Disproportionation Phenomena on Free and Strained $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}(111)$ and $\mathrm{S}\mathrm{n}/\mathrm{S}\mathrm{i}(111)$ Surfaces

Distortions of the $\sqrt{3}\times \sqrt{3}$ $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}(111)$ and $\mathrm{S}\mathrm{n}/\mathrm{S}\mathrm{i}(111)$ surfaces are shown to reflect a disproportionation of an integer pseudocharge, $Q$, related to the surface band occupancy. A novel understanding of the $(3\times 3)$-1U (“1 up, 2 down”) and 2U (“2 up, 1 down”) distortions of $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}(111)$ is obtained by a theoretical study of the phase diagram under strain. Positive strain keeps the unstrained value $Q=3$ but removes distortions. Negative strain attracts pseudocharge from the valence band causing first a $(3\times 3)$-2U distortion ($Q=4$) on both $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}$ and $\mathrm{S}\mathrm{n}/\mathrm{S}\mathrm{i}$, and eventually a $(\sqrt{3}\times \sqrt{3})$-3U (“all up”) state with $Q=6$. The possibility of a fluctuating phase in unstrained $\mathrm{S}\mathrm{n}/\mathrm{S}\mathrm{i}(111)$ is discussed.

Figure 1

Energy gained by the $(3\times 3)$-1U and 2U distortions against the relaxed undistorted $\sqrt{3}\times \sqrt{3}$ phase, as a function of strain. (a) $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}\mathrm{(}\mathrm{111}\mathrm{)}$; (b) $\mathrm{S}\mathrm{n}/\mathrm{S}\mathrm{i}(111)$.

Figure 2

Adatom height $z$ as a function of strain in $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}\mathrm{(}\mathrm{111}\mathrm{)}$ (see inset). Open squares: “down” adatoms; filled squares: “up” adatoms; dashed lines: optimal height of an isolated charged Sn adatom (see text).

Figure 3

Electron bands of $\mathrm{S}\mathrm{n}/\mathrm{G}\mathrm{e}\mathrm{(}\mathrm{111}\mathrm{)}$ $3\times 3$. (a) 1U, zero strain; (b) 2U, strain $-4\%$. Dashed lines indicate the chemical potential. Its shift below the valence top is a finite-size effect.