From microstructure to function: spatial visualization of magnetic permeability

Understanding the physics of high-frequency magnetic response is essential for advancing electromagnetic materials. Here, a general strategy is presented for visualizing magnetic permeability as a spatially distributed, frequency-dependent field, instead of a uniform bulk parameter. Our framework reveals that high-frequency magnetic loss arises from coherent spin-wave propagation and interference, leading to spatially selective energy absorption and localized regions of negative imaginary permeability. Tailoring geometric connectivity to suppress spin-wave propagation enables the selective enhancement of localized resonances. Such resonance enables deterministic control over high-frequency response through structural design. The principle is further examined in nanoscale iron spheres. Their surface morphology plays a decisive role in reorganizing the internal resonance domains. This structural reconfiguration, in turn, reshapes the broadband magnetic permeability spectra. To navigate the high-dimensional space of surface defect configurations, machine learning models are integrated with micromagnetic simulations, allowing rapid prediction of magnetic performance across microwave bands. Computational modeling provides an ideal platform to complement experimental exploration. The physic-data framework reveals geometry-to-spectra design rules, validated on iron nanoparticles. This framework uniquely bridges the microscopic spin dynamics with electromagnetic response, enabling tunable high-frequency materials for communication and sensing. This work introduces a general framework describing permeability as a spatially varying, frequency-dependent field, showing high-frequency magnetic loss stems from spin-wave propagation and interference, establishing geometry-to-spectra design methods linking microscopic spin dynamics to tunable high-frequency electromagnetic properties.