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

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SECTION 9.2. CALCULATIONS PROCEDURE 221

and replace the time derivative by the finite difference

here is the step in time and are the values of in the initial and

subsequent time layers.

We divide the interval L into n parts, h being the space step and and

denoting the value of in discrete point j

Replacing the coordinate

derivative by the finite difference

we rewrite Eq. (9.23) in a lattice form

where

The indices in Eq. (9.27) numerate the interior points of the compu-

tation interval where Eq. (9.23) is valid. The values at the current moment must

be found (the upper index is omitted). The coefficients depend on

at the preceding moment and are assumed to be known.

The boundary conditions of the type (9.14) and (9.15) may be written in a form

analogous to (9.27):

where the points and correspond to the limits of the computation

interval.

9.2.3. Recurrence Relations

We seek the solution of the system (9.27) in the form

where the matrices are to be found. Inserting (9.31) twice into (9.27), one

finds the recurrent relations

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CHAPTER 9. PHASE-SLIP CENTERS

Expressions (9.31) to (9.33) provide the matrix generalization of the standard sweep

method

19,20

in solving the differential equations.

9.2.4.

Solution Procedure

The successive steps of finding the solutions are as follows. First we find from

the boundary conditions the matrices

The remaining matrices and (for are found further, taking into

account that Afterward, the functions are determined from

(9.31). Using the newly found values of one can calculate and also

the potential from (9.22). Having determined one can calculate a new

boundary value of the phase from the relation

where the superscript zero denotes the preceding moment of time. (From symmetry

considerations we have _ In the case of boundary condition (9.14) we

always have In the case of boundary condition (9.15), one should

use the relation [see (9.5)] to find the values of where

the current is found in terms of

and

, which is known from the

preceding step. A new value of the phase determined in this way can be used

in the boundary conditions (9.14) and (9.15), which are equivalent to

Writing down these conditions in a matrix form [(9.29) and (9.30)], one can repeat

the procedure described above once again to find the solution on a new time layer,

and so on. If R

(

)

and I(x) are known, one can calculate the values of

(9.22),

at any moment of time.

In the case of the boundary conditions (9.17), some complication arises owing

to the presence of the derivatives

SECTION 9.3. ANALYSIS OF RESULTS 223

If the derivatives are represented as a finite difference: then

the unknown value appears. To overcome this difficulty, the

count interval is extended by introducing a subsidiary point The

condition (9.17) may be written in the form

(the lower indices denote the values One finds from this

i.e., the values are expressed in terms of the as-yet unknown values and

and the known (from the preceding step) value One more relation between

is provided by the recurrence formula (9.41)

The subsidiary value is easily excluded from (9.40) and (9.41), and the value

may be determined. After that, the sweep procedure of (9.27) to (9.35) is applied

to find the remaining values

To guarantee the equivalence of both left and right halves of the calculation

interval, the left and right runs were alternated and the solution’s symmetry was

maintained at every step.

To secure the convergence of solutions and to attain a stable oscillation regime,

the calculations were carried out using mainly the space step about

and the time step about The oscillation periods were found

to the accuracy of three meaningful numbers. The phase increase for one oscillation

period with the same accuracy was found to be (relative to one PSC). Several

hundred oscillations were sometimes performed before reaching the stable oscilla-

tion regime, thus making evident the presence of long-lived and weakly damping

oscillation modes in the system.

9.3. ANALYSIS OF RESULTS

A numerical analysis shows that if the values of the transport current j along

the filament are small enough, then there exist only static solutions of Eq. (9.12)

with When j is increased, the solutions start to depend on time, the

electric field and normal current appear in a superconducting

filament. All the parameters describing the superconductor

begin to oscillate in the vicinity of certain active points (PSC). After a sufficiently

long time, these oscillations reach a stable regime, and only such stable oscillations

are considered in the following section. To elucidate the characteristic features of

224

CHAPTER 9. PHASE-SLIP CENTERS

these processes, let us consider first a short superconducting filament with only one

active center.

9.3.1. Single Active Center

Figure 9.3 shows the distributions of at different moments for a

filament with the parameters for the boundary conditions

(9.14). Figure 9.4 depicts the time dependence of at the PSC

point The numerals at the curve denote the characteristic moments,

which correspond to the consecutive stages of an oscillation cycle (see Fig. 9.4). At

the moment 5,6 the momentum

and the potential suffer discontinuity, and the

derivative of suffers a jump (see later discussion). The numeral 5 denotes the

moment just preceding and 6 just after the jump.

A detailed analysis of the numerical solutions has shown that the order

parameter and the superconducting current vanish simultaneously at the instant

of the jump (Fig. 9.4), but the superfluid momentum behaves singularly,

turning to This shows that the current in the vicinity of PSC at

some moments is negative. (An analogy with the vortex currents in Type II

superconductors and Josephson junctions is evident here.) Such characteristic

behavior is common to all time-dependent solutions of Eq. (9.11) investigated by

us and does not depend on the type of the boundary condition or the parameters

involved. (Negative values of were reported earlier, e.g., in Ref. 14). The

gauge-invariant potential also behaves singularly, taking values on

both sides of PSC at the moment of the jump (Fig. 9.5).

Figure 9.6 shows the behavior of the invariant Coulomb potential and of the

normal current at all

times.

SECTION 9.3. ANALYSIS OF RESULTS 225

226

CHAPTER 9. PHASE-SLIP CENTERS

Averaging the electric field relative to the coordinate x, we find a

voltage drop V at the filament ends

Here V is a function of time only. Using the relation one finds the

potential drop averaged over the period of oscillations:

where are time-averaged values of at the ends of filament and

is the phase difference gain between these points for one oscillation

In the case of the boundary condition (9.14) we have and

period

SECTION 9.3. ANALYSIS OF RESULTS 227

Our calculations show that the phase difference gain during the period of steady

oscillations is (related to one PSC), and consequently

(For unsteady oscillations this gain generally is not equal to

9.3.2.

Oscillation Frequency

The expression (9.45) for the oscillation frequency at the phase-slip center may

be also derived analytically from topological considerations, as was demonstrated

by Ivlev and Kopnin.

Assuming the oscillation process is stable and the positions

of PSC are equidistant along the coordinate x

one obtains in the two-dimensional

space-time the periodic lattice of points at which the modulus of the order

parameter vanishes (Fig. 9.7). Consider the two-dimensional vectors

in this space-time. Using a relation between the vectors (9.46):

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CHAPTER 9. PHASE-SUP CENTERS

which follows from the definitions in (7.82) and (7.85), and integrating (9.45) over

the closed contour going around PSC, we have

The integral on the left side of (9.48) vanishes if the integration contour is chosen

along the “elementary cell,” denoted by in Fig. 9.7 [in the case of a single PSC

this contour must be stretched over subject to the condition The

first integral on the right side of (9.48) may be expressed by Stocks theorem through

the integral of curl

over the area S

restricted by the contours so

Taking into account that the last integral in (9.48) provides a phase gain equal to

(

n is an integer) in going around the PSC, one has

and, accordingly,

where is the flux quantum (see Sect. 1.2). If

is a static quantity, then

the Josephson relation follows from (9.51) at

= 1:

in which Introducing the space and time-averaged value

we present (9.52) in the form:

SECTION 9.3. ANALYSIS OF RESULTS 229

which is a generalization of the Josephson relation (9.52).

9.3.3.

Temporal Behavior of the Phase Difference

In Fig. 9.8a the phase difference across the filament ends is

depicted as a function of time; the moments 1–6 are marked to correspond to the

notations in Fig. 9.4. The phase difference grows monotonically in time. Such

growth is a consequence, in particular, of the boundary condition from

which it follows that monotonically de-

creases and, analogously, monotonically increases in time. The superconducting

current always stays finite, i.e., in all regular points the phase

gradient is finite. However, the gradient of the continuous function cannot remain

limited if the function difference at the ends of the interval grows unlimitedly in

time. It follows then that should suffer a discontinuity at some point to preserve

the finite gradient at all other points.

Figure 9.8b shows the behavior of at different moments. For symmetry

reasons is an odd function relative to the middle point of the interval. If the

230 CHAPTER 9. PHASE-SLIP CENTERS

moment 1 corresponds to the minimal current j

in a system, then the phase

can be represented at this moment as a single-valued continuous function in the

interval In subsequent moments (2, 3, 4) decreases and the phase

gradient increases. Close to the moment of a jump, the phase gradient first turns to

(at the moment 5 immediately before the jump), and then changes (at the

moment immediately after the jump). At this moment the phase suffers a

discontinuity at the location of the value on both sides of PSC

being their difference is equal to at the moment of a jump. After

this the two branches of the phase diverge in the directions shown by the arrows,

smoothly deforming, and after the moment (which corresponds to the subsequent

minimum of the current they take the positions marked by the primed numbers

on the curves (the phase difference is at this stage). Further on the process is

repeated and the two branches of the phase continue to diverge from one another.

Because the phase is determined by the relation the resulting

picture can always be reduced to the one shown in Fig. 9.8b by shifting it along the

ordinate by

(

n is integer).*

The voltage-current characteristics of the short superconducting filaments with

the boundary conditions (9.14) are shown in Fig. 9.9a. The ratio of the voltage drop

(9.22) to the filament’s length is presented as a function of the transport

current j through the filament. The dashed straight line corresponds to Ohm’s law

in a normal filament. The curves corresponding to the resistive state of the

*The multivalued function can also be represented as a single-valued continuous function on a

cylindrical surface glued together along the element The phase difference

at the ends of the interval will change by at the moment of a jump, remaining always

within the limits