Physica C 153-155 (1988) 21 25 North-Holland, Amsterdam

THE CURRENT THEORETICAL SITUATION IN HIGH-To SUPERCONDUCTIVITY

J.R. SCHRIEFFER,X.-G. WEN and S.-C. ZHANG

Institute for Theoretical Physics, University of California Santa Barbara, California 93106

Proposals for the mechanism of high temperature superconductivity in layered oxides are discussed. Within the context of the pairing theory these include spin, charge and lattice vibration degrees of freedom. The question of whether a localized versus an itinerant de- scription is more appropriate is discussed including the possible separation of the charge and spin of excitations.

I. INTRODUCTION

The pairing theory (1) has given an accurate ac- count of the properties of conventional superconduc- tors, as well as superfluid 3He, and those properties of nuclear matter which depend on pairing correla- tions. This theory requires as input the spectrum of low lying quasiparticle excitations in the normal phase as well as the effective residual interactions between quasiparticles. In conventional metallic su- perconductors, the order parameter A k is found to be nodeless as k moves over the Fermi surface cor- responding to "s-wave" pairing or orbital angular momentum ~ = 0. The Pauli principle requires that for even e the total spin S of each pair is zero. Here the exchange of phonons is found to be the dominant pairing interaction.

Exceptions to this situation are found in 3He where the pairing condensate involves l = 1 and S = 1 pairing. ~ ~ 0 pairing apparently also oc- curs in aetinide superconductors. In both cases, spin flucutation excitations are believed to play a domi- nant role in the pairing interaction. If A k exhibits nodes or lines of nodes, the smallness of A near these singularities leads to power law temperature depen- dence in quantities such as the heat capacity, spin susceptibility, acoustic attenuation, magnetic reso- nance relaxation rates, etc., as opposed to an acti- vated behavior e-ZX/kBT characteristic of an l = 0 superconductor. Unfortunately, materials difficul- ties can cause spurious T n behavior in certain cases, masking the intrinsic behavior of a nodeless gap.

With the discovery of high the temperature oxide superconductors (2,3), the question has been raised whether the pairing theory continues to be appli- cable but with an exceptionally strong pairing in- teraction or whether a totally different framework must be developed. One such alternative approach is based on a Bose condensation of single boson ex- citations. Such excitations might be strongly bound bipolarons or spinless charged solitons. Because of experimental uncertainties as well as incompleteness of the theories, this question has not been fully an-

swered, although many features of oxide supercon- ductors are well accounted for by the pairing theory. For references we refer the reader to recent confer- ence prodeedings since several hundred papers have been published on the theory of high Tc (4).

Concerning the pairing interaction, one can group the theoretical effort in several broad classes, those based on (a) lattice vibrations, (b) charge fluctua- tions (excitons, plasmons) and (c) spin fluctuations (antiparamegnons, spin polarons, spin bags). In each case, probes such as neutron scattering, give infor- mation on the energy spectrum, spatial variation and lifetime of these objects. If one can calculate or otherwise determine the coupling constants between these excitations and the electronic quasiparticles, a perturbative estimate of the pairing interaction can be made, as with the conventional electron phonon interaction. However, the nesting property of the Fermi surface can lead to divergences in perturbation theory which require more complete analysis with the possibility of qualitatively new effects occurring.

Another central question is whether supercon- ductivity in the layered oxide materials is essentially 2-dimensional in character with a high mean field transition temperature and a weak Josephson-like coupling between copper oxide planes; or is super- conductivity intrinisically 3-dimensional in nature. It appears that substantial superconducting order exists in the 2d plane regardless of its coupling to neighboring planes, as in other layered superconduc- tors such as TaS 2.

Finally, there remains the important question of whether the effective Hubbard interaction U be- tween electrons on a single site is large, small or of order the band width W of the hole band. The issue here is whether the electrons are best described to zero order in terms of localized states or band states. While arguments have been advanced that U >> W, the situation remains open, with the possiblity that U -- W a distinct possibility.

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22 J.R. Schrieffer et al. / Theoretical situation in high-T,, superconductivity

2. PAIRING INTERACTIONS

2.1 Phonons

A number of studies have attempted to account for the large value of Tc for oxide superconductors in terms of the exchange of phonons between elec- trons (5). It is generally agreed that the conventional second order perturbation approach to this process cannot lead to Tc greater than 40-50 °K for these materials. This small value of Tc is due to the fact that one finds T¢ <~ 0.20, where 0 is the Debye tem- perature. The reduced or vanishing isotope effect in the oxides is consistent with the large value of Tc arising primarily from another interaction.

Nevertheless, several authors have invoked the Jahn-Teller effect (6), going beyond perturbation theory to account for high To, either through the pairing theory or through strong bipolar formation with subsequent Bose condensation. In the former (7), it is shown how a reduced isotope effect occurs because the interaction cutoff energy wc is the width of the electronic resonance, which is smaller than the characteristic phonon energy in the strong Jahn- Teller limit.

2.2 Charge Fluctuations

By replacing a phonon by an exciton or plasmon, one might hope to raise Tc since the frequency scale of these excitations is larger than that of a phonon by an order of magnitude or more. The central ques- tion, however, is how strong is the electron-charge fluctuation coupling. As in the electron-phonon case, when the coupling increases the collective modes soften, ultimately going to zero frequency, followed by a static charge distortion, i.e., a charge density wave. In second order perturbation theory one is in- terested in the dynamic dielectric matric e~.l(q, q +

G, w), where G is a reciprocal lattice vector.

While an optical anomaly has been observed in the oxides near 0.4 eV in YBa2Cu307_ 6 the peak disappears in single crystal materials and is appar- ently not related to the pairing interaction (9). Sug- gestions that charge fluctuations between Cu ++ and O - - are responsible for high Tc have been made (9), however no direct evidence exists at present for such a mechanism.

Since the above topics have been discussed exten- sively in the recent literature, for further information concerning the phonon and charge fluctuation mech- anisms, the reader is referred to the literature (4).

2.3 Spin Fluctuations

La2CuO4(214 ) and YBa2Cu306(123 ) have been observed to be commensurate antiferromagnets with large Nell temperatures (10,11). The ordered mo- ment is found to be of order 1 ~#B. On doping of the 214 compound with Sr and the 123 compound with added oxygen, T N falls to zero and superconductiv- ity onsets at still higher concentration. Neutron and Raman (12) scattering studies show that strong 2d

antiferromagnetic spin correlations extend into the superconducting phase with a spin-spin correlation length L falling from ~ 200/~ above TlV in the un- doped 214 material to ~ 10 - 20/~ in the supercon- ducting phase. These observations have stimulated a great deal of theoretical work on the relation be- tween antiferromagnetic correlations and supercon- ductivity.

As mentioned above, the appropriate language for describing the electronic structure depends on whether U < W (itinerant limit) or U > W (localized limit). Here we consider U < W and discuss U > W in the next section.

In the band picture, the valence band of these materials is in essence a partially filled antibonding Cu-dx2_y2 and Op~ band. For La2CuO 4 this band has one electron per primative unit cell (i.e., con- taining one Cu atom). Band structure calculations (13,14) show that the Fermi surface has roughly a square shape with large portions nesting with other portions when translated by the nesting wave vector

~r ~r 0 Qo = (~, ~, ) in the CuO 2 plane. Because of this nesting the system is unstable with respect to spin density wave (SDW) formation with the SDW period being locked to the lattice period by the commensu- rability energy. Mathematically, this locking is due to the coherent addition of direct and umklapp scat- tering of electrons from the spin and crystal lattices. The SDW opens a gap ASDW, k at the Fermi surface, with ASD w presumably being nodeless. This is con- sistent with the insulating behavior of the undoped materials. As holes are added to the filled negative energy states below the gap in La2CuO4 the SDW period remains locked at (a, a, 0) and a positive Hall coefficient is observed. Since doping pulls the Fermi surface away from the nesting surface, the SDW amplitude and TN fall until T N vanishes at doping x ~ 2 - 3%. This remarkably sensitivity of T N to x suggests that a single hole decreases the antiferro- magnetic order over a region surrounding the hole, i.e., the hole has an extended antiferromagnetic form factor.

Since strong antiferromagnetic spin fluctuations exist in the 2d superconducting CuO 2 planes, it is reasonable to investigate the pairing interaction which occurs in second order perturbation theory on ex- change of a spin fluctuation. In analogy with the charge case, this involves the spin susceptibility x(q, w) (15). Since x(q,w) is large for q ~ Qo, one expects a strong pairing potential for large momentum trans- fer. However, x(q, 0) is positive (repulsive)for all q so that a nonzero superconductor order parameter ASC,k is expected only when this quantity changes sign over the fermi surface. This is what is found in studies of the pairing gap equation, i.e., d-wave spin zero pairing is found to be the most stable conden- sate (15). It appears that Tc is likely to be low in this mechanism. Other models favor p-wave S = 1 pairing.

J.R. Schrieffer et al. / Theoretical situation in high-T~ superconductivity 23

A nonperturbative approach to pairing involv- ing 2d spin correlations is the spin bag scheme (16). Here one takes into account the SDW gap, or rather pseudo gap in the absence of long range spin order, in zero order. Thus, relative to midgap, the energy required to insert a hole without local relaxation of the SDW in ASDWk. However, the presence of the hole pulls the local fermi surface away from the nesting condition and thereby reduces the amplitude of the SDW and the order parameter ASD w in the vicinity of the hole. As opposed to an antiferromag- netic polaron for which the spins are aligned ferro- magnetically in the region surrounding the hole, the spin bag remains antiferromagnetic inside the bag. In addition, the bag is formed by a reduction of the band structure nesting while the spin polaron arises from an exchange interaction between the hole and the localized Heisenberg spins, an effect independent of band structure effects.

Since ASDW(r ) is reduced inside the bag the en- ergy to create a hole is also reduced. The bag shape can be estimated by a variational calculation for a static hole or by a diagram scheme in which fluc- tuations of the SDW order parameter around its mean field value are taken into account. One finds the bag to be cigar shaped, its long axis being of length ~SDW = V F / ~ A s D w where V F is the Fermi velocity, while the transverse dimensions are of order the lattice spacing. Since ASD w is reduced in the bag, a second hole will be attracted to the first hole by sharing the region of depressed A. This attractive potential appears to lead to a nodeless gap, i.e., s- wave singlet pairing and potentially a large value of To.

3. STRONG CORRELATION LIMIT

If the repulsion U between electrons located on the same site is large compared to the band width W -~ St, the electrons are in essence localized if there is one electron per site. There remain the spin degrees of freedom which are coupled by an effec- tive nearest neighbor antiferromagnetic interaction J ~ - t 2 / U . Anderson proposed that in the square CuO 2 planar lattice, the spins are described by a resonating valence bond (RVB) state (17) in which pairs of spins of varying separation are coupled to angular momentum zero, with all possible pairings admixed. While antiferromagnetic long range order is observed in La2CuO 4 and YBa2CuaO6, doping reduces T N to zero and only finite range spin corre- lations exist. Anderson proposed that this phase is well described by the RVB state.

Starting from the Hubbard model with one elec- tron per site it is argued (18) that the low lying elec- tronic excitations are of two types, holons of charge -t-e and spin zero (bosons) and spinons of charge zero and spin 1 (fermions). Since only holes, that is a holon plus a spinon, can hop between planes, it is proposed that the superconducting phase is de- scribed by a pairing condensate of holons since the

accompanying spinons couple to spin zero and can be viewed as being absent. At present it is found that the holons repel each other in the plane and the pair- ing is stabilized by the interplanar hopping. While the theory is able to account for a number of features seen in the experimental data (19), such as the large linear specific heat "/T and the linear electrical resis- tivity A T in the normal phase, Tc would appear to be small and depend sensitively on the interplanar hopping, contrary to experiment.

There are a number of alternative formulations of the large U theory. By writing a trial wavefunction in the form

= Pd~

where Pd projects out any admixture of doubly oc- cupied sites, one can vary the form of • to minimize the energy. Rice et al. (20) have found that taking

to describe either a spin density wave or a pairing type state leads to comparably low energies, d-wave superconductivity is found to have the lowest energy.

Starting with the large U limit, several authors have investigated the interaction between a pair of holes. Shraiman and Siggia (21) find binding with an energy of order t only in p and d states. Alter- natively, Trugman (22) finds essentially no binding because of a localization of a pair of holes due to a many-body frustration effect.

Laughlin (23) has argued that the holons of the RVB theory obey fractional statistics. This leads to an effective attractive interaction between these excitations when they are viewed as fermions. One knows that particles of a given statistics can be rep- resented in 2d as particles of another statistics each carrying a pseudo magnetic flux tube perpendicular to the plane, the flux being proportional to the dif- ference of statics.

Recently, the role of topological excitations has been investigated by Dzyaloshinskii, Polyakov and Wiegmann (24).

4. DISCUSSION

The present theoretical situation in oxide super- conductivity remains one of intense activity with many new ideas and new results appearing. The discovery of antiferromagnetism in La2CuO 4 and YBa2Cu306 and the behavior of spin order with doping into the superconducting phase has focussed interest on spin excitations as a likely candidate for causing superconductivity.

The description starting from the small U / W and large U / W limits are at first sight totally different. However, antiferromagnetic spin order in the small U case leads to a gap at the Fermi surface which plays a similar role to the Mo.tt Hubbard gap. Spin waves of the SDW are analogous to spin waves of the Heisenberg antiferromagnetic of the large U regime.

24 J.R. Schrieffer et al. / Theoretical situation in high-T, superconductivity

Anderson's RVB state (19) as presently formu- lated is different from the small U theory in that a hole splits its spin and charge degrees of freedom into separate excitations. This is just as in the lin- ear polymer polyacetylene where it is known that an injected hole decays into two solitons, separating the spin 1 from the charge +e (25). This issue is in- t imately related to the nature of the ground state. For a state exhibiting local antiferromagnetic order, it would appear to be favorable for the spin 1 of the hole to remain bound to the hole since the exchange energy is likely minimized when the spin 1 is in a re- gion of lowered spin coordination number. Hopefully, experiment will be able to probe the spin order in the doped oxides and determine whether the RVB or local antiferromagnetic order provides a more accu- rate description of the system and whether the spin and charge degrees of freedom of injected holes are in fact separated in space.

While a great deal of progress has been made in our understanding in the year since oxide super- conductivity was discovered, the critical experiments which unambiguously determine whether spin, charge or lattice vibration degrees of freedom dominate in determining Tc remain to be done. Also, a clear ex- perimental measure of the interelectronic charge cor- relation (Mott Hubbard gap) as opposed to the spin correlation (SDW gap) ' remains for the future. Pos- sibly the oxide materials are in the intermediate cou- pling regime U / W ~ 1 where these distinctions are blurred.

Numerical studies on finite size lattices (26) are addressing situations which begin to approach those involved in these oxides, where the coherence length ~ab in the plane may be only 4-5 lattice spacings. If the excitations are complex, like the holon, spinon or spin bag, it will be necessary to sample the Monte Carlo data in special ways to bring out the physics in the problem.

Finally, high Tc has provided the theorist a mar- velous new arena in which to develop concepts and methods which span the major areas of condensed matter physics, insulators, semiconductors, metals, magnetism and superconductivity. The continued discovery of new materials promises to keep the chal- lenge alive for some time to come.

This work was supported in part by the National Science Foundation, grant DMR85-17276.

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