1 Introduction

Research on magnetoelectric (ME) materials has been growing dramatically owing to the material’s potential for high frequency applications such as magnetic sensing in memory or storage devices and current-to-voltage conversion Vopsaroiu et al. (2007); Palneedi et al. (2006); Choowitsakunlert et al. (2014); Nan et al. (2008). The ME material is made of multiferroic composites that possess direct and converse ME coupling effects, which are the induction of electric polarization by the magnetic field and the induction of magnetization by the electric field, respectively Palneedi et al. (2006); Saengow et al. (2018); Wang et al. (2010). The ME material can be formed as a single-phase structure or a heterostructure. The heterostructure consists of alternate layers of ferromagnetic (FM) and ferroelectric (FE) materials, which exhibit magnetostrictive and piezoelectric properties, respectively. The ME coupling behaviour of an ME structure is commonly analyzed by a parameter known as ME coefficient. The ME coefficient is a measure of the ability of an ME material to generate the electric field from the input magnetic field and vice versa.

There were several studies on developing mathematical models of the ME coefficient of the nano heterostructures with different input-output field orientations and operating modes such as L-T (longitudinal-transverse) and T-T (transverse-transverse) modes with bar and plate structures, respectively Vopsaroiu et al. (2007); Saengow et al. (2018); Wang et al. (2010); Saengow and Silapunt (2020). These models were developed under the low frequency (f\(\rightarrow \)0) approach, to reduce complexity of the modeling Vopsaroiu et al. (2007); Saengow et al. (2018). It was shown that that the L-T mode bar structure exhibits higher ME coefficient and offers more flexibility in the structure design, compared to the plate structure Saengow and Silapunt (2020). The low frequency approach, however, shadows the actual ME coupling behaviour of the defined structure.

This paper is aimed on understanding the actual ME coupling behaviour of the nano bi-layer L-T mode bar structure through the ME coefficient mathematical model that is developed in high frequency regime. The ME coefficient model will then be tested on the bi-layer of Terfenol-D and lead zirconate titanate (PZT). The magnetostrictive effect that defines the ME mechanism in the proposed structure will be discussed. Finally, the demonstration of the proposed structure as the nano read sensor for a hard disk drive is performed. The readout voltage is determined and the feasibility of the structure for the magnetic sensing application is assessed.

2 Mathematical modeling of the magneto electric coefficient of the nano bi-layer L-T mode bar structure

The MF composites of the nano-bilayer in a shape of bar structure is considered with length (l), width (w), and total thickness (t) as shown in Fig. 1. Note that, the design condition is l \(\gg \)w, t Vopsaroiu et al. (2007). The magnetostrictive constitutive equation, piezoelectric constitutive equation, and Newton’s second law are employed altogether for the modeling of the ME coefficient of the proposed structure.

Magnetostrictive constitutive equation is a relationship between magnetic field, stress, and strain as shown in Eq. (1). For the L-T mode, the external magnetic field is applied to the FM layer in a longitudinal direction in order to induce the mechanical strain in the same direction. This strain is then coupled to the FE layer.

$$\begin{aligned} \chi _1^m=s_{11}^m \sigma _1^m + d_{11}^mH_1, \end{aligned}$$
(1)

where \(\chi \) is strain, s is elastic compliance, \(\sigma \) is stress, dm is the ferromagnetic constant, and H is magnetic field. Superscript ‘m’ refers to a ferromagnetic material and subscript ‘1’ refers to the longitudinal direction.

The piezoelectric constitutive equation is used to explain the relationship between stress, strain and electric field as shown in Eq. (2) and between stress, electric field, and electric displacement as shown in Eq. (3).

$$\begin{aligned} \chi _1^e= & {} s_{11}^e \sigma _1^e + d_{31}^eE_3 \end{aligned}$$
(2)
$$\begin{aligned} D_3= & {} d_{31}^e\sigma _1^e + \epsilon _{33}E_3, \end{aligned}$$
(3)

where de is piezoelectric constant, E is electric field, D is electric displacement, and \(\epsilon \) is electric permittivity. Superscript ‘e’ refers to a ferroelectric material and subscript ‘3’ refers to the thickness direction. Note that, H, E, \(\chi \), \(\sigma \) and D are dependent variables that are functions of a point along its designated direction.

Fig. 1
figure 1

Multiferroic composites of the nano bi-layer L-T mode bar structure

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For the proposed nano bi-layer structure, perfect physical bonding between the FM layer and the FE layer and uniformly applied magnetic field are assumed. This external magnetic field induces the mechanical strain in the longitudinal direction that results in equal elastic displacements (\(\xi \)) in both layers. The elastic displacement at a point along the longitudinal direction is related to the mechanical strain at that point by \(\chi _1 = \frac{d\xi _1}{dx}\).

The Newton’s second law and the relation between force and stress shown in Eqs. (4) and (5) respectively, are used to explain the relation between stress and elastic displacement in Eq. (6).

$$\begin{aligned} dF= & {} M\frac{{d^2}\xi _1}{dt^2}={\rho }dV\frac{{d^2}\xi _1}{dt^2}={\rho }dxdydz\frac{{d^2}\xi _1}{dt^2} \end{aligned}$$
(4)
$$\begin{aligned} dF= & {} \sigma _1{dA}=\sigma _1{dydz} \end{aligned}$$
(5)
$$\begin{aligned} \frac{\sigma _1}{dx}dydz= & {} \rho {dydz}\frac{{d^2}\xi _1}{dt^2}, \end{aligned}$$
(6)

where F is the mechanical force, M is the mass of material, \(\xi _1\) is the elastic displacement at a point along the longitudinal direction, V is the volume of the material, A is the area, and \(\rho \) is the material density.

The piezoelectric effect induced by the mechanical coupling generates the uniform electric field in the transverse direction across the FE layer. By substituting Eqs. (1) and (2) into Eq. (6), the relationships of strains and elastic displacements in the FM and FE layers can be expressed as

$$\begin{aligned} \frac{1}{s_{11}^m}\frac{d\chi _1^m}{dx}w\times {t_m}= & {} \rho _m\times {w}\times {t_m}\frac{{d^2}\xi _1}{dt^2} \end{aligned}$$
(7)
$$\begin{aligned} \frac{1}{s_{11}^e}\frac{d\chi _1^e}{dx}w\times {t_e}= & {} \rho _e\times {w}\times {t_e}\frac{{d^2}\xi _1}{dt^2} \end{aligned}$$
(8)

respectively, where tm and te are the thickness of FM and FE layers, respectively.

The total force of these system is obtained by combining Eqs. (7) and (8). The result is shown in Eq. (9) as

$$\begin{aligned} \frac{1}{s_{11}^m}\frac{d\chi _1^m}{dx}{t_m} + \frac{1}{s_{11}^e}\frac{d\chi _1^e}{dx}{t_e}=\left( {\rho _m}{t_m} + {\rho _e}{t_e}\right) \frac{{d^2}\xi _1}{dt^2}. \end{aligned}$$
(9)

Eq. (9) is then rearranged and simplified by defining t as the total thickness or t = te + tm and \(\eta \) as the thickness ratio or \(\eta = \frac{t_{e}}{t_{e} + t_{m}}\). The result is expressed as

$$\begin{aligned} \left[ \frac{\eta {s_{11}^m} + \left( 1-\eta \right) s_{11}^e}{s_{11}^ms_{11}^e}\right] \frac{{d^2}\xi _1}{dx^2}=\left[ \left( 1-\eta \right) \rho _m + \eta \rho _e\right] \frac{{d^2}\xi _1}{dt^2}. \end{aligned}$$
(10)

Since the objective of this research is to investigate the ME effect of the nano bi-layer L-T mode bar structure in high frequency environment, therefore Eq. (10) is transformed into the frequency domain as

$$\begin{aligned} \left[ \frac{\eta {s_{11}^m} + \left( 1-\eta \right) s_{11}^e}{s_{11}^ms_{11}^e}\right] \frac{{d^2}\xi _1}{dx^2}=-\omega ^2\left[ \left( 1-\eta \right) \rho _m + \eta \rho _e\right] \xi _1, \end{aligned}$$
(11)

where \(\omega \) is the angular frequency, which equals 2\(\pi \)f, and f is the operating frequency.

Rewriting Eq. (11) in the second order linear differential equation form yields

$$\begin{aligned} \frac{{d^2}\xi _1}{dx^2} + k^2\xi _1=0, \end{aligned}$$
(12)

where k2 = \({\omega }\)2\(\frac{[(1-\eta )\rho _m + {\eta }{\rho _e}]s_{11}^ms_{11}^e}{{\eta }s_{11}^m + (1-\eta )s_{11}^e}\).

The second order linear differential equation gives a solution in a sinusoidal form as

$$\begin{aligned} \xi _1\left( x\right) =Acos\left( kx\right) + Bsin \left( kx\right) , \end{aligned}$$
(13)

where A and B are constants, and x is a point along the longitudinal direction.

By substituting the elastic displacement in Eq. (13) into Eqs. (1) and (2), the stresses as a function of x in both layers can be expressed as

$$\begin{aligned} \sigma _1^m\left( x\right)= & {} \frac{1}{s_{11}^m}\left[ k\left( Bcos\left( kx\right) - Asin\left( kx\right) \right) - d_{11}^mH_1\right] \end{aligned}$$
(14)
$$\begin{aligned} \sigma _1^e\left( x\right)= & {} \frac{1}{s_{11}^e}\left[ k\left( Bcos\left( kx\right) - Asin\left( kx\right) \right) - d_{31}^eE_3\right] . \end{aligned}$$
(15)

The boundary condition is then applied to determine the constant A and B at x = \(\pm {\frac{l}{2}}\), where the stress is zero due to zero external force. The result shows that

A = 0 and B = \(\frac{{\eta }s_{11}^md_{31}^eE_3 + (1-\eta )s_{11}^ed_{11}^mH_1}{k[{\eta }s_{11}^m + (1-\eta )s_{11}^e]cos(kl/2)}\).

Substituting A, B, and Eq. (15) into Eq. (3) yields

$$\begin{aligned} D_3=\frac{d_{31}^e}{s_{11}^e}\left[ \frac{{\delta }d_{31}^eE_3 + {\beta }d_{11}^mH_1}{\left( \delta + \beta \right) cos\left( kl/2\right) }cos\left( kx\right) -d_{31}^eE_3\right] + \epsilon _{33}E_3, \end{aligned}$$
(16)

where \(\delta \) = \(\eta {s_{11}^m}\) and \(\beta \) = \((1-\eta )s_{11}^e\)

In order to calculate the ME coefficient, the evaluation of electric field across the FE layer is necessary. This is accomplished by determining the current flow through the FE layer and zeroing it out under the open circuit condition. First, the relations between the current and electric charge and the electric displacement and electric charge as shown in Eqs. (17) and (18) respectively, must be defined.

$$\begin{aligned} I_3= & {} \frac{dQ}{dt} \end{aligned}$$
(17)
$$\begin{aligned} D_3= & {} \frac{dQ}{dA}, \end{aligned}$$
(18)

where dA = wdx, the differential area through which the current flows.

The total current is found by substituting Eq. (18) into Eq. (17) and performing the integration as shown in Eq. (19).

$$\begin{aligned} I_3=\int _{-\frac{l}{2}}^{\frac{l}{2}}\frac{dD_3}{dt}dA \end{aligned}$$
(19)

Transferring Eq. (19) to the frequency domain yields

$$\begin{aligned} I_3=\int _{-\frac{l}{2}}^{\frac{l}{2}}i\omega {D_3}wdx=i{\omega }w\int _{-\frac{l}{2}}^{\frac{l}{2}}D_3dx. \end{aligned}$$
(20)

It can be seen that the total current, I3, is now a function of the electric displacement, D3, which is shown in Eq. (16). Substituting Eq. (16) intoEq. (20) thus yields

$$\begin{aligned} I_3=i{\omega }l\left[ \frac{d_{31}^e}{s_{11}^e}\left( \frac{{\delta }d_{31}^eE_3 + {\beta }d_{11}^mH_1}{\left( \delta + \beta \right) }\frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}\right) - \frac{{d_{31}^e}^2}{s_{11}^e}E_3 + {\epsilon _{33}E_3}\right] . \end{aligned}$$
(21)

Under the open circuit condition, the current between the electrode is equal to zero and Eq. (21) is simply reduced to

$$\begin{aligned} \frac{{\beta }d_{31}^ed_{11}^m}{s_{11}^e\left( \delta + \beta \right) }\frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}H_1=\left[ \frac{{d_{31}^e}^2}{s_{11}^e} - \frac{{\delta }{d_{31}^e}^2}{s_{11}^e\left( \delta + \beta \right) }\frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}} - \epsilon _{33}\right] E_3. \end{aligned}$$
(22)

By rearranging Eq. (22), the ME coefficient, which is the ratio between the induced electric field and the applied magnetic field, can be found as shown in Eq. (23).

$$\begin{aligned} \alpha _{31}^H=\frac{E_3}{H_1}=\frac{{\beta }d_{31}^ed_{11}^m\frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}}{{d_{31}^e}^2\left[ {\delta }\left( 1 - \frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}\right) + {\beta }\right] - \epsilon _{33}s_{11}^e\left( \delta + \beta \right) }. \end{aligned}$$
(23)

3 ME coupling behavior of Terfenol-D and PZT bi-layer bar structure

The ME coefficient model is tested on the bi-layer of Terfenol-D and PZT. The material parameters are shown as follows. \(d_{11}^m\) = \(11\times 10\)-9 m/A, \(s_{11}^m\) = \(17.9\times 10\)-12 m2/N, \(\rho _m\) = 9,250 kg/m3, \(d_{31}^e\) = \(-0.6\times 10\)-10 m/V, \(s_{11}^e\) = \(1.07\times 10\)-11 ms2/kg, \(\rho _e\) = 7,700 kg/m3, \(\epsilon _{33}\) and = 617 \(\epsilon _o\)Colussi et al. (2017); Guo et al. (2013). The ME coupling behaviour influenced by structure dimension and operating frequency is investigated. It is found that the value of fl is maximum at around 1,228 at \(\eta \) = 0.19 limit. The range of fl is then simply selected between 0.1 and 1,200. Fig. 2 shows the ME coefficient as a function of thickness ratio at fl = 0.1, 1, 10, 100, 200, 500, 1,000 , 1,100, 1,200, and f \(\rightarrow \) 0. It can be seen that all ME coefficient traces exhibit a parabolic-like shape with the overall highest ME coefficient at fl = 1,200. For the proposed bi-layer structure, the high frequency environment and/or long structure are then desirable in promoting the ME effect.

Fig. 2
figure 2

The ME coefficietn as a function of thickness ratio for fl = 0.1-1,200

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Fig. 3 shows the plot of the optimal thickness ratio and corresponding peak ME coefficient as a function of fl. The optimal thickness ratio decreases and its corresponding peak ME coefficient increases with fl. The observed exponential trend can be explained by the dependence on the tan(kl/2)/(kl/2) term. This result indicates the trade-off that needs to be considered in the ME sensor design. The minimum and maximum peak ME coefficients at fl = 0.1 and 1,200 respectively, are approximately 1,756 and 5,617 mV/Oe\(\cdot \)cm with \(\eta \) = 0.43 and 0.19, respectively.

Fig. 3
figure 3

The peak ME coefficient and optimal thickness ratio as a function of fl

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The mechanism responsible for the observed ME behavior is known as the magnetostrictive effect, which addresses mechanical stress and strain induced by the applied magnetic field in the FM layer. Knowing the variation of both parameters will certainly provide insights on the ME behavior. Since FM and FE layers are perfectly bonded, the mechanical stress and strain in both layers are assumed equal. Equations (1) and (14) are then used to evaluate the stress and strain as a function of a point along the longitudinal direction as shown in Eqs. (24) and (25), respectively.

$$\begin{aligned} \sigma _1^m\left( x\right)= & {} \frac{1}{s_{11}^m}\left[ \frac{{\delta }d_{31}^eE_3 + {\beta }d_{11}^mH_1}{\left( {\delta } + {\beta }\right) cos\left( \frac{kl}{2}\right) }cos\left( kx\right) - d_{11}^mH_1\right] \end{aligned}$$
(24)
$$\begin{aligned} \chi _1^m\left( x\right)= & {} \frac{{\delta }d_{31}^eE_3 + {\beta }d_{11}^mH_1}{\left( {\delta } + {\beta }\right) cos\left( \frac{kl}{2}\right) }cos\left( kx\right) \end{aligned}$$
(25)

To determine the stress and strain, the average stress and strain per structure length are evaluated and can be expressed as

$$\begin{aligned} \overline{\sigma }_1^m= & {} \frac{1}{l}\int _{0}^{l}\sigma _1^m\left( x\right) dx=\frac{1}{s_{11}^m}\left[ \frac{{\delta }d_{31}^e\alpha _{31}^HH_1 + {\beta }d_{11}^mH_1}{\left( {\delta } + {\beta }\right) cos\left( \frac{kl}{2}\right) }\frac{sin\left( kl\right) }{kl} - d_{11}^mH_1\right] \end{aligned}$$
(26)
$$\begin{aligned} \overline{\chi }_1^m= & {} \frac{1}{l}\int _{0}^{l}\chi _1^m\left( x\right) dx=\frac{{\delta }d_{31}^e\alpha _{31}^HH_1 + {\beta }d_{11}^mH_1}{\left( {\delta } + {\beta } \right) cos\left( \frac{kl}{2}\right) }\frac{sin\left( kl\right) }{kl}. \end{aligned}$$
(27)

It is clearly seen that both stress and strain are linearly proportional to the applied magnetic field. To further observe their characteristics, a sample magnetic field of 190 Oe is applied to Eqs. (26) and (27). Note that, this magnetic field strength is a typical value obtained from a magnetic bit of a hard disk Choowitsakunlert et al. (2017). Figures 4 and 5 show the average stress and strain induced in the Terfenol-D as a function of \(\eta \) for different fl value, respectively. It can be seen that their profiles are similar to that of the ME coefficient. Both stress and strain increase with fl and at each fl, reach the peak values exactly at the same thickness ratio as that of the ME coefficient. These findings are sufficient to confirm the magnetostrictive effect as a key mechanism in promoting the ME effect. The maximum and minimum stress and strain are around 4.84 and 1.42 MPa, and 0.087 and 0.0025, with \(\eta \) = 0.19 and 0.43, respectively.

Fig. 4
figure 4

The stress as a function of external magnetic field for fl = 0.1 - 1,200

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Fig. 5
figure 5

The strain as a function of external magnetic field for fl = 0.1 - 1,200

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4 Demotration of the nano bi-layer structure as a magnetic read head for HDD

In this section, the demonstration of the ME nano bi-layer structure as the magnetic read sensor for the hard disk drive (HDD) is presented. The read sensor currently used in a typical HDD is the tunneling magnetoresistive (TMR) type. The TMR sensor detects a magnetic signal and changes its resistance accordingly. It requires a current to read out the output signal in forms of the voltage. The proposed structure is mechanized by direct conversion of the magnetic field into the electric field. The current is thus not required. This is believed to help lower power consumption of the HDD.

The output voltage across the FE layer can be determined by multiplying H3 from Eq. (23) with the FE thickness as shown in Eq. (28)

$$\begin{aligned} V_{out}=\alpha _{31}^HH_1t_e=\frac{{\beta }d_{31}^ed_{11}^m\frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}H_1{\eta }t}{{d_{31}^e}^2\left[ {\delta }\left( 1 - \frac{tan\left( \frac{kl}{2}\right) }{\frac{kl}{2}}\right) + {\beta }\right] - \epsilon _{33}s_{11}^e\left( {\delta } + {\beta }\right) }. \end{aligned}$$
(28)

The magnetic read sensor with the total thickness of 4 nm Vopsaroiu et al. (2008); Chong et al. (2011) is assumed. It is expected to read 2 Tbit/in2 areal density and still provides sufficient readback voltage. Using Eq. (28) for different fl yields the voltage profiles shown in Fig. 6. The plot shows that the voltage increases linearly with the external magnetic field and at the same magnetic field, it increases with fl. The profile slopes are higher for higher fl and more prominent at fl = 1,200. At the magnetic field strength of 190 Oe/bit, the output voltage is around 0.43 mV, considerably high for the raw readback signal. All results indicate that the improvement of sensing capability can be accomplished by increasing the applied frequency or the structure length or both.

Fig. 6
figure 6

The voltage output as a function of external magnetic field for fl = 0.1 - 1,200

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5 Conclusions

The ME coupling behaviour coefficient of the nano bi-layer L-T mode bar structure is analysed using the mathematical model of the ME that is developed in high frequency regime. The model is tested on the bi-layer of Terfenol-D/PZT at fl = 0.1–1,200. It is shown that all ME coefficient traces exhibit the parabolic-like shape. The optimal thickness ratio decreases exponentially and its corresponding peak ME coefficient increases exponentially with fl. The minimum and maximum peak ME coefficients at fl = 0.1 and 1,200 respectively, are approximately 1,756 and 5,617 mV/Oe\(\cdot \)cm with \(\eta \) = 0.43 and 0.19, respectively. Both stress and strain profiles are similar to that of the ME coefficient. Stress and strain increase with fl and their peaks occur at the same thickness ratio as that of the ME coefficient. These findings thus confirm the magnetostrictive effect as a key mechanism responsible for the observed ME behavior. Then, it is found that the output voltage of the proposed structure increases linearly with the magnetic field and also increases with fl. The demonstration of the proposed structure as the read sensor for the HDD containing a 2 Tbit/in2 areal density shows the readback voltage of 0.43 mV at fl = 1,200. This voltage value is considerably high for a raw readback signal. All results indicate that the sensing capability can be improved by increasing the applied frequency or the structure length or both.