**2. Principles of magnetoelectricity**

The thermodynamic consideration of magnetoelectric effect is obtained from the expansion of free energy of the system in terms of magnetic and electric field, such as.

$$G\left(\overrightarrow{E}, \overrightarrow{H}\right) = G\_o - P\_{s\\_}E - M\_{s\\_}H - \frac{1}{2}\chi\_{\vec{\eta}}(E)\_{-}E\_{i\\_}E\_{j} - \frac{1}{2}\chi\_{\vec{\eta}}(H)\_{-}H\_{i\\_}H\_{j} - a\_{\vec{\eta}\\_}E\_{i\\_}H\_{j} \tag{2}$$

where and are the electric field and magnetic field respectively. Differentiation of Eq. (2) gives us polarization and magnetization as following:

$$P\_i = -\left(\frac{\delta G}{\delta E}\right) = P\_s + \chi\_{\vec{\eta}} E\_j + a\_{\vec{\eta}} H\_j \tag{3}$$

$$\mathbf{M}\_i = -\left(\frac{\delta G}{\delta H}\right) = \mathbf{M}\_s + \chi\_{i\bar{j}} H\_{\bar{j}} + a\_{i\bar{j}} E\_{\bar{j}} \tag{4}$$

Here *αij* is the magnetoelectric tensor. Magnetoelectric effect combines two important materials property, permittivity and permeability, and for a single-phase material they define the upper limit of *αij* as following.

$$a\_{\vec{\text{ij}}} < \sqrt{\varepsilon\_{\vec{\text{ij}}} \mu\_{\vec{\text{ij}}}} \tag{5}$$

Single phase multiferroic materials shows either low permeability or low permittivity or both. As a result, the magnetoelectric coupling is small. For high response dual phase magnetoelectric materials, combination between ferroelectric and ferromagnetic phase need to be established via strain. Piezoelectric coefficient (d33/d31) defines the materials property that converts applied stress in to proportional electric charge. The linear equations for a piezoelectricity and magnetostrictions are given as:

$$D\_3 = \epsilon\_{33}^T E\_3 + d\_{33} T\_3 \tag{6}$$

$$S\_3 = d\_{33}E\_3 + s\_{33}^E T\_3 \tag{7}$$

$$\mathbf{S} = \mathbf{s}^{H}T + qH \tag{8}$$

$$B = qT + \mu^T H \tag{9}$$

where


T = stress,


Magnetoelectric coefficient of a composite can be described in direct notation of tensors as:

$$T = c\mathbf{S} - e^T \mathbf{E} - a\mathbf{S}^{ms} \tag{10}$$

$$D = e\mathbf{S} + e\mathbf{E} + aH\tag{11}$$

$$B = \mu(\varepsilon, E, H)H \tag{12}$$

where σ, c and K are the stress, stiffness constant at constant field and dielectric constant at constant strain respectively. It was found in the literatures that ME coefficient can be varied by piezoelectric and piezomagnetic coefficients. Elastic compliances (s) of piezoelectric and magnetostrictive phases are found to be another critical parameter that affects the ME coefficient. According to Srinivasan et al., the ME coefficient can be written as:

$$\frac{\delta E\_3}{\delta H\_1} = \frac{-2d\_{31}^p q\_{11}^m v^m}{(s\_{11}^m + s\_{12}^m) \varepsilon\_{33}^{T,P} \upsilon^p + (s\_{11}^p + s\_{12}^p) \varepsilon\_{33}^{T,P} \upsilon^m - 2(d\_{31}^p)^2 \upsilon\_m} \tag{13}$$

where

*d p* <sup>31</sup> = piezoelectric coefficient, *vm*and *v<sup>p</sup>*= volume of magnetic and piezoelectric phase, *t <sup>m</sup>*and *t <sup>p</sup>*= thickness of magnetic and piezoelectric phase, *s p* 11, *s p* 12= the elastic compliances for piezoelectric phase, *s m* 11, *s m* <sup>12</sup> = elastic compliances for magnetostrictive phase, q11 = piezomagnetic coefficient of the magnetic phase and. *ε T*,*P* <sup>33</sup> = permittivity of the piezoelectric phase.

Further derivation for the magnetoelectric coefficient in T – T mode of Eq. (13) was done by Dong et al. and was expressed as:

$$\left| \frac{dV}{dH} \right|\_{T-T} = \beta \frac{n(1-n) A d\_{33,m} d\_{31,p} g\_{31,p}}{\mathbf{S}\_{11}^{\mathrm{E}} \left[ n \mathbf{S}\_{11}^{\mathrm{E}} \left( 1 - k\_{31}^2 \right) + (1-n) \mathbf{S}\_{11}^{H} \right]} \tag{14}$$

where

*β* = a constant related to DC magnetic field (<1),


It is quite clear from the Eq. 14 that the ME coefficient is directly related to piezoelectric constant (d31) and piezomagnetic coefficient (q11). *d<sup>p</sup>* <sup>31</sup> which is related to dielectric permittivity [*d*<sup>2</sup> <sup>31</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> <sup>31</sup> *s E* 11*ε<sup>T</sup>* <sup>33</sup> ] and q11 is related to permeability [q11,m = μ33.s33.λ33].

*Doping Effect on Piezoelectric, Magnetic and Magnetoelectric Properties of Perovskite… DOI: http://dx.doi.org/10.5772/intechopen.95604*

### **3. Effect of doping**

As is well known that compared to BaTiO3, PZT has stronger piezoelectric and dielectric properties, higher Curie temperature, higher resistivity and lower sintering temperature. Doping in PZT can be done by adding acceptor dopants (Fe, Mn, Ni, Co) or donor dopants (La, Sb, Bi, W, Nb) in order to make it piezoelectrically hard or a soft. Hard piezoelectric materials can be characterized as decreased dielectric constant and loss, lower elastic compliance, lower electromechanical coupling factor, and lower electromechanical losses compared to undoped PZT. Soft piezoelectric materials exhibit increased dielectric constant, dielectric loss, elastic compliance, electromechanical coupling factor, and electromechanical losses. **Table 1** shows a comparison chart how the physical, dielectric and piezoelectric properties vary between soft and hard piezoelectric materials.

The open circuit output voltage (V), under an applied force of a ceramic is given as:

$$V = E \cdot t = -\mathbf{g} \cdot X \cdot t = -\frac{\mathbf{g} \cdot F \cdot t}{A} \tag{15}$$

where

t = the thickness of the ceramic,

E = the electric field, and.

g = the piezoelectric voltage coefficient given as:

$$\mathbf{g} = \frac{d}{\varepsilon\_o \varepsilon^X} \tag{16}$$

where ε <sup>X</sup> is the dielectric constant under constant stress condition. The charge (Q) generated on the piezoelectric ceramic is given by the relation:

$$Q\!\!/\_{V} = \frac{\varepsilon^{\mathcal{X}}\varepsilon\_{\theta}\mathcal{A}}{t} = \mathcal{C} \tag{17}$$

where

$$\mathbf{C} = \text{the capacitance and}\tag{18}$$


**Table 1.**

*Comparison of Dielectric and Piezoelectric properties between Soft and Hard Piezoelectric Materials.*

It can be inferred from Eq. (17) that a piezoelectric plate can behave like a parallel plate capacitor at low frequencies. From here it can be derived that under ac stress, electric energy generated is given as:

$$U = \frac{1}{2}CV^2$$

or energy per unit volume,

$$
\mu = \frac{1}{2} (d\_{\hat{\mathbf{g}}} \mathbf{g}) \cdot \left(\frac{F}{A}\right)^2 \tag{19}
$$

Eq. (15) and (19) conclude that under a fixed AC mechanical stress, piezoelectric material with high (d.g) product and high piezoelectric voltage (g) constant will generate high voltage and high power for a fixed area and thickness. In the case of magnetoelectric composite, the force is applied on the piezoelectric phase due to magnetostriction through elastic coupling, therefore the high energy density piezoelectric material will lead to higher response.

Nickel and cobalt ferrites have the advantage of higher resistivity and increased permeability. Cobalt ferrite has higher magnetization but also has higher coercivity compared to nickel ferrite. In order to increase the resistivity, permeability, and magnetization, doping of zinc in to ferrite is beneficial but it also reduces its magnetic Curie temperature. The theory behind this is, Zn+2 replaces Fe+3 on the tetrahedral sites as it is added to the spinel structure and Fe+3 occupies the vacant octahedral sites emptied by Co+2. As a result, there will be no unpaired electrons for Zn+2, Co+2 has one and Fe+3 has five. Hence the outcome of it is increase in magnetization of Zn-doped ferrites.

### **4. Synthesis and fabrication**

Pb(Zr0.52Ti0.48)O3 (PZT), 0.85[Pb(Zr0.52Ti0.48)O3] – 0.15[Pb(Zn1/3Nb2/3)O3 [PZT (soft)], Pb(Zr0.56Ti0.44)O3 ─ 0.1 Pb[(Zn0.8/3 Ni0.2/3)Nb2/3]O3 + 2 (mol %) MnO2 [PZT (hard)], Ni(1-x)ZnxFe2O4 [NZF] (where x varies from 0 to 0.5) and Co(1-y)ZnyFe2O4 [CZF], were synthesized using mixed oxide route. PZT, NZF and CZF powders were calcined at 750°C for 2 hrs and 1000°C for 5 hrs, respectively in order to make sure that inorganic oxides react to each other. Powder X-ray diffractions patterns were taken using Siemens Krystalloflex 810 D500 diffractometer to make sure that the pure perovskite and pure spinel structure was formed out of PZT/PZT (soft)/PZT (hard) and NZF/CFO/CZF. The PZT and NZF powders were then mixed together as 0.8 PZT – 0.2 NZF and compacted. CFO/CZF powders were mixed in PZT with stoichiometric ratio of 3, 5, 10, 15 and 20 mole percents. After homogeneous mixing using ball mill, powder was pressed using a hardened steel die having diameter of 12.7 mm under a pressure of 2 ksi and then cold isostatically pressed under pressure of 40 ksi. This was followed by pressure-less sintering in air at 1150°C for 2 hrs, resulting in consolidated ceramic composites. XRD patterns of sintered samples showed only two phases (PZT and CFO/CZF). Ag/Pd paste was painted manually on top and bottom of the sintered disc using a paint brush and heated for an hour at 825<sup>o</sup> C. The polarization process was done in a heated (120°C) silicone oil bath. D.C. electrical field of 2.5 kV/mm for 20 minutes was applied for the poling process. Dielectric constant as a function of temperature was measured using HP 4274A LCR meter (Hewlett Packard Co. USA). Magnetization as a function of temperature was measured using Quantum Design physical properties

*Doping Effect on Piezoelectric, Magnetic and Magnetoelectric Properties of Perovskite… DOI: http://dx.doi.org/10.5772/intechopen.95604*

measurement system from room temperature to 900 K. Transmission electron microscopy (TEM) was conducted by using JEOL 1200EX machine with an accelerated voltage of 120 kV.
