Quantum magnetotransport properties of silicene: Influence of the acoustic phonon correction

We study the transport properties of silicene in a perpendicular magnetic field by evaluating the Hall and longitudinal conductivities and resistivities when the acoustic phonon correction to the Landau level (LL) energy is taken into account. The acoustic phonons are considered by three modes: the transverse (TA), longitudinal, and out-of-plane ones. Under the influence of the acoustic phonon correction, the quantum Hall effect plateaus occur at higher values of the magnetic field, where the TA phonon displays the strongest effect. The combined effects of the strong spin-orbit coupling in silicene, the external electric field, and the Zeeman field on the transport parameters are investigated. These combined effects lift the spin and valley degeneracy of the LLs, leading to the additional plateaus in the Hall conductivity with the sequence being found as ${\sigma }_{yx}=(4e^2/h)(n/4+1/2)$. The temperature has a significant effect on the width of the Hall conductivity plateaus. We also appraise the longitudinal conductivity ${\sigma }_{xx}$ and the Hall, ${\rho }_{xy}$, and longitudinal, ${\rho }_{xx}$, resistivities and show the difference between the present results and those for graphene as well as for silicene without the Zeeman field effect. The combined effects of the electric and Zeeman fields lead to the quadrupled peaks of ${\rho }_{xx}$ and the sequence of $(h/e^2)/(n+2)$ in the height of the plateaus in ${\rho }_{xy}$.

Figure 1

The exchange self-energy corrections and the splitting of energy levels: The real part of the exchange self-energy versus the magnetic field for (a) $n=0$ and (b) the Landau level. (c) The splitting of the energy level for $K\uparrow$ due to acoustic phonon interaction versus the magnetic field; the green and black curves show the chemical potential vs $B$ with and without phonon corrections, respectively. (d) The dependence of $E_{\lambda }$ on the magnetic field. In all panels, the electron concentration is $n_e=5\times {10}^{11}{\mathrm{cm}}^{-2}$, and $T=2$ K.

Figure 2

Hall conductivity as a function of magnetic field with and without phonon corrections at different values of ${\mathrm{\Delta }}_z$ and $M_z$: (a) ${\mathrm{\Delta }}_z=0, M_z=0$, (b) ${\mathrm{\Delta }}_z\ne 0, M_z=0$, (c) ${\mathrm{\Delta }}_z=0, M_z\ne 0$, and (d) ${\mathrm{\Delta }}_z\ne 0, M_z\ne 0$. The electron concentration is $n_e=5\times {10}^{11}{\mathrm{cm}}^{-2}$, and $T=2$ K.

Figure 3

Hall conductivity as a function of electron concentration $n_e$ at $B=5$ T and $T=2$ K: (a) for different types of acoustic phonons at ${\mathrm{\Delta }}_z=0, M_z=0$, (b) with and without phonon corrections for $M_z\ne 0$ at different values of ${\mathrm{\Delta }}_z$, (c) with phonon correction for $M_z=0$ at different values of ${\mathrm{\Delta }}_z$, and (d) the same as in (c), but for $M_z\ne 0$.

Figure 4

Hall conductivity as a function of the applied voltage $V_g^0$ with and without phonon corrections at $T=2$ K: (a) for ${\mathrm{\Delta }}_z=0, M_z=0$, and $B=5$ T with different types of acoustic phonons, (b) for $M_z=0$ and $B=5$ T at different values of ${\mathrm{\Delta }}_z$, (c) for ${\mathrm{\Delta }}_z=0, M_z=0$ at different values of $B$, and (d) for ${\mathrm{\Delta }}_z=4{\mathrm{\Delta }}_{\mathrm{SO}}, M_z\ne 0$ at different values of $B$.

Figure 5

Hall conductivity versus the electron concentration for different temperatures at $B=5$ T, ${\mathrm{\Delta }}_z=4{\mathrm{\Delta }}_{\text{SO}}$, and $M_z\ne 0$. (a) and (b) are different only in the range of the axes.

Figure 6

Spin- and valley-Hall conductivities versus the perpendicular electric field $E_z$ both with and without phonon corrections at $B=5$ T and $T=2$ K. (a) $M_z=0$ and $n_e=0$, (b) $M_z=0$ and $n_e=5\times {10}^{11}{\mathrm{cm}}^{-2}$, (c) $M_z\ne 0$ and $n_e=0$, and (d) $M_z\ne 0$ and $n_e=5\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 7

The longitudinal conductivity versus the electron concentration for $N_i=1.3\times {10}^9{\mathrm{cm}}^{-2}, B=5$ T, and $T=2$ K: (a) for different types of acoustic phonons at ${\mathrm{\Delta }}_z=0$ and $M_z=0$, (b) with and without phonon correction for $M_z=0$ at different values of ${\mathrm{\Delta }}_z$, (c) with and without phonon correction for ${\mathrm{\Delta }}_z=0$ at different values of $M_z$, and (d) the same as in (b), but for $M_z\ne 0$.

Figure 8

The Hall, ${\rho }_{xy}$, and longitudinal, ${\rho }_{xx}$, resistivities versus the electron concentration for $N_i=1.3\times {10}^9 {\mathrm{cm}}^{-2}, B=5$ T, and $T=2$ K: (a) for different types of acoustic phonons at ${\mathrm{\Delta }}_z=0$ and $M_z=0$, (b) with and without phonon correction for $M_z=0$ and ${\mathrm{\Delta }}_z=4{\mathrm{\Delta }}_{\mathrm{SO}}$, (c) with and without phonon correction for ${\mathrm{\Delta }}_z=0$ and $M_z\ne 0$, and (d) the same as in (b), but for $M_z\ne 0$.

Figure 9

The real part of the exchange self-energy versus the electron concentration for $n=0$ at $B=5$ T and $T=2$ K.

Figure 10

Hall conductivity versus the magnetic field for different LL broadenings at ${\mathrm{\Delta }}_z=0$ and $M_z=0$: (a) no phonon and (b) total phonon. The electron concentration is $n_e=5\times {10}^{11}{\mathrm{cm}}^{-2}$ and $T=2$ K.