Unfortunately, superconductors with \(p\)-wave pairing are very rare, with mainly one material being a good candidate. But instead waiting for nature to help us, we can try to be ingenious.

As Carlo mentioned, Fu and Kane realized that one could obtain an effective \(p\)-wave superconductor and Majoranas on the surface of a 3D TI.

We already know how to make Majoranas with a 2D topological insulator. Let us now consider an interface between a magnet and a superconductor on the surface of a 3D topological insulator. Since the surface of the 3D TI is two dimensional, such an interface will be a one dimensional structure and not a point defect as in the quantum spin-Hall case.

The Hamiltonian of the surface is a very simple extension of the edge Hamiltonian, \(\sigma_x k_x + \sigma_y k_y\) instead of just \(\sigma_x k_x\). We can imagine that \(k_y\) is the momentum along the interface between the magnet and the superconductor, and it is conserved. The effective Bogoliubov-de Gennes Hamiltonian is

\[H_\textrm{BdG}=(-i\sigma_x \partial_x+ \sigma_y k_y-\mu)\tau_z+m(x)\,\sigma_z+\Delta(x) \tau_x.\]

What is the dispersion \(E(k_y)\) of states along the interface resulting from this Hamiltonian? Well, for \(k_y=0\) we have exactly the Hamiltonian of the magnet/superconductor interface in the quantum spin-Hall case, which had a zero mode. So we know that the interface is gapless. The magnet breaks time-reversal symmetry, so we will have a chiral edge state, with energy \(E\) proportional to \(k_y\). Just like in the \(p\)-wave superconductor case!

At this point, analyzing the case of a vortex is very simple. We just have to reproduce the geometry we analyzed before. That is, we imagine an \(s\)-wave superconductor disk with a vortex in the middle, surrounded by a magnetic insulator, all on the surface of a 3D topological insulator:

The introduction of a vortex changes the boundary conditions for the momentum at the edge, like in the \(p\)-wave case, and thus affects the spectrum of the chiral edge states going around the disk.

Following the same argument as in the \(p\)-wave case, particle-hole symmetry dictates that there is a Majorana mode in the vortex core on a 3D TI. Interestingly, the vortex core is spatially separated from the magnet - so the vortex should contain a Majorana mode irrespective of the magnet that was used to create the chiral edge mode.

In fact, the magnet was only a crutch that we used to make our argument. We can now throw it away and consider a vortex in a superconductor which covers the entire surface of the topological insulator.

To confirm this conclusion, below we show the result of a simulation of a 3D BHZ model in a cube geometry, with a vortex line passing through the middle of the cube. To make things simple, we have added superconductivity everywhere in the cube, and not just on the surface (nothing prevents us from doing this, even though in real life materials like Bi\(_2\)Te\(_3\) are not naturally superconducting).

In the right panel, you can see a plot of the wavefunction of the lowest energy state. You see that it is very well localized at the end points of the vortex line passing through the cube. These are precisely the two Majorana modes that Carlo Beenakker explained at the end of his introductory video.