Complex Low Energy Tetrahedral Polymorphs of Group IV Elements from First Principles

The energy landscape of carbon is exceedingly complex, hosting diverse and important metastable phases, including diamond, fullerenes, nanotubes, and graphene. Searching for structures, especially those with large unit cells, in this landscape is challenging. Here we use a combined stochastic search strategy employing two algorithms (ab initio random structure search and random sampling strategy combined with space group and graph theory) to apply connectivity constraints to unit cells containing up to 100 carbon atoms. We uncover three low energy carbon polymorphs ($Pbam\text{−}32$, $P6/mmm$, and $I\overline{4}3d$) with new topologies, containing 32, 36, and 94 atoms in their primitive cells, respectively. Their energies relative to diamond are 96, 131, and $112\mathrm{meV}/\mathrm{atom}$, respectively, which suggests potential metastability. These three carbon allotropes are mechanically and dynamically stable, insulating carbon crystals with superhard mechanical properties. The $I\overline{4}3d$ structure possesses a direct band gap of 7.25 eV, which is the widest gap in the carbon allotrope family. Silicon, germanium, and tin versions of $Pbam\text{−}32$, $P6/mmm$, and $I\overline{4}3d$ also show energetic, dynamical, and mechanical stability. The computed electronic properties show that they are potential materials for semiconductor and photovoltaic applications.

Figure 1

The perspective crystalline views of the optimized carbon-version $Pbam\text{−}32$ (a) and $P6/mmm$, (b), as well as those of $I\overline{4}3d$ in both crystalline cell (c) and primitive cell (d). Balls in different colors indicate different inequivalent carbon atoms.

Figure 2

Scatterplot of the PBE-based relative average energy (eV/atom) against equilibrium volume (${\AA }^3$) of the carbon version $Pbam\text{−}32$, $P6/mmm$, and $I\overline{4}3d$ predicted in this work (red squares), the $s{p}^3$ carbons previously predicted in the past decades [4] (black five-point stars), as well as those $s{p}^3$ carbons discovered in our recent stochastic search [5] (blue circles). Some well-known carbon phases (green triangles) are also plotted in for comparison.

Figure 3

The HSE06-based electronic band structures and density of states (DOS) of the newly discovered $Pbam\text{−}32$, $P6/mmm$, and $I\overline{4}3d$ structures as carbon (a) and silicon (b) at zero pressure.