Symmetry-driven valleytronics in the single-layer tin chalcogenides SnS and SnSe

The concept of valleytronics has recently gained considerable research attention due to its intriguing physical phenomena and practical applications in optoelectronics and quantum information. In this study, by employing the Green function-Bethe-Salpeter equation (GW-BSE) calculations and symmetry analysis, we demonstrate that single-layer orthorhombic SnS and SnSe possess remarkable carrier mobility and exceptional excitonic effects. These materials display spontaneous linearly polarized optical selectivity, a behavior that differs from the valley-selective circular dichroism observed in the hexagonal lattices. Specifically, when the two-dimensional tin chalcogenide is subjected to a zigzag polarization of light, only the A exciton becomes optically active, while the B exciton remains dark. The armchair-polarized light triggers the opposite behavior. This selective optical excitation arises from the symmetry of the bands under mirror symmetry. Additionally, the study reveals a strong coupling between valley physics and ferroelectricity in layered tin chalcogenides, enabling the manipulation of exciton polarization. Layered tin chalcogenides thus emerge as promising candidates for valleytronic materials.

Figure 1

Electronic properties of SnS and SnSe monolayers: Electronic band structure of (a) SnS and (b) SnSe monolayers. The black line indicates the band dispersion under the DFT level of theory, while the red-dashed one shows the band structure of studied materials under G0W0 correction. The G0W0 quasiparticle band structure in this work was achieved under WANNIER90 codes [28]. (c) Top view of the single-layer SnS and side view of its bulk counterpart. The red square indicates the computed unit cell. (d) Electronic wave functions of single-layer SnS at the critical points. The green and red isosurfaces indicate the positive and negative phases of the wave function, respectively. The black-dashed lines denote the $M_z$ mirror plane and $(+)$ and $(-)$ signs above each panel indicate the even and odd parity, respectively.

Figure 2

The absorption spectrum of monolayer (a) SnS and (b) SnSe as a function of polarization of light. The spectra are achieved with a Lorentzian width of 20 meV. (c) The polarization of incident light could be effectively controlled via the polarizing filter. By manipulating the orientation of incident light, we can arbitrarily modulate the exciton of which valley can be optically excited.

Figure 3

The excitonic properties of single-layer SnS: Exciton energy spectrum of the SnS monolayer for the polarization of light along the (a) zigzag and (b) armchair directions. Excitonic amplitudes (color scale in the bottom panels) are normalized with respect to the brightest exciton among all of the exciton states from blue (dark exciton) to red (bright exciton). (c) the k-space distribution of the envelope functions for the first six bought excitons. The exciton states are labeled with primary and azimuthal quantum numbers according to their nodal structure in the radial and azimuthal directions, respectively. The valence-to-conduction band transition is indicated below each exciton wave function. (d) 3D electronic energy spectra of single-layer SnS with two emerging valleys X' and Y' in the frequency of interest. The modulus of the optical transition dipole moment $\left(\right|{\mathbf{p}}_{vc\mathbf{k}}^{\mathbf{e}}\left|\right)$ for the polarization of light along (e) zigzag and (f) armchair directions. For the polarization of light along the zigzag direction, $|{\mathbf{p}}_{vc\mathbf{k}}^{\mathbf{e}}|$ gets a significant value at the X' valley, while it almost vanished at the $\mathrm{\Gamma }$ and Y' valleys. The opposite behavior is true for the polarization of light along the armchair direction.

Figure 4

Top and side views of the spatial Exciton amplitudes for (a) A-exciton and (b) B-exciton in monolayer SnS. (c) and (d) depict analogous plots for single-layer SnSe. The green sphere denotes the position of the hole. The visualization of the real-space exciton wave functions was generated using the exciting code [51], employing identical criteria as in VASP, but restricted to 15 $\times$ 15 $\times$ 1 KPOINTS.

Figure 5

(a) The calculated out-of-plane Berry curvature (BC) of SnS monolayer mapped in the 2D Brillouin zone. The BC was computed using WANNIER90 codes [28], with the Fermi energy set at the midpoint of the band gap. (b) Double-well potential as a function of the polarization in monolayers SnS and SnSe. Two ferroelectric phases with spontaneous polarization along the x and y directions and the paraelectric configuration, respectively, are denoted as $P_x, P_y$, and $P_0$. The circles indicate the calculated values achieved from the Modern Theory of Polarization [54], while the solid lines present the fitting function using Landau-Ginzburg expansion [34].

Figure 6

Convergence of the G0W0 electronic band gap of SnS monolayer with respect to (a) number of empty bands, (b) k-grid, and (c) cutoff frequency. (d) the evolution of the first exciton states as functions of KPOINTS.

Figure 7

The real (a) and the reciprocal space (b) of single-layer tin chalcogenides. Quasiparticle band structures of (c) SnS monolayer and (d) SnSe monolayer were calculated by using the G0W0 functionals. The black (red) color stands for the electronic band structure without (with) spin-orbit coupling. The orbital projected electronic band structure for (e) and (f) SnS monolayer, and (g) and (h) SnSe monolayer.

Figure 8

Valley characteristics of the ferroelectric (a) $P_y$ and (c) $P_x$ phases, and (b) the paraelectric $P_0$ phase. Ferroelectricity breaks the fourfold rotational symmetry, making the valleys at X' and Y' no longer identical, which is the critical sign of spontaneous valley polarization. The global band gap of $P_x$ and $P_y$ exhibits the same characteristic, only the polarity of valley polarization is reversed. The valley splitting in the paraelectric phase, on the other hand, is absent due to its centrosymmetric structure.