# Modeling the Composites for Magnetoelectric Microwave Devices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ME Effect in Ferromagnetic Metal—Piezoelectric

_{0}, then it has components (H

_{0}, 0, 0), and the equilibrium magnetization has components (M

_{0}, 0, 0). In this case, the 1 (x) axis is directed along the easy magnetization axis of the nickel film.

_{0}under the condition h << H

_{0}and considering the dissipation:

_{0}is a magnetic constant, ω is a cyclic oscillation frequency, and α is a dissipation parameter.

_{0}for ${{\chi}^{\u2033}}_{22},{{\chi}^{\u2033}}_{33}$, which were numerically found in the Maple program for the frequency f = 10 GHz, coincide with each other and with the value found from the resonance condition to within 1 A/m:

_{0}:

_{0}must be positive, then from the 2 roots of the quadratic equation we choose the one with a plus sign in front of the root of the discriminant:

_{0}for other FM and alloys for resonant frequency f = 10 GHz:

#### FMR Line Shift in a Two-Layer ME Structure of Ferromagnetic Metal/Piezoelectric

^{m}T

_{j}are the components of the stress tensor of the magnetostrictive phase,

^{p}T

_{j}are the components of the stress tensor of the piezoelectric phase,

^{m}t is a thickness of the magnetostrictive phase, and

^{p}t is a thickness of the piezoelectric phase.

_{i}are the strain tensor components of the magnetostrictive and piezoelectric phases,

^{m}s

_{ij}are the compliance coefficients of the magnetostrictive phase,

^{p}s

_{ij}are the compliance coefficients of the piezoelectric phase, and d

_{ki}are the piezoelectric modules.

_{100}is the magnetostriction coefficient.

^{m}t = 5 × 10

^{−8}m and the piezoelectric one is

^{p}t = 5 × 10

^{−4}m.

^{m}T

_{1},

^{m}T

_{2}, which, in turn, also depend not quite trivially on the corresponding components of the piezoelectric tensor d

_{31}, d

_{32}. Thus, the sign of the resonant line shift, depending on the total action of all the above factors, can be either positive or negative. As can be seen from the graphs in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, the largest FMR line shift is observed in the structure with PZN-PT, and the smallest one is observed in the structure with the Quartz X-cut, which is related to the magnitude of the piezocoefficients. The results obtained show that in order to clearly observe the FMR line shifts in an electric field, it is necessary to have narrow FMR lines for magnetic phases of ME composite.

## 3. ME Effect in Ferrite—Piezoelectric

_{0}is applied along the plane of the plate. We will assume that the value of this field is sufficiently large and the YIG plate is uniformly magnetized to saturation. Axis 2 (y) is directed along H

_{0}, then it has components (0, H

_{0}, 0), and the equilibrium magnetization has components (0, M

_{0}, 0). Axis 1 (x) is directed along the edge of the YIG plate, which coincides with the crystallographic direction [100]. The motion equation of magnetization in a thin YIG plate under the action of h in the presence of H

_{0}under the condition h << H

_{0}, considering the dissipation, is given in the previous section. Furthermore, we consider the case when the bias field is directed perpendicular to the plane of the YIG plate. Axis 3 (z) is directed along H

_{0}, then it has components (0, 0, H

_{0}), and the equilibrium magnetization has components (0, 0, M

_{0}). Using the equation of motion of the magnetization in the YIG layer, the necessary imaginary parts of the complex components of the high-frequency magnetic susceptibility ${{\chi}^{\u2033}}_{11},{{\chi}^{\u2033}}_{33}$ were found.

_{0}for ${{\chi}^{\u2033}}_{11},{{\chi}^{\u2033}}_{33}$, which were numerically found in the Maple program for the frequency f = 10 GHz, coincide with each other with an accuracy of 1 A/m and with the value found from the resonance condition ω

^{2}= ω

_{0}

^{2}, where:

#### FMR Line Shift in a Two-Layer Magnetoelectric Structure of YIG/Piezoelectric

_{0}is directed along the axis 2 (y):

^{1}δH

_{E}associated with small effective demagnetizing factors arising from the action of the electric field E:

^{1}δH

_{E}, N

_{11}

^{E}, N

_{22}

^{E}, N

_{33}

^{E}:

^{m}t = 2 · 10

^{−5}m,

^{p}t = 5 · 10

^{−4}m, respectively.

_{32}-d

_{31}), which is proportional to the FMR line shift, has a maximum value. The decrease in the FMR line shift for PZT is because its piezoelectric coefficients are smaller than those of PMN-PT. In addition, due to the symmetry of the PZT, its values d

_{31}and d

_{32}coincide. Since the piezoelectric coefficients of quartz and langatate are approximately two orders of magnitude lower than those of PZT and PMN-PT, the FMR line shift in an electric field is relatively small for these materials.

## 4. ME Effect in Piezoelectric –YIG–GGG–“Substrate Effect”

_{0}is applied along the plane of the YIG film. We will assume that the value of this field is sufficiently large and the thin YIG film is uniformly magnetized to saturation. Axis 2 (y) is directed along H

_{0}, then H

_{0}has components (0, H

_{0}, 0), and the equilibrium magnetization has components (0, M

_{0}, 0). Let us direct the axis 1 (x) along the edge of the GGG plate, which coincides with the crystallographic direction of the YIG [100].

_{0}for a thin YIG film for two orientations of the bias field are the same as in the previous section.

_{1}, A

_{2}and B

_{1}, B

_{2}are unknown constants associated with the planar and bending modes, respectively.

_{31}, h

_{32}are piezo coefficients.

_{1}, A

_{2}, B

_{1}, B

_{2}, and the electric field E. Furthermore, we substitute the obtained longitudinal forces and bending moments into Equations (43) and (44):

_{2}in front of the unknown constant B

_{2}for the longitudinal force F

_{1}, which is also the multiplier in front of the unknown constants A

_{1}, A

_{2}, B

_{1}for M

_{2}, M

_{1}and for F

_{2}, respectively.

_{1}, A

_{2}, B

_{1}, B

_{2}(49), looks cumbersome. Therefore, we do not present it here but use it only in further calculations to obtain the final result. From Equation (51), we find the distance from the neutral line of the composite to the interface between the YIG and the piezoelectric:

_{1}, A

_{2}, B

_{1}, B

_{2}into Equations (40) and (41), and then we substitute the obtained strain tensor components along axes 1 and 2 into the YIG stress components

^{m}T

_{1},

^{m}T

_{2}from Equation (46).

^{m}T

_{1},

^{m}T

_{2}), in which the influence of the substrate and bending vibrations are now taken into account, are substituted into Equation (38), which gives the FMR line shift at a bias field directed along the plane of the YIG thin film. Similarly, to calculate the FMR line shift for the case when the bias field is directed perpendicular to the thin film, the found YIG stress components

^{m}T

_{1},

^{m}T

_{2}are substituted into Equation (39). These calculations were used to plot the dependence of the FMR line shift on the magnitude of the applied electric field to the ME composite of piezoelectric/YIG/GGG for two cases of the bias field orientation: a bias field directed in the plane of the plate YIG along axis 2 (y) and perpendicular to the YIG plate along axis 3 (z). For ease of comparison, in Figure 9 the corresponding graphs for the ME composite of piezoelectric/YIG are shown.

^{s}s

_{11}= 4.4 · 10

^{−12}m

^{2}/N,

^{s}s

_{12}= −1.2 · 10

^{−12}m

^{2}/N, t = 5 · 10

^{−4}m.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Nan, C.-W.; Bichurin, M.; Dong, S.; Viehland, D.; Srinivasan, G. Multiferroic magnetoelectric composites: Historical perspectives, status, and future directions. J. Appl. Phys.
**2008**, 103, 031101. [Google Scholar] [CrossRef] - Bichurin, M.I.; Viehland, D. (Eds.) Magnetoelectricity in Composites; Pan Stanford Publshing: Singapore, 2012; p. 286. [Google Scholar]
- Bichurin, M.; Petrov, V.; Petrov, R.; Tatarenko, A. Magnetoelectric Composites; Pan Stanford Publishing Pte. Ltd.: Singapore, 2019; p. 280. [Google Scholar]
- Bichurin, M.; Petrov, V. Magnetic resonance in layered ferrite-ferroelectric structures. Sov. Phys. JETP
**1988**, 58, 2277. [Google Scholar] - Bichurin, M.; Kornev, I.; Petrov, V.; Lisnevskaya, I. Investigation of magnetoelectric interaction in composite. Ferroelectrics
**1997**, 204, 289. [Google Scholar] [CrossRef] - Bichurin, M.; Kornev, I.; Petrov, V.; Tatarenko, A.; Kiliba, Y.V.; Srinivasan, G. Theory of magnetoelectric effects at microwave frequencies in a piezoelectric/magnetostrictive multilayer composite. Phys. Rev. B
**2001**, 64, 094409. [Google Scholar] [CrossRef] - Ustinov, A.; Tiberkevich, V.; Srinivasan, G.; Slavin, A.; Semenov, A.; Karmanenko, S.F.; Kalinikos, B.A.; Mantese, J.V.; Ramer, R. Electric field tunable ferrite-ferroelectric hybrid wave microwave resonators: Experiment and theory. J. Appl. Phys.
**2006**, 100, 093905. [Google Scholar] [CrossRef] - Bichurin, M.; Petrov, V.; Ryabkov, O.V.; Averkin, S.V.; Srinivasan, G. Theory of magnetoelectric effects at magnetoacoustic resonance in single-crystal ferromagnetic-ferroelectric heterostructures. Phys. Rev. B
**2005**, 72, 060408. [Google Scholar] [CrossRef] - Nan, T.; Zhou, Z.; Liu, M.; Yang, X.; Gao, Y.; Assaf, B.A.; Lin, H.; Velu, S.; Wang, X.; Luo, H.; et al. Quantification of strain and charge co-mediated magnetoelectric coupling on ultra-thin Permalloy/PMN-PT interface. Sci. Rep.
**2014**, 4, 3688. [Google Scholar] [CrossRef] - Xue, X.; Zhou, Z.; Dong, G.; Feng, M.; Zhang, Y.; Zhao, S.; Hu, Z.; Ren, W.; Ye, Z.-G.; Liu, Y.; et al. Discovery of Enhanced Magnetoelectric Coupling through Electric Field Control of Two-Magnon Scattering within Distorted Nanostructures. ACS Nano
**2017**, 11, 9286. [Google Scholar] [CrossRef] - Srivastava, A.; Hurben, M.; Wittenauer, M.; Kabos, P.; Patton, C.E.; Ramesh, R.; Dorsey, P.C.; Chrisey, D.B. Angle Dependence of the Ferromagnetic Resonance Linewidth and Two Magnon Losses in Pulsed Laser Deposited Films of Yttrium Iron Garnet, MnZn Ferrite, and NiZn Ferrite. J. Appl. Phys.
**1999**, 85, 7838. [Google Scholar] [CrossRef] - Duan, C.-G.; Velev, J.P.; Sabirianov, R.F.; Zhu, Z.; Chu, J.; Jaswal, S.S.; Tsymbal, E. Surface Magnetoelectric Effect in Ferromagnetic Metal Films. Phys. Rev. Lett.
**2008**, 101, 137201. [Google Scholar] [CrossRef] [PubMed] - Hou, W.; Zhou, Z.; Xue, X.; Guan, M.; Hu, Z.; Liu, M. Voltage Control of Two-Magnon Scattering in Multiferroic Layers for Tunable Magnetoelectric Devices. IEEE Trans. Magn.
**2018**, 54, 1–4. [Google Scholar] [CrossRef] - Bichurin, M.; Petrov, R.; Kiliba, Y. Magnetoelectric microwave phase shifters. Ferroelectrics
**1997**, 204, 311. [Google Scholar] [CrossRef] - Bichurin, M.; Petrov, V.; Kapralov, G.; Petrov, R.; Kiliba, Y.; Bikashev, F.; Smirnov, A.Y.; Tatarenko, A. Magnetoelectric microwave devices. Ferroelectrics
**2002**, 280, 211. [Google Scholar] [CrossRef] - Tatarenko, A.; Gheevarughese, V.; Srinivasan, G. Magnetoelectric microwave bandpass filter. Electron. Lett.
**2006**, 42, 540–541. [Google Scholar] [CrossRef] - Fetisov, Y.K.; Srinivasan, G. Electric field tuning characteristics of a ferrite-piezoelectric microwave resonator. Appl. Phys. Lett.
**2006**, 88, 143503. [Google Scholar] [CrossRef] - Srinivasan, G.; Tatarenko, A.; Bichurin, M. Electrically tunable microwave filters based on ferromagnetic resonance in ferrite-ferroelectric bilayers. Electron. Lett.
**2005**, 41, 596–598. [Google Scholar] [CrossRef] - Tatarenko, A.S.; Gheevarughese, V.; Srinivasan, G.; Antonenkov, O.V.; Bichurin, M.I. Microwave magnetoelectronic effects in ferrite-piezoelectric composites and dual electric and magnetic field tunable filters. J. Electroceram.
**2010**, 24, 5. [Google Scholar] [CrossRef] - Yang, X.; Liu, M.; Peng, B.; Zhou, Z.Y.; Nan, T.X.; Sun, H.J.; Sun, N.X. A wide -band magnetic tunable bandstop filter prototype with FeGaB/Al2O3 multilayer films. APL
**2015**, 107, 122408. [Google Scholar] [CrossRef] - Tatarenko, A.S.; Srinivasan, G.; Bichurin, M.I. Magnetoelectric microwave phase shifter. Appl. Phys. Lett.
**2006**, 88, 183507. [Google Scholar] [CrossRef] - Ustinov, A.B.; Srinivasan, G.; Kalinikos, B.A. Ferrite-ferroelectric hybrid wave phase shifters. Appl. Phys. Lett.
**2007**, 90, 031913. [Google Scholar] [CrossRef] - Tatarenko, A.S.; Srinivasan, G. A strain engineered voltage tunable millimeter-wave ferrite phase shifter. Microw. Opt. Technol. Lett.
**2010**, 53, 261–264. [Google Scholar] [CrossRef] - Tatarenko, A.; Srinivasan, G.; Filippov, D. Magnetoelectric microwave attenuator. Electron. Lett.
**2007**, 43, 674–675. [Google Scholar] [CrossRef] - Tatarenko, A.; Snisarenko, D.; Bichurin, M. Modeling of magnetoelectric microwave devices. Facta Univ. Ser. Electron. Energetics
**2017**, 30, 285–293. [Google Scholar] [CrossRef] - Tatarenko, A.; Bichurin, M. Microwave Magnetoelectric Devices. Adv. Condens. Matter Phys.
**2012**, 2012, 1–10. [Google Scholar] [CrossRef] - Petrov, R.V.; Tatarenko, A.S.; Srinivasan, G.; Mantese, J.V. Antenna miniaturization with ferrite ferroelectric composites. Microw. Opt. Technol. Lett.
**2008**, 50, 3154–3157. [Google Scholar] [CrossRef] - Petrov, R.; Tatarenko, A.; Pandey, S.; Srinivasan, G.; Mantese, J.; Azadegan, R. Miniature antenna based on magnetoelectric composites. Electron. Lett.
**2008**, 44, 506–508. [Google Scholar] [CrossRef] - Lin, H.; Page, M.R.; McConney, M.; Jones, J.; Howe, B.; Sun, N.X. Integrated magnetoelectric devices: Filters, pico-Tesla magnetometers, and ultracompact acoustic antennas. MRS Bull.
**2018**, 43, 841. [Google Scholar] [CrossRef] - Sun, N.; Srinivasan, G. Voltage control of magnetism in multiferroic heterostructures and devices. Spin
**2012**, 2, 1240004. [Google Scholar] [CrossRef] - Liang, X.; Chen, H.; Sun, N.X. Magnetoelectric materials and devices. APL Mater.
**2021**, 9, 041114. [Google Scholar] [CrossRef] - Mallinson, J. On damped gyromagnetic precession. IEEE Trans. Magn.
**1987**, 23, 2003–2004. [Google Scholar] [CrossRef] - Tatarenko, A.; Sokolov, O.; Ivanov, S.; Bichurin, M.; Wang, Y. Microwave magnetoelectric effect in structures based on ferromagnetic metals. ITM Web Conf.
**2019**, 30, 07013. [Google Scholar] [CrossRef] - Andreev, I. Single crystals of the langasite family: An intriguing combination of properties promising for acoustoelectronics. Tech. Phys.
**2006**, 51, 758. [Google Scholar] [CrossRef] - Lou, J.; Liu, M.; Reed, D.; Ren, Y.; Sun, N.X. Giant Electric Field Tuning of Magnetism in Novel Multiferroic FeGaB/Lead Zinc Niobate-Lead Titanate (PZN-PT) Heterostructures. Adv. Mater.
**2009**, 21, 4711–4715. [Google Scholar] [CrossRef] - Pertsev, N.A.; Zembilgotov, A.G.; Tagantsev, A.K. Effect of Mechanical Boundary Conditions on Phase Diagrams of Epitaxial Ferroelectric Thin Films. Phys. Rev. Lett.
**1998**, 80, 1988–1991. [Google Scholar] [CrossRef]

**Figure 2.**FMR line shift as a function of electric field for Ni/PZN-PT, Ni/PMN-PT, Ni/PZT, Ni/Quartz structures.

**Figure 3.**FMR line shifts as a function of electric field for Fe/PZN-PT, Fe/PMN-PT, Fe/PZT, Fe/Quartz structures.

**Figure 4.**FMR line shifts as a function of electric field for Co/PZN-PT, Co/PMN-PT, Co/PZT, Co/Quartz structures.

**Figure 5.**FMR line shifts as a function of electric field for NiFe/PZN-PT, NiFe/PMN-PT, NiFe/PZT, NiFe/Quartz structures.

**Figure 6.**FMR line shifts as a function of electric field for FeGaB / PZN-PT, FeGaB / PMN-PT, FeGaB/PZT, FeGaB/Quartz structures.

**Figure 7.**Dependence of the FMR line shift on the electric field in the structures: (

**a**) YIG/PMN-PT, (

**b**) YIG/PZT, (

**c**) YIG/Quartz, (

**d**) YIG/Langatate, (

**e**) YIG/PZN-PT. Solid line is the bias field directed in the plane of the plate YIG along axis 2 (y), dash line is the bias field is directed perpendicular to the YIG plate along axis 3 (z).

**Figure 9.**Dependence of the FMR line shift on the electric field in the structures: (

**a**) YIG/PMN-PT is blue line, PMN-PT/YIG/GGG is red line, (

**b**) YIG/PZT is blue line, PZT/YIG/GGG is red line, (

**c**) YIG/Quartz is blue line, Quartz/YIG/GGG is red line, (

**d**) YIG/Langatate is blue line, Langatate/YIG/GGG is red line, (

**e**) YIG/PZN-PT is blue line, PZN-PT/YIG/GGG is red line. Solid line is a bias field directed in the plane of the plate YIG along axis 2 (y), dash line is a bias field is directed perpendicular to the YIG plate along axis 3 (z).

**Table 1.**Material parameters of ferromagnetic metals and alloys [3].

Material | Ni | Fe | Co | NiFe | FeGaB |
---|---|---|---|---|---|

s_{11} (10^{−12} m^{2}/N) | 20 | 5 | 4.7 | 6.67 | 18.2 |

s_{12} (10^{−12} m^{2}/N) | −7 | −1.45 | −2.3 | −1.93 | −4 |

M_{0} (A/m) | 262,698 | 101,587 | 1,153,968 | 1,230,158 | 1,110,000 |

γ (m/C) | 236,803 | 228,910 | 236,803 | 220,000 | 221,017 |

λ_{100} (10^{−6}) | −35 | −8 | −50 | 4.6 | 70 |

H_{a} (A/m) | 15,900 | 15,662 | 1401 | 0 | 1990 |

Material | Cut (011) of PZN-PT | Cut (011) of 0.67PMN-0.33PT (PMN-PT) | PZT | X-cut of Quartz |
---|---|---|---|---|

s_{11} (10^{−12} m^{2}/N) | 54 | 69 | 15.3 | 12.8 |

s_{12} (10^{−12} m^{2}/N) | −41.6 | −33.4 | −5 | −1.22 |

s_{22} (10^{−12} m^{2}/N) | 180 | 22.9 | 15.3 | 9.6 |

d_{31} (10^{−12} m/V) | 1100 | −940 | −175 | 0 |

d_{32} (10^{−12} m/V) | −3000 | 475 | −175 | −2.29 |

Material | YIG (001) | Langatate X-Cut |
---|---|---|

s_{11} (10^{−12} m^{2}/N) | 4.8 | 5.27 |

s_{12} (10^{−12} m^{2}/N) | −1.4 | −1.84 |

s_{22} (10^{−12} m^{2}/N) | - | 9.13 |

M_{0} (A/m) | 142,143 | - |

λ_{100} (10^{−6}) | −1.4 | - |

H_{a} (A/m) | −3343 | - |

d_{31} (10^{−12} m/V) | - | 0 |

d_{32} (10^{−12} m/V) | - | −6.54 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bichurin, M.; Sokolov, O.; Ivanov, S.; Ivasheva, E.; Leontiev, V.; Lobekin, V.; Semenov, G.
Modeling the Composites for Magnetoelectric Microwave Devices. *Sensors* **2023**, *23*, 1780.
https://doi.org/10.3390/s23041780

**AMA Style**

Bichurin M, Sokolov O, Ivanov S, Ivasheva E, Leontiev V, Lobekin V, Semenov G.
Modeling the Composites for Magnetoelectric Microwave Devices. *Sensors*. 2023; 23(4):1780.
https://doi.org/10.3390/s23041780

**Chicago/Turabian Style**

Bichurin, Mirza, Oleg Sokolov, Sergey Ivanov, Elena Ivasheva, Viktor Leontiev, Vyacheslav Lobekin, and Gennady Semenov.
2023. "Modeling the Composites for Magnetoelectric Microwave Devices" *Sensors* 23, no. 4: 1780.
https://doi.org/10.3390/s23041780