Optimizing proximitized magnetic topological insulator nanoribbons for Majorana bound states

TL;DR

This work analyzes Majorana bound states in proximitized MTI thin-film nanoribbons by combining a thin-film effective Hamiltonian with superconducting proximity and disorder modeling. It introduces a figure of merit to maximize both the topological regime size and the proximity-induced gap, revealing how electron-hole asymmetry shifts Dirac-point energies and shapes edge-state localization. The key finding is that normal-insulator MTI thin films (m0 < 0) with magnetization near NI–QAHI or QSHI–QAHI boundaries and strong asymmetry (D < 0) yield the most robust, well-separated MBSs, even in the presence of moderate disorder. These results provide material- and geometry-specific guidance for designing MTI-based devices that realize topological superconductivity and MBSs, highlighting the critical roles of thin-film hybridization and asymmetry in setting gap scales and state localization.

Abstract

Heterostructures comprised of a magnetic topological insulator (MTI) placed in the proximity of an $s$-wave superconductor have emerged as a platform for the practical realization of Majorana bound states (MBSs). More specifically, it has been theoretically predicted that MBS can appear in proximitized MTI nanoribbons (PNRs) in the quantum anomalous Hall regime. As with all MBS platforms, disorder and device imperfections can be detrimental to the formation of robust and well-separated MBSs that are suitable for fusion and braiding experiments. Here, we identify the optimal conditions for obtaining a topological superconducting gap that is robust against disorder, with spatially separated stable MBSs in PNRs, and introduce a figure of merit that encompasses these conditions. Particular attention is given to the thin-film limit of magnetic topological insulators (MTIs), where the hybridization of the surface states cannot be neglected, and to the role of electron-hole asymmetry in the low-energy physics of the system. Based on our numerical results, we find that (1) MTI thin films that are normal (rather than quantum spin Hall) insulators for zero magnetization are favorable, (2) strong electron-hole asymmetry causes the stability and robustness of MBS to be very different for chemical potentials above or below the Dirac point, and (3) the magnetization strength should preferably be comparable to the hybridization or confinement energy of the surface states, whichever is largest.

Figures (9)

• Figure 1: The wave function density $|\psi|^2$ at $k_x=0$, and the dispersion of the low-energy states $\text{LES}_1$ (green continuous line) and $\text{LES}_2$ (orange dashed line), for different values of the hybridization parameter $m_0$ and magnetization $M_z$, without electron-hole asymmetry $D=0$. In the insets, $|\psi|^2$ is shown as a function of the transverse coordinate $y$, and the label of the state that can be used to analytically describe the state. On the right side, we show the effect of the electron-hole asymmetry parameter $D$ on the dispersion of the NR. The different background colors represent the possible topological phases, with the NI ($0<M_z<-m_0$), QSHI ($0<M_z<m_0$) and QAHI ($|m_0|<M_z$) phases shown with a white, orange and blue background, respectively.
• Figure 2: Low-energy states at $k_x=0$ in a 50 nm-wide MTI nanoribbon, as a function of the hybridization parameter $m_0$ or the magnetization $M_z$, with blue-to-red color scale indicating the (up-to-down) spin $z$ polarization of the states: (a) the nonmagnetic case ($M_z=0$) as a function of hybridization parameter $m_0$, with the sign of $m_0$ controlling whether the system is in the QSHI or NI phase; (b),(c) the magnetic case ($|m_0| = 40~\mathrm{meV}$) as a function of magnetization $M_z$, with (b) $m_0>0$, giving rise to a QSHI-to-QAHI crossover, and (c) $m_0<0$, giving rise to a NI-to-QAHI crossover. The dashed orange and yellow lines represent the approximate energy of the edge states (Eq. \ref{['eq:E_edge']}) and the continuous black lines represent the approximate energy of the confined surface states (Eq. \ref{['eq:E_surf']}). The inset in (a) shows the different parameter sweeps that are evaluated in $M_z\text{-}m_0$ space. The inset in (b) shows the two lowest energy states on a logarithmic scale in the QSHI phase. The background colors signify the topological phase, as already introduced in Fig. \ref{['fig:wf_map']}. We consider $v_\mathrm{F}=3~\text{eV\AA}$, $m_1=15~\mathrm{eV\AA^2}$, $D=0~\mathrm{eV\AA^2}$.
• Figure 3: (a),(b) The energy $\varepsilon_\text{e}^+$ and decay length $\tilde{\lambda}^+$ of the edge state in the QAHI phase as a function of $k_x$, with (a) no electron-hole asymmetry ($D=0$) and (b) with electron-hole asymmetry ($D=-10~\mathrm{eV\AA^2}$). The gap edges are shown with dashed lines and the Dirac point energy $E_\text{DP}^+$ is indicated. (c) The spin $z$ polarization of the edge state as a function of the asymmetry parameter $D$. (d) The decay length $\tilde{\lambda}^+$ as a function of asymmetry parameter $D$ for different energies, as listed in Table \ref{['tab:lambdas']} and indicated in (b). We consider $v_\mathrm{F}=3~\text{eV\AA}$, $m_1=15~\mathrm{eV\AA^2}$, $m_0=-10~\mathrm{meV}$, and $M_z=40~\mathrm{meV}$.
• Figure 4: Schematic showing the realization of a PNR with MBS in steps, starting from a TI nanoribbon. (a) The dispersion of a TI nanoribbon in the NI phase has spin-degenerate bands, which in the presence of nonzero magnetization $M_z\neq0$ are split up (grey lines). (b) When $M_z>|m_0|$ the nanoribbon enters the QAHI phase, with the lowest energy state being an edge state. (c) The chemical potential $\mu$ is tuned to the center of the topological regime. The extent of the topological regime ($E_\text{a}$), indicated with orange shading, is an important quantity for the robustness of topological superconductivity throughout the PNR. (d) When proximitized by an $s$-wave superconductor, a superconducting gap ($\Delta_\text{a}$) opens up around zero energy. The size of this gap will determine the stability of the MBSs. The BdG spectrum is shown, with red (blue) lines corresponding to electron (hole)-like states, and the induced superconducting gap indicated with green shading.
• Figure 5: The size and topology of the proximity-induced superconducting gap $\Delta_\text{ind}$, determined numerically for a $W=50$ nm-wide PNR. The red (black) color map shows the size of the topological (trivial) gap in the case of (a) no asymmetry $D=0$ and (b) strong asymmetry $D=-14~\mathrm{eV\AA^2}$. In (c) we show a cross section of (b) taken at $M_z=40~\text{meV}$, and we compare the numerical result with the analytical formula in Eq. \ref{['eq:Dind_edge']}. We use $v_\mathrm{F}=3~\text{eV\AA}$, $m_1=15~\mathrm{eV\AA^2}$, $m_0=-5~\mathrm{meV}$Burke2024, and $\Delta_0=1~\mathrm{meV}$.