From pseudo-direct hexagonal germanium to direct silicon-germanium alloys

We present ab initio calculations of the electronic and optical properties of hexagonal ${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ alloys in the lonsdaleite structure. Lattice constants and electronic band structures in excellent agreement with experiment are obtained using density-functional theory. Hexagonal Si has an indirect band gap, whereas hexagonal Ge has a pseudo-direct gap, i.e., the optical transitions at the minimum direct band gap are very weak. The pseudo-direct character of pure hexagonal Ge is efficiently lifted by alloying. Already for a small admixture of Si, symmetry reduction enhances the oscillator strength of the lowest direct optical transitions. The band gap is direct for a Si content below 45 %. We validate lonsdaleite group-IV alloys to be efficient optical emitters, suitable for integrated optoelectronic applications.

Figure 1

Alloy-averaged excess energy $\langle E^{\mathrm{excess}}\rangle \left(x\right)$ of hex-${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ as a function of composition (solid line). Dots indicate the excess energies $E_j^{\mathrm{excess}}$ of the individual clusters with the spread of values at each composition highlighted by the shaded region.

Figure 2

Alloy-averaged lattice parameters $\langle a\rangle \left(x\right)$ and $\langle c\rangle \left(x\right)$ of hex-${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$. The very narrow shaded regions indicate the range of lattice constants obtained for the individual clusters at a given composition. Solid squares and triangles represent experimental values [5, 8] for $a$ and $c$, respectively. Experimental error bars are smaller than the used symbols.

Figure 3

Band structures of hex-${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ clusters for Ge-rich stoichiometries. We plot the unfolded band structure of the eight-atom cluster cell with the lowest gap (blue dots) for each composition as well as the band structures of hex-Si (green) and hex-Ge (red) as a guide to the eye [(a)–(d)]. Additionally, for $n_j/n=3/8$, also the unfolded band structure of the cluster with the lowest direct gap is shown (e).

Figure 4

(a) Alloy-averaged band gaps as a function of composition. Shaded regions show the range of gap values obtained for the individual clusters at a given composition. Vertical dashed (dotted) lines indicate the direct-to-indirect gap transition for the random alloy (the lowest-gap clusters). (b) Radiative lifetime as a function of composition (solid line). Dots indicate the results for the individual clusters with the range at each composition highlighted by the shaded region.