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.

💡 Research Summary

This paper addresses the puzzling discrepancy between recent reports of strong room‑temperature photoluminescence (PL) from hexagonal‑diamond (2H) germanium and the longstanding theoretical view that 2H‑Ge possesses only a very weak direct‑gap transition. The authors perform a comprehensive first‑principles study of the excitonic properties and intrinsic radiative lifetimes of pristine 2H‑Ge, uniaxially strained 2H‑Ge, and a series of 2H‑SiₓGe₁₋ₓ alloys (x = 1/6, 1/4, 1/2). Calculations are carried out using density‑functional theory (DFT) with a J‑parameter correction (DFT + J) calibrated to reproduce HSE06 hybrid‑functional band gaps, including spin‑orbit coupling. The resulting band structures serve as the basis for solving the Bethe‑Salpeter equation (BSE) with the Yambo code, yielding exciton energies, binding energies, dipole moments, and oscillator strengths for both in‑plane (⊥ c) and out‑of‑plane (∥ c) polarizations.

Key findings are as follows. Pristine 2H‑Ge exhibits a sizable exciton binding energy of ≈30 meV, considerably larger than that of cubic Ge (≈4 meV), owing to a larger reduced electron‑hole mass along the c‑axis and slightly lower dielectric constants (ε⊥c ≈ 14.35, ε∥c ≈ 14.84). However, the dipole matrix elements are extremely small (|µ|² ≈ 10⁻⁴ a₀²), leading to oscillator strengths below 10⁻⁶ and radiative lifetimes exceeding 10⁻⁴ s at low temperature and remaining on the order of 10⁻³ s at 300 K. This confirms the “pseudo‑direct” nature of 2H‑Ge: the transition is formally direct but optically dark.

Alloying with Si lifts symmetry‑imposed selection rules and raises the CBM+1 band, thereby enhancing the dipole moments by roughly one order of magnitude for x = 1/6 and up to two orders for x = 1/2. Consequently, the calculated radiative lifetimes drop to the 10⁻⁵ s range, still far longer than typical direct‑gap emitters.

Applying a modest 2 % uniaxial compressive strain along the c‑axis induces a band crossing between the Γ‑7ᶜ (original CBM+1) and Γ‑8ᶜ (original CBM) states. Although the fundamental gap changes only slightly, the lowest‑energy exciton now derives from a much stronger optical transition. The in‑plane dipole moment increases to |µ⊥c|² ≈ 10⁻² a₀², and the radiative lifetime collapses to the nanosecond regime (≈1 ns). This strained system approaches the radiative efficiency of wurtzite GaN (used here as a benchmark), albeit with a much lower exciton energy.

By thermally averaging over exciton states, the authors demonstrate that the temperature dependence of the radiative lifetime follows a T³⁄² law, reflecting the light‑cone population of exciton center‑of‑mass momenta. They also discuss the low‑temperature limit where momentum‑conserving recombination becomes frozen, a regime not captured by the present model.

The overarching conclusion is that the strong PL observed experimentally cannot be explained by intrinsic radiative recombination in an ideal 2H‑Ge crystal, whether pristine, Si‑alloyed, or modestly strained. Extrinsic factors—such as defects, surface states, localized strain fields, or quantum‑confinement effects—must be responsible for the reported brightness. Nevertheless, the study identifies strain engineering as a powerful route to dramatically improve the intrinsic light‑emission efficiency of 2H‑Ge‑based materials, potentially enabling practical optoelectronic devices based on group‑IV hexagonal phases.