First-principles insight in structure-property relationships of hexagonal Si and Ge polytypes

Hexagonal SiGe is a promising material for combining electronic and photonic technologies. In this paper, the energetic, structural, elastic, and electronic properties of the hexagonal polytypes ($2H, 4H$, and $6H$) of silicon and germanium are thoroughly analyzed under equilibrium conditions. For this purpose, we apply state-of-the-art density functional theory. The phase diagram, obtained in the framework of a generalized Ising model, shows that the diamond structure is the most stable under ambient conditions, but hexagonal modifications are close to the phase boundary, especially for Si. Our band structure calculations using the modified-Becke-Johnson–local-density-approximation (MBJLDA) and Heyd-Scuseria-Ernzerhof (HSE06) exchange-correlation functionals predict significant changes in electronic states with hexagonality. While Si crystals are always semiconductors with indirect band gaps, the hexagonal Ge polytypes have direct band gaps. The branch-point energies of the Si polytypes appear in the fundamental gaps, while for the Ge crystals they are below the valence band maxima. Band alignment based on the branch-point energy leads to type-I heterocrystalline interfaces between Ge polytypes, where electrons and holes can be trapped in the layer with the higher hexagonality.

Figure 1

Bond stacking along the [0001] direction in the hexagonal unit cells of the four polytypes. Group-IV atoms are displayed as red spheres. Bonds in the ($11\overline{2}0$) plane are indicated by thick black lines. The cubic ($c$) or hexagonal ($h$) character of each bilayer is defined by the nonparallel bond in the plane. The signs $+$ and $-$ denote the orientation of the bilayer. For a more detailed explanation, see Ref. [45]. The primitive basis vectors ${\mathit{a}}_i$ ($i=1,2,3$) are also shown.

Figure 2

Dependence on the hexagonality of (a) structural and (b) elastic properties of Si (dashed lines) and Ge (solid lines) polytypes.

Figure 3

(a) Total energies of the polytypes vs hexagonality with respect to $3C$. (b) Phase diagram of polytypes from the ANNNI model. The solid lines represent the phase boundaries between the parameter regions where the indicated phases are the most stable. The blue and red dots represent the group-IV materials that favor the $3C$ polytype. For comparison the $4H$ equilibrium structure of SiC [45] is also displayed.

Figure 4

Approximate QP band structures of the four Si polytypes (a) $2H$, (b) $4H$, (c) $6H$, and (d) $3C$ along high-symmetry lines, calculated within MBJLDA (red dashed lines) and HSE06 (black lines). The right panels show zooms on an energy interval around the fundamental gap and the BZ center, as indicated in the left panels by a black rectangle. The BP is used as energy zero, and BPs are indicated by black horizontal lines in the right panels of (b) and (c). The insets show the lowest conduction bands in MBJLDA and HSE06 on the $\mathrm{\Gamma }-M-L$ lines. The irreducible representations of high-symmetry states around the lowest conduction bands and highest valence bands are given in the double-group notation of Koster et al. [107] for $2H$. Because of zone-folding arguments the denotation of $2H$ is also applied in the cases of $4H$ and $6H$. Ab initio band parities at $\mathrm{\Gamma }$ are also displayed for all hexagonal polytypes. The gap regions are shaded in gray.

Figure 5

Approximate QP band structures of the four Ge polytypes (a) $2H$, (b) $4H$, (c) $6H$, and (d) $3C$ along high-symmetry lines, calculated within MBJLDA (red dashed lines) and HSE06 (black lines). The right panels show zooms on an energy interval around the fundamental gap and the BZ center, as indicated in the left panels by a black rectangle. The BP is used as energy zero, and BPs are indicated by black horizontal lines in the right panels of (b) and (c). The insets show the lowest conduction bands in MBJLDA and HSE06 on the $M-L$ line. The irreducible representations of the high-symmetry states around the lowest conduction bands and highest valence bands are given in the double-group notation of Koster et al. [107] for $2H$. Because of zone-folding arguments the denotation of $2H$ is also applied in the cases of $4H$ and $6H$. Ab initio band parities at $\mathrm{\Gamma }$ are also displayed for all hexagonal polytypes. The gap regions are shaded in gray.

Figure 6

Valence band parameters (a) ${\mathrm{\Delta }}_{cf}$, (b) ${\mathrm{\Delta }}_{\mathrm{SO}\parallel }$, and (c) ${\mathrm{\Delta }}_{\mathrm{SO}\perp }$ vs hexagonality of the polytypes in MBJLDA (solid lines) and HSE06 (dashed lines) for Si (red) and Ge (blue).

Figure 7

Conduction [direct (dir) in red and indirect (ind) in gray] and valence (blue) band edges of hexagonal Si (a) and Ge (b) polytypes aligned by their branch-point energies (black horizontal lines). Dashed (solid) lines are computed using the MBJLDA (HSE06) functional.