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 nontrivial 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.

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• Majorana bound states
• Heterostructures
• Topological materials