Effects of strong spin-orbit coupling on Shiba states from magnetic adatoms using first-principles theory

Abstract

Magnetic impurities at surfaces of superconductors can induce bound states referred to as Yu-Shiba-Rusinov states (i.e., Shiba states) within superconducting (SC) gaps. For superconductors with strong spin-orbit coupling (SOC), Shiba states arising from even single magnetic adatoms are too complex to be fully understood using effective models alone because SOC cannot be treated perturbatively and multiple orbitals are strongly mixed with spin projections. Here we investigate Shiba states of single magnetic adatoms at the surface of strongly spin-orbit coupled SC Pb, by solving the fully relativistic Dirac-Bogoliubov-de Gennes equations using multiple scattering Green's function methods. For Fe and Co adatoms on Pb(110), we show that the Shiba states are better characterized by total angular momentum, $J$, and its projections on the $z$ axis, $m_J$. As a hallmark of the SOC effect, the Shiba states show a strong dependence of the orientation of the adatom moment. As the orientation of the Fe/Co moment changes, the deepest Shiba states merge at zero energy. This zero-energy state disappears with an additional non-magnetic adatom next to the magnetic adatom, although the other Shiba states unchange. For a Mn adatom on Pb, our Shiba states overall agree with experiments. The characteristics of our Shiba states are also observed with the similar energies and characters in the experiments. The deepest Shiba states that we compute, however, do not appear as close to the Fermi level as the experimental data. It would be interesting to compute the Shiba states with continuously varying vertical distances of the Mn adatom from the surface or with varying the charge state of the adatom, and to calculate the spatial dependence of the spectral density. Our findings will be also useful for understanding of Shiba states for dimers and longer spin chains on the Pb surface considering noncollinear magnetic structures in them.

Topological superconductivity from first principles. II. Effects from manipulation of spin spirals: Topological fragmentation, braiding, and quasi-Majorana bound states

“…The introduction of a phase difference moves the energy of the inverted YSR states away from the edge of the minigap, closer to zero energy, and pushes them further toward zero as the phase difference increases. This effect is similar to the shift of single-impurity YSR states with the canting angle of the magnetic moment [27]. This also leaves the remaining YSR states at the minigap edge around the middle of the nanowire noninverted [Fig.…”

“…In Ref. [27] it was shown that a single magnetic impurity can host a 134513-6 zero-energy state by imposing certain canting angles. Here one should remember how Shiba bands are formed.…”

Topological superconductivity from first principles. I. Shiba band structure and topological edge states of artificial spin chains

“…But it has to be emphasized that even though there are states at zero energy at some spiraling angle, this does not necessarily mean that MZMs are found. As has been demonstrated before, even in the case of a single magnetic impurity on the surface of a superconducting host, it is possible to obtain a state at zero energy by imposing a canting angle [62], even though it is not possible to obtain MZMs for a single impurity. Such states are just YSR states that are accidentally shifted to zero energy as a result of the canting angle.…”

“…This is allowed by the Green's-function-based solution of the Kohn-Sham-Dirac-Bogoliubov-de Gennes (KSDBdG) equations [59,60]. On the one hand, it has been demonstrated previously that many aspects of STM experiments are reproducible by such calculations [60][61][62][63], while on the other hand, it allows us to calculate other quantities, such as spin-polarization and the superconducting order parameter (OP) [64], which are important to understand the nature of these states. Furthermore, some of their properties can be further explored and tested by computational experiments which go beyond the capabilities of conventional experimental techniques.…”