Disproportionation Phenomena on Free and Strained Sn/Ge(111) and   Sn/Si(111) Surfaces
  Distortions of the $\sqrt3\times\sqrt3$ Sn/Ge(111) and Sn/Si(111) surfaces are shown to reflect a disproportionation of an integer pseudocharge, $Q$, related to the surface band occupancy. A novel understanding of the $(3\times3)$-1U (``1 up, 2 down'') and 2U (``2 up, 1 down'') distortions of Sn/Ge(111) is obtained by a theoretical study of the phase diagram under strain. Positive strain keeps the unstrained value Q=3 but removes distorsions. Negative strain attracts pseudocharge from the valence band causing first a $(3\times3)$-2U distortion (Q=4) on both Sn/Ge and Sn/Si, and eventually a $(\sqrt3\times\sqrt3)$-3U (``all up'') state with Q=6. The possibility of a fluctuating phase in unstrained Sn/Si(111) is discussed. 
G.Ballabio (1,5), G.Profeta (2), S.de Gironcoli (1), S.Scandolo (1,3),   G.E.Santoro (1), and E.Tosatti (1, 4) ((1) SISSA an...