THEORY OF NOISE IN A MULTIDIMENSIONAL SEMICONDUCTOR WITH A P-N JUNCTION

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R002800140012-9
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 54
• Document Creation Date: December 23, 2016
• Document Release Date: January 16, 2014
• Sequence Number: 12
• Publication Date: June 24, 1957
• Content Type: REPORT

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NAVORD REPORT
5762
STAT
THEORY OF NOISE IN A MULTIDIMENSIONAL SEMICONDUCTOR WITH A P ? N JUNCTION
=1' ttiT OF N
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24 JUNE 1957
U. S. NAVAL ORDNANCE LABORATORY
WHITE OAK, MARYLAND
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NAVORD Report 5762
THEORY OF NOISE IN A MULTIDIMENSIONAL SEMICONDUCTOR
WITH A P-N JUNCTION
ABSTRACT: This thesis discusses the fluctuations of noise
in a two and three dimensional semiconductor containing a
p-n junction. We consider a rectangular parallelepiped
single crystal. It is bisected in the longest dimension by
a p-n junction. Since this dimension is several diffusion
lengths it can be considered infinite. In the transverse
plane we investigate the case where both dimensions are
finite, and then the case where one is finite and the other
infinite. In the p-n junction the noise is the result of
fluctuations in the minority carrier density. In a p-n
junction there are two classes of minority carriers: 1.
holes in the n-type material, 2. electrons in the p-type
material. Since both hole and electron density fluctuations
are similar, we discuss only the former in detail. We
investigate the differential equations for a two and three
dimensional semiconductor with a p-n junction and find the
inhomogeneous form of these equations. These equations are
solved with the help of the scalar and tensor Green's
function. The noise problem is solved by using these
equations as Langevin equations and interpreting the dis-
tributed sources as random forces. Then the noise current
spectrum is determined with stochastic process theory after
deriving the sources from basic physical models and the
theory of stationary, ergodic, Markovian processes. We
U. S. NAVAL ORDNANCE LABORATORY
WHITE OAK, MARYLAND
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consider two cases of surface recombination velocity on the
transverse surfaces: infinite s and finite s. For the
infinite case, we get the exact solution which provides an
upper bound for the noise spectrum for large s. For an
arbitrary s we get a solution but have confidence in the
solution for only small s.
Therefore we have obtained a complete solution for the two
cases of practical interest: large and small surface
recombination velocity. These cases should prove of interest
in the analysis of noise phenomena in semiconductors.
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CONTENTS
CHAPTER
Page
I. INTRODUCTION 1
II. PROBLEM AND METHOD OF SOLUTION 3
III. THREE-DIMENSIONAL NOISE SOURCES
FOR A p-n JUNCTION 7
IV. SCALAR INHOMOGENEOUS SEMICONDUCTOR
EQUATION AND GREEN'S FUNCTION 22
V. TENSOR GREEN'S FUNCTION FOR THE SEMI-
CONDUCTOR WITH A p-n JUNCTION .??? OOOO 33
VI. NOISE CURRENT SPECTRUM IN THE p-n JUNCTION WITH
INFINITE SURFACE RECOMBINATION VELOCITY 46
VII. NOISE IN A p-n JUNCTION WITH ARBITRARY
SURFACE RECOMBINATION VELOCITY 62
VIII. CURRENT SPECTRUM IN A TWO-DIMENSIONAL
SEMICONDUCTOR WITH A p-n JUNCTION 75
IX. SUMMARY AND CONCLUSIONS 81
APPENDIX
A. DISCUSSION OF GREENS FUNCTIONS 85
B. EXCESS CHARGE DENSITY AND p-n JUNCTION INPUT
ADMITTANCE 90
BIBLIOGRAPHY 96
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RAVORD Report 5762
CHAPTER I
INTRODUCTION
Semiconductor noise studies give useful
information about the basic pnysical micros-
copic processes in semiconductors and in the
solid state. Furthermore, noise becomes very
inportant when a semiconductor device such as
a transistor is used with signal levels com-
parable to the noise.
A semiconductor crystal which is p-type
at one end and n-type at the other has a
transitiop zone whlch is called a p-n junction
(Shockley"-, Kittel). Current is carried
across the;pnction by minority carriers; that
is, electrons in the p-type region and holes
in the n-type mion.
Petritz3,t has shown that noise in a
p-n junction arises from fluctuations in the
concentration of minority carriers. Consid-
ering a p-n junction as an ideal one dimen-
sional structure, he has derived expressions
for this noise. Van der Ziel2 has extended
the solution to the one-dimensional p-n-p
transistor structure. In both studies the
effects of surfaces were considered in an ap-
proximate manner.
However, surface conditions have been
found to influence markedly the performance
p-n junction diodes and transistors (Kingston u).
Considerable theoretical work has been done to
understand the signal properties (voltage,. cur-
rent, -frequency relations) of p-n junction
devices, consiaered as three-ditgensional struct-
ures (Shockley(, Van Roosbroecku). ' It is the
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purpose of this thesis to develop a theory of
noise which considers the p-n junction as a
three dimensional system, and which treats the
effects of surfaces in an exact manner.
A second objective ofthe thesis is to
test and extend a powerful method developed by
Petritz304 for studying complicated random
processes. This aspect of the work is of inter-
bst in the general theory of random processes.
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CHAPTER II
PROBLEM AND METHOD OF SOLUTION
2.1 Introduction
We assume that semiconductor noise is a
stationary, ergodic and Markovian random process.3
Considering the local hole density, pt(xly,z,t),
as a random variable, this is a three-fold infin-
ite random process. In order to solve such a
complicated problem, we have generalized a method
used originally by Petritz.3 This method employs
the Kolmogorov,-Fokker-Planck (KFP) and the
Langevin techniques to describe the noise (Fe..-
lex', Chandrasekhar101 Uhlenbeck and Ornstein-'l,
Wang and Uhlenbeck12).
2.2 The Kolmogorov-Fokker-Planck Equation Approach
The KFP equations12 for the three dimensional
semiconductor are given by-
P (r) 111(111 = ? P (Tn.l-t91m(r),t)E (TnIT9i c(r))
lc,ic *To.
t p(rns(r)lk(r),t) QNMI-rn(r))
x;x*Tri
P(mo/m1t) is the conditional probability of
finding the random variable with a value m after
the time t, if at zero time the random variable
had a value mo. The random variable is the hole
density in the n-type semiconductor. The symbol
r represents r(xly,z)la function of the three
rectangular coordinates. q is a transition
probability and is defined) by the equation,
gm at.) = Q(xl-m) At t Graer (0)t . (2)
got
(1)
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Q describes how the system changes in an infin-
itesimal interval of time, At, and characterizes
the stochastic process. For the semiconductor
problem, Q is independent of time and the pro-
cess is stationary. Q is non-zero and less than
unity and the process is ergodic.
The interpretation of equation (1) is that
the rate at which the conditional probability
P(m0/m1t) changes with time results from transi-
tions away from and to the desired state. Equa-
tion (1) is subject to the boundary condition
13(m.1171,0)=- , (3)
where 8?.,is the Kronecker delta.
SinCe a random process is characterized by
transition probabilities, we list them for the
p-n junction:
rn(r)-1)
Gli-m(r)1,rn(r)+
((Nn (r)lin(r) =n1 (r) ,
-m(r) =
bulk recombination; (4)
bulk emission;
(5)
bulk diffusion (6)
decrease;
bulk diffusion
increase.
At the transverse surfaces, the surface
tion probabilities are
Qs(rn Vs) -1.11 = mc.r,V-cs ,
Qs(in (r,11-rnK + 9 is evaluated
by multiplying Eq. (42) by 111 and summing; the
result is
IN1z)e. (t(N> + (N) ? 2045. +
at
(62)
When the time of observation of N goes to infiri-
ity,
since
1112.)= ova> - clot = (63)
At civ),
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(65)
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where 4pt(r)? is given by Eq. (B11). This i6
the desired noise source for bulk recombination.
.3.9 The Diffusion Noise Source
Solving Eqs. (13) and (14) for p,
letting 8r be zero and using the model for the
diffusion noise source, Section (3.7), we get
3 ?Az?
arr_. D =
a t -33
(66)
where xi is x, y, or z. We assume that diffusion
in the three directions is statistically inde-
pendent. Equation (66) becomes a set of three:
9?T
?Dr (67)
ST,
(67)
where u stands for x, y or z.
To write equation (66) in the Langevin
form, the spacial term is transformed to contain
a time constant. We write the second derivative
for the finite but small cube and use the densi-
ties
p(0)= 0 % ? ,p(--1',2--)=p? (68)
In this differentiation the diffusing 'direction
is u and the length of the cube in this direction
is Au. The second derivative is
and equation (67) becomes
51)
18
(69)
(7o)
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where rt-D11,_ eu.272D (71)
With these time constants the transition
probabilities for the u direction are
Q.Du.(N1)11413-1)=NDirt Du. loss by diffusion; (72)
Qt.u(K+1)?=41%1?keDu.,
gain by diffusion. (73)
Here ND is the total number of holes in the cube
with dimensions Ax, Ay, Az and with the diffu-
sion boundary conditions:
Sp cist = bas IDA .
(74)
Using the techniques of the previous sec-
tion, the noise source for diffusion in the u
direction is
vi(I$D,11z) = 11(414 pt(r)>/
(75)
3.10 The Surface Recombination Noise Source
With the model for the surface recombina-
tion noise source, Section 3.7, the Langevin
equation (10) becomes
-0
- te '
The pig is evaluated at the semiconductor
boundary and the ifs is the current flowing in
(76)
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the direction normal to the surface i. These
surface recombination time constants define the
transition probabilities:
Q$0$111e-1)=. Ns/51
by recombination at the surface;
CI (14
Ntis= > [(De) +I. w/1DJ , and
identity for large positive integers m and
Agr
the
(4- -11- Y1L .NYI VI (188)
the series in Eq. (B28) becomes
b.\JA K . (189)
b cm
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As m and n individually go to infinity the
terms go to zero because (Cie.tror (fien)z' goes
to zero. The value of the input admittance is
finite except when AGI, is 17121 when it is
,infinite. When szco the surface of the semi-
conductor seems to be covered with a perfectly
conducting layer which short-circuits the semi-
conductor. Therefore the Nyquist noise current
spectrum becomes infinite when the surface
recombination velocity is infinite.
A convergent expression for the case of
arbitrary s can be written down directly from
the Nyquist Law, Eq. (180) and the real part of
the admittance, Eq. (B31). This is discussed
further in Chapter VII for the case of arbitrary
s. Now we examine this expression for the case
of large but finite s.
From the boundary conditions, Eqs. (96)
and (97),
= .541./D =R:zx tdne. (190)
When V is large we can write 4a( as 7
(191)
Y
where r is odd. Substituting Eq. (B31) into
(180), letting f be large and the dc voltage at
x=0 be zero, we get for the range of small Er,
W(IVA = 9-127AD?"-A
x]rt cos (m7r4/25i) cos (nirp;isc) 14 (192)
071r/X/
tti,n
-t -1
onrrz I [1 Sir Di5C).)
M r DA I
(i+ T- n:ir 14
t D/SC)
X ift (D r/34 Ctrn 1r/Z01 -D/5 Of+ Eirrizc )(1--D/sort+ id1/414
(Dtylf unip-A9/(!-1)/s8)t + [ (Pi 1ttzc)(1-D/5c)J2 19.
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If we let sb/D and sc/b become very large
while (,,and en remain small, Eq. (192) becomes
identical with the Nyquist current spectrum with
Eq. (179); to make the two equations agree
the conversion factors between Figures 2 and 3
are used:
13-zb; C=2c, (193)
6.7 Convergence of the Series for the
Excess Noise Spectrum
To discuss the convergence of the excess
noise spectrum, we investigate Eq. (181). In
the denominator of this equation there are two
factors which contain minus signs, one in m and
one in n. Taking the factor in m (identical
results are obtained with n) the indices m, m:
and m" are related so that Imtmlvel must be odd.
For the whole factor to be zero
( - le1)1(a nine' (19))
Solving we get
771?t77';_trn"--r-o. (195)
Since zero is an even number, the factors cannot
vanish. Furthermore the factor in m is nega-
tive whenever m