Surface impedance of superconductors with magnetic impurities
Maxim Kharitonov, Thomas Proslier, Andreas Glatz, Michael J. Pellin
SSurface impedance of superconductors with magnetic impurities
Maxim Kharitonov , Thomas Proslier , Andreas Glatz , and Michael J. Pellin Center for Materials Theory, Rutgers University, Piscataway, NJ 08854, USA Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (Dated: November 7, 2018)Motivated by the problem of the residual surface resistance of the superconducting radio-frequency(SRF) cavities, we develop a microscopic theory of the surface impedance of s -wave superconduc-tors with magnetic impurities. We analytically calculate the current response function and surfaceimpedance for a sample with spatially uniform distribution of impurities, treating magnetic im-purities in the framework of the Shiba theory. The obtained general expressions hold in a widerange of parameter values, such as temperature, frequency, mean free path, and exchange couplingstrength. This generality, on the one hand, allows for direct numerical implementation of our resultsto describe experimental systems (SRF cavities, superconducting qubits) under various practicallyrelevant conditions. On the other hand, explicit analytical expressions can be obtained in a num-ber of limiting cases, which makes possible further theoretical investigation of certain regimes. Asa feature of key relevance to SRF cavities, we show that in the regime of “gapless superconduc-tivity” the surface resistance exhibits saturation at zero temperature. Our theory thus explicitlydemonstrates that magnetic impurities, presumably contained in the oxide surface layer of the SRFcavities, provide a microscopic mechanism for the residual resistance. PACS numbers:
I. INTRODUCTION
Magnetic impurities in s -wave superconductors havebeen a subject of interest for a long time. Shortly afterthe development of the Bardeen-Cooper-Schrieffer (BCS)theory of superconductivity , Abrikosov and Gor’kov(AG) demonstrated that magnetic impurities intro-duced into the sample in moderate concentrations leadto the suppression of superconductivity. If the magneticscattering rate 1 /τ s > /τ ∗ s exceeds the critical value1 /τ ∗ s ≈ . T c , where T c is the transition tempera-ture of a sample without magnetic impurities (we set (cid:126) = 1 throughout the paper), the superconductivity iscompletely suppressed at all temperatures. In contrast,much stronger nonmagnetic disorder is required to sup-press superconductivity: the scattering rate 1 /τ ∼ (cid:15) F must be on the order of the Fermi energy (cid:15) F ∼ T c .According to the AG and subsequent theories, evenbelow the critical value, 1 /τ s < /τ ∗ s , the presence ofmagnetic impurities can result in the regime of “gap-less superconductivity” (GSC), where the superconduct-ing order parameter ∆ is nonzero, yet the single-particledensity of states (DOS) ν ( (cid:15) ) does not vanish down to theFermi level (cid:15) = 0, Fig. 1. The GSC regime is predictedto occur quite generically, although the magnitude of the“subgap” DOS is parameter-dependent. Even for lowscattering rate 1 /τ s (cid:28) /τ ∗ s and weak exchange coupling J , ν F J (cid:28) ν F is the normal state DOS at the Fermilevel per one spin projection), optimal fluctuations in theimpurity distribution produce “tails” in the DOS be-low the “hard gap” predicted by the AG theory, Fig. 1(a).The GSC regime becomes much more pronounced withincreasing the exchange coupling and/or scattering rate,Fig. 1(b), as the Shiba theory demonstrates. This isalso supported by a recent numerical study of a differ- ent, but mathematically equivalent model.In the GSC regime, gapless quasiparticle excitationsgive rise to dissipation even at zero temperature . Al-though this dissipation mechanism (caused either bythe natural presence of magnetic impurities or unin-tentional/unavoidable contamination of the sample withthem) may be negligible for most practical applicationsof superconductors, it could play an important role indevices that require high quality performance. One ex-ample of such systems are the superconducting radio-frequency (SRF) cavities, widely used in particle accel-erators (see Ref. 13 for a review and references therein;another notable system is superconducting qubits). TheSRF cavities are characterized by exceptional quality fac-tors, which are, however, limited to a finite residual value ∼ at temperatures T (cid:28) T c much smaller than thesuperconducting transition temperature T c , where thecontribution from thermally excited quasiparticles van-ishes.Despite the high practical relevance of the problem,there is no commonly accepted theoretical explanationof the origin of the residual Ohmic losses in SRF cavi-ties. Given the above properties, it was recently arguedin Ref. 16 that they could indeed be attributed to thepresence of magnetic impurities in the system. Althoughthe bulk of Nb samples used for SRF cavities is typicallyvery clean, a disordered oxide surface layer forms dueto exposure to atmosphere , Fig. 2. Most importantly,magnetic moments can develop in the oxygen vacan-cies of the sub-stoichiometric Nb O layer of thickness ∼ − ∼ a r X i v : . [ c ond - m a t . s up r- c on ] J u l Ε (cid:144) DΝ H Ε L(cid:144) Ν F H a L Ε (cid:144) DΝ H Ε L(cid:144) Ν F H b L Ε (cid:144) D Q H Ε ,k L H c L H d L Ε (cid:144) D Q H Ε ,k L S u r f a c e R e s i s t an c e R () [ n ] (T=0)/T (e) (f) S u r f a c e R e s i s t an c e R () [ n ] (T=0)/T FIG. 1: (Color online) Regimes of “gapped” [(a), (c), (e), weaker exchange coupling ν F J and lower scattering rate 1 /τ s ;parameters of the Shiba theory used: 1 / ( τ s ∆) = 0 . γ = 0 .
95] and “gapless” [(b), (d), (f), stronger exchange coupling and/orhigher scattering rate; parameters used: 1 / ( τ s ∆) = 0 . γ = 0] superconductivity. (a) and (b) The single-particle density ofstates (DOS) ν ( (cid:15) ), obtained from the Shiba theory Eqs. (4.8)-(4.12), and (6.1). The inset in (a) schematically shows (in red)the “tail” of the DOS produced by optimal fluctuations of impurity distribution – exponentially small nonperturbative effect,not captured by the AG-Shiba theory and not considered in the present paper. (c) and (d) The function ¯ Q ( (cid:15), k ) [Eq. (5.3)]describing the dissipative contribution to the current response from a given energy (cid:15) , see Secs. V and VI for details. The plotsare presented for k = 0; the inset in (d) shows the full range of ¯ Q ( (cid:15), k ). The function ¯ Q ( (cid:15), k ) is nonzero if and only if ν ( (cid:15) )is nonzero. (e) and (f) The temperature dependence of the surface resistance R ( ω ), obtained from the main Eqs. (3.1), (3.3),(4.1), (4.15), (4.16), and (4.17) of the paper by numerically calculating the integrals over (cid:15) and k . In the gapped regime (e), R ( ω ) ∝ exp( − ∆ ∗ /T ) is exponential at lower temperatures and vanishes at T = 0. In the gapless regime (f), for moderate“subgap” DOS, the surface resistance R ( ω ) is exponential at lower but finite temperatures and saturates to a nonzero value at T = 0. The latter case reproduces the commonly observed experimental behavior . DOS with appreciable “subgap” contribution, consider-ably greater than one would expect from a high-purityNb material. Combined with good fits to the Shiba the-ory, these data suggested magnetic impurities in the ox-ide surface layer as an important contributing factor tothe dissipation in SRF cavities.To support this idea, in the present work, we developa microscopic theory of the surface impedance of s-wavesuperconductors with magnetic impurities. According tothe surface chemistry of the air-exposed Nb samples ,Fig. 2, the real SRF cavity material is most appropriately described by a model of disordered surface layer that con-tains both magnetic and nonmagnetic impurities, whilethe rest of the sample is weakly disordered or pure. Inprinciple, such model can be studied in the framework ofthe quasiclassical approach to superconductivity basedon the Eilenberger equation . However, this involvessolving a self-consistency problem for a system of differ-ential equations, which, for realistic parameter values, ischallenging even using numerical methods.Instead, here we consider a simpler model of a super-conducting sample with uniform in space distribution of Inclusions, Hydride precipitates Surface oxide Nb O Interface: sub oxides NbO, NbO not crystalline ~ 2 nm Interstitials dissolved in niobium (mainly O, some C, N, H) Grain boundaries Residue from chemical processing Clean niobium λ N b = n m FIG. 2: (Color online) Typical structure of the surface layerof the air-exposed Nb samples used for SRF cavities. Lo-calized magnetic moments (shown as green arrows) can form(Ref. 15) in oxygen vacancies of the Nb O layer of thickness ∼ ∼ magnetic and nonmagnetic impurities, Fig. 3. The mainpractical advantage is that for this model we are able toanalytically obtain the general expressions for the currentresponse function and surface impedance. The expres-sions are valid, within the approximations of the theory,in a wide range of parameter values and, in the generalcase, only the resulting integrals need to be calculatednumerically. This generality allows for the applicationof our theory to the description of experimental systems,such as SRF cavities and superconducting qubits, in var-ious practically relevant regimes. On the other hand, ifnecessary, explicit analytical expressions can be obtainedin numerous limiting cases.The current response function and surface impedanceof superconductors without magnetic impurities are pro-vided by the Mattis-Bardeen and Abrikosov-Gor’kov-Khalatnikov theories (see also Ref. 21,22), in the pres-ence and absence of nonmagnetic disorder, respectively.The surface impedance of superconductors with magneticimpurities was previously studied in Refs. 23,24 in thelimit ν F J (cid:28) ν F J , treating theinteractions of conduction electrons with magnetic im- purities within the framework of the Shiba theory. Thissufficiently widens the range of impurity concentrations,where the GSC regime (of particular interest to us) withappreciable DOS at the Fermi level occurs.As the feature of key relevance to SRF cavities, wedemonstrate that the presence of magnetic impuritiesdoes lead to the saturation of the surface resistance atzero temperature in the GSC regime.Our theory employs the linear response formalism andis therefore valid as long as the superconducting stateis not appreciably suppressed by the magnetic field H .For type-II superconductors in thermal equilibrium, theupper bound for this is set by the first critical field H c ,above which the system becomes unstable towards cre-ation of vortices. Real SRF cavities, however, are knownto operate in a metastable vortex-free state that per-sists up to a higher “superheating” field H sh > H c .Thus, our theory should be applicable in the range H (cid:46) H sh .At higher fields H ∼ H sh one could, in fact, expecta cooperative effect of the two dissipation mechanisms:magnetic disorder could create “hot spot” regions oflocally suppressed superconductivity at the surface andthus trigger proliferation of vortices. Such regime de-serves a separate study.The rest of the paper is organized as follows. In Sec. II,the studied system is presented and the main approxima-tions are formulated. In Sec. III, the surface impedanceand current response function are introduced. In Sec. IV,the current response function is calculated. In Sec. V, thelow-frequency expansion is performed. In Sec. VI, the keyresult pertaining to the presence of magnetic impurities– finite residual surface resistance in the GSC regime –is demonstrated. Concluding remarks are presented inSec. VII. II. MODEL
We assume the superconducting sample occupies thehalf-space z > n and n s , respectively, Fig. 3.Within the framework of the BCS theory , the Hamil-tonian of the system can be written asˆ H = (cid:90) z> d r (cid:110) ˆ ψ † σ (cid:104) E (cid:0)(cid:12)(cid:12) ˆ p − ec A (cid:12)(cid:12)(cid:1) − (cid:15) F + (cid:88) a uδ ( r − r a ) (cid:105) ˆ ψ σ + (cid:88) b J s b ( ˆ ψ † σ σ σσ (cid:48) ˆ ψ σ (cid:48) ) δ ( r − r b ) + ∆[ ˆ ψ ↑ ˆ ψ ↓ + ˆ ψ †↓ ˆ ψ †↑ ] (cid:111) Here, ˆ ψ σ = ˆ ψ σ ( r ) is the electron field operator, σ, σ (cid:48) = ↑ , ↓ are the spin indices, σ = ( σ x , σ y , σ z ) is the vector of Paulimatrices, and summation over repeated spin indices isimplied; E ( p ) is the electron spectrum, which we assume isotropic in momentum p , p = | p | , ˆ p = − i ∇ ; A = A ( t, r )is the vector potential of the electromagnetic field pene-trating the sample; ∆ is the superconducting order pa-rameter, which has to be found self-consistently in the FIG. 3: (Color online) Studied system: a half-infinite ( z > k , electric E and magnetic H fields, and the vector potential A is shown. presence of magnetic impurities.Next, r a and r b are random positions of nonmagneticand magnetic impurities, respectively. We use the con-ventional disorder averaging technique (“noncrossing”approximation) and, to keep calculations simpler, assumecontact interaction potential of impurities (“point disor-der”). We (i) treat magnetic impurities as classical spinsdescribed by the unit vectors s b and assume them unpo-larized, (ii) consider arbitrary exchange coupling strength ν F J , summing the full perturbation series for a single im-purity. These are the approximations of the Shiba the-ory .Note that the exponentially small subgap contribution(“tail”) to the DOS arising from the optimal fluctuationsof magnetic disorder is not captured within the non-crossing approximation. This nonperturbative effect isdominant only in the limit of weak exchange coupling ν F J (cid:28) / ( τ s T c ) (cid:28)
1. In this work, we concentrate on more sig-nificant contributions to the DOS that arise at larger im-purity concentration and/or stronger exchange coupling.
III. SURFACE IMPEDANCE
The complex surface impedance Z ( ω ) = R ( ω ) + i X ( ω ) (3.1)(see, e.g., Refs. 22,31) relates the electric E ( z )e − i ωt andmagnetic H ( z )e − i ωt fields at frequency ω at the interface z = 0 between the vacuum and sample as E (0) = c π Z ( ω )[ H (0) × n ] , (3.2)where n = (0 , ,
1) is the unit vector normal to the sur-face pointing into the sample, Fig. 3. The real part R ( ω )of the impedance (3.1) determines the energy flux (aver-aged over the oscillation period) W = 12 (cid:16) c π (cid:17) R ( ω ) | H (0) | of the electro-magnetic field per unit area from the vac-uum into the sample and is referred to as the “surfaceresistance”.The general expression for the surface impedance ofa uniformly disordered system reads Z ( ω ) = − i 4 πω λ ( ω ) c , λ ( ω ) = 2 π (cid:90) + ∞ d kk + 4 πQ ( ω, k ) /c . (3.3)Here, Q ( ω, k ) is the linear current response function ofan infinite sample in Fourier representation, dependenton the frequency ω and the absolute value k = | k | of thewave-vector k . In Eq. (3.3), we also introduce the com-plex penetration depth λ ( ω ): its real part Re λ ( ω ) is theactual penetration depth that determines the decay scaleof the electromagnetic field into the bulk. The Ohmicdissipation is determined by the imaginary part Q ( ω, k )of the current response function Q ( ω, k ) = Q ( ω, k ) − i Q ( ω, k ) . (3.4)According to Eq. (3.3), the surface resistance R ( ω ) isfinite, only if Q ( ω, k ) is nonzero.The response function Q ( ω, k ) defines the relation j ( ω, k ) = − c Q ( ω, k ) A ( ω, k ) (3.5)in the Fourier representation between the electric cur-rent j ( ω, k ) and the electro-magnetic field, described bythe vector potential A ( ω, k ). The trivial tensor struc-ture of Eq. (3.5) holds for cubic crystal symmetry and,in particular, for the isotropic electron spectrum E ( | p | )assumed here. It is convenient to work in the gauge ofabsent scalar potential ϕ ( ω, k ) = 0. Additionally, thevector potential A ( ω, k ) may be assumed to satisfy theconstraint A ( ω, k ) k = 0 , (3.6)which significantly simplifies the calculations. This isequivalent to the local electroneutrality condition, whichis a very good approximation for superconductors. Thegeometric relation between the wave-vector k , vector po-tential A ( ω, k ), and electric E ( ω, k ) = i ωc A ( ω, k ) andmagnetic H ( ω, k ) = [i k × A ( ω, k )] fields is shown inFig. 3.In the next section, we calculate the current responsefunction Q ( ω, k ), which fully determines the surfaceimpedance (3.3). IV. CURRENT RESPONSE FUNCTION
According to the general Kubo formalism , theexpression for the linear current response function[Eq. (3.5)] can be written down as Q ( ω, k ) = Q − (cid:90) + ∞−∞ d (cid:15) (cid:104)(cid:16) tanh (cid:15) + T − tanh (cid:15) − T (cid:17) (cid:104) jj (cid:105) RA ( (cid:15), ω, k ) + tanh (cid:15) − T (cid:104) jj (cid:105) RR ( (cid:15), ω, k ) − tanh (cid:15) + T (cid:104) jj (cid:105) AA ( (cid:15), ω, k ) (cid:105) , (4.1) (cid:104) jj (cid:105) ab ( (cid:15), ω, k ) = (cid:28) n α (cid:90) d ξ π (cid:2) G a ( (cid:15) + , p + ) G b ( (cid:15) − , p − ) + F a ( (cid:15) + , p + ) F b ( (cid:15) − , p − ) (cid:3)(cid:29) n , a, b = R, A. (4.2)In Eq. (4.1), we introduced Q = 2( ev F ) ν F / , (4.3)where v F = (d E/ d p ) p = p F is the Fermi velocity, p F is theFermi momentum, E ( p F ) = (cid:15) F , and ν F = p F / (2 π v F ).The quantity Q = Q (0 , | T =0 , clean is the value of thecurrent response function for a clean system at ω = 0, k = 0, and T = 0; it can be related to the formal Lon-don penetration depth λ L (which can be introduced as acharacterization parameter, regardless whether the Lon-don limit actually applies) as1 /λ L = 4 πQ /c . (4.4)The quantities Q and λ L describe the correspondingclean system and are determined only by the band struc-ture parameters.In Eq. (4.2), G R ( (cid:15), p ) = ˜ (cid:15) + ξ ˜ (cid:15) − ξ − ˜∆ , F R ( (cid:15), p ) = ˜∆˜ (cid:15) − ξ − ˜∆ , (4.5) G A ( (cid:15), p ) = [ G R ( (cid:15), p )] ∗ , F A ( (cid:15), p ) = [ F R ( (cid:15), p )] ∗ , are the retarded ( R ) and advanced ( A ) “normal” ( G ) and“anomalous” ( F ) Green’s functions, averaged over disor-der, where the conventionally introduced functions ˜ (cid:15) and ˜∆ are defined below. For point disorder and due tothe property (3.6), the “ladder” contribution vanishes,and the current-current correlation functions (4.2) aredetermined by the products of disorder-averaged Green’sfunctions.Further, in Eqs. (4.1), (4.2), and (4.5), (cid:15) is the energyrelative to the Fermi level (cid:15) F , and (cid:15) ± = (cid:15) ± ω/ p ± = p ± k /
2. We split the integration over momentum (cid:90) d p (2 π ) . . . = ν F (cid:28)(cid:90) d ξ... (cid:29) n , (cid:104) . . . (cid:105) n = (cid:90) | n | =1 d n π . . . , in a standard way into the integration over its absolutevalue p = | p | , expressed in terms of ξ = E ( p ) − (cid:15) F , andaveraging over its direction, expressed in terms of theunit vector n = p /p .Equation (4.2) defines the correlation functions of thecurrent components perpendicular to k , see Eq. (3.6),and so, n α are the components of n = ( n x , n y , n z ) per-pendicular to k : if k = (0 , , k ), as in Fig. 3, then α = x, y . Since the spectrum is isotropic, the integral I ( nk ) = (cid:82) d ξ π . . . in Eq. (4.2) depends just on nk = n z k and angular averaging is reduced to calculating the inte-gral (cid:104) n α I ( nk ) (cid:105) n = 14 (cid:90) − d n z (1 − n z ) I ( n z k ) . (4.6)Note that for the integration order as in Eqs. (4.1) and(4.2) – first over ξ and then over (cid:15) – the contributionto Q ( ω, k ) arising from the dependence of the currentoperator on the vector potential is already compensatedfor.It is convenient to introduce the (retarded) quasiclas-sical Green’s functions { g, f } ( (cid:15) ) = i π (cid:90) d ξ { G R , F R } ( (cid:15), p ) , (4.7)which, according to Eq. (4.5), equal g ( (cid:15) ) = ˜ (cid:15) (cid:112) ˜ (cid:15) − ˜∆ = v √ v − , (4.8) f ( (cid:15) ) = ˜∆ (cid:112) ˜ (cid:15) − ˜∆ = 1 √ v − . (4.9)Within the Shiba theory , the function v = v ( (cid:15) ) ≡ ˜ (cid:15) ˜∆ = g ( (cid:15) ) f ( (cid:15) ) (4.10)satisfies the equation v ∆ = (cid:15) + 1 τ s √ − v γ − v v. (4.11)Here, γ = 1 − ( πν F J ) πν F J ) (4.12)is the parameter of the Shiba theory characterizing ex-change coupling strength and τ s is the scattering time onmagnetic impurities,1 τ s = n s πν F (1 − γ ) = 2 πν F n s J [1 + ( πν F J ) ] . In the weak coupling limit ν F J (cid:28) , γ = 1.We also introduce the function h ( (cid:15) ) as (cid:112) ˜ (cid:15) − ˜∆ = h ( (cid:15) ) + i2 τ , (4.13)which is related to g ( (cid:15) ) and f ( (cid:15) ) as h ( (cid:15) ) = 12 (cid:18) (cid:15)g ( (cid:15) ) + ∆ f ( (cid:15) ) (cid:19) . (4.14) In Eq. (4.13), τ is the scattering time on nonmagneticimpurities, 1 τ = 2 πν F nu . Solving Eq. (4.11) for v (in the general case – numeri-cally), one obtains g ( (cid:15) ), f ( (cid:15) ), and h ( (cid:15) ). Integration over ξ in Eq. (4.2) is straightforward and we obtain (cid:104) jj (cid:105) RA ( (cid:15), ω, k ) = 12 [ g ( (cid:15) + ) g ∗ ( (cid:15) − )+ f ( (cid:15) + ) f ∗ ( (cid:15) − )+1] (cid:104) jj (cid:105) RA ( (cid:15), ω, k ) , (cid:104) jj (cid:105) RA ( (cid:15), ω, k ) = (cid:28) i n α h ( (cid:15) + ) − h ∗ ( (cid:15) − ) + i /τ − v F nk (cid:29) n , (4.15) (cid:104) jj (cid:105) RR ( (cid:15), ω, k ) = 12 [1 − g ( (cid:15) + ) g ( (cid:15) − ) − f ( (cid:15) + ) f ( (cid:15) − )] (cid:104) jj (cid:105) RR ( (cid:15), ω, k ) , (cid:104) jj (cid:105) RR ( (cid:15), ω, k ) = (cid:28) i n α h ( (cid:15) + ) + h ( (cid:15) − ) + i /τ − v F nk (cid:29) n , (4.16)and (cid:104) jj (cid:105) AA ( (cid:15), ω, k ) = [ (cid:104) jj (cid:105) RR ( (cid:15), ω, k )] ∗ , (cid:104) jj (cid:105) AA ( (cid:15), ω, k ) =[ (cid:104) jj (cid:105) RR ( (cid:15), ω, k )] ∗ . Angular averaging in Eq. (4.15) and(4.16) can also be performed explicitly according toEq. (4.6), (cid:104) jj (cid:105) RR,RA ( (cid:15), ω, k ) = − i4 v F k (cid:20)(cid:0) − l (cid:1) ln l − l + 1 − l (cid:21) , (4.17)where l = 1 v F k × (cid:26) h ( (cid:15) + ) − h ∗ ( (cid:15) − ) + i /τ, for RA,h ( (cid:15) + ) + h ( (cid:15) − ) + i /τ, for RR.
Equations (4.1), (4.15), (4.16), and (4.17), combinedwith Eqs. (4.8)-(4.14) of the Shiba theory, provide the an-swer for the current response function Q ( ω, k ) and consti-tute the main result of our work. Within the approxima-tions of the theory, these equations are valid at arbitraryvalues of frequency ω , temperature T , and six micro-scopic parameters characterizing the system. Three stan-dard parameters describe the clean system: (i) thesuperconducting transition temperature T c or, equiva-lently, the superconducting order parameter ∆ at T = 0for a system without magnetic impurities; (ii) the currentresponse Q [Eq. (4.3)] or, equivalently, the formally in-troduced London penetration depth λ L [Eq. (4.4)] for aclean system at T = 0; and (iii) the Fermi velocity v F .The other three parameters describe disorder: (i) thenonmagnetic scattering time τ ; (ii) the magnetic scatter-ing time τ s ; and (iii) the exchange coupling strength ν F J or, equivalently, the Shiba parameter γ [Eq. (4.12)].In the general case, Eqs. (4.1), (4.15), (4.16), and(4.17) provide the most explicit analytical form of thecurrent response function Q ( ω, k ) possible. The func-tion is given by the integral over energy (cid:15) in Eq. (4.1),where the dependence of the integrand on the absolute value of momentum k is explicit in Eqs. (4.17), whilethe dependence on (cid:15) is obtained from the well-knownShiba equation (4.11), the solution to which determinesthe functions g ( (cid:15) ), f ( (cid:15) ), and h ( (cid:15) ).For arbitrary values of parameters, the solution to theShiba equation and the integrations over (cid:15) for the currentresponse Q ( ω, k ) and over k for the surface impedance Z ( ω ) [Eq. (3.3)] need to be carried out numerically. Thegenerality of the obtained results, however, should makethem applicable to a variety of realistic experimentalregimes.On the other hand, in a number of limiting cases, thegeneral formulas can be further simplified and in manycases explicit analytical expressions for the current re-sponse function and surface impedance can be obtained.Analysis of such limiting cases is straightforward and wedo not present it here.In the weak coupling limit ν F J (cid:28) ( γ = 1), the results of Refs. 23,24 are (presumably)recovered, and in the complete absence of magnetic im-purities [1 /τ s = 0, h ( (cid:15) ) = √ (cid:15) − ∆ , g ( (cid:15) ) = (cid:15)/h ( (cid:15) ), f ( (cid:15) ) = ∆ /h ( (cid:15) )] the Mattis-Bardeen theory is repro-duced.In the next two sections, we consider the low-frequencylimit ω (cid:28) ∆, most relevant for practical applicationsto SRF cavities, and concentrate on the key propertypertaining to the presence of magnetic impurities – finiteresidual surface resistance in the GSC regime. V. LOW FREQUENCY EXPANSION
The typical operating frequencies of the SRF cavitiesare ω ∼ c/L ∼ − meV (cid:28) ∆ ∼ L ∼ ω (cid:28) ∆. Separating the real (“nondissipa-tive”) and imaginary (“dissipative”) parts [Eq. (3.4)] of Q ( ω, k ) [Eq. (4.1)], in the leading order in ω for each, weobtain, Q ( k ) = Q (cid:90) + ∞−∞ d (cid:15) tanh (cid:15) T (cid:8) [ f ( (cid:15) )] (cid:104) jj (cid:105) RR ( (cid:15), , k ) − [ f ∗ ( (cid:15) )] (cid:104) jj (cid:105) AA ( (cid:15), , k ) (cid:9) , (5.1) Q ( ω, k ) = Q ω (cid:90) + ∞−∞ d (cid:15) (cid:18) − d n ( (cid:15) )d (cid:15) (cid:19) ¯ Q ( (cid:15), k ) , (5.2)¯ Q ( (cid:15), k ) = 32 (cid:8) [ f ( (cid:15) )] (cid:104) jj (cid:105) RR ( (cid:15), , k ) + [ f ∗ ( (cid:15) )] (cid:104) jj (cid:105) AA ( (cid:15), , k ) + [1 + | g ( (cid:15) ) | + | f ( (cid:15) ) | ] (cid:104) jj (cid:105) RA ( (cid:15), , k ) (cid:9) . (5.3)Here, n ( (cid:15) ) = 1 / [exp( (cid:15)/T ) + 1] is the Fermi distributionfunction.The real part Q ( k ) [Eq. (5.1)] is finite at ω = 0 anddetermines the penetration depth λ ( ω = 0) [Eq. (3.3)] ofthe quasistatic magnetic field (Meissner effect). On theother hand, the imaginary part Q ( ω, k ) ∝ ω [Eqs. (5.2)and (5.3)], which determines the dissipation, is nonzeroonly at finite frequency and is linear in it at 1 /τ s (cid:29) ω .Since Q ( ω, k ) is smaller than Q ( k ) at least in ω/ ∆, onemay also expand Eq. (3.3) in Q ( ω, k ) to obtain Z ( ω ) = 32 πωc (cid:90) + ∞ d k Q ( ω, k )[ k + 4 πQ ( k ) /c ] . (5.4)Thus the surface resistance R ( ω ) ∝ ω is quadratic in fre-quency, which is the most common dependence observedexperimentally in SRF cavities . VI. RESIDUAL SURFACE RESISTANCE
We now turn to the key finding of our work. The func-tion ¯ Q ( (cid:15), k ) [Eqs. (5.3)] describes the contribution tothe dissipative part Q ( ω, k ) [Eq. (5.2)] of the current re-sponse function from quasiparticles with a given energy (cid:15) , while the derivative − d n ( (cid:15) ) / d (cid:15) constrains their distri-bution to the range | (cid:15) | (cid:46) T around the Fermi level. As iswell known , the real part g ( (cid:15) ) of the normal Green’sfunction g ( (cid:15) ) = g ( (cid:15) ) − i g ( (cid:15) ) [Eq. (4.8)] determines theDOS ν ( (cid:15) ) = ν F g ( (cid:15) ) . (6.1)Inspecting Eqs. (5.2) and (5.3), we notice that¯ Q ( (cid:15), k ) = 0 ⇔ ν ( (cid:15) ) = 0 . (6.2)Indeed, if g ( (cid:15) ) = 0, i.e., g ( (cid:15) ) = − i g ( (cid:15) ) is imaginary,then, according to Eqs. (4.8), (4.9), and (4.14), so are f ( (cid:15) ) = − i f ( (cid:15) ) = − i (cid:112) g ( (cid:15) ) and h ( (cid:15) ). In this case, (cid:104) jj (cid:105) RA ( (cid:15), , k ) = (cid:104) jj (cid:105) RR ( (cid:15), , k ) = (cid:104) jj (cid:105) AA ( (cid:15), , k ) and thefunction (5.3) does vanish,¯ Q ( (cid:15), k ) = 32 (cid:104) jj (cid:105) RA ( (cid:15), , k )[ − f ( (cid:15) )+1+ g ( (cid:15) )+ f ( (cid:15) )] = 0 . We do not present a more cumbersome rigorous proof ofthe converse here. Instead, the property (6.2) is clearlyillustrated in Figs. 1(a), (b), (c), (d).Thus, as one would intuitively expect, only the energies (cid:15) at which the DOS ν ( (cid:15) ) is nonzero contribute to dissipa-tion. At T = 0 the envelope function − d n ( (cid:15) ) / d (cid:15) → δ ( (cid:15) )becomes a delta-function and only the excitations at theFermi level (cid:15) = 0 contribute, Q ( ω, k ) | T =0 = Q ω ¯ Q ( (cid:15) = 0 , k ) . (6.3)According to (6.2), as the central result, we obtain thatthe system exhibits finite surface resistance (5.4) at T = 0if and only if the DOS at the Fermi level is nonvanishing,i.e. the system is in the GSC regime. R ( ω ) | T =0 > ⇔ ν ( (cid:15) = 0) > . The result is illustrated in Figs. 1. If the spectrum hasa gap ¯∆ (DOS “tails” are not captured by the AG-Shiba theory), Figs. 1(a),(c),(e), the surface resistanceobeys an activation law R ( ω ) ∝ e − ¯∆ /T at temperatures T (cid:28) ¯∆, eventually vanishing at T = 0.On the other hand, in the gapless regime,Figs. 1(b),(d),(f), the surface resistance R ( ω ) satu-rates to a finite value at T = 0. We note that inexperiments it is quite typical for R ( ω ) to exhibitboth the saturation at T = 0 and an activation behavior R ( ω ) ∝ exp( − ∆ ∗ /T ) at finite but low temperatures T (cid:46) ∆ ∗ , with ∆ ∗ close to the value of the superconduct-ing order parameter ∆ of a clean sample. Such behaviorcan be realized, if the finite DOS ν ( (cid:15) = 0) (cid:28) ν F at theFermi level is much smaller than the DOS ν ( (cid:15) (cid:38) ∆) ∼ ν F above the “nominal” gap. In the framework of the Shibatheory, this is possible in the limit of low magneticscattering rate 1 /τ s (cid:28) ∆ and stronger exchange cou-pling ν F J ∼
1, the case shown in Figs. 1(b),(d),(f).Thus, our microscopic model can reproduce the typicalexperimental temperature dependence of the surfaceresistance.
VII. CONCLUSION
In conclusion, we developed a microscopic analyticaltheory of the surface impedance of s -wave superconduc-tors with magnetic impurities. The theory can poten-tially be applied to a variety of superconducting systemsand is of direct relevance to the problem of the residualsurface resistance of SRF cavities. We explicitly demon-strated that, in the regime of gapless superconductivity,the system exhibits saturation of the surface resistance atzero temperature – a routinely observed, but largely un-explained experimental feature. This substantiates therecent conjecture that magnetic impurities, formed atthe surface of the oxide surface layer, could be the dom-inant dissipation mechanism limiting the performance of the SRF cavities. Our theory is valid in the wide range ofparameter values and can be used for direct comparisonwith experimental data, as will be presented elsewhere . Note added.
After a preprint of the present work be-came available, a paper came out, in which the sameproblem was studied in the diffusive limit ( τ ∆ (cid:28)
1) andfor weak exchange coupling ( ν F J (cid:28) VIII. ACKNOWLEDGEMENTS
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