In this paper we discuss the theory of microwave absorption in superconductors. In linear response to the microwave field E(t ) = E ω cos(ωt ), and in the limit of low frequencies ω, the current density in a superconductor may be written as
Here N s is the superfluid density, e and m are, respectively, the charge and the mass of the electron, and the superfluid momentum is defined by p s = h 2 ∇χ -e c A, with χ being the phase of the order parameter, and A the vector potential. The second term in Eq. (1), characterized by the conductivity σ , represents the dissipative part of the current.
The microwave absorption coefficient is controlled by the conductivity σ . The value of σ is determined by the quasiparticle scattering processes in the superconductor, which are generally characterized by two relaxation times: elastic, τ el , and inelastic, τ in , ones. In a typical situation, which we assume below, τ in τ el . The theory of transport phenomena in conventional superconductors was developed long ago, see, for example, Refs. [1][2][3][4][5]. The conventional result is that the conductivity, and consequently the microwave absorption coefficient, are proportional to the elastic relaxation time τ el . For example, at temperatures T near the critical temperature T c the conductivity of a superconductor σ is [3,4]
where σ D = e 2 ν n D is the Drude conductivity in the normal state, ν n is the normal state density of states at the Fermi level, D = v 2 F τ el /3 being the diffusion coefficient, and v F is the Fermi velocity. 1 Measurements of microwave absorption in s-wave superconductors in the absence of an applied dc supercurrent generally agree with the aforementioned theory [6,7]. However, there is a marked lack of experimental literature on measurements of microwave absorption in the presence of an applied dc supercurrent. The recent paper by Santavicca et al. [8] shows that the dependence of the microwave absorption in an s-wave superconductor on an applied dc supercurrent is not described by the conventional theory, as their measurements are several orders of magnitude larger than the conventional theory would suggest. We believe this dependence of the microwave absorption coefficient on an applied dc supercurrent could be described by our new contribution discussed below.
In this paper we discuss a novel contribution to the conductivity, σ DB , which is proportional to the inelastic relaxation time τ in . As a result, it may exceed the conventional contribution by orders of magnitude. This contribution to the linear conductivity exists only in the presence of a dc supercurrent. Furthermore, it is strongly anisotropic and depends on the relative orientation between E ω and the supercurrent. Even in situations where this contribution is small in comparison to the conventional result, it determines the dependence of the conductivity on both the magnitude and direction of the dc supercurrent. This enables determination of τ in , which is difficult to measure by other methods.
In the case of the nonlinear conductivity, a contribution proportional to τ in exists even at zero dc supercurrent. Consequently, the nonlinear threshold for microwave absorption turns out to be anomalously low.
The physical mechanism of this contribution to the conductivity is similar to the Debye mechanism of microwave absorption in gases [9], Mandelstam-Leontovich mechanism of the second viscosity in liquids [10], and Pollak-Geballe mechanism of microwave absorption in the hopping conductivity regime [11]. It can be qualitatively understood as follows. Let us separate the superfluid momentum p s (t ) = ps + δ p s (t ) into the dc part ps and the ac part δ p s (t ) whose time evolution is determined by the microwave field δ ṗs (t ) = eE(t ).
(
Below we assume ω , where is the pairing gap. In this regime the quasiparticles may be characterized by the instantaneous energy spectrum. Furthermore, for ω τ -1 el , the quasiparticle distribution function n depends only on the energy . Importantly, the density of states ν( , p s ) depends on the instantaneous value of the superfluid momentum p s . In other words, as the value of p s changes, individual quasiparticle levels move in energy space. At a nonzero temperature the quasiparticles occupying these levels travel in energy space as well. This motion creates a nonequilibrium quasiparticle distribution. Relaxation of the latter due to inelastic scattering causes entropy production and energy dissipation. The corresponding contribution to the conductivity is proportional to τ in . The reason why the Debye contribution to the linear conductivity exists only at ps = 0 is the following. The density of states is invariant under time reversal and thus can depend only on the magnitude of the condensate momentum p s = |p s |. As a result, in the linear in E approximation ν( ) changes in time proportionally to δ p s (t ) • ps .
The paper is organized as follows. In Sec. II we obtain a general formula for the Debye contribution to the microwave conductivity, σ DB , in terms of the superfluid-momentumdependent quasiparticle density of states ν( , p s ). In Sec. III we apply this result to evaluate the Debye contribution to the linear conductivity in s-wave superconductors at temperatures near the critical temperature. The results depend on the microwave frequency and the degree of disorder in the superconductor. In Sec. III A we focus on the clean case where the quasiparticles responsible for microwave absorption move ballistically. In Sec. III B we discuss the regime of diffusive motion of the relevant quasiparticles, which may be realized in both clean and dirty superconductors. In Sec. IV we study the nonlinear Debye conductivity in s-wave superconductors in the absence of a dc supercurrent. Technical details of the derivations are presented in Appendixes A and B. We discuss our results in Sec. V.
In this section we show that under very general conditions the Debye contribution to the microwave conductivity, σ DB , may be expressed in terms the quasiparticle density of states and its dependence on the superfluid momentum p s .
As noted above, under the condition ω quasiparticles may be characterized by an instantaneous energy spectrum. To describe the motion of the quasiparticle energy levels we note that the number of levels in the system is conserved. Therefore the density of states ν[ , p s (t )] is subject to the continuity equation in energy space ∂ t ν( , p s ) + ∂ [v ν ( )ν( , p s )] = 0, where v ν ( , p s ) is the level velocity in energy space. Using the condensate acceleration equation (3) we can express the latter in the form v ν ( , p s ) = eE • V ( , p s ), where
characterizes the sensitivity of the energy levels to changes of p s . The quasiparticle distribution function n( , t ) describes the occupancy of energy levels. In the absence of inelastic scattering its time evolution due to the spectral flow is described by the continuity equation ∂ t (νn) + ∂ (v ν νn) = 0. Combining this with the continuity equation for ν( , p s ) and allowing for inelastic collisions we obtain the kinetic equation
where I{n} is the collision integral describing inelastic scattering of quasiparticles.
The power W of microwave radiation absorbed per unit volume of the superconductor may be obtained by evaluating the rate of work performed by the microwave field on the quasiparticles, which is given by
Here . . . denotes time averaging. Below we characterize the absorption power by the dissipative part of the conductivity σ DB defined by
Linear regime. For an equilibrium distribution the integrand in Eq. ( 6) is a total derivative and W = 0. At small microwave fields we can linearize the kinetic equation ( 5) in E(t ) and the deviation of the quasiparticle distribution function from equilibrium, δn( ,
Below we assume that the temperature is near the critical temperature, |T -T c | T c . In this case the density of states is affected by the condensate momentum in a narrow energy window | -| T . Since the energy transfer in a typical inelastic collision is of order T the inelastic collision integral in Eq. ( 5) may be written in the relaxation time approximation,
where the inelastic relaxation time τ in (T ) depends only on the temperature T . For an isotropic spectrum, which we assume below for simplicity, the vector V ( , p s ) is parallel to p s . In this case only the longitudinal conductivity, which corresponds to E ω ps , is affected by inelastic relaxation. For a monochromatic electric field, the solution of the linearized kinetic equation ( 5), with the collision integral in the form Eq. ( 8), is given by
Next we substitute this expression into Eq. ( 6) for the rate of energy dissipation and note that in the relevant energy interval where the density of states depends appreciably on p s we may approximate -∂ n F ( ) ≈ 1 4T . Then, using Eq. ( 7) we obtain
Equation ( 10) expresses the Debye contribution to the conductivity in terms of the density of states in a currentcarrying superconductor. It applies to superconductors with arbitrary symmetry of the order parameter.
Below we focus on s-wave superconductors with an isotropic spectrum and assume v F ps . In this case the density of states is most strongly affected by the supercurrent at energies near the gap . Namely at ps = 0 the peak in the BCS density of states, ν( , 0) → ν n 2( -) at → , is broadened. The width and the shape of the broadening depends on the magnitude of the condensate momentum ps and the strength of disorder.
In the ballistic regime, v F ps τ 2 el 1, (which can be realized only in clean superconductors, τ el 1) the density of states can be found from a simple consideration. In this case one can use the standard expression for the quasiparticle spectrum [1,12,13]
where k is the quasiparticle momentum, ξ k is the electron energy measured from the Fermi level, and v k = dξ k /dk is the electron velocity. The density of states at | -| is given by
where z = ( -)/v F p s , and θ (z) is the Heavyside step function. The width of the broadening of the BCS peak is δ ∼ v F ps . The shape of the broadening is illustrated in Fig. 1.
Using Eqs. ( 4) and (10) we obtain for the Debye contribution to the conductivity in the ballistic regime
where the numerical coefficient I b = 8 45 is given by a definite integral defined in Eq. (A16) of Appendix A. The power-law dependence of σ DB on the condensate momentum ps follows from the simple scaling form of the density of states in Eq. (11). The exponent of this power-law dependence, σ DB ∝ √ ps , can be understood from the following consideration. The quasiparticle states whose energies are affected by the supercurrent lie in a narrow energy window of width δ ∼ v F ps near = . The number of such states per unit volume may be estimated as ν n √ v F p s . Since the characteristic level displacement in the microwave field is given by v F δ p s ∼ v F eE 0 /ω one obtains an estimate for the absorption power consistent with Eq. (12).
The above consideration of the density of states is valid as long the condition of ballistic motion v F ps τ el ( ) 1 is satisfied for most of the quasiparticles in the relevant energy interval | -| v F ps . Here τ el ( ) is the energy-dependent quasiparticle mean-free time, which for | -| is given by the standard expression τ -1 el ( ) ≈ τ -1 el 2( -) (see, for example, Ref. [14]). Therefore the regime of ballistic motion of quasiparticles participating in the Debye mechanism of microwave absorption is realized at relatively large supercurrent densities, where v F ps τ 2 el 1.
To study the Debye contribution to the conductivity outside the ballistic regime we express the density of states in terms of the disorder-averaged Green's functions. This enables us to utilize the standard theoretical methods developed in the theory of disordered superconductors [2,15]. We show in Appendix A that the density of states can be expressed as
where the variable x satisfies the quintic equation (A9a).
In the ballistic regime, v F p s τ 2 el 1, the latter reduces to the biquadratic equation (A11) whose solution, when substituted into Eq. ( 13), reproduces Eq. ( 11). In the opposite regime v F p s τ 2 el 1, which corresponds to diffusive motion of quasiparticles in the relevant energy interval, the quintic equation (A9a) simplifies to
Here ζ = v F p s / , γ = (τ el ) -1 , and w = ( -)/ . The solutions of this equation can be written in the scaling form x = ζ 2/3 γ 1/3 x( wγ 2/3 ζ 4/3 ), where the explicit form of x( wγ 2/3 ζ 4/3 ) is given by the Cardano formula, Eq. (A18). Substituting this form into Eq. ( 13) [the corresponding ν( , p s ) is plotted in Fig. 1], and using Eqs. ( 4) and (10), we obtain
where the numerical coefficient I d ≈ 0.0549 is given by a definite integral defined in Eq. (A23) of Appendix A. This expression is consistent with the result obtained in Ref. [4] by a different method.
The exponent of the power-law dependence σ DB ∝ p4/3 s in Eq. ( 15) and its order of magnitude can be obtained by noting that the width of the broadening of the BCS singularity in the diffusive regime is δ ∼ ( D 2 p4 s ) 1/3 and the number per unit volume of levels that participate in microwave absorption is ∼ν n ( 2 D p2 s ) 1/3 . It is worth noting that the diffusive regime can be realized in both clean, τ el 1, and dirty τ el 1, superconductors. Accordingly Eqs. ( 13) and ( 14) for the density of states and the resulting conductivity (15) can be obtained using either the Gor'kov equations or the Usadel equation, see Appendix A.
Let us now consider the situation in which the dc supercurrent is absent, ps = 0. In the presence of the microwave field the oscillation amplitude of the condensate momentum is given by δ p s = eE ω /ω. Since the Debye contribution to the nonlinear conductivity defined by Eq. ( 7) is proportional to τ in the nonlinear threshold for the microwave absorption is anomalously low. To evaluate microwave absorption in the nonlinear regime it is convenient to introduce the integrated over energy density of states
and consider the quasiparticle distribution function not as a function of energy and time t but rather as a function of N and t. The change of variables n( , t ) → n(N, t ) is equivalent to the transformation from Eulerian to Lagrangian variables in fluid mechanics [10]. In this representation the kinetic equation ( 5) acquires a very simple form,
Note that the electric field enters this equation only via the time dependence of the quasiparticle energy level (N, t ).
In the presence of the microwave field the latter undergoes nonlinear oscillations (N, t ) = 0 (N ) + δ (N, t ) whose form is determined by Eq. ( 16). Note that the linearization of the collision integral is justified in the nonlinear regime as long as the amplitude of δ (N, t ) is small as compared to T . The solution of Eq. ( 17) can be written as n(N, t
τ in δ (N, t -τ ). The absorption power per unit volume in this representation is given by W = ∞ 0 dN n(N, t )∂ t (N, t ) . Substituting the solution for n(N, t ) into this expression we get
Here the level velocity is given by ∂ t (N, t ) = v ν ( , t ) = eE • V ( , p s ), with V ( , p s ) defined in Eq. ( 4). This can be shown by taking the time derivative of (16). Equation ( 18) expresses the power of nonlinear microwave absorption in terms of the correlation function of level velocities ∂ t (N, t ), which are defined by Eq. ( 16). Similarly to the linear regime, the results depend of the degree of disorder, the amplitude of the microwave field E ω and the frequency of radiation ω. They simplify in the ballistic regime eE ω v F τ 2 el ω and in the diffusive regime eE ω v F τ 2 el ω where the nonlinear conductivity has a simple power-law dependence on the amplitude of the microwave field E ω . In the ballistic regime we obtain
while in the diffusive regime we find
The functions F b (ωτ in ) and F d (ωτ in ) that describe the frequency dependence of nonlinear microwave conductivity are given Eqs. (B5) and (B11) in Appendix B. Although, in contrast to Eqs. ( 12) and ( 15), they do not have a simple Lorentzian form, their high-and low-frequency asymptotic behavior is similar; at low frequency F b (0) ≈ 0.10848 and F d (0) ≈ 0.10909 while at high frequencies they behave as 1/(ωτ in ) 2 .
In summary, we have identified a mechanism of microwave absorption in superconductors, which arises from the motion of quasiparticle energy levels in the presence of microwave radiation. The corresponding contributions to both the nonlinear microwave conductivity, and to the linear conductivity in the presence of a dc supercurrent are proportional to the energy relaxation time τ in . Therefore, they can be several orders of magnitude larger than the conventional ones. Let us now discuss conditions of applicability of our main results.
The nonanalytic dependences of σ DB on ps in Eqs. ( 12) and ( 15), and of σ nl DB on E ω and ω in Eqs. ( 19) and ( 20) are related to the divergence of the BCS density of states at = . In real superconductors this divergence is smeared by the anisotropy of the order parameter in k space and by pair-breaking processes, which are characterized by a broadening parameter
. Consequently, at δ ( ps ) the ps dependence of the linear conductivity should become analytic, σ DB = c p2 s . The magnitude of the coefficient c can be estimated by matching this expression to Eqs. ( 12) and ( 15) at the values of ps determined by the condition that the energy broadening of the BCS singularity, δ ( ps ) be of order .
So far we focused our consideration on the temperature interval near T c . At low temperatures, T , the dimensionless quasiparticle concentration x = (ν n ) -1 ∞ 0 d ν( )n F ( ) in s-wave superconductors is exponentially small; x ∼ T exp(-/T ). Consequently the conventional contribution to the microwave absorption coefficient, which is proportional to τ el , is exponentially small as well. It is interesting that the Debye contribution to the microwave absorption coefficient in this regime does not contain the exponentially small factor and is, roughly speaking, comparable to that near T c . Indeed, at T there are two quasiparticle inelastic relaxation rates in superconductors. The quasiparticle-phonon relaxation processes that conserve the number of quasiparticles are characterized by the rate 1/τ (st) in (T ) (which is of the same order as a electron-phonon relaxation rate in normal metals, see, for example, Ref. [3]). The second type of inelastic relaxation processes correspond to recombination, which changes the total number of quasiparticles. The recombination rate is proportional to the quasiparticle concentration 1/τ r ∼ x/τ (0) r τ (st) in , where τ (0) r ∼ τ (st) in ( ). The Debye contribution to the dissipative kinetic coefficients is proportional to the longest relaxation time in a system (see, for example, Ref. [10]). At T it is τ r . Therefore, in the low-frequency limit, ωτ r 1, the exponentially small factor exp(-/T ) is canceled from the expression for the conductivity. Below we illustrate this fact for the linear Debye conductivity in the diffusive regime, and at δ ( ps ) T . In this case the magnitude of the level velocity in the interval of energies of order T is V ∼ 1 ps δ ∼ ( D 2 ps ) 1/3 and we get from Eq. ( 10)
The situation with a spatially uniform supercurrent density that was considered above can only be realized in thin superconducting films. In bulk superconductors in the presence of a magnetic field H < H c1 that is parallel to the surface ps is nonzero only within the London penetration depth λ L near the surface. The mechanism of microwave absorption discussed above will still apply in this geometry and the presented above results still hold up to a numerical factor of order unity. The reason for this is that the quasiparticles that give the main contribution to microwave absorption have energies that lie in a narrow interval near the gap, | -| δ , where δ = v F ps in the ballistic regime and δ = ( D 2 p4 s ) 1/3 in the diffusive regime. Roughly half of these quasiparticles have energies below and therefore they are trapped near the surface within a distance of order λ L .
The microwave absorption coefficient in thin films of dwave superconductors should also be proportional to τ in . However, its dependence on ps is expected to be different from those in Eqs. ( 12), ( 15), (19), and (20). Moreover, in bulk samples of d-wave superconductors in the presence of a magnetic field parallel to the surface the situation is substantially different. In a gapless superconductor the quasiparticles in the relevant energy interval can diffuse into the bulk. Therefore in this case the characteristic relaxation time is given by the minimum between the inelastic relaxation time and the time of diffusion from the surface layer of thickness λ L .
Finally, we would like to note that the considered above mechanism of the microwave absorption is closely related to the mechanism of ac conductivity of SNS junctions discussed in Refs. [16][17][18]. conductivity with I b given by
Substituting Eq. (A15) into Eq. (A16) we obtain
At small supercurrent densities, ζ /γ 2 1, the relevant root of Eq. (A9a) satisfies the condition |x| γ . In this regime, which corresponds to diffusive motion of quasiparticles participating in microwave absorption, Eq. (A9a) simplifies to the cubic equation
Using the Cardano formula and substituting the root with x -1 0 into Eq. (A10) we can express the density of states in terms of rescaled variables η ≡ 2ζ 2 3γ , and w = wη -2/3 in the form ν( , p s ) = ν n νd ( w)
where the function α( w) is defined by α( w) = (4 w3 + 27 + √ 27 8 w3 + 27) 1/3 . (A21)
Substituting this form into Eq. ( 4) we obtain
where D is the diffusion coefficient and I ( w) is given by
Using Eqs. (A22) and (A19) in Eq. ( 10) we get Eq. ( 15) for the longitudinal Debye conductivity where the definite integral I d is given by
In dirty superconductors, when τ el 1, the density of states may be evaluated using the Usadel equation [20], which has the following form for the retarded Green's function
Here [ , ] denotes the commutator and the hat indicates 2 × 2 matrices in Gor'kov-Nambu space, which are given by τ3
The Green's function satisfies the nonlinear constraint ĝR s • ĝR s = 1, and can be expressed in terms of the angles θ and χ in the form g R s = cos θ, F R s = e iχ sin θ . The density of states is given by ν
and Eq. (A24) reduces to
Here p s = ∇χ . As we are considering a thin film we assume p s to be spatially uniform. Then Eq. (A28b) yields ∇θ = 0, while Eq. (A28a) reduces to
where w ≡ ( -)/ as before, η ≡ / , and we have used the fact that we are concerned only with the energy range the nonlinear conductivity are complicated functions of E ω and ω. The situation simplifies dramatically in the limiting regimes v F eE ω τ 2 el /ω 1, and v F eE ω τ 2 el /ω 1. In these regimes the nonlinear conductivity has a simple power-law dependence on E ω and ω.
For v F eE ω τ 2 el /ω 1 the quasiparticles contributing to the Debye conductivity are in the ballistic regime and the density of states may be described by Eq. (A13). In this case the time-dependent width of the energy window in which the density of states is affected by microwave radiation is δ (t ) ∼ v F p s (t ). The characteristic density of quasiparticle levels within the energy window where ν(ε, t ) is affected by microwave radiation may be estimated as δN ∼ ν n √ v F eE ω /ω. Therefore it is convenient to introduce a rescaled level density
and express the time-dependent energy (N, t ) in terms of a new function z N b (ωt ) as
Substituting the change of variables (B2), (B3) into Eq. (B1) and using the form of the density of states in the ballistic regime, Eqs. (A13) and (A14), we find that the dependence of z N (φ) on the phase of the microwave field φ = ωt is determined by the equation
which does not contain the amplitude E ω . Substituting change of variables (B3) into Eq. ( 18) we find that the nonlinear conductivity has a simple power law dependence on E ω given by Eq. ( 19) with the function F b (x) given by , which can most readily be seen by Fourier transforming Eq. ( 18).
At v F eE ω τ 2 el /ω 1 the quasiparticles contributing to the Debye conductivity are in the diffusive regime, and the density of states may be described by Eq. (A19). In this case the timedependent broadening of the singularity in the quasiparticle density of states is δ (t ) ∼ [ D where . . . denotes averaging over half the oscillation period, and I d is given by Eq. (A23). In the high-frequency limit, ωτ in 1, F d (ωτ in ) ∼ 1 ω 2 τ 2 in , which is most easily seen by Fourier transforming Eq. (18).
At small frequencies the logarithmic divergence in Eq. (2) is cut off by taking into account inelastic quasiparticle scattering, the gap anisotropy, and nonuniformity of the electron interaction constant.