Debye mechanism of giant microwave absorption in superconductors

TL;DR

This paper proposes a Debye-like mechanism for microwave absorption in superconductors, showing it depends on inelastic quasiparticle relaxation time and can significantly enhance absorption, especially in nonlinear regimes.

Contribution

It introduces a novel microwave absorption mechanism in superconductors linked to inelastic quasiparticle relaxation, differing from conventional elastic-based models.

Findings

Absorption is proportional to inelastic relaxation time $ au_{in}$.

The mechanism depends on the orientation of microwave field relative to supercurrent.

Nonlinear absorption threshold is unusually low due to $ au_{in}$ dependence.

Abstract

We discuss a mechanism of microwave absorption in conventional superconductors which is similar to the Debye absorption mechanism in molecular gases. The contribution of this mechanism to the \emph{ac} conductivity is proportional to the inelastic quasiparticle relaxation time $\tau_\mathrm{\mathrm{in}}$ rather than the elastic one $\tau_{\mathrm{el}}$ and therefore it can be much larger than the conventional one. The Debye contribution to the linear conductivity arises only in the presence of a \emph{dc} supercurrent in the system and its magnitude depends strongly on the orientation of the microwave field relative to the supercurrent. The Debye contribution to the nonlinear conductivity exists even in the absence of \emph{dc} supercurrent. Since it is proportional to $\tau_{\mathrm{in}}$ the nonlinear threshold is anomalously low. Microwave absorption measurements may provide direct information about $\tau_\mathrm{in}$ in superconductors.

Figures

Equations

$${\bf j}=\frac{e}{m}N_{\mathrm{s}}\,\mbox{\boldmath{$p$}}_{\mathrm{s}}+\sigma{\bf E}.$$

$$\sigma ={\sigma }_{\mathrm{D}}\ln \frac{T_c}{\omega },$$

$$\delta\dot{\mbox{\boldmath{$p$}}}_{s}(t)=e{\bf E}(t).$$

$$\mbox{\boldmath{$V$}}(\epsilon,\mbox{\boldmath{$p$}}_{\mathrm{s}})=-\frac{1}{\nu(\epsilon,p_{\mathrm{s}})}\int_{0}^{\epsilon}d\tilde{\epsilon}\frac{\partial\nu(\tilde{\epsilon},p_{\mathrm{s}})}{\partial\mbox{\boldmath{$p$}}_{\mathrm{s}}}$$

$$\partial_{t}n(\epsilon,t)+e\mbox{\boldmath{$E$}}(t)\cdot\mbox{\boldmath{$V$}}(\epsilon,\mbox{\boldmath{$p$}}_{\mathrm{s}})\,\partial_{\epsilon}n(\epsilon,t)=I\{n\},$$

$$W=\int_{0}^{\infty}d\epsilon\big{\langle}\nu(\epsilon,p_{s}(t))n(\epsilon,t)e\mbox{\boldmath{$E$}}(t)\cdot\mbox{\boldmath{$V$}}(\epsilon,\mbox{\boldmath{$p$}}_{\mathrm{s}}(t))\big{\rangle}.$$

$$\frac{{\sigma }_{\mathrm{DB}}}{2}{E}_{\omega }^2=W.$$

$$I\{n\}=-\frac{\delta n(ϵ,t)}{{\tau }_{\mathrm{in}}},$$

$$\frac{{\sigma }_{\mathrm{DB}}}{{\sigma }_{\mathrm{D}}}=\frac{3{\tau }_{\mathrm{in}}}{4{\tau }_{\mathrm{el}}}\frac{1}{\left[1+{\left(\omega {\tau }_{\mathrm{in}}\right)}^2\right]}{\int }_0^{\infty }\frac{dϵ}{T}\frac{\nu (ϵ,{\bar{p}}_{\mathrm{s}})V^2(ϵ,{\bar{p}}_{\mathrm{s}})}{{\nu }_n{v}_{\mathrm{F}}^2}.$$

$$\frac{\nu (ϵ,p_s)}{{\nu }_n}=\sqrt{\frac{\Delta }{2v_{\mathrm{F}}{p}_{\mathrm{s}}}}\left[\theta (z+1)\sqrt{z+1}-\theta (z-1)\sqrt{z-1}\right],$$

$$\frac{{\sigma }_{\mathrm{DB}}}{{\sigma }_{\mathrm{D}}}=I_{\mathrm{b}}\frac{{\tau }_{\mathrm{in}}}{{\tau }_{\mathrm{el}}\left[1+(\omega {\tau }_{\mathrm{in}}{)}^2\right]}\frac{\Delta }{T}\sqrt{\frac{v_{\mathrm{F}}{\bar{p}}_{\mathrm{s}}}{\Delta }}$$

$$\frac{\nu (ϵ,p_{\mathrm{s}})}{{\nu }_n}=\frac{1}{\sqrt{2}}\Im x^{-1},$$

$$x\left(x^2+w\right)+\frac{\sqrt{2}{\zeta }^2}{3\gamma }=0.$$

$$\frac{{\sigma }_{\mathrm{DB}}}{{\sigma }_{\mathrm{D}}}=I_{\mathrm{d}}\frac{{\tau }_{\mathrm{in}}}{{\tau }_{\mathrm{el}}}\frac{\Delta }{T}\frac{{\tau }_{\mathrm{el}}{\left(\Delta D^2{\bar{p}}_{\mathrm{s}}^4\right)}^{1/3}}{1+{\left(\omega {\tau }_{\mathrm{in}}\right)}^2},$$

$$N(ϵ,t)={\int }_0^{ϵ}d\epsilon \nu (\epsilon ,t)$$

$${\partial }_{t}n(N,t)=I\{n\}=-\frac{n(N,t)-n_F(ϵ(N,t))}{{\tau }_{\mathrm{in}}}.$$

$$W={\int }_0^{\infty }dN{\int }_0^{\infty }\frac{d\tau }{4T}{e}^{-\frac{\tau }{{\tau }_{\mathrm{in}}}}⟨{\partial }_tϵ(N,t){\partial }_tϵ(N,t-\tau )⟩.$$

$$\frac{{\sigma }_{\mathrm{DB}}^{\mathrm{nl}}}{{\sigma }_{\mathrm{D}}}=\frac{{\tau }_{\mathrm{in}}}{{\tau }_{\mathrm{el}}}\frac{\Delta }{T}\sqrt{\frac{v_{\mathrm{F}}e{E}_{\omega }}{\omega \Delta }}{F}_{\mathrm{b}}(\omega {\tau }_{\mathrm{in}}),$$

$$\frac{{\sigma }_{\mathrm{DB}}^{\mathrm{nl}}}{{\sigma }_{\mathrm{D}}}=\frac{{\tau }_{\mathrm{in}}}{{\tau }_{\mathrm{el}}}\frac{\Delta }{T}\frac{{\Delta }^{1/3}{D}^{5/3}|e{E}_{\omega }{|}^{4/3}}{v_F^2{\omega }^{4/3}}{F}_{\mathrm{d}}(\omega {\tau }_{\mathrm{in}}).$$

$$\frac{{\sigma }_{\mathrm{DB}}}{{\sigma }_{\mathrm{D}}}\sim \frac{{\tau }_r^{(0)}}{{\tau }_{\mathrm{el}}}{\left[\left(\Delta {\tau }_{el}\right)\left(v_F{p}_s{\tau }_{\mathrm{el}}\right)\right]}^{2/3}.$$

Full text

In this article we discuss the theory of microwave absorption in superconductors. In linear response to the microwave field \mbox{\boldmath{E}}(t)=\mbox{\boldmath{E}}_{\omega}\cos(\omega t), and in the limit of low frequencies $\omega$, the current density in a superconductor may be written as

[TABLE]

Here $N_{\mathrm{s}}$ is the superfluid density, $e$ and $m$ are, respectively, the charge and the mass of the electron, and the superfluid momentum is defined by \mbox{\boldmath{p}}_{s}=\frac{\hbar}{2}\mbox{\boldmath{\nabla}}\chi-\frac{e}{c}\mbox{\boldmath{A}}, with $\chi$ being the phase of the order parameter, and $A$ the vector potential. The second term in Eq. (1), characterized by the conductivity $\sigma$, represents the dissipative part of the current.

The microwave absorption coefficient is controlled by the conductivity $\sigma$. The value of $\sigma$ is determined by the quasiparticle scattering processes in the superconductor, which are generally characterized by two relaxation times: elastic, $\tau_{\mathrm{el}}$, and inelastic, $\tau_{\mathrm{in}}$, ones. In a typical situation, which we assume below, $\tau_{\mathrm{in}}\gg\tau_{\mathrm{el}}$. The theory of transport phenomena in conventional superconductors was developed long ago, see for example  Schrieffer (1964); Larkin and Ovchinnikov (1977); Aronov et al. (1981); Ovchinnikov and Isaakyan (1978); Mattis and Bardeen (1958). The conventional result is that the conductivity, and consequently the microwave absorption coefficient, are proportional to the elastic relaxation time $\tau_{\mathrm{el}}$. For example, at temperatures $T$ near the critical temperature $T_{c}$ the conductivity of a superconductor $\sigma$ is Aronov et al. (1981); Ovchinnikov and Isaakyan (1978)

[TABLE]

where $\sigma_{\mathrm{D}}$ is Drude conductivity of a normal metal 111Equation (2) applies at sufficiently high frequencies.

In this article we discuss another contribution to the conductivity, $\sigma_{\mathrm{DB}}$, that is proportional to the inelastic relaxation time $\tau_{\mathrm{in}}$. 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, this contribution is strongly anisotropic and depends on the relative orientation between \mbox{\boldmath{E}}_{\omega} 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 $\tau_{\mathrm{in}}$ from measurements of microwave absorption. A contribution to the nonlinear conductivity which is proportional to $\tau_{in}$ 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 Debye (1970), Mandelstam-Leontovich mechanism of the second viscosity in liquids Landau and Lifshitz (2013), and Pollak-Geballe mechanism of microwave absorption in the hopping conductivity regime Pollak and Geballe (1961). It can be qualitatively understood as follows. Let us separate the superfluid momentum \mbox{\boldmath{p}}_{\mathrm{s}}(t)=\bar{\mbox{\boldmath{p}}}_{\mathrm{s}}+\delta\mbox{\boldmath{p}}_{\mathrm{s}}(t) into a dc part \bar{\mbox{\boldmath{p}}}_{\mathrm{s}} and the ac part \delta\mbox{\boldmath{p}}_{\mathrm{s}}(t) whose time evolution is determined by the microwave field

[TABLE]

At low frequencies, $\omega\ll\tau_{\mathrm{el}}^{-1}$, the quasiparticle distribution function $n$ depends only on the energy $\epsilon$. Importantly, the density of states per unit energy, $\nu(\epsilon,p_{\mathrm{s}})$, depends on the instantaneous value of the superfluid momentum $p_{\mathrm{s}}$. In other words, as the value of ${\bf p}_{s}$ changes individual quasiparticle levels move in energy space. At finite temperature the quasiparticles occupying these levels travel in energy space as well. This motion creates a non-equilibrium quasiparticle distribution. The relaxation of the latter due to inelastic scattering causes entropy production and energy dissipation. The corresponding contribution to the conductivity is proportional to $\tau_{in}$. The reason why the Debye contribution to the linear conductivity exists only at \bar{\mbox{\boldmath{p}}}_{\mathrm{s}}\neq 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_{\mathrm{s}}=|\mbox{\boldmath{p}}_{\mathrm{s}}|. As a result, in the linear in $E$ approximation $\nu(\epsilon)$ changes in time proportionally to \delta\mbox{\boldmath{p}}_{\mathrm{s}}(t)\cdot\bar{\mbox{\boldmath{p}}}_{\mathrm{s}}.

To describe the motion of energy levels we note that the number of levels in the system is conserved. Therefore the density of states $\nu(\epsilon,p_{\mathrm{s}}(t))$ is subject to the continuity equation in energy space $\partial_{t}\nu(\epsilon,p_{s})+\partial_{\epsilon}[v_{\nu}(\epsilon)\nu(\epsilon,p_{s})]$=0, where $v_{\nu}(\epsilon,p_{s})$ is the level “velocity” in energy space. Using the condensate acceleration equation (3) we can express the latter in the form v_{\nu}(\epsilon,p_{s})=e\mbox{\boldmath{E}}\cdot\mbox{\boldmath{V}}(\epsilon,\mbox{\boldmath{p}}_{\mathrm{s}}), where

[TABLE]

characterizes the sensitivity of the energy levels to changes of ${\bf p}_{s}$. The quasiparticle distribution function $n(\epsilon,t)$ 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 $\partial_{t}(\nu n)+\partial_{\epsilon}(v_{\nu}\nu n)=0$. Combining this with the continuity equation for $\nu(\epsilon,p_{s})$ and allowing for inelastic collisions we obtain the kinetic equation

[TABLE]

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

[TABLE]

Here $\langle\ldots\rangle$ denotes time averaging. Below we characterize the absorption power by the dissipative part of the conductivity $\sigma_{\mathrm{DB}}$ defined by

[TABLE]

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 ${\bf E}(t)$ and the deviation of the quasiparticle distribution function from equilibrium, $\delta n(\epsilon,t)=n(\epsilon,t)-n_{F}(\epsilon)\ll 1$ (here $n_{F}(\epsilon)=[\exp(\epsilon/T)+1]^{-1}$ is the Fermi function). Below we assume that the temperature is near the critical temperature, $|T-T_{c}|\ll T_{c}$. In this case the density of states is affected by the condensate momentum in a narrow energy window $|\epsilon-\Delta|\ll T$. 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,

[TABLE]

where the inelastic relaxation time $\tau_{\mathrm{in}}(T)$ depends only on the temperature $T$.

For an isotropic spectrum, which we assume below for simplicity, the vector \mbox{\boldmath{V}}(\epsilon,\mbox{\boldmath{p}}_{\mathrm{s}}) is parallel to \mbox{\boldmath{p}}_{\mathrm{s}}. In this case only the longitudinal conductivity, which corresponds to \mbox{\boldmath{E}}_{\omega}\parallel\bar{\mbox{\boldmath{p}}}_{\mathrm{s}}, is affected by inelastic relaxation. Solving the linearized kinetic equation in the relaxation time approximation (8), substituting the result into Eq. (6), and using the relation (7) we obtain the ratio of the longitudinal conductivity $\sigma_{\mathrm{DB}}$ to the Drude conductivity $\sigma_{\mathrm{D}}$ at $|T-T_{c}|\ll T_{c}$

[TABLE]

Here $\nu_{n}$ is the normal state density of states at the Fermi level, $v_{\mathrm{F}}$ is the Fermi velocity, and $\sigma_{\mathrm{D}}=e^{2}\nu_{n}D$, with $D=v_{F}^{2}\tau_{\mathrm{el}}/3$ being the diffusion coefficient. Equation (9) expresses the Debye contribution to the conductivity in terms of the density of states in a current-carrying 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 $v_{F}\bar{p}_{\mathrm{s}}\ll\Delta$. In this case the density of states is most strongly affected by the supercurrent at energies near the gap $\Delta$. Namely at $\bar{p}_{\mathrm{s}}\neq 0$ the peak in the BCS density of states, $\nu(\epsilon,0)\to\nu_{n}\sqrt{\frac{\Delta}{2(\epsilon-\Delta)}}$ at $\epsilon\to\Delta$, is broadened. The width and the shape of the broadening depends on the magnitude of the condensate momentum $\bar{p}_{s}$ and the strength of disorder.

In the ballistic regime, $v_{\mathrm{F}}\bar{p}_{\mathrm{s}}\tau_{\mathrm{el}}^{2}\Delta\gg 1$, (which can be realized only in clean superconductors, $\Delta\tau_{\mathrm{el}}\gg 1$) the density of states can be found from a simple consideration. In this case one can use the standard expression for the quasiparticle spectrum Schrieffer (1964); Tinkham (2004); Gennes (2018), \epsilon(\mbox{\boldmath{k}})=\sqrt{\xi_{\mbox{\boldmath{k}}}^{2}+|\Delta|^{2}}+\mbox{\boldmath{v}}_{\mbox{\boldmath{k}}}\cdot\bar{\mbox{\boldmath{p}}}_{\mathrm{s}}, where $k$ is the quasiparticle momentum, \xi_{\mbox{\boldmath{k}}} is the electron energy measured from the Fermi level, and \mbox{\boldmath{v}}_{\mbox{\boldmath{k}}}=d\xi_{\mbox{\boldmath{k}}}/d\mbox{\boldmath{k}} is the electron velocity. The density of states at $|\epsilon-\Delta|\ll\Delta$ is given by

[TABLE]

where $z=(\epsilon-\Delta)/v_{F}p_{\mathrm{s}}$, and $\theta(z)$ is the Heavyside step-function. The width of the broadening of the BCS peak is $\delta\epsilon\sim v_{F}\bar{p}_{\mathrm{s}}$. The shape of the broadening is illustrated in Fig. 1. Using Eq. (4) and Eq. (9) we obtain for the Debye contribution to the conductivity in the ballistic regime

[TABLE]

where $I_{\mathrm{b}}=\frac{8}{45}$ is a dimensionless integral defined in equation (S.16) of the Supplementary Material 222See Supplementary Material at (URL) for derivations of Eqs. (11), (14), (18), and (19).. The powerlaw dependence of $\sigma_{\mathrm{DB}}$ on the condensate momentum $\bar{p}_{\mathrm{s}}$ follows from the simple scaling form of the density of states in Eq. (10). The exponent of this power law dependence, $\sigma_{\mathrm{DB}}\propto\sqrt{\bar{p}_{\mathrm{s}}}$, 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 $\delta\epsilon\sim v_{\mathrm{F}}\bar{p}_{\mathrm{s}}$ near $\epsilon=\Delta$. The number of such states per unit volume may be estimated as $\nu_{n}\sqrt{\Delta v_{\mathrm{F}}p_{\mathrm{s}}}$. Since the characteristic level displacement in the microwave field is given by $v_{\mathrm{F}}\delta p_{\mathrm{s}}\sim v_{\mathrm{F}}eE_{0}/\omega$ one obtains an estimate for the absorption power consistent with Eq. (11).

The above consideration of the density of states is valid as long the condition of ballistic motion $v_{\mathrm{F}}\bar{p}_{\mathrm{s}}\tau_{\mathrm{el}}(\epsilon)\gg 1$ is satisfied for most of the quasiparticles in the relevant energy interval $|\epsilon-\Delta|\lesssim v_{\mathrm{F}}\bar{p}_{\mathrm{s}}$. Here $\tau_{\mathrm{el}}(\epsilon)$ is the energy-dependent quasiparticle mean free time, which for $|\epsilon-\Delta|\ll\Delta$ is given by the standard expression $\tau^{-1}_{\mathrm{el}}(\epsilon)\approx\tau^{-1}_{\mathrm{el}}\sqrt{\frac{2(\epsilon-\Delta)}{\Delta}}$ (see for example Ref. Mineev and Samokhin, 1999). Therefore the regime of ballistic motion of quasiparticles participating in the Debye mechanism of microwave absorption is realized at relatively large supercurrent densities, where $v_{\mathrm{F}}\bar{p}_{\mathrm{s}}\tau_{\mathrm{el}}^{2}\Delta\gg 1$.

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 Larkin and Ovchinnikov (1977); Maki (1969). We show in the Supplementary Material that the density of states can be expressed as

[TABLE]

where the variable $x$ satisfies the quintic equation (S.9a). In the ballistic regime, $v_{\mathrm{F}}p_{\mathrm{s}}\tau_{\mathrm{el}}^{2}\Delta\gg 1$, the latter reduces to the biquadratic equation (S.11) whose solution, when substituted into Eq. (12), reproduces Eq. (10). In the opposite regime $v_{\mathrm{F}}p_{\mathrm{s}}\tau_{\mathrm{el}}^{2}\Delta\ll 1$, which corresponds to diffusive motion of quasiparticles in the relevant energy interval, the quintic equation (S.9a) simplifies to

[TABLE]

Here $\zeta=v_{\mathrm{F}}p_{\mathrm{s}}/\Delta$, $\gamma=(\tau_{\mathrm{el}}\Delta)^{-1}$, and $w=(\epsilon-\Delta)/\Delta$. The solutions of this equation can be written in the scaling form $x=\frac{\zeta^{2/3}}{\gamma^{1/3}}\,\tilde{x}\left(\frac{w\gamma^{2/3}}{\zeta^{4/3}}\right)$, where the explicit form of $\tilde{x}\left(\frac{w\gamma^{2/3}}{\zeta^{4/3}}\right)$ is given by the Cardano formula, Eq. (S.17). Substituting this form into Eq. (12) (the corresponding $\nu(\epsilon,p_{\mathrm{s}})$ is plotted in Fig. 1), and using Eqs. (4) and (9), we obtain

[TABLE]

where $I_{\mathrm{d}}\approx 0.0549$ is defined in Eq. (S.22) in the Supplementary Material. This expression is consistent with the result obtained in Ref. Ovchinnikov and Isaakyan (1978) by a different method.

The exponent of the powerlaw dependence $\sigma_{\mathrm{D}B}\propto\bar{p}_{\mathrm{s}}^{4/3}$ in Eq. (14) 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 $\delta\epsilon\sim(\Delta D^{2}\bar{p}_{\mathrm{s}}^{4})^{1/3}$ and the number per unit volume of levels that participate in microwave absorption is $\sim\nu_{n}\left(\Delta^{2}D\bar{p}_{\mathrm{s}}^{2}\right)^{1/3}$.

It is worth noting that the diffusive regime can be realized in both clean, $\Delta\tau_{\mathrm{el}}\ll 1$, and dirty $\Delta\tau_{\mathrm{el}}\gg 1$, superconductors. Accordingly Eqs. (12) and (13) for the density of states and the resulting conductivity (14) can be obtained using either the Gorkov equations or the Usadel equation, see Supplementary Material.

Nonlinear regime. Let us now consider the situation in which the dc supercurrent is absent, $\bar{p}_{\mathrm{s}}=0$. In the presence of the microwave field the oscillation amplitude of the condensate momentum is given by $\delta p_{\mathrm{s}}=eE_{\omega}/\omega$. Since the Debye contribution to the nonlinear conductivity defined by Eq. (7) is proportional to $\tau_{in}$ 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

[TABLE]

and consider the quasiparticle distribution function not as a function of energy $\epsilon$ and time $t$ but rather as a function of $N$ and $t$. The change of variables $n(\epsilon,t)\to n(N,t)$ is equivalent to the transformation from Eulerian to Lagrangian variables in fluid mechanics Landau and Lifshitz (2013). In this representation the kinetic equation (5) acquires a very simple form,

[TABLE]

Note that the electric field enters this equation only via the time-dependence of the quasiparticle energy level $\epsilon(N,t)$. In the presence of the microwave field the latter undergoes nonlinear oscillations $\epsilon(N,t)=\epsilon_{0}(N)+\delta\epsilon(N,t)$ whose form is determined by Eq. (15). Note that the linearization of the collision integral is justified in the nonlinear regime as long as the amplitude of $\delta\epsilon(N,t)$ is small as compared to $T$. The solution of Eq. (16) can be written as $n(N,t)=n_{F}(\epsilon_{0}(N))+\frac{dn_{F}(\epsilon_{0}(N))}{d\epsilon_{0}(N)}\int_{0}^{\infty}\frac{d\tau}{\tau_{\mathrm{in}}}e^{-\frac{\tau}{\tau_{\mathrm{in}}}}\delta\epsilon(N,t-\tau)$. The absorption power per unit volume in this representation is given by $W=\int_{0}^{\infty}dN\left\langle n(N,t)\partial_{t}\epsilon(N,t)\right\rangle$. Substituting the solution for $n(N,t)$ into this expression we get

[TABLE]

Here the level velocity is given by \partial_{t}\epsilon(N,t)=v_{\nu}(\epsilon,t)=e\mbox{\boldmath{E}}\cdot\mbox{\boldmath{V}}(\epsilon,\mbox{\boldmath{p}}_{\mathrm{s}}), with \mbox{\boldmath{V}}(\epsilon,\mbox{\boldmath{p}}_{\mathrm{s}}) defined in Eq. (4). This can be shown by taking the time derivative of (15).

Equation (17) expresses the power of nonlinear microwave absorption in terms of the correlation function of level velocities $\partial_{t}\epsilon(N,t)$, which are defined by Eq. (15). Similarly to the linear regime, the results depend of the degree of disorder, the amplitude of the microwave field $E_{\omega}$ and the frequency of radiation $\omega$. They simplify in the ballistic regime $eE_{\omega}v_{\mathrm{F}}\Delta\tau_{\mathrm{el}}^{2}\gg\omega$ and in the diffusive regime $eE_{\omega}v_{\mathrm{F}}\Delta\tau_{\mathrm{el}}^{2}\ll\omega$ where the nonlinear conductivity has a simple powerlaw dependence on the amplitude of the microwave field $E_{\omega}$. In the ballistic regime we obtain

[TABLE]

while in the diffusive regime we find

[TABLE]

The functions $F_{\mathrm{b}}(\omega\tau_{\mathrm{in}})$ and $F_{\mathrm{d}}(\omega\tau_{\mathrm{in}})$ that describe the frequency dependence of nonlinear microwave conductivity are given Eqs. (S.35) and (S.41) in the Supplementary Material. Although, in contrast to Eqs. (11) and (14), they do not have a simple Lorentzian form, their high- and low-frequency asymptotic behavior is similar; at low frequency $F_{\mathrm{b}}(0)\approx 0.10848$ and $F_{\mathrm{d}}(0)\approx 0.10909$ while at high frequencies they behave as $1/(\omega\tau_{\mathrm{in}})^{2}$.

The nonanalytic dependences of $\sigma_{\mathrm{DB}}$ on $\bar{p}_{\mathrm{s}}$ in Eqs. (11) and (14), and of $\sigma_{\mathrm{DB}}^{\mathrm{nl}}$ on $E_{\omega}$ and $\omega$ in Eqs. (18) and (19) are related to the divergence of the BCS density of states at $\epsilon=\Delta$. In real superconductors this divergence is smeared by the anisotropy of the order parameter in ${\bf k}$ space and by pairbreaking processes, which are characterized by a broadening parameter $\Gamma\ll|\Delta|$. Consequently, at $\bar{p}_{\mathrm{s}}\to 0$ the $\bar{p}_{\mathrm{s}}$-dependence of the conductivity should become analytic, $\sigma_{DB}=c\,\bar{p}_{\mathrm{s}}^{2}$. The manitude of the coefficient $c$ can be estimated by matching this expression to Eqs. (11) and (14) at the values of $\bar{p}_{\mathrm{s}}$ determined by the condition that the energy broadening of the BCS singularity, $\delta\epsilon(\bar{p}_{\mathrm{s}})$ be of order $\Gamma$.

So far we focused our consideration on the temperature interval near $T_{c}$. At low temperatures, $T\ll\Delta$, the dimensionless quasiparticle concentration $x=(\nu_{n}\Delta)^{-1}\int_{0}^{\infty}d\epsilon\nu(\epsilon)n_{F}(\epsilon)$ in s-wave superconductors is exponentially small; $x\sim\sqrt{\frac{T}{\Delta}}\exp(-\Delta/T)$ . Consequently the conventional contribution to the microwave absorption coefficient, which is proportional to $\tau_{\mathrm{el}}$, 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 $T\ll\Delta$ 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 $1/\tau_{\mathrm{in}}^{(st)}(T)$ (which is of the same order as a electron-phonon relaxation rate in normal metals, see for example Aronov et al. (1981)). 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 $1/\tau_{r}\sim x/\tau_{r}^{(0)}\ll\tau_{\mathrm{in}}^{(st)}$, where $\tau_{r}^{(0)}\sim\tau_{\mathrm{in}}^{(st)}(\Delta)$. The Debye contribution to the dissipative kinetic coefficients is proportional to the longest relaxation time in a system (see for example Landau and Lifshitz (2013). In our case it is $\tau_{r}$. Therefore, in the low frequency limit, $\omega\tau_{r}\ll 1$, the exponentially small factor $\exp(-\Delta/T)$ is canceled from the expression for the conductivity. Below we illustrate this fact in the diffusive regime, and at $\Delta\gg\delta\epsilon(p_{s})\gg T$. In this case the magnitude of the level velocity in the interval of energies of order $T$ is $V\sim\frac{1}{\bar{p}_{\mathrm{s}}}\delta\epsilon\sim\left(\Delta D^{2}\bar{p}_{\mathrm{s}}\right)^{1/3}$ and we get from Eq. (9)

[TABLE]

We would like to note that at very low temperatures the microwave absorption in gapped superconductors is controlled by two level systems (see for example Ref. Zhang et al., 2019 and references therein).

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 $\bar{p}_{\mathrm{s}}$ is nonzero only within the London penetration depth $\lambda_{L}$ 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, $|\epsilon-\Delta|\lesssim\delta\epsilon$, where $\delta\epsilon=v_{\mathrm{F}}\bar{p}_{\mathrm{s}}$ in the ballistic regime and $\delta\epsilon=\left(\Delta D^{2}\bar{p}_{\mathrm{s}}^{4}\right)^{1/3}$ in the diffusive regime. Roughly half of these quasiparticles have energies below $\Delta$ and therefore they are trapped near the surface within a distance of order $\lambda_{L}$.

The microwave absorption coefficient in thin films of d-wave superconductors should be also proportional to $\tau_{\mathrm{in}}$. However, its dependence on $\bar{p}_{\mathrm{s}}$ is expected to be different from those in Eqs. (11), (14), (18), and (19). 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 $\lambda_{L}$.

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 Ref. Zhou and Spivak, 1997.