Energy-band structure of Ge, Si, and GaAs: A thirty-band $\mathbf{k}\mathbf{\bullet }\mathbf{p}$ method

A 30-band $\mathbf{k}\mathbf{\bullet }\mathbf{p}$ method taking into account spin-orbit coupling is used to describe the band diagram of Ge, Si, and GaAs over the whole Brillouin zone on an extent of $5\mathrm{eV}$ above and $6\mathrm{eV}$ under the top of the valence band. The band diagrams provide effective masses in agreement with experimental data both for direct gap semiconductors (GaAs) and for indirect gap semiconductors (Ge, Si). This method also gives explicit expressions for Luttinger parameters and effective masses in the $\Gamma$ valley.

Figure 1

Wave functions in the 30-band $\mathbf{k}\mathbf{\bullet }\mathbf{p}$ model at $k=0$ for $O_h$ group. $T_d$ simple group notations are reported on the left, and $T_d$ double group notations on the right. Only ten matrix elements are enough to describe the band diagram.

Figure 2

$T_d$ group: Additional matrix elements and spin-orbit coupling due to the lack of symmetry in $T_d$ group. For the simulation of GaAs, all these elements are taken null except $P^{\prime }$ and ${\Delta }^{\prime }$. ${\Delta }_j^{\prime }$ are interband spin-orbit couplings.

Figure 3

Band diagram of bulk Si at $T=0\mathrm{K}$. Spin-orbit coupling is taken into account even if it does not appear on the diagram because of its scale. ${\Delta }_{\mathrm{so}}=44\mathrm{meV}$ in Si.

Figure 4

Band diagram of Ge at $T=0\mathrm{K}$.

Figure 5

Band diagram of GaAs at $T=0\mathrm{K}$.

Figure 6

Density of states in the valence band for Si. This density of states is compared to empirical pseudopotential (Ref. 19) and $s{p}^3{d}^5$ tight binding (Ref. 23) methods.

Figure 7

Comparison of Si valence band obtained by Empirical Pseudopotential (Ref. 19), $s{p}^3{d}^5$ tight binding (Ref. 23), and 30-band $\mathbf{k}\mathbf{\bullet }\mathbf{p}$ methods.