Universal Quantized Spin-Hall Conductance Fluctuation in Graphene

We report theoretical investigations of the quantized spin-Hall conductance fluctuation of graphene in the presence of disorder. Two graphene models that exhibit the quantized spin-Hall effect (QSHE) are analyzed. Model I is with unitary symmetry under an external magnetic field $B\ne 0$ but with a zero spin-orbit interaction, $t_{\mathrm{SO}}=0$. Model II is with symplectic symmetry where $B=0$ but $t_{\mathrm{SO}}\ne 0$. The two models give exactly the same universal QSHE conductance fluctuation value $0.285\pm 0.005e/4\pi$ regardless of symmetry. We also examined a third model that exhibits QSHE but with quadratic dispersion and obtained the same results. Finally, all three models of QSHE have a one-sided log-normal distribution for spin-Hall conductance. Our results strongly suggest that the quantized spin-Hall conductance fluctuation belongs to a new universality class.

Figure 1

(a)–(c) Spin-Hall conductance and its fluctuation versus disorder strength at different energies and magnetic fluxes for the first model. (d) Schematic plot of the four-terminal graphene device.

Figure 2

(a)–(c) Spin-Hall conductance and its fluctuation versus disorder strength at different energies and magnetic fluxes for the second model. The parameters and symbols used in (a) and (b) are the same.

Figure 3

(a),(b) Spin-Hall conductance and its fluctuation versus disorder strength at different energies for the third model for a $40\times 40$ square lattice with 5000 configurations for each data. (c) The transmission coefficient $T_{31}$ versus disorders for the first model [Fig. 1c].

Figure 4

(a)–(d) The distribution of spin-Hall conductance at different disorder strengths for the second model. (e)–(h) The distribution of $\ln (G_{\mathrm{SH}})$.