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In this contribution, we analyze the properties of two-phase magneto-electric (ME) composites. Such ME composite materials have raised scientific attention in the last decades due to many possible applications in a wide range of technical devices. Since the effective magneto-electric properties significantly depend on the microscopic morphology, we investigate in more detail the changes in the in-plane polarization due to an applied magnetic field. It was shown in previous works that it is possible to grow vertically aligned nanopillars of magnetostrictive cobalt ferrite in a piezoelectric barium titanate matrix by pulsed laser deposition. Based on x-ray linear dichroism, the displacements of titanate ions in the matrix material can be measured due to an applied magnetic field near the boundary of the interface between the matrix and the nanopillars. Here, we focus on (1–3) fiber-induced composites, based on previous experiments, where cobalt ferrite nanopillars are embedded in a barium titanate matrix. In the numerical simulations, we adjusted the boundary value problem to match the experimental setup and compare the results with previously made assumptions of the in-plane polarizations. A further focus is taken on the deformation behavior of the nanopillar over its whole height. Such considerations validate the assumption of the distortion of the nanopillars under an external magnetic field. Furthermore, we analyze the resulting magneto-electric coupling coefficient.
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Arch Appl Mech
https://doi.org/10.1007/s00419-019-01534-z
ORIGINAL
Matthias Labusch ·Veronica Lemke ·Carolin Schmitz-Antoniak ·Jörg Schröder ·
Samira Webers ·Heiko Wende
FEM analysis of a multiferroic nanocomposite: comparison
of experimental data and numerical simulation
Received: 9 June 2018 / Accepted: 22 February 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this contribution, we analyze the properties of two-phase magneto-electric (ME) composites. Such
ME composite materials have raised scientific attention in the last decades due to many possible applications
in a wide range of technical devices. Since the effective magneto-electric properties significantly depend on
the microscopic morphology, we investigate in more detail the changes in the in-plane polarization due to an
applied magnetic field. It was shown in previous works that it is possible to grow vertically aligned nanopillars
of magnetostrictive cobalt ferrite in a piezoelectric barium titanate matrix by pulsed laser deposition. Based
on x-ray linear dichroism, the displacements of titanate ions in the matrix material can be measured due to
an applied magnetic field near the boundary of the interface between the matrix and the nanopillars. Here, we
focus on (1–3) fiber-induced composites, based on previous experiments, where cobalt ferrite nanopillars are
embedded in a barium titanate matrix. In the numerical simulations, we adjusted the boundary value problem
to match the experimental setup and compare the results with previously made assumptions of the in-plane
polarizations. A further focus is taken on the deformation behavior of the nanopillar over its whole height.
Such considerations validate the assumption of the distortion of the nanopillars under an external magnetic
field. Furthermore, we analyze the resulting magneto-electric coupling coefficient.
Keywords Magneto-electro-mechanical coupling ·FEM analysis ·Nanopillars ·Multiferroics
1 Introduction
Ferromagnetic and ferroelectric materials are used in all kinds of modern technology. Since many of these
materials additionally show a coupling with ferroelasticity, which means that they change their shape due to an
applied magnetic and electric field, they can transform magnetic and electric signals in deformations and are
therefore used in sensor and actuator devices. A combination of ferromagnetism and ferroelectricity, resulting
in a magneto-electric (ME) coupling, enables the improvement in many technical applications. Examples are
M. Labusch (B
)·V. Le m k e ·J. Schröder
Department of Civil Engineering, Institute of Mechanics, Faculty of Engineering, University Duisburg-Essen, Universitätsstr. 15,
45141 Essen, Germany
E-mail: matthias.labusch@uni-due.de
Tel: +49 201 183 2679
Fax: +49 201 183 2680
C. Schmitz-Antoniak
Research Center Jülich, Peter-Grünberg-Institute (PGI-6), 52425 Jülich, Germany
S. Webers ·H. Wende
Faculty of Physics/Experimental Physics - AG Wende and Center for Nanointegration Duisburg-Essen (CENIDE), University
Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany
M. Labusch et al.
electric-write/magnetic-read-memories (Magneto-Electric Random Access Memory; MERAM) or electrical
magnetic field sensors, see [10,35]or[3]. Materials which are characterized by two or more ferroic properties
are called multiferroic. Magneto-electric multiferroics have been investigated comprehensively for example
in the papers of [7,13,17,22,23,26,29,39,42]or[43]. However, the magneto-electric coupling property of
single-phase ME materials becomes apparent mostly at a temperature which is far underneath the room tem-
perature. The influence of their crystallographic structure and characteristics of this phenomenon is explored
and explained in the works of [1,8,13,27,28,38]. Additionally, most of them have comparatively low coupling
coefficients, such as lithium manganese phosphate (LiCoPO4), bismuth-ferrite (BiFeO3) or chromium(III)-
oxide (Cr2O3), which therefore show boundaries in technical realizations, see [6] for the upper bound of the
linear magneto-electric coefficient. In order to use an enhanced ME property at room temperature, composite
materials consisting of magneto-mechanically and electro-mechanically coupled phases are used. Hereby, the
magneto-electric effect will be generated via the mechanical interaction of these phases as a strain-induced
product property. A product property can be explained as a composite property which is not available in the
particular phases but which manifests effectively in consequence of the synergy between the components, see
[37].
In the field of ME composites, two different magneto-electric effects can be differentiated, the direct and
the converse ME effect. On the one hand, the direct ME effect refers to an electric polarization evoked by an
applied magnetic field. The magnetic field causes a deformation of the ferromagnetic part of the composite
which is transmitted to the ferroelectric part of the composite. As a consequence, the strain yields a change
in the electric polarization in the electro-mechanically coupled phase. On the other hand, the converse ME
effect describes an electrically induced change in magnetization. Here, the applied electric field produces the
deformation of the ferroelectric phase which is transferred to the ferromagnetic phase. The strain induces a
magnetization in this material. Besides the impact of the different phase properties on the ME coupling, the
microscopic structure of the composite highly influences the effective properties. For example in the works
of [4,5,11,12,1416,19,24,25,34,36], analytical methods to determine the effective properties of composites
have been demonstrated.
In this contribution, we present a numerical characterization of a magneto-electric two-phase composite,
consisting of vertically aligned nanopillars of magnetostrictive cobalt ferrite in a piezoelectric barium titanate
matrix, which was synthesized by pulsed laser deposition. The synthesization and the numerical simulations
were performed by two different projects within a research group. Finite element simulations are performed for
the determination of magnetically induced in-plane polarizations on microscopic level and the determination
of the magneto-electric coupling coefficient. These results serve as a basis for the validation of a rough
estimation of the in-plane polarization. The underlying experimental setup is described in Sect. 2.Forthe
numerical characterization of this composite, the continuum mechanical formulations for the magneto-electro-
mechanically coupled boundary value problem are given in Sect. 3. In Sect. 4, the numerical simulations of
the nanocomposite are performed. Resulting in-plane polarizations and magneto-electric coupling coefficients
are compared with rough estimations. Finally, Sect. 5gives an outlook of the comparison between numerical
simulations and experimental measurements.
2 Experimental setup and estimation of effective properties
As it was shown in the publication of [40], it is possible to grow vertically aligned nanopillars of magnetostrictive
cobalt ferrite (CFO) in a piezoelectric barium titanate (BTO) matrix by pulsed laser deposition. Those nanoscale
vertical heterostructures are magneto-electrically coupled via stress–strain mediation. Due to the geometry,
they exhibit larger interfaces and thus the coupling between the ferromagnetic and ferroelectric phase is larger
compared to horizontally aligned heterostructures. The formation of the self-assembled nanostructures, see
Fig. 1a, can be obtained under certain conditions. By selecting the appropriate laser parameter and substrate
orientation, nanopillars can be grown in a matrix. Those CFO nanopillars in a BTO matrix were prepared by
a pulsed laser deposition system. The main setup of the pulsed laser deposition (PLD) system consists of a
ultra-high vacuum (UHV) chamber with a sample holder, a target rotation, a shutter and a quartz oscillator
for the thickness calibration. The biggest advantage of the PLD is the preparation of high-quality multi-layer
systems as well as stoichiometrically complicated composition of different materials, like in the paper of
[31] of self-organized CFO nanopillars in BTO matrix. For the ablation, the laser beam of an excimer laser
(COMPEXPro from Lambda Physik) providing ultraviolet radiation with a wavelength of 248 nm, a repetition
time of 10 Hz, a pulse energy of 100 mJ and a pulse duration of around 20 ns, is focused via a lens on the target.
FEM analysis of a multiferroic nanocomposite
(a)Scheme of CFO nanopillars in BTO matrix. (b) Topview scanning electron microscope
(SEM) image of the CFO nanopillars (bright
contrast
)
in BTO matrix
(
dark contrast
)
Fig. 1 a Schematic illustration and ba topview SEM image of CFO nanopillars in BTO matrix (with courtesy of [31])
A mixed target material consisting of 35 mol% CFO and 65 mol% BTO is used to provide those self-organized
structures. For oxide films, the ablation process occurs under an oxygen atmosphere in a range of 101mbar
and directly determines the energy of the ablated particles. To prepare the self-organized nanopillars on top of
the substrate, it can be heated up to enable the diffusion process and to optimize the deposition conditions.
For studying the magneto-electric coupling between the interfaces on the microscale, a combination of
x-ray magnetic circular dichroism (XMCD) at the Co edge and x-ray linear dichroism (XLD) at Ti edge can
be used.
Since the nanopillars are clamped to the strontium titanate substrate, an external magnetic field yields to an
asymmetric distortion of the nanopillars and, therefore, also in the BTO matrix, which is schematically shown
in Fig. 2(top). The deformation of the BTO matrix results in an electric polarization, which originates from the
displacement of the Ti4+ions along the crystallographic c-axis. That displacement changes the hybridization
3d-state of the Ti cations with the 2p states of the surrounded oxygen anions. The charge anisotropy and the
changes of the surrounding of the Ti atom can be studied by XLD at the Ti L3,2edge. The XLD originates
from the charge anisotropy in the vicinity of the Ti ions. This connects to the distortion of the Ti ions from
their symmetric positions and thereby links to a macroscopic electric polarization of the BTO matrix. The
investigation of the XLD shown in Fig. 3shows the following: If the magnetic field is aligned in the direction
of the nanopillars, the XLD signal vanishes (Fig. 3, left) demonstrating that there is no charge anisotropy
when comparing x-and y-directions. However, if the magnetic field is oriented perpendicular to the pillars a
clear XLD signal can be identified (Fig. 3, right). This demonstrates that a charge anisotropy for the x-andy-
directions is detected and hence an electric polarization can be concluded. For studying the charge anisotropy at
the surface, the magnetic field was applied perpendicular to the nanopillars and the XLD signal was measured
under vertically incident x-ray beam, see Fig. 3. For example, Fig. 2a shows the displacement of the nanopillars
with an external field parallel to the yaxis. Due to the magnetostriction along the field direction, the nanopillars
contract in y-direction and expand in x-direction. The deformation of the nanopillars is transferred to the BTO
matrix yielding into an XLD signal at the Ti L3,2edge. If the magnetic field is applied along the nanopillars in
z-direction, the electric polarization anisotropy can be canceled which is schematically illustrated in Fig. 2b.
Further, the x-ray absorption spectroscopy (XAS) reveals information about crystal field parameter, which
describes the splitting of Ti orbitals. The adaptation of the XAS spectra with multiplet calculations gain also
information on the displacement of the Ti ion and, therefore, the electric polarization. This electric polarization
from the XAS spectra is in the following compared to FEM analysis.
For a rough estimation of the in-plane component of the electric displacement from the magnetic field-
induced XLD measurements, the geometry of the compressed BTO matrix needs to be taken into account. The
starting point is schematically depicted in Fig. 2a. Under the neglection of the electric polarization varying at the
{110}interface and due to clamping effects, the BTO matrix is compressed in xand zdirection near the surface
by expanding CFO pillars under an applied magnetic field Hy. The resulting surface displacements uxand
M. Labusch et al.
x
y
z
y. z.
x
z
(c) along the (xz) plane
z
y
(a)Displacement of nanopillars with H (b) Displacement of nanopillars with H
(d)along the (yz) plane
Fig. 2 a,bSchematic picture of distorted nanopillars under an external magnetic field with different field orientations. c,d
Deformation of nanopillars and the transferred deformation to the BTO matrix to a magnetic field along the ydirection for two
different cross sections (adopted from [31])
uyare calculated with help of the in-plane magnetostriction of the adjoining CFO pillars λx=λ1·104
and λy=λ≈−2·104[41]:
ux=λxx0,uy=λyy0.(1)
In the following, we denote with 1,2,3thex,y,zdirection, respectively. In general, the average electric
displacement of the in-plane BTO can be calculated by
D1d113 σ13 and D2d223 σ23 (2)
with the piezoelectric coefficient d113 and d223, and the components of the Cauchy stress tensor σ13 and σ23.
By using the nanopillar height of z0400 nm and the surface displacement uxand uy, we can estimate the
electric displacement Dby the following equations:
Dx=d113 c1313
ux
z0
and Dy=d223 c2323
uy
z0
.(3)
For the tetragonal BTO with the piezoelectric coefficient d113 =d223 3.9·1010 mV1[2], the average
edge length of the CFO pillars of 2x0=2y0100 nm and c1313 =c2323 =1.1·1011 Nm2[9] an electric
displacement of Dx5.4·104Cm2and Dy1.1·103Cm2can be estimated.
FEM analysis of a multiferroic nanocomposite
Fig. 3 XLD measurement of displacement of Ti ions (with courtesy of [31]). (left) The magnetic field is aligned in the direction of
the nanopillar and the XLD signal vanishes (red curve), since there is no charge anisotropy when comparing x-andy-directions.
(right) If the magnetic field is oriented perpendicular to the nanopillar, an XLD signal can be measured due to the charge anisotropy
in x-andy-directions. (Color figure online)
Tabl e 1 Effective properties: average dielectric displacement Din Cm2(taken from [31]) and an estimated ME coefficient α
in s/m
Dx5.4×104Dy1.1×103
α12 9.0×1015 α22 1.83 ×1014
Tabl e 2 Magneto-electro-mechanical quantities and corresponding SI-Units
Symbol Continuum mechanical description SI-Unit
uDisplacement vector m
εLinear strain tensor 1
σCauchy stress tensor kg/s2m(=N/m2)
tTraction vector kg/s2m(=N/m2)
fMechanical body forces kg/s2m2(=N/m3)
ϕMagnetic potential A
HMagnetic field A/m
BMagnetic flux density kg/s2A(=Vs/m2)
QmMagnetic surface flux density kg/s2A(=Vs/m2)
φElectric potential m2kg/s3A(=V)
EElectric field vector m kg/s3A(=V/m)
DElectric displacement vector As/m2(= C/m2)
QeElectric surface flux density As/m2(= C/m2)
qDensity of free charge carriers As/m3(= C/m3)
The ME-coefficients have been estimated by αi2Di/H,where Hwas set to 8.95 A/100 nm as been
used in the simulation.
3 Continuum mechanics
In the following chapter, the coupled magneto-electro-mechanical boundary value problem is defined and the
underlying fundamental continuum balance equations as well as the magnetic, electric and kinematic quantities
are expressed. The important basic magneto-electro-mechanical quantities and their corresponding SI-Units
are given in Table 2.
The strain tensor εcan be written as the symmetric part of the gradient of the displacement uand the
magnetic as well as the electric field Hand Eare defined as the negative gradient of the respective magnetic
and electric potential as
M. Labusch et al.
ε=1
2[grad u+(grad u)T],H=−grad ϕand E=−grad φ. (4)
The balance equations for the underlying boundary value problem are the balance of linear momentum, Gauss’s
law of magnetostatics and Gauss’s law of electrostatics
div σ+f=0,div B=0 and div D=q,(5)
respectively. For the following analysis, we neglect mechanical body forces ( f=0) and free charge carriers
(q=0). The analyzed problem consists of a ferromagnetic inclusion represented as a nanopillar in the center
of a ferroelectric matrix. For the material phases, we assume linear transversely isotropic behavior, where the
preferred direction is denoted by a. The coordinate-invariant representation of the constitutive framework is
formulated in accordance with [32] and expanded to the magneto-electro-mechanical coupling, see Appendix
for details.
For the magnetic and electric phase, we assume the existence of the thermodynamical functions ψmand
ψe,see[18]. The piezomagnetic phase is expressed by
ψm=1
2ε:Cm:εH·qm:ε1
2H·μm·H1
2E·m·E,(6)
and the piezoelectric phase is characterized by
ψe=1
2ε:Ce:εE·ee:ε1
2E·e·E1
2H·μe·H.(7)
In these formulas, C,q,μ,and edescribe the tensors of elasticity, piezomagnetic coupling, magnetic
permeability, dielectric permittivity and piezoelectric coupling given for the separate magnetoactive (m)and
electroactive (e)phases, respectively. The constitutive equations for both material phases in terms of the
generalized vector S, containing the stress, dielectric displacement, and magnetic induction, the generalized
vector D, including the strain, electric and magnetic field, as well as the generalized matrix Ccontaining the
material tangent moduli, appear as
Si=CiDi,with Si=
σi
Di
Bi
and Di=
εi
Ei
Hi
,for i={e,m}.(8)
For the piezoelectric and piezomagnetic phases, we obtain the tangent moduli as
Ce=
CeeT
e0
eee0
00μe
,Cm=
Cm0qT
m
0m0
qm0μm
,(9)
where the piezoelectric tensor is active in the electric phase and the piezomagnetic tensor in the magnetic
phase. The individual phases have no magneto-electric coupling (αi=0). However, the composite exhibits
an effective ME coupling which results out of the strain coupling of the individual phases across the interface
between the magnetic and electric phases.
In order to characterize the overall behavior of the composite, we formally define the effective thermody-
namic function
ψ=1
2ε:¯
C:εH·q:εE·e:ε1
2E··E1
2H·μ·H+ψME .(10)
where the quantities with an overline define the associated effective quantities. Here, the additional term
ψME := −H·α·E,(11)
represents the effective ME coupling. The overall magneto-electric modulus αis given by
α=B
E=D
HT
,with D=−
ψ
Eand B=−ψ
H.(12)
In order to determine the magneto-electric coupling of the multiferroic nanocomposite, we perform a finite
element analysis of the experimental setup described in Sect. 2. The main goals of the following analysis are
the verification of the assumed kinematical assumptions, see Fig. 2and the determination of the change of the
dielectric displacement in the matrix material due to an applied macroscopic magnetic field.
FEM analysis of a multiferroic nanocomposite
(a) (b)
Fig. 4 a Boundary value problem of the RVE, where the corner nodes at the top surface are fixed with respect to the in-plane
displacements and the magnetic potential. The vertical opposite sides of the RVE are linked periodically. At the bottom surface,
all degrees of freedom are fixed at the corner nodes. Additionally the bottom surface is mechanically and electrically fixed due to
the clamping on the substrate. The initial polarization of the electric matrix material is denoted by Pz. bTopview of the schematic
displacement of the nanopillar is with εy≈−2×104nm ([41])
4 In silico characterization of the nanocomposite
Tabl e 1summarizes the estimations of the effective in-plane polarizations and the magneto-electric coupling
coefficients. For this rough estimations, the deformation geometry, shown in Fig. 2, is assumed. Thereby, the
deformation of the nanopillar from the surface displacements to the bottom clamping is linearly interpolated.
With the following numerical simulations, the efficiency and accuracy of such estimations will be discussed.
In the experiments of [30], CoFe2O4nanopillars grew in a BaTiO3matrix on a SrTiO3substrate, compared
to the topview SEM image in Fig. 1b and the schematical illustration of the composite in Fig. 1a. For the
numerical characterization, we assume a periodic structure of the nanopillars and consider a representative
volume element (RVE) consisting of one particular cobalt ferrite (CoFe2O4) nanopillar in a barium titanate
(BaTiO3) matrix. Periodic boundary conditions are used for all fields at the vertical boundaries of the RVE,
since a periodic distribution of the nanopillars is assumed. However, no periodic boundary conditions are
applied at the top and bottom surface. The bottom surface is clamped due to the strontium titanate (SrTiO3)
substrate, see Fig. 4a for a depiction of the magneto-electro-mechanical boundary value problem. The surface
on the top can deform unrestricted in vertical direction.
The dimensions of this single nanopillar are 100 nm×100nm ×400 nm centered in the 200 nm×200 nm ×
400 nm BaTiO3matrix, compared to [30]and[31]. The preferred direction of the magnetic material is oriented
in y-direction whereas the electric material is polarized in z-direction. The material parameters used for the
two different ferroic materials are taken from [20]and[21] and given in Table 3.
In order to match the assumed in-plane magnetostriction εy≈−2×104of the CFO pillar, an overall
magnetic field Hy=6 A/100 nm is applied in y-direction, such that the displacement of the nanopillar at
the upper surface is adjusted to this assumption, see Fig. 4b. Since a linear material model is used and the
saturation of the nonlinear magnetostriction cannot be captured within this approach, the magnitude of the
magnetic field has to be adjusted to obtain the assumed magnetostriction. Initially, the macroscopic magnetic
field is applied in each integration point of the RVE, equally to an externally applied field. Due to the different
magnetic material properties of both phases, we obtain the fluctuations of the magnetic potential.
The applied field Hy=6 A/100 nm yields a magnetic potential distribution with the corresponding maxi-
mum values at the boundaries of the nanopillar, see Fig. 5. At the highest magnitudes of the magnetic potential,
the maximum values of the magnetic field Hyare obtained, see Fig. 5b. Since the magnetic field lines tend to
M. Labusch et al.
Tabl e 3 Material parameters of BaTiO3and CoFe2O4(taken from [20,21,33])
Parameter Unit BaTiO3CoFe2O4Para. Unit BaTiO3CoFe2O4
C1111 N/mm221.1×10425.71 ×104e311 C/m23.88 0.0
C1122 N/mm210.7×10415.00 ×104e333 C/m25.48 0.0
C1133 N/mm211.4×10415.00 ×104e123 C/m232.60.0
C3333 N/mm216.0×10425.71 ×104q311 N/Am 0.0 580.3
C1212 N/mm25.62 ×1048.53 ×104q333 N/Am 0.0699.7
11 mC/kVm 0.0175 8 ×105q123 N/Am 0.0 550.0
33 mC/kVm 0.000425 9.3×105α11 s/m 0 0
μ11 N/kA21.26 157.0α33 s/m 0 0
μ33 N/kA21.26 157.0
(a)
ϕ
(b)
Hy
(c)
By
(d)
uy
Fig. 5 Distribution of amagnetic potential ϕin A, bmagnetic field Hyin A/100 nm, cmagnetic induction Byin T, and d
displacement uyin 100 nm. The deformation of the RVE is scaled with a factor of 200
pass through the magnetic material, a magnetic field concentration can be observed in the matrix material at the
boundaries to the nanopillar in y-direction. A similar distribution can be observed for the magnetic induction
By, with the maximum values in the magnetic pillar, see Fig. 5c. Figure 5d depicts the displacements uyin
100 nm. The obtained displacements of the pillar differ from the expected linear deformation.
The maximum displacements are slightly below the upper surface, where no mechanical clamping in
vertical z-direction is applied. However, the displacements uyremain nearly constant in the mid-area of the
pillar, in contrast to the approximations of a linear interpolation. The behavior of the displacements in x-and
y-direction at the vertical edges at the nanopillar surface is depicted in Fig. 6in more detail. Based on the
homogeneously applied magnetic field, the nanopillar tends to contract across the whole height. However, due
to the clamping on the SrTiO3substrate at the bottom and the not fixed top surface, the contraction of the
nanopillar in both regions is highly inhomogeneous. In the mid-area, between approximately 100 nm and 300
nm, the contraction is more or less constant in contrast to the assumptions of a linear interpolation.
Figure 6a shows the expansion of the pillar in x-direction with a used finite element discretization of 11,520
tetrahedrons with quadratic shape functions. In the region between about 80 nm and 370 nm, the displacements
have nearly the same magnitude due to the homogeneously applied magnetic field. At the clamping and the
free upper surface, inhomogeneous deformations occur. Figure 6b depicts the displacements at the edge of the
nanopillar in y-direction. A nearly constant contraction occurs in the mid-area of the pillar height, whereas at
the lower clamping the contraction nonlinearly increases. The displacement at the upper surface matches the
magnetostriction used for the estimated effective properties in Table 1. However, below the upper surface, the
magnetostriction nonlinearly increases to a maximum value instead of an assumed linear decrease to the lower
clamping. These results demonstrate that the before-mentioned assumption is too simplistic for the complex
deformation behavior. The strain distribution εyin y-direction over the RVE is illustrated in Fig. 7a. This
strain distribution demonstrates the maximum deformations below the upper surface and the nearly constant
deformation across an area of about 200 nm in the mid-area of the nanopillar height. Due to the transferred
strains from the nanopillar to the electric phase and the piezoelectric coupling modulus of the polarized matrix,
an electric potential arises in the RVE, which is shown in Fig. 7b. The nanopillar elongates in vertical z-direction
and the highest vertical deformations and therefore electric potentials occur at the upper region of the RVE.
FEM analysis of a multiferroic nanocomposite
(a)
0
0.5
1
1.5
2
2.5
3
3.5
4x10+2 nm
00.511.522.53
(b)
0
0.5
1
1.5
2
2.5
3
3.5
4x10+2 nm
00.511.522.53
x10401x
mn
4nm
Fig. 6 Displacement of a middle edge of the whole height of the nanopillar of the finer mesh ain x-direction and bin y-direction.
The dots denote the nodal displacements at the edge of the nanopillar, whereas the solid line is a nonlinear interpolation function
(a)
εy
(b)
φ
(c)
Ez
(d)
Dz
Fig. 7 Distribution of the astrain εy,belectric potential φin V, celectric field Ezin V/100 nm, and ddielectric displacement
Dzin C/(100 nm)2. The deformation of the RVE is scaled 200 times
This results in magnetically induced electric fields Ez, which concentrate at the interface between both phases
near the upper surface, see Fig. 7c. Electric field concentrations also appear in the lower regions near the
strontium titanate clamping. The resulting dielectric displacement Dzis depicted in Fig. 7d. Again, the highest
absolute values occur in the upper regions of the RVE, where the maximum strains εyoccur, compared to the
strain distribution in Fig. 7a.
In order to compare the results of the numerical simulations with the estimated effective properties, we
determine the in-plane polarizations in different horizontal layers of the RVE. Therefore, the variable ξ(z)is
introduced, which denotes a function of depth with the upper surface as a starting area, see Fig. 4a. In Table 4,
the effective dielectric displacement in x-direction of one quarter of the matrix material Dmat
x(1/4),seeFig.8for a
depiction of the considered quarter, is listed for a layer thickness of ξnin nm. The dielectric displacements of the
different layers are summarized with 1
nn
i=1Dmat
x(1/4)and listed in the fourth column of Table 4. Furthermore,
the magneto-electric coupling coefficient α12 in the corresponding layers is determined. It can be seen that
the nonlinear deformation behavior results in an inhomogeneous distribution of the dielectric displacement in
z-direction, which is also observable in Fig. 7d. The absolute quantities of Dxin different layers vary in values
of two orders of magnitude. In order to match the estimated effective value of |Dx|=5.4×104Cm2,the
summarized average dielectric displacement in the third layer (below 40nm) has to be taken into account. In
order to determine the coupling coefficient α, we compute layerwise the "linear" coupling coefficient α12 and
α22 via the approximation
M. Labusch et al.
Tabl e 4 Average electric polarization in x-direction and α12 in s/m of different layers over a quarter (1/4) (see Fig. 8b) of the
whole sample for a finer mesh of 11,520 elements Dmat
(1/4)=ξnADmatdA dξin Cm2,wherenis the number of the layer
nξnin nm Dmat
x(1/4)
1
nn
i=1Dmat
x(1/4)α12 Dmat
x(1/4)/Hy
1 0–20 1.41 ×1031.41 ×1032.36 ×1014
2 20–40 1.08 ×1046.54 ×1041.09 ×1014
3 40–60 2.45 ×1043.54 ×1045.91 ×1015
4 60–80 1.98 ×1042.16 ×1043.60 ×1015
5 80–100 1.15 ×1041.50 ×1042.50 ×1015
6 100–120 4.84 ×1051.17 ×1041.96 ×1015
7 120–140 9.54 ×1059.87 ×1051.65 ×1015
8 140–160 6.67 ×1068.72 ×1051.45 ×1015
(a)(b)
Fig. 8 a Displacement of the composite in z-direction from above with a 200 times scaled overall deformation and bquarter area
for the calculation of the average electric polarization
Tabl e 5 Average electric polarization in y-direction and α22 of different layers over a quarter (1/4) (see Fig. 8b) of the whole
sample for a finer mesh of 11,520 elements Dmat
(1/4)=ξnADmatdA dξin Cm2,wherenis the number of the layer
nξnin nm Dmat
y(1/4)
1
nn
i=1Dmat
y(1/4)α22 Dmat
x(1/4)/Hy
1 0–20 1.57 ×1031.57 ×1032.62 ×1014
2 20–40 1.94 ×1048.83 ×1041.47 ×1014
3 40–60 2.75 ×1044.97 ×1048.28 ×1015
4 60–80 3.54 ×1042.84 ×1044.74 ×1015
5 80–100 2.78 ×1041.72 ×1042.86 ×1015
6 100–120 1.68 ×1041.15 ×1041.92 ×1015
7 120–140 7.88 ×1058.74 ×1051.46 ×1015
8 140–160 2.34 ×1057.35 ×1051.23 ×1015
α12 Dx
Hy
and α22 Dy
Hy
,(13)
where Hycorresponds to the applied field. For the comparison of the magneto-electric coefficient α12 with
the estimated values, the third layer has to be considered again. In this layer, the ME coupling rises to absolute
values between 1.09 ×1014 s/m and 5.91 ×1015 s/m, compared to the estimated value of 9.0×1015 s/m.
FEM analysis of a multiferroic nanocomposite
Tabl e 5summarizes the effective dielectric displacement in y-direction in different layers of one quarter of
the RVE Dmat
y(1/4), the summation of the dielectric displacement over multiple layers 1
nn
i=1Dmat
y(1/4)as well
as the overall magneto-electric coefficient α22 in these layers. Here, the estimated effective properties of |Dy|
match with the simulated values in a depth of 40nm. In the case of α22, the numerical simulations of the second
layer can be compared with the estimated value of 1.83 ×1014. In this layer, the ME coefficient gets values
between 2.62 ×1014 s/m and 1.47 ×1014 s/m.
5Conclusion
We have presented a numerical characterization of a magneto-electric two-phase composite, consisting of
vertically aligned nanopillars of magnetostrictive cobalt ferrite in a piezoelectric barium titanate matrix, which
was synthesized by pulsed laser deposition. The magnetically induced in-plane polarizations in the matrix
material were first estimated by an assumed linear interpolation of the vertical magnetostriction behavior
of the nanopillars. However, the numerical simulations demonstrated a more complex deformation response
and showed that a linear interpolation of the deformation behavior between the bottom and top surface is too
simplistic. For a detailed investigation of the real change of the in-plane polarization and a comparison with the
numerical simulations, XLD measurements of the displacements of the Ti ions could be used for a determination
of the polarizations at the interface between both materials. Furthermore, the local magneto-electric coupling
coefficient can be compared on a length scale of several nanometers.
Acknowledgements We gratefully acknowledge the financial support by the “Deutsche Forschungsgemeinschaft” (DFG),
research group “Ferroische Funktionsmaterialien - Mehrskalige Modellierung und experimentelle Charakterisierung”, Project 1
(SCHR 570/12-2) and Project 2 (WE 2623/13-2).
A Appendix
The equations for the stress field σ, dielectric displacement Dand magnetic induction Bfor piezoelectricity
and piezomagnetism in direct and index notation
σ=C:εeT·EqT·H
ij =Cijklεkl ekij Ekqki j Hk,(14)
D=e:ε+·E+α·H,Di=eijkεjk +ik Ek+αik Hk,(15)
and
B=q:ε+α·E+μ·H,Bi=qijkεjk +αik Ek+μik Hk.(16)
They can be reformulated to
ε=C1σ+C1:eT

de
·E+C1:qT

dq
·H.(17)
With (17), Eqs. (15)and(16) can be reformulated to
D=de·σ+(e:de+)·E+(e:dq+α)·H,(18)
and
B=dq·σ+(q:de+α)·E+(q:dq+μ)·H.(19)
In our approach, we use a coordinate-invariant setting based on the following invariants for the transversal
isotropic material law
I1=tr ε,I2=tr ε2,I4=tr (εm), I5=tr(ε2m),
Je
1=tr (EE), Je
2=tr (Ea), Ke
1=tr (ε(Ea)) ,
Jm
1=tr (HH), Jm
2=tr (Ha), Km
1=tr (ε(Ha)) , (20)
M. Labusch et al.
with the second-order structural tensor m:= aa, then the quadratic electric enthalpy function ψis given
with
ψ=1
2λI2
1+μI2+ω1I5+ω2I2
4+ω3I1I4
+β1I1Je
2+β2I4Je
2+β3Ke
1+γ1Je
1+γ2(Je
2)2
+κ1I1Jm
2+κ2I4Jm
2+κ3Km
1+ξ1Jm
1+ξ2(Jm
2)2.(21)
Then, σ=εψ,D=−Eψand B=−Hψappear as
σ=2ω2I4+ω3I1+β2Je
2+κ2Jm
2m+2με
+λI1+ω3I4+β1Je
2+κ1Jm
21+ω1(aε·a+a·εa)
+1
2β3(aE+Ea)+1
2κ3(aH+Ha),(22)
D=−(2γ2Je
2+β1I1+β2I4)a2γ1Eβ3a·ε,(23)
and
B=−(2ξ2Jm
2+κ1I1+κ2I4)a2ξ1Hκ3a·ε.(24)
With the second derivatives of Eqs. (22), (23)and(24), C=εσ,e=εD,q=εB,=EDand μ=HB,
we get
C=λ11+ω3(1m+m1)+2ω2mm+2μI+ω1, (25)
e=−β1a1β2amβ3θ,(26)
q=−κ1a1κ2amκ3θ,(27)
=−2γ11+2γ2m,(28)
μ=−2ξ112ξ2m,(29)
with the fourth-order unit tensor I,=a1a+a1aand θ=1
2(a1+a1).Takingthese
equations written in Voigt notation and renaming the parameters which are equal, we get with the preferred
direction a=x3of the transversal isotropic material
σ11
σ22
σ33
σ12
σ23
σ13
=
c1111 c1122 c1133 000
c1122 c2222 c2233 000
c1133 c2233 c3333 000
000c1313 00
0000c1212 0
00000c1212
ε11
ε22
ε33
2ε12
2ε23
2ε13
00e311
00e311
00e333
000
0e123 0
e123 00
E1
E2
E3
,(30)
and
D1
D2
D3
=
00000e123
0000e123 0
e311 e311 e333 00 0
ε11
ε22
ε33
2ε12
2ε23
2ε13
+
11 00
011 0
0033
E1
E2
E3
.(31)
We can identify
λ=c1122 =1
2(c1111 c1122),ω
1=2c1212 +c1122 c1111 ,
ω2=1
2(c1111 +c3333)2c1212 c1133
3=c1133 c1122
1=−e311 ,
β2=e311 e333 +2e123
3=−2e123
1=−
1
211 ,
γ2=1
2(11 33), κ
1=−q311
2=q311 q333 +2q123 ,
κ3=−2q123
1=−
1
2μ11
2=1
211 μ33).
(32)
FEM analysis of a multiferroic nanocomposite
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