A nonlinear model for magnetocapacitance effect in PZT-ring/Terfenol-D-strip magnetoelectric composites

The controllable capacitance or dielectric permittivity in magnetoelectric (ME) composites with external DC magnetic field,^1–5 which is known as magnetocapacitance (MC) effect, has a widely potential practical application, such as tunable spin filters, storage devices, and magnetic sensor.^4,6–8 As reported in previous study, multiferroic ME materials possesses ferromagnetism, ferroelectricity and ferroelasticity simultaneously,^9,10 the direct method to assess the degree of ME coupling is based on ME effect, and another widely used characterization method is an indirect measurement of dielectric response to an applied field, which is called MC method.³ The ME effect is calculated under open circuit electric boundary condition, and often require an AC magnetic field and a DC magnetic field, moreover, it can be affected by the frequency of AC magnetic field significantly.^11–15 Furthermore, the ME effect is driven by two magnetic fields, which limited the practical use in some fields. But the MC effect is estimated under close circuit condition and can be achieved with a single DC magnetic field. And the magnetic field detection sensor based on MC effect is more sensitive than Hall detector, moreover, the configuration is simpler and the cost is lower.^16,17

The MC effect in single phase material often needs a very strong magnetic field and is extremely weak at room temperature.^18,19 It is not easily available in practice applications. Huge MC effect can be expected in ME composites because of the resonant frequency with the enhanced ME properties, and a small change near resonance affect capacitance dramatically.^2,20,21 In order to make it possible for practical application within the room temperature range, many attempts have been done, such as searching for new materials, fabricating heterostructures, then big resonance shift achieved with a relatively low magnetic field.^22–25 Besides, most researchers conducted experiments and proposed some theoretical models to explain the MC resonance in ME composites. For example, Israel et al. detected largest resonant frequency shift in multilayer capacitors, and developed a model for effective permittivity of piezoelectric layer.²¹ Later on, Yao et al. proposed a model to investigate the influence of temperature on the dielectric permittivity of Terfenol-D/PZT ME composites with considering the mechanical energy loss.²⁶ In addition, Wang et al. carried out a series of experiments to improve resonance shift, and built a linear constitutive model to describe the MC effect in composite structures.^{5,24,25,27–29} Although, those models can describe the behavior qualitatively, nevertheless, the magnetostrictive materials possess a strong nonlinearity, and the resonant frequencies of magnetostrictive materials will be changed under the action of external magnetic field and pre-stress.^2,30 The differences between Wang et al.’s theoretical results and experiment data range from 12.82% (in high resonance frequency) to 29.77% (in low resonance frequency), which can be attributed to the linear constitutive relation of magnetostrictive material in their model for PZT-ring/Terfenol-D-strip structure, thus it’s urgent to establish a nonlinear model to describe the MC effect in this structure. Meanwhile, a series of nonlinear models on ME effect for the layered magnetoelectric composites have been proposed,^31–33 which also help us to study the nonlinear MC effect in the ring-strip structure conveniently.

In the present study, utilizing a nonlinear constitutive relation of magnetostrictive material and a linear constitutive relation of piezoelectric material, a nonlinear model describing MC effect in ring-strip structure has been established. By adopting proper material parameters, the proposed model in this paper can describe the MC effect more accurately, and the predicted results may be useful in the designing of new multifunction devices. The paper is arranged as follows. In second section, we derive an expression of the capacitance of PZT-ring/Terfenol-D-strip ME composites. In third section, firstly, comparing the theoretical calculations with existing experiments, then we investigate the effects of some factors on the MC effect of the ring-strip ME structure, such as external magnetic fields, pre-stresses and geometric sizes. Finally, some conclusions are drawn in the last section.

As shown in Fig. 1, a magnetostrictive strip is inserted into the piezoelectric ring to form the ring-strip structure. In order to study the MC effect of the structure, we build two sets of coordinate systems, the cylindrical coordinate system for the piezoelectric ring and the orthogonal coordinate system for magnetostrictive strip. The origin of the cylindrical coordinate system is the center of piezoelectric ring and the origin of the orthogonal coordinate system is the right end of magnetostrictive strip.²⁸ The bias magnetic field is applied along x-axis, and the structure is symmetrical, therefore, the piezoelectric and magnetostrictive phases are coupled through normal stresses.

The linear constitutive relation of the piezoelectric materiel is,

The nonlinear constitutive relation of magnetostrictive materials is Zheng-Liu model,³⁴ expressed as,

We rewrite the above nonlinear constitutive equations of magnetostrictive material into the form of similar linear constitutive equations,

where B is the magnetic induction, φ is the magnetic potential. $m s 11 H , m σ 1$⁠, $m d 11 H , m σ 1$ and $m μ 11 H , m σ 1$ are the elastic compliance coefficient, piezomagnetic coefficient and magnetic permeability of magnetostrictive material, respectively. According to the definitions of elastic compliance coefficient, piezomagnetic coefficient and magnetic permeability, we can obtain³⁵ 

Assuming that the magnetostrictive strain is small, and for simplicity the motion equations of piezoelectric ring and magnetostrictive strip can be summarized as,

In order to make a comparison with the linear model proposed in Ref. 28, the boundary conditions are the same with them, which can be written as,

Only considering the motion along the radial and hoop directions, ^pu_r and ^pu_θ are merely functions of r. Then the general solution of Eq. (13) has the following form as,

where $k p = p ρ 1 − ξ 2 p s 11 ω$⁠, $ξ=− p s 12 p s 11$⁠, J₁(•) and Y₁(•) are the first and second kind Bessel functions of order 1. The constants C₁ and C₂ are determined by the boundary conditions.

The electrically induced displacement can be expressed as,

According to Eqs. 6–8, we can obtain

By substituting Eqs. (6) and (8) into Eq. (14), the latter can be rewritten as,

Integrating Eq. (21) with respect to x and substituting the obtained formula into Eq. (22), we obtain

In which $c ̃ = m μ 11 / m s 11 m μ 11 − m d 11 2$ is the stiffened elastic constant. Because the material parameters ^ms₁₁, ^md₁₁ and ^mμ₁₁ are the functions of magnetic field H and pre-stress ^mσ₁, $c ̃$ also is a function of the magnetic field H and pre-stress ^mσ₁. For these general cases, it is difficult to obtain the analytical expression of ^mu_x due to the complexity of $c ̃$⁠. For the sake of simplicity, we consider that the initial stress $m σ 1 0$ is approximately equal to pre-stress ^mσ₁, and then make the problem simple. Thus, the solution of the Eq. (23) has the following form as,

where $k m = m ρ m s 11 ω$⁠, the constants C₃ and C₄ are determined by the boundary conditions.

in which

According to the Gaussian theorem of Maxwell’s equations, the quantity of electric charge of the composites can be defined as,

Then substituting Eq. (20) into Eq. (33), Q has the following form,

According to the original definition of the capacitance C, which can be obtained by $C= ∂ Q ∂ V$⁠, here the voltage V = E₃h.

In this section, based on the nonlinear MC effect model, we will present some numerical results for the PZT-ring/Terfenol-D-strip ME composites. The materials parameters of piezoelectric and magnetostrictive components are listed in Table I:

Figs. 2(a)–2(c) display the variation of effective material parameters for Terfenol-D as a function of external magnetic field and the compressive pre-stress. Obviously, the variations of effective material parameters of Terfenol-D (such as ^ms₁₁, ^md₁₁ and ^mμ₁₁) are nonlinear with external magnetic field, which are different from ordinary ferromagnetic materials, such as ferrite. Thus, in order to analyze physical behaviors of the composite structure consisting of magnetostrictive materials, it’s very necessary to adopt a nonlinear constitutive relation.

When the compressive stress is a constant, with the increase of external magnetic field, the effective compliance coefficient increases firstly, then decreases, finally reaches a steady state (see Fig. 2(a)). The effective piezomagnetic coefficient firstly increases, reaches a peak, then decreases till it approximates to zero (see Fig. 2(b)). The effective magnetic permeability shows a monotonic decreasing tendency as magnetic field increases, and finally closes to zero (see Fig. 2(c)). Meanwhile, the increase of compressive stress results in a decrease in maximum of compliance coefficient, piezomagnetic coefficient and magnetic permeability. All the curves of effective material parameters shift to right as compressive stress increases, and tend to a relative stable value as magnetic fields are large enough. It’s suggested that the final saturation magnetization state cannot be changed by compressive stresses. Those tendencies are similar with Refs. 35 and 36. Fig. 2(d) shows the magnetic field dependence of magnetostriction of Terfenol-D strip with various compressive stresses. When the magnetic field is lower than 2000Oe, the magnetostriction increases rapidly, then grows slowly and approaches to saturation. Besides, the magnetostrictive strain curves will approach different saturation values with different compressive pre-stresses, because compressive stresses change the initial distribution of magnetic domain.³⁴ 

In order to prove the validity of our nonlinear theoretical model, we have conducted some calculations on the capacitance of the structure by our theoretical model, as shown in Figs. 3. Here the parameters are taken from Ref. 28. From Figs. 3, we can see that the capacitances at resonant points experience a peak and a valley. It denotes that the impedance of the sample changes from positive capacitance to negative inductance around the resonance.^29,37

Then the comparisons of our theoretical results, the experimental data and the theoretical results of Ref. 28 are presented in Table II. Moreover, the differences between numerical simulations and experimental results are listed in Table II. The results show that the error range is 12.82% ∼ 29.77% in Ref. 28 and 0.23% ∼ 6.39% in this paper. Apparently, a large error exists between the linear model in Ref. 28 and experimental data, but the error will be reduced significantly by utilizing the nonlinear model built in this paper. It also confirms the reliability and validity of our theoretical analysis and numerical code. Therefore, it’s very necessary to account for the nonlinear properties in Terfenol-D.

After that, we study the effects of the external magnetic field on resonance frequencies of $f r 1$ and $f r 2$⁠, as plotted in Fig. 4. With the increase of the magnetic field, the resonance frequencies decrease at the beginning, then increase gradually till it closes to a relatively stable value. Then the curve of the elastic modulus ^mc₁₁ (^mc₁₁ = 1/^ms₁₁) versus magnetic fields is present in Fig. 4. Here the compressive stress is −10MPa. It can be seen that the tendencies of the resonance frequencies curves is same as that of elastic modulus. Because the resonance frequency is linked to elastic modulus.³⁸ 

The frequency dependences of capacitance under different DC magnetic fields and compressive stresses are illustrated in Figs. 5(a) and 5(b), respectively. By setting the frequency from 60 kHz to 120 kHz. In Fig. 5(a), we take the pre-stress as −10MPa, with the enhancement of magnetic field intensities, the first resonance frequency shifts to left. However, the effects of compressive stress on resonance frequencies are invert to that of magnetic field intensities, when compressive stress increases (see in Fig. 5(b), H = 100Oe), the first resonance frequency shifts towards right.

It’s obvious to see that four resonance frequencies appear in all of them, but the locations are various under different magnetic fields and pre-stresses. Because the capacitance is related to the effective coefficients of Terfenol-D (such as ^ms₁₁, ^md₁₁ and ^mμ₁₁), and the capacitance is codetermined by those effective coefficients, meanwhile, the variations of each effective coefficient are various with different magnetic fields and pre-stresses (see Figs. 2).

In the following, we investigate the magnetic field intensity for resonance at a certain frequency, as plotted in Fig. 6. When the magnetic field intensities range from 0Oe to 3500Oe, the impedance of the sample changes from capacitive to inductive ones, thus the curves of capacitance appear lots of peaks and valleys in Fig. 6. When the given frequency is low (for example f = 50 kHz), the ring-strip structure needs a larger driven magnetic field for first resonance, with the given frequency is higher, the driven magnetic field is relatively lower. The resonant points of the capacitances will be changed with the increase of magnetic field. According to the theory proposed in section II and the results plotted in Figs. 2, the effective coefficients (such as ^ms₁₁, ^md₁₁ and ^mμ₁₁) are important variables for calculating the capacitance, however, the effective coefficients are affected by external magnetic fields when the pre-stress is a constant. Therefore, it’s reasonable to say that the external magnetic field is responsible for the resonant frequency shift.

Then wee analyze the influence of pre-stresses on the capacitance of the ring-strip structure, as displayed in Fig. 7. It’s the same tendency with the results mentioned above. When the magnetic field is low (less than 200Oe), the resonant frequencies appear only in the tensile stress range, but begin to emerge in compressive range when magnetic field intensity is higher than 200Oe. That is to say, the resonant frequencies of the structure can be controlled by the pre-stress when magnetic field is a constant.

In addition, Fig. 8 shows the resonant frequencies of the ring-strip structure under the action of magnetic field and compressive stress, the frequencies range from 80 kHz to 89 kHz. Here the inner radius of the PZT ring is fixed at 1.35 × 10⁻² m, then the outer radius is expanded gradually. It can be observed that the resonant frequency is sensitive to the ratio of outer and inner radii of the PZT ring, actually, which is related to the volume fraction. The resonant frequency decreases as outer radius of PZT ring increases. Compare with the case of H = 100Oe and ^mσ₁ = − 10MPa, the resonant frequency is downward when the magnetic field increases to 200Oe, but goes up when the pre-stress increases to −20MPa. The results are consistent with those in Figs. 5. The result can provide an alternative way to change the resonant frequency of the structure by changing geometrical dimensions, which may be significant for designing tunable devices.

In practical application, there are many kinds of piezoelectric materials available, so we can choose different piezoelectric materials to obtain different performances of the ring-strip structure. And the effect of different piezoelectric materials on the MC effect of the structure is investigated. Here three kinds of piezoelectric materials are chosen, they are PZT, PMN-PT, BTO. Based on the theoretical model, we predict the tendency of the capacitance versus magnetic field with three piezoelectric materials. The compressive stress is −10MPa, and the frequency is 100 kHz. The material parameters of piezoelectric materials come from Refs. 11, 21, 39, and 40. From Fig. 9, it’s observed that the response points of capacitance are different for different piezoelectric materials. That is to say, the resonance frequencies of the structure can be changed by selecting different materials when the magnetic field and pre-stress are set to stable values. For PMN-PT-ring/Terfenol-D-strip, the resonance will occur when the magnetic fields are 275.72, 466.75, 704.27, 1015.11, 1397.42 and 1896.82Oe, respectively. For further analysis of the six resonance peaks, we find that the first three of driven magnetic fields make the sample from inductive to capacitance, and the tendency is opposite under the later three of driven magnetic fields, with the increase of magnetic field, the difference of adjacent driven magnetic field increases.

In Figs. 10, the changes of Young’s modulus and density of piezoelectric material (the other material parameters are the same with PZT) on capacitance are presented respectively. As presented in Fig. 10(a), when magnetic field intensities are 100Oe and 200Oe, the capacitance decreases as the Young’s modulus of piezoelectric material increases, but shows an opposite tendency when changing the frequency and pre-stress, the first two curves can be regarded as part of the curve of capacitive, the later two curves are parts of the curve of inductive. The capacitance is a linear increase when the density of piezoelectric material increases (see Fig. 10(b)). It’s revealed that the resonant frequency is associated with the composites material parameters.

In this paper, we proposed a nonlinear model for MC effect in ring-strip ME composites, compared with previous model, our theoretical results are consistent with experimental data better, thus it’s necessary to consider the nonlinear constitutive relations of magnetostrictive materials. Based on this nonlinear model, the influence of external magnetic fields and pre-stresses on the MC effect of the ME structures is investigated. The results indicate that the resonance frequencies of the structure can be controlled by the magnetic fields and pre-stresses when the material is selected. In addition, the characteristics of the capacitances under different geometric sizes and material parameters are also discussed. The results show that the resonance frequencies of the structure can be changed by select different materials when the magnetic field and pre-stress remain unchanged. Those theoretical results can provide fundamental basis in the design and fabrication of new multifunction devices based on the MC effect of ME composites.

This work was supported of the National Natural Science Foundation of China (11372120, 11421062 and 11572143). The authors gratefully acknowledge these supports.