The adsorption of the two group-III metals Ga and In on Si(112) has strong influence on the morphology of the intrinsically faceted Si(112) surface. Upon Ga or In adsorption, the Si(112) surface is smoothed, and quasi-1D adsorbate structures along the $[1\overline{1}0]$ direction are observed. These structures consist of ($N\ifmmode\times\else\texttimes\fi{}1$) building blocks of different sizes, the periodicity of which can be controlled by surface coverage and deposition temperature, as revealed by spot profile analysis low-energy electron diffraction. From x-ray standing-wave measurements, building blocks consisting of two parallel rows of adsorbate atoms are identified for both Ga/Si(112) and In/Si(112). One adsorption site is identified as a terrace (substitutional) site and the other one as a step-edge (adatom) site. These experimental results are compared to several relaxed model structural configurations obtained from density functional theory calculations. In the case of Ga/Si(112), a previously reported structural model by Snijders et al. [Phys. Rev. B 72, 125343 (2005)], including two Ga vacancies per unit cell is corroborated, while for In/Si(112), existing models by Gai et al. [Phys. Rev. B 61, 9928 (2000)] and by Bentmann et al. [Phys. Rev. B 80, 085311 (2009)] can be ruled out, and a new structural model including only one In vacancy per unit cell in the step-edge site is concluded on, similar to the Al/Si(112) model introduced by Gupta and Batra [Phys. Rev. B 72, 165352 (2005)].
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