Ferroelectric topological superconductor: $\alpha \text{−}{\mathrm{In}}_2{\mathrm{Se}}_3$

A two-dimensional topological superconductor (TSC) represents an exotic quantum material with quasiparticle excitation manifesting in dispersive Majorana modes (DMMs) at the boundaries. A domain-wall DMM can arise at the boundary between two TSC domains with opposite Chern numbers or with a π-phase shift in their pairing gap, which can only be tuned by a magnetic field. Here we propose the concept of a ferroelectric (FE) TSC, which not only enriches the domain-wall DMMs but also significantly makes them electrically tunable. The π-phase shift of the pairing gap is shown to be attained between two TSC domains of opposite FE polarization, and switchable by reversing FE polarizations. In combination with ferromagnetic (FM) polarization, the domain wall can host helical, doubled chiral, and fused DMMs, which can be transferred into each other by changing the direction of the electrical and/or magnetic field. Furthermore, based on first-principles calculations, we demonstrate $\alpha \text{−}{\mathrm{In}}_2{\mathrm{Se}}_3$ to be a promising FE TSC candidate in proximity with a FM layer and a superconductor substrate. We envision that a FE TSC will significantly ease the manipulation of a DMM by electrical field to realize fault-tolerant quantum computation.

Figure 1

Representative TSC platforms with tunable DMMs. (a) The helical DMMs realized by adding a π-phase shift to the pairing gap $\mathrm{\Delta }$ of SC. (b) The fused and chiral DMMs realized by opposite FM fields under a uniform pairing gap. (c) The helical and chiral DMMs realized by opposite FE polarizations under a uniform pairing gap and FM field. (d) The doubled and chiral DMMs realized by opposite FE polarizations and opposite FE fields under a uniform pairing gap. The DMMs are plotted by red and blue lines with the arrow indicating the transport direction. The dot with a yellow arrow shows the direction of the FM field and the bar with a purple arrow shows the direction of the FE polarization. The gray circle with arrows represents the spin-momentum-locked states with specific chirality. The wrapped DMMs at the FM-FM domain wall in (b) denotes the fused DMMs.

Figure 2

FE TSC TB model analysis. (a) Schematics of the TB model on a square lattice. (b) FM field $V_z$-dependent pairing gap at $\mathbf{k}=0$ calculated by diagonalizing $H_{\mathrm{BdG}}^{h\alpha }\left(\mathbf{k}\right)$ with different $\eta$. Inset is the magnified view near the FE TSC phase transition. The used parameters are $\mu =4t, \mathrm{\Delta }=0.1t$, and $\lambda =0.5t$. $V_z$ is in the unit of $\mathrm{\Delta }$. (c) FE TSC phase diagram in the parameter space of $V_z$ and $4t-\mu$. The left and right insets show the distribution of Berry curvature with $V_z<0$ and $V_z>0$, respectively. (d) Four FE TSC domains with different sign combinations of OPP and $C$.

Figure 3

DMMs with high tunability. (a) Schematics of a nanoribbon, with ($V_z^l,{\lambda }^l$) and ($V_z^r,{\lambda }^r$) marking the left and right domains, respectively. The width of the left (right) domain is 40 (60) unit cells and the arrow $y$ indicates the periodic direction. (b)–(d) Nanoribbon BdG quasiparticle band structure (left panel) and real-space DMM distribution ${|{\psi }_{\mathrm{R}}|}^2$ (right panel) calculated for the left (right) domain configurations of (b) $\left\{V_{z+}^l{\lambda }_{+}^l\right\}\left(\left\{V_{z-}^r{\lambda }_{-}^r\right\}\right)$ or $\left\{V_{z+}^l{\lambda }_{-}^l\right\}\left(\left\{V_{z-}^r{\lambda }_{+}^r\right\}\right)$, (c) $\left\{V_{z+}^l{\lambda }_{+}^l\right\}\left(\left\{V_{z+}^r{\lambda }_{-}^r\right\}\right)$ or $\left\{V_{z+}^l{\lambda }_{-}^l\right\}\left(\left\{V_{z+}^r{\lambda }_{+}^r\right\}\right)$, and (d) $\left\{V_{z+}^l{\lambda }_{+}^l\right\}\left(\left\{V_{z-}^r{\lambda }_{+}^r\right\}\right)$ or $\left\{V_{z+}^l{\lambda }_{-}^l\right\}\left(\left\{V_{z-}^r{\lambda }_{-}^r\right\}\right)$, respectively. We use curly braces to show the combinations of $V_z$ and $\lambda$, whose sign is indicated by the subscript $+$ or $-$. The DMMs inside the pairing gap (left panels) and the corresponding real-space distributions (right panels) are plotted in the same color. Crosses and circles indicate the DMMs transporting in the $+y$ and $-y$ direction, respectively. Insets are the distributions of the DMMs in a square island of FE TSC with a domain wall in the middle.

Figure 4

Characterization of $\alpha \text{−}{\mathrm{In}}_2{\mathrm{Se}}_3$-based FE TSC. (a) Schematic diagram of $\alpha \text{−}{\mathrm{In}}_2{\mathrm{Se}}_3$-based FE TSC nanoribbon with opposite FE polarizations at the left and right domains. (b),(c) Nanoribbon BdG quasiparticle band structure (upper panel) and real-space DMM distribution ${|{\psi }_{\mathrm{R}}|}^2$ (lower panel) calculated for the nanoribbon with (b) opposite and (c) uniform FM fields at left and right domains. The DMMs inside the pairing gap (upper panels) and the corresponding real-space distributions (lower panels) are plotted using the same color. ${|{\psi }_{\mathrm{R}}|}^2$ peaks around the outer edges (position 1 or 300) and around the inner domain wall (position 150). Symbols of crosses and circles indicate the DMMs transporting in the $+y$ and $-y$ direction, respectively.