Structure and lattice dynamics ofGe1−yCyalloys using anharmonic Keating modeling

Abstract

The incorporation of substitutional C into Ge is studied theoretically with an anharmonic Keating model fitted to recent experimental data and specifically adapted to the computation of the structural properties and the lattice dynamics of ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloys. In the range of physically realistic concentrations $(y<3%),$ the change of lattice parameter due to substitutional carbon is found to agree with Vegard's law of linear interpolation between germanium and diamond with a maximum relative deviation less than 0.03%. This result is obtained in the case of a ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloy either random, diluted, or with C mostly arranged in third nearest neighbors. In the case of a pseudomorphic alloy on germanium (001), the tetragonal distortion of the lattice is well described (error $<0.13%)$ if the elastic constants of the alloy are linearly interpolated between the corresponding parameters of bulk Ge and C in the framework of macroscopic linear elastic theory. The position of the localized vibrational mode of substitutional C in Ge depends on the distribution of carbon, and new phonon modes appear for certain atomic arrangements. Carbon atoms in first nearest neighbor positions give new modes around 425, 480, and $610{\mathrm{cm}}^{\ensuremath{-}1},$ while a new GeC-like mode occurs at $487{\mathrm{cm}}^{\ensuremath{-}1}.$ C in third nearest neighbor (most stable) arrangements induce an additional mode around $550{\mathrm{cm}}^{\ensuremath{-}1}.$ A signature of the fifth nearest neighbor configuration is calculated around $510{\mathrm{cm}}^{\ensuremath{-}1}.$ This study suggests that the content of substitutional carbon in ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloys can be estimated from the lattice constant by x-ray diffraction for $y<3%,$ and that the local order around C can be probed by Raman and infrared absorption spectroscopy.