Magnetoconductance of the quantum spin Hall state

We study numerically the edge magnetoconductance of a quantum spin Hall insulator in the presence of quenched nonmagnetic disorder. For a finite magnetic field $B$ and disorder strength $W$ on the order of the bulk gap $E_g$, the conductance deviates from its quantized value in a manner which appears to be linear in $\left|B\right|$ at small $B$. The observed behavior is in qualitative agreement with the cusplike features observed in recent magnetotransport measurements on HgTe quantum wells. We propose a dimensional crossover scenario as a function of $W$, in which for weak disorder $W<E_g$ the edge liquid is analogous to a disordered spinless one-dimensional (1D) quantum wire, while for strong disorder $W>E_g$, the disorder causes frequent virtual transitions to the two-dimensional (2D) bulk, where the originally 1D edge electrons can undergo 2D diffusive motion and 2D antilocalization.

Figure 1

(Color online) Magnetoconductance $G$ of a QSH edge: (a) TB model with asymmetric edge states ${\lambda }_2⪡{\lambda }_1$ to study a single disordered edge; (b) dependence of $G$ on sample width $L_y$ for disorder strength $W=55\text{meV}$ larger than the bulk gap, length $L_x=2.4\mu \text{m}$, fixed clean width $L_y-L_{\text{dis}}=0.03\mu \text{m}$, and local mass term $\delta M=-70\text{meV}$, with error bars (plotted for $L_y=0.12\mu \text{m}$ and $B>0$ only) corresponding to conductance fluctuations $\delta G$; (c) dependence of $G$ on chemical potential $\mu$; and (d) quasi-1D spectrum of the device illustrated in a) for zero $W,B$, showing bulk states (blue thin lines), top edge states (green outer thick lines, solid and dashed), and bottom edge states (red inner thick lines).

Figure 2

(Color online) (a) Magnetoconductance for various disorder strengths $W$ and (b) small-$B$ slope of the magnetoconductance (obtained by linear regression for $0<B<15\text{mT}$). Device size is $\left(L_x\times L_y\right)=\left(2.4\times 0.12\right)\mu {\text{m}}^2$.

Figure 3

(Color online) Dependence of the magnetoconductance $G$ on (a) strength of the $\mathbf{k}$-independent BIA term ${\Delta }_z$ with ${\Delta }_e={\Delta }_h=0$; (b) magnetic field orientation; and (c) Dirac mass term $M<0$. Sample size is $\left(L_x\times L_y\right)=\left(2.4\times 0.12\right)\mu {\text{m}}^2$, disorder strength is $W=55\text{meV}$ for (a) and (b), and $W=30\text{meV}$ for (c).