Topological superconductivity in tripartite superconductor-ferromagnet-semiconductor nanowires

Motivated by recent experiments searching for Majorana zero modes in tripartite semiconductor nanowires with epitaxial superconductor and ferromagnetic-insulator layers, we explore the emergence of topological superconductivity in such devices for paradigmatic arrangements of the three constituents. Accounting for the competition between magnetism and superconductivity, we treat superconductivity self-consistently and describe the electronic properties, including the superconducting and ferromagnetic proximity effects within a direct wave-function approach. We conclude that the most viable mechanism for topological superconductivity relies on a superconductor-semiconductor-ferromagnet arrangement of the constituents in which spin splitting and superconductivity are independently induced in the semiconductor by proximity and superconductivity is only weakly affected by the ferromagnetic insulator.

Figure 1

Top left: Schematic of the nanowire geometry (cross section) employed in the experiments in Refs. [17, 18]. Bottom left: Enlarged view of the region, presumably most important for the emergence of topological superconductivity where the semiconductor (N), the superconductor (SC), and the ferromagnetic insulator (F) meet. Right: Three paradigmatic stackings of N, SC, and F, which we investigate to explore the emergence of topological superconductivity: (a) SC-N-F, (b) N-SC-F, and (c) N-F-SC. In experiment, the diameter of the nanowire is on the order of 100 nm with epitaxial SC and F layers of thickness $\sim 5\mathrm{nm}$.

Figure 2

Phase diagram for the SC-N-F arrangement. The magnitude of the overall gap $E_{\mathrm{gap}}$, multiplied by the topological invariant $\mathcal{Q}=\pm 1$, is color coded as a function of the phase difference $\mathrm{\Delta }{\phi }_{\mathrm{N}}$ and the number $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi$ of occupied transverse modes in the normal layer. In addition to the joint $\mathrm{\Delta }{\phi }_{\mathrm{N}}$ axis on the left, we also include (nonlinear) $\delta V$ axes for all four panels for reference. The four panels (a)–(d) focus on those regions where the first four modes of the N layer begin to be populated. Phase-transition lines are indicated by blue dashed lines. The inset in panel (a) enlarges the apex of the topological region and includes the semiclassical estimate Eq. (3) for the minimal spin-dependent phase difference required to enter the topological superconducting phase (red dashed line). The symbol in the top right corner of the panels indicates the parameter choice as shown in Fig. 6. The number of transverse modes in S was fixed at $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.52$ for all panels. For other parameters, see Table 1.

Figure 3

Results of a self-consistent approach to superconductivity in a SC-F nanowire, including the self-consistent pairing correlations $\mathrm{\Delta }$ and the spectral gap ${\mathrm{\Delta }}_{\mathrm{spec}}$ as a function of the scattering phase difference $\mathrm{\Delta }{\phi }_{\mathrm{S}}.{\mathrm{\Delta }}_{\mathrm{spec}}$ vanishes close to $\mathrm{\Delta }{\phi }_{\mathrm{S}}\sim 4d_{\mathrm{S}}/\xi$, whereas the pairing correlations $\mathrm{\Delta }$ remain finite. The condensation energy $\mathrm{\Delta }E$ goes to zero before the pairing correlations $\mathrm{\Delta }$ vanish, corresponding to a weakly first-order phase transition. Calculations were performed for $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.13$ and a F layer of infinite thickness. For other parameters, see Table 1.

Figure 4

Phase diagram for the N-SC-F arrangement. The magnitude of the overall gap $E_{\mathrm{gap}}$, multiplied by the topological invariant $\mathcal{Q}=\pm 1$, is color coded as a function of the phase difference $\mathrm{\Delta }{\phi }_{\mathrm{S}}$ and the number $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi$ of occupied transverse modes in the normal layer. In addition to the $\mathrm{\Delta }{\phi }_{\mathrm{N}}$ axis on the left, we also include a $\delta V$ axis on the right for reference. The four panels highlight the parameter ranges where the first four transverse modes in N become populated. At the optimal chemical potentials, the onset of the topological phase takes place close to the semiclassical estimate $\mathrm{\Delta }{\phi }_{\mathrm{S}}=4d_{\mathrm{S}}/\xi$ as indicated by the red dashed line. The gray dashed line labels the magnitude of $\mathrm{\Delta }{\phi }_{\mathrm{S}}$ where the superconducting condensation energy changes sign and superconductivity is fully suppressed by the adjacent ferromagnetic insulator. Phase transitions are denoted by a blue dashed line. The symbol in the top right corner of the panels indicates the parameter choice as shown in Fig. 6. The number of transverse modes in S was fixed at $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.16$ for all panels. For other parameters, see Table 1.

Figure 5

Phase diagram for the N-F-SC arrangement. The magnitude of the overall gap $E_{\mathrm{gap}}$, multiplied by the topological invariant $\mathcal{Q}=\pm 1$, is color coded as a function of the phase difference $\mathrm{\Delta }{\phi }_{\mathrm{S}}$ and the number $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi$ of occupied transverse modes in the normal layer. In addition to the $\mathrm{\Delta }{\phi }_{\mathrm{N}}$ axis on the left, we also include a $\delta V$ axis on the right for reference. The four panels highlight the parameter ranges where the first four transverse modes in N become populated. At the optimal chemical potentials, the onset of the topological phase takes place close to the semiclassical estimate $\mathrm{\Delta }{\phi }_{\mathrm{S}}=4d_{\mathrm{S}}/\xi$ as indicated by the red dashed line. The gray dashed line labels the magnitude of $\mathrm{\Delta }{\phi }_{\mathrm{S}}$ where the superconducting condensation energy changes sign, and superconductivity is fully suppressed by the adjacent ferromagnetic insulator. Due to the finite thickness of F, this occurs at a higher $\mathrm{\Delta }{\phi }_{\mathrm{S}}$ value than in Fig. 4. Phase transitions are denoted by a blue dashed line. The symbol in the top right corner of the panels indicates the parameter choice as shown in Fig. 6. In all panels, the number of transverse modes in S was fixed at $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.61$, and the thickness of F was chosen to satisfy ${\kappa }_{\mathrm{F}}{d}_{\mathrm{F}}=1.22$. For other parameters, see Table 1.

Figure 6

Induced gap ${\mathrm{\Delta }}_{\mathrm{ind}}$ in the absence of spin splitting as a function of the mode occupations $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi$ and $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi$ in the N and SC layers, respectively. The insets show line cuts along the dashed horizontal lines with vertical dashed lines corresponding between the main panels and the insets. Note that the position of the resonance peak in $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi$ shifts slightly with $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi$. (a) SC-N-F arrangement: Resonance at $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi =1.78$ and $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.52$. (b) N-SC-F arrangement: Resonance at $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi =1.78$ and $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.16$. (c) SC-F-N arrangement: Resonance at $k_{\mathrm{N}}{d}_{\mathrm{N}}/\pi =2.23$ and $k_{\mathrm{S}}{d}_{\mathrm{S}}/\pi =27.61$. The symbols marking the topological regions indicate the choice of parameter values in the panels in Figs. 2, 4, and 5, which are labeled by corresponding symbols. For other parameters, see Table 1.