Computation of X-Ray Absorption
Multiple scattering (MS) theory is widely used to calculate physical properties of solids, ranging from electronic structure to optical and X-ray response. X-ray absorption spectroscopy (XAS) uses energy-dependent modulations of photoelectron scattering to probe excited states and thus is important for determining electronic and chemical information from X-ray spectra. XAS is usually divided into the extended X-ray absorption fine structure (EXAFS) with photoelectron energies above ~ 70 eV, and the X-ray absorption near edge structure (XANES) in the 0-70 eV range.
Theoretical calculations of photoelectron scattering are now an integral part of both EXAFS and XANES analysis. These theoretical calculations have grown in sophistication and complexity over the past twenty years. Fortunately computing power has increased dramatically (in accordance with Moore's law) during the same time period, and as a result EXAFS calculations are now fast, accurate and easily executed on inexpensive desktop computers.
We parallelize the X-Ray Absorption code because XANES calculations are extremely time consuming for many materials. The photoelectron mean free path is large at the low photoelectron energies of the XANES region, so accurate XANES calculations require large atomic clusters and remain challenging on even the fastest single processor machines. Furthermore, the photoelectron scattering is strong for low energies, so that full multiple scattering calculations are required. These calculations require repeated inversions of large matrices which scale as the cube of the size of the atomic cluster. Fortunately, parallel processing using the message passing interface (MPI) standard, combined with modern Lanczos type MS algorithms, can speed real-space XANES and electonic structure calculations by about two orders of magnitude. In particular, FEFF, one of the most commonly used programs for XAS analysis (developed at the University of Washington) has been improved in this manner, leading to a parallel version, FeffMPI.
Additional Technical Details:
A series of similar MS calculations must be done at a large number (typically of order 100) of energy points to obtain a complete XANES spectrum. This number is determined by the natural energy resolution (due to lifetime broadening and inelastic losses) and the range of the XANES region (typically below 30 eV of threshold) for which full MS calculations are needed. Thus it is reasonable to consider doing these similar MS calculations in parallel. Since we aim to model the physical process of x-ray absorption, it is natural to exploit the intrinsic task parallelism (or physical parallelism) in this problem, namely that the x-ray absorption at a given x-ray energy is independent of the absorption at other energies, assuming they are separated by the inherent energy resolution (typically a fraction of an eV). Thus a natural way to parallelize the full spectral calculation is simply to distribute the energy points among an available set of processors. We can then assemble the results to obtain the full absorption spectrum.
To this end we have developed a parallel version of the ab initio full MS XANES code Feff8 using the Message Passing Interface (MPI) standard. This parallel code (here dubbed FeffMPI) compiles and runs without changes on all currently available operating systems tried to date (e.g., Linux, Windows NT, Apple Os X, IBM-AIX, SGI, CRAY ....). The starting point for parallelizing any code is to determine which parts of the calculation are the most time consuming. Profiling tests showed that only a small section (about 100 lines that call matrix inversion routines) of the code accounted for about 97% of the total runtime in typical calculations. Altering these calculations to run in parallel is straightforward with MPI, because each step involves similar calculations and utilizes identical matrix inversion routines.
FeffMPI runs on PCs running Windows and Linux as well as most commercial UNIX vendor machines. The list of machines supported has recently been extended to include the Apple Macintosh running the new OS X operating system. FeffMPI is now also operating on parallel processing clusters at the University of Washington and at DoE's National Energy Research Scientific Computing Center (NERSC). In fact, FeffMPI compiles and runs without changes on all currently available operating systems tried to date (e.g., Linux, Windows NT, Apple Os X, IBM-AIX, SGI, CRAY ....) with speedups of up to 30. A speedup of 30 makes it possible for researchers to do calculations they only dreamed about before. One NERSC researcher has reported doing a calculation in 18 minutes using FeffMPI on the NERSC IBM SP2 cluster that would have taken 10 hours before. In 10 hours this researcher can now do a run that would have taken months before, and hence would not have been even attempted.
With the improved efficiency of FeffMPI now in hand, it is feasible to carry out XANES calculations which otherwise would have been impractical. For example, a few days of calculations on a 48 processor Linux cluster can now complete a calculation that would take a year on a current single processor. Systems such as complex minerals, oxide compounds, biological structures and other nano-scale systems are obvious targets for this type of improved capability. The improved speed should be very useful, for example, for magnetic materials, which often have a large number of inequivalent sites of absorbing atoms, requiring many separate calculations to produce a full XANES or XMCD (X-ray magnetic circular dichroism) spectrum. Finally, the availibility of rapid calculations now permits closed loop fitting of XANES spectra both to physical and chemical phenomena.
As one example of these calculations, we show how XANES calculations can be used in the study of amorphous germanium (aGe). It is well known that the structure of amorphous tetrahedral semiconductors can be modeled well by an approach called a continuous random network (CRN). In this method, the amorphous semiconductor retains the parent structure of the crystal, but various degrees of freedom, the interatomic distance, the bond angle and the dihedral angle, are allowed to become statistically disordered. Originally built by hand with ball-and-stick models, CRNs have been generated by computer, and the degree of disorder in the structural parameters is determined by energy minimization methods. Comparisons of CRN models with EXAFS data have been done, but these comparisons were not extended into the XANES region because of the inability to perform ab initio XANES calculations, and even in the EXAFS region the calculations were limited to a simple single scattering theory. Here we show that we can use FeffMPI and a CRN model to reproduce the main features in the XANES of crystalline and amorphous germanium.
As a starting point for the XANES calculation of aGe, we first modeled the XANES of crystalline germanium to determine the cluster size needed to accurately reproduce the XANES. We found that a cluster of 87 atoms, which includes the first 7 coordination shells, out to a distance of approximately 0.78 nm is sufficient to reproduce the main features of the experimental data. The aGe XANES calcuations were then carried out using similar clusters of 87 atoms that had nearly the same size as the crystalline cluster, because the CRN yields a structure that has the same density as crystalline Ge, to within a few percent. In order to get a good ensemble average over the inequivalent sites of the CRN, we ran the same FEFF calculation over a total of 20 sites in the CRN. We tested the calculation on a single processor desktop machine, where a single run took approximately one hour. We then used a 16 processor cluster where each calculation took about 3 minutes. Using FeffMPI and this fairly modest cluster size thus reduced the total calculation time from what would have been 20 hours on the desktop machine to 1 hour, for a 20-fold improvement over the single processor desktop system and in agreement with our previous result that moderate-sized clusters (approximately 33 processors) typically give a 20-fold spped increase compared with an equivalent single-processor system, and up to a 50-fold increase compared with typical single-processor desktop systems.
In Figure 1(a) we show the 87 atom cluster used to calculate the XANES of crystalline Ge. In Figure 1(b) we show a similar cluster of 87 atoms of aGe from the CRN displayed with the same length scale. As shown in the figure, each cluster is about 1.5 nm across. In Figure 2 we show the full 519 atom cluster of aGe from the CRN with a typical cluster of 87 atoms highlighted in the interior. Although there are several hundred atoms in the interior of the 519 atom cluster that are fully coordinated by 87 atoms, we obtain an accurate ensemble average using just 20-30 atoms near the center of the cluster. The convergence occurs quickly since averaging over N sites includes 4N first neighbor atoms, 12N second neighbor atoms, etc. The disorder in the CRN is large enough that the separation of the neighboring atoms into separate coordination shells breaks down by the third or fourth shell.
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