In order to obtain the above expression we have noted that

’ ) = [D(r, r , ’ )]— ,

D(r, r , (11.273)

which follows from the relationship between the retarded and advanced propagators.

At zero temperature, the Fermi functions set the energy variables in the propagators

in the conductance loops to the Fermi energy, and the Di¬uson frequency to zero. At

zero temperature we therefore get for the considered Di¬uson diagram the following

analytical expression, D(r, r ) ≡ D(r, r , 0),

2

e2 2 u 2 dp dp

’4

G±β Gγδ = L p± pγ pδ pβ

D

4πm2 (2π )3 (2π )3

— [GR (p)GA (p)GR (p )GA (p )]2 dr dr |D(r, r )|2 . (11.274)

F F F F

440 11. Disordered conductors

It is important to note that the same Di¬uson appears twice. This is the leading

singularity we need to keep track of. If we try to construct variance diagrams con-

taining, say, three Di¬usons, we will observe that they cannot carry the same wave

vector, and will give a contribution smaller by the factor / F „ . The momentum

integrations at the current vertices can easily be performed by the residue method

(recall Eq. (11.27))

4π p2 N0 3

dp

p± pγ [GR (p)GA (p)]2 = F

j±γ = „ δ±γ (11.275)

3 3

(2π ) 3

F F

and for the considered Di¬uson diagram we obtain the expression

2

e2 D0 „

’4

δ±γ δδβ dr dr |D(r, r )|2 .

G±β Gγδ =L (11.276)

D

2π

To calculate the Di¬uson integrals we need to address the ¬nite size of the sample

and its attachment to the current leads, since the Di¬uson has no inherent length

scale cut-o¬. At the surface where the sample is attached to the leads, the Di¬uson

vanishes

D(r, r ) = 0 or r on lead surfaces (11.277)

r

in accordance with the assumption that once an electron reaches the lead it never

returns to the disordered region phase coherently. On the other surfaces the current

vanishes; i.e. the normal derivative of the Di¬uson must vanish (recall Eq. (7.96) and

Eq. (7.97))

‚D(r, r )

=0 r or r on non-lead surfaces with surface normal n .

‚n

(11.278)

74

We assume that the leads have the same size as the sample surface. Therefore by

solving the di¬usion equation for the Di¬uson, with the above mixed (Dirichlet“von

Neumann) boundary condition, we obtain the expression

2

1/„

dr dr |D(r, r )|2 = , (11.279)

2

D0 qn

n

where n ≡ (nx , ny , nz ) is the eigenvalue index in the three-dimensional case

π

qn± = n± n± = nx , ny , nz (11.280)

L

where

nx = 1, 2, ..., ny,z = 0, 1, 2, ... (11.281)

74 This “thick lead” assumption is not of importance. Because of the relationship between the

¬‚uctuations in the density of states and the time scale for di¬using out of the sample, the result

will be the same for any kind of lead attachment [106].

11.6. Mesoscopic ¬‚uctuations 441

and we have assumed that the current leads are along the x-axis. Less than three

dimensions corresponds to neglecting the ny and nz . We therefore obtain from the

considered Di¬uson diagram the contribution to the conductance ¬‚uctuations75

2

e2

G±β Gγδ = cd δ±,γ δδ,β , (11.282)

D

4π

where the constant cd depends on the sample dimension. The summation in Eq.

(11.279) should, in accordance with the validity of the di¬usion regime, be restricted

to values satisfying n2 + n2 + n2 ¤ N , where N is of the order of (L/l)2 . However,

x y z

the sum converges rapidly and the constants cd are seen to be of order unity. The

dimensionality criterion is essentially the same as in the theory of weak localization,

as we shall show in the discussion below of the physical origin of the ¬‚uctuation

e¬ects. The important thing to notice is that the long-range nature of the Di¬uson

provides the L4 factor that makes the variance, average of the squared conductance,

independent of sample size (recall Eq. (6.57)). The diagram depicted in Figure 11.7

is only one of the two possible pairings of the current vertices, and we obtain an

additional contribution from the diagram where, say, current vertices γ and δ are

interchanged.

In addition to the contribution from the diagram in Figure 11.7 there is the other

possible singular Di¬uson contribution to the variance from the diagram depicted in

Figure 11.8.

Rp

r r1

r r1

p p

R R

Ap

A A

p p

± γ δ β

p

R

A

p

Figure 11.8 The other possible conductance ¬‚uctuation diagram.

75 Because of these inherent mesoscopic ¬‚uctuations, we realize that the conductance discussed in

the scaling theory of localization is the average conductance.

442 11. Disordered conductors

This diagram contributes the same amount as the one in Figure 11.7, but with a

di¬erent pairing of the current vertices. We note that the diagram in Figure 11.8

allows for only one assignment of current vertices.76

Reversing the direction in one of the loops gives rise to similar diagrams, but

now with the Cooperon appearing instead of the Di¬uson. Because the boundary

conditions on the Cooperon are the same as for the Di¬uson, in the absence of a

magnetic ¬eld, the Cooperon contributes an equal amount. For the total contribution

to the variance of the conductance, we therefore have (allowing for the spin degree

of freedom of the electron would quadruple the value) at zero temperature

2

e2

”G±β,γδ = cd (δ±γ δδβ + δ±δ δγβ + δ±,β δγ,δ ) . (11.283)

2π

The variance of the conductance at zero temperature, and for the chosen geometry

of a hypercube, is seen to be independent of size and dimension of the sample and

degree of disorder, and the conductance ¬‚uctuations appear in the metallic regime

described above to be universal.77

Since the average classical conductance is proportional to Ld’2 , Ohm™s law, we

¬nd that the relative variance, ”G G ’2 , is proportional to L4’2d . This result should

be contrasted with the behavior L’2d of thermodynamic ¬‚uctuations, compared with

which the quantum-interference-induced mesoscopic ¬‚uctuations are huge, re¬‚ecting

the absence of self-averaging.

The dominating role of the lowest eigenvalue in Eq. (11.279) indicates that meso-

scopic ¬‚uctuations, studied in situations with less-invasive probes than the current

leads necessary for studying conductance ¬‚uctuations, can be enhanced compared

to the universal value. In the case of the conductance ¬‚uctuations, the necessary

connection of the disordered region to the leads, which cut o¬ the singularity in the

Di¬uson by the lowest eigenvalue, nx = 1, re¬‚ecting the fact that because of the

physical boundary conditions at the interface between sample and leads, the max-

imal time for quantum interference processes to occur uninterrupted is the time it

takes the electron to di¬use across the sample, L2 /D0 . When considering other ways

of observing mesoscopic ¬‚uctuations, the way of observation will in turn introduce

the destruction of phase coherence necessary for rendering the ¬‚uctuations ¬nite.

In order to understand the origin of the conductance ¬‚uctuations, we note that,

just as the conductance essentially is given by the probability for di¬using between

points in a sample, the variance is likewise the product of two such probabilities.

When we perform the impurity average, certain of the quantum interference terms

will not be averaged away, since certain pairs of paths are coherent. This is similar to

the case of coherence involved in the weak-localization e¬ect, but in the present case

of the variance of quite a di¬erent nature. For example, the quantum interference

76 Thecontribution from the diagram in Figure 11.7 can, through the Einstein relation, be ascribed

to ¬‚uctuations in the di¬usion constant, whereas the diagram in Figure 11.8 gives the contribution

from the ¬‚uctuations in the density of states, the two types of ¬‚uctuation being independent [107].

77 However, for a non-cubic sample, the variance will be geometry dependent [108, 109].

11.6. Mesoscopic ¬‚uctuations 443

process described by the diagram in Figure 11.7 is depicted in Figure 11.9, where the

solid line corresponds to the outer conductance loop, and the dashed line corresponds

to the inner conductance loop. The wavy portion of the lines corresponds to the long-

range di¬usion process.

R

R—

R

R—

Figure 11.9 Statistical correlation described by the diagram in Figure 11.7.

When one takes the impurity average of the variance, the quantum interference terms

can pair up for each di¬usive path in the random potential, but now they correspond

to amplitudes for propagation in di¬erent samples. The diagrams for the variance,

therefore, do not describe any physical quantum interference process, since we are not

describing a probability but a product of probabilities. The variance gives the statis-

tical correlation between amplitudes in di¬erent samples. The interference process

corresponding to the diagram in Figure 11.8 is likewise depicted in Figure 11.10.

444 11. Disordered conductors

R

R—

R—

R

Figure 11.10 Statistical correlation described by the diagram in Figure 11.8.

When a speci¬c mesoscopic sample is considered, no impurity average is e¬ec-

tively performed as in the macroscopic case. The quantum interference terms in the

conductance, which for a macroscopic sample average to zero if we neglect the weak-

localization e¬ect, are therefore responsible for the mesoscopic ¬‚uctuations. In the

weak-disorder regime the conductivity (or equivalently the di¬usivity by Einstein™s

relation) is speci¬ed by the probability for the particle to propagate between points

in space. According to Eq. (11.95)

|Ac Ac | cos(φc ’ φc )

P = Pcl + 2 (11.284)

c,c

as

1

Ac = |Ac | eiφc , φc = S[xc (t)] , (11.285)

where |Ac | speci¬es the probability for the classical path c, and its phase is speci¬ed

by the action. When the points in space in questions are farther apart than the mean

free path, the ensemble average of the quantum interference term in the probability

vanishes. The weak localization can be neglected because for random phases we

have cos(φc ’ φc ) imp = 0. However, for the mean square of the probability, we

encounter cos2 (φc ’ φc ) imp = 1/2, and obtain

|Ac | |Ac | .

P2 2

= P +2 (11.286)

imp imp

c,c

Because of quantum interference there is thus a di¬erence between P 2 and

imp

11.6. Mesoscopic ¬‚uctuations 445

P 2 resulting in mesoscopic ¬‚uctuations. Since the e¬ect is determined by the

imp

phases of paths, it is nonlocal.

The result in Eq. (11.283) is valid in the metallic regime, where the average

conductance is larger than e2 / . To go beyond the metallic regime would neces-

sitate introducing the quantum corrections to di¬usion, the ¬rst of which is the

weak-localization type, which diagrammatically corresponds to inserting Cooperons

in between Di¬usons. Such an analysis is necessary for a study of the ¬‚uctuations in

the strongly disordered regime, as performed in reference [84].

The Di¬uson and Cooperon in the conductance ¬‚uctuation diagrams do not de-

scribe di¬usion and return probability, respectively, in a given sample, but quantum-

statistical correlations between motion in di¬erent samples, i.e. di¬erent impurity

con¬gurations, as each conductance loop in the Figures 11.7 and 11.8 corresponds

to di¬erent samples. In order to stress this important distinction, we shall in the

following mark with a tilde the Di¬usons and Cooperons appearing in ¬‚uctuation

diagrams.

We now assess the e¬ects of ¬nite temperature on the conductance ¬‚uctuations.

Besides the explicit temperature dependence due to the Fermi functions appearing

in Eq. (11.272), the ladder diagrams will be modi¬ed by interaction e¬ects. The

presence of the Fermi functions corresponds to an energy average over the thermal

layer near the Fermi surface, and through the energy dependence of the Di¬uson

and Cooperon introduces the temperature-dependent length scale LT = D0 /kT .

Since the loops in the ¬‚uctuation diagrams correspond to di¬erent conductivity mea-

surements, i.e. di¬erent samples, interaction lines (for example caused by electron“

phonon or electron“electron interaction) are not allowed to connect the loops in a

¬‚uctuation diagram. The di¬usion pole of the Di¬uson appearing in a ¬‚uctuation

diagram is therefore not immune to interaction e¬ects. This was only the case when

the Di¬uson describes di¬usion within a sample, since then the di¬usion pole is a con-

sequence of particle conservation and therefore una¬ected by interaction e¬ects. The

consequence is that, just as in the case for the Cooperon, inelastic scattering will lead

to a cut-o¬ given by the phase-breaking rate 1/„• . In short, the temperature e¬ects

will therefore ensure that up to the length scale of the order of the phase-coherence

length, the conductance ¬‚uctuations are determined by the zero-temperature expres-

sion, and beyond this scale the conductance of such phase-incoherent volumes add

as in the classical case.78 A sample is therefore said to be mesoscopic when its size

is in between the microscopic scale, set by the mean free path, and the macroscopic

scale, set by the phase-coherence length, l < L < L• . A sample is therefore self-

averaging only with respect to the impurity scattering for samples of size larger than

the phase-coherence length.79 A sample will therefore exhibit the weak-localization

e¬ect only when its size is much larger than the phase-coherence length but much

smaller than the localization length L• < L < ξ.

An important way to reveal the conductance ¬‚uctuations experimentally is to

measure the magneto-resistance of a mesoscopic sample. To study the ¬‚uctuation

e¬ects in magnetic ¬elds, we must study the dependence of the variance on the

78 Forexample for a wire we have g(L) = g(L• ) L/L• .

79 Theconductance entering the scaling theory of localization is thus assumed averaged over phase-

incoherent volumes.

446 11. Disordered conductors

magnetic ¬elds ”G±β (B+ , B’ ) , where B+ is the sum and B’ is the di¬erence in

the magnetic ¬elds in¬‚uencing the outer and inner loops. Since the conductance

loops can correspond to samples placed in di¬erent ¬eld strengths, the di¬usion pole

appearing in a ¬‚uctuation diagram will not be immune to the presence of magnetic

¬elds, as in the case when the Di¬uson describes di¬usion within a given sample,

since particle conservation is, of course, una¬ected by the presence of a magnetic ¬eld.

According to the low-¬eld prescription for inclusion of magnetic ¬elds, Eq. (11.105),

we get for the Di¬uson

e 1

˜

(’i∇x ’ δ(x ’ x ) ,

A’ (x))2 + 1/„•

D0 D(x, x ) = (11.287)

„

where A’ is the vector potential corresponding to the di¬erence in magnetic ¬elds,

B’ = ∇x — A’ , and we have introduced the phase-breaking rate in view of the

above consideration. In the case of the Di¬uson, the magnetic ¬eld induced phases

subtract, accounting for the appearance of the di¬erence of the vector potentials A’ .

For the case of the Cooperon, the two phases add, and we obtain

e 1

˜

(’i∇x ’ δ(x ’ x ) ,

A+ (x))2 + 1/„•

D0 C(x, x ) = (11.288)

„

where A+ is the vector potential corresponding to the sum of the ¬elds, B+ =

∇ — A+ .

The magneto-¬ngerprint of a given sample, i.e. the dependence of its conductance

on an external magnetic will show an erratic pattern with a given peak to valley ratio

and a correlation ¬eld strength Bc . This, however, is not immediately the information

we obtain by calculating the variance

[G±β (B1 ) ’ G±β (B1 ) ][Gγδ (B2 ) ’ Gγδ (B2 ) ] , (11.289)

”G±β,γδ (B+ , B’ ) =

where B1 is the ¬eld in, say, the inner loop, B1 = (B+ + B’ )/2, and B2 is the

¬eld in the outer loop, B2 = (B+ ’ B’ )/2. In the variance, the magnetic ¬elds are

¬xed in the two samples, and we are averaging over di¬erent impurity con¬gurations,

thus describing a situation in which the actual impurity con¬guration is changed, a

hardly controllable endeavor from an experimental point of view. However, if the

magneto-conductance of a given sample, G(B), varies randomly with magnetic ¬eld,

the two types of average “ one with respect to magnetic ¬eld and the other with

respect to impurity con¬guration “ are equivalent, and the characteristics of the

magneto-¬ngerprint can be extracted from the correlation function in Eq. (11.289).

The physical reason for the validity of such an ergodic hypothesis [110, 111], that

changing magnetic ¬eld is equivalent to changing impurity con¬guration, is that

since the electronic motion in the sample is quantum mechanically coherent the wave

function pattern is sensitive to the position of all the impurities in the sample, just

as the presence of the magnetic ¬eld is felt throughout the sample by the electron.80

The extreme sensitivity to impurity con¬guration is also witnessed by the fact that

changing the position of one impurity by an atomic distance, 1/kF , is equivalent to

shifting all the impurities by arbitrary amounts, i.e. to create a completely di¬erent

sample [113, 114].

80 The validity of the ergodic hypothesis has been substantiated in reference [112].

11.6. Mesoscopic ¬‚uctuations 447

The ergodic hypothesis can be elucidated by the following consideration. In the

mean square of the probability for propagating between two points in space we en-

counter the correlation function

cos(φc (B1 ) ’ φc (B1 )) cos(φc (B2 ) ’ φc (B2 )) (11.290)

imp

where (φc (B) ’ φc (B)) depends on the phases picked up due to the magnetic ¬eld,

i.e. the ¬‚ux through the area enclosed by the trajectories c and c . When the

magnetic ¬eld B1 changes its value to B2 (where the correlation function equals one

half), the phase factor changes by 2π times the ¬‚ux through the area enclosed by

the trajectories c and c in units of the ¬‚ux quantum. This change, however, is

equivalent to what happens when changing to a di¬erent impurity con¬guration for

¬xed magnetic ¬eld, i.e. the quantity we calculate.81

In order to calculate the variance in Eq. (11.289) we must solve Eq. (11.287) and

Eq. (11.288) with the appropriate mixed boundary value conditions in the presence of

magnetic ¬elds, and insert the solutions into contributions like that in Eq. (11.276).

However, determination of the characteristic correlations of the aperiodic magneto-

conductance ¬‚uctuations can be done by inspection of Eq. (11.287) and Eq. (11.288).

The correlation ¬eld Bc is determined by the sample-to-sample change in the mag-