Andreev reflection in two-dimensional topological insulators with either conserved or broken time-reversal symmetry

We investigate Andreev reflection in two-dimensional heterojunctions formed by a superconductor in contact with a topological insulator ribbon either possessing or breaking time-reversal symmetry. Both classes of topological insulators exhibit perfect Andreev reflection, which is robust against disorder. This is assigned to topologically protected edge states. In the time-reversal symmetric case we show that doping one of the ribbon edges with magnetic impurities suppresses one Andreev channel, while no such suppression is seen in the broken symmetry situation. Based on this observation, we suggest a tabletop transport experiment able to distinguish between the two types of topological insulators, which does not involve the direct measurement of the material band structure.

Figure 1

Setup for calculating the two-terminal transmission. Region SC is proximity coupled to a superconducting electrode while region TI is the topological insulator described by the two chosen single-particle models. The rectangle marks the region at the TI/SC interface where disorder is introduced.

Figure 2

Andreev reflection coefficient for (a) $Z_2$ and (b) Chern insulators showing perfect Andreev reflection for electron energies smaller than the superconducting gap. The insets show the band structure for the two models solved in a ribbon geometry. Here we choose $t_2=0.33$, $\lambda /t_2=2.0$, $\gamma =0.20$, $\beta =-0.11$, and $\Delta =0.50$. The Fermi level $E_{\mathrm{F}}$ is set at zero.

Figure 3

Effect of on-site disorder on the Andreev reflection coefficient for (a) $Z_2$ and (b) Chern insulators. The Andreev process is highly robust against on-site disorder and the crossover to dissipative transport occurs for $W\approx 3.0t$ for the $Z_2$ insulator and $W\approx 2.0t$ for the Chern insulator. Here again we set $t_2=0.33$, $\lambda /t_2=2.0$, $\gamma =0.20$, $\beta =-0.11$, and $\Delta =0.50$, and the Fermi level $E_{\mathrm{F}}$ is taken at zero. The curves are averaged over 960 random configurations.

Figure 4

Andreev reflection coefficient in presence of magnetic impurities located at the edge of a TI ribbon: (a) $Z_2$ insulator, (b) Chern insulator. The suppression of one of the edge channels in the time-reversal symmetric case $R^{\mathrm{A}}$ produces a drop in $R^{\mathrm{A}}$ from 2 to 1. Magnetic impurities have no effect on the Andreev reflection for a time-reversal symmetry broken insulator. Here we have chosen $J_z=J_{\parallel }=0.50$ and $\left|S\right|=2$. The other parameters are the same as before.

Figure 5

Andreev reflection coefficient in the presence of magnetic impurities for the $Z_2$ insulator as a function of the spin inclination angle $\theta$ for various values of the exchange coupling. The relaxation of the spins leads to $R^{\mathrm{A}}$ reverting back towards unity.