Band structure and optical transitions in atomic layers of hexagonal gallium chalcogenides

We report density-functional-theory calculations of the electronic band structures and optical absorption spectra of two-dimensional crystals of Ga${}_2{X}_2$ ($X=$ S, Se, and Te). Our calculations show that all three two-dimensional materials are dynamically stable indirect-band-gap semiconductors with a sombrero dispersion of holes near the top of the valence band. We predict the existence of Lifshitz transitions—changes in the Fermi-surface topology of hole-doped Ga${}_2{X}_2$—at hole concentrations $n_S=7.96\times {10}^{13}$ cm${}^{-2}$, $n_{Se}=6.13\times {10}^{13}$ cm${}^{-2}$, and $n_{Te}=3.54\times {10}^{13}$ cm${}^{-2}$.

Figure 1

Structure of monolayer gallium chalcogenides Ga${}_2{X}_2$ ($X=$ S, Se, Te) in top and side views.

Figure 2

HSE06 band structures (solid red lines) for Ga${}_2$S${}_2$ (top), Ga${}_2$Se${}_2$ (middle), and Ga${}_2$Te${}_2$ (bottom). The zero of energy is taken to be the Fermi level and the bottom of the conduction band is marked with a horizontal line. For comparison, the semilocal band structures are also shown. In the case of the LDA the two symmetric bands that give rise to the main feature in the optical absorption spectrum are marked in bold.

Figure 3

Energy contours of Ga$X$ in the valence band using the fitted formula of Eq. (1). The separation of contours is 2 meV in all figures and the contour corresponding to the energy of the saddle point along the $\Gamma$-$M$ line is highlighted.

Figure 4

Absorption coefficient (the imaginary part of the dielectric function $\varepsilon$) of various Ga${}_2{X}_2$ two-dimensional crystals normalized to absolute units after it was compared to Im$\left(\varepsilon \right)$ evaluated for graphene in the range 0.8–1.5 eV, where monolayer graphene absorbs 2.3$\%$ of light (see text for details). Vertical lines mark the full width at half maximum for the main peak. The insets illustrate the relevant part of k-space for the transitions between bands marked by circle and diamond in Fig. 2 which give rise to the main peak (light red circle) and the low-energy shoulder (black circle).

Figure 5

Comparison between the LDA band structures of Ga${}_2{X}_2$ with and without spin-orbit coupling (SOC).

Figure 6

LDA phonon dispersion curves for Ga${}_2$S${}_2$, Ga${}_2$Se${}_2$, and Ga${}_2$Te${}_2$.