Geometry-induced topological superconductivity

Intrinsic topological superconductors with $p$-wave pairing are rare in nature. Its underlying reason is due to the fact that it is usually difficult to change the relative strength between the singlet and triplet channels for the electron-electron interaction in material. Here we show that by considering superconductivity occurring on surfaces of topological insulators (TIs), the relative strength between the singlet and triplet channels can be changed by geometry and sizes of TIs. Specifically, we show that pairing of electrons at different locations on the surface of a topological insulator generally tends to favor the triplet pairing and can induce topological superconductivity by controlling the surface curvature and size of the topological insulator. We illustrate the effects in two configurations, thin-film geometry and the spherical geometry with a sphere or a hemisphere, and find that topological superconductivity arises with the $p\pm ip$ pairing symmetry dominated in nanoscale size of the TI. As a consequence, vortices can spontaneously form on surfaces of topological insulators with roughness of appropriate curvature. These vortices support a Majorana zero mode inside each core and can be used as a platform to host Majorana zero modes without invoking real magnetic fields. Our theoretical discovery opens a new route to realize topological superconductivity in material.

Figure 1

(a) Schematic plot of Dirac cone states on top ($t$) and bottom ($b$) surfaces in a thin film of thickness $L$. (b) Pairing of electrons in the same surface, ${\mathrm{\Delta }}_s$ (top surface) and $-{\mathrm{\Delta }}_s$ (bottom surfaces), is singlet. (c) Pairing of electrons between top and bottom surfaces ${\mathrm{\Delta }}_t$ is triplet.

Figure 2

Thickness dependence of electron-phonon coupling ${\lambda }_{\alpha }$, strength of Coulomb interaction ${\mu }_{\alpha }^{*}$, and $T_c$ for three different Fermi energies [32]: (a) $0.5{\varepsilon }_F$, (b) $0.84{\varepsilon }_F$, and (c) ${\varepsilon }_F$, where ${\varepsilon }_F$ is the bulk Fermi energy of ${\text{Sb}}_2{\text{Te}}_3$ [28]. Here $L$ is number of quintuple layers (QL, 1 QL $\sim$ 1 nm). Subscripts $s$ and $t$ denote intrasurface and intersurface.

Figure 3

Phase diagram of superconducting orders for thin-film geometry with the Fermi energy being in the lower surface Dirac cone. Here $L$ is the thickness of the thin film in the unit of number of quintuple layers, $n_h$ is the surface hole density, and $n_{h,0}$ is the bulk hole density of ${\text{Sb}}_2{\text{Te}}_3$.

Figure 4

The total magnetic flux due to the spin-connection gauge field that passes through a surface $S$ is a topological invariant, which depends on the genus $g$ of the surface as ${\oint }_S{\vec{B}}_{\text{eff}}·d\vec{a}=\pm 2\pi (1-g)$, where $\pm$ are the local spin directions (down and up) of electrons.

Figure 5

Self-consistent vortex solution on the surface of a spherical TI and a semispherical TI. Here ${\lambda }_{\text{so}}=35$ meV nm, ${\xi }_C=4$ nm, ${\xi }_{\text{ph}}=50$ nm, $R=50$ nm, $V_{\text{ph}}=7.76$ meV, $V_C=50$ meV nm, $\mu =-9.5$ meV, the cutoff $\mathrm{\Lambda }=2.1$ meV, and the electron density is $2.7\times {10}^{-3}{\mathrm{nm}}^{-2}$ [44]. (a) The superconducting amplitude ${\mathrm{\Delta }}^{++}$ (meV) (pairing symmetry $p_x+i{p}_y$) on a sphere for the vortex solution with vortex charge $Q=1$ in the vortex phase. Here the color scale decreases from 0.001 meV (yellow) to 0 (blue). Contours and arrows mark directions of supercurrents. (b) The superconducting amplitude ${\mathrm{\Delta }}^{++}$ (meV) for a vortex solution on a semisphere. (c) The quasiparticle energy spectrum (in the unit of meV) inside the vortex core exhibits the Majorana zero mode (indicated by the arrow, tolerance is 0.02 meV). Here $m$ and $n$ are $z$ component of angular momentum of the quasiparticle. (d) Relative strength of the singlet pairing to the triplet pairing (${\mathrm{\Delta }}_{-}^{++}$) versus $\mathrm{\Omega }-{\mathrm{\Omega }}^{\prime }$ in the Meissner phase. Here $V_C=70$ meV and $T=1.24$ meV. (e) and (f) Pairing components in ${\mathrm{\Delta }}^{++}$ and ${\mathrm{\Delta }}^{--}$ for vortex phase (e) and vortex* phase (f). Here $\theta$ is the azimuthal angle relative to the north pole, red circles represent ${\mathrm{\Delta }}_{+}^{++}$, blue squares represent ${\mathrm{\Delta }}_{-}^{--}$, and black dots represent ${\mathrm{\Delta }}_{-}^{++}$ (=${\mathrm{\Delta }}_{+}^{--}$).

Figure 6

(a) The phase diagram of superconducting states on a sphere: $1/R^2$ versus temperature $T$. Here $1/R^2$ acts similar as the local magnetic field. (b) The phase diagram of superconducting states on a sphere: $V_C$ versus $T$. Here ${\xi }_C=4$ nm, ${\xi }_{\text{ph}}=50$ nm, $V_{\text{ph}}=7.76$ meV, $\mu =-9.5$ meV, and the cutoff $\mathrm{\Lambda }=2.1$ meV. Note that vortex phases diminish as $R\to \infty$, which can be accessed in (b) with $V_C\to 0$.