1
Supporting Information for:
Superstructures of Self-assembled Multiferroic Core-Shell Nanoparticles and
Studies on Magneto-Electric Interactions
G. Sreenivasulu,1 V. M. Petrov1,2, F. A. Chavez,3 and G. Srinivasan1a)
1
2
Physics Department, Oakland University, Rochester, MI 48309-4401
Institute of Electronic and Information Systems, Novgorod State University,
Veliky Novgorod 173003, Russia
3
Chemistry Department, Oakland University, Rochester, MI 48309-4401
a) Email: Srinivas@oakland.edu
2
Theory of Magneto-electric Effects in Magnetic Field Assembled Core-Shell ferriteFerroelectric Nanoparticles
A model for low-frequency direct magneto-electric effect for core-shell nanoparticles and
films consisting of particle arrays is provided here. The ME effect in an isolated particle of
nickel ferrite (NFO)-core and barium titanate (BTO)-shell is considered first. The shell is
assumed to be a close packed structure of BTO particles. Each core-shell particle is assumed to
be in a single domain (both ferroelectric and ferromagnetic) state. The magnetic exchange
interaction between adjacent particles is not significant for this case. The ME voltage coefficient
for an isolated core-shell particle due to magnetostrictive and piezoelectric coupling is estimated
as a function of the core and shell radius. The theory is then extended to the case of a film
containing linear array of particles of uniform diameter and the ME voltage coefficient is
estimated.
We consider first the ME coupling in an isolated core shell particle consisting of a
(ferromagnetic) core of radius mR and a (ferroelectric) shell with an outer radius sR.
The
spherical coordinate system (r,,) with the origin at the center of the core is assumed. The total
free energy density of the particle is expressed by
F = vFp + (1-v) Fm
(1)
with Fp, Fm and v being the free energy densities of the ferromagnetic (FM) core and ferroelectric
(FE shell), and volume fraction of the FE phase, respectively.
The free energy of the FE shell is defined by
3
F p   Ffi + FEi + Feli + Fesi + Fedi + Fdpi 
(2)
i
where Ffi, FEi, Feli, Fesi, Fedi, and Fdpi are the ferroelectric bulk free energy, energy in an electric
field, elastic energy, electrostriction energy, and inter-dipole electrostatic energy between the
BTO particles of the shell, and depolarization energy for ith-particle of the shell, respectively. We
use the following definitions for the various terms in the energy density: FEi = - EPicos φi with E,
p
Pi and φi denoting the electric field, polarization and angle between the two, Feli 
ckj p

Sk i  p S j i
2
with pSk and pcik denoting the strain tensor component and elastic stiffness coefficients,
Fesi  p q jkl  p S j i Pki Pli ),
Fedi 
1
80
 ( Pi  Pj )

j i

3
ij
r
where

p
qjkl
are
the
electrostriction
coefficients,
3( Pi  rij )( Pj  rij ) 
 where Pi and Pj are polarizations of ferroelectric particles
rij5

in the shell, rij is the separation between the two particles, ε0 is the permittivity of free space, Fdp
= ½ Pi2Ncos2 φ with N being depolarization factors along electric field direction. Similarly, the
free energy of the FM core can be written as
Fm = Fm +FH+ Fml + Fms + Fdm,
(3)
where Fm, FH, Fml, Fms, and Fdm are the ferromagnetic bulk free energy, Zeeman energy in a
magnetic field, elastic energy, magnetostriction energy, and demagnetization energy,
respectively. Magnetic energy terms are given by: FH = - HMcos ψ with H, M and ψ denoting the
4
magnetic field and magnetization and angle between the two, Fml 
m
cik m m
Si S j with mSi and mcik
2
denoting the strain tensor component and elastic stiffness coefficients, Fes mbikl m Si M k M l , where
bikl are the magnetoelastic coefficients, Fdm = ½ M2N'cos2 φ with N' being demagnetization
m
factor along magnetic field direction.
The stress in the FM and FE is assumed to be produced by applied magnetic field. Strain and
stress tensor components can be calculated by solving the elastostatic equations and using the
boundary conditions
F
F

  p S r i  mS r
at r=mR and i=1..N with N and mR being the number of
ferroelectric particles and the ferrite core radius, respectively. Finding the stress components is
carried out by using the procedure in Ref. [1]. The BTO particles in the shell are assumed to be
in contact with each other that implies pTθi=pTθ(i+1) at the points of contact.
The expressions for strain components should be substituted into Eq. (1) that enables
obtaining the equilibrium polarization of the BTO particles in the shell and magnetization of the
NFO core using the following conditions:
susceptibility,  
F
F
 0 . Finally, we find the ME
 0 and
 i

P
with P denoting the average polarization over the core-shell particle
H
volume. The polarization vector is assumed to be directed along applied magnetic field due to
symmetry of the structure. ME voltage coefficient is obtained from open circuit condition:
E 

. Shell radius-to- core radius ratio r dependence of ME voltage coefficient for an isolated particle is

shown in Fig.4 (a). The following material parameters (in SI units) of bulk materials of NFO
and BTO were used for the estimates [4-5]: Ps=0.3,
p
p
c11=1.09·1011,
q11=1.203·1010, pq12=-1.878·109, mc11=2.2·1011, mc12=1.73·1011, pq11=1.203·1010,
p
c12=8.464·1010,
5
p
q12=-1.878·109, mb11 =7.45·104, mb44 =5.65·104, Ms=0.32. Figure 4 (a) reveals that ME voltage
coefficient reaches the peak value of 240 mV/(cm Oe) at mR/sR=1.17.
Next we consider a linear chain which is made up of n core-shell particles. Each particle is
considered as a single domain particle with polarization Pj and magnetization Mj. The electric
and magnetic dipoles are oriented along the surface of cones that make angles θpi and θmj with
the vector joining the particles. We limit the discussion to an array with n = 10. For simplicity
we assume θm = θm1 =−θm2 = θm3 =−θm4 and θp = θp1 =−θp2 = θp3 =−θp4 .. [2]. The angular
dependence part of energy density of core-shell particle is as follows
F
1 

 FHi + FEi + Fedi + Femi    Pi H 

0
n i 

(4)
with n being the number of particles in the chain.
The
Fmd 
0
8
magnetic
 (Mi  M j )

i j

3
ij
r

dipole-dipole
interaction
energy
is
defined
by
3( M i  rij )( M j  rij ) 
 with Mi and Mj denoting the magnetic moments of the
rij5

particles. To simplify the calculation of the magnetostatic interactions between the particles in
the sample, we will use an effective magnetostatic energy density as a combination of the
limiting cases of an isolated particle with demagnetization tensor N and a homogeneously
magnetized
Fmd 
ellipsoidal
sample
with
the
demagnetization
tensor
N'
[3],
 0 (1   )

M  N  M  0 M  N   M , where N1=N2=N3=1/3 and ω is the volumetric
2
2
packing fraction of magnetic particles in the sample. The first limiting case of an isolated particle
is obtained for ω →0, the second one for ω= 1. Electric dipole interaction can be treated in a
6
similar way. Using expression Fed 
1 

PN P 
P  N   P enables one to calculate
2 0
2 0
effective electrostatic energy density. To derive expressions for the magnetization and
polarization, the total energy, Eq. 4 of the system is minimized with respect to the angle
parameters (θm and θp).
Consider a 2D film that is made up of chains consisting of the core-shell particles. Both the
magnetic and electric dipole-dipole interaction between the arrays is ignored. For the case of inplane magnetic field and poling direction along the chain, ME voltage coefficient is obtained
from open circuit condition:  E  
 P( H )ds
A
0
where P is average polarization over the sample
volume and A is film sectional area. Figure 4(b) shows the MEVC as a function of the ratio of
core-to-shell radius. The overall ME coupling is smaller in the film made up of the array than for
an isolated particle. This is due to the dipole-dipole interactions in the array particles that lead to
a reduction in the net magnetization and polarization and weakening of the magnetostriction and
piezoelectric effects.
It is noteworthy here that the porosity of the film must be taken into account in the estimates
of ME coupling. For finding the effective parameters of porous sample, it must be considered as
a structure consisting of core-shell matrix with embedded air pores. Thus the porous core-shell
sample can be modeled as a composite with 3-0 connectivity type. To take into account
depolarizing and demagnetizing fields, the electric and magnetic fields E and H components that
enter into material equations should be substituted for by E-N1P and H-N1M with N1 denoting the
form factor dependent on the pore shape. Estimates of effective parameters and ME coefficient
for porous sample could be performed based on the procedure reported recently [3].
7
References
1. C.W. Lim, Z.R. Li, L.H. He, International Journal of Solids and Structures 43, 5055
(2006)
2. I.S. Jacobs, C.P. Bean, Phys. Rev. 100, 1060 (1955)
3. V. M. Petrov, G. Srinivasan, U. Laletsin, M. I. Bichurin, D. S. Tuskov, and N.
Paddubnaya, Phys. Rev. B 75, 174422 (2007)
4. A. Petraru, N. A. Pertsev, H. Kohlstedt, U. Poppe, R. Waser, A. Solbach and U.
Klemradt, J. Appl. Phys. 101, 114106 (2007)
5. M.I. Bichurin, I. A. Kornev, V. M. Petrov, A. S. Tatarenko, Yu. V. Kiliba, and G.
Srinivasan, Phys. Rev. B 64, 094409 (2001).