Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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10 Fluctuation Phenomena in Superconductors 379

Fig. 10.2. Examples of diagrams for the

ﬂuctuation contribution to b

is sufficient to sum the relatively simple “parquet”

type diagrammatic series. Such a summation results

in the substitution of the “bare” vertex b by some

effective interaction



b, which diminishes and tends

to zero when the temperature approaches the tran-

sition point. Such a method was originally worked

out in quantum field theory [15–17]. For the prob-

lem of a phase transition such a summation was first

accomplished in [18].

Instead of a direct summation of the diagrams it

is more convenient and physically obvious to use the

method of the renormalization group. In the case

of quantum field theory this method was known

long ago [19,20]. For phase transition theory it was

proposed in [18,21], but the most simple and evi-

dent formulation was presented by Wilson [22]. The

idea of the renormalization group method consists

in separating the functional integration over “fast”

(

|k|

)and“slow”(

|k|<

)ﬂuctuationmodes.Ifthe

cut off  is large enough the fast mode contribution

is small and the integration over them is Gaussian.

After the first integration over fast modes the func-

tional obtained depends on the slow ones only. They

can, in turn, be divided into slow (|k| < 

)andfast

(

< |k| < ), and the procedure can be repeated.

Moving step by step ahead in this way one can calcu-

late the complete partition function.

As an example of the first step of renormalization,

the partition functioncalculationbelow T

can be re-

called. There we separated the order parameter into

the space-independent part



¦ (“slow”mode) and the

ﬂuctuation part (r) (“fast” mode), which was be-

lieved to be small in magnitude. Being in the GL re-

gion it was enough to average over the fast variables

just once,while in the critical region therenormaliza-

tion procedure requires subsequent approximations.

The cornerstone of the method consists in the fact

that in the critical region at any subsequent step the

free energy functional has the same form.ForD close

to 4 this form coincides with the initial free energy

GL functional butwith the coefficientsa



and b



de-

pending on .We will perform these calculations by

the method of mathematical induction. Let us sup-

pose that after the (n − 1)-st step the free energy

functional has the form:

F [





n−1

]=F

N,

n−1







n−1





n−1



n−1





n−1

|∇





n−1



. (10.40)

Writing





n−1

in the form





n−1





and

choosing 

close enough to 

n−1

it is possible to

make



so small that one can restrict the func-

tional to the quadratic terms in



only and per-

form the Gaussian integration in complete analogy

with (10.34). The important property of the spaces

with dimensionalities close to 4 is the possibility to

choose 

 

n−1

and still to have









.In

this case





canbetakenasanindependentcoor-

dinate, and one can use the result directly following

from (10.34):

380 A.I. Larkin and A.A.Varlamov

F [





]=F

N,

n−1















|∇







−





|k|

n−1

T

(3b







+ a



)

+ln

T







+ a



.(10.41)

Expanding the last term in (10.41) in a series in





one can get for F[





] the same expression (10.40)

with the substitution of 

n−1

→ 

. From (10.41) it

follows that

n,

= F

n,

n−1

− T





|k|

n−1

T









= a



n−1

+2T





|k|

n−1









, (10.42)



= b



n−1

−5T





|k|

n−1









Passing to a continuous variable 

→  one can

rewrite these recursion equations as the set of differ-

ential equations:

@F()

@

=−T



D−1

T



a()+





, (10.43)

⎧

⎪

⎨

⎪

⎩

@a()

@

=−2T

b()

D−1



a()+





@b()

@

=5T

()

D−1



a()+





. (10.44)

These renormalization group equationsareevidently

valid for small enough  only, where the transition

from discrete tocontinuousvariablesisjustified.This

means at least



/4m  T

(D)

(10.45)

in order to move away from the first approximation.

Let us recall that in the framework of the Landau

theory of phase transitions the coefficienta(T

)=0

at the transition point and this can be considered as

the MFA definition of the critical temperature T

The same statement for the function a = a()in

the framework of the renormalization group method

canbewrittenasthea(T

,  ∼ 

−1

) = 0. With the

decrease of  the effect of critical ﬂuctuations is in-

creasingly taken into account and the renormalized

value of the critical temperature decreases, being de-

fined by the equation: a(T

(), ) = 0. Finally, after

the application of the complete renormalization pro-

cedure, one can define the real critical temperature

, shifted down with respect to T

due to the effect

of ﬂuctuations, from the equation:

a(T

,  =0)=0. (10.46)

It is easy to find this shift in the first approximation.

Indeed, let us integrate the first equation of (10.44)

over  in limits [0, 

−1

]. The main contribution to

the integral will be determined by the region where

a() 



.Being far from the criticalpoint one can

assume that the coefficient b =const. and then

a(

−1

)=˛ıT

a(

−1

)



a(0)

da =−8mT

1/





D−3

d .

For the 3D case this gives the shift of the criticaltem-

perature ıT

due to ﬂuctuations

In the same way we can also analyze the shift of the critical temperature in the 2D case and obtain:

ıT

(2)

=−2Gi

(2)

4Gi

(2)

. (10.47)

As we will show below both 3D and 2D results for ıT

coincide with those obtained by the analysis of the effect of

ﬂuctuations on superconducting density in the perturbation approach.

10 Fluctuation Phenomena in Superconductors 381

ıT

(3)

∼ −

2mb

˛

=−

2T



(10.48)

=−

7(3)

16

 T



=−





(3)

Let us comeback to study of properties of the system

of equations (10.44).Onecan find its partial solution

at T = T

in the form:

a(T

, )=

4−D

4m[5 + (4 − D)]



, (10.49)

b(T

, )=

16m

T

4−D

[5 + (4 − D)]



4−D

These power solutions are correct in the domain of

validity of the system (10.44) itself, i.e. for small

enough  defined by the condition (10.45). Nev-

ertheless, in a space of dimensionality close to 4

(D =4−", "  1) it is possible to extend their valid-

ity up to the GL region and to observe their crossover

to the GL results: a(T

)=0;b(T

)=b

=const. In-

deed, in this case, due to proportionality of a()to

" → 0, one can omit it in the denominators of the

system (10.44) and to write down the solution for

b() in the form

−1

, )=b

−1

80m

(4 − D)

T

(

D−4

− 

4−D

) .

(10.50)

We have chosen the constant of integration as b

−1

−

80m

(4−D)

T



4−D

in order to match the renormaliza-

tion group and GL solutions at the value of  =



max

∼ 

−1

. Now let us pass to study of the func-

tion a(T, ) for the same interesting case of space

dimensionality D → 4 for temperatures slightly dif-

ferent (but still close enough) from T

, where one

can write

a(T, )=a(T

, )+˛(T

, )T

 .

The firsttermonthe righthand sideisdetermined by

(10.49).In order to determine ˛(T

, ) let us expand

the first equation in (10.44) in terms of 

@˛(T

, )

@

=2T

b(T

, )

D−1



a(T

, )+





˛(T

, ) .

(10.51)

For



/4m  ˛(T

, )T

 (10.52)

we can again use the solution (10.50) for b(T

, )

and omit a(T

, ) in the denominator of (10.51).

The constant of integration,appearing in the process

of solution of (10.51) is chosen in accordance with

the condition that for  = 

max

∼ 

−1

we match

˛(T

, )=˛(T

, 

−1

)=˛

with the GL theory:

˛(T

, )=˛



80m

(4 − D)

T

(

D−4

− 

4−D

)



−2/5

(10.53)

The condition (10.52) can be written as   

−1

(T),

where  (T) is the generalized coherence length, de-

termined by the equation:



−2

(T)=4m˛



, 

−1

(T)



 .

Such a definition is valid at any temperature. For

example, far enough from the critical point, in the

GL region, ˛



, 

−1

(T)



= ˛

and one reproduces

the result (10.3). Vice versa, in the critical region

the main contribution on the right hand side of the

(10.53)results from the second term containing 

D−4

so, putting 

−1

(T)=, one can rewrite the self-

consistent equation for (T) and get

(T)=(4m)

(1−D)/2

4−D

20b

T

√



−

, (10.54)

where 2 = [1 − (4 − D)/5]

−1

. As has already been

mentioned, strictly speaking this result was carried

out for " =4−D  1, so it is confident up to the

first in " expansion only:  =1/2+"/10. Neverthe-

less, extending it to " =1(D = 3) one can obtain



=3/5.

Let us pass to the calculation of the critical expo-

nent of the heat capacity in the immediate vicinityof

the transition. Forthis purpose one can calculate the

second derivative of equation (10.43) with respect

to :

@C()

@

= T





D−1

()



˛()T

 +





. (10.55)

The heat capacity renormalized by ﬂuctuations has

the value C( = 0), which is the result of integration

over all ﬂuctuationdegrees of freedom.Carrying out

382 A.I. Larkin and A.A.Varlamov

the integrationof (10.55)over all   

−1

one can di-

vide the domain of integration on the right hand side

in two:   

−1

(T)and

−1

(T)    

−1

.Inthecal-

culation of the integral over the region   

−1

(T)

the inequality ˛()T

 



holds, and the func-

tion ˛() can be omitted in the denominator. In the

numerator of (10.55) one can use for ˛()theso-

lution (10.53). In the region   

−1

(T)onehasto

use the partial solution (10.49) for˛() and can find

that the contribution of this domain has the same

singularity as that from the region   

−1

(T), but

with a coefficient proportional to (4−D)

= "

,hence

negligible in our approximation. The result is

C( =0)=˛



(4mT)







(4 − D)



4/5

4−D



4−D

(T) .

Substituting the expression for (T) one can finally

find

C =2

12/5



5

(4 − D)





−˛

, (10.56)

confirmingthe validity of the scaling hypothesis and

the relation(10.39).Thecriticalexponentin (10.56) is

˛ =

(4 − D)

10[1 − (4 − D)/5]

≈ "/10 .

Onecanseethat,generallyspeaking,thecriticalex-

ponents  and ˛ appear in the form of series in pow-

ers of ".Morecumbersome calculations permit find-

ing the next approximations for them in " =4−D.

Nevertheless it is worth mentioning that even the

first approximation, giving 

=3/5and˛

=1/10

for " = 1, is already weakly affected by the following

steps of the expansion in powers of " [23].

One can notice that the exercise performed in this

section has more of an academic than a practical

character.Indeed, the results obtained turn out to be

applicableto the analysis of thecriticalregion ofa3D

superconductor only if Gi is so small that the theo-

retical predictionsare hardly experimentally observ-

able. Nevertheless, we demonstrated the RG method,

which helps one to see the complete picture of the

ﬂuctuations manifestation in the vicinity of the crit-

ical temperature.

10.2.4 Fluctuation Diamagnetism

Qualitative Preliminaries

In this section we discuss the effectof ﬂuctuations on

the magnetization and the susceptibility of a super-

conductor above the transition temperature. Being

the precursor effect for the Meissner diamagnetism,

the ﬂuctuation induced magnetic susceptibility has

to be a small correction with respect to the diamag-

netism of a superconductor,butit can be comparable

to or even exceed the value of the normal metal dia-

magnetic or paramagnetic susceptibility and can be

easily measured experimentally.As was already men-

tioned in the Introduction the temperature depen-

dence of the ﬂuctuation induced diamagnetic sus-

ceptibility can be qualitatively analyzed on the basis

of the Langevinformula,butsome precautionsin the

case of low dimensional samples have to be taken.

As regards the 3D case we would like to mention

here that expression (10.12), presented in terms of

(T), has a wider region of applicability than the

GL one. Namely, the scaling arguments are also valid

for diamagnetic susceptibility and one can write the

general relation



(3)

∼ −e

T(T) ∼ −



−1/2

1,   Gi







1/2−

,   Gi ,

(10.57)

which is also valid in the region of critical ﬂuctua-

tions in the immediate vicinity of the transition tem-

perature. Here, in order to define the scale of ﬂuc-

tuation effects, we have introduced the Pauli para-

magnetic susceptibility 

= e

/4

.Moreover,

the Langevin formula permits us to extend the esti-

mation of the ﬂuctuation diamagnetic effect to the

other side beyond the GL region: to high tempera-

tures T  T

. The coherence length far from the

transition becomes a slow function of temperature.

In a clean superconductor, farfrom T

, (T) ∼ v

/T,

so one can write



(3c)

(T  T

) ∼ −e

T(T) ∼ −

(10.58)

and see that the ﬂuctuation diamagnetism turns out

to be of the order of the Pauli paramagnetism even

10 Fluctuation Phenomena in Superconductors 383

far fromthetransition.Moreprecisemicroscopic cal-

culations of 

(3)

(T  T

) lead to the appearance of

(T/T

) in the denominator of (10.58).

In the 2D case, (10.12) is applicable for the esti-

mation of 

(2)

in the case when the magnetic field

is applied perpendicular to the plane, permitting 2D

rotations of ﬂuctuation Cooper pairs in it:



(2c)

(T) ∼ e

R

∼e

T

(T) ∼ −

T − T

(10.59)

This result is valid for a wide range of temperatures

and can exceed the Pauli paramagnetism by factor

even far from the critical point (we consider the

clean case here).

For a thin film (d  (T)) perpendicular to the

magnetic field the ﬂuctuation Cooper pairs behave

likeeffective2Drotators,and(10.12)canstill beused,

though one has to take into account that the suscep-

tibilityin thiscase is calculated per unit square of the

film. So for the realistic case (from the experimen-

tal point of view) of the dirty film, one just has to

use in (10.59)the expression (10.6) forthe coherence

length:



(2d)

∼



(T) ∼ −





T − T

. (10.60)

Now let us discuss the important case of a layered

superconductor(for example,a high temperature su-

perconductor). It is usually supposed that the elec-

trons move freely in conducting planes separated by

a distance s. Their motion in the perpendicular di-

rection has a tunneling character, with effective en-

ergy J.The related velocity and coherence length can

be estimated as v

= @E(p)/@p

⊥

∼ J/p

⊥

∼ sJ and



z,(c)

∼ sJ /T for the clean case. In the dirty case the

anisotropy can be taken into account in the spirit of

(10.6), yielding 

z,(d)

∼

√

⊥

/T ∼ sJ

√

/T.

We start from the case of a weak magnetic field

applied perpendicular to layers. The effective area of

a rotating ﬂuctuation pair is 

()

(). The density

of Cooper pairs in the conducting layers (10.10) has

to be modified for the anisotropic case. Its isotropic

3D value is proportional to 1/(), which now has to

be read as ∼ 1/





()

(). The anisotropy of the

electron motion leads to a concentration of ﬂuctua-

tion Cooper pairs in the conducting layers,and hence

to an effectiveincrease of the Cooper pairs density of





()

()/

() times its isotropic value. This in-

crease is saturated when 

() reaches the interlayer

distance s, so finally the anisotropy factor appears in

the form





()

()/ max{s, 

()},andthesquare

root in its numerator is removed in the Langevin for-

mula (10.12), which is rewritten for this case as



(layer,⊥)

(, H → 0) ∼ −e



()

()

max{s, 

()}

. (10.61)

The existence of a crossover between the 2D and 3D

temperature regimes in this formula is evident: as

the temperature tends to T

the diamagnetic suscep-

tibility temperature dependence changes from 1/ to

√

. This happens when the reduced temperature

reaches its crossover value 

= r





(

) ∼ s



.The

anisotropy parameter

r =

4

(0)

⎧

⎪

⎨

⎪

⎩



, T  1

7(3)

8

, T  1

(10.62)

plays an important role in the theory of layered su-

perconductors.

It isinterestingtonotethatthisintrinsiccrossover,

related to the spectrum anisotropy, has an oppo-

site character to the geometric crossover that occurs

in thick enough films when (T) reaches d.Inthe

latter case the characteristic 3D 1/

√

-dependence

takes place far enough from T

(where  (T)  d),

is changed to the 2D 1/ law (see (10.60)) in the

immediate vicinity of transition (where (T) 

d) [24]. It is worth mentioning that in a strongly

anisotropic layered superconductor the ﬂuctuation-

induced susceptibility may considerably exceed the

normal metal diamagentic and paramagnetic effects

even relatively far from T

[25,26].

Let us consider a magnetic field applied along the

layers. First it is necessary to mention that the ﬂuc-

tuation diamagnetic effect disappears in the limit

J ∼ 

→ 0.Indeed,fortheformationof a circulating

We use here a definition of r following from microscopic theory (see Sect. 10.6).

384 A.I. Larkin and A.A.Varlamov

current it is necessary to tunnel twice, so



(layer,)

∼ −e



max{s, 

(T)}

∼ −





J/T

√

 max{

√

, J/T}

In the general case of an anisotropic superconductor,

choosing the z-axis along the direction of magnetic

field H, the following extrapolation of the results ob-

tained may be written:

 ∼ − e

T (10.63)



()

()

max{a, 

()}max{b, 

()}max{s, 

()}

This general formula is useful for the analysis of the

ﬂuctuation diamagnetism of anisotropic supercon-

ductors or samples of some specific shape: granular,

quasi-1D,quasi-2D,and3D.It is also applicable to the

case of a thin film (d  

()) placed perpendicular

to the magnetic field: it is enough to replace 

()by

d in (10.64). Nevertheless, (10.64) cannot be applied

to the cases of thin films in parallel fields, wires and

granules.In those cases theLangevin formula (10.12)

can still be used with the replacement of

→ d



(D)

∼ −







2−D

D−1

∼ 

D/2−1

The magnetic field dependence of the ﬂuctuation

part of free energy in these cases is reduced only

to account for the quadratic shift of the critical tem-

perature versus magnetic field.

For 3D systems or in the case of a film in a per-

pendicular magnetic field the critical temperature

depends on H linearly, while the magnetic field de-

pendent part of the free energy for H  H

∗

(−)(the

line H

∗

(−) is mirror-symmetric to the H

()with

respect to y-axis passing through T = T

) is propor-

tional to H

. This is why the magnetic susceptibility

is determined by (10.64) for weak enough magnetic

fields H  ¥

/[

()

()] = H

()  H

(0) only.

In the vicinity of T

these fields are small enough.

Zero-Dimensional Diamagnetic Suscept ibility

For quantitative analysis of the ﬂuctuation diamag-

netism we start by writing down the GL functional

for the free energy (see (10.16)) in the presence of

the magnetic field

F [¦ (r)] = F





a|¦ (r)|

|¦ (r)|

(

−i∇−2eA

)

¦ (r)|

8

−

H · B

4



(10.64)

where A is vector potential.As long as ﬂuctuation ef-

fects are comparatively small, the average magnetic

field in the metal B may be assumed to be equal to

the external field H. Thus we omit the last two terms

in (10.64) (see later on).

The ﬂuctuation contribution to the diamag-

netic susceptibility in the simplest case of a “zero-

dimensional” superconductor (spherical supercon-

ducting granule of diameter d  ()) was consid-

ered by V. Shmidt [11]. In this case the order param-

eter does not depend on the space variables and the

free energy can be calculated exactly for all temper-

atures including the critical region in the same way

aswasdoneforthecaseoftheheatcapacityinthe

absence of a magnetic field. Formally the effect of a

magnetic field in this case is reduced to the renor-

malization of the coefficient a, or, in other words, to

the suppression of the critical temperature. This is

why one can use the same formula (10.23) for the

partition function with the critical temperature T

shifted by magnetic field as:

(H)=T

(0)



1−

4



A





. (10.65)

Here ¥



is the magnetic ﬂux quantum and ...

means the averaging over the sample volume.

Such a trivial dependence of the properties of 0D

samples on magnetic field immediately allows one to

understand its effect on the heat capacity of a gran-

ular sample. Indeed, with the growth of the field the

Let us stress the difference between the H

shift of the critical temperature for a zero-dimensional granule and the

linear shift in the case of bulk material.

10 Fluctuation Phenomena in Superconductors 385

temperature dependence of the heat capacity pre-

sented in Fig. 10.1 just moves in the direction of

lower temperatures.

In the GL region Gi

(0)

  one can write the

asymptotic expression (10.23) for the free energy:

(0)

(, H)=−T ln





 +

4



A





In the case of a spherical particle the relation A

 =

(

H × r)

 =

can be used in full analogy

with the calculation of the moment of inertia of a

solid sphere.In this way an expression for the0D ﬂuc-

tuation magnetization valid forall fieldsH  H

(0)

can be found:

(0)

(, H)=−

(0)

(, H)

=−T

6

5¥



 +





10¥



H . (10.66)

Onecanseethattheﬂuctuationmagnetizationturns

out to be negative and linear up to some crossover

field, which can be called the temperature dependent

upper critical field of the granule H

c2(0)

() ∼

d()



(0)

√

 at which it reaches a minimum.At higher

fields H

c2(0)

()  H  H

(0) the ﬂuctuation mag-

netization of the 0D granule decreases as 1/H.In

the weak field region H  H

c2(0)

() the diamagnetic

susceptibility is:



(0)

(, H)=−

6T

5¥



≈ − · 10











which coincides with our previous estimate in its

temperature dependence but the numerical factor

found is very large.Let us underline that the temper-

ature dependence of the 0D ﬂuctuation diamagnetic

susceptibility turns out to be less singular than the

0D heat capacity correction: 

−1

instead of 

−2

The expression for the ﬂuctuation part of free en-

ergy (10.28) is also applicable to the cases of a wire or

a film placed in a parallel field; as was already men-

tioned above all its dependence on magnetic field

is manifested by the shift of the critical tempera-

ture (10.65). In the case of the wire in a parallel field

one has to choose the gauge of the vector-potential

A =

H × r yielding

(wire,)

(the calcula-

tion of this average is analogous to that of the mo-

ment of inertia of a solid sphere). For a wire in a

perpendicular field, or a film in a parallel field, the

gauge has to be chosen in the form A =(0, Hx, 0) (to

avoid the appearance of currents perpendicular to

surface). One can find

(wire,⊥)

for a wire

and

(film,)

for a film.

Calculating the second derivative of (10.28) with

the appropriate magnetic field dependencies of the

critical temperature one can find the following ex-

pressions for the diamagnetic susceptibility:



(D)

()=−2

T

(10.67)

× 

⎧

⎪

⎨

⎪

⎩

√



, wire in parallel field

√



, wire in perpendicular field

3



, film in parallel field.

GL Treatment of Fluctuation Magnetization

Let us analyze quantitatively, on the basis of the GL

functional, the temperature and field dependencies

of the ﬂuctuation magnetization. We will carry on

the discussion for a layered superconductor. As has

already been mentioned this system has great prac-

tical importance because of its direct applicability to

high temperature superconductors,wherethe ﬂuctu-

ation effects are very noticeable. Moreover, the gen-

eral results obtained will allow us to analyze 3D and

2D situations as limiting cases. The effects of a mag-

netic field are more pronounced for perpendicular

orientation, so let us consider this case first.

The generalization of the GL functional for a lay-

ered superconductor (Lawrence–Doniach (LD) func-

tional [27]) in a perpendicular magnetic field can be

written as

386 A.I. Larkin and A.A.Varlamov

[

]







a |¦

|¦

(10.68)





∇



−2ieA





|¦



+ J |¦

l+1

− ¦



where ¦

is the order parameter of the l-th supercon-

ducting layer and the phenomenological constant J

is proportional to the Josephson coupling between

adjacent planes. The gauge with A

=0ischosen

in (10.69). In the immediate vicinity of T

the LD

functional is reduced to the GL one with the effec-

tive mass M =(4J s

)

−1

along c-direction, where s

is the inter-layer spacing. One can relate the value

of J to the coherence length along the c-direction:

J =2˛T



. Since we are dealing with the GL re-

gion the fourth order term in (10.69) can be omitted.

As it is well known the Landau representation is

the most appropriate one for problems related with

the motion of a charged particle in a uniform mag-

netic field. Basic wave functionscan be chosen as the

products of a plane wave propagating along the mag-

netic field direction and a Landau state wave func-

tions 

(r). Let us expand the order parameter ¦

(r)

on the basis of these eigenfunctions:

(r)=



n,k



(r)exp(ik

l). (10.69)

The factor SH/¥

takes into account the degeneracy

of Landau state n while k

is the momentum compo-

nent along the direction of the magnetic field.Substi-

tuting this expansion into (10.69)onecan find the LD

free energy as a functional of the ¦

n,k

coefficients:



n,k





n,k



˛T

 +

2m¥



n +



(10.70)

+ J

(

1−cos(k

)



|¦

n,k

In complete analogy with the case of an isotropic

spectrum the functional integral over the order pa-

rameter configurations ¦

n,k

in the partition func-

tion can be reduced to a product of ordinary Gaus-

sian integrals, and the ﬂuctuation part of the free

energy in a magnetic field takes the form:

F (, H)=−



n,k

ln (10.71)

T

˛T

 +

2m¥



n +



+ J

(

1−cos(k

)

The sum in (10.72) is evidently divergent and in or-

der to regularize it we introduce a formal cut-off pa-

rameter n

, the number of the last Landau level, at

which the summation is interrupted. Its value can

be evaluated by appealing to the restrictions of the

GL theory, which breaks down for short wave length

ﬂuctuations and is valid for k  k

max

,where

max

/4m ∼ ˛T . (10.72)

In terms of the Landau level summation this condi-

tioncanberewrittenas:

∼ ˛T/!

∼ ˛Tm¥

∼ ¥







∼ H

(0)/H .

(10.73)

The formal divergence of the GL free energy does not

effect on such observable properties as the heat ca-

pacity and the diamagnetic susceptibility. Being the

second derivatives of the free energy these physical

values are well defined, the momentum (or Landau

levels) summation converges well for them. Never-

theless, the problem of regularization arises for ex-

ample in calculation of the 3D ﬂuctuation magne-

tization. Being only the first derivative of the free

energy it turns out to be also divergent and must be

cut off.Let us stress that the ultraviolet divergence of

the ﬂuctuation part of the GL free energy is the re-

sult of the limitedness of the GL functionalapproach,

unapplicable to the description of short wavelength

ﬂuctuations. This problem does not appear at all in

the consistent microscopic theory, which correctly

accounts for the short wavelength ﬂuctuations and

does not require any renormalization procedure.

In the limit of weak fields one can carry out the

summation over the Landau states by means of the

Euler–Maclaurin transformation

10 Fluctuation Phenomena in Superconductors 387



n=0

f (n)=

N+1/2



−1/2

f (n)dn

−





(N +1/2) − f



(−1/2)



and obtain

F(, H)=F(, 0) +

STH

24m¥

/s



−/s

N sdk

2

(10.74)



˛T

+J

(

1−cos(k

)



Here N is the total number of layers. After the mo-

mentum integration one gets:

F(, H)=F(, 0) +

VH

24m˛s¥

√

( + r)

with the anisotropy parameter defined as

r =

˛T

4

(0)

. (10.75)

The magnetic susceptibility in a weak field turns

out [28,29] to be



(layer,⊥)

=−

3s



√

( + r)

. (10.76)

These results confirm the qualitative estimation

(10.61), additionally providing the exact value of the

numerical coefficient and the temperature depen-

dence in the crossover region. In the limit r  

(10.76) transforms into the diamagnetic susceptibil-

ity of the 3D anisotropic superconductor [30].

For a film of thickness d the integral over k

in ex-

pression (10.75) has to be replaced by a summation

over the discrete k

and when 

(T)  d only the

term with k

= 0 has to be taken into account:



(film,⊥)

=−

3d





. (10.77)

Note that these formulas predict a nontrivial increase

of diamagnetic susceptibility for clean metals [29].

The usual statement that ﬂuctuations are most im-

portant in dirty superconductors with a short elec-

tronic mean free path does not hold in the particular

case of susceptibility,because here  turns out to be

in the numerator of the ﬂuctuation correction.

Now we will demonstrate that, besides the

crossovers in its temperature dependence, the ﬂuc-

tuation induced magnetization is also a nonlinear

function of magnetic field, and these nonlinearities,

different for various dimensionalities, take place at

relatively low fields. This manifestation of the non-

linear regime in ﬂuctuation magnetization, which

is strong in comparison with the expected scale of

(0), and hence, field-dependent ﬂuctuation sus-

ceptibility,wasthe subject of the intensive debates in

early 1970s [30–39] (see also the older excellent re-

view of W.J.Skocpol and M. Tinkham [40]) and after

the discovery of HTS [41–44] (see also the very re-

cent detailed essay of T. Mishonov and E. Penev [45]

and references therein). We will mainly follow here

the paper of Buzdin et al. [46], dealing with the ﬂuc-

tuation magnetization of a layered superconductor,

whichpermitsobserving in a unique way all varieties

of the crossover phenomena in temperature and the

magnetic field.

Let us go back to the general expression (10.72)

and evaluate it without taking the magnetic field to

small.The difficulty in dealing with it consists in the

divergence of the sum over Landau levels n.Thisdi-

vergence can be regularized (see [45,47]), but let us

observe that in order to calculate the magnetization

we must know the magnetic field dependent part of

the free energy only.A very convenient method to by-

passthedivergenceproblem[48]istocalculatethe

difference F(H)−F(0), turning the sum over Landau

states in F(, 0) into an integral and then, in turn,

turning this integral into a sum of integrals over the

Let us stress the difference between J and J in the two definitions (10.62) and (10.75) of the anisotropy parameter r.

The first one was introduced as the electron tunneling matrix element, while the second one enters in the LD functional

as the characteristic Josephson energy for the order parameter. Later on, in the framework of the microscopic theory, it

will be demonstrated that,in accordance with our qualitative definition, r ∼J

,whileJ turns out to be proportional to

too.In the dirty case it depends on the relaxation time of the electron scatteringon impurities: J ∼ ˛J

max{, 1/T}.

Hence both definitions (10.62), appearing in the qualitative consideration, and (10.75), following from the LD model,

are consistent.

388 A.I. Larkin and A.A.Varlamov

Fig. 10.3. Schematic representation of the dif-

ferent regimes for ﬂuctuation magnetization in

the (H, T)diagram.ThelineH

∗

(T) is mirror-

symmetric to the H

(T) line with respect to a

y-axis passing through T = T

.Thislinede-

fines the crossover between linear and nonlin-

ear behavior of the ﬂuctuation magnetization

above T

unit length intervals x ∈ [n −1/2, n +1/2]. Then

F(, 0) = − lim

H→0

/s



−/s

2

∞



n=0

1/2



−1/2

×ln

T

˛T

 +

2m¥



x + n +



(

1−cos(k

)

and by introducing the dimensionless variable

h =

(0)

, H

(0) = 2m˛T

/e = ¥

/2

(10.78)

one can write

F (, H)−F(, 0) = −

2s





−



n=0

1/2



−1/2

dx ln

(2n +1+2x)h + r/2(1 − cosz)+

(2n +1)h + r/2(1 − cosz)+

. (10.79)

Performing the integrations over z and x in (10.79)

and differentiating with respect to h we finally obtain

a very convenient general expression for the ﬂuctua-

tion magnetization in a layered superconductor:

M(, H)=

∞



n=0

n ln

'(R

+1)

'(R

)

(10.80)

+ln

'(R

+1)

'(R

+1/2)

−

n +1/2



+1/2)

− 

with '(x)=x +



− 

, R

= n + /2h +  and

 = r/2h. The sum in (10.81) converges as 1/n

and

it provides a volume magnetization expression that

can be compared with experiment.

Let us comment on the different crossovers in the

M(, H) field dependence analyzing the general for-

mula (10.81). Let us fix the temperature   r.In

this case the c-axis coherence length exceeds the in-

terlayer distance (

 s)andintheabsenceofa

magnetic field the ﬂuctuation Cooper pairs motion

has a 3D character. Supposing the magnetic field to

be not too high (h  r) we may perform an expan-

sion in (10.81) in 1/ and obtain

(3)

(  r, h  r)=

√

1/2



(10.81)





(

n +1

)



n +1+





1/2

−n



n +





1/2

−

(n +1+

2

)



n +





For weak fields (h  ) the magnetization grows lin-

early with magnetic field, justifying our preliminary

qualitative results. Nevertheless, this linear growth

is changed to the nonlinear 3D high field regime

M ∼

√

H already in the region of a relatively small

fields H

()  H (  h). The further increase of

magnetic field leads to the next crossover in the mag-

netization field dependence at h ∼ r. However the

expression (10.81) was obtained in the assumption

Let us recall that the exact definition of H

(0) contains the numerical coefficient A(0) (see footnote3).