Helical and topological phase detection based on nonlocal conductance measurements in a three-terminal junction

The helical state is a fundamental prerequisite for many spintronics applications and Majorana zero mode engineering in nanoscopic semiconductors. Its existence in quasi-one-dimensional nanowires was predicted to be detectable as a characteristic reentrant behavior in the conductance, which in a typical two-terminal architecture may be difficult to distinguish from other possible phenomena such as Fabry-Perot oscillations. Here we present an alternative method of helical gap detection free of the mentioned ambiguity, based on the nonlocal conductance measurements in a three-terminal junction. We find that the interplay between the spin-orbit coupling and the perpendicular magnetic field leads to a spin-dependent trajectory of electrons and as a consequence a preferential injection of electrons in one of the arms. This causes a remarkable enhancement of nonlocal conductance in the helical gap regime. We show that this phenomenon can be also used to detect the topological superconducting phase when the junction is partially proximitized by an s-wave superconductor.

Figure 1

Sketch of the $T$ junction with one arm partially covered by a superconductor. Electrons are injected into the left lead 1 and can flow out from the structure by all the leads $1,2,3$ as electrons or holes. The yellow shells correspond to a superconductor. In the sketch, the typical trace of charge in the helical gap regime is depicted for the configuration considered in Sec. 3b.

Figure 2

(a) Local $G_{11}$ and nonlocal $G_{21\left(31\right)}$ conductance as a function the incoming electron energy $E$. In the helical gap energy range, $G_{21}$ (red line) is dominant due to the injection of electrons into the upper arm, preferred due to the interplay between the SO interaction and the Zeeman effect. (b) Dispersion relations (two lowest in energy subbands) $E\left(k_{x\left(y\right)}\right)$ in the horizontal (vertical) lead. Green dots in panel (b) correspond to the points in panel (a). Red dashed line marks the value of energy $E$ chosen for a further analysis. Panels (c) and (d) presents the same as (a) and (b) correspondingly, but with the inclusion of the orbital effects of the magnetic field. We can see that orbital effects does not affect significantly the results. Here, we set $\mu =0, B=0.15$ T.

Figure 3

(a) Current and (b) electron density in the $T$ junction with a clear preference of injection into the upper arm. Panels (c)–(e) present the spin densities $s_x, s_y$ and $s_z$, respectively. Results for $E=2.45$ meV and $B=0.15$ T.

Figure 4

(a) Nonlocal conductance asymmetry defined as $G_{21}-G_{31}$ vs energy of the incoming electron and the magnetic field $B$. Panel (b) presents the nonlocal conductance $G_{21}\left(E\right)$ and $G_{31}\left(E\right)$ calculated without SO coupling at $B=0.3$ T.

Figure 5

(a), (c) Electron and (b), (d) hole current density for $B=0.4$ T. In (a), (b) the chemical potential in the whole system is set to $\mu =2.45$ meV such the normal leads are in the helical regime, while the proximitized region is in the topological state. In (c), (d) $\mu =4$ meV when the proximitized region is in the trivial range and there is no helical state in normal leads. Dashed rectangles mark the superconducting shell.

Figure 6

(a) Zero energy transmission coefficients and (b) the local and nonlocal conductance as a function of the chemical potential for the $T$-shaped junction partially proximitized by a superconductor. Results for $B=0.4$ T. Gray horizontal lines present values of $\mu$ chosen for the analysis; see Fig. 5.

Figure 7

(a) Zero-energy transmission coefficients and (b) the local and nonlocal conductance as a function of the chemical potential ${\mu }_{\mathrm{SC}}$ in the proximitized region while in the rest of the system it is kept constant at $\mu =2.45$ meV. Results for $B=0.4$ T.

Figure 8

Zero-energy nonlocal conductance asymmetry $G_{21}-G_{31}$ as a function of the chemical potential $\mu$ and the magnetic field $B$. Results for the case (a) when the chemical potential is assumed to be constant throughout the nanostructure and (b) when the chemical potential in the horizontal nanowire is fixed at the energy $\mu =4$ meV, beyond the helical gap regime. Black lines denote the topological phase transition calculated for a single nanowire with parameters corresponding to the superconducting part of the $T$ junction.

Figure 9

Nonlocal conductance asymmetry $G_{21}-G_{31}$ as a function of the chemical potential in the input horizontal nanowire $\mu$ and in the superconducting vertical nanowire ${\mu }_{\mathrm{SC}}$. Results for $B=0.4$ T.