Optical absorption in monolayer ${\mathrm{SnO}}_2$

Since the discovery of graphene, considerable research efforts have focused on understanding the properties of other two-dimensional materials. Herein, based on ab initio many-body calculations, we report the optical properties of monolayer ${\mathrm{SnO}}_2$. First, we apply the first-principles density functional theory, self-consistent quasiparticle Green's function, and screened Coulomb method to determine the quasiparticle electronic structure. Second, we solve the Bethe-Salpeter equation to obtain the absorption spectra. The quasiparticle band structure reveals an indirect quasiparticle band gap. The absorption spectra show that the direct optical excitation is characterized by an optical band gap of $\sim 5.36$ eV, which is dominated by strongly bound excitons.

Figure 1

(a) Top, side, and perspective views of the $D_{3d}$ hexagonal crystal structure of monolayer ${\mathrm{SnO}}_2$. The blue colored arrows highlight the unit cell defined by the lattice vectors ${\vec{a}}_1$ and ${\vec{a}}_2$. (b) A comparison plot of the electronic properties of monolayer ${\mathrm{SnO}}_2$ obtained using the density functional theory (DFT) with (solid red line) and without (black dashed line) the effects of spin-orbit coupling (SOC).

Figure 2

Convergence of the fundamental energy band gap $E_g^{\mathrm{ind}}$ calculated at the ${\mathrm{G}}_0{\mathrm{W}}_0$ level as a function of (a) energy cutoff $E_s$ with grid and vacuum size fixed at 144 $k$ points and 20 $\AA$, respectively, (b) out-of-plane lattice constant (size of vacuum) with $E_s$ and grid size fixed at 550 eV and 144 $k$ points, respectively, and (c) grid size (Brillouin zone sampling grid) with $E_s$ and vacuum size fixed at 550 eV and 20 $\AA$, respectively. A compromised input parameter of $E_s\sim 550$ eV, $c\sim 20\AA$, and BZ sampling of 225 $k$ points are converged enough and have been used for the results presented herein.

Figure 3

The electronic properties of monolayer ${\mathrm{SnO}}_2$ showing the quasiparticle band structure (left panel) and the total and partial density of states (right panel). The spectra exhibit an indirect quasiparticle bandgap $E_q^{\mathrm{ind}}\approx 6.51$ eV along the $\mathrm{\Gamma }-K$ of the $k$ space. The horizontal dashed black line is the Fermi level $E_F$, which has been set to the top of the valence band.

Figure 4

The optical properties of monolayer ${\mathrm{SnO}}_2$ showing the (a) imaginary part ${\varepsilon }_2$ and (b) real part ${\varepsilon }_1$ of the dynamical dielectric function as a function of the photon energy $\hslash \omega$. The spectra is obtained using DFT+sqGW+BSE calculations. $E_q^{\mathrm{ind}}\sim 6.51$ eV is the quasiparticle band gap obtained from DFT+sqGW and $E_b=1.15$ is the exciton binding energy. The structure at I in Fig. 4 denotes the first direct excitation energy $\approx 5.36$ eV. The dashed brown line is the absorption spectra obtained using DFT+${\mathrm{G}}_0{\mathrm{W}}_0$+BSE. We also present the absorption spectra without electron-hole interactions obtained using the random-phase approximation (solid black line), which show no structure below the minimal excitation energy (I).