NATL INST OF STAND & TfCH

A111Q7 117674

*z J

NBS SPECIAL PUBLICATION 614
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ol Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their eftective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the Naiional N4easurement Laboratory, the National Engineering Laboratory, and

the Institute lor Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities- — Radiation Research — Thermodynamics and

Molecular Science — Analytical Chemistry — Materials Science. THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering- — Mechanical

Engineering and Process Technology- — Building Technology — Fire Research —

Consumer Product Technology — Field Methods.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant txecutive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwi.se noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
Sixth International Conference

on Noise in Physical Systems

Proceedings of a conference held

at the National Bureau of Standards,

Gaithersburg, MD, April 6-10, 1981

Edited by:

P.H.E. Meijer

R.D. Mountain

R.J. Soulen, Jr.

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

Sponsored by.

National Bureau of Standards Washington, DC 20234

and

The Cathohc University of America

Washington, DC 20064

and

National Science Foundation

Washington. DC 20550

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued September 1981
Library of Congress Catalog Card Number: 81-600084

National Bureau of Standards Special Publication 614

Nat. Bur. Stand. (U.S.), Spec. Publ. 614, 419 pages (Sept. 1981) CODEN; XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON : 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

' Price $8.50

(Add 25 percent for other than U.S. mailing)

.
FOREWORD

The study of fluctuations (or noise) in a physical system provides insights, not available by any other technique, into the microscopic dynamic behavior of that system.

Besides being a source of information, noise can also be a source of irritation, in that it

limits the performance of numerous devices. The study of noise is of prime importance for

the testing of physical theories as well as for the development of improved physical

measurements and improved performance of devices. Therefore, the Conference has as one of

its goals an improved understanding of noise in devices and its influence on the error budget of a measurement. Indeed, progress in relieving or minimizing noise in some devices

was reported (e.g., the relationship of "burst noise" to the metallurgical condition of the

sampl e)

.

Strong emphasis was given in this Conference to new topics for which the noise spectra

proved to be particularly helpful in characterizing the underlying system dynamics.

Papers discussed, for example, the transition from periodic to chaotic behavior in chemical

systems and turbulent fluid flow, entropy generation in the computer process, the existence

and implications of quantum mechanical noise, and noise spectra occurring in electrochemical

processes.

Judging from the number of contributions and the intensity of the discussions following their presentations, the topic of 1/f noise remains as a very interesting one. It has

resisted most, if not all theoretical attempts to explain it. An invited paper by T. Musha

gave even more evidence to its ubiquity in nature. One of the most interesting developments

here has been the connection between 1/f noise and human comfort. Extending beyond the observation that noise exhibiting a 1/f spectrum is pleasing to the listener, clinical evidence now suggests that electronic alleviation of pain in humans is improved when the

electrical shocks are given a 1/f component.

We have to come to appreciate that studies of 1/f noise may benefit from techniques

found useful in several other activities at NBS. That is, controversies have often been

resolved in metrology by round robin exchanges among several laboratories of well-characterized samples. We suggest that a similar program (i.e., exchange of samples exhibiting

1/f noise) might help establish the level of accuracy of the magnitude of the observed noise,

the precise exponent, and other sample parameters (e.g., number of surface states).

The Conference was held at the National Bureau of Standards, Gaithersburg , MD,

from April 6-10, 1981. It was sponsored by the National Measurement Laboratory of NBS, The

Catholic University of America, and the National Science Foundation. The NBS Conference

was the sixth in a sequence of conferences previously held in Bad Nauheim, F.R. Germany

(1978), Noordwi jkerhout, Netherlands (1975), Gainesville, FL, USA (1973), Toulouse, France

(1971), and Nottingham, England (1968).

An International Advisory Board, the members of which are listed herein, assisted in the program selection and organization. We thank them for their generous contributions and

cooperation. We would also like to thank the NBS staff for their assistance in the smooth

operation of the Conference.

April 1981 P. H. E. Meijer

R. D. Mountain

R. J. Soulen

iii
Sixth International Conference on Noise in Physical Systems

International Advisory Board

Prof. A. Ambrozy

Department of Electronics Technology

Technical University

Budapest H-1521, Hungary

Prof. John Clarke

Department of Physics

University of California, Berkeley

Berkeley, CA 94720 USA

Prof. Brian K. Jones

Department of Physics

University of Lancaster

Lancaster LA 1 4XB, England

Prof. P. Mazzetti

Pol i teen i CO di Torino

Corse Duce degli Abruzzi

Torino 10125, Italy

Prof. T. Musha

Department of Applied Electronics

Tokyo Institute of Technology

Nagatsuta, Midorika

Yokohama 227 Japan

Prof. M. Savelli

Universite des Sciences et Techniques

Centre d'Etudes d' Electronique des Solides

34060 Montpeller Cedex France

Prof. F. Schogl

Institut fiir Theoretische Physik - RWTH Aachen

Erweiterungsgelande Seffent - Melaten

5100 Aachen, Germany

Prof. H. Thomas

Institut fiir Physik der Universitat Basel

Kl ingeberstrasse 82

4056 Basel Switzerland

Prof. K. M. van Vliet

Centre de Recherches Mathematiques

Universite de Montreal

CP 6128

Montreal H3 C 3J7 Canada

Prof. A. van der Ziel

Electrical Engineering Department

University of Florida

Gainesville, FL 32611 USA

Prof. R. J. J. Zijlstra

Fysisch Laboratorium Ri jksuni versiteit

Princetonplein 5 Ultrecht, The Netherlands

/

iv
ABSTRACT

This document contains the full text of papers submitted to the

Sixth International Conference on Noise in Physical Systems.

There are six categories of papers: THEORY; DEVICES; 1/f NOISE;

APPLICATIONS AND MEASUREMENT TECHNIQUES; QUANTUM NOISE; HOT

CARRIER NOISE.

Key Words: Hot carrier noise; noise; noise in devices; noise

measurement; quantum noise; 1/f noise.

Papers in this volume, except those by National Bureau of Standards authors, have not been edited or altered by the National Bureau of Standards. Opinions expressed in non-NBS papers are those of the authors, and not necessarily those of the National Bureau of Standards. Non-NBS authors are solely responsible for the content and quality of their submissions. The mention of trade names in the volume is in no sense an endorsement or recommendation by the National Bureau of

Standards.

V
TABLE OF CONTENTS

FOREWORD iii

INTERNATIONAL ADVISORY BOARD iv

ABSTRACT v

I. THEORY 1

Classification of Noise Phenomena by K. M. van Vliet 3

Fundamental Physical Limitations of the Computational Process by Rolf Landauer . . 12

Fractal Random Walks by Michael F. Shlesinger, Barry D. Hughes, and

Elliott W. Montroll 18

Transport Fluctuations Around Non-Equilibrium Steady States in Discrete Systems

by Eckart Frehland 22

Multiplicative Noise in Stochastic Differential Equations: A Numerical Study

by J. M. Sancho, M. San Miguel, S. Katz, and J. D. Gunton 26

Level-Crossing Intervals of a Gaussian Noise by Tadashi Mimaki , Hiroshi Sato,

and Masami Tanabe 30

Study of Disordered Structures as a Spatial Noise Problem by M. Celasco,

P. Mazzetti , and A. Stepanescu 34

On the Theory of Ambient Acoustic Noise in the Shallow Ocean

by Michael J. Buckingham 38

Mathematical Formulation of the Impedance Field Method. Application to the Noise

of the Channel of Field Effect Transistors by J. P. Nougier, J. C. Vaissiere, and

D. Gasquet 42

Time Domain Large Signal Noise Modelling in Microwave Oscillators by

J. F. Sautereau, J. Graffeuil, and J. C. Martin 47

Spectral Superposition and Irreversibility by Alle A. Walma 51

Classical Noise and Non-Linearity by Alle A. Walma 54

Fluctuations and Transition Rates in Time Dependent Dissipative Steady States

by David J. Bergman, Eshel Ben-Jacob, and Zeev Schuss 57

II. DEVICES 61

A New Approach to Model Channel "Thermal" Noise of the JG.FET Considered as an Active R-C Line by D. Sodini and D. Rigaud 63

Noise and Scaling Properties of Subthreshold MOSFETs by S. T. Liu and

J. S. T. Huang 67

The Dependence of the Low Frequency Noise of JFETs on Device Parameters and

Operating Conditions by C. E. Cox and K. Kandiah 71

The Characteristics of Noise Due to Individual Defects in JFETs by K. Kandiah,

M. 0. Deighton, and F. B. Whiting 75

Noise Due to Fast Surface States in MOSFETs by H. Mehta and K. M. van Vliet ... 79

vi
TABLE OF CONTENTS (Continued)

Low Frequency Noise in GaAs FET's at Low Drain Voltages by J. Graffeuil,

J. F. Sautereau, and K. Tantrarongroj 84

Thermal Noise in the Semiconductor Regime of Double-Injection Diodes by

A. van der Ziel 88

Photocurrent Noise Caused by Impact Ionization of Neutral Donors, and Free and

Bound Excitons in n-GaAs by K. Aoki , K. Miyamae, T. Kobayashi , and K. Yamamoto . . 93

Low Energy Electron Beam Investigation of Planar Transistors Using Scanning

Electron Microscopy with Particular Reference to Burst Noise

by Kenneth F. Knott 97

Burst Noise in Diodes by J. Sikula, M. Sikulova, P. Vasina, and B. Koktavy .... 100

Non-White Multiplication Noise in the Dark Current Regime of A P^nPflN* Avalanche Diode by R. Alabedra, A. A. Walma, C. Maille, and G. Lecoy 105

Theory and Experiment of Avalanche Noise in the Weak Multiplication Regime by

C. Maille, R. Alabedra, G. Lecoy, and M. Savelli 109

Noise Measurements on Photo Avalanche Diodes by J. Gong and K. M. van Vliet . . . 113

Noise in Heterojunction Transistors by G. Blasquez, F. Lavernhe, A. Marty, and

J. P. Bailbe 118

Digital Analysis of Superimposed Fluctuations on a Pulse Train by A. Touboul

,

A. A. Walma, and G. Lecoy 121

Noise in Dielectric Materials by J. Sikula, A. Cermakova, M. Cermak, and

P. Vasina 125

III. 1/f NOISE 131

Survey of Recent 1/f Noise Theories by M. B. Weissman 133

1/f Fluctuations in Biological Systems by Toshimitsu Musha 143

1/f Noise in Continuous Metal Films is Not Due to Temperature Fluctuations by

J. H. Scofield, D. H. Darling, and W. W. Webb 147

Temperature Response and Correlation of 1/f Noise in Transistors by J. Kilmer,

K. M. van Vliet, E. R. Chenette, and P. H. Handel 151

Lack of Spatial Cross-Correlation in 1/f Noise in Chrome Films by R. D. Black,

M. B. Weissman, and F. M. Fliegle 152

Temperature Fluctuations in a Steady State by Igor Pavlin and Stanley Engelsberg . 153

Earthquakes, Thunderstorms, and Other 1/f Noises by Stefan Machlup 157

1/f Diffusion Noise by K. L. Ngai and Fu-sui Liu 161

Temperature Dependence of the Excess 1/f Noise in Metals by K. L. Ngai 165

The Role of Mobility in 1/f Noise by D. A. Bell 169

A Model for 1/f Mobility Fluctuations in Elemental Semiconductors by R. P. Jindal

and A. Van der Ziel 173

vii
TABLE OF CONTENTS (Continued)

Common Mechanism of 1/f Noise and Burst Noise by A. M. Zaklikiewicz 178

1/f Voltage Noise Caused by Scattering of Electrons in Metals by

Leendert M. Bliek 181

Complements of 1/f Noise Theory by Bruno Pellegrini 185

Fluctuations of the Relaxation Time as a Source of 1/f Noise in Macroscopic

Physical Systems by J. Uebersfeld, P. H. Handel, and J. J. Gagnepain 189

1/f Noises and the Riemann-Liouvi 1 le Fractional Integral/Derivative of the

Brownian Motion by Claudio Maccone 192

Non-Gaussian Amplitude Distribution of Thermal Noise in Resistors with 1/f Noise

by P. H. Handel, D. Wolf, and K. M. van Vliet 196

l/f*^ Noise Behavior in Infrared Photoconducti ve Polycrystal 1 ine Thin Films of

Pb^Sn-|_^Te by A. D'Amico, G. Cappuccio, G. Petrocco, C. Fotinos, and G. Stochino . 199

Temperature Dependence of 1/f Noise in Thick-Film Resistors by Maria Prudenziati,

Bruno Morten, and Aldo Masoero 202

Excess Conductance Noise in Silicon Resistors by B. K. Jones 206

1/f Noise in Gate-Controlled Implanted Resistors by K. Amberiadis, A. van der Ziel,

and L. M. Rucker 210

Low Frequency Voltage Noise in Small Area Josephson Junctions by L. Krusin-Elbaum

and R. F. Voss 213

Concentration Fluctuations in Small Volumes of Ionic Solutions by

Rutgeris J. van den Berg, Arie de Vos, and Jacob de Goede 217

1/f Noise in MOS Transistors in Ohmic Region Under Strong Inversion by

Hong S. Min 221

1/f Noise Models in MOSFETs by H. S. Park and A. van der Ziel 226

1/f Noise in Short Channel MOSTs by Pierre Gentil and Ahmed Mounib 230

Theoretical Interpretation of Low 1/f Noise in JFETs by C. F. Hiatt,

A. van der Ziel, and R. J. J. Zijlstra 234

1/f Noise in GaAs MESFETs by C. H. Suh, A. van der Ziel, and R. P. Jindal .... 236

1/f Noise of Hot Carriers by T. G. M. Kleinpenning 240

Bias-Temperature Treatment, Surface State Density^ and 1/f Noise in MOSTs by

L. K. J. Vandamme and L. S. H. Dik 244

1/f Noise in Schottky Barrier Diodes by J. Sikula, P. Vasina, B. Koktavy, and

Z. Chobola 248

IV. APPLICATIONS AND MEASUREMENT TECHNIQUES 251

Josephson Junctions, Plasma Physics, and Electron Temperatures by

Bruce T. Ulrich 253

Survey, Applications, and Prospects of Johnson Noise Thermometry by

T. Vaughn Blalock and Robert L. Shepard 260

V i i i
TABLE OF CONTENTS (Continued)

Determination of the Thermodynamic Temperature of Fixed Points by Means of a Highly Accurate Noise Thermometer by G. Klempt and L. Storm 269

A Precision Noise Thermometer for Temperature Scale Studies by C. P. Pickup . . . 272

Electro-Acoustical Noise by R. J. J. Zijlstra and W. Westera 276

Experimental Studies of Noise in a Chemical Reaction and in a Fluid Flow by

Harry L. Swinney, J. C. Roux, and G. P. King 283

Dynamical Noise in Tunnel Diode Oscillator Systems by J. P. Gollub 288

Noise in Resonant Gravitational Wave Detectors by R. P. Giffard, P. F. Michelson,

and R. C. Taber 292

Measurement of the Brownian Noise of a Harmonic Oscillator with Mass M=389 Kg

by P. Bonifazi, F. Bordoni , G. V. Pallottino, and G. Pizzella 298

Two-Port Noise Characterization by Means of a Feedback Method by

Gabriel Blasquez 302

New Method for Wide Band Measurement of Noise Temperature of One-Port Networks

at High Pulsed Bias by D. Gasquet, J. C. Vaissiere, and J. P. Nougier 305

Application of the Theory of Random Matrices to a Reactor Noise Problem by

Felix C. Difilippo and J. J. Pineyro 309

Burst Noise Measurements o Op Amp's by R.F. SQUID by Arnaldo D'Amico 314

Domain Walls in Ferromagnetic Material and Irreversible Magnetization Processes

by Wilhelm Grosse-Nobis 317

Barkhausen Noise and Domain Structure Dynamics in 3% SiFe Single Crystals by

Giorgio Bertotti , Fausto Fiorillo, and Maria Paola Sassi 324

Noise in Electrochemical Systems by U. Bertocci 328

Ion Noise in ISFETs by Andre Haemmerl i, Jiri Janata, and James J. Brophy 332

Quasielastic Light Scattering from Macromolecules and Micelles by Mario Corti . . 335

Frank Elastic Constant Ratios Evaluation in MBBA by Intensity Measurement of the

Scattered Light by E. Miraldi, L. Trossi , P. Taverna Valabrega, and C. Oldano . . 339

The Determination of Elastic Constants of Nematic Liquid Crystals from Noise

Measurements of Scattered Laser Light by Jitze P. van der Meulen,

Rijke J. J. Zijlstra, and Jaap J. van Kooten 343

V. QUANTUM NOISE 347

Quantum Noise Limited Detection in Tunneling Devices at Millimeter Wavelengths by

John R. Tucker 349

Quantum Tunnelling and Noise in SQUIDs by A. J. Leggett 355

Quantum Noise in Josephson Junctions and SQUIDs by Roger H. Koch,

D. J. Van Harlingen, and John Clarke 359

Quantum Noise Effects at High Frequencies and Low Temperatures by

A. van der Ziel 364

ix
TABLE OF CONTENTS (Continued)

A Thermal Activation Model for the DC-SQUID by Claudia D. Tesche 369

Ultimate Sensitivity of an AC-SQUID by Juhani Kurkijarvi 373

The Mean Lifetimes of the Meta-Stable States of the DC-SQUID and its I-V Characteristics by Eshel Ben-Jacob and Zeev Schuss 376

External Field Effect on Particle Diffusion Above a Potential Barrier by

Eshel Ben-Jacob, David Bergman, Benny Carmeli, and Abraham Nitzan 381

Flux Flow Noise During Inhomogeneous Vortex-Motion. Dependence on DC-Velocity

Field by Heinrich M. Dirks, Klaus F. Beckstette, and C. Heiden 384

Flux Flow Noise During Inhomogeneous Vortex-Motion. Magnetic Field and Temperature Dependence by Klaus F. Beckstette, Heinrich M. Dirks, and C. Heiden 388

Vortex Noise in Two Dimensional Superconducting Films by C. M. Knoedler,

R. F. Voss, and P. M. Horn 391

VI. HOT CARRIER NOISE 395

Modeling Noise of Devices Working Under Hot Carrier Conditions by J. P. Nougier . 397

Noise of Hot Holes in Ge Due to Predominant Inelastic Scattering by V. Bareikis,

A. Galdikas, R. Miliusyte, and V. Viktoravicius 406

Hot Carrier Transport Noise in p-Type Silicon by G. Bosman, R. J. J. Zijlstra, and

A. van Rheenen 409

Monte Carlo Calculation of Hot-Electron Noise in Si at 77 K by Lino Reggiani,

Rossella Brunetti , and Carlo Jacoboni 414

X
I. THEORY

1

CLASSIFICATION OF NOISE PHENOMENA

K. M. van Vliet

Department of Electrical Engineering

University of Florida, Gainesville, FL 32611

and

Centre de Recherches Mathematiques

University of Montreal, Canada

I. INTRODUCTION

The purpose of this paper is to give some broad guidelines on how to classify noise » phenomena. In many respects this looks like an impossible task. First of all: Should we classify noise causes or noise manifestations? The more dogmatic oriented researcher

would opt for noise causes, while the morphologist would seek to classify the various pheno- mena as they occur. The latter easily leads to a chaos; what, e.g., to make of 1/f noise

(formerly often called current noise) observed in the Nyquist noise of a resistor by Clarke

and Voss? But a classification solely in terms of noise causes is equally ubiquitous; for

many phenomena we do not know the proper causes yet, and for others we may have the wrong

causes in mind—as will perhaps be borne out by future research. So, I believe a systematic classification of all observed noise phenomena is impossible. All we can hope to do

is to clarify certain principles, on account of which we can divide the phenomena in broad

groups. First, in Section II of this paper I will present four such principles. They are:

(1) The principle of thermal equilibrium. This allows noise to be characterized as thermal or nonthermal .

(2) The principle of microscopic versus mesoscopic fluctuations. The term "mesoscopic"

is due to van Kampen. It is used for quantities which are governed by the master equation

for coarse-grained variables, these quantities being themselves fluctuations of macroscopic

variables

.

(3) The principle of lumped versus distributed representation. While we borrow these

terms from the electrical engineer, the more mathematical-oriented researcher may want to speak of discrete and continuous random processes or of finite dimensional and infinite

dimensional (Markovian) random processes.

(4) The principle of classical versus quantum nature of the noise. Of course, this

writer adheres to the view that only quantum processes exist in nature. However, some allow for a classical description and others do not. However, in quantum statistical

mechanics there is both statistical and quantum mechanical uncertainty. If both occur, we speak of quantum noise.

The elaboration of these principles will be set forth in Section II. It is to be

understood that the various divisions are by no means mutually exclusive. Thus, on account

of these principles there are not 2^=16 classes of noise; rather, a given noise phenomenon

is to be classified in four different ways. If we call A the first part of each

principle and B its antithesis, then shot noise in tunnel diodes is (I)B-(2)A-(3)A-(4)B.

Secondly, I must introduce a super distinction which occurs over and above the principle just mentioned. It has only recently become clear to me that not all noise phenomena

can be reduced to manifestations of noise ^OuACU which describe the effects of c/ementOAt/

2VQ,YVtf>. In more specific terms, there are not always Langevin sources, there need not be a Campbell-Carson elementary event, and there may not be characteristic time constants associated with such events. In this respect I should like to introduce the name CHARACTERISTIC

3
NOISE PHENOMENA for all phenomena which are reducible to noise sources and which thereby

have characteristic time constants due to these sources, while I propose the name NONCHARACTERISTIC NOISE PHENOMENA for those fluctuation processes which are not reducible to noise sources in the sense of Langevin, Campbell-Carson^ or other elementary event pictures.

Examples of characteristic noise are g-r noise and flux-flow noise in super conductors;

examples of non-characteristic noise are wave interaction noise in photon statistics and

quantum 1/f noise according to Handel.

In Section III we will describe some of the tenets of characteristic noise. In particular we will urge that these noise phenomena are named according to the nature of the source of the noise and not in view of the response behavior of the system. In Section IV

we point out some aspects of non-characteristic noise.

II. VARIOUS PRINCIPLES OF CLASSIFICATION

The earliest noise phenomena discovered were thermal noise (Einstein 1906, de Haas

Lorentz 1913, Johnson 1925, Nyquist 1928) and shot noise (Schottky 1918, 1922>. In our opinion, these two noise processes are still the prototypes of all observed characteristic

noise phenomena in and out of thermal equilibrium. Thermal noise in resistors is due to

fehe thermal motion of the constituent electron and phonon gases; shot noise is due to the

corpuscular nature of transport.

A. Thermal Equilibrium versus the Steady State

In a closed system thermal equilibrium prevails if the extensive thermodynamic parameters do not depend on the time explicitly, i.e., other than through the p's and q's on

ivhich they depend. In an open system—subject to interaction with an energy or particle

reservoir—thermal equilibrium entails that there is no net transport of particles or energy across its boundaries; both definitions can be shown to be equivalent.

We define generalized thermal noise as fluctuations in any dissipative medium, i.e.,

any fluctuations associated with entropy production. For extensive variables in a lumped

system the variance is given by

<Aa.Aa.> = k f^^^A ^ (1)

1 1 V. , da . / \ 1 j/eq

The spectral density of <Aa.Aa. > is now given by the generalized Nyquist relation or fluctuation-dissipation theorem""^ (Cal len and Welton 1951, Callen and Greene 1952, Kubo 1967, van Vliet 1978) :

S; • =48(0), T) [l: (co)]^ + i[L;' (03)]^ (2)

1 J

where a. e J is the flow associated with the extensive thermodynamic variable a,

1 a L = L' + iL" is the generalized conductivity tensor, the superscripts s and a denote symmetric and antisymmetric combinations of L.., whereas " (a),T) is the energy of a harmonic

oscillator of frequency w,

^

E(w,T) = (f>a)/2) cosh (fiui/2kT) . (3)

One also writes ^(a),T) = kTp(co) ; then p(w) is the "quantum correction factor;" if a = Q,

(2) reduces to the ordinary Nyquist relation for electrical current fluctuation, J = dQ/dt.

Many examples of generalized thermal noise exist, the most important ones being

Brownian motion noise (Uhlenbeck and Ornstein 1930, M. C. Wang and Uhlenbeck 1945), generation-recombination noise (van Vliet 1958) , temperature noise (Milatz and van der Velden

1943), and blackbody radiation or thermal photon noise (R. C. Jones 1953). Two remarks on

the thermal nature of these phenomena are to be made.

First, these phenomena may have an "analytical continuation" outside thermal equilibrium proper. The nonequilibrium state may or may not affect the nature and magnitude of
the noise. For example, generation-recombination noise, which is due to the thermal fluctuations in the transition rates and the associated Nyquist noise of the quasi Fermi levels,

usually requires the passage of electrical current in order to detect the conductivity fluctuations due to g-r noise. When this current is small, we still have a quasi-equilibrium

state and the noise is thermal noise. (Since this current does not produce the noise, we once more emphasize that g-r noise is not a form of shot noise as thought in the early fifties.) The deviation from thermal equilibrium may also be large as in hot electron noise

or in photo-induced g-r noise. In this case the physical phenomena may still be similar as

in thermal equilibrium, but the noise should not be called generalized thermal noise (a

term we reserved for the noise as in eq (2)); the phenomenon should be called nonthermal g-r

noise or photo-induced g-r noise as we mentioned above. Noise in resistors outside thermal

equilibrium should be called hot carrier noise, nonthermal Johnson noise, or nonthermal

diffusion noise (see Section III)

.

Secondly, we note that shot noise methods may apply to thermal noise sources, but this

does not make the noise shot noise. In this regard we note that all thermal equilibria are maintained by opposing flows of particles or heat according to the principle of detailed

balance. Quite often the right thermal noise source is obtained by attributing full shot

noise to these opposing flows. Thus it is well-known that in a diode at thermal equilibrium

we have

S^(aj) = 4elQ = 4kTg . (4)

since the differential conductance is given by

p eV/kT ^

This argument was first given as a derivation of thermal noise by Weiszkopf (1943). Sometimes, however, this argument is phony and only by luck one obtains the right result.

For the sources of g-r noise one often writes the Langevin equation

^ = g(n) - r(n) + " 5^ • . (6)

Attributing full shot noise to the transition rates E, and E, , we have 5 g r

S + S = 2<g> + 2<r> = 4<g> . (7) g r

Fourier analysis of (6) now shows

<An2> = (S, + s ^ T = <g>T = <g>/r4^-?l (8) ^5 5' ^ LdndnJ n=<n> 3 g r

which is Burgess' g-r theorem (1954). However, a correct corpuscular treatment involving

phonon or photon emission or absorption easily shows that S ^ 2<g>, S ^ 2<r>, and

r r

S f

Q (van Vliet and Zijlstra 1977, van Vliet , Zijlstra and van Kampen 1978). Thus,

allfiough the detailed balance picture plus shot noise argument often gives the right thermal noise source, we emphasize that the result is often fortuitous.

Shot noise correctly is associated with a situation in which there is no detailed balance, but in which net transport occurs. Similarly, as in our distinction of thermal noise

proper (Nyquist) and generalized thermal noise (fluct, diss, theorem), we can distinguish

between shot noise proper (noise in electrical current flows) and generalized shot noise

(noise in energy flow, entropy flow, etc.). In all cases shot noise is proportional to the

average flow under consideration, <J> —if it is not, it is not shot noise. The noise also

possesses quantum attributes. Though the quantum noise correction factor was reported as early as 1955 by Epstein and Rostoker (Epstein and Rostoker 1955) , and also at one of the

Armour Research Foundation meetings organized by J. J. Brophy, no correct detailed publication of this quantum effect appeared until 1979 (Tucker 1979) . For tunnel junctions

5
Tucker finds

S^(a)) = e^coth [ (eVQ+hw) /2kT] KV^+fiw/e) + coth [ (eV^-fiu)) /2kT] I(VQ-fiu/e)

J. (9)

Here I(V) is the current-voltage characteristic. For VQ>>tloj/e and eVQ>>kT one finds 2el.

For Vq-K) one obtains thermal noise, 4g £ (co,T) with £ as in (3), noticing that the quantum

conductance is given by

I(V +hoj/e) - I(V -ftco/e) =

2tl./e • (10)

These equations apply to tunnel diodes and metal-oxide-metal diodes. For Schottky barrier,

pn diodes and vacuum diodes, they may need modification.

Generalized shot noise finds, as examples, particle flow noise, heat flow noise, and

photon noise (van Vliet 1976). Thus, if m is the number of emitted photons per second, and

if the line shape is Lorentzian,

r 2mn(T /2)

S (to) = 2S 1 + — — , (11)

L l-h^2(^^/2)2 J

where T is the coherence time and X] is the detection efficiency; the last factor is due to wave-interaction noise (Section V) . For noise due to an incoherent thermal source of radiative temperature T^, filtered with a monochromator of slit width Av:

S (oj) = 2m [1 + nB(v„) (1 - co/27tAv)1-(0 < to< 27tAv) , • (12) m 0 ^ 3 - - J

where „ \ (hv /kT ) B(Vq) = 1 / [e - 1] , (13)

is the Boson factor.

Stilly a word is in order about noise which is proportional to the square of the current, as is the case for 1/f noise and g-r noise. In the early days (Bernamont 1937) these

phenomena were called current noise . It is appropriate that this name has been abandoned.

It should not be used for phenomena in which the current is only a vehicle to detect conductivity fluctuations, as with g-r noise. For 1/f noise the situation is dubious. Granted, also here one can write S-|-/<I>2 = Sg/<g>2 ^ where Sg denotes conductivity fluctuations.

If the fluctuations are number fluctuations, then Sg/<g>'^ = S / <n>^ , and the current flow

n - ; 9 is nonessential. If, however, we have mobility fluctuations Sg/<g>^ = Sp/<y>^, then one may argue that the existence of a drift field is essential; yet the mobility itself is a thermal equilibrium quantity. In Handel's theory the noise is due to fluctuations in scattering cross section: S^/<\i>^ = S^/<a>^, and the presence of a current is essential for

the description.

The name excess noise is often used to denote all noise above proper thermal noise.

It serves no useful purpose.

B . Microscopic versus Mesoscopic Fluctuations

It is often said that noise involves the microscopic description of electronic or other

processes. However, this is usually not the case. The microscopic description in classical statistical mechanics refers to the motion in terms of particle positions and momenta;

in quantum statistical mechanics it involves the occupancies of many body or one particle

quantum states, e.g., Bloch states or momentum states in solids. Statistical mechanics has

a great deal to say about the microscopic fluctuations. The two particle correlation function <An(p^ q^t^) An(p2q2t2)> can be obtained from the BBGKY hierarchy; in quantum statistical mechanics <An^(t)An i)>, where (^) is a one particle state, can be obtained from the

Pauli-van Hove quantum faster equation (Charbonneau, van Vliet, Vasilopoulos 1981; van

Vliet 1978).
Usually, however, in noise problems we do not deal with the microscopic fluctuations.

On the contrary, the fluctuations involve coarse graining over a volume in phase space or over a range of quantum states. Thus, pressure fluctuations in a microphone involve the

momenta of a large number of particles in a certain subvolume of the phase space; in g-r

noise we are interested in the occupancy of a large number of states in the conduction or valence band; etc. Such fluctuations, being themselves still small with respect to the

corresponding macroscopic variables, are called mesoscopic. Markovian mesoscopic variables

are governed by a master equation for P(a,t alt^) in £-space (van Vliet and Fassett 1965).

Though this master equation is easily derived from the Smoluchowski or Chapman-Kolmogorof

f

equation, this constitutes no physical proof. In our opinion, this equation should be

derived from the Pauli-van Hove master equation; a satisfactory proof has not yet been

given in our opinion. However, fruitful applications to Brownian motion, g-r noise, laser

noise, nonlinear processes, etc., abound.

C . Lumped versus Distributed Noise

Irreversible thermodynamics these days is divided in the theory for discrete thermodynamic variables pertaining to homogeneous systems and the theory for continuous variables

involving, e.g., particle densities in inhomogeneous systems (De Groot and Mazur 1962).

Likewise, in noise we can distinguish processes involving a discrete set a^...a of

(Markovian) variables, and those involving densities in (k,j^) -space (microscopic continuous

processes) or in r-space (mesoscopic continuous processes) . Examples for these cases are

g-r noise in bulk solids, spontaneous emission noise in lasers and masers (discrete)^ and

density fluctuations in inhomogeneous solids or devices, such as semiconductor samples

with surface recombination or noise in bipolar transistors (continuous) . From a mathematical statistical point of view such processes are finite dimensional processes or infinitedimensional Markov processes. The theory for discrete processes is well-known (van Vliet

and Fassett 1965) ; they can often be represented by a lumped equivalent network (Champlin

1960, a) . The continuous processes form the domain of transport noise . They can often be

represented by a transmission line model (van der Ziel 1955,1957; Champlin 1960, b) . The

master equation level description for these processes is too cumbersome, since it involves

the functional P[a(r,t) |a(r_,tQ)] . One therefore uses a Langevin description, i.e., one supplements the phenomenological transport equation with Langevin sources. The general

form for a mesoscopic transport process is

+A}a(r,t) = C(r,t) (14)

where A is a spatial integral or differential operator. In case of diffusion, A = -DV^

,

eq (14) has to be supplemented with deterministic or stochastic boundary conditions. For a recent survey paper, see van Vliet and Mehta (1981). The noise solution of (14) requires

eigenfunction expansions or Green's function procedures.

The name transport noise denotes a category of noise processes, but individual processes have their own designation. See Section III for proposed nomenclature.

D. Classical versus Quantum Noise

The name quantum noise has come into use in recent years for a number of processes in

which the quantum mechanical nature is imperative. Thus we have quantum thermal noise,

eq (2); quantum shot noise, eq (9); quantum 1/f noise (Section IV). Further, the name

seems to be specifically used for noise in lasers, masers, Josephson junctions (see Wednesday p.m. section of the Conference). However, all g-r noise is also quantum noise. On the

contrary, for Brownian motion of a colloidal suspension, purely classical methods suffice;

so this is an example of classical noise.

III. CHARACTERISTIC NOISE

All noise processes which are governed by a master equation allow a Langevin description, which means that the phenomenological equations are amended by noise sources:

Ci(t) (discrete case), C-^(^,t) (mesoscopic distributed case). The problem now involves two parts. First, the noise source spectra must be known. Secondly, one must solve the

7
Langevin equation for the

the response will result

response of the system. If the noise source spec

in a sum of Lorentzians,

tra are white.

S^(lo) = E \Tj^/(l + (15)

k

For lumped processes the number of modes and the number of relaxation times is finite;

for distributed processes this number is infinite. For finite physical systems (however

large) this will always result in a constant spectrum for by>-0 and an oj *^(a>l) spectrum for

uy^. Thus 1/f noise for very low and very high frequencies cannot arise as characteristic

noise.

In naming the noise phenomenon, one has now the choice of characterizing it by the source function or by the response operator A. This leads to conflicting statements.

In nonhomogeneous solids the source is due to g-r fluctuations, while the response is

due to diffusion; in single injection diodes, the source is due to diffusion fluctuations,

while the response represents mainly drift; in bipolar transistors the source is due to g-r and diffusion fluctuations and the response represents recombination and diffusion.

We recommend that the process be characterized by its sources . Or, one can describe the

process by names like "diffusion-aided g-r noise," drift-aided diffusion noise, etc. In particular, we would like to reserve the name "diffusion noise" for processes which have

the source spectra

S = 4e2Dn(r)6(r-r') (16)

which in thermal equilibrium gives the thermal noise source 4kTa (^) 6 (r-r^' ) by Einstein's

relation. Hot electron noise generally is diffusion noise. The name diffusion noise

should not be used for transport noise in which A = -DV^ but the source is g-r.

We still mention that in the case of characteristic noise, often the noise can also be

treated with the Campbell-Carson method of elementary events. Clearly, these events are what make up the noise source. However, one incorporates the response in terms of the

Fourier transform of these events, rather than on the collective level. Discrete Markov

processes can be shown to be equivalent with the Campbell-Carson method applied to multistable random telegraph signals. For a two-level process the equivalence with a bistable

random telegraph signal is clear from the work of Machlup and Onsager (1954).

IV. NONCHARACTERISTIC NOISE

When no noise sounds can be indicated (and also no response part to the problem can be

separated off), we propose the classification "noncharacteristic noise." Such processes

lack a Master-, Langevin-, or Campbell-Carson description. Since there are no "elementary

events," the noise basically is a collective phenomenon. Examples are afforded by wave interaction noise and quantum 1/f noise (Handel 1975; 1980). Recently, 1/f noise has also

been described in terms of a wave self -interference phenomenon, similar to wave interaction

noise (van Vliet, Handel, and van der Ziel 1981). We will here briefly describe both

effects

.

The emission of photons is governed by compound Poisson statistics for emission of

M photons in time :

oo W(M,50 = / ^^^^ e W(U)dU (17)

0 Ml

t+^ where ^ ~

-^^ I(t')dt' is the fluctuating light intensity. The fluctuations of I(t) , which

affect the mean emission rate, are collective as is clear. (Formerly (van Vliet 1958b) we labeled such fluctuations as "modulation noise.") Finding the covariance <AM^ ^> and

using MacDonald's theorem, (17) yields for the spectrum of the rate m = M/^*:

S^(a)) = 2m [1 + I

S S^j(a))] (18)

8
where S is the normalized light-intensity noise. If S ^.(v) is the distribution of the spectral line width, one has by convolution

°° ^

(oj) = / S _^(v) S ^(f-v)dv , (19)

-°o E E

where OJ = 2TTf is the frequency of the beats, i.e., of the noise. For a Lorentzian line

width

S ,(v) = ~ , r<Sl/T J (20) E 4tt2(v-Vq)2 +

^

eq (11) is easily obtained; is the coherence time. This coherence time is, however, not

the characteristic time constant of an elementary event, as is clear since T is the line

width due to photon bunching (corpuscular picture) or due to collision and Doppler broadening of the light signal (wave picture) ; thus r represents a collective feature of the photons or of the light wave packet.

We now describe 1/f noise in terms of self-interference. Equations (18) and (19) are

still valid, but the analytic signal S ^.(v) must be replaced by the Schrodinger field S Hence, E

CO

^Al " ^ ^ +^^^ ^ ^(f-v)dv (21)

+ +

where i|j is the local creation operator of the quantized field; I = ij; ijj is the particle

density of the scattered field, which yields the cross section fluctuations. We assume that

\\j represents both elastic (or rather "subthreshold inelastic" in Handel's terminology) and

inelastic scattering (with the lattice. as occurring in mobility fluctuations) due to the

emission of infraquanta. Then (Handel 1980),

^ rvA -1

S _^(e) = C [6(e) + aA (e/Cq) e ] (22)

where otA is the infrared exponent, a is the fine structure constant if the infraquanta are photons, and C is a normalization constant; e = hv and £q is determined by the low frequency

detection limit allowed in the system (as indicated above, the elastic contribution "6 (e)

incorporates all e < e^) . From (21) and (22) one obtains for the spectrum of the beats

S^^«6(f)+ 2aA(f/fQ)°'^ f"^ Q^f-^O^ ^^^^

where 0 is the unit step function. Note, however, that f_ can be taken arbitrarily low,

there being no lower limit. Since 2aA is Hooge's constant <<1, (f/f^)^^^A. The current

noise, taking into account incoherence of N contributions is by (18) and (23) —noting

i = em,

S. = 2<i>2aA/N f , (24)

1 c which is Hooge's law if N = N j the number of carriers involved in the scattering process^

(Hooge 1969, 1972) .

The problem with this version of quantum 1/f noise is not with the mathematical derivation (a fuller account of (23) and the ensuing mobility fluctuations both for lattice and

impurity scattering will be presented elsewhere) . The main obstacle is the physical interpretation of aA. Handel has presented a number of processes which lead to emission of infraquanta. The values of 2aA seem, however, to be below Hooge's constant. However, it is

not clear whether the coherence factor N represents all carriers as in Hooge's formula.

c If N «N, the results could well be reconciled. The fact that there are no characteristic

c time constants in the theory of quantum 1/f noise, and consequently no lower limit appears

in the theory, is the attractive feature of this theory. Lack of space prevents us from

presenting other theories of this nature, like Ngai's theory of collective states

(Ngai 1980)

.
CONCLUSIONS

We have presented a number of principles to distinguish fluctuation phenomena. Various names and concepts have been introduced which will hopefully lead to a sharper classification of noise processes.

REFERENCES

J. Bernamont (1937), Ann. de. Pky^-tquz 71.

R. E. Burgess (1954), Phy^lca 20, 1007.

H. B. Callen and T. A. Welton (1951), Phy6 . Rzv. S^, 34

.

H. B. Callen and R. F. Greene (1952), . R2.V , 86, 702.

K. S. Champlin (1960a), Phy^-ica 26^, 751.

K. S. Champlin (1960b), IRE TfULn^ . E-V ]_, 29.

M. Charbonneau, K. M. van Vliet and P. Vasilopoulos , "LcneoA ReJ^pon^Q. JhdOfiij Rdv-U^zd 111,"

submitted to J. Math. Phy6.

S. R. deGroot and P. Mazur (1962), hlon-Eqalttbfujm ThM.modym.rfU.CJ> (North Holland, Amsterdam).

G. H. de Haas-Lorentz (1913), Viz BfioMmchd BoMizgang and exnx^ge viZAWandte. Euchzinungdn

(Vieweg Braunschweig)

.

A. Einstein (1906), Ann. doA Phy^lk 19_, 371.

H. Ekstein and N. Rostoker (1955), Phy6. R&v. 100^, 1023.

P. H. Handel (1975), Phyi . Rzv. LeJXeAi 34, 1492.

P. H. Handel (1980), Phy6. Reu. A22, 745.

F. N. Hooge (1969), Phy6. LoXtZAJ, 29A, 139.

F. N. Hooge (1972), Phyi,lca 60^, 130.

J. B. Johnson (1925), Phyi, . Rev. 26, 70.

R. C. Jones (1953), in Advances In EltcXKOwicA , Vol. V, Ed. L. Martin (Acad. Press, N. Y.)

.

R. Kubo (1957), J. Phyi> . Soc. Japan. VI, 570.

S. Machlup (1954), J. Kppl. Phyilu 25, 341.

J. M. W. Milatz and H. A. van der Velden (1943), Phy^locL 10^, 369.

K. L. Ngai (1980), in Pnoc. 2nd IntdHn. Symp. T / UoiAZ, Ed. K. M. van Vliet, p. 445.

H. Nyquist (1928), Phyi> . Reu. 32^, 110.

W. Schottky (1918), Ann. de/i Phyilk 57^, 541.

W. Schottky (1922), Ann. deA Pfiy^Xfe 68, 157.

J. R. Tucker (1979), IEEE J. Quantum EI&cXa. 15, 1234.

G. E. Uhlenbeck and L. S. Ornstein (1930), P/lt/-i . Reu. 36, 823.

A. van der Ziel (1955), Paoc. IRE 43, 1639.

A. van der Ziel (1957), PAOC. IRE 45, 1011.

K. M. van Vliet (1958a), Phyi . Reu. UO, 50.

K. M. van Vliet (1958b), Pfioc. IRE 46, 1004.

K. M. van Vliet and J. R. Fassett (1965), in TZucXuatlon Ph&noimna in Sotidi, , Ed. R. E. Burgess (Acad. Press, New York), p. 267.

K. M. van Vliet (1976), Phy^lca 8^ B & C, 52.

K. M. van Vliet and R. J. J. Zijlstra (1977), Physlca 89k, 353.

K. M. van Vliet (1978), J. Math. Pkyi> . J_9, 1345.

K. M. van Vliet, R. J. J. Zijlstra, and N. G. van Kampen (1978), in Wo-c4e In Phyiilcol

Syitmi, Ed. D. Wolf (Springer), p. 309.

K. M. van Vliet and H. Mehta (1981), "ThzoAy 0^ TAanApont Mo^e," Phy^lca Status Sotidt (b) , in press.

K. M. van Vliet, P. H. Handel and A. van der Ziel (1981), "SupeA Statistical EmlH,lon

Wo-c6e," submitted to Phy^iica.

M. C. Wang and G. E. Uhlenbeck (1945), Reu. Mod. Phyi . 17_, 323.

V. Weiskopf (1943), Mat. Pei^enie ReieoAch CoancJl. Rupont.

10
POSTSCRIPT

1. Altogether, it seems to us that there are four broad classes of noise phenomena,

viz . :

(i) noise due to thermal statistical uncertainty, as described by equilibrium statistical mechanics, and labeled here as generalized thermal noise;

(ii) similar processes outside thermal equilibrium, as described by nonequilibrium

statistical mechanics, labeled here as "analytical continuations" of generalized thermal

noise

;

(iii) noise due to the corpuscular nature of transport, labeled generalized shot noise,

being either of a classical nature (tubes) or of a quantum mechanical nature (photon noise,

shot noise in tunnel diodes)

;

(iv) noise due to quantum mechanical uncertainty, labeled "quantum mechanical noise"

(this noise does not contain kT in contrast to quantum statistical noise) ; quantum 1/f

noise (Handel-Ngai) is an example.

2. All noise phenomena can further be classified according to the four principles of

Section II and in addition as characteristic or noncharacteristic noise. In response to some papers of the Conference, to the distinction "microscopic-mesoscopic , " we would like

to add "macroscopic." The coupled oscillator noise reported by Gollup et al. is a macroscopic noise manifestation; in this category also comes noise due to turbulence.

11
FUNDAMENTAL PHYSICAL LIMITATIONS

OF THE COMPUTATIONAL PROCESS

Rolf Landauer

IBM Thomas J. Watson Research Center

P. O. Box 218, Yorktown Heights, New York 10598

INTRODUCTION

Thermodynamics arose out of attempts to understand the limits on the efficiency of steam engines.

Information theory arose out of the effort to understand channel capacity limitations. These models have

motivated an attempt to do the same thing for the computer. A first conference on the subject follows this noise conference by a few weeks, and the proceedings of the later conference will be published in the International Journal of Theoretical Physics [1]. It is easier to ask questions about computer limitations than

to answer them, and we can claim only very modest progress.

There has been enough progress, however, so that within the severe space limitation imposed by this conference we can only allude, in a very general way, to what has been done, and provide a road map of sorts. At the outset, let me emphasize the distinction between this field, and more well known areas.

Information theory deals with the behavior of linear channels where we hopefully take out information

just as we put it in. A computer is clearly different, the logic process creates a nonlinear interaction between

two or more information streams. The one part of a computer that is very much like a communications

channel is the memory, where one hopes to find things unchanged.

The attempt to do scaling calculations, in which we ask how a computer changes as we reduce the components in size, including both active components and transmission Hnes, is another well established field. Our concern differs; we are attempting to describe the computer in phase space, without invoking a particular

set of devices. A recent review paper by Keyes [2] discusses both fundamental limitations and scahng theories

for the computer.

In contrast to our historical models, information theory and thermodynamics, the fundamental limitations of the computer, to the extent they are understood today, are far too many powers of ten away from the technological limits, to be a guide for further technological progress. Then why is the subject important or

interesting? We are attempting to provide a physical basis for mathematics. The mathematician has given us

the impression that it makes sense to talk about an arbitrarily long sequence of information handling steps, with each step guaranteed to be unquestionably correct. Now we come back and ask whether nature really allows that.

The kind of questions we are tempted to ask:

1. How many degrees of freedom are there available in the universe, to handle information? How many of these can really be brought together, to interact effectively, in a computer?

2. What is the minimum energy consumption required by the computational process?

3. How immune to noise can the computational process be made?

4. The preceding question refers to degradation of information, by noise, in a fixed physical structure. We can

also ask how much can be done to offset the inevitable physical degradation arising from diffusion, corrosion,

electromigration, etc. Does this structural deterioration establish a limit to the number of computational steps?

12
5. How many photons, or more generally signal quanta, are required to bring a bit into a logic stage?

Of these questions, only 2 and 3 have been answered, to some reasonable extent. It is often hard to understand the difference between fundamental limits, and those that are less fundamental and circumventable.

Take the question about noise immunity as an example. We normally think of thermal equihbrium noise, i.e. Nyquist noise, as an inevitable minimum. It is determined by the temperature. Is there a minimum achievable

temperature? Can additional sources of noise, e.g. 1/f noise, be reduced to an arbitrary extent? (The answer,

for 1/f noise, is likely to be affirmative since, as we will show, computing can be carried out arbitrarily close

to thermal equilibrium.) In recent years we have become very conscious of the fact that secondary electrons,

produced by particle radiation, can act as a source of noise [3], and destroy information stored in the form of

electrical charge. Does this radiation have an unavoidable minimum level? Even if it does, information does

not need to be handled in the form of charges. We can use superconducting currents, magnetization, hydraulic

flow, or many other degrees of freedom. But while some of these may interact less strongly with particle

radiation, they undoubtedly all still interact with it to some extent. THE UNIVERSE HAS A FINITE MEMORY CAPACITY

One of the earliest and most perceptive treatments of memory limitations is due to Swanson [4] and is summarized in a later review paper [5]. Swanson asks how finely should we subdivide a given amount of

memory material, to obtain the maximum storage capacity, given a period T over which we want to preserve

information. If we make our elements too small, they are too susceptible to noise, and lose their information

too quickly. If we make the elements very large they become immune to noise, but we don't have much

capacity, either. In between these extremes there must be a maximum, and Swanson utilized information

theory, i.e. the possibility of redundancy to protect information, to find the optimum element size. Swanson's

considerations are not completely general and model independent, but have a broad degree of applicability.

The result: The individual elements must be large enough to be relatively reliable. We lose if we depend too much on redundancy. Swanson's considerations do not allow, explicitly, for readout, after time intervals short compared to T, and then utilizing the information protection provided by redundancy, to restore all the bits to their initial state. To do this sensibly, we must understand the probability of error in the readout and

restoration process, and thus we become even more model dependent.

Information is sent into storage, and retrieved from it, via a channel. Therefore one can invoke channel

capacity theory to describe storage limitations, without regard to the physical storage mechanism. If we

invoke classical channel capacity theory [6], a minimal energy of kT/«2 is required per transmitted bit. I do

not know what value of T characterizes the universe in some average way, but 3°K is (certainly within a few powers of ten) a reasonable guess. We can then divide the energy of the universe by kT/«2 to get an

upper limit on the amount of information that can be sent into a memory [6]. Alternatively, we can invoke

quantum channel capacity theory [7].

While the aforementioned channel capacity considerations yield a quick and easy way to point to the existence of physical limitations, unfortunately, we believe they are wrong. One could apply the above

reasoning equally to all interconnections between successive logic stages in a computer, and thus determine

channel capacity energy requirements for each of these channels. As we shall see in the next section, however, we have hypothetical computers whose energy requirements are far lower. The answer to this apparent

paradox: The energies required by channel capacity considerations represent energy dissipation only if the message is destroyed at the receiving end, and a computer need not do that. Consider, for example, a high

density reel of storage tape sent physically through space, at high velocity. Motion of the reel can be

controlled through a few degrees of freedom, related to the tape's position and orientation, and unrelated to the huge number of information bits. The required dissipation is at most a few kT, or perhaps, even zero,

since the kinetic energy of tape motion need not be dissipated. Shipment of the tape does, however, constitute transmission of information. Alternatively we could receive a message in electromagnetic form and store it between reflecting mirrors, for eventual later controlled release. Similar confusion underlies many discussions of the measurement process [8]. Szilard's pioneering analysis [9] of the measurement process taught us that measurement requires energy expenditure. But it does not seem to be adequately recognized that coupUng a meter to an object to be measured, and letting the object thus influence the meter does not require dissipation.

The dissipation arises from the need to reset the meter after subsequent decouphng, in order to prepare the meter for further use. Again it is the destruction of information which is associated with the real need for

13
energy dissipation. This relationship to information destruction has been emphasized only in our own field, and not equally in the related analyses of the measurement process and of channel capacity requirements.

Finally we make a different point, related to the fact that infomation can be stored in bistable dissipative

systems, e.g. active circuits, not only in dissipationless systems, such as ferromagnets. Since 1962 there has

been analysis [10] to see how small such structures can be, and still preserve information in the presence of

fluctuations. We do not discuss this subject, here, in further detail, for two reasons. First of all, no very fundamental model independent conclusions have been reached. Furthermore, analyses of that sort have

become very fashionable. This author has, in fact, taken the liberty in one recent publication [11] of poking

fun at the proliferation, and at some of the new labels, e.g. multiplicative noise that have been produced.

Another paper [12] points to the unnecessary complexity of some recent discussions of the noise activated escape from a metastable state.

REVERSIBLE COMPUTATION

The fact that an elementary logic event is, in some vague sense, associated with a minimal energy of the order of kT, was a widespread supposition in the 1950's, but not generally articulated in print [13]. Subsequently it became clear [14] that only logical processes which throw away information require energy

dissipation. This viewpoint has, since then, been elaborated on a number of occasions, and we cite only one

of the more recent discussions [15]. It was, initially, unclear whether loss of information was essential to the computational process, but the obvious supposition was that intermediate scrap results had to be discarded.

Ref. 14 pointed out that individual logic operations could be made logically reversible by the addition of unnecessary outputs to each logic stage. Consider a particular three-input, three-output device. Let p, q, and

r be the input variables. The truth function under consideration is one which replaces r by p«q if r = 0, and

replaces r by p»q if r = 1. The variables p and q are fed through the device unchanged. All logic can

be performed by iterations of this logically reversible operation [16]. To perform a simple negation p, for example, we would feed fixed bias variables q = 1 and r = 1 into the device, r would be replaced by the desired output, p. To duplicate a variable, i.e. generate fan-out, we feed in the variable p, and fixed inputs q = 1, r = 0, thus obtaining the variable p in two of the output positions. The unnecessary outputs would

then be saved, for example, by feeding into a shift register; otherwise their destruction would require energy

loss. It was originally presumed that this was a useless complication: We avoided the need to discard

information by the addition of many shift registers, whose content would eventually have to be erased anyway.

Bennett, in 1973 [17], finally pointed out that this erasure was not necessary. If the logical reversibility is properly utilized in a physical device, then erasure is not needed; we clean out the shift registers by running

the machine backwards. Properly utilized in the preceding sentence means that the device can be pushed in the reverse direction to perform the inverse of the original logical operation. We do not require the device to be strictly physically reversible, i.e. lossless, but only as reversible as automobiles or locomotives, which can back up along their path. We also assume that the frictional forces are proportional to velocity and, thus, the accompanying energy losses, per logic step, can be made as small as desired, by sufficiently slow computation.

The reader is referred to other discussions [8,15,17] for further details. A logic device which provides one-to-one mapping, and is logically reversible, does not necessarily have

the physical reversibility specified above. A number of Gedanken model computers have been devised to show

that physical reversibility can actually be achieved. Note that the reversibihty, at a local level, insures total reversibihty: every initial state is mapped into a unique final state, if the computation comes to a termination.

That does not mean such computers are simply table look-up devices where the designer has to anticipate all possible computations. The design of these computers can be simpler and more straightforward than that. We

list some of the proposed models, without attempting a real description, and also briefly comment on their

relative advantages and disadvantages.

1. Bennett's springless clockwork [8] Turing machine. In many ways the most satisfactory and most

developed model. Does not assume perfect parts, or the absence of backlash. Does assume hard components

whose shape and size is not subject to fluctuation.

2. Bennett's enzyme controlled DNA type Turing machine [8]. As a result of the molecular scale of the parts

thermally induced errors in the information content, as well as in the basic structure of the machinery, will occur.

14
3. Toffoli's meshed gear computer [18]. A rigid linkage in which the information has to propagate from one

stage to the next, instantaneously. This is not without further elaboration, a Turing machine capable of computations of arbitrary length. This is also a proposal in which, at least without further modification, errors accumulate through successive stages. To avoid that, logic devices must continually restandardize signals, i.e. push the signal level back toward favored states [19].

4. Fredkin's ballistic colliding billiard ball computer [20]. Information is represented by the presence or absence of particles which have a hard-core repulsion, no internal degrees of freedom, and are guided by

reflecting mirrors. Logic is accomplished through the deflection produced by particle collisions. This scheme,

in unmodified form, requires perfect mirror placement and exact initial velocities. Consider, instead, a modified version in which the particles move along guiding tracks and have a short range repulsion. The

tracks are viscous, and also expose the particles to thermal agitation. The tracks have branches where there is a fork between a straightforward continuation of the original track and a branching track, off at some angle.

(There are also compensating inverse branches, bringing tracks together, just as in the unmodified scheme

[20].) We also invoke (as Fredkin does, in some versions of this model) moving potential wells, i.e. time

dependent fields, which pace the particles in their motion along the tracks. Let two adjoining tracks have

forks at adjoining positions, along their lengths. Then if a particle arrives along only one of the tracks it will continue undisturbed. If two particles arrive simultaneously, they will repel and move into their respective

branching tracks. The strength of the repulsive force between particles will determine the probability that

particles will take the desired branching tracks. For any given repulsive force there will be a nonvanishing

residual error probability determined by thermal agitation. To insure that particles, in the absence of the

interacting repulsive force, stay on the straightforward continuation, we need a static bias field near the bifurcation point, which, in turn, will be overcome by the repulsive interaction, when that is present.

Additionally, of course, this system requires some elaboration to describe how the information bearing

particles can move in and out of storage in a memory, or Turing machine tape.

5. I have described a Turing machine [15,21] whose active logic elements were proposed by Fredkin and

whose overall organization follows Bennett's ideas [17], and have called this the Bennett-Fredkin-Turing

(BFT) machine. It resembles Bennett's springless clockwork Turing machine in its characteristics. But it has

the disadvantage that it invokes springs with a function closely related to that of the bias fields discussed

above, in connection with the interacting charged particle scheme. Thus the BFT machine has an error probabihty, per step, which can be made arbitrarily small by a suitable choice of parameters. But for any

given design there will be a maximal length of relatively error free computation.

NOISE IMMUNITY

Computers of the type described above consist of a series of one-to-one logical mappings. Thus the computation is a progression through a sequence of states, with only one antecedent state, and one consequent

state, for each intermediate state of the computation. If the computation is driven forward, with a very small

force, then fluctuations along that path, whether they are thermal equilibrium noise or some other form of

noise, do not induce errors. In the presence of noise and a small driving force, a computation will proceed

diffusively, going backwards almost as often as forwards. That, however, is irrelevant as long as it proceeds

forward on the average. When the computation reaches its final state a modest one time energy loss is required to trap it there, and to prevent diffusion back toward earlier states.

If we want a machine which, at each step of the computation, proceeds forward rather than backward,

then we must drive the mechanism sufficiently strongly to dissipate a few kT per step. Note, however, that one of these steps can involve a great many logic operations in parallel, and can be far less than a dissipation of kT per elementary logic operation. This dissipation, of only a few kT per machine cycle, requires that all of

the machine's logic operations are coupled together rigidly, and that we are not dealing with many relatively

independent degrees of freedom.

The discussion in the preceding paragraph assumes that the computation is constrained to follow the desired one dimensional path, and that noise can only affect motion along that path. Noise, in principle, can do more than that. First of all, fluctuations can have an effect on the actual physical computer structure, not

just on its information content. (Point 4, Introduction). If the structure is massive enough, this can be made

small, but once again, any given design will leave a nonvanishing probability, per unit time, for computer

failure. Additionally, we must ask whether it is reasonable to invoke an unlimited structure (as in a Turing

15
machine) of arbitrarily massive elements. We must also ask whether there are fluctuations which do not ruin the structure, but take us from the track corresponding to one program, to that of another, e.g. by changing one bit in the program. If the information bearing degrees of freedom are massive, e.g. if information is denoted by massive balls, or their absence, as in the BFT machine, this will be very improbable, but can

happen. The probability for fluctuations which mess up the machinery, e.g. a change of the ball into a shape

which cannot move along the tracks intended for it, will be very much larger. If, however, the information

bearing degree of freedom is very easily altered by fluctuations, as in Bennett's DNA model of a Turing

machine, then the unintentionally induced transitions from one track to another become a genuine possibility.

The key point: It is easy to find models where, if the probability of permanent machine failure is negligible, then the probability of track jumping is even more negligible, and thus the results of the computation becomes immune to fluctuations along the allowed track. This is independent of the computational

velocity; slow computation is not necessarily more reliable. Indeed a slow computation affords a greater

opportunity for fluctuations which deteriorate the physical machinery. The availability of almost perfect noise

immunity, for computations which are not extremely slow, may be somewhat counterintuitive. It is certainly in contradiction to some of the more easily analyzed models, including one discussed by this author [22]. On the other hand, the availability of computing which can be as free of thermally induced errors, as desired, at slow

enough computing rates, was known before the existence of reversible computing was clearly understood [23].

QUANTUM EFFECTS

Several authors have casually invoked the uncertainty principle, AEAt~^, to yield an energy dissipation

for fast switching. At is typically equated with the duration of a switching event. While we cannot guarantee

that this is an incorrect conclusion, it is certainly an unsubstantiated conclusion. The uncertainty principle

refers to a spread in energy measurements, not to an energy dissipation. Futhermore, a fast event does not

require an energy spread; an electron can pass an atomic nucleus in a crystal, or an interface between two

crystals, very rapidly, and still be in an eigenstate. Only if we try to measure the time when the fast event

occurred, do we meet the energy spread. Other authors have invoked quantum channel capacity considerations, instead of appeaUng directly to the uncertainty principle. But if a classical computer does not have to expend of the order of kT per bit, between successive logic stages, why should we expect channel capacity

considerations to apply to a quantum computer? The thoughts on quantum effects, presented here, are discussed in more detail elsewhere [21].

RELATION TO REAL COMPUTERS

Real systems require an energy dissipation, per elementary logic step, many powers of ten above kT. Why

are they so far from our limits? There are a number of reasons for this, probably not all of them understood.

A key contribution, however, comes from the fact that our simplified models treat the machine's internal coupling mechanisms as structures whose own degrees of freedom do not have to be taken into account. Genuinely practical machines, however, are not periodic structures in which interaction is limited to neighboring elements. Real machines require long transmission lines [15], and lines which differ greatly from each

other in their length. These lines have to be energized to voltages which are tied to the scale of the nonlinearities available in transistors [2], and cannot be made smaller than that, by going to very slow computation.

We have not yet seen a way to save and reutilize that transmission line energy. REFERENCES

[1] Proceedings of the Conference on Physics of Computation, MIT, Cambridge, Massachusetts (May

6-8, 1981), to be pubHshed in Int. J. Theor. Phys.

[2] R. W. Keyes, Proc. IEEE 69, 267 (1981).

[3] J. F. Ziegler, and W. A. Lanford, Science 206, 776 (1979).

[4] J. A. Swanson, IBM J. Res. Dev. 4, 305 (1960).

[5] R. Landauer, and J. W. F. Woo, in Synergetics, Ed. H. Haken (Teubner, Stuttgart, 1973).

16
[6] H. Marko, Kybernetik 2, 21 A (1965).

[7] H. J. Bremermann, in The Encyclopaedia of Ignorance, Eds. R. Duncan, and M. Weston-Smith

(Pergamon, New York, 1977) p. 167; J. D. Bekenstein, Phys. Rev. Lett. 46, 623 (1981); See also Ref. 6.

[8] C. H. Bennett, in Ref. 1.

[9] L. Szilard, Z. 53,840 (1929).

[10] R. Landauer, J. A ppl. Phys. 33, 2209 (1962).

[11] R. Landauer, in Self-Organizing Systems: The Emergence of Order, Eds. F. E. Yates, D. O. Walter, G. B. Yates (Plenum, New York, 1981).

[12] M. Biittiker, and R. Landauer, in Nonlinear Phenomena at Phase Transitions and Instabilities, Ed. T. Riste (Plenum, New York, to be published) see Appendix.

[13] J. von Neumann, Theory of Self-Reproducing Automata (Univ. of Illinois Press, Urbana, 1966).

[14] R. Landauer, /5My. Z)ev. 5, 183 (1961).

[15] R. Landauer, Ber. Bunsenges 80, 1048 (1976).

[16] T. ToffoH, Techn. Memo MIT/LCS/TM-151, Lab. for Comp. Sci., MIT (1980); T. ToffoH, in Automata, Languages, and Programming, Eds. J. W. deBakker and J. van Leeuwen (Springer, Berlin,

1980) p. 632.

[17] C. H. Bennett, Z)ev. 17, 525 (1973).

[18] T. Toffoli, Math. Syst. Theory 14, 13 (1981).

[19] A. W. Lo, in Microwave Electronics, Ed. E. Koenjian (Macmillan, New York, 1961) p. 19

[20] E. Fredkin in Ref. 1.

[21] R. Landauer in Ref. 1.

[22] R. Landauer, and J. W. F. Woo, J.Appl. Phys. 42, 2301 (1971).

[23] R. W. Keyes, and R. Landauer, IBM J. Res. Dev. 14, 152 (1970).

17
FRACTAL RANDOM WALKS

Michael F. Shlesinger, Barry D. Hughes and Elliott W. Montroll

Institute for Physical Science and Technology

University of Maryland

College Park, MD 20742

INTRODUCTION

Mandelbrot [1] has discussed properties of the Weierstrass function

00 W(a),Y,t) = Z oj exp(27riY'^t) , (1)

n=0

where 0<a)<l,Y>0 and -oo < t < <». Although the series is uniformly convergent for

-00 < t < 0° and therefore represents a continuous function of t, it is nowhere dif f erentiable

i^^wY > 1. For the frequency f = Y ( n = Inf/lny ) there is a spectral lin^ wjth energy

(jj Th^ tjtal energy in the^set of frequencies exceeding f = y is oj (l-w ) f (1-0) ) , where H = Zn(a) )/ZnY and 0 < H < 1 in the non-differentiable regime. This

form of the cumulative noise spectrum is similar to that of many physical noise processes.

THE WEIERSTRASS RANDOM WALK (WRW)

To gain further insight into the nature of the Weierstrass function, we use it to generate a random walk on a perfect one-dimensional lattice [2]. Our walk starts at the

origin and makes jumps of I sites according to the probability distribution

00 pa) = n + 6 ,n], (2)

n=0

where a > 1, b > 1 and b is integral. The walker makes about "a" jumps of unit length in a region, forming a cluster of points visited, before jumping a distance "b" to begin a new

cluster. After about a jumps occur, a jump of distance b occurs, etc. While this argument

is not precise, it shows that we have built a self-similar clustering property into

so that for a walk of a modest number of steps, a clustering of the sites visited may be

expected. If 0 < Ina/lnh < 1, then the walk is transient and distinguishable clusters

remain as the number of steps taken tends to infinity, giving non-Gaussian behaviour.

We first note that

00 ^(k) 5 Z exp(ik5,)p(^) = — Z a"\os(b\) (3)

£ ^ n=0

is a multiple of the Weierstrass function, with oo = a""*" and y t>. It is also related to the lacunary Taylor series [3]

°° n b'^ S(z) = E c z (c > 0, b an integer > 1). (4)

n=0

We should expect interesting behaviour from eq (3) since S(z) cannot be analytically

18
continued beyond its circle of convergence |z| =1 because it has a dense set of singular

points on |z| = 1 at z = exp(27rih/b ), for all integers h and m.

Except for the initial position, p(k) contains all the information about the random

walk, since

p a) = I P„(a')p(£-£') (5) n+i ^, n

implies that

P^(k) = [pCk)]"" P^Ck), (6)

where 'P^i^) is the probability of reaching site I at the n-th step. If the mean squared

displacement per step <£^> = E£2p(£) is finite, then

p(k) = 1 - j<l^>k^ + O(k^) (7)

and (for large n) P (I) will be a Gaussian centered around £ = 0, with variance n<£2>, i.e. no clustering occurs.

For the WRW

00 <£2> = ^ z (b^/a)'' = 00 if b^ > a (8) ^ n=0 , "

and eq (7)no~16Qger holds. The small k behaviour may be found using a Mellin transform

technique or Poisson's summation formula [2] to be

^(k) = 1 - |k|"Q(|k|) + 0(k^) (9)

if 0 < H = Ina/lnh < 2, where Q(|k|) is a periodic function of Zn|k| with period Inh. We

note that H corresponds to the fractal dimension of the self-similar clusters ("a" subclusters per cluster, scaled down by a factor "b") . The smaller H is, the more separated

will be the clusters,

A similar expression for p(k) without self-similar clustering may be obtained by

combining two incommensurate WRW's, i.e.

00 P(k) = Z [^^""cos(b"k) + ^^""cos(e\)], (10)

n=0

with H = Ina/lnh = Ina/lnB, but a and a and b and g incommensurate. The fractal dimension

H does not contain sufficient information to characterize the clusters completely. We

conjecture that for a higher dimensional generalization of eq (10) an infinite percolating

cluster may exist.

If the WRW is generalized to continuous space [2], with allowed steps ±Ab" (A a length

scale and b no longer restricted to integral values), and the time interval between steps

is T, then a L6vy (stable) distribution of^order H is obtained in the continuum limit A,t->-0 if a-1 ~ ctA, b-1 ~gA (0<ct<2£) and A /t ~ constant. The WRW is a discrete analog of

a L^vy flight [1]. An effective dimension for the WRW may be defined in terms of the small

k behaviour of p(k) [2]. In one dimension.

19
1 < effective dimension = 3-H < 3.

If the effective dimension exceeds 2, the walk is transient.

RENORMALIZATION GROUP TRANSFORMATIONS

Under a (real space) renormalization group transformation for the interaction parameters

of a system K' = K' (K) , the free energy F satisfies the scaling equation [4]

F(K) = «.''^F(K') + G(K), (11)

where d is the dimension, £ a decimation length and G an analytic function. In terms of a scaling field u, eq (11) becomes

F(u) = £"'^F(Au) + G(u) (12)

, n-1 _.,

= lim [£ F(a'^u) + Z £ ^ G(X^u)]. (13)

n-KX) j=0

It is usually assumed as a boundary condition that the first term in brackets on the right

hand side of eq (13) vanishes and all singular behaviour resides in the sum involving G. Eq (12) admits singular behaviour of the form

F . (u) = A(u) lul"^^^, (14) sing ^'^'11'

with

y = InXllnl, (15)

where

A(u) = A(Au) = Z A exp(2Trin Znu/lnX), (16) n

The eigenvalue X is chosen to depend on £ so that the critical exponent will not. Only the

n=0 term in eq (16) is kept, since otherwise the free energy would depend on I through X(£).

For the Weierstrass random walk

_i

^(k) = a TM^(b"V) + — Z a ^cos(b^k), (17) ^ j=0

in analogy with eq (13). When m^°° the boundary condition in eq (13) holds. We have

p . (k) = k"Q(k), (18) sing ^ '

-< ^

' '

where Q(k) = Q(bk), in analogy with eqs (14) - (16), implying a geometrical interpretation

for critical exponents. These matters will be discussed more fully elsewhere.

20
REFERENCES

[1] B. B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco , 1977

p. 327.

[2] B. D-. Hughes, M. F. Shies inger, and E. W. Montroll, Proc. Nat. Acad. Sci. USA (in press)

[3] P. Dienes, The Taylor Series {Clarendon Press, Oxford, 1931) p. 227.

[4] Th. Niemeijer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenomena, vol

6, Ed. C. Domb and M. S. Green (Academic Press, London, 1976).

21
TRANSPORT FLUCTUATIONS AROUND NON-EQUILIBRIUM STEADY STATES IN DISCRETE SYSTEMS.

Eckart Frehland

Universitat Konstanz,

Fakultat fiir Physik,

Postfach 5560,

D-TT50 Konstanz, ERG

We recently have developed a general theoretical approach to transport fluctuations around

steady states in discrete transport systems [1 ,2.] . The discrete description of transport

seems to be adequate for systems with discontinuous structures and coupling hetween transport and other processes as e.g. chemical reactions. It has heen especially successful for

the theoretical description of complex ion transport mechanisms through biological membranes, e.g. carrier mediated transport [3.1 or hopping diffusion through narrow channels

[^sS] (single-file transport), which additionally may assume different conductivity states

(e.g. open and closed). Especially in those systems, where the transport process takes

place in microscopic dimensions, the discrete description is more appropriate than continuous models. Applications to modern technological microstructures as e.g. highly integrated electronic circuity might also be possible.

TRANSPORT IN DISCRETE SYSTEMS

The basic idea leading to the concept of transport in discrete systems is that the state o the system can be described by a discrete set of variables, each of which stands for a cer- tain state. The time dependent probabilistic behaviour is given by a master equation for

the probabilities of the different states

ap ^

y > (M P - M P ) (1) = ^ yv V vy y V ,y+v

M (y+v) : transition rate per unit time,

yv ^

The description of transport observables (e.g. of electric current J) is based on the tran

sitions (fluxes) between different states of the system. Transport observables are defined

as a linear mapping of the fluxes (f)^^ between states y and v

T = ^ Y (f" (2) y yv yv

y ,v y+v

Equilibrium and nonequilibrium stationary states are distinguished by the validity or non- validity of detailed balance

^ = (|) ^ for all v,y (A ^: stationary fluxes),

yv vy yv

22
The underlying idea is that transport is determined by changes of the state of the system

and a special transition v u is coupled with a special, well defined contribution to T. Obviously, in principle there are no difficulties concerning the approximation of continuum

models by discrete models. In an appropriate limit to an infinite number of states, the discrete systems go over into continuum systems. As a consequence the general results e.g. for

the nonequilibrium transport fluctuations, can be applied also for continuum transport processes .

STEADY STATE TRANSPORT FLUCTUATIONS

By studying the time correlations between individual fluxes the autocorrelation function

and spectral density G _i(to) characterizing the steady state fluctuations of T can be derived [l],[2l,rT]

2 yv

p ,v= 1

+ h > Y y (j) ^ M I (t) cos tot dt (3)

K,y,v,p

with the fundamental solutions H (t) of (l)

Our general approach to transport noise has been used in order to investigate general properties of nonequilibrium fluctuations. It can be shown that the Nyquist or fluctuations dissipation theorem, by which at equilibrium the macroscopic admittance Y(a)) can be

expressed in terms of fluctuation properties of the system, breaks down at nonequilibrium.

The spectral density of steady state transport noise may be decomposed into one term con- taining the macroscopic admittance Y(tj) and a second term bilinear in the fluxes:

G^^((ij) = 1+ kgT^ Re Y(a))

+ 1+ \ Y Y :^ ((j) ^ - (j) ^) M \ (t) cos ut dt . (5) yv KP 2 yv vy Kp I py

K ; V ,irrp" I

The second term is decisive for the occurrence of "excess noise". In the case of electric

current fluctuations this typical nonequilibrium noise ( 1 /f noise (?), l/f^ noise, carrier

noise) has been measured in different systems in agreement with the theory (e.g. [ 3j )

.

DIFFERENT BEHAVIOUR OF SCALAR AND VECTORIAL FLUXES

An apparent discrepancy between our result and a number of classical papers (e.g. f6j

)

showing possibilities of generalization of the Nyquist theorem to nonequilibrium states can

be clarified in the following way: These papers are concerned with fluctuations of scalar

fluxes, i.e. the time derivatives of the state variables, while the treatment of (directed)

transport must be based on vectorial fluxes. It cam be proven [T] that transport observables defined by (2) can be represented by (a linear combination of) the time derivatives of

the state variables only in cases where the stationary value T^ of the transport observable vanishes, i.e. in equilibrium situations. Thus a main result of our contribution is

the essentially different noise behaviour of scalar and vectorial quantities at nonequilibrium.

23
y\PPLICATIONS

Apart from these general aspects of nonequilibrium fluctuations, we have performed a numter

of model calculations in order to investigate the influence of the internal structure of

transport systems (e.g. the potential profile for ionic hopping diffusion in membrane channels) on the properties of current noise. We mention the following two aspects:

a) the influence of nonequilihrium oscillatory transport on the current noise | 5 I and

b) the dependence of the intensity of transport noise on the transport parameters, e.g. special interactions.

In the latter case we have been able to show that under special conditions this dependence

may be very strong. At the example of single-file transport through narrow pores we could

demonstrate that the (low frequency) current noise is drastically reduced as a consequence o the ionic interactions within the channels in comparison with situations, where the ionic

interactions may be neglected.

As one example numerical resiilts are given in Fig. 1

Fig. 1: Current noise in ionic channels for high

applied voltage. Dependence of noise intensity on ionic interactions and on the number of ionic binding sites within the channels. —: interactions neglected, : single

file transport. At each line the assumed

potential profiles within the channels are indicated. The minima indicate the niimber

of ionic binding sites (0,1,2,3,6, and 9) • The spectral density is given in units of

2 J ze , i.e. the expected low frequency

shot noise . Frequency is given in arbitrary units

.

It is assumed that a high voltage is applied to the membrane, so that ionic jumps mainly

occur in one direction. In the low frequency region, i.e. for times long compared with the

characteristic ion passage times through the channels, the resulting current noise is usual

shot noise 2J ze^ (z: valency, e : elementary charge) in those cases where interactions can

be neglected. This occurrence of shot noise at low frequencies for systems with negligible

interactions can quite generally be derived from eq (3) . In the single-file transport

through narrow channels ionic interactions within the channels are taken into account in

so far as each ionic binding site within the channel can be occupied by only one ion [5).

The results in Fig. 1 clearly show a dependence of (low frequency) noise intensity on the

interactions and on the number of binding sites within the channels. Furthermore in the

single-file casej a peaking of spectral density (resonance) comes in as a consequence of the

damped oscillatory behaviour of transport (cf . ^5 1).

24
The possibilities of minimization of nonequilibrium transport noise should be further

investigated. It will be interesting to see if and how biological transport systems during

their evolution have reached states of minimum fluctuations. Furthermore technological

application should be discussed.

REFERENCES

[^] E. Frehland, Biophysical Chemistry 8 (1978), 255; 10' (1979) 128,

[2] E. Frehland, Biophysical Chemistry 12_ (198O) ; 63.

pj H.-A. Kolb, E. Frehland, Biophysical Chemistry 12_ (I98O) ; 21.

Ih] E. Frehland, Biophys . Struct. Mechanism ^ (1979); 91.

l^] E. Frehland, W. Stephaji, Biochim. Biophys. Acta 553. (1979); 326.

[6J M. Lax, Rev. Mod. Phys . 32_ (196O) ; 25-

f7J E. Frehland (198I, in preparation).

25
MULTIPLICATIVE NOISE IN STOCHASTIC DIFFERENTIAL EQUATIONS: A NUMERICAL STUDY

J. M. Sancho and M. San Miguel

Departamento Fisica Teorica, Universidad de Barcelona

Diagonal 647, Barcelona 28, Spain

S. Katz and J. D. Gunton

Physics Department, Temple University

Philadelphia, Pa. 19122 U.S.A.

Stochastic differential equations of the Langevin type for a finite set of variables

are a common tool to study a variety of physical, chemical, and biological systems. Interest in these types of equations has recently increased because of their success in describing nonequilibrium situations and in particular nonequil ibrium steady states. From a mathematical point of vie\A(, remarkable novel features appear when considering the case in which

the random term depends on the variables of the equation ("multiplicative noise"). There

are at least two main sources of such multiplicative noise. The first one is the usual procedure of adiabatic elimination of the fast variables of a system. The second one is modelling a fluctuating environment or a superimposed external fluctuation by letting a parameter

in a phenomenological equation of motion become a random variable with prescribed statistics.

This external noise situation has been considered experimentally in illuminated chemical re- actions tl3 , electrical circuits {2^ , and liquid crystals [Sl . The theoretical analysis

of these situations predicts [4l a kind of nonequilibrium phase transition in which the ex- ternal parameters governing the transition are the noise parameters.

In this paper we present some preliminary results of a numerical study of the effects

of external multiplicative noise on such nonequilibrium phase transitions. The full details of our work will be presented elsewhere C53 . We consider a prototype equation of motion for a single variable q of the form

q = q

- + q ^(t). (1)

We have chosen dimensionless units such that the usual coefficients of the linear and cubic

terms in this Ginzburg-Landau like model have been included in the noise term of (1). The

random variable ^{t) is assumed to be Gaussian distributed with a zero mean value. We

have considered two different models for the correlation ^^(t) ^(t')>, the first being

the so-called "colored noise" case in which

<^(t) ^ (t')> = ^ ^ > (2)

with two independent noise parameters, the correlation timeT and the intensity D. The sec- ond model is the conventional white noise limit of (2) in which t-*0 with D finite so that

the correlation becomes 2DS"(t - t'). We have investigated the dependence of the stationary

state solution of (1) on the parameters D and T . In particular, we have computed the sta- tionary state probability distribution function Ps(q) which would correspond to the steady

state solution of the Fokker-Planck equation associated with (1). We have also computed

the first and second moments, ^q^ and <!^q^)>, in this steady state. In addition to these

time independent properties we have studied the time dependent correlation function in the

steady state and its relaxation time T defined as

26
oo

I This relaxation time is usually used to characterize "critical slowing down" in the dynamics

of second order equilibrium phase transitions and is also of interest for nonequilibrium

phase transitions.

Our results for the time independent properties for the white noise model are in very

good agreement with the available exact analytical results, which gives us confidence in the algorithm used in our study. Our results for <'q> , <q2> , Pstql, and T for the colored

noise case show an interesting dependence on D and t and are in qualitative agreement with

an approximate analytical theory. We now turn to an explicit discussion of our investigation, beginning with the white noise model.

We first summarize the existing theory for the white noise model. The time independent properties in the stationary state are completely known, since the stationary solution

Pg(q) of the Markovian Fokker-Planck equation associated with (1) is exactly known {6] . Of

particular interest to us is the position of the most probable value qp (the maximum of Pg)

which is zero if D > 1 and is equal to (1-D)'2 if D £ 1. Thus, although the moments of Ps show no significant change as a function D, Ps itself is qualitatively different for D > 1 and D< 1, since for D<1 the most probable value qp becomes different from zero. If we identify qo as the macroscopic state C7j , and define a transition point (or threshold value

of D) as one for which the most probable value becomes different from zero Z^, 7, 8, 9l , then D = 1 is a transition point. There is no divergence of fluctuations in q at this point,

however, since the moments of Ps remain finite.

The dynamical properties of (1) for the white noise case can be analyzed in terms of

the eigenvalue problem associated with the Fokker-Planck equation for this model. The

eigenvalue spectrum has been discussed in references Z^l and C93 and has been shown to con- tain a continuous as well as a discrete part, with the discrete spectrum given by

= 4 n D(1/2D - n), for n i. Uq. Although there is a disagreement about the actual

value of nQ, no discrete spectrum exists for D>l/2. This means that the transition point

D = 1 lies in the region where only the continuous spectrum exists. The description of the

time dependence of the correlation function is therefore quite complicated near this threshold value. Thus we have studied this function in our computer simulation for both the white

and colored noise models. Another important aspect of the discrete spectrum given above is

that well below threshold, in the region where this discrete spectrum exists, the damping

constants of the system decrease linearly with increasing noise intensity D. Hence the approach of the system to equilibrium should be slowed down by increasing D. Our numerical

results for the relaxation time T, shown below in Figure 1, support this conclusion.

Oo

Fig. 1 Numerical results

for the inverse relaxation

time T-1 as a function

of noise intensity D. The

open circles correspond to white noise. The crosses correspond to a colored noise

with X = 1/3.

0.4 0.8 1.2 1.6 D

27
It is clear that for the white noise case T increases monotonically with D, so that there

is indeed a slowing down with increasing D in the entire range of values we have explored.

This is in agreement with the results for the discrete spectrum. Our analysis shows, how- ever, that the slowing down persists in the domain in which only a continuous spectrum exists. It should also be noted that it appears that T-1 decreases more slowly than as a lin- ear function of D for larger D. If so, this would be qualitatively different from the linear behavior obtained from the discrete spectrum for small D. Finally we note that we ob- serve no "critical slowing down" (!-«>) as D-*l.

One issue which we leave for future discussion UbI is the relationship between our com- puter studies of (1) and the closely related experimental studies by Kabashima et al . \2\

on a parametric oscillator under the influence of a wide band external noise.

We now turn to the more difficult theoretical problem of colored noise. Here there is an approximate theory [5],[lO] to compare with, which is based on a first order expansion

in the correlation time 'C , We begin by displaying our results for the mean values, <q>

and <q2> in Table 1

.

Table 1

D T <q> (<q> -<q>J/q <:q^> - <q2>o D-C

2 2

0.25 1/15 0.9458 0.0063 0.0069 0.0062

1.5 ^/uI 0.7545 0.035 0.042 0.038

0.5 1/3 0.9280 1 .005 0.0472 0.0047 0.083 0.05

0.75 1/3 0.8946 1.003 0.066 0.0028 0.125 0.075

0.90 1/3 0.8734 0.9988 0.095 -0.0012 0.15 0.09

1.5 1/3 0.8113 1.011 0.113 0.011 0.25 0.15

0,90 1/2 5 0.8836 1 .006 0.0856 0.006 0.18 0.100

1.50 1/2 5 0.8232 1 .007 0.129 0.007 0.30 0.167

2.00 1/2 5 0.7869 1.0147 0.164 0.0147 0.40 0.222

Here <q >o denotes the white noise value. The last two columns show two theoretical predictions based on the first order T analysis for the value of(<q> - K<^\}/^<^\ . As can

be seen^ the better fit with our data is obtained using a phenomenological "renormalized"

correlation time Tp^= 'C/( 1+211 ) . Figure 2 shows the dependence of <q> on D for fixed

X = 1/3. There is a clear difference between the white and colored noise results. In addition the first order theory for the colored noise (solid line) seems to qualitatively ex- plain our results, although there is clearly a need for a more accurate theory.

Fig. 2 The first moment q as

a function of D for X= 1/3.

The full circles are the numerical results. The solid and dashed lines are the theoretical and

white noise predictions, respectively.

28
Another prediction of the existing theory is that <q^> - <q^>o should be independent of

both D and t . As can be seen from Table 1, this seems to be confirmed by our results.

An interesting qualitative feature that emerges from our study is the development of a new "phase transition" due to the colored noise. This can be seen from our results for Ps(q)

obtained for different values of D and x . In Figure 3 below we show a typical curve for PjCq).

This shows a distinct and interesting difference between the white noise and colored case

for D = 1.5. The "colored noise" model for this value of 'C = (1/2.1) shows a new phase

transition not seen in the white noise case, at least in the sense of a qualitatively different Ps(q) vis-a-vis two extrema. The significance of this transition is not completely

clear, but it is in qualitative agreement with the first order t theory, as discussed in {.b].

The effect of T is less significant in other regions of the (D,x) domain, with the results

for Ps(q) being very similar to the white noise distribution function. Finally, we note that

the effect of colored noise can also be seen in dynamical properties such as the relaxation

time. In Figure 1 we see that there is a slowing down associated with increasing noise intensity D for fixed T . In addition our preliminary results indicate that T increases with

X for fixed D, although more extensive investigation is needed of this particular point.

0.0 0.4 n.3 1.2 1.6 2.0 2.4 2.8

References

1. P. deKepper and W. Horsthemke, C. R. Acad. Sci . Ser. C 287 , 251 (1978)

2. S. Kabashima, S. Kogure, T. Kawakubo and T. Okada, J. Appl . Phys. 50, 6296 (1979)

3. S. Kai, T. Kai , M. Takata and K. Hirakawa, J. Phys. Soc. (Japan) 47_, 1379 (1979)

4. W. Horsthemke in Dynamics of Synergetic Systems, p 67, ed. H. Haken, Springer Verlag

(Berlin, Heidelberg, New York 1980)

5. J. M. Sancho, M. San Miguel, S. Katz and J. D. Gunton (unpublished)

6. R. L. Stratonovich, Topics in the Theory of Random Noise. Vol. 2, Gordon and Breach

(New York 1967)

7. W. Horsthemke in Stochastic Nonlinear Systems in Physics , Chemistry and Biology,

Synergetic Series, Springer Verlag (Berlin 19811

8. A. Schenzle and H. Brand, Phys. Rev. A20, 1628 (1979)

9. M. Suzuki, "Fluctuations and Relaxation in Stochastic Systems" preprint (1980)

10. J. M. Sancho and M. San Miguel, Z. Phys. B36, 357 (1980)

Acknowledgements

Work supported by grants from the National Science Foundation, the U.S. -Spanish Committee for

Scientific Cooperation and the Research Corporation.

29
LEVEL-CROSSING INTERVALS OF A GAUSSIAN NOISE

Tadashi Mimaki and Hiroshi Sato

University of Electro-Communications

Chofu, Tokyo, Japan 182

Masami Tanabe

Kumamoto Radio Technical College

Kikuchi-Gun, Kumamoto, Japan 861-11

INTRODUCTION

The present paper treats the distribution density Pri+(T,I) of the sum

of (n+1) successive level-crossing interval lengths of a Gaussian process,

where T and I denote time length and crossing level, respectively. The

suffix "+" means that the intervals start at an upcrossing, i.e. crossing

with positive slope. This density is not determined yet in analytical

form. Therefore, considerable effort has been directed in the past toward

the numerical determination of it both theoretically and empirically [1-9].

It was found that for Gaussian process having eighth order Butterworth

spectrum the densities Po+(T,I), Pi+{T,I) and P2+(T,I) show a plural number

of peaks if the crossing level I is negative (10]. The number and height

of the peaks depend on the spectrum bandwidth and on the crossing level I.

The first peak is not necessarily the highest peak. Occasionally even P3+(T,I), P4+(T,I)^ and P^^Ct,!) show several peaks, too. The dependence

of this property, called multi-peak property, on the spectrum bandwidth is

not as strong as that of the correlation property between the level-crossing intervals, which changes remarkably with the spectra of low-pass, broad

and narrow band-pass types. The multi-peak property appears for all of

the spectrum bandwidths.

In order to explain the multi-peak property we introduced the random

excursion model, which was originally proposed by Sato et al. [11] to interpret the behaviour of the variance of the number of the upcrossings in a long fixed period. Since the model treats only upcrossings, we restrict

ourselves to the density Pjj+(T,I) with odd n in the present paper. It is

found that the distribution density P]^+{T,I) of arbitrary crossing level I itself can approximately be expressed in terms of Pj^+(t,Iq) at a reference

level Iq by utilizing this model. The expression is compared with the

experimental data. Although the model is very simple, the agreement is

surprisingly good and the multi-peak property of the level-crossing intervals

is explained excellently at least for the low-pass case. INTERVAL DISTRIBUTION

At first we introduce the notion of random excursion model: If only

the part of the waveform of a random process that lies above an arbitrary

reference level Iq is observed, that part is seen to consist of a succession of excursions, where an excursion means a part of the sample function

that starts from the level Iq and stays above it until it finally returns

to the original level Iq. Now the following assumptions will be made:

(a) A Gaussian process consists of a succession of random excursions, each

30
of which hasn't any minimum; (b) For any level I OIq) excursion is

said to be in state zero or one, according to whether the excursion attains

the level I or not. The change of the state of the successive excursion

obeys a simple Markov law. Let q and r be the transition probabilities

from zero to one and from one to zero, respectively. Let p be the probability that an excursion reaches the level I. Then we have a relation,

pq=r(l-p). If the heights of the successive excursions are mutually independent, q=l-p and r=p. For a Gaussian process and Io=0, then p=exp(-I^

/2); (c) The interval distribution at reference level 1q is independent

of the d is t r ibu t ion of the heights of excursions.

Let ffj(T) be the distribution density of the sum of the lengths of

successive n intervals of level Iq defined only by upcrossings. Since

the distribution density fj^(T) of n intervals contributes to the distribution density Pj^+Ct,!) of level I, only when the intermediate (n-1) excursions don't attain the level I and finally n-th excursion attains the

level I, we obtain,

. (T,I)=(l-q)f , (T)+qr I ( 1 -r

) ^ f ^ ( T ) . ' (1) . i+ i n=2

This expression indicates that the distribution density Pj^^.(T,I) of arbitrary level I can be given by the distribution densities fi{T), f2(T),

f3(T),...at reference level Iq. The closeness of the approximation

depends on the validity of the above assumptions. Let C(C) and F^(E,) be

the characteristic functions of Pj^+{T,I) and f^{x), respectively. We

have from eq (1),

C(?)=E[e^^'^]= (l-q)FT (C)+qr Z ( 1 -r ) ^F^ ( C ) , (2)

J- n = 2 n

where E indicates the expectation. If the crossing intervals at reference

level Iq are mutually independent, we obtain,

f^(T)=fT nil(T)*f, (T)*...*f- (T) or F {5)=f"(C),

1 n 1

where "*" means the convolution process. Thus, eq (2) is reduced to

C(C)=F3^(C) [l-q+qrFj^CC) {l-?Fj^(C) }"^] , (3)

where r=l-r. Furthermore, if the heights of the excursions are mutually

independent, we obtain q=l-p and r=p, with the result that

Pi+(T,I)= p{l-p)""^f (T), (4)

^ n=l n

and correspondingly

oo . C{C)= E, p(l-p)"" F (C)=pF (?) [1-pF (C) ]"} (5) n=l 11 1

where p=l-p. According to above expression, the distribution density

Pj^^(T,I) of arbitrary level I is given only by the distribution density

fj^{T) of the single interval at reference level Iq.

31
Heretofore, three independence assumptions, that is, interval independence, excursion independence^ and independence between interval and

excursion sequences, are individually treated. However, in the actual

situation they may appear in a mutually related way. The present paper

mainly concerns itself with the case where they are simultaneously valid,

should be noted that the above expressions can be applied not only to

the Gaussian process, but also to most of the general continuous stationary

random processes.

VARIANCE OF LEVEL-CROSSING INTERVALS

Let Mq Oq and K? be the expected length, variance of intervals and

correlation coefficient between zeroth and i-th crossing intervals at

Also, let U, and be the corresNow, we expand Fj^(^) in power series in 5,

reference level Iq, respectively,

ponding quantities at level I. substitute it into eqs (2), (3)^ and (5), and then compare it term by term

with an expression.

C(5)=l + iE (T)C-E (T^)CV2 + ,

From term of E, , we obtain

E (T ) = y =\io/p (6)

From term of E, , we obtain the following expressions:

From eq (2) P r ° P P i=l ^ 0*

From eq (3) ; E (T'^ )= (

a"=E (tM-eMt)= — Ip { ^ - ^

)\i^ + {l+2p .lK°r^'^}af. ] p r p " 1=1 1 ^

2 0 p 0

P 'K^'o''

From eq ( 5 ) ; Ed'')2 x =

P D 0 0

0 = p p

'J ^

(7)

(8)

(9)

(10)

(11)

(12)

The equation (12) shows that the quntity pa is linear in p. The equation

(10) can also be obtained from the expression of the variance of the number

of the upcrossing points during constant time period.

COMPARISON WITH EXPERIMENT

Figures 1(a) and (b) show the examples of the calculated results based

on eq (4) together with the experimental densities Pj^+(T,I), where only

three densities fj^(T), f2(T), and f^(T) experimentally determined at zero level are used. The power spectrum of the original Gaussian process is 8-th

order Butterworth low-pass. The crossing levels I are 1.0 in Fig. 1(a) and

2.23 in Fig. 1(b), respectively. Although the model is very simple, the

agreement is surprisingly good and the multi-peak property of the levelcrossing intervals is explained excellently at least for the low-pass case.

The following table shows the comparison of the calculated result

based on eq (12) with the experimental data for the Gaussian processes

having 8-th order Butterworth low-pass and band-pass spectra. Since the

32
coincidence between theoretical and experimental values is excellent for

low-pass spectrum, the level-crossing intervals deem to be mutually independent for this spectrum. For band-pass spectrum, the coincidence is

poor, except for the high crossing level.

Pl+(T,I)

4^

P M theo . exper i

.

eq (12)

0^

experi

.

0.0

1.0

2.0

3.0

0.606

0 .135

0.011

17.5

78.3

954

17.7

77.5

970

168

5690

9.40x10^

165

5641

9.33x10^

0.1

1.0

2.0

3.0

0.606

0.135

0.011

16.6

74.4

905

16.8

75.8

930

146

5108

8.48x10^

105

4387

8.37x10^

Table 1 Comparison of calculated

from eq (12) with experimental a^.

REFERENCES

[I] S. O. Rice, BSTJ 2_3, 282 (1944); 2±, 46 (1945).

[2] M. S. Longuet-Higgins , Phil. Trans. Roy. Soc. A254 , 557 (1962).

[3] J. A. McFadden, IRE Trans. IT IT-2 , 146 (1956); IT-4 , 14 (1958).

[4] D. Wolf and H. Brehm, AEU 2J_, 477 (1973).

[5] A. J. Rainal, IRE Trans. IT IT-8 , 372 (1962).

[6] T. Mimaki, JAP 44, 477 (1973).

[7] S. 0. Rice, BSTJ 3J_, 581 (1958).

[8] H. Brehm and D. Wolf, AEU £9, 415 (1975).

(9] T. Mimaki and T. Munakata, IEEE Trans. IT IT-24 , 515 (1978).

[10] T. Mimaki, M. Tanabe and D. Wolf, to appear on IEEE Trans, on IT (1981)

[II] H. Sato and M. Tanabe, IEEE Trans, on IT IT-25 , 127 (1979),

33
STUDY OF DISORDERED STRUCTURES AS A SPATIAL NOISE PROBLEM

M. Celasco"", P. Mazzetti and A. Stepanescu

Istituto Elettrotecnico Nazionale Galileo Ferraris

Gruppo Nazionale Struttura della Materia - U.R. 24 - Torino (Italy)

INTRODUCTION

The technique used to characterize noises can be conveniently applied to the study of

disordered structures constituted by the superposition of spatially distributed elementary

events, representing some local property of a physical system.

In the case of three-dimensional structures of atoms or molecules, the type of order

(long-range or short-range order, type of crystallographic order, range of directional order

etc.) is essential in determining many properties of the system, like electrical and

magnetic properties. X-rays, and neutron scattering, etc. In many cases a direct physical

meaning can be associated to the three-dimensional power spectrum. For instance, if in an atomic structure the elementary events considered are the atomic form factors, then the

power spectrum (}) (k) describes the scattered_^intensity with wave vector k'=ko+k when an electromagnetic radiation with wave vector kg incides on the structure. Another important

case is the one where the elementary events are localized fields or currents, like fluxons

in superconductors or local eddy currents due to Barkhausen jumps in ferromagnets : the power

spectrum can then be used to describe the excess energy or power loss in the system in

relation to the spatial distribution of such events.

More generally, the whole electronic structure of a condensed body may be visualized in

terms of the scattering properties towards the plane wave components of the electronic wave function in a very similar way as for X— ray scattering. In the following, to make the

language easier, we shall make reference to a system whose elementary events are simply

called "atoms" and shall use the same terminology found in X—ray scattering problems from

atomic systems.

The description of a partially disordered three-dimensional structure is in general a difficult problem, which involves the introduction of the local type of crystallographic

order (b.c.c, f.c.c, h.c.p. . . .

) , its range, the range of the directional order, and the

distribution functions describing the displacements of the neighbours of each particular

atom or elementary event in the structure. Two different types of approach to the problem

are found in literature. The first one [^1,2|^ makes use of computer simulated structures, on which relevant statistical quantities as the autocorrelation function and the structure

factor, are numerically calculated. This approach is particularly suitable to explain the

experimental results about X-ray scattering for amorphous substances, but has the big

disadvantage that it does not allow to obtain analytical expression describing these

quantities in terms of the elements characterizing the structure from the statistical point

of view. The second type of approach starts from a perfectly ordered structure and introduces the disorder through an external perturbation of the atom positions characterized by

a given fluctuation spectrum |_]3^. This type of approach is suitable to describe , for instance,

defects like dislocations, vacancies and interstitials in real crystals, but does not allow

to represent the structure from a microscopic point of view introducing atom pairs correlations effects.

The present approach has the advantage that the results are expressed in terms of very

general analytical expressions which can be used to describe the scattering properties of

structures having any type of local crystallographic order. Further, the range of local

order is described in terms of distribution functions taking into account the interactions

between nearest neighbour pairs, and thus they can be given some clear physical meaning.

* Permanent address: Istituto di Fisica Generale dell ' Universita di Torino.

Permanent address: Istituto di Fisica Sperimentale del Politecnico di Torino.

34
MODEL DESCRIPTION AND POWER SPECTRUM CALCULATION

The Starting point is a perfectly ordered lattice having the crystallographic ti'pe of

order characterizing the short-range order of the system under study.

The disorder is introduced by displacing the atoms from their crystallographic

positions through three distribution functions, which give the probability for the position

of the nearest neighbour atoms of each atom along the three axes of a primitive reference

system. The whole structure is self-consistently described, from the statistical point of

view, by a convolution technique, which allows to obtain the probability distribution for

the relative position of any atom in the structure with respect to any other atom chosen as the origin of a reference system bound to the local crystallographic order.

Thus, choosing any atom A in the structure and introducing a local primitive reference

system with origin in A and consistent with the local crystallographic type of order, the

probability distribution for the position vector rg of any other atom B, which in a perfectly ordered structure would have crystallographic indexes 1, m, n, is given by

Pir^) =

)P^(?) p^(?).....,.p^(?)| ,. )p^(?) =:= p^(?)=:c....,p^^(?)| ,: |p^(?) p^ ..^p^ (?)

| (1)

1 terms m terms n terms

where * means convolution product:

p (r) =:= p (r) = / p (r) p ir-r) dr (2) a a a a B

-> ^

and Pa(r) , P]-,(r) , Pj,(r) are the three distribution functions representing the probability

for the position vectors of the nearest neighbour atoms of A which, in a perfectly ordered

lattice, would be displaced with respect to A of the primitive vectors a, b, c respectively.

Starting from eq.(l) it is possible to represent in terms of these local distribution

functions the autocorrelation function "fd) of the whole structure. Details of calculations

are given in another paper [j^Zl- '^ important to note here that upon Fourier transform

of 4'(t), the convolution products become ordinary products, and thus the expression of the

power spectrum (pir) assumes a rather simple analytical form:

*(?) = { [<]s(k) |2> - |<S(k) >| 2~] + |<S(k) >|2 r (l + 2e ) • (1 + 26, ) • (1 + 26 } , (3)

where N represents the average number of atoms per unit volume and

^ Z. (k) ^

e.(k) = R (

^

) ; Z.{k) = /p.(r) exp(i k-r) dr Fi = a, b, c] . (4)

i — —

l-Z.(k)

1 In eq. (4) R^ means the real part of the expression within brackets.

Equation (2) hap. singular points when one, two, or all of the three distribution functdons

Pj_(r) are Dirac 6-functions. This corresponds to the case where there is long rang^e order in

one, two/or three dimensions respectively, and consequently the power spectrum <i> (k) has a line component as well as a continuous one. The line positions are given by the equations

Z^(k) =1 Q = a, b, cj , (5)

while the line intensities can be calculated by using the mathematical technique described

in paper [^s], and are simply given by

A, = N I <S (k, ) > I , (6) l,m,n ' l,m,n '

-> where k^^j^^n is a k-vector satisfying eq. (5) .Obviously, when all three Pi(r) distribution functions are Dirac 5-functions and all the atomic form factors are identical, the

well known Bragg result for X-ray scattering from crystals is obtained.

When the primitive axes are not an orthogonal set, it is convenient to use controvariant coordinates for r-vectors and covariant coordinates for k-vectors (=:=) . In this case

(=1=) This is equivalent, but more convenient, than the introduction of a reciprocal space

for k-vectors, as is usually done in X-ray scattering textbooks.

35
eq. (4) simplifies to

Z (k) = exp(k a) Q (k) . (7) a a ^a

-y -> Similar equations are true for Z, (k) and Z (k)

.

b c ^

In eq. (7) k^ is the covariant coordinate of k alonq the a-axis of the reference system,

a the lenqth of the primitive_^vector a and Q^lk) the three-dimensional Fourier transform of

the distribution function qa(^)/ defined in terms of Pa'^) by the equations:

q^{r') = p^(a+r') Qi = a, b, c] . (8)

These functions give the probability density for the displacements r', of the nearest

neighbours of each atom with respect to the positions they should have in a perfectly

ordered lattice having the chosen atom in one of its reticular points.

Equation (3) gives the power spectrum for a structure which may have short-range as well as long-range crystallographic order, but on which a long-range directional order is

always assumed. This means that the crystallographic axes associated with the local type of

order are the same in every point of the structure instead of being randomly rotated, as it

is expected in isotropic disordered structures. To describe such types of structures it is

thus necessary to perform a spectrum averaging over a sphere S of a radius k:

*0(k) = ff^ Mk) dk . (9)

For atomic systems, the three-dimensional Fourier transform of cjjg (k) is directly related

to the pair distribution function (P.D.F.). This quantity represents the three-dimensional

autocorrelation function of the system in the case where the atomic form factor is a Dirac

6-function

.

In the case where some sort of anisotropy concerning the directional order exists in

the structure, as is the case for many disordered structures, averaging should be made

by introducing a suitable distribution function in eq. (9) . Simplified expression for power

spectrum and autocorrelation function calculations in the case of directional isotropy, when

the distribution fractions qi(r') are spherical gaussians, are given in [^4^. In the same paper some extensions and generalizations of eq.(3), taking into account the symmetry of the

autocorrelation function, are also given. There we report few typical results concerning the

pair distribution function of amorphous structures with b.c.c. and f.c.c. short-range order.

The parameter H represents the inverse of the "correlation length" characterizing the local

order in terms of the side a of the conventional cells of the b.c.c. and f.c.c. lattices

(see figure 1) . For comparison^ some results obtained by Ichikawa by using numerical

calculations in a microcrystallite model, are reported in figure 2. It is seen that what he

calls "heavily distorted structure" corresponds closely to a short-range order having a "correlation length" of the order of 6-7 times the interatomic distance a. Figure 3 shows

the reduced interference function I (k) = (jjg (k) / | <So (k) > | ^ -N for a f.n.c. structure with

H=0.05. An interesting aspect of this quantity is its typical behaviour in the low k range.

In the case of a completely disordered structure with point-like atones, T(k) is a constant

equal to 1, while in a perfectly ordered structure it should be zero until the appearance

of the first peak. In the case of short-range order it is seen that I(k) remains lower

than 1 (i.e. the structure scatters less than a completely disordered one) from k = 0 up

to the first peak, but for very small k values there is also a peak which a direct analysis

through eq. (3) and eq. (9) shows to be of the type I (k) k ^. This aspect of I (k) , which is generally not found in other types of calculations where

a large but finite number of atoms are considered, is very important both for what concerns

the scattering of long wavelength radiations from atomic systems and, in the case of other

types of physical systems, like ferromagnetics and superconductors, for the computation of

the excess power loss during cycling. It is worth noting that such a behaviour of I (k) is

identical with the one expected for systems undergoing an order-disorder phase transition.

36
0

2

0

2

0

PDF BCC H=ai

—I~fv7"

1 1

- FCC 1 \^ H-ni 1 1 — \J.\

/

[\ BCC H=Q15

FCC H=Q15

> I I 1 0 Ar/a

Figure 1 - Pai.r distribution function for

isotropically disordered structures having a B.C.C. and F.C.C. short-range order. H represents the inverse of the correlation

length expressed in terras of the conventional cell side a.

10 r(A)

Figure 2 - Some results obtained by

Ichikawa [^6^ for heavily distorted struc- tures having a B.C.C. and a F.C.C. local

order (Microcrystallite model).

0

Kk) FCC H=Q05 U

0 1 3 k/(2n/a)

Figure 3 - Reduced interference function

for a isotropically disordered structure

having a F.C.C. short-range order , computed

through eq.(3) H=.05 corresponds to a correlation length of 20 a.

REFERENCES

JjT] G. S. Cargill III, Solid State Phus. , 30_, 227 (1975).

\jr\ J. Graczyk and P. Chaudhari, Phys. Stat. Sol. B. , 58_, 501 (1973), J. Non-Crist. Solids

17, 299 (1975) .

]23^ M. A. Krivoglas, "Theorie de la diffusion des rayons X et des neutrons thermiques par

les oristaux reels", Maddon et Cie (1969).

1^4^ M. Celasco, P. Mazzetti, A. stepanescu, "Power spectrum and scattering properties of

three-dimensional disordered systems having a variable degree of short-range order" (to

be published)

.

\^ P. Mazzetti, c. Oldano, J. Appl. Phys., 49, 5357 (1978).

nell I. Ichikawa, Phys. Stat. Sol. A, 19_, 707 (1973).

37
ON THE THEORY OF AMBIENT

ACOUSTIC NOISE IN THE SHALLOW OCEAN

Michael J. Buckingham

Radio and Navigation Department

Royal Aircraft Establishment

Farnborough, Hampshire GUI 4 6TD

England

INTRODUCTION

In the present context, 'shallow' water is defined as that part of the ocean which

overlies the continental shelf and shows a depth of 200 m or less. In general, acoustic

propagation in shallow-water channels is a complicated phenomenon due to the proximity of

the boundaries, the nature of the bottom, and the variation of the speed of sound with depth.

The usual approach is to treat the channel as a waveguide in which the' sound field is a sum

of normal modes. Considerable simplification is achieved if isovelocity (te sound-speed

independent of depth) water is assumed, and if the boundaries are treated as plane and

parallel. Then the Fourier transform of the velocity potential at depth z due to a source

at range r and depth is

M

,, X ^ A exp(jkr) Y"' . ( 0\ . /mirzN

^ m=l

where A is a constant independent of r, z and z_ , h is the depth of the channel, m is the mode number, M is the total number of modes the channel can support at angular

frequency o) , and k = w/c is the wave number (c = speed of sound in the channel). For

simplicity, bottom losses have been ignored in this expression, the bottom interface has

been treated as a pressure-release boundary, and a factor (equal to the reciprocal of the

square root of the horizontal wave number), which is essentially constant over the grazing

angles of interest, has been omitted. The total number of modes in the channel is

M = (!^] sin a , (2) Xttc/ c

where is the critical grazing angle of the bottom.

Equations (1) and (2) are the basis for the model of wind-generated noise in the shallow ocean described below. A full analysis of the problem is given in Ref fO* The model is

valid for frequencies up to about 1 kHz. At higher frequencies, the acoustic wavelength is

comparable with the surface wave height, surface scattering is then important and eq (1) is no longer a reasonable approximation.

THE NOISE MODEL

The source of wind-generated noise in the ocean can be modelled as a very large number

of independent, impulsive monopoles located in a plane immediately below the surface ^1,2^.

These sources give rise to a continuous noise field and a modal noise field. The former is

38
associated with near-field sources, and the latter with the very large number of far field

sources. Assuming low losses in the bottom, the modal field will dominate and the

continuous-radiation component of the noise field can be safely neglected. Only the modal

contribution is discussed here.

The cross-spectral density of the noise field at two vertically separated sensors due

to those sources in an annulus on the surface of radius r and thickness dr , centred on

the vertical axis of the sensors, is obtained from eq (1) in conjunction with an extension

of Carson's theorem. On integrating over the surface, the vertical cross-spectral density

of the noise field due to all the sources is obtained.

The result for the cross-spectral density can be expressed as the sum of two terms, one of which depends on the diffevenoe in depth between the two sensors and another which depends

on the sum of the depths of the sensors. If either sensor is on a boundary, these two terms sum to zero, in accord with the pressure-release boundary conditions. But away from the

boundaries, in the centre portion of the water column, the term containing the sensor separation predominates. Thus, to a good approximation, the cross-spectral density in this region

of the channel is independent of the mean depth of the sensors and depends only on their

separation. This establishes the noise field as essentially spatially homogeneous away from

the boundaries of the channel. The region of quasi-homogeneity is indicated in figure 1, which shows the power spectral density of the noise as a function of depth for two different

0.7 0.8 0.9 1 1.1 1.2 1.3 0.7 0.8 0.9 1 1.1 1.2 1.3

Fig 1 The power spectral

density of the noise as a function of depth, at two different frequencies, in

a water column 100 m deep

f = 200 Hz, M = 13 f = 500 Hz, M = 33

frequencies. The expression for the power spectral density in figure 1 is

sin[(2M + l)TTz/h]

| fy.

(2M + 1) sin(TTz/h)(

'

Note that the oscillatory component of the power spectral density, representing the non- homogeneous component of the noise field, is less than 10% of the depth-independent component

over the quasi-homogeneous region.

If the noise in the quasi-homogeneous region is treated as truly homogeneous, a considerable simplification can be achieved in representing the spatial properties of the

noise field: the concept of a unique directional density function can be introduced to describe the noise power incident at a point in the field from a given direction. The

directional density function of a homogeneous noise field is uniquely related to the crossspectral density through a Fourier transform relationship. (The Fourier variables are the sensor separation and the angle of incidence of the radiation.) The directional density

function for the shallow water noise field is

39
F(e) = (i) ^6(cose-I^)- 6(cose)

s=-M

(4)

where 0 is the vertical angle measured from the zenith and 6 (cos 6) is the Dirac delta

function. A sketch of F(e) is shown in figure 2.

F(9) shows some interesting features. It varies discretely with vertical angle, showing

'rays' in the directions of the eigenrays associated with the modes in the channel. There is no horizontal noise 'ray' because the pressure-release boundary condition precludes the

existence of a zeroth-order mode. At grazing angles greater than the critical grazing angle

there is no modal noise because, at these relatively high, grazing angles, total internal

reflection at the bottom boundary no longer occurs and the modes cannot propagate. The

noise 'rays' all have the same intensity, as indicated in the figure by their equal length.

At first sight this is perhaps an unexpected result, since the outer 'rays', associated with

the higher-order modes, undergo more encounters with the (lossy) bottom per unit distance

than those 'rays' closer to the horizontal. Thus, if this 'mode stripping' phenomenon were

the only mechanism acting, the outer 'rays' should be attenuated relative to the inner 'rays'.

However, this effect is exactly compensated by the generation mechanism: the sources lie in

a plane immediately below the surface and, as can be seen from eq (1), excite the higherorder modes more strongly than the lower-order modes. This accounts for the uniformity of

the noise 'rays' in figure 2.

The angular separation between the noise 'rays' in figure 2 depends on frequency. At

100 Hz, for example, it is about 4 . In order to resolve the 'rays', an array of acoustic

sensors with an angular resolution better than the angular distance between the 'rays' is

required. A vevtiaal line array would seem to be the obvious choice for an experimental investigation of the vertical directionality of the noise field; but such an array is unlikely

to have sufficient angular resolution because its aperture is limited in extent to the depth

of the channel

.

An alternative approach would be to use a horizontal line array, which could be made as long as necessary in order to achieve the required angular resolution. The (theoretical)

response [s] of a limiting form of such an array, in which the aperture is infinite and the

beam is of infinitesimal angular width, to the discrete noise field represented in figure 2,

is shown in figure 3. The figure shows the noise gain, , versus the steering angle, 6 ,

40
2. Or

Fig. 3 The theoretical noise

gain versus steering angle

for a very long horizontal

line array in shallow water

measured from the axis of the array. Note that for steering angles less than the critical

grazing angle, the noise gain goes through a series of zeros, occurring when the beam is

pointing directly at a noise 'ray'. Thus, this 'fine structure' on the curve is ascribed

to the modal nature of the noise. When the steering angle is greater than a^, , the noise

gain is monotonic increasing up to a steering angle of it/2 , corresponding to the broadside

condition. The fine structure around endfire in figure 3 could provide the means whereby

the modal character of the ambient noise in shallow water is experimentally detected.

REFERENCES

[l] M. J. Buckingham, J. Aaoust. Soc. Am. 67(4) , 1186 (1980)

[2] W. A. Kuperman and F. Ingenito, J. Aaoust. Soc. Am. 67(6) , 1988 (1980)

[3] M. J. Buckingham, submitted to the J. Acoust. Soc. Am.

Copyright ©, Controller HMSO, London 1981

41
MATHEMATICAL FORMULATION OF THE IMPEDANCE FIELD METHOD.

APPLICATION TO THE NOISE OF THE CHANNEL OF FIELD EFFECT TRANSISTORS

J. P. Nougier, J.C. Vaissiere, D. Gasquet

Centre d'Etudes d ' Elect ronique des Solides '^^^^ Universite des Sciences

et Techniques du Languedoc, 34060 Montpellier Cedex, France.

1 . MATHEMATICAL FORMULATION OF THE IMPEDANCE FIELD :

The impedance field method of Shockley et al. Pi ]

, for one dimensional devices, gives

the noise voltage spectral density Sy at the terminals of the device, as a sum of the

weighted contributions of the noise sources K(x) of the different slices at point x, of cross section A(x) :

S = \ A(x') K(x') I VZ(x') 1^ dx' (1)

^0

K(x') being supposed to be known, the critical step is the calculation of the impedance

field VZ(x'). One method for determining it, suitable when the basic parameter is the local

electric field E(x), is the transfer impedance method 1^2"] . The purpose of this paper is to develop a method suitable when the basic parameter is the local voltage V(x) , and to apply

it for modelling the noise of field effect transistors (FET)

.

Let us consider a device (Fig. 1) with a d.c. bias applied to the electrodes M and N. The noise is studied between the electrodes (or probes) M' and N', which may be identical

with, or different from^M and N. Let us apply, between M' and N', a small a.c. voltage, at frequency oj , superimposed to the d.c. one between M and N. This results in a set of a.c. equipotential surfaces, and a.c. current lines orthogonal to them (see Fig. 1). Let us con- sider such a line, labelled P, of total length L between M' and N', and an equipotential

surface S crossing P at point of abscissa x along P. The value 6V(x) exp icot of

the a.c. potential at point x is related to the a.c. current 6l(x) exp itot across S through a relation obtained by linearizing the conduction equations around the bias point.

One thus gets

£&ViK) = 6l(x) (2)

where ^ is a linear operator. Let 2(x,x',(ji3) be the Green function of j6 , defined as : £ Z (x,x' ,L0) = 6(x - x') (3)

where 6(x-x') is the Dirac function. The a.c. voltages 6V(x) at x and 6V(L) between M' and N' are :

6V(x) 2(x,x',(ja) 6l(x') dx' (4)

Jo

6V(L) ( Z(L,x',to) 6l(x') dx' V (5) Jo

1) Laboratoire associe au C.N.R.S. LA 21 and Greco Microondes.

2) This work was partially supported by D.G.R.S.T. (contract n° 79.7.0726) and G.CIS.

42
Fig. 1 : a.c. equipotential Fig. 2 : Impedance field method

surfaces and current lines

Let us introduce (Fig. 2) a current 6l(x') null everywhere except in the slice Ax where its value is AI : this results in a voltage AV at the probes M' and N', and the impedance field [l ] is VZ(x,w) given by VZ(x,lo) Ax = AV/AI , eq. (5) shows that

VZ(x',co) = Z(L, x',oj) (6)

Of course the impedance of the device between M' and N' is given by :

Z(a)) = 3 (L, x'.oo) dx' " (7)

J 0

The device can be considered as being one dimensional between M' and N', when the cur- vature radius of the a.c. equipotential surfaces is large compared with the transverse dimensions of the device, that is when the areas of the equipotential surfaces are approximate

ly equal to the cross sections of the device. Then the noise source term is constant every

where in the slice dx, thus depends only on x, so that eq . (1) can be used.

2. IMPEDANCE FIELD OF FIELD EFFECT TRANSISTORS :

The noise of the channel of FETs was recently calculated by Van Vliet f]3] , using the

transfer impedance method [2] , at low drain bias when the mobility y is constant. However,

at higher drain bias, hot carriers are involved and y = ij(E). The theory of section 1 will

be used in order to get the impedance field. Here M' = M is the source electrode, and N'=N

is the drain electrode. We suppose that the frequency is low enough so that no leakage drain

current occurs through the gate-channel capacitance. Furthermore the free electron density

will be supposed to be constant, equal to the impurity concentration Nj) (this is no more true in the saturation region of the I~Vp characteristics). The conduction equation then writes :

I = q v(E) A [V(x)] . (8)

This gives, for a.c. values (the subscript 0 means the d.c. values), using fiE =-d6V/dx :

-q A(V )(dv/dE) dfiV/dx + Do q N„ v(E ) (dA/dV) 6V = 61 . (9) o Do o The Green function £(x,x') is readily obtained (the integration constant vanishes

since 5V(x=0) HO implies through eq. (4) that 2 (x=0, x') = 0) :

H (x - x')

vIe (u)l (dA/dV) du

^ o o ,u £ (x.x') = V r . ^'^^

q a[v^(x')] (dv/dE)^^^, J ^, (dv/dE)^^^ a[(V^(u)]

H(x-x') is the step function. By setting x=L in eq . (10), one gets VZ (x ' ) =2 (x=L ,x ' ) . Generally speaking, v(E) has no analytical form, so that VZ (x ' ) is obtained numerically, thus

getting the impedance and the noise of the FET (provided the source term is known) . In the

ohmic range (low drain bias), when v(E) = Vq E, eq. (10) gives the expression obtained by

Van Vliet [3 ]

: VZ(x') = {q N^^y^ a|v(L)]}-1.

43
3. NOISE OF THE CHANNEL OF FETs

In order to get further information on the noise of FETs, we approximate v(E) using

an hyperbolic law f4j :

v(a) = - e/[i - (E/E^)J (11)

The signs choosen in eqs. (8) and (11) correspond to a n-type channel FET, with the source

at X = 0, the drain at x = L >0, q > 0, V(x) > 0, E(x) < 0, v[e(x)] > 0 and I > 0, E > 0. Equation (11), carried into eq. (8), gives then, for the d.c. and a.c. values:

I (1 - E /E ) = - q N n E A

o o c ^ D'^o o o ^_V^_V^/dA]

1-E (x)/E ^'^W.

o c

q N u A fv (x)l - I /E d6V(x)

6V(x) + = oI(x)

o,x 1 - E (x)/E dx

o c In the same way as in section 2, one gets the Green function.

[l - E (x')/E ]h(x-x')

Z(x,x') = exp

q N^y a[v (x')]- I /E ^Do'-o oc

q N U E (u) (dA/dV) Poo o ,u

q N U aFv (u)]- I /E ^ D o ^ o oc

du

(12)

(13)

The integral in eq. (14) can be carried out using V as a new variable: E = - d V /dx : o o o

Z (x,x')

1 - E (x')/E

o c

q N U AfV (x)]- I /E Do ^ <^ ' r, r H(x - x')

o o c Finally, by setting x = L, and taking into account eq. (12), one gets

VZ(x') =U (L)/ll[l - E (x')/E ] L o / oj'- o c-" (15)

The impedance Z is readily obtained. Carrying eq. (15) into eq. (1) and dividing by | Z | gives the noise current Sj, which is more meaningful than Sy because Eo(L) is not well

known for actual geometries. This gives, where V(L) is the drain voltage :

^I = L-2

1 + V (L) o E L

c

-2 L r K(x')

E (x-)

o A(x') dx' (16)

When E^ tends towards infinity, eq. (16) gives back the result of Van Vliet [3]. Equation (16) shows that the noise current is the sum of the contributions of the different

slides of the FET, the contribution of the slice at abscissa x' being proportional to its volume A(x')dx', to the noise source in it K(x'), and depending on the field through

the quantity [1 - Eq (x' ) /E^, ] ^ . Note that this theory includes the channel as well as then

regions between the source and the gate and between the drain and the gate.

The noise can also be expressed using the local noise temperature Tj^(x'), related to K(x') through the following equation ] , where the minus sign takes into account the opposite directions of v and E, and kg is the Boltzmann constant :

K(x') = - 4 k^ q N T (x')(dv/dE) = 4 k^ q N^y T (x')/[] - E (x')/E 1^ BUn B Don/^ o c' This relation carried into eq. (16) gives

L [l + V(L)/E l]' T (x') A(x') dx' n

(17)

This formula shows that, at low drain bias, the contribution, to the noise, of the channel

itself gives, since A(x') = A = constant, T^(x')-»-T and V(L) ^- 0 : S-|. = 4 kgT I/V.

a) Diffusion noise :

For diffusion noise^ T^ increases, more or less linearly with increasing E []6], and

44
therefore is maximum at the end of the channel, but the cross section A(x') is then minimum.

As a consequence, it can be expected that the contributions of all the slices of the FET are

of comparable importance.lt has been noticed that Si°^l/V at low bias, that is Sj - constant; at higher bias the increase of the noise in the channel versus V(L) in eq. (17) is counter balanced by the factor V(L)/Ej,L, so that Sj does not vary much.

b) Hot carrier generation-recombination (G-R) noise : in some devices, at low temperature,

all the impurities are not thermally ionized. Under the effect of high field, the neutral

impurities are ionized, via Poole Frenkel [?] or impact ionization effect, thus leading to an additionnal noise of G-R type, the parameters of which depend on E. Then, at the end of the

channel, T^j increases steeply, so that the noise is mainly due to the contribution of this

part of the FET, the total noise thus being much higher (40 to 100 times) than expected [(>~\. c) Thermal G.R. noise : For G.R. noise involving the conduction band and one impurity level, one

has K(x') = 4 q^ n(x') v2(x') aj/ (\ + uji^t^) . This expression, carried into eq. (16) gives,

taking into account eq. (11).

4 q^ N y

^ S^TM = -n ^-^ . 1 E (x') A(x') dx' (18) L fl +V(L)/E l]M 1 + 0)^2

Now the quasi one dimensional Poisson's equation writes : 9[e(x) A(x)]/ 9x = q A(x) [Nj^ - n(x)J/c . (19)

The hypothesis n(x) = N implies E(x) A(x) = constant = E(o) A(o) . Since the electric

field at the source electrode is weak, eq. (12) gives - Iq = q NoyQE(o)A(o)=qNQyQE (x)A(x)

.

This relation carried into eq. (18) gives : - 4 q y I ^ o o / ax

L^[l + V(L)/E l] 2 J

1 + a)V

E(x') dx' (20)

For hot carrier G.R. noise, a and t depend on E, thus on x' (see section 3b above). As

a consequence, the frequency spectrum is a sum of wei ghted 1/(1 + co2^2) curves, and thus

does not vary as 1/(1 + co^t^) : the noise spectrum, even for a single impurity level,

behaves as the G.R noise of an infinite set of uncorrelated levels lying in a sub-band with

a distribution of relaxation times.

For thermal G.R noise, a and t are constants, and eq. (20) writes :

4 q y^

L^[l + V(L)/E l]'^ (1 + u)^T^)

'lGR= 2r, 1.,...2 ^'2

2. ^o^(^>

^ (21)

Therefore, at low bias, such as V(L)/E L<<1, the thermal G.R noise is proportional to the

d.c. power IqV(L) displayed in the FET. This holds until saturation for long FETs , but for

very short FETs one would expect a decrease of S at high drain bias.

2 2 d) 1 /f bulk noise : Then, K(x') = a^^ n(x') q v^(x')/f , where ttj^ is the Hooge constant.

Carried into eq. (16) in the same way as previously, one gets

(22)

q y a I V(L)

O ti O

I Bulk 1/f L fl + V(L)/E l1

c - One gets 1/f noise, proportional to loV(L) at least at low bias. This is also

the case in MES FETs for 1/f noise due to interface states between the epitaxial layer and

the wafer, but with a constant different from (provided that this constant is actually

independent on the electric field)

.

e) 1/f contact noise : when the 1/f noise takes place at the interface at the source electrode, one may write K(x') = An(x') q2 v2 (x ' ) 6 (x ' ) /f where 6(x') is the Dirac function.

This gives, carried into eq. (16) : 2 2 2 ^Nqy T^2...,. -, I D

^ o E (o) A(o) ^ A o L^[l + V(L)/E^l]^ ^ A(o) L^[l + V(L)/E^l]^

45
This noise behaves differently from the bulk 1/f noise, since the numerator involves Iq

instead of IqV(L). Hence the measurement of the 1/f noise current versus the drain bias

allows one to localize the 1/f noise source.

4. CONCLUSION : In this paper, we have settled a new formulation of the impedance field method, particularly well suited for the cases when the main parameter is the local potential. This method has been applied to the theoretical determination of the noise of field effect transistors, which may be JG FETs, MOS FETs or MES FETs. A general formula has been given[eq. (10)]

and, under some simplifying hypotheses, complete results have been carried out, leading to expressions very simple to use, and providing quite general features of the noise in a wide

range of operating conditions.

REFERENCES

[ 1 J W. Shockley, J. A. Copeland and R.P. James, in "Quantum theory of Atoms, Molecules

and the Solid State, P. Lowdin ed. , Acad. Press, New York, 1966.

[2 ] K.M. Van Vliet, A. Friedmann, R.J.J. Zijlstra, A. Gisolf and A. Van der Ziel,

J. Appl. Phys. 46^, 1804 (1975) and J. Appl. Phys. 46, 1814 (1975).

[ 3 ] K.M. Van Vliet, Sol. State Electron. 2_i» 233 (1979).

[4] F.N. Trof imenkof f , Proc. IEEE 53, 1765 (1965)

5 ^ J. P. Nougier, J. Appl. Phys., to be published.

[ 6 ] J. P. Nougier, in Physics of nonlinear transport in semiconductors, ed. by D.K.Ferry,

J.R. Barker, C. Jacoboni, Plenum Publi. Corp. (1980), p. 415-465.

I 7 ] A. Van der Ziel, R. Jindal, S.K. Kim, H. Park, J. P. Nougier, Solid State Electron.

22, 177 (1979).

46
TIME DOMAIN LARGE SIGNAL NOISE

MODELLING IN MICROWAVE OSCILLATORS

J.F. Sautereau - J. Graffeuil - J.c. Martin

Laboratoire d' Automat ique et d'Analyse des Systemes

Universite Paul Sabatier

Greco nicroondes - Centre National de la Recherche Scientifique

7 Ave. Roche, 31400 Toulouse, France

I INTRODUCTION

Noise in microwave oscillators has in the past been investigated mainly in the frequency

domain and the oscillator represented by an equivalent lumped circuit with the voltage or current noise source in the appropriate section of the circuit. The variations of all the

elements of the circuit versus frequency and ac voltage or current was then needed to derive the phase or amplitude noise performances of the oscillator.

Such a goal may be impossible to reach when many non linear elements are involved

in the equivalent circuit, such as in oscillators built with two-port devices.

We report here an alternative approach based in a time domain large signal analysis

which allows the obtaining of the up converted amplitude modulation (AM) and frequency

modulation (FM) noise as well as the frequency and output power. Consequences for the design of low noise oscillators are outlined.

II LARGE SIGNAL TIME DOMAIN APPROACH IN MICROWAVE OSCILLATOR NOISE

The active device is described by an appropriate network involving frequency and

instantaneous voltage dependent elements, frequency dependent and voltage independent

elements. This equivalent circuit is implemented into a time domain analysis program IMAG

III. A feedback circuit is then added to provide oscillations. The theoretical output wave

of the oscillator is consequently available versus time (figure 1). In the present work

two identical oscillators were simulated at the same time ; The only difference consists

of a low frequencv CLF) source (frequency f^j^) being added in one of the two equivalent

networks. Consequently the two oscillators deliver two output waves :

The first called V^q is noise free and corresponds to an noiseless oscillator. The

other, called VI 1 , is noisy and corresponds to an oscillator with a low frequency noise

source added. The LF noise source may be introduced anywhere in the oscillator network.

VIO and VI 1 can be expressed in the following way :

In most solid state oscillators FM noise near the carrier is many orders of magnitude

higher than AM noise. Then, it may be assumed that the difference between the two output

47
voltages (VIO noiseless, VI 1 noisy) is only due to a phase fluctuation induced by an LF noise

source

.

Thus

the phase fluctuation is given by

A is the amplitude maximum value of the noiseless output wave. In a strip of bandwith

i^f located at a frequency fAN^

^(^(f-)may be written : ^ (^ff-) = ^Cn C^^^ ^ J

As to calculate a supplementary network was included in the model to suppress the

O.C component and to remove the small parasitic amplitude modulation of the output waves.

The circuit used is shown in Fig (2) ; it is made up of a DC suppression network and

a diode limiter.

Ill RESULTS

Typical theoretical result for FM noise obtained with an X band GaAs FET oscillator

is given in Figure (3). fo is 10 GHx , f^^ is ^GWx. and the amplitude of the low frequency

ac generator located in the active region of the device is IO~'^A. This curve shows that

there is an up conversion of the LF signal to FM fluctuations.

We deduced from the noise measurements the ratio k^ between the FM noise of several

X band frafls FET oscillators in the 1 Hz. bandwith at 10^ Hz off the carrier and the spectral

intensity of the input noise generator measured at a frequency equal to .

Results are given in table 1 ; it must be concluded that the larger the low frequency

noise of the device, the noisier the oscillator.

In order to better understand this phenomenon we measured the increase between a c LF fluctuations of the gate voltage of an oscillator and the FM fluctuation resulting

when an a c LF signal of frequency f .„ was applied in conjunction with the DC supply on the

48
FET gate.

Frequency deviations were measured by a carrier null method on a microwave spectrum

analyzer. Aratio k between the a c LF signal amplitude and FM fluctuations was then deduced;

this ratio k was compared both to the theoretical one k^ deduced from the large signal time

domain simulation and to the experimental one kl deduced from the noise measurements. Results

are given in table 2 for an x band G-oRfYET oscillator using a RAYTHEON LNC 832 D device in

chip presentation. Oscillation frequency is 10.2 6H2 . Theoretical time-domain simulation was limited by computation time to frequencies higher than 100 MHz^ and the experimental procedure did not allow operations at frequencies greater than 30 MHz off the carrier.

However, an extrapolation of the measurements to 1 00 MH z gives a good agreement between the three coefficients. This extrapolation is possible because the amplitude and the

frequency of the oscillator output wave settle Lhe value of the network elements. It was verified that ratios k and kl were constant for frequencies near the carrier (f < Bt^MHaJ^

CONCLUSION

It was found that the FM noise of a FET oscillator depends both on the low frequency

input noise resistance of the device and on the up conversion of this low frequency noise.

The up conversion coefficient was measured and also theoretically investigated using a new

large signal time-domain approach. We believe that the up conversion coefficient could be

decreased with a proper choice of the oscillator feedback circuit and of the device biasing

conditions. We also found that checking the low frequency noise and rejecting noisier LF devices is an important condition in obtaining good FM noise performances.

This work was supported by DAII-France.

1 J.F. SAUTERFAU - J. GRAFFEUIL - K. TANTRARONGROJ - P. ROJSEL. High efficiency GaAs

schottky-barrier gate FET oscillator.

Electronics letters 19 th june 1980 volb n° 13 p 490-491

2 J.F. SAUTEREAU - J. GRAFFEUIL - K. TANTRARONGROJ - P. ROSSEL. X Band GaAs FET oscillator large signal Design. 10 th European Microwave conference. (8-11 th) Sept 80.

VARSAW.

FIGURE 1 : MICROWAVE OSCILLATOR LARGE SIGNAL OUTPUT WAVE.

49
R2

11

'16

DC SUPPRESSION NETWORK

DIODE LIMITER

FIGURE 2 : OSCILLATOR NOISE ANALYSIS NETWORK

J.fk).

i.2»..£-»5r

^

\ 0. £1 tE-OS^

4.5 V*

FIGURE 3 : THEORETICAL PM NOISE OF X BAND GaAs FET OSCILLATOR,

kl

3.B lo' 2 10^ 3.5 10^

TABLE Z : COMPARfsON OF THE UP-CONVERSION COEFFICIENT

(^: theoretical : from noise measurements,

k: from carrier null method) FOR AN X-BAND

RAYTHEON 832D GaAs FET OSCILLATOR

[2)Device ^^vg' in 1 HzBw

(V/ Vli?)

^L.F NOISE

^^'^^in 1 HzBw iHz/VHz)

fM NOISE

Up-conversio

coeff icien

G 7. 10'^ 25 3. 5 10^

A 4.5 lO''^ 45 10 .10^

D 1.5 lO"^ 170 11. 3.10^

C 5.5 10"^ 100 3. 6.10^

D 1. 10"^ 160 16 .10^

F 2. 10'^ 80 4. 10^

TABLE 1 : COMPARISON OF THE UP-CONVERSION COEFFICIENT DEDUCED FROM L.F AND F.M

NOISE MEASUREMENTS ON SEVERAL X-BAND GaAs FET OSCILLATORS

50
SPECTRAL SUPERPOSITION AND IRREVERSIBILITY

Alle A.Walma

Centre d 'Etudes d ' Electronique des Solides,associe au CNRS,

Universite des Sciences et Techniques du Languedoc,

34060 Montpellier-Cedex, France.

INTRODUCTION

Irreversibility is deeply rooted in daily experience since work can only be done whenever two systems are not in thermal equilibrium with each other .The reestablishment of

equilibrium is closely related to fluctuations on a microscopic level. For a more general

theory of non-equilibrium Statistical Mechanics there are two lines of attack; (i)The Boltzmann

and Pauli equation in classical and quantum mechanical dynamics respectively , and (ii)The timecorrelation method or linear response theory. Their range of validity remains unclear however.

Generally speaking, satisfactory theories exist only in those cases where a macroscopic

characterisation is sufficient (Optics , Classical Thermodynamics, Hydrodynamics etc.) .When this

is not allowed, our knowledge is not satisfactory (Quantum Mechanics , Nuclear Physics , Solid

State Physics etc.). It would then be interesting to know at what degree of coarse graining

of the state variables and time the superposition of Lorentzian spectra is allowed (1/ t idea).

The least we can do is to recognise certain strong concepts such as the Einstein relation

or the Fluctuation Dissipation theorem. In this case a conceptional difficulty arises with

respect to existing 1/f treatments.

It is suggested that in the single variable case the Fluctuation Dissipation theorem

excludes the existence of 1/f noise in the thermodynamic limit. A rigorous proof is still

lacking, but some evidence in favour of this goes as follows. The Equipar tition Theorem requires

Ergodicity and Einstein's interpretation of the Boltzmann principle shows the effectiveness

of the central limit theorem. Since it can be linearised in the thermodynamic limit, the relevant

variable can be modelled by an Ornstein-Uhlenbeck process | l| and therefore does not yield

1/f noise (exponential relaxation) .Going a step further, one might ask whether Kolmogorovte

axiomatic description of an experiment can explain 1/f noise in the one dimensional case

for homogeneous processes, or equivalently, whether an isolated and weakly interacting (van Hove

limit) system does incorporate 1/f noise. We think that it does not, since a Master Equation

expansion |2 | yields in the thermodynamic limit a (non) homogeneous Ornstein-Uhlenbeck process.

Markov systems tend to propagate their equilibrium distribution in time and converge to some Fokker-Planck description .This becomes easily a representative Ornstein-Uhlenbeck process

for times that are relevant to a 1/f spectrum.

The 1/f theories in physical devices all incorporate the Fluctuation-Dissipation idea.

This means that on their way to the desired spectrum some "nonthermodynamic trick"must have

been used. This is to our opinion the superposition, averaging, or integration of the basic

spectra (l/xidea).It is therefore unrealistic since:

(i) The spectrum as a function of a random variable implies a fluctuating variance, which is

incompatible with the ergodicity requirement of the Fluctuation-Dissipation theorem.

(ii) It has been shown in | 3

| that the variable under concern becomes non-Mar kovian and

therefore does not satisfy the Onsager-Machlup theory of Irreversible Thermodynamics.

Roughly speaking, noise theory in physical devices as it stands today is based upon the

Ornstein-Uhlenbeck process and therefore of a phenomenological character .The more microscopic

theories for irreversibility (Boltzmann, BBGKY hierarchy , linear response and generalised

Langevin,Pr igogine-Glansdorf f and van Hove's theory) all have problems with the stochastics

at an atomic scale. In this contribution it is therefore started with a macroscopic

character isation. An attempt is made to explain the superposition idea within the context of

the Onsager-Machlup theory by adding enough variables in order to make the process Markovian.

This coarse grained framework includes necessarily ( i) Seperation of timescales (ii) Molecular

Chaos (Boltzmann) (iii)Random Phase Approximation (Pauli) ( iiii) Ergodicity (Liouville) (iiiii)

Locality (Prigogine) (iiiiii)Timereversibility (Onsager).It remains to be seen whether these

approximations are allowed for a real explanation of 1/f noise. As for surface related theories

this can be listed as follows (i)Adiabacy of the free density ( ii) Boltzmann Approximation

( iii) Linear isation (iiii) Detailed Balance. They are a direct consequence of the superposition

principle | 3 | and of the abovementioned assumptions from Irreversible Therm.odynamics

.

51
THEORY

A large literature exists about the n-dimensional Ornstein-Uhlenbeck process. For physical

devices the standard texts are | 4

|

| 5 |. Assuming detailed balance, the spectrum can be written

as: S{f) = 4 Re { M + Doil } C (1)

where M is the relaxation,! the identity and C the covariance matrix. From the two and three

variable case onwards, the expressions become hopelessly complicated and usually the spectra

are therefore computed numer ically .The simple 1/t structure appearently does not exist. This

maybe the reason why 1/f theories use so oft ad-hoc assumptions as a distribution of relaxation times and independent trapping centers for instance.lt is shown in |3| that the

covariance matrix can be calculated explicitly .The variance of the i-th subsystem, denoted by

An^fCan be expressed into the grandcanonical variances of the other systems: —2 1 rest 1 1 1-1 1+1 k ' ^ ,-, ^ An. = =, ; , a, = f, (1-f, N, 2

1 o.+o (a,+...a. +o.+a. ,+...o,+o ) k k k k

1 rest ^ 1 1-1 1 1+1 k

'

where a is the grandcanonical variance of the system of interest (which is the free density

in our case) and f^^ the occupation of the k-th subsystem with N^^ being the total number of

states. On the basis of the particle constraint, the covariances are:

An. An, =

1 : o.+...a.

-o.o.

1 J

. , +0 . +a . , + . . .a, + a 1-1 1 1+1 k

(3)

Applying the adiabatic assumption a»a . ("G-R linearisation of SHR statistics") and working

with identical subsystems ("Uniform traps") with a =a =....=a : 12 o

2 _2 Oq _2

An.1013a ^ a , An. An. ^ - — , An - ko (4) o

The covariance matrix can now be written as:

The Phenomenological Relaxation Matrix is obtained in the usual way |4| p] ,and is based

upon the configuration which is shown on the right:

52
M =

a^N

^3^

{a^-a^)N, (a^-a^)N (a,-a,_2)N, {a^-a^_^) N,^a^N^+ (a^N)

(6)

where a^^ is the rate of transitions per particle from the k-th subsystem to the heatbath and

where N =N2=...=N^ implies identical subsystems .The interactions take place at the same energy "("Direct communication of conduction band with the oxide") .This assumption can be relaxed, which will be shown elsewhere.lt is the diagonal character of the covariance matrix

that allows an explicit result. The spectral density of the particles in the heatbath

becomes 1 3 I

:

^n'f^ = ^ % Jl

a^N + 0)

It will be clear that numerous subsystems and a suitable choice of a^^can account for the

desired 1/f type of spectrum. For an exponential rate dependency a pure 1/f spectrum is

obtained | 3

|, where the thermodynamically unrealistic l/x idea is avoided.

CONCLUSION

More important than the final result are the underlying conditions .They are often not

satisfied in surface related theor ies , which will be shown elsewhere. Concerning the origins

of 1/f noise, the classification of the superposition idea in terms of well established theories of Irreversible Thermodynamics maybe helpfull.lt shows that this principle has a phenomenological character and implies (i)A seperation of timescales ( ii) Macroscopic linearity

( iii) Microscopic additivity ,and ( iiii) Adiabacy .The theories based upon this idea offer an explanation in the coarse grained sense on the basis of averaged quantities and cannot be

considered as microscopic.

REFERENCES

111 L.Onsager ,S.Machlup,Phys.Rev.91,1505 (1953).

I2I N.G.v.Kampen,In:Stochastic processes in Nonequilibr ium Systems, Springer Verlag,1978.

|3| A.A.Walma,Revue Phys.Appl.l6, (1981)

.

|4| K.M.v.Vliet,J.R.Fasset,In:Fluctuation Phenomena in Solids ,Ch. 7 , Ac .Press , 1965

.

is] M.Lax,Rev.Mod.Phys.22,25 (1960).

53
CLASSICAL NOISE AND NON-LINEARITY

Alle A.Walma

Centre d'Etudes d ' Electronique des Solides ,associe' au CNRS.

Univer site' des Sciences et Techniques du Languedoc

34060 MONTPELLIER CEDEX, FRANCE

INTRODUCTION

Noise theory in Physical Devices |l||2| relies upon the Ornstein-Uhlenbeck process l3| . This includes Rice's method, the simple Langevin procedure, Burgess'variance theorem, G-R theory

and the use of Greens' functions.lt affords (i)Additive microscopic processes (ii)A separation

of timescales ( iii) Processes that are linear in their state variables or that can be

linearised, and (iiii)Time homogeneous phenomena in the majority of cases. The analysis is of a phenomenological character.

As far as the fluctuations are concerned , the nearest one can get to the axiomatic propositions of Kolmogorov |4| is by means of his differential equations , usually referred to as the Master Equation. No complete solution is known yet of these equations and one generally

introduces a locality assumption. The latter enables a complete solution of linear processes.

In the non-linear case an infinite hierarchy of moment equations arises, which has not been

solved yet. It is therefore looked for a suitable parameter that allows an expansion, by means

of which the hierarchy can be closed effectively . In this contribution the closure is carried

out through the large system assumption. A quasi exact solution is compared with the linearised

version of a simple non-linear problem.

THEORY

Our macrovar iable is the free density in a uniform semiconductor .The recombination

process can then be modelled by:

= a - bn (1)

where the rate a is a function of temperature and lattice properties and b of the thermal

velocity and recombination crossection .The linear noise procedures yield for the autocovariance |1 |

| 2 ] : _t

C(t) = An^ e

^

, T~^= , An^ = V| (2)

2 b

The question is now whether these linearised results are correct. The following assumption

is madeiThe macroscopic differential equation yields the evolution with respect to time of

the mean of the process in the thermodynamic limit.

We have in mind a homogeneous and stable Markov process that is stationary and not far

from equilibrium. At first sight, the above assumption seems to be unrealistic in view of eq (1) It is quite strong however in equilibrium statistical mechanics and phenomenological thermodynamics. Not only can it be seen as an Onsager relation, but it also constitutes the basis of

thermostatics.lt enables definite statements about thermodynamic variables because the mean

values can be identified with the deterministic ones. It also legalizes the molecular chaos

and random phase approximation in classical and quantum mechanical dynamics respectively.

This independent particle idea is possible on a macroscopic scale where the large system

assumption can be made. It is suggested in our case that the obtained result is exact in the

thermodynamic limit. Our confidence is based upon the systematic expansion of the Master

Equation from |5 | where it is shown that in the limit the most probable path is determined

by the deterministic law of evolution (pathological processes that do not satisfy the central

limit theorem are excluded)

.

The exact equilibrium statistics of our problem can be obtained from the general

equation of Markov processes |6| (see also for further references) .The generating function

of the random variable n: 54
P{s,t) = Z p (t) (3) n=o

satisfies:

2 ll

= a(s-l)P + b(l-s){|| + s^} , p = P(s,t) (4) 3s

In equilibrium:

bsP" + bp' - aP = O (5)

This yields:

I (z/s)

P(s,-) = — , z = 2A (6) b

13^(2)

with I and I, being modified bessel functions |7| : o 1 'I

1 \) CO ( —z ) I^{z) = (^z)'' E

' (7)

k! r (v+k+1)

Transforming back:

(|)"

P„(«') = 5 (8) (n!)^I (z) o This shows that the normal assumption of linear noise theories does not hold. With (2) it

becomes highly accurate however in the thermodynamic limit. This follows straightforwardly

from Stirling's approximation for n! and by means of the following expansions l?]:

I (z) = 7-^— {1 + + —- + } (9) ''^.z 8z 2.(8z)2

T (z) = ^=i=- {1 - - r-....} (10)

1 /JztT 8z ^ , ,„ > 2 21 (8z)

The exact mean is obtained from (6)

a ^'^^

o

and the variance:

—2 2 a ^1^^^

An = P"(l) + P' (1) - P'tl) = r il z } (12)

I (z) o These values reduce in the limit to those from eq (2)

.

For the timedependency the solution of eq (1) is identified with the average regression

upon the initial value n(o):

55
, -— , n o + /rtgh>/abt —n(o) /a b

^ n(t) '= /- —2 (13) + n{o)tghv^bt

For obtaining the autocovariance,this conditional mean is averaged over the exact equilibrium

distribution given by eq (8) .Denoting n(o) by x,one is thus interested in xf(x), which can be

written as:

C^(t) = xf(x) = xf(x) + -V^{xf{x)}" _ + -M^{xf(K)} _+ .... (14) x=x x=x

where it is expanded around the equilibrium value. The moments y , 1<=2 , 3 , . . . ,are taken from (6) The third %^j;m on the right can be neglected .This follows from a straightforward calculation

of {xf(x)} - and because: x=x

= 2n - 3nn + n (15)

The autocovar iance becomes finally: ,

O O 1 (I +I,A) o 1 with I =I,(z),I =1 (z> ,z=2/rfand A=tgh/abt.In the limit (z-^)

:

11 o o b

^ a 1/a, -2/abt -4/abt,

where x is replaced by n.This should be compared with the result from linear noise theory, as

is given by eq (2)

:

C(t) = ^ + i/^ e-^''^^^ (18) b 2 b

Note that eq (2) represents A n (t) An (t+x) , whereas eq (17) equals n (t) n (t+x) .For t=o,both (17)

t^nd (18) yield the same equilibrium value.

CONCLUSION

A diffusion limit of a stochastic process (as is given by G-R theory for instance) may be

accurate in the thermodynamic limit for a nonlinear process as far as its equilibrium

properties are concerned, but difficulties arise with respect to the timedependency .The rapid

vanishing of the influence of the higher moments means that a nonlinear stationary and

homogeneous Markov process is not a suitable candidate for 1/f noise. The appearance of a second relaxation in our quadratic case suggests a reinterpretation of SHR statistics in

physical devices.

REFERENCES

M.Lax,Rev.Mod.Phys.32, 25 (1960).

K.M.v.Vliet,J.R.Fasset,In:Fluctuation Phenomena in Solids,Acad. Press, 1965.

A.A.Walma,Revue Phys.Appl.15,1435 (1980).

A . Kolmogorov , Math .Ann

.

104 ,451 (1931).

N.G.v.Kampen,Can.J.Phys.2i/551 (1961) .

A.A.Walma,In:Noise in Physical Systems, Spr inger Verlag,1978.

7 I M.Abramowitz, J. A. Stegun, Handbook of Mathematical Functions, Dover Publ.,1965.

56
FLUCTUATIONS AND TRANSITION RATES IN TIME DEPENDENT DISSIPATIVE STEADY STATES

David J. Bergman and Eshel Ben-Jacob .

Department of Physics and Astronomy

Tel -Aviv University, 69978 Tel -Aviv, Israel

Zeev Schuss

Department of Applied Mathematics,

School of Mathematics, Tel -Aviv University 699T8 Tel-Aviv Israel

1. INTRODUCTION

It is well known that a particle moving in a potential well under the influence of random

forces exhibits fluctuations in its position x

and velocity x. Moreover, if the well is finite,

the particle will have a finite probability per

unit time for leaving the well. An old paper by

Kramers^ discusses the one dimensional version

of this problem in detail for the case where the

distribution function in phase space p(x,x,t)

satisfies a Fokker-Planck equation, and derives

simple expressions for the transition rate in various limits. (The fluctuations about the

equilibrium state are given, as usual, by the

Boltzmann distribution. ) The calculation is achieved by first transforming to a Smoluchowskitype diffusion equation, for which a stationary

diffusion solution is then sought. This approach,

which is very useful for discussing transitions

between two equilibrium states, cannot be applied

to the equally interesting probelm of transitions

involving one or more nonequilibrium steady states Moreover, in the case of steady states with low

dissipation, there has not existed, until now, a general prescription even for calculating the

spontaneous fluctuations about the steady state.

In Section 2 we propose a rather general approach

to such problems that enables us to obtain very

simple expressions for both the fluctuations and the transition rates of nonequilibrium steady

states. In Section 3 we apply our method to the case of a hysteretic Josephson junction^.

In Section h we present a general formal

approach to the problem based on the results of

ref.3. An extension of this approach can be

used to discuss potential wells in higher dimensions.

2. THE METHOD OF EFFECTIVE ACTION-ANGLE VARIABLES

We start from the usual Fokker-Planck

equation for a particle of unit mass

- 1>(f OX T^| ) :i)

where G is the dissipation coefficient and T is the temperature in energy units. The force Fq

acting on the particle may depend explicitly on

time. We will focus our attention on the cases

where, in the absence of noise, the equations of

motion have a time-dependent, steady state solution in which at least x is periodic. Examples of

motion of this type are: a) A particle moving

in a potential well with a barrier of finite

height, under the influence of friction as well

as a periodic driving force"*, b) A hysteretic

Josephson junction with a constant driving cur

rent (see fig.l).

The main idea of our approach is to find a family of steady-state trajectories in phase

space and use a trajectory index A as the

appropriate slow variable of the problem. The

fast variable n is taken to be the position along

such a trajectory, measured in units that are proportional to the time. One way to define

such a trajectory index is by

(2)

where the integration is over one period of x. Although this definition appears identical to the

action variable used by Kramers in his discussion

of low dissipation equilibrium states, we stress

that in our case the variable A has a completely

different meaning. We now assume, following

Kramers, that when p is defined in terms of A,ri, it is independent of n. In order to do away with

the streaming terms in eq.(l), we first rewrite

it as follows

TJX +

(3)

57

where AF=AF(x,x,t) is chosen so that the point x,x lies upon the steady state trajectory corres- ponding to the force Fq+AF. If this equation is now averaged over x and x for a fixed trajectory,

i.e., for a fixed value of A, the streaming terms

(i.e., the first two terms on the r.h.s. of eq.(3))

vanish. If we know the steady state solutions

of the noise-free equation of motion, we can in principle calculate A(x,x) as well as AF, and use

these results to calculate the average of the

last two terms on the r.h.s. of eq.(3) over a
fixed trajectory. The final result is then a Smoluchowski-type equation for p(A,t ) ,froin which

lx)th the fluctuations and the transition rates

in the steady state can be deduced. The example

of a particle moving in a potential well in the presence of an external oscillating force is discussed in ref .^4, in the next section we discuss the example of a hysteretic Josephson junction.

3. HYSTERETIC JOSEPHSOW JUNCTION

Assuming the RSJ model for the junction,

the equation of motion in the absence of noise

is

The Fokker-Planck equation for this problem

has the following form

nit oe ^ °

(5)

Restricting our discussion to the case G<<I, we use the approximate form of the steady-state

trajectory.

9 + ^6+ SluO = I

Gr T

^ '

in order to get

(6)

Here 0 is the phase difference across the junction, R and C are the resistance and capacitance

of the junction, I is the externally driven

current in units of the critical current Ij, and time is measured in units of u)j-'-,the reciprocal

Josephson plasma frequency. In the underdamped

regime G<1, the junction is hysteretic. The I-V characteristic consists of two overlapping

branches as shown in fig. 2. The typical motion in phase space for

Ij]j^jj<I<l is shown in Fig.l(ljujjj is defined in

fig. 2): There is a non-equilibrium, time-dependent,

dissipative, steady-state trajectory correspondirg

to the finite voltage branch, as well as a set of

metastable equilibrium states that correspond to the zero voltage branch and that are time independent and the different domains of attraction

are separated by separatrices'' (see fig.l). At

nonzero temperature, thermal noise will cause

transitions from the steady state to one of the

equilibrium states and vice versa. By contrast, when I^^Imin only the equilibrium

states are possible, and thermal transitions take place between neighboring states. Finally,when

I>1 only the steady state is possible, and thermal noise will only cause fluctuations about that

state, without any transitions to a different,

long lived state.

Fig. 1. States and trajectories in phase space

for G=0.2, Io=0.5- The points A and B represent equilibrium (V=0) states. The line d rep- resents the steady-state.

The lines a and b are the separatrices of

the states A and B, respectively. The line

c is both the steady-state traje'-tory and part

of the separatrix corresponding to the current

[^ji^S0.26, {see fig. 2). Below that line there

are no steady-state trajectories.

^

IIT 1_

We can therefore choose I to be the slow variable,

while the fast variable is the position along the

trajectory, which is roughly the same thing as the

phase 0. Rewriting the Fokker-Planck equation as follows

(8)

We average over 0 for fixed I=Io+AI, assuming

that p depends only on I. In this way we get

V/RI,

50 0.2 04 06 08 10 12

Fig. 2. The I-V characteristics with (solid

line) and without (dashed line) thermal noise

for G=0.5, r =0.1. For comparison, we plot also the I-V characteristic calculated in Ref. 9 from numerical solutions of the Lang- evin equation (dotted line). 58
the following equation

new.

7) €

^ '01/

(9)

predict the average voltage across the Junction.

In Fig, 2 we show a typical computed I-V characte)?- istic in the presence of noise, which is in

excellent quantitative agreement with the numerical simulations of ref.9.

h. KPFECTIVE ENERGY APPROACH

which is just the Smoluchowski equation for a particle moving in a harmonic oscillator

potential ^[Al]^, with a friction coefficient 1/G, at a temperature G^T. This description is valid in the region of

phase space that includes the Iq steady state

traj ectory,but does not extend all the way to the

separatrices(see fig.l). Since Eq.(9) is valid

in the vicinity of the Iq trajectory, we can obtain from it the probability distribution pgg for thermal fluctuations about the steady state.

Al'

(10)

As mentioned earlier there are cases where

a more general approach to the problem is needed.

This, for example, is the case of the Josephson

junction in which the distance in phase space between the T-xain 'brajectory and the separatrix is large.

The method we present in this section is based on a generalization of the form of the

Boltzmann distribution about equilibrium states,

which is given by

e ilk)

In this approach, we define an effective energy

Eg such that the stationary distribution about

the steady state is given by an infinite series

The fluctuations in the voltage AV across the

junction are related to AI according to

(11)

where Rp is the dynamic resistance of the junction

(Rd=R for I>> G).^ Thus the distribution function

of Eq.(lO)can be compared with a measurable

quantity. The transition rate from the steady

state to points on the I^ifj trajectory can be calculated by following the procedure of Kramers,

and this can be shown to yield a good approximation to the transition rate l/Tgg out of the

steady state provided the Imin trajectory is close

enough to the separatrix^. The result is

(12)

where AU is an effective potential barrier height

given by

(13)

- £& do

P

_ e~ 'E- ST' (15)

Using the Fokker-Planck equation, we obtain,

at low temperatures, the following equations for Eg and pg.

(l6a)

1^ - O

X 21 + G-^ +

(i6b) = O

V/e require Eg=o on the trajectory of the steady

state solution and Eg>0 away from it. On this

trajectory we define Pq to be the non trivial

periodic solution of eq.(l6b) with the periodicity of the trajectory.

It can be shown^'' that the transition rate

is given by

With the aid of the lifetime igg and the

lifetime of the static solution, which can also

be calculated using Kramers approach, we can

59
(17:

where AEg is the minimum of on the separatrix

at the lowest saddlepoint, and the effective

frequency fle is given by fie=wjii<a)>/27r(jor[i, where

<io> is the frequency of the oscillations in Eg

normal to the steady state trajectory, averaged

with respect to over this trajectory. The

quantities ojjj and ooip are the frequencies of the

normal modes of Eg at the lowest saddlepoint in

the directions perpendicular and parallel to the

separatrix, respectively. (A more general case

is discussed in refs. 3,10.)

It can be shown that when I/G >> 1 in the

Josephson junction problem. Eg is given approximately by AI^/2G^. This is consistent with the

results obtained in section 3. If G is large

then Eg^, the energy. This is consistent with

the usual Smoluchowski approximation , which is the right one for this limit

.

REFERENCES

* Partially supported by the US-Israel

Binational Science Foxmdation BSF.

** Partially supported by A.F.O.S.R. AFOSR78-

362OA.

[1] H. A. Kramers, Physica 7^,284(1940).

[ 2 ] E . Ben- Jacob , D . J . Bergman, Z . Schuss, Submitted

to Phys. Rev. Lett.

[3] B. J. Mathkowsky.Z. Schuss, SIAM J.Appl. Math.

(in press)

[4 ] E. Ben- Jacob, D .J .Bergman, B .Carmely ,A.Nitzan.

This proceedings

[5] D.E.McCumber, J.Appl .Phys .39., 311 3 (1968).

[6] W. C.Stewart, Appl. Phys. Lett .12, 277 (1968).

[7 ] M.Levi , F .C .Hoppensteadt ,W.L .Miranker, Quart

.

Appl .Math, 26^ 167 (1978).

[8] A.J .Dahm, A .Dennenstein,D .N .Langenberg,

W.H.Parker ,D .Rgovin,D . J .Scalapino, Phys .Rev

.

Lett.^,1416(1969) .

[9] J . Kur ki j arvi , V . Ambegaokar , Phys .Lett. 31A , 314 (1970)

.

[10] "Theory and Applications of Stochastic

Differential Equations "by Z . Schuss, John

Wiley and Sons NY (1980)

.
II. DEVICES

i 61

A NEW APPROACH TO MODEL CHANNEL

"THERMAL" NOISE OF THE JG. FET CONSIDERED

AS AN ACTIVE R-C LINE

D. Sodini, D. Rigaud

Centre d'Etudes d'Electronique des Solides, associe au C.N.R.S.

Universite des Sciences et Techniques du Languedoc

34060 MONTPELLIER CEDEX, FRANCE

1 . INTRODUCTION

The theory of the "thermal" noise in the channel of JG.FET was first developed by

A. Van der Ziel | 1

| assuming that ohmic conditions hold inside the channel and that the

hypotheses of Shockley are valid. Later, introducing hot carrier phenomena, new noise expres

sions have been carried out |2,3] but till now, in particular at 77 K, no satisfactory theory

for the device noise calculation has been convincing.

The purpose of this paper is not to determine if the origin of the increase of noise

at low temperature is due to hot carrier phenomena or can be produced by other noise sources such as generation-recombination | 4 | but to show how it is possible to introduce in a computed model the experimental Si bulk data for the noise calculation of an active device

as JG.FET 's.

2. ANALYSIS OF THE NOISE MODEL

The channel noise of an epitaxial channel and diffused gate FET has been calculated

from a quasi-unidimensional numerical analysis.

The model takes into account the electrical data (mobility and diffusion laws versus electric field | 5 | | 6 |) obtained on Si bulk material and the technological data given by

the manufacturer (Thomson-Sescosem, France) such as geometry, resistivity, diffusions...

This computer simulation is carried out by sampling in slice the whole channel of the FET in common gate configuration from the source to the drain contacts which first allows

us to obtain the static and dynamic behaviour of the device | 7 | . The equivalent circuit

of each slice considered as an elementary transistor is given on Fig. 1

.

c ,/ I s d I / Q

——

<

1—

n

g IkI w 1 K - 1 1 —

m '

' '

' __ (

Aw|k|

- c| k|

•

63
The noise of this elementary transistor is represented by the generator AW | K |. (The equivalent circuit needs one generator

since only g^ | K | is noisy). With the aid

of the transfer matrix of the elementary

transistors located between the K+ l^h cell

and the drain contact we obtain the output

noise contribution of the Kth slice : AWD K and AJD K

elementary generators

, Assuming that all the AW I K I are decorrelated, the output noise of the whole transistor is :

y AW^ K and

Is. U (Fig. 2. a)

Then the input noise generators Ej^ and can be obtained in the common sourc configuration (see (Fig. 2.b)) and we deduce with the admittance matrix the channel noise i^, and the induced gate noise

i (see (Fig.2.c)) :

c 2 1 n

i = y, , E + I g

•' 1 1 n n

(a)

Noiseless

transistor

(c)

Figure 2

3, EXPRESSION FOR THE ELEMENTARY NOISE VOLTAGE GENERATOR

The noise voltage spectral density ASy | K | of the K*"^ slice must be expressed as | 3

T I K I AS„ I K I = 4 k

Sdl^l

9lp

where g, I K I = -r;^ and T I K | = T (E I K I ) is the noise temperature which is

d 3V , ,7 ^ n '

' n '

' ds V =cte

different from the lattice temperature Tq, since electric field is high.

This expression differs from the one used by Van der Ziel in that : i) the lattice temperature Tq is replaced by the noise temperature T^.

ii) we use the differential conductance instead of the d.c. conductance,

iii) we include in the g^j calculation the mobility law given by Jacoboni et al .| 5 |.

The noise temperature Tn(E) of hot carriers in n-Si has been measured on homogeneous

bars at 300 K and 77 K | 3 |

| 8 | . From these measurements we have deduced two empirical

laws :

T (E) = T (1 + 1.72 10~^

I E \^'^^) for 300 K

n o II'

and

T (E) = T (1 + 3.66 lO"^ I E

|'*^^) no for 7; ' 64
4. RESULTS

The thermal channel noise has been measured at 100 kHz using a pulse technique (pulses

of 5 ms duration at repetition rate of 0.1 s ) in order to avoid heating effects of the

device.

Figure 3 shows a comparison between the experimental and computed data at 300 K and 77 K.

21 10

X /

T = 77 K

•Computed data

22

^ Experimental data (100 kHz)

10 A A —

T = 300 K

2 V^(V)

Figure 3

In the saturation range, the discrepancy with the experimental data may be due to the

failure of the unidimensional approach for modelling (gradual channel hypothesis). Furthermore at 77 K because of the ionization effect of phosphorus impurities, the characteristics

modelling is less accurate | 4 | and it must be remembered that the excess noise temperature

can increase by a factor of two according to the direction of measurement so that the agreement

of the computed curve is quite satisfactory.

This model allowed us also to compute the behaviour at high frequencies of the channel

noise, the induced gate noise, their correlation and the noise figure.

CONCLUSION

In spite of the roughness of the approximations used for modelling the I.V. characteristics and the noise temperature the agreement between experimental and computed data is

satisfactory. Especially at 77 K, the present theory gives results 30 % lower than the

experimental curve but compared with other noise calculations! 3 | this result is better. To

improve this model we need experimental data for the mobility and the noise temperature at very high electric field for usual doping levels . Nevertheless this paper gives a way for

65
modelling every one dimensional device where hot carriers play an important part.

ACKNOWLEDGMENT

The authors wish to thank SESCOSEM France for supplying the devices used in this study

and Prof. J. P. Nougier for many helpful suggestions.

REFERENCES

1

1 1 AA Van oer ^lei, rroc. i.k.ej. du p. louo loiz ^^i^dz^

1 / A Van der /,iei, b.b.fc. \h, p. jh/ jjU

1 3 J P. Nougier et al . , S.S.E. 2_1_, 133-138 (1978)

1 ^ A Van der Ziel et al . , S.S.E. 22, 177-179 (1979)

1 5 C Jacoboni et al . , S.S.E. 20, 77-89 (1977)

1 6 J P. Nougier et al.. Physical Review B, 11, 1497-1502 (1975)

1 7 D Sodini, S.S.E. to be published

1 8 M. Rolland, J. P. Nougier, Proc. 13'"'^ Int. Conf. Phys

.

Semicond

66
NOISE AND SCALING PROPERTIES OF

SUBTHRESHOLD MOSFETS

S.T. Liu

Honeywell Corporate Technology Center

Bloomington, MN 55420

J. S.T. Huang

Honeywell Solid State Division

Plymouth, MN 55441

INTRODUCTION

The high frequency noise in subthreshold (weakly inverted) MOSFET was recently reported

to be thermal noise given by Liu and van der Ziel [1]

S = 4kTR = 4kT V n 61 (1)

where a = 2/3, n* and 3 are parameters defined in the subthreshold drain current equation.

The drain current of an n-channel MOSFET in subthreshold region has been given by van

Overstraeten, et.al.[2]

l-exp m*BV

n*

. V < V ' G T

(2)

and C* is the depletion capacitance per unit area at

id

= 3())_/2 and 6 = q/kT.

S r The constants

n* and m* are capacitance ratios evaluated at the surface potential <() =• 3<i>^/2 and defined

as n* = (1 + C*/C + qN /C ) and m* = (1 + C*/C ) , where C is tie oxide capacitance

D ox ss ox D ox ox per unit area and N^^ the surface state density. Tne constant V* is the gate voltage at

<t>g = 3<^-p/2 and is the threshold voltage.

In this paper, we present both low frequency and high frequency noise data and

correlate the results with the parameters n* and m* defined in the current equation. We

also assess the scaling parameters which affect the performance of these subthreshold

MOSFETS.

EXPERIMENTAL RESULTS AND DISCUSSIONS

Figure 1 shows the I-V characteristics of an n-channel MOSFET in subthreshold regions

This device has an oxide thickness of 1100 ga te area of 2.56 X lO'^cm^ and threshold

implantation and the junction depths are about 0.35Mm. This device has a threshold voltage

of 0.95 V. In the exponential region, the parameter n* was evaluated to be 2.5 from the

slope of the I-V characteristics.

67
10.0

SAMPLE 382-2

• 0 8 I lO'^ cm^

0.2 0.4 0.6 0.8 10 1.2 Vg (VOLTS)

,P 10

Sample 382 - 2

Fig. 1 I-V^ of a subthreshold MOSFET F ig . 2 Noise resistance of a subthreshold MOSFET

Figure 2 shows the noise resistance R as a function of frequency for this device at a subthreshold current of 4.9 X 10"^ Ampere and a drain bias of 1 V. The result is fitted to

R = 8 .75 X 10° ^ n V in6 u n —r-—— + 9.06 X 10 ohms. (3)

From eq (1) the high frequency noise resistance is

2 n* 6 R =^ = 8.8X10 ohms

n 3 61

which is close to the observed value of 9.06 X 10 .

According to van der Ziel, the limiting 1/f noise is set by the dielectric loss in the

oxide and is given by [3]

R = tan 6 n ojC (4)

where C is the gate capacitance and tan 5 is the loss tangent of the oxide. If we take C and tan 6 as the effective gate capacitance and the effective loss tangent, we have

C WL

ox

(5)

2 —8 —2

where WL is the gate_^rea. For this device, WL = 256 (ym) , C^^ = 3.13 X 10 F cm , C* = 3.6 X 10 F cm . From 1/f noise data, we evaluate tan 6'^to be 2.4 X 10"^. The loss

tangent of 2.4 X 10"'^ is not unreasonable because the value of tan <5 is 0.7 X lO"'^ for

quartz [4] . Equation (4) can be rewritten in terms of m* as:

68
tan WL (4a)

where ^ = 3.9 is the dielectric constant of Si02. We see from eq (4a) that the limiting

1/f noise is proportional to t^^ and inversely proportional to the gate area WL, which is seen in samples with tan ^ <10~4. Further, it is related to the parameter m''= as m*/ (m*-l) . It should be pointed out that m* may be frequency dependent since the low frequency noise

data varies slower than 1/f in some devices. Another point is that the Ngg contribution

is absorbed in the tan 6 term. It means the trapping and detrapping in the interface would

give rise to the ac loss.

The dependence of the noise on gate area is shown in Fig. 3 for two sets of devices at

f = 40 Hz. One set of devices has an oxide thickness of 1100 % and the other 540 %.. It is

clear from the figure that the noise spectrum is inversely proportional to the gate area as expected, except for devices with small channel length. The noise in short channel devices

is observed to be higher than the long channel devices. This may be due to the high field

effect which was not investigated here. It is clear also from the figure that the noise in

thinner oxide devices is smaller than those devices with thicker oxide for the same gate

area.

10

> -12 ^ 10

10

lo = 20 nA

' t = / ( 0 0 A

10

10' w Ltm»?)

10

Fig. 3 Gate area dependence of 1/f noise

The effect of oxide thickness scaling on I-V characteristics is further examined in

Figure 4. In the exponential region the parameter n* is evaluated to be 2.5 for to^ = 1100

R and 1.9 for to^ = 540 R for the same threshold doping density. It is clear that scaling

down tox improves the parameter n*. We do not scale upward the doping density N^, because

n* and m* would increase through Cg, which scales slower than the square root of ratio.

Scaling up would certainly degrade the I-V (eq (2)) and the noise as given by eq (1) and

(4a). It is often necessary to scale up as channel length L is scaled down in order to avoid punch through [5]. From the figures, the parameters Vg arc vg = 0.584 V for tox = 1100 & and = 0.152 V for t^^ = 540 The threshold voltages are respectively 0.95 V

and 0.55 V. Therefore, we see (Vg -,Vq*) - (Vrj. - V^) . We can therefore maintain the

operating voltage by letting (Vg - Vq) = (Vrj. - V^) while maintaining the same subthreshold

current level.

69
CONCLUSION

It is clear from the 1/f noise results discussed that we should not scale down the gate

area or scale up doping density for linear circuit application where noise is an important

consideration. We can, however, scale down the oxide thickness to improve the noise

performance. In so doing we maintain the operating voltage difference as the difference of

the threshold voltages between the unsealed and scaled devices. Further, we have related

the high frequency noise to the parameter n* and the low frequency noise to the parameter

m*.

ACKNOWLEDGEMENT

We would like to thank Prof. A. van der Ziel of the University of Minnesota for a fruitful

discussion and Mr. M. Vogt for the measurement.

REFERENCES

[1] S.T. Liu and A. van der Ziel, Appl. phys. Lett. 37, 950 (1980).

[2] R.J. van Overstraeten , G.J. Declerck, and P. A. Muls, IEEE Trans. Electron. Devices

ED-22, 282 (1975).

[3] A van der Ziel, Solid St. Electron. 18, 1031 (1975).

[4] W. Espe, Werkstof f kunde der Hochvakuumtechnik, Band II, p463, VEB Deutscher Verlag der

Wissenshaf ten, Berlin (1960).

[5] R.W. Keyes, Proceedings of the IEEE 63, 740 (1975).

70
THE DEPENDENCE OF THE LOW FREQUENCY NOISE OF JFETs

ON DEVICE PARAMETERS AND OPERATING CONDITIONS

C.E. Cox and K. Kandiah

U.K. Atomic Energy Authority

Harwell, Didcot, Oxon

1. INTRODUCTION

The important physical mechanisms which generate noise in the drain current of JFETs

are fluctuations of carrier velocity and density in the channel and modulation of the

current by charge state fluctuations at defects situated in the depletion or Debye regions

adjoining the channel. The power spectrum of the noise is dependent on temperature and

device parameters. Papers on noise generally deal with individual mechanisms and sometimes

fall short of practical information relating to device parameters.

Recent work [1,2] on low frequency noise in JFETs has given a better understanding of

the responsible mechanisms. In this paper we expand those studies to cover variable

operating conditions. The emphasis is on the performance of 4-terminal silicon JFETs at

frequencies below a few MHz. In section 2 we briefly review current understanding of the

major noise sources. In the later sections we study the dependence of noise on temperature,

bias conditions and device gate width for a fixed manufacturing technology.

2. MAJOR NOISE SOURCES

Noise due to thermal agitation of the carriers in the channel is always present and is

well understood. It has a white noise spectrum and the equivalent noise resistance Rj^ in

series with the gate for a three terminal device is inversely proportional to the mutual

conductance g. In 4-terminal JFETs it will be more convenient to consider the drain current

noise 1^^ since the signal is applied only to one gate. We have

(I^^)^ = 4kTgr ' (1)

where I^^ is the thermal drain noise current, T is the temperature, k is Boltzmann's

constant and F is a constant.

Fluctuations of carrier density due to incomplete ionisation of the dopant atoms in the

channel [2] cause an increase in the noise at lower temperatures. In the case of phosphorous doped silicon the effect is observable at temperatures below 125K and is independent of

frequency for temperatures greater than 75K.

Low frequency noise due to defects present in the Debye region adjacent to the channel

gives noise peaks in certain temperature bands depending on the energy level of the defect.

This is usually the most significant source of low frequency noise. At temperatures

greater than about 220K low frequency noise can be generated by defects with near midband

energy levels even when they are situated in the fully depleted region between the gate

and the channel [4], is the total drain noise current due to defects.

3. TEMPERATURE AND SUBSTRATE BIAS DEPENDENCE AT CONSTANT DRAIN CURRENT

Figure 1 shows Ij^ at lOOkHz as a function of temperature at fixed drain current for

three very different substrate voltages. It is seen that above 125K, where Ij^^. is the major

component at this frequency, the noise is independent of substrate bias and temperature.

This behaviour is consistent with eq (1) since g is approximately inversely proportional to temperature and independent of substrate voltage. If the noise had been expressed as it

71
Temper Bturo(Degs.K) Temperature (Degs . K)

Fig 1 The dependence of noise current on Fig 2 The dependence of low frequency

temperature and substrate bias measured noise on temperature.

at lOOkHz. Device XSOl. 620jj.m gate width Device PSOl. 620)im gate width

would have shown both a temperature and substrate bias dependence. Below 125K the predicted

increase in noise due to carrier density fluctuations [3] is observed.

In figure 2 Ij^^j is seen to exhibit peaks as a function of temperature at the lower

frequencies, the temperature for peak noise varying with frequency. In most devices studied

three temperatures corresponding to noise peaks at lOHz are seen at about 85K, 170K and 270K.

The magnitude of the two higher temperature peaks has been shown to be strongly dependent on

the substrate bias [2]. The degree of substrate bias dependence of the peak at 85K varies

widely with the device. The magnitude of the noise in the valley regions falls at bias

voltages at which the noise at the peaks is very low. However the valley noise does not fall

below a level I^p' called the plateau noise level, which consists of a component with a 1/f

spectrum adding to the thermal noise. 1^^ increases smoothly by a few percent from lOOK to 300K but is seen to decrease at temperatures above BOOK.

Figure 3 illustrates the dependence of Ij^^ on substrate bias at a fixed temperature for

a device of 620|am gate width and shows the generation of noise peaks when a point defect is

positioned in the Debye region. For any given defect at a fixed drain current the magnitude

of the noise peak at the optimum activation temperature for each frequency has a 1/f

spectrum.

4. DRAIN CURRENT AND DRAIN VOLTAGE DEPENDENCE

The movement of a low frequency noise peak with respect to substrate bias as the drain

current is varied at constant temperature and drain voltage is shown in figure 3. The

defect generating this peak is situated in the Debye region nearest the top gate electrode

thus making the position of the peak dependent upon drain current. As expected, no such

movement with drain current is observed for a defect positioned in the other Debye region

nearest the substrate.

The magnitudes of I^f ^nd 'np ^^^ dependent upon drain current for fixed

temperature and drain voltage. It can be seen in figure 3 that the magnitude of 1^^^ at

lOHz saturates at some drain current. A similar saturation effect is observed for Ij^p, but

not for which continues to increase very slowly at high drain currents.

It is well known that 1^^^ is invariant at drain voltages at and above the pinch-off

voltage. Little or no dependence of 1^^^ on drain voltage is observed, as the movement of

the Debye region on the source side of the pinch-down region with drain voltage is

negligible.

72
XS01 ; Vd-4V ; F-75HZ J T- 1 84K

SUBSTRATE VOLTS

125 175 225 275 TofnperBture(Dea8.K)

325

Fig 3 The dependence of low frequency

noise on substrate bias and drain current.

Device XSOl. 620)a.m gate width

Fig 4 Dependence of low frequency noise on temperature for a large gate width device.

Device yHA84. 13600)ain gate width

5. DISCUSSION OF THE OBSERVED BEHAVIOUR AND DEPENDENCE UPON GATE WIDTH

Preliminary calculations [5] indicate that the effective modulation of the drain current

by a point defect in the Debye region is approximately proportional to the local mean drift

velocity u. It is believed that noise peaks are only observed from defects near the pinchdown region. A statistical analysis of the noise peak magnitudes in a large number of

devices at lOHz and 167K shows similar distributions for devices of 620(am and 1800)im gate

width when operated with the same channel current density. It is found that the number of

peaks observed per unit noise magnitude ^nterval is independent of the magnitude of the peak

(expressed as Ij^) up to about 200pA(Hz) . A very small number of peaks are observed at higher magnitudes and are probably the result of the chance coincidence at some substrate

bias of two or more peaks.

At sufficiently small drain currents and fixed frequency and temperature the magnitude

of a noise peak generated by a defect near the pinch-down region is seen to decrease with

decreasing drain current (figure 2) . It is believed that at very low drain currents the

width of the channel is so small that insufficient charge is available in the adjacent

region of the channel for maximum modulation of the drain current to take place. A saturation of the peak magnitude is seen at higher drain currents as more than adequate charge is

available. This effect is also observed for I^p*

The noise mechanism responsible for the generation of 1^^ is not fully understood but

is thought to be connected with surface states near the pinch-down region. When devices are operated at similar channel current densities it is found that the maximum magnitude of

due to a single defect near the pinch-down region is independent of gate width, and that Ij^^

and Ij^p (when expressed as an equivalent saturated diode current) are approximately proportional to the gate width. A statistical analysis of noise peaks shows that the number of

defects observed per device (which is about six for a device of 620|am gate width) is also

approximately proportional to the gate width. Figure 4 illustrates the dependence of I^^ on temperature at low frequencies for a device of 13600|am gate width. Assuming a defect density

independent of device size many defects are expected to be present in the Debye region in

this device at any one substrate bias and the summation of noise peaks at the peak temperatures can be seen.

We have measured the noise in a number of three-terminal JFETs as a function of drain

current and temperature. At any measurement frequency the peaks of noise occurred at the same temperatures as in the four terminal devices. The magnitude of the noise peaks varied

randomly by large amounts between samples of the same type and as a function of drain

current for the same device. This is consistent with the present model [2],

73
Fig 5 Dependence of low frequency noise

on substrate bias at very low temperatures

for 3 typical devices of 620M.in gate width

Fig 6 Waveform of drain current modulation

at very low temperatures. Device PSOl,

T = 90K, Vertical lOnA/cm, Horizontal lOms/cm

6. EFFECT AT TEMPERATURES BELOW 120K

The noise at 75Hz and 85K for three devices of 620)J.m gate width is shown in figure 5. Note that in some devices one large, broad peak is observed rather than many individual

peaks. It is seen in figure 2 that there are characteristic temperatures at which the

noise has a maximum at each frequency. An analysis of these peaks suggests the existence

of a shallow defect with an energy level of about 190meV below the conduction band. If the

increase in noise below 120K was entirely due to carrier density fluctuations the noise

would be independent of frequency below a few MHz.

In many devices studied a three-level random telegraph signal drain current waveform

has been observed directly as shown in figure 6 especially when the temperature and bias

conditions correspond to the broad peaks in figure 5. An adequate explanation of this

behaviour has not been found.

ACKNOWLEDGEMENTS

The authors are indebted to M.O. Deighton for discussions on the mechanism of drain

current modulation in a JFET and F.B. Whiting and D. Lloyd for assistance with the measurements. Some of the devices used in our work were produced under a programme funded by

M.O.D. (DCVD)

.

REFERENCES

[1] K. Kandiah and F.B. Whiting, Solid State Electronics , 2\_, pp 1079-1088, April 1978

[2] K. Kandiah, M.O. Deighton and F.B. Whiting, this conference

/37 A. Van der Ziel, Proc. IEEE, 5J., p. 1670, November 1963

[A] C.T. Sah, Proc. IEEE, 52^, pp 795-814, July 1964

[5] M.O. Deighton, private communication

74
THE CHARACTERISTICS OF NOISE DUE TO INDIVIDUAL DEFECTS IN JFETs

K. Kandiah, M.O. Deighton and F.B. Whiting

U.K. Atomic Energy Authority

Harwell, Didcot, Oxon

1. INTRODUCTION

Low-frequency noise in JFETs has been attributed [1] to charge fluctuations at electrically active centres (defects) situated in the gate-to-channel depletion region. The

spectrum of this component of noise is strongly dependent on temperature and frequency [2,3],

varying as f~^ over a large part of the spectrum where f is the frequency and n lies between

1 and 2. Sah's model [I] accounts for the noise due to defects with an energy level near midband at temperatures above about 220K. Experimental confirmation of this model comes from noise measurements in which a high density of defects with a deep level, such as gold

impurities, were introduced into the device. Most of the published work on low-frequency

noise deals with defects with energy levels near midband owing to the interest in the use of

the devices at room temperature. Measurements on some neutron irradiated JFETs [4] and

normal devices [2,3] showed the presence of shallower level defects but no model describing

the mechanism in detail has been published.

This paper discusses noise in JFETs caused by point defects with arbitrary energy levels

and physical location in the device as a function of temperature and frequency. In section 2 we present some noise measurements on low noise 4-terminal JFETs and introduce the principles

of a model which explains these results. In section 3 we examine charge transitions at point

defects. Finally the drain current noise as a function of temperature and carrier density at

the defect is discussed in section 4.

2. NOISE IN 4-TERMINAL JFETs

The JFETs used in our experiments were silicon n-channel 4-terminal types in which

separate connections are available to the gates on either side of the channel. The channel

is in an epitaxial layer with a dopant concentration of about 10 cm on a p+ substrate

which acts as one gate. The other gate (top gate) is formed with a p+ diffusion into the

top of the epitaxial layer giving a gate length of about 2.5|J.m. All the results reported

in this paper are on JFETs with gate widths of 620)im but devices with a range of gate

widths are discussed in another paper [5],

In each set of measurements the drain voltage V.^ and current I,^ were maintained at predetermined values and the drain current noise was measured as functions of substrate bias

voltage V33 and temperature. When the V^^ is varied the measurement system automatically

adjusts the top gate bias voltage to maintain the desired constant and I^. This variation of Vgg therefore shifts the position of the channel in a direction normal to the substrate. Each noise current reading was calibrated measuring the response to a current from

a pseudo-random telegraph signal generator, with an accurately known amplitude and clock

frequency, injected into the drain. The filters used for noise measurements had about 20%

bandwidth with centre frequencies in the range lOHz to lOOkHz. A servo system was used to maintain the device temperature within ±0.5K of any temperature in the range 79K to 335K.

Typical results showing the variation of noise with temperature for fixed bias conditions are shown in figure 1. These results indicate the presence of 3 types of defects

which generate noise in three (overlapping) temperature ranges. Referring to the noise at

lOHz the ranges are A (<70K to 130K), B (llOK to 230K) and C (200K to 350K) . We have

observed that the heights of the noise peaks in the temperature ranges B and C vary

independently with bias.

75
The variation of noise with substrate bias in two different JFETs, at various fixed

temperatures in regions B and C, is shown in figure 2 and figure 3. In general the positions

of the peaks (in terms of substrate voltage) in the different temperature ranges do not

correspond for the same drain current and drain voltage. The peaks are wider (expressed in

substrata volts) for the larger values of substrate voltage but are of nearly constant width

when expressed in terms of the distance moved by the channel. The peak shapes are not

dependent on device type when there is no overlap between peaks.

The interpretation of these results is that each noise peak in figure 2 and figure 3 is

generated by a point defect when it is situated in the Debye region between the neutral

channel and the fully depleted region. The term 'Debye region' is used in preference to 'transition region' in order to avoid confusion, since we often refer to charge transitions.

The significance of the peak positions in figure 2 and figure 3 is that the noise is maximum

at a given frequency when the siibstrate bias places the defect at a position in the Debye

region, where the carrier capture rate and the emission rate match the frequency. A typical

waveform of drain current modulation in this case is seen in figure 4.

}l I I I I I I 1 I I I I I I 1 L 1 I I I I I I I I I I I 1 I I 1 I 8 1 2 3 4 5 B SUBSTRATE VOLTS

Fig 3 Drain current noise vs substrate bias in JFET type PSOl

Fig 4 Waveform of drain current modulation

vertical 5nA/cm, horizontal 5ms/cm

76
3. CHARGE TRANSITIONS AT POINT DEFECTS

Charge transitions due to electron capture and emission take place with a characteristic capture time and emission time given by

T„ = (1/N v^, a) exp (E/kT) (2) 2 c th

where v^j^ is the mean thermal velocity and n the density of free electrons, 0 is the capture

cross-section of the defect, N^ is the density of states in the conduction band, E the

defect energy level, T the temperature and k Boltzmann's constant. Similar expressions

apply for the capture and emission of holes with appropriate values for the parameters. In uncompensated n-channel JFETs hole capture can only take place when holes are generated by

avalanche multiplication in high-field regions of the channel. We ignore this possibility,

leaving electron emission as the only possible route for a charge transition to the more positive state. A defect in the depletion region can acquire an electron only by hole

emission, owing to a virtual absence of any carriers in this region. On the other hand, a defect in the channel or Debye regions can undergo such a transition either by hole emission

or electron capture. We therefore consider a Markov chain in which electron emission is

followed either by electron capture or hole emission.

In general, acquisition of an electron by a defect is governed by two characteristic

times, and Tj^ representing the respective probabilities of the two competing processes,

T^^ is given by eq (1) , while T\q is a version of eq (2) modified for holes. The two

probabilities are simply additive and we have a net electron acquisition time X\, where

1/T, = 1/T. + 1/T , - (3)

1 1 c le , .

Thus, tends to a finite upper limit (T^g) as the defect moves into the fully

depleted region where T^^ is nearly infinite.

4. DRAIN CURRENT NOISE AS A FUNCTION OF TEMPERATURE AND BIAS

The two distinct aspects of the present theory of drain current noise i^^ (f) are:

(i) the mechanism by which a charge increment ±q (the electron charge)

at a defect causes a change iAlj^ in drain current of a JFET

(this is the subject of a separate study)

and (ii) the influence of the independent time parameters and T2 on the

frequency spectrum of the noise, and hence on the amount passing

through the measurement filter.

In this paper we are chiefly concerned with (ii) . The noise current waveform at the

drain consists of alternate positive and negative steps Alp, assumed to be constant and

separated by randomly variable intervals having mean values at one level and at the

other. Such an asymmetric random telegraph signal has been shown by Machlup [6] to have a well-known spectrum shape 1/(1 + co^T^^,) with a corner frequency (or time constant T^,) given by:

1/T = 1/T, + 1/T- (4) c 1 z

If this noise is passed through a narrow-band filter, centred at frequency f (= 1/2ttt)

and of bandwidth Af , the resulting output noise can conveniently be written in terms of the

normalised parameters x(= t^/t) and y(= T2/T)

:

77
2 2 2 4x^v^

I = i (f) Af = T(Al„) Af..—.v i 2,—I—T- (5) n n D (x + y) + x y'^ (x + y)

The last fraction has maximum possible value 0.5 when x = y = 2. If either x or y is held

constant at a value other than 2, while the other is varied, then the filtered noise plotted

as a function of this variable shows a smaller peak displaced to one side or the other of

the main peak. Since is related to local electron density i.e. to substrate bias, while

T2 depends mainly on temperature and defect energy level, one can interpret many features of

the measurements directly as properties of the function of x, y above.

The observed behaviour of the peaks of noise is in line with this model. The measured

widths and shapes of the noise peaks as a function of substrate bias agree with changes in

carrier concentration at the defect estimated from the movement of channel position. The

expected asymmetry of the peaks and the higher level of noise on one side due to hole

emission (at T > 250K) , as discussed in section 3, are also observed as in the noise at 247K

in figure 3. The movement of the position of the highest peaks (when x = y = 2) , found to be about 0.35v for a change in frequency from lOOOHz to lOOHz for the temperature range C,

also agrees with the predictions.

In our theoretical model we have assumed that AIq is independent of and T2 over the

limited range where the defect is most active. We have observed that the magnitude of the

step in the drain ciirrent waveform, as seen in figure 4, does not change significantly as the substrate bias and temperature are varied although the mean dwell times and T2 in the

two states behave as predicted.

Measurements of the temperatures at which the noise is a maximum for various values of

T will yield activation energies for the defects. We have found a scatter in these measured

energy levels which cannot be attributed to experimental error. Possible explanations could

be the variation in T2 due to the large varying electric fields around the pinched down

region of the channel or the dependence of on the crystal orientation in relation to the

direction of maximum gradient of carrier velocity and density at the defect. The three

energy levels for the temperatures ranges A, B and C are .15 ± .04eV, .28 ± .02eV and

— Ill 2 *~17 2 0.56 ± .03eV respectively with 0 having values in the range 10 cm to 10 cm .

5. ACKNOWLEDGEMENTS

The JFETs used in our work were produced as part of a programme funded by M.O.D. (DCVD)

.

V^e are also indebted to Mr. D. Colman for valuable discussions during the earlier

measurements

.

REFERENCES

/I7 C.T. Sah, Proc. IEEE, 52_, pp 795-814, July 1964

[2] J.W. Haslett and E.J.M. Kendall, IEEE Trans Electron Devices, ED-19 , pp 943-950,

August 1971

13] J. Llacer and D.F. Meier, IEEE Trans. Nucl . Sci , NS-24 , pp 317-326, February 1977

i"47 K.K. Wang, A. Van der Ziel and E. Chennette, IEEE Trans. Electron Devices, ED-22 , pp 591-593, August 1975

/5J C.E. Cox and K. Kandiah, this conference

[6] S. Machlup, Jnl. App. Phys., 25_, pp 341-343, March 1954

78
NOISE DUE TO FAST SURFACE STATES IN MOSFETs

H. Mehta and K. M. van Vliet

Department of Electrical Engineering

University of Florida

Gainesville, Florida 32611

INTRODUCTION

McWhorter's theory of 1/f noise due to tunneling of conduction carriers (say electrons

in an n-type channel) via interface states to oxide traps is very suitable for explaining

low-frequency 1/f noise in MOSFETs [1]. A detailed investigation of the applicability of

this theory to MOSFETs has recently been made by van der Ziel [2] . Though basically this

theory involves density fluctuations, the fluctuating trap occupancy also causes surface

mobility f luctuationSdue to ionized impurity scattering. The combined effect is accounted

for by a modulation factor [ 1 + (N/y ) (dy /dN) ] . In order to obtain 1/f noise, one invokes

the well-known distribution of time constants g(T)°=l/T, which results from a uniform distribution in tunneling widths. However, there is a limit to the lowest possible relaxation time, set by the interaction of the channel carriers with the fast surface states at the oxide channel interface, since no recombination can be faster than this interaction, even if the tunneling time becomes very small. Also, the bulk lifetime plays a role. If

the bulk lifetime is smaller than the time to recombine via fast surface states, then this

sets an upper limit to the occurring of 1/f noise due to the McWhorter-van der Ziel model.

Yet, experimentally the 1/f noise may occur over several decades beyond this limit.

In this paper we will show that, where the McWhorter-van der Ziel spectrum ends, a new

1/f noise regime occurs due to the stochastic generation-recombination of carriers via the

fast surface states, and the associated carrier diffusion to these states. In this respect

we point out that all older theories involving surface generation-recombination are incomplete (Champlin [3], Hyde [4], van Vliet-Fassett [5], Lax-Mengert [6]). These theories

have correctly dealt with the effect of the fi^^poniZ modification of transport noise due to the surface generation-recombination path; usually the response changes from l/w^ to l/u^'^

typical for the carrier diffusion to the surface states; however, these theories have not

considered the noise -iOUAC^ associated with this surface generation-recombination process.

This was for the first time considered by Mehta and van Vliet [7,8].

We basically consider three models: (1) We formulate the equations for a bulk semiconductor with surface recombination at both sides and with a stochastic surface source. We set the bulk noise source equal to zero. Thus this noise must be added to the spectra

due to references [3 to 6] stemming from the volume noise source—modified by the boundary

conditions involving surface recombination as explained above. (2) We consider a MOSFET

with an inversion layer serving as the channel, with surface recombination both on the

oxide side and at the bulk semiconductor side. The noise concerns here the carrier population in the carrier enriched channel only; we take the mobility and lifetime to be uniform

over the entire channel-semiconductor region. (3) Finally, we consider a MOSFET model with

surface recombination at the oxice side and reflection of the current at the semiconductor

side, with the mobility and carrier lifetime in the channel being different from those in

the bulk semiconductor.

COMPUTATION OF THE NOISE

The transport problem is described by the following equation:

3An(x,t) _ ^ 8^An(x,t) _^ An(x,t) ^ ^(^^t)

9t

(1) 3x^ 79
where is the bulk lifetime and C is the sum of volume g-r source and diffusion source,

X is the coordinate perpendicular to the oxide surface interface. The boundary conditions

are taken to be

D ^ I - oAn I = C(t) (2a)

x=0 x=0

D I + oAn I = cCt) (2b)

x=W x=W

where D is the diffusion constant and a is the surface recombination velocity; ? is the

surface Langevin noise source. Since we are interested in the noise due to the surface

noise source, we set E, = 0 . The surface noise source c, has the spectrum

S CO= 4MaN a/A (3)

where N^is the number of surface carriers and where M is a modulation factor, which accounts

for the fact that the noise is enhanced due to the effect of band bending caused by the

surface charge. on the recombination velocity a (see also McWhorter's theory)^ a = <An^>/n^-

The noise is calculated using the Green's function procedure. The response of An(x)

due to the surface source is characterized by the Fourier-Laplace transformed surface

Green's function [10] H (x,0,icj); then for the carrier density fluctuation spectrum of

<An(x,t)An(x' ,t) > we have

S, (x,x',w) = H(x,0,ia)) H(x',0,-ia)) . (4) An c,

For the first problem the carrier density fluctuations in the whole sample are of importance. Thus, if n is the average number of carriers in the sample, then from (4)

S W

S._(a)) = I ^ / H(x,0,ico)dx P , (5) An o w

o

where W is the thickness of the sample. Thus for the noise spectrum we must compute the

response function

W

f(ia)) = ^ / H(x,0,iw) . (6) w

o

With the b.c. (2a) one easily shows that H(x,0,iaj) = (oi/D) G (x,0,i(jj), where

Dico

• a =

ia)+l/T

(7)

and where G is the volume Green's function defined as in [5]. Explicitly, we obtained

, , -L[l-cosh W/L-(D/aL)sinh W/L]

f(ioj) = — . (OJ

W[l+(D2/a2L2)] sinhW/L + 2(DW/aL) cosh W/L

80
The nomalized spectrum S(u)) is |f(ia))|2; L = Va/iw . At sufficiently low frequencies we

find

constant (9)

Here = W^/D is the diffusion time and = W/o is the surface lifetime. At high frequencies we obtain

1 S(to) ~

0)^ (uT )^

s

(10)

The spectrum closely resembles a Lorentzian.

Next we consider a MOSFET with inversion layer. For our purpose we shall approximate

the carrier distribution such that

n=n , 0<x<W^ (surface region)

n = n, ,W < x<W (bulk region)

t> 1 (11)

where n >>n, and where W = W + W , W being the width of the bulk region. Though the

s b 12 2 carrier density in the bulk n is much less than the channel density n , carriers diffuse

s s into the bulk and undergo bulk recombination and surface recombination at the other end.

For simplicity we assume tha't the surface recombination velocity in the oxide and at the

free surface of the semiconductor are the same. Thus the b.c. are again eqs (2a) and (2b).

Also, we assume y, D, and t to be the same in the channel as in the bulk. The normalized

spectrum is now obtained by integrating H(x,0,ia)) up to W (instead of up to W as in eq (6);

hence, we obtain

S (o)) = 2L sinh

/ww\ w ^ /ww\ W-i

Vlf^ "li) ^^^^ + 21/ ^^'"^2P

(

2 \ WW WW

1+^^—\ fsinh-^osh-z^ + cosh-^sinh-^-Jf — (R ) ^2l2 j

L l L L L aL

(12)

where

, W W W W \ = ^cosh-^-'-cosh-^ + sinh-^-'-sinh-^ j. (12a)

At sufficiently low frequencies we find

S (oj) ^ constant

0)^ ). T T d'b

(13)

At intermediate frequencies we set |w /l|<<1 and |w /l|>>1. In a straightforward manner we obtain 1 2

81
S (03)

1 + (iU) +l/T)

1 + Da' "2(iaj+l/T) + 2 VDo-2(ia)+lT)

If in addition w>1/t, we have

(14)

S(a)) ^

w->int T ' ^0)

(15)

where now t

' = W ^/d and t

' = W /a . The spectrum shows 1/f behavior in this frequency range.

d J s 1 For very high frequencies (W^/L>>1, W^/L>>1) we find

S(w) - (16)

1

The high frequency range is in practice never realized for we find that the 1/f and l/f^

region meet at w

= D/W^2_ „ith D = 25 cm^/sec and = 400A° , this yields o)^ = 1.6 x 10 rad/sec or f 25 x 10^°hz. If t = lo"*^ sec, then the spectrum goes as 1/f for

o b . 2 X 103<f<25 X 10^°, i.e., over eight decades!. Likely, however, the noise drowns m the

white thermal noise at a much lower frequency. A computer plot of the full equation (12)

is given in figure 1.

FREQUENCY

82
Finally, we consider a model in which D, y, and T are different in the channel and for

the bulk, and we take the b.c. at x = W to be DVn =0. In this form the results are mathematically very close to the results for temperature fluctuations in a thin film supported by

a substrate, published recently [11]. The 1/f noise range is again found to be as in (15),

with t' now being W ^/D , Since D , ~ (1/5)D, , the noise is enhanced by a

d 1 channel channel bulk

factor five. Also, the noise is apparently insensitive to the b.c. at the far end of the

bulk (x = W) . Further, we note that, in contrast to the case of temperature fluctuations

[11], it is likely that the surface noise contribution (due to S^) as computed here dominates the volume noise contribution (due to S^) , since as a rule M>>1, particularly for

strong inversion. Altogether, the models presented here for MOSFETs give a very realistic

model for 1/f noise over many decades above the reciprocal lifetime 1/^^. This explains

that the 1/f noise in some MOSFETs extends up to many megahertz. Since the noise source S^

is essentially the same as in McWhorter's model (but not in van der Zlel's model which

misses the modulation factor M) , the noise spectra from slow and fast surface states should

be continuous, though possibly there is a small 'wiggle' in the spectrum near to = 1/t, which

should be searched for experimentally.

Finally, we note that the carrier density fluctuations in the channel are coupled to the gate in the same way as the low frequency 1/f noise. Thus one has for near zero drain

bias

= ^^^^

ea C dL ^o ox

where d and L are the breadth and length of the channel, C is the capacitance per unit

_ ox length of channel. Since S-(f) is proportional to n, with ii = e"^C (V -V ) , we find as for n ox g i Klaassen's result [12],

S^ (f) - '^""^tant w^(v^-v^). ^ (18)

eq

The dependence on drain bias is the same as that calculated by van der Ziel [2] .

REFERENCES

[I] A. L. McWhorter, 7/^ Woi^e and R2Zat^d SuA{^ac2, lUzoZi, in GMmayiitm, M.I.T. Lincoln

Lab Report No. 80 (1955).

[2] A. van der Ziel, in Advanc&i -in EttcViovlicJi and EZ^ctAon ?hLji>lu , Eds. L. Martin and

C. Martin (Academic Press, New York, 1979) Vol. 49, 225-297.

[3] K. S. Champlin, ?kyi>lca 2b_, 751 (1960).

[4] F. J. Hyde, Rupt. Con^. Phy6. Soc on Semicondactou , Rugby, England (1956), 57.

[5] K. M. van Vliet and J. R. Fassett, in Fluctuation Vhtnomma In SoUjii>, Ed. R. E. Burgess (Academic Press, New York, 1965) 267, Sections V and VI.

[6] M. Lax and P. Mengert, Vhiji, . Chm. Solidi, 14, 248 (1960).

[7] H. Mehta, Tmnipo^ Hoi^z knJj>lnq (^nom Vmui,lon and Bulk on S,uh.{^a.cQ, GmeAotionRzcombtnation In SmtaonductofU , Ph.D. thesis. University of Florida (1981).

[8] H. Mehta and K. M. van Vliet, Phyi . Stat. Sotidt (to be published).

[9] K. M. van Vliet, Solid Statz llzctyLOnic^ 15, 1033, Appendix C (1972).

[10] P. Morse and H. Feshbach, lAzthodU, o{, Thzo^Ztical ?ky6icJ> (McGraw-Hill, New York, 1953),

Vol. 1, Chapter 7.

[II] K. M. van Vliet, A. van der Ziel, and R.R. Schmidt, J. Appl. Vhiji> . _51 (6), 2947 (1980).

[12] F. M. Klaassen, IEEE Jh.an^. El. Vzv., ED-18, 887 (1971).

83
LOW FREQUENCY NOISE IN GaAs FET's AT LOW DRAIN VOLTAGES

J. GRAFFEUIL, J.F. SAUTEREAU, K. TANTRARONGROJ

Laboratoire d 'Automat ique et d 'Analyse des Systemes

Toulouse, France

Universite Paul Sabatier, Toulouse, France

I - INTRODUCTION

Low frequency noise sources in field effect transistors have already been extensively investigated mainly on silicon J. FET. The dominant low frequency noise sources in these devices were found to be charge fluctuation of generation centers in the depletion region (1). Subsequently it was reported that carrier density fluctuations in the channel may also contribute, to the overall low frequency noise (Z).Low frequency noise in GaAs MESFETs was attributed

to trapping in the channel C3,43, l/£ noise sources in the semi-conductor-oxyde interface

(5) and to a process governed by a distribution in time constant (6)

.

The aim of this paper is first to theoretically investigate carrier density fluctuation noise

and charge density fluctuation noise in GaAs MESFETs at low drain voltages in the one dimensionnal FET model. Secondly, it will be shown that the experimentally observed noise is the

consequence of these two physical mechanisms previously investigated. Finally, an original

technique to derive noise coefficients which depend only on the material used for processing

the device is proposed.

II - CARRIER DENSITY FLUCTUATION NOISE

Carrier density fluctuation may arise in the transistor's channel from generation and recom- bination on trapping. Let us assume that only single time constant processes are involved,

According to Van Der Ziel's theory (2) the spectral intensity of the short circuit channel

noise current (nonsaturated) is given by :

where is the potentiel between source and drain ends of the active channel, yU the low

field mobility, L the gate length,*^ the carrier density fluctuation time constant, CO is

equal to 277? where f is the frequency and a constant given by

where A/ is the number of free carriers in the channel and AN^ the variance.

The output conductance in the one-dimensionnal model is given by

V^^ = S^d = God-^) ^=

Woo

^

where Ip is the drain D.C. current, Go the conductance of the open channel, V^^ the

gate junction built-in voltage, Vq the gate voltage and Woo the pinch-off potential. The

spectral intensity of the open circuit channel noise voltage is :

84
In GaAs FETs the influence of regions between source and gate (length Ls(j) and gate and

drain (length Lgp ) may be important. Let us assume that epilayer thickness is constant

between source and drain and that noise sources are uniformly distributed in it : the lateral

regions will also contribute to the overall low frequency noise. The spectral,

intensity of the open circuit lateral regions, voltage noise is found from 1 and 2 where

LsG and Lqd are substituted from L a.nd$j[^^= . ^dap = are substituted fromgd • Let the two lateral regions noise sources and the channel noise source -been v*:orreLated

so that the three spectral intensities add together, and let_ the total output conductance be

$d — ^^/^^ where Vjj is the drain voltage, the total short circuit noise current contributed

by carrier density fluctuation is

it

^As a consequencethe plot of the product of the spectral intensity at a given frequency

by Vp/lp against ^1-7^^3 should be a straight line. The slope of this line should provide the

noise coefficient ffc'Cc/Ci+w*'C*)and the linear extrapolation of the plot to the ordinate should

give the ratio (.1-sg+ Lg,,)/ 1.

Ill - CHARGE FLUCTUATION NOISE .

According to SAH' s theory (7), the random generation of electrons and holes from

centers in the transition region of the gate induces a charge fluctuation on the gate

contact. Subsequently an equivalent voltage fluctuation on the gate is derived using the

relationship) between voltage, charge and gate-to channel capacitance (7). The spectral

intensity ot this fluctuation is given by :

where the time constant "^f is given in (7), and where is equal to '^fftO~^t)/Nl>if "^t is

the fraction of filled traps. The total transconductance ^„ including series resistances is

given (one-dimensionally) by :

A' =x.i2. _L _ 6

and the spectral intensity of the short circuit channel noise current contributed by charge

fluctuations is :

The plot of the product of the spectral intensity at a given frequency by V^/Ip against

Vx/Cl-Vx )^ should also be a straight line. The slope should provide the noise coefficient

y^'^^/( H.a)*>f^') and the intercept point with the axis should be at the origin.

~ 3 CGoL)*

7

IV - EXPERIMENTS

The equivalent short circuit noise current was measured on about ten short gate length

devices {0.5jun to lyum ) available from most of the major manufacturers. For a given ga

voltage these measurements were performed between 10* and 10^ for different drain

currents and drain voltages selected so that the one dimensionnal model holds, i.e» Ip and

must satis fy (8)

Go l-Vx 6

where R3 and Rp are source and drain series resistances,

From the plot of Si ( f ) versus f (figure 1) we can observe spectra of various forms

which may be attributed to either a process governed by a distribution in time constants (6) or to the superposition of a few single time processes (4i So as to generalize the previous

theoretical analysis we must substitute frequency dependent noise coefficients NcC^) and

/^gtf) for the coefficients Tj^TTc /C 1 +Co't| ) and IJgTg /CI +6J*Tg ) previously

introduced

85
The experimental results from all the samples indicate that the product of S;(f ) byVp/l^

is a constant at a given gate voltage and a given frequency as long as equality 8 holds.

Furthermore, once the pinch-off voltage has been carefully determined (9), Sj(f)-Vp/lp

was plotted versus (1 — y>(')~*^ • Two typical plots of devices A and G are given in

Figure 2.

86
I . . . Plot A clearly shows following 4 , that carrier density fluctuation is the major

source of noise in device A. Moreover values of at 10^ and of C'-S&+ l-cio)/!- evaluated

from this plot are 1.7'lo''s. and 2.5 . The latter value compares well with the one found by microscopic observation of the device.

Plot G is a two sections plot : the first section is a straight line and shows that the

carrier density fluctuation noise predominates at low gate voltages (Nc= 8 10 S. and

CI-SG + l-(ip)/L <^ 4)* 1^^^ difference between the second section and the linear extrapolation

of the first section to high was plotted versus V></C1~/X )^ . We obtained a straight

line, which fairly matches relationship 7 with = 4-10'^ S . It must therefore be con-- cluded that both carrier density fluctuation and charge fluctuation will contribute to the

overall noise - The major contribution is carrier density fluctuation at low gate voltages

and charge fluctuations at large gate voltages - We believe that this conclusion is valid

for most of the devices, but in some cases (for example, device A), the carrier density fluctuation noise is too large so that charge density fluctuation noise cannot be detected.

Moreover since the spectrum of device Or exhibits a quite perfect 1/f behaviour, the open

channel noise current spectral intensitycalculated from Hoodge ' s formula should have been :

where Wh is Hoodge's constant ,V the epilayer volume and n = the mean free carrier

density in the epilayer. From the extrapolated plot G at X = O it can be deduced thatO(n

should be approximately two orders of magnitude smaller than 0(h measured in bulk materials.

V - CONCLUSION

A new method was used to distinguish between carrier density fluctuation noise and

charge density fluctuation noise in GaAs MESFETs at low frequencies and low drain voltages.

This technique provides noise coefficients independent of bias conditions and other

parameters of the devices. The technique appears to be useful as a means of monitoring material used for devices. For example we found that the carrier density fluctuation noise

coefficient at 10^ Hz is usually lower in LPE devices than in VPE ones (10). Finally, since

the low frequency of the FET under normal operating conditions is closely related to the low

bias noise coefficients Nc and (10), the technique could also provide an efficient

means of evaluating devices for microwave low noise large signals applications as oscillators or mixers (10).

This work was supported by a DAII-CNET contract. We wish to thank Dr BLASQUEZ for his

useful comments.

(1) P.O. LAURITZEN - Solid State Electron. 8, 41 (1965)

(2) C.F. HIATT, A. VAN DER ZIEL, K.M. VAN VLIET IEEE trans ED 614 (1955)

(3) J. GRAFFEUIL, J. CAMINADE - EL. Letters 10-13 (1974)

(4) D. SODINI, A. TOUBOUL, G. LECOY, M. SAVELLI EL. Letters 12-2 (1976)

(5) K. TAKAGI, A. VAN DER ZIEL Solid State ElectroYi22-285 22-285 (1979)

(6) C.H. SUH, A. VAN DER ZIEL Appl. Phys. Lett 37-565 (1980)

(7) C.T. SAH Proc IEEE 52-795 (1964)

(8) J. GRAFFEUIL - These Universite Paul Sabatier (1974)

(9) H. FUKUI Bell Syst. Tech. J. 771 (1979)

(10) J. GRAFFEUIL (....) to be published.

87
THERMAL NOISE IN THE SEMICONDUCTOR REGIME OF DOUBLE-INJECTION DIODES

A. van der Ziel

Electrical Engineering Department

University of Minnesota

Minneapolis, MN 55455

SUMMARY

In an earlier paper Huang and van der Ziel could not fully evaluate the thermal noise

in the semiconductor regime of a double-injection diode. By a careful application of the

boundary conditions and by making the approximation for the semiconductor regime only at the end of the calculation this problem has now been solved. The final result consists of

thermal noise of the d.c. resistance V^/l^ at all frequencies plus a g-r like noise term

that disappears at high frequencies and is small for long diodes.

In their calculation of the noise in double injection diodes operating in the semiconductor regime Huang and van der Ziel-'- ran into difficulties because the approximation

for the semiconductor regime was introduced early in the calculation. This removed one

of the boundary conditions, which, in turn, made a solution impossible. By applying the

approximation at the end of the calculation these difficulties can be avoided.

The basic equations for the double-injection diode are [1]

J = ey (p+p^)E - eD 9p/3x + eh • (1)

P P T p p

J = ep (n+n„)E + eD 3n/9x - eh (2) n n i n n

9p/9t = -R - (l/e)9Jp/9x - r (3)

9n/9t = -R + (l/e)9J /9x - r (A) n

9E/9x = (e/ee ) (p-n) (5)

o

Here Jp and Jj^ are the hole and electron current densities, n and p the injected hole and

electron densities, n^j- and pr^ the equilibrium electron and hole densities, R = n/x the

recombination rate per unit volume and T the lifetime of the injected carriers, E the field

strength, hp and h^ the particle current densities for holes and electrons describing the

diffusion noise sources and r the generation-recombination noise source.

If we now manipulate eqs. (3) and (4) properly, neglect the diffusion terms except

h and h and ignore r, we obtain

p n ^ '

o 9 ,„ dE, , , , 9E p n ,9n , n. 9 e 9x 9x' T 9x u u 9t t 9x

p n

h^(x) h (x)

_P + ^

P n

(6)

If the small terms with pj and n-j. are neglected, and if also the diffusion terms except hp

and h^ are neglected, and if furthermore the approximate space charge neutrality condition

n - p is introduced, the total current density J(t) is

88
J(t) = J + J = e(y + y )nE + e[h (x) - h (x) ] (7)

p n p n p n

We now put J=J +J, ,n=no+n^,E=E +E, ,V=V +V, , where V is the

potential; the subscripts 0-1-o denote d.c.

1quantitiesoland the subscripts ol 1 denote small signal

a.c. quantities. We next make a Fourier analysis for the a.c. quantities for 0 < t < T, where T is sufficiently large, and introduce the Fourier coefficients E., ,n-, ,V, .J, ,h

, '

, , , . . . . , ^ In' In! In' In' pn and hjjjj. We make the assumption that the device is open-circuited for a.c, i.e.,

J^^ = 0. This yields the equations

ee , dE dE y +1J o d , o, , - o p n

- — d^ (^o d^> ("t-pt> d^ = ^nrr % P n

J = e(y +y )n E (9) o p n o o

££ ,2 dE., y +y , h h

o d ,^ ^ . , ,

s. In P n , ,

. d , pn , nn , , - —7T (^o^m) + (^T-Pt) - ilVr (1 + J'^^^^ln = d^ (y —) ^^^^ dx P n P n

0 = e(y +y ) (n-, E +n E-, ) + e(h -h ) (11)

p n In o o In pn nn

Eliminating in (8) with the help of (9) yields

J ee J dE dE

° E„ (E„ -j^) + (n -p,)E^ -j^ . - (12)

ey y T e o dx o dx T '^T o dx

P n

Equation (11) may be written

n, E-, e(h -h ) In _ In pn nn

n ~

~

E ~ J

o o o

(13)

Substituting for n^^/n^ with the help of (13) and for n^ with the help of (9) yields

ee „ ,2 „ dE.,^ J (l+jtox)

dx p n

= _ a±ji^ E (h -h ) + E^ A (!^ + Ji^) (14) y y T o pn nn o dx y y

p n p n

which must be solved with the boundary conditions E^ = 0 at x = 0 and x = d.

We now introduce the spectra

(x,x',f) = lim 2T h h* = 4D n (x' ) 6 (x'-x) /A (15)

hp' ' ' pn pn p o

89
S, (x,x',f) = lim 2T h h = 4D n (x' ) 6 (x'-x) /A

nn „ nn nn n o (16)

where A is the cross-sectional area of the device. We also bear in mind that [hp^(x) - h (x) ] and [hpjj(x')/yp + h (x')/iJjj] have a zero cross-correlation spectrum, as is found

by applying (15) and (16) an9 the Einstein relation ^^/v^ =

^n''^n

~

Up to here we have not yet discriminated between the semiconductor regime and the

insulator regime. We do so by introducing^

J = J , (x) + J ,(x) ' (17)

O Ol OS

where

J .(x) Ol

P n

ee , dE

E f (E -r^)

e o dx o dx (17a)

and

J (x) OS

ey M T P n

dE

^"t-Pt^^o dF (17b)

Here J . (x) and J (x) refer to the insulator term and the semiconductor term in (12)

,

respectively. In°addition there is a diffusion current density, here neglected, which is

small everywhere except near the points x = 0 and x = d. In the semiconductor regime

J^^. (x) « for most of the region 0 < x < d, whereas for the insulator regime J (x) << The important point to make, however, is that these

The reason is that in the approximation for the semiconductor regime one of the boundary conditions E^ = 0 at x = 0, E =

0 at X = d is violated [3].

^o 1

~ ' o ^ , . ^ , J for most of the region 0 < x < d. o = = approximations are made at the end of the calculation.

We now multiply Eq. (14) on both sides by dx, integrate by parts from 0 to d and apply

the boundary conditions at x = 0 and x = d. This yields [4]

E^

^

(EE-, ) dx = 2 0,2 o In dx

, dE

E f- (E ^T-^) E^ dx

o dx o dx In

(18)

In J

„ E —-— dx = -2

o dx

dE

E -;— Et dx

o dx In

(19)

„ J h h

E^ f + ^ ^ _2

J o dx y y^

o

dE h h

o , pn . nn. , E -— (-^ + )dx

o dx 'y^ y^

P n

(20)

Substituting all this into the integral of (14) yields

90
^ J^.(x)(3+>T)-J^^(x)(l-ja)T)

_ ^ 3^ d

eu u T P n

dx = n p

E (h -h )dx

o pn nn

-2

J (x)(h /y +h /y ) OS pn ^p nn

dx (21)

We now apply this to the semiconductor regime by putting J . (x) = 0 and J (x) = J and bearing in mind that °

E, dx = - V, (d) In In(21a)

where ^-|^j^(d) is the Fourier coefficient of the open-circuit voltage at d. This yields

o E (h -h )dx - 7——

s /-, . \ o pn nn (n^-p^) (I-jlot)

h h

(22)

Defining

(f) = lim 2T v-^^(d)v^^(d)

a

(23)

yields after some manipulations

V V

(f ) = 4kT ^ + 4kT Ya a a l+OJ T (24)

where the anode current I = -J A xs taken positive (J flows from cathode to anode and

• • s „ a o ^ o hence is negative) . Here

T, = 4d^/(3y V ) and T, = 4d^/(3y V ) dp pa dn n a (24a)

are the drift times for holes and electrons, respectively.

In the evaluation of the integrals use was made of the fact that in the semiconductor

regime [3]

V (X) = V [d^/^ . _ ,)3/2^/^3/2

o a

(25)

when n > p . Note that here the condition dV /dx = 0 is violated at x = 0. Since this was

done only in the last step of the calculation, °the error thus introduced is small.

91
REFERENCES

[1] C. H. Huang and A. van der Ziel, Physica, 78 (1974) 220.

[2] R. J. J. Zijlstra and A. Gisolf, Solid State Electron. 15^ (1972) 877.

[3] A. van der Ziel, in Semiconductors and Semimetals . Vol. 14 (1979) 195-247.

[4] A. van der Ziel, Physics (1981) in the press.

92
PHOTOCURRENT NOISE CAUSED BY IMPACT IONIZATION

OF NEUTRAL DONORS, AND FREE AND BOUND EXCITONS IN n-GaAs

K. Aoki , K. Miyamae, T. Kobayashi and K. Yamamoto

Department of Electrical and Electronic Engineering, Faculty of Engineering, Kobe University

Rokko, Nada, Kobe, Japan

Abstract

In this paper, we have investigated in detail the photocurrent noise or current-filament instability caused by impact ionization of shallow neutral donors, and free and bound

excitons in n-GaAs at 4.2 K. From the power spectra of the observed photocurrent noise, it was found that the current-filament instability can be described by the bifurcation theory

based on the first-order phase transition.

1. Introduction

Electrical breakdown of the low temperature freeze-out of shallow neutral donors in the

compensated materials of semiconductors [1-3] causes the current-control ed negative resistance or the current-filament instability. Many works have been reported concerning

the impact ionization mechanisms [4] and the energy relaxation mechanisms of hot electrons

[5], while little attention has been made on the characterization of the current-filament

instability based upon the phase transition regime. The purpose of this paper is to investigate the photocurrent noise (i.e., the current-filament instability under the weak photoexcitation) caused by impact ionization of shallow neutral donors, and free and bound

excitons at 4.2 K in n-GaAs. Our final goal is to characterize the current-filament

instability based upon the phase transition regime of the current noise and to speculate the

microscopic origin of the fluctuating force.

2. Experimental Procedure

14 2 Samples used were epitaxial n-GaAs (n= 2x10 /cm at 300 K) grown on Cr-doped substrate

Thickness of the epitaxial layer was about 12 ym. Planar-type ohmic contacts were formed

by alloying Sn on the sample surfaces. Typical dimensions of the sample surfaces were 5.5 mm in length and 4 mm in width, respectively. The light sources used were a 20 mW He-Ne

laser, and also a standard halogen lamp for the illumination of monochromatic light around

band gap energy of GaAs. All measurements were done at 4.2 K.

3. Experimental Results and Discussions

3.1 Power spectra

The electrical breakdown of the shallow

neutral donors occurs at around the critical

electric field of 4 ^ 6 V/cm under the dark

and the illumination. The critical electric

field differs slightly from sample to sample.

Figure 1 shows a typical current-voltage

characteristics under the dark. Above the

onset voltage of the electrical breakdown

with dc-applied electric field (E > 4 V/cm),

the spiky noise with the duration of 10 ^40

ms was observed, and was randomly distributed

in time.

Fig.l Typical current-voltage characteristics under the dark, obtained by the continuous motor-drive. Speed of the motor-drive

was about 30 mV/s.

0 3 6

E (V/cm)

93
Each spike contains superimposed oscillations [6] and was explained well by the currentfilament instability, as is discussed below. Under the weak photoexcitation, the breakdown voltage shifts to lower electric field, and the number of spikes per unit time

increases with increasing the photoexcitation density Op.

Figure 2 shows the power spectrum around 0 Hz observed by He-Ne laser excitation, with

Jp= 3.2 mW/cm2 and E= 4.5 V/cm, respectively. The spectrum in the low frequency range is

very similar with those of the dielectric relaxation in various semiconductors [7] and also

with that of random telegraph signals. Assuming the Lorentzian shape (broken line in Fig.

2), the effective relaxation time was obtained to be 2 ms. The behaviors of the currentfilament correspond to the soft-mode excitation in the first-order phase transition, and are rather a static property of the hysterysis loop in the S-shape of J-E curve with the

frequency around 0 Hz. Furthermore, the photocurrent noise contains a discrete line spectrum in the higher

frequency range (0 ^ 1 MHz), which corresponds to the noise structures superimposed on each

current-filament. The number of discrete lines increases at the same frequency range when

the electric field and/or photoexcitation density is increased. Figure 3 shows the

discrete line spectra as a function of photoexcitation density with E= 4.5 V/cm, indicating

existence of many periodic states of the current filament. From the detailed analysis, it was found that the observed line spectrum seems to be incommensurate [8], i.e., containing

several discrete lines expressed by sum and difference of the higher harmonics of two frequencies fi and f2. Above the critical photoexcitation density of Jp= 40 mW/cm^, the

discrete line spectrum changes abruptly into the continuous spectrum, as is shown in Fig.

3

(c). All of the results mentioned above greatly suggest that both the soft- and hardmodes' excitations occur at the same time and that the continuous bifurcation occurs as a function of photoexcitation density and/or electric field.

Qualitatively, the current-filament instability can be explained by the feedback

mechanism of the impact ionization of neutral donors. At the onset of the electrical

breakdown, the impact ionization of the neutral donors enhances the impact ionization rate

and thus increases the carrier density due

to the carrier heating, while the excess energy of the hot electrons is relaxed into

acoustic phonon system due to the heat

balance, and in a little while the carrier

density begins to decrease with a short time

lag. Again, the impact ionization takes

place due to the applied electric field as soon as the cooling of the carrier gas

facilitate the current filament to disappear.

n-GoAs

n-Ga As

1 4.2 K

U)dB

-io\

\\

//

/

"20 \\

1 > > -0.6 (MHz)

2 -I O 2 f(KHz) Fig.3

Fig.

2

Power spectrum of the photocurrent

noise observed with Jp= 3.2 mW/cm^ and E= 4.5 V/cm, respectively. The broken line

is calculated Lorentzian shape with T=2ms.

Change of the power spectra as a function of photoexcitation density; Jp=3.2

mW/cm2 in (a), 32 mW/cm^ in (b), and 80 mW/

cm2 in (c), respectively. The electric

field was fixed to be 4.5 V/cm. The line

spectra around 1 MHz are stray signals from

radio frequencies.

94
A couple of nonlinear differential equations can be obtained as follows,

8T. at8T, 9x ^ 9x

- n + neyE'^ + P(Jp) + Pg.g + fi{Jp,t) (1)

|n_ = D ^ + n(Np - - n)Bj + (N^ - - n)B^ - nA^(N^ + n) + f2(Jp,t) (2)

9x

where one dimensional diffusion is assumed for simplicity, and the x-direction is perpendicular to the current flow. In eq.(l), Tg and Tl are the electron temperature of the current

filament and the lattice temperature, respectively, C is the heat capacity, k is the thermal

conductivity, is the energy relaxation time, n is the carrier density, y is the electron

mobility, E is applied electric field, P(Jp) is the energy added to the carrier gas by the

photoexcitation , Pg_g is the energy gain due to electron-electron scattering if it is important, and fi(Jn»t) is the random fluctuating force or small agitation for the heating

process, respectively. In eq.(2), D is the diffusion constant of the majority carriers in

the filament, Bj and Bj are the impact ionization rate per carrier and the thermal ionization rate of the neutral donors, respectively, Aj is the thermal recombination rate of the

majority carriers, Nn and N/\ are the densities of the shallow donors and acceptors,

respectively, and f2(Jp,t) is also the random fluctuating force. A couple of nonlinear

INn ctliu ni\ f2(Jp,t) i: equations (1) and (2) are very similar with those in the fluid system and the chemicalreaction system [8], and suggest that the diffusions of the electron temperature and the

carrier gas play important roles for the current-filament instability (diffusion-induced

instability). If we assume that the solution for the current f ilamentation has mode

patterns in space and time, the linear stability analysis can be easily investigated.

The necessary condition for the soft-mode excitation must be satisfied as follows.

(3)

The relation describes that the diffusion constant of the electron temperature or the heat

flow defined by eq.(3) should be larger than the diffusion constant of the majority carriers

for the soft-mode excitation. At low temperature where the heat capacity changes by the

T"3 rule, the mean-free path is very long and

competitive with the sample size. Therefore,

eq.(3) seems to be fully satisfied at the n-GaAs

temperatures as low as 4.2 K. a o

^•^

(Cf.X)

Fig.

4

Photocurrent noise (PCN) spectra as

a function of the excitation wavelength, with

Jp'\^ 2 yW/cm2 and E= 4 V/cm. Spectrum (a) was obtained by applying the pulse voltage with

the width of 500 ns and the repetition of 100

kHz, and was detected by a lock-in amplifier.

Spectrum (b) was obtained by dc voltage, and

was directly recorded on the recorder. The

scanning speed of the monochromator was 10 A/

min. For comparison, photo! uminescence

spectrum observed by He-Ne laser excitation

with Jp = 80 mW/cm2 is also shown.

(b)

-I u 820 819 818 817 816

X(nm)

95
3.2 Fluctuating Force

3

<

O

c

'55 z

o

Q.

n-6aAs

4.2 K

Xn«l

_jii(i(ilrtlfC_JIW

E=4.33(V/cm)

E = 4.38(V/cm)

4.44(V/cm)

822 820 818 816

X(nm)

814

In order to obtain a clear evidence

for the microscopic origin of the fluctuating force for the trigger of the

current filamentation, we investigated

the excitation-wavelength dependence of

the photocurrent noise (PCN spectrum)

with using resonant photoexcitation

around band gap energy. Figure 4 shows the PCN spectra observed with

J ~ 2 yW/cm2 and E= 4 V/cm. It is

cTear that the current-filament can be

selectively generated at the excitation

wavelength of the bound excitons and

the excited donor state for the resonant formation of the bound excitons

(we denote as (D0,X)p=2)- Figure 5 shows the PCN spectra as a function of

electric field. The results in Figs.

4

and 5 are very suggestive of the

microscopic origin of the fluctuating

force; (a) the current filamentation

originates from the localized donor

sites in the microscopic region and

(b) some preferential dissociationprocess of bound excitons acts as a trigger of the current filamentation [9]. The number

of spikes or density of the current filament per unit time greatly depends upon the photoexcitation density. The generation mechanism was found to be the Poisson process, which

gives a clear evidence that the photon noise and the chance process of the bound-exciton

formation and dissociation in the microscopic region act as a seed for the trigger of the

current filamentation. Concerning the preferential dissociation of the bound excitons,

Auger recombination process is inferred to be very effective for the trigger mechanism,

since Auger recombination is autoionization process releasing the majority carrier ener- getically into conduction band.

The current-filament instability discussed here is very analogous with the oscillatory

chemical reaction [8] and the turbulence in the fluid system [10], revealing an unique

problem of the "turbulence" in semiconductors.

Fig.

5

PCN spectra as a function of

electric field. Photoexcitation density

is ~ 1 yW/cm2.

References

[1] W. Schairer and N. Stath, J. Appl . Phys. 43, 447 (1972)

[2] T. Yao, K. Inagaki, and S. Maekawa, Solid State Commun. 13, 533 (1973)

[3] W. Bludau and E. Wagner, Phys. Rev. BJ[3, 5410 (1976)

[4] R. P. Khosla, J.R. Fischer and B.C. Burkey, Phys. Rev. BT^, 2551 (1973)

[5] T. Kurosawa, J. Phys. Soc. Japan 20, 1405 (1965)

[6] K. Aoki , K. Miyamae, T. Kobayashi and K. Yamamoto, Jpn. J. Appl. Phys. 19, 1657(1980)

[7] M. Buttiker and H. Thomas, Noise in Physical Systems ed.

Berlin, Heidelberg and New York, 1978) p. 115

D. Wolf (Springer-Verlag,

[8] H. Thomas, ibid ref.[7] p. 278

[9] K. Aoki, K. Miyamae, T. Kobayashi and K. Yamamoto, J. Phys. Soc. Japan 50, 357 (1981)

[10] G. Ahlers and R.P. Behringer, Phys. Rev. Letters 40, 712 (1978)

96
LOW ENERGY ELECTRON BEAM INVESTIGATION OF PLANAR TRANSISTORS

USING SCANNING ELECTRON MICROSCOPY WITH PARTICULAR REFERENCE

TO BURST NOISE

Kenneth F. Knott

Department of Electrical Enpineering

University of Salford, England M5 4WT

INTRODUCTION

Several workers |l| have attempted to correlate burst noise and dislocations on a statistical basis whilst others |2| have studied the effect on burst noise of gate voltage

in tetrode transistors. In the work to be presently described an attempt has been made to obtain 100% correlation between burst noise and a specific dislocation or dislocation

network in the transistor. In order to do this the operation of the transistor needs to be influenced on a microscopic scale. One way of doing this is to subject the transistor to an electron beam either in the form of a spot or in the form of a small raster in a scanning electron microscope (SEM) . At low incident energy (2-6 keV) the effect of the

beam is to charge up positively the Sj^-Si02 interface with results similar to the application of gate potential in a tetrode structure but on a much smaller geometric scale. At higher energies the beam can be used to generate hole-electron pairs in the bulk of the

transistor and hence artificially increase the leakage current in specific parts of the

device. If this induced current is monitored the SEM is said to be operating in the

Electron-Beam Induced Current (EBIC) modeo

EXPERIMENTAL PROCEDURE

In all experiments the transistor was connected in the circuit shown in Figo 1, with

V zero if unbiassed oneration was being studied, e.go in the EBIC mode.

BE

Figure 1. Test circuit.

Balance

Before carrying out detailed investigations of particular devices several transistors were

•sacrificed' to demonstrate the general feasibility of the experimental method. Firstly,

the effect of the beam on base current, Ig was tested by subjecting transistors to successive scans of the whole of their base and emitter areas. Some results are shown in

Fig. 2 where the ratio of final value of Ig to initial value of Ig is plotted against

estimated aggregate electron dose. These effects were long-term, the changes in Ig

persisting for periods ranging from several days to several weeks.

97
Figure 2. Change in I with electron dose

for CA2018 transistors.

Electrons/cm'

Secondly, ten transistors exhibiting burst noise were subjected to scanning of their

complete base and emitter areas. Three of these showed changes in burst noise characteristics at low incident energy. Five showed no change in burst noise even when scanned at 20 keV, an energy high enough to increase 1^ by 50 times. In the remaining two, burst

noise disappeared completely after scanning at 20 keV.

Having demonstrated the possibility of altering burst noise with the electron beam

a more rigorous and detailed investigation was initiated. Two approaches were adopted for

the next steps in the study „ Firstly, the SEM raster was reduced in size so that the

exposed area on the specimen was small compared with the emitter area. The raster was then

stepped around the region where the emitter-base junction intersected the slirface. Each

time a new area was exposed the beam was shut off and the burst noise observed. By

controlling the time of exposure and incident energy known increments in I could be

impressed^ Secondly, the SEM was set to 'spot' scan and the noise of the EBIC signal was observed as the spot was moved over the specimen

„

RESULTS - ELECTRICAL

Results have been obtained with three transistors up to date. These transistors have

been subsequently etched and re-examined in the SEM„

The first transistor was examined with a small raster whilst maintaining V constant

at 510 mV, the initial value of I being 40 nAo At 5 keV a particular area of the

transistor was showing signs of modifying the burst noise. This area was then scanned at 10 keV with a marked effect on the burst noise as Ig went through the range 250-300 nA. The change in form of the noise is indicated in Figs. 3a and b. When allowed to drift

back the original form of the noise reappeared at Ig = 265 nA.

The second and third transistors were examined with a spot scan„ Burst noise was induced in both devices at sites on the collector-base junction perimeters.

1 min

Figure 3a. Burst noise, 1^ < 250 nA. Figure 3b. Burst noise, 1^ > 300 nAo

RESULTS - ETCH PIT STUDY

Having noted the positions where burst noise was either affected or induced in the

electrical tests the devices were then etched with Dash solution. In all three devices

etch pits were found at these positions, indicating the presence of dislocations. Fig. 4

is a micrograph of one of the etched devices. The bright line running across the centre

is the intersection of the c-b junction with the surface. Many etch pits are in evidence

but the group arranged in a triangular pattern near the centre of the figure are in a 98
position where burst noise was induced

Figure 4. Micrograph of an etched CA3018

transistor

.

4 ym

CONCLUSIONS

It has been demonstrated that areas of a transistor associated with burst noise

activity have crystallographic damage although the converse is not necessarily true. More

work needs to be done in the future on the effects of parameters such as incident energy

and electron dose on the burst noise characteristics since induced burst appears to be a transitory phenomenon,

ACKNOWLEDGEMENTS

Thanks are expressed to ToJ. Brady and J.R„ Hunter who carried out some of the

experiments described here as part of their final year project at Salford University,

REFERENCES

1 G. Blasquez and J„ Carainade, 5th Into Conf„ Noise in Physical Systems, Bad Nauheim,

(1978)

,

2 R„ Radenheber, M„ Stoisiek and D„ Wolf, 4th Int. Conf„ Noise in Physical Systems,

Noordwi jkerhout , (1975)

„

99
BURST NOISE IN DIODES

O.Sikula, M.Sikulova, P.Vasina and B.Koktavy

Department of Physics

Technical University of Brno, Barvicova 85

Czechoslovakia

INTRODUCTION

According to S.T.Hsu, R.D.Whittier and C.A.Maed [l], the burst noise is

a random process caused by the gene rat ion- recombinat ion (g-r) process of

capture and emission of a carrier by a trap controlling the current through

the defect. The experimental results cannot, however, be explained satis - factorily by their model, even if the modification of K.B.Cook and A.D.Brodersen [2j is taken into account. Remarkable discrepancies are observed in

the voltage and temperature dependencies of the mean impulse duration as well as the capture cross section. The heart of the problem is in the as - sumption that the set is "pure" and thus the probability density of the ith stele occupation time is a single exponential function.

We have studied the bursit noise of LED 's , transistors, Schottky diodes

and we found out that the mentioned assumption is fulfilled only in some particular cases. The occupation time probability density is, in general,

expressed as a su\'.- of two exponential distributions.

THEORY

In the present paper we assume that the X(t) process of the trap occu- pation by a carrier is a three-state one. The state of the systeii; is determined by the number of carriers in the band, on the trap, or at the g-r cen

ter. The following table gives the system states for an accep t c r- 1 ike g-r

center. The: X-state denotes the X(t) process state.

Table 1.

X-st6te cond . band g-r center valence b. Y-state

0 n - 1 1 P a, 1 n 0 P /3

2 n - 1 0 P - 1 /3

The X(t) primary process of the trap occupation induces the Y(t) secondary

process of the current modulation. This process is a two-state g-r process.

The single zeroth state of the X(t) process is transformed into the oo-state of the secondary process, while the two states, i.e., 1 and 2 of the X(t

process are transformed into one state of the Y(t) process, which is denoted /3 . To find the absolute probability distribution we assume that the primary process states can be projected on the set I=(0,l,2), We suppose that

X(t) is stationary with constant transition probability intensHies ^a/j

(i,j=0,l,2) defined by

100
= 1 im (1)

where pjj is the transition probability at the time s from the i-state to

the j-state at the time t. If Y(t) process is such a function of the

ry process that it holds

primaY( t) = oc for X( t ) = 0 Y(t) =/3 for X(t) = 1 or 2 (2)

the we can find out the distribution function of the Y(t) process as well

as its other characteristics. The absolute probability distribution of the

ni'(t) = P{X(t) = ij i c I, can be determined by

s differential equations

X(t) process, i.e.,

the Kolmogcrov

soldn.(t)

dt

= z n,(t)L(.

.

(3)

With respect to

nj(t) = ni(o) = in the i-state:

the

n,- stationarity of

The.n we obtain

the primary process we have dJli/dt = 0

the absolute probability distribution

(4)

where M,. is minor of the Uji term in the probability intensity matrix. It

holds ^ ^ ^ . .

(5)

The absolute probability distribution of the Y(t) process, i.e., /7,( t ) = P[Y(t) = ij for i€I, I = (c«:,/3), is then

0 }

(6)

The Y(t) process is stationary. It is not, however, identical with a homogeneous markovian two-state process, even though it has the same absolute

distribution of probability. The Y(t) process differs from the homogeneous

markovian process by the distribution of the/3-state occupation time.

Now we find the matrix of the transition probability intensities for a two-charge accept o r- 1 ike center through which the excess carriers recombine

The intensity of the system transition from the i-state into the j-state is

determined by means the probability intensity of quatum transitions from

the center to the allowed energy bands and vice versa. In the Shockley- Read

model terms we get the matrix of probability intensities of transition of

the system from the i-state into the j-state

101
CD. + C p n 1

p'^

c n

n

c p. p^l

c n ^ c P

n 1

0

0

(8)

where Cp , Cn are the capture coefficients of the hole and electron, respectively , p, , n| are the hole and electron concent rait iors corresponding to

the Fermi energy equal to the trap level.

The absolute probability distribution in the oc-state is then

-1

In conditions of the t he rirodynamical equilibrium it is n^ /n

klj and n^/n = p/p^ , so that

n^= (1 + 2 exp [(E^ - )/I<t]3

- 1

(9)

exp [ (E^- Ep/

(10)

The distribution of the oC state occupation time is

get (t) = exp ( -/tot) = —- exp {-t/Tf^ )

'iX, -1

where Tgc = (c^n^ + c p)

(11)

(12)

The distribution of the /3-state occupation time (for detailed deriva -

t ion see is

9/3(t) = (-V'^i) + exp (-t/^- ) ,

where = ( ^ *^pP/^n'^p ' 82 ~ ^ ~ ^

(13)

(14)

l/c n

' n ^2 - VCpP,

The mean occupation time of the oc state is E{Tp^}

the /5 state is E {7^3 = S/'Z"/ + '^2 "'^'^-^ From eq . (14)-(16) we get

=<T^> exp { (E^- J/kTj + exp { (E^ - E^p)/l<Tj

(15)

= <T^> = V)C • and that of

(16)

(17)

The mean occupation time of the/3-state differs substantially from that derived in [1] and [2] . The expected value of the/3-state occupation time is

determined by the electron capture as well as the hole emission. The coef102
ficients and a^ express the relative participation of the two mentioned

p rocesses

.

If the capture probability intensity of a hole is lower than the probability intensity of emission of an electron from a g- r center to the con- ductivity band, than r^= 1/cnn, ; a, The mean occupation time of the

/J-state is given by the probability intensity of an electron capture only in

the case that it simultaneously holds p^ » p . This g- r center makes a trap

and the equations for Tp<, and a re identical with those derived in [ l]

(see eqs (9), (10)). When increasing the injection the hole concentration

grows up and the condition Pi» p is no longer satisfied. Then Toc drops with

increasing the forward voltage and the mean /3-state occupation time is determined also by the process of hole capture and emission.

EXPERIMENT AND DISCUSSION

We measured the burst noise on p-n juction diodes and on Schottky diodes. The noise voltage was amplified by means of a wide-band amplifier and

fed to a low-pass filter to reject other noise components than those of the

burst noise. The durations of the pulses Tpc , T^/jwere measured directly by a universal counter. The pulse duration distributions got, g/^ were evaluated

by a computer.

The probability density of the occupation of theot-state is represented

in fig.l. The mean pulse length pertinent to the cc - st at e <'tr> was found as aritmetic average of all pulse lenghts. This average value <T(^>is an estimate of the V^, which determines according to (11) the theoretical distribution of the 06 state occupation time. The teoretical distribution gcc( t ) is

represented by the heavy line in fig.l. and is in a good agreement with the

measured distribution.

The /3-state distribution consists of two components which both correspond to exponential distributions with different time constants , T'^(Fig.

2 and 4). The time constants Tj , and cannot be determined from the mean pulse durations in the /3-state. To estimate them the second and third mo- mentum of the pulse durations is to be determined. The estimate method will

be published in the next paper.

103
The frequency dependence of the current fluctuation-spectral density

(Fig, 3) is a superposition of the 1/f-type noise and the g- r noise. The ex- perimental dependence Sj can be approximated by a g- r noise spectral density with a single relaxation time fp = 156/is, This value is near the time con- stant 'f^ value found from the probability of the /3-state occupation time.

The components of the spectral density Sj corresponding to the time con - stants and <To(,> should be within the frequency range f<500 Hz and pro - bably are masked by the 1/f-noise,

CONCLUSION

The gene rat ion- recombinat ion process through the traps which have two

charge states makes a tnree-state primary process. The secondary process

representing the current modulation is a two-state process which, unlike

the models presented up to now [1,2], features two states of the primary

process projected into one state of the secondary process. Owing to this

fact the system cannot be desribed by a markovian process. Experimental study of the oO and /3 state occupation time distribution on p-n juction and

Schottky diodes indicate that the /3 state occupation time set is not pure

and consists of at least two subsets. These subsets are observed on the g^ distribution only in the case that the transition intensities of the primary process from the 0 state to the 1 and 2 state are not too different.

REFERENCES

1. S.T,Hsu, R.G.Whittier and C. A, Mead, Solid-State Electronics 1^ , 1055(1970)

2. K.B.Cook and A . D , Brode rsen , Solid-State Electronics _14 , 1237 ^1971)

3. 3,§ikula, M,'§ikulova, B.Koktavy, P,\/asina, Burst Noise in LED s, in the

Proc, of the International Conf , Radiat ive Recombination and Related Phenomena in III-V Compound Semiconductors, Prague Sept. 4-7 (1979).

104
NON-WHITE MULTIPLICATION NOISE IN THE DARK CURRENT REGIME

OF A P+npnN+ AVALANCHE DIODE

R. Alabedra, A. A. Walma, C. Maille, G. Lecoy

Centre d'Etudes d 'Electronique des Solides, associe au C.N.R.S.

Universite des Sciences et Techniques du Languedoc

34060 Montpellier Cedex, France

The object of this communication is to give experimental evidence of non-white noise

appearing in the multiplication regime of avalanche photodiodes (R.A.P.D.) for frequencies

lower than transit time effects. The darkness behaviour of the device was used to show this

effect.

1 . DEVICE UNDER ILLUMINATION CONDITION

In figure 1 is represented a crossection view of the device under test^'^

250pr

Fig. 1 Crossection view

of the n''"'n"pTTp+ diode.

5pr

I P implanted

b^fjm 7

,220pm

280pr

320tjm

830pm

28pm

p''"

( -2^2 10"^n.cm)

Guard Ring

(POCI3')

When illuminated on the n"^ side at A = 827 nm, the multiplication regime is initiated

by three current components | 1 | which are : + - The diffusion current of holes created m the n region.

- The primary current of electrons created in the outside active multiplication region.

- The primary photocurrent of carriers generated in this active region.

Devices provided by C.G.E. Marcoussis, France.

105
without any multiplication, shot noise 2 q 1^^ is associated to this photocurrent Ip^

(see figure 2)

.

10

-2 4

. f

Fig. 2 Shot noise of the

photocurrent without multiplication. 3

16"

X=8 27 n m

5j« Experiment — S. f = 2 q I ph

f = 1 0 KHz

1 0-8

I I I I I \— \ III I II.

10i-7 10"6 10

ph^ '

The statistics of the avalanche process amplifies the shot noise without any modification

of its frequency dependence | 2 |

| 3 | . This result is represented in Fig. 3.

Fig. 3 Current spectral

density as a function of

frequency at M = 40.

-21

10

<

10

-2 2

IVI = 40

X = 827nm

^ - ^ Oo

1 bill 1 III 1 1 1 1 1 1 1 III 1 III

10^ 104 10^ 10^

f (Hz) _^

10

1

The law describing the multiplication noise can be experimentally derived as :

S. (f) = 2 q I. . M with 2 < x < 4

1 mj (0

where I. .is the macroscopic incident current.

M the macroscopic multiplication coefficient due to the three components of

^^^^j

Equation 1 can be written as :

S.(f) = 2 q I. / l"" 1 mj (2)

106
where M = -—

inj

Measurements of the parameter x are performed for important bias current in order to avoid space charge spreading | 4 |. Fig, 4 gives the plot of S^Cf) versus the bias current I. The deduced value of x is 2.25, which leads to an excess noise factor of 0.25

(this result is in good agreement with those published by J.J. Goedbloed and al. I 5 I ).

-23

10

Fig. 4 Current spectral density

as a function of the bias current

at A = 82 7 nm. <C 10

-24

2. DEVICE UNDER DARKNESS CONDITION

-25 10

-26 10

Without illuminatioi^ experimental difficulties arise with respect to the extremely

high impedance of the diode, the input capacity of the amplifier and low noise level. The

nearest we can get for the moment with respect to the real dark current regime is by allowing

a multiplication of about 40. In the obscurity condition the incident current is not controlled, nevertheless a similar noise variation was expected. Noise spectra represented in Fig. 5 are not white whereas

the experimental variation of S£(f) versus bias current follow nearly the same law given by

equation 2 (see figure 6)

.

-24

10

-25 10

10-2«f

AAA

1=610-9 I=510-9a

1=410" '^A r

o o o o o o O

1=3 10"=" A

1=210

= 10 -26 310

10"^Fig.

lO** 10-^ HZ

Sj:(f) versus frequency.

'6 Fig. 6

10 Current spectral density versus

bias current.

107
These experimental results without any attempt of physical interpretation confirm the

remark of Van Vliet and al . | 3 | stating that it is not necessary to commit oneself to Poisson statistics of the primary particles.

One can notice in Fig. 5 that noise spectra are white when the bias current increases.

This fact can be explained because the conduction regime is near of the avalanche process

in which the incident current does not play any part . This behaviour starts for values of

the bias current as low as 6.10"^ A. We propose an interpretation which consists of asso- ciating to the incident current a G-R noise instead of shot noise. The impurities can be

identified as Au. This non-white noise is found but for weak darkness current (I<4.]0~^ A). As soon as a photocurrent is injected in the device the results of the first paragraph are obtained again.

REFERENCES

I 1I R. Alabedra, C. Maille, D. Ratsira et G. Lecoy, Revue de Physique Appliquee, 15, pp. 1201-1208, Juillet 1980.

I 2 I R.J. Mc Intyre, IEEE Trans. El. Dev. 13, 164 (1966).

I 3 I K. Van Vliet, A. Friedman, Z. Rucher, Part I and II, IEEE Trans. El. Dev. 26,

pp. 746-764, May 1979.

til

I 4 I G. Lecoy, C. Maille, D. Ratsira, R. Alabedra, V International Conference on Noise

in Physical Systems, Bad-Nauheim (1978).

I 5 I J.J. Goedbloed and E.T.J.M. Smeets, Elect. Lett. Vol. 14, pp. 67-69, (1978).

i08
THEORY AND EXPERIMENT OF AVALANCHE NOISE

IN THE WEAK MULTIPLICATION REGIME

C. Maille, R. Alabedra, G. Lecoy, M. Savelli

Centre d'Etudes d 'Electronique des Solides, associe au C.N.R.S.

Universite des Sciences et Techniques du Languedoc

34060 Montpellier Cedex, France.

INTRODUCTION

The purpose of this work is to verify the noise theories presented successively by

Mc Intyre | 1 | and Van Vliet ( 2 | on the basis of experimental results obtained on R.A.P.D. photodiodes (silicon diodes n''"TTpTTp"'" described in a previous work by R. Alabedra

and al. | 3 |).

The interpretation of these results will be carried out with the help of the real struc- ture modeling leading to the determination of the different currents created in the device

and of the spectral intensity of the related noise currents.

MODELING

From the experimental capacity-voltage characteristics and values of the technological

data given by the manufacturer, it has been possible to determine the doping profile of the

carriers (ionic implantation) and the distribution of the electric field and the potential

in the device | 4 |

.

The exact profile being known it is then easy with the help of Poisson, transport and

continuity equations to compute the different primary currents flowing through the device

under illumination.

In the avalanche photodiode, the primary photocurrent is the sum of three components : the injected electron and hole currents (Ion ^op^ generated current in the high

field region.

1=1+1+1 (I) o on op ow

Using numerical data from Sze | 5 | for the calculation of the electron and hole ionization coefficients a and 3 » we have computed the electron and the hole multiplication

factors Mj^ and Mp as M(x) which is related to the electron-hole pairs generated in the

strong electric field region as a function of the x direction. The last parameter M(x) was computed on the basis of relations derived by Mc Intyre | I | . The knowledge of M(x) allows

the definition of a multiplication factor M,^ associated to the primary current Ion generated in the multiplication zone.

The expression of the multiplied total current is then : I=MI+MI+MI=MI (2) n on p op w ow o

where M is the overall multiplication.

109
Figure 1 gives the variations of the

three multiplication factors as a function

of M.

In Figure 2 are represented the related

contributions to the total photocurrent I of

the components I =M I and I =M I for

n n on w w ow

an illumination on the n"*" side at A = 827 nm

and a primary photocurrent Iq of 10~7 Amperes,

The contribution of Mp Igp was found

negligible ( 10~9 Ampere).

a

E

If) I-z

O

o

o

I

a.

UJa

z

[diode 1-278-3

\ = 827nm

/

/

/

/ - / /

THEORY

I 1 I I . 10'

* If

* 'v

' ' ' ' 10^

TOTAL PHOTOCURRENT l(Amp)—

VlquJie. 1 ; E^emetT-toAxe^ photocu/uiznt compomnt{> veA^ijbi total pkotocuAAtnt.

The good agreement of the experimental

current-voltage characteristics under different illumination conditions (at X = 827 nm)

with the data given by the described model

is verified in Figure 3- The effects of the dark current can be non negligible if the injected photocurrent

is low (i.e. for I, 10-9 Ampere it is possible to observe an increase of the current

in the I-V characteristic).

10

10

[DIODE 1-278-3 I

827nm

-/

JO V

THEORY

f M„

X Mn

I ..I 10 10^

ftguAt 7 : MacAo^cop^c muIXA^pLLccition ^actoK

M veAMii ml(iAo6copi.c multxpticjcution ^acto^

M , M , M . n p w

10

E<^ .7 " 10 r

t- 4.040^04 +<>-f -to 4"+* 0

"

q:

q:o -8 O 10 O

Ia

10

DIODE 1-278-3

A= 827nr

+ E X per i ment

o Theory

4 o u> o

Breakd own

1(5' VOLTAGE V(Volts)-

10

ftguAt 3 : Photocu/iAzyvt-voltaQQ. chcvuxcJizKJjitA,CL.

110
NOISE

With the three current components 1^^, Ip and I^^ are associated uncorrelated noise

components defined by their current spectral intensity S-„(f), S. (f) and S. (f) such as

ip iw

S.(f) = S. (f) + S. (f) + S. (f)

1 m ip iw (5)

On the basis of the relations proposed by Mc Intyre | I | , it is possible to determine

these three components and the total spectral intensity Sj^(f). The above mentioned spectral

intensities are plotted versus the bias photocurrent in Figure A.

If they are compared to the experimental

results, we must point out the following re- marks :

1) From both theoretical and experimental

point of view, S^(f) can but be expressed

in an empiric way by the relation 2 q Iq

in a restricted range of the multiplication

factor.

2) The contribution of S^pCf) to the total

noise is negligible.

3) On the other hand the contribution of

Si„(f) cannot be neglected because it takes

into account the simultaneous fluctuations of

electrons and holes.

4) The theoretical results deduced from

Mc Intyre 's model | 1

| are greater than the

experimental data.

Van Vliet | 2 | has theoreticaly shown

yet that as soon as the possible number of

ionizations by primary carrier transit was

small (N < 8) , Mc Intyre 's results were over- estimated.

: 827nm • a EXPERIMENT

I IjP Mc Intyre

,» S: Van Vhet

f.l J I'll"

10" PHOTOCURRENT I [Amp].

^ks.028

10

Value given In the

litterature

[diode I-278-3|

M = 2.6

\ I 10

VIqvjlz 4 : NoXaz veAMU, pkotocuAAznt.

We have then applied Van Vliet 's variance method in which the constant values of kgj

(0.028 for silicon) has been replaced by its

computed values as a function of M (see Figure 5) I 4 I .

In Figure 4 are reported the corresponding results. They show a good agreement with

experimental datas. Goedbloed and Smeets [ 6 | have obtained the same results on the basis of

noise measurements performed in similar struc tures. This confirms that in n^'^IIpIIp''' type

photodiode, the number of ionizations per

primary carrier transit is small as soon as M <100.

F^guAe 5 ; Computed vahiz fe^^^ veA^vu,

muZtipLiccution ^olcXok M.

Ill
REFERENCES

I 1I R.J. Mc Intyre, IEEE Trans. Electron. Devices, Vol. ED-13, J_64, (1966).

R.J. Mc Intyre, IEEE Trans. Electron. Devices, Vol. ED-19, 703, (1972).

R.J. Mc Intyre, IEEE Trans. Electron. Devices, Vol. 20, 637, (1973).

I 2 I K. Van Vliet, A. Friedman, Z. Rucher, Part. I and II, IEEE Trans. Electron. Devices

Vol. ED-26, 746, (1979).

I 3 I R. Alabedra, A. A. Walma, C. Maille, G. Lecoy, This issue.

I 4 I C. Maille, These de Doctorat d'Etat, Montpellier, Juillet (1980).

I 5 I G.M. Sze, Phys. of Semiconductor Devices, Wiley, (1969).

I 6 I Goedbloed and J.M. Smeets, Electronic Letters, Vol. 14, 67^, (1978).

112
NOISE MEASUREMENTS ON PHOTO AVALANCHE DIODES

Jeng Gong and K. M. van Vliet

Department of Electrical Engineering

University of Florida

Gainesville, Florida 32611

.1. INTRODUCTION

A new theory based on "discrete device physics" was presented recently by van Vliet

et al. [1], [2]. The salient features of this theory were: (1) The noise, when plotted as

a function of the mean multiplication factor or gain M, shows breakpoints when the regime

changes from N possible ionizations per carrier transit to N+1 possible ionizations per

carrier transit; (2) the noise is always lower than the Mclntyre limit, the latter being

approached to within 5% for gains of the order of 100 or higher.

In this paper we present data on three types of diodesj RCA, Philips, and Nippon Electric, which closely confirm the new theory.

II. RESULTS OF THE THEORY

The main results of the theory for discrete ionization processes [2] , obtained with

the "method of recurrent generating functions," devised in that paper, are contained in the

following two formulas [eqs (2-1) and (2-2)]. The gain or mean multiplication factor,

M=<X>, for the regime in which N ionizations can take place in the path length W of the

high field region, is given by:

M = ^'^'ly-^^ ,

'

. (2-1)

(l+kA)^+^-k(l+A)N+^

where A is the a priori probability for ionization by an electron after the length I, necessary to gain sufficient energy to ionize, has been covered; p is similarly the

a priori probability for ionization by a hole and k=\i/\. The variance of the multiplication process was found to be, var X^=<AX^> :

M(M-l) (1-k) r , _^ o var = l-kx2 1+A , 1 A

2+A+kA + ^ [Mk— + —]| . (2-2)

We now consider the results (2-1) and (2-2) in more detail. At the onset of ionization the field is just high enough to sustain one possible ionization in the avalanche

region W; thus N=l. With increasing field, the value of the probability, denoted as A(l),

increases and so the gain increases according to eq, (2-1) . The length 5,^ necessary to gain

the ionizing energy simultaneously decreases, until two ionizations per carrier transit are

possible. The value of A(l) just prior to this is denoted as When two ionizations per carrier transit are possible, the diode switches over from the regime N=l to the

regime N=2. To realize the gain M of that operating point, the value of A for the new

regime, denoted as A(2), is considerably less than the value of ^(D^^,^ prior to the switch

over, as is found by inverting eq (2-1) for A with M fixed, taking N=l and N=2 respectively

We also illustrate this behavior in Figure 1

.

Equation (2-2) indicates that the decrease in A causes var X to make a positive jump

at this fixed M, which is called a break-point value. Figure 2 shows an overall curve of var X versus M. 113
III. NOISE REPRESENTATION

According to the variance theorem the total noise, spectral intensity , due to primary current noise and multiplication noise, is given by d

S^ = M^S^ + 2ql var XS:2qI (m2 + var X) , (3-1)

I , I pr pr ^

' d pr

where we have assumed full shot noise for the primary current noise. For the equivalent

saturated diode current of the noise we have

I = I (m2 + var X) = I <x2> . (3-2) eq pr pr

The excess noise factor is described by I =1 M^F so that

eq pr

F = 1 + (var X)/m2 . (3-3)

Instead of F we are interested in FM 2. The variance theorem result (3-2) is written

as

I /I = m2 + var X = FM2

eq pr

I, /I = M . (3-4)

d pr

Theoretically, we have plotted log (M^ + var X) versus M (see figure 3). we. nOM ptot

{ffiom thz zxpoAAjmntat data log il^^) v2AMjj> tog (I^), thzn by eq (3-4) tkii> plot ma6t be

M&ntical WAXh the. thzofioXldol auAve log (M^ + vo/i X) veA6u6 log (M), thd onLgtn. o^ the. tMo cuAvu boA-ng (il6place.d along a loauA at an angle, o^ 45° mXh fieJ^pect to the axeJ>, the. X-aXAJ) and the. y-axJj, dJj) placements beting log (^p^l- Thus, one simply needs to superimpose the experimental curve over the theoretical one, displacing the origin of the former

along a 45° line from that of the latter, until the two curves superimpose. The horizontal

(and vertical) displacement required yields I . Once I has been found, we then plot

, ^ pr pr

I /I versus I, /I on linear paper, and we compare it with the theoretical curves. From

eq pr d pr

the magnitude as well as the break points in the curve, k is obtained for the best fit.

Here we only give one log (I^ ) versus log (I^) curve (see figure 4). The curve is very

close to that of figure 3. The displacement of the origin gives 1^^ = 96 nA.

IV. EXPERHIENTAL RESULTS

In the data to be presented we show the noise as expressed in equivalent saturated

diode current I = ST-j/2q, as a function of diode current I,. Figure 5 shows I versus

eq Id ^ d eq

I, for an RCA 3393 diode labeled RCA-3. The noise I shows distinct breaks (vertical

d eq inflection points) indicated by arrows in the graph, corresponding to the transitions

N->N+1

.

Data for Philips diode (labeled PHI) are shown in figure 6; the regimes N=l to 9 are clearly discernible.

The I^^ versus 1^ curve for Nippon Electric diode (labeled N63) yielded similar

results, as shown in Figure 7. The break points are somewhat less pronounced than for the

previous diodes.

114
V. DISCUSSION OF THE RESULTS

A. RCA DIODES

For the diode RCA-3 the plot of log (I^ ) versus log (1^) » corresponding to figure 5,

is given in figure 8; this yielded I^^ = 96 nA.

We will now make a rough estimate of the ionization coefficient a, where a=X/J, and

ll=W/N. We then compute a for the various break points. The X's are inferred from the

M break-point values and (2-1). For simplicity, we replace the triangular top of the

field profile, see figure 9, by a rectangle of width W = 1.2y and a field E = 3.5 x lO^V/cm.

Now we have the values of H and a. Also, we calculate as, where s is the phonon scattering mean free path (63A at room temperature [3]) and E^/eEs, where E^ is the ionization energy

(1.6 ev) and E is proportional to applied bias V, , with E being 3.5 x 10^ at V. = 100 volts.

b b The results are plotted in figure 10. Also plotted is the theoretical Baraff curve [4]

for the parameters E /E = 0.04 where E is the optical phonon energy of 0.063 eV [5] and

R 1 R

r = 0.4. (Baraff 's r is equivalent to the mean X for the curve.)

We note that the data are of the right order of magnitude, which lends strong support

to our interpretation of the break points and the small N-numbers involved. The results

also compare favorably with the data of Lee et al. [6]. For E = 3.4 x lO^V/cm, they give

a = 2 X lO'* cm-1 (their figure 12). We have (V = 97 volts), a = 1.3 x 10'+ cm~^

.

B. OTHER DIODES

For the Philips data of figure 6, the primary current was found to be 1^^ = 10 nA. Converting the data of figure 6 to a plot of + var X versus M, the curve of figure 11 is obtained. The results are in excellent agreement with the theoretical plot for k=0.018.

The Nippon Electric diode data of figure 7 also fit the theory quite well, as shown

in figure 12, with a primary current of 1^^ = 5.1 nA, and for k=0.028.

REFERENCES

[1] K. M. van Vliet and L. M. Rucker, IEEE JKavii,. lldctfion VdvlcU ED-2_6, 746 (1979).

[2] K. M. van Vliet and L. M. Rucker, IEEE JfiaM. ElzcXHOn VzvlceJ> ED-26, 752 (1979).

[3] J. Conradi, SoUd State. ElzatA. 17. (1967).

[4] G. A. Baraff, Pfit/i . Rtvl^M) 128, 2507 (1962).

[5] C. R. Crowell and S. M. Sze, Appl. Phyi . LtttoA^ 9, 242 (1966).

[6] C. A. Lee R. A. Logan, R. L. Batdorf, J. J. Kleimach, and W. Wiegmann, Vkyi . ReviOM)

34, A761 (1964).

115

117
NOISE IN HETEROJUNCTION TRANSISTORS

G. Blasquez, F. Lavemhe, A. Marty and J. P. Bailbe

Laboratoire d 'Automat ique et d'Analyse des Systemes

Centre National de la Recherche Scientifique

7 Ave. Roche, 31400 Toulouse, France

THEORETICAL MODEL

In the majority of studies of hetero junctions it is considered that carrier transport

can be carried out by diffusion, by tunneling and by thermoionic effect. The noise associated with these three mechanisms is fundamentally shot noise. If in addition, recombination

centres exist in the hetero junction space charge, some of the carriers recombine by creating an extra noise source. Since the mean squared value of this noise is hardly different

to that of the shot noise, the total noise associated with the sum of the current components

is described, as a first approximation, by the Schottky relationship. Consequently the

noise behavior of a heterojunction transistor (H.T .) must be analogous to that of a junction

transistor (J.T.).

EXPERIMENTAL RESULTS

In order to investigate the real behaviour of these devices a mesa-type transistor has

been processed by liquid-phase epitaxy in our laboratory [ij . The growth procedure starts

first with a substrate etching by a nondoned unsaturated melt ; the substrate material used

is < 1 00 > oriented and tellurium doped n"*" _ Ga As. Next comes the deposition of an n - type

collector layer from the same melt subject to a temperature decrease. The following p - Ga

As type base layer has a thickness ranging between 0. 5 and Ijamand is strongly Ge doped

(N^g of about 10' 9 cm ~ . Finally the emitter region, which is Sn-doped GaAlAs, consists

of two layers : the first one with an Al concentration x 0.4 is weakly doped, the second

with a smaller aluminium content (x 0.2) but with a much more higher doping level. This

last layer is intended to facilitate the deposition of good quality ohmic contacts.

A study of D.C. and A.C. behavior showed that the h^^ gain varied from ten or more

to some hundreds, that the breakdown voltage BV was equal to or higher than 25 volts

and that the gain bandwidth product was of the of3er of some GHz [l] , C2^.

The noise study consisted of measuring the input equivalent voltage e^^ and the current

ijj noise generators. Measurements were made by the direct method described in [.3j .Biasing

conditions were 0.3|J.ft^ Ic <-3 m A ^ 3V<. V^t 4 'lOV . The frequency range analyzed was from lOH

to IMHz. The accuracy of the measurements was estimated at about 10 %. At low frequency

1/f excess noise has been observed. In the present state of our technology its value is

dispersed. In the frequency range where this excess noise did not predominate (typically

f >10 kHz) the spectra are white and independent of the collector base voltage. In such

conditions, and for a collector current I ^ 30 jiA , an examination of Figs. 1 and 2 shows

that the behaviour of the H.T. is analogous to that of the J.T. for 1 ^ 30 jiA the generator e has a mean squared value. Which goes through a minimum and then slowly increases.

CorreYatively the generator i has its m.s.v. which increases more rapidly than 2 I Af.

118
Figure 3 gives the equivalent input current noise generator in the common base configuration when the input is short-circuited.

119
For I ^2>0 p. A we have < ll-nol > 2.q Iq Af as expected. For Ic > ZO fx f\ an excess noise is again observed. By substituting < | i. |^>into the theoretical expressions of <( ] and<^|Ln l^^t appears that the excess noise is essentially due to that of

^1 'riol^ > • With the aim of localizing the origin of this excess, a study of i^flo as a function of the collector base voltage was carried out. The results obtained show that

^I'nol^^ is independent of V^^,. This suggests that the origin of the excess noise is due

to a generation recombination mechanism.

CONCLUSION

To summarize : simple theoretical considerations suggest that heterojunction bipolar

transistor noise is described by the same exnressions as those for junction transistors.

An experimental study of GaAlAs/GaAs heterojunction transistors confirmed this analysis for

low biasing levels.

We thank the technology center of LAAS for fabrication of the experimental transistors ; more specially, Mr G. Pierrel, Mr Pham Huu Hiep, Mrs J. Chevalier, G. Lacoste,

C. Solano and P. Fadel for their assistance.

REFERENCES

[l] J. P. Bailbe, A. Marty, Pham Huu Hiep and G. Rey, IEEE Trans, on Electron. Dev

ED27, 6, 1 160 (1980)

.

C2] A. Marty, G. Rey and J. P. Bailbe, Solid State Electronics 22, 549 (1979).

L 33 G. Blasquez, Comptes Rendus a I'Academie des Sciences, Paris, t 289, Serie B, 219 (1979).

120
DIGITAL ANALYSIS OF SUPERIMPOSED FLUCTUATIONS ON A PULSE TRAIN

A. Touboul, A. A. Walma, G. Lecoy

Centre d'Etudes d'Electronique des Solides, associe au C.N.R.S.

Universite des Sciences et Techniques du Languedoc

34060 Montpellier Cedex, France

INTRODUCTION

The output signal of a Charge-Coupled Device (Surface and Buried Channel) consists of

pulses determined by the clock pulses frequency with superimposed fluctuations due to the

non-ideal behaviour of the transfer and the storage of the charge packet. These devices play an important role in signal processing and imaging applications. Noise measurements can be

used as a means of electrical characterization. A well-knovm figure of merit for a CCD. is

the r.m.s. number of fluctuating carriers associated to the charge packet. We propose a digital method of noise measurements. Results obtained on surface-channel CC.D.'s will be

discussed.

COMMENTS ABOUT THE ANALOG METHOD

The usual procedure is described widely in literature 1 JjQ 2 3 1 • It consists of

filtering out the clock pulses (at half the clock frequency) and performing a spectral density analysis. For this purpose, the output signal is supposed to be stationary and the

fluctuations should follow a gaussian law. The weaknesses of this method are the following.

This analog method multiplies the measured spectral density at half the clock frequency

with the relevant bandwidth in order to obtain the variance of the number of fluctuating

carriers in the well. An important error arises from the fact that the measured noise is not white. Moreover, the output signal consists of pulses the amplitudes of which are related to a reference (or "reset") level. During the read-out and the reset phases, the output

amplifier is biased under different conditions. So the output amplifier gives an "average"

of the noise from which informations about the CCD. operation should be extracted. Another

limitation of this method could be mentioned if the "sampling effect" of the CCD. operation is considered. Indeed for a 3-phase CCD., a numerical factor has to be determined

according to calculations given in reference 4 J

.

DIGITAL ANALYSIS OF THE NOISE

In order to overcome all the above-mentioned difficulties, we propose a different method described in figure 1. The output signal is sampled and A/D converted after amplifying

the superimposed fluctuations by using an artificial threshold as a reference. The latter

avoids any saturation of the amplifier by the clock pulses. The data acquisition is performed at the clock frequency, but any sequence of data acquisition can be realized.

This procedure is no longer limited by the filter-response and by the noise standard

precision. Moreover it is unsensitive to drifts of the CCD. bias system. (For a 50 kHz

clock frequency data are stored in 20 m sec )

.

Before calculating the fluctuations parameters, the following tests are performed on

the data considered as time series :

- The stationarity is checked by means of a "rank test" giving the "Kendall's t" [Js]].

- The randomness of the data is verified by using the "turning-point" method 5 ] .

121
- Trend analysis can be performed to suppress a possible parasitic signal present

in the data by means of averaging the data.

Then, the following statistical parameters are calculated for each file of 1024

data : the mean value M, the variance V,

the skewness S and the kurtosis K of the

amplitude distribution, the autocorrelation

factor C.

The spectral intensity of the noise

voltage is obtained by a F.F.T. program

including a Hamming-window to avoid aliasing effects.

CCD. Threshold Fast J A/D

Amplifier Sx H 1 Conv.

Synchronization

H.P. Interface

X- Y Plotter

IHP.9872 A

Computer

HP. 9825 A

Data

Acquisition

HP. 3455 A

Fig.l : Block diagram of the experimental

set-up

.

The calculated variance is easily converted into a r.m.s. number of carriers with the

help of a numerical factor including the amplifier gain and the total output capacitance

of the on-chip read-out circuit.

RESULTS ON SURFACE-CHANNEL C.C.D.'s

'

3-phase C.C.D.'s with one level of Al gates have been tested. The rather large gate

areas allow the processing of charge packets in the order of 3 x 10^ carriers.

The stationary and random properties of the CCD. noise have been verified through

the parameters described in the previous section. The r.m.s. number of noise carriers are in the range 4400 -» 5600 for a standard delay line operation. They confirm the low noise

properties of C.C.D.'s and lead to S/N ratios better than 75 dB's. As a comparison, the

fluctuations measured on a clock pulse necessary to drive the C.C.D.'s under test have

given an equivalent number of noise carriers of 120.

To eliminate the effect of a possible pick-up (50 Hz and its harmonics) or of a deterministic trend in the noise, the data can be "smoothed" by realizing an averaging over at least 10 files (10 x 1024 samples). So, parasitic signals vanish as their phase is random.

Thornber's calculations [] 6 3>[] '^'^ '^he transfer have shown the existence of correlations between neighbouring charge packets. The calculations of the autocorrelation coefficient did not point out such a mechanism (see figure 2)

.

The frequency dependence of the output noise is represented in figure 3 ; at lower frequencies, the noise follows a 1/f law and reaches a constant level till the Nyquist frequency (f(;/2). The (1 - cos 27rf/f(,) behaviour is not observed. This result could be expected on

the basis of the correlation coefficient represented in figure 2.

Carnes et a 1. [37] had de rived a general expression of the charge variance by considering an "incomplete transfer" noise source :

= 2 n £ N (1)

s

where n is the number of transfers, e the transfer inefficiency and N the number of

carriers in the well. This expression can also be obtained by assuming t^at the trapping

on Fast Interface States 8 ^ dominates the noise.

The measured noise is in good agreement with values given by eq. 1 as long as Ng is

less than 75 to 80 % of the full well. For greater values of the charge packet, the measured noise decreases (see figure 4) . This same variation has been found for e measurements

versus the packet size.

Devices provided by LETI/MEA - CENG - FRANCE.

122
% of the full well'

J _J I I 1 L 20 40 60 80 100 Fig. 4 : Variation of the rms number of noise carriers vs. the packet size.

This last behaviour can be explained

within the F.I.S. and the S.H.R. statistics

context. Indeed the minimum value of the fluctuations of the number of trapped carriers occurs for a full well when the product ft(l-ft)

becomes null. So the value of Nr .m. s . obtained

for a full well is mainly due to the extrinsic

noise sources (charge injection, output amplifier...). The calculated value of the number

of noise carriers due to trapping mechanisms

is compared to the product result fj-Cl-ft)-

A good agreement with the theoretical law is

obtained (see figure 5).

0.03

5 Fig. 5 Comparison of f(-(I-f(-)

ted values of (Nj-^g)2.

80 95 and calcula123
CONCLUSION

The flexibility of this method of measuring fluctuations in C.C.D.'s has been checked

by applying it to various kinds of devices. These means of electrical characterization re- present a powerful tool to predict the performances of devices with reduced geometry gates

(and small packet sizes). They can give information on the lower limit of the storage cell

dimension when considering the V. L.S.I, development of C.C.D.'s. However their low noise

properties are such that other limitations as hot electrons effects may be more restrictive.

REFERENCES

[| 1 ] J.E. Carnes, W.F. Kosonocky, RCA Rev. 33., 327 (1972)

Q2 ] A.M. Mohsen, M.F. Tompsett et al . , IEEE Trans, on Electron Dev. ED-22 , 209 (1975)

1^3 21 R-W. Brodersen, S.P. Emmons et al., IEEE Trans, on Electron Dev. ED-23 , 215 (1976)

[| 4 J

D. Middleton in "An Introd. to Stat. Comm. Theory" Ed. Mc Graw-Hill (New-York,

1960) Chap. 4.

5 3 S.M. Kendall in "Time-Series" Ed. C. Griffin (London, 1976), Chap. 3->7.

[6] K.K. Thornber, Proc. IEEE 60, 1113 (1972)

[7|] K.K. Thornber, B. S.T.J. 53., 1211 (1974)

j^S J

M.F. Tompsett, IEEE Trans, on Electron. Dev. ED-2

1

, 701 (1974)

124
NOISE IN DIELECTRIC MATERIALS

O.Sikula, A ,£e rmakova , M.Germak, P.Vasina

Department of Physics

Technical University of Brno, Barvicova 85.

Czechoslovakia

INTRODUCTION

During the process of the dielectric pola riza t ion , t he time fluctuation

of the polarization vector P takes place. Consequently, there appears a noise current in the circuit. We studied the current fluctuations in a dielectric produced by a) a steady electric field, b) varying electric field.

In a capacitor in the thermal equilibrium there is a noise the origin

of which is in the stochastic nature of the interaction among the electric

dipoles and the phonon field. This is the steady state thermal noise [1.2] . In the second case, i.e., when a varying electric field is applied a non- stationary noise is produced due to the stochastic process of dipole orientation in the external electrical field and in the case of ferroelectric ma- terials due to the domain nucleation, decay and their motion.

In the short circuit the polarization vector fluctuation can be

measured [1 ] . The current fluctuations are due to the polarization fluctuations

and it holds

where ip is the polarization current, A-the area, P-the polarization vector.

Simultaneously with the polarization current there flows also a conductivity current both due to the DC sample conductivity and due to the carrier

injection from the electrode. Along with spectral density of the current

noise we measured the U-I curve and the time dependence of the DC component

of sample current.

EXPERIMENT

We studied samples of three industrially important insulating materials,

namely the paper base laminated sheets whose thickness was 0,6 mm and the

electrode diameter was 3 cm;the polyethylene terephtalate (40 ffm thichness)

and softened PVC coaxial cable with a cylindrical internal electrode of a diameter 0,8 mm and external dielectric diameter 1.6frmThe electrical parameters of PVC were as follows: the relative pe rmittivity <*> = l>.5;tg6= 0.08

at 800 Hz and the breakdown electral field strength 4.10 V/m. The heat re- sistance of this material is TO^after Martens. The U-I curves and the time

dependence of the DC current were measured by conventional methods. Fig.l

shows the U-I curve of the PVC-1 sample and the aging effect (the time cycle 28 days, the temperature SO^C). The original state is represented by the

curve 1.1. The current can approximated by the function

where for the PVC- 1 . 1 sample we have I = 3.10 A, a = 5,5 . 10 V The U-I curve of the polyethylene terephtalate sample has not been measured owing

(1)

I = Ip exp (a/u) , (2)

125
to a very high sample resistance (more than 10 ohms). The U-I curve of the

paper base laminated sheets can be expressed in the form I-vU", where the

Ohm's law region (n = 1) passes into the square law (n = 2) region at the

electric field intensity of about B.lO'^V/m.

UW

SfJQ I—I 1 1 1 1 1 1 1—I 1—I—I—I' l l—I I I 0,S 1,0 9.0 3,0 S,0 7,0 10

U/ktff ^

Fig.l. The U-I curve of the

PVC-1 sample

ssmp/c

Fig. 2. Apparatus for the mea- surement of the dielectric noise

If we neglect the transient effect of the capacitor charging (the

time constant is less than 1 msjthen

the time dependence of the DC component is due to the polarization relaxation current and it holds

I = 1^+ Vexp (-t/T-), (3)

where we have for the paper base laminated sheet = 3.6 s, for PVC

30 ms at U = 1 kV and2^= 16 ms at 6

kv. The time necessary for reaching

the steady state decreases when the

applied voltage increases.

The experimental study of the dielectric noise was carried out as follows. The sample with the measured dielectric was connected in the

circuit drawn in Fig. 2., where A is

the statistical analyzer, Z-the supply, C^R^-the filter, R^-the load re- sistor, and the coupling capacitor.

The sample was shielded and the electrode connected to the supply was protected by means of a guard ring.

The resistance of R^^was more then

100 times as less as the real component of the impedance of the measured capacitor, so that for the AC com- ponent the sample was short circuited

The spectral density S,- of the

current fluctuations depends on the

time dependence of the electrical

field strength. In Fig. 3 the time dependence of the spectral density S at

f = 800 Hz is shown for the PVC-1.

4

sample for the step voltage of 8 kV.

If we neglect the transients in the

measuring apparatus, the spectral

density of the current fluctuation

decreases to its steady state value

with a time constant of about 42s . For the polyethylene terephtalate the

time constant is of the same order,

while for the paper base laminated

sheets the speed of reaching the steady state value is by more than one order of magnitude higher.

For the paper base laminated

sheets^ the steady state spectral density vs. voltage plot at f = 800 Hz is

in Fig. 4. Within the limits of the measuring error, this result is in agreement with the value of S,- obtained in the conditions of the ramp voltage, the

slope of which is less than 10 V/s.

The spectral density S; of the PVC and the polyethylene terephtalate de126
Fig. 6. The spectral density vs.

Fig. 5, The S- vs. U plot. the ramp voltage time

rate diagram.

127
pends on the ramp voltage slope. Fig. 5 shows the spectral density S, ; for

the PVC-1.4 sample at f = 800 Hz and the ramp voltage slope 5 V/s . In Fig.

6

we bring the spectral densities at U = 8 kv for various slopes v of the ramp

voltage (2.5£ v< 250 V/s), The steady state value of the spectral density is

indicated by the dotted line.

Fig. 8. The spectral density vs. voltage plot for linearly

decreasing sample voltage

In Fig, 7 is the dependence of the current fluctuation steady state spectral density on the current for PVC-1,3 at f = 800 Hz. From this experiment

it follows that the spectral density is proportional to the square of the

current

.

The spectral density vs. voltage plot when linearly decreasing the sample field strength for PVC-1.4 is in Fig, 8.

Fig. 7. The spectral vs. current

plot for the PVC-1.3 sample

DISCUSSION

For our experiments we chose insulating materials with a wide range of

the electrical resistivity. The lower value of resistance was exhibited by

the paper base laminated sheets (the sample resistance being of the order

of 10'^j7 ), while the resistance of the PVC samples was about 10^-^_r2and that

of the polyethylene terephtalate was higher than 10'"'J7.

The S, vs. voltage plcts show a saturation both for the steady state and

the linearly increasing voltage (Figs. 4 and 5). The spectral density Sj is an exponential function of the voltage (see Figs. 5^8). At lower values of

the voltage^ the noise of the sample is lower than the apparatus backround

which is 2.4 . 10'^^ A^.

On samples which have lower resistivity (paper base laminated sheets)we

observed that increasing the electric field time rate up to 10 V/s does not

influence substantially the measured value of the spectral density which is

128
the same as in the steady state conditions (see Fig. 4).

Samples of PVC and polyethylene terephtalate have their spectral density pronouncedly dependent on the electrical field strength time rate. This

is illustrated by Fig. 6, It is seen that the spectral density vs. electric

field strength time rate v plot follows the power law, Sj ~ v We suppose that this effect is due to the process of orientation of the

dipole quasi- domains , similarly to domains in crystallic materials. When an external electric field is applied the probability of the quasi-domain to

set into a new position with a lower energy follows the exponential law. The

energy of the quasi- domains drops when the external electric field is applied

which gives rise to an electric impulse in the circuit. When all quasi-domains are oriented into the external field direction the noise saturation

takes place. To make sure that the measured noise is not due to breakdown

(which is the case supposed by Pender and Wintle [3]) we loaded same PVC sam- ples until the breakdown occurred which happened at voltage 40-50 kV , so that

in all our measurements we were far below the breakdown voltage.

The only means to pick up the polarization vector fluctuation is the current in the circuit. This is why the spectral density for the PVC- 1 .3( Fig .7)

is proportional to the square of the conductivity current and for PVC- 14

(Fig. 6) approximately to the square of the polarization current Ip = v.C , where C is the capacitance.

When reversing the polarization current at a given voltage the spectral

density for PVC and polyethylene terephtalate decreases substantially( Fig .8

,

curve a). For example, for the PVC-1.4 , at the voltage 5 kV reversing the sense of the voltage time rate, the magnitude remaining unchanged, i.e., 5 V/s , makes the spectral density decrease by one order. The same procedure

applied to the polyethylene terephtalate makes the noise decrease under the

apparatus background. Removing the external electric field from the oriented

quasi-domains gives rise to a random orientation of the dipole quasi- domains

,

thus producing noise and discharching the surface charge in the case of the

polyethylene terphalate. This case is represented by the curve b in Fig. 8.

CONCLUSION

The noise of dielectric materials of reasonably low resistivity is mea- surable both in the steady state and in the conditions of the time varying

electric field strength, while ir the case of dielectrics with high resistivity only the time varying field strength method can be used.

The spectral density vs. voltage dependence shows saturation and is proportional to the square of the total current. From the point of view cf noise the PVC and terephtalate samples behave as if they had a domain s^-ructure . To reach the f e rroelec t ric- 1 ike properties of these materials a certain

threshold value of the field strength must be applied.

REFEIRENCES

[l] D.Brophy in: Fluctuation Phenomena in Solids, ed. by R.E. Surges,

Ac. Press, N.Y. (1965) p.l.

[2] H.Bittel, Physics 83 B , 6, (1975)

[3] L.F.Pender, H.D. Wintle, . Appl . Phys , , 50 (1), 361, (1979)

129

III. 1/f NOISE

131

SURVEY OF RECENT 1/f NOISE THEORIES

M. B. Weissman

Department of Physics

University of Illinois at Urbana-Champalgn

1110 West Green Street

Urbana, IL 61801

INTRODUCTION

When current flows in many types of resistors, an excess voltage noise appears which

has a power spectrum, S(f), inversely proportional to frequency, f, or to f with a 1

.

Typically this scaling holds up to frequencies at which the excess noise is lost in Johnson

noise and down to frequencies limited by the time available for an experiment. This phenomenon, 1/f noise, has been known for over fifty years, and such progress as there has been

in explaining it has been reviewed several times [1,2,3,4,5]. In this review I shall

emphasize recent theories and those experiments which are needed for their evaluation. Due

to space limitations, I shall concentrate on relatively simple devices operating near equilibrium. For a good review including much fascinating work in more complicated systems,

see Van der Ziel [4].

With rare exceptions [6] the voltage noise appears to result directly from resistance

fluctuations which are present without current. Van der Ziel has reviewed the evidence for

this conclusion [4], of which the most elegant remains the detection of fluctuations in t^e

Johnson noise envelope [7,8]. Since the value of a resistance is given by ((y*n^q^lj )c) , where n^, p , and q^ are the concentration mean mobility, and charge of the i ^ carrier

species, and c depends on the size and shape of the resistor. Most models of 1/f noise

involve fluctuations in n or p, although for some surface effects c fluctuations may also

be involved.

NUMBER FLUCTUATIONS

The effective carrier concentration can fluctuate by two general mechanisms —

exchange with an external bath or exchange with immobile internal states. For materials

with carriers of only one sign of charge, such as metals and extrinsic semiconductors,

charge neutrality essentially eliminates the first mechanism, so that significant carrier

number fluctuations require trapping states.

In many metal film and semiconductor samples, resistance fluctuations have been found

to be roughly described by Hooge's relation [3]:

Sj^(f)/R^ = o^/Nf (1)

-3

with 0^ = 2 • 10 , where N is the total number of carriers. Integrating this exp^es^ion

over ten decades (a reasonably conservative estimate of the 1/f range) gives <(6R) /R > 1/(20 N) . This large magnitude would require that the number of traps be comparable with

the number of carriers, which is out of the question for metals. Therefore carrier number

fluctuations cannot account for the 1/f noise in metal filns [7,9,10] and whiskers

[11,12]. Kleinpenning [13] has oberved noise in space-charge limited diodes for which

magnitude arguments also appear to rule out carrier number fluctuations.

In semiconductors the number of carriers is not comparable to the number of atoms, so that one cannot immediately exclude the possibility that the 1/f noise results from trap

occupation fluctuations. Each independent trap contributes an amount p(l-p) to the mean square fluctuations, where p is the occupation probability, as for the variance in any two state system. The ordinary donors in an extrinsic bulk material have a very small p at room temperature, and it is very temperature sensitive. Neither the magnitude nor temperature spectrum of 1/f noise can be accounted for by trapping by such shallow donors. What

133
would be required is a range of trapping depths (to produce approximately temperature-independent noise) and a range of characteristic trapping times for each energy (to account for

the 1/f spectrum). Partially occupied traps imply temperature dependent carrier concentrations. An approximate relation may be shown [14]:

where N is the number of carriers, n is the quantum concentration and < = <(6N) >/N »

£.n(n /n) » 6 (this analysis only applies for non-degenerate semiconductors), we find

9JlnN?8£nT = 0.4. So far as I know, no such large temperature dependences of the number of

carriers are ordinarily found in noisy samples, although there seems to have been little

effort to find them. Thus it is unlikely that samples obeying equation (1) or having more noise are showing simple number fluctuations. This restriction applies regardless of

whether the traps are in the bulk or near the surface. Smaller noise magnitudes may of course be due to simple number fluctuations.

Pelligrini [15] has recently suggested that the trapping and release of charge

carriers by defect clusters may be a process with a natural spread of relaxation times and

trap depths due to random variations in cluster sizes and shapes. As with any number fluctuation theory, the island trap theory cannot quantitatively account for the noise in

metals and in many semiconductors.

P. H. Handel and coworkers have set forth one of the most intriguing general theories

of 1/f noise, which was recently summarized [6]. The 1/f noise phenomenom is attributed to the infrared divergence found in cross-sections for inelastic scattering [bremsstrahlung]

.

Semi-classical calculations indicate that interference terms between elastically scattered

and inelastically scattered waves can cause 1/f fluctuations in the current density. Ngai

[7] has proposed that the same effect may arise when the energy lost in the inelastic scattering goes into certain "correlated states" of the system rather than into photons.

Tremblay, however, has made a fully quantum-mechanical version of Handel's calculations

without finding a predicted 1/f noise [18].

Regardless of the resolution of that dispute, it is clear that actual 1/f noise does

not result from the infrared divergence mechanism. In that theory, one may easily show

that the square of the wave function itself exhibits the same 1/f fluctuations as the

current density. Thus I consider these theories to be number fluctuation theories. This

is not surprising, since the theory explicitly neglects any momentum difference between the

elastic and inelastic wave. However, the integral over the entire wave packet of the

square of the wave function is a constant, so that, if present, the local 1/f fluctuations

would integrate to zero over the wave packet.

MOBILITY FLUCTUATIONS

Hooge and coworkers [3] have proposed that equation (1) applies to all homogeneous

materials, with the conductivity fluctuations resulting from independent fluctuations of

the mability of each carrier. This formulation has since been modified to include only

fluctuations in lattice scattering [19], not impurity scattering, so that the numerical

prefactor is material dependent. In ionic solutions it has been proposed [20] that this

factor is proportional to ionic concentration; in effect this leaves a formula in which the

noise is inversely proportional to volume, with no direct dependence on the number of

One obvious objection to any ncbility fluctuation description of 1/f noise in semiconductors is that the Hall coefficient has been found to fluctuate by about as much as the

resistance [21,22], while for mobility fluctuations equally affecting all carriers no Hall

coefficient fluctuations would result. Kleinpenning and Bell [23] have pointed out, however, that if the noise is inversely proportional to the number of carriers the carriers

would experience independent mobility fluctuations. If, as in a non-degenerate semicon-

(2)

carriers

.

134
ductor, there are a range of mobilities Hall coefficient fluctuations would be predicted

[22,23]. However, attempts to show that the magnitude of the Hall effect noise fits such a model even better than a simple number fluctuation model [22] have been brought into

question [24 ]

.

Kleinpenning has found that the relative magnitudes of ordinary resistance fluctuations and noise in the thermoelectric effects in intrinsic and extrinsic semiconductors can be well described by the independent carrier fluctuation description, but not by simple

number fluctuations [25] . More importantly, he found a rather low coherence between thermoelectric and resistance noises. Since the spatial weighting functions [26] for both heat

flow and current flow follow the same pattern in a homogeneous material, this observation

is inconsistent with any model, including number fluctuations, in which the fluctuations

are locally describable by a single parameter, which would give a coherence of one or minus

one. However, since this experiment used point-contact resistors one cannot completely

exclude the possibility that surface effects were involved, in which case different spatial

weighting patterns for heat flow and electrical current could give a relatively uninteresting explanation of the lack of coherence.

Despite its popularity the independent carrier ncbility fluctuation model must be

rejected on simple physical grounds. The explanations of the Hall effect and thermoelectric noise observations [23,25] would require the fluctuations to remain associated with

carriers of a particular kinetic energy for times on the order of a fluctuation lifetime —

at least a second. Since the carrier scattering time is about a picosecond this would be

impossible if each carrier fluctuated independently [27]. Furthermore, 1/f noise in

samples with carrier transit times in the microsecond to millisecond range looks no different from that in the relatively rare samples with longer transit times. Obviously low

frequency fluctuations are not tied to individual carriers which hang around for less than

a millisecond.

If Hooge's relation held in many types of materials, it would be difficult to explain

the 1/N behavior except by independent carrier fluctuations. However, very great deviations of the numerical prefactor both toward larger and smaller values may be found, as summarized by Van der Ziel [4]. The recent results of VandeVoorde et al . [28] on InSb

crystals are a particularly good example, since the sample was apparently quite well characterized. With some surface treatments, the noise was more than two orders of magnitude

smaller than relation (1) would predict. In one of the best known measurements of very low

frequency excess noise in Ge, the frequency dependence was f"^*"^, which obviously cannot

agree with equation (1) [29].

Hooge and Vandamme [19] found that in semiconductor samples so heavily doped as to give significant impurity scattering, ci , decreased as the inverse square of the mobility.

This effect was interpreted to mean that only phonon scattering cross-sections fluctuated.

It was subsequently shown [27] that their data did not fit the independent carrier model,

and thus they remain unexplained.

The results of Hooge and Gaal [20] who obtained 1/f noise of about the same magnitude

in the resistance and concentration-gradient voltage of ionic cells remain to be explained,

if the unphysical hypothesis of independent long-lived mobility fluctuations is abandoned.

Van der Ziel [30] has suggested that surface trapping of ions could account for independent

fluctuations in the number of mobile anions and cations, which would produce comparable

magnitudes of resistance and concentration-gradient noise. However, the observed magnitude

of the noise could only be accounted for by assuming strong positive cooperativity in the

binding for each sign of ion separately, with little cooperativity between the anion and

cation binding [30]. This would of course be exactly the opposite of what vrould be expected from thermodynamics for ordinary binding processes, given the electrostatic interactions .

There is one well-known mechanism, loosely referred to as temperature fluctuations,

which does cause mobility fluctuations. Voss and Clarke [7] obtained evidence that the 1/f

noise in metal films results from temperature fluctuations. The principal evidence was a similarity of noise magnitudes in various films except for manganin, which has a low tem- perature coefficient of resistivity; and a strong positive correlation between the low

135
frequency noise in adjacent portions of a strip of Bi . The noise was attributed to equilibrium temperature fluctuations, although the spectral shape could not be explained [7].

The expected magnitude of equilibrium temperature fluctuations is determined by thermodynamics [31]. So long as the heat capacity and thermal coefficient of resistivity are uniform throughout a conductor, regardless of the function describing the space-time evolution of temperature fluctuations, a simple relation exists between the autocorrelation

function of the resistance fluctuations due to equilibrium temperature fluctuations and the

Green's function for the resistance charge produced by Joule heating [32,33,34]. This

relation has been confirmed experimentally in two disparate systems with high temperature

coefficients of resistivity — an ionic resistor [34] and (with a somewhat adjustable magnitude) metal films at the superconducting transition [35] . The low frequency limit of the

thermodynamic prediction is particularly simple [33]:

^v^°^ ^ AR 3toR . . 4kTR R SinT ^

where is the spectrum due to temperature fluctuations and AR is the change in resistance

due to Joule heating. Except for materials with exceptionally large values of 3X,nR/9J!.nT,

spontaneous temperature fluctuation noise is always small compared with Nyquist noise. In those systems where spontaneous temperature fluctuations have been observed, it has not had

a 1/f form. Very low frequency temperature fluctuations associated with the thermostat

rather than the sample have also been observed [28,36], but these are neither 1/f noise nor of much theoretical interest.

We have not yet accounted for the Voss and Clarke data. The absence of noise in manganln has been disputed [37], although the contrary data were obtained in a point-contact

sample which might exhibit surface effects. Manganin might also be distinguished by its

relatively small fraction of phonon scattering. Horn's group has shown that the temperature dependence of the noise magnitude does not resemble that predicted for spontaneous

temperature fluctuations [10]. The spatial cross-correlation experiments are most crucial.

Recent work in my lab [38] on Cr films of dimensions similar to those of the origifial Bi

samples, except with the distance between the two observed regions reduced by a factor

of ~ 6 has revealed no cross-correlations. Work in Webb's lab on Au films in good thermal

but not electrical contact has also revealed no cross-correlations [39]. J. Clarke has

suggested that the possibility of some ar tifactually correlated signal resulting from instability of the Bi samples cannot be eliminated [40] . Therefore we conclude that there is no longer any reason to suspect any role for equilibrium temperature fluctuations in metal

films

.

Several theoretical papers have appeared to explain why spontaneous temperature fluctuations can be Important for low frequency excess noise [41,42,43,44]. One [41] has been

shown to be inconsistent [45], while the others are not altogether convincing. In view of

the experimental situation, I shall not discuss them.

Min [46] has argued that mobility fluctuations in nondegenerate semiconductors can only be found in systems not in quasi-equilibr iium. The argument is essentially that

Fermi-level fluctuations do not change the average mobility [46]. However, spontaneous

occupation number fluctuations of different momentum states will give mobility fluctuations, but on a picosecond time scale.

To avoid these difficulties while retaining a 1/f noise model in which only electronic, not lattice, properties fluctuate, Min postulates [47] that the presence of traps

may disturb the quasi-equilibr ium distribution of electrons. In order to obtain slow fluctuations, Min makes two additional claims. First, that free electron trapped electron

processes are fast compared with intraband scattering; and second, that the rate of these

trapping processes scales as a power of the energy above the band minimum. The result is that there is a range of trapping times caused not by a range of trap parameters but by the

inevitable spread of carrier momenta. By adjusting the rate-energy scaling parameter one

may obtain a 1/f prediction [47]. (A somewhat simlar model was once put forth by Burgess

[48] and criticized by McWhorter [1].)

136
The assumption that Intraband scattering may be neglected, however, can apply only in

the momentum range for which the trapping process is very fast. The low frequency noise in

the model comes from low-energy electrons, for which intraband scattering would still dominate. The characteristic rate for occupation number fluctuations of these low-energy

states is not slowed by adding a parallel trapping pathway. No 1/f noise would actually be

predicted if the intraband scattering were not neglected. In view of this, I shall not

consider other difficulties with the model.

SURFACE EFFECTS

A very substantial amount of experimental evidence, well summarized by Van der Ziel,

indicates that surface effects are responsible for much of the 1/f noise in semiconductors

[4]. The situation in thick metal films is less clear because most experiments involve

similar surfaces and surface to volume ratios.

McWhorter proposed that number fluctuations due to trapping in surface oxide states

could cause 1/f noise [1]. The 1/f spectrum may be predicted as a natural consequence of a tunneling rate distribution resulting from a uniform distribution of traps in a thick oxide

layer [1]. Part of the appeal of this proposal is that the spectrum thus appeats from

simple, almost first-principles considerations rather than from ad-hoc constructions.

Unfortunately, experimental data generally indicate that almost any surface treatment

affects the magnitude of the 1/f noise without changing its spectrum [1,34]. That is, each

layer appears to contribute a 1/f spectrum itself, so that the spectral shape may even for

surface effects require some ad hoc assumptions. One piece of experimental data shows

signs of preferential removal of low frequency noise sources by surface cleaning, but the

spectra are still not consistent with the original McWhorter explanation [28].

Van der Ziel and others [4] have pointed out that in thin-channel devices such as MOSFET's the. fluctuating surface trap occupancy can have a greater effect on carrier

mobility than on carrier concentration. One may also view such fluctuations as noise in

the boundary-condition parameter. It would be of great interest to investigate whether, in

simple semiconductor and metal resistors, fluctuating net surface charges or fluctuating

patterns of surface charge could produce significant mobility fluctuations. A fluctuating

pattern of surface charges would produce fluctuating interference terms in the Coulomb

scattering, analogous to the flicker pattern seen in laser light-scattering off solutions.

In addition such a fluctuating pattern can cause per eolation- like effects [50].

The fluctuating scattering cross section is Inherently wave vector dependent and hence

could give some of the experimental results which require multiparameter fluctuations. A

simple dependence of mobility on total occupied trap concentration in a region, such a Van

der Ziel has suggested [4] can produce Hall effect fluctuations due to the energy-dependent

proportional contribution of lattice scattering, but remainas a locally single-parameter

model. Contrary to Van der Zlel's speculation [51] no plausible mobility fluctuation model

not based on independent carriers can fully reproduce Kleinpenning' s model. I have discussed some of the experimental implications elsewhere [27].

McWhorter [1] also considered the effect of trap occupation on the band gap near the

surface. In near-intrinsic materials this effect can be large, but it cannot be Important

in strongly extrinsic materials and of course plays no role in metals. Such net carrier

number fluctuations are electrophoretically transported with the minority carriers — this

effect was observed experimentally [49]

.

DEFECT DIFFUSION

Horn's group has extensively studied the temperature dependence of the noise magnitude

In several types of metal films on several types of substrate [10,5]. Although they have

not set forth a specific model, they have pointed to some likely features of one. The most

striking result [52] was the correspondence noted between the temperature dependence of the

noise magnitude in a particular frequency range and deviations from an f ^ spectral shape

in the same range. If one assumes that the net fractional resistance fluctuations are temperature independent but that their rate (spectral distribution) does depend on temperature, one finds that the flat, low frequency part of the spectrum drops as the tem- perature is raised. The sharp, high frequency part increases.

137
Dutta et al . [52] assume that the noise is expressible as the sum of many Lorentzians

with characteristic times given by the same Arrhenius prefactor t with a range of activation energies. They obtain the experimentally checkable result

where 32.nS(aj,T)/8Jin(jj = -a(a),T). This prediction qualitatively fits their data, except for

a feature in the noise magnitude near 550°C. A roughly comparable fit could be obtained by

assuming a constant activation energy with a range of prefactors [14]. The peak magnitude

of the noise occurred at a temperature that was a monotonically increasing function of the

lattice cohesive energy [52].

The strong implication is that defect diffusion is involved in generating the noise.

(Arguments against such mechanisms in semiconductors [1] do not apply to these metal

films.) Reproducibility of the noise magnitudes by different labs [7,9,10,38] suggests

that impurities are not necessary. However, defect creation and annihilation would show

far too great a temperature dependence, so a non-equilibrium defect concentration is likely

to be involved. The question arises — why should the resistivity depend on the positions

of defects?

Macroscopic inhomogeneities in the current density due to sample geometry are unlikely

to supply the answer, because diffusion is too slow to give noise in the right frequency

range. Diffusion very close to surface irregularities is also not likely to explain the

effect, because the defects which can diffuse near the surface reach an equilibrium con- centration. The explanation may lie in the dependence of the conductivity on the relative

positions of defects [53]. One such dependence arises from interference terms in the scattering. Yakota [54] has calculated the magnitude of this effect, finding that it is not necessarily tiny. The relevant distance for diffusion to affect conductance in this

mechanism is an inverse Fermi wave vector,, about one lattice spacing [14], not the typical

nearest-neighbor defect spacing assumed by Yakota, so that essentially all the defects are mobile in the time-temperature range of interest. The range of activation energies might

itself be due to defect-defect interactions.

A major difficulty with Yakota 's explanation is that if the noise was due to residual

non-phonon scattering, one would expect its net magnitude to decrease rather sharply as the

temperature increased. Perhaps the alteration of the phonon dispersion relation by the

presence of the defects causes the phonon scattering rates themselves to fluctuate. Likewise the carriers see a modified lattice which will affect their dispersion relation.

These effects have not been calculated, so far as I know.

It is interesting to note that all of these effects, as well as some surface effcts,

predict tensor, not scalar, fluctuations in the conductivity. Although one may have a cubic lattice with scalar conductivity and even have equal mean-square fluctuations along

any projection, the instantaneous conductivity will not be a scalar. Techniques for

measuring this effect have been proposed [24]. This property contrasts sharply with the

inherently scalar nature of any fluctuations resulting from simple number fluctuation processes .

Lest it seem that 1/f noise in metals is nearly understood, I should point out that

two groups have now reported 1/f noise in metal whiskers about 2-3 orders of magnitude

larger than in metal films of similar volume [11,12]. Equations (1,2,3 and 4) were not obeyed. Reasonable, if not airtight, precautions against contact noise were taken. The

whiskers have less surface area and many less defects than the films. No theory appears to begin to explain these results.

WEIGHTING FUNCTION SINGULARITIES

It has long been known [55] that diffusion does not produce 1/f noise in any variable,

B, which affects resistance through an equation of the form

(4)

138
6R(t) = /6B(r ,t)W(r)d'^r (5)

-»• >• so long as W(r) is non-singular. For the ordinarily considered W(r)'s, which have a delta

function singularity in the gradient, an f~^'^ high frequency limmit of S(f) is found —

the "universal" three-halves power law for diffusion.

In an idealized contact resistor a singularity occurs in the electric field [56].

This singularity gives a predicted 1/f noise spectrum, with a low frequency cutoff determined by the size of the contact and a high frequency cutoff determined by the smallest

distance over which the macroscopic idealizations hold [57,58]. For diffusion, frequency

scales as the inverse square of distance, so a distance scale range of a factor of 10^

would give about six decades of 1/f noise. Very few plausible geometries could give more

than about eight decades by such a mechanism.

The singularity theory predicts that the W(r) singularities which give 1/f noise for

diffusion will also do so for other fluctuation transport mechanisms [58]. The theory has

been confirmed in a somewhat contrived experimental arrangement in which light scatterers

were flowed past a cylindr ically focussed beam [59].

In the ionic contact resistors studied by Hooge and Gaal [20] , the 1/f law was only

approximately obeyed over a frequency range of one or two decades. Simple carrier number

fluctuations would give 1/f resistance fluctuation in about the right frequency range, but

with a magnitude slightly lower than the smallest found. They would not give noise in

concentration-cell voltage. However, electrophoresis or diffusion of small charged contaminant particles could give noise in about the right frequency range. As in Van der Ziel's

model [30] it vrould appear as largely uncorrelated fluctuations in the anion and cation

mobility, but with the large, concentration independent, somewhat irreproducible magnitude

appearing naturally. The absence (reduction by at least two orders of magnitude from the

prediction of [20]) of 1/f noise in long pores filled with meticulously filtered solutions

[60] supports this interpretation. So does the wide range of amplitudes found by another

group [61].

In lipid bilayers conductance usually occurs in discrete channels which are free to diffuse in two dimensions. The conductance of a pair of channels is reduced as they

approach each other, due to field-line overlap, with the effect scaling inversely with

distance. As a result one may predict 1/f noise over about eight decades centered around

one Hz, with a magnitude determined by the single-channel conductance [53] . The observed

magnitudes are larger than predicted by this effect, being apparently dominated by gating

mechanisms with a range of activation energies [62,63]. Nevertheless, this case remains

interesting because the 1/f prediction follows directly from well known fundamental

features of the system, with no ad hoc assumptions and with no adjustable parameters in the

magnitude.

It is tempting to consider the possibility that defect diffusion in metals might give

1/f noise directly through singular defect-defect interaction terms. However, in three

dimensions the required singularity would have an r~ ' functional form, which does not correspond to anything I know of. A 1/r dependence for point like objects diffusing on a surface could produce a limited range of 1/f noise. So could a 1/r dependence for pointlike defects diffusing in the vicinity of stationary line- like defects [57]. Overall,

however, a range of time constant logarithms as in tunneling or activated processes seems much more likely to account for very broad 1/f spectra, as found in semiconductors, than

does a natural distance scaling.

STATISTICAL CONSIDERATIONS

The simplicity of the 1/f spectrum encourages speculation (e.g. [2]) that some general

statistical considerations may account for it. The central limit theorem predicts that

when an activation energy, barrier height, or tunneling distance appearing in an expression

for the logarithm of the characteristic frequency is affected by many random parameters,

its distribution function will approach a Gaussian shape. Since near the maximum such a distribution is flat, a range of 1/f noise is expected. However, the wings of the Gaussian

139
also give power-law spectra. In fact, if the homogeneous lineshape is Lorentzian, there are fewer decades approximating I/f than any other power law between f° and f ^. It appears that the prevalence of 1/f noise is nearly equivalent to the assertion that distribution functions for logarithms of rates tend not only to be broad but to have wings that

fall off much less sharply than Gaussians. Except in special cases this remains unexplained.

CONCLUSIONS

Despite the progress that has been made in accounting for the 1/f noise in some devices, particularly MOSFETS, the substantial noise very nearly following the 1/f law that

appears in such simple materials as metal films, metal whiskers, semiconductor filaments,

and liquid metals [64,65] has not been satisfactorily explained. Even the relative roles

of surface vs volume effects or of mobility vs carrier number fluctuations are not well

established. Much of the theoretical work has been wrong, some merely irrelevant.

McWhorter's words still apply [1]: "The similarity of the 1/f noise from the various

devices mentioned earlier certainly leads one to look for a common mechanism, but so far it

has been difficult to get a satisfactory explanation for even one device.".

Nevertheless, we may make some tentative judgements. Some variant of the McWhorter

model including mobility fluctuations at internal noise sources at grain boundaries [66] seems to be the most likely explanation for the noise in semiconductors. The dominant

mechanism in metal films may involve a rate-limiting defect diffusion step. Metal whiskers

and liquid metals are very poorly understood although it is possible that in the liquid

metal experiments a fluctuating charge on the oxide surface of the contacts produced the

noise.

Recently there has been a renewed emphasis on studying the relations between resistance fluctuations and other fluctuating electrical properties such as the Hall coefficient

and thermoelectric EMF's. While the initial results have not been as clear cut as one might hope, this type of experiment, closely tied to theory, appears to offer the best road

toward understanding 1/f noise.

This work was supported by NSF grant DMR 80-07057 and, through the Materials Research

Laboratory by DMR 77-23999. I think J. Clarke, P. M. Horn, A. M. Tremblay, G. Feher and W.

W. Webb for helpful conversations at one stage or another of my work in this field. None,

of course, bears any responsibility for the controversial views expressed here.

REFERENCES

[1] A. L. McWhorter, in Semiconductor Surface Physics , Ed. R. H. Kingston (Univ. of

Penn., Philadelphia, 1957) p. 207.

[2] D. A. Bell, Electrical Noise (Van Nostrand, London, 1960) p. 140.

[3] F. N. Hooge, Physica 83B, 14 (1976).

[4] A. Van der Ziel, in Adv. in Elect, and Elect. Phys. , V. h9_, Ed. L. Mar ton and C. Marton (Academic Press, New York, 1979) p. 225.

[5] D. A. Bell, Proc. 2nd Int. Conf. on 1/f Noise (1980) p. 19 (unpublished).

[6] S. Lifson, B. Gavish and S. Reich, Biophys. Struct. Mechanism 4, 53 (1978)

[7] R. F. Voss and J. Clarke, Phys. Rev. B13 , 556 (1976).

[8] H. G. E. Beck and W. P. Spruitt, J. Appl. Phys. 49, 3384 (1978).

[9] F. N. Hooge and A. M. H. Hoppenbrouwers , Physica 45 , 386 (1969).

140
[10] J. W. Eberhard and P. M. Horn, Phys . Rev. B18 , 6681 (1978).

[11] P. Dutta, J. W. Eberhard and P. M. Horn, Sol. St. Comm. 21, 679 (1977).

[12] C. Leemann, M. J. Skove and E. P. Stillwell, Sol. St. Comm. 35 , 97 (1980).

[13] T. G. M. Kleinpennlng, Physlca 94B, 458 (1978).

[14] M. B. Weissman, unpublished.

[15] B. Pellegrini, Phys. Rev. B22, 4684 (1980).

[16] P. H. Handel, Phys. Rev. A22 , 745 (1980).

[17] K. L. Ngai, Phys. Rev. B22, 2066 (1980).

[18] A. M. Tremblay, Thesis, M.I.T. (1978).

[19] F. N. Hooge and L. K. J. Vandamme, Phys. Lett. 66A. 315 (1978).

[20] F. N. Hooge and J. L. M. Gaal, Philips Res. Repts . 26 , 77 (1971).

[21] J. J. Brophy, Phys. Rev. 106, 675 (1957).

[22] T. G. M. Klelnpenning, J. Appl. Phys. 51 , 3438 (1980).

[23] T. G. M. Klelnpenning and D. A. Bell, Physica 81B, 301 (1976).

[24] M. B. Weissman, J. Appl. Phys. 51 , 5872 (1980).

[25] T. G. M. Kleinpennlng, Physica 77, 78 (1974).

[26] J. M. Richardson, Bell. Syst. Tech. J. 29 , 117 (1950).

[27] M. B. Weissman, Physica lOOB, 157 (1980).

[28] P. VandeVoorde, C. K. Iddings , W. F. Love and D. Halford, Phys. Rev.

B19, 4121 (1979)

[29] M. A. Caloyannides, J. Appl. Phys. 45 , 307 (1974).

[30] A. Van der Ziel, Physica 96B, 81 (1979).

[31] J. W. Gibbs, Elementary Principles In Statistical Mechanics (Yale, New Haven, 1902)

pp. 68-86.

[32] M. B. Weissman, Appl. Phys. Lett. 32 , 193 (1978).

[33] M. B. Weissman, Phys. Rev. Lett. 41, 1 (1979).

[34] M. B. Weissman and G. D. Dollinger, J. Appl. Phys. (in press).

[35] M. B. Ketchen and J. Clarke, Phys. Rev. B17, 114 (19 78).

[36] J. Clarke and T. Hsiang, Phys. Rev. B13 , 4790 (1976).

[37] T. G. M. Klelnpenning, reported in Ref. 3.

[38] R. D. Black, F. Fllegle and M. B. Weissman, in these Proceedings.

[39] W. W. Webb,, private communication.

141
J. Clarke, private communication.

S. H. Liu, Phys. Rev. B16, 4218 (1977).

S. Putterman, Phys. Rev. Lett. 39 , 585 (1977).

I. Pavlin and S. Englesberg, 2nd Int. Symp. on 1/f Noise (1980) (unpublished).

M. Mikulinsky and S. Starobinets, Phys. Rev. B21 , 5558 (1980).

M. B. Weissman, Phys. Rev. B18, 4538 (1978).

H. S. Min, J. Appl. Phys. 50 , 4461 (1979). ^

H. S. Min, J. Appl. Phys. 51, 1637 (1980).

R. E. Burgess, Brit. J. Appl. Phys. 6, 185 (1955).

H. C. Montgomery, Bell Syst. Tech. J. 31, 950 (1952).

M. Aoki, H. Katto, E. Yamado, J. Appl. Phys. 48 , 5135 (1977).

A. Van der Ziel, Appl. Phys. Lett. 33, 883 (1978).

P. Dutta, P. Dimon and P. J. Horn, Phys. Rev. Lett. 43 , 646 (19 79).

M. B. Weissman, Biophys. J. 21, 87 (1978).

M. Yakota, Proc. 2nd Int. Conf. on 1/f Noise, (1980) p. 484 (unpublished).

K. M. Van Vliet and J. R. Fassett, in Fluctuation Phenomena in Solids, Ed. R. E. Burgess (Academic Press, New York, 1975) pp. 267-354.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon, Oxford, 1904) Vol.

I, p. 328-329.

M. B. Weissman, Phys. Rev. Lett. 35 , 689 (1975).

M. B. Weissman, J. Appl. Phys. 48 , 1705 (1977).

M. B. Weissman, R. A. Isaacson and G. Feher, Phys. Rev. Lett. 43 , 733, (1979).

G. Feher and M. Weissman, Proc. Nat. Acad. Sci. USA 70 , 870 (1973).

D. L. Dorset and H. M. Fishman, J. Memb. Biol. 21 , 291 (1975).

R. Sauve, , private communication.

S. M. Bezrukov, G. M. Drabkin, L. A. Fonina, A. I. Irkhin, E. I. Meluik and A. I. Sibilev, preprint 598 of the Acad, of Sci. USSR, Leningrad Nuc. Phys. Inst. (1980).

J. T. M. Stroeken and T. G. M. Kleinpenning , J. Appl. Phys. 47 , 4691 (1976).

J. Kedzia and L. K. J. Vandamme, Phys. Lett A66, 313 (1978).

H. I. Hanafi and A. Van der Ziel, Sol. State Elect. 21, 1019 (1978).

142
l/f Fluctuations in Biological Systems

Toshimitsu Musha

Department of Applied Electronics

Tokyo Institute of Technology

Nagatsuta, Midoriku, Yokohama 227, JAPAN

INTRODUCTION

Since Johnson reported in 1925 the first observation of enhancement of a low-frequency

tail of the spectral density of shot noise passing a vacuum tube, a number of papers have

been published that report dc electric current through (or dc voltage across) an electric

resistor have the so-called l/f spectral density regardless of the kind of material composing the resistor. At the moment it is believed that l/f noise is a result of conductivity

f luctuations[ 1 - 5 ]• The mechanism for l/f fluctuations is not fully understood as yet.

On the other hand, l/f fluctuations have also been found in various other fields; typical

examples are frequency fluctuations of well-regulated quartz oscillators and of atomic

clocks [6 - 8], and cellular membrane potential fluctuations [ 9 - 12]. The present paper

aims at suggesting some relationship of human perception to l/f-like fluctuations. Firstly,

some examples will be mentioned that show bodily l/f-like fluctuations and then external stimuli of l/f spectral densities that are accepted

by the human body are described; finally l/f-like noise

that appears in biological information transmission will

be discussed.

BODILY FLUCTUATIONS

EEG alpha wave : The electroencephalogram (EEG) is

a potential fluctuation observed across electrodes placed

on the specified locations on the head, which reflects mental

activities. The EEG is divided in four components according

to the frequency. The alpha wave is a component between

8 and 13 Hz and its amplitude is large as compared with

other components. Suzuki et al. [13] counted the number

of zero crossings of the alpha wave in 1 sec to estimate

the spectral density of frequency fluctuations. 15 Subjects

were examined. Figure la refers to a subject placed in

a dark, quiet room, and figure lb refers to a subject who

was hearing audio impulses at 1 kHz at large sound levels.

The subject in a quiet environment extends the l/f spectrum

down to 0.02 Hz but the subject in a noisy environments

limits the l/f spectrum above 0.1 Hz. When a patient comes round from narcosis after a surgical operation, spectral

density of the alpha wave is almost white. After taking

a pain killer, however, the l/f spectrum is gradually recovered [14]. Extension of the l/f spectrum toward lower frequencies is an evidence of pleasing.

Heart beat period : The cardiac sinus has pace-making cells that make spontaneous electric oscillations. We measured the heart beat period fluctuations of a normal subject.

The beat period was defined as a time interval between the R peaks of the electrocardiogram

and a spectral density estimated is shown in figure 2. A peak at 0.3 Hz was caused by

breathing; for lower frequencies the spectral density is approximately proportional to l/f.

The body temperature was measured simultaneously with the heart beat period, and a coherence

function was estimated. It was 0.3 to 0.5 and hence their correlation is very weak.

143

10'^ 10"^ Hz

FREQUENOf

Fig. 1 Spectral densities of

the alpha wave of a subject

in a quiet (a) and noisy (b)

environment. The vertical

axis is arbitrary scale.
Body sway : When one stands upright, the body sways

laterally and back and forth. The horizontal position

of the center-of-gravity of the body was measured with

GRAVICORDER that has three strain guages on a trangle

supporting a metallic plate on which one stands. The

spectral density of its lateral fluctuation when one stood on a single foot is plotted in figure; this is

proportional to l/f below 1 Hz. If eyes are closed

the fluctuation amplitude would be larger but the shape

of the spectrum would remain the same. We have found

two types of the spectrum; the other has a wavy structure above 1 Hz; we tentatively call them types A and

B. The spectral shape is suggesting mechanism of con- trolling the posture of the body.

EXTERNAL STMULI

Music: The spectral density of fluctuations in

the loudness of (especially classic) music is found

to be l/f type down to 5 x 10 Hz, and the frequency

fluctuations of music also have the l/f spectrum down

to the inverse of the length of the piece of music [15].

Character of the composer is hidden in the phase relations between sinusoidal vibrations with audio frequencies. Voss [15] presented three pieces of music

composed of random numbers that we^e sampled from fluctuations with white, l/f, and l/f spectral densities.

They were performed at the 1st Symposium on l/f Fluctuations in Tokyo; the white music was irritating, the

l/f 2 music was boresome, and the l/f music sounded nice.

10

10

10

10

10

10

10'

%V'

10

-4 10' 10 F R E

-2

u

10"

E N

10°Hz

C Y

Picture

;

We enjoy good paintings and photos.

Fig. 2 Spectral density of heart

beat period of a normal subject.

The period is normalized to the mean value, and the unit in the

vertical axis is /Hz.

What kind of stochastic characters do they have in com- mon as is the case in music? Kosugi and I evaluated

spectral densities of pictures with respect to space

frequency, in which original pictures were reformed

as black-and-white photos of the size 10 cm wide and

15 cm long, and darkness of the picture elements was measured along horizontal and vertical lines with a line-scanning camera. ^iteralistic paintings and ordinary photos are of l/f type and cartoons are of l/f

type. All the pictures we examined were between these.

Three typical examples are shown in figure 4-

Pain-relieving stimuli : Various kinds of chronic,

intractable pain can be relieved by transcutaneous electrical nerve stimulation (TENS) through transcutaneous

electrodes or acupuncture needles. It was reported

that TENS was effective approximately 25 to 45% of patients [l6 - 19] • Takakura et al. [20] applied rectangular electric current impulses of 0.1 ms in width to patients. The impulse repetition frequerncy and

the duration t^ were determined by random numbers that were generated by l/f fluctuations. Furthermore, they

used fluctuations of frequency and audio power of music

to determine the values of f^ and , respectively,

and gave patients music performance in synchronism with

the transcutaneous stimuli. They examined 220 patients

and found that TENS was effective for 35% of 60 patients

treated with constant pulse repetition rates, JOfo of

91 patients treated by l/f stimulations without music,

and 7A%> of 69 patients treated by l/f stimulations with

synchronized music performance. Effectiveness was judged by the degree of recovery of

the l/f spectral density of the frequency fluctuation of the EEC alpha wave.

144

10' 10 10'

FREQUENCY

Hz

Fig. 3 Spectral density of body

sway in the lateral direction

when one stands on single foot.

The vertical scale is arbitrary.
DENSITY MODULATION OF ACTION POTENTIALS ON NERVE AXON

Biological information is carried by impulses of

the action potential that propagate on a nerve axon. The action potential takes the same shape under a given

environmental condition regardless of the stimulus waveform, and, therefore, biological information must be

coded in the time relation of the action potential [21 ,22]. It is naturally expected that biological information

will be distorted or acquire noise if the time relation

of the action potentials is modulated or distorted sto- chastically during their propagation on the axon. There

are three possible origins for the distortion and noise.

Propagation speed of the action potential on a nerve axon is functions of ionic concentrations outside and

inside the axon, the environmental temperature, and the

time relations to the foregoing action potentials. The

former two would cause very little effect on the speed

of the action potential under well-regulated experimental

conditions as compared with the last one. Modulation

of the time relations of the action potentials was experimentally examined and the following results have been

obtained

.

Speed of the action potential : Giant axons of squid

(doryteuthis bleekeri) were used in the experiment.

An axon was placed in a nerve chamber filled with natural

sea water at about 10° C. The axon about 6 to 7 cm long

was tied at the ends with threads, the diameter being

0.5 to 0.7 mm. A silver wire electrode, 0.025 nun in

diameter, bare for 7 mm length and the other portion

coated with enamel, was inserted into the axon near the

end and used as intracellular electrode for stimulation,

ceived by a wire electrode placed outside the axon near the intracellular electrode. The

action potential was excited with a rectangular electric current impulse of 0.1 ms wide

that was twice as high as the threshold level. The four action potentials were successively

excited and their propagation speeds were measured. The result is plotted in figure 5- The first action potential travels at 25 m/s, but the following three action potentials

travel at slower speeds because the axon gets into a refractory state right after an action

potential is excited. This effect is equivalent to existence of some kind of repulsive

interactions between the action potentials, where it acts only onto the following ones.

FREQUENCY (/m)

Fig. 4 Spectral densities of a catoon 'Yasuda'(a), Picasso's

'Le trois femmes'(b), and

Renoir's 'Femme au chapeau'(c).

The stimulation current was reImpulse interval distributions : Sequences of random impulses were generated by a com- puter whose time intervals were in Gaussian distributions. The propagation length was 44

mm, and the output impulses were reformed in 0.1-ms rectangular impulses, which were again

applied to the stimulation electrode; thus four runs were tried. Figure 6 shows interval

distributions; the bottom refers to the computer-generated impulses that are Gaussian with

a mean of 3 ms and a standard deviation of 1 ms. Intervals shorter than 4 ms were cut off

because of the absolute refractory period of the axon.

Spectral densities of the impulse trains : Spectral densities of fluctuations of the

impulse density were plotted in figure 7, where the bottom refers to the computer-generated

impulses that is 'white' except for a peak at 300 Hz that has been caused by a mean interval

of 3 ms. The repulsive interactions of the impulses gave rise to modulation of the impulse

density that results in the l/f rise in the spectrum. The four sepctral densities have

almost the same shape, suggesting that the time relations of the impulses have reached a stable state after a single run on the axon. Cross-correlations between the input and output impulse trains, where action potentials were reformed in 0.1-ms rectagular impulses,

has a peak at a lag of 4 ms, and its width agreed with spread of the speed of the action

potential.

Resemblance to automobile current fluctuations : This density modulation of the action

potentials that is acquired during propagation on the axon[23] resembles that of automobiles

which was first observed by Musha and Higuchi [24] . The spectral density consists of shot

145
noise and a l/f part that is associated with bunching

of the cars. This spectrum was derived by a hydrodynamical model with an assumption that the car speed

is a linearly decreasing function of the car concen- tration 25 . The observed speed-concentration relations of the action potential as shown in figure 5 is well approximated by the same model. Therefore,

the density modulation of the action potentials on the axon has the same mathematical background as that

of the cars on the expressway.

CONCLUSIONS

The l/f spectrum is often found in fluctuations

in biological control and biological oscillations.

It seems that the l/f fluctuation is accepted as a pleasing sensation. Biological information has been

found to acquire the l/f fluctuation as a result of

the refractory nature of the nerve axon.

REFERENCES

m/ s 25

20

15

10

. < 2nd response

* 3 rd response

04th response

0 100 200 300/s

DENSITY OF ACTION POTENTIALS

Fig. 5 Speed of action potential vs. density of action potential when 4

action potentials are excited.

:i]

:2]

:3]

A]

:5]

:6]

;?]

:io]

;ii]

:i2]

:i3]

:i4]

:i5]

:i6]

:i7]

:i8]

:i9]

:2o]

:2i]

:22]

:23]

:24]

[25]

F. N. Hooge, Physica, 60^ 13O (1972).

T. G. M. Kleinpenning, Physica 77, 78 (1978).

H. M. J. Vaes and T. G. M. Kleinpenning, J. Appl.

Phys. 48, 5131 (1977).

T. G. M. Kleinpenning, J. Appl. Phys. 51^, 3438

(1980).

F. N. Hooge and L. K. J. Vandamme, Phys. Letters

66A, 315 (1978).

D.W.Allan, Proc. 1st Symp. l/f (1977), p. 158.

J . J .Gagnepain and J .Uebersf eld , ibid. p. 173

J . J .Gagnepain, P.Handel, and J.Uebersfeld Proc. 2nd Symp. l/f (198O), p. 550. R.J. van den Berg, E.Siebenga, J.de Goede, W.H.Moolenaar

and A.A.Verveen, Proc. 1st Symp. l/f (1977), p.97.

L.DeFelice, Proc. 2nd Symp. l/f (198O), p. 112.

H.M.Fushman, ibid., p. 132.

F.F.Offner, ibid., p.l40.

M.Suzuki, et al.. Tech. Rep. lECE Japan, MBE 80-59.

M.Suzuki, Master's thesis, Tokyo Inst. Tech. (I981)

R.F.Voss, Proc. 1st Symp. l/f (1977), p.l99.

J.D.Loeser, et al., J. Neurosurgery 42 March (1975)

D. Burton, Postgraduate Medicine 59, June (1976).

C.N.Shealey, Clin. Neurosurgery 21, 269 (1974).

L.Augustinson, et al., Lakartid 73, 4205 (I976).

K.Takakura, K.Sano, Y.Kosugi, and J.Ikebe, Appl.

Neurophysiology 42, 314 (1979).

Y.Lass and M.Abeles, Biological Cybernetics

19, 61 (1975).

M.Abeles and Y.Lass, Biological Cybernetics

19, 121 (1975).

T. Musha, Y. Kosugi, and G. Matsumoto, Proc. 2nd

Symp. l/f (1980), p. 176.

T. Musha and H. Higuchi, Jap. J

0 5 10 ms

Fig. 6 Cross-correlation between

stimuli and responses.

1271 (1976).

T. Musha and

811 (1978).

H. Higuchi, Jap. J

Appl . Phys

.

Appl. Phys.

15'

17

10

10'

10

10

10

*>. V.

A

fi

'

10 10 10 10 Hz FREQUENCY

146

Fig. 7 Spectral density of density

modulation of action potentials as they travel down the axon. The

bottom refers to the random stimuli.
1/F NOISE IN CONTINUOUS METAL FILMS IS NOT DUE TO TEMPERATURE FLUCTUATIONS

J. H. Scofield, D. H. Darling^ and W. W. Webb

Applied Physics

Cornell University

Ithaca, New York 14853

The possibility that 1/f conductance noise (also referred to as "excess", or "1/f"

noise) in thin continuous metal films is caused by temperature fluctuations has been con- sidered by several authors [1-6]. The power spectral density (PSD) of the voltage fluctuations S (f ) , if generated by temperature fluctuations is supposed to be S (f)=3 V^S (f)

,

where S Yf) is the PSD of the fluctuations of the average temperature of tKe film, V is the mean voltage across the film, and 6 = (l/R)(dR/dT) is the film's temperature coefficient of

resistance. Whatever the origin of temperature fluctuations, their propagation is governed

by the diffusion equation so that the resulting excess noise would exhibit a freauency dependent correlation length characteristic of a diffusion process, A(f) = {D/7Tf } ^ ^ where

D is the thermal diffusivity of the material. Spatial correlations of the excess noise

along metal films have been observed [1,2].

Equilibrium temperature fluctuations do account for the magnitude and spectrum of

measured conductance noise in freely suspended tin films near their superconducting transition temperature (where BT>1000) [3]. However, for substrate supported metal films various

theoretical temperature fluctuation models predict spectra with a low frequency cut off, a limited 1/f range, if any, and a monotonic temperature dependence. In contrast, experiments

reveal a wide range of 1/f spectrum and complex temperature dependence [1,4]. Thus, equilibrium temperature fluctuations do not appear to account for most observed 1/f noise in

metal films. Voss and Clarke have proposed a thermal fluctuation model with a 1/f regime

[1] . Such alternative temperature fluctuations of yet unrecognized origin might generate

the observed 1/f noise [1,5]. Van Vliet, et al. have calculated that the spectrum of the

fluctuations of the average temperature of a metal film on an insulating substrate with a uniform surface noise source having a white spectrum spatially coherent across the top of

the substrate would yield a 1/f spectrum [6]. However, they find that the magnitude of the

resulting temperature fluctuations would be negligible for thermal radiation fluctuations.

Nevertheless, temperature fluctuations in conducting metal films near room temperature has

remained in contention as a seductively simple mechanism to account for 1/f conductance

noise.

We have designed an experiment to detect the possible existence of temperature fluctuations large enough to account for excess noise in thin metal films. Calculations and ex- periments showed that the temperatures of superimposed films separated by a thin electrically insulating layer are closely coupled over the relevant frequency range. Thus temperature fluctuations in one film will be correlated with the temperature fluctuations of the

other. Therefore, if the 1/f noise in a film were due to temperature fluctuations this

noise would be correlated with the 1/f noise in the neighboring film. We have fabricated

suitable superimposed gold films, confirmed the calculated thermal coupling by modulated

heat input experiments, and have looked for a correlation in their 1/f noise. Our experiments show that the coherence (defined below) between the excess noise of the two films

is less than 1/lOOth the value anticipated were the measured 1/f noise in each film due to temperature fluctuations.

1 This research was supported primarily by the NSF through the Materials Science Center

at Cornell

2 Present address: Lawrence Livermore National Laboratory, Livermore, CA 94550

147
oThe superimposed electrically insulated film bridges were prepared by evaporating a 600 A layer of gold onto a 0.6 mm thick single crystal sapphire substrate and photo-etching

it to a 1 mm long, 80 ym wide bridge. Next, a 6000 A layer of SiO was evaporated on top of

the gold bridge and surrounding sapphire substrate, and a second bridge similar to the first was evaporated and formed on top of the SiO layer. Finally, a thicker gold 4-probe electrical contact super-structure was evaporated onto the sample for electrical connections.

Noise free contacts of negligible resistance were made using either pressed indium wire or gold wire soldered with indium. The substrate was then mounted onto a large copper heat

sink. Bridge resistances ranged between 11^ and 19Q due to differences in geometry. Calculated resistivities were typically 6 yfi-cm (about twice the bulk value) and the measured

thermal coefficients of resistance were about 6 = 0.003°C~-'- (similar to the bulk value).

Conductance and Nyquist noise were measured with a constant current I < 40 mA derived

from lead-acid batteries in series with a 1 Kfi wire wound ballast resistor. Due to the

large thermal conductivity (K=0.1 cal/C s cm) of the single crystal sapphire substrate, the

relatively large current densities (10^ A-cm"^) generated less than a 1°C film temperature

rise as measured by I-V characteristics and confirmed by heat flow calculations.

Voltages across each sample were amplified with either an Ithaco 1201 or a PAR 113 lownoise preamplifier, impedance matched to the films with a PAR 190 low-noise transformer,

and fed into an HP 5420A spectrum analyzer. The PSD of the excess noise for each film was obtained by measuring the PSD of its voltage fluctuations Sg(f) and subtracting from this

the PSD of the background (mostly Nyquist) noise Sg(f) , obtained by replacing the film with

an equivalent wire-wound resistor. For frequencies between 1 Hz and 5 Hz it was necessary

to correct up to 8 dB for the transfer characteristics of the instrumentation. Within the

frequency range measured, 1 Hz <f< lOOHz, both films showed excess noise roughly consistent

with Hooge's empirical formula, S (f)/V = a/{N f^ }, with 1.0 <b< 1.1 and 0.005 <a< 0.014,

where N is the number of carriers [71.

c We have modeled the three-dimensional thermal coupling problem by considering diffusion into the substrate. Our model calculations indicate that the average temperatures of

those portions of each of the two films that are directly superimposed upon one another are virtually identical for frequencies f <<

^/''^c.i^o' ^^^^^ ^SiO

~ ^^^^^SiO' ^ thickness,

and Dg-Q '^^^ diffusivity of the SiO layer [8j. Since < 1 ys this result holds true

for ali frequencies of interest. In order to confirm the validity of our model and to test

for possible thermal barriers at the interfaces we have calculated the amplitude of the

sin (27Tft) component of the average temperature of the lower film that results when the top

film dissipates a power P^ sin(2Trft) . We measured this same quantity by passing a current

i^ sin(iTft + 6) through the top film and a steady current I through the bottom film. The

amplitude of the sin (27Tft) component of the voltage across the bottom film is then a measure of its average temperature modulation amplitude. Measurements of the ratio of the modulation amplitude of the average temperature of the bottom film AT„(f) to the amplitude of

the power dissipated in the top film P^, for frequencies 0.2 Hz <f< 24 KHz are plotted in

figure 1. The modulation amplitude was proportional to li^ as expected for thermal coupling. Measurements taken with the roles of the two bridges reversed gave identical results.

The calculated values of AT2(f) /P-j^ are plotted as the solid curve in figure 1. Theory and

experiment agree at all frequencies within the factor of two uncertainty due to 3, the

temperature coefficient of resistance of the film, K, the thermal conductivity of the substrate, and the transfer function calibration at low frequency. The excellent agreement

without adjustable parameters indicates that the model incorporates the important features

of the system, and gives us confidence in its prediction for strong thermal coupling between the two films.

The correlation between fluctuations of the average temperatures of the two films can be expressed by the coherence function, Y^(f) = I^t12^^^^ /{S^.j^(f) 'S ^(f) ) » where S^^^^ the cross-PSD between the average temperatures of the two films, and S^, and S^2 their

individual PSD' s [9]. The thermal coupling calculation indicates that if the two films were perfectly superimposed the coherence between fluctuations of their average temperatures

would be unity for f << . However, since they are not perfectly aligned Yr|,(f) is

slightly reduced by a factor that depends on the fraction of the ^rea of one film which

overlaps the other film. For our film geometry we expect 0.9 < y^(i) < 1.0.

Consequently, if temperature fluctuations were the cause of the 1/f noise, essentially

148
^ 3 o

cn

log,o (f/IHz)

Fig. 1 Thermal response hT^(,f)/7^. Data

points are representative of several different runs; omitted points superimpose; O

and O observed with power into top film,

with power into bottom film. Error bars

represent low frequency uncertainties in the

transfer function. Solid line is the calculated value of the thermal response

AT2(f)/P^.

the same temperature fluctuations would appear in both films and the 1/f noise would be

correlated. In this case we expect Y^(f) = Y^(f) » where Y^(f) is the coherence between the

1/f noise of the two films. Since the measured voltage fluctuations across a film include

both the 1/f noise and (mostly incoherent) background noise, the coherence ^g(f) of the

measured voltage fluctuations of the two films is not in general equal to y Xf) . Instead,

Y^(f) and Y^(f) are related by ^

V ij

Y^(f) = Y^(f)/{[1+Sg3^ <f>/Svl (f)Hl+Sg2 (f)/S^2 (1)

where S , and S „ are the PSD's of the excess noise of each film and S„, and S„„ are the

PSD's o¥ their background noise. Using measured values of 5^^, 5^2^ ^Bl' B has been calculated and is plotted as curve a in figure 2 assuming that temperat

ations are completely responsible for the observed 1/f noise of a single film (i.e.

2' ^E^^^ ure IluctuY. = Y^)

r/(f)io

Fig. 2 Y^(f) = V

and Sg2

coherence YT,(f) E

(a) Calculated Yg(f) assuming

Y^(f) from measured S^^, S^2' ^gi

(b) Upper limit of experimental

149
Measurements of Yp(f) for frequencies 1 Hz <f< 100 Hz discovered no coherence above

the instrumental sensitivity indicated by curve b in figure 2. In this frequency range 0 <Yg(f)<5 X 10"^. At the low frequency the coherence is more than a factor of 10 lower

than would be expected if the 1/f noise^were due to temperature fluctuations. At higher

frequencies the Johnson noise reduces y (f) , but the discrepancy is still large. Since

the correlation length for temperature rluctuations X(f) = {D/irf} at IHz is already very

much longer than the separation between the two films, and increases with decreasing frequency, this result should extend to lower frequencies. Therefore, we must conclude that

the 1/f noise in these gold films is not due to temperature fluctuations.

Voss and Clarke [1] and Zhigal'skiy, et al. [2] have previously reported the observations of frequency dependent spacial correlations in the room temperature 1/f noise

along metal films, as would be expected if the noise were due to thermal fluctuations.

However, elsewhere in this conference Weissman reports similar experiments in which spacial

correlations were not observed. Van Vliet and Chenette also report in private communication the lack of correlation of the 1/f noise between two thermally coupled semiconductor

j unctions

.

In summary, we have measured the coherence between the 1/f conductance noise of two substrate mounted thin gold films that have been demonstrated to be in strong thermal contact, and have calculated the coherence between the 1/f noise that would be expected if

temperature fluctuations were responsible for the observed 1/f noise. The measured co- herence is several orders of magnitude lower than predicted for temperature fluctuations.

Therefore, we conclude that the 1/f noise is not caused by temperature fluctuations.

We would like to thank Joe Mantese and Dr. Mark Nelkin for helpful discussions.

REFERENCES

[1] R. F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976); Phys. Rev. Lett. 33, 24 (1974)

[2] G. P. (1979)

Zhigal'skiy and A. V. Karev, Radio Engin. and Electr. Phys. (USA) 22(12) , 112

[3] M. B. Ketchen and J. Clarke, Phys. Rev. B 17_, 114 (1978).

[4] J. W. Dimon,

Eberhard and P. M. Horn, Phys. Rev. B 18, 6681 (1978); see also P and P. M. Horn, Phys. Rev. Lett. 43, 646 (1979).

Dutta, P.

[5] M. B. Weissman, Phys. Rev. Lett. 41, 1 (1978).

[6] K. M. Van Vliet, A. Van der Ziel, and R. R. Schmidt, J. Appl. Phys. 51, 2947 (1980).

[7] F. N. Hooge and A. M. H. Hoppenbrouwers , Physica 45, 386 (1969).

[8] J. H. Scofield, D. H. Darling, and W. W. Webb, to be published.

[9] J. S. Bendat and G. Piersol, Random Data: Analysis and Measurement Procedures (John

Wiley and Sons, Inc., 1971), p. 141.

150
TEMPERATURE RESPONSE AND COBIRELATION OF

1/f NOISE IN TRANSISTORS

J. Kilmer, K. M. van Vliet, E. R. Chenette,

and P. H. Handel

Electrical Engineering Department

University of Florida

Gainesville, Florida 32611

In order to investigate the possibility of temperature fluctuations as the origin of

1/f noise in thin films and in devices, we examined the thermal coupling properties of

several bipolar transistors on an integrated circuit chip (CA3018NPN) . The transistors

chosen for investigation were thermally coupled by the substrate but electrically isolated;

they were 10 to 200 microns apart. One transistor was used as a "heater" and another one nearby as a "sensor." The heater power frequency was varied between a few hertz and about

20 kilohertz. The response of the other transistor was measured by monitoring V , which

oE

is a sensitive function of temperature. The thermal response was quite well observable.

The transfer function was nearly flat below 100 hertz, and above this varied as 1/V^, in

accord with the solution of the one dimensional heat diffusion equation. In order to rule

out capacitive coupling effects, the response was also determined by using two frequencies

for the heater input signal, and by measuring the response for the sum or difference frequencies; the latter only can occur due to the quadratic function of the Joule heating.

Thus the occurrence of a strong thermal coupling was ascertained.

Next, we measured the noise of both isolated transistors in the range 5 raillihertz to 25 kilohertz; in most of this range the noise was pure 1/f. Using a Hewlett Packard fast

Fourier spectrum analyzer in this frequency range, we then determined the cross-correlation

noise of the two transistors. Within experimental error this yielded zero for the coherence

coefficient in the entire frequency range.

From the measured heat transfer response we could in principle calculate how strong the

noise correlation should be, if the noise were due to spontaneous temperature fluctuations.

However, this requires knowledge of the heat coupling sensitivity factors, like Vgg(T) , etc.,

the determination of which may involve relatively large errors. Therefore, we modified the

heat transfer experiment as follows. We selected three electrically isolated transistors

on the integrated circuit chip, and used one transistor as the "heater" and two others as "sensors." A good heat transfer response between heater and sensors was obtained. We then

measured the correlation between the two sensors when an a.c. signal was fed into the heater.

to be 0.6, and at 40 hz the coherence coefficient was found to be 0.4. Tnis experiment

allows a direct comparison with the measured coherence of the noise, which was 0+0.02. We

thus conclude that the measured 1/f noise in these transistors could not have been due to temperature fluctuations of the transistors or the substrate.

For a heater frequency of 12.5 hz the coherence coefficient. found

151
LACK OF SPATIAL CROSS-CORRELATION IN 1/f NOISE IN CHROME FILMS

* * t ^

R. D. Black , M. B. Weissman and F. M. Fliegle

Department of Physics, Electrical Engineering Department

University of Illinois at Urbana-Champaign

Urbana, IL 61801

Several years ago Voss and Clarke [1] observed correlations in the low frequency

components of the excess noise in two adjacent lengths of a strip of bismuth film. From

the characteristic frequency of the effect and from the center-to-center spacing of the

regions, a diffusion coefficient was calculated which seemed too large to be associated

with most variables affecting the resistance except for temperature. As a result, much

theoretical effort has gone into explaining the connection between temperature fluctuations

and 1/f noise. We have performed similar experiments on chrome films and have found no spatial cross-correlations in the 1/f noise.

The films were obtained from Basic Microelectronics Industries. The specified

thickness was 1000 ± 100 A on a soda-lime glass substrate. Assay by eraitted-photon

spectroscopy using an SEM showed no significant contaminants. Patterns were etched using

optical lithography. The observed stretches were ~ 10 ym wide and 500 \m long, each. A

seven probe contact method was used, with separate pairs of current-carrying contacts on each region. One voltage-sensing contact was shared by the two regions, but no current

flowed through the shared path. Three similarly prepared samples showed resistances and

noise magnitudes matched to within ~ 20%. One sample was observed in an SEIi; it showed

shallow (~ 1 pm) scallops on the edges of the film and some shadowing of the edges,

indicating a bit of edge-thinning during etching.

The resistance of the film was larger than expected for pure Cr by about an order of

magnitude, and the temperature coefficient of resistivity was about -3 • 10~ /°C. Although

tViis indicates significant disorder, we observed no grains at a resolution of several

hundred angstroms.

Leads were attached by compressing blobs of indium very tightly onto the contact pads

using steel screws. The standard checks for battery and contact noise were run, revealing

none above one Herz. Below one Herz there were intermittent current fluctuations much

smaller than sample noise. Ue used a PAR 114-185 amplifier for one channel and two series

PAR 113' s for the other. Spectra were taken with a PAR 4520 dual channel FFT analyzer.

The 1/f noise magnitude was within the range found in other metal film and semimetal

film samples of comparable volume [1]. No spatially correlated 1/f noise components were

found, to an accuracy of ~ ±.05 of the sample l/f noise in the one to ten Herz range

and ~ ±0.1 in the one-eighth to one Herz range. A slight correlated Johnson noise

component due to the shared part of the tv70 voltage lead paths was detectable at higher

frequencies. In our samples, there is no reason to suspect any fast-diffusing variable,

whether associated with the lattice or the carriers, of playing a role in the 1/f noise.

We have also made measurements in which only one current source was used, connected to the outer current pads. Som.e intermittent low-frequency coherence (~ 10% below 0.5 Hz)

appeared, but a similar correlated noise component could also be found with one region

replaced by a wire-wound resistor, so this small correlation was due to current

fluctuations. Since the time for carriers to move from one stretch to the other vjas less

than 10 msec, fluctuations transported with the carriers would be fully correlated

below ~ 15 Hz. We believe such fluctuations can be ruled out. Data on one non-annealed

bismuth sample using independent current paths also show no cross-spectra down to 0.12 Hz.

This v7ork was supported by NSF grant DMR 80-07057 and, through the teterials Research

Laboratory, by NSF DKR 77-23999.

REFERENCES

[1] R. F. Voss and J. Clarke, Phys. Rev. B13 , 556 (1976).

152
TEMPERATURE FLUCTUATIONS IN A STEADY STATE

Igor Pavlin

Department of Physics and Astronomy

University of Massachusetts

Amherst, MA 01003, USA

and

Institute "Boris Kidric"

Belgrade, Yugoslavia

Stanley Engelsberg

Department of Physics and Astronomy

University of Massachusetts

Amherst, MA 01003, USA

INTRODUCTION

Are temperature fluctuations responsible for the 1/f noise of voltage fluctuations in

thin metal films? This problem, posed by experiments of Voss and Clarke [1], is still not

resolved. The authors assume that temperature fluctuations are spatially correlated and use equilibrium statistical mechanics in order to obtain satisfactory theoretical explanation.

Due to the large currents used in experiments, one might suggest that temperature fluctuations in a steady state cause 1/f noise. One of the arguments might be that steady state

fluctuations would lead to 1/f noise without the additional assumption of correlated fluctuations. However, we argue that the assiunption of spatially uncorrelated temperature fluctuations in a steady state cannot be responsible for 1/f noise. It is a spatial correlation

either in the equilibrium or in a non-pquillbrium which could be the cause of 1/f noise.

Our results should provide an excellent opportunity for experimental verification of

some questions related to temperature fluctuations. First, they can check indirectly whether

or not temperature fluctuations cause the voltage noise seen in thin metal films. Second,

the applicability of microscopic reversibility, Onsager [2], to temperature fluctuations in

a steady state could be verified. Third, one can test the concept of local equilibrium for

temperature fluctuations (see, for example, Balescu [3]).

In the next section we outline our theoretical predictions. Than we give two one-dimensional examples. One example is of a thin rod of length 1, whose lateral sides are thermally

isolated. The ends of the rod are maintained at different temperatures. Fluctuations around

the steady state are examined. The exact solution for the spectrum is found. In the second case we consider an infinite homogeneous rod. A steady state with a smooth spatial dependence in the rod is considered. The spectrum of temperature fluctuations coming from a small

part of length 21 is found. In both cases we conclude that spatial dependence of the steady

state temperature distribution cannot be the cause of correlated temperature fluctuations,

TEMPERATURE FLUCTUATIONS AROUND A STEADY STATE

The formalism for temperature fluctuations around a steady state temperature distribution is derived on the assumptions of the linearity and microscopic reversibility of the

system. The probability of a temperature deviation Si from some mean temperature in a steady

state T„(r)

153
is given by Prigogine ['4]. Let us interpret eq (1). A fluctuation cTt as a whole is composed

of many independent deviations of temperature from the mean temperature T^(r). In a very

small volume ^CL containing the point we assume that the system behaves as if it is in

equilibrium at temperature T„=To(r). The distribution of a deviation from the mean temperature at that particular point l' is Gaussian

The probability of a deviation at some other point r' would be as in eq (2) with r replaced

by T' . When equal-time temperature fluctuations are assumed to be independent, the joint

probability distribution of cTTCt ,t) and cTKT'' ,t) is the product of their distributions.

The probability of a fluctuation cTt as a whole in a system is then the product of the

individu al probabilities for any small volume £^Sl contained in 12. . The change to the

continuum results in the integral over the considered volume £L.

The expected value of ItfTC r^,t) 1^ at the point r can be obtained in a similar way as in

the case of the equilibrium. It is the variance of the random variable <f'lCr,t) having the

Gaussian distribution and the mean zero. Using the assumption of uncorrelated temperature

fluctuations, the equal-time two-point correlation function, P(?,7'), is than assumed to be

In the case of T^(r)=Tp we get the standard equilibrium result. Similar considerations are given to the case of density fluctuations in a nonhomogeneous gas, Cohen [6].

Using the Green's function formalism, van Vliet [5], and the correlation functions

PCi'.r') we find the spectrum Sy(co) of temperature fluctuations around a steady state T^(r)

where

The noise spectrum S (W) originates from the volume XZ- D is the diffusion constant. )^ , Ajh

154
are the eigenfunctions and eigenvalues of the related Sturm-Liouville problem, which satisfy

the relevant boundary conditions. Cy is the heat capacity per unit volume of the sample.

When T^(r)=T^ we obtain the coefficients A^^(T, ) for uncorrelated temperature fluctuations in the equilibrium at The coefficients A^_are in many cases bounded between

A^^dmin) i A^^5 A^CTmax) where Tmin<T(r)i Tmax . The spectrum of any steady state

temperature distribution which is contained between Tmin and Tmax is bounded above and below

by the spectra of uncorrelated equilibrium fluctuations taken at Tmax and Tmin. Since the

spectra of uncorrelated equilibrium temperature fluctuations do not contain a 1/f part in

its spectrum, the spectra of fluctuations around steady state, assuming uncorrelated random

flows, do not contain it either.

TWO ONE-DIMENSIONAL EXAMPLES

Thin rod

.

We denote the tempe ratures at the ends of a rod of length 1, T^^ , T^ . The

eigenfunctions are ^ = ]/ 2/1 sin( Xrtt. x) , with the corresponding eigenvalues A^,=mX/l;

m=1,2,3... The stationary temperature distribution is T^( x) =T^ +(T, -T^ ) x/1 . Using eq (4) and

(5) we find that the spectrum is

(' denotes summation over odd integers, 0 /(Of^) , ^g=D7r/41 ). The spectrum (6) reduces

as T^-»Tq to that obtained by Ketchen and Clarke [?] for uncorrelated temperature

fluctuations in equilibrium. Using complex analysis we find the analytic form of the

spectrum (6). The spectrum (6) retains the shape of the spectrum of uncorrelated equilibrium

temperature fluctuations and no significant change is introduced by the spatial dependence

of the steady state temperature distribution T^Cx) , as long as the linearity of the system

is maintained. The results are plotted in figure 1.

3

-3

^-5

^7

Fig . 1 Spectrum of uncorrelated

fluctuations for a thin rod in

the steady state. Temperatures of

the rod's end are T^, and T, . The

parameter ^ =T| /lijis a measure of

the steady state temperature gradient. The spectra are obtained

according to eq (6) in the text.

-2 -i

155
Smooth steady state temperature distribution

.

We consider a slowly varying spatial

distribution of temperature over a length much longer than the length 21 of a rod. The rod

itself is a part of an infinite homogeneous system. To simplify, we write

where ^ =^J/'T^«^ and a>>l. The eigensolutions will be plane waves.

If the assumption a>>l is used, the spectrum is approximated using Taylor expansion

Jo anci j, are spherical Bessel functions. The integrals can be evaluated using a contour

integration. After that, asymptotic expansion of the spectrum gives

For a>>l, the results in eq (9) are only a slight modification of spectra for uncorrelated

temperature fluctuations in a part of the homogeneous rod which is in thermal equilibrium

with its surroundings.

CONCLUSIONS

Non-equilibrium fluctuations in a system with spatially uncorrelated temperature

fluctuations cannot be responsible for a 1/f spectrum. Thus, theoretical support is given to the intuitive assumption of experimentalists that the introduction of a steady state does

not enhance spectrum significantly.

Derived expressions for temperature fluctuations in a thin rod having different end

temperatures are suitable for experimental verification of our theory. The verification of

the theory would imply the applicability of Onsager's hypothesis of microscopic

reversibility to temperature fluctuations in non-equilibrium.

REFERENCES

[1] Voss, R. F. and Clarke, J. Phys. Rev. B J13, 556-573 (1976).

[2] Onsager, L. Phys. Rev. 38, 2265-2279 (1931).

[3] Balescu, R. Equilibrium and nonequilibrium statistical mechanics. New York;

John Wiley & So . , 1975.

[4] Prigogine, I. Introduction to thermodynamics of irreversible processes.

Third edition. New York: John Wiley & Sons, Inc., 1967.

[5] Van Vliet, K. M. J. Math. Phys. J2, 1981-1998 (1971).

[6] Cohen, E. G. D. The Boltzmann equation and fluctuations. In Fundamental

Problems in Statistical Mechanics V. Ed. E.G.D. Cohen, North Holland, 1980.

[7] Ketchen, M.B. and Clarke, J. Phys. Rev. B 17, 11^-121 (1978).

156
EARTHQUAKES, THUNDERSTORMS, AND OTHER 1/F NOISES

Stefan Machlup

Department of Physics

Case Institute of Technology

Case Western Reserve University

Cleveland, OH 44106

ABSTRACT

Improving an experiment may mean getting better signal-to-noise ratio, perhaps

by filtering out a specific source of noise. Each such noise source is likely

to have a characteristic frequency or time scale. The noises most difficult

to filter are the catastrophic events, fortunately rare, whatever their cause. The ensemble of such events has no characteristic time scale . There are

as many repeat times in the interval between 1 and 2 seconds as between 1 and

2 minutes, etc. The distribution is scale-invariant, i.e. logarithmic.

Differentiating a logarithmic cumulative distribution results in a 1/f spectral density. Filtering the noise out in any frequency band means the 1/f

shape obtains only for frequencies below that band. If you have not found

the 1/f spectrum, it is because you have not waited long enough. You have

not looked at low enough frequencies.

In designing experiments, one tries to foresee the possible sources of noise and,

of course, to eliminate them. Johnson (thermal) noise and shot noise set a floor, a minimum below which noise can not be reduced. Thermal noise and shot noise both have

white spectra, i.e. they are flat up to very high frequencies — what I like to call

quantum frequencies — where the spectra bend down in a Lorentz shape. In this

gathering I don't have to demonstrate what white noise sounds like. It is the shshsh

sound of rushing water. l^That characterizes white noise is that, for any given frequency, there is more noise power in the frequency bands above the given frequency

than in the bands below the given frequency.

Here I also don't have to demonstrate what flicker noise sounds like. Flicker

noise is also called pink noise, because its spectrum is proportional to 1/frequency.

There is more noise power at low frequencies than at high. There is twice as much

noise power in a 1-Hz band near 1 kHz as in a IHz band near 2kHz. Well, I'll do it

anyway. In a transistor, the 1/f noise sounds like pshsh, ktshsh, pdk, kshsh... You

can hear the individual events. Big events are less frequent than little events.

Twice as big occurs half as often. That is the meaning of 1/f spectrum.

UNLIKELY EVENTS

Mandelbrot uses the term Joseph Effect for unlikely events having low characteristic frequencies. The name refers to the biblical Joseph, who predicted the 7 years of plenty followed by 7 years of famine. Clearly the remarkable feature was not

the existence of aworld-shaking phenomenon with a 14-year period (a frequency of

2 X 10 Hz). The remarkable feature was the accuracy of Joseph's prediction.

1/f spectra are so common that there is a large literature exploring noise mechanisms having such a spectral shape over many decades. This paper is not intended to be

a contribution to that literature. It takes the opposite point of view. It asks the

question: What can the ubiquity of 1/f spectra in widely different situations tell us about unexpected events? The answer is that the ensemble of unforeseen events has

characteristic repetition times which have a scale- invariant distribution.

157
SCALE-INVARlANCE

This Is not arcane mathematics. It is elementary. A 1/f spectral shape is the only

shape that has no characteristic time, or, if you prefer, no characteristic frequency.

If all frequencies are multiplied by a constant, the spectrum does not change: df/f is

independent of the units of f. That means that the statistical distribution governing

the events that produce that spectrum also has to be scale-invariant. You may find the

argument more convincing if we talk about the integrated power spectrum. For a 1/f

spectrum the total power in a frequency interval (f^,f2) is proportional to logCf^/f,) —

the integral^df /f . The only function of f that is dimensionless is log f. Only this

spectral shape has the property that there is the same amount of power between 1 Hz and

2 Hz as between 10 Hz and 20 Hz..,

Readers of the Scientific American know by now that th^-,height of the floods of the

Nile has a 1/f spectrum . They have read that the music of Bach and Mozart and even

jazz is 1/f. Designers of experiments know that the banging of doors is 1/f, and they

design their measuring devices to block out such interruptions. By this I don't mean

that the spectrum of one bang is 1/f. Indeed, one bang has a lot of high-frequency com- ponents. What the 1/f shape of door-banging noise says is that hard slams are scarcer

than soft slams and the whole phenomenon does not have a characteristic repetition time.

If the soft slams are not recorded, all that remains is the lower-frequency hard slams.

Some noises are too big and/or too infrequent to discriminate against. These catastrophic events get through our filters, our vibration-free mountings, our isolation

transformers. The term "earthquakes and thunderstorms" in the title is a euphemism for

the rare, unforeseen and cataclysmic events that constitute low-frequency noise you just

can't block out. They would include the truck crashing into your lab. The better you

are at eliminating noise, the more nearly the spectrum of the remaining noise is likely

to be 1/f, i.e. scale-invariant.

You conclude that a badly designed experiment may have a 1/f noise spectrum simply

because there is a large ensemble of sources of big noise, so large an ensemble that the

statistical distribution of characteristic times is scale-invariant. As these noises are removed by better design, what noise remains is likely to be dominated by one or two mechanisms, and to have a characteristic time — a Lorentzian spectrum. When the predictable

noise source has been removed, what remains are the unpredictable cataclysmic events —

the earthquakes and thunderstorms. They are likely to have myriad causes, so again, to have a scale-invariant distribution of characteristic times. Both the very raw experiment

and the very mature experiments are likely to have 1/f noise spectra — at different

levels, to be sure.

HIGH AND LOW-FREQUENCY BENDPOINTS

Back in 1952 I was assigned the 1/f noise problem in transistors by William Shockley,

who headed my research group at Bell Telephone Laboratories. One of my early worries was the infinities, the divergences at both high and low frequencies. I resolved the highfrequency-divergence worry quickly. Scale-invariance breaks down at high frequencies

because Planck's constant determines a2scale. The high characteristic frequencies that cause 1/f spectra to curve down to 1/f shapes I have called quantum frequencies. The

existence of atoms determines both a length scale and a time scale onrthe short end. The

low frequency divergence of a 1/f spectrum was resolved by Mandelbrot for the theorists.

The experimenters were never bothered by it. They know intuitively that t^ measure noise

at 10 Hz they have to measure for ^t least 10 seconds; to measure at 10 Hz the

measuring time has to be at least 10 seconds; etc. They know the bandwidth theorem.

I was always looking for a low-frequency cut-off. In other words, I thought there

ought to exist a very low frequency below which the flicker noise would be white. I was looking for a time scale on the long-time end. If such a characteristic time were to exist, i.e., if there were a longest characteristic repeat time for random events, then

the 1/f spectra would have that shape for only a limited number of decades in frequency.

That would imply the existence of a pure number characteristic of the universe, giving

the ratio of the quantum frequency bendpoint to that low-frequency bendpoint. The

persistence of 1/f spectra to ever lower frequencies can be taken as evidence that nature

does not possess such a ratio. If, for example, the age of the universe turns out to 158
set a low-frequency bound, then 1/f spectra might span as much as 342^ecades in frequency. We would be talking about a pure number somewhere around 10 , There are, of course, some big dimensionless numbers in nature. For example, the ratio o^^the electrical repulsion to the gravitational attraction of two electrons Is 4 x 10 . To have

a 1/f noise spectrum covering 42 decades seems almost mindboggling.

No attempt is made here at a model for a low-frequency bendpoint. The question at issue is whether scale-lnvariance might be one of the approximate symmetries of nature.

We know it breaks down at the short-time end. We do not know how far into the long-time

region it might extend. The ubiquity of 1/f noise spectra is our evidence for conjecturing this symmetry property.

What kinds of experimental evidence would support or contradict this conjecture?

To be sure, no matter how many decades long an observed 1/f spectrum might be, not even

dozens of such spectra would constitute proof. But if a universal low-frequency bendpoint were to be found, with no 1/f noise at lower frequencies, that would be sufficient

to disprove the conjecture.

ENSEMBLE OF PURELY RANDOM PROCESSES

It might be useful to write down the connection between the power spectrum and the

statistical distribution of repeat times for purely random processes. A purely random

process is one with an autocorrelation function of the form-,e . The power spectrum

of such a process has a Lorentz (also called Debye) shape : -t/x

S(oj) Fourier Transform of e

- t/(1 + W^T^) (1)

If we have a large collection of different random processes, each with its own correlation time T, then the power spectrum of the whole ensemble depends on the statistical

distribution p(t) of thesecorrelation times. If these processes have not been filtered,

then our conjecture is that the weighting function is scale-invariant:

p(t) dt °= dx/x (2)

This gives a power spectrum

"2 J2

j

S^(w)p(x)dx cc j

dT

1 J. 2 2 X

Xj^ 1 + OJ X

tan "'"djx "2

"l

•

(3)

If the scale- invariance extends over many orders of magnitude, i.e^gif

'^yj'^'l ^ large

ratio, then the spectrum is l/w over a correspondingly large range . For many years we have been scratching our heads to find special mechanisms that would have that special

distribution of time constants over many decades. But we do not need special mechanisms.

We need simply a large ensemble of mechanisms with no prejudice about scale. The con- jecture is that nature is sufficiently chaotic to possess this lack of prejudice.

Some of my friends refuse to call this sort of thing physical theory. They feel it

belongs more properly in the realm of philosophy, metaphysics perhaps. It is important

to note that whatever the status of this approximate symmetry principle, it does not

detract in any way from the work of those who have explored individual noise mechanisms

with 1/f spectra.

159
ACKNOWLEDGEMENTS

This paper grew out of conversations with my colleague Professor T. Hoshiko of the

Case Western Reserve University School of Medicine. Our work is partially supported by

the National Institutes of Health under Grant AM-05865-13.

REFERENCES

[1] Benoit Mandelbrot, Fractals : Form , Chance , and Dimension (W. H. Freeman and Co.,

San Francisco, 1977)

[2] Martin Gardner, Scientific American 238 , No. 4, 16 (April 1978).

[3] R. F. Voss and J. Clarke, J. Acoust. Soc. Am. 63, 258 (1978).

[4] Benoit Mandelbrot, I.E.E.E. Transactions on Information Theory jL3, 289 (1967).

[5] S. Machlup, Journal of Applied Physics 25, 341 (1953).

[6] A. Van der Ziel, Fluctuation Phenomena in Semiconductors (Academic Press, New York

1958) p. 56.

160
1/f DIFFUSION NOISE*

K. L. Ngai

Naval Research Laboratory

Washington, D. C. 20375

Fu-sui Liu

Department of Physics

Beijing University, Beijing, China

A physical description of low frequency fluctuation, dissipationj and relaxation properties of condensed matter that departs from the conventional ideas has recently been proposed by one of us [1-7]. In this model we have explored and found the nature and some universal

features of excitations in condensed matter when the excitation energy ho) can be arbitrarily

small. This new class of states and associated excitations between them have been discussed

in detail [4] . To distinguish them from well known elementary excitations in solid state

physics which offer no help to an arbitrary low-energy regime, these states are called cor- related states (CS) and the excitations between them correlated-state excitations (CSE)

.

By nature a CS involves for its description a large number of coordinates which, dependent

on the particular material, can be nuclear, bonding, ionic, electronic, etc., or their com- posite in origin. These coordinates are themselves engaged in complex interactions and

their nvmiber that is required to describe a CS increases as the excitation energy for the

CSE it gives rise to decreases. This complex situation makes a conventional Hamiltonian or Lagrangian modeling approach to CS properties, if not impossible, at least so cumbersome

that it would be futile to try to extract from the results any well-defined behaviors. Fortunately, Wigner's statistical approach [8] to quantum mechanics cones to the rescue [4] and

enables us to deduce some general properties of CS and CSE. One of these invariant properties is that the density N(E) of CSE with energy E will be linear in E provided E is small

enough. In any low frequency fluctuation-dissipation (F-D) phenomenon, a primary species

responsible for it can be identified. They can be dipolar groups, charged particles, etc. F-D is caused by transition of the primary species [4] that has a characteristic time t . The primary species is coupled to the CS. Its transition, considered as sudden for times

t»TQ, will switch on a new potential V that induces excitation and deexcitation of correlated states, and the corresponding matrix element of V is uncorrelated with and independent

of the CSE energy E. Thus E - dependence of |v|^N(E) comes from N(E) alone and is hence

linear in E. Written as |vpN(E) = nE, this condition guarantees that the CS excitations

and deexcitations that trail the primary transition will be infrared divergent. That is,

there is increasingly high probability of exciting (deexciting) decreasingly small energy

ho) CSE. This causes a power law aj^~-'-(t~^) dependence of the response in the frequency

(time) domain at low frequencies (long times). Here 0<n<l, and n will be referred to as the

infrared divergence exponent. Explicitly, the transition rate of the primary species Tq^

is modified to be time dependent as exp(-nY) (E^t)~^ where y~0'577 and E^, the upper cutoff energy for CSE such that, for E>Ej,, N(E) is no longer linear in E. It is important to note that we are interested in low frequencies v<10lO U2 and ambient temperatures T such

that hu)/kT«l. Unlike electron-hole pairs in the X-ray edge problem where the infrared

divergence is removed for hto^kT by thermal factors, such a problem does not arise in the

case of correlated state excitations with haj/kT<<l and infrared divergence continue to arbitrary low hw.

When addressing any specific F-D properties, a physical quantity Q pertaining to the

primary species is considered [4] . For dielectric relaxation, Q is the electric polarization of dipoles or charges; and for particle hopping transport and diffusion, Q is the

probability density that the particle remain on a particular site. The transition rate

equation for Q is modified from dQ/dt=-Q/T to

161
dQ/dt = -t"-"- e (E t)~"Q

o c

(1)

by the infrared divergence. Of central importance is the function ^^Ct) = -dQ/dt which, from

eq (1), is

i< (t) = (l-n) a t~" exp(-a t^~") : a = e~"V(l-n)T e" (2) n n n n o c

In dipolar dielectric relaxation the polarization correlation function - <P (t) (t-r)>^

is simply P^'I'n^t) with p being the dipolar polarization. In particle hopping transport and

diffusion, ^Jn^t) is the topping time probability density (distribution function). That is,

the particle if initially at a site will hop between t and t+>St with probability

The fundamental process of infrared divergent excitations (deexcitations) of CS that is

proposed to occur in all condensed matter always introduces a distribution of event time

"constants" and stochasticity in a natural and well-defined manner according to eq (2). The

distribution is characterized mainly by one parameter n, the infrared divergence exponent.

The limit as n->o of eq (2) recaptures the classical Markov process.

Predictions of our model based on 'f'jj(t) in a number of areas including dielectric

relaxation, mechanical and viscoelastic relaxations, internal friction, voltage noise, g-r

noise, 1/f noise from mobility fluctuations, NMR spin- lattice relaxation, transient electrical transport, and transient capacitance have been worked out [4]. We have then a unified

theory [4]. Good agreement of the predictions with experimental data is achieved for these

diverse low frequency responses and for materials of different physical and chemical structures. The most accurate and direct observation of our 'I'n^'^) mechanical relaxations

by photon-correlation spectroscopy [9]. Over a great nun\ber of decades of t, the data are

in excellent agreement with i|>n(t) of eq (2). The unified theory offers predictions on quantities such as activation energies [4] as well as dispersion.

In this work we have extended the unified theory to the study of non-Markovian diffusion transport and noise caused by 4'n(t) . For non-Markovian processes the starting point

is the generalized master equation. We use the form given by Kenkre, et al. [10]:

dP(.e,t)/dt=/^4,^(t-T)S^, [p(.t,C,')P(£',T)-p(.r,.OP(.t,T)]dT (3)

where P(\I,t) is the probability that the primary species is in a state £. at time t, p(.«,,.(!') is the probability of the transition's being from to .<!. and ifin^*-) '^ memory or relaxation function that is derived from dynamics specified by 4in('^) • The Laplace transform (LT)

(fi*(u) of 4>n(t) is given in terms of ^^(u) the LT of \l)^it) by^°

4)*(u) = u^*(u)/[l-4'*(u)] (4) n n n

One can easily check from eqs (2) and (4) for the limiting case n=0, 4)o(t) = 2aQ6(t) and the

non-Markovian master eq (3) becomes the Markovian master equation. We now specialize eq (3) to hopping in a lattice network (where C is lattice site) with the possible presence of a constant electric field E in the x-direction. In the limit of small jump distance Aq, a continuum form of eq (3) useful for further discussion is^^

9p(x,t)/3t = a2/6 ^ (t-x) [32p(x,T) / 3x2]dT - 2A bE ^ (t-t) [ 3p (x, x) /3x]dT (5) o o n o o n

where b is a constant such that bE is the increment of the probability p for hopping in the

direction of the field. The first term on the right hand side of eq (5) is due to diffusion, and the second term is the drift by the electric field when present.

The noise spectrum resulting from spatially-dependent stochastic processes such as diffusion has been a subject of great interest in the past three decades. The motivation behind

these activities in diffusion noise is the hope of deriving a 1/f spectrum over a wide frequency range and of explaining the ubiquitous 1/f noise which accompanies all transport and

162
diffusion processes. Earlier works that employed specific models have claimed to derive a 1/f spectrum from classical Markovian diffusion transport [12](n=0 case of eq (5)). However,

it has been convincingly proved that infinite systems classical Markovian transport by diffusion or by drift will in general not have a 1/f noise spectrum [13]. Specifically for diffusion, one always finds a high frequency u~ ' behavior, and for sufficiently low frequencies one finds a)"-^/^, log cj and w'^ behavior for one, two and three dimensional diffusion systems [14,15]. In a recent work through the assumption of a volume noise called the "P-source,"

it is possible to obtain a 1/f spectrum [16] over a frequency range. However, the "P-source"

implies long-range spatial correlations, which is unphysical [17]. Thus classical Markovian

diffusion will not give rise to 1/f noise. The physical picture in our model is that diffusion in condensed matter in general is non-Markovian (0<^n<l) in a well-defined way [eqs (2),

(4) and (5)]. The question that naturally arises is: will a 1/f noise spectrum emerge from

diffusion transport with n^O? We proceed to answer this.

We calculate the diffusion noise spectra by the Green's function procedure [14]. The

Green's function G(r, r', t) is the probability that a carrier will be found at r at time t if it is at r' at time 0. For diffusion transport in one-dimension described by the 1-dim.

counterpart of eq (5), G satisfies 9G(x,t)/2t = a2/2 (j)n(t-T) [ a^GCx, x) /3x^] dx+o (t) 6 (x) .

domain in which one-dimensional diffusion occurs in a sample of length 2£, N(t) the total

number of carriers in and <An2> the variance of__N. G''(x,x',iw) is the LT in t ofG(x,x',t)

with u=ico. _In terms of Laplace-Fourier transform G"(k,u), Sj^(w) = 4<An2>^^~-'- Re/^dv/Qdv'

(l/27r) /dke-""^^^"^ )G*(k,iaj). Examine first the high frequency behavior of the noise spectrum. High frequencies or equivalently earlier times behavior is governed by the earlier

time dependence of ^(t) of eq (2) which is ii^(t) = (l-n)ant~'^ for t«an with LT given

by iCn^u) ~ (l-n)a^r(l-n)u^~"'" H Xj^u^" where r(z) is the Gamma function. From eq (4) we have

(i)n(u).= Anu'^[l-XnU"^+n]-l. Substituting this into Sj^Cw) gives Sjg(w) = 4(<AN2>/fi) Re/dk/2Tr 1 f-^ e^Wxp {ioj + (XnA§/2)k2(iw)'^/[l-Xn(iuj)'^-l] }. For high frequencies such that

it can be simplified to Sj^(w) = (8<AN2>/TTf^) Re /dk(sin\5./k^) [iw+(XnA§/2)k2(iaj)^]"l. The

integral is evaluated by contour integration in the complex k-plane. On introduction of a natural

(<An2>/

sin$)

]

for several representative values of n and the results can be seen in Fig. 1. -3/2 When n=0, we recapture the results of Markovian 1-dimensional diffusion: the u low

at (jj » (xiq, and the ^~ ]_ow at i^«uiQ, and coo=Do/2J^^ with Dq, the 1-dimensional diffusion

constant, equals AqAq/2.

-3/2 -1/2 For n^O both the high frequency to law and the low frequency w law are modified

as evident from Fig. (1). Analytically those modified asymptotic behaviors for Sj^Cw) are

" (f2n/2) (co/co^)'^^^^^^ ; oj«a)n

where ^ = 2(X^I\q/2)^^'^ cos(nTr/2) <AN2>/ncos4> . As n increases continuously from zero to unity, both (3-n)/2 and (l-hi)/2 approaches one. Thus we have found 1/f spectra of many

decades in our model. These predictions now await detailed comparisons with experimental

data.

It is worthwhile to emphasize that the present dispersive diffusion model is just one extension among many of our unified theory of low frequency fluctuation-dissipation properties of condensed matter [1-7]. In earlier works we have shown that the unified theory

enjoys remarkable success in its application to other areas [4] including dielectric relaxation, mechanical and viscoelastic relaxation, NMR spin-lattice relaxations, voltage noise,

1/f mobility fluctuations and dispersive transient transport. Non-zero value of n for the

description of these phenomena in laboratory condensed matter systems is found to be the

rule rather than the exception. Hence the general occurrence of diffusion-associated 1/f

noise in condensed matter can be understood in our present model that is based on the unified theory [4]. Most intimately related to dispersive diffusion is dispersive transient

transport [4]. Their relationship can be seen through eq (5) which gives both phenomena

163
with a connnon memory function <^ (t). Experimental data of dispersive transient transport

gives direct evidence for non-zero values of n. In Si02, the value of n for hole hopping [4]

is 0.7 to 0.8. In a-As2Se2 n has the value [4] of 0.46. These independent measurements in

related phenomena should enhance the credibility of our model of diffusion noise.

REFERENCES

* Work supported in part by ONR. K.L. Ngai, Bull. Am. Phys. Soc. 24B, 284 (1979)

.

K.L. Ngai, A.K. Jonscher and C.T. White, Nature, 277 , 185 (1979).

K.L. Ngai and C.T. White, Phys. Rev. 20B, 2475 (1979). K.L. Ngai, Comments Solid State Phys. % 127 (1979) Part I; 9^, 141 (1980) Part II. K.L. Ngai, in Fast Ion Transport in Solids , ed. Vashista, Mundy and Shenoy (Elsevier,

North Holland, 1979), p. 203.

S.T. Liu, K.L. Ngai and S. Teitler, Ferroelectric 28, 369 (1980). K.L. Ngai, Naval Research Laboratory, Memorandum Report #39]7, Jan., 1979; Phys. Rev.,

B22, 2066 (1980).

E.P. Wigner, Gatlinberg Conf. on Neutron Physics, Oak Ridge National Lab Report No. ORNL-2309, p. 59; M.L. Mehta, Random Matrices , Academic Press, N.Y. (1967).

C.C. Lai, P.B. Macedo and C.J. Montrose, J. Am. Ceram. Soc. 58, 120 (1975);

J. A. Bucaro, H.D. Dardy and R.D. Corsaro, J. Appl. Phys. 47, 741 (1975); and

CP. Lindsey, G.D. Patterson and J.R. Stevens, J. Polymer Sci. PPE. V.M. Kenkre, E.W. Montroll and M.F. Shlesinger, J. Stat. Phys. 9^, 45 (1973).

G.F. Leal Ferreira, Phys. Rev. B 1^, 4719 (1977); G.F. Leal Ferreira and L.E. Carrano

de Almeida, Phys. Rev. E 19^, 6601 (1979).

J.R. Richardson, Bell Syst. Techn. J., 29_, 117 (1950); C.G. MacFarlane, Proc. Phys.

Soc. (London, B63, 807 (1950)).

K.M. van Vliet and A. van der Ziel, Physica, 24, 415, 556 (1958).

K.M, van Vliet and J.R. Fassett, Fluctuation Phenomena in Solids , ed. R.E. Burgess

(Acad. Press. N.Y. , 1965), p. 267.

M. Lax, Rev. Mod. Phys. 32, 25 (1960).

R.F. Voss and J. Clarke, Phys. Rev. B13, 556 (1976).

K.M. van Vliet, Proceedings of Second International Symposium on 1/f Noise, Orlando,

Florida (1980), p. 305.

164
TEMPERATURE DEPENDENCE OF THE EXCESS 1/f NOISE IN METALS

K. L. Ngai

Naval Research Laboratory

Washington, D. C. 20375

INTRODUCTION

The temperature dependence of the excess (1/f) noise is a subject of some interest since

the discovery of Eberhard and Horn [1] of a strong temperature dependence of the excess (1/f)

voltage noise in thin films of the group IB transition metals and in Ni. This strong temperature dependence of the matnigude of 1/f noise is accompanied also by the appearance of

a high temperature peak in Ag and Cu. These behaviors for thin metal films should be con- trasted with the rather weak temperature dependences of 1/f noise observed in semiconductors

and carbon resistors. Any of the current models of 1/f noise in their respective present

forms either do not predict or have not yet made a connection with this strong temperature

dependence. Dutta, Dimon and Horn [2] have nevertheless made progress in the understanding

of the temperature dependence of the magnitude and of the frequency spectrum of the 1/f

noise in metal films by postulating a distribution of activation energies D(E) centered in

the vicinity of 1 eV. They envisage an activated process with activation energy E has a characteristic time t = Tq exp(E/kT) will contribute to resistance fluctuations and voltage noxse Sv(a)): t/ (0)2x2+1). Then the distribution of activation energies will give rise to a total voltage noise of

Proceeding from eq (1) and assuming the width of D(E) is much larger than kT, Dutta et al. then demonstrated that a given D(E) can explain both the strong dependence of S on T and

the variation of the exponent a with 0.8<a<1.4 in the frequency spectrum or the excess

noise. These results do follow from some key assumptions that have to be made. To this

author's mind the important question that has not yet been addressed to in Dutta et al.'s

model is how the noise of resistance fluctuations comes from the fluctuating quantity the

characteristic time of which is thermally activated and its activation energy is distributed

according to D(E) . Dutta and coworker [2] certainly are aware of this problem but, in

order to proceed along the lines of their model, they simply assume that the resistance is

linearly coupled to the fluctuating quantity. In this short note I would like to propose a specific mechanism for resistance fluctuation noise as caused by the fluctuating quantity.

The fluctuating quantity can be diffusing vacancies as suggested by Dutta et al. or diffusing impurity atoms. Both possibilities will collectively be referred to as diffusing

species

.

We consider excess 1/f noise contribution in metal films that is caused by electron

scattering from hopping and/or diffusing species. Hopping/diffusion of vacancies or impurities in metal films are generally thermally activated. The autocorrelation function

C(t) of the current fluctuation associated with electron scattering is to be calculated.

Electrons are scattered by static impurities, phonons and imperfections of the metallic

film as well as from the hopping/diffusing species on which we focus our attention in this

paper. In a unified model approach [3,4], I have shown in an earlier published work on 1/f

noise that electron scattering from the static impurities, phonons and imperfections will

give rise to resistance fluctuations that have the power spectral function with the frequency dependence of

S,,(u),T) CC /[t/(u)^t^ + 1)] D(E) dE (1)

MODEL

S(f) 1/f 1-n

(2) CC

165
and its magnitude is weakly temperature dependent. The quantity n, called the infrared

divergence exponent, is a positive number that is expected to be small compared with unity

for metal films. The mechanism for the mobility fluctuations from electron scattering is

from infrared divergent excitation and deexcitation of correlated states. The name "correlated states" was coined [5-7] to describe the states of any condensed matter that excitations and deexcitations between any two of them are of arbitrary low energy. It is

worth emphasizing that for the correlated states, the infrared divergence condition holds

down to arbitrary low frequencies for any ambient temperature. This follows from the

properties of the correlated states.

The unified theory emerges from an inquiry into the nature and properties of excitations of low energy (say below 10^ Hz) and the corresponding states that, through transitions between them, give rise to these excitations and deexcitations. For an excitation of

low energy, the two states that define it must each involve for its description a very

large number of coordinates (atomic, bonding, electronic or their composite). In fact the

number of coordinates that must be included to describe the states must increase as the

excitation energy decreases towards zero. This can be proved by reduction to absurdum.

Let us assume on the contrary that the number of coordinates that describe the states do

not increase as the excitation energy decreases towards zero. Then for the finite number

of coordinates, there are only a finite number of admissible states and their excitation

spectrum will have a lower cut-off in energy, contradicting the assumption that the excitation energy decreases towards zero. Returning to our discussion, the Hamiltonian relevant

for the description of the states cannot be specified due to complex interactions and the

increasing large number of coordinates. However, there is a way out of this apparently

hopeless situation by adopting the statistical approach to quantum mechanics of Wigner,

also known as random matrices, to our problem. Wigner 's approach enables us to deduce some very general properties of the low energy excitations that hold for any material. Among

these is one that the density of excitation N(E) with energy E is linear in E provided E is

smaller than a characteristic energy Ec i.e. N(E) = kE for The physics behind this

is level repulsion. The proportionality constant k and E^, are material dependent however.

For crystalline solid state material both k and E^ are expected to be small. This is also

the case for liquids which are simple, such as water, non-glass-forming liquids or even non-polymeric glass-forming liquids such as B2O3 at temperatures far above the glass transition. Proceeding from crystalline to disordered, amorphous or glassy solid state materials

the Wigner analogue states (referred to from now on as correlated states) as well as their

low energy excitations increase in density, hence both k and E^ increase. For the same reason for liquids on going from simple to more complex molecular ones including glassforming polymeric liquids and supercooled non-polymeric liquids, both k and Eq will

increase.

In most if not all considerations of the low frequency responses, a primary species can be identified to be responsible for any particular response. There can be dipolar groups

of atoms or molecules, chain segment, charged and uncharged particles. Response and fluctuations are triggered by transitions of the primary species. The primary species does

interact with the correlated states however, and its sudden transition will switch on a new

potential V which causes both excitations and deexcitations of correlated states. The

matrix element of for excitation or deexcitation of energy E connecting two correlated

states are uncorrelated with and independent of E itself if E is low enough. This follows

from the local extent of the primary species limiting its interaction with only a small

fraction of the totality of large numbers of coordinates that define the two correlation

states and detemine their energy separation E. Recalling that N(E) = kE, the independence

of V-j^ on E then implies that |v|2n(E) = nE. Here n is a dimensionless constant and |v|^

is the average of the matrix elements |Vi|^ and independent of E. The last equation turns

out to be the condition for infrared divergence and n is called the infrared divergence

exponent. "Infrared" carries the meaning of low frequency and has nothing to do in our case with the infrared optical spectrum. Its usage in infrared divergence comes from the

first known example of such phenomena in Cerenkov radiation or Bremstrahlung where there is an increasingly high probability of exciting low energy (infrared) photons emitted after a charged particle impacts on a target. In our case, the primary species transition will be

trailed by an infrared divergent excitation and deexcitations of the correlated states.

That is there is an increasingly high probability of exciting (or deexciting) decreasingly

small energy correlated state excitations, and this causes a power law divergence at small w of the response in the frequency domain and the algebraic t"'^ time dependence of

166
the response function at large times with 0 ^ n < 1. This is the fundamental result that comes out of an adaptation of Wigner's formulation of complex quantum systems and the infrared divergence principle.

TEMPERATURE DEPENDENCE OF EXCESS NOISE

For electron scattering from the species that is hopping/diffusing in the metallic film,

the autocorrelation function C(t) of the current fluctuation is now a convolution of C^Ct),

the autocorrelation function of the electron current fluctuation due to scattering of electrons from the species, with C2(t) the correlation function of the hopping/diffusive motion

of the species. We have seen in the last paragraphs C-^(,t) will contribute a S-]^ (f )o^l/f °' resistance fluctuation noise with a = l-n2_, and n-|^ is small compared with unity. C2(t) can be calculated also from the unified model of low frequency fluctuation, dissipation and

relaxation properties of condensed matter [7] . Consider hopping motion of the species with

a jump time t = Tq exp(E^/kT). Again infrared divergent excitation and deexcitation of

correlated states by the hopping species modifies the correlation function from the familiar

form C2(.t) = expC-t/x) to the form of

C^Ct) = exp[-e"''2^ t-^"''2/ (l-n^) tE^2] C (3)

where n2 with 0 <^ n2 < 1 is the infrared divergence exponent for the hopping motion of the

species; y is the Euler's constant and E^ is the upper cut-off energy for the correlated

state excitations [7J. Note that for n2 = 0, the expression for C2(t) that appears on the

right hand side of eq (3) reduces to expC-t/x). The correlation function C^(t) of eq (3)

alone will give rise to a noise 82 (f) which has a peak at a certain temperature Tjjj^^^. This

can be readily seen from the similar results on the spin-lattice relaxation time Tj^ in

nuclear magnetic resonance as discussed in Ref. (7). The noise power spectrum S2(f), except

for a constant, has the same temperature and frequency dependence as the inverse of T]^. It has no frequency dependence for T >^ "^maK ^'^'^ ^ frequency dependence l/f^ for T < T^^^^^ (Fig. 1). Note that g increases monotonically from g=0 at T=Tjj,gj^ to the value of 6=2-n2 at

Fig. 1. Frequency dependence of S2(f) for

several values of n2 = 0, 0.2, 0.4, 0.6

and 0.8. The ordinate as labelled in the

figure is l/T^ which has the same functional dependence on f as S2(f).

NORMALIZED FREQUENCY

167
T<<Tjjj^jj. Here n2 is expected for a significant fraction of unity. Since C(t) is a convolution of Cj(t) and C2(t), the voltage or current noise spectrum S(f) is the product S]^(f)

S2(f)- The result we find are then consistent with several striking features of the excess

(1/f) noise observed by Eberhard and Horn [1]. These include: (1) the strong temperature

dependence and the appearance of high-temperature peak in S(f) for a fix f due to S2(f);

(2) the predicted 1/f spectrum with y

~ ot+3 which decreases continuously from a value

larger than unity at T<Tjjj^^ to Y=o'=l~n]^<l as T approaches T^^^^ and beyond; and (3) the shape

of the calculated peak structure is approximately the same as was observed by Eberhard and

Horn [1] for some chosen value of n2. Although we have discussed only the hopping motion,

the results are similar for diffusive motion. Thus the contribution to resistance fluctuation from electron scattering by hopping/diffusing species when taken together with the

weakly temperature dependent l/f"^ noise caused by electron scattering from other static

potentials and from phonons etc. , can explain the data of Eberhard and Horn on metal thin

films. Nevertheless the overall mechanisms involved in the noise observed in metal thin

films can be much more complicated than the simple specific model proposed here. In spite

of the limited success of the present model, this work should be viewed as a first step in

incorporating a fundamental process of 1/f noise into a model that attributes the strong

temperature dependence of the noise magnitude to fluctuating quantities that have characteristic times which are thermally activated.

REFERENCES

[1] J. W. Eberhard and P. M. Horn, Phys. Rev. B18, 6681 (1978).

[2] P. Dutta, P. Dimon and P. M. Horn, Phys. Rev. Lett. 43, 646 (1979).

[3] K. L. Ngai, Bull. Am. Phys. Soc. 24, 235 (1979); 284 (1979); K. L. Ngai and C.T. White,

NRL Memo Report 3863 (1978).

[4] K. L. Ngai, NRL Memo Report 3917 (1978); Phys. Rev. B22^, 2066 (1980).

[5] K. L. Ngai, A. K. Jonscher and C. T. White, Nature 277, 185 (1979); K. L. Ngai, Bull.

Am. Phys. Soc. 24., 465 (1979).

[6] K. L. Ngai and C. T. White, Phys. Rev. B20, 2475 (1979).

[7] K. L. Ngai, Comments Solid State Phys. 9, 127 (1979); 9, 141 (1980).

168
THE ROLE OF MOBILITY IN. 1/f NOISE

D. A. Bell

Professor Emeritus of Electronic Engineering

University of Hull

Hull HU6 7RX, U.K.

THE NUMBER OR MOBILITY QUESTION

The usual characteristic of 1/f noise, a mean-square voltage proportional to the square

of steady current, suggests a fluctuation in conductance; and this has been confirmed by

several tests, notably the correlation experiment of Jones and Francis [1]. Since conductance is proportional to the product of number and mobility of carriers, the next question

to be asked was whether the 1/f noise represented a fluctuation in number or in mobility

of the charge carriers. Early work favoured a fluctuation in number [2], but there has been

accumulating evidence that it is the mobility which is involved. The elegant experiments

of Kleinpenning [3] on 1/f fluctuations in thermo-e .m. f. seem to have established that 1/f

noise is associated with fluctuations in mobility rather than in number of charge carriers.

Meanwhile Hooge had proposed [4], and demonstrated for a metal [5], the attractive

hypothesis that 1/f noise can be described by the formula

AR^ = ^. Af (1) R N f

where N is the total number of charge carriers taking part in the conduction. Hooge originally suggested that ot would be a universal constant, but this has since been found

unsatisfactory, e.g. for the results of Hooge and Gaal on electrolytic conductors [6].

Equation (1) is not necessarily in conflict with the finding that fluctuations in conductance, the left-hand side of eq (1), are due to fluctuations in mobility. If each charge

carrier or group of carriers makes an independent contribution to the total fluctuation,

then usual statistical principles indicate that the mean-square relative fluctuation in the

combined effect may be expected to be inversely proportional to the number of individual

contributions, i.e. inversely proportional to N. On the other hand the spatial correlation

of 1/f noise in semiconductors is found to be limited to a distance of less than 1 ym

which is even less than the mean free path of electrons in some materials [7].

MOBILITY OR DIFFUSION ?

The electrons in a conductor have always a random (thermal) motion, but a directed

drift velocity only when a field is applied; and usually the random motion involves much

higher velocities, of the order of 10 m/sec, than the drift velocity. Now "mobility"

normally refers to drift velocity, by means of which it may most directly be measured.

"Diffusion," on the other hand, normally implies a concentration gradient and rate of

transport of particles down this gradient. Mobility and diffusion are difficult to distinguish because the coefficient or diffusion is proportional to the mobility according

to the equation.

^bs'^T/^

In semiconductors one often uses Hall mobility, which may be determined by comparing Hall

coefficient with conductivity. Both terms in the comparison involve drift velocities.

169

i
Here Vi ^j^g

is the absolute mobility, with the applied field measured in absolute e.s.u.

It is clearly a constant multiple of the "conventional" mobility expressed in cm^V "^sec But now suppose there is neither applied field nor concentration gradient: a particle will

still move from an initial position, and given long enough time will move through any given

distance, however large. Since it depends on the random motion of thermal agitation, this

movement is akin to diffusion and in this paper will be regarded as diffusion, in spite of

the absence of a concentration gradient. It is in this sense that it is queried whether

1/f noise is a diffusion rather than mobility phenomenon, since it is most probably related

to the random motions of individual charge carriers. (This is entirely different from the

diffusion theories of the 1950' s, in which it was supposed that the conductance of a semiconductor was modulated by the diffusion through it of some "foreign" particle, probably

in the neighbourhood of the terminal contacts.)

THE ZERO-CURRENT EVIDENCE

It has been questioned whether 1/f noise is caused or revealed by the passage of current.

Hooge and Gaal [8] observed 1/f noise in a closed circuit at zero current, this being obtained

by balancing a thermo-e .m. f . against an e.m.f. of opposite sign from an external source. There was then no obvious drift velocity and the 1/f noise appears to have been associated

with the random motions of electrons, which can be regarded as diffusion. There is, however,

a serious difficulty about this interpretation: when there is a current the addition of a small drift velocity to a large random velocity will make little difference to the mean- square velocity. The excitation of 1/f noise by a steady current is of a different order

of magnitude than "hot electron" effects. Alternatively one can say that the observations

of Hooge and Gaal were made at zero net current, but there were two equal currents of

oppositely charged particles. In the end one cannot say definitely that 1/f noise has been

observed in the absence of current. Min [9] deduced that in non-degenerate semiconductors

the 1/f noise could not be due to scattering. Experimentally there is 1/f noise in highly

conducting metals, which are certainly degenerate. No indication has been recorded of

the state of degeneracy of semiconductors used for 1/f experiments; but most of them, of small or moderate degree of doping, are probably non- degenerate at room temperature. There does

not seem to be any clear distinction between degenerate and non-degenerate conductors as generators of 1/f noise.

EXPERIMENTAL EVIDENCE ON MOBILITY

Taking a very broad sweep, from the measurements of Van de Voorde et_ al . [10] on indium

antimonide through the Hooge average for silicon to the results of Hooge and Gaal for

electrolytes, the data are consistent (within the variability of observations) with a simple

hypothesis that a is inversely proportional to the 2/3 power of mobility, as shown in figure

1. The noise coefficients for metals are based on the work of Hoppenbrouwers and Hooge

[11] but with the following reservations:

(i) Only the hemispherical samples of the "good" metals have been considered.

(ii) The numbers of conduction electrons per atom have been taken from Ehrenberg [12].

These are based on Hall coefficients, not valency.

(iii) Values of cx have been estimated by drawing a line of appropriate slope and good

fit through the experimental points and noting its displacement from the line for

a = 2x10"-^. It may be a coincidence that a good fit to the 2/3 power law for

a is only obtained for silver, the one of the three metals which is uniquely

monovalent.

But the relationship breaks down when one examines a particular limited group, for

example the metals in figure 1. Hooge' s first publication of his hypothesis [4] included

a diagram showing noise versus total number of carriers for various previously published

experiments. This shows both n-InSb (his reference 3) and p-InSb (his reference 10) near

the line established for n-Si, although the n-InSb had an electron mobility between 100 and 350 times as great as that in silicon and the hole mobility in InSb, which should be

relevant to the p-InSb of Hooge' s reference 10, is about half that of electrons in silicon.

Some dispersion is to be expected on account of the powerful influence of the surface state,

but one can only say that the data collected by Hooge in 1969 give no positive evidence in

support of a hypothesis relating 1/f noise to mobility.

170
cx

Fig. 1. Mobility and relative

noise coefficients for various

materials: (1) InSb; (2) n-Si;

(3) Ag; (4) Au; (5) Cu;

(6) hatched area, electrolytes.

2x10 The line is of

slope a°: y-2/3

10 10^ 10'

MOBILITY cm^v''s"''

The hypothesis that a is inversely proportional to mobility appears to be clearly

contradicted by the results of Hooge and Gaal on electrolytes [6], where they found a to be linearly proportional to the molar concentration. But the variation of mobility with

concentration is small and far from linear. A further difficulty is that the mobility of

protons (hydrogen ions) is some seven times greater than that of most other ions in water

[13] and this would be expected to make the behaviour of HCl significantly different from

that of salts: yet no significant difference appears in figure 6 of [6]. A complicating

factor, however, is that the transport of protons through an aqueous solution is not by

drift but by exchange between neighbouring water molecules, so it might not make the same contribution to 1/f noise. The results for electrolytes also cast doubt on the usual

assumption that 1/f noise is the resultant of numerous current pulses corresponding to the movement of individual particles, because the contribution of each to the mean-square

fluctuation would be proportional to the square of the charge which it carries. Thus

+ . . , . . .. . „ .2 ,2 AgNO^ ^ Ag + NO^ would make a relative contribution of 1 + 1 2 while CuCNO^)^

Cu"^* + 2im^') would give 2^ + 2x1^ 6. There is no indication of such a distinction by

valency in [6]. This leads to the suggestion that 1/f noise may be a collective phenomenon,

i.e. a modulation of mobility rather than a diffusion effect. Some clue to the distinction

between mobility and diffusion might come from the effect of temperature, since eq (2)

shows that for constant mobility the diffusion coefficient is proportional to temperature.

But temperature also changes the number of charge carriers in a semiconductor, and may in

fact have some effect on mobility.

LATTICE SCATTERING

It has been suggested [14] that 1/f noise is produced only by lattice scattering and

2 -3

that a in eq (1) ought to be given the value (y/y, ) x2.10 where the effective mobility

1 diX-T. y is a combination of lattice mobility and the component y. due to impurity scattering

according to the formula.

+ 1 latt

C3)

This was supported by measurements with ohmic metallic point contacts on specimens of

germanium and gallium arsenside having low resistivities in the range 3xl0"2 to 4.5x10"^

ohn-cm and later by measurements on films of bismuth having thicknesses between 0.08 and

8 times the electron free path [15]. The major difficulty with this hypothesis is that it

predicts only a decrease in a below its standard value, whereas the data originally collected

171
by Hooge [4] had 11 examples of a substantially higher, against 6 substantially lower. The

position of some metals in figure 1 causes difficulty since one would expect only lattice

scattering and therefore a of standard value in bulk metals^: Presumably electrolytes can be

excluded from the argioment by qualifying the hypothesis as applicable only to homogeneous

solid or electronic conductors, other mechanisms being present elsewhere. (It would be

interesting to know whether 1/f noise is generated in solid electrolytes, and if so, how

it compares with that in aqueous electrolytes.)

CONCLUSION

It does appear that 1/f noise depends in a very general way on the number of charge

carriers involved and on their mobility. But when one limits investigation to a particular

type of conductor and assumes an inverse proportionality to the number of charge carriers,

there is no clear relationship between noise and mobility. The attribution of 1/f noise

solely to lattice scattering has so far been confirmed for a few specimens only: there is

not sufficient evidence that 1/f noise in general can be so attributed. Whatever the role

of mobility and of scattering there is still no general mechanism for the 1/f shape of

spectrum and there still seems to be a need for different mechanisms in different types of

conductor.

REFERENCES

[1] B. K. Jones and J. D. Francis, J. Phys. 8, 1172 (1975).

[2] J. J. Brophy and N. Rostoker, Phys. Rev 100, 745 (1955).

[3] T. G. M. Kleinpenning, Physica 7, 78 (1974).

[4] F. N. Hooge, Phys. Lett. A. 29, 139 (1969).

[5] F. N. Hooge and A. M. H. Hoppenbrouwers , Physiaa 45, 386 (1969).

[6] F. N. Hooge and J. L. M. Gaal, Philips Res. Repts. 26, 77 (1971).

[7] T. G. M. Kleinpenning, J. Appl. Phys. 48, 2946 (1977).

[81 F. N. Hooge and J. L. M. Gaal, Philips Res. Repts. 26, 77 (1971).

[9] H. S. Min, Jowm. Appl. Phys. 50, 4461 (1979).

[10] P. Van de Voorde, C. K. Iddings, W. F. Love and D. Halford, Phys.

19, 4121 (1979).

[11] A. M. H. Hoppenbrouwers and F. N. Hooge, Philips Res. Repts. 2S_, 69 (1970).

[12] W. Ehrenberg, Eleatrio Conduction in Semiconductors and Metals (Clarendon Press,

Oxford, 1958).

[13] J. O'M. Bockris and A. K. N. Reddy, Modem Electrochemistry: Volvme 1 (Plenum

Press, New York, 1970).

[14] F. N. Hooge and L. K. J. Vandamme, Phys. Lett. 66A, 315 (19''8).

[15] F. N. Hooge, J. Kedziah and L. K. J. Vandamme, Joum. Appl. Phys. 50, 8087 (1979).

172
A MODEL FOR 1/f MOBILITY FLUCTUATIONS IN ELEMENTAL SEMICONDUCTORS

R. P. Jindal* and A. Van der Ziel

Electrical Engineering Department

University of Minnesota

Minneapolis, Minnesota 55455

INTRODUCTION

1// noise in semiconductor devices has been explained using the number fluctuation model

[1-3] as well as the mobility fluctuation formula [4,5]. The first experimental hint towards a connection between lattice scattering and mobility fluctuation was by Hooge and Vandamme [6]. Using the idea of phonon population fluctuations [7,8] we shall establish this connection in detail. Also, the eff"ect of a finite electric field will be examined.

ESTABLISHMENT OF g-r SPECTRUM FOR PHONON POPULATION FLUCTUATIONS

Consider phonons with wave vector g_. The number of such phonons = n^. From statisti- cal considerations, we have

«7= 7/[(exp(^a;g//s:r) - 1]. (1)

Also we have

A/ig2 = „-(i . (2)

Both eqs (1) and (2) do not refer to any specific mechanism for this phonon population

fluctuation. In general, there are a number of mechanisms which can give rise to phonon scatter- ing and hence population fluctuations. From the experimental data [9,10], for germanium and sil- icon, it can be inferred that thermal conductivity, above 30K is dominated by isotope scattering

for fairly pure samples. For samples with higher doping, phonon scattering from chemical impurities lowers the thermal conductivity further. However, whatever be the mechanism, the existence

of time constant gives,

ESTIMATION OF FOR ISOTOPE SCATTERING

From [9], we get

47r M T Pm

(4)

m = atomic mass, AM^ = 2 XjiAMjY

j where Xj is the fraction of isotopes with mass Mj and p = density of crystal.

Substituting typical numbers for silicon, i.e., AM^ = 0.209, M = 28, p = 2.33 g/cm-',

s = 8X10'' m/sec, m = 28 atomic mass units we get

Now with Bell Laboratories, Murray Hill, N. J. 07974.

173
{X^^Xi.lXloy sec. (5)

Then for = 10 ^ sec, X = 86A, and for = 10^ sec, X = 8.6 fim. Thus for a range of 12

decades, the corresponding phonon wave lengths are quite meaningful.

ABSOLUTE LIMITS FOR 1/f SPECTRUM

The limits of are the limits of the l/f spectrum. An upper bound can be derived by tak- ing X = lattice spacing.

.-. for X = 5A, /^ax = 1-4X10"^ Hz

Also, since the ^ = 0 mode always exists (if not for the sample then for the noise cage as a whole), it would seem that there is no lower limit for the l/f spectrum. However, bounds more

restrictive than those derived above, might exist, from other considerations.

PHONON MEAN-FREE PATH AND SAMPLE LENGTH

Note that even for = 10~^ sec, the phonon has a mean-free path of 0.8 cm. It therefore seems likely that for a sample whose dimensions are smaller than 0.8 cm, boundary scattering [11]

should become important.

However, one should note that this is true only for "perfectly rough" surfaces. From

[12,13], it is clear, that the roughness of the surface depends upon the phonon wave length. Thus

for surface roughness <<86A (corresponding to = 10~^ sec), we will have specular reflection

which does not give rise to scattering. Hence the estimate of mean-free path and relaxation

time Tg are valid. MEAN-FREE PATH FORMULATION FOR CARRIERS

In order to determine the momentum relaxation time, we shall evaluate the collision term in Boltzmann Transport Equation.

From time dependent perturbation theory, we get

dt

1

coll (27r)^

J -^[(«s + l)/(^)(l -f{k')

-nj(k') [\-f{k) ^ik) - e{k') 3^' (6)

To be able to extract momentum relaxation time under low electric field conditions, we have

to assume < 1

or for rtg » 1, we have

Now

(7)

An.

a bctual a Therefore, let

«„(l+«g)

9 lallowed = A/7.

9 bctual*^"

«g(l + n^a^, 0 < a„ < 1. (8)

174
We shall refer to this as the small fluctuation approximation. Then under the small fluctuation

approximation and assuming elastic collisions \foik) = fqOc')], we get after some manipulation — = —= — ff ^- {I - cos d)sin 0 dd d(l> (9)

(1(0,0) = average distance travelled by carriers before they are scattered in (0,0) direction. In this case, it is independent of the electron energy. FLUCTUATIONS IN CARRIER MEAN-FREE PATH

1 me2, 2

(1(0,0) TTn Using (3) and (8), we get

(10)

• •• 5j(0,0)(/) =(1(0,0) al (11)

We shall now average this over electrons travelling along the z axis with diff"erent speeds (Fig. 1).

A

9 /

Fig. 1 Coordinate axes showing the

direction of motion of the carrier along

the vector k before scattering and along

]c after scattering.

Now sin — =

2 Ik

Hence, as we vary k, keeping 0 and q constant, we are in eff"ect varying the magnitude q. However, q is related to r. Thus averaging over energy is eff'ectively an average over a time constant

distribution. We shall presently show that,

1 g(T)dT=

T /«(ti/to)

Now

(1(0,0)

me2,2

KT.

(12)

(13)

This is independent of q. Therefore, averaging over energy and dropping the subscript q, we get

from (11)

175
^«(M)C/-)=7^)^4 (14)

'''^ /«(ti/to) / To Ti From (9) and (14) we get

Sjif) _ 1 t lnir,/To) f

(15)

Now n =

• -^mC/) _ «i 1 ^2 ln(Ti/To) /

Since d for different carriers fluctuate independently, we get for A'^ carriers SM) al 1 „

^2 ln(ri/ro) fN fN

where

,2 aa ln(Ti/To)

(17)

(18)

(19)

Note for —= 10*^ and ~ 0.2 we get a = 1.4X10 ^ which is close to Hooge's parameter

^0

au = 2X10 ^. A detailed determination of and its dependence on doping is still to be investigated.

DETERMINATION OF THE TIME CONSTANT DISTRIBUTION FUNCTION

The averaging over the magnitude of a is done for q_ pointing in a particular direction.

Hence we are concerned with phonons with g pointing along a specific direction. Hence the pho- non distribution can be treated to be one dimensional.

The number of states which are responsible for generating the modulation with time constant Tq is proportional to the number of phonons with wave vector ij. Number of phonons with wave vector g

KT = «7=^ iKT»1i<^g) (20)

••• g(-)dran,d,=-^^. (21)

Now, for point defect scattering in general, we have

roc-!- (22)

This yields,

dq 1_ dr

q 4 T

'

Comparing with (21) and normalizing, we get

^(T)^/r = -—^—

(23)

ln(Ti/To) T

as assumed in (12).

176
NOISE IN PRESENCE OF OTHER SCATTERING MECHANISMS

According to the model presented here, 1// noise is generated by fluctuations in the population of acoustic phonons and hence in the zero electric field acoustic mobility Maco- pres- ence of other noiseless scattering mechanisms, following Hooge and Vandamme [6], we get

^2 a = a. (24)

ELECTRIC FIELD DEPENDENCE OF 1/f NOISE

Experimentally it has been shown [16] that

a{E) = a(0)/[l + [E/E^)^]. (25)

where mo^c = speed of sound in the medium. This can be explained [17] in terms of the

present model by considering the acoustic and optical mode scattering jointly and letting only ^^co

generate 1/f noise. The field dependence for that fits with the above analysis is given by

Mac = Macotl + (^/^C)']- (26)

REFERENCES

[1] A. van der Ziel, Physica 16, 359 (1950).

[2] F. K. Du Pre', Phys. Rev. 78, 615 (1950).

[3] A. L. McWhorter, M.I.T., Lincoln Laboratory Report, No. 80 (1955).

[4] F. N. Hooge, Phys. Lett. 29A, 139 (1969).

[5] F. N. Hooge, PAyjica 60, 130 (1972).

[6] F. N. Hooge and L. K. J. Vandamme, Phys. Lett.. 66A,. 315 (1978).

[7] R. P. Jindal and A. van der Ziel, Appl. Phys. Lett, to be published.

[8] R. P. Jindal and A. van der Zel, Jour. Appl. Phys., to be published.

[9] R. Herman, E. L. Foster and J. M. Ziman, Proc. Roy. Soc. A. 237, 344 (1956).

[10] J. A. Carruthers, T. H. Geballe, H. M. Rosenberg and J. M. Ziman, Proc. Roy. Soc. A.

238, 501 (1956).

[11] H. B. G. Casimir, Physica 5, 595 (1938).

[12] R. Berman, F. E. Simon and J. M. Ziman, proc. Roy. Soc. A., 220, 171 (1953).

[13] R. Berman, E. L. Foster and J. M. Ziman, proc. Roy. Soc. A. 231, 130 (1955).

[14] See for example, J. Callaway, Quantum Theory of Solid State, part B, Chapter 7, Academic

Press, New York and London (1974).

[15] See for example, K. Seager, Semiconductor Physics, Chapter 6, Springer-Verlag, New York

(1973).

[16] G. Bosman, R. J. J. Zijlstra and A. van Rheenen, Phys. lett. 78A, 385 (1980).

[17] R. P. Jindal and A. van der Ziel, in process with S.S.E.

177
COr-KOK M30IIANI3M OP 1 /f NCI3E AND BURST NOISE

A.M. Zakliki ewi c 2

Ins ty tut Technologli iilektronowe j 03MI, 02-668 '//arszawa, Poland.

1 . INTRODUCTION 2. PI-IY3IGAL MODEL

It is a known fact that 1/f noise

generation is related to the gr?in

boundaries in resistive materials and

is dependent on the defects level in

semiconductor materials and devices.

One assumes that the local potential

barriers on the metal precipitations

are responsible for burst noise generation in semiconductor devices [1].

The author assumes that there

exists a common source of 1/f noise

and burst noise. This source is related to the potential barriers on a different defects and discontinuities

in resistive and semiconductor materials.

It is obvious that if there ex- ists a discharging mechanism of the

local potential barriers, which ex- plains burst noise generation, then

with it a charging process should be

associated. A charging relaxation

processes of the local potential barriers can result in 1/f noise spectrum [3] in the limited low frequencyrange .

We assume that there exists a kind of random distribution of the

local potential barriers in the con- sidered element. The proposed equivalent circuit of such a single barrier is shown in Pig.1, where C is

the capacity'- of the barrier, R

charging resistance, - discharging

resistance, I^R - resistance of that

part of the element which is parallel

to the barrier.

0^

9

Fig.1. The equivalent circuit of the

single potential barrier

We can surmise that key K is the

generation-recombination center asso- ciated with the local potential barrier as it is done for the case of the

178
proposed burst noise mechanism [l] , Such a potential barrier is charged

v;ith the time constant GR^ v/hen the

external electrical field is applied,

Pig. 2. The processes at the external

connections of the single potential barrier

The discharging process takes place

in the G, R^, K circuit when the switch K is closed. Some amount of discharging current flows in the C, QR,

R^ circuit and can give rectangular

pulses at the external connections of

the considered element. After the discharging current pulse is finished,

the circuit is charged again - the

cycle is repeated. As a result of this

mechanism we receive a series of periodical rectangular and exponential

pulses for any single barrier, as it

is shown in Pig, 2.

Similar model has been proposed

previously [2] to explain 1/f noise

phenomenon but discharging processes

were disregarded at that time.

3. NOISE SP3GTRUM

If we consider only charging process for the single potential barrier

then the power spectrum is [2] :

A(w]

w T

where angular frequency

2Trk

(1

(x) = 2Jrf T » k - ^ * 2, 3,...

A single relaxation process is

periodical and cannot be named noise.

For random distribution of such

relaxation processes coming from many

local potential barriers located in

the considered element we can apply

the relation given by M.Agu [3]. For

the limited low frequency range noisepower spectirum is

|A{co)| = f PIO) IacI'i^ 2)

where: A^ - the initial value of each

individual relaxation

process,

P(0)- the density function of

the random process P(C)

for random variable G—O.

Por a common case of the multilevel burst noise we have got theequation describing noise power spectrum 1 4] .[5] :

|A{co)| = UBNO X

•n2 ,2 A + (O

(5)

where : U

BNO the square mean value ' of bursts amplitude,

the mean number of level changes in the unit

of time

179
The relation given here [3] is

valid also for two-level and threelevel burst noise, which are particular cases of burst noise, most frequently observed in practice.

Burst noise can give some deviations in typical 1/f noise spectrum

14). [5] . It is impossible, however,

to calculate the common spectrum for

l/f noise and burst noise without the

knowledge of the elements of equivalent circuit shown in Fig,1.

4. DISCUSSION

The common mechanism of 1/f noise

and burst noise, presented here, makes

it possible to explain the fundamental

difference between these kinds of noise, namely universal character of 1/f

noise and limited presence of burst

noise. Considering equivalent circuit

of the 1/f noise and burst noise source /Pig.l/, we can notice that the

exponential processes of the barrier

charging are a.lvjajs present at the a-a connections and thus at the external connections of the considered element. To the contrary discharging processes take place mainly in the C, R^,

E circuit and they can be observed at

the external connections over background noise level only for sufficient

value of the / ( l?R + R„ ) rati o

.

C In the conclusion v/e may state

that burst noise should be recognized

as the associated, side effect of 1/f

noise.

RE?3RSKCSS

h ] 3 . T .Hsu, R. J.'Vhi ttier, C . A.. Mead,

Solid-state Electronics V5* 1055-

1071, 1970.

[2] A.M.Zaklikiev/icz in Proc. Golloque

international sur le biniit de fond

des composants active semiconducteurs, Toulouse, 1971, pp. 27-29.

[3lM,Agu, Journal of the Phj'-slcal

Society of Japan 40, 1510-1511,

1976.

[4] A.H.Zaklikiewicz, Doctorate Thesis,

Warszawa, 1978 /in Polish/.

l5]A,M.Zaklikiewlcz, Solid-State Electronics 24., 1-5, 1981.

180
1/f VOLTAGE NOISE CAUSED BY SCATTERING OF ELECTRONS IN METALS

Leendert M. Bliek

Physikalisch-Technische Bundesanstalt

D 3300 Braunschweig, Fed. Rep. Germany

INTRODUCTION

The spectral intensity Sy(f) of the voltage noise in current carirying electric conductors is often found to depend on the frequency f as 1/f. Experimentally, Hooge 's empirical

relation

= f (1) U N

where U is the average potential difference, ^is the number of electrons or holes involved,

and the constanta is normally close to 2. 10 , holds in most cases, but it is found difficult

to give a conclusive explanation for the effect [l]

.

In the following, thermo e.m.f. 's, due to temperature fluctuations, caused by scattering

of the electrons, are considered as a possible mechanism for generating 1/f noise. For electrically conducting rectangular bars, thermally isolated from their surroundings, the noise

power spectrum can be calculated analytically and is found to agree closely with eq (1) over

a wide frequency range.

POWER SPECTRUM OF TEMPERATURE FLUCTUATIONS WHICH ARE DUE TO ELECTRON SCATTERING

As is well known from resistance theory, only electrons with energies, within a few kT from

the Fermi energy 5, take part in scattering processes. Let their nxjmber be N' and let kT<<?,

as in normal metals near room temperature. Before being scattered, the electrons are accelerated by the electric field and obtain an extra kinetic energy W and an extra momentum p in

addition to the Fermi momentum p :

By the scattering process, this energy W is transferred to the lattice as Joule heat and con- sequently one finds for the average value of W

2 „2 2 _ e E T N

W = —p (3) m N

Here e is the elementary charge, E is the applied electric field, m is the effective electron

mass, T is the mean free time between two scattering processes, and N is the total number of

conduction electrons. Since the electrons move in all directions, p . p is zero, apart from

fluctuations, and hence N'

|P1 = e E (2 N / N' ) (4) 181
Therefore, the_amount of energy transferred in each scattering process fluctuates around the

average value W, given by eq (3) , with an average amplitude

e E T , AW = p^.

^ . (2 N / N') ^ ' (5)

As a result, local temperature fluctuations around the average temperature T arise. The

temperature equalization process being governed by the equation

f

= a T (6)

where a is the thermal diffusivity, these fluctuations will have the form

o V ^^(t-^o^

T - T = T e (7) ° ^ 2 / a (t-t ) 3 o

with an average value of AW/C for the amplitude T . Here, C and V are the heat capacity and

the volume, respectively, of the conductor. The individual scattering process considered, is

supposed to have taken place at (r ,t ). On the average, N'/x such fluctuations will arise

per second in the conductor; half of ?hem negative and half of them positive.

A suitable method, to calculate the resulting power spectrum for thermally isolated con- ductors in the form of rectangular bars, has been published previously [2] . For the present

purpose, one needs to know the power spectrum (f) of fluctuations in the temperature difference At between two points of the sample which are a distance 1 apart, rather than the

power spectrum for the temperature at a given point. Using eq (7) to calculate AT, S^^(f) can be obtained in the same manner, described in reference [2], as S^{f) . One finds:

n 1 n 1 n 1

^ / X X y y z z>

1 - cos 2Tr ( ; + ^ ^ + ; ) Li L Li

2 2

2

, 2 2 \ 2 ^ 2,2

16 IT a (— + 2 + 4Tr f L L L

X y z

where d is the lattice constant and L >^ L >^ L are the dimensions of the conductor. Unlike

S^(f) , from which it differs by the aiditi^nal cosine term, S (f) remains finite for f=0, even, if the fluctuations are completely uncorrelated. As for (f ^

, ^a^correl^t^og time T of

the kind discussed in |2|, would introduce an additional factor 4Tr f t /(1+4tt f t ) on the

left-hand side of eq (8)

.

Temperature fluctuations, of course, affect the distribution of the electrons. Energy

equilibrium requires that a temperature difference AT should be accompanied by a difference

in electrostatic potential U, such that

G . AT / N = e U 182

(9)
where C is the heat capacity of the N electrons ( see, e. g. , reference [3] , p. 732 and p. 148) . Hince,

Sy(f) = ( / e N )^

^AT^^^

L

->- ^ X For the special case, where the electric field points in the x direction and 1={:^, 0,0) one finds , upon inserting U= L E / 2 and T = Aw/C :

S^(f)

L

X "x= 2d Y d

64 C T e s

2 2 N C L m

X L

X X 2d Y d

z d

1

2 2

16tt a

2 2 2 -»

2

(2n +1) n n^

2 2^2 L L L

X y z

(11)

^ 2^2

+ 477 f

2 2 2 2 If 277a/L , 277a/L << f <<2Tra/L , 2T7a/d , the sums over n and n can be replaced by integrals, and out of tKe sum over n^ only the term with n =0 Jias to ^e retained. This leads

to a 1/f spectrijm as given by eq (f ) , with ^

2 L T ? C Y s e

77 L m a C

X

(12)

Depending on the dimensions of the conducting bar, eq (11) predicts up to four further

regions with simple frequency dependencies of the form

S(f)/U^ = fTM/ N

u m

(13)

2 3 If f>>277a/d , one finds m = -2 and a_. = a.l6aL /(77d ).

2 2 2 2 ^ 1/2 If 277a/L$^2^2 , 277a/L , 277a/L << f <<277a/d , one finds m = -1/2 and a , ,= a..2L /(77a)

"^/^ ^ 1/2 If 277a/L << f «277a/L , 277a/L , one finds m = -3/2 and a ^ a. (2/L ) . (a/77) ^2 2^ 2^ 3 3 ^

If f <<277a/L , 277a/L , 277a/L , one finds m = O and =a . 2L / (77 L a) . X y z 0 X y

g FOR ROOM-TEMPERATURE METALS

For metals at room temperature, the thermal and electric conductivities are related by the

Wiedemann-Franz law ( see, e.g., [3], p. 761) which leads to

a = 2^T C / (3mC) s e

(14)

Inserting eq (14), C = 3N k, where N is the number of atoms in the conductor,

a a

= N77^ T / (2 ^) (15)

2 2 2/3

(v., e.g., reference [3], p. 148) and ? = (t /2m) . (377 N/V) into eq (12) leads to

183
2 L d m k T 2 1 y ,17 N . 72- ' ( i-TT > L TT n a

X

(16)

As an example, figure 1 shows the calculated noise power spectrum for three gold samples

with different dimensions. For this material, complete agreement in the 1/f region with

Hooge 's empirical relation, i. e. , a value of 0.002 for a, is obtained at O C for L /L =7.

To facilitate comparison with the results presented in [2], an arbitrarily choosen correlation time of 100 s was assumed. Without it, the spectrum would have remained flat at low

frequencies.

Fig. 1 Calculated noise power spectrum accordDISCUSSION

The calculations, presented above, show that scattering of the conduction electrons which is

responsible for the electric resistance, can also generate 1/f noise. For thermally isolated

metal bars, this 1/f noise has the experimentally observed magnitude. Further calculations,

treating other boundary conditions, such as those, corresponding to thin metal or semiconductor layers on thermally conducting substrates, would be desirable.

The author is endebted to Prof. Dr. V. Kose (PTB, Braunschweig) for drawing his attention to the interesting phenomenon of 1/f noise and to him, to Prof. F. N. Hooge and his group (T U Eindhoven) , to Prof. P. H. Handel (U. St. Louis) and to many of his colleagues at PTB for

helpful discussions on the subject.

REFERENCES

[1] F. N. Hooge, Physica 84B , 14 (1976)

[2] L. Bliek, in Proc. 2^'^ Int. Symposium on 1/f Noise, Orlando and Gainesville, Fla. (USA)

(1980). Ed. E. R. Chenette, Univ. of Florida, Gainesville, Fla. (USA), 1980, p. 343.

[3] W. Jones and N. M. March, Theoretical Solid State Physics (John Wiley & Sons Ltd.,

London, New York, Sidney, Toronto, 1973)

184
COMPLEMENTS OF 1/f NOISE THEORY

Bruno Pellegrini

Istituto di Elettronica e Telecomunicazioni

Universita di Pisa, Via Diotisalvi 2, 56100 Pisa, Italy

A new and general model of the flicker

noise has been proposed in a recent paper [l]. The physical origin and mechanism of the phe- nomenon have been found in islands and microdefects, of any nature and size, of the con- ducting medium which are enclosed by an energy barrier or which contain an energy well for

the carriers. The essential characteristic of

these is that the relaxation time of their e- lectric charge is an exponential function of

a random variable energy and that, accordingly, it has a huge dispersion able to account

for the flicker noise spectrum on many frequency decades and down to any, however low,

frequency

.

Some developments and new results of the

theory are now presented. They are carried out

by taking into account, other than the electric potential fluctuations, the ones of the

quasi Fermi level both inside and outside the

islands

.

ISLAND OUTSIDE

ConduLcXance,

Let

#=EE^/q , (3)

is the electric field.

The fluctuations AE =-qAv and AE =-qAu

C ^ F of E and E , originated by whichever cause,

C . F

^ produce, according to (l)-(3), the fluctuations

An=(qn/kT) (Av-Au)

AJ=(J /h)An-oVAu ~o

(4)

(5)

of n and J, respectively. The symbols y (Y ) o and Ay (AY) represent the time average and

the fluctuations, respectively , of the quantity y (vector Y) and o-=qun is the conductivity;

n is assumed to be constant in the island

neighborhood.

On the other hand, for frequencies smaller

than the reciprocal of the dielectric relaxation time ,the current divergency is null,

i.e. y .J =0 and V . A J=0, so that, for J values which makel J .VAn/n| «: lo-V^uLfrom (5) it

I o

' follows that

F Au = O (6)

n=N expT- (E -E ) /kTl C C F J=q (^n^+DVn)

(1)

(2)

be the electron concentration and the current

density, respectively, in the conducting me- dium outside the island. The symbols have the

usual meaning: is the effective density of

the states of the conduction band and E is

C its edge, E is the quasi Fermi level, k is

F the Boltzmann constant, T is the absolute

temperature, q is the electron charge, is

electron mobility, D=kT^</q is the diffusion

constant and

For spherical islands of radius r^, to which one refers for simplicity, and forAu =

= Au =constant on the island surface A =47rr , from

E

(6) one has that

II

Au=(r /r) Au

I E

where r is the distance from the island center. According to V. AJ=0, the current Ai=

=-j , dA. A J, entering any surface Aj enclosing -'• the island is independent of Aj- itself

so that by chosing surfaces A^ on which An/n

is constant from (5), (7) and V.J =0 one ob-

. . o tains

185
Ai=-G Au

E E

(8)

where G = a A /r is the conductance of the me- E II

dium surrounding the island.

- CapacAXancu -

which differs from the one obtained in the

preceding analysis [l] for the fact that it

holds for any current type J . ~o

ISLAND INSIDE

JitcincLs SuAJiOLLnd&d by an EnoAQij BoAAi&fL -

The Poisson equation for the fluctuations

V . Ag=-{q/ £ ) An , (9)

where e is the dielectric constant, and the

eqs (3) , (4) and (6) give

V { Av- Au) = { Av- Au) /A (10)

, 2_ 1/2 where A = ( f kT/q n) is the Debye length.

For the boundary condition ( Av- Au)=

=( Av - Au )=constant on the island surface,

E E the solution of (10) becomes

When the island sizes are much greater

than the wavelength of the states localized

in it, the current density which crosses the

energy barrier surrounding the island itself

becomes proportional to [exp(- <p^/k.T) -exp

(- (^^/kT)] where ip^ and f are the energy

barrier heights evaluated irom the Fermi levels inside and outside the island, respectively [l] . Therefore, since ( f^- o^)=

=- q( Au - Au ) where -qAu is the Fermi le- ^ E I I vel fluctuation inside the island, the fluctuation Ai^ of the current entering the island becomes

( Av- Au) = (r /r)expr-(r-r )/A1c Av -Au ) . (11)

I I E E

Then, according to (7) , (11) and the

Gauss theorem, the fluctuation AQ of the island charge becomes

Aq=C Av -C Au

E E F E

(12)

where the capacitances are given by C^=£h /A

and by

Ai =G( Au - I E Au^) (16)

where the island conductance G is given by

the eqs (6) and (13) of the Ref . [l]

.

The variation Au and the one Av of

I I the mean electric potential inside the island

produce a variation AQ of its charge

Q=-q Q^n given by

AQ=C ( Au - I I Av ) I (17)

C = £A (1/r +1/A ) E I I

CixnAtwt Vlpolz VzcXon -

(13)

The charge fluctuation ^AQ induce a cur- rent dipole vector AP=/AJd X [ij which) ac- cording to (5), (7)-(9) and (11)-(13) and on

the assumption that J /n be constant in a zoo ne a few A large surrounding the island, bewhere, according to (1) ,the internal capacitance is

C^=q n 0^/kT (IB)

Q being the island volume.

I On the other hand the electric field

fluctuation AS=( Av - Av )/w across the barI E rier, of width w, according to the Gauss

theorem leads to

AP=(J /qn) ( AQ -T Ai) — ~0 D

(14) Aq=c ( Av - Av ) W I E

(19)

where t = £ / a is the dielectric relaxation

D time

.

Since in the following part it will be

shown that Ai=- AQ/t , where the island re- laxation time T is much greater than r^^.from

(14) one obtains the relationship

where the barrier capacitance C , for w<?:r , W I is given by

C = e / (l/w)dA

T^lancU Containing an EneAgy Well

(20)

AP=(J /qn) AQ — ~o

(15) When the sizes of the islands are comparable with the wavelength of the states loca186
lised in it, the current i entering the isI land itself may be computed by means of the

theory of the generation-recombination processes. By taking into account the effects of the

fluctuations of Fermi level and of the electric potential inside and outside the island

on the flow of the electron emitted and captured by the island, since now it is w=0

and Av = Av , one obtains again eq (15) where the conductance G is given yet by eqs (14)

and (16) of the preceding analysis \_\\ ,i.e.

G=(q /kT) y nv a N (1-f ) , (21) m j j j

D where v is the electron thermal velocity, a m j and N . are the capture cross section and the

number of the states of energy E . , respectively, and ^

f =|l+exp Re -E )/kT|}

j

I '- j I

-1

(22)

is their mean occupation factor, E^ being the

Fermi level inside the island.

Also (17) and (19) continue to hold, where,

however, now it is

and

C={q /kT) y N.f . (1-f .; I 3 3 3

3

C =00

W

(23)

(24)

RELAXATION TIME AND NOISE

- Rzlax-CLtLon time. -

From the current continuity Ai= Ai^ on

the island surface and from (8), (12), (16),

(17) and (19) one obtains

-1 -1 -1 -1 -1

Ai=- C +C +C )(1+tGC ) GAQ , (25)

I W E D E

so that the island charge conservation leads

to the stochastic equation (dzi Q/dt)=-AQ/T + r/ where tj is the stochastic component of the

current crossing the island surface and t is

the relaxation time which, according to (25)

where normally it is «; G c ^ t , is gi- ven again by

T =C/G (26)

in which now, however, the island capacitance

-1 -1 -1 -1

C=(C^ % -^S ^ '

is the series of the three capacitance C ,C

I W and C . T , instead, according to (13) and

E

(25) remains independent of the conductance

G =ctA /r of medium surrounding the island

E I I -1

because t <SC G C ^ t . D E

- MOAJ,t -

The more general and detailed analysis,

which takes into account the effects on the

currents and the charges of the fluctuations

of the electric potential and of the Fermi

level, both inside and outside the island,

leads to the new expression (27) of the island capacitance C which intervenes in the

relaxation time and in the noise equations obtained in the preceding model . In fact now C is the series of C , C and C .whereas

I W E

in the previous case it was C=CWEor C=C . However, if one excludes the case of islands which contain energy wells and have si- zes so small as to make

c «: c

I E

(28)

one obtains again the results of the preceding

approach [ij because C and G remain largely

independent each other and G is the quantity

which depends exponentially on a random variable energy (J) . In particular, by taking into account also the more general expression (15) of the

current dipole vector, the coefficient /3 of

the noise spectrum S^= /3 V /Nf^ of a homogeneous device, according to (14) of the Ref.

, may be written in the more simple form

(29)

2 -1 .

/3 = (k'l/q) f* n / CD ( (p* ,C) dC

F

where V is the average voltage across the device, N is the its carrier number, 5 is a proper exponent (see (6.6) of the Ref.j^J),

y =1+ S , f* is a given reference value of

the frequency f, <I)*=-kTln( Tof*) , D is the

F island density in the space (r,C,<P).

Instead when (28) is verified, according

to (21) , (23) , (24) and (27), one has that G

tightly depends on C through f j so that, from

also (1) , (22) and (26), for (E -E )£kT

jmax jmin one gets

f E -E E -E

j r F C m C-,-1

T = =^= T Lexp( ——)+exp{—; ) van , o"- kT kT-' mm (30)

187

where t =1/n v cr ,E and a are mean values

o C m m m m
of Ej and CTj , respectively. According to (30),

T is an exponential function either of (E -E ) ^ _ C F or of the random variable (E -E ), according — ^ _ _ Cm

to (E -E ) ? (E -E ) . Therefore in both case t Cm C F may have a great dispersion only in the device depletion-regions, both on the surface and

in the bulk, because only in them (E -E ) may

C F reach great values.

However, since the island contribute

one obtains r*=0.22, 1.54 and 10.6 nm for

Ie -E 1=0, 0.1 and 0.2 eV, respectively.

F m'Therefore, in any case, microdefects

with sizes greater than a few nm generate

flicker noise with a spectrum S^= F /f^ characterised by y value near one, whereas that

does not happen for the point defects.

CONCLUSIONS

S = J . VI '-0 7z |2 ,2-22 I (4kTG)/(qn) (t +w ) (31)

to the noise spectrum [l] , owing to the (21)

and (30) and to the dependence on TT of Jq>

t

and -

VZ, has not the form S oc t / (1+t^ a»^) ,

' VI such an exponential dependence of t on a ran- dom variable energy now leads to a total noise spectrum S^= F/f ^ with an exponent y which

may be rather different from 1 and may change

appreciably with the frequency even on narrow

bands

.

- Inland Size.

When (E -E ) < kT and the state num-

^ jmax jmin

J Nj- Q^/ Q IS proportional to the island volume Q_^, form (23) one obtains

ber

C^^ q Q/kT Q

I I e

(32)

where Q = Q exp (E -E ) /kT and Q / Q

' is

e s F m I e the average number of electron or holes con- tained in the island. In this case, according

to (13) and (32) , the condition (28) is verfied for

Same complements and new results , which

generalize and strengthen the preceding flicker noise theory [l] , have been carried out

by taking into account the effects, on the

carrier and current densities, of the electric potential and Fermi level fluctuations

both inside and outside the islands.

A more accurate and general expression

of the current dipole vector has been so obtained.

Above all a new expression of the island

relaxation time has been achieved according

to which it is proportional to the series of

the internal external and barrier capacitances of the island itself.

Finally from such a result it has been

deduced that the noise spectrum differs appreciably from the 1/f shape when the sizes of

the islands are so small as to make its cur- rent depending on the capture-emission pro- cesses and the internal capacitance smaller

than the external one.

2 1/2

r «r*=( D /2 Al) ri+(l+4L A / O) 1 , (33)

I e

' e where L=q /3 e kT ^ 20nm, for T=300 K and f = =10-^°F/m.

3 _ ^3

For instance for Q =1 (nm) and n=10

-3 s cm , which leads to A = 130 nm, from (33)

REFERENCES

1 B.Pellegrini, Phys.Rev. B22, 4684 (1980)

188
FLUCTUATIONS OF THE RELAXATION TIME AS A SOURCE

OF 1/f NOISE IN MACROSCOPIC PHYSICAL SYSTEMS

J. Uebersfeld

University of Paris VI "Pierre et Marie Curie"

75005 Paris, France

P. H. Handel

Physics Department

University of Missouri

St. Louis, MO 63121

J. J. Gagnepain

Laboratoire de Physique et Metrologie

des Oscillateurs du C.N.R.S. 25000 Besancon, France

As was pointed out in the theory of quantum 1/f noise, cross sections of many elementary processes are infrared quasi-divergent and consequently exhibit low frequency 1/f fluctuations.

In large systems, characterized by relaxation times T, the fluctuations of the elementary cross sections are likely to produce 1/f fluctuations of T.

In many cases, the system is characterized by a single, dominant, T. This will be reflected in

the behavior of the susceptibilities of the macroscopic system. If X is a force-field applied to the system, Y the causal and linear response of the system, and W = XY the stored energy in the system

caused by the X-field, any fluctuation of the loss rate of the system will produce a fluctuation in the stored energy. The quantitative connection between these two fluctuations is given by the Kramers-Kronig relations, applied to the complex susceptibility K, which expresses the causal and

linear relation between X and Y.

This method has been applied to quartz crystal resonators and easily yields a theoretical interpretation of the experimental spectrum of the flicker of frequency noise

Sy(f) = ^ , (0)

^ Q f

where y is the fractional frequency fluctuation, Q the quality factor of the resonator, and A a dimensionless constant of about 1 for quartz resonators.

It is known that the interaction between the sound wave and the thermal phonons in a dielectric

crystal is at the origin of the attenuation of the wave. Such an interaction being a nonlinear process, the wave velocity, and therefore the resonator frequency, are also affected [1]. In a first

approach three-phonon collisions are considered, involving a sound wave phonon and two thermal phonons. Three different types of collisions must be distinguished:

1) Normal processes (N) , in which the total energy of the phonons and the total quasi-momentum

are conserved.

2) Umklapp processes (U) , in which only the total energy is conserved, but not the quasi-momentum.

3) Elastic scattering (E) , corresponding to collisions of phonons with impurities, defects, etc. The scattering being elastic, the frequency does not change, but the wave vector and the polarisation

can be altered. The total energy is conserved. However, the number of phonons of any particular

frequency cannot change.

The phonon has a finite lifetime T, which makes its energy uncertain by an amount Ti/T. The

effect on the attenuation and velocity will be more or less important if this uncertainty is larger

or smaller than tia), the energy of a sound wave phonon, of angular frequency o). 189
At high temperature, i.e., (jJT« 1, the previous processes N, U and E and/or combinations like N with some U and E processes, or E with some N and U processes, lead to attenuations a and velocity

changes V-V^ which all take the general form [1]:

2pV^ l+u)^T^

V_v = -CT. y2 JuflL , .2)

0 2pV *

*^ , ^ 2 2 0 l-Ho T

where Y is an effective Grueneisen constant depending on the nature of the processes involved, C the

specific heat, T the temperature, p the specific mass, and the sound wave phase velocity.

Introducing the low frequency Young modulus E = pV^ , the quantity AE = CT.Y^, and the relation

between attenuation and Q-factor a = to/2QV^, one obtains:

Q

" % Hto2T2

V-V . O O 0 AE ui^T^ /•,^

A fluctuation 6t in T will produce a velocity fluctuation 6V, and therefore a fractional fluctua

tion y of the resonator frequency (o^

2 2

6a) ,^ w

= __I = M r 6t

r 0 d+w^T'^)^ T

Introducing the power spectral densities S (f), and using in eq (5) the expression eq (3) for

Q~^, one obtains immediately ^

^(^> =^ (ae) I At/ h.l.^'^

which exhibits the Q~ -law.

If it is now assumed that t has a power spectral density S^^^^(f) = A'/f in the spirit of the quantum 1/f noise approach [2-3], eq (6) coincides with the experimental law (0).

A rough evaluation of the Grueneisen constant y can be made either by means of its relation with

the thermal expansion constant, or by means of the third order elastic constants. In both cases it

is found that y is of the order of 10, for quartz. Therefore (AE/E^)"^- 10^. This means, since

Awl in eq (0), that A' is approximately equal to 10~^, which is a reasonable value if considering

that 5t/t = -6Q/Q.

A similar formula can be derived for electromagnetic resonators filled with a lossy dielectric

material.

In each case, the constant A depends either on the mechanical, or the electrical properties of the material. The exact behavior of the mechanical (or the electrical) susceptibility is irrelevant,

if the susceptibility depends on only one relaxation time t, and if the condition u^t<<1 is satisfied

(u^ being the resonance frequency of the electrical or mechanical resonator)

.

Although the method has been applied only to the electrical and mechanical case, it is anticipated that it would work for other systems.

Finally, we would like to discuss the experimental basis of the Q -law. Measurements of 1/f

frequency noise were performed on a large number of quartz resonators, at different frequencies, and

with different quality factors. The measuring system is composed of a phase bridge, doubly balanced

190
Fig . 1 Spectral density S of fractional frequency fluctuations

y at 1 Hz Fourier frequency as a function of the quality factor Q for various quartz resonators. The resonance frequency of the quartz is indicated for each point. The solid line has a slope

of -4.

in order to reject the frequency noise of the source itself. Fig. 1 gives the noise levels, measured

at 1 Hz Fourier frequency, as a function of the unloaded Q-factor. It can be observed that the dependence of the noise power on Q is well described by a Q~'*-law.

We have some indications that the Q~'*-law is more general and can also be applied to frequency

standards other than quartz.

REFERENCES

[1] H, J. Maris, Physical AcOLi6ticU>, Vol. (/Ill, Eds. W. P. Mason and R. N. Thurston (Academic Press, New York, 1971) .

[2] P. H. Handel, Solid Statz EldcXAOViicJ, .Tl, 875 (1979).

[3] P. H. Handel, ?ky6. Rev. Letter 34, 1492,1495 (1975).

[4] P. H. Handel, ?hyi>. Rev. 22A, 745 (1980).

191
l/f* NOISES AND THE RIEKASN - LIOOVILLE FHACTIONAL

IHTBORAL/ DERIVATIVE OP TBE BROWHIAN MOTION

Claudio Kaccone

(Private address : Via Martorelli 43 -

- 10155 Torino (Turin) - Italy)

INTRODUCTION

Let B(^) denote the Brownian notion. The present paper is devoted to the study

of its Riemann- Liouville fraotiooal integral / derivative of order , whioh is

formally defined

J

r(H*4-)

BH(t)«

J

\ \^ <IB(s) (t^o). (1)

0 ^

This process is a fractional integral of the Brownian motion for H > , and a fractional derivative for H'^^* Actually it is a one •> parameter class of processes

(the peo'ameter being H ), and will be called the fractional Brownian motions in the

sequel. Its definition seems to go back to Paul L^vy ^1] . Benoit B. Mandelbrot [2]

provides a full list of the references to the subject up to 1977. is evident from

eq (l) that the Brownian motion oan be regarded as the particular ease H^-^ of the

fractional Brownian motions, since the initial condition B||(0)sO holds good.

Moreover, remembering the scaling property fulfilled by the Brownian motion

B(rt)=T^B(*) (••>0) (2)

it is easy to prove that the fractional Brownian motions fulfill the scaling property

BH{rt)= r" B„(t) (3)

sometimes called the H self - similarity property.

LINEAR SYSTEM WHITE NOISE « FRACTIONAL BROWNIAN MOTIONS AND CONSEQUENCES

The fractional Brownian motions can be regarded as the output of a suitable linear

system of which the input is the (uniform) white noise t in fact eq (l) is already in a convolution form, so that it suffices to replace dB(i) by "B^t) dt , where 6 (t)

is the (uniform) white noise. Thus, we find at once the impulse response of the system

r(H-|-) *

192
and, "tiy oomputing the lattar'e Fourier transforn, tha aystam function H(iw)

1 I \i 1

Since the white noiae autocorrelation is the delta function ^C^i'^g) > use of eq (4)

and of the standard formula ^t'^(^ii^t) ^ ^(j^i^^t) ^ ^i^t^ yields the cross- -correlation 1^B'B^{'^i > ^2) white noise - fractional Brownian motions

The autocorrelation ^b^Bh^^i ) ^t) ft>aotional Brownian motions can then be found

from eqs (4) and (6) by virtue of the standard formula 'Rb„Bh(*1»'^«) * '^b'BhC** > *

'^^^ reads (the symbol '^i^t^^ denoting the minimum among and )

1^BH(*i'*t)«{B„(*i) B„M = . (7) ^ ' 2"[r("*^)]

Setting t^^t^=ti *9 (7) yields the veu^iance of the fractional Brownian motions

(the mean being zero)

= — o)

whence

2H[r(H + 4-)]*

*^<*> itir r(H+4-)

is the standard deviation of the fractional Brownian motions, and is evidently H

self-similar. Also, eq (9) shows that H must be positive, and, by plotting the curves

^^ifj^y on the t - B^(t'} plane, that H must be less than one (otherwise the curves would be "divergent"). In conclusion, the important restriction 0<H<1 arises.

POWER SPECTRA OP THE FRACTIONAL BROWNIAN MOTIONS

From eq (3) the power spectra of the fractional Brownian motions follow upon

resorting to the non-stationary spectral theory, as outlined, for instance, in [3] • The result is . — /normalization\ /.«\ S(^)= V constant )^ ^'^^

Hence the fractional Brownian motions are an example of a rigorous mathematical theory

leading to a spectral law of the type l/f , in which 1<X< 3 . Though the l/f noise

is not covered by the theory, it can be approximated indefinitely from above. This fact seems to suggest that further investigations along these lines might lead to a model

for the l/f noise. From eq (IO) it also appears that the fractional Brownian motions

are necessarily a class of band-limited noises, a realistic feature for their usage.

193
OAUSSIAN RESCALINQ AND CONSEQUENCES

It is not difficult to' prove, by virtue of eqe (2) and (7), that the fornula holds

(11)

that is, the autocorrelation of the process on the right equals that of the fractional

Brownian motions. Since both processes have equal mean (zero), we conclude that

B„(«) (12)

Thus, the fractional Brownian motions are just a rescaled form of the Brownian motion,

and of course they are Gaussian processes. For H ^"J~ , that is for the fractional

integral, the diffusion is faster than the Brownian diffusion, whereas for H<^~ > that is for the fractional derivative, the diffusion is slower. The density of ]^(()

reads

ViT t(»*-t)

exp

and

4H[r(H*f)]'

{-"[4*4-)]'-^} (13)

(14)

Is its characteristic function (that is, the Fourier transform of eq (13))> A number

of results can be derived from eqs (13) and (14) by following up the procedures that are well-known from the theory of the Brownian motion, but we have little space here.

For instance, the density of the first-passage time at any abscissa a>0 reads

first-^passage

time dens (15)

Another interesting result is the diffusion partial differential equation

2

[r(H*i-)]

at

(16)

fulfilled by the fractional Brownian motion density (13) with the initial condition

J^(0,x) s S(x). The left side of eq (16) reflects the time rescaling, and its right

side the Oaussian character of the density (I3). Therefore it would be easy to consider

higher-dimensional fractional Brownian motions simply by inserting the Laplacian on the

right side of eq (I6). If expressed in polar coordinates, only the radial part would be

left, and the radial density would be found. These developments are left to the reader.

194
BICBaiFUKCTIOH EXPANSION (KARHUNEH - LOEVB EXPAKSIOl)

The present author proved in [4J the elgenfunotlon expanelon of the fraotional

Brownlan notions whioh will presently be stated without proof. The expansion reads

' *" Z ^" J''(«"-$:t) '''^

Here t I) The tine t ranges from zero to a fixed positive instant T t O^^^T.

2) The order V of the Bessel functions of the first kind, Jy(x), is a one-to-one function of K given by

3) The constants axe the real positive zeros of the Bessel function

Jy.|^X^ f arranged in ascending order of magnitude. These can be computed

numerically as soon as a value of y (that is, a value of H , by virtue of

o<I (is)) has been fixed.

4) The normalisation constants are given by

1

N„* . ^ • 09)

yyX[JvU)r*(xS-0[Jv(yH)]^

5) And finally the Zn are a set of orthogonal Qaueeian random variables,

having each zero mean and variance given by the eigenvalue Xy| , that is

x2 [r(H*i-)]'

There is only one eigenfunction (given by the two last terms under the

summation in eq (17)) corresponding to the eigenvalue X^f , that is,

there is no degeneracy. And, since the eigenvalues \^ form a decreasing

sequence as n^oo , the series (17) converges.

The author thanks Prof. J.O. Taylor of the University of London (England) King's College

and Prof. P. Mazzetti of the Istituto Elettrotecnico Nazionale Galileo Ferraris, Torino,

Italy, for their cooperation.

REFERENCES

[1] Paul L6vy, Univ. California Publ. Statis. ±, 357 (1953).

[2] Benoit B. Mandelbrot, in "Fractals: form, chance, and dimension" (Freeman, San

Francisco, 1977).

[3] Athanasioe Papoulis, in "Probability, random variables and stochastic processes**

(KcOraw-Hill, New York, I965).

[4] Claudio Maccone, II Nuovo Cimento B (to appear spring 1981).

195
NON-GAUSSIAN AMPLITUDE DISTRIBUTION OF

THERMAL NOISE IN RESISTORS WITH 1/f NOISE

P. H. Handel

University of Missouri

St. Louis, MO 63121

D. Wolf

University of Frankfurt

Frankfurt a.M., F.R. Germany

K. M. van Vliet

University of Florida, Gainesville FL 32611

and

University of Montreal, Canada

It is well-known that resistors of any nature which contain a relatively small number

of carriers are also characterized in general by the presence of resistance fluctuations

with a 1/f spectrum. These, apparently fundamental, 1/f resistance fluctuations are not

due to temperature fluctuations, but are present both in thermal equilibrium and in the

presence of an applied voltage. From a theoretical point of view, a certain level of such

fundamental 1/f fluctuations is to be expected as a consequence of quantum 1/f noise at the

level of the scattering cross sections, from various systems of infraquanta, both in equilibrium and nonequilibrium. In general, the fundamental 1/f fluctuations of the resistance

should cause fluctuations in the spectral power level N = 4kTG of thermal current fluctuations, and in the power level N' = 4kTR of voltage fluctuations:

<(6N)2>^ <(6N')2>^ <(6G)2>^

^ <N>2 <N'>2

f

' (1)

where C is the 1/f-coeff icient observed also in the presence of an applied voltage. However, there should not be 1/f fluctuations in the available power level N" = kT

<(6N")2>^ = 0, (2)

Experimentally, Clarke and Voss [1] have first observed the presence of the 1/f fluctuations (1), although they interpreted their results in terms of temperature fluctuations,

rather than true 1/f noise. Beck and Spruit [2] have observed the Nyquist-level fluctuations subsequently in carbon resistors. It would be useful if Eq. (2) could also be verified experimentally.

The purpose of the present communication is to derive the amplitude distribution for

the 1/f power-modulated thermal noise described by eqs (1) and (2).

Let us denote by x(t) the (unmodulated) Nyquist noise variable, and by C the resulting

thermal noise variable. We interpret the resulting thermal noise by writing

K = xCG/G^)-"-^^ = x(l + 6G/G^)^^^ = x + xy , (3)

196
where G is the average conductance, and

0

^ 2G

o

is half the relative conductance fluctuation, (^)

Notice that 5 will have a white spectrum up to high frequencies, described by the

Nyquist-Planck formula. Indeed, we know that x is white, and therefore has a 6-function

autocorrelation. Since x and y are independent, the xy-term has an autocorrelation function

which is the product of the corresponding autocorrelation functions, i.e., also a 6-function.

This means that xy is white. Since x and xy are uncorrelated (although they are not independent) their autocorrelation functions and spectral densities must be additive. Consequently, 5 is white, as claimed.

In order to derive the amplitude distribution of z = xy, we introduce x] = in z, a = Jin X and 3 E £n y. From n = a + 6 we obtain the distribution

" ' • 2dn " r e2" 1 r e^^*^""^ 1 P(ri)dn = dn / P (a) ^ (ri-a)da = / exp 1- a exp + n-a da

1 2 2Tra^a^_ L 2 J L 2 J 2a ^ -J L 2a

1 2

e"dTi

2Tra a

^ / exp cosh e de = K (

) (5) a laa J ira a ol aa; 12"°° 12 12^12/

where P (a) and P„(3) are derived from Gaussians of dispersion a and a respectively,

e = 2a-ri

1 + £n(a /a),

2 and K is the modified zero-order Bessel function.

12

Returning to the

2 1 o „ real variable z, we obtain from P(n)dri = P(z)dz

a a I

1 2

^

ira a o\

1 2

The corresponding characteristic function is elementary:

X(v) = (1 + a2a2v2)-l/^ (7)

1 2

Due to the mutual dependence of x and z, the characteristic function of 5 differs from the

product of the characteristic functions of x and z. We obtain instead the amplitude p.d.f,

00 ~l/2a^ °° r 2 2 1 1(0 = / P (x)P (| - lUdx = ^ / expf - ^— + Idvlacl

-00 1 2X l» Z^OO _^ L 9n2 9^2„2 ^„2 J

1 o -» - 2a 2 2x2a ^ xa ^

1 2 2

^-l/2a^2 " r .2 .2

/ expr-^-^i-]cosh(-^)dx/X.

0 2a 2 2x2a 2-1 \ xa 2 /

ira a 12 2

(8)

This distribution tends to a Gaussian in the limit a„->0, but differs slightly from a Gaussian in general. The skewness is zero, but the kurtosis exhibits the small deviation

197
from the Gaussian amplitude p.d.f. The necessity of a non-Gaussian behavior of thermal

noise has been mentioned recently by Tremblay and Nelkin [3].

Finally, we mention that for the current fluctuations ^ in the presence of an applied

voltage we can write the sjrmmetrical expression

0 0

I where y E 2G V y. The corresponding amplitude distribution will be closer to a Gaussian,

and will not be discussed here.

The authors acknowledge support from Contract N00014-79-0405 and AFOSR.

REFFERENCES

[1] R. F. Voss and J. Clarke, Phy6. Rev. B13, 556 (1976).

[2] H. G. E. Beck and W. P. Spruit, J. App£. Phy6 . 49, 3384 (1978).

[3] M. Nelkin and A. M. Tremblay, P/LOC. II Tnt. Con{,. on 1 / Moi^e, Orlando, 1980, p. 71.

198
l/fC NOISE BEHAVIOR IN INFRARED PHOTOCONDUCTIVE

POLYCRYSTALLINE THIN FILMS OF Pb Sn, Te

X 1-x

A. D'Amico, G. Cappuccio, G. Petrocco, C. Fotinos

Istituto di Elettronica dello Stato Solido del

Consiglio Nazionale delle Ricerche

Via Cineto Romano, 42

00156 Roma

G- Stochino

Laboratorio Circuiti Analogici FATME

Via Anagnina, Roma

Italy

We report experimental results

concerning the 1/f noise behavior

f^l-5] in semiconductor polycrystalline thin films of Pb Sn^ Te used as photocOnductors in infrarei applications. Such a noise has been seen in

relation to the spread mosaic variation of the films due to suitable an- nealing processes.

These films have been obtained

by the radio frequency sputtering technique using a mixed AT- C2 gaseous

plasma with the following partial

pressure: p(A2) = 3 x 10~-^torr,

p(02) = 9 X 10"^torr. The surface substrate temperature during the deposition has been maintained close to

150°C.

BaF2 [lOO"] , NaCl[l00"], CaF2l3l00'^

oriented samples with 1.6, 1.8, 1.4

degrees respectively of spread mosaic

were used as substrates for the polycrystalline growth of Pb Sn, Te . DiX X ~ X mension of the deposited films were

o thickness about 8500 A and contact2 less area equal to (lx2)inm . The mosaic spread of the substrates and films was measured by

X-ray diffraction techniques by using

a Bragg-Breantano single-axis diffractometer. All the measurements were performed by the 9-scan technique [6],

Here the X-ray source and the

film are set up in a fixed position

in order to select the Bragg angle of

a specific reflection: the specimen

turns around the goniometer axis, in

order to bring the differently oriented grains toward reflection. The re- flection intensity as a function of

rotation angle 0 (rocking-curve ) gives'

a direct measurement of both the preferential orientation of the film and

the mosaic spread around the preferential orientation. To overcome the problem related to the low intensity typical of X-ray characterization of

thin films, quite a strong, low angle

reflection was used: the [200I reflection at a Bragg angle of about 16°

(Kd radiation of Co).

Anisotropy measurements were re- peated along two orthogonal directions

and no valuable variation of the mosaic spread have been observed. By

using the Au/Pbjj,Sn]^_jjTe/Au (with gold

contacts under the film) structure

with two concentration values of x:

Xj^ = 0.20, X2=0.16) noise measurements

have been carried out, before and

after annealing, close to liquid nitrogen temperature: (79i0.3)°K in two

different ways. First by considering

films as grown, second by using the same Au/Pb^Sn^.^Te/Au structure after

a suitable annealing whose characteristics are the following: hold temperature close to (180+2)°K for 25 hours

199
and dT/dt equal to +10°C/h from^and

towards room temperature.

Experimental results indicate

both a reduction of the spread mosaic

in annealed Pb^Sn^^.^^Te films and va- riations of the l/f°' noise behavior.

As a matter of fact, films showed

as-grown a spread mosaic of about 12,

8, 11 degrees respectively on the a- bove mentioned substrates and, after

annealing a spread mosaic of 5, 2 and

3.5 de gree s .

been used after a check by a suitable

calibrated low noise metal film resistor of 10.05 nQ at 81°K.

The electronic chain used in . n such measurements was formed by a low noise preamplifier made up by a low noise junction field effect transistor (2N 6550) and a thermalized

low drift operational amplifier,

(0P05) followed by a digital signal

anal yzer .

Ae (degrees)

Fig. 1. Reflection intensity as ^ function of AG = 03 - 0 for: (1)

NaCl substrate . [lOOj , (2) Pb gSn 2Te

film after annealing, and (3*) Pfc g Sn 2Te film as grown . Figure 1 shows as an example, the relative reflection intensity of X-ray as

a function of AG = 03 - 0 for the Au/

/P bj^Sn j^_^Te /Au film as grown and after

annealing on a [lOO] oriented NaCl

substrates. Similar behavior has been

observed for the other two types of

substrates. Noise measurements were carried out leaving the structure in

a liquid nitrogen cooled tail of a window-less cryostat in order to avoid

any influence of external light.

Furthermore^ sample s were surrounded

by a copper shield whose average temperature was maintained at about 80°K.

In this way the internal surface of

the shield was the only source of

noise radiation.

To test the linearity of our system the electronic chain has always

Hz

10^

10°H

10 -

-2 10

1) Pb.jSn .iTe before annt&linj

3) Pb^St. before

79 K 'DC'-125^A

10-^ 10 10' 10' 10-= 10

Fig. 2. Power spectral density in

Au/Pb^Sn^^.^Te/Au structures at 79°K

before and after annealing.

As far as the 1/f noise is concerned,

Fig. 2 summarizes results in the case

of the film grown on a NaCl substrates

at 79°K with a DC polarization current

of 125ijA and two different concentrations of lead. As we see from Fig. 2_, a lead concentration from .2 to .16 produces a lower excess 1/ f^' noise, and

during the annealing process a ranges

from 1.3 down to .9.

From Fig. 2 and from data obtained taking into considertaion the spectral noise power density vs. the DC

polarization current (at a fixed frequency) we can summarize and write

the following results:

200
before annealing

V

9 2.1

S (f) <^l/t • I

' after annealing

V

which confirm the flicker noise behavior and indicate a connection between

the soectral noise former density and

the spread mosaic variation caused by

annealing processes.

Further investigations are going

on to better understand a possible cor- relation between the l/f*^ noise and

the structural variations of the films

under thermal conditioning.

ACNOWLEDGEMENTS

Many thanks are due to Mr.Grilli

for his assistance in carrying out figures and to Dr. G. Spissu for her skilled technical assistance.

REFERENCES

[1] F.N. hooge, Phisica, 42 , 331(1969)

[2] S.T. Hsu, D.J. Fitzgerald and A.S. Grove, Appl. Phys. Lett. 12, 287

(1968)

[3] R.F. Voss and J. Clarke, Phys.

Rev. B, 13 556 (1973)

[4] Noise in Physical Systems, Proceedings of the Fifth International

Conference on Noise, Bad Nauheim,

March 13-16, 1978

[5] A. van der Ziel "Flicker noise in

semiconductors: not a true bulk

effect" Appl. Phys. Lett. 33(10)

15 Nov 1978

[6] G. Balestrino, A. D'Amico, G. Petrocco and A. Grilli: Infrared

Phys., 19 245 (1979)

201
TEMPERATURE DEPENDENCE OF 1/f ^^OISE IN THICK-FILM RESISTORS

Maria Prudenziati and Bruno Morten

Gruppo Nazionale Struttura della Materia

Istituto di Fisica dell'Universita, Modena, Italy

Aldo Masoero

Gruppo Nazionale Struttura della Materia

Istituto Elettrotecnico Nazionale Galileo Ferraris, Torino, Italy

INTRODUCTION

Thick-film resistors (TFRs) are a particular class of cermet materials with a relevant

role in microelectronics [l] . The conduction mechanisms consistent with their microstructure

[2] and electrical properties have been analyzed and include different possibilities i.e.

hopping of electrons from and to conductive grains and localized states in the glassy matrix

direct tunneling of electrons between near-neighboring grains assisted by resonant centres

in the intergranular material, and conduction in a narrow band of nearly delocalized states

in the intergranular material [3] . In any case a relevant role in the conduction mechanisms

of TFRs seems to be played by deep energy levels in the glass.

One method of investigating the nature of these energy levels and the conduction mechanism is by a study of the current noise of these materials. In this framework we have under

taken an extensive study of the noise effects in TFRs. The results of the temperature depend

ence of 1/f noise are here reported and discussed. Although presently limited to a narrow

range of resistivities the study gives interesting results for further progress in our knowl

edge of electrical properties and conduction mechanisms in TFRs.

EXPERIMENTAL PROCEDURE

We included in the study three commercial ink series based on different metal-oxides

and a prototype RuOa-based ink series [4], see Table I. Resistors were prepared with the well-known techniques [l] on 96% alumina and were provided with prefired thick-film PtAu-based terminations in order to minimize the effect of

metal migration in TFRs [5] . The connections for the TFR from the metal terminations towards

the measuring circuit were obtained with platinum pressure contacts, which enabled us to extend the measurements up to high temperatures. Patterns of resistors of constant width (2 mm

and variable length (from 0.5 mm to 10 mm) were prepared in order to evaluate the effect of

contact (at the interface between terminations and TFR) on the detected noise [6]. Experimen

tal procedures for characterizing the composition of the samples were described in [?]

.

Temperature dependence of resistance in the samples was measured with a digital multimeter with a relative accuracy of 10-^. The temperature of the samples was changed from 77 to 650 K in ultrahigh vacuum (10~® mbar) and was measured with a platinum sensor capable of

Table I. Data on composition and electrical properties of TFRs.

Sample —————————— Nom. Sheet Resist . (K^^/n) Metal-Oxide T (K) —mr

Du Pont 1741 10 Ir02 180

Du Pont 1441 10 BiaRuaOy 350

ESL 2914 10 RUO2 303

Prototype 90 RuOa 465

an accuracy and reproducibility of 1 K. In order to measure the excess noise, a d.c. current was supplied to the sample through

a low-pass filter from fresh dry cells. The voltage fluctuations across the sample were

202
amplified by a low noise preamplifier ( PAR

Mod. 113) and then fed to a spectrum analyzer

to measure the spectral power density in a frequency range from 0.1 Hz to 20 KHz; spectra were measured at different fixed temperatures. In order to measure the noise integral, the amplified noise signal in the bandwidth from 2 Hz to 10 KHz was fed to a true RMS voltmeter, giving a d.c. output signal

which was continuously recorded together with

the temperature during its variations in the

sample chamber.

The experimental results on both resistance and noise of TFRs were reproducible

even after many cycles of temperature changes over the whole range investigated.

EXPERIMENTAL RESULTS AND DISCUSSION

^ig.l Normalized resistance vs temper- Figure 1 shows the temperature dependature for Du Pont 1741 resistors. ence of resistance in Du Pont 1741 resistors;

resistance is normalized to the value R at which it exhibits a minimum according to the

typical behaviour of TFRs. The temperature T corresponding to this minimum is in this case quite low. Table I reports values for T in other samples investigated.

According to the literature [s] the excess noise in TFRs should exhibit a noise power

spectral density S^ of the type:

S^= <AV^>^^/Af = KI^R^/f (1)

where "^^^^-^^^ is the mean square voltage noise found in the bandwidth Af, I the steady state

current flowing in the sample of resistance R, f the center of the bandwidth Af and K a constant which depends on the composition of the sample and is inversely proportional to the

sample volume v ( v=Lwt where L is the length, w the width and t the thickness of the resistor) for TFRs having the same intrinsic resistivity.

1.020

1.01S

1 010

1 005

1 000

100 200 300 400 500 600 TEMPERATURE IKI

Fig. 2 Normalized noise power spectral density S /<V>^ vs frequency in Du Pont 1741 resistors

at different temperatures. The sample was 2 mm wide and 8 mm long.

203
Relation (1) implies that: i) a plot of S vs L, with the other parameters fixed, is a straightline which crosses the zero-point in aKsence of any contact noise [6] ; ii) the

quantity S /<V>^ is independent of the bias conditions of the resistor; iii) the noise spectral density is proportional to 1/f. These three conditions have been checked in our samples.

Measurements of S vs L show that the contribution due to contacts to the total noise

is negligible in our samples, since S vs L plots cross the zero-point within the experimental errors. We found that at a fixed temperature S is proportional to the square of the applied d.

c. voltage used for noise measurements ( up to some tens of volts).

Figure 2 shows typical noise power spectra of Du Pont 1741 (10 KCl/o) resistors, at various temperatures. The quantity S /<V>^ is reported as a function of frequency; the spectra

have the form of 1/f with values of the frequency exponent a very close to the unity near

room temperature, but a<l at lower temperatures and a>l at higher temperatures. This is quite

general behaviour for resistors investigated, as shown by the data reported in Fig. 3. It is

worthnoticing that the values of a are quite similar in resistors of different composition , little differences appearing at the highest temperatures.

Since the noise spectra at both 84 and 650 K in Fig. 2 are higher than that at 300 K,

data in the figure support for the first time the evidence of a minimum of the power spectral

density at the intermediate temperatures. The existence of this minimum is well evidenced in

Fig. 4 where the relative integral noise <AV^>/<V>^ is given for different TFRs . As already

reported [9] starting from liquid nitrogen temperature the noise decreases by increasing the

temperature, but it reaches a minimvrai after which it increases again. This behaviour recalls

the analogous temperature dependence of resistance of TFRs (Fig.l); on this topic it should

be noted that: i) the relative change of noise can be much larger ( up to some hundred times)

than that of resistance in the same temperature range (compare Fig.l with Fig. 4 and note log.

scale in Fig. 4); ii) the temperature T at which the minimum of resistance occurs (Tab. I)

does not always coincide with the temperature T at which the minimum of noise is observed

in the same sample (Fig. 4); in some cases T <T , but, at least intuitively, a correlation

between these minima naturally arises. This TM^finSings could suggest a common physical origin

for the temperature dependence of resistance and noise.

In conclusion we have collected new experimental evidence for the noise behaviour in

different TFR systems. We observed that the exponent a characterizing the frequency dependence of excess noise is temperature dependent, making the power spectrum more white at low

temperatures, in the line reported in [9]. A minimum of the noise was observed in analogy

with the minimum of resistance, even if the temperatures where these minima occur are not

necessarily the same in the same sample. The relative changes of noise are much larger than

that of resistance in the same sample. The integral noise scales with the sample resistance

and it is contained in half a decade in resistors with the same nominal resistivity but different composition.

Presently there is no comprehensive model which satisfactorily describes conductivity

and current noise behaviours in TFRs. A recently proposed model for excess noise in TFRs [9] is based on the assumption of

direct electron tunneling between conductive grains. This hypothesis holds true only if reso- nant centres strongly enhance the tunneling probability [3,10]. In this picture the excess

• ESL 2914

» DU PONT 1 741

• DU PONT 1441 • PROTOTYPE

B = 33.8

R = 4 1.2 R = 67.2

R = 360

BANDWIDTH 2Hz-10 kHz

Fig. 4 Relative integral noise <AV^>/<V>^

in the bandwidth from 2 Hz to 10 KHz as a function of temperature. The samples were

2 mm wide and 8 mm long.

TEMPERATURE IkI

204
noise may be generated by barrier height fluctuations due to trapping-detrapping effects in

the intergranular material in analogy to discontinuous metal thin films [ll] . This leads to simple suggestions for frequency exponent a near to unity but does not easily predict the ex- istence of a minimum of the noise at intermediate temperatures.

An alternative model, which can be of ^nterest, follows the results of [l2] for variable

range hopping (VRH) . Also in this case 1/f noise should be expected with exponent a around

the unity and with a noise level which might be much larger than that in free electron con- duction [13] as found in TFRs . Of course the original theory [l2] must be further elaborated

to make explicit the temperature dependence of noise in the case of VRH.

AKNOWLEDGEMENTS

This work was partially supported by Consiglio Nazionale delle Ricerche of Italy. Stimulating discussions of Prof .P .Mazzetti and Dr .L.Reggiani are aknowledged. Moreover we thank

Drs. G.Aprilesi and T.Garofano for making available their electronic systems for some measurements .

REFERENCES

G.A.Harper Ed., Handbook of thiok film miavoeleatronias O^c Graw Hills Inc., New York, 1977)

T.V. Nordstrom and C.R. Hills, in Proa. Int. Miaroeleatronias Symposium, Los Angeles

(1979), p. 40.

M.Prudenziati, in Proa. Srd Europ. ISHM Conference, Avignon (1981).

M.Prudenziati, Alta Frequenza 46, 287 (1977); M.Prudenziati and U.Costa in Proa. 76a

Riunione Annuale AEI, Bari (197S') n.l8.

A. Cattaneo,F.Forlani,M.Cocito and M.Prudenziati, EleatroComp. Soi.Teahnol.^ ,205 (1977)

T.M.Chen, in Proo. 5th Conf. on Noise in Physical Systems , Ed. D. Vol f (Springer Verlag , New York, 1978) p. 134.

C.Canali, D.Malavasi, B.Morten, M.Prudenziati and A. Taroni, J. Appl.Phys. 51_, 3282 (1980)

C.Y.Kuo and H.G. Blank, in Proa. ISHM Symposium (1968) p. 163.

T.M.Chen and D.Smith, in Proo. 2nd Symposium on 1/f Noise, Florida (1980) p. 376.

G.E.Pike and C.H.Seager, J. Appl.Phys. 48 , 5152 (1977).

M.Celasco, A.Masoero, P.Mazzetti and A.Stepanescu, Phys.ReV. B-17 , 2553 and 2564 (1978)

B. I.Shklovskii, Solid State Comm., 33 , 273 (1980)

F.N.Hooge ,Phys. Letters, 29 A , 139 (1969).

205
EXCESS CONDUCTANCE NOISE IN SILICON RESISTORS

B, K.Jones

Department of Physics

University of Lancaster

Lancaster LAI 1+YB England

ABSTRACT

Excess noise has been measured on a numher of silicon resistors over a wide range of

variables such as carrier number, dimensions, surface condition and temperature. In all cases the noise has a near 1/f spectrum and can be interpreted as a number fluctuation,

n^^ = g^^/y^, located at the boundary of the conducting channel, where \i is the carrier mobility at the boundary and g

^ is the measured conductance noise. This number fluctuation

changes little with the variables.

INTRODUCTION

The results pre_sented here extend the interpretation of the measurements presented in

full previously [l,2_|. The specimens were commercial silicon field-effect transistors biased

in their ohmic condition, that is with very small drain-source voltage, so that the conducting channel has uniform dimensions, and other properties, along its length. This condition

was rigorously met in the experiments so that the noise was often of low intensity. The

specimens were of the categories, junction fet (jfet), enhancement MOS fet (enhmost) and

depletion MOS fet (deplmost). Where possible n and p-channel devices have been selected

with separate contacts to the oxide gate (G) and the diffusion gate (sub) so that separate

biases and V^^^ can be applied. The measurements have been carried out over a range of

gate bias and temperature (100 K - 38O Kl

.

In this way the following influences on the noise have been studied:- presence of oxide

interface, number of carriers, width of channel, temperature, proportions of lattice,

impurity and surface scattering, transverse quantum effects, carrier freeze-out, doping

density and the presence of a space charge. Measurements have been made of the excess voltage noise, that is the increase in noise on applying a dc current, and the resistance as a function of the gate bias voltages. At room temperature additional measurements have been

made of the carrier density profile (from 100 KHz C-V^^ or C-Vg^^) across the channel and the

local mobility at the channel edge (from 100 KHz transconductance )

.

ANALYSIS

It is now well established that in homogeneous conductors the excess noise is a resistance fluctuation. To compare the results with the three basic categories of noise the

reduced noise, or noisiness r^^/R^ , has been plotted as a function of R as the resistance has

been altered by the gate voltages. Simple considerations suggest that a true bulk effect

should give no R-dependence , an effect based on Hooge's empirical formula in a uniform speci- men should give an R-dependence and a boundary effect should give an R^ dependence at each

temperature. The results show that a near-R^ variation is followed in all cases. This is

illustrated in figure 1 for a typical enhmost. The results for deplmost and jfets are less

direct but follow a general upward trend as in figures 2, 3, ^- The deviations from the

exact R2 variation are discussed later.

206
1 1—TT-TTTT] 1

1 1 1 1 ITT 7 a r 80 0

• -"-^^ ° 28^ + J r */ ' st J Ji.\

9 o / /A/ p nj ///

9 / / y//

/ /

- //

«/*///

Slope : i + 2.0

/DdL'

/

- ByTV

3N 163(B)

+/

1 1—1—

L

, ...1 , . 1 . . .iT 1000 1000

Fig.l Fig. 2

Figs.l and 2 Noisiness (r^^/R^) against resistance (R) for the p-channel enhmost 3N l63(l

The vertical lines indicate conditions of constant inversion. Also for p-channel jfet

2K 3820(3) at temperatures 130, I90, 280, 36O K.

I 1 I—I I I I I

I

1 1 rVCRTN(F)

J I I I I I 1

1

Fig. 3 10,000 Fig.

4

Figs. 3 and \ Noisiness (r ^/R^) against resistance (R) for the n-channel jfet VCRTN(F) at temperatures ll+l, I62, 19^^ 221, 310, 322 and 386 K and for n-channel jfet 2N 38l9(J) at

temperatures 187, I98, 207, 227, 262, 323 K.

Since no other model has "been suggested to give such a general behaviour we assume

(1)

to give a constant g

^ as the slope of the R^ variation. A constant noise conductance as the

channel is depleted'^suggests that it is associated with the boundary.

207
The temperature dependence shown in figures 1, 2, 3, ^ indicates that the curves are translated along the R axis as the average channel mobility changes. The vertical lines in

figure 1 and the lowest resistance data points in figures 2, 3, h show conditions of constant

inversion. Thus

r^VR^ = fCyR) (2a)

or approximately

Vr^ a (uR)^ or g

^ a (2b)

A model which satisfies the data is a number fluctuation occurring at the channel boundary due to some trapping or generation-recombination process. In this case the mobility in

the second part of equation 2 should be the local mobility at the carrier creation point.

Then

To investigate the validity of this assumption mobilities have been measured. For the

enhmost of figure 1 the slope, or conductance noise, is almost constant. The channel is very

narrow so that the local and average mobilities are similar. The mobility increases at small

inversion (large R) by an amount siifficient to give a constant at each temperature.

Using C-V^^^, G-V^^-jj S'lid = dl^/dV^^^ measurements the local mobility and carrier

density have been determined for the jfets at room temperature and are shown in figures 5 and

6. The carrier density increases into the channel as might be expected from a gate diffused

into an epi-layer channel and hence producing compensation. The mobility also rises as the

damage and impurity concentration decreases.

VCRTN(F)

Figs.^^and 6 Carrier density (N^) in units of 10''^ cm ^, and mobility (y) in ujiits of

10 cm /V sec against depletion layer depth (x^) in units of ym for the n-channel jfets

VCRTNCF) and 2N 38l9(J).

208
The values of the iiumDer fluctuation, n , calculated from the experimental values of

eiCS) for the specimens of figures 2, 3, h are shown in figure 7 against d/d.Q which is an approximate measure of the position of the channel boundary relative to the position at pinchoff. Figure 8 shows similar results for other jfet specimens.

-1 1 1 r

2N 3820(B)

o 00

o o 2N 38l9(J) °

- VCRTN(F)

_i I I 1- 0.1

Fig.T

0.5

d/d 0.3

Fig. 8

2N 1+220 (B)

o 00

o o

2N 3823(B)

I I I I I 1_ 0.9 d/d

Figs . 7 and 8 Experimental values of Cr ^/R^)y(y^R^) proportional to n ^ against relative

position of the boundary across the channel d/d^ for jfets 2N 3820(3),

VCR7N(.F); and for jfets 2N U220(B) and 2N 3823(.B).

2N 38l9(j;

It can be seen that whereas the value of g

^ from the original noise measurements is not constant, the values of n^^^ are almost independent of position as might be expected for well

made devices. The specimens of figure 8 show a 'general increase at low and high values of d. These can be interpreted as the increase in the density of g-r centres near the gate diffusion and an increase in the noise as a space charge develops in the channel.

CONCLUSION

These results suggest that the excess noise is generated by a carrier number fluctuation

located at the edge of the conducting channel. This number fluctuation is relatively independent of temperature but is influenced to some extent by the quality of the specimen at the

position of the boundary. Experiments on MOST suggest that the presence of an oxide interface produces a lar^ number fluctuation as expected by a modified McWhorter model. In jfets

the model of Sah [^4J seems appropriate. In this traps within the gate depletion region fill

and empty causing a fluctuation in the channel resistance. Since the resiilts suggest that the

effect is localised close to the channel boundary, a more detailed account of the influence

of the Fermi level in the occupancy, f , and hence the efficiency of the process given by the

factor f (l.-f ) is needed. It has been suggested by Kandiah and Whiting Qs] that this

results in the noise being concentrated in the transition region between the channel and the

depletion region. The near^l/f spectrum probably therefore results from the distribution of

the traps in space, energy and occupancy through this region to give the l/f spectrum by a suitably weighted summation of generation-recombination processes with their individual

characteristic times.

REFERENCES

[ij B K Jones, Proc Second Int Symp on l/f Noise, Orlando 198O, p 2I+5-5O. [2^ B K Jones, J Phys D: Appl Phys 11+ C1981) 1+71-90.

Csll K Kandiah and F B Whiting, Solid-State Electron 21 (1978) 1079-88.

[1+] C T Sah, Proc IEEE 52 (196I+) 795-81 1+. 209
1/F NOISE IN GATE-CONTROLLED IMPLANTED RESISTORS

K. Amberiadis and A. van der Ziel

EE Dept., University of Minnesota, Minneapolis, MN 55455

L. M. Rucker

Bell Laboratories, Reading, PA

SUMIARY: We measured 1/f noise in gate-controlled p-type Implanted resistors in which the

surface could be brought from accumulation to strong inversion. The noise increased by a factor of 150 when the surface was brought from accumulation to strong inversion. The

results indicate strongly that the noise is due to the interaction of electrons in the

inversion layer with the surface oxide. This gives rise to a fluctuating surface

potential, which in turn gives a 1/f modulation of the surface mobility.

There are two rival models for explaining 1/f noise in MOSFETs: 1) The number

fluctuation model based on McWhorter ' s^"*" -l work and 2) The mobility fluctuation model,

based on the work of Hooge and his coworkers f ^1

.

In order to discriminate between these two models, the noise equivalent current

SjCf)

Igq(Igq = —j^— , where S-[-(f) is the noise spectral intensity and q the electronic charge)

of a gate-controlled p-type resistor was measured as a function of gate bias. The

resistance R was linear and the MOS device had a transconductance of 5 ymhos. Figure 1 gives the I-V characteristics of the device for a gate bias, Vg, of 12 volts. Figure 2 gives the I versus Vg characteristic curve of the device for a constant device bias of

15 volts. The resistance of the device changes about 16% with a change of gate voltage

from 0 to 16 volts.

2 The noise had a 1/f spectrum and, at a given frequency varied as I , indicating

resistance fluctuation noise. In Figure 3 is sho\jn the parameter Igq as a function of

the current I at a frequency of 20 Hz and at zero gate bias. Changing the gate bias from

0 to 19.5 volts and with constant current through the device of 0.5 mA, gives the curve

of Fig. 4. We observe that the noise increases by about a factor of 150 when the surface

is brought from accumulation to strong inversion. The onset of noise is at 6 volts where

the inversion starts, as indicated by the C-V characteristics of the device (Vg positive

only, Fig. 5).

If the 1/f noise was due to interaction of holes with the surface oxide, the noise

should decrease strongly at strong inversion, since it eliminates that interaction. The

increase in noise can also not be explained as bulk mobility noise, for then the noise

would increase as (R/Rg)-^ and with R/Rq < 1.2 this would amount to an increase in noise of at most 75%. We therefore believe that the noise is due to the interaction of electrons

in the inversion layer with the surface oxide. This gives rise to a fluctuating surface

potential, which, in turn, gives a 1/f modulation of the surface mobility. The increase

in noise is thus surface mobility noise and is not of the bulk variety.

ACKNOWLEDGEMENT

The work was supported by the National Science Foundation and the measured devices were experimental devices provided by the Bell Laboratories.

210
KmA)

V(Volts

)

Rg. I I-V choractrislic of the gole controlled p-type

resistor at a gate bias of VL= 12 Volts

Fig. 4 Noise equivalent current 1^^ of the device os o

function of gate voltage at f = 20 Hz.

' ' ' ' '

I '

I I I 'I 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 Vg (Volts)

Fig. 5 Copocitonce vs gate voltage of the device

at 10 kHz ond positive gate voltage.

211
REFERENCES

1. A. L, McWhorter, "1/f Noise and Related Surface Effects in Germanium", M.I.T. Lincoln

Laboratory Report, 80^ (1955). Also in Semiconductor Surface Physics, U of Pennsylvania

Press, (1957).

2. F. N. Hooge, Physica (Utrecht), 60, 130 (1972). - Ibid, 83B , 14 (1976).

212
LOW FREQUENCY VOLTAGE NOISE IN SMALL AREA JOSEPHSON JUNCTIONS

L. Krusin-Elbaum and R. F. Voss

IBM Thomas J. Watson Research Center

Yorktown Heights, NY 10598

INTRODUCTION

Recently there has been considerable interest in low frequency voltage noise in Josephson

tunnel junctions. Both white [1] and 1/f [2] noise has been studied experimentally and various

models have been proposed to explain the origin of the observed fluctuations. Such investigations

are of importance because noise limits the performance of devices utilizing Josephson junctions

such as SQUIDs (Superconducting QUantum Interference Devices). Moreover, these studies yield

insight into the mechanisms responsible for 1/f noise and the manner in which quantum mechanics

limits device performance at low temperatures.

The work of Clarke and Hawkins [2] on Nb-NbOj-Pb Josephson junctions with areas « 3x 10^

jLim^ and biased at a current I^ greater then the critical current Iq gave strong evidence that the excess noise, Sv(f) 1/f, arises from fluctuations in Iq modulating the voltage across the junction.

They suggested that equilibrium temperature fluctuations were responsible for the fluctuations in

Iq. Their interpretation was an extension of the semiempirical approach developed by Voss and

Clarke [3] for thin metal films. With the assumption that the effective volume of the junction for

temperature fluctuations extended a Ginzburg-Landau coherence length into each electrode, they

found good agreement between the predictions of this model and the measured Sv(f)- A similar

agreement was found by Ketchen and Tsuei [4] in their investigation of 1/f noise in the dc

SQUID. At sufficiently high frequencies Sv(f) becomes white (independent of f) with a crossover frequency that depends on the experimental parameters. This white noise regime was recently

investigated by Koch et al. [1] who suggest that it originates from equilibrium noise currents in the

shunt resistor and produces Sy Rd > where Rp is the dynamic resistance about the operating

point.

SAMPLE >C

MATCHING

CIRCUIT AND

FILTER

rf SQUID PREAMPLIFIER

Figure 1. Measuring circuit using rf SQUID as

a preamplifier with Rq = 25 fi.

213
EXPERIMENTAL PROCEDURE

We have performed a detailed investigation of the low frequency voltage noise in well

characterized small-area resistively shunted Josephson junctions. The junctions were fabricated

with the Pb-alloy technology developed at IBM for computer applications [5]. The junction areas were defined by 2.5 nm diameter SiO windows and were shunted with a normal resistor R w 4-7 S2. Iq was typically 15-30 iiA and the junction parameter jS^ = 27rlQR^C/<|)o ~ 0.3 - 0.7, where C is the junction capacitance and (pQ is the flux quantum. For (i^. < 1 the junction I-V characteristics are non-hysteretic.

An rf SQUID with an impedance matching transformer was used as a low noise preamplifier, in a scheme similar to that used by Ketchen and Tsuei. The sensitivity of the measuring system was

limited by the Johnson noise in the 10 resistor in the primary of the impedance matching

transformer. A simplified circuit diagram is shown in Fig. 1.

Both the junction and the rf SQUID were immersed in liquid He inside a superconducting

shield. The spectral density of the voltage noise Sv(f) at constant Ij, and T was obtained from a Fast Fourier Transform (FFT) of the digitized rf SQUID output by an IBM Series/ 1 computer.

f!

N >

CO

10"

10" ' -

10 -

io-"|-

10 -

10"" -

10-^H

10

-2110 10" 10" 10'

f(Hz)

10' 10^ 10

Figure 2. Spectral density of the voltage fluctuations Sv(f) of a Pb-alloy shunted junction at T=3.7K for several values of bias current I^,.

Figure 3. (a) Junction <V> vs lb at 3.7K. (b)

Temperature sensitivity | dV/dT | ^ and dynamic resistance R^^. (c) fxSv(f) for the i /f- like noise, (d) White noise at lOkHz.

214
Alternately, I^, was slowly varied by a continuous capacitor discharge while both <V> and Sy at some fixed frequency fo were digitally recorded. In the latter case SyCfg) was estimated using a PAR 124A lockin amplifier with a constant Q filter in the ac voltmeter mode.

RESULTS

Typical power spectral densities for various I5 at 3.7K are shown in Fig. 2 for a junction with

Iq ~ 34 jiiA. The rolloff around 2 kHz is due to the response of the measuring system. There are two distinct regimes separated by a crossover region at about 100 - 200 Hz: a white noise regime

at high frequencies and a low frequency excess noise regime in which the noise can be adequately

described by a power law Sv(f) oc 1/f^. The magnitude of SyCf) in both regimes as well as the

exponent A are strong functions of I^.

By sweeping Ij, and simultaneously measuring various junction parameters, it is possible to

clarify this dependence. Figure 3(a) shows <V> across the junction as a function of I5 as measured by the rf SQUID. Fig. 3(b) shows Rj)^ from a numerical derivative of the data in Fig.

3(a). Rjy exhibits structure including that corresponding to a resonance at w 120/iV. Figure 3(b)

also shows the temperature sensitivity | dV/dT | of the junction which was measured by comparing I-V characteristics at two slightly different temperatures. As an indication of the power law

behavior at low frequencies. Fig. 3(c) shows fxSv(f) at 10 Hz and 100 Hz with background

subtracted. An exact 1/f spectrum (A=l) would correspond to coincidence of the two curves. This coincidence occurs only for a small range of I5. Figure 3(d) shows Sv(lOkHz) in the white

noise regime corrected for the measuring system response. The variation of A with bias current is shown in Fig. 4 obtained from a fit of the data in Fig. 2 to the form 1/f^. Also shown is the

variation in A with at 4.2K.

DISCUSSION

As shown in Figs. 2-4, there is an obvious correlation between Sv(f) and the detailed characteristics of the junction. Large increases in the noise are found as I5 is increased above Iq until the

characteristic finally becomes linear at high I5. This is just the region in which superconducting

Figure 4. Power law fit of Sv(f) to the

form 1/f^ vs lb at 3.7K and 4.2K.

Figure 5. Comparison of fSv(f) at low f with

prediction of thermal fluctuation model.

215
effects are important. Moreover, the influence of the junction resonance at 120 juV is clearly seen

in Sy The behavior above and below the resonance seems different. The amplitude of the 1/f-like

noise is clearly related to both and |dV/dT| indicating a connection with the thermal

properties of the junction. The fact that | dV/dT | is not simply proportional to Rj) indicates that

the temperature senisitivity is not entirely due to changes in Iq as assumed earlier [2]. By estimating

the total heat capacity (« 2xlO'^'^J/K) of our junction area [2,4] we can make a direct

comparison with the equilibrium thermal fluctuation model. Figure 5 shows the ratio of fSyCf) to

that expected from equilibrium temperature fluctuations, | dV/dT | ^keT/SCv. The agreement in magnitude is reasonably good although the simple model of a junction voltage responding to equilibrium fluctuations cannot explain the changes in exponent A with 1^,. Presumably, cannot

change the time decay of an equiUbrium fluctuation although nonequilibrium thermal feedback

effects might have this effect.

2 The magnitude of the white noise shown in Fig. 3(d) is certainly related to as predicted by

most theories [1] but the correspondence is not exact. Increased noise appears in the regions of

large | dRp/dl | where nonlinear mixing of noise at higher frequencies is expected to be important.

The amplitude of the noise appears to be greater than that expected due to Johnson noise

(including mixing) from the shunt [1] indicating that other noise sources may be important.

Although junctions were studied in the temperature range 1.5 - 4.2 K, the large unexplained

dependences on I5 preclude any quantitative statements about the T dependence. In general, as T

is lowered the structure in the I-V characteristics becomes sharper and the white noise shows

significant changes in amplitude (both increases and decreases depending on I5) while there is only

a slight decrease in the 1/f-like regime. In some cases, Lorentzian-like shoulders were observed in Sv(f) at specific T and Ij,. Both these shoulders and the 1/f-like behavior varied with changes in the thermal coupHng of the junction to the bath.

As with previous work [2,4] we find good agreement of the measured 1/f-like noise with that

expected from thermal fluctuations although the detailed dependence of the exponent A and

amplitude on Ij, cannot be explained. Further investigation of both the white and 1/f regimes is needed.

ACKNOWLEDGEMENT

The authors are indebted to M. Ketchen and the Josephson fabrication group headed by J. Greiner for helpful discussions and well characterized samples.

REFERENCES

[1] R. H. Koch, D. J. van Harlingen, and J. Clarke, Phys. Rev. Lett. 45, 2132 (1980); also R. H. Koch, D. J. van Harlingen, and J. Clarke, Bull. Am. Phys. Soc. 76, 382 (1981).

[2] J. Clarke and G. Hawkins, Phys. Rev. B14, 2826 (1976).

[3] R. F. Voss and J. Clarke, Phys. Rev. B13, 556 (1976).

[4] M. Ketchen and C. C. Tsuei, in the Proc. of the 2nd Int. Conf. on SQUID's, Berlin, May

6-9, 1980.

[5] J. H. Greiner et ai, IBM J. of Res. and Dev. 24, No. 2, (1980).

216
CONCENTRATION FLUCTUATIONS IN SMALL VOLUMES OF IONIC SOLUTIONS

Rutgeris J. van den Berg, Arie de Vos and Jacob de Goede

Laboratory of Physiology and Physiological Physics

Wassenaarseweg 62, 2333 AL Leiden

The Netherlands

INTRODUCTION

In small well defined volumes of ionic solutions^ 1/f fluctuations in the resistance

have been reported [1-4]. However the results on the intensity of this noise are controversial [1,2]. A quantitative description of 1/f noise in electrolytes is certainly a prerequisite in the search for the underlying physical mechanism. Furthemiore its relative intensity plays an important role in the use of 1/f fluctuations in nerve membranes [5] and

lipid bilayer membranes [6] to obtain information about ionic channels.

Therefore we reinvestigated the electrical noise in short and narrow capillaries, filled

with KCl solutions. Contrary to our expectations a Lorentzian excess noise component was

found instead of 1/f noise.

EXPERIMENTAL METHODS

Single capillaries were made in thin membranes (parafilm "M" , American Can Company).

Their lengths and diameters were about 10 ym. The membranes separated two thoroughly cleaned

compartments, which were connected to a pressure source. Ultrafiltration (0.02 ym PTFE-filter, Schleicher and Schull) and degassing of the ionic solution proved to be essential to minimize noise, induced by particulate impurities. To avoid vapour biibbles the compartments

were filled under a low pressure (2 Tor) . Thereafter the pressure was brought back to the

atmospheric level.

The electrical measurements were made in a six or four point arrangement with Ag-AgCl electrodes (Clark) . One pair was used to inject a constant current and two other pairs for

measuring the voltage and the voltage fluctuations. This set-up allowed to eliminate the

contribution of electrode and amplifier noise from the noise generated in the capillary by

estimation of the cross spectral density. This procedure was not utilized when capillary

resistances exceeded 2 mH. In a number of cases the current and the current fluctuations,

under conditions of constant voltage, were measured.

Capillaries were accepted for fluctuation measurements whenever the electrical and flow

resistances, within the experimental uncertainty, were in agreement with the theoretical

values based on their dimensions.

RESULTS

The thermal noise levels of the capillaries were given by Nyquist's formula:

S(f) = 4 k T Re {Z(f) }

' (1)

in which Re {z(f)} is the real part of the membrane impedance, k the Boltzmann factor and T the absolute temperature. The d.c. measured capillary resistance was equal within 10% to the

resistance {z(+0)} as obtained from eq (1). Current flow with or without an applied pressure

(Ap <_ 0.5 mm H^O) may raise the intensity of the noise by several orders of magnitude. The

frequency dependency of the excess noise could be described by:

S(f) =_^(10)_

1 + (f/f )^

217

(2)
500 5000

Fig. 1 The spectral density of the

noise of a capillary under (a) current

clamp and (b) voltage clamp conditions.

The length and radius of the capillary

were 6.0 ym and 3.5 ym respectively

and it was filled with 10~ M KCl

solution. In (a) the current was 0,576

yA, producing a mean voltage <V > of

2.05 V. In (b) the voltage was ffeld constant at 2.00 V. The open symbols

represent the noise levels at thermodynamic equilibrium. The continuous

lines are obtained by a least-squares

curve fitting procedure on 300 frequency points. The arrows correspond,

to the corner frequencies.

where S(+0) is the low frequency asymptote, f represents the corner frequency and x is a constant. Since the noise generated in the capillary is filtered by the membrane impedance,

the noise voltage should be multiplied by |z(+0)/ Z(f) |^, which requires the measurement of

|z(f) |. To circumvent this complication, current fluctuations under the condition of constant

voltage were analyzed (Fig. 1) . The value of x estimated by curve fitting was found to be

close to 2. The low frequency asymptote was proportional to the square of the mean voltage

<V>. No systematic dependence of f on the voltage could be found. S(+0) turned out to be

inversely proportional to the ionic concentration. The corner frequency was not measurably

dependent on the concentrations used.

Application of a pressure difference (AP> 0.5 mm H^O) caused a volume flow (Q) , which is

consistent with the Hagen-Poiseuille relation:

Q = AP (3)

where r and 1 are the radius and length of the capillary respectively and T] the viscosity

of the solution. Volume flow at a certain level of applied voltage changed both S(+0) and

f (Fig. 2) . An increase in volume flow, independent of its direction, led to a decrease of

S?+0) and an increase of f . The relative noise variance, however, was independent of AP.

10-"

0

vm- 3 0

V

200

t(Hzl

Fig. 2 The spectral density of the voltage fluctuations

at various levels of an applied pressure difference, as indicated. A constant current of 1.89 yA produced a mem- brane voltage V of 3.04 V. The corner frequencies indicated by arrows from left to right, correspond to the four

spectral densities from top to bottom. The lowest curve represents the thermal noise level, which is predicted by

eq (1). Note that at AP = 200 mm H^O where S(+0) has been

decreased more than an order of magnitude with respect to

its level at AP = 0 detectable. The capillary (( 5.5 ym) was filled with 10~^ M KCl solution. The Q-AP

relation was measured to be linear in a AP-range from 5 mm

to 500 mm at a temperature of 20 C. The normalized

standard error of S(f) is 0.1.

mm H^O there still is no 1/f noise " , 0 ym long and a radius of

DISCUSSION AND CONCLUSIONS

It will be assumed that the voltage fluctuations are linearly related to the fluctuations in the capillary resistance. These fluctuations are likely to originate from fluctuations in the concentration, the temperature or both. Pressure fluctuations are neglected

218 .
(cf. [4]), while association - dissociation noise [7] is very unlikely to occur in diluted

aqueous KCl solution. We omitted generation - recombination noise at the walls of the capillary. Since in short capillaries the electric field (E) is nonuniform, the spatial variation

of E must be taken into account. The correlation fiinction of the voltage fluctuations, Av(t)

under a constant current (I) can be written [8,9] as:

<AV(0)AV(T)> = —-— / dr / dr' <Ap (r ,0) Ap (r

,t) > |E(r) |^ |E(r')|^ (4)

In this expression Ap(r,t) represents the fluctuation of the resistivity p. The integrations

are carried out over the conducting volume {^) of our system. The resistivity correlation

function can be calculated in terms of fluctuations of the charge carrier density An(r,t)

around the average density n and temperature fluctuations AT(r,t) , viz.

<Ap(r,0) Ap(r' ,T) > = -—2_ <An (r, 0) An (r

,t) > + -—^ <AT(r,0) AT(r' ,t)>

n^

(5)

with X = ^ ^ and e = ^

p dn p,T p 9t p,n c.

It has been assumed that An and AT are not correlated. Neglecting possible contributions to the variances, due to the fact that the system is not at equilibrium, we find from thermodynamic fluctuation theory, also using electro-neutrality, for the relative variance of the

voltage fluctuations

<V>^ " ^p ^eff

in which c is the heat capacity at constant pressure per unit volume and V^^^ is an effective voSume, expressed as:

^

p^<V>^ I^ / dr |E(r)

I

For the geometry of our capillaries it is reasonable to take as the effective volume (9)

:

where $ = 1/r.

The two terms of the right hand side of eq 6 can be calculated. In the concentration range

covered the relative variance of the temperature fluctuations is one to three orders of magnitude smaller than the relative variance of the concentration noise. Therefore we estimated

V from eq (6) by neglecting the contribution of the temperature fluctuations. Fig. 3 s^ows that the noise at different concentrations is concentrated in the same effective

volume, which can be described by eq (8). This result justifies the use of eq (8) to calculate V^,^ for different capillaries. In all capillaries studied the variance of the noise

turned out to be in good agreement with the variance predicted by concentration fluctuations

as the sole noise source.

In general the spectral density will be determined by the kinetics of the decay of the con- centration fluctuations, which is governed by diffusion, modified by volume flow. As we could not manipulate pressure differences better than 0.5 mm H2O, we cannot rule out that

our spectral densities at no applied pressure are dominated by Poiseuille flow. This type

of noise has been demonstrated in narrow, but long capillaries and was described by a Lorentzian function [7]. Volume flow, which could also originate from electro-osmosis,

should lead to a spectral density proportional to <V> [4], contrary to our observations. At

present it is hard to explain convincingly the Lorentzian frequency dependence of the noise.

Since Poiseuille flow leads to a Lorentzian in both short and long capillaries, this sug219
Fig. 3 For one capillary <N>

<v> obtained from

<Av^>

the fluctuation measurements is plotted against the total

charge carrier density n. The variance of the noise was measured under the condition of no applied pressure at voltages ranging between 0.2 to 1.0 V. The value of X was calculated from the Debije - Huckel - Onsager parameters,

using She^lovsky's approximation method for concentrations

up to 10 M. For higher concentrations X was obtained

graphically from the resistivity data given in [10]. From

eq (6) it follows that for concentration fluctuations:

<N> = V^^^ n. Drawn is the regression line: log <N> = < log V + log n, with V^^^ estimated as 1.5 x 10^ ym^.

V calculated from the capillary dimensions (eq 8) was

Q T* t o g 1.9 X 10 ym . Since the relative error in V may amount

eff 0.4, due to the uncertainty in r, these values agree satisfactorily.

gests that the fringing electric field (eq 4) is not an important factor in determining the

shape of the spectral density.

In all experiments we could not detect a clear 1/f noise component. Evidently, if 1/f noise

exists in electrolytes it can not be explained by concentration fluctuations in a linear

noise theory. In order to get a rough measure of the 1/f noise intensity possible we take

as an upper limit twice the standard deviation of the low frequency asymptote. Expressing

the 1/f noise intensity as [1]:

S(f) = <V>

V ^- n eff

(9)

we calculate as an upper limit for a:

a < 4 x

10~^

-3

This value approaches 2 x 10 , as found for homogeneous metals and semiconductors. It would

be extremely interesting to determine whether the actual value of a in electrolytes indeed

has this magic value.

We thank Kees Versluis for his indispensible assistance with the electronics. This research

was supported by the Netherlands Organization for the Advancement of Pure Research (ZWO)

.

REFERENCES

[1] F.N. Hooge, and J.L.M. Gaal, Philips Res. Rep. 26, 77 (1971).

[2] D.L. Dorset, and H.M. Fishman, J. Membr. Biol. 2_1_, 291 (1975).

[3] M.J. Strassfeld, and M.E. Green, in: Proc. Second Intern. Symp. on 1/f noise, Univ. of

Florida, Gainesville (1980).

[4] M.W. Kim, Y.C. Chou, W.I. Goldburg, and A. Kumar, Phys. Rev. A. 22_, 22 (1980) and

references cited therein.

[5] R.J. van den Berg, Electrical fluctuations in myelinated nerve membrane. Thesis.

State Univ. of Leiden (1978).

[6] S.M. Bezrukov, G.M. Drabkin, L.A. Fonina, A.I. Irkhin, E.I. Melnik, and A.I. Sibilev,

Acad. Sci. USSR, Leningrad Nuclear Phys. Inst. 598 (1980).

[7] G. Feher,and M.B. Weissman, Proc. Natl. Acad. Sci. 70.' (1973).

[8] M.B. Weissman, Phys. Rev. Lett. 35./689 (1975).

[9] L.K.J. VanDamme, Appl. Phys. _n, 89 (1976).

[10] R.A. Robinson, and R.H. Stokes, Electrolyte Solutions, Butterworths , London (1959).

220
1/f NOISE IN MOS TRANSISTORS IN OmilC REGION UNDER STRONG INVERSION

Hong S. Min

Department of Electrical Engineering

University of Mnnesota

Minneapolis, MN 55455

1. INTRODUCTION

A theoretical 1/f noise model for homogeneous semiconductors given by Min 111, where 1/f

noise is shown to be caused by dominance of interband scattering over intraband scattering,

which gives mobility fluctuations, and where it is shoim that 1/f noise in homogeneous nondegenerate semiconductors can be described by Hooge's relation 1.21, will be applied to 1/f

noise calculation in MOS transistors. Assuming that 1/f noise occurs only in the defect-concentrated regions v/here interband scattering is dominant over intraband scattering, and that

laider strong inversion electron-electron scattering becomes an important scattering mechanism,

it will be shown that the main difficulty encountered in applying Hooge's relation to a derivation of 1/f noise formulas for MOS transistors, which is that the calculated 1/f noise is

10^ to lO'^ times as large as the experimentally determined noise f2] , can be solved. It will

be also shown that an introduction of a nonuniformity into the channel can explain the con- flicting experimental results between n-channel and p-channel devices f3'J .

2, 1/f NOISE FOR MOS TRANSISTORS WITH UNIFORM CHANNELS IN THE OHMIC REGION

A diagram of an n-channel MOS transistor with uniform channel is given in Fig. la. The

channel will be divided into many sublayers parallel to the interface. V/e assume that in

each sublayer the electron concentration profile n(x,t) and the electron mobility profile

p(x,t) are homogeneous in y-z plane in the ohmic region, v/here x is the distance from the

Si-Si02 interface, amd t is time.

Now Hooge's relation will be applied to each sublayer to calculate the short-circuited

1/f noise current for each sublayer. When noise current in each sublayer is assumed to be

mutually independent, the spectral intensity of the total 1/f noise current through the channel denoted by Sj^(f) can be written as

Si^(f) =E<<(x^)/JI^g(x^)^/[wMxng(x^)f:, (1)

i where the summation is taken all over the sublayers, the subscript s denotes the steady

state values, ^I(jg(x^) is the dc current through the i-th sublayer, f is the frequency, and

o((Xj^) is Hooge's constant for i-th sublayer. o((x^) has been believed to be a constant with

a magnitude of 2X10"'^, but there are many experimental evidences [41 that d{x^) may vary in

different samples. Min [1] has also shown that c((x^) should be a function of the density of

defect centers. For MOS transistors the defect-concentrated regions will be located near the

Si-Si02 interface' with a thickness of x^, which will be much smaller than the channel thickness St So we will assume that

c<i(Xj^) = 0(j^ for 0<_x^<_x^, and (^(x^) = 0 elsewhere, (2)

From eqs (1) and (2) we can show that

Si^(f)/Ids^ = (qR/L2f)o(jjp^^^, (3)

with (compare with P^^^ in reference 5)

f^f 2 J dxpg(x) n^Cx)/ \ dx;ag(x)n^(x), (4)

0 0

where -q is the electron charge, 5 is the channel thickness, R is the dc channel resistance

221
in ohmic region, and 1^^ is the total do current through the channel given by

= (WDV^^N^^:^^^. (5)

v;ith

N = Vdxn (x), = ( dx)a (x)n (x)/ ( dxn (x) . (6) 1 Q s eii s s >Q s Now to evaluate eq (3), the functional form of yx (x) and n (x) should be given. For n (x) we will use an approximate quantummechanical solution of tfie inversion layer carrier ^ distribution [5,63 given by

n^(x) = n^exp(-x/h) for 0<_x<_4h=S, and n^(x)=0 elsewhere, (7)

where n^ is the carrier concentration at the interface, and h is a constant.

Since it seems that there is no conclusive theoretical model for the local mobility ^^(x),

we will Just assume that when the electron concentration gets larger, electron-electron scattering becomes an important scattering mechanism. If this is the case, p. (x) can be v;ritten as C5,7]

^

>i^(x) = ;iQ/(l+Jng(x)/n^), (8)

where n^, is a critical electron concentration, yx is the electron mobility when ng(x)«n , and in u the effect on the mobility of the electric field perpendicular to the interface is

assumed to be taken into account quantummechanically even though it is not known how the

electric field in the x-direction reduces the mobility. Here we also assume tacitly that

once the quantummechanical effect gets important, an increase of the electric field in the

x-direction doesn't have much effects on reduction in jJq,

Using eqs (3), (4), (7) and (8), it can be shown that

Si^(f) = Ids^(qPo/fL2)Rc(^^/h = ( qp^/fl2 ) I^^V^^oCj^x^/h for and Xf./h«l, (9a)

Sj^(f) = l^/{q)iJfh^)R^Q^(i^^ = (q2p^VfL3)V^^2^j^ncXf for no/nc» 1 and Xf/h« 1 (9b)

with G£.=qwx^ u^/L and V^^ being the dc drain voltage, where eq (9a) is same as Klaassen's

formula [8] if o'-^x^/h is replaced by Klaassen's 5, and eq (9b) is same as the formula for the

number fluctuation model [33 if ^'^jjn x^ can be replaced by lNr[,( ) j g^„/£ according to van der

Ziel. The facts that x^/h in eq (9) is believed to be much smaller than 1, and that

Sj^(f)/I^g2 is proportional to r2 when R is small, can solve the main difficulty encountered

in applying Hooge's relation to a derivation of 1/f noise formulas for MOS transistors.

To find the dependence of Sj (f) on the dc gate voltage V^^, and to obtain the spectral

intensity of the equivalent input noise voltage at the gate defined by

with gjn being the transconductance given by gm='^Ids''^^gs » functional dependence of N-j. in

eq (6) on Vg^ should be known. Empirically we may assume the following relationship

^ox ( Vg3-V^ )"^ with Y > 1 and for V^< Vg^iVg^

,

qN^

I

" • - ^° ' - - (11)

C (V -V„) for V >V , ox^ gs t' gs- go'

. where Vy and V are the critical gate voltages chosen empirically, ky and Tare constants,

C^^ is the oxiae capacitance per unit area, and is the threshold voltage.

To derive the noise formulas, first we will put

= ^Co^(Vg^-A^)"^ for all V^^V^, (12)

and then we v/ill return to the case of eq (11) later. Using eqs (5), (9), (10) and (12), we can show that

222
Id

r = (qwk,).io2/fL3)CQ^V 2(v v^) ^ x^/h for no/n^«l,

(f) £ n

(f) = (q/fLwlvY2)CQ^-l(V -V^)2-"*'o<j^jXf./h for all n^. (14)

Curves of eqs (9), (13) and (14) for the cases of eq (12) with T=l and r=2, and eq (11) with

Y=2 are shown in Fig, 2, Experimentally p-channel devices usually show the noise characteristics of Fig. 2 t31

,

3. 1/f NOISE IN MPS TRANSISTORS WITH NONUNIFORM CHANNELS IN THE OHMIC REGION

V/e consider an n-channel MOS transistor with a nonuniform channel with a narrow defect-

-concentrated region (such as misfits or interfacial dislocations [9]) lying perpendicular to the interface as shoim in Fig, lb, vihere the region betv/een L^<y<L-i+L2 is the newly introduced

defect-concentrated region. Let R-^, Rg and R^ be the dc resistances for regions I, II and

III, respectively (see Fig, lb), V/e assume that L2« L-|^+L3 and R2«R-]^+R3. We also assume that

ng(x) is given by eq (7) throughout the channel, and that Hooge's constant C<.(x,y) and the

mobility jj(x,y) are given by

= o(^^ for 0<x<x^ in regions I and III,

c(ix,y) \ =^H2 ^^'^ 0<x< S" in region II, (15)

0 elsewhere.

>i(x,y) f

" >iol/(l+^ns(x)/n^i) for regions I and III,

^^^^ ^ = y^^/ (.l+\n^{x) /n^^) for region II,

where we usually have n^i<n^2» '^oi>>*o2 ''^Hl^^'^12* Wnen

y^i/f-^

is P-2./f regions I and III defined by eq (4), and Pi/f2 is for region II given

^ (-5 2 >*l/f2 ) "^x^s^^ty) n^(x,y)/ dxp^(x,y)n^(x,y) , (17)

the spectral intensity of the total noise current S-|-^(f) can be shown to be

^Id^^^ = C^IdsW^^^l+4)^3Kl^l/fl + <^2>^l/f2tV(4+S)^ (Peffl/>>eff2)^}. (18)

where y^ffi and }ieff2 ^® 'the effective electron mobilities for regions I and II, respectively, which are defined by eq (6),

If the first term of eq (18) is dominant, v/e will have the same result for Sj^(f) as obtained for the MOS transistors with uniform channels.

When the second term of eq (18) is dominant, to interpret eq (18) we can show that the

following rough approximations can be made for Pgffi, >^eff2' •'^1/fl ^"^^ >^l/f2 using the

exact expressions of them:

Peffl = >iol/U+(2/3)fK7K^l], = -^o2/I^l+(2/3)]IV^]. (19)

Hl/fl = Moi(V^)/[!+( 4/3) 3' Hi/f2 = p^2^[l+i^/^)\no/n^2]' (20)

When v/e assume eq (12) for noise calculations, we have from eqs (18)-(20)

Si^(f) = [q^^L2Hol\^2/f(Ll+L3)W]CoxVds^(V-''t)''

X [i+e2(V-^r)'^^^]^/[i+^i(\s-^y)^'^^3^' ^21)

where we usually have ©^^O^j ^\ ®2 given by ^ = (2/3)JlvC^^/qn^^h, 9^ = (2/3)K^ox/^"c2h- ^^2)

Now from eqs (5), (10) and (21) it can be shown that

for the ranges of V„ with 6i (V„^-V^ )^/2^<; 1 ^ (denoted by region A in Fig. 3)

223
'^d^'^ = ,n , ,4 2 ^ox^ds (\s-^ ) =

, ,3 2 ^ds'(%-%) . (23)

f(4+L3)X2 f(Li+L3)-^>iQ2

S^g(f) = [qL2>ioi2e<^2/f (4+^3)^^02V^JC^x~^(Vg^-y^)2-r (24)

for the ranges of V with ^2^"^gs'^-i^'^^'^'^'^ '^^'^^1^^gs-^^"^^^ ^ (denoted by region B in Fig. 3)

81qwLgHoi^(q"clh)^H2 ^ _i„ 2,„ ,_y _ ,27,2 q^%Hoi^(q"clh)^H2

v4

•-d ifi^-^T T ^4 2, ox ds gs Y 8 t \6 2 ds a 16f(L^+L3)>Q2 ^ ^( X2

X (R^+Rg)"^, (25)

BlqL p^l2(qn h)ol

'Vg^^^ = ..... .2... 2. I2 ^ox-'( W (26)

16f ( L^+L3 ) '^wPq2 ^

'^'^

and for the ranges of V with 6„(V -V^)^^^»l, (denoted by region C in Fig. 3) gs id. gs

S 9qwL2Poi^(qn^.^h)n^^0(H2 2 ^^^"'^Hol^cl^^^cA2 ^,

-r f =

^ V, = I, ^ Rt+R„^, 27)

4f(4+L3) '^02 nc2 4f(4+L3) >io2 "c2

S (f) = (9/4)[qL2H^^2^^^^^/f(L_^^L3)Vo2^"c2lVY2jC^^-l(Vg^-V^)2-\ (28)

tb Fig. 3 shows the curves of eqs (23)-(28). Similar types of curves as in Fig. 3a have been

observed by Park [10]. Also similar curves as in Fig. 3b have been observed by Vandamme et al

till in the devices with low surface state densities, where the second term of eq (18) may

easily become dominant over the first term. Since the curves in Fig. 3c have been found more often in n-channel devices 13], we may say that n-channel devices have more possibilities to

have nonuniform channels than p-channel devices.

4. FINAL DISCUSSION

We have derived various theoretical expressions of 1/f noise characteristics of MOS transistors. These noise characteristics depend on the dc characteristics (specially N-j- vs V^^,

and ^g(x) vs ^^(x) characteristics) and geometries (specially uniformity, of channels) of ^ devices, and we have shown that one noise characteristic can not predict other noise characteristics without knowing the dc characteristics and geometries. It has been quite common

that interpretations of experimental 1/f noise characteristics of MOS transistors are con- fined to one or two specific types of noise datas with assumptions of qNj=CQ^( Vgg-Vrp) and

uniform channels. But as we have shown above, to understand the 1/f noise characteristics of

MOS transistors we must analyze all the noise characteristics discussed in this paper as well

as the dc characteristics simultaneously without interpreting each noise data separately.

This vrork was supported by the government of R.O.K. I am gre.teful to Dr. A. van der Ziel

for his encouragement.

REFERENCES

1. H.S. Min, J. Appl. Phys. 51, 1637 (1980), and in Proc. Symposium on 1/f Fluctuations,

Orlando, Florida (1980).

2. F.N. Hooge, Physica 83B, 14 (1976).

3. A. van der Ziel, in Proc. Symposium on 1/f Fluctuations, Orlando, Florida (1980).

4. H.I. Hanafi and A. van der Ziel, in Proc. Symposium on 1/f Fluctuations, Tokyo (1977).

5. L.K.J, Vandamme, in Proc. Symposium on 1/f Fluctuations, Orlando, Florida (1980), and

Solid-state Electron. 23, 317 (1980).

6. C.T. Hsing, Ph.D. Thesis, University of Florida (1977).

7. F. Dannhauser, Solid-State Electron. 15, 1371 (1972).

8. F.M. Klaassen, IEEE Trans. Electron Devices ED-18 . 887 (1971).

224
9. G.F. Neumark, Phys. Rev. Lett. 21, 1252 (1968).

10. H.S. Park, Ph.D. Thesis, University of Mnnesota (1981).

11. L.K.J. Vandamme and A.H. de Kuijper, in Proc. Symposium on Noise in Physical Systems,

Bad Nauheim, Germany (Springer-Verlag, Berlin, 1978).

(a)

SiO.

(b)

^ y

SiOc 2

>^Region . I ; II t III

Fig. 1 Diagrams of MOS transistors with coordinates. (a) A MOS transistor with an uniform channel. (b) A MOS transistor with a nonuniform channel.

(a)

Si.

(b)

(c)

case oi eq

(12) v;ith r=2. —.: case of eq

(12) with T=l.

gs

i fS

1 ^ \

In

I

InR 1

-ds

case of

^^eq (12) v/ith Y=2.

case of

eq(12) with Y=l. — : case of

eq (11) with y=2.

gc

gs

Fig. 2 Asymtotic curves of noise characteristics of MOS transistors with uniform

channels. (a) S^ (f) vs V

(b) ln(fS^ /I^ 2jd (c) ds

gs

^„ vs V

V„ gs

(relative magnitudes are not important)

case of eq

(12) with r=2. — : case of eq

(12) with Y=l.

(q) : case of

eq(12) withY=2

. ^ : case of

eq(12) with r=l

: case of

eq(ll) with Y=2

Fig. 3 Asymtotic curves of noise characteristics of MOS transistors with nonuniform channels, (a) Sj,(f) vs V . (b) ln(fSiyid32), svg(f) #s Vg^.

(relative magnitudes are not important)

225
1/f NOISE MODELS IN MOSFETs

H, S. Park and A. van der Ziel

E. E. Dept., University of Minnesota, Minneapolis, MN 55455, U.S.A.

SUI'/II4ARY

In the number fluctuation model of 1/f noise, the effective oxide trap density at Fermi

level, %(%)eff evaluated as a function of the gate bias voltage Vg, In the mobility

fluctuation model, Hooge's parameter a is found at the same condition as in the niimber

fluctuation model. The values for N'j(Ep)g£-£ and OJ from the experimental results are reasonable. The Nrp(Ep)gj-£- is approximately proportional to the (V - V^j-) and the Ct is almost

independent of the (V^ - Vip). ViJhere Vrji is the threshold voltage of the device. But we can not

discriminate between the models from the data.

To discriminate, v;e measure the drain current spectrum Sj^jCf) versus drain bias voltage

V(j, In the number fluctuation model, Si,j(f) increases monotonically with increasing V^j until

saturation. In the mobility fluctuation model, because of the electric field dependence ofot,

Sj(j(f) goes through a maximum well below saturation and level off to a lower value.

Also the noise curves calculated numerically are presented for both of the models.

MEASUREMENT OF Nx(Ep)gff AND Ct

In the number fluctuation model of 1/f noise fl] , the drain noise spectrum at low drain

bias V(3 under strong inversion is given as

Sj^(f) = (qVw V^^/fL^) N^(Ep)^^^ /€ (1)

whereas for the mobility fluctuation model of 1/f noise CsJ^fl]

Si^(f) = (q/VfL^) I/^a (2)

Here w and L are the width and the length of the conducting channel, € is a tunneling

parameter which has a value of about 10^ cm"-^, ^ is the mobility of the carrier, q is the

electron charge and I^ is the drain current.

To evaluate the parameters, NT(EF)gff and a,from the data, first we observe the drain

conductance g^. In the case of low V^j and under strong inversion, gd is

' % = V^d= (^^^^ox/^)(\- V

Knowing the dimensions of the device and evaluating the threshold voltage yield mobility

as a fuction of V , Next from the measured value of S-j-,(f), eq (1) and eq (2), we can

find the values of nI (Ep)^^^ and a . Figiare 1 shows Ny(Ep)/€ versus for various devices.

Figure 2 shows 01 as a function of V„ for three devices.

226
NylE, )*«/«

t 7720(S0S N.CHJ. RCA, fMyi

o BP-3(N.CH} BELL LAB. L*a^

& BP-3(N.CH) BELL LAB. L«4^

2249(N.CH; HONEYWELL, L*8^

0 2 4 6 6 O

Fig« 1 Nx(Ep)gr.^/ € versus the gate voltage.

At larger Vo-jNipCEp) varies almost linearly

with Vg, For the silicon-on-sapphire device

the parameter shows that the value is about

a factor 200 larger.

a

1 -X 10 V

-»

ID

M 7720 (SOS, N.CK) RCA, L'90|t

o BP3(N.CHJ BELL LAB L>8^

A BP4(N.CH) BELL LAB. L>4^

-4

10

V

k -8

10

.6'0 2 4 6 8

V,(V)

Fig, 2 Hooge's parameter a versus Vg, « is practically independent of V„,

For the SOS device the parameter snows that

the value is larger than the others by factor

200,

Since the two models describe the same phenomenon , their parameters are formally

related, so that

^T^^F^eff = « "^ox^ \ - V^"^

Here C is the oxide capacitance per unit area, ox DISCRIMINATION BETWEEN TVJO NOISE MODELS

In order to discriminate betv/een the models, we measure Sj^(f) versus V^j from low V^j to saturation. According to van der Ziel [JsJ , if N(x) is the carrier concentration for

unit length at arbitrary point x, then its spectrum is

. , n

^T^^F'-^eff ,^ S^(x,f) = w (5

Here N.p(Ep,x)gff is the effective trap density at Fermi level at point x.

In the mobility fluctuation model, the mobility fluctuation spectrum S^(x,f) for unit

length was postulated such that

S„(x,f) a(x) —2 = (6

/I (x) f N(x)

Here fi (x) is the average mobility at x, anda(x) is Hooge's parameter at x. If E(x) is the

227
electric field strength at x and the mobility fl is in the hot electron regime, such that

1 + E/E

c then 01 must be replaced [4] by

a(o,x)

(7)

( 1 E/E')'

c

(8)

Here flo is the low field mobility and is a critical field strength, such that

uc = UqS-c is the limiting velocity of the carriers, and is another critical field strength,

such that Ug =/'o^c sound velocity in the sample. Then eq (6) and eq (8) yield

S^(x,f) a (o,x)

f N(x)( 1 + E/E')'

c If g(x) = q/i(x)N(x) is conductance for unit length at x, then

(9)

and

Sg(x,f) SN(x,f) N^(Ef,x)^^^ w

g^(x) N^(x) fN^(x)

(10)

Sg(x,f)

g^(x)

S^(x,f)

/i^(x)

0(0, x)

fN(x)( 1 + E/Ei )

After some maniplation, one can obtain for the number fluctuation model QsJ ,

(Q^w/fL^) /u/(x) [NT(Ep.x)gff./€j dx

and for the mobility fluctuation model,

, L (ql^/fL^) / u/(x)-

a (o,x)

( 1 + E/E' )' dx

(11)

(12)

(13)

Here u^(x) =/i(x)E^(x) is the drift velocity at x, van der Ziel et al, solved the integrals eq (12) and eq (13) numerically QsJ ,

The Fig. 3 and Fig, 4 show the experimental results of S-[-^(f) versus in the

several MOSFETs. And the Fig, 5 and Fig, 6 show the numerical results of the integrals.

CONCLUSION

In the number fluctuation model, Si(j(f) increase monotonically with increasing

until saturation. In the mobility fluctuation model, because of the strong field dependence

of Hooge's parameter a » S-[-^(f) goes through a maximum well before saturation and level off

to a lower value. Some MOSFETs agree better with the number fluctuation model and others

agree with the mobility fluctuation model.

228
(A) icf

10

a 2N435I(N CH) Vg»3V

/o /

a 3NI70(N.CH) V,'2V

« 7720 (SOS) V,-5V

D BP-3(N.CH) Vg«2V

1 1

1*1 KHz

1

Fig. 3 Most of the devices show that a mono- tonic increase of Sj^Cf) with increasing

imtil saturation sets in. They agree better

with the number fluctuation model.

Vg.O V

10' cm/s

o,« 10® cm/s

0.2 0.3 07 1

(A)

10

10

1 1 1

D f2M(2249) A V,.25V — ^ V,'0.8V / \°

ao

—

' L«8/i(2249)

Y Vt«0.9V

O >ft_Xt

^

L'20;i(2249)

1

1 1

<v) Fig. 4 A number of Sj^(f) vs V^j curves show

a significant maximum m Si(j(f) before

saturation. Therefor these devices agree

better with the mobility fluctuation model.

L*10 |im

4 |im Vg.O V -v^.aov

Uq»600 cm^/vs

Uj, 10' cm/s

Uj, 10® cm/s

L»1.6 iim

Fig, 5 The dependence of the 1/f noise in the

number fluctuation model of MOSFET on the drain

voltage, with L as a parameter.

Fig.

0.2 03 C.7 1 6 The dependence of the noise in the

mobility fluctuation model of MOSFET on the

drain voltage, with L as a parameter.

REFERENCES

1. H,S, Park, A. van der Ziel and S.T. Liu, Solid State Electronics, in the press.

2. F.N. Hooge, Pliysica, 60, 130, 1972, 83B, 14, 1976.

3, A, van der Ziel, IEEE Conf, Record, Third Biennial University, Industry, Government

Microelectronics Symposium, Texas Tech, U, Texas, 102-106, may 21-23, 1979,

4, G, Bosman, R,j.J. Zijlstra and A, van Rheenen, Phys, Lett, 78A, 385, 1980,

5, H,S, park, A, van der Ziel, R.J, J, Zijlstra and S.T, Liu, J, Appl. Physics, in the press.

6. A. van der Ziel, R.J.J Zijlstra and H.S. Park, J. Appl. Physics, in the press.

229
1/F NOISE IN SHORT CHANNEL MOSTs

Pierre Gentil and Ahmed Mounit

Laboratoire P.C.S. ERA CNRS 659

ENSERG 23 Avenue des Martyrs

38031 Grenoble Cedex FRANCE

INTRODUCTION

Short channel length MOSTs exhibit large variations of the mobility with the gate bias

voltage. Consequences of this property on 1/f noise of a short channel MOST is examined

below. First, we discuss whether noise physics of short channel MOSTs can be evaluated from

the measured spectrum Sve of an equivalent gate fluctuation or not. Van Der Ziel and Park

[1] have shown that the elementary theory of a MOST which connects Sy^ to the drain current

spectrum Sjj^ cannot apply in all cases. Secondly we compare experimental noise results

of the devices with the surface trapping noise model and with the mobility noise model in

the inverted layer closed to the Si-Si02 interface. 1/f noise in MOSTs is explained for a long time by help of the trapping of the channel carriers by traps into the oxide near the

Si-Si02 interface. In a recent work [2], Vandamme attempts to apply the universal empirical

model proposed by Hooge to the case of the MOST. The latter model considers that mobility

fluctuations are the origin of flicker noise. The empirical relation of Hooge is adapted to

the MOST by taking into account in the mobility, the different scattering mechanisms of the

channel carriers. We apply these theories to short channel (# lym) MOSTs biased in the

strong inversion ohmic region.

Devices were supplied by the LETI (Grenoble, France). The enhancement mode MOSTs have a 20 ym channel width (W) and an oxide thickness of 320 A. Silicon substrate is not implanted ; doping is lO^^cm"^. Drain and source resistances are inferior to 20 Thermal noise

and parasitic noise were subtracted from total noise when not negligible. On figures, curves

are labelled by the effective channel length L (in ym) of the devices, absolute values of

currents and voltages are considered.

STATIC PARAMETERS

The following current -voltage relationship of a MOST working in the strong inversion

non-saturated region is used :

^D = MeffCox^W/L) (Vg-Vt-V2/2) Vd , (D

where : V^ff = Mo d + G(Vq-Vt)0'^ (2)

Threshold voltage Viji, effective channel length L, effective mobility at small gate voltage yo and the parameter 0 are carefiilly determined : a- Vij. is determined from the 1-^ Vg. Vq plot at a very low drain voltage. As shown in table

1 , Viji varies with the channel length and variations agree with the theory of short channel

devices [3].

% (ym) 2.9 3.7 14.7 5.5

Vt (v) 1.7 1.85 1 .90 1 .92 L (ym) 0.9 1.7 2.7 3.5

0 (VM 0.15 0.12 0.08 0.08

Table 1 : measured parameters of short

P-channel MOS transistors. The four

transistors are located closed one to the

other on the same chip.

230
b- L is determined by plotting Ij) vs. L^- Lm is the mask channel length as measured

optically Lm = L + AL. The plot gives a straight line from which AL is obtained. For this

experiment Vq-Vt is kept constant and sufficiently small so that 1 + 0(Vg-Vt) does not vary

greatly with L. Also Yj) is taken very inferior to Vq-Vt.

c- 0 and yo are determined by plotting [ (Vg-Vt-Vd/2)/Id] vs. Vq-Vt for each transistor. The

straight lines thus obtained agree with yo independent of L and allow to determine 0 for each

transistor. As shown in table 1, 0 becomes important for the smallest channel length ; ©(Vq-Vt) may become greater than unity for a few volts gate voltage. The hole mobility in the case of P channel, yo = 150cm^v ^s ^ is small but agrees with the high doping of the substrate. With the values of the parameters obtained above, good agreement is obtained between . drain c"urrent model and experiments below saturation of the MOST. Uniformity of channel

lengths and threshold voltages is not very good on the whole wafer so that it is necessary to

determine these parameters for each transistor before applying noise models.

DRAIN CURRENT NOISE AND EQUIVALENT INPUT NOISE '

'

N and P short channel MOSTs exhibit an experimental spectral density which varies like

1/f in all cases from 10~^Hz to 10'*Hz. Furthermore, equivalent input noise varies about like

1/L. Noise in short channel MOSTs behave like large MOSTs do. Drain current noise Sj-q is more easily connected to the physics of the device, however, the equivalent gate noise S^g

is the usual way for evaluating noise in MOST. Figure 1 gives the ratio Sjj~j/S^g as a function

of Vg-Vrp for the P channel MOSTs. Sjj^/S-^g varies greatly with the gate voltage. This ratio

must correspond to gj^^ and can be evaluated from the static model described above. From (l)

and (2), the transconductance g^ is given by : gm.= ^0 Cox(W/L)Vd (1+0 Vp/2) [1 + 0(Vg-Vt)]"^ (3) As seen from figure 1^ good agreement exists between the measured Sj^-^/Syg and the calculated

gm^ • Consequently, on short channel MOSTs, Sye cannot be a direct measure of the noise spectrum of the cnannel carrier density when interpreted with the help of the trapping theory.

If Syg is measured, relation (3) must be taken into account before interpreting data in

terms of carrier trapping fluctuations.

231
TRAPPING NOISE MODEL AND MOBILITY NOISE MODEL

In the trapping model, very different variations of the noise spectrum as a fiinction of

the gate voltage can be obtained by varying the energy level of the traps, by introducing two

or more discrete energy trap levels or by using different continuous energy trap distributions. The noise density level can be independently adjusted by varying the density of each

type of traps.

For single energy traps distributed into the oxide, the trapping model gives the following Sj-Q relation under sma^l drain voltage Lh2

SiD = Ivg-vtJ

f+n (1-f^n) A WL Cox f where : Nt is the trap level density, ftothe equilibrium Fermi factor of the traps, X the

tunnel transition parameter, X = [ti^/(8m*(j))]V^ where (b is the tiinnel barrier height. For

relation (l) and (2) are used. For a trap below the Fermi level, fto^^'^to) increases with

Vq-V<p. For a trap above the Fermi level, fto^l'^to) is almost independent of Vq-Vij.

Experimentally, Sjj^ varies slowly with Vq-Vt on short channel MOSTs as shown in figure 2. However, Sjj^ increases with Vq-Vt at small and high Vq while Sip is almost constant for moderate Vq. We have used three discrete trap levels for modelling this noise as shown in table

2. The trap density near the valence band decreases when the channel length decreases. This

effect could be attributed to a more important influence of the drain and source doped re- gions on the channel region in short channel MOSTs.

Ei-Et (eV) o,H 0,^7 0,62 '

L = 0,9ym ; Nt(cm"')= k,3 10'' 9 10'^ 1,3 10'''

L = l,Tym ; Nt(cm"')= 11 10'^ 12 10'^ 2,3 10'^

L = 2,Tym ; Nt(cm'^)= 10 10'^ 10 lO''' k,k 10''

L = 3,5ym ; Nt(cm'')= 5,5 lO'^* 10 10'^ 6 10"

Table 2 : trap energies and trap densities

used for the model in fig. 2.

Figure 2 : oa, + +.oo,^A : Sj-^ as measured under = 50mV and f = 1 KHz.

: theoretical curves obtained from

{h) , using (l) and (2) and the trap parameters given in table 2.

0,5 1 2 Vg-Vt (V) In order to confront trapping noise model with the mobility noise model, we have also plotted

in figure 3 fSy/Vi)^ as a function of Bq for the experimental results and the theoretical curves of the figure 2. Sy is the noise power spectrum at the drain terminal and Rc is the

channel resistance. Good agreement is obtained in the whole range of investigation.

For the itDbility model, we have used the theory of Vandamme [2] which proposes

to take into accxjunt various scattering mechanisms of the carriers. Carrier-carrier

scattering gives a direct dependence of the effective mobility on . Surface scattering

relation used in [2] leads to a gate voltage dependent effective moBility by considering

the channel shrinkage effect. Noise models taking into account each scattering mechanism

separately and the ccnibined mechanism were compared to noise results on short channel

MOSTs. The parameters were deduced by fitting conductance vs. V^^ of the MOST under test

with the Vandaitme's model. The theoretical noise curves were then calculated using these

parameters. As can be seen on figure 4, theoretical noise spectrum curves disagree with

measurements. The form of the curves is not very good and theoretical noise level is too

232
10

10

10

Q>

>

CO

Cm-9

10

•1 0

10

2,7 3.5

10^

-I I !

10^ 10- J-^J

Tigure 3 : trapping model, all curves are deducted from fig. 2 under \53=50mV and

f=1 KHz. and Sy are calculated from

relations (l) to (U) for theoretical curves

( ) and evaluated from measurements for

experimental points (d , + , o , )

The same experimental points are used in

fig. 3 and h.

Figure U : Mobility model. : Theoretical curves which are deducted from

Vandamme's model [2]. Using the carriercarrier scattering mobility. Parameters

were deducted by fits of conductance

vs. (vg-vt) . t = 100 a

Pg (0.9) = U.2 lO^^cm"

Pe (1.7) =6 lO^^cm"^

pe (2.7) = 5.10^^cm'^

Pe (3.5) = i+.3 lO^^cm"

yi = 500cm^v"^s'

high. The fluctuation model cannot be applied to the short channel MOST at least when using

the scattering expressions of reference [2].

ACKNOWLEDGEMENTS

We wish to thank Dr Gautier and Dr de Pontcharra for supplying the devices. This work

was supported in part by GCIS (LETI-CNRS) contract.

REFERENCES

[1] A. Van Der Ziel and H.S. Park, Appl. Phys. Lett., 35, 9k2 (1979).

[2] L.K.J. Vandamme, Solid-State Electr. , 23, 317 (1980).

C3] L.D. Yau, Solid-State Electr., 17, 1059 (197^).

P. Gentil, These d'Etat, ENSERG, GRENOBLE ( 1979).

233
THEORETICAL INTERPRETATION OF LOW 1/f NOISE IN JFETs

C. F. Hiatt*

E.E. Dept. U of Florida, Gainesville, FL 32611

A. van der Ziel

E.E. Dept. Univ. of Minnesota, Minneapolis, MN 55455

and

R. J. J. Zijlstra

Laboratory Experimental Physics, Princetonplein 5, Utrecht, The Netherlands

SUMMARY

It is shown that the absence of 1/f noise in low-noise JFETs, as found by Hiatt, can

be explained by a mobility 1/f noise source as long as the low-field value of Hooge's

parameter a is less than 0.4 x 10"-''.

Van der Ziel recently concluded''' that the absence of 1/f noise in low-noise n-channel

JFETs, found by Hiatt^, could not be explained by a mobility fluctuation 1/f noise source,

if the Hooge parameter-^, a, of this source had a low-field value of 2 x lO"-^, Recently,

n-channel MOSFETs have been found^ that had a - 10"^ at low drain voltages. Bosman^ found

for 1/f noise in n-type space-charge-limited solid-state diodes that a - 2 x 10"^ at low

fields. Hanafi and van der Ziel^ found for HgCdTe photoresistors that a had a value of

1.5-2.0 X 10"^ in selected units and a value as low as 10~^ was reported in one unit^.

Hence much lower values for a than 2 x 10"-^ are not uncommon,

g Bosman et al found that the high-field value of a decreased strongly with increasing

field, and van der Ziel at al.^ deduced from this that the 1/f noise in JFETs at saturation

would be strongly reduced by this effect. We therefore reexamined Hiatt 's data to come up

with a more accurate limit for a.

The laboratory-built device used in the measurements had a 5 y meter gate window and

a 2,0 y meter undercut on each side, so the channel length L was 5.0 + 2 x 2.0 = 9.0

y meter. Van der Ziel et al.^ found that for L = «>, where the elementary theory was valid,

the normalized drain 1/f noise was 0.148, whereas its value at 9.0 y meter channel length was 0.015 , so that the a-reduction factor g(L) at saturation is (0.015) /(0.148) = 0.10^

Hiatt measured at I^ — 0.62 mA and gjj^ — 6.0 m mhos at V^ — 0.4; the effective gate

voltage (Vg - V ) was about 0.20 V. According to Hiatt R^^ was about 30 kOhm at 3 Hz, so we take for the 1/f noise at saturation (Fig. 1)

R^(l/f) ^ 2.5 X 10'^ /f Ohm (1)

According to the elementary theory, as corrected by van der Ziel et al., the drain noise at saturation is

eya I (V - V^)g(L)

Sj(f) = 4kTR^(l/f) P

(2) L f

A Now at Honeywell Solid State Electronics Division, Plymouth, MN 55441,

234
so that the low-field value a of a is

o

4kTR (l/f)g^L^f

% - eyI^(Vg - Vp)g(L) ^

If we substitute the numerical values just quoted we obtain < 0-4 x 10 ^.

Hence a mobility fluctuation 1/f noise source with a low-field value < 0.4 x 10

^ of Hooge's parameter would not give any observable 1/f noise in Hiatt's JFET. Since this

is well within the observed limits of a^, the mobility fluctuation 1/f noise model can no longer be excluded as an explanation of Hiatt's experiments.

REFERENCES

1. A. van der Ziel, Appl. Phys, Lett., 33 (1978) 883.

2. C. F. Hiatt, Ph.D. Thesis, U of Florida (1974); C. F. Hiatt, A. van der Ziel and

K, M. van Vliet, IEEE Trans. El. Devices, ED- 2

2

(1975) 614.

3. F. N. Hooge, Phys. Lett., A-29 (1969) 139.

4. H. S. Park, A. van der Ziel and S. T. Liu, Solid State Electron. (1981) in the press.

5. G. Bosman, private communication.

6. H. I. Hanafi and A. van der Ziel, Physics, 94B (1978) 351.

7. R. N. Broudy, Honeywell Radiation Center, Lexington, Mass. Final Tech. Report

Contract NASA CR-132512 (1973).

8. G. Bosman, R. J. J. Zijlstra and A. van Rheenen, Phys. Lett., 78A (1980) )385.

9. A. van der Ziel, R. J. J. Zijlstra and H. S. Park, J. Appl. Phys., June 1981, in the

press.

Freq.(Hz)

235
1/f NOISE IN GaAs MESFETs.

C.H. Suh, A. van der Ziel and R.P. Jindal

Electrical Engineering Department

University of Minnesota

Minneapolis, Minnesota 55^55

SUMMARY

Measurements on l^f noise in GaAs MESFETs are reported. The spectra are of the form

(A/f) [l - (2/Tr) tan (ojTq)], where A and Tq are constants. While both the number and the

mobility fluctuation models can qualitatively explain the data, the latter explanation is

somewhat more likely.

We report here on l/f noise measurements on NEC GaAs MESFETs. As shown earlier^ these

devices have spectra of the form

(A/f)[l - (2/Tr) tan'-'-CcoT^)] (1)

where A and Tq are constants that depend somewhat on the gate bias. Figure 1 shows that by

proper choice of A and Tq a perfect fit can be made between curve (l) and the experimental

data. A superposition of two Lorentzian spectra cannot produce such a fit.

This formula can be interpreted in terms of the following distribution of time constants

= mill ) forT^<T<T^ (2)

1 o

and zero otherwise. Such a distribution gives a spectrum of (1) for cut _> i and a l/f

spectrum for I/t^ < w < l/xp* The perfect fit between the data and Eq. (l; indicates that a distribution of the form (2; is the most likely explanation of the spectra.

As a preliminary for discriminating between possible l/f noise mechanisms^ we measured

the drain noise spectrum Sj (f) at Vg = -1.50 V at very low drain bias (Vd = 0.10 V) and

near saturation (V(j = I.5OV), found that the constant Tq depended hardly on V(j whereas the

parameter A did. This is shown in Fig. 2, which plots Igq = Sj (f)/2q, the equivalent

saturated diode current of the drain noise. Since Tq does not depend on Y^, the dependence

leq on must come from the dependence of A upon Vcj.

Figure 3 shows In Igq plotted versus In V(j at Vg = -I.50 V and f = 1 kHz; this frequency was so chosen that the device operated in the l/f part of the spectrum. At low drain bias

the value of Ig^ varies as V,^. At higher values of V(j the value of Igq passes through

maximum at V^j = O.65 V, a minimum at = I.50 V, and levels off to a constant value above

2.0 V. This behavior will now be explained.

According to Park et al^ the drain noise spectrum for the number fluctuation model is

h -f^ />d [N,(E,,x)]^^^dx (3)

236
whereas for the mDbility fluctuation model

(f) =^/V(x) a (E ,x) dx . (U) ^2 Jo d o

Here v is the device width, L the device length, q the electron charge, f the frequency,

Eq(x) the d.c. field strength at x, u<j(x) the drift velocity at x^ and the drain current

at the drain voltage V^. Furthermore [NT(Ef ,x)]eff is the effective trap density at the

Fermi level at x, e is a tunneling parameter and a(Eo,x) is Hooge's parameter at x at the

field strength EqCx).

The characteristic I(j(V(j) increased monotonically with increasing V(j, reaching its maxi- mum value at saturation, as expected. Since for low drain bias V(j the drain current 1^ and

the drift velocity (x) are both proportional to Y^, whereas [Nt( Ef ,x ) ] gff and a(Eo,x) are now constants, we see that Igq should be proportional to V(j for low values of Vd, in agreement with our data.

The effective mobility y in GaAs may be written

p = 3y^ + (1 - B)y2 (5)

where y^ and yg are the mobilities in the lower and the upper valleys, respectively, and

P(Eq) is the relative occupany of the lower valley. Hence the average drift velocity is

u, = yE = 3y-E + (l - e)y^E (6) d o 1 o 2 o

Therefore increases linearly with Eq for low Eq, passes through a maximum and then

decreases when intervalley transfer of carriers becomes important; finally it increases

again when mDst of the carriers are in the upper valley. We see from Fig. 2 that

[Niii(E(i,x)] eff decreases with increasing Vg - Vp, so that [Nt Ef,x)]eff increases with

increasing V(j at a given x and with increasing x at a given V^. Putting it eill into Eq. (3) we see that the nvimber fluctuation model can qualitatively explain the data.

We now turn to the mobility fluctuation model. We have for fluctuations 6y^ and Sy^ in

y^ and y^, respectively

6y = 06y^ + (l - 0) (7)

from which follows

Sy(f)/y^ = 3^Sy^(f)/y2 + (1 - Q)^SM^(f) /v^ (8)

3 2 According to Bosman et al the field dependence of Sy(f)/y in silicon may be written

Sy(f)/y^ = a(0,x)/{fN[l + (y^E^/uJ^]} (9)

where a(0,x) is the low-field value of Hooge's parameter a, yQ he low-field mobility and

Ug the velocity of sound. Hence in GaAs one would expect

SyJf)/y^ = a.(0,x)/|6fN[l + (y ' E /u f] } i i 1 loos

237

(9a)
SPg^f)/^? = a^(0,x)/{(l - e)fN [l + (p^^E^/uJ ]} (9b)

so that

e(p /u)^ (i-e)(p /u)^

a(E ,x) = a (0,x)

^ + a (0,x)^ , . ° 1 1 + (U E /u )2 2 1 + (u E /u )2

lo o s 2o o s

One would thus expect a to be constant for low values of Eq, followed by a decrease when

UIq Eq/us becomes comparable to unity, followed by a plateau when the second term becomes

significant, and finally by another decrease when W2oEo/us becomes comparable to unity. It thus seems that the mobility fluctuation model can also qualitatively explain the data.

While both models seem to be able to qualitatively explain the data, the mobility fluctuation model is here more likely, since the field dependence of a(EQ,x) occurs at much

lower fields than where the dependence of U(i(Eo,x) upon Eq ceases to be linear. Hence the

field dependence of a(EQ,x) should predominate.

ACKNOWLEDGEMENT

This work was performed under ARO and NSF contracts. Professor Suh is now back at Hong-Ik University, Seoul, Korea, and Dr. Jindal is at Bell Laboratories, Murray Hill, N.J.

REFERENCES

[l] C.H. Suh and A. van der Ziel, Appl. Phys. Lett, 37 (l98o) 565.

[2I H.S. Park, A. van der Ziel, R. J. J. Zijlstra and S.T. Liu, J. Applied Physics, in the

press.

[3] J. Bosman, R.J.J. Zijlstra and A. van Rheenen, Phys. Lett. 8OA (1980) 57.

238

1/f NOISE OF HOT CARRIERS

T.G.M. Kleinpenning

Eindhoven University of Technology

Eindhoven, Netherlands

INTRODUCTION

On the basis of experimental f actSy the 1/f noise in the conductance G of homogeneous

semiconductors and metals can be described by Hooge's empirical relation ^l]

Sg = aG^/(fN) (1)

. . -3

Here, a is an empirical constant of the order of 10 , f the frequency, and N the number of

free carriers. Recent investigations have proved that 1/f noise is due to mobility

fluctuations [2]. Relation (1) holds for semiconductors in the case of low electric fields,

where mobility is field independent. This paper presents the results of 1/f noise in

semiconductors at high electric-field strengths where the mobility is field dependent.

CALCULATIONS (PLANAR GEOMETRY)

Consider an n-type semiconductor with ohmic contacts. The contact spacing is L and

the cross-section is A. The current density J(t) at time t is

J(t) = qny(x,t)E(x,t) = qnp[E(x,t) ,y^(x,t)]E(x,t) (2)

where q is the elementary charge, E(x,t) the electric-field strength at x(0 $ x ^ L)

.

The chordal mobility y = J/ (qnE) is a function of the low-field mobility y and E. At low fields we have y = y , and at high fields yE = v , the saturation velocity [3,4].

At high fields two scattering mechanisms are simultaneously active: the low-field

scattering and the scattering by the optical modes of the lattice vibrations, which are beginning to occur at high fields, where the electron energy is comparable with the optical

phonon energy U . As the energy of an electron approaches Up its velocity is likely to be

reduced by exciting an optical phonon. In order to calculate the 1/f voltage noise we assume that y^ fluctuates according to eq (1) and that the optical -phonon excitation

process shows no 1/f noise. At constant current we have

L

HT <dJ/dy > <f>AE(x,t) . <f^>Ay^(x.t) = 0 ; AV(t) = ^^jj^

f Ay^(x,t)dx (3)

With the help of the cross-correlation spectral density in the mobility [2,5]

2

= fA^^(--') -

and N = nAL, the 1/f voltage noise is found to be

<dJ/dy >^ ay^L^ <y dJ/dy >^ „2

<dJ/dE>2 fN

<EdJ/dE>2

240
To evaluate eq (5) we have to know the relation between p, \i and E. For n-silicon

this relation can be approximated by ['^jSj

y(E) = y /(1 + y E/v^ .) (6)

.o o sat

2 Using eqs (2,6) we find S^(f) = aV /fN. The 1/f current noise is found to be

S^(f) = S^(f) Idv) fN yidvy fN V y(E)/ fN \^l+y^E/v^^^ ) ^

^

where y,(E) is the differential mobility d(y(E)E)/dE.

-Q If we assumed that 1/f fluctuations were caused by number fluctuations (McWhorter

model) , then the result wouId be [5] S^(f) = al /fN.

From flicker noise measurements performed on n'''nn''' 78K, Bosman et al. [6,73 found experimentallyperformed on n'''nn''' and p^irp^ silicon planar devices at

Sj/I^ = I's^^g^/y^E)] = a(E)/fN ; a(E) = a (0)/[l + (y^E/v^) ^] (8)

s • Since experiments have to be performed at high electric fields. Joule heating creates

where v is the sound velocity

„ . s . , Since experiments have to serious problems. Therefore, we investigated samples with hemispherical geometry.

CALCULATIONS AND EXPERIMENTS (HEMISPHERICAL GEOMETRY)

Consider a sample, where the radius of the point-contact a is small with respect to

the other dimensions. The current through a hemispherical surface with radius r is

I(t) = qny[E(r,t),y (r,t)]E(r,t)2Trr^ - (9)

With the help of eq (6) we find

i>0O V = E(r)dr =

a-*

°(p^l/2Tr)dr

r2-l/(2.qnv^^^)

'^o''" z , /z+A

(10)

where Pq is the low-field resistivity, z = /Ij^/l', and I = 2TTa q^^'^g^j. the maximum current.

At constant current, we find for low-field mobility fluctuations Ay-

<§>AE(r,t) .<f->Ay,(r.t) = 0 ; S^(f) = |f

[^''

^2

dE

S^ (r,r')drdr' (11)

The source term S (r,r') can be obtained from (4) putting A = 2i:r and x = r, so we find

^o az

2iTfna"' 1+ ^

z 2 "I^^

2(z -1) (5f) (12)

If we ascribed the 1/f noise to free-carrier density fluctuations, we would find [5]

,2 az

5 r

2Trfna'

z _ z _ J 2 3^ 2 2 2 32

6(z -1) 24(z -1) 16(z -1) (13)

Assuming that eq (8) is correct, we have

241
41 - 0 . <g>4E(r,t) *<fi>A„(r,t) .<^^sat->AE(r,t) + <^>Ay(r,t)

Thus

S^(f) = E^(r) 1 + ^sat -I

S^(r,r')

drdr' =

2iTfn

E (r) 1+y E(r)/v_^

I o sat

E (r)/v^

J ~ . 2 by putting A = 2iTr , x = r,

dr

(14)

(15)

where the source term S (r,r') is obtained from (4) Po =2l^(E) =

y, and a = a(E) given b^ (8). If Bosman at al. [6,7] had found a(E) = a/(i+y e/v ) instead of eq (8), then eq (15) wou^d have led to the same result as eq (11).

At low fields both V/I and Sy/I are current independent. At higher fields V/l

increases with increasing current and S^/I^ also becomes current dependent. Figure 1 shows

the relation between deviations from Ohm's law and deviations from the normal current

dependence of the noise ('\'l2). The deviations are given in normalized quantities, R and

(S^/I^)^ representing the low-field values. The upper curve represents the relation

assuming N-f luctuations [eq (13)], the solid line that for yQ-f luctuations [eq (12)], the

broken line being based on Bosman's result [eq (15)]. The circles are experimental results

obtained from n-Si samples with a = 15 ym at 77K. For more details see [5].

The conclusion is that 1/f noise of hot carriers must be interpreted in terms of lowfield mobility fluctuations. Ay , characterized by the empirical relations (1) and (4).

MPS TRANSISTOR

Let us consider an n-channel MOST biased at a drain-source voltage V and an effective

gate voltage Vq = ^q~^T> where Vrj, is the threshold voltage and V << V*. The distance sourcedrain is L. The direction source-drain is the x-axis, and the y-axis is perpendicular to the surface. Let us assume that the electric field in the y-direction Ey is approximately

independent of y, at least in that part of the channel where the free electrons are located.

The source-drain current is then I = qNjM(E)Ejj, with N] the free electron density per unit

length. The open-circuit 1/f voltage fluctuations due to Ay are given by

AI = 0 = <dI/dE >AE (x,t) + <dl/dy >Ay (x,t) X X ' o o (16)

Using eqs (4,6) the 1/f voltage noise is found to be

S^(f) =

2 2 2 <dl/dy > ay L

o o <dI/dE > X

2

• fN

1 + o_y_

2 2 V ,(E +E ^) sat X y

-2 gV

* fN

At low drain-source voltages where E^ is small, eq (17) can be approximated by

'V^^^ "(l+y e'/v J fN ' (I ) \ o y sat/ \ o/

^2 2 fN

(17)

(IB)

The total number of free electrons in the channel N is approximately given by [8]

N = C^(Vc-V^)/q = C^V^/q (19)

with C the oxide capacitance. To find the relation between S^ and V^, we have to know

the reiation between E^ and Vq. This relation can be approximated by [9]

E = E (1+0V*)^

y o G

(20)

242
"7—

r

Fig. 1 Relation between deviations from the

current-dependence of the 1/f noise (normally '\' l2) at high electric-field strengths and

deviations from Ohm's law. based on eq

(13); based on eq (12); based on eq

(15); o experimental results. The subscript o denotes low-field values.

Fig. 2 Experimental data for R, Sy and y

(eq (21)) versus Vq of the M 100-type MOST

[8] . Calculated results for the same quantities are given by solid, broken and dotted

lines, respectively.

with and 0 empirically fitted parameters and 0.5 ^ k g 1. Since S^ y^/N, the channel resistance R 'v- l/(yN), and N V*, the quantity y defined by

Y 5 f(S /V^)rV^ = aqL^/(y ^) (21) V u O O

is independent of y, N, Vq. In Fig. 2 are plotted the experimental results of the M lOO^^

type MOST in Ref. [8]. The dotted line gives the calculated quantity y using a = 2 x 10 The same figure shows Sy and R versus Vq. The solid lines give the high and low electricfield limits, the broken lines are the best fits for k = 0.5.

From the above we conclude that the 1/f noise in MOSTs can be interpreted in terms of

yg-f luctuations if hot carrier effects are taken into account. Furthermore, in our model

we have assumed that the low-field mobility is determined by the bulk material, so inelastic

surface scattering effects are neglected.

REFERENCES

1 F.N. Hooge, Physica 83B, 14 (1976).

2 T.G.M. Kleinpenning, in Proc. Second International Symposium on 1/f Noise, Univ. of

Florida, Gainesville, Florida (1980), pp. 1-17.

3 R.A. Smith, Semiconductors, Cambridge Univ. Press, Cambridge (1980).

4 J.C. Jacoboni and L. Reggiani, Advances in Physics 28, 493 (1979).

5 T.G.M. Kleinpenning, Physica 103B , 340 (1981).

6 G. Bosman, R.J.J. Zijlstra and A. van Rheenen, Phys. Letters A 78., 385 (1980).

7 G. Bosman, R.J.J. Zijlstra and A. van Rheenen, Phys. Letters A 80, 57 (1980).

8 L.K.J. Vandamme, Solid-State Electron. 23, 317 (1980).

9 T.G.M. Kleinpenning and L.K.J. Vandamme, J. Appl. Phys. in press.

243
BIAS-TEMPERATURE TREATMENT, SURFACE STATE DENSITY^

AND 1/f NOISE IN MOSTs

L.K.J. Vandanme and L.S.H. Dik,

Eindhoven University of Technology

Eindhoven, Netherlands

INTRODUCTION v

There was a generally accepted relation between the surface state density Ngg and the

equivalent input noise S^^^[l,2] in MOSTs. The empirical relationship is given by

Ko. = >o'"n^ [v^s] (1)

vgs ss

at 1 kHz, 1 mA drain current and for an unspecified channel area. The proportionality

S g « N^g seems to illustrate that 1/f noise in a MOST is a surface effect. This is due to fluctuations in the occupation of interface traps and thereby in the number of free charge

carriers.

There is another school of thought who believe that 1/f noise is due to mobility

fluctuations [3,5] . The 1/f noise in a MOST can also be interpreted in terms of mobility

fluctuations [6,7,8]. Here we report on our investigations of the effect of bias "tempera"

ture treatment on the interface state density and the equivalent input 1/f noise. This

enables us to discriminate between mobility fluctuations and concentration fluctuation

models.

EXPERIMENTAL PROCEDURE

The surface state density and the 1/f noise were measured prior to and after bias"

temperature heat treatment. The interface state density was measured on the same MOSTs on which the 1/f noise was measured. From the shape of the 1^ versus curve below threshold,

Ngg near midgap can be calculated [9] . In the subthreshold the I^ - relation at a constant

gate voltage and temperature is given by [9]

^d ''dmax ^l-e-^1^d/^T^ (2)

C +C,(V*)

_ ^ ox d g

^ C +C^(V»)+N q ox d g ss

where q is the elementary charge, k the Boltzmann constant, and T the temperature in Kelvin,

n is an ideality factor given by the oxide capacitance C^^, the depletion capacitance

Cjj(V*) and the interface state capacitance per unit area qNgg. The MOST is biased below

threshold when a typical gate voltage V* is used so that at a drain source voltage of 50 mV

the drain current is 10"^ A. For this typical gate voltage V| the series connection of the

oxide capacitance and depletion capacitance is measured by a threepoint procedure [10].

N^^ is then given by

N = -(C +C^(V*) ) ( - " 1 I (4) ss qV ox d^ ^

/ V'^ /

where t\ is obtained from the 1^ versus curve measured at Vg = V*. Owing to channel lenght

reduction at large V^, problems can occur in the determination of ^^^g^^' Here we propose a

244
quick determination of Ngg by measuring the drain current at two chosen voltages: I^j at V^jj and 1^2 ^d2 ~ ^ ^dl * nq/kT we find

-ln(I,„/I, -1)

dl dl _ _r|£

dl kT (5)

We choose V^, =30 mV and V^„ = 60 mV. At 300 K we find

dl dl

n = -0.87 ln(I^2/^dr'^ (6)

From eqs. (6) and (4) and observed values of ^dl/^dl' ^ox '^d^^G^ ^ss ^^^^^

calculated. These results are in good agreement with computer fitting results. To avoid

measuring errors due to the leakage current in the p-n junction between substrate and drain,

the current 1^ was measured in series with the source, the substrate being grounded.

All heat treatments were carried out on encapsulated MOSTs. The bias -temperature

conditions were | hour at a temperature of about 600 K with Vg^ = Vg^ = Vgd half the breakdown voltage, as specified by the manufacturur at 300 K. A positive bias -temperature

treatment means that the gate voltage is positive with respect to the substrate. The bias

voltages correspond to 15 V to 20 V per 1000 S gate oxide thickness. After bias -temperature treatment the instabilities caused by mobile ions, were cured out during 5 hour at room temperature by biasing the MOST in saturation at a drain current of | ^dmax* '^^^ equivalent input noise voltage at the gate was measured prior to and after-bias temperature treatment using a Hewlett and Packard type 4470 A transistor noise analyzer.

RESULTS

The experimentally observed trends are: (i) bias -temperature treatment leads

small increases in Ngg for commercial MOSTs and to larger increases in Ngg for a laboratory

made sample, (ii) bias -temperature heat treatment with a polarity leading to strong inversion

results in an increase in Ngg and decrease in S-^^gg . This trend contradicts the rule

S^gg a Ngg as proposed in surface state models [1,2], (iii) bias-temperature treatment

resulting in accumulation layers lead to increases in both Ngg and S^gg of at least a factor

two, (iv) after a bias-temperature heat treatment, the dependence of S^gg on Vg is reduced.

A summary is presented in table 1 below.

Table 1. Change of Surface State Density and l/f Noise in MOSTs after

a Bias-Temperature Treatment.

bias-temperature treatment

polarity

N

ss

Svgs

n-channel + >0 <0

>0 >0

p-channel + >0 >0

>0 <0

Ten n-channel depletion MOSTs have been investigated (M 100 and 3 N 128), 29 p-channel MOSTs

(M 108 and university made samples).

The survey of our experimental results on the dependence of l/f noise in MOSTs on Ngg

before a bias-temperature treatment is presented in the next figure. In this survey the

observed noise is expressed in a relative way in order to compare the results for different

channel areas and bias conditions. When the noise is measured in the ohmic region the

results are denoted by 0 and the observed relative noise is normalized as (S^/V^) (f C V ) /q

where C is the total gate oxide capacitance. When the equivalent input noise S^gg it

measure! under saturation conditions, then the noise results are denoted by s and normalized

as SYggX(2fCg)/ (q.Vg) . The height of the bars have nothing to do with inaccuracies in the

measurements but with the fact that the normalized noise decreases with increasing gate

245
voltage. This trend is in agreement with the reduction in a due to a reduction in the

mobility with increasing gate voltage.

The lowest and highest values of the gate voltages Vg used are given along the bars in

the figure. The numbers correspond to n-channels and the letters to p-channels.

The broken line follows S N . The experiments contradict this rule.

vgs ss

10 .1

10 .3

10"

10'

10

.6

T r

0.5V

G.17V |0.16 05V 0.2V

s c

5V

2V

/

/OSV

1.7V

0.3V /

,er.3V /V

~1

7\ / /

/

/

0.1V

22V

1.5V

4V

1 J L

0.5V

0

8.55V

1.65V

0 Nss(cm-^eV-b

^5.6V

2

J L. J L

10

10

10 10

12 10

Fig. 1 . Observed noise and

surface state density from

measurements on MOSTs.

The code used in the figure is as follows.

p-channels n-channels

b. 3N155 Texas Instruments

c. 3N163 Siliconix

d. 3N174 Texas Instruments

e. L IV, 1 University made

f. M 108 Siliconix

1 . LMOA 1 before biastemperature treatment

2. LMOA 1 after biastemperature treatment

3. 3N128

4. 40467A

5. M 100

6. 3N142

university made

university made

R.C.A.

R.C.A.

Siliconix

R.C.A.

Experimental results which can be explained by mobility fluctuation show normalized noise

between 2 x 10 and 5 x 10~^. There are MOSTs for which the experimental results agree

with both the predictions by the trapping model and by the mobility fluctuation model.

There are also MOSTs for which the noise is in agreement neither with the Ay nor the An

model. The final conclusion is that a proportionality between S^gg and Ngg does not exist.

Even after changing Ngg and S.^gg on the same sample by a bias-heat treatment the proportionality S «e N is not observed.

vgs ss %

246
REFERENCES

1 F.M. Klaassen, IEEE Trans. Electr. Dev. ED-18 , 887 (1971).

2 G. Broux, R. van Overstraeten, G. Declerck, Electron. Lett. n_, 97 (1975).

3 F.N. Hooge, Physica 83B, 14 (1976).

4 T.G.M. Kleinpenning, Physica 77, 78 (1974).

5 F.N. Hooge, L.K.J. Vandannne, Phys. Lett. 66A, 315 (1978).

6 L.K.J. Vandannne, Solid-St. Electron 23, 317 (1980).

7 L.K.J. Vandamme, H.M.M. de Werd, Solid-St. Electron 23, 325 (1980).

8 T.G.M. Kleinpenning and L.K.J. Vandamme, J. Appl. Phys. in Press.

9 R.J. van Overstraeten, G.J. Declerck, P. A. Muls, lEE Trans. Elect. Dev. ED-22, 282 (1975)

10 I.M. Bateman, J. A. Magowan, Electron. Lett. 6_, 669 (1970).

247
1/f NOISE IN SCHOTTKY BARRIER DIODES

O.Sikula, P.VaSina, B.Koktavy and Z.Chobola

Department of Physics

Technical University of Brno, Barvicova 85

Czechoslovakia

INTRODUCTION

The dominating component of the low- f requency noise of the Schottky

barrier diodes is that of the 1/f type. The model of the 1/f noise in Schottky diodes elaborated by S.T.Hsu tl»2] assumes that the current modulation

is due to the barrier height modulation which in turn is due to the fluctuation in the trap or g- r center occupation. Furthermore, it is necessary to

assume the tunnelling mechanism.

T.G.M.Kleinpenning [3] elaborated another model of the noise generation,

He assumes that the 1/f noise is due to the mobility fluctuation and the va- lue of the 1/f noise spectral density S^ decreases rapidly if the ideality

factor tends to unity. As the effective carrier concentration in the depletion region is inversely proportional to the current, the S\ vs. I plot in

[3] follows the linear law.

EXPERIMENT

The papers hitherto published studied either the open circuit or the

short circuit Schottky diode noise. In the former case the voltage spectral

density S„ is measured, while in the latter one measures the current spectral density S^ . In the both cases the measuring apparatus introduces an error which can be allowed for only if the equivalent circuit of the measured diode is known.

'A/ CZh

We assume the equivalent diode circuit drawn in Fig.l. We modified the

existing methods by measuring the voltage fluctuation spectral density S^l

across the load resistor R^within

a large range of the diode voltage

or current. The spectral density S„l

reaches the maximum value from which

we get information about the noise

source resistance. At low diode voltages, when Rji^ Rl , the value of the

voltage fluctuation spectral density

SuL is proportional to the current

spectral density Sj due to that the

circuit is shorted and it holds SyL=

I

Fig.l The equivalent diode circuit

S; R ?. When the diode voltage is so high that R^i*^ j then we measure the

voltage fluctuation spectral density

in the open circuit and S^l ^ Su

.

The current spectral density S," vs. the forward current Ip- plot for GaAs

Schottky diode is in Fig. 2. The position of the S,' vs. Up or Ip-plot depends

on the value of the load resistance R^. Decreasing the value of R^ makes the

maximum shift towards the higher currents or voltages across the diodes. The

asymptote of the measured Sj[_ curves makes the short circuit spectral density S, and it holds

5, = S,o , exp (/32Up)

248

(1)
From our experiments on GaAs Schottky diodes we found out that /9j < /?2 < where /3,= e/nkT ; n-ideality factor, e-elementary charge. Eq,(l) is in accordance with the resultg of other authors. Kleinpenning [3]gives for the spectral density of the current fluctuation S, , where l<y<2 ; as I = IqSxp ('O^Up.), we get S, exp (jr/3,Uf-).

Fig. 2. The current spectral density Fig .3 . The voltage spectral density

Sj and SjL vs. If. S^l vs. Up.

The spectral density vs . the forward diode voltage Up plot is in Fig. 3.

Increasing the load resistor results in shifting the maximum of the SyL cur- ves towards the lower voltages. The asymptote of the measured curves S^g can

be expressed in the form

Sua = S^o^xp (-/SjUp) (2)

where for our set of GaAs Schottky diodes it holds 0 </33</3i. If the series

diode resistance Rg is negligible with respect to the load resistance , the spectral density S^g follows the equation

S.a = Si . (3)

Taking into account the dynamic diode resistance R^ = 1/[/3,Iq exp (/3,Up ) ] we

get using (1)

Sua= exp (^Up)/[/3jloexp(2/3,Up)] (4)

and S^a= S^^ exp[(/32- 2/J,)Up] (5)

2 2 where S^JQ = Siq/^^Iq . We find that for a Schottky diode where the current

fluctuation generator is conected in parallel to the diode dynamic resistance and the load resistance it holds /J^ = 2/3^-/32 . 249
Only a part of our set of Schottky diodes exhibits the noise characteristics in accordance with the suggested model. The current generator of

the fluctuation is connected across the parallel combination of the dynamic

diode resistance and the load resistance. The equivalent circuit is in Fig,

1, We suppose that the source of noise - in accordance with [3]- is made by

the fluctuations in the space charge region of the semiconductor.

to

~o^i 5p 5j '^'f 5^*^ 5^ '0/7

We observed that another phenomenon may be the source of the noise,

namely the inhomogenit ies in the me- tal-semiconductor structure. In this

group of the diodes the spectral density of the voltage fluctuation is

not a monotounous function of the diode current. Fig. 4 shows the voltage

fluctuation spectral density Sy^ vs

.

the diode voltage plot for two values

of the load resistance Rl(Rl= 10 kll,

Rl= 200 kfl).The spectral density S^l

has a local maximum, the absolute value of which depends on the match of

the noise generator and the load. Increasing the load resistance makes the

extremum of the spectral density shift

to lower values of the diode voltage.

The position of local extremum of the

spectral density is also slightly influenced by the value of load reslstan

ce. This fact is also due to the match

of the noise generator and the load.

When taking the influence of the

measuring circuit into account, we get

the following extremum voltages Ujfor

some GaAs Schottky diodes (table 1.).

The predominant noise component is

the 1/f noise for some samples even

at a frequency of 1 kHz. Only in the

proximity of the spectral density ex- tremum we observe the burst noise.

The spectral density of the fluctuation Su|_reaches the minimum value

at a voltage Up«0,6 M. At this voltage the noise spectral density of the

Schottky barrier equals the spectral

density of the b-type (see the b-component in Figs. 3 and 4). In accordance with Kleinpenning [3]we attribute

the b-component to the Schottky diode series resistance noise.

Fig. 4. The voltage spectral density

Syj_ vs.U .

Table 1.

Diode Nq U, /V Ug/V U3/V

SO- 640 0 , 14 0,25 0,51

SD-943 0 , 13 0,27 0,50

SD-743 0 , 13 0 ,30 0,52

REFERENCES

fl] S. T. Hsu, IEEE ED- 17 , 496 (1970)

[2j S. T. Hsu, IEEE ED- 18 , 882 (1971)

[3J T. G. M. Kleinpenning, Sol. State Electron., 22, 121 (1979)

250
IV. APPLICATIONS AND

MEASUREMENT

TECHNIQUES

251

JOSEPHSON JUNCTIONS, PLASMA PHYSICS, AND ELECTRON TEMPERATURES

Bruce T. Ulrich

Max-Planck-Institute for Plasma Physics

EURATOM-Association

D-8046 Garching bei Mtlnchen

Federal Republic of Germany

GOALS OF FUSION PLASMA RESEARCH

Energy is set free not only by fission of heavy nuclei, but also when the lightest

nuclei fuse to form heavier ones. Such fusion processes are the energy source of our sun

and other stars. The goal of fusion research in many laboratories all over the world is to make this energy source available for the future energy needs of mankind. The goal is an ambitious one. To build a fusion reactor would rank as one of the great human engineering

achievements of this century, or of the next. Conditions in the center of the sun in many

ways are less extreme than in a fusion reactor. The sun burns hydrogen at about 15 million

degrees, but the temperature in a fusion reactor which burns a deuterium-tritium fuel mixture should be about 100 million degrees. The sun confines its hot ionized gas (plasma)

with its gravitational field. A fusion device confines its plasma with a magnetic field,

or for a brief instant through the inertia of the plasma during implosion. The sun generates 300 Watts per cubic meter in its core; a 1 GW (1 GW = 1000 MW) tokamak fusion reactor

4 must generate 10 times more power per cubic meter. The temperature in the sun decreases

5 from its peak over a distance scale of 10 m, but the temperature in a magnetic confinement

fusion reactor must decrease over a scale of 1 m, so the temperature gradients in the fusion

reactor are 10^ times times higher than in the sun.

A nearly limitless supply of energy on earth could be available from deuterium in water and from tritium with thermonuclear fusion via the reaction

D + T —^ He^ + n + (17.6 MeV).

Tritium does not occur naturally on earth, but can be formed by the neutrons in the fusion

reactor via a reaction with lithium. The term "thermonuclear fusion" means that the temperature of the DT fuel mixture should be high enough that the thermal energy of the D and T is high enough for this reaction to take place. In practice, a temperature of about 10 keV

(10 K) would be required. The goal of fusion energy research is to create this temperature

at high enough density for a time long enough that a useable amount of energy is released by

the DT reaction. The problem of fusion research, then, is to confine the DT fuel while

heating it to ignition, in a reactor which is small enough to be built by humans. The size

of fusion device experiments has changed from table top experiments in a university laboratory to international collaborations. The time scale has expanded from one year to twenty.

Extrapolation from present fusion devices to a full scale reactor stretches the limits of

present technology.

The problem of plasma confinement is so difficult that many methods are being tried in

Ij parallel. The two basic approaches are called magnetic confinement fusion (MCF) and

inertial confinement fusion. In MCF, the electrons and ions in a plasma spiral around a line of magnetic field and can be held away from a wall. Two difficulties arise with this

approach. The first arises because a charged particle experiences no force when it moves

parallel to a magnetic line. Thus, the lines of magnetic field confining the plasma should

253
not cross a material surface. A doughnut shaped (toroidal) magnetic field satisfies this

criterion. The second difficulty is that a simple toroidal magnetic field alone does not

confine the plasma in equilibrium. To retain the plasma, the magnetic field should twist as

it goes around the torus. One way to introduce this twist is with an additional magnetic

field produced by a current circulating around the torus in the plasma itself. Such a device Is called a tokamak, a word derived from the Russian word "tok" for current. The

current in the toroidal shaped plasma is induced by a transformer, in which the plasma serves as a single turn secondary winding. The circulating current simultaneously produces

the stabilizing twisted field, and heats the plasma ohmically through the resistance of the

plasma. The tokamak suffers from the important drawback that because of the transformer, it

must operate in pulses. In addition, because the resistance of the plasma drops as the

temperature increases, the circulating current cannot heat the plasma enough to reach fusion

temperatures, so additional heating is needed. The main technique is to inject beams of

neutral atoms which can cross the magnetic field lines and deposit their energy in the

plasma, heating it to temperatures high enough for fusion.

The early success of tokamaks in confining a plasma for longer times and in reaching

higher temperatures than other techniques has led to a much greater investment in this

fusion device. Over 100 tokamaks have been built, and the largest fusion devices presently

built, and under construction are tokamaks. Tokamaks have come closer to achieving conditions for a MCF reactor than other approaches, and parameters can be extrapolated more confidently for tokamaks than for other MCF devices to the domain of reactor operation. Other

approaches, called "alternative lines" are pursued because of potential advantages such as steady state, rather than pulsed operation, and in some cases, greater simplicity.

Because parameters generally have improved as the size of the MCF device has increased, it

is difficult to compare results of alternative lines directly with the results for presentday tokamaks, because the alternative lines are generally at a much earlier stage of

development.

In addition to confining the plasma in a stable way, the MCF device also must retain

the thermal energy of the plasma to maintain the plasma temperature at a high value. A plasma can cool by radiation or by conduction. Atoms of heavy impurities such as titanium,

tungsten, and iron are not completely ionized at the temperature of several keV in the

center of a plasma, and can radiate energy directly from the center as X-rays. Light impurities such as carbon and oxygen radiate energy from the cooler edge regions of the plasma.

Radiation from impurities has such an important influence on the conditions of the plasma

that the ability to control the parameters of a plasma is to a large degree the ability to control impurities. One technique to control impurities is to divert the magnetic field

lines near the edge of the plasma to ^a cooler region. Impurities that enter from the walls

are diverted by the lines to a target where they are caught before they can enter the center

of the plasma. Such divertors require an expensive increase in the total volume of the magnetic field region. The divertors appear to extract heavy impurities, but not light ones. Other techniques, such as discharge cleaning, and titanium getter pumping also are effective

to reduce light impurities. It appears that in present and future experiments in thermonuclear fusion, effective methods can be developed to control impurities without a divertor,

but that a divertor may be required for a fusion reactor.

A plasma cools also by conducting heat. Here lies the great unsolved problem of plasma

physics. The electrons in a plasma conduct heat 1000 times faster than they would if the

magnetic field lines remained rigid and the electrons collided only in pairs. The ion heat

conductivity seems in fair agreement with theoretical predictions. Although several mechanisms have been proposed for the high electron heat conductivity and for other anomalously

high transport processes, the underlying physics is not well enough understood to calculate

important parameters of a tokamak, such as the energy confinement time. In default of

physical understanding, the measured confinement time is plotted versus other experimental

parameters such as density, temperature, major and minor radius of different tokamak experiments in an effort to find empirical interpolation schemes. These interpolation schemes are called "scaling laws", but they do not represent physical laws in the usual meaning of the

word. They are then used to extrapolate from present experiments to future ones. Different

interpolation schemes extrapolate differently. An important goal is "ignition", at which

the thermonuclear reaction releases enough energy to sustain itself. An example of the

degree of uncertainty is indicated by the amount of additional heating required to reach

ignition in the Joint European Torus now under construction. Estimates for the amount of

required additional heating range from 15 MW to 100 MW, depending on the scaling law chosen.

254
Clearly there is a need for further physical understanding, but the pressure to make engineering choices for the next generation of devices dictates that empirical extrapolation be

used in default of physical understanding.

Significant progress in alternative approaches to MCF occurred in the first true operation of a stellarator in a current-free mode. A reactor built from a stellarator could in principle operate in a steady state, rather than in a pulsed mode. A stellarator, like a tokamak, has a doughnut shaped magnetic field, but the twist in the magnetic field is obtained with additional external magnetic field coils, rather than with a circulating

current as in a tokamak. Until the recent results [1] at the Wendelstein (W VII-A)

Stellarator in West Germany, stellarators had achieved high temperatures and densities only

through use of a circulating current to heat the plasma, so, in fact, they operated like

tokamaks, although with a lower value of current circulating in the plasma. At W VII-A, the

plasma was heated with 1 MW of power from neutral injectors while in the current free mode.

The plasma remained stable in the current free mode, and the energy confinement time increas

ed. This result was surprising, and very encouraging, because the magnetic field of W VII-A

twists uniformly without shear, and generally, it was thought by theoreticians that shear in the magnetic field was necessary to stabilize the plasma. An additional result from W VII-A

and from CLEO (England) was that the electron heat conductivity decreased in the experiments

as T"*^"^, where T is the electron temperature of the plasma. This result indicates that in operation at higher temperatures in future stellarator experiments, the heat losses via

electron thermal conductivity should decrease. Further important progress in the understand

ing of fusion plasmas can be made through better measurements of the electron temperature

distributions in MCF experiments.

ELECTRON TEMPERATURES

The temperature of the electrons in a fusion plasma can be measured only indirectly.

Important techniques include Thomson scattering, soft X-ray emission, and electron cyclotron

emission measurements. In the first, light from a laser is scattered from the plasma, and

the Doppler broadening of the scattered light by the thermal motion of the electrons is

measured to determine the electron temperature. Such lasers generally provide temperature

information only once during a plasma discharge, and do not give the time evolution of the

temperature. In the second, the ratio of thermal X-ray radiation at different photon

energies gives an effective electron temperature along the line of sight. Time resolution

is high, but spatial detail difficult to resolve without a tomograph-type inversion nliethod. The third method, electron cyclotron emission, offers the advantages of both high time and

high spatial resolution.

Electrons in a plasma spiral around magnetic field lines, and radiate electromagnetic

radiation at the electron cyclotron frequency

= ]- ^ B = 28 ,^ GHz ce Ztt m Tesla

and its harmonics. Such radiation is called "electron cyclotron emission". In toroidal

plasma machines, such as stellarators and tokamaks, typical plasma conditions are such that

the plasma is optically thick at the second harmonic 2f^^ so the electrons in the plasma

radiate like a black body at this frequency [2,3]. Thus, the intensity of the radiation is proportional to the electron temperature T^. In a toroidal plasma machine, the magnetic

field decreases in a kfiown way with the major radius from the inside to the outside of the

torus. For temperatures below about 1 keV, the Doppler broadening of the cyclotron emission

due to thermal motion is small, so the emission at a particular frequency corresponds to a particular location in the torus. This location can be determined because the magnetic

field fJrofile is known in advance. Thus, when electron cyclotron emission is observed from

the plasma in the plane of the torus, the intensity of the radiation as a function of frequency determines the electron temperature as a function of position. The width of the

peak in the frequency spectrum is determined by the fractional variation of the magnetic

field over the torus, and in W VII-A is 10 %. Under typical conditions, the peak is 20 GHz

wide, centered at 200 GHz. In other present and future plasma machines, the width of the

peak ranges up to 30 or 40 %. The entire peak needs to be scanned to determine the

255
temperature profile. In a heterodyne receiver based on conventional microwave techniques,

it is already difficult to scan the local oscillator over 10%, and nearly impossible over a wider range. In addition, it is important to make rapid observations of the temperature

profile during each plasma discharge to determine the temporal structure of plasma instabilities and oscillatory modes. In the W VII-A stellarator, as observed by X-ray

emission, the plasma electron temperature oscillates with a sawtooth waveform with a period

ranging from one to a few milliseconds under typical conditions. The X-ray observations

have not given the spatial profile of these temperature oscillations, whereas the observation of electron cyclotron emission can give this spatial dependence provided the

frequency profile is scanned in less than 1 ms, A typical plasma electron temperature is

500 eV = 6 X 10^ K, and experimental results for self-oscillating Josephson mixers have

been reported [4] with mixer noise temperatures of 600 K, so a sufficiently low receiver

noise temperature appears realistically obtainable, The main advantage of a Josephson

effect heterodyne receiver with internal local oscillator is the ease of changing the

frequency of observation rapidly. Such a receiver is under development in Garching. [5,6],

JOSEPHSON EFFECT

When two superconductors are weakly connected together, such as via a point conjact a few ym in size, or via quantum mechanical tunneling through an oxide layer about 20 A thick

between the two superconductors, a supercurrent 1^, where

Is = Ic sin i> (1)

can flow between the two superconductors, The characteristic current I is proportional to the strength of the coupling between the two superconductors, and typicSlly ranges from a few yA to a few mA, The variable i> is the difference between the phases of the quantum

wave functions in the two superconductors on opposite sides of the contact region. The

phase (() evolves in time according to time dependent quantum mechanics as

J_

2tt dt

2e

(2)

where V is the voltage difference across the contact. The other quantities are the fundamental constants, e, the charge of the electron, and h, Planck's constant. The ratio

2e/h = 0.483594 GHz/yV.

The phenomena which arise from (1) and (2) are collectively called the Josephson

effect, and the weakly connected superconductors which exhibit them are called a Josephson

junction [7]. For a constant bias voltage V, when (2) is integrated and substituted into

(1) it can be seen that the supercurrent I oscillates at frequency

(2e/h) V, (3)

and can serve as the local oscillator in a mixer for a heterodyne receiver. When a signal

voltage v cos 2ir ft, such as from electron cyclotron emission, adds to the constant bias

voltage V , the total voltage across the junction is

V + V cos 2tt f^t. (4)

The signal voltage produces a frequency modulation of the local oscillator frequency as can

be seen by substituting (4) into (2), with mixing currents at frequencies f^ + nf , where

n = ±1 , ±2, ±3, . , , , The component of interest for a heterodyne receiver is that at the

intermediate frequency fg ' ^s* combined, the current at the

intermediate frequency is ,

i,f.

= I^ Ji(2ev/hf5) cos 2Tr(fQ - f^) t (5)

where Jiii is the .... first Bessel function. ..... For ... small values of its argument, ^ , 2 ev /hf , this

Bessel function is linear in its argument, so the Josephson junction acts as a mixer with

output at the i, f, linear in the input signal voltage. The above discussion illustrates

the origin of mixing in a voltage biased Josephson junction. In practice, a Josephson

256
junction is more nearly current biased, and the analysis is somewhat more complex but the

physical mechanism of ;mixing and the results remain essentially the same [8],

To discuss the sensitivity of a Josephson effect mixer, and the dependence of the

sensitivity on frequency, we need to introduce the idea of the characteristic frequency f of a Josephson junction. When a voltage is applied across a Josephson junction, in addition

to the supercurrent I , a normal current also can flow. Within the resistively shunted

model [9], this current can be represented by V/R, so the total current through the

junction can be represented by

I = sin (}. + V/R, (6)

The normal resistance R of the junction is, for example, the resistance determined just

above the superconducting transition temperature. The junction model now contains an intrinsic characteristic frequency

2eRI

^c = -Ti^ (7)

which in typical junctions ranges from 100 to 500 GHz, and in especially good junctions can

be above 1 THz. The characteristic frequency divides the low frequency properties of a Josephson junction from the high frequency properties as follows. For small input signals,

it can be shown that the supercurrent component i responds to the signal voltage v as if

the junction were an inductive reactance. At f , the impedance for the current to flow as

a supercurrent is exactly R, so the current produced by the signal divides equally a supercurrent and as a normal current. Only the supercurrent component interacts nonlinearly to produce output at the intermediate frequency. At higher frequencies f » f more current

flows as a normal current, and the performance of the junction as a mixer fails off. The

low and high frequency limits of the mixer noise temperature (DSB, referred to the output)

are [10] r T . f « fc

^"^N^min ^ \8T Lf/f^)^, f » f^ (8)

where T is the temperature of the Josephson junction, typically 4.2 K.

For magnetic fields of most present and planned tokamaks and stellarators, at the

usual frequency of observation, 2 f , where f is the electron cyclotron frequency, the

Josephson junction would be oeprating in the l8w frequency limit, and mixer noise temperatures less than 10 K are predicted. These ultimate values have never been realized in

practice because of contradictory requirements on the resistance R of the junction. On one

hand, R should be close to the signal source resistance R which typically is R > 50 ohms.

On the other hand, the line width of the internal oscillation is directly proportional to R,

and is 0.5 GHz for each ohm of normal resistance at 4.2 K. It is this line width which determines the frequency resolution of the receiver. For this reason, the minimum noise

temperatures of the Josephson junction oscillator-mixer which have been obtained experimentally are about two orders of magnitude higher than the predicted minimum. A Josephson

junction oscillator-mixer has been operated over the frequency range of 40 to 260 GHz,

with the best 600 K at 80 GHz, and values of Tj^ ranging up to a factor of 3 higher over

the entire frequency range of 40 to 260 GHz [4]. For use as a receiver for observation of

electron cyclotron emission, a receiver noise temperature Tr of 6000 K ("NEP" = kTp = 10"^^ W/Hz) would be adequate. The method of operation is to sweep the

junction bias voltage V over a range corresponding the frequency range of interest. For example, a 20 GHz wide peak at 200 GHz would be covered by a sweep of amplitude 41 yV

centered at 414 yV. The speed at which the bias voltage, and thus the frequency can be

swept is not limited by the properties of the Josephson junction, and would be determined by the required signal to noise. For example, assuming a receiver noise temperature

of Tr = 6000 K, and a comparable source brightness temperature, a frequency resolution of

1 GHz, a 100 ys observation time for a single frequency element, the signal to noise ratio

would be about 150.

EXPERIMENTAL PROGRAM

A Josephson effect heterodyne receiver has been used in the frequency scanning mode

257
to observe electron cyclotron emission from W vn ^ A, A preliminary version of this re^

ceiver, not optimized for input coupling, or for coupling to the i,f, amplifier was used.

The purpose of the initial experiments was to gain information about the effects of the

electrical disturbances present near the plasma device and to determine whether it was possible to shield the junction from these disturbances. Oversize circular waveguide 2,5 cm

in diameter with a polarization rotator was used to transmit the radiation from one end

which served as a microv/ave horn outside the access port of W VI UA, to the junction after a taper transition to the 5 mm dia circular waveguide leading down the neck of the helium

dewar, The radiation was coupled to the junction at the bottom of a circular cone tapering

from 5 mm to 1 mm diameter. The junction consisted of a Nb wire which entered the side of

the cone where it was bent to contact the other superconducting electrode attached to the

center conductor of the coaxial line that carried the intermediate frequency signal from

the junction. The receiver operated as a double sideband receiver with an intermediate

frequency amplifier of bandwidth 10 MHz to 500 MHz. The line width of the internal

oscillation of the Josephson junction determined the observed resolution of 8.5 GHz CFWHM)

as measured with a klystron source at 156 GHz. In laboratory experiments, frequency

resolutions of 2 GHz have been obtained,

In the vicinity of the helium dewar, the magnetic field was of order 50 gauss, and the

junction was surrounded by a lead supershield, but without external mu metal shielding.

The bias leads from the metal dewar were carried to a shielded electronics enclosure inside

a flexible 10 GHz waveguide which served as shielding. Battery powered bias electronics

inside the enclosure were used to sweep the bias voltage across the junction with an offset

triangle wave of 30 ms period to scan the frequency of observation. The bias voltage is shown in the middle curve of Figure 1. The lower curve shows the plasma current as a function of time. During the rising part of the plasma current, the bias voltage is

strongly disturbed. It is not yet known whether the variations in the bias voltage sweep

during the plasma discharge occur across the junction, or whether they are picked up in the

amplifier which measures the bias voltage. The upper curve shows the output from the i.f.

amplifiers, The absolute frequency scale of the receiver was calibrated by observing the

signals from a 156 GHz klystron (turned off in Figure 1) to avoid uncertainties in the

exact bias voltage which could be caused by thermoelectric potentials in the leads to the

junction. The frequency of electron cyclotron emission at 2 f^-g was 169 GHz. The signals

observed occur at the expected bias voltage, A signal to noise about ten times higher than

that presently attained is required before the results of this technique can be used for

electron temperature measurements. Through improvements in the input signal coupling, in

the coupling to the i.f. amplifier, and in the noise temperature of the i.f, amplifier, it

is expected that this requirement improvement in signal to noise can be achieved.

Middle curve; Swept bias voltage V across Josephson junction.

Fig. 1 . Lower curve; Ring plasma current

Ip during discharge in W Vll-A stellarator,

Ol Upper curve: Power output P. ^ at the

,T. intermediate frequency from the Josephson

effect heterodyne receiver with electron

cyclotron emission profiles.

20-

Q. 0 100 200

t (ms)

258
DISCUSSION

The present measurements demonstrate that a Josephson junction heterodyne receiver can

be used in the vicinity of a plasma machine to observe electron cyclotron emission at 2 f Additional work is required to improve the signal to noise by about a factor of 10 and to shield the junction and electronics more completely from electrical disturbances. The

simplicity and ease in changing the frequency of observation suggest that such a Josephson

receiver can be useful in electron cyclotron emission measurements. It can be used in a- scanning mode to obtain spectra, or in a fixed frequency mode to obtain measurements with

high time resolution,

REFERENCES

[1] W VII-A Team, Procs, of the IAEA Eighth International Conference on Plasma Physics and

Controlled Nuclear Fusion Research, Brussels, July 1980,

[2J Engelmann, I,, Curatolo, M. , Nucl . Fusion 1_3, 497 0973),

[3] Meservey, E, B,, Schlesinger, S, P., Phys. of Fluids 8, 500 (1965),

[4] Avakjan, R, S., et al . , IEEE Trans. Magnetics MAG^ll . 838 (1975),

[5] B, T. Ulrich and M. Tutter, in Proc. of the Second International Conference on Superconducting Quantum Devices, Berlin (West), May 1980, Eds. H. D. Halbohm, et al .

,

Walter de Gruyter, Berlin, 1980.

[6j B, T, Ulrich and M. Tutter, in Proc. of the Joint Workshop on ECE and ECRH,

Oxford, July 1980, Culham Laboratory, Report No. CLM-ECR.

[7] See, for example, M, Tinkham, "Introduction to Superconductivity",

McGraw-Hill, New York 1975.

[8] K. K. Likharev and B. T. Ulrich, "Systems with Josephson Junctions", Moscow

University Press, Moscow 1978 (English edition to be published).

[9] L. G, Aslamasov and A. I. Larkin, JETP Letters 15^, 349 (1972),

[10] V, P, Zavaleev and K. K. Likharev, Radiotechni ka i Elektronika, 23, 1268 (1978).

259
SURVEY, APPLICATIONS, AND PROSPECTS

OF JOHNSON NOISE THERMOMETRY*

T. Vaughn Blalock

Electrical Engineering Department

University of Tennessee

Knoxville, Tennessee 37916

Robert L. Shepard

Instrumentation and Controls Division

Oak Ridge National Laboratory

Oak Ridge, Tennesee 37830

INTRODUCTION

Significant progress in the field of Johnson noise thermometry has occurred since the

1971 survey of Kamper [1]. This paper will review the foundation work of Johnson noise

thermometry, survey several basic methods of noise thermometry which use conventional

electronic signal-processors, and present some applications of noise thermometry in tem- perature scale metrology and process temperature instrumentation. The important methods

of cryogenic noise thermometry [1] which use quantum devices such as Josephson junctions

are not included in this survey.

THEORETICAL FOUNDATION FOR JOHNSON NOISE THERMOMETRY

Johnson noise thermometry is based on the early work of Johnson [2] and Nyquist [3].

The noise-voltage power density spectrum (S^) in positive frequency space appearing across

an unloaded resistor of value R ohms was first measured by Johnson [2], and a few months

later it was shown by Nyquist [3] to be given by

S^ = 4hfR/[exp (hf/kT) - 1] , (1)

2 where S^ has units of V /Hz, h is Planck's constant, k is Boltzmann's constant, f is the

frequency in Hz, and T is the absolute temperature in K. If the frequency is sufficiently

low that f < kT/h (kT/h = 6.25 GHz at T = 300 K) , then to a very good approximation

S^ = 4kTR . (2)

One circuit model for Eq. (2) is a noise voltage generator in series with a resistor of

value R ohms. An alternate circuit model is a resistor R in parallel with a noise current

generator of power density (A^/Hz) expressed by

* Research sponsored by the Division of Reactor Research and Technology, U.S. Department

of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

260
Equations (2) and (3) suggest three methods of measuring absolute temperature. In the

first method, the resistor's noise voltage over a bandwidth Af^ is amplified by a signal

processor using a high input-impedance, voltage-sensitive preamplifier to obtain an output

rms voltage given by

r—-1(1/2) , e = K (4kTAf R) ' , (4)

[_ nvj v V

where is the gain constant of the signal processor which is assumed to be ideal (noiseless) . Calibration of the signal processor and an auxiliary measurement of R would allow a determination of the absolute temperature. The second method uses a low input- impedance,

current-sensitive preamplifier with a gain K_j^ and bandwidth Af^, to obtain T from

r—-1(1/2) 1/2 . hni =

'^i (^l^TAf,/R)^'^ . (5)1

The third method requires the measurement of Johnson noise power by multiplying Eqs. (4)

and (5) to obtain the result

P = 4kK K^(Af Af.)"'"''^ T , (6) n V i V 1

which is independent of R. By dividing Eq. (4) by (5), the value of R can be found independently of T.

The output noise voltages of Eqs. (4) and (5) are fluctuating quantities that are subject to a fundamental statistical uncertainty. This uncertainty in the averaged signal

appearing at the output of an integrator or low-pass network following a detector was given by Rice [4] for noise described by a Gaussian probability density function as

a = 100 (cAf^t)"-""^^ , (7)

where t is the integration time or a characteristic time-constant of the low-pass network,

Afj^ is the noise bandwidth of the noise at the input to the detector, c is a constant that

depends on the noise detector and the type of averaging, and a is the percent standard

deviation of the output signal from the integrator or low-pass network. A value of c = 1 is appropriate for narrow-band white noise and pure integration of the output signal from a quadratic detector.

Equations (4)-(7) provide the theoretical basis for Johnson noise thermometry.

BASIC METHODS OF PRACTICAL JOHNSON NOISE THERMOMETRY

The first reported practical noise thermometer, developed by Garrison and Lawson [5]

for high temperature measurement (990-1340 K) , employed a technique (Fig, 1) of comparing

the noise voltage of a reference resistor R-^ at temperature T-^ to the noise voltage of a sensing resistor R^ at unknown temperature Tg. The two noise voltages are made equal in

the low-frequency channel by adjusting R-]^. The effect of capacitance across the resistors

on noise bandwidths is compensated by adjusting capacitors Cg and to obtain equal noise

voltages in the high-frequency channel so that RiC^ = RgCg- This technique eliminates

errors due to signal processor nonlinearities and long-term transfer function drifts. The

error caused by preamplifier noise is also reduced by the averaging provided by the lowpass filter. The unknown temperature is found from

261
VOLTAGESENSITIVE PREAMPLIFIER

' BANDPASS

AMPLIFIER 455-475 kHz

BANDPASS

AMPLIFIER

10-50 kHz

DETECTOR OSCILLOSCOPE

DETECTOR OSCILLOSCOPE

Fig. 1. Noise thermometer of Garrison and Lawson [5].

T =

s (8)

Commutation switches allow the same measurement system to interrogate both the reference

and sensing resistors. To obtain the value of the temperature by this comparison technique,

three quantities are measured: the reference temperature Tj^, the value of the reference

resistor R-^, and the value of the sensing resistor R^. Garrison and Lawson claimed uncertainties as low as ±0.1%, but Hogue [6] estimated that their results were subject to errors

up to ±0.5%. A major contribution to the error is the dependence of noise generated in the

preamplifier on source impedance. Pursey and Pyatt [7] developed a new thermometer that uses reference and sensor resistors of equal impedance. In their design, balance is

achieved by attenuating the noise voltage from the resistor which is at the higher temperature. The authors [7] concluded that, with further research, errors due to the preamplifier

noise could be reduced to ±0.1% and, perhaps, lower.

Several other noise thermometry systems have been published [8-11] , all based on the

voltage comparison method. All are equipped with commutation switches, except the system

developed by Fink [9]. His system (suggested by Garrison and Lawson) uses a cross- correlation scheme which incorporates a sensing resistor and its associated capacitance in an R-C, iT-section, two-port network. The noise voltage at each port is measured, and,

after proper adjustment of the -rr-section parameters, the unknown temperature is determined

from the values of two reference resistors, the value of the sensing resistor and the

temperatures of the reference resistors. Fink reported an uncertainty of ±1% between 1.3

and 4.2 K, and ±0.2% in the range from 77 to 90 K.

A simpler correlation scheme for noise thermometry was suggested by Shore and

Williamson [12]; it uses the correlator-amplifier system (Fig. 2) of Brophy, Epstein, and

Webb [13] to reduce the noise contribution of the preamplifiers, which is a major source of error in noise thermometry—especially at low temperatures. The outputs from two voltage

amplification channels, driven by the noise voltage from a single sensor are multiplied and

averaged (Fig. 2) to reduce the uncorrelated noise from the two channels while emphasizing

the correlated noise generated by the sensing resistor. The scheme is still subject to the

statistical uncertainty expressed by Eq. (7) and it requires careful matching of the transfer functions of the two channels. Practical realizations of the scheme were published by

Wagner and Bertman [14], Storm [15], and Brixy [16].

262
AMPLIFIER & BANDPASS

FILTER

AMPLIFIER & BANDPASS

FILTER

MULTIPLIER LOW-PASS

FILTER

Fig. 2. Correlator-amplifier system of Brophy, Epstein, and Webb [13]

The correlator-amplifier system (Fig. 2) is most useful at low temperatures and with

noise voltage preamplifiers having high equivalent noise resistance, R^. Over the years,

values of have been decreased from a value of 1000 Q in 1959 for a vacuum-tube preamplifier [8] to 44 Q in 1974 for a junction field effect transistor (JFET) input stage [17],

and to 24 Q in 1979 for a differential preamplifier with parallel JFETs in the input [18].

The lowest published value of is 10.6 Q for a wideband, feedback, single-ended, voltage

preamplifier designed by Blalock [19]. The recent availability of low-noise preamplifiers

may eliminate the more complex correlation techniques in future noise thermometers, except

perhaps for high-accuracy thermometers used for temperature scale metrology at low temperatures.

A different approach to noise thermometry signal processing was reported by Brodskii

and Savateev [20] in 1960 and by Maninger [21] in 1961. The temperature of a resistor is

measured by counting the number of times N that the noise voltage levels exceed a reference

level V in a unit time. This level-crossing process is described by the equation

N = A exp [- V^/(B + CT)] , (9)

where the constants A, B, and C depend on the value of the sensing resistor, the transfer

function of the signal processor, and the noise of the signal processor. The sensitivity

of this method varies with the temperature T being measured, since the output N is a very

nonlinear function of T; consequently, a rather complex calibration procedure is required.

Although this method does not need an rms-to-dc converter with its inherent nonlinearities

,

it requires a special-purpose pulse height discriminator, which is no easier to implement

than a high-linearity rms-to-dc converter. Unpublished work at the Oak Ridge National

Laboratory (ORNL) on noise pulse counting techniques shows no improvement in accuracy over

the rms-to-dc conversion technique for the same measurement time, and no significant simplification or improvement in the signal processor. However, the counting technique

allows greater discrimination against some types of nonthermal noise.

All of the preceeding methods of noise thermometry involve a measurement of opencircuit noise voltage and an auxiliary measurement of resistance. The quantity defined by

Eq. (6) —termed "virtual power" by Johnson in his classic paper [2] —is the product of the

open-circuit noise voltage and the short-circuit noise current of a resistor and is independent of the value of the resistor. A Johnson noise power thermometer (JNPT) based on

Eq. (6) was developed by Borkowski and Blalock [19] in 1974. The system (Fig. 3) uses a single resistor, a voltage preamplifier with a high input-impedance and a current preamplifier with a low input-impedance to alternately sample the resistor noise through commutating reed relays at the preamplifier inputs. A stable noise source with a level large

compared to the preamplifier noise was developed [22] to facilitate calibration of the JNPT

amplifiers. For field use, the JNPT (Fig. 3) is equipped to perform the sample-and-hold,

multiplication, and data correction functions with a minicomputer which also controls both

the switching and the automatic calibration functions.

263
ORNL-OWC T2-I23ea

Fig. 3. Johnson noise power thermometer system of Borkowski and

Blalock [19].

The JNPT was also implemented [19, 23] in pure analog form, using a probe containing

two sensing resistors that maintain a constant resistance ratio as their temperature varies.

The noise voltage of one resistor and the noise current of the other resistor are processed

simultaneously. These signals are multiplied by an analog multiplier, and the product

signal, after resistance-capacitance low-pass filtering, is displayed by a digital panel

meter for direct indication of absolute temperature.

APPLICATIONS OF NOISE THERMOMETRY IN TEMPERATURE SCALE METROLOGY

Several high-accuracy noise thermometers have been reported for applications in the

determination of the temperature scale.

Crovini and Actis, following extensive contributions [11, 24, 25] to noise thermometry

in the temperature range above 670 K, have recently reported [26] the development and use

of a noise thermometer to measure differences between the thermodynamic scale and the IPTS68 (International Practical Temperature Scale of 1968). They carefully measured absolute

temperature referenced to the ice point (273.15 K) with two different realizations of a noise voltage thermometer using a signal processor similar to the low-frequency channel of

the Garrison-Lawson (GL) system (Fig. 1). In the first realization, the rms voltages of two equal resistors at different temperatures were compared (method of [7]). Balance of

the two output voltages was achieved with a precision attenuator switched into the signal

processing chain when the, higher noise voltage was measured. In the second realization, a modified version of the classical GL system, the balance was achieved in two steps: first,

the resistance ratio [Eq. (8)] was preadjusted to match the IPTS-68 kelvin temperature

ratio; after which an attenuator in the higher level noise measurement was adjusted to achieve an accurate balance which is described by

R R

^s = ^ V^1 + - 1) r7 ^1 • S 1 ;

2 Since a , the power attenuator ratio, is in practice so near unity, a measurement of Rj^ to within a few percent is adequate. Resistors R]^ and Rg were alumina insulated, platinum

resistance thermometers with resistance values near 600 9. at 1230 K. In both realizations,

the noise voltage measurements were made in the frequency band from 20 to 120 kHz and a low-noise (R^j 120 fi) preamplifier with selected JFETs was used. The average uncertainty

of the experimental results of Crovini and Actis was ±0.034% (99% confidence limits).

Their results showed that the IPTS-68 is lower than the thermodynamic scale between 900 and

1230 K; the maximum difference is 0.56 ± 0. 20 K (99% confidence limits) near 1093 K.

A high-resolution noise thermometer for measurements in the temperature range from 90

to 100 K was reported by Pickup [27]. The measurement system employed was a modified GL

system; the preamplifier input circuit consisted of a JFET cascoded with a bipolar transistor second stage, yielding an R^ of '\'300 9.. The signal processor bandpass extends from

'\'10 kHz to 200 kHz for the low-frequency channel and the sensing resistor is '\^10 kfi. The

temperature of a 3-kf2 reference resistor was '^^25°C as indicated by a platinum resistance

thermometer calibrated against the triple point of water. Pickup estimates that the overall

uncertainty of his results is ± 0.0078% at 90 K and ± 0.0096% at 97 K (99% confidence

264
limits) for 7-h integration times. His results show that the IPTS-68 is lower than the

thermodynamic scale by 3 ± 6.99 mK at 90.17 K and 8 ± 9. 32 mK at 97 K (99% confidence).

Another high-resolution noise thermometer for measurement of temperatures near 4 K was developed by Klein, Klempt, and Storm [18]. Their thermometer system measures noise voltage

in a frequency band from 2 to 15 kHz and employs a correlator-amplifier system with a low

noise (R^ = 24 Q) differential preamplifier achieved by a parallel JFET input stage. The

temperature sensing resistor is about 10 kfi. A thorough analysis of error sources led the

authors to conclude that the total uncertainty in their measurements was 0.30 mK (±0.35 mK,

99% confidence)

.

Some of the latest developments in precision noise thermometry for temperature scale

metrology are reported in this conference by Klempt and Storm [28] and by Pickup [29].

APPLICATIONS OF NOISE THERMOMETRY IN PROCESS TEMPERATURE INSTRUMENTATION

Brixy applied the GL method, modified by using the correlator-amplifier technique

[13], to measurements of temperatures in nuclear reactors [30-32]. His preamplifier has a JFET-bipolar transistor cascode input with R^ 50 fi. Brixy suggests some techniques [32]

for solving problems arising from the use of long cables to connect the sensing resistor to the preamplifier and that it is possible to make accurate temperature measurements even with cables as long as 100 m. He reported that the measurement inaccuracy was <0.1% over a range from 300 to 1200 K under laboratory conditions, and <0.5% over a range from 300 to 500 K in high radiation fields in a nuclear reactor. Brixy concluded that noise thermometry

is especially suitable for the measurement of temperature in nuclear reactors, and that no additional noise would be contributed from ionization produced in the sensing resistor by

gamma rays and neutrons.

Development of the JNPT [19] at ORNL has continued since the early feasibility studies

of 1971 and this work has resulted in several practical applications in process temperature

measurements. Temperatures and sensor resistances are routinely measured by Johnson noise

with uncertainties less than ±0.5% (99% confidence) for sensing resistors from 50 to 300 Q,

and temperature range of from 273 to 1000 K, using signal cables as long as 18 m. Calibration of the signal processor independently of the sensing resistor and connecting cable is

accomplished by injection of a high-level noise signal of known spectral density directly

into the inputs of the voltage and current preamplifiers [22]. The Johnson noise generated

in the cable itself and the effect of the cable on the system transfer function are accounted for by applying data reduction algorithms. One algorithm, developed by Blalock

[34], uses a Iximped-element model for the cable and has been employed successfully for most

cables <20^m in length. For longer cables, a different algorithm based on a distributed

model was developed by Agouridis [34].

The JNPT was used to measure temperatures of uranium fuel during irradiation in a nuclear reactor. The JNPT sensor, made of rhenium wire with a resistance value near 80 Q, at 1670 K, was installed in the centerline of the fuel irradiated in ORNL High Flux Isotope

Reactor (HFIR) . After 4500 hours of high- temperature (1570-1770 K) high-radiation exposure,

the decalibration of the JNPT was negligible even though 80% of the rhenium transmuted to osmium [35], Occasional bursts of microphonic noise caused by vibrations of the loosely

supported rhenium coil contaminated the JNPT output. This defect was corrected by swaging

the sensor sheaths to form a more rigid containment.

Recently the ORNL group applied the JNPT to an in situ calibration of reactor power

plant platinum resistance thermometers [34]. The JNPT can independently measure both the

temperature and the resistance of the plant thermometers without removal of the sensor. JNPT measurements in two power plants [34] show great promise for in situ calibration where

access to the sensor can be achieved through short cables (<18 m) . Long cables, already

installed in power plants for dc measurements, may receive large amounts of nonthermal

noise from the wide variety of noise sources. The major challenge in this application

appears to be the characterization of both the noise contribution and the signal attenuation produced by long, industrial signal cables using measurements made only at the accessible end of the cable.

265
CONCLUSIONS AND PROSPECTS

Much progress has been made in Johnson noise thermometry since the early pioneering

work of Garrison and Lawson in 1949. Progress has accelerated in the last 10 years due to rapid improvements in signal processor components and the increasing availability and power

of digital computation. The difficulties noted by Kamper [1] of eliminating noise of

nonthermal origin still remain and provide challenging opportunities in the design of

grounding and shielding systems for successful noise thermometry.

The successful applications of noise thermometry to temperature scale metrology by

Crovini, Actis, Pickup, Klein, Klempt, and Storm show the advantages of using a fundamentally linear, absolute thermometer for interpolation and extrapolation of the defining

fixed points of the temperature scale.

The process temperature measurements of Brixy and the ORNL group indicate encouraging

prospects for such applications. Major problems are the effects of extraneous noise pickup

and random noise generated in the cable, and the effect of the cable signal transfer characteristics on the overall transfer function of the signal processor. Solution of these

problems will be provided by specification of special cables for noise thermometry, development of effective techniques for measuring relevant cable parameters, and design of

appropriate digital computer algorithms for data reduction and correction.

REFERENCES

[1] R. A. Kamper, "Survey of Noise Thermometry," Temperaturey Its Measurement and Control

in Science and Industry, Instrument Society of America, Vol. IV (1), 349 (1972).

[2] J., B. Johnson, "Thermal Agitation of Electricity in Conductors," Phys. Rev. 32, 97 (1928).

[3] H. Nyquist, "Thermal Agitation of Electric Charge in Conductors," Phys. Rev. 32^, 110

(1928)

.

[4] S. 0. Rice, "Mathematical Analysis of Random Noise," Bell Sys. Tech. J. 23^, 282 (1944).

[5] J. B. Garrison and A. W. Lawson, "An Absolute Noise Thermometer for High Temperatures

and High Pressures," Rev. Sci. Instrum. 20, 7813 (1949).

[6] E. W. Hogue, "Factors Affecting the Precision and Accuracy of an Absolute Noise

Thermometer," NBS Report 3471 (1954).

[7] H. Pursey and E. C. Pyatt, "Measurement of Equivalent Noise Resistance of a Noise

Thermometry Amplifier," J. Sci. Instrum. 36, 260 (1959).

8] E. T. Patronis, Jr., H. Marshak et al. , "Low Temperature Thermal Noise Thermometer,"

Rev. Sci. Instrum. 30, 578 (1959).

[9] H. J. Fink, "A New Absolute Thermometer at Low Temperatures," Can. J. Phys. 3_7, 1397

(1959)

.

[10] M. H. Tillinger, "Noise Thermometer at High Pressure," Technical Report No. 1,

Polytechnic Institute of Brooklyn (1965).

[11] L. Crovini, "Absolute Temperature Measurement by Means of Thermal Noise," La Ricenca

Scientifica 37(12) , 1238 (1967).

[12] F. J. Shore and R. S. Williamson, "Suggested Thermometer for Low Temperatures Using

Nyquist Noise and Correlator-Amplifier," Rev. Sci. Instrum. J7 (6) , 787 (1966).

[13] J. J. Brophy, M. Epstein, and S. L. Webb, "Correlator-Amplifier for Very Low Level

Signals," Rev. Sci. Instrum. ^(12), 1803 (1965).

266
[14] R. Wagner and B. Bertman, paper T-2, Commission 1, International Institute of

Refrigeration, Tokyo (1970)

.

[15] L. Storm, Z. Ang. Phys. 28, 331 (1970).

[16] H. G. Brixy, "Temperature Measurement in Nuclear Reactors by Noise Thermometry," Nucl.

Instrum. Methods 97, 75 (1971).

[17] P. T. Anderson and P. B. Pipes, "A Low Noise Amplifier with Application to Noise

Thermometry Between 300 and 4 K," Rev. Sci. Instrum. 45(1), 42 (1974).

[18] H. Klein, G. Klempt, and L. Storm, "Measurement of ^He at Various Vapour Pressures by

a Noise Thermometer," Metrologia 15, 143 (1979).

[19] C. J. Borkowski and T. V. Blalock, "A New Method of Johnson Noise Thermometry," Rev.

Sci. Instrum. 45(2), 151 (1974).

[20] A. D. Brodskii and A. V. Savateev, "A New Method of Absolute Temperature Measurements,

Izmeritel'naya Teknika No. 5, 21 (1960).

[21] R. C. Maninger, "Thermal Noise Investigation," U.S.A. Patent 3,181,365, May 4, 1965.

Filed January 9, 1961.

[22] T. V. Blalock and C. J. Borkowski, "High-level Noise Source for the Calibration of

Johnson Noise Power Thermometers," Rev. Sci. Instrum. 4_9 (8) , 1046 (1978).

[23] T. V. Blalock and R. L. Shepard, "Johnson Noise Power Thermometer," ORNL internal

report (1977).

[24] A. Actis, A. Cibrario, and L. Crovini, "Methods of Noise Thermometry Above 400°C,"

Temperature^ Its Measurement and Control in Soienae and Industry, Instrument Society

of America, Vol. IV(1), 355 (1972).

[25] L. Crovini and A. Actis, "Systematic Errors in High Temperature Noise Thermometry,"

Temperature Measurement (Eds. , B. F. Billings and T, J. Quinn) Inst. Phys. Cont. Ser., 26. 398 (1975).

[26] L. Crovini and A. Actis, "Noise Thermometry in the Range 630-962°C," Metrologia 14, 69

(1978).

[27] C. P. Pickup, "A High Resolution Noise Thermometer for the Temperature Range 90-100

K," Metrologia U, 151 (1975).

[28] G. Klempt and L. Storm, "Determination of the Thermodynamic Temperature of Fixed

Points by Means of a Highly Accurate Noise Thermometry," Proc. 6th Int. Symp. Noise in

Physical Systems, April 1981.

[29] C. P. Pickup, "A Precision Noise Thermometry for Temperature Scale Studies," Proc. 6th

Int. Symp. Noise in Physical Systems, April 1981.

[30] H. G. Brixy, "Temperature Measurement in Nuclear Reactors by Noise Thermometry," Nucl.

Instrvim. Methods 97^, 75 (1971)

.

[31] H. Brixy, "Die Rauschthermometrie als Temperaturmeszmethode in Kernreaktoren,"

Dissertation, JU1-885-RG (1972)

.

[32] H. Brixy, R. Hecker, and T. Overhoff, "Development of the Noise Thermometer and Its Test as an In-Core Temperature Detector," Proc. Sjnnp. Nuclear Power Plants Control and

Instrumentation, International Atomic Energy Agency Report No. IAEA/SM-168/F-1, Vienna

1973.

267
[33] H. Brixy, K. Rittinghaus, K. GSrtner, and R. Hecker, "Developments in the Area of

Noise Thermometry at KFA-JULICH," Proc. Int. High Temperature In-Pile Thermometry

Colloquium, Petten, The Netherlands, December 1974.

[34] R. L. Shepard, T. V. Blalock, J. L. Horton, and D. C. Agouridis, "In-Situ Calibration

of Reactor Plant Platinum Resistance Thermometers Using Johnson Noise Methods," Final

Report, EPRI Contract RP-1440-1 (1980).

[35] R. L. Shepard, C. J. Borkowski, J. K. East, R. J. Fox, J. L. Horton, and T. V. Blalock,

"Ultrasonic and Johnson Noise Fuel Centerline Thermometry," Proc. Int. High Temperature

In-Pile Thermometry Colloquium, Petten, Netherlands, 1974.

268
DETERMINATION OF THE THERMODYNAMIC TEMPERATURE OF FIXED

POINTS BY MEANS OF A HIGHLY ACCURATE NOISE THERMOMETER

G. Klempt, L. Storm

Institut fur Angewandte Physik der Universitat Miinster,

Roxeler Str. 70/72, D-4400 Munster, Fed. Rep. of Germany

The standard device for the measurement of thermodynamic temperaturexin

the region from 2 K up to the melting point of gold is the gas thermometer.

Since several systematic errors have been discovered in the last decades,

the developement of an alternative thermometer with comparable accuracy

seemed to be desirable. The noise thermometer is a suitable alternative,

since the thermal noise voltage U^ of an impedance Z at the temperature T is

determined by a fluctuation-dissipation theorem, i.e. the Nyquist formula:

<^u^^^ = J'

4 k T ReZ df ' ' (1)"

^In 1976 a device was set up to measure temperatures in the liquid He-region with an accuracy of about 10"'^, s. t^1 . Particularly at low

temperatures^ the specific error sources of the noise thermometer could be

investigated. A measuring proceedure was developed, which gives the thermodynamic temperature without any additional corrections. To measure the temperature of several thermometric fixed points up to the water boiling point

within a few mK, an improved noise thermometer with an accuracy of about

10"^ has been set up during the last years.

The figure shows the block diagram of the noise thermometer, operating

by a correlation method. The temperature-sensing resistor 2R, with centre

point at earth potential, is connected in parallel to two identical highly

linear and stable amplifier- and filter channels, using a 4-wire measurement

technique. The output voltages are digitalized by 16 bit ADC's with a rate

Temperature sensing

resistors

Low noise Symmetrical Diff Filter 16 bit

differential ohmic ampli- 2 kHz ADC

preamplifier voltage divider f^er 25kHz

Digital multiplier

Accumulator

0

X

Block diagram of the noise thermometer

269
of 50 kHz, then multiplied and added up. The output voltage U of the cross spectrometer - averaged over a finite time B - represents a stochastic process. Its expectation value is given by eq (2)

:

<u> = K.or^^. / Re [vf V, a b W

uaub. df (2)

^uaub i'^^i'^^tes the complex cross spectrum of the preamplifier input voltages u^ and u^. K is determined by the ADC's properties. The attenuatj

factored of the resistive voltage divider is presumed to be independent of

the frequency. To avoid nonlinearity errors of the ADC ' s, the attenuators

are adjusted in such a way that the ADC's are equally modulated at the measurement of the unknown temperature T and at the calibration of the whole

device at the water triple point temperature T^^ = 273.16 K, which defines

the temperature scale. V and V, are the gains of the channels a and b. V*^

means the conjugate complex number of V . If both channels have an identical

phase response, '^^ 'becomes a real number and only the real part of W . contributes to <^tf> . A thorough theoretical analysis of the correlator inpu^

circuit leads to the following equation for the real part of W ^j^:

^^^uaub = 2 4kR' rT+A(f ,T) ;R1

1 + (27ffRC)^ (3)

4kRT / (1+ (2irfRC)

temperature sensing

the lead capacity C

the current noise s fiers and the tempe

of the leads. With

very low-loss diele

of some mK/ki2. . Thi

value of R because

) means the power spectrum of the Nyquist noise of the

resistors, filtered by the low pass consisting of R and

in parallel to R. The error term A-R is mainly caused by ources i^^ including the noise current of the preamplirature dependent noise due to the insulating dielectric

ultra low noise FETs in the preamplifier input stages and

ctrics in the whole measuring input circuit, A has a value

s error term cannot be made neglible by decreasing the

of the statistical fluctuations.

In this correlation method the statistically independent noise sources

u - including the inherent noise voltage of the preamplifiers and the temperature dependent thermal noise of the leads - contribute practically nothing

towards <U>

formula:

Its relative standard deviation is given by a modified Ricei<V>)~' 2 1+ (1 + W

un , 4kTR' 6)Af

(4)

W

un « 4kTR^

Even in the best case ( W <«< 4kTR ) a measuring time 0 of 5 • 1 0 s and a bandwith C^f of 2-10'^ Hz is needed f or c^/ffl) = 1 0~^ , which equals an accuracy of some mK in the measurement of medium temperatures. 4kTR should not be made

smaller than W , otherwise the measuring time would become inadmissibly high,

This implies values of R in the order of some ki^with conventional transistor

amplifiers. Thus eq (4) determines a lower limit of R and hence of the error term AR. As the theoretical estimation is not accurate enough, AR must be

eliminated by experimental means taking advantage of the fact that A is independent of R.

270
For the measurement of an unknown temperature T, the same sensing

resistors are brought alternatingly to T and the water triple point temperature T^^ = 27 3.16 K. We measure the output voltages <U> and > , the re- sistances R and R,^ and the attenuation factors ocg and c^b . With the measured

values for eq (2) and eq (3) for both temperatures we compute the magnitude F

= T • ( 1 + PR )

which is a linear function of R. If one measures F at the same temperature T

for different values of R, the unknown temperature can be obtained by extrapolation of R to 0. The variations AR in R and AC in C can be kept so small

at the changeover from T to T , that the correction term V'AIRC) is of the

order of 1 0 with R below 5 kSs . The described method has the advantage that all the disturbing noise is

eliminated. But for an accuracy of 1 0 in T one needs 10 s for the measurement of <U> and <U^^>and for each value of R. Therefore a long-term

stability of some 10 of all electronic components and the temperature baths

is demanded. The temperature sensor 2R is located together with four platinum

resistance thermometers in a cylindrical block of copper. Its temperature is

kept constant within some 100 /cK by a heater controlled by one of the platinum resistors. The dominant error sources are the statistical error and the

gain stability of the correlator.

For the momentary series of measurement the four platinum resistance

temperatures have been calibrated at the Physikalisch-Technische Bundesanstalt in Braunschweig at the argon triple point temperature. At the

Symposium on Noise in April 1981, we will present the value of the argon

triple point temperature measured by the noise thermometer in Miinster.

LI] H.-H. Klein, G. Klempt, L. Storm: Metrologia 1_5, 143 ( 1 979)

271
A PRECISION NOISE THERMOMETER FOR

TEMPERATURE SCALE STUDIES

C. P. PICKUP

CSIRO Division of Applied Physics

Sydney, Australia 2070

INTRODUCTION

Practically all of our knowledge of thermodynainic temperatures has been obtained from

gas thermometry either by direct pressure versus volume measurements or more indirectly

from isotherm or acoustic thermometry, all of which are difficult and tedious experiments

subject to many corrections. In particular the non-ideal properties of all real gases have

a serious effect which is difficult to determine accurately.

In recent years a program of high precision gas thermometry here at NBS [1] from room temperature upwards has shown a serious systematic error in the International Practical

Temperature Scale, amounting to approximately 30 mK at the steam point and increasing at

higher temperatures.

It is unlikely that any comparable gas thermometer experiment will be undertaken elsewhere in the near future^ and noise thermometry is one of the very few alternative techniques

which might be capable of yielding confirmatory data. We at the Division of Applied Physics

(DAP) have accordingly constructed a noise thermometer operating in the range 100-150°C

which is designed to be accurate to within a few mK.

PRINCIPLE OF THE NOISE THERMOMETER

The noise thermometer is based fairly closely on a successful earlier design [2] which

operated at the oxygen boiling point using the switching principle pioneered by Garrison

and Lawson [3] more than thirty years ago in which a noise measuring channel is switched

alternately between two resistors at different temperatures, one of which is known. The

noise of the measuring system is constant (ideally) and thus a balanced condition can be

detected in which the resistors produce the same noise. An unknown temperature can then be

determined from the other, which would usually be 273.15 K, and the ratio of the two resistances.

DETAILED DESCRIPTION

A block diagram of the noise thermometer is shown in figure 1. The resistor Rj is

maintained at the temperature to be measured (Tj) whilst R2 is in an ice bath (T2= 273.15 K)

.

Either resistor may be connected via mercury-wetted reed switches S^ and S2 to a fairly low

noise pre-amplifier with a type BFW 11 JFET cascode input stage. This was selected mainly

for high input resistance, and its equivalent noise resistance is 450 ohm.

The amplified noise signals are fed through the shield surrounding the noise thermometer by a screened transformer and split by filters into two frequency ranges, 10-100 kHz

and 100-200 kHz. The mean amplitudes of each filter output are determined by linear

detectors, V/F converters and frequency counters. The data are collected and processed by

| a micro-computer system.

A crystal-controlled clock and counter chain provides timing control signals for the

system. In normal operation the switches change over each 15 s with 5 s being allowed after

each switching for settling before integration begins. In addition an early stage of the

pre-amplifier is gated off during the switching period. Signals for this gating and the

reed switch operation are transferred through the shield by optical links with light pipes.

272
rOI

I RESISTANCE I RATIO MEASUREMENT

Fig. 1 Block diagram of switching noise thermometer.

The unselected noise resistor is normally short-circuited; however, with the shield

raised and both switches operated, it is possible to measure the ratio R1/R2 at 1 kHz using

a 7 decade inductive divider bridge. This is normally done at the beginning and end of each

run.

The temperature of resistance is maintained by heated oil circulated from an external bath with the returning oil providing thermal shielding for an isothermal column

extending into the electromagnetically shielded area. An IPTS calibrated platinum resistance thermometer is provided for both measurement and control of the oil temperature.

This temperature exhibits rapid fluctuations of the order of 10 mK, due presumably to the

turbulent processes in the circulating pump; however, the mean value is stable to within

1 mK indefinitely.

Resistance R2 is surrounded by high purity ice with the melt water drained from the

bottom of the container. Tests conducted with a standard platinum resistance thermometer in

this ice bath showed variations generally less than 1 mK.

THERMOMETRIC RESISTANCES

Desirable properties for the thermometric resistors include

a. only thermal noise should be produced

b. the conductance should be independent of frequency to at least 200 kHz

c. the conductance should be adequately stable at temperatures approaching 150°C.

Metal film resistors (Philips MR25) were used. Any excess noise from these should be

well below the 10 kHz low frequency cut-off of the system and would presumably be proportional to the dc current which is only 2 pA.

Present technology does not permit the convenient measurement of the conductance ratio

with sufficient precision at frequencies as high as 200 kHz, and thus extrapolation of audio

frequency measurement is necessary. The resistive film is too thin for skin effect to be

significant and any change of conductance with frequency would have to be due to some effect

such as loss in the substrate or protective lacquer or extreme spiral grooving.

In order to provide even greater confidence in the ratio determination it was decided

to use initially groups of identical resistors (6.8 kfi) in parallel, three for Ri and two

for R2. This forces a 3/2 ratio of thermodynamic temperatures for balance, i.e. Tj 135°C.

This is a reasonably convenient temperature for the purpose of checking the scale and

a few hours after each warm-up the resistance ratio becomes stable to within a few PPM

although there is a small change each time the hot resistors are cycled to room temperature.

MEASUREMENT PROCEDURE

Before the series of measuring runs was commenced (and also after) calibrating tests

were performed with short circuits substituted for the noise resistors. This process provides a direct measurement of the difference in noise of the reed switches and the leads to 273
the resistors at their normal operating temperatures. In principle it also provides reasonable corrections for any (possibly non-Gaussian) extraneous noise signals arising from

mechanical vibration or inductive coupling which fall within the pass-band. When the resistors are replaced in the circuit there will be some attenuation of this measured difference

due to the shunt capacitances but this is less than 1% at 200 kHz and can be neglected.

Also at this time the actual resistances of the leads at their operating temperatures

can be measured. These resistances, of the order of 1 ohm, are needed to correct the

measured resistance ratio which includes them.

In normal operation the bath temperature and the capacitance are varied until

balance is obtained in both frequency ranges, thus ensuring that there will be no errors caused by the attenuation of the RiCi and R2C2 time constants. The results of each 10 s integration are squared (because the detector is of the linear type) and the differences

between those from and R2 (called LOWER and UPPER respectively) are stored by a dedicated

micro-computer over the period of each run which normally lasts just under 24 hours.

At the end of each run, after the correction for the noise of the leads has been subtracted, there is normally a small residual unbalance U-L in both channels and the

temperature Tj is calculated from the formula [2]

Rp

P(

7 U-L

LF

1 U-L

6 U HE

(1)

Here U represents the total mean square noise from the system when switched to the

UPPER resistor and the factor p, which is taken as 1.15, reflects the reduction in the

relative unbalance caused by the noise of the amplifier.

ERRORS DUE TO GATE NOISE CURRENTS

Since the resistors Rj and R2 have different values the switching noise thermometer is

sxjbject to errors due to noise currents originating in the input circuit of the amplifier

and also, of course, to any systematic dependence of gain on source impedance resulting

from feedback. The latter is effectively eliminated by the cascode arrangement.

Fig. 2 Equivalent circuit in terms of noise current generators. Johnson noise in the

thermometric resistances is represented by ii and i2, Rg is the shunt resistance of the

input circuit with noise ig and iL, ig are respectively the shot noise due to the gate

leakage current and the induced gate noise.

The important noise current generators are shown in figure 2. The shunt resistance Rg

arising from the input conductances of the FET and the insulation losses of the circuit was measured using a special digital Q-meter at 150 kHz to be approximately 1000 Mfi. This may

be considered to produce ordinary thermal noise corresponding to room temperature and causes an error in the present system of only 0.2 mK. This error would, however, be more serious in a cryogenic noise thermometer.

274
The gate leakage current Ig is 2 pA and its shot noise component given by di-^ = 2elgdf

may also be shown to give a completely insignificant error in this case.

The induced gate noise current is capacitively coupled from the channel and may be re- presented by dig = Bf^df where B is a constant. For the frequencies and impedances involved here the correlation with the channel noise is pure imaginary and the noise voltages at

the gate may be simply added in quadrature. The effect of this additional noise current is

largest for the higher value resistance in the higher frequency range when it represents

an apparent temperature increase of approximately 5 mK. This would be partly off-set by a proportionately smaller increase in the apparent temperature of R2; however, it was shown in

[2] that errors due to capacitively-coupled noise currents at the gate, for the noise

thermometer balanced simultaneously in two frequency ranges, cancel out completely in a second order analysis.

RESULTS

The standard deviations of the differences in both frequency ranges throughout all the runs were consistent with the integration period and pre-detection bandwidth. For a single

24 hour run, after combining the high and low frequency means, the standard deviation of

the temperature should be approximately 12 mK.

For 62 runs the mean difference Tjprjig - Tj^Qjgg was 12 mK, the standard deviation 19 mK

and the S.D. of the mean 2.4 mK. The largest positive and negative difference from the mean were 30 mK and -39 mK respectively with rather more than the expected number of large

differences.

This result is in the same direction but otherwise not in particularly good agreement

with the recent gas thermometry; however, a number of break-downs towards the end of the

series of runs raises the possibility of undetected intermittent electronic problems earlier.

A further series of tests are planned during the next few months after the detectors and

counting systems have been re-built.

REFERENCES

[1] L. A. Guildner and R. E. Edsinger, J. Res. NBS 80A , 703 (1976).

[2] c. P. Pickup, Metvologia 151 (1975).

[3] J. B. Garrison and A. W. Lawson, Rev. Sai. Instr. 20, 785 (1949)

.

275
ELECTRO-ACOUSTICAL NOISE

ii

E

3

R.J.J. Zijlstra and W. Westera E

Fysisch Laboratorium, Rijksuniversiteit Utrecht, The Netherlands •

1

.

INTRODUCTION

Electro-acoustical noise is the noise associated with the interaction of free carriers

with acoustic waves in piezoelectric semiconductors. Increased interest in these physical phe- nomena was triggered by the experimental results on acoustic amplification in CdS reported in

1961 by Hutson, McFee and White [1]. The electric field accompanying an acoustic wave, be it ' a bulk or a surface wave, in a piezoelectric semiconductor produces periodic variation in the s electric potential. Free electrons, interacting with these waves, tend to seek the potential :

i minima and in doing so get dragged along by the propagating wave. If the drift velocity of the

carriers is smaller than the sound velocity, the acoustically trapped carriers receive a net ! accelerating force from the wave. Acoustic energy is thereby transferred from the wave to free

electrons and the wave is attenuated. However, if an applied electric field causes the drift

velocity of electrons to become larger than the sound velocity, then energy is transferred

from electrons to the wave while it is travelling in the direction of the drifting carriers, s i As a result the acoustic waves are amplified. White [2] in 1962 gave a linear description of i this electro-acoustic effect, known as the linear small signal gain theory. This effect also i applies to spontaneously generated waves with thermal amplitudes. These amplified acoustic waves also interact with the free carriers. This gives rise to essentially non-linear effects,-

which may include parametric interaction of acoustic waves, current saturation, large current

noise and domain formation [3,4]. These effects cannot be described by White's theory. Several;

authors [5,6] have tried to describe these phenomena. Since, however, it is very difficult to describe these non-linear effects when starting from basic principles, many of these phenomena^ i are not yet understood quantitatively. Moore realized that the observed current saturation and , current fluctuations are caused by the trapping of bunches of free charge carriers in potential troughs that are associated with acoustic waves amplified from the thermal background.

Accordingly, he ascribed the observed current noise in CdS to fluctuations in the creationannihilation processes of these potential troughs [7]. The expression for the noise spectrum

thus obtained gave a reasonable explanation for Moore's experimental data. Subsequently « Friedman [8] and Nakamura [9,10] using different approaches were less successful in describing! ti these experimental data. ii ii In our opinion Moore's model is essentially correct and scored highest in the description ii of experimental data. Zijlstra and Gielen [11] extended Moore's theory by accounting for « a transit time effects in a local description. However, they neglected the displacement and the

diffusion current in the expression for the current density and assumed that the creation and ' i[ annihilation of troughs are independent of electric field strength. Their calculation resulted

among other things in a frequency independent impedance. Experimentally, however, the observed

ac impedance of electro-acoustically active CdS crystals turned out to be frequency dependent

[12]. The observed frequency dependence of the ac impedance could be explained, however, by I taking into account diffusion, space charge, displacement current and the electric field de- j

pendence of the trough creation and annihilation rates [13,14].

In section 2 we present theoretical results along the lines indicated. Section 3 is con- l cerned with experimental results on electro-acoustical noise in CdS.

1, 2. THEORY ,

•

Essentially we shall use Moore's model but extend it as indicated in the introduction. ' i In addition we shall use the so-called phenomenological , hydrodynamic theory of electron ab- \< |

sorption and emission of sound in semiconductors. Usually the criterion for the validity of ) (f this approach is written as ql << 1 and wx << 1 [14], where q is wave number of acoustic • t( waves, 1 is free carrier mean free path, t is their collision time and u is angular frequencyj; \\ Gulguaev and Kozorezov [15] argue, however, that the mean free path 1 should be replaced by I [tj /Dt^ and the collision time by x , where x^ is the energy relaxation time and D is the free ! carrier diffusion constant. 276
2.1. DESCRIPTION OF THE MODEL AND BASIC EQUATIONS

We consider an n-type homogeneous piezoelectric semiconducting crystal, where the electric field is applied along a symmetry axis, the x-axis. The sample is provided with two ohmic

contacts with spacing L. It can be shown for this case that the formulas can be put in a one- dimensional form [14]. Therefore we shall start straightaway with a one-dimensional description. In the analysis we use the following sign convention, E < 0 and I < 0; then V > 0, where

E is the electric field strength, I is the electric current and V is the applied voltage.

Let nj be the local density of the free electrons in the conduction band and ng the density of electrons trapped in potential troughs that are associated with acoustic waves amplified from the thermal background. The total density of electrons in the conduction band is

then given by n = n^j + ng . Gauss's equation yields

3D/9x = - q(n - n) , (1)

where D is the dielect_ric displacement, -q the electron charge and where steady-state values

are indicated by bar; n is assumed to be equal to the thermal equilibrium density of free

carriers. The total current density then is

j = - qn.v - qn V ' + qD On/8x) + 3D/8t , (2) Q Q s s n

where D^^ is the diffusion constant of electrons, v^ is the drift velocity of free electrons

and Vg' is the x-component of the sound velocity Vg. Since j is solenoidal, we have 9j/9x = 0. The piezoelectric relations read

T = cS - eE ; D = eE + eS . (3)

According to Newton's second law

9T/9X = pO^u/gt^) (4)

with S = 3u/9x, where T and S are the stress and strain, whereas c, e and e in eq (3) are the

elastic, piezoelectric and dielectric constants respectively, p is the mass density and u the

displacement. It should be borne in mind that the constants in the piezoelectric relations are

related to elastic, piezoelectric and dielectric tensor elements valid for an anisotropic

solid [14].

When p and b are the creation and annihilation rates of troughs per unit volume re- spectively, the master equation for the trough density nj- reads

3n^/9t = p - b - Vg' (9n^/9x) . (5)

In addition we assume that each trough contains N electrons, independent of position x and e- lectric field strength E. Then the density of trapped electrons becomes ng = N n^.

2.2. THE STATIONARY STATE

From eq (1), p) and (4) it follows for the steady state that 9D/3x = (1 + Kg^) (9E/9x) =

0

where Kg = (e^/ec)z is t^he electromechanical coupling constant, which is usually much smaller

than one. Thence E = - V/L and v^ = V/L is independent of position. It follows from eq (3) and

(4) that Vg'(3ng/9x) = v^(dxig/dx) and since v^j / Vg' in the electroa£oustically active regime

we have (9ng/9x) = (9n^/9x) = 0. Finally for the current we obtain I = - qAngVg' - qAn^jCuV/L),

where A is the contact area. Note that the mobility y as well as rig and n^j may depend on electric field strength at high applied fields.

277
2.3. AC BEHAVIOUR AND NOISE

In order to describe the ac behaviour and fluctuations around a steady state^we lineariz*

the equations from section 2.1. Then the linearized current equation becomes

Aj = a AE . q(v^ -v; )An^ ^ q\ |f " ^ |f (2a)

where o = qprf^j. The linearized rate equation for the troughs extended by a Langevin source function H, which formally describes the fluctuations in the creation and annihilation, reads;

3An An 3An

+ = gAE - V • + H (5a) dt T S 3x

Where t'^ = - [3 (p - b) /3n B =[3 (p - b)/3E]^^^Q and H E A(p-b)^^^^^

t t s

Note that we finally have 11 linearized equations with 14 variables. We are interested in a spectral decomposition of the fluctuations. We Fourier transform the equations with respect

to time. Then 11 equations result with 13 variables. These 11 equations allow one in principle to eliminate 10 variables so that 3 remain. If we do not consider fluctuation's (H = 0) 2 variables remain, say j and E (Fourier transforms are denoted by tilde). Thence j can be exL pressed m E and in the ac voltage V = -

J Edx. As a result the ac impedance can be calculated. 0 „ „ If one considers an ac open circuit by putting j =0, then V can be expressed in H and

the voltage fluctuations can be calculated in terms of Langevin source terms.

2.4. THE AC IMPEDANCE

Carrying out the programme for the cal^ulatioj^ of the ac impedance as indicated in sec- tion 2.1 and using the boundary conditions T(0) = T(L) = 0, corresponding to free end surfaces one finds [13, 14]

Z(co) = {1 + K 2f((,)} , (6)

A(-;—; + 0 + loje) 1+lCJT

where a = q(vj-Vg')N3 and f(u)) is a complicated function of frequency [13,14]. Substituting

reasonable values for the unknowns in eq (6) we found that the second term between brackets

only contributes significantly at frequencies given by

f = (2m + l)Vg'/2L m = 0, 1, 2, (7)

'

Four limiting cases are of interest:

(i) If we put a = 0 and a = qyn, in other words we neglect the presence of troughs, we have

the linear small signal approximation, and eq (6) reduces to an expression derived by Greebe

[17].

(ii) If CO 0, then eq (6) reduces to the differential resistance: Z(0) = L/A(aT + a).

(iii) If 0) >> |a + aT/(l + icoT)|/e, then Z((jj) ^ L/iAwe , corresponding to the geometrical

capacitance of the device.

(iv) If a >> I icoe + aT/(l + iux) | , then we find Z(u) L/Ao.

Note that a < 0 if the number of troughs increases with increasing applied voltage.

It follows that the low frequency plateau value of the impedance is a factor o/ia+aj) larger

than the intermediate frequency plateau value.

In addition to the ac impedance the following expression could be found for the imaginary part of the wavenumber of waves travelling in the direction of drifting particles:

278
, CO * „ = 4 K 2 ^ (8)

e 2 e V CO to S ^ (_ + ^)

2

where Vg = sound velocity, co^ = Vg^/D^j and co^, = (aT + a)/E; y* = 1 - (^d'^s^ ^^s' These ex- pressions reduce to White's result for the linear attenuation coefficient if the differential

conductivity is replaced by the ohmic one. The frequency of maximum amplification, u^^, is

given by

i^iJ = u^co (9) m c U

For a discussion of the influence of traps and magnetic fields on the electro-acoustic effect we refer to the literature [3,18],

2.5. NOISE

The noise problem has been solved only by ignoring diffusion and displacement current,

by assuming space charge neutrality and by assuming that the number of electrons, N, trapped

per trough is independent of position and electric field strength. By Fourier transforming eq

(5a) with respect to time and using ng = N n^. one then finds

'\< dn 1 IT . 3— + r (1 + iuT)n = E + —r H (5b) dx TV ' s V ' V

' s s s

The Fourier transformed eq (2a) reads

J = oE + q(v, - V ')n , (2b)

d s s

'V a. If one considers an ac open circuit by putting j = 0, ng can be eliminated from these equations and the following differential equation in E results:

f.A^ = -qNll^ (10) dx a

where A = (tj"'^ + iw)/Vg', tj"^ = a/a and n = (v^j - Vg')/Vg'. If we assume^that t^e end

surfaces are kept free, then T(0) = T(L) = 0, which for this case also implies^E^O) = E(L).

Using these boundary conditions and remembering that S^(f) is proportional to <V«V*>, where

L ^ = -

J ^ dx, one finds by integrating eq (10)

0 L L

^v(f) = aF [ ^ I S^(Xj,X2,f)dXjdx2 (11)

0 0

If it is assumed that creation and annihilation processes occur spontaneously and that a_6- _

function space correlation exists, we have Sjj(xj,X2,f) = (4b/A)6(xj- X2), where we used p = b. Substituting this into eq (11) yields:

4bLv '^T 2 q2N2Ti2

S.^(f) = ^ !— (12)

^ A(l + co^T 2) a2

1 279
The outcome of the calculations is a Lorentzian noise spectrum. It should be remembered, however, that the outcome depends on the boundary conditions and on the rather stringent restriction on the space correlation of the fluctuations in the creation and annihilation processes.

The latter condition is obviously included for mathematical convenience. It seems reasonable

to assume that physically these conditions are met if u) << ui^. If the ac impedance Z is calculated using the same conditions as for the noise calculations one finds

(13)

If we introduce the trough transit time Tj- = L/vg' and assume that b = n^/i, then the saturation current Ig = AqNbxVg ' and then it follows from eqs (12) and (13) that

Sj(f) = 4NqI^

1 + 0)2x2

(14)

3. EXPERIMENTAL RESULTS

Some experimental results obtained recently with CdS [16]

crystal bars will be presented. The small faces, typically of

the order of 1 mm2 , were provided with indium evaporated

ohmic contacts; the lengths of the bars varied from 1.4-

2.8 mm. The c-axis of the hexagonal crystals was in the

length direction. In this shape the crystal bars could be

used for Brillouin scattering as well as for electrical

measurements. The crystals were semiconducting with re- sistivities of the order of 1 fim. The length of the bars

was kept smaller than 3 ram in order to suppress travelling

electro-acoustic domain formation and the accompanying

oscillatory behaviour. To avoid excessive Joule heating of

the samples the high voltage was applied in voltage pulses

of 40 ys duration with a repetition rate of 4 Hz. Figure 1 shows a typical current-voltage (I-V)

characteristic. At low applied voltage the sample is ohmic,

whereas at higher voltages non-ohmic conduction occurs because electrons are trapped in potential troughs as- sociated with acoustic waves moving at the sound velocity.

In figure 2 the spectral intensity of the ac short-circuited

current noise, Sj, at 10 MHz is plotted double logarithmically versus applied voltage. One distinguishes a rise

with slope 2 at low voltages due to generation-recombination

noise, followed by a very sharp rise because of the occurrence

of the electro-acoustic effect and finally a less sharp rise

which is even followed by a decrease with increasing voltage.

The threshold voltage for the occurrence of electroacoustic current fluctuations can be determined [12] by an extrapolation of the steep curve down to the thermal current

noise level. The thus obtained values for invariably

turned out to be somewhat lower than the knee voltage of the

I-V characteristic due to the fact that in the transition

region between ohmic and non-ohmic linear behaviour the

density of trapped electrons is field dependent. In the past

there has been a controversy about whether longitudinal or transverse waves are responsible for the onset of the

electro-acoustic effect. Particularly^ electrical data alone

left some room for speculation. This was partly due to the

wide spread in the electron mobility values reported in

the literature and to the occurrence of trapping effects,

which were not taken into account [7,12].

280

Fig. 1 Current versus applied

voltage for a CdS crystal

with the electric field applied along the c-axis.

10

10"

L = 1.76 mm

f = 10 MHz r

Slope 2

10' 10' lO-" Fig. 2 The spectral current

noise intensity Sj at 10 MHz versus applied voltage. The

critical voltage for the

onset of the electro-acoustic

effect is indicated by an arrow.
Fig. 3 The current noise intensity SjiCf) and

the absolute value of the ac impedance Z vs frequency. Resonances in |z| are indicated by

arrows

.

1.0 - ^

I

i

. =#= 2 O

0.8

i

0.6

^

t

0.4 .?

i

i j

1 < 1 Y —" v(v)

Fig. 4 The frequency where the relative acoustic energy density is at its maximum is plotted

versus applied voltage. The voltage V^, marking

the onset of the electro-acoustic effect is

indicated by an arrow.

By combining results of electrical measurements on relatively low ohmic samples with good

ohmic contacts with Brillouin scattering data on the same samples^ Westera [14,16] settled the

dispute by proving that transverse waves were responsible for the onset of the electro-acoustic effect. The measured threshold voltage (cf. figure 2) corresponded to a sound velocity of

transversal on-axis waves of 1.77 x 10^ ms~^ and a room temperature mobility of 2.20 x 10~^

m^V^'-^jin agreement with a value for the mobility reported by Spear and Mort [19]. Figure 3 shows Sx and the absolute value of the ac impedance |z[ versus frequency for the same sample

at (V - Vc)/V^ = 0.19. The spectrum has a Lorentzian frequency dependence with some modulations and can, apart from these modulations, be interpreted with eq (14). The observed crossover frequency corresponds to an average life time for troughs of 5.3 x 10"^ s. From the low

frequency plateau and a saturation current of 160 mA (deduced from the IV-characteristic) one

then finds with the help of eq (14) that N is about 10^. |z| is almost frequency independent

for frequencies larger than 10 MHz, whereas with decreasing frequency it gradually rises

towards the value of the differential resistance. In addition resonances are observed at the

odd harmonics of 450 Hz. The impedance data can be well interpreted with eqs (6) and (13) with

tj/t = 2.4 and provided that the velocity occurring in eq (7) can be interpreted as the x- component of a group velocity that corresponds to a phase velocity with an off-axis angle of

25" of a transverse wave, which is in good agreement with Keller [20] and San'ya et al. [21].

The very sharp peaks predicted by eq (6) are apparently smoothed out because of the occurrence

of a distribution of waves with different off-axis angles. It should be noted that minima

occur in the noise spectrum at the resonance frequencies of the impedance. However, spectral

noise intensities often differ markedly from a Lorentzian. This is supposedly due to the fact

that the 6-function space correlation for the trough creation and annihilation fluctuations no longer holds. Finally in figure 4 the frequency fj^, where the ratio of electro-acoustic to thermal acoustic energy has a maximum, as determined from Brillouin scattering data, is

plotted versus applied voltage. It was found that the behaviour of fjjj is in good agreement

with eq (9)

.

REFERENCES

[1] A.R. Hutson, J.M. McFee and D.L. White, Phys. Rev. Lett. 1_, 231 (1961).

[2] D.L. White, J. Appl. Phys. 33, 2547 (1962),

[3] N.I. Meyer and M.H. J^rgensen, Festkdrperprobleme X, Ed. 0. Madelung, Pergamon Vieweg

(1970) p. 21.

[4] H. Kuzmany, Phys. Stat. Sol. 25a, 9 (1974).

[5] G. Johri and H.N. Spector, Phys. Rev. B14 , 4955 (1977).

[6] P.K. Tien, Phys. Rev. m_, 970 (1968).

[7] A.R. Moore, J. Appl. Phys. 38, 2327 (1967).

[8] L. Friedman, Phys. Rev. j_74, 737 (1968).

[9] K. Nakamura, J. Phys. Soc. Japan 29, 860 (1975).

[10] K. Nakamura, J. Phys. Soa. Japan 40, 350 (1976).

[11] R.J.J. Zijlstra and P. A. Gielen, Physiaa 95B, 190 (1978).

[12] P. A. Gielen and R.J.J. Zijlstra, Physiaa 95B, 347 (1978).

281
[13] W. Westera, R.J.J. Zijlstra and M.A. van Dijk, Phys. Lett. 78A, 371 (1980).-

[14] W. Westera and R.J.J. Zijlstra, Physiaa 106B , to be published June 1981.

[15] Y.V. Gulguaev and A.G. Kozorezov, Sol. State Corrmun. 27_, 375 (1978).

[16] W. Westera and R.J.J. Zijlstra, Physica, to be published.

[17] C.A.A.J. Greebe, Philips Res. Repts. 20, 1 (1965).

[18] R.M. White, Eigh-Frequency Ultrasonic Devices, Topics in Solid State and Quantum Electronics, Ed. W.D. Hershberger, Wiley, London, (1972).

[19] W.W. Spear and J. Mort, Proc. Phys. Soa. 8\_, 130 (1963).

[20] D. Keller, Phys. Stat. Sol. 16a , 87 (1973).

[21] M. San'ya. M- Yamada, C. Hamaguchi and J. Nakai, Jap. J. Appl. Phys. \2_, 611 (1974).

282
EXPERIMENTAL STUDIES OF NOISE IN A CHEMICAL REACTION AND IN A FLUID FLOW

Harry L. Swinney, J. C. Roux*, and G. P. King

Department of Physics

The University of Texas

Austin, TX 78712

We are investigating noise that appears in two quite different physical nonlinear

systems when they are taken away from equilibrium. In each system two types of noise

can be distinguished, at least qualitatively: intrinsic noise and extrinsic noise.

The noise that is a consequence of random fluctuations in the boundary conditions is

termed here extrinsic noise , while intrinsic noise , which is the principal object of

our studies, is fundamentally different--it is the chaotic (unpredictable) behavior

that occurs in nonlinear deterministic systems.

Nearly two decades ago Lorenz [1] discovered that a simple nonlinear deterministic

model system could exhibit complex behavior that was completely unpredictable. It is now realized that the Lorenz model, which consists of three coupled ordinary differential

equations, is not an isolated example, but in fact nonlinear systems, which are common

in all sciences, frequently have chaotic regimes [2],

We will describe briefly the two systems we have studied and will give examples

of the noisy behavior that has been observed. Details of this work are described

elsewhere [3-5].

NOISE IN THE BELOUSOV-ZHABOTINSKI I REACTION

About twenty years ago Belousov discovered an oscillating chemical reaction that

has become the prototype system for the study of oscillations and complex dynamics in

chemical and biological systems. Soon thereafter another Soviet scientist, Zhabotinskii

,

extended Belousov's work, and subsequently the "BZ" reaction has been investigated by

many workers, notably Noyes and collaborators [6].

In our experiments the four input chemicals of the BZ reaction are continuously

fed into a reactor, which is vigorously stirred so the system is essentially homogeneous.

The resultant reaction produces more than 30 reaction products or intermediate

species [6]. The reactor has a fixed volume--the outflow rate for the reacting

mixture is the same as the input flow rate. In the experiments the distance away from

equilibrium is varied by varying the rate at which the chemicals are pumped through the

reactor (the concentrations are held fixed).

The concentration B(t) of one of the reaction products, the bromide ion, is

measured as a function of time and recorded in a computer. From these digital time

series records we determine (a) the power spectral density and (b) a two-dimensional

phase space portrait, obtained by plotting points [B(t), B(t + t)] for a suitably

chosen time delay t.

The time series, power spectra, and phase portraits are shown in Fig. 1 for two rates of flow of the chemicals through the reactor: in (a) - (c) the mean residence

time of the chemicals in the reactor is 62 minutes, and the system is essentially

* Permanent Address: Centre de Recherche Paul Pascal, Domaine Universitaire, 33405

Talence Cedex, France

283
Fig. 1 The Belousov-Zhabotinskii reaction in a periodic state [(a)-(c)] and in a chaotic

state [(d)-(f)]. The time dependence of the bromide ion concentration is shown in (a) and

(d), the power spectral density in (b) and (e), and two-dimensional phase plots in (c) and (f)

.

284
periodic , characterized by sharp peaks in the power spectrum and a limit cycle phase

space attractor.

In contrast, when the residence time is decreased to 54 minutes, the behavior is

qualitatively different, as shown in Fig. 1 (d) - (f): it is characterized by a broadband spectrum and a "strange attractor" [2] in phase space. This chaotic behavior has

been found by Turner [5] to be modeled accurately by a deterministic four-variable

model that contains the principal chemical mechanisms.

Noise is also present in the periodic state: the amplitude of the small peak

fluctuates [see the inner loop of the attractor in Fig. 1 (c)]. The small amplitude

peak is sensitive to external perturbations, and in fact the small amplitude peak

disappears at shorter chemical residence times (< 50 minutes); hence it seems likely

that the small amount of noise observed for this state is caused by extrinsic noise.

NOISE IN CIRCULAR COUETTE FLOW

In the circular Couette system a fluid is contained between concentric cylinders.

In our apparatus the inner cylinder rotates and the outer cylinder is stationary. The

distance away from equilibrium is given by the Reynolds number, which is proportional

to angular frequency of the inner cylinder.

At sufficiently small Reynolds number the flow is purely azimuthal, except near

the ends. Experiments [4] have revealed the following transitions as the Reynolds

number R is increased: (1) To Taylor vortex flow (at R = R^), in which toroidal

vortices encircle the inner cylinder^ (2) From Taylor vortex flow to wavy vortex

flow, in which there are traveling azimuthal waves on the vortices^ (This flow is

periodic--the velocity power spectrum consists of a single sharp frequency component

and its harmonics. )j (3) From wavy vortex to a doubly periodic modulated wavy vortex

flow, in which the spectrum contains two sharp fundamental frequencies and their

combinations; (4) From modulated wavy vortex flow to noisy modulated wavy vortex

flow, in which the spectrum has a broadband component in addition to the sharp

frequencies.

A velocity power spectrum for noisy modulated wavy vortex flow is shown in

Fig. 2 (a), and a photograph of the flow is shown in Fig. 2 (b). This flow appears

to be another example of intrinisic noise: the spectrum is not changed qualitatively

by imposing small amounts of external noise, and essentially the same spectrum has

been obtained in another laboratory by Walden and Donnelly [7]. Also, in a deterministic model of circular Couette flow (based on the Navier-Stokes equations),

noise appears at around 22R-, in qualitative agreement with the observed onset at

12Rc [8].

The transitions in Couette flow described above were observed for a system with

twenty or fewer vortices. Donnelly et al_. [9] found that when the annulus length was increased beyond about 80 times the size of the gap between the cylinders, a different

kind of unpredictability appeared, and this "noise" was observed in the periodic wavy

vortex flow regime even at very low Reynolds number (=2 R(.). This type of noise

is illustrated in Fig. 2 (c) and (d), obtained in our laboratory. The spectral component

corresponding to the azimuthal traveling waves is broadened [Fig. 2(c)] and this

broadening is due to wandering dislocations [Fig. 2 (d)].

Our preliminary studies of the dislocations indicate that they occur for conditions

under which the system is particulary sensitive to external perturbations. At a given

Reynolds number wavy vortex flows can be obtained for a wide range of axial wavelengths,

but the dislocations are much more numerous for wavelengths near the extremes of the

range. In fact, for wavelengths in the middle of the range, the system can be free

of dislocations even when it is very tall. Thus the dislocations appear to be an example of extrinsic noise.

CONCLUSIONS

We have described preliminary results that indicate that it is possible to distinguish between two types of noise in the systems we have studied: extrinsic

285
(b)

>-

I—

00

o.00

(c)

-2 o

C3O

2a)j

1 1*^1 '

3a)j

1 1 ' 1 ^ M

0 1 2 3

(d)

Fig. 2 Circular Couette Flow: (a) and (b)--a chaotic or turbulent flow at R/Rq = 15.1 in an annul us with a ratio of height to the gap between the cylinders equal

to 20. (c) and (d)--a chaotic flow at R/Rq = 1.6 caused by vortex boundary dislocations

in a tall annul us (height to gap ratio equal to 82). Power spectral density plots are shown in (a) and (c), where n is the frequency of rotation of the inner cylinder. The

frequency component labeled (nji is the frequency of traveling azimuthal waves. This

component is instrumental ly sharp in (a) but significantly broadened in (c). The fluid

motion that corresponds to the broad component ojg in (a) is not known. Photographs of

the flow corresponding to the power spectra in (a) and (c) are shown in (b) and (d).

The flow pattern is made visible by seeding the flow with small platelets that align

with the flow. Only the central portion of the tall annulus is shown in (d); note

the dislocations in the middle of the photograph.

286
noise, which is caused by external stochastic driving forces, and intrinsic noise, which

is the chaotic or unpredictable behavior that can occur in deterministic systems.

We should emphasize that the intrinsic noise discussed here does not arise from

thermally excited fluctuations. Chaotic behavior occurs in deterministic continuum

systems in the absence of fluctuations [2]. Studies of thermal noise in chemical

and hydrodynamic systems have shown that thermally excited fluctuations will be

difficult to observe in experiments such as ours.

How can a deterministic system exhibit apparently unpredictable behavior? It is

true that if the initial conditions and boundary conditions are specified, then the

behavior is determined for all times. However, in a chaotic regime, two sets of

initial conditions that are only infinitesimally different can evolve quite differently

at large times. Since in practice the initial conditions can never be specified with

arbitrarily high accuracy, it follows that the behavior of a chaotic deterministic

system cannot be predicted far into the future. This is called sensitive dependence

on initial conditions .

Our distinction between intrinsic and extrinsic noise in the experiments has been

a heuristic one. In the future we intend to be more quantitative. The data will be

analyzed using tests that have been developed to distinguish between intrinsic and

extrinsic noise processes.

The distinction between extrinsic and intrinsic noise is helpful in developing an understanding of physical phenomena, even though no absolute distinction can ever be

made. That is, a given set of noisy data might be simply and elegantly described by a deterministic model, yet one could in principle always devise a stochastic process,

albeit a complex one, that would fit the data.

We thank Professor Mark Kac for the discussion that led to the last paragraph. This

research was conducted in collaboration with J. Turner, W. D. McCormick, M. Kilgore,

D. Andereck, M. Gorman, and L. Reith. This work was supported by National Science

Foundation Grants CME 79-09585 and CHE 79-23627 and by The Robert A. Welch Foundation.

REFERENCES

[1] E. Lorenz, J. Atm. Sci . 20, 130 (1963).

[2] See, e.g., proceedings of recent conferences: Nonlinear Nonequil ibrium

Statistical Mechanics , Supplement 64, Prog. Theor. Phys. (1978); Nonlinear

Dynamics , ed. by R. G. Helleman, Vol. 357, Ann. N. Y. Acad. Sci. (1980)

.

[3] H. L. Swinney and J. P. Gollub, Physics Today 31 (8), 41 (1978).

[4] P. R. Fenstermacher, H. L. Swinney, and J. P. Gollub, J. Fluid Mech. 94, 103 (1979).

[5] J. C. Roux, J. Turner, W. D. McCormick, M. Kilgore, and H. L. Swinney, to be

published; J. Turner, to be published.

[6] See, e.g., R. J. Field and R. M. Noyes, Accounts in Chem. Res. 10, 214, 273 (1977).

[7] R. Walden and R. J. Donnelly, Phys. Rev. Lett. 42, 301 (1979).

[8] H. Yahata, Prog. Theor. Phys. 61, 791 (1979).

[9] R. J. Donnelly, K. Park, R. Shaw, and R. Walden, Phys. Rev. Lett. 44, 987 (1980).

287
DYNAMICAL NOISE IN TUNNEL DIQDE OSCILLATOR SYSTEMS

J. P. Gollub

Physics Department

Haverford Col 1 ege

Haverford, PA 190i»l

INTRODUCTION

Most of the noise phenomena exhibited by electronic devices have their origin in microscopic processes such as thermal fluctuations or the discrete nature of the charge

carriers. It is not widely appreciated that macroscopic nonlinearity can also produce

noise in an electronic circuit, and that this noise can be understood without reference

to noise sources. In this paper we briefly describe laboratory experiments and models

of the noisy behavior of coupled tunnel diode relaxation oscillators. This noise is far

larger than that caused by shot noise in the diodes, and is not appreciably affected by

microscopic processes. The noise is caused by instability of trajectories in the phase

space of the system. A more extensive description has been published elsewhere [1,2].

EXPERIMENTS

The circuit we have studied experimentally is shown in Fig. 1. (For details, see

Ref. 1.) It consists of two tunnel diode oscillators coupled by a resistance R,-. Because of the negative resistance of part of the current/voltage characteristic of the

diodes (Fig. lb), the diode voltages Vq] and \Iq2 and the corresponding currents \q] and

1 132 oscillate at frequencies determined by the inductances (and resistances). The voltage

signals are pulse-like, while the current signals are composed of segments of exponentials.

These signals can be either periodic or nonperiodic (noisy), depending on the way in which

each oscillator influences the switching times of the other oscillator.

Vd (volts)

Fig. 1 (a) Tunnel diode oscillator circuit. Each oscillator consists of an inductor, a resistor, and a tunnel diode; the two oscillators are coupled by the resistor R^. (b) Current/voltage characteristic of the tunnel diode. Arrows indicate the path of oscillation.

288
In order to demonstrate that this circuit exhibits broadband noise, we show power

spectra of the diode voltages for two different values of the natural frequency ratio Fq of

the uncoupled oscillators (Fig. 2a, b). In one case (Fig. 2a) the spectrum is composed of

sharp peaks at multiples of a fundamental frequency. This complex but periodic behavior is due to the frequency modulation of one oscillator by the other. The modulation is such that

the two oscillators phase lock; their frequencies are in the ratio of two integers.

This periodic spectrum is to be contrasted with that of Fig. 2b obtained by varying one of the inductances and hence changing the natural frequency ratio of the two oscillators. This spectrum contains broadband noise, and the peaks that remain are in fact

broadened as well. The behavior is hence nonperiodic or noisy under these conditions.

These two qualitatively different spectra, one of which is broadband, both result from a deterministic process. To prove this, we generated a discrete sequence of current values

{ ( I [)2) n ^ by cond i t ional sampl i ng whenever = 0 .kl \l , Id1=5 ma, and \/[)2<0.2 V. These

conditions enable us to reduce the dynamics to a single discrete dynamical variable. We

plot (lD2)n+l versus (lD2)n '° obtain Fig. 2c, d. These diagrams are known as return maps.

The use of conditional sampling to reduce the dimensionality of a dynamical system was

I 1 1 1 1 L_ 1 1 I I 0 0.5 1.0 1.5 2.0/0 0.5 1.0 1.5 2.0 / (kHz) / (kHz)

(/o2)„ (0.021 mA) iIo2)„ (0.021 mA)

Fig . 2 (a) Power spectrum of the diode voltage \Jq2 for 3 phase-locked periodic state. The peaks are instrumental ly sharp. (b) Power spectrum for a nonperiodic (broadband)

state obtained by changing the uncoupled frequency ratio Fq. (c) and (d) Corresponding

return maps obtained by conditional sampling of the current 1 02 to reduce the dynamics to

a single discrete variable. These maps demonstrate that the system is deterministic even

when it is noisy.

289
invented by Poincare. In the periodic case, the return map consists of a finite number of

discrete points which are traversed repetitively, as one might expect. In the noisy case, the relationship between (l[)2)n+l ^'oz^n still one-to-one, although it is no longer

a finite set of points. Thus, the process is determi n i st i c , in the sense that each value

is sufficient to predict the next one.

How can the process be deterministic and yet noisy? The answer lies in the steepness

of the mapping of Fig. 2b. When the slope of this mapping has magnitude greater than one, then two currents (lD2^n ^^'^'^ ^''^ separated by A] will be separated by A2>Ai at the next

sampling. More precisely, if the quantity

N

h = lim I I Jin|f((lD2)n)l - (D

N->-oo n=l

is positive, where f is the local slope of the return map, then separations are magnified

by a factor e^ per iteration, on the average. In Fig. 2b, h = 0 . 'tO ± 0 . 05 • As a result,

trajectories in a phase space spanned by Ip], \q2' ^D1» ^D2 ^""^ unstable, and this is the fundamental cause of the noise. The instability is common in many non 1 i near systems

and model s [3]

.

In order to further support the above explanation, we mapped out the behavior of the

system in a parameter space composed of the coupling conductance G(- = 1/Rj. and the uncoupled

frequency ratio Fq. Our goal was to compare the behavior of the experimental system and an appropriate numerical model. The experimental behavior is shown in Fig. 3a. The non- periodic (broadband) regions are labeled N, while periodic (phase- locked) regions are

labeled by the frequency ratio, or by the letter P where the integers are large. The phase

diagram is quite complex, with many different interlaced regions.

1.5 1.6 1.7 1.8 1.9 2.0 i.s 1.6 1.7 1.8 1.9 2.0 ^0

Fig . 3 Parameter space defined by the coupling conductance Gj. = l/R(. and the uncoupleu frequency ratio Fq of the two oscillators for (a) experiment and (b) numerical model. (See

text for legend.)

NUMERICAL MODEL

In order to demonstrate that noise sources are not required to reproduce this behavior,

we constructed a simple numerical model [l] using the standard circuit equations and a rectangular approximation to the I -V characteristic of the tunnel diodes. With this

290
approximation, the diode voltages change only in discrete steps, and the equations become

linear in the intervals between the steps. The solutions can hence be obtained by computing

the switching times for 1000 or more oscillations with very little cumulative error. Return

maps and spectra were then used to construct a diagram of the parameter space of the model

(Fig. 3b) to compare with the experimental behavior (Fig. 3a). The agreement is reasonably

good. Many of the periodic regions are accurately simulated, and the model correctly

reproduces the large region of nonperiodic behavior (with positive divergence rate h) near

the center of the diagram. There are also some significant differences which we believe

arise from the approximations of the model.

CONCLUSIONS

The fact that the numerical model correctly reproduces the noisy behavior of the

circuit, and the fact that the return maps are one-to-one even in the noisy state, provide

strong evidence that the observed noise is not due to thermal fluctuations or other microscopic processes. Rather, it arises solely from macroscopic non 1 i near i ty , and the resulting

nonperiodic orbits in phase space. Similar conclusions have also been reached by other

investigators [3.'+] > and processes of this type may be responsible for the onset of turbu1 ence in fluids [5]

•

Dynamical noise may occur in any system of nonlinear oscillators; the tunnel diodes

may be replaced by other negative resistance devices. Furthermore, if the inductances are

stray components, the noise will be at very high frequency and its source might not be

easily recognized. Noise levels beyond thermal values in oscillatory systems may be due to macroscopic nonlinearity more often than is generally recognized.

The experiments were performed by J.E. Socolar and E.J. Romer. This work was supported by

the National Science Foundation.

REFERENCES

[1] J. P. Gollub, E.J. Romer, and J.E. Socolar, J. Stat. Phys. 23, 321 (I98O).

[2] J. P. Gollub, T.O. Brunner, and B.C. Danly, Science 200 , (1978).

[3] M.l. Rabinovitch, Sov. Phys. - Usp. 2J_, hk3 (1978).

[k] Y. Ueda and N. Akamatsu, IEEE Trans. CAS-28 , 217 (1981).

[5] J. P. Gollub and S.V. Benson, J. Fluid Mech. 100, hk3 (I98O).

291
NOISE IN RESONANT GRAVITATIONAL WAVE DETECTORS

R. P. Giffard"*", P. F. Michelson and R. C. Taber

Department of Physics and High Energy Physics Laboratory

Stanford University

Stanford, CA 94305

INTRODUCTION ^

A striking feature of Einstein's general theory of relativity is the predicted existence of gravitational waves which can transport energy with the velocity of light. In fact,

various modes of gravitational radiation are allowed in all contemporary theories of gravitation. Gravity is a very weak force, and a significant flux of energy is only radiated by an

event of astrophysical scale in which objects of near solar mass move with a significant

fraction of the velocity of light. Efforts to demonstrate the existence of the radiation

thus depend on extraterrestrial sources. The careful study of waves from such sources might eventually provide a unique observational probe of physical processes involving

strongly non-linear gravitational interactions.

The pioneeering work of J. Weber in the sixties encouraged several groups of experimenters to build and operate sensitive gravitational wave detectors. In spite of a vigorous

program, the evidence for the detection of gravitational waves is not yet conclusive. This

situation is initially surprising in view of the enormous energy flux which is predicted from

the strongest sources. For example, current estimates of the gravitational wave luminosity

associated with the collapse of a spinning star of about 5 solar masses suggest a peak

energy spectral density of 10^^ wHz~l in the range 1 to 10 kHz. Such an event, occurring

once in the lifetime of a star and lasting about 1 ms, is expected to occur in a galaxy like our own between once every 10 and once every 75 years. For observations at a reasonable

rate, one would need to be able to detect such events in all galaxies within about 6xlO^-^m,

the distance characterizing the MlOl cluster containing several hundred galaxies. These

events result in about one pulse per month with a flux density of 2 x 10~^ wm ^ Hz"-*- at the

earth. In view of approximate sensitivity limits for astronomy in the visible —perhaps

4 X 10"-'-^ wm~^ , and for radioastronomy — considerably less than 1 Jy or 10"^^ wm~^ Hz"-'- , it

is clear that gravitational wave detection must involve special difficulties. In this paper

we consider the way in which various forms of random noise limit the sensitivity of existing resonant -mass detectors. The problem is unusual in the fact that thermal noise of both

electrical and mechanical origin is important. As an example to illustrate some of the

unconventional technology involved in obtaining a low noise level, we describe a refrigerated 4800 kg detector now operating at Stanford.

RESONANT-MASS DETECTORS

The problem of detecting gravity waves has been the subject of several reviews. ^"^ It is well known that the signal available from an ideal electromagnetic antenna irradiated

by an incident flux E is given by the product of E, the ideal antenna cross section X'^/^'n, and a directionality factor gj. which is equal to unity for an isotropic antenna. For

detection in bandwidth B, this power must be of the order of k^T^B, where T-^ is the receiver

noise temperature. Although physical arguments show that this relationship is also valid

for an ideal gravity wav^ detector, two factors reduce the performance of practical antennas

dramatically. Since the lowest order mass multipole which can emit gravitational radiation

is the quadrupole, the most promising receiving antenna is also a quadrupole. A typical

antenna, described as a "Weber Bar", consists of a massive aluminum cylinder oscillating in

its fundamental longitudinal acoustic mode, whose eigen-frequency must be within the

expected signal spectrum. The antenna oscillations are monitored by a readout system sensitive to mechanical signals. The mechanical quantity analogous to the radiation resistance

for such an antenna may readily be calculated by finding the gravitational radiation power

emitted for a given stored energy. An idea of the magnitude of this quantity can be grasped

292
ISOLATION ENCLOSURE

TRANSDUCER

ANTENNA

OPTIMUM

FILTER

DETECTION

SYSTEM

OUTPUT

SIGNALS V

Fig . 1 Typical arrangement of a resonantmass gravitational wave detector. The

antenna motion is converted by a transducer

into an electrical signal which is detected

by a sensitive amplifier. The transducer

may use either electric or magnetic field

modulation and may be parametric. In order

to obtain a suitable impedance match, resonant structures may be used in both the

mechanical and electrical branches.

from the fact that if a typical resonant mass antenna such as that described in Section 4 were to lose energy only to gravitational radition, its amplitude would decay with a time

constant Xg of the order of lO"'-^ times the age of the universe.

Obviously the bandwidth of a radiation-dominated gravity wave antenna would be ridiculously small. In practice, however, the antenna possesses mechanical dissipation mechanisms

which limit the decay time to a value x^j, typically of the order of 10^ s for a low temperature antenna. Thus in the most general sense, the efficiency of a resonant mass gravitational wave antenna (equal to the ratio T^/Xg) is exceedingly small, 10"^'^ in the case considered. If the overall detector noise level is dominated in bandwidth B by dissipation

in the antenna, assumed to be at temperature T^, the condition for detection of a continuous gravity wave by an isotropic antenna is given by

2

- EX /4Tr > k-,T B (x /x,), (1) — D a g d

where E is the signal flux in bandwidth B. For the example given, assuming T^ = 4K and

B = 1 Hz, the minimum detectable flux would be about 0.8 \m~^

.

PRACTICAL ANTENNAS

A typical arrangement for a resonant-mass detector is shown schematically in Figure 1. The low signal level part of the detector may be exactly represented by an electrical equivalent circuit in which a shunt resonator representing the antenna Is connected by a cascade

of linear twoports to the amplifier. The gravitational wave signal is correctly represented

by a current generator shunting the resonator. Since the signal of Interest is a pulse, it

Is best described by its Fourier Transform M(a)) . The relationship between M(a3) and the

energy spectral density E(a)) in the incoming wave may be obtained by an extension of the

antenna theory given in Section 2. It is easy to show that M(&)) can be expressed in the form

I

a)ZQ|M(a3)|^ = E(a))|^x2g^/2uxJ , (2)

where co = 27rc/A is the frequency, and Zq = /L/ C is the characteristic impedance of the

equivalent resonator. Equation (2) is most easily understood In terms of the response of a noiseless antenna to a very short signal pulse. The quantity Jsoj^ZqImCo)^) | ^, where cjj^ is

the antenna eigenf requency , is then equal to the available signal energy induced by the

pulse. The quantity multiplying E(a)^) on the right-hand side of the equation is known as

the Ajot^QfiaXod antenna CA-O^i isZCJUvn a. In the case of the antenna discussed In section 4,

0 is about 2 X 10 25 ^2^^ so that a pulse with a flux of 50 Jm~2Hz~-'- gives an available

signal energy of 10-23 J. 293
Each element in the equivalent circuit potentially contributes noise which can always be

modelled by placing a noise generator in parallel with each port. The total system noise

current referred to the input is given, as usual, by

S (CO) = 4k„T /a)Z„T, + I h. (a))/G. (co) , (3) n BaUdjjj

where hj (co) is the spectrum of the j'"'^ noise source and Gj (co) is a transfer function connecting it with the resonator input termainals. The first term on the right of the equation

represents thermal noise.

With optimum filtration the maximum signal-to-noise ratio pQp^ which can be achieved is

well known to be

00 Popt = 2F J^lM(") 1^/8^(03)} dco . (4)

~oo

A practical approximation is obtained by assuming that the noise level is dominated by the

antenna thermal noise over a bandwidth near the antenna frequency, and is much larger

elsewhere. Equations 2-4 then give the conditions for detection with unity signal-to-noise

ratio

E(a)al°rgj )\x'^g /2UT 1 = 4Trk--T Band /B t , . (5)

Fig. 2 Superconducting transducer. A thin

circular superconducting diaphragm is

clamped at its edge to one end-face of the

antenna. The mechanical resonance of the

diaphragm matches that of the antenna.

Facing the diaphragm and separated from it

by small gaps are two flat superconducting

pickup coils carrying a persistent d.c.

supercurrent

.

SCALE I I I II I 0 1 2 3 4 5cm

294
-SQ CUR

NOI

V-TRANSDUCER

ICAL UID \ \

»| / /-TRANSDl RRENT \ • \l I I MECHAN

ISE \ \i' I NOISE

I \ M/ /I

Fig. 3 Measured output spectrum of the

detector. Also shown are the calculated

contributions from various sources. Antenna thermal noise dominates in a region

of about 0.5 Hz close to the uncoupled

antenna frequency. Elsewhere transducer

electrical noise is most significant.

The optimization of the transducer design

requires attention to all of these noise

sources; with the available technology, the

parameters can be adjusted to cause any one

of the noise sources indicated here to domi- nate. The only exception to this is the

SQUID voltage noise; it has the same frequency dependence as the transducer voltage

noise, and for electrical Q's<2xl0', the

transducer voltage noise exceeds the SQUID

voltage noise.

840 842 FREQUENCY (Hz)

Another interesting limit obtains if the noise is entirely due to an amplifier of noise temperature T„. The detection condition becomes

N

E(co^) \ g /2ttt r g

(6)

which may be described as the amplifier limit of sensitivity [6]

It is clear from Equations 2, 3, and 4 that one task of the antenna designer is to maximize the quantity on the right of Equation (2) . The only free parameter for a given

frequency is which is minimized by building the heaviest possible antenna. The second

objective is to minimize the quantity S^C^jJ) over the largest possible bandwidth. In practice this involves using an antenna of high Q, at a low temperature, matched over the widest

possible bandwidth to a low noise amplifier. Losse," in the matching structure — the trans- ducer — must of course be minimized. An example of a working detector embodying these

principles will now be presented.

STANFORD DETECTOR

The cryogenic resonant-mass detector now operating at Stanford follows the form shown

schematically in figure 1. The cylindrical aluminum antenna weighing 4800 kg is suspended

by a multi-stage vibration isolation system inside a vacuum tank cooled to 4.2 K by liquid

helium. The unloaded Q of the antenna is 2 x 106.

The transducer shown in figure 2 makes use of the low mechanical and electrical losses

in superconducting materials. Oscillations of the antenna end-face are coupled to the fundamental mechanical eigenmode of a superconducting diaphragm, which modulates the inductance

295
2 -

>o

200

FILTERED OUTPUT

JoAlnAii-il L^vlvJU^^ jJ»jiAA.JiJi.

UNFILTERED OUTPUT

30 sec

I 1

1^

TIME

Fig . 4 Time-domain response of the detector to input pulses of known strengths applied by

a piezoelectric calibrator. The upper trace shows the detector output after filtration:

five calibration pulses with available signal energies of 2 x lO"^^ j are easily identified.

The lower trace shows the mean energy in the antenna modes: only a single large pulse of

energy 2 xlO'^-'- J can be positively identified.

of current-carrying pick-up coils, causing an a.c, voltage proportional to the velocity of

the diaphragm to appear at the output terminals . l^J The transducer output signal is fed to a Josephson junction parametric amplifier or SQUID. [^] The function of the diaphragm resonance is to provide a suitable noise match between the impedance of the antenna and the

much higher impedance characterizing the electromechanical transduction process. The

present transformation ratio is about 300, resulting in a useful bandwidth of about 1 Hz.

A detailed analysis of the signal-to-noise ratio of the detector will appear elsewhere . The main contributions of the overall system noise presently come from mechanical dissipation in the antenna and the transducer, and from a.c. losses in the transducer

output ciruit. Further contributions from the SQUID voltage and current noise are of minor

significance. Figure 3 shows the output noise spectrum of the detector and the calculated

contributions from the various sources. Figure 4 shows the response of the complete

detector in its present form to calibration pulses. The measured noise level after signal

processing corresponds to an available signal energy of 3x10"^^ J.

CONCLUSIONS

The analysis of resonant antennas is completely supported by the measurements discussed

above. The 4800 kg detector is now in almost continuous use for observations, which are being carried out in collaboration with other experimental groups.

The authors have collaborated with W. M. Fairbank, C. Chun, S. Boughn, J. Hollenhorst,

E. Mapoles, M. S. McAshan, and H. J. Paik on the development and construction of the

detector

.

296
REFERENCES

[1] J. A. Tyson and R. P. Giffard, Ann. Rev. Astron. Astrophys. 16, 521 (1978).

[2] D. H. Douglass and V. B. Braglnsky, in GzneAxit RoJboutbJAXy: An EA,yU>tZA.n Cznte.mAy

SiVLVZy, edited by S. W. Hawking and W. Israel (Cambridge University, Cambridge,

England, 1979).

[3.] G. Pizzella, Rev. Nuovo Cimento 5^, 369 (1975).

[4] G. Papini, Can. J. Phys . 52, 880 (1974).

[5] K. S. Thorne, Rev. Mod. Phys. 52., 285 (1980).

[6] R. P. Giffard, Phys. Rev. D. 14, 2478 (1976).

[7] H. J. Paik, J. Appl. Phys. 47, 1168 (1976).

J. N. Hollenhorst and R. P. Giffard, IEEE Trans. Mag. MAG-15 , 474 (1979),

[9] P. F. Michelson and R. C. Taber, to be published in J. Appl. Phys., June, 1981.

+ Present address: Hewlett-Packard Laboratories, 3500 Deer Creek Road, Palo Alto, CA

297
MEASUREMENT OF THE BROWNIAN NOISE OF A HARMONIC

OSCILLATOR WITH MASS M=389 kg

P.Bonifazi, F.Bordoni, G. V. Pallottino, G.Pizzella

Istituto Plasma Spazio del CNR, Frascati

Istituto di Fisica dell ' Universita, Roma

1. INTRODUCTION

We report experimental data on the Brownian noise at the fundamental longitudinal mode (Vj^=1795 Hz) of an aluminum (6061 alloy) cylinder (M=389 kg) in a cryostat at the temperature of the liquid helium.

The cylinder is part of an antenna system ^l"] designed for the detection of the gravi~

tational waves emitted by celestial bodies according to Einstein's general relativity theory [A.

The sensitivity of the antenna system is limited by the Brovmian noise of the mechanical oscillator and by the electronic noise of the instrumentation.

The measurement of the Brownian motion is very important for the gravitational experiment. It must be performed with accuracy over several decades of energy in order to observe possible small disturbances due to the environment, which affect the performance of the

antenna as a detector of gravitational waves.

2. THE EXPERIMENTAL APPARATUS

According to the energy equipartition principle, the Brownian energy of the oscillator is

M cu ^ X ^/4 = kT/2 (1) R

where M/2 is the reduced mass of the alxaminum bar and x(t) the displacement of the bar ends

measured relative to the center of mass. The temperature of the oscillator is increased from

2 T to T by the heating effect of the current noise, with spectral density I , due to the

e n preamplifier and to the electrical losses of the input node.

It can be shown that the equivalent temperature is

2 2 2 T = T+AT=(1+JX Q I /2 M CJ k) (2) e n R

where Q is the merit factor of the oscillator and the response of the piezoelectric transducer is

v(t) = ckxit) (3)

298
Our transducer consists of two Gulton G-1408 piezoelectric ceramics located in a slot

cut around the median plane of the aluminum cylinder. The preamplifier ^3,4^ has an input

stage with four FET's BF 817 connected in parallel and is followed by a selective amplifier

tuned at V with Q=10. The output signal is then applied to a pair of phase-sensitive deR

tectors driven in phase and in quadrature by a synthesizer operating at y^, in order to ex- tract the components p(t) and q(t) of the signal in a narrow band near y . The output signals are converted in digital form by a 12 bit analog-to-digital converter, with sampling time At which can be adjusted from 1 ms to 10 s, and stored on magnetic

tape. The block diagram of the experimental apparatus is given in Fig. 1. We also have an electronic analog chain composed by squarers, summer and integrators (t=l min - 3 h) , whose

output is applied to a chart recorder that provides quick-look information on the status of

the system.

cosa>t

FET

PP EAMPLIFIER

EZOELECTRIC

ERAMIC

ANTENNA

BANDPASS

AMPLIFIER

MAGNETIC

o

TAPE

PHASE SENSITIVE

DETECTORS

Fig.

The two signals p(t) and q(t) are independent and with equal variance 6

the probability density function of the variable

2 2 2 r (t) = p (t)+q (t)

Therefore

(4)

is given by the Boltzmemn function

2. < 2 . 2i,„ 2 f(r ) = exp|-r /iB^/2e> . (5)

We remark here that, since our transducer is wide-band, our antenna is a multimode de- tector; in fact, its vibration can be observed in the various odd longitudinal modes, that

are coupled with the gravitational radiation, and also in the flexural modes.

3. EXPERIMENTAL RESULTS

We report experimental results obtained in March 1980 with the M=389 kg

operating in Frascati j^s"]

.

antenna

Run T Q o( T T Duration

6 e e

(K) (10 ) (MV/m) (K)comp. (K)meas. of run

1 4.4+0.1 .667 57+1

2 5.2+0.1 .621 52+1

h m

5.9+.1 6.1+.1 35 36

5.5+.

1

5.6+.

1

5^6"^

) 299
Fig. 2

Fig. 2 gives a semilog plot of the hystogram of the variable r (t) in two different

operating conditions as specified in the table; N is the nxamber of samples, obtained with

At=l s.During run 1 we used a preamplifier with T =0.11 K; during run 2 we used a preamplifier

[a] with T =36 mK with input stage cooled at 140 K, but. the temperature of the bar was slightly higher due to a lower level of the liquid helium in the cryostat.

300
The scale of Fig. 2 is given both in volt and in K, as ccsnputed from the calibration

of the measuring system. The calibration is based on the knowledge of the parameter and

of the gain of the electronics; it has been also checked independently by means of capacitor plates which are used to inject a calculable amount of energy into the mechanical oscillator. Note that this system allows one to perform mechanical noise thermometry on objects

of large size, but requires a very careful isolation from the mechanical and electrical disturbances due to the environment.

The attenuation provided by the filters used at present amounts to 240 db at 1795Hz.

As regards the operation of our system as a gravitational wave antenna_^we recall here

that the quantity of interest is the sensitivity, which depends on the effective temperature T rather than the equivalent temperature T^.

The effective ten^eratxire is a measure of the detectable fluctuations at the input of

the antenna, when the data analysis algorithms are applied to the data. It has been shown

that when the antenna system is properly designed (optimum coupling)^ the effective temperature is equal to twice the noise temperature of the preamplifier [?] . In the two runs mentioned above , by using the Wiener-Kolmogoroff optimxim filter, we have obtained T =.42 K

eff (first run) and T , =.28 K (second run). The last value can be further reduced by modifyeff ing the apparatus, which was designed for an amplifier with T =.11 K. n

REFERENCES .-

-

1^1^ E. Amaldi et al. "Initial operation of the M=389 kg cryogenic gravitational wave einten- na" "Nuovo Cimento, Nov. -Dec. 1978, pp. 497-509.

[2] J. Weber "General Relativity and Gravitational Waves" Interscience, New York, 1961.

[3^ F. Bordoni, G. V. Pallottino "Low noise amplifier for gravitational research" Rev. Sci.

Instr., July 1977, pp. 757-761.

[^4^ F. Bordoni, G. V. Pallottino "Very low noise cooled audiofrequency amplifier for gravitational research" in press. Rev. Sci. Instr., 1981.

[5] E. Amaldi et al. "Information on the operation of the M=389 kg gravitational wave an- tenna of the Roma group at an effective noise temperature of '^^Qff~^ '-^ ^" Lett. Nuovo

Cimento, 5 July 1980, pp. 366-366.

[&] P. Bonifazi et al. "Data Analysis Algorithms for Gravitational-Wave Experiments" Nuovo

Cimento, vol. Ic, nov.-dec. 1978, pp. 465-487.

[7] R. P. Giffard "Ultimate sensitivity limit of a resonant gravitational wave antenna

using a linear motion detector" Phys. Rev. D, vol. 14, 15 nov. 1976, pp. 2478-2486.

301
TWO-PORT NOISE CHARACTERIZATION BY MEANS

OF A FEEDBACK METHOD

Gabriel Blasquez

Laboratoire d'Automatique et d'Analyse des Systemes

Centre National de la Recherche Scientifique

7 Ave. Roche, 31400 Toulouse, France

A BRIEF OUTLINE OF THE METHOD

Let us consider an operational amplifier set up as a voltage follower with non-inverting input connected to a voltage source E. It is known that the output voltage is equal

to E. This circuit is similar to the one given in figure 1.

Fig. 1 Equivalent circuit of a generator

E connected to the non inverting input of

a voltage follower.

By applying this property, the equivalent input voltage noise generator E plifying two-port X can be simply and directly obtained. To this end, the input

output of the two-port must be short-circuited, and the output noise voltage V We then have

of an amand the

measured.

(1)

Let us now consider an electrometer^ the input of which is connected to a current source

I. If R is the feedback resistance, the voltage output V^nK^I . By applying this property

to the two-ports having a sufficient gain, the equivalent input I current noise generator

can be simply and directly obtained. To this end a resistance R of a high value must be

placed between the input and the output of the two-port, and the output voltage noise V must be measured. We then have

(2)

Finally, let us consider figure 2 where a resistance R is connected in the way just

indicated, and where E is the equivalent noise source of R . ns s .

Fig. 2 Noise equivalent circuit of twoport X connected to a resistance R . E and I^^ are the equivalent noise generators of the two-port.

nx

302
It follows from the preceding that

With

<lEnsl% 2^ 4 RsZif (A)

and where k, T, Af have their usual meanings.

Let F be the noise factor of the two-port X connected to a resistance R at a temperature T. It is known that

Substituting (3) into (5) yields

<lEn5l>

If we assume that the output of X is loaded by Y^^ and that the noise of can be disregarded^ the noise power measurable at the output of the feedback circuit, is equal to :

where Re ) is used for the real part of the quantity between brackets.

Following (7) and (6) we have

P

Fc:! Ik (8)

4 f^T £FL£ CYl) A-f

We can see that in order to obtain F of X connected to a resistance R it is necessary

and sufficient to measure I ^nol^-^O"" '^nL- ^ more detailed analysis would s^ow that relations

(0> (2), (6) and (8) are only correct if the open-loop transfer function of the circuit

(voltage gain) in figure 2 is high.

EXPERIMENTAL RESULTS

In order to test the validity of the method under discussion, experiments were carried

out on integrated operational amplifiers, field effect transistors^and bipolar transistors.

The parameters measured were <| £1^,^ 1 > T^^l^> and F. The experimental procedure consisted

of estimating these parameters using a classical technique as well as using the method under

discussion. For the classical technique we constituted and amplifier stage with the device

303
under test, its load and the source resistance R .. The input noise was obtained from the

measurement of the low frequency voltage gain (1 KH^) , An example of the circuits used and

the results obtained are given in figures 3 to 5.

*fffth

Fig. 3 Indirect noise

measurement-sine wave method

10

-7

> 10"

X

c

UJ 10"

777; 7777

Fig. 4 Direct noise

measurement-voltage follower configuration

Fig. 5 Equivalent input noise genera- tor of a JFET (BIFET technology)

feedback method

indirect method. '

Vq = ov

10" 10

F{Hz )

10^

A satisfactory agreement is seen up to 20 KHz. Beyond it a decrease in the noise determined by the classical technique is observed. This effect is due to two causes : (i) the

decrease in voltage gain in the high frequency range which had not been taken into account,

and (ii) the attenuation by the output impedance of the device under test and the preamplifier input capacitance of the measurement system. This effect appears • at higher frequencies

in the method under discussion because feedback decreases (a) circuit sensitivity to two- port gain variations, and (b) output impedance (as long as the open-loop gain remains high).

CONCLUSION

By using the properties of the feedback systems it is possible to directly measure the

input noise of amplifying two-ports. This method has the advantage, when compared to classical techniques, of not requiring the measurement of the two-port gain. It follows that the

total number of measurements to be carried out is reduced from 3 to 1 , and the duration of

the experiments considerably decreased. The method is considerably faster, more accurate

and, in addition, is more economical to implement because it does not use sine wave or noise

generators. Finally, it can easily be automated.

The author wishes to thank J. Borreil for his measurement assistance and valuable experimental contribution.

REFERENCE

[1] G. BLASQUEZ, Revue de Physique Appliquce, 15 (1980) 1007.

304
NEW METHOD FOR WIDE BAND MEASUREMENT OF NOISE TEMPERATURE

OF ONE-PORT NETWORKS AT HIGH PULSED BIAS

D. Gasquet, J.C. Vaissiere, J. P. Nougier

Universite des Sciences et Techniques du Languedoc

Centre d 'Etudes d 'Electronique des Solides"^

34060 Montpellier Cedex, France

1. INTRODUCTION

Measurement of noise temperature of non linear one-port networks under high bias conditions must be performed using pulsed techniques in order to avoid heating of the device. The

usual technique [_ ] ] f

2 ^ consists in comparing the output signal Xn corresponding to the

studied noise temperature T^(V) of the device at the bias voltage V, to the output signals

xq and Xq corresponding to input noise temperatures Tq and Tq. Tq and Tq are known values

given by a standard noise generator. These three measurements give then 1 ] J

Tno- T = (T OonoGo - T ) (x - X )/(x - X ) (1)

if X and T^ are respectively the gain and the noise temperature of the apparatus, eq. (1)

follows from the fact that each noise temperature T^ gives an output signal xi = X (T£ + T^^) . The above equations are valid under the conditions that A and T^ are kept unchanged

during the three measurements, which implies that the load impedance at the input of the

apparatus remains constant (generally equal, for convenience, to its characteristic impedance Rq) • Since the device studied may be non linear, its impedance is bias dependent , so that

it must be matched to Rq, using a tuner without losses, which then keeps unchanged the noise temperature of the device [ ' 1[ 2 ^. It can be shown [ 1 "| that the finite duration to

of the measurement during the pulsed bias results in a statistical fluctuation of the output

signal of average value x, such that

/ X = \/2/mt^ Av) (2)

where Ox is the r.m.s. deviation of x, Av is the bandwidth and N is the number of pulses

over which the result is averaged (N to is the total duration of the measurement)

.

Since to must be as short as possible in order to avoid thermal heating, Av has to be

increased for increasing the accuracy. Unfortunately the impedance matching is very selective, and when impedances are matched at the central frequency of the band, they are mismatched at the edges, thus producing standing waves, hence losses.

The purpose of this paper is to evaluate these losses, which will be shown to be important (section 2)^ then to describe a new method allowing to take these losses into account

(section 3). The uncertainty on the measurement^ and example of results obtained^will be gi- ven in sections 4 and 5.

2. EVALUATION OF THE LOSSES

Using a computer simulation, the losses have been determined at each frequency for a given real input impedance R matched to 50 ^ at frequency Vq.'

^ Laboratoire associe au C.N.R.S., LA 21 et Greco Microondes. This work was partially

supported by D.G.R.S.T. (France).

305
Figure 1 , shows^for a stub stretcher tuner, the percentage of losses versus frequency V (plotted in relative units (v - Vo)/Vq , for various values of R in the range 50 J^- 10 kfi.

Figure 2 shows the overall losses, integrated in the bandwidth Av centered on Vo>

plotted versus R. The curves are given for a stub stretcher. The losses were found to be

much higher for a double stub tuner which is much more selective : for example the losses

corresponding to Av-0.1 Vq for a double stub are about the same as those corresponding to Av = 0.25 Vq for a stub stretcher.

Figures 1 and 2 clearly show that the matching impedance technique requires a very narrow bandwidth and hence very long or numerous pulses (see eq. 2) : each experiment is

then very long and tedious, and matching impedances at low repetition rate is much tricky.

Losses(%)

3. PRINCIPLE OF. THE METHOD WITHOUT MATCHING IMPEDANCES :

In this method we suppress the impedance matching and we measure the losses on the whole bandwith. For this purpose we use a circulator (see figure 3). The impedance at the port

2 is always kept equal to Rq, so that the input load of the amplifier is always equal to Rq hence X and T^ will be constant. Now we have to determine four values Tn, X , Ta, r, where r is the reflection coefficient of the device studied. Four measurements are performed;

1/ Port 2 : Rq at room temperature Tq. Port 1 : device of impedance Z, and noise temperature Tn we want to measure. The reflexion coefficient is r, and the signal x at the output

of the apparatus is

X = X[(\ -

I r|^) T + |r|^ T + T.I (3)

2/ Port 2 : Rq at a temperature Tq (noise generator). Port 1 : unchanged.

x'^ = A[(I - |r|^ T^ + |r|2 Tg + tJ (4)

3/ Port 2 : unchanged. Port 1 : a short circuit (r = 1)

= ^ (^G ^ V

306
4/ Port 2 : Rq at room temperature Tq. Port 1 : unchanged.

X =X(T + TJ

o o A

equations (3) to (6) give

|r|^= (x',-x^)/(x^-x^)

G o no Go

(6)

(7)

(8)

Note that eqs. (5) and (6) give A and T^. Eq. (1) is now the particular case of eq. (8)

when impedances are matched in the whole bandwidth, since then r = 0.

Circulator

|

Apparatus

fig 3

100

I

r —I

1—T—I I I I

I ^

New method • Old method (double stub)

4. Old method (corrected)

>) = 850 MHz O /^'j=142MHz

5 50|

I

, c

20-

IOL

fig.

4

N^=2.7«10 cm"

J I I 1 I L_l_ 5 10 E (KV/cm)

4. UNCERTAINTY ON THE MEASUREMENT

Equations (7) and (8) give :

A(T -T ) A(T -T ) (Ax +Ax )(x ~x' ) Ax' + Ax

n o _ " ^ + S

o G n n G

T -T n o G O (xnonoGn -X ) (x -X + x_-x' ) xnoGi - X + x_-x'

(9)

From eq. (2) , we define the uncertainty Ax as

Ax = a a , with a = 1.96 since N > 30. X

(10)

The number of measurements performed for getting x , x , x^ and x'^ will be set equal

to No, Nq, Nn and Nn respectively. Eqs (9), (10) and (25* to (6)"give : ^

A(T -T ) = n o

A(T^-T^)

V^o ^ (l-lrl^) M'oA^/skjl

(T -T ) + n o

(11)

I We can see that the absolute uncertainty is the sum of a constant term and a term proI portional to (T^-Tq) . In order to increase the accuracy, one should increase No and N^ ,

|i 307
decrease the reflection coefficient (R near Ro) , increase the bandwith Av , decrease T^.

If the impedance of the sample is matched, the uncertainty is given by (11) using r = 0.

5. RESULTS

We have built two apparatus, the first one works between 220 MHz and 850 MHz with five

points of frequency (220 MHz, Av/Vq = 10 % ; 360 MHz, Av/v^ = 10 % ; 460 MHz, Av/Vq = 10 % ; 650 MHz Av/Vq = 5 % ; 850 MHz, Av/Vq = 1 %, Av/Vq = 5 % and Av/Vq = 15 %), the second

one works at 10.5 GHz, Av/Vq = 10 %.

Figure 4 shows the excess longitudinal noise temperature Tji(E)-To measured, Vq = 850MHz^

for p-type silicon bars (N^ = 2.7 x lO'^ cm'^, Tq = 300 K, // <111> , R = 184 ^ at Eo=0) versus the electric field Eq up to Eq = 12 kV cm"'. The triangles represent experimental

points measured using the new technique described in the present paper, although the dots

represent results obtained using the old method (impedance matching) . The conditions were

such that a great difference could be exhibited, as one can see on figure 4 : thus, we used

a double stub tuner (which is more selective than a stub stretcher) and a relatively large

bandwith of 142 MHz ( 0.15 Vq) • Of course, it has been verified that the noise was white

at that frequency. The crosses on Fig. 4 were obtained from the dots by applying a computed correction using a simulation analogous to that mentioned earlier, but performed for a double stub tuner. The losses are thus shown to be quite important. As comes out from the

theory here above, only the impedance mismatching was taken into account, the line was supposed to be perfectly non dissipative : indeed, the results represented by the crosses should be raised by the amount of the power displayed in the resistive part of the line and

of the insertion losses of the connectors, which could not be evaluated : as can be shown

on Fig. 4, losses a low as 1.3 dB are sufficient to take into account the difference bet- ween the triangles and the crosses.

6. CONCLUSION

The new method, presented here, needs four measurements. Its advantages are the following :

(i) All the losses (mismatching as well as insertion losses) are taken into account

on the whole bandwidth.

(ii) The bandwidth may be wide, thus allowing short pulses.

(iii) The matching impedance, which is tricky and time consuming at low repetition

rate pulses, is suppressed.

(iv) It allows automation of the measurements.

The mismatching must be low enough so that the losses be lower than about 90 % of the

noise power, so as to reach an acceptable accuracy : this corresponds to an a.c. impedance

of the device of about 2 kiL Note that, for such devices, the noise level is in many cases

so high that the accuracy on the 10 % noise power remaining is good enough.

REFERENCES

[1 1

J. P. Nougier, J. Comallonga and M. Rolland, J. Phys. E 7_. 287 (1974).

J. P. Nougier, Proc. 5*''^ Int. Conf. on Noise in Physical Systems, Springer Series

in electrophysics 2, Springer Verlag, 1978, p. 72.

308
APPLICATION OF THE THEORY OF

RANDOM MATRICES TO A REACTOR NOISE PROBLEM

Felix C. Dlfllippo

Department of Nuclear Engineering

University of Tennessee, Knoxville, TN 37916

J. J. Pinieyro

Departamento de Reactores

Comision Nacional de Energfa Atomica

Buenos Aires, Argentina

INTRODUCTION

There are neutron reacting systems where a degree of randomness exists in the spatial

distribution of materials. Fluctuations in composition appear, for example, as a consequence of the operational characteristic of boiling water and pebble-bed reactors.

Traditionally these type of systems have been approximated by homogenizing methods that use average values for the distribution of materials; this deterministic approach gives values

of reactivities and flux distributions that do not agree with a stochastic treatment of the

problem [1, 2, 3, 4]. According to ref [1] a knowledge of the statistical laws obeyed by

the system are necessary in order to obtain an equivalent homogeneous system. In this , context, a model has previously been developed which gives the first two moments of the

reactivity distribution in an ensemble of reactors having a specified random distribution

of macroscopic absorption and fission cross sections. An important result of this previous

work is that the use of higher order perturbation theory gives a positive value for the

average reactivity as opposed to a value of zero when calculated by ordinary perturbation

theory. Recent work [3] confirms this conclusion but these same authors [4] stress that

the sign of the average reactivity depends on the fluctuations in the moderation process.

In the present work we have recast the equations derived in ref [2] in a matrix form.

In this way the distribution of reactivities can be described by the distribution of the

highest eigenvalues of random matrices, with the result that all the mathematical

techniques developed for the study of the statistical laws of excited nuclei [5J can be

applied to the similar reactor physics problem. Following a presentation of the theory, a numerical example is given, for which a Monte Carlo simulation of several ensembles of reactors is made.

Using the standard nomenclature of reactor theory, the reactivity, p, of each member

of an ensemble of reactors is given by the highest eigenvalue of the equation

THEORY

(1)

where the matrix elements of [C] are given b

(2)

309
and the components of vector a are the coefficients of the expantion of the neutron flux in

a basis Tt given by the Helmholtz equation. In eq (2),52Jf and 5-?<a. are, respectively,

the fluctuations of the macroscopic fission and absorption cross sections, is the

reactivity of the ith mode in the reference homogeneous reactor, and S'c^ is the Kronecker

delta functions. Equations (1) and (2) were obtained with the following assumptions: (a)

each member of the ensemble is a bare reactor; (b) one energy group theory applies and (c)

random dispersal of materials does not affect the diffusion coefficient, which is assumed a constant throughout the reactor volume.

Due to the linear relation, eq (2), between and the fluctuating cross sections, the

matrix element are Gaussian distributed. If, in addition, we assume that the ensemble

average of and are zero for all v~ , the mean value of CjLt is equal to (P/ at\

and the variance is given by

^ ^ i

where R is the reactor volume and pj; is the correlation function.

where o5= j) J'>^/; — ,

If we make the additional assumptions that the fluctuations are correlated only within

a distance X and (b) the ensemble average ^J^^'^ is independent of , then the variance

can be approximated, in the case of a slab reactor, by

where the thickness CL is much greater than A . Under these conditions the distribution of

reactivities, p(p), is given by the distribution of the highest eigenvalues of random

matrices which have the following properties: (1) they are real and symmetric; (2) the

elements are Gaussian distributed; (3) the off-diagonal elements have zero mean; (4) the

diagonal elements have mean values equal to PZ ; (5) all off-diagonal elements have the same variance, O^p ; (6) all diagonal elements have the same variance , and (7) defining

^^X^up/c^ this parameter has a value 4/3 in our case.

The probability distributions are known for particular and restricted cases. For

example, if the diagonal elements have zero mean and |k ^ I , the probability distribution is

given by the Wishart law [6], a distribution well known for its application to the analysis

of level distribution in excited nuclei [7]. A relaxation of these conditions, namely

diagonal elements with zero mean was studied [8] in the case of 2 x 2 matrices.

We will now derive a more gene.ral expression for p(p), in the case of 2 x 2 matrices having

properties 1 to 7 above. The joint distribution H of the 3 independent matrix elements of

a symmetric 2x2 matrix is given by

(6)

310
where G is the Gauss distribution; the highest eigenvalue is given by

(7)

If we ncfw select C^^ , C|| and O as independent variables the corresponding distribution,

P, is given by

P(C<„c,„f) = 2U[c„<.,c„,c„,(p)]

^ , (8)

where the factor 2 appears because, given Coo, Ci/ , and p , two values of C^; satisfy eq

(7). Integration of eq (8) with respect to Coo gives p(p), namely

OO

-2 -{7:7'

(10)

where we have assumed that the reference reactor is in the critical state ((^=0). However

in the case Pi^O and ^ ^£ , no closed expression can be obtained for p(p). Expanding

the exponential in eq (10) as a power series and Integrating term by term, we have

(11)

where 1^ are the modified Bessel functions of order n.

Introducing eq (11) into eq (9), an asymptotic expression for p(p) can be obtained

when ^j^^4jp,j^» ^o'^ values of p near the average^p^, this is

OO

(12)

311
where the value has been substituted. However, due to the approximations, p(p)

vanishes when p--/,S?/Pt/ whereas we should obtain a smooth behaviour for p_^o®. The

corresponding moments are

(13)

which implies that the mean value and the skewness of the distribution are positive and

that the distribution has a kurtosls less than 3.

MONTE CARLO SIMULATION

Due to the availability of diffusion theory simulations of ensembles of reactors, we chose to study the same ensembles as in ref {2] • In this way we compared our matrix

simulations with simulations based on the direct solution of the diffusion equation. The

properties assumed for the reference reactor and the stochastic parameters that describe

each ensemble of reactors are given in Table I.

Table I. Data for the reference reactor and stochastic parameters

of different ensembles of reactors

Reference reactor Stochastic parameters

Slab width (cm) 100

0.05

0.1

2.01974

D(cm) 1

Po 0.0

P. -0.02932

Ensemble Xcm)

1 4. 3.33 E-3 0 0.0140

2 4. 0 3.33 E-3 0.0141

3 4. 8.33 E-4 0 0.00700

4 4. 0 8.33 E-4 0.00707

5 10. 8.33 E-4 0 0.0111

6 10. 0 8.33 E-4 0.0112

As in ref [2] rectangular probability distributions for « and O-^/C were assumed

for each ensemble without reference to any particular variation of composition. These

distributions were symmetric about zero and were scaled to provide relative variations of

±10% (ensembles 1 and 2) and ±5% (ensembles 3, 4, 5 and 6). For each ensemble we ran a sampling of 1,000 reactors, in each case the order of the matrix was varied so as to see

the effect of expanding the flux with a limited number of harmonics In all cases an symptotic behaviour was achieved, for matrix order greater than a certain minimum, the two simulations agreed, and for small perturbations (cases 3, 4, 5, and 6) even

simulations based on low order matrices gave reasonable results.

Other aspects of the theory were also studied such as the Gaussian distribution of

matrix elements and the distribution given by eq (12) for the highest eigenvalues of 2 x 2 matrices having the seven properties listed previously. In particular, we compared mean

value, variance, skewness and kurtosls from Monte Carlo simulation with the theoretical

results. In all cases the agreement was within the statistical uncertainties attributable

to finite sampling. However, we found that while the shape of the distribution of

reactivities from simulations with the appropriate matrix order is reproduced reasonably

well by eq (12), the mean values tend to be preserved by this equation when the perturbations are small.

312
CONCLUSION

We have related a space-domain reactor noise problem to the theory of random matrices.

Simulations based on random matrices have been shown to give good results in comparison to simulations based on numerical solution of the diffusion equation, thereby allowing us to apply mathematical techniques developed for nuclear physics problems to the calculation of

the probability distribution of reactivities. This distribution is not Gaussian and Illustrates the non-linear relationship between variations of reactor materials composition and

reactivity. The shape of the theoretical distribution ' obtained compares quite well with a Monte Carlo simulation, but the theoretical averages reproduce the numerical results in cases with small perturbation.

The probability distribution of reactivities was related in an explicit way with a stochastic parameter (*^o) which is determined by the random distribution of . materials in a reactor. In this way it is possible to relate the probability to find the system in a given reactivity range with the degree of randomness, this relationship might be Important

in the safety analysis of nuclear reacting systems with inherent randomness. An important

limitation of the results presented in this report is the use of one energy group

approximation, further investigations are needed in order to Include fluctuations effects

in the moderation process in our model.

REFERENCES

[1] M. M. R. Williams, Random Processes in Nuclear Reactors (Pergamon Press, Oxford, 1974)

Chapter 8.

[2] H. van Dam, in Progress in Nuclear Energy, Vol 1, Numbers 2-4, Edited by M. M. R. Williams and R. Sher (Pergemon Press, Oxford, 1977), p. 273,

[3] S. Yamada, M. Nlshimura, and K. Sumita, Annals of Nuclear Energy, T_, 561(1980).

[4] S. Yamada, C. Yamagoe, and K. Sumita, Annals of Nuclear Energy, 7_, 655(1980)

[5] C. E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New

York, 1965).

[6] S. S. Wilks, Mathematical Statistics (Princeton University Press, New Jersey, 1943),

Chapter XI.

[7] J. E. Lynn, Theory of Neutron Resonance Reactions, (Clarendon Press, 1968) Chapter V.

[8] C. Porter, N. Rosenzweig, Suomolalsen Tiedeakatemlan Tolmituksia (Ann. Acad. Sci.

Fennicae) AVI, No. 44(1960).

313
BURST NOISE MEASUREMENTS ON OP AMP'S BY R.F. SQUID

Arnaldo D'Amico

Istituto di Elettronica dello Stato Solido del

Consiglio Nazionale delle Ricerche

Via Cineto Romano, 42

0015 6 Roma

During the past ten years burst

noise has been extensively studied in

P/N j unc t i ons [ 1- 3] , transistors [^-^J

but not sufficiently in operational

amplifiers which are used in many [8j

fields of application such as:low noiSe

AC pre amp lifier'Iow noise^low drift

low offset DC instrumentation amplifier^etc. At both low and intermediate frequencies^ they arethe most important and utilized electronic devices.

Here we report preliminary burst

noise measurements performed on com- mercially available op amp ' s by a new technique. The measurement technique

considered makes use of a high sensitive low noise radiof requency superconducting quantum interference device

(SQUID) which has allowed both the detection and the measure of the very

low magnetic field intensity due to

current variations into the devices

associated to the burst noise mechanisms .A superconducting magnetometer

coupled to a second derivative gradiometer used as spatial discriminating

pick-up coil has been utilized. By u- sing such a gradiome ter^ the SQUID system becomes practically insensitive

to both spatial uniform magnetic field

and gradients^ and it can be used to

detect the weak magnetic field of a nearby source also in the presence of

a stronger but uniform field at distant sources.

The same apparatus is almost daily used for biomagnetic measurements

in our Institute by the biomagnetic

314

group and it has been extensively described in a recent paper [9].

The main performances associated

to the SQUID system and related to the

noise measurements are the following:

the r.m.s. noise is about 40fT//Hz , i.e. 40 X lO"-*- 'gauss //Hz above its

l/f*^ corner frequency which is located at about .5 Hz (see Fig. 1).

101

1^2

-14 10 ' 1(52 10^

Fig. 1. Noise spectrum of the 2nd de*

rivative gradiometer used for burst

noise measurements

The overall gain is 12.5 x lO^V/

gauss and the bandwidth used is:

BW = 700 Hz (with this value the minimum detectable pulse width is about

500 fis) . The SQUID system is actually

limited, due to a sensible deviation

from the linearity only at 8KHz. Measurements have been carried out by

putting the head of different kinds

of bipolar op amp's as close as possible to the pick-up coil of the SQUID
used.

The first network configuration

considered round the op amp is shown

in Fig. 2 where we see a noninverting

unity gain configuration with a feed

back low noise resistor value Rf and

a load low noise resistor Rc.

Fig. 2. The non inverting configuration

The second external network used (see

Fig. 3) refers to an op amp in the

differential configuration with a DC

battery used to change the common mode DC input voltage.

90 inV which means that the SQUID detected a magnetic field intensity of

about 10"^ gauss. Taking into account

as a crude estimation of the decoupling between the magnetometer and the

o op amp, a factor 10 , current variations, due to burst noise in the output load of about 10 p.k have been mea

sure d

.

Photo 1 shows the burst noise in

two traces: the upper one represents

the output we can observe by using a low noise AC amplifier chain, with a low cut off frequency of 5 Hz, the

lower one is the response (inverted)

of the superconductive magnetometer

chain. The good agreement suggests to

continue to improve this method in or der to make it more promis ing^ al though at present the high cost of the ap

paratus compared to the conventional

one seems to be the only remarkable

inconvenience.

Fig. 3. The differential configuration

The two circuits have been utilized

to investigate burst noise amplitude

duration and rate of occurence at a nearby constant temperature and in

different conditions as far as the common mode input DC voltage values

are concerned.

All measurements have been done

at 15°C with a tempearture stability

of .5°C by using a cooling system based on the Peltier effect. This value

of temperature has been selected because it is the highest (for most of our samples) which can permit a sufficient number of bursts to be statistically analyzed in a reasonable

time

.

In this condition^ a typical

burst noise amplitude measurement

gave us an overall response of about

Photo 1 : upper trace is the response

of the electronic chainj

lower trace is the response

of the SQUID

Fig. 4 shows experimental data of the

burst noise amplitude behaviour obtained by considering the schematic

of Fig. 3 where the common mode DC

voltage has been gradually changed.

M2'-'

4-

315

~~2

4 6 8 lOwlJ

Fig. 4 Burst noise current measured by the

SQUID versus the DC common voltage ,
Our future developments and studies on this subject wi-11 be polarized towards the low temperature dynamic dependence of both burst noise am- plitude and occurrence, for different

closed loop gain values in order to

collect sufficient data to validate

the current interpretation on its both

physic origin and location |lO,ll|.

Finally we list some features which

can be associated to this non touching techniques

:

a) the posssibility to measure 1 /

,

g-r and burst noise because of

the intrinsic 1 / f low noise of

the SQUID system^

b) the possibility by a suitable

pick up coil with reduced dimension coupled to the magnetometer

to have "local" noise information

in scanned surface investigation^

c) the possibility to measure either

the excess noise or other kind of

noise in very high impedence devices where the matching problem

between source under investigation and conventional low noise

preamplifier could be a problem^

d) the possibility to check the

noise behaviour in working active

and passive electronic chains.

The author wishes to thank Dr. P. Carelli. Dr. R. Leoni and Dr. G.L. Romani for their assistance and discussions. Particularly, he wi,she& to

thank Dr. G. Spissu for her skilled

technical assistance.

REFERENCES

[l] D.S. Tan, "Current pulses during

breakdown in silicon p-n junctions"'

Proc. IEEE (Corresp.) Vol. 53,

Feb. 1965, pp. 210-211

J. Appl. Phys. Vol,

p. 189-192

38, Jan. 1967

[2] J.F. Schenk, "Burst noise and

walkout in degraded silicon devices", Proc. 6th. Annu. Reliability of Physics Symp., Nov. 6-8,

1967, pp. 31-35

[3] D. Wolf and E. Holber, "Bistable

current fluctuations in reverse biased p-n junctions of germanium

[a] S.Y. Hsu and R.J. Whitter, "Characterization of burst noise in

silicon devices". Solid State

Electron., Vol. 12 , 1969, pp. 867

878;

[5] G. Ginalt, J.C. Martin and F.X.

Ma ten-P ae r i z , "Burst noise of silicon devices", Solid-State Electron., Vol. 12, June 1956 , pp. 228-

229.

[6] R. C. Jaeger and A. J. Broderson

"Low-frequency noise sources in

bipolar junction transistors",

IEEE Trans. Electron Devices,

Vol. ED-17, Feb. 1970 pp. 128-133

[7] S.T. Hsu, R.J. Whittier, and C.A.

Mead, "Physical model for burst

noise in semiconductor devices".

Solid State Electron., Vol. 13,

1970, pp. 1055-1071

[8] Sheng. T. Hsu, "Bistable noise in

operational amplifier" IEEE Journal of Solid-state circuit. Vol.

SC-6, No. 6, Dec. 1971

[9] S. Barbanera, et al., "Use of su- perconducting instrumentation for

biomagnetic measurements performed in a h ospi tal "pre sen te d at

A.S.C. Sept. 29, 1980, Santa Fe

,

New Mexico, USA

[10] K.F. Knott, "Anomalous 1/V and

burst noise characteristics as- sociated with surface channel in an NPN integrated transistors"

Solid State Electronics, Vol. 23,

pp. 727-733, Nov. 1979

[11] K.F. Knott, "Evidence of collector base junction burst noise".

Electronics Lett., Vol. 15, No. 6 pp. 184-198, Mar. 15, 1979.

316
DOMAIN WALLS IN PEEHOMAGHETIC MATERIAL

AND lEEEVERSIBLE MAGNETIZATION PROCESSES

Wilhela Grosse-Nobis

Institut fxir Angewandte Physik, Universitat Miinster

D-4400 Miinster, Fed. Rep. Germany

INTRODUCTION

The concept of domains and displaceable domain walls bad been successful in the early years to explain the macroscopic featxires of ferromagnetism

as the magnetization ciirve and the Barkhausen noise. The first measurements

on moving domain walls were made by Sixtus and Tonks [1] in 1931 • Since then

walls became by further experimental and theoretical work objects which now

are fundamental in xinderstanding the behavior of most of the varied phenomena

in applied f erromagnetism and ferrimagnetism.

The wall displacement is the process which makes the easy remagnetization of both soft crystalline and amorphous solids possible. In case of large

excursions; walls are mainly displaced irreversibly because of randomly distributed defects of the magnetic solid which act as pinning centers. This may

result in sudden macrosopic wall displacements, the Barkhausen Jumps. This

report deals mainly with the space-time behavior of domain walls in soft ferromagnetic material and is confined to measurements performed by electrical

methods. Regarding this subject, the monograph of Stierstadt [2] and the re- view article of McClxire et al. [5] are to be mentioned as well as the articles by Bittel [4] and Montalenti [5]» Wiegman [6] gave a report on Barkhausen noise in thin films.

ORIGIN OF WALLS

Crystalline and amorphous solids exhibit f erromagnetism because of the

exchange interaction which aligns electronic moments against thermal agitation and thus defines a direction by the macroscopic saturation polarization vector J . The other main property is the crystal anisotropy which defines easy axes of lowest energy content for the polarization vector being

parallel with them. These easy axes are well defined in an ideal crystal lattice, but exist also in a metallic glass. These two features alone may cause

the solid to split into several Weiss domains magnetized each in one of the

317
easy directions and which are separated by the above mentioned interfaces*

e. g. the Bloch walls. Their interfacial energjy density of order 1 mJ/m^ in

soft magnetic material raises the energy content of the solid, and hence

walls are stable \inder internal or external restrictions. These inhomogeneous

states of finite solids are result of the other main sotirces which raise the

total energy of the body: the stray field energy due to divergencies of the

magnetic polarization, the magnetostrictive stress energy and the field ener- gy due to an applied external field. The static micromagnetic and Maxwell's

equations determine the possible states of the finite solid under given boundary conditions [7] if noise is excluded.

WALL DISPLACEMEMT

Bloch walls are in most cases thick (about 100 nm) compared with the

lattice constant and could easily be moved in an ideal crystal by changing

the external field. But the real structiire of magnetic material is given by

random fluctuations of compositional and structural defects and internal

stresses. These distiirbances determine energetically favored positions of

the walls with characteristic distances well comparable with the wall thickness and hence are very effective as pinning centers for the walls [8]. For

this reason and because of the flexibility of the interfaces, which introduces a large number of internal degrees of freedom, as well as the nonlinearity of the micromagnetic equations a large set of stable solutions of the

magnetic state is generated. Some of them may have a limited range of existence as function of, say, the external field. Hence, if the system of domain

walls occupies a certain metastable state and the field is changed quasistatically to a certain value, the domain wall system will change discontinuously to a configuration given by another possible solution. These transitions

correspond to the observed Barkhausen discontinuities. The configuration

which is actually reached depends on the exact dynamics of the domain walls

and in some cases of the pinning defects if they are also mobile, e. g. by

thermal agitation. Moreover, ;just before walls are released from pinning centers^ the noise in the solid is able to influence very effectively the branching into the different states and thus determines the behavior on a macroscopic scale. Lieneweg [9] found an increased Barkhausen noise spectrum below

10 kHz depending on the rate of field change for a hard-drawn 50% NiPe-wire

at temperatures below 77 ^ which he attibuted to the lack of thermal agitation with the consequence that only large segments of the Bloch walls can

move coherently. This means that large groups of Barkhausen Jumps occiir and

determine the Barkhausen noise spectrum [10]. This grouping is of great importance in the theory of losses of magnetic material [11,12]. t The thermal activation theory has been reviewed by Gaunt [13] and the

book by Lambeck [14]. The influence of thermally activated unpinning of walls

318
from defects was recently investigated in case of thin films and massive material [15-17]. Some of the papers discuss also the possibility of quantum

mechanical tunneling which could be the main process below some Kelvin. In

the case of narrow domain walls degenerated to socalled monatomic walls the

thermally activated wall displacement determines the available coercive field

of Co-B£ intermetallic compounds [18] and hence applies to hard magnetic material.

SIGNIFICANCE OF DOMAIN WALLS

The existence of displaceable domain walls is the reason that soft magnetic material is magnetized by low fields. This is due to the fact that the

wall displacement process prevents the polarization vector from attaining

transitory other than the easy directions in the major parts of the solid and

thus from dissipating the large amount of anisotropy energy stored when the

polarization passes the direction of extreme anisotropy energy. Existence of

walls on the other hand means that the process of remagnetization is very inhomogeneous: the rate of polarization change is high at the interfaces and

otherwise very low. This concentration of the externally impressed flux

change causes near and inside the walls high dissipation by eddy current and

spin relaxation damping because of the considerably high wall velocities. The

internal degrees of freedom of the interfaces enhance this effect, e. g. by

micro eddy currents and emission of sound waves. Hence, the existence of

domain walls as the nonlinear elementary excitations of a ferromagnet with

localized energy leads far away from thermodynamic equilibrium, even when we try to magnetize the solid quasistatically. On the other hand^ further walls

may be produced if a given sample is magnetized at high rates [19»20], and

there exists a close relation between the ntimber of walls and the magnetizing

frequency [21,22]. At low rates there might be evidence that for a given 180°

Bloch wall the average simultaneously moved wall area depends on the mean

wall velocity [23].

The dynamics of wall displacement is thus an important contribution in

understanding losses associated with the widespread applications of soft

magnetic material. The method of Sixtus and Tonks [I] is nowadays still applied experimentally [2^,25], and much theoretical work has been done to include the flexibility of the walls [25»26].

fiANDOM MAGNETIZATION PROCESSES

The Sixtus-Tonks experiment needs the shape of the moving wall to be

known and is a deterministic experiment which can only be performed under

certain ideal conditions. These are not met by the usual magnetization of

ferromagnets in which process mobile wall segments are generated and annihi319
lated. When slowly magnetized by an external homogeneous field the polarization change proceeds by starting the wall movement at a certain position

where a wall is most probably released from pinning centers over a long

enotigh distance so that this segment might serve as a nucleus for Barkhausen

discontinuities which sometimes cover a considerable part of the sample. The

nucleus grows in consideration of a sufficiently low stray field energy along

the direction of the local polarization and at right angles to it. The process is modified by the random local pinning forces and the already mentioned

damping mechanisms. Numerous processes of that kind form a space-time random

process. We call the resulting rate ^(z,t) of the magnetic flxoc through the

sample at position z (measured along the sample axis) at time t the stochastic

Barkhausen field [27,28]. It is hoped that the experimental analysis of this

random one-dimensional field could support the tinderstanding of magnetization

noise and power loss [11] and stimulate calculations on irreversible Bloch

wall movement.

The first step could be calculations similar to those of refs. [25,29,

30] which apply to the one-dimensional wall curvattire in the plane perpendicular to the polarization vector. An important contribution to these questions

with full account also of bowing in the other direction was given by Hilzinger

et al. [31]<» But this consideration has to be worked out to yield the time

development of a wall which is passing randomly distributed defects. There

have been some efforts to include dynamical effects into the model of Neel

[32]. This was done by adding a Rayleigh dissipation term to the generalized

forces corresponding to the Poxirier coefficients of the wall distortion as the generalized coordinates by Baldwin [53] which work was one of the starting considerations for his nonconservative spring model [3^]« Wagner [35]

used the more general interaction energy due to the perturbation of the material to derive the spectrum of the magnetic field fluctuations needed to

maintain a constant wall velocity in a pictxire frame single crystal [36]* The

equations of motion of this model clearly show at least with the simplifications to be made that it is superior to impress the average wall velocity instead of the magnetic field. One main problem which is yet tinsolved is to

determine the coefficients of the Eayleigh dissipation function for each of

the PoTirier components of the wall distortion. Concerning the measurements

made with frame-type single crystals and their consequences Porteseil [37]

and Vergne et al. [38] have published valuable concepts.

The stochastic Barkhausen field should be easier to be interpreted when

only one wall is involved. Hence, we meastired the normalized cross power

spectrum of a single crystal, cf. figure 1 and [27] for the measuring method

and definition. The data apply to a 180^ Bloch wall running round the frame

and moving with a mean wall velocity of 0.2 mm/s with exc\irsions of ± 0.5 sun around the middle of the legs (width 2 mm). The Barkhausen Jximps were observed as flux ;jumps with the usual spectral density [36] above the cutoff

320
•o«o*o«o«o*o«o«o*o aoa^*^.

^.8 7o SiFe single crystal °

o•Ki3

11mm

o

•o

Pig. 1 Hormalized cross spectrum of the Barkhausen noise

due to one 180° wall around the

shown single crystal fraae.

Thickness 0.283 nm, width of

the legs 2.0 am, specific electric resistance 0.65 rQb,

J a 1.92 Tesla.

10^ 10^ 10^ f/Hz 10^

frequency of the control circuit of 20 Hz. Their correlation is given by

K. . for different legs of the crystal as indicated in the figure. (For

^54 ^ ^'5 this quantity is much effected by the noise of the control circuit

which could not be detected with s\ifficient accxiracy.) The wall shows a good

correlation (K. . > 0.7) in the direction parallel with the polarization

(main stiffness by stray field effect) up to 11 kHz along a distance of 18 mm. and up to 26 kHz along 9 mm. Recently measurements of the cross spectra were made by Jansen [39] on

long thin samples with the method of ref. [27]. The wavenumber spectrum

i-ao -too rCk^u;) = J

Jc(^ , r ) exp[o(k!f -a;-tr)]d^dr (1)

-05 -co of the COvariance

t+T , C(^, t) = T-^ J<:^(z + , f +r) (J(z.t')>dt' (2)

t was determined for slowly varying external field of period T. P decays with

increasing k and tu as is shown in [28]. The cutoff wavenumber kg for a given

angular frequency tu = 2jrf defined by P(k , 6o)/P(0,i^) = 1/2 is shown in

figures 2 and 3 for the indicated materials, a^^ is the cutoff wavenumber for

the slow irreversible processes. The range a^^ ^^irr ^"^^^ [27] equals

( "VS" - 1)'^^^aj^) is by the factor of A larger than the reversible range a^.

The figures show the dispersion of the irreversible range beginning at about

f = 1 to 10 Hz. Figure 2 indicates also that of the reversible propagation

above t« = 10^ s"'^ which had been compared [27] to linear calculations [^O]

.

a^^ = 80 mm, cf. figure 2, is determined by the finite length of the sample.

An estimate of the irreversible range for an infinitely long sample is

a^ = 214 mm. The full lines of the figures are determined from interpolation

functions similar to those given in [27], Jansen [39]. They have not been

calculated from first principles.
Fig. 2 Dispersion relation of the cut- Fig. 3 Same as figure 2 for riboff wavenumber of a harddravm iron wire. bons. Width/thickness 2.0/0.10 mm

(PeSi, Goss texture); 0.95/0.033

mm (metallic glass as quenched).

We are indebted to Fried. Krupp GmbH, Forschomgsinstitut , D-4300 Essen for

providing the sample material of the metallic glass. The author thanks K.

Jansen and N. Hangmann for experimental work.

REFERENCES

12

3

4

5

6

7

8

9

10

K. J. Sixtus and L. Tonks, Phys. Rev. 42, 419 (1932).

K. Stierstadt, Der Magnetische Barkhausen-Eff ekt , Springer Tracts

Modern Physics 40, 1 (1966).

J. C. McClure, Jr. and K. Schroder, ORG Grit .Rev. Sol. St .Sc. 6,45 (1976)

H. Bittel, IEEE Trans. Magn. MAG-5 , 359 (1969).

G. Montalenti, Z. angew. Phys. 28, 295 (1970).

N. J. Wiegman, Appl. Phys. 12, 157 (1977).

W. F. Brown, Jr., Micromagnetics (Wiley, New York, 1963).

H. Trauble, in Moderne Probleme der Metallphysik, vol. 2, Ed. A. Seege

(Springer, Berlin, Heidelberg, New York, 1966) p. 157.

U. Lieneweg, J. Mag. Magn. Mat. 4, 242 (1977).

P. Mazzetti and C. Oldano, J. Appl. Phys. 42., 5351 and 5357 (1978).

322
p. Mazzetti, in Noise in Physical Systems, Springer Ser. Electrophys.

»

vol. 2, Ed. D. Wolf (Springer, Berlin, Heidelberg, New York, 1978) and

IEEE Trans. Magn. 758 (1978).

P. Mazzetti, A. Ferro and G. Montalenti, IEEE Trans. Magn. MAG-13 . 1514 (1977).

P. Gaunt, J. Appl. Phys. 48, 3W (1977).

M. Lafflbeck, Barkbausen-Effekt und Nacbwirktmg in Ferromagnetika

(de Gruyter, Berlin, 1971)

.

J. A. Baldwin, Jr. and St. L. Senesac, IEEE Trans. Magn. MAG-15 * 999

(1979).

0. Bostanjoglo and H. P. Gemiind, pbys. stat. sol. (a) A;8, 481 (1978).

D. Siemers and E. Nembacb, J. Appl. Phys. 4895 (1979).

M. Fahnle, B. Barbara and H. EronmiUler, phys. stat. sol. (b) 96 , 343 (1979).

T. R. Haller and J. J. Kramer, J. Appl. Phys. 41, 1036 (1970).

G. Hellmiss, Wiss. Ber. AEG-Telefunken 4^, 77 (1970).

A. Peu, phys. stat. sol. (a) 41, 361 (1977).

1. Sakaki, IEEE Trans. Magn. MAG-16 , 569 (1980).

¥. Grosse-Nobis and G. Rauscher, NTG-Pachber. 26, 233 (1980).

fi. C. O'Handley, J. Appl. Phys. 46, 4996 (1975).

P. Williams and J. E. L. Bishop, J. Mag. Magn. Mat. 20, 245 (1980).

W. J. Carr, Jr., J. Appl. Phys. 42, 4176 (1976).

W. Grosse-Nobis and K. Jansen, in Noise in Physical Systems, Ed. D.

Wolf (Springer, Berlin, Heidelberg, New York, 1978), p. 204.

W. Grosse-Nobis, K. Jansen and J. Tolkmitt-Pudlich, J. Mag. Magn.

Mat. 1i, 380 (1980).

J. E. L. Bishop, phys. stat. sol. (a) 11_, K75 (1972).

E. Schlomann, IEEE Trans. Magn. MAG-10 , 11 (1974).

H. -R. Hilzinger and H. Kronmiiller, J. Mag. Magn. Mat. 2, 11 (1976).

L. Neel, Ann. Univ. Grenoble 22, 299 (1946).

J. A. Baldwin, Jr., J. Appl. Phys. 5982 (1968).

J. A. Baldwin, Jr. and G. J. Culler, J. Appl. Phys. 40, 2828 (1969).

W. Wagner, Diploma work, Univ. Miinster 1977* unpublished.

W. Grosse-Nobis, 4th Int. Conf. Phys. Aspects of Noise in Sol. St. Dev.,

Noordwijkerhout 1975 and J. Mag. Magn. Mat. 4, 247 (1977).

J. L. Porteseil, Physica 93B, 201 (1978).

R. Vergne, J. L. Porteseil and J. C. Cotillard, J. Mag. Magn. Mat. 1^.,

1470 (1980).

K. Jansen, impublished.

H. Eifrig, W. Grosse-Nobis and P. Schulz, J. Mag. Magn. Mat. I^,,

383 (1980).

323
BARKHAUSEN NOISE AND DOMAIN STRUCTURE DYNAMICS

IN 3% SiFe SINGLE CRYSTALS

Giorgio Bertotti, Fausto Fiorillo, knd Maria Paola Sassi

Istituto Elettrotecnico Nazionale Galileo Ferraris

Gruppo Nazionale Struttura della Materia del C. N. R. 1-10125 Torino, Italy

INTRODUCTION

The magnetization process in ferromagnetic materials is stochastic in nature. The

domain walls overcome, under the action of the applied field, the random hindering

forces of lattice defects through a sequence of jumps, which can be sensed upon a pick-up

coil as induced voltage fluctuations (Barkhausen noise). Barkhausen (B. ) noise experiments put in evidence the tendency of the domain wall jumps to cluster into large avalanches, under the action of internal coupling fields. The statistical investigation of such a correlation effect gives information on the dynamics of the domain walls at a microscopic

level.

In a recent paper [llit has been shown that the microscopic behavior of the domsdn

walls is strictly related to the macroscopic magnetic properties of the material (e. g. the

shape of the magnetization curve), as well as to the evolution of the domain structure as a whole during the magnetization process. This has been clearly put in evidence by means

of a new experimental procedure, whose main features are: a) cyclic magnetization of the

sample at a constant rate of induction variation; b) analysis of the statistical properties

of the B. noise corresponding to different points of the magnetization curve. Actually,

owing to the strong non-stationarity of the noise along the magnetization curve, this is the

proper method of investigation, if one aims at a clear picture of the domain wall dynamic

behavior along the hysteresis half- loop. In the present work the above method has been used in the study of a (110) fooij

3% SiFe single crystal, characterized by a system of longitudinal antiparallel magnetic

domains. In this case, due to the simple domain structure involved, the search for a connection: B. noise - domain structure evolution - macroscopic magnetic behavior, is of

great interest from a fundamental point of view. The investigation has been made in different conditions of applied stress. The modifications of the magnetization curve and of

the domain structure as a function of the stress are reflected in a predictable way into

changes of the B, noise features. It is found in fact that the behavior of the noise power

<P>abng the hysteresis half-loop is in all cases consistent with a law of the type:

<P> = a/u + b|d(w/dB| , where /u and d;u/dB are the irreversible permeability and its derivative with respect to the induction respectively, with a and b constants. The predicted

contribution of d^ /dB to the noise power is a most interesting feature of the law, not yet

put in evidence so far. The term |d;u/dB| is related to the processes of enucleation and

annihilation of the longitudinal domains, which take place about the knees of the magnetization curve. Here the permeability derivative exhibits sharp peaks and the same occurs

to the noise power intensity. The term a/u in the law is related to the simple motion of

the domain walls and predominates near the coercive field, v/here a stable domain

structure exists. This is the term usually taken into account in the experiments reported

in the literature (2, 3] . 324
RESULTS AND DISCUSSION

A (110) [OOl] 3% SiFe single crystal, 150 mm long, 5 mm wide and 0. 29 mm thick was

investigated. Through the use of a feedback technique, a cyclic suitably shaped field wave- form was applied to the sample, ^n order to obtain a constant rate of magnetization. The

actual rate was dB/dt = 0. 14 Ts , along the whole investigated induction Interval ± 1. 75 T.

A detailed discussion of the experimental procedure is given in paper [ 1] . In figure 1 the

measured behavior of the noise power <P>is shown, compared with the behavior of the

irreversible permeability n , for different values of the applied tensile stress a ( the magnetization is assumed to proceed from -1. 75 T to + 1. 75 T). A striking correspondence

between the<P>and fi curves can be noticed. In fact, the noise peaks at high induction levels always take place where /u changes steeply, that is at the knees of the magnetization

curve. More quantitatively, the noise intensity behavior is found to be always consistent

with the law:

<P>= aft + b d/u/dB (1)

where d/^/dB is the derivative of permeability with respect to the induction B. a and b are

constant along the magnetization half- loop. This is a quite general law, which has been

found to hold also for polycrystalline samples [ l] . A typical fitting of the<Pl^curve

through eq (1) is shown in figure la (dashed line). What turns out particularly new and

T 1 1 I r

a- SOMPa.

o 5MRa

Fig. 1 a) Behavior of the noise power

<P>measured at different val- ues of the applied stress a in a (110) [001] 3% SiFe single

crystal during magnetization

at a constant rate dB/dt = 0. 14

Ts" (B = ± 1. 75 T). Fitting of <^l^rough the law (1)

in the case a = 5MPa is

shown by the dashed line.

b) Corresponding behavior of

the irreversible permeability

/" ( vacuum permeabiity).

It can be noticed the correspondence between the drop of

permeability at the knees of

the magnetization curve, increasingly steep with increasing !stress, and the peaks of

the noise intensity.

0.5 1 1.5 INDUCTION (T)

325
interesting from eq (1) is the contribution b|d/u/dB| to the noise power, which accounts for

the peaks occurring at the knees of the magnetization curve. Actually, early results obtained in 3% SiFe polycrystalline samples [2, 3] , approximately agree with a law of the

type : <P> «

Insight in the physical meaning of eq (1) can be obtained through direct domain observation, by using the magneto-optical Kerr effect technique [4] . One can see that the domain structure evolution along the magnetization curve is characterized by two types of

phenomena. First, in the central portion of the curve, where the permeability is approximately constant, a well defined system of antiparallel domains is observed, which evolves

through simple motion of the walls. Correspondingly, the noise power <P>is low and depends weakly on B. On.the contrary, strong domain rearrangements take place about the

knees of the magnetization curve. They are related to the enucleation and annihilation of

the longitudinal domains. In this region the permeability changes very rapidly. d|U/dB is

strongly peaked and the same occurs to<P>.

Having traced a phenomenological correspondence between noise intensity behavior,

shape of the magnetization curve and domain structure, one can try,through the help of

the statistical theory of the B. noise [s, 6, ij , to seek an explanation in terms of microscopic dynamic properties of the domain walls. The theory, which takes into account the

tendency of domain wall jumps to correlate in large clusters, predicts that<P>=<I' ( ^/^<^^

= <[> ' <Dfy , being o the flux rate upon the sample cross-section, p and p<T^>the average flux

variation and the average duration of a B. cluster respectively and the average flux

rate associated to a cluster. Since <i) is kept constant in the present measurements, we obtain that the processes associated to domain creation and annihilation are characterized by

B. clusters carrying out on the average a much greater flux rate than processes occur10 10^ 10' 102

FREQUENCY (Hz)

103 lO*

Fig. 2 a) Noise power spectra measured upon three representative

points of the magnetization curve in the case = 5 MPa; b) Normalized

cross spectra measured at the same points, upon windings 2 0 mm apart.

326
I

ring in the region B~0, where simple domain wall motion takes place. This means that

much stronger internal coupling fields act on the domaia walls in the regions where d/u/dB

j is high. Further insight can be obtained by analysing the spectral properties of the noise

I at different induction levels (points A, B, . , , , G in figure 2). Typically found power spectra

! S(f) and normalized cross-spectra K(f) [2] , related to a couple of narrow windings placed

i 20 mm apart, are shown in figure 2. Two main results arise from spectra inspection;

First, if we consider K(f) in the limit f-»0, which represents the flux fraction propagating on the average during a B. reversal upon the coil distance, we obtain that in the region

B~0 correlation on such a sample length is complete. This must be obviously connected

to the presence of a well defined longitudinal domain pattern. On the contrary, one can see

that a short correlation range is associated to domain enucleation and annihilation processes (K(0)~0. 6). For what concerns S(f), we get immediately that the cut-off frequency f , proportional to l/^<%>, is higher at the knees of the magnetization curve. In such regions

the time needed for a B. cluster to fully develop is then the shortest.

When the applied stress is increased, the noise non-stationarity at the knees of the

magnetization curve becomes nearly catastrophic, according to the approximately rectangular shape assumed by the permeability curve. Such a trend can be qualitatively explained by considering the role played by the flux-closure domains present at the sample edges

and surface. Let us consider, for instance, the domain structure in the material when,

starting from the nearly saturated state, magnetization begins to reverse. One has to deal

with a flux-closing domain structure (spike domains [4] ), which can evolve through two

basic mechanisms: a) growth of the existing spikes and formation of new ones, b) transformation of the spikes into longitudinal domains. Process b) in particular is expected to reqviire higher effective driving fields, compared to the case of growth through wall motion

of a definite domain structure. This is also what is strongly suggested by the noise results,

where peaks of<P> (that is of %) are found to be associated with the domain structure rear- rangements. It has been in fact previously remarked that the intensity of the noise is directly related to the strength of the internal coupling fields. At low stress levels, mechanisms a) and b) mix up along a substantial induction interval. The ordered longitudinal

domain structure forms at a low pace and, as figure 1 shows, the rise of permeability is

slow. Correspondingly, the noise intensity maximum is wide. When the tensile stress is

increased, the formation and growth of spike domains, which would imply an increasingly

high magnetoelastic energy contribution, are progressively inhibited. Thus, the longitudinal domain formation, originating at the few remaining heavily pinned spikes, takes place

at the very early stages of the magnetization reversal, associated with a very steep rise

of the permeability. The fact that the noise peaks become increasingly sharp, gives an

indication on how strongly the local driving field has to rise in order to accomplish the

transformation from the spike to the longitudinal domain structure.

, REFERENCES

[l] G. Bertotti, F. Fiorillo, and M. P. Sassi, J. Magn. Magn. Mat. 23^ (1981), in press.

[2] L. Storm, C. Heiden, and W. Grosse- Nobis, IEEE Trans. Mag. MAG-2. 434 (1966).

[3] H. Bittel, IEEE Trans. Mag. MAG- 5, 359 (1969).

[4] J.W. Shilling and G. L. Houze, Jr., IEEE Trans. Mag. MAG-10. 195 (1974).

[5] P. Mazzetti and G. Montalenti, in Proc. International Conference on Magnetism,

Nottingham (1964), p. 701.

[s] M. Celasco, F. Fiorillo, and P. Mazzetti, Nuovo Cimento 23B, 376 (1974).

327
NOISE IN ELECTROCHEMICAL SYSTEMS

U. Bertocci

Center for Materials Science

National Bureau of Standards

Washington, D.C. 20234

INTRODUCTION

The analysis of electrochemical noise, that is, of the fluctuations of the current and

potential of an electrochemical cell, can give useful information about the rate and the

nature of the chemical processes taking place at the electrodes. This technique has been

used to study the equilibrium properties of redox reactions [1] and homogeneous processes

in solution [2]; recent work has centered on electrocrystallization [3] and on corrosion

[4,5,6], particularly localized corrosion and the properties of passive films.

There are two applications of noise measurements of interest in electrochemistry.

The first and more conventional consists of applying a broadband noise signal to an electrochemical cell, and in recording its current output [7]. In this way the electrode imped- ance as a function of frequency is obtained, which gives important kinetic information. A second and less common way of employing noise measurements consists in detecting and analyzing the fluctuations of the steady state voltage and current of an electrochemical cell.

This paper will survey the experimental techniques used in electrochemical noise

studies, discussing some of the problems encountered and ways to overcome them. Finally,

some recent experimental results will be presented as an example of application of these

techniques to corrosion science.

MEASUREMENT TECHNIQUES

a) Galvanostatic measurements

One approach to noise measurements

spectrum of the voltage fluctuations at done by observing the total noise power

10

* to

to<5 10

10

o NO CHROMATE

• CHROMATE ADDED

1 FREQUENCY. Hz

in electrochemistry is that of recording the

the open circuit potential. Some early work was

by means of an ac amplifier connected to a loudspeaker or a recorder [8], studying the

effect of adding corrosion inhibitors to the

solution upon the amplitude of the signals.

In recent work, however, the time records

are transformed in the frequency domain

using either a spectrum analyzer [9] or a correlator [10].

An example of the use of this measurement method is shown in Fig. 1, for aluminum

in a chloride solution. A qualitative

correlation can be observed between corro- sion rate and noise amplitude: the noise

decreases when a corrosion inhibitor,

sodium chromate, is added to the solution.

Fig. 1 Noise spectrum at open circuit of

Al in borate buffer with and without

corrosion inhibitor.

b) Potentiostatic measurements

A more refined method consists of con- trolling the electrode potential by means

of a potentiostat and observing the spectrum

328
)f the current fluctuations. In this way, it was found, for aluminum in a neutral solution,

that relatively little change in the noise spectrum occurred over a fairly large potential

span if chlorides were not present in solution. However, if 0.01 M NaCl was added and the

potential was moved from a value at which the protective film is stable, to the pitting

potential, at which the oxide film breaks down the noise increased by two or three orders of

magnitude [6,11], marking quite clearly the onset of pitting before it could be detected

with other methods.

With the potentiostatic method, however, a problem arises because instrumental noise is

being injected into the electrochemical cell by the potentiostat, and its level can interfere with the measurements. In order to obviate this problem, a low-noise potentiostat has

been developed and built at NBS [12], and care is being taken to minimize the instrumental

noise, as well as external interference, electrical or mechanical

.

The interpretation of the experimental data, in spite of the low level of noise in the

input, may become difficult because, in many instances, it is impossible to know whether

the spectrum is mainly the response to the input noise or is an indication of the random

fluctuations of the electrode characteristics, and only in clear-cut cases can it be attributed unequivocally to one or the other of its possible causes. On the other hand, since

both parts contain useful information, it would be desirable to be able to separate them so that both can be measured at the same time.

c) Measurement of the cross-power spectrum

The use of a two-channel spectrum

analyzer allows one to obtain the cross-power

spectrum of the two signals, providing the means to establish the origin of the noise

observed. When the potentiostat is connected

to the cell, the cross-power spectrum of the

input noise contained in the control voltage

and of the current fluctuations gives an indication of the degree of causality between

input and output.

Instrumental noise still plays a role

and limits the overall sensitivity of the

system, but if the cross-power spectrum is

measured, a scheme for the separation of the

deterministic and stochastic part of the

current noise can be developed [13]. As far

as the deterministic part is concerned, which

yields the electrode impedance, the very low

level of the input and output (typically less

than 1 yV//Hz, and less than 1 nV//Hz) is quite desirable, since electrochemical

systems are non-linear, and only when the

signals are so small the deviations from linearity are negligible.

The layout of the instrumentation for potentiostatic noise measurements in use at NBS

is shown in Fig. 2. A two-channel spectrum analyzer is used to measure the noise spectrum

of the input (voltage) and of the output (current) as well as the cross-power spectrum.

The auxiliary equipment shown in the figure records the d.c. values of the electrode potential and of the cell current. The signal generator can be used if one desires to modulate

the control voltage.

The acquisition of the cross-power spectrum can be very useful also in the galvanostatic case: if two reference electrodes are employed in measuring the potential fluctuations of an electrode, the coherence function can be employed to estimate what fraction of

the observed signal originates from the only common part of the two measuring circuits, so that one can discriminate between the electrode fluctuations and the noise produced by the

reference electrodes and by the amplifying circuitry. This method is being tested in our

laboratory for the monitoring of the corrosion of underground electrical cables.

1 I A.C. CufTwit^ I turdmrn I

Fig. 2 Block diagram of instrumentation

for electrochemical noise measurements.

329
SOME RECENT RESULTS

An important instance of the application of the analysis of noise to corrosion science

Is the study of the stability of passive films. Some authors have suggested that passive

films are in a condition of dynamic equilibrium, so that breakdown and repair occur continuously [14]. If this is the case, such a condition could lead to observable fluctuations of

the passive current, whose amplitude and frequency may depend on the degree of uniformity

of the film and the time necessary for repairing the breaks, as well as on the potential

applied. In our laboratory a study has been made of the passive current noise of a ternary

amorphous alloy Fe-Cr-Ni alloy.

s

I.M V

' O.M V

0.34,V

1 ••;

\ 1

AMORPHOUS F«]illi](CiHP|2B|

SI MO ISt FREOUIHCf. Hi

Fig. 3 Spectrum of current noise of

amorphous alloy in 1 mol/L H2S0it at

various potentials.

The spectra of Fig. 3, taken with a single channel spectrum analyzer, show a gradual increase in noise by increasing an- odic polarization, limited, however, to the

low frequencies, suggesting a long relaxation,

time for the repair process. In order to confirm this interpretation, it was important

to separate the stochastic part of the signal,

generated by the breakdown -repair mechanism,

from the effect of the change in electrode

impedance caused by anodic polarization.

The results of the separation carried

out with a two-channel analyzer are reported

in Figs. 4 and 5. The electrode impedance,

shown in Fig. 4, can be understood as being

constituted by a capacitance, which varies

little with potential, in parallel with a reaction resistance which decreases with increasing anodic polarization. The power

spectrum of the random noise, shown in Fig. 5,

however, indicates that a substantial increase

in noise with anodic potential occurs at higher frequencies than could be inferred

from the single channel data of Fig. 3, being

FREQUENCY, Hi

Fig. 4 Absolute value of impedance vs. frequency for amorphous alloy in 1 mol/L

H2SO1,.

Fig. 5 Power spectrum of random current

fluctuations for amorphous alloy in 1 mol/L

HjSO^

.

330
more prominent above one Hz. The frequency range of the noise is similar to that found

in studies of the passive behavior of stainless steel [4].

REFERENCES

[1] V. A. Tyagai, N. B. Lukyanchikova , Electrokhim. , 3, 316 (1967).

V. A. Tyagai, ibid., 3, 1331 (1967).

G. C. Barker, J. Electroanal. Chem., 82, 145 (1977).

[2] M. Fleischmann, J. W. Oldfield, J. Electroanal. Chem., ^7, 207 (1970).

[3] C. Gabrielli, M. Ksouri , R. Wiart, J. Electroanal. Chem., 86, 223 (1978).

G. Blanc, C- Gabrielli, M. Ksouri, R. Wiart, Electrochim. Acta, 23, 337 (1978).

[4] G. Okamoto, T. Sugita, S. Nishiyama, K. Tachibana, Boshoku Gijutsu, 23, 439, 445

(1974).

[5] G. Blanc, I. Epelboin, C. Gabrielli, M. Keddam, J. Electroanal. Chem., 75, 97 (1977)

G. Blanc, C. Gabrielli, M. Keddam, C. R. Acad. Sci . , Paris, 283, Serie C, 107 (1976)

[6] U. Bertocci , Seventh International Congress on Metallic Corrosion, Rio de Janiero

1978, p. 2010, ABRACO (1979).

[7] G. Blanc, I. Epelboin, C. Gabrielli, M. Keddam, Electrochim. Acta, 20, 599 (1975).

S. K. Rangarajan, J. Electroanal. Chem., 62, 43 (1975).

[8] W. P. Iverson, J. Electrochem. Soc, US, 617 (1968).

[9] U. Bertocci, J. Kruger, Surf. Sci., lOi, 608 (1980).

[10] G. Blanc, C. Gabrielli, M. Keddam, Electrochim. Acta, 20, 687 (1975).

G. Blanc, I. Epelboin, C. Gabrielli, M. Keddam, J. Electroanal . Chem., 62, 59 (1975).

[11] U. Bertocci, J. Electrochem. Soc, 127, 1931 (1980).

[12] R. W. Shideler, U. Bertocci, NBS J. Res., 85, 211 (1980).

[13] U. Bertocci, J. Electrochem. Soc, 128, 520 (1981).

[14] K. Videm, Kjeller Report Dr-149. Institutt for Atomenergi , Kjeller, Norway (1974).

J. Kruger, in "Passivity and its Breakdown on Iron and Iron-Base Alloys," p. 94,

R. W. Staehle, Ed., NACE, Houston (1976).

1

I' 331
Ion Noise in ISFETs

Andre Haemmerli, Jiri Janata, and James J. Brophy

University o£ Utah

Salt Lake City, Utah 84112

Ion fluctuations associated with the ion- sensitive membranes of ISFET sensors are observed experimentally by measuring short-circuit channel noise currents. The ISFET is a NDSFET in which the metal gate is replaced by an electrochemical structure consisting of

a reference electrode, electrolyte solution, and the ion-sensitive membrane [1] . Changes

in the chemical environment of the membrane induce corresponding changes in the electrical potential at the membrane/semiconductor interface which are observed as changes in the

channel current. The favorable geometry of an ISFET permits detection of ion fluctuations

which are not possible with conventional ion-sensitive electrodes [2]

.

These ISFET sensors studied are fabricated by integrated circuit techniques and con- sist of two ISFET devices and two conventional MOSFETs on the same p-type silicon chip.

This configuration is designed to facilitate balanced circuitry and to provide an electrical reference capability. All four devices have the same geometry with channels 20 pm

long by 400 ym wide. The gate area of the ISFET devices is covered by a silicon nitride

layer (a so-called pH ISFET) or by calcium or sodium ion- selective polymeric membranes.

The transconductance of the MOSFET units is 1.2xlO"^mho at standard operating potentials,

Vgs=2V, V(js=3V. A silver/silver chloride reference electrode was used for all ISFET

measurements

.

Channel current noise in all devices is determined by measuring voltage fluctuations

across a lO'^ohm resistor in series with the source using a conventional amplifier in con- junction with a General Radio Model 1900A wave analyzer. In the absence of current, the

system accurately measures the Nyquist noise spectrum of the series resistor over a frequency interval from lOOHz to 50kHz. The MDSFET noise spectra are consistent from unit to

unit and agree with results reported in the literature [3]. For all measurements,

the channel current was set at 0.1 mA by adjusting the gate bias or the reference electrode

potential.

The current noise spectra of pH ISFETs, shown in Fig. 1, have noise levels greater

than MOSFETs which are independent of pH over the range examined, pH4 to pHlO. The

spectral noise power varies as f"^/^, and there is a suggestion of a low frequency plateau

below 150Hz. Conversely, Ca^"*" ISfJeTs do exhibit a pH effect and have a markedly weaker

frequency dependence. There is, however, little variation of noise magnitude with con- centration, even though the noise is two orders of magnitude greater than that of the

MOSFETs. The sodium devices appear to show 1/f noise over the upper frequency range.

A simple model for noise in a pH ISFET arising from adsorption/desorption of ions on

the gate insulator characterized by a single time constant x gives for the spectral

density [4]

412 X

where I is the channel current and Nq is the average number of adsorbed ions. According

to Fig. 1, the low frequency value of Sj, 4x10" 2° A^/Hz, and t, 8x10" '*s, suggests a value

for No of about 10^ ions, independent of pH. Furthermore, if Nq/t is treated as a generation rate, the equivalent exchange current density calculated from the observed noise and

the ISFET geometry is lmA/cm2, in agreement with accepted values. The comparable value

for the Wasserman membrane is about lO'^A/cm^, which has not previously been measured. /

332
ION NOISE IN ISFETS

MX

CM

cr

10

rl9

10

-20

o .^-21

10

ifJ 10

<X

o

r22

10

-23

WASSERMAN

KFET ,

. TSFETo0J4M/'*°^'

^Ca^* +PH4

ISFET 7

ApH 10

• O.IM CaClg ' ISFET

MOSFETs

10

r24

100 1,000 lopoo

FREQUENCY IN HERTZ

100,000

Figure 1. Channel current noise spectra of pH ISFETs and

ISFETs with Na"*" and Ca^"*" membranes compared to MOSFET with some geometry.

333
The absence o£ an ion concentration effect can be accounted for by the adjustment of

the reference electrode potential for constant channel current, although this is not anticipated on the basis of the reactions at the electrolyte/membrane interface. In any event,

these preliminary results illustrate a variety of noise effects in ion-selective sensors, further experiments to characterize the noise mechanisms are currently underway.

REFERENCES

[1] J. Janata and R. J. Huber, lon-Sel-Electrode Rev., 1, 31 (1979).

[2] L, J. DeFelice and D. R. Firth, IEEE Trans. Bio-Med. Engr., H^-18, 339 (1971).

[3] A. van der Ziel and E. R. Chenett, Adv. Elect, and Electron Phys., 46, 313 (1978).

[4] A. van der Ziel, "Fluctuation Phenomena in Semiconductors", Academic Press, Inc.,

New York, C1959) p. 24,

334
QUASIELASTIC LIGHT SCATTERING .FROM MACROMOLECULES AND MICELLES

Mario Corti

CISE S.p.A.

P.0.B.1208J., 20100 Milano, Italy

The advent of lasers has greatly stimulated the study of matter by the scattering of

light. Laser light, powerful, highly monocromatic and amplitude stabilized, is the ideal ra- diation source to probe properties of liquids and solutions of macromolecules by Rayleigh

scattering. Spectral measurements are often simplified by modern techniques of intensity flu

tuations and heterodyne analysis, which are easier to use and allow higher resolution than

conventional interf erometric methods. A number of excellent reviews and some books [l-S] now

exist dealing with many applications in physics, physical chemistry, biology, biochemistry^

and engineering.

The physical origin of the scattered light is the spatial inhomogeneity of the optical

polarizability (i.e. the dielectric constant or the refractive index) of the medium. In a pure fluid in thermodynamic equilibrium at a temperature T, local fluctuations in polariza

bility are caused by fluctuations in the thermodynamic parameters describing the state of

the system. In a macromolecular solution there is an extra contribution to the fluctuations

in polarizability, due to the fluctuations in concentration of the macromolecules inside the

scattering volume.

The energy transfer in the Rayleigh scattering process is quite small; hence its denomi

nation is quasi elastic light scattering. In practical situations the Rayleigh linewidth is

indeed never larger than 10~^ Vq where is the optical frequency of the incident field.

The scattered field Eg is linearly related to the incident light field E^ through a scattering amplitude which is a random function of space and time. Therefore Es is a random process

By measuring the statistical properties of Eg and knowing Eq, it is possible to characterize

the statistical properties of the scattering medium.

It is well known that a complete description of the process is given by an infinite set

of field correlation functions, which for a given point in space can be written as

V^l ' ^2m) = <^s(V ^s (V K (Vl^ ^s* (^2m»

where m runs from 1 to infinity. An infinite number of increasingly complex measurements is

therefore needed. Fortunately great simplification occurs in the large majority of practical situations where the scattered electric field follows Gaussian statistics. In this case

the field statistics is described in terms of a single quantity, the first order (m=l) correlation function G^(t^,t„) = <E (t..)E*(t„)> which for t,=t„ reduces to the average scattered intensity.

Gaussian statistics is found whenever the scattering volume contains a large number of

independent scatterers. The field correlation function is then related to the intensity cor- relation function G2(t^,t2) = ^^s^^;,^) Ig(t2)> by the relation G2(t^,t2) = <Ig>^ + I G-,^ (t^ , t2) !

This relation has a great practical importance. It is in fact much easier to measure

the correlation function of the intensity output from a photodetector , than a field correlation function by optical techniques. For a given scattering field the task is to measure the

scattered intensity as a function of time to form the intensity correlation function. Then

from the above relation it is simple to evaluate the first order field correlation function

which is just the Fourier Transform of the optical spectrum.

335
Non Gaussian statistics may be found in experiments where the scattering particles are

large, like bacteria or long viruses, and there are few of them in the scattering volume.

Number fluctuation effects will arise besides the usual Gaussian fluctuations (see the articles by P.Pusey in Refs.7 and 8).

Laser light scattering has been successfully applied to a large variety of problems including the study of pure fluids, liquid mixtures, viscous liquids, liquid crystals, macromolecules in solution and motility of microorganisms.

For each problem the average scattered intensity gives a measure of the static properties of the system, while the time behaviour of the correlation function reflects the dynamics of the spatial inhomogeneities which are responsible of the scattering process. For

example, in a pure fluid in thermodynamical equilibrium density fluctuations, due to entropy

fluctuations, decay with a characteristic time determined by the thermal diffusivity of the

fluid, while the intensity of the scattered light is proportional to the isothermal compressibility of the fluid. For a macromolecular solution the intensity of the scattered light

gives the molecular weight. The temporal fluctuations in the scattered light intensity contain detailed information about the Brownian movement of the macromolecules . Hence the mutual

translational diffusion coefficient can be measured.

Let us summarize the information obtainable by a laser light scattering experiment from

a solution of macromolecules, whose linear dimension is smaller than the wavelength of

light [7,8].

From the average intensity < I^> we obtain:

Molecular weight of small non interacting particles,

<Ig> c M P(q)

where c is the solute concentration and M the molecular weight in Daltons. The function

P(q) = 1 in the limit qL<<l, where q = (47t/A) sen (9/2) is the modulus of the scattering vector, L the linear dimension of the particle, X the wavelength of light and 9 the observation

angle.

Gyration radius of particles by intensity measurements at different angles. The small

q expansion of P(q) depends on the gyration radius

2 „ 2 P(q) = 1 -

3 ^ +

Interparticle interactions , by the nonlinear dependence of the average intensity versus concentration

<I>-!-cri/ + 2Bc + 3Cc +...] s 1- M

-1

where B,C are the coefficients of the virial expansion of the excess chemical potential

of the solution with solute concentration c. The coefficients B,C,,.. are zero for non interacting particles.

From the intensity correlation function G2(t), with T = ^2~^l* "® obtain:

Translational diffusion coefficient of particles.

with f(T) = exp [- q^ D t] for a monodisperse solution of small particles. The diffusion coef- ficient D is then related to the particle hydrodynamic radius R^^ by | 336
° "

6Trn Rjj

with kg the Boltzmann constant, T the absolute temperature and n the solvent viscosity. This

formula is valid for non interacting particles.

Polydispersity of the solute particles . A polynomial fit of the logarithm of f(T) determines the deviation from the single exponential benhaviour of f(T) and hence gives information on the particle size distribution function.

Interparticle interactions from the diffusion coefficient dependence on solute concentration [11] .

Combination of intensity and correlation function data gives also information on the

shape, hydration and the interaction potential (eventually the macromolecular charge)

.

As an interesting example of application of light scattering techniques, we shall discuss

the aggregation phenomena of amphiphilic molecules in solution which leads to formation of agi

gregates, named micelles [9] . Aggregation of amphiphilic molecules is an important subject

of research since it is one of the basic aspects of living matter. Amphiphilic molecules, like detergents, are made of an hydrophobic portion, normally an hydrocarbon chain, and an hydrophilic group either polar or ionic. Above a certain critical concentration, globular aggregates are formed in water with an hydrocarbon core and a polar or ionic surface which

shields the core from the surrounding water. An optimum size is reached by a self aggregation

process, since the hydrophobic attraction is contrasted by the electrostatic repulsion among

the polar or ionic groups on the micelle surface. Typical aggregation numbers are of the order of one hundred and the micelle radius of 20-30 A. Rather interesting physical problems

can be studied in micellar water solutions. Let us consider a representative phase diagram

337
Phase I . The concentration is so low that detergent amphiphilic molecules are in solution as monomers.

Phase II . In a narrow concentration region, called critical micelle concentration, at the

boundary I-II molecules aggregates to form micelles. Light scattering gives here information

about aggregation number, molecular weight, diffusion coefficient (hence hydrodynamic radius)

and shape of micelles at their formation

to study micelle-micelle interactions [ll.

[lO] . Then inside phase II it is quite interesting

The interaction potential can be charged by acting on the ionic strength of the solution with some added salt (micelles may be changed)

and by varying the type of detergent. So micelles may offer a simple and flexible model to study interaction effects in solution like those found for proteins.

Phase III . Detergent precipitates to form very hydrated crystals. At the boundary II-III a first order melting transition can be observed.

Phase IV . A peculiar transition, mostly for non-ionic detergents, occurs at the boundary

III-IV. The homogeneous micelle phase breaks into separate phases. There is a lower critical

consolution point, called cloud point, and the transition is of the second order type like

in binary mixtures. Approaching the transition from the homogeneous phase one finds critical

exponents predicted by the mean fie Id theory [l2] .

At large concentrations, not shown in the figure, other interesting transitions may be

found, like transitions to gels or to liquid crystal type phases.

REFERENCES

|l| G.B.Benedek, in Polarization^ MaHere et Rayonnementy (Presses Universitaires de France,

1969) p. 49.

2 I H.Z. Cummins, and H.L.Swinney, in Vvogvess in OptioSj Ed .E .Wolf , 1970, Vol. 8, p. 133.

3| P.A.Fleury, and J. P. Boon, in Advances in Chemioal P?2z/sieSjEd.I.Prigogine and S.A.Rice

(Wiley Interscience,New York, 1973) Vol.24, p.l.

Photon Correlation and Light Beating Speotrosaopy^ Ed. H.Z. Cummins and R.E.Pike (Plenum

Press, New York, 1974).

5| B.Chu, in Laser Light Scattering^ A.cademic Press, New York, 1974).

61 B.J. Berne, and R.Pecora,in Dynamic Light Scattering, (John Wiley, New York, 1976).

7 I Photon Correlation Spectroscopy and Velocimetry, Ed. H.Z. Cummins and R.E.Pike (Plenum

Press, New York, 1977).

Light Scattering in Liquids and Macromoleoular Solutions, Ed.V.Degiorgio.M.Corti and

M.Giglio (Plenum Press, New York, 1980).

C.Tanford, in The Hydrophobia Effect, (John Wiley, New York, 1980).

10| M.Corti,and V.Degiorgio, Chem.Phys . Lett . 53_, 232 (1978).

ll| M.Corti,and V.Degiorgio, J. Phys. Chem. 85_, 711 (1981).

12| M. Corti, and V.Degiorgio, P/Z2/S. /fey. Lett. 45, 1045 (1980).

338
FRANK ELASTIC CONSTANT RATIOS EVALUATION

IN MBBA BY INTENSITY MEASUREMENT OF THE SCATTERED LIGHT

E.Miraldi, L.Trossi and P.Taverna Valabrega

Istituto di Fisica Sperimentale

Politecnico - Torino

COldano

Istituto di Fisica Sperimentale

Politecnico - Torino

Gruppo Nazionale Struttura della Materia

del CN.R. , U.R. 24

The predominant contribution to the light scat-l;ering in nematics comes from the thermally excited orientation fluctuations of the director n. To each ^-component of the fluctuation

there are two uncoupled modes, whose mean amplitudes, in the presence of a magnetic field

TT // TT, are respectively given by [l] :

where: • " 6n^, 6 n^ are the components of 6n respectively parallel and normal to the (rT, '5) plane,

<> means thermal average, is the sample volume, kg the Boltzmann constant, T the absolute

temperature, 0 the (^, n) angle, = X//- Xx the diamagnetic anisotropy and ^^22' ^33

are the Frank elastic constants corresponding to splay, twist and bend deformation respectively. Both modes are purely relaxational , with a relaxation time [2] , [s]

:

(2)

where '^^ ( ^ ) is a viscosity coefficient.

The intensity of the light scattered by each mode is proportional to <( 6nj ) )> through

a known coefficient '^o,^^ which depends on the optical geometry and on the polarization states (X and of the incident and scattered beams; the spectral line-width is proportional to l/t^ [2] . By light scattering experiments one can measure in principle all the constants

which determine thehydrodinamic behaviour of the nematic, i.e. Kjj, X a and the five independent viscosity parameters. More precisely the intensity of the scattered light can give the

constants K-. and X^, its line-width can give the ratio between any pair of constants.

J J a

Actually, while on line-width measurements several papers are found in literature, in339
tensity measurements aiming to the determination of K^^^/Kj^ ratios are reported only in two recent papers [4, s] . Indeed these experiments are rather difficult for the following reasons :

a) The intensity measurements are always affected by many errors, due to multiple scattering

and to the scattering from crystal defects. Particularly important is the scattering due

to the distorsion of the director field near the crystal surfaces.

b) In general a small error in the intensity measurements gives a very large error in the

Kj^j^/Kjj ratios evaluation. For instance in paper [4] a standard deviation of 0. 2% of the

experimental points from the best fitting curve gives an error of about 5% in the K /K

ii Jj ratios

.

For what concerns point a), an optimal sample thickness must be chosen. Besides the use of homeotropically aligned samples has some advantages with respect to the planar ones. The

most important improvement, however, concerns point b), with the choice of suitable geometric

conditions. One must select the incidence and scattering angles t which allow the

best separation of the scattering effect of pure splay, twist and bend fluctuations. This can- not be obtained with the normally used scattering geometries, i.e. with the incident beam parallel or perpendicular to the director. Oblique incidence gives the further advantage of re- ducing the background due to diffuse multiple scattering, as discussed in a previous paper [5].

In the present note we report some results concerning the K;a/K ratios evaluation, 1 j j carried out in the following conditions.

The sample is a mono-domain cell of nematic MBBA, homeotropically aligned between glass

plates. Its thickness is 50 yum, such that the double scattering never exceedes 2% of the sin-,

gle scattering, and that the higher order effects are negligible. In order to reduce the forward scattering the sample is kept in a magnetic field of 3000 gauss. The shape of the magnet ; is such as to allow a large variation of both the incidence and scattering angles. The light source is a 3 mW He-Ne laser. The irradiated area is about 2 mm in diameter. A fraction of

the incident beam is sent to a photodiode.and the scattered light to a single-photon photomultiplier, who^e effective area is seen from the sample under an angle of 2 mrad. The ratio

between the measured intensities is done by using an electronic analogic divider.

The intensities have been corrected taking into account:

- the attenuation of the incident and scattered beams inside the nematic;

- the double scattering contribution;

- the transmission factors at nematic-glass and glass-air surfaces;

- the solid angle change of the scattered beam when it goes from inside the crystal out. The

correction factors are given in ref. [sj ,

;

All measurements are carried out with the director lying in the scattering plane, because in this condition the light scattered by the two independent distortion modes can be se- parated by simply selecting the polarization states of the incident and scattered beams. 1 When both beams are polarized in the scattering plane [ E - E geometry one selects the splay

+ bend mode; with crossed polarizers [e -OorO-E geometry] one selects the twist + bend

mode.

In fig. 1 are reported the intensities measured at 22 "C for an incident external angle I 9^(e) = 45 o

f-Qj, different scattering angles and polarization states. The experimental points are divided in three separated sets which correspond respectively to:

i a) nearly pure splay ( ~ 90% splay and '-^10% bend), E - E geometry;

340
b) nearly pure twist (90 + 100 % twist and 0 + 10% bend), E - 0 geometry;

c) predominantly bend mode (55 + 75% bend and 25 + 45% twist), E - 0 geometry.

The full lines are the best fitting curves for E - E and E - 0 geometries, and have been

obtained with refractive indices n^ = 1.544 and n^ = 1.758 [6j , and with Kq^]^ : K22= ^23 ~

= 5/ 4/7.5.

The standard deviation of the experimental points from the best fitting curves is '^'3%,

corresponding to a standard deviation of 'v 5% for the ^11/^22 ^'^^ °^

ratio.

-

100

External scattering angle [degreej

Fig. 1 Experimental points of the relative intensity of scattered light versus external scattering angle for an external incidence angle of 45°,

and theoretical best fitting curves for E - E (a), E - 0 (b and c) geometries. Points labelled T and B correspond to pure Twist and pure Bend

respectively.

We are indebted to Prof. P.Mazzetti for many stimulating and helpful discussions.

REFERENCES

[1] P.O. de Gennes, The Physics of liquid crystals, Claredon Press, Oxford.

[2] Group d'Etude des Cristaux Liquides (Orsay), J.Chem. Phys. 5; 816 (1969).

[3] D.Forster, T.C. Lubensky, P.O. Martin, J.Swift and P.S. Pershan, Phys. Rev. Lett. 26,

1016 (1971).

341
[a] H.Usui, H.Takezoe, A.Fukuda and E, Kuze, Jpn. J. Appl. Phys. 18 1599 (1979).

[5] E.Miraldi, C.Oldano, L.Trossi and P.Taverna Valabrega, Nuovo Cimento SOB , 165 (1980).

[e] E.B.Priestley et al .

, Introduction to Liquid Crystal, Plenum Press, New York, London,

1975.

342
THE DETERMINATION OF ELASTIC CONSTANTS OF NEMATIC LIQUID CRYSTALS

FROM NOISE MEASUREMENTS OF SCATTERED LASER LIGHT

Jitze P. van der Meulen, Rijke J.J. Zijlstra and Jaap J. van Kooten

Fysisch Laboratorium, Rijksuniversiteit, Princetonplein 5,

3584 CC Utrecht, The Netherlands

ABSTRACT

A new optical technique is introduced for determining the elastic constants of nematic

liquid crystals by measuring the spectral noise intensity of scattered laser light.

I. INTRODUCTION

In a nematic liquid crystal three types of distortion can be distinguished, each of

which is characterized by a specific elastic constant [I]. These static quantities can be determined via light scattering experiments by measuring the angular dependence of the total

cross section [2]. However, experimental problems will arise because of refraction phenomena

at the walls of the sample. These problems are caused mainly by poorly defined angular dependent reflection, interference phenomena and a change in the subtended solid angle [3] . In addition the measurements may be impaired by the occurrence of stray light.

In a liquid crystal the elongated molecules fluctuate around a mean direction called the

director. It is because of these orientation fluctuations that light is strongly scattered.

Relaxation effects that are coupled with the three types of distortion give rise to Lorentzian

line broadening of the scattered light, from which a relaxation time can be determined. This

relaxation time provides us with direct information about the dynamics of the nematics,

namely the distortion viscosity constants [4]

.

The experimental problems that one normally associates with total cross section measurements are of little consequence in measurements of the spectral linewidth of the scattered

light. Because of the three types of distortion one generally measures a spectrum of a sum of

Lorentzian broadened lines each with an intensity that is weighted by functions depending on

optical parameters and on the elastic constants. The linewidth as determined from noise data

has been published earlier [5, 6]. As this paper will show we can obtain the three bandwidths separately and the ratio of

the elastic constants by making a computer analysis of noise data obtained as a function of

the scattering angle.

shall demonstrate this for the case of an optical configuration in which only the

splay and twist deformation phenomena are observed.

II. THEORY

The expression for the noise intensity spectrum of scattered laser light is given by [5,

6]:

r 2Us (Ug + u ) 1 s^,U)=m2 I r2 + 2r r„ ^2 ^ 1 a=l,2 [0.2 , (2u3^)2] ' 2 ^(.2 -H (U3^+ U32)2] J

where r EG ^/k (q) , with G E (f i. + f„i )

; M is a constant.

a a a a 3 3 a The vectors T and f are the polarization vectors of the incoming and scattered light respectively. The vector components are considered with respecjt to^an orthonormal coordinate

system defined by the following base vectors e^ = n^, e^ = (nQxq)/|nQX q|, gj E 62 x e^,

343
where q is the scattering wave vector and nQ represents the director. The function K(j((q) is equal to K^^^qJ^^ + K-^q/ (a = 1, 2), where Kj , K2 and K3 represent the splay, twist and bend

elastic constants respectively [2]. The components q/ / and qj^ of the scattering wave vector

are parallel and normal to the director. In our case we took the polarization of the incoming light to be 1 = [0 1 0] ; therefore q// = 0 and k^^^ =

K^qJ^^. Hence denoting splay and

twist by a = I , 2 respectively we observe that the spectral intensity S^j can be considered

to be a sum of three Lorentzians, associated with pure splay, pure twist and a splay/twist

cross term, the half-bandwidths of which are related to the viscoelastic properties [5], i.e

(to,) = 2u^ = 2k /n . In our configuration it holds that u„ = (K /n )q|^.

2 a Sot a a °a a a j. Although the measured noise spectrum itself is not a pure Lorentzian, but is a sum of three

Lorentzians, it is still possible to determine its half-width.

This result depends on the well-known optical parameters Ga and qj^ as well as on three

unknown ratios, i.e. m/K], ^2/^2 ^^'^ ^]/^2' ^'^^^^ constants (see eq (1)).

By measuring the noise spectrum at several different scattering angles, the half-bandwidths can be found as a function of G^j and qj^, because the latter parameters are functions

of the scattering angle. With the help of a computer one can obtain a best fit to these data

by using the following ratios as fitting parameters: ti]/K], 112/^2 Kj/K2.

Therefore not only one can obtain the viscoelastic ratios from noise measurements [5] but one can also find the ratio of the elastic constants.

III. EXPERIMENTAL RESULTS

In figure 1 results are shown for OHMBBA (O-hydroxy-p-methoxybenzylidene-p'-butylaniline)

i(a)p exp. at a temperature t = 60.00 C. In the figure the experimental data for Ug

plotted on a log-log scale versus q|^. The solid line is the best fit to data. This best fit is obtained witTT the help of a least squares method c The fitted values for the viscoelastic and elastic ratios are indicated in figure 1. Figure 2 shows results for K]/K2 at different temperatures. In addition we present the results obtained

from measurements of scattered light intensity distributions [7] and from measurements of

Freedericksz transitions [8]

.

40

20

10

-I

1 OHMBBA

tz eo oo'c

1 1 1 1 1 1 1 1 1 1 10

10

11 m40

Fig. 1 An example of Ug versus q^ . The crosses represent the experimental data; the

straight curve is the best fit with

m/Ki = 7.2- 10^ sm-2, n2/K2 = 1.45- lO^O sm"^ and K1/K2 = 1.8.

Fig. 2 Results of K1/K2 versus temperature.

The crosses are from the fitting procedure;

the dots are from light intensity distributions [7]; the dashed curve is the best fit

for data obtained from Freedericksz transitions [8].

344
IV. CONCLUSION

It has been shown that spectral noise intensity measurements not only give us information

about dynamical quantities but they also tell us about static quantities such as the splay/

twist elastic ratio, K]/K2. The error in Kj/K2 is mainly caused by experimental errors in

(ojpgxp.* shown in figure 2, this statistical error is rather large compared to the

statistical errors associated with the other two methods. It should be noted, however, that

the different methods yield results which differ by more than can be accounted for by statistical errors. Apparently the errors involved are of a systematic nature.

We thank Sheila McNab for indicating and correcting the grammatical mistakes and Greetje

Hollander for typing this manuscript.

This work was performed as a part of the research program of the "Stichting voor Fundamenteel

Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor

Zuiver Wetenschappelijk Onderzoek" (ZWO).

REFERENCES

[1] M.J. Stephen, and J. P. Straley, Rev. Mod. Phys. 46, 617 (1974).

[2] P.G, de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974).

[3] M. Lax, and D.F. Nelson, in Coherence and Quantum Optics, Ed. L. Mandel and E. Wolf

(Plenum Press, New York, 1973) p. 415.

[4] D.C. van Eck, and R.J.J. Zijlstra, J. Physique 4J_, 351 (1980).

[5] D.C. van Eck, and R.J.J. Zijlstra, in Noise in Physical Systems, Ed. D. Wolf (Springer

Verlag, Berlin, 1978) p. 270.

[6] D.C. van Eck, and W. Westera, Mol. Cryst. Liq. Cryst. 38, 319 (1977).

[7] J. P. van der Meulen, and R.J.J. Zijlstra, to be published.

[8] F. Leenhouts, H.J. Roebers, A.J. Dekker, and J.J. Jonker, J. Physique 40, C3-291 (1979).

345

V. QUANTUM NOISE

347

QUANTUM NOISE LIMITED DETECTION IN TUNNELING

DEVICES AT MILLIMETER WAVELENGTHS

John R. Tucker

Department of Electrical Engineering

and Coordinated Science Laboratory

" 1406 West Green Street

University of Illinois

Urbana, Illinois 61801

INTRODUCTION

Detectors and heterodyne receivers designed to approach quantum limited sensitivity in

the millimeter wave spectral region are currently being developed. This new class of quasiparticle mixer utilizes the extraordinary nonlinearities that occur in single-particle tunneling of electrons into a superconductor on one or both sides of a tunnel junction. These

nonlinear devices perform the same detector functions as standard Schottky diodes and other

resistive mixers, but with an important physical difference. At frequencies where the

photon energy exceeds the voltage scale of the dc nonlinearity , a single-particle tunnel

junction ceases to respond classically and becomes capable of detecting individual quanta.

The results of this transformation are very dramatic, and appear certain to revolutionize

low noise receiver technology in the millimeter wave region.

The phenomenon of photon-assisted tunneling was first observed almost twenty years ago

by Dayem and Martin [1]. A theoretical interpretation proposed shortly thereafter by Tien

and Gordon [2] correctly described the basic effect illustrated in figure la. Microwave

power applied to a superconductor-insulating oxide-superconductor (SIS) tunnel barrier re- sults in the appearance of well-defined step structures on the quasiparticle portion of the

dc I-V curve. This type of SIS diode is actually a form of Josephson junction in which the

capacitance is large enough to effectively shunt the oscillating pair current at frequencies

(jj = 2eVo/'fi near the gap voltage Vg = 2A/e. Under these conditions, the tunneling of single

electron quasiparticles will determine the shape of the dc I-V curve over a substantial

range of bias voltage. The sharp onset of quasiparticle current at Vg = 2A/e is a direct

reflection of the gap opened up in the single-particle excitation spectrum for identical

superconductors on opposite sides of the barrier, as schematically illustrated in figure lb.

At the bias voltage in this diagram, the absorption of a single millimeter wave quantum fiui

Fig. 1 (a) Photon-assisted

tunneling steps (dashed curve)

induced by an applied microwave

field on the dc I-V characteristic (solid curve) of an SIS

tunnel junction, (b) Densities

of states vs. energy of singleparticle excitations for identical superconductors on either

side of a tunnel barrier.

I . I 1 I A 0 2A/e Vq •>(«) D(«)

(a) (b)

349
can supply the energy required to tunnel from occupied single-particle states on the left

into unoccupied states above the energy gap on the right. The sequence of steps below the

gap voltage on the driven I-V curve in figure la reflects the series of bias thresholds for

which the absorption of at least n = 1,2,3 ... photons is required to provide sufficient

energy for single-particle tunneling. The amplitudes of these steps are directly related

to the Tnagnitude of the applied ac field.

Although this effect has been known for many years, the relationship between photonassisted tunneling and the potential for quantum limited detection has been understood and

applied only recently. The first device to utilize the extreme single-particle nonlinearity

of a superconductive tunnel junction as a detector was the superconductor-semiconductor or super-Schottky diode [3] . The super-Schottky demonstrated extraordinary sensitivity as both a direct detector and heterodyne mixer at 10 GHz [4], and subsequently at 31 GHz [5]. It also stimulated a substantial body of theoretical work aimed at characterizing the potential impact of photon-assisted tunneling on the detection process [6,7,8]. Quantum effects

in this type of resistive mixer were predicted at frequencies where the photon energy -ftco/e is large compared to the voltage scale of the dc nonlinearity, as in figure la. The nonlinearity of the super-Schottky diode is less spectacular than that of an SIS junction,

however, and parasitic effects have thus far prevented its operation at frequencies high

enough to induce significant quantum effects. In the spring of 1979, three groups reported

successful mixing experiments using the sharp quasiparticle nonlinearity in small area SIS

tunnel junctions [9,10,11]. Since that time, substantial progress has been made toward

utilizing photon-assisted tunneling as a technique for approaching quantum noise limited

detection in the millimeter wave region.

THEORY

A physical feeling for photon-assisted tunneling and its impact on detection may be

obtained by considering the response to an applied sinusoidal potential:

V(t) = + cosojt (1)

The result derived by Tien and Gordon [2] for the dc component of the tunneling current is

then given by:

00

^dc

" ^ J^(eV^/1ico) I^^CVq + nWe) (2) n=-oo Here the dc current in the presence of applied radiation is expressed in terms of the static

dc I-V characteristic evaluated at a series of bias voltages reflecting the absorption or emission of integral numbers of quanta in the tunneling process. Equation (2) provides a quantitative account of the step structure observed in figure la.

In small ac fields, the following result for incremental change in the dc current is

obtained by expanding these Bessel functions to lowest order:

+ -Ka)/e) - 21, (VJ + I, - 1iw/e) "1 AT =^ Vf i

" ^ y (3) dc 4 1 1 /« / n2

(ti(jj/e) J

The expression in brackets is a second difference which reduces to the second derivative of

the I-V curve in the classical limit. The dissipative component of current induced at the

applied frequency is similarly found to be [7,8]:

}

f I,^(V_ +^tM/e) - I, (V„ -fiw/e)

oj 1 I 2ti(jL)/e

The conductance of the junction at high frequencies is thus determined by a first difference

form representing the absorption or emission of a single photon. The current responsivity

350
of the tunnel junction, defined as the change in dc current per unit power absorbed, may be

obtained from eqs. (3) and (4) in the form:

i dc 1 0)

e_ \\c^\ ^^/"> - ^^dc^^O> ^dc^^O - ^^/^^l

tico 1 I^^(VQ+We) - I^^CVq -1ia)/e)

J

2 2 . (5)

d Ijjj./dVQ

:r — — — » Classical limit

2 dl, /dV„

dc 0

e_

'hoi , Quantum limit

The standard classical result is recovered when the slope of the dc I-V curve changes slowly

on a voltage scale "fiw/e. The quantum limit, on the other hand, is realized at high frequencies where the nonlinear conductance varies rapidly on the photon energy scale. This

limit would be approached in figure la, for example, at values of dc bias voltage lying in

the region of the first step below the gap. The basic process in this case is illustrated

in figure lb. The quantum limit for current responsivity , e/tico, represents the tunneling

of one additional electron for each signal photon absorbed.

The transition to quantum limited response in a single-particle tunnel junction produces

a variety of new phenomena which can be utilized to achieve low noise detection. The com- plete theoretical framework developed in Ref. [8] characterizes all properties of such

devices in terms of their static dc I-V characteristic. The resistive portion of the re- sponse is directly related to the dc current at voltages corresponding to the absorption

or emission of specific number of photons, as in eq. (2). At high frequencies, additional

quantum reactances appear which are expressed in terms of the Kramers-Kronig transform of

the dc I-V curve. The noise properties of these tunnel junctions may also be related to the dc current through a fluctuation-dissipation theorem. In the direct detection mode

described above, the noise equivalent power will be limited by shot noise due to the bias

current. Heterodjme detection may be utilized, however, to approach quantum noise limited

sensitivity. Photon-assisted tunneling theory makes the following major predictions for

such heterodyne mixers under appropriate operating conditions [12]

:

(1) . Quantum noise limited mixer temperatures: T^^ ~ "Rw/k.

(2) . Conversion gain : simultaneous amplification accompanying frequency downconversion of the incoming signal.

The potential for conversion gain is a matter of crucial practical importance. Since photon

energies correspond to only a few degrees Kelvin in the millimeter wave region, noise from

amplifiers following the mixer would otherwise dominate the total receiver temperature.

Modest amounts of conversion gain, however, will reduce these amplifier noise contributions

sufficiently that quantum noise limited performance may be approached in practical re- ceivers. The theory also predicts that very large conversion gain associated with a negative output impedance may be obtained under certain conditions.

EXPERIMENT

The initial experiments on SIS quasiparticle mixers [9,10,11] all reported very low

mixer noise temperatures, but with conversion losses in the range ~ 6-10 db. These re- sults were comparable to those previously achieved with the super-Schottky diodes, except

that distinct evidence of quantum effects due to photon-assisted tunneling were observed [9,10]. The direct detection properties of SIS junctions were also measured at

36 GHz, yielding an NEP ~ 2.6xl0~16 W/Hz^ and a current responsivity in virtually exact

agreement with eq. (5) [13].

351
Subsequent work on SIS heterodyne mixers has produced dramatic improvements. The

Caltech-Bell Labs group has constructed complete 115 GHz astronomical receivers that are already competitive with the best available cooled Schottky diode instruments [14]. Conversion efficiencies in excess of the theoretical maximum for classical resistive mixers

were first demonstrated by the Berkeley group [15]. A single-sideband (SSB) conversion

efficiency of 0.90, and a double-sideband gain of 1.40 were reported using Pb(Bi) alloy

junctions whose mixer noise temperatures ~ 1.5K equaled the quantum noise limit within

experimental accuracy at 36 GHz. Their measurement is shown in figure 2. The peaks in

mixer output correspond to the positions of the photon steps on the driven dc I-V curve, and are separated by -Roj/e ~ 0.15 mV. The theoretical prediction indicates that even greater conversion efficiencies could be realized on steps closer to the gap voltage

2A/e ~ 3 mV if higher harmonics, not included in the calculation, were suppressed by a larger junction capacitance. Below about 2 mV, Josephson pair tunneling currents disrupt

the low-noise quasiparticle mixer performance. This effect may be partially suppressed by

application of a constant dc magnetic field.

t 1

I I I I L 0 12 3

V (mV)

Fig. 2 I-V curves at T = 1.5K for a Pb(Bi) alloy junction (a) without and

(b) with applied 36 GHz local oscillator

power. Plots of IF mixer output power

are shown in the lower portion of the

figure as a function of bias voltage.

Point (c) and curve (d) represent thermal

sources, and are used to deduce mixer

noise temperature. Curve (e) is the

mixer signal obtained from a calibrated

36 GHz source, and measures conversion

efficiency. The dashed curve (f) is

the prediction of 3-port quantum mixer

theory with no adjustable parameters [15].

The group at Chalmers University in Sweden recently reported excellent results using

six-junction Pb(In) series arrays at 73 GHz [16]. Their maximum SSB conversion efficiency

was approximately 0.65 (L^ =2.0 db) as illustrated in figure 3, again with mixer noise

temperatures near the quantum limit. Note that optimum efficiency occurs for dc bias

voltages near the middle of the first and second photon steps below the gap. Because this

is a six-junction array, the width of these steps and the spacing between conversion peaks

becomes 6tiaj/e. The entire voltage scale is thus enlarged by a factor of six, because six

times as many photons must be absorbed in passing a single electron through the array com- pared with a single junction. The use of arrays, particularly for high frequency applications, appears very promising based on these experiments. Larger area junctions can be

combined in series to achieve a total impedance appropriate to meet circuit matching re- quirements, and these larger areas alleviate constraints on both fabrication requirements

and the magnetic field strengths required to suppress Josephson effects. Dynamic range

will be increased for an array, although the quantum limited mixer noise temperature will

also increase to Tj^ ~ Ntito/e for N junctions in series.

Very recently, the Berkeley group has achieved quite large SSB conversion gains of

approximately 5.7 db at 36 GHz using Sn junctions with very sharp I-V curves [17]. In- finite available conversion gain has been observed by Kerr, et al. [18] at 115 GHz, and

the predicted negative differential output impedance seen for the first time. The

352
0 Fig. 3 I-V curve (a) without and (b)

with applied local oscillator power at 73.5 GHz, and (c) mixer conversion loss

for a six element series array of Pb(In)

alloy junctions at T = 1.5K [16].

CD

T3

1.0 <

E

c 05 2

- 0

0 10 20

Bias voltage (mV)

development of these SIS quasiparticle mixers is proceeding very rapidly, and practical

millimeter wave receivers with system noise temperatures approaching the quantum limit

should be realized in the near future.

This work was supported in part by the Joint Services Electronics Program (U.S. Army, U.S.

Navy, U.S. Air Force) under contract number N00014-C-79-0424

.

REFERENCES

[I] A. H. Dayem and R. J. Martin, Phys. Rev. Lett. 246 (1962).

[2] P. K. Tien and J. P. Gordon, Phys. Rev. 129 , 647 (1963).

[3] M. McColl, M. F. Millea, and A. H. Silver, Appl. Phys. Lett. 23, 263 (1973).

[4] M. McColl, M. F. Millea, A. H. Silver, M. F. Bottjer, R. J. Pedersen, and F. L. Vernon, Jr., IEEE Trans. Magn. MAG-13 , 221 (1977).

[5] A. H. Silver, R. J. Pedersen, M. McColl, R. L. Dickman, and W. J. Wilson, IEEE Trans. Magn. MAG-17 , 698 (1981).

[6] J. R. Tucker, in Proo. 14th Int. Conf. on Low Temperature Physios LT-14, Eds. M. Krusius and M. Vuorio (North Holland, Amsterdam, 1975), vol. 4, p. 180.

[7] J. R. Tucker and M. F. Millea, Appl. Phys. Lett. 21, 611 (1978).

[8] J. R. Tucker, IEEE J. Quantum Electron. QE-15 , 1234 (1979).

[9] P. L. Richards, T. M. Shen, R. E. Harris, and F. L. Lloyd, Appl. Phys. Lett. 34,

345 (1979).

[10] G. J. Dolan, T. G. Phillips, and D. P. Woody, Appl. Phys. Lett. 34, 347 (1979).

[II] S. Rudner and T. Claeson, Appl. Phys. Lett. 34, 711 (1979).

[12] J. R. Tucker, Appl. Phys. Lett. 36^, 477 (1980).

[13] P. L. Richards, T. M. Shen, R. E. Harris, and F. L. Lloyd, Appl. Phys. Lett. 36,

480 (1980).

353
[14] T. G. Phillips, D. P. Woody, G. J. Dolan, R. E. Miller, and R. A. Linke, IEEE

Trans. Magn. MAG-17 . 684 (1981).

[15] T. M. Shen, P. L. Richards, R. E. Harris, and F, L. Lloyd, Appl. Fhys. Lett. 36_, 777 (1980).

[16] S. Rudner, M. J. Feldman, E. Kollberg, and T. Claeson, IEEE Trans. Magn. MAG-17 , 690 (1981).

[17] W. R. McGrath, P. L. Richards, A. D. Smith, H. Van Kempen, D. E. Prober, and P. Santhanan, Bui. Am. Phys. Soo. 26, 307 (1981).

[18] A. R. Kerr, S. K. Pan, M. J. Feldman, and A. Davidson, paper submitted to IBth

Int. Conf. on Low Temperature Physios LT-16j Los Angeles, August 1981.

354
QUANTUM TUNNELLING AND NOISE IN SQUIDS

A.J. Leggett

School of Mathematical and Physical Sciences,

University of Sussex, Falmer, Brighton, BNl 9QH,

Sussex, U.K.

The dynamics of the flux ^ threading a SQUID ring is usually described by the so-called

RSJC ("resistively shunted junction with capacitance") model (see e.g. ref []l_|) which leads

to the following classical equation of motion

:

c$ + ^/R + 6u/6(}) = 0 (1)

U(()):(i)^) E (()) - (J),.)^+ I„<))„ /l - cos /27t4)W (2)

Here C is the effective capacitance of the Josephson junction (cf. below) and R the

resistance shunting it, which under the assumptions of the model can be inferred []221 from the

asynptote of the experimental I-V characteristic. (^^^ is the externally imposed flux threading the ring, which in the present context will be taken as a c-nximber parameter controlled

by the experimenter, I^ the junction critical current, L the self-inductance of the ring and

= h/2e the flux quantum. I shall be concerned in this paper exclusively with "hysteretic"

SQUIDS (2'^L1^/(^Q > 1) for which the "potential" U((f)) may have one of more metastable minima;

I shall refer to a well (minimum) afe "deep" if the energy barriers separating it from the

neighbouring wells are approximately I^(()q/tt (a necessary, but not sufficient, condition for

this is evidently that LI(^/())o >> D •

If the flux is trapped in a well which is metastable with respect to its neighbours,

thermal noise may drive it over the energy barrier; this process is of particular practical

significance when it occurs near the classical "break point" of a SQUID operated in the rf

mode, since it will then limit the accuracy with which the external flux <t>x can be measured.

The standard theory of this process is due to Kurkijarvi |^lj , and for present purposes it is

useful to note how the main factors in his expression arise. As is well known, for any

classical nucleation process, the rate can be expressed, apart from a prefactor of order

unity, in the form

P = uj exp - Au/k„T (3) cl o B where U) is the frequency of small oscillations around the metastable equilibrium (but see below) and Au is the height of the free energy barrier. Note that the (dominant) expone^nt

is independent of both the capacitance C and the resistance R of the junction (The

prefactor depends on both, and in fact for the highly damped case is proportional to

(j0q^/2tty rather than Ij^q, where Y = 1/CR: thus in this limit it is proportional to R and

independent of C) . The order of magnitude of the rms fluctuation, Oj^nof the value of external

flux at which the transition to the neigtibouring well occurs can be obtained by setting 0 ^ equal to the value of 6(j) E (j) - 4> at which the P , is of the order of T ^ ; here T is X XC X cl the time available for the transition to occur rf period) and 6 is the values of 6 at

XC X the classical break point (i.e. AU((J) ) = O) . In the case of a strongly hysteretic SQUiD

^"c^^o -"-^ ^® ^^^^ C-'-^- AU(6<1) ) =^^2/2 <}) ^. /<}) / 6(}) \

3rf L \hlJ \^ /

(4)

and also (for future reference)

cj ox{&<t> ) =/i {2W2)V2 (LI /(J) )V''v6()) \yi. (5) „ _ I .. 1 :/i y/2 (2Tr/2) V2 (Li^/(j)^) y-y^^^Y
Note that cOq as well as Au tends to zero as we approach the classical break point. From (4) we obtain the result [^1_|

("sir /k^T f3/LI \ V, (6)

where the constant depends logarithmically on the sweep rate and is of order 1. The

formula (5) agrees well with the published data [^3~|.

It was long ago suggested []4j that at sufficently low temperature the dominant flux

transition mechanism in a SQUID should be not thermal fluctuations but quantum tunnelling

through the energy barrier. Such a possibility is of interest not merely in the context of

the ultimate attainable sensitivity of SQUID magnetometers but also because, since the

tunnelling variable (the flux) is recognizably macroscopic, the observation of this phenomenon

would constitute circumstantial evidence for the hypothesis that the linear laws of quantvim

mechanics can be extrapolated to arbitrarily complex systems [^5, 6J - an assumption which,

while often made in the context of discussions of the quantum theory of measurement, has at

present no experimental basis (see ref . [|6 |) . In the remainder of this paper I review the

theoretical and experimental status of this problem. Although similar considerations can be

given for the case of a single current-biased Josephson junction (^7J, I shall confine the

explicit discussion to the case of SQUID rings, where the tunnelling process is in some ways

conceptually simpler and also moj;e unarguably "macroscopic". (though see below).

If we for the moment forget about the dissipative terms in eqn. (1) , this equation can

be derived from a Hamiltonian consisting of the "potential" energy U((})) (eqn. (2)) plus a "kinetic" energy p^^/2C = hC^^ , where the junction capacitance C plays the role of the particle mass in a mechanical problem. Application of the standard WKB techniques to the problem

of tunnelling between neighbouring minima of U(())) (at zero temperature) then leads to the

result

P. = const. 03 exp - aAu/ri03 (7) QM o o when the dimensionless constant a depends on the shape (but not the scale) of the energy

barrier: we note that a is 36/5 for a cubic potential ^Sj (appropriate to tunnelling near

the break point) and 8v^ for a pure cosine potential (roughly appropriate to tunnelling

between "deep" wells) . Comparing eqn. (7) with eqn. (2) we see that for a fixed potential

shape quantum tunnelling is likely to overtake thermal fluctuation as the dominant flux

transition mechanism when kgT< a ti03 ; however, by this time both processes will have

negligible probability in a laboratory time-scale unless aAU/fi03Q is less than about 30. In particular, the observation of tunnelling between "deep" wells would appear to require

CIc < 10 in SI units.

For transitions occuring near the break point of a strongly hysteretic SQUID we use

the same order-of-magnitude argument as above to estimate the rms uncertainty Ogm in the

value of 4) at the transition which is due to quantum tunnelling (at zero temperature) . This

gives

V5 <h (8) o

where (jJ^^^ = (LC) Vz and 03 = (2tti^/C(J)^) (03 is the Josephson plasma frequency of the isolated

junction at zero current Bias) . More quantitative treatments of this problem are given in

refs. ^8^1 and ^9 |. Comparing (8) with (6) we would conclude that tunnelling is likely to become the dominant mechanism at temperatures well below a characteristic temperature Tq

given by

k„T '\^o.l5cTi03 , c S (fi03,/l (}) )°'^ (9) B o = J J CO

The factor c is fairly insensitive to the parameters and is generally of order 0.2. Thus

for 03 lO^^ we would expect timnelling to be dominant at temperatures well below lOO mK.

= cons QM 5 \VI ^ fi03^^.Ti03j LI^

L O o.

However, the above results are all based on a simple WKB approach in which the

356
dissipative term in the equation of motion (1) is completely neglected. Since practical

SQUIDS are often quite heavily overdamped, it is of crucial importance to know how the

dissipation will affect the quantum tunnelling probability. This is_a special case of a general problem considered recently by Caldeira and the author f8,10 |; transposing our

results to the SQUID case, we find that if the motion in the classically accessible region

is well described by eqn. (1) , then the tunnelling probability ^QM suppressed relative

to the value Pqm^ calculated in the simple WKB approximation by an exponential factor:

P^/Pq^^"^ exp - f.(A(}))VRri (10)

Here Ac}) is the distance in flux space that must be traversed "under the barrier", and f is

a niomerical factor which must be found from the solution of a nonlinear integrodifferential

equation and depends on the shape of the barrier and, weakly, the degree of damping. Without

obtaining an exact solution, it can be shown |~8^ that f is in general of order unity and

for heavy damping is bounded by the inequalities > f > ^2/^^ (Y/w ) where y = 1/RC is the

damping coefficient and the numbers Bj» depend on the shape of tEe potential barrier.

In this limit, therefore, the "effective WKB exponent" is dominated by the shunting

resistance R and is almost independent of the junction capacitance.

Two consequences are immediately evident. First, tunnelling between "deep" wells (for

which Acf) (J)^) should be essentially forbidden, whatever the value of capacitance, if the

order of magnitude of the shunting resistance is much less than the characteristic value

(t>^^/1i '^^ 40 kO,. Secondly, the uncertainty in tha value of at which the transition

occurs in an rf SQUID should be changed from eqn (8) to an expression which in the overdamped

limit varies approximately as RLI;^, with only a very weak dependence on C. For reasonable

values of the parameters Og^^ can be very small indeed, and Kurkijarvi |^9 | has argued on this

basis that it_is not relevant to the ultimate flux sensitivity of a realistic rf SQUID (cf

also ref . [nj) .

Let us nov turm to the experimental situation. We note, first, that two experiments

have so far looked for the related phenomenon of quantum tunnelling of the phase in a simple

current-biassed Josephson junction; one, several years ago by Fulton and Dunkleberger ^12 |, observed some suggestive departures at low temperatures from the classically predicted

behaviour which they thought might have been due to quantum effects, while a similar very

recent experiment by Jackel and co-workers |^13_| found no deviation from the classical

predictions down to temperatures of the order of 0.25 hw^/kg. In the case of SQUID rings

proper, a rather direct experiment was carried out by den Boer and de Bruyn Ouboter in

Leiden, using Mb point contact SQUIDS for which the parameter LI /cj) was not large compared

to 1 (and to which, therefore, not all the above formulae would §e expected to apply

quantitatively) ; they varied the externally imposed flux (j) and monitored the trapped flux cj) as it jumped between the two available minima. At ambient temperatures of the order of IK

the rate of jumping was several orders of magnitude faster than the classical fluctuation

theory would allow, and they show that it is compatible with the quantum tunnelling rate as given_by the simple WKB calculation without damping if one assumes a capacitance of the order

of 10 ^^F. (cf below) . A second, rather less direct, experiment was carried out by

T.D. Clark and co-workers at Sussex, who measured the current-voltage characteristic of a low-noise 430 MHz rf SQUID down to 1.8K; at this temperature they observed a number of

intriguing departures from the conventionally expected I-V characteristic which they believe

"are caused by the onset of QE behaviour (i.e. flux tunnelling) in the SgUID ring". It should be observed that the interpretation of the data given in ref. |^14_| appears to require

an appreciable probability of tunnelling between "deep" wells.

Two questions obviously arise: first, how far are these two experiments actually evidence

for quantum tunnelling of the flux, and secondly, if such tunnelling is indeed occurring, how

far is this compatible with the theoretical framework outlined above? I shall not discuss the

first question here, since while in the Leiden experiments the only obvious question concerns

the possible presence and effect of non-thermal external noise (cf below) , a proper discussion

of the Sussex data would require a detailed analysis of the dynamics of the tank circuit for

this somewhat novel situation. Turning to the second question, it is clear that a great

deal hinges on what we take to be the effective junction capacitance C. In the Sussex

experiments the above WKB type considerations, even without account of damping, would predict

negligible tionnelling unless C is less than 10 ^®F (The theory actually used in ref. l^l^J to interpret the Sussex data, which would allow tunnelling for values of C several orders of

magnitude larger, appears not to be equivalent to any simple version of the WKB theory) . For

357
such ultra-small values of C it is not clear that the model used here would be even a good

zeroth-order description. (since, inter alia, the capacatitive energy of a single pair,

2e^/C, is larger than both hto^ and the bulk energy gap) . In the Leiden experiments compatibility with the WKB predictions ignoring dam;^ing [[S | requires that we assign to C not its

bulk"geometrical"value (presumably 10 ^ F) but a smaller value, say 10 ^^F. It is

argued in ref fs | that this is plausible since the capacitance of the junction itself may be

effectively in series with a rather large inductance L', say 10 ^°H, the whole being presumably shunted by the much larger geometrical capacitance Cj^; the absence of hysteresis in

the lunction I-V characteristic is used to put an upper limit on C which is indeed of order

10 ^ F (cf also ref. . A preliminary study of the "undamped" WKB prediction for this

effectively two-dimensional model seems to indicate that tunnelling is indeed possible

provided that L' is large enough, but that the tunnelling event itself is localized in the

region very close to the junction and the flux change then propagates through the macroscopic

ring by a purely classical process. It is however less c].ear that such a mechanism would

necessarily give the observed lack of hysteresis. The effect of damping also needs a detailed study, since the junction resistances in this experiment are sufficiently large that

a crude order-of-magnitude estimate leaves it unclear whether it would suppress the

tunnelling sufficiently to render it unobservable . Clearly, in the light of the above, it

would be highly desirable in principle to do tunnelling experiments in SQUIDS using junctions

whose capacitance can be independently measured, e.g. by observing the Josephson plasma

resonance

.

In conclusion I_note that if the RSJ description is indeed applicable and the theory

developed in ref flO | to take account of damping is correct, then it should be relatively

easy to discriminate genuine quantum tunnelling in SQUIDS from extrinsic noise effects by

investigating the effect of varying the shunting resistnace: the very strong suppression

predicted by eqn flO | should be an unambiguous signature of quantum tunnelling.

I am grateful to A.O. Caldeira, T.D. Clark, J. Clarke, R. de Bruyn Ouboter,

T.A. Fulton, J. Kurkijarvi and L.D. Jackel for helpful discussions on these topics and for

various preprints.

References

J. Kurkijarvi, Phys. Rev. B6, 832 (1972).

[2] D.E. McCiimber, J. Appl. Phys. 39, 3113 (1968)

[3] L.D. Jackel, W.W. Webb, J.E. Lukens and S.S. Pel, Phys. Rev. b9, 115 (1974)

[4] D.J. Scalapino, in Tunnelling Phenomena in Solids, ed. E. Burstein and

(Plenum, New York, 1969), p. 477.

S. Lundqvist

[511 W. den Boer and R. de Bruyn Ouboter, Physica 98 B+C, 185 (1980)

A.J. Leggett, Progress of Theoretical Physics, special issue in honour

(in press)

of R. Kiabo, 1981

1

— ^11

—

i

1

1

1

1

Yu M. Ivanchenko and L.A. Sil'bersman, Sov. Phys. JETP 28, 1272 (1969)

A.O. Caldeira, Ph. D. Thesis, University of Sussex, 1980 (unpublished)

[911 J. Kurkijarvi, submitted to J. Low Temp. Phys.

A.L. Caldeira and A.J. Leggett, Phys. Rev. Letters 46, 211 (1981)

[[ll^l R.H. Koch, D.J, van Harlingen and J. Clarke, Phys. Rev. Letters 45, 2132 (1980)

|^12~1 T.A. Fulton and L.N. Dunkleberger , Bull. Am. Phys. Soc. 19,205 (1975): cf Phys. Rev.

B9, 4760 (1974)

|^13j L.D. Jackel et al, abstract submitted for March 1981 APS meeting.

Ql4~| R.J. Prance et al, Nature 289, 542 (1981)

358
QUANTUM NOISE IN JOSEPHSON JUNCTIONS AND SQUIDS

Roger H. Koch, D. J. Van Harllngen and John Clarke

Department of Physics y University of California, and

Materials and Molecular Research Division, Lawrence

Berkeley Laboratory, Berkeley, California 94720

INTRODUCTION

The effects of thermal noise on a current-biased resistively shunted Josephson junction

(RSJ) have been extensively studied. The noise source is assumed to be the Nyquist noise

with spectral density AkgT/R developed in the shunt resistor R at a temperature T. This noise

has two effects. First, it induces transitions from the zero voltage state to the non-zero

voltage state at bias currents below the thermodynamic critical current, Iq, thereby inducing

"noise rounding" of the current-voltage (I-V) characteristic [1]. Second, the Nyquist noise

generates a voltage noise across the junction that, according to the theory of Likharev and

Semenov [2] , has a spectral density

4k_TR^

at frequencies much less than the Josephson frequency. In Eq. (1), - 8V/8I is the dynamic

resistance of the junction.

The results of the theory have been well established experimentally [3,4]. One application of the theory is to predict the sensitivity of the dc SQUID [5] for a given set of ex- perimental parameters. The predictions have been in quite good agreement with results obtained experimentally [6]. As the sensitivity of these devices is improved, however, the

theory will eventually break down as quantum processes become important and set a limit on

the performance. In this paper, we briefly outline a theory for noise in the RSJ in the

quantum limit [7], and extend it to the dc SQUID [8]. Experimental results for both the

single junction and the SQUID are reported.

QUANTUM NOISE THEORY FOR A RESISTIVELY SHUNTED JUNCTION

We consider a Josephson tunnel junction with critical current Iq and capacitance C shunted with a resistance R. We assume that the voltage is less than 2A/e, where A is the

energy gap, and that T is well below the transition temperature, so that the quasiparticle

tunneling conductance is small compared with the shunt conductance. Our central assumption

is that the current noise in the shunt resistor has a spectral density

„ , - 2hv / hv \_ 4hv / 1 . 1 \ S^(v) = — coth^—j= -^(2+^hv/kBT_J

at frequency v. The equation of motion of the junction can be written in the form

1^ 6 + ^ 6 + I sin 6 = I + T (t) (3) 2e 2eR o N

where 6 is the phase difference across the junction, and InC^) has the spectral density of

Eq. (2).

359
We first assume that the capacitance is small (g^, E 2irIoR^C/$Q<<l, where $q = h/2e is the

flux quantum), and that the noise is small so that noise-rounding can be neglected. The I-V

characteristic is then [9] V = R(l^-lo^)i , and Eq. (3) may be solved analytically using the

techniques of Likharev and Semenov [2]. For frequencies much less than the Josephson frequency 2eV/h one finds

^(0) = V

2eV / I cotheV

V

(4)

In the limit eV<<kBT, the hyperbolic cotangent may be expanded and one recovers the thermal

noise limit, Eq. (1). The first term on the right hand side of Eq. (4) represents noise

generated in the shunt at the measurement frequency, while the second term arises from noise

generated at frequencies near the Josephson frequency that is mixed down by the non+linearity

of the junction. Equation (4) is plotted in Fig. 1 for 5 values of the parameter k= el^jR/kgl.

When K<<1, the curve becomes indistinguishable from that of the thermal limit, but as k in- creases, the voltage noise is significantly enhanced above the thermal result. In the ex- treme quantum limit (T=0) , one finds

S (0) = 2eV(I /I)^ rJ/R

V o D

2 3 2el R /V.

o

(5)

The physical interpretation of the noise is as follows. Equation (3) also describes the

motion of a particle on a tilted washboard. Then, C represents the mass of the particle,

1/R the viscous damping, and Iq the amplitude of the oscillating potential. The average

slope of the washboard is proportional to I. At bias currents above the critical current,

the effect of I^Ct) is to induce random variations in the slope, and thus produce random

noise in the velocity of the mass, 6, that

represents the voltage in the case of the

junction. If the current is reduced to a value below Iq, in the absence of noise the

ball no longer runs freely, but remains trapped in one of the potential wells. However,

if I is sufficiently close to 1^, the noise

may instantaneously tilt the washboard sufficiently so that the particle rolls into the

next well. This effect produces noise roundof the I-V characteristic. In our model,

this effect occurs even at T=0, where the

noise current is due to the zero point fluctuations in the large collection of harmonic

oscillators representing the resistor. Alternatively, one may view the current fluctuations as supplying pulses of energy that

"activate" the particle over a stationary potential barrier. These fluctuations are virtual: The energy in the pulse, AE, and the

Fig. 1 Low-frequency spectral density 3^(0)

of the voltage noise vs current for 5 values

length of the pulse. At, satisfy AEAt;^h. The

dissipation that occurs when the phase changes

by 2i; is supplied by the bias current, not by

the noise current. (On the other hand, in

the thermal limit in which the noise currents have a spectral density 4kBT/R, AEAt'^'kBTAt

,

and the fluctuations are real).

of K 5 eloR/kgT with

We emphasize that the description of the junction by a Langevin equation assumes that

the wave packet representing the phase of the junction is highly localized so that a particle-like representation is appropriate. When the wave packet representing the phase is

significantly broadened, a full quantum mechanical description is essential, and use of the

Langevin equation may lead to unphysical results. Furthermore, we have neglected the effects

of macroscopic quantum tunneling [10] which we believe aire relatively unimportant for over- damped junctions.

360
EXPERIMENTS ON SINGLE JUNCTIONS

To observe quantum effects we require a junction with k= el^R/k^T = (e/kgT) (B^^qJ 2^/2Trc)i

»1, where ji is the critical current density and c is the capacitance per unit area. At IK with 21 = 10 Acm"2, c = 0.04 pF ym"^ and 3^, = 0.1 we find k«=<3, a value at which quantum

corrections should be readily observable.

The Pbln-Ox-Pb junctions were fabricated using photolithographic techniques. The junction area was about 6 ym^ , and the critical currents ranged from 0.1 to 15 mA (0.2 to 30x10^* Acm~^). The CuAl shunt resistance was typically 0.10, and had an inductance of about 0.2 pH.

The low frequency voltage noise of each junction was amplified with a cooled 30- or 100-kHz

LC-circuit coupled to a room-temperature low-noise preamplifier. The noise was mixed down

to frequencies below 100 Hz, and its power spectrum was measured with a computer. Any 1/f

noise contribution was estimated from the measurements at 30 and 100 kHz, and, when significant, was subtracted from the 100 kHz-measurement

.

We first investigated junctions with k<<1, and found excellent agreement with Eq. (1).

These results demonstrate the accuracy of our measuring techniques, and give some assurance

that we have taken adequate precautions to eliminate external noise. We then investigated

junctions with k>1, and discovered the resonant structure illustrated in Fig. 2. The

Fig. 2 I-V characteristics and dynamic

resistance for a junction at 1.4K with

Iq = 1.53 mA, R = 0.0920, k = 1.15, = 0.03,

and 27rLsIo/'I>o ~ 1.0. Inset shows equivalent

circuit of junction.

0.2

0.1 Q?

0 ICQ _ 200 300 ' '

' V (/^V)

resonances occur when the Josephson frequency is near a subharmonic of the resonant circuit

formed by the shunt inductance Lg and the junction capacitance; the equivalent circuit is

shown inset in Fig. 2. In addition to modifying the I-V characteristic, the increased non- linearity of the phase evolution enhances the mixing down of noise from multiples of the

Josephson frequency. The spectral density of the noise measured on one junction is shown

in Fig. 3. The correction due to amplifier voltage and current noise is negligible. The

1/f noise was at most 20% of the total noise and was subtracted out. If we assume that the

error due to an inadequate knowledge of the exact spectral density of the flicker noise

is ±25%, the overall error resulting is ±5% of the total measured noise. We have subtracted

the noise from the resistor at the measurement frequency, AkgTRp^/R, from the total noise to determine the mixed down noise; the latter is also plotted in Fig. 3.

Fig. 3 Measured and computed spectral densities of the voltage noise at 1.4K for the

junction in Fig, 2.

361
We have computed the noise expected from the circuit shown in Fig. 2 using the equations

of motion

1 = 1 sin (S + CV + I , o s

and V=IR+IL + V„.

s s s N

(6)

(7)

Here Ig is the current through the shunt inductance, Lg, and Vjj has a spectral density

2hvRcoth(hv/2k3T) at frequency v. The computed mixed-down noise is plotted in Fig. 3. For

comparison, the computed mixed-down noise using a noise spectral density for the resistor

without the zero point term, 4hvR/[exp(hv/kgT)-l] , is also shown. The latter result clearly

underestimates the measured noise. The agreement between the computed and measured noise is

quite good below about 120 yV, but at higher voltages the measured noise lies somewhat above

the predicted value. We believe this discrepancy arises from self-heating in the junction,

which increases the magnitude of the noise at the measurement frequency. Furthermore, the

measured and predicted resonances are misaligned, indicating that our choices of Lg and C were slightly incorrect.

These results suggest strongly that zero point fluctuations in the shunt resistor are

the source of the limiting noise in a resistively shunted Josephson junction in the quantum

limit

.

QUANTUM NOISE IN THE dc SQUID

The dc SQUID consists of two Josephson junctions in a superconducting loop of indue- ' tance L. The critical current of the SQUID oscillates as a function of the applied magnetic

flux $ threading the loop with a period Thus, when the SQUID is current-biased in the non-zero voltage regime, the voltage also oscillates as a function of the applied flux. A

small change 60 (<<$q) produces a corresponding change in voltage 6V = (3V/9$)6$. The

smallest detectable change in flux that can be detected is determined by the voltage noise

across the SQUID. To characterize the performance, it is convenient to define the noise

energy per Hz

e S. S IhI " 2L " 20V/9'i>)^L '

where S-^ is the spectral density of the voltage noise, and S^ is the spectral density of the

equivalent flux noise in the SQUID.

Tesche and Clarke [5] computed the noise energy assuming that the Nyquist noises in the

two resistive shunts were the only sources of noise. They found that the SQUID was optimized when and 3 = 2LlQ/<i>o 1 > and that the optimum noise energy was given by

^lOk^T (LC)^. (9)

This result adequately predicts the measured performance of SQUIDs over a wide range of

values of L and C. The most sensitive devices reported to date have noise energies of

'v^6h [11, 12, 13], in reasonable agreement with Eq. (9). When T, L, and/or C are lowered

sufficiently, quantum corrections to the noise become important. The theory described

earlier in this paper can be extended to the case of the SQUID, and one finds that at T=0

for an optimized device

At higher temperatures the performance is degraded: At 4K, one expects e/lHz«=3h.

362
We have made SQUIDs that should approach the quantum limit in the He^ temperature range.

The fabrication procedures were similar to those used for the single junction, and the following values are typical: L'* 2 pH, C«0,5 pF, I^^^CS mA, R«lfi. Thus and 3 1. The SQUID was connected across the cooled LC resonant circuits (typically 30, 100, 300 kHz)

and the noise measured as for the single junction as a function of bias current and applied

flux. The transfer function 9V/3$ was also measured by modulating the flux through the

SQUID and lock- in detecting the resulting voltage. The best result so far was obtained at 3.4k, where the measured noise at 118 kHz was 3.8h. By measuring the noise at higher and

lower frequencies, however, we estimated that approximately one-half of this noise was 1/f

noise, so that the white noise contribution was approximately 2i , in rather good agreement

with the predictions of the model.

CONCLUDING REMARKS

The predictions of the theory for the noise in a single junction are in good agreement

with experiment when the effects of resonances are taken into account. The measured noise

in dc SQUIDs also agrees well with the predictions of the theory at 3.4K, although it should

be realized that the quantum corrections to the noise are small at this temperature for the

parameters used. Finally, a more detailed investigation of the theory is necessary to es- tablish the limits of validity of the Langevin approach.

This work was supported by the Director, Office of Energy Research, Office of Basic Energy

Sciences, Materials Science Division, U. S. Department of Energy under contract No. W-7405-ENG-48.

REFERENCES

[1] V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 (1969).

[2] K. K. Likharev and V. K. Semenov, Pis 'ma Zh. Eksp. Tear. Fiz. 15, 625 (1972) [JETP Lett.

15, 442 (1972)].

[3] CM. Falco, W. H. Parker, S. E. Trullinger, and P. K. Hansma, Phys. Rev. B10_, 1865

(1974).

[4] R. P. Giffard, P. F. Michelson, and R. J. Soulen, Jr., IEEE Trans. Mag. MAG-15 , 276

(1979).

[5] C. D. Tesche and J. Clarke, J. Lew Temp. Phys. 29, 301 (1977).

[6] J. Clarke, IEEE Trans. Electron Devices ED-27 , 1896 (1980).

[7] R. H. Koch, D. J. Van Har lingen, and J. Clarke, Phys. Rev. Lett. 45, 2132 (1980).

[8] R. H. -Koch, D. J. Van Harlingen, and J. Clarke, Appl. Phys. Lett. 38^, 380 (1981).

[9] D. E. McCumber, J. Appl. Phys. 39, 3113 (1968); W. C. Stewart, Appl. Phys. Lett. 12,

111 (1968).

[10] A. 0. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1980).

[11] R. F. Voss, R. B. Laibowitz, A. N. Broers, S. I. Raider, C. M. Knoedler, and J. M. Viggiano, IEEE Trans. Mag. MAG-17 , 395 (1981).

[12] P. Carelli and M. W. Cromar, 1980 Applied Superconductivity Conference, Santa Fe,

New Mexico (unpublished)

.

[13] R. H. Koch, D. J. Van Harlingen, and J. Clarke, 1980 Applied Superconductivity Conference, Santa Fe, New Mexico (unpublished).

363
QUANTUM NOISE EFFECTS AT HIGH FREQUENCIES AND LOW TEMPERATURES

A. van der Ziel

Electrical Engineering Department

University of Minnesota

Minneapolis, MN 55455

SUMMARY

It is shown how noise measurements at 100 GHz can discriminate between the various

thermal noise expressions found in the literature. It is indicated how the noise in

tunnel junctions transforms from thermal noise at zero bias to shot noise at sufficiently

forward bias; this leads to measurable effects at 100 GHz and 2°K in the 0-1 mV bias

range, both for the metal-oxide-metal (MOM) diode and for the Josephson junction. An

improved version of the maser noise theory is presented that takes the zero point energy

noise term in the thermal noise seriously; omission of this term would result in an

earlier, erroneous expression for the maser noise.

1. THERMAL NOISE

According to Nyquist L-LJ the high-rrequency noise os a resistance K kept at the

temperature T can be represented by a series emf [Sg(f )Af J"*"' ^ in a frequency interval Af,

where

Sg(f) = 4R E (1)

and E is the average energy of an harmonic oscillator of frequency f kept at the temperature

T. At low frequencies E = kT and (1) reduces to the well-known expression

Sg(f) = 4kTR (2)

(Nyquist 's low-frequency theorem). At high frequencies quantum effects occur and

E = 1/2 hf + hf/[exp(hf/kT) - 1] (3)

where 1/2 hf is the zero point energy term. Nyquist omitted this term and thus obtained

Sg(f) = 4hfR/[exp(hf/kT) - 1] . (4)

When it is taken into account, however, one has instead

S_(f) = 2hfR + 4hfR/[exp(hf/kT) - 1] = 2hfk coth(hf/2kT) , (5)

It is interesting to note that the latter expression may be derived by considering

emission and absorption of quanta hf by the electrons of the resistor without any

reference to zero point energy considerations [2]. The name "zero point energy noise" for

the term 2hfR is thus a misnomer.

At sufficiently high frequencies deviations between Eqs. (2), (4) and (5) occur, and

it becomes interesting to determine experimentally which of these expressions is correct.

364
Efforts are on the way at the University of Florida at 100 GHz, using a Hanbury Brown-Twiss

type microwave circuit being developed under the direction of A. D. Sutherland.

To discuss the measurement in more detail, we equate

Sg(f) = 4kT^R (6)

and so define an "equivalent noise temperature" T^. If Eq, (2) is correct we have

T = T , = T (6a) n nl

at all frequencies; if Eq. (5) is correct we have

= ° hf/k + (hf/k)/[exp(hf/kT) - 1] (7)

whereas if Eq. (4) is correct

T = T _ = (hf/k)/[exp(hf/kT) - 1] = T „ - hf/2k . (8) n nj nz

It is now easily shown that " negligible if hf/kT « 1. But at 100 GHz we

have hf/2k = 2.40°K and with Hanbury Brown-Twiss type circuits one can measure T with an accuracy of 0.10°K. Hence at 100 GHz and T = 77 °K and 300°K one can discriminate between

(6a) and (7) on the one hand and (8) on the other hand. By going down to T = 2.00°K and

f = 100 GHz, one has T „ = 2.88''K; so the difference T „-T , = 0.88°K, which can again be

n2 n2 nl ° measured.

2. NOISE IN TUNNEL JUNCTIONS

According to Tucker [3] the h.f. conductance and the shot noise spectrum S^(f) of a tunnel junction are given by

g(f) = [I(V^ + hf/e) - I(V^ - hf/e)]/(2hf/e) (9)

Sj(f) = e{coth[(eV^+hf)/2kT]I(V^+hf/e) + coth[ (eV^-hf ) /2kT] I (V^-hf /e) } (10)

where I(V) is the current at the bias V = ± hf/e.

At zero bias (V = 0) this reduces to

o

g(f) = g (f) = [I(hf/e) - I(-hf/e)]/(2hf/e) (9a) o

S^(f) = e coth(hf/2kT)[I(hf/e) - I(-hf/e)] = 2hfg coth(hf/2kT) (10a)

as expected for thermal noise. For (eV -hf)/2kT > 2.5 the hyperbolic tangents are unity

and the noise is

365
I(V +hf/e) + I(V -hf/e)

Sj(f) = 2eI(V^) iTvrr^ 2eI(V^)r(V^) (11)

corresponding to shot noise with a quantum correction factor r(V ); the factor differs

from unity if the characteristic is strongly curved. °

In metal-oxide metal (MOM) diodes near zero bias the characteristic is linear,

I = g^V, where = (dl/dV)^ is the l.f. conductance at zero bias. Hence g(f) = g^ and

SjCf) = go{(eV^+hf) coth[(eV^+hf)/2kT] + (eV^^hf) coth [ (eV^-hf ) /2kT] } (12)

so that the available power per unit bandwidth is

P^^(f) = Sj(f)/4g^ = 1/2 kT[l/2(x+y)coth l/2(x+y) + l/2(x-y)coth l/2(x-y)] (13)

where x = eV^/kT and y = hf/kT. For low frequencies y = 0 and

P^^(O) = kT(l/2 X coth 1/2 x) (13a)

as expected for shot noise. For zero bias x = 0 and

P^^(f) = kT(l/2 y coth 1/2 y) (13b)

as expected for thermal noise, whereas for 1/2 (x-y) > 2.5

P (f) = kT(l/2 x) (13c) av

as again expected for shot noise. At 100 GHz and T = 2.00°K the transition from thermal

noise to shot noise lies in the 0-1 mV bias range. We plan to measure it with an MOM

diode having a contact diameter of 1 y meter matched to the waveguide and feeding into a Hanbury Brown- Twiss microwave circuit.

3, NOISE IN JOSEPHSON JUNCTIONS

The Josephson junction is a much more complex device, since it is also an oscillator

that produces complicated mixing effects when terminated into a non-zero load. If I(V)

is the d.c. short-circuit current at the bias V, then

KV) = I (y) + I CV) (14) n .p

where In(V) and I (V) are the currents caused by single electrons and by Cooper pairs,

respectively. If^Sj^(f) and S^ (f) are the corresponding spectra, then the total noise

spectrum is ^ . ,

Sj(f) - Sj^(f) + Sjp(f) . (14a)

According to Rogovin and Scalapino [4], for a junction driven by a constant voltage

366

source V , o'
Sj^(f) = e{coth[(eV^+hf)/2kT]I^(V^+hf/e) + coth[ (eV^-hf )/2kT]I^(V^-hf/e) } (15)

Sjp(f) = 2e{coth[(2eV^+hf)/2kT]Ip(V^+hf/2e) + coth [ (2eV^-hf /2kT] 1^ (V^-hf /2e) } (16)

which is an extension of Tucker's expression (10).

It is interesting to note that Sj.^(f) behaves normally at the bias V = hf/e. But

Sj (f) blows up at the bias = hf/2e, since Ip (V^^-hf /2e) does not approach zero when

Vq approaches hf/2e from the upper side. Since this is also the bias at which the

Josephson pair current oscillates at the frequency f, complicated mixing processes will

occur when the junction is matched to a waveguide. At 100 GHz the required bias is

0.207 mV. Interesting noise effects should occur and they should be measurable.

4. MASER NOISE

In the early theory of maser noise [5] an error was made, in that Nyquist's

expression (4) for the thermal noise was used. We shall now correct this error for a cavity maser and shall show that the resulting expression is more easily understood. The

traveling wave maser can be treated similarly.

Let the maser be a two-level system with energies E-|^ and E2 and populations N-]^ and

The emitted and absorbed quanta are then hf = E^-E^, and the conductance of the maser system is

g = C(N^ - N2) (17)

where C is a coupling factor.

With zero pump signal the noise is thermal noise of g, or

i^ = 2ghf Af + 4ghf Af/[exp(hf/kT) - 1] • (18)

But in equilibrivrai

N^/N2 = exp[(E2 - E^)/kT] = exp(hf/kT) ' (19)

Substituting for g and for exp(hf/kT) yields

i^ = 2Chf Af (N, - N„) + 4Chf AfN„ (20) a 1 z I •

The first term is due to the zero point energy noise term in (18) . It was missing in the

earlier theory, so that in that case

±1 = 4Chf AfN„ (20a)

d 2

Equation (20) may be rewritten as

367
= 2Chf Af (Nj^ + = 2Chf Af N (20b)

where N = N^^ + N2 is the total ntimber of active atoms. Since this Is Independent of N^^ and Eq. (20b) should be valid for arbitrary pump signals.

When the total noise power of the system is evaluated. It Is found to be the same for

the new and the old theories. But the distribution between the source noise and the

device noise is different and hence the two theories give different noise figures and

different system noise temperatures.

The first half of Eq. (20b) has a very simple interpretation in that 2Chf AfN- is

the true spontaneous emission noise and 2Chf AfN^ is the true absorption noise. They are polssonian noise terms that come about because the elementary emission and absorption

events occur independently and at random. Since Eq. (20a) has no such simple interpretation, it should be obvious that Eq. (20b) is correct.

This has ramifications for the thermal noise expression. Since the correct

expression for the maser noise came about by taking the zero point energy noise term in

the thermal noise seriously, it follows that the term actually belongs in the thermal

noise expression. Eliminating it would give the wrong maser noise.

This work was done in preparation for a quantum noise project initiated at the University

of Florida. The author is indebted to Professors A. D. Sutherland, E. R. Chenette and

K. M. van Vllet for their interest and support. For a list of papers see references

[6] - [10].

REFERENCES

[1] H. Nyqulst, Phys. Rev., 32 (1928) 110.

[2] H. B. Callen and T. E. Welton, Phys. Rev., 83 (1951) 34.

[3] J. R. Tucker, IEEE J. Quantum Electron., 11 (1979) 1234.

[4] D, Rogovln and D. J. Scalapino, Ann. Physics, 86^ (1974) 1.

[5] See, e.g., N. Bloembergen, Solid State Masers , In "Progress in Low Temperature

Physics", North Holland, Amsterdam, Vol. 3 (1961) 396-429.

[6J A. van der Ziel, J. Gong, W. A. Krull, H. Metha, R. R. Schmidt and M. W. Trlppe,

Physics, lOOB (1980) 97. .

[7] A. van der Ziel, J. Appl. Phys., 51 (1980) 2292.

[8J A. van der Ziel, J. Appl. Phys., 51 (1980) 4329.

[9 J A. van der Ziel, Physica, lOlB (1980) 107.

[10] A. van der Ziel, Physica, 96B (1979) 325; A. van der Ziel, submitted to J. Applied

Physics.

368
A THERMAL ACTIVATION MODEL FOR THE DC-SQUID

Claudia D. Tesche

LuTech , Inc

.

P.O. Box 1263

Berkeley, CA 94701

A thermal activation model is described for the dc-SQUID. The device characteristics

are derived directly from the features of a two dimensional potential field analogous to the washboard potential for a single current biased junction.

We describe a simple model for the dc-SQUID analogous to the thermal activation models

for single junction devices. The dc- SQUID consists of a superconducting loop of inductance

L interrupted by two Josephson junctions. The junction parameters are critical current Iq , capacitance C and shunt resistance R. The voltage developed across the junctions, V = vIqR,

is a function of the dc bias current I = ±1q and the applied flux $a = <t>a^o. The equations

of motion can be written in the form [1] , [2]

2

at k

where the dimensionless time t is In units of <b /2ttI R and the potential satisfies

o o

uiS^^S^) = - cos 6^ - cos 62 - (5^+62) i/2 + 7TBj^/2 (2)

and

j = (6^ - 62 - 2TT(f)^)M6. , (3)

In these equations, 3 = 2LIq/$q and Be = 2itIqR C/$q. The random currents i^^ are generated by the Johnson noise in the shunt resistances.

Equations (1-3) are a set of classical equations of motion for the macroscopic variables 5]^ and 62. The form of the potential U (61,62) is determined by the macroscopic

quantum effects of superconductivity. The phase differences 6^ and 62 are classical variables analogous to the coordinates of a particle moving in a two dimensional potential

field. This description is adequate provided the wave function for the position/phase

difference is well localized (A6-j^ « 2tt) .

An example of the potential is plotted in Figure 1 for i = 0, <})^ = 0.5 and B = 1.0.

The potential consists of a chain of wells (L) separated by saddle points (S) , all situated

within a low potential trough. In the thermal limit (kBt >> 2eIoR) the random forces ij^ cause the particle to oscillate within the well. Occasionally, the particle makes a classical transition over the saddle point to an adjacent well [3]. The probability P of making

a single transition is

^wl*^w2^sl

's2i

f Auv exp (- —

)

(4)

369
where C^i , C^2' ^sl » l^szl curvatures of the well and saddle expressed in nor- mal coordinates. The factor Au is the barrier height and T = l-nk-QT /Iq^q.

Fig. 1 Contour plot of potential energy

u as a function of 6^ "^2 i = 0>

A = 0.5 and 6 = 1.0.

The escape of the particle from one well to the next corresponds to the generation of

both a voltage pulse across the SQUID, and a circulating current pulse around the SQUID

loop. In the limit « I, only one barrier is transversed with transit time T. As a result, the motion of the particle in the periodic potential is analogous to a renewal counting process for a non-paralyzable counter. For a chain of identical wells, the average

voltage V = (IqR) v, where

- ^

2TTp (1+pT) (5)

Fluctuations in the number of pulses produce a voltage noise with low frequency spectral

density 8° = S°(kgTR/r) where

S° = 2 (^^>^P

(l4pT)3

(6)

The circulating current noise, Sj, arises from two sources. A current pulse, jp, is

generated during the transition from one well to the next. Fluctuations in the number of

pulses per unit time produce a current noise

2<j >^P

(1+PT)3

This contribution to the total current noise is strongly correlated with the voltage noise

generated by the voltage pulses. In addition, motion within the well generates a contribution to the current noise which can be readily calculated from the first order correlation

theory [4]. The low frequency component is

370
^ (a+b-HTBab)^

where a = cos b = cos ^21' (8)

Note that this contribution is appreciable even at low bias currents, where the pulse rate

p is very small (Figure 2),

Fig. 2 Low Frequency noise spectral

10.0 -

^'2.0

3.0

2.0

1.0

density as a function of loop inductance L. The dashed curve is from

a numerical simulation [5]. The smooth

curves are from the thermal activation

model.

BIAS CURREhTT. I/Iq

If the potential consists of a chain of dissimilar wells, the analysis is more involved.

In particular, if the circulating current in successive wells varies (see Figure 1), the low

frequency current noise may increase dramatically [5]. If the circulating current in the wells

alternates between values and 22* switching noise will be observed. For escape probabilities p]^ and P2 the current noise at frequency f is

^js P1P2

// 2^2^ 2,

(4tt f +y )

(9)

where jg _ (jj^ - and y = pi + P2. At frequencies f « y, the spectral density is white.

^ (Pl-4>2)^

(10)

Near zero bias current, pj^ and P2 are small. Thus the switching noise dominates the well

and pulse noise sources. As an example, we plot Sj as a function of applied flux in Figure

3 for i = 0, (()^ = 0.5. Note that the current noise scales with the shunt resistance,

j Sj a 1/R. The critical current asymmetry:" parameter a = (Iqi - Io2)/2lo.

The results quoted above have been derived for a dc SQUID with junctions in the thermal

j limit. In general, for a lumped circuit element model, the Johnson noise currents are generated by fluctuations in the voltage across the shunt resistances. These fluctuations in the

electromagnetic field are in thermal equilibrium with the resistors at temperature T. The

power spectral density in this case is white. However, for junction parameters kgT ^

2elQR,

the power spectral density of the thermal fluctuations near the junction resonances will roll

off like Sg ^ hco exp (-Bco/kBT) [6]. As a result, the formalism developed by Kramers (Eqn. 4) cannot be used. In this regime, the interaction between the electromagnetic field ("zero point

term") and the junction currents must be treated with care. It is not obvious that the purely

classical expression for the junction phase differences (Eqs. 1-3) are adequate in this case. Furthermore, the assumption that the wave functions for &i and 62 are well localized may break

down, and macroscopic tunneling effects may be observed.

371
REFERENCES

[1] Claudia D. Tesche, Ph.D. Thesis, University of California, Berkeley (1978).

[2] CD. Tesche and J. Clarke, J. Low Temp. Phys. 29, 301 (1977).

[3] H,A. Kramers, Physica 7, 284 (1940).

[4] CD. Tesche, A Thermal Activation Model for the DC SQUID, accepted for publication in

J. Low Temp. Phys.

[5] CD. Tesche and J. Clarke, J. Low Temp. Phys. 37, 397, (1979).

[6] Herbert B. Callen and Theodore A. Welton, Phys. Rev. 83, 34 (1951).

APPLED FLUX. a/*o

372
ULTIMATE SENSITIVITY OF AN AC-SQUID

Juhani Kurkijarvi

School of Mathenatical and Physical Sciences

University of Sussex

Brighton BNl 9QH

Sussex UK

If one could cool an ac-SQUID to the tenperature T = 0, it still would not become a

perfect magnetoneter . There is an absolute limit on the noise flux of the instrument, the

superconducting condensation energy lost in excluding the flux, v/hich cannot be measured

better than within the limits of the uncertainty principle

2L ^ - 2 - (1)

fl^ere L is the inductance of the SQUID ring, the external flux to be measured, and t the

time used for the measurement or the inverse frequency band width. It is suggested in this

paper that a limit of a similar form

^ ^ - RC (2)

arises from the zero point uncertainty of the flux admitted through the ring in conjunction

with a construction requiranent of the instrument, the necessity of making the SQUID ring

Josephson junction "overdartped" . Here C is the capacitance of the junction and R the normal

current resistance across the junction, which is assumed to make sense even at T = 0.

There is the possibility, in principle, that collective quantijm tunneling of the flux

variable through a potential barrier [ij could play a role in this context, but this seeiis

improbable [2} after the recent work on the effect of friction on quantum tunneling [s] .

The sensitivity of the ac-SQUID is determined by the accuracy with which the hysteresis

loop in the external flux c})^ versus acinitted flux plane is always triggered at the same

value of or at higher frequencies, how precisely the sonewtiat deformed loop is reproduced

at each ac cycle of the instrument. The flux moves, at a given 4)^, in the potential

V(t!)) =2L^^-^x^ - ^c 2^F ^o (3)

vAiich has a hollow with a minimum at a very snail value of (p till <i)-^ reaches a critical value

<i>xc at which the hollow disappears. Here i^ is the critical current of the junction and (})q the flux quantum. When (p^ is swept sinus§)idally , 4) either always stays in the (metastable)

hollow and ranains small or rolls down into a deeper minimum triggering a hysteresis loop if

the anplitude of (Jj^ is large enough. It is the uncertainty in the tri^ering amplitude of (^t^ which determines the intrinsic noise of the SQUID. At high temperatiires chance scaling of

the barrier with the aid of thermal fluctuations leads to an uncertainty in the measured

fl\ax [4^ . In this paper we are interested in the uncertainty coming from the fact that the

value of <p at the bottom of the hollow is uncertain by roughly the width of its harmonic

oscillator ground state. We assume that there is no tunneling or thermal barrier penetration.

373
Then clearly a small uncertainty arises simply from the possibility that the flux may return

into the metastable valley upon reduction of although it could have escaped but happened to

be "on the wrong side" and did not have time to take advantage of the momentarily vanishing

barrier at the peak value of (j) . One understands irrmediately that such a process is

influenced by the frequency of the sinusoidally driven c^)^. A scaling argument is given at the end of this paper showing that the increasing uncertainty fron this source at higher

frequencies is exactly cancelled by the statistical factor (fT)l/2,the square root of the

frequency multiplied by the measuring time, ie the number of measuranents performed. Thus

the maximum flux sensitivity implied by the uncertainty in <p is independent of the rffrequency of the SQUID.

Why should the flux be uncertain at the bottom of the potential trough? If one writes

down the well known resistively shunted junction differential equation for (j) in a SQUID ring,

linearises the potential term (ie considers the harmonic motion at the botton of a potential

valley) and imagines the system driven by the noise current at T = 0

^n ('^^ = ^ 2k^ = ^ . (4)

2 one can solve for (J) (w) and integrate this quantity over w to get the mean square spread of

the flux. The result is exactly the harmonic oscillator gromd state width ^1/(200^) if the

damping term is small and a narrower peak by the factor

1 '^o , 1

in the case of large damping

3= 0/ = RCw « 1,

(jJq being the oscillation frequency in the trough. Caldeira [6 J has investigated an explicit

model of friction for the harmonic oscillator with identical results.

One can now use the independence on the frequency of the noise considered in this paper

to estimate this noise at a particular frequency where it is best known. Such a frequency is

(jJq^/^ at w4iich one expects the uncertainty in (^-^ to be roughly the same as the uncertainty of

4). LOo^/ri '^ frequency at which the damped flux' (v classically) in the trough can just follow

the external frequency leading to parallel trajectories of <}> in the (<t>xA) plane under the

sinusoidal drive of <^^. Estimating the harmonic grouni state spread of (}> as

_5_

2C

a sonewhat pessimistic assumption in the overdamped case, one gets eq (2) for the ultimate

energy sensitivity of a SQUID. At frequencies where the parameter [4}

X = 0.1

CO r.h rf

2kgTL

c'^o 2TTkgT

1/3

becones canparable to unity or smaller, the present argument can be used at finite ternperature as well, with the result

-2^ T > 2Tr| kgT (5)

374
vihere k^T has been substrituted for zero point energy. Ah experimental value at this limit

has been claimed ["7] but judged controversial [sj. Theoretical estimates similar to the present

results have been given for the dc-SQUID [9j.

We now return to the scaling argument which establishes that the noise originating in the

uncertainty of is independent of the rf-frequency of the SQUID. The argument rests on the

observation that the equilibrium (}> versias 4)^ ciarve of the SQUID ring

* = (j) - L. sin -^r-^ " *o (6)

looks like (})^ - <{) = /(^^^ - in the vicinity of (J)^^. If instead of a given range of we

observe a larger range ^<^^, ie- instead of the point (f)^ pick the point k^tji^. soad scale 4) by

the factor ]sP-7^, the shape of the trajectory remains the same. When (()x is driven sinusoidally

(|) follows roughly the curve (6) but tends to have difficulties in the ever sharper and sharper

turn before ^>^. The deviation occasioned by the "centrifugal force" must be conpensated by

the balancing narroonic force in the potential trough. The corresponding equation of motion

reads

d^* A. , .N. 2-1 dd) 2 . , 2.

,

2 l?^^x ^ ^) V\f] -

( Wf) =

**o - (7)

fliiere Acjjx = ^Jxc ~ 'I'x ^J^d A is the overshoot of the drive amplitude beyond the point (j)xc, A = (i)A - (fixe A(j) is the deviation of (p frcm the equilibrium position. The quantity ojq is proportional to i^<^)'^/^. The scaling suggested above together with scaling ojj.f by k

leaves the equation of motion unchanged if a = 4/3. Therefore, after this scaling, the

trajectories in the (({)x, <i>)-plaxie are similar at different frequencies. In the particular

expanded frame at each frequency we can safely assume that the uncertainty of (i)x,6$ xi^^

proportional to the uncertainty 6(}) in cj) as observed in that frame. Then the harmonic

oscillator ground state width of 4 scales like ko^/^/kct/S = k^a/S and 5^x~'5(t)~k-'-/^ where the

tilda on underlines that the statistical factor (fT)~l/2 has not been included yet.

After dividing with (fT)-'-/^ we have 6({)x"'k'^.

REFERENCES

[l] A J Leggett, Journal de Physique Colloque C6 (1978)

[2] J Kurkijarvi, submitted to JLTP

[3] A 0 Caldeira and A J Leggett, Phys Rev Lett 46, 211 (1981)

(4] J Kurkijarvi and W W Webb, Proceeding of the 1972 Applied Superconductivity Conference,

Annapolis, IEEE Conf Rec No 72CHo282-5TABSC

(5I E E H Shin and B B Schwartz, Phys Rev 152, 207 (1966)

jfe] AO Caldeira, PhD thesis. University of Sussex, unpublished

[7] A P Long, T D Clark and R J Prance, Rev Sci Instrum 51, 8 (1980)

[sj J Clarke in SQUID '80, Proceedings of the Second International Conference on Superconducting Devices, de Gruyter, Berlin, New York 1980

[9] C D Tesche and J Clarke, J Low Tenp Phys 29, 301 (1977)

375
The Mean Lifetimes of the Meta-Stable States of the DC-SQUID and its I-V Characteristics

Eshel Ben-Jacob*

Department of Physics and Astronomy, Tel Aviv University 69978 Tel Aviv Israel

Zeev Schuss**

Department of Applied Mathematics, School of Mathematics, Tel Aviv University

69978 Tel Aviv Israel

1. Introduction

The DC-SQUID with small coupling

(LI > h/2e, where L is the self inductance

J and Ij is the junctions critical current) has

several types of meta-stable states^ and

therefore it can be used as a logic element. At a finite temperature the thermal noise

causes spontaneous transitions between the

various states. Hence, the meta-stable states have finite mean lifetimes. It is of interest to know the dependence of these lifetimes on the DC-SQUID parameters, on the external driving current I and on the external magnetic

flux (t) . ex Here we find this dependence for the

shunted DC-SQUID (6 < l). In this case we

c use the Smoluchowski approximation of the

Fokker-Planck equation.^ The resulting problem is equivalent to the exit problem of par- ticle out of two dimensional potential well,

a case for which Kramers' results'? can not be

used and have to be extended. Such an extension was done by Landauer and Swanson.^ Here,

however, we present a new method based on the

results of Matkowsky and Schuss. ^'^ Using

this method we compute the mean lifetime in a two dimensional potential well with several

saddle points on its boundary and the relative

probability of exit through each saddle point. We use these results to construct the I-V characteristics of the DC-SQUID. We find sat- isfactory agreement between our analytical results and the numerical simulations of Teshe

and Clarke. We note that the method we present can be applied to elements containing more than two coupled Josephson junctions.

2. The Meta-stable States

We consider DC-SQUID which consists of two identical Josephson junction and which is driven by an external current source I and an external magnetic flux d) , as shown in

ex fig. 1. Assuming the RSJ model^'^ for the

junctions we obtain the following equations of

motion for 0^'s, the phase differences across

the junctions^:

02 ^i^®2 = ir ^ '^^®l-®2*0ex^ J

(2-1)

where

G i (ujRC)'

K = '0 2TrLI

^c ' '^J

2ir<ti ex

*c

(2-2)

J

^0

_h

2e

R and C are the resistance and capacitance

of the junctions and time is measured in units

of 0)"^. Assuming the ring is made of a superJ conductor whose thickness is bigger than

the London penetration depth, we have

(2-3)

where ^ is the total magnetic flux through

the ring.

Equations ( 2-1 ) can be interpreted as the

equations of motion of a particle with unit mass and dissipation G in a two dimensional

field derived from the potential U(e^,92)

given by -, p U(e^,02) = 2^(0i-V®ex^ "

'^°^®1 (2-M

COS0.

J

Note that the energy is measured in units of

*qIj/2..

Fig. 1. The

DC-SQUID model.

376
For a sufficiently small external current

I there exist meta-stable states which are the

static solutions = 0, corresponding to those

values of 02'®2 ^'^^^^ "the potential has local

minima. The meta-stable states are located on

lines parallel to the main diagonal 0

1 Such lines are characterized by the integer

number of fluxons passing through the ring,

when the ring is at one of the meta-stable

states located on a particular line. For ex- ample, at the meta-stable states along lines

a in fig. 2, no fluxons pass through the

ring. At points of local minimum along lines

b in that figure one fluxon passes through

the ring in the direction of (j) in fig. 1 (thus the integer value of '0 is +1).

Along lines c the value is -1 and along line d

in fig. 2b - +2. The number of lines of

meta-stable states increases as K decreases

and for K < 1/2tt there are no lines outside the main diagonal (fig. 2a). For

ex the range

0 and K < l/2Tr for values of I

21,

> 1 - 2-nK (2-5)

there are no meta-stable states outside the

main diagonal. For any K l/2Tr and any

I

g-j. < l-2irK there are no lines of 4>/!j>Q

if 0 is

KG > 1 21

2-wK

±1

(2-6) J

respectively. From the foregoing description

it can be seen that the DC-SQUID can be used

as a logic element whose values are the different meta-stable states and the control variables are I and (\>

IQ 12

li( C

0 5 10 15 ^

Fig. 2. Equipotential curves for U(0 ,^p)

for I/I^ = 0.5 (a) K = I/tt, Q

- - ^ 'ex

0, no meta-stable states outside the main diagonal line a. (b) K = 0.02, Q^^ = 0, metastable states appear on diagonal lines b,c

etc., see discussion in text. The domain 0

enclosed by the dashed line is the domain of

attraction of the point M, which is the mini- mum of uO^jOg) in The point S is a sad- Ide point connecting two adjacent domains of

attraction, (c) K = 0.1, Q = ti/2. No metastable states on line c

.

3. Thermal Noise and the Smoluchowski Equation

At finite temperature, spontaneous transitions between the meta-stable states will occur, thus these states will have finite lifetimes.

Our purpose is to study the dependence of the

expected lifetimes on the DC-SQUID parameters

and I and (j) . We start by assuming that the

377
thermal fluctuations can be described by an 'additive Langevin noise term in the equations

of motion2'7(eq. (2-1))

:3-i)

h. Exit Probabilities and Mean Lifetime of Two Dimensional Potential Well .

One of our purposes is to calculate the exit probability density through the boundary 8D of D, the domain of attraction of some metastable point (see fig. 2 ), It can be proved

that this probability P is the Green's function solution of the following boundary value

problem^

02 + + sinGg = + V®2^®ex^

J

where j^(t) represent independent Gaussian

white noises.

<i^it)> = <j^{t)j2(t)> = 0

<J.(t)j.(t') = 2Gr 6{t-t') (3-2)

S*P = 0 in D

P = 6(6-^^) on 3D

where 0 indicates a point e,,02,

ik-i)

and S* is the operator conjugate to S. The other purpose is to calculate the mean first passage time to the boundary t of

a particle which starts inside the domain D. It can be proved that t is the solution of the

following boundary problem^

K^T /

2lT

S*T = -1 in D

T = 0 on 3D (U-2)

The lifetimes as well as the I-V characteristics

can be calculated by numerical simulations of

equations (3-1). Such an approach for the calculation of the I-V characteristics is presented in ref. 7- However, in this reference the

case K > tt" (no meta-stable states outside

the main diagonal) was considered.

In order to calculate the lifetime analytically we would like to write an equation for

the distribution function. Since we consider

the case of large dissipation ( shimted junctions, G>>1) we can use the Smoluchowski ap- proximation^ to write the following equation

for the probability density p(0^,02,t)

f

= a.p . -div ?

s

= 2 2 L 1 O.

« 302 G

1 1

1 W 3_

G 30. 90.

1 1 1,2

(3-3)

^ _ 1 3U ^i = - G 307 P

1 I 9P_

G 30.

1

Kramer's approach for calculating the exit

time from the Smoluchowski equation was applied

in ref. 10 to the case of overdamped single

Junction. However, this approach fails in two

does not imply 3p -> dimensions as -rf- = -div J = 0 ot ->J = const. The first result in higher dimensions for the mean life time in a potential

well with single saddle point was given in ref.

5. Here we shall use the method of ref. 6 which will enable us to calculate the mean lifetime of potential well with number of saddle

points and the relative exit probability through

each of these points.

First we consider eq. (l*-l), and use the

following observation. If u satisfies

S*u = 0 in D

u = f on 3D then u = f«p d0

3D

(U-3)

Thus, we shall construct an asymptotic solution

to (i*-3) and recover P from (U-U).

The solution is based on the observation

that u is approximately constant inside D and changes rapidly near the boundary to satisfy the boundary condition. This observation can

be understood as follows; Using a power series

expansion in the temperature T we see that

the leading term u^ which is the solution of

eq. (U-3) for zero temperature has to be a con- stant along the characteristics of the equation.

Now, since all characteristics converge to the

meta-stable state in D (the minimum of the potential in the domain) u^ should have the same value throughout D.^ At finite small temperature there must be a boundary layer to connect

Uq with the boundary value f. Using local

coordinates near the boundary we obtain

(U-5)

378

u(^) = + (f(0')-UQ)[l-ERF(y(-Y^

where 0' is the boi^ndary point closest to 0, y is the distance |0-5'| and -u^ is the sec-
ond normal derivative of the potential at ^'

.

(erf indicates error function!.}^ To determine

Uq we use the quasi-equilihrium distribution

p = Pq exp(-U/r) inside D to write eq. {h-3) in the form

VpVu = 0

and cOg are the resonant frequencies at the

minimum and the summation is over all lowest

saddle points. The contribution of the higher saddle points can be incorporated into

Ct-ll) by adding exponentials with the appro- priate values of AU and to. However for AU-AU . >> r the contribution of such a sad- min

die point is negligible.

Note that in the one dimensional case eq. ih-6)

yields pVu = const, so that Uq can be easily

determined. In the two dimensional case we

integrate eq. (U-6) over D and using the divergence theorem we obtain

p|jd^=0 (U-T)

8D Evaluating the normal derivative 3u/3n with eq. (U-5) we obtain

U U

3D 3D

Hence using eq. {h-h) we have

-u/r , f -u/r,^

e / (t ujjje d0

3D

(l^-8)

{h-9)

We see that the exit probability density is concentrated about the lowest saddle points

on the boundary 3D and have a Gaussian

shape of width (v^/iu^) where oj^ is the

frequency of vibration in the saddle point

in the direction tangent to 3D. Next we compute the mean lifetime by

considering eq. (U-2). Since t grows ex- ponentially in 1/r for small temperature

we rescale by setting

max(u) = 1 and c, and

Cp/r

c £ae where

are constants

1 "2

to be determined. Constructing a boundary

layer for u as above (i.e. ^j^^

in Ct-J) we immediately obtain

T = c^e BRF[y(-y^n

In order to determine c

(U-10)

; ^, i; 2 we multiply eq. (h-2) by the quasi-equilibrium distribution p and integrate over D. proceeding as above we obtain

>u„/a)„ min/r

=

^ N T e

T 217 (1), 0)^ (1+-11)

where AU

12

U

sad

U ^ and U . sad mm

are the values of U at the lowest saddle

point and at the minimum respectively, u.

5 . Computations of the Mean Lifetimes and the

I-V Characteristics

We use eq. (U-ll) for the computation of

the mean lifetime of the meta-stable states described in section 2. The saddle points

AU, (Ojj, a)|p, 0)^ and were evaluated num- erically from the potential equation (2-U).

The dependence of t on I, thus obtained,

is shown in fig. 3. for various values of K

and 0 and for temperature V = 0.05. For other values of the temperature the corres- ponding graph can be obtained from fig. 3. since Und/r) is linear in 1/T. We see in fig. 3. that for smaller values of the coupling constant K the mean

lifetime of the meta-stable states on the

main diagonal (lines a in fig. 2) becomes

shorter (see the dotted curve 0 relative

to the solid curve 0). On the other hand

the mean lifetime of the meta-stable states

on the diagonal 'I'/'I'q - ±1 (lines b, c in

fig. 2) becomes much longer (See the dotted

curve 1 relative to the solide curve 1). For 0 > 0 the mean lifetime on the diaex gonal (ti/(ti„ = -1 is shorter than the mean '0 lifetime on the diagonal

0

^0 is sufficiently large (fig.

+1.

2c)

If the

Fig. 3. 1/t vs I/Iji r = 0.05. The curves

0 correspond to metastable states on the main

diagonal a of fig. 2. Curves 1 correspond

to the diagonal '^0 = 1 (lines b in Fig. 2)

.

The solid curves correspond to K = 0.1,

0 =0. The dashed curves correspond -(rO n/2. The dotted curves corex , K = 0.1, eex respond to K 0.02, 9 =0.

''ex . 379
meta-stable states on the line 4'/<t'Q - -1 disappear altogether. In this case fig. 3 also shows the dependence of the mean lifetime of the meta-stable states '^/'^q ~ and di/tt. = 0 on 0 (see the dashed curves ^0 ex

1 and 0 relative to the solid curves 1 and

0). For convenient comparison with experi- ments or with existing numerical simulations

of eq. (3-1), we have constructed the I-V characteristics. This was done with the aid

of the relative probabilities of exit through

the various saddle points (eq. (U-9)). The I-V characteristics for three cases is shown in

fig. k. It can be seen that our analytical

results are in good quantitative agreement

with the numerical simulations of ref. 7-

,10 .025 .05

[6] B. Matkowsky, Z. Schuss, SIAM J. Appl. Math. 33 230 (1977).

[7] C.D. Tesche, J. Clarke, J. Low Temp.

Phys. 29 301 (1977).

[8] W.G. Stewart, Appl. Phys. Lett. 12 277 (1968).

[9] D.E. McCumber, J. Appl, Phys. 32 3113 (1968).

[10] V. Ambegaokar, B.I. Halperin, Phys.

Rev. Lett 22 136U (I969).

Fig. li. The I-V characteristics in presence

of thermal noise r = O.O5. (a) K = l/n,

^)ex = ° K = ^ex =

° (c) K = 0.1, 0 = 7r/2. ex

The authors wish to express their gratitude to Mb . R. Shainsky for carrying out the

numerical computations.

References

* Partially supported by Grant from the US-ISRAEL

Binational Science Foundation BSF.

** Partially supported by Grant from the 4.F.<^'S>R»

[1] T.A. Fulton, L.N. Dunkleberger , R.C. Dynes, Phys. Rev. 6:^6 855 (1972).

[2] E. Ben-Jacob, Y. Imry, Submitted to J. Appl. Phys.

[3] "Theory and Applications of Stochastic

Differential Equations" by Z. Schuss,

John Wiley and Sons, N.Y. (198O).

[k] H.A. Kramers, Physica ]_ 28h (I9U0)

[5] R. Landauer, J. A. Swanson, Phys. Rev. 121 1668 (1961).

380
EXTERML FIELD EFFITT ON PARTICLE DIFFUSION

ABOVE A POTENTIAL BARRIER*

Eshel Ben-Jacob and David Bergman

Department of Physics and Astronony, Tel Aviv University,

Tel Aviv, 69978, Israel.

Benny Carmeli and Abraham Nitzan

Department of Chemistry, Tel Aviv University,

Tel Aviv, 69978, Israel.

INTRODUCTION

Kramers' treatment of chemical reactions as

a diffusion of a brownian particle above a potential barrier^ ' ^ has long become a starting i>oint for many studies of thermally activated processes. Chanical reactions '•> ^, surface desorption"* , diffusion of atoms in solids and on solid surfaces^

and transition between the equilibrium states of

a hysteretic Josephson junction^ are but a few examples for the applicability of this model. While Kramers' treatement focuses on thermal

activation alone, recent experimental developments

have made it increasingly desirable to consider

this model in the presence of an external periodic force. Thus multiphoton dissociation and isomei>- ization of molecules in high pressure gas or in

condensed phases^ the possibility of laser assisted desorption^and transitions in current driven

Josephson junctions under the influence of microwaves^ can all be modelled as a diffusion of a particle above a potential barrier under the in- fluence of an externally induced oscillating force

.

In the present work, we solve this problem for the model depicted in Fig.lTa particle of mass m

moving in a one dimensional harmonic potential

well (characterized by the frequency Wq ) with a given transition threshold( E^^=%iia)Qxl^ )under the

influence of an oscillating force aCt)=A sin ut

and of thermal dissipation and noise, Our aim is

to derive an expression for the steady state rate of transition out of the well. The equation of motion is

X + GX + a)^X = A sinujt + R(t) (l)

fx < ^thj

where (kg being the Boltzmann constant and T the

temperature)

<R(tj)R(t2)> = 2Gr6(ti- ti);r=kgT/m (2)

In the absence of noise(T=Q) the steady state solution of Eq.(l) is

1 2 X(t) = X(t) = — A sinuit - —Acoscot

(3a)

where

= Q^J^ (3b)

when noise is present the distribution P(x,x,t) may be obtained exactly by standard mentods . The

result is

;

P(:!9ct) = exP^-t'^o -

+ (x - i(t))M/2r} (U)

where x(t) = dx/dt. At log time, x and x are given by Eq, (3) and the distribution is independent of the initial condition. (The result (1+) is valid at all times where x and 5 correspond

to the motion in the absence of noise). We shall be interested in the low viscosity (slow damping,

G <<ai,a)o) limit. In this limit Eq.(l+) may be

simplified by transforming x and x into action

angle variables -

-1-

Figure 1 A potential surface with a realistic

barrier and its truncated harmonic approximations . Figure 2 Steady state trajectories in phase

space. Shown are the trajectories corresponding to A, A = A + AA, and A^j^. Note that every point (x,x)lies upon a unique trajectory charact- erized by a definite value of A.

381
X = (2J/ni(0()) ^ cos

X = ( 2Ja)n/m) ^ sin ((>

(5)

and Integrating over the phase ()>. This leads to a time independent probability distribution for the action J

lOg 2wo/jJ CJ- J) P(J) = 7^ Io( T ^ ^"^f ^

(6) where Iq is the modified Bessel function of

order zero and where

= m(wg <x^ >+<v>'' )/a)n=>r;^-^^ A (7J

is the time averaged steady state action of the

particle in the absence of random noise. The

result (6) is obtained for | (to-cjj )/a)* | << 1. For

2u)o/jJ >> kgT Eq.(6) becomes (using I j

( x

) SiE^^.^

Here we substituted E=(jOqJ and E=(OqJ.

Eq.(^) contains in principle all the needed

information on the processes. However, for the

purpose of obtaining the transition rate it is easier to take a different route. To this end, we proceed in the spirit of Kramers by first

identifying the slow variable of the system and

then deriving a Smoluchowski type equation for

its probability distribution.^

The Fokker-Planck equation corresponding to

Eq.(l) is

We have added and substructed a term On%)(bh'^^ P). M may be chosen at will. A con- venient choice is suggested by the observation

that a stationary orbit(in phase space(x,x) )in

the absence of noise is determined by A. Indeed

Eq. (3a)implies

(10)

This equation describes a closed trajectory in phase space(x,x) as shown in Fig. 2. In this

noiseless case, the stationary probability distribution satisfies

{11)

which expresses the fact that the probability

current remains on the stationary orbit. The ther- mal noise causes fluctuations so that at a given

instant x=x+Ax and x=x+Ax . AA is defined such that

the point(x,x)lies on a stationary orbit corres- ponding to A=A+AA (i.e. satisfies Eq. ( Ifd )below; see Fig.2). The phase defining the position of

this point along this orbit is(in the low viscosity limit )a rapidly evolving variable while A may

be taken(in this limit )as the slowly changing

variable. This definition of A implies that the

first two terms on the r.h.s. of Eq.(9) have the

form of the divergence of a current in phase aps.ce such that at every point x,x this current is tangent to the stationary orbit through that point.

Up to this point no approximation has been

made. We next change variables from (x,x) to

(A,()>)where (fi denotes the phase angle along the closed stationary trajectory in the (x,x)plane.

We further assume that on the interesting timescale P is independent of (|).(More rigorously, this

is expected to hold after coarse grained averaging over time intervals large compared with or

a)o~^ but small compared with G~^.) We then in- tegrate Eq.(9) over ^ for a fixed value of A. This has the effect of making the divergence term

vanish, and we get a nonzero contribution only

from the last two terms on the r.h.s. of Eq.(9). We thus obtain a Smoluchowski-type equation for

(12)

In performing the integration of Eq.(9) the follow- ing relations were used

sih(u)t)-j^('<3r)<+(c^^c;i'))C)

O)c>=0

(lUa)

(H4b)

(lUc)

(litd)

ilhe)

where < > denotes time (phase )average. Eq.(l3)

yields the steady state probability distribution

(15)

382
where is the approriate normalization constant. The result(l5) may be rewril^ten in terms

of the average energy £=0)5 J = '"C'^n^ '> ^^2 (c.f. eg.(U), This leads to

3 (16)

which agrees with the result (8) obtained for

i/eS >> kgT and for u>-u^o . (the presence of E'^''in the preexponential term of (8) is numerically

insignificant )

.

It is worthwhile to mention that the same

resxilt may be obtained in a seemingly more

direct fashion by making an appropriate trans- formation on the Langevin equation(l). Intro- ducing the transformation(x,x) —»(A,(j))

X = iCi^-u)^)5i'^ t - G CO t3 (it )

we can derive a pair of Langevin equations for A and (|) from which a Fokker-Planck equation can

be obtained using the standard procedure. By

assuming that P(A,(}>)is independent of (fi and by

averaging over the fast time scale (e.g. replacing sin^(j) by \, etc.) we may again derive Eq.Q-3).

Eq.(l3)may now be used to derive the transition rate using an approximate procedure similar

to Kramers' . The threshold value of A is found

from Eq.(l2d) to be

as

)

The transition rate, which is the inverse

^ean first passage time t"-^ for A to attain the

value A-tji when the initial probability distribution is given by (13), is found to be

In terms of the average energy we obtain( putting

a.2/(co2+a)92)= 1)

C20)

It should be kept in mind that this result is valid only provided that (/E^ij-/E)2 >> k3T. A

more rigorous treatment yields

E

Y

(21)

which is valid for all values of E < E^j^. However, when E is close to E^jj our model, which

replaces a physical barrier by the model of Fig.l,

breakes down.

Eqs . (18 ), (20) or (21) provide our final

expression for the transition rate out of a harmonic potential well in the low viscosity

limit under the influence of an oscillating ex- ternal field. Significantly, the activation en- ergy Eth which appears in the expressions for the rate in the absence of the external force is replaced by (/E^jj-i/E)^ and not by E^j^ - E as could have been naively expected. The dependence

on the strength and frequency of the external

force enters through the value of E=(jJqJ as given by Eq. (j)

.

In the above treatment we have focused on

Kramers low viscosity limit. It should be kept

in mind that for real potential barriers (the

dashed line of Fig.l) the low viscosity picture

breaks down close to the threshold and a different procedure(e .g. , transition state theory)may

have to be invoked. If the difference between

the two curves of Fig.l is significant only in

an energy range smaller than kgT, the correction

, is expected to be small. Finally, we note that for the high viscosity

limit where G >> Uq , an external force with w<<G

corresponds to an adiabatic change of the potential surface. In this case the transition

rate is time dependent and is given by the usual

Kramers' result for this limit with time dependent activation energy and force constants.

REFERENCES

* Partially supported by the U.S. -Israel Binational Science Foundation (BSF)

.

[1] H. A. Kramers, Physica ^ 28A (1940)

[2] S. Chandrasekhar , Rev. Mod. Phys. 15 1 (1943)

[3] D. Chandler, J. Chem. Phys. 68 2959 (1978);

J. A. Montgomery, D. Chandler and B. J. Berne, J. Chem. Phys. lH 4056 (1970);

J. L. Skinner and P. G. Wolllner, J. Chem. Phys. 69 2143 (1978) and U 4913 (1980).

[4] G. Iche and Ph. Nozieres, J. de Physique

37 85 (1976)

.

[5] Physics of Superiounic Conductors, Edited

by M. B. Salamon, Springer, Berlin (1979)

[6] V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22 1364 (1969).

[7] V. N. Sazonov and S. V. Zatzepin, Chem. Phys. 52 305 (1980).

[8] K. F. Freed and A. Metiu, to be published.

[9] E. Ben-Jacob and D. J. Bergman to be pres- ented at the LT-16 LA (1980).

[10] D. J. Bergman, E. Ben-Jacob and Z. Schuss

this proceedings.

383
FLUX FLOW NOISE DURING INHOMOGENEOUS VORTEX-MOTION.

DEPENDENCE ON DC-VELOCITY FIELD*

Heinrich M. Dirks**, Klaus F. Beckstette, C. Heiden

Institut fiir Angewandte Physik der Justus-Liebig-Universitat Giessen,

Heinrich-Buf f-Ring 16, D-6300 Giessen, Germany.

INTRODUCTION

The ac-component of flux flow voltage in type-II-superconductors often

is observed as a random noise signal SV [1]. A convenient setup for the investigation of flux flow noise uses thin films or foils as samples in a magnetic field perpendicular to the large surface of the specimen, (see fig. 1)

V=V+ 6V

o

ts

Fig . 1 Experimental setup for

flux flow measurements.

the barrier can be overcome, resulti:

whole process repeating in a more or

Vortices are moved via the

Lorentz force exerted by a sufficiently large transport current. The

resulting electric field, which cor- responds to the vortex velocity field

is analyzed by a pair of movable volt

age probes.

Local investigations performed

in this way showed that flux flow

noise originates at or near pinning

sites, such as grain boundaries [2].

A stop and go mechanism has been

proposed [ 3] to explain the underlying fluctuations of flux flow.

Vortices come to a halt in front of

a pinning site until due to the collective pressure of many flux lines

in a sudden rush of vortices, the • ess random way.

Whereas this mechanism involves a direct interaction of vortices with

pinning sites, there appears to be also another possibility, in which the

action of pinning with respect to flux flow noise is a more indirect one. NOISE GENERATION DUE TO INHOMOGENEOUS VORTEX MOTION

Let us consider a sample with inhomogeneous pinning, in which domains

of high critical current density without flux flow are separated by other

regions with negligible pinning, where vortex motion takes place. The asso- ciated velocity field in general will not be homogeneous. Therefore the

motion of a perfect vortex lattice is not possible . Local rearrangements in

the relative position of vortices instead will take place. Due to the elastic properties of the flux line lattice, these rearrangement processes

should be associated with a change of elastic energy, and this change may

take place in a relaxation process involving plastic deformation of the vortex lattice, triggered by the randomly changing pattern of vortex positions.

* Supported by the Deutsche Forschungsgemeinschaft

.

** This work represents parts of a Ph.D. thesis of one of the authors (HMD)

.

384
EXPERIMENTAL

To test this possibility of noise generation, a special technique was developed allowing [ 4 1 to prepare single crystalline niobium foils with very

low intrinsic volume pinning. Typical values of critical current density

are below 100A/cm^ at B=0.1T. Nbj Sn-pinning structures of well defined geometry then were deposited on such foils by means of a photoplithographic

lift-off-technique. Pinning structures realized in this way, together with

their typical micros tructure are shown in fig. 2.

Fig,

5 mm

a) Dipole ("twin club") and

quadrupole geometry

b) SEM micrograph of the border

between NbjSn layer (lower part)

and niobium surface (upper part)

RESULTS

The mean square flux flow noise voltage 6V^ , obtained in a sufficiently

wide frequency band (typically 0.1 ... 10^ Hz), and the dc-flux flow voltage

for a sample with "twin club" structure are shown in fig. 3. The data is

taken for positions of the contact pair along the dashed line. Before entering and again after leaving the channel between the clubs, vortices pass a zone of convergent or divergent dc-velocity field, respectively. Since our

samples have a demagnetization factor near one, local deviations from mean

vortex density n=B/$ are small. The two zones of divergent dc-velocity

field therefore are associated with a vortex rearrangement, which may

lead to flux flow noise as outlined above.

Such a feature can indeed be seen from the local behavior of 6V^ shown

in fig. 3. There are two separate regions of maximum noise, which correspond

to locations of maximum divergence in the dc-flow. The channel region exhibits less noise, the minimum value of however being noticeably higher

(more than an order of magnitude) than for the niobium sample without Nbj Snstructure

.

These results however are not quite unambiguous. Measurements along a line in x-direction show that considerable noise is seen for all positions,

for which one voltage probe comes close to the border of the pinning structure. The observed noise in fig. 3 therefore could at least in part result

from contributions of the border zone along the Nbj Sn-structure . To remove

this ambiguity, a pinning structure is needed, for which the location of

maximum noise also has maximum distance from all Nbj Sn-domains

.

385
Fig. 3 Total noise power 6^V^ and dcflux flow voltage V for different y-positions of midpoint

between the voltage probes

(separation 0.4 mm), which are aligned in x-direction. The

gradient in the DC-velocity

field corresponds to the slope

of the V (y) -prof ile . Inset:

NbjSn structure as black

area. Vortex flow indicated

by two equipotential lines.

Field: 0.175T, temperature:

4.2K, transport current 4A.

Fig. 4 V and 6V^ measured at the

quadrupole structure along

the dashed line (see inset)

.

Voltage probes aligned in

y-direction for measurement

of 6V^ , in x-direction for

V- measurement . B= . 1 5T, T=4.2K,

total transport current I=4A.
As a suitable geometry we prepared the quadrupole-arrangement of NbjSn areas

shown in the inset of fig. 4. At the center between the electrodes there is

zero_net f lux flow but maximum noise, as can be seen from a comparison of

the V and 5V^ characteristics in fig. 4. Switching the polarity of two of the

electrodes changes the velocity field as depicted in fig. 5.

y [ mm

Local pinning properties of

the sample are not affected by

this procedure, of course. If

however the observed noise is

due to plastic deformation of

the flux line lattice

,

the

local dependence of JW should

be changed drastically. Indeed,

a minimum of 6V^ is now observed at the center between the

electrodes, instead of the

maximum in fig. 4. Also, as expected, there are two zones with enhanced noise adjacent to

the minimum of 6V^ [ 5 I

.

Concluding^ we see strong

evidence for a noise generation

mechanism involving random plastic deformation of the vortex

lattice [ 6 ] . This result is

further supported by investigating the dependence of 6V^ on magnetic field and temperature

[71.

Fig . 5 Same measurement as in fig. 4,

but with reversed polarity of two of

the electrodes.

REFERENCES

I 1 ] A good review of recent theoretical and experimental work on flux flow

noise can be found in J.R. Clem. Proc . 5th Intern. Conf. on Noise (Bad

Nauheim, Germany) Springer, Berlin, (1978) p. 214.

[21 D. Kohake and C. Heiden, J. Low Temp. Physics 40, 531 (1980)

[3] J.D. Thompson and W.C.H. Joiner, Phys . Rev. B20, 91 (1979)

[4] H.M. Dirks, Ph.D. thesis, Univ. of Giessen, Germany (1981)

t 5] The residual resitivity of the sample changed somewhere in the time interval between the measurements of figs. 4 and 5, although no variation

in the volume pinning of the niobium foil was observed. This may be the cause for the overall higher noise intensity seen in fig. 5 (c.F. Ph. D.

thesis of K.F. Beckstette, Univ. of Giessen, Germany (1981).

( 6 ] A more detailed report will be published elsewhere.

[7] See following paper, this conference.

387
FLUX FLOW NOISE DURING INHOMOGENEOUS VORTEX-MOTION.

MAGNETIC FIELD AND TEMPERATURE DEPENDENCE*

Klaus F. Beckstette** , Heinrich M. Dirks, C. Heiden

Institut ftir Angewandte Physik der Justus-Liebig-Universitat Giessen,

Heinrich-Buf f-Ring 16, D-63oo Giessen, Germany.

INTRODUCTION

Results presented in the preceeding paper [l3 suggest that one rSle of

pinning concerning the generation of flux flow noise is to provide guided in

homogeneous vortex motion such that plastic deformation of the flux line lat

tice will take place [23 . The geometrical arrangement of pinning sites and

their individual pinning strength therefore should be important factors for

the observed noise. This may be one reason for the lack of reproducibility

in the magnetic field dependence of flux flow noise which is encountered for

different polycrystalline samples with average grain diameters comparable to or larger than the distance between the voltage probes. This also provides

an explanation for the fact that reproducibility in these investigations becomes better, and a smoother field dependence of the mean square noise voltage is obtained, if the grain size is reduced such that the voltage probes see noise contributions from many grains in their neighbourhood [3]

Good reproducibility on the other hand is observed for carefully prepared

single crystalline samples with isolated pinning sites of defined geometry

as used in the preceeding paper. In the following we present measurements of

the dependence of 5V^ on flux density B and temperature T.

EXPERIMENTAL

The sample used is a single crystalline niobium foil with "twin club"'

shaped dipole structure of a high pinning Nbj Sn-layer. All data reported

here is taken for a fixed position of the voltage probes as depicted in

fig. 1 . Broad band measurements again are performed with the same frequency

band as in the previous paper. The sample is submerged in a liquid helium

bath, whose temperature can be varied by pumping. The magnetic field is

supplied by a superconducting magnet. The sample is doped with nitrogen in

order to increase sufficiently the normal state resistivity such that flux

flow voltages of measurable magnitude are obtained. This doping is carried

out in such a way that it does not increase the volume pinning force of the

niobium foil from its very low value, it had before doping except for the

occurence of peak effect in the critical current at a field B just below

the upper critical field B . ^

RESULTS

Fig. 1 shows the typical behaviour of the field dependence of 6V^ and

the volume pinning force Fp = jc*B of the niobium foil (not of the Nbj Snstructure) obtained at T=4,2 K. Fp turns out to be almost independent of B

whereas goes through a maximum. This may be taken again as a sign that

* Supported by the Deutsche Forschungsgemeinschaf

t

** This work represents parts of a Ph.D. thesis of one of the authors (KFB) .

388
there is no direct connection between pinning and flux flow noise.

1 1 r

T=^.2 K

BIT]

Fig .

1

Typical field dependence of mean square flux flow noise

voltage for low pinning single crystalline niobium foil with high pinning twin club Nb^Sn-layer deposited

on it. Contact positions as indicated by two dots. Transport current

is I = 2Ai Also shown is the field

dependence of the volume pinning

force.

T[K] BcplT

^.2 0.39

o 3.5 o./»o

A 3.0 0.^6

2.6 0.^9

0 1.9 0.51

Fig.

2

Dependence of mean

square noise voltage

on reduced field b*=B/B obtained at five dif ferent^temperatures under the same measuring conditions as in fig.1,

Solid line: field dependence

of the shear modulus.

389
Measurements at other temperatures yield analogous results. Fig. 2 shows data obtained at five different temperatures, plotted as function of

the reduced field b*=B/B_p [4] . Obviously, the data can be represented well

by a scaling law of the form

= f(T) • h(b*)

where f(T) is nearly constant for our sample. Measurements at other positions of the voltage probes in the region of inhomogeneous flux flow yield

similar results, such that a function of position g(r) may be separated:

= f (T) • h(b*) • g(r) . DISCUSSION

The equation of motion for the vortices contains as essential parameters the driving Lorentz force determined by the transport current

_

density, the pinning force, given by material inhomogeneities , a viscous

force, which is proportional to a viscosity coefficient ti=B^2 'I' o /Pn ' where

i}) 0 is the flux quantum, and Pj^ the normal state resistivity and interaction

forces between the flux lines, which are controlled by the elastic moduli

C^- of the flux line lattice. Intrinsic volume pinning force, and viscous

friction are essentially independent of magnetic field for our sample. This

suggests to relate the behaviour of 6V^ (B) to the field dependence of the

elastic moduli. A major contribution during the plastic deformations of the

vortex lattice certainly will arise from shear-effects. We therefore consider the shear modulus Cge/ whose field dependence is given by [5]

r -If i.2d^H H S H

Cee -

2 ) B —- dB ci

0 dB

yo (2<l - 1) g C66'= o,48 ^ (1 - H S H , [l + 6^(2Kf-1)j^ ^C2

The behaviour of C6e(b*) and Cee' (b*) , calculated for our sample, also is

shown- in fig. 2. Fitted only in magnitude, these curves describe rather v/ell

the field dependence of 6V^ . These results again give strong support for

noise generation by plastic deformation of the flux line lattice.

REFERENCES

[l] H.M. Dirks, K.F. Beckstette and C. Heiden, preceeding paper, this con- ference

[2] K.F. Beckstette, H.M. Dirks and C. Heiden, Phys. Letters 81A, 351(1981)

L3] see for instance: C. Heiden, D. Kohake/ W. Krings and L. Ratke, Journal

Low Temp. Phys. 27, 1 (1977)

[4] B is used instead of B , since the latter could not be measured as accurate as B__,.

^

[5] R. Labusch, Phys. Stat. Sol. J_9'

'^''^ (1967), Phys. Stat. Sol. 3_2/ 439

(1969)

390
VORTEX NOISE IN TWO DIMENSIONAL SUPERCONDUCTING FILMS

C.M.Knoedler, R.F.Voss, and P.M.Horn

IBM T.J.Watson Research Center, Yorktown Heights, N.Y. 10598

INTRODUCTION

In 1969 Maul, Strandberg, and Kyhl [1] observed a large excess of noise in the power spectrum

of a thin film tin bolometer. While they were unable to draw any conclusions as to its origin they

noted that the noise was only observed in the transition region of the film and speculated that it might have a connection with the superconducting to normal phase transition. It is well known [2]

that free vortices (or flux bundles) induced in superconductors by a magnetic field cause phase

slips which result in finite electrical resistance and flux flow noise. Recent theories [3,4,5]

postulate that equal numbers of positive and negative vortices exist in thin superconducting films in the absence of an apphed magnetic field. Below some characteristic temperature Tj^j they exist

only as bound pairs. Above Ty^j the pairs unbind to form a plasma of free vortices that is responsible for the resistive transition. In order to probe the nature of such superconductingnormal transitions we have systematically studied the voltage noise and I-V characteristics of

granular superconducting films. In this case the noise measurements provide a unique probe of the

properties of thermally excited vortices in the vicinity of the transition.

EXPERIMENTAL PROCEDURES

The aluminum and tin samples were flash evaporated in an oxygen atmosphere onto glass

substrates using procedures described by Abeles, Cohen, and CuUen [6]. This technique produces

films with small grains and high sheet resistances R|--|« 30-50000/ . The aluminum films

typically had 50A grains and thicknesses less than 100A while the tin films had grain sizes greater

than 1500A and thicknesses on the order of 400A.

Individually shielded, twisted pair leads were connected to the sample in a four probe configuration. The samples were immersed in liquid helium in a cryostat surrounded by a double mu-metal

shield with an ambient magnetic field of less than 1.7xlO''*Oe. The leads from the sample were

connected to a special low noise field-effect-transistor preamphfier and to external current and

voltage connections. The preamplifier output was connected to a bandpass filter and an ac voltmeter to allow estimation of the voltage noise spectral density in the frequency range 10-

lOOkHz. A dc current Ij, « (0.25-200/iA) was supplied to the sample and the dc voltage V^j^, as well as the rms noise voltage were digitally recorded and averaged.

EXPERIMENTAL RESULTS

Typical results for a 380fi/n Al sample are shown in figure 1. The resistive transition as well

as the noise spectral density at lOOkHz, Sy(lOOkHz) (with the preamplifier background of

2.5xlO"^^V^/Hz subtracted from Sy) are plotted versus T. The curve with 1^=0 depicts the

decrease in the Johnson (thermal) noise as the sample becomes superconducting. With increasing

391
Fig. 1 (a) Resistive transition for a 380S2/n Al

film, (b) Noise spectral density SyClOOkHz) vs. T for various

2.00 2.05 2.10 2.15 2.20 2.25 T (K)

lyj an increase in the magnitude of the noise is seen at the low T end of the transition. Both the

transition and the noise were found to be functions of Rj-j, with the broader noise "peaks" and

transitions occuring for large R^. In the frequency range investigated (10-lOOkHz) the noise

spectral density was independent of frequency, indicating that the characteristic times of these

processes were <0.1 msec. The application of a perpendicular magnetic field was found to strongly suppress Sy, simiHar to the results seen by Maul, Strandberg, and Kyhl [1].

The voltage and current dependence of the noise is better illustrated in figure 2 where the excess noise spectral density is plotted versus V^^, for samples of Al and Sn at various l^. This excess noise was obtained by subtracting both the preamplifier background and the calculated

Johnson noise from Sy. V^j^. was varied by changing T and thus R at constant I^^. For low and

low Vjj^. the noise is proportional to V^j^,, with the proportionaUty constant independent of 1^. At

sufficiently large the magnitude of Sy increases at low V^j^, possibly indicating a different type of

noise process. The linear relation between Sy and V^j^, strongly indicates that Sy is a voltage shot

noise arising from independent voltage pulses. Traditional noise sources such as equilibrium

resistance fluctuations [7] would lead to SyocV^dc

-17 10 >

N ..^-18

-19

-20

10

-21

1 1

(a) Al X^^^ -

• 36mA

1 1

° ll/iA

* 3.6/iA

] 1 i"

-1— I

(b) Sn —7—

•••••••

1 1

• ll/tA

° 5.5mA

* 3.6mA

] i i" 10"^ lO"* 10 ^ 10"^ 10

^ 10 10

-3 10 10 10"

Fig. 2 (a) Sy vs V^^, for the Al film in Figure 1. (b) Sy vs V^j. for a 4000S2/n Sn fikn. Both

show the dependence of Sy on

392
Van Ooijen and Van Gurp [8] showed that the motion of flux across a sample causes a change

in the voltage at the sample electrodes. If the jumps A<|) in the superconducting phase occur

independently at a rate r then, W^^=<^Qri!^<^/2'n, where <^>q is the flux quantum and

S^(J) = 2<t>Qi^<i>/2'!T)V^^ (1)

-1 at frequencies /<<t' , where t is the characteristic frequency for these phase jumps. Similar

results are expected for one dimensional (weak link) superconductors and Josephson junctions with

A^=27r as the phase jump due to vortex motion across the entire sample [9]. This 1-D prediction

of eq (1) is plotted as a solid line in figure 2. Sy is seen to parallel this prediction for low Ij, and

low Vjjj, but with A(^)<< 277.

Figure 3 depicts data for a narrow 350fi/n Al film with a width of 27/im, at various Ij,, and

with an applied magnetic field. As the width of the sample decreases the magnitude of Sy is larger

than that shown in figure 2 for similar Ij, and approaches the 1-D prediction of eq (1). At the same Ij, the magnetic field tends to suppress the noise at low V^j^,.

>

10

-21

OOe

6/iA OOe -

° 3.6fiA 4 Oe

' 3.6/iA 16 Oe

Fig. 3 Sy vs y^f, for a narrow (275/im by 27/xm)

3500/ Al sample showing the dependence of

Sy on both bias current and magnetic field.

10 10" 10" 10

-2 10

In figure 4 we display data derived from measurements on tin films in which the temperature in the cryostat was held constant to 1 mK. The current through the sample was slowly ramped using

a capacitive discharge technique while V^j^, and Sy(lOOkHz) were recorded. The data for several T

is displayed. As expected near the transition at T«3.81K Sy is linear with respect to V^^, at low

voltages. As the temperature is lowered the magnitude of Sy tends to increase and approach the

1-D prediction of eq (1).

Fig. 4 Sy vs V^ic for a 30S2/n Sn sample at various temperatures. As T is lowered the magnitude of the noise increases and approaches the eq

(1) prediction.

10

10 10" 10 10

393
DISCUSSION

We have observed a white noise spectrum in the range of frequencies monitored, with Sy

varying as V^^ at the onset of the resistive transition. The noise is current induced and is inversely

dependent on the width of the film. These findings suggest that we are observing voltage (phase

slip) shot noise in analogy with electrical shot noise. In this case the charge carrier corresponds to

a flux quantum moving a distance i across the film producing a phase sHp = 2ir/'/w where w is the film width. In addition it should be noted that as the voltage (temperature) is increased the excess noise tends to level off and then decrease. This effect can be explained by analogy to

electrical shot noise. As the temperature (and voltage) increase the vortices interact more strongly

and their motion is no longer independent. Thus, the phase sHps become correlated and the noise

is reduced similar to shot noise in space-charge limited diodes [10]. The imposed magnetic field produces a similar result (reduction of Sy) with large numbers of field induced vortices causing a correlation in the vortex motion. It should be emphasized that these results are quite different

from the usual flux flow noise in type II superconductors [8] where «10^ flux quanta are in motion, B=1000Oe, J«500A/cm., and T«5msec. In these experiments BaO, Jw0.05A/cm,

T<0.1msec and we are observing the motion of individual quanta moving a fraction of the distance across the sample.

SUMMARY

We have used noise measurements in conjunction with standard electrical measurements to yield information about the nature and dynamics of the two dimensional superconducting transition. The excess current induced noise at the onset of resistance in granular Al and Sn samples can be explained as a phase slip shot noise arising from the independent motion of vortices. The

amplitude of the noise provides a measure of the mean free path of the vortices.

REFERENCES

[1] M.K. Maul, M. Strandberg, and R. Kyhl, Phys. Rev. 182, 522 (1969).

[2] Y.B. Kim and M.J. Stephen, in Superconductivity, R.D. Parks, ed. (Marcel Dekker, New York,

1969) Chapter 19.

[3] J.M. Kosterlitz and D.J. Thouless, Jour. Phys. C. 6, 1181 (1973).

[4] M.R. Beasley, J. Mooij, and T. Orlando, Phys. Rev. Lett. 42, 1165 (1979).

[5] B.J. Halperin and D.R. Nelson, Jour, of Low Temp. Phys. 36, 599 (1979).

[6] B.A. Abeles, R.W. Cohen, and G.W.Cullen, Phys. Rev. Lett. 17, 632 (1966).

[7] R.F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976)

[8] D.J. Van Ooijen and G.J. Van Gurp, Phys. Lett. 17, 230 (1965).

[9] J.E. Lukens, R.J. Warburton, and W.W. Webb, Phys. Rev. Lett. 25, 1180 (1970).

[10] J.B. Johnson, IEEE Spectrum 8, 42 (1971).

394
HOT CARRIER NOISE

395

MODELING NOISE OF DEVICES WORKING UNDER HOT CARRIER CONDITIONS

J. P. Nougier

Universite des Sciences et Techniques du Languedoc

Centre d'Etudes d 'Electronique des Solides+

34060 Montpellier Cedex, France

1 . INTRODUCTION

Let us consider a device (Fig. 1), with a d.c. bias Vq^n between the electrodes M and

N. The measurements are perfomed at the probes M' and N', which may be identical with or different from the bias electrodes M and N. Let VqM'N' be the d.c. voltage between the

probes. A d.c. current IqMN flows through the device, corresponding to an average flow of

electrons. In fact the electrons move randomly around their average trajectory, thus producing at each point of the device a random current superimposed to the d.c. current, which

results in a random voltage superimposed to the d.c. voltage between every couple of points.

Of course, the local noise source at every point gives a contribution to the noise measured

between M' and N', the total noise being the sum of the contributions of each local noise

source.

This device can be schematically represented by a pool of water, crossed by a creek

arriving at point N and going out at point M. When one drops a bunch of stones somewhere

in the pool, a local noise source is created, producing noise at the terminals M' and N'. Obviously this noise depends both on the local noise source (studied in section 2) and on

the way in which its effect is transmitted to the probes M' and N' : one way for studying

this effect is to consider the noise source as a random signal and to study the signal re- sulting between the probes : such are the macroscopic models : Langevin (section 3)

,

impedance field (section 4) and salami (section 5) methods. The other way is to consider

the noise as a random motion of- each particle and to derive it from the fluctuations of

their velocities, known at every time (microscopic models, section 6). All these theories

are linear around the bias point, and therefore apply either to large or small random signals of linear devices, or to small random signals of non linear devices.

N

N'

(a.c)

Ax Ax'

Laboratoire associe au Centre National de la Recherche Scientif ique, LA 2 1 et Greco

Microondes.

397
2. LOCAL NOISE SOURCES OF HOT CARRIERS

2.1. Definition

Let us label j(r,t) the instantaneous local current density, of d.c. value i (r) : o

Aj ir,t) = J ir,t) - j^U) (1)

Aj(r,t) is randomly distributed; if AjjjQj(r,f) denotes the current density noise source along the direction a, i.e. the component of AjQ^(r,t) at frequency f within the bandwidth

Af, one defines the noise source term K^g(r,f) as [ 1 ]| :

where * means the complex conjugate, the bar means average value over the time. Indeed, the

spectral density (s.d.) S^j^^g of Aj is the Fourier transfrom of eq. (1). Eq. (2) means that

the noise sources at two different points are uncorrelated. Integration of eq. (2) over the cross section AA(x') of a slice Ax' located at abscissa x' along an a.c. current path (see

Fig. 2) gives, if AI^ denotes the noise current through the slice :

AIf^(x,f) AI*(x',f) = AA(x) K^(r,f) 6(x-x') Af . (3)

This relation had already been given for particular types of noise sources [2 ] 3 ] . Integrating eq. (3^ once more over the domain x'6 Ax gives the relation between the local noise source K (r) and the s.d. of the noise current S.^ (r.f) : X ^ AI X K (r,f) = S.^ (^,f) Ax/AA(x) . (4) X Alx

2.2. Experimental determination of the noise source term : the experimental way for determining the noise source term obviously follows from eq. (4) : one has only to determine the

s.d^ of the noise current of an homogeneous bar in a uniform field, from which one gets

Kx(E,f), and then in_^a non homogeneous device with the same impurity concentration, orientation,

etc..., one has Kj,(r,f) = Kx[E(r)yf]. However, hot carrier experiments require short pulses in order to avoid thermal heating of the semiconductors, and thus high frequency analysis : at those frequencies, because of matching impedance problems, the experimental parameter which can be measured is the noise temperature T^ (an extended bibliography will

be found in refs.[ ^

] [] 5 ] 6 ]] ) . Since T^ can be related to ^^it K. can be related to T^

[] 7 ] . Let AYx be the differential admittance of the slice Ax. One has [ ^ ] [ ^ 3'

I

S^^^ (r,f) =4k^T^^(?,f) Re {AY^} (5)

( Ay^ = AA 6j^(r)/tAx 6E^(r)] (6)

when k„ is the Boltzmann constant. Eqs. (5) and (6), carried into eq. (4), give : D

:^ -> ->- ->- where a (r,f) is the a.c. conductivity tensor : 6j = a 6E. Eq. (7) shows that the noise

source can be determined experimentally by measuring the noise temperature and the conductivity, without needing any knowledge on the physical nature of the noise (an example is

given in ref. [ 7 ]). Note that the direction a may take different values for a given direction o f the d.c. field [ 7 ]

.

2.3. Expression of the noise sources for various physical mechanisms : For deriving the

expression of the local noise sources of hot carriers, one starts with the expression of

the current through the slice Ax (AN is the average total number of carriers in the slice, n the carrier density, v their average velocity) :

I(x) = q n(x) v(x) AA(x) = q v(x) . (8)

a) Diffusion noise : one assumes that the contribution to the noise is mainly due to the

fluctuations of the velocities of the carriers, the fluctuations of n being negligible.

Eq. (8) gives then :

398
S^^(x,f)=qV(x) AA^(x)S^^(x,f) = q^n^(x)AA^(x)[s^^.(x,f)/n(x)AA(x) Ax] (9)

where S^^^ (x,f) describes the fluctuations of the velocity of one given carrier, although

S^^ is related to the fluctuation of the average velocity of the carriers in the slice Ax. Since, by definition ][ 5 ], S^^^(x,f) = 4 D (x,f) when D is the diffusion coefficient,

eqs. (9) and (4) give :

T^^^^{r,f) = 4 n(^) D (^,f) . (10)

This result was already given in Refs.[ l] to [3 j. Comparison of eqs. (7) and (10) give

the relation [sj between the noise temperature and the diffusion coefficient •

w T (?,f)/q = q n(r) D (r,f)/Re {a (r,f)} . (11)

D na cx otct This relation should replace in space charge regimes of hot carriers, the one previously

obtained [^4 ]

5 ] for homogeneous bars.

b) Generation-recombination (G.R) noise : in that case, the noise is mainly produced by the

fluctuations of the number AN of the carriers in the slice. Eq. (8) gives then

= qV(x) S^^/(Ax)2 = j^x)AA^ ^^/^^ • ('2)

For G.R noise involving only two levels 9 ] ~ aT(var AN)/(1 + w^t^) , and due to the

quasi Poissonian character of the fluctuations, var AN =aN ; this gives for eq. (12)

carried into eq. (4) : 2 -y -> ^ j (r) a(r) T(r)

^aGR^^'^^ = ^"T 2 2 2 ^ -1 ' ^'^^

""^^ nir) [l + 4/f^^r)]

One thus gets an expression analogous to that obtained for thermal G-R nois^e , except

that j is not linear with respect to the d.c. electric field, and that a(r) and T(r) may

depend on the local d.c. field. Such a situation occurs for example in n-Si at 77 K : then,

at thermal equilibrium, only part" of the donor impurities are ionized : when a high d.c.

field is applied, a fraction of the neutral impurities is ionized, due to Poole Frenkel

and/or impact ionization effect, thus leading to hot carrier G-R noise, since this fraction

depends on the d.c. field, so as a and x 10 ].

c) 1/f noise : The 1/f noise can be represented as if it were the sum or G-R noises on a subband of impurities with a continuous distribution of relaxation times. As a consequence,

it may be inferred from eq. (13) that the bulk 1/f noise source is of the form :

^al/fbulk<^'f> = ja -h(^)A"(^^ ('^^

where tends towards the Hooge's constant [ 1

1 ]] at low field. In the same way, the 1/f

noise arising at an interface at abscissa Xq may be described by a noise source term :

K , , (r,f) =Aj^ (x^) 6(x-x )/Cn(x )f^) , (15)

al/f contact a o o ^ o

•

3. THE LANGEVIN METHOD

This method [l2] has been extensively described in a review paper of Nicolet et al.

[ island applied to single and double injection diodes. According to this theory : (i) one first writes the usual equations (conduction eqs., continuity eqs, Poisson

eq.). For example, for a one dimensional device j = + jp + e9 E/8t ; jp = q P Vp-q Dp 3p/9x ; 9p/9t = - rp + gp - q"' 9jp/9x ; etc...

(ii) Every variable Y involved is then set equal to Y

Yq + 6Y exp iwt. The zero order

terms give the static characteristics. The first order terms give the a.c. equations (i.e.

linear variations around the bias point).

(iii) to each a.c. equation one adds the appropriate noise source. One then gets a set of linear equations. For example : 6j = 6jn+ <5jp """^ i- '5E ; 6 jp=qvpo6p + qPo (dVp/dE) q6E - q Dp9 6p/9x + 6Hp(x) ; icofip = - 6rp + 6gp - q" 1 (95j p/ 9x) + 6HQR^p(x) ; etc... Here 6Hp is a 399
diffusion noise source term for the holes, 6Hqj^p is a G-R noise source term for the holes,

etc . .

.

(iv) the auj^liary variables are eliminated so^as to get a relation between 6l and 6E,

of the form 61 =^(^6E, 6Hp, 6HQg^p , . . . . where ^ is a linear operator and 5l = A6j . This

eq. is solved, which gives 6E as a function of 6l, 6Hp, SHg^p, ..^ Then, integrating over

the length L of the device, one gets the voltage at is terminals

6V = C ?'[6l(x'), 6H (x'), 6Hg^(x'), ...] dx' (16)

Jo

"

where )C' is a linear operator. When the noise sources 6Hp, 6HQ^p, are set equal to zero, eq. (16) gives the a.c. impedance. When 61 = 0, multiplying eq. (16) by 6V gives

the open circuit noise voltage around the bias point. For further details, see [[l3] . This

method was applied to single inj ection diodes f^l'^J[] 15 16^.

Obviously, an alternate possibility would be to write the usual eqs.^then to set

Y = Yo+6Y, then to take the first order linear eqs. as previously. But, instead of writing

the noise s^ources, one could first eliminate the auxiliary variables so as to get the equation 61 = ^ 6 Ej^ The local noise sources could be introduced only now, leading to an eq. of

the form 61 = 5S6E - ^ ^^Ni where the subscripts i stands for the various noise sources. When 61 = 0, the solution of this eq., after integration over x, gives

rL 6V = i|j(x',f) ? AI .(x') dx' (17)

1 1 Ni

Jo

where 4^(x') is some function of x', depending on which operator i is involved. Multiplying

eq. (17) by its complex conjugate, and taking into account eq. (3) gives :

S^^(f) = A ( |i|j(x',f)|^

I K.(x',f) dx* (18)

J o the K. are the noise sources studied in section 2 above.

1

4. THE IMPEDANCE FIELD METHOD :

The impedance field method was originated on one dimensional devices for diffusion

noise by Shockley et al [] 17 J, and was later on also applied to G-R noise 18 3' Its purpose is to relate the local noise sources to the effect they produce at the terminals

M' and N', through the "impedance field".

4.1. Expression for the noise : A three dimensional statement will be given here. Let us introduce a small a.c. current 6l exp iwt at point r (see Fig. !), superimposed to the

d.c. current^. The a.c. voltage appearing between M' and N' is 6V(M' N',r) = Z(N',r,f)6I

5jhere Z(N',r,f) is the a.c. impedance between r and N'. If this current is taken out at r-dr, it produces an a.c. voltage 6V(M' ,N ' , r-dr) = Z (N ' ,r-dr , f ) 61 . The overal voltage is

thus :

6V(M' N',r) = 61 ?Z (N',r,f).dr • (19)

Here a matrix notation has been adopted, in which the tilde means "transposed" : hence a vector such as dr is represented by a column matrix, and d? by a row matri^x.VZ^N' ,"r,f

)

is the impedance field. If 6j^(?) is the current density at r, one has : 6l dr = 6j d3r

so that eq. (19) after integration over the volume of the device, gives the a.c. voltage

between M' and N', labelled 6V(N') :

6V(N') =

\\\ VZ(N',?,f). 6T(?) d^r . (20)

Let us label a the component along every direction x, y or z : eq. (20) writes :

6V V ... ,3 (N') = E JjJ

d-^r 6j^(r)[3Z(N',r,f)/9 a] . (21)

For getting the noise, one multiplies 6V(N') by 6V (N'), which gives, using eq. (2) :

400
= E E||j[3Z(N',r,f)/9a] [8Z*(N' ,?,f)/9B ] K^^(?) d^r . (22)

This eq. can be written using a matrix notation :

S^^ =

III d^r V*Z(N',^,f) f'(^) VZ* (N',r,f) . (23)

For one dimensional devices, every quantity is a scalar and depends on only one variable labelled x, so that eq. (23) gives eq. (24), which reduces to the formula of Shockley

et al 1^173 for diffusion noise (see eq. (10)) :

2 (f) =\ A(x) K(x,f) |VZ(x,f)| dx . . (24)

Jo

As one can see, by comparing eqs. (18) and (24), the impedance field method and the

Langevin method are formally identical, as was previously pointed out |] 19 ~| [ 20 "|. The noi

se voltage can also be expressed as a function of the local noise temperature by replacing

K(r,f) by its value[eq. (7)]. For one dimensional devices, one gets :

? S^^(f) = 4 kg \ T^(x,f) Re { 6I/6E} I VZ(x,f) r dx . (25)

Jo ^ ^

The noise sources being supposed to be known, one then needs to determine VZ. This wil

be studied in subsections 4.2 and 4.3 below.

4.2. Green function of the electric field ; transfer impedance method : this method was dedevelopedby Van Vliet et al. T 1 ] and applies easily when the fundamental electric variable

is the local electric field E (r) . Let us apply an a.c. voltage at the terminals M' and N '

.

This ^e^ults in a local a.c. field and in a_^lo^al cur^eni^ density of complex values 6E(r)

and 6j (r) ^ superimposed to the d.c. values Eo(r) and Jo(^)* '^^^ master_^e^uation^ aje written

the auxiliary vari^ablefare_^eleminated so as^t^ get^a relati^n_^between E(r) and j (r) . One

then sets E(r) = Eo(r) + 5E(r) exp icot and j (r) = Jo(^^)^+ ^ (r) exp iojt. The zero order

terms give the d.c. characteristics linking Eo(r) and Jo(r). The first order termsgive an equation of the form :

L 6E(r) = 6j(r) (26)

where L a linear operator. The Green matrix z(r,r ,f) of the operator L is defined

as (here I means the unitary matrix and 6(r-r') the Dirac function) : Lf(r,T\f) = l&(v-v') (27)

eq. (26) and (27) give then, integrating over the whole volume of the device :

6E(0 =

jjj

d^r- (?,?', f) sT (r'.f) . (28)

Integrating on a line (generally a current line) of length L from M' to N', on gets

the a.c. voltage at the terminals :

6V(N')=-^^'S 6E(r) = - ^^(((d^r' d? f (?,?',f) 6T (r',f) . (29)

Comparison between eqs. (20) and (29) shows that

VZ(N',?') = - ( d^ z (?,^',f) . (30)

3o

For one dimensional devices, z(x,x',f) is the Green function of the equation linking 6E(x)

and 6l(x) (and not 6j(x)), as follows from eq. (28), Eq. (30) writes then :

Vz(x') = - I dx z (x,x',f) . (31)

Jo

401
This method was used for various modelling, among them single injection diodes [ 1 ~I and

FET's at low drain bias [21 ] .

4.3. Green function of the voltage : This method [^22 J[ 23^ applies easily when the fundamental electrical variable is the voltage. As previously, an a.c. voltage is applied to the terminals M' and N' , superimposed to the d.c. voltage between M and N. This results in

a set of orthogonal equipotential surfaces and current lines. Let 6v(x) be the a.c. voltage

of the equipotential surface crossing a given current line, of length L, at abscissa x, and let 6l(x) be the a.c. current through this equipotential surface. As previously, 6v and

6l are obtained by writing the usual eqs., which give :

5"6v(x) = 6l(x) (32)

where '£ is a linear operator. The Green function ?S(x,x',f) of eq. (32) is a solution of : ^2(x,x'.f) = 5(x - x') (33)

and gives

6V(x) = ( 2(x,x',f)5l(x') dx' . (34) Jo

By setting x=L one gets the a.c. voltage 6V(L) = 6V(N') at the terminals. When AI is introduced at X and withdrawn at x + dx, being null elsewhere, eq. (34) leads to 6V(L) = 2(L>x)M(x)dx

which gives the impedance field initially defined by Shockley et al. [l7] , which in this

formulation thus takes the form (see eq. (19)) :

VZ (N,x') =2(L,x') . (35)

->• When K(r,f) only depends on x, that is does not vary on an equipotential surface, the device is then one dimensional, and the noise can be expressed using eq. (24) or (25). This method was successfully applied for deriving quite general behaviour of the noise produced at the drain [22] and at the gate [23] of FETs resulting from noise sources in the channel.

4.4. Numerical solutions : very often, the expressions of K(r,f) and VZ(r,f) are so complicated that one is unable to carry out the calculations so as to get analytical results. Then

the^calculation has to be performed numerically [l5"] [ 16] , Furthermore ^(r,f) and

VZ(r,f) may even have no analytical expression, and in that case, the entire calculation

has to be performed on a computer. For example, it is possible to determine numerically the

voltage produced by the introduction of a current at a mesh point, and by withdrawning it at an other mesh point : this gives a numerical value of VZ which is then used for getting

nimerically the noise [24] . The impedance field method can even be applied without writing

the master eqs. : this was done in FETs [25 J , where the transistor was considered as an active line, being made of a chain of elementary FETs^ with progressively varying bias conditions^ each one representing a slice of the actual FET. The noise is then considered as the sum of the effects, at the drain and at the gate, of the noises produced by the various

slices, which is the basic idea of the impedance field method, applied to a representation

using circuits instead of master eqs.

5. THE SALAMI METHOD

In the salami method, the device is devided into slices supposed to be uncorrelated,

so that the total noise voltage S/\y of the device is taken as being the sum of those of the

slices d S/^Y • was mentioned by Thornber [ 26 ] and later on by Van Vliet et al. [ 1 2»

this method fails because, although the noise sources of two different slices are indeed

uncorrelated (see eq. (2)), the noises are correlated since the noise of the slice x depends on the noise induced in it by all the other slices through the transfer impedance bet- ween slices. However, this method gave satisfactory or even exact results when applied to single or double injection diodes (see refs. [ 9 ] and [67 ] to [ 73] of the review paper

[ 13] of Nicolet et al.), and was also applied to FETs [ 27 ].

The conditions under which the Salami method gives correct results have been recently

studied [ 7 ]. From eqs. (5) and (6) one gets the noise voltage dS^y of the slice dx :

402
so that the noise voltage S/^y of the whole device is, according to the salami hypothesis :

= 4 ^

(x) Re dx . ( ^n"

J o 61' (35)

This is to be compared with the correct result given by eq. (25). Comparison of these

two expressions allows us to determine the conditions under which the salami approximation

works. In particular, eqs . (25) and (35) give identical results when the integrals are equal,

that is when

I 6E/6I I = I VZ(x) (36)

This gives, according to eqs. (31 )( where VZ should be taken at point x and not x')

and (28).

z(x,x') dx' z (x ' ,x) dx

(37)

For bars in a uniform (even high) field, z(x,x') = z(x',x) and eq. (31) is valid : the

salami method gives rigourous results in that particular case.

6. MICROSCOPIC MODELS

In the microscopic models (or particle models), the current is not considered as a signal, but rather as a motion of carriers : this is a mechanical point of view instead of an electrical point of view. In these models, the trajectories of N particles inside the

device are studied as a function of time, so that at each time (more exactly at multiples

nAt, of a time step A(t), the position and the velocity of each particle is known. Since

the actual number of carriers in a device is very large, each particle indeed represents a number of carriers.

The basic method is the Monte Carlo technique (see for example [^28 "[ [ 29 ] _ 30 _

. Since

this method is much time consuming, simplified models are now being studied 31 . Although the Monte Carlo technique is well suited for studying local noise [ 32 ] and diffusion, even transient [ 33 ]

, the application of particle models for studying noise in devices is

not well established. The only one efficient tentative, up to my knowledge, has been performed by J. Zimmermann [| 34 ]

, who solves the problem in the following way : the current con- servation in a one dimensional device of uniform cross section writes :

I(t) = q n(x,t) v(x,t) A + eA 9E(x,t)/9t (38)

After integration from x=0 to x=L of eq. (38), one gets

- 9V(t)/3t = e^' "

' [a \ I(t) - q { n(x,t) v(x,t) dx ] . ' o

(39)

Now let us consider the 6N(t) particles located in the slice dx at time t. One has : -1 n(x,t) = 6N(t)/A dx and v(x,t) = (6N)

i th carrier in the slice. Thus, eq. (39) writes :

6n

.Z, v.(x,t) where v.(x,t) is the velocity of the

1= 1 L 1

- 3V (t)/9t = (eA) ' f LI(t) - q 1 v.(x,t)]

i=l ^ (40)

where N is the total number of particles. Substracting to eq. (39) its average value

gives the instantaneous deviations : N

- 96v(t)/9t = (eA) ' {L5l(t) - q I [v.(x,t) - v(x)]j

I i=l ^ (41)

where v(x) is the average velocity of the carriers in the slice dx. Two situations can then be investigated C

34 ]

:

403
a) Noise voltage of an a.c. open circuit :6l(t) = 0 in eq. (41), gives

6V(t) =

e A i=i

ft [v^(x,t) - v(x)J dt (42)

Since the velocity of each particle is known as a function of time, 6V(t) can easily been

computed, so as 6V(t + x) , and hence the noise voltage

00 Sg^(f) = 4 \ 6V(t) 6V(t + T) cos 2TTfT dx

b) Noise current of an a.c. short circuit : 6V(t) = 0 in eq. (41) gives

6l(t) = J

! [v.(x,t) - v(x)]

i=l (43)

In the same way as previously, this allows determining the noise current S|5-]-(f) =

4

J 6l(t) 6l(t +x) cos 2Trfx dx.

Once the noise Sgy or Sgj of the particles has been obtained, the noise of the carriers

is deduced by dividing the result by the number of carriers represented by each particle.

In spite of its simplicity, this model is questionnable. Eq. (38) is valid only for

average values, since it is deduced from the Boltzmann eq. which involves probabilities,

that is average number of particules in each state. Thus, using eq. (38), when describing

the noise, is probably erroneous, since obviously, due to the local fluctuations, I should

depend on x, being randomly distributed around its x independent value : in the Langevin

formulation for example, this is taken into account by the addition of the x dependent noise term.

7. CONCLUSION

In the last few years, powerful methods for modelling the noise of devices have been

built (Langevin^ impedance field), providing formulations which can be applied to a great

variety of situations, and to analytical as well as to numerical models.

However these methods are valid for usual situations, where the noise is considered as

a signal, that is for devices long enough, such that each carrier inside is in a steady

state. For devices short enough (submicron devices) such that the distance between the electrodes is of the order of magnitude of the mean free path, the carriers have no time to un- dergo enough collisions so as to reach their steady state : even in the steady state regime, each carrier is in a transient state. A microscopic model is then required, and the applicability of the previous methods for modelling noise then becomes questionnable. An

important effort should be devoted in thee next years in the formulation of methods suited

to such situations.

REFERENCES

[ 1 ] K.M. Van Vliet, A. Friedmann, R.J.J. Zijlstra, A. Gisolf and A. Van der Ziel, J. Appl. Phys. 46, 1804 (1975) and J. Appl. Phys. 46, 1814 (1975).

[ 2 ] A. Van der Ziel and K.M. Van Vliet, Solid St. Electron. 508 (1968).

[ 3 ] K.M. Van Vliet, Solid St. Electron. _I3.r 649 (1970).

[4] J. P. Nougier, Proc. 5 Int. Conf . on Noise in Physical Systems (Bad—Nauheim,

FRG, 1978), Springer Series in Electrophysics 2, Springer Verlag ed., p. 72 (1978).

[ 5 ] J. P. Nougier, in Physics of Non Linear Transport in Semiconductors, ed. by D.K.Ferry, J.R. Barker and C. Jacoboni, Plenum Publ. Corp. p. 415 (1980).

404
Y. Pozhela, hot electron diffusion coefficient, to be published.

J. P. Nougier, J.C. Vaissiere, D. Gasquet and A. Moatadid, to be published,

J. Appl. Phys.

J, P. Nougier, J.C. Vaissiere, D. Gasquet, to be published.

A. Van der Ziel, "Noise : Sources, Characterization, Measurement", Prentice Hall

ed. , Englewood Cliffs, New Jersey, 1970.

A. Van der Ziel, R. Jindal, S.K. Kim, H. Park and J. P. Nougier, Solid St. Electron.

22, 177 (1979).

F. N. Hooge, Phys. Lett. _29 A, 139 (1969).

P. Langevin, Comptes Rend. Ac. Sc. Paris 146 , 530 (1908). M.A. Nicolet, H.R. Bilger and R.J.J. Zijlstra, Phys. Stat. Sol. b 70 , 9 and 415 (1975)

.

A. Gisolf and R.J.J. Zijlstra, Solid. St. Electron. J_7, 839 (1974).

G. Bosman and R.J.J. Zijlstra, Phys. Lett. 71A , 464 (1979).

J. p. Nougier, D. Gasquet, J.C. Vaissiere and H.R. Bilger, Proc. 5 Int. Conf.

Noise in Phys. Systems, Springer Series in Electrophysics 2, Springer Verlag ed.,

p. 110 (1978).

W. Shockley, J. A. Copeland and R.R. James, in "Quantum Theory of Atoms, Molecules

and the Solid State, P. Lowdin ed. ,Acad. Press, New York, 1966.

A. Rigaud, M.A. Nicolet and M. Savelli, Phys. Stat. Sol. a 18 , 531 (1973).

T.C. Mc Gill, M.A. Nicolet and K.K. Thornber, Solid St. Electron. \7_, 107 (1974) K.K. Thornber, T.C. Mc Gill and M.A. Nicolet, Solid St. Electron. 17^, 587 (1974) K.M. Van Vliet, Solid St. Electron. 22, 233 (1979)

J. P. Nougier, J.C. Vaissiere and D. Gasquet, Proc. 6^"^ Int. Conf. on Noise in

Physical Systems, Washington (USA), 6-10 April 1981.

P. Hesto, J.C. Vaissiere, D. Gasquet, R. Castagne and J. P. Nougier, to be published.

til C. R. Brewitt-Taylor , P.N. Robson and J.E. Sitch, Proc. 8 Europ. Microwave Conf.,

Paris, 1978, p. 415.

D. Sodini, D. Rigaud, Proc. 6^'^ Int. Conf. on Noise in Physical Systems,

Washington (USA), 6-10 April 1981. K.K. Thornber, Solid St. Electron. 17^, 95 (1973).

J. P. Nougier, D. Sodini, M. Rolland, D. Gasquet and G. Lecoy, Solid St. Electron.

2J_, 133 (1978).

C. Moglestue, Solid. St. El. Dev. 2. '33 (1979)

J. Zimmermann, R. Fauquembergue, M. Charef and E. Constant, Electron. Lett. 1

6

, 664 (1980).

B. Carnez, A. Cappy, A. Kaszynski, E. Constant and G. Salmer, J. Appl. Phys. 5_1_, 784 (1980).

P. Bonjour, R. Castagne, J. P. Pone, J. P. Courat, G. Bert, G. Nuzillat and

M. Peltier, IEEE Trans. El. Dev. ED 27 , 1019 (1980).

C. Moglestue, Computer simulated noise current in semiconducting material, to be

published.

R. Fauquembergue, J. Zimmermann, A. Kaszynski and E. Constant, J. Appl. Phys. 5

1

, 1065 (1980).

J. Zimmermann, These de Doctorat es Sciences, Universite des Sciences et Techniques de Lille, 59655 Villeneuve d'Ascq Cedex-France (available on request) ( 1 980)

.

405
NOISE OP HOT HOLES IN Ge DUE TO PREDOMINANT

INELASTIC SCATTERING

V. Bareikis, A. Galdikas, R.^ MiliuSyt^ , V,. Viktoraviiius

Institute of Semiconductor Physics of the Academy

of Sciences of the Lithuanian SSR,^ K, Pozhelos 52,

232600 VUnius. USSR

INTRODUCTION

It is well established that the spectral density of current noise in isotropic

semiconductors is anisotropic in respect to the direction of high electric field E [l,2}.

Two limiting cases in the study of hot current carrier fluctuations are of

particular interest. The first one, when the scattering is quasielastic and the second

one, when the scattering is strongly inelastic^

Por the quasielastic scattering theory predicts [l] that along the electric

field direction in adition to the thermal noise/ the convective noise due to energy

fluctuations exists, Por this reason in the case of sublinear current-voltage dependence transverse (

J- ) spectral density of current noise exceeds the longitudinal

( II ) one (s(co )^ > S(co )„ ). The theory for highly inelastic scattering by optical phonons gives that

S(co )|| has peaks at frequencies equal to the multiple reciprocal time necessary

to accelerate the current carriers from energy & = O to the energy of optical

phonon 6q [4] .. S(co for such scattering mechanism is not investigated yet^

RESULTS

So far, the available experimental investigations on hot current carrier noise

[2] and closely related phenomenon-diffusion ^3] have confirmed the main conclusions of the theory for quasielastic scattering mechanisms,'

The purpose of this work is to study experimentally the anisotropy of

fluctuations for predominant inelastic scattering,. Por this the noise temperature T^

and small signal conductivity 6" (cO ) were measured in microwave range (9,8 GHz)

in p-type Ge with concentration of holes '^300K ~ 8,.7.10^'^ cm"'' at the lattice

temperature lO K,. At low temperatures in Ge in high electric field the inelastic

scattering of holes by optical phonons becomes predominant [5] .< The technique

used for Tj^ and Q (co) measurement have been described elsewhere [6] The

electric field was applied along <110> direction. Noise in transverse <110> and

longitudinal direction was investigated,' ^ The data of T^ , d.c,» conductivity 0" , differential conductivity (3 and

G" (co ) for longitudinal and transverse direction are plotted in Pig,» 1,» It was found

that at low temperatures as weU as at BO K and 300 K Tj.,,, > T^j^ but anisotropy of Tf-j at 10 K is stronger.

At electric fields E < ISO V/cm the difference between <S and G as

well as of <3 ('*')|| and G is observed,. This effect is caused by the inertia

of the heating process of holes,-

along oC direction was determined from T^^ and CJ" (<^)oC measurements using the relation

S(w )^ = 4kT^^ G i^U y where k - is the Boltzmann constant.-

S(,co )^ are plotted as a functions E in Pig. 2a, The results indicate that

in the electric field region up to ~ 140 V/cm s(co)|, > S(co , Consequently, the

sign of anisotropy of S(co )^ is opposite to that obtained for quasielastic

scattering mechanisms. At higher electric fields (E > 140 V/cm) s(co )|| < s(co

406
Pig. 2, s(to )^ vs E. (a) experimental, (b) calculated.. 1 - longitudinal; 2 - transverse,

DISCUSSION

Calculations on S(co )^ were carried out by Monte Carlo technique for the

simple model of p-Ge including only isotropic and parabolic heavy hole band,

acoustic and nonpolar optical scattering.. Calculated S(co vs E are plotted in

Pig. 2b. Pig». 3 reports the dependence of s(oj )^ on the frequency as well as

autocorrelation function of hole velocity ijr (t )^ vs time^ It is seen that at certain

resonant frequency s(co )„ has a maximum,. This frequency corresponds to the

reciprocal time necessary for a hole with fi = O to get the energy equal to that

407
0.1 1 10 100 0 0.6 1.2 1.8 f,GHz TlO^,s

Fifi^ 3'' Calculated s(co )^ vs frequence Fig. 4. Calculated autocorrelation

1,3 - transverse, 2,4 - longitudinal; function ^ (<t: )^ vs time^ 1 - longi1,2 - E = 50 V/cm; 3,4 - E = 200 V/cm; tudinal; 2 - transverse,

1 - longitudinal, 2 - transverse.

of optical phonon.^ In the range of high frequencies at the vicinity of this maximum

s(co )|| > s{co )j_ . This effect suggests the existance of the excess noise caused

by the hole transit resonance due to predominant inelastic scattering by optical

phonons. The resonant character of s( co ) „ and oscillations of ("T shows

the regular motion of holes in the longitudinal direction^ Wliile increasing E the

maximum of S(co ),| shifts to the higher frequencies.' Por this reason anisotropy

of S(oo depends on the frequency used in measurement.* Por the fixed frequency

(9,8 GHz) for E = 50 V/cm s(co )„ > S(<^ )j_ while for E = 200 V/cm S(co \

< s(co )^ These results agree with experimental data.< REFERENCES

1 J,P,. Price, in Fluctuation Phenomena in Solids, Ed.' R.E.. Burgess (Academic

Press, New York, 1965), p.> 355; SA/.' G-antsevich, V.Iv. Gurevich and R.' Katilius,

Rivista del Nouvo Cimento 2, 1 (1979).

2 V. Bareikis, Jf Pozhela, I. Matulionene, in Proc.^ International Conference on the

Physics in Semiconductors, Moscow (l968), p. 760; J.P. Nougier and M,» Rolland,

Phys. Rev.. B 8, 5728 (l973).

3 C Canali, G. Gavioli, P. Nava, Gr Ottaviani, L.- Reggiani, in Proc.. International

Conherence on the Physics in Semiconductors, Rome (1976), p. 1231.

4 I,B. Levinson, A.J. Matulis, Zurn. Exp.. Teor.. Piz. _54, 1466 (l968)..

5 T. Kurosawa and H, Maeda, Journ.. of the Phys. See. of Japan, 31 , 668 (l97l).

6 V. Bareikis, R. ^altis, J.. Pozhela, Lietuvos fizikos rinkinys, 6^, 99 (1966);

J. Vindervoghel, ,y,. Leroy, C.. Bruneel and J. Zimmermann, Rev." Sci.' Instr.* 45 . 920 (1974).

408
HOT CARRIER TRANSPORT NOISE IN p-TYPE SILICON

G. Bosman, R.J.J. Zijlstra and A. van Rheenen

Fysisch Laboratorium, Rijksuniversiteit Utrecht, The Netherlands

ABSTRACT

The electric field dependence of the longitudinal diffusion coefficient D, the Hooge

parameter a and the mobility \i were determined from thermal noise, 1/f noise^ and d.c. current

measurements respectively on p^ir p'^ planar silicon devices. Generation-recombination noise

measurements revealed the presence and the properties of four different deep-level impurities.

1. INTRODUCTION

In this paper we report on the electric transport noise of p'^-n p* planar silicon devices

operating at low bias voltages in the ohmic regime and at sufficiently high bias voltages in

a regime where space charge injection and carrier heating occur. Three types of noise could be

distinguished in the measured voltage noise spectra (1 < f < 10^ Hz):

(i) 1/f noise at low frequencies;

(ii) frequency independent diffusion noise caused by charge carrier velocity fluctuations at high frequencies (f > 10^ Hz);

(iii) generation-recombination (g-r) noise caused by the trapping of charge carriers by deeplevel impurities at intermediate frequencies.

From diffusion noise measurements versus bias voltage, we calculated the longitudinal diffusion

coefficient D of holes as a function of the electric field strength. The field dependence of

the Hooge parameter a was derived from 1/f noise measurements versus bias voltage and the

field dependence of the mobility followed from d.c. current measurements versus bias voltage.

From g-r noise measurements (in the ohmic regime) versus temperature we were able to determine

the properties of four deep-level impurities.

2. THEORY '

2.1. THERMAL AND 1/f NOISE

Neglecting the displacement- and the dif fusion-current^ Zij Istra derived the following ex- pression for the spectral intensity of the a.c. open-circuited voltage fluctuations of a planar

semiconductor device operating in a regime where space charge injection and carrier heating

occur [ 1

]

, , f\ D(E)(E - E)2dE EqA r\ a(E)(E - E)2dE

Iq^

i

(1 - qAp^Ey(E)/lQ)3 £1^2 (, _ qAp^Eu (E) /Iq) = (1)

In eq (1) p^ = density of ionized acceptors, I = current, L = contact spacing. El = electric

field at the collecting contact at x = L, A = cross-sectional area, e = permittivity of the

material, -q = electron charge. Steady-state values are denoted by subscript 0. Note that D, y and a may depend on the electric field strength. In addition, Zijlstra derived

for the current-voltage characteristic [1]

E2y(E)dE

(1 - qAp^EyCE)/!^)

and

0

Ey(E)dE

(1 - qAp^EyCE)/!^) (2)

409
In eqs (1) and (2) the following sign convention holds: IqjVqjE > 0. On the assumption that

a, y and D do not change appreciably within a small electric field interval AE, eqs (1) and

(2) were used to calculate a, y and D as a function of E, with the help of a numerical decomposition procedure [2].

2.2. GENERATION-RECOMBINATION (g-r) NOISE

We consider a semiconductor at thermal equilibrium with s discrete energy levels i at energies e^; where i = 1 refers to the bottom of the conduction band at Ej =0 eV, i = 2 refers to the top of the valence band at ^2 ~ ^gJ ~^g the band gap energy. The number of

available states at level i which are occupied by one electron is N^, whereas the number of

vacant states is P^, In thermal equilibrium N^ and follow from Fermi-Dirac statistics. The

rate of the electron transitions from level i to level j is given by: P;[j = a^jN^P-, where

a^j = transition parameter. Note that a£2'vol = Vp5£, which is the hole capture coefficient

of the impurities at level i. In this expression vol = sample volume, Vp = average thermal

velocity of the holes and 6^ = hole capture cross section of the impurities at level i. Due to electron transitions between levels the number of electrons occupying a level becomes

time dependent. The rate equation for N£ becomes

J-th = jj Pli - Pil

where p^^ =0. In our g-r noise measurements we consider a time scale that is large with

respect to the dielectric relaxation time. Then space charge fluctuations may be neglected

and consequently it holds that

s^ AN. = 0. This indicates that in the case of a system with s energy levels we are dealing

i=l

^ with (s - 1) independent variables N-j^. We linearize eq (3) with respect to these independent

variables and obtain

5- AN. = - y M. . AN. (4) dt L >j ij J

r ^Pil ^^li 0 where M^j =

^ (—^— - — —) is an element of the so-called relaxation matrix M. The super1=1 j j script 0 denotes thermal equilibrium values. Assuming that the electron transitions occur at random and are uncorrelated Klaassen [3] and van Vliet and Fassett [4] found the following ex- pression for the spectral cross intensity

1TM k=ll=l /, ^ c \2\ (1 + (27rfT^)'')

where the tj^'s are the reciprocal eigenvalues of M and C^j's are elements of matrix C which

transforms M into a diagonal matrix M' with elements M[j = Ti~^6£j: M' = CMC~'^

.

Note that S^j^^^j^ (f) consists of a sum of (s - I) Lorentzian-shapea spectra, each of which is

characterized by a time x-^ and a low-frequency plateau level.

For the spectral intensity of the a.c. open-circuited voltage fluctuations in the ohmic

regime we obtain with the help of eq (5)

hv^'^l\' = ^^n^AN,AN/f) ' ^p^AN^AN^^^^ " ^Vp^AN.AN^^^^ ' V

where y^^ and y are the electron and hole mobility respectively.

410
Fig . 1 The low frequency g-r noise plateau

values divided by Vq^ versus 1000/T. The open

circles represent the measured values. The

solid curves show the results of the computer

calculation. The labels 3-6 refer to the impurity level mainly responsible for the maximum.

Fig. 2 The characteristic times of the different g-r noise components versus 1000/T. The

open circles represent the measured values.

The solid curves show the results of the com- puter calculation. The labels 3-6 refer to the

impurity level mainly responsible for the curve

.

3. MEASUREMENTS

We used p^TT p"*" (Boron-doped) planar silicon devices which were provided with plane

parallel contacts having a cross-sectional area of I.O'IO"^ m^. The contact spacing was 40 pm

and the room-temperature resistivity of the it region was 100 fim. The electric field could be

applied along the <100> crystallographic direction.

The experimental set-up permitted spectral noise analysis to be carried out at frequencies

between 2 and 10^ Hz and at temperatures between 78 and 300 K [5]. Thermal noise and Iq were measured as a function of Vq by Gisolf [5] at T = 100, 145 and 210 K under pulsed bias con- ditions. Using d.c. bias we measured 1/f noise and Iq as a function of Vq at T = 78 K [6].

The 1/f frequency dependence was observed for at least two decades of frequency, depending on

the voltage applied. The frequency exponent was -1 within the accuracy of our measurements

(10%). We checked that no excessive Joule heating occurred.

We found that in the ohmic regime the magnitude

of the 1/f noise was proportional to Vq^. However,

at bias voltages where carrier heating occurred substantial deviations from this proportionality became

apparent

.

Generation-recombination noise was measured as a function of temperature in the ohmic range. We observed that the number of g-r noise components, their

low frequency plateau level and their characteristic

time all varied with temperature. In figure 1 we have

plotted the values of the low frequency plateau levels

of the various g-r noise components divided by Vq^ versus 1000/T for 78 ^ T 1 300 K. The corresponding

characteristic times are shown in figure 2.

Fig. 3 The energy level positions of our model for p-type

silicon versus the temperature. The solid lines represent

the energy level positions. The dashed line represents the

position of the Fermi level. Levels 1 and 2 represent the

bottom of the conduction band and the top of the valence

band respectively. The levels labelled 3-7 are associated

with impurities.

411

T(K) »

-1.2 I .. I .... I .... I .... I .... I 100 150 200 250 300
The relative error in the experimental data is about 20%. We observe four maxima in the noise

(cf. figure 1) located at T = 270, 200, 113 and 91 K, respectively.

4. DISCUSSION AND INTERPRETATION

The solid lines in figure 1 and 2 show the results of computer calculations [9] based on

the theory given in section 2.2 and applied to the seven-level model depicted in figure 3. In this figure the energy positions of the different levels and the Fermi level are given as a function of temperature. Note that levels 3-6, which are responsible for the measured g-r

noise, are traversed by the Fermi level at different temperatures. In table 1 we have listed

the trap parameters which followed from the computer calculations and have compared them with

literature values. We assumed that the spin degeneracy of the impurity levels equals four.

Table 1. Calculated impurity parameters and values collected from the literature.

Level

Lit.

Energy position (eV)

Exp. Lit.

Hole capture coefficient (m^s ^) Exp. Lit.

Concentration (m ^) Lit.

3 Fe [11]

4 Pt [12]

5 Mn [13]

6 Cr [14]

eg+0.40

eg+0.325

,+0.

:+o. 16513

Eg+O

e +0

Eg+O

Ecr+O

.40 9.0-10-^5 5 .10-15

,32 2.3- 10"l'*(T/300)2 9.1-10-15 ^t; T=135 K

,17 1.9-10-15 2 -10-15

,11 6 .10-16 —

8.0-

4.9-

8.6-

6.7-

1617 1015

lOl^

From the thermal noise data we calculated y and D as a function of Eq. The mobility results

are in very good agreement with those obtained by Ottaviani et al. [7] , who used the Time of

Flight technique. The results for D are plotted in figure 4 for different temperatures. The curves at T = 100 and 145 K show a pronounced structure which we ascribe to the effect of

light and heavy holes. A more quantitative explanation of this structure is given elsewhere

[2] . In figure 5 we show y and a as a function of Eq at T = 78 K. We found that a. can be

described by a(EQ) = a(0)/(l + (E/E^i)^) , where E^i is the field where the drift velocity of

the light holes equals the sound velocity.

5. CONCLUSIONS

(1) It is clear from D(E) curves that light as well as heavy holes contribute to charge trans- port . (2) Light and heavy holes become hot when their respective drift velocities are equal to the

sound velocity.

(3) At T = 78 K oi(E) can be described for E i 105 V/m by a(E) = a(0)/(l + ECE^i)^), where

Ej,]^ = Vg/y]^(0). This indicates that there is a connection between 1/f noise apd the scattering

of light holes by acoustic phonons.

412
(4) Our experimental results for 1/f noise at T = 78 K are at variance with the expression

for hot carrier 1/f noise derived by Kleinpenning [10] S^y/vg^ = a(0)/fNQ.

(5) The g-r noise measurement technique is a powerful tool for the spectroscopic study of deeplevel impurities even in multi-level systems.

[1] R.J.J. Zijlstra, in Noise in Physical Systems, Ed. D. Wolf (Springer-Verlag, Berlin,

1978) Chapter II, p. 90.

[2] G. Bosman and R.J.J. Zijlstra, Phys. Lett. TJA,, 464 (1979).

[3] F.M. Klaassen, thesis, Free University of Amsterdam, The Netherlands (1961).

[4] K.M. van Vliet and J.R. Fassett, in Fluctuation Phenomena in Solids , Ed. R.E. Burgess

(Academic Press, New York, 1965).

[5] A. Gisolf and R.J.J. Zijlstra, J. Appl. Phys. 47, 2727 (1976).

[6] G. Bosman, R.J.J. Zijlstra and A. van Rheenen, Phys. Lett. 80A, 57 (1980).

[7] G. Ottaviani, L. Reggiani, C. Canali, F. Nava and A. Alberigi Quaranta, Phys. Rev. B 12 , 3318 (1975).

[8] P.J. Price, J. Appl. Phys. 31_, 949 (1960).

[9] G. Bosman, R.J.J. Zijlstra, to be published.

[10] T.G.M. Kleinpenning, Physica 103B , 340 (1981).

[11] C.B. Collins, R.O. Carlson, Phys. Rev. 108 , 1409 (1957).

[12] J. A. Pals, Solid-state Electron. \]_, 1139 (1974).

[13] A.O. Evwaraye, J. Appl. Phys. 48, 3813 (1977).

[14] H.H. Woodburg and G.W. Ludwig, Phys. Rev. jn_7, 102 (1960). /

REFERENCES

.1

413
MONTE CARLO CALCULATION OF HOT-ELECTRON NOISE IN SI AT 77 K

Lino Reggiani, Rossella Brunetti and Carlo Jacoboni

Gruppo Nazionale di Struttura della Materia, Istituto di Fisica

Universita di Modena, Via Canpi 213/A, 41100 Modena, Italy.

INTRODUCTION

This paper reports on a Monte Carlo calculation of the hot-electron noise temperature

associated with velocity fluctuations for the case of electrons in Si at 77 K. The main objective is to present a microscopic theory v^ch interprets three main aspects of transport

phenomena, namely: drift, diffusion and v\diite-noise. Furthemore, results will throw light on a recently controversial interpretation of noise-temperature measurements [1,2] about the

importance of the most relevant sources of noise (i.e. velocity fluctuations and generation

recombination processes)

.

THEORY AND RESULTS ^

The transport approach follows a Monte Carlo procedure of standard type [3,4] . The

parameters modelling the Si band

Table 1 - Physical parameters used in calculations.

Longitudinal effective mass

Transverse effective mass m^

m^ = 0.916

9.04 10

2.33 gr/cm'

0.190

cm/ sec

3

Sound velocity v

s Crystal density

p

Non-parabolic ity parameter 0( = 0.5 eV

Acoustic deformation potential E = 9.0 eV

-1

Type of Phonon Eq. Temp. Coup. Cons.

intervalley

scattering

mode

(K) g

( 10 eV/cm)

^1 (TA) 220 0.3

f

2 (LA) 550 2.0

^3 (TO) 685 2.0

^1 (TA) 140 0.5

^2 (LA) 215 0.8

^3 (LO) 720 11.0

structure and scattering mechanisms were obtained by fitting existing drift and diffusion data in a

wide range of field strengths and

temperatures [5] and are summarized in Table 1. The theoretical longitudinal

noise temperature T^ is obtained

by substituting computed values of

the longitudinal diffusion coefficient D^ and of the differential

longitudinal mobility ^u|^ , nume- rically interpolated from results

of drift velocity versus field,

through the formula [6]

Dg (E)

/U. (E)

(1)

e being the unit charge and K the B Boltzmann constant.

Fig. 1 reports the results at

77 K for the two directions <"100>

and <111> : (a) refers to the drift

velocity, (b) to the longitudinal

414
diffusion coefficient, (c) to the mean carrier energy and longitudinal

noise both measured in equivalent

temperature units. Drift and diffusion results are compared with available experimental data obtained with

two complementary techniques (time-

-of-flight and noise-conductivity)

and the agreement, v\*iich is found to

be within experimental and theoretical uncertainties, guarantees the reliability of the microscopic model.

Concerning the longitudinal noise temperature, values calculated

with an uncertainty of about 15% are reported up to about 15 kV/cm; in

fact the saturation tendency of the

drift velocity above about these field strengths, leading to a zero value of ^u'£ , sets a physical and a numerical limit to the application

of Eq.(l). From Fig. 1(c) it is seen

that: i) noise values systematically

exceed those of the mean energy; ii) a strong anisotropy shows up, with

values along ^100^ direction higher

than along (IH^ by about a factor

three. This latter fact is known to be due to the presence of interval ley

noise resulting from different drift

velocities in different valleys vihen the field is along a <^100^ direction

[5,7] . Fig. 1 Drift velocity (a), longitudinal diffusion Fig. 2 shows the comparison becoefficient (b) , mean energy and longitudinal noise tween theory and available experi-

(c) in Si as a function of field strength at 77 K. ments on the noise temperature.

Circles indicate time-of-flight data, triangles Data of Bareikis et al [8] obnoise-conductivity data, lines the results of Monte tained for 30Iicm samples at a fre- Carlo calculations. quency of 10 Ofe confirm the anisotropy effects, and are found to agree

with theory within about 40%. In

view of an experimental uncertainty

estimated around 20% [9] this result has to be considered acceptable. Thus, the above agreement supports that the noise measured in Ref.(8) is due to velocity fluctuations source. Data of Nougier et al [1] obtained for 15Acm and 6Jlcm samples in a frequency range

250 - 950 MHz disagree with present calculations; in particular, they are higher by more than a factor three with respect to the theory at the highest field strengths. Furthermore, by

decreasing the sample resistivity from 15 to 6 and 2Jlcm the noise temperature was found to increase systematically. Consequently other sources of noise, in addition to the velocity

fluctuations one, should be taken into account for the microscopic interpretation. A plausible source of additional noise V(*iich has been suggested [2] is generation recombination due

to traps, and a calculation perfomied in the 1-3 kV/cm region succeds in totally attributing the noise measured to this mechanism [2] . From present results it is found that in

the 1-3 kV/cm region and for 6A cm samples, the contribution to the noise measurements of

415

E (Vcm"M
Ref.(l) due to velocity fluctuations is

negligible only v*ien E is along a<lll>

direction, vMle v^en E is along a<lC)C^

direction the contribution is about 35%

at 1 kV/cm and reduces to about 2CP/o at

3 kV/cm.

CONCLUSIONS

Hot-electron noise due to velocity

fluctuations in Si at 77 K is calculated

on the basis of a reliable microscopic

model. Results show that for a perfect

crystal, hot-electron effects become evident at field strengths above about 50

V/cm. Furthermore, the noise temperature exhibits a strong anisotropy with respect to the field direction with values

along a <100)> direction higher by about a factor three than those along a^lll^

direction. Comparison with experiments

confirms theoretical expectations in the

most favourable case of Ref.(8) (i.e. higher resistivity samples, measurements

performed at 10 GHz). Thus present calculations provide useful information for a discussion

on the relative importance of other sources of noise, such as generation recombination [2] .

ACKNOWLEDO^IENTS

We are grateful to the Computer Center of the Modena University for providing computer facilities. One of the authors (L.R.) wishes to thank the European Research Office for providing a travel grant to the U.S.A..

REFERENCES ' '

[l] J. P. Nougier, D. Sodini, M. Rolland, D. Gasquet and G. Lecoy, Solid-St. Electron., 21,

133 (1978).

[2] A. van der Ziel, R. Jindal, S.K. Kim, H. Park and J. P. Nougier, Solid-St. Electron., 22,

177 (1979).

[3] C. Canali, C. Jacoboni, F. Nava, G. Ottaviani and A. Alberigi-Quaranta, Phys. Rev. B12 , 2265 (1975).

[4] C. Canali, C. Jacoboni, G. Ottaviani and A. Alberigi-Quaranta, Appl. Phys. Lett., 27,

278 (1975).

[5] R. Brunetti, C. Jacoboni, F. Nava, L. Reggiani, G. Bosman and R.J.J. Zijlstra, to be

published.

i6| J. P. Nougier in "Physics of Nonlinear Transport in Semiconductors", Eds. D.K. Ferry, J.

R. Barker and C. Jacoboni: Plenum Press (New York, 1980).

[7] P.J. Price, J. Appl. Phys., 31, 124 (1960).

[8] V. Bareikis, V. Viktoravichyus , A. Gal'dikas and R. Milyushite, Sov. Phys. Sol. State

20, 85 (1978).

[9] V. Bareikis, V. Viktoravichyus, A. Gal'dikas and R. Milyushite, Sov. Phys. Semicon. , 14,

847 (1980).

10 10^ 10^ lO"

E (V cm"')

Fig. 2 Comparison between theoretical and

experimental longitudinal noise temperature

of electrons in Si at 77 K. Triangles (a

and v refer to 15 and 6ii. cm sample resi_s

tivity, respectively) indicate data of Ref

.

(1). Circles indicate data of Ref. (2).

416
NBS-lUA IREV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1, PUBLICATION OR

REPORT NO. NBS SP 614

2. Performing Organ. Report No. 3. Publication Date

ocpLtjiiiDcr ^ o n*!" Qm 1^ 1 QQT 1 yo 1

4. TITLE AND SUBTITLE

Sixth International Conference on Noise in Physical Systems

Proceedings of a conference held at the National Bureau of

Standards, Gai thersburg, MD, April 6-10, 1981

5. /^uTiji^R(^)^ Meijer, R. D. Mountain, and R. J. Soulen, Jr. , Editors

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D C 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

The Catholic University of America National Science Foundation

Washington, DC 20064 Washington, DC 20550

no. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 81-600084

1^21 Document describes a computer program; SF-185, FlPS Software Summary, is attached. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

This document contains the full text of papers submitted to the Sixth International

Conference on Noise in Physical Systems. There are six categories of papers:

THEORY; DEVICES; 1/f NOISE; APPLICATIONS AND MEASUREMENT TECHNIQUES; QUANTUM NOISE;

HOT CARRIER NOISE.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Hot carrier noise; noise; noise in devices; noise measurement; quantum noise;

1/f noise

|3. AVAILABILITY

Unlimited

I I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington. D C ^ 20402. Q Order From National Technical Information Service (NTIS), Springfield. VA. 22161

14. NO. OF

PRINTED PAGES

419

15. Price

$8.50

USCOMM-DC 6043-P80

<tU.S. GOVERNMENT PRINTING OFFICEi 19 8 1-340-997/1780





NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to time are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to all recent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription: domestic

$13; foreign $16.25. Single copy, $3 domestic; $3.75 foreign. NOTE: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical

Sciences." DIMENSIONS/NBS—This monthly magazine is published to in- form scientists, engineers, business and industry leaders, teachers, students, and consumers of the latest advances in science and

technology, with primary emphasis on work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution abatement,

health and safety, and consumer product performance. In addition, it reports the results of Bureau programs in measurement

standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and

automatic data processing. Annual subscription: domestic $11; foreign $13.75. NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE: The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's technological marketplace. Order the above NBS publications from: Superintendent of Documents, Government Printing Office. Washington, DC 20402. Order the following NBS publications—FIPS and NBSlR's—from

the National Technical Information Services. Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amende^.

Public Law 89-306 (79 Stat. 1127), and as implemented by F ,- ecuti ve Order I1717(38FRI2315, dated May 1 1

, 1 973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. D.C, 20234

OFFICIAL BUSINESS

Penalty for Private Use, S300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

COM-215

SPECIAL FOURTH-CLASS RATE

BOOK