Schottky-Barrier Electroreflectance: Application to GaAs

We describe a Schottky-barrier electroreflectance (ER) technique for making high-resolution optical spectroscopic measurements on semiconducting materials. When combined with recent line-shape theories of low-field ER spectra, the method provides order-of-magnitude improvement in resolution of structure and accuracy in the determination of critical-point energies and broadening parameters as compared to previous spectroscopic work on higher interband transitions. The Schottky-barrier technique is applied to GaAs, where separate critical-point contributions of $\ensuremath{\Gamma}$ and $\ensuremath{\Delta}$ symmetry in the ${{E}^{\ensuremath{'}}}_{0}$ triplet are resolved for the first time, together with all members of the quadruplet at $X$. We find the values of critical-point energies ${E}_{g}$ and broadening parameters $\ensuremath{\Gamma}$ for the following transitions at 4.2 \ifmmode^\circ\else\textdegree\fi{}K (all energies are in meV): ${E}_{0}(1517.7\ifmmode\pm\else\textpm\fi{}0.5,<0.3)$; ${E}_{0}+{\ensuremath{\Delta}}_{0}(1859\ifmmode\pm\else\textpm\fi{}1,6\ifmmode\pm\else\textpm\fi{}2)$; ${E}_{1}(3043.9\ifmmode\pm\else\textpm\fi{}1,28\ifmmode\pm\else\textpm\fi{}1)$; ${E}_{1}+{\ensuremath{\Delta}}_{1}(3263.6\ifmmode\pm\else\textpm\fi{}1,38\ifmmode\pm\else\textpm\fi{}2)$; ${{E}^{\ensuremath{'}}}_{0}$ triplet, $\ensuremath{\Gamma}$ symmetry: (4488 \ifmmode\pm\else\textpm\fi{} 10, 40 \ifmmode\pm\else\textpm\fi{} 5), (4659 \ifmmode\pm\else\textpm\fi{} 10, 30 \ifmmode\pm\else\textpm\fi{} 5), (5014 \ifmmode\pm\else\textpm\fi{} 15, 47 \ifmmode\pm\else\textpm\fi{} 10); ${{E}^{\ensuremath{'}}}_{0}$ triplet, ${M}_{1}$ transitions, $\ensuremath{\Delta}$ symmetry: (4529 \ifmmode\pm\else\textpm\fi{} 10, 36 \ifmmode\pm\else\textpm\fi{} 5) and (4712 \ifmmode\pm\else\textpm\fi{} 10, 34 \ifmmode\pm\else\textpm\fi{} 5); ${E}_{2}$ complex, $\ensuremath{\Sigma}: (5137\ifmmode\pm\else\textpm\fi{}10,104\ifmmode\pm\else\textpm\fi{}10)$; ${E}_{2}$ complex, $X$ quadruplet: (4937 \ifmmode\pm\else\textpm\fi{} 10, 47 \ifmmode\pm\else\textpm\fi{} 10), (5014 \ifmmode\pm\else\textpm\fi{} 10, 47 \ifmmode\pm\else\textpm\fi{} 10), (5339 \ifmmode\pm\else\textpm\fi{} 10, 48 \ifmmode\pm\else\textpm\fi{} 10), (5415 \ifmmode\pm\else\textpm\fi{} 15, 50 \ifmmode\pm\else\textpm\fi{} 15). These values enable us to determine the following spin-orbit-splitting energies: ${\ensuremath{\Delta}}_{0}=341\ifmmode\pm\else\textpm\fi{}2$ meV, ${\ensuremath{\Delta}}_{1}=220\ifmmode\pm\else\textpm\fi{}2$ meV, ${{\ensuremath{\Delta}}^{\ensuremath{'}}}_{0} (\mathrm{at} \ensuremath{\Gamma})=171\ifmmode\pm\else\textpm\fi{}15$ meV, ${{\ensuremath{\Delta}}^{\ensuremath{'}\ensuremath{'}}}_{0}(\mathrm{at} \ensuremath{\Delta})=183\ifmmode\pm\else\textpm\fi{}15$ meV, and ${\ensuremath{\Delta}}_{2}=77\ifmmode\pm\else\textpm\fi{}10$ meV. The splitting of the lower conduction bands at $X$ due to the antisymmetric potential is ${{\ensuremath{\Delta}}^{\ensuremath{'}}}_{2}=402\ifmmode\pm\else\textpm\fi{}10$ meV. The ${{E}^{\ensuremath{'}}}_{0}$ transitions of $\ensuremath{\Delta}$ symmetry are shown to lie about 10% of the way from $\ensuremath{\Gamma}$ to $X$. By comparing the period of the large number of Franz-Keldysh oscillations observed at the ${E}_{1}+{\ensuremath{\Delta}}_{1}$ transition with those of the ${E}_{0}+{\ensuremath{\Delta}}_{0}$ transitions observed in the high-field measurements, we determine a value ${\ensuremath{\mu}}_{T}=(0.055\ifmmode\pm\else\textpm\fi{}008){m}_{e}$ for the transverse reduced mass at ${E}_{1}+{\ensuremath{\Delta}}_{1}$. These results are compared to previous experimental measurements and to calculated energy-band structures for GaAs. The determination of critical-point symmetry in surface-barrier geometries in terms of the transformation properties of the third- and fourth-rank low-field ER line-shape tensors is also discussed. Finally, the vanishing of an ER spectrum at a hyperbolic critical point, a reduced-mass effect predicted by the general theory of the Franz-Keldysh effect, is observed for the first time.