Permeability stops being one bulk number: cell-by-cell maps show spin-wave interference carving out local negative loss

High-frequency permeability is computed as a map over position and frequency rather than as one averaged value for the whole sample. Broadband methods such as ferromagnetic resonance, vector network analysis and impedance spectroscopy give frequency information without spatial resolution, while magnetic imaging gives local spin structure without spectral signatures, leaving the two disconnected. In a 200×200×20 nm planar square model with a 4×4×4 nm mesh, the imaginary part of permeability at the strongest resonance, 24.70 GHz, is markedly higher in the central region than elsewhere, placing the spin resonances at the plane's center.

Locally negative imaginary permeability arises from spin-wave propagation and interference, not from the pulsed excitation itself. Traditional magnetism treats magnetic dissipation as an irreversible absorptive process, so the imaginary part of permeability should stay strictly positive. Under 14.10 GHz excitation, magnetization precession lags the field by π/10 in the four corner regions where the imaginary part is positive, and leads the field by 9π/10 in the central side regions where it is negative; the corresponding M(H) loops switch from counterclockwise absorption to clockwise energy release.

Geometric cuts rewrite the resonance landscape, but the rewriting is globally coupled rather than a one-to-one mapping. The uncut sample shows five principal magnetostatic modes across 1–50 GHz, centered on the left and right edges, the corners, the off-center sides, the central core, and the upper and lower boundaries. At a 14.70 GHz reference, a 20 nm cut depth turns the imaginary permeability at the central edge region from negative to positive while suppressing model-1 and generating new modes model-1′, model-2′ and model-4′; at 30 nm, model-3 is also suppressed.

Combining micromagnetic simulation with machine learning enables rapid prediction of microwave-band magnetic performance across a high-dimensional space of defect configurations. High-resolution spatial reconstruction demands fine grids, repeated Fourier transforms and LLG solver tuning; a representative 500,000-cell simulation takes about 300 s, while full spatial permeability reconstruction takes about 45 min and more than 50 GB of RAM, making broad structural parameter sweeps infeasible with conventional micromagnetic simulation alone. On a dataset of 210 iron sphere configurations, ensemble methods such as GBDT and EGB consistently outperformed the other tested algorithms in accuracy and robustness; predicted trends for the C-, X-, Ku- and K-bands and the full 2–18 GHz range agree with broadband vector network analysis on four types of etched iron particles.

The demonstrated scope is a planar square model and 400 nm-diameter nanoscale iron spheres in micromagnetic simulation, with broadband vector network analysis spectra from four types of etched carbonyl iron powder samples as indirect experimental comparison; no experimental technique currently exists to directly map the spatial distribution of magnetic permeability.

Local permeability is explicitly defined as a mesoscale dynamical response quantity under weak excitation, relying on linear response and local quasi-uniformity within each cell, rather than a macroscopic constitutive parameter in the conventional homogenized sense; the geometry-spectrum relationship is a learnable many-parameter response landscape, so enhancing one mode may accompany suppression, shifting or hybridization of others.