Charged-impurity-induced Majorana fermions in topological superconductors

We study numerically Majorana fermions (MFs) induced by a charged impurity in topological superconductors. It is revealed from the relevant Bogoliubov–de Gennes equations that (i) for quasi-one-dimensional systems, a pair of MFs are bounded at the two sides of one charged impurity and are well-separated; and (ii) for a two-dimensional square lattice, the charged-impurity-induced MFs are similar to the known pair of vortex-induced MFs, in which one MF is bounded by the impurity while the other appears at the boundary. Moreover, the corresponding local density of states is explored, demonstrating that the presence of MF states may be tested experimentally.

Figure 1

Schematic illustration of an off-plane charged impurity, with $d$ being the distance between the plane and the impurity.

Figure 2

The renormalized normal state band dispersion in momentum space [Eq. (6)].

Figure 3

The numerical results for a $400\times 3$ lattice. (a) The self-consistent results of the order parameter $\Delta$ and the site-dependent particle number $n$. (b) The eigenvalues of the Hamiltonian. (c) The spatial distributions of the two MFs. (d) The intensity plot of the LDOS at zero energy.

Figure 4

The numerical results for a $48\times 48$ lattice. (a) The intensity plot of the order parameter. (b) The intensity plot of the site-dependent particle number. (c) The two-dimensional cuts of the order parameter and the particle number along $x=24$. (d) The eigenvalues of the Hamiltonian in the presence of a charge-impurity. Inset of (d): The eigenvalues of the Hamiltonian with the periodic boundary and without the impurity.

Figure 5

The numerical results for the $48\times 48$ lattice. (a) The spatial distribution of ${\gamma }_1$. (b) The spatial distribution of ${\gamma }_2$. (c) The intensity plot of the zero-energy LDOS. (d) The two-dimensional cuts of panels (a)–(c) along $y=24$.