Low-temperature linear transport of two-dimensional massive Dirac fermions in silicene: Residual conductivity and spin/valley Hall effects

Considering finite-temperature screened electron-impurity scattering, we present a kinetic equation approach to investigate transport properties of two-dimensional massive fermions in silicene. We find that the longitudinal conductivity is always nonvanishing when chemical potential lies within the energy gap. This residual conductivity arises from interband correlation and strongly depends on strength of electron-impurity scattering. We also clarify that the electron-impurity interaction makes substantial contributions to the spin and valley Hall conductivities, which, however, are almost independent of impurity density. The dependencies of longitudinal conductivity as well as of spin and valley Hall conductivities on chemical potential, temperature, and gap energy are analyzed.

Figure 1

(a) Dependencies of longitudinal conductivity on chemical potential at various temperatures. From top to bottom, the lattice temperatures are $T=10.0$, $8.0$, $6.0$, $4.0$, $1.0$, $0.5$, and $0.1$ K. (b) Enlarged version of (a) for chemical potential lying within energy gap and near the bottom of conduction band. In these figures, ${\Delta }_{\mathrm{SO}}=3.9$ meV, ${\Delta }_z=2.0$ meV, and hence the bottom of conduction band is at the energy $1.9$ meV, which is denoted by the vertical dash lines. The impurity density is assumed to be $n_i=1\times {10}^{-12}$ ${\mathrm{m}}^{-2}$ and the cut-off parameter is chosen to be $\Lambda =2$ meV. In inset of (b) ${\sigma }_{xx}$ versus ${\mu }_0$ at lattice temperature $T=0.001$ K are plotted for various impurity densities: $n_i=1\times {10}^{12}$, $5\times {10}^{12}$, and $1\times {10}^{13}$ ${\mathrm{m}}^{-2}$.

Figure 2

Dependencies of longitudinal conductivity on temperature for various chemical potentials: ${\mu }_0=0$, 0.5, 1.0, 1.5, 1.85, 1.9, and 2.0 meV. Other parameters are the same as in Fig. 1.

Figure 3

${\sigma }_{xx}$ versus ${\Delta }_z$ at various lattice temperatures: $T=0.1$, 2, 4, 6, and 10 K. Other parameters are the same as in Fig. 1.

Figure 4

Intrinsic (a) as well as intrinsic and extrinsic (b) spin Hall conductivities versus ${\Delta }_z$ at various lattice temperatures: $T=0.1$, 2, 4, 6, and 10 K. (c) ${\sigma }_{xy}^{\left(s\right)}$ versus ${\Delta }_z$ at $T=0.1$ K for various impurity densities: $n_i=1\times {10}^{12}$, $5\times {10}^{12}$, and $1\times {10}^{13}$ ${\mathrm{m}}^{-2}$. Other parameters in (a)–(c) are the same as in Fig. 1.

Figure 5

Intrinsic (a) as well as intrinsic and extrinsic (b) valley Hall conductivities versus ${\Delta }_z$ at various lattice temperatures. (c) ${\sigma }_{xy}^{\left(v\right)}$ versus ${\Delta }_z$ at $T=0.1$ K for various impurity densities. Other parameters in (a)–(c) are the same as in Fig. 1.

Figure 6

Temperature dependencies of total spin (a) and valley Hall (b) conductivities for various chemical potentials. In (a), from top to bottom, the chemical potentials are ${\mu }_0=0$, $0.5$, $1.0$, $1.5$, $1.85$, $1.9$, $2.0$, $2.1$, and $2.2$ meV, respectively. Other parameters in (a) and (b) are the same as in Fig. 1.