Valley selectivity of soft x-ray excitations of core electrons in two-dimensional transition metal dichalcogenides

Optical properties of semiconducting monolayer transition metal dichalcogenides have received a lot of attention in recent years, following the discovery of the valley selective optical population of either ${\text{K}}_{+}$ or ${\text{K}}_{-}$ valleys at the direct band gap, depending on the polarization of the incoming light. We use group theoretical selection rules, as well as ab initio DFT calculations, to investigate whether this valley selectivity effect is also present in x-ray optical transitions from the flat core level of the transition metal atom to the valence and conduction band K valleys. Valley selectivity is predicted for $s, p_{1/2}$, and $p_{3/2}$ edges in transitions to and from the valence band edges with circularly polarized radiation. Possible novel applications to the diagnostics of valleytronic properties and intervalley dynamics are investigated and the feasibility of ultrafast pump-probe and Kerr rotations experiments with suitable soft-x-ray free-electron laser sources is discussed.

Figure 1

Structure of MTMDs. Panel (a) shows a perspective side view and panel (b) a top view, including the two unit vectors ${\mathbf{a}}_{\mathbf{1}}, {\mathbf{a}}_{\mathbf{2}}$ and an orthogonal cartesian frame to describe all vectors in real and reciprocal space. (c) Brillouin zone for the two-dimensional lattice structure, with the unit reciprocal lattice vectors ${\mathbf{b}}_{\mathbf{1}}, {\mathbf{b}}_{\mathbf{2}}$ and with the two wave vectors of the band structure “valleys,” ${\mathbf{K}}_{+}, {\mathbf{K}}_{-}$. A vector equivalent to ${\mathbf{K}}_{+}$ is shown for illustration purposes.

Figure 2

The middle plane of the monolayer structure, showing the position of the metal atoms and the unit vectors ${\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{2}}$. Some atoms are labeled with their $j_1,j_2$ coordinates as an example. The three vectors ${\delta }_{\mathbf{i}},i=1,2,3$ are also shown. The dotted diagonal lines are the wavefronts of the ${\mathbf{K}}_{+}$ Bloch waves, where $(2j_1+j_2-1)$ is constant (see text). The integers at the top end of the dotted lines are $(2j_1+j_2\text{mod}3).$

Figure 3

Valley selectivity for transitions from the Mo fourfold degenerate $3p_{3/2}$ level to the uppermost valence bands. (a), (c) Sketches of the valence to conduction band transition (a) and the core to valence band transition (c); (b), (d) Computed normalized degree of optical polarization $\eta$ (see text) as a function of quasi-momentum k in the relevant portion of the Brillouin zone for the valence to conduction and core to valence band transitions, respectively; (e) Partial absorption cross-section ($\sigma$) obtained through integration over a circular area around ${\text{K}}_{\pm }$ points for left ($-$) and right ($+$) circular polarization of the incoming x-ray photons [see also text and illustration of the absorption process in the bottom of panel (e)].

Figure 4

Valley selectivity for transitions from the Mo fourfold degenerate $3p_{3/2}$ level to the lowest conduction band. (a), (c) Sketches of the valence to conduction band transition (a) and the core to conduction band transition (c); (b), (d) computed normalized degree of optical polarization $\eta$ (see text) as a function of quasi-momentum k in the relevant portion of the Brillouin zone for the valence to conduction and core to conduction band transitions, respectively; (e) partial absorption cross-section ($\sigma$) obtained through the integration over circular area around ${\mathbf{K}}_{\pm }$ points for the left ($-$) and right ($+$) circular polarization of the incoming x-ray photons [see also text and illustration of the absorption process in the bottom of panel (e)].

Figure 5

Band structure of ${\text{MoS}}_2$ monolayer, where we highlight the projection of the Kohn-Sham wave function onto the $4d_{m}m=\{0,\pm 2\}$ and $3p_{j=3/2}$ orbitals of the molybdenum atom.

Figure 6

Valley selectivity of ${\text{WS}}_2$ and schematic of pump-probe experiment. (a), (b) Normalized degree of optical polarization as a function of quasi-momentum for transitions from $\text{W}5p_{3/2}$ levels to the uppermost valence band (a), and from the uppermost valence band to the lowest conduction band (b). (f), (j) [(e), (i)] Partial absorption cross-sections calculated in vicinity of ${\mathbf{K}}_{\pm }$ points for the valence to conduction (core to valence) band transitions. (c), (d) Schematic representation of the core to valence and valence to conduction bands transitions for the optical and x-ray photons with the same (c), and opposite (d), circular polarization. The vertical red arrows denote transitions from IR/visible laser pump pulses, the blue arrows transitions from core levels. (g), (h) Schematic figures of the pump pulse (in red) and the transient probe absorption (in blue) as a function of pump-probe delay $\mathrm{\Delta }\tau$ for the configuration of polarization presented in panels (c) and (d), respectively. See text for the definition of the various characteristic times $\tau$.

Figure 7

(a) Computed normalized degree of optical polarization $\eta$ (see text) for transitions from the hBN highest valence band the lowest conduction band, as a function of quasi-momentum k in the relevant portion of the Brillouin zone. (b) Partial absorption cross-section calculated for the circular area around ${\mathbf{K}}_{\pm }$ points for left ($-$) and right ($+$) circular polarization of the incoming photons (see also Sec. 3).

Figure 8

Schematics of the valence and conduction band edges and their symmetry labels for hBN at the ${\mathbf{K}}_{\pm }$ points.

Figure 9

Computed normalized degree of optical polarization $\eta$ for transitions from the fourfold degenerate $5_{p_{3/2}}$ levels of tungsten in monolayer ${\text{WS}}_2$ to the lowest conduction band, as a function of quasi-momentum k in the relevant portion of the Brillouin zone (a). Partial absorption cross-section calculated for the circular area around ${\mathbf{K}}_{\pm }$ points for left ($-$) and right ($+$) circular polarization polarization of the incoming photons (see also Sec. 3).