1 Introduction

The ΔE effect, a change of the Young's modulus in presence of a magnetic field, is directly influencing the vibrational behavior of resonators and used for sensing of magnetic fields. In the past two decades, many material combinations have been investigated focusing on the optimization of the characteristics. Often, lead-based piezoelectrics are used^[1-4] in combination with Fe-based alloys as the magnetostrictive material^{[1, 5, 6]} within the laminate due to its respective outstanding material properties. However, lead-based compounds are environmentally critical and iron is an element that is usually avoided in semiconductor technology as it is forming deep traps within the band structure. In a recent study,^[7] magnetoelectric sensors based on a TiN/AlN/Ni composite were investigated, avoiding the disadvantages of the aforementioned materials. Experimentally, such MEMS sensors are often based on sputtered layers due to cost effectiveness in a high-volume production. As a consequence, the resulting material stack is usually poly-crystalline^[8] leading to a mixture of different crystalline orientations in the magnetostrictive layer. The polycrystalline character of the experimental thin films is depicted in Figure 1a. Recently, a finite-element study was carried out determining the anisotropic sensitivity and its limits on basis of a single-crystalline approximation of the material stack.^[9] It was found that the sensitivity is strongly affected by the crystal orientation and maximum along the (110) direction and minimum for (111). Due to the polycrystalline nature of the functional layers, experimental conditions cannot be adequately reproduced with this approach. In this work, the recently derived ΔE effect model is extended to investigate the specific impacts of the poly-crystalline nature in the limit of an untextured magnetostrictive nickel. Using the eigenfrequency change arising from the anisotropic ΔE effect contributions of individual crystallites, different nickel layer thicknesses and sensor geometries are studied. The results are divided in two parts. First, a study to investigate the sensitivity of the sensor in terms of geometrical aspects is presented. On the one hand, a moving ΔE effect region along the cantilever is used to determine the lateral sensitivity distribution within a single-crystalline material stack. On the other hand, the influence of partial nickel layer coverage on the sensor sensitivity is studied. The second study treats the normalized eigenfrequency response f_B/f_sat in the case of polycrystalline nickel of different but similar crystalline variations and thicknesses within the model limits. As the underlying model is derived from the magnetocrystalline anisotropy,^{[10, 11]} other anisotropy effects are neglected, like the surface^[12] and shape anisotropy.^[13] Further limits of the model are discussed in the appropriate section. Apart from these limits, the subsequent analysis can help to understand experimental characteristics or to improve specific aspects of the sensor design.

2 Model Details

Based on the experimentally realized magnetoelectric sensors, the simulation model consists of a three-layer laminate with TiN (90 nm) as back electrode, AlN (450 nm) as piezoelectric, and Ni (100, 300, and 500 nm) as the magnetostrictive material (see Figure 1a). The length of the analyzed cantilever(s) is 25 μm. The investigation of the vibrational behavior is carried out using Comsol 5.6 (Comsol Multiphysics GmbH, D-37 073 Göttingen, Germany). The underling model is using coupled multiphysics (solid mechanics and magnetostriction) and is briefly described in recently published work^[9] investigating the vibrational properties of a pure single-crystalline material stack. Hence, the following details are mainly focused on the extensions made to the original model. The geometry is kept 2D in favor of short calculation times while maintaining a high accuracy throughout the different crystallite size variations. The simulations are focused on the natural mode of the cantilevers. The static mesh was adjusted to the respective layers with a quad mesh for TiN (mesh size of ≈131 000 elements), a triangular mesh for AlN ($\lesssim 250000$ elements, depending on the nickel layer), and a second triangular mesh for nickel (mesh size of $\lesssim 150000$ elements, depending on the variation and t_Ni). The minimum and maximum element sizes were 1 nm/20 nm, 10 nm/100 nm, and 1 nm/(t_Ni/20) for TiN, AlN, and nickel, respectively. The element growth rate was 1.1 and constant for all layers. The model was solved using a linear, fully coupled (stationary and eigenfrequency), direct MUMPS solver with tolerances ≤10⁻⁶.

2.1 ΔE Effect in Polycrystalline Layers

The anisotropic ΔE effect as a change of Young's modulus of a magnetostrictive material in dependence of the magnetic field can be described analytically for a nickel crystal grain i^{[9, 14]}, so that

$$\frac{1}{E_{\text{hkl,}i}^{\text{Ni}}}=\frac{1}{E_{\text{hkl,sat},i}^{\text{Ni}}}+\frac{1}{\Delta E_{\text{hkl},i}^{\text{Ni}}}$$(1)

where $\Delta E_{\text{hkl,}i}$ is reproducing the ΔE effect curve of the respective orientation of a nickel grain. A grain i is modeled to appear along any of the three principal axes of the face centered cubic (fcc) crystal, whereas each ΔE effect curve is derived from the magnetostriction ${\lambda }_{\text{hkl,i}}$ and the magnetization $M_{\text{hkl,}i}$ of bulk nickel

$$\frac{1}{\Delta E_{\text{hkl},i}^{\text{Ni}}}=\frac{{(\partial {\lambda }_{\text{hkl},i}/\partial H)}^2}{{\mu }_0\partial M_{\text{hkl},i}/\partial H}$$(2)

The respective ΔE effect curves gained from a single-crystalline cantilever of the same dimensions as used in this work are shown in Figure 1b. The sensitivity $S_{\mathrm{B}}$ of a magnetoelectric sensor is then given by

$$S_{\mathrm{B}}=\frac{1}{f_{\text{sat}}}\frac{\partial f}{\partial B}$$(3)

As a consequence of the anisotropic ΔE effect, the sensitivity is also anisotropic which is shown in Figure 1c with individual peak sensitivities and saturation thicknesses of the nickel layer at the given working points. Therefore, three different thicknesses t_Ni of 100, 300, and 500 nm are used to investigate the vibrational response of polycrystalline nickel under magnetic flux, covering the thin film, the transition, and the saturation region defined by the sensitivity plot.

The bulk properties used for the derivation of the ΔE effect are changing drastically for grain sizes lower than 20 nm^[15] during the transition from a ferromagnetic to a superparamagnetic state. Thus, this is set as the lower crystallite size limit to ensure the applicability of the model. The polycrystalline geometry for the nickel layers is based on a Voronoi net, adapted from ref. 16 and generated using MATLAB (MATLAB R2021b, The Mathworks, Inc., Natick, MA, USA). The size distributions of the generated polycrystals are depicted in Figure 2a for the three different nickel layer thicknesses and two different crystal counts of 350 (blue) and 700 (orange). For each t_Ni, three slightly different distributions are simulated with randomly assigned orientations of either (100), (110), or (111) type. The shown log-normal distributions are fits to the data depending on the square equivalent diameter d_seq derived from the square root of the grain cross sections. Examples of the final generated cantilever geometry of the three thicknesses are presented in Figure 2b for the grain count of 350. For the 100 nm thick nickel layer, a grain count of 700 was not calculated due to crossing the lower size limit.

3 Results

In Figure 3a, sensitivity curves for the lateral variation using a moving ΔE effect region are shown for the case of (100)-oriented nickel. The used simulation model is illustrated in the inset. The moving region is 0.5 μm wide while its position is varied from the fixed end at a position of 0 μm to the free end of the cantilever at 24.5 μm. The step width is 0.5 μm. The three investigated thicknesses exhibit a similar behavior showing a monotonic increase while at the cantilever boundaries a saturation appears. The minimum of $|S_{\mathrm{B}}|$ at the fixed end is to be expected as the curvature of the natural mode is maximum here. At this point, a softening in the nickel layer is affecting the eigenfrequency at most. For increasing position, the sensitivity converges to zero. The minimum of $|S_{\mathrm{B}}|$ is decreasing with increasing t_Ni, reproducing the trend for the absolute sensitivity in Figure 1c. Above 15 μm, the effect of the moving region on S_B is negligible. Hence, for a fully covering nickel layer, the first approximately 60% are contributing to the total sensitivity of the sensor structure whereas the last 40% are mainly affecting the eigenfrequency simply acting as an additional mass. In a second analysis on basis of the same single-crystalline model, the effect of a discontinuous (100)-oriented nickel layer is investigated. In the first case, an increasing nickel layer is simulated as indicated in Figure 3c whereas in the second case the nickel layer is decreasing in direction of the cantilever tip as shown in Figure 3d. At the fixed end, the nickel layer is either set to a fixed (Figure c) or free (Figure d) boundary condition. The step width is 0.5 μm. The absolute sensitivity curves are presented logarithmically scaled for the respective t_Ni in Figure 3b as a function of the edge position of the nickel layer. For the case of increasing coverage, the resulting curves show a monotonic increase for each thickness t_Ni while $|S_{\mathrm{B}}|$ saturates at approximately 15 μm similar to Figure 1a. In saturation, the sensitivity reflects the thickness dependency of Figure 1c. However, in the range between 2 and 10 μm, a decrease of the sensitivity for t_Ni = 500 nm is observable with values even lower than for t_Ni = 300 nm. This is originated in the stepped cantilever configuration with two sections of different bending stiffnesses (also depending on B), reduced masses, and fractional lengths. The frequency response and thus the derived sensitivity of such a cantilever can be quite complex while varying these parameters^[20] why a single cause cannot be identified in the present case. In the second case, the nickel layer coverage is decreased toward the cantilever tip with the same step size and edge positions as before. With increasing edge position (decreasing nickel coverage), $|S_{\mathrm{B}}|$ is monotonically decreasing as expected as the edge position is following the sensitivity curves in Figure 3a. However, S_B behaves different for $t_{\text{Ni}}>100\text{nm}$. The sensitivity curve for 500 nm is lower on average than for 300 nm. With increasing edge position (or nickel thickness), the highest curvature appears at the fixed cantilever end, hence in the TiN/AlN stack. Consequently, the thick nickel layer acts more like a magnetically insensitive mass. For full nickel coverage, the determined peak sensitivities are summarized in Table 1. According to the results, two edge cases arise. For thin layers ($t_{\text{Ni}}<<t_{\text{AlN}}+t_{\text{TiN}}$), the difference between the fix and free boundary condition is decreasing/negligible. Thus, the boundary condition is irrelevant then leaving room for further design optimization of the material stack. For $t_{\text{Ni}}>100\text{nm}$, a fixed boundary condition is favorable as a higher $|S_{\mathrm{B}}|$ can be achieved.

In contrast to the single-crystalline case of the sensor investigated so far, a polycrystalline nickel layer is affected by contributions of all principal axes within the investigated model. As a consequence, the ΔE effect characteristic changes according to the respective fractions of each orientation. Hence, three variations of untextured nickel are investigated subsequently, each with a slightly different distribution biased approximately 5–10% for a specific orientation. Figure 4a,b presents the normalized eigenfrequency curves for a stack containing 350 and 700 nickel crystals of varying orientation. The size distributions of the crystals are given in Figure 2 for comparison. In Figure 4a, the ΔE effect magnitude reflects the general thickness dependence. However, the characteristics are exhibiting differences depending on the variation and the respective bias, whereas under consideration of Figure 1b, a single-dominant orientation cannot be found. With increasing nickel thickness, the curves appear to converge to a specific characteristic. Comparing Figure 4a,b, the generated characteristics for 300 and 500 nm are very similar and independent of the amount of crystals. To explain the observed behavior, ΔE effect curves for different compositions of orientations are derived by averaging the single-crystalline solutions of Figure 1b. Figure 4c illustrates the averaged curves for all possible combinations of two or three orientations labeled as the fractions of (100):(110):(111). Accordingly, the generated polycrystalline characteristics for 300 and 500 nm appear to reflect a combination that lies between pure 1:1:1 and 0:1:1. As the (100) orientation exhibits the smallest ΔE effect magnitude of the three orientations, it appears to be underrepresented in combination with (110)- or (111)-oriented crystals. Following the lateral sensitivity curves of Figure 3a, the specific orientations near the clamping of the cantilever have the highest impact on the ΔE effect characteristic. As a consequence, the variance of different polycrystalline variations should be highest in case of large crystals relative to the nickel layer thickness. This effect can be observed in Figure 4a, where the variance for $t_{\text{Ni}}=100\text{nm}$ is higher than for the thicker layers. To quantify this effect, the deviation from the untextured case is derived from the average of $|{(f\text{/}{s}_{\text{sat}})}_{\text{poly}}/{(f\text{/}{s}_{\text{sat}})}_{1:1:1}-1|$ in the range between 1 and 50 mT. This is shown in Figure 4d for all simulated variations in dependency of the nickel layer thickness. Due to averaging, the peak deviations are higher. Two effects can be seen here. First, the deviation decreases on average with increasing $t_{\text{Ni}}$ for both, 350 (blue) and 700 (orange) crystals. Second and more interesting, the smaller crystals (orange) exhibit a higher deviation than the larger ones. The reason lies in the model itself in respect to the ideal grain boundaries. These have no effect on mechanical or magnetic properties. Hence, neighboring crystals of the same orientation act as a single one. To avoid this, the underlying model has to be extended by a correct description of grain boundaries. However, the types of boundaries are manifold^[21] as well as their effects on the magnetic properties.^[22] Additionally, only their influence on the saturation magnetization is usually investigated why a full dependency of the magnetic field and hence their impact on the ΔE effect is not accessible.

The found results support the optimization process for magnetoelectric MEMS cantilevers in respect of eigenfrequency tuning without affecting the ΔE effect characteristics. The analysis of the polycrystalline nickel layers delivers the eigenfrequency response to be expected in the limit of polycrystalline untextured nickel in contrast to the fully single-crystalline approach recently.^[9] As the experimental data is usually found within these two limits, an estimation can be made regarding the (poly-)crystalline degree of such a sensor on basis of the measured characteristics. As the basic model is derived from the magnetocrystalline anisotropy, other magnetic anisotropy types may have a large impact on the presented results.

4 Conclusion

In this work, the magnetocrystalline properties of magnetoelectric MEMS cantilever sensors were investigated via a finite-element study. Based on an experimental template, a 2D model consisting of a single-crystalline TiN/AlN/Ni laminate was analyzed in terms of the influence of a localized ΔE effect region as well as a partial nickel layer coverage on the magnetic sensitivity. The lateral sensitivity dependency along the cantilever structure was derived for three different nickel layer thicknesses using a moving ΔE effect region. The gained dependencies show that the first $\approx 60\%$ of the cantilever length are contributing to the sensitivity, while the last $\approx 40\%$ are acting as an additional mass only. The investigation of partial nickel layer coverage revealed an increasing sensitivity for increasing coverage while saturating above 60%. For a decreasing coverage toward the cantilever tip, the sensitivity is decreasing. The boundary condition of the magnetostrictive layer was found to be negligible for thinning nickel layers whereas a fixed condition is preferable for thick layers. To study the influence of a polycrystalline untextured nickel layer, different variations based on 350 and 700 crystals were generated on basis of a Voronoi net. The results indicate a higher deviation of the ΔE effect characteristic for thin nickel layers arising from only few crystals within the laterally high-sensitivity region. Thicker layers were found to exhibit a characteristic similar to an untextured composition with a tendency toward a stronger appearance of the (110) and (111) orientations. Though other magnetic anisotropies are not considered, the gained results might not be directly applicable to experimental data but suited to understand experimental characteristics or helping to improve aspects of the sensor design.

Conflict of Interest

The authors declare no conflict of interest.