interaction of the electrons with the lattice vibrations. Without this compensation,

each of the mechanisms would appear to be enhanced in an impure metal, and would

lead to an enhanced phase-breaking rate proportional to (kT )2 /(nM c3 l).

l

±βγδ

For the case of transverse vibrations, we note that DT is of similar form as

ˆ± kγ has to be replaced by (δ±γ ’ k± kγ ) and an addi-

ˆ ˆˆ

Eq. (11.154) where, however, k

tional factor of 2, which accounts for the multiplicity of transverse modes. We then

obtain a phase-breaking rate due to interaction with transverse phonons, „•,t , which

is similar to the expression in Eq. (11.158) with cl and φL replaced by ct and

3 2k 3 l3 + 3kl ’ 3(k 2 l2 + 1) arctan kl

¦T (kl) = (11.160)

k4 l4

π

respectively. In particular, we obtain the limiting behaviors for the phase-breaking

rate due to transverse phonons

§ 2 (kT )2

⎪ π mMc3 l ct /l kT ct kD

⎨2

1 t

= (11.161)

⎪ π4 (kT )4

„•,t ©

20 l 2 mMc5 kT ct /l .

t

We note that in the high-temperature region, ct /l kT ct kD , the transverse

contribution is negligible in comparison with the longitudinal one if ct cl . But the

transverse rate dominates in the case where the transverse sound velocity is much

smaller than the longitudinal one. Such a situation may quite well be realized in

some amorphous metals; then, it is possible to observe a phase-breaking rate of the

’1

form „• ∝ T 2 /l at higher, but not too high, temperatures.49 The predictions of the

theory are in good agreement with magneto-resistance measurements and carefully

conducted experiments of the temperature dependence of the resistance [92].

49 A quadratic temperature dependence of the phase-breaking rate is often observed experimen-

tally.

416 11. Disordered conductors

The physical meaning of the second term in Eq. (11.149) is as follows. It is

appreciable only if the lattice deformation stays approximately constant during the

time the electron spends on its path and leads, in this case, to a cancellation of the

¬rst term. Equivalently, electron“phonon interactions with small energy transfers do

not lead to destruction of phase coherence. The e¬ect of this term is thus e¬ectively

to introduce a lower cut-o¬ in the integral of Eq. (11.158) at wave vector k0 = 1/cl „•,l .

However, there are no realistic models of phonon spectra where this e¬ect is of

importance. We therefore have the relationships ω kT / ω 1/„• . It is

therefore no surprise that the calculated phase-breaking rates are identical to the

inelastic electron“phonon collision rates in a dirty metal [93]. When considering

phase breaking due to electron“electron interaction, which we now turn to, the small

energy transfer interactions are of importance.

11.3.2 Electron“electron interaction

In this section we consider the temperature dependence of the phase-breaking rate

due to electron“electron interaction.50 As already discussed at the beginning of this

section, special attention to electron“electron interaction with small energy transfer

must be exercised due to the di¬usion enhancement. In diagrammatic terms we

therefore need to take into account diagrams where the electron“electron interaction

connects also the upper and lower particle lines in the Cooperon.

In Section 11.5 we shall show that the e¬ective electron“electron interaction at low

energies can be represented by a ¬‚uctuating ¬eld. Its correlation function in a dirty

metal will be given by the expression in Eq. (11.269), which we henceforth employ.

We can therefore obtain the e¬ect on the Cooperon of the quasi-elastic electron“

electron interaction by averaging the Cooperon with respect to a time-dependent

electromagnetic ¬eld using the proper correlator. We therefore consider the equation

for the Cooperon in the presence of an electromagnetic ¬eld, Eq. (11.81),

2

‚ ie 1 1

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

T

+ Ct,t (x, x ) =

„ e’e

‚t „

(11.162)

where we have chosen a gauge in which the scalar potential vanishes, and 1/„ e’e is

the energy relaxation rate due to high-energy electron“electron interaction processes,

i.e. processes with energy transfers ∼ kT .51

To account for the electron“electron interaction with small energy transfers, we

must perform the Gaussian average of the Cooperon with respect to the ¬‚uctuating

¬eld. This is facilitated by writing the solution of the Cooperon equation as the path

integral

xt =x

1

Dxt e’S[xt ] ,

T

Ct,t (R, R ) = (11.163)

2„

xt =x

50 We follow reference [94].

51 As will become clear in the following, the separation in high- and low-energy transfers takes

place at energies of the order of the temperature. However, in the following we shall not need to

specify the separation explicitly.

11.3. Phase breaking in weak localization 417

where the Euclidean action consists of two terms

S = S0 + SA , (11.164)

where

t

x21

™t 1

S0 [xt ] = dt1 + e’e (11.165)

4D0 „

t

and

t

ie

dt1 xt1 · AT (xt1 , t1 ) .

™

SA [xt ] = (11.166)

t

In terms of diagrams, the Gaussian average corresponds to connecting the external

¬eld lines pairwise in all possible ways by the correlator of the ¬eld ¬‚uctuations,

thereby producing the e¬ect of the low-energy electron“electron interaction. Since the

¬‚uctuating vector potential appears linearly in the exponential Cooperon expression,

the Gaussian average with respect to the ¬‚uctuating ¬eld is readily done

rt =R

1

Drt e’( S0 [xt ] +

T SA [xt ] )

Ct,t (R, R ) = (11.167)

2„

rt =R

where the averaged action SA is expressed in terms of the correlator of the vector

potential

t t

e2

dt1 dt2 xμ (t1 ) xν (t2 ) AT (xt2 , t1 )AT (xt2 , t2 ) .

SA [xt ] = ™ ™ (11.168)

μ ν

2

2

t t

If we recall the de¬nition of AT (xt , t), Eq. (11.77), we have

dq dω iq·(xt ’xt )

AT (xt2 , t1 )AT (xt2 , t2 ) = 2 e Aμ Aν

1 2

qω

μ ν

(2π)d 2π

t1 ’ t2

t 1 + t2

— cos ω + cos ω , (11.169)

2 2

where we have introduced the notation

≡ Aμ (q, ω)Aν (’q, ’ω) .

Aμ Aν (11.170)

qω

The electric ¬eld ¬‚uctuations could equally well have been represented by a scalar

potential

1

Aμ (q, ω)Aν (’q, ’ω) Eμ (q, ω)Eν (’q, ’ω)

=

ω2

qμ qν

φ(q, ω)φ(’q, ’ω) .

= (11.171)

ω2

418 11. Disordered conductors

In Section 11.5 we show that the electron“electron interaction with small energy

transfers, ω kT , is determined by the temperature, T , and the conductivity of

the sample, σ0 , according to52

2kT qμ qν

Aμ Aν = . (11.172)

qω

ω 2 σ0 q 2

Upon partial integration we notice the identity (the boundary terms are seen to

vanish as x’t = xt )

t t

ω(t1 ’ t2 )

ω(t1 + t2 )

dt1 dt2 qμ qν xμ (t1 ) xν (t2 )eiq·(xt 1 ’xt 2 ) cos

™ ™ + cos

2 2

t t

t t

ω(t1 ’ t2 )

2

iq·(xt 1 ’xt 2 ) ω ω(t1 + t2 )

= ’ dt1 dt2 e ’ cos

cos (11.173)

4 2 2

t t

and obtain

t t

eiq·(xt 1 ’xt 2 ) ω(t1 ’ t2 )

e2 kT dq dω ω(t1 + t2 )

[xt ] = ’ ’ cos

SA dt1 dt2 cos .

2σ0 (2π)d 2π q2 2 2

t t

(11.174)

Performing the integration over ω and t2 , the expression for the Cooperon becomes

xt =x t xt

™ 2 dq

q’2 (1’cos(q·(xt 1 ’x’t 1 )))

’ dt1 + „ e’e + 2eσ k T

1

1

1 (2π )2

4D 0

Dxt e

0

T ’t

Ct,’t (x, x ) = .

2„

x’t =x

(11.175)

The singular term is regularized by remembering that in Eq. (11.174) the ω-integra-

tion actually should have been terminated, in the present context, at the large fre-

quency kT / . The factor exp{iq · (xt1 ’ xt2 )} does therefore not reduce strictly to

1 for the ¬rst term in the parenthesis in Eq. (11.174) as |xt1 ’ xt2 | ≥ (D0 /kT )1/2 ,

and this oscillating phase factor provides the convergence of the integral. We should

therefore cut o¬ the q-integral at the wave vector satisfying q = (kT / D0 )1/2 ≡ L’1 ,

T

as indicated by the prime on the q-integration in the two previous equations.

Introducing new variables

xt ’ x’t

xt + x’t

√ √

Rt = , rt = (11.176)

2 2

52 Since the time label T now has disappeared, no confusion should arise in the following where T

denotes the temperature. We recall Section 6.5, and note that the relation Eq. (11.172) is equivalent

to the statement that the low-frequency electron“electron interaction in a disordered conductor is

identical to the Nyquist noise in the electromagnetic ¬eld ¬‚uctuations.

11.3. Phase breaking in weak localization 419

the path integral separates in two parts53

√

Rt = 2R

∞ R2

t ™

’ dt t

1

√

2D 0

DRt e

Ct,’t (R, R) = dR0 0

2 2„

’∞ Rt =0 =R0

rt =0 √

r2

t ™

2e 2 k T dq

q’2 (1’cos( 2q·rt ))

’ dt 2

t + + (2π )2

„ e’e

4D 0 σ0

— Drt e (11.177)

0

r0 =0

The path integral with respect to Rt gives the probability that a particle started at

√

position R0 at time t = 0 by di¬usion reaches the point 2 R (recall Eq. (7.103)).

Integrating this probability over all possible starting points is identical to integrating

over all ¬nal points and by normalization gives unity. We are thus left with the

expression for the Cooperon

ρt =0 t

r™¯2

’ dt

¯ t

1 + V (rt )

¯

=√

4D 0

Drt e

Ct,’t , (11.178)

0

¯

2 2„

ρ0 =0

where we have introduced the notation

√

2e2 kT

2 dq

q ’2 1 ’ cos( 2 q · r)

V (r) = + . (11.179)

„ e’e (2π)d

σ0

As expected from translational invariance, the Cooperon is independent of position.

We have thus reduced the problem of calculating the quantum correction to the

conductivity,

∞

2

4e D0 „

δσ(ω) = ’ dt eiωt Ct,’t (r, r) , (11.180)

π

’∞

in the presence of electron“electron interaction, to solving for the Green™s function

the imaginary time Schr¨dinger problem

o

1

√

{‚t ’ D0 δ(r ’ r ) δ(t ’ t ) .

+ V (r)} Ct,t (r, r ) = (11.181)

r

2 2„

In the three-dimensional case the ¬rst term in the integrand of Eq. (11.179) gives

rise to a temperature dependence of the form T 3/2 . This is the same form as the one

we shall ¬nd in Section 11.5 for the inelastic scattering rate due to electron“electron

interaction in a dirty metal. This term can thus be joined with the ¬rst term of

Eq. (11.179). We note that the description of the low-energy behavior thus joins up

smoothly with the description of the high-energy behavior, as it should.

53 This is immediately obtained by using the standard discretized representation of a path integral.

420 11. Disordered conductors

We thus have for the potential in the three-dimensional case

2 ˜

V3 (r) = + V3 (r) (11.182)

„ e’e

§

where

1

⎨ r LT

’e 2 r

˜3 (r) = √ kT

V (11.183)

√

2π 2 σ0 © L’1

22

r LT .

T

π

Fourier-transforming Eq. (11.181) with respect to time and taking the static limit we

obtain

1

{’D0 r + V3 (r)} Cω=0 (r, r ) = √ δ(r ’ r ) . (11.184)

2 2„

˜

Solving this equation to ¬rst order in the potential V3 gives

√L

√L

e2 kT L ’2 2 L

C1 (0, 0, ω = 0) = ’ ’1 + Ei ’2 2

e T

2

4π 2 „ D0 σ0 πLT LT

(11.185)

where Ei is the exponential integral54 and

D0 „ e’e .

L= (11.186)

In accordance with the calculation of the inelastic lifetime in section 11.5 we have

( kT „ )1/4

LT

∼ . (11.187)

L kF l

We can therefore expand the expression in Eq. (11.185), and obtain for the quantum

correction to the conductivity

e2 4πe2 kT L LT

δσ = 1+ ln , (11.188)

2π 2 L 2D σ L

00

where the second term is the correction due to collisions with small energy transfer,

proportional to T 1/4 ln T . In the two-dimensional case we obtain from Eq. (11.179)

for the potential

L’1 √

T

1 ’ J0 ( 2 qr)

2

2 e kT

V2 (r) = + dq , (11.189)

„ e’e π 2 σ0 q

0

where J0 denotes the Bessel function. We observe the limiting behavior of the po-

tential

§ 2

⎪

⎪ 1 r

r LT

⎪

⎪ 4 LT

⎨

2

2 e kT

V2 (r) = e’e +

π 2 σ0 ⎪