Magnetic Edge States in Transition Metal Dichalcogenide Monolayers

We developed theory of the magnetic edge states in a half-plane of transition metal dichalcogenide monolayer. The main qualitative feature of such system that distinguishes it from an unbounded plane is “tau-splitting”; i.e., lifting of the valley degeneracy in an energy spectrum. Interband optical transitions are characterized by the violation of the selection rules established for the same transitions in an unbounded monolayer.

APPENDIX

Let us make, in Eq. (13), transformation $E,\tau \to - E, - \tau$:

$$\begin{gathered} {{D}_{{q - \theta ( - \tau )}}}(z) \\ + \;\frac{\Omega }{{\Delta {\text{/}}2 - E}}{{D}_{{q - \theta (\tau )}}}(z)[\theta ( - \tau ) - q\theta (\tau )] = 0. \\ \end{gathered} $$

(17)

The energy spectrum corresponding to Eq. (10) consists of two bands: conduction band and valence band. We will consider Eq. (13) in the conduction band (i.e., for $E > 0$) and Eq. (13) in the valence band (i.e., for $E < 0$). Further, one can rewrite Eqs. (13) and (17) by explicitly introducing the band and valley indices:

$$\begin{gathered} {{D}_{{{{q}^{{({\text{c}},\tau )}}} - \theta ( - \tau )}}}(z) + \frac{\Omega }{{\Delta {\text{/}}2 + {{E}^{{({\text{c}},\tau )}}}}}{{D}_{{{{q}^{{({\text{c}},\tau )}}} - \theta ( - \tau )}}}(z) \\ \times \;\left[ {\theta (\tau ) - {{q}^{{({\text{c}},\tau )}}}\theta ( - \tau )} \right] = 0; \\ \end{gathered} $$

(18)

$$\begin{gathered} {{D}_{{{{q}^{{({\text{v}}, - \tau )}}} - \theta (\tau )}}}(z) + \frac{\Omega }{{\Delta {\text{/}}2 - {{E}^{{({\text{v}}, - \tau )}}}}}{{D}_{{{{q}^{{({\text{v}}, - \tau )}}} - \theta (\tau )}}}(z) \\ \times \;\left[ {\theta ( - \tau ) - {{q}^{{({\text{v}}, - \tau )}}}\theta (\tau )} \right] = 0. \\ \end{gathered} $$

(19)

Energies can be expressed in terms of q as ${{E}^{{({\text{c}},\tau )}}} =$ $\sqrt {{{\Delta }^{2}}{\text{/}}4 + {{\Omega }^{2}}{{q}^{{({\text{c}},\tau )}}}}$ and ${{E}^{{({\text{v}},\tau )}}} = - \sqrt {{{\Delta }^{2}}{\text{/}}4 + {{\Omega }^{2}}{{q}^{{({\text{v}},\tau )}}}}$. Comparing Eq. (18) with Eq. (19) we see that there are relations ${{E}^{{({\text{v}},\tau )}}} = - {{E}^{{({\text{c}}, - \tau )}}}$ and ${{q}^{{({\text{v}},\tau )}}} = {{q}^{{({\text{c}}, - \tau )}}}$.