Magnetoelectric effects in bilayer multiferroic core-shell composites

1 Introduction

Recently, the multiferroic composite materials have drawn increasing interest due to their strong magnetoelectric (ME) coupling effect at the room temperature. The multiferroic composite materials are known as a mechanical mixture of piezoelectric (PE) and piezomagnetic (PM) components and can exhibit the special magneto-electro-elastic coupling effect. The ME effect in multiferroic composite materials is generated as a product property of PE and PM subsystems and this coupling effect opens new opportunities for various important applications [1–3].

In the past decades, many investigations have been done on the topic of ME effect in multiferroic composite materials. The laminated multiferroic plates are common configuration and easy to be fabricated. Up to now, many theoretical and experimental investigations were carried out at both low-frequency and electromechanical resonance ranges [4–14].

In most investigations on multiferroic composite materials, an important issue is to enhance their ME effect. Several ways have been proposed, such as using functionally graded materials[15–17], choosing curved structures–annular bilayer structures [18] or cylindrical layered composites [19–22], introducing nanocomposites [23–26], taking advantage of material nonlinearity [27, 28], among others.

To the authors’ knowledge, however, ME effect in spherical multiferroic composites has not been studied so far. Inspired by the fact that the ME effect can be strongly affected by the curvature of composite structures, this investigation focuses on a theoretical understanding of ME effect in bilayer PE/PM core-shell multiferroic cylindrical and spherical structures under radial deformation. Linear magneto-electro-elastic theory is employed to complete this theoretical analysis. Numerical results are presented to illustrate the effect of geometric configuration and mechanical boundary conditions on the ME effect in CoFe₂O₄/PZT-5A core-shell composites.

2 Three-dimensional equations for magneto-electro-elastic media

A unified form for magneto-electro-elastic theory [29] is employed to analyze the deformation problem of PE/PM multiferroic composites. Here, we summarize the three-dimensional equations of magneto-electro-elastic media. In the absence of body force, electric charge and electric current density, the equations of motion and Maxwell equations are represented as

where σij, Di and Bi are the stress, electric displacement and magnetic flux, respectively. ρ is the mass density and ui the mechanical displacement. The general strain-displacement relations are

where εij, Ek and Hk are the strain, electric field and magnetic field, respectively. Φ and Ψ are the electric potential and magnetic potential. The constitutive equations are

where cijkl, ∈ij and μij are the elastic, dielectric and magnetic permeability coefficients, respectively; ekij, qkij and gik are the piezoelectric, piezomagnetic and magnetoelectric coefficients, respectively. It is noted that we have, for PE layer, qkij=0 andgik=0, and for PM layer, ekij=0 and gik=0.

3 Basic equations of magneto-electro-elastic media for radial vibration mode

Generally, to solve analytically any three-dimensional problem in magneto-electro-elastic media would be dIfficult if not impossible. In this investigation, a relatively simple case – radial vibration mode is considered. Based on the three-dimensional linear theory for magneto-electro-elastic media, we further present the governing equations of magneto-electro-elastic media for radial vibration mode. In the following analysis, we introduce spherical coordinate system (r,θ,φ) to analyze the spherical composites and cylindrical coordinate system (r,θ,z) for cylindrical composites. For harmonic excitation, all the physical fields have the same time dependence factor exp(iωt)(i=−1 is imaginary unit and ω is the driving circular frequency) and thus we drop it for the sake of brevity. For radial vibration, all the physical fields depend on the radial position r and time t only. Then eq. 1 can be simplified as

where α is a structural configuration parameter. We have α=1 for cylindrical composites and α=2 for spherical composites, and therefore, both core-shell composite structures can be analyzed simply in a mathematically uniform manner. It should be mentioned though that the cylindrical composite is studied in the cylindrical coordinate system and the spherical composite in the spherical coordinate system. Similarly, eq. 2 can be simplified as

For cylindrically isotropic (plane strain) and spherically isotropic media, eq. 3 is simplified as

4 Governing equations of PE and PM layers for radial vibration mode

For a PE layer, we have q31=q33=g33=0. The solution of eq. 4b can be written in the form as Dr=D0r−α, where D0 is an unknown constant. If the PE layer is insulated at its outer surface, we have D0=0. Then from the third equations in eq. 6, we have

.

Substituting eq. 7 into the expressions for σθθ and σrr in the first two equations in eq. 6 and utilizing eq. 5, we obtain

where

With the aid of eq. 8, eq. 4a can be rewritten as

where

In eq. 11, ρe is the mass density of the PE layer.

For the PM layer, we have e31=e33=g33=0. When it is driven by a time-harmonic uniform radial magnetic field Hr=H0exp(iωt), where H0 is the given amplitude, the expressions of σθθ and σrr in eq. 6 can be rewritten as

Substitution of eq. 12 into eq. 4a gives

where

In eq. 14, ρm denotes the mass density of the PM layer.

5 ME effect in multiferroic core-shell composites

In the PE layer, the general solution of eq. 10 is

where Ae and Be are two undetermined constants and

In eq. 16, Jμe(⋅) and Yμe(⋅) are the first and second kind Bessel functions of order μe.

Similarly, for the PM layer, the general solution of eq. 13 can be obtained as

where Am and Bm are two undetermined constants and

where Jμm(⋅) and Yμm(⋅) are the first and second kind Bessel functions of order μm, and S12(α−1),μm(kmr) is the Lommel function and can be expressed in terms of Bessel functions as

Substituting eq. 15 into the second part of eq. 8, we obtain the expression of radial stress for PE layer:

where

Similarly, substituting eq. 17 into the second part of eq. 12, we obtain the expression of radial stress for PM layer:

where

Now we consider a bilayer multiferroic core-shell structure driven by the time-harmonic uniform radial magnetic field Hr=H0exp(iωt). The radii of the most inner, interface between PE and PM layers, and outer surface of the composite are denoted by a, b and c, respectively, see Figure 1. The inner layer (a≤r≤b) is made of PM material and the outer layer (b≤r≤c) is PE. The PE layer is polarized in radial direction. In this study, the ME effect in multiferroic core-shell composites for four sets of mechanical boundary conditions (MBCs) will be considered. These MBCs are listed here as:

1. Both the inner and outer surfaces are traction-free (F-F)

   (24a)σrr(a)=σrr(c)=0.

2. The inner surface is traction-free while the outer surface is clamped (F-C)

   (24b)σrr(a)=ur(c)=0,

3. The inner surface is clamped while the outer surface is traction-free (C-F)

   (24c)ur(a)=σrr(c)=0,

4. Both the inner and outer surfaces are clamped (C-C)

   (24d)ur(a)=ur(c)=0.

Figure 1:

Schematic of a bilayer core-shell multiferroic composite.

For a perfectly bonded bilayer multiferroic core-shell structure, the continuity conditions can be expressed as

where b+ denotes the interface on the PE side and b− the same interface on the PM side.

By applying any one set of the MBCs and the continuity conditions to the general solutions obtained for the displacement and radial stress, the involved four unknown constants Ae, Be, Am and Bm can then be determined. Consequently, all the physical fields in the multiferroic composites can be obtained. Particularly, with the aid of eq. 5, eq. 7 can be rewritten as

Integrating eq. 26 over the spatial interval [b, c], the voltage difference between the inner and outer surfaces of the PE layer is determined as

The ME effect is then expressed as

where tc=c−a is the total thickness of the composite. eq. 28 is the expression of the frequency-dependent ME effect in bilayer PE/PM multiferroic core-shell composites. If ω≈0, η denotes the ME effect under low-frequency range.

6 Numerical results

The ME effect in bilayer PE/PM multiferroic core-shell composites will be investigated numerically in this section. As illustrative examples, the PM material is taken as CoFe₂O₄ (CFO) [30] and the PE as PZT-5A (PZT) [31]. In all the following analysis, the thickness of the multiferroic composites is fixed at tc=c−a= 2 mm.

We first consider the ME effect in the multiferroic core-shell composites under low-frequency range ω≈0. Figure 2 shows the variation of ME effect η versus thickness ratio m for the four different boundary conditions in the multiferroic core-shell composites with the inner radius a = 10 mm. In Figure 2, the thickness ratio m is defined as m=te/tc, in which te=c−b is the thicknesses of the PE layer. From Figure 2, some important features can be observed: (1) For both two core-shell composites, a larger ME effect can be attained by applying F-F or C-C MBCs. The ME effect for F-C and C-F MBCs is relatively weak; (2) For both two core-shell composites, the largest ME effect for F-C and C-F MBCs appears around the thickness ratio m = 0.5, whilst for F-F MBCs around m = 0.6 (m 0.5) and for C-C MBCs around m = 0.4 (m 0.5); (3) Except for C-C MBCs, the ME effect in spherical multiferroic composites is always larger than that in cylindrical ones; (4) For C-C MBCs, the ME effects in spherical and cylindrical core-shell composites are almost the same, which are the strongest among the four MBCs.

Figure 2:

Dependence of ME effect η on thickness ratio m (thickness of PE layer over total thickness of the composite) for four different boundary conditions. (a) F-F; (b) F-C; (c) C-F; (d) C-C. The inner radius of the multiferroic composites is fixed at a = 10 mm, and the unit of η is (V/m)(A/m)⁻¹.

Figure 3 depicts the dependence of ME effect η on nondimensional inner radius R for four different boundary conditions. In this example, the thickness ratio of the multiferroic core-shell composites is fixed at m = 0.5 and the nondimensional inner radius R is defined as R=a/tc. The effect of the curvature on the ME effect in the multiferroic core-shell composites is clearly captured. (1) For F-F, F-C and C-F MBCs, the ME effect in cylindrical multiferroic core-shell composites decreases monotonously with increasing inner radius. But it is not the case for spherical multiferroic core-shell composites. For all these three sets of MBCs, a peak value is observed at R = 0.4~0.5 for the spherical composites. For F-F and C-F MBCs, the ME effect in spherical composites is always larger than that in cylindrical ones. (2) For C-C MBCs, the ME effect in both spherical and cylindrical multiferroic composites increases monotonously with increasing inner radius. When R is small, the ME effect in cylindrical composites is slightly larger than that in spherical ones. On the other hand, when R is large, the ME effect in cylindrical composites is almost the same as that in spherical ones. (3) Physically, when R is very large, the ME effect approaches gradually those in the polarized bilayer multiferroic plate. For C-C MBCs, the ME effect for both spherical and cylindrical multiferroic composites approaches 0.95(V/m)(A/m)⁻¹. This value agrees very well with that reported by Pan and Wang (the ME effect is measured as 0.95(V/m)(A/m)⁻¹ for V_f= 0.5 from Figure 3 by the finite element method [14].

Figure 3:

Dependence of ME effect η on nondimensional inner radius R (inner radius over total thickness of the composite) for four different boundary conditions. (a) F-F; (b) F-C; (c) C-F; (d) C-C. The thickness ratio of the multiferroic composites is fixed at m = 0.5 and the unit of η is (V/m)(A/m)⁻¹.

We next consider the ME effect in the bilayer multiferroic core-shell composites around the electromechanical resonance frequencies. To take into consideration of the energy loss in practical devices, we multiply the elastic constants of PE and PM layers by a complex factor (1+Qi) to introduce slight damping into the system [13]. Generally, Q is in the order of 10⁻²~10⁻³. Here we take Q = 0.02. In this demonstration, the geometric parameters are taken as a = 10 mm and m = 0.5 (the thickness of both PE and PM layers is 1 mm).

The ME effect as a function of the driving frequency is plotted in Figure 4. As we expected, all ME effect curves exhibit peaks at the resonance frequencies. It can be seen that, for both cylindrical and spherical multiferroic composites, the lowest fundamental resonance frequency appears for F-F MBCs. The fundamental resonance frequency for spherical multiferroic composites is much higher than that for cylindrical one (65.7kHz for cylindrical composites and 110.1kHz for spherical one). Under the fundamental resonance frequencies, the ME effect in spherical composites is stronger than that in cylindrical one. For the other three sets of MBCs (C-F, F-C and C-C), the fundamental resonance frequencies for spherical composites are slightly higher than those for cylindrical ones, while the dynamic behaviors of the ME effect in spherical composites are similar to those in cylindrical ones.

Figure 4:

ME effect η as a function of driving frequency for four different boundary conditions. (a) F-F; (b) F-C; (c) C-F; (d) C-C. The unit of η is (V/m)(A/m)⁻¹, and the geometry is fixed at inner radius a = 10 mm and thickness ratio m = 0.5.

7 Conclusions

Based on the three-dimensional magneto-electro-elastic theory, an analytical solution is developed to investigate the ME effect in bilayer multiferroic core-shell composites under radial deformation/vibration. Two core-shell configurations are considered in a concise and uniform manner mathematically. One is spherical and the other is cylindrical. It can be seen that both the geometric configuration and boundary condition play important roles on the ME effect in bilayer multiferroic core-shell composites. As such, it is possible and also practical to enhance the ME effect in multiferroic composites by selecting a suitable or optimal geometric configuration and/or a suitable or optimal mechanical boundary condition.

Our numerical results demonstrate clearly that one can always achieve a high ME effect by exciting the bilayer multiferroic core-shell composites near its resonance frequencies, as the peak values of the ME effect are closely associated with and greatly affected by the material damping factor. Furthermore, the presented model could also be employed to study the ME effect in triple-layer or multilayer multiferroic core-shell composites.