Proximity-induced superconductivity in magnetic topological insulator films

Abstract

Inducing superconducting correlations in magnetic topological insulators (MTIs) is emerging as a promising route toward the realization of topological superconductivity and Majorana modes. Here, we develop an analytical model for the proximity effect induced by an ordinary s-wave superconductor (SC) placed on top of a MTI thin film with finite thickness. Using a perturbative approach with respect to the electron tunneling between MTI and SC, we derive the leading-order correction to the anomalous Green's function and evaluate the position-dependent induced pairing as a function of all the system parameters. This framework allows us to resolve the spatial, spin, and momentum structure of the induced superconducting order parameter. In particular, we derive an explicit expression for the decay length of the pairing amplitude at the $k_x=k_y=0$ point, and show that increasing magnetization enhances the spin-polarized triplet components and the p-wave contributions of the anomalous Green's function. These findings highlight the interplay between topology, magnetism, and superconductivity in MTI films, providing analytical insight into the emergence of unconventional pairing symmetries relevant for the realization of Majorana modes in finite geometries.

Figures (5)

• Figure 1: Feynman diagram illustrating the second-order correction to the MTI anomalous GF $\mathcal{F}^{\dagger , (2)}_{\mathrm{MTI}}(z,z'; \omega)$, as given in Eq. \ref{['eq:second_order_F']}. The blue (red) arrows represent the normal (anomalous) propagator in the MTI (SC), while the wavy lines stand for the tunneling interaction.
• Figure 2: Band structure and norm of the anomalous propagator in the $(k_x, z)$ plane at $k_y = 0$. The first, second, and third columns correspond to Bi$_2$Se$_3$, Bi$_2$Te$_3$, and Sb$_2$Te$_3$, respectively. Panels (a–c) show the spin-resolved band dispersion, with blue and red denoting spin-down and spin-up states, respectively. Panels (d–f) display the magnitude of the induced anomalous GF in the surface states near the Dirac point at $\bar{\Gamma}$, corresponding to the dotted green line in the band structure. Panels (g–i) show the proximity effect at higher energies within the linear Dirac dispersion, corresponding to the dashed green line. Panels (j–l) show the induced pairing in proximitized bulk states, evaluated at the energy marked by the solid green line. All quantities are given in units of $|\gamma_0|^2$ (see Eq. \ref{['eq:tunneling']}). Calculations are performed for a 10-nm-thick film of MTI with magnetization $\Lambda = 20$ meV, proximitized by a superconductor with $\mu = 10$ meV and $\Delta = 5$ meV. All remaining material parameters are taken from Ref. Edi_fit.
• Figure 3: (a) Induced pairing amplitude at the $\bar{\Gamma}$ point as a function of distance $z$ from the MTI--SC interface. The curves are normalized to their value at the interface $z=0$. (b) Probability density of the lowest-energy eigenstate in the unperturbed MTI film, normalized to unity over the film thickness. (c) Decay length $\xi_{\mathrm{ind}}$ and (d) interface amplitude of the anomalous propagator as functions of the magnetization $\Lambda$. In panel (d), the superconducting proximity effect acts on the surface states near the $\bar{\Gamma}$ point, corresponding to the green dotted line in Fig. \ref{['fig:pairing_spectrum']}. The decay length is evaluated using Eq. \ref{['eq:decay_length']}, while the pairing amplitude corresponds to the norm of the anomalous GF computed at the Fermi momentum $k_F$ and averaged over the polar angle $\theta$. Both quantities are normalized to their zero-magnetization values at $\Lambda = 0$. The film thickness and material parameters are chosen consistently with Fig. \ref{['fig:pairing_spectrum']}.
• Figure 4: Magnitude of the spin--singlet and spin--triplet components of the induced superconducting pairing as functions of the magnetization $\Lambda$ for (a,d,g) Bi$_2$Se$_3$, (b,e,h) Bi$_2$Te$_3$, and (c,f,i) Sb$_2$Te$_3$. Each curve shows the norm of the corresponding component, averaged over the in--plane momentum. All quantities are evaluated at the $z=0$ interface, and their sum is normalized to unity for every $\Lambda$. Panels (a–c), (d–f), and (g–i) correspond to the pairing induced in the low-energy surface states near $\bar{\Gamma}$, in the higher-energy surface states within the Dirac cone, and in the bulk states, respectively. These three regimes correspond to the dotted, dashed, and solid horizontal green lines in the energy spectra in Fig. \ref{['fig:pairing_spectrum']}. Material parameters are chosen consistently.
• Figure 5: Relative weights of the lowest angular momentum components of the induced anomalous GF as functions of the magnetization $\Lambda$ for (a,d,g) Bi$_2$Se$_3$, (b,e,h) Bi$_2$Te$_3$, and (c,f,i) Sb$_2$Te$_3$. Each curve is obtained by evaluating the weights defined in Eq. \ref{['eq:wm_weights']}. The sum of the lowest components with $|m| \leq 3$ is normalized to unity, and only the $m = 0, \pm 1$ terms are shown, since the weights of higher-order components are negligible. Panels (a–c), (d–f), and (g–i) correspond to the pairing induced in the low-energy surface states near $\bar{\Gamma}$, in the higher-energy surface states within the Dirac cone, and in the bulk states, respectively. These three regimes correspond to the dotted, dashed, and solid horizontal green lines in the energy spectra in Fig. \ref{['fig:pairing_spectrum']}. Material parameters are chosen consistently.