Temperature-dependent dielectric function of intrinsic silicon: Analytic models and atom-surface potentials

The optical properties of monocrystalline, intrinsic silicon are of interest for technological applications as well as fundamental studies of atom-surface interactions. For an enhanced understanding, it is of great interest to explore analytic models which are able to fit the experimentally determined dielectric function $\ensuremath{\epsilon}({T}_{\mathrm{\ensuremath{\Delta}}},\ensuremath{\omega})$, over a wide range of frequencies and a wide range of the temperature parameter ${T}_{\mathrm{\ensuremath{\Delta}}}=(T\ensuremath{-}{T}₀)/{T}₀$, where ${T}₀=293\phantom{\rule{4pt}{0ex}}\mathrm{K}$ represents room temperature. Here, we find that a convenient functional form for the fitting of the dielectric function of silicon involves a Lorentz-Dirac curve with a complex, frequency-dependent amplitude parameter, which describes radiation reaction. We apply this functional form to the expression $[\ensuremath{\epsilon}({T}_{\mathrm{\ensuremath{\Delta}}},\ensuremath{\omega})\ensuremath{-}1]/[\ensuremath{\epsilon}({T}_{\mathrm{\ensuremath{\Delta}}},\ensuremath{\omega})+2]$, inspired by the Clausius-Mossotti relation. With a very limited set of fitting parameters, we are able to represent, to excellent accuracy, experimental data in the (angular) frequency range $0<\ensuremath{\omega}<0.16\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{u}.$ and $0<{T}_{\mathrm{\ensuremath{\Delta}}}<2.83$, corresponding to the temperature range $293\phantom{\rule{4pt}{0ex}}\mathrm{K}<T<1123\phantom{\rule{4pt}{0ex}}\mathrm{K}$. Using our approach, we evaluate the short-range ${C}₃$ and the long-range ${C}₄$ coefficients for the interaction of helium atoms with the silicon surface. In order to validate our results, we compare to a separate temperature-dependent direct fit of $\ensuremath{\epsilon}({T}_{\mathrm{\ensuremath{\Delta}}},\ensuremath{\omega})$ to the Lorentz-Dirac model.