Structural, vibrational, and electronic properties of single-layer hexagonal crystals of group IV and V elements

Using first-principles density functional theory calculations, we investigate a family of stable two-dimensional crystals with chemical formula $A_2{B}_2$, where $A$ and $B$ belong to groups IV and V, respectively ($A=\mathrm{C}$, Si, Ge, Sn, Pb; $B=\mathrm{N}$, P, As, Sb, Bi). Two structural symmetries of hexagonal lattices $P\overline{6}m2$ and $P\overline{3}m1$ are shown to be dynamically stable, named as $\alpha$- and $\beta$ -phases correspondingly. Both phases have similar cohesive energies, and the $\alpha$ phase is found to be energetically favorable for structures except CP, CAs, CSb, and CBi, for which the $\beta$ phase is favored. The effects of spin-orbit coupling and Hartree-Fock corrections to exchange correlation are included to elucidate the electronic structures. All structures are semiconductors except CBi and PbN, which have metallic character. SiBi, GeBi, and SnBi have direct band gaps, whereas the remaining semiconductor structures have indirect band gaps. All structures have quartic dispersion in their valence bands, some of which make the valence band maximum and resemble a mexican-hat shape. SnAs and PbAs have purely quartic valence band edges, i.e., $E-\alpha k^4$, a property reported for the first time. The predicted materials are candidates for a variety of applications. Owing to their wide band gaps, CP, SiN, SiP, SiAs, GeN, GeP can find their applications in optoelectronics. The relative band positions qualify a number of the structures as suitable for water splitting, where CN and SiAs are favorable at all pH values. Structures with quartic band edges are expected to be efficient for thermoelectric applications.

Figure 1

Top and side views of the crystal lattice structure of group IV-V systems in two different space groups. (a) $P\overline{6}m2$ ($\alpha$ phase) and (b) $P\overline{3}m1$ ($\beta$ phase). (c) Reciprocal lattice of the hexagonal symmetry and the high symmetry points. Lattice vectors are represented by ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ which are equal in length $a$. Primitive unit cell comprises of four atoms, each of two are from either species. Group IV and V atoms are shown in blue and orange, respectively. Related structural parameters are detailed in Table 1 for $P\overline{6}m2$ and Table 2 for $P\overline{3}m1$.

Figure 2

The change in the cohesive energy with the lattice constant of structures in the $\alpha$ phase. Group V elements are shown in their respective colors.

Figure 3

The change in total energies with lattice constants of the two configurations ($P\overline{6}m2$ and $P\overline{3}m1$) shown in solid (red) and dashed (blue), respectively (top panel). Phonon dispersion relations of $\beta$-CP, $\beta$-CAs, $\beta$-CSb, $\beta$-CBi belonging to the $P\overline{3}m1$ symmetry (bottom panel). See Table 2 for structural and electronic properties.

Figure 4

Phonon dispersion relations of $\alpha$ phases of group IV-V monolayers. Absence of negative frequencies is an indication for the dynamic stability of these compounds.

Figure 5

Constant volume vibrational heat capacities ($C_v$) at 100 K, 300 K, 500 K and 800 K, from a) to d), respectively.

Figure 6

Electronic band diagrams of the $\alpha$ phases as calculated with PBE (solid red) and hybrid HSE06 (dashed blue) functionals. Energy gaps are given in Table 1. Conduction band minima and the valence band maxima are joined to indicate the band gaps.

Figure 7

Electronic band structures of CP, CAs, CSb, CBi in $\mathrm{P}\overline{3}\mathrm{m}1$ ($\beta$) symmetry calculated with both PBE and HSE06 functionals, given in solid red and dashed blue lines, respectively. See Table 2 for structural and electronic properties.

Figure 8

Band edge positions of all the compounds studied vs the redox potential of water. Dashed green, dashdotted black and dotted cyan horizontal lines depict the corresponding potentials at pH 0, pH 7 and pH 14, respectively. The compounds starting with $\beta$- are in $P\overline{3}m1$ space group.