Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors

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SECTION 1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY 11

which enables one to obtain the gauge-invariant equations. Here

is the magnetic

field's vector potential and e

is the charge of the carrier, represented by the wave

function . Thus the free energy density may be written in the form

Demanding the minimum for the total free energy

(

is the system’s volume), one can obtain the equation for Varying (1.46) by

gives

(

S is the metal’s surface). Because is arbitrary, we find from (1.47) an equation

for the order parameter

and also the boundary condition

(here n is a vector normal to the metal surface). The variation of (1.46) by A yields

the Maxwell equation

where the current

has a typical quantum-mechanical form [cf. (1.3)].

12 CHAPTER 1. BASIC EQUILIBRIUM PROPERTIES

Expressions (1.48) and (1.51) comprise the Ginzburg–Landau system of

equations describing the behavior of superconductors in a static magnetic field.

Presenting the complex function in the form

one can rewrite the expression for the current:

Here is the density of superconducting electrons in the Ginzburg–Landau

normalization, and the superconducting velocity v

is equal to

Putting B = curl A, one can easily prove that Eq. (1.53) coincides with the London

equation [ef. (1.2)]

Substituting into (1.55) the Maxwell equation

and taking into account that

we obtain the equation

1.2.2.

London Penetration Depth

Equation (1.58) subject to condition (1.57) describes the expulsion of a

magnetic field from superconductor’s interior (the Meissner effect). Let us consider

the distribution of a magnetic field in a superconductor near its surface, assuming

the latter to be a plane. The characteristic parameter, which has the dimension of a

length, in this situation is

SECTION 1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY 13

as may be seen from expression (1.58). In the case considered here, the field

distribution depends on one (say, x) coordinate only. Then

with the boundary condition

which follows from Eq. (1.57). From the expressions (1.61) and (1.57) one can

conclude that the vector of induction B in the depth of the superconductor has the

form

where is a tangential component of the external field. The characteristic length

(1.59) is called the London penetration depth.

1.2.3. Coherence Length

The Ginzburg-Landau set of equations has one more characteristic scale,

which has the dimensionality of the length. Its value [usually marked as may

be seen from Eq. (1.48), characterizes the scale of spatial evolution of the -func-

tion and is given by

The temperature dependence of in the vicinity of is found using the

formula (1.37):

We are able also to obtain the temperature dependence of Indeed,

in the vicinity of one can substitute Eq. (1.39) into (1.59) with the result:

14 CHAPTER 1. BASIC EQUILIBRIUM PROPERTIES

The ratio of these two characteristic lengths

is temperature independent and, as we will now see, is an important parameter of

a superconductor, defining its behavior in a magnetic field.

1.2.4. Sign of Surface Energy

Let us consider the surface energy of the flat boundary between normal and

superconducting phases, which may exist in a magnetic field. In the normal phase,

the free energy density, including the field energy, is equal to In the

region where the free energy density is [Eq. (1.42)]. Near the boundary

one must take into account the energy associated with the magnetization of a

superconductor

In the depth of the normal phase, the equation holds (the second of these

equations is also valid in the superconducting region, because

Thus the surface energy

taking into account Eq. (1.45), is equal to

[according to Eq. (1.41), In Eq. (1.69), the -function was

assumed to be real, because the term vanishes owing to the condition

The analogous term also disappears from Eq. (1.48) for the order parameter,

so Eq. (1.53) for the current acquires the form

It is expedient to use the variables

SECTION 1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY 15

Removing the bars above the symbols to simplify the notation, we present Eq. (1.48)

in the form

and write expression (1.69) as

Integrating the first term in (1.73) by parts, using the condition

one can reduce Eq. (1.73) to the form

Expression (1.74) vanishes if the integrand is identical to zero: . Since

and

must decrease with increasing then

From Eqs. (1.50) and (1.70) it follows that

Taking the derivative of (1.75) and introducing it into (1.76), we find

Substituting '(1.77) and A

(1.75) into Eq. (1.72) shows that Eq. (1.72) is fulfilled

identically at One can also see that the condition (1.75), which was used

earlier does not contradict the boundary conditions:

and at

Thus we arrive at a very important conclusion: at the solution of the

order parameter equation causes the surface energy to vanish (this criterion was

established numerically by Ginzburg and Landau

and proved analytically

by the

Sarma method; the alternative method we used here is by Lifshitz and Pitaevski

Generally, may have an arbitrary sign. To see this we will once again use the

expression (1.73), as well as the first integral of Eq. (1.72) subject to (1.76), which

has the form

The last identity here follows from Eq. (1.76) and boundary conditions

16 CHAPTER 1. BASIC EQUILIBRIUM PROPERTIES

As a result we find

Note that the second term in (1.79) is always negative, because the field

which penetrates into the superconductor, is always smaller than the critical

one: The first term in Eq. (1.79) is always positive, but its value and

consequently the sign of are determined by the magnitude of [see Eq. (1.66)].

Superconductors in which

are called Type I superconductors, or Pippard superconductors. In these supercon-

ductors, as we have seen, The superconductors in which

are called Type II superconductors, or London superconductors.

1.2.5.

Superheating in a Magnetic Field

Consider a film of a Type I superconductor with a thickness d,

placed in the magnetic field H parallel to its surface.

29,32

Let the film occupy the

space in the appropriate reference frame (so that formally the

problem is one dimensional). To simplify calculations we will made In this

case Eq. (1.72) reduces to Accounting for the fact that

(according to Eq. 1.49), one obtains = const. In addition,

integrating the Maxwell equation (1.56) and taking into account that

one gets

i.e., the mean value of the superfluid velocity (averaged over the film’s thickness)

is zero. Substituting the value of (still unknown) into Eq. (1.58), subject to the

above-mentioned boundary condition for

(

), we find

Here denotes the value of in the absence of the

magnetic field (see Eq. 1.39). For the superfluid velocity, we obtain on the basis of

Eqs. (1.53), (1.56), and (1.83),

SECTION 1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY 17

Averaging over the film’s thickness, we arrive at

Omitting in Eq. (1.48) the term with a spatial derivative we find

or, using Eq. (1.34),

On the basis of Eqs. (1.41), (1.59), and (1.39), the relation (1.87) can be

represented in the form

The critical value of the field is reached when the Gibbs potentials of supercon-

ducting and normal states equalize. The Gibbs potential density in a supercon-

ducting state is

where is expressed by Eqs. (1.40) and (1.31) with the spatial derivatives omitted

in view of Using the relation (1.86), one arrives at

To proceed further we must integrate Eq. (1.90) over the specimen’s volume, which

is equivalent to averaging over the film thickness. Using Eq. (1.83) and calculating

the averaged values and

18 CHAPTER 1. BASIC EQUILIBRIUM PROPERTIES

one finally gets

In the normal phase To obtain the critical value of a

magnetic field one must equalize and which leads to the equation

Another relation between and is supplied by Eq. (1.88). As follows from

these equations, the critical field depends on the film’s thickness d.

We consider the two limiting cases of thin and thick films. In the case

, Eq.(1.88)is

i.e., the reduced-order parameter decreases monotonically, with the field ampli-

tude H increasing. This behavior corresponds to the second-order phase transition.

Evidently the value corresponds to the critical field. Substituting into

Eq. (1.95), one finds

where is the thin film critical field.

In the opposite limiting case of thick film , Eq. (1.88) reduces to

Substitution of this into Eq. (1.93) yields the equation

The presence of a large multiplier on the left side means that i.e., the

thick-film’s critical field is approximately equal to because of Eqs. (1.17) and

(1.98). Thus the phase transition in thick films is of the first order, because it occurs

abruptly at finite values of

SECTION 1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY 19

More detailed study (see, e.g., that in Ref. 32) shows that the first-order

transition changes to the second order at the film's thickness The

physical reason for the first-order transition is the negative value of the surface

energy . Contrary to this, in Type II superconductors is positive and this leads

to the appearance of the layered structure of superconductors in the external

magnetic field (i.e., the coexistence of normal and superconducting regions).

1.2.6. Flux Quantization

In their basic paper

Ginzburg and Landau assumed that e

was equal to the

electron's charge e. As was later established microscopically by Bardeen, Cooper,

and Schrieffer the current in superconductors being transmitted by the

Cooper pairs, and Gor’kov was able to provethat (seeSection 1.4). We stress

that in contrast to the mass m

, which in the above system of equations may be

changed by simple renormalization of the charge e

enters Eqs. (1.48) and (1.51)

additively and its magnitude is important. To demonstrate this we consider the

phenomenon of magnetic flux quantization.

Let us imagine a massive superconductor with a cylindrical cavity placed in a

magnetic field H, which is parallel to the cylinder’s axis. Consider a contour C (see

Fig. 1.2) that encloses the cavity and lies entirely within the depth of the supercon-

ductor. Owing to the Meissner effect, the superconducting current vanishes at

distances from the surface much greater than the London penetration depth

Thus, as follows from Eq. (1.51), on the contour C one has

20 CHAPTER 1. BASIC EQUILIBRIUM PROPERTIES

Integrating j along this contour and using Eq. (1.99), we find

It must be taken into account that

where is the magnetic flux captured in the cavity. The first of the integrals (1.100)

is the phase difference acquired while going around the contour C and it must be a

multiple of :

because the -function is single valued [n is an integer in Eq. (1.102)]. As a result,

one finds from Eqs. (1.100) to (1.102)

or, in other words, the magnetic flux in the cavity of a superconductor is quantized

and may change only in portions This phenomenon was predicted first

by F. London

and later confirmed experimentally by Deaver and Fairbank

and

also by Doll and Näbauer,

who found the value of e

in (1.103) to be equal to

twice the electronic charge: Had the doubling of the carrier’s charge been

known in 1950, the analysis of the Ginzburg–Landau equation for the quantum

mechanical function of superconducting electrons might significantly accelerate

the subsequent development of the microscopic theory of superconductivity.

1.3. BCS–GOR'KOV THEORY

The basic cornerstone of the microscopic theory of superconductivity was laid

down by Cooper

in 1956. Cooper considered the indirect(mediated by the phonon

exchange) interaction between electrons in metals; this is a process of the second

order in electron–phonon interaction. As is known from perturbation theory, the

second-order correction to the energy of the ground state is always negative (see,

e.g., Ref. 37), i.e., the Cooper interaction is attractive. Because the Fermi sphere at