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