a r X i v :c o n d -m a t /9501118v 1 25 J a n 1995s -wave superconductivity from antiferromagnetic
spin-fluctuation model for bilayer materials
A.I.Liechtenstein a ,I.I.Mazin a,b ,and O.K.Andern a
a Max-Planck-Institut f¨u r Festk¨o rperforschung,Heinbergstr.1,D-70569Stuttgart,FRG.
b Geophysical Laboratory,Carnegie Institution of Washington,5251Broad Branch Rd.,NW,Washington,DC 20015.Abstract It is usually believed that the spin-fluctuation mechanism for high-temperature superconduc-tivity results in d -wave pairing,and that it is de-structive for the conventional phonon-mediated pairing.We show that in bilayer materials,due to nearly perfect antiferromagneti
c spin correla-tions between the planes,the stronger instabil-ity is with respect to a superconducting state
who order parameters in the even and odd
plane-bands have opposite signs,while having
both two-dimensional s -symmetry.The interac-
tion of electrons with Raman-(infrared-)active
phonons enhances (suppress)the instability.
71.10.+x,74.20.Mn,74.72.Bk
Typet using REVT E X The currently most exciting discussion about high-T c superconductivity deals with
the symmetry of the pairing state[1].In-timately related to this,is the question of whether the superconductivity is due to an-tiferromagnetic spinfluctuations(e e.g. Monthoux and Pines(MP)Ref.[2],and also Refs.[3,4]),to electron-phonon(EP)interac-tion enhanced by inter-layer pair tunnelling [5],or to neither of two.In this discus-sion,it is indirectly assumed that the antifer-romagnetic spin-fluctuation(AFSF)mecha-nism necessarily leads to d-wave pairing,and that the AFSF and EP mechanisms cannot coexist.
In this Letter we point out that,whereas the AFSF mechanism leads to d-pairing for one layer,it may lead to(two-dimensional) s-symmetry for a bilayer.The condition for that is existence of strong antiferr
omagnetic correlations between the two layers in a bi-layer,as found experimentally in YBa2Cu3O7 [6,7,8].Wefind that,for a given coupling strength,T c(s)is about twice as high as T c(d)thus making it easier to achieve the obrved values of T c∼100K.Esntial for the positive influence of layer-doubling is the single-particle tunneling which splits the one-electron plane-bands into even and odd with respect to the mirror-plane between the lay-ers.In this aspect our mechanism is very different from the interlayer pair-tunnelling (IPT)mechanism discusd by Anderson et al.[5].Nevertheless,similar to the IPT model,any attractive interaction between electrons in the same band,such as the one mediated by even(Raman-active)phonons, enhances T c.This is opposite to the previ-ously considered single-layer AFSF models in which such interactions are mutually destruc-tive.
Support for such an enhancement mech-anism may be found in the experimental Refs.[9,10])that some mem-bers of the cuprate family(Nd2−x Ce x CuO4, HgBa2CuO4)behave as conventional s-wave EP superconductors.MP AFSF theory,on the other hand,would have to imply princi-pally different mechanisms for this compound and for tho with high T c’s.Another ex-perimental fact which suggests a constructive interplay between phonon-and non-phonon mechanisms is that in YBa2Cu3O7,the iso-tope effect increas smoothly when the su-perconductivi
ty is suppresd[11].Finally, the most impressive argument is that in all high-T c materials T c is anticorrelated with the in-plane antiferromagnetic correlation length
ξ.In particular,in YBa2Cu3O7.0,ξis about one lattice parameter,which would make the single-layer AFSF superconductivity virtu-ally inoperative.To the contrary,as we shall argue below,the propod bilayer model is barely nsitive to the in-plane AF correla-tion lengthξat all.
In the following we shall assume a con-ventional picture in the n that the one-electron tunneling between the planes is al-lowed both in the normal and in the super-conducting states.This is in contradiction with the IPT scenario[5],but in agreement with some photoemission experiments[12]. In this ca,the single-particle eigenstates for a bilayer are the even|+,k and odd|−,k combinations of the individual plane states and k is the2D Bloch-vector.The properties of the even and odd bands are discusd in de-tail in Ref.[14],but,for the purpo of com-parison with the MP model,we u the same band model as they did.We neglect com-pletely the k z dispersion due to small inter-cell c-hopping,which can lead to interesting effects(,Ref.[13])but which are how-ever beyound the scope of this Letter.Ac-cordingly,in the following the term“bands”always means“two-dimensional bands”.As regards the interplane hopping inside the unit
cell,t⊥(k),we assume that it is sufficiently
large to t even and odd symmetry of the
two-dimensional bands,but we neglect,for simplicity,in the following numerical calcula-
tions the even-odd splittingǫ−(k)−ǫ+(k)=
2t⊥(k).
The generalization of the MP AFSF
model to two-bands is straightforward;one
has only to take into account that the effec-
tive vertex for scattering of an electron from
band i to band j by a spin-fluctuation de-
pends on i,j,while the spectrum of thefluc-tuationsχis the same as in MP.Then,Eqs.
(6-8)of MP become
Σij(k,iωn)=T q m
kl
V ik,lj(k−q,iωn−iωm)
×G kl(q,iωm)
G−1ij(k,iωn)=[iωn−ǫ(k)+µ]δij−Σij(k,iωn)
Φij(k,iωn)=−T q m
klst
V ik,tj(k−q,iωn−iωm)(1)×G kl(−q,−iωm)Φls(q,iωm)G st(q,iωm) whereΣandΦare respectively the normal
and anomalous lf-energies,G is the single-particle Green function,ǫthe bare electron energy,and q m denotes the average over
the Brillouin zone plus the sum over the Mat-subara frequencies.V is AFSF pairing inter-action,deter
mined by the exchange interac-
tion of electrons with the AFSF’s,V ij,kl= d R d R′ αβγδ iα|J(r−R)σαβ|jβ
טχ(R−R′) kγ|J(r−R)σγδ|lδ where J is exchange interaction and˜χ= S(R)S(R′) is spin-spin correlation function.For a bi-layer,one can let R be a two-dimensional vec-tor and introduce˜χ=χ(R−R′)I uv,where u,v=1,2label layers,and I accounts for in-terplane correlations,if any.Then the func-tionχis the same as in MP.
The key to the our bilayer AFSF model is the experimental fact that the spinfluc-tuations in the bilayer of YBa2Cu3O7−x are always antiferromagnetically correlated be-tween the planes[6,?,8].Even fully oxy-genated samples,where the in-plane corre-lation length is already of the order of the lattice parameter,show nearly perfect inter-layer correlation[8].The exchange potential t up by such a spin-fluctuation is there-fore odd with respect to the mid-layer mir-ror plane and,correspondingly,couples ex-clusively even and odd electron states(but neither odd to odd,nor even to even).In other words,I u=v=−I uu=−1.In this ca, after summation over u,v in the expression for V ij,kl and defining the appropriate cou-pling constant g,Eqs.1become:Σ−(k,iωn)=T g2 q mχ(k−q,iωn−iωm)G+(q,iωm) G−1+(k,iωn)=iωn−ǫ(k)+µ−Σ+(k,iωn)
Φ+(k,iωn)=−T g2 q mχ(k−q,iωn−iωm)(2)×G−(−q,−iωm)Φ−(q,iωm)G−(q,iωm)
and the same with+and−subscripts in-terchanged,and g andχare the same as in
MP.
For reasons of symmetry,the solution of
the equations must have the form G+=
G−,Φ+=±Φ−.For the upper choice of the
sign,the Eqs.(2)reduce precily to the orig-
inal MP pairing state.For the lower choice,
the Eqs.(2)again reduce to the one-plane
ca,but now the interaction in the equa-
tion forΦis effectively attractive.In other
words,now the order parameter has the op-
posite sign in the two bands,and therefore
the last Eq.in(2)can be rewritten in terms
of|Φ|,and with plus instead of minus on the
right-hand side.
The concept of a superconducting state
where two distinctive bands had the order parameters of the opposite signs wasfirst dis-
cusd in1973in connection with mimet-
als[16].More recently,in a two-layer Hub-
bard model,such a solution was found by
Bulut et al[17](which they labeled as“d z”
state)and in the conventional superconduc-tivity theory[18],where it appears in ca of strongly anisotropic electron-phonon and/or Coulomb interaction,or becau of a strong interband scattering by magnetic impurities. In all cas,order parameter has s-symmetry inside each band and changes sign between the bands.
From Eqs.(2)it is quite plausible that such an instability is stronger than the d x2−y2 one,and will occur at a higher T c.Below we shall prove this numerically,but before go-ing to numerical results,it is instructive to get a conceptual understanding about the two different solutions.The physical reason for having d-symmetry in the one-plane ca, is that the AFSF interaction makes pair-ing energetically favorable only when it cou-ples parts of the Fermi surface which have opposite signs of the order parameter[19]. In Y123the AFSF interaction is peaked at Q=(π/a,π/a).The shape of the Fermi sur-face is such that the condition is satisfied only for d x2−y2symmetry.On the other hand,the small-q interaction couples parts of the Fermi surface where the order parameter has the same sign.This makes pairing unfavorable. Sinceχ(q≈Q)≫χ(q≈0),nevertheless,more is lost by making an s-state than by making a d-state(which has been found nu-merically by MP),becau the latter loss is the difference between the small-q loss and the large-q gain,while in an s-state one los over the whole Fermi surface[20].
Now,coming to the bilayer ca,we ob-rve that there is no conflict between the small and the large q’s any more.The AFSF interaction spans two different sheets of the Fermi surface,which always have or-der parameters of the opposite signs.Thus the AFSF interaction is as attractive for s-pairing in a bilayer as it is repulsive in a sin-gle plane,and conquently more attractive than d-pairing in a single plane.Of cour, the resulting s-state is likely to be highly anisotropic,to take better advantage of the largeχ(q)at q≈Q.This is similar to An-derson’s model[5].To demonstrate this ef-fect numerically,we have solved Eqs.2with the parameters from MP paper,and using the same numerical technique.As expected, the maximal eigenvalue of the last Eq.2is larger than that for the MP d-pairing(about 1.5compared to1).Fig.1shows the plot of T c as a function of the interaction constant g for both cas.To test the numerics,we
have also solved the original MP equations and obtained the similar results as MP.
From Fig.1,one immediately obrves that the value (0.69eV)of the coupling con-stant g which yields T c ≈90K for two planes and s -symmetry,is much smaller than the corresponding value (1.24eV)for one plane and d -symmetry.Actually,T c (s )∼2T c (d )for g up to about 1eV.At stronger cou-plings,T c (d )saturates faster than T c (s )due to stronger effect of mass renormalization.Similarly,as we shall e below,the ratio of the maximal gap to T c tends to be larger for the one-plane model,for the same
T c .One can also obtain the lf-consistent so-lution for Φat T ≪T c .To do that,one has to include higher-order terms (,Ref.[3]).In this ca one of the Green functions in the Eq.2for Φshould be replaced by:
˜G −1−
(k ,iωn )=iωn −(ǫk −µ)−Σ−(k ,iωn ),(3)G −1−(k ,iωn )=˜G −1−(k ,iωn )−Φ−(k ,iωn )˜G −(−k ,−iωn )Φ−(−q ,−iωn )].This new t of equations can be solved iter-atively,starting with G =˜G
(which is cor-rect to first order in Φ).The actual solution for T =T c /2,shown in Fig.2,was achieved by making two iterations of Eqs.3.The frequency-dependent superconducting gap is related to Φas ∆(p ,iωn )=Φ(p ,iωn )
1−Im Σ(p ,iωn )/ωn .
From Fig.2we obrve that the absolute value of ∆behaves similarly in both cas,having a minimum along (11)directions and a max-imum along (10)directions.Furthermore,|∆|’s in both cas differ by less than 10%on two-thirds of the whole Fermi surface,thus making it extremely difficult to distinguish between the two in an experiment which does not probe the relative phas of ∆.In other words,the order parameter,formally having s −symmetry,is still strongly anisotropic,but nodeless.
Now we shall briefly discuss some exper-
imental conquences of the bilayer AFSF superconductivity model.It turns out that many difficulties associated with the original AFSF superconductivity model disappear in
the prent version.(1)In-plane vs.perpendicular-to-the-
planes Jophson tunneling.Recent arches
for the d -pairing in YBa 2Cu 3O 7(Refs.[21,22]and others)still do not give a definite an-swer.Experiments probing the angular de-pendence of the order parameter,as well
