The conventional approach to solving for the eigenstates of a Hamiltonian containing a mean-field potential is to minimize the expectation value of the Hamiltonian with respect to the expansion coefficients of the wavefunction (such as plane-wave coefficients). The first eigenstate obtained is then the lowest energy eigenstate of that Hamiltonian. To find higher states, one needs to orthogonalise each wavefunction to all the previously converged states. This orthogonalization process scales as N3, and therefore only small systems (no. atoms <100) can be studied. The central point of the folded spectrum approach is that eigenfunctions of the standard Hamiltonian, H, are also eigenfunctions of (H-Eref)2. This is illustrated in the figure below where the spectrum of H has been folded at the reference energy Eref into the spectrum of (H-Eref)2. Now the lowest eigenstate of the folded spectrum is the eigenstate with energy closest to Eref. Hence, by placing Eref in the physically interesting range , one transforms an arbitrarily high eigensolution into the lowest one, thus removing the need for the costly orthogonalization step. For example, if one places Eref within the band gap the lowest eigensolution of the folded Hamiltonian is either the valence band maximum or the conduction band minimum.
The code proceeds to solve for these few interesting eigensolutions of the folded Hamiltonian, by minimizing the expectation value of the folded Hamiltonian with respect to the plane-wave expansion coefficients of the wavefunctions. A conjugate gradient algorithm is used to perform this minimization.
Here we studied wavefunction localization in (AlAs)n(GaAs)m(AlAs)p... superlattices, where the periods (n,m,p....) were random. We discovered that disordering enhances the transition probabilities.
There has recently been considerable interest in the electronic, optical, transport and structural properties of semiconductor quantum dots. This interest follows from the rich, novel physical properties exhibited by these systems (Coulomb Blockade, quantum confinement, exchange enhancement and shape dependent spectroscopy) and their promise for applications such as lasers. We have used our FSM code,as implemented in ESCAN, to study the near-edge states of GaAs[4], InAs, InP[1,3] and CdSe free standing quantum dots. Such calculations have enabled us to obtain band gaps[1,3,4,6], transition probabilities[1,3], exchange splittings[4] and red shifts[3] for these dots.
Self Assembled or "Stranski-Krastonow" (SK) dots provide the possibility of using standard MBE and MOVPE techniques to manufacture high quality, dislocation free quantum dots with narrow (<10%) size distributions. The most popular system of such kind is InAs dots embedded within GaAs dots. Our solid state theory group is carrying out active reseach on the properties of such dots. In particulari, we are interested in the effects on the electronic structure of altering the size and shape of the embedded InAs dots or applying hydrostatic pressure to the system containing the dots.
This figure is taken from the front cover of the February 1998 issue of the MRS Bulletin. It shows the electronic structure of a 45-Angstrom high, 90-Angstrom base, strained InAs pyramidal quantum dot embedded within GaAs. The strain-modified potential offsets in a (001) plane through the center of the pyramid are shown above the atomic structure. They exhibit a quantum well for both heavy holes and electrons. These are localized within the pyramid and wetting layer, as shown by the blue raised (lowered) triangle and ridge(trough) respectively.
Isosurface plots of the 4 highest hole states and 4 lowest electron states, as obtained from semiempirical pseudopotential calculations are shown on the left and right. The lowest electron state (CBM) is s-like, whereas the next 2 states (CBM+1 and CBM+2) are non-degenerate p-like. Refer to the work by J. Kim, L.W. Wang, A.J. Williamson and A. Zunger. See also the article by A. Zunger in the MRS Bulletin.
Auger rates are calculated for CdSe colloidal quantum dots using atomistic empirical pseudopotential wave functions.We[13] predict the dependence of Auger electron cooling on size, on correlation effects (included via configuration interaction), and on the presence of a spectator exciton. Auger multiexciton recombination rates are predicted for biexcitons as well as for triexcitons. The results agree quantitatively with recent measurements and offer new predictions.
Our former group members, Andrew Canning(LBL), Andrew Williamson(LLNL) and L.W. Wang(LBL) ported the Folded Spectrum codes (PESCAN and LCBB) to the Cray T3E and later on to IBM SP Seaborg at NERSC. There were two main parts of the parallelization process. (i) Efficiently distributing the set of reciprocal lattice vectors across all the nodes so that operations performed in reciprocal space, such as calculating the kinetic energy can then be performed in parallel. (ii) Implementing Fast Fourier Transforms (FFT) that work in parallel so that the wavefunctions can be efficiently transformed between real and reciprocal space. On all the systems we have studied the code demonstrated excellent scaling with the number of processing elements used. This is illustrated in the figure below, which shows the total wallclock time and the inverse of this time for a nanostructure calculation.
The above system was stored on a real space grid of 480x480x240 and 20 line minimizations of the conjugate gradient routine were used in the timing. We estimate that the parallel FSM code performs at approximately 110-130Mflops per node on the T3E900.