Accurate electronic and optical properties of hexagonal germanium for optoelectronic applications

High-quality defect-free lonsdaleite Si and Ge can now be grown on hexagonal nanowire substrates. These hexagonal phases of group-IV semiconductors have been predicted to exhibit improved electronic and optical properties for optoelectronic applications. While lonsdaleite Si is a well-characterized indirect semiconductor, experimental data and reliable calculations on lonsdaleite Ge are scarce and not consistent regarding the nature of its gap. Using ab initio density-functional theory, we calculate accurate structural, electronic, and optical properties for hexagonal Ge. Given the well-known sensitivity of electronic-structure calculations for Ge to the underlying approximations, we systematically test the performance of several exchange-correlation functionals, including meta-GGA and hybrid functionals. We first validate our approach for cubic Ge, obtaining atomic geometries and band structures in excellent agreement with available experimental data. Then, the same approach is applied to predict electronic and optical properties of lonsdaleite Ge. We portray lonsdaleite Ge as a direct semiconductor with only weakly dipole-active lowest optical transitions, small band gap, huge crystal-field splitting, and strongly anisotropic effective masses. The unexpectedly small direct gap and the oscillator strengths of the lowest optical transitions are explained in terms of symmetry and backfolding of energy bands of the diamond structure.

Figure 1

Atomic bilayer stacking in the lonsdaleite (2H) and diamond-structure (3C) phase of Ge. The figures of the crystal structures have been produced with Vesta [14].

Figure 2

Band structure of 3C-Ge computed with the HSE06 and MBJLDA functionals (a). The irreducible representations of relevant high-symmetry states in the band-gap region are given in the double-group notation of Koster et al. [73] (b). The VBM is set to zero.

Figure 3

Band structure of 2H-Ge computed with the HSE06 and MBJLDA functionals (a). The irreducible representations of relevant high-symmetry states in the band-gap region are given in the double-group notation of Koster et al. [73] (b). The VBM is set to zero.

Figure 4

Optical transition matrix elements $\left|p\right|$ of the lowest band-to-band transitions in 2H-Ge along high-symmetry lines in the vicinity of $\mathrm{\Gamma }$. The matrix elements have been calculated for ordinary ($\mathbf{E}\perp \mathbf{c}$) and extraordinary ($\mathbf{E}\left|\right|\mathbf{c}$) light polarization using the HSE06 and the MBJLDA functionals, respectively.

Figure 5

Radiative lifetime $\tau$ versus temperature for 3C-Ge and 2H-Ge calculated at the $\mathrm{\Gamma }$ point only (dashed lines) or integrating over the whole BZ (solid lines). Results are given for the HSE06 and MBJLDA functionals.