Range-Separated Approach to the RPA Correlation Applied to the van der Waals Bond and to Diffusion of Defects

The random-phase approximation (RPA) is a promising approximation to the exchange-correlation energy of density functional theory, since it contains the van der Waals (vdW) interaction and yields a potential with the correct band gap. However, its calculation is computationally very demanding. We apply a range-separation concept to RPA and demonstrate how it drastically speeds up the calculations without loss of accuracy. The scheme is then successfully applied to a layered system subjected to weak vdW attraction and is used to address the controversy of the self-diffusion in silicon. We calculate the formation and migration energies of self-interstitials and vacancies taking into account atomic relaxations. The obtained activation energies deviate significantly from the earlier calculations and challenge some of the experimental interpretations: the diffusion of vacancies and interstitials has almost the same activation energy.

Figure 1

LR-RPA (filled symbols) and SR-RPA (open symbols) correlation energies of bulk silicon for lattice constant $a=10.26\mathrm{bohr}$, as a function of the cutoff radius $r_c=1/\omega$. The SR-RPA correlation energy is obtained as the RPA correlation energy minus the LR-RPA correlation energy. The explicit calculation is displayed with squares and LDA with circles. The total correlation is displayed with crosses. The horizontal line emphasizes the calculated RPA correlation energies with full Coulomb interaction. Each point is associated with the corresponding convergence parameters necessary to achieve a $2\mathrm{meV}/\mathrm{atom}$ accuracy: first, the cutoff energy for the polarizability (Ha), and second, the number of states to be included in the expression of the polarizability.

Figure 2

Energy as a function of the interlayer spacing $d=c/2$ of hexagonal BN within LDA (dashed line), PBE (solid line), RPA with $r_c=4$, 2, or 1 (respectively, circles, squares, or diamonds). The equilibrium spacing $d_0$ and elastic constant $C_{33}$ are shown and compared to the modelized method vdW DF [31] and to experiment [41, 42].

Figure 3

Silicon vacancy $V_{\mathrm{Si}}$ formation energy in a 63-atom supercell as a function of the nearest neighbor atom distances. In the left-hand panel, the longest distance $r_{\mathrm{long}}$ is fixed and the shortest one $r_{\mathrm{short}}$ is varied. In the right-hand panel, $r_{\mathrm{long}}$ is fixed and $r_{\mathrm{short}}$ is varied. The absence of Jahn-Teller distortion corresponds to $r_{\mathrm{long}}=r_{\mathrm{short}}$. The atomic configuration is displayed in the inset: The cube stands in the empty lattice site.