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

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SECTION 14.1. VORTEX ORIGINATION BY A MAGNETIC FIELD 371

The expression for in Eq. (14.17) holds for whereas at smaller this

formula is quite inaccurate.

Note that Eq. (14.17) has been derived taking into

account the gradient of the order parameter near the vortex axis.

1–6

In the following

paragraphs we will use Eq. (14.16) for assuming and taking into

account Eq. (14.17) for

The field at some point in a semi-infinite superconductor generated

by a vortex at distance from a plane boundary is calculated using the

mirror-reflection technique as a sum of two solutions of Eq. (14.16) for the vortex

at point and its mirror reflection [the antivortex ] at with respect to the

boundary:

We have added the index to the field B to indicate that this field is generated by

the vortex. Here is the radius vector connecting the center of the vortex with

the observation point and is the radius vector connecting the center of the

antivortex with the observation point where 2x is the vector

connecting the centers of the vortex and antivortex and is perpendicular to the

superconductor boundary). It may be shown that at any point on the interface

between the superconductor and vacuum the field satisfies

as follows from Eq. (14.19).

If there is a certain external field on the interface an exponentially

decaying function should be added to Eq. (14.19). In this case, the

condition is satisfied. By setting and in Eq. (14.19)

(the point is on the vortex axis), we obtain at the axis of the vortex

at distance from the interface:

For a superconducting plate of a finite thickness d in an external magnetic field

the solution is expressed as a sum of repeated mirror reflections from

the two interfaces. If one can take into account only the nearest reflections:

[Note that since both these functions determine the field on the

vortex axis.] The solution (14.21) satisfies Eqs. (14.7) and (14.8), and the boundary

conditions

372 CHAPTER 14. VORTICES AND THERMOELECTRIC FLUX

moreover, if

The total magnetic flux in the system is the sum of two terms:

where is the flux without a vortex and is the flux associated with the

vortex. The latter decreases as the vortex approaches an interface:

so the vortex placed on any interface does not contribute to the total flux in a plate,

since

The Gibbs free energy of a superconducting plate containing a vortex for

(or ) is derived from Eq. (14.15):

As a result, we derive from Eq. (14.25), with due account of Eqs. (14.21) and

(14.24),

where is the Gibbs free energy of the plate without vortices,

and is the

contribution due to the vortex. Equation (14.26) is a generalization of the result

presented in Ref. 1 and 2 for a plate with a finite thickness

Note that when and it follows from Eq. (14.26) that

therefore the equilibrium critical field can be defined as When

holds, a vortex added to the system does not change the total energy:

for i.e., the vortex inside the superconductor is in thermodynamic

equilibrium.

The behavior of the function defined by Eq. (14.26) is illustrated in Fig.

14.2. The surface barrier vanishes when the condition

is fulfilled, from which the threshold field (i.e., the maximum “superheating”) is

derived using Eqs. (14.16) and (14.17):

SECTION 14.1. VORTEX ORIGINATION BY A MAGNETIC FIELD 373

where is the thermodynamic critical field of the superconductor. When

vortices should move from the interface into the superconductor interior.

14.1.5. Penetration into a Hollow Cylinder

The Gibbs free energy of a hollow cylinder containing a vortex is determined

by the general formula (14.15), which requires the total magnetic flux the

field in the cavity, and the magnetic field on the vortex axis. These

parameters are usually determined using the mirror-reflection method, but in the

case of a hollow cylinder of arbitrary dimensions it is difficult to find an exact

solution like (14.19) because of the surface curvature. In the following paragraphs

we consider the case of a circular cylinder characterized by radii and with a

large cavity radius: This allows us to neglect the effect of the surface

curvature and use the formulas for the field generated by a vortex in the case of a

plane interface.

When the total magnetic flux is calculated using Eq. (14.23) (the integration

is performed over the entire cross section of the sample, including the inner cavity),

one should take into account that, unlike the case of Eq. (14.24), the flux associated

with a vortex is now

374 CHAPTER 14. VORTICES AND THERMOELECTRIC FLUX

since only the fraction of the flux dissipates into the environment,

whereas the other fraction which also dissipates into the environment in

the case of a planar plate, now remains in the inner cavity and contributes to the

trapped field.

The field in the cavity is a sum of the field in the absence of the vortex

and added by the vortex:

The field corresponds to m flux quanta contained in the cavity and is

expressed as follows

where and are modified Bessel functions, and

The factors and in Eq. (14.32) are functions of temperature and system

dimensions, and describe shielding properties of a superconducting cylinder of a

finite thickness. In the specific case the shielding factor

at all realistic temperatures, and the factor is exponentially small. In the

limiting case of a temperature very close to when and the

factors and . Thus, the shielding factor in Eq. (14.32) accounts

for the fact that the trapped flux as i.e., as the cylinder

becomes transparent for a magnetic field, although the number m of trapped flux

quanta remains fixed. In this case We assume that the condition

holds, therefore and

The additional flux in the system and the additional field in the cavity

due to the vortex are

respectively.

When using Eq. (14.15), one must know the total flux in the system given by

Eq. (14.23), the field in the cavity given by Eq. (14.31), and the field

on the vortex axis. Instead of Eq. (14.21), the field on the vortex axis is

determined now by the equation [here, as in previous calculations,

SECTION 14.1. VORTEX ORIGINATION BY A MAGNETIC FIELD 375

Using Eqs. (14.30) to (14.35), let us express the Gibbs free energy (14.15) as

where is the system energy in the absence of vortices

and is the energy due to the vortex:

For we have i.e., a vortex placed on the outside surface does not

affect the system’s energy. If the vortex is placed on the inside surface this

means, in fact, that the system contains quanta in the cavity. Therefore the

following condition should hold:

where is the Gibbs free energy of the system without vortices. It can be shown

that this condition is satisfied exactly:

Note that a vortex placed on the inside surface is just a current encircling

the cavity and maintaining an additional flux quantum.

The behavior of the function given by Eq. (14.37) in the case of a hollow

cylinder with a wall thickness d is illustrated by Fig. 14.3a (we consider the case

of ). The barrier preventing penetration of a vortex from the cavity into the

superconductor vanishes when the condition is satisfied,

i.e., with due account of Eq. (14.17),

for which we determine either the maximum number of flux quanta that can be

trapped in the cavity at a given temperature:

376 CHAPTER 14. VORTICES AND THERMOELECTRIC FLUX

or the maximum “overheat” temperature above which a field corresponding to

m trapped flux quanta cannot be confined in the cavity:

At a flux quantum should be ejected from the cavity (Fig. 14.3b, curve 3).

14.2. VORTEX ORIGINATION BY THERMOELECTRIC CURRENT

The giant thermoelectric effect observed in hollow bimetallic cylinders

13,14

was investigated previously

by solving a model problem of a homogeneous

cylinder carrying a normal current circulating around a cavity to simulate the real

thermoelectric current (see Fig. 14.1). We now generalize the formulas of the

preceding section to include the thermoelectric current The energy conservation

is expressed in this case (we assume that the external field is zero) by

SECTION 14.2. VORTEX ORIGINATION BY THERMOELECTRIC CURRENT 377

where the last term on the right is the work done by the electric field

on the given current We assume that this current flows in

the plane perpendicular to the cylinder axis, where

The amplitude of total current around the cavity in the normal state is

and the related field in the plane

14.2.1. Free Energy Barrier

The Gibbs free energy of the system (provided that and )

is expressed similarly to Eq. (14.5):

where is given by Eq. (14.6), and the variation yields the equation

where is expressed by Eq. (14.8). Equation (14.44) is easily transformed to a form

similar to Eq. (14.10) (recall that

where is the vector potential in the normal state, i.e., and, using

Eqs. (14.12), (14.13), and (14.45), to the form similar to Eq. (14.15):

where Equation (14.47) is exact, but

after substituting and one has to use approximate expressions simi-

lar to those of the mirror-reflection method.

As a result, we obtain where is identical to

the expression in Ref. 15:

where the field in the cavity containing m flux quanta is given by

378 CHAPTER 14. VORTICES AND THERMOELECTRIC FLUX

the shielding factor is

By setting and we obtain

The contribution to the Gibbs free energy due to the vortex is

For x

d we have as expected. At x

0 the condition

is satisfied exactly.

The expression for has the form

Comparison with Eq. (14.40) demonstrates that the right-hand side of Eq. (14.53)

contains the additional term which can be either positive or negative,

depending on the direction of current and field generated by this current.

The behavior of the Gibbs free energy as a function of the thermoelectric

current when the cavity contains the largest possible magnetic flux (Eq. 14.41)

is illustrated by Fig. 14.4. Curve 1 corresponds to the case when

and the derivative (Eq. 14.40); curves 2 and 3 correspond to when

and and curve 4 to when and

This means that in the presence of a thermoelectric current generating magnetic

field in the same direction as the captured magnetic flux in the cavity (Eq.

14.41), the derivative since The force acting on the vortex

is directed toward the cavity at Thus the thermoelectric

current effectively blocks the trapped magnetic flux in the cavity and prevents

ejection of magnetic vortices from the superconductor (curve 3 in Fig. 14.4), which

would take place at higher temperatures in the absence of the thermoelectric current

(Fig. 14.3b).

SECTION 14.3. VORTEX-ANTIVORTEX PAIR GENERATION 379

If the current is directed oppositely to (curve 4 in Fig. 14.4), then

the trapped magnetic flux cannot be confined in the cavity; and flux

lines should be formed and ejected from the superconductor. Whether this predicted

effect really takes place can be determined experimentally.

Note also that the function has a minimum at (curve 2 in Fig.

14.4). This means that a vortex can occupy a metastable position at some distance

from the cavity. Here we do not discuss this effect in detail.

14.3. VORTEX–ANTIVORTEX PAIR GENERATION

Now compare the behavior of the vortex and antivortex (i.e., a vortex contain-

ing the magnetic flux of the opposite sign) inside a superconductor with a thermo-

electric current It follows from Eqs. (14.52) and (14.53) that and

in the case of a vortex and in a sufficiently strong

thermoelectric field (curves 3 and 4 in Fig. 14.5a). This means that a force

directed toward the cavity acts on a vortex near the inside surface. In the case of an

antivortex we have and i.e., the driving

force is directed from the cavity (curves 3 and 4 in Fig. 14.5b). Thus, if there is a

vortex–antivortex couple, its components can be driven apart by thermocurrent

380 CHAPTER 14. VORTICES AND THERMOELECTRIC FLUX

under certain conditions. This effect can lead to important consequences, which are

discussed in this section.

14.3.1. Two-Vortice Free Energy

Here we calculate the Gibbs free energy when the system shown in Fig. 14.1

contains two vortices, and located along the radius at distances

and from the cavity surface Using the same technique as in the preceding

section, one easily obtains an exact formula that is a generalization of Eq. (14.47):

Magnetic fields on the vortex axes and are determined using the mirror-re-

flection technique: