The Cooperon exhibits singular infrared behavior.14

In the singular region the prefactor in Eq. (11.16) equals the constant u2 as

R

+

F p+

u2 ζ(Q, ω) u2

= (11.24)

A

Q’p+

F

˜

i.e. in the region of interest we thus have C = u2 C. As far as regards the singular

behavior we could equally well have de¬ned the Cooperon by the set of diagrams

R

p+ p+

+

F p+

˜

Cω (Q) = +

A

p’ p’

Q’p+

F

R R

p+ p+

+ +

F p+ F p+

+ + ... (11.25)

A A

p’ p’

Q’p+ Q’p+

F F

as adding a constant to a singular function does not change the singular behavior,

and immediately the result of Eq. (11.23) is obtained.

14 The Di¬uson, the impurity particle“hole ladder diagrams, also exhibits this singular infrared

behavior, which leads to di¬usion enhancement of interactions in a disordered conductor as discussed

in Section 11.5.

11.1. Localization 383

Changing in the conductivity expression, Eq. (11.13), one of the integration vari-

ables, p = ’p + Q, we get for the contribution of the maximally crossed diagrams

u2 /„

e dp dQ

2

δσ±β (q, ω) = p± (’pβ + Qβ )

’iω + D0 Q2

(2π )d (2π)d

m π

— GR (p+ ) GR (’p+ + Q) GA (’p’ + Q) GA (p’ ) , (11.26)

+ +

F F

F F

where the prime on the Q-integration signi¬es that we need only to integrate over

the region Ql < 1 from which the large contribution is obtained. Everywhere except

in the Cooperon we can therefore neglect Q as |p ’ Q| ∼ p ∼ pF . Assuming a

kF ,15 we can perform the

smoothly varying external ¬eld on the atomic scale, q

momentum integration, and obtain to leading order in /pF l

4π„ 3 Nd ( F )p2

dp R R A A F

p p G F (p+ )G F (’p+ ) G F (’p’ )G F (p’ ) = δ±β ,

d ±β 3d

(2π )

(11.27)

where we have also safely neglected the ω dependence in the propagators as for the

integration region giving the large contribution, we have ω < 1/„ F/ .

At zero frequency we have for the ¬rst quantum correction to the conductivity of

an electron gas

2e2 D0 dQ 1

δσ(L) = ’ . (11.28)

(2π)d D0 Q2

π

In the one- and two-dimensional case the integral diverges for small Q, and we need

to assess the lower cut-o¬.16 In order to understand the lower cut-o¬ we note that

the maximally crossed diagrams lend themselves to a simple physical interpretation.

The R-line in the Cooperon describes the amplitude for the scattering sequence of an

electron (all momenta being near the Fermi surface as the contribution is otherwise

small)

p ’ p1 ’ · · · ’ pN ’ p ’p (11.29)

whereas the A-line describes the complex conjugate amplitude for the opposite, i.e.

time-reversed, scattering sequence

p ’ ’pN ’ · · · ’ ’p1 ’ p ’p (11.30)

i.e. the Cooperon describes a quantum interference process: the quantum interference

between time-reversed scattering sequences. The physical process responsible for the

15 In a conductor a spatially varying electric ¬eld will, owing to the mobile charges, be screened.

In a metal, say, an applied electric ¬eld is smoothly varying on the atomic scale, q kF , and we can

set q equal to zero as it appears in combination with large momenta, p, p ∼ pF . For a discussion of

the phenomena of screening, we refer the reader to Section 11.5 and chapter 10 of reference [1].

16 Langer and Neal [80] were the ¬rst to study the maximally crossed diagrams, and noted that

they give a divergent result at zero temperature. However, in their analysis they did not assess the

lower cut-o¬ correctly.

384 11. Disordered conductors

quantum correction is thus coherent backscattering.17 The random potential acts as

sets of mirrors such that an electron in momentum state p ends up backscattered into

momentum state ’p. The quantum correction to the conductivity is thus negative

as the conductivity is a measure of the initial and ¬nal correlation of the velocities

as re¬‚ected in the factor p · p in the conductivity expression.

The quantum interference process described by the above scattering sequences

corresponds in real space to the quantum interference between the two alternatives for

a particle to traverse a closed loop in opposite (time-reversed) directions as depicted

in Figure 11.2.18

Figure 11.2 Coherent backscattering process.

We are considering the phenomenon of conductivity, where currents through con-

necting leads are taken in and out of a sample, say, at opposing faces of a hypercube.

The maximal size of a loop allowed to contribute to the coherent backscattering pro-

cess is thus the linear size of the system, as we assume that an electron reaching the

end of the sample is irreversibly lost to the environment (leads and battery).19 For a

system of linear size L we then have for the quantum correction to the conductivity

1/l

2e2 D0 dQ 1

δσ(L) = ’ . (11.31)

d D Q2

π (2π) 0

1/L

17 The coherent backscattering e¬ect was considered for light waves in 1968 [81]. It is amusing

that a quantitative handling of the phenomena had to await the study of the analogous e¬ect in

solid-state physics, and the diagrammatic treatment of electronic transport in metals a decade later.

Here we reap the bene¬ts of employing the proper physical representation of Green™s functions in

the diagrammatic non-equilibrium perturbation theory, leading directly to a physical interpretation

of the summed sub-class of diagrams.

18 We will take advantage of this all-important observation of the physical origin of the quantum

correction to the conductivity (originally expressed in references [82, 83]) in Section 11.2, where the

real space treatment of weak localization is done in detail.

19 An electron is assumed never to reenter from the leads phase coherently, and the Cooperon

equation should be solved with the boundary condition that the Cooperon vanishes on the lead

boundaries, thereby cutting o¬ the singularity. For details we refer the reader to chapter 11 of

reference [1].

11.1. Localization 385

Performing the integral in the two-dimensional case gives for the ¬rst quantum

correction to the dimensionless conductance20

1 L

δg(L) = ’ ln . (11.32)

2

π l

We note that the ¬rst quantum correction to the conductivity indeed is negative,

describing the precursor e¬ect of localization. For the asymptotic scaling function

we then obtain

1

β(g) = ’ 2 , g 1, (11.33)

πg

and the ¬rst quantum correction to the scaling function is thus seen to be negative

in concordance with the scaling picture.

Exercise 11.1. Show that, in dimensions one and three, the ¬rst quantum correction

to the dimensionless conductance is

§

⎨ ’ π2 (1 ’ L )

1 l

d=1

δg(L) = (11.34)

©

’ π3 ( l ’ 1)

1L

d=3

and thereby for the scaling function to lowest order in 1/g

a

β(g) = (d ’ 2) ’ , (11.35)

g

§

where

⎨ 2

d=1

π2

a= (11.36)

© 1

d=3 .

π3

We can introduce the length scale characterizing localization, the localization

length, qualitatively as follows: for a sample much larger than the localization

length, L ξ, the sample is in the localized regime and we have g(L) 0. To

estimate the localization length, we equate it to the length for which g(ξ) g0 , i.e.

the length scale, where the scale-dependent part of the conductance is comparable

to the Boltzmann conductance. The lowest-order perturbative estimate based on

Eq. (11.32) and Eq. (11.34) gives in two and one dimensions the localization lengths

ξ (2) l exp πkF l/2 and ξ (1) l, respectively.

The one-parameter scaling hypothesis has been shown to be valid for the aver-

age conductance in the model considered above [73]. Whether the one-parameter

scaling picture for the disorder model studied is true for higher-order cumulants of

20 The precise magnitudes of the cut-o¬s are irrelevant for the scaling function in the two-

dimensional case, as a change can produce only the logarithm of a constant in the dimensionless

conductance.

386 11. Disordered conductors

the conductance, g n , is a di¬cult question that seems to have been answered in

the negative in reference [84]. However, a di¬erent question is whether deviations

from one-parameter scaling are observable, in the sense that a sample has to be so

close to the metal“insulator transition that real systems cannot be made homoge-

neous enough. Furthermore, electron“electron interaction can play a profound role in

real materials invalidating the model studied, and leaving room for a metal“insulator

transition in low-dimensional systems.21

We can also calculate the zero-temperature frequency dependence of the ¬rst

D0 /ω ≡ Lω .

quantum correction to the conductivity for a sample of large size, L

From Eq. (11.26) we have

δσ±β (ω) = δσ(ω) δ±β , (11.37)

where

1/l

2e2 D0 dQ 1

’

δσ(ω) = . (11.38)

(2π)d ’iω + D0 Q2

π

0

Calculating the integral, we get for the frequency dependence of the quantum cor-

rection to the conductivity in, say, two dimensions [86]

δσ(ω) 1 1

=’ ln . (11.39)

σ0 πkF l ω„

We note that for the perturbation theory to remain valid the frequency can not be

too small, ω„ 1.

The quantum correction to the conductivity in two dimensions is seen to be

universal

1 e2 1

δσ(ω) = ’ 2 ln . (11.40)

2π ω„

Let us calculate the ¬rst quantum correction to the current density response to a

spatially homogeneous electric pulse

δj(t) = δσ(t) E0 , (11.41)

where

∞ 1/l 1/l

2e2 D0 2e2 D0

dω ’iωt dQ 1 dQ ’iD0 Q2 t

δσ(t) = ’ =’

e e

(2π)d ’iω + D0 Q2 (2π)d

π 2π π

’∞ 1/L 1/L

(11.42)

which in the two-dimensional case becomes

e2 D 0t

e’ 2„ ’ e’ L 2

t

δσ(t) = . (11.43)

2π 2 t

21 For a review on interaction e¬ects, see for example [85].

11.1. Localization 387

After the short time „ the classical contribution and the above quantum contribution

in the direction of the force on the electron dies out, and an echo in the current due

to coherent backscattering occurs

e2 ’t/„D

j(t) = ’ e E0 . (11.44)

2π 2 t

on the large time scale „D ≡ L2 /D0 , the time it takes an electron to di¬use across the

sample (for even larger times t „D quantum corrections beyond the ¬rst dominates

the current).

Exercise 11.2. Show that, in dimensions one and three, the frequency dependence

of the ¬rst quantum correction to the conductivity is

§

’ 2√2 √1

⎪ 1+i

d=1

⎨ ω„

δσ(ω)

= (11.45)

√√

⎪

σ0 © (1 ’ i) 3√3 ω„2 d=3.

2 2 (kF l)

In dimension d the quantum correction to the conductivity is thus of relative order

1/(kF l)d’1 . In strictly one dimension the weak localization regime is thus absent;

i.e. there is no regime where the ¬rst quantum correction is small compared with the

Boltzmann result, we are always in the strong localization regime.

From the formulas, Eq. (6.57) and Eq. (11.40), we ¬nd that in a quasi-two-

dimensional system, where the thickness of the ¬lm is much smaller than the length

scale introduced by the frequency of the time-dependent external ¬eld, Lω = D0 /ω,

the quantum correction to the conductance exhibits the singular frequency behavior

e2 1

=’

δ G±β (ω) δ±β ln . (11.46)

2π 2 ω„

The quantum correction to the conductance is in the limit of a large two-dimensional

system only ¬nite because we consider a time-dependent external ¬eld, and the con-

ductance increases with the frequency. This feature can be understood in terms of

the coherent backscattering picture. In the presence of the time-dependent electric

¬eld the electron can at arbitrary times exchange a quantum of energy ω with the

¬eld, and the coherence between two otherwise coherent alternatives will be partially

disrupted. The more ω increases, the more the coherence of the backscattering pro-

cess is suppressed, and consequently the tendency to localization, as a result of which

the conductivity increases.

The ¬rst quantum correction plays a role even at ¬nite temperatures, and in

Section 11.2 we show that from an experimental point of view there are important

quantum corrections to the Boltzmann conductivity even at weak disorder. We

have realized that if the time-reversal invariance for the electron dynamics can be

388 11. Disordered conductors

broken, the coherence in the backscattering process is disrupted, and localization is

suppressed. The interaction of an electron with its environment invariably breaks

the coherence, and we discuss the e¬ects of electron“phonon and electron“electron

interaction in Section 11.3. A more distinct probe for in¬‚uencing localization is to

apply a magnetic ¬eld, which we discuss in Section 11.4.

We have realized that the precursor e¬ect of localization, weak localization, is

caused by coherent backscattering. The constructive interference between propaga-

tion along time-reversed loops increases the probability for a particle to return to its

starting position. The phenomenon of localization can be understood qualitatively

as follows. The main amplitude of the electronic wave function incipient on the ¬rst

impurity in Figure 11.2 is not scattered into the loop depicted, but continues in its

forward direction. However, this part of the wave also encounters coherent backscat-

tering along another closed loop feeding constructively back into the original loop,

and thereby increasing the probability of return. This process repeats at any impu-

rity, and the random potential acts as a mirror, making it impossible for a particle to

di¬use away from its starting point. This is the physics behind how the singularity

in the Cooperon drives the Anderson metal“insulator transition.22

11.2 Weak localization

We start this section by discussing the weak-localization contribution to the con-

ductivity in the position representation, before turning to discuss the e¬ects of in-

teractions on the weak-localization e¬ect, the destruction of the phase coherence of

the wave function due to electron“phonon and electron“electron interaction. Then

anomalous magneto-resistance is considered; this is an important diagnostic tool in

material science. Finally we discuss mesoscopic ¬‚uctuations.

The theory of weak localization dates back to the seminal work on the scaling

theory of localization [72], and developed rapidly into a comprehensive understand-

ing of the quantum corrections to the Boltzmann conductivity. Based on the insight

provided by the diagrammatic technique, the ¬rst quantum correction, the weak-

localization e¬ect, was soon realized to be the result of a simple type of quantum

mechanical interference (as already noted in Section 11.1.2), and the resulting phys-

ical insight eventually led to a quantitative understanding of mesoscopic phenomena

in disordered conductors. In order to develop physical intuition of the phenomena,

we shall use the quantum interference picture in parallel with the quantitative dia-

grammatic technique, to discuss the weak-localization phenomenon.

11.2.1 Quantum correction to conductivity

In Section 7.4 we derived the Boltzmann expression for the classical conductivity

as the weak-disorder limiting case where the quantum mechanical wave nature of

the motion of an electron is neglected. In terms of diagrams this corresponded

to neglecting conductivity diagrams where impurity correlators cross, because such

22 For a quantitative discussion of strong localization we refer the reader to chapter 9 of reference

[1].

11.2. Weak localization 389

contributions are smaller by the factor »F /l, and thus constitute quantum corrections

to the classical conductivity.

A special class of diagrams where impurity correlators crossed a maximal number

of times was seen, in Section 11.1.2, in the time-reversal invariant situation, to exhibit

singular behavior although the diagrams nominally are of order /pF l.23

p+ R

p+ p+

+ + ...

qω qω qω qω

p’ p’ A

Q’p+ (11.47)

We shall consider the explicitly time-dependent situation where the frequency ω of the

external ¬eld is not equal to zero, in order to cut o¬ the singular behavior. In this case

(and others to be studied shortly) the ¬rst quantum correction to the conductivity

in the parameter »F /l is a small correction to the Boltzmann conductivity (recall

Eq. (11.39)), and we speak of the weak-localization e¬ect.

In the discussion of interaction e¬ects and magneto-resistance it will be convenient

to use the spatial representation for the conductivity. The free-electron model and a

delta-correlated random potential, Eq. (11.15), will be used for convenience.

In the position representation the impurity-averaged current density

j± (x, ω) ≡ j± (x, ω) = dx σ±β (x, x , ω) Eβ (x , ω) (11.48)

β

is, besides regular corrections of order O( /pF l), speci¬ed by the conductivity tensor

∞

2

f0 (E) ’ f0 (E + ω)

1 e

σ±β (x ’ x , ω) ≡ σ±β (x, x , ω) = dE

π m ω

’∞

” ”

— GR (x, x ; E + ω) ∇x± ∇xβ GA (x , x; E) . (11.49)

The contribution to the conductivity from the maximally crossed diagrams is conve-

niently exhibited in twisted form where they become ladder-type diagrams.

23 In addition to these maximally crossed diagrams, there are additional diagrams of the same

order of magnitude (also coming from the regular terms). However, they give contributions to the

conductivity which are insensitive to low magnetic ¬elds and temperatures in comparison to the

contribution from the maximally crossed diagrams.

390 11. Disordered conductors

rr

rRr

A x x

x x

+ ...

+

± β

± β

rr

˜

C

x x

=

± β

. (11.50)

˜

The sum of the maximally crossed diagrams, the Cooperon Cω (r, r ; E), is in the

position representation speci¬ed by the diagrams

R

r r r

˜

r r . (11.51)

+ + ... = C

r

A

The analytical expression for the quantum correction to the conductivity is therefore

(E+ ≡ E + ω)

∞

2

f0 (E) ’ f0 (E + ω) ˜

1 e

δσ±β (x ’ x , ω) = dr dr dE Cω (r, r ; E)

π m ω

’∞

” ”

— GR+ (x ’ r)GR+ (r ’ x ) ∇x± ∇xβ GA (x ’ r)GA (r ’ x) . (11.52)

E E E E

11.2. Weak localization 391

The impurity-averaged propagator decays exponentially as a function of its spatial

variable with the scale set by the impurity mean free path. The spatial scale of

variation of the sum of the maximally crossed diagrams is typically much larger.

For the present case where we neglect e¬ects of inelastic interactions, we recall from

Eq. (11.23) that the spatial range of the Cooperon is Lω = D0 /ω, which for ω „ 1

is much larger than the mean free path, since D0 = vF l/d is the di¬usion constant

in d dimensions.24 The impurity-averaged propagators attached to the maximally

˜

crossed diagrams will therefore require the starting and end points of Cω (r, r , E) to

be within the distance of a mean free path, in order for a non-vanishing contribution

to the integral. On the scale of variation of the Cooperon this amounts to setting its

arguments equal, and we can therefore substitute r ’ x, r ’ x, and obtain

∞

2

f0 (E’ ) ’ f0 (E+ ) ˜

1 e

δσ±β (x ’ x , ω) = dE Cω (x, x; E) dr dr

π m ω

’∞

” ”

— GR+ (x ’ r)GR+ (r ’ x ) ∇x± ∇xβ GA’ (x ’ r)GA’ (r ’ x) . 11.53)

(

E E E E

The gate combination of the Fermi functions renders for the degenerate case, ω, kT

F , the energy variable in the thermal layer around the Fermi surface, and we have

for the ¬rst quantum correction to the conductivity of a degenerate electron gas

e 2

˜

δσ±β (x ’ x , ω) = ¦± , β (x ’ x ) ,

Cω (x, x; F) (11.54)

π m

where

” ”

¦± , β (x ’ x ) ≡ dr dr GR (x’r)GR (r ’x ) ∇x±∇xβ GA (x ’r)GA (r ’x). (11.55)

F F F F

Clearly this function is local with the scale of the mean free path, and to lowest order

in /pF l we have25

(2πN0 „ )2 (x ’ x )± (x ’ x )β ’|x’x |/l

¦±,β (x ’ x ) = ’ cos2 kF |x ’ x | .

e (11.56)

|x ’ x |

2 4

2

Since the function ¦±,β (x’x ) decays on the scale of the mean free path, and appears

in connection with the Cooperon, which is a smooth function on this scale, it acts

e¬ectively as a delta function

(2πN0 „ )2 l

¦±,β (x ’ x ) ’ δ±β δ(x ’ x ) .

= (11.57)

32

We therefore obtain the fact that the ¬rst quantum correction, the weak-localization

contribution, to the conductivity is local

δσ±β (x ’ x , ω) = δσ(x, ω) δ±β δ(x ’ x ) (11.58)

24 For samples of size larger than the mean free path, L > l, the di¬usion process is e¬ectively

three-dimensional, so that one should use the value d = 3 in the expression for the di¬usion con-

stant. In strictly two-dimensional systems, such as for the electron gas in the inversion layer in a

heterostructure at low temperatures, the value d = 2 should be used.

25 For details we refer the reader to chapter 11 of reference [1].

392 11. Disordered conductors

and speci¬ed by26

2e2 D0 „

δσ(x, ω) = ’ Cω (x, x) . (11.59)

π

As we already noted in Section 11.1.2 the Cooperon is independent of the energy

of the electron (here the Fermi energy since only electrons at the Fermi surface

contribute to the conductivity) Cω (x, x ) ≡ Cω (x, x , F ), and we have introduced

Cω (x, x ) ≡ u’2 Cω (x, x ).

˜

The quantum correction to the conductance of a disordered degenerate electron

gas is

= L’2 dx dx δσ±β (x, x , ω)

δG±β (ω) ≡ δG±β (ω)

2e2 D0 „ ’2

’

= L δ±β dx Cω (x, x) . (11.60)

π

11.2.2 Cooperon equation

Many important results in the theory of weak localization can be obtained once the ef-

fect on the Cooperon of a time-dependent external ¬eld is obtained. Later we present

the derivation of the Cooperon equation in the presence of a time-dependent elec-

tromagnetic ¬eld based on the quantum interference picture of the weak-localization

e¬ect. But ¬rst we provide the quantitative derivation of this result by employing the

equation obeyed by the quasi-classical Green™s function in Nambu or particle“hole

space in the dirty, i.e. di¬usive limit, Eq. (8.197). This will, in addition, extend

our awareness of the information contained in the various components of the matrix

Green™s function in Nambu or particle“hole space.27

The goal is to generate the equation for the Cooperon by functional di¬erentiation

of the quasi-classical Green™s function, and we therefore add a two-particle source to

the Nambu space Hamiltonian, Ψ denoting the Nambu ¬eld, Eq. (8.32),

dx1 Ψ† (x1 , t1 ) V (x1 , t1 , t1 ) Ψ(x1 , t1 ) ,

V (t1 , t1 ) = (11.61)

which therefore, according to Section 8.1.1, needs only o¬-diagonal Nambu matrix

elements

0 V12 (x1 , t1 , t1 )

V (x1 , t1 , t1 ) = . (11.62)

V21 (x1 , t1 , t1 ) 0

In linear response to the two-particle source we thus encounter the two-particle

Green™s function in the form of the particle“particle impurity ladder, and the Cooperon

can be obtained by di¬erentiation with respect to the source, which therefore is taken

local in the space variable.

26 We could also have evaluated the conductivity, Eq. (11.52), directly by Fourier-transforming

the propagators, and recalling Eq. (11.27).

27 This provides an alternative derivation to the ones in the literature. We follow the derivation

in reference [9].

11.2. Weak localization 393

For the retarded component of Eq. (8.197) we have (we leave out the subscript

indicating it is the s-wave, local in space part of the quasi-classical Green™s function,

gs )

’1

[g0 + iV R ’ D0 ‚ —¦ g R —¦ ‚ —¦ g R ]’ = 0 ,

, (11.63)

where the scalar potential enters in

’1

g0 (x1 , t1 , t1 ) = („3 ‚t1 + ieφ(x1 , t1 ))δ(t1 ’ t1 ) (11.64)

and the vector potential through the di¬usive term according to

‚ = (∇x1 ’ ie„3 A(x1 , t1 )) . (11.65)

The equation of motion, which is homogeneous, is supplemented by the normalization

condition, Eq. (8.182),

g R —¦ g R = δ(t1 ’ t1 ) . (11.66)

The self-energy term associated with superconductivity has been expelled from Eq.

(8.185) since for our case of interest the conductor is assumed in the normal state.

Instead a source-term, V R , has been introduced, a matrix in Nambu-space. Taking

the functional derivative of the 12-component of g R with respect to the Nambu

R

component V12 is seen to generate the Cooperon

R

1 δg12 (R, t1 , t1 )

C(R, R , t1 , t1 , t2 , t2 ) = (11.67)

R

2i„ δV12 (R , t2 , t2 )

since the o¬-diagonal Nambu components of the source term add or subtract pairs

of particles, and in the di¬usive limit only ladder diagrams are considered. By con-

struction, the functional derivative on the right in Eq. (11.67) is the ξ-integrated

particle“particle ladder (including external legs) with the in¬‚uence of the electro-

magnetic ¬eld fully included in the quasi-classical approximation.28 The result of

the functional derivative operation involved in Eq. (11.67) is depicted diagrammati-

cally in Figure 11.3.

R t

t2 = T +

t

t1 = T + 2

2

R R

t1 = T ’ t2 = T ’

t t

2 2

A

Figure 11.3 Cooperon obtained as derivative with respect to the two-particle source.

In order to obtain the equation satis¬ed by the functional derivative, the equation

of motion is linearized with respect to the solution in the absence of the source term,

R

g0 . We thus write

g R = g0 + δg R

R

(11.68)

28 This is usually no restriction since interest is in weak ¬elds.

394 11. Disordered conductors

and use our knowledge that the normal state solution in the absence of the source

term is

g0 = „3 δ(t1 ’ t1 ) .

R

(11.69)

Inserting into Eq. (11.63) and linearizing the equation with respect to the source

gives

’ ’ ’ ’

’1

[g0 ’ D0 ‚ —¦ g0 —¦ ‚ —¦ δg R ]’ + i[V R —¦ g0 ]’ ’ D0 [ ‚ —¦ δg R —¦ ‚ —¦ g0 ]’ = 0 .

R

,R ,R

,

(11.70)

Taking the 12-Nambu component gives

‚t1 ’ ‚t1 + ie(φt1 ’ φt1 ) ’ D0 (∇x ’ ie(At1 + At1 ))2 δg R (x, t1 , t1 )

R

= 2iV12 (x, t1 , t1 ) , (11.71)

where the spatial dependence x of the ¬elds has been suppressed. Taking the func-

tional derivative with respect to the 12-Nambu component of the source we get

R

δg12 (x, t1 , t1 )

‚t1 ’ ‚t1 + ie(φt1 ’ φt1 ) ’ D0 (∇x ’ ie(At1 + At1 )) 2

R

δV12 (x , t2 , t2 )

2i δ(x ’ x ) δ(t1 ’ t2 ) δ(t1 ’ t2 ) .

= (11.72)

Because of the double time dependence of the external ¬eld, the functional deriva-

tive and the Cooperon have the time labeling depicted in Figure 11.3 and the following

diagram

˜ ˜

t1 t1

x t1 =T + 2 x t2 =T + t2

t

C =

x t1 =T ’ 2 x t2 =T ’ t2

t

˜ ˜

t2 t2

···

+ + .

(11.73)

Introducing new time variables

1 1

t = t1 ’t1 t = t2 ’t2 , (11.74)

T= (t1 +t1 ) , T= (t2 +t2 ) , ,

2 2

11.2. Weak localization 395

we get

2 R

‚ ie δg12 (x, T, t)

2 + ieφT (x, t) ’D0 ∇x ’ AT (x, t) R

‚t δV12 (x , T , t )

1

δ(x ’ x ) δ(t ’ t ) δ(T ’ T ) ,

= (11.75)

„

where we have introduced the abbreviations

φT (x, t) = φ(x, T + t/2) ’ φ(x, T ’ t/2) (11.76)

and

AT (x, t) = A(x, T + t/2) + A(x, T ’ t/2) . (11.77)

Accordingly for the Cooperon we get the equation

2

‚ ie

2 + ieφT (x, t) ’ D0 ∇x ’ AT (x, t) T,T

Ct,t (x, x )

‚t

1

δ(x ’ x ) δ(t ’ t ) δ(T ’ T ) ,

= (11.78)

„

where we have introduced

T,T

Ct,t (x, x ) ≡ C(x, x ; t1 , t1 , t2 , t2 ) . (11.79)

Since there is no di¬erentiation with respect to the variable T in Eq. (11.78), it is

only a parameter in the Cooperon equation, and we have

T,T

Ct,t (x, x) = Ct,t (x, x ) δ(T ’ T ) ,

T

(11.80)

T

where Ct,t (x, x ) satis¬es the equation

2

‚ ie 1

2 ’D0 ∇x ’ AT (x, t) δ(x ’ x ) δ(t ’ t ) .

T

Ct t (x, x ) = (11.81)

‚t „

Here we have left out the e¬ect of a time-dependent scalar potential on the Cooperon

since in the following we represent the electromagnetic ¬eld solely by the vector

potential. We note that it can be restored by invoking the gauge co-variance property

of the Cooperon.

We now derive the conductivity formula relevant for the case in question. We

are here beyond linear response since we are taking into account to all orders how

the Cooperon is in¬‚uenced by the electromagnetic ¬eld. In the case of an external

electromagnetic ¬eld represented by a vector potential in¬‚uencing the Cooperon as

well we consider the quantum correction to the kinetic propagator which is given by

the contributions speci¬ed in the following diagram

396 11. Disordered conductors

x1 t1

δGK (x1 , t1 , x1 , t1 ) = (11.82)

x1 t1

where the summation sign indicates the summation of all maximally crossed dia-

grams. For the quantum correction to the current we then have

e ‚ ‚

’ δGK (x, t, x , t)

δj(x, t) = . (11.83)

2im ‚x ‚x

x =x

The structure of the general maximally crossed diagram with n impurity correlators

is

2n+1

K

(GR )j GK (GA )2n’j .

δG = (11.84)

j=0

If the equilibrium kinetic propagator GK occurs in the above diagram at a place

0

di¬erent from the ones indicated by circles, the contribution vanishes to the order

of accuracy. In that case, viz. we encounter the product of two retarded or two

advanced propagators sharing the same momentum integration variable, and since

the impurity correlator e¬ectively decouples the momentum integrations, such terms

are smaller by the factor / F „ .

Displaying a maximally crossed kinetic propagator diagram on twisted form we

have (we use the notation 1 ≡ (x1 , t1 ) etc.); the diagram in depicted in Figure 11.4.

1 t10 t9 t8 t7

6

5 4 3 2 1

Figure 11.4 Twisted maximally crossed kinetic propagator diagram.

Because of the four di¬erent places where the kinetic propagator can occur we

explicitly keep the four outermost impurity correlators, and obtain for the quantum

11.2. Weak localization 397

correction to the kinetic propagator

eu8

K

d“ GR (x1 , t1 ; x5 , t10 ) GR (x5 , t10 ; x4 , t9 ) G0 (x3 , t8 ; x2 , t7 )

δG (x1 , t1 , x1 , t1 ) =

2im

”

— A(x6 , t6 ) · G0 (x2 , t7 ; x6 , t6 ) ∇x6 G0 (x6 , t6 ; x5 , t5 )

R

δg12 (x4 , t9 , t4 )

— 0

G (x5 , t5 ; x4 , t4 ) R

δV12 (x3 , t8 , t3 )

— GA (x3 , t3 ; x2 , t2 ) GA (x2 , t2 ; x1 , t1 ) , (11.85)

where the propagators labeled by a zero as the superscript index indicate where the

kinetic propagator can appear (i.e. we have a sum of four terms, and the kinetic

propagator is always sandwiched in between retarded propagators to the left and

advanced propagators to the right), and we have introduced the abbreviation

6

d“ = dt7 dt8 dt9 dt10 dxi dti . (11.86)

i=1

Since the propagators carry the large momentum pF , we can take for the explicitly

appearing linear response vector potential

A(t) = Aω1 e’iω1 t . (11.87)

The eight exhibited propagators in Eq. (11.85) can be taken to be the equilibrium

ones, and by Fourier transforming the propagators, and performing the integration

over the momenta, we obtain for the quantum correction to the current density at

±

frequency ω2 , E1 = E1 ± ω1 /2,

∞

4e2 D0 „ dE1 ’

f0 (E1 ) ’ f0 (E1 ) dt1 dt1 dt2 dt2 δ(t2 ’ t1 )

+

δj(x, ω2 ) = A(ω1 )

iπ 2π

’∞

’ +

(E1 t1 ’E1 t2 ’ ω2 t1 )

i

— e C(x, x; t1 , t1 , t2 , t2 ) (11.88)

or equivalently

∞

4e2 D0 „ dE1 ’

f0 (E1 ) ’ f0 (E1 )

+

δj(x, ω2 ) = A(ω1 )

iπ 2π

’∞

∞∞

dt dT Ct,’t (x, x) eiT (ω1 ’ω2 )+i 2 (ω1 +ω2 ) .

t

— T

(11.89)

’∞ ’∞

398 11. Disordered conductors

For the quantum correction to the conductivity in the presence of a time-dependent

electromagnetic ¬eld

δj(x, ω2 ) = δσ(x, ω2 , ω1 ) E(ω1 ) (11.90)

we therefore obtain29

∞

4e2 D0 „ dE1 ’

δσ(x, ω2 , ω1 ) = ’ f0 (E1 ) ’ f0 (E1 )

+

πω 2π

’∞

∞∞

dt dT eiT (ω2 ’ω1 )+ 2 t(ω1 +ω2 ) Ct,’t (x, x) .

i

— T

(11.91)

’∞ ’∞

In the degenerate case we have

∞∞

4e2 D0 „

dt dT Ct,’t (x, x) eiT (ω1 ’ω2 )+i 2 (ω1 +ω2 ) .

t

δσ(x, ω2 , ω1 ) = ’ T

(11.92)

π

’∞ ’∞

In the event that the included e¬ect of an electromagnetic ¬eld on the Cooperon is

caused by a time-independent magnetic ¬eld, we recover the expression Eq. (11.59)

for the quantum correction to the conductivity.

We shall exploit the derived formula when we consider the in¬‚uence of electron“

electron interaction on the quantum correction to the conductivity.

11.2.3 Quantum interference and the Cooperon

In this section, we shall elucidate in more detail than in Section 11.1.2 the physical

process in real space described by the maximally crossed diagrams, and in addition

consider the in¬‚uence of external ¬elds. The weak-localization e¬ect can be under-

stood in terms of a simple kind of quantum mechanical interference. By following the

scattering sequences appearing in the diagrammatic representation of the Cooperon

contribution to the conductivity, see Eq. (11.47), we realize that the quantum cor-

rection to the conductivity consists of products of the form “amplitude for scattering

sequence of an electron o¬ impurities in real space times the complex conjugate of the

amplitude for the opposite scattering sequence.” The quantum correction to the con-

ductivity is thus the result of quantum mechanical interference between amplitudes

for an electron traversing a loop in opposite directions. To lowest order in »F /l we

need to include only the stationary, i.e. classical, paths determined by the electron

bumping into impurities, as illustrated in Figure 11.2 where the trajectories involved

in the weak-localization quantum interference process are depicted. The solid line,

say, in Figure 11.2 corresponds to the propagation of the electron represented by

29 For an electron gas in thermal equilibrium f0 is the Fermi function, but in principle we could

at this stage have any distribution not violating Pauli™s exclusion principle. However, that would

then necessitate a discussion of energy relaxation processes tending to drive the system toward the

equilibrium distribution.

11.2. Weak localization 399

the retarded propagator in the conductivity diagram, and the dashed line to the

propagation represented by the advanced propagator, the complex conjugate of the

amplitude for scattering o¬ impurities in the opposite sequence. The starting and

end points refer to the points x and x in Eq. (11.52), respectively.30 According to

the formula, Eq. (11.59), for the quantum correction to the conductivity, we need to

consider only scattering sequences which start and end at the same point on the scale

of the mean free path, as demanded by the impurity-averaged propagators attached

to the maximally crossed diagrams in Eq. (11.52).

In the time-reversal invariant situation, the contribution to the return probabil-

ity from the maximally crossed diagrams equals the contribution from the ladder

diagrams, and the return probability including the weak-localization contribution is

thus twice the classical result31

d/2

1

Pcl+wl (x, t; x, t ) = 2 Pcl (x, t; x, t ) = 2 , (11.93)

4πD0 (t ’ t )

where the last expression is valid in the di¬usive limit. To see how this comes about

in the interference picture, let us consider the return probability in general. The

quantity of interest is therefore the amplitude K for an electron to arrive at a given

space point x at time t/2 when initially it started at the same space point at time

’t/2. According to Feynman, this amplitude is given by the path integral expression

xt / 2 =x

i

K(x, t/2; x, ’t/2) = Dxt e ≡

S[xt ]

Ac (11.94)

c

x’t / 2 =x

where the path integral includes all paths which start and end at the same point.

For the return probability we have

Ac A—

P = |K|2 = | Ac |2 = |Ac |2 + (11.95)

c

c c c=c

where Ac is the amplitude for the path c. In the sum over paths we only need to

include to order »F /l the stationary, i.e. classical, paths determined by the electron

bumping into impurities. The sum of the absolute squares is then the classical con-

tribution to the return probability, and the other terms are quantum interference

terms. In the event that the particle only experiences the impurity potential, we

have for the amplitude for the particle to traverse the path c,

t

2 dt { 1 mx2 (t) ’ V (xc (t))}

¯ ™c ¯ ¯

i

’t 2

Ac = e . (11.96)

2

30 The angle between initial and ¬nal velocities is exaggerated since we recall that in order for the

Cooperon to give a large contribution the angle must be less than 1/kF l.

31 The fact that impurity lines cross, does not per se make a diagram of order 1/k l relative to a

F

non-crossed diagram. In case of the conductivity diagrams this is indeed the case for the maximally

crossed diagrams because the circumstances needed for a large contribution set a constrain on the

correlation of the initial and ¬nal velocity, p ’p + Q (recall also when estimating self-energy

diagrams the importance of the incoming and outgoing momenta being equal, see reference [1]).

However, in the quantity of interest here the position is ¬xed.

400 11. Disordered conductors

Owing to the impurity potential, the amplitude has a random phase. A ¬rst con-

jecture would be to expect that, upon impurity averaging, the interference terms in

general average to an insigni¬cant small value, and we would be left with the clas-

sical contribution to the conductivity. However, there are certain interference terms

which are resilient to the impurity average. It is clear that impurity averaging can

not destroy the interference between time-reversed trajectories since we have for the

amplitude for traversing the time-reversed trajectory, xc (t) = xc (’t),

¯

t t

2 dt { 1 mx2 (t) ’ V (xc (t))} 2 dt { 1 m[’x (’t)]2 ’ V (xc (’t))}

¯ ™c ¯ ¯¯ ¯ ¯ ¯

i i

™c

¯

’t ’t

2 2

Ac = e =e = Ac .(11.97)

2 2

¯

In this time-reversal invariant situation the amplitudes for traversing a closed loop in

opposite directions are identical, Ac = Ac , and the corresponding interference term

¯

contribution to the return probability is independent of the disorder, Ac A— = 1!c

¯

The two amplitudes for the time-reversed electronic trajectories which return to the

starting point thus interfere constructively in case of time-reversal invariance. In

correspondence to this enhanced localization, there is a decrease in conductivity

which can be calculated according to Eq. (11.59).

The foregoing discussion based on the physical understanding of the weak lo-

calization e¬ect will now be substantiated by deriving the equation satis¬ed by the

Cooperon. The Cooperon Cω (x, x ) is generated by the iterative equation

R

x x x x

x x=

Cω + + + ...

x x

A

x x

x x

= + Cω (11.98)

x

where we have introduced the diagrammatic notation

x

≡ δ(x ’ x ) . (11.99)

x

The Cooperon equation, Eq. (11.98), is most easily obtained by adding the term

11.2. Weak localization 401

x

u2 δ(x ’ x )

= (11.100)

x

˜

to the in¬nite sum of terms represented by the function C, Eq. (11.51). Alterna-

tively, one can proceed as in Section 11.1.2, now exploiting the local character of the

˜

propagators. In any event, we have in the singular region C u2 C.

The Cooperon equation in the spatial representation is

˜C

Cω (x, x ) = δ(x ’ x ) + dx Jω (x, x ) Cω (x , x ) , (11.101)

where according to the Feynman rules the insertion is given by

˜C

Jω (x, x ) = u2 GR + ) GA (x, x ) .

ω (x, x (11.102)

F F

The Cooperon is slowly varying on the scale of the mean free path, the spatial range

˜C

of the function Jω (x, x ), and a low-order Taylor-expansion of the Cooperon on the

right-hand side of Eq. (11.101) is therefore su¬cient. Upon partial integration, the

integral equation then becomes, for a second-order Taylor expansion, a di¬erential

equation for the Cooperon

1

’ iω ’ D0 ∇2 Cω (x, x ) = δ(x ’ x ) . (11.103)

x

„

This equation is of course simply the position representation of the equation for the

Cooperon already derived in the momentum representation, Eq. (11.23). Indeed we

recover, that the Cooperon only varies on the large length scale Lω = (D0 /ω)1/2 .

The typical size of an interference loop is much larger than the mean free path, and

we only need the large-scale behavior of the Boltzmannian paths of Figure 11.2, the

smooth di¬usive loops of Figure 11.5.

The fact that in the time-reversal invariant case we have obtained that the

Cooperon satis¬es the same di¬usion-type equation as the Di¬uson is not surprising.

The Di¬uson is determined by a similar integral equation as the Cooperon, however,

with the important di¬erence that one of the particle lines, say the advanced one, is

reversed (recall Exercise 7.8 on page 197). The Di¬uson will therefore be determined

˜C

by the same integral equation as the Cooperon, except for Jω now being substituted

˜D

by the di¬usion insertion Jω , given by

˜D

Jω (x, x ) = u2 GR + ) GA (x , x) .

ω (x, x (11.104)

F F

˜C ˜D

In a time-reversal invariant situation the two insertions are equal, Jω = Jω , and we

recover that the Di¬uson and the Cooperon satisfy the same equation (and we have

hereby re-derived the result, Eq. (11.93)).

402 11. Disordered conductors

R—

R

Figure 11.5 Di¬usive loops.

In the time-reversal invariant situation the amplitudes for traversing a closed loop

in opposite directions are identical, and in such a coherent situation one must trace

the complete interference pattern of wave re¬‚ection in a random medium, and one

encounters the phenomenon of localization discussed earlier. However, we also realize

that the interference e¬ect is sensitive to the breaking of time-reversal invariance. By

breaking the coherence between the amplitudes for traversing time-reversed loops the

tendency to localization of an electron can be suppressed.32 In moderately disordered

conductors we can therefore arrange for conditions so that the tendency to localiza-

tion of the electronic wave function has only a weak though measurable in¬‚uence on

the conductivity. The ¬rst quantum correction then gives the dominating contribu-

tion in the parameter »F /l, and we speak of the so-called weak-localization regime.

The destruction of phase coherence is the result of the interaction of the electron with

its environment, such as electron“electron interaction, electron“phonon interaction,

interaction with magnetic impurities, or interaction with an external magnetic ¬eld.

From an experimental point of view the breaking of coherence between time-reversed

trajectories by an external magnetic ¬eld is of special importance, and we start by

discussing this case.

11.2.4 Quantum interference in a magnetic ¬eld

The in¬‚uence of a magnetic ¬eld on the quantum interference process described by

the Cooperon is readily established in view of the already presented formulas. In the

weak magnetic ¬eld limit, l2 < lB , where lB = ( /2eB)1/2 is the magnetic length, or

2

equivalently ωc „ < / F „ , the bending of a classical trajectory with energy F can be

32 By disturbance, the coherence can be disrupted, and the tendency to localization can be sup-

pressed, thereby decreasing the resistance. Normally, disturbances increase the resistance.

11.2. Weak localization 403

neglected on the scale of the mean free path. Classical magneto-resistance e¬ects are

then negligible, because they are of importance only when ωc „ ≥ 1. The amplitude

for propagation along a straight-line classical path determined by the impurities is

then changed only because of the presence of the magnetic ¬eld by the additional

phase picked up along the straight line, the line integral along the path of the vector

potential A describing the magnetic ¬eld. In the presence of such a weak static

magnetic ¬eld the propagator is thus changed according to

x

ie

GR (x, x ) ’ GR (x, x ) exp d¯ · A(¯) . (11.105)

x x

E E

x

The resulting change in the Cooperon insertion is then

x

2ie

˜C ˜C

Jω (x, x ) ’ Jω (x, x ) exp d¯ · A(¯)

x x

x

2ie

˜C (x ’ x ) · A(x)

= Jω (x, x ) exp . (11.106)

The factor of 2 re¬‚ects the fact that in weak-localization interference terms between

time-reversed trajectories, the additional phases due to the magnetic ¬eld add.

Repeating the Taylor-expansion leading to Eq. (11.103), we now obtain in the

Cooperon equation additional terms due to the presence of the magnetic ¬eld

2

2ie 1

’iω ’ D0 ∇x ’ δ(x ’ x ) .

Cω (x, x ) = (11.107)

A(x)

„

Introducing the Fourier transform

dω ’iω(t’t )

Ct,t (x, x ) = e Cω (x, x ) (11.108)

2π

we obtain in the space-time representation the Cooperon equation33

2

‚ 2ie 1

’ D0 ∇x ’ δ(x ’ x ) δ(t ’ t ) .

Ct,t (x, x ) = (11.109)

A(x)

‚t „

We note that this equation is formally identical to the imaginary-time Green™s func-

tion equation for a particle of mass /2D0 and charge 2e moving in the magnetic ¬eld

described by the vector potential A. The solution of this equation can be expressed

as the path integral

xt =x x2

t ™¯

’ dt xt ·A(xt )

¯ ie

t ™¯

+

1 ¯

4D 0

Dxt e

Ct,t (x, x ) = . (11.110)

t

„

xt =x

33 Thisis of course just a special case of the general equation, Eq. (11.81), the case of a time-

independent magnetic ¬eld

404 11. Disordered conductors

11.2.5 Quantum interference in a time-dependent ¬eld

Let us now obtain the equation satis¬ed by the Cooperon when the particle interacts

with an environment as described by the Lagrangian L1 . The total Lagrangian is

then L = L0 + L1 , where34

1

mx2 ’ V (x)

™ ™

L0 (x, x) = (11.111)

2

describes the particle in the impurity potential. We ¬rst present a derivation of the

Cooperon equation based on the interference picture of the weak-localization e¬ect,

before presenting the diagrammatic derivation.35

The conditional probability density for an electron to arrive at position x at time t

given it was at position x at time t is given by the absolute square of the propagator

P (x, t; x , t ) = |K(x, t; x , t )|2 . (11.112)

In the quasi-classical limit, which is the one of interest, »F l, we can, in the path

integral expression for the propagator, replace the path integral by the sum over

classical paths

xt =x

S[xcl ]

i i

Dxt e S[xt ]

A[xcl ] e

K(x, t; x, t ) = (11.113)

t

t

xcl

xt =x t

where the prefactor takes into account the Gaussian ¬‚uctuations around the classical

path. We assume that we may neglect the in¬‚uence of L1 on the motion of the

electrons, and the classical paths are determined by L0 , i.e. by the large kinetic

energy and the strong impurity scattering. The paths in the summation are therefore

solutions of the classical equation of motion

m¨ cl = ’∇V (xcl ) . (11.114)

xt t

The quantum interference contribution to the return probability in time span t from

the time-reversed loops is in the quasi-classical limit

t t (S[xcl ]’S[xcl ])

i

P x, ; x, ’ |A[xcl ] |2 e

= , (11.115)

’t

t

t

2 2

xcl

t

where xcl = x = xcl . We are interested in the return probability for an electron

’t/2 t/2

constrained to move on the Fermi surface, i.e. its energy is equal to the Fermi

energy F . For the weak-localization quantum interference contribution to the return

probability we therefore obtain

1 2 cl

ei•[xt ] δ( [xcl ] ’

A[xcl ]

C(t) = F) , (11.116)

t t

N0

xcl

t

34 A possible dynamics of the environment plays no role for the present discussion, and its La-

grangian is suppressed.

35 We follow the presentation of reference [87].

11.2. Weak localization 405

where the sum is over classical trajectories of duration t that start and end at the

same point, and

1

[xcl ] = m [xcl ]2 + V (xcl )

™t (11.117)

t t

2

is the energy of the electron on a classical trajectory. The normalization factor

follows from the fact that the density of classical paths in the quasi-classical limit

equals the density of states.36 We have introduced the phase di¬erence between a

pair of time-reversed paths

1

S[xcl ] ’ S[xcl ] .

•[xcl ] = (11.118)

’t

t t

As noted previously in Section 11.2.4, a substantial cancellation occurs in the phase

di¬erence since L0 is an even function of the velocity and the quenched disorder

potential is independent of time. Hence, the phase di¬erence is a small quantity

given by

t/2

1

dt {L1 (xcl , xcl , t) ’ L1 (xcl t , ’xcl t , t)}

¯ ¯¯ ™ ’¯ ¯

•[xcl ] = t ™t ’¯

¯

t

’t/2

t/2

1

dt {L1 (xcl , xcl , t) ’ L1 (xcl , ’xcl , ’t)}

¯ ¯¯ ¯

t ™t ™t

= ¯ ¯ ¯

t

’t/2

t/2

≡ ¯˜ t

dt •(xcl ) , (11.119)

’t/2

where in the last term in the second equality we have replaced the integration variable

t by ’t. We recognize that L1 though small, plays an important role here since it

¯ ¯

destroys the phase coherence between the time-reversed trajectories.

We must now average the quantum interference term with respect to the impurity

potential. Since the dependence on the impurity potential in Eq. (11.116) is only

implicit through its determination of the classical paths, averaging with respect to

the random impurity potential is identical to averaging with respect to the probability

functional for the classical paths in the random potential. In view of the expression

appearing in Eq. (11.116), we thus encounter the probability of ¬nding a classical

path xt of duration t which start and end at the same point, and for which the

particle has the energy F

1 2 •=0

δ( [xcl ] ’ δ[xcl ’ xt ]

A[xcl ]

Pt [xt ] = F) = C(t) .

t t t imp

N0

xcl

t imp

(11.120)

36 The Bohr“Sommerfeld quantization rule.

406 11. Disordered conductors

The second delta function, as indicated, in the functional sense, allows only the

classical path in question to contribute to the path integral. The classical probability

of return in time t of a particle with energy F is given by

xt / 2 =x

(cl)

Dxt Pt [xt ] .

PR (t) = (11.121)

x’t / 2 =x

We therefore get, according to Eq. (11.116), for the impurity average of the weak-

localization quantum interference term, the Cooperon,

xt / 2 =x

cl

Dxt Pt [xt ] ei•[xt ] .

C 2 ,’ 2 (x, x) = C(t) = (11.122)

t t imp

x’t / 2 =x

In many situations of interest, an adequate expression for the probability density

of classical paths in a random potential, Pt [xt ], is obtained by considering the classical

paths as realizations of Brownian motion;37 i.e. the classical motion is assumed a

di¬usion process, and the probability distribution of paths is given by Eq. (7.103).