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Second, the term ’k 2 v2 /3 is connected with ’mv2 ∇ · u/3, and originates from the
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

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)


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)

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 ⎪

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