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p’ p’
Q’p+ Q’p+

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
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)
+ +

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 )
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

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

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
2e2 D0 dQ 1
δσ(L) = ’ . (11.31)
d D Q2
π (2π) 0
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)
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
β(g) = ’ 2 , g 1, (11.33)
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
δg(L) = (11.34)
’ π3 ( l ’ 1)

and thereby for the scaling function to lowest order in 1/g
β(g) = (d ’ 2) ’ , (11.35)

⎨ 2
a= (11.36)
© 1
d=3 .

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
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)

2e2 D0 dQ 1

δσ(ω) = . (11.38)
(2π)d ’iω + D0 Q2

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
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)

∞ 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
which in the two-dimensional case becomes
e2 D 0t
e’ 2„ ’ e’ L 2
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
⎨ 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
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)

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

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


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