Physica B 162 (1990) 344-352
North-Holland
THE RELATION BETWEEN l/f NOISE AND NUMBER OF ELECTRONS
F.N. HOOGE
Department of Electrical Engineering,
Eindhoven
University of Technology,
Eindhoven,
The Netherlands
Received 5 February 1990
The validity of the empirical relation for l/f noise, S,IR* = alNf, is studied. An alternative version with a factor l/A
instead of the factor l/N is considered. N is the number of electrons and A the number of atoms. Experimental and
theoretical arguments for the relation in its original form are put forward.
1. Introduction
The electrical conductance of semiconductors
and metals shows spontaneous
fluctuations
around the equilibrium value. One type of noise,
which is always present, has a spectral power
density that is inversely proportional to the frequency, hence its name, l/f noise.
Twenty years ago [l] an empirical relation was
proposed for l/f noise in homogeneous samples,
that is
s
Rz-
L-Y
R2
Nf'
where S, is the spectral power density of the
fluctuations in the resistivity R, f the frequency
and N the total number of electrons (or holes).
The dimensionless parameter (Yturned out to be
of the order of 10e3. For a long time it was
thought that (Ycould be a constant [2].
The introduction of N was not based on a
theoretical model. At that time we did not know
what the electrons were doing while producing
l/f noise. Even today that is a problem. But we
may follow the naive reasoning that, whatever
the electrons do, they do it independently. Then,
for purely statistical reasons, we may expect the
factor l/N. Considering relation (1) as the relative noise of an ensemble of N electrons, (Y is
nothing else but a measure of the noise at a
frequency of 1 Hz and normalized to one electron. There has always been severe scattering in
0921-4526/90/$03.50 0
the reported a-values. This was not too alarming
so long as the deviating a-values could be
blamed on faulty contacts, inhomogeneous material or ill-understood devices. But when silicon
samples made by modern I.C. technology were
studied, cu-values of quite different orders of
magnitude
were found. Modern technology
means two things:
(i) higher quality of the lattice, because of epitaxial growth techniques, and
(ii) very small, sometimes even submicron, dimensions.
Measurements on noise in samples of different
size, made on the same wafer, showed that small
dimensions brought no extra complications [3].
Measurements
on semiconductor
or metal
samples whose degree of crystal lattice perfectness was changed showed a very systematic variation in (Y.The lattice was damaged by radiation
or mechanical means and then partly repaired by
annealing [4-61. The results of the many applications of relation (1) to metals [5-61 can be
summarized in two statements.
(i) The value of Q in metals decreases systematically with the degree of perfectness of
the crystal lattice.
(ii) The range of a-values is narrower than in
semiconductors.
The results of noise investigations in metals are
often presented in plots where the vertical axis is
in units of 10e3. All in all, the success of relation
(1) is greater in metals than in semiconductors.
Yet, in many publications on noise in metals, the
1990 - Elsevier Science Publishers B.V. (North-Holland)
F.N.
Hooge
authors consider the fact that (Yis not a constant
in the strict sense as proof of the absolute failure
of relation (1).
There is one well-understood reason why (Yis
low in some cases [2]. There is good evidence
that l/f noise is a fluctuation in mobility. These
mobility fluctuations are in the contribution of
the lattice scattering. The observed mobility /L is
given by the relation
1
1
-=-++
CL Pl,tt
1
PX *
When there are two scattering mechanisms: lattice scattering and another one, labelled x, relation (1) becomes
s
s
R2
p2
R=l!
a
=
Ff.
If the noise is in the lattice scattering only,
s l/&L= SllPllatt3
345
I 1 If noise and number of electrons
(4)
(5)
from which it follows that
(6)
To summarize this part: LX,that looked like a
constant at the beginning, is no constant, but a
parameter. It can be lowered by the presence of
scattering mechanisms other than lattice scattering. alatt is of the order of 10e3 to 10m4in normal
materials. But, in perfect crystal lattices, (Ycan
be very low, 10m6or lower. There is no explanation for this dependence on the lattice quality.
2. The factor l/N
When we interpret noise data with the help of
relation (l), we express the experimental findings as S,R-*f. This measured quantity is then
written as cw/N. So long as the substitution of N
led to constant or nearly constant values of (Y,we
had the safe feeling that LYhad a physical meaning. Obviously, it was right to use the normalizing factor l! N. But since it appeared that cy may
vary by more than a factor of 1000, one may well
ask whether it makes sense to decompose S,Rm2f
into two factors, (Yand l/N. The widely scattering values of (Ycan hardly be used as an argument to choose l/N as a parameter that brings
order into experimental data. Therefore, without
further discussion of the numerical value of LY,
we now concentrate on the essential problem: Is
there a factor l/N in the relation for l/f noise?
The objections raised against the factor l/N are
twofold:
(i) statistical arguments,
(ii) arguments of a more physical nature based
on the long characteristic times needed for
the low-frequency part of the spectrum.
(i) There has been a long discussion on statistical arguments against the factor l/N. This discussion has been definitely settled 7 years ago
[7-lo]. Making a theoretical model for l/f noise
means making the calculations for an electron i
which is representative
of the electrons in the
sample. One thus discusses the behaviour of this
“average” electron, finds pi and Api, and no
factor N appears in the analysis. But, in the
experiments, one always observes a group of N
electrons. The experimental findings on N electrons (e.g. fiN) can be expressed as properties of
one electron. The essential steps are
(7)
(8)
(G”)’ = $
=
(slA*,)’
=$ i
$ (Api)”
.
(APi)*
(9)
(ii) The more fundamental objection is that a
factor l/N cannot be reconciled with the long
times required for the low frequencies. Arguments of this type have often been brought
346
F. N. Hooge
I 1 If noise and number of electrons
forward and have been summarized by Weissman in a review paper [5], in which he concludes
that it is absurd even to try a relation containing
a factor l/N. Our assumption behind the factor
l/N is that we add the uncorrelated effects of N
independent electrons. If we could follow one
individual electron we would find that it stays in
the sample only for a short time. When it moves
by Brownian motion it will stay in a sample
measuring 1 cm in length for 0.1 s, since its diffusion coefficient D is of the order of lo-’ m2/s.
This time interval of 0.1 s creates problems when
we compare it to the low frequencies at which
l/f noise is observed. The l/f spectrum has been
observed even at 10m6Hz [ll]. We consider the
l/f spectrum as the summation of Lorentzian
spectra r( 1 + 0 2r2)-1 with r-values between 7,
and T* and statistical weight g(r) dr = 7-l dT.
The l/f spectrum is then found between the
frequencies ril and ~1’) as follows from
(IO)
Now the problem is: how can the electrons that
are present for only such short times contribute
to fluctuations with those very long time constants?
These arguments have been countered by
pointing out that the slow processes are in the
crystal lattice. We have good evidence that the
l/f noise is in the lattice scattering [2]. Somehow, the cross-sections for the electrons fluctuate
slowly with a l/f spectrum. There are always N
electrons to probe the cross-sections of the lattice modes. We do not follow individual electrons; but the group of - on the average - N
electrons that is continuously present, measures
the slowly varying lattice properties. Individual
electrons may move in and out; the lattice is
permanent. However, it is not that easy to reformulate these verbal arguments in mathematical language. At the Budapest Conference [12] a
crude model was presented that suggested that
the normalizing factor should not be N but the
number of atoms A. At the end of this paper an
improved model will be presented that advocates
again the use of N. Nevertheless, it is worthwhile
trying A as the normalizing factor and comparing
the results with those of normalization on N.
3. The factor l/A
The choice of A as the relevant parameter can
be justified by several arguments.
(1) The experiments with samples of different
sizes and the same dopant demonstrated not
so much the dependence on N, but rather on
the volume of the sample. Volume can be
expressed by the number of atoms A.
(2) Since the noise is in the lattice mobility, the
fluctuating entitities are the modes of the
longitudinal
acoustic-lattice
vibrations
[2,13]. Their number equals the number of
atoms.
(3) With metals, it is difficult to estimate the
effective number of valence electrons. Here
one has always used A as an approximation
of N.
In order to avoid old (Y’Sbased on N and new (Y’S
based on A we now introduce a quantity y,
honouring the tradition followed in noise research on metals. The definitions of (Yand y are:
S
2 =
&
Y
A
__=-=
(Y N
[eq. (l)]
3
5 x 1O22cm-3
n
= -+ ,
(11)
(12)
The numerical value 5 x 1O22cm-3 is sufficiently
accurate for most semiconductors
and metals.
This value is used for constructing the framework of fig. 1. In fig. 1 we have plotted the
experimental results obtained in homogeneous
materials. They were published since 1985. We
included all published data without discrimination. Some older data are included, mainly from
systematic investigations of the dependence of (Y
on volume or on N. The values of (Y derived
from the data of electronic devices are not included. Their derivation is much too indirect,
requiring too complicated reasoning and too
much interpretation. An earlier version of fig. 1
F. N. Hooge
nN=
13
Id"
I 1 if noise and number of electrons
<
lo**
1oz3
1O24
II
n
IO6
12
IO4
IO3
IO2
16'
Fig. 1. a- and y-values of semiconductors and metals according to the relations (1) and (11): (0) Si; (0) others;
(am) amorphous.
was published in the proceedings of the Budapest Noise Conference [14], where also the references to the original publications were given.
Two groups of materials are worth mentioning.
(i) Semiconductors. We must keep in mind that
a-values may be reduced by a factor of lo3
347
or so in high-quality silicon. When the quality of the lattice has been improved by annealing in a series of experiments, then we use
the a-value of the lowest-quality material of
this series. Several semiconductors
other
than silicon are represented in the plot. The
quality of their crystal lattice is that of
silicon before the I.C. era. Here we see a
return to (Y2: 10e3 that was normal for
silicon long ago.
(ii) Metals. In old publications, a-values were
calculated using the number of valence electrons per atom. Those old (Y’Swill now be
called y’s in our new notation. The number
of electrons that effectively take part in the
conduction is smaller than the number of
valence electrons. It is the group of electrons
that are within a few kT of the Fermi level.
For a calculation of CY,we must use the
effective number of free electrons. Calculations on simple band models show that the
reduction factor is about 20 [15]. Therefore
we use n=1/20x5x1022cm-3=2.5X
1021cme3.
If one believes that the 1 lf noise in metals and in
semiconductors has the same origin then fig. 1
provides a strong argument for preferring (YlN to
y/A. There are, however, good theoretical models for 1 lf noise in metals [5,6] that do not apply
to semiconductors.
That is a good reason to
restrict ourselves to semiconductors.
But even
then, comparing the spread in (Yand in y, there
is no need to reject N as the normalizing factor.
4. Experimental evidence for l/N
The problem with fig. 1 is that the noise data
were collected from samples with different degrees of perfectness, and therefore varying values of CLWe need experiments where N is varied
in the same part of the sample, so that cr keeps
the same numerical value at different values of
N. There are two groups of publications that
describe experiments where N has been varied in
the same volume. (i) Injection of electrons in
forward diodes, (ii) photoconductivity.
F. N. Hooge
348
I 1 If noise and number of electrons
(i) Forward diodes, Kleinpenning [ 161.
In general one must prefer homogeneous samples to complicated devices, but the analysis of
this simple device is so straightforward that we
run no great risk by accepting it as good evidence. Now let us first consider a simplified
model that shows all the essential features. We
consider a p+-n junction in forward bias. The
current Z injects holes into the N region. There
they recombine in recombination time r. A concentration of holes builds up in the N region.
The total number of holes P is given by
z
P
-=_
4
7’
(13)
The diode shows l/f noise and shot noise
a
sI
=
2
(14)
Ff z + 2qz.
The frequency-independent
shot noise and the
1 lf noise are equal at a crossing point f, that lies
in the region 100 Hz to 1 kHz. The crossing point
f, is given by
fx_az2 1 _L?Lx_a.
P
2qz
Zrlq 2qz
(15)
27 ’
f, is independent
of 1. This was confirmed by the
experiments in which the diode current Z was
varied by a factor of 1000. If a/P were replaced
by y/A then f, would go with I.
Kleinpenning [16] gave a detailed analysis of
the forward diode. His results for different situations are presented in table 1. The a-values in
the last column follow from his expressions for
the corner frequency and his own or previous
measurements by others. Here the presence of
the factor 1 lN was demonstrated by the con-
stancy of the corner frequency. There are other
investigations on diodes that indirectly show that
l/N is the correct normalizing factor; however,
without explicitly giving the constant corner frequency. All studies lead to normal a-values
10-4-10-3. For Si diodes, see [17-201, for
Cd,Hg,_,Te
diodes, see [21,22].
(ii) Photoconductivity, Hofman and Zijlstra [23]
An elegant way of varying the number of
electrons in a given volume is provided by the
“presistent photoconductivity effect”. Deep centres are present in Al,Ga,_,As.
These, socalled, DX centres have a small capture crosssection. Therefore the optically excited electrons
remain in the conduction band for long times. A
homogeneous electron excitation is realized since
the absorption is weak. These experiments resulted in normal a-values: (Y= 3.4 x 10m4 at a
temperature of 79 K.
5. A model for mobility fluctuations
Finally we present a simple model for mobility
fluctuations. It is based on the treatment of
lattice scattering as given in many textbooks. We
follow here Chapter XI of Spenke’s book [24]
that treats lattice scattering in great detail. As a
new element we introduce fluctuations in this
formalism of average mobility. This results in a
proportionality factor l/N in the relative noise.
In order to keep the expressions as simple as
possible, we write explicitly only the quantities
E, k, (kT), f and L3 = volume of the crystal with
dimensions L x L x L. Other quantities and numerical constants will be included in the constants B, . . . B, and C, . . . C,, , since our only
interest is whether we end up with 1 lN or 1 /A.
Table 1
Type of p’-n
diode
Simplified (15)
Long, g-r region
Long, low injection
Long, high injection
Short
f,
Lyl27
al37
al87
al47
(aD/2w2) In(kJpw)
(Yin units of 10m3
0.3
1.2
0.9
0.3
F, N. Hooge
I 1 If noise and number
Boltzmann’s constant will be denoted by k,
which will not lead to confusion since it always
occurs together with T; we shall write kT in
parenthesis as (kT). We chose E = 0 at the
bottom of the conduction band, therefore we
have for the Fermi level E, < 0. The subscript e
applies to electrons, p to phonons.
The probability that an electron is scattered by
one longitudinal acoustic phonon is P,:
P, = B, (kT)
L
(16)
,
independent
of k,, k, or the angle between
them. Since the electron has approximately the
same energy after the scattering, as before, the
final k,-values of the electron lie on a sphere
with radius kk. See fig. 2. The phonons that can
scatter the electron will be called here relevant
phonons. The relevant phonons have k-vectors
that obey
k; + k, = ka .
(17)
There are as many relevant phonons as there are
allowed k-states on the sphere. The spacing
there is 21r/L. Thus, the number of relevant
phonons is given by
=$
k2dk.
349
Here we have counted the number of discrete
states in a shell with thickness dk, = 27rrlL. The
probability P that an electron is scattered by
acoustic phonons is
(kT) = B,(kT)ka
P=iP,=RBIT
L
dk,
1
= B3(kT)E;12 dE, = dE, ,
r(E)
T(E) dE, = B,(kT)-1E,“2
(19)
dE, .
(20)
An electric field E, in the x-direction gives a
current i = ix. We use spherical coordinates; 13is
the angle with the z-axis and 4 the angle with the
x-axis in the xy-plane. One electron gives a
contribution to the current
4
i=-vv,,
L
(21)
j=+u,.
(22)
There are dN, electrons in a volume element dV,
of k-space:
dN, =f(k)
3
R = 4nkt dk,
of electrons
dV, $
.
(18)
k'
The spacing
of the two
k-state. f(k)
The volume
rent density
j=
3
is 27~/L; there is a factor 2 because
possible spin orientations in each
is the Fermi occupancy probability.
element dV, contributes to the cur-
v,f(k)
dl/, -$
.
(23)
All electrons with the same Ikl contribute:
97
j=
2%
Y$
If
u,
f(k)k%in
8 dtI dr$ dk
.
-$
,
e=o r#J=o
(24)
Fig. 2. Relevant phonons. Such a phonon scatters an electron from kc into one of the states kf on the sphere with
radius /CL.
2rm
u, = v sin 8 cos t$ = - h
k sin 13cos 4 ,
(25)
F. N. Hooge
I 1 if noise and number of electrons
C, -$ k sin20 cos 4
0=0 +=o
x f(k)k*
-$
d4 do dk .
(26)
The occupancy f(k) of a state deviates little from
the equilibrium Fermi factor f,(k) because of the
influence of the field e,. Using the Boltzmann
approximation, we obtain
f(k) = f,(k)
df(k)
+ q&x do
=.6(k) +
C,
&
=i,tk)+
‘2
-(kT)
UT(k)
kT(k)
fo
k sin 13cos Q,T(k) .
(27)
The term f, does not contribute to the integral
giving the current, but substituting the second
term into eq. (26) yields
B=O
+=a
c
jk
=
k2+V
sin38 cos’r$ d9 d+
3L3(kT)
x [$
fo(k)k2 dk] .
1
(28)
The latter factor between square brackets is dN,,
the number of electrons with the same value of
lkl:
dN, = L’
n=
fo(k)k2
dk ,
and the fluctuations therein. This is the essential
difference from the oversimplified model of ref.
[12]. There we did not consider individual electrons. In that model it is as if an electron is
smeared out over AIN states. According to the
present model this gives an extra factor A lN in
the noise, since the AIN states fluctuate together, because fractions of an electron cannot
be scattered. The noise becomes proportional to
(A/N)’
instead of to A lN. The final result 1 /A
will therefore become l/N, as in the present
treatment.
We try to find an expression for Sjlj2. Therefore we calculate j from eq. (30) and Sj from eq.
(31). It is no surprise that the first calculation
leads only to well-known results. It is the second
one that is important. Relation (30) together
with (20) leads to
jk = C&)
=
&
(kTt\/E
(k\)eeEIkTE2 da,
ewkT
C6
f(W4dk
(32)
r
j
=
C,
eEFikT
e-EIkT
0
d (k?)
= c, eWkT
Comparing
(33)
(33) with (34) and (35) gives ( I_L
):
(34)
(29)
(35)
(p) - (T) - (kT)-3’2
@jk12
= [C,
T3$!)]*
[dNkl.
(kt)
.
(36)
The noise follows from (31):
(31)
The noise is in the scattering, expressed here by
AT(k). We do not consider the number fluctuations, i.e. the fast fluctuations in the occupancies
of the states.
Separation of the factor dN, in (30) and (31)
is the basis of our treatment of average mobility
f(W6 dk ,
sjk = C,
x
eEFIkTy
(37)
(kT)3’2
emEIkT(
g)“’ d(&) ,
(3%
F. N. Hooge
Sj = C, e
E,IkT
w)3’2
-yT-7
I 1 If noise and number of electrons
Ias
(E)
0
X
emE,xT(~)5’2d(j$). (39)
Sj is proportional to 1 lf; this does not necessarily
mean that each S,(E) is proportional to 1 lf. But
if this is so, and if S,(E) can be written as
S,(E) = y
EX(kT)Y ,
(40)
then eq. (39) can easily be worked out:
351
tion volume [25]. So we might end up with some
power of the volume, since R is proportional to
volume (eq. (16)). In principle, it could be possible that the factor l/N we just found (eq. (43))
should be multiplied by AP. But there certainly is
no risk that N should be multiplied by a factor
containing some power of N. So, instead of the
factor l/N we could have ApIN. However, all
experimental evidence (e.g. fig. 1) shows that we
could have either l/N or 1 /A, but not some
combination of N and A like ApIN with p deviating appreciably from zero. Therefore, it seems
correct to assume that S,(E) does not depend on
L3.
6. Conclusions
(41)
=f
C,C,,
eEFikT
y?(W
(42)
The magnitude of l/f noise in homogeneous
samples can be expressed by the empirical relation
s
Ai=_
(Y
From (33) and (42) we then obtain
R2
s, _ c,, eE~‘k=(q3’2-x+y~-3
where N is the total number of electrons.
(1) Theory does not provide arguments against
the use of the normalizing factor l/N.
(2) There are several experimental
investigations that directly verify the dependence on
1 IN.
(3) (Y is not a constant but a parameter. The
values of (Yare of the order of 10m4 to 10e3
in normal materials. Perfect crystal lattices
go together with very low a-values.
.2
I
e2EFlkT
-
f
=
C(kT)3-“+y
f
1
77.
(43)
If S,(E) cannot be written as (40) we still find
the factor 1 lN, provided S,(E) does not depend
on L3. From theory - e.g. eq. (20) following
from eq. (16) - and experiments, we know that
T(E) does not depend on volume. Is this also
true for S,(E)?
In the discussion of the relevant phonons
(around fig. 2) we considered Pi, the probability
of an electron being scattered by one phonon
(eq. (16)). We obtained the total probability P
by summation over R phonons (eq. (19)). If we
do the same for the fluctuations in P, and P of
one electron, there is no way for f(k) to come in,
and therefore not for N. But AP, could depend
on L3 and there could be correlation effects to
be taken into account in the summation over the
R relevant phonons. There could be a correla-
Nf'
References
PI F.N. Hooge, Phys. Lett. A 29 (1969) 139.
121F.N. Hooge, T.G.M. Kleinpenning and L.K.J. Vandamme, Rep. Prog. Phys. 44 (1981) 479.
131 R.H.M. Clevers, Physica B 154 (1989) 214.
[41 L.K.J. Vandamme and S. Oosterhoff, J. Appl. Phys. 59
(1986) 3169.
M.B. Weissman, Rev. Mod. Phys. 60 (1988) 537.
;:; N. Giordano, Rev. Solid State Sci. 3 (1989) 27.
[71 K.M. van Vliet and R.J.J. Zijlstra, Physica B 111 (1981)
321.
352
F. N. Hooge
/ 1 If noise and number of electrons
[8] A. van der Ziel and K.M. van Vliet, Physica B 113
(1982) 15.
[9] F.N. Hooge, Physica B 114 (1982) 391.
[lo] A. van der Ziel, C.M. van Vliet, R.J.J. Zijlstra and
R.P. Jindal, Physica B 121 (1983) 420.
[ll] T.G.M. Kleinpenning and A.H. de Kuijper, J. Appl.
Phys. 63 (1988) 43.
[12] F.N. Hooge, in ref. [14].
[13] T. Musha, G. BozbCly and M. Shoji, in ref. [14] p. 385.
[14] 10th Int. Conf. on Noise in Physical Systems, Budapest
(1989), A. Ambrozy, ed. (Akademiai Kiad6, Budapest,
1990).
[15] T.G.M. Kleinpenning and J. Bisschop, Physica B 128
(1985) 84.
[16] T.G.M. Kleinpenning, Physica B 98 (1980) 289.
(171 T.G.M. Kleinpenning, Physica B 94 (1978) 141.
[18] T.G.M. Kleinpenning, J. Vat. Sci. Techn. A 3 (1985)
176.
[19] T.G.M. Kleinpenning, Physica B 154 (1988) 27.
[20] P. Fang, L. He, A.D. van Rheenen, A. van der Ziel and
Q. Peng, Solid State El. 32 (1989) 345.
[21] X. Wu, J.B. Anderson and A. van der Ziel, IEEE
Trans. Electron Devices ED-34 (1987) 1971.
[22] L.K.J. Vandamme and B. Orsal, IEEE Trans. Electron
Devices ED-35 (1988) 502.
[23] F. Hofman and R.J.J. Zijlstra, Solid State Commun. 72
(1989) 1163.
[24] E. Spenke, Elektronische Halbleiter (Springer, Berlin,
1965) chap. XI.
[25] L.B. Kiss and T.G.M. Kleinpenning, Physica B 145
(1987) 185.