Transport through a quantum spin Hall antidot as a spectroscopic probe of spin textures

We investigate electron transport through an antidot embedded in a narrow strip of a two-dimensional topological insulator. We focus on the most generic and experimentally relevant case with broken axial spin symmetry. Spin-nonconservation allows additional scattering processes, which change the transport properties profoundly. We start from an analytical model for noninteracting transport, which we also compare with a numerical tight-binding simulation. We then extend this model by including Coulomb repulsion on the antidot, and we study the transport in the Coulomb-blockade limit. We investigate sequential tunneling and cotunneling regimes, and we find that the current-voltage characteristic allows a spectroscopic measurement of the edge-state spin textures.

Figure 1

Sketch of the setup. The light gray area is the TI embedding an antidot of radius $R$. The dark gray areas are the leads. The tunneling processes (black dashed line) occur at $x=0,r=0$ and $x=0,r=\pi R$. The chirality of the edge states is indicated by different colors: “$+$” is red and “$-$” is green.

Figure 2

Representative example of the electron densities (green dots) for scattering states originating from lead 1. Panel (a) shows the up-spin propagation and panel (b) shows the down-spin propagation. The lattice is 60 lattice constants long and 34 lattice constants wide. The leads, shown in red, are 9 lattice constants wide. The model parameters are set to $t=1, {\lambda }_{\mathrm{SO}}=0.2t, {\lambda }_{\mathrm{R}}=0.1t$, and $\mu =0.157t$.

Figure 3

Transmission coefficient from lead 1 to lead 2 on panel (a), to lead 3 on panel (b), and to lead 4 on panel (c), for several values of RSOC. The lattice setup is the same as in Fig. 2. The model parameters are set to $t=1$ and ${\lambda }_{\mathrm{SO}}=0.2t$.

Figure 4

${\theta }_T$ at the resonance energies for several values of (a) RSOC in the KM model and (b) bulk inversion asymmetry in the BHZ model. The white crosses indicate the points of transport resonance that were used to evaluate ${\theta }_T$. The model parameters are set to $t=1$ and ${\lambda }_{\mathrm{SO}}=0.2t$ for the KM model, and $A=3, B=-1, M=-2$, and $C=D=0$ for the BHZ model [12].

Figure 5

Nonlocal resistance $R_{14,14}$ in unit of $G_0^{-1}$. The lattice setup is the same as in Fig. 2. The model parameters are set to $t=1, {\lambda }_{\mathrm{SO}}=0.2t$, and ${\lambda }_{\mathrm{R}}=0.05t$. The continuous (blue) line is the computed nonlocal resistance for the lattice. The (green) dashed line is the fit of our model assuming that ${\theta }_T\left(\mu \right)\approx \alpha {\mu }^2+\beta$.

Figure 6

Sketch of a possible cotunneling process between lead 1 and lead 4. The energy level on the antidot is fully occupied. One of the two electrons escapes the antidot toward lead 4, creating a virtual state with an additional hole. The final state is reached when the electron in lead 1 tunnels to the antidot.

Figure 7

Sketch of the two contributions to the tunneling current $I_{N\mathrm{odd}}^{++}$ and $I_{N\mathrm{odd}}^{+-}$ in the case in which only lead 1 is biased, that is, ${\mu }_{U+}=\mu +eV$ and ${\mu }_{U-}={\mu }_{L+}={\mu }_{L-}=\mu$. Panel (a) corresponds to weak Rashba interactions, so that spin is almost conserved: therefore, the contribution $I_{N\mathrm{odd}}^{+-}$ due to spin-flip is strongly suppressed. In the opposite scenario of strong Rashba interactions, spin-flip contributions can even become dominant compared to the spin-preserving one, leading to the scenario depicted in panel (b).