Inverse-photoemission study of unoccupied electronic states in Ge and Si: Bulk energy bands

We have measured k-resolved inverse-photoemission spectra of the Ge(111)2×1 and Si(111)2×1 surfaces. The main spectral features are well described in terms of direct transitions to the three lowest conduction bands along ΓL and a transition to a surface resonance near the conduction-band minimum via a surface umklapp process. An extra feature reflects a maximum in the density of empty states. Several critical points are determined, e.g., for Ge, $L_1$ at 0.7 eV, $L_3$ at 4.2 eV, and $L_2^{\mathcal{’}}$ at 7.9 eV and a higher L point at 11 eV, and for Si, $L_1$ at 2.4 eV and $L_3$ at 4.15 eV above the valence-band maximum. By comparing our inverse-photoemission results with those from photoemission and optical spectroscopy we find that the $E_1$ transition could be lowered by the electron-hole interaction by up to 0.15 eV for Ge and 0.5 eV for Si. A comparison with first-principles and empirical band calculations is made.