Stability and electronic structure of two-dimensional allotropes of group-IV materials

We study six different two-dimensional (2D) allotropes of carbon, silicon, germanium, and tin by means of the ab initio density functional theory for the ground state and approximate methods to calculate their electronic structures, including quasiparticle effects. Four of the investigated allotropes are based on dumbbell geometries, one on a kagome lattice, and one on the graphenelike hexagonal structure for comparison. Concerning carbon, our calculations of the cohesive energies clearly show that the hexagonal structure (graphene) is most stable. However, in the case of Si and Ge, the dumbbell structures, particularly the large honeycomb dumbbell (LHD) geometries, are energetically favored compared to the $s{p}^2/s{p}^3$-bonded hexagonal lattice (i.e., silicene and germanene). The main reason for this is the opening of a band gap in the honeycomb dumbbell arrangements. The LHD sheet crystals represent indirect semiconductors with a $K\to \mathrm{\Gamma }$ gap of about 0.5 eV. In the Sn case we predict the ${\mathrm{MoS}}_2$-like symmetry to be more stable, in contrast to the stanene and LHD geometries predicted in literature. Our results for freestanding group-IV layers shine new light on recent experimental studies of group-IV overlayers on various substrates.

Figure 1

Top and side view of the six 2D bonding geometries with hexagonal or trigonal symmetry HEX, LHD, TD, HD, ${\mathrm{MoS}}_2$, and KL studied in this work. The lattice parameters $a, {\delta }_1$, and ${\delta }_2$ indicate the unit cell size and the largest and smallest layer buckling, respectively. Group-IV atoms out of the plane are indicated by yellow spheres.

Figure 2

Cohesive energies of group-IV materials of different 2D allotropes compared with bulk diamond geometries.

Figure 3

GGA (dotted blue lines) and GGA-1/2 (solid black lines) band structures of carbon, silicon, germanium, and tin for the allotropes diamond, HEX, LHD, HD, and KL. The top of the valence bands or the Fermi level is taken as zero energy (red horizontal lines).