Analysis of Magneto-Mechanical Response for Magnetization-Graded Ferromagnetic Material in Magnetoelectric Laminate

Abstract

This paper analyzes the dynamic magneto-mechanical response in magnetization-graded ferromagnetic materials (MGFM) comprised of high-permeability Finemet and traditional magnetostrictive materials. The theoretical modeling of the piezomagnetic coefficient that depends on the bias magnetic field of MGFM is proposed by using the nonlinear constitutive model of a piezomagnetic material, the magnetoelectric equivalent circuit method, and the simulation software Ansoft. The theoretical variation of piezomagnetic coefficients of MGFM on the bias magnetic field is in good agreement with the experiment. Using the piezomagnetic coefficient in the magnetoelectric voltage model, the theoretical longitudinal resonant magnetoelectric voltage coefficients have also been calculated, which are consistent with the experimental values. This theoretical analysis is beneficial to comprehensively understand the self-biased piezomagnetic response of MGFM, and to design magnetoelectric devices with MGFM.

1. Introduction

Magnetoelectric (ME) laminate composites comprising of piezomagnetic and piezoelectric phases have been extensively investigated in recent years for use in potential smart devices due to their low cost, lightweight, and flexible features [1,2,3]. The principal of ME response can be determined from the product properties of the piezomagnetic and piezoelectric phases. The ME coefficient is expressed as

$${\alpha }_{ME}=k_c\times d_m\times d_p$$

(1)

where k_c is a coupling factor (0 ≤ k_c ≤ 1) between the two phases, d_m is the piezomagnetic coefficient, and d_p is the piezoelectric coefficient. The d_m and d_p are decided by the choice of piezomagnetic materials (Terfenol-D, Metglas, Gafenol, etc.) and piezoelectric materials (Pb(Zr_1−xTi_x)O₃ ceramics, PMN-PT single crystal, AlN films, etc.), respectively [1,2,3]. The coupling factor k_c, according to previous reports, is mainly influenced by configuration [4,5,6], boundary conditions [7], and composite mode (epoxy bonding [8], magnetron sputtering [9], and spin-wave interactions [10,11,12]). The configuration and boundary conditions determine the mechanical structure of ME composites. The composite mode determines the interface coupling between two phases.

For most magnetostrictive materials with a small magnetic hysteresis, the value of d_m is near zero without an external magnetic field [1,2,3,4,5,6,7,8,9]. Thus, a magnetoelectric composite requires a DC biased magnetic field (H_dc) to obtain an enhanced value of d_m [13]. Generally, the H_dc is provided by a pair of permanent magnets resulting in a possible noise source in the sensor array, a reduction of the spatial resolution, and an increase in the device volume.

For weakening the dependence of H_dc, the self-biased piezomagnetic effect has been presented by researchers. Using the inherent remanence of magnetic material is an effective way [14,15,16], but the self-biased ME coupling is still weak. The antiferromagnetic–ferromagnetic exchange coupling effect is the other effective method [17,18], but the processing technology is complex. Beyond that, the ME laminate with magnetization-graded ferromagnetic materials (MGFM) consisting of high-permeability ribbons and traditional magnetostrictive materials also has an effective self-biased ME response [19,20,21,22,23]. Up until now, the experimental self-biased ME response for the asymmetric piezoelectric/MGFM laminate and the symmetric MGFM/piezoelectric/MGFM structure have been studied [19,20,21,22,23]. However, up until now, few studies have focused on the theoretical magneto-mechanical response for MGFM.

In this paper, we analyze the magneto-mechanical response of MGFM consisting of Finemet and traditional piezomagnetic layers (Ni or FeNi alloy). The theoretical modeling of the magneto-mechanical behavior of MGFM is established using the nonlinear piezomagnetic constitutive model and the method of the equivalent circuit. Compared with the experimental data, the bias field dependences of piezomagnetic coefficients are in good agreement with the calculation from the presented model.

2. Theoretical Analysis

2.1. Dynamic Effective Piezomagnetic Coefficient d_33,m of MGFM

The MGFM FeCuNbSiB/Ni is shown in Figure 1. The phase m₁ is Finemet (FeCuNbSiB), phase m₂ is traditional magnetostrictive material (Ni or FeNi). According to the principle of wave mechanics, the displacement direction of the laminate is the same as the propagation direction of the vibration, so the wave formed by elastic vibration is a longitudinal wave. Under the free boundary conditions, the MGFM vibrates freely.

Then, the mechanical vibration equation can be derived as follows [24,25]

$$\{\begin{array}{l}{F}_1=Z_1{\stackrel{•}{u}}_1+Z_2({\stackrel{•}{u}}_1-{\stackrel{•}{u}}_2)+({\phi }_{m1}+{\phi }_{m2})H_3\\ F_2=-Z_1{\stackrel{•}{u}}_2+Z_2({\stackrel{•}{u}}_1-{\stackrel{•}{u}}_2)+({\phi }_{m1}+{\phi }_{m2})H_3\end{array}$$

(2)

where $Z_1=j\stackrel{-}{\rho }\nu A\tan \frac{kl}{2}$, $Z_2=\frac{\stackrel{-}{\rho }\nu A}{j\sin kl}$, ${\phi }_{m1}=\frac{A_{m1}{d}_{33,m1}}{s_{33,m1}^H}$, ${\phi }_{m2}=\frac{A_{m2}{d}_{33,m2}}{s_{33,m2}^H}$, F_l and F₂ are forces at the two end faces of the structure, and the ${\dot{u}}_1$ and ${\dot{u}}_2$ are the corresponding displacements. $\overline{\rho }=\left({\rho }_{m1}{t}_{m1}+{\rho }_{m2}{t}_{m2}\right)/t$ is the average density of the laminate. ${\phi }_{m1}$ and ${\phi }_{m2}$ are the magneto-elastic coupling factor. The d_33,m1 and d_33,m2 is the longitudinal piezomagnetic coefficient of FeCuNbSiB and Ni, respectively. $\overline{\rho }$ is the average density of the laminate. The k is the wavenumber and v is the sound velocity. $s_{33,m1}^H$ and $s_{33,m2}^H$ is the elastic compliance under constant H for FeCuNbSiB and Ni, respectively.

In this paper, the method of equivalent circuit is used to deal with the magneto-elastic interaction, as shown in Figure 2.

From the equivalent circuit, the relationship for φ_m of FeCuNbSiB/Ni, φ_m1 of FeCuNbSiB phase and φ_m2 of Ni phase is

$${\phi }_m={\phi }_{m1}+{\phi }_{m2}$$

(3)

Then

$$\frac{(A_{m1}+A_{m2})d_{33,m}}{\overline{s}}=\frac{A_{m1}{d}_{33,m1}}{s_{33,m1}^H}+\frac{A_{m2}{d}_{33,m2}}{s_{33,m2}^H}$$

(4)

where $\overline{s}$ is the average compliant coefficient and $\frac{1}{\overline{s}}=\frac{t_{m1}}{(t_{m1}+t_{m2})s_{33,m1}^H}+\frac{t_{m2}}{(t_{m1}+t_{m2})s_{33,m2}^H}$. t_m1 and t_m2 are the thicknesses of FeCuNbSiB and Ni, respectively. So, the d_33,m of MGFM is

$$d_{33,m}=\frac{t_{m1}{d}_{33,m1}{s}_{33,m2}^H+t_{m2}{d}_{33,m2}{s}_{33,m1}^H}{t_{m1}{s}_{33,m2}^H+t_{m2}{s}_{33,m1}^H}$$

(5)

2.2. Effective Dynamic Piezomagnetic Coefficient d_33,m1 of FeCuNbSiB in MGFM

According to the nonlinear magnetostrictive constitutive relationship, the d_33,m of magnetostrictive material is [26]

$$d_{33,m}=\frac{\partial \epsilon }{\partial H}=\frac{\partial \epsilon }{\partial M}\frac{\partial M}{\partial H}=2{\lambda }_s(\coth (\eta H)-\frac{1}{\eta H})\times [\eta (1-\coth {(\eta H)}^2)+\frac{1}{\eta H^2}]$$

(6)

where M is the magnetization, ε is the stain, λ_s is the saturation magnetostrictive coefficient and H is the external magnetic field. η = 3χ_m/M_s χ_m is the initial magnetic susceptibility, M_s is the saturation magnetization.

Due to the remanent magnetism of Ni, an additional magnetic field H_f in FeCuNbSiB is generated. Thus, the effective magnetic field of the FeCuNbSiB ribbons in the FeCuNbSiB/Ni laminate (H_eff,m1) is

$$H_{eff,m1}=\frac{H_{\mathrm{dc}}+H_{\mathrm{ac}}+H_f}{1+N_{d,m1}({\mu }_{r,m1}-1)}$$

(7)

where N_d,m1 is the demagnetizing factor of FeCuNbSiB. H_dc is the bias magnetic field. H_ac is the alternating current magnetic field. Therefore, the effective dynamic effective piezomagnetic coefficient d_33,m1 of FeCuNbSiB in the FeCuNbSiB/Ni laminate is

$$d_{33,m1}=2{\lambda }_{s,m1}(\coth ({\eta }_{m1}{H}_{eff,m1})-\frac{1}{{\eta }_{m1}{H}_{eff,m1}})\times [{\eta }_{m1}(1-\coth {({\eta }_{m1}{H}_{eff,m1})}^2)+\frac{1}{{\eta }_{m1}{(H_{eff,m1})}^2}]$$

(8)

In order to obtain the value of H_f, we use the magnetic field simulation software, Ansoft 11.0. In the simulation, the residual magnetization of Ni is assumed as the parameter input, so Ni provides the magnetic field H_f. For magnetostrictive Ni with cubic magnetocrystalline anisotropy and $\left|K_1\right|<M_s^2$, the remanent magnetization is M_r = 0.866M_s, μ₀M_s = 0.616 T. The simulation settings are as follows; Ni: μ₀M_r = 0.534 T, μ_r = 220; dimensions are 12 × 6 × 1 mm³; FeCuNbSiB: μ_r = 30,000; dimensions are 12 × 6 × 0.120 mm³.

Figure 3a,b illustrates the distributions of magnetic fields of the FeCuNbSiB/Ni laminate and the FeCuNbSiB ribbon in FeCuNbSiB/Ni, respectively. In the absence of H_dc, it can be clearly displayed from Figure 3 that the Ni plate provided a H_dc to FeCuNbSiB due to the residual magnetization. When the layer of FeCuNbSiB ribbon L = 1, H_f = 276.78 (A/m).

2.3. Dynamic Effective Piezomagnetic Coefficient d_33,m2 of Ni Plate in MGFM

For FeCuNbSiB as a soft magnetic material, the permeability μ_rf = 30,000, saturation magnetization μ₀M_s = 1.45 T, and the anisotropic constant is −30,000 J/m³. Comparatively, for pure Ni, the permeability (μ_r = 60) and saturation magnetization (μ₀M_s = 0.616 T) are lower, and the anisotropic constant (100 J/m³) is smaller. The magnetic properties of the FeCuNbSiB ribbons and the Ni plate are significantly different, which results in an additional magnetic field being generated in the Ni plate due to the flux concentration effect of the high-permeability FeCuNbSiB ribbons [9,23]. According to previous reports [9,23], a magnetic material with high-permeability can be assumed as a static magnetic source, which results in the flux concentration effect and generates an effective magnetic field around it. Thus, the effective magnetic field in Ni of FeCuNbSiB/Ni (H_eff,m2f) is larger than that of a single Ni plate without FeCuNbSiB (H_eff,m2n) due to the flux concentration effect of FeCuNbSiB. Here, we assume that the coefficient of the flux concentration effect is δ. So, both H_dc and H_ac in Ni are amplified δ times. In general, H_dc is far larger than the H_ac. Furthermore, the internal magnetic field in the Ni plate is influenced by the demagnetizing field H_d which is completely opposite to the magnetization. The effective magnetic field H_eff,m2n of the single Ni plate and H_eff,m2f of Ni in the FeCuNbSiB/Ni laminate are

$$H_{eff,m2n}=\frac{H_{\mathrm{dc}}+H_{\mathrm{ac}}}{1+N_{d,m2}({\mu }_{r,m2}-1)}$$

(9a)

$$H_{eff,m2\mathrm{f}}=\frac{\delta (H_{\mathrm{dc}}+H_{\mathrm{ac}})}{1+N_{d,m2}({\mu }_{r,m2}-1)}$$

(9b)

From Equation (9a) and (9b), the piezomagnetic coefficient of single Ni d_33,m2n, and Ni in FeCuNbSiB/Ni d_33,m2f are

$$d_{33eff,m2n}=2{\lambda }_{s,m2}(\coth ({\eta }_{{}_{m2}}{H}_{eff,m2n})-\frac{1}{{\eta }_{{}_{m2}}{H}_{eff,m2n}})\times [{\eta }_{m2}(1-\coth {({\eta }_{m2}{H}_{eff,m2n})}^2)+\frac{1}{{\eta }_{{}_{m2}}{(H_{eff,m2n})}^2}]$$

(10a)

$$d_{33eff,m2f}=2\delta {\lambda }_{s,mf}(\coth ({\eta }_{{}_{m2}}{H}_{eff,m2f})-\frac{1}{{\eta }_{{}_{m2}}{H}_{eff,m2f}})\times [{\eta }_{m2}(1-\coth {({\eta }_{m2}{H}_{eff,m2f})}^2)+\frac{1}{{\eta }_{{}_{m2}}{(H_{eff,m2f})}^2}]$$

(10b)

According to the principle of magnetic charge, the magnetic charges of the Ni plate are gathered and distributed at the two end faces. So the amplitude of the magnetic field at the face ends of the Ni plate can reflect its magnetization state, assuming that the amplitude of the magnetic field at the end of the single Ni plate is H_Ni and Ni in FeCuNbSiB/Ni is H_F-Ni. In order to obtain the values of H_Ni and H_F-Ni, we use the Ansoft 11.0 software package (2D magnetostatic simulation) so we can obtain δ = H_F-Ni/H_Ni. The two-dimensional simulation parameters are set as follows: permanent magnet NdFeB-N35 (75 mm length × 5 mm width, the distance of two magnets is 90 mm); Ni (μ = 220, 12 mm length × 1 mm height); and FeCuNbSiB (μ = 3 × 10₄, 75 mm length × 0.03 mm height). When the layers of FeCuNbSiB ribbon L = 1, the distribution of flux lines of FeCuNbSiB/Ni; the distribution of the magnetic field of the FeCuNbSiB/Ni laminate are shown in Figure 4.

As shown in Figure 5, when centerline x changes, the magnetic field H curves along the x-direction near the two ends of the Ni plate. Clearly, the magnetic field of the Ni plate in FeCuNbSiB/Ni is far greater than that in the single Ni plate, which is attributed to the flux concentration effect of FeCuNbSiB. For a single Ni plate, H_Ni is 908 (A/m) at the end face. When the FeCuNbSiB layer number L = 1–5, H_F-Ni at the end faces of the Ni plate in FeCuNbSiB/Ni are 1631.15395 (A/m), 1653.96385 (A/m), 1658.66092 (A/m), 1662.78705 (A/m), and 1657.69218 (A/m), respectively. When L = 4, the end face magnetic field intensity is the maximum. Therefore, according to the simulation results, the magnetic field convergence factor is δ = H_F-Ni/H_Ni, which is about 1.83 times.

3. Results and Discussion

In the present work, we compare the calculated results to the experiments to prove the effectiveness of our model for the magneto-mechanical and ME coupling characteristics. Substituting the material’s parameters shown in

Table 1. Material parameters of the FeCuNbSiB ribbon, Ni plate, FeNi plate, and the Pb(Zr_1−xTi_x)O₃ (PZT) plate.

Material	χm	μ0Ms (T)	λs (ppm)	L × w × t (mm3)	s33 (10–12 m2/N)	ε33/ε0	ρ (g/cm3)	Qmech
Ni	200	0.616	40	12 × 6 × 1	4.9		8.9	150
FeCuNbSiB	4 × 105	1.25	2.7	12 × 6×0.03	5.2		7.25	1000
FeNi	700	0.75	20	12 × 6 × 0.6	5		8	200
PZT				12 × 6 × 1	15.3	1750	7.75	1000

into Equations (5), (8), (10a) and (10b), the theoretical results of d_33,m versus H_dc for FeCuNbSiB/Ni can be obtained, as shown in Figure 6. The FeCuNbSiB is a positive magnetostrictive material, and Ni is a negative magnetostrictive material. In the calculation, d_33,m1 and d_33,m2 in Equation (5) are converse.

In experiments, the Doppler vibrometer (Polytec OFV-5000) was used to measure the vibration velocity v at the end faces of the FeCuNbSiB/Ni laminate, then the d_33,m was calculated by d_33,m = dλ/dH = v/(πƒlH_ac). The H_ac is the external AC magnetic field, ƒ is the vibration frequency, and l is the length. It shows clearly that the calculated d_33,m agrees well with the experimental data, and a large zero-biased d_33,m of FeCuNbSiB/Ni can be achieved. The errors in experimental data originally from the magnetization state of piezomagnetic material is inconsistent in each measurement during all the processes of repeated measurements.

From Figure 6, there is an error between the calculation result and the experimental values. The reason for this phenomenon is that the experimental device was not fully demagnetized in the actual experimental process, and some simulation parameters are used in the theoretical calculation. We can obtain the results:

(i)

For the outstanding soft magnetic performance of FeCuNbSiB, the magneto-mechanical coupling of FeCuNbSiB is larger than Ni in Section 1 (d_33,m1 > d_33,m2). As H_dc increases, the d_33,m1 increases rapidly, and the maximum value P_I = 12.9 nm/A is obtained under the optimal H_dc = 1184 A/m.

(ii)

When H_dc increases to a certain value, the magneto-mechanical effects of FeCuNbSiB and Ni cancel each other out, and the dynamic d_33,m appears at the zero-cross point as shown in Figure 6.

(iii)

Due to the flux concentrate effect of FeCuNbSiB, the internal magnetic moments of Ni rotate rapidly. Then, the d_33,m2 becomes larger. In Section 2 of Figure 6, the d_33,m is originated from the magneto-mechanical coupling of Ni with the flux concentrate effect. After the zero-crossing point, the d_33,m rapidly increases to the maximum value P_II. However, when H_dc increases to a certain value, FeCuNbSiB gradually saturates, the flux concentration effect is abate, which results in the appearance of the downward trend in the II area.

(iv)

With the increase of H_dc, the flux concentration effect gradually decreases, so in Section 3 of Figure 6, the d_33,m is attributed to the magneto-mechanical coupling of Ni itself. From Equation (10a) and (10b), we assume that d_33,m2f = d_33,m2n when H_dc = H_n, andd_33,m2f is far larger than d_33,m2n when H_dc < H_n. As H_dc increases gradually, the flux concentration effect disappears when H_dc > H_n. So for the d_33,m of the Ni plate in the theoretical calculation, d_33,m2 =d_33,m2f when H_dc < H_n and d_33,m2 = d_33,m2n when H_dc > H_n. From Figure 6, the magnetic field H_n for d_33,m2f = d_33,m2n in Equation (10a) and (10b) is ~12,800 (A/m) in experiments, and is ~15,600 (A/m) in theory.

Using the method of the magneto-mechanical-electric equivalent circuit, the longitudinal resonant ME voltage coefficient α_ME,r for L–T mode laminate structure is given by [24,25]

$${\alpha }_{ME,\mathrm{r}}=\left|\frac{dV}{td{H}_3}\right|=\frac{8Q_m}{{\pi }^2}\frac{n(1-n)d_{33,m}{d}_{31,p}}{{\epsilon }_{33}[f(1-k_{31}^2)s_{11}^E+(1-n)s_{33}^H]}$$

(11)

where Q_m is the mechanical quality factor, n is the volume ratio of the magnetostrictive phase, ε₃₃ is the permittivity tensor, d_33,m is the piezomagnetic coefficient, d_31,p is the piezoelectric coefficient, k₃₁ is the electromechanical coupling coefficient of the piezoelectric material, $s_{11}^E$ is the elastic compliance of the piezoelectric phase, and $s_{33}^H$ is the elastic compliance of the piezomagnetic phase.

Using Pb(Zr_1−xTi_x)O₃ (PZT) ceramic as the piezoelectric material, the ME composite FeCuNbSiB/Ni/PZT can be obtained. Taking the material’s parameters in Table 1 and Equation (5) into Equation (11), the theoretical α_ME,r can be obtained, as shown in Figure 7. For comparison, the experimental data for FeCuNbSiB/Ni/PZT with L = 1 are used, as shown in Figure 7. It can be seen from the figure that the theoretical calculation and experimental value of α_ME,r is consistent with the variation trend of the H_dc. From Equation (11), one can derive α_ME,r $\propto$ d_33,m. Therefore, the curve of α_ME,r vs. H_dc in Figure 7 is consistent with that of d_33,m vs. H_dc in Figure 6. The theoretical calculation result is different from the experimental result. The reason is that the experimental device is not demagnetized in the actual experimental process. In addition, the composite structure of FeCuNbSiB/Ni/PZT has a certain thickness of glue, which was not considered in the theoretical calculation results.

In previous work, FeCuNbSiB shows positive magnetostriction and Ni shows negative magnetostriction. In order to check the validity of the theoretical model, the other magnetostrictive material FeNi with positive magnetostrictive properties was chosen to replace Ni. Then, we also compared the theoretical results with the experimental data using MGFM FeCuNbSiB/FeNi and ME laminate FeCuNbSiB/FeNi/PZT.

Substituting the material’s parameters, shown in Table 1, into Equations (5), (8), (10a) and (10b), the calculated d_33,m versus H_dc for FeCuNbSiB/FeNi is achieved, as shown in Figure 8. The result in Figure 8 is different from that of Figure 6. The reason is that the Ni shows negative magnetostriction and FeNi shows positive magnetostriction. In Section 1, the d_33,m of FeCuNbSiB/FeNi originates from d_33,m1 of FeCuNbSiB and d_33,mf (Equation (10b)) of FeNi with the flux concentration effect. In Section 2, the d_33,m of FeCuNbSiB/FeNi originates from d_33,mn (Equation (10a)) of FeNi.

To make further efforts to verify the proposed theoretical model, the FeCuNbSiB/FeNi/PZT composite is used, and the calculated and experimental data are shown in Figure 9. The curve of α_ME,r vs. H_dc in Figure 9 is consistent with that of d_33,m vs. H_dc in Figure 8. The theoretical result for d_33,m, and α_ME,r agrees well with those of experiments. Clearly, the proposed theoretical models are also proven to be useful for the other MGFM and ME laminated composite with MGFM.

4. Conclusions

According to the nonlinear constitutive model of piezomagnetic material, magnetoelectric equivalent circuit method, and simulation software Ansoft, a theoretical model about d_33,m vs. H_dc is effectively and accurately established for MGFM. The theoretical d_33,m of FeCuNbSiB/Ni, or FeCuNbSiB/FeNi has been compared with the experimental data. By using the presented piezomagnetic model of MGFM, the resonant ME voltage coefficient α_ME,r as a function of H_dc for FeCuNbSiB/Ni/PZT or FeCuNbSiB/FeNi/PZT composite has been investigated. The theoretical calculated result agrees well with that of the experiments. Therefore, the proposed piezomagnetic model will provide an effective tool to fully understand the dynamic magneto-mechanical behavior of MGFM and self-biased ME coupling of composites with MGFM. It is useful to guide the designing of ME devices with MGFM, such as magnetic sensors and energy harvesters.