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).

Performing the impurity average gives in the di¬usive limit for the weak-localization

interference term38

xt / 2 =x t/ 2

x2

™

’ ’ i•(xcl ))

t

dt ( 4D ˜t

0

Dxt e ’t / 2

C t , ’t (x, x) = , (11.123)

2 2

x’t / 2 =x

2

where D0 is the di¬usion constant for a particle with energy F , D0 = vF „ /d.

Let us now obtain the equation satis¬ed by the Cooperon in the presence of a

time-dependent electromagnetic ¬eld. In that case we have for the interaction the

Lagrangian

L1 (xt , xt , t) = ext · A(xt , t) ’ eφ(xt , t) .

™ ™ (11.124)

Since the coherence between time-reversed trajectories is partially upset, it is con-

venient to introduce arbitrary initial and ¬nal times, and we have for the phase

di¬erence between a pair of time-reversed paths

1

{S[xcl ] ’ S[xcl+tf ’t ]}

•[xcl ] =

t t ti

tf

dt L1 (xcl , xcl , t) ’ L1 (xcl+tf ’t , xcl+tf ’t , t)

t ™t ™ ti

= (11.125)

ti

ti

37 An exception to this is discussed in Section 11.3.1.

38 In case the classical motion in the random potential is adequately described as the di¬usion

process, we immediately recover the result Eq. (11.93) for the return probability.

11.2. Weak localization 407

as the contributions to the phase di¬erence from L0 cancels, and we are left with

tf

e

dt xcl (t)·A(xcl (t), t) + φ(xcl (ti + tf ’t), ti + tf ’t) ’ φ(xcl (t), t)

•[xcl ] = ™

t

ti

’ xcl (ti + tf ’ t) · A(xcl (ti + tf ’ t), ti + tf ’ t) .

™ (11.126)

Introducing the shift in the time variable

1

t ≡t’T , T≡ (tf + ti ) (11.127)

2

we get

t f ’t i

2

e

xcl (t + T ) · A(xcl (t + T ), t + T )

•[xcl ] = ™

dt

t

t i ’t f

2

’ xcl (T ’ t ) · A(xcl (T ’ t ), T ’ t )

™

’ φ(xcl (t + T ), t + t) + φ(xcl (T ’ t ), T ’ t ) . (11.128)

The electromagnetic ¬eld is assumed to have a negligible e¬ect on determining the

classical paths, and we can shift the time argument specifying the position on the

path to be symmetric about the moment in time T , and thereby rewrite the phase

di¬erence, t ≡ tf ’ ti ,

t

2

e

dt xcl · AT (xcl , t) ’ φ(xcl , t) ,

¯ ™t ¯¯ ¯¯

•[xcl ] = (11.129)

¯

t t t

’2

t

where

φT (x, t) = φ(x, T + t) ’ φ(x, T ’ t) (11.130)

and

AT (x, t) = A(x, T + t) + A(x, T ’ t) . (11.131)

An electric ¬eld can be represented solely by a scalar potential, and we imme-

diately conclude that only if the ¬eld is di¬erent on time-reversed trajectories can

it lead to destruction of phase coherence. In particular, an electric ¬eld constant in

time does not a¬ect the phase coherence, and thereby does not in¬‚uence the weak-

localization e¬ect.

The di¬erential equation corresponding to the path integral, Eq. (11.123), there-

fore gives for the Cooperon equation for the case of a time-dependent electromagnetic

408 11. Disordered conductors

¬eld

2

‚ e ie

+ φT (xt , t) ’D0 ∇x ’ AT (x, t) Ct,t (x, x ) = δ(x ’ x ) δ(t ’ t ) .

T

‚t

(11.132)

When the sample is exposed to a time-independent magnetic ¬eld, we recover the

static Cooperon equation, Eq. (11.107).

11.3 Phase breaking in weak localization

The phase coherence between the amplitudes for pairs of time-reversed trajectories

is interrupted when the environment of the electron, besides the dominating random

potential, is taken into account. At nonzero temperatures, energy exchange due

to the interaction with the environment will partially upset the coherence between

time-reversed paths involved in the weak-localization phenomenon. The constructive

interference is then partially destroyed.

Quantitatively the e¬ect on weak localization by inelastic interactions with energy

transfers ”E of the order of the temperature, ”E ∼ kT , strongly inelastic processes,

can be understood by the observation that the single-particle Green™s function will

be additionally damped owing to interactions. If in addition to disorder we have an

interaction, say with phonons, the self-energy will in lowest order in the interaction

be changed according to

R

pE pE

pE

R R

’ pE pE pE pE

+

pE pE

(11.133)

and we will get an additional contribution to the imaginary part of the self-energy

mΣR = ’ ’ . (11.134)

2„ 2„in

11.3. Phase breaking in weak localization 409

Upon redoing the calculation leading to Eq. (11.22) for the case in question, we obtain

in the limit „in „

„

ζ(Q, ω) = 1 ’ + iω„ + D0 „ Q2 . (11.135)

„in

This will in turn lead to the change in the Cooperon equation, ω ’ ω + i/„in , and

we get the real space Cooperon equation39

1 1

’ iω ’ D0 ∇2 + δ(x ’ x ) .

Cω (x, x ) = (11.136)

x

„in „

The e¬ect on weak localization of electron“electron interaction and electron“

phonon interaction have been studied in detail experimentally [88, 89], and can phe-

nomenologically be accounted for adequately by introducing a temperature-dependent

phase-breaking rate 1/„• in the Cooperon equation, describing the temporal expo-

nential decay C(t) ’ C(t) exp{’t/„• } of phase coherence. In many cases the in-

elastic scattering rate, 1/„in , is identical to the phase-breaking rate, 1/„• . This is

for example the case for electron“phonon interaction, as we shortly demonstrate.

However, one should keep in mind that the inelastic scattering rate is de¬ned as the

damping of an energy state for the case where all scattering processes are weighted

equally, irrespective of the amount of energy transfer. In a clean metal the energy

relaxation rate due to electron“phonon or electron“electron interaction is determined