Tunable proximity effects and topological superconductivity in ferromagnetic hybrid nanowires

Hybrid semiconducting nanowire devices combining epitaxial superconductor and ferromagnetic insulator layers have been recently explored experimentally as an alternative platform for topological superconductivity at zero applied magnetic field. In this proof-of-principle work we show that the topological regime can be reached in actual devices depending on some geometrical constraints. To this end, we perform numerical simulations of InAs wires in which we explicitly include the superconducting Al and magnetic EuS shells, as well as the interaction with the electrostatic environment at a self-consistent mean-field level. Our calculations show that both the magnetic and the superconducting proximity effects on the nanowire can be tuned by nearby gates thanks to their ability to move the wavefunction across the wire section. We find that the topological phase is achieved in significant portions of the phase diagram only in configurations where the Al and EuS layers overlap on some wire facet, due to the rather local direct induced spin polarization and the appearance of an extra indirect exchange field through the superconductor. While of obvious relevance for the explanation of recent experiments, tunable proximity effects are of interest in the broader field of superconducting spintronics.

Figure 1

Hybrid nanowire geometries. (a, b) Sketches of the devices studied in this work: a hexagonal cross-section InAs nanowire (green) is simultaneously proximitized by an Al superconductor layer (light grey) and an EuS magnetic insulator layer (blue). Two side gates and one back gate (dark grey) allow to tune the chemical potential and control the position of the wavefunction inside the heterostructure. Different dielectrics are used in the experiments [8] to allow gating ($\mathrm{Si}{\mathrm{O}}_2$, in purple) and to avoid the oxidation of the EuS layer ($\mathrm{Hf}{\mathrm{O}}_2$ and $\mathrm{Al}{\mathrm{O}}_2$, in orange and yellow, respectively). In the overlapping device, (a), the Al and EuS layers overlap on one facet, while in the nonoverlapping device, (b), they are grown on different facets. (c) Diagram of the conduction- and valence-band edge positions (in red and blue, respectively) across the heterostructure, spanning the three different materials. In the Al and InAs, the Fermi energy is located in the conduction band (close to the band bottom in InAs), whereas in the EuS it is inside its insulating gap. The EuS conduction band is spin splitted (being $h_{\mathrm{ex}}$ the exchange coupling). For this simulation we fix all gate voltages to zero.

Figure 2

Full model results. (a–c) Spin-resolved partial density of states (pDOS) for the overlapping device integrated over the InAs wire volume (top row) and the Al layer volume (bottom row) when (a) the exchange field $h_{\mathrm{ex}}$ in the EuS layer and the Rashba SOC ${\alpha }_{\mathrm{R}}$ in the InAs wire are set to zero, (b) only the exchange field is turned on, and (c) both are present. Red and blue correspond to the pDOS for different spin orientations along the $z$ axis (wire's direction). (d) Low-energy band structure versus $k_z$ for the hybrid-wire parameters in (c). The color bar represents the relative weight $W$ of a given state in the Al layer (black) and in the InAs wire (yellow). The wavefunction weight in the EuS layer is negligible since it is an insulating material. The ${\mathbb{Z}}_2$ topological invariant is $\mathcal{Q}=-1$, signaling a topological phase. We take here $V_{\mathrm{bg}}=-0.95$ V and $V_{\mathrm{sg}}^{(\mathrm{L},\mathrm{R})}=0$ V. Other parameters can be found in Table I of the SM.

Figure 3

Same as in Fig. 2 but for the nonoverlapping device. The ${\mathbb{Z}}_2$ topological invariant in (d) is now $\mathcal{Q}=1$, signaling a trivial phase.

Figure 4

Simplified-model results. The Al and EuS layers are integrated out and their respective effective induced pairing amplitude and exchange field on the InAs wire are included within the streaked regions shown in the sketches of (a) the overlapping device and (b) the nonoverlapping one. We take ${\mathrm{\Delta }}^{\left(\mathrm{Al}\right)}=0.23$ meV and $h_{\mathrm{ex}}^{\left(\mathrm{Al}\right)}=0.06$ meV over a wide region of 45 nm near the Al interface. $h_{\mathrm{ex}}^{\left(\mathrm{Al}\right)}$ is only present in (a) where the Al and the EuS are in contact. We include $h_{\mathrm{ex}}^{\left(\mathrm{EuS}\right)}$ over a thin region of 1 nm close to the EuS layer. (c) Topological phase diagram of the overlapping device versus back-gate potential, $V_{\mathrm{bg}}$, and exchange field at the EuS-InAs interface, $h_{\mathrm{ex}}^{\left(\mathrm{EuS}\right)}$, for $V_{\mathrm{sg}}^{(\mathrm{L},\mathrm{R})}=0$. In the topological regions, we show with colors the expectation value of the induced exchange field (left panel), the topological minigap (middle panel), and the wavefunction profile (right panel), all of them for the transverse subband closest to the Fermi energy. The parameters for which the wavefunctions are plotted are indicated with arrows. (d) Same as (c) but for the nonoverlapping device and fixing $V_{\mathrm{sg}}^{\left(\mathrm{R}\right)}=2$ V and $V_{\mathrm{sg}}^{\left(\mathrm{L}\right)}=0$. The values of $V_{\mathrm{sg}}^{(\mathrm{L},\mathrm{R})}$ in (c) and (d) are taken to maximize the topological regions in each case. Other parameters can be found in Table I of the SM [20]. The extension of the topological phase is very much reduced with respect to (c) (almost negligible for some subbands) both in the $V_{\mathrm{bg}}$ and $h_{\mathrm{ex}}^{\left(\mathrm{EuS}\right)}$ axes. Moreover, in the regions where it is present, the topological gap is small. In contrast, large topological regions with stronger minigaps are found for the overlapping device. The reason is twofold: (i) the wavefunction can be pushed simultaneously close to the Al and EuS layers due to the electrostatics of the overlapped shells, which increases the superconducting and magnetic proximity effects; (ii) the induced exchange field feeds both from the direct and indirect contributions in this case.