Majorana edge states in superconductor-noncollinear magnet interfaces

Through $s-d$ coupling, a superconducting thin film interfaced to a noncollinear magnetic insulator inherits its magnetic order, which may induce unconventional superconductivity that hosts Majorana edge states. We present a unified formalism that covers the cycloidal, helical, and tilted conical order discovered in multiferroics, as well as Bloch and Neel domain walls of ferromagnetic insulators, and show that they induce $\left(p_x+p_y\right)$-wave pairing that supports Majorana edge modes. The advantages over one-dimensional proposals are that the Majorana states can exist without fine tuning of the chemical potential, can be stabilized in a much larger parameter space, and can be separated over the distance of long-range noncollinear order that is known to reach a macroscopic scale. A skyrmion spin texture, on the other hand, induces a nonuniform $\left(p_r+i{p}_{\phi }\right)$-wave-like pairing under the influence of an emergent electromagnetic field, yielding a vortex state that displays both a bulk persistent current and a topological edge current.

Figure 1

(a) Topological phase diagram of the 2D model, given by Eq. (1), as a function of $s-d$ coupling $B_0$ and pitch angle $\alpha \equiv {\overline{\alpha }}_a=cos(\mathbf{Q}·\mathbf{a}/2)$ for an in-plane spiral (i.e., $B_{\perp }=0$) propagating along $\mathbf{Q}\propto (1,0)$ and at $\mu =0$. The topologically nontrivial phase corresponds to the blue region. For comparison, the nontrivial phase of the 1D model with $\mu =-0.2$ and $-0.4$ is shown as the region inside the light blue and white dots, respectively. Other noncollinear orders, such as helical and tilted conical orders propagating along various directions, exhibit similar phase diagrams. (b) Energy levels of a system with size $L=80a$ subject to an in-plane spiral of wave length $\lambda =2\pi /\left|\mathbf{Q}\right|=16a$ with $B_0=0.3$ and $\mathbf{Q}\propto (1,0)$. The orange regions are the $q_y$'s at which the topological criterion given by Eq. (8) is satisfied, where one also sees the Majorana zero-energy edge states. (c) LDOS along $\widehat{\mathbf{x}}$ for different values of $\lambda /L$. Red, green, and blue lines are the first, second, and third sites away from the edge, respectively. The $\lambda /L=2$ case corresponds to a Neel or Bloch domain wall in a ferromagnetic insulator. Other parameters are $t=-1$ and $\mathrm{\Delta }=0.05$.

Figure 2

(a) Spin texture of the pedagogical model, which is closely related to the spin texture of a single nonchiral skyrmion on (b) a polar lattice and (c) a square lattice. (d),(e) Energy bands of the pedagogical model with OBC in $\widehat{\mathbf{r}}$ and PBC in $\widehat{\mathbf{\phi }}$, with $N_r=N_{\phi }=40$ and $B_0=0.24$, in (d) the presence and (e) the absence of the emergent EM field. (f)–(i) Spontaneous current patterns at the interface between an $s$-wave SC and a single nonchiral skyrmion on a square lattice with size $9\times 9$ for (f),(g) the normal state $\left(\mathrm{\Delta }=0\right)$ and (h),(i) the superconducting state $\left(\mathrm{\Delta }=0.2\right)$. Parameters are $t=-1$ and $\mu =-0.1$.