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

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SECTION 8.3. STABILITY AND BREAKING OF COOPER PAIRS 211

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ductor with a finite difference between the populations of electron- and hole-like spectral branches,

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Zh. Eksp. i Teor. Fiz. 81 [6(12)], 2118–2125 (1981)].

19. Yu. M. Gal’perin, V. I. Kozub, and B. Z. Spivak, Tunnel injection and tunnel stimulation of

superconductivity: The role of branch imbalance, J. Low Temp. Phys. 50(3/4), 185–199 (1983).

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strong magnetic fields, Phys. Rev. B 33(4), 6068–6068 (1986).

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221–311 (1983).

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Phase-Slip Centers

Time-dependent Ginzburg–Landau equations provide a rather deep understanding

of the nature of resistive states arising in narrow superconductive filaments. If a dc

current passes through such a filament, then, according to the thermodynamic

equilibrium GL theory (Sect. 1.2), the filament should pass into its normal state

when the current achieves the critical value (Fig. 9.1). But in reality, other types of

curves have been registered.

Moreover, electromagnetic radiation was detected,

which resembles the Josephson effect. These facts demonstrate directly that the

resistive state in narrow superconducting filaments is of a nonequilibrium and

nonstationary nature.

9.1. ONE-DIMENSIONAL APPROACH

Because in reality the filament is narrow [its cross-sectional dimensions should

be less than the resistive state cannot be explained by the motion of

Abrikosov’s vortices (in distinction to the situation in wide films in Sect. 7.3). The

search for new mechanisms that can adequately describe the resistive state has

become necessary.

9.1.1. Phase Slippage

Probably, Little was the first to use the idea of a “phase slippage” while

analyzing the stability problem of electric current in quasi-one-dimensional struc-

tures.* This idea was developed further in a number of papers, particularly by

Langer and Ambegaokar,

McCumber and Halperin,

Skochpol et al.,

and others

(a detailed bibliography, that can be used to reconstruct the history of the problem

is contained in the reviews in Refs. 7–9).

*The expression “phase slippage” was introduced earlier in the theory of superfluids by Anderson.

213

214 CHAPTER 9. PHASE-SLIP CENTERS

To clarify this idea, let us consider two points, of a superconducting

filament (Fig. 9.2) with a potential V applied between them. Ignoring for a moment

nonequilibrium effects, we conclude that the phase difference of the supercon-

ducting order parameter between the points increases linearly in time

[according to Eq. (8.9), if in equilibrium This indicates the

thickening in time of the spiral shown in Fig. 9.2. But such a thickening cannot

occur for too long a time, because the phase gradient determines the velocity of the

superfluid motion of electrons, which is restricted by the critical value. To decrease

the number of loops, the order parameter modulus should vanish at certain moments

of time. At such moments the phase can dump the excessive as it is shown in

Fig. 9.2.

The behavior of phase-slip centers (PSCs) was investigated rather extensively

on the basis of the Ginzburg–Landau-type dynamic equations (see in particular

Refs. 10–18). It was found that in the region of a PSC the order parameter modulus

oscillates, periodically turning to zero. In the time-averaged picture, the order

SECTION 9.1. ONE-DIMENSIONAL APPROACH 215

parameter modulus in the region of a PSC effectively decreases. Something like a

“weak link” arises between superconducting “banks” and the situation as a whole

resembles the nonstationary picture in weakly coupled superconductors, where the

Josephson oscillations appear if a voltage is applied between the banks.

In the next section we present the numerical solutions of dynamic equations,

which illustrate in detail the behavior of all physical quantities of interest (such as

the order parameter modulus and phase, superconducting and normal currents,

superconducting velocity, the scalar and gauge-invariant potentials the

charge appearing in the filament, the current-voltage characteristics, and the

periods of the stable oscillations). The results will also illustrate the influence of

the interference current component (Chap. 7) on the solutions of the dynamic

equations. The spectral characteristics of phonon fluxes emitted from a supercon-

ductor in a resistive state will also be discussed.

9.1.2. Initial Dimensionless Equations

Equations (7.45), (7.52), and (7.111) to (7.114) may be written in a convenient

compact form introducing dimensionless variables [in (9.1) they are underlined]

216 CHAPTER 9. PHASE-SLIP CENTERS

Here

In a dimensionless notation, Eq. (7.45) takes the form* [we will deal further with

dimensionless quantities only, dropping the underlines below the symbols in (9.1)]:

Note that and in this chapter relate to the charge and differ from

quantities introduced earlier (which relate to the electron charge e

)

by a

factor of 2. Representing the complex parameter where

is the modulus and is the phase of the order parameter, we find separately the real

and imaginary parts of Eq. (9.3)**

where

*The term in Eq. (7.45) is neglected in the derivation of (9.3). Correspondingly, the equation for

the inequilibrium phonon subsystem is not used.

**We drop the term with in Eq. (9.3).

SECTION 9.1. ONE-DIMENSIONAL APPROACH 217

Before attempting to solve Eqs. (9.4) and (9.5), we will make one remark about

method. Equations (9.4) and (9.5) [as well as (9.3)] are gauge invariant, i.e., they

do not change under the transformation

where is an arbitrary gauge function. Usually the following method of

handling Eq. (9.3) is applied. One considers the “current state” and chooses the

specific gauge of the vector potential in Eq. (9.3), putting and thus neglecting

the magnetic field in a narrow superconducting filament. The phase 6 now acquires

the meaning of the superconducting velocity potential, Solving then Eq.

(9.3) [or (9.4) and (9.5)] relative to one arrives at the usual picture of the

resistive state, where the order parameter modulus oscillates in time and the phase

suffers jumps (more properly, it “slips,” see the following discussion). As men-

tioned, such active regions are named the phase-slip centers.

This procedure, however, is not completely satisfactory. Indeed, the gauge

leading to strictly speaking, does not exist, wherever there exist a current j

and magnetic field in the system. Besides, as one can see from (9.10),

the superconducting phase depends on the choice of gauge and, consequently, the

physical quantities cannot depend on In fact, Eqs. (9.4) and (9.5) written in terms

of the invariant variables , and

do not depend on In particular, these

equations conserve their form in the case also, i.e., in the case of a real order

parameter. The question then arises of how one should interpret the expression

“phase-slip center” if the phase everywhere is identical to zero.

To answer this question, we will split the vector potential into longitudinal and

transverse components:

218 CHAPTER 9. PHASE-SUP CENTERS

where is some scalar function and

is a vector function. Under the gauge

transformation (9.10), only the longitudinal component is transformed, but the

transverse part remains unchanged. Introducing into (9.3) instead of the

invariant complex function is the invariant

superconducting phase, we can rewrite (9.3) in terms of invariant variables

The superconducting velocity

and potential (9.6) can be written now in the

form

where the values are invariant under the transformations of (9.10).

Considering a narrow superconducting filament and neglecting within it the mag-

netic field The invariant quantity has a meaning

of the potential part of the superfluid velocity is the invariant

potential of normal motion. Indeed, the electric field E in dimensionless units has

the form

Equation (9.12) has the same form as (9.3), if we use However the

invariant potential rather than the gauge-variant quantity enters into (9.12), and

the invariant phase instead of the gauge-variant superconducting phase In

particular, Eq. (9.12) is valid also at In further analysis we will use the

equation in the form (9.12), assuming This form coincides in fact with the

commonly used one.

9.1.3. Boundary Conditions

Several types of boundary conditions for Eqs.(9.12) and (9.8) may be formu-

lated, depending on the physical state assumed at the ends of the superconducting

filament. If a superconducting filament having a length

connects two (identical,

for simplicity) bulk superconductors, one can use the following boundary condi-

tions (at the points

where is some constant characterizing the bulk banks. The boundary condition

at the ends corresponds to the value of the bulk superconductor in

equilibrium (such boundary conditions were considered in Refs. 12 and 13). The

*The specifics of the spatial one-dimensional case emerge here. In general, it is necessary to use the full

system of Maxwell equations [(7.119) to (7.122)] to calculate the transverse component

SECTION 9.2. CALCULATIONS PROCEDURE 219

condition at the filament end is equivalent to the condition and

One can also choose different boundary conditions

accepting that in a bulk superconductor the total charge is transported by the

superconducting current. (The condition 9.15 for is equivalent to the condition

= 0 at the ends.) The boundary condition for can be written in a more general

form

where is some constant (see Sect. 7.2).

In the case of a long superconducting filament, when the phase-slip centers are

created evenly along the filament’s length, one can use the cyclic boundary

conditions

Here the points and correspond to the positions of the neighboring

extrema of (which can coincide with the phase-slip centers or lie between

them). Because the extrema of the superconducting current and of should

coincide, then on these points we evidently have If the supercon-

ducting filament connects two normal banks, the following boundary conditions

are possible

in accordance with the depression of the order parameter at the ends of the filament

due to the proximity effect, so that near the ends the total charge is transferred by

the normal current. We will use mainly the boundary conditions (9.14) and (9.15),

though the results will not depend substantially on the character of boundary

conditions.

9.2. CALCULATIONS PROCEDURE

9.2.1. Nonsingular Representation

The solutions of Eqs. (9.4) and (9.5) or (9.11) are singular and their numerical

evaluation demands certain precautions. We present in this section a rather full

description of the calculation scheme.

The singularity of Eqs. (9.4) and (9.5) written in terms of the invariant variables

and

, is due to the term, which hampers the calculations by turning infinite

220 CHAPTER 9. PHASE-SLIP CENTERS

at the moments when vanishes. A more convenient form of these equations is

attained by using a complex function

Introducing the real and imaginary parts of by the relation substituting

this in Eq. (9.19), and solving the resulting system of equations with respect to

and we find

where

The integration constant in (9.22) is chosen from symmetry considerations to ensure

that (the solution does not depend on the value of this constant).

Equations (9.20) and (9.21), written in terms of the Cartesian variables

differ advantageously from the representation in Eq. (9.12) of the

polar variables in that the point is no longer singular and

the process of its intersecting the phase trajectory {R

I} can be calculated

9.2.2. Matrix Representation for "Sweeping" Method

We present the system (9.20) and (9.21) in a matrix form

where

normally.