Magnetic Sensitivity of Bending-Mode Delta-E-Effect Sensors

In recent years, magnetoelectric Delta-E-effect sensors have proven the potential to detect low-frequency and low-amplitude magnetic fields, while being fully integrable. The magnetic field sensors use the magnetization-induced change of the effective Young's modulus, referred to as the Delta-E effect. Although several sensor types have been demonstrated, little is reported about the dependence of the design parameters and the resulting sensor characteristics, such as sensitivity and noise. In this Paper, we comprehensively analyze and discuss theoretically and experimentally the magnetic sensitivity and the Delta-E effect of bending-mode Delta-E-effect sensors. Analytical expressions are derived for the magnetic sensitivity and compared with the results of a more elaborate numerical mean-field model. The mean-field model additionally considers spatial variations of the magnetic properties, including the internal stray field of the real shape. Measurements are performed on state-of-the-art bending-mode Delta-E-effect sensors. We find that the Delta-E effect significantly depends on the resonance mode and, moreover, second-order bending modes increase the magnetic sensitivity compared to first-order modes. This can be explained well by the spatial variation of the internal stray field in agreement with the numerical simulations and magneto-optical measurements. Overall, we provide a comprehensive description of the magnetic sensitivity for the improvement of bending-mode Delta-E-effect sensors. We show that the weighting of local properties by the mode shape is not only key for modeling and understanding the Delta-E effect of different bending modes, but also an important factor for improving magnetic sensitivities.

Figure 1

Cantilever design analyzed in this study: (a) sketch of the top view with top electrodes $E_1$ and $E_2$ and the in-plane dimensions of the beam. The counter electrode is located between the $\mathrm{Al}\mathrm{N}$ layer and the poly-$\mathrm{Si}$ substrate; (b) illustration of the layer structures including the respective thicknesses; (c) photograph of the MEMS sensor from above. More information and process details can be found in Ref. [21].

Figure 2

Results from the analytical calculations for a hard-axis magnetization process. (a) Young's modulus $\mathrm{E}(H)$ [Eq. (6)] normalized to its value $E_m$ at fixed magnetization and its derivative $|{\mathrm{\partial }}_{E,H}|:=|dE/dH|$ [Eq. (8)] normalized to its maximum ${\mathrm{\partial }}_{E,H}^{\max }\approx 68\mathrm{G}\mathrm{Pa}/\mathrm{mT}$. (b) Geometry factor G of the magnetic sensitivity $S_H$ [Eq. (16)] as a function of the ratio $t_r=t/h$ of film thickness t to total thickness h of the cantilever beam for different ratios $E_r=E_s/E$ of the two Young's moduli E of the film and $E_s$ of the substrate.

Figure 3

(a) Absolutes $|{\kappa }_1|$ and $|{\kappa }_2|$ of the normalized curvature $\kappa$ for the first bending mode (BM1) and the second bending mode (BM2) of a cantilever beam. The effective mechanical properties are locally weighted by the weighting factors $w_1$ (BM1) and $w_2$ (BM2) indicated as bar diagrams. (b) Top view of the magnetic layer, with local shape-anisotropy energy-density constant $K_s\propto D_{11}-D_{22}$ calculated from the ballistic demagnetizing factors in x and y direction. The six regions (R1–R6) of spatial discretization for the mechanical calculations are indicated.

Figure 4

Delta-E effect calculated for a single-domain region: (a) for different angles ${\phi }_{\mathrm{EA}}$ of the easy axis between 90° and 45° and (b) between 45° and 0° to the magnetic field direction. The calculations are started at positive magnetic fields close to magnetic saturation.

Figure 5

(a) Influence of the tilt of the mean easy ${\overline{\phi }}_{\mathrm{EA}}$ from its ideal 90° orientation, with fixed dispersion parameters ${\delta }_K=15\mathrm{\%}$ and ${\delta }_{\mathrm{EA}}=0.6\mathrm{\%}.$The two maximum slopes ${\mathrm{\partial }}_{E,H}^1$ and ${\mathrm{\partial }}_{E,H}^2$ of the magnetic field $(H)$ dependent Young's modulus $\mathrm{E}(H)$ are plotted with a red line. (b) Absolute of the slopes ${\mathrm{\partial }}_{E,H}^1$ and ${\mathrm{\partial }}_{E,H}^2$ for different ${\delta }_K$ and fixed ${\delta }_{\mathrm{EA}}=0.6\mathrm{\%}$ plotted over the tilt angle ${\overline{\phi }}_{\mathrm{EA}}$ of the magnetic easy axis. The slope ${\mathrm{\partial }}_{E,H}^0$ of $E(H)$ at $H=0$ is indicated with a red line. The analytical solution from Eq. (9) is shown for comparison with a red dot.

Figure 6

Measurements and models of the cantilever sensor in three different regions (P1, P2, P3). The magnetic easy axis (EA) is induced perpendicular to the short axis. (a) Hysteresis curves for three regions along the beam compared with the results from the numerical model (red). The resulting distribution that entered the simulation via the spatial shape anisotropy is shown as a histogram of all $H_a$ values [Eq. (5)]. (b) Example of magneto-optical Kerr effect measurements along the cantilever at 0.7 mT and (c) the domain width (DW) extracted from (b) and plotted over the cantilever length, compared with the calculated demagnetizing anisotropy energy density $K_s$.

Figure 7

Comparison of measured resonance frequency and relative magnetic sensitivity $S_{H,r}$ with results from the combined mean-field FE model and mean-field Euler-Bernoulli approximation for (a) the first bending mode (BM1) and (b) the second bending mode (BM2). Apart from the local stray field, no additional distributions are used for these calculations.

Figure 8

(a) Maximum of the relative magnetic sensitivity $S_{H,r}$ calculated as a function of the magnetic layer thickness t for the second bending mode. (b) Examples of resonance frequency curves calculated for the three different layer thicknesses indicated in (a).