Transport noise arising from diffusion and bulk or surface generation-recombination in semiconductors

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Transport noise arising from diffusion and bulk or surface generation-recombination in semiconductors
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Thesis (Ph. D.)--University of Florida, 1981.
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Includes bibliographical references (leaves 128-129).
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by Harshad Mehta.
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TRANSPORT NOISE ARISING FROM
DIFFUSION AND BULK OR SURFACE
GENERATION-RECOMBINATION IN SEMICONDUCTORS














BY

HARSHAD MEHTA


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


UNIVERSITY


OF FLORIDA


1981

































To

Dear mother and (late) father












ACKNOWLEDGMENTS


I am greatly indebted to Professor K.M. van Vliet for his most

valuable guidance, suggestions and time; without his help this work

would not have been possible. I am very thankful and highly obliged

to Professor and Mrs. van Vliet and family for their generosity,

kindness and greatness.

I am very thankful to Professor A.D.Sutherland and Professor

E.R.Chenette for all their help, guidance and helpful suggestions

during this work. I am highly thankful to Professor Cherrington,

Chairman, Electrical Engineering Department for his help from time

to time. I am very thankful to Professor Neugroschel and Professor

Ihas for serving on my advisory committee and their valuable sugges-

tions. I am very grateful to Professor A.D.Sutherland for his continued

help and support whenever I needed.

I am also very much thankful to my brother, Dr Jitu Mehta, for

all his moral, social and financial support. I am also highly

thankful to some of my best friends Kelly, Pankaj, Satish, Pramod

and Kirit for their cooperation and help whenever I needed.

I am highly thankful to the Electrical Engineering Department

of the University of Florida and to the Centre de recherche de

math6matiques appliques of the University of Montreal, for providing

the financial support during this work.

Last but not least I am highly thankful to Francine Houle-Miller

for her excellent work in typing this manuscript.












TABLE OF CONTENTS


ACKNOWLEDGEMENTS iii

ABSTRACT vii

Chapter Page

1 INTRODUCTION 1

1.1 Object 1

1.2 Flicker Noise 1

1.3 Transport Noise 2

1.4 Surface Noise 4

1.5 Embedded and Nonembedded Bodies 5

PART A: MATHEMATICAL METHODS AND PHYSICAL NOISE SOURCES 8

2 MATHEMATICAL METHODS FOR TRANSPORT NOISE 8

2.1 Methods 8

2.2 Langevin Equation and Source Methods 9

2.3 The Phenomenological Equation and 19
Correlation Methods

3 PHYSICAL NOISE SOURCES 28

3.1 Introduction 28

3.2 Volume Noise Sources 28

3.3 Surface Noise Sources 34












Chapter Page

PART B: NOISE DUE TO VOLUME NOISE SOURCES 37

4 SYMMETRICAL EMBEDDED BODIES 37

4.1 Introduction 37

4.2 Survey of Older Work 37

4.3 Previous Work Using the Green's Function Method 39

4.4 Present Work: Carrier Fluctuations in Sphere 42
Due to Diffusion and Generation-Recombination

5 NONSYMMETRICAL EMBEDDED BODIES 52

5.1 Introduction 52

5.2 Voss and Clarke's 'P' Sources 52

5.3 Physical Diffusion Source 58

6 SYMMETRICAL NONEMBEDDED BODIES 68

6.1 Introduction 68

6.2 Summary of Previous Work 68

6.3 Present Work: Sphere 71

7 NONSYMMETRICAL NONEMBEDDED BODIES 90

7.1 Introduction 90

7.2 Linear Case 90

7.3 Rectangular Nonembedded Bar 92

PART C: NOISE DUE TO SURFACE NOISE SOURCES 98

8 NOISE DUE TO STOCHASTIC BOUNDARY CONDITIONS 98

8.1 Introduction 98

8.2 Summary of Previous Work 98

8.3 Noise Caused by Surface Generation-Recombination 99













APPENDIX 119

REFERENCES 128

BIOGRAPHICAL SKETCH 130











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


TRANSPORT NOISE ARISING FROM
DIFFUSION AND BULK OR SURFACE
GENERATION-RECOMBINATION IN SEMICONDUCTORS


By

Harshad Mehta

March 1981

Chairman: Professor K.M. van Vliet
Cochairman: Professor A.D.Sutherland
Major department: Electrical Engineering

The purpose of this thesis is to present a systematic study of the

type of noise spectra in solids or solid state devices, which can arise

from "transport noise," which stands for noise due to combination of

causes, such as diffusion of carriers, generation and recombination (g-r)

processes of carriers in the bulk and at the surface. The techniques to

be described apply equally well to the problem of heat diffusion, with

heat transfer or reflection at the boundaries of the system; we have,

however, mainly the carrier noise problem in mind.

After an introduction we describe the various mathematical techniques,

applicable to transport noise. The methods studied are (1) the eigen-

functions expansion method, (2) the Green's function method. Each of these

methods can be applied to two physically distinct procedures, (a) the

sources or Schottky method, (b) the correlation method.











In chapter 3 we discuss the physical noise sources,viz. the diffusion

source, g-r source and surface source, as well as Clarke and Voss' P-

source.

In the next few chapters we discuss: transport noise in symmetrical

embedded bodies, in nonsymmetrical embedded bodies, in symmetrical non-

embedded bodies and in nonsymmetrical rnmmbedded bodies. The concept

"embedded" body refers to a system which is part of a very large system,

with which it exchanges particles or heat, such as a volume between

probes on a large semiconductor. The concept embeddedd body" refers

to a body inside which there is transport noise, with deterministic

boundary conditions on the surface of the body.

In chapter 4 we first review the older work on the circular patch

and the cylinder (MacFarlane, Burgess, Richardson, van Vliet, Chenette).

Next we solve for the noise of an embedded sphere, caused by diffusion

and volume g-r processes. For infinite carrier lifetime, the results are

found to be in agreement with those of Fassett. For finite lifetime an ex-

tension of his results is obtained. The high frequency 3/2law is

confirmed.

In chapter 5 we discuss the results for a rectangular embedded bar,

disc, or cube. Because of the lesser symmetry, the spectrum cannot

be obtained in closed form, but only in a triple integral :form.

The result has been numerically evaluated by computer techniques. Not

all the ranges predicted by Clarke and Voss are visible, however.

For the P-source the noise has also been evaluated. Contrary to Clarke


vili











and Voss' claim, no 1/f range is visible. Thus, their heat diffusion

noise model cannot account for the observed 1/f noise in metal films.

In chapter 6 symmetrical nonembedded bodies are discussed. Parti-

cular attention is given to the noise in a sphere, with boundary condi-

tions due to surface recombination. A closed form is obtained via

an expansion of the Green's function in spherical polynomials.

In chapter 7 the nonsymmetrical bar is discussed. The results

given previously by Lax and Mengert and others were evaluated by

computer.

In chapter 8, finally, we discuss noise due to the surface g-r

source. This noise can be dominant due to the surface modulation

factor; it has not been considered in all older theories. We discuss

three examples: (a) a semiconductor single crystal sample with surface

g-r noise; (b) the noise in the channel of a MOSFET, caused by diffusion

into the adjacent bulk and by g-r (surface) processes near the oxide

layer; (c) the same model but with special care for the alteration

of the diffusion constant and mobility in the channel due to surface

scattering. For models (b) and (c) it is shown that a realistic large

1/f range (104 hz up to 1011 hz) can occur. Thus this model can account

for much of the high frequency 1/f noise in MOS devices.

The other examples and models considered in this thesis firmly

indicate that volume g-r and diffusion processes give rather smooth

spectra, but without a 1/f range. The high frequency asymptote is
a3/2 -2
always c (embedded case) or 1 (nonembedded case).













CHAPTER 1
INTRODUCTION




1.1 Object

The purpose of this study is to make a systematical investiga-

tion of diffusion and generation-recombination noise spectra. Such

studies may aid in understanding the physical mechanisms underlying

'1/f' noise. The noise spectra obtainable from transport processes

are analyzed and the possibility of contributions to '1/f' noise is

studied. It will be shown that most transport processes caused by

volume sources do not give rise to 1/f noise.

In addition, a detailed investigation has been undertaken

of surface noise. This is noise caused by surface sources which

give rise to stochastic boundary conditions at the surface. Surface

noise contributions are studied in detail, since the older theories

due to Hydel, Champlin2, Lax-Mengert3,and van Vliet4 only considered

the volume noise contribution. This analysis gives for the first

time a transport noise model which results in 1/f noise; however,

the frequency range for which it occurs is often too high to ac-

count for the experimental data, unless very long lifetimes of the

carriers can be shown to exist.



1.2 Flicker Noise

The phenomenon of 1/f noise or 'flicker noise' started with

the study of J.B.Johnson5 on noise in vacuum tubes in 1925. He












noted, that besides shot noise, which has a white frequency spectrum,

at low frequencies another noise component appears, which varies

roughly inversely proportional to the frequency. The first interpre-

tation of this noise was given by W.Schottky6 in 1926. Schottky

attributed the effect to fluctuations in the work function of the

cathode material, caused by the random arrival and leaving of foreign

atoms at the cathode surface. The spectrum computed by Schottky

was not exactly of the form S(f) = A/f; rather, his effect results
2 2
in a Lorentzian spectrum, S(f) = B/(l+o T ).

Since Schottky's mechanism failed to interpret Johnson's 1/f

data, as well as the flicker noise later on observed in a large number

of solids and devices, various authors have proposed a great variety

of detailed studies to understand the physical mechanisms behind 1/f

noise. In the subsequent chapters we shall summarize some of these

efforts and extensively study the contribution of transport and

surface noise to the origin of flicker noise.



1.3 Transport Noise

Transport noise is caused by quantities subject to stochastic

transport in the sample. The most notable examples involve particle

diffusion and heat diffusion. Both are described by a linear partial

differential equation. The noise can be obtained in two ways. In

the first procedure, the 'sources' or 'Langevin' procedure, one adds











noise source terms to the phenomenological transport equation, which

then becomes of the form


( 2-+ A)a (,t) = ((,t) (1.3.1)

where A = -DV2 and a = AT or An for heat or particle diffusion

respectively, and is the volume noise source. The spectrum of the

latter must be known.

In the correlation procedure, on the other hand, one uses

the ensemble-averaged stochastic equation, which by Onsager's7 prin-

ciple, is the phenomenological equation. Thus we must then solve


t A) (aCr,t) aC(,0)) = 0 (1.3.2)

where (<) denotes the ensemble average of a(E,t), conditional

to a given value a(Zr,0) at time t = 0. In this case the covari-

ance function

r Cz,') = (Aa(C,t)Aa(' ,t)) (1.3.3)

must be known.

The connection between both procedures is given by A-theorem

due to van Vliet8, which connects the source spectra to the cova-

riance function

CA ,+A )rCr,') = 2C,'). (1.3.4)

To solve either (1.3.1) or (1.3.2) one can use two mathematical

formalisms, viz.the eigenfunction expansion method and the Green's












function description. Thus, in total, there are four methods to

obtain transport noise. The equivalence of all these methods has

been shown by van Vliet and Fassett9'S0. The details of these

methods are described in the next chapter.



1.4 Surface Noise

This is noise caused by stochastic boundary conditions at

the surface. An example occurs when a semiconductor sample is

subject to surface generation-recombination, with the surface re-

combination velocity S being unequal to zero or infinity.

In the experiments on modulation of conductance by surface

charges, it was found that only 10 percent of the induced charges

were effective in changing the conductance. J.Bardeen11 proposed

that the ineffective portion of the induced charges is lodged in

states localized at the surface. Since the work of McWhorter12

and others, it is held that the traps in the oxide modulate the

generation-recombination rate and this gives rise to flicker

noise in the device. Hooge 13 explained the excess noise by an

empirical model, though in his case the noise is a volume rather

than a surface effect.

Recently van Vliet, van der Ziel and Schmidt14 considered

heat conduction in a thin film supported by a substrate and found

that surface sources can give rise to 1/f noise. A sample with a

small inversion layer near the surface which is coupled to larger












bulk volume is considered in this thesis. The model is mathematically

similar to that considered by van Vliet et al., 4 and, as for the thin film

case, the surface noise gives rise to 1/f type of noise behavior with

some theoretical restrictions.



1.5 Embedded and Non-embedded Bodies

From the theory it emerges that one must distinguish between

models involving embedded and non-embedded domains. In the former

case, one supposes that the volume of interest for the noise, Vs, is

part of a larger domain V. Fluctuations in the particle number or heat

content of Vs arise due to sinks and sources in Vs (g-r contributions)

and due to transport across the boundary into the rest of V (diffusion-

drift contributions). An example is afforded by the case of field

emission from a metal tip. Here ions within a, say circular, patch

of the active area tip can modulate the electron emission; these ions

therefore affect the emission, when they are within this circular

patch Vs, which is a part of the larger cathode area V over which

ions diffuse.

The non-embedded case applies when we monitor the particle or

heat content of a domain V which does not communicate with the ex-
s
terior but on the boundary of which we have deterministicc boundary

conditions', usually of the Dirichlet, Neumann or mixed type. An

example is when we measure noise due to carrier fluctuations in a

semiconductor sample Vs, subject to surface generation-recombination

(mixed boundary conditions) on the free surface.












This dissertation is divided into three parts. The first

part, A, contains the mathematical and physical basis; in particular

in chapters 2 and 3 we discuss the various mathematical methods for

transport noise and the physical noise sources, respectively. The

second part, B, gives the noise due to volume sources. In chapter 4

we discuss the noise for symmetrical embedded bodies. Chapter 5

is devoted to noise from nonsymmetrical embedded bodies. Chapters

6 and 7 consider the noise from symmetrical and nonsymmetrical non-

embedded bodies, respectively. Finally, in the third and last part,

C, we consider the noise due to stochastic boundary conditions.

In chapter 8 we study in detail the noise caused by surface generation-

recombination processes in bulk samples as well as in MOSFETs.


































PART A
MATHEMATICAL METHODS AND PHYSICAL NOISE SOURCES













CHAPTER 2
MATHEMATICAL METHODS FOR TRANSPORT NOISE



2.1 Methods

As indicated in the introduction, there are basically two

ways to attack the problem of transport noise. The first class of

procedures considers (a) transport equation(s) in the sense of

Langevin, that is, one adds source terms ((,t) which themselves

have no memory, to the phenomenological equations) for the fluc-

tuating variables) aQ(,t); the variables ar,,t) are Markovian

random of infinite dimensionality since C is a continuous parameter

like a position in some domain V. The boundary conditions on the

surface S of V may be deterministic or stochastic; in the latter

case surface Langevin sources (,t) must be specified (o E S).

The "Langevin procedure" or "sources procedure" is still the most

versatile one. The stochastic Langevin equation can be solved with

Green's functions, orthogonal or biorthogonal expansions or Fourier

methods.

The second class of procedures is based on the solution of

the transport equation for the ensemble averaged transport quanti-

ties (a(r,t)|a(r,0)> -= a(E,t)>a(), in which the fluctuations

Aa(r,t=O) are specified. The solution involves the "correlation

procedure." The noise is obtained by the Fourier-Laplace trans-

formed Green's function of the transport equation, providing the

covariance function, Fr(,,r') = (Aa(Z,t)Aa(Z',t)>s is known;











here ()s denotes a stationary nonconditional average. This leads

to the form suggested by van Vliet and Fassett9. This procedure

is mathematically much faster than the Langevin procedure; however,

it has the disadvantage that stochastic boundary conditions cannot

be included. Nevertheless, if the transport equation is linear and

if the surface sources are uncorrelated with the volume sources,

the former can be separately considered with the sources method, due

to the superposition principle. Also, despite the fact that the

covariance function in the presence of surface sources is often not

unique, it has been shown by van Vliet10 that the spectra resulting

from the Green's function procedure are unique. We will discuss both

classes of methods and their prospects for obtaining 1/f noise.



2.2 Langevin Equation and Source Methods

The stochastic transport equation is supposed to be of the form


La(r,t) = ( + A )a(r,t) = (,t), (2.2.1)


where L is a linear differential or integral operator. For the case

of diffusion:

a = An (particle diffusion)

A = -DV 2, (2.2.2)

a = AT Cheat diffusion)


In the diffusion model A is a self adjoint operator which simplifies

the mathematics considerably. The boundary conditions must be

specified at the boundary S of V. The noise may concern the entire











domain V or subdomain Vs c V. If V is infinite, the only condition is

that aCx,t) goes sufficiently fast to zero. If on the surface S of V

we have

maCro,t) = 0, 'o E S, (2.2.3)

m being a linear surface operator, the boundary conditions are deter-

ministic; if

ma(~o,t) = C(o,t), o E S, (2.2.4)

the boundary conditions are stochastic.


2.2.1 The Bulk Noise

Because of the superposition principle, noise associated with

F and ( can be separately considered. Thus first setting Y = 0,

the boundary conditions are deterministic (2.2.3). Various methods

will be given to solve from (2.2.1)-(2.2.3).


2.2.1Ca) Eigenfunction expansion

Let k( ) be the eigenfunctions of A subject to b.c.(2.2.3)

that is

Ask ) = Xk k(U; (2.2.5)

let fk(z) be the eigenfunctions of the adjoint operator At, then

At *k( = )1k4k). (2.2.6)

The functions 4 and are biorthogonal, i.e.

(Ck^ ) = f d3r k(,) (y) = 5k. (2.2.7)
V











Expanding aX,,t) and Cbt,t) in the functions k and denoting the

expansion coefficients by ak(t) and Yk(t), then (2.2.1) gives

Z[a(t)+kak(t)]4k = Z k (t)q( (2.2.8)
k k

Multiplying with 4/(r) and integrating over V and with (2.2.7) one

obtains

+ X )ae(t) = e(t), (2.2.9)

which is a standard one variable Langevin equation. In the usual

. method of solution truncated Fourier series in the time interval

(0,T) are used. Fourier integrals are not suitable since the
Langevin equation (2.2.9) does not apply for negative T; thus

.o in t
a (t) = Z acne n = 2nn/T, n=0,l,..., (2.2.10)
n=--

and similarly for z (t), one obtains


(Ekn Ens
(aknan) s = (2.2.11)
(Ls +Xk) (-iW+X*)
(i+n ) (-iYXn)


Now the essence of the sources method is that the white spectra

of (E(C,t)~((',t)), denoted as S(,,X'), are known. Thus according

to the Wiener-Khintchine theorem, the sources correlation function is

(r,t)('*X ',t')>s = SC,,Z')6(t-t') (2.2.12)

(E=C* is real), in which S must be known. Expanding in E.F., we
have












S(k(t)5e(t')> sk(9 ~ ) = S C,,,' )6 (t-t'). (2.2.13)
ke

Multiplying this with ,C) \V,C('), integrating over d3rd3r',

and using biorthogonality, it is found that

1 3 3
( *k t) (t')>s = -JJd3rd3r' SE !,,') k [ Ct ') 5(t-t').(2.2.14)

Representing the ('s by truncated Fourier series we further have,

i(O t-m t') 1 3 3 i (t-t')
Z (knem>s e = Tfd rd r'S ,') 'e n
nm
(2.2.15)
from which we obtain


(Eknm >s = ffd3rd3r'S( ')k '). (2.2.16)


The spectrum is given by


S a*(n) = lim 2T(ana s. (2.2.17)
k t T-ro


Thus from (2.2.11) and (2.2.16)

1 3 3
Saa* = (iX+kk) -i+X*) ffd rd r'S(r,')*kT)*t(,').(2.2.18)
K z ('w+kk) (-i

Finally for the spectra of (a(g,t)a(c',t)> one obtains for the noise

due to the bulk source :

bulk
S Cbu r',rc) = Z S (o)k )'), (2.2.19)
a k ak Za

or substituting from (2.2.18) one obtains

Sbulk Zk 3 3
s (Xi)(-i) ff d rld r2Sg C1 ( 2kr l) 2 .)
a (X(+i)( i) .2.20)
(2.2.20)











To show the connection with the correlation procedure to be

discussed in section 2.3, we write

1 = 1 f 1 1 .
+i) + -i (2.2.21)
(Xk+ia)(w -ia) k+j X*-im '

For the covariance from (2.2.20) and (2.2.21)

bulkk d (-'. d) bulk ,Rb
rbu lk,t') = j d( )ReSa ( w', )
0 a

1 Y k C) ') d 3 3
2 (,ri 'Z2) *k (i) *k 1 )
= ~ k d rd r2S ( )1'2~ k1 C2Cr)

(2.2.22)
Operating on the LHS with A +A we note with (2.2.5)

(A )bulk 1C ffd) 1 d 3 3r,
(^ +,)b lk('') = 12 2 dk1 d 2S )1 T2 k1) /(X2)

(2.2.23)
Observing the closure property

k 0 kC)l) = 6f-r1) (2.2.24a)
k

2 0Cr')QC2 = 6(r'-2) (2.2.24b)

and integrating out the delta functions we find

(A +A ,)Fbulk ,z') = ). (2.2.25)

This is the A-theorem due to van Vliet8. This theorem gives the

connection between the sources spectra and the covariance function.
Also it warns that sources spectra cannot be postulated ad hoc, if
bulk ) is known from statistical mechanics.
F (%, ') is known from statistical mechanics.












Substituting (2.2.25) in equation (2.2.20) and using equation

(2.2.21) one obtains


bulk
a


hk C)Wfd')
=2 k k &
kt


1


x ffd3rld3r2 ( +A 2 bulk (, ';)2) (I 2)


Introducing Green's theorem for the operators A and A :


(Af,g)-(f,Atg) = #C[f,g*].da,


. (2.2.26)


(2.2.27)


where #C-do is the bilinear concomitant involving a surface integral.

Thus in (2.2.26)


fA bulk 3,
ATF (1"Z2 'k~ 1)d rl


,bulk t 3
jr C1', )Ar k C1)d rl
~1


+d2r0C[Frbulk Zr2 ) ] k
2 bulk


= +kfd3rl rCbulk 1 2 1)


+d2r 0C[rbulk rZO,;2) '* rO)].


Doing the same for


and keeping in mind the closure property,


we get


bulk ,
Sa ( '


1 3
=2 Zkk ) k+i fd
k k k ,r)


+ hcj + surface terms,


(2.2.28)


AZ2'


(2.2.29)













where hcj stands for Hermitian conjugate, r r', io 4* -io,

X X*; the surface term involves the bilinear concomitant. The

first part of (2.2.29) is called the volume contribution. It will

be denoted by S When the bilinear concomitant is zero as is
a
often the case the volume contribution equals the bulk contribu-

tion (2.2.29).

Often the interesting part is to know the spectra of the total

value or spatial average of variable a(r,t) in V Thus let
s


A(t) = f a(r,t)d3r.
V


(2.2.30)


The integration yields for (2.2.20)


Sources form:


bulk
S (c) =
A


3 3 \kW^W
f f d rd r' -F. k Z
v v (V k+io) -i)
s s


3 3
x f f d r1d 3r2S (1~ 2)k1* IF/(l) 2)-
VV


The correlation

Let us suppose,


(2.2.31)


form, obtainable from (2.2.29) contains fd r'rT(,",e').

as is often the case,


= [(/V5 ]5 ct(z"-


(2.2.32)


Then we find












Correlation form:


vol 4(AA2) 3 3 k
S Vol =4 Re f f d3rd3r' Z k() (2.2.33)
A V+=
s V V k Xk+i+
s s

In general \ = Pk+ivk is complex; for self adjoint case vk = 0

and (2.2.33) reduces to


ol = 4AA2> pk fd3 rk( 12
S (W) = Z (2.2.34)
a V 2 2
s k pk+o


Finally, note that the eigenfunction method has several

drawbacks. If the domain V is finite, the results occur in a series form

which is not easily evaluated. If the eigenfunctions can be taken

to be plane waves, then we end up with the integrals over ,-space

which are not trivial. We therefore prefer the Green's function

method, whenever the Green's function can be evaluated in closed

form, as is the case for suitable geometries (one dimensional bar,

cylinder,sphere). The limiting spectral ranges can then be obtained.



2.2.1(b) Green's function procedure

The Green's function procedure is much more direct. The

Langevin's equation (2.2.1) is Fourier transformed with
io t
n 2nn
aC(,t) = Z an () e ,n T-, n=0,+l,+2,... (2.2.35a)
n
iW t
(SC,t) = Z, %() e (2.2.35b)
n












This gives

Cic +A )an( = n(.). (2.2.36)


The Green's function G(Z,Z',in ) is defined by


(in +A )GC,,',i n) = 65(-I'), (2.2.37)


and satisfies homogeneous boundary conditions corresponding to equa-

tion (2.2.3). Using (2.2.37), the solution of (2.2.36) is found

by using the standard procedure


and() = J drlG ,,' n) ). (2.2.38)
V

The spectrumis found by the result analogous to (2.2.17)

bulk
Sa (ug,',o) = lim 2T(an()a*((')>
T-

= f d rld r2G R~1 i)G ( ,-i)S (1 2) ; (2.2.39)
VV

bulk
for S we find

bulk 3 3 3 3
SbA k() = f / d rd r'f f d3rldr2Gc', ,io)G(', 'E -i)S ( l'2)
Sv V V V
(2.2.40)

These are the sources Green's function forms.

It is well known that Green's functions can be expanded in a

biorthogonal series


G ',i = k k+i (2.2.41)












When (2.2.41) is substituted in (2.2.39), the result (2.2.20) is

recovered.



2.2.2 Surface Noise

Now we consider the transport equation (2.2.1) with ( 0

and while we have stochastic boundary conditions (2.2.4); we also

assume that and are uncorrelated. (If this is not the case there
cross
will be a contribution S .)


2.2.2(a) Eigenfunction expansion

This method has not been tried due to the complexity of a series

expansion in eigenfunctions; however a response method without Green's

functions can be derived.


2.2.2(b) Green's function procedure

Again, we make a Fourier analysis of the (homogeneous) Langevin

equation and of the stochastic boundary condition (2.2.4)

(ice +A )a (Z) = 0, (2.2.42)

man () = n(). (2.2.43)


The solution of these equations leads to

an ) = fdaoH( ,i0 n o) (2.2.44)


(do = d2 r ), where H is the surface Green's function. For the

spectra we now obtain due to the surface sources












,Z )= jd0 H (2.2.45)
surfa IL',W ) = fdao doHCC(,o,ii)H( ',o,-ic)S(o ,) (2.2.45)

surf
while for S one obtains
A

Ssurf(] = f j d3rd3r'l f do do.H( ,jO,io)H(1' ,,-iw)St oo).
V V SS
s s C2.2.46)

In particular, let S = Zr)6(2) CT-). Then

surf,) = j 2r Z(C )I f d3rHCX,o,iw) 2. (2.2.47)
S V
s
These are the sources Green's function forms. The covariance

function associated with the surface noise is

surf ) surf
,surf ') = Re Saurf, ). (2.2.48)
0

It can also be shown that rsurfr') satisfies the homogeneous

A-theorem

(Ar+Ar) surff) = 0 (,r' E V). (2.2.49)


Total covariance r = bulk +surf, therefore,satisfies the A-theorem

which is

(A +A ,)r ') = ') = (,), (2.2.50)

where E is the correlation strength of the volume sources.



2.3 The Phenomenological Equation and Correlation Methods

The phenomenological equation corresponding to (2.2.1) is

L(a,t)>a ,0) +( +A r)<(a(t)>a (O,) = 0, (2.3.1)
a~r;, OP









where >al,0) means a conditional ensemble average for systems
ar,0)
with fixed fluctuations at time t=0. The boundary conditions cor-

responding to (2.2.3) or (2.2.4) is

m(a(x, 't)>a(,0o) = 0. (2.3.2)



2.3.1 The Volume Noise

Since the surface sources average to zero, it is unlikely

that the noise found by correlation method includes surface noise.

Moreover, the correlation method does not give the complete bulk

noise; in fact, it only gives what is identified as the volume noise

in section 2.1. The reason for this is that the covariance func-

tion F, which figures centrally in this method, only represents the

solution of the inhomogeneous A-theorem. The surface source and

other boundary spectra give covariances which satisfy the homo-

geneous A-theorem.


2.3.1(a) Eigenfunction expansion

The stochastic variable is once more expanded in eigenfunc-

tions 0k(r), which satisfy equation (2.3.2), this yields for

for equation (2.3.1)


ak (t)>{ak(O)} + k(ak(t)>{ak(0)} = 0, (2.3.3)

which gives
-kt
ak(t))>ak(O)} =e (2.3.3a)

and furthermore

k(t) = <(a(t)a(0)>s = a(t)) (2.3.3b)











Finally for the solution one obtains


Svol
aka.


= 2 /dt[e-ict(ak(t)a*(O)>s +
0

=2( s 2(a ^ +


itt))]
e ]


(2.3.4)


We have used here the transposition property for the correlation

function 4kt(t) = 'Sk(-t). With an expression similar to (2.2.19)

we find


vol 1i
Sa k t,z' ,) = 2Z i t/ ) aka s ).


Next we consider the expansion for the covariance function


r;E1,2) -=

= (aka sk l)1 (2).
ke


(2.3.5)


(2.3.6)


Multiplying with k(C ) 1 2) and integrating over all 1 and X2'
one obtains


(ak, a,>


= ffd3rd 3r2 F(C1r2)4, -l)d*, C2).


(2.3.7)


Substituting this into equation (2.3.5) we obtain


Svol
a


=2 Id rd3r Vk d)) (1)) ( )
kt V V


1 1
X[Xk+ico 1 2).
Xk~iwXC-e


(2.3.8)


Carrying out the required integration, the result is obtained as

Sa(,,',w) =2 Z fd3r"k )3(r") (E )F(",r') + hcj, (2.3.9)
a k V k











which is the same result as obtained previously via the sources method

in equation (2.2.29). For the result of the noise of A(t) = f d3r a(,t),
V
using this relation one obtains (2.2.33) or (2.2.34). s



2.3.1(b) Green's function procedure

For this procedure we need the solution of equation (2.3.1);

forthis purpose we Fourier-Laplace transform equation (2.3.1), i.e.

let


0
aCr, i) = J e- (aFt))ad,.0) dt. (2.3.10)

Then equation (2.3.1) yields

Ci+Ar )a(,ia) = a(~,0). (2.3.11)


The Green's function is given by

Cic+A )G(X,,r',iw) = (6-,'), (2.3.12)

with the boundary condition mGxo,,r' ,i) = 0 corresponding to

equation (2.3.2). By the standard procedure we find the solution as


aC,id) = f d3r"G(,~",io)a(V",0). (2.3.13)
V

Therefore, by inversion of the Fourier-Laplace transform


(aCt)>)a, ) = jideit fd3r"G(Z,,e,,i-)aC(",0) (2.3.14)
t))aO) 2 C V

where c is a complex frequency and C is contour encircling all poles.

For the correlation function we have (t ? 0)












,(I,.',t) =- s = <(a(r,t))a> )a(r',0)>s


= dicot f d3r"G(C,",i )F 'bulk ,Z"). (2.3.15)
C V

For negative time (-t) we use the transposition property

C,*' ,-t) = 4~(,',,X,t). For the spectrum the Wiener-Khintchine

theorem then yields,

W1 i(c-o)t 3 bulk
Sol(,r,) = -fdt dei(-)t fd3r"G,",i)bulkr',r") + hcj.
a ,, 0 C V
(2.3.16)

Interchanging the time and contour integral we have


f df dt ei(m- )tG,", i) = -f ds[i- +68(G-o)]G(,a",ic)
C 0 C -)

=-2nG(,r,", io+0)(by Cauchy's theorem) (2.3.17)

So now substituting the integral in equation (2.3.16) we get the

result


Svol ,', ) = 2 fd r"GC ,",i) ~b1','") + hcj. (2.3.18)
a V

This is van-Vliet-Fassett form. The result is in full accord

with equation (2.3.9) if we use the series form of the Green's

function, equation (2.2.41).


2.3.2 The Boundary Noise

The spectrum found by the above procedure does not include

surface noise. However, it also does not comprise all the bulk noise,












which is to some extent surprising. The reason is that the transport

equation is not Markovian in the usual sense. The fact that the di-

mensionality is infinite means that the usual Markov property


W2(a(Xt),a'(X',t')) = P(a(,,t) |a'(',t'))Wl(a'(X',t')), (2.3.19)


which is the basis for the evaluation of the correlation function in

(2.3.15) must be doubted. In other words, the process cannot be

Markovian unless boundary behaviour is also specified.

The total noise can be broken up in two ways. The first way

is
Total = Sbulk + Ssurface (+Scross). (2.3.20)


This breaking up is according to the considerations of section 2,

the terms representing effects due to the bulk sources S.r,t), the

surface sources C(;0,t) and due to their possible correlation. In

the second way we evaluate volume noise according to section (2.3.1)

based on (2.3.15). The remaining part to be evaluated in this section

is called Sboundary. Thus

Total = Svolume + Sboundary(+Scross) (2.3.21)


To obtain Sboundary we start with the response expression

(2.2.39). In the RHS we substitute for S the A-theorem, and obtain

bulk 3 3
Sba uE ',l) = 2 / f/dd r d r2GC,,iw)GF(',,2,-i)(A +Ar) F~,-).
VV 2 2.3
(2.3.22)












We indicate by Fl,j12) here any (interior domain) particular solution

of the A theorem, which may differ from physical covariance function

by a solution of the homogeneous A theorem. The RHS of (2.3.22) is

rewritten as
RHS 2 d3 r G (r, X 3(-
RHS = 2 / drl G( 1,im) f d r2 G(r',2,-im)(-i- Ar ) r(1' r2)
V V 2

+2 f d3r2 G( ,', ,-icw) J d3rl GC(br,ic)(; (c& )rl'r2)
V V -1

3 3 ^
2 f d rl Gg,'1, ico){f d r2 r2)Lr GCr',2,-iw)
V V -2

2
+# d roC V ixC l0),G('xg0, -ic)]}
S

+2 f d3r2G( ',~2,-i){f d rF(~ r1 2)L G(rr1,ic)
V V 1

4+ d2r0CI r0F2) ,GCr0,iw)]}, (2.3.23)
S

where we have used Green's theorem in the form


(L f,g)-(f, Ltg) = (Af,g)-(f tg) = d7 C[f,g*]. (2.3.24)

According to the definitional equation for the Green's function

t t t


t* t*
LX GC,,io) = Lr G* E1,,l ) = 5 1) (2.3.25b)

Carrying out the delta function integration we are left with

Sa ulk ') = 2 i dr" G(,r,",ici)rI,r ,r') + hcj
a V
+2fd3r" G,,r",im) f d2r C[F(r",ro),G(r',ro,-ic)] + hcj. (2.3.26)
S













vol
The first two terms comprise S The last two terms,

together with S comprise ndary. The Green's function

form for surf was given in (2.2.45). We then have
a


V S
3 3 2 2
+ hcj + j dr d3 r' d2r0 d2r' H(,~0,iC)H(g', 0,-ic). (2.3.27)
V V SS
s s

We note that all terms in (2.3.27) are quadratic in the Green's

function. Thus the division of S into Svol and Sbound separates

the spectra in expressions linear and quadratic in G. The merit of

this separation is that in many instances all terms of Sundary are

zero for homogeneous systems and delta covariance function. For in-

homogeneous systems such as transitors, the boundary terms are of great

importance, however;when Sboundary = 0, the evaluation of the noise

by Svol as given in section (2.3.1(b)) is by far the fastest pro-

cedure.

In equation (2.3.26) we used a particular solution r Another
vol
choice of this solution, i.e., r', leads to another division of Sv

and Sboundary. However, the total result (2.3.26) is unique. Usually

one chooses r such that the spectrum Sy vanishes, if possible.

If on physical grounds it can be argued that there is no boundary

spectrum contribution, then this choice of r is clearly the physical

covariance function.





27







So far we have discussed the mathematical methods to obtain

transport noise spectra. In the next chapter we shall discuss and

briefly explain the different physical noise sources, their spectra

and importance in calculating overall noise spectra of the physical

system.












CHAPTER 3
PHYSICAL NOISE SOURCES





3.1 Introduction

We have seen in the previous chapter that a calculation of

noise spectra with the Langevin or the sources method requires that

the spectra of the Langevin noise source must be known. There are

various physical noise sources with known spectra. Furthermore, we have

also seen that there are two different types of noise, viz. first

volume or bulk noise and second surface or boundary noise stemming

from volume noise sources and surface noise sources, respectively.

In this chapter we shall try to explain both types of noise sources

as we encounter them in physical systems.



3.2 Volume Noise Sources15



3.2.1 Particle Diffusion Noise Source

To explain the diffusion noise source let us assume that the

fluctuations in number of particles (electrons or holes) are governed

by the diffusion equation, say for electrons


(An) + DV(An) = 0 (3.2.1)

where An is the fluctuation in densityof electrons.
where An is the fluctuation in densityof electrons.












The Langevin noise source representation of this diffusion equation

in the form of the standard diffusion noise source (,t) is given by


a (An) + DV2An) = ) (,,t). (3.2.2)


The source r follows from the stochastic current as follows. For

(D)diff we have the randomness

diff = -qDV(An) + qT(C,t). (3.2.3)
P-0 diff ^

On a microscopic scale, '(3,t) represents the thermal scattering or

Brownian motion velocity fluctuations of the carriers. This -q source

has a spectrum15

S = 4D(rTC.))68 -,X')I (3.2.4)

where I is the unit tensor.

We also write (n(Z)) = n0o). The conservation theorem for

charge reads

an 1
V d(AJ) = 0. (3.2.4a)
at q ~ diff

Substituting (3.2.3) we obtain


n DV2 (n) = ..C,t). (3.2.5)
Comparing (3.2.) with 3.2.2), we note
Comparing (3.2.5) with (3.2.2), we note


t) = -C ,t)












Hence for the spectrum


S=(,'') = : SzCz,') (3.2.6a)


= 4DV ., {no0)68C-')}. (3.2.6b)


This diffusion noise source is the proper noise source of sta-

tistical mechanics. As we can see by the A-theorem, due to van Vliet8,

which connects Langevin's noise source spectra to the covariance

function,
1
(A + )rCr,') = 2 E8 CX ') (3.2.7)

the standard diffusion noise source gives for the covariance function

a delta function:

C,') = no(r)6(r-r'). (3.2.8)

All the previous calculations of diffusion noise by McFarlane16,

Burgess Richardson 1, van Vliet and Fassett9, Lax and Mengert3,are

based on the assumption that the spatial covariance function is a delta

function given as constant x 6(z-r'). This result can also be proven

from first principles, see ref.9, page 329.



3.2.2 Heat Diffusion Noise Source

The heat diffusion noise source can be explained by considering

the relation between the heat conductivity a and the heat current K,

which is


K = -aVT + -3Cr,t)


(3.2.9)











where 11 is a Langevin source. Heat conservation requires that

aT
cd a + V., = 0 (3.2.10)


where c is specific heat per gram, and d is the density. We can

find the stochastic heat diffusion equation by substituting (3.2.8) into

(3.2.9) and by using T = (T>+AT,


BAT 2 -Vg~gt)
aA_ CV (AT) = 1,t) = (,t) (3.2.11)
at cd

where a = a/cd is the thermal diffusion constant. The correlation

strength or white noise spectrum, of the source Ti has been found by

van Vliet, using the Boltzmann transport equation, to be


S ,r') = 4Ik T() 2C )5r-r'), (3.2.12)

where T(C) is the time or ensemble averaged temperature at position

x and I is the unit tensor. Clearly

SC,') = (4k/c2d2) V .V Tr 2oar)6Fr')} (3.2.13)

which is the spectrum of the standard heat diffusion noise source.14



3.2.3 Voss and Clarke's P-Source19

Voss and Clarke introduced another noise source, which has a

delta function type correlation function and a white spectrum given by

SV-ci ( PO')= -r' (3.2.14)
where P0 is a constant.











As we have seen previously noise sources cannot be chosen ad hoc

due to limitations imposed by the A-theorem and statistical mechanics.

Using the A-theorem the three dimensional spatial covariance function

corresponding to the P noise source is found to be
p2
rz') 16 D (r-r (3.2.15)


Such a long-range correlation function is highly unphysical.



3.2.4 Particle Volume Generation-Recombination Source

Usually noise due to generation and recombination of carriers

results in fluctuations in the resistance of the specimen under

consideration. The noise can be described by the fluctuating number

N of the carriers, say for electrons; then the equation governing

the fluctuations in N is


d = g(N) r(N) + Ag(t) Ar(t) (3.2.16)

where g(N) and r(N) are generation and recombination rates and Ag(t)

and Ar(t) are the randomness in these rates.

Substituting N = N +AN, where NO is the equilibrium number

of carriers and neglecting higher-order terms in AN yields the

equilibrium condition g(N0) = r(N0) and the linearized Langevin

equation, given by

ddN AN AN
dA + Ag(t) Ar(t) +- + (t) (3.2.17)
dt T gr
where
1 dr d
T (dN dN) NO











The Langevin source gr(t) can be obtained by assigning shot

noise to the random transitions Ag and Ar:

SAg = 2g(N0) 2g0 (3.2.18)


SA = 2r(N0) 2r0 = 2g0 (3.2.19)


SAg,r = 0 (no correlation).

Hence

SEgr = S g + SAr 2SAg,Ar = 4gO. (3.2.20)

Making a Fourier analysis of (3.2.17) one finds for the spectral connec-

tion 2

SNW) = .2 (3.2.21)
SA(+l+o-T ) l+CO

Integrating over all frequencies, we find

2 1 go
(AN2) = SAN() d = g [r'-g C(3.2.22a)


which is Burgess'20,21 g-r-theorem.

In transport systems the spatial dependence must be added.

Thus, for the density n(Z,t) subject to diffusion and generation

recombination processes we have


aAn(r,t) + DV2 An (, t) + An,t)t).
at T diff('t) +agr 't)"

The g-r noise source now becomes

S~gr ( ') = 4g056C-'). (3.2.22b)
gr












3.3 Surface Noise Sources



3.3.1 Particle Surface Generation-Recombination Source

Just like generation-recombination processes in the bulk give

rise to a bulk Langevin noise source, so generation-recombination

processes at the surface can be represented by a surface noise source;

this latter source causes surface transport noise, at the surface

of the sample.

For example, for the case of non-zero surface recombination

velocity, the stochastic boundary condition can be written as

DV 0(An) + SAn = (m(0,t) (3.3.1)

where D is thediffusion constant for electrons V0 is the surface
n ~0
concentration gradient and K(~0,t) is the surface stochastic Langevin

noise source. For the threedimensional case this surface noise

source has a spectrum

S =4MaSn()5 (2) 0r-) (3.3.2)

and for the one dimensional case

S = 4MaSn(X0)/A

where M is the modulation factor > 1, S is the surface recombination

velocity, A is the area of surface, a = (AN2>/N0 and (2)C0_)

is the two dimensional delta function.











3.3.2 Heat Transfer Source

The heat transfer source is a stochastic source due to ab-

sorption and emission of blackbody radiation at the surface of a

body. For example, this occurs in the case of a thin film supported

by a substrate with the front face of the body being blackened,

so that it sees the blackbody radiation field. The boundary condition

in this case becomes

oCO) aAT gAT(O) = E(t). (3.3.3)
axlx=0

Here (Ct) is the stochastic heat transfer source. The spectrum of

this source is given by the generalized Nyquist's theorem

S = 4kT2g = 16cBkT5 (3.3.4)


where g = 4a T3 is the heat transfer conductance, TO is the tempera-

ture of the environment and the mean temperature of the body, and

cB = 2 5k4/15h3 c2 is the Stefan-Boltzmann constant.


































PART B
NOISE DUE TO VOLUME NOISE SOURCES














CHAPTER 4
SYMMETRICAL EMBEDDED BODIES





4.1 Introduction

In this chapter we shall study the noise spectra resulting from

symmetrical geometries which are part of an infinite expanse i.e. the

"embedded" case. We have already seen in a previous chapter that there

are two volume noise sources, namely the standard diffusion noise

source and Voss and Clarke's 'P' source. Further we have also seen

that the 'P' source is highly unphysical. In this chapter we con-

sider only noise stemming from the diffusion noise source.



4.2 Survey of Older Work

In this section we briefly summarize the results of previous

work for obtaining noise spectra from symmetrical embedded bodies

like the linear bar, the cylinder and the sphere, prior to the dis-

covery of the Green's function method.

MacFarlane16 in 1950, studied the problem of the power density

spectra of noise current resulting from contact noise in semiconductors

or from emission noise in field emitter diodes. In these theories

low frequency noise is attributed to the random movement of absorbed

ions on the contact surface between two granules of a carbon resistor

or on the emission tip of a field emitter tube. The current through











the tube or the emission of electrons is assumed to take place only at

localized patches on the surface and the absorbed ions are assumed to give

rise to a dipole layer, which modulates the work function in such

a way that AW is linearly related to the concentration of ions in

that patch. This concentration fluctuation is due to diffusion of

ions over the surface of which the conducting patch is a small part.

MacFarlane found, by first calculating the autocorrelation

function and then using the Wiener-Khintchine theorem, that the

spectral power density of the noise current is given by


RC)) {l-exp(-x1/4)[I0 ()x-1/ 1(x-/4)]cos(px) dx (4.2.1)
0
2
where p = wrOP, T = (2r) /D, r is the radius of the patch, and

D is the diffusion constant of ions, and I's are the modified Bessel

functions. In the limiting cases it was shown that, for a circular

patch, (using probability theory results22.)

R(c) = 0.2345 p3/4, p << 1 (4.2.2a)

and

R(o) = 0.1466 p9 p > 1 (4.2.2b)

while for a long thin strip

RC) = (2p)-12, p << 1 (4.2.3a)

and


R(w) (2p)-3/2


C4.2.3b)


p >> 1.












For the long thin strip this shows the occurrence of the 3/2 power law

at high frequencies. This universal law was violated by the result

C4.2.2b); Burgess showed MacFarlane's result to be in error; when

properly integrated, a 3/2 power law at high frequencies also result

for the circular patch.



4.3 Previous Work Using the Green's Function Method



4.3.1 Linear Bar

Van Vliet-Fassett9 considered similar problems, as discussed

above, using the Green's function procedure. For the one-dimensional

diffusion equation


A D a2 (4.3.1)
at ax2
ax

where Ap is a fluctuation variable and D is a diffusion constant,

the Green's function is found by standard procedure to be


GCx,ico,x') = p-expCiu0 x-x'l) C4.3.2)
2u0D

where u0 = Vw5U e3i/4.


The spectrum for the fluctuations AP, where P is the total

number of particles in a segment L is found to be


SAp( ) = D Cl-e (coso+sin6)) (4.3.3)
D8











where e = L() /2, in agreement with Burgess' solution17 of the

integral given by MacFarlane16. The low frequency spectrum goes as
-1/2 -3/2
1/2 and the high frequency asymptote is again -3/2 The latter

behavior is characteristic for all diffusion spectra.



4.3.2 Cylinder or Circular Patch

Van Vliet and Chenette23 studied the noise spectra resulting

from diffusion processes in cylindrical geometries. The spectral

densities are derived from the relevant Green's functions. Also

extensions to MacFarlane's16 spectrum for the circular patch were

obtained.

The noise spectrum for the fluctuating variable X, where X

is the total number of particles in the cylindrical volume Vs, in

terms of the relevant Green's function, is given by

4varX
S M(c) = 4vX f G(Z,ic,X')dv dv' (4.3.4)
s DD
s s

where Vs is the volume of domain Ds (cylinder or circle), which is

part of an expanse V > Vs.

The Green's function was found to be the Hankel function of

the first kind of order zero; the spectrum is given by

S C)= 8CyarX)a2 1 1 ii (1)
S D) = aD Re I- [- 1 + (xa)J1(xa)]
Cxa)
C4.3.5)
where H and J are Hankel and Bessel functions, a is the

radius of the cylinder, D is the diffusion constant and x = ivo .
D












It was found that for low frequencies the spectrum has loga-

rithmic behavior and at high frequencies the -3/2 behavior takes

over.



4.3.3 Sphere

The problem of three dimensional diffusion has been solved by

van Vliet and Fassett9. Using the method of reciprocity, the
2-
spectrum was found to be (for fluctuating variable AP, with AP = Po),

16TPR 5
GAP16P) o= 5 y 2-2+e [y Ccosy+siny) +4ycosy +2(cosy-siny)]},
DY
(4.3.6)

where

PO average number of particles in the sphere

RO radius of sphere

D diffusion constant

and

Y = 2ROVAC/2D)


At low and high frequencies the spectrum was found to have the limiting

values:
32TnPOR
GA ) = 15D (4.3.7a)

00

P0(2D) 4nR
G p( ) = 3/2 (4.3.7b)

CO-W

Again, at high frequencies the spectrum follows the 'universal'

3/2 power law.












4.4 Present Work: Carrier Fluctuations in a Sphere
Due to Diffusion and Generation-Recombination

In this section we investigate the noise spectra resulting from

fluctuations in the carriersin a sphere, due to diffusion across the

boundaries and due to sinks or sources inside the sphere. This problem

has not been solved hitherto.

We consider a sphere of radius 'a' (volume Vs) which is part

of a very large expanse. The fluctuating variable is x, the concentra-

tion of particles in the spherical volume under consideration; the total

number of particles in the sphere is X. The boundary conditions on

the surface Ds of the sphere are 'fictitious' i.e. x and its

derivatives are continuous across the boundary.

We consider the equation

ax 2
+ ax DV x = 0 (4.4.1)
at
-1
where a = T is the volume lifetime and D is a diffusion constant.

The Green's function g(r,tJl',t') satisfies the equation


- (F,t|r',t') + agC(,t[t',t') DV2gC,t l',t') = 6(t-t')6r-,').
(4.4.2)

Assuming t' = 0 as initial time, with g(r,Or',0)=0, and taking the

Laplace transform of equation (4.3.2) we obtain,

SG(Z,s,r') + aGCr,s,r') DV2G(Cs,g') = 6(1,-'). (4.4.3)

This can be written as

-sG a
--,sc') G(z,s,E,') + V2G(T,s, ') = 1-6-') (4.4.4)
















where


(x2+V2)GZs,') = -')



x = i .F


(4.4.5)


(4.4.6)


Since the boundary conditions on D are continuity of x and its

derivative and the domain D is assumed to be very large, the ultimate

boundary condition on the boundary of D is that G should behave

properly at infinity, i.e. go sufficiently fast to zero.

Adopting a spherical coordinate system we have
2
2 1 a 2 a 1 6 a 1
V r 2-) + + '- -(sine -).(4.4.7)
2 sr r 2 2 2 i 2 ae as .
r r sin a(p r sinO


Following Morse

infinite domain

only depends on

with R = 0 as


and Feshbach page 808, one can show that for the

the Green's functions in spherical coordinates

-r'| = R. So, evaluating V2 in spherical coordinates

origin, we find


72 = 1 a CR2 ~a
S2 aR l R
R
2
a2 R aR
aR2 R R

Also we can write

GC(,s,Z') = Gx(R).

So, the Green's function equation (4.4.5) becomes

S2Gx(R) 2 aGx(R) 1 (R)
R + R R + x2G(R) = 16R
R2 R aR x D 4R2
8R 4nR


(4.4.8)




(4.4.9)




(4.4.10)











Following Morse and Feshbach26, page 809, we need a solution for the above

equation which goes as

Sfor R 0. (4.4.11)
R
Therefore we require the solution to satisfy the following conditions

1
i) G (R) -I as R- 0. (4.4.12a)
x
In addition, the noise spectrum for high frequencies c -* should go

to zero, or

ii) lim Gx(R) = 0. (4.4.12b)
x -'+im

The only solution for equation (4.4.10), satisfying the above

two conditions is
ixR
e
Gx(R) = 4RD C4.4.13)

The proof of the solution is as follows.

+ixR +ixR
a +ixeixR eixR a 1
x(R) = 4TRD D R- C4.4.14)

2 .+ixR +ixR 2
2R) 2ixe ai 1 e x a2
(R) = -x G + D R 4R 2 4R (4.4.15)
aR aR

+ixR +ixR
2 a 2ixei 2e a 1
4---G (R) = (4.4.16)
RaR 42 DR aR 4nR .


The second term of (4.4.15) can be shown to not contain a delta function;

it cancels the first term of (4.4.16). The third term of (4.4.15) and

the second term of (4.4.16) are combined to yield

+ixR 2 +ixR
e a 2a 1 e 2 1
a + ----7 ( C4.4.17)
D a2 RaR 4R D 4KR
aR


For the latter we have










+ixR +ixR
ix 2 1 e ixR 6(R)
D 41TR D 6(z-') = 2.2 (4.4.18)
4wDR

To prove (4.4.18) we must show

R22 = 0 if R / 0. (4.4.19)

and

/ V2 d3R1 = -4r. (4.4.20)
small sphere

The first statement follows from substitution of (4.4.8). For the latter

statement we use Gansz' theorem:

213 1
f 2 1 d = grad r dS. (4.4.21)
AV Tr-rl Ir'-r '

Now grad(1/|r-r' ) = -1/R2 and dS = 4ffR2, which proves (4.4.20).

We notice that the solutions
+ixR
GCR -
x ) e4TRD

satisfy the equation

(V2+x2)G (R) = 1- (,-').

Using the condition of equations(4.4.12a) and (4.4.12b), we find

that the condition (4.4.12a) is satisfied by both solutions but using

equation (4.4.12b) we find
+ixR R(-i--)
l e e
lim RD (4.4.22)
4nRD 4nRD
X -++~+a.

and the condition is satisfied only by the solution
+ixR
GxR = eR (4.4.23)
which is the required solution for the Green's function.
which is the required solution for the Green's function.












For the fluctuating variable X, the noise spectrum is given


Sx (c)


4varX
= vf G(r,i d,r') dV dV'
s D D
s s


(4.4.24)


The required double integration sought for the noise spectrum is


I = f f d3r d3r' G (R).
V V
s s

Further R can be written as


R = Vr2+r'2-2rr'cose


(4.4.25)


(4.4.26)


where 6 is the angle between r and r'. Using polar coordinates with

r' as polar axis and integrating (4.4.25) first over r we need the

integral (denoting by Il)


ix r2 +r' -2rr'cose
= fff r dr sine de dip
4nDVr+r' -2rr'cos


(4.4.27)


Further we write this integral as


12 2
SixVr +r' -2rr'cose
= -fr dr fdpfd(cosO) e
4nD r2+r'2-2rr'cose


(4.4.28)


Realizing that integration over cp contributes 2n and further that


2 2
d ixVr +r' -2rr'cose
d(cose) e


'2 2
ixVr +r' -2rr'cose
-ixrr' e
2' c2
r +r' -2rr'cos9


(4.4.29)


we find that (4.4.28) reduces to

S 1 a ix(r+r') 'ixl r',r
I1 2xD r'i rdr [ei( e
0


(4.4.30)












As to the double integration (4.4.25), we can write

aa a r' a a
f f f(r,r')dr dr' = f dr' f dr f(r,r') + f dr' f dr f(r,r')
0 0 0 0 0 r'

and it can be shown that the second contribution equals the first one, i.e.
aa a r'
f f dr dr' f(r,r') = 2 f dr' f dr f(r,r') (4.4.31)
00 0 0

because of the symmetry of the integrand f(r,r'). We can thus restrict

ourselves to r 5 r' and omit the absolute sign in (4.4.30).

Realizing the fact that integration over the solid angle d2 con-

tributes 4n, the required double integration yields

8 a reix(r+r') eix(r+r'r)(r'-r) ix(r-r')
r'8dr re + e re e r'
I = f r'dr'------- +-
2xDi rdr ix 2 ix 2
0 x x 0
(4.4.32)
or

4TT r' 1 2ixr' r' 1
I = J r'dr' [(- e2 + -+e +( ]. (4.4.33)
xDi ix 2 ix
0 x x

Using the following relations due to integration by parts

ir 2ixr' 2ixr'
2ixr' r'e e
fr'dr' e = + (4.4.34)
2ix 2
4x
and

2 2ixr' 2ixr'
r2 2ixr' r' e r'e 1 2eixr'
r' dr'e i 2 e, (4.4.35)
2x 4ix











the required integration is finally found to be


41i a 2 2ixa ae2ixa e2ixa 3 a2
xDi 2 .3 4 3i (4.4.35)
i -2x ix 2x 3i 2x 2x

Using for the noise spectrum equation (4.4.24) and representing the
AP l' P
fluctuating variable as p, with = = p, the number of holes
s s
per unit volume, we get

Sp(o)=Re64pOa [eP ( p2-4p+4)+( + 2-4)] (4.4.36)
D(p

where p = 2ixa.

Equation (4.4.36) can be rewritten as
5
64np a 3
S Ap(c)=Re-- 5 (4e -4(pe +pe + 2 4]. (4.4.37)


For the required noise spectrum we need to evaluate the real

part of expression (4.4.37). We shall discuss two different cases.



4.4.1 Pure Diffusion (a=O)

If no sinks or sources are present then a=0. In this case

the required expression for the noise spectrum, taking the real part

of equation (4.4.37), becomes

16ta p 2
SAp (c) = 5 _(-2+y2+e [y2 (siny+cosy)+4ycosy+2(cosy-siny)]},
DY
(4.4.38)

where y = 2a- '
2D











The result (4.4.38) is the same as previously obtained by

van Vliet and Fassett9 for the case a=O. The low and high frequency

asymptotes are found to be (expanding sin and cos terms)
5
32npoa
S (M) 15D (4.4.39a)

and 1/2 2
p0(2D) 1/24a2 (4.4.39b)
S p(c) 3/2 (4.4.39b)

The plot for the noise spectrum is shown in the rn=0 curve of fig.1.

Spectrum is constant at low frequencies and at high frequencies the

spectrum goes as -3/2. The computer program used to plot the

spectrum is given in the Appendix. (u=wTd, n=aTd).


4.4.2 Diffusion With Sinks (O0)

When sinks are present a is not equal to zero. The required

noise spectrum expression,taking real part of (4.4.37) becomes

16na5p0(p2-62) 16Tap0 K
S Cp ) = 2 + (4.4.40)
3D(p2+6 2 D( 2 +6 5

where


K = (10p263-5p46-55)[2p6+26e- cos6(p+2)+e-.sin6(62_ 2-4-4p)]


-(P5-lOp 35+5p64 ) 2-52265e sin(p+2)+e=cos6(P 22+4-4p)-4]
(4.4.41)

= 2a [v + ]a + (4.4.42)

and -
6 d 2a [ Zi + oJav+c ]. (4.4.43)
Vff



































































*--U


Fig. 1

Noise spectra for embedded sphere (n=0, r~0)












Furthermore using the following dimensionless quantities


Td = a /D, diffusion time

u = Wrd
d (4.4.44)
and

) = "d/yv = d

we get for p and 6

p = "TV + I+u2 (4.4.45a)

and

6 = V--n+ V + (4.4.45b)


If in the above expression q = 0, we have p = 6 = y and

equation (4.4.40) reduces to equation (4.4.38). The general case

a 0 has been considered for the first time as far as we know. The

low and high frequency asymptotes are found to be the same as for

a= 0
5
32Trp0a
S -) 5 (4.4.46)
AP 15D

and
P (2D) 1/24ra2
S (-) 2 (4.4.47)
WAP W3/2

The noise spectrum has been plotted in fig.1 For low fre-

quencies the spectrum is constant and at high frequencies, the spectrum

goes as c-3/2. For q >> 1 if a range can be found such that

Ts u < ) 3,then the spectrum goes as c-2 and at higher frequencies

W-3/2 behavior takes over. The computer program used to plot the

spectrum is given in the Appendix.













CHAPTER 5
NONSYMMETRICAL EMBEDDED BODIES



5.1 Introduction

We have seen in chapter 2 that for nonsymmetrical geometries

Green's function cannot be found in closed form and therefore eigen-

function expansion methods should be used. In this chapter we shall

study the noise spectra arising from nonsymmetrical embedded bodies

for both Voss-Clarke's19 'P' source and the physical diffusion noise15

source.



5.2 Voss and Clarke's 'P' Source



5.2.1 Nonsymmetrical Bar

We consider the diffusion in an infinite v-dimensional

domain with fictitious boundary conditions i.e. embedded case. In

all integration of chapter 2 d3r is replaced by dVr. The norm-

alized eigenfunctions of (2.2.2) are


(Pk) = k /) e (5.2.1)

with = (kl,...,k) and
2nn.
ki = L (5.2.2)
1
v
n. = 0,l,2,... and V = 7 L. ; (5.2.3)
1 i=l

for embedded case V goes eventually to -. The eigenvalues are
2 2 2 2 2
=k = = Dk where k kl+k2+...+k (5.2.4)
1 2 (.2.4











Since k-values are dense, we can write

Z fdk Z(~), with Z(J =- (5.2.5)
X (21Y)

being the density of states for a domain of dimension v. Since all

noise expressions are bilinear in (p or *, V cancels out for any sum Z

in the limit V -+o. Then, equation (5.2.5) is exact and the

eigenfunction expansions simply represent Fourier integrals. Thus,

(2.2.31) results in, with f dvk denoting integration over infinite

v dimensional )-space, .

I vdr 2
bulk. 1 ( C dvk dvk s
A (2 ,) (Dk2+ico)(Dk'2-ico)


A ( d1rV ri 2vr
x fJ dVrl dVr2 S IC; Z2) e l (5.2.6)


For Voss and Clarke's P source


S(' = p56(1-i ) (5.2.7)

So for (5.2.6) we now obtain

i. ik.r v 2
2 le e e d r
P V
bulk 0 v s
bulk dvk (O) = (5.2.8)
SA =2 fd k 4 2 (5.2.8)
"A (2n) D2k +C2
L. L.
Now considering V to be a rectangular bar of dimensions-- xi -
2 2
bulk
i=l,2,...,v, the result for SA becomes


bulk 2 2 dvk v sin21 (iLi)
A (o) PO(V s 2 4 2 2 (5.2.9)
D k +w i=l (k.L.)
21 1












For a three dimensional rectangular bar v = 3, and

3 3 sin2 (.L.)
bulk 2 2 d k 2 1 1
s () P(Vs) / ni
A 0 s k i2 l 2 kiL)


For a three dimensional
2
at = D/L., and for
1 1i'
the following frequency

a << e1,

1 << W < 02'

'2 << << W3%
COQ CO)


rectangular bar the spectrum shows breakpoints

the case L>> L2 >> L3 (nonsymmetrical)

ranges are claimed by Voss and Clarke9,

c .. ..-1/2


-1
SC() 2 c

S -3/2

S O) C-2
S (c) -- e


(5.2.11)


Voss and Clarke1' noted in particular the 1/c range between limits

01 and c2.

In an effort to find the breakpoints and the spectral frequency

ranges the method of numerical integration was used. The complete

analysis is given below and the listing of the computer program which uses

the trapezoidal rule for numerical integration is given in the Appendix.

Explicitly, for the three dimensional bar we have


bulk
SA (C) = const. f f J
--


dk dk dk
x y z
2 2 2 2 2
D (k +k .+k ) +o
x y z

sin2 (kxL) sin2( L sin2(kL
1 2 1 2 1 2
=x -k L-- --x L x---k-L --
SkxL1 2 L2 2 1 2


(5.2.10)


(5.2.12)












Let us call

kL =

k L2

k L3 =
kz L3


Using the above transformation, we get as. integrand


dt dr dr X
L xL2xL
123


D 2 D 2 D 22 2
L L2 L
1 2 3


1
sin 2
x C-- )
2


1
sin 2q
x C-) x


1
sin 2
( 1
2--->


(5.2.14)


Furthermore call


S= D/L
C1 = D/


(5.2.15)


S= D/L, = D/L2
I2 = L2, C = D/L 3


3
O = O/V 12 3 ,
C O = F w 2 0 ) 3




_______ (5.2.16)

"2= 2'1


3 = p/W 12 *

We then obtain the integral in the following dimensionless form,


with F(c) = const. S(o):


(5.2.13)































Fig. 2

Noise spectra for nonsymmetrical bar
CVoss and Clarke's 'P' noise source)





































slope -1.5


slope -2.0


- 401 I
10' 10-'


10"' I


I0 100


+ 40




+ 30


* 20




+ 10


- 20


- 30


0.3 ria

\ 4










1 1. 1
F d dg d sinsin si2 1 2 n- 2
F( f 2 2 2-2 1
1 (1 2 +3 +2 2

(5.2.17)

Finally we choose as parameters L1 = 109m, L2 = 1cm and L = 0.1cm,

D = 25cm2/sec (corresponding to mobility of 1000 cm /volt-sec.)

Then we have wl = 0.25rad, 2 = 25rad, w3 = 2500rad; and

C1 = 102 2 = 1, and W3 = 10 These are the normalized break-

point frequencies. (Units for w's should read as Rad/sec.)

Substituting all the above constants and using the symmetry

property of the integrand we have for (5.2.17)

1 1 1
F d8 d df in n 2 sin -2
2 2 22-2 Cl17--
0 0 0 (.01+ + lo +100o2 ) +c
2-2

(5.2.18)


The plot of the spectrum is given in fig. 2. We have not

been able to verify results as in (5.2.11). The spectrum has no 1/f

range and goes much steeper at low frequencies; only the break-

points at C2 and c3 are visible.



5.3 Physical Diffusion Source

We have seen in chapter 3 that the physical diffusion noise

source has a spectrum

S 4 = 4D V-'n(X)6(X-,'). (5.3.1)

When this is substituted in equation (5.2.6), which is the general

result for nonsymmetrical embedded diffusion, one finds for the

spectrum










2 1
v 2 v sin2 (-Li)
bulk d k Dk2 sn2 i (5
S (m) = C 2 4 2 (5.3.2)
"A D2k4+co i=l C.kiLi)


which differs from equation (5.2.9) by the factor Dk2 in the

numerator; C is a proportionality constant. (For particle diffusion

C = 4n0/(2n) .)



5.3.1 Rectangular Bar

For a three dimensional rectangular bar equation (5.3.2) becomes
2 1
3 2 3 sin2 (-kLi)
-bulk ( d3k Dk2 i 2i (5..3)
SA (C) = C f D2 n42 2- (5.3.3)
A 24 2 1 2
SDk+Co i=1 (k.L.)

Again for the case L1 >> L2 >> L3 (nonsymmetrical) Voss

and Clarke claimed the spectral ranges as

f<< fl S(o) = constant

f1<< f f2, S(c) = -log f
f2 << f << f3, S(o) = f-1/2

f >> f3, S(w) = f-3/2 (5.3.4)

Using the same transformation as in section 5.2, the additional

factor Dk2 is found to be


= D(k2+k2+k2)
x y z
D 2 D 2 D 2
L L L
1 2 3
1/3 2 2- 2
1O2C3 ( 0 2 3n + ) (5.3.5)











The overall noise spectrum is given as

-i2 2 2
M M 00 ddid&(wl +m 2 qCO3
F(c) = C f 2 2f 2
0 0 0 (1+2 +0 3 ) +m

1 1 1
in 2 sin 2 sin, 2
sin2 2) 2 2 -2 2 (1- --)
x ) x ( x--- ) (5.3.6)
2I


For the values of L = 10cm, L = 1cm, L3 = O.lcm, and D = 25cm /sec,

this becomes

ddrddt (.012O +T2+100o 2)
FC) = Cf f 2 2 2 2 -2
00 (.012 +2 +100 ) +W2
1 1 1
sin-( 1 2 sin2 (5.3.6a)
x(---) x (-1----) x (1---) (5.3.6a)



The listing of the computer program is given in the Appendix.

We have not been able to verify the results as claimed by Voss and

Clarke's equation (5.3.4). The spectrum is given in fig. 3.

The high frequency slope is -3/2 as predicted by Lax and Mengert3

Breakpoints 2 and w3 are visible and at low frequency (not shown)

the spectrum levels off. Also there is no obvious 1/f range in

the spectrum.

Furthermore we can also see that in the noise spectrum for

the nonsymmetrical bar for both the noise sources (namely 'P' source

and diffusion source), figs.2 and 3, at the low frequencies there is

a "wiggle" in the spectrum. The presence of such a "wiggle" must be

attributed to the nonsymmetry of the volume under consideration.
















20




10

slope-0.38



slope.-0.5

- 0 .

















-40 I I I II
-20 0




-30 -





I0-' 10-2 10"- 10O 10' 10' 10a



Fig. 3

Noise spectra for nonsymmetrical bar
CPhysical diffusion noise source)












The "wiggle" appears on the noise spectrum towards the longest dimension

of the specimen. As we shall see in the next subsection, for more sym-

metrical geometries like square bar and cube, the noise spectrum is smooth

and the low frequency "wiggle" disappears (figs. 4 and 5 ).



5.3.2 Square Disc

The noise spectrum arising from a square disc has been analyzed.

In equation (5.3.6) the parameters have been changed. The longest dimen-

sion has been eliminated and a square disc of lcm2 cross section and

thickness of 0.032cm has been considered. The noise spectrum is given as

2 2 2
F(c) = C f f d dt(0. 12+0.112+100l2)
2 2 2 -2
0 (0.12+0.1 r2 +100t )+C


1 1 1
sin2 2 sinql sin- 2
x (----) x (-----) x (-- --) (5.3.7a)
2- 2


The computer program listing is given in the Appendix. The

plot for the spectrum is shown in fig. 4. We can see that the spectrum

varies smoothly and the low frequency "wiggle" disappears. Further

at high frequency the spectrum goes as f3/2. Breakpoints atl=c 2 and

C3 are visible and at low frequencies (not shown) the spectrum

levels off.
































Fig. 4

Noise spectra for square disc and square bar
CPhysical diffusion noise source)


































































10a 10


I 10


10
































ITS











5.3.3 Square Rectangular Bar

The noise spectrum resulting from a square rectangular embedded

bar has been studied. The smallest dimension has been eliminated

and a square bar of crosssection 0.lcm2 and length 3.16cm has been

considered. Equation (5.3.6) for the noise spectrum is


d~drjdt(olE2 2 2
F(C) = C f dgdd (0.02+102102 )
0 o0.0t +10ri +102 )+2


1 1 1
sin2 2 sini 2 sin- 2
S) 1 -r2 (5.3.7)
2 2 2


The computer program listing is given in the Appendix. The plot

for the spectrum is shown in fig.4 We see that the spectrum varies

smoothly andagain the low frequency "wiggle" disappears. At high
-3/2
frequencies the spectrum goes as Breakpoints at w=1 2 and &3

are visible and at low frequencies (not shown) the spectrum levels

off.



5.3.4 Cube

The noise spectra obtainable from a less asymmetrical embedded

geometry like a cube has been studied. The longest and the smallest

dimensions of the nonsymmetrical bar have been eliminated and we

considered a cube of dimension 1cm3. The noise spectra is given by

bulk 2 2Cf ddp 2 2 2 ssin2 2 sin2- 2 sin 2

S(E +1 + C + 5.3.8)
(5.3.8)












The plot for the noise spectrum is shown in fig. 5. We can

see that the spectrum is very smooth, the spectrum levels off at
-1.6
low frequencies and the high frequency slope is f-.6 The break-

point at W2 is quite visible and the roll off is much steeper than

for the square bar. The computer program listing is given in the

Appendix.

Thus in this chapter we have analyzed the noise spectra

obtainable from nonsymmetrical embedded bodies for 'P' source and

nonsymmetrical as well as less asymmetrical embedded bodies for

the diffusion noise source. We note that the spectral ranges claimed

by Voss and Clarke are not verified. For the 'P' source their claim

for a 1/f noise spectrum is far from true. Further we have seen that

the noise spectra for the physical diffusion noise source results

in f3/2 type of behavior at high frequencies. There may be a quasi

frequency range where 1/f kind of behavior is observed, but this

range is small.

In the next chapters we shall study the effect of the boundary

conditions on the noise spectra, both for the symmetrical and non-

symmetrical unembedded geometries.




































-a \








-30-





-40-





-50 I I II
10"' Ifra 10" I 10'


Fig. 5

(Slope indicated is per decade of frequency)

Noise spectra for cube
(Physical diffusion noise source)












CHAPTER 6
SYMMETRICAL NONEMBEDDED BODIES



6.1 Introduction

In this chapter we shall study the noise spectra resulting

from symmetrical nonembedded bodies. We already have discussed in

chapter 1 the difference between embedded and nonembedded cases,

the latter being subject to boundary conditions at the surface of

the body. We thus investigate the effects of the boundary condi-

tions on the noise spectra for symmetrical geometries. For example,

in the case of surface generation-recombination, we have a mixed

Dirichlet-Neumann boundary condition for the fluctuating carrier

density An,

aAn + bv(An) = 0 (6.1.1)

where a = s, surface recombination velocity and b = D, diffusion

constant. The presence of boundary conditions results in (i) altering

the shape of the spectra from that of the "infinite mediuid and

(ii) correlation effects may occur.



6.2 Summary of Previous Work

In this section we briefly review the results of the work done

so far to obtain noise spectra from symmetrical nonembedded bodies

such as linear bar and cylinder.












6.2.1 Linear Bar

The problem of one dimensional diffusion in a finite domain

was studied by van Vliet and Fassett9. The one dimensional diffusion

in the region -L < x f L with perfectly reflecting barriers at

-L and +L is considered, where L > d. The spectral density for

the fluctuating variable AP, S p(o) in the subregion -d 5 x f d

is found to be

3
S ) P0(2L) sine+sinhe-F(a,be) (6.2.
9 (coshe-cose)

where pO is the average number of particles per unit length,


F(ae,be) = sinae coshbe + cosae sinhbe + sinbe coshae + cosbe sinhae

and

S= 2LV-/2D, a = b = L-
L L

Equation (6.2.1) reveals that at sufficiently high frequencies the
3/2
spectrum varies as 32, whereas at low frequencies the spectrum

becomes constant

(2D)3/2
S C2 = 3 3/2 (6.2.2a)
(o-r (2L) 0
and
d d2 (L-d)2
SAp (o) -)= 4 (6.2.2b)
So0 3L

Thus the presence of reflecting boundaries limits the low

frequency noise, whereas without walls, as in the embedded case the spec-

trum rises indefinitely as c -* 0 according to 1/2











6.2.2 Cylinder

The case of fluctuationsin the finite cylindrical domain with

mixed Dirichlet-Neumann conditions was investigated by van Vliet-

Chenette23 in 1965. Following Morse and Feshbach24, it was shown

that the Green's function or consequently the total spectra can be

written as
total A
Stta S + S (6.2.3)

where S is the spectammas in the embedded case and SA is due to

the presence of boundary conditions. Following this procedure the

total spectra for the nonembedded cylinder was found to be

2
total A a
St S + S = 4(varX)2- -Re P(co) (6.2.4)
x x D
where
1 2a J1Ca)
PCo) = {1 + 2o
(xa)2 xa DxJ1 (xa) -oJ (xa)


J's are the Bessel functions

x = i a is the radius

and a is a constant;in the case of surface recombination, a is the

surface recombination velocity.

The spectrum is constant at low frequencies and at high fre-

quencies the spectrum goes as W-2, with the presence of a diffusion
-3/2
range -3/2) depending on related time constants. Thus due to the

presence of boundary conditions the logarithmic low frequency behavior,

as in the embedded case disappears and the spectrum becomes flat.











6.3 Present Work: Sphere

In this section we shall consider the case that the domain

p = V is a spherical domain of radius 'a' with homogeneous boundary

conditions on the boundary of the domain


an G ,s, ) + oGCa,s,. ) = 0 (6.3.1)
a.
where T is a surface position coordinate, n is the outer normal

and a is the surface recombination velocity. The differential equa-

tion for the Green's function is


d2G 2 dG 2 1 a a 1 82 1
2 dr + x Gsine a .2 ]G = '),
dr r sin 0 acp
(6.3.2)

where Gx(,;') G(;,ico,r'), and x is the same as in equation (6.2.4)

The polar axis we take along ;,'.

Following the treatment as in Methods of Theoretical Physics,
24
vol.1, by Morse and Feshbach page 825, we expand the Green's

function in terms of a complete set of functions involving all but

one of the coordinates (0 and cp) with coefficients which are undeter-

mined functions of the uninvolved coordinate. Thus let

Gx ') Z ,m(r,r')Pm (cose)eim(P-') (6.3.3)
S,m
= 0,1,..., ; m = -4,...,+ .

Substituting equation (6.3.3) in (6.3.2) for the first three terms on

the left hand side,we get
,2 &,m ,m
Sp x 2 dpx 2 E,m m ,im((p-(p')
2 [d2 r dr + x Px ]p(cos@)e (6.3.4)
Z,m dr











and the remaining terms on the left hand side of (6.3.2) give

(Quantum Mechanics, Schiff25, page 75)

2
1 1 8 8 1 a m im(P-(P')
2 sin (sin + 2 ]p(cose)e
r sin 8 a8p

--(5+1) 5 im((p-p') (3
= 2 p (cose)e (6.3.5)
r

Now expanding the delta function on the right hand side of equation

(6.3.2), we get

8 = C8-r') 6(p-(p')56(-9')
2 sine (6.3.6)
r

Let

8e-9 = Z Nm F(',cp')Pm(cos)eimp (6.3.7)
m,e

-2 25
where (Nm) is the normalization constant; according to25

(Quantum Mechanics, Schiff, page 73, equation (14.16))


N = 2 (6.3.8)
Sm 4r (+ m )1 (6.3.8)

Then by inversion from (6.3.7) we get


F(e','p) = f 6(e-e' N mP p(cose)e-imed (6.3.9)


where dS = sine dedp ;thus

FC',(p') = NmP (cos')e im

Hence we get
_mI = Z m im(mi-pl') 6(r-r')
6-') = N2m Z Pm(cose)Pm(cose')e 2 (6.3.10)
Sm m,t r











Since the polar axis is along r', 9' = 0 and

Pm(cose') = 1. (6.3.11)

The result of the full substitution and equating term by term,

yields

2 Z,m t,,m
Px 2 dpx 2 t(t+1) Z,m 5(r-r')
d 2+dp2 + x2[l- +l) ,m= (6.3.12)
-2 r dr 2 -x 2 2
ddr r Dr N2
tm

The solutions of the homogeneous equation corresponding to (6.3.12)

are given by spherical Bessel functions of first and second kind

j,(xr) and y,(xr) respectively.

The complete solution of equation (6.3.12) is given as

Fw (r)wt (r'), r 5 r'
-N
p,m Sm
p = r2 (6.3.13)
r D[W(w i,wi)] wi (r')w II(r), r > r'


where W is the Wronskian evaluated at r=r' and we and wiI are

two independent solutions which satisfy the boundary conditions at

r=0 and r=a, respectively. For r=0 the solution must be regular. We
thus have:
wt = j (xr) (6.3.14)

and

w = Ajt(xr) + By(xr). (6.3.15)

wSII satisfies the boundary condition (6.3.1) which is

aw
S II
D ) = -w (a). (6.3.16)
ar a uII











Substituting wiI in equation (6.3.16) we get

aj (Cxr) Dy (xr)l
DA[--r ]ra D-L = -a[Aj(xa)+By (xa)]
r=a

or
ay (xa) aj (xa)
DB ar + aBye (xa) = -DA -r x Aj (xa).


Substituting the recurrence relations, see ref. 26,


ay (xa)
ar -= t(xa) xy+1(xa)
and
aj (xa) t
r a jt(xa) xj+(xa)

in equation (6.3.18) and simplifying, we obtain


B D[a j(xa)-xj,+,(xa)] +oj (xa)
A a
D[- y (xa)-xyL+1(xa)] +ay (xa)

A suitable choice for the coefficients A and B is:


A = i(o+ a a) Dxy+1(xa)]

and

B = -if(+ ) (xa) Dxj-+1(xa)].

Substituting A and B in equation (6.3.15) we obtain for coI as
S (xa) I a)] I
W~I = j (xr)I(a a) y (xa)-Dxiy 1(xa)]
D-
-iy (xr)[ (o+-) j (xa) -Dxj+ 1 (xa)] .
=a ee+jx)


(6.3.17)


(6.3.18)





(6.3.19)



(6.3.20)





(6.3.21)





(6.3.22)


(6.3.23)


(6.3.24)










Now,
iy1(xa) = h )(xa) j (xa) (6.3.25)

where he (xa) is the spherical Hankel function of the first kind.

Thus we obtain

D= (1) (1) (1) D.
w1 = j (xr)[(C+ h (xa-Dxh+1 (xa)]-h (xr)(a+ -j)j (xa)-Dxj+l1(xa)]
(6.3.26)
which can also be written as

wmI = h(1 (xr) je(xr)Qe(xa) (6.3.27)

where
(1) Da (1)
Dxh (xa)- (a+-)h xa)
Q (Cxa) = D. (6.3.28)
Dxj + 1 (xa)- (+D) j (xa)

The Wronskian as defined by


wn1 wn,
weI WeI'
W =

nII "mII'


(' denotes first derivative) evaluated at r = r' is found to be


W(wIw ii) = 2. (6.3.29)
xr'

Substituting for r f r' in equation (6.3.13) we get

xiN2
emr m (1)
pm (r,r') = m [j(xr)(h (xr') j Cxr)je(xr')Qe(xa)]. (6.3.30)

Finally, substituting equation (6.3.30) and equation (6.3.8) in

equation (6.3.2) we find for the Green's function, for r 5 r'












SIm)!. (1)
GxC 2 D (2 l)( I- [jZ (Cxr)hh ( 'j (r)j(xr)j.tCxr')QE(xa)]
S 4,m


(6.3.31)


For the noise spectrum we must evaluate the six fold integral


S(o) = 2Re f/-/ r dr d(cos9)dpr'2dr'd(cose')dp'GxC(,'))
6 fold
reduced
domain

where the integral involves the reduced domain 0 < r < r',

Integration over (p gives 2n for m=0; it gives zero for m#O.

use the addition theorem, ref.25 which gives the relation


ixR
e
i R


(1)
Z (2Z+l)j,(xr)h1 (xr)P (cose).
t=0


(6.3.32)




0 5 r' I a.

Next we


(6.3.33)


Substituting in equation (6.3.31) and separating the noise spectrum

into the two parts, indicated by the [ ] of 6.3.31, we find for the

first part


ixR
2 2 e i
S (c)= 4T Re f'"* r dr(dcosO)dcp' r' dr'd(cos9') 4w R
5 fold

and


(6.3.34)


4var X
S () = V Is O).
s


The equation (6.3.34) for SI(w) is exactly the same as for the spectrum

of the embedded sphere computed in section 4.3; this spectrum will

now be denoted as SX(c).


x Pm(cos9)eim(1 )











The second part of the spectrum has an integrand which is

regular for r = r', the solution being the same for r r' as

r > r'. Hence, we can at once integrate over the full domain

0 r < a, 0 < r' 5 a. Thus,


2xi 2 2
S (o) = -Re 24-D r2drd(cose)d(cose')r'2dr'd(p'
r 5 fold,
full
domain

x Z (21+l)j (xr)j (xr')Q Cxa)Pe(cose). (6.3.35)
0=O


The integration over d(cos6')dp' gives -4v. Further due to the

orthogonal property of Legendre functions, integration over d(cos6)

contributes -2 only for 4=0:

T 1
f P (cose)d(cos) = -f Pt(z)PO(z)dz = -2650.
0 -1

Thus we have

4xi 2 2
Si(co) =-Re- --ff r dr r' dr' j0(xr)j0(xr')Qo(xa) (6.3.36)


(Qo(xa), substituting &=0 in equation (6.3.28). We can separate

the integration over r and r'; thus evaluating the integral

fr2drJ(r) = fr2dr J1/2(xr) (6.3.37)


In equation (6.3.37) we have used the relation which connects regular

and spherical Bessel functions J and j, which is26

in(z) = Jn+/2 (z) (6.3.38)












We can write equation (6.3.37) as

xa 2 2
fr drj0(xri= f X d(xr)J/2(xr) (6.3.39)
0 2 x3


= xa (xr)3/2d(xr)J1/2(xr) (6.3.40)
x 0

We can also use the derivative formula for the-Bessel function which is


d-[zVJ(z)] = zVJvl(z) (6.3.41)

this gives for v = 3/2 the required integral as

f z3/2J1/2(z)dz = z3/2 3/2(z) (6.3.42)


which finally gives for the integral


fr2drj(xr)= a3/2 J3/2(a) (6.3.43)
(xa)

And similarly we get for

fr'2dr'j(xr') = J 3/2(xa). (6.3.44)
(xa)3/2 2 3/2

Substituting equations(6.3.43) and (6.3.44) in equation (6.3.36)

we get

SI ()=-Re ia3 2 J/(xa)] 2Q(xa) (6.3.45)
Dx

and for the spectrum we obtain

4varX
SI() = SII()
s











4 3
Taking V =s 4
Taking V a ,we obtain for the fluctuating variable X,

6 rivarX
S () = -Re ivar [J 3 xa) J2Q0(xa). (6.3.46)
Dx

Denoting the spectrum associated with second term by S (o), we

can write the total spectrum for nonembedded sphere as

total O a
S (o ) = S=(;) + S A(). (6.3.47)


Now we shall derive another form for S(co) in terms of Bessel
A
functions more suitable for combining with S (co). From equation (6.3.34)

we can write

8varX 2 2
S = Re 8 r dr r' dr' d(cose)d(cos6') depdp'
s 6 fold,
reduced
domain


x x j,(xr)h t(xr')Pm(cos)eim(-') (6.3.48)
,m DN (6..48)
tm

integration over dp gives 2n for m=0 only,

integration over d(cose) gives -2 for Z=0 only and

integration over d(cose')dp' gives -4r.

Thus

a r2
8varX 4nix a (1) r'2
SI = Re vT- ---f D r' dr'h) (xr') f r2drj0(xr). (6.3.49)
na 0 0


These integrals are similar as found in equation (6.3.37), thas


r' 3/2 --
f rdrj0(xr) = 372- 4 j3/ (xr'). (6.3.50)
0 x











So we get for SI
S8varX 4rwix n a (xr')2d(')(xr')3 ) x (61)
S3 H xr')J d2(xr') (6.3.51)
I Re Vs D 2 6r)3/2 xt 1/2 xr)3/2


8varX 4ni 1 Ta 3 a)
= Re 8V D 15 2 (xr) d(xr)H 1/2 CxrJ Cxr). (6.3.52)
s x 0 3/2

The required integral was calculated by Luke27
a =3 Cl (xa4(1) (4a 1)
f (xr )3d(xr )H1 Cxr )J3(xr ) = (xa) J3a) + J (xa) H xa).
0 1T 3
2 2 2 2 2 2
And using25 (equation 9.1.27) which is (6.3.53)


J5/2Cxa) = -J1/2(xa) + x3a 3/2(xa), (6.3.54)

and further using the Wronskian relation we found that


J (xaH a)- J/2(xa)H2(xa) = 2 (6.3.55)

Substituting equations (6.3.54) and (6.3.55) we found that for

equation (6.3.53) (also using (6.3.38)):

(xa) 2i 6 (1)
S6 (xa 4 + jl(xa)h1 (xa) (6.3.56)


For SI we obtain finally

S (w) = ReA [- a+ i j(a)h 1)(xa)] (6.3.57)

where
A 12(varX) (6.3.58)
D (6.3.58)











The spectrum for SII from equation (6.3.46) is

S =-Re 6 2ivar [J32(xa)] 2Q(xa)
Dx

= -12(varX) a2 (xa 2 x
RDxa [2xa 3/2(xa)] Q0(xa)
Ai 2 0(xa


= -Re[ l(xa)] 2Q0(xa)


where Q0(xa) from equation (6.3.27) for Z = 0


Q Cxa)


Dxh1l) (xa) -ch01) (xa)
Dxj(xa) (a)
DxjI(xa)-cj0(xa)


For the total spectrum we now find


total A 1 (1) .2
X (-RKa) 3- 3 i + 1(xa) 1 xa) j1(xa)
xx) 3x)1


Dxhl (xa)-Ch01)(xa)
Dxj1 (xa)-oj 0 (xa)


(6.3.62)


Solving for { } quantity first we get


-ah1 (xa)j0(xa)+oh0) xa)jl(xa)
Dxj (xa)-aj 0xa)


Using ref. 25 (equation (10.1.31)) we have


h0)(xa)j (xa) h(1 (xa)j0(xa)

So we get for { } quantity


i
(a-- .2
(xa)


S 1 j1 (xa)
(xa)2 (Dxj (xa) -o 0 (xa)


Thus we get for the total spectrum:


(6.3.59)


(6.3.60)


(6.3.61)


i{ }


= i 1Cxa)


(6.3.63)


i{ }


(6.3.64)


(6.3.65)












total
X (c) = -ARe
x


1 2 1 + 1a(xa)
(xa)2 3 xa (Dxj1(xa)-oj0(xa))
(xa)


Further we know that


xa = iaV-


(xa) = -a 2-i ).

Introducing the following time constants

1
S = (volume lifetime)
v a
2
a
Td = (diffusion time)
d D -


a
T = a-
S T


(surface lifetime) ,


and also introducing the dimensionless quantities.


u = Ord, T = Td/v


S= /s, d


we can write
2
(xa) = -(iu+*).
total
We can rewrite equation (6.3.66) for SX as
A


total
S (C) = ARe


where = A 4(var X)a2
3- D


1 -_ J,(xa)


iu+|-[1
iu+r


L+ j (xa ] (6.3.74)
Dx2a (xa)-(o/Dx)j(xa)

Using the time constants and dimension-


less quantities defined in equations (6.3.69) to (6.3.72), we have


Dx



a 1
2 a(r~+iu)
Dx a


(6.3.66)


(6.3.67)


(6.3.68)


(6.3.69)


(6.3.70)


(6.3.71)


(6.3.72)


(6.3.73)


and


(6.3.75)


(6.3.76)











Using equations (6.3.75) and (6.3.76), the spectrum finally becomes
2
total 4(var X)a ReP2() (6.3.77)
SX ) = D


where


1 3j1(ivi u) 1
P( C) = r+iu -- (T'+iu)j l(iVT-+iu) +iV+iu j0(iVrT+iu) J (6.3.78)


Now we shall discuss various cases.


6.3.1 No Volume Sinks

If there are no volume sinks present,then ri = 0. For this case


1 r 3jlCiuT) ]
ReP(co) = Re j .. (6.3.79)
hu i e iu)jt(iV1fu)+iVn je (iVonu)s fr

The first term does not contribute. Using the relations for spherical


Bessel functions26 we have

sinz
j0(z) = z
z


sinz cosz
z) = 2 z
z


(6.3.80)


(6.3.81)


Thus we obtain


Re P(c) = 3AD+BC)
4a 4(C2+D2)

where a =
2


A = (-sina cosha+a cosa cosha+a sina sinha)

B = (cosa sinha + asina sinha ~ :osa cosha)


(6.3.82)


(6.3.83)

(6.3.84)












C = (cosa sinha+a sina sinha-a cosa cosha cosa sinha (6.3.85)

D = (-sina cosha+a cosa cosha+a sina sinha + Sifa cosha (6.3.86)


At low frequencies, u << 1, and the spectrum reaches a limiting

value. For finite domain a plateau is always reached. Expanding the

trigonometric terms, we find canceling terms up to order a4:

1+55
Re P(c) = 15 (6.3.87)


At high frequencies, u >> 1, and the spectrum reaches the

asymptote

3
Re P(c) = (6.3.88)


For a closer investigation we must distinguish between two cases.

6.3.1.(a) First, let t << 1, or Ts < td This option

corresponds to the "diffusion limited" case. In equation (6.3.87)

the terms in t are now irrelevant and the spectrum reaches the low
1
frequency limit Re P(0) = At high frequencies there is another
-2 2
region, prior to the u2 region, in the range 1 << u < /2.

6.3.1.(b) The surface limited case occurs when >> 1,

or Ts > .d In this case the diffusion process is practically

instantaneous and the noise is only due to surface sinks. At high

frequencies the limiting behavior is 2- for any u >> 1. However,
tu2
this spectrum is also found at small u, since in the range
1 3
< u<< 1, we also found the limiting value of. the spectrum to be --
1 ou
This behavior occurs therefore for all u > -. The turnover occurs












1 1
for u or co The plateau that then is reached has the
Ts
value -
3.
The full spectrum for various E has been plotted and is shown

in fig.6. We see for >> 1, or for the surface limited case,the

turnover frequency is u = 2/E. For E << 1, or for the diffusion

limited case, for E = .01, the diffusion type behavior is observed,

in the range 10 < u < 7x103. The computer program used to plot

the noise spectrum is given in the Appendix.



6.3.2 No Surface Sinks
aG
If c = 0, = 0. This corresponds to E being infinite.
an
The spectrum despite the possible presence of microscopic diffusion

exhibits only a regular relaxation spectrum.

Re P(w) = T2 (6.3.89)
T) +u

or the spectrum


S(o) = 4(var X) (6.3.90)
2 2
v

This is the "volume limited case".



6.3.3 Both Surface and Volume Sinks Present

When both surface and volume sinks are present we have

that 710. We again use the relations for j and j0 as given in equa-

tions (6.3.80) and (6.3.81) and after much algebra the real part is

found to be





86
















































-.Lo


(*10.0\



8-*
I IO 10a 10o 10' 10 I








Fig. 6


Noise Spectra for Nonembedded Sphere (n=0)

(Slopes indicated are u-3/2 and u-2)











Re P (w) = (MP+NQ)-3(AM+BN) (6.3.91
V{(P P2p ) 2+(p2Q+ 2Q) 2

where

S= + +Vu2 (6.3.92)


6 = / + v)+u (6.3.93)


A = (-sin6coshp+6cos6coshp+psin6sinhp) (6.3.94)

B = (cos6sinhp-pcos6coshp+6snin5sinh) (6.3.95)

C = A+ incosh (6.3.96)

D = B cos6sinh
D = B (6.3.97)


P = (p2-62)C-2p6D (6.3.98)

Q = (p2- 2)D+2p6C (6.3.99)

M = C(2 -)P 2PBQ (6.3.100)

N = (262 )Q+2p6P. (6.3.101)

At low frequencies in the above equation (6.3.91) p = v

and 6 = 0. The low frequency plateau value for small n is found

to be

Re P(o) = 1 (6.3.102)

as in the previous subsection.

At high frequencies p = 6 = 5 The high frequency limiting

value is found to be for any T

3
Re P(o) = (6.3.103)
cwu 2u




































































- u


Fig. 7

Noise spectra for nonembedded sphere (r#O)












For n = 0 equation (6.3.91) reduces to equation (6.3.82). The

noise spectrum is plotted for different values of and r, and is

shown in fig.7. We distinguish the following options.

6.3.3(a). For >> 1, or Ts d, we have a surface limited

spectrum. The low frequency plateau is

3
Re P(0) .
-2
And, at high frequencies the spectrum goes as u-2 and the high frequency

limiting value is found to be

3
Re P () w 2

as predicted.


6.3.3(b). For << 1 in addition to T << 1 we found a

diffusion limited range. The low frequency limit is

1+55
Re P(co) = 15
15

1
And in the range 1 << u < 3- we see that the spectrum goes as u

reflecting the typical diffusion behavior.
-2
For high frequencies the spectra go as u The compute

program used to plot the spectrum is given in the Appendix.


-3/2


er











CHAPTER 7
NONSYMMETRICAL NONEMBEDDED BODIES



7.1 Introduction

In this chapter we shall analyze the noise spectra resulting

from diffusion and generation-recombination in nonsymmetrical non-

embedded geometries. We have already seen in Chapter 1 the difference

between embedded and nonembedded bodies,the latter being subjected

to various boundary conditions. We shall use the boundary conditions

stemming from a finite surface recombination velocity at either one pair or

two pairs of surfaces of the semiconductor sample under consideration.

A three dimensional case is of academic value only,since one always

has an ohmic contact in one of the directions.



7.2 Linear Case

The Green's function procedure has been used to find a closed

form expression for the noise spectra resulting from samples being

subject to a finite surface recombination velocity in the z-direction.

D(CAP) = tSAp (7.2.1)
z C

where we have assumed that the sample is infinite in x and y direction. D is

the diffusion constant, S is the surface recombination velocity and

we have considered the sample dimension to be 2C in the z-direction;Ap being

the fluctuation in the numberof particles, say holes. This problem

finds application in noise resulting from buried layers as studied

by Hsieh28. The diffusion equation is












L(AP) = 5-t(Ap) + a= 0 (7.2.2)
az

where T is the carrier lifetime; the Laplace transformed Green's

function satisfies the equation


G(z,s,z')(s+ 1) D Gzsz = (z-z (7.2.3)
az

According to the standard techniques the Green's may be found

by setting

S u(z) u(z') z < z'
G(z,zl) = DW(u,v) u(z')v(z) z z' (7.2.4)


where W is the Wronskian of two independent solutions u(z) and v(z)

which satisfy the left and right side boundary conditions, respectively,

evaluated at z = z'. The solution is found to be [by Lax and Mengert3]
-I

G(z,jo,z') = (2yDa) (coshyz+asinhyz)(coshyz'-asinhyz') (7.2.5)

for z 5 z'

and
-1
G(z,jc,z') = (2yTDa) (coshyz'+asinhyz')(coshyz-asinhyz) (7.2.6)
for z > z'
where

Y = V (l+jo'r)/DT (7.2.6)

and
sinyCi- (-) coshyC
a = (7.2.7)
coshyC+ --) sinhyC

Introducing the surface lifetime and diffusion time by

Ts = C/S and Td = C2/D (7.2.8)