Fractional parametric resonance in spintronic diodes

Parametric pumping is a powerful tool for the excitation, amplification, and processing of oscillations and waves of different nature. In general, parametric resonance can occur when the pumping frequency $f_p$ and eigenfrequency of a linear mode (or wave) $f_0$ satisfy the relation $f_p$=2$f_0$/n (n=1,2,3,...). While such parametric resonance is well known in mechanical, superconductive, and quantum systems, in magnetic and spintronic systems only the lowest (n=1) parametric resonance at double the spin wave mode frequency $f_p$=2$f_0$ was thoroughly studied and explored. Here, using a theoretical analysis based on both micromagnetic simulations and an analytical model, we show the emergence of resonances at fractional frequencies $f_p$=2$f_0$/n (with n>10) in spintronic diodes driven by the simultaneous action of ac spin-transfer torque (STT, current densities < $10^6$ A/cm2) and voltage-controlled magnetic anisotropy (VCMA, effective anisotropy fields < 50 mT). The analytical model shows that parametric magnetization dynamics is irreducible to the standard Mathieu model of a parametric oscillator and demonstrates the crucial role of VCMA-driven mode frequency modulation: together with parametric coupling, it results in higher-order odd (n=3,5,7,...) fractional resonances, observed above certain VCMA pumping threshold, while simultaneous action with linear STT drive produces thresholdless even (n=4,6,8,...) resonances. This higher-order parametric dynamics is not restricted to VCMA pumping and opens new directions for the application of spintronic diodes for nonlinear signal processing and electromagnetic energy harvesting.