Majorana zero modes in superconductor-magnet heterostructures with d-wave order

TL;DR

This work analyzes Majorana zero modes in two-dimensional heterostructures where a Néel skyrmion in a magnetic layer couples to fully gapped $d{+}is$ or $d{+}id$ superconductors. Using exact diagonalization of BdG tight-binding models, the authors show that skyrmion-induced spin–orbit coupling and texture-induced Zeeman fields can stabilize MZMs, but unlike the $s$-wave case, stronger $d$-wave components or larger spin twisting can destroy topology due to induced odd-angular-momentum triplet channels. They develop an analytical rotated-basis framework showing that the transformed pairing decomposes into competing singlet and triplet parts, with topology governed by the projected lower-band gap $\Delta_{--}$; phase boundaries follow from the balance $|\text{triplet}| \sim |\text{singlet}|$, and finite-size effects set additional transitions at small $p$ or small $\Delta_d$. These results delineate the parameter regime where MZMs can be realized in unconventional superconductors interfaced with skyrmions, with implications for twisted cuprate systems and magnetically driven topological superconductivity.

Abstract

Magnetic skyrmions in proximity to superconductors offer a route to engineering topological superconductivity due to the synthetic spin-orbit coupling engendered by the spin twist of the skyrmion texture. Previous theoretical works show that this leads to Majorana zero modes (MZMs) in skyrmion-vortex pairs for s-wave superconductors. Here we investigate this mechanism in fully gapped d+is and d+id superconductors. We find the surprising result that while stable MZMs are found in large parts of the phase diagram, strongly enhanced d-wave pairing or stronger skyrmion-induced spin twisting can in fact destroy topology unlike in s-wave superconductors. This effect can be understood from the non-trivial spatial structure of the d-wave pairing, and mixing of odd and even angular-momentum pairing channels in a rotated frame which untwists the skyrmion texture. Our results inform the feasibility of realizing MZMs with unconventional superconductors in such heterostructures.

Figures (6)

• Figure 1: Magnetization profile $\mathbf{N}(r,\theta)$ for a Néel skyrmion with $p=1$.
• Figure 2: Schematic heterostructure consisting of a $d+is$ or $d+id$ superconductor coupled to a magnetic layer hosting a Néel skyrmion with $p=1$.
• Figure 3: Topological phase diagrams for the $d+is$ and $d+id$ skyrmion–superconductor models. Left column: Majorana overlap criterion (low overlap = topological). Right column: energy-ratio criterion. In all panels the inner and outer radii of the annulus are $r_0=2a$ and $R=175.5a$, respectively. The red curve in the $d+is$ panel indicates a heuristic estimate of the phase boundary based on competition between singlet and skyrmion-induced triplet terms. The blue dashed curve for the $d+id$ case is a transition arising from finite size effects, as discussed in the text, with the topological phase extending leftwards on larger system sizes.
• Figure 4: Radial probability distributions of the two edge Majorana modes for the $d+is$ model. (a) A point deep in the topological phase ($p=16$, $\Delta_d=0.1$) showing localization on opposite boundaries. (b) A point in the trivial phase ($p=30$, $\Delta_d=0.75$), where both modes localize at the outer edge.
• Figure 5: Topological phases predictions for the d+is skyrmion model. (a) numerical overlap of left and right Majorana for the states closest to zero energy.(b) are the d+is phase diagrams using the ratio of energy closest to zero energy and the next energy level as an order parameter. The base of the logarithm for the energy plot is $10$. Points $1$ and $2$ on the phase diagram can be considered topological or non-topological, respectively, looking at the energy, but are clearly both in the topological regime when considering the overlap method. Point $3$ on the phase diagram is correctly judged to be in the non-topological phase via both methods.