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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.
.... 191
.... 192
212
218
222
223
226
227
228
229
230
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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