Department of Physics Seminar Magnetoelectric effect The challenge of coupling magnetism and ferroelectricity Luka Vidovič Mentor: prof. dr. Denis Arčon Ljubljana, december 2009 Abstract Magnetism and ferroelectricity are widely used in current technology. However, they tend to be mutually exclusive and interact weakly when coexist. In multiferroic materials magnetism and ferroelectricity do coexist and their mutual coupling is described by magnetoelectric effect. Such magnetic ferroelectricity occurs in frustrated magnets as a result of competing spin interaction. These compounds have great potential in many areas as tunable multifunctional devices.
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Department of Physics
Seminar
Magnetoelectric effect
The challenge of coupling magnetism and ferroelectricity
Luka Vidovič
Mentor: prof. dr. Denis Arčon
Ljubljana, december 2009
Abstract
Magnetism and ferroelectricity are widely used in current technology. However, they
tend to be mutually exclusive and interact weakly when coexist. In multiferroic
materials magnetism and ferroelectricity do coexist and their mutual coupling is
described by magnetoelectric effect. Such magnetic ferroelectricity occurs in frustrated
magnets as a result of competing spin interaction. These compounds have great
potential in many areas as tunable multifunctional devices.
For description of extended lattice the Heisenberg model applies:
ℋ = −2 𝐽𝑖 ,𝑗𝑆 𝑖𝑆 𝑗𝑖>𝑗
(1.5)
Where 𝐽𝑖 ,𝑗 is the exchange constant between the 𝑖𝑡ℎ and 𝑗𝑡ℎ spin.
2.2 Ferroelectrics
Ferroelectric material undergoes a phase transition from a high-temperature phase that behaves
as an ordinary dielectric to a low-temperature phase that has spontaneous polarization. Many
properties of ferroelectric materials are analogous to ferromagnets but with corresponding
electric parameters. Ferroelectric materials also have domains and show a hysteresis response.
The most widely studied and used ferroelectrics are perovskite-structure oxides, 𝐴𝐵𝑂3 , which
have cubic structure shown in Figure 4. A structure is characterized by small a cation, 𝐵, at the
centre of an octahendron of oxygen anions with large cations, 𝐴, at the unit cell corner. Below
the Curie temperature, there is a structural distortion to a lower-symmetry phase accompanied
by the shift off-centre of small cations. The spontaneous polarization derives largely from
electric dipole moment created by this shift.
2.2.1 What makes ferroelectrics ferroelectric?
The existence or absence of ferroelectricity is determined by a balance between short-range
repulsion, which favor the nonferrelectric symmetry structure, and additional bonding
consideration, which might stabilize the ferroelectric phase. In ferroelectric materials, the short-
range repulsions dominate at high temperature, resulting in the symmetric, unpolarized state.
As the temperature is decreased, the unit cell undergoes a series of phase transitions. In low-
temperature phase the 𝐵 3𝑑 − 𝑂 2𝑝 hybridization is responsible for stabilization of the
ferroelectric distortion along preferred diagonal.
Also significant is the observation that 𝐵 cation is formally in a 𝑑0 lowest unoccupied state.
Typical representative compound is 𝐵𝑎𝑇𝑖𝑂3. A simple charge count gives 𝐵𝑎2+𝑇𝑖4+𝑂32−.
Evidently, 𝑇𝑖4+ ion is in a 𝑑0 state.
Basic principles
7
In contrast, 𝑑-orbital occupancy is a requirement for existence of magnetic moments and
consequently of magnetic ordering. This brings us to a conflicting situation: ferroelectric
materials favor 𝑑0 while magnetic order require 𝑑𝑛 , thus excluding coexistence of both orders.
Figure 4: Cubic perovskite structure. The small B cation (in black) is at the center of an octahedron of
oxygen anions (in gray). The large A cations (white) occupy the unit cell corners (reproduced from [1]).
2.3 Multiferroism
The term multiferroism is used to describe materials where ferroelectricity and ferromagnetism
occur in the same phase. This means that they have the spontaneous magnetization controlled
by the applied magnetic field and the spontaneous polarization controlled by applied electric
field. In addition, the ability to control charges by applied magnetic fields and spins by applied
voltage offers an extra degree of freedom. However, this proved to be a difficult problem, as this
order parameters turn out to be mutually exclusive. Furthermore, simultaneous presence does
not guarantee strong coupling, as microscopic mechanisms of ferroelecticity and magnetism are
quite different.
2.3.1 Requirements for magnetoelectric multiferroics
Multiferroic physical, structural and electronic properties are restricted to those that occur both
in ferromagnetic and ferroelectric materials. We will analyze a range of properties and discuss
how they limit choice of potential materials.
Basic principles
8
2.3.1.1 Symmetry
From the point of view of symmetry consideration, ferroelectricity needs the broken spatial
inverse symmetry while the time symmetry can be invariant. A structural distortion from
symmetry structure is needed for spontaneous polarization to emerge. In contrast, ferromagnets
possess broken time symmetry and spatial symmetry is invariant. Thus multiferroics have
neither symmetry. The time-reversal and spatial-reversal symmetry in ferroic materials is
illustrated in Figure 5.
Figure 5: 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 thus m. 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(reproduced from [5]).
2.3.1.2 Electric properties
By definition, a ferroelectric material must be an insulator. Ferromagnets, on the other hand, are
often metals. One could assume that small number o multiferroics is due to lack of magnetic
insulators. However, there are also very few antiferromagnetic ferroelectrics, despite
antiferromagnets are usually insulators.
2.3.1.3 Chemistry: 𝒅𝟎 band
Ferroelectric materials have a formal charge corresponding to the 𝑑0 electron configuration on
the 𝐵 cation. But if there are no 𝑑 electrons creating localized magnetic moments, then there can
be no magnetic ordering of any kind. It appears however, that in most cases, as soon as the 𝑑
shell of cation is partially occupied, the tendency to shift and remove the centre of symmetry.
Approaches to coexistence
9
3 Approaches to coexistence
3.1 Magnetoelectric coupling
The magnetoelectric effect in a crystal is traditionally described in Landau theory by writing the
free energy 𝐹 of the system in terms of an applied magnetic field 𝐻 and an applied electric field
𝐸 . Using Einstein summation convention 𝐹 can be written as
−𝐹 𝐸, 𝐻 =1
2𝜀0𝜀𝑖𝑗 𝐸𝑖𝐸𝑗 +
1
2𝜇0𝜇𝑖𝑗 𝐻𝑖𝐻𝑗 + 𝛼𝑖𝑗 𝐸𝑖𝐻𝑗 +
𝛽𝑖𝑗𝑘
2𝐸𝑖𝐻𝑗 𝐻𝑘 +
𝛾𝑖𝑗𝑘
2𝐻𝑖𝐸𝑗𝐸𝑘 + ⋯ . (1.6)
The first term on the right hand side describes the contribution resulting from the electrical
response to an electric field, where 𝜀𝑖𝑗 (𝑇) is relative permittivity. The second term is the
magnetic equivalent of the first term, where 𝜇𝑖𝑗 (𝑇) is relative permeability. The third term
describes linear magnetoelctric coupling via 𝛼𝑖𝑗 (𝑇). Other terms represent higher-order
magnetoelectric coupling coefficients.
The magnetoelectric effect can be established in the form 𝑃𝑖 𝐻𝑗 or 𝑀𝑖 𝐸𝑗 by differentiating 𝐹.
One obtains
𝑃𝑖 𝐻𝑗 = 𝛼𝑖𝑗 𝐻𝑗 +𝛽𝑖𝑗𝑘
2𝐻𝑗 𝐻𝑘 + ⋯ ,
𝑀𝑖 𝐸𝑗 = 𝛼𝑖𝑗 𝐸𝑖 +𝛾𝑖𝑗𝑘
2𝐸𝑗𝐸𝑘 + ⋯ .
(1.7)
Term 𝛼𝑖𝑗 is designated as the linear magnetoelectric effect and corresponds to the induction of
polarization by a magnetic field or a magnetization by an electric field. Materials exhibiting ME
effect are 𝐶𝑟2𝑂3 , 𝐵𝑖𝑀𝑛𝑂3, 𝐵𝑖𝐹𝑒𝑂3. Unfortunately, the magnetoelectric effect is usually too small
to be practically applicable as term 𝛼𝑖𝑗 is limited by the relation
𝛼𝑖𝑗
2 ≤ 𝜀𝑖𝑖𝜇𝑗𝑗 . (1.8)
Approaches to coexistence
10
3.2 Frustrated magnets
Recent discovery report coexistence and gigantic coupling of ferroelectricity and
antiferromagnetism in spin frustrated systems. The key questions are how it is possible that
magnetic ordering can induce ferroelectricity and what the role of frustration is.
The importance of the frustration is illustrated in Figure 6. In some lattices it is not possible to
satisfy all exchange interactions and energy can not be minimized. On the square lattice it is
possible to satisfy the requirement of antiparallel ordering. However, on a triangular lattice
things are not so straightforward. If two neighboring spins are placed antiparallel, the third spin
is faced with a dilemma. In any case the one of two neighbors will not have their energy
minimized. As a result the system is frustrated and tends to release this frustration by forming
unusual magnetic order where magnetization is inhomogeneous.
Figure 6: Frustration of spins.
The coupling between electric polarization and magnetization is governed by the symmetries of
these two order parameters. As we already mentioned, the polarization 𝑃 and electric field 𝐸
change sign on the inversion of all coordinates, 𝑟 → −𝑟 , but remain invariant on time reversal,
𝑡 → −𝑡. The magnetization 𝑀 and magnetic field 𝐻 transform precisely the opposite way.
Because of this difference in transformation properties, the linear coupling between 𝑃, 𝐸 and
𝑀, 𝐻 described by Maxwell’s equations is only possible when these vectors vary both in space
and time; spatial derivatives of 𝐸 are proportional to the time derivative of 𝐻 and vice versa.
This is where frustration comes into play. Its role is to induce spatial variation of magnetization.
The period of magnetic states in frustrated systems depends on strengths of competing
interactions and is often incommensurate (out of proportion) with period of crystal lattice. For
example, a spin chain with a ferromagnetic interaction 𝐽′ > 0 between neighbouring spins has
uniform ground state with all spins parallel. An antiferromagnetic next-nearest-neighbour
interaction 𝐽 < 0 frustrates this simple ordering, and when sufficiently strong stabilizes a spiral
magnetic state:
𝑆 𝑛 = 𝑆[𝑒 1 cos 𝑄𝑥𝑛 + 𝑒 2 sin𝑄𝑥𝑛 ] , (1.9)
where 𝑒 1 and 𝑒 2 are two orthogonal unit vectors and wavevector 𝑄 is given by
Approaches to coexistence
11
cos𝑄
2= −𝐽′ 4𝐽 . (1.10)
Like any other magnetic ordering, the magnetic spiral (Figure 7) spontaneously breaks time-
reversal symmetry. In addition it breaks inversion symmetry, because the change of the sign of
all coordinates inverts the direction of the rotation of spins in the spiral. Thus, the symmetry of
the spiral state allows for a simultaneous presence of electric polarization
Figure 7: Frustrated spin chains with the nearest-neighbour FM and next-nearest-neighbour AFM
interactions J and J´ (reproduced from [4]).
Ferroelectricity is induced by lattice relaxation in a
magnetically ordered state. The exchange between
spins of transition metal ions is usually mediated by
ligands, for example oxygen ions, forming bonds
between pair of transition metals. The effect is
shown in Figure 8. Interaction between spins 𝑆 𝑛 and
𝑆 𝑛+1 pushes negative oxygen ions in one direction
perpendicular to the spin chain formed by positive
magnetic ions, thus inducing electric polarization
perpendicular to the chain.
Figure 8: Ferroelectricity induced by the
exchange striction in a magnetic spiral state(reproduced from [4]).
The perovskite manganite 𝑇𝑏𝑀𝑛𝑂3 is an example of described mechanism. The spin structure is
a sinusoidal antiferromagnetic ordering of the 𝑀𝑛3+ moments that takes place below 𝑇𝑁 ≈ 41𝐾 .
A rough sketch of the magnetic order on 𝑀𝑛 moments is illustrated in Figure 9. With further
decreasing of temperature below 𝑇𝑁 ≈ 27𝐾 the ferroelectric phase with spontaneous
polarization emerges. The polarization along 𝑐 axis at ~10𝐾 is about 8 ∗ 10−4 𝐶 𝑚−2 which is
still rather small compared with that of conventional ferroelectrics (~2,6 ∗ 10−2 𝐶 𝑚−2 at 296𝐾
in 𝐵𝑎𝑇𝑖𝑂3). Dependence of polarization upon temperature in magnetic field is revealed in Figure
10. As the magnetic field 𝐵 is applied along the b-axes the magnetic 𝑄 vector changes and the
individual magnetic moments change their direction. As a result, the direction of polarization
changes and becomes zero in c-direction, while increases in a-direction. Temperature versus
magnetic field phase diagram for compound 𝑇𝑏𝑀𝑛𝑂3 is shown in Figure 11.
Approaches to coexistence
12
Figure 9: A rough sketch of the magnetic order on 𝑀𝑛 moments (reproduced from [3]).
Figure 10: electric polarization along the c and a axes (c and d), respectively, at various magnetic fields in single crystals of 𝑇𝑏𝑀𝑛𝑂3 . Magnetic fields are applied along the b axis (reproduced from [3]).
Figure 11: Temperature versus magnetic field phase diagram for 𝑇𝑏𝑀𝑛𝑂3 for magnetic field applied along
the b axis. The shaded areas show magnetic field hysteresis regions (reproduced from [3]).
Example material – Slovenian contribution
13
4 Example material – Slovenian contribution
Research field of multiferroic materials is fast evolving and discoveries of compounds such as
𝑇𝑏𝑀𝑛𝑂3 ignited extensive research activities. I will present the magnetic and ferroelectric
properties of the 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 system, studied by team of Slovene and Swiss researches (Ref. [7]).
The crystal structure (Figure 13) of compound implies both magnetic frustration and reduced
dimensionality where magnetoelectric effect is probable. The system has monoclinic unit cell
and adopts a layered structure. The layers consist of triangularly arranged 𝐹𝑒4𝑂16 20− clusters
linked by [𝑇𝑒4𝑂10𝐵𝑟2]6− units. Clusters contain two crystallographically non-equivalent 𝐹𝑒3+
ions (labeled as 𝐹𝑒1 and 𝐹𝑒2 in Figure 12).
Figure 13: Crystal structure of the 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 system (reproduced from [7]).
The exchange interaction between 𝐹𝑒1 and
𝐹𝑒2 moments, 𝐽1, is antiferromagnetic and
interaction between 𝐹𝑒1 and 𝐹𝑒1 moments, 𝐽2,
is ferromagnetic. The structure of the iron
tetrameter suggests competition between 𝐽1
and 𝐽2. Furthermore, 𝑇𝑒4+ cations possess
active lone-pare electrons (pared electrons
not used in chemical bonding). All inter-
cluster magnetic exchange paths go through
𝑇𝑒4+ sites, thus providing intimate link
between the 𝐹𝑒3+ magnetic moments and
𝑇𝑒4+ lone pairs.
Figure 12: 𝐹𝑒4𝑂16 20− cluster with denoted 𝐽1 and 𝐽2 exchange interactions (reproduced from [7]).
Example material – Slovenian contribution
14
The reduced dimensionality and the magnetic frustration within the iron tetramers suggest
interesting magnetic properties of the compound. Studies of magnetic susceptibility 𝜒 showed a
distinct change from Curie-Weiss law, 𝜒 =𝐶
𝑇−𝑇𝐶, in temperature dependence at 𝑇𝑁 = 10,6𝐾
(Figure 14). Neutron diffraction measurements revealed magnetic ordering, suddenly emerging
at 𝑇𝑁 (inset Figure 14). Magnetic structure is incommensurate along the crystallographic 𝑏-axis
which arises from the amplitude (Figure 15), rather than directional modulation of the magnetic
characteristics of spiral ordering (Figure 7). The structure is characterized by a periodic
modulation of 𝐹𝑒3+ magnetic moment amplitudes
𝑆 𝑖, 𝑘 = 𝑆0 cos 𝑄 𝑟 𝑖 + 𝜓𝑘 . (1.11)
Where 𝑆0is the amplitude and 𝜓𝑘 is the phase of the modulation associated with different 𝐹𝑒3+
magnetic ions within the unit cell.
Figure 14: Magnetic susceptibility and neutron diffraction in inset (reproduced from [7]).
Figure 15: Incommensurate magnetic structure which arises from the amplitude. Red dots
represent 𝐹𝑒3+ ions and arrows represent magnetic moments.
Example material – Slovenian contribution
15
We discuss the possible origin of the multiferroic behavior in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟. The studied
compound is highly frustrated which is responsible for the complex magnetic ordering. The
ferroelectric order develops simultaneously within the low-temperature magnetic phase. In this
phase the shift of 𝑇𝑒4+ ions will take place, leading to the polarization of the 𝑇𝑒4+ lone pair
electrons. It is important to stress that all inter-cluster exchange interactions go through 𝑇𝑒4+
ions, thus providing a natural way to couple magnetic and ferroelectric order. The compound
𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 offers a new type of magnetic structure and new type of magnetoelectric coupling.
The unique interaction between magnetic frustration and 𝑇𝑒4+ lone pairs bridging magnetic
inter-cluster exchange appears vital for the occurrence of magnetoelectric effect.
Figure 17: Coupling between magnetic and polar order in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 . Parameter 𝐶–𝐶0 is proportional to
𝜖(𝑇) and shows great field dependence in all directions (reproduced from [7]).
In low-temperature incommensurate
magnetic structure none of the crystal
symmetry elements is preserved. This
opens the possibility for coexistence of
ferroelectric order. The measurements of
electric polarization, 𝑃, confirmed
existence of spontaneous polarization
(Figure 16).
The existence of magnetoelectric effect is
presented in Figure 17 where parameter
𝐶– 𝐶0 is proportional to 𝜖(𝑇). The FE
transition is strongly field dependent,
ambiguously proving a coupling between
magnetic and polar order in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟.
Figure 16: Electric polarization hysteresis loop (reproduced from [7]).
Applications
16
5 Applications
Most of the research in multiferroics has been curiosity-driven basic research, but there are a
number of fresh ideas for device applications based on multiferroic materials. Multiferroics
combine application favorable properties of magnetic and ferroelectric materials and blend
them together, opening a new window of opportunities for application usage.
One of the most popular ideas is that multiferroic bits may be used to store information in the
magnetization 𝑀 and polarization 𝑃. This type of encoding information in such four-state
memory has recently been demonstrated (Figure 18). Such memory does not require the
coupling between ferroelectricity and magnetism; a strong cross coupling would be even
disastrous. If magnetoelectric coupling is present, device application could be realized where
information is written electrically and read magnetically. This is attractive, given that it would
exploit the best of aspects of ferroelectric random access memory (FeRAM) and magnetic data
storage, while avoiding the problems associated with reading FeRAM and generating the large
local magnetic field needed to write. However, significant materials developments will be
required to develop magnetoelectric materials that could make a real contribution to the data
storage industry.
The direct application of multiferroics is development of a magnetic field sensor. Multiferroics
possess great potential value due sensitivity to both electric and magnetic field. External
magnetic field could be detected and immediately transformed into electric polarization –
voltage. Multiferroics could offer an alternative to modern field sensors, but sensitivity still has
to be investigated.
Figure 18: Schematic representation of four-state memory.
Conclusions
17
6 Conclusions
In this paper we reviewed the challenge of coupling magnetism and ferroelectricity in a
magnetoelectric effect. The magnetic and electric properties of compound have been presented.
By analyzing characteristics of both type of materials we defined necessary attributes for
multiferroic material. However, we found the requirements for ferrroelectricity and magnetism
to be in contradiction.
We demonstrated a way to get around this problem by introducing frustrated systems.
Frustration removes symmetry and induces spin incommensurate spin ordering which makes
ferroelecticity possible. Example material has been introduced and analyzed.
Slovenian contribution in this exciting field of research was also presented. Compound
𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 was characterized as frustrated system where magnetic structure is
incommensurate. The origin of ferroelectricity was attributed to the shift of 𝑇𝑒4+ ions which are
also vital for occurrence of magnetoelectric effect. By analysis of the characteristics of selected
compound new materials could be engineered and take a step forward in exploiting
magnetoelectric effect.
As multiferroics possess properties of magnets and ferroelectrics their applicational value is
vast. The magetoelectric effecy offers a whole new dimension in application possibilities.
Already, ideas of four-state memory, spintronics and magnetic field sensors are being under
intense development.
References
18
7 References
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(2000)
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[3] Kimura, T. et al: Magnetic control of ferroelectric polarization, Nature 426, 55-58 (2003)
[4] Cheong S.-W. and Mostovoy M.: Multiferroics: A magnetic twist for ferroelectricity, Nature
Materials 6, 13 - 20 (2007)
[5] Eerenstein W., Mathur N. D. and Scott J. F.: Multiferroic and magnetoelectric materials, Nature
442, 759-765 (2006)
[6] Wang K.F., Liu J.-M. and Ren Z.F.: Multiferroicity: the coupling between magnetic and
polarization orders, Advances in Physics 58, 321-448 (2009)
[7] Pregelj, M. et al: Spin amplitude modulation driven magnetoelectric coupling in the new