$2\mathbf{\times }1$ reconstructed Si(111) surface: STM experiments versus ab initio calculations

The electronic structure of the Si(111)-$2\times 1$ surface has been studied with scanning tunneling spectroscopy (STS). A large experimental local density of states data set of $\mathrm{LDOS}(x,y,E)$ with subatomic resolution has been compared with ab initio calculated LDOS distributions. The influence of the tunneling tip DOS has been eliminated by repeated measurements with different tips. The experimentally determined shape of the $\mathrm{LDOS}(x,y,E)$ agrees very well with the calculated results based on the $\pi$-bonded-chain model for both the surface valence and the conduction band. The good agreement with ab initio calculations of the electronic structure of the Si(111)-$2\times 1$ surface shows that STS provides reliable information of the sample LDOS even with subatomic resolution.

Figure 1

Pandey model of the $2\times 1$-reconstruction on Si(111). (a) Sketched top view of the (111) surface with numbers denoting different atom positions in the surface: 1, up atoms; 2, down atoms in the $\pi$-bonded chains; 3 and 4, atoms in the second layer row in between. The dotted line shows the $2\times 1$ unit cell $(6.65\times 3.84{\mathrm{\AA }}^2)$. The real space lattice vectors correspond to the directions in $k$ space as marked in the image. The crystallographic orientation with $\left[0\overline{1}1\right]$ along the $x$ axis and $\left[\overline{2}11\right]$ along the $y$ axis will be maintained in all following figures. (b) Calculated electronic surface dispersion with dangling bond states (black lines) and projected volume bands (hatched areas); $\pi C$ denotes the $\pi$-chain states; the indices CB and VB denote states of conduction- and valence-band type.

Figure 2

Constant current images of ${(45\times 45\mathrm{\AA }}^2)$ on Si(111)-$2\times 1$ at different set points: $+1.4\mathrm{V}$; 0.5 nA (a), $+0.6\mathrm{V}$; 0.5nA (b), $-0.3\mathrm{V}$; 0.1 nA (c) and $-1.1\mathrm{V}$; 0.2 nA (d). Gray scaled with black, denoting minimal, and white, denoting maximal, tip height (the $z$ range varies between the images). The black (white) cross marks the maximum (minimum) position as referred to in Fig. 3b.

Figure 3

LDOS on Si(111)-$2\times 1$ (a) ab initio calculated $\mathrm{DOS}\left(E\right)$ and (b) experimental spectra taken on (solid line) and between (dashed line) the maxima of the corresponding topography. Examples for both positions are marked in Fig. 2d. The curves coincide expect for small differencies. (c) The difference between the dashed and the solid LDOS curve from subset (b) is plotted in the same scale and highlights the small variation inside the unit cell.

Figure 4

(a) Constant current topography $Z(x,y)$ at ${\mathrm{V}}_{\text{sample}}=-1.0\mathrm{V}$ and $I_T=0.4\mathrm{nA}$ and (b) apparent work function $\Phi (x,y)$ taken simultaneously. The corrugation amplitude $Z(x,y)$ due to the reconstruction pattern is about 1 Å. $\Phi (x,y)$ is modulated within the surface unit cell between 180 meV and 350 meV. The data are gray-scaled; light indicates high and dark indicates low values of $\Phi$ and $Z(x,y)$. Referring to the text and Fig. 5 we present a surface unit cell at the up atoms by the dashed rectangle. For crystallographic orientations see Fig. 1a.

Figure 5

Laterally resolved LDOS structure $\mathrm{LDOS}(x,y)$ of the empty states from theory (right, $t_i$) and experiment (left, $e_i$) at typical energies (white, high LDOS; black, low LDOS). In the calculated images the underlying atomic lattice is shown by large (small) open circles marking the up (down) atoms of the $\pi$-bonded chains and the gray filled circles marking the zigzag rows in between. The different surface atoms are referred to as numbers 1 to 4 in Fig. 1a. We present a surface unit cell connecting at the up atoms (dashed rectangle). The characteristical shifts of the LDOS maxima relative to the atom positions is discussed in the text. For crystallographic orientations see Fig. 1a.

Figure 6

$\mathrm{LDOS}(x,y)$ at the band onsets from experimental (left, $e_i$) and calculated (right, $t_i$) data. The LDOS maxima characteristically shift relative to the surface unit cell when the energy is switched from $\pi C_{\mathrm{VB}}({\mathrm{e}}_1,{\mathrm{t}}_1)$ to $\pi C_{\mathrm{CB}}({\mathrm{e}}_2,{\mathrm{t}}_2)$. For crystallographic orientations see Fig. 1a; lattice positions and gray scale, see Fig. 5.

Figure 7

Comparison of experimentally measured (left, $e_i$) and calculated (right, $t_i$) LDOS distributions in the occupied states. At high negative energies the structure fades out. For crystallographic orientations, see Fig. 1a; lattice positions and gray-scale, see Fig. 5.