Zeeman and Orbital Effects of an in-Plane Magnetic Field in Cuprate Superconductors

a r X i v :c o n d -ma t /9910418v 1 [c o n d -m a t .s u p r -c o n ] 26 O c t 1999Zeeman and Orbital Effects of an in-Plane Magnetic Field in Cuprate SuperconductorsKun YangNational High Magnetic Field Laboratory and Department of Physics,Florida State University,Tallahassee,Florida 32310S.L.SondhiDepartment of Physics,Princeton University,Princeton,New Jersey 08544(February 1,2008)We discuss the effects of a magnetic field applied parallel to the Cu-O (ab )plane of the high T c cuprate superconductors.After briefly reviewing the Zeeman effect of the field,we study the orbital effects,using the Lawrence-Doniach model for layered superconductors as a guide to the physics.We argue that the orbital effect is qualitatively different for in-plane and inter-layer mechanisms for superconductivity.In the case of in-plane mechanisms,interlayer couplings may be modeled as a weak interlayer Josephson coupling,whose effects disappear as H →∞;in this case Zeeman domi-nates the effect of the field.In contrast,in the inter-layer mechanism the Josephson coupling is the driving force of superconductivity,and we argue that the in-plane field suppresses superconductivity and provides an upper bound for H c 2which we estimate very crudely.One of the most important milestones in the study of the cuprate superconductors is the identification of the predominantly d x 2−y 2symmetry of their pairing order parameters.This discovery has motivated a tremendous amount of theoretical work on the physical properties of superconductors with unconventional pairing symmetry.Among them,it was recently pointed out 1that the re-sponse of a two-dimensional (2D)d x 2−y 2superconductor to the Zeeman coupling between the spins of the elec-trons and an external magnetic field is much stronger than that in an s -wave superconductor.This is so,for unlike the fully gapped s -wave case,in a d x 2−y 2su-perconductor there exists nodal points in the order pa-rameter and therefore gapless quasiparticle excitations.Consequently,an arbitrarily weak Zeeman field gener-ates a finite density of spin-polarized quasiparticles in the ground state 1.It was also shown that the Zeeman field enhances the low-T specific heat,thermal conductivity,and tunneling density of states,while suppressing the superfluid density 1,2.At strong fields,the system sup-ports a Fulde-Ferrell-Larkin-Ovchinnikov type supercon-ducting state.3,1Eventually,a sufficiently strong Zeeman field suppresses superconductivity completely,leading to a “Pauli limit”on the upper critical field.The results mentioned above are expected to be partic-ularly relevant to the cuprate superconductors when the external field is oriented parallel to the Cu-O (ab )planes (for possible caveats on the influence of disorder see [4]),as the cuprates are quasi-2D systems with electrons pre-dominantly moving in the Cu-O planes.In a truly 2D system,the orbital motions of the electrons would not sense the existence of a parallel magnetic field,and the Zeeman effect would be the only effect of the field.Ex-perimentally,there is some evidence 5suggesting that the upper critical field for the parallel orientation is indeed Pauli limited in YBCO at low temperatures.However for smaller fields there do exist orbital effects,even when the field is parallel to the planes,due to the presence of inter-plane couplings.It is the purpose of this paper to report results of studies of such orbital ef-fects,and to compare them with the Zeeman effect in field ranges that are relevant to current experiments.We will argue that the nature and importance of the orbital effects depend sensitively on whether superconductivity is driven by in-plane correlations or inter-plane hopping.Thus their study may provide a way to distinguish be-tween in-plane and inter-plane mechanisms for supercon-ductivity in the cuprates.The orbital effects of the parallel field may be sum-marized qualitatively as the following.The field gener-ates screening currents both within and across the planes.The current within the plane generates a “Doppler shift”of the quasiparticle spectrum 6,and therefore a finite den-sity of states (DOS)for the quasiparticles.This current-induced DOS is proportional to√2|ψn |4+¯h2¯h c A n )ψn |2+t |ψn +1−ψn |2}.(1)Here n is the layer index,α∝T −T c ,and other param-eters may be assumed to be T independent.For T veryclose to T c ,the coherence length becomes very large,so1that continuum approximation can be used along the cdirection,and(1)reduces to the usual Ginsburg-Landau theory with anisotropic mass,and the mass along the c direction is m c=¯h2H.Here we computethe answer at highfields.We assume the externalfield H=H(1,0,0)is along theˆa direction.It is known that H c1is very low(<100Gauss)for a parallelfield,andforfields of order H∼1T(which is the range of interest in the present paper and typical for experimental stud-ies),thefield basically penetrates the system uniformly and the screening effect is negligible.(This justifies ne-glecting thefield energy term in(1)).Thus we can as-sume thefield inside the superconductor is the same as that of the externalfield and uniform.We use the gauge A n=nsH(0,1,0).We need to solve the equation forψn:δF2m ab (∇−2ie−α/βe i(φn+k n y),(3) where k n=2ensH−αγ2(ℓ¯h c/2eH is the magnetic length for Cooper pairs,ηn=1for n=4j or4j+1,and −1otherwise(j is an integer).In order for the pertur-bative approach to be valid,we need to have1s )4≪1,(5)which is always valid in the strongfield limit.The typical in-plane superfluid velocity is thusv s≈¯h∆k s)4,(6)and the Doppler shift energy is6,7E D∼¯h v s k F=ǫF kFs3∝1/H.(7)Since the Zeeman energy E Z=µB H∝H,it alwaysdominates the Doppler shift at sufficiently strongfield.Atfield H=10T,usingǫF∼2eV,k F∼1˚A−1ands∼10˚A,we obtain for BSCCO(in which(5)is satis-fied)E D/E Z∼0.4,indicating Zeeman effect starts todominate the orbital effect in thisfield range in BSCCO.(5)is not satisfied in YBCO at thisfield due to lessanisotropy;but using the above results anyway(whichresults in an overestimate of the in-plane current)wefind E D/E Z∼150,suggesting that the orbital effectwill dominate.In the above discussion we have assumed that the in-terlayer Josephson couplings are secondary to supercon-ductivity in cuprates,and may be treated as weak per-turbations.This is appropriate for in-plane mechanismsfor superconductivity in cuprates.However it was pro-posed by Anderson and coworkers12,13that the interlayerJosephson coupling(or pair-hopping)may actually bethe driving force behind superconductivity in cuprates.In this case,the interlayer Josephson coupling can nolonger be treated as a weak perturbation as above,andwe expect an in-plane magneticfield to have a much moredramatic orbital effect.This is because the in-planefieldtends to suppress interlayer phase coherence,and there-fore Josephson energy and superconductivity in this case.Among other things,the orbital effect of thefield pro-vides a mechanism for the upper criticalfield H c2.Whilefor in-plane mechanisms,thefield simply decouples thelayers,and H c2can only come from the Zeeman effect1,3.In the inter-layer pair hopping model12,13,the multi-layer structure of the cuprates is important.In particu-lar,the Josephson coupling among layers within the sameunit cell might be much stronger than between differentcells,thus the large anisotropy could largely reflect theweakness of the inter-unit cell Josephson coupling only.For simplicity,we study in the following in a single bi-layer(as in Ref.13),and neglect the coupling betweenthis pair of layers and the rest of the system.A propertreatment of even this problem requires a microscopictheory that does not really exist.So here we will contentourselves with a speculative illustration of what such atreatment might produce.To this end we again use theLawrence-Doniach model,in a slightly different form andwith a crucial reinterpretation of the parameters:F= dxdy{α′(|ψ1|2+|ψ2|2)+β+¯h2¯h c+ψ∗2ψ1e−2ieHsx(t−α′)/β,and there is no obvious distinc-tion between these two cases;the system is phase coher-ent both within each individual layer and between the two layers.The situation is very different when H=0 and is sufficiently large.This is because for a loop that encloses afinite amount offlux,one must accumulate a finite amount of gauge-invariant phase difference propor-tional to theflux(see Fig.1a).It is therefore no-longer possible to maintain in-plane and interlayer phase coher-ence simultaneously14.Forα′<0(in-plane mechanism),the situation is basically the same as a Josephson junction in a mag-neticfield10;we haveψj=e iφj¯h c +φ1−φ2),while the in-planecurrent goes to zero as1/H at large H(see Fig.1b). The Josephson coupling term in(8)averages to zero in this solution.In short,the system gives up interlayer co-herence(as manifested by the loss of Josephson coupling energy and appearance of Josephson current),in order to maintain in-plane coherence(therefore no in-plane cur-rent and kinetic energy).Forα′>0(interlayer mechanism),on the other hand, the system will do everything possible to maintain inter-layer coherence,as the Josephson energy is the origin of superconductivity.Thus the solution in this case is ψ1,2= 2m ab c2)/βe±iesHx/¯h c.(9) In this solution the interlayer coherence is maintained and there in no interlayer Josephson current induced by thefield.As a price,the system loses in-plane coherence, and there is diamagnetic current(∝H)and in-plane ki-netic energy loss(∝H2)(see Fig.1c).Clearly,as H increases,at some point the loss of in-plane kinetic en-ergy will overwhelm the gain of Josephson energy,and the system will cease to superconduct.This is the or-bital upper criticalfield H o c2(to be distinguished from the Zeeman upper criticalfield H Z c2),as determined by the point whereψvanishes in(9).For in-plane mech-anisms,H o c2→∞.If we assume that interlayer mech-anism is responsible for superconductivity in cuprates, a crude estimate yields in BSCCO2212H o c2∼200T at T=0.Unfortunately this is comparable to H Z c21,3.How-ever,in principle,the Zeeman effect can be suppressed by introducing spin-orbital scatterers to the system which could enable the two to be distinguished.The qualitative difference in the orbital response to a parallel magneticfield can provide a way to experimen-tally distinguish between in-plane and interlayer mech-anisms for cuprate superconductivity.We note the in-terpretation of previous experimental attempts to distin-guish between the two based on c-axis penetration depth is controversial15.This work was supported by NSF DMR-9971541and the Sloan Foundation(KY),and NSF DMR-9632690, and Sloan and Packard Foundations(SLS).10M.Tinkham,Introduction to Superconductivity,second edi-tion,McGraw-Hill,New York(1996).11Our results below differ from a previous study(S.Theodor-akis,Phys.Rev.B42,10172(1990))which appears to usea similar approach.12P.W.Anderson,The Theory of Superconductivity in the High-T c Cuprates,Princeton Univ.Press,Princeton(1997). 13S.Chakravarty,A.Sudbø,P.W.Anderson,and S.Strong, Science261,337(1993).14A similar situation happens in double-layered quantum Hall systems with spontaneous interlayer phase coherence: K.Yang et al.,Phys.Rev.Lett.72,732(1994);K.Moon et al.,Phys.Rev.B51,5138(1995);K.Yang et al.,Phys.Rev.B54,11644(1996).15S.Chakravarty,H.-Y.Kee and E.Abrahams,Phys.Rev.Lett.82,2366(1999),and references therein.(a)Φ(b)α’ < 0(c)α’ > 0FIG.1.(a)A closed path that encloses afinite amountofflux.(b)The current pattern in a bilayer system in the presence of strong parallel magneticfield,for in-plane mech-anism of superconductivity.(c)Same as(b),for inter-layer pair hopping mechanism of superconductivity.4。