Large-scale atomic effective pseudopotential program including an efficient spin-orbit coupling treatment in real space

Within the scheme of the large-scale atomic effective pseudopotential program (LATEPP), the Schrödinger equation of an electronic system is solved within an effective single-particle approach. Although not limited to, it focuses on the recently introduced atomic effective pseudopotentials derived from screened local effective crystal potentials as obtained from self-consistent density functional theory calculations. The problem can be solved in both real (real-space grid) and reciprocal space (plane-wave basis functions). Following the idea of atomic effective pseudopotentials, the density, and hence a self-consistent cycle, is not required and not implemented. An iterative solver is implemented to deliver the eigenstates close to a selected reference energy, e.g., around the band gap of a semiconductor. This approach is particularly well suited for theoretical investigations of the electronic structure of semiconductor nanostructures and we demonstrate linear scaling with the system size up to around $100000$ atoms on a single standard compute node. Moreover, an efficient real-space treatment of spin-orbit coupling within the pseudopotential framework is proposed in this work allowing for a fully relativistic description.

Figure 1

Number of iterations of the arpack solver requesting 14 states with respect to the utilized reference energy. The shaded region marks the range of reference energies for which the same solutions are expected. Data points supplied with a circle identify calculations that indeed contain these solutions. The blue lines correspond to the four blocks of eigenvalues of the requested states. The red line corresponds to the middle of the band gap.

Figure 2

Number of iterations required to achieve convergence with respect to the number of requested states for a GaAs bulk structure consisting of 1728 atoms, for which two different distortions are applied to a single Ga atom of the structure.

Figure 3

Convergence of the band-gap energy of a bulk GaAs supercell containing eight atoms for two plane-wave energy cutoffs (green: 22 Ha, blue: 30 Ha) with respect to the grid size in one dimension in terms of the minimum grid (M); the full grid is indicated by (F). Results are shown for calculations in the reciprocal-space basis with the nonlocal potential being evaluated in real space (dashed line) and reciprocal space (solid line).

Figure 4

Convergence of the band-gap energy of a bulk GaAs supercell containing eight atoms with respect to the plane-wave cutoff energy. For the full real-space calculations, the equi-valent cutoff energy is calculated according to Eq. (46). The kinetic energy is calculated in real (blue) or reciprocal space (red or green) and the nonlocal potential in real (blue or green) or reciprocal (red) space.

Figure 5

Convergence of the band-gap energy of a bulk GaAs supercell containing eight atoms with respect to the polynomial order of the finite difference scheme in the fully real-space calculations. Results for different grid sizes are shown.

Figure 6

Excerpt of the electronic band structure of the highest occupied $p$ and the lowest unoccupied $s$ state of GaAs (left) and AlAs (right) bulk structures.

Figure 7

Squared wave function in real space of the highest occupied (top, red) and lowest unoccupied (bottom, blue) states of a GaAs QD embedded in AlAs. Green, tan, and silver spheres correspond to Ga, Al, and As atoms. The structure consists of $97336$ atoms.

Figure 8

Total calculation time for a QD structure consisting of $4\times 4\times 4$ lattice constants containing 512 atoms with respect to the number of threads. Results of four different combinations of libraries utilized in latepp are shown.

Figure 9

Speedup of latepp gained by parallelization with respect to the number of threads. Here, FFTE [38] is used for the transformations.

Figure 10

Time for one iteration (black lines) for bulk GaAs structures with respect to the number of atoms. Results are shown for fully real-space calculations (dashed lines) as well as for calculations in the reciprocal-space basis with the nonlocal potential being evaluated in real space (solid lines). The total time is composed of the time of the $H\psi$ evaluation (blue lines) and the arpack routine (orange lines). The treatment in reciprocal space requires FFTs (red line) while a more computationally intensive kinetic energy evaluation (green line) is used in the real-space treatment. A minimum grid (see Fig. 3) and 22 Ha cutoff energy for the plane waves are used.

Figure 11

Time for the first iteration of fully real-space calculations (dashed line) and calculations in the reciprocal-space basis with the nonlocal potential being evaluated in real space (solid line) for different sizes of bulk GaAs structures. The insets show magnified regions of the plot. The contribution of the $H\psi$ evaluation and the arpack routine is shown by the blue and orange lines, respectively. The contributions of the FFT and kinetic energy calculation within the $H\psi$ evaluation are highlighted by red and green colors.

Figure 12

Electronic band structure of GaAs illustrating the lifting of the degeneracy of the split-off and heavy-/light-hole band at the $\Gamma$ point as well as the characteristic splitting of the heavy- and light-hole bands approaching the $L$ and $X$ points due to spin-orbit interaction. In addition, the spin splitting of the two spin components (dashed and solid lines) between the $X$ and $\Gamma$ points along the [1 1 0] can be seen. The results of latepp (blue lines) are compared to DFT results using abinit employing the HGH [56] pseudopotentials (black lines).

Figure 13

Lifting of the light-/heavy-hole degeneracy in GaAs due to spin-orbit interaction. The results of latepp (solid line) are compared to a DFT result using abinit employing the HGH [56] pseudopotentials (dashed line).

Figure 14

Spin splitting in the valence and conduction bands of GaAs between the $\Gamma$ and $X$ points along the [1 1 0] direction due to spin-orbit interaction. Results of latepp (solid lines) are compared to DFT results using abinit employing the HGH [56] potentials (dashed lines).

Figure 15

Spin splitting in the heavy-hole band of GaAs between the $L$ and $\Gamma$ points along the $\Lambda$ direction due to spin-orbit interaction. The result of latepp (solid line) is compared to the DFT result using abinit employing the HGH [56] potentials (dashed line).

Figure 16

Spin splitting of the GaAs heavy-hole band from $\Gamma$ to $X$ along $\Sigma$ direction. The inset shows a magnified region near the $\Gamma$ point. Results of latepp (solid line) are compared to results of abinit employing the HGH [56] potentials (dashed line).

Figure 17

Spin splitting in the valence bands of GaAs between the $\Gamma$ and $X$ points along the [1 1 0] direction due to spin-orbit interaction. Results of latepp (solid lines) are compared to DFT results using abinit employing the HGH [56] potentials (dashed lines). In addition, results with the wrong normalization as given in Refs. [47, 48, 49] are shown (blue lines).