Low-energy tetrahedral polymorphs of carbon, silicon, and germanium

Searches for low-energy tetrahedral polymorphs of carbon and silicon have been performed using density functional theory computations and the ab initio random structure searching approach. Several of the hypothetical phases obtained in our searches have enthalpies that are lower or comparable to those of other polymorphs of group 14 elements that have either been experimentally synthesized or recently proposed as the structure of unknown phases obtained in experiments, and should thus be considered as particularly interesting candidates. A structure of $Pbam$ symmetry with 24 atoms in the unit cell was found to be a low-energy, low-density metastable polymorph in carbon, silicon, and germanium. In silicon, $Pbam$ is found to have a direct band gap at the zone center with an estimated value of 1.4 eV, which suggests applications as a photovoltaic material. We have also found a low-energy chiral framework structure of $P{4}_1{2}_{1}2$ symmetry with 20 atoms per cell containing fivefold spirals of atoms, whose projected topology is that of the so-called Cairo-type two-dimensional pentagonal tiling. We suggest that $P{4}_1{2}_{1}2$ is a likely candidate for the structure of the unknown phase XIII of silicon. We discuss $Pbam$ and $P{4}_1{2}_{1}2$ in detail, contrasting their energetics and structures with those of other group 14 elements, particularly the recently proposed $P{4}_2/ncm$ structure, for which we also provide a detailed interpretation as a network of tilted diamondlike tetrahedra.

Figure 1

Enthalpy-pressure relations for various structural phases of carbon calculated using the PBE. The enthalpies are given with respect to the zero-pressure C-cd phase.

Figure 2

Enthalpy-pressure relations for various structural phases of silicon calculated using the PBE. The enthalpies are given with respect to the zero-pressure Si-cd phase.

Figure 3

Enthalpy-pressure relations for various structural phases of germanium calculated using the PBE. The enthalpies are given with respect to the zero-pressure Ge-cd phase.

Figure 4

Simulated powder XRD patterns of the $P{4}_2/ncm,P{4}_1{2}_{1}2$, and $Pbam$ phases of Si at zero pressure, calculated within the PBE approximation. The x-ray wavelength used was 0.44864 Å.

Figure 5

Construction of $P{4}_2/ncm$ as a network of tilted diamondlike tetrahedra. Panel (a) shows a single slab of corner-sharing regular tetrahedra of the diamond structure viewed along the (001) direction. The projected tetrahedra and conventional cubic cell are shown in black. The centers of the tetrahedra are shown in red and their corners in blue, with the different sizes of the atoms and varying thickness of the bonds used to give a sense of perspective. Panels (b) and (c) show two adjacent slabs (lower and upper, respectively) of tilted tetrahedra that alternate stacked along the $z$ axis to give the $P{4}_2/ncm$ structure, as shown in (d) [80]. Note the opposite orientation of the tetrahedral bonds and the tilting of the tetrahedra in the two adjacent slabs. Intraslab bonds (parallel to the plane of projection), resulting from the tilting of the tetrahedral units within each slab, are represented by blue rods, while interslab bonds, arising from the stacking of the slabs, are represented by green rods. Intraslab bonding leads to characteristic fivefold rings, one of which is depicted in blue in (b), while interslab bonding leads to buckled sixfold rings, two of which are shown in profile in (d).

Figure 6

A slab of the $Pbam$ structure viewed along the $z$ axis, with unit cells shown in black and a modicum of perspective so as to reveal the connectivity among sites. The six inequivalent sites are labeled and shown in different colors: the 4g-type sites are depicted in three different hues of red, while blue is used for the three 4h-type sites. The fivefold and sevenfold rings of the structure (as well as sixfold diamondlike rings) are all visible in this projection. Note that the red 4g sites and the blue 4h sites are each in different projection planes a distance $c/2$ apart.

Figure 7

A slab of the $P{4}_1{2}_{1}2$ structure of Si at zero pressure, viewed along the direction of its fourfold rotation axis. A modicum of perspective has been introduced to facilitate visualization of the connectivity among sites. The three inequivalent sites are labeled and shown in different colors.

Figure 8

Electronic band structures and densities of states of the $Pbam,P{4}_1{2}_{1}2$, and $P{4}_2/ncm$ structures of Si at zero pressure. The valence bands are shown in red and the conduction bands in blue, and the band-gap regions are indicated by parallel black lines. The zeros of energy are at the top of the valence band at the zone center $\mathrm{\Gamma }$ point. The panel on the right of each plot shows the total density of states.

Figure 9

Calculated phonon dispersion relations of the $Pbam,P{4}_1{2}_{1}2$, and $P{4}_2/ncm$ structures of Si at zero pressure. The panel on the right of each plot shows the total phonon density of states.

Figure 10

Calculated Raman spectra of the $P{4}_2/ncm,P{4}_1{2}_{1}2$, and $Pbam$ structures of Si at zero pressure and 300 K, calculated within the PBE. The vertical dotted line indicates the calculated frequency of the single intense Raman mode of the Si-cd phase.