BCS-BEC model of high-T superconductivity in layered cuprates with c unconventional pairing C. Villarreal1 and M. de Llano2 1Instituto de F´ısica, Universidad Nacional Auto´noma de M´exico Ciudad Universitaria, 04510 M´exico, DF, MEXICO 2Physics Department, University of Connecticut, Storrs, CT 06269 USA and ∗Instituto de Investigaciones en Materiales, Universidad Nacional Auto´noma de M´exico Apdo.Postal 70-360, 04510 M´exico, DF, MEXICO (Dated: January 6, 2010) 0 High-TcsuperconductivityinlayeredcupratesisdescribedinaBCS-BECformalismwithlinearly- 1 dispersives-andd-waveCooperpairsmovinginquasi-2Dfinite-widthlayersabouttheCuO planes. 0 2 2 Thisyieldsaclosed formulaforTc determinedbythelayerwidth,theDebyefrequency,thepairing energy, and the in-plane penetration depth. The new formula reasonably reproduces empirical n values of superconducting Tcs for seven different compounds among the LSCO, YBCO, BSCCO a and TBCCO layered cuprates. J 6 PACSnumbers: 05.30.Fk,74.20.-z,74.72.-h ] n I. INTRODUCTION phonon interaction and the Fermi surface with a con- o comitant shift from d- to s-type coupling as doping in- c - It seems to be well-established that central to creases. The strongest evidence for an s-wave order pa- r rameterinacuprateisreviewedinRef.[12]whereseveral p high-Tc superconductivity (HTSC) is the layered two- c-axis twist experiments on BSCCO along with earlier u dimensional(2D)structureofcopperoxidesandthatsu- s perconductingpairingoccursmainlyontheCuO planes. c-axis tunneling between BSCCO/Pb junctions are sur- . 2 veyed. Ref.[15] summarizes many of the problems with t However,theprecisenatureofthepairingisstillthesub- a theso-called“phase-sensitive”tests[16]inYBCO.Addi- m ject of intense research. Recent experiments based on tionally,predictionsmadeinRef.[17]thatavortexinad- angle-resolvedphotonemissionspectroscopy(ARPES)of - wavesuperconductorwouldexhibitameasurabledensity d underdopedcupratessuggestthatboundfermionCooper of states in a four-fold pattern emanating from the core n pairs (CPs) form already at and below temperatures o higher than the critical transition temperature T [1– have not been observed[18] in either YBCO or BSCCO. c c 4]. Furthermore,ARPES studies of the electronspectral On the contrary, the vortex cores appear consistent [18] [ with isotropic s-waves. function of optimally doped Bi2212 samples show that 1 the magnitude of the isotope effect correlates with the Replacing the characteristic phonon-exchange Debye v superconducting gap [5], thus suggesting a role of lat- temperature ΘD ≡ ~ωD/kB of around 400K (with kB 7 tice phonons in the superconducting pairing. ARPES theBoltzmannconstantandωD theDebyefrequency)by 5 data also suggest that the energy gap ∆ (a measure of acharacteristicmagnon-exchangetemperatureofaround 9 the energy needed to break a CP) displays an unconven- 1000K canleadtoasimplemodelinteractionsuchasthe 0 tionald orbitalpairingsymmetry,withafunctional BCS one but associated with spin-fluctuation-mediated . x2−y2 1 dependence ∆ = ∆ cos2θ where θ = tan−1K /K is pairing [19]. 0 the angle between t0he total or center-of-mass my omxen- The so-called “Uemura plot” [20] of data from µSR, 0 tum (CMM)~K =(~K ,~K )ofpairedelectronsinthe neutron and Raman scattering, and ARPES measure- 1 x y : CuO2 plane andthe a- (or x-) axis while ∆0 is the value ments exhibit Tc vs Fermi temperatures TF ≡ EF/kB v of the superconducting gap at the antinode (θ =0,π/2) where EF the Fermi energy and kB the Boltzmann con- i X [6]. This behavior is also apparent in studies based on stant. EmpiricalTcsofmanycupratesstraddleastraight electronic Raman scattering [7] and in determinations of line parallel to the Uemura-plot line associated with the r a the in-plane magnetic penetration depth λ [8, 9]. simple BEC formula T ≃3.31~2n2/3/m k ≃0.218T ab B B B B F Although the majority viewpoint in the high-T com- corresponding to an ideal gas of bosons of mass m = c B munity seems to argue for such non-s-wave pairing sym- 2m∗ and number density nB = ns/2 where m∗ is the metrytherearecompellingdissentingviews,particularly effective mass and n the number density of individual s workwithinthepastfewyears,byMu¨ller[10],Harshman chargecarriers. TheparallellineisshifteddownfromTB [11],Klemm[12],andmanyothers. Inparticular,Mu¨ller by a factor 4-5. This has been judged [21] as a “funda- concludesthat“...recentexperimentsprobingthesurface mental importance of the BEC concept in cuprates.” and bulk of cuprate superconductors [show that their] Previous theoretical papers on the possible origin of character is d on the surface and substantially s in the HTSC [14, 22–24] proposed that the phenomenon might bulk.” This conclusion has been bolstered by muon-spin be rooted in a 2D Bose-Einstein condensate (BEC) of relaxation (µSR) experiments with YBCO reported and CPs pre-existing above T and coupled through a BCS- c interpretedinRef.[11]. Severalauthors[13,14]havepro- like phonon mechanism [25], originally taken as s-wave. posedthatthedopingprocesscouldmodifytheelectron- AsapparentlyfirstreportedbySchrieffer[26],theCooper 2 model interaction [27] leads to an approximate linear II. BCS THEORY WITH l-WAVE PAIRS energy-vs-CMM relation 1v ~K for excited CPs propa- 2 F gatingintheFermiseain3D.Thiskindofdispersionre- Some aspects of the l-wave BCS theory [6, 26, 34] rel- lationisnotuniquetotheCoopermodelinteraction. For evant to our HTSC model follow. Consider a system example,anattractiveinterfermiondeltapotential[28]in of electron- (or hole-) pairs formed via a two-fermion 2D (imagined regularized [29] to support a single bound isotropic potential V near the Fermi surface and with state of binding energy B2 > 0) leads [30] (for a review kinetic energies ǫk ≡ ~2(k2 − kF2)/2m∗ (with ~kF the see Ref.[24]) to the general CP dispersion relation EK = Fermi momentum) taken relative to the Fermi energy. E0+c1~K+ 1−(2−16/π2)EF/B2 ~2K2/4m∗+O(K3), ThePauliprinciplepreventsbackgroundfermionsinelec- wherec1 =2vF/πpreciselyaswiththeCoopermodelin- tron states just below (above) the Fermi surface from (cid:2) (cid:3) teraction [31]. Hence, the leading-order linearity is not participating in the interaction. In the absence of ex- induced by the particular interfermion interaction bind- ternal forces each pair propagates freely within a layer ing the CPs but is a consequence of the Fermi sea with of finite width δ along the z direction and infinite ex- vF 6= 0 and in which a CP by definition propagates. tent on the x − y plane so that its total momentum Only in the vacuum limit vF → 0 ⇒ EF ≡ 21m∗vF2 → 0 ~K=(~Kk,~Kz)isaconstantofmotion. By neglecting doesthatgeneraldispersionrelationreducebyinspection spin-dependentinteractionsthe totalspinS is conserved to the expected quadratic form EK = E0 +~2K2/4m∗ too and for a spin singlet S = 0 configuration the Pauli for a composite object of mass 2m∗. For either inter- principlerequiresthatthe orbitalwavefunctionbeofthe electron interaction model, the linear term is a conse- form Ψ(r ,r )=exp(iK ·R )cos(K z)Φ(r), where the 1 2 k k z quence of the presence of the Fermi sea. The forma- relative coordinate r = r − r , R is the horizontal 1 2 k tion of a BEC of CPs in 2D does not violate Hohen- projection of the CM coordinate R = (r +r )/2, and 1 2 berg’s theorem [32] as this holds only for quadratically- K = nπ/δ (with n integer). The z-dependence of the z dispersiveparticles. Thepredicted2DBECtemperature wavefunction ensures that the vertical flux of the elec- is T ∝ n2D 1/2 ∝ (Θ T )1/2 where n2D is the CP tron (hole) pair across the layer boundary is null. Since c D F number per unit area. This leads to values of T that the relative-coordinate problem is isotropic then Φ(r) is c (cid:0) (cid:1) are too high compared with empirical values. However, an eigenfunction of angular momentum with quantum these schemesprovidea correctdescriptionofotherrele- numbers l = 0,1,2,···. The total spin S = 0 singlet vantphysicalpropertiesofHTSCs suchasashortcoher- eigenstates of the system satisfy the Schr¨odinger equa- ence length, a type II magnetic behavior, and the tem- tion perature dependence of the electronic heat capacity[14]. They also lead to excellent fits of the condensate frac- (H0−V)Ψ(r1,r2)=EΨ(r1,r2) (1) tion curves for quasi-2D cuprates just below T [33], as c where H is the free Hamiltonian V the interaction po- well as for 3D and even quasi-1D SCs. To go beyond 0 tential and E the energy eigenvalue. For a given CMM the simple s-wave interaction, an l-wave formulation of wavevectorK, we may expand the wave function as BCS theory was discussed by Schrieffer [26] himself and studied in considerable detail by Anderson and Morel [34] in the weak-coupling limit. This has been success- Ψ(r1,r2)=exp(iKk·Rk)cos(Kzz) akexp(ik·r). (2) k fully employed [6, 35] to describe thermodynamic and X transportpropertiesofhigh-Tc cuprates. Thed-waveex- In momentum space (1) thus becomes tension in strong-coupling Eliashberg theory is reported in Refs.[36]. E −ǫk+K/2−ǫk−K/2 ak = Vkk′ak′ (3) k′ (cid:0) (cid:1) X with Vkk′ = < k,−k|V|k′,−k′ >. Since the interaction potential V depends only on r it admits the expansion Here we develop a general l-wave BCS-type theory ∞ l which is then applied in a quasi-2D BEC picture with Vkk′ = Vl(|k|,|k′|)Ylm(Ωk)Yl−m(Ωk′). (4) eitherl=0orl=2pairingsymmetry. In§II thel-wave l=0m=−l BCStheorywithintheframeworkofthepresentmodelis X X discussed. In § III we study a quasi-2D BEC of linearly- For small coupling amplitudes V (|k|,|k′|) the contribu- l dispersive,massless-likeCPsandweevaluatethenumber tionsofdifferentlsphericalharmonicsYm(Ωk)in(4)can l density. In§IVthearealdensityn2D ofchargecarriersis withgoodaccuracybe consideredrelativelyindependent estimatedbycalculatingthemagneticpenetrationdepth [34]. In that case (3) yields an analytical solution by as- arisingfromthe CPs. In§ Vananalyticalexpressionfor suming that V is separable, i.e., V (|k|,|k′|)=V(l)flfl∗ l l 0 k k′ the critical BEC temperature is derived, which is then so that (3) becomes appliedin § VI for varioussuperconductingmaterialsin- cluding YBCO under different doping levels. Discussion E −ǫk+K/2−ǫk−K/2 ak =V0(l)fkl ak′fkl∗′ (5) and conclusions are given in § VII. k′ (cid:0) (cid:1) X 3 where ak =akYlm(Ωk). Eq.(5) can be now rewritten as The dispersion relation (11) is linear in leading order rather than quadratic as would be expected in vacuo. V(l)fl As a consequence, all excited CPs behave like a gas of a(l) =C(l) 0 k (6) k EK(l)−ǫk+K/2−ǫk−K/2 fitryeecm=as~sl−es1sd-Eli(kl)e/dbKos,onbsutwaitvharaiabcolememneorngygrdoeutpermveilnoecd- 1 K wahBeCreS-tykp′eakin′ftkel∗′gr≡alCre(ll)atisioancfoonrsataCntP. Oinnethtehueisgoenbstataintes tbhyatthienirorCdMerMfor~KaC. PThtoerdeimspaeirnsiboonurnedla(tii.oen.,(E1(1l))<im0p)liietss K charactPerized by (l,m) maximum CMM wavenumbermust not exceed the value |E(l)|/c ≡K sinceCPs withK >K haveE(l) >0and 0 1 0 0 K 1=V(l) |fkl|2 . (7) thus break up [26]. 0 k EK(l)−ǫk+K/2−ǫk−K/2 Explicit expressions of relevant thermodynamic vari- X ables and transport coefficients evaluated within the Following Schrieffer [26] we assume that the angular- weak-coupling limit of the l-waveBCS theory have been independent l component of the generalized BCS model derived in Refs.[6, 34, 35]. In these papers it is shown interaction (4) is given by thattheaverage behaviorofmostofthesequantitiesover thecylindricalFermisurfaceexhibitssmallvariationdue V0(l)fklfkl∗′ =−V0 (8) to the explicit realization of an l = 0 or l = 2 symme- try[6,35]. Inparticular,thetemperature-dependentgap with V >0 for CPs with relative momenta (k,k′) lying equation is given by [6, 26, 34] 0 in the neighborhood of the Fermi surface k <|k+K/2|,|k−K/2|<K (9) ~ωD tanh 21β ǫ2k+∆(l)2|g(l)(k)|2 F max 1=N0V0 dǫk (cid:16) q (cid:17) and V(l)flfl∗ = 0 otherwise. Here K = k2 +k2 Z0 D ǫ2k+∆(l)2|g(l)(k)|2 EF with k0 dkefikn′edin terms ofthe DebyemenaexrgyviaF~ω ≡D q (13) ~2k2/2Dm∗. A straightforward analysis [26] repveals tDhat where β = 1/kBT, g(0)(k) = 1 for l = 0, and g(2)(k) = D cos(2θ)forl=2. The criticaltemperatureisdetermined (7)yieldsaboundstatewithenergyE(l) <0forarbitrar- from (13) by the condition ∆(l)(T ) = 0. In the weak- ily weak couplingso longas the potentialis attractivein c coupling limit ~ω /k T ≫ 1 it can be calculated an- the region (9) in k-space. Then, a bosonic CP can form D B c alytically and it follows that T is independent of the only if the tip of vector k lies within the intersection of c l−state [34]: the two spherical shells defined by (9) whose center-to- center separation is K; fermions with wave vectors lying 2eγ outside this overlap are unpairable [24]. kBTc = ~ωDexp(−1/N0V0)≃1.13~ωDexp(−1/N0V0) π In the quasi-2D limit the fundamental expression (7) (14) can be evaluated by substituting the summation over k with γ ≃ 0.577··· Euler’s constant. In the zero- by a 2D integration. In addition, for small δ the only temperature limit ∆(l) ≡ ∆(l)(T = 0) the energy inte- term in K that yields a finite contribution is n=0. By 0 z gration in (13) leads to the gap relation [34] assuming a 2D cylindrical Fermi surface we obtain V0 2π k2 kdk ∆(0l) =2Γ(l)~ωDexp(−1/N0V0(l)) (15) 1 = dθ (10) (2π)2 Z0 Zk1 |EK(l)|+ǫk+K/2+ǫk−K/2 where Γ(l) = exp[− |g(l)|2ln|g(l)| ]. For l = 0, Γ(0) = F m∗V |E(l)|+2~ω −v ~Kcosθ 1, while for l = 2, Γ(2) = 2exp(−1/2) ≃ 1.213, so that ≃ 0 ln K D F (cid:10) (cid:11) 4π |E(l)|+v ~Kcosθ F combining(14)and(15)weareledtothegap-to-Tcratios D (cid:12) K F (cid:12)E (cid:12) (cid:12) wherek =k +(K(cid:12)/2)cosθ,k =k +k −(K(cid:12)/2)cosθ, 2∆(0) 2∆(2) 1 F 2 F D 0 ≃3.53 0 ≃4.28. (16) and the approximate equality in the second row holds k T k T B c B c up to terms of order (k /k )2 ≡ Θ /T . The angular D F D F brackets denote an average over a 2D cylindrical Fermi For l = 0 one recovers the standard BCS result [25] and 2π thesomewhathighervalueforthel =2d-wavecase. We surface < ... > → (1/2π) dθ. The Fermi average then gives the enFergy spectru0m of excited CPs [26]: note that the quantity ∆(l)/Γ(l) has the same functional 0 R dependence asthe zero-temperaturegapofthe BCSthe- E(l) ≃E(l)+c ~K+O(K2) (11) ory [25]. Considering that measurements of the energy K 0 1 gapforanygivencuprateshowsomescatteraboutacen- where c1 ≡ 2vF/π in 2D and E0(l) is the binding energy tralvalue∆e0xp [38]inthe followingweshallassumethat of the CP ground state (~K =0) [27] ∆exp ≃∆(2)/Γ(2) ≃∆(0) ≡∆ . 0 0 0 0 On the other hand, the average superfluid density E(l) =−2~ω /[exp(2/N V(l))−1]. (12) ρ (T) ≡ λ2 (0)/λ2 (T) exhibits a more pronounced an- 0 D 0 0 s ab ab 4 gular momentum dependence. This is given by [6] ∞ |g(l)(k)|2 ρ(sl) =1−β dǫk . Z0 Dcosh2 21β ǫ2k+∆(l)2|g(l)(k)|2 EF (cid:16) q (cid:17)(17) In the low-temperature limit (17) yields for l = 0 an exponential T-dependence (0) 1/2 2π∆ ρ(0)(T)≃1− 0 exp(−∆(0)/k T) (18) s k T 0 B B ! while for l=2 it gives the linear T-dependence (2ln2)k T ρ(2)(T)≃1− B . (19) s (2) ∆ 0 Experiments [8, 9] on the temperature variation of the magnetic penetration depth λ (T) are consistent with ab the quasi-linear behavior (19) which is a signature of d- wave symmetry. However,its asymptotic value λ (T → ab 0) is independent of l, a result that we apply below. FIG. 1: Comparison of experimental Tcs vs. theoretical pre- dictions(29)asfunctionofzero-temperatureinversepenetra- III. BOSE-EINSTEIN CONDENSATION tion length λ−1 for YBCO compounds with different doping ab degrees. Square datapoints are taken from Ref.[37], except We assume that charge carriers are an ideal binary foruppermostsquarereferringtotheoptimallydopedregime mixtureofnon-interactingunpairedfermionsplusbreak- [38]. Vertical “error bars” represent full widths of σ1 peaks, where σ is the real part of the conductivity σ employed in able bosonic linearly-dispersive CPs [14, 22, 23, 26]. Let 1 Ref.[37] to determine λ−1. the fermion number per unit area be n = n + n ab f f1 f2 wheren andn arethenumberdensitiesofunpairable f1 f2 andpairable fermions, respectively. Unpairable fermions c =2v /π andK =k (1+k2/k2), the upper inte- lieoutsidetheinteractionregionof(8)unlikethepairable 1 F max F D F gration limit x in (21) is then be very large, namely max fermions whose T-dependent density nf2(T) is x = β~v k = 2E /k T ≫ 1. The last inequality max F F F B is consistent with the maximum empirical value for the n (T)=2 n2D(T)+n2D (T) +nu (T). (20) f2 0 0<K≤K0 f2 ratio k T /E ≤ 0.05 reported [20] for all SCs includ- B c F Here n2D is (cid:2)the bosonic number den(cid:3)sity of CPs with ing cuprate SCs. Given the rapid convergence of Bose 0 CMM wavenumberK =0, n2D that with 0<K < integrals the upper integration limit xmax may safely be K ,andnu thenumberdens0i<tyKo<fKp0airablebutunpaired takenas infinite in (21) sothat the integralscanthen be 0 f2 evaluated exactly by expanding the integrand in powers fermions. By assertingthatin thermalequilibriumthese of zexp(−x) and integratingterm by term. The number kindsoffermionsarisepreciselyfrombrokenCPs[22]we identifynu (T)=2n2D (T). Ontheotherhand, density (21) becomes f2 K0<K<Kmax at T =0 all pairable fermions should belong to the con- (k T)2 ∞ zn 2dnen2Dsatweh(eRreef.[n328D], pis. t1h2e2)tostoalthbaotsonnf2n(0u)m=ber2np20Der(0u)n≡it n2D =n20D(T)+ 2πB~2c21 n=1n2 (22) X area. The number equation for pairable fermions may The critical BEC temperature T is now determined by thus be reexpressed in terms of boson quantities alone, c solving (22) for n2D(T )=0 andz(T )=1. One obtains namelyn2D =n2D(T)+n2D (T)+n2D (T) 0 c c 0 0<K≤K0 K0<K≤Kmax ≡n20D(T)+n20D<K≤Kmax(T). Thus ~c1 2πn2D 1/2 T = (23) c k ζ(2) Kmax d2K 1 B (cid:18) (cid:19) n2D =n2D(T)+ (21) 0 Z0+ (2π)2z−1expβEK(l)−1 where ζ(2)=π2/6. where β ≡ 1/k T, µ the boson chemical potential and B z ≡ expβµ is the fugacity (0 ≤ z ≤ 1). On in- IV. CHARGE CARRIER DENSITY troducing (21) the energy-shifted boson dispersion re- lation E(l) = ~c K for K > 0 the integral can eval- The areal density of charge carriers was formerly es- K 1 uated by changing to the variable x ≡ β~c K. Since timated from measurements of the London penetration 1 5 superconductor ΘD (K)a ∆0 (meV)b λab (nm)c δ(A˚)d Tcexp(K)e Tcth (K) (2∆0/kBTc)exp(f) (2∆0/kBTc)th (La.925Sr.075)2CuO4 360 6.5 250 4.43g 36 36.4 4.3 4.14 YBa2Cu3O6.60 410 15.0 240 2.15h 59 56.0 5.90 6.09 YBa2Cu3O6.95 410 15.0 145 2.15g 93.2 92.6 4.0 3.68 Bi Sr CaCu O 250 16.0 250 2.24g 80 72.2 4.64 4.85 2 2 2 8 Bi Sr Ca Cu O 260 26.5 252 2.24i 108 109.2 5.7 4.99 2 2 2 3 10 Tl Ba Ca Cu O 260 22.0 221 2.14i 110 104.1 4.5 4.47 2 2 2 2 8 Tl Ba Ca Cu O 280 14.0 200 4.30i 125 105.5 3.1 2.96 2 2 2 3 10 TABLE I: Physical parameters of cupratesuperconductors and predicted values for Tc, and the ratio 2∆0/kBTc according to (28). Debye temperature is ΘD ≡ ~ωD/kB. Parameters taken from from Ref.[38] (see also references cited therein): a) table 4.1,b)table6.1,c)tableA.1,d)tableA.2,e)tableA.1,f)table6.1,g)estimated fromband-structurecalculations [39,40],h) estimated as δ =0.64 cint, and i) estimated as δ =0.68 cint, where cint is theCuO2 interlayer separation for a given cuprate. For YBCO ΘD, ∆0, and δ are assumed the same for different dopings. depth λ which is the distance over which an external λ−1(0) = λ−1(0)+λ−1(0) is the geometric mean of this L ab a b magnetic field decays within the superconductor. For parameter measured along crystallographic in-plane di- superelectrons with a 3D density n , chargee, and effec- rections a and b. As shown in § II, this parameter is in- s tive mass m∗, one has the well-known relation 1/λ2 = dependentoftheexplicitvalueoftheangularmomentum L 4πe2n /m∗c2. By introducing [20] the averageinterlayer l. By substituting the dispersion relation (11) to elimi- s spacing c between CuO planes in HTSCs it follows nate K from λ and imposing the relation n2D = δn3D int 2 0 0 that n2D ≃ c n . Penetration-depth data spanning a the 2D charge carrier density becomes int s wide range of critical temperatures are consistent with the phenomenological Uemura relation T ∝ 1/λ2 ∝ e2δ|E(l)| 1 c L n2D = 0 . (26) n2D/m∗ [20]. c2 16πc2 λ2 1 ab Within the present model we evaluate the magnetic penetration depth due to linearly-dispersive CPs with Thislatterexpressioncanbereformulatedbyconsidering charge2e,andconstrainedtomovewithinathinlayerof the relation (15) and the weak-coupling limit of (12). It widthδ withauniformCMspeedc1. Thus,wefirstcon- follows that |E0(l)|=(∆(0l))2/2~ωD so that sider the expression for the 3D supercurrent of excited CPs [14] Js = n3D(2e)c1Kˆ with Kˆ ≡ K/K. We now n2D = e2 δ∆20 1 (27) introduce the contour integral of the CP wavefunction 32πc2c2~ω λ2 phase within a homogeneous medium and in the pres- 1 D ab ence of an external magnetic field B = ∇ × A. The where the approximate relation ∆(l)/Γ(l) ≃ ∆ as justi- following integral along any closed path vanishes 0 0 fied in § II was used. 2e ~K+ A ·dr=0 (24) c I (cid:18) (cid:19) V. CRITICAL TEMPERATURE where c is the speed of light in vacuum. By expressing K in terms of J and using Stoke’s theorem to evalu- s The final explicit expression for the critical BEC tem- ate (24) we get a modified version of London’s equation peratureT is now obtainedby substituting (27) in(23). J = −Λ A where Λ ≡ 4e2c n3D/~cK. Taking now c s p p 1 This leaves the curl of this modified London equation and introduc- ing Ampere’s law ∇×B = (4π/c)Js, it follows that the ~c 3δ 1/2 ∆0 magnetic induction B satisfies the Helmholtz equation Tc = 2πk e 2~ω λ (28) ∇2B=λ−2B, where B (cid:18) D(cid:19) ab which is independent of the CP speed c . We observe 1 (2e)24πc n3D 1 λ2 ≡ c2 ~1K . (25) pthearattfuorrefiixnecdrevaaseluselsinoefaωrlDy,w∆it0h, aλn−d1.δ,Tthheiscdrietpiceanldteenmce- ab NotethatLondon’sresultisrecoveredforquasi-particles has been observed by Zuev et al.[37] in experiments in with density n3D → n /2, momentum ~K → 2m∗c , underdopedYBCOfilmswithT srangingfrom6to50K. s 1 c and charge 2e → e. This expression for λ varies be- Theyconcludethat,withinsomenoise,theirdatafallon tween its minimum value λ = 0 when K = 0 (perfect the same curve ρ ∝ λ−2 ∝ T2.3±0.4, irrespective of an- s ab c diamagnetism), and its maximum, say λ , for K = K nealingprocedure,oxygencontent, etc. Thus,by assum- 0 0 (CP breakup). It seems natural to identify λ with the ing that except for λ the other YBCO parameters are 0 ab experimentally observed value of the in-plane penetra- approximatelyindependentofthe dopinglevel,weintro- tion depth at T = 0, namely λ = λ (T = 0). Here, duced in (28) the values: Θ = 410K [38], ∆ = 14.5 0 ab D 0 6 meV [38], and δ =2.15 A˚[39, 40] to get the relation holds also for the thallium compounds. The former esti- mations are congruent with Uemura’s surmise [20] that 16.79[(µm)−1K] SC charge carriers in layered cuprates are concentrated T = . (29) c λab within slabs of width δ =cint. Table I shows results obtained using the foregoing as- Figure 1 is adapted from Ref.[37] and compares theoret- sumptions, together with the physical parameters in- ical predictions (29) with experimental data, as well as volved in the calculation. In most cases we find rather with data pertaining to higher doping regimes. We see satisfactory agreement between predicted and measured that (29) gives an excellent fit to the experimental data. values of T . We also find very good agreement c The same functional dependence has been observed in between theoretical and experimental gap-to-T ratios c single YBCO crystals near the optimally-doped regime 2∆ /k T . Average theoretical and experimental such 0 B c [41]. More recently, Broun et al. [9] found that their ratios presented in Table I are (2∆ /k T )th ≃4.45 and 0 B c samples of high-purity single-crystalYBCO followedthe (2∆ /k T )exp ≃ 4.59, respectively. Both are consis- 0 B c rule Tc ∝ λ−ab1 ∝ n1s/2 ∝ (p−pc)1/2 where the doping tent with the ratio 2∆(02)/kBTc ≃ 4.28 predicted by the p is the number of holes per copper atom in the CuO2 l = 2 BCS theory in (16). We have not attempted es- planes and pc the minimal doping for superconductivity timate uncertainties of our theoretical results since the onset. The measured value of the penetration length in accumulateddata of the physical parametersinvolvedin YBCO crystals is an order of magnitude bigger than in the calculation, particularly ∆ and λ , show a wide 0 ab thin films [9, 41], so that the specific values of Tcs de- scatter. rivedfrom(28) arenot in suchgoodagreementasin the YBCO films. However, one should expect variations of parameters such as the energy gap associated to crys- VI. DISCUSSION AND CONCLUSIONS tals and film systems. It has been pointed out [37] that YBCO films seem to behave more like other cuprates suchasBiSrCaCuOorLaSrCuOthandoYBaCuOcrys- We haveshownthatlayered-cuprateHTSC canbe de- tals. Furthermore, a different approach [42] based on scribedbymeansofanl-waveBCStheoryforaquasi-2D BEC of Cooper pairs. The theory involves a linear, as measurementsofthe lowercriticalmagneticfieldH (T) c1 opposed to quadratic, dispersion relation in their total for highly underdoped YBCO indicates that experimen- or CM momenta. The theory yields a simple formula for tal data may be consistently described only by assuming T ∝ n0.61, in close agreement with studies mentioned the critical transition temperature Tc with a functional abcove. s relation Tc ∝ 1/λab ∝ n1s/2 which applies to a variety of Theoretical values of T for superconducting cuprateSCs overawide rangeofdopings. Althoughthis c cuprates with different compositions have been behaviorapparentlydisagreeswiththephenomenological also calculated using (28). Here we report on Uemura relation Tc ∝ 1/λ2ab [20], different experimen- these seven layered-cuprate superconducting com- tal studies [9, 37, 41] show consistency with the inverse pounds: (La.925Sr.075)2CuO4; YBa2Cu3O6.60; linear dependence of Tc. Additional consistency is also YBa2Cu3O6.95; Tl2Ba2Ca2Cu2O8; Tl2Ba2Ca2Cu3O10; seen with the reported dependence Tc ∝ n0s.61 arising Bi Sr Ca Cu O ; and Bi Sr CaCu O . Characteristic from measurements of the lower critical magnetic field 2 2 2 3 10 2 2 2 8 parameters for these materials were taken from tables [42]. When averaged over a cylindrical Fermi surface, compiled in Ref.[38] (see also [43–45]). Concerning the physicalquantitiesinvolvedinthe theoryshowsmall the layer width δ no direct experimental data are dependence onthe angularmomentumstate l. However, available. We haveemployedresultsderivedfromenergy the gap-to-Tc ratio 2∆0/kBTc is closer to that predicted band-structure calculations for cuprates. Contour by the extended BCS theory for l = 2 than for l = 0. It plots [39, 40] of the charge distribution for La CuO , is shown elsewhere [46] that all relevant 2D expressions 2 4 YBa2Cu3O7, and BiCa2SrCu2O8 suggest that charge derived here arise in the limit kBTδ/~c1 → 0 of a more carriers in each of these systems are concentrated general 3D BCS-BEC theory for layered materials. within slabs of average width δ ≃ 2.61A˚, 2.15A˚, and ∗Permanentaddress. 2.28A˚, respectively, about their CuO planes. As c Acknowledgments We thank M. Fortes, S. Fu- 2 int denotes the average separation between adjacent CuO jita, L.A. P´erez, and M.A. Sol´ıs for fruitful discus- 2 planes, it follows from crystallographic data [38] that sions. MdeLl thanks UNAM-DGAPA-PAPIIT (Mexico) the yttrium and bismuth compounds give δ ≃ 0.64 c IN106908aswellasCONACyT(Mexico)forpartialsup- int and 0.68 c , respectively. Taking into account that port. He thanks D.M. Eagles and R.A. Klemm for e- int BiSr Ca Cu O compounds possess the same correspondence and is grateful to W.C. 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