Dependence of electronic properties of germanium on the in-plane biaxial tensile strains

The hybrid HSE06 functional with the spin-orbit coupling effects is used to calculate the habituation of the electronic properties of Ge on the (0 0 1), (1 1 1), (1 0 1) in-plane biaxial tensile strains (IPBTSs). Our motivation is to explore the nature of electronic properties of tensile-strained Ge on different substrate orientations. The calculated results demonstrate that one of the most effective and practical approaches for transforming Ge into a direct transition semiconductor is to introduce (0 0 1) IPBTS to Ge. At 2.3% (0 0 1) IPBTS, Ge becomes a direct bandgap semiconductor with 0.53 eV band gap, in good agreement with the previous theoretical and experimental results. We find that the (1 1 1) and (1 0 1) IPBTSs are not efficient since the shear strain and inner displacement of atoms introduced by them quickly decrease the indirect gap of Ge. By investigating the dependence of valence band spin-orbit splitting on strain, we prove that the dependency relationship and the coupled ways between the valence-band states of tensile-strained Ge are closely related to the symmetry of strain tensor, i.e., the symmetry of the substrate orientation. The first- and second-order coefficients describing the dependence of indirect gap, direct gap, the valence band spin-orbit coupling splitting, and heavy-hole-light-hole splitting of Ge on IPBTSs have been obtained by the least squares polynomial fitting. These coefficients are significant to quantitatively modulate the electronic properties of Ge by tensile strain and design tensile-strained Ge devices by semiconductor epitaxial technique.