6-1 6.1 Introduction e magnetoelectric (ME) effect is defined as an induced elec- tric polarization in a material when it is subjected to a magnetic field, or an induced magnetization when in an electric field. e term “magnetoelectric” was first used by Debye and was marked by two major independent discoveries [1]. e first in discovery by Röntgen in 1888 was that a moving dielectric became mag- netized when an electric field was applied to it [2]. e second discovery was the realization of the possibility of intrinsic ME behavior of stationary crystals on the basis of symmetry con- siderations by Pierre Curie in 1894 [3]. e reverse effect of the first event, i.e., the polarization of a moving dielectric in a mag- netic field, was observed 17 years aſter the discovery in 1888. Even though Curie recognized that symmetry was an important issue in ME behavior, it took many decades to demonstrate that the ME response was allowed only in time-asymmetric media. However, aſter the two discoveries that formed the founda- tion of ME effect, no work was done until 1958, when Landau and Lifshitz verified the possibility of the ME effect in certain crystals on the basis of the crystal symmetry [4]. In 1959, the symmetry argument was applied by Dzyaloshinskii to an anti- ferromagnetic Cr 2 O 3 [5] to show the violation of time-reversal symmetry for this particular system and suggesting that the ME effect can be seen in Cr 2 O 3 . In 1960 and 1961, Astrov confirmed this experimentally by measuring the electrically induced ME effect in Cr 2 O 3 for the temperature range of 80–330 K [6]. In materials that are magnetoelectric, the induced polariza- tion P is related to the field H by the expression, P = αH, where α is the second rank ME-susceptibility tensor and is expressed in the units of s/m in SI units (in Gaussian units α = 4πP/H = 4πM/E is dimensionless). One generally determines α by mea- suring P z for an applied field H along the z-axis. Another param- eter of importance is the ME voltage coefficient α E = E/H which is related to α by the expression α = ε o ε r α E , where ε r is the relative permittivity of the material. Several single-phase materials that are either antiferromagnetic, weak ferromagnetic, or ferrimag- netic, such as Cr 2 O 3 , TbFeO 3 , and Fe x Ga 2−x O 3 , show rather weak ME effects [7–12]. 6.2 ME Effects in Composites Composites are of interest for the engineering of materials either with desired properties, or new characteristics that are absent in single-phase materials (for details see [13,14]). Composite materials can be divided into two categories as proposed by van Suchtelen [15]: (a) sum properties and (b) product properties. A sum property of a composite is the weighted sum of the con- tributions from the constituent phases that is proportional to the volume or weight fractions of these phases. Physical quantities such as density and resistivity are examples of sum properties. For product properties, consider a composite material with two component phases. e first phase has a property A → B with a proportionality tensor dB/dA = X; and the second phase has a property B → C with a proportionality tensor dC/dB = Y . en the composite will have the property A → C with a proportional- ity tensor dC/dA = Z, where Z = Y ∙ Z; hence, the name “product property.” e product property is achieved in a composite but not seen in the individual phases. An ME composite can be realized with piezomagnetic and piezoelectric components [15–48]. When a magnetic field is applied to a piezomagnetic material, it is strained; the strain in turn causes stress on the piezoelectric material, which becomes electrically polarized as a consequence. e reverse effect is also feasible; but the consequence here will be magnetization of piezomagnetic material. Such ME composites could also be made with magnetostrictive and piezoelectric phases and theories predict ME voltage coefficients as high as 3.2 V/cm Oe [22–25]. 6 Magnetoelectric Interactions in Multiferroic Nanocomposites 6.1 Introduction ............................................................................................................................. 6-1 6.2 ME Effects in Composites ...................................................................................................... 6-1 6.3 Device Applications for ME Composites ............................................................................. 6-3 6.4 eory of Low-Frequency ME Coupling in a Nanobilayer ............................................... 6-3 6.5 Conclusion ................................................................................................................................6-6 References.............................................................................................................................................6-6 Vladimir M. Petrov Oakland University and Novgorod State University Gopalan Srinivasan Oakland University
Transcript
6-1
6.1 Introduction
The magnetoelectric (ME) effect is defined as an induced elec-tric polarization in a material when it is subjected to a magnetic field, or an induced magnetization when in an electric field. The term “magnetoelectric” was first used by Debye and was marked by two major independent discoveries [1]. The first in discovery by Röntgen in 1888 was that a moving dielectric became mag-netized when an electric field was applied to it [2]. The second discovery was the realization of the possibility of intrinsic ME behavior of stationary crystals on the basis of symmetry con-siderations by Pierre Curie in 1894 [3]. The reverse effect of the first event, i.e., the polarization of a moving dielectric in a mag-netic field, was observed 17 years after the discovery in 1888. Even though Curie recognized that symmetry was an important issue in ME behavior, it took many decades to demonstrate that the ME response was allowed only in time-asymmetric media. However, after the two discoveries that formed the founda-tion of ME effect, no work was done until 1958, when Landau and Lifshitz verified the possibility of the ME effect in certain crystals on the basis of the crystal symmetry [4]. In 1959, the symmetry argument was applied by Dzyaloshinskii to an anti-ferromagnetic Cr2O3 [5] to show the violation of time-reversal symmetry for this particular system and suggesting that the ME effect can be seen in Cr2O3. In 1960 and 1961, Astrov confirmed this experimentally by measuring the electrically induced ME effect in Cr2O3 for the temperature range of 80–330 K [6].
In materials that are magnetoelectric, the induced polariza-tion P is related to the field H by the expression, P = αH, where α is the second rank ME-susceptibility tensor and is expressed in the units of s/m in SI units (in Gaussian units α = 4πP/H = 4πM/E is dimensionless). One generally determines α by mea-suring Pz for an applied field H along the z-axis. Another param-eter of importance is the ME voltage coefficient αE = E/H which
is related to α by the expression α = εoεrαE, where εr is the relative permittivity of the material. Several single-phase materials that are either antiferromagnetic, weak ferromagnetic, or ferrimag-netic, such as Cr2O3, TbFeO3, and FexGa2−xO3, show rather weak ME effects [7–12].
6.2 ME Effects in Composites
Composites are of interest for the engineering of materials either with desired properties, or new characteristics that are absent in single-phase materials (for details see [13,14]). Composite materials can be divided into two categories as proposed by van Suchtelen [15]: (a) sum properties and (b) product properties. A sum property of a composite is the weighted sum of the con-tributions from the constituent phases that is proportional to the volume or weight fractions of these phases. Physical quantities such as density and resistivity are examples of sum properties. For product properties, consider a composite material with two component phases. The first phase has a property A → B with a proportionality tensor dB/dA = X; and the second phase has a property B → C with a proportionality tensor dC/dB = Y. Then the composite will have the property A → C with a proportional-ity tensor dC/dA = Z, where Z = Y ∙ Z; hence, the name “product property.” The product property is achieved in a composite but not seen in the individual phases.
An ME composite can be realized with piezomagnetic and piezoelectric components [15–48]. When a magnetic field is applied to a piezomagnetic material, it is strained; the strain in turn causes stress on the piezoelectric material, which becomes electrically polarized as a consequence. The reverse effect is also feasible; but the consequence here will be magnetization of piezomagnetic material. Such ME composites could also be made with magnetostrictive and piezoelectric phases and theories predict ME voltage coefficients as high as 3.2 V/cm Oe [22–25].
6Magnetoelectric Interactions
in Multiferroic Nanocomposites
6.1 Introduction .............................................................................................................................6-16.2 ME Effects in Composites ......................................................................................................6-16.3 Device Applications for ME Composites .............................................................................6-36.4 Theory of Low-Frequency ME Coupling in a Nanobilayer ...............................................6-36.5 Conclusion ................................................................................................................................6-6References .............................................................................................................................................6-6
Vladimir M. PetrovOaklandUniversity
and
NovgorodStateUniversity
Gopalan SrinivasanOaklandUniversity
6-2 HandbookofNanophysics:FunctionalNanomaterials
Most attempts in the past to realize strong ME effects were on composites of magnetostrictive cobalt ferrite and piezoelec-tric barium titanate or lead zirconate titanate were unsuccess-ful [21]. Van den Boomgaard first synthesized composites of CoFe2O4 and BaTiO3 by two methods: sintering and unidirec-tional solidification of eutectic melts [16–19]. Both composites developed microcracks due to thermal expansion mismatch and yielded ME coefficients that were a factor 40–60 smaller than calculated values.
Harshe, Dougherty, and Newnham, in their pioneering work on ME composites (a) proposed a theoretical model for multi-layer heterostructures with alternating layers of magnetostric-tive and piezoelectric phases and (b) fabricated such structures [20,21]. A multilayer structure is expected to be far superior to bulk composites for the following reasons: (a) An essential con-dition for maximizing ME effects is a large dielectric constant (and piezoelectricity). In bulk composites, the leakage currents due to low-resistivity cobalt ferrite inclusions reduce the overall dielectric constant. A similar decrease in the dielectric constant of piezoelectric layers is also possible in a multilayer composite. But such undesirable effects can easily be eliminated by short-ing electrically the magnetostrictive layers. (b) The piezoelectric layer in a layered structure can easily be poled electrically to fur-ther enhance the piezoelectricity. They fabricated the ML struc-tures by sintering tape-cast ribbons [21]. The ME coefficients did show an improvement over bulk sintered composites, but were a factor of 2–30 smaller than the theoretical values. This could be due to (a) incompatible structural and thermal properties of the two phases leading to poor coupling at the interface; (b) further weakening of the mechanical bonding due to platinum electrodes at the interface, which makes the composite a trilayer structure; (c) possible hexagonal hard magnetic phases at the interface due to high temperature processing; and (d) porosity. But with proper choice for magnetostrictive and piezoelectric phases and a multilayer composite geometry, a much higher α-value has been reported in recent studies [23–25].
A composite with alternate layers of magnetostrictive and piezoelectric phases is called a 2–2 composite since the two phases have mechanical connectivity only in the plane of the layers. Composites of interest in recent years have been NiFe2O4 (NFO) or CoFe2O4 (CFO) with lead zirconate titanate (PZT), terfenol-D and PZT, and similar systems [21–33]. Samples of NFO-PZT and CFO-PZT are made by sintering thick films obtained by tape-casting. Thin disks of the samples are polar-ized with an electric field perpendicularly to its plane. The ME coefficient αE = δE/δH, where δH is the applied ac magnetic field and δE is the measured ac electric field, and is measured with a setup as shown in Figure 6.1 and for two conditions: (a) trans-verse or αE,31 for bias magnetic H and ac field δH parallel to each other and to the disk plane (1,2) and perpendicular to δE (direc-tion-3) and (b) longitudinal or αE,33 for all the three fields paral-lel to each other and perpendicular to sample plane.
Figure 6.2 shows the static magnetic field dependence of the transverse ME coefficient αE,31 and the longitudinal coefficient αE,33 for a composite of nickel ferrite (NFO) and lead zirconate
titanate (PZT) [24]. The data at room temperature are for a fre-quency of 1 kHz and for unit thickness of the piezoelectric phase. As H is increased from zero, αE,31 increases, reaches a maximum value of 400 mV/cm Oe at 300 Oe, and then drops rapidly to zero above 2 k Oe. We observed a phase difference of 180° between the induced voltages for +H and −H. Similarly, the longitudinal coefficient αE,33 for all the fields along direction-3 has a relatively small magnitude compared to the transverse case and peaks at a much higher H. These observations can be understood in terms of the demagnetizing field. The magnitude and the field depen-dence in Figure 6.2 are related to the variation of magnetostric-tion λ with H. The coefficients are directly proportional to the piezomagnetic coupling q = dλ/dH, and the H-dependence of αE tracks the slope of λ vs. H. The saturation of λ at high field leads to αE = 0. We succeeded in achieving giant ME coefficients on the order of 400–1500 mV/cm Oe predicted by theory [24].
The variation of αE,31 with frequency and temperature are also shown in Figure 6.2. Upon increasing the frequency from 20 Hz to 10 kHz, one observes an overall increase of 25% in the ME voltage coefficient, but a substantial fraction of the increase occurs over the 1–10 kHz range (except at 353 K). These varia-tions are most likely due to frequency dependence of the dielec-tric constant for the constituent phases and the piezoelectric coefficient for PZT. Data on temperature dependence of αE,31 at 100 Hz are shown in Figure 6.2. A peak in αE,31 is observed at room temperature and it decreases when T is increased from room temperature.
Composite
Helmholtzcoils
12
Electromagnet
FerromagneticPiezoelectric
FIGURE 6.1 Schematic diagram showing a setup for the measurements of low-frequency ME effects in a ferromagnetic–piezoelectric composite. The sample is initially poled in an electric field and subjected to a bias magnetic field H and ac magnetic field dH. The ac electric field δE gener-ated due to ME coupling is measured across the piezoelectric layer.
The ME effect has provided a tool for converting energy from electric to magnetic form or vice versa. In 1948, Tellegen suggested a gyrator using an ME material even before the indi-cation of any ME material [47]. In 1965, O’Dell proposed an ME memory device application [48]. Other possible applications of the ME materials have been proposed by Wood and Austin in 1975 including devices for (a) modulation of amplitudes, polar-izations and phases of optical waves, (b) ME data storage and switching, (c) optical diodes, (d) spin-wave generation, (e) ampli-fication, and (f) frequency conversion [47]. Recently magneto-electric magnetic field sensors have been demonstrated for pico Tesla fields [29].
Other ME phenomena of fundamental and technological interests are the coupling when the electrical or the magnetic subsystem shows a resonant behavior, i.e., electromechanical resonance (EMR) for PZT and ferromagnetic resonance (FMR) for the ferrite. The resonance ME effect is similar in nature to the standard effect, i.e., an induced polarization under the action of an ac magnetic field. But the ac field here is tuned to the electro-mechanical resonance frequency. As the dynamic magnetostric-tion is responsible for the electromagnetic coupling, EMR leads to significant increase in the ME voltage coefficients [44,45]. Measurements of ME effects at EMR on ferrite-PZT show a two to three orders of magnitude increase in αE compared to low frequency values. We developed the theory for ME coupling at EMR at radial and thickness modes for a ferromagnetic–piezo-electric bilayer [44]. There are several device possibilities based on resonance ME effects, such as dual electric and magnetic field tunable microwave and millimeter wave resonators, filters, and phase shifters [49,50].
6.4 theory of Low-Frequency ME Coupling in a Nanobilayer
The main focus of this chapter is the ME effect in nanobilayers. Nanostructures in the shape of wires, pillars, and films are important for increased functionality in miniature devices. Theories for low frequency ME effects in magnetostrictive-piezoelectric nanostructures were reported recently [46]. The modeling was based on the homogeneous longitudinal strain approach. However, configurational asymmetry of a bilayer implies the presence of a bending strain in the sample in an applied magnetic or electric field. The principal objective of present work is modeling of the ME interaction in a magne-tostrictive-piezoelectric nanobilayer on a substrate, taking into account such bending or flexural strains. We concentrate on ME effect in low frequency. Cobalt ferrite (CFO)–barium titanate grown on SrTiO3 substrate is chosen as the model sys-tem for numerical estimations. This nanobilayer was recently prepared by Zheng [30]. The ME voltage coefficients αE have been calculated for field orientations that provide minimum demagnetizing fields and maximum αE. The effect of substrate clamping has been described in terms of dependence of αE on substrate dimensions. The ME voltage coefficient is shown to drop with increase in substrate volume. Variations in piezo-electric and piezomagnetic coefficients and permittivity of components due to lattice mismatch are taken into account. We consider here only piezoelectric and ferrite films with thickness much larger than the ferroelectric and ferromagnetic correlation length for both phases. Under these conditions, the size effects may be neglected.
For piezoelectric and magnetostrictive phases and substrate, the following equations can be written for the strain, electric, and magnetic displacements:
400
200
Mag
neto
elec
tric
vol
tage
coef
ficie
nt (m
V/cm
Oe)
–200
–400–2000 –1000
Longitudinal
Transverse
Static magnetic field H (Oe)0
Frequency (Hz)
1000
500450400350300
10 100 1000
353 K313 K
T = 293 K
333 K
104
250200
α E,31
(mV/
cm O
e)
2000
0
FIGURE 6.2 Representative data on bias field H dependence of transverse and longitudinal ME voltage coefficients for a layered sample of NFO and PZT. The inset shows the frequency and temperature dependence of the transverse ME coefficient.
6-4 HandbookofNanophysics:FunctionalNanomaterials
p p p p p
p p p p p
m m m m m
m
S s T d E
D d T E
S s T q H
i ij j ki k
k ki i kn n
i ij j ki k
= +
= +
= +
ε
BB q T H
S s T
k ki i kn n
i ij j
= +
=
m m m m
s s s
µ
(6.1)
whereSi and Tj are strain and stress tensor componentsEk and Dk are the vector components of electric field and elec-
tric displacementHk and Bk are the vector components of magnetic field and
magnetic inductionsij, qki, and dki are compliance, piezomagnetic, and piezoelec-
tric coefficientsεkn is the permittivity matrixμkn is the permeability matrixthe superscripts p, m, and s correspond to piezoelectric and
piezomagnetic phases and substrate, respectively.
We assume the symmetry of piezoelectric to be ∞m and that of piezomagnetic to be cubic. As shown in Figure 6.3, xm, xp, and xs are the neutral axes that are located at the horizontal mid-plane of piezomagnetic, piezoelectric, and substrate lay-ers, respectively, and separated by distances hm and hp. The thickness of the plate is assumed small compared to remaining dimensions.
Assuming that the longitudinal axial strains of each layer are linear functions of the vertical coordinate zi, it can be shown that
m m m
p p p
s s s
m m m
p p
S S zR
S SzR
S S zR
S S zR
S S
1 101
1 101
1 101
2 202
2
= +
= +
= +
= +
= 2202
2 202
+
= +
zR
S S zR
p
s s s
(6.2)
whereiS10 and iS20 are the centroidal strains along the x- and y-axes
at zi = 0R1 and R2 are the radiuses of curvature
From geometric considerations, it can be shown that
m p m
p s p
m p m
p s p
S S hR
S ShR
S S hR
S ShR
10 101
10 101
20 202
20 202
− =
− =
− =
− =
(6.3)
For the transverse field orientation (ac electric field perpendicular to the sample plane and ac magnetic and bias fields in the sample plane) providing the minimum demagnetizing fields, Equation 6.2 can then be rewritten using Equations 6.3 and 6.1 as
m m m m m m m m
m p m p p p
S zR
s T s T q H
S z hR
s T s
101
11 1 12 2 11 1
101
11 1
+ = + +
+−
= +( )
112 2 31 3
101
11 1 12 2
202
p p p
m p m p s s s s
m m m
T d E
S z h hR
s T s T
S zR
+
+− −
= +
+ =
( )
ss T s T q H
Sz hR
S T S T d
12 1 11 2 12 1
202
12 1 11 2 31
m m m m m
m p m p p p p p
+ +
+−
= + +( ) pp
m p m p s s s s
E
Sz h h
Rs T s T
3
202
12 1 11 2+− −
= +( )
(6.4)
The axial forces in the three layers must add up to zero to preserve force equilibrium, that is,
mm
pp
ss
m
m
m
p
p
s
s
m
T dz T dz T dz
T
t
t
t
t
t
t
t
1
2
2
1
2
2
1
2
2
2
0− − −
−
∫ ∫ ∫+ + =/
/
/
/
/
/
//
/
/
/
/
/
2
2
2
2
2
2
2
2
0m
p
p
s
s
mp
ps
s
t
t
t
t
t
dz T dz T dz∫ ∫ ∫+ + =− −
(6.5)
SrTiO3
xs
xp
xm
st
pthp
hmmt
zs
ys
yp
ym
zp
zm
PZT
NFO
FIGURE 6.3 Schematic diagram showing an NFO PZT bilayer on a strontium titanate substrate.
where mt, pt, and st are the thicknesses of piezomagnetic, piezo-electric, and substrate layers.
Further, Equation 6.4 should be solved for iTj and substi-tuted into Equation 6.5. Using Equation 6.5 and taking into account Equations 6.2 through 6.4 and 6.6 enables finding mS10 and mS20:
m m m p p ms
s m pS s V Y q H V Y d E hR
V Yh h
R10 1 11 1 31 31 1
1= − + +
++
( )
= − + +
++
m m m p p m
ss m pS s V Y q H V Y d E h
RV Y
h hR20 1 12 1 31 3
2 21( )
(6.6)
whereE3 and H1 are electric field induced across the piezoelectric
layer and applied magnetic fields1 = t (mt mY + pt pY + st sY)−1
V = pt/tVs = st/tt = mt + ptmY, pY, and sY are the modules of elasticity of piezomagnetic,
piezoelectric components, and substrate, respectively.
To conserve moment equilibrium, the rotating moments of axial forces in the three layers are counteracted by resultant bending moments Mmj, Mpj, and Msj, induced in piezomagnetic, piezo-electric, and substrate layers. That is,
F h + F h + h = M + M + M
F h + F h + h = M + M + M
m1 m p1 m p m1 p1 s1
m2 m p2 m p m2 p2 s2
( )
( ) (6.7)
where
F T dz F T dz M z Tii
t
t
ii
t
t
i ii
t
t
i
i
i
i
i
i
1 1
2
2
1 2 2
2
2
1 1 1
2
= = =− − −∫ ∫
/
/
/
/
/
, ,//
/
/
,2
1
2 2
2
2
1
∫
∫=−
dz
M z T dzi ii
t
t
i
i
and
Taking into account Equations 6.3, 6.4, and 6.6, the equilibrium condition (6.7) can be solved for R1 and R2. The expressions for R1 and R2 are not given here because of their complicated nature. The values of these radii of curvature can then be sub-stituted into Equation 6.6 to obtain the centroidal strains. Once the centroidal strains are determined, the axial stress iT1 can be found from Equation 6.4. To obtain the expression for ME voltage coefficient, we use the open circuit condition on the boundary:
D3 0= . (6.8)
Since electric induction is divergence free and has only one com-ponent, it is evident that D3 is equal to zero for any z. In this case, Equations 6.1 and 6.8 result in the expression for ME voltage coefficient
α
εE31
p p p
pp
p
= = −+
−∫EH
d T T dz
t Ht
t
3
1
31 1 22
2
1 33
( )/
/
(6.9)
where pT1 and pT2 are determined by Equation 6.4 taking into account Equation 6.6.
In case of small flexural strain, the radius of curvature in Equations 6.3, 6.4, and 6.6 must tend to infinity. It is easy to show that expression for ME voltage coefficient reduces in this case to well-known expression, which was obtained with the assump-tion of homogeneous longitudinal strains [46].
It is well known that the lattice mismatch between the sub-strate and piezoelectric layers results in variation of piezoelec-tric coefficients and permittivity. This variation can be found using the Landau–Ginsburg–Devonshire phenomenological thermodynamic theory [51]. According to this approach, the thermodynamic potential G′ of a thin film on a thick substrate is defined as [52]
′ = + + +G G S S S1 1 2 2 6 6σ σ σ (6.10)
whereG is the elastic Gibbs function for the barium titanate layer
without substrateS1, S2, and S6 are in-plane strains at the film/substrate inter-
face arising from lattice mismatchσ1, σ2, and σ6 are stress components
In case of a cubic (001) substrate, S6 = 0 and S1 = S2 = Sm, where the misfit strain Sm = (b − a0)/b can be calculated using the substrate lattice parameter b and the equivalent cubic cell con-stant a0 of the freestanding film. For the barium titanate film on SrTiO3 substrate a0 = 0.397 nm and b = 0.393 nm. Using these equations with parameters of the Gibbs function enables determining the dielectric constant and piezoelectric coef-ficients of PZT: pε11/ε0 = 51, pd31 = −18 pm/V [52]. Similarly, the piezomagnetic parameters of the NFO film should be less compared to bulk NFO. Nevertheless, in the NFO/PZT/STO composite film, the bottom PZT layer acts as a buffer layer and effectively reduces constrains from the STO substrate and compressive strains in the NFO layer are almost released. In what follows, we neglect the contribution of residual strains on NFO parameters arising from lattice mismatch. Substituting the appropriate material property values and the dimensions of the structure into Equation 6.9, we can find the ME volt-age coefficient with regard to axial and bending stresses for the bilayer on a substrate.
6-6 HandbookofNanophysics:FunctionalNanomaterials
Next, we apply the model to the case of NFO-PZT on STO substrates. Material parameters used for calculations are listed in Table 6.1. Figure 6.4 shows the PZT volume fraction v dependence of the ME voltage coefficient. For a freestanding bilayer, graph of αE,31 vs. v is a double-peaked curve. Such a dependence structure is stipulated by a decrease in the longitudinal strain of PZT due
to bending strain in the sample. Figure 6.4 also shows the same dependence for the case when we neglect the flexural strains. That is the case for a freestanding trilayer of NFO-PZT-NFO or PZT-NFO-PZT. This dependence coincides with our earlier model [46]. Our model predicts a decrease in the ME voltage coefficient for the freestanding bilayer compared to the case when neglecting the flexural strains. Placing the bilayer on a substrate of equal thick-ness gives rise to an increase of ME output due to sign reversal in the contribution of flexural strain to the total strain of PZT in comparison with the freestanding bilayer.
Figure 6.5 shows the ME voltage coefficient as a function of sub-strate-to-bilayer thickness ratio. Estimates based on this model and our previous model [42] are compared. Increase in the substrate thickness leads to a substantial decrease in ME coupling, as seen in Figure 6.5. However, the rate of change of αE,31 is considerably lower than that for the case when neglecting the flexural strains.
6.5 Conclusion
A brief review of ME interactions in layered composites is pro-vided. We then extended the discussion to include low-frequency ME interactions in nanocomposites, with specific focus on a fer-rite-piezoelectric bilayer on a dielectric substrate. Expressions have been obtained for the ME voltage coefficients, taking into consideration the substrate clamping effects.
references
1. P. Debye, Bemerkung zu einigen neuen Versuchen uber einen magneto-elektrischen Richteffekt, Z. Phys. 36, 300 (1926).
2. W. C. Rontgen, XLI. Experiments on the electromagnetic action of dielectric polarization, Phil. Mag. 19, 385 (1885).
3. P. Curie, Sur la symetrie dans les phenomenes physiques, J. Phys. 3, 393 (1894).
4. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, U.K. (1960), p. 119. (Translation of Russian Edition, 1958.)
5. I. E. Dzyaloshinskii, On the magneto-electrical effect in antiferromagnets, Sov. Phys. JETP 10, 628 (1959).
6. D. N. Astrov, Magnetoelectric effect in chromium oxide, Sov. Phys. JETP 13, 729 (1961).
7. G. T. Rado and V. J. Folen, Observation of magnetically induced magnetoelectric effect and evidence for antiferro-magnetic domains, Phys. Rev. Lett. 7, 310 (1961); S. Foner and M. Hanabusa, Magnetoelectric effects in Cr2O3 and (Cr2O3)0.8 (Al2O3)0.2, J. Appl. Phys. 34, 1246 (1963).
TABLE 6.1 Material Parameters (Compliance Coefficient s, Piezomagnetic Coupling q, Piezoelectric Coefficient d, and Permittivity ε) for NFO, PZT, and SrTiO3 Used for Theoretical Estimates
FIGURE 6.4 PZT volume fraction dependence of ME voltage coefficient αE,31 as a function of substrate-to-bilayer thickness ratio R = st/(pt + mt).
0 2 4 6 8 100.00
0.05
0.10
0.15
0.20
0.25
ME
volta
ge co
e�ci
ent (
V/cm
Oe)
Thickness ratio R
Current model
Previous model
FIGURE 6.5 Dependence of ME voltage coefficient αE,31 on substrate-to-bilayer thickness ratio R for PZT volume fraction of v = 0.6. The estimates based on present model and the previous model of Ref. [46] are shown.
8. E. F. Bertaut and M. Mercier, Magnetoelectricity in theory and experiment, Mater. Res. Bull. 6, 907 (1971).
9. G. T. Rado, Linear magnetoelectric effects in gallium iron oxide at low temperatures and in high magnetic fields, J. Appl. Phys. 37, 1403 (1966).
10. E. Kita, S. Takano, A. Tasaki, K. Siratori, K. Kohn, and S. Kimura, Low temperature phase of yttrium iron garnet (YIG) and its first-order magnetoelectric effect, J. Appl. Phys. 64, 5659 (1988).
11. E. Fischer, G. Gorodetsky, and R. M. Hornreich, A new family of magnetoelectric materials: A2M4O9 (A = Ta,Nb; M = Mn,Co), Solid State Commun. 10, 1127 (1972).
12. H. Tsujino and K. Kohn, Magnetoelectric effect of GdMn2O5 single crystal, Solid State Commun. 83, 639 (1992).
13. M. D. Sacks, Advanced Composite Materials: Processing, Microstructures, Bulk and Interface Properties, Characterization Methods, and Applications Proceedings, vol. 19. American Ceramic Society, Westerville, OH (1991).
14. K. V. Logan, Z. A. Munir, and R. M. Spriggs, Advanced Synthesis and Processing of Composites and Advanced Ceramics II, Ceramic Transactions, vol. 79. American Ceramic Society, Westerville, OH (1997).
15. J. van Suchtelen, Product properties: A new application of composite materials, Philips Res. Rep. 27, 28 (1972).
16. J. van den Boomgaard, D. R. Terrell, and R. A. J. Born, An in situ grown eutectic magnetoelectric composite mate-rial: Part 1: Composition and unidirectional solidification, J. Mater. Sci. 9, 1705 (1974).
17. A. M. J. G. van Run, D. R. Terrell, and J. H. Scholing, An in situ grown eutectic magnetoelectric composite material: Part 2: Physical properties, J. Mater. Sci. 9, 1710 (1974).
18. J. van den Boomgaard, A. M. J. G. van Run, and J. van Suchtelen, Piezoelectric-piezomagnetic composites with magnetoelectric effect, Ferroelectrics 14, 727 (1976).
19. J. van den Boomgaard and R. A. J. Born, A sintered magne-tostrictive composite material BaTiO3-Ni(Co,Mn)Fe2O4, J. Mater. Sci. 13, 1538 (1978).
20. G. Harshe, J. P. Dougherty, and R. E. Newnham, Theoretical modelling of multilayer magnetoelectric composites, Int. J. Appl. Electromag. Mater. 4, 145 (1993); M. Avellaneda and G. Harshe, Magnetoelectric effect in piezoelectric/ magne-tostrictive multilayer (2–2)composites, J. Intell. Mater. Syst. Struct. 5, 501 (1994).
21. G. Harshe, Magnetoelectric effect in piezoelectric-magne-tostrictive composites, PhD thesis, The Pennsylvania State University, College Park, PA, 1991.
22. Ce-Wen Nan and D. R. Clarke, Effective properties of ferroelectric and/or ferromagnetic composites: A unified approach and its application, J. Am. Ceram. Soc. 80, 1333 (1997).
23. G. Srinivasan, E. T. Rasmussen, B. J. Levin, and R. Hayes, Magnetoelectric effects in bilayers and multilayers of mag-netostrictive and piezoelectric perovskite oxides, Phys. Rev. B 65, 134402 (2002).
24. G. Srinivasan, E. T. Rasmussen, J. Gallegos, R. Srinivasan, Yu. I. Bokhan, and V. M. Laletin, Magnetoelectric bilayer and multilayer structures of magnetostrictive and piezo-electric oxides, Phys. Rev. B 64, 214408 (2001).
25. G. Srinivasan, E. T. Rasmussen, and R. Hayes, Magnetoelectric effects in ferrite-lead zirconate titanate lay-ered composites: Studies on the influence of zinc substitu-tion in ferrites, Phys. Rev. B 67, 014418 (2003).
26. N. A. Spaldin and M. Fiebig, The renaissance of magneto-electric multiferroics, Science 309, 391 (2005).
27. H. Schmid, Introduction to Complex Mediums for Optics and Electromagnetics, Eds. W. S. Weiglhofer and A. Lakhtakia, SPIE Prsee, Bellingham, WA (2003), pp. 167–195.
28. M. Fiebig, Revival of the magnetoelectric effect, J. Phys. D: Appl. Phys. 38, R123 (2005).
29. Ce-Wen Nan, M. I. Bichurin, S. Dong, D. Viehland, and G. Srinivasan, Multiferroic magnetoelectric composites: Historical perspective, status and future directions, J. Appl. Phys. 103, 031101 (2008).
30. H. Zheng, J. Wang, S. E. Lofland, Z. Ma, L. Mohaddes-Ardabili, T. Zhao, L. Salamanca-Riba et al., Multiferroic BaTiO3-CoFe2O4 nanostructures, Science 303, 661–663 (2004).
31. H. Zheng, J. Wang, L. Mohaddes-Ardabili, M. Wuttig, L. Salamanca-Riba, D. G. Schlom, and R. Ramesh, Three-dimensional heteroepitaxy in self-assembled BaTiO3-CoFe2O4 nanostructures, Appl. Phys. Lett. 85, 2035 (2004).
32. J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland, Detection of pico-Tesla magnetic field using magneto-electric sen-sors at room temperature, Appl. Phys. Lett. 88, 062510 (2006).
33. J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland, Magnetoelectric laminate composites: An overview, J. Am. Ceram. Soc. 91, 351 (2008).
34. Z. P. Xing, J. Y. Zhai, S. X. Dong, J. F. Li, and D. Viehland, Modeling and detection of quasi-static nanotesla magnetic field variations using magnetoelectric laminate sensors, Meas. Sci. Technol. 19, 015206 (2008).
35. S. Priya, R. Bergs, and R. A. Islam, Magnetic field anomaly detector using magnetoelectric composites, J. Appl. Phys. 101, 024108 (2007).
36. R. A. Islam, Y. Ni, A. G. Khachaturyan, and S. Priya, Giant magnetoelectric effect in sintered multilayered composite structures, J. Appl. Phys. 104, 044103 (2008).
37. M. Liu, X. Li, J. Lou, S. Zheng, K. Dui, and N. X. Sun, A modified sol-gel process for multiferroic nanocompos-ites, J. Appl. Phys. 102, 083911 (2007).
38. Y. Chen, J. Wang, M. Liu, N. X. Sun, C. Vittorai, and V. Harris, Giant magnetoelectric coupling and E-field ten-ability in a laminate Ni2MnGa/PMN-PT multiferroic het-erostructure, Appl. Phys. Lett. 93, 112502 (2008).
39. J. Zhai, S. Dong, Z. Xing, J. Li, and D. Viehland, Geomagnetic sensor based on magnetoelectric effect, Appl. Phys. Lett. 91, 123513 (2007).
6-8 HandbookofNanophysics:FunctionalNanomaterials
40. J. Ryu, A. V. Carazo, K. Uchino, and H. Kim, Piezoelectric and magnetoelectric properties of lead zirconate titanate/Ni-ferrite particulate composites, J. Electroceram. 7, 17 (2001).
41. K. Mori and M. Wuttig, Magnetoelectric coupling in terfenol-D/polyvinylidenedifluoride composites, Appl. Phys. Lett. 81, 100 (2002).
42. S. Dong, J. F. Li, and D. Viehland, Giant magnetoelectric effect in laminate composites, Philos. Mag. Lett. 83, 769 (2003).
43. S. Dong, J. Zhai, Z. Xing, J. F. Li, and D. Viehland, Extremely low frequency response of magnetoelectric multilayer com-posite, Appl. Phys. Lett. 86, 102901 (2005).
44. M. I. Bichurin, D. A. Fillipov, V. M. Petrov, U. Laletsin, and G. Srinivasan, Resonance magnetoelectric effects in layered magnetostrictive-piezoelectric composites, Phys. Rev. B 68, 132408 (2003).
45. U. Laletsin, N. Paddubnaya, G. Srinivasan, and C. P. DeVreugd, Frequency dependence of magnetoelectric interactions in layered structures of ferromagnetic alloys and piezoelectric oxides, Appl. Phys. A 78, 33 (2004).
46. V. M. Petrov, G. Srinivasan, M. I. Bichurin, and A. Gupta, Theory of magnetoelectric effects in ferrite-piezoelectric nanocomposites, Phys. Rev. B 75, 224407, (2007).
47. V. E. Wood and A. E. Austin, Possible applications for magnetoelectric materials, Proceeding of Symposium on Magnetoelectric Interaction Phenomena in Crystals, Seattle, WA, May 21–24, 1973, eds. A. J. Freeman and H. Schmid, Gordon & Breach Science Publishers, New York (1975), p. 181.
48. T. H. O’Dell, The Electrodynamics of Magneto-Electric Media, North-Holland Pub. Co., Amsterdam, the Netherlands (1970).
49. G. Srinivasan and Y. K. Fetisov, Microwave magnetoelectric effects and signal processing devices, Integr. Ferroelectr. 83, 89 (2006).
50. M. I. Bichurin, D. Viehland, and G. Srinivasan, Magnetoelectric interactions in a ferromagnetic-piezo-electric layered structures: Phenomena and devices, J. Electroceram. 19, 243 (2007).
51. N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Effect of mechanical boundary conditions on diagrams of epitaxial ferroelectric thin films, Phys. Rev. Lett. 80, 1988 (1998).
52. N. A. Pertsev, V. G. Kukhar, H. Rohlstedt, and R. Waser, Phase diagrams and physical properties of single domain epitaxial Pb (Zr1−xTix)O3 thin films, Phys. Rev. B 67, 054107 (2003).
