# A Review of Magnetoelectric Composites Based on ZnO Nanostructures
## 2 Magnetoelectric (ME) Effect  
The magnetoelectric effect (ME) originates from the coupling between electrical and magnetic phenomena in bulk, composite or multiphase materials. In particular, the direct magnetoelectric effect (DME) is defined as the emergence of an electric polarization P under the application of an external magnetic field H, whereas the converse ME effect (CME) is described as the emergence of or change in magnetization M when an external electric field E is applied [20]. ME composites rely on the interaction between a ferromagnetic (FM) phase exhibiting strain when subjected to a magnetic field due to magnetostriction and a piezoelectric (PE) phase that exhibits electric polarization via the application of mechanical stress/strain. The coupling between the electrical, magnetic and mechanical domains is given by the constitutive relations below in direct tensor notation [20]:  
\[ \begin{array}{l}\boldsymbol{\sigma}=\boldsymbol{\epsilon}\mathbf{S}- \mathbf{e}^{\mathrm{T}}\mathbf{E}-\mathbf{q}^{\mathrm{T}}\mathbf{H}\,,\\ \mathbf{D}=\boldsymbol{\epsilon}\mathbf{S}+\epsilon\mathbf{E}+\alpha\mathbf{H }\,,\\ \mathbf{B}=\boldsymbol{\mathrm{q}}\mathbf{S}+\boldsymbol{\alpha}^{\mathrm{T}} \mathbf{E}+\mu\mathbf{H}\,,\end{array}  \tag{1}\]  
where \(\sigma\), \(\mathrm{S}\) and \(\mathrm{c}\) are the stress, strain and stiffness tensors, respectively. D, E, \(\epsilon\) and \(\mathrm{e}\) denote the electric displacement field, electric field, electrical permittivity and piezoelectric coefficient tensors in the stress-charge form respectively. B, H, \(\mu\) and \(\mathrm{q}\) are the magnetic induction, magnetic field intensity, permeability and piezomagnetic coefficient, respectively, and \(\alpha\) is the magnitude of the linear ME effect. Superscript T corresponds to the transpose of the tensor. The coefficients for the direct (\(\alpha_{\mathrm{E}}\)) and converse (\(\alpha_{\mathrm{H}}\) or \(\alpha_{\mathrm{B}}\)) ME effects are conventionally used in the literature since they adequately describe the magnitude of the respective effects according to the \"product property\", defined by the combination of PE and FM phases, and are used as a figure of merit describing the strength of the ME coupling [21]. Based on the \"product property\" definition, the direct ME coefficient is given by:  
\[ \begin{array}{l}\alpha_{\mathrm{E}}=\frac{\delta\mathrm{E}}{\delta\mathrm{H }}\approx\!\!\frac{\delta\mathrm{E}}{\delta\mathrm{S}}\frac{\delta\mathrm{S}}{ \delta\mathrm{H}}\,,\\ \alpha_{\mathrm{E}}=\frac{\mathrm{V}}{\delta\mathrm{H}}\,,\\ \alpha_{\mathrm{E}}=\mathrm{k_{c}}\;\mathrm{q}\,\mathrm{e}\,,\end{array}  \tag{2}\]  
where V is the piezoelectric voltage generated across a PE material with thickness t under the application of magnetic field H with \(\mathrm{k_{c}}\) representing the electromechanical coupling between the FM and PE phases. The evaluation of the ME coefficient is usually performed in dynamic conditions by the application of alternating magnetic fields, thus, \(\delta\mathrm{E}\) and \(\delta\mathrm{S}\) are the amplitudes of the ac electric field and strain in a varying magnetic field \(\delta\mathrm{H}\). In addition, the ME coefficient includes the geometrical effects stemming from the configuration of the composite, the dependence of the magnetostriction \(\lambda\) of the ferromagnetic phase on H and the piezoelectric properties of the PE phase [16]. An important factor that greatly influences the ME response of a composite is its connectivity, which essentially specifies the way the FM and PE phases are interacting with each other, presented schematically in Figure 1. The connectivity is defined by two numbers, with the first one representing the FM phase and the second one the PE phase, indicating the number of coordinates along which strain transfer (i.e., deformation) occurs, from the magnetostrictive to the piezoelectric phase [22].