Hexagonal indium double layer on $\mathrm{Si}\left(111\right)-\sqrt{7}\times \sqrt{3}$

Density-functional calculations are used to verify the atomic structure of the hexagonal $\mathrm{In}/\mathrm{Si}\left(111\right)-(\sqrt{7}\times \sqrt{3})$ surface, which has been considered to represent an ultimate two-dimensional (2D) limit of metallic In overlayers. Contrary to the prevailing assumption, this surface consists of not a single layer but a double layer of In atoms, which corresponds to a hexagonal deformation of the well-established rectangular In double layer formed on $\mathrm{Si}\left(111\right)-(\sqrt{7}\times \sqrt{3})$ [Park and Kang, Phys. Rev. Lett. 109, 166102 (2012)]. The same double-layer thickness accounts well for the typical coexistence of the hexagonal and rectangular phases and their similar 2D electronic structures. It is thus conclusive that, regardless of rectangular or hexagonal, the $\mathrm{In}/\mathrm{Si}\left(111\right)-(\sqrt{7}\times \sqrt{3})$ surface does not represent a one-atom-thick In overlayer.

Figure 1

Double-layer models for In/Si(111)$-(\sqrt{7}\times \sqrt{3})$. (a)–(c) One rectangular (Ref. [13]) and two hexagonal models. Large (small) balls denote In (Si) atoms. Solid and dashed lines denote $(\sqrt{7}\times \sqrt{3})$ and effective In-($1\times 1$) unit cells, respectively. Numbers denote optimized bond lengths and layer spacings. (d) Phase boundary between $\sqrt{7}$-rect and HEX2.

Figure 2

STM comparison for the HEX2 model. (a) STM image taken from Ref. [8]. (b) Simulated image representing the surface of constant density with $\rho =5\times {10}^{-4}e/{\AA }^3$. For better comparison with the low-bias STM image, the calculated Fermi level was shifted up by 0.4 eV. The inset shows a contour map drawn with a uniform increment of $0.05\AA$ where filled circles denote the topmost In atoms. (c) Experimental (Ref. [25]) and simulated line profiles taken along the arrow marked in (b).

Figure 3

(a) DOS for the present $\sqrt{7}$-hex model, the $\sqrt{7}$-rect model of Ref. [13], and the 1.2-ML model of Ref. [18]. (b) Projected DOS of the In $p$ and $s$ orbitals for $\sqrt{7}$-hex and $\sqrt{7}$-rect.

Figure 4

(a) Band structure of $\sqrt{7}$-hex. Circles denote the In states that contain more than 50% of charge in the In double layer: The states containing more than 40% of charge in the top In layer are emphasized by filled circles. Solid (green) curves denote the band structure of the ($1\times 1$) freestanding In double layer. A dashed (red) curve denotes a parabolic fitting to the bottom part of the In-derived free-electron-like band. Here, the Fermi level was set to zero. (b) Fermi contours. The $(\sqrt{7}\times \sqrt{3})$ and In-($1\times 1$) Brillouin zones are denoted by solid and dashed lines, respectively. The arrow denotes the line for the band-structure calculation in (a). The large (red) circle represents the 2D Fermi circle constructed from the parabolic bands described in (a). In the lower panel, the $(\sqrt{7}\times \sqrt{3})$ zone folding of the Fermi circle accounts for most of the complex Fermi contours.