Excitonic magneto-optics in monolayer transition metal dichalcogenides: From nanoribbons to two-dimensional response

The magneto-optical response of monolayer transition metal dichalcogenides, including excitonic effects, is studied using a nanoribbon geometry. We compute the diagonal optical conductivity and the Hall conductivity. Comparing the excitonic optical Hall conductivity to results obtained in the independent-particle approximation, we find an increase in the amplitude corresponding to one order of magnitude when excitonic effects are included. The Hall conductivities are used to calculate Faraday rotation spectra for ${\mathrm{MoS}}_2$ and ${\mathrm{WSe}}_2$. Finally, we have also calculated the diamagnetic shift of the exciton states of ${\mathrm{WSe}}_2$ in different dielectric environments. Comparing the calculated diamagnetic shift to recent experimental measurements, we find a very good agreement between the two.

Figure 1

(a) Schematic of the tight-binding couplings in the NNN-TB model for monolayer ${MX}_2$ TMDs. (b) Unit cell of a ${MX}_2$ armchair nanoribbon of width $(N-1)a/2$ and length $\sqrt{3}a$, where $a$ is the lattice constant. (c) Band structure of ${\mathrm{WSe}}_2$ along the path in the Brillouin zone specified by the letters. The blue and red lines are the spin-up and spin-down bands, respectively. (d) Brillouin zone of monolayer TMD.

Figure 2

Single-particle linear optical conductivities versus photon energy. The spectra are calculated for $B=130$ T and $\hslash \mathrm{\Gamma }=25$ meV. The factor of ${10}^{-2}$ should be multiplied with the $y$ axis directly underneath. The blue lines refer to the 2D conductivities and red and yellow lines to the nanoribbon case.

Figure 3

Many-body diagonal conductivities versus photon energy, calculated for $B=0$ T, $\hslash \mathrm{\Gamma }=50$ meV, and $\kappa =1$. The blue lines refer to the 2D conductivities and yellow and red lines to the nanoribbon spectra.

Figure 4

Excitonic optical conductivity versus photon energy, for different magnetic field strengths. The first row illustrates $\mathrm{\Delta }\mathrm{Re}{\sigma }_{xx}\left(\omega \right)$, which is the difference between the diagonal conductivity at a finite magnetic field strength and at 0 T. The dashed gray lines show the unperturbed spectra. The second row shows the Hall conductivities at different magnetic field strengths. Spectra are for nanoribbons with $N=100$ and $\kappa =1$. The factor ${10}^{-1}$ should be multiplied with the $y$ axis directly underneath.

Figure 5

Plot of the Verdet constant versus photon energy for normal incident light on a TMD monolayer in vacuum.

Figure 6

Excitonic optical conductivity versus photon energy, calculated for ${\mathrm{WSe}}_2$ in different dielectric environments. The top panel shows the diagonal conductivity in the unperturbed case. The middle panel shows the change in the diagonal conductivity from the unperturbed case to the $B=30$ T case. The bottom panel shows the Hall conductivities calculated at $B=30$ T. The factors of ${10}^{-1}$ should be multiplied with the $y$ axes directly underneath.

Figure 7

Diamagnetic shift of the peak associated with the $1s$ state of the $A$ exciton in ${\mathrm{WSe}}_2$. The vertical dashed lines indicate the peak position at different magnetic field strengths. The upper panel illustrates the fit of the function $E_0+\sigma B^2$ (blue line) to the peak positions marked by the red dots.

Figure 8

DOS versus energy for ${\mathrm{WSe}}_2$ plotted for a 2D monolayer and different nanoribbon widths. The inset shows a zoom of the DOS in the vicinity of the conduction edge states around 3.47 eV.