BCS theory of nodal superconductors Hyekyung Won,1 Stephan Haas,2 David Parker,2 Sachin Telang,2 Andra´s Va´nyolos,3 and Kazumi Maki2 1Department of Physics, Hallym University, Chuncheon, 200-702, South Korea 2Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484, USA 3Department of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary 5 (Dated: February 2, 2008) 0 0 This course has a dual purpose. First we review the successes of the weak-coupling BCS theory 2 indescribingnewclassesofsuperconductorsdiscoveredsince1979. Theyincludetheheavy-fermion n superconductors,organic superconductors,high-Tc cupratesuperconductors,Sr2RuO4 etc. Second, a wepresentthequasiclassicalapproximationintroducedbyVolovik,whichweextendtodescribethe J thermodynamics and the thermal conductivity of the vortex state in nodal superconductors. This 9 approach provides the most powerful tool in identifying the symmetry of the energy gap function 1 ∆(k) in these new superconductors. ] n I. INTRODUCTION o c - Question: What is the difference between a Fermi liquid and a non-Fermi liquid? r p Answer: The difference is the same as the one between bananas and non-bananas. u s . Boris Altschuler (2001) t a m Unconventionalor“nodal”superconductorsappearedonthescenein1979,whentheheavy-fermionsuperconductor - CeCu Si and the organic superconducting Bechgaard salts (TMTSF) PF were discovered. Since then, many more 2 2 2 6 d heavy-fermion superconductors [1] and organic superconductors [2] have been synthesized. This development was n followed by the epoch making discovery of high-T cuprate superconductor La Ba CuO by Bednorz and Mu¨ller o c 2−x x 4 c [3] with the superconducting transition temperature Tc = 35K in 1986. Within a few years new classes of high-Tc [ cupratesemerged,includingYBa Cu O ,La Sr CuO ,Bi Sr Ca Y Cu O (Bi2212);andHgBa CaCu O 2 3 6+δ 2−x x 4 2 2 1−x x 2 8+δ 2 2 6 with T = 145K. The subsequent enthusiasm and confusion are well documented in an early review by Enz [4]. 1 c Confusion? Yes, initially it was thought that Landau’s Fermi liquid theory [5] and the BCS theory [6] were no longer v 3 applicable. 6 AmongmanyproposalsoneofthemostinfluentialwereAnderson’sdogmas[7],whichcanbesummarizedasfollows: 4 1 a. The action takes place in the CuO2 plane common to all high-Tc cuprate superconductors. 0 5 b. The undoped state is a Mott insulator with antiferromagnetic (AF) order. Upon doping superconductivity 0 appears. Therefore the simplest Hamiltonian is the two-dimensional (2D) one-band Hubbard model: / t a m - H =−t (c+iαcjα+h.c)+U ni(cid:2)ni(cid:3), (1) d hXi,jiα Xi n co owpheerraethoir,jfoircothnenehcotlsetshaetntehaeressittenieiwghitbhorsspiinntαhean2dDnsqi(cid:2)u=arec+il(cid:2)actit(cid:2)i.ce. Thec+iαandciα arecreationandannihilation : v c. As a possible ground state of Eq. (1) Anderson proposed the resonating valence bond (RVB) state: i X r a Ψ= (1 d )BCS , (2) i − | i i Y where |BCSi is the BCS state for s-wave superconductors [6], and i(1−di) with di = ni(cid:2)ni(cid:3) is called the Gutzwiller operator. (1 d ) annihilates all doubly occupied states. i − i Q InspiteoftremendouseffQortsspentonbothEqs.(1)and (2)ithasbeendifficulttofindsolutionsintwodimensions. On the other hand the 1D version of Eq. (1) is now completely understood [8,9]. In the meantime the perturbative analyses based on Eq. (1) predict BCS d-wave superconductivity in high-T cuprates [10,11,12,13]. c 2 FIG. 1: Local densityof states around a single vortex for d-wave(left) and s-wavesuperconductivity. High-quality single crystals of YBCO, LSCO and thin films of Bi2212 became available around 1992. The d-wave superconductivity in these high-T cuprates was established in 1994. Among many experiments, angle-resolved pho- c toemission spectroscopy (ARPES) [14] and Josephson interferometry [15,16] played a crucial role. Around this time several theoretical groups started to analyze the physical properties of d-wave superconductivity within the BCS framework [17,18,19,20]. In 1993 Patrick Lee [21] demonstrated the universal heat conduction in d-wave supercon- ductors. Furthermore the thermal conductivity was shown to increase with increasing impurity scattering [19]. This counterintuitive behavior was observed in the Zn-substituted YBCO in [22]. The electronic contribution to the ther- mal conductivity in d-wave superconductors is proportional to T at low temperatures (i.e. T ∆ where ∆ is the ≪ maximum value of the energy gap). Here we assume ∆(k)=∆cos(2φ) and φ is the angle k in the a b plane makes − from the a axis. Then in the limit of no impurity scattering (i.e. Γ 0 where Γ is the quasiparticle scattering rate → in the normal state) the thermal conductivity is given by κ k2 v 00 = B n, (3) T 3~ v 2 where κ /T = lim κ/T, and v/v = E /∆ and n is the quasiparticle density or the hole density. The velocities 00 Γ→0 2 F v and v are defined from the quasiparticle energy at the Dirac cone 2 Ek = v2(kk−kF)2+v22k⊥2, (4) q where v is the Fermi velocity, k is the radial component of the wave vector and k the component perpendicular to k ⊥ the Fermi surface. Later we shall derive Eq.(3) in Section IV. The thermal conductivity in single crystals of optimally doped YBCO andBi2212belowT =1Kwasmeasuredby May Chiaoet al[23,24]. Makinguse ofEq.(3)they found∆/E =1/10 F and 1/14 for Bi2212 and YBCO respectively. These remarkable ratios imply: a. High-T cuprates are described by the BCS theory of d-wave superconductivity. They are far away from the c Bose-Einstein condensation limit which requires ∆ E [25]. F ∼ b. According to the Ginzburg criterion, fluctuation effects are of the order ∆/E (p ξ)−1. In other words they F F ∼ shouldbe at most 10%. This appears to exclude large phase fluctuations andstripe phases discussedin [26,27]. c. For ∆/E =1/10there are hundreds of quasiparticlebound states aroundthe coreof a single vortexin d-wave F superconductors [28,29]. The radial (r) dependence of the local quasiparticle density of states is very similar to the one obtained for s-wave superconductivity [30]. We show in Fig. 1 the local density of states around a single vortex for a d-wave and s-wave superconductor. These are well-known bound states first discovered by Caroli, de-Gennes and Matricon [31,32] for s-wave superconductors. In earlier works [33,34,35,36] it was asserted that there would be no bound state around a single vortex in d-wave superconductors. However in these works it was assumed ∆ E in order to facilitate the numerical analysis F ≃ 3 T ∗ T pseudogap phase 500 AF Tc d-waveSC 100 0 0.05 0.15 0.25 x FIG. 2: The phase diagram of hole-doped high-Tc cuprate superconductors. based on the lattice version of the Bogoliubov de Gennes equation. This assumption (∆ E ) eliminates the F ≃ bound states in these numerical analyses. If we limit ourselves to the experiments on single crystals of high-T c cuprate superconductors, we find that most of the behaviors observed are consistent with the BCS theory of d- wave superconductivity [37]. Also it is better to use the continuum version of the Bogoliubov de Gennes equation. which is proposed in [38] and used in [28,29]. More recently a similar analysis is extended for a vortex in an f-wave superconductor [39]. FromatheoreticalpointofviewtheuniversalityoftheLandauFermiliquidin2Dsystemswasdemonstratedwithin the renormalization group analysis [40,41,42]. The quasiparticles in the normal state of high-T cuprates appear to c be a Fermi liquid state. Furthermore the quasiparticles in d-wave superconductors are in a BCS-Fermi liquid state with the quasiparticle energy Ek = v2(kk kF)2+∆2cos2(2φ). (5) − q In the vicinity of the Dirac cone Eq. (5) reduces to Eq. (4). Another consequence of the renormalization group analysis is that the instability of the normal Fermi liquid is relatedtotheinfrareddivergenceintheparticle-particle(andthehole-hole)channelortheparticle-holechannel. The former results in conventional or unconventional superconductivity and the latter in conventional or unconventional density wave states. Therefore it is of great interest to look at the phase diagram of the high-T cuprates from this c point of view. We show a schematic phase diagram in Fig. 2. Let us considerthe hole-dopedregion. In the vicinity ofx=0 there is anantiferromagnetic(AF) insulating phase, whichis aMottinsulator(MI). As the hole-dopingxis increased,the AF orderis rapidlysuppressedaroundx=3%. Then a superconducting region develops for 5%<x<25%. This is sometimes called “the superconducting dome”. Also in the underdoped region there is the pseudogap (PG) regime. The nature of the pseudogap is still hotly debated. Thereisevidencethatitisad-wavedensitywave(dDW)[43,44,45,46]withanenergygap∆(k)=∆cos(2φ). Earlier proposals [43,45] have only considered the commensurate case with Z symmetry something similar to a flux 2 phase [47,48]. On the contrary,we consider the incommensurate d-wave density wave [44,46] case where the condensate has U(1) symmetry as in conventional charge-density wave systems (CDW). Here the phase vortex [49] is the most common topological defect. Moreover the phase diagram suggests a coexistence region of dDW and d-wave superconductivity [43,45]. Then the phenomenological gap introduced by Tallon and Loram [50] should be the energy gap associated with dDW. Recently Laughlin [51] has pointed out that the wave function (2) is impractical, since the Gutzwiller operator has no inverse. Instead, he proposed to analyze Eq. (2) with a modified Gutzwiller operator which has an inverse. For example (1 d ) (1 αd ), (6) i i − → − i i Y Y 4 FIG. 3: From top left: 2D f-wave in Sr2RuO4,dx2−y2-wavein CeCoIn5 and in κ-(ET)2Cu(NCS)2, (s+g)-wavein YNi2B2C, (p+h)-wave in the A phase of PrOs4Sb12, (p+h)-wavein theB phase of PrOs4Sb12. withα<1. He calledthis “gossamersuperconductivity”. “Gossamer”meansfilmycobweborsomethinglight,fragile but strong. If we look at this new wave function with Eq. (6), we realize that this is an example of competing order parameters [52,53]. We shall come back to this question at the end of our course. In a seminal paper Volovik [54] has shown that the quasiparticle density of states in the vortex state of the d-wave superconductors is calculable within a semiclassical approximation. The predicted √H dependence of the specific heat has been confirmed experimentally in single crystals of YBCO [55,56], LSCO [57] and Sr RuO [58,59]. This 2 4 semiclassical approach has been extended in a variety of directions [60,61,62,63,64,65]. When high-quality single crystalsinthe extremelycleanlimit (i.e. l ξ wherel is the quasiparticlemeanfree pathandξ the superconducting ≫ coherence length) are available,the angle- dependent thermal conductivity in the vortex state providesunique access to the gap symmetry ∆(k). In the last few years Izawa et al. have determined ∆(k) in Sr RuO [66], CeCoIn [67], 2 4 5 κ-(ET) Cu(NCS) [68], YNi B C [69] and PrOs Sb [70,71]. These gap functions are shown in Fig. 3. 2 2 2 2 4 12 Inthefollowing,wefirstfocusonthequasiparticlespectruminavarietyofnodalsuperconductors. Thentheeffectof impurity scatteringandthe universalheatconductionis brieflysummarized. The quasiclassicalapproximationin the vortexstateinnodalsuperconductorsisthecentralpartofthiscourse. Alsothepropertiesofnodalsuperconductivity in YNi B C and PrOs Sb are briefly summarized. In the last chapter we discuss unconventional density wave and 2 2 4 12 gossamer superconductivity, which indicate new directions to follow in new materials. 5 II. QUASIPARTICLE SPECTRUM IN NODAL SUPERCONDUCTORS Following the BCS paper [5] we consider the effective Hamiltonian 1 H = ξ(k)c+kαckα+ 2 v(k,k′)c+k′α′c+−k′,−α′c−k,−αckα, (7) k,α k,α,k′,α′ X X with v(k,k′)= f 2 −1Vf(k)f(k′), (8) −h| | i and 1 f 2 = dΩf(k)2 3D (9) h| | i 4π | | Z 1 = dχdφf(k)2 2D (10) (2π)2 | | Z dependingonwhetherthesystemis3Dorquasi-2D.Inthefollowing,weconsideragroupofquasi-2Dsuperconductors, whose quasiparticle density of states, thermodynamics etc. are identical [64]. We define quasi 2D systems by a cylindrical Fermi surface, as in high-T cuprates, Sr RuO , CeCoIn , κ-(ET) Cu(NCS) etc. Also we consider a c 2 4 5 2 2 group of nodal superconductors with f(k) = cos(2φ),sin(2φ) (d-wave), e±iφcos(χ) (f-wave), cosχ (d-wave), sinχ (p-wave), e±iφsin(χ) (d-wave), cos(2χ) (g-wave) etc. Here χ = ck . Then it can be readily shown (Exercise 1) that z these superconductors have an identical quasiparticle density of states (DOS). Within the mean-field approximation, i.e. the BCS approximation, Eq. (7) is transformed as ∆(k)2 H = Ψ+k,α(ξ(k)ρ3+∆(k)ρ1)Ψk,α− | v | (11) k,α k X X The corresponding Nambu-Gor’kov Green function [72,73] is given by G−1(k,ω)=ω ξ(k)ρ ∆(k)ρ , (12) 3 1 − − where the ρ ’s are Pauli matrices operating on the Nambu spinor space. For simplicity we consider here only spin i singlet pairing and f as a real function. Then the poles of the Green function Eq. (12) give the quasiparticle energy ω = ξ2(k)+∆2(k) v2(k k )2+v2k2. (13) ± ≃± k− F 2 ⊥ p q The last expression is an approximation near the Dirac cone. From Eq. (13) the quasiparticle density of states is obtained as [17] 2 xK(x) for x <1, E π| | | | g(E)=Re | | = (14) * E2 ∆2 f 2+  2K(1/x) for x >1, − | |  π | | p where x = E /∆ and K(k) is the complete elliptic integral of the first kind. We show in Fig. 4 the quasiparticle | | density of states versus x, compared with the one for s-wave superconductor with a full energy gap. For small energies, the density of states can be expanded as g(E) E /∆ (15) ≃| | for E /∆ 1. | | ≪ 6 4 3 0 N )/ 2 E ( N 1 0 0 0.5 1 1.5 2 E/∆ FIG.4: Thequasiparticledensityofstatesfornodalsuperconductors(solidline)andfors-wavesuperconductors(dashedline). The mean-field approximation also gives the gap equation ′ f2 λ−1 =2πT f 2 −1 h| | i n * ωn2 +f2+ X (16) E0 p E E = f 2 −1 dERe | | tanh( ), h| | i Z0 * E2−∆2|f|2+ 2T which we have written in 2 alternative forms. Here λ =pνN is the dimensionless coupling constant, ω is the 0 n Matsubarafrequency andE isthe cut-offenergy. Also the ω suminthe firstequationhas tobe cut offatω E . 0 n n 0 ≃ Then within the weak coupling limit we find [74] ∆(0)/T =2.14, (17) c ∆(t)/∆(0) 1 t3, (18) ≃ − where t=T/T . In Fig. 5 we show ∆(t)/∆(0) versus t togethper with the approximate expression Eq.(18). c As to the thermodynamics, it is convenient to start with the entropy S given by ∞ S = 4N dEg(E)[flnf +(1 f)ln(1 f)] 0 − − − Z0 (19) ∞ =4N dEg(E)[ln(1+e−βE)+βE(1+eβE)]. 0 Z0 Here N is the quasiparticle density of states at the Fermi surface in the normal state and f = (1+eβE)−1 is the 0 Fermi function. From S, the specific heat and the thermodynamic critical field are obtained by ∂S C /T = , (20) s ∂T and 1 Tc H2(T)=F (T) F (T)= dT′(S (T′) S (T′)). (21) 8π c n − s n − s ZT 7 1 0.8 ) 0.6 0 ( ∆ / ) t (0.4 ∆ 0.2 0 0 0.2 0.4 0.6 0.8 1 t FIG. 5: The temperature dependenceof theorder parameter (dashed - approximate, solid - exact Using the density of states g(E), we obtain C 27ζ(3)T s = for T ∆, (22) γ T 2π2 ∆ ≪ s and H (0)= 2πN ∆(0). (23) c 0 Similarly the superfluid density within the a b planepis given by − ∆2f2 ρ (T)/ρ (0)=2πT . (24) s s (ω2 +∆2f2)3/2 n (cid:28) n (cid:29) X In the limit T 0 this reduces to → T ρ (T)/ρ (0)=1 2ln2 . (25) s s − ∆(0) In other words one can obtain ∆(0) from the T linear slope of the superfluid density. There is still a controversy as to the correct expression of the c axis superfluid density. The simplest assumptions [75] give π ∆(t) ρ (T)/ρ (0)= ftanh(∆f/(2T)) s,c s,c 2∆(0)h i (26) π2 T 2 1 . ≃ − 6 ∆(0) (cid:18) (cid:19) Finally, the spin susceptibility and the nuclear spin lattice relaxation rate are given by χ /χ =1 ρ (T)/ρ (0), (27) s n s s − 8 and ∞ dE E T−1/T−1 = g2(E)sech2( ) 1 1n 2T 2T Z0 (28) π2 T 2 . ≃ 3 ∆ (cid:18) (cid:19) We stress again that these expressions are not only valid for d-wave superconductors in the weak-coupling limit as in high-T cuprates, but for all nodal superconductors with f(k) given above. Therefore in order to explore the c individual gap symmetry, we have to look at other properties. It is somewhat surprising that all the energy gaps of the identified nodal superconductors in the quasi-2D systems belong to the above class of f’s. Exercises 1. 1.1 Within the weak-coupling theory calculate the jump in the specific heat at T =T c a. for s-wave superconductivity. (Answer: ∆C/C =12/(2ζ(3))=1.43) n b. for nodal superconductivity with ∆(k)=∆f(k). (Answer: ∆C/C =12/(7ζ(3)) f 4 / f 2 2) n h| | i h| | i 1.2 Evaluate the ratio ∆(0)/T within the weak-coupling theory c a. for s-wave superconductivity. (Answer: ∆(0)/T =π/γ =1.76) c b. for nodal superconductivity. (Answer: ∆(0)/T =(π/γ)exp( f2 −1 f2lnf )) c −h i h i 1.3 Show that the quasiparticle density of states for a group of f’s discussed is the same as given by Eq. (14). 1.4 Express Re f2/ x2 f2 in terms of complete elliptic integrals. Answer: h − i p f2 π2(K(x)−E(x)) for x<1, Re = (29) * x2 f2+  2x(K(1/x) E(1/x)) for x>1. −  π − p  III. EFFECT OF IMPURITY SCATTERING In metals the presence of impurities or foreign atoms is unavoidable. Also, they provide the simplest agents of quasiparticle relaxation. Therefore the study of impurity scattering is crucial to understand quasiparticle transport such as electric conductivity and thermal conductivity. In the early sixties the effect of impurity scattering in s-wave superconductivity was systematically studied in [76,77,78]. As is well known, nonmagnetic impurity scattering has little effect in s-wave superconductors. The superconducting transition temperature and the thermodynamics are almost unaffected. The most dominant effect is a reduction of the quasiparticle mean-free path, as in the normal state, and its consequence on the magnetic penetration depth. On the other hand, magnetic impurities have a profound effect on s-wavesuperconductivity. The superconducting transition temperature is sharply reduced. Also in some cases gapless superconductivity is induced [78,79]. Incontrast,thenonmagneticimpuritieshaveprofoundeffectsonnodalsuperconductorsaspointedoutin[80,81,82]. In particular, resonant impurity scattering appears to be prevalent. The extreme limit is the unitary limit where a resonance occurs at E = 0. Such a model has been discussed for d-wave superconductivity in high-T cuprates in c [83,84]. Thefirstself-consistentstudiesofimpurityscatteringind-wavesuperconductorswereperformedin[18,19,20]. Forsimplicityweshalllimitourselvestotheunitarylimit. Alsoweassumethattheimpurityispointlike,i.e. ithas only s-wave scattering amplitude. Then the effect of impurity scattering can be incorporated by ω ω˜ in Eq. (12) → where ω˜ is the renormalized frequency given by [18] π √1 x˜2 1 −1 ω˜ =ω Γ − K , (30) − 2 x˜ √1 x˜2 (cid:18) (cid:18) − (cid:19)(cid:19) 9 FIG. 6: The quasiparticle density of states in thepresence of impurities, for several impurity concentrations. and x˜ = ω˜/∆ and Γ = n (πN )−1 is the quasi-particle scattering rate in the normal state. Then the quasiparticle i 0 density of states in the presence of impurities is given by 2 x˜ 1 g(E,Γ)= Re K . (31) π √1 x˜2 √1 x˜2 (cid:26) − (cid:18) − (cid:19)(cid:27) The quasiparticle density of states in the presence of impurities is shown in Fig. 6. We note a rapid appearance of disorder-induced spectral weight at E =0. Indeed g(0,Γ) is given by C 1 0 g(0,Γ)= K , (32) 1+C2 1+C2! 0 0 p p where C is obtained from 0 C2 π Γ 1 0 = K−1 . (33) 1+C2 2∆ 1+C2! 0 0 p p In the limit Γ/∆ 0, both Eq. (32) and Eq. (33) reduce to → 1/2 4 πΓ 2∆ g(0,Γ)=C ln( ) ln−1 4 , (34) 0 C0 ≃ 2∆ rπΓ!! and πΓ C2ln(4/C ) . (35) 0 0 ≃ 2∆ 10 FIG.7: ∆(0,Γ)/∆00 (dashedline), Tc/Tc0 (solid line) and g(0,Γ)/N0 (dashed-dottedline) areshown asafunction of Γ/Γc in theunitary limit. The gap equation in the presence of impurity scattering is given by ′ f2 λ−1 =2πT f2 −1 h i n * ω˜n2 +∆2f2+ X (36) 8T ′ p 1 x˜2 1 = 1+x˜2 E n K , ∆ n n 1+x˜2n!− 1+x˜2n 1+x˜2n!! X p p p where x˜ = ω˜ /∆ and ω˜ is the renormalized Matsubara frequency. Then in the limit ∆ 0, we obtain the n n n → Abrikosov-Gor’kovformula: T 1 Γ 1 c ln =Ψ + Ψ , (37) − T 2 2πT − 2 (cid:18) c0(cid:19) (cid:18) c(cid:19) (cid:18) (cid:19) where T (T ) is the superconducting transition temperature in the presence (absence) of impurities. Here Ψ(z) is c c0 the digamma function. Note that Eq. (37) is the same as for s-wave superconductors in the presence of magnetic impurities [78]. In nodal superconductorsΓisduetononmagneticimpurities,whereasins-wavesuperconductorsΓisassociatedwithmagnetic scattering which involves spin flipping. At T =0K Eq. (36) reduces to ∆(Γ,0) 2Γ ∞ ln =2 f2ln(C + C2+f2) dxx2(1 E/K)[(1+x2)E K], (38) − ∆(0,0) 0 0 − ∆ − − (cid:18) (cid:19) (cid:28) q (cid:29) ZC0 where E = E(1/√1+x2) and K = K(1/√1+x2) and C has already been defined in Eq. (33). We show in Fig. 7 0 T /T , ∆(0,Γ)/∆(0,0,) and g(0) versus Γ/Γ where Γ =πT /(2γ) 0.8819T . c c0 c c c0 c0 ≃ Surprisingly, ∆(0,Γ)/∆(0,0) follows very closely T /T . Also g(0,Γ) increases very rapidly with Γ. This rapid c c0 increase in the DOS has been measured by the low temperature specific heat in doped LSCO [85]. Note C /(γ T)= s s g(0). As seen from FIG. 8 the experimental data agree very well with the theoretical prediction. Especially the agreementisalmostperfectinthevicinityoftheoptimallydopedLSCO.Thissuggestsstronglythattheweak-coupling BCS theory for d-wave superconductivity is adequate for LSCO. We have already mentioned that ∆(0)/T =2.14 in c