Test of long-range exchange-correlation kernels of time-dependent density functional theory at surfaces: Application to $\text{Si}\left(111\right)2\times 1$

We present an application of time-dependent density-functional theory (TDDFT) to the study of the optical properties of the $\text{Si}\left(111\right)2\times 1$ surface from the infrared to the ultraviolet. We have carried out ab initio calculations using different methods, from DFT to Bethe-Salpeter equation (BSE) and, within TDDFT, we have tested the ability of different kernels to describe the optical features in a wide range of energies. We find good agreement between TDDFT and BSE results, by using in TDDFT a long-range frequency-dependent exchange-correlation kernel derived from the many-body formalism (the MB kernel). The agreement between theory and experiment is very good in the whole frequency range studied. Excitonic effects, important in the infrared part of the spectrum, are less pronounced in the visible and UV ranges.

Figure 1

(Color online) Experimental SDR spectrum of $\text{Si}\left(111\right)2\times 1$ for light polarized along the silicon surface chains ($y$ polarization, red dashed line) and perpendicular to the chains ($x$ polarization, black line) (Ref. 26); $y$ is the $\left[1\overline{1}0\right]$ direction, $x$ is the $\left[11\overline{2}\right]$.

Figure 2

Experimental RAS spectrum of $\text{Si}\left(111\right)2\times 1$ by Goletti et al. (Ref. 28), $y$ is the $\left[1\overline{1}0\right]$ direction (parallel to the chains), $x$ is the $\left[11\overline{2}\right]$ (perpendicular to the chains).

Figure 3

(Color online) Calculated RAS for $\text{Si}\left(111\right)2\times 1$ within GW and BSE. $y$ is the $\left[1\overline{1}0\right]$ direction (parallel to the chains), $x$ is the $\left[11\overline{2}\right]$ (perpendicular to the chains).

Figure 4

(Color online) RAS calculated within DFT-LDA and TDLDA, compared with experiment (Ref. 25). A broadening of 0.05 eV has been used in the calculated spectra.

Figure 5

(Color online) Calculated surface contribution to reflectance for light polarized along $x$ (perpendicular to the chains) and along $y$ (parallel to the chains) within TDDFT (many-body kernel) and BSE.

Figure 6

(Color online) Calculated RAS for $\text{Si}\left(111\right)2\times 1$ within TDDFT (many-body kernel), TDDFT-RORO kernel $\left(\alpha =+0.2\right)$, and BS equation.