Odd-frequency triplet superconductivity at the helical edge of a topological insulator

Nonlocal pairing processes at the edge of a two-dimensional topological insulator in proximity to an $s$-wave superconductor are usually suppressed by helicity. However, the additional proximity of a ferromagnetic insulator can substantially influence the helical constraint and therefore open a new conduction channel by allowing for crossed Andreev reflection (CAR) processes. We show a one-to-one correspondence between CAR and the emergence of odd-frequency triplet superconductivity. Hence, nonlocal transport experiments that identify CAR in helical liquids yield smoking-gun evidence for unconventional superconductivity. Interestingly, we identify a setup—composed of a superconductor flanked by two ferromagnetic insulators—that allows us to favor CAR over electron cotunneling, which is known to be a difficult but essential task to be able to measure CAR.

Figure 1

Examples of hybrid junctions in a helical liquid. A possible scattering process is represented, with a spin up right moving electron (solid red line), a spin down left moving electron (solid blue line), a spin down right moving hole (red dashed line), and a spin up left moving hole (blue dashed line).

Figure 2

Pairing amplitudes $f_0(x,x,\omega )$ (blue dotted line), $f_3(x,x,\omega )$ (red dashed line), and $f_{+}(x,x,\omega )$ (black solid line), at the NS interface $x=0$ [see Fig. 1]. The impurity is located at $x_1=-3\xi$ and its strength is $m_0=0.5$. The positions of the peaks correspond to the localized Andreev states in the junction and signal an enhancement of triplet pairing.

Figure 3

Nonlocal conductance $G_{21}=-\partial I_2/\partial V_1=T_{\text{CAR}}-T_{\text{EC}}$, at zero temperature, in the setup of Figs. 1 (blue dashed line) and 1(b) (red solid line). A positive signal indicates an excess of crossed Andreev reflection over electron tunneling. We have taken $d=2\xi , \delta =0.5\xi , m_1=1.5{\mathrm{\Delta }}_0, m_{z,1}=0, x_1=-2\xi , x_2=d+\xi$, and a nonzero chemical potential $\mu =0.5{\mathrm{\Delta }}_0$. In the first setup (blue dashed line) there is no ferromagnetic impurity in the right lead, $m_2=m_{z,2}=0$, while in the second setup (red solid line) $m_2=m_1$ and $m_{z,2}=m_{z,1}=0$. The black dotted line shows the conductance in the absence of any ferromagnetic impurity.