Topological s-Wave Pairing Superconductivity with Spatial Inhomogeneity: Midgap-State Appearance and Robustness of Superconductivity

Abstract

We study the quasiparticle spectrum of 2D topological s-wave superconductors with the Zeeman magnetic field and the Rashba spin-orbit coupling in the presence of spatial inhomogeneity. Solving the real-space Bogoliubov-de Gennes equations, we focus on the excitations within the superconducting gap amplitude, i.e., the appearance of mid-gap states. Two kinds of potential functions, line-type (a chain of impurities) and point-type (a single impurity) ones are examined to take spatial inhomogeneity into account. The line setting shows a link of the mid-gap states with the gapless surface modes indicated by the bulk-boundary correspondence in topological superfluid. The point one shows that the quasiparticles with mid-gap energy are much easily excited by an impurity when the Zeeman magnetic field increases within the topological number to be unchanged. Thus, we obtain insights into the robustness of a topological superconductor against non-magnetic impurities. Moreover, we derive an effective theory applicable to high magnetic fields. The effective gap is the mixture of the chiral p-wave and s-wave characters. The former is predominant when the magnetic field increases. Therefore, we claim that a chiral p-wave character of the effective gap function creates the mid-gap states.

Orbital angular momentum in a nonchiral topological superconductor

“…For large α, even in the topological phase, L z /N does not reach −1/2, which is the intrinsic value in chiral p-wave SCs. We have explained the impurity effects on the topological SC: the appearance of the midgap bound states 31,33 , the reduction of the critical temperature 32 , and the vorticity dependence of the vortex bound state 34 .…”

“…In contrast to the Chern number which characterizes a topological SC, the bulk orbital AM may characterize how close it is to a spinless chiral SC with the same Chern number and how fragile it is to nonmagnetic impurities. Topological SCs are not always robust against nonmagnetic impurities, namely, the midgap bound states appear and the critical temperature T c is suppressed [31][32][33] . These impurity effects in the Rashba+Zeeman+s-wave model depend on h and are similar to those in a chiral p-wave SC for large h.…”

Self-consistent study of Abelian and non-Abelian order in a two-dimensional topological superconductor

“…For example, in the study of topological superconductivity the interest usually lies in the detection of Majorana zero modes and the surrounding lowest-energy eigenvalues. For these reasons, the SS method is an excellent tool in the study of superconductivity as demonstrated by Nagai et al [50,70] and has proven to be extremely useful for the research presented in this thesis.…”

“…Within this model, we perform microscopic mean-field calculations and self-consistently obtain converged solutions for both the superconducting order parameter and the Hartree potential. The tight-binding Hamiltonian for such an s-wave topological superconductor is given by [70] …”

“…It is straightforward to extend the model to describe different band structures by including next-nearest-neighbour hopping, or by using different lattice structures such as a hexagonal lattice. Lastly, although Hamiltonian (5.1) is written in the notation of [70], the authors did not consider the effects of the Hartree potential within their study.…”