Three-dimensional superconductors with hybrid higher-order topology

We consider three-dimensional superconductors in class DIII with a fourfold rotation axis and inversion symmetry. It is shown that such systems can exhibit higher-order topology with helical Majorana hinge modes. In the case of even-parity superconductors, we show that higher-order topological superconductors can be obtained by adding a small pairing with the appropriate C-4 symmetry implementation to a topological insulator. We also show that a hybrid case is possible, where Majorana surface cones resulting from nontrivial strong topology coexist with helical hinge modes. We propose a bulk invariant detecting this hybrid scenario, and numerically analyze a tight binding model exhibiting both Majorana cones and hinge modes.