Helical Hinge Majorana Modes in Iron-Based Superconductors

Motivated by recent experiments on ${\mathrm{FeTe}}_{1-x}{\mathrm{Se}}_x$, we construct an explicit minimal model of an iron-based superconductor with band inversion at the $Z$ point and nontopological bulk $s_{\pm }$ pairing. While there has been considerable interest in Majorana zero modes localized at vortices in such systems, we find that our model—without any vortices—intrinsically supports 1D helical Majorana modes localized at the hinges between (001) and (100) or (010) surfaces, suggesting that this is a viable platform for observing “higher-order” topological superconductivity. We provide a general theory for these hinge modes and discuss their stability and experimental manifestation. Our work indicates the possible experimental observability of hinge Majorana modes in iron-based topological superconductors.

Figure 1

(a) Schematic plot of the helical hinge Majorana modes on the edges between top and bottom surfaces and side surfaces. Red and blue contours indicate the left- and right-moving Majorana modes, respectively. (b) $|\psi (r){|}^2$ for the lowest-energy eigenstate of the lattice model on a cube geometry with open boundary conditions in all directions has support exponentially localized to the hinges.

Figure 2

(a) To describe an arbitrary facet $\mathrm{\Sigma }(\varphi ,\theta )$ (purple surface), we consider a spherical geometry and use the Euler angles $\varphi$ and $\theta$ to define a normal vector ${\mathbf{n}}_{\mathrm{\Sigma }}(\varphi ,\theta )$ (red arrow) that uniquely labels $\mathrm{\Sigma }(\varphi ,\theta )$. (b) A schematic plot of the surface gap evolution as a function of $\theta$ when ${\theta }_c$ exists. The blue and red color denote surface pairings with different signs.

Figure 3

(a) Evolution of the (001) and (010) surface gaps. The green dashed and red solid lines use Eq. (15). Results for both surfaces agree well with numerical calculations on the full lattice model (dots). The red region denotes the phase with helical hinge Majorana modes. In (b) and (c), we plot the surface spectrum of both (001) and (010) surfaces using the iterative Green function method. (d) Energy spectrum in a wire geometry along $x$ with open boundary conditions on both $y$ and $z$ with 20 lattice sites along each direction. The linear modes inside the surface gap are the helical hinge Majorana modes. (e) Spatial profile of the eigenstates at the red dot in (d).

Figure 4

Topological phase diagram in ${\mathrm{\Delta }}_0$ (in unit of ${\mathrm{\Delta }}_1$) and chemical potential $\mu$ (in unit of $m_2$). The bulk conduction band starts at $\mu \simeq 0.8$ (dashed line), which is also where the hinge Majorana phase vanishes.