Optical properties of real surfaces: Local-field effects at oxidized $\mathrm{Si}\left(100\right)(2\times 2)$ computed with an efficient numerical scheme

We show the application of an efficient numerical scheme to obtain the independent-particle dynamic polarizability matrix ${\chi }^{\left(0\right)}(\mathbf{r},{\mathbf{r}}^{\prime },\omega )$, a key quantity in modern ab initio excited-state calculations. The method has been applied to the study of the optical response of a realistic oxidized silicon surface, including the effects of crystal local fields. The latter are shown to substantially increase the surface optical anisotropy in the energy range below the bulk band gap. Our implementation in a large-scale ab initio computational code allows us to make a quantitative study of the CPU time scaling with respect to the system size, and demonstrates the real potential of the method for the study of excited states in large systems.

Figure 1

(Color online) (a) Accuracy test for the HT-based algorithm, shown for the real part of ${\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}^{\left(0\right)}\left(\omega \right)$ of a model system (see text). The two curves turn out to be indistinguishable on the scale of the plot (maximum error less than 0.5%). (b) Computational load requested to evaluate ${\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}^{\left(0\right)}\left(\omega \right)$ on a model system, as a function of the number of transitions, for both the traditional and the HT-based methods.

Figure 2

(Color online) Surface structure of $\mathrm{Si}\left(100\right)(2\times 2):\mathrm{O}$ at 1 ML coverage with oxidation of Si dimers and backbonds. Oxygen atoms are depicted in dark gray (red), while light gray (yellow) circles represent bulk and surface Si atoms. Dimer chains are oriented along the $x$ direction. (a) Top view of the surface ($xy$ plane), with the surface unit cell; (b) lateral view of the half slab ($yz$ plane).

Figure 3

(Color online) Imaginary part of the diagonal components of the slab dielectric tensor, calculated with (LF) and without (IP-RPA) local-field effects, for the three polarizations: (a) parallel to the surface, along the dimer axis; (b) parallel to the surface, along the dimer chains; and (c) perpendicular to the surface. The spectra presented in this figure are not fully converged in the $\mathbf{k}$-point sampling.

Figure 4

(Color online) Calculated RAS spectrum of $\mathrm{Si}\left(100\right)(2\times 2):\mathrm{O}$ at convergence, for the structural model shown in Fig. 2. Results including (LF) or neglecting (IP-RPA) the local-field effects are very similar, except for the region between 0.8 and $1.8\mathrm{eV}$, where the LF effects strongly enhance the RAS signal. The energy scale has been shifted by $0.6\mathrm{eV}$ to compensate for the neglect of self-energy effects.

Figure 5

(Color online) Calculated SDR spectrum (unpolarized light) of $\mathrm{Si}\left(100\right)(2\times 2):\mathrm{O}$, for the structural model shown in Fig. 2. Results including (LF) or neglecting (IP-RPA) the local fields are almost indistinguishable, showing that the effects visible in the low-energy part of Fig. 3 are canceling each other in the SDR spectrum. The same energy shift as in Fig. 4 has been applied. In polarized SDR, the local-field effects would be of the same size as for the RAS spectra.

Figure 6

(Color online) Quantitative study of the computational load required to evaluate the ${\chi }^{\left(0\right)}$ matrix for the 56-atom slab representing a $\mathrm{Si}\left(100\right)(2\times 2):\mathrm{O}$ surface. The effects of the three main parameters determining the numerical convergence of the theoretical spectra are studied separately. (a) Number of valence-conduction transitions, determined by the energy cutoff on the empty (conduction) bands included in Eq. (9), the number of occupied bands being fixed; (b) number of frequency intervals taken on the $\omega$ axis, determined by the requested spectral resolution; and (c) size of the ${\chi }^{\left(0\right)}$ matrix in reciprocal space, roughly proportional to the system size (for a fixed real-space resolution).