In recent years the delta-E effect has been used for detecting low frequency and low amplitude magnetic fields. Delta-E effect sensors utilize a forced mechanical resonator that is detuned by the delta-E effect upon application of a magnetic field. Typical frequencies of operation are from several kHz to the upper MHz regime. Different models have been used to describe the delta-E effect in those devices, but the frequency dependency has mainly been neglected. With this work we present a simple description of the delta-E effect as a function of the differential magnetic susceptibility $\chi$ of the magnetic material. We derive an analytical expression for $\chi$ that permits describing the frequency dependency of the delta-E effect of the Young’s modulus and the magnetic sensitivity. Calculations are compared with measurements on soft-magnetic ${\left({\mathrm{Fe}}_{90}{\mathrm{Co}}_{10}\right)}_{78}{\mathrm{Si}}_{12}{\mathrm{B}}_{10}$ thin films. We show that the frequency of operation can have a strong influence on the delta-E effect and the magnetic sensitivity of delta-E effect sensors. Overall, the delta-E effect reduces with increasing frequency and results in a stiffening of the Young’s modulus above the ferromagnetic resonance frequency. The details depend on the Gilbert damping. Whereas for large Gilbert damping the sensitivity continuously decreases with frequency, typical damping values result in an amplification close to the ferromagnetic resonance frequency.

Magnetoelectric resonators have been studied for the detection of small amplitude and low frequency magnetic fields via the delta-E effect, mainly in fundamental bending or bulk resonance modes. Here, we present an experimental and theoretical investigation of magnetoelectric thin-film cantilevers that can be operated in bending modes (BMs) and torsion modes (TMs) as a magnetic field sensor. A magnetoelastic macrospin model is combined with an electromechanical finite element model and a general description of the delta-E effect of all stiffness tensor components C_ij is derived. Simulations confirm quantitatively that the delta-E effect of the C₆₆ component has the promising potential of significantly increasing the magnetic sensitivity and the maximum normalized frequency change $\Delta f_{\mathrm{r}}$. However, the electrical excitation of TMs remains challenging and is found to significantly diminish the gain in sensitivity. Experiments reveal the dependency of the sensitivity and $\Delta f_{\mathrm{r}}$ of TMs on the mode number, which differs fundamentally from BMs and is well explained by our model. Because the contribution of C₁₁ to the TMs increases with the mode number, the first-order TM yields the highest magnetic sensitivity. Overall, general insights are gained for the design of high-sensitivity delta-E effect sensors, as well as for frequency tunable devices based on the delta-E effect.

Magnetoelectric thin-film cantilevers consisting of strain-coupled magnetostrictive and piezoelectric layers are promising candidates for magnetic field measurements in biomedical applications. In this study, we investigate magnetoelectric cantilevers that are electrically excited and operated in a special mechanical mode with resonance frequencies above 500 kHz. In this particular mode, the cantilever bends in the short axis, forming a distinctive U-shape and exhibiting high-quality factors and a promising limit of detection of 70$\mathrm{p}$$\mathrm{T}$/$\mathrm{Hz}$^1/2 at 10 $\mathrm{Hz}$. Despite this U mode, the sensors show a superimposed mechanical oscillation along the long axis. The induced local mechanical strain in the magnetostrictive layer results in magnetic domain activity. Due to this, the mechanical oscillation may cause additional magnetic noise, deteriorating the limit of detection of such sensors. We compare finite element method simulations with measurements of magnetoelectric cantilevers in order to understand the presence of oscillations. From this, we identify strategies for eliminating the external effects that affect sensor operation. Furthermore, we investigate the influence of different design parameters, in particular the cantilever length, material parameters and the type of clamping, on the amplitude of the undesired superimposed oscillations. We propose design guidelines to minimize the unwanted oscillations.