Intrinsic first and higher-order topological superconductivity in a doped topological insulator

We explore higher order topological superconductivity in an artificial Dirac material with intrinsic spin-orbit coupling, which is a doped $\mathbb{Z}_2$ topological insulator in the normal state. A mechanism for superconductivity due to repulsive interactions -- $\mathit{pseudospin}$ $\mathit{pairing}$ -- has recently been shown to naturally result in higher-order topology in Dirac systems past a minimum chemical potential \cite{Li2021}. Here we apply this theory through microscopic modeling of a superlattice potential imposed on an inversion symmetric hole-doped semiconductor heterostructure, known as hole-based semiconductor artificial graphene, and extend previous work to include the effects of spin-orbit coupling. We find that spin-orbit coupling enhances interaction effects, providing an experimental handle to increase the efficiency of the superconducting mechanism. We show that the phase diagram of these systems, as a function of chemical potential and interaction strength, contains three superconducting states -- a first-order topological $p+ip$ state, a second-order topological spatially modulated $p+i\tau p$ state, and a second-order topological extended $s$-wave state, $s_\tau$. We calculate the symmetry-based indicators for the $p+i\tau p$ and $s_\tau$ states, which prove these states possess second-order topology. Exact diagonalisation results are presented which illustrate the interplay between the boundary physics and spin orbit interaction. We argue that this class of systems offer an experimental platform to engineer and explore first and higher-order topological superconducting states.