Second-harmonic and reflectance-anisotropy spectroscopy of vicinalSi(001)∕SiO2interfaces: Experiment and simplified microscopic model

Abstract

We directly compare experimental second-harmonic generation (SHG) spectra and reflectance-anisotropy spectra (RAS) of native-oxidized vicinal Si(001) interfaces with off-cut angles $\ensuremath{\zeta}=0\ifmmode^\circ\else\textdegree\fi{}$, 4\ifmmode^\circ\else\textdegree\fi{}, 6\ifmmode^\circ\else\textdegree\fi{}, 8\ifmmode^\circ\else\textdegree\fi{}, and 10\ifmmode^\circ\else\textdegree\fi{} from (001) toward [110]. We fit the measured azimuthal rotational-dependence of $p$-in/$p$-out SHG at two-photon energies $2.8\phantom{\rule{0.3em}{0ex}}\mathrm{eV}l\ensuremath{\hbar}\ensuremath{\omega}l3.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ including the ${E}_{1}$ critical point resonance using a simplified bond hyperpolarizability model. The model includes dipolar contributions from tetrahedrally coordinated interfacial bonds as well as quadrupolar contributions from the Si bulk. The fit yields complex axial hyperpolarizability spectra ${\ensuremath{\beta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$ and averaged bond angles of the interfacial back-edge, terrace-edge and step-edge bonds, and bulk quadrupolar hyperpolarizability ${\ensuremath{\delta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$. The fitted microscopic model accurately reproduces measured $s$-in/$p$-out SHG spectra. We then used a Miller's rule approximation to generate axial linear polarizability spectra ${\ensuremath{\alpha}}_{\ensuremath{\Vert}}(\ensuremath{\omega})$ from the fitted ${\ensuremath{\beta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$. The measured RAS was satisfactorily reproduced within the same photon energy range. Including the bulk quadrupole contribution to SHG was crucial to accurately retrieving the RAS and to achieving fitted real and imaginary $\ensuremath{\beta}(2\ensuremath{\omega})$ spectra consistent with a nonlinear Kramers-Kronig relation. The results demonstrate the possibility of formulating a common microscopic model of SHG and RAS responses of complicated interfaces.