Improved ab initio calculation of surface second-harmonic generation from Si(111)($1\times 1$):H

We carry out an improved ab initio calculation of surface second-harmonic generation (SSHG) from the Si(111)($1\times 1$):H surface. This calculation includes three new features in one formulation: (i) the scissors correction, (ii) the contribution of the nonlocal part of the pseudopotentials, and (iii) the inclusion of a cut function to extract the surface response, all within the independent particle approximation. We apply these improvements on the Si(111)($1\times 1$):H surface and compare with various experimental spectra from several different sources. We also revisit the three-layer model for the SSHG yield and demonstrate that it provides more accurate results over several, more common, two-layer models. We demonstrate the importance of using properly relaxed coordinates for the theoretical calculations. We conclude that this approach to the calculation of the second-harmonic spectra is versatile and accurate within this level of approximation. This well-characterized surface offers an excellent platform for comparison with theory and allows us to offer this study as an efficient benchmark for this type of calculation.

Figure 1

Representation of the three-layer model for SSHG. Vacuum is on top with ${\varepsilon }_v=1$; the layer with nonlinear polarization ${\mathcal{P}}^{\mathrm{a}}\left(2\omega \right)={\chi }^{\mathrm{abc}}(-2\omega ;\omega ,\omega )E^{\mathrm{b}}\left(\omega \right)E^{\mathrm{c}}\left(\omega \right)$ is characterized with ${\varepsilon }_{\ell }\left(\omega \right)$ and the bulk is characterized with ${\varepsilon }_b\left(\omega \right)$. In the dipole approximation, the bulk does not radiate second harmonic.

Figure 2

Comparison of ${\chi }_{\parallel \parallel \parallel }(-2\omega ;\omega ,\omega )$ calculated using relaxed and unrelaxed atomic positions, with the experimental data presented in Ref. [23]. Theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data were taken at 80 K.

Figure 3

Comparison between theoretical models (see Table 1) and experiment for ${\mathcal{R}}_{pS}$, for $\theta ={65}^{\circ }$. We use a scissors value of $\hslash \mathrm{\Delta }=0.7\text{eV}$. All theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data are taken from Ref. [16], measured at room temperature.

Figure 4

Comparison between theoretical models (see Table 1) and experiment for ${\mathcal{R}}_{pS}$, for $\theta ={45}^{\circ }$. We use a scissors value of $\hslash \mathrm{\Delta }=0.7\text{eV}$. All theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data are taken from Ref. [21], measured at room temperature.

Figure 5

Calculated results for ${\mathcal{R}}_{pS}$ for the different levels of approximation proposed in this paper. All curves were calculated using the three-layer model. We take $\theta ={65}^{\circ }$ for this plot. See text for full details. All curves are broadened with $\sigma =0.075\text{eV}$.

Figure 6

Comparison between theoretical models (see Table 1) and experiment for ${\mathcal{R}}_{sP}$, for $\theta ={65}^{\circ }$. We use a scissors value of $\hslash \mathrm{\Delta }=0.7\text{eV}$. All theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data are taken from Ref. [16], measured at room temperature.

Figure 7

Comparison between theoretical models (see Table 1) and experiment for ${\mathcal{R}}_{pP}$, for $\theta ={65}^{\circ }$. We use a scissors value of $\hslash \mathrm{\Delta }=0.7\text{eV}$. All theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data are taken from Ref. [16], measured at room temperature.

Figure 8

Comparison between theoretical models (see Table 1) and experiment for ${\mathcal{R}}_{pP}$, for $\theta ={45}^{\circ }$. We use a scissors value of $\hslash \mathrm{\Delta }=0.7\text{eV}$. All theoretical curves are broadened with $\sigma =0.075\text{eV}$. Experimental data are taken from Ref. [21], measured at room temperature.