' 2006 Nature Publishing Group

Multiferroic and magnetoelectric materials
W. Eerenstein
1
, N. D. Mathur
1
& J. F. Scott
2
A ferroelectric crystal exhibits a stable and switchable electrical polarization that is manifested in the form of
cooperative atomic displacements. A ferromagnetic crystal exhibits a stable and switchable magnetization that arises
through the quantum mechanical phenomenon of exchange. There are very few ‘multiferroic’ materials that exhibit both
of these properties, but the ‘magnetoelectric’ coupling of magnetic and electrical properties is a more general and
widespread phenomenon. Although work in this area can be traced back to pioneering research in the 1950s and 1960s,
there has been a recent resurgence of interest driven by long-term technological aspirations.
S
ince its discovery less than one century ago, the phenomenon
of ferroelectricity
1
, like superconductivity, has been considered
in relation to the ancient phenomenon of magnetism. Just as
recent work has shown that magnetic order can create super-
conductivity
2
, it has also been shown that magnetic order can create
(weak) ferroelectricity
3
and vice versa
4,5
. Single-phase materials in
which ferromagnetism and ferroelectricity arise independently also
exist, but are rare
6
. As this new century unfolds, the study ofmaterials
possessing coupled magnetic and electrical order parameters has
been revitalized. In this Review we set recent developments in the
context of the pioneering works of the 1950s–1960s.
Thefieldofresearchthatwearedescribinghasatortuous
taxonomy and typically involves terms such as ‘multiferroic’ and
‘magnetoelectric’, whose overlap is incomplete (Fig. 1). By the
original definition, a single-phase multiferroic
7
material is one that
possesses two—or all three—of the so-called ‘ferroic’ properties:
ferroelectricity, ferromagnetism and ferroelasticity (Fig. 2 and
Table 1; see Box 1 for a glossary of terms). However, the current
trend is to exclude the requirement for ferroelasticity in practice, but
to include the possibility of ferrotoroidic order (Box 1) in principle.
Moreover, the classification of a multiferroic has been broadened to
include antiferroic order (Box 1). Magnetoelectric coupling (Box 1),
on the other hand, may exist whatever the nature of magnetic and
electrical order parameters, and can for example occur in para-
magnetic ferroelectrics
8
(Fig. 1). Magnetoelectric coupling may arise
directly between the two order parameters, or indirectly via strain.
We also consider here strain-mediated indirect magnetoelectric
coupling in materials where the magnetic and electrical order
parameters arise in separate but intimately connected phases (Fig. 3).
A confluence of three factors explains the current high level of
interest in magnetoelectrics and multiferroics. First, in 2000, Hill
(now Spaldin) discussed the conditions required for ferroelectricity
and ferromagnetism to be compatible in oxides, and declared them to
be rarely met
6
. Her paper in effect issued a grand materials develop-
ment challenge that was taken up because empirically there are
indeed few multiferroic materials, whatever the microscopic reasons.
Second, the experimental machinery for the synthesis and study of
various contenders was already in place when this happened. Third,
the relentless drive towards ever better technology is aided by the
study of novel materials. Aspirations here include transducers and
magnetic field sensors, but tend to centre on the information storage
industry.
It was initially suggested that both magnetization and polarization
could independently encode information in a single multiferroic bit.
Four-state memory has recently been demonstrated
9
, but in practice
it is likely that the two order parameters are coupled
10,11
. Coupling
could in principle permit data to be written electrically and read
magnetically. This is attractive, given that it would exploit the best
REVIEWS
Figure 1 | The relationship between multiferroic and magnetoelectric
materials. Ferromagnets (ferroelectrics) form a subset of magnetically
(electrically) polarizable materials such as paramagnets and
antiferromagnets (paraelectrics and antiferroelectrics). The intersection
(red hatching) represents materials that are multiferroic. Magnetoelectric
coupling (blue hatching) is an independent phenomenon that can, but need
not, arise in any of the materials that are both magnetically and electrically
polarizable. In practice, it is likely to arise in all suchmaterials, either directly
or via strain.
1
Department of Materials Science, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK.
2
Centre for Ferroics, Earth Sciences Department, University of
Cambridge, Downing Street, Cambridge CB2 3EQ, UK.
Vol 442|17 August 2006|doi:10.1038/nature05023
759

' 2006 Nature Publishing Group

aspects of ferroelectric random access memory (FeRAM) and mag-
netic data storage, while avoiding the problems associated with
reading FeRAMand generating the large local magnetic fields needed
to write. Unfortunately, significant materials developments will be
required to generate magnetoelectric materials that couldmake a real
contribution to the data storage industry. But given the paucity of
serious competitors to contemporary memory technologies, the
study of novel materials remains important if disruptive technologies
are ultimately to emerge. In the shorter term, niche applications are
more likely to emerge in strain coupled two-phase systems of the type
that we describe later.
The purpose of this Review is to assess the current state of the field,
to remind readers of the relevant work performed in the latter half of
the twentieth century, and to discuss matters of scientific ‘hygiene’
pertaining to accurate measurements and analyses. For further
details we refer the reader to three reviews written at different stages
of this re-emerging field
12–14
.
Magnetoelectric coupling
The magnetoelectric effect in a single-phase crystal is traditionally
described
13,15
in Landau theory by writing the free energy F of the
system in terms of an applied magnetic fieldHwhose ith component
is denoted H
i
, and an applied electric field Ewhose ith component is
denoted E
i
. Note that this convention is unambiguous in free space,
but that E
i
within a material encodes the resultant field that a test
particle would experience. Let us consider a non-ferroic material,
where both the temperature-dependent electrical polarization P
i
(T)
(mCcm
22
) and the magnetization M
i
(T)(m
B
per formula unit,
where m
B
is the Bohr magneton) are zero in the absence of applied
fields and there is no hysteresis. It may be represented as an infinite,
homogeneous and stress-free medium by writing F under the
Einstein summation convention in S.I. units as:
2FðE;HÞ¼
1
2
1
0
1
ij
E
i
E
j
þ
1
2
m
0
m
ij
H
i
H
j
þa
ij
E
i
H
j
þ
b
ijk
2
E
i
H
j
H
k
þ
g
ijk
2
H
i
E
j
E
k
þ ··· ð1Þ
The first term on the right hand side describes the contribution
resulting from the electrical response to an electric field, where the
permittivity of free space is denoted 1
0
, and the relative permittivity
1
ij
(T) is a second-rank tensor that is typically independent of E
i
in
non-ferroic materials. The second term is the magnetic equivalent of
the first term, where m
ij
(T) is the relative permeability and m
0
is the
permeability of free space. The third term describes linear magneto-
electric coupling viaa
ij
(T); the third-rank tensors b
ijk
(T) and g
ijk
(T)
represent higher-order (quadratic) magnetoelectric coefficients.
In the present scheme, all magnetoelectric coefficients incorporate
the field independent material response functions 1
ij
(T) and m
ij
(T).
The magnetoelectric effects can then easily be established in the form
P
i
(H
j
)orM
i
(E
j
). The former is obtained by differentiating F with
respect to E
i
, and then setting E
i
¼ 0. A complementary operation
involving H
i
establishes the latter. One obtains:
P
i
¼ a
ij
H
j
þ
b
ijk
2
H
j
H
k
þ ··· ð2Þ
and
m
0
M
i
¼ a
ji
E
j
þ
g
ijk
2
E
j
E
k
þ ··· ð3Þ
In ferroic materials, the above analysis is less rigorous because 1
ij
(T)
and m
ij
(T) display field hysteresis. Moreover, ferroics are better
parameterized in terms of resultant rather than applied fields
16
.
This is because it is then possible to account for the potentially
significantdepolarizing/demagnetizing factors infinitemedia, and also
because the coupling constants would then be functions of tempera-
ture alone, as in standard Landau theory. In practice, resultant
electric and magnetic fields may sometimes be approximated
17
by
the polarization and magnetization respectively.
A multiferroic that is ferromagnetic and ferroelectric is liable to
display large linear magnetoelectric effects. This follows because
ferroelectric and ferromagnetic materials often (but not always)
possess a large permittivity and permeability respectively, and a
ij
is
bounded by the geometric mean of the diagonalized tensors 1
ii
and
m
jj
such that
18
:
a
2
ij
# 1
0
m
0
1
ii
m
jj
ð4Þ
Equation (4) is obtained from equation (1) by forcing the sum of the
first three terms to be greater than zero, that is, ignoring higher-order
coupling terms. It represents a stability condition on 1
ij
and m
ij
, but if
the coupling becomes so strong that it drives a phase transition to a
more stable state, then a
ij
, 1
ij
and m
ij
take on new values in the new
phase. Note that a large 1
ij
is not a prerequisite for a material to be
ferroelectric (or vice versa); and similarly ferromagnets do not
necessarily possess large m
ij
. For example, the ferroelectric KNO
3
possesses a small 1 ¼ 25 near its Curie temperature of 120 8C (ref. 19),
Table 1 | Spatial-inversion and time-reversal symmetry in ferroics
Characteristic symmetry Spatial-inversion symmetry? Time-reversal symmetry?
Ferroelastic Yes Yes
Ferroelectric No Yes
Ferromagnetic Yes No
Multiferroic* No No
*A multiferroic that is both ferromagnetic and ferroelectric possesses neither symmetry.
Figure 2 | Time-reversal and spatial-inversion symmetry in ferroics.
a, Ferromagnets. The local magnetic moment m may be represented
classically by a charge that dynamically traces an orbit, as indicated by the
arrowheads. A spatial inversion produces no change, but time reversal
switches the orbit and thusm. b, Ferroelectrics. The local dipole moment p
may be represented by a positive point charge that lies asymmetrically
within a crystallographic unit cell that has no net charge. There is no net time
dependence, but spatial inversion reverses p. c, Multiferroics that are both
ferromagnetic and ferroelectric possess neither symmetry.
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