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U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Introduction to Noise in

Solid State Devices

-QC —

100

.U5753

1169

1932
NATIONAL BUREAU OF STANDARDS

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OF STANDARDS

LIBRARY

Introduction to Noise in Solid State Devices H m fi

Julius Cohen

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1982
National Bureau of Standards Technical Note 1169

Natl. Bur. Stand. (U.S.), Tech. Note 1169, 63 pages (Dec. 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982

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

Price $4.75

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

Noise is the expression of fluctuations in the numbers or velocities of an ensemble of mobile electrons. Although it may be a fundamental physical phenomenon sometimes beneficiently exploited, as

in noise thermometry, it is its harmful aspects which account for so much activity aimed at its reduction. In electronics, for example,

noise establishes the lower limit of a signal which can be detected or amplified; in radiometry, noise sets the lower limit of radiative

power which can be detected with a sensor and adversely affects the

accuracy of measurements.

Its importance having been long recognized, there is a considerable body of published literature dealing with noise. Notwithstanding,

the theoretical reviews are by nature quite mathematical and tend to

be esoteric and abstract; the qualitative reviews generally are more assertive than explanatory.

This is a didactic monograph intended to be readable by physicists

and electronic engineers with only general background, and should be

of particular value to beginners of applied work in the field of

noise. Others who wish to gain an understanding of this important

phenomenon also will profit. The attempt has been made here to simplify

a complex subject, and the text is mainly qualitative. An aim is to

impart a "feel for the subject", and while the treatment generally is

not rigorous, it is not misleading. Although noise reduction as an ultimate goal is a main concern, the premise is that a basic understanding of noise processes forms a necessary base for the applications

engineer or scientist as well as for the device designer.

This monograph is divided into two parts. The first, Theory,

deals in detail with each of the four principal noises found in solid

state devices. The emphasis here is on phenomenological aspects of

noise, its origins, and its manifestations.

The second part, Applications, is a systematic treatment of noise

in selected solid state devices. Analyses progress from a single

noise source in a circuit element to four noise sources in a device;

concomitantly equivalent circuits are developed to facilitate the

solution of various complex noise problems. The devices considered

here were chosen as much for illustrative value as for commercial

importance or currentness. They reveal the evolutionary nature of

electron devices and the transfer of concepts from the earlier to the

later devices, thus providing a simplifying coherence. Because of the

way this work is structured, it should be read successively.

Thanks are due Jon Geist for suggesting that I write such a monograph and for his continued support. I appreciate also Edward

Zalewski's scrutinous review of the manuscript and helpful comments.

in
CONTENTS

Page

Preface iii

PART I. THEORY

1. Elementary Statistical Considerations 1

1 .

1

Terms 1 1.2 The One-dimensional Random Walk 2

2. Designations of Noise 3

3. The Four Principal Noises in Solid State Devices 4

3.1 Shot Noise 4

3.1.1 Operation of a Vacuum Diode 4

3.1.1.1 Thermionic Emission 4

3.1.1.2 The Three Ranges of Emission Current

Characteristics 6

3.1.2 Full Shot Noise 8

3.1.3 Partial Shot Noise 9

3.1.4 Equivalent Circuits: Shot Noise and Partial

Shot Noise 9

3.2 Thermal Noise (a.k.a. Johnson noise; Brownian noise) 9

3.2.1 Brownian Movement in Fluids 9

3.2.2 Einstein's Prediction of "Brownian Noise" in Solids . ... 10

3.2.3 Johnson's Discovery of Thermal Noise 13

3.2.4 Nyquist's Formula 13

3.2.5 Two Paradoxes 14

3.2.6 Comparison of Thermal- and Shot Noises 14

3.2.7 Equivalent Circuits of a Noisy Resistor 15

3.3 1/f Noise (a.k.a. flicker noise) 16

3.4 Generation-Recombination Noise 16

3.4.1 Mechanisms 16

3.4.1.1 The Two-level System 17

3.4.1.2 The Three-level System 18

PART II. APPLICATIONS

4. Noise Power, Noise Temperature, and Noise Factor 20

4.1 Noise Power 20

4.2 Noise Temperature 20

4.3 Noise Factor 21

5. Noise Theorems 21

TV
Page

6. Crystal Rectifiers 22

6.1 Operation 22

6.2 Shot- and Thermal Noise 23

6.3 Comparison of Theory and Experiment 25

7. Junction Transistors and Amplifiers 27

7.1 Operation of Junction Transistors 27

7.2 Noise in Junction Transistors 29

7.2.1 Frequency Spectrum 29

7.2.2 Equivalent-circuit Analysis of White Noise 30

7.3 Transistor Amplifier 32

7.3.1 Types of Connection 32

7.3.2 Noise in Transistor Amplifiers 32

8. Solid-State Photon Detectors 33

8.1 General Types of Photodetector 33

8.2 The Photoconductive Process 35

8.3 Measurements of Photoconductivity 37

8.4 Infrared Detectors 37

8.4.1 Detectivity 37

8.4.2 Small-signal Circuit Analysis 38

8.4.3 The Photovoltaic Effect 39

8.4.4 Noise in Photoconductive Detectors 39

8.4.5 Noise in Photovoltaic Detectors 42

8.4.6 The Ideal Detector 43

8.4.7 Amplifier-limited Noise 45

9. Concluding Remarks 45

10. References 47

Appendix , 49

A. Derivation of Thermal Noise 49

B. Derivation of Shot Noise 51
List of Figures

Page

1. Maxwellian velocity distribution of the emitted

electrons (Herrmann and Wagener [4]) 5

2. The three ranges of the characteristics. Electron

densities j e as a function of anode voltage for

different temperatures of the cathode; cathode work

function ^ 1.4 eV (Herrmann and Wagener [4]) 7

3. Constant-current generator equivalent of a diode

with temperature-limited current (Spangenberg [7]) 10

4(a). Constant-current generator equivalent of a diode

with space-charge-limited current; > p « 1 11

4(b). Resistor equivalent to a diode with space-chargelimited current (Spangenberg [7]) 11

5. Illustration of free path 12

6. Equivalent circuits of a noisy resistor

(Spangenberg [7]) 15

7. Two-level g-r problem (van Vliet and Fassett [19]) 17

8. (a) Three-level semiconductor;

(b) a phonon-assisted process 18

9. Schematic diagram of the equivalent circuit of a crystal rectifier; case of uniform contact potential 24

10. Schematic diagram of the equivalent circuit of a crystal rectifier; case of a distributed contact

potential (Torrey and Whitmer [21]) 26

11. Characteristics of typical n-type semiconductor

point-contact transistor (Hunter [23]) 28

12. Schematic diagram of two-junction PNP transistor 29

13. Noise factor vs. frequency (Woll and Herscher [22]) 30

14. Equivalent circuit for white-noise sources

(Woll and Herscher [22]) 30

VI
Page

15. Various types of photocells 34

16. Energy band diagrams illustrating photoexcitation in

intrinsic (a) and extrinsic (b) and (c) semiconductors 35

17. Illustration of the contribution of various noise

sources to the total noise spectrum (Bratt [26]) 40

18. The dependence of the detectivity of an ideal background-limited detector on cutoff wavelength (Bode [25]) .... 44

Al . Equivalent circuit of two noisy resistors,

R and R, connected by a transmission line 50

Bl . Thermionic diode circuit with plate resistor 52

vn

INTRODUCTION TO NOISE IN SOLID STATE DEVICES

Julius Cohen

Radiometric Physics Division

Center for Radiation Research

This is a short didactic monograph on electronic noise which

aims to impart a "feel for the subject". The work is divided into

two parts. The first, Theory, deals in detail with the principal

noises found in solid state devices, namely shot noise, thermal

noise, 1/f noise, and generation-recombination noise. The second

part, Applications, is a systematic treatment of noise in selected

solid state devices. Analyses progress from a single noise source

in a circuit element to four noises in a device; concomittantly

equivalent circuits are developed to facilitate the solution of

various complex noise problems. Examples treated in this part

include resistors, rectifiers, transistors, and photo-detectors.

The work concludes with a recapitulation and useful references.

Key words: electronics; noise; photon detector; rectifier; solid

state devices; transistor.

Part I. THEORY

1. Elementary Statistical Considerations

1.1 Terms

A few statistical terms first need to be defined:

A deviation is the difference between the arithmetic mean of a series of

statistical data and any single datum belonging to the series.

Fluctuations consist of deviations, and connote irregularities, an uncertain

shifting back and forth; e.g., temporal or spatial randomness of a physical entity

or attribute. Thus, fluctuations are irregular, or non-systematic, deviations. A

vivid example of fluctuations would be Brownian movement, or motion (to be described

in sec. 3.2.1).

When dealing qualitatively with fluctuation phenomena involving large numbers

of events, of which noise is an example, it is convenient to express data collectively in terms of a mean and some mean deviation from the mean. But the mean deviations (the arithmetic sum of the individual deviations in the series divided

by the total number of deviations or data belonging to the series) is nearly

zero, which per se carries little useful information and none whatsoever as regards the average magnitude of the deviations.

On the other hand, the mean square deviation, e.g., (y-y)

2 or alternatively,

the root mean square deviation, e.g., ((y-y)V^, does carry the desired useful

1
information, and therefore is used extensively in expressing fluctuations. To

express the degree of fluctuation (i.e., the percentage of fluctuation about the mean value) one need only divide the mean square deviation by the square of the

mean, e.g., ((y-y)

2 )/(y2

) •

1.2 The One-dimensional Random Walk

The following example, the one-dimensional random walk, should help to

make these statistical concepts more concrete. Picture a straight line with

centerpoint x=0 and let the opposite directions of the line with respect to the

centerpoint be designated the positive and the negative directions. Assume a particle, constrained to the line, to be located at the centerpoint at time t=0,

and thereafter to be struck repeatedly with a constant impact such that each

impact moves the particle randomly as to direction along the line, a fixed distance

± I. Then,

ln

= ± I, (1)

where I denotes the n-th step,

n Let x denote the total displacement after N steps. Then,

_N

n"l

«.<£ V (2)

The symbol < > will be used to signify the average taken over an infinitely large

group of many similar observations of the displacement x. Then

N

<x> =^<Zn > = 0. (3)

n=l

Now consider instead, the quantity, x

2

.

N

x

2 = (Z, + l 2 + .. + l H

) Ux + l 2 + .. + l

H

) = ^2 V + X Z nV (4)

n=l n^m
N

n=l n^m

N But,

and

J2 <V> = N l\ (6)

n=l

E<ln l> = 0*. (7)

n m

Therefore, <x

2 > = N I2

, (8)

which is an important fundamental equation relating the average magnitude of

displacement with the total number of individual random events.

2. Designations of Noise

Although there are but few fundamental sources of noise, there is a large

variety of noise processes and manifestations, and in this respect many particular

kinds of noises have been encountered. Their designations have stemmed, not from

systematics, but from natural linguistic processes; thus, particular noises may be

named for a person, a source, a process, or a manifestation.

The lack of a taxonomy 1 can obscure identity or relationship, as when a single noise has multiple designations, or a noise is a derivative, extension, or

special case of a more general noise. It is not the purview of this report to

treat the large variety of noises which has been reported; instead, this report is

confined to the common noises, specifically those which are prominent in solid

state devices. For these noises, where alternative designations are in use, they

will be indicated; also relationships will be given.

*Because each step is random as to direction, these products will occur with equal

probability. Hence <l I > = 0.

*An attempt to classify noise was made recently by van Vliet; see ref. 1, p 3.
3. The Four Principal Noises in Solid State Devices

The four principal types of noise in solid state devices are shot noise,

thermal noise, 1/f noise, and generation-recombination noise. They are explained

below.

3.1 Shot Noise

Shot 1 noise and thermal noise constitute the two most fundamental noises

and are the progenitors of most other noises. Both are intimately bound with the

structure and nature of matter: the former arises from the fact that electrons are discrete particles (governed by statistical laws); the latter is a kinetics phenomenon involving collisions between the free electrons of a substance and its lattice.

Shot noise will be treated first because of its historical precedence.

Vacuum tubes and solid state devices are more closely related than is generally

appreciated, and although shot noise was initially observed over 60 years ago in a thermionic vacuum diode, it is a common occurrence in modern solid state devices.

To gain an understanding of why shot noise exists, where it should be expected to

be found, and how it might be modified, it is instructive to consider the operation

of a simple vacuum tube as it relates to shot noise.

3.1.1 Operation of a Vacuum Diode

A vacuum diode consists of two elements in an evacuated envelope: the

cathode, the source of electrons; and the anode, the receiver, or collector of

electrons. The cathode is the "heart" of the tube; sometimes it is a metal,

usually it is a semiconductor. At the interface between cathode and vacuum there

exists a potential barrier which may be thought of as a force which keeps the

electrons from otherwise spilling out of the cathode. Electrons may escape from

the cathode in one of two ways: by going over the barrier or by tunneling through

the barrier, and both processes require energy to be put into the cathode. For the

former, either heat (thermionic emission) or light (photoelectric emission) may be

used; for the latter, an intense electric field (field emission) between cathode

and anode may be applied which reduces barrier width (and height).

3.1.1.1 Thermionic Emission

The case of thermionic emission, wherein the cathode is heated, will

be treated: Electrons are discrete particles, the collective properties of which

: The discoverer, W. Schottky, likened this noise to the audible effect made by an impinging stream of shot (schrot, in German).
are described by statistical laws. Thus, their velocity (and energy) distribution

in a solid normally is given by Fermi -Dirac statistics for a metal, while classical

Maxwell ian statistics may be used for a semiconductor [2]

1

. Heating the cathode

liberates electrons internally and imparts energy to free electrons through the

process of random atomic collisions by the lattice, and only those electrons which

reach the vacuum interface with energy component normal to the cathode surface

sufficiently high to surmount the barrier, will emerge. The emission is random in

time as it is inconceivable that equally large numbers of electrons could reach the

surface possessing the required energy and direction, in equally minute increments

of time. And regardless of what the form of the electron distribution was inside

of the cathode, the electrons emerging into free space will assume a Maxwell ian

distribution of velocities [3].

%

0.5

0.4

N

i

e 0.3

0.2

0.1

\t = 1000K

Vr = 2500

K

100 200 300 400 500 km/s

Figure 1. Maxwell ian velocity distribution of the emitted electrons

(Herrmann and Wagener [4]).

Figure 1 shows plots of the Maxwelliam velocity distribution of the emitted

electrons for two different cathode temperatures; N(v) is the number of emitted

electrons having velocity v, and N is the total number of emitted electrons.

figures in brackets indicate the literature references at the end of this paper,

5
Thus, the ordinate gives the fraction of emitted electrons possessing a velocity

corresponding to v, or alternatively, the probability1 of an emitted electron

having any particular velocity. The curves show that the probability falls some- what rapidly either side of the maximum probability, a characteristic which becomes

more pronounced the lower the temperature. Hence, the totality of emitted electrons

may be treated approximately as being endowed with the most probable velocity. In

addition, the fact that the emitted electrons obey Maxwell's distribution law

constitutes a proof that they are free; i.e., each emitted electron is independent

of the others.

So far, only the emission of electrons from the cathode has been considered,

above. To obtain an emission current, however, requires that the electrons be

collected at the anode, and the role of the anode in determining emission current

characteristics will be considered next; after which the shot noise may be treated.

3.1.1.2 The Three Ranges of Emission Current Characteristics

First, in order to obtain an emission current, a voltage is required

to be applied to the anode. This determines the distribution of potential in the

interspace, thereby influencing the collection of the emitted electrons. Thus, the

operating conditions of a diode are cathode temperature, which (for a given cathode

material) establishes the rate of electron emission, and anode voltage, which

determines the fraction of emitted electrons collected. The operating conditions

determine the operating characteristics (of the emission current) of the diode, and

three different regions of characteristics may be distinguished, as shown in figure

2: (1) Saturation, or temperature-limited region — If an adequate positive voltage

is applied to the anode, all of the electrons emerging from the cathode surface

will reach the anode. As shown in the figure, the emission current at fixed temperature increases slowly with increasing anode voltage; i.e., saturates, and the

magnitude of the emission current is essentially determined by the cathode temperature. (2) Space-charge-limited region -- If the anode voltage is not sufficient

for saturation, only a fraction of the emitted electrons can reach the anode, with

the remainder forming a negative space charge in front of the cathode. This causes

the potential barrier in the interspace to increase; the less energetically emitted

electrons then cannot surmount the barrier and fall back to the cathode, thus

reducing the emission current. The curves of figure 2 show the space-chargelimited currents to be quite sensitive to anode voltage.

Maxwell's law was originally deduced purely from considerations of probability;

see ref. 5.
+ 6 VOLTS

«~VA

Figure 2. The three ranges of the characteristics. Electron

densities j as a function of anode voltage for

different temperatures of the cathode; cathode work

function ^ 1.4 eV (Herrmann and Wagener [4]).
(3) Retarding field region — If the anode voltage is negative or only weakly

positive, a substantial potential barrier is presented to the emitted electrons, of

which only the very energetic can reach the anode, and the emission current is

drastically reduced. This region is mentioned only for completeness; it is of

little practical importance and need not be discussed further.

3.1.2 Full Shot Noise

As concerns noise, the saturated region is of particular significance

because only in this region can "pure", or full shot noise be obtained. The

reason, alluded to above, is that only in this region do the emitted electrons

behave as totally free, random particles. In other words, only the saturated

current is comprised of electrons which have been randomly emitted in time and

which remain free, from emergence from the cathode to collection at the anode. For

an assembly of particles so described, a well-known theorem in statistics is applicable, which states that the mean square of the fluctuations in a number of independent, random particles is equal to the mean number of particles; i.e.,

<(n - N) 2 > = N, (9)

where n is the number of electrons emitted in a time interval t, and N is the mean

value of n. (Note the similarity of this equation with that for the 1-dimensional

random walk, previously derived.)

Fluctuations in current are equivalent to noise. Current, by definition, is

proportional to the number of electrons and the (transit) velocity; however, as argued earlier, all of the emitted electrons may be assumed to have approximately

the same velocity. Therefore, the fluctuations in current are essentially fluctuations only in the numbers of electrons, and the source of the noise is the random

emission of individual electrons from the cathode. The electrons, emitted from the

cathode in pulses, are collected at the anode in pulses. Further, from the above

statistical theorem, the fluctuations in current--hence, the noise--should be

proportional to the square root of the average, or steady, current.

An equation for calculating the shot current/noise can be found by expanding

the current pulse associated with the passage of an electron into a Fourier series,

and adding the terms of this series lying in a certain bandwidth 1. Schottky [6],

examining the noise in the saturated region, derived the following formula for the

*For a detailed derivation of shot noise, see section B of the Appendix,

8
mean shot current within a bandwidth Af (equal to the frequency band f to f+Af, and

during measurement determined by the bandwidth of the amplifier):

T

sh

=

( 2eI

s Af) ^' (10)

where I is the saturated current of the diode (also, the steady, average emission

current in the saturated region).

Thus, the shot current is independent of the frequency 1 (as expected, because

of its random nature), and, as expected, determined only by the magnitude of the

saturation current and the bandwidth. The dependence of the shot noise on the

square root of the current is a unique characteristic.

3.1.3 Partial Shot Noise

The validity of Schottky's equation has been experimentally verified

and is well established [7]. Nevertheless, vacuum tubes are seldom used as temperature-limited current devices; the space-charge-limited mode is normal. When

a diode is operated in the space-charge-limited mode, it exhibits much lower noise

[7] than it would if it were operated in the temperature-limited mode passing the same emission current2

; as in shot noise, the noise current is frequency independent.

The mechanism whereby space charge causes reduction in noise is not well understood; a possible explanation which is rather involved is given elsewhere [7].

3.1.4 Equivalent Circuits: Shot Noise and Partial Shot Noise

Finally, some equivalent circuits for a diode with temperature-limited

current (shot noise) and a diode with space-charge-limited current (reduced shot

noise), respectively, are given in figure 3 and figure 4 (a) and (b).

3.2 Thermal Noise (a.k.a. Johnson noise; Brownian noise)

3.2.1 Brownian Movement in Fluids

In 1827, the English botanist, Robert Brown [8], observed a phenomenon

which was to have great significance. Although Brown initially looked at microspores (pollen) in water, the phenomenon can be readily observed if any small

particles suspended in a liquid are examined with a powerful microscope. The

particles are seen to be darting about to and fro, spontaneously, randomly, and

continuously. Each particle spins, rises, sinks, and rises again. This irregular

frequency-independent noise is commonly termed white noise.

2 Strictly, shot noise refers to the noise accompanying temperature-limited, or saturated, emission current; however, the reduced noise is referred to as "partial

shot noise" and "reduced shot noise".
rms =V2eiQAf

INFINITE IMPEDANCE

CONSTANT CURRENT

GENERATOR

TEMPERATURE

LIMITED

CURRENT

Figure 3. Constant-current generator equivalent of a diode

with temperature-limited current (Spangenberg [7])

motion is referred to as the Brown i an movement/motion of the particle. It is

independent of the nature, or source, of the suspended particles, all particles of

the same size being equally in motion; smaller particles, however, move more

vigorously than larger ones.

A valid explanation for the phenomenon was not forthcoming for nearly eight

decades after its discovery, in spite of the many attempts made by various investigators. Indeed, Brown himself misinterpreted the origin of the phenomenon he had

discovered. It was Einstein [9] who proved in a series of classical papers that

Brownian movement arose directly from the incessant and random bombardment of the

particles by the molecules of the surrounding fluid. Thus, the phenomenon of

Brownian movement is a direct proof of the existence of random molecular motion,

and thereby of the discrete molecular structure of matter.

3.2.2 Einstein's Prediction of "Brownian Noise" in Solids

Einstein proceeded to argue that an analagous Brownian movement must

exist inside a solid by virtue of collisions between the free electron particles of

the solid and the atoms comprising its lattice. Two further assumptions were made:

10
rms = V2eloAf0

INFINITE IMPEDANCE

CONSTANT CURRENT

GENERATOR

SPACE CHARGE

LIMITED CURRENT

Figure 4(a). Constant-current generator equivalent of a diode

with space-charge-limited current; > B « 1

R

d\ [ -*- av

erms =V4kRTcAf

(0(0

N ZERO

IMPEDANCE

CONSTANT

VOLTAGE

GENERATOR

SPACE CHARGE

LIMITED CURRENT

Figure 4(b). Resistor equivalent to a diode with space-charge-limited

current (Spangenberg [7]).

11
1) that the free electrons behaved like a perfect gas--thus investing otherwise

sterile particles with pressure or force on the one hand and temperature or energy

on the other; 2) that the degree of thermal agitation of the electrons was influenced by a "viscosity" of the medium (here, its conductance—the reciprocal of

resistance). Then, by extension of his random-walk analysis for the Brownian

motion in liquids, Einstein derived the following equation for the fluctuations in

charge 6q, in a material of conductance G, during an interval t, thus predicting

the existence of thermal noise in a solid:

<6q 2 > = 2kTGt. (11)

Although this equation is not in a practicable form, owing to a time rather

than a frequency interval, it is nevertheless correct (and is consistent with the

usual and useful form that was derived years later by others).

Brownian movement is spontaneous and eternal; so too, thermal noise. It may

be reduced, but short of destroying the matter which it infests, it cannot be

eliminated.

Figure 5. Illustration of free path.

One can visual ize--simply, yet physically justifiably—hordes of miniscule

particles, free electrons suspended in interstitial space, falling into collosal,

sluggishly vibrating atoms from which they bounce off wildly in any and all directions, repeating the process, ad infinitum. A schematic diagram of the Brownian

movement of a single particle is depicted in figure 5. The path traversed between

two successive collisions is a straight line (described with constant velocity),

and the course consists of a series of zigzag paths of various lengths. The

average length of a path is the mean free path, defined as the sum of the lengths

12
of a large number of paths divided by the total number. (This quantity is of great

importance in transport phenomena, in general.) Just as in the 1-dimensional

random walk, the result after many collisions will be that there is no net current--

each motion by an electron being compensated by an equivalent opposite motion by

another electron--but there is a fluctuating current, attributable primarily to the

paths or directional changes in velocity, not to the number of free electrons,

which remains essentially constant.

3.2.3 Johnson's Discovery of Thermal Noise

Some 20 years after Einstein's prediction of thermal noise--perhaps

detection was not technologically feasible earlier--J. B. Johnson [10], working at

The Bell Telephone Laboratories, measured the thermal noise in a wide variety of

conductors, and determined empirically that the mean-square potential fluctuation 1

across the conductor is proportional to the electrical resistance (the inverse of

conductance) and the absolute temperature of the conductor; it is independent of

the size, shape, and material of the conductor. These findings are in full agreement with Einstein's prediction: the "viscosity" of the conductor shows up here as resistance, the inverse of conductance, because fluctuations in voltage, not in

current, are described; and just as the nature or origin of the particles was

immaterial in their Brownian movement in fluids, so where the electrons came from

is irrelevant here.

3.2.4 Nyquist's Formula

Diverse disciplines such as thermodynamics, statistics, and statistical

mechanics have each been used to derive the (same) equation for thermal noise

[11,12,13]. Nyquist's [11] pithy deduction of the formula using thermodynamical

reasoning is, however, renowned for its elegance 2. The well-known expression for

thermal voltage noise in a conductor of pure resistance R and of temperature T is

v

2" = 4kTRAf, (12)

where v

2 is the mean square of the voltage fluctuation within the frequency range

Af and k is Boltzmann's constant.

Actually, noise phenomena originate as fluctuations in current, not voltage; however, it is more convenient to speak of a disturbance to an amplifier as a voltage

fluctuation at the input rather than as a current fluctuation at the output.

2 For a detailed derivation of thermal noise, after Nyquist, see section A of the

Appendix.

13
It will be noted that such quantities as charge, number, and velocity of

electrons do npt appear explicitly in the formula for voltage or noise; they are,

however, implicit in R.

The above noise formula refers to the open-circuit voltage across the resistor

R; alternatively (by Ohm's law), the noise may be expressed as the short-circuit

current through the resistor R:

~~

2 _

i = 4kTGAf. (13)

3.2.5 Two Paradoxes

It is instructive to consider the resolution of two paradoxes relating

to measurements in thermal noise. The first, with metaphysical overtones: If,

indeed, there is no net current in a passive conductor in thermal equilibrium and

the amount of current drawn by the measurement instrument is negligible, how is the

noise current/ voltage able to be measured? The second paradox: The formula for

thermal noise is derived for a passive conductor in thermal equilibrium. Why

doesn't the passage of a steady (small) current through a resistance increase the

fluctuation voltage (it remains the same)?

The answer to the first question is to be found in Nyquist's theorem [11]; it

is because thermodynamically there must be a transfer of power from the "noisy"

resistor to the input terminal of the instrument, considered initially noiseless.

Alternatively, from a circuitry viewpoint, the 'noisy' source resistor may be

thought of as becoming incorporated into the circuitry of the measurement instrument.

The answer to the second question was given by Bell [14], and is restated

here. It is that within a (solid) conductor, collisions between free electrons and

atoms are extremely frequent, and the drift velocity acquired under an electric

field is small compared with the thermal agitation velocity. Thus, the number of

random events, fluctuations, is insensitive to the average current passing through

the conductor. The effect of passing a current may be likened to walking while

holding a screen on which a motion picture is being shown; the movie doesn't

change, it merely translates.

3.2.6 Comparison of Thermal- and Shot Noises

Thermal noise and shot noise have been seen to have commonalities as well as significant differences. On the one hand, both are of fundamental nature,

derive from the random events of free electrons, and (because of the randomness)

14
are frequency independent. On the other, thermal noise occurs in a passive (or

active) resistor and is unaffected by passage of a steady current; shot noise

occurs only if a current is passed, and the noise is proportional to the square

root of the steady current. Thermal noise is thermal agitation noise, originating

in collisions between free electrons and lattice atoms at temperature T which

produce fluctuations in the velocity of free electrons; shot noise originates in

the fluctuations in the number of free electrons. Notwithstanding the contrary

assertions of early investigators [15], thermal noise and shot noise are two

distinctly different noises (and attempts to unify them result in physically

unpalatable interpretations).

3.2.7 Equivalent Circuits of a Noisy Resistor

The equivalent circuits of a resistor with thermal noise, of which

there are two, are shown in figure 6. As shown, the effect of thermal noise in a resistor can be expressed either as an emf in series with the resistor, considered

noiseless, or as a constant-current generator in parallel with the resistor,

considered noiseless.

ZERO IMPEDANCE

CONSTANT VOLTAGE

GENERATOR

'rms =OvSkTVRAf

INFINITE IMPEDANCE

CONSTANT CURRENT

GENERATOR

rms = V4kTrAf/R

Figure 6. Equivalent circuits of a noisy resistor (Spangenberg [7])

15
3.3 1/f Noise (a.k.a. flicker noise)

The distinguishing characteristic of 1/f (one-over-f) noise is its

frequency response; the square of the noise current is approximately proportional

to the inverse of electrical frequency. It is also approximately proportional to

the square of the steady current. These fluctuations were discovered by Johnson

[16] while studying vacuum tubes, and have been investigated in detail by others

[17,18]. If (atomic film- and oxide) cathodes are examined at low frequencies

(< 10 kHz), fluctuations are observed which are (often considerably) larger than

that given by the shot-current equation, and as for shot noise from vacuum diodes,

1/f noise is lower in the space-charge current region than in the saturated

current region. Further, experiments suggest that 1/f noise increases with

increasing temperature. Johnson attributed the 1/f noise to fluctuations in

cathode work function.

Unlike thermal noise and shot noise which are fundamental and intrinsic, 1/f

noise is apparently extrinsic, being particularly sensitive to surface phenomena

and imperfections. Thus, thin film- and polycrystalline devices are especially

susceptible. The unwanted phenomenon is commonly encountered in widely used solid

state devices such as photodiodes and the various mutants of field-effect transistors, and is being intensively investigated. Previously, 1/f noise had been

observed in many other devices such as metal and carbon resistors, crystal rectifiers, vacuum tubes, and photoconductors 1

. The phenomenon has been attributed

variously, as appropriate, to variation in contact potential, ionic conduction/

migration, instabilities of (metal point) contacts, surface states/conduction,

tunneling into traps, dislocations, diffusion of defects, defect-defect scattering,

and fluctuations in mobility. Notwithstanding, the mechanisms—apparently there

are more than one--have not yet been adequately explained.

3.4 Generation-Recombination Noise

3.4.1 Mechanisms

Generation-recombination noise, commonly referred to as g-r noise 2 is

caused by fluctuations in the generation, recombination, and trapping of carriers

*As an example of the universality of 1/f phenomena, T. Musha has shown that a [Japanese] man standing on only one leg generates a 1/f spectrum; see ref. 1,

p 143.

2 Sometimes, especially in the early literature, g-r noise is referred to as shot

noise.

16
in homogeneous semiconductors (and insulators), particularly photoconductors,

which in turn cause the number of carriers to fluctuate. Hence, the conductance

of the sample fluctuates. The mechanism underlying these processes is the quantummechanical transitions of electrons between designated energy states of the con- duction band, the valence band, localized impurity levels, traps, or recombination

centers. The transitions of concern in g-r noise are specifically transitions

between the conduction band and the valence band, and between impurity levels (or

traps, or recombination centers) and the conduction- or valence band. The electrons making transitions spend various amounts of time in each state they occupy,

according to the designation of the state (i.e., the conduction band, the valence

band, etc.) and the number of states per energy level, and the occupancy of the

states depends on the particular system studied.

G-r noise may be regarded as a generalized form of thermal noise in which the

Fermi level is caused to fluctuate. In g-r noise, however, the conductance of the

solid fluctuates because of the fluctuations in the number of carriers, while in

thermal noise the number of carriers is essentially constant, and the noise stems

from fluctuations in the velocity of the carriers.

3.4.1.1 The Two-level System

Formulas for calculating g-r noise are usually derived with the use of probability statistics, and the simplest system to consider is the two-level

system in a semiconductor where carrier transitions can occur only between two

energy levels, as previously specified; e.g., between the valence band and the

conduction band or between the conduction band and a set of impurity levels. This case is depicted schematically in figure 7. Here E and E

l denote two energy

N ELECTRONS

ENERGY, £-*- -, **. •- f- -*- -•- Z STATES

RECOMBINATION

N + N' = NT Z,Z'>NT /-N ' ELECTRONS

ENERGY, £ — SC -» -*- --•- —Z' STATES

Figure 7. Two-level g-r problem (van Vliet and Fassett [19])

17
levels containing respectively Z and V possible states; the numbers of electrons

occupying these «states are denoted respectively by N and N', their sum N

t being

constant. Only one electron is allowed per state (in accordance with the Pauli

exclusion principle), and not e\/ery state is occupied by an electron. A transition from E\ to E is a generation, and that from E to E

% is a recombination.

The probability of a transition is usually assumed to be proportional to the

product of the number of electrons available to make the transition and the number

of unoccupied states in the other level which can accept the electrons. Thus,

expressions for the generation- and recombination probabilities can be derived,

and an expression for the variance in the number of carriers <A N

2 > obtained, as

well as noise expressions for particular cases [19].

The solutions are usually of academic interest, however, because of the

apparent lack of experimental confirmation for the existence of the two-level

model 1. Indeed, many experimental studies [20] of g-r noise in various systems

suggest that either the two-level model is an unrealistic simplification or the

noise phenomena are obscured by other processes; e.g., space charge, impact

ionization, or surface recombination.

3.4.1.2 The Three- level System

It is desirable therefore to consider a semiconductor system wherein

electronic transitions take place between three or more energy levels, and two

examples are depicted schematically in figures 8(a) and (b).

CONDUCTION

BAND i i

1

' 1 :

i CONDUCTION

BAND

' '

\

E/ALENCEIAND

VAL

BAM

ENCE

ID

(a) (b)

Figure 8. (a) Three-level semiconductor; (b) a phonon-assisted process,

*In some semiconductors; e.g., GaAs, band-to-band recombination can be important;

in Ge and Si this process is improbable.

18
Figure 8(a), illustrating a three-level system, depicts an extrinsic semiconductor with electrons being generated from both the valence band and the impurity

level and recombination taking place from the conduction band to the impurity

level

.

Figure 8(b), illustrating a more complicated many-level system, depicts a semiconductor with an electron making a transition to a lower energy level releasing

phonon energy in the process which is absorbed by another electron, in another

energy level, which then makes a transition up into the conduction band; i.e., the

latter transition is a phonon-assisted transition.

The derivation of equations for fluctuations in a three-level system is already

complex, and the complexity increases inordinately with each additional level

assumed to be involved in the transition process.

Van Vliet and Fassett [19] have solved a three-level system problem where

transitions are assumed to occur between the two bands and one set of impurity

levels (or traps, or recombination centers). This can represent, for example, the case of indirect recombination in a semiconductor with generation taking place from

the valence band to the conduction band, and recombination taking place via recom- bination centers. (In contrast, only direct recombination can occur in an assumed

two-level system.) Using mathematical methods similar to those used for the twolevel system, outlined above, for this particular case of indirect recombination,

an expression of the following form is obtained for the noise spectra G(oj):

G(u>) «f^4 T— + 6, (14)

n + p 1 + o) 2 t

2

where go is (angular) frequency; n and p denote respectively the average number of

electrons in the conduction band and holes in the valence band; t is a composite

relaxation time for the carriers; i.e., a complicated function of the numbers of

each type of carrier and the relaxation times of each, and 6 is a small term. If

6 were neglected, the above equation would resemble that for g-r noise spectra in an intrinsic, two-level semiconductor.

19
PART II. APPLICATIONS

4. Noise Power, Noise Temperature, and Noise Factor

In the measurement of small signals, electron devices are normally used in

conjunction with amplifiers, and it is often desirable to know the noise limitation

imposed by each. The noise of a device or component may be characterized by its

noise power, noise temperature, or simply, noise, while it is convenient to characterize the noise of an amplifier (which depends on the signal) by its noise

factor (a.k.a. noise figure). These specialized terms are defined forthwith.

4.1 Noise Power

Nyquist [11] showed that the available noise power P from a resistor R

(into a "cold", or "noiseless", load) in an incremental band of frequency Af is

given by P=kTAf.

Comparison of this expression with those for thermal noise [eqs. (12) and

(13)] indicates that the available noise power may be given also by

and P = ^ . (16)

The two expressions immediately above have been shown [21] to have general

validity; i.e., they hold for any impedance, not just a pure resistance, and for

any type of noise, not just thermal noise.

4.2 Noise Temperature

Noise temperature t

N is defined as the ratio of the available noise power

output of a device or component to that of a resistor at room temperature; i.e.,

l

H

= kT^Af

' ( 17)

It is convenient to choose as a standard temperature T = 290°K, in which case

kT = 4 x 10~ 2 joule.

Noise temperature is actually a ratio, not a temperature. The rationale for

the name is that the product t»T is the temperature a passive resistor would have

to have in order to generate as much thermal noise as the noise output of the

device or component in question.

20
4.3 Noise Factor

Noise factor, or figure, is a quantity which compares the noise from an actual amplifier with that from an ideal, i.e. noiseless, amplifier. This ideal

amplifier is assumed to be identical to the actual amplifier, with the sole exception that it does not add any noise to the signal. Thus, if a source resistance is

connected to the input terminals of a noiseless amplifier, the noise power output

of the noiseless amplifier will be due only to the thermal noise in the source

resistance.

The noise factor NF of an actual amplifier is defined as the ratio of the

noise power output of the actual amplifier to the noise power output of the ideal

(noiseless) amplifier.

Equivalently:

The noise factor NF of an amplifier is the ratio of the noise power at the

output of the amplifier to the available thermal noise power of the source resistance; i.e.,

NF w •

< 18 >

where T is the absolute temperature of the source resistance and P is the noise

power measured at the output (corrected for gain) of the amplifier.

Noise factor depends on the value of source resistance, so any specification

of NF must include the value of the source resistance used for the measurement. It

is customary to express NF in decibels.

Methods of measuring NF of an amplifier, along with pitfalls in measurement,

are given elsewhere [22].

5. Noise Theorems

To calculate the total noise or the total available noise power from a combination of noise -sources, the following theorems are used:

Theorem 1. If a number of noise sources of differential (dynamic) resistance

Pk are connected in parallel, the mean squares of the noise currents are additive;

i.e.

,

^=1? (parallel), (19)

k K

21
and the total available noise power P is

p - pE i

k p

k

p

k P

k

(parallel). (20)

Theorem 2. If a number of noise sources are in series, the mean squares of

the emf's are additive; i.e.,

v1* = z2 v. 2 (series), (21)

k

K

and the total available noise power is given by

F ?

p

k pk

k K

(series). (22)

6. Crystal Rectifiers

6.1 Operation

Owing to its historical importance, the early "cat whisker" type of

crystal rectifier serves here as a first example for theoretical analysis of noise

in a solid state device. This simple device consists essentially of a semiconductor; e.g., germanium or silicon, contacted by a sharp metal point, or whisker.

As for any two dissimilar materials when a metal and a semiconductor are brought into contact there will be an initial transfer of electrons from the

material of lower work function (in this case, the semiconductor) to that of higher

work function in order to establish equilibrium, as required by thermodynamics. A

result of this process is the establishment of a potential barrier at the interface

of the metal and semiconductor. This potential barrier is called the contact

potential barrier and its height, the contact potential difference, will be given

roughly by the difference in the two work functions. It is a characteristic of

22
this barrier that when a voltage is applied across it, the current passed in one direction will be larger than that passed in the other, a property regarded as rectification. Regardless of the polarity of the applied voltage with respect to

metal and semiconductor, an electron current will flow from metal to semiconductor

and another from semiconductor to metal, and the net current will be the difference

in the two currents of opposite directions.

The width of the contact potential barrier is quite small; in fact, less than

the mean free path of the electrons, so collisions (by the carriers) in the barrier

layer can be neglected. Carriers having sufficient kinetic energy swiftly cross

the barrier unimpeded and are immediately collected, so there is no interfering

space-charge build-up.

The process wherein electrons traverse a potential barrier is termed emission,

irrespective of whether the electrons emerge into a vacuum or another region of the

solid state device. The latter process is called internal electron emission to

distinguish it from the former. In the particular case of the crystal rectifier

under discussion, the nature of the emission current is analagous to that of the

temperature-limited vacuum diode, previously treated: the electrons are emitted

randomly and remain free during transit. Hence just as for the vacuum diode, shot

noise is expected to occur in the crystal rectifier. (In general shot noise may be

associated with solid-state junctions; e.g., pn junctions and planar metal-semiconductor junctions.)

In addition to the rectifying barrier, there is a spreading resistance r in

the semiconductor resulting from the constriction of current-flow lines in the

semiconductor adjacent to the contact. When a voltage is applied across the

device, there will be an Ir drop across this resistance, with the remainder of the

applied voltage falling across the barrier.

6.2 Shot- and Thermal Noise

In the present analysis 1

, for simplicity, only shot noise produced by the

barrier and thermal noise caused by the ohmic spreading resistance, will be considered; further, the contact potential barrier will be assumed to be uniform, and

the excitation will be assumed to be d.c. Figure 9 represents the equivalent

circuit of the white noise components of the crystal rectifier.

1 Based on the treatment of Torrey and Whitmer, loc. cit.

23
Figure 9. Schematic diagram of the equivalent circuit of a crystal

rectifier; case of uniform contact potential.

The shot noise equation for the barrier is

V = 2 e I Af

,

(23)

where I = Ii + I 2 , (24)

Ii and I 2 being respectively the electron current from the semiconductor to the

metal, and from the metal to the semiconductor. The currents add because noise is

produced each way.

The available noise power P. from the barrier alone is, by eq. (16)

P b

= h eIRAf, (25)

where R is the differential resistance of the barrier.

The available noise power P from the spreading resistance r is

P

p

= kTAf. (26)

The total available noise power P, obtained by using eq. (22), is

= h eIR* + kTr

f (27)

R + r

The noise temperature t

N is, by definition,

*N

= kT~Kf ' (28)

o

where T = 290K. Therefore,

24
t», =

2kT

Q

IR 2 + r(J ) o

R + r (29)

The net current I' traversing the barrier is given by the approximation [22]

I' « A [e

aeV/kT

-1] , (30)

where A and a are constants incorporating the contact potential difference, and V

is the voltage across the barrier .

On the other hand, I is given by

I * A [e aeV/kT

+ 1] . (31)

The differential resistance R of the barrier is

R VdV/ (32)

Therefore, from eqs. (30), (31), (32), and (25), the following formula is

obtained for the available noise power from the barrier:

P b

= h kTAf [1 + e"

eV/kT

] . (33)

Three cases may be demarcated

1) V = 0;

2) V > and large;

3) V < and large;

P b

= kTAf

P k = h kTAf

P b * h kTAf[e" eV/kT

]

(34)

(35)

(36)

For these formulas to hold, V must be less than the contact potential difference; otherwise the barrier no longer contributes noise.

6.3 Comparison of Theory and Experiment

The equations for t

N and P. (noise temperature and noise power) are only

in qualitative agreement with experiment, the measured noise being higher than that

predicted by theory.

25
A simple model was used initially to avoid undue complexities in the mathematical derivations, but now the need for extending the model has been indicated.

Previously, the effect of image force on barrier lowering was neglected; however,

if this effect is included, the agreement is only slightly better. The assumption

of a uniform contact potential was another simplification. Owing to unavoidable

surface contamination and defects, the work functions and the contact area are not

uniform. Hence, the equivalent circuit of the white noise components is more

realistically represented by a distributed contact potential, as shown in figure

10.

_I

Figure 10. Schematic diagram of the equivalent circuit of a crystal rectifier; case of a distributed contact

potential (Torrey and Whitmer [21]).

Now the total noise power P. of the barrier must be found by using both noise

theorems (sec. 5) to sum the contribution of the rectifying elements of different

contact potential with their associated spreading resistance. Thus theory can be

reconciled with experiment, but the elemental contacts are not unique, and their

parameters difficult to verify. The solution to the distributed contact potential

problem is complicated and its presentation here not justified.

Even the distributed contact potential model, already complicated, is a crude

representation because the contact potential barrier has been tacitly assumed to

remain invariant with time. Actually, there are electrons within the barrier layer

constituting its space charge; since they are discrete particles they cause fluctuations in the height and width of the barrier. Current fluctuations, or noise,

produced in a fluctuating barrier is expected to be enhanced compared to that

produced in a static barrier, which perhaps can account for the discrepancy between

existing theory and experiment.

26
7. Junction Transistors and Amplifiers

7.1 Operation of Junction Transistors

This section treats three aspects of a junction transistor: first, a rough description of the operation of a junction transistor is given; next, noise

in a (two) junction transistor; finally, noise in a one-stage, low-frequency,

low-noise amplifier made with this transistor.

The transistor, or solid-state triode, evolved from the point contact

(whisker) crystal rectifier. In its initial form, the transistor consisted of two

closely-spaced rectifying point contacts on a semiconductor with a large-area,

non-rectifying contact across the base. In operation, one point emits charge

(minority carriers) into the semiconductor crystal, while the other point collects

the charge; the collector current I is controlled by the emitter current I , an effect known as transistor action. Figure 1.1 shows a set of typical characteristics of an n-type semiconductor point-contact transistor; V is the collector

voltage.

A key parameter of the transistor is alpha (a), the grounded-base current

gain, defined as a = 91 c/3Ip> at constant V . Transistors can be used to produce

an amplifier exhibiting current gain, voltage gain, and power gain (as the dynamic

resistance of the collector remains high even though the total current in the

collector is markedly increased).

Rectifying contacts can be obtained also in a single chip of semiconductor

crystal without the use of metal points. Here, the junction of two contiguous

regions of opposite conductivity type (n and p) in a single crystal of semiconductor forms a rectifier with diode characteristics similar to that of the

point-contact rectifiers. Thus, junction diodes can be made with a single pn

junction, and junction transistors with a pair of back-to-back junctions (npn or

pnp) on a single crystal—one conductivity type being sandwiched between the ends

of the other conductivity type. Owing to their superior performance, stability,

and quality control, junction devices have replaced most point-contact devices;

nevertheless, point contacts remain viable, being found in the latest exotic

devices.

27
+ VC (VOLTS)

o r_

-o

-o*

EMITTER

POINT

COLLECTOR

POINT

CRYSTAL

OFigure 11. Characteristics of typical n-type semiconductor

point-contact transistor (Hunter [23]).

28
JUNCTIONS

O +

-O 1 oFigure 12. Schematic diagram of two-junction PNP transistor.

Figure 12 is a schematic diagram of a two-junction pnp transistor, and an ohmic contact is made to each region. Such a device may be viewed as two back-toback diodes, or rectifiers, separated by a thin base region. The current-voltage

characteristics of this transistor are similar to that of the point-contact transistor.

Junctions can be formed by any of several different methods; e.g., during

growth of the semiconductor crystal, by alloying, or by diffusion of impurities

into the crystal. The method of preparation affects the configuration of the

junction—e.g., alloy junctions are abrupt and grown junctions tend to be diffuse--

and thus affects the performance of the device, including the noise.

7.2 Noise in Junction Transistors

As for junction devices in general, three types of noise are manifest in

junction transistors: 1/f noise, shot noise, and thermal noise. Shot noise and

thermal noise are present at all frequencies, and 1/f noise is present at low

frequencies; the type of noise which predominates depends on the frequency.

7.2.1 Frequency Spectrum

Figure 13 depicts the frequency spectrum of the noise factor NF, and

three regions may be demarcated relative to frequency. In the low-frequency

region 1/f noise predominates and the NF is seen to decrease linearly with

increasing log frequency. At intermediate frequencies, the (frequency-independent)

white noise, comprised of shot noise in the junctions and thermal noise in the

base region, is predominant. At high frequencies, shot noise in the collector

junction predominates. The white noise region is the region of minimum noise.

29
cc?i

o

<

u.

UJ(0

O

1/f FLICKER

NOISE

COLLECTOR NOISE

PREDOMINATES

DUE TO DECREASE

IN GAIN

WHITE

^JIOISE

LOG FREQUENCY

Figure 13. Noise factor vs. frequency (Woll and Herscher [22]).

7.2.2 Equivalent-circuit Analysis of White Noise

The equivalent circuit for the white-noise components of the junction

transistor is represented by figure 14. The shot-noise components may be attributed to two correlated current generators, one at the emitter and one at the

collector; however, these generators may be assumed to be independent, as it

simplifies the mathematics and does not result in significant error when dealing

with low- noise circuits. Thermal noise in the base is represented by an independent voltage generator in series with a noiseless base resistance r. .

, (includes

y/i

e

afblE

A Wv • o C

W Zc

>v/EJb= V4KTAfrbbi

Figure 14. Equivalent circuit for white-noise sources (Woll and Herscher [22])

30
the contact resistance to the base region as well as the innate resistance of the

base region itself); cf figure 6.

The thermal-noise component is given by the familiar expression [eq. (12)]

v1" = 4kTRAf (37)

where R is the differential resistance.

The shot-noise components are shown [24] to be

= 2eI cAf (38)

and V=2eAf [I

E (aQ

- |afb | 2

) + I co

] (39)

afb|\ | a

fb|

2 = 2e Af [I r (1- -L-L^J-) + J_Lii.

L i ] (40)

C a ' a„ CO x ' o o

where i is the shot noise generated in the emitter,

i is the shot noise generated in the collector,

In is the direct emitter current,

I c is the direct collector current,

I cois the reverse col lector- to-base current with the emitter

open-circuited,

a is the zero- frequency grounded-, or common-base current gain,

and a

f

. is the short-circuit common-base current gain.

Also, because

a

(41)

°fb

~~ ] + " f/f

ab>

'

where f . is the frequency at which a

f

. = 0.707a , it can be shown from the above

equations, that for f large, 1 2 increases with frequency. At low frequencies,

a

fb

H a

o

• (42)

31
and i c

2 - 2eAf [I c (l-a

Q

) + l QQ a

Q

] . (43)

Thus, for low noise, transistors having low 1/f cutoff frequency, high a , and high f . should be used (I CQ

« I ).

7.3 Transistor Amplifier

7.3.1 Types of Connection

A transistor can be used to build a single-stage amplifier, and the

transistor can be connected in three different ways:

1) common-emitter, where the signal is applied to the base and taken from

the collector;

2) common -base, where the signal is applied to the emitter and taken from

the collector;

3) common-collector, or emitter follower, where the signal is applied to the

base and taken from the emitter.

The type of connection made will determine the characteristics of the amplifier, and is chosen on the basis of intended application; e.g., the common-emitter

connection gives the highest power gain of the three, making it generally the most

suitable for use as an audio amplifier; the common-base connection has the lowest

input resistance and the highest output resistance; the common-collector connection

has approximately unity voltage gain, high input resistance, and low output

resistance (making it analagous to a cathode follower).

An amplifier has an impedance associated with its input, called the input

impedance, and an impedance associated with its output, called the output impedance,

These two impedances are integrated within and internal to the amplifier. The

external impedance which is connected to the input of the amplifier is called the

source impedance and the external impedance which is connected to the output of

the amplifier is called the load impedance.

Besides selecting low-noise transistors; i.e., those having the appropriate

parameters, one wishes to know the optimum type of connection, operating point,

and source impedance for minimal noise in a transistor amplifier.

7.3.2 Noise in Transistor Amplifiers

In general, below the alpha-cutoff frequency, the NF is found to be

essentially independent of collector voltage and of type of connection; but is

affected by the source impedance [22]. Provided sufficient gain can be obtained

32
in the first stage of a low-noise amplifier, the noise contributions from any

succeeding stages can be neglected; because the source impedance yielding the

minimum NF is nearly the same as that which provides maximum common-emitter gain,

the common-emitter configuration usually is preferable.

As alluded to earlier, it is usually tacit in the term, noise factor, that the source impedance is a pure resistance. If the source impedance is not a pure

resistance, the noise of the amplifier will be a function of frequency. If the source impedance is complex, the signal-to-noise ratio at the output is usually

given, rather than the NF. If the source impedance is a pure reactance, the

concept of NF is meaningless.

8. Solid-State Photon Detectors

8.1 General Types of Photodetector

There are two general types of solid-state photon detector; photoconductive

and photovoltaic. In a photoconductive detector, light is used to produce a current in addition to that which would flow in the dark (i.e., with no illumination), and the device must be biased in order to operate. In a photovoltaic

detector, light is used to produce a voltage, and no bias, or applied voltage, is

required. The properties of these two types of detector differ, and the choice is

governed by intended application; e.g., photovoltaic cells tend to have larger

active areas and may be used directly to power devices.

Both photoconductive- and photovoltaic detectors generally utilize semiconductors (sometimes insulators are used), but differ in configuration. Examples of

several photoconductive- and photovoltaic configurations are shown in figure 15.

A photoconductive detector may be a single crystal slab of semiconductor (or

insulator) with electrical contacts at two opposite faces, such as depicted in

(a), or a polycrystalline thin film of semiconductor deposited on an insulating

substrate (e.g., by evaporation, sputtering, or electrolysis), as depicted in (b).

Photovoltaic detectors are either metal -semiconductor barrier devices (c) or junction devices (d) of planar geometry. For the barrier devices, a "semi transparent" metal film, usually of gold, deposited on the surface of the semiconductor

crystal, makes electrical contact to the crystal while transmitting roughly half

the radiation which impinges on it. The interface of the gold film and the semiconductor forms a rectifying barrier (somewhat similar to that formed by a metal

point contact). Such a device is called a Schottky barrier photodiode 1

.

^ome authors reserve the term photodiode to signify an externally-biased photovoltaic detector.

33
z

ILLUMINATION \ HIGH RESISTIVITY BAR

PHOTOCELL

(a)

ILLUMINATION

\\\u

POLYCRYSTALLINE FILM

PHOTOCELL

(b)

ILLUMINATION

METAL FILM

SEMICONDUCTOR

T

SCHOTTKY BARRIER

PHOTODIODE

(C)

ILLUMINATION

PLANAR JUNCTION

PHOTODIODE

(d)

Figure 15. Various types of photocells.

34
Alternatively, a (rectifying) planar pn junction near the surface of a single

crystal of semiconductor, as shown in (d), may be fabricated. Such a device is

called a planar junction photodiode.

8.2 The Photoconductive Process .

The photoconductive process was alluded to earlier in the treatment of

g-r noise, as both processes originate in the transitions of electrons between

allowed energy levels. The essential differences in the two phenomena are that

photoconductivity relates to the average rates of generation and recombination of

the carriers and the generation transitions are signal photon induced; whereas g-r

noise relates to fluctuations , or deviations, from the average generation and

recombination rates, and the generation transitions may be either photon induced

or thermally induced.

Photoconductivity may be more readily understood with the aid of figure 16, a schematic representation of the pertinent energy levels in the two broad classes

of semiconductor, intrinsic and extrinsic. In intrinsic semiconductors the con- centration of impurities which can act as either donors or acceptors of electrons

is negligible; in extrinsic semiconductors this impurity concentration is preponderant. In figure 16, (a) relates to an intrinsic semiconductor, (b) to an n-type, or donor, extrinsic semiconductor, and (c) to a p-type, or acceptor,

extrinsic semiconductor.

CONDUCTION

BAND

i

i BAND

GAP

VALENCE

BAND

C.B. ~~f

1 1

(a)

B.G.

V.B.

(b)

C.B.

B.G.

1

'

V.B.

(c)

Figure 16. Energy band diagrams illustrating photoexcitation in

intrinsic (a) and extrinsic (b) and (c) semiconductors

35
A semiconductor at, say, room temperature and in the dark already has

carriers--free electrons and holes—owing to thermal excitation of electrons and

background radiation. In the intrinsic semiconductor electrons are thermally

excited by the lattice vibrations, from mainly the top of the valence band to the

bottom of the conduction band. The electrons in the conduction band are mobile,

as are the electron vacancies—the holes—left behind. Thus, when a voltage is

applied across the semiconductor a (dark) current flows. The situation for an extrinsic semiconductor, figure 16 (b) and (c), is somewhat similar, except that

the electrons make transitions between a band and impurity levels, rather than

between bands and only one type of carrier (the majority carrier) need be considered. In either case recombination of carriers ensues, the rate of generation

being equal to the rate of recombination as required for equilibrium.

Shining light of sufficient energy on the semiconductor will excite additional electrons into making transitions, either interband or between an impurity

level and a band, thus creating additional (mobile) charge carriers— hole-electron

pairs in intrinsic material and mobile holes or. electrons in extrinsic material —

which in turn increases the current flowing in the semiconductor. Recombination

of these additional carriers also ensues. The vertical arrows in figure 16

indicate the direction and location of the photon- induced electron transitions.

The energy of a photon is hv, where h is Planck's constant and v is the frequency

of the light. Thus, if E. denotes the difference in energy between the two levels

involved in a transition, the frequency of the light required to effect the transition when absorbed in the material will be given by v. = E./h. In the figure,

the short arrows relate to the effective minimum frequencies (or maximum wavelengths) and the long arrows to the effective maximum frequencies (or minimum

wavelengths) required to obtain photoconductivity.

As for the dark case, the rates of electron generation and hole generation

produced by the light are equal for intrinsic material, and the rate of generation

of a hole or electron is equal to the rate of recombination of a respective hole or electron.

After the light has been turned on sufficiently long for the equilibrium

photocurrent to be established and the light is turned off "instantaneously",

there will be an exponential decay in the photocurrent. The time after turn-off

for the photocurrent to decay to 1/e (37 percent) of its steady-state value is the

lifetime, and the corresponding distance traversed by the carriers is the diffusion length. The values of these parameters do not depend on turning off the

36
light. They remain the same when the light is on; turning off the light is used

here merely for illustrative purposes.

8.3 Measurements of Photoconductivity

Photoconductive detectors of visible radiation ("visible detectors")

have relatively large band gaps, or activation energies, so the number of carriers

in the dark tends to be small and the number of photon-excited carriers large--

hence photoconductivity may be observed with steady irradiation and with the

detector at room temperature.

Infrared photoconductors tend to have small band gaps (if intrinsic) or small

impurity activation energies (if extrinsic), and the number of carriers in the

dark tends to be large, with correspondingly large dark current. On the other

hand, the number of photon-excited carriers is small, and their presence probably

would not be detectable if special measures were not taken. These include cooling

the detector in order to reduce the dark current and the noise, and/or modulating

the radiation in order to produce a modulated photocurrent which then may be

measured by an amplifier sharply tuned to the frequency at which the radiation is

being modulated.

An important performance parameter of a photodetector is its responsivity,

which expresses its output signal current or voltage per watt of incident radiant

power. For detectors of visible radiation this parameter is usually adequate

because the signal current tends to be large compared to the dark current, and

noise is negligible. At low levels of radiant power, however, the noise becomes

appreciable relative to the small-signal output from the detector and signal-tonoise ratio is an additional, and sometimes more useful, performance parameter.

8.4 Infrared Detectors

8.4.1 Detectivity

In infrared detectors, where the signal outputs are small, noise is

always an important consideration, and the ratio of responsivity to noise defines

the detectivity of the detector. For convenience, detectivity is usually nor- malized to a detector area of 1 cm

2 and an amplifier bandwidth of 1 Hz; the

normalized detectivity is written D* (pronounced Dee-star). Detectivity measurements of photoconductive infrared detectors are made by connecting the detector in

series with a load resistor and a battery (photovoltaic detectors may be measured

by connecting them directly to the preamplifier); a blackbody simulator, usually

at 500 K, provides the signal radiation, which is then modulated sinusoidal ly with

a mechanical chopper assembly, usually at 900 Hz. Thus D* may be written

37
s H

D*(500K, 900 Hz) = , N ^ LV

*

, (44)

H(A) 2

where S is the signal measured across the load resistor with signal radiation on the detector, N is the noise measured across the load resistor in the dark (i.e.,

without signal radiation on the detector), Af is the band pass of the amplifier,

H is the power density of the signal radiation in Wcm" 2

, and A is the area of the

detector in cm

2

.

8.4.2 Small-signal Circuit Analysis

The equivalent small-signal circuit of the detector may be given by a current generator shunted by both the detector resistance R

D

and the load resistor

R, , where the detector has been represented by a short-circuit current generator

in shunt with the detector resistance. Then, the short-circuit current responsivity i? sci can be calculated from the following relation [25]:

[|CAV3 , *.;* =-1^(1 +w2 t

2 )" 2 (45)

where ri is the quantum efficiency, e the charge of the electron, X the wavelength

of irradiation, h Planck's constant, c the velocity of light, G the photoconductive

gain, oo = 2iTf where f is the modulation frequency, and x the carrier lifetime. At

low frequencies or dc where u) 2 x

2 « 1.

neXGn *sci " Tc^ ' < 46 >

The photoconductive gain G is given by

G

p

= x/t = yxE/Z . (47)

where x is the carrier lifetime, t is the carrier transit time, u is the carrier

mobility, E is the electric field strength, and I is the interelectrode spacing.

The measured signal voltage across the load resistor S

y

is

V P «sci(v\)' < 48 »

where P is the radiant power on the detector in watts.

38
This circuit analysis is applicable to a photovoltaic detector as well as a photoconductive detector, except that the gain G of a photovoltaic detector is

unity.

8.4.3 The Photovoltaic Effect

The electronic processes which occur in a solid when light interacts

with a semiconductor are determined by the characteristics of the light (energy

and spectral distribution) and the material parameters (e.g., light absorption

coefficient, band gap energy, impurity concentration). Thus the physics of photoconductive- and photovoltaic detectors have much in common. Nevertheless, there

are some notable differences in the two types of device which arise from the

difference in their compositions: photoconductive detectors are homogeneous and

may use intrinsic or extrinsic semiconductors; photovoltaic detectors are heterogeneous, incorporating a (rectifying) contact potential barrier in the form of a metal -semiconductor barrier or a pn junction; only intrinsic semiconductors may

be used for Schottky photodiodes. Owing to the presence of this barrier/junction,

no external bias is required for operation (as is the case for photoconductive

detectors). Assuming zero bias for the photovoltaic detector, there will still be

a thermal noise. An illuminated barrier/junction with zero bias applied may be

likened to an unilluminated barrier with bias applied (see sec. 6); i.e., illuminating the barrier/junction is tantamount to applying an external voltage across

the barrier/junction. The barrier/junction serves as a charge separator. For

example, in the junction device, when hole-electron pairs are created through the

absorption of light energy, the electrons at the junction will pass easily only

in one direction and be collected virtually without recombination having occurred,

while the holes will pass easily only in the opposite direction and be collected

also without recombination having occurred. Thus it is important that the junction

thickness be small compared to the mean free path of the carriers to prevent

current-retarding space-charge buildup in the junction and that the light energy

be absorbed within a diffusion length of the junction. Details of the current

flow in a Schottky barrier photodiode differ 1

, but the same restrictions apply to

barrier width.

8.4.4 Noise in Photoconductive Detectors

Photoconductive detectors exhibit three of the four types of noise

which were identified as principal noises: 1/f noise, thermal noise, and g-r

^ee ref. 23, section 5.

39
1 1 -| 1 » 'T-

/

//^ ^^ ^"ISO

noise 26]). — i uj / — O i i j

T" to3 +* O

UJ

CO

O 4-> •i- to

S- $- z (0 CO

- UJ 1 CO \s

o

z

z^

O

T"

N

X n

of

v

trum (

O I Z l Z

o CO

^

>-

•1- d)

4-> a.

=3 tO —

TOTAL

r-o

X

Cl

z

X

o-> X UJ

x to

o

o

o

z

UJ

o ;he

contrib

ital

noise

z

<

COo

1—

CM

UJ

DC

LL

ration

of

t

s

to

the

tc

V uj

2

DC

UJ

X

o

Illust source

Ii ^o rT" i--. t/i /

is

z o

CD

i-3

*/ Ll_ o i

' III o

o

mo o

CO

o

CMooo

(31V0S 3AI1V13U)

iNaaano ssiON asavnos nvsiai

40
noise. Figure 17, a plot of the mean-squared noise current vs. (amplifier)

frequency, illustrates how the various noises may be manifest 1

.

Shown individually are the 1/f noise, prominent at low frequencies, but

decreasing with increasing frequency2

; the white Johnson, or thermal noise, present

at all frequencies, but overshadowed by other noises; and the two forms of g-r

noise--that which is produced by background radiation (photons) is commonly

referred to as (background) photon noise to distinguish it from that produced by

thermal vibrations of the lattice, referred to simply as g-r noise. An equivalent

circuit would have each noise current source mutually parallel; thus the total

noise spectrum is formed by addition of the individual noise currents. Three

regions may be demarcated according to frequency range: At low frequencies 1/f

noise predominates; at intermediate frequencies the noise is essentially photon

noise and this is the frequency region of normal operation; at high frequencies

the photon noise decreases and approaches thermal noise.

That the g-r noises are flat may be understood from the following: the

spectral density of the noise current S. (f) is defined by the relationship

"•I*(f) df (49)

where i 2 is the mean squared short-circuit noise current and f is frequency. (The

frequency dependence of the noise current is implicit in S.(f); e.g., for thermal

noise, S.(f) = 4kTG.)

For the general case of g-r noise (thermal) in an extrinsic semiconductor

containing both donor impurity atoms and acceptor impurity atoms, but with one in

the majority, S.(f) is a function of the total number of free electrons N, the

total number of acceptor impurity atoms N . and the total number of donor impurity

a atoms N, [19,27]. Extrinsic photoconductive detectors are normally operated

cooled, the result being that N « N , N

d

- Then S

i (f) is given by

x This example represents cooled, extrinsic silicon and germanium detectors; for

other detectors or other operating conditions, the relative magnitudes of the

noise sources may differ.

2 l/f noise in germanium devices is attributed to an inversion layer at the surface,

while in silicon devices it is associated with the oxide passivating layer.

41
s

* (f) = 4 HrrWl * (50)

where I is the steady, or average, current and x is the lifetime of the free

carriers. Thus at low frequencies where 2iTfT « 1, the noise is flat, or frequency

independent; when 2irfx » 1, the noise decreases. At high temperatures where

N * N . - N (N. has been assumed to be the major impurity), the g-r noise asso- ciated with impurities approaches zero, but g-r noise from the now intrinsic

semiconductor becomes significant and will escalate with increasing temperature.

In most cases photon noise will be equivalent to (thermal) g-r noise, and eq.

(50) derived for the latter is applicable too for the former when N is replaced by

N. , the total number of free electrons generated by the background radiation. In

those rare cases where detectors are operated at room temperature and used for

long wavelengths > 20 ym, however, eq. (50) would have to be modified [26].

8.4.5 Noise in Photovoltaic Detectors

Photovoltaic detectors may exhibit all four principal noises, the

additional noise being shot noise, if the detector is biased. The shot noise

arises because of the contact potential barrier associated with either the Schottky

metal-semiconductor barrier or the pn junction of the photovoltaic detector. If

the contact potential barrier width is small--i.e., less than the mean free path

of the conducting electrons—virtually all of these electrons will traverse the

barrier without suffering collisions within the barrier and there will be no resultant space charge buildup. The current through the barrier will be the

saturated current I , and full shot noise will be generated in the barrier. The

equation for shot noise, repeated here for convenience, is

V = 2eI

$

Af , (51)

and the equivalent circuit, a current generator (shunted by an infinite impedance),

has already been shown in figure 3 in connection with the thermionic diode. On

the other hand, if the barrier width is not less than the mean free path, there

will be a space-charge buildup in the barrier and the current through the barrier

will be space-charge limited and reduced. As for the analogous thermionic diode,

previously treated, the noise will be a partial shot noise substantially lower in

magnitude than the full shot noise as would be given by the equation. And as for

the thermonic diode, the magnitude of partial shot noise is unpredictable; the

equivalent circuits of figure 4 are applicable here too.

42
There is a further important distinction between the photon noise in a photoconductive detector and in a biased photodiode. In a properly designed and

constructed photodiode as described in the preceding section— i.e. , one where the

radiation is absorbed in a distance from the barrier/junction which is much less

than the diffusion length, and the thickness of the barrier is less than the mean

free path—there will be negligible recombination taking place 1

. Hence there is no recombination noise, just generation noise. In a photoconductive detector,

however, the rates of generation and recombination are equal, both contributing

equally to the noise. Thus the generation noise (g noise) power in a biased

photodiode will be one-half the g-r noise power in the corresponding photoconductive detector, and the detectivity for a given incident energy flux for a photovoltaic background (photon) limited detector is therefore greater by a factor

of 2 than for a corresponding photoconductive detector prepared from the same material. Otherwise, the noise spectrum of a cooled photodiode, particularly one of silicon or germanium, would be somewhat similar to that shown in figure 17 for

a photoconductive detector.

8.4.6 The Ideal Detector

The higher the detectivity of an IR detector, the smaller the amount

of radiant energy it is capable of detecting, and for a detector to realize its

optimal detectivity, the limiting noise is required to be background photon noise;

i.e., all other noises must be suppressed (e.g., by cooling the detector, modulating the radiation at high enough frequency to squelch 1/f noise, and using a large load resistance). Even so, no actual detector is perfect so the maximum

detectivity achievable is always less than the maximum value of detectivity

theoretically possible with an assumed perfect, or ideal, detector.

The fictitious ideal detector is background photon noise limited and has a quantum efficiency of unity up to the long wavelength cutoff and zero beyond

(the latter stipulation is unrealistic). Detectivities may be calculated as function of wavelength for ideal detectors, and curves generated therefrom, an example of which is shown in figure 18. Such curves are useful for determining

the highest value of detectivity theoretically possible at a given wavelength.

Thus for example, an actual detector may approach, but not attain, a D*(X) of

1 x 10 11 cm Hz^ W

-1 at a wavelength X = 5 um, with conditions as specified in the

figure.

x At zero-bias, however, there is no net current so the rates of recombination and

generation must be equal

.

43
100

WAVELENGTH M

Figure 18. The dependence of the detectivity of an ideal backgroundlimited detector on cutoff wavelength (Bode [25]).

44
Background-limited infrared photodetectors are said to be in the blip con- dition, blip being the acronym. Photon noise may be reduced (thus increasing D*)

by reducing the field of view of the detector and by reducing the temperature of

the background; e.g., by use of a cooled apertured baffle. Other ideal detector

detectivity vs. wavelength curves covering various situations may be found elsewhere [26,28,29].

8.4.7 Amplifier- limited Noise

If the background noise of a detector in the blip condition is sufficiently reduced, the detectivity will become limited by noise in the preamplifier.

This condition is referred to in the jargon as plip, the acronym for preamplifier

limited infrared photodetector. If the amplifier noise is caused mainly by

thermal noise in the source resistance, increasing its value will reduce the

noise. This follows because in an equivalent circuit this noise source is represented by a current generator and noiseless source resistance in shunt with the

amplifier input; the thermal noise current in the source resistance is given by

eq. (13). See also figure 6 and section 3.2.7.

9. Concluding Remarks

Most of the ongoing work involving noise is aimed at its reduction or minimization, and either of two approaches may be taken: 1) device design, including new and improved materials; (2) optimal operation of given devices. The former,

however, is an esoteric pursuit, and the latter accounts by far for most of the

activity.

Throughout this work various operational means for reducing or minimizing

noise have been given explicitly, while more may be easily inferred. To recapitulate: Obviously 1/f noise may be reduced by operating at higher frequencies, and

by reducing current; shot noise in junction or barrier devices in general may be

reduced by reducing current, and avoided or minimized in photodiodes by zero-bias

operation; thermal noise may be reduced by reducing temperature and by increasing

resistance; thermal g-r noise may reduced by reducing temperature; photon noise in

IR detectors may be reduced by reducing field of view and background temperature;

noise in general may be reduced by reducing bandwidth. Further, section 7.7.2

discusses requirements for low-noise junction transistors and amplifiers.

Sometimes it may not be feasible or desirable to employ one or another of

these methods because of the consequences. For example, reducing bandwidth will

reduce noise, but also may block out desired sidebands; or a device may be operated

advantageously in a mode which increases noise, but which also increases the

45
signal-to-noise ratio, as when a photodiode is reverse-biased. In other words, in

practical situations it is necessary to consider the trade-offs, if any.

A word about noise identification: The principal noises have distinguishing

characteristics which may be used for such purpose.

Recapitulating: Thermal noise is the only noise which occurs in a passive resistance and which is (explicitly) independent of current. All other noises require

the existence of steady current and depend on the current noise, increasing with

increasing current. Of these other noises, only shot noise is proportional to the

square root of the current, the remainder being proportional to the current.

Thermal noise and shot noise are white noises. On the other hand, the distinguishing characteristic of 1/f noise is its frequency dependence (i 2 a 1/f).

Photon noise is sensitive to background temperature and field of view. Identification of particular noises will be facilitated if the particular noise has

predominance; otherwise it may be inextricable.

Finally, for those readers going on to investigate noise in electron/solid

state devices: van der Ziel's recent historical survey of noise [30] provides an extensive source of references; the (out-of-print) book by Kruse et al [29]

contains an anomalously lucid treatment of noise; and, of course, a detailed

knowledge of the operation of the device of concern is requisite.

46
10. References

[1] Proc. 6th International Conference on Noise in Physical Systems,

NBS Special Publications 614, Wash'. D.C., (1981).

[2] G. Herrmann and S. Wagener, The Oxide Coated Cathode, Vol. 2,

pp 10, 11, Chapman & Hall Ltd., London (1951).

[3] 0. W. Richardson, Emission of Electricity from Hot Bodies,

2nd edition, London (1921).

[4] G. Herrmann and S. Wagener, The Oxide-Coated Cathode, Vol. 2,

p 17, Chapman & Hall, London (1951).

[5] The Scientific Papers of J. Clerk Maxwell, W. D. Niven, ed.,

Vol. I, p 380, Cambridge Univ. Press (1890).

[6] W. Schottky, Ann. Phys. 5_7_, 541 (1918).

[7] K. R. Spangenberg, Vacuum Tubes, p 306, McGraw-Hill Book Co., Inc. (1948)

[8] R. Brown, Phil. Mag. 4, 161 (1828).

[9] A. Einstein, Ann. Phys. 1_7, 549 (1905); Ann. Phys. T9, 289, 371 (1906).

^10] J. B. Johnson, Nature H9, 50 (Jan. 8, 1927); Phys. Rev. 29, 367

(Feb., 1927); Phys. Rev. 32, 97 (July 1928).

11] H. Nyquist, Phys. Rev. 32, 110 (July 1928).

12] J. Bernamont, Ann. Physique 7_, 71 (1937).

13] C. J. Bakker and G. Heller, Physica VI, no. 3, 262 (March 1939).

14] D. A. Bell, J. Instn. Elect. Eng. 82, 522 (1938).

15] D. A. Bell, loc. cit. ; see also its appended Discussion.

16] J. B. Johnson, Phys. Rev. 26, 71 (1925).

17] H. N. Kozanowski and N. H. Williams, Phys. Rev. 36, 1314 (1930).

18] W. Graffunder, Telefunken-Rohre 15, 41 (1939).

19] K. M. van Vliet and J. R. Fassett, in Fluctuation Phenomena in

Solids, R. E. Burgess, ed., Academic Press, New York, Chap. 7 (1965).

20] References (59) through (66) in K. M. van Vliet and J. R. Fassett,

loc. cit.

21] H. C. Torrey and C. A. Whitmer, Crystal Rectifiers, p 26,

McGraw-Hill Book Co., Inc. (1948).

22] H. J. Woll and M. B. Herscher in Handbook of Semiconductor Electronics,

L. P. Hunter, ed. , 2d ed., pp 11-83, McGraw-Hill Book Co., Inc.,

New York (1962).

47
[23] L. P. Hunter in Handbook of Semiconductor Electronics, loc. cit.,

Section 1.

[24] A. van der Ziel, Proc. IRE 46, no. 6, pp 1019-1038 (June, 1958).

[25] D. E. Bode, "Infrared Detectors" in Applied Optics and Optical

Engineering, Vol. VI, R. Kingslake and B. J. Thompson, eds.,

Chap. 8, Academic Press, New York (1980).

[26] P. R. Bratt, "Impurity Germanium and Silicon Detectors", in

Semiconductors and Semimetals, Vol. 12, P. K. Willardson and

A. C. Beer, eds., Chap. 2, Academic Press, New York (1977).

[27] K. M. van Vliet, Proc. IRE 46, 1005 (1958).

[28] K. M. van Vliet, Appl. Optics 6, 1145 (1967).

[29] P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, Elements of

Infrared Technology, Chapters 7 and 9, Wiley, New York (1962).

[30] A. van der Ziel, "History of Noise Research", in Advances in

Electronics and Electron Physics, Vol. 50, L. Marton and

C. Marton, eds., Academic Press, New York (1980).

[31] Cf S. 0. Rice, Bell Syst. Tech. J. 23, 282 (1944).

[32] Cf R. V. Churchill, Fourier Series and Boundary Value Problems,

p 43, McGraw-Hill Book Co., Inc., New York (1941).

48
APPENDIX

This appendix contains derivations of the thermal noise equation and the shot

noise equation. Each is derivable by more than one way; the treatments here are based on Nyquist's [11] deduction for the thermal noise and on Schottky's [6]

method for the shot noise. The former is mainly physical --although Nyquist

assumed familiarity with properties of transmission lines—and gives insight into

the nature of noise. On the other hand, the latter is mainly of mathematical

interest.

A. Derivation of Thermal Noise

Nyquist's calculation of the electromotive force (emf) owing to thermal

noise in a conductor cleverly couples principles in thermodynamics and statistical

mechanics with principles in electricity, specifically those governing electrical

transmission lines. In the present treatment properties of lines are first

reviewed, after which the thermal noise emf is derived.

A line is a pair of long parallel conductors, each of uniform cross section,

used to carry ac. A line has a resistance R. and an inductance L per unit length

(both sides of the line are included), and a leakage conductance G and a capacitance C between conductors per unit length. Thus the series impedance per unit

length along the line is given by Z, = R, + Loo and the shunt impedance per unit

length across the line by Z = 1/(G + iCoj). The characteristic impedance Z., of a line refers to the impedance of the line if it is infinitely long. An infinite

line can contain no reflected waves because the original transmitted wave is

attenuated to zero before it reaches the output end, and

zK =(z

L LC

} l_G + iCoo J

* (Al)

Assuming R. and G to be negligible (the line will be distortionless), Z

K

reduces

to the pure resistance

R

K

= (L/C)' (A2)

If some line of arbitrary length I, terminated at one end in a load resistance

R, is constructed such that (L/C) 2 = R, there will be no reflection at the terminating end. Thus such a line is like an infinite line and has the characteristic

impedance R which is a pure resistance. If such a line is thought of as a filter

49
it becomes apparent that the line will pass all frequencies because a pure resist- ance passessall frequencies. If this line is terminated at both ends in a resist- ance R, there will be no reflection at either end and all the power transmitted

from one end will be received at the other end, provided the line is non-dissipative. Further, this matching of termination resistances is the condition for

maximum transfer of power. With this as background, Nyquist's deduction of

thermal noise equations can be readily understood, and we proceed to calculate the

emf due to thermal noise.

Consider two noisy resistors each of value R and having uniform temperature T

to be connected to each other by a long non-dissipative ideal transmission line as described immediately above. The equivalent circuit for this model is given in

figure Al where the noisy resistors are assumed to be representable by a voltage

generator in series with R (noiseless).

Figure Al . Equivalent circuit of two noisy resistors,

R and R, connected by a transmission line.

The noise voltage v generated in one_ of the resistors, say the one at the

left, induces a current flow in the circuit whose value is given by i = v/2R.

This current flowing through the other resistor causes it to heat up or absorb

power, the absorbed power being given by P = i 2 R = v

2 /4R. That is, power is

transferred from the one resistor to the other. The choice of resistor was arbitrary, and obviously there must be an identical transfer of power from the

right resistor to the left, otherwise the temperature of each would not remain

equal and unchanged, as required.

So far our ideal transmission line has provided the means for perfect—

unrestricted—transfer of power between resistor. Now the line will also allow

the means for reckoning the effect of frequency (bandwidth) on noise power and

emf.

50
The line is non-reflecting and at any instant can be thought of as vibrating

at its natural frequencies. If the length of line I is taken to be sufficiently

large, the number of modes of vibration, or degrees of freedom, lying within a frequency range Af is 21 Af/y, where v is the velocity of propagation. Then, the

energy 'frozen' on the line at this instant is given by (the average energy per

degree of freedom) X (the degrees of freedom). The average energy per degree of

freedom is kT, where k is Boltzmann's constant. Thus the total energy of the

vibrations within the frequency interval Af is kT2ZAf/t>.

But since there is no reflection^, this is the energy within that frequency

interval which was transferred from the two noisy resistors to the line during the

transit time l/v.

Power is defined as energy per unit time. Thus the average power transferred

from each resistor to the line within the frequency interval Af during the time

interval l/v is

p = i ozm. - kTAf

.

2 l/v

(A3)

Thus the available thermal noise power—or maximum noise power which can be

transferred by a resistor--is independent of the resistance value.

Earlier we derived from the equivalent circuit another relationship for the

noise power, viz. P = v

2 /4R. Therefore,

k - kTAf (A4)

and v

2 = 4kTRAf . (A5)

Although the above equation was derived for R assumed to be pure resistance,

similar reasoning shows it holds also for an impedance R(f) + X(f), when R in Eq.

(A5) is repaced by R(f).

B. Derivation of Shot Noise

This derivation is made, as was the original, for a thermionic diode having

temperature-limited, or saturated, emission (cf sec. 3.1.1). Nevertheless, the

results hold equally well for any transport current which is by randomly produced

free carriers; e.g., junction/barrier devices in general.

51
6

#-

Figure Bl . Thermionic diode circuit with plate resistor.

Figure Bl shows a thermionic diode circuit with plate (anode) resistor. The

electrons emerge from the cathode randomly in time, and being free are distributed

according to Maxwell's law with the minimum velocity just large enough to overcome

the potential barrier at the surface (i.e., the work function); most electrons

have velocities around the average value. The constant voltage applied to the

anode establishes a constant electric field across the interspace between cathode

and anode causing the emerging electron to be transported from cathode to anode

with constant acceleration. The velocity of the electron is thus increased from

its initial value v. to some final value y

f

, which is the velocity of the electron

as it just reaches the anode. The effect of the electric field on velocity is

s/ery large so that y

f

» v

. Thus the average velocity v = {v. + vAll - vJ2.

Further, the applied voltage is sufficiently large so that the emission

current is saturated. In saturated emission all emitted electrons reach the

anode. Because the electrons are free the fluctuations in emission reach the

anode unchanged. The random emergence of electrons from the cathode results in

their random arrival at the anode.

The derivation of the shot noise equation is heuristically inspired in that

the emission current arriving at the anode is perceived as a series of pulses.

Owing to the corpuscular nature of the electron and the randomness in emission,

the emission, and thus the plate current, varies from instant to instant. Measured

over a long interval of time, however, the emission current averages out to an essentially constant value. Thus there is an instantaneous time-varying current

fluctuating about an average, or direct current, in this case the saturated

emission current.

52
Associated with the average current I are n, the average number of electrons

emitted per unit time and a constant; v, the average velocity; and t, the transit

time, given by t = d/v, where d is the cathode-to-anode spacing.

Associated with the instantaneous current I are n, the instantaneous rate of

emission; v(t), the instantaneous velocity; and t, the time at which the electron

acquires a velocity v. A statistical theorem relates n and n (cf sees. 1.2 and

3.1.2), viz.:

(n-n) 2 = n . (Bl)

The fluctuations in current i are defined by

i 2 = (I-T) (B2)

The derivation of the shot noise current equation--!. e. , putting i 2 into a practicable form—consists, in a nutshell, of 1) relating the average current and

the instantaneous current at the plate by means of Campbell's (mathematical)

theorem [31]; 2) introducing Fourier transforms to transform events from the time

domain to the frequency domain, aided by Parseval's (mathematical) theorem [32];

3) solving for i 2 (f). Details of the mathematics are given below.

Applying Campbell's theorem:

r..f F(t) dt (B3)

and from eq. (B2)

2 _ ^ l = •/ F

2 (t) dt (B4)

53
The Fourier transforms are now introduced:

1 -^

J

G(o)) exp (Iwt) da) . (B5)

(2n) 2

''-co

(2n)^2 jf

G(a>) = -1—r / F(t) exp(-iwt) dt , (B6)

The Fourier transform of F

2 (t), obtained through use of Parseval's theorem, is

F

2 (t) dt = / FMt) dt . / 6(a)) G*(a>) dco , (B7)

y

where G*(u>) is the complex conjugate of G(a>). In the present example, G(w) G*(oo)

is an even function, so

i 2 = ir I ffG(a)) G*(w) dw , (B8)

where e is the electronic charge.

The function F(t) is just the instantaneous current reaching the anode. The

instantaneous current is given by

I = ey/d , (B9)

where e is the electronic charge.

But d = v

f

tII , (BIO)

and v = (t/x)v

f

. (Bll)

Therefore, I = 2et/x 2

. (B12)

54
Substituting this into eq. (B6) gives

T

G(oj) = ——- — / t exp(-ioot) dt . Un)"2 x

2 /

(B13)

Denoting 8 e wt and integrating, gives

2f2 e

G(u>) = l^J p [(6 sine + cose - 1) -i (sine - ecose)]. (B14)

Then

G(a>) G*(u>) 2ei 1 n e"

1 [e 2 + 2(i-cose-esine)] (B15)

For saturated emission at low frequencies 6*0, and eq. (B15) reduces to

G(u) G*((o) = e

2 /2n . (B16)

Integrating eq. (B8) over the bandwidth Af gives

i 2 = (eI/n)Au) = 2eIAf , (B17)

which is the expression for the mean square shot noise, when I is the saturated

current.

55
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN 1169

2. Performing Organ. Report No. 3. Publication Date

December 1982

4. TITLE AND SUBTITLE

Introduction to Noise in Solid State Devices

5. AUTHOR(S)

Julius Cohen

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as item 6.

10. SUPPLEMENTARY NOTES

] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This is a short didactic monograph on electronic noise which

aims to impart a "feel for the subject". The work is divided into

two parts. The first, Theory, deals in detail with the principal

noises found in solid state devices, namely shot noise, thermal

noise, 1/f noise, and generation-recombination noise. The second

part, Applications, is a systematic treatment of noise in selected

solid state devices. Analyses progress from a single noise source

in a circuit element to four noises in a device; concomittantly

equivalent circuits are developed to facilitate the solution of

various complex noise problems. Examples treated in this part

include resistors, rectifiers, transistors, and photo-detectors.

The work concludes with a recapitulation and useful references.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

electronics; noise; photon detector; rectifier; solid state devices; transistor.

13. AVAILABILITY

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PRINTED PAGES

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15. Price

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