Electronic Structure of Organic Superconductors

D. D. Koelling

Materials Science Division

Argonne National Laboratory

9700 South Cass Ave.

Argonne, IL 60439-4845

ABSTRACT

An attractive feature of organic superconductors is their potential tailorability. However, the detailed consequences of structural variations and atomic substitutions are not well understood. Ab initio electronic structure calculations can begin to provide such understanding. Self-consistent local density functional theory calculations , within the pseudopotential framework, have recently been performed by Benedek , et al., for the electronic structure of layered organic superconductors derived from the ET molecule (See Fig.1). The simplest such system

, which contains 55 atoms in a triclinic unit cell, is already highly complex, and represents a formidable computational challenge. Calculations have also been begun for the monoclinic

, which has a unit cell twice as large. The size of the (electronic orbital) data for the latter system has driven the calculation from the Cray C-90 to the Thinking Machines CM-5 even more than the CPU-time requirements.

PROLOG

In selecting the work to present at this meeting, I have somewhat reluctantly decided that it is of greater utility to discuss some work in which I am only very tangentially involved rather than that which is my most direct concern. (Not able to resist at least a brief mention, my more immediate interest deals with the question of relativistic effects in those high Tc superconductors where one or more of the constituents has a large atomic number. Such problems easily qualify for power usage status: The crystal structures of these materials already dictate large computational effort so when one fully incorporates relativistic effects--most calculations do not--the already very large secular matrices are doubled in size and can no longer be made real. The result is that the calculations are about 30 times more computationally demanding. However, these calculations are sweethearts for the system: They involve lots of calculation without serious demands for memory or system storage.)

Instead, I will here discuss work done within our group by R. Benedek, L.H.Yang, A.P.Smith, and M. Minkoff on organic superconductors. The science is exciting but requires that resources be pushed hard, especially in the area of memory availability. These calculations deal with very large internal datasets which modify traditional expectations focussed on the power of the processor. Consider that a MIPS R4000 based machine runs at about 6% the speed of a Cray 2 for these problems. For this problem, one can exploit virtual memory without undue disk thrashing if about a quarter of the problem can be kept in real memory. Thus, one achieves about 1/30 a Cray 2 AT FULL MEMORY if the workstation is configured with 200Mb memory and 600Mb swap space. Note that the slower "memory" access does also slow the calculation down. Nonetheless, one has to bid with a very expensive (low) nice value to get large memory--and certainly not the whole memory--on the Cray 2 and access it 1/30th of the time. The Cray C-90 greatly tips the balance, not only due to its faster processor but also due to its faster memory. The C-90 made the results to be discussed here feasible. But, on the other hand, it has only twice the memory of the Cray 2. In order to proceed with the next (kappa) configuration, one needs at least five times the memory. That memory is only available on a parallel machine--neither the workstation nor the Crays can offer it. The kappa configuration requires about 1.5 Gigawords (6Gbytes) of memory which, under todays standard configurations, implies committing to using 128-256 nodes on the basis of memory considerations alone.

INTRODUCTION

Carbon-bearing superconductors are an emerging field with considerable promise. The alkali doped fullerenes have, for example, recently received much attention. The more traditional organic superconductors offer more intriguing possibilities for specific tailoring since organic chemists can build the system to specification. ie, given a hypothesis that a particular variation of the system will have a salutary effect, one can expect that it will eventually be achieved. The problem lies in the fact that a detailed set of hypotheses is lacking (we don't know what to ask for). Consequently, a detailed understanding of the electronic structure of the known organic superconductors can have far reaching effects by providing clues to what features should be enhanced. Whangbo[1] has extensively studied these systems using Extended Huckel Theory. Using this rough theory, he has been able to consider many aspects of the problem relatively quickly. One of the motivations of the current study was to determine just how well that theory does. The Augmented Spherical Wave method[2] has also been applied to the material but suffers from using an overlapping atomic sphere approximation in this highly anisotropic system.Further, to keep the calculation within range, a further approximation was made that identical potential factors were used on inequivalent atoms. Recently, a Linear Combination of Atomic Orbital (LCAO) calculation[3] has been reported. Such a calculation would have restricted variational freedom compared to those discussed here but would have the advantage of simplifying the discussion of the wavefunction properties. (The authors indicate drastic differences between their calculations and the more empirical schemes which implies the need for further scrutiny before proceeding). For the calculations being discussed here, R. Benedek, L. H. Yang, A. P. Smith, and M. Minkoff have applied a plane-wave pseudopotential technique which has the variational freedom to account for orbital polarization and interstitial charge build up that would not be present in the LCAO calculation. This, unfortunately, is at the price of greater computational effort and less direct interpretation of the wavefunction data.

METHOD

The calculational technique is a plane wave basis set applied with norm conserving pseudopotentials[4] (the pseudowavefunction has the same integrated charge within the range of the pseudopotential as the all-electron wavefunction). The (non-linear)variation of the total energy to determinethe wavefunctions and charge density was performed using the Teter-Payne-Allan[5] band by band conjugate gradient algorithm. This was modified to incorporate the charge density mixing of Benedek, et al [6] since it is necessary to stabilize the self-consistency search because of the metallic character of these materials. Because many of the atoms are quite light (low atomic number), the pseudopotential must be strong forcing the plane wave energy cutoff to be made quite high: 60 Ry. This implies about 45000 planewaves for the simpler

material and many more for

. The optimization techniques employed require frequent evaluations of the potential multiplying the wavefunction so they must be done quickly. Fast Fourier Transforms are used to this end and thus are the major critical kernel of the calculation. For

, a 32x45x72 grid was used while for

, the grid had to be extended to a 64x64x128 grid to accommodate the greater structure in the unit cell. It is to be noted here that the large number of basis functions and the large grid size imply very large amounts of data which must be kept available. The Brillouin zone (reciprocal space unit cell) was sampled at only 4 points to achieve the self consistent results. While this is an extremely small number for a metal, it is hoped that the large number of atoms, and thus small Brillouin zone, would make it adequate. A standard technique of Gaussian broadening the eigenvalues by 10 mRy. was used. Test calculations using a single point were found to yield a density and potential that gave rise to similar band dispersion. Another consequence of the materials being metallic is that one must reorthonormalize states and reoccupy states at each step.

RESULTS

The simplest possible ET systems are containing halogen anions and occurring in the beta phase. Thus, the material chosen for initial study was

. In this material, superconductivity is significantly enhanced when a pressure of 1 kbar is applied to suppress the competing charge density wave. This geometrically simplest of the ET-based superconductors has "only" 55 atoms in the (triclinic) unit cell. Its 213 valence electrons arrange themselves into 106 filled bands plus the half filled band 107. At least, that is what is found by both the EHT and pseudopotential calculations. (The ASW results do not agree but remember that additional severe approximations were applied in that calculation.) Shubnikov-deHaas[6] and tilted field magnetoresistance[7] data strongly support the simpler picture of the EHT and pseudopotential calculations. Although the measurements were actually taken on a material with a different anion, the anion has relatively little effect on the Fermi Surface. Of course, the next question is how do the EHT and pseudopotential calculations compare? The answer is seen in Fig.2. The current SCF calculation and the EHT calculation agree fairly well although the SCF calculation has a slightly wider bandwidth and accordingly smaller electronic mass. Also, the band of the SCF calculation more closely approaches the Fermi energy near the M point. This will be of some interest as photoemission results become available[8]. Fig.3 shows the charge density. The ET molecule (Fig.1) can be clearly discerned along with the charge of the anions in the plane below. The analysis of these results continues. The first issue is to get a more numerically rigorous alignment of the Fermi energy --- an "engineering detail" that is important so that reliable numerical masses can be determined and enhancement factors estimated. Next is to dissect the wavefunction character associated with the band that crosses the Fermi energy. Conventional wisdom is that the in-plane conduction proceeds as sulfur to sulfur hopping. It will be useful to take a close look at that in this model as well as the distribution of the state over the ET molecule. One would like to know about how much polarization effects are influencing the behavior of the state --- expected to be small due to the close agreement of the EHT and SCF calculations --- and whether it can be represented by simple models amenable to further manipulation.As photoemission measurements[9] become available, not only will we learn more about the anisotropy of the Fermi surface but gain information about relaxation and correlation in these systems through comparison to calculations. There is a rich field of questions that can be explored by performing further calculations within this structure but varying the anions since, although the Fermi surface will probably not be found to vary greatly, the superconducting properties do vary. However, the real touchstone will be to exploit a very different structure. Since one is working at the limits of computational ability, the choice is dictated: the next simplest structure is the k- phase. This variation drastically changes the positioning of the ET molecules so the similarities and differences can be insightful. The calculations are well underway but would be premature to discuss as this time other than in terms of their resource requirements.

THE NEXT HURDLE(S)

As stated in the prolog, the C-90 enabled the calculations on the b- phase. However, progress is slow because 40Mw jobs traverse the regular queues very slowly. Although the codes have been converted to CM-Fortran and F90 and parallelized for other machines, a parallel realization for the C-90 has not been created. Thus, the SPP program does nothing for these calculations. (Another problem of concern for these calculations is that they want only a sixth of the memory and the multitasking is not fully operational in the multiprocessing environment.) However, SPP could prove a useful accelerator for further work on the b- phased materials. These calculations are arduous, but feasible, and the more critical development effort is in the area of interpretation.

The situation is quite different for the k- phased materials. Such calculations require 1.5 Gw of memory (the largest memory available on a C-90 is 1 Gw at a doubling of the cost of the machine). It is this requirement even more than the processor time which drives the use of a parallel machine. To that end, the code has been converted to CM-Fortran and parallelized as guided by CMAX. It has run on several Thinking Machines computers but is now focussed on the CM-5. The first major task to accommodate to the parallel architecture is reorganization of the data distribution. As mentioned, the memory associated with between 128 and 256 nodes is needed to run the job. The number is larger than a simple division of the memory between nodes. This is, in part, because one needs to expand memory usage by about a factor of three to operate efficiently as a parallel application: so much for seamless computing!. On the bright side, this is not a bad match of node count for appropriate processor effort thereby giving some level of balance. The next step is efficient implementation of the two critical kernels: FFT's and eigensystem analysis; both of which one expects to get from program libraries on any production level machine. A more memory and computationally efficient scheme could be achieved with differently organized FFT routines. These will come with greater maturity of libraries but are inappropriate efforts for the application programmers (ie physicists). Thereafter, one has to deal cleverly with different features of the problem than on a serial implementation: Global masked sums and global dot products being the natural examples.

Another aspect which sneaks into the picture is I/O. The I/O is significantly slower on the parallel machines. Since the restart file is about 0.5 Gw in size, this is a serious issue. On the C-90, a restart file is written after each iteration. On the parallel machines, that is too expensive and one chooses to write that file only every n'th iteration. That leaves one with a lot of good work exposed and unprotected for much longer times.

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