[modern problems in condensed matter sciences] mesoscopic phenomena in solids volume 30 ||...

C H A P T E R 2

Mesoscopic Fluctuations of Current Density in

Disordered Conductors

B.Z. SP IVAK

Science and Technology Corporation of the Academy of Sciences

Institute of Analytical Instrumentation

Ogorodnikova 26, Leningrad, USSR

and

A.Yu. Z Y U Z I N

A.F. Ioffe Physical-Technical Institute of the Academy of Sciences

Leningrad, USSR

© Elsevier Science Publishers B.V., 1991

37

Mesoscopic Phenomena in Solids Edited by

B.L. Altshuler, P.A. Lee and R.A. Webb

Contents 1. Introduction 39

2. Mesoscopic fluctuations of transmission coefficient of a medium that scatters

elastically 41

2.1. The microscopic fluctuations of current density arising in the course of diffusion

through the media 41

2.2. The Langevin approach to calculation of the mesoscopic fluctuations of the

transmission coefficient 44

2.3. Mesoscopic fluctuations of the transmission coefficient of a medium that scatters

elastically 46

3. Mesoscopic current fluctuations in disordered metals 50

3.1. Mesoscopic fluctuations of current density in disordered metals 50

3.2. The Langevin description of mesoscopic fluctuations in disordered metals . . 54

3.3. Methods of measuring of mesoscopic fluctuations in disordered metals . . . . 57

3.4. Mesoscopic fluctuations in metals caused by a magnetic field variation . . . . 61

4. Friedel oscillations in disordered metals 63

5. Mesoscopic fluctuations of current density in disordered superconductors 66

5.1. Mesoscopic fluctuations of the density in superconductors 66

5.2. Mesoscopic fluctuations of the normal current 70

5.3. Can the ground state of a disordered superconductor correspond to a non-zero

supercurrent state? 72

6. Sensitivity of mesoscopic fluctuations to the motion of a single scattering centre . . 73

7. Conclusions 76

References 78

38

L Introduction

The theory of transport phenomena in metals based on the classical Boltzmann equation assumes that the electron moves along a classical path between the scattering events. This is valid, however, only provided interference of scattering from different centres can be neglected. The criterion for the validity of such an approach is the small size of the electron wavelength λ compared to the mean free path Z.

The momentum relaxation of electrons in disordered conductors at sufficiently low temperatures is controlled by their elastic scattering by impurities and structural defects. In this case, at I > λ there exist the quantum interference corrections to the classical expressions for transport coefficients (the so-called weak-localization corrections) that are responsible, e.g., for negative magnetoresistance and for the Aharonov-Bohm effect in disordered metals. These phenomena are reviewed in the papers of Altshuler and Aronov (1985), Lee and Ramakrishnan (1985), and Aronov and Sharvin (1987).

The theory of the phenomena mentioned above was developed for infinite samples where the kinetic coefficients were calculated by averaging over the ensemble of the random scattering potential realizations or over the infinite volume of the sample. However, the spatial microscopic fluctuations of the current density arising in a metal due to the random distribution of the impurities can considerably exceed the average current density. That is why finite samples, being macroscopically similar, display nevertheless very different conductances. We shall call these fluctuations mesoscopic.

The experimental investigations of mesoscopic fluctuations in metals started with the pioneering papers by Umbach et al. (1984) and Webb et al. (1985). It has been revealed in these papers that the resistance of a small metallic film at low temperature undergoes random time-independent fluctuations as a function of magnetic field that vary from sample to sample but are reproducible within a given sample. For a recent review of experiments in this field see the paper by Washburn and Webb (1986).

Stone (1985) was the first to give the qualitative explanation of this phenomenon, which is based on the random interference of electrons as they travel through the sample along different diffusion paths. The magnetic field changes the relative phases of the probability amplitudes for the electron to travel along

39

40 B.Z. Spivak and A.Yu. Zyuzin

different paths so the total probability to diffuse through the sample oscillates randomly as a function of magnetic field.

Altshuler (1985) and Lee and Stone (1985) have shown that the zero-temperature conductance of small metallic samples measured by the two-probe method exhibits fluctuations from sample to sample that have a 'universal' magnitude of the order of e2/h independent of the sample size and the degree of disorder, provided λ < Z, e being the electron charge.

Finally, Lee and Stone (1985) put forward the 'ergodic' hypothesis that an average of the quantities of interest over an ensemble of the random-potential realizations at a given magnetic field is equivalent to an average over many values of magnetic field in a given sample.

By the present time similar phenomena have been discussed in elastic scattering media, in superconductors and in metallic spin glasses. All these phenomena take place in systems with extended quasiparticle wavefunctions.

In this paper we present a review of the theory of these phenomena. In the second section we consider the mesoscopic fluctuations of the transmission coefficient of disordered media through which the coherent waves (electron or light) propagate. The third section is devoted to a consideration of the mesoscopic fluctuations of the current density produced in a metal by the electric

It is well known that the non-local conductivity in a disordered metal <70(r, r'\ averaged over the random potential realizations, decays as

It is important, however, that σ 0( ι \ r') is not a self-averaging quantity and the dispersion of < 7 y(r , r ') falls off as \r — r ' | ~ 2, i.e., much more slowly than its average (Zyuzin and Spivak 1986). It means that the mesoscopic fluctuations of current density caused by the electric field in metals can drastically exceed its average.

Any method of mesoscopic fluctuations measurement in a metallic sample is associated with the averaging of microscopic fluctuations over the size of the measuring probes, so different methods of measurement yield different results for the mesoscopic fluctuations of the conductance.

The fourth section deals with the Friedel oscillations in disordered metals. The well known consequence of these oscillations is the Ruderman-Kittel exchange interaction I(r) between localized spins, r being the distance between the spins. At r > I the value of I(r) averaged over the random-potential realizations decays as exp(— r/l) (de Gennes 1962, Mattis 1965).

It has been shown, however, that due to the large mesoscopic fluctuations I(r) is not a self-averaging quantity and the variance of I(r) falls off according to the power law r~6 (Zyuzin and Spivak 1986, Bulaevskii and Panyukov 1986).

field.

Mesoscopic fluctuations of current density 41

In the fifth section we consider the mesoscopic fluctuations of both the normal current and the supercurrent densities in superconductors. It will be shown that the current density and the superfluid velocity in disordered superconductors undergo time-independent mesoscopic fluctuations.

In the sixth section we discuss the sensitivity of all the above mentioned quantities to a small change of the scattering potential. It appears that the quantum interference effects make all the quantities of interest very sensitive to the detailed microscopic arrangement of the disorder (Altshuler and Spivak 1985, Feng et al. 1986). For example, the conductance of a two-dimensional sample at zero temperature will change by an order of e2/h, provided one impurity atom is shifted by a distance of the order of λ.

In the conclusion (section 7) we discuss some unresolved problems.

2. Mesoscopic fluctuations of transmission coefficient of a medium that scatters elastically

2.1. The microscopic fluctuations of current density arising in the course of diffusion through the media

Consider an elastic scattering medium, all dimensions being greater than the mean free path I. The surface is uniformly illuminated with the coherent electron wave of the wavevector p. In this case the current density J(r) and the electron density n(r) in the medium, averaged over random-potential realizations satisfy the diffusion equations at / > λ = Inhjp

Here D = lv/3 is the diffusion coefficient, υ = p/m, m is the mass of an electron and the angular brackets < > denote averaging over an ensemble of the random-potential realizations. However, the microscopic values of J(r) and n(r) can differ drastically from their averages due to the spatial mesoscopic fluctuations. The physical origin of these fluctuations may be understood in terms of the random interference of electrons when they travel from the sample boundary to the observation point r (see fig. la).

The total probability W(r) for an electron to travel from the sample boundary to the observation point r is given by the Golden rule of quantum mechanics:

d i v < / ( r ) > = 0, < / ( r ) > = - * D P < n ( r ) > . (2.1)

|2

W(r)oc Σ Λ = Σ Ι Α Ι 2 + Σ Ak*t, k k k*k'

(2.2)

Ak being the amplitude of the probability for an electron to pass along the feth path.


Fig. l . (a ) Schematic illustration of the electron wave interference that gives rise to microscopic fluctuations of current density at point #·. Crosses indicate the locations of scatterers. (b) Self-intersecting paths that give rise to the weak localization quantum correction to the conductivity,

(c) Schematic illustration of the local conductivity correlations, eq. (3.17).

The first term in eq. (2.2) represents the sum of probabilities for the electron to travel by any path to the point r and corresponds to the classical result given by eq. (2.1).

The second term in eq. (2.2) represents the interference between the various amplitudes.

The different diffusion paths that the electron can take can be characterized by the scattering sites ra that it visits (see fig. la). For most paths their lengths differ strongly, and hence the phases of the amplitudes Ak also differ substantially. If we ignore weak localization corrections (which are connected with the self-intersecting paths shown in fig. lb ) then the averaging over all the impurity positions turns the interference term in eq. (2.2) to zero due to its rapid phase dependence.

However, if we do not perform the average, then it is this term that is responsible for the mesoscopic fluctuations of the electron density and the current density.

The scatterers situated near the point r in the region \r — ι · α| < / can be considered as sources of coherent spherical waves that propagate without scattering between ra and r, so one can rewrite the electron wavefunction at r as

The phase φα and amplitude Ba corresponding to the ath impurity are determined by the sum of a large number of amplitudes Ak(ra) with random phases, so φΛ are randomly distributed over the interval [0, 2π]. The amplitudes ΒΛ are of the order of

B * * ^ ( < n ( r J > ) 1 /2 (2.4)

and fluctuate by an amount of the order of its average, Μ being the amplitude of electron scattering by the impurity.

Using eqs. (2.3) and (2.4) and quantum mechanical expressions for J(r\ and n{r)

\eh

m^^wwrw-rirwtm, « ( r > = ι ^ ι 2, (2.5)

one can calculate the dispersions of the mesoscopic fluctuations

5/(r) = / ( r ) - < / ( r ) > , 6»(r) = »(r) - <n(r)>,

< ( 5 « ( r ) ) 2> = <n(r)> 2, (2.6)

< ( 5 / ( r ) ) 2> = i [ e i > < » M > ] 2, (2.7)

< 6 Ι ^ ) ^ ( ^ ) 2 λ \ ^ - ' \ X<\r-r'\<l, (2.8)

<δη(Ι·)δη(^)> = ^ ^ ( ^ Γ | 7| ) 2 , Α < | γ - γ Ί < / . (2.9)

It follows from the eqs. (2.6)-(2.9) that the electron density fluctuates by an amount of the order of its average δη « <n> while the fluctuations of the current density \bJ\&ev(ri) have random directions and always considerably exceed their average. For example, in the case of a semifinite medium the incident electron beam is completely reflected from the medium so that </> = 0; <n> « J0/ev while |δ/| &J09J0 being the incident electron current density.

Another important example is the electron emitted by the coherent point source, when

<(5/(r))2> = Kev<n(r))Y = l- > </(r)>2 (2.10)

and (Shapiro 1986)

<(5n(r))2> = <n(r)>2 = (j^ J , (2.11)

while

(J(r)} = eQr/4nr\ (2.12)

< * ( r ) > = β / 4 π ΰ Γ , (2.13)

where Q is the power of the source and r the distance from the source.

2.2. The Langevin approach to calculation of the mesoscopic fluctuations of the transmission coefficient

To calculate the correlation functions of the electron density and the current density at \r — r'\ > Ζ one can use the following equations (Zyuzin and Spivak 1987):

div8/(r) = 0, (2.14)

§J(r) = -eDVhn{r) + / L( r ) , (2.15)

similar to the well known Langevin equations (Curevich 1962, Landau and Lifshitz 1978), which describe the thermodynamic fluctuations. However, the expression we use for the correlation function of Langevin random currents (or Langevin sources) JL(r),

<J\{r)J){r')y = <η(ή)2δ^ - r'); |r - r'\> /, (2.16)

differs from that in the thermodynamic fluctuations theory. The eqs. (2.14, 2.15, 2.16) are valid for any sample geometry and coordinate

dependence of the diffusion coefficient D(r\ provided \r — r'\ > /. The boundary conditions for these equations are the same as for the diffusion equation.

The qualitative interpretation of the eqs. (2.14)—(2.16) is as follows. The 'microscopic' current density correlation function eq. (2.8) averaged over a spatial scale of the order of I results in the correlation function eq. (2.16) of the Langevin sources JL{r). These Langevin sources have to be added to the expression for the diffusion current eq. (2.15) that describes the current fluctuations on the 'macroscopic' \r — r'\ > Ζ scales.

Thus the eqs. (2.14), (2.15) and (2.16) allow us to distinguish two phenomena:

(i) The random Langevin sources JL(r) resulting from random electron interference are extremely sensitive to the variations of magnetic field and impurity configurations.

(ii) The classical currents arising in a non-uniform media due to the current conservation law obey the conventional diffusion equation and have nothing to do with quantum interference.

One can prove the eqs. (2.14)—(2.16) by summing the diagrams shown in figs. 2a,b. The correlation function of the Langevin sources eq. (2.16) corresponds to the diagrams shown in fig. 2a.

We use the standard impurity averaging technique (Abrikosov et al. 1965) where the solid lines correspond to the electron Green function, and the dashed lines correspond to the correlation function <w(r)u(r')> of the random scattering potential u{r). We assume that u(r\ in the disordered medium, obeys a white-noise Gaussian statistics (outside the medium u(r) = 0)

<W(r)> = 0, <u(rW)y = <u2)d(r - r'). (2.17)

x = 0

ι ι I I I I I I I I I I " Μ η !

( d )

Fig. 2. (a) Feynman diagram for the Langevin sources of eq. (2.16). (b) Diagrams that lead to eqs. (2.14). (c) Diagrams for the Langevin sources, eqs. (2.29)-(2.32). (d) Diagram for the diffusion

ladder.

In the Born approximation for the electron interaction with the short-range impurity potential,

dr V{(r)

where N{ is the impurity concentration and V{(r) is the potential of a given impurity. When averaged over the impurity concentration, the Green function depends only on the difference r — r' and can be written in the main approximation in hi pi < 1 in the form

<G«A>(r, r')> = <GeR<A>d>)> e x p [ i / > . ( r - r ' ) ] , (2.18)

« * » w > - i . - £ ± £ (2.19)

Here G R ( A) are the retarded (advanced) electron Green function, τ - 1 = πν<ι*2> and ν is the density of states. The important element of the diagrams in fig. 2 is the so-called diffusion function

P D( r , f )oc<G? ( r , r')G${r\ r)>

shown in fig. 2d, that obeys the equation

(2.20)

2.3. Mesoscopic fluctuations of the transmission coefficient of a medium that scatters elactically

Consider the transmission coefficient Τ of the sample in the form of the slab of area LyLz and thickness Lx for an incident electron. Using the eqs. (2.1) and the boundary conditions (n}\x=Lx = 0, and </±> = 0 on the insulating sample boundaries y = +\Ly, ζ = +\LZ one obtains

<n> = ^ ^ , (2.21) ev Lx

< f > =< J J M > = l _ ( 1 2 2)

Here JL is the component of the current density perpendicular to the boundary, J°L = \JQ ' N\ a n (l n is the normal to the sample boundary χ = 0.

Using the eqs. (2.14) and (2.15) one obtains

δι* = J2x{r) dr, (2.23)

< ( δ ϋ 2> = ^ f < ^ ( r ) J i ( r ' ) > dr dr'. (2.24)

Here i? = LxLyLz is the sample volume and ix is the total current through the sample in the x-direction.

Substituting eq. (2.16) into eq. (2.24) and using eq. (2.21) one obtains (Stephen and Gwilich 1987, Zyuzin and Spivak 1987)

 = J ! ^ _ (225)

The qualitative interpretation of the eq. (2.25) is as follows: According to Thouless (1977), the sample transmittance is determined by energy levels lying in the energy band centred at the incident beam energy and with a width of the order of h/t0; t0 = L2X\D being the time it takes an electron to diffuse out of the sample.

The number of these levels is of the order of Ν « w(h/t0) « lLyLzjX2Lx. The contribution of each specific level into t fluctuates by an amount of the order of its average. Therefore

< ( 8 f ) 2> / < f > 2^ i V - 1 (2.26)

in agreement with eq. (2.25). The correlation function <δΤ(0)δΤ(0 ' )> can be calculated using eqs. (2.16)

and (2.15) as well, 0 = /o/|/o| being the unit vector along the incident current density J0 (see fig. la). The corresponding correlation function of Langevin sources can be obtained by summing the diagrams in figs. 2a and c, where the outer and inner solid lines correspond to the electron Green functions for beam directions 0 and 0', respectively.

Taking into account the fact that a change in 0 leads to a change in the phase of the incident wave,

δφ(ρ) = 2πρ ,

at the sample boundary χ = 0 one obtains for Lx~ Ly~ Lz (p is the coordinate at the plane χ = 0)

<Jftr, θ)3){ν\ 0')> = <Jftr, 0)J){r\ 0 ' ) > i + <J\(r9 0)J;(f, 0 ' ) > 2, (2.27)

(J\{r, e)J){r', 0')>x = f JllXHl}d{r-r') exp ( 2* 2τι V L\0_0>\

(2.28)

Lx>L -2n\9-e'\>h

/2eD\2

(Jt(r, e)J){r\ fl')>2 = I — J (K, +K2 + K3), (2.29)

K t = V i ' - O J d' i C p > ' Ί ) ] 2 ^ » ^ , β ) >^< » ( · Ί , « ' ) > , (2.30)

X 2 = F,<n(r, β ' )>Ρ;<η(ι· ' , 0 ) > [ P > , r ' ) ] 2, (2.31)

X 3 = F,<»(»•, β ' ) > [ Ρ > , Ό ] 2 · (2.32)

/ L( r , 0) is the Langevin source corresponding to the direction of incidence 0. Using eqs. (2.24) and (2.27) one obtains (Lx « Ly& Lz) (Stephen and Cwilich

1987, Zyuzin and Spivak 1987)

f 3A 1 λ 1Λ Λ„ λ , — <\θ-θ' <-,

<δΓ (0 )£Γ (0 ' ) ) 1 4 τ γ ^ | β - 0 Ί L x /

«">•> % |<ι«-η The correlation function < J\(r, 0)J)(r\ # ' )> ! corresponds to diagrams in fig. 2a and gives the main contribution to eq. (2.33) at |0 — 0'| < λ/l. While in the case \θ — θ'\> λ/Ι the main contribution into eq. (2.33) is given by the correlation function <J\{r, 0)J){r\ 0 ' )> 2 corresponding to the diagrams in fig. 2c. The interpretation of <J^(r, θ)3)(#·', 0 ' )> 2 will be discussed in the next section.

Note that it is the diagrams in fig. 2c and the corresponding correlation function <J\(r, 0)J)(r\ 0 ' )> 2 that are the analogues of the diagrams summed by Altshuler (1985) and by Lee and Stone (1985) to calculate the mesoscopic conductance fluctuations G.

It follows from the Landauer formula that

< ( 5 G ) 2> * e2LvL

ηλ2 άθ άθ' < δ ί ( 0 ) 5 f (0')>. (2.34)

Taking into account the fact that it is the function (J\{r, 0)J)(r\ 0 ' )> 2 that gives the main contribution to eq. (2.34), one obtains the famous result of Altshuler (1985) and of Lee and Stone (1985):

((6G)2}xe4/h2.

Obviously, the results obtained above can be applied as well to the case when the coherent light propagates through the elastic scattering medium. In

this case one can study the statistics of T(0) varying the external magnetic field. Because of Faraday rotation the magnetic field changes the relative polarization angles for the light waves travelling via different diffusion paths through the sample. If this rotation angle is of the order of π then T(0) changes completely, so that T(0) is the random function of Η with the typical period AHC ~ //L2 β; β being the Werde constant.

If the incident light pulse has a duration t <t0 = L2X\D then (Spivak and Zyuzin 1988b)

Here tt is the transmission coefficient averaged over the time. The factor t/t0

appears because the width of the essential energy band is of the order of h/t instead h/t0.

Equation (2.25) holds provided the width of the incident beam exceeds Lx. The opposite case can be studied with the help of eqs. (2.14)—(2.16) as well (Pnini and Shapiro 1989).

By now we have dealt with the transmission coefficient integrated over the angle of beam scattering 0X (see fig. la).

A detailed study of transmission coefficient as a function of both 0 and 0X

has been performed by Feng et al. (1988). They pointed out some sort of spatial memory in δΤ(/>, q) where ρ and q are the components of the incident and final momenta of light that are parallel to the sample boundaries:

Here pt and p\ are the coordinates at the planes χ = 0 and χ = Lx. App> and W(p, ρ') are the amplitude of probability for the electron to travel between ρ

< ( δη ) 2> < ( δ Τ ) 2) t < f > 2 ~ < f > 2 t 0"

(2.35)

<δΤ (/>, q)f(p\ q')} oc dp dPl dp' dp', dp2 dp'2 dp3 dp'3

x e x p i i ^ - f p - p j + i - i p ' - p i )

+ />' (P3-P2) + 0 , ( P 3 - P 2) ] }

(2.36)

and ρ'. According to the diagram in fig. 2a

<ΑΡΡ,ΑΪ1Ρ,ΑΡ2Ρ,ΑΪ3Ρ,} oc W(p, P')W(P2, Pi)

x ίδ(ρ - Pi)δ(ρ' - p\ )δ(ρ2 - ρ3)δ(ρ'2 - p'3)

+ HP - Ρτ>)δ(ρ' - PiWpi - ρ2)<5(ρΊ - Pi)\

(2.37)

Thus the second term in (2.37) leads to the contribution in <δΤ (/>, q)ht(p\ q')) that is proportional to δ(ρ -ρ' - q + q) (Feng et al. 1988) at Ly,Lz>Lx.

In addition to the fluctuations of δη and 8J considered above, which are of an interference nature, there exist classical fluctuations connected with the fluctuations of the impurity concentration, but we confine our attention to the mesoscopic fluctuations of an interference nature because of their extreme sensitivity to the external magnetic and electric fields and to variations of scattering potential.

Moreover, the fluctuations of 8J and δη of an interference nature exceed the classical ones at sufficiently low temperatures and for large sample dimensions.

3. Mesoscopic current fluctuations in disordered metals

3.1. Microscopic fluctuations of the current density in disordered metals

We start with the investigation of the microscopic spatial fluctuations of the current density in disordered metals at low temperature, where the conductivity is determined by elastic scattering of electrons by impurities.

The current passing through the sample heats it so the entropy rises. On the other hand, in a particular realization of a random potential u(r) over distances less than the inelastic mean free path the current flow can be described by the exact solution of the Schrodinger equation. The Schrodinger equation, however, as well as the Liouville one, conserves the phase volume (i.e., it conserves the entropy).

In classical mechanics this contradiction is solved in the following way. As an electron undergoes a complicated motion through a random potential, there is a mixing of states in phase space, which is apparently equivalent to an averaging over the ensemble of random realizations of the scattering potential. Thus, the concept of a mean free path arises after an average is taken over the realization of the random potential.

In the quantum mechanical theory of a disordered metal the microscopic current density does not depend on time. It is plausible to assume that the complication of electron motion manifests itself in large spatial fluctuations of current density.

Consider the non-local conductivity σ0·(ι·, r') determined by the equation

E(r) being the electric field. The Chambers formula for the ensemble-averaged

(3.1)

conductivity

ζ , , , N 3 aO(r-r>W-r>)j_ J \r-r'\\

implies that the electron loses the memory of its initial direction of motion in a distance of the order of I. Here σΌ = e2Dv is the Drude conductivity and ν is the density of states at the Fermi level in a metal. Zyuzin and Spivak (1986) have shown that the dispersion <(cri7(r, r ' ) ) 2> falls off much more slowly than eq. (3.2) as a function of \r — r'|, and for example at T= 0 (T is a temperature) in the infinite sample at \r — r'\ > /

<(a y(r , r ' ) ) 2> 1 /2 * > ')>· (3-3)

Thus the mesoscopic fluctuations of current response at r due to an electric field at r' exceed greatly its average and have a random direction. Thus, the non-local conductivity <7f</-(r, r') appears to be a non-self-averaging quantity. Let us note the analogy between eq. (3.3) and eq. (2.10).

One can directly prove eqs. (3.2) and (3.3) by the summation of the diagrams shown in figs. 3a,b, which represent the conductivity bubble and two conductiv-

( g ) ( h )

Fig. 3. (a) Diagram for the conductivity, (b) - (d) Diagrams for the Langevin sources, eqs. (3.10)-(3.13). (e)-(g) Example of the diagrams that lead to eq. (3.8). (h) Diagrams for the screening of the

electron density fluctuations.

ity bubbles correspondingly averaged over the ensemble of impurity configurations.

The qualitative explanation of the eqs. (3.3) and (2.10) can be done as follows. According to the eq. (2.2),

< W 2( r , r')> - < W(r, r ' )> 2 « {Σ AtAf AkAf j * < W(r, ι>')>2· (3.4)

In the infinite sample

<W(r, r')> * X <|^(r, Ο Ι 2 > (3.5)

and we arrive at the inverse proportionality to |r — r'|, eq. (3.3). At finite temperature there are two reasons for <[δσ0·(*% r ' ) ] 2> 1 /2 to fall off faster than eq.(3.3).

First, the difference in electron energy, of the order of T, leads to a momentum difference Ap~ T/vF, so the electrons with the energy difference Τ travelling on the very same diffusion path acquire the phase difference π on the length LT = (hD/T)1/2 so the expressions for <(σ0·(ι% r ' ) 2> at \r — r'\ > LT acquire the extra factor (LT/\r — r'\)2 and vF is the Fermi velocity.

Second, at |r — r'\ > Εφ = (ϋτφ)1/2 the inelastic processes destroy the electron phase memory and the expression for <(σ0·(ι% r'))2} acquires an extra factor

( 2 | r - i exp —

Here τφ = min [ r i n, t s ] is the time for which the electron wavefunction retains its coherence; t i n and t s are the corresponding inelastic and spin flip times.

In a bulk sample under a uniform electric field E=(EX, 0, 0) summation of the diagrams in fig. 3b gives (Aronov et al. 1986)

while

ι, 3 π ^ / ΐ 7ν he\2L0 φη(ή)2} = - [ ν ( Ε )

8 V PFJ I

Here </(r)> = a D< £ > and pF is the Fermi momentum. The results for L 0 at different temperatures and sample dimensions are summarized in table 1. The different expressions for L 0 correspond to different \r — r'\ power laws in <σ? > for different relations between L^, LT and the sample dimensions. Thus at L 0 > l(pFl/h2 we have <(δ/)2> > </>2 and only a small fraction of the current flows along the electric field direction.

Table 1 Expressions for L 0, Hc and /Jx ) for different values of the sample dimensions, L T, Lv and

LAH = (hc/2e\AH\).

Condition L0 Hc

Lx>L LT m ) LT Ψο 2 ^ 2 - 1

Ly>Lz> LT; LyLz > L rL ^ L\ 12x

L x ^ L y> L ( p> L T> Lz

Ψο

if LT < LAH < Ly

1η(8πχ)"2

In ( L„/LT)

Lx>> Ly> L L T> Lz

Ψο

if LAH <LT<L(p

1

16x In ( L , / L T)

Lx > L„ > Ly > Lz 4π L9L\ Ψο LjLv > LyLz\ > LT 3 LyLz LyLy 2πχ

Lx ^ Ly ~>> Lz > ^ Ψο ( 2 >/ 2 - l ) « i ) LT>LLy>L(p>Lz i £ Ψο π

π Lz Ll

Lx>L(p>Ly>: Lz Ψο LT>L(p LyLZ LyLy \2πχ)

L T, L(p>Lx>Ly>; Lz Ll

^ LyL,

Ψο

LXLy ~ χ " 4

The large fluctuations of current density 5/(r), eq. (3.6), is the consequence of the long-range behavior of < ( σ 0) 2> given by eq. (3.3). The presence of these currents, random in magnitude and direction, gives rise to fluctuations of the magnetic moment density m , so that we have (Aronov et al. 1986)

< m 2 >* { $ < E > 21 7 ' L* * L Y ~ L * > LT ' <3-7)

This effect is closely related to the lowering of the symmetry in mesoscopic samples (Altshuler and Khmelnitskii 1985) and to the appearance of a magnetization in low-symmetry crystals under an electric field (Levitov et al. 1985). Another consequence of the symmetry lowering in mesoscopic samples is the spin density fluctuations 8S(r) so at LT > L s o,

< (δ5( » · ) ) 2>«< (δη( . · ) ) 2> .

Here L so = ( D t s o) 1 / 2, with L so the spin orbital scattering time.

dr, P^r, r^PfJrt, ή{Ε(^)}2 (3.11)

K2 = < £ (( r ) > < £ / r ' ) > P f l 2( r , r')P<;Jr', ή (3.12)

K3 = < £ , ( γ ) > < £ ^ ) >

x i R e IPfJr, r')P^y, ή + P?Jr, r')P?Jr', r ) ] . (3.13)

Here diffuson P?oc <G* (r, r , ) G * +, ( i /, r ) > and cooperon P^oc <G*(»-, r') Gf 1 + £( i - , »·')> obey the equations (Lee and Stone 1985), ε 12 = 6 t — ε 2

-D j p - - c lA{r) + A'(r)lJ + i - i e 1 2) P?12 = 5{r - r'), (3.14)

-D j p - * [>ί(ι·) - ^ ' ( r ) ] J 2 + ^ - i e 1 2) P ° 2 = 5 ( r - r'), (3.15)

JL( r , ε,//) being the Langevin source at given energy ε, /(ε) = [εχρ ( ε/Τ )+ l ] - 1 being the Fermi function, Η being the magnetic field and A(r) being the vector potential of the magnetic field.

Equations (3.8)—(3.15) can be proved directly from the Kubo formula by summing the diagrams shown in figs. 3b-h, moreover the correlation function

3.2. The Langevin description of mesoscopic fluctuations in disordered metals

To describe the correlation functions of mesoscopic fluctuations of the current density in disordered metals at \r — r' | > / > h/pF we shall use the set of equations (Zyuzin and Spivak 1987)

div 5/(r) = 0; bJ{r) = -eDvVhn + / L( r , H) (3.8)

with the boundary conditions δ/ ± = 0 at the insulating boundaries. Here δ/ ±

is the component of the current density normal to the boundary; η = eq) + μ, <p(r) and μ{τ) are the electrochemical, electrical and chemical potentials, respectively. In many respects this approach is similar to that proposed by Maekawa et al. (1987) and Kane et al. (1988a,b). The origin of the eqs. (3.8) is the same as the eqs.(2.14) and (2.15) discussed in the section 2. One has to average the microscopic density current fluctuations over a length scale of the order of /, and then substitute it into the diffusion equation as Langevin sources:

(J\(r, H)J){r', / / ' ) > = J ds, d e 2/ ' ( e 1) / ' ( 8 2)

χ (J\(r,zu H)J)(r', e 2, J5T)>, (3.9)

(J\{r, 8 l, H)JW, e2, #')> = (j£J(Ki + K 2 + K ( 3· 1 0)

of the Langevin sources < J{'(r)Jj'(r,)> correspond to the diagrams of figs. 3c,d only. Here the wavy line corresponds to the Coulomb propagator. The thermoelectric current density fluctuations can also be described by eqs. (3.8)—(3.15), with the substitution in eq. (3.8) E(r) (e/eT)FT(Anisovich et al. 1987, Esposito et al. 1987). Equations (3.8)—(3.15) can be applied to any sample geometry with an arbitrary electric field distribution and spatial dependences D(r) and τφ(ή provided |r —r'| > /. We have broken up the correlation function of Langevin sources eq. (3.9) into three terms corresponding to different physical mechanisms and represented by the diagrams shown in figs. 3b,c,d.

Let us discuss the qualitative interpretation of the terms in eq. (3.10). Unlike eq. (2.16), only the first term in eq. (3.10) K1 is ^-correlated. The interpretation of K^r, r') is the same as for <σ?·> (see eqs. (3.3), (3.4)). The value K2 describes the long-range correlation between the current response at r due to the electric field at *·', Ir' — r\ > / and the current response at r' due to the same field at r. To explain this phenomenon one can verify the equation

with the help of eq. (2.2). The value K3 describes the long-range correlation of the local (on a size scale

of order /) conductivities. One can explain this phenomenon with the help of fig. lc. Consider the processes of electron transport between two couples of points r -> r1 and r' r\, provided \r — rx \ < /, \rt — r\ | < /, \r — r' \ /. Using eq. (2.2) one obtains

Here A0(r, r') is the probability amplitude for an electron to travel between r and r' without scattering.

In the conclusion of this section we present expressions for the correlation functions of the current density and the electrochemical potential imposed by the uniform electric field in the bulk samples. At |r — r'\ < LT one can omit the second and the third terms in eq. (3.10), so one has

(W(r, r')W{r\ r)> - (W(r, r')}2 » <W(r, r')}2 (3.16)

* \A0(r, rMoV, r\)\W(r, r')W{r\ r)

oc<^(r, r ' )>2. (3.17)

(3.18)

|r - r'\< LX9 Ly9 Lx, G(r, r') = ^\r-r'\~\

In the cases of the film Lz<^Lx,Lp and the strip Ly, 4 < Lx we present the correlation functions averaged over the film thickness Lz and the strip cross-section L yL z, respectively:

(3.21)

Ly~Lx>\p-p'\>Lz,

/R < \κ ( >\\ Lo fe<E}\2 1 fxx' x + x ' u \ (δη(Χ)δη(Χ )> = ~~z ~7Γ~ γ-γ- ~ — + \\x - X | ;

(Dv)2 \ 2nh J LyLz \LX 2 ) (3.22) Lx > \x — x'\ > Ly, L z,

p, p' and x, x' being the coordinates in the plane of the film and along the strip, respectively 0 < x , x' <LX. One should stress that the power and logarithmic laws in eqs. (3.19)—(3.22) are of a purely classical nature, so L 0 is the only quantity that is determined by Εφ and LT.

The spin density fluctuations hS(r) induced by the electric field behave like 8 i K r )a t L r > L s o, \r-r'\<Lso

<8Si(r)8Si(r')> * Suv2 (δη(ήδη(/)}9

while at \r — r'| > L s o,

' I r - V l <8Si(r)8S/r ' )>ocexp -

^ s o

Using the Poisson equation div Ε=4πβνδμ one can obtain that

If \r — r'| > L r, one has to take into account the second and the third terms in eq. (3.10). This leads to the additional factor of the order of unity in eq. (3.18).

Using eqs. (3.8) and (3.18) one obtains (Zyuzin and Spivak 1987, Kane et al. 1988a)

<δΜή^)>=u ( ^ p j ( V ( ' - η+|r | j G ^ ' ) ) > ( 3· 1 9)

< 8 , W8 ^ ) > = ^ ( ^ ) 2 G ( , , 0 . (3.20)

Here G(r, r') is the Green function of the diffusion equation subject to the appropriate boundary conditions. In the case

at |r — r'\5>K, κ being the screening length. Long ago Landauer (1957, 1987) discussed these fluctuations of electric potential qualitatively. Averaging eq. (3.20) over the region \r — r'\ % / one can show that so long as the inequality

holds it is possible to neglect in eqs. (3.10), (3.11), (3.12) and (3.13) and the diagrams in fig. 3 the difference between the averaged, <£(*·)>, and the acting, E(r\ fields. For sufficiently low temperature when the opposite inequality holds, the theory of mesoscopic fluctuations has not been developed but fortunately eq. (3.23) is valid in most experiments.

We have been concerned so far with the mesoscopic fluctuation dispersions of the quantities of interest. Lerner (1989) has shown that the Langevin approach eq. (3.8) can also be applied to the calculation of the higher moments of 8/(r) and has calculated the distribution function of JL(r) which appeared to have a non-Gaussian form.

3.3. Methods of measuring the mesoscopic fluctuations in disordered metals

In this section we apply eqs. (3.8)—(3.15) to the calculation of the mesoscopic voltage fluctuations in some devices.

The original calculations of the conductance fluctuations (Altshuler 1985, Lee and Stone 1985) were carried out for a two-lead geometry shown in fig. 4a in which the voltage is measured between the same leads through which the

3

r Ύ Ύ 4

• χ

1 3 4

( a ) ( b ) (c )

Fig. 4. Different devices for the measurement of mesoscopic fluctuations.

current passes, using eq. (3.8) one obtains

< ( δ ϋ 2> = ρ J <JLx(r)JLx(r')> dr dr\ (3.24)

ix being the total current passing through the sample. Substituting eq. (3.9) into eq. (3.24) one obtains the famous result (Altshuler 1985, Lee and Stone 1985)

i y T=o-< ( 8 G ) 2> *

e2 \2L0LyL2

1 3 ' {\2nhJ L

(3.25)

7 V 0 ;

that the fluctuations of conductance 6G = 8iJ C/<£J C>Lx measured by the two-lead method at Τ = 0 have the universal value.

The qualitative interpretation of eq. (3.25) is as follows: Ohm's law for the conductance G can be rewritten as

G = ^N; N = vv-, (3.26) h t0

t0 = L 2 /D being the time it takes for an electron to diffuse out of the sample. Thus the only electron energy levels in the sample lying in the energy band centred at the chemical potential level and with the width of the order of ft/i0

are responsible for the value of G (Thouless 1977). Altshuler and Shklovskii (1986) pointed out that at T = 0 the fluctuations of Ν are drastically reduced due to the level repulsion and according to Dyson (1962) δ Ν % 1, so

« δ ο 2 > - , 1 ί 3 2 7)

< G >2 ~ N 2 > (3-27)

contrary to the relative fluctuations of the transmission coefficients eq. (2.25), which are proportional to i V - 1. This is quite natural because the conductance of the sample is proportional to its transmittance averaged over the beam incidence angles.

There is, of course, no reason to expect the resistance of a quantum-mechanical system to be independent of the way it is measured. Figure 4 displays the multilead devices conventionally used for the measurement of voltage fluctuations. Apparently, in any device shown in fig. 4 the voltage leads average the large microscopic fluctuations of electrochemical potential discussed above over the lead dimensions. That is why the voltage lead dimensions play a very important role in determining the fluctuations of voltage measured in multilead devices. Another circumstance is that the leads are themselves disordered conductors and the electrons can diffuse into them and lose their coherence memory.

Thus, generally speaking, one has to study the mesoscopic system as a whole, including the measuring leads and macroscopic reservoir.

Suppose that the measuring leads used in devices shown in fig. 4 are ideal, i.e. have an infinite conductivity. In this case one can use the boundary conditions

δη(ή\8 = δη* = const, (3.28)

S being the plane dividing the sample and the leads. In the case of the four-lead device shown in fig. 4b the current passes through the leads 3, 4 and the voltage bias Δ ^ 12 = δ^ι — δη\ is measured between the leads 1, 2.

Strictly speaking, to calculate < (Af/1 2) 2> it is necessary to average the correlation functions <6^(r)5?y(r,)> in the finite sample with the boundary conditions eq. (3.28) over the lead size a, but instead of this we shall use the eqs. (3.20) and (3.21). That is why we shall drop numerical factors in the expression for < ( Δ ^ 1 2) 2> .

To average eqs. (3.20) and (3.21) over the lead size one has to substitute a for |r — r'| or \ p — p'\ and the expression for the conductance dispersion

measured by the four-lead method has a form, G 0 = a0LyLz/L, L being a distance between the voltage leads, (Zyuzin and Spivak 1987):

e 2 \2 L z, L x, Ly>L>a,

' — In - ; L x, Ly L>a, Lz. Lz a

Thus 6G1 2,34 can be dramatically enhanced over the 'universal' value of the order of e2/h, provided L 0 > a, and the large magnitude of the fluctuations is connected with the small lead dimensions rather than with the sample size. The only difference between the four- and six-lead devices shown in fig. 4c is the unused potential leads. Averaging the eqs. (3.20) and (3.21) over the lead size one can obtain the correlation function of the voltages measured by the 1, 2 and 3,4 probes, respectively,

< Δ *7 1 2Δ>73 4> :

Here L and r0 are the distances between leads (see fig. 5c).

We have presented the results of eqs. (3.29) and (3.30) for the two- and three-dimensional cases because only in this case do the leads affect the electrical field and Langevin sources insignificantly in the bulk of the sample.

Up to this point we have been considering the devices with the ideal leads. Figure 4d displays the lead geometry that is closer to real life. In this case the electrochemical potential is measured by a probe with the form of a one-dimensional strip that has the same conductivity as the bulk of the sample. Using the eqs. (3.8), (3.11) and (3.15) one can obtain LT>a,b>L9> a (Benoit et al. 1987, Buttiker 1987, Maekawa et al. 1987, Kane et al. 1988b)

where S is the cross-section of the strip. Figure 4e displays the device proposed by Washburn and Webb (1986), where

the mesoscopic fluctuations are themselves manifested most surprisingly. The current passes through leads 3,4 and the electrochemical potential difference is measured by leads 1, 2. If b > a (see fig. 4e) the electric field penetrates only into the bulk of the sample and according to classical expectations, the averaged voltage between leads 1, 2 (η\ — η°2} tends to zero as exp(— b/a). Therefore the value Δί/12 = η\ — ηΙ is of P u re mesoscopic nature and is directly connected with the long-range behaviour of <(al 7(r, r ' ) ) 2> , e cl - (3.3).

The Langevin sources JL(r) at r are caused by the electric field E(r) distributed in the region \r — r'\<LT. In the case b>a they are determined by the first term in eq. (3.10). Integrating eq. (3.8) over the sample region between leads 1 and 2 shaded in fig. 4e, one obtains

u and S being the volume and the cross-sectional area, respectively, of the region shaded in fig. 4e.

Substituting eq. (3.10) into eq. (3.32) at b < L v, and L> Εφ one arrives at the same expression for < ( Δ ^ 1 2) 2 > as eq. (3.31) (Spivak and Zyuzin 1988c). If b > Εφ

the expression for < ( Δ η 1 2) 2> differs from eq. (3.31) by the factor exp(— 2b\L^. If the device has the geometry shown in fig. 4f, then the current i flowing

through the loop cross-section has a random sign and at b < L^, 2nR > Εφ

(3.32)

(3.33)

R being the radius of the loop in fig. 4f. < j > is the average current flowing through the bulk, S being the cross-sectional area of the loop.

In conclusion let us note that the above-mentioned spin density fluctuations 8S(r) induced in the metal by the electric field can be measured by ferromagnetic point contacts. In this case the voltage fluctuations should depend on the directions of the current and the magnetization of the contacts.

3.4. Mesoscopic fluctuations in metals caused by the magnetic field variation

We have studied so far the way in which the various quantities of interest fluctuate from sample to sample because of different microscopic impurity distributions. In this subsection we consider the mesoscopic fluctuations of the conductance of a given sample caused by the magnetic field variation. Umbach et al. (1984) have observed time-independent fluctuations of magnetoresistance as a function of Η (magnetofingerprints), which vary between samples but are reproducible (at a given temperature) within a given sample.

Stone (1985) gave the qualitative explanation of this experimental result in terms of the 'diffusive' interference of electrons as they travel through the sample. The transmission coefficient of the sample can be obtained by squaring the modulus of the sum of all the amplitudes of the probability for an electron to pass through the sample (see eq. (2.2)). Each interfering path Ak in eq. (2.2) acquires in magnetic field H=(0,0, Hz) a phase shift of the order of

A(r) being the vector potential of the magnetic field. The integral in eq. (3.34) is taken along the T t h path.

This makes the transmission coefficient an oscillating function of Η. The characteristic period of oscillations Hc at T = 0 can be obtained from the condition that the magnetic flux through the area between two interfering paths (of the order of LyLx) is of the order of the flux quantum φ0 = hc/e. Thus Hc*(p0/LxLy.

Lee and Stone (1985) have put forward the 'ergodic' hypothesis that at sufficiently large Η the ensemble average at fixed H, AH is equivalent to an average over many values of Η for fixed AH

(3.34)

(δΧ(Η)δΧ(Η + AH)) = lim -Ή

δΧ(Η)δΧ(Η + AH) dH, (3.35) ο

i.e., (δΧ(Η)δΧ(Η + AH)) depends only on AH for large H. Here X denotes any fluctuating quantity of interest discussed above (G; G 1 2, 34 ; δ/; δη; δη).

Altshuler et al. (1986) were able to prove for G e2/h the identity

lim ( ( <δΧ(Η)δΑ ' (Η + Δ Η ) > - - ^ Η \2

δΧ(Η)δΧ(Η + Δ Η ) άΗ ) )= 0, ο ,

(3.36)

which is equivalent to eq. (3.35). To calculate the correlation function <5X(r, H)bX(r'9 Η + AH)} for

\r — r'\<LT one can drop the second and the third terms in eq. (3.10):

< 5 X ( r , f l ) 8 WH + A H ) > = // A f l \ ^

(δΧ{ήδΧ(/)} JL\H(

so only the field difference AH remains in eq. (3.37). Here f^y) falls off for y>\.

For \r — r' | > LT one can not drop the second and the third terms in eq. (3.10), so (8X(r9 Η)δΧ(/9 Η + AH)} does not depend on Η only if H>HC. In this case the corresponding expression differs from eq. (3.37) by a constant of the order of unity. The results for Hc and the asymptotics of fx (y) at finite temperature are summarized in table 1 (Lee and Stone 1985, Lee et al. 1987).

It is important that ((8J(r,H))2) and <(5iy(r, H))2) depend slowly on Η while, e.g., {(8G(H))2} is reduced in half at H>HC in comparison with <(6G(0))2> (Lee and Stone 1985). (We do not take into account the Zeeman splitting of the electron energy in a metal.)

Let us discuss the applications of the Onsager-Casimir (Casimir 1945) symmetry relations

aiJ(r9/9H) = aJi{i'9r9-H) (3.38)

for the mesoscopic system. It should be emphasized that the relations eq. (3.38) hold for arbitrary conditions. In the absence of the electron-electron interaction eq. (3.38) can be directly proven for a mesoscopic system by summing the diagrams shown in figs. 3b,c. The eq. (3.38) leads to the Onsager-Casimir symmetry relation for conductance measured by the two-probe method,

G(H) = G(-H). (3.39)

To drive the relation

< [ 8 G ( » ) - 6G ( -// ) ] 2> = 0 (3.40)

from the eqs. (3.8)—(3.15) it is necessary to take into account all the three terms in eq. (3.10) (Altshuler and Khmelnitskii 1985).

It should be noted that the microscopic current density and the electron density at given E(r) do not possess any symmetry with respect to the reversal of the magnetic field, because the only quantum-mechanical requirement that

has to be satisfied is

A(r, σ; r\ σ') = A(r\ — σ'; r, — σ).

A(r, σ; r\ σ') is the probability amplitude of transition between the states r, σ and r\ σ'; σ is the spin index.

Therefore, there is no reason to expect that the conductance measured by any methods obeys the symmetry relation analogous to eq. (3.39). In the case of the four-lead geometry at a <^ LT the second and the third terms in eq. (3.10) can be omitted and one obtains at H>HC (Ma and Lee 1987, Zyuzin and Spivak 1987)

from the multichannel Landauer formula (Biittiker et al. 1985). Spal (1980) and Sample et al. (1987) give the derivation of the eq. (3.42) from the local symmetry relations, eq. (3.38).

It is necessary to keep in mind that in addition to the mesoscopic fluctuations the conductance of a given sample exhibits the average magnetoresistance due to the weak-localization corrections (Altshuler and Aronov 1985, Lee and Ramakrishnan 1985) that are connected with the self-crossing paths shown in fig. lb. The relative size of the average and fluctuation effects in the magnetoresistance of a small sample depends on the temperature, sample parameters and the method of conductance measurement, so no general statement can be made about which effect will dominate. This question was considered in detail by Lee et al. (1987).

4. Friedel oscillations in disordered metals

In this section we address the properties of Friedel oscillations in disordered metals resembling the current oscillations considered above.

It is well known that the existence of the Fermi surface in metals results in a power law oscillating response of the electron wavefunction to a local perturbation. The important example is the Ruderman-Kittel exchange interaction of two localized spins I0(r) in a pure metal, separated by a distance r (Ruderman and Kittel 1954):

<[6G 1 2 e 34 ( i50 - 5 G 1 2, 3 4( - / / ) ] 2> * < ( 6 G 1 2, 3 4 ( # ) ) 2>

Biittiker (1986) has derived the identity

G l 2 , 3 4 ( # ) = G 34 i l 2( — H)

(3.41)

(3.42)

lo(r)=j2&(r,0)=j2 — , 2 ν cos(2pFr/fc)

' Ait ? (4.1)

Here the index Ό ' corresponds to a pure metal, and j is the constant of the interaction of the impurity spin with the conduction electrons, # i m(r , r') is the

spin susceptibility of the metal, given by the formula

r') = Τ Σ Sp&Gfr r, ω)σ™(?(*·', r, ω ) } , (4.2)

ω where σ* = σ[β are the Pauli matrices and ω = πΤ(2η Η-1); G = Ga /3 is the electron Green function, and α and β are spin indexes.

Another consequence of the Friedel oscillations is the fact that the electric field of the charged impurity in a metal is screened at large r as

De Gennes (1962) and Mattis (1965) have shown that in a weak disordered metal eqs. (4.1) and (4.3) become modified at \r — r'\ > I:

(φ(ή) = φ0(ή exp(-r//).

The eq. (4.4) has often been used to calculate the spin glass transition temperature 7^G.

We shall show that it is incorrect to use eq. (4.4) for this purpose (Zyuzin and Spivak 1986, Bulaevskii and Panyukov 1986). A qualitative explanation of this fact is the following. In a disordered metal eq. (4.1) becomes modified:

where C(r) is a smooth but otherwise random function, and <5(r) is the phase shift that is associated with the scattering of an electron by impurities. This shift becomes nearly random at r > I When we take an average over the realization of the random potential at r > /, i.e., over the random phase shifts (5(r), we find eq. (4.4).

It is clear, however, that the critical temperature 7^G oc </2 > 1 /2 is determined by the typical values of I(r\ regardless of its sign. It is also obvious that we have < / 2> o c r " 6 and this quantity does not possess an exponentially small parameter. The estimate TSG&j2v/rs, which we find proceeding from these arguments, differs by a factor of exp (— rjl) from a standard estimate, rs being the average distance between the paramagnetic impurities. If the random scattering potential can be considered as a white noise one can use the standard diagram technique for # i m.

To find (xfm} we need to sum the Feynman diagrams shown in figs. 5a,b, similar to those describing the conductance fluctuations. Here are the results

(4.3)

(4.4)

(4.5)

Mesoscopic fluctuations of current density 6 5

for the three-dimensional case at T = 0 (Zyuzin and Spivak 1986):

/ 2 , „ x J ν λ 2 [4<5i m; / < | r - r ' | < L s o, <xL(r, ')> = 3 J - : ^ < , , 4.6

In the limiting case Z< |r — r ' | <^Lso and LT > |r — ι*'| ρ L s o, the quantity <#2m(r,#·')> has a universal form, independent of Z. Furthermore, at \r — r'\}Lso

the quantity </2m> does not depend on the spin indexes i and m. This means that the exchange between the localized spins becomes anisotropic and is described by a random non-Heisenberg matrix of the Dzyaloshinskii-Moriya type. This fact has been used in explaining the /-dependence of the hysteresis in the magnetization of spin glasses (Levy and Fert 1981). The calculations that have been carried out, however, have been restricted to first order in r/Z<^ 1. Calculations analogous to eq. (4.6) indicate that the dispersion <(p2(r)> ocr" 6

significantly exceeds the average, eq. (4.4). Note that large fluctuations of cp(r) have been observed in a study of the electric field screening in the insulating phase of doped semiconductors (Baranovskii et al. 1984).

Stephen and Abrahams (1988) and Jagannathan et al. (1988) have carried out detailed investigation of the higher moments of # i m and the anisotropy of the spin interaction due to spin-orbit scattering. They have showed that all the odd moments of xim(r, r') fall off exponentionally while all the even ones are of the order of (xf£(r, r ' )> oc < / 2m> w-

Bulaevskii and Panyukov (1986) have considered another model in which the electrons move in a random potential slowly varying in comparison with h/pF. In this case one can use the quasiclassical approximation for the Green function and in fact in this case the random phase in eq. (4.5) is due to slow spatial fluctuations of the Fermi momentum p F, but unfortunately, no one has been able to calculate <#2(r, *·')> in this model for \r — r'\ > Z.

In the conclusion of this section we would like to stress that long ago much of the essential physics of the disordered Ruderman-Kittel interaction was discussed qualitatively by de Chatel (1981).

Fig. 5. (a) particle-hole and (b) particle-particle diagrams for the second moment of the non-local spin susceptibility.

5. Mesoscopic fluctuations of current density in superconductors

<0 = <0 sin χ, χ = χ1-χ2, (5.1)

LX>LT. (5.2)

s Ν S

X

(α)

c

s Ν S

( b )

Fig. 6. (a) Superconductor-normal-metal-superconductor (SNS) junction. Schematic illustration of the self-intersecting paths that give rise to the χΧ - χ2 dependence of the conductance in the normal

metal region, (b) Device for the measurement of the ΐγ—χ2 conductivity dependence.

We shall consider in this section the mesoscopic fluctuations of the current density in superconductors. This phenomenon resembles both the fluctuations of current density in the normal metal considered in section 3 and the Friedel oscillations in the disordered metals considered in section 4.

5.1. Mesoscopic fluctuations of supercurrent density

(a) To begin with let us consider the mesoscopic fluctuations in the superconductor-normal-metal-superconductor (SNS) junction shown in fig. 6a. It has been shown above that in a disordered normal metal the phase of the electron wavefunction remains coherent over distances much larger than the electron mean free path /. This coherence also determines the properties of a disordered SNS junction.

The supercurrent of the SNS junction averaged over the random impurity distribution for Τ<ζ Δ has the form (Aslamazov et al. 1968)

Here is is the superfluid current through the junction, ic is the critical current; ^> Xi(2) a re the modulus and the phases of the order parameters of the opposite superconducting banks of the junction and G N is the conductance of the N-region of the junction.

It is clear that the critical current of an actual disordered sample differs noticeably from its means value (Altshuler and Spivak 1987)

<02 ~ < G N > 2 ' ( J

This effect can be qualitatively attributed to the sensitivity of the electron energy levels in the N-region to the change of χ = Χι — fo- Such a sensitivity results from the Andreev reflection (Andreev 1964) of the electron from an NS boundary. In the course of this reflection the electron is transformed into a hole and acquires an extraphase χη2). The hole, reflected from the NS boundary is transformed into an electron and acquires an extraphase — χ1{2).

The change of χ leads to the change of the boundary conditions on the NS interface and to a random shift of the electron energy levels in the Ν region by the amount δΕ ~ Ε0δχ (Thouless 1977), where Ec « \iD\L\. This is accompanied by the changes in the density of states near the Fermi level and in the free energy Ω of the Ν region of the SNS junction, so Ω appears to be a sample-specific periodic function of χ. Following the relation (Byers and Yang 1961)

^ 9 ? ( 5· 4 )

one can obtain the corresponding fluctuations in is and ic. Obviously these oscillations in is are an analog of the diamagnetic currents

induced in the normal metal by the magnetic flux φ penetrating a solenoid of Aharonov-Bohm geometry. (The metallic cylinder encloses a solenoid so that everywhere outside the solenoid the magnetic field is zero. We believe that all cylinder dimensions exceed /). Thus, the magnitude of these diamagnetic currents can be obtained from eq. (5.4) by substitution χ 2πφ/φ0, φ being the magnetic flux penetrating the solenoid. Without interaction between electrons the χ dependence of Ω arises only due to the mesoscopic effect. However, in the presence of interactions even the average <Ω> depends on χ with the period π (Altshuler et al. 1983).

(b) Consider now the spatial mesocopic fluctuations of supercurrent density in the bulk of the disordered superconductor. The starting point of our calculations is the expression for the supercurrent density 7s(r),

Stj(r9 r')vsj(r') dr'. (5.5)

Fig. 7. Diagram that describes the supercurrent density mesoscopic fluctuations.

Here

is a superfluid velocity, the resolvent kernel S0(r, **') is the random sample-specific function.

Averaging eq. (5.5) over the random-potential realizations leads at ξ(0) = [Dft/zl(0)]1 /2 >l>h/pF to the well known results (de Gennes 1966):

< Λ > = * < # . > < » . > , (5.7)

<iVs> = lJd ,- '<S1, (^ ' )> = f ( ^ ) 2 t h ( ^ ) , (5.8)

where Ne is the electron concentration.

In a given disordered superconductor the value Js(r) differs from its average value of </s>, because of spatial mesoscopic fluctuations 8/s = /s — </s>.

According to eq. (5.5) the dispersion <(5/s(r))2> may be expressed in terms of the correlation function:

Kijtir, rl9 r, r[) = <8Sv(r, r^ S^ r , r')>; SSU = StJ - <S„>.

One can obtain Kijh(r9 rl9 r9 *·') by summation of the diagrams in fig. 7, where the solid lines correspond to the electron Green function in superconductors.

At ξ(0) > / one obtains (Spivak and Zyuzin 1988a)

<(5/»)2>*</s>2^5. (5.9) Thus the supercurrent density fluctuations exceed the average supercurrent <(δ/.)2> > <Λ>2 provided {(0) > l(pFl/h)2.

The large value of <6J s( i * ) ) 2> is the consequence of the long-range behaviour of bSu(r,r'):

(\r-r'\-2; 1<\,-/\<ξ(0),

<(5S f, ( i-,r ' )2>oc / 2\r-r'\\ , „ „ m (5.10)

It follows from the eqs. (5.5) and (5.6) that the average value </s(r)> is caused by the superfluid velocity vs(rf) distributed throughout the region \r — r'| </; meanwhile the dispersion <(5Js(r ) )2> is determined by vs(r) distributed over a much larger region l<\r — r'\< ξ(0).

As in a normal metal, a lowering of the symmetry in disordered superconductors leads to the existence of spin density mesoscopic fluctuations

induced by the supercurrent ( L so < £(0)). Let us emphasize that there exists a close analogy between eqs. (5.9) and

(5.10), and the corresponding expressions for the density current fluctuations in a normal metal, eqs. (3.3) and (3.6). One can measure the mesoscopic fluctuations of the supercurrent density eq. (5.9) and the corresponding fluctuations of δχ with the help of a tunnelling microscope.

In the case \r — r'\ > ξ(0) the mesoscopic fluctuations of the superfluid current density can be described by the fluctuations of the superfluid density (Spivak and Zyuzin 1988a), which may be determined from the equation

δJsi(r) = β(ΝΒ}δν5ί + ebN'M^j}, (5.11)

<δΝΪ(ήδΝ?(Γ'))^φΝΪ)2) exp( | r - f | > { ( 0 ) , (5.12)

ξ(0){γ) . (5.13) < ( δ ^ ) 2> _ < (5G , ) 2) _ Ζ /pgA-2

<Nsy ~ <G<>2 - \ a O )h 2

Here ϋξ is the conductance of a cube with side ξ(0). The fluctuations of the superfluid density leads to fluctuations of the critical

current ic in a quasi-one-dimensional superconducting channel with the small transverse dimensions Ly~ Lz £(0),

h2 < ( 6 i c) 2> * 2U ( 0 ) - £ . (5.14)

In the long channel Lx > ξ(0) the critical currents of sections of length ξ(0) fluctuate independently.

Figure 4e displays a device, proposed by Washburn and Webb (1986) where the mesoscopic fluctuations of the supercurrent and the phase of the order parameter manifest themselves most surprisingly.

The supercurrent passes through the leads 3,4 and the phase difference Αχ = Χι—Χι is measured between the leads 1 and 2. The value of <Δχ>(χ exp(— b/a) decays exponentionally and Δ χ ^ Ο is of pure mesoscopic nature provided b$> a>l (see fig. 4e). In this case the superfluid velocity (vs(r)} penetrates the bulk region of the sample only.

The origin of Δχ Φ 0 is a consequence of the long range behaviour of δ50·(ι·, r'\ Integrating <8S0(r, r')5Sij(rl,r'l)> over r, rl in the shaded region in fig. 4e and over r\ r[ in the bulk of the sample, as in eq. (3.32), one obtains for L> ξ(0) > a

( 1 , ^ , ^ > ξ ( 0 ) > ^ ,

The eq. (5.15) holds if b< ξ(0). In the case b > ξ(0) one acquires an extra factor exp(-2fc/£(0))in eq. (5.15).

If the sample has the geometry shown in fig. 4f, then there is a supercurrent is passing through the cross-section of the loop. This current has a random direction (depending on impurity configuration) and at b < ξ(0) < 2nR,

(2nR)2 \ p¥l

L „ L ^ ( 0 ) > L z ,

(5.16)

</s> is an average supercurrent in the bulk of the sample, S being the cross-section area of the shaded region in fig. 4e, and the loop in fig. 4f, respectively.

5.2. Mesoscopic fluctuations of the normal current

(a) To begin with consider mesoscopic fluctuations of the conductance G N of the SNS junction normal region.

Andreev reflection from SN boundaries makes G N sensitive to the change of χ so the conductance of a given junction has the form;

GN = <GN(X)> + ?>GNg(x). (5.17)

Equation (5.17) is an analog fo the φ-dependence of the conductance of a normal metal sample with Aharonov-Bohm geometry. The origin of the χ-dependence in <GN(/)> is due to weak-localization corrections to the averaged metal conductivity (Spivak and Khmelnitskii 1982).

The qualitative interpretation of this correction can be made with the help of fig. 6a where the solid lines correspond to the electron paths and the dashed lines correspond to hole paths. Each self-intersecting path (see fig. 6a) can be assigned a pair of amplitudes A\l) and A\2) in eq. (2.2) corresponding to the passage of the loop clockwise and counterclockwise. At χ = 0 these amplitudes are coherent and A\1] = A\2). As has been mentioned in subsection 5.1, at χ Φ 0 these amplitudes acquire additional phase factors

A\1](x) = A\»(0) exp fa ) , Α\2\χ) = A\2\0) e x p ( - i Z) . (5.18)

Thus <GN(# )> can be obtained from the corresponding expression for the Aharonov-Bohm disordered-metal conductance oscillations with the half-flux quantum ^φ0 period (Altshuler et al. 1981, Sharvin and Sharvin 1981) with a substitution 2πφ/φ0 - » χ , i.e., <G N(# )> has the period π.

The second term in eq. (5.17) is the analog of the mesoscopic Aharonov-Bohm oscillations in a finite sample with the normal flux quanta φ0 period (Washburn and Webb 1986), so the random sample-specific function g(x) has period In and < (5G N) 2> has the same value as eq. (3.25). It is important that the weak-localization oscillations <GN(# )> survive even at LT<^LX<^ when both the Josephson oscillations, eq. (5.1), and the mesoscopic oscillations bGNg(x) have already decayed to zero. Therefore to observe the mesoscopic oscillations in G N from the background of the weak-localization oscillations <G N(x )> one needs the small N-region (Lx ^ LT) samples. Note that the mesoscopic oscillations are not suppressed by the magnetic field and the magnetic ordering in the Ν region contrary to <GN(# )> .

There are two methods for the experimental measurement of GN(# ) (Spivak and Khmelnitskii 1982). If a voltage u is applied to the opposite banks of a SNS junction, then

9/ 2ne

i=T- (519)

and the normal current i N = GN(x(t))u passing through the junction is a periodic function of the time t. Moreover, one can use the additional Josephson junction V, see fig. 6b, to produce the phase difference χ between the banks of the SNS junction. In this case one can measure, e.g., the resistance of the point contact ' C shown in fig. 6b as a function of χ.

(b) Another example of normal-current mesoscopic fluctuations in a superconductor is the mesoscopic fluctuation of the thermoelectric power.

It is well known that the thermoelectric power in bulk uniform superconductors is absent because a thermoelectric current of normal excitations caused by a temperature gradient F T is exactly offset by a supercurrent (Ginsburg 1944, Galperin et al. 1978). However, it is actually only the normal current averaged over impurity configurations that is offset by the averaged supercurrent. There-

fore, in a single sample with a given impurity distribution there exist spatial mesoscopic fluctuations of the electrochemical potential proportional to VT. At \ T— Tc \ < Tc, the amplitude of these fluctuations is the same as that of the corresponding fluctuations in a normal metal discussed in section 3. Here Tc is the critical temperature of the superconductor.

For example, in the case of an infinite two-dimensional film, LZ<^LT, the dispersion of the electrochemical difference measured by the two-point contact has the form (Spivak and Zyuzin 1989);

< ( Δ ^ 1 2) 2> ^ h2VT\2fL

In h

In (5.20)

Here a is the measuring contact size and L is the distance between contacts.

5.3. Can the ground state of a disordered superconductor correspond to a nonzero supercurrent state?

Generally speaking the ground state of a disordered superconductor always corresponds to the same distribution of supercurrent density Js(r) and order parameter phase χ{ν\ provided there is no time-reversal symmetry in the superconductor (e.g., due to magnetic ordering in antiferromagnetics or spin glasses). In this case χ(ι·) can be measured, e.g. using a superconducting interferometer.

The question arises: can the supercurrent exist in the ground state of a disordered superconductor in the presence of the time-reversal symmetry?

In this case the superconducting density Ns(r) in a bulk superconductor and the critical current in the SNS junction are real quantities but generally they can be both positive and negative.

It follows from eqs. (5.3) and (5.13) that <(5JVs(r))2> < < N S> 2, <(5*c)2> < 2

provided pFl>h, so Ns(r) > 0 and ic > 0 with a high probability. Altshuler and Spivak (1987) have speculated that for pFl^h when

<(6fc)2> ~ 2 it is possible for a small SNS junction to have a negative critical current ic < 0. Speaking more generally, the supercurrent density /s(r, vs) in a disordered superconductor is a random sample-specific function of r and

(o) (b) (c) Fig. 8. Schematic illustration of the several supercurrent dependences on velocity, which are possible

in 'strongly disordered' superconductors.

vs. The several possible dependences Js(vs) for pFl^h are shown qualitatively in fig. 8, so one can find the regions in a disordered superconductor with Ns < 0.

As far as we know, exact evidence for the possibility that

d V* vs = 0

does not exist*. The existence of regions of the size of ξ(0) with Ns(r) would make the zero

supercurrent state unstable with respect to the same random sample-specific distribution of the supercurrent density.

6. Sensitivity of mesoscopic fluctuations to the motion of a single scattering centre

It has been shown in the previous sections that different samples, which are macroscopically identical (i.e., have the same density of defects etc.) can have quite different conductances and other quantities of interest because the microscopic arrangement of the impurities is different. The question naturally arises: how sensitive is the conductance of a given sample to a small change in the impurity arrangement?

In this section we provide an answer to this question and the main result can be summarized as follows. The interference effects make all quantities of interest, X , considered in previous sections extremely sensitive to the detailed microscopic arrangement of the disorder. We denote by X any quantity considered above (G, G 1 2, 34 , η, n, /s, JVS, etc.) because their sensitivity to the scatterers motion can be described by the same equations.

Suppose we have a small disordered metallic sample of dimensions less then LT (or less then £(0) in the case of the superconducting sample). Then, e.g., in one and two dimensions the fluctuations of X induced by a movement of a single scattering centre are as large as those in eqs. (2.10), (2.11), (3.6), (3.25), (3.29), (3.33), (4.6) and (5.8) etc. produced by changing the entire sample (Altshuler and Spivak 1985, Feng et al. 1986). These results can be understood with the help of eq. (2.2).

•Kulik (1965), Shiba and Soda (1969) and Bulaevskii et al. (1977) have calculated the critical supercurrent of a tunnel junction with paramagnetic impurities inserted into the dielectric film. The averaging over the spin directions of the impurity recovered the time-reversal symmetry so ic

appeared to be real and changed its sign at a sufficiently large amplitude for electron scattering by paramagnetic impurities. Andreev (1987) proposed that ic < 0 to explain the magnetic properties of high-Tc superconductors with twinning planes.

The volume of a diffusion Feynman path that crosses the sample is of the order of v0 = L2xl2\\. We assume that the cross-section of the Feynman paths is of the order of λ2.

The motion of a single strong scatterer (with the cross-section of the order of i 2 ) o n a distance of the order of λ ~ h/pF will completely alter the phases of all the Feynman paths passing through that site. The number of moving scatterers visited by a Feynman path is of the order of ν0δΝι; 5JVft; being the total number of scatterers that change their locations or cross-sections, ν = LxLyLz.

Thus the resulting interference in eq. (2.2) will be completely altered provided (Altshuler and Spivak 1985, Feng et al. 1986)

ν IL„L^ 8 ^ 1 ) ^ 0 = - = - ^ . (6.1)

Vq A L·x

Thus movement of a single impurity provides a change in X of the order of

δ Χ * < ( δ Χ ) 2> 1 / 2- ^ 1 = . (6.2)

In a two-dimensional case (Lx ~ Ly5> Lz) the typical Feynman path visits a finite fraction of impurities in the sample so the movement of a single impurity produces the same effect at / ~ λ ~ Lz as altering the entire impurity configuration.

For L x, L y, Lz > LT the sample can be thought of as n0 = LxLyLz\L\ independent subsystems, which is of the order of LT on a side, and the fluctuations of X are then just the incoherent sum of the fluctuations of the individual subsystem.

The theory presented above allows the measurement of extremely small diffusion coefficients of impurities, Dh in a metal. For example, at Lx~Ly~Lz~LT~ 10 " 4cm , / ~ 1 0 _ 6c m . One can measure D ^ I C T 1 9

c m 2/ s _1 provided the accuracy of the conductance measurement is about e2/h. Another reason for altering X is the relaxation of the two-level systems in metals at low temperatures.

It is necessary to take into account that, in addition to changes in the conductance of an interference nature, the motion of single impurities induces a change in the conductance of a classical nature of the order of

5 G c l < G > ^ , (6.3)

so the contribution of the interference effect exceeds the classical one provided

(XL Y/4

Note, however, that the interference effects can be distinguished from the classical ones with the help of a magnetic field study of X.

An obvious application of eq. (6.2) is the phenomenon of flicker noise (Feng et al. 1986). A standard model for l/ω noise is that one measures the spectrum of the resistance fluctuation, which is thought to be due to the thermally activated motions of defects with a broad distribution of activation energies. What is new is that one can relate the magnitude of the noise to eqs. (6.2) and (6.4).

One can distinguish between the interference eq. (6.2) and the classic eq. (6.4) effects measuring the magnitude of l/ω noise against magnetic field. The classic contribution is insensitive to Η while the interference tends to decrease by nearly one half at Η > φ 0/^τ ·

There is another side of the problem discussed above, which concerns mesoscopic fluctuations of conductance of metallic spin glasses (Altshuler and Spivak 1985). In metallic spin glasses at T<^TSG the localized spins are in an ordered phase, i.e., the spin-flip process is effectively frozen out, and as far as conductance is concerned, one can regard the spins as fixed vectors. Thus, the exchange potential between localized spins and conducting electrons may be considered as an effective scattering potential for conductance electrons in addition to the static impurity scattering. Thus, the conductance of the system is sensitive to the configurations of the spins and undergoes fluctuations related directly to the corresponding spin fluctuations in the localized-spin subsystem.

The proof of the results presented above can be obtained by calculating the quantity (6X{u(r)}bX{u'(r)}y. Here u' = u(r) + 8w(r); u(r) and hu{r) are the scattering potential and the change of the scattering potential, respectively. The brackets < > mean in this case an averaging over random potentials u(r) and u'(r). The Feynman rules in this case have to be modified, so the dashed lines connecting the two conductivity loops in fig. 3 correspond now to the value

As a result, to obtain <5X{u(r) }8X{w'(r) }> one has to make the substitution in eqs. (3.9)-(3.15) (Altshuler and Spivak 1985, Feng et al. 1986)

ν ' - τ ^ + τ Γ 1 , (6.5)

τ Γ = //ιν& β = \ - < ^ . (6.6)

If the scattering cross-section of electron by an impurity is of order λ2, then β ~ SNJNi.

In the metallic spin glasses at T4:TSG,zt has the form (Altshuler and Spivak 1985)

(6.7)

Here t s = ls/vF, ls is the mean free path with respect to scattering of free electrons by the localized spins and S is the value of the localized spin.

Thus eq. (6.7) relates to the conductance of the metallic spin glass sample with localized spin correlation. Using eq. (6.5) one can easily obtain the results eqs. (6.1) and (6.2).

Up to now we have considered the sensitivity of X averaged over the positions of the moving scatterers. If in a disordered sample under an electric field a single scatterer at ra changes, e.g., its scattering amplitude, then evidently the response of the potential in the sample,

Αη{ή = η{Γ, u) - η{ν, ιι'},

will be a random quantity. We present here the expressions for the correlation function of Δη(ν) at L x, Ly > Εφ, LT > L z, \r — ra\ < L r,

< ( M W ) 2 > ^ < ( 6 ^ ) ) 2 > l n ( ^ ) , (6.8)

< ( δ , / ( . · ) ) 2 > = £ ^ < £ > Μ η ^ . (6.9)

Another aspect of the problem is the //-dependence of the hopping rate of the impurity in the two-level system. A change of the scattering amplitude leads to a change of the density of electron states near the impurity and to a change of the magnetization energy under a magnetic field. Therefore, the correction to the activation energy 8Ea for impurity hopping has a random sign and a magnitude of the order of (coc is the cyclotron frequency) (Altshuler and Spivak 1989);

6Eaoch(oc(l/Lz)il2 (6.10)

at L x, Ly> LT> L z; LT>LH = (ch/2eH)112 > I Thus at 8Ea > Τ the //-dependence of the hopping rate at a given point is exp(8£f l/T).

7. Conclusions

We believe that the main unsolved problem is how to take into account the difference between the averaged (E(r)} and the local E(r) electric fields.

Calculating the Langevin sources eq. (3.8) in metals (section 3) we have used <£(#·)> instead of E(r). However, according to eq. (3.20), microscopic electric field fluctuations rise as LT increases so the results obtained in section 3 hold only provided condition (3.23) is fulfilled (i.e., only for sufficiently high temperature).

In the case condition (3.23) is not fulfilled, the theory of mesoscopic fluctuations of current in metals in absent.

The problem is that if E(r) in eq. (3.1) means the local field (which is created both by external sources and by fluctuations of electron density) then σ0·(ι·, r') in eq. (3.1) can not be obtained with the help of the Kubo formula or with the help of summation of the diagrams of perturbation theory. In this case V^ijir, r') Φ 0, so eq. (3.1) and the current conservation law div J(r) = 0 form a close set of equations for the determination of E(r). See the opposite point of view in the paper (Kane et al. 1988a).

If this set of equations were used to obtain the correction to the average conductivity arising due to the electric field redistribution, then (Landau and Lifshitz 1982) at |£|>|δ£|

* e f f = < σ > + 1 3<σ> J

<δσο·(0, ι·)δσΜ(ι·, #·')>

8 2

Gfcr jd r df! dr'. (7.1)

Here the effective conductivity aeff is the proportionality coefficient between </(r)> and <£(r)>,

<*> = ^ |<*«(r, O ) dr',

G(r, rx) is the Green function of the Laplace equation, δΕ = E— <£>. It has been noted that the diagrams of fig. 3d and eq. (3.13) describe the long-

range correlation of local conductivities a£j-(r);

σ^, r,) = tfii(r)<5(r-r').

In the three-dimensional case

<MW ' i ) > ^ ( 0 j ^J2 . Ι ' - Ί Ι > 1 · (7-2)

Using eqs. (7.1) and (7.2) one obtains a correction to the average conductivity of the order of ( β 4/ * 2) ( / 2< σ » _ 1.

However, at sufficiently large LT the main contribution into eq. (7.1) corresponds to the diagrams of fig. 3c and eq. (3.12), so averaging over a region of the order of I one obtains for an infinite three-dimensional sample (G(r,r') = ( l / ^ k - r T 1 )

One must stress that eq. (7.3) drastically contradicts the calculations of the quantum corrections to the conductivity due to the electron-electron interaction (Altshuler and Aronov 1985).

Another side of the problem is the 'fluctuational' mechanism of magnetoresistance. If the dispersion of σ0·(ι·, r') depends on H, then according to eq. (7.1), ae(( also depends on H.

Note that it is this mechanism that gives the main contribution to the magnetoresistance of the strong disordered conductors possess hopping conductivity (Nguen et al. 1985).

In metals (pFl>h) the diagrams of fig. 3d lead to 'fluctuational' magnetoresistance

Equation (7.4) exceeds the estimation of Altshuler (1985) by the factor LT/l and is less than the weak-localization correction by the factor (ft/pF/)2.

However, the diagrams of fig. 3c give a huge negative magnetoresistance, so ° " e f f ( # ) — ^effiO) is of the order of eq. (7.3) for Η > q>§\L\.

However, until now no one has presented correct calculations of these effects using the diagramatic technique.

Another unsolved problem is the relation between the enormous current density fluctuations eq. (3.6) and the entropy production. Let us note the difference between the current density fluctuations in a normal metal and in a superconductor: eq. (3.6) diverges as T ->0 while eq. (5.9) remains finite.

Acknowledgement

We are grateful for discussions with B.L. Altshuler, B.I. Shklovskii and V.L. Gurevich.

References

Abrikosov, Α.Α., L.P. Gor'kov and I.E. Dzyaloshinskii, 1965, Quantum Field Theoretical Methods in Statistical Physics, 2nd Ed. (Pergamon Press, New York).

Altshuler, B.L., 1985, Pis'ma Zh. Eksp. & Teor. Fiz. 42, 530 [JETP Lett. 41, 648]. Altshuler, B.L., and A.G. Aronov, 1985, in: Electron-Electron Interactions in Disordered Systems,

eds A.L. Efros and M. Pollak (Elsevier, Amsterdam) p. 1. Altshuler, B.L., and D.E. Khmelnitskii, 1985, Pis'ma Zh. Eksp. & Teor. Fiz. 42, 291 [JETP Lett.

42, 359]. Altshuler, B.L., and B.I. Shklovskii, 1986, Zh. Eksp. & Teor. Fiz. 91, 220 [Sov. Phys.-JETP 64,

127]. Altshuler, B.L., and B.Z. Spivak, 1985, Pis'ma Zh. Eksp. & Teor. Fiz. 42, 363 [JETP Lett. 42, 447]. Altshuler, B.L., and B.Z. Spivak, 1987, Zh. Eksp. & Teor. Fiz. 92, 609 [Sov. Phys.-JETP 65, 343]. Altshuler, B.L, and B.Z. Spivak, 1989, Pis'ma Zh. Eksp. & Teor. Fiz. 49, 671. Altshuler, B.L, A.G. Aronov and B.Z. Spivak, 1981, Pis'ma Zh. Eksp. & Teor. Fiz. 33, 103 [JETP

Lett. 33, 94].

' i r ( Η ) - <σ(Η)> - σ „ , (0) + <σ(0)> oc h\a)LTV

(7.4)


Altshuler, B.L., D.E. Khmelnitskii and B.Z. Spivak, 1983, Solid State Commun. 48, 8. Altshuler, B.L., V.E. Kravtsov and I.V. Lerner, 1986, Pis'ma Zh. Eksp. & Teor. Fiz. 43, 342 [JETP

Lett. 43, 441]. Andreev, A.F., 1964, Zh. Eksp. & Teor. Fiz. 46, 1823 [Sov. Phys.-JETP 19, 1228]. Andreev, A.F., 1987, Pis'ma Zh. Eksp. & Teor. Fiz. 46, 463. Anisovich, A.V., B.L. Altshuler, A.G. Aronov and A.Yu. Zyuzin, 1987, Pis'ma Zh. Eksp. & Teor.

Fiz. 45, 237 [JETP Lett. 45, 245]. Aronov, A.G., and Yu.V. Sharvin, 1987, Rev. Mod. Phys. 59, 755. Aronov, A . G , A.Yu. Zyuzin and B.Z. Spivak, 1986, Pis'ma Zh. Eksp. & Teor. Fiz. 43, 431 [JETP

Lett. 43, 555]. Aslamazov, L .G , A.I. Larkin and Yu.N. Ovchinnikov, 1968, Zh. Eksp. & Teor. Fiz. 55, 323. Baranovskii, C D , B.I. Shklovskii and A.L. Efros, 1984, Zh. Eksp. & Teor. Fiz. 87, 1793 [Sov.

Phys.-JETP 60, 1031]. Benoit, A , C P . Umbach, R.B. Laibowitz and R.A. Webb, 1987, Phys. Rev. Lett. 58, 2343. Bulaevskii, L . N , and S.V. Panyukov, 1986, Pis'ma Zh. Eksp. & Teor. Fiz. 43, 190 [JETP Lett. 43,

240]. Bulaevskii, L . N , V.V. Kuzii and A.A. Sobianin, 1977, Pis'ma Zh. Eksp. & Teor. Fiz. 25, 314 [JETP

Lett. 25, 7]. Biittiker, M , 1986, Phys. Rev. Lett. 57, 1761. Biittiker, M , 1987, Phys. Rev. Β 35, 4123. Biittiker, Μ , Y. Imry, R. Landauer and S. Pinhas, 1985, Phys. Rev. Β 31, 6207. Byers, N , and C.N. Yang, 1961, Phys. Rev. Lett. 7, 46. Casimir, H.B.G, 1945, Rev. Mod. Phys. 17, 343. de Chatel, P.F, 1981, J. Magn. Magn. Mat. 23, 28. de Gennes, P .G , 1962, J. Phys. Rad. 23, 630. de Gennes, P .G , 1966, Superconductivity of Metals and Alloys (Benjamin, New York). Dyson, F.J, 1962, J. Math. Phys. 3, 140, 157, 166. Esposito, F.P, B. Goodman and M. Ma, 1987, Phys. Rev. Β 36, 4507. Feng, S, P.A. Lee and A.D. Stone, 1986, Phys. Rev. Lett. 56, 1560. Feng, S, C L . Kane, P.A. Lee and A.D. Stone, 1988, Phys. Rev. Lett. 61, 834. Galperin, Y . M , V.L. Gurevich and V.I. Kozub, 1978, Phys. Rev. Β 18, 5116. Ginzburg, V .L , 1944, Zh. Eksp. & Teor. Fiz. 14, 177. Gurevich, V .L , 1962, Zh. Eksp. & Teor. Fiz. 43, 1771 [Sov. Phys.-JETP 16, 1252]. Jagannathan, A , E. Abrahams and M.J. Stephen, 1988, Phys. Rev. Β 37, 436. Kane, C L , R.A. Serota and P.A. Lee, 1988a, Phys. Rev. Β 37, 6701. Kane, C L , P.A. Lee and D.P. Divincenzo, 1988b, Phys. Rev. Β 38, 2995. Kulik, I .O , 1965, Zh. Eksp. & Teor. Fiz. 49, 1211. Landau, L . D , and E.M. Lifshitz, 1978, Statistical Physics, Part 2 (Nauka, Moscow). Landau, L .D , and E.M. Lifshitz, 1982, Contenoius Media, Moscow, Nauka. Landauer, R , 1957, I B M J. Res. Dev. 1, 223. Landauer, R , 1987, Z. Phys. Β 68, 217. Lee, P .A, and T.V. Ramakrishnan, 1985, Rev. Mod. Phys. 57, 287. Lee, P .A, and A.D. Stone, 1985, Phys. Rev. Lett. 55, 1622. Lee, P .A, A.D. Stone and H. Fukuyama, 1987, Phys. Rev. Β 35, 1039. Lerner, I .V, 1989, Zh. Eksp. & Teor. Fiz. 95, No . 1, 253. Levitov, P.S, Yu.V. Nazarov and G .M. Eliashberg, 1985, Zh. Eksp. & Teor. Fiz. 88, 229 [Sov.

Phys.-JETP 61, 133]. Levy, P.M., and A. Fert, 1981, Phys. Rev. Β 23, 4667. Ma, M , and P.A. Lee, 1987, Phys. Rev. Β 35, 1448. Maekawa, S, Y. Isawa and H. Ebisawa, 1987, J. Phys. Soc. Jpn. 56, 25. Mattis, D .C , 1965, Theory of Magnetism, New York.


Nguen, V .L , B.Z. Spivak and B.I. Shklovskii, 1985, Zh. Eksp. & Teor. Fiz. 89, 1770 [Sov. Phys.-JETP 62, 1021].

Pnini, R , and B. Shapiro, 1989, Phys. Rev. Β 39, 6986. Ruderman, M.A., and C. Kittel, 1954, Phys. Rev. 96, 99. Sample, H . H , W.Y. Bruno, S.B. Sample and E.K. Sichel, 1987, J. Appl. Phys. 61, 1079. Shapiro, B , 1986, Phys. Rev. Lett. 57, 2168. Sharvin, D .Yu , and Yu.V. Sharvin, 1981, Pis'ma Zh. Eksp. & Teor. Fiz. 34, 285 [JETP Lett. 34,

272]. Shiba, H., and T. Soda, 1969, Prog. Theor. Phys. 41, 25. Spal, R., 1980, J. Appl. Phys. 51, 4221. Spivak, B.Z, and D.E. Khmelnitskii, 1982, Pis'ma Zh. Eksp. & Teor. Fiz. 35, 334 [JETP Lett. 35,

412]. Spivak, B.Z, and A.Yu. Zyuzin, 1988a, Pis'ma Zh. Eksp. & Teor. Fiz. 47, 221 [JETP Lett. 47,

268]. Spivak, B.Z, and A.Yu. Zyuzin, 1988b, Solid State Commun. 65, 311. Spivak, B.Z, and A.Yu. Zyuzin, 1988c, Solid State Commun. 67, 75. Spivak, B.Z, and A.Yu. Zyuzin, 1989, Europhys. Lett. 8, 669. Stephen, M.J, and E. Abrahams, 1988, Solid State Commun. 65, 1423. Stephen, M.J, and G. Cwilich, 1987, Phys. Rev. Lett. 59, 285. Stone, A . D , 1985, Phys. Rev. Lett. 54, 2692. Thouless, D.J, 1977, Phys. Rev. Lett. 39, 1167. Umbach, C P , S. Washburn, R.B. Laibowitz and R.A. Webb, 1984, Phys. Rev. Β 30, 4048. Washburn, S, and R.A. Webb, 1986, Adv. Phys. 35, 375. Webb, R.A, S. Washburn, C P . Umbach and R.B. Laibowitz, 1985, Phys. Rev. Lett. 54, 2696. Zyuzin, A .Yu , and B.Z. Spivak, 1986, Pis'ma Zh. Eksp. & Teor. Fiz. 43, 185 [JETP Lett. 43, 234]. Zyuzin, A .Yu , and B.Z. Spivak, 1987, Zh. Eksp. & Teor. Fiz. 93, 994 [Sov. Phys.-JETP 66, 560].

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