A x x

x x

+ ...

+

± β

± β

rr

˜

C

x x

=

± β

. (11.50)

˜

The sum of the maximally crossed diagrams, the Cooperon Cω (r, r ; E), is in the

position representation speci¬ed by the diagrams

R

r r r

˜

r r . (11.51)

+ + ... = C

r

A

The analytical expression for the quantum correction to the conductivity is therefore

(E+ ≡ E + ω)

∞

2

f0 (E) ’ f0 (E + ω) ˜

1 e

δσ±β (x ’ x , ω) = dr dr dE Cω (r, r ; E)

π m ω

’∞

” ”

— GR+ (x ’ r)GR+ (r ’ x ) ∇x± ∇xβ GA (x ’ r)GA (r ’ x) . (11.52)

E E E E

11.2. Weak localization 391

The impurity-averaged propagator decays exponentially as a function of its spatial

variable with the scale set by the impurity mean free path. The spatial scale of

variation of the sum of the maximally crossed diagrams is typically much larger.

For the present case where we neglect e¬ects of inelastic interactions, we recall from

Eq. (11.23) that the spatial range of the Cooperon is Lω = D0 /ω, which for ω „ 1

is much larger than the mean free path, since D0 = vF l/d is the di¬usion constant

in d dimensions.24 The impurity-averaged propagators attached to the maximally

˜

crossed diagrams will therefore require the starting and end points of Cω (r, r , E) to

be within the distance of a mean free path, in order for a non-vanishing contribution

to the integral. On the scale of variation of the Cooperon this amounts to setting its

arguments equal, and we can therefore substitute r ’ x, r ’ x, and obtain

∞

2

f0 (E’ ) ’ f0 (E+ ) ˜

1 e

δσ±β (x ’ x , ω) = dE Cω (x, x; E) dr dr

π m ω

’∞

” ”

— GR+ (x ’ r)GR+ (r ’ x ) ∇x± ∇xβ GA’ (x ’ r)GA’ (r ’ x) . 11.53)

(

E E E E

The gate combination of the Fermi functions renders for the degenerate case, ω, kT

F , the energy variable in the thermal layer around the Fermi surface, and we have

for the ¬rst quantum correction to the conductivity of a degenerate electron gas

e 2

˜

δσ±β (x ’ x , ω) = ¦± , β (x ’ x ) ,

Cω (x, x; F) (11.54)

π m

where

” ”

¦± , β (x ’ x ) ≡ dr dr GR (x’r)GR (r ’x ) ∇x±∇xβ GA (x ’r)GA (r ’x). (11.55)

F F F F

Clearly this function is local with the scale of the mean free path, and to lowest order

in /pF l we have25

(2πN0 „ )2 (x ’ x )± (x ’ x )β ’|x’x |/l

¦±,β (x ’ x ) = ’ cos2 kF |x ’ x | .

e (11.56)

|x ’ x |

2 4

2

Since the function ¦±,β (x’x ) decays on the scale of the mean free path, and appears

in connection with the Cooperon, which is a smooth function on this scale, it acts

e¬ectively as a delta function

(2πN0 „ )2 l

¦±,β (x ’ x ) ’ δ±β δ(x ’ x ) .

= (11.57)

32

We therefore obtain the fact that the ¬rst quantum correction, the weak-localization

contribution, to the conductivity is local

δσ±β (x ’ x , ω) = δσ(x, ω) δ±β δ(x ’ x ) (11.58)

24 For samples of size larger than the mean free path, L > l, the di¬usion process is e¬ectively

three-dimensional, so that one should use the value d = 3 in the expression for the di¬usion con-

stant. In strictly two-dimensional systems, such as for the electron gas in the inversion layer in a

heterostructure at low temperatures, the value d = 2 should be used.

25 For details we refer the reader to chapter 11 of reference [1].

392 11. Disordered conductors

and speci¬ed by26

2e2 D0 „

δσ(x, ω) = ’ Cω (x, x) . (11.59)

π

As we already noted in Section 11.1.2 the Cooperon is independent of the energy

of the electron (here the Fermi energy since only electrons at the Fermi surface

contribute to the conductivity) Cω (x, x ) ≡ Cω (x, x , F ), and we have introduced

Cω (x, x ) ≡ u’2 Cω (x, x ).

˜

The quantum correction to the conductance of a disordered degenerate electron

gas is

= L’2 dx dx δσ±β (x, x , ω)

δG±β (ω) ≡ δG±β (ω)

2e2 D0 „ ’2

’

= L δ±β dx Cω (x, x) . (11.60)

π

11.2.2 Cooperon equation

Many important results in the theory of weak localization can be obtained once the ef-

fect on the Cooperon of a time-dependent external ¬eld is obtained. Later we present

the derivation of the Cooperon equation in the presence of a time-dependent elec-

tromagnetic ¬eld based on the quantum interference picture of the weak-localization

e¬ect. But ¬rst we provide the quantitative derivation of this result by employing the

equation obeyed by the quasi-classical Green™s function in Nambu or particle“hole

space in the dirty, i.e. di¬usive limit, Eq. (8.197). This will, in addition, extend

our awareness of the information contained in the various components of the matrix

Green™s function in Nambu or particle“hole space.27

The goal is to generate the equation for the Cooperon by functional di¬erentiation

of the quasi-classical Green™s function, and we therefore add a two-particle source to

the Nambu space Hamiltonian, Ψ denoting the Nambu ¬eld, Eq. (8.32),

dx1 Ψ† (x1 , t1 ) V (x1 , t1 , t1 ) Ψ(x1 , t1 ) ,

V (t1 , t1 ) = (11.61)

which therefore, according to Section 8.1.1, needs only o¬-diagonal Nambu matrix

elements

0 V12 (x1 , t1 , t1 )

V (x1 , t1 , t1 ) = . (11.62)

V21 (x1 , t1 , t1 ) 0

In linear response to the two-particle source we thus encounter the two-particle

Green™s function in the form of the particle“particle impurity ladder, and the Cooperon

can be obtained by di¬erentiation with respect to the source, which therefore is taken

local in the space variable.

26 We could also have evaluated the conductivity, Eq. (11.52), directly by Fourier-transforming

the propagators, and recalling Eq. (11.27).

27 This provides an alternative derivation to the ones in the literature. We follow the derivation

in reference [9].

11.2. Weak localization 393

For the retarded component of Eq. (8.197) we have (we leave out the subscript

indicating it is the s-wave, local in space part of the quasi-classical Green™s function,

gs )

’1

[g0 + iV R ’ D0 ‚ —¦ g R —¦ ‚ —¦ g R ]’ = 0 ,

, (11.63)

where the scalar potential enters in

’1

g0 (x1 , t1 , t1 ) = („3 ‚t1 + ieφ(x1 , t1 ))δ(t1 ’ t1 ) (11.64)

and the vector potential through the di¬usive term according to

‚ = (∇x1 ’ ie„3 A(x1 , t1 )) . (11.65)

The equation of motion, which is homogeneous, is supplemented by the normalization

condition, Eq. (8.182),

g R —¦ g R = δ(t1 ’ t1 ) . (11.66)

The self-energy term associated with superconductivity has been expelled from Eq.

(8.185) since for our case of interest the conductor is assumed in the normal state.

Instead a source-term, V R , has been introduced, a matrix in Nambu-space. Taking

the functional derivative of the 12-component of g R with respect to the Nambu

R

component V12 is seen to generate the Cooperon

R

1 δg12 (R, t1 , t1 )

C(R, R , t1 , t1 , t2 , t2 ) = (11.67)

R

2i„ δV12 (R , t2 , t2 )

since the o¬-diagonal Nambu components of the source term add or subtract pairs

of particles, and in the di¬usive limit only ladder diagrams are considered. By con-

struction, the functional derivative on the right in Eq. (11.67) is the ξ-integrated

particle“particle ladder (including external legs) with the in¬‚uence of the electro-

magnetic ¬eld fully included in the quasi-classical approximation.28 The result of

the functional derivative operation involved in Eq. (11.67) is depicted diagrammati-

cally in Figure 11.3.

R t

t2 = T +

t

t1 = T + 2

2

R R

t1 = T ’ t2 = T ’

t t

2 2

A

Figure 11.3 Cooperon obtained as derivative with respect to the two-particle source.

In order to obtain the equation satis¬ed by the functional derivative, the equation

of motion is linearized with respect to the solution in the absence of the source term,

R

g0 . We thus write

g R = g0 + δg R

R

(11.68)

28 This is usually no restriction since interest is in weak ¬elds.

394 11. Disordered conductors

and use our knowledge that the normal state solution in the absence of the source

term is

g0 = „3 δ(t1 ’ t1 ) .

R

(11.69)

Inserting into Eq. (11.63) and linearizing the equation with respect to the source

gives

’ ’ ’ ’

’1

[g0 ’ D0 ‚ —¦ g0 —¦ ‚ —¦ δg R ]’ + i[V R —¦ g0 ]’ ’ D0 [ ‚ —¦ δg R —¦ ‚ —¦ g0 ]’ = 0 .

R

,R ,R

,

(11.70)

Taking the 12-Nambu component gives

‚t1 ’ ‚t1 + ie(φt1 ’ φt1 ) ’ D0 (∇x ’ ie(At1 + At1 ))2 δg R (x, t1 , t1 )

R

= 2iV12 (x, t1 , t1 ) , (11.71)

where the spatial dependence x of the ¬elds has been suppressed. Taking the func-

tional derivative with respect to the 12-Nambu component of the source we get

R

δg12 (x, t1 , t1 )

‚t1 ’ ‚t1 + ie(φt1 ’ φt1 ) ’ D0 (∇x ’ ie(At1 + At1 )) 2

R

δV12 (x , t2 , t2 )

2i δ(x ’ x ) δ(t1 ’ t2 ) δ(t1 ’ t2 ) .

= (11.72)

Because of the double time dependence of the external ¬eld, the functional deriva-

tive and the Cooperon have the time labeling depicted in Figure 11.3 and the following

diagram

˜ ˜

t1 t1

x t1 =T + 2 x t2 =T + t2

t

C =

x t1 =T ’ 2 x t2 =T ’ t2

t

˜ ˜

t2 t2

···

+ + .

(11.73)

Introducing new time variables

1 1

t = t1 ’t1 t = t2 ’t2 , (11.74)

T= (t1 +t1 ) , T= (t2 +t2 ) , ,

2 2

11.2. Weak localization 395

we get

2 R

‚ ie δg12 (x, T, t)

2 + ieφT (x, t) ’D0 ∇x ’ AT (x, t) R

‚t δV12 (x , T , t )

1

δ(x ’ x ) δ(t ’ t ) δ(T ’ T ) ,

= (11.75)

„

where we have introduced the abbreviations

φT (x, t) = φ(x, T + t/2) ’ φ(x, T ’ t/2) (11.76)

and

AT (x, t) = A(x, T + t/2) + A(x, T ’ t/2) . (11.77)

Accordingly for the Cooperon we get the equation

2

‚ ie

2 + ieφT (x, t) ’ D0 ∇x ’ AT (x, t) T,T

Ct,t (x, x )

‚t

1

δ(x ’ x ) δ(t ’ t ) δ(T ’ T ) ,

= (11.78)

„

where we have introduced

T,T

Ct,t (x, x ) ≡ C(x, x ; t1 , t1 , t2 , t2 ) . (11.79)

Since there is no di¬erentiation with respect to the variable T in Eq. (11.78), it is

only a parameter in the Cooperon equation, and we have

T,T

Ct,t (x, x) = Ct,t (x, x ) δ(T ’ T ) ,

T

(11.80)

T

where Ct,t (x, x ) satis¬es the equation

2

‚ ie 1

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

T

Ct t (x, x ) = (11.81)

‚t „

Here we have left out the e¬ect of a time-dependent scalar potential on the Cooperon

since in the following we represent the electromagnetic ¬eld solely by the vector

potential. We note that it can be restored by invoking the gauge co-variance property

of the Cooperon.

We now derive the conductivity formula relevant for the case in question. We

are here beyond linear response since we are taking into account to all orders how

the Cooperon is in¬‚uenced by the electromagnetic ¬eld. In the case of an external

electromagnetic ¬eld represented by a vector potential in¬‚uencing the Cooperon as

well we consider the quantum correction to the kinetic propagator which is given by

the contributions speci¬ed in the following diagram

396 11. Disordered conductors

x1 t1

δGK (x1 , t1 , x1 , t1 ) = (11.82)

x1 t1

where the summation sign indicates the summation of all maximally crossed dia-

grams. For the quantum correction to the current we then have

e ‚ ‚

’ δGK (x, t, x , t)

δj(x, t) = . (11.83)

2im ‚x ‚x

x =x

The structure of the general maximally crossed diagram with n impurity correlators

is

2n+1

K

(GR )j GK (GA )2n’j .

δG = (11.84)

j=0

If the equilibrium kinetic propagator GK occurs in the above diagram at a place

0

di¬erent from the ones indicated by circles, the contribution vanishes to the order

of accuracy. In that case, viz. we encounter the product of two retarded or two

advanced propagators sharing the same momentum integration variable, and since