Oscillations of the electron energy loss rate in two-dimensional transition-metal dichalcogenides in the presence of a quantizing magnetic field

We study the hot electron energy-loss rate (ELR) induced by acoustic ${P}^{\mathrm{ac}}$ and optical ${P}^{\mathrm{op}}$ phonons, in two-dimensional transition-metal dichalcogenides (TMDCs), in presence of quantizing magnetic field $B$, including the hot-phonon effect. At the low-temperature regime, the ELR is found to display a quantum oscillations with their amplitude being found to increase with the increase of the magnetic field. The unscreened TA phonon due to deformation potential (DP) coupling is dominant over the other screened acoustic phonon contributions. In the extreme Bloch-Gr\"uneisen (BG) region, the ELR displays a pronounced ${P}^{\mathrm{ac}}\ensuremath{\propto}{T}^{4}\phantom{\rule{0.16em}{0ex}}({T}^{6})$ temperature dependence for unscreened TA-DP phonons (other screened mechanisms). When the temperature increases, ${T}_{e}g{T}_{\mathrm{BG}}$, the ELR shows a linear ${P}^{\mathrm{ac}}\ensuremath{\propto}T$ behavior in the equipartition region. At higher temperatures, there is a crossover from ${P}^{\mathrm{ac}}$ dominated ELR to ${P}^{\mathrm{op}}$ by zeroth-order DP coupling dominated ELR with the cross-over temperature being about ${T}_{e}\ensuremath{\sim}50\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The hot phonon effect is demonstrated to reduce ELR significantly. The effect of the magnetic field is found to enhance the ELR significantly, making $B$ as its another tuning knob. Among the four TMDC materials, ${\mathrm{MoS}}_{2}$ (${\mathrm{MoSe}}_{2}$) displays the biggest ELR due to acoustic (optical) phonon scattering while ${\mathrm{WSe}}_{2}$ (${\mathrm{WS}}_{2}$) shows the smallest ELR. Our results for the ${\mathrm{MoS}}_{2}$ material are compared with those of the zero-field ELR.