Flicker Noise in Electronic Devices

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 49

Flicker Noise in Electronic Devices A. VAN DER ZIEL Electrical Engineering Department University of Minnesota Minneapolis, Minnesota

..

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Statement of the Problem. History of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . B. Divergent Integrals . . . . . . . . . 11. Unusual Examples of l/f Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Open-Circuit Noise Voltage of a Lossy Capacitor . . . . . . . . . . . . . . . . . . . . . . B. Flux Flow Noise in Type I1 Superconductors. . . . . . . . . . . . . . . . . . . . . . . . 111. Flicker Noise in Vacuum Tubes . . . . . . . . . . A. Flicker Noise in Field Emission Diodes B. Flicker Noise in Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... C. Flicker Noise in Secondary Emission D. Flicker Noise in Vacuum Diodes and E. Flicker Partition Noise in Positive Grid Triodes and Pentodes . . . . . . . . . . . . . . . IV. Flicker Noise in Resistors A. Fundamental Experim B. Noise in Metal-Meta C. Flicker Noise in Semi D. Miscellaneous Topics on l/f Noise in Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. l/f Noise in Solid-state Devices . . . . . . A. l/f Noise in Tunneling Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Flicker Noise in Schottky Barrier Diodes C. 1 Noise in Junction Devices . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Flicker Noise in JFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Flicker Noise in MOSFETs

................................

VI. Miscellaneous Problems A. Formal Explanation

........................

225 225 226 23 1 23 1 233 234 235 231 238 239 243 244 245 256 258 268 269 269 212 214 219 28 1 289 290 290 29 1 292 292

I. INTROPUCTION A . Statement of the Problem. Hiktory of the Problem

The name “flicker noise” refers to noise phenomena with a spectrum of the form AZB/p, where A is a constant, Z is the current, f is the frequency, 225

Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014649-5

226

A.. VAN DER ZIEL

and the q o n e n t s j? and y are also constants. In its most restricted form one requires y = 1 and speaks of true l/fnoise, but often this requirement is relaxed; for example, flicker noise in saturated vacuum diodes with a tungsten filament has y N 2. The exponent j? often has a value of about 2, though deviations are not uncommon. For example, flicker noise in saturated vacuum diodes with a tungsten filament has p N 2, which is in contrast with the shot noise in these diodes, whose spectra show a linear current dependence. Flicker noise in vacuum tubes was discovered by Johnson (1) in 1925, whereas Schottky (2) gave the first interpretation in 1926. In 1937 Schottky (3) showed that flicker noise was more strongly suppressed by space charge than shot noise. Flicker noise in carbon microphones and in carbon resistors was first studied in detail by Christenson and Pearson (4). Here the noise shows roughly a quadratic voltage (or current) dependence, and y IT 1. Later it was found that flicker noise in one form or the other occurred in most electronic devices. This almost universal occurrence (with a few notable exceptions, however!) of flicker noise has raised the question whether there might be a universal flicker noise mechanism behind these various flicker noise manifestations. In the opinion of this reviewer, such a universal physical mechanism is rather unlikely. The flicker noise processes have in common related spectra of the form l/fy with y 21 1; this relationship is mathematical rather than physical. In order to make progress in the understanding of flicker noise, one must try to understand the physics behind it and so comprehend the reasons why different physical mechanisms can give rise to the same type of spectrum. The remainder of this section deals with divergent integrals occurring in l/f spectra. Section I1 discusses two processes that have a llfspectrum and so qualify as llfnoise; but they are usually not recognized as such. Section I11 deals with flicker noise in vacuum tubes; while these devices have been almost completely replaced by solid-state devices, their flicker noise mechanisms are of interest in that they shed light on the general problem of flicker noise. Section IV discusses flicker noise in resistors and photoconductors, whereas Section V deals with flicker noise in solid-state devices. Finally, Section VI discusses deviations from y = 1 and a fundamental noise mechanism that may give rise to llfspectra. B. Divergent Integrals

Spectra of the form l/fy for 0 1) or the upper limit (for y < l), or at both limits (for y = 1, true l/fnoise). We shall

227

FLICKER NOISE IN ELECTRONIC DEVICES

see that this violates some commonsense notions about fluctuating qualities. To demonstrate this, we start with the discussion of the Wiener-Khintchine theorem, which can be stated as follows (5) : Let the autocorrelation function X(u)X(u+ s) of a stationary random variable X(t),having x(t)= 0, be absolutely integrable for the interval 0 Is I 00, and let the spectrum S , ( f ) of X(t)be defined as

[ Jo

m

S,(f) = 4

X(u)X(u+ s) cos ws ds

Then S , ( f ) exists and the inverse transformation is equal to X(u)X(u+ s); that is

X(U)X(U + S) =

r

S , ( f ) cos osdf

(la)

+

The condition of absolute integrability of X(t)X(t s) is a sufficient condition for the simultaneous validity of the pair of equations (1) and (la). If the condition of absolute integrability is not satisfied, then the pair of equations (1) and (la) may not be simultaneously valid; that is, even if the integral in (1) exists, Eq. (la) may not be valid. Let us now investigate the convergence of the integral

in more detail. A sufficient condition for convergence at the upper limit (sl + 00) is that for large s IX(u)X(u+ s)l goes to zero faster than l/d with 6 > 1. A sufficient condition for convergence at the lower limit (so+O) is that for small s IX(u)X(u s)l goes to infinity more slowly than l/se with E < 1. Note that for 0 < E < 1, we have p = 00 ; this is mathematically allowed since the integral (2) converges at the lower limit. If the integral (2) diverges at the lower limit (so 0), then the definition (1) is invalid since the right-hand side of (1) diverges at the lower limit for co),then the all w. If the integral (2) diverges only at the upper limit (sl integral of the right-hand side of (1) exists for o # 0 as long as X(u)X(u+ s) goes to zero for s + 00, so that Eq. (1) can be used as a definition for S , ( f ) . If then X(u)X(u s) 2 0 for all s, S,(O) = co. This may in itself be allowed, but there is now no absolute guarantee that the transform (la) of S , ( f ) exists and is equal to X(u)X(u s); that needs to be investigated in each individual case. Let us now investigate what we find in practice. Actually, X(u)is always quite small, so that p is finite and X(u)X(u+ s) is of bounded variation.

+

-+

-+

+

+

228

A. VAN DER ZIEL

There can be, therefore, no doubt that the integral (2) will converge at the lower limit. Moreover, all physical systems are finite and hence they have finite time constants, including a largest one. There can therefore be no doubt that IX(u)X(u + s)J will go to zero faster than 1/sd with 6 > 1 for large s. Only in idealized models, such as diffusion into a half space, can the latter rule be violated. In practice, therefore, the integral (2) will always converge at both limits. An infinite value of in itself does not cause mathematical difficulties, provided that the integral (2) converges at the lower limit. As an example, consider the case where X(u)X(u + s) = a d(s), where a is a constant and 6(s) is the Dirac delta function. We then have' that S,(f) = 2 4 and Eq. (1) is valid because X(U)X(U+ S ) = 2a

cos osdf = u 6(s)

(3)

according to a well-known definition of 6(s). Let us now investigate the integral

j;

Sx(f)df

(4)

If it converges at both limits, we can use Eq. (la) as definition for X(u)X(u s); X(u)X(u + s) then exists and its transform (1) is equal to S,(f). If the integral diverges at one or both limits, difficulties may arise. Divergence of (4)at the upper limit may in itself not be harmful. As an example, consider the case X(u)X(u + s) = a 6(s); then S,(f) = 2u and the integral (4)diverges at the upper limit. Nevertheless Eq. (3) is valid, so that the inverse transform of S,(f) is equal to a 6(s), as required by (la). Divergence of (4)at the lower limit is much more serious, for in that case the integral on the right-hand side of (la) diverges at the lower limit for all s and therefore (la) cannot be used to define X(u)X(u + s). This should be avoided. While the divergence of (4) at the upper limit may not always cause difficulty, it violates the commonsense rule that must be finite. In reality, therefore, integral (4)should converge at both limits. Hence in all noise spectra encountered in practice, there will be a lowfrequency f,, however small but not zero, below which the spectrum S(f) varies more slowly than l/f, so that the integral (4)converges at the lower limit (fo +0). There must also be a high-frequency fb, however large but

+

' 4s:

6(s) ds = 2jTm 6(s)ds = 2, according to the definition of 6(s).

FLICKER NOISE IN ELECTRONIC DEVICES

229

not infinite, above which the spectrum varies faster than l/f; so that the integral (4)converges at the upper limit (fi + 00). The integral

then exists, the Wiener-Khintchine theorem is certainly valid, and our commonsense notions about noise quantities are satisfied. We can arrive at the same result from the point of view of measurements. Suppose we have a spectrum of the form a/f. At very high frequencies the noise is so small that it drowns in the noise background of either the device itself or the measuring equipment. Hence the existence of an a/f spectrum cannot be proved above a high-frequency fd that is a characteristic for either the device or for the measuring equipment. At very low frequencies, when recording the noise and using fast Fourier transform techniques, a measurement at the frequency f needs a time of the order l/J Hence it takes 28 hr to record and measure down to lo-’ Hz; measuring down to zero frequency would require an infinite measuring time. There is, therefore, a lower frequency f,’ below which the existence of a l/f spectrum cannot be proved; this frequency depends on the patience of the investigator and on the stability of the device and the measuring equipment. Beyond the range f,’ If < fd we know nothing about the spectrum; statements like “I believe that l/f noise persists for 0 If I00” are nonverifiable and hence do not belong to the realm of science. It is perfectly compatible with the measurements to assume that S x ( f ) has a frequency dependence for f 4 f,’ and f % fd such that the integral (4)converges at both limits. Another important theorem used in noise calculations is Carson’s theorem (51 which can be formulated as follows. Let a stationary random variable Y(t) be written as Y(t) =

c F(t i

ti)

(5)

where the F(t - ti)% are pulses of the same shape occurring at random instants ti, and F(t - t i ) = 0 for t < ti. Let F(u) be absolutely integrable for the interval 0 5 u I; 00 and let the pulses occur at the average rate 1; then the spectrum of Y(t) may be written as S,(f) = Ul$(f)12

where

(54

230

A. VAN DER ZIEL

The condition of the absolute integrability of F(u) is again a sufficient condition for the validity of the theorem, If that condition is not satisfied, Eq. (5a) might not be valid, even if the integral (5b) exists. We illustrate this with an example. Schonfeld has observed that the pulse (6) F(u) = l/u112 for u > 0

and

F(u) = 0 for u < 0

(6)

gives l/f noise. For if we put u = mu, @(f) becomes

where the integral I exists. Applying Carson’s theorem yields S,(f) = 2Jl@(f)l2= (2I/w)11l2

(74

But since F(u) is not absolutely integrable for the interval 0 Iu I 00, the last step is in doubt, and hence the conclusion is in doubt. We may also illustrate this as follows. A series of random pulses of the form (6) should have an autocorrelation function (5)

+

Y(t)Y(t s) - (p>2 =

u”*(u

+ s)1/2

Unfortunately this integral diverges at the upper limit for all s and at the lower limit for s = 0, i.e., Y(t)has no well-defined autocorrelation function. We can eliminate the difficulty by modifying F(u). If

F(u) = 1/u1I2 for 0 Iu I ul,

F(u) = 0 otherwise

(9)

then for large but finite ul, the function F(u) is absolutely integrable and S,(f) varies as l/f for f % fl, where fl is related to u1 and is very small if u1 is sufficiently large. Moreover, S,(O) is finite. The new model violates only the commonsense rule that Y 2 - (p>’ must be finite. We can eliminate the latter difficulty by further modifying F(u). If

F(u) =

for 0 Iu Iuo,

F(u) = 1/u’I2 for uo Iu Iul,

F(u) = 0 otherwise

(10)

then F(u) is absolutely integrable, the integral over the spectrum exists, and - (Y)’is finite. For uo Q ul there is now a large frequency range l/u, Q f 4 l/uo for which the spectrum viries as l/f.

231

FLICKER NOISE IN ELECTRONIC DEVICES

11. UNUSUAL EXAMPLES OF 1/ f NOISE A . The Open-Circuit Noise Voltage of a Lossy Capacitor

We shall show that the open-circuit noise voltage of a lossy capacitor shows a llfspectrum. This is rather interesting, for we know that in this case the noise is thermal noise of the dielectric losses. Let E = E' - jd' be the complex dielectric constant of the dielectric used in the capacitor; then its loss tangent is defined as tan 6 = t n / d

(1 1)

The capacitor can now be represented by a capacitance C = dc,A/d and a resistance r = tan 6/wC in series; here E~ is the permittivity of free space, A the electrode area of the capacitor, and d the thickness of the dielectric. Hence, since the noise is thermal noise of the resistance r, the open-circuit noise voltage has a spectrum

S,y) = 4kTr

=

4kT(tan6/wC)

(12)

Since in normal dielectrics C and tan 6 are practically independent of frequency, S,y) represents a 1/f spectrum. This is a case of 1/f noise in which no dc voltage needs to be applied to the device to detect it. In order to explain why tan 6 is practically independent of frequency, one usually introduces a dielectric aftereffect with a continuous distribution g(z) dz in relaxation times such that

jr

g(z)dz = (normalization)

For an aftereffect with a single relaxation time z, one finds

where 4 is the low-frequency dielectric constant (at w = 0) and E, the highfrequency dielectric constant (for wz % 1). One finds this behavior for polar molecules dissolved in nonpolar dielectric liquids. In 1907 von Schweidler (7) proposed to introduce a distribution in time constants in order to explain dielectric properties of solids. In this case Eq. (14) must be written El = E m

+ (E, -

Em)

1

+ w2z2'

Eff

= (E,

- Em)

wzg(z)dz 1 w2z2 (15)

+

232

A. VAN DEX ZIEL

For the case of small losses

[(E,

- 8,)

< E,]

In the particular case that

we obtain

For l/z,

Q w

4 l/t1 this reduces to

which is independent of frequency. For w % 1/z, and w -4 l/z,, tan6 goes to zero. It is not quite clear who first introduced this distribution in z ; for details see Gever's papers (8).Van der Ziel(9) and du Pr6 (10) applied this distribution to explain l/f noise in various devices and materials. One can understand Eq. (16) when one attributes the losses to impurity ions trapped in a potential well, such that it takes an activation energy En to bring it out of the well. In that case one can write for the time constant at the absolute temperature T z = toexp(E,/kT)

(17)

since the ion has a probability B exp( - E , / k T ) to have an energy E,; here B is a constant and zo is a time constant of the order of sec. Converting (16) to a distribution function g(E,) dE, in activation energies, the equation may be written

otherwise, corresponding to a uniform distribution in E, for En, < Ea < E,, . In view of the fact that in most noncrystalline dielectrics there will be a large number of different potential wells in which the ions can be trapped, this is a very reasonable distribution function for En. In Eq. (17a) E,, and E n , are the activation energies of z, and zl, respectively: En, = kT ln(zz/zo),

En, = kT ln(zl/zo)

(17b)

FLICKER NOISE IN ELECTRONIC DEVICES

233

Gevers (8) has applied this model very successfully to the interpretation of dielectric loss measurements as a function of frequency and temperature in various noncrystalline dielectrics. He obtained a number of interesting results of which we mention two. (1) The temperature coefficient of the capacitance C of a dielectric capacitor of loss tangent tan6 is

1 dC

2

where a, is the linear expansion coefficient of the material. Gevers found that the coefficient in front of tan 6 depended only very slightly on the nature of the dielectric. This is not so surprising, for zo does not differ too much for different materials, and the coefficient only varies as In zo. It should be observed that wzo Q 1 for all frequencies of practical interest. Equation (18) agreed quite well with experiment in a number of cases. (2) Tan 6 / T is a function of Tln(l/wz,) only. By measuring tan 6 as a function of T and as a function of w, Gevers was able to evaluate the time sec for glass. constant z,. He found zo N The results obtained by Gevers can be considered as an experimental verification of the validity of the distribution function g(z) dz. The application of this model to the derivation of the lifcharacter of S,y) thus has an experimental basis, and there is no need to invoke a universal l/f noise mechanism in this case. B. Flux Flow Noise in Type II Superconductors

It will now be shown that flux flow noise in superconductors has a spectrum that is constant at low frequencies and varies as l/fat higher frequencies, and that this spectrum comes about by an appropriate superposition of independent random pulses. In a flux flow noise experiment dc current is passed through a type I1 superconducting foil, a magnetic field is directed perpendicular to the face of the foil, and the noise is measured between two probe point contacts made to the foil. In the superconductor the magnetic field breaks up into elementary fluxoids, and due to the flow of current, a Lorentz force acts on the fluxoids, causing them to move across the foil with a uniform velocity v , thereby inducing voltage pulses between the probe contacts. Since the fluxoids are generated at random on the one side of the foil and disappear at random on the other side of the foil, the motion of fluxoids produces a kind of shot noise. It was first assumed that all pulses had the same shape (II), but a more

234

A. VAN DER ZIEL

careful examination showed that the pulse shape depended strongly on the path taken by the fluxoids, so that the noise is a superposition of independent pulses of different shape (12-14). Actually, the problem is somewhat more complicated than that. Usually the fluxoids do not occur as single flux quanta c$o but as bundles of fluxoids of size nc$O.This is caused by the fact that fluxoids can be pinned at pinning centers and are only released when a sufficiently large force is applied. For an infinite foil of width w with probes at a distance 2 4 the spectrum is of the form

{: ( A) +

S ( f ) = S(0) -exp -2n-

[1+

4n;/jx

-27+)]}

(19)

where f, = u/2u and S(0) is the spectral intensity at low frequencies: S(o) = (2nc$0/c)vdc

(20)

Here vd, is the dc voltage generated between the probes, c the velocity of light, and b0 the fluxoid quantum. This equation takes into account that the fluxoids occur in bundles of size nc$o. The calculation applies Carson’s theorem to fluxoids moving through the foil at x and integrates the resulting expression from x = -a to x = +a. We see that the spectrum is constant at low frequencies but varies as l/f at higher frequencies. This agrees with experiment (15, 16). At the higher frequencies Eq. (19) becomes invalid and S ( f ) decreases much faster than l/f because the probes are not “point probes,” as assumed in the derivation of (19), but have a finite size [probe size effect (16)]. Consequently S ( f ) can be integrated for the interval 0 I f I00. A phenomenon that is caused by helium bubbles on the sample due to the boiling of helium, and that results in small temperature fluctuations of the sample, gives rise to spurious noise with a l/f2 spectrum (17).Somewhat confusingly, this is called flicker noise. The importance of this process is that it gives rise to l/f noise as a consequence of the superposition of differently shaped independent pulses.

111. FLICKER NOISE IN VACUUM TUBES

We discuss here 1/f noise in various vacuum tubes, not because these devices are so important in themselves, but because the discussion sheds light on various l/f noise mechanisms.

FLICKER NOISE IN ELECTRONIC DEVICES

235

A. Flicker Noise in Field Emission Diodes

Noise studies in these diodes were performed by Kleint and his coworkers (18), by Timm and van der Ziel(19), and by Shen and van der Ziel (20). Kleint and his co-workers found l/f noise and shot noise; the l/f noise increased strongly with residual gas pressure. Timm and van der Ziel found shot noise and const/fY noise with y N 3 for clean tungsten surfaces ; they attributed this to diffusion noise (see below). Shen and van der Ziel found in some cases y N 1. The current I of a point emitter is of the form

where A and B are constants, F is the field strength at the field emitter, and is the work function. The only parameter that can possibly fluctuate is the work function 4. Hence’

4

But 4 depends on the density n of foreign atoms on the surface. Let the active surface be represented by a circular patch of radius a (the atoms diffuse in and out of this patch), and let ii be the average density of atoms on the surface; then the average number R of atoms on the patch is R = (naz)ii and

But according to Burgess (21) and van Vliet and Chenette (22) a2 ber,(u’/’) kei,(u”’)

SN(f)= 8 var N-

D

+ bei,(u’/’) ker,(u’/’) -U

(24)

where D is the diffusion constant of atoms along the surface, u = oa’/D, and ber,, bei,, ker,, and kei, are Kelvin functions of the first order. Substituting (24) into (23), one finds for the low-frequency limit of the spectrum Sn(o+0) =

-(Pfi/nD)ln(oa’/D)

(244

We assume here for local fluctuations that 6 d/q5 6 1 and 3(Bd1’2/F) 64 6 1 so that the Taylor expansion of (21) is valid. We may then average over the active region of the point and obtain Eq. (22).

236

A. VAN DER ZIEL

and for the high-frequency limit of the spectrum S,(O

8pi

--+

c

co) = nD (oa2/D)3/2

where C is a constant that does not depend on ii, a, and D and is defined by var N = N. The turnover frequency fl occurs at about ln(wa2/D) = 1. Experimentally Timm and van der Ziel found a turnover frequency of the cm2/sec order of 1 Hz This would agree with theory for D = 2 x and a = cm (100 A). The high-frequency form (24b) gives indeed the experimental f - 3 / 2 spectrum. If one defines the equivalent saturated diode current of the noise by equating

SAf) = 2qIeq

(25)

where q is the electron charge, Timm and van der Ziel found various expressions for clean tungsten surfaces. Sometimes I,, could be represented as

I,, = I + (254 for frequencies above 30 Hz.The first term is a shot noise term, the second

term a diffusion noise term. Sometimes, however, the following puzzling expression was found

I,, = ( 1 1 / 2 + A1/2f

- 9 2

(25b)

It is difficult to explain the latter theoretically. For cathodes contaminated by residual gases they found

I,,

=

I

+ B/f

(254

where y has a value slightly above 1 at low current levels, increasing to about $ at higher current levels. When a tungsten point was cleaned by flash-heating and then left in contact with the residual gases (gas pressure N lo-'' torr), the noise increased strongly with time, going through a rather sharp maximum after about 5000 min. When the point was contaminated by barium, the noise was again of the form (2%) with y N 1.2 at a low current and y N 1.4 at a higher current. The noise was about two orders of magnitude larger than for an uncontaminated point. Noise of the form (25c) could be caused by superposition of spectra of the form (24) with different work functions and diffusion constants. This could result from patchy surfaces or from faces on the point with different work functions and diffusion constants. What is not quite clear is how y can now depend on the current.

FLICKER NOISE IN ELECTRONIC DEVICES

237

Separate experiments on clean tungsten surfaces indicated that the noise varied as the square of the current, as expected from Eq. (22). Kleint found that the field emission noise increased rather strongly with increasing temperature. Timm and van der Ziel explained this in terms of the temperature dependence of the diffusion constant D in Eq. (24). Taking Kleint’s data for a Ba-covered surface at 295 and 410°K they obtained an activation energy of the diffusion process slightly above 0.5 eV, which is not an unreasonable value. Kleint assumed that the fluctuation 6N of the number of surface atoms was caused by the random arrival and evaporation of impurity atoms at the emitter. If this process has a time constant z, then

S , ( f ) = 4 var N[z/(l

+ w’z’)]

He then introduced a distribution of time constants g(z)dz that was normalized and finally assumed

as could be expected from a distribution in activation energies. This yields

which varies as l/f for l/z2 4 w 4 l/zl. In view of what was said before, this may be only a formal exercise without physical meaning. Timm and van der Ziel have argued that the diffusion mechanism should be the predominant noise source. The measurements are performed in so short a time interval that uery f a v atoms will leave or arrive at the surface during that time. But all the atoms present on the surface will diffuse over the surface and so contribute to the noise. Finally, the model does not readily explain the l/fy noise spectra with 1 < y < $. According to Eq. (23) the noise would be zero if a+/an = 0. Such an effect has never been found. Apparently then, there is still something missing in both theories.

B. Flicker Noise in Photoemission Work function fluctuations such as encountered in field emission diodes might also occur in photocathodes (23) and give rise to l/f7 noise with l$4$. Let Zp be the photoemission current and 4 the work function of the emitting surface. We may then write, if n is the number of photons arriving

238

A. VAN DER ZIEL

per second, n, the number of photoelectrons emitted per second, and q the quantum efficiency of the photocathode,

I, = qn,

= qnq

Fluctuations An, in n, may occur for three reasons: (a) Fluctuations An in n: If they obey a Poisson process, they give a contribution q2q2 = q2q2ii to and hence a contribution

z2

2q2q2ii

(2W

to the spectral intensity Sr( f ) of AZ,. (b) Spontaneous fluctuations in n, : According to partition noise theory and hence the contribution to they give a contribution q2iiq(1 - q) to S r ( f ) is

A T

2q2W(1(c) Induced fluctuations due to fluctuations A$ in qii(dq/d$) A$, the contribution to Sr(f) is

(28b)

4:

Since AI, =

where S,( f ) is the spectral intensity of A$. Adding all expressions yields

The first term gives the well-known shot noise term of I,, whereas the second term is a flicker noise term. Since it varies as I;, it should not show up at low levels of I,, as, e.g., encountered in photomultipliers. Smit et al. (24) found flicker noise in the dark current of photomultipliers by counting techniques; Young discusses the problem of the occurrence of flicker noise in the photocurrent of these devices (25). C. Flicker Noise in Secondary Emission (23,26)

In secondary emitters the anode current I, may be written as I , = 61,

(30)

where I, is the primary current and 6 the secondary emission factor. Now 6 depends on the work function of the secondary emitter; a fluctuation

FLICKER NOISE IN ELECTRONIC DEVICES

239

A$ in 4 gives rise to a fluctuation A6 in 6 such that A6 = (a6/a4)A+. Consequently, if both I , and 6 fluctuate, AI, = 6 AI,

+I

as A4 a4

-

or

Here the last term represents the secondary emission flicker noise. This effect was observed in a Philips EFP 60 tube, which was a tetrode with one stage of secondary emission multiplication. The experiment was most easily performed by inserting a resistance R, between cathode and ground and HF connecting the screen grid to the cathode. The cathode shot noise, cathode flicker noise, partition noise, and partition flicker noise were then all suppressed by negative feedback, whereas the secondary emission flicker noise, which flows from the secondary emitter to the anode, was not affected. By measuring the noise as a function of the resistance value R, , it could be conclusively determined that secondary emission flicker noise did exist and that it had a l/f' spectrum with y N 1. D. Flicker Noise in Vucuum Diodes and Triodes (27) 1. Schottky's Theory of the Effect

Schottky (2)ascribed the flicker noise observed by Johnson (I)to foreign atoms arriving at and departing from the surface at random. He assumed that these atoms spent an average time zo on the surface and calculated the noise spectrum as follows. A single foreign atom, arriving at a point of the cathode and adsorbed as an ion, produces an electric moment p and causes a change 6' in potential ; the effect will extend nearly uniformly over a circle of area CJ around the , co = 8.85 x lo-'' F/m is the dielectric point, so that 6' = P / C J E ~where constant of free space. Let the current flow be temperature limited (saturated diode) and let Jo be the average current density; then the current density in the area CJ changes by a factor exp(qO/kT,) N (1 + qO/kT,), where T, is the cathode temperature. The atom thus produces a current pulse of amplitude i , = aJoq6'/kT, = FJo,

I; = qpc/kT,e,

(31)

of average duration z. If N foreign ions are located on the surface, then the fluctuating current is i(t) = ( N -

mi,

(32)

We deviate here from Schottky's derivation in order to simplify the calculation. The Wiener-Khintchine theorem was not yet known in 1926.

240

A. VAN DEX ZIEL

where N is the average value of N . Assuming a Poisson distribution for N, we have (N - N)’ = N,and the autocorrelation function is [ ~ ( t-) iTJ[N(t

+ s) - iTJ = Nexp(-s/zo)

(33)

Hence the spectrum, as given by the Wiener-Khintchine theorem, is

sr(f)= 4NFZJ;To/(l 4- 0’7;)

(34)

The spectrum thus varies as J;, in reasonable agreement with Johnson’s experiments for o’z; >> 1. The frequency dependence agreed quite well with Johnson’s data, and for oxide-coated cathodes the agreement was thought to be reasonable, since Johnson’s data gave some indication of leveling off at low frequencies. Later it was found that the spectrum usually did not level off and that it was of the form l/f with y close to unity. This can be explained by a wide distribution in time constants, as discussed earlier.

2. Space-Charge Suppression of Flicker Noise Flicker noise in space-charge-limited vacuum diodes was usually thought of as being caused by fluctuations in the emission current I,. As a consequence

Here SI, is the fluctuation in I,, and 61, the resulting fluctuation in the anode current I,. According to Langmuir’s theory of the diode (28)

aI,/ar, = kT,g/qI,

for I,

+ I,

(354

where T, is again the cathode temperature and g is the small-signal conductance. If the emission current fluctuation is interpreted as a fluctuation 8 4 in the work function (Schottky’s microscopic model is easily transformed into this macroscopic model !), then, since 1 s = AoSTZexp(-qWT,),

S r . ( f ) = (I,/kT,)’S+(f)

(36)

Here A. is Richardson’s constant S in the cathode area and S , ( f ) the spectrum of &$. Then S r , ( f ) = g2S,(f)

(364

or, by introducing an equivalent noise resistance Rn with the help of the definition, 4kT0Rn = S,,(f)/g2 = S,(f)

(36b)

FLICKER NOISE IN ELECTRONIC DEVICES

24 1

where To is room temperature. Measurement of R, as a function off thus gives S , ( f ) directly. We can compare flicker effect and shot effect in the case of space-charge limitation. The shot noise then has a spectrum (29) where 9 = 3(1 - 4 4 ) = 0.644. Since the shot noise in the emission current has a spectrum we see that

Therefore, the space-charge suppression factor for shot noise is 29 aIJaIs, whereas for flicker noise it is (aIJaI,)2. Since usually aIJaI, < 1, flicker noise is much more strongly suppressed by space charge than shot noise, as Schottky (3) already suggested in 1937. 3. Flicker Noise in Tubes with Tungsten Cathodes, Thoriated Tungsten Cathodes, and Thin-Film Cathodes In good vacuum diodes with tungsten cathodes, the noise under saturated conditions is shot noise down to about 30 Hz and varies as P /f below that frequency. In poorer tubes the noise can be improved considerably by flashheating the filament and thereby driving off foreign atoms from the surface. This indicates that the noise is caused by the arrival at and the departure from the cathode surface of foreign atoms. It cannot be caused by a diffusion process, since in contrast with the field emission cathode, where only the tip of the field emitter was active, the whole cathode is an active region, so that foreign atoms cannot diffuse out of the active region but can only evaporate from the active region. In vacuum diodes with thoriated tungsten filaments ( 2 3 , the noise can be quite low and is of the l/fy type with y N 1. Here the noise can be caused by a diffusion mechanism, since the thorium is contained in pores and thorium ions can diffuse over the surface from pore to pore, covering the surface with a near monolayer of thorium ions on tungsten. This problem has not been solved. The same is true for thin Ba-film cathodes (27). Here barium atoms come out of pores and diffuse over the surface as ions, forming a near monolayer of barium on oxygen on tungsten.

242

A. VAN DER ZIEL

Under space-charge-limited conditions diodes with tungsten cathodes show anomalous Bicker effect having a constant zo/(l + W’T;) spectrum. The effect is caused by the emission of bursts of positive ions from pockets of dirt in the filament. These ions are trapped in the space-charge region, raise the potential minimum, and so increase the noise. As a consequence the effect gives rise to pulses of the form (30) Z(t) = 0

for t < to,

i(t) = io exp[-(t - to)/zo]

for t > to (38)

where zo is the lifetime of the ions in the space charge. The effect disappears when an electrode with negative potential with respect to the cathode is inserted in the tube, indicating that the effect is indeed caused by positive ions. This means that the effect is absent in triodes with a negative grid.

4. Flicker Noise in Tubes with Oxide-Coated Cathodes (27, 30a) Lindemann’s experiments (31) on planar diodes and triodes indicate that SI,f) at constant anode current I, is independent of the cathode-grid distance d. This has three important consequences : (a) Other things being equal, tubes with small cathode-grid distance have the lowest noise resistance R, . The reason is simple, since R, is defined as 4kTRn =

~Ilf)/s:

(39)

and g, increases strongly with decreasing d (approximately as l/d4/3 at constant Za). This rule gives a good guide for selecting low-noise tubes. (b) The theory of flicker noise as suppressed emission fluctuations cannot be valid, for in that case R, would be independent of d. Replacing g by gm in (35a) and substituting (35) and (35a) into (39) gives that R, is a constant, independent of d, whereas in fact SIaf)at constant I, is independent of d. This also excludes Schottky’s flicker noise mechanism as a direct cause of the noise. (c) Neither can the noise be generated deep in the coating as resistance noise. For in that case the noise can be represented by an emf e in series with the coating;2 may now depend on I, but not on the cathode-grid distance d. Therefore, since in this model -

e2 = 4kTRn Af

(394

R, at constant Z, should be independent of d, in contrast with experiment. The latter effect does indeed exist. If the cathode nickel contains a few percent of silicon, the silicon reduces part of the coating under the formation of barium, leaving a high-resistance layer of Ba,SiO, behind at the interface

FLICKER NOISE IN ELECTRONIC DEVICES

243

between nickel and coating. The current flowing through this high-resistance layer generates a huge amount of flicker noise (32). What then is the source of the noise? Further experiments by Lindemann (31,33)suggested that the noise was generated in a thin surface layer and that this surface layer contained an appreciable voltage drop. Fluctuations in this voltage drop could be the source of flicker noise. This brings us to a theory of flicker noise proposed by Johnson and van Vliet and backed up by experimental evidence provided by Johnson (34). The basic assumptions of the theory are as follows. 1. A thin surface layer exists with possibly an appreciable voltage drop in that layer. 2. Donor centers (oxygen vacancies) can diffuse through this layer. 3. Barium ions arrive at random at the surface of the grains, resulting in an increase of donor sites in the area adjacent to the surface. Various kinds of spectra can be obtained : (a) Difusion spectra, which are flat at very low frequencies and vary as l/pIz at higher frequencies; they occur if the field in the surface layer is relatively small. (b) Low field spectra, which are somewhat different in shape but also vary as l/f3I2 at higher frequencies; they occur if the field in the surface layer is relatively small. (c) Strong field spectra, which are similar but vary as l/f” at higher frequencies; they occur if the field in the surface layer is sufficiently large. These predictions were verified for particular diodes. Note that none of these spectra show l/fnoise; neither did the experiments. What is not quite clear is how l/p spectra with y N 1 can be explained. Probably a nonuniformity in the thickness of the surface layer can give rise to an appropriate distribution of time constants ; this might explain these spectra. E. Flicker Partition Noise in Positive Grid Triodes and Pentodes

In 1954 Tomlinson (35) observed a l/f component in partition noise in pentodes. Schwantes and van der Ziel further studied the effect both in positive grid triodes (36) and in pentodes (37). In pentodes one can reduce the cathode current fluctuations by negative feedback by inserting a resistance R, in the cathode lead; the true partition noise, which flows from screen grid to anode, is not affected by the feedback (37). In positive grid triodes one finds shot noise, partition noise, cathode flicker noise, and flicker partition noise. Flicker partition noise indicates that the rate at which the current divides

244

A. VAN DER ZIEL

between the positive electrodes is randomly modulated by a l/f-type process. We can look at this as follows. If I,, I,,, and I , are the cathode, anode, and screen grid currents, respectively, and A is the partition parameter, then I, = I,A,

I , = Z,(1

-

A)

(40)

so that for fluctuations in a small frequency interval AJ Ai, = A Ai, -k I, 81,

Ai2 = (1 -

A> Ai, - I, AA

(404

and Ai, Ai, = A(1 - A)

@ - I:

where @ = S,,m AL = SAWAJ and S I E mand SAY>are the spectra of I, and A. Experiments indicate that S A Y )has a l/fspectrum and that hi, and Ai2 are positively correlated; the latter means that A(1 - A) @ > I:

Iv. FLICKER NOISE IN RESISTORS When dc current is passed through a carbon microphone, a carbon resistor, a thin-film metal resistor, or a semiconductor resistor, noise is generated that is proportional to the square of the current and has a l/f-type spectrum (4).When ac current of frequency& is passed through these devices, noise is generated that is proportional to the square of the rms current and that shows two sideband noise spectra around fo with a spectral intensity (IlAfnoise) (38, 39). Since the current is kept constant, varying as l/lfo the only parameter of the system that can fluctuate is the resistance of the device. One can therefore distinguish between contact noise, in which the fluctuating parameter is the contact resistance, and true resistance noise, in which the fluctuating parameter is the bulk resistance. Fluctuations 6R in the resistance R can readily explain the current dependence of the noise. Let I be the dc current; then this produces a fluctuating voltage 6 V = I 6 R , and hence the spectrum is

-fl

=

IZsRY)

(41)

so that SOY)vanes indeed as P , as found experimentally. Small deviations from the quadratic law can be expected when the resistance R depends slightly on the current, e.g., because current heats the sample, or because R is slightly nonlinear. Strong deviations may be expected when the resistance is strongly nonlinear (40). The problem of explaining the spectral dependence of S u mhas now been moved one step back; it now consists in explaining the spectrum of SRY). The response to ac currents is also easily understood. Let I . cos mot be

245

FLICKER NOISE IN ELECTRONIC DEVICES

the ac current and let 6R(t)be represented by a series representation

c m

6R(t) = -

a, COS(U,,~

n=O

+ 4")

where %a,* is proportional to l/fn; then W ( t )= I , cos o o t

c m

=$Io

c ancos(ont+ 4") m

n=O

{a,, cos[(oo

n=O

+ on)t+ 4,,] + a,,cos[(wo - on)t-

(42a)

so that two noise sidebands of frequencies fo k fn and carrier frequency = w0/2x are generated having a spectrum varying as l/lfo - f).

fo

A . Fundamental Experiments

1. Clarke and Voss's Fundamental Experiments (41-43)

Clarke and Voss have demonstrated that resistance fluctuations were indeed responsible for the observed noise. They did this by detecting such resistance fluctuations at zero current. The circuit used is the RC circuit of Fig. 1 and C is so chosen that C R = z has a certain value. The amplified noise is now filtered by a square filter of passband fl - fo so that noise over a frequency band fo < f < fl is passed; usually f, % fo . The filtered signal is then squared. The output of the squaring device may now be written as

Here the first term represents the subensemble average of P(t) for a given T and a given R, and the second term represents the residual fluctuation in each element of the subensemble. If now T and (or) R slowly fluctuates with time, and P(t) is Fourier analyzed for very low frequencies, the contribution of Po(t)becomes negligibly small. So for 2nfz 4 1 we see the slow fluctuations in the first term only.

R

==c

--

FILTER fo< f < f,

-

AMPL

2

SQUARING DEVICE

'('I

LOW FREQUENCY FOURIER ANALYZER

-

FIG.1. Schematic diagram for the experiment of Clarke and Voss.

-

246

A. VAN DER ZIEL

We now have three possibilities. (a) 4n2f:z2 g 1 (bandpass completely below the knee): P(t) = 4kTR

jf:

df = 4kTRCf, - fo)

(44)

If T and R both fluctuate (fluctuations AT and AR, respectively) and the resistance R has a temperature coefficient B = (1/R) dR/dT, then if Ro and To are equilibrium values, A(TR) = Ro(l

+ BT0)AT + ToAR

or

A(TN (1 TOR0

AR + BTo) AT -+ To Ro

so that

(b) 4nzfiz2 3 1 (bandpass completely above knee):

or

so that

) ~f: (passband straddles knee): (c) f2, g 1 / ( 2 n ~4

In cases (a) and (b) one obtains a response from S,(f), as well as from S R ( f ) , whereas in case (c) one only obtains a response from S,(f). Clarke

FLICKER NOISE IN ELECTRONIC DEVICES

247

and Voss measured a 20-Mn InSb bridge and found a l / f noise spectrum. Four methods of measurement were used: ( 1 ) Measurement by dc technique. (2) Measurement by ac square-wave bias and subsequent phase-sensitive detection to measure the noise sidebands of the carrier frequency fo. (3) Measurement by positive pulse current bias to reduce heating. (4) By thermal noise measurement [case (b)].

In all cases the same value of SR(f)/R2was found. In a separate calculation it was shown that the noise in these bridges was a factor lo3 larger than predicted by a thermal fluctuation noise theory (see below). Therefore the noise is indeed resistance noise. A similar conclusion was reached by Beck and Spruit (43). 2. Hooge's Fundamental Formula

Hooge and his co-workers (44-53) have shown that for many thin-film resistors made from different materials, and also for many semiconductor resistors, SR(f)/Rzmay be written as SR(f

)/R2= a / N f

(47)

where a is a universal dimensionless constant that is only a slow function of the temperature T and has a value of about 2 x N is the total number of carriers in the resistor, and f is the frequency. This fundamental formula provided the first breakthrough in the characterization and understanding of flicker noise. It is usually interpreted as a noise mechanism due to a volume effect. It also holds for point contacts (54-59). There is no difference to speak of between a for solid and molten gallium (60). Clarke and Voss (42,42) observed, however, that thin manganin films, which had a practically zero temperature coefficient of the resistance, did not show a measurable l/f noise; that is, a is much smaller in that case. They felt that this indicated that the l / f noise could be caused by spontaneous temperature fluctuations of the resistor. This is easily seen as follows. Let R be the resistance of the sample at the temperature T, and let 6T be a spontaneous fluctuation in temperature ; then, since R = f ( T ) , dR 6R=-aT, dT

6R R

-=--

1 dR 6T = /36T RdT

where /3 = (1/R)dR/dT is the temperature coefficient of the resistance of the material. Consequently SR(f)/RZ= B"S,(f)

(484

248

A. VAN DER ZIEL

where S,( f)is the spectral intensity of the temperature fluctuations. Hence for small p, SR(f ) / R Zshould be negligible if the noise was caused by spontaneous temperature fluctuations. Any other mechanism would not give zero noise in manganin. Clarke and Voss (42) also observed that for thin bismuth films the relative noise spectrum was about the same as for thin metal films, despite the fact that the carrier density in bismuth was about a factor lo5 smaller than in metals. According to them, this indicates that the noise was inversely proportional to the volume of the resistor rather than to the number of carriers. This can readily be explained by the temperature fluctuation mechanism, since S,( f)should be inversely proportional to the heat capacity C, (see next section). Hooge’s formula (47) is valid for those metal films for which the conductivity is close to the conductivity of the bulk metal. For very thin films the electrical conduction is probably practically via a hopping process ; in that case the films exhibit much more noise than expected from Eq. (47) (61). The discussion of flicker noise in semiconductor materials is given in Section IV,C. One can now ask how resistance fluctuations can occur. The resistance R of a sample of length L is given by (49)

R = Lz/qpN

where p is the carrier mobility and N the number of carriers. Fluctuations may thus occur either in N or in p. (a) Fluctuations in N : In this case dR/R = - d N / N ,

SR( f ) / R 2 =

sN(f)/N2

(494

s,(f )/pz

(49b)

(b) Fluctuations in p : In this case dR/R = -sp//cl,

SR(f

)IR2

=

Fluctuations in N would be compatible with Eq. (47) if S,( f)were proportional to N . However, in that case there would be no guarantee that a would be a universal constant. We come back to that problem in the discussion of flicker noise in semiconductors. Fluctuations in p would not be so easily compatible with Eq. (47) unless S,( f)/p2 were independent of p and proportional to 1/N. There is no good general argument why this should be the case. Nevertheless, Hooge, for noise in ionic cells (44, and Kleinpenning, for noise in the thermoelectric emf of intrinsic and extrinsic semiconductors (53),both conclude that the

FLICKER NOISE IN ELECTRONIC DEVICES

249

noise arises from mobility fluctuations, and they therefore postulated S,(f)/P2 = d f N

(49c)

We come back to this problem in Section IV,C. Fluctuations in mobility would be easily understandable if the primary fluctuation were a spontaneous fluctuation in temperature. Consider, for example, the case that p = CTY

(50)

Then Sp/p = yST/T and hence

where y N 1 for metals and y that in the next section.

1:

3 for semiconductors. We come back to

3. Clarke and Voss’s Temperature Fluctuation Mechanism (41,42)

It has sometimes been tried to invoke diffusion mechanisms as a source of l/fy noise (62-64). This diffusion may either be particle diffusion or heat diffusion. For example a three-dimensional particle flow out of a small active sphere into a much larger inactive region gives a constant spectrum at low frequencies, and a l/f3I2 spectrum at high frequencies. For other three-dimensional geometries there might be an intermediate region with a l/f spectrum For example, in a thin film of width w and length L the characteristic time constants are z1 = w 2 / 2 n and z2 = c/2D,where D is the diffusion constant. Clarke and Voss (41,42) have shown that there will then be a region 1/z2 < w < l/z, for which the spectrum is about l/f; for w L this can be quite a wide region. Now the question is “What is diffusing?” Carrier diffusion is out of the question, since the time constants are far too short. Diffusion of heat would be much more likely, since it gives time constants that are the right order of magnitude. But it should also be taken into account that the films are usually deposited on a substrate. This means that there is not only heat exchange with the other parts of the film but also with the substrate. This will have the tendency of changing the temperature fluctuations. In addition, it will give rise to much longer time constants, which, in turn, will extend the l/f spectrum to lower frequencies. It might therefore be perfectly feasible that the temperature fluctuation spectrum could be approximated by the piecewise linear graph on a logar-

+

250

A. VAN DER ZIEL

ithmic scale.

In that case

where k is Boltzmann’s constant and C, is the effective heat capacity of the sample. Consequently, for the intermediate frequency region Clarke and Voss (41,42) propose a temperature fluctuation spectrum of the form

For a free-bearing thin film this can be modified to

since f2/f1 = (L/w)’. Clarke and Hsiang (65) found excellent agreement with experiment for the l/f noise of free-bearing tin and lead films at the superconducting transition. Ketchen and Clarke (66) found equally good agreement between theory and experiment and even were able to detect the flattening of the spectrum at low frequencies and the change in slope at higher frequencies expected from Eq. (51). Table I sums up the results obtained by Voss and Clarke (41)for various TABLE I MEASURED b, MEASUREDS , ( f ) / V 2 ,AND CALCULATED S , ( f ) / V 2 FOR VARIOUSMATERIALS AT 10 Hz

Material cu Ag Au Sn Bi Manganin

Measured

B

0.0038 0.0035 0.0012 0.0036 -0.0029

l p ~ <10-4

Measured

v2

S”(f)/

(Hz-’)

6.4 10-14 6.4 x 0.6 x 7.7 x 13 x <7 x 10-19

Calculated S”(f)/vz (Hz-’) 16 x 2x 0.76 x 7.7 x 9.3 x <3.5 x

10-19

FLICKER NOISE IN ELECTRONIC DEVICES

251

materials. The agreement is quite good, indicating that spontaneous temperature fluctuations give a plausible explanation of l/f noise. However, in InSb layers the measured noise was several orders of magnitude larger than that calculated from (521 indicating that in this case the l/f noise was of a different origin. If the noise is due to spontaneous temperature fluctuations, there should be a correlation between the noise in different parts of the film at low frequencies. Voss and Clarke (42) and Clark and Hsiang (65) demonstrated that this was the case with the help of the arrangement of Fig. 2. Let Vl(t) and V&) be the two noise sources; then the arrangement allows for rneasuring the spectra S+(f) of Vl(t) V,(t) and S-(f) and K ( t ) - Vz(t).The correlation coefficient c between Vl(t) may then be expressed as

+

Good agreement was obtained, indicating that the heat coupling between the films (1) and (2) was the cause of the correlation. However, in InSb layers no such correlation could be observed, indicating that spontaneous temperature fluctuations are not the cause of the l/f noise in this material. For a further theoretical discussion of temperature fluctuation noise see Kleinpenning (67).

POP-II

.j

so

~

0.25 0

0.25

f (Hz)

( b)

FIG.2. (a) Experimental configuration for correlation measurement; (b) fractional correlation for two samples (R. F. Voss and J. Clarke, Phys. Rev. E 13,556 (1976)).

252

A. VAN DER ZIEL

We shall now demonstrate that Eq. (52) is formally equivalent to Hooge’s equation (47)and that it yields the right order of magnitude for Hooge’s constant a. To that end we observe that C, is proportional to the effective number N, of atoms in the sample. According to Dulong and Petit’s law, the heat content Q = 3N,kTy and hence C, = 3N,k, since each atom has 6 degrees of freedom. Therefore T Z

and hence, according to Eq. (50a)

where n is the number of conduction electrons donated per atom. This is of the form (47). For metal films y N 1 ; taking fz/fl = lo5 and n = 1 yields a = 25 x which is the right order of magnitude. Therefore Clarke and Voss’s temperature fluctuation mechanism is compatible with Hooge’s formula. Improved agreement would have been obtained if we had taken n somewhat smaller. For example one would expect n < 1 in Ag, Au, and Cu. More important, due to the effect of the substrate, N, might be larger than the number of atoms in the film, and that would also result in R lower n (68). The latter possibility finds support from measurements discussed in the next section. Since spontaneous temperature fluctuations give rise to fluctuations in the mobility p, the temperature fluctuation noise mechanism is compatible with the mobility fluctuation mechanism, and the latter finds here a natural explanation. One can now also understand why bismuth (41,42)behaved anomalously, in that it had a value for the parameter a that was several orders of magnitude According to Clarke and Voss, their bismuth film smaller than 2 x had N = lO-’N,; nevertheless, the noise was of the same order of magnitude as for the other metal films. This is understandable, since the first half of Eq. (55) should be valid and hence a should be quite small because n is so small. Since N, is proportional to V, one can now see (41, 42) why S,y)is proportional to 1/ V rather than to 1/N. 4. Experiments by the Chicago Group (69-72)

Dutta, Eberhard, and Horn (69) measured l/fnoise in copper whiskers (diameter = 3 pm) ranging in length from 0.15 to 1.20 cm. They found the noise to be proportional to the square of the applied dc voltage, and the noise

KICKER NOISE IN ELECTRONIC DEVICES

253

had a power spectrum of the form constlj”, with y = 1.05 L- 0.05. But the noise power was lo2 to lo3 times larger than predicted by Hooge’s formula (47). They did not find any low-frequency turnover, even for the shortest samples, as would have been expected from Clarke and Voss’s theory (42, 42). While the temperature fluctuation should be somewhat larger than for a thin film on a substrate, they find it hard to explain the large difference in noise magnitude. They also found a rather strong temperature dependence. Gold contact wires were connected to the whisker by silver conducting paint, and the driving dc current was about 20 mA. Since silver paint contacts are sometimes noisy, it was deemed necessary to measure the noise as a function of the sample length. They found indeed that S v y ) / V 2 varied as 1/N as expected from Hooge’s and from Clarke and Voss’s formulas. Eberhard and Horn (70- 72) measured the temperature dependence of l/f noise in silver, gold, copper, and nickel films on sapphire substrates having conductivities comparable to those in bulk material. In contrast with Hooge’s and Clarke and Voss’s formulas, they found a rather strong temperature dependence of S,cf)/V2. They expressed this dependence over a wide temperature range by

S,V)N/ V 2 N A

+ A’ exp( - E,/kT)

(56)

where A and A‘ are constants and E, is a kind of activation energy, which is 1750°K for Ag, 1400°K for Au, and 1250°K for Cu. The noise showed a peak at a higher temperature (400°K in Cu, 500°K in Ag, no peak in Au below 500”K,no primary peak in Ni but a slight secondary peak at the Curie temperature of 625°K). According to these authors there are two types of noise : Type A : Noise that is weakly temperature dependent and strongly substrate dependent, possibly caused by the temperature fluctuation mechanism proposed by Clarke and Voss. Type B: Noise that is strongly temperature dependent and only very weakly substrate dependent, whose origin is unknown. The free-bearing copper whiskers had noise of this type.

These statements are demonstrated in Figs. 3 and 4. Figure 3 shows S;cf)N/V2 at 20 Hz for 800-A Ag films on sapphire and on fused silica substrates as functions of temperature. Also shown is the temperature fluctuation noise deduced from the Clarke and Voss formula, which is found from (52a), and the temperature fluctuation noise as expected from measured temperature fluctuations on a fused silica substrate and on a sapphire substrate (72). There is limited agreement at low temperatures for a fused silica substrate, but the temperature fluctuation noise for the sapphire substrate was too low to show up in the graphs. Figure 4 shows similar data

254

A. VAN D W ZIEL

g f l

200

n-x

I

__--

100

200

300

- -I

400

I

500

T (K)

FIG.3. (i) Voltage noise S,(20)N/V2 versus temperature for 800-A Ag films on sapphire (open circles) and fused silica (open triangles) substrates. Error bars do not reflect calibration errors due to uncertainty of N. (ii) Temperature-fluctuation-induced noise Su,,,(20)N/ V2 versus temperature from direct measurement of temperature fluctuations in 800-A Ag films for sapphire (filled circles) and fused silica (filled triangles) substrates according to Dutta et al. (J. B. Eberhard and P. M. Horn, to be published).

for 800-A Cu films; here the measured noise shows reasonable agreement with the measured temperature fluctuations at low temperatures, both for the fused silica substrate and for the sapphire substrate. The “measured” temperature fluctuation noise is decreased from the Clarke-Voss formula by about a factor of 8 for metals on fused silica and by a factor of 200 for metals on sapphire, independent of the metal. This suggests that the substrate lowers the spontaneous temperature fluctuations, which is not unreasonable, and that the difference between substrates is due to differences in the heat properties of the substrates (68). The fact that the Clarke-Voss formula agrees reasonably well at room temperature is probably accidental, since the measured and predicted temperature dependences vary widely (see Figs. 3 and 4). Eberhard and Horn (71) speculate about the temperature-dependent l/fnoise that is independent of the substrate. They assume primary spectra of the form z/(l + 02z2) with a distribution g(z) dz, as introduced in Eq.

FLICKER NOISE IN ELECTRONIC DEVICES

100

300

200

400

255

500

T (K)

FIG. 4. (i) Voltage noise Su(20)N/V2versus temperature for 800-A Cu films on sapphire (open circles) and fused silica (open triangles) substrates. Errors do not reflect calibration errors due to uncertainty in N. (ii)Temperature-fluctuation-induced noise Su,,,(20)N/ V 2 versus temperature from direct measurement of temperature fluctuations in 800-A Cu films for sapphire (filled circles) and fused silica (filled triangles) substrates according to Dutta et al. (From J. W. Eberhard and P. M. Horn, to be published).

(6), which is caused by a uniform distribution in activation energies. They assume that vacancies are created at the surface or at any crystal deformation, with the equilibrium number of vacancies given by n

=

const exp( -E , / k T )

(57)

where E, is the energy required to create a vacancy. Once created, these vacancies diffuse through the crystal in a process characterized by a diffusion constant D given by

D

-ED/kT] (58) where ED is the energy barrier presented to the diffusing vacancy. If ED is uniformly distributed in energy and 0.15 5 ED 5 1.1 eV, the observed l/f spectrum results. The temperature dependence of the magnitude of the noise would then be governed by E,. The measured values of E, are a factor 8 smaller than in bulk material, but since they increase with increasing film thickness, this mechanism cannot be ruled out. Eberhard and Horn also investigated the voltage dependence of the noise. They corrected for the heating effects by measuring the resistance R versus T a t zero current; then they measured R at the driving current and deduced the device temperature from it. Strange enough, they found that S u m ,even = D,[exp(

256

A. VAN DER ZIEL

after this correction, varied as Vo with b > 2 (typically p = 2.26 k 0.13). They offer no satisfactory explanation for this result. The important conclusions to be drawn from these experiments are the existence of two noise mechanisms: type A and type B, and the effect of the substrate on the noise of type A but not on the noise of type B. B. Noise in Metal-Metal and Semiconductor-Semiconductor Contacts

The work on this problem was mainly performed by Hooge’s group

(54-59). A very good summary is given in Vandamme’s Ph.D. thesis (73).

The resistance and the noise have been calculated for the contact resistance R of a half constriction (point contact of radius a on conducting metal plate) and for a symmetric constriction of radius a. They found in the first case R = p/2na, and in the second case

S,y)

=

ap2/40n3na5f

(59)

R = plna, S R y ) = ap2/20n3na5f (60) Here p is the resistivity, n is the carrier density, and a is Hooge’s constant. These formulas were derived under the assumption that Eq. (47) S R y ) / R 2 = a/Nf

is valid for any spherical sheet between two equipotential surfaces a distance dx apart. This leads to a resistance dR between the surface and a noise SdRO

dR = p dx/2nx2,

SdRy)= up2 d ~ l ( 2 n x ) ~ n f

(61)

from which Eqs. (59) and (60) follow by simple integration. It is common practice to eliminate a, which is not so easily determined, , by expressing a in terms of the total resistance R, evaluating S R ( f ) / R 2 and comparing it with the experimental data. This yields for the case of Eq. (60)

Here R was varied by varying the pressure with which the two pieces were pressed together. Reasonable agreement was obtained, indicating that Eq. (47) forms a reasonable basis for noise in point contacts. One should, of course, not expect perfect agreement, for a point contact is not as stable a configuration as a metal thin-film resistor. Moreover, the surface may be covered with a thin oxide layer and that can have an in-

257

FLICKER NOISE IN ELECTRONIC DEVICES

fluence. For example, Vandamme finds that manganin point contacts satisfied Eq. (62), whereas Clarke and Voss found no measurable flicker noise in thin manganin films. It could be that the noise in Vandamme’s case was either due to an oxide film or due to the fact that his manganin material showed noise of type B (in the notation of the Chicago group). This could be decided by an experiment of the type described below. Equations (59) and (60) also hold for semiconductor-semiconductor contacts (73,74). Here it is especially important to take into account the effect of a possible oxide film between the materials. To that end a parameter 4 W a s introduced : 4 = R f i d R c o n s t r i c t i o n . Since Rconstriction = Pbuik/na, and Rfilm= tpfilm/na2,where pfih is the resistivity of the film, t the thickness of the film, and a the contact radius,

4

=

tPfilm/aPbulk

(63)

Generally t 4 a and the total resistance R is

+

= Pbulk(l

+ 4)/na = P b u l d n a p

where a,, = a/(l 4) is the “apparent” contact radius (ap c a). For the parameter C is

(634

4

=0

fMf) - m 2 R 3 C=-fS”(f) =-VZ RZ 20nbulk PkIk

as follows directly from Eq. (62). For film + oxide, Vandamme finds

Here nfilmand pfilmrefer to the carrier density and the resistivity of the oxide film. It is assumed that the oxide satisfies the relation

CIf

(65) where Nfilmis the effective number of carriers in the oxide film. If the resistance and the noise is film dominated, then since Rfilmis proportional to l/Nfilm, the constant C is proportional to R. These predictions were well varified by experiments on Ge contacts (73, 74) and InSb contacts at 77 and 300°K (74, 75). In particular, the factor C was found to be proportional to the contact resistance when the properties of the contacts were film dominated. To lower the flicker noise in carbon resistors, which consist of carbon grains with the resistance mainly in the contacts between the grains, one can put n resistors R in series and n strings of such resistors in parallel. The total SuY)IV2 =

aiflfilm

=

258

A. VAN DER ZIEL

resistance is then the same, but the noise has been reduced as l/n2, since the volume V of the resistor is n2 times as large. To prove this, consider that a total current I passes through the sample. The current in each string is then l/n times as large, and hence S,y) for each string is l/n times the Soy) as for the single-resistor case. Since the resistance of each string is nR, S r y ) for a single string is l/n3 times the Sly) for the single-resistor case. But there are n such strings, and hence the total value of S r y ) is l/n2 times the value for the single-resistor case; since R has not changed, Soy) varies as l/n2 also. If we shrink all the dimensions of the grains, including the contact radii, by a factor p while leaving the total volume V intact, then there are p 2 as many contact strings and p times as many contacts per string. Then the total resistance R has not changed, and the noise for a given current I has not either. For the current through each string is p 2 as small, and hence S,y) for each contact is p s / p 4 = p times as large, so that Soy) for each string is p 2 times as large. But since the resistance per string is p 2 times as large, S,y) for each string is p 2 / p 4 = lip2 times as large. Since there are p 2 as many strings, the total S,y) has not changed; neither has the total S,y). For a study of l/f noise in HgCdTe devices with current-carrying nonohmic contacts, see Hanafi and van der Ziel(40).

C. Flicker Noise in Semiconductors 1. Validity of Eq. (47) According to Hooge’s compilation of the literature ( 4 3 , Eq. (47) is also correct for semiconductors with CI of the order of 2 x This seems to qualify it as a bulk effect. One has to be somewhat careful about this conclusion. On the basis of a suggestion by Klaassen (76), van der Ziel (77) showed that Eq. (47) could also be derived from the McWhorter model (78), that is, from a surfacecontrolled density fluctuation model. The basic assumptions are

~

1. The electrons interact with oxide traps via surface states with a time constant z. 2. There is a distribution g(z) dz in time constants of the form (6). 3. If 6N is the fluctuation in the number N of carriers in the sample, then 6N2 = PN, where P is proportional to the surface-state density at the interface between semiconductor and oxide. In that case one obtains Eq. (47) (see next section), with = P/ln(~Z/zl)

(66)

FLICKER NOISE IN ELECTRONIC DEVICES

259

For samples in which no special measures are taken, it is quite possible that a lies in the range 1-10 x in fair agreement with CI 2: 2 x 10-j. Hanafi and van der Zief (79), for example, found that for many samples of CdHgTe the value of a lay between 2-9 x I0-j. Those samples that had undergone a surface treatment, however, had values of a that were more than a factor of I0 smaller (1 -2 x 1 0-4) and Broudy (80) even produced a sample that had CI N This speaks in favor of a surface-controlled density fluctuation mechanism. For silicon and germanium samples no such measurements have been reported, but McWhorter’s model was introduced to explain his noise data in germanium filaments; in those filaments he found sound evidence for the existence of a wide distribution in surface time constants (see Section IV,C,2). Moreover, the measurements of flicker noise in p-n junctions and in transistors (Section V) strongly suggest that the noise is caused by fluctuations in the surface recombination velocity s, which parameter is directly proportional to the surface-state density N,, of the semiconductor-oxide interface. Finally, noise measurements in MOSFETs strongly suggest that the noise is proportional to the surface-state density N,, (Section V). All this evidence points again to a surface-controlled density fluctuation mechanism. Nevertheless, there is also strong evidence that the noise is caused by a mobility fluctuation mechanism, in which the mobility in subbands of width AE and Energy E fluctuates independently. We come back to that problem in Section IV-C3. Hanafi and van der Ziel (40, 81) made two current-carrying contacts 1 and 2 and several noise-measuring contacts a, b, c, . . . on a CdHgTe sample and measured the l/f noise between the contact pairs ab, bc. and ac. If S&Cf),s b , y ) , and S,,Cf) are the spectra, they found (67) s ~ ~ C f=) s a b Y ) -k s b c C f ) This remained true even if contact b was a noisy contact when carrying current (40). In other words the noise voltages generated in the adjacent regions ab and bc were independent, in agreement with the Clarke-Voss experiments on InSb (41,42). The same was true if one of the regions ab or bc contained a grain boundary, but the noise in the grain boundary region was about 4 times larger than in the region not containing a grain boundary (81). We can, therefore, draw the conclusion that grain boundaries are a source of llfnoise. This is understandable, for a grain boundary is a conglomeration of dislocation lines containing l/f generating centers with a distribution g(z) dz in time constants of the form (6). Bess (82) has developed a detailed theory for the effect. It should be noted that Montgomery found a different effect (83). He

260

A. VAN DER ZIEL

measured llfnoise generated between adjacent contact pairs ab and bc on germanium and found correlation between the noise in sections ab and bc; in other words, Eq. (67) was not valid. This effect can be explained by assuming that the surface states not only interact with free electrons and with oxide traps, but also generate hole-electron pairs (see next section). The carriers trapped in the oxide then randomly modulate the hole-electron pair emission of the surface states ; this random modulation contributes to the l / f noise. The randomly generated hole-electron pairs were swept through the device by the current and so produced the correlation. Apparently this effect did not take place in the CdHgTe samples. The most clear-cut indication that Eq. (47) was not valid for HgCdTe came from a field effect experiment (79). When providing the surface of a thin HgCdTe sample, epoxied onto a germanium substrate, with an insulating layer and a field plate and then applying a variable dc voltage to the field plate, one can bring the HgCdTe surface from accumulation to strong inversion. The sample was so thin that the inversion layer reduced the effective height of the conducting channel and so increased the resistance R of the sample by about 25%. According to Eq. (47), S,(f) is proportional to R3 and hence should change by about a factor of 2 when going from accumulation to strong inversion. Hanafi and van der Ziel (79) observed, however, that S,(f) was independent of the dc bias on the field plate. This rules out the validity of Eq. (47). Unfortunately, not enough was known about the sample to decide whether a surface effect could fully explain the observations. 2. Mc Whorter’s Surface Model (78,84) McWhorter’s surface model was developed for germanium. Figure 5a shows the energy band structure of a weakly n-type germanium sample and a weaklyp-type inversion layer caused by interface states at or near the semiconductor-oxide interface. There are two kinds of states: fast states at the interface and slow states 0-40 A inside the oxide. A metal layer was deposited on the oxide and an ac voltage was applied between metal and semiconductor; this modulates the conductivity of the sample. Experiments indicated that the conductivity modulation response could be approximated as S,(w) = a In bw

for w 4 wmaXr

S,(w) = S,,

for w % w,,,

(68) This can be explained as follows. Consider first that there is only one type of slow surface state with a time constant z. These surface states store charge and thus can be represented by a capacitance C,/cm2; the excess charge leaks away and this can be represented by a resistance R,/cm2 such that z = C,R, (Fig. 5b). The ac charge SQ, on the capacitance C,,due to the

FLICKER NOISE IN ELECTRONIC DEVICES

26 1

(b)

FIG.5 . (a) Energy band structure at the surface of a semiconductor; (b) equivalent circuit of the field effect.

application of the ac voltage 6 V between metal and interface via the oxide capacitance C/cm2, can for C, B C be written as SQ, = C,SV, = CSV

jwCsR, = C 6 V - j w z 1 joC,R, 1 jwz

+

+

The charge SQs induces an excess charge -SQs in the germanium near the interface, and this gives rise to a sample admittance 6Y = ps,6Q,E/6VL, where is the surface mobility, which is a function of the surface voltage depicted in Fig. 5a; E is the field strength parallel to the surface and L is the device length. Therefore

SY

jwz jwz

= const ___

1

+

262

A. VAN DER ZIEL

This does not resemble (68) at all. Agreement with (68) is obtained by introducing a distribution of time constants : dz/z g(7) d z = ___ W,/z,)

for z1 < z < z,

g(z)dz

=0

This yields 6Y = const

r2

j o z g ( z ) d z = -In( const

(1 + joz) ln(z,/zl)

+

otherwise (71)

)

1 jwz, 1 + jwz,

(72)

or

6Y = const

for ozl% 1

(7W

so that om, = l/zl. But that is exactly the distribution that is needed to produce a noise spectrum with a frequency dependence of the form constlf. McWhorter assumed that the distribution (71) in time constants, needed to explain l/f noise, was due to tunneling from the surface to oxide traps at a depth y. Assuming a uniform distribution in traps for 0 < y < y l , this yields B(Y) dY = dY/Yl

for 0 < Y < Y l

dY)dY = 0

otherwise

(73)

But z = zo exp(ay)

(734

where a N 10’ cm-l; this yields immediately Eq. (71), and hence, if we put 6N2 = PN

This agrees with Eq. (47) with a = fi/[ln(z,/z,)], as stated earlier. McWhorter introduced the following refinement into the theory. If the carrier is trapped by an oxide trap, the surface recombination centers adjacent to the trap begin to emit hole-electron pairs; this increases the conductance, whereas the trapping of an electron reduces the conductance. If M hole-electron pairs are emitted, the noise must be multiplied by the factor [-pn M(pp pn)l2/pi. The effect is usually ignored in other

+

+

FLICKER NOISE IN ELECTRONIC DEVICES

263

semiconductors, but it explains the correlation found by Montgomery for l/f noise in his germanium samples.

3. 'Ihe Case for Mobility Fluctuations The assumption of mobility fluctuations as an explanation of l/f noise was first introduced by Hooge (44)in order to explain a peculiar result found by Hooge and Gaal (85) in electrolytic concentration cells. They measured S,(f)/G2 in electrolytic resistors, where G represents the conductance of the electrolytic device, and they determined S,(f)/V', where V is the terminal voltage, for open-circuit noise in electrolytic concentration cells, and they found

where c2 is the lowest concentration of the cell. Hooge showed that the latter relationship could not be explained by fully correlated ion density fluctuations (that is, full correlation for the positive and negative ion density fluctuations, or 6n = 6 p ) ; such a correlation would be expected from space-charge neutrality considerations. However, if the mobilities of the positive and negative ions fluctuated independently so that Eq. (49c) was satisfied for each, then Eq. (75) could be explained. Van der Ziel (86) has shown, however, that this relationship could also be proved if the density fluctuations of the positive and negative ions were partially correlated. The space-charge neutrality argument can here be overcome if the ions are temporarily adsorbed at the walls, each immobilized positive ion surrounding itself with mobile negative ions, and vice versa. The experiments seem to require that positive and negative ions be adsorbed in bunches. Hence density fluctuations are still a viable option for explaining Hooge and Gaal's results. In semiconductor resistors the situation is similar. Equation (49c) shows how Hooge's formula (47) can be explained in terms of mobility fluctuations. But Section V,C,2 shows how the McWhorter theory can explain the same data. Again this does not allow the conclusion that one or the other explanation must be excluded. The situation is different, however, in thermoelectric cells. Experiments by Brophy (87) showed that the thermoelectric emf showed 1/f noise. This was interpreted as follows. The thermoelectric emf per degree dB/dT, set up by a temperature gradient in the semiconductor, is

264

A. VAN DER ZIEL

when lattice scattering predominates. Here AE, is the distance between the Fermi level and the band edge, T is the absolute temperature, k is Boltzmann's constant, and q is the electron charge; the plus sign holds for n-type and the minus sign for p-type material. Fluctuations in the carrier density would show up as fluctuations in the Fermi level (or in AE,), and this shows up as thermoelectric noise. Kleinpenning (53)repeated the experiments and found a somewhat different result. He found that the current noise and the thermoelectric noise in thermoelectric cells were practically uncorrelated, as required by the mobility fluctuation model, whereas according to the carrier density fluctuation hypothesis they should be fully correlated. In addition, the thermoelectric noise data in intrinsic materials could not be explained by carrier density fluctuations. In this case mobility fluctuations seem to offer the only explanation. But now the hypothesis had to be extended. The conduction band had to be divided up into a large number of subbands, and in each subband the mobility fluctuated independently such that where c1 N 2 x and the subscript s refers to the subband. This does not involve fluctuations in E,. A similar situation occurred in single-injection space-charge-limited solid-state diodes (88).At low voltages the characteristic is linear (ohmic regime) and at high voltages it is quadratic (space-charge regime). For plane-parallel geometry the mobility fluctuation theory gives

Sdf) = a V 2 / N f ,

I = (aA/L)V

(78)

for the ohmic regime, whereas for the space-charge regime aqLV

Wf)= S A E E O f Y

9 AV2 I = - &&& 8 L3

(79)

Here a N 2 x N is the number of carriers for low voltage, o the lowvoltage conductance, L the device length, A the cross-sectional area, V the applied voltage, q the electron charge, E~ = 8.85 x F/m, and the relative dielectric constant. Under the assumption of density fluctuations, similar noise expressions were found, but the breakpoint in the noise curve occurred at a different voltage than for mobility fluctuation noise. Here again the latter assumption seems to be required. Equations (78) and (79) seem to agree reasonably well with experiment. We now turn to the problem of l/f Hall noise. Brophy (89), Brophy and Rostoker (90), and Bess (92) found that the open-circuit Hall voltage showed l/f noise. This was easily explained in terms of carrier density

FLICKER NOISE IN ELECTRONIC DEVICES

265

fluctuations. If a current I flows in the X direction and a magnetic field B, is applied in the 2 direction, then the Hall voltage across terminals in the Y direction is

for an n-type bar of carrier density n and dimensions w,, w,,, w,. The noise is thus generated by majority carriers, except in intrinsic semiconductors where both types of carriers contribute. In the latter case the experimental data fitted best with the idea of fully correlated hole and electron density fluctuations ; i.e.

An/n = -ApJp

(81)

This would be expected for slow fluctuations where equilibrium is maintained at all times so that p n = n’ = const. However, in the frequency region where generation-recombination noise predominated, they found that the Hall noise was due to correlated carrier density fluctuations with An = Ap. Vaes and Kleinpenning (92)and Kleinpenning and Bell (93) have shown, however, that mobility fluctuations can explain the Hall noise equally well. Hence Hall experiments cannot discriminate between the mobility and carrier density fluctuation theories. Kleinpenning (88) has summed up the mobility fluctuation theory in the following formula for the conduction fluctuation spectrum :

Here CI is Hooge’s constant, E the kinetic energy of the carriers, n k , E ) the density of carriers of energy E at the position r, and ak, E ) the conductivity of carriers of energy E at the position z. The delta function Sk - r’) indicates that fluctuations at r and r’ are independent, whereas the delta function 6(E - E’) indicates that fluctuations at the energies E and E’ are independent. Kleinpenning (88) has also integrated Eq. (82) with respect to E and E’ and obtained S,k,

i,f) = [ ~ ~ a ~ k ) / f n6kk )-] r’)

(83)

This result was obtained by using pb, E ) 1: E-’/’, and n k , E ) N x exp(-E/kT). He used this expression to explain flicker noise in spacecharge-limited diodes. Zijlstra used Eq. (83) directly in order to prove Eq. (79) (934.

266

A. VAN DER ZIEL

It should be emphasized that for space-charge-limited diodes Kleinpenning used Eq. (83) in order to derive the formula

where A is the cross-sectional area of the device. For the space-charge regime this yields Eq. (79). The mobility fluctuation hypothesis is thus needed for explaining the noise data in thermoelectric cells and in single-injection space-chargelimited solid-state diodes. In all other cases the density fluctuation hypothesis explains the data equally well. There is one further piece of evidence in favor of mobility fluctuations. Hooge and Vandamme (94) found from contact measurements on heavily doped Ge and GaAs that the parameter a decreased with increasing doping and could be expressed as a =2 x

lo-3(~/~latt)2

(85)

Here p is the total mobility, phttthe lattice mobility, and pimpthe mobility due to impurity scattering, so that

1 / = ~ l/PIatt

+ VPimp

(85a)

Such a relationship would be expected if lattice scattering were the source of l / f noise and impurity scattering were noiseless. This could be the most direct evidence so far in favor of the mobility fluctuation hypothesis, and could point to phonon scattering as the source of noise. Compare, however, van der Ziel (94a). 4. Integrated and Zon-Implanted Resistors Hsieh has investigated l / f noise in integrated surface resistors and in ion-implanted resistors (95). The results were published in two papers (96,97).A third paper (98) gives the theory. Hsieh et al. expressed the l / f noise in terms of the noise ratio n, defined as

where S;(f) = S,(f) - 4kT/R is the l/f noise generated in the resistor, S,(f) the total noise spectrum, and 4kT/R the thermal noise spectrum. Their data do not allow the evaluation of S R ( f ) / R zand hence of Hooge’s noise parameter a. They found for integrated surface resistors (95) that (n - 1) varied as

FLICKER NOISE IN ELECTRONIC DEVICES

267

12/fw2, where I is the current, f the frequency, and w the device width; also, (n - 1) was independent of the device length L. This is compatible with Eq. (47). noise at low doping In ion-implanted resistors (97) they found l/f levels and l/f noise at higher doping levels. Bilger et al. found l/f noise (99). The latter changed the value of the resistance R by changing the voltage of the epitaxial layer surrounding the resistor. Hooge found that in the expression (100) sR(f)

Af/ R 2 = Af/f

(87)

the constant C varied as R2,whereas Eq. (47) would give C = a/N= (aqp/L?)R. This discrepancy is easily explained by the McWhorter model. We saw in Section IV,C,2 that Eq. (47) resulted if 6 p = /IN, with a = /I/[ln(z2/zl)]. If instead the device satisfied the relationship

6NZ= PN,,

(88)

where N,, is independent of N, then

-sR(f) R2

/I

.%

1n(z2/zl)f N 2

which varies as R 2 ; now Eq. (47) is numerically valid for N = Neq.This agrees with Hooge's plot of their data shown in Fig. 6. Assumption (88) is not an ad hoc assumption made in order to force agreement with experiment but must also be introduced in n-channel MOSFETs to explain the flicker noise data in those devices. Equation (88) says that if N is varied by varying the bias of the epitaxy, then the meansquare value of the fluctuation 6 N is independent of N.

-

-

FIG.6. llfnoise of ion-implanted resistor. The resistance R is changed by the depletion voltage. Points R2: experimental results; line R : calculated according to Hooge's relation (47) (F. N. Hooge, Proc. Symp.IlfFluctuations, 1977 Conf. Rep., p. 88 (1977)).

268

A. VAN DER ZIEL

5 . Noise in Conductors by the Four-Probe Method Hawkins and Bloodworth (101) measured l/f noise in thick-film resistors. They found experimentally that 1/f noise arises between probes placed both at right angles to the current flow on opposite sides of the film and parallel to the current flow at one side of the film. In both cases the l/f noise was of the same magnitude. The noise between the probes on opposite sides of the film is called transverse noise. While the measurements were performed on thick-film resistors, it should be clear that the experiment works equally well for semiconductor resistors. The noise is caused by conductivity fluctuations. A theory of the transverse as well as the longitudinal effect was given by Kleinpenning (102), who also gives a list of references on the subject. D . Miscellaneous Topics on I /f Noise in Resistors

There have been many reports on the excess l /f noise generated in carbon resistors by ac excitation. The noise intensity is proportional to V,., with 2 < n < 4 (103-110). When Kc increases, n decreases from 4 to 2. The intensity is several percent of the intensity of the l/Af spectrum (i.e., the noise sidebands a t f k f , ) . The effect can be explained if one assumes that the contacts between the grains are not perfectly ohmic. The exciting sinusoidal voltage will then lead to a sinusoidal current with a very small extra dc current through each contact. Averaged over all contacts, the dc currents will more or less compensate each other so that the I- V characteristic of the resistor is fairly ohmic. Since the l/f noise generated in a contact does not depend on the direction of the current, the l/f noises of all contacts will add. In addition, noise is generated at the second harmonic at frequencies (2f ff,). The correlation between the noises at f, ,f +f, , and (2f ff,) has been investigated by Jones and Francis (110). The statistical properties of l/f noise have been extensively studied (111-121). The overall amplitude distribution was Gaussian, but the variance noise was not. With variance noise one means the fact that in some devices the variance of one time sample of duration T may be different from the variance of another time sample of duration z. One would expect such a phenomenon, e.g., if the sample gave noise at a high level at some times and at a low level at other times with random transitions from the one level to the other. Of course, more complicated models are possible. Voss (122) measured (V(t)l V(0) = Vo) for various devices showing l/f noise; this is the average behavior of the noise before and after a fluctuation of amplitude Vo. He found that some noise sources were linear and others

FLICKER NOISE IN ELECTRONIC DEVICES

269

slightly nonlinear, whereas systems like p-n junctions showing burst noise revealed a nonlinear mechanism.

V. l/f NOISEIN SOLID-STATE DEVICES A . l/fNoise in Tunneling Devices

The most important tunneling devices are the Josephson junction, the metal-oxide-metal diode, and the semiconductor tunnel diode.

1. l/f Noise in the Josephson Junction

As suggested by Clarke and his co-workers (123,124), the l/f noise in these devices is caused by spontaneous temperature fluctuations at the junction. The device voltage V depends on the critical current I,, such that at a current I > I,, according to the Stewart-McCumber model (125),

where R is a constant equal to the resistance at high current. A temperature fluctuation 6T produces fluctuations in I,, and this, in turn, produces fluctuations in V according to the relation

But according to Voss and Clarke (42) [our Eq. (52a)]

where C, is the heat capacity of the active volume of the junction, and w1 and w2 the width of the film making up the junction (in their case Nb and Pb), so that

In view of Eq. (89), S , ( f ) decreases with increasing current I . As Fig. 7 shows, the agreement between theory and experiment is reasonably good, which is somewhat surprising since Eq. (52a) gives a

270

A. VAN DER ZIEL

-2

-19

-

\

-=8 -20-

N . ,

c

CS -

-21-

01

-22

I

BACKGROUND I

log f

I

I

(Hrl

FIG.7. Voltage spectra for three values of Josephson junction bias current I. Note that the fluctuations decrease with increasing I (J.Clarke and G . Hawkins, IEEE Trans. Magne. M A G 11, 841 (1975)).

1/f noise regime only if w1 and w 2 differ appreciably (in these experiments w1 and w 2 were of the same order of magnitude). This l/f noise sets a lower limit to the low-frequency application of Josephson junctions (126-128). l/f Noise Noise inin Metal-Oxide-Metal Metal-Oxide-Metal Diodes Diodes 2.2. l/f Flicker noise measurements have been reported by Zijlstra (129) on Ta-Ta,O,-Ta structures and by Liu and van der Ziel(130) on A1-A1,O3-AI structures. The noise was shot noise at high frequencies, as expected, and l/f noise at low frequencies with S,(nvarying as P . In the Ta-Ta,O,-Ta structures the shot noise was suppressed because the electron moving from metal to metal hops from metal to trap to trap * * to metal, and at high frequencies each movement gives rise to independent pulses carrying a fractional charge (131). In the A1-A1,O-Al3 structures the oxide was much thinner (1.20 A), and the shot noise was not suppressed because multiple hopping is a very rare event. The noise was attributed to temporarily trapped carriers. The trapped carriers have two effects :

(a) They modulate the barrier height E,, and this gives rise to noise (see also Section V,B). (b) The random distribution of traps gives rise to a wide distribution in time constants.

FLICKER NOISE IN ELECTRONIC DEVICES

27 1

We now investigate this model. Since the current density J is given by the Fowler-Nordheim equation,

where Eb is the potential barrier height, F the electric field strength, h = h/2n (where h is Planck’s constant), q the electron charge, m* the effective mass, f ( y ) Nordheim’s elliptic function, y = (qFo/n&&o)’/2/Eb, E~ the permittivity of free space, and E the relative dielectric constant of the oxide. Differentiating (92) with respect to E,, assuming that f ( y ) is slow function of E,, yields

6J = - J[l

6Eb + $BE;/’f(y)] Eb

with B =

so that S,( f ) = 2qDJ2

where D =

[l

4(2m*/h2)l12 (4F)

+ %BE;’2f(y)]2&,(f1 2qEb’

where S , , ( f ) has a l/f spectrum because of a wide distribution in time constants. The two types of devices measured gave about the same value of D at low frequencies. For an alternate theory see Kleinpenning ( 1 3 2 ~ ) . 3 . The Semiconductor Tunnel Diode

The (Z- V) characteristic of a tunnel diode has three distinct regimes : (a) the low-voltage regime with positive derivative; (b) the intermediate-voltage regime with negative derivative; and (c) the high-voltage regime with positive derivative. In regimes (a) and (b) the device is a majority carrier device due to tunneling through the barrier. In regime (c) the device is a minority carrier device due to diffusion or thermionic emission over the barrier; this corresponds to what one finds in normal p-n junctions. In regime (a) Agouridis (232) found no flicker noise above 100 kHz; he did no measurements in regime (b) and found a considerable amount of flicker noise in regime (c), all for germanium tunnel diodes. Yajima and Esaki (233) found no flicker noise in regimes (a) and (b) down to 10 Hz for Ge tunnel diodes, whereas they also found a large amount of flicker noise in region (c). The llfnoise mechanism in part (c) of the characteristic should be similar to the noise in p-n junction diodes, except for the fact that the p

272

A. VAN DER ZIEL

and n regions are much more heavily doped. We refer to Section V,C,l for details. The llfnoise in regions (a) and (b) is difficult to measure because of the low impedance levels involved. For example, in the negative conductance region the device may have to be shunted by a small load resistance R, to prevent oscillations. Any possible flicker noise may then drown in the thermal noise background of R , . Since the device is a majority carrier device in regimes (a) and (b), one would expect only a relatively small amount of flicker noise, if any. The basic tunneling mechanism should be noiseless, but some l/f noise might occur due to tunneling via traps in the forbidden gap. See Section V,B,l for details. It would be worthwhile to verify the absence of flicker noise in regimes (a) and (b) more carefully, since it could have important theoretical consequences. B. Flicker Noise in Schottky Barrier Diodes

The Schottky barrier diode is a majority carrier device. Zettler and Cowley (134) have demonstrated that by using a p-n junction guard ring structure it is possible to obtain Schottky diodes with forward currentvoltage characteristics that approximately follow the expected Richardson equation. The guard ring structure also lowers the low-frequency noise because it reduces the current flow across the surface. In the ideal case the llfnoise should be caused by the current flow from bulk to contact and should be inversely proportional to the contact area. This surface effect was corroborated by Wall (135) and by Hovatter (136). Wall first measured the noise in a planar structure and then changed the structure to a mesa structure without changing the contact area and remeasured the noise. He found a decrease in noise by more than one order of magnitude. He concluded from this result that the llfnoise was generated by edge currents. Hovatter found for point contact Schottky barrier diodes made on heavily doped material that the flicker noise at a given current was inversely proportional to the current diameter rather than to the contact area. Hsu (137) measured Schottky barrier diodes with a gate structure. He found less noise at flatband conditions under the gate than when the region under the gate was strongly accumulated. This puts the main source of the noise in the edge (or surface) current. The carriers contributing to this current interact with the surface oxide via interface states. If that were the case, S,y) should indeed vary as l/J would be proportional to the surface state density N,, and could be reduced by appropriate surface treatment.

FLICKER NOISE IN ELECTRONIC DEVICES

273

Hsu (138) has given the theory for the case in which surface effects are absent. The mechanism discussed is a two-step tunneling process in which the electron tunnels from the conduction band of the semiconductor to a trap state and from the trap state to the metal. The fluctuating trap occupancy gives rise to a fluctuation in barrier height, and this produces noise. The tunneling distances depend on the energy of the carriers and decrease with increasing carrier energy, so that a wide range of tunneling times is present. He found for the case of a parabolic band structure and a triangular potential profile,

where

(934 Here Nt is the trap density, T~ the tunneling time constant at the top of the barrier, L, the width of the lower limit of the potential barrier where twostep tunneling occurs, L, the width of the upper limit of the potential barrier where two-step tunneling occurs, w the width of the space-charge region, A the contact area, T the device temperature, Nd the donor density, and E the relative dielectric constant of the material. The spectrum is constant at low frequencies ( 0 7 , < 1) and varies as l/f at intermediate frequencies (l/z2 < w < l/zJ and as l/f2 at high frequencies (wrl > 1). According to this equation, S r ( f ) is proportional to 1/A, as expected. The corner frequency f,, defined as the frequency for which the flicker noise equals the shot noise, is given by

But

is independent of N, so that the corner frequency at constant current density I/A is practically independent of the donor concentration N,. Here vd, is the diffusion potential. According to Grant (139) this flicker noise model is operating in InP Schottky barrier diodes.

274

A. V A N DER ZIEL

C. l/f Noise in Junction Devices

1. l/f Noise in p-n Junctions

If we consider a p+-n junction, then the current flowing in the forward direction is due to holes. In long junctions it comes about because the holes recombine with electrons at the surface of the n region (in germanium and silicon) or at the surface of the space-charge region (in silicon). The noise comes about because the surface recombination velocity s shows fluctuations 6s with a spectrum S,(f). Such a fluctuation would be expected because the surface recombination velocity is modulated by the trapping and detrapping of carriers in traps in the oxide adjacent to the oxide-semiconductor interface (McWhorter model). One would expect 6s to be proportional to SNt, and hence S,(f) should be proportional to S N , ( f ) . But, S N , ( f ) is proportional to q, and hence, since Tt is proportional to the surface-state density N,,, proportional to N,. Since s is also proportional to (or to N,,), we may thus write S J f ) = CS/f

(94)

where C may be a slow function of the surface current density J,. This is essentially Fonger’s result (140). We now turn to the case where the recombination occurs at the surface of the n region. If p’ is the excess hole density at a surface element dA, J, =

6 J , = 4P’dSY

qp’s,

SJp(f)= (qP’)2S,(f)

(95)

whereas the junction current I, =

s s J, dA =

qsp‘ d A = qsp’(O)A,,

(96)

when the n region is so long that practically no holes reach the ohmic contact to the n region. Here p‘(0) is the hole concentration at the beginning of the n region and A,, is an effective recombination area. Moreover, since all surface elements fluctuate independently,

f

cs CZ; A:, p ” d A = q2-[p‘(0)12A:, = -f fs A&

(97)

where AiE is another effective area. We thus see that S,(f) is proportional to If. This agrees reasonably well in some germanium p-n junctions at low currents, but there is often a large dip in the noise (sometimes an actual zero) at higher currents (Z4Z). Fonger attributed it to the current dependence of the series resistance r, of the n region.

FLICKER NOISE IN ELECTRONIC DEVICES

27 5

OHMIC RING CONTACT

OHMIC CONTACT

FIG. 8. Watkins’ arrangement for studying flicker noise in Ge p-n junctions.

Watkins (142) had a different explanation. He used a geometry shown in Fig. 8. The surface to the n region parallel to the junction was so treated that a high recombination velocity was obtained and the width w was so chosen that most recombination occurred at the treated surface. He then expressed the junction voltage V in terms of the hole concentrations p(0) at x = 0 and p(w) at x = w. The zero in the noise occurred if.dV/ds = 0. Van der Ziel (143) calculated dV/ds more accurately and showed that the condition for a zero value in dV/ds was 2p(w)

+ Nd(2 - In 2) = (sw/D,)p(w)

(98)

where D, is the hole diffusion constant. This gives a positive solution for p(w) when sw/Dp > 2, which is the case if s is sufficiently large. As shown from Eq. (981 this can only occur at relatively high injection. For small s the zero should not occur. Gutkov (144) found for germanium p-n diodes that S , ( f ) = const (Z2/szf)

(99)

and concluded that this contradicted Fonger’s equation (94), since it contradicts Eq. (97).Van der Ziel(143) was able to show, however, that the same treatment given for the Watkins experiment yielded for his geometry S,(f) = const

I2

sf(1

+ SW/D,)~

If one now takes the n region sufficiently long, w changes along the surface and one must average over w. This can give a wide range of s for which S , ( f ) varies as l/s2, especially if s is sufficiently large. Hsu et al. (145) found for silicon diodes that the noise spectrum S , ( f ) at constant current was proportional to the surface recombination velocity s, in contradiction with Eq. (97). By assuming that in this case the noise is generated in the junction space-charge region, van der Ziel (143) reconciled this result with Eq. (94). This recombination occurs in a well-defined part of the space-charge region, characterized by the coordinate xl. Let for an applied voltage

276

A. VAN DER ZIEL

V the potential at x1 change by an amount V, and let p(xl) and p1 be the hole concentrations at x1 for applied voltages V and zero, respectively; then the recombination current I, is given by 1, = qSp(x1)Aefi = qsp1 exP(eVl/kT)Ae,

(101)

where Aeff is the effective area of the recombination region. If s now fluctuates by an amount as, then the fluctuation 81, in I, and the fluctuation 61 in I is 81 = 61, = qp1 exp(qI/,/kTMe,ds,

S d f ) = (qP1Ae,)* exp(2qVdkT)Ss(f)

(102) Substituting (94) for Ss(f) and bearing in mind that I = I. exp(qV/mkT), we have S,(f) = const(I)zmY1/Ys/f

(103)

in agreement with Hsu et al.’s data. If Vl N- 1/2V and rn N 1, Sz(f) would be proportional to 1. This current dependence of S,(f) agrees reasonably well with the results obtained by Plumb and Chenette (146) for noise generated in the emitter space-charge region of silicon transistors. For Vl/V > 3, Sl(f) goes faster than linear with the current I. The final conclusion of this discussion is that the noise is due to fluctuations in the surface recombination velocity s, and that there is direct experimental evidence in favor of it. North (147) had a similar theory. He assumed that the noise is caused by fluctuations in the surface potential 4s.These fluctuations are thermal and the fluctuations in the surface recombination velocity s follow from

where Re, is the real part of the equivalent impedance into which 4, looks. To calculate Req, North developed an equivalent network for the surface traps similar to the one shown in Fig. 5b. This is an alternate way of looking at the noise. Hsu (148) has developed a theory for the l/f noise observed in a gatecontrolled diode in terms of surface-state effects. He finds that the noise is proportional to the density of these states and the square of the transconductance gm = aZ/aV of the device, where I is the device current and V the gate voltage. For diodes without gate gm must be replaced by aI/a4,, where 4sis the surface potential. There is then a similarity with North’s theory.

FLICKER NOISE IN ELECTRONIC DEVICES

277

2. Flicker Noise in Transistors

The theory of flicker noise in p-n junctions can be directly applied to flicker noise in transistors. That means that in silicon transistors most of the flicker noise comes from recombination in the emitter-base spacecharge region. In general the flicker noise can be represented by two current generators, i f , and if2,in parallel to the emitter-base junction and the collector-base junction, respectively. In view of what was just said, one would expect if, and ifz to be fully correlated, since they come from the same noise source, and i f , should far predominate over i f z , since the noise is a fluctuation in the emitter-base recombination current IR that is not transmitted to the collector. This was demonstrated by Chenette (149) and further substantiated by Gibbons (150). Probably the clearest demonstration came in a classical paper by Plumb and Chenette (146). Their experimental setup is shown in Fig. 9a. Here a large resistance RE was inserted into the emitter lead and a variable resistor RB was inserted between base and ground. The noise was measured between emitter and ground, and RB was so adjusted that the measured noise was a minimum. The noise voltage u appearing across the emitter terminal is

FIG.9. (a) Plumb and Chenette'scircuitfor studying flicker noise in transistors (IEEE Trans. Electron Devices 4-10, 304 (1963)). (b) Equivalent circuit of flicker noise in transistors, incorporating surface noise and dislocation noise.

278

A. VAN DER ZIEL

where reO = kT/& and if, has been split into two parts, i;z, fully correlated with ifl, and i;,, uncorrelated with ifl, respectively. We see that 3 will go through a deep minimum if

(105a)

By plotting (RB),,,in versus reo/ao = kT/qI,, one should obtain a straight line intercepting the vertical axis at -rb, whereas the slope gives [l c2/aoifl)]. This agreed very well with the experimental data and the values found for i;, were only a few percent of i f l , so that if, can be neglected. Measurements of i& were obtained for small values of Z E , by omitting RB altogether; they indicated that izl varied as Zt,with B somewhat smaller than unity in some units. This is compatible with Eq. (103). Measurements by Viner (151) on silicon transistors operated at elevated temperatures showed another interesting feature. At elevated temperatures ZB = -ICBo + 4 where -ICBois the base current for zero emitter current and rBthe injected base current. Viner showed that the currents ICBoand Zh fluctuated independently, that each showed flicker noise, and that the flicker noise spectrum of TBvaried as (4)1.5 in his units (TI483). One might now ask the question whether the flicker noise in transistors can be completely eliminated by fully passifying the surface. The answer is that this is not the case (252, 253). After fully removing the source of noise at the surface, one is left with llfnoise due to dislocations. The dislocations then act as flicker noise generating centers just as the surface traps did, and the mechanism is probably similar. Since the dislocations are distributed through the bulk, this is a true bulk effect. The flicker noise can be further reduced by reducing the dislocation density. This can be done in two ways

+

(153) :

(a) One starts from dislocation-free material (perfect crystal technology). (b) In n-p-n transistors one deposits P/As mixed doped oxide onto Si using PH, and ASH, in a flow ratio of 4: I and one then diffuses in. This reduces stress in the emitter and so prevents the introduction of dislocations by the diffusion process. The two noise sources, surface l/f noise and dislocation l/f noise, are located somewhat differently in the equivalent circuit. This is shown in Fig. 9b, where ifl represents the surface flicker noise and iil represents the dis-

279

FLICKER NOISE IN ELECTRONIC DEVICES

location flicker noise, and rb = R,, + R,, is the total base resistance. Brodersen et al. (154) have the two noise sources interchanged, however. Mueller (155) has suggested that the flicker noise in p-n junctions and transistors is caused by temperature fluctuations described by a thermal circuit with a wide distribution in time constants (62); this transforms the shot noise into llfnoise. Van der Ziel(156) was able to show that this could only occur when the device was operating close to a thermal instability. Shacter et al. (157) demonstrated that such a situation did occur in second breakdown in power transistors and that it transformed the normal llfnoise into l/f’ noise as expected theoretically. 3. Burst Noise (158)

Burst noise has been observed in planar silicon and germanium diodes and transistors. The phenomenon consists of a random turning off and on of a current pulse of 10-8-10-9 8.It can be described by a random telegraph signal approach (159), and leads to a spectrum const/(l 02r2), where z is the time constant associated with the pulse. The noise can be practically eliminated by using a low source resistance in common base amplifiers. This indicates that the current generator describing the burst noise must be located much closer to the external base contact than the current generator describing the flicker noise generated at the surface. Since burst noise is not a flicker noise phenomenon, we refrain from a more detailed discussion.

+

D . Flicker Noise in JFETs

Good silicon JFETs at room temperature do not show any flicker noise to speak of (160), whereas GaAs FETs show a relatively large amount of it (161). We shall see that this hangs together with the structure of these devices. We shall also see that the absence of l/fnoise in silicon JFETs has important consequences for our understanding of semiconductor noise in general and that it can help pinpoint the source of this noise (162). Let us first assume that flicker noise is a bulk effect and that Hooge’s formula

Mf )/NZ= ./fN

(106)

is valid. Following Klaassen’s theory of flicker noise in MOSFETs (76), we then obtain for the spectrum of the equivalent flicker noise emf 6V in series with the gate

280

A. V A N DER ZIEL

for V, V,, where V, is the pinch-off voltage, Zo the current, V, the drain voltage, gm the transconductance, q the electron charge, p the mobility, and L the device length. We substitute Hooge's constant a = 2 x and further put q = 1.6 x lo-'' C, p = 1.4 x cm2/v sec, k = 1.38 x J/"K, T = 30O0K,L = 10-3~m,Z= 5mA, V, = 2Vygm= 5 x mho, which are very reasonable values. We then obtain for the noise resistance at room temperature R, N lO'O/f ohms. In fact one finds (160) for good JFETs at room temperature R , N 105/(1 + 02z2) ohms. We thus conclude that the noise at room temperature is generation-recombination (8-r) noise and that flicker noise is absent. This means that flicker noise in semconductors cannot be a bulk effect, for if it were, it would be present in JFETs also. Conversely, we may say that if Eq. (106) were valid, then a < 2 x lo-*, in clear violation of the bulk noise hypothesis. But we can go one step further. The only difference between good silicon JFETs and all other semiconductor devices is that the former have no semiconductor-oxide interface, whereas all other semiconductor devices do. This points to the semiconductor-oxide interface as the source of the noise. This leads to the following model. Carriers interact with traps in the oxide near the interface and this produces three effects : 1. The fluctuating occupancy of the traps modulates the surface recombination velociiy s of injected carriers in structures involving minority carrier flow; this explains the l/f noise in p-n junctions and transistors (Fonger model) (140). 2. The fluctuating occupancy of the traps gives rise to fluctuations in the carrier density of the semiconductor material or device; this explains the McWhorter model of flicker noise (78). 3. The fluctuating occupancy of the traps modulates the surface potential of the interface and so gives rise to fluctuations in the (surface) mobility of the carriers in the semiconductor. This explains the Kleinpenning model of flicker noise (53,88).

We wish to point out here that we have now, in principle, unified all the important models of flicker noise in semiconductor materials and devices. We shall come back to this problem in Section V,F. We can now also understand why there is a large amount of flicker noise in GaAs FETs. These devices have a gate width that is much smaller than the source-drain length. Hence there is a large semiconductor-oxide interface area between gate and source as well as between gate and drain. This gives a full contribution to flicker noise, since one would expect Eq. (106) to be approximately valid in this case with a value of a determined by the interaction between the carriers and the oxide traps via surface states.

FLICKER NOISE IN ELECTRONIC DEVICES

28 1

The ramifications of this insight for the theory of flicker noise in MOSFETs will be discussed in the next section. E. Flicker Noise in MOSFETs 1. General Theory

Flicker noise in MOSFETs is usually thought to be caused by carrier density fluctuations, brought about by interaction of free carriers with oxide traps via interface states (76,163-172). In order to clarify the problem, we start here the theory from first principles, rather than applying previously developed formulas. Let ( x , y , z ) be the coordinate system; here x is in the direction of the length L of the sample and y is the direction of the width w , whereas z is perpendicular to the oxide and pointing into it. We consider a surface element AxAy and a volume element AxAyAz in the oxide. Let AN, =n,(E)AEAxAyAz by the number of traps in the volume element AxAyAx with an energy between E and E + AE. Let ANs = n,(E)AEAxAy be the number of surface states in the surface element AxAy with an energy between E and E + AE. Let ANt and AN, be the number of carriers in those states, respectively, and let the interaction occur by tunneling at constant energy E ; then (170) dAN,/dt = g(AN,) - r(AN,),

g(AN,) = aAN,(AN, - AN,),

r(AN,) = PAN,(ANs - AN,)

( 108)

The interaction between the channel carriers and the surface states is so fast that we can ignore it. In equilibrium ANs = ANsa = ANsfi,

AN, = ANta = AN,fi

(108a)

wherefi is the fractional occupancy of the surface states and the traps, and dAN,/dt = 0. Substituting (108a) into this equation yields a = P. Now we look for slow fluctuations GAN, in AN, and slow fluctuations GAN, in AN, that are driven by the fluctuations in g(AN,) - r(AN,). Substituting AN, = ANsa + GAN, and AN, = AN,, + GAN, yields dGAN,/dt = uAN,GAN, - aANsSAN,

+ Ag(t) - Ar(t)

(109)

We now substitute JAN, = -yGAN, and bear in mind that AN, is proportional to AxAyAz and ANs to AxAy. In a subsequent integration process with respect to z, we let Az go to zero, so that the first term in (109) becomes negligible. We may thus put as dominant lifetime t =

l/(aANs)

(109a)

282

A. VAN DER ZIEL

According to the theory of carrier density fluctuations (172)

46AN;z S d f ) = 1 + mzz2

6AN: = g(ANto)t= ANTft(1

and

- ft)

(110)

Hence

where

f, = ( 1

+ exp[(E - E,)/kT]}-'

(1 12)

and E, is the Fermi level, so that ft(l - fd has a very sharp peak at E = E,. We now integrate with respect to z. To that end we bear in mind that in a tunneling model z = zo exp(ez)

(113)

where E is of the order of lo8 cm- '. We now assume a uniform trap distribution for 0 c z c z1 and zero traps outside; this is allowed if z1 is so chosen that traps for z > z1 have so long a time constant z that their effect on the noise cannot be measured. We then have a normalized distribution

for zo < z < z1 and zero outside that interval. Replacing Az by z,g(z)Az, and integrating (111) with respect to z between the limits 70 and zl yields

We next integrate this expression with respect to the energy E. Since f ( l - ft) has a sharp peak at E = E,, we introduce the parameter

J-

W

nr(Ef)eff =

n,(~)ft(l- ft) d~

(115a)

If we also integrate with respect to y between the limits 0 and w, we obtain, since ln(zl/zo) = E Z ~ ,

where S,,(f) is defined for unit area. Hence, z1 has disappeared.

283

FLICKER NOISE IN ELECTRONIC DEVICES

We next illustrate the meaning of Eq. (115a). Assuming &(E) for E not too far from Ef , nT(Ef)eff

=

N

n@f)

(115b)

nT(Ef)kT

For the case in which %(E) depends strongly on E near E = E,, the full definition (115a) must be used. We now bear in mind that %(Ef)is usually thought of as being proportional to the surface-state density ns(Ef) so that nT(Ef)e& is proportional to the effective surface-state density N,, eff. If C is the proportionality factor, we obtain (116a)

It will be shown shortly that for very strong inversion the fluctuation

SAn in the number of free carriers AN for area wAx is equal to -SANt, or = -SANt,

S,(f)

= S,,(f),

S"(f)= S"Jf)

(117)

where S , ( f ) is again defined per unit area. Next we consider the case that the device is operated at a very low drain voltage Vd(Vd 4 5 - b).The device is then nearly uniform and the spectrum S N ( f )of the fluctuation in the N carriers in the sample is 'N(f)

(118)

= Sn(f)wL

Since the charge fluctuation SQ in the channel may be written SQ = q S N = C,,wLSV,,,

or

SVeq 0 =-

'

COXWL

SN

where SV,,, is the equivalent emf in series with the gate at near zero drain bias, the spectrum of SV,, , is S"@(f) = (q2/c:;ww"(f)

(1 19)

The equivalent noise at the input is thus a direct measure for S,,(f). This is essentially Katto's result (268, 169). If S , , ( f ) is independent of the applied voltage (V, - b),where V, is the gate voltage and VT the turn-on voltage, as seems to be the case in many n-channel MOSFETs, SVeqO(f) is independent of (V, - b);this is Katto's result (168,272). If, however, S,,(f) is proportional to (V, - b),as seems to be the case in many p-channel MOSFETs, SVeq,(f)is proportional to (V, - VT); this is essentially Klaassen's result (76, 272). It should be borne in mind, however, that Klaassen assumed that S , ( f )

284

A. VAN DER ZIEL

was proportional to n, rather than to (V, - V,) only. He put S,,(f) = an/f, where a is Hooge's parameter. Since qn = Cox(&- V,), this yields

(119a) This gives a slight difference in the dependence of the equivalent input noise on the oxide thickness t in both cases. Since Cox= E E , / ~ , Eq. (119) varies as t2 if S,,(f) does not contain the parameter Cox;whereas Eq. (119) varies as t if S,,(f) is proportional to Cox.Equation (119a) varies as t also. Both cases seem to have been found; for the first see Katto (169),for the latter case see Kaassen (76) and Berz (165). See also a paper by Vandamme and de Kuijper ( 1 6 9 ~ ) . We now want to evaluate Sv0,(V,, V, - V,,f) for arbitrary drain voltage V, (170).According to the Klaassen-Prim approach, as formulated by van der Ziel (173), S,(f) =

$J

0

where F ( x , f ) = S r ( x , f ) Ax

F ( u , f )du

(120)

and Sr(x, f ) is the spectrum of the primary current fluctuation in the section Ax at x. But dx

where o(V0) = pwCox(Q- V, - V,) is the conductance for unit length at x, p the carrier mobility, V, the gate voltage, and V, the turn-on voltage of the channel. Hence

):();(

~ I ( x t), =

-

- 6AN(x, t )

and

so that

where we have switched to V, as a new variable.

FLICKER NOISE IN ELECTRONIC DEVICES

285

We now define Now and (123b) Substituting into (123) yields

- V, -

h - Vd/2Jo"d V,

sue,~(V,

- - VO,f)dVO V,-v,-V, (124)

The first half corresponds essentially to Christensson et al.'s Eq. (9) (163) and the second half to van der Ziel's Eq. (10) (170). If Sueqo( V, - 5,f ) is proportional to V, - V,, we have

which decreases from Sucq ,JV, - V,, f ) at V, = 0 to half this value at saturation (Vd = V, - G).This agrees with experiments for many p-type samples. If Svoq0(V, - h,f) is independent of (V, - V,), the integral in Eq. (124) diverges logarithmically at saturation. This is overcome because the condition of strong inversion is violated when the channel is at or near cutoff; - V,, f ) must go to zero when V, - V,;0' we shall consequently Sueqo(V, see that this comes about because (- 6AN/6ANt)-+ 0 at pinchoff. Both results presuppose, of course, that the channel has uniform doping and a uniform surface. 2. The Correction for Arbitrary Inversion Up to here we assumed that 6AN = -6ANt. Jindal and van der Ziel (174) have shown that this is true in the region of very strong inversion. If this condition is not satisfied, the above results must be multiplied by the

286

A. VAN DER ZIEL

rI

'2

FIG.10. Equivalent circuit of an MOS structure.

factor R2, where R = -6AN/6ANt; this parameter decreases from unity to a very small quantity when the gate voltage changes from the condition of very strong inversion to the condition of very weak inversion. They found from first principles that

This has a very simple equivalent circuit interpretation. Q, is the total charge density, Qinvthe inversion layer charge density, and Qsd = Q, - Qinv the depletion layer charge density; whereas us = q+,/kT, the normalized surface potential. We now introduce the capacitances per unit area : &&

cox= 0 t '

cN -

q2

aQinv

kT au, '

c

W

= - - -q2

aQsd

dT au,

(125a)

where is the relative dielectric constant of the material and t the thickness. This leads to the equivalent circuit of Fig. 10. Since the induced fluctuation is shared by Cox,C,, CNin parallel and only the charge on C, is effective (174, =

cN/(cN

+ cw + cox)

(125b)

For a numerical evaluation of R these capacitances must be evaluated numerically (174). 3. The Effect of Mobility Fluctuations The fluctuations in the occupancy of oxide t r a p not only produce density fluctuations, they also modulate the surface potential and so produce mobility fluctuations. We investigate this problem for a MOSFET at low-drain bias. Let N , be the number of trapped electrons, N the number of free carriers, and 6Nt and 6N the respective fluctuations such that 6N/6Nt = -R. Since

FLICKER NOISE IN ELECTRONIC DEVICES

287

the fluctuation SNi drives the fluctuation 6p in p, Sp and SN will be fully correlated. The relative current fluctuation SZ/Z may then be written

so that the noise calculated previously must be multiplied by the factor (1

+;$

(126a)

a factor that was first proposed by Berz (165).But dp/dN = (dp/d4,)(d$,/dNi) (dNJdN) and SN = SN, (dN/dN,).If we now put dN/dN, = - R, we have (126b) so that the noise calculated in Section KE,l must be multiplied by the factor

(R-fg.2) 2

(126c)

Since d$,/dNi < 0, the two terms add if dp/d$, > 0. Katto (169) essentially added the two terms in (126b) quadratically rather than linearly. This does not make much difference. When the one term predominates over the other, the error is negligible, and the maximum error (of a factor 2) occurs when the two terms are equal. For weak inversion R is extremely small, but the second term can still be quite significant; in that case the mobility fluctuations will predominate. The same conclusion was drawn by Katto.

4. Experimental Data Klaassen (76) and Berz (165) find the noise resistance R,, at low drain voltage to be proportional to the oxide thickness t, whereas Katto (169), Mantena and Lucas (176),Hsu (166) and Christensson and Lundstrom (177)find a t 2 dependence.We have already seen how that can be explained; the t 2 dependence is probably more fundamental. Klaassen (76) finds RnOproportional to - V, forp-channel MOSFETs, whereas Katto (169) and Pai (171) find R,, to be practically independent of - & for n-channel MOSFETs. We saw already how that can be interpreted. Fu and Sah (167) find a more complicated behavior in especially constructed units.

<

<

288

A. VAN DER ZIEL

Takagi and van der Ziel (178) find R, to be rather independent of temperature for p-channel devices and more strongly dependent of temperature for n-channel devices. Christensson and Lundstrom (177) also report temperature data, whereas Rogers (179) has measured down to even lower temperatures. This must be interpreted in terms of the temperature dependence of N,, Klaassen (76) and van der Ziel (180) have shown that R,, is inversely proportional to wL, but the L dependence may no longer be true for the noise resistance R, of short-channel devices at saturation because of the occurrence of hot electron effects (180). Various authors (76, 169, 181) report R, to be proportional to the surface-state density N,, (Fig. 11). Leuenberger (182) finds an N: dependence, but the evidence for the latter is not clear, since it seems that a linear plot would fit the data equally well. D-MOS devices were investigated by Huang and van der Ziel (183) and

lijla

/

7

-

/

I 3

I

L

I

1

I

I I

I

1

I

I

FLICKER NOISE IN ELECTRONIC DEVICES

289

by Takagi and van der Ziel(184). They find a behavior different from other devices. Not only is the current dependence of R, different, but also the spectrum is different: n+-p-p--n+ structures, with the gate over p - p - , have a 1iY.O dependence at lower frequencies and a 1/y.6 dependence at higher frequencies, whereas n+-p-n--n+ structures, with the gate over p-n-, have a l/f'.6 spectrum throughout. This indicates that the 1/f'.6 spectrum comes from the p region and the l/fnoise from the p - region, whereas the n- region gives no measurable noise. Various papers have been written about surface treatment. Yau and Sah (185) used phosphorus gettering, Cheng et al. (186) used the HCl process, whereas Leuenberger (187) claims to have found a process that gives only g-r noise. Hielscher and End (188) have studied the effects of oxygen-reduction treatment on l/f noise. For further experimental papers see the list of references (189-202). For further theories see also references (203-205). Van der Ziel (206) has suggested a limiting flicker noise in MOSFETs due to the dielectric losses of the oxide; in modern devices this limit is not yet reached, however. F. Semiconductor Noise Revisited

The discussion in Section V,D has indicated the possibility of surfacecontrolled mobility fluctuations, and the discussion of Section V,C has indicated that in certain MOSFET cases, especially those involving weak inversion, the mobility fluctuations may predominate. This verifies Kleinpenning's model to a certain extent. Now, it should be taken into account that the MOSFET is a very particular device in which an inverted layer forms a conducting path between source and drain. In most other devices, e.g., in semiconductor resistors, this is not the case; rather, majority carriers in the bulk interact with carriers in a somewhat accumulated or depleted surface, and the carriers near the surface interact with oxide traps. It may well be that in this case the mobility fluctuations also predominate; that needs to be proved by a detailed calculation. There are two experiments for which the above mobility fluctuation model must be carefully scrutinized : (a) Can the model explain the experiment by Hooge and Vandamme (94)? (b) Can the model explain the thermoelectric experiment (53)? For one would, in this case, expect the carrier fluctuations in each subband to be fully correlated with the fluctuating occupancy of the traps.

290

A. VAN DER ZIBL

The space-charge-limitedflow experiment (88) is much less critical, since it only needs predominance of the mobility fluctuations. VI. MISCELLANEOUS PROBLEMS A . Formal Explanation of llfy Spectra with y # 1

In most of the theories developed so far, the spectrum was exactly l/f over a wide frequency range. What one finds experimentally, however, is a l/pspectrum with y close to unity, but not exactly equal to unity. We shall see that an appropriately chosen distribution function in z can formally explain this behavior. Consider a number fluctuation with a single time constant z and a spectrum -

S,(f) = 46N2 [2/(1

+ o"z")]

where 6Ni does not depend on z. Let now the following distribution function be introduced : g(z) dz = A

=0

dz

for zo < z c z1

zfl

otherwise

(128)

with j? # 1, where A is so chosen that

["g(z) dz = 1

J To Then for l/zl c o c l/zo

which yields

(128a)

For zo 4 z1 this gives a l/f2-fl spectrum over a wide frequency range. Butz (207) has given an extension of this model using very rigorous mathematical methods. We now transform back to the y domain by putting 7/70

This yields

= e"y

(130)

FLICKER NOISE IN ELECTRONIC DEVICES

29 1

We then see that the distribution function in y is no longer uniform. It thus seems that a nonuniform distribution function in y can explain deviations from the exact l / f spectrum. Since there is no particularly plausible argument in favor of an exactly uniform distribution in y , a nonuniform distribution seems quite reasonable. In our example we considered fluctuations in the number N of free carriers. Actually the same spectrum would be found if we had considered carriers trapped in the oxide. We could then have explained l/y spectra in the fluctuating surface recombination velocity, in the fluctuating number of free carriers, or in the fluctuating free-carrier mobility. That covers most of the practical cases of l/y noise in semiconductor devices.

B. Handel’s Theory of l / f Noise (208-211)

Handel has proposed a quantum theory of l / f noise caused by a selfinterference effect. In a noise experiment charge carriers pass from one terminal of the circuit element to the other. The corresponding quantum transition amplitude will contain a large term corresponding to the transition without emission of any photon above the low-frequency threshold of the noise measurement. But it will also contain a small term which is the amplitude of the same transition with emission of a bremsstrahlung energy between E and E d&. Consequently, the wave function that describes the coherent propagation of the charge carriers through the circuit element will contain a small part whose frequency is reduced by an amount between E/h and ( E dE)/h. The interference of this small part with the large part corresponding to the absence of bremsstrahlung represents beats with frequencies between E/h and ( E d )/h. This interference appears as a noise current having a quantum expectation value of the spectrum that is nonzero. This spectrum duplicates the number spectrum of the emitted photons. Since the bremsstrahlung radiation power per unit frequency interval is constant at low frequencies, the number spectrum is proportional to l/o. The exact shape of the spectrum is made convergent by radiative corrections, but the exponent of the frequency in the spectrum remains very close to - 1 . In contrast with the theories presented so far, Handel’s theory is a fundamental theory based directly on general physical principles. If correct, it should therefore set an absolute lower limit to the l / f noise process occurring in physical systems. Other l / f noise processes may be present, and we have seen what kind of processes they are, but they will give noise over and above the limit set by Handel’s theory. Tremblay has failed to verify Handel’s theory (211) in detail. The latest information seems to be, however, that this controversy is being resolved by a set of joint papers (212).

+

+

+

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A. VAN DER ZIEL

C . Other l / f Noise Theories

There have been a large number of formal theories of l / f noise in which the main purpose is to produce a l / f spectrum without much regard to physical reality. It is beyond the scope of this paper to discuss them.

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