Shallow-band superconductors: pushing superconductivity types apart
SShallow-band superconductors: pushing superconductivity types apart
S. Wolf, A. Vagov, A. A. Shanenko, V. M. Axt, and J. Albino Aguiar Institut f¨ur Theoretische Physik III, Bayreuth Universit¨at, Bayreuth, Germany Departamento de F´ısica, Universidade Federal de Pernambuco, Recife, PE, Brazil (Dated: November 7, 2018)Magnetic response is a fundamental property of superconducting materials, which helps to distin-guish two superconductivity types: the ideally diamagnetic type I and type II, which can developthe mixed state with Abrikosov vortices. We demonstrate that multi-band superconductors withone shallow band, that have recently attracted much attention for their high critical temperaturesand unusual properties, often stay apart of this simple classification. In a wide range of microscopicparameters such systems fall into the inter-type or transitional interval in between the standardtypes and display unconventional mixed state configurations.
The ongoing search for superconductors with highertransition temperatures ignited intense research on mate-rials with many carrier bands (multi-band systems) withso-called shallow bands [1]. It has long been known thatsuperconductivity properties in multi-band materials arevery sensitive to the position of the chemical potential.As was first predicted in 1970’s if a chemical potentialtouches the lowest energy point of a band with a largedensity of states (DOS) the superconductivity may enterthe regime of the BCS-BEC crossover [2]. Evidences ofthis regime have been recently observed in multi-bandFeSe x Te − x [3]. However, the research on the role ofshallow bands is, at present, limited to a few microscopiccharacteristics of the BCS-BEC crossover, such as in-creased fluctuations and the appearance of the pseudogap(for recent developments see [4, 5] and references therein).Until now, manifestations of the BCS-BEC crossover ona macroscopic level, in particular possible modificationsof the magnetic properties, remain largely unknown. Inthis Letter we demonstrate that many multi-band super-conductors with a shallow band are in a mixed state andreveal magnetic properties that cannot be attributed tostandard superconductivity types.We start by recalling the well known fact that theGinzburg-Landau (GL) theory predicts that the mag-netic response of a superconductor is determined by theGL parameter κ = λ/ξ , where λ is the magnetic pene-tration depth and ξ is the GL coherence length [6–8].The superconductivity types interchange sharply at thecritical GL parameter κ = 1 / √
2, so that at κ < κ or κ > κ a superconductor belongs, respectively, to type Ior II. More elaborate theoretical analysis [9] as well as ex-perimental studies [10–13] have led to the conclusion thatthe type interchange takes place over a finite interval of κ ’s, here referred to as inter-type interval or inter-type do-main , if considered in the entire ( κ, T )-plane. In conven-tional single-band materials this domain is narrow and isusually ignored in most discussions of the superconduc-tivity types. Superconductivity properties in this intervalare investigated in detail neither theoretically nor experi-mentally. The studies presented so far suggest that it hasmany non-conventional phenomena, not found in stan- dard bulk superconductors. In particular, unlike type IImaterials such inter-type superconductors demonstratethe first order first-order phase transition between theMeissner and the mixed states, which is associated withnon-monotonic vortex-vortex interactions [14] (schemati-cal magnetization curves are shown in Fig. 1). They alsodemonstrate stable multi-quantum vortices [15, 16] anda giant paramagnetic Meissner effect [17].In this work, using a two-band prototype model of asuperconductor, we demonstrate that the presence of ashallow band with strong pairing interaction dramati-cally enlarges the inter-type domain and thus changesqualitatively the magnetic properties of such systems.To this end we study details of the interchange of su-perconductivity types. A standard description of the in-terchange in the GL theory is done by introducing a crit-ical GL parameter κ ∗ , which marks the appearance ofthe mixed state at the thermodynamic critical field H c ,so that at κ > κ ∗ it wins energetically over the uniformMeissner state. It is obtained by solving the equation: G ( κ ∗ , T ) = 0 , G = (cid:90) g d r , g = f + H c π − H c B π (1)where G is the Gibbs energy difference between the mixedand the Meissner states calculated at H c and f is thecondensate free-energy density. The magnetic induction B is assumed parallel to the external field H = H c .In the GL theory a particular choice of the non-uniformmixed state is not important for criterion (1): one obtainsthe same critical parameter κ ∗ = κ for all possible fluxconfigurations. The reason is a special degeneracy of theGL equations at κ , often referred to as the Bogomolnypoint [18]. However, beyond the GL theory at T < T c this degeneracy is removed and κ ∗ = κ ∗ i depends on theflux configuration i . The number of topologically differ-ent configurations i is infinite and so is the number ofcritical parameters κ ∗ i . This defines a finite inter-typeinterval [ κ ∗ min , κ ∗ max ], where superconductivity types in-terchange gradually by a sequential appearance of fluxconfigurations i when κ ∗ i is crossed. Previous researchindicates that the lower boundary of this interval is de-fined by the start of superconductivity nucleation at H c a r X i v : . [ c ond - m a t . s up r- c on ] A p r (this is equivalent to the condition H c = H c ), while theupper boundary if determined by the appearance of along-range attraction between two Abrikosov vortices.We now calculate the inter-type boundaries for a modelwith two carrier bands, one of which is a shallow 2D bandwith a minimal energy that coincides with the chemi-cal potential. The lower dimensionality of the band in-creases its DOS, which is crucial for reaching the BCS-BEC crossover regime [19]. The other band is deep. Itsdimensionality is not so important, although a 2D modelrequires more attention in view of thermal fluctuations.However, once it is assumed that the superconductivityin the deep band can be described within the mean fieldapproach, details of the model are not critical. For sim-plicity we assume that both bands are two dimensional.According to the BCS theory the free-energy densityof the condensate state in a two-band system writes as: f = B π + ∆ † ˇ g − ∆ + (cid:88) ν =1 , f ν , (2)where ∆ † = (cid:0) ∆ ∗ ( r ) , ∆ ∗ ( r ) (cid:1) is the band gap functionand ˇ g − is the inverted 2 × g ij = g ji . We calculate the free energy using the pertur-bation expansion with the small parameter τ = 1 − T /T c ,which leads to the so-called extended GL (EGL) theory[20, 21]. For a two-band system the lowest order of theexpansion yields the standard GL theory with a single or-der parameter [21, 22]. A far-reaching consequence of thisfact is that two-band superconductors follow the stan-dard classification, where types I and II are separatedby the Bogomolny point at T c and by a finite inter-typeinterval at T < T c .The perturbation expansion is derived similarly to Ref.[21], with the difference that here the bands are qualita-tively different (shallow and deep), which leads to ad-ditional contributions to the series expansion. Here, wepresents a sketch of the derivation which highlights thedifferences with Ref. [21]. Expanding f ν in Eq. (2) inpowers of ∆ ν and its gradients yields: f ν = − a ,ν | ∆ ν | + a ,ν | D ∆ ν | − a ,ν (cid:16) | D ∆ ν | + rot B · i ν e (cid:126) c B | ∆ ν | (cid:17) + a ,ν B | ∆ ν | + b ,ν | ∆ ν | − b ,ν (cid:16) L ν | ∆ ν | | D ∆ ν | + l ν (cid:2) (∆ ∗ ν ) ( D ∆ ν ) + c . c . (cid:3)(cid:17) − c ,ν | ∆ ν | , (3)where the band coefficients a n,ν , b n,ν , c n,ν are T -dependent. The constants L ν and l ν are introduced tocapture the differences between the bands and i ν = 4 e (cid:126) c Im (cid:2) ∆ ν D ∗ ∆ ∗ ν (cid:3) , D = ∇ − i e (cid:126) c A . (4) The τ -expansion is obtained by representing all quanti-ties in Eq. (3) as τ -series:∆ ν = τ / (cid:2) ∆ (0) ν + τ ∆ (1) ν (cid:3) , A = τ / (cid:2) A (0) + τ A (1) (cid:3) , B = τ (cid:2) B (0) + τ B (1) (cid:3) , H c = τ (cid:2) H (0) c + τ H (1) c (cid:3) , (5)where the two lowest orders, needed to derive the lead-ing order corrections to the GL theory, are kept. Wetake into account the τ -scaling of the coordinates [20],which introduces an additional factor √ τ for each gradi-ent in the series. Expanding the temperature-dependentcoefficients in Eq. (3) yields: a ,ν = A ν − τ (cid:2) a (0) ν + τ a (1) ν (cid:3) , a ,ν = K (0) ν + τ K (1) ν ,a ,ν = Q (0) ν , a ,ν = r (0) ν , b ,ν = b (0) ν + τ b (1) ν ,b ,ν L ν = L (0) ν , b ,ν l ν = (cid:96) (0) ν , c ,ν = c (0) ν , (6)where the coefficients are calculated from the chosenmicroscopic model for the band states. SubstitutingEqs. (5), (6) and the gradient scaling into Eq. (3) andthen calculating the integrals in Eq. (1) one derives the τ -expansion for G .It is important that the leading correction to the GLfree energy requires only ∆ (0)1 , and B (0) while ∆ (1)1 , and B (1) are not needed [21]. Thus, the corrected free energycan be evaluated from the knowledge of the solution ofthe GL equations alone. The latter exhibits a single orderparameter Ψ, which determines both gaps by: (cid:32) ∆ (0)1 ∆ (0)2 (cid:33) = (cid:18) S − / S / (cid:19) Ψ( r ) . (7)The band weight factor S is obtained by solving the lin-earized gap equation for T c which yields: S = 1 g (cid:0) g − G A (cid:1) = g g − G A , (8)where G = det[ g ] = g g − g . The integration inEq. (1) is simplified with the help of the GL equationsand the final result for G depends only on the integrals: I = (cid:90) | Ψ | (cid:0) − | Ψ | (cid:1) d r , J = (cid:90) | Ψ | (cid:0) − | Ψ | (cid:1) d r . (9)Solving Eq. (1) up to the leading order corrections of theGL theory we obtain: κ ∗ = κ + τ κ ∗ (1) (10)with κ ∗ (1) κ = ¯ K − ¯ c + 2 ¯ Q + ¯ G ¯ β (cid:0) α − ¯ β (cid:1) + JI (cid:18) ¯ L − ¯ c −
53 ¯
Q − ¯ G ¯ β (cid:19) , (11) ν M (0) b,ν M (0) c,ν M (0) K ,ν M (0) Q ,ν M (0) L ,ν M (1) a,ν M (1) b,ν M (1) K ,ν ζ (3) / (8 π ) 93 ζ (5) / (128 π ) 7 ζ (3) / (32 π ) 93 ζ (5) / (2048 π ) 31 ζ (5) / (32 π ) 1/2 2 22 7 ζ (3) / (8 π ) 93 ζ (5) / (128 π ) 3 ζ (2) / (8 π ) 7 ζ (3) / (512 π ) 25 ζ (4) / (16 π ) 1/2 2 1TABLE I. Numerical factors M (0) w,ν (with w = b, c, K , Q , K ) and M (1) w,ν (with w = a, b, K ) for the deep ( ν = 1) and shallow( ν = 2) bands, see Eqs. (14) and (15). In the table ζ ( x ) is the Riemann zeta function of x . where the dimensionless constants read as:¯ K = K (1) K − b (1) b , ¯ c = ca b , ¯ Q = a QK , ¯ L = a L b K , ¯ G = Ga g , ¯ α = αa − Γ K , ¯ β = βb − Γ K . (12)The coefficients are defined by the band contributions as: ω = ω (0)1 S p + S p w (0)2 , ω (1) = ω (1)1 S p + S p w (1)2 ,α = a (0)1 S − Sa (0)2 , β = b (0)1 S − S b (0)2 , Γ = K (0)1 S − S K (0)2 . (13)where ω = { a, K , Q , r, b, L , c } , ω (1) = {K (1) , b (1) } , w (0) ν = { a (0) ν , K (0) ν , Q (0) ν , r (0) ν , b (0) ν , L (0) ν , c (0) ν } and values p = { , , } appear respectively for coefficients a n,ν , b n,ν and c n,ν .The band coefficients in Eq. (6) are calculated for amodel with 2D quadratic dispersion for both bands. Wenote that in the EGL formalism the band dimensionalityaffects the results mainly via the ratio between the bandDOSs. The calculations are done in the clean limit. Forthe deep band ( ν = 1) the inequality ∆ (cid:28) µ − ε ,k =0 means that we can use standard approximations em-ployed in the derivations of the EGL theory for the 3Dcase [20]. For the shallow band ( ν = 2) the chemical po-tential is assumed to coincide with the band minimum, µ = ε ,k =0 . Then the leading order coefficients in Eq. (6)are: A ν = N ν ln (cid:16) e γ (cid:126) ω c πT c (cid:17) , a (0) ν = − N ν , b (0) ν = N ν M (0) b,ν T c ,c (0) ν = N ν M (0) c,ν T c , K (0) ν = N ν M (0) K ,ν (cid:126) v ν T c , Q (0) ν = N ν M (0) Q ,ν (cid:126) v ν T c , L (0) ν = N ν M (0) L ,ν (cid:126) v ν T c , (14)where (cid:126) ω c is the cut-off energy, γ is the Euler constant, N ν is the band DOS, v ν denotes the characteristic bandvelocity, i.e., the Fermi velocity v F = (cid:112) µ/m ν for thedeep band and the temperature velocity v T = (cid:112) T c /m ν for the shallow band. The additional numerical fac-tors M (0 , w,ν are listed in Tab. I. The band DOSs are N ν = ˜ N ν m ν / (2 π (cid:126) ) with ˜ N ν being an additional fac-tor that accounts for the density of states in z-direction(this quantity accounts for the 3D character of the entiresystem and does not affect the final conclusions). Thenext-order coefficients in Eq. (6) are given by: w (1) ν = M (1) w,ν w (0) ν , (15)where w = { a, K , b } .Knowing the ratio η = N /N of the DOSs and thecoupling constants λ ij = g ij N ( N = N + N ) one ob-tains the critical temperature T c from the linearized gapequation and then κ ∗ from Eq. (11). It is important thatapart from T c the final expression for κ ∗ depends only on η and v /v , but not on N , and v , separately. Theratio J / I in Eq. (11) is calculated using the solution ofthe GL equations at κ , which at this point reduces tothe pair of self-dual Sarma-Bogomolny equations [6, 18].As mentioned before, the lowest boundary of the inter-type interval κ ∗ min is calculated from the condition thatthe inhomogeneous mixed state disappears at H c . It fol-lows from Eq. (9) that in the limit of a vanishing mixedstate one has J / I = 0. The upper boundary κ ∗ max isdefined by the condition that the sign in the long-rangevortex-vortex interaction changes. In this case we needto calculate the long-range asymptote of J ( R ) / I ( R ) forthe two-vortex solution as a function of the distance R between the vortices. This can be done analytically yield-ing the exact asymptote J ( R ) / I ( R ) = 2 at R → ∞ .Although the final expression for κ ∗ is a complicatedalgebraic function of the microscopic model parameters,it can be considerably simplified for the case of a two-band system with v /v ∼ (cid:112) T c /µ (cid:28) S (cid:38) κ (1) ∗ κ ≈ ˜ Q (cid:18) − JI (cid:19) S η, ˜ Q = a (0)1 Q (0)1 K (0)21 , (16)where the bracket gives 2 for κ ∗ and − / κ ∗ li , if weuse the above results for the J / I ratio. According to thissimplified expression the width of the inter-type domainis governed by the ratio of the DOSs η , the couplings (via S ) and the dimensionless constant ˜ Q . Equations (3) and(6) show that ˜ Q controls the contribution of the fourth- inter-typetype Itype II - M - M - M H c H c1 H c2 H* H c2 a) b) c) FIG. 1. Phase diagram of a two-band superconductor on the ( κ, T )-plane. The left panels illustrates schematically themagnetization field dependence of types I and II and of the inter-type domain. Panels a), b) and c) correspond to η = 0, 1 and2, respectively, demonstrating progressive widening of the inter-type domain. For comparison, dots in the left panel representnumerical results for the inter-type boundaries (squares for κ ∗ max = κ ∗ li and stars for κ ∗ min = κ ∗ ), obtained by solving theEilenberger equation [23]. order gradient term in the free-energy expansion for thedeep band.Taking into account that S increases with η , when thesystem is close to the crossover [19], one concludes thatthe inter-type interval increases with η . This wideningof the inter-type domain in the ( κ, T )-plane is illustratedin Fig. 1, where three panels show κ ∗ and κ ∗ li as func-tions of temperature (this is of cause a linear dependencein the EGL theory) calculated at η = 0 , , κ ∗ . We note that κ ∗ de-pends only modestly on the coupling constants but isindeed very sensitive to the value of η . We also note thatwhen η = 0 (left panel) only the deep band is involved inthe condensate formation so that our results should becomparable with those obtained earlier for a single-bandmodel. Indeed, a comparison with microscopic numeri-cal calculations for a 2D system [23], shown by dots inthe left panel of Fig. 1, reveals a very good quantita-tive agreement down to temperatures 0 . T c . At larger η the contribution of the shallow band to the condensateincreases and the inter-type domain widens sharply, asshown in the middle and right panels of Fig. 1.We now address the question whether thermal fluctua-tions invalidate the mean field foundations of the GL andEGL approaches. It is well known that the existence ofshallow bands strongly enhances fluctuations. This canbe seen by considering the fluctuation-related contribu-tion to the heat capacity. The GL theory for a single-band system yields for this quantity δC V ∼ ( L/ξ ) τ − ,where L is the sample length and ξ = −K /a is thezero-temperature GL coherence length [8]. Here, thislength is determined by the coefficients of the GL equa-tion, but it is also related to the microscopic BCS co-herence length or the Cooper-pair size. One notices thatthe coherence length calculated separately for the shal-low band, ξ , ∼ v , is rather small and, therefore, thecorresponding Ginzburg-Levanjuk parameter Gi (tem-perature interval around T c , where the fluctuations inthis band are important) may become comparable with the temperature interval where the EGL theory can beused. However, in a two-band system the fluctuations arescreened due to the interactions with the deep band. Thisfollows from the calculations of the full two-band coher-ence length, which yields ξ = (cid:80) ν ρ ν ξ ,ν with ρ ν beingthe band weight factors. The large coherence length ofthe deep band, ξ , , ensures that ξ is not very small(unless ρ (cid:28) ρ ). In the same limit, that was used toderive Eq. (16), one obtains ξ ≈ ξ , / ( S √ η ) as an es-timation for the coherence length and Gi ≈ Gi S η forthe corresponding Ginzburg-Levanjuk parameter of thetwo-band system, where Gi is this quantity calculatedseparately for the deep band. Thus, the enlargement ofthe inter-type domain in Eq. (16) and the increase of Gi is controlled by the same factor S η . One concludes thata notable enlargement of the inter-type domain can beachieved without compromising the validity of the meanfield calculations if Gi is small enough. We also notethat the above relations suggest an interesting inversedependence κ ∗ li − κ ∗ ∝ ξ − of the inter-type interval onthe correlation length, or the Cooper-pair size, ξ .In summary we predict that the inter-type domain be-tween the two standard superconductivity types is con-siderably enlarged in multi-band materials with a shallowband when the latter yields a measurable contribution tothe condensate state. Thus for a wide range of micro-scopic parameters a multi-band superconductor can fallinto this domain. This will reveal itself in many notablechanges in the system’s magnetic properties, which re-semble the type I or II superconductivity only in the closevicinity of the critical temperature T c . At lower tempera-tures such superconductors enter the inter-type domain.Although a comprehensive description of the the inter-type domain has not been achieved yet it is possible topredict that the mixed state in such systems will exhibitmany unusual spatial vortex configurations not observedin standard type II superconductors.The work was supported by Brazilian CNPq (grants307552/2012-8 and 141911/2012-3) and FACEPE (grantAPQ-0589-1.05/08). [1] I. Bozovic and C. Ahn, Nat. Phys. , 892 (2014).[2] D. M. Eagles, Phys. Rev. , 456 (1969).[3] Y. Lubashevsky, E. Lahoud, K. Chashka, D. Podolsky,and A. Kanigel, Nat. Phys. , 30 (2012).[4] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[5] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A.Perali, P. Pieri, and G. C. Strinati, Nat. Phys. , 569(2010).[6] P. G. de Gennes, Superconductivity of Metals and Alloys ,(Benjamin, New York, 1966).[7] E. M. Lifshitz and L. P. Pitaevskii,
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