High-resolution electron energy loss spectra of reconstructed Si(100) surfaces: First-principles study

We present ab initio calculations of electron energy loss spectroscopy in the reflection geometry (REELS) for the Si(100) surface for which several experimental data are available. The standard surface models [$p\left(2\times 1\right)$, $c\left(4\times 2\right)$, and $p\left(2\times 2\right)$] are structurally very similar in nature, and precise calculations are necessary to differentiate between them. Starting from optimized geometries we compute REELS spectra within the framework of the three-layer model. We adopt several methodologies to ensure a realistic model of the experiment, including a precise partitioning of the surface and bulk dielectric functions and a numerical integration over the detector aperture. We obtain good agreement with the various available experimental energy loss and reflectance anisotropy spectra. The calculations allow us to definitively rule out the presence of the $p\left(2\times 1\right)$ reconstruction. We interpret the $S_0$ peak observed by Farrell et al. [Phys. Rev. B 30, 721 (1984)] in high resolution REELS. Furthermore, we explain the observed dependence of the spectra on temperature by inferring the presence of dimer flipping at room temperature.

Figure 1

(Color online) Ball and stick model of Si(100) $p\left(2\times 1\right)$, $p\left(2\times 2\right)$, and $c\left(4\times 2\right)$ surface reconstructions. Large circles indicate “up” silicon dimer atoms. Unit cells are indicated by shaded regions.

Figure 2

Schematic representation of the constituent parts of the three-layer model of the surface.

Figure 3

(Color online) Left: schematic diagram of the $\text{Si}\left(100\right)\text{−}c\left(4\times 2\right)$ slab and calculated layer averaged charge density; atomic layers are marked with horizontal lines. Right: dependence of a typical REEL spectrum on the value of surface layer thickness $d$ using the two methods explained in the text. A mean value detector sampling approach has been used here (see Sec. 3B).

Figure 4

(Color online) Dependence of typical REEL spectra on detector integration method: numerical Monte Carlo, mean value scheme, and single point sampling. Example shown for $c\left(4\times 2\right)$ surface, averaged ${\mathbf{q}}_{\parallel }$; $E_0=40\text{eV}$; ${\theta }_0=60°$; and ${\theta }_{\text{det}}=1°$.

Figure 5

(Color online) RA spectra of the $c\left(4\times 2\right)$, $p\left(2\times 2\right)$, and $p\left(2\times 1\right)$ reconstructions of clean Si(100), compared with the experimental spectrum of the nominal surface (scaled by a factor of 5). Experimental data are taken from Ref. 41.

Figure 6

(Color online) REEL spectra of $c\left(4\times 2\right)$, $p\left(2\times 2\right)$, and $p\left(2\times 1\right)$ reconstructions of clean Si(100), and comparison with experiment (see Ref. 27): $E_0=40\text{eV}$; ${\theta }_0=60°$. The surface thickness is assumed to be $d=8$ layers (plus one vacuum as shown in Fig. 3), i.e., the half slab. A background signal has been subtracted from the experimental spectrum (see text).

Figure 7

(Color online) REEL spectra of $\text{Si}\left(100\right)\text{−}c\left(4\times 2\right)$ calculated using a mixture of long-range reflection and short-range transmission loss functions [Eq. (8)], compared with experiment (see Ref. 27): $E_0=40\text{eV}$; ${\theta }_0=60°$.

Figure 8

(Color online) REEL spectra of $p\left(2\times 1\right)$, $p\left(2\times 2\right)$, and $c\left(4\times 2\right)$ reconstructions of clean Si(100) for low energy incident beam ($E_0=7\text{eV}$; ${\theta }_0=60°$). Surface thickness $d=2$ atomic layers.

Figure 9

(Color online) HREELS along $x$ and $y$ (top) and surface dielectric functions (along $x$, $y$, and $z$ directions) for $E_0=7\text{eV}$, ${\theta }_0=60°$; a scissor shift of $+0.5\text{eV}$ is applied.

Figure 10

(Color online) Total oscillator strength ${\overline{P}}_E\left(\mathbf{k}\right)$ (in Hartree) corresponding to the $S_0$ and $S_1$ HREELS peaks as a function of $\mathbf{k}$ within the irreducible part of the $c\left(4\times 2\right)$ Brillouin zone. ${\overline{P}}_E\left(\mathbf{k}\right)$ is shown explicitly for the components $S_{0,A}$, $S_{1,A}$, etc., indicated in Fig. 9. The polarization of the transition is indicated in each case.

Figure 11

(Color online) Band structure of the $c\left(4\times 2\right)$ surface: surface localized states are indicated by the heavier dots. A rigid scissor shift of $+0.5\text{eV}$ has been applied to unoccupied bands.

Figure 12

(Color online) Isosurface plots of ${\left|{\psi }_{n\mathbf{k}}\left(r\right)\right|}^2$ for representative states involved in transitions giving rise to the $S_0$ (top) and $S_1$ (bottom) HREELS peaks. Plots were obtained using the XCRYSDEN (Ref. 52) package.

Figure 13

(Color online) Top: computed HREELS spectra for a 1:1 mixture of the $c\left(4\times 2\right)$ and $p\left(2\times 2\right)$ reconstructions (a), along with the same spectrum containing a 2% average fraction of the symmetric dimer $p\left(2\times 1\right)$ spectrum (b). Bottom: temperature dependent experimental data of Gavioli et al. (Ref. 28), stacked vertically for clarity. Kinematic parameters: $E_0=7\text{eV}$; ${\theta }_0=60°$; surface thickness $d=2$ atomic layers.

Figure 14

(Color online) Top: computed ${\epsilon }_s$ (averaged over $q$) at the $\text{RPA}+\text{scissors}$ and $GW+\text{BSE}$ levels. Bottom: experimental data for similar experimental kinematic conditions, taken from Gavioli et al. (Ref. 28) and Farrell et al. (Ref. 27).