Exchange stiffness constant determination using multiple-mode FMR perpendicular standing spin waves

The exchange stiffness constant is recognized as one of the fundamental properties of magnetic materials, though its accurate experimental determination remains a particular challenge. In thin films, resonance measurements exploiting perpendicular standing spin waves (PSSWs) are increasingly used to extract this parameter, typically through a determination of the first-order PSSW mode. Here, we present a systematic study of multiple PSSW modes in NiFe films, where both the sample thickness and the cap layer material are varied. The results show that a simple analysis based on the Kittel rigid pinning model yields an exchange stiffness constant that varies with thickness, mode number, and capping layer material. This finding is clearly inconsistent with physical expectation that the exchange stiffness constant of a material is single valued for a particular set of thermodynamic conditions. Using a more general exchange boundary condition, we show, through a comprehensive set of micromagnetic simulations, that a dynamic pinning mechanism originally proposed by Wigen is able to reproduce the experimental results using a single value of Aex. Our findings support the utility of short wavelength, higher order PSSWs to determine the Aex of thin films and show that the value of Aex obtained has a weak dependency on the material immediately adjacent to the magnetic layer.