Conductance Fluctuations and Low-Frequency Noise in Small Disordered Systems: Experiment

CHAPTER 5

Conductance Fluctuations and Low-Frequency Noise in Small Disordered Systems: Experiment N. G I O R D A N O Department

of Physics,

Purdue

West Lafayette,

Mesoscopic

IN

University

47907,

Phenomena

USA

in Solids Edited

©

Elsevier

Science

Publishers

B.V.,

B.L.

1991 131

Altshuler,

P.A.

Lee and R.A.

by

Webb

Contents 1.

Introduction

133

2.

Theory

135

2.1. Nature and magnitude of the fluctuations

135

2.2. Low-frequency noise

139

3.

Fluctuations due to the motion of single defects

143

4.

Low-frequency U C F noise

152

5.

N o n - U C F fluctuations - The local interference model

160

5.1. Theory

161

5.2. Experiments

163

Conclusions and outlook

169

6.

References

170

132

1. Introduction A great deal of theoretical and experimental work during the past decade has demonstrated the crucial importance of phase coherent effects in disordered systems (for recent reviews of this subject see, e.g., Al'tshuler et al. 1982, Bergmann 1984, Lee and Ramakrishnan 1985, Al'tshuler and Aronov, 1985, Washburn and W e b b 1986, Lee 1987, Al'tshuler 1987). Indeed, this is a central theme of many of the articles in this volume. Initially, phase coherent effects were studied within the context of weak localization (Thouless 1977), which is manifest by its effect on the conductance and magnetoconductance of a disordered system at low temperatures. As the understanding of weak localization has grown, a number of closely related phase coherent effects have been discovered. G o o d examples are the A h a r o n o v - B o h m effect in multiply connected geometries (Washburn and W e b b 1986), and the phenomenon of universal conductance fluctuations, U C F (Lee 1987, Al'tshuler 1987). The latter will be the main focus of the present paper. The importance of conductance fluctuations among members of an ensemble of disordered systems was first noted by Landauer (1970), in his discussion of the conductance of a strictly one-dimensional disordered system. As shown by Landauer, and by subsequent workers (see e.g., Stone and Joannopoulos 1981, 1982a,b, Azbel 1983, Lee 1984, Serota et al. 1986), these fluctuations can have non-Gaussian distributions, and other surprising statistical properties. While some of these effects arise only for strictly one-dimensional systems, other interesting types of fluctuations, in particular, U C F , occur in higher dimensional systems as well. As will be discussed in more detail in the next section, U C F are fluctuations of the conductance of members of an ensemble of weakly disordered systems. Different members of such an ensemble can be generated in several different ways. First, one can consider systems which correspond to different realizations of the random potential. T w o random systems are usually considered to be 'identical' when the randomness is statistically the same, even though it is, of course, different in detail; i.e., the precise positions of the scattering centers are different, even though their number, strength, etc., are all, on average, the same. Usually, the properties of such 'identical' systems are the same, but it turns out that as far as mesoscopic properties, and especially U C F , are concerned this is 133

134

Ν. Giordano

not necessarily the case. A second way to generate a different member of the relevant ensemble is to begin with a particular ensemble member, and apply a magnetic field or impose a shift of the Fermi energy. If the magnitude of the magnetic field or the shift of EF is sufficiently large, then the system is effectively transformed into a statistically 'independent' member of the ensemble. Ensembles of disordered systems generated in these ways exhibit some unu­ sual properties. Of particular interest to us here is the distribution of the conductance. It turns out that as .long as the disorder is weak*, the fluctuations of the conductance, 5G = (G2 — G 2 ) 1 /2 (where G is the conductance), have a universal value, which is of order e2/h (Stone 1985, Lee and Stone 1985, Al'tshuler 1985, Al'tshuler and Khmel'nitskii 1985, Lee 1986, Lee et al. 1987). The value of 8G is independent of both the size of the system, and the degree of disorder. Given the presence of large fluctuations when the entire distribution of impurities is altered, it is natural to consider the magnitude of the fluctuaiton produced if only a small change is made in this distribution. As will be discussed in the following section, the movement of only a single impurity, i.e., changing the position or scattering strength of only one scattering center, can also yield a large change in the conductance (Al'tshuler and Spivak 1985, Feng et al. 1986a), although in this case 5G is generally smaller than the fluctuation produced by a complete change in the impurity distribution. In this paper we will discuss experiments in which fluctuations of the conduc­ tance due to the motion of single scattering centers have been observed. These experiments can be grouped into two categories. First, under certain conditions, it is possible to observe the individual fluctuations, separated in time. This makes it possible to study the fluctuation magnitude as a function of temper­ ature, sample size, etc., and also obtain information concerning the kinetics of the defect motion. Complementary information is obtained from experiments which measure the low-frequency conductance noise produced by many, over­ lapping fluctuations. It is found that this noise has a spectral density which is approximately proportional to 1/f, where / is the frequency. The experiments indicate that, as first predicted theoretically by Feng et al. (1986a), this mecha­ nism can be a large source of 1/f noise in disordered systems, especially at low temperatures. W e begin in the next section with a qualitative description of the theory; a much more complete and rigorous discussion is given in the article by Feng (1991) in this volume. In section 3 we consider experiments in which fluctuations due to the motion of individual scattering centers are resolved as a function of time. Section 4 contains a discussion of experiments in which U C F are observed by way of their contribution to the low-frequency conductance noise. In sections 3 and 4 we restrict our discussion to experiments in which the fluctuations are due to U C F . In section 5 we describe studies of a related type of fluctuation, * Quantitatively, the weak disorder limit corresponds to the regime G ^ e2/h.

Fluctuations and noise: Experiment

135

that is also due to the motion of single defects, but for which the mechanism through which this motion actually couples to the conductance is somewhat different from that of U C F . Section 6 contains an outlook for future work in this area.

2. Theory 2.1. Nature and magnitude of the fluctuations Since the accompanying article by Feng (1991) contains an in-depth discussion of the theory of U C F with regards to single-atom motion, we will limit ourselves in this section to a qualitative discussion of the phenomenon. Our aim here is to emphasize those aspects which are of special importance in understanding the experiments. W e will first consider fluctuations in small systems at low temperatures. The meaning of the terms 'small' and 'low' as used here will become clear later in this section, when the effects of system size and finite temperature will be treated. W e will assume that our system contains impurities and defects, which provide a source of elastic scattering. Because it is elastic, this scattering will not directly affect the electron phase coherence. This coherence will be limited only by processes such as inelastic scattering, and scattering from the fluctuating spins of magnetic impurities (Thouless 1977, Bergmann 1984, Lee and Ramakrishnan 1985). The length scale over which phase coherence is maintained, Εφ, is the distance an electron travels between the scattering events which disrupt the phase. If we assume, as is generally the case in the systems we will be considering, that the elastic mean-free-path is much shorter than Εφ, then the electronic motion will be diffusive, and Εφ = ^/ϋτφ, where D = vFx2/3 is the diffusion constant, vF is the Fermi velocity, τ 6 is the elastic scattering time, and τφ is the phase breaking time; e.g., τφ is the appropriate combination of the inelastic and spin scattering times.* N o t e that our assumption that the motion is diffusive also implies that the disorder is sufficiently weak that we are far from the metal-insulator transition (Lee and Ramakrishnan 1985). Let us first consider a system whose dimensions are all small compared to Εφ. A t low temperatures, Εφ can, in practice, be of order 5000 A or even larger, hence the realization of such structures is straightforward with modern fabrica­ tion techniques. Such a system could contain a large number of atoms, Ν « Ι Ο 10 or more, and one would ordinarily expect that changes which involve only one atom, such as moving a single scattering center, would alter the measurable properties by an amount which is at most of order N~1/2 & 1 0 ~ 5. However, this conclusion is based implicitly on the assumption that the effects of all of • O n e has τ ^ 1 = τ , -1 + 2τ~/ η. See, e.g., Al'tshuler et al. (1982).

136

Ν. Giordano

the atoms are independent. If L 0 is comparable to the size of the system, this assumption is not justified. In this case, one is dealing with a single coherent system, and phase dependent effects must be treated carefully. That is, one has a mesoscopic system. This lesson becomes clear when one considers conductance fluctuations due to changes in the random potential. Detailed calculations show that the fluctuations of the conductance among members of an ensemble of statistically similar systems yield (Lee and Stone 1985, Al'tshuler 1985, Al'tshuler and Khmel'nitskii 1985, Lee 1986, 1987, Lee et al. 1987) hG = Ce2IK

(1)

where C is a constant of order unity, which is independent of N. The constant C depends only weakly on dimensionality, and has the values 0.73, 0.86, and 1.09 in one, two, and three dimensions respectively (Lee and Stone, 1985). These fluctuations are known as universal conductance fluctuations ( U C F ) . The unu­ sual properties of U C F , i.e., the fact that 5G is independent of N9 system size, etc., are due to the phase coherent effects noted above. This 'universality' is in sharp contrast with the simple result that one obtains from standard statistical arguments. The central-limit theorem generally leads to fluctuations which scale as N~1/2, but this is only found when one is dealing with Ν random components whose effects are independent. This is not true in a mesoscopic system; in this case the number of independent 'components' is of order unity, i.e., the appropri­ ate coherence length is the size of the system. For this reason, the magnitude of the fluctuations is scale invariant, and is of order unity when measured in the appropriate units, namely the quantum unit of conductance, e2/h. The fluctuations (1) are produced by a complete change in the random potential. One can use eq. (1) to obtain a qualitative estimate of the effect of making a small change in this potential, i.e., moving only a single scattering center, or changing the strength of just one scattering center, from the following argument (Feng et al. 1986a). Consider a d-dimensional hypercube of size L. The number of scattering centers in such a system will be N c e ns ~t Led. r An electron which travels through the system will move diffusively, hence its classical trajectory will be a random walk. A typical walk will consist of N s t e p s ~ ( L / L J 2, where the average step length is L e, the elastic mean free path. Thus we have Nsteps

^j2-d

^centers

Since the number of scattering centers visited by a particular random walk is equal to AT s t e ,p sthe result (2) means that the fraction of scattering centers visited by a typical electron will also vary as L2~D. Equivalently, each center will be visited by a fraction ~LD~2 of the electrons. W e know from eq. (1) that the motion of all of the scattering centers affects all of the electron trajectories, and

Fluctuations and noise: Experiment

137

results in a fluctuation 8G ~ e2/h. Hence, we would naively expect that since the motion of one center will affect a fraction ~Ld~2 of these trajectories, it should produce a conductance fluctuation which is smaller than eq. (1), with the result hG^e^Ud-2)i2.

(3)

2 Here 8G1 is proportional to L ( d _ 2 ), / rather than the factor Ld~2 in eq. (2), because G is obtained by combining many trajectories, the effects of which are assumed to be independent. It is not obvious that this assumption of 'independent' trajectories is justified (Feng et al. 1986a), especially in view of the coherent effects characteristic of mesoscopic systems. Nevertheless, it seems that this assumption is reasonable in this case. From eq. (3) we expect there to be a large change in the conductance even if only a single scattering center is moved, especially in one or two dimensions. The fluctuations become larger as the dimensionality is reduced, and even appear to diverge as L - > oo in one dimension. However, the result (3) is only applicable in the diffusive regime, which corresponds to G ^ e2/h. As L - > oo in one dimension, this condition is violated, and eq. (3) is no longer applicable. If more than one scattering center moves, the total fluctuation is obtained by simply multiplying ( 6 G J 2 from eq. (3) by the number of centers which are moved, with the constraint that the total fluctuation cannot exceed ~e2/h9 since the fluctuation magnitude cannot exceed the value found for a complete change in the random potential, eq. (1). This upper bound is known as the saturation limit. The precise result for 8GX depends on the dimensionality, as well as on kF and L e, through factors which are not included explicitly in eq. (3). In three dimensions, we have (Feng et al. 1986b)

(4) Here L is the size of the system (which in this case is assumed to be a cube). The factor c(x) = 1 — [sin 2(x/2)]/(x/2) 2 in eq. (4) takes into account the distance through which the scattering center moves, and is %1 when this distance is « kF

1 or

larger, and vanishes in the opposite limit. N o t included in eq. (4) is a

term corresponding to what is known as energy averaging. This is an effect due to thermal averaging of the energy dependent conductance (Stone and Imry 1986, Lee et al. 1987). The effect of energy averaging depends on the relative sizes of L4 and the thermal length LT = y/Dh/kB T. This effect is generally not too large in metals (i.e., the factor which would enter eq. (4) is usually of order unity), so for simplicity we will not include it explicitly in our discussions below.

138

Ν. Giordano

While it is possible to study U C F effects in systems which are sufficiently small that eq. (4), or the analogous result in one or two dimensions, applies, it is also important to consider these fluctuations under more general conditions. In particular, we will now consider what happens when the temperature and system size are such that the phase breaking length is smaller than one or more dimensions of the system. In this case, the behavior will be 'coherent' on length scales less than L4, but phase breaking effects will prevent coherence over longer scales, and it is useful to view such a system as composed of coherent subsystems (Feng et al. 1986a). If the system is large in all three dimensions, then the subsystems are regions of order L j in size, etc., if one has a thin film or thin wire geometry. Qualitatively, the behavior within a subsystem will be the same as discussed in the preceding paragraphs, hence the conductance of a subsystem will exhibit fluctuations described by eqs. (1) and (3). Because of phase breaking effects, the behavior of different subsystems will be uncorrelated, and the fluctuations of the entire system are obtained by combining those of the individual subsystems incoherently, according to the usual series-parallel rules for adding fluctuations. T w o situations are of primary interest. If only a single scattering center in the entire system moves, then one finds (Feng et al. 1986a) (5) where ( 8 G ) 2 is the (mean square) fluctuation of the entire system, ( 8 G X) 2 is the fluctuation of a single subsystem produced by the motion of one impurity, eq. (3), and Lz is the length of the system in the current carrying direction. The result (5) is applicable in one, two, and three dimensions. Another situation we will encounter is one in which there are many scattering centers moving in each subsystem. In this case one finds (Feng et al. 1986a) (6) where G and Ω are the conductance and volume of the entire system, respectively, G s and Qs are the analogous quantities for a subsystem, and ( 8 G S) 2 is the mean square magnitude of the fluctuation of a single subsystem. The shape of a subsystem will depend on the dimensionality, i.e., £2S= L | in three dimen­ sions, etc. Since we assume that many scattering centers are moving within each subsystem, ( 8 G S) 2 is given by N^SG^2, where Ns is the number of centers which move in a subsystem, and ( 5 G X) 2 is given by eq. (3) [or eq. ( 4 ) ] . The restriction that the fluctuation for a single subsystem cannot exceed e2/h still applies, so if iV s is large enough that Nl/2bGx ^ e2/h9 the fluctuation magnitude is saturated, and we have ( 8 G S) 2 « ( e 2/ h ) 2 in eq. (6).

Fluctuations and noise: Experiment 2.2. Low-frequency

139

noise

In the preceding section we distinguished two different regimes concerning the 'dynamics' of the fluctuations. In the first case, the movement of a scattering center was assumed to be sufficiently rare that the individual conductance fluctuations are well separated in time, and hence can be treated as isolated events. In the opposite limit, the fluctuations due to many scattering centers were assumed to overlap in time. While there was no problem, in principle, in estimating the magnitude of the fluctuations in the latter case, it is also interest­ ing to consider the frequency spectrum of the fluctuations Feng et al. (1986a) predicted that this spectrum should vary with frequency as l/f. This result follows from ideas developed to understand the general problem of low-fre­ quency noise (Dutta and Horn 1981). If one has a random process which is characterized by a single characteristic time scale, τ, then one finds a Lorentzian spectrum

τ 2( 2 π / ) 2 +

(7)

Γ

where S is the power spectral density of a quantity such as the conductance. However, in a system such as a disordered metal, one would expect to find a distribution of characteristic times, so that < °

τW )

W 2

+

1

(8)

where in our case, D(z) characterizes the distribution of time scales for defect motion. W e expect that this motion will involve the motion of atoms, or clusters of atoms, over potential barriers which separate two (or more) metastable states. In general, this motion will be thermally activated, hence T

=

eT£/o * B T

(9)

where Ε is the height of the potential barrier. In a disordered system there will be many different defect configurations, resulting in a broad distribution of barrier heights. If this distribution D(E) is uniform, then eqs. (8) and (9) lead to a spectral density which is proportional to l/f (Bernamount 1937, du Pre 1950, van der Ziel 1950). O f course, D(E) will never be perfectly uniform, but Dutta and Horn (1981) have shown that, in general, the spectrum has the form S~l/f,

(10)

where the spectral slope, a, is close to unity, as long as the variations of D(E) occur over scales large compared to kBT. Since kBT is small compared to typical energies associated with defect motion in a solid, this assumption is quite plausible. Hence, one would expect U C F to be a source of l/f noise.

140

Ν. Giordano

This connection between low-frequency noise and U C F was first noted by Feng et al. (1986a). They also pointed out that this noise would not be restricted to small (i.e., mesoscopic) systems at low temperatures, and suggested that it could be a major source of low-frequency noise in metal films even at room temperature. This proposal has attracted much attention for the following reason. The problem of 1/f noise in metals (and indeed, in many other systems) has been of continuing interest for many years. The work of Dutta and Horn, and others (see, e.g., Weissman 1988a), showed how a 1/f spectrum could arise in a fairly natural way in a system with even a small degree of disorder (i.e., inhomogeneity). However, while that work explained the form of the spectrum, the actual microscopic origin of the noise, and in particular an explanation of its magnitude, has remained uncertain. General arguments lead one to expect a connection with defect motion (Weissman 1981, 1988a, K o g a n and Nagaev 1982, 1984, K o g a n 1985, Black et al. 1983, Robinson 1983), and this is also indicated by a number of experiments conducted in the past five years or so (Weissman 1988a, Giordano 1989). However, prior to the work of Feng et al., there were no quantitative calculations of the size of conductance fluctuations, and thus the low frequency noise, produced by defect motion. Hence, their work provided a possible answer to an important part of the 1/f noise problem, at least for the strongly disordered systems to which the theory applies (see below). W e should note that another, related, mechanism appears to be respon­ sible for the noise in weakly disordered systems, and this is discussed in section 5. Of course, it is extremely important to test this idea experimentally. In addition, it is quite conceivable that there could be other microscopic mechanisms by which defect motion couples to the conductance. W e will briefly discuss another such mechanism in section 5. Let us next consider the magnitude of SG to be expected from U C F (here and below we denote the power spectral density of fluctuations of a quantity χ by Sx). One can employ eqs. (6) and (4) to show that (Pelz and Clarke 1987a, Feng et al. 1987)*

(ii) where Ν is the total number of conduction electrons in the sample, η is the electron density, and ns is the density of mobile scatterers. The derivation of eq. (11) assumes that the conductivity is given by the usual free-electron ex­ pression, ne2ze/m, where τ 6 is again the elastic scattering time, and m is the electron mass. If the number of moving defects is large, then one can have saturation, as discussed above in connection with eq. (3). This occurs when

* Since eq. (11) is obtained using eq. (4), it is only applicable in three dimensions. However, a similar result is found for other dimensionalities.

Fluctuations and noise: Experiment n^libG^2

N

141

^ e*/h2, and in this case eq. (11) must be replaced by


(l2)

where 3π

L„

2

kFLi

(13)

A very important feature of U C F noise, eqs. (11) and (12), is that if all other parameters remain fixed, the magnitude of the fluctuations, and hence also the noise, becomes larger as Εφ increases, that is, as the temperature is reduced. Let us now evaluate eqs. (11) and (12) for a typical case. W e consider a fairly disordered system with L e = 10 A. At temperatures above % 10-20 Κ the phase breaking is generally due to electron-phonon scattering, and in this case τ φ ~ δ / & ΒΓ (Feng et al. 1987, Giordano 1989). Taking the typical values kF= 1 χ 10 8 c m " 1, and vF = l χ 10 8 cm/s, we find at room temperature (n s/n) 0 = 0.06. If we assume that all of these impurities are moving, this yields [ i V ( 8 G ) 2/ G 2] t h e yo ~ r 0.3. Here we have calculated the total, integrated magnitude of the fluctuations, while noise measurements usually yield the spectral density, SG(f). In general, the spectral density can be written in the form (Dutta and Horn 1981, Weissman 1988a)

2 - * , , ·

G2

(14)

Nf

This form is convenient, since the dependence on Ν is explicit, so that γ is independent of Ν (and hence also the volume of the system). W e will therefore often refer to γ as the 'magnitude' of the noise. W e can integrate this expression to obtain (SG)2, via the relation for SG(f) (5G)2 =

SG(f)

df.

(15)

Since S~l/f, performing the integral is no problem, the only difficulty is in determining the limits of integration. Fortunately, the integral depends logarith­ mically on the values chosen for the limits, so only a rough estimate is required. A reasonable upper limit would be τφ9 while we might take ~ 1 0 ~ 10 H z for the lower limit. Assuming a l/f spectrum over 20 decades of frequency, and that the remaining frequency ranges contribute negligibly to the integral, yields N ^ * 4 6 y .

(16)

Experiments find that the amplitude γ is typically in the range from 1 χ 1 0 " 4 to 1 χ 10"2, which gives [ N ( 8 G ) 2/ G 2] e xp = 5 χ 1 0 " 3 to 0.5. This is quite consis-

142

Ν. Giordano

tent with the theoretical upper limit of 0.3 obtained above. N o t e that in obtaining this prediction for the absolute magnitude of the noise, we have assumed that the number of mobile impurities is 6% of the total number of conduction electrons (i.e., of order 6% of the total number of atoms in the sample), which at first sight seems to be quite high. However, we must also consider the time scales involved. If we assume the lower cutoff to be at 1 0 " 10 H z (recall that we assumed the spectrum to have a l/f form for 20 decades of frequency), then we require only that these impurities be mobile on the scale of 1 0 10 seconds, i.e., 300 years, which does not seem at all unreasonable. In addition to comparing the predicted magnitude of U C F noise with the magnitude of the l/f noise observed in real systems, it is important to consider if there are any features of U C F noise which could be used to distinguish it from other microscopic sources. Since becomes larger as the temperature is reduced, the number of subsystems becomes smaller. This in turn makes the change in G produced by the motion of a single scattering center larger. Hence, if all other factors remain fixed, eqs. (11) and (12) predict that U C F will become larger as Τ is lowered. This is opposite to the behavior produced by other types of fluctuations, and is a clear signature of U C F noise. However, U C F noise does not always become larger as Τ is decreased. Other factors can be temper­ ature dependent, and dominate the variation of L 0. In particular, the density of mobile scatterers, n s, will usually become smaller as Τ reduced, and this will tend to offset (or even dominate) the effect of changes in L 0. Another distinctive feature of U C F noise is its dependence on magnetic field. It has been shown that the application of relatively small magnetic fields has a large effect on U C F (Al'tshuler and Shklovskii 1986, Lee et al. 1987). This has to do with the effect of time reversal symmetry on the statistics of U C F . Magnetic fields which produce a flux of order or greater than one flux quantum through an area « Ll are predicted to reduce the mean square U C F fluctuation, and hence the magnitude of U C F noise, by a factor of precisely two (Stone 1989, Al'tshuler and Shklovskii 1986, Lee et al. 1987). The fields required for this reduction of γ can be relatively small, of order 10 3 G or less, which is much smaller than the fields which are usually necessary to have a significant effect on conduction in metals. Thus, the observation of this effect is a clear and unambiguous signature of U C F noise. Throughout our discussion of U C F noise, we have also assumed that the D u t t a - H o r n model applies, and hence that the spectrum will have a l/f form. In deriving the frequency dependence of the spectrum, this model essentially assumes only that the motion of the individual fluctuators is thermally activated, and that their effects combine independently. Weissman (1987) has pointed out that this last assumption is not necessarily satisfied by U C F . The key idea is that the fluctuations produced by the motion of different scattering centers are not independent, because of the overall condition that 8G cannot be larger than ~e2/h. Consider, for simplicity, a case with only two types of fluctuators,

Fluctuations and noise: Experiment

143

fast ones and slow ones. The fast fluctuators will cause the system to partially 'forget' the effects of the slow ones for the following reason. Suppose a slow fluctuator switches from one state to the other, and then stays 'switched' for a period of time. During this time, many fast fluctuations will occur, and if the conductance changes they produce saturates 8G, i.e., if 5G « e2/h in that subsys­ tem, then when the slow fluctuator switches back, the conductance will be completely uncorrelated with respect to the situation when it originally switched. Roughly speaking, the different fluctuators are not uncorrelated because the universal conductance fluctuations involve the positions of all of the scattering centers. Weissman (1987) has shown that this makes the slow fluctuators less effective in producing noise, leading to a decrease in the fre­ quency exponent a, [eq. (10)]. The amount by which α is reduced depends on how close one is to saturation, with shifts of ^ 0 . 1 being typical. In addition, the overall magnitude of the noise is also reduced. It is unclear just how important this effect is in a real system. It may also be difficult to detect, since the experimentally observed values of α are often as low as 0.8, and it is hard to accurately predict the magnitude of U C F noise. In any case, we would expect this to be a significant effect only near saturation. Despite this complication with regards to the spectrum of the U C F noise, it seems clear that for highly disordered metals, and especially at temperatures well below room temperature, this mechanism is capable of accounting for all, or certainly a large part, of the low-frequency noise commonly observed in these systems. W e will discuss this problem further in connection with the experiments in section 4, and when we consider another noise mechanism in section 5.

3. Fluctuations due to the motion of single defects In this section we consider experiments involving metal films at very low temperatures. The experiments are, in principle, quite simple; the conductance is simply measured as a function time, with the temperature held fixed. The initial experiments* (Beutler 1986, Beutler et al. 1987, Meisenheimer et al. 1987a,b) involved Bi films « 1 0 0 - 2 0 0 A thick, and ^ 1 0 μηι on a side, and very narrow Bi strips (i.e., wires) whose width and thickness were both ^ 4 0 0 A. Some typical results are given in fig. 1, which shows results for the resistance of a Bi film as a function of time. The rapid fluctuations which have a magnitude of approximately 10 Ω (peak-to-peak) are due to the measuring electronics. In addition to this system noise, there are also 'slower' fluctuations, two of which are visible in fig. 1. The first one occurs at ^ 1100 s on the time scale. Here the * Similar U C F fluctuations due to single atom motion have also been observed, but not studied extensively, in inversion layers at low temperatures (Skocpol et al. 1986).

144

Ν. Giordano

7830 1000

2000

3000

4000

TIME(s) Fig. 1. Resistance of a thin Bi film as a function of time at 195 m K . The sample was 16 μπι long and 8 μπι wide. After Beutler et al. (1987).

resistance jumps abruptly to a value significantly higher than it had at earlier times, stays at this new value for about 60 s, and then jumps abruptly back to its original value. There are clearly many possible interpretations of this beha­ vior, but the one which seems most plausible (especially in view of further results discussed below) is that a scattering center moved to a new position, stayed there for a period of time, and then moved back to its original location. This behavior is repeated at ^2900 s in fig. 1, although the 'fluctuator' switched back to its initial state more quickly in this instance. The fact that the fluctuation magnitudes are the same in the two cases suggests that the same defect was responsible for both fluctuations. Later measurements of similar Bi samples with improved electronics and analysis methods yielded results like those in fig. 2. These results are similar to those in fig. 1. However, several instrumental changes were significant, the most important being that the results in fig. 2 were obtained from a time scan much longer than that of fig. 1, and that the level of system noise was reduced by digital filtering. This filtering was accom­ plished by simply averaging adjacent resistance values; for the results in fig. 2 the data were taken approximately once every minute, and every 10 adjacent values were averaged. In addition, in these experiments the measuring electron­ ics was frequently switched between several different samples; that is, data was accumulated in a 'multiplexed' fashion. The fluctuations observed for different

Fluctuations and noise: Experiment

145

38500

f

38000

~

%

37500

U.AJ

U UAJLJU

37000

36500

5000

10000

15000

20000

TIME(s) Fig. 2. Resistance of a thin Bi film as a function of time at 70 m K . The sample was 10 μπι long and 4 μπι wide. Different fluctuators are indicated by the arrows. The numbers labeling the arrows are used to indicate what appear to be repetitions of the same fluctuator. After Meisenheimer et al. (1987b).

samples were found to be uncorrected, demonstrating that they were not due to any sort of instrumental effect, but were indeed properties of the samples. Returning to fig. 2, fluctuations of several different sizes are evident. It seems natural to assume that fluctuations of the same magnitude are due to motion of the same scattering center, and with this assumption we have identified several different fluctuators. It can be seen that the fluctuators are, at least in this case, independent; i.e., the associated resistance changes appear to be simply additive. Qualitatively similar results have been found for Pt and A g films (Meisen­ heimer et al. 1987a,b, Meisenheimer and Giordano 1989). In all cases the number and sizes of the observed fluctuators varied greatly from sample to sample, and even from day to day for a given sample. This implies that the structural units responsible for the noise are extremely sensitive to their environ­ ment. Unfortunately, these variations make it difficult to perform a quantitative analysis of these results. Nevertheless, it was possible to draw several conclu­ sions from these experiments. First, the vast majority of the fluctuations con­ sisted of only abrupt resistances changes. The best time resolution in these experiments was generally no better than a few seconds, so this is certainly not surprising given our picture of the fluctuations as being due to the motion of

Ν. Giordano

146

75.6

75.4

75.2

α 75.0

74.8

74.6

0

1000

2000

3000

4000

TIME(s) Fig. 3. Resistance of a 460 Λ diameter Bi wire as a function of time at 116 m K . After Beutler (1986).

single defects. However, in some of our first experiments, much slower fluctua­ tions were observed on several occasions. An example of a slow fluctuation is shown in fig. 3, which gives results for a 460 A diameter Bi wire (Beutler 1986). Only one 'fluctuation' is shown here, and it is seen to take several minutes to 'switch' from one state to the other. It is hard to see how such behavior could be due to any sort of reversible motion of a single defect or cluster of defects. Rather, it would seem more likely that this is due to an irreversible rearrange­ ment of atoms. This view is supported by the fact that, in contrast to the fluctuations seen in figs. 1 and 2, fluctuators like those in fig. 3 were never observed to switch back to their initial state. As our studies of U C F have progressed, it has become clear that behavior like that shown in fig. 3 is extremely rare; the abrupt changes observed in figs. 1 and 2 are far more common. A second general result is that measurements as a function of temperature (Meisenheimer 1989) suggest that the motion of at least some of the fluctuators are thermally activated. This is concluded from the observation that the switch­ ing rate for a given fluctuator is often found to increase as the temperature is increased. The switching times in Bi, Ag, and Pt generally become extremely long below about 70 m K . A third general feature of these experiments is that the magnitude of the resistance change due to an individual fluctuator increases as the temperature is lowered. This is a unique signature of the U C F mechanism. Some early results for the magnitude in the fluctuations in Bi films and wires as a function of temperature are shown in fig. 4. The time traces for these samples (e.g., fig. 1) strongly suggest that these are individual fluctuators (as opposed to many

Fluctuations and noise: Experiment

147

T(K) Fig. 4. Results for 6G for the thin Bi film considered in fig. 1 (open circles, left scale), and the thin Bi wire considered in fig. 3 (filled circles, right scale). The dashed and solid lines are the theoretical predictions for the two cases, respectively, and are discussed in the text. After Beutler et al. (1987).

overlapping ones), and fig. 4 also shows the predictions of the theory (Feng et al. 1986b) for this case. The theoretical predictions are [see also eq. ( 5 ) ]

/M e i 2 (/c " LV)%U- g L CC(/C f6T)

(ss)2 =

F

4

4

e

a?)

in one dimension (the solid line in fig. 4), where L is again the length of the sample, and L ec(/c F8r) / L A 4 e4 (8G)2 = 4 8 7 ^ r ^ ( - ^ ) - I , (kFLjt \ Lj h2

(18)

in two dimensions (the dashed line in fig. 4), where t is the film thickness. While the predicted temperature dependence agrees with the data, the predicted magnitude is significantly smaller than observed. There appear to be several possible reasons for this discrepancy. One is that the values used for the parameters which enter eqs. (17) and (18) are in error. However, it is hard to see how the estimated values* of fcF, L e, i, L 0, etc., could account for more than about half of the error (the total discrepancy is approximately a factor of 10) in 5G seen in fig. 4. The most likely explanation of the remaining discrepancy concerns the manner in which 8G was estimated from the measurements. * T h e phase breaking length, L^,, was found from measurements of the weak localization magnetoresistance on these and similarly prepared samples (Beutler and G i o r d a n o 1988).

148

Ν. Giordano 60500,

1

1

—

58500h

580001 0

1 5000

1

10000 TIME(s)

15000

1

1

20000

Fig. 5. Resistance as a function of time for a 10 A thick Pt film at 250 m K . After Meisenheimer et al. (1987b).

Typically only a few different fluctuators were ever observed at any given temperature. Since the magnitude of the noise due to the measuring system was not negligible, the small fluctuators were lost in the system noise, and their contribution to 8G could not be obtained. Hence, the estimates of 6G were unavoidably biased towards the largest fluctuators, and the value of 8G derived from the experiments was almost certainly larger than the correct value.* It seems quite plausible that at least half the discrepancy seen in fig. 4 could arise in this way, and this would make the results at least consistent with the theory. W e will return to this problem below. The results in fig. 2 suggest that the fluctuators move independently. Occa­ sionally however, the motion of different fluctuators appears to be correlated. Figure 5 shows such results for a thin Pt film. Here there is a slow fluctuator which has a relatively large magnitude. When this fluctuator is in its low resistance state, a second fluctuator is seen to switch quite rapidly, but when the slow fluctuator is in its high resistance state, the fast fluctuator becomes inactive. This clearly implies that the two fluctuators interact, and hence that they must be near each other, presumably within a distance of order of the •These estimates of 5G were obtained by simply averaging the magnitudes of the

fluctuators

observed at a given temperature. If a large number of fluctuators could be observed, it should be possible to analyze the distribution of their magnitudes, and obtain an improved estimate of 6G which would be less sensitive to this problem. However, in these experiments, only a few fluctuators were ever observed at any given time (see, however, the results for A g below).

Fluctuations and noise: Experiment

149

screening length. Similar behavior has also been found occasionally in Bi. This behavior was, on the whole, quite rare, which is not surprising since one would expect it to be unusual to find fluctuators sufficiently close together. In more recent work on A g films (Meisenheimer and Giordano 1989, Meisen­ heimer 1989), it has been possible to address some of the shortcomings of the earliest experiments involving Bi. The A g samples were patterned using optical projection lithography to obtain a microbridge-like geometry. They were typi­ cally 0.5 μιη on a side, which is about an order of magnitude smaller than the Bi and Pt samples for which results are given above, which were made by mechanical scribing. The dimensions of the A g samples were comparable to L^, and hence they consisted of only a small number of independent subsystems ( % l - 5 , depending on the sample) at 100 m K . * Some typical results for the resistance as a function of time for a small A g film are shown in figs. 6 and 7. Here we show the raw data (fig. 6) and also the results after digital filtering to average out the most rapid fluctuations (fig. 7). T w o features of these results are of particular interest. First, there are occasional slow fluctuators, indicated by the arrows in the filtered data in fig. 7. These slow fluctuations occur over long time scales, are similar to those seen in figs. 1 and 2, and are clearest after filtering has been used to reduce the magnitude of the faster fluctuations. Second, comparison of the noise 'envelopes', i.e., the size of the 'fast' fluctuations in the unfiltered data (see fig. 6) indicates the presence of rapid fluctuations whose magnitude is strongly temperature depen­ dent. Presumably the slow and fast fluctuations both arise from the motion of defects, but with different time scales involved in the two cases. The slow fluctuations allow us to examine the motion of individual defects, since the individual slow fluctuations are well separated and resolved. The fast fluctua­ tions cannot be individually resolved, but since they occur quite frequently they make it possible to obtain a more accurate measure of the magnitude of the 'average' fluctuation than was possible in the experiments with Bi and Pt discussed above. Figure 8 shows results for the rms fluctuation, which we will refer to as 8G, as a function of temperature, obtained from measurements like those in fig. 6. It is seen that 8G increases as the temperature is reduced, as predicted by U C F theory. The magnitude of 5G at the lowest temperatures is &0.3e2/h. Magnetoresistance measurements (Meisenheimer and Giordano 1989), and also theoretical estimates (Al'tshuler and Aronov 1985), indicate that at the lowest temperatures, Εφ is of order 4000 A for this sample. Since the sample length and width were only slightly larger than L 0, this sample was, at * Another difference between the samples made by optical lithography and those made by scribing, was that in the latter case, silver paint was used to partially define the length, and attach leads. However, it does not appear that this caused any problems. In particular, the experiments with thin Bi wires employed samples which were typically 50 μπι in length, much longer than the thin film samples. To the extent that the two types of samples exhibited similar results (fig. 4, etc.), one can conclude that the A g paint was not a source of problems.

150

Ν. Giordano

90.4 Γ

90.2

90.0

89.8

<3

90.0

0.250 Κ

90.0 0.509 Κ I

89.8 20000

40000 TIME

60000

80000

( s )

Fig. 6. Resistance of a thin A g film as a function of time at several temperatures. The sample was approximately 1.0 μπι long and 0.5 μπι wide. After Meisenheimer and Giordano (1989).

the lowest temepratures, composed of only a few independent subsystems. Hence, according to U C F theory, if we are near saturation, we expect the magnitude of 5G to be within a factor of « 3 or so of e2/h. The observed value of 8G is thus in very good quantitative agreement with U C F theory. The decrease of 8G with increasing temperature is also expected, since becomes smaller as Τ increases, and this increases the number of subsystems, resulting in smaller fluctuations for the overall sample. Returning to the slow fluctuations evident in fig. 7, let us consider the magnitude. At 79 m K this magnitude is &0.2e2/h, for one such fluctuator. According to the theory, at temperatures such that the effective sample size*, L, is equal to L^, the fluctuation due to a single fluctuator, 8 G 1? is given in * I f the actual sample length is less than L^, then it turns out that the coherent region extends out into the sample leads, and the leads effectively become part of the sample insofar as the fluctuations are concerned. Hence the smallest sample length is in a sense just L^. See, e.g., Lee et al. (1987).

Fluctuations and noise: Experiment

J

151

0.05 Π

79m Κ

250mK

509mK

20000

40000 TIME

60000

80000

(S)

Fig. 7 Same data as in fig. 3, but after digital filtering with a time constant of « 5 min. The arrows indicate slow fluctuators, as discussed in the text. After Meisenheimer and G i o r d a n o (1989).

0.6

(3 CO

Fig. 8 Temperature dependence of 6G, in units of e2/h for an A g film at low temperature, as obtained from data like that in fig. 6. After Meisenheimer and G i o r d a n o (1989).

two dimensions by (Feng et al. 1986b) (5G,)2

L e c(kF8r) = 1.44-5 L (kPLe)2

e* h2'

(19)

Using our best estimates for kF, etc., in eq. (19) we find 8Gi κ 0.07e2/h, which

152

Ν. Giordano

is a factor of about 3 smaller than found from fig. 7. When one realizes that the experimental value will (as in the analysis of the Bi data) be biased towards larger values of 5 G 1? this level of agreement is quite satisfactory. W e therefore conclude that the theory provides a good picture of these experiments, especially for Ag. The early results for Bi and Pt are in good qualitative agreement, but the absolute magnitude of the fluctuations found for Bi was somewhat larger than predicted. N e w experiments with smaller Bi samples made using projection lithography are currently in progress, and preliminary results (Meisenheimer 1989) are in agreement with the theory. M o r e work is needed to understand better why the earlier experiments do not agree, but it seems safe to conclude that these experiments give strong support to U C F theory.

4. Low-frequency UCF noise As discussed by Weissman (1988a), very general arguments lead one to expect that low-frequency noise in metals is likely to be due to the motion of impurities. Quantitative support for this idea was provided by Feng et al. (1986a), who pointed out how U C F could give rise to 1/f noise. W e next discuss several experiments for which it appears that U C F are the dominant source of lowfrequency noise. Koch and co-workers (Koch et al. 1985, 1988, K o c h 1988) have measured the noise of very thin, highly disordered, A u films. By using ion-milling to vary the thickness, it was possible to measure the variation of the noise magnitude as a function of G for a single sample, thus eliminating many sources of systematic error. This noise was found to have a power spectrum which varied as 1/f, and fig. 9 shows results for the magnitude of the resistance noise, SR, as a function of the sheet resistance, R. Here we see that SR/R2 was proportional to R2, as R was varied over 3 decades. This behavior can be understood in terms of U C F theory in the following way. Using the fact that SR/R2 = SG/G2, together with eqs. (6) and (15), one can show that in two dimensions the average fluctuation of a single subsystem, 5G S, is given by (20) here we have used the fact that G x = G = 1/R for a thin film. W e also have Ω/Ωγ = A/Al9 where A is the area of the entire sample, and Ax is the area of a coherent subsystem, respectively. Using Ax = L\ we find (21)

Fluctuations and noise: Experiment 10

153

-8

1 0 "

•V 10

Ν

X

10

-10

/A

slope = 2 . 0 11

/

h A/

oc

QC

CO

Τ (Κ)

4

12

io' h • A

io" 1 3h

-12 ^

-13 QC

in - 1 4 14

ίο" h

4

ο

-15 50

10

150

250

350

-15

1(T

10°

10

10

10

10

10

R e s i s t a n c e (Ω) Fig. 9 Magnitude of the noise of a thin A u film as a function of resistance, R, which is approximately the sheet resistance. The resistance was varied by ion milling. The different symbols denote data for different samples. After Koch et al. (1988).

The experiments (fig. 9) show that SR varies as R 4, hence eq. (21) implies that 5G S is independent of R. This is precisely what is expected for U C F noise. A quantitative comparison with U C F theory requires a consideration of L e and Εφ. K o c h and co-workers estimated that L e ~ 38 A, while Εφ was about a factor of 3 larger, so these samples should indeed be in the U C F regime. Using eq. (21) to obtain ( 8 G S) 2 from the measured SR, one finds the results shown in fig. 10. W e see that the magnitude of the fluctuation for a single subsystem, 8G S, is approximately e2/h, which is in very good agreement with the U C F model, assuming that the noise is saturated. This provides further strong evidence in favor of the U C F model. This result for 8G S is quite striking, and difficult to understand without the U C F model. It is also useful to consider the temperature dependence of the noise found by K o c h and co-workers. They found (see the inset in fig. 9) that for the highconductance samples, the noise magnitude increased as the temperature was increased, typically by about two orders of magnitude as the temperature varied from 77 Κ to room temperature. As noted in our discussion of U C F noise, if all other parameters are kept fixed the magnitude of this noise should decrease as the temperature increases, since Εφ decreases in this case. However, this assumes that the concentration of mobile impurities, /is, remains the same,

Ν. Giordano

154

1(Γ

10 \

CM

Ο)

4**

Ο 10

10

-2

1(T

1CT

10

10

1CT

10

RESISTANCE (2) Fig. 10. Magnitude of the fluctuations of a single subsystem (i.e., a region of size

χ L^), ( 6 G S) 2

[see eq. (21)], as a function of Κ for thin A u films. The different symbols denote data for different samples. Note that ( 6 G S) 2 is measured in units of e4/h2. After Koch et al. (1988).

which certainly might not be the case. The temperature dependence found by K o c h and co-workers implies that ns increases rapidly as the temperature increases. The results in fig. 10 were obtained at room temperature, hence the observed temperature dependence indicates that below room temperature the conductance fluctuations in a single subsystem, must become smaller than e2/h. This is consistent with U C F theory, which requires only that 6G S cannot become larger than e2/h. However, the fact that 5G S at room temperature (fig. 10) is &e2/h also means that the noise is saturated, and hence 8G S cannot become any larger at higher temperatures. Since L4 is a monotonically decreas­ ing function of T, this implies that above room temperature the noise magnitude must decrease with increasing temperature. Unfortunately the measurements of K o c h and co-workers stop at room temperature, although there is some hint that SR does begin to flatten at the highest temperatures studied (see again the inset in fig. 9). Further experiments at higher temperatures would certainly be of interest, and should provide a strong test of U C F theory. A complication encountered in trying to account for all aspects of these results is that, since one appears to be near or at the saturation limit, the arguments of Weissman (1987) concerning the spectrum of U C F noise should apply. Hence, we would presumably not expect a 1/f spectrum in this case, although the deviations from this form might not be large (i.e., the spectral slope α could still be fairly close to unity). While it is not clear how to reconcile all of these results with the U C F model, we do not see how to account for the constancy of ( 8 G S) 2 seen in fig. 10 without appealing to U C F theory.

Fluctuations and noise: Experiment

155

Fleetwood and Giordano (1985) have studied the noise of A u P d alloys which are extremely disordered, with typical resistivities in the range 100-400 μΩ cm. These authors measured the temperature dependence of the noise after various stages of annealing, and noted that the overall behavior was consistent with the general picture that defects are responsible for the noise. Here we analyze quantitatively the magnitude and variation of the noise with annealing, and also discuss the temperature dependence, in light of U C F theory. Figure 11 shows typical results for the noise magnitude as a function of temperature before and after two different anneals. The behavior above 300 Κ is complicated by the fact that the sample was annealing during the measure­ ments, so we will consider only the results below this temperature. First we consider the magnitude of the noise. For data set A in fig. 11, the resistivity was 180 μΩ cm at 300 K. Using Matthiessen's rule together with the fact that the resistivity of clean bulk A u P d is 20 μΩ cm at 300 K, we obtain L e « 4 A. From U C F theory [eq. (12)] we then find saturation when (ns/n)^(ns/n)0 = 6 χ 10 ~ 2. This would correspond to a maximum possible U C F noise of N(8G)2/G2 = 1.1, where we have used the same relations for τφ, etc., as in section 2. If we assume again that the l/f spectrum extends over 20 decades of frequency, this gives a maximum spectral magnitude y % 2 x 1 0 ~ 2, which is about a factor of 2 larger than observed. Hence, U C F theory can account for the magnitude of the noise if we assume that the density of mobile defects is about 3 χ 10 _ 2/atom. The total defect density can be estimated from the 10"

_j 100

ι

I

I

200

I 300

L

400

T(K) Fig. 11. Noise magnitude, γ, as a function of temperature for A u P d after various stages of annealing. Data set A was taken first, then B, etc. After Fleetwood and Giordano (1985).

156

Ν. Giordano

resistivity as 4 χ 10 2 2/cm 3 ^0.5/atom. Hence, only six percent of the defects need to be mobile in order for U C F theory to account for the noise magnitude, which seems quite reasonable. It is also interesting to consider the variation of the noise magnitude with the resistivity, p, as this quantity is varied with annealing. Data sets A, B, and C in fig. 11 correspond to ρ = 180,160, and 125 μΩ cm, respectively. Let us first assume that annealing does not change the fraction of defects which are mobile. W e can then use eq. (11) to calculate the variation of the noise magnitude y with p. Since ~ y/D ~ y/r~e, and the resistivity is proportional to τ~1, U C F theory predicts that y should vary as p 3 / .2 In the present case, this yields (y A/yc)theor y ~ 2 (the subscripts here refer to the different data sets), which is somewhat smaller than the factor of 10 which is observed at 300 K. However, this difference is not surprising. In data set A in fig. 11, γ increases rapidly above about 200 K, suggesting that above this temperature some defects, which were immobile at lower temperatures, are now able to move about and hence contribute to the noise. Such behavior is not as dramatic in data set C, presumably because many of these defects have been removed by the previous annealing steps. This suggests that it may be more appropriate to compare with the theory at lower temper­ atures, i.e., 200 K. Indeed, the ratio ( y A / y c ) e x P~ 4 at 200 K, in better agreement with the theory. Another reason the theory might underestimate yA/yc is that the annealing would almost certainly preferentially remove the mobile defects which are reponsible for the noise, as opposed to the immobile defects. Without more information, it is impossible to estimate how large this effect might be, so a more quantitative comparison with the theory is not possible at this time. In any event, it seems fair to conclude that the \/f noise of A u P d at room temper­ ature and below is quite consistent with the U C F model. Garfunkel et al. (1988) have studied the noise of C - C u composites. This is an attractive system for studying U C F noise, since by tuning the concentration one can make the conductivity fairly low (but still 'metallic' enough so that U C F theory applies), which makes the universal fluctuations a larger fraction of the total conductance. Figure 12 shows data for C - C u samples with several different concentrations. The magnitude of the noise y = NfSv/V2 [eq. (14)] is seen to increase as the temperature is reduced below about 40 K, a clear signature of U C F noise. For this system, G is relatively low, so the results derived for U C F noise in section 2 do not apply. In this regime U C F theory predicts (Garfunkel et al. 1988) y^nn%CL^L\

(22)

where ns is again the concentration of mobile defects, which are presumable two-level systems ( T L S ) in this case (Anderson et al. 1972, Phillips 1972). Here C is a factor which accounts for the shift of characteristic times with temperature

Fluctuations and noise: Experiment

102

I

,

1

10 Temperature

157

, 10 0

,

, , 100

0

(K)

Fig. 12. Results for several different C - C u composite samples, with a concentration of Cu near 15%. (a) Conductivity, σ, as a function of T. (b) Noise magnitude, γ, as a function of Τ for the same samples as in (a), (c) γσ2 as a function of Τ for the same samples. After Garfunkel et al. (1988).

Ν. Giordano

158

(as derived from the spectral slope α via the D u t t a - H o r n model), σ is the conductivity, and LT = yJhD/kB Τ arises from energy averaging effects, which we have now included explicitly. Theoretically one expects L 0 ~ T ~ 3 /4 and ns ~ T 1 , 3 -± 30. This relation for ns is obtained assuming that the T L S density of states exhibits the usual behavior found in other materials (Phillips 1987), which corresponds to a slight variation with energy. The measurements of α yield 4 putting this all together we expect that yo2 should be essentially C ~ T°A±0 , so independent of T, and this is precisely what is found at low temperatures, fig. 12c. The absolute magnitude of the noise is also quite reasonable. Expressing the measured noise in terms of the total conductance fluctuation, and assuming which 20 decades of l/f noise, Garfunkel et al. found ( 5 G S) 2 ~ 5 χ I0~2e*/h2, means that this is about 5% of the maximum possible U C F noise. The variation of ya2 with σ is harder to predict, since ns in eq. (22) presumably increases slightly as σ is reduced (i.e., as more disorder is introduced), while L4 and LT will decrease. Nevertheless, one can make rough estimates of these competing effects, which suggest that the more than order of magnitude varia­ tion of yo2 seen in fig. 12c, when comparing the results for different samples, is too large to be understood on this basis.* It may be possible to account for this variation in terms of deviations from the U C F theory which are expected to occur when σ becomes so small that the system is no longer 'metallic' (Weissman 1988b). U C F theory assumes that the electronic motion is diffusive, but if the elastic mean-free-path becomes very short, i.e., if σ becomes very small, corrections to this diffusive description become important. While a detailed calculation of the fluctuations due to the motion of a single impurity in the low conductance limit have not yet been carried out, there have been numerical calculations of the fluctuations produced by complete changes in the impurity configuration as a function of G (Giordano 1988). These calculations indicate that in both one and two dimensions, the magnitude of the fluctuations due to a complete change in the impurity potential, 8G, varies linearly with G. The results in fig. 12c yield ( 5 G ) 2 % G 0 7, with uncertainties (Weissman 1988b) large enough to be consistent with the linear variation seen in the numerical work (Giordano 1988). * T h e problem here is to estimate how ns^L\

varies with ns. Generally, one has

=

^/ϋτφ,

while LT = y/hD/kB T. These relations may be questionable in this case, since C - C u is near the metal-insulator transition, but we will nevertheless assume that they are applicable. D is propor­ tional to the 'bare' conductivity, i.e., the conductivity which would be observed if localization and electron-electron interaction effects were not present. These effects cause the temperature depen­ dence of σ seen in fig. 12a. The bare conductivity is presumably proportional to n'1, D ~ h s _ . 1 If we assume that the phase-breaking scattering,

then τφ~

is due to (three dimensional)

D 3 2/ ~ n ~ 3 /, 2and we find that ns^Ll

so we have

electron-electron

varies as n s~ 5 /. 4It is hard to estimate

ns from the data given by Garfunkel et al. (1988), but it appears that ns varied relatively little from sample to sample, probably only a few percent, and certainly much less than a factor of 3. If so, then ns did not change by an amount sufficient to account for the variation seen in fig. 12c in terms of the theory [eq. (22)].

Fluctuations and noise: Experiment

159

A comparison of α and the temperature dependence of y for C - C u using the D u t t a - H o r n model indicates that U C F noise dominates up to about 50 Κ (at much higher temperatures it appears that noise due to the local interference mechanism, discussed in section 5 below, is important). In addition, studies of the effect of a magnetic field, while not definitive, are consistent with the factor of two reduction of γ which has been predicted theoretically (Stone 1989). Very recently Birge et al. (1989) have reported noise measurements on Bi films at low temperatures. They have found that below about 70 K, the magni­ tude of the noise increases as the temperature is reduced, as shown in fig. 13. This behavior is easily explained in terms of the U C F model. For a thin film (i.e., a film whose thickness is less than L4) the size of a coherent region will vary as « L j . The overall temperature dependence of U C F noise results from a competition between L^, which becomes larger at lower temperatures, result­ ing in a smaller number of subsystems, and the number of mobile defects, n s, which one would expect to become smaller at low temperatures. Evidently, in the Bi films studied by Birge et al., the temperature dependence of Εφ dominates, and it is found that the magnitude of the noise, γ, varies approximately as T " 1 . The absolute magnitude of γ is consistent with U C F theory [eq. (11)], and is much smaller than the saturation limit (12). An especially interesting result of the experiments is that y decreases by a factor of two in the presence of a magnetic field, fig. 14, in very good agreement with the theory (Stone 1989, Altshuler and Shklovskii 1986), providing further proof that this is indeed U C F noise. This experiment provides very strong evidence in favor of the U C F model. It appears that the slow fluctuations seen in Bi by Beutler et al. (1987)

10-11

10"15' 0.1

I 1

I 10

I 100

T(K) Fig. 13. Temperature dependence of the noise of several Bi films at low temperatures (the different symbols denote data for different samples). After Birge et al. (1989).

Ν. Giordano

160

0.0

0.4

0.2

0.6

Η(Τ) Fig. 14. Variation of the noise magnitude of Bi as a function of magnetic field, at two temperatures; •

1.5 K, ·

0.5 K. After Birge et al. (1989).

were not present in the samples studied by Birge et al. However, the two experiments were carried out at very different temperatures, and it also seems plausible that films produced by different deposition techniques (Beutler et al. used sputtered films, while Birge et al. employed thermal evaporation) will possess different types of defects, etc. Thus, it is not surprising that the two experiments observed different behavior, although it would certainly be worth­ while to obtain a better understanding of this difference.

5. Non-UCF fluctuations - The local interference model There has recently been much progress in understanding n o n - U C F types of fluctuations in disordered metals. It turns out that the microscopic origin of these fluctuations has some interesting similarities to the U C F mechanism, and it therefore seems appropriate to briefly discuss the pertinent theory and experiments here (see also Weissman 1988a, Giordano 1989).*

* In addition to U C F and L I fluctuations, a theory based on the scattering of electrons from two-level systems ( T L S ) has been worked out (Ludviksson et al. 1984, Kree 1987). T L S noise can be viewed as a special case of L I noise. Since there does not appear to be any firm experimental evidence for T L S noise, we will not consider it here. For a discussion of T L S noise see Giordano (1989).

Fluctuations and noise: Experiment 5.7.

161

Theory

Implicit throughout our discussion of U C F was the assumption that the electron motion is diffusive; i.e., that the elastic mean-free-path is shorter than the phase breaking length. This assumption is justified in very disordered metals at room temperature and below. However, in weakly disordered systems, this condition is often not satisfied. In systems in which L e is longer than L^, it turns out that the motion of a single defect can, under the right conditions, again lead to a significant change in the conductance. In addition, the micro­ scopic origin of the fluctuation is similar, in some respects, to U C F . One can consider qualitatively a calculation of the contribution of a defect to the resistance (or equivalently, the conductance) as sketched in fig. 15b. Here we show a plane wave incident on a portion of a lattice which contains two defects, which scatter the wave elastically. The resistance due to these defects

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Two

neighboring vacancies. The dashed arrow signifies the motion of one vacancy. Here the initial and final configurations are not related by a simple translation, and hence some of the components of the resistivity tensor will change. The solid arrows indicate schematically incident and scattered waves, as would be considered in a calculation of either the resistivity or the structure factor.

162

Ν. Giordano

will be related to the scattering amplitude, and it can be seen that the calculation of the amplitude of the outgoing wave is analogous to the calculation of an X-ray structure factor. The simplest possible case is shown in fig. 15a. Here one can see that the motion of a single, isolated, vacancy from one location to another will not affect the scattering amplitude, since in this case the two lattice configurations are related by a translation through one lattice vector. Hence, in this particular example, motion of the defect will not change the resistivity. However, in many cases the lattice configurations before and after the defect motion are not related by a simple translation. An example of this is shown in fig. 15b, which shows a 'cluster' of two vacancies. The motion of one element of the cluster, as indicated in fig. 15b, will change the amplitude of the scattered wave, and hence will change several elements of the resistivity tensor. This mechanism through which defect motion couples to the conductance is known as the local interference ( L I ) model. The essential ideas were discussed by Kogan and Nagaev (1982, 1984), by K o g a n (1985), by Black et al. (1983), and by Robinson (1983). A quantitative treatment was first given by Pelz and Clarke (1987b). Like the U C F model, the L I mechanism is based on the interference of waves scattered from defects. In U C F , a large number of multiply scattered waves are involved, while L I fluctuations are due to a relatively small number of such waves. Clearly, the two mechanisms are closely related, and Hershfield (1988) has given a careful discussion of the two theories from a unified point of view. The first quantitative estimate of the magnitude of L I noise was made by Pelz and Clarke (1987b). They pointed out that the relevant calculations of the changes of the resistivity tensor had been performed previously by Martin (1971, 1972), in the context of a similar problem, and they used Martin's results to make quantitative predictions of the L I effects. Their final result can be cast in the form

= Ρ2

(nLmfp pac)2^, η

(23)

where ρ is the resistivity, L m pf is the total mean-free-path (due to both elastic and inelastic scattering), ac is the average scattering cross-section of a defect, and β = ( δ σ 0) 2/ σ 2 is the mean square relative change of σ0 produced when the defect moves. A quantitative comparison of eq. (23) with eq. (11) (Pelz and Clarke 1987a, Feng et al. 1987, Giordano 1989) shows that if n s, etc., are comparable, then L I noise dominates when the disorder is weak, i.e., when L e is long in comparison with L^, while U C F noise dominates in the opposite limit. Hence, we expect L I fluctuations to be important in weakly disordered systems and at high temperatures. In the next section we consider several experiments in which L I fluctuations have been observed.

Fluctuations and noise: Experiment

163

While we have discussed the different models separately, it seems clear that they are, as noted above, intimately related (Hershfield 1988). The U C F and L I models are both based on the fact that the interference 'patterns' of waves scattered by defects depend on the detailed arrangement of the defects. In the U C F model, the interference depends crucially on waves that are multiply scattered from many (i.e., 'all') of the impurities, while in the L I model the important interference terms involve only defects which are near neighbors. One could clearly imagine a 'unified' theory which would include all such terms, although it is not clear that such a theory would lead to the intuitive understandings which one obtains from the U C F and L I theories in their present forms. In this spirit, it might be worthwhile to consider corrections to the L I model by systematically including terms corresponding to higher order scattering (Weissman 1988b). 5.2.

Experiments

W e first discuss experiments by Ralls and Buhrman (1988) in which individual L I fluctuations were observed. They examined conductance fluctuations of structures with a point-contact like geometry. These structures were produced by first ion etching a small conical hole in a thin (500 A) free-standing film of S i 3N 4, then coating both sides with an evaporated layer of Cu. The result was a Cu point-contact, with the diameter of the constriction being in the neighborhood of 100 A. The experiments were performed at relatively high temperatures ( ^ 1 5 0 K ) , and the elastic mean-free-path was long (^1800 A for co-evaporated films), hence the L I model should be appropriate here. Some typical results for the resistance as a function of time are shown in fig. 16. The most common type of behavior which was observed is shown in figs. 16a and b. In fig. 16a the resistance is seen to fluctuate* between two reproducible 'states'. As in the experiments of Beutler et al. (1987) discussed above (figs. 1 and 2), the interpretation is that a single defect is moving back and forth between two potential minima. Figure 16b is slightly more complicated; the behavior here is consistent with what one would expect from two different fluctuators which move independently between two different positions. Other interpretations (e.g., a single fluctuator with more than two states) cannot be ruled out, but seem less likely. Figures 16c and d show fluctuators which appear interact with each other. In fig. 16c the amplitude of the resistance change due to the faster fluctuator is smaller when the slow fluctuator is in its 'up' state than when it is in its 'down' state. In fig. 16d the motion of the rapid fluctuator is seen to be suppressed when the slow fluctuator is in its 'down' state. This behavior is similar to that seen in Pt films, see fig. 5. •Similar fluctuations have been observed in tunnel junctions (Rogers and Buhrman 1984) and in M O S devices (Ralls et al. 1984). In those cases the fluctuations are believed to be due to fluctuations in the occupancy of electron trap states.

Ν. Giordano

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fluctuators;

here the fast fluctuator is turned O n ' when the slow fluctuator is in its 'up' state, and turned 'off' otherwise. The total time interval in each trace is approximately 30 s. After Ralls and Buhrman (1988).

The fluctuating defects responsible for the behavior in fig. 16 appear to be, on the average, stationary within the sample. That is, while the defects do move, they do not generally enter or leave the sample, so that the total number of defects is constant. However, Ralls and Buhrman have also observed cases in which a defect appears to be diffusing through the sample, as illustrated in fig. 17. Here, the defect seems to be fluctuating about while at the same time it is moving through the sample. The effect of these fluctuations on the resistance is seen to first grow, and then become smaller with time, which appears to correspond to the defect moving towards, and then away from, the center of the sample. From these measurements, Ralls and Buhrman were able to show that the defect motion is thermally activated, and that the attempt frequencies are of order 1 0 13 s, which is a typical phonon frequency. The activation energies are found to be %0.1-0.3eV, which again is quite reasonable for defect motion. One surprising result concerns the magnitude of the resistance fluctuations. Ralls and Buhrman estimate that a typical resistance fluctuation corresponds to a change in the scattering cross-section in the rank 1-30 A 2. Martin's calculation of the cross-section changes which enter the L I predictions (25) and take kF = yields Δσ0/σ0 ^ 0.5. If we use the standard relation oc = 4nkF2,

Fluctuations and noise: Experiment

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TIME Fig. 17. Resistance of C u microcontacts as a function of time. This trace shows a defect moving through the contact region. This defect gives rise to very rapid fluctuations of the resistance. Top - the raw data, which shows an ordinary fluctuator (i.e., fig. 12a), which switches on a relatively slow time scale, together with very fast fluctuations which are due to the atom which is moving through the sample. Bottom - same data with the effect of the ordinary fluctuator subtracted off, so that the behavior of the fast fluctuator is now isolated. The total time interval is approximately 30 s. After Ralls and Buhrman (1988).

1.0 A ~ x, then Martin's results suggest that Aac should be less than 6 A 2. Hence, the changes observed by Ralls and Buhrman seem much too large to be explained by this theory. It is not clear if the uncertainties in the experimental result are large enough to account for this discrepancy, or if defects not considered by Martin might yield larger values of Aac. This is a matter that certainly needs to be resolved. The experiments of Ralls and Buhrman provide important qualitative insight concerning the motion of defects. This work has also shown that the activation energies, attempt frequencies, etc., are in the range expected for defect motion, and hence provide strong support for the L I model. A key quantity in the L I and U C F theories in ns, the concentration of mobile defects. This is clearly a very difficult quantity to determine experimentally. It is possible, however, to have some quantitative control over the total number of defects. Experiments along this line have been performed by Pelz and Clarke (1985), and Pelz et al. (1987, 1988). They have studied how the noise of Cu films is affected by irradiation, which introduces defects, and subsequent anneal­ ing so as to remove these defects. In their initial experiments, the irradiation was with 500 keV electrons at 90 K, a situation believed to produce vacancyinterstitial pairs, in which the resulting vacancy is frozen in place, and the interstitial is able to move until it is trapped at any of various possible sites. Results for the noise spectral density after several irradiations are shown in fig. 18. The noise magnitude is seen to increase significantly with irradiation, and the spectral slope, a, also increases slightly. Figure 19 shows results for the increase in the magnitude of the noise, Ay, as a function of Ap, the increase of the resistivity due to the irradiation. It is known that the total concentration

166

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Fluctuations and noise: Experiment

167

of added defects, nt, is proportional to Ap (and the constant of proportionality is also known), so fig. 19 demonstrates that there is a well defined relation between the noise magnitude and nx. While the number of mobile defects, ns, is the quantity of primary interest, nt does provide an important upper bound on ns. These results can thus be used to estimate the absolute magnitude of L I noise. Pelz et al. showed that the noise magnitude predicted by the L I model is reasonably consistent with their results (see below). In addition, annealing experiments (Pelz and Clarke 1985, Pelz et al. 1988) were used to show that it is probably the motion of a vacancy which plays the dominant role in the noise process in this system. The work of Pelz et al. has provided very detailed information linking defect motion and 1/f noise, and has also shed light on precisely which types of defects are (and are not) important. However, comparison with the L I model was complicated by the fact that the motion of an isolated vacancy should not, in this case, change the resistivity. This is because the point symmetry of a monovacancy is the same as that of the lattice (see the discussion in connection with fig. 15a). Such a defect must, according to L I theory, be within a few lattice spacings of some other defect, such as a grain boundary, dislocation, surface, etc., which breaks this symmetry. In addition, it turns out that in the films of Pelz et al., all of the vacancies which were near other defects must have been mobile in order for the Li model to correctly predict the magnitude of the noise. As pointed out by these authors, this seems rather unlikely. However, the uncertainties in the estimated defect densities are probably large enough that perhaps only 10% of the vacancies need be mobile (assuming of course that the errors here are in the 'favorable' direction). Hopefully this problem can be addressed in future experiments. In any case, despite these difficulties in constructing an explanation of the results in terms of the L I model, this work has demonstrated unambiguously that mobile defects produce 1/f noise, and has shown that the L I model provides a reasonably accurate estimate of the amount of noise which is produced by an individual defect. Zimmerman and W e b b (1988) have recently reported an experiment in which the effect of a known number and type of impurity is observed. The system is Pd, and the impurities are H + , i.e., protons. Protons in Pd are extremely mobile, as they readily diffuse through the lattice. In fact, at room temperature this diffusion is so rapid, that on the experimental time scale there are significant fluctuations in the total number of protons in the sample. This leads to diffusion noise; the spectrum in this case is quite different from 1/f, and the form of the spectrum is a function of sample size (Schofield and W e b b 1985). Since proton diffusion is thermally activated, the fluctuations in the number of protons, i.e., defects, in the sample become negligible at low temperatures, and the diffusion noise vanishes. In its place 1/f noise is observed. In this lowtemperature regime, the Η + motion is believed to be primarily hopping between adjacent sites, and hence the ideas of the U C F and L I models should apply.

168

Ν.

Giordano

Fig. 20. Noise of P d doped with H +. Relative magnitude of the noise at / = 1 H z as a function of temperature. Different symbols correspond to different H + concentrations, x H, as indicated. The inset shows the magnitude of the noise as a function of xH.

After Zimmerman and W e b b (1988).

For the Pd samples studied by Zimmerman and Webb, the resistance ratios (i.e., the ratio of the resistance at room temperature to that at low temperatures) were in the range 2.5-7.5, which implies a fairly long elastic mean-free-path. Even at the lowest temperatures studied, the estimated elastic mean-free-path was longer than the phase-breaking length, so that the L I model should apply in this case. Analysis of the l/f noise in terms of the L I model is greatly aided by the fact that the (average) number of mobile impurities can be determined from the diffusion noise at high temperatures. As shown in the inset of fig. 20, the magnitude of the l/f noise varies linearly with the H + concentration, x H. In addition, the temperature dependence of the noise magnitude at / = 1 H z is found to exhibit a maximum at 120 Κ (fig. 20). This is close to the temperature at which the frequency for nearest neighbor hopping of the protons is 1 Hz, which supports the assumption that this is the relevant defect motion. The L I model can also account for the magnitude of the noise if one assumes [see eq. (23)] that β « 0.3 (which is consistent with Martin's calculations), and if all of the protons are assumed to be within one lattice spacing of another impurity or defect. However, the nature of these 'other' defects is not yet known. In any case, the L I model seems to provide a consistent picture of the noise in this system.

Fluctuations and noise: Experiment

169

6. Conclusions and outlook In this article we have reviewed a number of experimental studies of U C F in disordered metals. The experiments are in good qualitative and quantitative agreement with the theory, demonstrating that individual U C F fluctuators can be readily observed, and that they can also be a significant source of lowfrequency, 1//, noise in disordered metals. Since U C F can only occur when L e < L^, U C F do not appear to be responsible for low-frequency noise in weakly disordered systems near room temperature. However, a closely related model, based on the L I mechanism, appears to give a good account of the behavior in this regime. This is not to say, however, that the U C F and L I mechanisms can explain all of the low-frequency noise observed in metals. While this may indeed turn out to be the case, further work is needed on this question. In any event, it is clear from the experiments discussed here that U C F do certainly exist, and that the theory is in very good agreement with the available experiments. However, a number of questions remain open. In particular, the types of defects which are involved are completely unknown. In the experiments of Beutler et al. on metal films at low temperatures, the hopping time for simple point defects is many orders of magnitude too long for these to be the relevant defects in experiments like those shown in figs. 1 and 2. This suggests that the active defects are probably near a dislocation or grain boundary, etc., so that their activation energies are much smaller than would be the case for, e.g., an isolated point defect. It would clearly be very interesting to determine just what types of defects are involved, and where they are located. This kind of information could lead the way to the use of U C F as a unique and powerful tool for the study of the motion of individual defects in metals.

Acknowledgements I thank D.E. Beutler, N . O . Birge, R.A. Buhrman, J. Clarke, S. Feng, D . M . Fleetwood, B. Golding, R.H. Koch, J. Liu, T.L. Meisenheimer, P. Muzikar, P.A. Lee, W . Smith, A . D . Stone, R.A. Webb, W . W . Webb, M.B. Weissman, and N . M . Zimmerman for useful discussions, advice, and correspondence, which was of invaluable help in putting together this article. I am particularly indebted to D.E. Beutler, D . M . Fleetwood, and T.L. Meisenheimer, for their collaboration on the work discussed in this article, and, along with M.B. Weissman, for many useful comments on this manuscript. M y work in this area has been supported by the U.S. National Science Foundation through grant DMR-8614862, which is gratefully acknowledged.

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