van der Waals Interactions in Ionic and Semiconductor Solids

van der Waals (vdW) energy corrected density-functional theory [Phys. Rev. Lett. 102, 073005 (2009)] is applied to study the cohesive properties of ionic and semiconductor solids (C, Si, Ge, GaAs, NaCl, and MgO). The required polarizability and dispersion coefficients are calculated using the dielectric function obtained from time-dependent density-functional theory. Coefficients for “atoms in the solid” are then calculated from the Hirshfeld partitioning of the electron density. It is shown that the Clausius-Mossotti equation that relates the polarizability and the dielectric function is accurate even for covalently-bonded semiconductors. We find an overall improvement in the cohesive properties of Si, Ge, GaAs, NaCl, and MgO, when vdW interactions are included on top of the Perdew-Burke-Ernzerhof or Heyd-Scuseria-Ernzerhof functionals. The relevance of our findings for other solids is discussed.

Figure 1

Least-squares fitting results for the polarizability (top, in ${\mathrm{bohr}}^3$) and ${\mathrm{C}}_6$ (bottom, in $\mathrm{hartree}{\mathrm{bohr}}^6$) per atom to a set of $N$ hydrogen-saturated Si clusters. The solid black curve shows the fitting results starting from the smallest cluster (${\mathrm{SiH}}_4$). The dashed red curve starts from the ${\mathrm{Si}}_{17}{\mathrm{H}}_{36}$ cluster. The crossed blue curve starts from the largest ${\mathrm{Si}}_{172}{\mathrm{H}}_{120}$ cluster.

Figure 2

The Si-Si ${\mathrm{C}}_6$ coefficient (in $\mathrm{hartree}{\mathrm{bohr}}^6$) as a function of the crystal lattice constant. For TDLDA, the results from both the Clausius-Mossotti (CM) equation and the cluster extrapolation (CE) are shown. The experimental equilibrium lattice constant is marked with a dashed vertical line.

Figure 3

Mean absolute relative error (in %) for PBE, HSE06, $\mathrm{PBE}+\mathrm{vdW}$, and $\mathrm{HSE}06+\mathrm{vdW}$ compared to experimental data for the lattice constant, cohesive energy, and bulk modulus as an average over all (C, Si, Ge, GaAs, NaCl, and MgO) solids. The quasiharmonic zero-point energy is calculated using the phonon spectrum for every functional, except the HSE06 scheme, where the PBE phonon spectrum is used. The experimental data is taken from Refs. 17, 36, 37.