A compact analytical expression for the maximum tunneling magneto-dielectric response in nanogranular films

The tunneling magneto-dielectric (TMD) effect is a magnetic-field-induced spin-dependent enhancement of the dielectric relaxation rate. This effect provides critical insights into tunneling electronic oscillations in disordered nanogranular systems and offers substantial benefits for fast-response, low-loss energy-conversion materials and magnetic sensing devices. However, developing an analytical framework that links fundamental material properties to the TMD response remains a considerable challenge. In this study, we introduce a simple figure of merit for the maximum TMD response, together with a compact analytical expression that directly relates it to dielectric properties, magnetization, and spin polarization. The expression identifies spin polarization as a key parameter for enhancing the maximum TMD response and predicts an intrinsic upper bound of 300% in the ideal limit. For experimental validation, Co/(Co–Fe)/(Co–Ni)–SrF 2 nanogranular films were prepared, with spin polarization tunable via nanogranule compositional control. A maximum TMD response of 11.2% is achieved at room temperature in the CoFe-containing film, representing, to the best of our knowledge, the first TMD response exceeding 10%. These results establish a direct analytical link between fundamental material properties and TMD performance and provide a materials-design guideline for high-performance nanogranular magneto-dielectric materials.