• TABLE OF CONTENTS
HIDE
 Title Page
 Copyright
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Theoretical development
 Experimental procedures and...
 Discussion of results
 Conclusions
 Appendix I
 Appendix II
 Appendix III
 Bibliography
 Biographical sketch
 Permission form






Title: Noise in bipolar junction transistors at cryogenic temperatures
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Permanent Link: http://ufdc.ufl.edu/UF00084183/00001
 Material Information
Title: Noise in bipolar junction transistors at cryogenic temperatures
Physical Description: xvii, 237 leaves. : illus. ; 28 cm.
Language: English
Creator: Wade, Thomas Edward, 1943-
Publisher: Thomas Wade
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Junction transistors   ( lcsh )
Transistors -- Noise   ( lcsh )
Electronic noise   ( lcsh )
Low temperatures   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 231-236.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00084183
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 14089421
alephbibnum - 000580844
notis - ADA8949

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Table of Contents
    Title Page
        Page i
    Copyright
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
        Page xiv
    Abstract
        Page xv
        Page xvi
        Page xvii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Theoretical development
        Page 5
        Page 6
        Page 7
        Page 8
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    Experimental procedures and results
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    Discussion of results
        Page 205
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    Conclusions
        Page 209
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    Appendix I
        Page 211
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    Appendix II
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    Appendix III
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    Bibliography
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    Biographical sketch
        Page 237
        Page 238
        Page 239
    Permission form
        Permision form
Full Text











NOISE IN BIPOLAR JUNCTION TRANSISTORS
AT CRYOGENIC TEMPERATURES











By

THOMAS EDWARD WADE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA

1974


























Copyright, 1974


by



Thomas Edward Wade


1


























To


Ann















ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation

to Dr. E. R. Chenette for suggesting this research topic

and to both Dr. Chenette and Dr. A. van der Ziel for their

assistance throughout its development. The author wishes

also to express his gratitude to Dr. A. J. Brodersen,

Dr. S. S. Li, Dr. T. Hodgson, Dr. J. R. O'Malley, and

Dr. K. M. van Vliet for their assistance.

To his fellow colleagues and undergraduates of the

Noise Research Group, the author wishes to say simply

"thanks."

And lastly, the author wishes to thank his devoted

wife Ann who typed this dissertation and provided the

correct atmosphere and faith for attaining this goal.

This investigation was supported by the Advanced

Research Projects Agency, U. S. Department of Defense

(and monitored by the Air Force Cambridge Research

Laboratories), and also by the National Science Foundation.














TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS..................................... iv

LIST OF TABLES....................... ........... ..... vii

LIST OF FIGURES..................................... viii

ABSTRACT ............................................ xv

CHAPTER

I. INTRODUCTION.................................. 1

II. THEORETICAL DEVELOPMENT...................... 5

Introduction.............................. 5
Basic Transistor Theory ................... 7
Basic Noise Sources in Transistors........ 30
Electrical Noise Characterization...... 30
Theory of Noise in Transistors at
Room Temperature ....................... 33
Low Temperature Effects on Basic
Transistor Theory........................... 51
Low Temperature Effects on the Noise in
Transistors............................... 64
Generation-Recombination Noise at
Low Temperatures....................... 70
Noise Parameter Determination at Low
Temperature ............................. 76
Noise at High-Injection................. 79

III. EXPERIMENTAL PROCEDURE AND RESULTS........... 89

Introduction..................... ........ 89
Variable Temperature Apparatus ......... 92
Device Parameter Determination........ 95
Equivalent Noise Resistance Measurements., 116
Determination of Equivalent Saturated
Diode Current ............................ 174

IV. DISCUSSION OF RESULTS.......................... 205

V. CONCLUSIONS ............... ................. . 209








Page


APPENDICES

I. DERIVATION OF EQUATION FOR MEASUREMENT OF
EQUIVALENT NOISE RESISTANCE REFERRED TO THE
INPUT FOR COMMON-BASE CONFIGURATION.......... 211

II. DERIVATION OF EQUATION FOR MEASUREMENT OF THE
EQUIVALENT SATURATED DIODE CURRENT AT THE
OUTPUT OF THE COMMON-BASE CONFIGURATION...... 217

III. AUXILIARY EQUIPMENT............................ 221

Description of Low Noise Preamplifier..... 221
Description of Dual Monitoring Noise
System ..................................... 224

BIBLIOGRAPHY... ......... ........................... 231

BIOGRAPHICAL SKETCH................................. 237













LIST OF TABLES


Table Page


I. TYPICAL NUMERICAL RESULTS FOR DETERMINING
R NB. ....... ........................... .... ...... 162

II. IEQ COMPARISON FOR 2N4401 NO. 3................ 193

III. DETERMINATION OF C FOR 2N4403 NO. 1............. 199

IV. COMPARISON OF gsl FOR 2N4062 NO. 12............. 200


vii













LIST OF FIGURES


Figure Page


1. a) CURRENT FLOW DUE TO FOUR GROUPS OF HOLES.... 9
b) CURRENT FLOW DUE TO FOUR GROUPS OF ELECTRONS 9

2. a) CURRENT FLOW DUE TO "SAH-NOYCE-SHOCKLEY"
RECOMBINATION CURRENTS. ...................... 9
b) CURRENT CARRYING GROUPS RESULTING FROM
HIGH FREQUENCY AND/OR RECOMBINATION EFFECTS ..... 9

3. HIGH FREQUENCY EQUIVALENT OF SIMPLIFIED CIRCUITS
(A) COMMON-BASE CIRCUIT (B) COMMON-EMITTER
CIRCUIT........... ............................... 23

4. FULL EQUIVALENT CIRCUIT FOR (A) COMMON-BASE
CIRCUIT AND (B) COMMON-'i.lTTER CIRCUIT.......... 23

5. NOISE EQUIVALENT CIRCUIT FOR A BIPOLAR JUNCTION
TRANSISTOR....... .... ... ......................... 36

6. MODIFIED NOISE EQUIVALENT CIRCUIT FOR COMPARISON
BETWEEN THEORETICAL AND EXPERIMENTAL PERFORMANCE 37

7. NOISE MODELS USED TO CALCULATE EQUIVALENT NOISE
RESISTANCE REFERRED TO THE INPUT ............... 3

8. NOISE EQUIVALENT CIRCUIT INCLUDING EFFECTS OF
GENERATION-RECOMBINATION PROCESSES IN THE BASE
AT LOW TEMPERATURES..... ............. ......... 77

9. TRANSISTOR EFFECTS AT HIGH-INJECTION LEVELS..... 87

10. TYPICAL V-I CHARACTERISTICS AT 77 AND 3000 KELVIN 91

11. VARIABLE TEMPERATURE JIG FOR DEVICE UNDER TEST,
770K TO 3000K.................................. 93

12. HEATER DRIVE AND COMPARATOR CIRCUIT ............ 96

13. VARIABLE TEMPERATURE JIG FOR DEVICE UNDER TEST,
APPROXIMATELY 600K TO 770K........................ 97

14. BIAS MONITORING SCHEME FOR DEVICE UNDER TEST.,.. 99


viii








Figure


15. BLOCK DIAGRAM--LOW FREQUENCY BASE RESISTANCE
MEASUREMENTS............................... ....... 100

16. TEST CIRCUIT FOR MEASURING LOW FREQUENCY BASE
RESISTANCE................................. .... 101

17. BASE RESISTANCE AS A FUNCTION OF V APPLIED
FOR 2N5089 NO. 5............. ............ .... 105

18(a). A-C AND D-C ALPHA FOR 2N4062 NO. 12, T = 1800K
TO 3000K............... ......... ................ 106

18(b). A-C AND D-C ALPHA FOR 2N4062 NO. 12, T = 770K
TO 1800K ............. ..... ............ ......... 107

18(c). A-C AND D-C ALPHA FOR 2N5089 NO. 11, T 1400K
TO 3000K .................... ......... ... ...... 108

18'(d). A-C AND D-C ALPHA FOR 2N5089 NO. 11, T = 620K
TO 1400K..... ..... ..................... ......... 109

18(e). A-C AND D-C ALPHA FOR 2N4403 NO. 1, T = 1600K
TO 3000K ...... ............ ...... ..... .. ...... . 110

18(f). A-C AND D-C ALPHA FOR 2N4403 NO. 1, T = 770K
TO 1600K.......... ....... ........................ ll

18(g). A-C AND D-C ALPHA FOR 2N5087 NO. 7, T = 1600K
TO 3000K............... .. ......... .... ......... 112

18(h). A-C AND D-C ALPHA FOR 2N5087 NO. 7, T = 770K
TO 1400K ....... ............. .................... 113

18(i). A-C AND D-C ALPHA FOR 2N3711 NO. 7, T = 1200K
TO 3000K .............. .... ...... ......... .... 114

18(j). A-C AND D-C ALPHA FOR 2N3711 NO. 7, T = 77K
TO 1000K .............. .......................... 115

19(a). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR THE 2N4062, NO. 12, T = 770K
TO 1400K .................. ..................... 117

19(b). LOW FREQUENCY BASE RESISTANCE DATA AS MEASURED
BY THE LOW FREQUENCY BRIDGE TECHNIQUE FOR THE
2N4062 NO. 12, T = 160K TO 3000K .............. 118


Page








Figure


19(c). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N5089 NO. 11, T = 620K TO
1400K .......................... .... ............

19(d). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N5089 NO. 11, T = 1600K TO
3000K .. ................... .... ........... ......

19(e). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N3711 NO. 7, T = 770K
TO 1600K ........................................

19(f). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N3711 NO. 7, T = 1600K
TO 3000K ........................................


19(g). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N4403, NO. 1, T = 770K
to 1400K ................................ ...

19(h). LOW FREQUENCY BASE RESISTANCE DATA AT LOW
TEMPERATURES FOR 2N4403 NO. 1, T = 1600K
TO 3000K ........................... ........


.... 123



.... 124


20(a). NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N5087 NO. 7................................


20(b), NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N4062 NO. 12 ...........................

20(c). NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N5089 NO. 11 ..... ......................

20(d). NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N3711 NO. 7............................


.... 127


128


.... 129


20(e). NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N4403 NO. 1................................

20(f). NOISE RESISTANCE AS A FUNCTION OF FREQUENCY
FOR 2N4401 NO. 3................................


21.


TEST CIRCUIT FOR MEASURING RI VS. RS, WHITE
NOISE REGION... .. .............................


22(a). BASE RESISTANCE COMPARISON AS A FUNCTION OF
TEMPERATURE FOR 2N4062 NO, 12,...................


119



120



121



122


126


130


131


.... 132


134


1


Page








Figure


22(b). BASE RESISTANCE COMPARISON AS A FUNCTION OF
TEMPERATURE FOR 2N3711 NO. 7.................... 135

22(c). BASE RESISTANCE COMPARISON AS A FUNCTION OF
TEMPERATURE FOR 2N5087 NO. 7 ................... 136

23. INPUT IMPEDANCE OF A GROUNDED EMITTER TRANSISTOR
WITH COLLECTOR SHORT-CIRCUITED................. 137

24. CIRCUIT USED AND LOW TEMPERATURE ADAPTER FOR
DETERMINING BASE RESISTANCE BY THE V.H.F.
BRIDGE TECHNIQUE...................... ..... ...... 139

25(a). BASE RESISTANCE DATA AS DEDUCED FROM THE NOISE
DATA FOR THE 2N4062 NO. 12........... ...... 140

25(b). BASE RESISTANCE DATA FOR 2N3711 NO. 7 AS
DEDUCED FROM NOISE DATA.......................... 141

26(a). NOISE RESISTANCE AS A FUNCTION OF TEMPERATURE
WITH I AS A PARAMETER FOR (A) 2N5089 NO. 11
AND (B 2N3711 NO. 7............................ 142

26(b). NOISE RESISTANCE AS A FUNCTION OF TEMPERATURE
WITH RS AS A PARAMETER FOR 2N5087 NO. 7......... 143

26(c). NOISE RESISTANCE AS A FUNCTION OF TEMPERATURE
WITH IE AS A PARAMETER FOR 2N4062 NO. 12........ 144

27(a). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 60 MICROAMPERES.......... 146

27(b). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 100 MICROAPERES............ 147

27(c). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 200 MICROAMPERES ......... 148

27(d). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 350 MICROAMPERES......... 149

27(e). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 600 MICROAMPERES......... 150

27(f). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 1000 MICROAMPERES...,.... 151

27(g). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 770K, IE = 1500 MICROAMPERES........ 152


1


Page









Figure


27(h). NOISE RESISTANCE AS A FUNCTION OF R, 2N4062
NO. 12, T = 900K, IE = 60 MICROAMPERES..........

27(i). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO, 12, T = 90K, IE = 100 MICROAMPERES.........

27(j). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 900K, IE = 200 MICROAMPERES..........

27(k). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 900K, IE = 350 MICROAMPERES .........

27(1). NOISE RESISTANCE AS A FUNCTION OF R 2N4062
NO. 12, T = 90'K, IE = 600 MICROAMPERES .........

27(n). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062 NO.
12, T = 900K, IE = 1000 MICROAMPERES.............

27(n). NOISE RESISTANCE AS A FUNCTION OF RS, 2N4062
NO. 12, T = 900K, IE = 1500 MICROAMPERES........

27(o). NOISE RESISTANCE AS A FUNCTION OF RS WITH
IE AS A PARAMETER, 2N4062 NO. 12, T = 1000K.....


Rf' AS A FUNCTION OF IE FOR 2N4062 NO. 12, FOR
(A) T = 900K AND (B) T = 770K...................


29(a). EXCESS NOISE RESISTANCE AS A FUNCTION OF IB2rb'b
FOR 2N4062 NO. 12 AT T 770K...................


29(b). EXCESS NOISE
1 2 2
B rb 'b FOR
29(c). EXCESS NOISE
B2 b'b2 FOR
29(d). EXCESS NOISE
IB2rb'b2 FOR

29(e). EXCESS NOISE
IB2rb'b2 FOR
29(f). EXCESS NOISE
2 2
IB rbb2 FOR
29(g). EXCESS NOISE
IB2rb'b2 FOR


RESISTANCE AS A FUNCTION OF
2N4062 NO. 12 AT T = 900...........

RESISTANCE AS A FUNCTION OF
2N4062 NO. 12 AT T = 1000K .........

RESISTANCE AS A FUNCTION OF
2N5089 NO, 11 AT T = 620K...........


RESISTANCE
2N5089 NO.


AS A FUNCTION OF
11 AT T = 770K ..........


RESISTANCE AS A FUNCTION OF
2N5089 NO. 11 AT t = 90K...........

RESISTANCE AS A FUNCTION OF
2N5089 NO. 11 AT T = 1000K.........


153


154


155


156


157


158


.159


160





163


164


165


166


167


168


169


xii


28.


Page








Figure


29(h). EXCESS NOISE RESISTANCE AS A FUNCTION OF
Ig2b'b2 FOR 2N4403 NO. 1 AT T = 770K........... 170
30(a). EXCESS NOISE RESISTANCE AS A FUNCTION OF
TEMPERATURE FOR (A) 2N4062 NO. 12 AND
(B) 2N5097 NO. 7................. .......... ..... 171

30(b). EXCESS NOISE RESISTANCE AS A FUNCTION OF TEMPERA-
TURE FOR 2N5089 NO. 11 .......................... 172

30(c). EXCESS NOISE RESISTANCE AS A FUNCTION OF
TEMPERATURE FOR (A) 2N3711 NO. 7 AND
(B) 2N4403 NO. 1 ........ ....................... 173

31. TEST CIRCUIT FOR MEASURING IEQ 1/f NOISE
REGION ........ ............. ...... ............ 175

32(a), IEQ AS A FUNCTION OF FREQUENCY FOR 2N4062 NO. 12 176

32(b). IEQ AS A FUNCTION OF FREQUENCY FOR 2N5087 NO. 7. 177

32(c). IEQ AS A FUNCTION OF FREQUENCY FOR 2N5089 NO. 11 178

32(d). IEQ AS A FUNCTION OF FREQUENCY FOR 2N4401 NO. 3. 179

32(e). IEQ AS A FUNCTION OF FREQUENCY FOR 2N3711 NO. 7. 180

32(f). IEQ AS A FUNCTION OF FREQUENCY FOR 2N4403 NO, 1, 181

33. TEST CIRCUIT FOR MEASURING IEQ--WHITE NOISE
RE .C ON ...... .................................. 182

34. NOISE DIODE BIAS AND HEATER CONTROL CIRCUIT...... 184

35. CIRCUIT USED TO MEASURE f AS A FUNCTION OF
TEMPERATURE ..................................... 186

36(a). B AS A FUNCTION OF FREQUENCY FOR 2N5087
N7 7 ........................................... 187

36(b). GA-C AS A FUNCTION OF FREQUENCY FOR 2N5089
NO. 11............................................. 188

36(c). RA-C AS A FUNCTION OF FREQUENCY FOR 2N4403
NO 1............................................ 189

37. MORE PRECISE CIRCUIT FOR MEASURING f. AS A
FUNCTION OF TEMPERATURE .......................... 190


xiii


I


Page







Figure


ALPHA AS A FUNCTION OF FREQUENCY FOR 2N4401
NO. 3 ......................................

IE AS A FUNCTION OF IE FOR 2N4401 NO. 3 AT
FREQUENCY = 850k HZ ............ .... .......


40(a). I


40(b). IEQ
FOR

40(c). IEO
FOR

40(d). IQ


41(a). IEQ

41(b). IEQ

41(c). IEQ


VS. I WITH TEMPERATURE AS A PARAMETER
2N406 NO. 12............................... 195

VS. IE WITH TEMPERATURE AS A PARAMETER
2N5089 NO. 11............................... 196

VS. I WITH TEMPERATURE AS A PARAMETER
2N371 NO. 7...................................... 197

VS. IE WITH TEMPERATURE AS A PARAMETER
2N4403 NO. 1.......................... ........ 198

AS A FUNCTION OF TEMPERATURE, 2N4062 NO. 12. 202

AS A FUNCTION OF TEMPERATURE, 2N5087 NO. 7.. 203

AS A FUNCTION OF TEMPERATURE, 2N5089 NO. 11. 204


APP-1. EQUIVALENT NOISE RESISTANCE MEASUREMENT SET-UP..

APP-2. EQUIVALENT SATURATED DIODE CURRENT MEASUREMENT
SET-UP ..........................................

APP-3. LOW NOISE PREAMP SCHEMATIC......................

APP-4. RN VS. FREQUENCY FOR LOW NOISE PREAMPLIFIER.....

APP-5. BLOCK DIAGRAM OF DUAL MONITORING NOISE
MEASURING SYSTEM....................................

APP-6. MASTER CLOCK AND LOGIC SYSTEM FOR SWITCHING
NETWORKS....................................... ..

APP-7. SWITCHING SEQUENCE FOR DUAL MONITORING NOISE
SYSTEM.......................................... ..

APP-8. VIDEO SWITCH FOR CALIBRATION SIGNAL, 80DB OF
ATTENUATION.....................................

APP-9. VIDEO SWITCHES FOR NOISE AND GAIN MONITORING
SIGNALS.........................................


xiv


38.


39.


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Page














Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in
Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



NOISE IN BIPOLAR JUNCTION TRANSISTORS
AT CRYOGENIC TEMPERATURES


By




Thomas Edward Wade

August, 1974

Chairman: E. R. Chenette
Co-Chairman: A. van der Ziel
Major Department: Electrical Engineering

This investigation involved the characterization of

the noise performance of modern silicon planar bipolar

junction transistors over a temperature spectrum of 600

Kelvin to 3000 Kelvin in twelve discrete intervals and for

seven different bias conditions. All noise measurements

were performed above the l/f noise frequency region.

Several conditions exist in bipolar junction transistors

at cryogenic temperatures which allow for accurate testing

of conventional theories used to characterize these devices

at room temperature. These include a drastic decrease in

the d-c and a-c current amplification factors; increased

effects of carrier recombination and trapping in the








emitter-base space-charge-region; a large tendency towards

high level injection operation (especially for room

temperature high gain devices); and a marked decrease in

cutoff frequency (at least for double diffused planar

devices).

For temperatures slightly below those of room

temperature, excellent agreement is obtained by the conven-

tional theories. However, at lower temperatures, a marked

increase in recombination and trapping of carriers in the

emitter-base space-charge-region is observed. This recom-

bination current component should exhibit less than full

shot noise since it represents a two-step process (i.e.,

hole capture followed by an electron capture). A theoretical

development including these effects is presented and is

verified at least partially by experimental results.

The conventional noise theory for transistors was

developed assuming low-level injection conditions. A

theoretical development is presented which demonstrates

that at least a major portion of this theory is also valid

under high level injection conditions. The accuracy of this

prediction is verified by experimental results.

At temperatures below approximately 1100 Kelvin, an

excess noise source as measured by the equivalent noise

resistance referred to the input of the device, common-base

configuration, is revealed. It is projected that this

source results from generation-recombination processes


xvi









within the base region of the device. A theoretical develop-

ment is presented which characterizes the dependence of this

noise emf on the base current-base resistance product

squared. Experimental results verify this projection.


xvii


i













CHAPTER I


INTRODUCTION


In all electronic circuits there is a lower limit to

the signals that can be processed because of the spontaneous

currents and voltages developed in the components. These

spurious, unwanted, but always present, signals give rise

to what is called "electronic noise," or simply "noise."

As a result, very small desired signals may thus be masked

or drowned in the noise background of the electronic equipment

used to process them.

By means of very accurate measurement and device modeling

techniques, the sources of noise for a given electronic

component can be fully characterized and hence predicted.

Also, since the noise of a device (or component) is the

direct result of physical processes or mechanisms inherent

in the device (or component), accurate noise characterizations

may be used to completely define a given device (or component).

One of the most commonly used electron devices today is

the bipolar junction transistor (B.J.T.). The noise of this

device, at room temperature and at frequencies where 1/f noise

is small, has been completely characterized.1-5 Its behavior

is determined via "shot noise" generators for each of its two

junctions which are also partially correlated, plus a thermal

noise generator located in the base region of the device.6

1








At ambient temperatures well below that of room temperature

however, the noise characteristics of bipolar junction

transistors have been shown to deviate considerably from

those at room temperature.7,8 It should be noted that very

little information is available in the literature concerning

B.J.T.'s at cryogenic temperatures. Only recently has a

thorough study of silicon B.J.T.'s at liquid nitrogen

temperature (770 Kelvin) been undertaken.9 The primary

results of these studies indicate that additional noise

sources within the B.J.T. model must be considered. These

additional sources at low temperature result primarily

from the effects of fluctuations in the number of partially

ionized impurities, a "generation-recombination" process,

which has its predominant effect in the vicinity of the

base region of the device.

In this study, an attempt to extend the noise theory

of silicon npn and pnp devices to include effects at cryogenic

temperatures (ranging from approximately 600K to 300K) is

undertaken. The common-base physical-T equivalent model

for B.J.T. is used almost exclusively. For this arrangement,

measurement of the equivalent noise resistance, RN, referred

to the input as a function of source resistance at cryogenic

temperatures reveals an excess noise emf. It is proposed

that this noise source results directly from generation-

recombination processes in the base region of the device.

Its mean-square value has a functional dependence given by

KIBrb, b where K is a proportionality constant, I is the







d-c base current and rb'b is the extrinsic base resistance

as seen by the noise sources. Experimental results are given

which confirm this proposal.

Measurement of the equivalent saturated diode current,

IEQ, at the output of the device with the input a-c open

circuited indicates that for some devices at low emitter

currents, there exists a reduction in IEQ due to the trapping

and recombination effects in the emitter-base space-charge-

region of the device. This effect becomes more predominant

with decreasing temperature.

The chapters which follow consist of first a summary

of the theory for noise in B.J.T.'s via the corpuscular

approach at room temperature. This is followed by a review

and an investigation of processes which take place in

semiconductors and B.J.T.'s at cryogenic temperatures. With

this information, theoretical derivations for the noise

sources at cryogenic temperatures are undertaken. A proposed

procedure for experimentally verifying these sources is

also given.

Chapter III presents the experimental methods and

results for characterizing the transistor small signal

parameters and noise sources. Chapter IV consists of a

discussion of results, comparing the experimental results

obtained with the projected theoretical results. In Chapter

V, a brief conclusion is given followed by several suggested

areas for further investigation. Theoretical derivations

for RN and IEQ in terms of measurable parameters are given




4


in the appendices, along with a description of some of the

more important auxiliary equipment used in Chapter III.













CHAPTER II


THEORETICAL DEVELOPMENT


Introduction

This chapter is divided into four separate sections.

each dealing with a different but inter-related aspect

of the theoretical development. The first section deals

with the basic transistor theory as defined by van der Ziel's

corpuscularr approach." The purpose of this section is

as follows: to introduce the various current carrying

groups which constitute the d-c terminal currents; to

define the basic parameters used in both the common-base

and the common-emitter equivalent circuits and the relation-

ship between these parameters; and to derive, via the corpus-

cular approach, equivalent circuits for the common-base and

the common-emitter configurations for small signal low-level

injection applications including high frequency effects.

The parameter definitions and their various relationships

along with the equivalent circuits will be used exclusively

throughout the remainder of this thesis.

The second section is directed toward the development

of the basic noise sources used to characterize the noise

in the bipolar junction transistor at room temperature.








The various current carrying groups defined in the previous

section are used to derive these sources. Elementary noise

definitions and parameters are defined and the noise equivalent

circuit for the common-base physical-T configuration is

derived. Methods for experimentally determining the various

noise sources are also given.

The third section deals with low temperature effects in

the basic transistor theory. Since the temperature dependence

of the various transistor parameters (as saturation currents,

amplification factors, etc.) are directly related to the

temperature dependence of the more fundamental semiconductor

properties (i.e., mobility, conductivity, recombination

mechanism, etc.), a brief review of the temperature dependence

of these properties is also included.

The last section is devoted to the low temperature

effects on the noise of transistors. Decreasing temperature

thas a direct effect on the flicker noise as well as the white

noise sources in the device. At rather low temperatures

(i.e., T < 1100K) an additional noise source due to generation-

recombination processes in the base region of the device must

be included. Also, the effects of recombination and trapping

in the emitter-base space-charge-region as well as the effects

of high level injection must be considered for temperatures

somewhat less than room temperature. The theoretical develop-

ments for these effects are given. Again, experimental

methods for determining the various noise sources of the device,

but at low temperatures, are also given.











Basic Transistor Theory


A bipolar junction transistor, hereafter referred to

as a B.J.T. or simply as a "transistor," can be considered

as a combination of two semiconductor junctions with a

common-base region. In this development, we shall discuss

the pnp transistor; the operation of the npn transistor is

quite analogous. Transistor action for the "forward-active-

region" of operation consists of carriers being emitted by

the emitter into the base, due to a forward bias emitter-

base junction, and collected by the collector, resulting

from a reverse biased collector-base junction. Also, since

most of the carriers injected by the forward biased, low

impedance level emitter junction are collected by the reverse

biased, high impedance level collector junction, the

resulting output power is much larger than the input power

and the device is said to exhibit amplification. Thus for a

pr overly biased common-base configuration (base common to

input and output), a small a-c signal (voltage) applied at

the emitter terminal will result in an amplified signal

(voltage) at the collector terminal. Likewise, for a properly

biased common-emitter configuration, a small a-c signal

(voltage and/or current) applied at the base terminal will

result in an amplified signal (voltage and/or current) at the

collector terminal.







To develop expressions for the transistor's terminal

currents in terms of its terminal voltages and device

parameters, the corpuscular approach, as presented by

van der Ziel,10 will be utilized in conjunction with those

parameters (and sign convention) as defined in Figure 1.

It is first assumed that practically all current is carried

by holes. Four groups of holes must be considered as

depicted in Figure l(a) as follows:

i) Holes injected by the emitter and partly

collected by the collector. This yields a

contribution IESexp(qVEB/kT) to the emitter

current IE and a contribution -afIESexp(qVEB/kT)

to the collector current, where

IES=the injected hole current for zero emitter bias

ca forward current amplification factor

ii) Holes injected by the collector and partly

collected by the emitter. This group yields a

contribution -alcYCSexp(qVCB/kT) to IE and a contri-

bution Icsexp(qVCB/kT) to IC, where

ICS=injected hole current for zero collector bias

ior=reverse current amplification factor

iii) Holes generated in the base and collected

by the emitter. This gives a contribution -IBE

to IE and is independent of bias.

iv) Holes generated in the base and collected by

the collector. This gives a contribution -IBC to

IC and is also independent of bias.



















VEw.e 4 a c VCB

B


FIGURE I.a)CURRENT FLOW DUE TO FOUR GROUPS OF HOLES

b)CURRENT FLOW DUE TO FOUR GROUPS OF ELECTRONS


IE


IE
-1_-


XJc


Ic
'( -*--


(b)
FIGURE 2. a)CURRENT FLOW DUE TO "SAH-NOYCE-SHOCKLEY"
RECOIMBINATION CURRENTS
b)CURRENT CARRYING GROUPS RESULTING FROM HIGH
FREQUENCY AND/OR RECOMBINATION EFFECTS


. _


q R


<09 -= =


@_ o'







Adding all these contributions, one obtains

E = IESexp(qVEB/kT)-arlcsexp(qVCB/kT)-IBE
(1)
IC = -aflESexp(qVEB/kT) + ICSexp(qVCB/kT)-IBC
The terms IBE and IBC may be eliminated by observing that

E = IC = 0 when VEB = CB = 0. Hence,


IBE = IES-arlCS and IBC = ICS-afIES (2)

Substitution back into equation (1) yields


IE = IES[exp(qVEB/kT)-l]-arlCS[exp(qVCB/kT)-lI
(3)
IC = -aflEs[exp(qVEB/kT)-l]+ICS[exp(qVcB/kT)-I]


Evaluation of the parameters IES, ICS and af and ar will

be undertaken shortly.

If the assumption that all current is carried by

holes is now relaxed, four groups of electrons must also

be taken into account as per Figure l(b):

i) Electrons injected from the base into the

emitter. Their contribution to IE is dependent

on emitter bias as exp(qVEB/kT).

ii) Electrons injected from the base into the

collector. Their contribution to the collector

current is dependent on the collector bias as

exp(qVCB/kT).

iii) Electrons generated in the emitter region

and collected by the base. They contribute to

IE independent of bias.






iv) Electrons generated in the collector region

and collected by the base. They contribute to

IC independent of bias.


Adding the contributions of the four groups of holes

and of the four groups of electrons to IE and IC yields

equation (3) again, with a different meaning and different

values for the four parameters IE I CS af and a c For

example, IES is now the sum of the hole current and the

electron current injected across the emitter junction for

zero emitter bias. And af is the part of the emitter

current collected by the collector for zero collector

bias. Equation (3) is thus of general validity and

is known as Ebers-Moll equation.11

The four parameters I ICS, af and a are not
ES' CS' f r
independent, but they must satisfy the reciprocity condition


fS = a (4)
I ES r CS

Now, for normal operation of a pnp transistor, the

collector bias VCB is negative to such an extent that
CB
exp(qVCB/kT) is negligible and VEB is positive. Equations (3)

may then be written

IE = IES[exp(qVEB/kT)-l + (5)
IC = -_f Es[exp(qVEB/kT)-l] ICS
Eliminating IS[exp(qV EB/kT)-l] in the above, the second

equation becomes










IC = "-alE-CS(l1-.raf) = -a"fE-ICO (6a)


where


ICO = ICS(1-arof) (6b)

represents the collector current for zero emitter current

(emitter open circuited) and is called the "collector

saturated current." From equation (6a), it is seen that

the parameter af is more appropriately called the d'd-c

current amplification factor in common-base connection";

thus af = adc-

The base current may be obtained via Kirchhoff's

current law and equation (6) as


IB -IE-IC = -( f)IE+ICo (7)

and from this expression, IE and IC may be found in terms

of IB as follows

IB CO
IE ---- + + (8)

-a 1-af

and via equation (6)


a I I
f CO CO
I I co I (9)
C B -- (9)
i-f I-af l-af








The factor


f= = dc (10a)
1-
f

Is called the "d-c forward current amplification factor

in the common-emitter connection." Hence, a small fluctuation

in the base current AIB will yield a corresponding fluctuation

AIC in the collector current such that


AIC = 3fAIB (10b)

Since af is close to unity in a good transistor, Bf is

large resulting in a large current amplification.

The parameters of equation (3) may be calculated for

a planar geometry in the following manner. First, to

calculate IES and -.f, we assume that almost all current

is carried by holes and that VCB = 0. Let the spatial

parameter x be equal to zero at the emitter side of the

base region and x = W at the collector side of the base

region, W being the width of the base region. Thus,


p(0) = Pnexp(qVEB/kT) and p(W) = pn (11)

where pn is the equilibrium hole concentration in the

n-type base region. In most modern transistors, very

little recombination in the base region occurs; thus

the hole gradient is practically constant and is approximately

given by -Ip(0)O)Pn/W. Now, if A is the junction area,





14


the hole emitter current IEp is


IEp


so that


IES


ap
= -qDpA

Sx=
x=0


qDpA[p(O)-p ]


qDpApn
S p exp(qVEB/kT)-1]

W





qDoApn
- P


(12)


(13)


To calculate af, the base hole current IBp must be
1 Bp


determined.


The excess hole charge QBp stored in the base
~bp


region is


QBp = qAW[p(0)-P]/2

and hence the base hole current is


qAW[p(O)-pn]


IBp


QBp
--QI
T
p


2Tp


S2

2 -Ep
L 2


where T is the hole lifetime and L = /D T is the hole

diffusion length. Consequently, the collector hole current


ICp is
Cp


(14)


(15)









ICp = -(IEp+ Bp) = -.(-W2/2Lp)IEp = -a IEp

since VCB = 0. Hence


af = 1 W/2L 2 (16)
p

From equations (13) and (16), for W small, IES is large and

af is close to unity if W << L p

For the determination of ICS and ar, set VEB = 0 and

repeat the calculation. For a planar geometry in which all

current is carried by holes, the problem is a symmetrical

one, and hence one obtains ICS = IES and a = In actual

transistors the junctions can have different areas and N
D
in the base may be nonuniform; hence asymmetry can result.

Now if the assumption that all current is carried by

holes is relaxed, the additional emitter current carried by

electrons will affect IES and a .f Putting VCB = 0, the

electrons' contribution to the emitter current is


I = (qD An /L ) [exp(qV E/kT) 1] (17)
En n p ne EB

where L is the electron diffusion length of electrons in the
ne
emitter region and the expression for IE remains the same.

Thus instead of equation (13),

qD Ap qD An qD Ap DnW
I p n + n n (1 +- n p (18)
ES W L W DpL
ne pn ne

Since only the holes are collected by the collector,

qD Ap
S =D p An (1 1 W ) (19)
af ES
W 2 L 2
p







according to equations (13) and (16). Hence,

1 W2/2L 2 D n W
acf = P -2 1 I W2 n p (20)
1+(D n W)/(D pL ne) 2 L 2 D L
np ne 2 DpPnLne

if (Dnn W)/(D pL e) << 1. To make "f close to unity, one

should make W << L as before, but in addition one should

choose W << Lne and n << p The latter can be achieved

by making the emitter region very strongly p-type. Note

that for heavily doped emitter, p = nib2/N and n = n.ie/N

where n. is the intrinsic carrier concentration, Now

n ie2 = nib2exp(qAE /kT), where AEg (> 0) is the difference in
ie lb g g
gap width between the base and emitter regions; AE increases

for increasing emitter doping,

To calculate I and a again put V = 0. Since there
CS r EB
is no guarantee that the electron's diffusion length L
nc
in the collector region is equal to Lne ICS and a may

differ slightly from IES and af, respectively. The actual

symmetries found in transistors are usually caused by

deviations from the planar geometry or by a distribution in

ND.

Now, the differential (small signal low frequency) input

conductance for the common-base connection may be found via

equation (5) as

dIE 1 +Eg E
dE ql qlE
eb S + rS (21)
EB





17


if -IES + r I is small in comparison with I The
ES r CS
corresponding differential conductance of the ideal common-

emitter connection is via equation (7)

dl dlB dIE
S = = (1- f) = (1-af)g (22)
dV dV dV
dBE dEB dVEB

for VBE = -VEB. Thus since (1-.a) << 1, the common-emitter

circuit presents a much smaller input conductance (or larger

input resistance) than the common-base circuit.

The transconductance, gm, of the device is defined as

dIC dIE qIC
gm a = ag f (23)
SEB dVEB e kT

This same expression is true for common-base and common-

emitter since dIl/dVBE -dlc/dVE. Expressing eb andg
/d E B Ceb Thbe
in terms of gm yields


r o. = f, = g / f (24)
'eb n- m be -- m m f (2

In some cases when the transistor is biased in the

normal region (VE > 0, VC < 0), generation and recombination

of hole-electron pairs in the emitter space-charge-region (SCR)

become important. This gives a contribution to E that is

not collected by the collector, and exhibits a different

dependence on VEB. Equation (6) is still correct however, thus

-I -I
C CO
f d e (25)
I
E
so that af is current dependent.







If a small emitter current fluctuation AIE is produced,

the corresponding collector current fluctuation AIC is
C


IC
AI = AIE = -oAIE where ac
C Io Eo
E


"Ic

IE


(26)


and from equation (6)


IC 8f
a a + I
o I f E
E E
Therefore if af increases with increasing IE, ao > a

if af decreases with increasing IE, a < af.

The corresponding base current fluctuation AIB is


AIB = -(AIC + AE) = -(1 ao)AIE


Expressing AIC in terms of AIB yields


(27)


or


(28)


AIC = aoAlE =
C


0o
AIB = L oAlB
1-ao


B =
l-a,


Thus the low frequency a-c current amplification factor for

the common-emitter circuit is 8 = a /(l-a ) instead of

Bf = cf/(l-af). This has an influence upon gm geb and g .

It remains true that g = Ic / SVBE- but

AI AIE AI

SVEB A C AVEB
and


AI

VBE


AIB AIC
B CAVBE
n^ nrE


(30)


where


(29a)


(29b)


gm/ o








Sah, Noyce and Shockleyl2 have shown that the observed

voltage-current characteristic of silicon p-n junctions

can be explained for a wide range of currents by including

the effects of generation-recombination centers in the

junction space-charge-region.* This effect can explain the

observed decrease in emitter efficiency and hence lend a

physical interpretation as to the difference in a and adc
o dc
Their derived expression for recombination current is

given by


I = AT kTW ni exp(qVE) = IRSexp(qVE) (31)
2 (Vd-VE) To 2kT 2kT

where A is the cross-sectional area of the junction, W is

the width of the space-charge-region, Vd is the applied

voltage, ni is the density of electrons or holes in the

intrinsic specimen, TO is the mean lifetime of electrons and

holes, and IRS is the equivalent slightly voltage dependent

"saturation current." Hence, the "Sah-Noyce-Shockley"

currents resulting from trapping and recombination of

current carriers in the transition region may be given

by the formal equations

IRE = IRES[ex(qVEB/2kT) 1]
and

IRC = IRCS[exp(qVCB/2kT) 1] (32)

*The terms "space-charge-region" and "transition region" will
be used synonymously throughout this chapter as is customary
in the literature.








where IR and IRC are the recombination currents of the
RE RC
emitter and collection transition regions, respectively, and

IRES and IRCS are the corresponding "saturation currents."

In terms of the groups of current carriers as given

in Figure 1, at least three groups of "Sah-Noyce-Shockley"

currents must be considered as depicted in Figure 2 and

having the following interpretation:

i) Holes from the emitter trapped in the emitter-

base transition region and recombining with electrons

coming from the base.

ii) Hole-electron pairs thermally generated at

trapping centers in the emitter-base transition

region. The holes move into the emitter and the

electrons move into the base.

iii) Hole-electron pairs thermally generated at

trapping centers in the collector-base transition

region. The holes move into the collector and

the electrons into the base.

The current dependence of 3o (and hence a since

o = a /(l-a_) ) is explained by Searle et al.13 as follo-ws.

At low current levels, this parameter increases somewhat with

current because of nonlinear effects in the base. The reason

for the increase is that the tendency of added minority and

majority carriers to recombine is progressively reduced as

the total carrier concentrations involved are increased

from their equilibrium values. This occurs because reco-ibina-

tion in the base region takes place at crystal imperfections








or impurities located within the junction space-charge-

layers, and within the bulk or on the surface. As the

bias is applied, causing excess carrier concentration by

injection, these imperfections get progressively filled

up on the average. Thus the effective lifetime of "increment-

ally" added minority carriers in the base increases with

bias current level, causing a corresponding increase in B.

At high current levels, however, 50 decreases with

increasing current. The principal reason for this action

(and several are actually involved) stems directly from the

increase of majority-carrier concentration in the base

at high-injection levels. For a pnp transistor, the electron

concentration in the base is normally much less than the

hole concentration in the emitter--so that as much as possible

of the emitter current shall be comprised of holes injected

into the base (rather than electrons injected bacikard from

base into emitter). At high bias current levels, the

increased concentration of majority electrons in the base

results in the enhancement of 'back' carrier injection from

base to emitter, thereby reducing the ratio of the incremental

current injected in the base to the total incremental

emitter current. This effect also increases base current

rather than collector current, and causes (and hence a )

to fall with increasing IC. ad will follow the behavior

described by equation (27) in relation to a

Since recombination occurs in the base region, electrons

must flow from the base contact to places where recombination







occurs. This electron current flow is restricted to the

narrow base region and causes a voltage drop in the base

layer. The voltage drop gives rise to a "distributed

resistance" in the base region which, to a first approximation,

may be characterized as a single effective resistance,

rb'b, in series with the base lead.

The constant effort to decrease base width, W, to achieve

high Bo transistors (via equation (20) ) as well as to

improve high frequency operation (to be discussed momentarily)

poses an interesting problem. Other things being equal,

rb'b is inversely proportional to the donor concentration ND,

the base width W and the electron mobility pn


rbb = constant (33)
1nNDW

Thus to avoid excessive values of rbfb, a decrease in base

width W should be accompanied by a corresponding increase

in ND. The higher the cutoff frequency of the transistor,

the larger the base conductivity should be.

Discussion of the high frequency effects of the

transistor requires the establishment of an "equivalent

circuit" for both the common-base and the common-emitter

configurations (for comparison) using the parameters derived

thus far. This is given in Figure 3. At high frequencies,

the space-charge-regions of the two junctions have associated

capacitances Ce for the emitter and Cj for the collector
je je
which must be taken into account. In addition, a diffusion

capacitance Cd due to the holes stored in the base region















Veb


FIGURE 3.




-4Vl


Vab
li-ve


HIGH FREQUENCY EQUIVALENT OF SIMPLIFIED CIRCUITS
(A) COPIMON-BASE CIRCUIT (B) COMMON-EMITTER
CIRCUIT


FIGURE 4,


FULL EQUIVALENT CIRCUIT FOR (A) COMMON-BASE
CIRCUIT AND (B) COMMON-EMITTER CIRCUIT
(The "hybrid-w" equivalent parameters have been
superimposed on the common-emitter equivalent
circuit for future reference.)








must be accounted for,

According to equations (14) and (12), the total hole

charge QBp stored in the base region is

I I
QB Ep E (3
2Dp/W2 2Dp /W2

if IEn is neglected in comparison with IEp. Hence the small

signal storage capacitance Cd is

dQ 1 dl
C Bp E g 1-W2 (35)
d dVE 2Dp/ dEB eb 2 Dp
EB dVEB p
The average time needed for the holes to diffuse over the base

width W is Td = W2/2D p (For one-dimensional diffusion the

mean-square distance x2 traveled during the time t is

x2 = 2D t.)

For the common-base transistor circuit, if an a-c

emitter current ie flows into the emitter, then the a-c

voltage veb developed between emitter and base is

Veb = ie/[geb + ji(Cd + Cje)] (36)
and the output current generator is

gmveb e gm i + aogm i =ai (37)
T77'/ao)+J(Cd+Cje) gm+iJ(Cd+Cje) o
The quantity

a a
a = (38)
l+jW(C +C. )a /g l+jf f
d je o m a
is known as the "high frequency current amplification factor

of the common-base circuit," Obviously


a = 2rf = m = geb :=1 Cd (39)
a 0(Cd je) CC e d dje







is the frequency where Jal has dropped 1//2 times its

low frequency value. The quantity fa is known as the

alpha-cutoff frequency. The same expression for a is

found if the diffusion through the base region is fully

taken into account.

If a field is built into the base region--as is

sometime the case--then this aids the minority-carrier

flow. Thus the motion of the minority-carriers through

the base region is partly by diffusion and partly by drift.

The expression for a must then be written as

a exp(-jcw )
a (40)
1i+j f/f

where T is a kind of transit time and the exponential term

represents the additional phase shift in alpha at high

frequencies due to the drift time of minority-carriers

across the base (as in graded base transistors4.

The actual frequency dependence of alpha is much more

complex in the case of a graded base transistor at high

level injection but equation (40) may be taken as a fair

approximation.15

Now, for the common-emitter equivalent circuit of

Figure 3, if an a-c current ib is flowing in the base and

if C. << C + C. the a-c voltage v is
Je d iJe' be

v = (41)
be g /o +ja(C +Cje )
m o d je
and hence the output current generator is








g
g v m i = ib (42)
m be g /B + j(C +C. ) b b
m o d je
The quantity
gm ,13
S= m0 (43)
E/Lo3 + jn(Cd+Cje) l+jf/fg
is called the "high frequency current amplification factor

of the common-emitter circuit." Thus,


w = 2rf gm = (1-a ) (44)
P o- (C +C. ) a
o d Je
is the frequency at which 131 has dropped to 1//7 times

the low frequency value.

Another interesting frequency is the frequency for

which f1 = 1. Since in equation (43), the quantity (gm/ 1)2

is very small, then at the W cutoff frequency,
T

(.T(Cd+Cj ) = g or vp = om (45)
S(Cd je m or C d+C. e a
d Je
At small emitter currents it may happen that Cje >> Cd.

In this case

cm q I-
C. kTC.
Je je
for which the cutoff frequency decreases linearly with

decreasing emitter current-IE. This can be important in

high frequency silicon transistors that have a rated cutoff

frequency of the order of a few hundred megacycles. The

cutoff frequency at low current levels can be several

orders of magnitude smaller than this.






For the case when C << C d
je d
g 2D 1
T = m p (47)
T _-
d d
where here wT is practically independent of current level

but is completely determined by the diffusion time Td of

the carriers through the base region. Thus a higher cutoff

frequency wT can be obtained by decreasing the base width.

In accordance with the corpuscular approach, at high

frequencies three additional groups of carriers must be

considered. These carriers do not contribute to the d-c

current of the device but do have an effect on its high

frequency a-c behavior, and hence its inherent noise proper-

ties at these frequencies. The groups of carriers, as

shown in Figure 2(b) are as follows:

i) holes which are injected by the emitter and

which return to the emitter

ii) electrons which are injected into the emitter-

region and which return to the base

iii) holes which are emitted by the emitter and

trapped in the space-charge-region but which are

then detrapped thermally and return to the emitter.

(Note: this group of carriers could be considered

under the recombination current carriers of

Figure 2(a) also.)

When the load impedance of the amplifier is such that

the incremental voltage gain of the transistor is large,

a second-order mechanism requires a modification of the







small signal circuit model. This mechanism, which is

referred to as "base-width modulation" or the "Early effect,"16

results from voltage dependence of the width of the collector

junction space-charge-region, which causes the effective base

width to be voltage dependent. The change in collector

junction voltage AVC clearly has three components:
CB
i) The collector current changes because the

slope of the minority-carrier distribution in the

base changes.

ii) The change in stored base charge must be

accompanied by a transient component of base

current which supplies the additional stored

majority-carriers.

iii) The component of base current which feeds

recombination must change to accommodate the change

in stored base charge.

These changes are accounted for in the equivalent

circuit models of Figure 3 by adding two conductances

gbc and g to the common-emitter circuit, and the conductance

gcb and voltage-dependent generator pvcb to the common-base

circuit. The parameters are given by17


ce = gm

bc = (l--f) (48)
(i g ioW 2 /L 2
cb Cm p
(if iB is due to hole recombination in the base solely) and







where

kT 1 DW AVE
=- = EB (49)
q W 8VcB AVCB

is the "inverse voltage feedback factor in common-base

connection." Because AW/AVCB is very small, p is a small

quantity and both gbc and gce are very small compared to

gm; consequently, these elements influence the circuit

performance appreciably only for large voltage swing of

the collector (i.e., for very large voltage gain).

The common-base and common-emitter equivalent circuits,

including secondary effects, are given in Figure 4. Notice

that a capacitance C has been added. Because of transverse

voltage drops in the base, it is frequently necessary to

split the collector junction space-charge capacitance into

two components; one which is charged through the base

resistance or impedance and one, C, which is not (this is

the so-called "overlap diode" capacitance). The relative

values assigned to these capacitors depend upon the geometrical

arrangement of the transistor.

It should be noted at this point that while the develop-

ment of the transistor noise theory and measurements will

be predominantly in the common-base connection, the common-

emitter as well as the common-base configurations will be

used in determining certain transistor parameters. This is

the reason for the simultaneous development of both of these

equivalent circuits.








Basic Noise Sources in Transistors


Electrical Noise Characterization

Electrical noise is commonly divided into three

distinct parts: thermal noise, flicker noise, and shot noise.

Thermal noise occurs in any conductor and is caused by

the random thermal motion of current carriers in the

conductor. Flicker noise is so called because of the

fact that its spectral distribution is similar to that of

the flicker noise in vacuum tubes being of the form, constant/

fI, where n is close to unity. Shot noise derives its

name from the resemblance to shot noise in vacuum tubes,

an important characteristic being that of a flat spectrum

at low frequencies. These names are more or less heuristic

and make no reference to the physical causes of the noise.

The following more precise terminology has been suggested

for noise in semiconductors and semiconductor devices;18

a) Transition noise

As an example of transition noise consider generation-

recombination noise. It results from fluctuations

in generation rates, recombination rates, trapping

rates, etc. It is the result of fluctuations in

"interband" transitions, i.e., transitions between

energy levels or between impurity levels and the

valence or conduction band. Thermal noise is an

example of noise caused by "intraband" transitions

which result in the scattering of carriers. The noise







of "intraband" transitions stems from velocity

fluctuations.

b) Transport noise

This noise results because the carrier

transport mechanism, whether drift or diffusion,

is a random process. In a temperature-limited

vacuum diode this transport noise has long been

called "shot noise." The name is frequently

applied to noise resulting from transport fluctua-

tions even though the transport mechanism may not

be drift. The most important characteristic about

transport noise, for our purpose, is that the

magnitude of the noise resulting is, at least at

low frequencies, about the same as that of the

shot noise of a. temoerature-limited vacuum diode.

c) Modulation noise

Modulation noise refers to noise which is

not caused by transition or transport fluctuations

directly but which, rather, is caused by some

modulation mechanism. A modulation mechanism

located at the device surface probably gives rise

to most flicker noise in transistors.

It has been shown19 that the noise of any arbitrary two-

part network in a small frequency interval, can be represented

by a noise current generator /T" of infinite impedance

in parallel with the impedance of the network or by a noise

emf /J of zero impedance in series with the impedance of the







network. The units for use in indicating the spectral

intensity of these two representations are amperes-squared--

per-unit-bandwidth (amp2-sec) and volts-squared-per-unit-

bandwidth (volts -sec). In addition the following units

are often used: (1) equivalent saturated diode current,

(2) equivalent noise resistance, (3) equivalent noise

conductance, (4) noise temperature, and (5) noise ratio.

Since (1) and (2) shall be used extensively later, they

will be defined as follows:

(1) Equivalent saturated diode current, IEQ. By

Schottky's theorem,20 the noise of a temperature saturated

diode in a frequency interval Af can be represented by a

current generator of infinite impedance in parallel with

the plate resistance of the diode. The magnitude of the

noise current is


T = qiAf (50)


where Id is the diode plate current and q is the electron

charge. Thus it is possible to equate the noise current

in terms of the plate current of this ideal equivalent

saturated diode. Hence,

IQ = f /2qAf (51)

(2) Equivalent noise resistance, RN. By Nyquist's

theorem,21 the noise of a resistance, R, in a frequency

interval Af may be represented by a zero impedance noise

emf, /e in series with the resistance. The magnitude of

the noise emf is









A = /4kTRAf (52)

where k is Boltzmann's constant and T is the temperature

in degrees Kelvin. Equating the noise voltage of an

arbitrary two-pole to that of an equivalent noise resistance

yields

RN = e /4kTAf (53)

Noise of four-pole networks differs from that of two-pole

in that two partially correlated noise sources are required

to represent the noise.22 This fact will be used in develop-

ing the theory for the noise of the common-base transistor

configuration.



Theory of Noise in Transistors at Room Temperature


There are two different approaches to the theory of

noise in transistors. One is the collective approach in

which the noise is attributed to the random diffusion of

minority-carriers and to the random generation and

recombination of hole-electron pairs.23,24 The other is

the corpuscular approach in which the noise is attributed

to a series of random and independent events such as the

crossing of the emitter and/or collector junctions by the

individual current carriers.25,26 The two apparently

different approaches were shown by van der Ziel to be

equivalent at low-injection,27 by deriving the corpuscular

approach with the help of the more fundamental collective

theory. Also, the corpuscular approach does not require







any geometrical assumptions, thus it is expected that the

results are independent of geometry. In addition, the

theory makes no reference to the mechanisms of carrier

transport and, therefore, is not confined to the diffusion

model.

Since these two approaches are supposedly equivalent,

a brief review of the simpler corpuscular approach only is

presented here.

The groups of current carriers crossing the emitter-

base as well as the collector-base space-charge-regions

are shown in Figures 1 and 2. Note that everytime a hole

crosses the emitter-base junction, there is a sharp pulse

of current in the external base and emitter terminals.

This is due to a charge +q entering the emitter terminal

to balance out t;he loss of the hole from the emitter region

and a charge -q entering the base terminal to counteract

the increase of charge brought by the hole into the base

region. Thus space-charge neutrality in the two regions is

restored immediately. By considering every transition of

hole and electron crossing the emitter-base and collector-

base junctions as an independent, random phenomenon, the

accompanying noise power can be evaluated via Schottky's

theorem (i.e., the spontaneous fluctuations in the rate of

injection and collection of minority-carriers can each be

represented by a shot noise current generator in parallel

with the admittance of the emitter-base and collector-base

junction, respectively),






The noise equivalent circuit for the transistor in

the common-base physical-T configuration is given in

Figure 5. This equivalent differs from that given in

Figure 4(a) in that the real and imaginary parts of the

emitter-base and collector-base junctions have been replaced

by complex admittances Ye and Yc respectively. Also, since

the noise of the device will be determined under "very small

applied signal" conditions, modulation of the collector-base

transition region (the Early effect) will be small; thus the

conductance gcb is small, as is the voltage feedback

parameter p. Even if the Early feedback emf, PVcb, is

significant, it does not change the noise figure of the

device and has little influence on the input impedance,

power gain, etc. Hence, it is not included. The parasitic

capacitance C can be thought of as belonging to the load.

Since the ratio of the signal-to-noise currents passing

through any load element is invarient under a change in the

load characteristics, the load may be removed entirely without

affecting the noise performance of the devices.28 The

capacitance C can thus be omitted in the noise analysis. Also

included in Figure 5 are two shot noise current generators

added in parallel with the two junctions of the device

(i.e., i1 and i2), as well as a thermal noise emf due to the

base impedance.

By combining equations (4), (5) and (32), the expressions

for emitter and collector currents for normal bias (i.e.,

VEB > 0, VCB << 0) are




























4 ~Yc


NOISE, EQUIVALENT CIRCUIT FOR A BIPOLAR JUNCTION
TRANSISTOR


T:CJ I: 5

















T e = il/ye
-+


Ye aCl. y ] i i2- ai,




rIfb


e,










FIGURE 6. MODIFIED NOISE EQUIVALENT CIRCUIT FOR COMPARISON
BETWEEN THEORETICAL AND EXPERIMENTAL PERFORMANCE









IE = IEsexp(qVEB/kT)-IES (-lf)+IRESexp(qVEB/2kT)-IRES

and

IC = -ffIESexp(qVEB/kT)-ICS(1-a )-IRCS (54)
These expressions account for the current carrying groups

1 through 7 of Figures 1 and 2. The magnitude of the noise

current generator 1i due to the contribution of groups

associated with the emitter-base junction is given by


i2 = 2qAf[IEexp(qVEB/kT)+IES(l-a))+IREexp(qVE/2kT)+IRE

(55a)

= 2qAf[IE + 2ES(1-tf) + 2IRES
= 2qAf[IE + 2IEE] (55b)

where

EE ES(1-af) + RES (56)

The carriers of groups 8 through 10 of Figure 2(b) must also

be taken into account. As already stated, these current

carrier components become important at relatively high

fr queI n cies. At higrh frequencies the emitter admittance

Y becomes complex and its real part geb increases with
eb
increasing frequency. The noise associated with these

diffusion components corresponds to thermal noise of the

incremrent (ge eo) in the emitter conductance since
-eb tebo
diffusion is a thermal process,29 Their contribution to

i is therefore, 4kT(ge eo )Af, where ge represents the

emitter junction conductance and geo, its low frequency








value. Therefore, the general expression for 1I becomes


I = 2qAf[IE+2IEE] + 4kT(ge-go)Af (57)

The low frequency conductance, defined by equation (21), for

the total emitter current of equation (54) is


geo q IESexp(qVEB /kT)+RESexp(qVEB/2kT)] (58)
kT
Thus,


2kT geo = 2IESexp(qVEB/kT)+IRESexp(qVEB/2kT) (59)
q
Taking equation (59) minus equation (54),

2kT go-IE I Eexp(qV/kT)+IE (-af)+IRES (60)

If equation (60) is now substituted into equation (55),

S=2qAf[ 2kT -I+I exp(qVE/2kT)]+kT(g )Af
= 2ceo E RES EB e-geo
q
= 4klP Af'-2qA f[IEIRESexp(qVEB/2kT)]
Sr

-7 = .!kTgeAf-2qAf (IE ) (61)
provided IR >> L

Therefore we have two expressions for i-; equation (57)

will prove to be more useful however.

Current transport across the collector-base junction
results from the carriers of groups 1, 4 and 7 of Figures 1

and 2. Adding the shot noise of each of these groups yields

? = 2qAf[aLi Sexp(qV E/kT)+ICS(1-)+IRCS

= 2qAflC (62)







Because the currents flowing through the emitter and

collector junctions have a component afIESexp(qVEB/kT)

in common, ii and i2 should be strongly correlated.

According to van der Ziel,30 the cross-correlation at

low frequencies is

il 2 = 2qAf[a IEsexp(qVEB/kT)] (63)

where the asterisk denotes the complex conjugate quantity.

If the low frequency transconductance defined by equation (23)

and the complete collector current given by equation (54) is

introduced,

aIC q
e = = 'Eexp(qV /kT) (64)
ceo f ES EB
c VBE kT

then equation (63) can be written in the form

il i2 = 2kTgceoAf (65)

At high frequencies, the signal transfer properties are

defined by a cor:nplx transfer admittance, Yce thus the

cross--correlation assumes the form

ili2 = 2kTY cef (66)

Signal transfer in the transistor is represented by a

current generator YceVeb in parallel with the collector

junction, where Veb is the a-c emitter voltage. If i

denotes the a-c emitter current, then the high frequency

behavior of the emitter diode is described by its admittance

Y = i /Veb

The high frequency a-c intrinsic current gain a can then be

defined in terms of the driving point and transfer admittances








by

a Y /Y
ce e
This permits the cross-correlation to be expressed in its

most convenient form

il-i2 = 2kTaY Af (67)

Account must now be taken of the thermal noise generated

in the base impedance Zbb. In accordance with Nyquist's

theorem, this is done by placing a random voltage emf

eb'b of mean-squared value,

eb'b = 4kTrb,bAf

in series with that impedance, where rb'b is the real part of

Zb'b'

An alternate equivalent circuit which enables a more

direct determination of the noise performance of the

transistor in common-base connection is given in Figure 6.

The two almost completely correlated noise current generators

in Figure 5 are replaced by an output noise current generator

S i2 ii and an inout noise emf e = ilZ = il/Y This

equivalent circuit is much more desirable in terms of comparing

the measured noise performance with that predicted by the

theory, since through measurements of the equivalent saturated

diode current, IE, at the output with the input a-c open

circuited, direct information of the noise current generator

i is obtained. Also, through measurements of the equivalent

noise resistance referred to the inout as a function of

source resistance, indirect information about ee and ei can
be ob nee eenras
be obtained. The mean-squared values of the generators in








Figure 6 are
e = 1 2
e 1 e

= {4kT(g e-go)Af+2qAf[IE+2IEE1]}Ze 2 (68)
= 2kTAf[2(gegeo)+q(I +2EE)/(kT)]Ze 2 (69)

and

1 + 121'I) 2-
i = (i2-ali 2-ail) = (i -a ilx)(i2-ail)

= 12 + ai 1 2Re(al i )

Now, from equations (57), (62) and (65),

o = 2qAf[I C+Ia2(IE+2IE)-2 a 2kTge/q]

= 2qlQAf (70)
Also the correlation between these two sources is given by


e To 1(ilZe)(i-ai) e= iZ 1(i2-il
= Z ( I2 al) (71)
SZe (i12
S2kPTaf[-l+2Z geo-Zeq(IE+21E)/(kT)] (72)
where equation (72) is the result of combining equations

(65) and (57) with equation (71). Now from equations

(58) and (54), the low frequency emitter conductance geo
becomes

geo = q(IE+EE -R)/(kT) (73)
for I >> IS By substituting equation (73) into equation

(70), a much simpler expression for IEQ may be obtained as

IEQ = IC-"I2[IE- IR (74)
The equivalent noise resistance RN as a function of

source resistance R is calculated by referring all noise
sources shown in Figure 7 to a single equivalent noise emf at

the input. This may be accomplished by first calculating the







ee






Z8 Z b' b


eeb'b




(a)



^|------j j-----1---^\;
Si, iout
ya fri ye



"n Zb'b







(b)



FIGURE 7. NOISE MODELS USED TO CALCULATE EQUIVALENT
NOISE RESISTANCE REFERRED TO THE INPUT.





44

short circuit current at the output resulting from the noise

sources of Figure 7(a) and equating this to the output current

resulting from a single equivalent source, en, of Figure 7(b).

For the circuit of Figure 7(a), the mean-square value of the

output current is given by


l H2(es + b e
- + e + (75)
out (Z + Z + Zbb 2 Z + Z + Zbb i
e s b'b s e b'b

where

es = 4kTRAf ebb = 4kTrb'bAf


Z = R+jXs Ze = re+jXe Zbb = r bb+jXbb
s s s e e e b'b b'b b'b
Here we have assumed no correlation between e and i However,
e o
following Hanson and van der Ziel, we now split the noise

emf ee into a part e' fully correlated with i (correspond-

ing to that portion of emitter current fully correlated with

collector current) and a part e': uncorrelated with i The
e o
following parameters are also introduced;

e = 4kTr, Af (76)



1 1
iY L 4kTgslAf (77)

and
ae' ae 1
e e o
s (78)
10 i2
0

where r,, is the noise resistance of the spectral density

of e" F g is the equivalent noise conductance of T'
e sl o
referred to the input; and Z, is the correlation impedance.







The mean-squared value for the output current of

Figure 7(b) is

i-- H2
nut (79)
out (Z +Z +Z+ )2
s e b'b

By splitting e into its correlated and uncorrelated parts
e
in equation (75) and equating this to equation (79) yields

for e
n e' 2
e = + e- + + 0 e +Z +Z +Z (80)
n s b'b r i s e b'b
0

Setting e' = 4kTRNAf, and from equations (76) through (78),


RN = Rs+rb'b+rsl+gsl Zs+Z +Zb1b+Zsc2 (81)

The last term in this relation can be expanded to yield


Rr = + R +r +r sl +r+r +r 2 +g X +X+X +Xs 2
N = R b'b si sl s e b'b scl sl s e bb sc
(82)

It follows directly from equation (82) that there exists

some X = X say, for which
s sm3
X + Xe + Xbb + Xsc = 0 (83)

Thus, a variation of X over a sufficient range should
s
produce a minimum in the noise figure at

X = -(X + Xb + X )
sm e b'b sc
In other words, series tuning at the input improves the

signal to noise ratio of the amplifier and is recommended

under mismatched conditions. This effect was studied by

Chenette.31 Thus, from equation (83), the emitter reactance

can be measured or calculated, and the source reactance







X can be measured. Therefore, X can be found.
s sc
Once the condition given by equation (83) is satisfied,

the expression for RN becomes

RN = R + rsl + r + + s (Rs + re + rsc + rbb)2 (84)
The (spot) noise figure, F, is found by normalizing

RN of equation (84) with respect to Rs, or
F = RN/Rs (85)

Now, Hanson and van der Ziell made the suggestion to determine

the remaining noise parameters from measurements of the

noise figure F, or of the equivalent noise resistance

referred to the input RN. If RN is written as

RN = A + BRs + CRs2 (86)

and by comparing equations (84) and (86), we obtain

A = rbb + rsl + sl (re + rb'b + rsc)2 (87)


B = 1 + 2gs1 (re + rb'b + rsc) (88)


C = gsl (89)

The measurement of RN as a function of Rs will generate

points that belong to a parabola. Therefore A, B, and C

can be determined from the shape of the parabola via normal

curve fitting techniques.

Having deduced the values of A, B, and C, the following

quantities may be expressed in terms of A, B, and C.


(re + rb'b + r sc B- (90)
2C

r b + r = 4AC (B-1)2 (91)
b'b --


(92)


gsl C







There are now two different ways to proceed. Either by

measuring rb'b independently and using the low frequency

value of r and then determining rsl and rsc; or, by using

the low frequency value of re and the theoretical expression

for rsl, and then determining rb'b and rsc.

The theoretical expressions for the noise parameters

can be obtained with the help of equations (76) through

(78) as follows:

i2 q I
g so 0 EQ (93)
sl kTAf; l 2 2kT T(9

since i2 2qI Af. Now since e = e + e, using equation
0 EQ e e eu
(78) with equation (76)

e sc 12
e e e e

(e*i )2
e o
= e2 (94)
e
o
Therefore by equation (76),

e e*i / e| ei | 2
Se e o o e e (5)
r (95)

4kTAf 4kTAf 4kTAff

Substituting equations (69), (72) and (93) into equation (95)


rsl = [2(ge geo) + q(IE + 2IEE)/(kT)l IZe 2 +

-[-1 + 2Ze g Z q(I + 2I )/(kT)]2/4gs (96)


Now if equations (72) and (93) are substituted into equation

(78),








Z = ae 1 / T2
sc e o o

= [-1 + 2 Zeo Z q(E + 2IEE)/(kT)] / 2gs1 (97)

Recall that a shot noise mechanism was attributed to both

the diffusion and recombination currents. Since this

investigation deals with noise in B,J.T.'s at cryogenic

temperatures, and since the diffusion and recombination

components are directly proportional to ni2 and ni,

respectively,*32 it can be seen that little error is made

if IES(l-af) and ICS(1-ar) are neglected at low

temperatures (i.e., T = 2000K). (Note that for high

temperatures (i.e., T = 3530K), by the same reasoning the

terms IRE and IRC can be neglected.32)

As discussed earlier in reference to equation (27),

when trapping-recombination of carriers in the emitter-

base transition region is important, distinction between the

d-c current amplification factor and the a-c current

amplification factor must be made. Recalling from equation

(26)
BIC C/ 9VEB
a = a= IE/ VEB (98)
0 IE IE/ 3VEB
where from equation (54)


IC q qlC
K = [IC + ICS(1-a ) + IRCS] (99)
VEB kT kT

at low temperatures, and
*Recall that n. is a strong function of temperature, as

ni = CT3/2exp(-E /2kT)








IE = [I + I IR]
kT E EE R


kT E
The neglecting of the saturation current terms in equations

(99) and (100) is equivalent to neglecting that part of

the emitter current due to hole-electron generation in the

base region,33 This is a very good assumption at room

temperatures and below, as already mentioned.

Putting equations (99) and (100) into equation (98)

yields
I
a = -C (101)
O -I-

Recalling that ad = IC/IE the recombination current can
dc C -
be found from equation (101) as

21 a a dc
R = 2IE ( o )
R ,o (102)

If this value of IR is now used in equation (74), the

expression for IEQ then becomes

IEQ =IE [d + JC2 21L2 a d/o

= [I C la12{(-a0 + 2a dc)/ }] (103)
where lai is defined by equation (40). It has been shown that

equation (103), though developed under low-level injection

conditions, is also accurate for high-injection levels.34

Now at low frequencies (i.e., f << f ), the current

amplification factor of equation (40) becomes a = a
0




50

For this case, equation (103) becomes

IEQo = IE[dc( adc) + (ao adc)2 (104)
From equation (93),

q I
= q EQo (105)
sl 2kT Ua

Also, from equations (100) and (102), Z for low frequencies

becomes
1 kT a
r = (106)
eo -
geo qIE adc
Similarly, equation (96) for rsl becomes

qlE (1 r qlE/kT)2
r -=r [ r (1 eo- ] (107)
1 e kT 2gsloreo

and from equations (105) and (106), this reduces to


rso e (108)
'dc( adc) + To adc J
Equation (97) for Z becomes

ao (a0 ad, )
sco o (109)
Sa.c a d) + (a adc

Now,i. for the case where trapping-recombination effects

in the emitter--base transition region can be neglected

(i.e., at high temperature), then I = 0. It is then

convenient to redefine ad by requiring a = a If I
dc ac d o EE
and ICS(I-lp) are not neglected, one then obtains

adc = = ICS ICS(1 ar)
(110)
IE +IEE
Since eo = q(IE + IEE)/kT, one obtains


(111)


IEQ = _C ll IE = (a I" )IE + ZCO







The expression for I and the corresponding expressions
EQ
for g Z and r coincide with expressions given
S sc sl
earlier by van der Ziel.30 The results of equations (110)

and (111) will not be used for this investigation but are

given here for completeness.



Low Temperature Effects on Basic Transistor Theory


The parameters and hence the operation of the bipolar

junction transistor are highly temperature dependent.

Generally speaking, a considerable degradation in device

performance is observed with decreasing temperature. This

results primarily from the fact that the current amplification

factor B (or a ) and the minority-carrier lifetime decrease

with temperature.

Insight into the temperature dependence of the transistor

operation may be gained by referring to the Ebers-Moll

equations given by equation (3). It is obvious that and
S
ICS are both proportional to the equilibrium hole concentration,

pn = n2/i, in the base region by equation (13). Consequently,

Cn Cn2 C n
I i. C 2 i. = (112)
ES N CS N CO
D D D
Because n2 depends so strongly upon temperature, the transistor
i
characteristics are strongly temperature dependent. This

dependence can be shown as follows. As already stated,

n2 = AT3exp(-qE/kT) (113)

where A is a constant and E the gap width of the forbidden
g







band in the semiconductor material. Hence, if at temperatures

T and T the current of IC0, say, has a value IC and
1 2 CO CO1
ICO2, then


I AT exp(--qE /kT ) qEg T2-T
C02 = g exp[ ] (114)
ICO AT{exp(-qEg/kT ) k TIT2
Hence, ICO IES and ICS depend on temperature in an

exponential fashion.

Also, it is shown by van der Ziel35 that the voltage

VEB required to maintain a constant emitter current, IE,

with changing temperature must shift in value, its magnitude

being approximately linearly proportional to T (i.e.,

2.5 m-volts/degree Kelvin). Thus at a lower temperature,

a larger VEB will be required to maintain IE constant.

Several authors have attempted to characterize the

B.J.T, at low te1mperatures.36-38 A brief review of the

temperature dependent parameters of semiconductors as iell

as transistors will be given at this time, Frequent

reference here is mad-e to Elad,39

The ene:;.- gap for silicon increases slightly with

decreasi ng- temperature, as40

E (Si) = 1.21 3.6 X 10'"T T < 5000K

where measured values of E at T = 3000K is 1.107 eV and

at T 4.2K is 1.153 eV. Also, the ionization energies

of impurities in silicon can be shown to be relatively

temperature independent via the theoretical expression for

the modified Bohr theory of the hydrogen atom. i Measured







ionization energies for typical dopants in silicon are:

phosphorous 0.044 eV, boron 0.046 eV and arsenic 0.049 eV.

The free carrier densities at cryogenic temperatures

differ somewhat from those at room temperature in that

some impurity atoms are not ionized. Therefore, the charge

neutrality condition takes the form:

n + nd + Na = p + na + Nd

where nd and na represent the concentration of un-ionized

donors and acceptors, respectively. These two quantities

are determined by Fermi-Dirac distribution, because at

cryogenic temperatures the Fermi level can come very close

to (and even cross) the impurity level. The explicit

neutrality condition follows:

E -Ef Nd
N exp(- ) + + N =
kT 1 + exp ED-Ef a
KT J

Ef-EV N
= e) + + Nd
kT 1 + f-exi E d
SkT j

This is a transcendental equation relating the Fermi level

Ef to the impurity concentrations Na, Nd and the temperature

T. The N and NV are the effective number of states in the

conduction and valence bands and E and E are the energy
a d
levels of the acceptor and donor impurities. Making some

simplifying assumptions, say for n-type material (i.e.,

E Ea > 0, therefore na = 0, etc.), the majority-carrier

concentration is given by39

E 2
n = (NdNC) = (52kT)






where Ed is the donor activation energy. The minority-

carrier concentration is

2Nc 8 2E -Ed
P = C NVexp g2 (116)
d.
where E is the forbidden energy gap, and N and NC have
g V C
3/2
a temperature dependence of T

Therefore we may conclude that the majority-carrier

concentration is strongly dependent upon the ratio of

activation energy to thermal energy. Thus at the lower

end of the cryogenic range, where the activation energies

are large in comparison with 2kT, the impurities become

de-ionized and consequently the free carriers tend to

"freeze-out." This constitutes an important limitation

in device application at cryogenic temperatures. From

equation (116), we may conclude that minority-carrier

densities in silicon are negligible at low temperatures.

The extent to which the free carriers are affected

by electric fields is determined by their mobility p.

,Mobility, in elernntal semiconductors, is determined

principally by two processes--scattering by lattice vibrations

and scattering by ionized impurities.42 The mobility, limited

by lattice vibration p, has the temperature dependence

Pi = Am*-"/2T-U for 1.5 < a < 2.5 (117)

where A is a numerical constant, m is the effective mass

and T is the absolute temperature. The component of mobility

limited by the scattering action of ionized impurities is







given by

pi = Bm N-T3/2 (118)

where B is a proportionality constant that varies very

slowly with temperature and N is the impurity concentration.

Finally, the total mobility p is given by


1 = 1 + 1 (119)
1 1l1 P1
The opposite temperature dependence of the two scattering

mechanisms points to the existence of a maximum in the

mobility-temperature curve. The temperature TM at which

the mobility peaks is found by substituting equations

(117) and (118) into equation (119) and setting dp/dT = 0.

(3/2 + a) 3A
T = N (120)
9 Bm

where TM is only weakly dependent on impurity concentration

N. From experimental data, 3 it is seen that for concentra-

tions of 1014 and 10 1 cm"3, Th{ is approximately 300K

and 500K, respectively.

The electrical conductivity u in the presence of n

electrons and p holes is given by

a = qlnn 4 ap p (121)

where, as stated, only the majority-carrier component is

significant at low temperatures. Now since the density of

free carriers is exponentially dependent on temperature, while

the mobility varies according to a lower power dependence,

it may be concluded that the conductivity is determined

mainly by the availability of free carriers.







The temperature dependence of the mobility and the

density of free carriers cause peaking of the conductivity-

temperature curve at a temperature decreasing with

impurity concentration of the sample. Experimental data

of conductivity vs. temperature for silicon indicate that

the temperature TM for maximum conductivity is observed

for concentrations of 1014 and 1017 cm-3 at 1000K and 1500K,

respectively. Also, for temperatures above about 600K,

reasonable conductivity is still obtainable. Below this

temperature, the conductivity decreases rapidly.

The equilibrium densities of free carriers at any

temperature are maintained by the balance of generation and

recombination rates. The thermal generation rate is very

much reduced at low temperatures due to a scarcity of energe-

tic phonons, but it is otherwise with recombination in the

presence of injected excess carrier densities. The

recombination processes may be classified into direct

transitions across the energy gap and transitions via deep

level trapping centers. In elemental semiconductors, having

the indirect energy band structure, the latter is the main

recombination process. According to the statistics of this

process derived by Shockley and Read, the rate of recom-

bination is

pn n
U =- (122)
-(n ) + n(p + p1)Tn
IP







where
E
n. = (NCNv)-exp(- (122a)
2kT

Et-E
n = N exp ( Et- ) (122b)
1 C kT
kT

T and T are minority-carrier lifetimes, Et is the energy
n pP

level of a trapping center and NC is the effective number

of states in the conductor band. Since ni, n1 and pl

become negligibly small at low temperature the recombination

rate will be determined by the value of p, n and the carrier

lifetime. Under nonequilibrium conditions p and n may well

reach values comparable with those obtainable under room

temperature operation. It should be noted that it is diffi-

cult to evaluate the applicability of Shockley-Read statis-

tics at cryogenic temperatures due to the inherent high

injection carrier densities. Unfortunately there is

virtually no information on lifetime, or the related

capture cross-section, in the cryogenic range of tempera-
tures.4546 The lifetime of minority-carrier electrons is

given by

S= 1 (123)
e V N1
Se g
where V is their thermal velocity, 7 their capture cross-

section and N the density of generation-recombination
g
centers. The concept of a constant lifetime is normally

defined for low injection levels; however, this low-level

lifetime may not be applicable at low temperatures where,

as mentioned, injection densities are invariably high in







comparison with equilibrium densities, Indirect conclusions

can be drawn about the temperature dependence of T from such

experimental evidence as the enhancement of luminescence at

low temperature in a wide range of materials, and also

from the performance of bipolar transistors (both Si and

Ger) at liquid nitrogen temperatures. The decrease in p

strongly suggests a decrease in minority-carrier lifetimes.

Also, similar conclusions can be derived from the increased

trapping of carriers in large area germanium p-i-n diodes

operating at or below liquid nitrogen. 7

For a constant N and since V increases with
g e
temperature, Ve = (3kT/m*)', a decrease in lifetime necessarily

implies an increase in capture cross-section with decreasing

temperature. It is known that, at least for comparatively

shallow centers, the capture cross-section increases very

rapidly with falling temperature.45 This is explained in

terms of the so-called "giant trap" model in which the

carriers are not captured directly into the ground state of

the capturing center but are first trapped in one of the

higher excited states and are then cascaded down to the

ground state. The excited states provide a much larger

capture cross-section than would be expected from the ground

state. This model applies, however, only to the shallow

"hydrogenic" impurities and it is not clear how the theory

can be extended to deeper levels which are normally associated

with recormbination.








As already mentioned, trapping of minority-carriers

becomes important at low temperatures.48 Once trapped,

a carrier can only be re-emitted if sufficient energy is

supplied to it, normally in the form of thermal excitation,

which becomes particularly difficult at very low temperatures

since it depends on the exponential factor exp(-Et/kT),

where Et is the activation energy of the trap. At cryogenic

temperatures any levels in the forbidden gap may act as

trapping levels from which the carrier may not be re-excited

thermally as long as the low temperature is maintained, This

may, in turn lead to space-charge effects and irreversible

behavior,

From the foregoing discussion, we see that p-n junctions

and bipolar junction transistors, which depend on the injection

of carriers and the diffusion transport mechanism, are

considerably degraded at low temperatures due to carrier

freeze-out and the decrease of carrier lifetime. In

addition, forward biased p-n junctions at low temperatures

suffer from problems with the injection of excess carriers,

an increase in junction barrier potential, and different

proportions between the diffusion and recombination

components of the current. For normal injection of carriers,

one side of the junction must be heavily doped (degenerate)

so as not to be affected by carrier freeze-out and the

other side must be fairly heavily doped (but not degenerate

or else we would have a tunnel diode) to have a measure of

impurity conduction, especially in the case of the base







region of a transistor, Also, the metal semiconductor

contacts must be ohmic.

The junction barrier potential increases with decreasing

temperature as the Fermi levels in both the p and n semicon-

ductors approach the doping impurity levels, Thus, the

turn-on voltage of silicon diodes is close to 1.15 volts at

500K and 0.95 volts at 770K,

The relative magnitude of the space-charge recombination

component and the diffusion component of the forward current

influences the current gain of the transistor, The ratio

of the two components is given by49


Jrec/diff = [KNTNC/qEL]exp[-(V Eg)/2kT] (124)

where N is the impurity concentration, NC the effective

density of states in the conduction band, E the electric

field in the space-charge-region, L the diffusion length

and V the applied voltage. From equation (124) we see that

at low temperatures the rate J re/Jdiff decreases,

For the reversed bias p-n junction, the reverse current

consists of a diffusion component, a bulk space-charge-layer

generation component and a surface generation component. All

three components decrease approximately exponentially with

temperature. The bulk components are given by the formula49


TI = B qn.WA / T + qnJLA /NTm (125)

where W is the width of the space-charge-layer, ni the






intrinsic carrier concentration, Tm the minority-carrier

lifetime and A the area of the junction. The surface

generation component is expressed by50

ISR = qAsni (126)

where A is the surface area of the junction and s the
S o
surface recombination velocity. At low temperatures,

the generation components dominate and therefore the

reverse current of the p-n junction is approximately

proportional to the volumn of the depletion layer.

Junction capacitance decreases with temperature

according to the relationship

So(T2) V
C(TI) = C(T2) [ ]a (127)
So(T1) -V

where V is the bias and 0o the contact potential.

The a = for an abrupt junction and 1/3 for a graded

junction though in practice the exponent normally lies

between these two values.

For transistors to operate effectively at cryogenic

temperatures the width of the base region must be much

smaller than the diffusion length, L.

L = (DT)2 (128)

where D is the diffusion coefficient and T the lifetime

of minority-carriers. From the Einstein relationship

D = kT u (129)
q

we see that the diffusion coefficient should increase slowly

as the temperature is lowered (due to p), but then decrease







faster as the temperature T of equation (120) is passed.
m
However equation (129) applies to low level injection and

at high level injection, present at cryogenic temperatures,

carrier-carrier scattering reduces the mobility; and the

temperature T, which is the temperature of the carriers,

may be higher than that of the lattice. As discussed

earlier, T decreases with temperature. Therefore, at

cryogenic temperatures, the diffusion length is small

and for transistor operation we need very thin base regions.

From here (as was observed experimentally) high ft

transistors will operate down to lower temperatures than

low frequency transistors.

The common-emitter current gain B at cryogenic

temperatures is determined by injection and transport

efficiencies. Its approximate value, assuming a constant

doping in the base (for a pnp transistor), is given by51

1 D Wn W
=ee + i( )2 (130)

D bLePbo b

where W is the width of the base region and ne and Pbo

are minor.ity-carrier densities in the emitter and base,

respectively. Substituting equation (128) into equation

(130) we obtain

1 De Wn eo
= e eo + (131)
S Db n2Pbo DbTp
The ratio (n eoP bo) is temperature independent and the

diffusion coefficients vary slowly with temperature.






Therefore, B decreases strongly with temperature following

the behavior of T.

A brief review of some of the other important parameters

of the hybrid-7 common-emitters equivalent circuit of

Figure 4(b) for low temperature operation follows. The

transconductance, gm = qI/KT, increases with decreasing

temperature as T-1. The emitter-base junction resistance,

r, = 6/gm, decreases sharply with temperature due to the

decrease in 6 and increase in gm. The junction capacitance,

C = gmW2/2Db, varies very little above TM of equation (120)

because of the counteracting temperature dependencies of

gm and Db, but should increase sharply below TM following

the behavior of Db (see equation (129)). The base resistance,

rx, should have a minimum in the vicinity of TMg and then

increase with further decrease in temperature as per

equation (121). The cutoff frequency, ft, defined by

equation (45) as [ = gm/2"iTC should increase slightly

with decreasing temperature (if C is in fact approximately

temperature independent above TM ) and then decrease sharply

below T .

From equation (84) it can be seen that the main advantage

in operating the transistor somewhat below room temperature

is the reduction of noise generated by rx and the leakage

currents. The main disadvantage in that temperature range

is the reduction in B.
52
It is argued2 that the fall-off of at high currents

is due to high level injection problems. At cryogenic




U -t


temperatures when freeze-out sets in, high-injection

conditions exist. This fact, at least to a degree, will

be substantiated with experimental data. It is also argued

that the fall-off of B in npn silicon transistors is due

to trapping of electrons by donors in the base region.53

A more recent publication attributes the decrease in B

with decreasing temperature on shallow planar bipolar

transistors to tunneling currents through the emitter-

base junction.37 It is expected that possibly all of these

mechanisms take place simultaneously within a given device

and are probably interdependent on each other to a degree.




Low Temperature Effects on the Noise in Transistors


Since many of the parameters of the bipolar junction

transistor are highly temperature dependent, one would

expect the resulting noise performance which depends on

the sme parameters to also be quite temperature dependent.

This is observed from experimental results.

Plumb and Chenette54 have shown that the low frequency

flicker noise of the device (characterized by a 1/f) noise

power spectral, 1 being close to unity) is directly propor-

tional to the base current I raised to some power F

(1 < F < 2). For a constant emitter current, as the

temperature of the device is decreased, the current

amplification factor Bdc decreases, thus the base current






increases. Therefore we may conclude that at lower transistor

temperatures, the magnitude of the flicker noise generator

increases causing the frequency range, over which the

i/f spectrum is observed, to increase.

Also, as the temperature is decreased, recombination

of current carriers in the emitter-base space-charge-region

and at the semiconductor surface increases for low emitter

currents. (This is evidenced by the fact that at low emitter

currents, ade becomes much more current dependent upon IE)

These surface and space-charge-region recombination components

of the incremental base currents are seen only at low

emitter currents because they are less strongly dependent on

the emitter bias voltage VEB than is the bulk recombination

current. The rate of recombination in the neutral base

region, which is proportional to the excess minority-carrier

concentration, thus varies as pnoexp(qVEB/kT 1) for a

pnp transistor, and the corresponding "incremental" component

of recombination current varies ,with operating point as

noexp(qV B/kT) The average recombination rate within the
55
space-charge-region (and on its surface) is seen to be

proportional to no2exp(qVEB/2kT). Consequently, at large

values of VEB (and thus IE), the surface and space-charge-

region recombination components of the incremental base

current are masked by the bulk recombination current.

The space-charge-region recombination current component

becomes more important at low temperatures for the following

reason. For fixed VEB, the bulk recombination components of
EB'




OO


the incremental base current are proportional to the

equilibrium minority-carrier concentration Pno' which is,

in turn, proportional to the square of the intrinsic carrier

concentration, n2. The surface and space-charge-region

component are proportional to /P, or to ni. Therefore

at lower temperatures, the bulk recombination component

decreases much more rapidly than the space-charge-region

recombination component.

According to equation (27), the difference between the

incremental current amplification factor a and the d-c

current amplification factor adc is related to the current

dependence of adc. Thus the magnitude of the space-charge-

region recombination current is directly related to the

difference between ,o and a dc

Sah, Noyce and Shockley12 have considered this decrease

in ad at low emitter currents. They attribute the decrease

solely to recombination process of the Shockley-Read-Hall

(SRH) mechanismsi1,56 in the emitter-base space-charge-

region. Their expression for dc is

W J tanh Wb
a = sech ( ){ 1 + L } (132)
de L ~-J +J
b rg d

where Wb is the base width; Lb the minority-carrier diffusion

length; Jd the injected current density flowing into the

base; Jd the injected current density flowing into the

emitter; and Jr the recombination current density in the

emitter-base space-charge-region. The following three cases

are of interest:







i) For I very small, ad approaches zero since
E dc
Jd < rg

ii) For large forward bias, adc approaches unity

because Jd > > Jrg J

iii) For very large IE, adc decreases because

Jd J? > > J
d' d rg
Lauritzen57 has shown that for relatively low emitter

currents, the emitter-base recombination component of IE

should show less than full shot noise by a factor

3/4 < F < 1. He also states that if the predominant region

of recombination occurs at the semiconductor surface that

this should not effect his calculations as long as the

model of a single level SRH recombination center is valid.

The development for noise in a bipolar junction

transistor which includes the effect of Lauritzen's for

the recombination current component follows. The expressions

for the total currents of equation (54) remain unaltered,

however the noise current generator i7 of equation (57)

becomes

1 2qAf[l+exp(qV E/kT) + T (l1-a.) + REexp(qV /2kT)+
L EB ES f RES EB


+ IRES + kT(ge-eo

=2qAf [IE + (-l)IR + 2!EE] + 4kT(ge-geo)Af (133)

where


IEE iES (-af) + IRES and 3/4 < < 1







The noise current generator iT7 of equation (62) becomes

2 = 2qAf[ofIEsexp(qVEB/kT) + ICS(1-cf) + IRCS1

= 2qAf[IC + (-1)IRCS] 2qAfTC (134)

provided generation of carriers in the collector-base

transition region is negligible. The expressions for

the cross-correlation 1112 remains the same as equations

(63) and (67). The expression for the input noise voltage

emf, ee, of the modified noise equivalent circuit given

by equation (69) becomes

ee = 2kTAf{2(ge-geo) + qclE + (_-)IR + 21EE]/(kT)}I Z 2

(135)
Also the output noise current generator, iT, of equation

(70) becomes

2q Af{Ic + a E+( -()I+2IEE-2 a 2kTgeo/q

S 2q ci ,Af (136)

The correlation between these two sources as given by

equation (72) becomes

S 2kTaAf{- +2Z -Zq[IE+(l-1)IR+2IEE/(kT)} (137)
eo eeo e E R I2kT/f- Z(kT)g
Now the expression for the low frequency emitter conductance,

geo, of equation (73) remains unchanged, and when this is
substituted into equation (136), 1EQ becomes

EQ9 = IC [IE R] (138)

The factor has no effect on the calculation of the







equivalent noise resistance, RN, given by equation (81),

but it does show up in the expressions for the various

parameters of RN.

The expression for the equivalent noise conductance

of i2o given by equation (77) remains the same as that given

in equation (93). However the noise resistance rs1 of

e" defined by equation (96) becomes

rsl = {2(ge-geo)+q[IE+(-1)IR+21EE]/(kT)}IZe2 +

-{-1+2Z geo-Zq[E+(l)) I 2IEE (kT)
I E(139)
gsl
and equation (97) for Z becomes

{-l+2Z g -Z q[IE+(-1)I +2IEE]/(kT)}
Z (140)
sc 2gsl

Equations (98) through (101) involving a remain unchanged,

hence when the expression for IR given by equation (102)

is substituted into equation (138), the expression for the

high frequency IEQ becomes

1 1 22- C(dc
I_ = IEa Fa|d + 2a((1 d )] (CL)
Co
and at low frequency where a is replaced by ao,

I = E dc a 2 + 2a (a dc )] (142)

Equations (141) and (142) replace equations (103) and (104)

when recombination currents in the emitter-base transition

region become important. For the case when 5 is equal to

unity, equation (142) reduces to equation (104) as expected.







The low frequency expressions for r gslo r and
eo' slo2 slo
rsc as given by equations (105) through (109) in terms of

the factor become

kT a
reo = 0 (143)
qIE adc


q IEpo 1 a a +2a E(a -a )
Sq IEQo dc o o o +2a -dc) ] (144)
gslo ---
2kT a, 2reo o adc

o ((l-c )
r = r dc (145)
slo 2 eo a-0- z + &-0F W -o--- dc
dc o 2o o dc
and

ao(ao dc )
r = -ro dc (146)
l dc "o o + 2 (o dcd



Generation-Recombination Noise at Low Temperatures


As discussed earlier, at low temperatures the impurity

levels are not fully ionized. Thus generation-recombination

processes of the type


neutral impurity + energy free carrier + ionized impurity


will produce a fluctuation rbb in the base resistance
b '
(rbb). Because the base resistance is the one of higher

resistance, the fluctuations in the series emitter and

collector resistances can be neglected.

The development for the dependence of this generation-

recombination (g-r) noise on the other device parameters is

given, following van der Ziel,58 at this time. It is assumed






71

that the electric field in the base region is negligible.

At low temperatures this assumption isn't completely correct.

A more complete theory would include the effects of the

field-dependent mobility.

The appearance and disappearance of carriers in a

semiconductor sample by the process of generation and

recombination is described by a differential equation of

the form


dAN + AN = H(t) (147)
dt T

where AN is the fluctuation in the number of carriers,

H(t) is a random noise term, and T is the lifetime of the

added carriers. Using the Langevin method,59 for 0 < t < T

we now substitute the following Fourier series

CO 00
H(t) = Za exp(jw t) AN = Zb exp(jw t) (148)
,n nn n

where w = 27n/T (n = 0, 1, 2,...). Substituting these

into equation (147) yields

anT
b = T (1-49)
n 1 + jO) T
n
since d/dt = j n'

Now, by definition,

SH(f) = li 2T S(f) = lim 2T b b( (150)
H Tn n AN n n

So that when equation (149) is substituted into equation (150),


S AN(f) = SH(f) 2 ] (151)
1 + NT T
n






Since H(t) is a white noise source, SH(f) = SH(0). The

mean-square value of the fluctuation in the number of carriers

is given by
ix) co
varN= A- = I SN(f)df = SH(0)T f T df = SH(0)T/4
o o i+w H
(152)

so that

S (f) = T (153)
N I + WZT2


= 4aNT = SN(0) for wT << 1 (154)


Therefore we must evaluate T and w for equation (153).

Van der Ziel illustrates that this may be accomplished in

terms of the differential equation for the probability of

finding N electrons in the conduction band, the so-called

"Master's equation.' This results in the following

expressions:

g(N )
AT7 = (N No)T7 ____ o (155)
r'(No) g'(No)

and

S= 1 (156)
r'TN g'I(N

where No is the most probable value (average value) of N,

g(N)dt is the probability that an electron is generated in the

sample during the time interval dt, r(N)dt is the probability

that an electron is taken out of the sample by recombination

during that time interval, and g'(N) and r'(N) denote the

derivatives of g(N) and r(N) with respect to N. Now if we

consider an n-type semiconductor with Nd deep-lying donors,







then g(N) is proportional to (Nd N), the number of neutral

donors, and r(N) is proportional to N2, since there are

N free electrons and N ionized donors. Hence

g(N) = y(Nd N) r(N) = pN2 (157)

where y and p are'constants, so that

N (Nd No)
aN d 0 (158)
2Nd N

and

Nd N
S 1 = d o (159)
y + 2pN pN (2Nd N )

If the trapping cross-section of the donors is a, the constant

p is given by

p = a < v >
V

where is the average velocity of the conduction

electrons, = (3kT/m*)2, V is the volumn of the sample

and m* is the harmonic average of three principal effective

masses. If equations (158) and (159) are now substituted

into equation (154),


SAN(0) = 4TT = d N (160)
p 2H1, d NoZ

Now, the base resistance rb'b can be expressed in

terms of N as follows

rbb C (161)
b 'b
quN,

where C is a constant which depends on the base region

geometry, and p is the mobility of the host carriers. If

at a given temperature we take p to be constant, then




74

rb'b = C'/N, where C' = C/qp and N fluctuates around the

equilibrium value N Let 6N = (N No) be the fluctuation

in the total number of carriers and let 6' = aNo; the

fluctuation in the base resistance is then


Srbb = (rb,b-rbbo ) = ( C' C' ) = C'( -N ) = -rbb 6N
N NO NN- No

(162)

Since there is a fluctuation in the base resistance, the

flow of the base current IB will give rise to a noise

emf 6V = IB6rbb over and above the thermal noise of the

base resistance. The mean-square value of 6V is given

by V = I 6r bb Therefore, if r b' is independent
B b b' s
of current, the spectral intensity of 6V' should be

proportional to the square of the base current. Also,

from equation (162)

I r
6V = IBrbb B bb 6N (163)
No

From i this, the spectral intensity of 6V, Sv(f) = (V6/Af,

and the spectral intensity of 6N, S (f) -= 6Y/Af are related

by

I2 r b
V(0) = b S (0) (164)
No N

and from equation (160) and from equation (161)

1 N N
SV(0) = GIB ( C )2 [ d ]2 (165)
qp p (2Nd No) No







Now, in equilibrium, g(N ) = r(N ) thus
O O
y(N N) = p(N )2 or p = N -Z No (166)
d 0
Y No
Substituting this into equation (164)


S (0) = 4I2( C 2 1 P (167)
qp- p(2Nd No) [ ]2

where (p/y)-1 is the equilibrium constant for the ionization

process and is proportional to exp[-E/kT] for an activation

energy E. Rearranging equation (167) with the help of

equation (161),

No
S (0) = 4lIB rb'b( ) _= 4kTRNBAf (168)
2Nd No

where RNB is an apparent noise resistance which characterizes

this generation-recombination process at low temperatures.

When this effect begins to take place, Nd N is still

small and T of equation (159) is so small that the spectrum of

the noise is practically white. In fact, since the time

constant of this g-r process is so small, a thorough character-

ization of excess g-r noise source should be conducted at

relatively high frequencies as a function of frequency.

Since (Nd N ) is still small at temperatures where this

effect begins to take place (i.e., 600 to 1000K), the term

N /2Nd N ) of equation (168) is approximately unity, thus

for a given temperature

RNB = KI 'b r 2 (169)
where K is a proportionality constant. Since this g-r

process is predominant in the base region of the transistor,







it may be characterized in the noise equivalent circuit by

placing its equivalent noise emf 6V in series with the thermal

noise emf Sebb as shown in Figure 8.


Noise Parameter Determination at Low Temperature


The equivalent noise resistance R for the noise

model shown in Figure 8 is given by

4kTdRNAf = 4KT R Af + 4kTdrbbAf + 4kTdrslAf +


+ 4kTRNBAf + 4kTdgslAfERs+rb'b+re+rsc]2 (170)

where Td represents the device temperature and T the room

temperature. If we let

4kTdRN Af = 4kT RAf + 4kT dRAf (171)

where from equation (170)

R' = b'b + r1 + RB + s(R + r'b + re + rsC)2(172)
Writing RN as

RN = A' + B'R + C'R 2 (173)
and comparing equations (172) and (173), it is found that

A' = r b + rs + RN + + (r + + rs )2 (174)
b'b s1 NB "sl b'b re s

B' = 2gsl(rb'b + r + rsc) (175)

C' = gsl (176)

From equation (171), it is found that

T
R' = R R (177)
N N T- s
d




















ale


bb


FIGURE 8. NOISE EQUIVALENT CIRCUIT INCLUDING EFFECTS
OF GENERATION-RECOMBINATION PROCESSES IN
THE BASE AT LOW TEMPERATURES


-ii-ai.




78

Therefore, R' can be deduced from measurements of the
N
equivalent noise resistance R as a function of the source

resistance Rs.

The system of equations (174) through (176) can be

rearranged as

rb'b + re + rsc = B'/2C' (178)


rbb + RNB + r = A' B' (179)
NBb sT


gsl= C' (180)

The base resistance rb'b can be measured independently

by means of bridge measurements. By using the low frequency

value reo of re given by equation (106) together with the

computed value of rslo given by equation (108), rsco and

RNB can be determined with the help of equations (178) and

(179).

r = B' r r (181)
sco 2C' b'b eo
and

R A' B 2 r r (182)
NB CT b'b slo

The coefficient C' can also be found from measurements

of the equivalent output saturated diode current IEQ since

by equation (93)

q IEQ
C' = = (183)
slo 2--TT








Noise at High-Injection


As stated earlier, at low temperatures where the

current amplification factors for the transistor are low,

conditions for high level injection of minority-carriers

into the base region exist (i.e., majority and minority-

carrier concentration are approximately equal over most of

the base region). High level injection operation leads

to several undesirable effects. These may be enumerated

as follows.

1) At high level injection, the conductivity of the

base material is increased. This increase in conductivity

may be related to the diffusion length Lb of minority-

carriers in the base region through the average minority-

carrier lifetime since Lb2 = DbTb. Now Tb, as applied to

minority-carriers lost by recombination, is inversely

proportional to the number of majority-carriers in the

base region and, hence, inversely proportional to the

conductivity of the base material.60

2) For homogeneous base transistors, high-injection

of minority-carriers sets up an electric field in the base.

This field results due to space-charge neutrality in the

base region. It serves to maintain a gradient of majority-

carriers (to neutralize the minority-carriers) and is in

such a direction as to aid the flow of minority-carriers

across the base region. This has the net effect of increasing

f of the transistor.
a







For double diffused transistors however, the gradient

of impurity concentration in the base region gives rise to an

electric field in equilibrium. This field, which causes

drift-aided transport of the minority-carriers under low-

level injection conditions, is reduced in magnitude and has

a proportionately less effect on the flow of minority-

carriers as the injection level increases. This occurs

because the impurity charge distribution is dominated at

high-injection levels by the charge distribution which

results from the small deviations between excess hole and

electron concentrations. Thus the flow of minority-carriers

is aided considerably less at high-injection levels. The

transit time for double diffused silicon transistors has

been shown to increase by a factor of 2 to 3 due to this

field lessening effect.61 This results in a proportional

decrease in .

3) The effective diffusion constant of minority-

carriers in the base region increases at high-injection,

approaching a value twice that at low-injection levels.60

Since the minority-carrier lifetime is inversely proportional

to this diffusion constant, it decreases at high-injection.

4) The emitter efficiency, defined as the ratio of the

minority-carrier component injected into the base region

to the total emitter current, decreases at high-injection

due to the tendency of the large majority-carrier concentration

in the base to diffuse into the emitter region. The net

effect of this decreasing emitter efficiency with increasing







IE is a decreasing ade with increasing IE. Webster shows

that adc should decrease less with increasing IE for npn

devices than for pnp devices.60

5) High-injection currents in the B.J.T. are not

proportional to exp(qVEB/kT) as in the low-injection case

due to non-linear transport mechanisms as well as carrier-

concentration-voltage relationships which include ohmic

voltage drops and conductivity modulation effects.62

These currents are instead dependent upon exp[qVE~/(lm)kT]

where m is a parameter which depends upon the ratio of hole

and electron mobilities in the base region and V'B is the

applied voltage less the ohmic drops. For npn devices, T
C
is proportional to exp[qVEB/(l+m)kT] and for pnp devices, it

is proportional to exp[qVEB/(l-m)kT]. The value of m for

Si is approximately 0.45.

6) For homogeneous base transistors, if the electric

field in the collector-base SCR is large enough (- 103 volt/cm)

so that carriers drift at their saturated drift velocity

(= 107 cm/sec)63 at high-injection levels, this has the

net effect of widening the effective base width. This

occurs because when the drift velocity is limited, increasing

IC requires increasing the number of minority-carriers

undergoing transit of the collector-base SCR, so that the

space-charge in transit becomes comparable to the space-charge

of the immobile donors and acceptors in the SCR. This

changes the charge distribution in the SCR since mobile space-

charge adds to immobile charge density in the base portion







of the SCR and subtracts from this density in the collector

portion. Thus the entire space-charge shifts towards the

collector causing the base width to increase.

There are conflicting reports of base width widening

effects for Si double diffused transistors.61,64

7) High level current flow in the base region leads

to "emitter crowding" effects. This restricts basic

transistor action to the edge of the emitter region and

portions of the extrinsic base region (as opposed to the

intrinsic base region).

Several investigators have considered noise in bipolar

junction transistors at high-injection levels.34,65,66 At

least two of these34,65 have shown that the low-level

injection theory for the equivalent saturated diode current

at the output of the common-base configuration with the

input a-c open circuited--as given by equation (103)--

to also be valid at high-injection levels. The validity of

equation (103) at high-injection levels is quite surprising

since it was derived in terms of recombination currents in

the emitter-base space-charge-regions which are significant

only at low bias conditions and hence relatively low levels

of injection. (The component of recombination current

was shown earlier to be proportional to the increase in

adc with IE at low current levels; this in turn being

proportional to the difference in ao and adc via equation

(27).) At room temperature and at moderate currents, this

recombination component is usually insignificant compared to








the current density flowing through the emitter-base SCR

and hence a ade and equation (104) reduces to


IEQ = IE[dc ( adc)] (184)

It will now be shown that equation (103) for IE at
EQ
low-level injection is in fact valid at high-injection

levels where adc decreases with increasing IE due to the

already mentioned increase in the conductivity of the

base material which lowers the emitter efficiency, (At

these current levels, recombination current components in

the emitter-base SCR have long since been negligible.)

This development goes as follows. Consider the pnp device

shown in Figure 9(a) to be biased in the normal region.

The following groups of current carriers must be considered:

i) Holes emitted by the emitter and collected

by the collector: aIlESexp(qVEB/kT)

ii) Holes emitted by the emitter and recombined

in the base: (l-af)I Eexp(qVEB/kT)

At high-injection levels, an additional current component

must be considered. This component gives rise to the

decrease in the emitter efficiency at high-injection levels

and is caused by the large majority-carrier concentration

in the base region which has a tendency to diffuse across

the emitter-base SCR into the heavily doped emitter region.

If it is assumed that the diffusion length of electrons

in the heavily doped emitter is much less than the width




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