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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)


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

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