# Anisotropy of the ΔE Effect in Ni-Based Magnetoelectric Cantilevers: A Finite Element Method Analysis

^{1}

^{®}), Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany

^{2}

^{®}), Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany

^{*}

## Abstract

**:**

^{−1}for (110) in-plane-oriented Nickel at a magnetic bias flux of 1.78 mT. The results achieved by FEM simulations are compared to the results calculated by the Euler–Bernoulli theory.

## 1. Introduction

## 2. Model Details

#### 2.1. Analytic Description of the ΔE Effect in Nickel

#### 2.2. Description of the Finite Element Model

#### 2.3. Limits of the Proposed Model

## 3. Results

#### 3.1. Magnetostriction and Bending

#### 3.2. Eigenfrequency Behavior in the Magnetic Field

^{−1}which is comparable to other references based on soft magnetic FeGa or FeCo compounds. Sensors based on FeCoSiB are able to reach higher sensitivities by a factor of 5–10 in combination with a high degree of optimization. The theoretical results still remain significantly higher but are similar between soft magnetic FeCoSiB and the (110)-oriented Nickel. The given simulated FeCoSiB sensitivity of 48 T

^{−1}is obtained for the second bending mode, which yields a higher value than the first/natural mode. The first bending mode should yield a sensitivity approximately 20% lower according to the data in [11], leading to an almost identical result as Ni(110). The direct growth of (110) in-plane-oriented Nickel is experimentally difficult on a hexagonal substrate such as AlN. However, there are approaches using 150 nm thick Au/Ge interfacial layers [10] for larger sensors. Additional interface engineering is needed to see whether this configuration can be scaled down to MEMS structures. Further similarities between FeCoSiB and Nickel apply to the saturation magnetostriction [45] or the density [46] leading to a similar mass inertness in the vibrational behavior, e.g., in passive operation. However, MEMS structures are less suitable for passive operation due to the size dependence of the limit of detection [47]. In actively operated sensors, the limit of detection plays a negligible role, which is why the sensitivity in combination with the dynamic range are the figures of merit to be used.

## 4. Conclusions

^{−1}at a magnetic bias flux of 1.78 mT. Such a high sensitivity is nearly identical to that of frequently used soft magnetic materials, such as FeCoSiB. However, the comparison between simulations and experiment was limited due to some assumption of the simulation that were in the general case not true in real sensor samples and partly not easy to achieve even if they would lead to an improved sensitivity. Especially, the magnetostrictive film is normally not single crystalline and it is more realistic to generate a polycrystalline microstructure with a strong preferential orientation. Other limitations, as listed in Section 2.3, were the 2D instead of 3D modeling, which changed the shape anisotropy behavior, and also of course no defects such as point defects or dislocation were considered, which would change the magnetization behavior. In any case, the simulation results gave a good indication of a high potential for further optimizations of the sensor performance, regardless of the used material.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Fit constants for the magnetostriction and magnetization curves according to Equation (A1).

Parameter | ${\mathsf{\kappa}}_{\mathit{hkl},\mathit{sat},1}$ | ${\mathsf{\kappa}}_{\mathit{hkl},\mathit{sat},2}$ | ${\mathsf{\alpha}}_{1}$ (${10}^{-4}$ m/A) | ${\mathsf{\alpha}}_{2}$ (${10}^{-4}$ m/A) | ${\mathit{H}}_{0,1}$ (A/m) | ${\mathit{H}}_{0,2}$ (A/m) |
---|---|---|---|---|---|---|

${\lambda}_{100}$ | −1.61$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-5}$ | −1.34$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 2.17 | 1.40 | 0 | −22,132 |

${\lambda}_{110}$ | −7.95$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ | −1.27$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | −52.4 | 2.80 | 0 | −11,500 |

${\lambda}_{111}$ | −2.24$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-5}$ | −7.16$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ | −26.2 | −5.35 | −200 | −500 |

${M}_{100}$ | 260,000 | 276,100 | −2.08 | 9.43 | −5000 | 0 |

${M}_{110}$ | 403,234 | 248,314 | 42.8 | −2.50 | 0 | −10,000 |

${M}_{111}$ | 10,621 | 483,372 | 8.86 | 37.9 | 0 | 0 |

**Table A2.**Summary of the elastic constants of Ni gathered from [32]. * Omitted values due to high deviation.

Reference | ${\mathit{C}}_{11}$ (${10}^{11}$Pa) | ${\mathit{C}}_{12}$ (${10}^{11}$Pa) | ${\mathit{C}}_{44}$ (${10}^{11}$Pa) |
---|---|---|---|

Honda et al. | 2.52 | 1.51 | 1.04 |

Bozorth1 et al. | 2.5 | 1.6 | 1.19 |

Bozorth2 et al. Saturated | 2.52 | 1.57 | 1.23 |

Neighbours et al. | 2.53 | 1.52 | 1.24 |

Yamamoto et al. | 2.44 | 1.58 | 1.02 |

Levy et al. | 2.47 | 1.52 | 1.21 |

DeKlerk et al. Saturated | 2.46 | 1.47 | 1.24 |

Shirakawa et al. | 2.55 | 1.69 | 0.90 * |

DeKlerk2 et al. Saturated | 2.46 | 1.48 | 1.22 |

Alers et al. | 2.51 | 1.5 | 1.24 |

Sakurai et al. | 2.51 | 1.53 | 1.24 |

Epstein et al. Saturated | 2.5 | 1.54 | 1.24 |

Vintaikin et al. | 2.47 | 1.44 | 1.24 |

Salama et al. | 2.52 | 1.54 | 1.22 |

Shirakawa2 et al. | 2.88 | 1.81 | 1.24 |

Average | 2.52 | 1.55 | 1.2 |

## Appendix B

## Appendix C

**Figure A2.**Convergence curves for the solution of the presented model with ${t}_{Ni}=$100 nm: (

**a**) magnetic potential, (

**b**) displacement field.

## Appendix D

**Figure A3.**Relative eigenfrequency change $f/{f}_{sat}$ of the three principal axes (

**a**) (100), (

**b**) (110) and (

**c**) (111) for Nickel layer thicknesses ${t}_{Ni}$ in the range of 50–1000 nm derived from the Euler–Bernoulli theory.

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**Figure 1.**(

**a**) SEM images of 4 $\mathsf{\mu}$$\mathrm{m}$ wide magnetoelectric cantilevers consisting of a TiN( 90 $\mathrm{n}$$\mathrm{m}$)/ AlN( 450 $\mathrm{n}$$\mathrm{m}$)/Ni( 100 $\mathrm{n}$$\mathrm{m}$) layer stack investigated in recent work [26]. (

**b**) Eigenfrequency characteristics in dependency of the magnetic flux of four 25 $\mathsf{\mu}$$\mathrm{m}$ long and identically aligned cantilevers as marked in (

**a**). (

**c**) Solution of the 2D model used for the simulation study with the layer configuration from (

**a**). The bending effect due to magnetostrictive strains in the 25 $\mathsf{\mu}$$\mathrm{m}$ cantilever is upscaled for better visibility.

**Figure 2.**(

**a**) Computed magnetic-field-dependent curves of Young’s modulus for the three principal axes. (

**b**) Eigenfrequencies of the simulated cantilevers in magnetic saturation as a function of the crystalline orientation of the Nickel layer and its thickness.

**Figure 3.**(

**a**) Deflection in magnetic saturation of the simulated cantilever for different thicknesses ${t}_{Ni}$. (

**b**) Influence of the cantilever curvature on the eigenfrequency. The curvature caused by magnetostriction is derived for comparison from (

**a**).

**Figure 4.**Relative eigenfrequency change $f/{f}_{sat}$ (left hand side) and the respective specific sensitivity $\partial f/\partial B$ (right hand side) of the three principal axes (

**a**) (100), (

**b**) (110) and (

**c**) (111) for Nickel layer thicknesses ${t}_{Ni}$ in the range of 50–1000 nm. The point of highest absolute sensitivity as well as the dynamic range is marked by the lines, respectively.

**Figure 5.**(

**a**) Absolute values of the maximum sensitivities derived from Figure 4 (FEM) and from the Euler–Bernoulli theory (EBT) for the three principal axes and ${t}_{Ni,sat}$ at the given offsets of the magnetic flux. The experimentally achieved sensitivity is added for comparison. The value for zero thickness is extrapolated. (

**b**) Extracted dynamic range in dependence of $|{S}_{H}|$ for the three orientations.

**Table 1.**Comparison of simulated and experimental sensitivities of the natural frequency of electromechanical system based on magnetoelectric sensors. (Values calculated according to Equation (10) if not given in the reference). * Sensitivity for the second eigenmode.

Material | Reference | Sensitivity (1/T) |
---|---|---|

Ni(100)/AlN/TiN${}^{sim}$ | this work | −14.9 |

Ni(110)/AlN/TiN${}^{sim}$ | this work | −41.3 |

Ni(111)/AlN/TiN${}^{sim}$ | this work | 8.8 |

poly-Ni/AlN/TiN${}^{exp}$ | [26] | −0.9 …−1.4 |

FeCoSiB/poly-Si/AlN${}^{exp}$ | [11] | 10 |

FeCoSiB/poly-Si/AlN${}^{exp}$ | [11] | 13 * |

FeCoSiB/poly-Si/AlN${}^{sim}$ | [11] | 48 * |

FeCoB/Al/AlN/Pt${}^{exp}$ | [19] | −0.7 |

FeGaB/AlN/Pt${}^{exp}$ | [48] | −2.2 |

FeGa/Ti/Diamond${}^{exp}$ | [20] | 0.5 |

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**MDPI and ACS Style**

Hähnlein, B.; Sagar, N.; Honig, H.; Krischok, S.; Tonisch, K. Anisotropy of the ΔE Effect in Ni-Based Magnetoelectric Cantilevers: A Finite Element Method Analysis. *Sensors* **2022**, *22*, 4958.
https://doi.org/10.3390/s22134958

**AMA Style**

Hähnlein B, Sagar N, Honig H, Krischok S, Tonisch K. Anisotropy of the ΔE Effect in Ni-Based Magnetoelectric Cantilevers: A Finite Element Method Analysis. *Sensors*. 2022; 22(13):4958.
https://doi.org/10.3390/s22134958

**Chicago/Turabian Style**

Hähnlein, Bernd, Neha Sagar, Hauke Honig, Stefan Krischok, and Katja Tonisch. 2022. "Anisotropy of the ΔE Effect in Ni-Based Magnetoelectric Cantilevers: A Finite Element Method Analysis" *Sensors* 22, no. 13: 4958.
https://doi.org/10.3390/s22134958