basic excitation of the bare Coulomb potential in an electron gas is the particle“

hole excitation, and it will lead to screening of the interaction. It is su¬cient to

use the random phase approximation where additional interaction decorations by

electron“electron interactions are negligible since the disorder e¬ects are driven by

the long ranged Di¬uson.66 Before averaging with respect to the random impu-

rity potential we thus have the diagrammatic matrix representation of the e¬ective

electron“electron interaction

p+ E+

= +

qω q q q

p’ E’

p+ E+ p+ E+

+ + ...

q q q

p’ E’ p’ E’

p+ E+

= + , (11.246)

q q qω

p’ E’

where the thick wiggly line represents the e¬ective Coulomb interaction, i.e. in the

triagonal representation the matrix

R K

V (q, ω) V (q, ω)

= (11.247)

A

qω 0 V (q, ω)

and similarly for the thin line representing the bare Coulomb interaction for which we

note V K (q) = 0. Analytically the Dyson equation for the matrix Coulomb propagator

has the form

V (q, ω) = V (q) + V (q) Π(q, ω) V (q, ω) , (11.248)

where the polarization, Π, in the triagonal representation has the form

ΠR (q, ω) ΠK (q, ω)

Π(q, ω) = (11.249)

ΠA (q, ω)

0

66 The random phase approximation can also be stated as the linearized mean-¬eld approximation

as discussed for example in chapter 10 of reference [1].

434 11. Disordered conductors

and in the random phase approximation speci¬ed in terms of the dynamical indices

according to

Πkk = ’2i˜ii Gi j Gji γj j .

γk k

(11.250)

Solving the Dyson equation for the e¬ective interaction in the random phase

approximation gives

1

= . (11.251)

qω p+ E+

’1 ’

q qω qω

p’ E’

According to our universal rules for boson“fermion coupling in the dynamical

indices, Eq. (5.51) and Eq. (5.52), the retarded polarization bubble is given by

∞

dp dE

ΠR (q, ω) = ’i GR (p) GK (p ’ q) ’ GK (p) GA (p ’ q) , (11.252)

(2π)3 2π

’∞

where p = (E, p) and q = (ω, q). In the diagrammatic expansion of the e¬ective

electron“electron interaction, we must then impurity average the electron“hole or

polarization bubble diagram. To lowest order in the disorder parameter 1/kF l, we

should insert the impurity ladder into the bubble diagram; i.e. we encounter the

diagrams of the type

R R

. (11.253)

qω qω

A A

The impurity-averaged bubble diagram is evaluated using the standard impurity

Green™s function technique, and we thus have in the di¬usive limit, ql, ω„ 1 (in

the three-dimensional case), for the dielectric function, ql, ω„ 1,

e2 2N0 D0 q 2 D 0 κ2

s

(q, ω) = 1 + = 1+ , (11.254)

2 ’iω + D q 2 ’iω + D0 q 2

0q 0

relating the bare Coulomb interaction to the e¬ective interaction.67 Inserting into

Eq. (11.244), we can calculate the inelastic collision rate.

67 The calculation is equivalent to the calculation of the density“density response function of a

disorder conductor giving the expression

2N0 D0 q 2

χ(q, ω) = .

’iω + D0 q 2

This is understandable since we note that a ¬‚uctuation in the density of electrons creates an electric

potential, which in turn is felt by an electron. Fluctuations in the density or current of the elec-

trons give rise to ¬‚uctuations in an electromagnetic ¬eld inside the electron gas, as discussed quite

generally in Section 6.5 in connection with the ¬‚uctuation“dissipation relations of linear response.

11.5. Coulomb interaction in a disordered conductor 435

We could also calculate the inelastic collision rate or energy relaxation rate in

the dirty limit by solving the Boltzmann equation with the two-particle interaction

modi¬ed by the impurity scattering

∞ ∞

‚f ( ) d

= 2π dω P (ω) R( , , ω) , (11.255)

‚t ’∞2π

’∞

where

f ( ) f ( ’ ω) (1 ’ f ( ’ ω)) (1 ’ f ( ))

R( , , ω) =

’ f ( ’ ω) f ( ) (1 ’ f ( )) (1 ’ f ( ’ ω)) (11.256)

and

2

2N0 „ 2 (D0 q)2

dq V (q)

P (ω) = (11.257)

| (q, ω)| ω 2 + (D0 q)2

(2π)3

π

is analogous to Eliashberg function, ±2 F , for the electron“phonon case. We notice

that we can rewrite

„ dq ζ(q, ω)

V R (q, ω)

P (ω) = m , (11.258)

1 ’ ζ(q, ω)

(2π)3

πω

where ζ is the insertion Eq. (11.20) (here the relevant case is the particle“hole channel,

but the result is identical to that of the particle“particle channel) and given (in two

and three dimensions) by

i ql + ω„ + i

q ≡ |q|

ζ(q, ω) = ln , (11.259)

’ql + ω„ + i

2ql

with the limiting behavior

§ π

⎨ ql > ω„, ql > 1

2ql

ζ(q, ω) = (11.260)

©

1 + iω„ ’ D0 „ q 2 ql, ω„ < 1 .

In the three-dimensional case we have, ω„ < 1,

ω ’1/2

√

P (ω) = . (11.261)

3/2

8 2π 2 N0 D0

We therefore get for an electron on the Fermi surface in a dirty metal the electron“

electron collision rate at temperatures kT < /„ 68

∞

„ 1/2 (kT )3/2

1 2ω

√

= dω P (ω) =c , (11.262)

ω

„e’e (T ) kF l

sinh kT F„

0

The dielectric function and the density and current response functions are thus all related

e2

iσ(q, ω)

(q, ω) = 1 + = 1+ χ(q, ω) .

2

ω0 0q

For a discussion we refer the reader to chapter 10 of reference [1].

68 From the region of large ω and q we get the clean limit rate, Eq. (7.206), which dominates at

temperatures kT /„ .

436 11. Disordered conductors

where c is a constant of order unity (ζ(3/2) 2.612)

√

√

3 3π

ζ(3/2)( 8 ’ 1) .

c= (11.263)

16

For an electron in energy state ξ, ξ < /„ , we get analogously in the dirty limit

for the electron“electron collision rate at zero temperature69

√

„ 1/2

1 6

ξ 3/2 .

= (11.264)

3/2 (k l)2

„e’e (ξ) 4 F

The scattering rate due to electron“electron interaction is thus enhanced in a

dirty metal compared with the clean case [103, 104, 105], di¬usion enhanced electron“

electron interaction.70 Equivalently, the screening is weakened owing to the di¬usive

motion of the electrons. The interpretation of this enhancement can be given in terms

of the previous phase space argument of Exercise 7.10 on page 214 for the relaxation

time and the breaking of translational invariance due to the presence of disorder.

The violation of momentum conservation in the virtual scattering processes due to

impurities gives more phase space for ¬nal states. Alternatively, viewing the collisions

in real space, owing to the motion being di¬usive instead of ballistic the electrons

spend more time close together where the interaction is strong, or, wave functions

of equal energy in a random potential are spatially correlated thereby leading to an

enhanced electron“electron interaction. The scattering process now includes quantum

interference between the elastic and inelastic processes as signi¬ed by the collision

rate /„e’e being dependent on .

We note that the expression for the energy relaxation rate in two dimensions

diverges in the infrared for a dirty metal in the above lowest-order perturbative

calculation. For the Coulomb potential for electrons constricted to movement in two

dimensions the bare Coulomb potential is

2πe2

V (q) = (11.265)

|q|

and for ω„ < 1

1 1

P2 (ω) = (11.266)

8 F„ ω

69 At temperatures and energies kT, ξ > /„ , the expressions for relaxation rates are those of the

clean limit, recall Exercise 7.10 on page 214.

70 In the case of electron“phonon interaction, local charge neutrality forces the electrons to follow

adiabatically the thermal motion of the ions, and because of the coherent motion with the lattice of

the ¬xed impurities, the interaction with the longitudinal phonons is in fact decreased owing to this

compensation mechanism. The imaginary part of the electron self-energy will therefore be given by

the results obtained in Section 11.3.1 for the phase-breaking rate. As shown there, the interaction

with transverse phonons are either enhanced or diminished depending on the temperature regime.

The in¬‚uence of impurities will not be universal for the case of interaction with phonons as will be

the case for the di¬usion enhanced electron“electron interaction.

11.6. Mesoscopic ¬‚uctuations 437

giving the divergent expression for the relaxation rate, kT < /„ ,71

∞

1 1 1

= dω . (11.267)

ω

„e’e (T ) 2kF l sinh kT

0

However, this is not alarming since we do not expect the relaxation rate to be the

relevant measurable quantity, as in this quantity scattering at all energies is weighted

equally. We do not expect such divergences in physically measurable rates, and indeed

the phase relaxation rate of the electronic wave function in a dirty two-dimensional

metallic ¬lm does not diverge because of collisions with small energy transfer, as

discussed in Section 11.3.2. There we made use of the expression for the e¬ective

electron“electron interaction at low energies and momenta in a dirty metal for which,

according to Eq. (11.243), we have72

’4ie2 kT

K

V (q, ω) = = . (11.268)

σ0 |q|2

The low frequency electron“electron interaction in a disordered conductor is thus

identical to the Nyquist noise in the electromagnetic ¬eld ¬‚uctuations, the correlator

we used in Section 11.3.2 (here represented by the scalar potential),

2kT

φ(q, ω)φ(’q, ’ω) = . (11.269)

σ0 q 2

We observe the generality of the result of Section 6.5.

11.6 Mesoscopic ¬‚uctuations

In the following we shall show that when the size of a sample becomes comparable to

the phase coherence length, L ∼ L• , the individuality of the sample will be manifest

in its transport properties. Such a sample is said to be mesoscopic. Characteristically

the conductance will exhibit sample-speci¬c, noise-like but reproduceable, aperiodic

oscillations as a function of, say, magnetic ¬eld or chemical potential (i.e. density

of electrons). The sample behavior is thus no longer characterized by its average

characteristics, such as the average conductance, i.e. the average impurity concen-

tration. The statistical assumption of phase-incoherent and therefore independent

subsystems, allowing for such an average description, is no longer valid when the

transport takes place quantum mechanically coherently throughout the whole sam-

ple. As a consequence, a mesoscopic sample does not possess the property of being

self-averaging; i.e. the relative ¬‚uctuations in the conductance do not vanish in a

71 Wenote that the relaxation rate due to processes with energy transfers of the order of the

temperature is

1 kT

∼ .

„T mD0

72 The factor of ’2i between Eq. (11.268) and Eq. (11.269) simply re¬‚ects our choice of Feynman

rules.

438 11. Disordered conductors

central limit fashion inversely proportional to the volume in the large-volume limit.

To describe the ¬‚uctuations from the average value we need to study the higher

moments of the conductance distribution such as the variance ”G±β,γδ . We shall

¬rst study the ¬‚uctuations in the conductance at zero temperature, and consider the

variance

”G±β,γδ = (G±β ’ G±β )(Gγδ ’ Gγδ ) . (11.270)

For the conductance ¬‚uctuations we have the expression

= (L’2 )2 dx2 dx2 dx1 dx1 σ±β (x2 , x2 ) σγδ (x1 , x1 ) .

G±β Gγδ (11.271)

The diagrams for the variance of the conductance ¬‚uctuations can still be managed

within the standard impurity diagram technique in the weak disorder limit, F „ ,

and a typical conductance ¬‚uctuation diagram is depicted in Figure 11.7 (here the

box denotes the Di¬uson).73

r1 r1

p p

R

R

Ap p A

± γ δ β

p p

R R

p p

A A

r r

Figure 11.7 Conductance ¬‚uctuation diagram.

The construction of the conductance ¬‚uctuation diagrams follows from impurity

averaging two conductivity diagrams. Draw two conductivity bubble diagrams, where

the propagators include the impurity scattering. Treating the impurity scattering

perturbatively, we get impurity vertices that we, upon impurity averaging as usual

have to pair in all possible ways. Since we subtract the squared average conductance

in forming the variance, ”G, the diagrams for the variance consist only of diagrams

where the two conductance loops are connected by impurity lines. As already noted

in the discussion of weak localization, the dominant contributions to such loop-type

diagrams are from the infrared and long-wavelength divergence of the Cooperon, and

here additionally from the Di¬uson.

73 The diagram is in the position representation, and the momentum labels should presently be

ignored, but will be explained shortly.

11.6. Mesoscopic ¬‚uctuations 439

To calculate the contribution to the variance from the Di¬uson diagram depicted

in Figure 11.7, we write the corresponding expression down in the spatial represen-

tation in accordance with the usual Feynman rules for conductivity diagrams. Let

us consider a hypercube of size L. If we assume that the sample size is bigger than

the impurity mean free path, L > l, the spatial extension of the integration over the

external, excitation and measuring, vertices can be extended to in¬nity, since the

propagators have the spatial extension of the mean free path. We can therefore in-

troduce the Fourier transform for the propagators since no reference to the ¬niteness

of the system is necessary for such local quantities. Furthermore, since the spatial

extension of the Di¬uson is long range compared with the mean free path, we can

set the spatial labels of the Di¬usons equal to each other, i.e. r1 = r and r 1 = r .

All the spatial integrations, except the ones determined by the Di¬uson, can then be

performed, leading to the momentum labels for the propagators as depicted in Figure

11.7 Let us study the ¬‚uctuations in the d.c. conductance, so that the frequency, ω,

of the external ¬eld is zero. The energy labels have for visual clarity been deleted

from Figure 11.7, since we only have elastic scattering and therefore one label, say

, for the outer ring and one for the inner, . According to the Feynman rules, we

obtain for the Di¬uson diagram the following analytical expression:

∞ ∞

2

e2 2 u 2 ‚f ( ) ‚f ( ) dp dp

’4

G±β Gγδ =L d d

D

4πm2 (2π )3 (2π )3

’∞ ‚ ‚

’∞

— GR (p )GA (p )GA (p )GR (p )GR (p)GA (p)GA (p)GR (p)