Conductivity of superconductors in the flux flow regime
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Conductivity of superconductors in the flux flow regime
M. Smith, A. V. Andreev,
2, 1, 3
M. V. Feigel’man,
3, 2 and B. Z. Spivak Department of Physics, University of Washington, Seattle, WA 98195, USA Skolkovo Institute of Science and Technology, Moscow, 143026, Russia L. D. Landau Institute for Theoretical Physics, Moscow, 119334 Russia (Dated: August 11, 2020)We develop a theory of conductivity of type-II superconductors in the flux flow regime takinginto account random spatial fluctuations of the system parameters, such as the gap magnitude ∆( r )and the diffusion coefficient D ( r ). We find a contribution to the conductivity that is proportionalto the inelastic relaxation time τ in , which is much longer than the elastic relaxation time. This newcontribution is due to Debye-type relaxation, and it can be much larger than the conventional fluxflow conductivity due to Bardeen and Stephen. The new contribution is expected to dominate inclean superconductors at low temperatures and in magnetic fields much smaller than H c2 . When a type-II superconductor is subject to a mag-netic field H in the mixed state interval, H c1 < H < H c2 ,the magnetic field penetrates into the sample in the formof vortices [1]. Here H = n v Φ is the average magneticfield, with n v being the flux line density and Φ = π ~ c/e - the flux quantum. Typically, defects and intrinsic dis-order of the underlying crystalline lattice induce inho-mogeneities in the superconducting order parameter. Asa result, the vortex lattice becomes pinned to the crys-talline lattice. For current densities j below some criti-cal value j c the vortices remain pinned, and the currentin this metastable state is dissipationless. However, at j > j c , or if the flux lattice is melted by thermal fluctua-tions, the vortices begin to move, generating dissipation,and the system acquires a finite conductivity σ . Thisphenomenon has been extensively studied both experi-mentally and theoretically (see, for example, Refs. 2–13,and references therein.).Near the critical current density j c this motion pro-ceeds by creep [14], but as the current density is increasedthe system enters the flux flow regime, in which the vor-tices move with a macroscopic velocity V . The latter isrelated to the macroscopic electric field E by the Joseph-son relation [15] E = − c [ V × H ] , (1)which implies that in the reference frame moving with thevortex lattice the electric field vanishes. The nonlinearconductivity σ in the flux flow regime can be expressedin terms of the energy dissipation rate as σE = n v W, (2)where W is the energy dissipation rate per unit length ofthe vortex.At relatively weak magnetic fields, H ≪ H c2 , the de-pendence of the conductivity on the magnetic field canbe established from rather general considerations. Theenergy dissipation in this case occurs in the vortex cores.In the ohmic regime the dissipation rate in each vortex is quadratic in V . From here, using Eqs. (1) and (2) one ar-rives at the conclusion that the conductivity is inverselyproportional to the magnetic field, σ = C/H . Evaluationof the coefficient C requires a microscopic theory.The problem of flux flow conductivity in superconduc-tors has been studied for a long time. It is generallyaccepted that in the regime where temperature is nottoo close to the critical temperature T c and the magneticfield is not too close to H c2 , the longitudinal conductivityin the flux flow regime is given by the Bardeen-Stephenrelation [2] (see also reviews 6 and 8): σ BS = ζσ n H c2 H , H c2 = Φ πξ . (3)Here, ζ is a number of order unity, ξ is the superconduct-ing coherence length, and σ n = e ν n D n is the conductiv-ity of normal metal, with ν n being the density of statesat the Fermi energy, and D n - the electron diffusion coef-ficient. The latter can be expressed in terms of the Fermivelocity v F and the elastic momentum relaxation time, τ el , as D n = v τ el /
3. Equation (3) reflects the fact thatthe core region of a vortex (of area πξ ) may be consid-ered, with respect to its electronic properties, as a normalmetal. It is important that the Bardeen-Stephen expres-sion for the conductivity is proportional to the elasticrelaxation time τ el , and is independent of the energy re-laxation time. This means that at T ≪ T c the flux flowconductivity Eq. (3) is temperature-independent.In the dirty limit, T c τ el ≪
1, the Bardeen-Stephenrelation (3) was confirmed by microscopic calculations inRefs. [4, 5, 7, 8] in the approximation neglecting pinningof vortices, which is valid at the current density j ≫ j c .It was also found [7] that up to a factor of order unity,the same formulas describe the flux flow conductivity ofsuperconductors in the clean limit, T c τ el ≫ τ in . Since typically τ in is ordersof magnitude larger than the elastic relaxation time [16],this contribution can significantly exceed the one given byEq. (3). At low temperatures this contribution is stronglytemperature dependent. The physical mechanism thatgives rise to this new contribution is similar to the De-bye mechanism of microwave absorption in gases [17], su-perconductors [18, 19], and the Mandelstam-Leontovichmechanism of second viscosity in liquids [20].Below we will adopt a model where, in the absence of amagnetic field, both the modulus of the order parameter∆( r ) = ¯∆ + δ ∆( r ) and the diffusion coefficient D n ( r ) =¯ D n + δD n ( r ) exhibit random spatial variations. Forbrevity we introduce a parameter α ( r ) ≡ (∆( r ) , D n ( r ))which denotes both the above parameters. We assumethat the spatial variations are small, δα ≪ ¯ α , and de-note their correlation function by h δα ( r ) δα ( r ′ ) i = (cid:10) ( δα ) (cid:11) g (cid:18) | r − r ′ | L c (cid:19) , (4)where h . . . i denotes averaging over random realizationsof α ( r ). For simplicity we assume the correlation radiusto be large, L c > ξ .We begin with the simplest case of a thin film of s -wave superconductor at H ≪ H c2 , where the distancebetween vortices exceeds the coherence length ξ , whilefilm thickness d ≤ ξ . In this case the modulus of theorder parameter changes from zero at the center of avortex, to its maximal value ∆ at | r | of order of the inter-vortex distance. Below we assume that the temperatureexceeds the mean level spacing in the core. We thereforeneglect discreteness of the quasiparticle energy spectrum,and introduce the density of states ν ( ǫ ) per vortex at ǫ < ∆ . At low energies, ǫ ≪ ∆ , the density of statesis ν ( ǫ ) ∼ ν n ξ d . It changes by a factor of order unity at ǫ ∼ ∆ , and dramatically increases as ǫ → ∆ .In the flux flow regime the vortices pass through sam-ple regions with different values of α ( r ), which changesthe spatial profile and amplitude of the order parameter∆( r ) near the vortex cores. As a result the density ofstates in the vortex core, ν ( ǫ, α ), changes in time. Sincethe number of energy levels is conserved, the time evolu-tion of the density of states is described by the continuityequation: ∂ν ( ǫ, α ) ∂t + ∂ [ v ν ( ǫ, α ) ν ( ǫ, α )] ∂ǫ = 0 , (5)where v ν ( ǫ, α ) is the level “velocity” in energy space. In-tegrating this equation over energy and bearing in mindthat the spectral flow vanishes at ǫ = 0 we can express v ν ( ǫ, α ) in the form v ν ( ǫ, α ) = − ˙ αν ( ǫ,α ( t )) R ǫ d ˜ ǫ∂ α ν (˜ ǫ, α ),where ˙ α denotes the time derivative of α along the tra-jectory of the vortex motion. To leading order in inho-mogeneity we have v ν ( ǫ, t ) = A ( ǫ ) ˙ α, (6)where A ( ǫ ) = − ν ( ǫ, ¯ α ) Z ǫ d ˜ ǫ ∂ α ν (˜ ǫ, α ) | α =¯ α . (7) characterizes the sensitivity of the density of states in thevortex cores to local variations of α . The level velocities v ν ( ǫ, t ) oscillate in time as the vortices move. The typicalfrequency of these oscillations is ω E ∼ cE/HL c .At T > ν ( ǫ, t ) caused by the vortex motion cre-ates a non-equilibrium quasiparticle distribution. At lowvortex velocities V , the quasiparticle distribution func-tion n ( ǫ, t ) depends only on the energy ǫ . In the ab-sence of inelastic scattering its time evolution due tothe spectral flow is described by the continuity equation ∂ t ( νn )+ ∂ ǫ ( v ν νn ) = 0. Combining this equation with thecontinuity equation (5) for ν ( ǫ, t ), allowing for inelasticcollisions, and working to lowest order in inhomogeneity,we obtain the following kinetic equation ∂ t δn ( ǫ, t ) + v ν ( ǫ, t ) dn F ( ǫ ) dǫ = I in { n } . (8)Here n F ( ǫ ) = ( e ǫ/T + 1) − is the Fermi function, δn ( ǫ ) = n ( ǫ ) − n F ( ǫ ) is the nonequilibrium part of the distributionfunctiton, and I in { n } is the linearized inelastic collisionintegral, which we write in the relaxation time approxi-mation, I in { n } = − δn ( ǫ, t ) /τ in .The rate of energy absorption per unit length due tothe quasiparticles in the vortex core in Eq. (2) is givenby [18, 19] W = d R ∞ dǫ ν ( ǫ, α ( t )) n ( ǫ, t ) v ν ( ǫ, t ), where · · · denotes time averaging along the vortex trajectory. If onereplaces the quasiparticle distribution function here bythe equilibrium distribution n F ( ǫ ), the energy dissipationrate vanishes as the integrand becomes a total derivative.Therefore, to lowest order in inhomogeneity we have W = 1 d Z ∞ dǫ ν ( ǫ, ¯ α ) δn ( ǫ, t ) v ν ( ǫ, t ) . (9)Substituting here the solution of the linearized kineticequation (8), and using Eqs. (6), (7) we get W = 1 d Z ∞ dǫ (cid:18) − dn F ( ǫ ) dǫ (cid:19) ν ( ǫ, ¯ α ) A ( ǫ ) C ( E ) , (10)where the dependence on the electric field is described bythe quantity C ( E ) defined as C ( E ) = Z ∞ e − ττ in dτ ˙ α ( t ) ˙ α ( t − τ ) . (11)The correlator of ˙ α in the integrand must be averagedover the trajectories of the vortex motion at a given elec-tric field E . Substituting Eq. (10) into (2) we obtain forthe Debye contribution to the nonlinear conductivity σ = n v d C ( E ) E Z ∞ dǫ T ν ( ǫ, ¯ α ) A ( ǫ )cosh (cid:0) ǫ T (cid:1) . (12)This expression, with C ( E ) in the form (11), applies toboth creep and flux flow regimes. The correlator in theintegrand of Eq. (11) depends on the statistical propertiesof vortex trajectories in the presence of disorder, and itsdependence on the electric field E is difficult to establishin the general case.The situation simplifies dramatically in the flux flowregime. In this case the vortices move with the velocity V = c [ E × H ] /H along straight lines, and thus α ( t ) = α ( r + V t ), where r is the initial position of the vortex.As a result, C ( E ) in Eq. (11) can be expressed in termsof the disorder correlation function in Eq. (4). Passing tothe Fourier representation (see Supplementary Material[21] for a detailed derivation) we obtain C ( E ) = h ( δα ) i τ in Z d ˜ ω π ˜ ω ˜ g (˜ ω ) (cid:0) E ∗ E (cid:1) + ˜ ω , E ∗ = HL c cτ in . (13)Here ˜ g (˜ ω ) = R dxg ( x ) e i ˜ ωx denotes the Fourier transformof the function g ( x ) in Eq. (4), and E ∗ is the characteris-tic electric field of the onset of nonlinearity for the Debyecontribution to the conductivity.At small electric fields, E < E ∗ , which corresponds tolow flow velocities, V τ in < L c , C ( E ) in Eq. (13) may beestimated as C ( E ) ∼ ( cE/H ) τ in h ( ∇ α ) i . Substitutingthis into Eq. (12) we obtain the following estimate for theDebye contribution to the linear flux flow conductivity, σ DB ∼ d e ~ τ in H c2 H h ( ∇ α ) i ξ Z ∞ dǫT ν ( ǫ, ¯ α ) A ( ǫ )cosh (cid:0) ǫ T (cid:1) . (14)This expression applies at an arbitrary value of the pa-rameter T c τ el . In the clean ( T c τ el ≫
1) and dirty( T c τ el ≪
1) limits the coherence length ξ here is givenby, respectively, ξ = ~ v F /π ∆ and ξ = p ~ D n /
2∆ .At low temperatures, T ≪ ∆, the integral in Eq. (14)is dominated by energies ǫ ∼ T . In this energy range A ( ǫ ) in Eq. (7) may be estimated as A ( ǫ ∼ T ) ≃ T / ¯ α .Taking into account that ∇ α ∼ δα/L c and ν ( ǫ, ¯ α ) ∼ ν n ξ d , we find the Debye-type contribution to the fluxflow conductivity: σ DB ∼ e ν n τ in H c2 H h ( δα ) i ¯ α ξ L (cid:18) ξT ~ (cid:19) , T ≪ T c . (15)The ratio between the Debye contribution to the conduc-tivity, Eq. (15), and the Bardeen-Stephen expression inEq. (3) is of the order of σ DB σ BS ∼ τ in τ el h ( δα ) i ¯ α ξ L (cid:18) T ξ ~ v F (cid:19) , T ≪ T c . (16)This ratio is proportional to a product of a very large fac-tor ( τ in /τ el ) ≫ τ in /τ el may reach many orders of magni-tude at low temperatures (some estimates are providedbelow), the whole ratio (16) may become large. Then theDebye contribution to the conductivity (14) is the dom-inant one. In this case the flux flow conductivity willexhibit strong temperature dependence.The estimates (14)-(16) are obtained under the condi-tion ω E τ in ≤
1, which corresponds to low electric fields
E < E ∗ . The maximal current density attainable in thelinear regime, j max ∼ σ DB E ∗ is independent of τ in , j max ∼ e ν n Φ cL c h ( δα ) i ¯ α (cid:18) ξT ~ (cid:19) . (17)The linear regime in the current-voltage characteristic(CVC) that is dominated by the Debye conductivity (15)exists provided j max exceeds the critical current density j c ≪ j max , which is determined by the strength of vortexpinning.If E & E ∗ the CVC becomes non-linear. FromEqs. (13), (10) and (2) it follows that at E ≫ E ∗ theDebye contribution to the current density is j ( E ) ∝ σ DB ( E ∗ ) /E . At arbitrary electric fields the current den-sity can be described by an interpolation formula j DB ( E ) = σ DB E a ( E/E ∗ ) , (18)where a is a number of order unity. The denominator inEq. (18) can be rewritten in the form (1+( ω E τ in ) ), whichis characteristic of the Debye absorption mechanism.Since at E > E ∗ the current density is a decreas-ing function of the electric field, in this regime spa-tially uniform flow becomes unstable. A similar sce-nario based on a thermal instability of the Bardeen-Stephen flux flow was proposed in Ref. [22], with thecharacteristic electric field E LO ∼ Hc p D n /τ in . The ratio E ∗ /E LO = L c / √ D n τ in is typically small due to the largevalue of τ in .If j max < j c , then upon depinning at j > j c the systemwould jump into the unstable branch of the CVC withthe negative differential conductance, − dj/dE ∝ /E .However, the depinning electric field may exceed the field E at which the Debye contribution becomes of order σ BS ; σ DB / [1 + ( E /E ∗ ) ] ∼ σ BS . In this case the insta-bility develops at E ∼ E LO . The interval E < E < E LO exists if E /E LO ∼ h ( δα ) i ¯ α ξ ( v F τ el ) (cid:18) T ξ ~ v F (cid:19) ≪ . (19)Consideration of the nonlinear regime is beyond the scopeof our article.Let us now discuss the physical processes that governthe inelastic relaxation rate 1 /τ in . The value of conduc-tivity Eq. (14) is controlled by energy relaxation pro-cesses for electrons with ǫ < ∆, which reside in the vor-tex cores. Because the core size ∼ ξ is smaller than thewavelength λ ph of thermal phonons the rate of electron-phonon scattering for such electrons is suppressed by anadditional factor ( ξ/λ ph ) , in comparison to the rate ofelectron-phonon scattering in the bulk. Since this factoris very small, in a wide temperature interval the relevantenergy relaxation rate is dominated by electron-electronscattering, 1 /τ in = 1 /τ (ee) . At T ∼ ∆ this rate is roughlythe same as the electron-electron scattering rate in nor-mal metals.At T ≪ ∆ the electron-electron relaxation processesare characterized by two relaxation times. The shortertime, τ ee , corresponds to relaxation processes involvingonly quasiparticles with typical thermal energies. Suchrelaxation processes conserve the total energy of quasi-particles in the vortex core and lead to the establishmentof a local electron temperature in the vortex core. Subse-quent relaxation to equilibrium characterized by a globalelectron temperature requires energy exchange betweendifferent cores and must involve quasiparticles with en-ergies ǫ > ∆ , which can propagate between differentvortices. As a result, the relaxation time associated withsuch processes is much longer, τ ee1 > τ ee . The Debyecontribution to the linear kinetic coefficient is propor-tional to the longest relaxation time in the system [20].Therefore, at T ≪ ∆ we must set τ in ∼ τ ee1 in Eq. (14).We also note that at T ≪ ∆ there are two nonlinear elec-tric field thresholds corresponding to the two relaxationtimes. The above estimates of relaxation times assumedthat quasiparticles with energies ǫ < ∆ are confined tothe vortex cores. However, in disordered superconductorsthe density of states in this energy range can be nonzeroeven outside the vortex cores. In this case the value of τ in in Eqs. (14), (15) will be decreased.The above results apply to the case of thin films wherethe quasiparticles with ǫ < ∆ are confined in the coresof the pancake vortices. In bulk superconductors non-equilibrium quasiparticles can diffuse along vortex lines,which effectively shortens the energy relaxation time. Toaccount for this effect we allow for the dependence of thequasiparticle distribution function on the coordinate z along the vortex, δn ( ǫ, z, t ), and modify the kinetic equa-tion Eq. (8) as follows (cid:20) ∂ t − D v ∂ + 1 τ in (cid:21) δn ( ǫ, z, t ) = − dn F ( ǫ ) dǫ v ν ( ǫ, α, z ) , (20)where D v ( ǫ ) is the diffusion coefficient of quasiparticlesinside the vortex core. In this case the z -dependent levelvelocity v ν ( ǫ, α, z ) is still described by Eqs. (6) and (7),but ν ( ǫ ) should be understood as the density of statesper unit length of the vortex. Finally, Eq. (9) for theenergy absorption rate should be modified as follows, W = L R dz R ∞ dǫ ν ( ǫ, ¯ α ) δn ( ǫ, z, t ) v ν ( ǫ, z, t ), where L isthe length of the vortex line. Using Eq. (20) and follow-ing the arguments that lead to Eq. (13) we obtain (seeSupplemental Material for the details): W = Re Z dqdω (2 π ) Z ∞ dǫν ( ǫ, ¯ α ) A ( ǫ )4 T cosh (cid:0) ǫ T (cid:1) τ in h ( δα ) i ω ˜ g ( q, ω )1 + D v q τ in − iωτ in , (21)where ˜ g ( q, ω ) = R dzdte iωt − iqz g (cid:16) √ z + V t L c (cid:17) .If D v τ in < L diffusion along the vortex is irrelevant,and the energy dissipation per unit length, and thus theconductivity are the same as those for thin films, whichare given by Eqs. (12), and (13). In the opposite limit, √ D v τ in ≫ L c , one finds for theDebye contribution to the conductivity σ (3 D )DB ∼ e ν n r τ in D v L c H c2 H h ( δα ) i ¯ α ξ L (cid:18) ξT ~ (cid:19) (22)which is smaller than the 2D result in Eq. (15) by a factorof order L c / √ D v τ in ≪
1. The physical reason for this isthat the fluctuations δα ( x ) are effectively averaged overa segment of the vortex with length ∼ √ D v τ in ≫ L c .In this case the Debye contribution may still exceed theBardeen-Stephen result, σ DB > σ BS . However, since j (3 D )max ∼ / √ τ in the range of current densities correspond-ing to the stable branch of the CVC ( j (3 D )max > j c ) turnsout to be much smaller than in the 2D case.The value of the diffusion coefficient D v depends onthe value of the parameter ∆ τ el . In isotropic dirty super-conductors, ∆ τ el ≪
1, it can be shown [23] with the aidof the Usadel equation that D v ≈ D n . In clean supercon-ductors the value of D v can be significantly smaller. Inthis case quasiparticle states inside a vortex are describedby the Caroli-deGennes-Matricon (CdGM) solution [24]with energy dispersion ǫ µ ( p z ) ≈ µω ∗ / p − p z /p F , where µ + 1 / ω ∗ = ∆ / ( k F ξ ) . At smallenergies, ǫ ≪ ∆, the quasiparticle velocities along thevortex are greatly reduced in comparison to the Fermivelocity, and may be estimated as v v ∼ v F ǫ ∆ ( k F ξ ) − ,where ǫ = µω ∗ . Determination of the elastic relaxationtime in the core, τ v el , requires a careful consideration ofquasiparticle wave functions in the core and is beyondthe scope of the present paper. Assuming no delicatecancellation of the scattering amplitude for electron- andhole-components of the quasiparticle wave functions oc-curs, τ v el may be estimated using the density of states inthe core as τ v el ∼ τ el . The corresponding diffusion coeffi-cient, D v ∼ D n k ξ ǫ ∆ ∼ D n k ξ T ∆ , may be several orders ofmagnitude smaller than that in the normal state. In sucha situation diffusion of quasiparticles along the vortex lineis inefficient and the 2D regime of inelastic relaxation isrealized.Finally, we mention a related effect. Microwave ab-sorption in type-II superconductors in a mixed state maybe greatly enhanced due to the Debye mechanism evenwithout depinning of vortices by a strong transport cur-rent. The microwave field will exert a time-dependentMagnus force on the vortices, which in turn cause themto oscillate about their equilibrium positions. Becauseof the inhomogeneity of α ( r ) the density of quasiparticlestates in the vortex cores will vary in time. Relaxationof quasiparticles to equilibrium will produce a contribu-tion to microwave absorption which is proportional tothe inelastic relaxation time τ in at low frequencies. Thusmicrowave absorption measurements in the mixed statecould be used to extract τ in for quasiparticles in vortexcores. The present mechanism relies on the inhomogene-ity of the sample parameters α ( r ) and produces a contri-bution to microwave absorption proportional to τ in evenin the absence of macroscopic supercurrent through thesample. In contrast, in the absence of inhomogeneity of α ( r ) the linear microwave absorption coefficient dependson τ in only in the presence of a macrosopic supercur-rent [18, 19, 25]. Conclusions.
We developed a theory of the Debyedissipation mechanism in the flux flow regime of type-II superconductors. The energy dissipation rate due tothis mechanism is controlled by the inelastic relaxationtime τ in , and becomes nonlinear at rather weak electricfields E ∼ E ∗ ∼ /τ in , see Eq. (13). At weak fields, E . E ∗ , the Debye contribution to the conductivity,Eqs. (15), (22), increases as τ in increases, and greatly ex-ceeds the Bardeen-Stephen result, the enhancement be-ing especially pronounced at low temperatures, T ≪ T c .In such a case the flux-flow resistivity ρ xx ( T ) ∝ /τ in ( T )is expected to be strongly temperature-dependent; theaccompanying Hall resistance ρ xy is small and scales as ρ xy ( T ) ∝ ρ ( T ) for the reasons outlined in Ref. [26].Currently, we are not aware of experimental results indi-cating significant enhancement of the conductivity com-pared to the Bardeen-Stephen value. We expect howeverthat the proposed mechanism may be observable at lowtemperatures in clean two-dimensional or layered mate-rials (such as NbSe and MoS ), and under magneticfields H ≪ H c2 perpendicular to the layers. It is im-portant to work under weak pinning conditions, wherethe critical depinning current density j c is much smallerthan the pair-breaking current density j . This conditioncan be satisfied for H ≪ H c2 in clean superconductors inthe regime of weak collective pinning [8, 9], where j c isproportional to a high power of the disorder parameter h δα i , while the maximal dissipative current, Eq. (17) isproportional to h δα i . We expect that in such materialsthe crossover to the unstable branch of the CVC shouldoccur at very weak electric fields E ∗ ∼ /τ in , see Eq. (13).In contrast, in dirty superconductors (e.g. [27–29]), whichexhibit the Bardeen-Stephen flux flow resistance (3) theinstability occurs at a much higher field, E LO ≫ E ∗ ,predicted by Larkin and Ovchinnikov [8, 22]. Finally, wenote that a similar Debye-type mechanism may accountfor giant microwave absorption in a pinned vortex state. Acknowledgements
The authors are grateful for help-ful conversations with D. Geshkenbein, A. Kapitulnik, S.Kivelson, E. Sonin and M. Skvortsov. M.S. and A.A.were supported by the U.S. Department of Energy Of-fice of Science, Basic Energy Sciences under Award No.DE-FG02-07ER46452 and by the National Science Foun-dation Grant MRSEC DMR-1719797. A.A. and M.F.were partly supported by the RSF grant 20-12-00361. [1] A.A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshin-ski,
Methods of Quantum Field Theory in StatisticalPhysics (Courier Corporation, 1975) google-Books-ID:E 9NtwNY7UcC.[2] John Bardeen and M. J. Stephen, “Theory of the Motionof Vortices in Superconductors,” Physical Review ,A1197–A1207 (1965), publisher: American Physical So-ciety.[3] P. Nozi`eres and W. F. Vinen, “The motion of flux linesin type II superconductors,” The Philosophical Maga-zine: A Journal of Theoretical Experimental and AppliedPhysics , 667–688 (1966), publisher: Taylor & Franciseprint: https://doi.org/10.1080/14786436608211964.[4] L. P. Gor’kov and N. B. Kopnin, “Some features of vis-cous flow of vortices in superconducting alloys near thecritical temperature,” Sov.Phys. JETP , 7 (1973).[5] A. I. Larkin and Yu. N. Ovchinnikov, “Resistance ofsuperconductors near the critical field strength Hc2,”Sov.Phys. JETP , 4 (1973).[6] L. P. Gor’kov and N. B. Kopnin, “Vortex motion andresistivity of type-ll superconductors in a magnetic field,”Soviet Physics Uspekhi , 496 (1975), publisher: IOPPublishing.[7] A. I. Larkin and Yu. N. Ovchinnikov, “Viscoscity ofvortices in pure superconductors,” Sov.Phys. JETP ,210–213 (1976).[8] A. I. Larkin and Yu. N. Ovchinnikov, “Vortex motion insuperconductors,” in Nonequilibrium Superconductivity ,edited by D. N. Langenberg and A.I. Larkin (ElsevierScience Publ. B. V, 1986) pp. 493–542.[9] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein,A. I. Larkin, and V. M. Vinokur, “Vortices inhigh-temperature superconductors,” Reviews of Mod-ern Physics , 1125–1388 (1994), publisher: AmericanPhysical Society.[10] Michael Tinkham, Introduction to Superconductiv-ity (Courier Corporation, 2004) google-Books-ID:VpUk3NfwDIkC.[11] S. Bhattacharya, M. J. Higgins, and T. V. Ramakrish-nan, “Anomalies in Free Flux-Flow Hall Effect,” PhysicalReview Letters , 1699–1702 (1994), publisher: Ameri-can Physical Society.[12] M. C. Hellerqvist, D. Ephron, W. R. White, M. R.Beasley, and A. Kapitulnik, “Vortex Dynamics in Two-Dimensional Amorphous Mo77Ge23 Films,” Physical Re-view Letters , 4022–4025 (1996), publisher: AmericanPhysical Society.[13] Avishai Benyamini, Dante M. Kennes, Evan Telford,Kenji Watanabe, Takashi Taniguchi, Andrew Millis,James Hone, Cory R. Dean, and Abhay Pasupa-thy, “Blockade of vortex flow by thermal fluctua-tions in atomically thin clean-limit superconductors,”arXiv:1909.08469 [cond-mat] (2019), arXiv: 1909.08469.[14] P. W. Anderson, “Theory of Flux Creep in Hard Super-conductors,” Physical Review Letters , 309–311 (1962),publisher: American Physical Society.[15] B. D. Josephson, “Potential differences in the mixed stateof type II superconductors,” Physics Letters , 242–243(1965).[16] Depending on parameters of the system and temperaturethe ratio τ in /τ el can be is big as 10 . See for example [30]. [17] Peter Debye, Polar molecules (Dover Publ., 1970) google-Books-ID: f70ingEACAAJ.[18] M. Smith, A. V. Andreev, and B. Z. Spivak, “Giant mi-crowave absorption in s- and d- wave superconductors,”Annals of Physics , 168105 (2020).[19] M. Smith, A. V. Andreev, and B. Z. Spivak, “Debyemechanism of giant microwave absorption in supercon-ductors,” Physical Review B , 134508 (2020), pub-lisher: American Physical Society.[20] L. D. Landau and E. M. Lifshitz,
Fluid Mechanics (Else-vier, 2013) google-Books-ID: CeBbAwAAQBAJ.[21] See Supplemental Material at [URL will be inserted bypublisher] for detailed derivations of Eqs. (13) and (21).[22] A. I. Larkin and Yu. N. Ovchinnikov, “Nonlinear conduc-tivity of superconductors in the mixed state,” Sov.Phys.JETP , 6 (1975).[23] R. Bundschuh, C. Cassanello, D. Serban, and M. R.Zirnbauer, “Localization of quasiparticles in a disorderedvortex,” Nuclear Physics B , 689–732 (1998).[24] C. Caroli, P. G. De Gennes, and J. Matricon, “BoundFermion states on a vortex line in a type II superconduc-tor,” Physics Letters , 307–309 (1964).[25] Yu. N. Ovchinnikov and A.R. Isaakyan, “Electromag- netic field absorption in superconducting films.” JETP , 178–184 (1978).[26] V. M. Vinokur, V. B. Geshkenbein, M. V. Feigel’man,and G. Blatter, “Scaling of the Hall resistivity in high-T c superconductors,” Physical Review Letters ,1242–1245 (1993).[27] L. E. Musienko, I. M. Dmitrenko, and V. G. Volotskaya,“Nonlinear conductivity of thin films in a mixed state,”JETP Letters , 4 (1980).[28] W. Klein, R. P. Huebener, S. Gauss, and J. Parisi,“Nonlinearity in the flux-flow behavior of thin-film su-perconductors,” Journal of Low Temperature Physics ,413–432 (1985).[29] A. V. Samoilov, M. Konczykowski, N. C. Yeh, S. Berry,and C. C. Tsuei, “Electric-Field-Induced Electronic In-stability in Amorphous Mo3Si Superconducting Films,”Physical Review Letters , 4118–4121 (1995), publisher:American Physical Society.[30] M. E. Gershenson, D. Gong, T. Sato, B. S. Karasik, andA. V. Sergeev, “Millisecond electron–phonon relaxationin ultrathin disordered metal films at millikelvin temper-atures,” Applied Physics Letters , 2049–2051 (2001),publisher: American Institute of Physics. r X i v : . [ c ond - m a t . s up r- c on ] A ug Supplementary Material to Conductivity of superconductors in the flux flow regime
M. Smith, A. V. Andreev,
2, 1, 3
M. V. Feigel’man,
3, 2 and B. Z. Spivak Department of Physics, University of Washington, Seattle, WA 98195, USA Skolkovo Institute of Science and Technology, Moscow, 143026, Russia L. D. Landau Institute for Theoretical Physics, Moscow, 119334 Russia (Dated: August 11, 2020)
Derivation of expressions for the energy dissipationrate in thin films
Here we provide a detailed derivation of Eqs. (9), (10),and (13) for the energy absorption rate in thin films.According to Ehrenfest’s theorem [1], the energy ab-sorption rate is given by ddt h ˆ H i = D ∂ ˆ H ( t ) ∂t E , where ˆ H isthe system Hamiltonian, and h . . . i denotes statistical av-eraging. Adapting this expression to quasiparticles in thevortex core we write the energy absorption rate per unitlength of the vortex in the form W = 1 d Z ∞ dǫ ν ( ǫ, α ( t )) n ( ǫ, t ) v ν ( ǫ, t ) , (S.1)where · · · denotes time averaging over the trajectory ofthe vortex motion. For the equilibrium quasiparticle dis-tribution, n ( ǫ, t ) = n F ( ǫ ) the integrand above is a to-tal derivative, and the energy dissipation rate vanishes.Therefore, to lowest order in inhomogeneity we may re-place n ( ǫ, t ) → δn ( ǫ, t ) in the above equation. This yieldsEq. (9).Writing the solution of the linearized kinetic equa-tion (8) in the form δn ( ǫ, t ) = (cid:18) − dn F ( ǫ ) dǫ (cid:19) A ( ǫ ) Z ∞ dτ e − ττ in ˙ α ( t − τ ) , and using Eqs. (6) and (7) we arrive at Eqs. (10) andEq. (11)Next, using the fact that in the flux flow regime thevortex trajectories are given by α ( t ) = α ( r + V t ), where r is the initial position of the vortex, and V = c [ E × H ] /H is the drift velocity of the lattice, we can converttime averaging into spatial averaging over inhomogeneityin Eq. (4). We thus express the quantity C ( E ) in Eq. (11)in the form C ( E ) = −h ( δα ) i Z ∞ d ˜ te − ˜ t/τ in d d ˜ t g (cid:18) cE | ˜ t | HL c (cid:19) . (S.2)Introducing the Fourier transform ˜ g (˜ ω ) ≡ R dxe i ˜ ωx g ( x )of the function g ( x ) in Eq. (4) we obtain C ( E ) = h ( δα ) i Z dω π L c HcE τ in ω ω τ ˜ g (cid:18) ωL c HcE (cid:19) . (S.3)Finally, introducing the dimensionless frequency ˜ ω = ωL c HcE and the characteristic electric field E ∗ = H L c cτ in we arrive at Eq. (13). Derivation of expressions for the energy dissipationrate in bulk superconductors