Phosphorene analogues: Isoelectronic two-dimensional group-IV monochalcogenides with orthorhombic structure

The group-IV monochalcogenides SnS, SnSe, GeS, and GeSe form a family within the wider group of semiconductor “phosphorene analogues.” Here, we used first-principles calculations to investigate systematically their structural, electronic, and optical properties, analyzing the changes associated with the reduction of dimensionality, from bulk to monolayer or bilayer form. We show that all those binary phosphorene analogues are semiconducting, with band-gap energies covering part of the infrared and visible range, and in most cases higher than phosphorene. Further, we found that they have multiple valleys in the valence and conduction band, the latter with spin-orbit splitting of the order of 19–86 meV.

Figure 1

Optimized structures of monolayers of group-IV monochalcogenides with phosphorenelike structure. (a) Side view of the $x-z$ plane for the four compounds and for phosphorene. (b) Side view of the $y-z$ plane of SnS and phosphorene. (c) Top view of the structures, with the lattice vectors $\mathbf{a}$ and $\mathbf{b}$ along the $x$ and $y$ directions. (d) The respective BZ and the high-symmetry points $\mathrm{\Gamma }, X, T$, and $Y$.

Figure 2

Electronic band structures for monolayer, bilayer, and bulk group-IV monochalcogenides calculated using the HSE hybrid functional. The VBM and CBM are highlighted by full circles. Dashed black arrows indicate possible direct transitions ($T_1$ and $T_2$) to points very close in energy to the VBM and CBM. Triangles indicate the position of the CBM when spin-orbit coupling effects are considered.

Figure 3

Electronic band structures for SnSe monolayer with (green and blue dashed lines) and without (continuous black lines) spin-orbit coupling effect.

Figure 4

Calculated SO splittings of the CB and VB for the valleys along $\mathrm{\Gamma }-Y$ in meV. The schematic picture shows in continuous black lines the bands without spin-orbit effects, which are taken into account in a fully relativistic calculation. The lifted conduction and valence bands are shown in dashed lines. The lowering in $E_g$ is given by ${\mathrm{\Delta }}_{E_g}^{\mathrm{SO}}$.

Figure 5

Real part of the optical conductivity ${\sigma }_r\left(\omega \right)$ for monolayer (full red lines), bilayer (full orange lines), and bulk (full blue lines) monochalcogenides. For comparison, experimental results are also included from previous works. As the experimental works report ${\epsilon }_i\left(\omega \right)$, we use relations (1) to obtain ${\sigma }_r\left(\omega \right)$. (a) Ref. [50]; (d) Ref. [48].

Figure 6

Real part of dielectric constants ${\epsilon }_r\left(\omega \right)$ for monolayer (full red lines), bilayer (full orange lines), and bulk (full blue lines) monochalcogenides. Experimental results from previous works are also included. (a) Ref. [50]; (b) Ref. [51]; (c) Ref. [31]; (d) Ref. [48].