In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an $s$-Wave Superconductor Heterostructure

Second-order topological superconductors host Majorana corner and hinge modes in contrast to conventional edge and surface modes in two and three dimensions. However, the realization of such second-order corner modes usually demands unconventional superconducting pairing or complicated junctions or layered structures. Here we show that Majorana corner modes could be realized using a 2D quantum spin Hall insulator in proximity contact with an $s$-wave superconductor and subject to an in-plane Zeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac masses between adjacent boundaries, leading to one Majorana mode at each corner. A similar paradigm also applies to 3D topological insulators with the emergence of Majorana hinge states. Avoiding complex superconductor pairing and material structure, our scheme provides an experimentally realistic platform for implementing Majorana corner and hinge states.

Figure 1

Illustration of a heterostructure composing of a quantum spin Hall insulator on top of an $s$-wave superconductor and subject to an in-plane Zeeman field. The spheres at four corners represent four Majorana zero energy modes.

Figure 2

(a) Quasiparticle bands with edge spectrum (red lines) for open boundary conditions along the $y$ direction. The gap for edge spectrum closes at $h_x={\mathrm{\Delta }}_0=0.4$. (b) Density distributions of the edge bound states (red dots in the inset) for a trivial superconductor, where we have chosen $h_x=0.0$, ${\mathrm{\Delta }}_0=0.4$. (c)–(d) Density distributions of MCMs in different geometries. The radii of the blue disks are proportional to local density. The insets show the energy levels for $h_x=0.8$ and ${\mathrm{\Delta }}_0=0.4$. In both Fig. 2 and Fig. 3, $t_x=t_y={\lambda }_x={\lambda }_y=1.0$ and ${\epsilon }_0=1.0$.

Figure 3

(a) Wannier spectra ${\nu }_x$ versus $h_x$. The inset showcases Wannier centers for different state indexes with $h_x=0.8$. (b) Majorana polarization distribution $p_x(i_y)$ versus lattice index $i_y$. (c) Majorana edge polarizations $p_x^{\mathrm{edge},y}$ and $p_y^{\mathrm{edge},x}$ along $y$-normal and $x$-normal edges, respectively. In (a)–(c), ${\mathrm{\Delta }}_0=0.4$ and $\mu =0.0$ are used. (d) Phase diagram with $\mu =0.2$. NSC denotes a trivial superconductor, and GSC denotes a gapless superconductor. MCM and NSC phases are separated by the phase boundary $h_x=\sqrt{{\mu }^2+{\mathrm{\Delta }}_0^2}$.

Figure 4

(a) Quasiparticle spectrum along $k_z$ with open boundary conditions along the $x$ and $y$ directions. (b) Majorana hinge excitations in a 3D second-order topological superconductor. Parameters are $t_x=t_y=t_z={\lambda }_x={\lambda }_y={\lambda }_z=1.0$, ${\mathrm{\Delta }}_0=0.3$, $h_x=0.6$, and $m_0=2.0$.