65, 72( I 17), 75( I I7), 77. 78 Braun, E., 7 I ( I 13). 78 Breimer, H., 251(91), 261

A

Abe, T., 298(29), 327(68), 328.330 Adams, K. M., 89,94, 143, 160, 177 Aggarwal, R. L., 1(2), 17(22c), 75, 76 Akasaka, Y.,285(16), 328 Amelio, G., 206(35), 208(40, 41), 225(63), 235(35), 236,260. 261 Anders. R.A., 197(24), 259 Andersen, J . U., 318(58), 323(62), 329 Ando, T., 191(18), 199(18). 259 Andreasen, 0..318(58), 329 Antoniou, A., 127, 128, 141, 177 Aoki, K., 309(50), 329 Apel, J. R.,41, 77 Arams, F. K., 25, 76 Argyres. P. N., 41(60), 42(60), 77 Aritome, H., 324(65), 329 Ark, G., 110, 177 Arnold, E., 200(31), 259 Arnold, G. W., 310, 329 Astrov, D. N., 87, 1 11, 177 Aubuchon, K. G., 325(66), 329 Auston, D. H., 17(22d), 76 Auyang. Y. C. S., 48(82), 77

E Babloyantz, A,, 82, 178 Baerstsch, R. D., 247(85). 26/ Baglin, J . E. E., 321(61), 329 Ball, G. L., 288(19), 328 Bangert, E., 73, 78 Barbe, D. F., 182(1), 246(1), 259 Barker, D. H., 23, 76 Baroff, G. A,, 53(91), 54(91), 78 Barton, J . B., 243(81), 247(86), 249(89). 261 Beadle, C. W.. 251(92), 261

33 I

332

AUTHOR INDEX

Bresler, M. S., 70(109), 78 Brice, D. K., 278(12), 328 Bridges, T. J., 17(22b), 76 Brodsky, M. H., 302(34), 329 Brower, K. L., 269(2, 3)- 324(63,64), 328. 329 Brown, F., 288(19), 328 Brown, R.N..17(16). 28(35), 48(83), 58, 65(16), 68(16), 76,77 Brueck, S. R.J., 48(80), 77 Brugler. J. S., 182(2). 197(2), 255(100), 256(100), 257(100), 259.262 Bruton, J. J., 101, 177 Buck, T. M., 184(9), 259 Buckley. R.R.,183(6), 186(6),206(6). 208(41), 216(51a), 229(6,66),230(6), 23 I(6). 259,260 Burgiel, J . C., 53,54,55. 78 Burke, H. K., 186(12. 12a), 191(12), 192, 200,201(12,I2a). 212(12), 256,259 Burstein. E., 32,76 Buss, D. D., 39,77,249(89),254(98), 261, 262 Button, K. J., 40,63(99, IOO), 72(117, I18), 73(118, 122),74(118,122,123),75,77. 78 C

Callahan. D. E., 197(24), 259 Campana, S. B., 182(1), 240,241.243,245, 246( I), 259,261 Cardona, M., I ( I ) , 75 Carlin, H. J., 96,102.114,177 Carnes, J. E., 214(46,47,47a). 215(46), 216(47), 217(52. 54,55). 218,222(55), 234(72), 240,241(60), 243,245,254(96), 260,261 Carter, D. L., 71(1 12). 78 Castriota, L.J., 96,177 Chadderton, L. T., 292,328 Chamberlain, J. M., 35,40,42,77 Chan, W. W., 243(81), 261 Chang, T. Y.,23,76 Chantry, G. W., 17,76 Cheek, T. F.. Jr., 247(86), 261 Cherry, E. M., 171,177 Cohen. M. H., 55(92), 78 Cohn, D. R.,38(55), 40(55), 77

Cole, J . W., 321(61), 329 Coleman, P. D., 23,76 Collins, D. R.,216(51), 243(81), 249(89), 260,261 Cooky, J. W., 20(26), 76 Couder, Y., 66( 103). 68(103). 7I , 72,73,74, 78 Cronburg, T. L., 74,78 Crowder, B. L., 302(33,34),303(37), 329 Crowell, M. H., 184(9), 200(31), 259 Curan, D. R.,I 1 I , 178

D

Dalton, J . V., 184(9), 259 Daniels, R.W., 121,177 Daverveld, L. A,, 190(15a), 259 Davies, J., 288,328 Davies, J. A., 288,289(20), 290(21), 296(25), 300(20), 302(36), 318(20, 58). 328.329 Davies, R. W., 48(81), 77 Dearnaley, G . , 273(8), 287(17), 288,291(17), 328 de Groot, S. R.. 82.93,177 de Hoop, A . T., 85,177 Dennis, R.B., 35(49), 40(49), 77 Deprette, E., 133, 176,177 DeTemple, T. A., 23,76 Dexter, R. N.,71(115), 72.78 Dicke, R. H., 6,102,75,177 Dickey, D. H., 33,34,39(56), 40,48,77 Dill, H. G., 325(66). 328(69), 329,330 Dimmock, J. 0.. 28(33b), 76 Doi,T.,68,69,70(110), 71(110), 72(110), 73( 110). 74( I lo), 75,78 Domeij, B., 288(19), 328 Donnelly, J. P.. 306(41), 308(45.48). 329 Dresselhaus, M . S., 39(56), 77 Dreybrodt, W., 40(58), 72,73,74(123),77. 78 Duinker, S., 85,93,102,155,175.176,177. I 78 Dunford, N.,85,177 Dyck, R.H., 191(17),197(17), 198(17), 226(65), 227(65),228,235(73), 236(73), 259.261 Dym, H., 190(15),259 Dyubko, S. F., 23,76

333

AUTHOR INDEX

E Eisen, F. H., 310(55), 318(59), 329 Emmons, S. P.,247(86), 254(98), 261.262 Engler, W. E.. 247(85), 261 Enstrom, R. W., 240(77, 78), 261 Eriksson, L., 289(20), 300(20), 302(36), 3 18(20), 328. 329 Erwin, E. L., 191(16), 259 Esser, L. J . M.,220,260 F Fan, H. Y., 43(71), 77 Farnsworth, D., 191(16), 259 Favrot, G., 17(22c), 76 Feher, G., 270.328 Fellgett, P.,22, 76 Fesenko, L. L., 23, 76 Fetterman, H. R.. 23, 35, 37.40. 76. 77 Fettweis, A., 109, 177 Finnila, R. M.,328(69), 330 Fischer, D., 73(122), 74(122), 78 Fletcher, R. C., 60, 78 Foltz, J . W., 190(15b), 259 Foner, S.. 35(45), 77 Fonstad, C. G., 40(58), 77 Forgue, S. V., 184(8), 259 Foyt, A . G., 306, 308(45,48), 329 Frantz, V. L., 197(26), 259 Freeman, J. H., 288(17), 328 Fujinuma, K., 327(68), 330 Fujiyasu, H., 60, 78 Furdyna. J . K., 2, 75 Furukawa, S.,284(/4, 15), 328

c Gamo, K.. 294(24), 296(26), 302(35), 308(43), 309(50), 328,329 Gantmacher, F. R.,154, 177 Card, G. A,, 288(17), 328 Gebbie, H. A,, 32(39), 76 Geffe, P. R., 95, 177 Geyser, R. D., 200(31), 259 Gibbons, J. F., 278(1 I), 308(46), 328, 329 Gibson, W. M.,233(62), 329 Glass, A. M.,17(22d), 76 Glotin, P. M.,298(28), 328

Goodrich, R. R., 184(8), 259 Gornik, E., 17(22e), 76 Gray, S., 251(92), 261 Grosse, R.,72(117), 75(117), 78 Groves, S. H., 17(16), 38(54), 46(77), 49(85), 65,68( 109, 76, 77, 78 Grynberg, M., 66( 103). 67( 104). 68( 103, 104), 78 Guldner, Y., 67, 68(104), 78 Gunsager, K. C., 219(58), 260 Gurgenishvilli, G. E., 43(66), 49(66), 5 1(66), 77 Guthmann, C., 71(114), 78 Gyulai, J., 313(57). 329 H

Hannan, W. J., 208(42), 260 Harman, T. C., 39(56), 77 Harris, J. S.; 310(55). 329 Hebel, L. C., 53, 54, 55, 78 Heiman, F. P., 185(10),259 Heinlein, W. E., 128, 141, 178 Hensel, J. C., 51(86), 5 2 , 55(93,94, 9 9 , 57(88,94,95), 60(88,93,94,95), 62.63, 77, 78 Hodby, J. W., 28(34), 34, 76 Hodges, D. T., 23, 76 Hofstein, S. R., 185(10), 259 Hogan. C. L.,87, 110. 177 Hogied, P.,320(60), 329 Holm. W., 251(91),26/ Homa, W. S., 195(22). 259 Hooper, D.E., 171, 177 Horie, K., 285(16), 328 Horton, J . W., 190(15), 259 Houghton, J., 17, 76 Hulin, M., 68. 69(111), 70, 73(120), 74(120). 78 Hunsperger, R. G., 306, 310(53), 329 I Ikegarni, T., 324(65), 329 Imada, A., 296(26), 328 Isaacson, R. A., 49(84), 77 Ishihara, S., 294(24), 296(24), 328 Ishiwara, H., 284( 14). 328 Itoh, T., 299(30), 329

334

AUTHOR INDEX

J Jefferson, H., 86, 178 Jespersgard, P., 296(25), 328 Johnson, E. J., 33, 34,40, 48, 77 Johnson, M. H., 6, 75 Johnson, W. S., 278(1 I), 328 Joseph, R. A,, 41(61), 77 Joyce, W . B., 237(74), 261

Kosonocky, W. F., 214(46, 47, 47a), 215(46, 50), 216(47), 217(52, 54, 5 5 ) , 218, 222(55, 96), 234(72), 240, 241(60), 243, 245,254(96), 260,261 Kovac, M. G., 191(20), 205(20), 207(20, 36) 208(20, 37, 38, 39). 210(38), 21 1(20,44), 232(36), 233(68), 234(72), 246(36), 249(36a), 25 1(36), 255(20, lo]), 256(36, 68), 259,260,261,262 Krambeck, R. H., 215(49), 219(57), 260

K L

Kachurin, G. A,, 308(47), 329 Kacman, D., 43(69), 51(69). 77 Kanzig, W., 270, 328 Kamimura, H., 68(107), 69(107), 70(1 lo), 71(1 lo), 72(1 lo), 73(l lo), 74(1 lo), 75(126), 78 Kanazawa, K., 298(29), 328 Kane, E. O., 28, 76 Kaplan, R., 28(33a), 41,42,46(73), 76, 77 Kaufrnan, I., 190(15b), 259 Kawabata, A,, 41,42, 77 Kawazu, S., 285,328 Kazan, B., 247(83), 261 Keen, A. W., 113, 178 Key, E. L., 170, 178 Keyes, R. J . , 32(41), 35, 74, 77 Kida, K., 102, 107, 108, 178 Kim. C . K., 217(53), 219(58), 226(65), 227(65), 228,260 Kimura. I . , 294(24), 296(24), 328 Kinch, M . A,, 28(33a), 39, 76. 77 Kishi, G., 102, 107, 108, 178 Kittel, C., 11, 14(14), 74 Kjeldaas, T., 13(12), 76 Klauss, E., 87, 178 Klein. M. V., 10, 75 Klein, W., 124, 178 Krnetz, A . R., 249(89), 261 Kneubuhl, F., 19(24), 76 Knoll, B., 247(83), 261 Kobayashi, K. L. I., 34, 77 Kohler, J. S., 292, 299(32), 328.329 Koehler, T., 204(33), 259 Kohn, W., 13(12), 56(11), 76 Kolm, H. H., 35(45), 77 Konaka, M., 298(29), 328 Korn, D. N., 23(28), 76

Labuda, E. F., 184(9), 259 Lampe, D. R., 255(99), 262 Landau, L. D., 6, 75 Landwehr, G., 71(113), 72(117), 73(122), 74(122, 123), 75(117), 78 Lankard, J. R., 17(22a), 76 Larsen, D. M., 38(55), 40(55), 77 Lax, B., 2, 17(22c), 32, 35(45), 38(55), 40(55), 41(60), 42(60), 63(99), 72(117, 118), 73(118), 74(118, 123), 75(117), 75, 76, 77. 78 Le Fur, P., 17(22d), 76 Leich, W., 167, 178 Lenpp, A. M., 328(69), 330 Lenzlinger, M., 217(53), 260 LeToullec, R.. 66(103), 68(103), 78 Leung, W., 65, 78 Levine, P. A . , 222(96), 234(72), 250(90), 254(96), 261 Lindhard, J., 273,328 Lindley, W. T., 306(41), 308(45, 48). 329 Lippmann, B. A., 6 , 75 List, W . F., 191(16), 197(24), 259 Litton, C. W.. 35, 40, 77 Liu, L., 65, 78 Luttinger, J . M., 13, 17(15),49(15), 56(11, 15), 76 M

McCann, D. H., 197(24), 259 McCombe, B. D., 41(62), 42(62), 43(70), 46(70), 47(78), 48,49(79), 77 McGee, J . D., 23, 76 Mack, I . A., 255(99), 262 McKenna, J., 219(57), 260

335

AUTHOR INDEX

McMahon, W. R., 2 16(5 I), 260 McMillan, E. M., 86, 178 McNamara, W. J., 183(6), 186(6), 206(6), 208(41), 229(6), 230(6), 231(6), 259 Manchester, K. E., 308(44), 329 Mankarious, R., 306(39), 329 Marsh, 0. J., 306, (39, 40). 329 Mashovets, D. V., 70( l09), 78 Masuda, K., 294(24), 296(24, 26), 302(35), 308(43), 309(50), 324(65), 328, 329 Mathur, D. P., 200(31), 259 Matuda, K., 268(1), 328 Mavroides. J. G., 2, 32(41), 39(56), 53(90), 75, 76. 77, 78 Mayer, J. W., 302(36), 306, 312(56), 313, 328.329 Mazur, P., 82,93,177 Mazza, R. V., 190(15), 259 Mead, C. A., 306(40), 329 Meindl, J. D., 182(2), 197(2), 256,259, 262 Meray-Horvath, L., 182(3), 186(13), 187(14), 189(13). 191(13), 194(14), 195(13,22), 199(13), 200(13), 201(13), 257(13), 259 Merritt, F. R., 60(97), 78 Mesner, M. H., 251(94), 261 Meyer, Jr., J. E., l87( l4), l94( l4), 259 Michon, G. J., 186(12, 12a). 191(12), 192, 200, 201(12, 12a), 212(12), 256,259 Mikkelsen, J. C., Jr., 208(41), 260 Millar, W., 81, 178 Mimura, S., 298(29), 328 Minamisono, T., 268(1), 328 Mitchell, D. L., 48(83), 77 Mitchell, I. V., 312(56), 313,329 Mitra, S. K., 123, 176, 178 Mizobuchi, A., 268(1), 328 Moller, K. D., 17, 18(23), 76 Morehead, F. F., 302(33), 329 Morris, F. J., 216(51a), 260 Miiller, J. H. W., 143, 178 Murakami, K., 302(35), 329 Murase, K., 60(98), 78 Murotani, T., 34, 77 Mycielski; A., 67(104), 68(104), 78

N Nakao, K., 68(107), 69(107),70(1 lo), 7l(I lo), 72(l lo), 73(110), 74(1 lo), 78

Nakazawa, K., 102,178 Namba, S., 294(24), 296(24, 26), 302(35), 308(43), 309, 324(65), 328,329 Neirynck, J., 101, 178 Ngai, L. H., 23, 76 Nguyen, V. T., 17(22b), 76 Nicolet, M. A., 312(56), 329 Nicolis, G., 82, I78 Nicoll, F. H., 246(82), 261 Nishimura, T., 324(65), 329 Nisida, Y., 34, 77 Noble, P. J. W., 191(19), 199(19),259

0 Ochs, S. A., 251(92), 261 O’Dell, T. H., 111, 178 Ohdomari, I., 299(30), 329 Ohlmann, R. C., 25(31), 76 Ohmura, Y., 298(29), 327(68), 328,330 Ohtsubo, H., 298(29), 328 Oono, Y., 91, 109, 178 Orchard, H. J., 100, 106, 108, 109, 119, 128, 141, 144, 171, 178 Otsuka, E., 34, 60(98), 77, 78

P Palik, E. D., 2, 17(21), 32, 35, 36, 75, 76, 77 Pars, L., 82, 178 Paul, W., 38(54), 77 Penfield, P., 85,93, 102, 155, 178 Penn, T. C., 216(51),260 Pettit, G. D., 302(34), 329 Pfister, J. C., 297(27), 328 Phillips, J. D., 219(58), 260 Picard, J. C., 71(112), 78 Pickar, K. R., 215(49), 260 Picraux, S. T., 302(36), 309(49), 329 Picus, G. S., 32(39,40), 76. 77 Pidgeon, C. R., 17(16), 28(35), 48, 58, 65(16), 68(16, 105), 76, 77, 78 Pike, W. S., 186(13), 187(14), 189(13), 191(13,20), 194(14), 195(13), 199(13), 200(13), 201(13), 205(20), 207(20, 36), 208(20, 37,38,39), 210(38), 21 1(20,44), 232(36), 233(68), 246(36), 249(36a), 251(36), 255(20, IOI), 256(36,68), 257(13, 103, 104). 259,260,261, 262

336

AUTHOR INDEX

Plant, T. K., 23, 76 Plummer, J. D., 182(2), 197(2), 256,259,262 Poehler, T. O . , 41(61), 77 Polder, D., 87, 178 Pressley, R. J., 23, 76 Pridachin, N. B., 308(47), 329 Prigogine, I., 82, 178 Prinz, G. A,, 24, 46(75), 76 Pritchard, D. H., 251(93), 261

R Radoff, P. L., 71(115), 72, 78 Ramberg, E. G., 217(52), 260 Ranvaud, R.,52, 64, 78 Rashba, E. I., 43(63). 77 Rayleigh, Baron (Strutt, J. W.), 81, 83, 85, I 78 Reed, R. D., 23, 76 Rhines, W. C . , 243(81), 261 Richards, P. L., 25(31), 76 Rigaux, C., 67(104), 68(104), 78 Riordan. R. H. S., 128, 141, 178 Robinson, G . A,, 240(77), 261 Robinson, L. C., 17, 65, 76 Rogers, K . T., 28(36), 76 Rodgers. R. L., 111.233, 234(72), 256(69), 26 I Rose, A., 239(76), 240(76), 261 Rothschild, W. G . , 17, 18(23), 76 Roughan, P. E., 308(44), 329 Rowell, J . M., 53(91), 54(91), 78 S

R., 95. 178 Sadasiv. G., 182(3), 186(13), 187(14), 189(13), 191(13), 194(14), 195(13, 22), 199(13), 200(13, 301, 201(13), 257(13, 103). 259,262 Sakamoto, T., 327(68), 330 Saleh, A . S., 43(71), 77 Sallen, R. P.,170, 178 Salsbury, P. J., 182(2), 197(2), 223(62), 246(62), 255,256(62), 259.261 Sangster, F. L. J., 183, 186(4), 202(32), 208, 212(45), 247(84), 259,260.261 Sansbury, J. D., 308(46), 329 Sato, K . , 327(68), 330 Sdal,

Schade, 0. H., Sr., 232(68), 237(67), 261 Schanne, J. F., 208(42), 260 Scharff, M., 273(9), 328 Schiott, H. E., 273(9, lo), 328 Schlossberg, H. R., 23, 76 Schockley, W., 299(31), 329 Schoeffler, J. D., 111. 178 Schroeder, J. E., 247(86), 249(89), 261 Schryer, N. L., 219(57), 260 Schuster, M. A., 197(24. 25), 259 Schwartz, J. T., 85, 177 Scott, W. C., 28(33a), 76 Sealer, D. A., 183(6), 186(6), 206(6), 208(41), 216(51a), 225(64), 229(6, 66), 230(6), 231(6), 251(95), 259,260. 261 Secombe, S. D., 23(28), 76 Seitz, F., 292, 301(32), 328, 329 Stquin, C . H., 183(6), 186(6), 206(6, 34), 216(5 la), 225(64), 229(6, 66). 230, 23 I , 234(71), 251(95), 259,260.261 Shallcross, F. V., 182(3), 186(13), 189(13), I9 I ( 13, 20), 195(13), 197(26), 199( I3), 200(13), 201(13), 205(20), 207(20, 36), 208(20, 37-39), 2 10(38), 2 I 1(20,44), 232(36), 233(68), 234(72), 246(36), 249(36a), 251(36), 255(20, 101). 256(36, 68), 257( l3), 259,260,261. 262 Shankoff, T. A., 183(6), 186(6), 206(6). 216(51a), 225(64), 229(6), 230(6), 231(6), 259,260.261 Sharpe, G. E., 124, 178 Sheahan, D. F., 106, 109, 128, 141, 144, 171, I 78 Sheka, V. I . , 43(63, 65). 45. 77 Shin, E. E. H . . 41, 42, 77 Shinno, H., 75(126), 78 Shortes, S. R., 216(51), 243(81),260, 261 Silverman, J . H., I 1 I , 178 Simmonds, P. E., 35(50), 40(50), 42(50). 65(101), 77, 78 Skwirzynski, J. K., 95, 178 Smirnov, L. S., 308(47), 329 Smith, G. E., 53(9l), 54, 183, 186(5), 213, 215(48), 219(57), 225(63), 78,259, 260, 26 f Smith, S. D., 17, 35(49), 40(49), 76, 77 Snoek, J. L., 110, 177 Snow, E. H., 191(21), 259 Soderstrand, A . M.,176, 178

337

AUTHOR INDEX

Sorokin, P. J., 17(22a), 76 Spence, R., 85,93, 102, 155, 178 Steckl, A. J., 204(33), 260 Stein, H. J., 309(51), 329 Stensgaard, I., 320(60), 329 Stevenson, J. R., 35(51), 36, 77 Stillman, G. E., 28(33b), 76 Stockton, J. R., 65(101), 78 Storm, L., 154, 177 Stout, M. B., 150, 178 Stradling, R . A,, 35(50), 40(50), 42(50), 65(101), 77, 78 Strain, R . J., 219(57), 260 Strauss, A. J., 39, 77 Strull, G., 191(16), 197(25), 259 Sugimoto, K., 268( I ) , 328 Suzuki, K., 52, 55(93,94,95), 57(88,94,95), 60(88, 93,94,95), 62, 63, 78 Svich, V. A,, 23, 76 Szentirmai, G., 176, 178

T Takai, M., 308(43), 329 Tan, S. L., 251(91), 261 Tanaka, S., 72, 73, 74,75(126), 78 Tannenwald, P. E., 38(55), 40(55), 77 Tasch, A . F., Jr., 247(86), 261 Teer, K., 183, 186(4), 208,259 Teitler, S., 32(40), 35(46), 76, 77 Tellegen, B. D. H., 86, 87, 93, 102, 1 1 I , 113, 114, 115, 178 Temes, G. C., 108, 178 Thiran, J . P., 101, 178 Thome, H., 66(103), 68(103), 73(120), 74( I20), 78 Thompson, H . C., 251(92), 261 Thompson, S. A,, 325(66), 329 Thornberger, J. K., 222,261 Thuiller, J. M., 71(114), 78 Tichovolsky, E. J., 53(90), 78 Tiemann, J. J., 247(85), 261 Tinkham, M., 25(31), 76 Title, R. S., 302(33, 34) 329 Tokuyama, T., 326(67), 330 Tompsett, M. F., 183(6), 186(6), 206(6), 208(40,41), 216(51a), 218(56), 222, 225(64), 229(6,66), 230(6), 231(6),

237(75), 238(75), 248(88), 251(95), 254(97), 259,260,261.262 Toombs, T. N., 328(69), 330 Trimmel, H. R., 128, 141, 178 Truesdell, C., 82, 179 Tuchendler, J., 65, 66,68(103), 78 Tukey, J. W., 20(26), 76 U

Uggerhoj, E., 318(58), 323(62), 329 V

van Bastelaer, P., 167, 178 van der Ziel, A,, 143, 179 van Looij, H. T., 143, 160, I79 Van Vechten, J . A,, 37(53), 77 von Ortenberg, M., 73, 74, 78 Vook, F. L., 269(4), 309(49), 324(64), 328, 329 Voorman, H. O., 115, 132, 133, 140, 154, 155, 169, 172,179 Voorman, J. O., 129, 143, 144, 154, 162. 169 I79 W

Wagner, R. J., 23, 24, 41(62), 42(62), 46(75), 47(78), 48, 76, 77 Walden, R. H., 215(49), 219(57), 260 Waldman, J . , 23, 35(52), 37(52), 38, 40(52), 76, 77 Wallis, R. F., 32(40), 35(46, 51). 57, 76, 77, 78 Walsh, E. J., I84(9), 259 Walsh, L., 235(73), 236(73), 261 Warabisako, T., 326(67), 330 Weckler, G. P., 186(11), 188(11), 191(17), 197(17), 198(11, 17,27),259 Weiler, M . H., 69(108), 70, 71(108), 73(108), 74( 108, 123). 78 Weimer, P. K., 182(3), 183(7), 184(8), 186(13), 187(14), 189(13), 191(13,20), 194(14), 195(13, 22). 196(23), 197(26), 199(13), 200, 201,205(20), 207(20, 36). 208(20, 37, 38, 39). 210(38), 211(20, 44), 232(36), 233(68), 240(7), 246(36), 247(87), 249(36a), 251(36, 92), 255(20,

338

AUTHOR INDEX

101). 256(36, 68). 257(13, 103). 259, 260, 261,262 Wen, D. D., 223(62), 246(62), 255, 256(62), 261 Westgate, C. R., 41(61), 77 Whan, R . E., 310,329 Whelan, M. V., 190(15a), 259 White, M . H., 255, 262 Wilkins, M. A , , 288(17), 328 Winterbon, K . B., 283,328 Wittke, J . P., 6, 75 Wolfe, C. M., 28(33b), 35(52), 37(52), 40(52), 76, 77 Wolf, C . M . , 308(48), 329 Wolf, E. D., 310(53), 329 Wolff, P., 43(68), 55(68), 77 Woodruff, T. 0.. 270(6), 328 Woolley. J . C., 37(53), 77 Woywood, D. J., 208(42), 260 Wynne, J . J . , 17(22a), 76

Y Yafet, Y . , 13(13), 28, 29(38), 43(64), 76. 77 Yager, W . A , , 60(97), 78 Yashida, I., 326(67), 330 Yasu-ura, K . , 91, 109, 178 Yoshihiro, N., 304(38), 329 Yoshizaki, R., 72, 73, 74, 75(126), 78 Youla, D., 96, 98, 114, 179 Young, W. T., 182(2), 197(2), 259 2

Zawadzki, W . , 43(69), 51(69), 77 Zeiger, H. J . , 32(41), 76 Zelano, A . J . . 23, 76 Zelevinskaya, V. M., 308(47), 329 Ziegler, J . F., 321(61), 329 Zimany, E. J.. Jr., 248(88), 261 Zwerdling, S., 35(45), 77

Subject Index A

Abstract spaces, theory of, 85 Adaptive gyrator, 172-174 Alkali halides, cyclotron resonances and, 28 Amplifier, ideal, 113-116 Amplifier circuit, nullor-resistor equivalents of, 128 Analog signal processors, charge transfer sensors as, 247-251 Antireciprocity, 91 origin of, 83-85 Artificial NPN and P N P transistors, 132-135

B Backscattering, in ion implantation, 311-314, 321-322 Bell Telephone Laboratories, 182, 225, 265, 270 n. Bismuth, spin-flip resonances of, 53-55 Bloch functions, 11-12 Bohr magneton, 9 Bolometer, as cryogenic detector, 26 Bowers-Yafet model conduction-band wavefunctions from, in electron cyclotron resonance, 28-32 Brillouin zone, for tellurium, 68 Bucket brigade camera, 209-210 Bucket brigade registers, 208-212, 255

C Camelback band, for tellurium, 69-73 CCD(s) (charged-coupled device), 193, 196, 199, 201, 203 see also Charge-coupled devices vs. bucket brigade registers, 211 buried-channel, 216-217, 219-220 characteristics of, 213-214 continuous operation of, 216 “fat-zero” current in, 217-219, 222, 245

fixed-pattern noise in, 224 general description of, 213-216 noise characteristics of, 220-224, 240 silicon gate in, 215 surface-channel, 217-219 transfer losses in, 216-220 two-phase silicon gate, 215 TV resolution in, 233 CCD page reader, 225 CCD registers, 206 CCD sensors, 225-227 ideal, 240, 243 single-line, 225-228 Channeling effect technique, 31 1-324 displaced atoms and, 320 Charge-coupled devices, see CCD(s) Charge-coupled image sensors, see Chargecoupled sensors Charge-coupled registers, 206, 214 see also CCD(s) Charge-coupled sensors see also CCD(s) experimental, 225-236 with illuminated registers, 229-234 with nonilluminated registers and interleaved photocells, 235-236 performance limitations of, 236-247 resolution of, 237-239 three-phase, 230-231 two-dimensional area-type, 229-236 Charge-injection sensor% 199-202 Charge pumping, clock voltages and, 219 Charge transfer scanning by, 202-208 three-phase register and, 213-214 Charge transfer delay lines, 247-249 Charge transfer registers, integrated, 202 Charge-transfer sensors see also Charge-coupled sensors as analog signal processors, 247-251 with bucket brigade registers, 208-212 output circuit design for, 254-256 signal-to-noise ratio in, 203 single-line, 202-204

339

340

SUBJECT INDEX

two-dimensional, 205-208 Chebyshev characteristics, gyrator and, 105, 108 Circulator, gyrator as, 98-99 Clock cycle, in CCDs, 216 Clock frequency, in CCDs, 217 Clock voltages or transients, in chargetransfer scanning, 203-206, 215,249,255 Color TV cameras, self-scanned sensors for, 251-253 see also Television (adj.) Complex dielectric function, 4 Complex refractive index, 4 Conjunctor, gyrator as, 174-176 Converter, negative-impedance, 126-127 COS/MOS threshold voltages, control of, 325 Cryogenic detectors, 26-27 Cyclotron, “classical” vs. “quantum,” 59 Cyclotron resonance for bismuth semiconductors, 53 of electrons, 28-42 FIR lasers in, 33-34, 39-40 of germanium, 57 of holes, 55-75 Landau levels and, 34-35,41 line shape studies in, 40-42 Luttinger effects in, 63-64 nonequilibrium electrons and, 34-35 stress dependence in, 61-62

D Diamond semiconductors degenerate bands in, 56-65 valence bands in, 49-53 Dielectric function, complex, 4 Ductility, in gyrator, 92-93

E EDE-ESR, see Electricdipole excited electron spin resonance Effective mass approximation, 11-16 Electrical engineering, gyrator concept in, 87 Electricdipole excited electron spin resonance (EDE-ESR), 45-49 Electricdipole interaction, 10

Electric-dipole transitions, matrix element for, .16 Electric field, equation for, 4 Electromagnetic field, equation for, 5 Electromagnetic radiation, optical properties and, 3-6 Electromagnetic selection rules, 10 Electromagnetic transitions, 3 Electron in periodic potential, 11-17 Schrodinger equation for, 11-12 Electron cyclotron resonance, 28-42 see also Cyclotron resonance line positions in, 32-40 Electronic circuitry, ‘‘ latch-up ” in, 140-141 Electronic gyrator, see Gyrator Electronic systems, gyrator in, 79- 177 see also Gyrator Electron spin, intrinsic, 8-9 Electron spin resonance (ESR) electricdipole excited (EDE-ESR), 45-49 in ion implantation, 324-325

F Fabry-Perot cavity, 23 Fairchild Camera and Instrument Corp., 182, 235 Faraday geometry, in hole cyclotron resonance, 66 Far infrared boundaries of, 2 experimental techniques in, 17-28 intraband transitions in, 1-2 see also FIR (adj.) Far infrared magneto-optical techniques, for zone-centered electrons in zinc blende semiconductors, 28-29 Fast Fourier Transform, 20 Fat-zero current, in CCDs, 217-219, 222,245 Ferromagnetic materials, gyromagnetic effect in, 87 Filter(s) defined, 94 gyrator as, 109, 167-169 power-handling capacity of, 169 resonance in, 101-107 Filter-network configuration, classical, 94-95

SUBJECT INDEX

FIR (far infrared) grating spectrometers, 18-19 FIR lasers, 17, 23-25 in cyclotron resonance of electrons, 33-34, 39-40 in hole cyclotron resonance, 65 FIR laser spectrometer, 24-25 FIR magneto-optics, detection methods for, 25-28 FIR magnetospectrometry, 17-23 FIR methods, description of, 17-28 FIR spectrometry, in cyclotron resonance, @4 1 Flux peaking, in ion implantation, 318-320 Fourier Transform spectrometers, 19-23 advantages of, 22 basic equation for, 20 modular, 21 Four-port, gyrator as, 145-147 Four-port admittance matrix, 147 Four-resistor network compared with two-resistor, 128 in signal path, 141-143 Free carrier resonances, 28-75 cyclotron resonances and, 28-40 spin-flip resonances and, 42-55 Free electron bolometer, 26 Free electrons, quantum theory of in magnetic field, 6-11 FTS, see Fourier Transform spectrometers

G Gallium arsenide, in ion implantation, 305-310 General Electric Co., 182, 270 n. Germanium bolometer, 26-27 Golay cell, 25 Gyrational resistance, 90 Gyrator(s) see also Nullor active circuits of, 111-1 13 adaptive, 172-174 admittance measurements for, 150 antiparallel connection in, 117 antireciprocal nature of, 155 applications of, 155-176 base-current compensation circuit for, 140 basic circuit for, 134

341

basic electronic design of, 128-145 basic measurements in, 145-169 basic properties of, 88-94 bias or offset in, 152-153 cascaded, 157 as circulator, 155-157 class B, 171-172 as conjunctor, 174-176 for consumer products, 170-171 design of, 128-145, 170-176 as device, 109 duality in, 92-93 in electrical engineering, 87 and electromechanical or mechanicomagnetic transducers, 110-1 11 in electronic systems, 79-177 equations for, 164 filters and, 94109, 167-169 future applications of, 175-176 gyroscopic forces and, 109-1 10 grounded and semifloating, 148-150 ideal, 86 ideal transformer and, 91 impedance inversion property of, 130-131 interchange of variables in, 92-93 intermodulation with, 164, 167-169 isolation and power transmission in, 89-90 as isolator, circulator, and transformer, 155-1 57 ladder filter and, 99-101 as lossless two-port, 98-99 as network element, 86-94 noise currents in, 143-145, 153-155 as nonenergic system, 88-89 as n-port, 146-155 nullar concept and, 113-116 offset in, 152-153 parasitic effect in, 123 passive circuits and, 109-111 physical effects of, 109-111 principle of realization of, 109-128 for professional use, 171-176 for radios and tape recorders, 170 reactive power, 93-94 scattering matrix and, 96-98 semifloating, 148- 149 sensitivity in, 107-108 stability of circuit in, 135 synthesis of from nullor and resistors, 125 Tellegen medium and, 111

342

SUBJECT INDEX

as three-port circulator, 157 as transformer, 155-157 Gyrator-capacitor circuits, 136- 137, 175 Gyrator-capacitor parallel resonant circuit, 157-167 Gyrator circuit basic, 134 frequency response of, 136 as grounded four-port, 145-147 noise in, 143-145, 153-155 stability of, 135 Gyratorless networks, magnetic and electrical energies in, 105 Gyrator resonant circuit, 157-167 Gyromagnetic effect, 87, 110 Gyroscopic forces, cryostat and, 109-1 10 Gyroscopic systems, nonreciprocity in, 82 Gyrostat antisymmetric relations in, 82 equation of motion for, 84 nonenergic system and, 84 spin direction in, 83

H Hall-effect measurement, in ion implantation, 300 Hamiltonian free-electron, 15 for single band, 14 for single electron, 6, 12-13 of system of charged particles plus radiation, 4-5 for two-dimensional simple harmonic oscillator, 7 Harmonic oscillator functions, 57-58 Hilbert spaces, theory of, 85 Hole cyclotron resonance, 55-75 FIR lasers in, 65 Luttinger effects in, 63-64

I

IBM Watson Research Center, 267 Ideal active network elements, 113 Ideal amplifier, nullor as, 113-114 Ideal gyrator, 86

Ideal sensor, resolution vs. illumination in. 240,243 Ideal transformers, from two gyrators, 91 Illuminated-register sensor, 203-204 Image sensor charge-coupled, see Charge-coupled sensor function of, 183-184 MOS transistor in, 185-186 multiplexed scanning in, 187-193 output circuit diagram and, 254257 single-line, 188-190 two-dimensional, 190 XY-addressed, see XY-addressed photodiode arrays; XY image sensors Immitance inversion, 91-92 IMPATT diodes, 325-328 Interband transition, vs. intraband, 17 International Conference on Ion Implantation in Semiconductors (1970), 266 Interstitial diffusion, in ion implantation, 295-296 Intraband magneto-optical studies, of semiconductors in far infrared, 1-75 Intraband transitions, 1 vs. interband, 17 Inverter, negative-impedance, 126 Ion implantation activation analysis in, 323-324 annealing and electrical properties in, 299-310 backscattering phenomena in, 3 11-314 in bipolar transistor, 326-327 channeled particle in, 288-289 channeling effect technique in, 31 1-323 concentrated profiles of, 271-289 concept of, 267 damage centers in, 292 defects in, 271-289 devices based on, 325-328 diffusion coefficient in, 285-286 diffusion through vacancies in, 291-295 electron spin resonance method in, 324 enhanced diffusion in, 289-299 exponential tail in, 295-296 flux peaking in, 318-320 foreign atom location in lattice, 296 gallium arsenide in, 305-310 gaussian distribution in, 283-284, 287 Hall-effect measurement in, 300 historical review of, 264267

343

SUBJECT INDEX impurities in, 267-271 interstitial diffusion mechanism for enhanced diffusion in, 295-296 lattice configuration in, 314318 lattice irregularities in, 320-321 Lindhard theory in, 267-268, 272-278, 284 measurement technique in, 310-325 nuclear reaction method in, 321-323 numerical calculations in, 278-283 radiation enhanced diffusion in, 297-299 in semiconductors, 263-328 silicon in, 300-305 single crystal target in, 285-289 thermal neutron beam method in, 321-323 threshold voltage control in, 326

J JFET (junction field effect transistor) devices, ion implantation in, 271

K Kinch-Rollin detector, 27 Kronecker delta, 12

L Ladder filters, 99- 101 Lamellar interferometer, 20 Landau level in cyclotron resonance, 41 lowest, 48 for tellurium, 71 Landau quantum number, 8, 16,60 spin change and, 45 Laser, FIR, see FIR lasers " Latch-up," in electronic circuitry, 140-141 Lattice, atom location in for ion implantation, 314318 Leap-frog circuit, gyrator in, 176 Light pipes, 25 Lindhard-Scharf-Schi$tt (LSS) theory and calculations, 267-268, 272-278, 284 depth distribution and, 289-290 Line positions, in electron cyclotron resonance,32-40

Line shape studies, in cyclofron resonance, 40-41 Lorentz force, 110 antisymmetry and, 83 Luttinger effects, in hole cyclotron resonance, 63-65

M Magnetic-dipole interaction, 11 Magnetic field, quantum theory of free electrons in, 6- 11 Magneto-optical transitions, in semiconductors, 2-6 Mercuric telluride, inverted band ordering in, 65-68 Michelson interferometer, 20 Modulation transfer function, 237 MOS (metal oxide silicon) bucket brigade circuit, 210 MOS capacitor, 186 MOSFET (metal oxide silicon field effect transistor), gate-masked ion implantation of, 325-327 MOS gates, in CCD line sensors, 225-227 MOS output amplifier, output register and, 210 MOS photodiode array(s), 199, 208 charge-transfer readout of, 21 1-212 MOS scan generator, 189 MOS transistor, 198 in image sensors, 185-186 ion implantation and, 271 in single-line sensors, 188 thin-film, 182 MTF (modulation transfer function) curves, 237 Multiplex scanning, in image sensors, 187-193

N

Negative-impedance converter, 126-1 27 Negative-impedance inverter, 126 Network(s) with four resistors, 124-128 with two resistors, 117-123 voltage-current reciprocity in, 8 1-82

344

SUBJECT INDEX

Network element gyrator as, 86-94 ideal active, 113 Noise in gyrator circuits, 143-145 in n-port gyrator, 153-155 Nonenergicness, of gyrator, 88-89 Nonenergic system, 84 Nonideal parallel resonant circuit, 158 Nonreciprocity, in physical systems, 82-83 Norator, 114 Nor-port, 114 Norton circuit, 95 NPN artificial transistor, 133 n-port gyrator as, 146-155 noise in, 153-154 n-port network, 112 Nullator, 114 paired, 121 Nullor approximation of with transistor, 129-130 current sources for, 139-140 electronic realization of, 128-129 four-nullor, four-resistor network and, 124 gyrator and, 113-1 16 improved design of, 131-134 noise source representations for, 143 in Riordan and Trimmel-Heinlein circuits, 142 three-nullor, two-resistor configurations and, 121 Null-port, 114 Nyquist diagram, of gyrator-capacitor circuit, 137

Photodiode arrays charge-injection sensors and, 199-202 MOS, 199 multiplexed, 198-202 XY-addressed, 199-202 Photodiode sensors, single-line, 198-199 Phototransistor arrays, 197-198 Physical systems nonreciprocity in, 82-83 reciprocity in, 80-86 PNP transistor, artificial, 132- 133 Power absorption coefficient, 5 Purdue University, 270 n.

Q Quantum theory, of free electrons in magnetic field, 6-1 1

R Radiation-enhanced diffusion, in ion implantation, 297-299 RCA Corp., 182, 233 RCA Laboratories, 232 Reciprocity misconceptions about, 87-88 in physical systems, 80-86 “ Reciprocity theorem,” 87-88 Refractive index, complex, 4 Resonance, in filters, 101-107 Resonant circuit, gyrator in, 157-167 Riordan circuit, 141-142 Room temperature detectors, 25-26

0

S

Optical thermocouple, 25-26 Optical transition and selection rules, 10-1 1

Sandia Laboratories, 270 n. Santa Fe Conference on Radiation Effects in Semiconductors, 269-270 Scanning by charge transfer, 202-208 in color TV camera, 25 1-253 solid-state, 202 Scattering matrix gyrator and, 96-99 nullor and, 114 two-port, 148

P Parallel resonant gyrator capacitor circuit, 175 Pauli spin matrices, 9 Photoconductive arrays, XY image sensors and, 194

345

SUBJECT INDEX

Scattering parameters, measurement of, 15&151 Schrodinger equation, for electron in periodic potential, 11-12 Self-scanned sensors, photoelements for, 183- 187 Semiconductors see also Ion implantation; Transistor diamond, 49-53, 56-65 dry fabrication of, 271 electronic states of, 2-3 interband magneto-optical studies of in far infrared, 1-75 ion implantation in, 263-328 magneto-optical transitions in, 2 new detectors for, 25-28 new devices utilizing ion implantation in, 325-328 optical properties of, 3-6 zinc blende, 28-29, 43-53, 56-65 Shot noise, in CCDs, 245 Silicon as detector for image sensors, 186-187 in ion implantation, 300-305 in solid state sensors, 258 Silicon gate, in charge-coupled devices, 215 Silicon photodiode array, 191 Single-line image sensors, 182, 188-190 Single-line photodiode sensors, 198-199 Small-signal operation, 112 Solids, microscopic optical properties of, 3 Solid state cameras see also Solid state sensors; Television cameras color, 251-253 image sensors for, 181-258 self-scanned sensors in, 251-253 video signal processing by signal recycling in, 249-25 1 Solid state sensors see also CCD(s); Charge-transfer sensors; Image sensors input circuit design in, 253-256 peripheral circuits for, 253-257 silicon in, 258 Spectrometer Fourier Transform, 19-23 grating, 18-19 Spectroscopy, wavelength-dependent, vs. magnetic-field dependent, 23

Spin-flip resonances, 42-55 Spin-flip transitions, in hole cyclotron resonance, 67

T Television cameras see also Solid-state cameras color, 251-253 image sensors for, 181-258 Television tubes, low light sensitivity type, 239-247 Tellegen medium, gyrator and, 111 Tellurium Brillouin zone in, 68 “camelback” band in, 69-73 in hole cyclotron resonance, 68-75 interference effects in, 74 YFT (thin film MOS transistor), 182 Thevenin circuit, 95 Transducers, electromechanical or mechanico-magnetic, 110-1 11 Transistor approximation of nullor by, 129-130 artificial NPN, 133 artificial PNP, 132-133 in nullor circuits, 139-140 super-a, 131- 132 as two-port, 112 Trimmel-Heinlein circuit, 141-142 TV cameras, see Television cameras; Solid state cameras Two-nullor realizations, 142 Two-port antireciprocal electrical, 86-87 gyrator as, 145 lossless, 97-99 nullor as, 114 synthesis of with admittance matrix, 118 time-invariant, 89 transistor as, 112 Two-port conductance matrix, 89 Two-port network, 117 Two-port scattering matrix, 148 Two-resistor network, compared with fourresistor, 128

346

SUBJECT INDEX

U

Unitor, as ideal amplifier, 113-114

V

Vacancies, in ion implantation, 291-295 Variables, interchange of, 92-93 Vertical frame transfer sensor, 206 Vidicon photoconductive camera tube, 184 equivalent circuit of, 184 silicon photodiode target for, 185 Voight geometry, in hole cyclotron resonance, 66 Voltage-current reciprocity, 81

W Westinghouse Corp., 182

X XY-addressed photodiode arrays, 182- 183, 189, 199-202 external multipliers and, 192 output circuits for, 256-257 signal-to-noise ratio in, 193 two-dimensional, 187-188 XY image sensors charge-transfer readout for, 207-208 early types of, 193-198 multiplexed scanning of, 190-193

z Zinc blende semiconductors conduction band of, 43-49 degenerate bands in, 56-65 valence bands of, 49-53 zone-centered electrons in, 28-29

A 5 8 6

c 7 D E F G H

8 9 a 1 2 1 3 J 4