# Size-Dependent and Multi-Field Coupling Behavior of Layered Multiferroic Nanocomposites

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## Abstract

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## 1. Introduction

## 2. Basic Equations for Constituent Materials

## 3. Modeling of Multiferroic Nanocomposites

## 4. Numerical Results and Discussions

_{dc}, or even realize an approximately zero-biased ME effect [40].

## 5. Conclusions

- (i)
- For the multiferroic composites which is thicker than a critical thickness (about 1 μm), the influence of surface layer and flexoelectricity cannot be negligible in accurately evaluating its ME performance. There may be ways and means of increasing the flexoelectric coefficient, or decreasing thickness, both of which can increase the value of the ME coefficient.
- (ii)
- The strain gradient in multiferroic composites depends strongly on the thickness, and is influenced by external multi-filed conditions. A medium magnetic field could improve strain gradients. Besides, applying a large compressive stress or reducing temperature increments in the range of low magnetic fields will be beneficial for improving the strain gradient. However, the corresponding opposite operations for stress and temperature should be performed under a high magnetic field.
- (iii)
- Dual-peak phenomena can be obtained in the ME coefficient of multiferroic nanostructures consisting of different FM materials. One can enhance a positive ME coefficient by increasing the flexoelectric coefficient or decreasing thickness, and optimize a negative ME coefficient by eliminating flexoelectricity.
- (iv)
- Multiferroic nanocomposites operating under multi-field conditions exhibit significant multi-field coupling characteristics. Appropriate compressive stress and temperature can improve the ME coefficient at a fixed bias magnetic field. In particular, large compressive stress or temperature increments promotes the advantage of the first peaks in the ME coefficient curves, thereby reducing the required magnetic field.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

Nomenclature | |||

${}^{s}B_{k}$ | Surface magnetic induction tensor | ${}^{p}p_{i}$ | Pyroelectric coefficient |

$C$ | Second-rank stiffness coefficient tensor | ${q}_{11}$ | Equivalent piezomagnetic coefficient |

${c}_{ijkl}^{m}$ | Equivalent elastic coefficient of FM phase | ${s}_{11}$ | Equivalent compliance coefficient |

${}^{s}c_{ijkl}^{m}$ | Surface elastic coefficient of FM phase | Greek symbols | |

${c}_{ijkl}^{p}$ | Elastic coefficient of FE phase | ${\alpha}^{p}$ | Thermal expansion coefficient of FE phase |

${}^{s}c_{ijkl}^{p}$ | Surface elastic coefficient of FE phase | ${}^{s}\alpha ^{p}$ | Surface thermal expansion coefficient |

${D}_{l}$ | Electric displacement vector | ${\beta}^{m}$ | Thermal expansion coefficient of FM phase |

${}^{s}D_{l}$ | Surface electric displacement vector | $\mathsf{\sigma}$ | Stress tensor |

${d}_{31}$ | Piezoelectric coefficient | ${\sigma}_{ij}$ | Stress tensor |

${E}_{k}$ | Electric field vector | ${}^{s}\sigma _{ij}$ | Surface stress tensor |

${E}_{s}$ | Saturated Young’s modulus | ${\sigma}_{s}$ | Saturation stress |

${}^{s}E_{j}$ | Surface electric field vector | ${\tau}_{ijk}$ | Higher order stress |

${e}_{ijk}^{p}$ | Piezoelectric coefficient | $\mathsf{\epsilon}$ | Strain tensor |

${}^{s}e_{ijk}^{p}$ | Surface piezoelectric coefficient | ${\epsilon}_{10}$ | Centroidal strain |

${g}_{kij}^{m}$ | Equivalent piezomagnetic coefficient | ${\epsilon}_{33}$ | Relative dielectric constant |

${}^{s}g_{kij}^{m}$ | Surface piezomagnetic coefficient | ${\epsilon}_{kl}^{s}$ | Surface strain tensor |

$H$ | First-rank magnetic field tensor | ${\lambda}_{s}$ | Saturated magnetostriction |

${}^{s}H_{k}$ | Surface magnetic field tensor | ${\chi}_{m}$ | Linear magnetic susceptibility |

$I$ | Second-rank unit tensor | ${\mu}_{0}$ | Vacuum permeability |

$k$ | Relaxation factor | ${\mu}_{ijkl}^{p}$ | Fexoelectric coefficient |

${l}_{s}$ | Material intrinsic length | ${\mu}_{ki}^{m}$ | Equivalent magnetic permeability |

$M$ | First-rank magnetization tensor | ${}^{s}\mu _{ki}^{m}$ | Surface magnetic permeability |

${M}_{s}$ | Saturated magnetization |

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**Figure 1.**Schematic sketch of multiferroic nanocomposites consisting of ferroelectric (FE) material and different ferromagnetic (FM) materials.

**Figure 3.**ME coefficients of Model B and Model A without flexoelectricity and temperature effect, only with flexoelectricity, and only with temperature effect.

**Figure 7.**The strain gradient varied with magnetic field under different (

**a**) stresses and (

**b**) temperatures.

**Figure 8.**The ME coefficient varied with the applied magnetic field for different flexoelectric coefficients.

**Figure 9.**The ME coefficient varied with the applied magnetic field for different total thicknesses.

**Figure 12.**The ratio of the first peak to the second peak varied with temperature under different pre-stresses.

**Figure 13.**The ME coefficient varied with stress under different (

**a**) magnetic fields and (

**b**) temperatures.

Constants | Terfenol-D | Ni | PZT |
---|---|---|---|

${E}_{s}$ (GPa) | 110 | 216 | — |

${\lambda}_{s}$ (ppm) | 1000 | −37 | — |

${\mu}_{0}{M}_{s}$ (T) | 0.8 | 0.6 | — |

${\sigma}_{s}$ (MPa) | 200 | −225 | — |

${\chi}_{m}$ | 20 | 39 | — |

$\beta $ (°C^{−1}) | 12 × 10^{−6} | 13 × 10^{−6} | 2 × 10^{−6} |

${s}_{11}$ (m^{2}/N) | — | — | 14.8 × 10^{−12} |

${d}_{31}$ (m V^{−1}) | — | — | −175 × 10^{−12} |

${\epsilon}_{33}$ (C N^{−1} m^{−2}) | — | — | 1.55 × 10^{−8} |

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Shi, Y.; Wang, Y.
Size-Dependent and Multi-Field Coupling Behavior of Layered Multiferroic Nanocomposites. *Materials* **2019**, *12*, 260.
https://doi.org/10.3390/ma12020260

**AMA Style**

Shi Y, Wang Y.
Size-Dependent and Multi-Field Coupling Behavior of Layered Multiferroic Nanocomposites. *Materials*. 2019; 12(2):260.
https://doi.org/10.3390/ma12020260

**Chicago/Turabian Style**

Shi, Yang, and Yongkun Wang.
2019. "Size-Dependent and Multi-Field Coupling Behavior of Layered Multiferroic Nanocomposites" *Materials* 12, no. 2: 260.
https://doi.org/10.3390/ma12020260