Nonanalyticity, Valley Quantum Phases, and Lightlike Exciton Dispersion in Monolayer Transition Metal Dichalcogenides: Theory and First-Principles Calculations

Exciton dispersion as a function of center-of-mass momentum $\mathbf{Q}$ is essential to the understanding of exciton dynamics. We use the ab initio $GW$-Bethe-Salpeter equation method to calculate the dispersion of excitons in monolayer ${\mathrm{MoS}}_2$ and find a nonanalytic lightlike dispersion. This behavior arises from an unusual $|Q|$-term in both the intra- and intervalley exchange of the electron-hole interaction, which concurrently gives rise to a valley quantum phase of winding number two. A simple effective Hamiltonian to $Q^2$ order with analytic solutions is derived to describe quantitatively these behaviors.

Figure 1

(a) QP band structure of monolayer ${\mathrm{MoS}}_2$. Lowest energy transitions corresponding to electron-hole excitations with momentum transfers of $\mathbf{Q}=0$ and $\mathbf{Q}=K$ are shown with the labeled arrows. (b)–(c) Schematics of interactions between two electron-hole pairs with momentum near $\mathbf{Q}=0$, corresponding to BSE matrix elements $⟨vc\mathbf{k}\mathbf{Q}|H^{\mathrm{BSE}}|v^{\prime }{c}^{\prime }{\mathbf{k}}^{\prime }\mathbf{Q}⟩$ for (b) two like-spin transitions within one valley and (c) two like-spin transitions from different valleys. (d) Schematic of two electron-hole pairs with momentum near $\mathbf{Q}=K$ giving rise to BSE matrix elements for like-spin transitions.

Figure 2

Exciton dispersion of monolayer ${\mathrm{MoS}}_2$ near (a) $\mathbf{Q}=0$ and (b) $\mathbf{Q}=K$ along the $K–\mathrm{\Gamma }–K^{\prime }$ direction. Red (blue) lines indicate states arising from like-spin (unlike-spin) transitions. The label $A$ refers to states involving transitions from the highest valence band at $K/K^{\prime }$. “$B$” states involving transitions from the second highest valence band are similar and shown in the Supplemental Material [27].

Figure 3

(a) Closeup of dispersion of $A_{1s}$ near $\mathbf{Q}=0$. Ab initio values are stars. Fit to effective Hamiltonian [Eq. (11)] are solid lines. Red (blue) lines indicate states arising from like-spin (unlike-spin) transitions. (b) Valley pseudospin texture of the (upper) nonanalytic like-spin transition band around $\mathbf{Q}=0$ for states of fixed energy in $\mathbf{Q}$ space. (c) Valley pseudospin texture of the (lower) parabolic like-spin transition band. (d) Optical absorbance of linearly polarized light at fixed incidence as the polarization vector $\widehat{\mathbf{e}}$ is rotated over 360°. The angle of the polarization vector, ${\theta }^{\prime }$, is defined with respect to the vector formed by the intersection of the polarization plane (blue) and the $xy$ plane. Red (black) indicates the absorbance of states arising from the lower (upper) like-spin band. (e) Energy difference between the upper and lower like-spin bands that is probed as $\varphi$, the angle between the wave vector of light and the $z$ axis is changed, for light of $\hslash \omega \approx 2\mathrm{eV}$. The inset shows how $\theta$, $\varphi$, and ${\theta }^{\prime }$ are defined.