Exciton radiative lifetimes in hexagonal diamond Ge and Si$_x$Ge$_{1-x}$ alloys
Recent reports of strong room-temperature photoluminescence in hexagonal diamond (2H) germanium stand in marked contrast to theoretical predictions of very weak band-edge optical transitions. Here we address radiative emission in 2H-Ge and related materials through a comprehensive investigation of their excitonic properties and radiative lifetimes, performing Bethe-Salpeter calculations on pristine and uniaxially strained 2H-Ge, 2H-Si$_x$Ge$_{1-x}$ alloys with $x=\frac{1}{6},\,\frac{1}{4},\,\frac{1}{2}$, and wurtzite GaN as a reference. Pristine 2H-Ge features sizable exciton binding energies ($\sim\!30$ meV) but extremely small dipole moments, yielding radiative lifetimes above $10^{-4}$ s. Alloying with Si reduces the lifetime by nearly two orders of magnitude, whereas a 2% uniaxial strain along the $c$ axis induces a band crossover that strongly enhances the in-plane dipole moment of the lowest-energy exciton and drives the lifetime down to the nanosecond scale. Although strained 2H-Ge approaches the radiative efficiency of GaN, its much lower exciton energy prevents a full match. These results provide the missing excitonic description of 2H-Ge and 2H-Si$_x$Ge$_{1-x}$, demonstrating that, even when excitonic effects are fully accounted for, the strong photoluminescence reported experimentally cannot originate from the ideal crystal.