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 b-ET2I3, 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 k-ET2I3, 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
alreadyvery 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 b-ET2I3 material and many more for
k-ET2I3 . 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 b-ET2I3 ,
a 32x45x72 grid was used while for k-ET2I3 , 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 b-ET2I3 . 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.
______________________________________________________
[1] M. H. Whangbo, et. al., in Organic Superconductivity, [ed] V. Z.
Kresin and W. A. Little, Plenum, New York, 1990.
[2] J. Kubler and C. B. Sommers, in The Physics and Chemistry of
Organic Superconductors, [ed] G. Saito and S. Kagoshima, Springer,
Berlin, 1990.
[3] W. Y. Ching, et. al., Bull. Am. Phys. Soc. 39,880(1994).
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[5] M. Teter, M. Payne, and D. G. Allan, Phys. Rev. B 40,12255(1989).
[6] R. Benedek, L. H. Yang, C. Woodward, and B. I. Min, Phys. Rev. B 45,
2607 (1992).
[7] W. Kang, et al, Phys. Rev. Lett. 62,2559 (1989).
[8] M. V. Kartsovnik et at, J. Phys. (France) 2,89 (1992).
[9] R. Liu, et al (unpublished).