Hartman effect for spin waves in exchange regime

Jarosław W Kłos,Józef Barnaś,Igor L Lyubchanskii,Yuliya S Dadoenkova,Nataliya N Dadoenkova,Justyna Rychły

doi:10.1038/s41598-018-35761-1

Abstract

Hartman effect for spin waves tunnelling through a barrier in a thin magnetic film is considered theoretically. The barrier is assumed to be created by a locally increased magnetic anisotropy field. The considerations are focused on a nanoscale system operating in the exchange-dominated regime. We derive the formula for group delay τgr of a spin wave packet and show that τgr saturates with increasing barrier width, which is a signature of the Hartman effect predicted earlier for photonic and electronic systems. In our calculations, we consider the general boundary conditions which take into account different strength of exchange coupling between the barrier and its surrounding. As a system suitable for experimental observation of the Hartman effect we propose a CoFeB layer with perpendicular magnetic anisotropy induced by a MgO overlayer.

Highlights

We have analyzed the Hartman effect for high-frequency spin waves tunnelling through a narrow magnetic barrier
We investigated a planar system where the barrier was formed by the local increase of a perpendicular magnetic anisotropy
Such an increase may appear due to the specific fabrication of the system – we propose to change the thickness of CoFeB layer covered by MgO to modify spatially the effective out-of-plane anisotropy field

Summary

Introduction

For the barrier characterized by a symmetric shape of the effective anisotropy field: Ha(L/2 + x) = Ha(L/2 − x) and identical boundary conditions at x = 0, L, the saturated values of the group delay for transmitted and reflected waves will be equal.

Results

Conclusion