Onsite matrix elements of the tight-binding Hamiltonian of a strained crystal: Application to silicon, germanium, and their alloys

We discuss a model for the onsite matrix elements of the $s{p}^3{d}^5{s}^{\ast }$ tight-binding Hamiltonian of a strained diamond or zinc-blende crystal or nanostructure. This model features onsite, off-diagonal couplings among the $s$, $p$, and $d$ orbitals and is able to reproduce the effects of arbitrary strains on the band energies and effective masses in the full Brillouin zone. It introduces only a few additional parameters and is free from any ambiguities that might arise from the definition of the macroscopic strains as a function of the atomic positions. We apply this model to silicon, germanium, and their alloys as an illustration. In particular, we make a detailed comparison of tight-binding and ab initio data on strained Si, Ge, and SiGe.

Figure 1

(Color online) Band structure of bulk, unstrained silicon in the $s{p}^3{d}^5{s}^{\ast }$ TB and $GW$ approximations.

Figure 2

(Color online) Band structure of bulk, unstrained germanium in the $s{p}^3{d}^5{s}^{\ast }$ TB and $GW$ approximations.

Figure 3

(Color online) Band structure of bulk, unstrained ${\text{Si}}_{0.5}{\text{Ge}}_{0.5}$ in the $s{p}^3{d}^5{s}^{\ast }$ TB and $GW$ approximations.

Figure 4

(Color online) (a) [001], (b) [111], and (c) [110] biaxial strain behaviors of bulk silicon (Ref. 71). The black diamonds are the ab initio data (Ref. 72).

Figure 5

(Color online) (a) [001], (b) [111], and (c) [110] biaxial strain behaviors of bulk germanium (Ref. 71). The (red) dots, (blue) squares, and (magenta) triangles are the $L$, $\Gamma$, and $\Delta$ valleys, respectively. The black diamonds are the ab initio data (Ref. 72).

Figure 6

(Color online) Lowest two conduction bands (plotted along the $\Gamma X$ axis) of bulk silicon under the [111] biaxial strain. The conduction-band minimum is marked with a dot. Its motion with strain is plotted as a dotted gray line. The crosses are the ab initio data (also see Fig. 7).

Figure 7

(Color online) Conduction-band, longitudinal effective mass of bulk silicon under the [111] biaxial strain. Inset: position of the conduction-band minimum along the $\Gamma X$ axis. The black diamonds are the ab initio data.

Figure 8

(Color online) Band-gap energy of random bulk ${\text{Si}}_{1-x}{\text{Ge}}_x$ alloys as a function of the Ge mole fraction $x$. Inset: lattice parameter of the alloy. The solid line is Dismukes’s law (Ref. 76) $a\left(x\right)=5.431+0.2x+0.027x^2\text{Å}$.

Figure 9

(Color online) (a) Band-gap energy of random ${\text{Si}}_{1-x}{\text{Ge}}_x$ alloys biaxially strained on Si [001] as a function of the Ge mole fraction $x$. The inset is the TB valence-band discontinuity $\Delta E_v$ (dots) compared to various experimental sources (diamonds) compiled in Ref. 81. (b) Band-gap energy of random ${\text{Si}}_{1-x}{\text{Ge}}_x$ alloys biaxially strained on Si [110].