Integer and half-integer quantum Hall effect in silicene: Influence of an external electric field and impurities

The influence of silicene's strong spin-orbit interaction and of an external electric field $E_z$ on the transport coefficients are investigated in the presence of a perpendicular magnetic field $B$. For finite $E_z$ the spin and valley degeneracy of the Landau levels is lifted and leads to additional plateaus in the Hall conductivity, at half-integer values of $4e^2/h$, due to spin intra-Landau-level transitions that are absent in graphene. These plateaus are more sensitive to disorder and thermal broadening than the main plateaus, occurring at integral values of $4e^2/h$, when the Fermi level passes through the Landau levels. We also evaluate the Hall and longitudinal resistivities and critically contrast the results with those for graphene on a substrate.

Figure 1

(a) Buckled structure of silicene. (b) The spin-split energy spectrum of silicene, for the $K$ valley, subject to a perpendicular electric field $E_z=12$ $\mathrm{mV}/\ell$. The bold solid (dashed) curves indicate the energy dispersion in the absence of a magnetic field for spins up (down). The horizontal lines show the split LLs for $B=2\mathrm{T}$.

Figure 2

Energy spectrum $E_{n{s}_z}^{\lambda \xi }$ as a function of the magnetic field $B$. Panel (a) is for $E_z=0$ and panel (b) for $E_z=12$ $\mathrm{mV}/\ell$. The black solid curve shows the Fermi level vs $B$. The additional intra-LL small jumps in (b) result from the lifting of the spin degeneracy in each valley; e.g., for the $K$ valley, the solid and dashed curves are for spin down and spin up, respectively, whereas for the $K^{\prime }$ valley they are interchanged. Notice also the lifting of the spin and valley degeneracy for the $n=0\mathrm{LL}$.

Figure 3

Density of states $D\left(E\right)$ as a function of the energy for $B=1\mathrm{T}$ and $\Gamma =0.1\hslash {\omega }_c$. The index $n$ above the peaks labels the LLs.

Figure 4

Hall conductivity vs magnetic field $B$ for $e\ell E_z=0$ (dashed curve) and $e\ell E_z=12\mathrm{meV}$ (solid curve); the electron density is $n_e=5\times {10}^{11}/{\mathrm{cm}}^2$ and $T=\Gamma =0$. The upper curve shows the Fermi level vs $B$ and the small intra-LL jumps in it signal transitions between the up and down spin states.

Figure 5

The same as in Fig. 4 but for $2\le B\le 7\mathrm{T}$ and the red solid curve being the $T=\Gamma =0$ result of Fig. 4. Panel (a) is for different temperatures and panel (b) for different $\Gamma$ values.

Figure 6

(a) Fermi level, density of states near the Fermi level, and longitudinal conductivity vs magnetic field $B$ for $E_z=0$ (dotted curves) and $E_z=12$ $\mathrm{mV}/\ell$ (solid curves). The broadening is $\Gamma =0.04\hslash {\omega }_c$, $T=2$ K, and the electron density $N_e=5\times {10}^{11}/{\mathrm{cm}}^2$. The doubling of the ${\sigma }_{xx}$ peaks for $E_z\ne 0$ directly correlates with that of the DOS and the intra-LL transitions of the Fermi level (see vertical lines). (b) Longitudinal conductivity for $\Gamma =0.04\hslash {\omega }_c$ and $\Gamma =0.1$ $\hslash {\omega }_c$ in the range 1.5 $\le B\le$ 3.5 T.

Figure 7

($B,n_e$) fan diagram of the longitudinal resistivity ${\rho }_{xx}$, in units of $\hslash /e^2$, and the curve of ${\rho }_{xx}$ (upper $x$ axis) vs field $B$ (left $y$ axis). Panel (a) is for $E_z=0$ and panel (b) for $e\ell E_z=12\mathrm{meV}$. Further, $\Gamma =0.1\hslash {\omega }_c$ and $N_e=5\times {10}^{11}/{\mathrm{cm}}^2$. In the bright region ${\rho }_{xx}$ diverges; the color scale appears on the right.

Figure 8

(a) The Hall conductivity ${\sigma }_{yx}$ and the longitudinal component ${\sigma }_{xx}$ as a function of the electron density for $T=2\mathrm{K}$, $\Gamma =0.1\hslash {\omega }_c$, $B=3\mathrm{T}$, and $E_z=0$. (b) The Hall (${\rho }_{yx}$) and longitudinal (${\rho }_{xx}$) resistivities for the same parameters as in (a). The dent in ${\rho }_{xx}$ at low electron density is due to the fact that ${\sigma }_{yx}<{\sigma }_{xx}$. The same dents are seen in the experimental results of Refs. [24] and [25].

Figure 9

The same as in Fig. 8 but for $e\ell E_z=12\mathrm{meV}$.