Locking of length scales in two-band superconductors
LLocking of length scales in two-band superconductors
M. Ichioka
Department of Physics, RIIS, Okayama University, Okayama 700-8530, Japan ∗ V. G. Kogan
Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA
J. Schmalian
Institut f¨ur Theorie der Kondensierten Materie und Institut f¨ur Festk¨orperphysik,Karlsruher Institut f¨ur Technologie, D-76131 Karlsruhe, Germany (Dated: November 6, 2018)A model of a clean two-band s-wave superconductor with cylindrical Fermi surfaces, differentFermi velocities v , , and a general 2 × V αβ is used to study the order param-eter distribution in vortex lattices. The Eilenberger weak coupling formalism is used to calculatenumerically the spatial distributions of the pairing amplitudes ∆ and ∆ of the two bands forvortices parallel to the Fermi cylinders. For generic values of the interband coupling V , it is shownthat, independently of the couplings V αβ , of the ratio v /v , of the temperature, and the appliedfield, the length scales of spatial variation of ∆ and of ∆ are the same within the accuracy of ourcalculations. The only exception from this single length-scale behavior is found for V →
0, i.e., fornearly decoupled bands.
PACS numbers: 74.20.-z, 74.25.Uv
I. INTRODUCTION
Just at the dawn of the theory of multiband super-conductors, it was established that near the critical tem-perature T c , the coherence lengths, which set the lengthscales of spatial variation of the pairing amplitudes of thebands, are in fact the same, notwithstanding differencesin zero- T BCS lengths ξ ,α ∝ v α /T c ( α is the band indexand v α is the Fermi velocity) [1]. This result has been“rediscovered” in the recent debate on the proper form ofGinzburg-Landau (GL) theory of two-band superconduc-tors [2, 3]. This debate was triggered by extensive studiesof multiband MgB which prompted the formulation oftwo order-parameter GL energy functionals to allow fordifferent length scales ξ (cid:54) = ξ associated with the twounderlying bands, see [4, 5] and references therein. Oneof the predictions of these models was an intervortex at-traction at distances large with respect to the Londonpenetration depth. The observation of vortex clusteringin MgB in very small fields was considered as evidencefor asymptotic intervortex attraction [6].While it is established [1–3] that near T c , where theGL-expansion is justified, any generic superconductorwith finite interband coupling is governed by a singlesuperconducting order parameter with one coherencelength, it was pointed out in Ref. [3] that this does nothave to be true away from T c . Novel behavior is expectedespecially in cases with different Fermi velocities of thebands and for very weak interband coupling; this requiresto turn to microscopic descriptions of superconductors ∗ [email protected] that are applicable at all temperatures. Interesting cal-culations of this kind were performed in Refs. [7, 8] andshowed that away from T c and for a very weak inter-bandcoupling the length scales ξ and ξ are indeed not equal,in particular for low temperatures and at small magneticfields.However, there are several reasons why in real materi-als the inter-band coupling is not as weak as was assumedin Refs. [7, 8]. First, the ever present Coulomb repulsionwill inevitably give rise to off-diagonal matrix elementsin band-representation, eventhough the usual renormal-ization of the Coulomb pseudopotential tends to reduceinterband interactions more strongly than intraband in-teractions [9]. For MgB the latter effect was analyzedand is rather moderate [9] : the bare interband Coulombinteraction is about half of the bare intraband interac-tion; renormalizations only reduce this ratio by anotherfactor of 2, yielding interband Coulomb interactions thatare approximately 25% of the intraband couplings. Sec-ond, the matrix elements of the electron-lattice couplingwithin and between electronic bands are for the impor-tant optical phonon branches a priori of the same orderof magnitude. Even for MgB , where the observation of aLeggett-mode in the Raman spectrum [10] is evidence forcomparatively weak interband coupling, a careful analy-sis of the inter- and intraband interactions reveals thatthe former is still about 20% of the larger and similar tothe smaller of the intraband interactions [9, 11–14]. Inother systems, such as the recently discussed iron-basedsuperconductors is even argued that the interband cou-pling is the dominant source of pairing, see e.g. Refs.[15, 16].Further support for comparatively large interband cou-pling comes from an analysis of recent Scanning Tunnel- a r X i v : . [ c ond - m a t . s up r- c on ] O c t ing Microscopy (STM) measurements of the density ofstates (DOS) distribution within the vortex lattice at lowtemperatures in several two-band compounds [17, 18].For a single-band material one can construct a phe-nomenological model to relate the measured zero-biasDOS distribution N ( r ) to the pairing amplitudes | ∆( r ) | in the lattice unit cell [17]. This procedure is readily ex-tended to a two-band situation, for which N ( r ) dependson both ξ and ξ . The fit to the STM data for NbSe andfor NbSe . S . showed that ξ ≈ ξ at T = 0 .
15 K (cid:28) T c .The same procedure has been applied to the novel su-perconductor CaKFe As with T c ≈
35 K and the zero-field tunneling spectrum having clearly two-gap features,again with the result ξ ≈ ξ at sub-Kelvin temperaturesand at all fields examined [18].These theoretical considerations and observations mo-tivated us to re-examine the question of the relative val-ues of ξ and ξ in two-band superconductors within a mi-croscopic approach that covers a broad temperature andmagnetic field regime. In particular, the analysis of theSTM-data suggests that the emergence of one commonlength-scale is a much more robust phenomenon than onewould expect for moderately coupled multi-band system.Thus, we aim at clarifying the issue of when the couplingbetween two superconducting bands becomes sufficientlystrong to give rise to a common length scale and underwhat conditions two separate length scales of the band-order parameters emerge.To this end, we use a “brute-force” numerical proce-dure of solving Eilenberger equations for a vortex latticein the two-band case developed in studies of MgB [19].We consider a weak-coupling model of a two-band super-conductor with two Fermi surface parts having differentFermi velocities and study the spatial variation of thepairing amplitudes ∆ , ( r ) of the two bands within thevortex lattice unit cell. While we analyze this model overa wide range of parameters, we do not focus on a specificapplication for a particular material. Rather, we intendto clarify general properties of the spatial dependency of∆ , ( r ). Substantially different values of the Fermi veloc-ities notwithstanding, the coherence lengths proportionalto the vortex core size defined as ξ ( c )1 , ∝ ( d | ∆ , | /dr ) − r → ( r is the distance from the vortex center) turn out nearlythe same for all choices of coupling constants V αβ exam-ined ( α, β = 1 ,
2) except the case of nearly decoupledcondensates V /V ≤ . V (cid:28) V our results agree with previouscalculations [7, 8]. However, as soon as V /V ≥ . ξ = ξ , insensitive to details of coupling V αβ , temperature, and field. Given the exponential de-pendence of the superconducting gap on the couplingconstants, comparatively weakly coupled systems with V /V ≥ . · · · . II. APPROACH
We consider two-band system with two cylindricalFermi surfaces ( α = 1 ,
2) both oriented parallel to thesame crystal axis (the c -axis) and with Fermi velocities v α ( k ) = v α (cos φ, sin φ ). k is the Fermi momentum and φ the corresponding azimuth. The magnetic field is ap-plied along c as well, i.e. the field is parallel to the axisof the cylinder. For simplicity, the bands normal densi-ties of states are assumed the same: N , = N , = N (the total DOS per spin N (0) = 2 N ). This assumptionwill not affect any of our results qualitatively and caneasily be dropped. It still allows for distinct values of theFermi velocities of the bands. We set v = 3 v to assuresubstantially different coherence lengths in the limit offully decoupled bands. The 2 × V αβ isassumed symmetric: V = V .Our approach is based on the quasiclassical versionof the weak-coupling BCS theory for anisotropic Fermisurfaces and order parameters [20]. This theory is for-mulated in terms of Eilenberger functions f, f + and g (Gor’kov’s Green’s functions averaged over the energy):(2 ω + v α · Π ) f α = 2∆ α g α , (1) g α = 1 − f α f + α , α = 1 , . (2)Here Π = ∇ + 2 πi A /φ with vector potential A and fluxquantum φ . ω = πT (2 n + 1) are fermionic Matsubarafrequencies with integer n ; hereafter ω and T are mea-sured in energy units, i.e. (cid:126) = k B = 1. The equationfor f + is obtained from Eq. (1) by taking the complexconjugate and replacing v → − v .The pairing amplitudes satisfy the self-consistency re-lations: ∆ α ( r ) = 4 πT N (cid:88) β, ω V αβ (cid:104) f ( ω, k , r ) (cid:105) β , (3)where the sum over positive Matsubara frequencies isextended up to ω D , the analog of Debuy frequency forelectro-phonon mechanism; (cid:104) f ( ω, k , r ) (cid:105) β stands for theaverage over the Fermi cylinder of the band β . The con-tribution of the α -band to the current density is J α ( r ) = − π | e | N T Im (cid:88) ω> (cid:104) v g ( ω, k , r ) (cid:105) α , (4)and the total current density is J = J + J = ∇ × ( ∇ × A ) c/ π . (5)The vector potential is taken in the form A ( r ) = ( B × r ) / A ( r ), where the magnetic induction B = (0 , , B )is the field averaged over the vortex lattice cell and ˜ A ( r )represents the variable part of the field which is periodicin the vortex lattice and has zero spatial average. Theunit vectors of the triangular vortex lattice are chosenas u = ( a , ,
0) and u = ( a , √ a / , a = (2 φ / √ B ) / . We use peri-odic boundary conditions for the unit cell of the vortexlattice and take into account the order parameter phasewinding around each vortex [21].Throughout the paper, we use Eilenberger units for thefirst band if it would have been single ( V = V = 0): R = (cid:126) v / πT c is taken as a unit length ( R ≈ . ξ where ξ is the zero- T BCS coherence length of the“bare” first band). Fermi velocities are normalized to v ,the magnetic field is measured in units of B = φ / πR and the current density in cB / πR , the energy unit is πT c , and T c is the transition temperature in the single-band limit. In these units, Eqs. (1) and (4) take the form:( ω + v α · ∇ ) f α = ∆ α g α − i v α · [( B × r ) / A ] f α , (6) J α ( r ) = − Tκ (cid:88) ω> (cid:104) v Im g ( ω, k , r ) (cid:105) α . (7)Hereafter we keep the same notation for dimensionlessquantities as for their dimensional counterparts; we willindicate explicitly if common units are needed.The quantity κ = φ T c /π (cid:126) v √ N has the sameorder of magnitude as the GL parameter for one-bandisotropic case, κ GL = 3 φ T c / (cid:126) v (cid:112) ζ (3) N (0). However, κ does not have the meaning of the penetration-depth-to-coherence-length ratio for the two-band system[2, 3],rather it is a convenient dimensionless material parame-ter.The dimensionless self-consistency equations take theform: ∆ α ( r ) = 2 tN V (cid:88) β, ω V αβ V (cid:104) f ( ω, k , r ) (cid:105) β , (8) πe − γ T c = 2 ω D exp( − /N V ) , t = T /T c (9)where γ is the Euler constant. In our calculations weset the cutoff frequency ω D = 40 T c and κ = 4. Thenumerical procedure is outlined in the Appendix.The profiles of the pairing amplitudes | ∆ α ( r ) | in realspace are fitted by a 5 th -order polynomial near the vortexcenter along the nearest neighbor vortex direction. Weestimate the vortex core size ξ ( c ) α from∆ α ( r ) = ∆ m ,α rξ ( c ) α + O ( r ) , j = 1 , m ,α is the maximum value of | ∆ α ( r ) | within the unit cell. III. V OF THE SAME ORDER AS V First, we present our results for V = 0 . V . In orderto see the effect of the coupling in the second band, weconsider two cases: V = 0 and V = 0 . V .The profiles of | ∆ ( r ) | and | ∆ ( r ) | are shown inFig. 1(a). Near the vortex center, both | ∆ ( r ) | and | ∆ ( r ) | recover over the same lengths; this is seenmost directly in panel (b) where nearly constant ratios | ∆ ( r ) | / | ∆ ( r ) | are shown. In the presence of finite in-traband coupling of the second band V , the ampli-tude of the pair potential of this band increases, with FIG. 1. (Color online) (a) Pairing amplitudes | ∆ ( r ) | and | ∆ ( r ) | (in units πT c ) vs distance r (in units of R = (cid:126) v / πT c ) from the vortex center to the midpoint betweennearest neighbor vortices. In this calculation, V /V = 0 . t = T /T c = 0 .
5, and B = 0 . φ / πR ). Solidlines are for V = 0, dashed lines are for V /V = 0 .
32. (b)Nearly constant ratios | ∆ ( r ) | / | ∆ ( r ) | imply the same lengthscales for both pairing amplitudes. | ∆ ( r ) | / | ∆ ( r ) | ∼ .
4, as expected. The spatial depen-dence of the two pair potentials is however the same.Temperature dependences of the core radii ξ ( c ) α andof the maximum value ∆ m ,α are given in Fig. 2. While∆ m ,α are slightly smaller than those in zero field (dottedline) as they should, the T -dependence of ∆ m ,α is similarto that at zero field. Nearly constant ratios ∆ m , / ∆ m , are ≈ . V = 0 and ≈ . V = 0 . V .As the temperature increases, this ratio changes little:from 0 .
291 to 0 .
295 for V = 0, and from 0 .
406 to 0 . V = 0 . V , respectively. Within our analysis wealso reproduce Kramer-Pesch shrinking of the vortex coresizes ξ ( c ) on cooling [22–24], see Fig. 2(c,d). Thus, weobtain ξ ( c )2 ≈ ξ ( c )1 in the whole temperature range. Whileit is expected [1–3] that ξ ( c )2 /ξ ( c )1 → T → T c , ourfinding of numerically very similar length scales over abroad temperature regime is rather surprising.The field dependencies of the pairing amplitudes anddeduced length scales are shown in Fig. 3. As expected,the ∆ m ,α are suppressed upon increasing the magneticfield, see Fig. 3(a). As shown in Fig. 3(b,c), after a slowdecrease at low B ’s, the core radii ξ ( c ) α are once againnearly constant over a wide range of field values. Mostimportantly however, we find at all fields that ξ ( c )1 ≈ ξ ( c )2 ,see panel (d) of Fig. 3. As B approaches the upper criticalfield H c , ξ ( c )2 /ξ ( c )1 →
1, see Fig. 3(d). This conclusionagrees with the two-band theory of H c [25], where ithas been shown that near a 2 nd order phase transitionat H c , the two pairing amplitudes satisfy the system ofequations − ξ Π ∆ α = ∆ α with the same ξ . FIG. 2. (Color online) (a) Temperature dependence of max-imum values ∆ m ,α of pairing amplitudes | ∆ α ( r ) | at B = 0 . V = 0 and V = 0 . V . Zero-field | ∆ α | are shown bydotted lines. (b) The same as (a) for V = 0 . V . (c,d) T dependences of core sizes ξ ( c ) α , and (e) of ξ ( c )2 /ξ ( c )1 for B = 0 . T is in units of T c . IV. DECOUPLING LIMIT V (cid:28) V Next we analyze the regime of almost decoupled band.In this limit, the two superconducting condensates arenearly independent. The vortex core radii can be dif-ferent and dependent on the characteristics of the bands[7, 8].We consider a weak inter-band coupling, V =0 . V , whereas V = 0 . V . The resulting | ∆ α ( r ) | are presented in Fig. 4(a). At a low field B = 0 . | ∆ ( r ) | with increasing r is indeed slow compared to | ∆ ( r ) | , and as a result wefind that ξ ( c )2 > ξ ( c )1 . This behavior can also be seen inthe r dependence of the ratio | ∆ ( r ) | / | ∆ ( r ) | , which isno longer constant, but decreases near the vortex core,see Fig. 4(b). For higher field, B = 0 . FIG. 3. (Color online) (a) Magnetic field dependence of ∆ m ,α , α = 1 ,
2. (b,c) B dependence of the core sizes ξ ( c ) α and (d)the ratio ξ ( c )2 /ξ ( c )1 . Inputs: t = 0 . V = 0 . V , solid linesare for V = 0, dashed lines for V = 0 . V .FIG. 4. (Color online) (a) | ∆ ( r ) | and | ∆ ( r ) | vs distance r from the vortex center to the midpoint between nearestneighbor vortices. (b) | ∆ ( r ) | / | ∆ ( r ) | . Input parameters are V = 0 . V , V = 0 . V , and t = 0 .
5; solid lines are for B = 0 .
1, dashed lines are for B = 0 . FIG. 5. (Color online) (a) Temperature dependence of ∆ m ,α at B = 0 (dotted lines), B = 0 .
03 (dashed lines), and B = 0 . T dependence of the vortex core radii ξ ( c )1 and ξ ( c )2 , and (c) the ratio ξ ( c )2 /ξ ( c )1 . V = 0 . V and V =0 . V . lines in Fig. 4), | ∆ ( r ) | within the core region doesnot change substantially compared to the low-field case,whereas | ∆ ( r ) | is suppressed strongly, as the intervortexdistance is too short for the recovery of | ∆ ( r ) | . In otherwords, since the “effective H c ” of the second band issmall due to a larger coherence length ( v = 3 v and ∆ is small), superconductivity of the second band is easilysuppressed by magnetic fields. Hence, at high fields, thecontribution to superconductivity of the second band isweak.The corresponding temperature dependence of thenearly decoupled band regime is shown in Fig. 5. ∆ m , has the typical T -dependence of the BCS theory. How-ever, ∆ m , ( T ) is different. At low T , the superconductiv-ity of the second band is enhanced, since it is caused hereby V = 0 . V . For B = 0, ∆ is very small at elevatedtemperatures. Above the intrinsic transition tempera-ture of the decoupled second band, superconductivity ofthis band is only induced by the weak interband coupling V , an observation that was made already shortly afterthe formulation of the BCS-theory [26]. With increasing B , the enhancement of ∆ m , at low T disappears andpractically vanishes at B = 0 .
1. The B -dependence ofthe pairing amplitudes are shown in Fig. 6. ∆ m , de-creases rapidly at low B reflecting small effective H c2 , FIG. 6. (Color online) (a) The field dependence of ∆ m ,α .(b) B -dependence of core radii ξ ( c )1 and ξ ( c )2 , and (c) the ratio ξ ( c )2 /ξ ( c )1 . Input parameters: t = 0 . V = 0 . V and V =0 . V . of the second band, and remains small at higher B dueto weak coupling V . In the high B range, ξ ( c )1 ≈ ξ ( c )2 .This combination of field and temperature variation ofnearly decoupled bands may serve as a tool to identifywhether one is indeed in this limit.We note that the Kramer-Pesch shrinking of ξ ( c )2 oncooling is weak compared to that of ξ ( c )1 , see Fig. 5(b).Thus, the ratio ξ ( c )2 /ξ ( c )1 increases upon lowering T . Onthe other hand, at higher T and for fields approaching H c , ξ ( c )2 /ξ ( c )1 → V. DISCUSSION
The issue of the spatial variation of the superconduct-ing order parameter in multi-band systems is interestingand relevant, in particular because of an increasing num-ber of physical systems that clearly display multi-bandbehavior in their superconducting properties. In additionto the description of the variation of the order parame-ter near vortex cores, the DOS distribution is related to∆( r ) and is measurable. Recent STM low- T data, inter-preted within a phenomenological model, suggest that ξ ( c )1 = ξ ( c )2 [17]. While such length-scale locking is tobe expected in the immeadiate vicinity of the transition FIG. 7. (Color online) (a) ξ ( c ) α vs interband coupling V /V for V = 0 . V at t = 0 . B = 0 .
03. (b) Ratios ξ ( c )2 /ξ ( c )1 and ∆ m , / ∆ m , vs V /V for the same parametersas (a). temperature, it is not obvious away from T c . Thus, amicroscopic analysis of this open question is timely andrelevant. It shows that, within the accuracy of our nu-merical routines, ξ ( c )1 ≈ ξ ( c )2 if the inter-band coupling isof the same order as intra-band ones. This conclusionturns out to be valid at all temperatures and fields. Inagreement with other microscopic calculations [7, 8], wefind this rule is violated for a very weak inter-band cou-pling when the system is close to the limit of nearly de-coupled condensates. The peculiar field and temperaturedependence of such nearly decoupled bands can easily beused to test, for a given material, whether the couplingbetween bands is weak or only moderate.To make this statement more quantitative, we showin Fig. 7 the ratio ξ ( c )2 /ξ ( c )1 as a function of the inter-band coupling V /V at fixed t = T /T c = 0 . B = 0 .
03. One sees that this ratio exceeds the value of2 only when roughly V /V < .
1. As discussed above,MgB can be very well described by V /V ≈ . T c ( B ), it is established [1–3, 25] that theemergence of one order parameter naturally implies thatthe spatial variation of this order parameter is governedby a single length scale. Away from T c it is howevernatural to expect that for a sufficiently weak couplingbetween the bands, distinct characteristic length scales for the respective pairing amplitudes emerge. However,what precisely is meant by sufficiently weak has not beeninvestigated thus far. Here we showed that such decou-pling of the length scales occurs for values of the inter-band pairing interaction V that is less than one order ofmagnitude of the largest intraband coupling. For largervalues of the interband coupling a common temperaturevariation of the length ξ ( c )1 and ξ ( c )2 of the pairing ampli-tudes sets in. What is most surprising about our resultsis that these two length scales not only follow a common T -dependence, they are essentially identical in their mag-nitude ξ ( c )1 ≈ ξ ( c )2 . In other words, we observe a robustlength scale locking of moderately coupled multiband su-perconductors. Whatever difference might there be inthe values of the length scales of the uncoupled system,our analysis shows that this difference is most likely todisappear everywhere below H c ( T ).In this work we only considered the situation of a cleantwo-band situation. Usually, the impurity scattering isexpected to cause an isotropization of superconductingcharacteristics. Hence, we do not expect scattering toamplify differences of the length scales ξ α . Still, as dis-cussed in Ref. [16], inter-band scattering can cause thesuperconductivity to become gapless with two bands ac-quiring substantially different DOSs in superconductingstate. The question of how this difference affects ξ α re-mains to be answered. ACKNOWLEDGEMENTS
We thank Lev Boulaevskii for illuminating comments.Work of V.K. was supported by the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences, Mate-rials Sciences and Engineering Division. The Ames Lab-oratory is operated for the U.S. DOE by Iowa State Uni-versity under Contract No. DE-AC02-07CH11358.
Appendix A: Numerical method
We briefly summarize the numerical approach to solvethe coupled Eilenberger equations Eqs. (1). For the nu-merical analysis, it is more convenient to employ insteadof the function f and g the functions a and b defined via f = 2 a ab , f + = 2 b ab , g = 1 − ab ab (A1)and transform the system (1)-(2) to Ricatti differentialequations, v · ∇ a = (cid:0) ∆ − ∆ ∗ a (cid:1) − ( ω + i v · A ) a, (A2) − v · ∇ b = (cid:0) ∆ ∗ − ∆ b (cid:1) − ( ω + i v · A ) b, (A3)for each band α [27]. Unlike the original Eqs. (1), theequations for a and b are decoupled. The Ricatti equa-tions are then solved by numerical integration along tra-jectories parallel to the vector v [28]. Choosing length FIG. 8. (Color online) Solving the first-order ordinary dif-ferential Eq. (A2) along the trajectory r (cid:48) = r + s ˆ v α for a at s = 0. Real and imaginary part of a are shown for start posi-tions s = − . − . − . a converges to the same solution at s = 0. Input parametersare φ = 1 . ◦ for k , α = 1, ω = πT and V = 0 in the caseof Fig. 1. r is near the midpoint ( − a / ,
0) between nearestneighbor vortices. | s | of these trajectories in Fig. 8, we confirm that thesolution does not change when this length is increased.We iterate the set of equations until self-consistent resultsare obtained. [1] B. T. Geilikman, R. O. Zaitsev, and V. Z. Kresin, Sov.Phys. Solid State , 642 (1967).[2] J. Geyer, R. M. Fernandes, V. G. Kogan, and J.Schmalian, Phys. Rev. B , 104521 (2010).[3] V. G. Kogan and J. Schmalian, Phys. Rev. B , 054515(2011).[4] E. Babaev and M. Speight, Phys. Rev. B , 180502(2005).[5] E. Babaev, J. Calstr¨om, M. Silaev, and J.M. Speight,arXiv:1608.02211.[6] V. Moshchalkov, M. Menghini, T. Nishio, Q. H. Chen,A.V. Silhanek, V. H. Dao, L. F. Chibotaru, N. D. Zhi-gadlo, and J. Karpinski, Phys. Rev. Lett. , 117001(2009).[7] M. Silaev and E. Babaev, Phys. Rev. B , 094515(2011).[8] L. Komendova, Yajiang Chen, A. A. Shanenko, M.V.Milosevic, and F. M. Peeters, Phys. Rev. Lett. ,207002 (2012).[9] I. I. Mazin and V. P. Antropov, Physica C , 49 (2003).[10] G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein,N. D. Zhigadlo, and J. Karpinski, Phys. Rev. Lett. ,227002 (2007).[11] A.Y. Liu, I. I. Mazin and J. Kortus, Phys. Rev. Lett., , 087005 (2001).[12] A. A. Golubov, J. Kortus, O. V. Dolgov, O. Jepsen, Y.Kong, O. K. Andersen, B. J. Gibson, K. Ahn and R. K.Kremer, J. Phys.: Condens. Matter , 1353 (2002).[13] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G.Louie, Phys. Rev. B , 020513 (2002). [14] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G.Louie, Nature , 758 (2002).[15] I. I. Mazin and J. Schmalian, Phys. C: Supercond. ,614 (1995).[16] V. G. Kogan and R. Prozorov, Phys. Rev. B , 224515(2016).[17] A. Fente, E. Herrera, I. Guillamon, H. Suderow, S.Ma˜nas-Valero, M. Galbiati, E. Coronado and V. G. Ko-gan, Phys. Rev. B , 014517 (2016).[18] A. Fente, W. R. Meier, T. Kong, V.G. Kogan, S.L. Bud’ko, P. C. Canfield, I. Guillamon, H. Suderow,arXiv:1608.00605.[19] M. Ichioka, K. Machida, N. Nakai, and P. Miranovi´c,Phys. Rev. B , 144508 (2004).[20] G. Eilenberger, Z. Phys. , 195 (1968).[21] M. Ichioka, N. Hayashi, and K. Machida, Phys. Rev. B , 6565 (1997).[22] L. Kramer and W. Pesch, Z. Phys. , 59 (1974).[23] M. Ichioka, N. Hayashi, N. Enomoto, and K. MachidaPhys. Rev. B , 15316 (1996).[24] A. Gumann, S. Graser, T. Dahm, and N. Schopohl, Phys.Rev. B , 104506 (2006).[25] V. G. Kogan and R. Prozorov, Rep. Prog. Phys. ,114502 (2012).[26] H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev.Lett. , 552 (1959).[27] N. Schopohl and K. Maki, Phys. Rev. B , 490 (1995).[28] P. Miranovi´c, M. Ichioka, and K. Machida, Phys. Rev. B70