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We present a comprehensive first-principles investigation of optical,transport, and thermoelectric properties of pure and doped hexagonalSi$_x$Ge$_{1-x}$ alloys based on density-functional theory calculations, theBoltzmann transport equation, and the generalized quasi-chemical approximationto obtain alloy averages of electronic properties. At low temperature, phasedecomposition into the hexagonal elementary crystals is thermodynamicallyfavored, but around and above room temperature random alloys are predicted tobe stable. While hexagonal Si has an indirect band gap, the gap of hexagonal Geis direct with very weak optical transitions at the absorption edge. The alloyband gap remains direct for a Si content below 45\,\% and the oscillatorstrength of the lowest optical transitions is efficiently enhanced by alloying.The optical spectra show clear trends and both absorption edges and prominentpeaks can be tuned with composition. The dependence of transport coefficientson carrier concentration and temperature is similar in cubic and hexagonalalloys. However, the latter display anisotropic response due to the reducedhexagonal symmetry. In particular, the transport mass exhibits a significantdirectional dependence. Seebeck coefficients and thermoelectric power factorsof $n$-doped alloys show non-monotonous variations with the Si contentindependently of temperature.

Ensemble averages of ab initio optical, transport, and thermoelectric properties of hexagonal Si$_x$Ge$_{1-x}$ alloys