Effective permeability tensor and double resonance of interacting bistable ferromagnetic nanowires

Abstract

We present a model for the effective permeability tensor of nonsaturated arrays of axially magnetized bistable ferromagnetic wires based on a Maxwell-Garnett formalism generalized to include the case of two oppositely magnetized wire populations. Explicit expressions for the complex diagonal and off-diagonal components of the effective permeability tensor are derived to describe the magnetic response of the array and its dependence upon microwave frequency and external magnetic field. The model accounts for the geometrical parameters and the static magnetic configuration of the array, as well as the shape and intrinsic properties of the wires. We incorporate the effect of the static and dynamic interwire dipolar interactions and obtain explicit expressions for the position and relative amplitude of the two ferromagnetic resonance peaks associated with the effective permeability of nonsaturated arrays. The two absorption peaks arise due to the presence of the up and down wire populations, which are predominantly excited by the right-handed and left-handed circularly polarized magnetic field components of the electromagnetic wave propagating inside the array. Excellent agreement is found between the model and experimental results obtained from $S$-parameter broadband microstrip line measurements. Small discrepancies when the array is magnetically unsaturated or when the peaks are close to each other are discussed and attributed to spatial variations of the local interaction field not accounted for by the present model.