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

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SECTION 12.1. "WIDE" PUMPING SOURCE 321

unique rule—the Pauli principle—thus regulates the filling of states lower than the

“energy horizon” , yielding (12.12).

12.1.3. Numerical Analysis for Realistic Spectral Function

Approximation (12.14) has relevance for some classes of superconductors,

such as alloys of transition metals

20,21

and high-temperature superconductors.

22,23

At the same time, more detailed information may be obtained by numerical analysis

of Eq. (12.4) subject to (12.5).

In this case it is useful to start with a model approximation for the spectral

function plotted in Fig. 12.1,a, and change the parameters

, and The results of numerical analysis for the case of

0 are shown in Figs. 12.2–12.8. They are different for longitudinal and transverse

phonon fields.

First we will discuss the case of longitudinal phonons. As is seen from these

results, the distribution function of quasi-particles may be inverted for certain

choices of parameters.

Changing the boundary values and , one finds that the most radical variation

in the shape of function (a decrease) takes place when the value of exceeds

(Fig. 12.2); this change is caused by the “switching on” of intermediate energy

phonons in the kinetic processes. We would like to mention here an

322 CHAPTER 12. INVERSE POPULATION INSTABILITIES

interesting feature: shifting of the maximum of when exceeds (Fig. 12.2,d).

The value of decreases noticeably when the value of increases (Fig. 12.3).

The change of localization boundaries of high-energy phonons also may

strongly influence the formation of function (Fig. 12.4). When the boundary

is near the critical value the distribution function is substantially depressed

(Fig. 12.4,a), and, in contrast, strongly increases when the group of high-energy

phonons is situated far from With respect to the parameters and , it may

be shown (Fig. 12.5) that the maximum of increases (being restricted by 1) when

the ratio increases.

The role of phonons of the lowest energies (described by the parameter ) is

negligible, as Fig. 12.6 illustrates. It is interesting to note that setting

one arrives at (Fig. 12.2,c) the results obtained in Refs. 17 and 18 (for

–1).

Let us consider now the case of a transverse phonon field. For such phonons a

very significant sign reversal

SECTION 12.1. "WIDE" PUMPING SOURCE 323

occurs in the coherence factors of the collision integral (12.2). Such an exchange

raises the intensity of relaxation processes and and diminishes the intensity

of recombination processes as may be demonstrated from (12.5). In accord-

ance with (12.6), this must yield an increase in . Figure 12.7 illustrates such a

conclusion. As may be found by a comparison of corresponding curves from Fig.

12.7 and Figs. 12.2-12.6, the function is indeed greater in the case of transverse

phonons than in the case of longitudinal phonons.

In real crystals, both longitudinal and transverse phonon modes are always

present. Two limiting curves corresponding to the cases of purely transverse and

purely longitudinal phonons are depicted in Fig. 12.8 (taken from Ref. 12). The

shape of the function was chosen to correspond to the data obtained

for high-temperature superconductors (in particular,

These superconductors as well as other representatives of high- families are highly anisotropic and

this raises the probability of transverse phonon field participation in electron–phonon interactions.

Dealing with this class of superconductors, one should bear in mind that there may be additional

interactions (e.g., paramagnon exchange), which can mediate the process of energy relaxation of

quasi-particles and change the whole picture.

Undoubtedly for more precise predictions, the mecha-

nism of superconductivity itself must become better known.

324

CHAPTER 12. INVERSE POPULATION INSTABILITIES

SECTION 12.2. "NARROW" PUMPING 325

12.2. "NARROW" PUMPING

In “narrow” pumping, the pumping source may be regarded as having a

resonant action on the system of electrons. Thus the term in (12.4) can no

longer be omitted. Taking this into account, the functional fraction (12.6) can be

presented in the form

On pumping by a electromagnetic field with the frequency

the quasi-particles will be created mainly in the narrow region above the

gap.

12.2.

1. Analytic Solution for Resonant Pumping Case

We will start our discussion with the case of relatively weak fields (cf. Chap. 5):

326 CHAPTER 12. INVERSE POPULATION INSTABILITIES

Dropping the components that are proportional to U-factors in the field term of

(5.21), Aronov and Spivak

11,26

found an analytic solution for (these factors are

responsible for the redistribution in the energy space of excess quasi-particles,

created by the electromagnetic field). Using the relaxation-time approximation for

the collision integral in (12.3), one can ascertain that under these conditions the

electron excitation's distribution function has the form

11,26

where

SECTION 12.2. "NARROW" PUMPING 327

The function at various values of and is plotted in Fig. 12.9.

The result

(12.21) indicates that the distribution function of the excess quasi-particles in a

superconductor pumped by an external electromagnetic field not only can exceed

the value 1/2 but can even attain the maximum value of This is connected

with the singularity in the superconductor's density of states. Owing to this, the

value may be attained at even for exceedingly weak pumping, i.e.,

even when (as is clear from 12.21). This result may be formulated in the

following manner: at the ground state of the superconductor (i.e., the state

without electron quasi-particles) is unstable with respect to the transition to the

nonequilibrium steady state, in which the value is attained, if a weak external

electromagnetic field with a frequency in resonance with the threshold frequency

is applied. We will consider the possible implications of this result in Sect. 12.3.

Here we ignore the insignificant “tail” of the excess quasi-particle distribution function at

Refs. 11 and 26).

328 CHAPTER 12. INVERSE POPULATION INSTABILITIES

12.2.2. Tunnel Injection of Electrons

Even stronger deviation from equilibrium is possible by tunnel injection of

quasi-particles at voltages At such voltages one can neglect the terms

and in (10.4): their influence average out. Because of that, the general

equations (12.3) and (12.5) may be significantly simplified. We will make these

simplifications assuming To make a more thorough analysis, we will add

to (12.4) the electron-electron collision term (4.36) in the form (below is

assumed):

SECTION 12.2. "NARROW" PUMPING 329

The quantity entering expressions (12.18) to (12.23) should be determined from

the self-consistency equation (12.5). The self-consistent system of Eqs. (12.18) to

(12.23) is highly nonlinear. However, the assumption of a “narrow” energy distri-

bution of the electron excitations makes it possible to effectively find the solution,

even taking into account (12.23).

12.2.3. Simplifications for "Narrow" Distributions

Introducing the variables and denoting

we present Eq. (12.24) in the form

where is the gap at in the absence of external perturbation. The “narrow”

distribution results [as is seen from (12.25)] in a small value of and hence

Consider now the collision integrals. If the function and is still

concentrated in the region immediately above the gap, the relaxation terms in

(12.23) are small in comparison with the recombination terms, and may be

simplified:

The electron–electron collisions may cause a significant change in the distri-

bution function if they are efficient. In our case, when the quasi-particles are

concentrated in a narrow layer near the Fermi surface and in this layer, we

must first account for the “collision pairing” processes [the second term in the

component of the collision integral when three colliding electron excitations

with energies . create a bound state (a Cooper pair) and a free quasi-particle with

the energy The opposite “collision breeding” processes in this case are not

efficient and hence may be reduced to the form

330 CHAPTER 12. INVERSE POPULATION INSTABILITIES

Comparing (12.27) and (12.28), one may ascertain that the electron-electron

collisions are not important if the following parameter:

is small (we set . For metals with relatively large Debye frequencies (such

as Al, see Table 10.1 in Sect. 10.2), the value of might be insufficiently small, so

it becomes necessary to account for (12.28). Unfortunately, the factors a and b in

(12.29) are not yet well established experimentally (as mentioned in Ref. 27), so in

the next section we consider the values to cover the limiting cases.

12.2.4. Analytic Solution for Symmetric Junctions

In the case of an SIS-junction, the quasi-particle source (10.4) has the form

(ignoring imbalance):

The analysis of kinetic equations proceeds in the same manner as for the case

of a UHF field. The nonequilibrium electron distribution function is given by the

expression

Even though we assumed to be small, the narrow injection source may drive

the distribution function to saturation

if We will consider the implications of (12.34) in Sect. 12.3.