Exciton radiative lifetimes in hexagonal diamond Ge and Si$_x$Ge$_{1-x}$ alloys

Abstract

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.

Figures (4)

• Figure 1: Electronic band structure of 2H-Ge near the band extrema, computed with the HSE06 hybrid functional (black lines) and within the DFT+$J$ method, with $J=23$ eV (red lines).
• Figure 2: Band structures of the studied materials: (a) pristine 2H-Ge, (b) 2H-Ge under $\epsilon_z=2\%$ uniaxial strain along the $c$ axis, (c) 2H-Si$_{1/6}$Ge$_{5/6}$ alloy, unfolded onto the 2H-Ge BZ, (d) 2H-Si$_{1/4}$Ge$_{3/4}$ alloy, unfolded, (e) 2H-Si$_{1/2}$Ge$_{1/2}$ alloy, unfolded, (f) GaN. The orange shading highlights the electronic states contributing to the lowest-energy exciton state. Horizontal lines mark the electronic bandgaps.
• Figure 3: Absorption spectra, $\mathrm{Im}(\varepsilon)$, and squared moduli of the exciton dipole moments, $|\mu_{S,\alpha}|^2$, in units of the Bohr radius squared, for 2H-Ge in panel (a), 2H-Ge under $\epsilon_z = 2\%$ uniaxial strain in (b), the 2H-Si$_{1/6}$Ge$_{5/6}$ alloy in (c), the 2H-Si$_{1/4}$Ge$_{3/4}$ alloy in (d), the 2H-Si$_{1/2}$Ge$_{1/2}$ alloy in (e), and GaN in (f). For each material, results are shown for in-plane ($\bot c$) and out-of-plane ($\| c$) light polarizations. The dashed vertical lines indicate the electronic bandgaps.
• Figure 4: Temperature-averaged exciton radiative lifetimes, Eq. (\ref{['eq:tau_avg']}), of the studied materials. Left panel, from top to bottom: temperature dependence of $\langle\tau\rangle(T)$ for pristine 2H-Ge, 2H-Si$_{1/6}$Ge$_{5/6}$ and 2H-Si$_{1/4}$Ge$_{3/4}$ alloys, uniaxially strained 2H-Ge and GaN. The dashed black line shows an exact $T^{3/2}$ trend. Right panel: comparison of the materials' average radiative lifetimes at $T=10$ K.