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

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SECTION 10.4. PHONON EMISSION FROM A TUNNEL JUNCTION 281

In conclusion, we wish to make a remark concerning the difference between

phonon emission from the tunnel junction and that from a film under UHF pumping.

As described in Sect. 10.1, the nonequilibrium tunnel source contains terms that

oscillate in time. These oscillations may reveal themselves in a system of single-

electron excitations, and modulate with the Josephson frequency the phonon

282 CHAPTER 10. JOSEPHSON JUNCTIONS

radiation emitted by the tunnel junction. The amplitude of this modulation however,

is small (at sufficiently large voltage V, it is proportional to and that allows

us to ignore this dynamic effect. Much more pronounced phenomena of this kind

exist in the phase-slip centers, as we saw in Chap. 9.

10.4.3. Microrefrigeration

Is it possible to use the phonon deficit effect to achieve refrigeration by having

the negative phonon fluxes yield a cooling effect? In the symmetric SIS junction

considered earlier, the answer is most likely “no,” since the negative fluxes are more

than compensated for by the positive ones. [Bear in mind that to consider the fluxes

in terms of energy, it is necessary to multiply the value of the phonon source in (6.4)

and (6.7) by the phonon density of the states in the Debye model) and the

energy factor which strongly enhances the relative contribution of

the positive “tail.”] Parmenter

was the first who argued the benefit of using an

SIS´ IS structure (the extraction of electron excitations from a smaller gap, thin-film

superconductor S´ by tunneling into the larger gap “banks” S) for electron cooling

in S´. This work pioneered the theory of order-parameter enhancement in conditions

of nonequilibrium superconductivity (see Chap. 5). It has also initiated subsequent

studies on the possibility of tunnel junction refrigeration,

which resulted in

practical realization of the effect.

20–24

Initially microrefrigeration studies focused mainly on the effective electronic

temperature of the smaller gap superconductor. (Note that in the ultimate case of

the smallest gap one deals with the most effective SINIS structure.) These

SECTION 10.4. PHONON EMISSION FROM A TUNNEL JUNCTION 283

studies were related to the order-parameter enhancement effect, which is based on

the effective cooling of the electrons (see Chap. 5). This intrinsic electron cooling

can have “on-chip” practical applications.

A wider application range is possible

when cooling also involves the crystalline lattice. In this case it will be possible to

refrigerate external objects. Consideration of the crystalline lattice must take into

account the phonon deficit effect, which counteracts Joule heating.

For SIS

junctions, both mechanisms are proportional to where V is the voltage across

the junction, and the heating is typically stronger than the cooling power of the

phonon-deficit mechanism. For asymmetric SIS´ junctions, the cooling power

284 CHAPTER 10. JOSEPHSON JUNCTIONS

becomes a slower function of V (linear in some range of values), so that for

sufficiently small voltages the net difference may result in cooling.

There is a close analogy here with the Peltier effect at thermoelectric cooling, where one has

competition between the cooling mechanism, which is proportional to the current j, and Joule heating,

which is proportional to At low values of j one has a net cooling.

SECTION 10.5. WEAKLY COUPLED BRIDGES 285

10.5. WEAKLY COUPLED BRIDGES

10.5.1. Modified Aslamasov–Larkin Model

We will consider now the resistive state arising in weakly coupled supercon-

ductors. To describe it, we will exploit the Aslamasov–Larkin model,

according

to which the order parameter within the weak link can be written in the form

We consider a one-dimensional problem; the coordinate x along the weak link varies

from const is the order parameter modulus of the bulk superconductor;

are, respectively, the time-dependent values of the order parameter phase

on the left and right “banks.”

Since we consider the one-dimensional case, it is sufficient to calculate the

current at the point It follows from (10.49) (in the gauge and at

that

286 CHAPTER 10. JOSEPHSON JUNCTIONS

In the bulk “bank,” one can set assuming , we find from (7.85) and

(7.86)

The condition indicate that an analysis based on an equation of the

Ginzburg-Landau type (7.45) (which assumes that the spatial derivatives are small)

cannot be applied. Nevertheless, the expression for the current in the form (7.98)

can still be used, since in (7.88) may be taken as an equilibrium function, owing

to the rapid diffusive dissipation of nonequilibrium excitations toward the bulk

superconductive banks. Substituting (10.50) to (10.52) into (7.101) subject to

(7.106), one finds

where is some constant phase difference and

In the region of the weak link, there exist a number of “captured” nonequili-

brium excitations with energies As Schmid and Tinkham showed,

the

presence of these excitations at voltages produces a current contribution

where is some non-negative even function of

period with a maximum value of 2/5. The magnitude of the same dissipative

phase-dependent structure could also be obtained formally by substitution of

(10.51) and an equilibrium distribution function into the second term in (7.88). The

interference term arising in this approach is not small in comparison with in

(10.53) As mentioned earlier, there should be no “local equilibrium”

contributions to the current in the case of short bridges because of the

fast diffusion processes toward the banks. In other cases, however, these terms may

be important.

10.5.2. Term Paradox

The interference current may determine qualitative features of the dynamic

characteristics of nonequilibrium superconducting bridges. In particular, it mani-

SECTION 10.5. WEAKLY COUPLED BRIDGES 287

fests itself in a so-called “ -term paradox” (see, e.g., Ref. 29). For weak-link

bridges, this paradox arises when one attempts to interpret the experimental data

on the basis of the standard expression for the current, found by the method of the

tunneling Hamiltonian.

It is expedient to reconsider this problem.

Let us examine, for example, the interpretation of the experiment by Falco et

al.,

who measured the fluctuating value of the derivative of the voltage with

respect to current. A theoretical analysis of that quantity is based on the Fokker–

Planck equation,

which can be written for fluctuations in weak-coupling super-

conductors in analogy with the motion of a Brownian particle in an external

potential field.

As a result, the expression for the quantity was found

to be

Here is the cross

section of the weak link and R is its normal state resistance; the parameters are:

In comparing the theory with experiment, Falco et al.

used the ordinary

Josephson expression for the current [in the latter case we should set

in (10.54)]. The experimental data were found to lie well below the theoretical

curve (they were close to the curve with , and this result was considered a

paradox (see Fig. 10.21). If instead one compares the experimental data with the

expression (10.54) for one would find that the theoretical curve of Eq.

(10.54) runs well below the one found when To demonstrate

this, consider the behavior of (10.54) in more detail, setting for a moment

We thus arrive at

The value for may be found if one retains in (10.53) the contribution to the current, which follows

from the last term in (7.83) and is approximately equal to Since incorporation

of this term into the expression for the current would involve going beyond the accuracy of our

treatment, there could be other contributions to of the same order of smallness, which are effectively

negligible.

288 CHAPTER 10. JOSEPHSON JUNCTIONS

We look for the dependence of (10.56) on Only the second integral in the

denominator of (10.56) depends on It may be decomposed into two terms, as is

illustrated in Fig. 10.3 (it is convenient to make ). The first term is proportional

to the area bounded by the solid line (dashed in Fig. 10.3). The second term is

proportional (up to the same multiple) to the area restricted by the line obtained

after multiplication of the solid and broken lines. as one may see

from Fig. 10.3, the second term is negative (and has a smaller absolute value

than the first term). For this reason, the denominator in (10.56) decreases though

remaining positive, and at positive the value of K(0) is smaller than

For a negative the contribution of the second term is always positive

(because the dashed curve in Fig. 10.22 should be reflected relative to the

abscissa) and for a given y we have In the same manner one can

see that the terms containing and diminish the values of (10.54). Thus

probably the paradox mentioned above might be explained in this particular case.

It would be premature to make a detailed comparison of theory and experiment

because of insufficient knowledge of the parameters that determine the voltage-

SECTION 10.5. WEAKLY COUPLED BRIDGES 289

current characteristic of the weak link. For example, the relation between and the

measured voltage in Ref. 30 is not known; this is an important question,

since it

determines the dominant term in (10.56).

10.5.3. Excess Current

Excess current has been observed in all the types of weakly coupled supercon-

ductors,

34–40

except Josephson junctions. Likharev and Jacobson

proposed a

model to account for the relaxation of the order parameter in the region of a weak

link. They tried to apply a simple generalization of the Ginzburg–Landau equations

to the nonstationary case. Though such a generalization may be justified for the

gapless superconductors only, nevertheless it was the first successful explanation

of the excess current. The problem was treated more precisely by Artemenko et al.

They have explained this phenomenon using the model of short bridges. The results

obtained in Ref. 42 were confirmed by experiments both at low and at

sufficiently high temperatures and at arbitrary bias voltages.

34–40

For a bridge

“superconductor–constriction–superconductor” at arbitrary temperatures and

the result of Artemenko et al. may be written as

this result may be rewritten as

290

CHAPTER 10. JOSEPHSON JUNCTIONS

where is the BCS value of the gap.

An expression of the same type follows from the formulas of Sect. 7.2, which

utilize the idea of interference current [see Eq. (7.104)]. For illustration we will use

the results obtained with the Aslamasov–Larkin model. From expression (10.53) it

follows that the time-averaged value of the interference current is not zero. Substi-

tuting into (7.104) the time-averaged value we have

For ordinary superconductors

so one can omit the logarithmic dependence on T [keeping only the dependence on

in the expression (10.59). Thus we arrive at a qualitative and even

quantitative coincidence of the results obtained by rather different techniques

(though in the vicinity of only). In particular, the temperature dependence is

essentially the same. In addition, some new aspects of this phenomenon emerge:

first, the excess current is periodic in time, as can be seen from (10.53). Furthermore,

while previously it was thought

that the excess current arises in the weak link

bridges because of the presence of massive banks (i.e., manifests itself as a boundary

effect), from the analysis presented here it follows that as a result of interference

between normal and superfluid motions, the excess current may reveal itself in bulk

samples also.

References

1. B. D. Josephson, Possible new effects in superconductive tunneling, Phys. Lett. 1(7), 251–253

(1962).

2. B. D. Josephson, in Superconductivity. R. D. Parks, ed., Vol. 1, p. 423, Marcel Dekker, New York

(1969).

3. A. I. Golovashkin, V. G. Eleonskii, and K. K. Likharev, Josephson Effect and Its Applications:

Bibliography, 1962–1980, pp. 1–222, Nauka, Moscow (1983) (in Russian).

Note that according to (10.59), the experiments on excess conductivity allow us to estimate (with a

logarithmic accuracy) the energy relaxation time of single-particle excitations in supercon-

ductors.