1926 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO.

11, NOVEMBER 1994

l/f Noise Sources

F. N. Hooge

Invited Paper

Abstract-This survey deals with l/f noise in homogeneous inhomogeneous and often nonlinear devices. For the same
semiconductor samples. A distinction is made between mobility reason, values obtained from devices are excluded from
noise and number noise. It is shown that there always is mobility Fig. 5 at the end of this paper. nere we present reliable data,
noise with an (Y value with a magnitude in the order of
Damaging the crystal has a strong influence on cy, cy may increase Obtained from homogenous
by orders of magnitude. Four types of noise are of importance in semiconductors.
Some theoretical models are briefly discussed; none of them 1) Thermal noise. Any resistance R shows spontaneous
can explain all experimental results. The cy values of several semi- current fluctuations or voltage fluctuations according to:
conductors are given. These values can be used in calculations of
l/f noise in devices.
Sv = 4kTR (1)

I. NOISE SOURCES

T HIS paper serves as an introduction to several papers in

this special issue, which is devoted to noise in devices.
One way of analyzing the noise of a device is to assume
SI = 4kT/R.

This white spectrum is always found, whatever the

(2)

noise sources in several parts on the device. The noise mea- nature of the conduction process, or the nature of the
sured at the output is the summation of many contributions mobile charge carriers. Thermal noise is generated in
from the different sources modified by the device character- any physical resistor that shows dissipation if a current
istics. The device may introduce coupling between different is passed through it. A mathematical property, which is
contributions. If one has a reliable model of the device and measured in 0, such as a dynamical resistance dV/dI
if one knows the noise sources, it is possible to calculate the of a nonlinear device must not be used in (1) or (2).
output noise. One then concludes that the noise of the device 2 ) Shot noise. The current carried by electrons emitted
is understood if agreement is observed between calculated and from a hot cathode in a vacuum diode, or by elec-
measured noise, as functions of voltages and currents. trons that cross a potential barrier in a semiconductor,
It is also possible to go the other way round. Starting from are randomly generated. Random generation leads to
the observed output noise one tries to determine the noise fluctuations around the average current I :
sources. One then estimates properties like concentrations,
cross-sections and a values of l/f noise. This procedure, from SI = 2qI. (3)
observed output to sources, is risky. Apart from the trivial
requirement that the model must be treated correctly-not The details of the emission process have no influence
too many simplifying approximations-there is the difficult on the noise, provided that there is no interaction be-
problem whether the model used, is a correct description of tween the electrons, and that the statistics is close to
the device under investigation. Noise is much more sensitive to Boltzmann.
details than the average voltages and currents. This procedure 3) Generation-recombination noise. The number of free
always leads to results, but are they reliable? If one incorrectly electrons in the conduction band may fluctuate because
assumes a bulk source where in fact a surface source is present, of generation and recombination processes between the
one finds an (Y value without any meaning. Such a values band and traps. The number fluctuations cause fluc-
are nevertheless used as an argument for or against certain tuations in the conductance G, and, therefore, in the
theories. The properties for the noise sources derived the resistance R.
second way need independent checking. One must have some
idea at least of what realistic values are. S R - SG
- - - SN -
- (AN)'
-~ 47
(4)
The properties of noise sources can, therefore, best be R2 G2 N2 N2 1+w2r2
studied on homogeneous samples. The numerical results can where r is a relaxation time, characteristic of the trap,
then be used in the much more complicated problem of the usually in the range of s to s. If there is one
Manuscript received October 29, 1993; revised June 17, 1994. The review type of trap only, then the variance (AN)2 is given by
of this paper was arranged by Editor-in-Chief R. P. Jindal.
The author is with the Department of Electrical Engineering, Eindhoven 1 = -1+ - +1- 1
University of Technology, 5600 MB Eindhoven, The Netherlands. (5)
IEEE Log Number 9405325. (AN)2 N Xn X p

0018-9383/94$04.00 0 1994 IEEE

HOOGE: NOISE SOURCES 1927

where X, is the average number of occupied traps samples, the carrier concentration, the frequency range of the
and X , the average number of empty traps. The van- measurements, etc., had to be eliminated. The only theoretical
ance thus approximates the smallest of the quantities idea behind the relation was, that whatever the electrons do
N , X , and X,. when producing l/f noise, they do it independently. Thus,
The complicated problem of a semiconductor with two a is a normalized measure for the relative noise in different
kinds of traps, X and Y , has been solved by van materials, at different temperatures, etc. There was no reason
Vliet and Fasset [l]. Later publications [2]-[4] deal to assume that a was a constant. On the contrary, we were
with practical questions like, under which conditions looking for factors influencing a. Given the inaccuracies of the
will the observed spectrum be the superposition of two individual experimental results-and with hindsight-given
Lorentzians, one which would have been found if only the rather poor qualities of the samples no systematic trend in
X were present, the other one for the case that only cy was found at the time. The a values were not too far apart,
Y were present. The condition for this, often naively and it seemed reasonable to take a = 2 x l W 3 as an average
assumed, situation is value. Later on it tumed out that a depends on the quality of
the crystal, and on the scattering mechanisms that determine
z<-+-<-+-
1
x,
1
xrl
1 1
yrl y p
1
(6) the mobility p. In perfect material a can be 2 or 3 orders of
magnitude lower than the 2 x originally proposed.
[3] gives simple procedures to determine cross sections Before we can further discuss dependences of cy on certain
and trap concentrations from observed GR spectra too. parameters, like temperature, dope, etc., or even before we can
4) I/f Noise. This is a fluctuation in the conductance with decide that it is meaningful to take an average of measured
a power spectral density proportional to f P Y , where values, we should have some idea of what we consider to
y = 1 , O f 0 , 1 in a wide frequency range, usually be essentially the same values. We need an idea of what
measured from 1 Hz to IO kHz. The spectrum cannot random errors are, and what may point to a systematic depen-
be exactly f -’ from f = 0 to f = 00,since neither the dence on some parameter. Noise very much depends on the
integral of the power density nor the Fourier transform physical conditions during growing, doping, intentional and
would be able to have finite values. At some higher unintentional surface treatment, and contacting. Fig. 1 gives
frequency f h the slope must be steeper than -1. This an example of the best results we can get. The points on
f h has never been observed for the simple reason that a vertical line (at the same value of p ) were measured on
at higher frequencies the 1/f noise disappears in white different samples made from the same wafer. A spread in a
thermal noise that is always present. Attempts to observe of a factor 1 . 5 is found. Each point is the average of several
the lower limit f l , below which the spectrum flattens, measurements, with different currents, on the same sample.
have always been in vain. Measurements down to lop6 The spread is wider if samples are made from different wafers
Hz showed that even there the spectrum still is f -’ [5]. with the same properties nominally. It is wider still if similar
Because of the restrictions on y CY 1 we do not consider samples from different laboratories are compared.
f - l I 2 and f - 3 / 2 as 1/f noise, like some people do. For a meaningful result we need the average over many
In semiconductor devices, such spectra usually follow samples. The best we can hope for is that samples from
from diffusion processes. different sources, measured under different conditions, give
Unlike the first three sources mentioned above, which a values with the same order of magnitude. As a result, if
are well understood, the origin of the l/f noise is still one wants to find certain numerical values to support or refute
open to debate, a debate full of vehement controversies a theoretical model one can always find them in literature.
[6]-[ 1I]. Therefore, from here on we will discuss 1/f Discussing systematic dependences of a on some parameter
noise only. Due to our present interest in devices, which requires a set of similar samples, expressly made for the
operate at room temperature, we concentrate on l/f purpose, such as the samples used for Fig. 1.
noise at T = 300 K. Temperature influence is only In spite of all uncertainties, Fig. 1 shows that the de-
discussed if it helps to elucidate the physical nature pendence of a on p can be found from a set of carefully
of l/f noise. For the same reason we need not deal prepared samples, where only one parameter is varied. It is
with the noise of hot electrons. In general we shall pay even possible to extrapolate the results to samples without
less attention to the theoretical problems of the physical impurity scattering, although such samples cannot be made.
model of the noise, than to reliable numerical values to The inaccuracy in the extrapolated value of alattis less than
be used in device models. lo%, despite the factor 1. 5 between individual a values. The
dimensions length, width, and thickness do not appear in (7),
11. THE FACTORa / N proving that 1/f noise is a bulk effect. Previous theories that
The relation considered l/f noise as a surface effect [ 131 were thus refuted,
at least that is for the homogeneous samples considered. The
(7) bulk l / f noise, which has been proven to exist beyond any
doubt in homogeneous samples, also occurs in devices where
was proposed 25 years ago [I21 in an effort to systemati- its magnitude can be estimated using the CY value determined
cally collect data on l/f noise from the literature. In that in homogeneous material. However, there is positive evidence
collection the influence of the size of the ohmic homogeneous that surface l/f noise exists too. Certainly in MOST’S, many
1928 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 11, NOVEMBER 1994

low-frequency Lorentzian distributions. However, even in a

sample with a length of 1 cm an electron only stays about 0.1
s as its diffusion coefficient D is of the order of lop3 m2/s.
How then can electrons, that stay in the sample for only such
a short while, produce noise at frequencies below 1 Hz? The
answer is as follows: We have evidence that the 1/ f noise is
in the lattice scattering (see Section 111). The lattice modes can
scatter electrons. The scattering cross-sections fluctuate slowly
with a l / f spectrum. There are permanently N electrons4n
average-that probe the slowly varying cross sections. We do
not follow individual electrons. Individual electrons move in
and out the sample; the lattice is permanent.
This idea has been worked out in more detail in [19]. A
model was considered where N electrons were scattered by A
lattice modes. This led to a relation for the noise in the current
density where S j / j 2 is proportional to 1/N and independent
of A.
Having answered theoretical objections against the factor
1/N, we would nevertheless like to investigate the experimen-
tal evidence at this point. We introduced N as a measure of the
size of the sample. In bigger samples the relative noise must
Fig. 1. Plot of log anleas versus log pmeasfor epitaxial n-GaAs at 300 K. average out. Another measure for the size may do as well, like
(Reprinted from: L. Ren and M. R. Leys, “I/f noise at room temperature in
N type GaAs grown by molecular epitaxy,” Physica B , vol. 172, pp. 319-323, the volume, the number of atoms A , or the number of lattice
1991.) modes, proportional to A. We therefore made a comparison
between N and A [19]. If the relative noise is written as
experimental data are better explained by surface effects than
by bulk effects. [ 141 The noise of MOST’S is discussed in the
paper by Vandamme, Li, and Rigaud, in this Special Issue.
it turns out that a is a better “constant” than y is. Values
If the same relation (7) holds good for metals, it is immedi-
of a are found between and 2 x lov3, whereas y varies
ately clear that for noise studies in metals very small samples
between 10 and lo5 in a group of some twenty semiconductors.
are required. Early experiments on point contacts [ 151 and thin
These experimental data give no argument for preferring r/A
films [ 161 showed that the relation does indeed hold good, and
to a/N.
that a has the same order of magnitude as in semiconductors.
Since it is doubtful whether a has the same value in different
The old idea that l/f noise is exclusively a semiconductor
materials, a more direct proof for a/N being the correct factor
effect was thereby proved to be incorrect.
is to study the injection of a varying number of electrons in a
Arguments against (7) have also been raised, not so much
given volume of a semiconductor, either by electrical injection
because of the difficulties with a (a constant or a parameter),
in a forward diode [20], or by persistent photoconductivity
but because of the factor N in the denominator. One argument
[21]. Such experiments show that the noise is proportional
is rather trivial. If one were to start the discussion of a
to 1/N. In each experiment a tums out to be a constant.
theoretical model by immediately considering an “average”
The observed constants from different experiments are in the
electron, then the number of electrons N would not appear in
normal range 1 0 - ~to
the final result. It has been concluded from such models that
relation (7) cannot be correct because of the factor 1/N. This
discussion is summarized in [19]. 111. EXPERIMENTAL DISTINCTION BETWEEN An AND A p
In experiments one always observes a group of N electrons. The l / f noise is a fluctuation in the conductivity. There
In the oversimplified models with an “average” electron, a is conclusive experimental evidence for this point of view.
single electron is considered [17]. A proper theory, however, Conductivity fluctuations lead to fluctuations in the resistance
yields the result averaged over an ensemble of identical R and thus to slow fluctuations in the power density of the
samples, electron systems, and so on in full agreement with thermal noise 4kTR.
experiment. Theory does not provide arguments against the
SS, SR. (9)
use of a normalizing factor 1/N.
Weissman [18] put forward a more serious argument that Because of (1) one can measure the noise in R by measuring
had already been brought up at some conferences. If one the noise in SV. In such measurements no current is passed
considers the l / f noise as a summation of Lorentzians (see through the sample [22]. These special measurements prove
Section IV) then the very low frequencies, where l/f noise is that 1/f noise is not generated by the current. In conventional
observed, require Lorentzians with a long characteristic time measurements the current is only necessary to transform
T >> 1 s. Therefore, the electrons must stay in the sample the already existing conductivity fluctuations into voltage
much longer than a few seconds, in order to produce these fluctuations that can be measured.
HOOGE: NOISE SOURCES 1929

So l/f noise is a fluctuation in the conductivity. This is of semiconductor samples, where two scattering mechanisms
the last point on which general agreement can be reached. determine the mobility. We consider a semiconductor in which
Any next step causes a great deal of controversy. As the two scattering mechanisms are active: lattice scattering and
conductivity impurity scattering. The mobility pmeasmeasured is given by
Matthiessen’s rule
= nqp (10)
1 - 1 1
contains the product of n and p the next question is: “What
~-
Pmeas
-.
PIatt
+ Pimp
(11)
is fluctuating .with a 1/f spectrum, n or p?”
In a series of experiments it was shown that there is a It is certain now that only the lattice scattering generates
type of l/f noise that is a fluctuation in the mobility. These l/f noise, whereas the impurity scattering has no appreciable
experiments were done on homogeneous samples, mainly contribution to the noise. In order to obtain simple relations
silicon. From these experiments it follows that this mobility we start by assuming &imp = 0. Later we shall introduce a
l/f noise is always present. It is described by (7), where small A p i m p > 0 and discuss the consequences thereof. From
the a value is in the order of magnitude of 10V4 in perfect (1 1) it follows that
material. In damaged material the mobility noise may be
considerably increased. On top of the mobility noise there
may be other types of l/f noise, e.g., number fluctuations
generated at surface states. Number fluctuations caused by
trapping processes at the surface play an important part in \ Platt /
MOST’S. It seems that we see An noise in an N-channel where a,,,, and alattare defined by relations corresponding
MOST, whereas in a P-channel Ap noise is observed. The to (7). From noise measurements on a series of samples with
very complicated situation of MOST noise is discussed in this different doping, and thus different contributions of pimp,we
Special Issue in the papers by Vandamme, Li, and Rigaud and find a straight line in a plot of log a,,,, versus log pmeas.
by Chang, Abedi, and Viswanathan. According to (13) the slope has the value 2. Fig. 1 shows
In this section we will concentrate on the fundamental how extrapolation to platt yields the value of alatt.This
mobility noise in good material. We will consider the noise situation was found in all cases studied. Plots like Fig. l - e v e n
in thermo EMF, Hall effect, etc. The principle of such an if the slope is not exactly 2 due to the approximations in
analysis is that in the same sample electrons move because of (1 1)aefinitely prove that, in the samples investigated, the
an applied electric field, and because of some other generalized l/f noise is mobility noise.
force, e.g., a temperature gradient. We will now discuss the situation in which there is l/f
In the experiment the ratio of the two generalized forces noise, both in the lattice scattering and in the impurity scat-
are varied and the change in magnitude of the l/f noise is tering. From Matthiessen’s rule we obtain
observed. In the analysis one first introduces a A p term in
the transport equation, and calculates the expected fluctuation
in the observed voltage or current. Then such a calculation
is done with a An source. The observed noise in voltage or From which follows
current always agrees with Ap fluctuations. In most cases the
observed noise is far off from the line for A n fluctuations.
Sometimes the results of the calculations for A n and A p are
ffmeas = (*) Platt ’Qlatt + (&)
Pimp ’aimp (15)
not that far apart, so that no distinction can be made. There
assuming that
was no case in which the differences between observed and
calculated A p values were so large that mobility fluctuations (Aplatt ‘ A p i m p ) = 0. (16)
had to be excluded in favor of An fluctuations. These experi-
ments have been discussed in a review paper in 1981 [22]. The plot of log a,,,, versus log pmeasnow is somewhat
At that time the result of noise measurements in the Hall more complicated than Fig. 1. If the term ( p m e a s / P I a t t ) ’ a l a t t
effect was indecisive. The theoretical lines of An and A p dominates in (15) we have the situation described by (12)
as functions of the magnetic induction B were not that far and (13). This corresponds to the line with slope +2 in
apart. Recently, results were published on the noise in the Hall Fig. 2: the points a to h. If the term (pmea,/pimP)’aimp
effect in nGaAs, where it is easy to reach high p B values dominates in (15) we have a situation where pmeasN pimp,
( p B >> 1). Under these conditions the theoretical lines of so that ameasN a i m p . The noise, characterized by aimpis
A n and A b versus B deviate widely; where the experimental proportional to the number of impurity centers, which in this
points follow the A p line nicely [23]. Such calculations are situation is inversely proportional to p i m p N pmeas.
rather complicated, since they have to start from individual
ameas Qimp -1 N Pmeas.
Pimp -1 (17)
levels in the conduction band. Integration over the whole band
then gives the noise magnitude that can be compared with the We find a line with slope -1 in the plot of log a,,,, versus
experimentally observed value. log pmeas.These are the points h to j. Fig. 2(b) gives the
A much more transparent proof for mobility fluctuations general situation at a constant temperature To. In all situations
is provided by the analysis of the noise in the conductance investigated only the right hand side of the figure, with the
1930 IEEE TRANSACHONS ON ELECTRON DEVICES, VOL. 41, NO. 11, NOVEMBER 1994

found that

with water:

where ( 4 ) is the mean phonon number. The essential factor

is the term l / f ; the value of the numerical constant is of less
importance. It should equal (lnfh/fl)-’, where fi and fh are
the lower and higher limits of the l/f spectrum. The numerical
values were calculated by assuming independent fluctuations
in the modes, and using a rough estimate of the number of
modes in the illuminated volume.
(a) (b) The consequence for electrical conductivity noise is that
Fig. 2. (a) log pmeas versus log T . The impurity scattering increases by
what is observed as mobility fluctuations of electrons are
equal factors in the series of samples a to j . (b) log ameas
versus log pmeas essentially number fluctuations of phonons.
at T = To. The noise of the samples d to h is determined by qatt although
Pmeas 2 Pimp.
IV. An-MODELS
slope +2, was observed. It is the familiar plot of log a versus All A n models are based on the same principle: the addition
of Lorentzian GR spectra with a special distribution of the
1% P*
Matthiessen’s rule is only an approximation. Therefore, the relaxation times r2.A very wide range of 7% values is required.
results for a,,, are also approximations. This does not raise A l/f spectrum is obtained between the frequencies 1 / 7 2 and
a serious problem. We must distinguish between slopes that 1/71 if 71 < T~ < 72. Below 1 / 7 2 the spectrum is white,
are close to -1 and +2. These values are so far apart that the above 1/71 the spectral density is proportional to f-’. We
distinction between lattice scattering and impurity scattering is use a normalized distribution function g ( r 2 ) :
always possible. Whether Tacano [24] is right in interpreting
his value a,,,, =3 x as the value of aimpof GaAs,
g(72) =0 for T~ < 71 (20)
1 1
depends on the branch his point is on. He assumes it is g(7-2) = ~- for 71 < T~ < 7 2 (21)
h / 7 1 7 2
like point j in Fig. 2. However, if this value corresponds to
point g, the same value a,,,, = 3 x then points to g(72) =0 for 7-2 < 7% (22)
alatt N 5 x which is quite a normal value in GaAs at
50 K. From values of a,,,, of samples with less or with more
impurity scattering, it can be decided whether the samples are
on the branch with slope -1, in which case Tacano is right,
4 Ti
or on the branch with slope +2. 2
=
In the case of number fluctuations, the following situations
occur:
a. a is proportional to ‘GR centers’ which create a l/f
spectrum, and which do not scatter electrons: a,,, 0:

PLeas. 1
b. a is proportional to centers that also are scatterers of w << 1 / 7 2 << 1/71: S N = (AN)’- .4~2 (24)
ln.r2/71
electrons: a c( p z p . If p,,,, 2~ pimp:a,,,, oc 1 1
c. like b, but now with N PIatt: ameasK pzp K
1/72 << w << I / T ~ SN
: = (AN)’-. - (25)
,,,U
,,
ln72/71 f
pi:,,, where k >> 1. 1 1
Number fluctuations will always lead to negative slopes in 1/72 << 1/71 << W : SN = (AN)’----. ~
(26)
lnT2/q r2~1f2
plots of log a versus log p; the slope might be close to zero.
Above, we have presented experimental evidence of the where the following approximations were used:
fact that lattice scattering is the source of l/f noise. This 1 7 r
evidence came exclusively from measurements of electrical arctan S 2~ S and arctan - = - - 6. (27)
6 2
noise. Evidence from other experiments than those on electri-
cal noise would, therefore, be most welcome. In a series of The trap distribution (21) only leads to a l/f spectrum if a
optical experiments, Musha et al. 1251-[27] showed that when necessary, but often overlooked, condition is fulfilled: there
photons are scattered at acoustic lattice waves the intensity of should be no transitions between traps w4th different T’S,
the scattered light also exhibits l/f noise. With quartz it was neither directly nor via the conduction band. See Fig. 3.
HOOGE: NOISE SOURCES 1931

-tlt2/T0

Fig. 3 . Isolated traps yield a l / f spectrum. Interacting traps yield a

Lorentzian spectrum.

A. Isolated Traps
The variance ( A X ) 2of the free electrons or of all trapped
electrons together is the sum of the variances of each individ-
ual kind of trap, characterized by its q.

AX)^ =S T 2 g ( r i ) m d r i . (28)
T1

Each individual spectrum is given by

Fig. 4. The Lorentzian and the l / f spectrum following from Fig. 3.
Q/TI = 5 x lo8 = eZo T O / T ~ = 20.

- 4ri
- (Ax)2g(.r;) 1 + (27rf7i)2d r i . (29) For instance, in McWhorters surface model, traps are as-
sumed to be homogeneously distributed in the oxide layer on
The summation of these spectra leads to
a semiconductor. The probability 1/r of an electron in the
semiconductor reaching a trap in the oxide layer by tunneling
is given by
r = rOex/d (33)
where z is the distance from the trap to the silicon-oxide
in agreement with (25) and with the variance J,"Sdf. interface, and d is a constant characteristic for the tunnel
process.
B . Transitions Between the Traps dN dN dx 1
g ( r ) = -= -. - 0: - (34)
A fluctuation in the number of free electrons now decays dr dx dr r
by interaction with all traps
N is the number of traps, dN/dx the concentration of traps
which is constant.
The St. Petersburg group [28] has proposed a special model
for GaAs, with a tail of states below the conduction band. The
densiy of states in the tail is assumed to be exponential
(31) N ( E ) = N(O)ePEIE* (35)
where we define
where E is the distance of the state to the bottom of the band.
ro 3 711nr2/r1. (32)
For the relaxation time of the state they assume
Thus one single Lorentzian is obtained with r1 close to 71. The
r ( E ) = r(0)e-E/kT. (36)
Lorentzian intersects the l / f spectrum at the frequencies f l =
71/47; and f 2 = l/7r2r1,which are a factor 2ro/7rr1 to the The result is again g(r) c( 1/71.
left and right from the characteristic frequency f o = 1/27rro. The required distribution g ( r ) 0: 1/r can be obtained with
Fig. 4 shows the two spectra, based on the same distribution processes that are thermally activated. If Ei is homogeneously
function g ( r i ) : a l/f spectrum for isolated traps, and a distributed between El and Ehk, and if g(E) is zero outside
Lorentzian for interacting traps. In this figure we used 7 2 / 7 1 = this interval, then an exact l/f spectrum is obtained at
5 x lo8 = e203r0/r1= 20. frequencies f between
Whatever g ( r ) one may choose, the result of interacting
1 1
traps will always be one average r , hence a single Lorentzian. - <2Tf < -. (37)
rh Tl
A distribution g ( r ) 0: 1/r is easily realized over a wide
range of r values, if r exponentially depends on a quantity In the Dutta Horn model [29], [30], g(E) need not be a
that is homogeneously distributed over a limited range. constant. A peak of E values a few lcT wide is good enough
1932 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 1 1 , NOVEMBER 1994

to produce a l/f-like spectrum: f - Y with y = 1 f 0,3. The Now the conductance between A and B is seen as a
slope will not be constant over the whole frequency range, summation of all possible paths from A to B , via all scattering
and there will be a relation between slope and temperature centers. The phase is preserved over a limited distance L+.
dependence of the spectral power: In the universal conductance fluctuation model (UCF), the
multiple scattering events of all defects in L; contribute to the
interference. After a defect has moved to a different position,
the conductance is different, which is interpreted as mobility
where TO N s, an attempt time in the order of an inverse noise. The defects have somewhat different activation energies
phonon frequency. for jumping, and therefore, different T values. The summation
The essence of the Dutta Hom model is that a wide of the individual Lorentzians yields a l/f spectrum, according
range of T ’ S results from a rather narrow peak of activation to the Dutta Hom model. The UCF noise is found in a crystal
energies. The width of the peak is determined by the disorder with a high degree of disorder at very low temperatures.
in the crystal. Our question now is: “Is this of interest for The local interference model (LI) applies to electron waves
semiconductors at room temperature?” singly scattered by a few neighboring defects. A special case
In applying the model directly to generation recombination of LI is the two-level system (TLS), where the scatterer
noise the activation energies of the traps will be 0.3 eV at moves from one position to an energetically equivalent second
most. The width of the peak could then be in the order of position, by tunneling through an energy barrier. Also in the
0.03 eV. Hence, AE/lcT is of the order 1. So there is no LI model, the defects move with nearly the same activation
appreciable frequency range in which a l/f spectrum can energy, giving a l/f spectrum by summation of Lorentzians.
be observed. The Dutta Hom model is of little interest to The LI model predicts l/f noise at room temperature in
number fluctuations at room temperature, but it can be applied weakly disordered metals.
to mobility fluctuations (to be discussed in the next section). One might think that the degree of disorder required is not to
There we need mobile lattice defects. The movement of lattice be found in nearly perfect epitaxial semiconductor films. Even
atoms requires activation energies in the order of magnitude though this may be true, it cannot be used as an objection
of binding energies being 1 or 2 eV. A high degree of disorder against the LI model. The noise intensity a, is proportional to
may give a spread in the activation energy of some 10%- 20%. mmd, where n is the electron density and 7Lmd the density of
Thus at room temperature AE/lcT is of the order of 10, which the mobile defects ((21) in [32]). In metals we find, therefore,
yields ~ h / 21~ lo5.
l The resulting l/f spectrum of the electron a c( 7L7Lmd 0: Anmd, where A is the number of atoms per
mobility could then be observed in a frequency range of 5 dec. cm3. In semiconductors we find a proportional to n, when
The Dutta Hom model can explain mobility noise, certainly we investigate differently doped samples made from the same
at low temperatures. host material, with the same nmd in all samples. So there
is no problem with the low value of n,d/A in high-quality
V. AP-MODELS semiconductors. The problem is rather that a would depend
on n, contrary to all experimental evidence. LI seems not to
We shall present two theoretical models for mobility l/f apply to semiconductors.
noise: 1) local interference noise, 2) quantum l/f noise. A second argument against LI is that in semiconductors the
No critical discussion of the theories will be attempted; the exponent of the spectrum is 1.O not 1fA, as would result from
emphasis is on results which can be used in a discussion of the use of the Dutta Hom procedure in the local interference
the noise of devices. model. We know of one interesting example where the local
interference model might apply to semiconductors: proton
A . Local Intei$erence Noise irradiated GaAs at low temperature (T < 150 K). Ren found
This is one of the three cases where the Dutta Hom model a practically temperature independent noise, proportional to
is applied to mobile defects that act as scattering centers the radiation dose. There is a small peak in lna versus l/T,
for the electrons [31]. The local interference model has been agreeing with the slope y, which is not exactly -1. Here (38)
very successful in the study of noise in metals, especially in holds [34]. The temperatures of the peaks correspond to 0.35
disordered metal films [32], [33]. Here our problem is whether eV.
it can be of interest for semiconductors at room temperature. Irradiation of samples that were originally doped with
We shall mainly be guided by Giordano’s review [32]. different donor concentrations show that this is mobility noise,
The principle can be sketched as follows. An electron since a c( & , 1351. However, the quadratic dependence is
retums to its original position after a random walk during characteristic of lattice scattering, whereas the local interfer-
which it has been scattered by several scattering centers, in ence model is based on impurity scattering.
this case lattice defects. Each scattering event gives a phase
shift. The electron arrives at the original position again with a
certain phase shift in its wave function. If it had travelled the B . Quantum l/f Noise
same path, but in the opposite direction, its final phase shift All three models UCF, LI, and TLS, deal with interfer-
would have been the same as for the original direction. So ence of waves scattered at many centers. When the spatial
there is constructive interference: the electron density at the arrangement of the centers changes, we observe a change in
original position is higher than for two uncorrelated functions. the conductance, noise. Handel’s model is more general: there
HOOGE: NOISE SOURCES

acoust.
phonon
polar opt.
phonon

wave function.
TABLE I

300-3000 20-50 -

are 1/f fluctuations in each scattering event at each individual

scattering center [36]. In an excellent review paper [37] van
Vliet does away with many later additions of the original
model, but the essential idea still stands: interference between
the Bremstrahlung and non-Bremstrahlung part of the electron

In the scattering process a low-frequency photon is absorbed

or emitted. The wave length of such a photon is much longer
than the dimensions of the samples and of Faraday cages. The
question is: can such photons be present or develop in this
limited space? This is the so-called cage effect, which is not
accepted as a serious problem by van Vliet [38]. This may be
correct, but then there is no way of coming to grips experimen-
tally with these photons. “They are lost to the universe.” The
model is so general that there are no specific features that lead
themselves to experimental confirmation. Due to this general
nature and because of the most characteristic participants-the
low-frequency photons+annot be studied, nothing else is
left to us than to compare numerical results of model and
experiments. Handel’s model predicts the following a values
(see Table I).
Handel cites some experimental cy values to support the the-
oretical results. All experimental values, but one, are derived
from noise studies of devices. The exception is Bisschop’s
work on polysilicon. His values lop9 - lop8 are, however,
not a values at all [42].
Comparison of the theoretical values with the experimental
values, as shown in Fig. 5 , leads to the conclusion that the
experimental results do not support Handel’s theory. It could
very well be that the theoretical model correctly predicts some
kind of 1/f noise, but then this type of 1/f noise is different
from the observed noise with a much higher a.

VI. EMPIRICAL
VALUES
In proposing a model for the l / f noise in devices, two
kinds of assumptions for the sources must be made. These
are assumptions about 1) their physical nature: A n or Ap,
isolated or interacting traps, bulk or surface states, etc., 2)
the numerical value of a (when mobility fluctuations are
considered).
Because of the latter reason we present experimental values
of a here in Fig. 5 taken from literature. Only results for
semiconductors, homogeneous samples and room temperature
are included. We did not reject any data that do not fit in the
general picture or that run against our own ideas about what
Q should be.
Open symbols are used for ameasr
are used for &Iatt.
whereas black symbols
A number next to a symbol refers to the
of references.
0
Si

nGaAs.9,

amas
a latt
9,
10

10,
43-

Ge
CdHgTe
m Ymixed
InSb
InP
p-GaAs
-

106

Fig. 5. Experimental values of anleas

given by the original author.

summarized in one single expression

of
alatt = 0.1 exp[-0.13 eV/kT]

can be 10 times lower: a,,,,

.102

+ 7 x lop5.
A similar study on Si is badly needed. (This study cannot be
expected from the Eindhoven group, since we concentrate on
111-V compounds in accordance with national science policy.)
Fig. 5 shows that alatt values are in the range of l o p 5 to
lop3. a,,,, is in the range
to lop3. If we now have to name one single value,
that can be said to be reasonably representative of alatt in
semiconductors at room temperature, then it is cy =
instead of the historical value 2 x lop3 of 25 years ago.
In many publications it is stated-ften triumphantly-that
the low values of a, which are reported there, prove the
high quality of the material used by the authors. This is not
necessarily correct when a,,,,
a,,,,
Pmeas < Platt.
The value of a,,,,
is considered. The value of
can be low for two reasons [43]: 1) alatt is low, 2 )

is low if scattering mechanisms other

than lattice scattering prevail. The factor (pL,,as/p~att)2
then is smaller than 1, and will give a low value of a,,,,,
in (13)

when qattis not low. Only if alattitself is low, it can safely

be concluded that the crystal lattice is perfect.
Damaging the crystal by mechanical stress or by irradi-
ation strongly increases the l/f noise. The electron con-
centration and the mobility hardly change; nevertheless the
noise increases by orders of magnitude. The first explanation
even
1933

and alatt of several semiconductors,

measured on homogeneous samples at 300 K. The numbers refer to the list

original literature, unless the value is included in an earlier

survey [9], [lo] and [43]. If no number is given at a point
for alatt,it means that alattwas calculated by us from a,,,,

Some of the Si data are much older than the n-GaAs data.
Many GaAs data stem from modem epitaxial material. Today
we are much better informed on n-GaAs than on Si. This is
largely due to the work of Ren [44], [45] on epitaxial n-GaAs,
where a large series of samples with the same geometry but
different doping levels and different scattering mechanisms
were investigated at different temperatures. All data could be

(39)
1934 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 41, NO. 11, NOVEMBER 1994

that comes to mind is that the defects act as generation- [3] F. N. Hooge and L. Ren, “On the generation-recombination noise,”
recombination centers which in some, as yet unexplained, way Physica, vol. B191, p. 220, 1993.
[4] F. N. Hooge and L. Ren, “On the variances of generation-recombination
generate l/f noise. If this is correct then the induced noise is noise in a three-level system,’’ Physica B , vol. B193, p. 31, 1994.
a fluctuation in the number of free carriers. However, it might [5] T. G. M. Kleinpenning and A. H. de Kuijper, “Relation between variance
also be possible that the defects act as scattering centers. If they and sample duration of 1/f noise signals,” J. Appl. Physics, vol. 63, p.
43, 1988.
are mobile, they will generate l/f noise according to the local [6] Proc. 7rh Int. Conf. on Noise in Physical Systems, and the 3rd Int. Conf.
interference model. Therefore, one would like to see further on l/f Noise, Montpellier, France, May 1983, M. Savelli, G. Lecoy,
investigations of the induced noise in damaged material, e.g., and J. P. Nougier, Eds. Amsterdam: North-Holland, 1983.
[7] Proc. 8th Int. Conf. on Noise in Physical Systems, and the 4th Int. Conf.
a plot of log (Yinduced versus log pmeas(like Fig. l), because on llf Noise, Rome, Sept. 1985; A. D’amico and P. Mazzetti, Eds.
that would decide whether the induced noise is mobility noise Amsterdam: North-Holland, 1986.
[8] Proc. 9th Int. Conf. on Noise in Physical Systems, MontrCal, May 1987,
or number noise. In case of mobility noise it is important C. M. van Vliet, Ed. Singapore: World Scientific, 1987.
to distinguish between impurity scattering-agreeing with the [9] Proc. 10th Int. Conf. on Noise in PhysicalSystems, Budapest, Aug. 1989,
LI model-and lattice scattering, as has possibly been found A. Ambrbzy, Ed. Budapest: AkadCmiai Kiad6, 1990.
[lo] Proc. 11th Int. Conf. on Noise in Physical Systems and IlfFluctuations,
with proton-irradiated GaAs [35]. Kyoto, Nov. 1991, T. Musha, S. Sato, and M. Yamamoto, Eds. Tokyo:
The relation between l/f noise and damage has been Ohmsha Ltd., 1991.
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tuations,” St. Louis, August 1993, P. H. Handel and A. L. Chung, Eds.
Levinshtein, and Rumyantsev [28]. The model favored by the New York: AIP Press, 1993.
authors is that the defects create the states in the tail below the [12] F. N. Hooge, “l/f noise is no surface effect,” Phys. Len., vol. 29A,
conduction band. There is experimental support for this model p. 139, 1969.
[13] A. L. McWhorter, “1/f noise and related surface effects in germanium,”
from measurements of photo conductivity. Ph.D. dissertation, MIT, Cambridge, MA, 1955.
Papers dealing with noise and damage, that appeared af- 1141 L. K. J. Vandamme, “Bulk and surface 1/f noise,” IEEE Trans. Electron
Devices, vol. 36, p. 987, 1989.
ter this survey, follow the lines of thinking of the survey 1151 A. M. H. Hoppenbrouwers and F. N. Hooge, “l/f noise of spreading
[46]-[52]. It is immediately assumed by all authors-except resistances,” Philips Res. Rep., vol. 25, p. 69, 1970.
one-that the induced noise is number noise. Based on this (161 F. N. Hooge and A. M. H. Hoppenbrouwers, “l/f noise in continuous
thin gold films,” Physica, vol. 45, p. 386, 1969.
assumption some model for the generation-recombination cen- 1171 F. N. Hooge, “On expressions for 1/ f noise in mobility,” Physica, vol.
ters is then presented without experimental evidence confirm- B114, p. 391, 1982.
ing the number fluctuations and excluding mobility fluctua- 1181 M. B. Weissman, “1/f noise and other slow, nonexponential kinetics
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[20] T. G. M. Kleinpenning, “1/f noise in p-n diodes,” Physica, vol. B98,
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1211 F. Hofman and R. J. J. Zijlstra, “The validity of Hooge’s law for 1/f
noise,” Solid State Commun., vol. 12, p. 1163, 1989.
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1) In all semiconductors there always is mobility l/f noise imental studies on 1/ f noise,” Rep. on Progress in Physics, vol. 44, p.
479, 1981.
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[25] T. Musha, G. Borkly and M. Shoji, “1/f phonon-number fluctuations
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1261 T. Musha and G. BorbCly, “1/ f fluctuations of phonon energy in water,”
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1271 T. Musha, K. Takada, and K. Nakagawa, “Fluctuations of light intensity
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Soviet Physics Semiconductors, vol. 25, p. 97, 1991. F. N. Hooge was bom in Amsterdam in 1930. He
N. V. D’yakonova, M. E. Levinshtein, and S. L. Rumyantsev, “Temper- studied chemistry at the University of Amsterdam,
ature dependence of low-frequency noise in structurally perfect GaAs where he received the Doctor’s degree in 1951.
and after destructive compression,” Sov. Phys. Semicond.. vol. 25, p. He then joined Philips Research Laboratory in
217, 1991. Eindhoven to work on semiconductors, and since
I. S. Bakshee, E. A. Salkov, and B. I. Khizhnyak, “1/f noise in HgCdTe 1968 he has been studying l / f noise. In 1971 he
converted from 1 1 - to p-type by native doping,” Solid State Commun., was appointed Professor of electronic materials at
vol. 81, p. 781, 1992. the Department of Electrical Engineering at Eind-
I. S. Bakshi, L. A. Karachevtseva, A. V. Lyubchenko, V. A. Petryakov, hoven University of Technology (EUT). From 1983
E. A. Salkov, and B. I. Khizhnyak, “Influence of compensating annealing to 1985 he was Dean of the department and from
on the 1/ f noise in CdHgTe,” Soviet Phys. Semicond., vol. 26, p. 97, 1985 to 1988 was Rector Mamificus of EUT.
1992. Dr. Hooge received the doctorate honoris causa from fbathclyde University,
G. M. Gusinskii, N. V. D’yakonova, M. E. Levinshtein, and S. L. Scotland, in 1989.