Temperature-dependent photoluminescence in Ge: Experiment and theory

We report a photoluminescence study of high-quality Ge samples at temperatures $12\phantom{\rule{0.16em}{0ex}}\mathrm{K}\ensuremath{\le}T\ensuremath{\le}295\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, over a spectral range that covers phonon-assisted emission from the indirect gap (between the lowest conduction band at the $L$ point of the Brillouin zone and the top of the valence band at the \ensuremath{\Gamma} point), as well as direct gap emission (from the local minimum of the conduction band at the \ensuremath{\Gamma} point). The spectra display a rich structure with a rapidly changing line shape as a function of $T$. A theory is developed to account for the experimental results using analytical expressions for the contributions from LA, TO, LO, and TA phonons. Coupling of states exactly at the \ensuremath{\Gamma} and $L$ points is forbidden by symmetry for the latter two phonon modes, but becomes allowed for nearby states and can be accounted for using wave-vector dependent deformation potentials. Excellent agreement is obtained between predicted and observed photoluminescence line shapes. A decomposition of the predicted signal in terms of the different phonon contributions implies that near-room temperature indirect optical absorption and emission are dominated by ``forbidden'' processes, and the deformation potentials for allowed processes are smaller than previously assumed.