Optical properties of surfaces with supercell ab initio calculations: Local-field effects

Surface optical and electronic properties are crucial for material science and have implications in fields as various as nanotechnology, nonlinear optics, and spectroscopies. In particular, the huge variation of electronic density perpendicular to the surface is expected to play a key role in absorption due to local-field effects. Numerous state-of-the-art theoretical and numerical ab initio formalisms developed for studying these properties are based on supercell approaches, in reciprocal space, due to their efficiency. In this paper, we show that the standard scheme fails for the out-of-plane optical response of the surface. This response is interpreted using the “effective-medium theory” with vacuum and also in terms of interaction between replicas, as the supercell approach implies a periodicity which is absent in the real system. We propose an alternative formulation, also based on the supercell, for computing the macroscopic dielectric function. Application to the clean Si(001) $2\times 1$ surface allows us to present the effect of the local fields for both peak positions and line shape on the bulk and surface contributions. It shows how local fields built up for the in-plane and out-of-plane dielectric responses of the surface. In addition to their conceptual impact, our results explain why the standard approach gives reliable predictions for the in-plane components, leading to correct reflectance anisotropy spectra. Our scheme can be further generalized to other low-dimensional geometries, such as clusters or nanowires, and open the way to nonlinear optics for surfaces.

Figure 1

Left: The supercell corresponding to void 1, as explained in the text. Right: Electronic density in the $x-z$ plane obtained from the DFT calculation. The $x$ and $z$ axes are defined as the crystallographic directions, $\left[1\overline{1}0\right]$ and [001], respectively. The density is more important in the dark regions. Red lines correspond to the limit of the matter, obtained for the supercell void 1.

Figure 2

Imaginary part (top panel) and real part (bottom panel) of the in-plane (${\epsilon }_{\left|\right|}$) and out-of-plane (${\epsilon }_{\perp }$) dielectric functions for three different sizes of the supercell. Local-field effects are not included.

Figure 3

Imaginary part (top panel) and real part (bottom panel) of ${\epsilon }_{\perp }$ for the three different sizes of the supercell, including local fields. The inset shows the bulk response for comparison.

Figure 4

Imaginary part of ${\epsilon }_{\left|\right|}$ and ${\epsilon }_{\perp }$ for the NLF standard supercell and LF mixed-space approaches.

Figure 5

Comparison between the imaginary part of ${\epsilon }_{\perp }$ obtained from the supercell approach (solid lines, as in Fig. 3) and the result obtained through the EMT (dashed lines), using bulk silicon and Eq. (11).

Figure 6

Comparison between the imaginary part of ${\epsilon }_{\mathrm{bulk}}$ and ${\widehat{\epsilon }}_{\mathrm{bulk}}$ obtained from the supercell approach and Eq. (13).

Figure 7

Selected-G method for $L_z=2L_z^{\mathrm{mat}}$.

Figure 8

Imaginary part of ${\epsilon }_{\left|\right|}$ (in plane) and ${\epsilon }_{\perp }$ (out of plane) for the surface, with and without local fields.

Figure 9

Imaginary part of ${\epsilon }_{\left|\right|}^{xx}, {\epsilon }_{\left|\right|}^{yy}$ (in plane) and ${\epsilon }_{\perp }$ (out of plane) for the surface and bulk responses, with local fields included.