Abstract
We theoretically demonstrate non-Hermitian indirect interaction between two magnetic impurities placed at the interface between a 3D topological insulator and a ferromagnetic metal. The coupling of topological insulator and the ferromagnet introduces not only Zeeman exchange field on the surface states but also broadening to transfer the charge and spin between the surface states of the topological insulator and the metallic states of the ferromagnet. While the former provides bandgap at the charge neutrality point, the latter causes non-Hermiticity. Using the Green’s function method, we calculate the range functions of magnetic impurity interactions. We show that the charge decay rate provides a coupling between evanescent modes near the bandgap and traveling modes near the band edge. However, the spin decay rate induces a stronger coupling than the charge decay rate so that higher energy traveling modes can be coupled to lower energy evanescent ones. This results in a non-monotonic behavior of the range functions in terms of distance and decay rates in the subgap regime. In the over gap regime, depending on the type of decay rate and on the distance, the amplitude of spatial oscillations would be damped or promoted.
Highlights
We theoretically demonstrate non-Hermitian indirect interaction between two magnetic impurities placed at the interface between a 3D topological insulator and a ferromagnetic metal
We develop the theory of indirect exchange interaction to a non-Hermitian case in such a way that magnetic adatoms are placed at the interface of a 3D topological insulator and a ferromagnetic metal
We have explored the effects of non-Hermiticity on the indirect exchange interaction mediated by the surface states of the topological insulator that is in contact with the ferromagnetic metal
Summary
Low-energy approximation has been applied on the metallic bands of ferromagnet similar to those of the topological insulator This causes that the non-Hermitian terms take constant values[24]. Due to the lack of reflection symmetry with respect to the plane of interface, we assume the interaction to be anisotropic such that Jx = Jy = Jz. Employing perturbation theory and treating Hint as a perturbation to H0 , up to second order of perturbation and at zero temperature, the indirect exchange interaction between the two magnetic impurities mediated by host fermions can be expressed b y32–34,56. On the right-hand side of Eq (12), the first and the second terms, describing the spin-frustrated interaction, cause an in-plane collinear magnetic ordering with spin orientation along and perpendicular to the line connecting the two impurities, respectively. The integrals of Eqs. (13)–(16) cannot be performed to obtain exact analytical range functions, so, in the following, we first evaluate them numerically and we use some approximations to get analytical expressions for the range functions in two extreme limits of the model
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