CRITICAL OVERWIEW OF THEORIES FOR HIGH-Tc SUPERCONDUCTORS

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CRITICAL OVERWIEW OF THEORIES FOR

HIGH-Tc SUPERCONDUCTORS

M. Cyrot

To cite this version:

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Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988

CRITICAL OVERWIEW OF THEORIES FOR HIGH-Tc SUPERCONDUCTORS

M. Cyrot

Laboratoire Louis Ne'el, C.N.R.S., 166 X , 38042 Grenoble Cedex, France

Abstract. - Theoretical models are discussed which have been proposed for the new high-Tc superconductors. Special attention is paid to electron-correlations in those systems. Their consequences for the superconducting state are reviewed critically.

Since the discovery of the copper oxide superconductors, hundreds of theoretical papers have appeared to propose ideas to explain their properties. I will try to make a critical overview of some of them. Other attempts already exist in the literature [I-31. The materials under discussions belong to the class of ionic perovskite and, except the last discovered materials Ba K Bi 0 3 , they all present CuOz planes stacked in

different manner. The number of different stacking which make different families and the possible sub- stitution in a particular family permits now to have general results not restricted to a single compound. After a first burst of theories which presents any kind of superconductivity, two clear experimental evidences make probably a loose framework. The first one is the flux quantization which clearly indicates that pairing of electrons are basic to these new superconductors. The second one is more constraining: the general phe- nomenological Ginsburg-Landau theory well explained the different results in the limit of high Kappa value. The observation of fluctuations near Tc are perfectly expected within this theory due to the very low value of the coherence length. Any theory must fit to this loose framework. The experimental properties of the non superconducting state (above Tc) or the effect of alloying or doping should strengthen this framework. However except some undisputed properties, many results are controversial and different groups are opposed. The first basic question, which seems in principle so easy to answer is the behaviour of the non superconducting state: is it a normal metal? This apparently simple question has no definite answer [4] but the answer would probably help very much to distinguish between theories.

The magnetic properties of the parent compounds are now well established and a good understanding of them, will probably clarify the situation about the importance of correlations. If it turns out that an Heisen- berg model is appropriate to describe them, it will be difficult not to consider correlations in the superconducting compounds. Another well established fact is that the transport properties is due to holes except perhaps for the 1 2 3 compound in the c-direction. This also h a important consequences.

As the superconducting state is concerned, at the moment of writing, we believe that it favours an usual B.C.S. state with a a-wave symmetry contrary to most model taking into account correlations. But this has to be confirmed.

1. T h e electronic s t r u c t u r e

The first basic ingredient of a theory is the, some- times implicitely assumed, electronic structure. De- tailed band structure calculations have appeared [5-'71: the basic structure of a CuOz plane is copper atom at the corner of a rectangle or a square and oxygen in the middle of the edge. Assigning a formal charge of -2 to the oxygen atoms and of +3 to Y or La leads t o copper two plus in the reference compounds i.e. a d9 config- uration or one hole in the d shell. This hole must be in the highest band. As the x2

-

y2 orbitals point to- wards the oxygen atoms, this band will be made from the antibonding combination of the Cu x2

-

y2 orbitals and p, or p, orbitals of the neighboring oxygen atoms. More generally in a rough approximation, we have a CuO bonding , a d antibonding band and an 0 non bonding band. However these ab-e'nitio calculations relie on more or less hidden approximations and are not able to predict an insulating state for these types of compounds except if an antiferromagnetic state is assumed. In short, correlations are not well taken into account if they are strong which is the next point that we will discuss. Secondly it is very difficult to han- dle ionic terms and the position of the different band are not so obvious. For instance some authors assume that ionic contribution can increase the energy of the 0 non bonding and that doping will lead to holes in this band. The location of holes is a controversial problem. Thank to the simplicity of the tight bonding method which now can be made as rigorous as abinitio calculation, it is possible t o study the repartition of holes in d-orbital of copper and the p-orbital of oxygen as a function of various parameters [S]. However important correlations are not taken into account in any of these calculations. A Gutzwiller approximation seems to localise holes on the oxygen sites [9].

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C8 - 2216 JOURNAL DE PHYSIQUE

2. Magnetism of parent compounds ing Cu-0 states. They are divided in two groups de- The antiferromagnetic behaviour of LazCu04 have

first been observed, then that of YBazC~306 [lo-131. It seems that when we have only cuzf in the plane, one obtains antiferromagnetism. This has not been observed in other compounds with Bi or T1. However copper has a valence which is greater than two although the given ,formula presupposes valence two. This has been first attributed to other oxygens not seen in the experiments but are more likely due to an under stoe- chiometry in strontium. One has to wait to obtain only copper two plus in the families and to search for magnetism. For the two families where magnetism has been observed, there is a controversy whether we have a magnetic Mott insulator or an itinerant band slater antiferromagnetism. In an ordinary band theory of a two dimensional square lattice, we would have perfect nesting and any small value of Coulomb interaction would give antiferromagnetism. We now list some experimental evidences which seem to us to favor the description by a Mott insulator:

i) the observed magnetic fluctuations above

TN

seem well described by an antiferromagnetic spin one half Heisenberg Hamiltonian;

ii) the small value of the magnetic moment some- times given as an indication of an itinerant character is well explained in an Heisenberg model by the quantum fluctuations;

iii) the value of the magnetic moment does not seem to depend on the doping although the N6el temperature decreases very strongly;

iv) it seems to exist a spin glass phase in between the antiferromagnetic and superconducting ones.

We also stress that usual theory of lattice of Jahn- Teller ions [14, 151 when applied to LazCu04 predicted antiferromagnetism [16]. This theory is based on large value of correlations. Moreover, the same theory predicted in pIane ferromagnetism for three dimensionnal copper perovskite. This can be an explanation [17] of the necessity of having materials with CuOz planes in order to obtain superconductivity if superconductivity is due to the antiferromagnetic interaction.

The question remains whether the itinerancy of the added carriers t o obtain superconductivity will screen the Coulomb correlation. We do not expect such a drastic reduction. Thus we believe that for the moment strong correlations are more likely. This would restrict the framework of a theory for superconductivity. However no such consensus exists. Thus we have to classify theories accordingly to the band structure they presuppose and the value of correlations they assume.

3. The different classes of theories

Most theories assume that holes are in the antibond-

pending on the importance they put on correlations. If correlations are assumed t o be small, one starts from an usual metal or a Fermi liquid description of the normal state. Then superconductivity can be due either to phonons [1&21] or excitons [21-241 or plas- mons [25] or magnetic interactions [26-301.

In the other limit, strong correlations [31, 35-43] we have the description in terms of the resonance valence bond model with weakly interacting excitations called holons and spinons. Superconductivity is due either to Bose-Einstein condensation of holons or interplane coupling between holons. There exists another description starting from the Mott insulator that we will call superexchange superconductivity where superconductivity is due directly to the antiferromagnetic interaction.

Other theories assume holes in the non bonding oxygen states [53-561.

3.1 THE MAGNETIC MECHANISMS IN THE WEAK COR- RELATIONS LIMIT [26-301.

-

Some authors start from the fact that an appropriate change in the composi- tion of the oxide superconductors turns them into an antiferromagnet. It leads to think that residual antiferromagnetic correlations in the metallic state can pro- mote superconductivity. This idea has been already invoked in heavy fermions [31-341. We discuss now only those papers in the small U limit i.e. which pos-

tulate that the undoped compound is a Slater itinerant antiferromagnet. The simplest version of the idea is to exchange spin fluctuations instead of phonons to pro- duce attractive interaction between electrons near the Fermi surface. These theories predict a d-wave symmetry superconductivity except the spin bag mechanism proposed by Schrieffer et al. [26].

The spin bag mechanism is a way t o obtain usual superconductivity. The authors assume that spin order- ing produces an electronic pseudogap which is locally suppressed by the addition of a hole. This suppres- sion forms a bag inside which the hole is trapped, two holes are attracted by sharing a common bag. How- ever this mechanism seems rather general and it is difficult to understand why it works only for CuO2 plans. Secondly the spin density wave have to persist in the superconducting state. thus above

T,,

one should see magnetic fluctuations which are not clearly observed for the moment.

3.2 THE LARGE U LIMIT. - In the large U limit, we will

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change superconductivity. In this approach superconductivity is directly linked to the superexchange interaction. This seems to be the original idea of Anderson but in a mean field approch, this was first proposed by Cyrot [38] and by Baskaran et al. 1391. This head- ing considers all the developments of this idea using various technics, the mean field approach [38, 391, the slave bosons one l40, 411, the functional integrals one [44], the variational one [45] with the wave function proposed in the Anderson's original paper.

The R.V.B. Model. - The large U Hubbard model was first proposed by Anderson [31]. He hypothesised that the insulating phase of the pure LazCu04 to be the resonating valence bond state. The ground state of a two dimensionnal Heisenberg antiferromagnet would not be the Nee1 state due to quantum fluctuations. On the contrary, the wave function would be a linear combination of product of pair wave fonction of singlet:

It turns out that pure LazCu04 is an antiferromagnet and that it is now believed that the ground state of the 2 dirnensionnal Heisenberg model is a N6el state even for the triangular lattice [46-481. However this could be the ground state of the doped Heisenberg model as holes will a priori favor two spins to make a singlet. Kivelson et al. [35] argue that the excitations of the R.V.B. state are holons or charged bosons holes with- out spins and spinons or neutral fermions. It was first assumed that the holes can Bose condensate and make a superconductors but later Anderson [37] proposed that interplane coupling would bind holons into pairs and this would be the reason of superconductivity.

Arguments in favor of the R.V.B. state or new type of quantum liquid were given through the explanation of the properties of the normal state. Hall effect and thermopower show that the charge carries are holes and proportional t o the doping. The resistivity is highly anomalous, linear in temperature on decades of temperature for in plane resistivity and p along the c-axis would be in 1 / T . The linear specific heat would be also the signature of a R.V.B. state. However the experimental results are not so sure. The resistivity along the c-axis becomes increasingly parallel to that in plane for better sample. The linear specific heat coefficient seems to bear little relation with Tc. In a theoretical point of view, the calculation of the resistivity due to holons spins scattering yields T~~~ in a 2-dimensions rather than T which were thought.

Superexchange superconductivity.

-

One starts from the Hubbard hamiltonian which in the large

U

limit is

an antiferromagnetic interaction:

The original papers [3&39] relies on two approximations a mean field one for the S+S- term of the Heisen- berg interaction and a Gutzwiller approximation for the kinetic energy.

This Gutzwiller approximation is equivalent to the mean field approximation done by authors 140-431 using the slave bosons technics. Both approximations for the kinetic energy replace a local constraint by a global one. This Gutzwiller approximation leads to some unphysical results. For instance, there leads to four states per site contrary to the physical results which would give only two as the double occupancy is forbidden as long as T

<

U. Moreover the suscep- tibility is wrong by a factor of two. The predicted Hall effect is negative contrary to the intuitive feel- ings considering the Hubbard subband pictures. Thus we think that this approximation breaks down because the number of current carrying states with low energy is 6 and not l+6. Using these two approximations, one finds d-wave superconductivity, also we have recently argued f8-491 that if the density of states of current carriers are going t o zero for 6 = 0 , one can find s-wave superconductivity with B.C.S. like properties.

The variational approach has been developed mainly by Rice and co-workers 1451. They start from An- derson's proposal. The wave function is a projected B.C.S. wave function (or a R.V.B. wave function):

where the operator Pd projects on not doubly occupied

sites:

P d = I T i ( 1

-

nitnil).

The projection operator makes difficulties and a Gutzwiller approach is taken wich introduces just a weighting factor in the expectation value. Thus the particle-particle amplitude is:

Gt

being the Gutzwiller narrowing of the band.

Gt = 26/1

+

6. Rice and coworkers argue that when

+ +

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C8 - 2218 JOURNAL DE PHYSIQUE We emphasize that this kind of approach must be

distinguished clearly from the RVB model although in the literature it is not because they use Anderson proposal for the wave function known as the R.V.B. wave function. The origin of superconductivity is directly in the exchange interaction and not in a Bose condensation of holons or in pairing of holons. How- ever the use of this wave function is an assumption on the description of the doped Mott insulator in the non superconducting phase. One can argue against this approach that superconductivity is obtained by hand either by a B.C.S. like variational wave function or a B.C.S. Gorkov linearisation.

Before concluding the subject, we want to raise the following question. Does a Hubbard hamiltonian per- mit to describe superconductivity? This is a two folded questions. First do the physics necessary t o explain the high T, materials is containing in this simple model? and second does such an hamiltonian lead to a superconducting ground state?

The first answer is related to the second one because some authors think that it is not possible to obtain superconductivity in such a repulsive hamiltonian. Nu- merical calculations [50-521 seem to go in that direc-

tion. However expansion in the large U limit or small

value of t seem to indicate possibility of superconduc- tivity. We think that a limiting factor is that the antiferromagnetic coupling is linked t o t and U. In a more general model, where the antiferromagnetic coupling is given by another mechanism, superexchange via the oxygen atom of magnetic interaction via the conduc- tion band in the heavy fermions compounds such a limitation does not occur and superconductivity must be possible.

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