Laser energy dependence of valley polarization in transition-metal dichalcogenides

Photoabsorption spectra by circular polarized light in transition-metal dichalcogenides are calculated as a function of laser excitation energy. Although the 100% valley polarization occurs at the $K$ point of the Brillouin zone, the difference of the absorption intensity for left-handed and right-handed circular polarized light becomes maximum at about 1 eV higher energy than the direct energy band gap. The maximum intensity difference corresponds to the so-called $\mathrm{\Lambda }$ valley in the Brillouin zone. In order to understand valley polarization, analytic formula of optical absorption is given by tight-binding method.

Figure 1

(a) The top and (b) side views of the crystal structure of monolayer TMDs. ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are primitive lattice vectors. ${\mathit{r}}_A^1,{\mathit{r}}_A^2$, and ${\mathit{r}}_A^3$ are the vectors pointing to the nearest transition-metal atoms from a chalcogen atom in two-dimensional atomic layer plane. (c) Schematic of the energy bands of monolayer TMDs near the $K$ and $K^{\prime }$ points. Main contributions of atomic orbitals for each band are also shown. $\ell$ and $m$ are azimuthal and magnetic quantum numbers, respectively.

Figure 2

(a), (b) Transition dipole moments near (a) the $K$ and (b) the $K^{\prime }$ points, analytically calculated by the tight-binding model (see Appendix). (c) Three-dimensional energy bands plot calculated by DFT method for monolayer ${\mathrm{MoS}}_2$ near the $K$ point. The positions of the $K$ and $\mathrm{\Lambda }$ points are also shown. Inset shows the two-dimensional Brillouin zone of monolayer TMDs.

Figure 3

(a)–(f) Electronic energy bands of six TMD materials [(a) ${\mathrm{MoS}}_2$, (b) ${\mathrm{MoSe}}_2$, (c) ${\mathrm{MoTe}}_2$, (d) ${\mathrm{WS}}_2$, (e) ${\mathrm{WSe}}_2$, (f) ${\mathrm{WTe}}_2]$ along to $M{K}^{\prime }\mathrm{\Gamma }KM$ line calculated by DFT method with spin-orbit interaction. Red (blue) color indicates spin-up (-down) state. The top of valence band is set as zero energy.

Figure 4

(a) Significant optical transition paths in TMDs in the low-energy region. (b)–(g) Optical absorption of six TMD materials. Excitation energy of $A,B,A^{\prime },B^{\prime }$, and $M$ paths in  (a) are also shown by arrows.

Figure 5

(a)–(d) Color plot of the absolute value of the matrix elements $|M_{\mathrm{opt}}^{\mathrm{cv}}\left(\mathit{k}\right)|$ in the two-dimensional Brillouin zone of monolayer ${\mathrm{MoS}}_2$ for RCP $\left({\sigma }_{-}\right)$ and LCP $\left({\sigma }_{+}\right)$ light, inducing the excitation energy $E_L$ of (a) 1.65 eV, (b) 1.82 eV, (c) 2.60 eV, and (d) 2.90 eV.

Figure 6

(a) Two-dimensional Brillouin zone which is divided to the two regions near the $K$ and $K^{\prime }$ valleys. (b)–(g) Laser energy dependence of the degree of the valley polarization of the optical transition $\left[{\rho }^K\left(E_L\right)\right]$, intensity of the optical absorption in the $K$ valley region excited by circular polarized light $\left[I_{{\sigma }_{\pm }}^K\left(E_L\right)\right]$, and difference of the intensity for RCP and LCP incident light in the $K$ valley region $[I_{{\sigma }_{-}}^K\left(E_L\right)-I_{{\sigma }_{+}}^K\left(E_L\right)]$. The excitation energies at the $K \left(K^{\prime }\right),\mathrm{\Lambda } \left({\mathrm{\Lambda }}^{\prime }\right)$, and $M$ points are labeled by $A,A^{\prime }$, and $M$, respectively. The energy position $M$ of ${\mathrm{WS}}_2$ is about 3.5 eV and out of the range of (e).

Figure 7

(a)–(d) Nonvanishing and (e)–(h) vanishing dipole parameters between $\widetilde{p_x}$ or $\widetilde{p_y}$ and $d_{xy}$ or $d_{x^2-y^2}$ orbitals. Red arrow indicates the direction of differential operator. (a) $D_{xy,x}^y=\langle \widetilde{p_x}|\frac{\partial }{\partial y}|d_{xy}\rangle$; (b) $D_{x^2-y^2,x}^x=\langle \widetilde{p_x}|\frac{\partial }{\partial x}|d_{x^2-y^2}\rangle$; (c) $D_{xy,y}^x=\langle \widetilde{p_y}|\frac{\partial }{\partial x}|d_{xy}\rangle$; (d) $D_{x^2-y^2,y}^y=\langle \widetilde{p_y}|\frac{\partial }{\partial y}|d_{x^2-y^2}\rangle$; (e) $D_{xy,x}^x=\langle \widetilde{p_x}|\frac{\partial }{\partial x}|d_{xy}\rangle$; (f) $D_{x^2-y^2,x}^y=\langle \widetilde{p_x}|\frac{\partial }{\partial y}|d_{x^2-y^2}\rangle$; (g) $D_{xy,y}^y=\langle \widetilde{p_y}|\frac{\partial }{\partial y}|d_{xy}\rangle$; and (h) $D_{x^2-y^2,y}^x=\langle \widetilde{p_y}|\frac{\partial }{\partial x}|d_{x^2-y^2}\rangle$.