Resonance line shape, strain and electric potential distributions of composite magnetoelectric sensors

Multiferroic composite magnetoelectric (ME) sensors are based on the elastic coupling of a magnetostrictive phase and a piezoelectric phase. A deformation of the magnetostrictive phase causes strain in the piezoelectric phase and thus an induced voltage. Such sensors may be applied both for static as well as for dynamic magnetic field measurements. Particularly high sensitivities are achieved for operation at a mechanical resonance. Here, the resonance line shape of layered (2-2 composite) cantilever ME sensors at the first bending-mode resonance is investigated theoretically. Finite element method (FEM) simulations using a linear material model reveal an asymmetric resonance profile and a zero-response frequency for the ME coefficient. Frequency-dependent strain and electric potential distributions inside the magnetoelectric composite are studied for the case of a magnetostrictive-piezoelectric bilayer. It is demonstrated that a positive or a negative voltage may be induced across the piezoelectric layer depending on the position of the neutral plane. The frequency-dependent induced electric potential is investigated for structured cantilevers that exhibit magnetostriction only at specific positions. For static operation an induced voltage is obtained locally at positions with magnetostriction. In addition to this direct effect a resonance-assisted effect is observed for dynamic operation. Magnetostriction in a limited area of the cantilever causes a global vibration of the cantilever. Thus, deformation of the piezoelectric layer and an induced electric potential also occur in areas of the cantilever without magnetostriction. The direct and the resonance-assisted pathway may induce voltages of equal or of opposite sign. The net induced voltage results from the superposition of the two effects. As the resonance-assisted induced voltage changes sign upon passing the resonance frequency, while the direct component is constant, an asymmetric line shape and a zero-response frequency result for the ME coefficient. The zero-response oscillator frequency may be below or above the resonance frequency. The calculated FEM resonance line shapes are fitted successfully to a superposition function of a constant component and a resonant component with a Lorentzian line shape. Equivalence of the superposition function line shape to a Fano resonance profile is derived for frequencies around the resonance. Fano resonances are ubiquitous in physics occurring due to the constructive and destructive quantum interference of two different scattering pathways, e.g., for photons or electrons. The superposition fit parameters describing the resonance line shape are calculated as a function of the cantilever substrate thickness. The inclusion of loss by adjustment of the damping parameter is discussed. The results derived here also are applicable to higher order modes or longitudinal resonance modes.