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 is interest is due both to the rich, novel physical properties exhibited by these systems (Coulomb Blockade, quantum confinement, exchange enhancement and shape dependent spectroscopy) and because of their promise for applications such as lasers. We have used our FSM code to study the near-edge states of GaAs[4], InAs, InP[1,3] and CdSe free standing quantum dots. These calculations have enabled us to calculate 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 offer the possibility of using standard MBE and MOVPE techniques to manufacture high quality, dislocation free quantum dots with a narrow (<10%) size distributions. The most popular system used for constructing such dots is InAs dots embedded within GaAs dots. The solid state theory group is actively researching the properties of such dots. In particular 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 45A high, 90A 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 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 pseudopotential calculations are shown on the left and right. The lowest electron state (CBM) is s-like, while the next 2 states (CBM+1 and CBM+2) are non-degenerate p-like. From J. Kim, L.W. Wang, A.J. Williamson and A. Zunger (unpublished). See also the article by A. Zunger in the MRS Bulletin.
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The above figure shows isosurface plots of the 4 highest energy hole states and the 4 lowest energy conduction states, calculated using the FSM.
This figure shows the strain modified electron, heavy hole and light hole potential offsets in the {010} plane through the center of a GaAs embedded (a) spherical InAs dot with 42.2A diameter and (b) a pyramidal InAs dot with 30.3AA height and 60.6\AA base. The scale bars show the maximum depth of the electron and hole wells in eV. The black lines mark the approximate edges of the electron and hole wells.
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The effect of the above offsets is to only weekly localize electrons and holes within the GaAs embedded quantum dots. This can be seen in the figure below which clearly shows how electron and hole wavefunctions are strongly localized in free standing InAs quantum dots, but only weakly localized in GaAs embedded quantum dots. For more information, see the publication A.J. Williamson and Alex Zunger, "InAs quantum dots: Predicted electronic structure of free-standing vs. GaAs embedded structures".
Contour plots of the electron and hole wavefunctions in the {001} plane through the center of a free standing InAs dot (top), the smae InAs dot embedded in GaAs(middle) and a pyramidal InAs dot with the same volume, embedded in GaAs (bottom).
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Andrew Canning(LBL), Andrew Williamson(NREL) and L.W. Wang(NREL) have ported the Folded Spectrum code to work on the Cray T3E 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.