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

Подождите немного. Документ загружается.

SECTION 6.2. NEGATIVE PHONON FLUXES 141

the “geometric-acoustical” approximation* can be used to study the phonon’s

propagation, whose frequency significantly exceeds the quantity u/d (where d is

the film’s thickness). In this approximation the phonon’s behavior at the boundary

is determined by the ratio of the “acoustic” densities of the media (

are the densities of the two media). Such phonons will lose all their energy upon

each collision with the wall (i.e., they will be freely emitting to the outside) if the

phonon “acoustic density” of the external medium coincides with the acoustical

density of the film is the mechanical density). After thus determining the lower

limit of the frequencies of the phonons of interest, we select a superconducting film

thickness so that the phonon escape time is less than its transit time, which is related

to the interaction with the electrons (in a bulk sample it is on the order of

Under such conditions, the feedback of the phonons on the electrons may be ignored

and the phonon heat-bath model becomes applicable for the electron system. For

example, let the film have a thickness d comparable to the coherence length

For such films it would mean that in the frequency range

the current scheme developed for phonons may be directly employed.

Note that in the preceding discussions phonon damping was ignored, i.e., the

phonon lifetime was assumed to be sufficiently long. Assuming that the phonons

are being absorbed in each collision with the wall, this time may be estimated as

and should satisfy the condition for our scheme to be

applicable. If we set then this condition takes the form

which virtually coincides with the applicability condition of

the “geometric-acoustical” approximation [see the left inequality in (6.6)]. Thus we

may indeed ignore the damping of the phonons with frequencies that lie in the

interval (6.6). In general, the problem of boundary conditions is not so simple (see,

e.g., the references mentioned in Chang and Scalapino

). When the reflection

processes are to be taken into account, one can envisage (6.4) as an intrinsic source

of phonons.

6.2.4.

Induced and Spontaneous Processes

The source [the right-hand side in expression (6.4)] determining phonon

kinetics may be classified in the following manner. We will call the first integral in

(6.4), which is related to the relaxation processes in the electron system, the

*It implies that the wavelength of the propagating phonon is much less than the characteristic dimensions

of the inhomogeneity of the system and consequently one can ignore the diffraction effects (in analogy

with the “geometric-optics” approximation for photons).

Such an estimate is appropriate in the vicinity of critical temperature when

142 CHAPTER 6, PHONON-DEFICIT EFFECT

relaxation source . The second integral is related to excitation recombination and

pair breaking. It may be called the recombination source Each of these sources

consists of a set of terms that are proportional to the occupation numbers of the

phonons, and of other terms that do not explicitly contain such proportionality. The

first set may be called the induced phonon source while the second one may

be called the spontaneous phonon source Thus the entire source may be

represented in the form

which is convenient for examination.

We emphasize that for frequencies near the transition temperature, the

following relation holds

Consequently, in this frequency range the spontaneous sources in (6.7) are small

and may be neglected. Of interest is the spectral dependence for various

frequencies of the external electromagnetic field. It turns out that in the spectral

range of primary interest, which is comparable to this dependence

can be found analytically. It is expedient to carry out further investigation of the

behavior of the relaxation and recombination sources separately.

6.2.5. Properties of the Recombination Channel

We consider first the recombination term It may be represented as

Using (6.3) we may write

which, after some simple transformations subject to (6.5), reduces to the form

SECTION 6.2. NEGATIVE PHONON FLUXES 143

where

while the function A is determined by the relation

Such notations are convenient for revealing the two most important spectral

features of the source First there is the minimum threshold frequency of the

nonequilibrium phonons (equal to twice the gap value), above which the source

(6.11) becomes nonzero. Second, since the quantity [which is a positive

quantity, as is clear from (6.12) and (6.13)] enters the source (6.11) with a minus

sign and is nonzero in a certain frequency range where the quantity is zero,

we may conclude that there exists a frequency range of where the value of

is negative.

At first glance this is an unexpected result. Hence it would be reasonable to

investigate in greater detail the cause of such an anomalous behavior of the

recombination source. To do this, we calculate first the integrals (6.12) and (6.13),

which may be done by approximately taking into account the smoothness of the

function in the integration ranges. This enables one to use the mean-

value law and represent (6.12) and (6.13) as

where and lie in the regions , while K (x) is the

first-order complete elliptical integral in a normal form, which appears during the

transition from (6.12) and (6.13) to (6.15) and (6.16). For this function we have

144 CHAPTER 6. PHONON-DEFICIT EFFECT

Of primary interest to us are the frequencies of the scale , Note that for

such frequencies the calculations displayed below, which utilize the mean-value

law, are exact in the asymptotic case . This is precisely the case we have in

mind. We set [here and find from (6.15)

and (6.16) the threshold value of the function As a result we

obtain

From this expression, which determines [in combination with (6.11)] the depth of

the dip in the spectral curve of the recombination source (Fig. 6. 1a), it follows that

the dip will become deeper as the external field frequency is increased. The

examination of expressions (6.15) and (6.16) shows that diminishes when

the frequency grows and at takes the value (to an accuracy of

(here we use the numerical values of the coefficients).

If grows further, the contribution from the term appears, its

threshold value coinciding (with accuracy up to with the corresponding

value of i.e.,

It follows from this, subject to (6.19), that the difference at

is positive and is equal to approximately one-fifth of the value of

Since a further increase in causes each of the quantities

and consequently their difference to vanish, we conclude that the function

possesses a “peak” (maximum). A study of the expression

shows that the half-width of the peak is of the order The results of the

calculations of the source are illustrated in Fig. 6. 1a.

SECTION 6.2. NEGATIVE PHONON FLUXES 145

146 CHAPTER 6. PHONON-DEFICIT EFFECT

6.2.6.

Comparison with Relaxation

We now consider the relaxation source which is represented in the

following form, based on Eq. (6.4):

Taking into account (6.3) and (6.50) and the relation as well as some

obvious transformations, one can write:

Further examination is completely analogous to the case of a recombination

source. Omitting the details of calculations, we present the results for the source

(6.23). The function has no threshold features. It vanishes (being positive)

at the beginning of the spectrum . At larger values of the argument,

increases and when it has a logarithmic singularity. Further

increasing the argument causes a decrease in , which initially is logarithmic

and then becomes exponential. For example, at values the source takes the

value This is illustrated in Fig. 6.1b. Thus the

behavior of the contributions is found in the whole range of phonon

frequencies of interest.

6.3. VIOLATION OF DETAILED BALANCE

The results obtained in Sect. 6.2 are summarized graphically in Fig. 6.2a. We

emphasize that the ranges of the relaxation and recombination sources in the

external radiation frequency range considered factually do not overlap.

For this reason the total phonon flux in the narrow frequency range

has a negative value (the “dip” in Fig. 6.2a). This means that a

superconducting film under the influence of a high-frequency electromagnetic field

would selectively absorb phonons.

SECTION 6.3. VIOLATION OF DETAILED BALANCE 147

6.3.1. Phonon Deficit and Order Parameter Enhancement

The origin of this effect is closely related to the mechanism that causes the

enhancement of superconductivity in a high-frequency electromagnetic field,

which was considered in Sect. 5.2. Indeed, at temperatures there are always

a certain number of electron excitations above the gap, which are in thermodynamic

equilibrium with the phonons. The phonons in the superconductor, having an

energy approximately equal to twice the gap value, effectively create quasi-particles

that recombine and emit phonons of the same frequency.

Note that according to the detailed balance principle, the probabilities of the

direct and reverse processes are identical. The situation changes when an external

high-frequency electromagnetic field is applied. As was shown in Chap. 5, if the

frequency of the field does not exceed the quasi-particle creation threshold, the

action of the high-frequency electromagnetic field is reduced to changing the

“center of gravity” of the quasi-particle distribution function toward the higher

energies. Then the number of excitations above the gap edge falls below its

thermodynamic equilibrium value [the function (6.3) is negative at

. This indicates a violation of the detailed equilibrium in electron–

phonon interaction in the presence of an external field. The probability of phonon

absorption at frequencies becomes greater than the probability of their

148 CHAPTER 6. PHONON-DEFICIT EFFECT

emission. As a result, at some frequency range near a deficit of phonons

is formed in a nonequilibrium superconductor, which should be compensated from

the outside of the system for the entire picture to remain stationary. Thus, phonon

fluxes flow from the heat-bath to the superconductor: they could be regarded as

negative fluxes.

6.3.2.

Comparison with Alternative Approaches

Note in this connection that the experimental detection of phonon absorption

could be considered not only as the direct confirmation of the theory developed

earlier, but also as an additional proof of the validity of the concepts utilized in

formulating the theory of superconductivity enhancement. At this point we offer a

comment on the study by Chang and Scalapino.

They incorporated an additional

term in the kinetic equation for the phonons, where is the escape time

of the nonequilibrium phonons from the film. This term accounts for the coupling

of the phonons to the external medium. Since in free exchange the phonon

deficit created by the external field is completely compensated by the flow from

the outside , they noted that the phonon distribution that arises within

the film is in equilibrium. However, this equilibrium is dynamic and is directly tied

to the existence of phonon fluxes of different signs. In Chang and Scalapino, these

phonon fluxes were not calculated. Figure 6.2b reproduces the curve they found by

numerical calculation for at from which the existence of negative

values of is implied. The form of this curve indicates the presence of a

phonon deficit within the film and also implicitly indicates the existence of negative

phonon fluxes.

The study by Dayem and Wiegand

is also of interest. In their study, the

behavior of the nonequilibrium electron–phonon system was investigated under the

conditions of phonon pumping. The phonon frequency in the incident flux was

considered to be much less than twice the gap value. These authors solved the

problem numerically, using their model of discrete levels, in which several dozen

levels were used to “construct” the energy region over the gap in the energy space

of single-electron states. Electron transitions involving phonon emission and ab-

sorption take place between these discrete levels. The authors focused on the

possibility of effectively increasing the frequency of the incident phonon flux

(“up-conversion,” which turned out to be not very effective). It is interesting to note

that the curve describing the spectral dependence of the integral phonon flux is

negative in a certain frequency range beginning at This also indicates the

existence of an analog of the phonon-deficit effect under external (in this case,

phonon) pumping.

SECTION 6.3. VIOLATION OF DETAILED BALANCE 149

6.3.3. Recombination Peak

As to the peak in the spectral curve (see Fig. 6.2a) at frequencies

its origin is determined by the recombination of quasi-particles,

shifted by an external field far from the gap edge on the energy distance with

the quasi-particles remaining at the edge. This is quite obvious in our examination.

Indeed, as was noted in Sect. 5.1, single-quantum transitions are characteristic of

this situation. Consequently, excess quasi-particles are created primarily at energies

As we see from Eq. (6.3), the contribution to the quasi-particle distri-

bution function is singular at such energies. However, since our study is restricted

by a linear (in intensity of the external field) approximation, it follows from (6.4)

that the main role is played by the recombination of nonequilibrium quasi-particles

(at energies with equilibrium quasi-particles (at energy level near

the gap edge, where the density of electron states is high. As a result of this

recombination process, the phonons with energy are produced.

6.3.4. Exclusion of Divergence

Concluding this section, we discuss the logarithmic divergence of the relaxa-

tion flux at frequencies (see Figs. 6.1 b and 6.2a). This formal divergence

is connected to the singularity in the density of electron levels, which is charac-

teristic for superconductors in the absence of external fields, and appears in the

approximations used. This logarithmic divergence vanishes if one accounts for the

smearing influence of the electromagnetic field on the electron BCS density of

states. Other methods for exclusion of such divergencies also exist [e.g., one can

bound Eq. (6.3) using the Pauli principle]. They do not, however, influence the

results in any significant way.

References

1. G. M. Eliashberg, Temperature Green’s function for electrons in a superconductor, Sov. Phys. JETP

(5), 1000–1002 (1960) [Zh. Eksp. i Teor. Phys. 39 [5(11)], 1437–1441 (1960)].

. A. M. Gulyan and G. F. Zharkov, Kinetics of phonons in nonequilibrium superconductors at

external electromagnetic field, Sov. Phys. JETP 53(1), 756-762 (1989) [Zh. Eksp. i Teor. Fiz.

(1),303–325(1981)].

3. G. M. Eliashberg, Film superconductivity stimulated by a high-frequency field, JETP Lett. 11(3),

114–117 (1970) [Pis’ma Zh. Eksp. i Teor. Fiz. 11(3), 186–188 (1970)].

4. J.-J. Chang and D. J. Scalapino, Kinetic equation approach to nonequilibrium superconductivity,

Phys. Rev. B 15 (5), 2651–2670 (1977).

5. A. H. Dayem and J. J. Wiegand, Behavior of thin film superconducting bridges in a microwave

field, Phys. Rev. 155 (2), 419–428 (1967).