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Magnetoelectric Multiferroic Composites

299

currents. The critical goal for the future is in the development of a wide class of efficient

wide band and low-loss electrically tunable magnetic film devices for battlefield radar,

signal processing, and secure and experimental evaluation of characteristics. The anticipated

advantages of ME devices are yet to be exploited.

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Coupling Between Spins and Phonons Towards

Ferroelectricity in Magnetoelectric Systems

J. Agostinho Moreira and A. Almeida

IFIMUP-IN, Institute of Nanosciences and Nanotechnologies

Departamento de Física e Astronomia da Faculdade de Ciências da Universidade do Porto

Portugal

1. Introduction

Materials, which are either electrically or magnetically polarizable, have attracted the

interest of the scientific community for long, due to both their potential technological

applications and the emergent fundamental questions associated with the mechanisms

tailoring their pertinent physical properties.

Among them, a class of materials exists, exhibiting at least two ferroic ground states, which

are known as multiferroic materials.

Another class is characterized by the emergence of a

magneto-electric coupling, where multiferroicity is not even necessary to be present. This

sort of materials yields the possibility to control the magnetic (polar) state through an

applied electric (magnetic field). One of the most relevant examples is its use to processing

of barriers and interfaces to spintronic devices, enabling to implement electrically controlled

spin transport, but other interesting applications are currently under investigation.

3-11

Unfortunately though, either magnetoelectric coupling coefficient is rather small, or the

range of temperatures it occurs is far way from room temperature, precisely where devices

are requested to have their working range. Based on theoretical and experimental results, an

intensive research to designing new magnetoelectric materials has been undertaken aimed

at overcoming the aforesaid drawbacks.

On designing new systems, their crystal structure

has been addressed concerning both lattice and geometrical arrangements of their atomic or

molecular constituents. This is particularly the case of the orthorhombic rare-earth

manganites belonging to the GdFeO

type structure.

In this sort of structure, rare-earth site

alteration drives the magnitude of the orthorhombic distortion, so-called cooperative

GdFeO

-type distortion, wherein the increase of the Mn-O1-Mn bond angle of the MnO

octahedra yields an increasing of the orthorhombic distortion, whenever a smaller-sized

rare-earth ion is used.

Thus, by handling the Mn-O1-Mn bonds angle between consecutive

MnO

octahedra, drastic changes of their phase diagrams can be reached.

As it will be

presented further below, this is highly predictable, since changing the Mn-O1-Mn bond

angle directly modifies the balance between the competitive ferromagnetic and

antiferromagnetic exchange interactions, existing in these materials. This is the background

mechanism of magnetically induced ferroelectricity, in some rare-earth manganites.

Contrarily to well know ferroelectrics, like BaTiO

, where hybridization between the

titanium 3d states and the oxygen 2p states is at the very origin of ferroelectricity, in

Ferroelectrics – Physical Effects

304

magnetoelectric rare-earth materials it seems to stem from the emergence of spin-lattice

interactions in modulated off-centre symmetric magnetic structures, which is in favour for

the improper nature of the ferroelectric ground state. For some magnetoelectric materials, it

has been proposed that ferroelectricity is originated from a variety of spiral magnetic

structures, and can be explained in terms of the inverse Dzyaloshinski-Morya model.

14-16

the grounds of this model, the interaction between spins, arranged in a modulated structure,

deform the crystal lattice, yielding the loss of the spatial inversion centre and, thus, enabling

the emergence of electric dipoles. As the electric polarization arises from lattice distortions,

the study of the coupling between spins and phonons is of most relevant matter in systems,

wherein magnetic and ferroelectric properties are coupled. Consequently, from both

fundamental and application point of views, a deeper understanding of spin-phonon

coupling remains a very important issue.

The most studied magnetoelectric materials are the orthorhombic rare-earth manganites,

such as EuMnO

, GdMnO

, TbMnO

and DyMnO

17-23

These systems exhibit rather

interesting phase diagrams, with magnetically driven ferroelectric phase transitions, for

compounds with the lighter rare-earth ions, or magnetically switching of electric

polarization, for those ones with the heavier rare-earth ions. These materials behave

differently since, by changing rare-earth ion size, the Mn-O1-Mn bond angle changes

accordingly and hence the balance between ferromagnetic and antiferromagnetic

interactions. Unfortunately, the change of rare-earth ions also brings changes to the

magnetic state through its own magnetic characteristics. If it is required to comprehend just

the effect of ion radius size, the solid solution obtained by A-site substitution in europium

manganite with a non-magnetic ion, with a radius smaller than europium will ensure the

aforesaid requisites. Since neither yttrium nor europium ions possess magnetic moment,

interchanging them does not change the total magnetic moment. It stems only from the

ions. From the reasoning presented to above, it is apparent why Y-doped EuMnO

system has been extensively studied to date.

24-32

Several studies of the lattice dynamics by using Raman and infrared spectroscopies in Y-

doped EuMnO

, have revealed a coupling between magnetic excitations and phonons,

through anomalous temperature behaviour of the phonon parameters or absorption

across the magnetic phase transitions.

33-38

However, an overview of the experimental

results in order to evidence the main mechanisms underlying the spin-phonon coupling is

still missing.

In this book chapter, we will present a detailed review of the main theoretical and

experimental results concerning the lattice dynamics and spin-phonon coupling in the case

study: Eu

1-x

MnO

for 0≤x≤0.5, where the structure is orthorhombic with the Pbnm space

group. The main goal is to understand the mechanisms of coupling between magnetic

excitations and phonons yielding the magnetically switching of ferroelectricity in spiral

magnets and, in this way, the very origin of the ferroelectric ground state in these

compounds.

2. Theoretical background

Important advances have been achieved towards a fundamental understanding of the

stabilization of ferroelectric ground states in magnetoelectrics through experimental

observation and investigation of the coupling mechanism between the ferroelectric and

magnetic orders. Some examples involve the use of second harmonic light generation,

Coupling Between Spins and Phonons Towards Ferroelectricity in Magnetoelectric Systems

305

dielectric and magnetic measurements or structural investigation. In contrast, very little is

known about the behaviour of phonons in magnetoelectrics, although studies of phonons

have in the past played a crucial role in the understanding of classic ferroelectrics. Phonons

are also known to be altered by spin correlations thus offering a complementary and maybe

even unique powerful tool to investigate coupling mechanism in magnetoelectrics.

39-41

Before going into details of phonon analysis and their coupling with spins, let us summarize

some aspects associated with the origin and main characteristics of spin structure in

orthorhombic rare-earth manganites. The main crystalline motif of their structure is the

existence of MnO

octahedra, which run over chains by sharing apical oxygen ions. The 3d

electronic orbitals of the transition Mn

ions yield a rather multifaceted correlation between

spin arrangements and crystalline structure, which is at the very origin of the broad range of

physical properties observed in rare-earth manganites. In an ideal cubic crystal field

environment, the 3d Mn

orbitals are splitted into two degenerate higher energy states,

called d(x

-y

) and d(3z

-z

), with e

symmetry, and three degenerate lower energy states,

, d

), with t

symmetry. Since e

orbitals are much closer to the 2p oxygen electronic

orbitals, Coulomb interaction yields a partial degeneracy lifting of the 3d orbitals. Moreover,

if the magnitude of the crystal field is much larger than spin-orbit coupling, a quenching of

the orbital angular momentum occurs, reflecting the actual spin structure in rare-earth

manganites.

42,43

Over Coulomb interaction, the 3d electrons of Mn

ions can assure different kinds of

interactions. Whilst Hund and direct interactions favour ferromagnetic ordering of spins,

Hubbard tends to align spins antiferromagnetically. Moreover, orbital degeneracy is at the

origin of some types of superexchange interactions. From the substantial overlapping of

these latter orbitals with 2p oxygen ones, a strong superexchange interaction is obtained.

Contrarily, due to the orthogonal spatial orientation of the t

orbitals against the 2p

oxygen

orbitals and just a reduced overlapping with the 2p

ones, weak superexchange interactions

are thus expected.

42,43

In orthorhombic rare-earth manganites a significant distortion occurs associated with MnO

octahedra, which is known as the Jahn-Teller effect.

44,45

This distortion, which occurs along

the molecular fourfold axes, has the effect of lifting both orbital and electronic degeneracies,

leading to a decrease of the total energy.

42,43

The coupling between spins and lattice has become one of the major mechanisms to

understand the interplay between magnetism and ferroelectricity occurring in rare-earth

manganites with intermediate-range magnitudes of the GdFeO

distortion, as it is the case of

ReMnO

, Re=Eu, Gd, Tb, and Dy. The so-called magnetoelectric effect has been understood

in the scope of both phenomenological and microscopic theories. In the former case,

contributions first from Landau and latter from Dzyaloshinskii and Moriya (DM) were

presented, wherein symmetry conditions imposed by the magnetic point group are essential

issues for understanding the emergence of the magnetoelectric effect.

46-48

In particular, both

time and inversion symmetries are key issues for controlling the ME effect. Microscopic

quantum theory of the ME effect has been developed by I. A. Sergienko et al

and Katsura et

al.

These latter authors showed that ME effect and spin current are related to one another

in non-linear spin structures, like spiral-like states. In fact, as both electric polarization and

spin current belong to the same symmetry class, it is expected that are coupled together. The

coupling was demonstrated by considering a cluster scheme with two transition metal ions

intermediated by an oxygen atom shown in Figure 1.

Ferroelectrics – Physical Effects

306

Fig. 1. The cluster model with two transition metal ions M1 and M2, and an oxygen atom

between them. Reprinted figure from Ref. 49. Copyright (2005) by the American Physical

Society.

By assuming the overlapping between the electronic wave functions of two adjacent sites i

and i+1 with mutually canted spins, 



and 



, the inverse Dzyaloshinskii-Moriya model

yields a local polarization p

, given by:







,









. (1)



,

stands for the unit vector linking both sites, being A a constant reflecting spin-orbit

interactions and spin-exchange. Thus, the cycloidal spin arrangement in rare-earth

manganites enables an electric polarization to emerge, obtained by summing all of the local

polarization 



. It comes out from equation (1) that the polarization lays in the cycloidal

plane. As a consequence, in rare-earth manganites, the ferroelectric polarizations expected

along a- and c-direction lay in the ab- and cb-cycloidal planes, respectively (Figure 2).

Fig. 2. Spatial scheme of the DM polarization against 





∑













for (a) ab-cycloidal

spin plane and (b) bc-cycloidal spin plane. Reprinted figure from Ref. 50. Copyright (2009)

by the American Physical Society.

Coupling Between Spins and Phonons Towards Ferroelectricity in Magnetoelectric Systems

307

The inverse DM model has been successfully tested both theoretically and experimentally in

a variety of magnetoelectric rare-earth manganite systems.

15,16,51,52

The emergence of inverse-

DM induced polarization in rare-earth manganites depends on the magnitude of the

GdFeO

distortion, characterized by the amplitude of the Mn-O1-Mn bond angle or the rare-

earth ionic radius size. The following examples show that increasing (decreasing) the

GdFeO

distortion (Mn-O1-Mn bond angle or the rare-earth radius size) enhances the

inverse-DM magnetically induced polarization. The first example is the emergence of an

electric polarization along the c-direction when changing from GdMnO

to TbMnO

and

DyMnO

, which is associated with the increase of the GdFeO

distortion due to ionic radius

size decrease.

53,54

Further examples can be found in Gd

1-x

MnO

55,56

and Eu

1-x

MnO

24,26,31,57,58

solid solutions.

Changes of the polarization state are obtained by continuously control the GdFeO

distortion by increasing dopant content, decreasing in this way the ionic radius size. In fact,

above a certain doping level a long-range ferroelectric ground state is stabilized at low

temperatures associated with an ab-cycloidal spin state, where electric polarization emerges

along the a-crystallographic direction. For larger dopant concentration the cycloidal cb-plane

spontaneously flops at higher temperature to the ab-plane at lower temperatures, along with

a rotation of the electric polarization from the c- to the a-direction.

The examples referred to above undoubtedly evidence the role of the GdFeO

distortion to

inducing electric polarization via inverse DM interaction. Moreover, it is has been

established that this distortion has a crucial effect on the balance between the ferromagnetic

superexchange interaction, associated with the next-nearest (NN) spins, and the

antiferromagnetic superexchange one of the next-next nearest (NNN) spins.

13,50

Increasing

the GdFeO

distortion enhances the antiferromagnetic against the ferroelectric interactions.

This mechanism apparently yields the reduction of the AFM(A) phase transition

temperature with decreasing Mn-O1-Mn bond angle in orthorhombic rare-earth manganites,

and the emergence of the AFM ordering or spiral spin ordering, where ferroelectric ground

states can be stabilized.

Several attempts have been made in order to develop microscopic spin models

to understand the phase diagrams of rare-earth manganites.

13,16,59-62

The essence

of magnetoelectric properties can be described if they are mapped onto a frustrated classical

S=2 Heisenberg model on a cubic lattice, with FM NN (J

), and AFM NNN (J

) exchange

interactions acting in the ab-plane, and AFM NN (J

) exchange interaction along

the c-direction, as it is shown in figure 3. In the Hamiltonian the Mn S=2 spins are

considered as classical vectors expressed as 





































,



, with

their components along a, b, and c axes of the Pbnm space group.

The Hamiltonian includes four terms involving superexchange and DM interactions, and

contributions from both single ion and cubic anisotropies, expressed as follows:

















, (2)

wherein:









∑





∙







∑





∙





,





∑





∙





,

,

,

, (3)







∑







∙









,

, (4)







∑









∑

1

























, (5)

Ferroelectrics – Physical Effects

308

(a)

(b)

Fig. 3. (a) Exchange interactions in rare-earth manganites. J

-FM interactions on the Mn-Mn

bonds along the cubic x- and y-axes, J

-AFM interactions on the in-plane diagonal Mn-Mn

bonds along the orthorhombic b-axis, and J

-AFM interactions on the Mn-Mn bonds along

the c-axis. (b) e

occupied orbitals configuration and both J

and J

interactions (solid lines)

in the ab-plane. A dashed line marks J

interaction path, via two oxygen 2p orbitals.

Reprinted figure from Ref. 50. Copyright (2009) by the American Physical Society.

Lallart M. (ed.) Ferroelectrics - Physical Effects

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