Where is the Cooper Force in High Temperature Superconductors?

Temmerman W. M., Szotek Z., Gyorffy B. L., Andersen O. K., Jepsen O.

Daresbury Laboratory University of Bristol;
Max-Planck-Institut für Festkörperforschung, Stuttgart

Superconductors are metals which can carry current without resistance below a certain temperature Tc. Until 1987 the technical usefulness of this remarkable property was limited by the fact that the highest Tc observed was only 23 K above absolute zero. Moreover, it had been a widely held belief that substantially higher Tc would be impossible. Thus it came as a great surprise when, in a dramatic succession of discoveries, materials with Tc as high as 150 K were found. Scientifically, these new superconductors (see cover) turned out to be spectacularly different from the conventional ones, and as such they attracted an enormous amount of interest. Their technological exploitation has however been frustrated by the lack of understanding of the basic physics which determines their behaviour.
As is well known, the phenomenon of superconductivity arises when in a metal electrons pair up and occupy a single quantum state. Since electrons normally repel each other, one of the principal questions in the case of any superconductor is "why do such Cooper pairs form?". Whilst for the conventional superconductors the answer is that the attraction is due to the electron-phonon coupling, in the case of the new, high temperature, superconductors, in spite of the unprecedented effort of the past nine years, the physical cause of the pairing remains a mystery.
Recently, we have proposed a new approach for making headway with the problem. With the aid of the Intel and IBM SP2 computers at Daresbury Laboratory, collaborating within an European Union Human Capital and Mobility Network (see box), we have developed a new strategy aiming to determine where in the crystal the Cooper pair has to be to experience the mysterious pairing force, without knowing its physical origin. It is based on Density Functional Theory, which is a commonly used first-principles method of describing quantum mechanical properties of solids. Its superconducting analogue, leading to the Bogolubov-de Gennes equations, involves the electron pairing interaction, the great unknown in the theory. Density Functional Theory differentiates between atoms and the s, p and d electron orbitals from which electrons forming the Cooper pair could originate. A natural follow-through of this is for the new strategy to express the electron pairing interaction in terms of the site- and orbital-dependent pairing constants. These pairing constants, treated as adjustable parameters, are determined by fitting to highly accurate data from such searching probes as e.g. photoemission or neutron scattering experiments.
What lends impetus to this semi-phenomenological approach is the conjunction of two fortunate circumstances. On one hand, a relatively small set of pairing constants describes a rather large class of mechanisms, and on the other, because these adjustable parameters are embedded in an otherwise parameter-free methodology, a very large number of observables can be calculated without further adjustable parameters. Remarkably, when a set of pairing constants has been identified as being consistent with a variety of experiments, it means that the orbital character and spatial location of the Cooper force has been determined without knowing its origin - a conclusion which could be of immense importance in describing its nature.
An important concept in the description of the electronic properties of solids is the Fermi surface. It is the constant energy surface in momentum space which separates the occupied one electron levels from the unoccupied ones. In a superconducting state this surface is associated with a superconducting gap. For normal superconductors this gap is isotropic over the Fermi surface. For the high Tc superconductors the gap is highly anisotropic. The Fermi surface and the superconducting gap can be 'seen' experimentally through angle resolved photoemission experiments. The accuracy required is immense, the equipment very specialised and expensive, limiting the experimental capability to only three groups world-wide, in Stanford, Argonne and Wisconsin. In spite of that, the available resolution is still not sufficient for the most crucial region, where the gap approaches zero, to make a definitive statement about the way in which zero is reached, if at all, which is vital to determine the gap symmetry.
With the strategy outlined above, we have been able to assign a specific pairing interaction to each of the different, experimentally observed, superconducting gap anisotropies. Through this we have been able to reproduce the richly structured photoemission data, and the observed transition temperatures seen by the Stanford and Argonne groups. In particular, the nearest neighbour copper d-copper d pairing interaction led to an anisotropic gap with a cusp (see Fig. 1), characteristic for the d-type symmetry. A more complex pairing, combining the nearest neighbour copper s-copper d interaction with the nearest neighbour copper d-oxygen p interaction, resulted in a more 'exotic' gap with a 'hump', as seen in Fig. 2. The excellent agreement with experimental data, originally published by the Argonne group, clearly illustrates the high potential of the present approach. Setting the nearest neighbour copper d-oxygen p interaction to zero caused the 'hump' to disappear, giving rise to an anisotropic gap with a minimum, in agreement with later results published by the same group.
Many cases have been studied by this approach, leading us to identify the nearest neighbour copper d-copper d pairing interaction as the most likely for YBa2Cu3O7. The results, a gap of d-type symmetry (seen in Fig. 1), and the power law behaviour of the calculated low temperature specific heat, agree well with the emerging consensus. It is still too early to draw any firm conclusions about the origin of the pairing. To resolve the issue more observables must be calculated-spin lattice relaxation rates and neutron scattering cross sections are particularly promising. Since no further adjustable parameters are required, this should decisively eliminate a number of contending interactions.
This is the first time such complex calculations of this sort have been carried out. While this is not a total solution to the problem of the superconducting mechanism, we believe that it is an important step toward elucidating the long standing and very complex theory of some of the most remarkable materials ever seen.

Fig. 1. Superconducting gap for the nearest neighbour copper d-copper d interaction, as a function of Fermi surface length for even (e) and odd (o) sheets of the Fermi surface.
Fig. 2. Superconducting gap for the nearest neighbour copper s-copper d interaction with the nearest neighbour copper d-oxygen p interaction, as a function of Fermi surface length for even sheet of the Fermi surface (left), in comparison with the experimental data from the Argonne group (right).

HPC Profile, 10 6-7, 1996.

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