${\mathit{sp}}^3$-hybridized framework structure of group-14 elements discovered by genetic algorithm

Group-14 elements, including C, Si, Ge, and Sn, can form various stable and metastable structures. Finding new metastable structures of group-14 elements with desirable physical properties for new technological applications has attracted a lot of interest. Using a genetic algorithm, we discovered a new low-energy metastable distorted $s{p}^3$-hybridized framework structure of the group-14 elements. It has $P$$4_2/\mathit{mnm}$ symmetry with 12 atoms per unit cell. The void volume of this structure is as large as ${139.7\AA }^3$ for Si $P$$4_2/\mathit{mnm}$, and it can be used for gas or metal-atom encapsulation. Band-structure calculations show that $P$$4_2/\mathit{mnm}$ structures of Si and Ge are semiconducting with energy band gaps close to the optimal values for optoelectronic or photovoltaic applications. With metal-atom encapsulation, the $P$$4_2/\mathit{mnm}$ structure would also be a candidate for rattling-mediated superconducting or used as thermoelectric materials.

Figure 1

Energy versus density of Si structures from a genetic algorithm search. ${\rho }_{\mathrm{dia}}$ and $E_{\mathrm{dia}}$ stand for the density and energy of the Si diamond structure. The known and proposed Si structures are marked for comparison.

Figure 2

(a) Crystal structure of the $P$$4_2/\mathit{mnm}$ structure with 12 atoms per unit cell. (b) The $X_{24}$ rugby-ball units of the $P$$4_2/\mathit{mnm}$ structure with their spatially relative positions. (c) The top view along lattice vector c of the repeated $P$$4_2/\mathit{mnm}$ structure with the directions of rugby balls indicated by (blue or red) arrows. The (blue or red) spheres are the group-14 element atoms and the thin (black) solid lines are the unit-cell boundaries.

Figure 3

Cohesive energies of the known and newly found $P$$4_2/\mathit{mnm}$ structures of Si in empirical potentials (Tersoff, Stillinger-Weber, and MEAM) and DFT calculations. Dia and Hex-Dia stand for diamond and hexagonal diamond.

Figure 4

The phonon band structures of $P$$4_2/\mathit{mnm}$ C, Si, Ge, and Sn structures.

Figure 5

The electronic band structures and the electronic density of states (states/eV/unit cell) of $P$$4_2/\mathit{mnm}$ C, Si, Ge, and Sn structures.

Figure 6

(a) Density of states of metal-atom-encapsulated $P$$4_2/\mathit{mnm}$ Si structures. (b) The potential energies of encapsulated metal atoms along the longest diameter of the rugby ball of the $P$$4_2/\mathit{mnm}$ Si structure.