Ab Initio Calculations for Large Dielectric Matrices of Confined Systems

Calculations for optical excitations in confined systems require knowledge of the inverse screening dielectric function ${ϵ}^{-1}(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime }})$, which plays a crucial role in determining exciton binding energies. We present a new efficient real-space method of inverting and storing large ab initio dielectric matrices of confined systems, which relies on the separability of $ϵ$ matrix in $\mathbf{r}$ and ${\mathbf{r}}^{\mathbf{\prime }}$. The method has allowed, for the first time, full ab initio calculation of ${ϵ}^{-1}(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime }})$ of dimension $N\sim 270000$, and for quantum dots as large as ${\mathrm{S}\mathrm{i}}_{35}{\mathrm{H}}_{36}$. The effective screening in Si quantum dots up to 1.1 nm in diameter is found to be very ineffective with average dielectric constants ranging from 1.1 to 1.4.

Figure 1

The unscreened (◇) and screened (□) exciton Coulomb energies of $\mathrm{S}\mathrm{i}{\mathrm{H}}_4$ as a function of the radius $R$ of the spherical domain. The Si atom is placed at the center of the sphere. For this calculation, the grid spacing $h$ and the number of unoccupied states $N_c$ were set to 0.8 a.u. and 16, respectively. The difference of the asymptotic values from the fully converged ones in Table  is due to the fact that the energies shown here are not converged with respect to $h$ and $N_c$.

Figure 2

The averaged screening functions $\overline{ϵ}(r)=1/{\widetilde{ϵ}}^{-1}(0,r)$ for the three quantum dots $\mathrm{S}\mathrm{i}{\mathrm{H}}_4$, ${\mathrm{S}\mathrm{i}}_5{\mathrm{H}}_{12}$, and ${\mathrm{S}\mathrm{i}}_{35}{\mathrm{H}}_{36}$. Their effective radii are 3.2, 5.4, and 10.4 a.u., respectively.