Electronic and optical properties of the monolayer group-IV monochalcogenides $MX$ ($M=\mathrm{Ge},\mathrm{Sn}$; $X=\mathrm{S},\mathrm{Se},\mathrm{Te}$)

By using density-functional theory and many-body perturbation theory based first-principles calculations, we have systematically investigated the electronic and optical properties of monolayer group-IV monochalcogenides $MX$ ($M=\mathrm{Ge},\mathrm{Sn}$; $X=\mathrm{S},\mathrm{Se},\mathrm{Te}$). All $MX$ monolayers are predicted to be indirect gap semiconductors, except the GeSe monolayer, which has a direct gap of 1.66 eV. The carrier mobilities of $MX$ monolayers are estimated to be on the order of ${10}^3$ to ${10}^5\mathrm{c}{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$, which is comparable to, and in some cases higher than, that of phosphorene using a phonon-limited scattering model. Moreover, the optical spectra of $MX$ monolayers obtained from GW-Bethe-Salpeter equation calculations are highly orientation dependent, especially for the GeS monolayer, suggesting their potential application as a linear polarizing filter. Our results reveal that the GeSe monolayer is an attractive candidate for optoelectronic applications as it is a semiconductor with a direct band gap, a relatively high carrier mobility, and an onset optical absorption energy in the visible light range. Finally, based on an effective-mass model with nonlocal Coulomb interaction included, we find that the excitonic effects of the GeSe monolayer can be effectively tuned by the presence of dielectric substrates. Our studies provide an improved understanding of electronic, optical, and excitonic properties of group-IV monochalcogenides monolayers and might shed light on their potential electronic and optoelectronic applications.

Figure 1

(a) Top view, (b) side view along the $x$ direction, and (c) side view along the $y$ direction of the $MX$ monolayer structure. Group-IV atom $M$ and chalcogenide atom $X$ are shown using green and yellow balls, respectively. The unit cell is indicated by the dashed rectangle in (a). (d) First Brillouin zone of the $MX$ monolayer with high-symmetry points.

Figure 2

Quasiparticle band structures of the $MX$ monolayers obtained by Wannier interpolation. The blue solid circles represent the $k$ points that are sampled in GW calculations. The top valance band and bottom conduction band derived from DFT-GGA are shown by shaded gray background. The first excitation transitions along the $x$ and $y$ direction are denoted by the red dashed arrows.

Figure 3

Optical spectra of the $MX$ monolayers calculated using GW-RPA (without $e$-h interaction) and GW-BSE (with $e$-h interaction) methods for linearly polarized light along the $x$ direction and the $y$ direction, respectively. The first optical gaps along the $x$ direction and the $y$ direction are denoted by blue and red vertical dotted lines, respectively.

Figure 4

Normalized squared exciton wave function calculated using GW-BSE for (a) the GeS monolayer and (c) the GeTe monolayer. The isovalue is set to $3\times {10}^{-6}e{\AA }^{-3}$, and the hole is fixed at the center. Comparison of the exciton wave-function distribution along the zigzag direction between GW-BSE and variational methods for (b) the GeS monolayer and (d) the GeTe monolayer.

Figure 5

(a) In-plane dielectric constants, ${\varepsilon }_x$ and ${\varepsilon }_y$, as functions of inverse interlayer separation 1/$L$ for the GeS monolayer. (b) Exciton binding energy $E_b$ and radius $a_x$ as a function of substrate dielectric constant ${\varepsilon }_2$ for the GeSe monolayer. (c)–(f) Squared wave functions $|{\psi }_n{|}^2$ for the four lowest energy exciton states of GeSe monolayer, shown in $200\AA \times 200\AA$.