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Russian Physics Journal, Vol. 56, No. 6, November, 2013 (Russian Original No. 6, June, 2013)

THEORY OF MAGNETOELECTRIC EFFECT IN A BILAYER MAGNETOSTRICTIVE-PIEZOELECTRIC STRUCTURE

D. A. Filippov and T. A. Galichyan UDC 537.9

A theory of the magnetoelectric effect in a bilayer magnetostrictive-piezoelectric structure is presented. As objects of research, structures in the form of nickel–lead zirconate–titanate and permendur–lead zirconate–titanate based plates are chosen. Joint solution of the motion equation for a magnetostrictive and piezoelectric medium and of the constitutive equations yields an expression for the magnetoelectric voltage coefficient in the region of electromechanical resonance.

Keywords: bilayer structure, magnetoelectric effect, magnetoelectric coefficient.

INTRODUCTION

The magnetoelectric (ME) effect in magnetostrictive-piezoelectric structures consists in mechanical interaction of the magnetostrictive and piezoelectric phases. When the structure is placed in a magnetic field, in the magnetostrictive component there appear mechanical stresses that are transferred to the piezoelectric phase, which leads to a change in the polarization of the sample. An advantage of bilayer magnetostrictive-piezoelectric structures is that the magnitude of the ME effect in them, as a rule, is higher than in bulk composites, so that they are of great interest for the creation of various devices based on solid-state electronics. The ME effect theory in bulk and multilayered composites in the region of electromechanical resonance was developed in [1–4]. In these works, based on the method of effective parameters, expression for the ME voltage coefficient was obtained and its frequency dependence was analyzed. A disadvantage of the method of effective parameters, on the one hand, is its limited applicability. It is applicable when the characteristic sizes of structure units of a composite are much less than the acoustic wavelength, and the composite can be considered as a homogeneous medium. On the other hand, one more disadvantage of the method is the complexity of finding the effective parameters; therefore, it seems expedient to derive an expression for the ME coefficient through the parameters characterizing the magnetostrictive and piezoelectric phases. In [5–8] the ME effect was studied in bilayer structures based on joint solution of the equation of motion and of the constitutive equation for the magnetostrictive and piezoelectric phases. However, in these works it was assumed that the amplitude of oscillations was identical throughout the thicknesses of both ferromagnetic and piezoelectric materials. This assumption can be used to describe the effect with this or that degree of accuracy only for sufficiently thin layers. In [9] the ME effect in the structure representing the ME film grown on a semi-infinite passive substrate was analyzed taking into account a change in the oscillation amplitude with the sample thickness. However, the structures representing mechanically connected magnetostrictive and piezoelectric layers of finite thicknesses are much more often encountered in practice. In the present work, a consecutive theory of the ME effect in such structures is presented. Elastic wave propagation in a bilayer magnetostrictive-piezoelectric structure is studied, and an expression for the ME voltage coefficient is derived taking into account that the oscillation amplitude changes with the thickness of the sample. Based on the limiting transition, it is demonstrated that the dispersion relations for elastic oscillations and the expression for the ME coefficient for thin layers are transformed into the expressions obtained in [5–8].

Yaroslav-the-Wise Novgorod State University, Velikii Novgorod, Russia, e-mail: Dmitry. Filippov@novsu.ru. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 74–79, June, 2013. Original article submitted February 25, 2013; revision submitted June 6, 2013.

1064-8887/13/5606-0686 ©2013 Springer Science+Business Media New York

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1. MODEL AND BASIC EQUATIONS

As a model, we consider a bilayer structure in the form of a rectangular plate of length L and width W consisting of mechanically interacting magnetostrictive and piezoelectric layers whose thicknesses mt and pt are not assumed to be small (Fig. 1). Thin metal contacts are deposited on the upper and lower sides of the plate. The reference frame is chosen so that its origin coincides with the interface between the layers, and the z axis is directed vertically upward perpendicularly to the interface.

The piezoelectric layer is preliminary polarized perpendicularly to the contact planes (along the z axis). We consider the longitudinal ME effect. In this case, magnetic fields (constant H0 and alternating H with angular frequency ω) coincide with the direction of the polarization vector P. In the magnetostrictive layer, elastic oscillations are excited by the alternating magnetic field with frequency ω. They are transferred through the interface to the piezoelectric layer via shear stresses, which results in coupled oscillations of the subsystems. Assuming that the plate is narrow, in the first approximation it is possible to consider that displacements along the y axis are homogeneous, and only the stress components xxT and xzT are nonzero. Since there is a sharp interface through which the magnetostrictive and piezoelectric layers interact, the strain magnitude will be inhomogeneous with the thickness of the sample perpendicularly to the interface. Taking this into account, the equation of motion for the x-projection of the displacement vector of the medium xuα is written in the form

2

2x xx xzu T T

x zt

α α αα ∂ ∂ ∂ρ = +

∂ ∂∂, (1)

where the superscript α is equal to m for the magnetostrictive layer and to p for the piezoelectric layer, αρ is the

density of the ferromagnetic or piezoelectric material, and ijTα is the stress tensor.

The equations for the strain tensor of the polarized piezoelectric phase pxxS and p

xzS and for the z-projections

of the electric induction vector pzD have the following forms:

,1p p p p

xx xx xx z zpS T d EY

= + , (2)

1

3

2

pt

mt x 0

z

Fig. 1. Schematical drawing of the structure composed of magnetostrictive layer 1 of thickness mt, piezoelectric layer 2 of thickness pt, and ohmic contact 3.

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1p pxz xzpS T

G= , (3)

,p p p p p

z zz z xx z xxD E d T= ε + . (4)

Here pxxT and p

xzT are the components of the stress tensor in the piezoelectric phase, andp pY G are the Young and

shear moduli of the piezoelectric material, ,p

xx zd is the piezoelectric tensor, pzzε is the dielectric permeability, and

pzE is the z-projection of the electric field strength vector.

For the magnetostrictive phase, analogous equations can be written in the following form:

,1m m m m

xx xx xx z zmS T q HY

= + , (5)

1m mxz xzmS T

G= , (6)

where mxxT and m

xzT are the components of the stress tensor in the magnetostrictive phase, andm mY G are the Young

and shear moduli of the ferromagnetic material, ,m

xx zq is the piezomagnetic coefficient, mzH is the z-projection of the

magnetic field strength vector. We now express a solution of the equation for the vector displacement of the medium in terms of the plane waves whose amplitude changes with the thickness of the sample:

( , ) ( )( cos( ) sin( ))u x z g z A t kx B t kxα α α α= ω − + ω − , (7)

where Aα and Bα are integration constants and ( )g zα is a certain function. Substitution of Eq. (7) into equation of

motion (1) yields the equation for the function ( )g zα . After simple transformations, the equations for the functions determining the change of the oscillation amplitude acquire the following form:

2

22''( ) 2(1 ) ( ) 0m m

mL

g z k g zV

⎡ ⎤ω+ + ν − =⎢ ⎥

⎣ ⎦, (8)

2

22''( ) 2(1 ) ( ) 0p p

pL

g z k g zV

⎡ ⎤ω+ + ν − =⎢ ⎥

⎣ ⎦, (9)

where 21 m

m mLV Y

ρ= ; 2

1 p

p pLV Y

ρ= ; m

LV and pLV are the longitudinal wave velocities in the ferromagnetic and

piezoelectric materials, respectively; and ν is the Poisson coefficient which is set equal in both media. The form of the functions ( )mg z and ( )p g z (exponential or trigonometric one) depends on the sign of the term in the square brackets in Eqs. (8) and (9) which, in its turn, is determined by the relationship between the sound velocities in the ferromagnetic and piezoelectric materials. For definiteness, we now choose the most widespread case when the velocity of elastic waves in the ferromagnetic material is greater than in the piezoelectric material. In particular, this is the case for nickel–lead zirconate–titanate (Ni–PZT) and permendur lead zirconate–titanate (Pe–PZT) structures. In this case, the coefficient in the square brackets in Eq. (8) will be less than zero, and in Eq. (9) it will be greater than zero. Taking this into account, solutions of the equations are written in the form

1 2( ) exp( ) exp( )m m mg z C z C z= χ + − χ , (10)

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3 4( ) cos( ) sin( )p p pg z C z C z= χ + χ , (11)

where we have used the following designations: 2

2 222(1 )m

mL

kV

⎡ ⎤ωχ = − + ν −⎢ ⎥

⎣ ⎦ and

22 2

22(1 )pp

Lk

V⎡ ⎤ω

χ = + ν −⎢ ⎥⎣ ⎦

.

2. DISPERSION RELATION

To find the dispersion relation, we take advantage of the boundary conditions. On the upper and lower free surfaces of the ferromagnetic and piezoelectric materials, that is, at points pz t= − and mz t= , the stress tensors are equal to zero; at the interface between the magnetostrictive and piezoelectric phases, the displacements of the first and second media are identical, and the shear stresses are also identical. These boundary conditions result in the system of equations the compatibility condition for which leads to the dispersion relation:

tanh( ) tan( )m m m p p pY Yχ κ = χ κ , (12)

where m m mtκ = χ and p p ptκ = χ are dimensionless variables. Equation (12) implicitly defines the dependence of the angular frequency ω on the wave number k for elastic

wave propagation through the bilayer magnetostrictive-piezoelectric structure. In the limiting case of thin layers of ferromagnetic and piezoelectric materials, that is, for 1m m mtκ = χ << and 1p p ptκ = χ << , expanding the functions in

Eq. (12) in small parameters mκ and pκ , we obtain

2 2( ) ( )m m m p p pY t Y tχ = χ . (13)

Substituting expressions for mχ and pχ into Eq. (13), after some transformations we obtain the dispersion relation in the form

m m p p

m m p pY t Y t k

t t+

ω =ρ + ρ

. (14)

Thus, the nonlinear relationship between the angular frequency and the wave number is generally observed, and the dispersion relation obtained previously in [8] is valid only for thin layers.

3. MAGNETOELECTRIC VOLTAGE COEFFICIENT

The magnetoelectric voltage coefficient is defined as a ratio of the arising electric field strength E to the strength of the magnetic field that created it, that is,

/E E Hα = . (15)

However, unlike bulk composites, the ME voltage coefficient for layered magnetostrictive-piezoelectric structures can be determined by two methods. The first method consists in the determination of the ratio of the electric field arising in the piezoelectric material to the strength of the magnetic field that creates it in the ferromagnetic material. In this case, the ME coefficient characterizes well the efficiency of ME field transformation; however, for such definition, the ME coefficient characterizes the efficiency of ME transformation of the structure not entirely qualitatively. Moreover, such definition of the ME voltage coefficient leads to unphysical result. In the limiting case when the thickness of the

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piezoelectric material tends to zero, the ME coefficient still has a finite value [10]; this result is incorrect. The second method of determining the ME voltage coefficient defines it as a ratio of the average electric field strength in the structure to the average strength of the external magnetic field that creates it, that is,

/E E H< α >=< > < > , (16)

where / ( )m pE U t t< >= + is the average electric field strength in the structure, and U is the potential difference arising between the electrodes. With such definition, the ME voltage coefficient characterizes the efficiency of ME transformation of the entire structure and has a maximum for a fixed relationship between the thicknesses of the magnetostrictive and piezoelectric layers.

To derive a theoretical expression for the ME voltage coefficient, we take advantage of the method developed in [1–4]. Conditions of mechanical equilibrium on the free lateral surfaces of the sample at points / 2x L= ∓ yield the following boundary conditions:

0

0( / 2, ) ( / 2) 0

m

p

tp m

xx xxt

T L z dz T L dz−

± + ± =∫ ∫ . (17)

Using these boundary conditions, we obtain the following expressions for integration constants A and B:

, ,0,tanh( ) tan( )cos( )(1 exp( 2 ))

m m m m p p p pxx z z xx z z

m pp m m p p

m p

Y t q H Y t d EA B

k Y t Y t

< > + < >= =

⎛ ⎞κ κκ + − κ +⎜ ⎟

κ κ⎝ ⎠

, (18)

where the dimensionless parameter / 2kLκ = has been introduced. Expressing from Eq. (2) the voltage tensor component through the strain tensor components and substituting

the expression obtained into the equation for the normal components of the electric induction vector, we obtain

2

,,

( )1

p p pxx zp p p p p x

z zz z xx zpzz

Y d uD E Y dx

⎛ ⎞ ∂= ε − +⎜ ⎟

⎜ ⎟ ∂ε⎝ ⎠. (19)

The electric current in the structure can be found from the equation

/2

0 /2

W Lp

zL

I dy D dxt −

∂=∂ ∫ ∫ . (20)

Substituting Eq. (19) into Eq. (20) and integrating, we reduce the equation for the electric current to the form

,2 2 sin( ) (1 exp( 2 ))tan( )(1 )p p p p

xx zpp z p p

zz

Y d BI i W K EL

⎛ ⎞κ + − κ κ= ω − < > +⎜ ⎟⎜ ⎟ε κ⎝ ⎠

, (21)

where 2

,2 ( )p pxx z

p pzz

Y dK =

ε is the squared electromechanical coupling coefficient.

The strength of the electric field pzЕ< > induced in the piezoelectric material can be found from Eq. (21)

using the condition of open circuit 0I = . Taking into account this condition and expression for B from Eq. (18), we obtain

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, , tan( ) tan( )tanh( ) tan( )

p p m m m m pxx z xx zp z

z p m p pm m p pzz a

m p

Y d q Y t HEY t Y t

< > κ κ< >=

κε Δ κ κ κ+

κ κ

, (22)

where we have introduced

2 tan( ) tan( )1 1tanh( ) tan( )

p p p

a p m p pm m p p

m p

Y tKY t Y t

⎛ ⎞⎜ ⎟κ κ⎜ ⎟Δ = − −

κ⎜ ⎟κ κ κ+⎜ ⎟

⎝ ⎠κ κ

. (23)

Taking into account that the arising potential difference between the electrodes is p pzU E t=< > , from Eq. (16) we

obtain the final expression for the ME voltage coefficient in the form

, , tan( ) tan( )tanh( ) tan( )

p p m m m p pxx z xx z

E p m p p m pm m p pzz a

m p

Y d q Y t tt tY t Y t

κ κα =

κε Δ κ κ κ ++

κ κ

. (24)

As can be seen from Eq. (24), the frequency dependence of the ME voltage coefficient is resonant in character. At antiresonance frequencies determined by the condition 0aΔ = , a resonant increase in the ME coefficient takes place. The value of the ME voltage coefficient equally depends on the parameters of both magnetostrictive and piezoelectric layers. The peak value of the coefficient depends significantly on the losses in the structure that can be taken into account through the attenuation coefficient if we represent the angular frequency in the form ω = ω' + iχ, where χ is the parameter characterizing attenuation.

For low frequencies, the ME voltage coefficient is practically independent of the frequency. Expanding expression (24) in a series in small parameters κ , mκ , and pκ and considering the first terms of the expansion, for the low-frequency value of the coefficient we obtain the expression

, ,low

21

p p m m m pxx z xx z

E m m p p m pm mp

zz p m m p p

Y d q Y t tY t Y t t tY tK

Y t Y t

< α >=⎛ ⎞⎛ ⎞ + +

ε −⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠

. (25)

From Eq. (25) we can determine the maximum value of the ME voltage coefficient for definite relationship between thicknesses of the magnetostrictive and piezoelectric layers. In the first approximation, the ME coefficient reaches its maximum value when thicknesses of the ferroelectric and piezoelectric materials are related by the expression

p p m mt Y t Y= . (26)

CONCLUSIONS

The layered magnetostrictive structures demonstrate the best ME properties compared to the bulk composites. Consideration of the inhomogeneous amplitude of oscillations with the sample thickness results in a nonlinear relationship between the angular frequency and the wave number, which in the limiting case of small thickness is transformed into a linear dependence. The maximum ME coefficient is observed for definite relationship between thicknesses of the magnetostrictive and piezoelectric layers whose value depends on the ratio of the Young moduli for the ferromagnetic and piezoelectric materials.

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