Electroreflectance and ellipsometry of silicon from 3 to 6 eV

Abstract

Using a five-band model, we calculate the effect of the $k$-linear intravalence band coupling term on dielectric function, third derivative, and low-field electroreflectance (ER) line shapes for Si. The $k$-linear term, which was recently shown by Cardona strongly to affect transverse-interband reduced-mass values on the $〈111〉$ symmetry axes, acts to increase the oscillator strength of the upper valence ($v$)—lower conduction ($c$) band transitions at the expense of that of the lower spin-orbit split valence bands and $c$. Using reasonable values of parameters, we show that the $E_1+{\Delta }_1$ transition in ER is weakened to the point where it appears only as a subsidiary oscillation, qualitatively in agreement with experiment. Other experimental evidence for the $k$-linear term includes the absence of a low-field limit for the $E_1$ structure, which is believed to arise from the strong nonparabolicity of the valence bands near the $〈111〉$ axes. All $E_1$ data, including in addition ER polarization anisotropies and a comparison between ER and numerically differentiated third-derivative spectra, are consistent with a three-dimensional $M_1$ critical point (or point set) at or near $L$ and a three-dimensional $M_0$ critical point at $\Gamma$. A third, non-$\Lambda$ critical-point structure is observed slightly above the $M_1$ structure, too close to be resolved. Both theory and experiment yield no evidence for the anomalously small transverse reduced mass reported by Grover and Handler: the $E_1$, not the $E_1+{\Delta }_1$, transition dominates, and as may be expected from the nonparabolicity the "apparent" mass obtained from the ratio of the magnitudes of the ER and third-derivative spectra, both theoretical and experimental, is larger than that calculated without intravalence coupling. Threshold energies are obtained for all observed critical points. For the $E_2$ region, a numerically differentiated second-derivative line shape gives the best fit to the ER spectrum for reasons which are not clear.