Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors

We propose models of two-dimensional paramagnetic semiconductors where the intrinsic spin Hall effect is exactly quantized in integer units of a topological charge. The model describes a topological insulator in the bulk and a “holographic metal” at the edge, where the number of extended edge states crossing the Fermi level is dictated by (exactly equal to) the bulk topological charge. We also demonstrate the spin Hall effect explicitly in terms of the spin accumulation caused by the adiabatic flux insertion.

Figure 1

(Color online) The Skyrmion configuration of $\widehat{\mathbf{d}}\left(\mathbf{k}\right)$ in the Brillouin zone of the system (7) with $c=1$, $e_s=3.7$, and $n=1$. The vector $\widehat{\mathbf{d}}\left(\mathbf{k}\right)$ starts from the north pole at the center of Brillouin zone and ends at the south pole at the zone boundary after covering the unit sphere once.

Figure 2

(Color online) (a) Energy spectrum of the QAHE system (7) when the open boundary condition is imposed in $x$-direction. The parameters are $c=1,t∕V=1∕3,e_s=0.5$. The solid and dashed lines between two energy bands stand for the edge states at the right and left edge, respectively. The horizontal dotted line shows a typical chemical potential inside the gap. The two arrows mark the two gapless edge excitation with momentum $k_y=\pm 0.16\pi$. (b) the density distribution of the two edge states at fermi surface, calculated for a $50\times 50$ lattice.

Figure 3

(Color online) (a) The energy spectrum in the case (18) with $t∕V=4$ and $e_s=0.5$. The isolated solid lines stand for the (doubly degenerate) edge states, and the dashed line indicates a typical in-gap Fermi energy (with $\mu =-4.2t$). Each crossing of the Fermi energy and the edge-state spectrum defines two edge states on left and right boundary with opposite value of ${\Gamma }^{12}$. The solid and open circles near a Fermi level mark the particle and hole edge excitations induced by adiabatic flux insertion. (See the text for details.) (b) Schematic picture of the edge states. Each red (blue) line stands for an edge state with ${\Gamma }^{12}=+1(-1)$. The double arrow shows the pseudo spin orientation of the current carried by the corresponding edge state when an electric field is applied in the $y$ direction. For simplicity, only one edge state with a definite ${\Gamma }^{12}$ eigenvalue on each edge is drawn.

Figure 4

(Color online) The schematic curve of the conductance $G$ versus Fermi level $\mu$ for a ballistic quantum well in the quantum spin Hall regime. The plateau with $G_{\mathrm{res}}=\frac{4e^2}{h}$ shows the residual conductance contributed by four edge states on two edges.