Magneto-optical transport properties of monolayer transition metal dichalcogenides

We study the optical transport properties of the monolayer transition metal dichalcogenides (TMDCs) such as ${\mathrm{MoS}}_2$, ${\mathrm{WS}}_2$, ${\mathrm{MoSe}}_2$, and ${\mathrm{WSe}}_2$ in the presence of a magnetic field. The TMDCs band structures are obtained and discussed by using the effective massive Dirac model, in which the spin and valley Zeeman effects as well as an external electric field are included. The magneto-optical absorption coefficient (MOAC) is derived as a function of absorbed photon energy when the carriers are scattered by random impurities combined with the intrinsic acoustic and optical phonons in TMDCs and the surface optical (SO) phonons of substrates. Our result shows that the spin-splitting feature appeared in all four TMDC materials. The combination of strong spin-orbit coupling (SOC) and Zeeman fields has doubled the Landau levels but has not changed the energy gap of the TMDCs monolayer, which can be controlled by the electric field. Because of their strong SOC effect, the absorption spectrum in monolayer TMDCs is separated into two separate peaks caused by spin up and down. At the low temperature, the MOAC intensity via impurity scattering is the biggest followed by that of the SO phonons while the intrinsic acoustic and optical phonon scatterings display the smallest. For the monolayer TMDCs on substrates, ${\mathrm{SiO}}_2$ always shows its superiority in comparison with the others. Among the four TMDC materials, ${\mathrm{MoSe}}_2$ shows the biggest MOAC intensity, while ${\mathrm{WS}}_2$ has the biggest value of the absorbed photon energy. The full-width at half-maximum (FWHM) via impurity scattering achieves its highest value in ${\mathrm{WS}}_2$, while this occurs in ${\mathrm{MoSe}}_2$ and ${\mathrm{MoS}}_2$ via intrinsic acoustic and optical phonon scatterings, respectively. Our estimation of mobility from FWHM gives good agreement with the experimental results in ${\mathrm{WS}}_2$.

Figure 1

The LLs of monolayer TMDCs as a function of the magnetic field $B$ including the electric, the spin, and valley Zeeman fields. The applied electric field is $e{E}_z=37.75$ meV/$d$. The top [(a), (c), (e), and (g)] and bottom [(b), (d), (f), and (h)] panels correspond to the conduction and valence bands of each material, respectively.

Figure 2

The LLs of monolayer ${\mathrm{MoS}}_2$ (conduction band) versus the magnetic field $B$. (a) $M_z=M_v=0$ and $e{E}_z=37.75$ meV/$d$, while (b) is for $M_z\ne 0$, $M_v\ne 0$, and $E_z=0$.

Figure 3

The MOAC in monolayer ${\mathrm{MoS}}_2$ versus photon energy including the electric, the spin, and valley Zeeman fields: (a) Transition from $n=0$ to $n^{\prime }=1$, (b) the diagram transition and spin separation, (c) transitions between different LLs for spin up, and (d) the same as those in (c) but for spin down. The results are evaluated at $B=10$ T, $T=4$ K, and $e{E}_z=37.75$ meV/$d$.

Figure 4

The MOAC caused by the first transition (from $n=0$ to $n^{\prime }=1$) in monolayer TMDCs versus photon energy for different magnetic fields at $T=4$ K, $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, and $M_z,M_v\ne 0$.

Figure 5

The FWHM of the peaks shown in Fig. 4 versus the magnetic field. The left (a) and right (b) consoles are for the one- and two-photon processes, respectively.

Figure 6

The MOAC due to the first transition in monolayer TMDCs versus photon energy for different electric fields at $B=10$ T, $T=4$ K, and $M_z,M_v\ne 0$.

Figure 7

The MOAC in monolayer TMDCs due to acoustic and optical phonon scattering is shown as a function of photon energy for different magnetic fields. The results are evaluated at $T=4$ K, $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, and $M_z,M_v\ne 0$. Symbols “ac” and “op” refer to the acoustic and optical phonons scattering, respectively.

Figure 8

The FWHM of the peaks shown in Fig. 7 as a function of magnetic field.

Figure 9

Temperature dependence of the peak value of MOAC at $\hslash \mathrm{\Omega }=E_{1,+1}^{+1,+1}-E_{0,+1}^{+1}+\hslash {\omega }_{\mathbf{q}}^{\beta }$. The results are evaluated for one-photon process, at $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, $M_z,M_v\ne 0$, and $B=10$ T. Symbols ac, op, and im refer to the acoustic phonon, optical phonon, and impurity scattering, respectively (note the factor of ${10}^4$ in the case of impurity scattering).

Figure 10

Temperature dependence of the FWHM due to phonon scattering in different materials. The results are evaluated for one-photon process at $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, $M_z,M_v\ne 0$, and $B=10$ T.

Figure 11

The MOAC due to carrier-SO-phonon in monolayer TMDCs on ${\mathrm{SiO}}_2$ substrate as a function of photon energy at $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, $M_z,M_v\ne 0$, $B=10$ T, $T=4$ K, and $d_i=0.4$ nm. (a) and (b) Two- and one-photon processes, respectively.

Figure 12

The MOAC due to carrier-SO-phonon in monolayer ${\mathrm{WS}}_2$ on different substrates versus photon energy for different magnetic fields. The results are evaluated for one-photon process, at $e{\mathrm{\Delta }}_z=37.75$ meV/$d$, $M_z,M_v\ne 0$, $T=4$ K, and $d_i=0.4$ nm.

Figure 13

The FWHM of the peaks shown in Fig. 12 versus the magnetic field. (a) and (b) ${\mathrm{SO}}_1$ and ${\mathrm{SO}}_2$ phonons, respectively.