bulk and surface polaritons in pml-type magnetoelectric...

Bulk and Surface Polaritons in PML-typeMagnetoelectric Multiferroics and The Resonance

Frequency Shift on Carrier-MediatedMultiferroics

Vincensius Gunawan Slamet Kadarrisman

BSc.,MSc.

This thesis is presented for the degree of Doctor of Philosophy of

The University of Western Australia

School of Physics

2012

Abstract

Polaritons are shown to be influenced by magnetoelectric coupling in a semi-infinite mul-

tiferroic material. The magnetoelectric coupling is considered as PML-type which allows

a small uniform canting in the magnetic sub-lattices, resulting in a weak ferromagnetism.

In this type of coupling, the electric polarisation P, the weak ferromagnetism M, and the

antiferromagnetic vector L are perpendicular to each other.

The case of transverse electric (TE) and transverse magnetic (TM) polarization are

obtained when the electric polarisation P and the weak ferromagnetism M are directed in

the plane parallel to the surface while the antiferromagnet vector L is directed out of the

plane, perpendicular to the surface. Using parameters appropriate for BaMnF4, Maxwell

equations are solved to obtain dispersion relations for bulk and surface modes. It is shown

that TE surface polaritons are non-reciprocal, such as ω(k) 6= ω(−k), where ω is the

frequency and k is the propagation vector. It is also shown that the non-reciprocity can

be controlled by an applied electric field. The magnetoelectric interaction also gives rise to

“leaky” surface modes in the case of TM polarisation, i.e. pseudosurface waves that exist

in the pass band, which dissipate energy into the bulk of material. These pseudosurface

mode frequencies and properties can be modified by temperature and by the application

of external electric or magnetic fields.

In the configuration where the electric polarisation P and the antiferromagnetic vector

L are in the plane parallel to the surface, and where the weak ferromagnetism is out of

the plane perpendicular to the surface, it is found that the surface modes are neither

TE nor TM. We term these modes “un-polarised”. Since in this configuration there are

two attenuation constants, a superposition of two plane waves is required to generate

the surface modes. It is also shown that surface modes are non-reciprocal due to the

magnetoelectric interaction. Additionally, we show that the strength of the non-reciprocity

depends on the strength of magnetoelectric coupling.

A different type of magnetoelectric effect that is based on charge transfer is also con-

sidered in this thesis. In a trilayer comprised of metallic ferromagnet, ferroelectric and

normal metal layers it is shown that an applied voltage is able to enhance the polar-

ization of the ferroelectric, and increase the magnetic moment at one interface (ferro-

magnet/ferroelectric) through spin polarization and charge transfer. The induced surface

magnetism results in shifts of the resonance and standing spin wave mode frequencies. A

new resonance peak is predicted, which is associated with a strong localized surface mo-

ment. Estimates are provided using parameters appropriate to the ferroelectric BaTiO3

iii

and four different ferromagnetic metals, including a Heusler alloy (Fe, CrO2, Permalloy

and Co2MnGe). The calculations use an entire-cell effective medium approximation that

takes into account the polarization profile in the ferroelectric. The metallic ferromag-

netic electrode is treated as a real metal, and the depolarization fields are included in the

determination of the polarization in ferroelectric.

iv

AcknowledgementsI would like to thank Prof. Robert L. Stamps as my PhD supervisor for his continuing

help, suggestion and support throughout my study in Australia. Thank you for this great

experience, Bob.

Thank you to A/Prof. Mikhail Kostylev for his help and advice. Thank you to my

great friends Dr Karen Livesey and Dr. Rhet Magaragia for the help, discussion and

feedback to my work. Thank you to Dr Peter Metaxas for proofreading this thesis. Thank

you to all the members of UWA Condensed Matter Group for being very good friends.

Thank you to all people in the School of Physics including Prof Ian MacArthur and

administration staff including Lydia Brazzale, Amanda Atkinson and Jeff Polard who

helped me resolve travel issues. Thank you to Ausaid international liaison officers including

Chris Kerin and Deborah Pyatt who helped me a lot when I came to Australia and also

with the visa issues of my family.

I would like to acknowledge Ausaid and Australia Goverment for providing me with a

full scholarship throughout my PhD.

Lastly, I would like to thank my long time friend, Angelika Riyandari for the love,

patient and support. Thank you Dito and Yogis for your laugh and smile.

v

Contents

1 Introduction 1

1.1 Introduction to Magnetoelectric Multiferroics . . . . . . . . . . . . . . . . . 1

1.1.1 Coupled magnetization and electric polarization . . . . . . . . . . . 2

1.1.2 Composite magnetoelectric multiferroics . . . . . . . . . . . . . . . . 4

1.2 Introduction to Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Polaritons in magnetic material . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Phonon polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Polaritons in magnetoelectric multiferroics . . . . . . . . . . . . . . . 12

1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 TE and TM Polarization 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Transverse Electric & Transverse Magnetic modes . . . . . . . . . . 15

2.1.2 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Geometry and Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Equation of motion and susceptibility . . . . . . . . . . . . . . . . . . . . . 18

2.4 Description for bulk bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.2 TE modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.3 TM modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Surface Polariton modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 TE surface modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Non-reciprocity of surface modes . . . . . . . . . . . . . . . . . . . . 27

2.5.3 Attenuated total reflection (ATR) . . . . . . . . . . . . . . . . . . . 28

2.5.4 TM surface modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Application to BaMnF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.1 TE modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

2.6.2 Effect of an applied field for TE modes . . . . . . . . . . . . . . . . . 35

2.6.3 TM modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.4 Effect of applied fields for TM modes . . . . . . . . . . . . . . . . . 39

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Un-polarized polaritons 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Dynamic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Theory for bulk bands and surface modes . . . . . . . . . . . . . . . . . . . 47

3.4.1 Dispersion relation for bulk modes . . . . . . . . . . . . . . . . . . . 47

3.4.2 Attenuation constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.3 Surface modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Application to BaMnF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Effect of external field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Ferromagnet resonance shift 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Geometry and screening charges . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Additional Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Theory of spin dependent screening . . . . . . . . . . . . . . . . . . 65

4.3.2 Application to Fe, CrO2, Permalloy and Heusler alloy . . . . . . . . 68

4.4 Effective medium: Susceptibility and Spin waves . . . . . . . . . . . . . . . 69

4.4.1 Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2 Standing spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Conclusion 81

A Energy density for ferroelectrics with electrodes 85

List of figures 88

Bibliography 93

viii

Chapter 1

Introduction

This thesis is divided into two main topics. The first topic, which is presented in chap-

ters 2 and 3, focuses on polaritons in magnetoelectric multiferroics with PML-type cou-

pling which allows for a canted spin system. Here, PML-type magnetoelectric coupling

is associated with the magnetoelectric energy density which is comprised of the electric

polarization P, the magnetization M and the antiferromagnetic vector L. In the second

topic, which is presented in chapter 4, we study composite multiferroics comprised of

metallic ferromagnets and ferroelectrics. In this type of multiferroic, the magnetoelectric

interaction is mediated by spin-dependent screening charges. We focus on the influence of

the magnetoelectric interaction on spin waves in such materials.

1.1 Introduction to Magnetoelectric Multiferroics

Multiferroic materials are those which have at least two ’ferroic’ properties such as ferro-

magnet-ferroelectric, ferromagnet-ferroelastic, etc. in the same phase [1]. In these materi-

als, the magnetoelectric interaction couples the magnetic and the electric responses, allow-

ing the manipulation of the electric polarization using magnetic fields, and vice versa [2].

The first experimental confirmation of magnetoelectric interaction [3,4] trigerred both the

theoretical and the experimental studies in this field. However, due to the lack of mate-

rials which possess magnetoelectric interaction and the weakness of the magnetoelectric

interaction, the interest in this field declined in the early of 1970’s [2]. Due to a better

understanding of the magnetoelectricity and also the development of the experimental

devices and techniques [5], there is again a growing interest in the study of multifer-

roics [2]. New ideas for designing a composite multiferroic had also been proposed where

the multiferroicity can be obtained through strain mediated [6–9] or carrier mediated

1

2 1. Introduction

coupling [10–14].

1.1.1 Coupled magnetization and electric polarization

Given that the ordering of magnetization and electric polarization which is symmetry al-

lowed in coupled system is mostly in the form of ferroelectric-antiferromagnet, the majority

of multiferroic materials found so far have ferroelectric-antiferromagnet ordering [15]. This

class of multiferroic materials was first predicted theoretically by Dzyaloshinskii [16] and

demonstrated experimentally by Astrov [4] in Cr2O3. In most of the previous studies, the

multiferroic model was represented by energy density in Landau formalism as an expansion

of the order parameters. For example, energy density for a ferroelectric-two sublattices

antiferromagnet can be written as [15]

F =λMa ·Mb − K

2

(

M2a,z +M2

b,z

)

−H · (Ma +Mb)

− α1P2 + α2P

4 −E ·P− γP 2 (Ma ·Mb) (1.1)

where the electric polarization P and the magnetization M are the order parameters. Here,

Ma andMb describe the magnetization of sub-lattices, λ represents the exchange constant,

K is the strength of the magnetic anisotropy which favors the magnetic alignment in the

z direction, H is an applied magnetic field, α1 and α2 describe the ferroelectric stiffnesses

and E is an applied electric field. The symmetry-allowed magnetoelectric coupling is

described by the last term of Eq.(1.1) with the strength γ. If we consider that TN is

the Neel temperature of antiferromagnet and TC is the Curie temperature and assuming

TN < TC , the interaction of two subsystems is present at T < TN . As a consequence

of the existence of magneroelectric interaction, The character of electromagnetic waves in

magnetoelectric multiferroic is determined by the electric susceptibility (χe), the magnetic

susceptibility (χm) and the magnetoelectric susceptibility (χme).

The more interesting model is magnetoelectric multiferroic with ferroelectric-antiferro-

magnetic ordering where a weak ferromagnetism is allowed. The weak ferromagnetism is a

phenomena where the spin systems in antiferromagnet cant and generate a residual mag-

netic moment. This phenomena in magnetic material is described by the Dzyaloshinskii-

Moriya (DM) interaction [17, 18] as EMEij = Dij · (si × sj) where si and sj represent

spin of the ion i and j and Dij is the DM tensor. The spin system in a ferroelectric-

antiferromagnet can be canted and result in a weak ferromagnetism if the energy density

of magnetoelectric interaction has the form [15,19] P · (Ma ×Mb). This form is basically

1. Introduction 3

a DM-type interaction with the electric polarization P induces canting in the magnetic

sub-system. In this type of energy density, a reversal of the electric polarization P by

applying an electric field E opposite to the spontaneous polarization leads to the reversal

of the canting angle and the weak ferromagnetism.

The condition for canting for magnetic ordering which is induced by the electric polar-

ization through the magnetoelectric effect has also been proposed for multiferroic BaMnF4.

The energy density of the magnetoelectric interaction which is symmetry-allowed for

BaMnF4 was proposed as [20]

FME =(

β1Py + β2P2y

)

MxLz (1.2)

where β1 and β2 describe the strength of the magnetoelectric couplings, M = Ma +Mb

and L = Ma −Mb represent the weak ferromagnetism and the antiferromagnetic vector.

In this type of coupling, the linear Py in the first term of the right hand side of Eq.(1.2) is

the most responsible term which generates canting of the magnetic sub-lattices. This term

contains the DM-type form P · (Ma ×Mb). The second term which has quadratic Py is

responsible in influencing the dielectric constant [20]. The canting of magnetic sub-lattices

leads to the weak ferromagnetism as Mc = 4πβ1P .

Later, by performing first principle calculation, Ederer [21] and Fennie [22] show that

the DM-type interaction in the form FPML ∝ P · (M × L) is responsible for the canting

of magnetic sub-lattices on multiferroic FeTiO3. This form of magnetoelectric energy

density is basically a general form of the first term of the energy density in Eq.(1.2)

which was previously proposed by Scott for BaMnF4. Weak ferromagnetism reversal can

be performed by reversing the polarization P while keeping the antiferromagnet vector

L unchanged [21]. However, in multiferroic BiFeO3 the cant of the magnetizations is

determined by the rotation of oxygen octahedra around the magnetic ions instead of DM-

type interaction [23].

In chapter 2 and 3 of this thesis, we will use the same type of magnetoelectric energy

density (FPML ∝ P · (M× L)) to study the influence of magnetoelectric effect on surface

modes of polaritons. We find some interesting results for polaritons in the magnetoelectric

multiferroic where canted spin systems are considered, including the effect of the weak

ferromagnetism reversal on surface polaritons.

4 1. Introduction

1.1.2 Composite magnetoelectric multiferroics

Due to enhanced experimental techniques and fabrications, the magnetoelectric effect on

artificially composite materials have been extensively studied during the past decade [2].

Multiferroicity on the composite materials can be tuned and optimized which leads to

a variety of potential applications. Multiferroic composite materials are able to achieve

much higher magnetoelectric coupling than that in single phase materials. For example,

multiferroic composite materials comprised of the piezoelectric ferroelectrics and mag-

netostrictive ferromagnets. These types of composite multiferroics have magnetoelectric

coupling 100 times higher than that in single phase multiferroics [6]. By setting the thick-

ness of the constituent materials, this kind of composite multiferroic can be made with

magnetoelectric constant around 5 V cm−1Oe [24].

There are three mechanisms in obtaining magnetoelectricity from the composite ma-

terials. The first mechanism is associated with the multiferroic composites comprising

of piezoelectric ferroelectric and magnetostrictive ferromagnetic [25, 26]. The magneto-

electric coupling involves ferroelectricity and magnetism through strain. An electric field

creates mechanical stress which is related to the strain by the material’s elastic properties.

The strain yields a magnetic moment through a magnetostrictive effect [25]. On the other

hand, the magnetic field creates strain which is related to the stress. Then, stress yields

an electric polarization through piezoelectric effect.

The second mechanism obtains a magnetoelectric effect through interface bonding [27].

Magnetoelectricc effect is mediated by the overlap between atomic orbitals at the interface.

For example, for Fe/BaTiO3 composites, the orbital of Fe(3d) overlap with the orbital of

Ti(3d). The displacement of ferroelectric atoms at the interface by means of applying

external fields influences the overlap between atomic orbitals, which may then induce the

change in the interface magnetic moment. Using density-functional calculation, It was

predicted that the change in the interface magnetization for Fe/BaTiO3 composite could

reach 120 G with the magnetoelectric coupling α ≈ 0.01 Gcm/V [27].

More recently, a new mechanism was proposed to obtain the magnetoelectric ef-

fect [11–13, 28, 29]. Composite materials which are comprised of a ferroelectric insulator

and ferromagnetic metals are able to show magnetoelectricity through screening carriers.

The conduction electrons in metal screen the electric field or electric polarization. When

the metal is ferromagnet, the screening electrons become spin-dependent because of the

exchange interaction. These spin dependent screening charges lead to the additional mag-

netization of ferromagnet at the ferromagnet/ferroelectric interface [11,13]. The additional

1. Introduction 5

magnetization can be manipulated by controlling the electric polarization using an exter-

nal bias. It was reported that the magnitude of magnetoelectric coupling was comparable

to that in the case strain-mediated magnetoelectric [12].

In chapter 4, we study the influence of magnetoelectricity of a carrier-mediated mul-

tiferroic on the spin wave resonance frequency. First, we calculate the polarization in

ferroelectric layer by considering depolarization effect generated by incomplete screening.

Then by using the calculated value of polarization, the additional magnetization is deter-

mined. In the next step, the ferromagnetic resonance frequency is calculated by employing

the entire-cell effective medium method.

1.2 Introduction to Polaritons

Polaritons are electromagnetic waves that travel in a material with dispersions and prop-

erties are modified through coupling to the elementary excitations of material [30, 31].

For example: in a magnetic medium, the magnetic components of electromagnetic waves

couples to the spin waves leading to the magnetic polaritons. In a dielectric medium, the

electric components of electromagnetic waves couples to the active phonons leading to the

phonon polaritons. Depending on the mechanism of the elementary excitations, polaritons

can be classified as phonon polaritons, exciton polaritons, plasmon polaritons or magnon

polaritons.

The propagation of electromagnetic waves is described with Maxwell equations. Then,

by assuming that the solution is a wave-like form, for example as E,H ∝ ei(k·x−ωt), the

Maxwell equations can be written as

k×E =ω

cB (1.3a)

k×H = −ω

cD. (1.3b)

Here, k is the propagation vector. E and H represent the electric and the magnetic field.

The B and D are the induced magnetic field and the displacement field. By employing the

constitutive equations as Bi = µijHj and Di = ǫijEj , the Maxwell Eqs.(1.3a) and (1.3b)

can be merged to obtain the wave equation such as [30]

k×[

µ−1 · (k×E)]

+ω2

c2ǫE = 0. (1.4)

In isotropic case, subtituting the form of electric field E into the wave equation Eq.(1.4)

6 1. Introduction

results in the explicit dispersion relation as

c2k2

ω2= ǫ(ω)µ(ω) (1.5)

which describes bulk polaritons. These bulk polaritons propagate within material.

In dielectric medium, where the permeability is generally set as one, then the properties

of bulk polaritons in Eq.(1.5) are determined by dielectric constant ǫ(ω). On the other

hand, in magnetic materials, the permeability µ(ω) determines the properties of magnetic

polaritons since the dielectric constant ǫ is usually set as a constant.

Polaritons can display a number of interesting and useful properties, one of which

is surface localization. These surface polaritons propagate along surfaces or interfaces,

decaying exponentially in the direction normal to the surface [32]. Polaritons (especially

surface polaritons) can also display non-reciprocity [33–37], whereby the frequency is not

symmetric under a reversal of the propagation direction: i.e., ω(k) 6= ω(−k). Surface

polaritons at optical frequencies have received much attention in recent years with potential

applications including detectors [38], biosensors [39] and microscopy [40].

The surface polaritons in magnetic system are generally having transverse electric

(TE) character, where the electric field normal to sagittal plane [35]. The transverse

magnetic (TM) modes with magnetic field normal to the sagittal plane is commonly found

on dielectric system. In the next sub-section, we will discuss the TE modes of the simple

magnetic polaritons. Then, we also discuss the TM modes of simple phonon polariton on

dielectric system.

1.2.1 Polaritons in magnetic material

In a magnetic system, the magnetic polaritons are obtained as a mixture of the spin

waves and the electromagnetic waves. Since permitivitty is assumed to be constant, then

permeability, which is strongly dependent on the frequency, has a vital role in determining

the properties of magnetic polaritons [30]. The properties of magnetic polaritons are

characterized by the resonance of permeability. In this sub-section, we consider two cases:

antiferromagnet and ferromagnet.

The dispersion relation for a ferromagnet was first derived theoretically by Walker [41]

in spherical system by using magnetostatic approximation which is valid in the region

where the propagation vector k is much greater than frequency ω, i.e.: k >> ωc . Later,

Auld [42] found a complete dispersion relation for bulk polaritons in a ferromagnet by in-

1. Introduction 7

Figure 1.1: A sketch of the geometry for a semi-infinite ferromagnet . Ferromagnetism ( ~M) along

z axis parallel to the surface. The external magnetic ~H0 is applied parallel to the magnetization~M . Propagation of the surface polaritons is along x axis with wave-number kx.

cluding the electromagnetic modes where the wavenumber k is comparable to the frequency

ω. We first consider a simple ferromagnet with the magnetization is directed parallel to

the z axis as it is illustrated in Fig.1.1. The external magnetic field is also applied parallel

to the z axis. It is assumed that anisotropy is negligible. The electromagnetic waves

equation can be derived from the Maxwell equations as

k×[

ǫ−1 · (k×H)]

+ω2

c2µ(ω)H = 0. (1.6)

By assuming the solution has the form H ∝ ei(k·x) and using magnetic permeability tensor

for the simple ferromagnet such as

µ =

µ1 iµ2 0

−iµ2 µ1 0

0 0 1

(1.7)

where µ1 = 1 + 4πγ2MsH

γ2H2−ω2 and µ2 = 4πγMsω

γ2H2−ω2 , the wave equation Eq(1.6) leads to the

explicit dispersion relation in the form [30,34]

c2k2

ω2= ǫµv = ǫ

(

1 +4πγ2MsB

γ2HB − ω2

)

(1.8)

where B = H + 4πMs. The permeability µv determines the properties of bulk polariton

since it divides the polariton into two branches at around resonance frequency. These bulk

modes are illustrated in Fig.1.2.

The lower branch starts at ω = 0, k = 0 and approaches the resonance frequency

ωr = γ√HB, which is associated to µv = ∞, as k → ∞. The upper branch starts at

8 1. Introduction

−6 −4 −2 0 2 4 6

0

1

2

k

ω

~ ~ L −L −L

SP ω SP

ωr

L

~

Figure 1.2: An illustration of the simple magnetic polariton in ferromagnet. Bulk and surfacemodes are shown. The shaded regions reprsent bulk bands, limited by frequencies ωr and ω. Surfacemodes are indicated by SP. The dashed lines indicated by L represent lightline with ω = ck. Thedashed lines which are indicated by L obey ω = ck√

µ∞

.

frequency ω = γB, which is associated to µv = 0, as k = 0. Then, it approaches the

dashed line L which is associated to ω = ck√µ∞

at high frequency. The permitivity µ∞

represents permitivity background.

The surface polaritons are calculated by considering that a semi-infinite ferromagnet

fills half the space with y < 0 while the other half is vacuum. Here we consider the case

where the surface modes have a pure TE character by considering the wave propagation

normal to the lattice magnetization. The solution of wave equation Eq.(1.6) is assumed

as a plane wave in x direction and decay in the y direction as H ∝ eβyei(kx−ωt). In

vacuum, the solution is assumed to be H ∝ e−βyei(kx−ωt). Here, β and β represent the

attenuation constant in magnetic material and vacuum. The surface localization requires

both β and β are positive. The attenuation constant is derived by subtituting the form

of H into the wave equation Eq.(1.6). Using Maxwell equations, the components of H and

E are determined for both magnetic material and vacuum. Then, by applying Maxwell

boundary condition at the surface, the implicit form of magnetic dispersion relation with

TE modes is derived as [34]

β + βµv − kµ2

µ1= 0. (1.9)

The surface modes were found in the gap of bulk bands where the attenuation constant

was positive. These surface modes are illustrated as thick lines in Fig.1.2. The appearance

of non-diagonal components of permeability (µ2) in Eq.(1.9) leads to the non-reciprocity

of surface polaritons. One branch (positive wave number) has Damon Eshbach modes

[33] where the branch is also exist at the region where k >> ωc . The other branch

(negative wave number) does not have DE modes and terminates at the upper bulk band

1. Introduction 9

[34]. The unique non-reciprocity of surface modes with respect to localization was found

by Karsono et al [43] for a ferromagnetic slab. Even though the dispersion relation is

reciprocal, the positive k branch is localized in the surface while the negative -k branch

propagates at the other interface. The first experimental observation of magnon polariton

in a ferromagnet had been reported by analyzing transmission and reflection spectra of

ferromagnetic resonance K2CuF4 [44].

−6 −4 −2 0 2 4 6

0

1

2

3

k

ω

ωr

SP

L

SP

−L

(a)

−8 −4 0 4 8

0

4

k

ω

ωra

−L

SP

SP

L

ωrb

(b)

Figure 1.3: The dispersion relation for a semi-infinite anti ferromagnet (AFM). In (a) the dis-persion relation is shown for the case without the application of external magnetic field. In (b)bulk and surface mode dispersions are shown with the application of magnetic field parallel to themagnetization M. The shaded regions represent bulk modes. The surface modes are illustratedby the thick lines. The dashed lines indicated by ”L” are light lines with ω = ck.

In the second case, we consider a semi-infinite two sub-lattices uniaxial antiferromag-

net. The exchange interaction between spins on different sub-lattices leads to the an-

tiparallel spins of the two sub-lattices. We assume that the anisotropy field Ha pins the

sub-lattices to the z direction. Then, without applied magnetic field, the resonance fre-

quency ωr is described by ωr = γ(

H2a + 2HaHe

)1/2. The permeability tensor has the

form as in Eq.(1.7),where the components are defined as: µ1 = 1 + 4πγ2HaMs

ω2r−ω2 and µ2 = 0.

Using a procedure similar to the ferromagnet case, the bulk and the surface dispersion

relations for antiferromagnet are derived. Assuming the propagation is perpendicular to

the magnetization, the dispersion relation for bulk is determined as [35]

c2k2

ω2= ǫµ1(ω). (1.10)

The permeability µ1 in Eq.(1.10) divides the dispersion relation into two branches around

resonance frequency ωr. The lower branch approaches resonance frequency ωr as k → ∞,

while the upper branch starts at frequency ω2 = ω2r + 8πγ2MsHa associated to µ1 = 0

10 1. Introduction

as illustrated in Fig1.3(a). The existence of magnon polariton in antiferromagnet was

confirmed experimentally for the first time in antiferromagnet FeF2 and FeF2:MnF2 by

Sanders et al. [45]. Through the study of the transmission spectrum, they found the

minimum transmission around resonance frequency which related to the bulk polariton

dispersion. Later using far-infrared reflectometry on CoF2, Haussler [46] also detected

their presence in the reflectivity spectrum.

In the gap between those two frequencies, the surface modes can be found. Here, we

also consider the surface modes have TE character. These surface modes are illustrated

as thick lines in Fig.1.3(a). The implicit dispersion relation for surface modes which

propagate perpendicular to the static magnetization was determined as [35]

µ1β + β = 0. (1.11)

Given that both the attenuation constant β and β are positive, then the surface polaritons

can be found where permeability µ1 is negative. The surface polaritons in this configura-

tion are reciprocal since the sign of propagation vector is not matter, i.e, ω(k) = ω(−k).

In the midle of 1990’s, the surface magnon polaritons were confirmed experimentally by

using attenuated total reflection (ATR) method on FeF2 [47]. By variating the angle of in-

cidence, the dispersion relation of surface polaritons were determined experimentally [48].

The application of an external magnetic field parallel to the static magnetization splits

degeneracy of spin modes into two components which change the resonance term in per-

meability as: 1ω2r−ω2 → 1

ω2r−(ω+γH)

2 + 1ω2r−(ω−γH)

2 . This splitting leads to the existence

of two gaps around the frequencies ωbr = ωr − γH and ωa

r = ωr + γH which divide the

bulk dispersion relation into three branches [49], as illustrated in Fig.1.3(b). An applied

magnetic field leads to the appearance of the non-diagonal components of the permeability

tensor (µ2 6= 0), which then modify the expression of dispersion relation for surface modes

into [35]

β +µ1β + µ2k

µ21 − µ2

2

= 0. (1.12)

The appearance of propagation vector k lead to the non-reciprocity of surface modes, i.e,

ω(k) 6= ω(−k). These surface modes are illustrated as thick lines in Fig.1.3(b). Using

ATR, this non-reciprocity had been probed experimentally [47,48].

1. Introduction 11

1.2.2 Phonon polaritons

Phonon polariton is a mixed modes between electromagnetic waves and phonon which

results from the lattice motions. The lattice motions are associated with the dielectric

constant, and therefore the properties of phonon polariton are strongly dependent on

the resonance characteristic of the dielectric constant. Phonon polariton can be found

in dielectric or ferroelectric system. The first theoretical work on phonon polariton was

provided by Huang [50] who derived dispersion relation of infinite isotropic diatomic crys-

tal. In isotropic dielectric media, dispersion relation for bulk was derived from the simple

relation Eq.(1.8) by setting the permeability µ → 1 as

c2k2

ω2= ǫ(ω). (1.13)

Here, the dielectric constant is described as ǫ(ω) = ǫ∞ω2LO−ω2

ω2TO−ω2 , where ωTO is transverse op-

tical phonon frequency, ωLO is longitudinal optical phonon frequency and ǫ∞ is background

permeability [51]. The bulk phonon polariton has two branches which are separated by a

gap between ωTO and ωLO where those frequencies are related by Lydanne-Sachs-Teller

relation ω2LO = ǫ∞

ǫ(0)ω2TO. The lower branch starts at wavenumber k = 0 at frequency ω = 0

and approaches asimptotically frequency ωTO at k → ∞ while the upper branch starts

at frequency ωLO and approches the line ck =√

ǫ(0)ω. These bulk bands can be seen

as shaded regions in Fig.1.4. The bulk modes of phonon polaritons had been confirmed

experimentally through Raman spectroscopy on α-quartz [52,53].

−8 −4 0 4 8

0

2

ωLo

ωTo

SP SP

L −L L −L ~ ~

Figure 1.4: An illustration of the simple magnetic polariton in ferromagnet. Bulk and surfacemodes are shown. The shaded regions reprsent bulk bands, limited by frequencies ωr and ω. Surfacemodes are indicated by SP. The dashed lines indicated by L represent lightline with ω = ck. Thedashed lines which are indicated by L obey ω = ck√

ǫ∞.

12 1. Introduction

The dispersion relation of surface phonon polariton in a semi-infinite isotropic dielectric

was derived using similar procedures as those for surface magnon polariton [30]. Solving

wave equation Eq.(1.4) and considering the boundary condition at surface, it was found

that the solution has magnetic field in-plane parallel to the surface (Transverse Magnetic

mode) [32].

The implicit dispersion relation for surface modes was expressed as

ǫ(ω)β + β = 0. (1.14)

The surface phonon-polaritons were found at the gap of bulk polariton between frequen-

cies ωTO and ωLO where the dielectric constant was negative. These surface modes are

illustrated as the thick lines in Fig.1.4. The surface modes start from the lightline at the

frequency ωTO and asymptotically aproach the limit ωs = ǫ+ǫǫ∞+ǫωTO. Since the surface

dispersion relation did not have a term with linear k, hence the surface phonon polari-

tons for isotropic case is reciprocal. By using ATR method, the experimental dispersion

relation for surface modes in GaP was probed [54].

1.2.3 Polaritons in magnetoelectric multiferroics

A most interesting class of polaritons occur in multiferroics. In magnetoelectric multi-

ferroics, both permitivity and permeability depend on the frequency, i.e, ǫ(ω) and µ(ω),

and the magnetoelectric interaction is described by the magnetoelectric susceptibility [55].

Hence, the characters of polaritons in magnetoelectric multiferroic are determined by the

electric susceptibility (χe), the magnetic susceptibility (χm) and the magnetoelectric sus-

ceptibility (χme). Then, it is expected that both phonon polaritons and magnon polaritons

can be found. The two gaps divide the bulk polaritons into three branches. The first gap

was located at around the optical transverse frequency ωTO while the second gap was

found around the magnetic resonance frequency [36, 56]. The bulk modes around the

phonon resonance frequency can be categorized as phonon polaritons while those near the

magnetic resonance frequency can be categorized as magnetic polaritons.

Since there is coupling between the electric and the magnetic system, the wave equa-

tions Eq.(1.4) or Eq.(1.6) can not be used to derive the dispersion relation for bulk. The

derivation of the dispersion relation for bulk modes starts by assuming the plane waves

E and H have the form as E,H ∝ ei(k·x−ωt). Then, subtituting E, D, B and H into the

Maxwell equations Eqs.(1.3a) and (1.3b) leads to a matrix equation. The bulk disper-

sion relation was determined by setting the matrix determinant to zero. In his series of

1. Introduction 13

papers [36,57–59], Barnas examined the bulk polaritons in magnetoelectric multiferroics.

Since the phonon resonance frequency is higher than the magnetic resonance frequency in

multiferroic BaMnF4, he showed that bulk modes in BaMnF4 consist of phonon polaritons

in the upper branch of dispersion relation and magnon polaritons in the lower branch [36].

The most interesting result was the appearance of the non-reciprocity of the bulk disper-

sion relation due to the magnetoelectric interaction in the propagation direction that is

parallel to the antiferromagnet axis. According to Barnas [36], the magnetoelectric energy

contribution change under time inversion, since this inversion changes the sign of magnetic

field but leave the electric field unchanged. Hence, the magnetoelectric energy contribu-

tion in two opposite directions of propagation will be different, leading to the different

frequency,ω(k) 6= ω(−k). If the phonon frequency is near the magnon frequency, there is

the possibility of using a static electric field to modify the polariton’s frequency to obtain

crossover between phonon and magnon polaritons [56].

Non-reciprocity has also been reported for surface modes in tetragonal magnetoelec-

tric antiferromagnets [60, 61]. Using symmetry allowed magnetoelectric energy density

as Fme = γijkMiLβPk, the surface dispersion relation which involves linear propagation

vector k was determined. The linear k in dispersion relation leads to the non-reciprocity.

However, if quadratic electric polarization and quadratic magnetization are used as mag-

netoelectric energy density [62, 63] such as Fme = P 2M2, the surface dispersion relation

is reciprocal. This is because quadratic P and quadratic M are not affected under time

inversion. Hence, the magnetoelectric energy contribution also remains unchange.

InChapter 2 and chapter 3, we study the polaritons in magnetoelectric multiferroics

with PML-type interaction by considering the canting angle in the calculation. In chapter

2, we focus on the transverse electric (TE) and transverse magnetic (TM) polarization

of surface modes which can be obtained by considering the configuration where the wave

vector is parallel to the polarization and perpendicular to the easy axis. Since neither the

permeability nor the permitivity are isotropic, different orientations of the wave vector

result in different polariton modes. In a certain wave vector direction, the surface modes

are neither TE nor TM polarized. This un-polarized condition is discussed in chapter

3. Un-polarized surface polaritons can be obtained in the configuration where the wave-

vector is parallel to the easy axis of magnetic sub-lattices and perpendicular to the electric

polarization. In this chapter we demonstrate that the superposition of two plane waves is

required to generate the un-polarized surface polaritons.

14 1. Introduction

1.3 Outline of Thesis

We begin chapter 2 by examining the dispersion relation for both the bulk and the

surface modes of polaritons in a magnetoelectric multiferroic with PML-type coupling. In

the calculation, we explicitly include the canting angle. Then, we apply the theory by

choosing BaMnF4 as an example material. In this chapter, we focus on transverse electric

(TE) and transverse magnetic (TM) modes. These modes can be obtained by considering

the wave number k parallel to the electric polarization P. The influence of both the

external electric and magnetic fields is also studied.

In chapter 3, by considering the wave number parallel to the antiferromagnet axis,

we study the surface polaritons which are neither TE nor TM polarized. Here, based

on the expression of the attenuation constant, it can be shown that the superposition

of two plane waves are needed. The dispersion relation expression for both bulk modes

and surface modes is derived. We also study the effect of an external field on the surface

polaritons.

In chapter 4 we study the carrier-mediated magnetoelectric effect in a composite

multiferroic comprised of ferroelectric (FE)/metallic ferromagnet (FM) heterostructure.

We start by calculating the additional magnetization due to screening carriers on metallic

ferromagnet. The BaTiO3 parameters are used for ferroelectric layers. For metallic ferro-

magnet, we use four example materials(Fe, CrO2 , Permalloy and a Heusler alloy). Then,

by using effective medium theory, we study the effect of additional magnetization on the

resonance frequency of spin waves in the metallic ferromagnetic layer.

A summary of the thesis is presented in chapter 5. We also provide an outlook

towards further work which could build upon this thesis.

Chapter 2

Surface and bulk polaritons in a

PML-type magnetoelectric

multiferroic with canted spins:

TE and TM polarizations

2.1 Introduction

2.1.1 Transverse Electric & Transverse Magnetic modes

The polarization of polaritons can be distinguished into two well-known groups [64]: trans-

verse electric (TE) and transverse magnetic (TM), with respect to the sagittal plane. Sag-

gital plane is defined as a plane where the normal vector of the surface and the propagation

vector lie on. In TE modes, the electric field is normal to the sagittal plane while the mag-

netic components lie on the sagittal plane. In TM modes, the magnetic component is

normal to the sagittal plane while the electric components lie on the sagittal plane. The

TE surface polaritons can be found in a magnetic system with the configuration where

the propagation vector is perpendicular to the applied field or static magnetization, which

is called Voigt geometry [64]. It was reported that TE modes can be found in ferromag-

nets [34] and antiferromagnets [35, 65]. The TM modes of surface polaritons is generally

found in the dielectric system [31,32,66].

The appearance of the polarization in polaritons can be recognized by examining the

magnetic H field and the electric E field in the Maxwell equations. Assuming H and the

E fields have the form of plane waves and using appropriate consecutive equations relating

15

16 2. TE and TM Polarization

D to E and B to H into the curl of H and E of the Maxwell equations, result in a matrix

equation. The polarized modes are obtained if the matrix equation can be separated into

several smaller matrix equations. For example, a matrix form of the Maxwell equation for

magnetic system in Voigt geometry can be separated into two smaller matrix equations

which represent the TE and the TM modes.

2.1.2 Outline of the chapter

In this chapter, the discussion is started by describing the geometry and also discussing the

energy density of the system in Section (2.2). The equation of motion and the susceptibility

of the system are presented in Section (2.3) and then used in the calculation of dispersion

relation for bulk and surface modes in Section (2.4). In this section we also discuss the

non-reciprocity of the TE surface modes and the attenuated total reflection (ATR) as

an effective method to probe surface modes. The results of bulk and surface polaritons

for both TE and TM polarizations are presented in Section (2.6) for the parameters of

BaMnF4 as a material sample. In this section, we also discuss the effect of an applied

external field on surface modes.

2.2 Geometry and Energy Density

The geometry of the system is illustrated in Fig.2.1. A semi-infinite multiferroic sample fills

the half space with z < 0. The electric polarization P lies in-plane parallel to y axis. The

two antiferromagnetic sub-lattices Ma and Mb which have dominant component along z

direction, lie perpendicular to the electric polarization. These two antiferromagnetic sub-

lattices are allowed to be cant with canting angle θ. We assume symmetric canting such

that |Ma| = |Mb| = Ms. The canting generates weak ferromagnetism M which lies in-

plane perpendicular to the polarization. We assume that the surface polaritons propagate

along the y direction. Thus the sagittal plane is parallel to the yz plane. Hence, the

TE mode has the electric component Ex normal to the sagittal plane while the magnetic

field has Hy and Hz components which lie in the sagittal plane. In the TM mode, it is

the component of magnetic Hx which is normal to the sagittal plane, while the electric

components Ey and Ez lie in saggital plane.

The energy density for the dielectric component is a fourth order Ginzburg-Landau

(G-L) energy density which is assumed to be

Fe =1

2ζ1P

2y +

1

4ζ2P

4y +

1

2∆1

(

P 2x + P 2

z

)

+1

4∆2

(

P 4x + P 4

z

)

− PyEy. (2.1)

2. TE and TM Polarization 17

The first and the second terms on the right hand side of the Eq.(2.1) represent the energy

density for the y component of polarization with dielectric stiffnesses ζ1 and ζ2. Here, the

dielectric stiffness ζ1 is a function of temperature as ζ1 = ζ(T − Tc) where ζ represents

dielectric stiffness background and Tc is Curie temperature. The third and the fourth

terms represent contributions of x and z components with dielectric stifnesses ∆1 and

∆2. The last term is a contribution from the external electric field applied parallel to the

spontaneous polarization P.

Figure 2.1: Canting of two magnetic sub-lattices (Ma and Mb by an angle θ produces a weak

ferromagnetism ( ~M) along x axis parallel to the surface. The spontaneous polarization (~P ) isassumed to lie in-plane parallel to the surface. Propagation of the surface polaritons is along yaxis with wave-number ky.

By restricting to long wavelength so that spatial dispersion can be neglected, the

contribution of magnetic component to the energy density is then assumed to be of the

form

FM = λMa ·Mb − K

2

[

(Ma · z)2 + (Mb · z)2]

− (Ma +Mb) ·H. (2.2)

Here, the first term on the right hand side represents exchange energy with a strength

λ > 0. The second term is anisotropy energy with anisotropy constant K. The final term

represents Zeeman energy of interaction with an external magnetic field.

The magnetoelectric coupling is assumed to be of the PML-type as discussed in chapter

1 which allows canting of the antiferromagnetic sub-lattices. We introduce the weak

magnetization Mx and the longitudinal component of magnetization Lz. In terms of

the canting angle θ, they can be expressed as Mx = 2Ms sin θ and Lz = 2Ms cos θ. Then,


the magnetoelectric coupling can be re-written as

FME = −2αPyM2s sin 2θ. (2.3)

Here, α is magnetoelectric constant which represents the strength of magnetoelectric in-

teraction. The magnetoeletric energy density FME will be high if the magnetoelectric

interaction is strong which is represented by the high value of magnetoelectric constant

α. Later, it will be shown that the magnetoelectric constant α also influence the canting

angle of magnetic sub-system.

The canting angle θ can be determined by minimizing the magnetic and magnetoelec-

tric energies density with respect to the canting angle, ∂(FM+FME)∂θ = 0. This results in

the condition

Ho cos θ −1

2KM s sin 2θ + 2αPyMs cos 2θ − λMs sin 2θ = 0. (2.4)

In the absence of an external magnetic field, equation (2.4) simplifies to

tan(2θ) =4αPy

K + 2λ. (2.5)

Note that if we use a negative magnetoelectric constant α, the canting angle will also

have a negative value describing a weak ferromagnetism Mx which is aligned along −x

direction. Later, the condition in Eq.(2.4) or (2.5) is also obtained as a static part in the

equation of motion. Since the polarization, the exchange field and the anisotropy can be

measured, the magnetoelectric coupling can be estimated by using Eq.(2.5).

2.3 Equation of motion and susceptibility

In order to solve the electromagnetic boundary value problem for the surface and bulk

polariton modes, we need constitutive relations for the dielectric and magnetic responses.

We consider a linear response and calculate the permitivitty and permeability using a

set of the equation of motion derived from the Eqs.(2.1), (2.2) and (2.3). We start by

calculating the magnetic and electric susceptibilities from the equation of motion for the

magnetic and the electric responses which are given by magnetic torque equation and

Landau-Khalatnikov equation of motion, as in Ref. [60–62]. The magnetic and the electric

susceptibilities provide information about the resonant response of the spin and electric

dipoles. The dynamic susceptibilities can be obtained from the magnetic and electric


equations of motion without damping. The magnetic torque equation are of the form

[35,37,51]

M = γM×( −∂

∂M(FM + FME)

)

(2.6)

and dielectric response is obtained from [60–62]:

P = −f∂

∂P(FE + FME) (2.7)

where γ and f are gyromagnetic ratio and the inverse of phonon mass. Here, the terms

−∂∂M(FM+FME) and

−∂∂P(FE+FME) represent the magnetic and the electric effective fields.

For small amplitude response, the magnetization and polarization can be written as

Ma =(

Ms sin θ +max,m

ay,Ms cos θ +ma

z

)

Mb =(

Ms sin θ +mbx,m

by,−Ms cos θ +mb

z

)

(2.8)

P = (px, Po + py, pz)

where mai ,m

bi and pi represent the dynamic components proportional to e−iωt. Here, the

dynamic components are assumed to be much smaller than the static part, mi << Ms

and pi << P.

The substitution of the set Eq.(2.8) into the equations of motion Eqs.(2.6) and (2.7)

yields a set of dynamic equations which after linearization yields

−iωmx = (ωa cos θ + 2ωme sin θ) ly, (2.9)

−iωmy = −2γMshz sin θ + (ωo − ωa sin θ + 2ωme cos θ)mz

− (ωa cos θ + 2ωme sin θ) lx, (2.10)

−iωmz = 2γMshy sin θ − (ωo + 2ωme cos θ)my, (2.11)

−iωlx = −2γMshy cos θ + (ωa cos θ + 2ωex cos θ + 2ωme sin θ)my, (2.12)

−iωly = 2γMshx cos θ − (2ωex cos θ + ωa cos θ + 4ωme sin θ)mx

+ (ω + 4ωme cos θ − ωa sin θ − 2ωex sin θ) lz

+ 4γαM2s cos 2θpy, (2.13)

−iωlz = − (2ωex sin θ + 2ωme cos θ + ωo) ly, (2.14)

ω2

fpx = ∆1px − ex, (2.15)

ω2

fpy =

(

ζ1 + 3P 2o ζ2)

py − 2αMs (mx cos θ + lz sin θ)− ey, (2.16)


and

ω2

fpz = ∆1pz − ez. (2.17)

The notations used above are in units of frequency and are defined by: ωa = γKMs as

the effective magnetic anisotropy field, ωex = γλMs as the exchange field, ωme = γαPoMs

as the magnetoelectric coupling and ωo = γHo as the external magnetic field.

There are two groups of coupled equations from the set of dynamic equations above.

The first group consists of Eqs.(2.10),(2.11) and (2.12). Their solution gives magnetic sus-

ceptibility χmyy, χ

mzz and χm

yz. Together with the electric susceptibility component χexx given

by electric dynamic equation Eq.(2.15) which is not coupled to the magnetic dynamics,

these susceptibilities are associated with the electric field Ex and magnetic components

Hy and Hz. These field components describe TE modes where the electric field is normal

to the sagittal plane and magnetic components lie in the sagittal plane.

The second group is comprised of coupled Eqs.(2.9), (2.13), (2.14), (2.16) and the

uncoupled equation (2.17). These give the magnetic susceptibility χmxx, the electric sus-

ceptibility χeyy and χe

zz and also the magnetoelectric susceptibilites χmexy and χem

yx . These

susceptibilities are associated with magnetic fieldHx, which is normal to the sagittal plane,

and electric field components Ey and Ez which lie in the sagittal plane and describe TM

modes.

For the TE modes, the equations of motion for the magnetic components do not coupled

directly to the equation of motion for the electric component, so that χem = χme = 0.

Hence the susceptibilities can be determined as

m = χmh and p = χee (2.18)

with magnetic components are given by

χmyy =

2γMs (ωa cos 2θ + 2ωme sin 2θ + ω sin θ)(

ω2afm cos2 θ +Ω2

me,TE +Ω2,TE − ω2

) (2.19a)

χmzz =

2γMs (ωme sin 2θ + ωo sin θ)(

ω2afm cos2 θ +Ω2


) (2.19b)

χmyz = −χm

zy =i2γMs (ω sin θ)

(

ω2afm cos2 θ +Ω2


) . (2.19c)

Here ω2afm = ωa (ωa + 2ωex ) represents the antiferromagnet (AFM) resonance frequency.


The contribution of magnetoelectric coupling to the resonance frequency is represented by

frequency Ωme,TE which is defined as

Ω2me,TE = 2ωme

(

2ωme +1

2ωa sin 2θ + ωex sin 2θ

)

. (2.20)

The frequency Ω,TE is the frequency shift due to an external magnetic field and is given

by the relation

Ω2,TE = ωo [ωo − ωa sin θ + 4ωme cos θ] . (2.21)

Even though there is no direct coupling between magnetic and electric parts rep-

resented by a ME susceptibility, the ME frequency ωme still appears in the magnetic

susceptibility, as seen in Eqs.(2.19a)-(2.19c). This results from the fact that the canting

condition involves the electric polarisation.

It should be noted here, that the susceptibility component χmyz describes the canting

condition. This component, which also appears in the case of canting generated by apply-

ing a magnetic field perpendicular to the sub-lattices magnetization [67], disappears when

canting angle vanishes (θ −→ 0).

Finally, by considering that the permitivitty is ǫ =ω2L−ω2

ω2T−ω2 [68], the electric susceptibility

for χexx can be written as

χexx =

1

4π

(

ωx2

L − ω2

ωx2

T − ω2− 1

)

(2.22)

where ωxL is the frequency of the longitudinal phonon mode, and ωx

T is the frequency of

the transverse phonon mode.

The relevant magnetic susceptibilitiy for TM modes is given by

χmxx =

1

2πωs

(

ωa cos2 θ + ωme sin 2θ

)

Cmx

(ω2m − ω2)

+Cex

(

ω2ey − ω2

)

(2.23)

where ωs = γ4πMs. The electric susceptibilities are

χeyy = f

Cmy

(ω2m − ω2)

+Cey

(

ω2ey − ω2

)

(2.24)

and

χezz =

f

(ω2ez − ω2)

(2.25)

where the frequency ωez is the phonon frequency along z direction. The magnetoelectric


susceptibility is

χmexy = χem

yx =Cme

(ω2m − ω2)

− Cme(

ω2ey − ω2

) . (2.26)

The frequencies ωey and ωm are defined as ω2ey = ω2

ey + δ and ω2m = ω2

m − δ, where δ is

frequency shift which is expressed in the form

δ =1

2

[

(

ω2ey − ω2

m

)2+ 4Ω4

C

]1/2−(

ω2ey − ω2

m

)

(2.27)

where

Ω4c = Ccωs cos 2θ

(

ωa cos2 θ + 2ωex sin

2 θ − ωo sin θ)

(2.28)

with Cc = 2πα

2M2Sf . Here, since the value of δ is positive, the magnetoelectric coupling

shifts the magnetic frequency down to a slightly lower value and shifts up the phonon

frequency to a slightly higher value, as it is also discussed in Ref. [55] for the case of

BaMnF4. Parameter ωey is the frequency of the soft phonon mode along the direction of

spontaneous polarization and ωm is the magnetic resonance frequency:

ω2m = ω2

afm +Ω2me,TM +Ω2

,TM . (2.29)

Here ωafm is resonance frequency of the the canted antiferromagnet,

ω2afm = ωa (ωa + 2ωex ) cos

2 θ + 2ωex (ωa + 2ωex ) sin2 θ, (2.30)

Ωme,TM is related to the magneto-electric interaction,

Ω2me,TM = 8ω2

me + 2ωme (ωa − 2.5ωex ) sin 2θ (2.31)

and Ωo,TM is related to the external magnetic field,

Ω2,TM = ω [ω + 6ωme cos θ + (4ωex − ωa) sin θ] . (2.32)

Other parameters in equations (2.23) to (2.26) are defined as

Cmx =(

ω2ey − ω2

m

)

/(

ω2ey − ω2

m

)

(2.33)

Cex = δ/(

ω2ey − ω2

m

)

(2.34)

Cey =(

ω2ey − ω2

m

)

/(

ω2ey − ω2

m

)

(2.35)

Cmy = δ/(

ω2ey − ω2

m

)

(2.36)


and

Cme = Cαfωs cos 2θ (ωa cos θ + 2ωme sin θ) /(

ω2ey − ω2

m

)

, (2.37)

where Cαf = 1παMsf .

From the expression for the susceptibilities above, it can be seen that the applied

magnetic field directly influences the susceptibilities through Ω. By way of contrast, the

applied electric field changes the susceptibilities indirectly by affecting the magnitude of

the spontaneous polarisation.

The permeability and permittivity of the medium can be obtained from the suscepti-

bilities above by using the relation µ = 1 + 4πχm and ǫ = ǫ∞(1 + 4πχe) which can be

written in the matrix form as

↔µ =

µxx 0 0

0 µyy iµyz

0 −iµyz µzz

(2.38)

and

↔ǫ =

ǫxx 0 0

0 ǫyy 0

0 0 ǫzz

(2.39)

with the tensor for magnetoelectric susceptibility given as

↔κme =

0 χmexy 0

χmexy 0 0

0 0 0

. (2.40)

The constitutive equations relating B,D with H,E for the magnetoelectric system are

defined as

B =↔µH+ 4π

↔κmeE and D =

↔ǫE+ 4π

↔κmeH. (2.41)

Inserting the permeability↔µ , the permitivity

↔ǫ and magnetoelectric susceptibility in

Eqs.(2.38)-(2.40) into constitutive equation, it can be seen that the magnetoelectric sus-

ceptibility χme is not involved in the constitutive equations for the field components which

are associated with TE modes. Hence for the TE modes, the magnetic induction B is re-

lated to the magnetic field H through the simple relation B = µH. Also the relation

between field D and E fields become D = ǫE.


2.4 Description for bulk bands

2.4.1 General description

The description of electromagnetic waves which propagate in the material is given by the

dispersion relation of the system. The calculation to derive dispersion relation starts with

the Maxwell equation for curls of H and E

∇×E = −1

c

∂B

∂t(2.42a)

∇×H =1

c

∂D

∂t. (2.42b)

The next step is evaluating Eqs.(2.42a) and (2.42b) for the solutions which have plane

wave character with the propagation vector k as E,H ∝ eik·x, hence the rotational of

Maxwell Eqs.(2.42a) and (2.42b) can be re-written in the form

k×E =1

c

∂B

∂t(2.43a)

k×H = −1

c

∂D

∂t. (2.43b)

Next, by considering constitutive Eq.(2.41) which describes the response of medium to the

fields, the Maxwell Eqs.(2.42a) and (2.42b) yield the relation in matrix form as

0 kz ky ǫxxωc 0 0

0 iµyyωc µyz

ωc ikz 0 0

0 −µyzωc µzz

ωc −iky 0 0

(

kz + 4πχme ωc

)

0 0 0 ǫyyωc 0

−ky 0 0 0 0 ǫzzωc

µxxωc 0 0 0 −

(

kz + 4πχme ωc

)

0

Hx

Hy

Hz

Ex

Ey

Ez

= 0.

(2.44)

Here, we consider the propagation vector as k = kyy + kz z. It is obvious from matrix

Eq.(2.44) that the equation can be separated into two groups. One group which is TE

modes is associated with the fields Ex, Hy and Hz. The second group which is involving

ME susceptibility is associated to TM mode with the magnetic field Hx is perpendicular

to the plane of incidence while the electric components Ey and Ez lie on that plane.


2.4.2 TE modes

The description for TE modes can be derived from the part of matrix Eq.(2.44) which is

associated to Ex, Hy and Hz fields. However, since the dynamic coupling does not exist

in TE modes, hence the dispersion relation for bulk polaritons can also be calculated from

the macroscopic electromagnetic wave equation as

∇2H−∇ (∇ ·H)− ǫxc2

↔µ ·∂2H/∂t2 = 0. (2.45)

For the TE modes, the permeability connecting B with H is defined by simple relation↔µ=

↔I +4π

↔χm, hence the implicit expression for the bulk mode frequency for the propa-

gation which is parallel to the spontaneous polarization is:

k2y = ǫxx

(ω

c

)2(

µyyµzz − µyz2)

µyy. (2.46)

Dispersion relation in Eq.(2.46) has two zeros, one is from the condition ǫxx = 0 and

the other from f (µ) = µyyµzz − µ2yz = 0. It also has two resonance poles from ǫxx and

another from the condition µyy = 0. These zeros and poles describe the gap between

bulk bands where the surface modes may exist. As a result, three bands of bulk polariton

modes are expected with two gaps separated between them. However, in the case where

the magnetoelectric coupling is absent, then the canting angle will vanish result in the

dissapearance of the permeabilities µyy and µyz which in turn give the expression for the

dispersion relation as k2y = ǫxxω2

c2 . Hence, without ME coupling the magnetic resonance

does not involve in the bulk’s dispersion relation in TE modes. Numerical results for bulk

and surface polaritons will be presented in Section 2.6.

2.4.3 TM modes

The electric and magnetic dynamics, especially permeability component µxx and permi-

tivity component ǫyy, are coupled for TM polarisation via ME interaction, and the dis-

persion relations for the bulk modes are obtained by solving the electromagnetic Maxwell

Eqs.(2.42a) and (2.42b) and assuming traveling modes of the form E,H ∼ ei(kyy+kzz−ωt).

By considering the configuration which is discussed in Section (2.2), the Maxwell equations


for TM modes can be taken from the TM part of Eq.(2.44) as

(

kz + 4πχme ωc

)

ǫyyωc 0

−ky 0 ǫzzωc

µxxωc −

(

kz + 4πχme ωc

)

0

Hx

Ey

Ez

= 0. (2.47)

The condition of nonvanishing solutions of Eq.(2.47) requires the matrix determinant

to be zero, leading to the expression of dispersion relation for the propagation vector

parallel to the spontaneous polarization as

ǫyyk2y =

[

ǫyyµxx − (4πχme)2]

ǫzzω2

c2. (2.48)

The dispersion relation has solutions determined by the three zeros, one from the dielectric

constant ǫzz and the other two zeros from the function f(µ, ǫ) = µxxǫyy − (4πχme)2. The

dispersion relation Eq.(2.48) diverges at the resonance poles of ǫzz and µxx. The third

pole is found from the condition ǫyy = 0. Since there are three zeros and three poles on

the expression of dispersion relation, it is expected there are three gaps separating four

bulk bands.

In the case where the ME interaction is absent, the magnetoelectric susceptibility

χme −→ 0, which in turn simplify the dispersion relation to k2y = ǫzzµxxω2

c2. Hence, there

will only two poles and two zeros associated to ǫzz and µxx. The gap around soft phonon

frequency ωey no longer exists.

2.5 Surface Polariton modes

2.5.1 TE surface modes

The expression for the dispersion relation for surface modes is obtained by assuming

solutions that propagate as a plane wave in the y direction and decay in the z direction

as:

H ∼ eβzei(kyy−ωt) for z < 0 (2.49a)

H ∼ e−βozei(kyy−ωt) for z > 0 (2.49b)

where β and βo are positive real attenuation constants for the sample and vacuum, re-

spectively. An implicit relation for the attenuation factor β of the medium is derived by


substituting Eq.(2.49a) into the wave Eq.(2.45):

µzzβ2 = µyyk

2y − ǫxx

(ω

c

)2 (µyyµzz − µ2

yz

)

. (2.50)

An explicit relation for the attenuation constant β (in vacuum) is given by

β2o = k2y −

(ω

c

)2. (2.51)

The attenuation constant is also required to determine the bulk region. The imaginary

value of the attenuation constant describes the bulk region. In the other hand, the real

value represents the region where the surface modes may exist.

An implicit surface mode dispersion relation is calculated by requiring continuity of

tangential H and normal B at the interface z = 0. Using Eqs.(2.50) and (2.51), we find

µzzβ +(

µyyµzz − µ2yz

)

βo + µyzky = 0. (2.52)

The dispersion implied by Eq.(2.52) describes magnetic surface polaritons for the weak

ferromagnet. From symmetry considerations, we know that magnetic surface polaritons

can be nonreciprocal [64]. The direction of ky matters in Eq.(2.52) and nonreciprocal

propagation arises as we will discuss later.

A key point is that the nonreciprocity of the surface modes depends strongly on the

canting angle. If the canting angle is zero, then µyz → 0 and µzz → 1, and the dispersion

relation in Eq.(2.52) becomes

β + µyyβo = 0. (2.53)

In that case, the surface modes are reciprocal under reversal of k. Note that in the cal-

culation, we exclude the damping factor, since the purpose of the study is to understand

on how the magnetoelectric coupling affecting the dispersion relation. Therefore, we focus

on how the electric and magnetic resonance frequencies limit the bulk and surface polari-

ton. However, it is not difficult to include damping factor on the calculation by changing

ω → Γω, where Γ describes damping factor. By including damping, we can calculate the

distance of the polaritons propagation.

2.5.2 Non-reciprocity of surface modes

The unique property of TE surface mode is non-reciprocity. In magnetic system, this

non-reciprocity could be found in ferromagnet [33], antiferromagnet [37] or canted antifer-


romagnet [69, 70]. The existence of non-reciprocity can be determine by using symmetry

argument which is proposed by Scott and Mills [71]. Non-reciprocity exist if by symmetry

operation, the final state, where the propagation vector is opposite to that in initial state,

has different configuration compare to the initial condition.

Figure 2.2: Illustration of symmetry argument for TE surface modes. In(a), the initial state withweak ferromagnetism along x direction and propagation vector along y. In (b), the system stateafter reflection through xz plane, while (c) illustrating the final state after it is reflected throughxy plane. The final state is not equivalent to the initial state describing non-reciprocity.

Following Camley [35], we examine the non-reciprocity for TE modes in PML-type

magnetoelectric multiferroic as illustrated in Fig.(2.2). In initial state, the weak ferro-

magnetism is along x perpendicular to the propagation vector ky. Since the TE surface

mode is in magnetic type which is dominated by magnetic susceptibility and the dispersion

curve is located near magnetic resonance, hence in constructing the configuration the po-

larization vector can be ignored. The symmetry operation which is reflection through xz

plane change the propagation vector into the opposite direction. Considering that magne-

tization is an axial vector, that reflection is also flip the magnetization into −x direction.

Then, the reflection through xy plane yield the final state which is different from initial

configuration. Although magnetization has the same direction as in initial state, however

the surface modes in the final state is localized at the bottom.

2.5.3 Attenuated total reflection (ATR)

The method that can be used to detect the existence of surface modes, especially the

non-reciprocity in this mode, is attenuated total reflection (ATR) [72] and has been used


to study surface modes of phonon polaritons [73,74], plasmon polariton [75] and magnon

polaritons on antiferromagnets [47,48].

Figure 2.3: Illustration of ATR configuration. Electromagnetic waves incident in high indexprism with incident angle θ larger than critical angle to obtain total internal reflection. The prismis placed above sample with the air gap of thickness d.

In this method, a high optical index prism is used to couple electromagnetic radiation

to surface excitations that lie off the vacuum light line as illustrated in Fig.(2.3). The

reflection in the bottom of prism generates the evanescent wave which extends into the

sample. If the frequency of incident waves is near the frequency of elementary excitation,

then there is absorption via evanescent wave to generate surface polariton. Hence, a sharp

dip of reflectivity indicates excitation of surface modes. Using results from Ref. [37], the

reflectivity in ATR is given by

R =

∣

∣

∣

∣

∣

kz(

1 + re−2βod)

− iβo(

1− re−2βod)

kz (1 + re−2βod) + iβo (1− re−2βod)

∣

∣

∣

∣

∣

2

(2.54)

where d represents the distance between prism and the sample, and r is defined as

r =βo − κ

βo + κ(2.55)

with

κ =µzzβ − µyzkyµyyµzz − µ2

yz

. (2.56)

Here, the wave-vector

ky = (ǫp)1/2ω

csin θ and kz = (ǫp)

1/2ω

ccos θ (2.57)

represents propagation along and normal to the surface, where ǫp is the prism dielectric


constant, and θ is the incident angle. Numerical results for the ATR spectroscopy will be

presented in Section 2.6.

2.5.4 TM surface modes

The dispersion relation for surface modes is calculated by assuming surface localized plane

wave solutions of the form:

E,H ∼ eβzei(kyy−ωt) for z < 0 (2.58a)

E,H ∼ e−βozei(kyy−ωt) for z > 0 (2.58b)

where β and βo are attenuation constant for sample and vacuum. Substitution of Eq.(2.58a)

into the Maxwell equation of the curl of E and H as in Eqs.(2.42a) and (2.42b) provide

an implicit relation for the attenuation constant in the material,

ǫzz

(

β + i4πχme ω

c

)2= ǫyyk

2y − ǫyyǫzµxx

(ω

c

)2. (2.59)

The existence of imaginary term in Eq.(2.59) above influences the attenuation constant β

to become complex. Hence, the solution will be a pseudosurface modes instead of normal

surface modes. The appearance of the term i4πχme ωc also make ky complex, result in the

attenuation of the wave as it is propagating. The attenuation constant in the vacuum, βo,

is given in Eq.(2.51).

Similar to the TE modes, an implicit solution for the surface wave frequencies is found

by matching the solutions in Eqs.(2.58a) and (2.58b) at z=0 using electromagnetic bound-

ary conditions. The unique conditions are continuity of tangential H, continuity of tan-

gential E and continuity of normal B and also continuity of normal D. When satisfied,

the following dispersion relation results:

(

β + i4πχme ω

c

)

+ ǫyyβo = 0. (2.60)

Since the dispersion relation does not involve the odd order of wave propagation k ex-

plicitly, the surface modes described by Eq.(3.31) should be reciprocal in the sense that

ω (k) = ω (−k). We also see that the existence of surface modes strongly depends on the

value of ǫyy, since the solution of Eq.(3.31) for surface modes can only be found when the

value of ǫyy is negative.


2.6 Application to BaMnF4

We now illustrate the preceding theory for the material BaMnF4. Parameters appropriate

for BaMnF4 are presented in Table 2.1.

Table 2.1: Material parameters for BaMnF4

Parameter value taken from reference

θ 3 mrad [76]λ 163.72 erg cm−3Oe−2 [77]1

Ms 3054 Oe estimated2

γ 9.33 × 10−5 cm−1Oe−1 estimated3

K 0.337 erg cm−3Oe−2 estimated4

P 3.45 statC cm−2 [78]f/(2πc)2 954.11 cm−2 [58]ωey/2πc 41 cm−1 [79]ωez/2πc 33.7 cm−1 [58]

Tc 1113 K [80]α 1.42 × 10−5 cm2statC−1 estimated5

ζ1 -10.528 estimated6

ζ2 1.934 × 10−8cm3erg−1 estimated6

ǫ∞x 8.2 [81]ǫ∞y 12 [81]

ǫ∞z 8.3 [81]

2.6.1 TE modes

In previous investigations [79], the dielectric constant in the x direction has been assumed

to be independent of frequency with the value ǫxx = 8.2. This assumption is valid for

magnetic polaritons if the dielectric and magnetic responses lie in very different frequency

ranges. As a result, only two bands for bulk magnetic polaritons can exist since only

one pole and one zero are contributed by the magnetic sub-system. We also make this

assumption in what follows, since BaMnF4 has a wide separation in frequency between the

dielectric and magnetic resonances. At the end of this section we discuss the consequences

of relaxing this assumption.

A positive value of real β is needed for the solution of surface modes in Eq.(2.49a).

1This parameter is estimated from the exchange field HE=50 T which was measured by Holmes et al.,λ can be calculated by using relation HE = λMs.

2Sub-lattice magnetization Ms is estimated using relation Mc = 2Ms sin θ where Mc = 18.347 Oerepresent weak ferromagnet magnetization [80].

3This parameter is estimated using relation γ = gµb/~ and set g = 2.4This parameter is calculated using antiferromagnet resonance ωr = 3 cm−1 [79], in the relation

ωr = γ2K(K − 2λ)M2s .

5The ME coupling constant is estimated using Eq.(2.5).6Parameters ζ1 and ζ2 are estimated by solving simultaneously ∂

∂Py(FE + FME) = 0

and(

ζ1 + 3ζ2P2o

)

f = ω2ey at T=4.2 K.


Using the above parameters, the bounding frequencies of the bulk modes can be seen by

plotting β as a function of ω for different values of k, such as k/2π = 0 cm-1 and k/2π=16

cm-1 as shown in Fig.2.4(a), and is indicated in Fig.2.4(b) by shading. Comparison of

Figs.2.4(a) and 2.4(b) shows that the resonance at ωm∼= 3 cm-1 is associated with a diver-

gence of the attenuation constant β due to the zero of µzz. From this zero condition, an

expression for ωm can be derived :

ωm =[(

ω2afm cos2 θ +Ω2

me,TE

)

+ (2ωsωme sin 2θ)]1/2

(2.61)

with ωs = γ4πMs.

2.9 3 3.1 3.20

10

20

30

ω/2πc (cm−1)

β/2π

(cm

−1 )

k/2π = 0 cm−1

ωm

k/2π = 16 cm−1

ωz ω

p

(a)

−20 −10 0 10 202.9

3

3.1

3.2

k/2π (cm−1)

ω/2

π c

(cm

−1 )

−A −L L A

ωz

ωp

SP

ωm

(b)

Figure 2.4: Attenuation constant and Dispersion relation. In (a) the attenuation constant isshown for two values of wavevector in the absence of external fields. The solid line representsk/2π = 0 cm−1, while the dashed line correspond to k/2π = 16 cm−1 . In (b) bulk and surfacemode dispersions are shown. Surface modes are indicated by SP. The shaded regions representbulk bands, limited by frequencies ωm, ωp and ωz. The vertical thin lines denote by vertical andhorizontal arrows represent ATR light lines for incident angles of 30o and 70o. The asymptoticboundaries and lightline are represented by vertical thin lines which are indicated as A and byL. The dashed lines represent surfaces branches when electric field (−5 × 109 V/m) is applied.Horisontal thin lines denote by ω′

m represent the magnetic resonance when electric field is applied.

The pole and zero frequencies of bulk modes, ωp and ωz, are related to the zeroes of

µyy and f(µ) = µyyµzz − µ2yz . These give

ωp =[(

ω2afm cos2 θ +Ω2

me,TE

)

+ (2ωsωa cos 2θ)]1/2

(2.62)

and

ωz =1√2

[

Ω2mp +

(

Ω4mp − 4ω2

mω2p

)1/2]1/2

(2.63)

where

Ω2mp = ω2

m + ω2p + 4ω2

s sin2 θ. (2.64)


These three frequencies, ωm, ωp and ωz divide the bulk region into the shaded regions

shown in Fig.2.4(b). Note that the lowest two bands touch each other, and are deformed

strongly near the resonance ωm.

Surface modes, which are indicated by “SP” in Fig.2.4(b) can exist in the gaps between

ωm and ωp. The surface branches are nonreciprocal with respect to propagation, with

ω(

~k)

6= ω(−~k). The surface modes does not have Damon-Esbach modes [33], since it

can not exist at propagation vector k −→ ∞ and terminate at certain value of k. Hence,

this surface mode can be categorized as virtual modes [82]. The virtual modes can also

be found in the case of magnetic multilayer comprise of ferromagnetic (FM) and non-

magnetic (NM) with the fraction of magnetic less than 50% [82,83]. The negative branch

begins from the intersection of the lightline with the ωm resonance, and is indicated by

“-L” . The branch terminates in the left middle bulk band. The positive branch begins

at the lightline with frequency

ω =

[

ǫxxω2m − ω2

afm cos2 θ +Ω2me

ǫxx − 1

]1/2

(2.65)

which is slightly higher than ωm, and then terminates at the upper bulk band. It is also

shown in Fig.2.4(b), that the positive and negative branch of surface modes intersect with

the 30 ATR lines at frequency slightly above and below 3.05 cm−1. However, there is no

intersection between surface modes and the 70 ATR lines.

2.9 3 3.1 3.20

0.5

1

ω/2πc(cm−1)

R

k −k

(a)

2.9 3 3.1 3.20.7

0.8

0.9

1

ω/2πc(cm−1)

R

−kk

(b)

Figure 2.5: ATR spectra with incident angles 30 (a) and 70 (b). In (a) two different sharp dipsillustrate the non-reciprocity of the surface modes. In (b) the absence of the sharp dips indicatethe absence of surface modes.

The ATR spectroscopy is illustrated in Figs.2.5(a) and 2.5(b) for incident angle 30

and 70. Here we use Si-prism with dielectric constant ǫp=11.56. The air gap between


prism and multiferroics is set at the value 10 µm. In the calculation, damping is introduced

by replacing ω → ω + iΓ. The damping parameter Γ is set at the value 0.01 cm−1. At

the incident angle 30, the existence of surface modes is represented by sharp dips. The

ATR spectra at 30 illustrate nicely the nonreciprocity of surface modes. Two sharp dips

correspond to surface polaritons traveling in the opposite directions and the frequencies

are the intersection points discussed above for the positive and negative surface mode

branches.

The shallow decrease of reflectance R represent the existence of bulk modes. At the

incident angle 70, the ATR lines do not intersect the surface modes, hence the sharp

dip does not appear and the ATR spectroscopy only describe the bulk modes. It can be

concluded that the surface modes in this region do not exist.

It is interesting to note that the surface modes are not Damon-Eshbach modes. They

do not exist in the region where the wave-vector k ≫ ωc . In this region, the surface

dispersion relation in Eq.(2.52) can be re-written in the form

µyyµzz − (µyz ∓ 1)2 = 0 (2.66)

for k = ±∞ . In terms of ωm, ωp and ωr Eq.(2.66) becomes

(

ω2m − ω2

) (

ω2p − ω2

)

−[

2ωsω sin θ ∓(

ω2r − ω2

)]2= 0 (2.67)

where ωr =(

ω2afm cos2 θ +Ω2

me,TE

)1/2represents the pole frequency of permeabilities

without external field. This pole is located slightly below ωm so that ωr < ωm < ωp.

Assuming that the asymptotic frequency for the surface mode is located in the region

above the pole frequency ωp, for k = +∞, the second term in Eq.(2.67) is always larger

than the first term. Hence, Eq.(2.67) is never satisfied and the surface modes do not exist

for large k.

A surface mode at k = −∞ may exist for some values of ωr, ωm and ωp. For a solution

to exist in this region, the function f(µ) of Eq.(2.67) should cross zero. This gives a

condition from Eq.(2.67) at the frequency ω = ωp for a surface mode solution to exist.

The requirement that f (µ) vanish means that

ωp ≤ ωa cos 2θ + 2ωme sin 2θ. (2.68)


−6 −4 −2 00.25

0.35

0.45

0.55

k/2π (cm−1)

ω/2

πc (

cm−

1 ) SP

−A −L

(a)

−6 −4 −2 00.15

0.25

0.35

k/2π (cm−1)

ω/2

πc (

cm−

1 )

SP

−A −L

(b)

Figure 2.6: The condition for surface modes in k >> ω/c. In (a), a solution of dispersion relationwhere λ/α = 1.5, the surface begin to overcome the bulk. In (d) the surface branch is slightlyhigher than bulk when the ratio increase to λ/α = 2.

2.6.2 Effect of an applied field for TE modes

In principle, application of an external field can be used to modify the surface mode fre-

quencies by changing the canting angle. Since there is no magnetoelectric susceptibility

in this mode and the electric part is coupled to the magnetic part only through the reso-

nance frequency, the application of electric field parallel to the spontaneous polarization

has only a very small effect on polariton frequency. However, an interesting result is ob-

tained when the external electric field is applied opposite to the spontaneous polarization

and flip the electric polarization into the opposite direction. For example, at a certain

value of an apllied electric field −5 × 109V/m, which is quite high , the polarisation can

be flipped into the opposite direction. As a consequence, the Dzyaloshinskii-Moriya (DM)

interaction requires the canting to be also flipped into the opposite direction. Hence the

position of surface branches will flip, as can be seen as dotted lines in Fig.2.4(b). Note

that we assumed here that the voltage is below the breakdown voltage.

The application of an external static magnetic field changes directly the canting angle

and thereby affects the frequencies. The canting angle as a function of applied magnetic

field shown in Fig.2.8(a). In fact, for a sufficiently strong magnetic field, the condition of

Eq.(2.66) can be satisfied and a surface mode extends to k = −∞. This occurs through

an increase in ωr via the field dependence of Ωo,TE(see Eq.(2.21)).

The frequencies ωm and pole frequency ωp also increase under an external magnetic

field. The difference between ωm and ωp is given approximately by

∆2 ≈(

ω2p − ω2

m

)

∝ (cos 2θ − sin 2θ) . (2.69)


−50 −30 −10 10 30 50−2

−1

0

1

2

E (× 108 V/m)

P (

µC/c

m2 )

Figure 2.7: The relation between the electric polarization P and the external electric field E isdescribed in the hgysteresis curve which is resulted from the calculation of the time dependentGinzburg Landau equation. It is shown that the electric polarization can be flipped by applyingthe external electric field opposite to the electric polarization araound 3×109 V/m.

As a consequence, the middle bulk band narrows in frequency with increasing magnetic

field. At a field of 15 T, the BaMnF4 middle bulk band is sufficiently narrow that a

surface mode rises above the middle band and extends to small wavelengths, as illustrated

in Fig.2.8(b).

0 5 10 150

4

8

12

H (T)

θ (

10−

2 rad

)

(a)

−700 −500 −300 −10015.22

15.24

15.26

k/2π (cm−1)

ω/2

π c

(cm

−1 )

−A−L SP

(b)

Figure 2.8: The influence of magnetic field. In (a)influence on canting angle. In (b), the appli-cation of a magnetic external field of 15 T increases the frequency of surface mode and allows thenegative surface branch to enter the magnetostatic region, where k >> ω

c.

2.6.3 TM modes

Solutions of Eqs.(2.48) and (2.60) for TM modes in BaMnF4 are plotted in Fig.(2.9(a)) for

the case with no applied fields present. There are two gaps in the bulk region created by

the poles and zeros of Eq.(2.48). First, as illustrated in Fig.(2.9(c)), a very narrow gap is


located at the frequency ωi around 41 cm−1, created by the magneto-electric interaction

[58]. This gap is associated with zeros in f(µ, ǫ) = µxxǫyy − (4πχme)2. The pole in

the bulk modes is due to a zero of the dielectric constant ǫyy. This gap is strongly

dependent on the ME susceptibility, χme, and disappears when χme=0. The width of the

gap is approximately proportional to (χme)2. Thus the gap becomes wider with larger

ME coupling. This increase is illustrated in Fig.(2.9(d)) where the ME coupling has been

increased by a factor of ten.

A second gap exists near the magnetic frequency ωm∼=3 cm-1 , and occurs at zero of

f(µ) = µxxǫyy − (4πχme)2 at the magnetic frequency ωm. The other boundaries for the

bulk regions are determined by the attenuation constant.

−400 −200 0 200 4000

10

20

30

40

k/2π (cm−1)

ω/2

πc (

cm−

1 )

ωez

ωoz

ω’ey

ω

i

ω’m

SP

−A A −L L

(a)

−200 0 20039

40

41

42

k/2π (cm−1)

ω/2

πc (

cm

−1 )

ω’ey

ωi

−A A −L L

(b)

175.782 175.784 175.78640.7525

40.7526

40.7527

k/2π (cm−1)

ω/2

πc (

cm−

1 )

ωi

SP

(c)

175.776 175.78 175.78440.7525

40.7526

40.7527

k/2π (cm−1)

ω/2

πc (

cm−

1 )

ωLy

(d)

Figure 2.9: Dispersion relation without external field. In (a) he dispersion relation without theexternal field is shown. The surface modes are indicated by“SP”. The shaded regions representbulk bands, which are limited by frequencies ωm, ωez , ωoz , ωey and ωi. In (b) The “window”where the surface modes exist is shown. In (c) a narrow gap around ωi is expanded. In (d) thenarrow gap is wider when the ME coupling is increased by a factor of 10.

Using the attenuation constants, we obtain a narrow window between transversal and

longitudinal phonon frequencies, ω′ey and ωLy, associated with the pole and zero value of


ǫyy (see Fig.2.9(b)). In this figure, since the ME coupling is weak, hence the frequency ωLy

is very slightly below the induced frequency ωi. Since the value of ǫy between these two

frequencies is negative, surface modes can be obtained inside this narrow window. The

surface modes start from the crossing between the lightline (ω = ck) and the resonance

frequency ωey and terminate at the longitudinal phonon frequency ωLy. The frequency

ωLy can be approximated as

ωLy =1

2

(

ω2ey + ω2

m + f)

+[

(

ω2ey + ω2

m + f)2 − 4

(

ω2eyω

2m + ω2

mf)

]1/2

(2.70)

and is indicated in Fig.2.9(d). Since the surface modes terminate at the longitudinal

phonon frequency, the gap around ωi does not influence the surface modes.

Interestingly, from the expression for the attenuation constant of Eq.(2.59), and the

dispersion relation of surface modes in Eq.(2.60), it was only possible to satisfy the bound-

ary conditions with a complex β. The resulting mode is therefore not a true surface mode

but is instead a pseudo-surface wave [84]. Because the solution for the attenuation con-

stant in Eq.(2.59) is complex, the attenuation constant for the material sample consists of

real and imaginary part as illustrated in Figs.2.10(a) and 2.10(b).

0 20 40 600

100

200

300

400

500

ω/2πc (cm−1)

Re(

β/2π

) (c

m−

1 )

0 20 40 600

100

200

300

400

500

ω’m

ω’

ey

ωoz

ωi

ωez

κ = 250 cm−1 κ = 0 cm−1

(a)

39 40 41 420

0.01

0.02

0.03

0.04

0.05

ω/2πc (cm−1)

Im(β

/2π)

(cm

−1 ) ω’ey

(b)

Figure 2.10: Attenuation constant as a function of frequency. In (a) the real part of attenuationconstant is shown for two values of wavevector in the absence of any external fields. The solidline represents k = 0 cm−1, while the dashed line corresponds to k = 250 cm−1 . In (b) theimaginary part of attenuation constant is shown for two values of ME coupling. The solid lineis for α = 1.42 × 10−5 cm2/statC, and the dashed line represents the coupling α = 1.42 × 10−4

cm2/statC.

The imaginary part of β is

βi = −i4πχme ω

c. (2.71)

The positive value of the real part defines regions where the surface modes can exist. The


existence of an imaginary part indicates that the solution in Eq.(2.58a) is a psuedosurface

mode, and not purely localized to the surface. Instead, energy ”leaks” into the bulk. The

wave is comprised of a localised component, that travels along the surface and decays

into the material according to the real part of β, and a component that travels into the

material with wave number equal to the imaginary part of β.

Values for the imaginary parts of β are plotted as a function of frequency in Fig.2.10(b)

for the case of no applied fields. Imaginary β depends linearly on the magnetoelectric

susceptibility and becomes large near the electric resonance frequency ωey . One can also

see from Fig.2.10(b), that the coupling directly influences the magnitude of imaginary β.

If the coupling is large, then the ME susceptibility will also be large and thereby increase

the imaginary part of β.

In the case where k >> ωc , the attenuation constants for the material and vacuum

regions can be approximated by ǫzzβ2 ≈ ǫyyk

2y and βo ≈ ky. The dispersion relation

Eq.(2.48) then reduces to:

ǫzzǫyy = 1. (2.72)

The surface modes require the permitivity ǫyy to be negative, and the longitudinal phonon

frequency polarized along z direction, ωoz, is lower than ω′ey. This means that the value of

ǫzz is positive in regions where surface modes may exist, and the requirement in equation

(2.72) is never satisfied. Therefore, surface modes do not exist in the limit k >> ωc .

2.6.4 Effect of applied fields for TM modes

We now study the influence of external fields on the band structures. Results for an

electric field value of 5× 108 V/m along the direction of spontaneous polarisation (but

with zero magnetic field) are shown in Fig.2.11(b). The effect of the electric field is to

shift the window where surface modes exist to higher frequencies. The upward frequency

shift is due to the increase of spontaneous polarisation, which directly increases the phonon

frequencies ω′ey and ωLy. However, the change in ωLy is smaller than the change in ω′

ey

(which is due to the third term in Eq.(2.70)) and so the surface mode window is narrowed.

The electric field increases the canting angle slightly, as shown in Fig.2.11(a), and the

effect on magnetic resonance is negligible.

Results for a magnetic field of 10 T (with zero electric field) are shown in Fig.2.11(c).

The magnetic field increases the canting angle (see Fig.2.8(a)) and shifts ωm to a lower

frequency (as shown in Fig.2.11(c)) but the effects on the bulk bands are negligible. This

shift can be understood if we consider the case where the ME coupling is neglected. Then


0 5 10x 10

9

2.5

3

3.5

4

E (V/m)

θ (r

ad)

(a)

−400 −200 0 200 4000

10

20

30

40

50

k/2π (cm−1)

ω/2

πc (

cm−

1 )

−A A −L L

SP ω’

m

ω’ey

ω’m

(b)

−30 −10 10 302.2

2.6

3

3.4

k/2π (cm−1)

ω/2

π c

(cm

−1)

−A −L L A

ω’m

(c)

Figure 2.11: Influence external field in the dispersion relation. In (a)the external electric field Eincrease the canting angle with small value. In (b) dispersion relation is shown with an externalelectric field along the spontaneous polarisation E = 5 × 108 V/m. In (c), magnetic resonancefrequency shift down when the external magnetic field, Ho = 10T is applied. The dotted line isthe magnetic frequency in the absence of magnetic field.

the frequency ωme −→ 0, and also Ωme,TM −→ 0. In this case, the magnetic resonance

frequency will take the form ω2m = ω2

afm+Ω2o,TM −→ ωa(ωa− 2ωex) cos

2 θ. It can be seen

from this expression that a magnetic field reduces the magnetic resonance frequency. We

note that this effect is mentioned in Ref. [67].

The surface modes can be modified if the magnetic resonance frequency can be shifted

into a surface wave “window”, as shown in Figs.2.12(b) and 2.13(b). In BaMnF4 where

the electric and magnetic resonance frequency are well separated around 37cm−1, it is

very difficult to arrange the magnetic resonance frequency inside this window. This may

be possible for a suitably prepared material (or artificially constructed material) whose

frequency separation between electric and magnetic resonance is smaller.

Lastly, we identify the key parameters affecting surface mode frequencies. In the first

case, changing the phonon mass to f/(2πc)2 = 3.18 cm−2 moves the magnetic resonance

frequency to 1 cm−1 above the the electric resonance frequency ωey. The result on surface


−30−20−10 0 10 20 301.7

2

2.3

2.6

2.9

3.2

k/2π (cm−1)

ω/2

πc (

cm−

1 )

−A

ω’ey

SP

ω’m

A L −L

(a)

−20 0 201.8

2

2.2

2.4

2.6

k/2π (cm−1)

ω/2

πc (

cm−

1 )

−A A L −L

SP ω’

ey

ω’m

(b)

Figure 2.12: Dispersion relation for a material with electric and magnetic resonances near oneanother in frequency. In (a) the dispersion relation without external magnetic field is shown. In(b), the magnetic resonance frequency is shifted into a surface mode window with the applicationof a large magnetic field, Ho = 12T.

and bulk polariton bands is shown in Fig.2.12(a). The dielectric constant background has

also been reduced to ǫ∞y = 2.6, in order to widen the surface mode window. Application of

an external magnetic field lowers the magnetic resonance frequency. Application of a large

external magnetic of 12 T places the magnetic resonance inside the window as illustrated

in Fig.2.12(b).

Inside the window, the magnetic resonance splits the surface mode into low and high

frequency branches for each direction of propagation. The properties of the upper part

are similar to that discussed in the previous section. However, the lower branch termi-

nates at the magnetic resonance frequency as illustrated in Fig.2.12(b). In this case, the

requirement that the dielectric constant ǫyy should be negative for surface modes prevents

both the upper and lower branches to exist in the region where k >> ωc .

In a second example, we consider if the exchange constant λ −→ −4000 erg cm−3 Oe−2

and the anisotropy constant is K −→ 4 erg cm−3 Oe−2. The ME coupling is also changed

to 1.33 ×10−4cm2/statC, which keeps the canting angle small. The dispersion relation at

temperature 150 K is presented in Fig.2.13(a). As temperature is increased to 250 K, the

polarisation will decrease while the magnetisation does not change significantly. Hence,

the electric resonance goes below the magnetic resonance frequency. Results are shown in

Fig.2.13(b). As in the first example, the magnetic resonance frequency then exists inside

the surface mode window and a similar splitting of the surface mode branches occurs.


−400 −200 0 200 400

32

34

36

38

k/2π (cm−1)

ω/2

π (c

m−

1 )−A

ω’ey

SP

A L −L

ω’m

(a)

−400 −200 0 200 40035

35.5

36

36.5

k/2π (cm−1)

ω/2

πc (

cm−

1 )

−A A L −L

ω’ey

ω’m

SP

(b)

Figure 2.13: Temperature dependence with modified exchange constant. (a) Dispersion relationat T=150 K. In (b), the magnetic resonance frequency shifts down at 250 K

2.7 Summary

We have shown how linear magnetoelectric coupling influences surface and bulk TE and

TM polaritons modes in a canted multiferroic. For TE polarisation, we find surface modes

associated with the weak ferromagnetism that are non-reciprocal with respect to propaga-

tion direction, such that ω(

~k)

6= ω(−~k). The non-reciprocity can be affected by an applied

electric field through changes in the weak magnetisation. For sufficiently large magnetic

fields, or different material parameters, surface modes may also exist for k ≫ ω/c. Appli-

cation of a static magnetic field can be used to modify the middle bulk band frequencies.

For TM polarisation, a narrow restrahl region forms in the bulk mode band at a

frequency near the longitudinal phonon frequency (along the spontaneous polarization).

Reciprocal surface excitations can exist in this region. In general the surface exciations in

this polarization are actually psuedosurface waves that are only partially localized to the

surface. The imaginary part of the decay constant is proportional to the magneto-electric

coupling. It also note that both TE and TM modes of surface modes are categorized as

virtual modes since it does not exist at the region k >> ωc .

Chapter 3

Un-polarized surface polaritons in

a PML-type magnetoelectric

multiferroic with canted spins

3.1 Introduction

Polaritons, which are electromagnetic waves that propagate in the medium, are strongly

depend on the direction of propagation if the medium is anisotropic. In anisotropic

medium, the components of susceptibility tensor have different values, such as the sus-

ceptibility of a PML-type magnetoelectric multiferroic discussed in section 2.3. The TE

surface polaritons on magnetic system generally were found in the configuration where the

propagation vector is perpendicular to the static magnetization [64]. In dielectric medium,

the TM surface polaritons were found in the configuration where the propagation vector

is parallel to the electric polarization. Hence, the previous studies of surface polaritons in

magnetic media are mostly focused on TE modes [34,35,63,85], or TM modes [56,62,85]

for dielectric materials.

In this chapter, we discuss theoretically the the un-polarized surface polaritons on

magnetoelectric coupled media with magnetoelectric coupling is in PML-type as in chap-

ter 2. Here, we focus on the case where the surface modes are not polarized into TE

or TM modes. This type of surface modes had been reported for magnetic system in

layered antiferromagnets by considering configuration where the propagation vector is di-

rected parallel to the static magnetization [86]. In this chapter, we show that by setting

the configuration where the propagation vector is directed along uniaxial easy axis, the

surface polariton is neither TE modes nor TM modes, hence the surface polarization is

43

44 3. Un-polarized polaritons

un-polarized. This un-polarized surface mode is characterized by the appearance of two

attenuation constants instead of one as in TE or TM modes. Unlike the un-polarized

surface polaritons on magnetic system which is reciprocal [86], here we found that surface

polaritons are non-reciprocal due to the magnetoelectric interaction.

This chapter is organized as follow. The required geometry to obtain un-polarized sur-

face modes is discussed in Section 3.2. The geometry involves the weak ferromagnetization

which is directed out of plane, then the demagnetization fields have to be considered. This

demagnetization field modifies the susceptibility which is discussed in Section 3.3. Then,

in Section 3.4 the expression for the surface modes is derived and applied to the mag-

netoelectric BaMnF4 in Section 3.5. In the last section, the effect of an external field is

discussed.

3.2 Geometry

The geometry is illustrated in Fig.3.1. A semi-infinite magnetoelectric multiferroic fills

the half space x < 0. The magnetic component of the multiferroic is a two sub-lattices

antiferromagnet with uniaxial magnetic anisotropy is assumed along z axis. The mag-

netoelectric coupling is assumed to be polarization induced Dzyaloshinkii-Moriya (DM)

type which allows the magnetic sub-lattices to be canted with canting angle θ producing

weak ferromagnetic moment M which is out of plane in the x direction. Here, we consider

symmetric canting where the magnitude of the magnetization in each sub-lattice is the

same,∣

∣

∣

~Ma

∣

∣

∣ =∣

∣

∣

~Mb

∣

∣

∣ = Ms. Spontaneous polarization is constrained to lie in yz plane. The

surface modes are assumed to be propagated along z axis parallel to the easy axis.

Since the weak ferromagnetism M is out of plane, then the demagnetization field which

is in opposite direction to the M should be considered. This demagnetization effect modify

the expression of magnetic density energy into

FM = λMa ·Mb − K

2

[

(Ma · z)2 + (Mb · z)2]

+ 2π[

(Ma · x)2 + (Mb · x)2]

− (Ma +Mb) ·H (3.1)

The first term on the Eq.(3.1) above represents exchange interaction with λ > 0 is ex-

change constant. The second term is anisotropy interaction with K represents anisotropy

constant. The demagnetization density energy is expressed in the third term while the

last term represent the interaction with the external magnetic field.

Since the demagnetization field is only affecting the magnetic system, hence the energy

3. Un-polarized polaritons 45

Figure 3.1: An illustration of the geometry. The weak ferromagnetism M which is out of planealong x is resulted from the canting of two sub-lattices antiferromagnetism ma and mb. The spon-taneous polarization (P) is assumed to lie in a plane parallel to the surface along y. Propagationof the surface mode is along the z perpendicular to the polarization.

density for ferroelectric (FE) and magnetoelectric (ME) system have the similar expression

as in Section 2.2. The equilibrium condition for canting angle, ∂(FM+FME)∂θ = 0, is then

modified into

Ho cos θ −1

2KM s sin 2θ + 2αPyMs cos 2θ − λMs sin 2θ − 2πMs sin 2θ = 0 (3.2)

which simplifies into the expression

tan(2θ) =4αPy

K + 2λ+ 4π(3.3)

when the external magnetic field is absent. The field in the x component from magneto-

electric energy density is derived asHME,x = ∂FME∂Mx

= 2αPyMs cos θ. The demagnetization

field is defined as Hd = 8πMs sin θ. Using parameters for BaMn4 from Table 2.1, we found

that HME,x ≈ 6.5Hd. Since the field from magnetoelectric interaction much stronger than

demagnetization field, then the canting angle is stable. From the Eq.(3.3), it can also be

seen that demagnetization gives only little effect on canting angle since λ >> 4π.

3.3 Dynamic susceptibility

In the purpose to evaluate the boundary value problem and obtain the dispersion relation

that represents the bulk and the surface modes, we need the expression of the dielectric

and magnetic response. Since the demagnetization field modifies the energy density of


magnetic part, then the demagnetization will also modifies the equations of motion of the

magnetic part in Section 2.3 into

−iωmy =(ω + 2ωme cos θ − ωa sin θ − ωd sin θ)mz

− (ωa cos θ + 2ωme cos θ) lx − 2γMshz sin θ, (3.4a)

−iωmz =(ωd sin θ − 2ωme cos θ − ω) + 2γMshy sin θ, (3.4b)

−iωly =− (ωa cos θ + 4ωme sin θ + 2ωex cos θ)mx

+ (ω + 4ωme cos θ − ωa sin θ − 2ωex sin θ − ωd sin θ) lz

+ 4γαM2s py cos 2θ + 2γMshx cos θ, (3.4c)

−iωlz =(ωd sin θ + 2ωex sin θ − ω − 2ωme cos θ) ly. (3.4d)

Then, equations of motion Eq.(3.4a)-(3.4d) lead to the expression of the susceptibility

components χmyy, χ

mzz and χm

yz as

χmyy =

2γMs

(

ωa cos 2θ + 2ωme sin 2θ − ωd sin2 θ + ω sin θ

)

(

ω2afm cos2 θ +Ω2

me +Ω2 +Ω2

d − ω2) , (3.5a)

χmz =

2γMs

(

ωme sin 2θ − ωd sin2 θ + ω sin θ

)

(

ω2afm cos2 θ +Ω2

me +Ω2 +Ω2

d − ω2) , (3.5b)

χmyz = −χm

zy =i2γMsω sin θ

(

ω2afm cos2 θ +Ω2

me +Ω2 +Ω2

d − ω2) . (3.5c)

Here the frequency ωd = γ4πMs represents a frequency from denagnetization field, while

the frequencies Ω and Ωd are now defined as

Ω2 = ω (ω + 4ωme cos θ − ωa sin θ − 2ωd sin θ) (3.6a)

Ω2d = ωd sin

2 θ (ωa + ωd) . (3.6b)

From the expression of susceptibility components above, it can be seen that without the

external magnetic field, the demagnetization tends to shift the magnetic resonance up.

The other susceptibility components (χmxx, χ

eyy and χme) have the same form as the

expression in Eqs.(2.23), (2.24) and (2.26). However, demagnetization field modifies the

expression of Ωc in Eq.(2.28) and magnetic resonance ωm in Eq.(2.29) into the forms:

Ω4c = 8γα2fM3

s cos 2θ(

ωa cos2 θ + 2ωex sin

2 θ + ωd sin2 θ − ω sin θ

)

(3.7)

ω2m = ω2

afm cos2 θ + 2ωex sin2 θ (ωa + 2ωex) + Ω2

d + Ω2 + Ω2

me (3.8)


where Ω2, Ω

2me and Ω2

d are the contributions from an external magnetic field, the magne-

toelectric coupling and the demagnetization field which now are described as

Ω2 = ω2

+ ω [6ωme cos θ − (4ωex − ωa) sin θ − 2ωd sin θ] , (3.9)

Ω2me = 2ωme(4ωme − ωa sin 2θ − 2ωex sin 2θ −

3

2ωd sin 2θ), (3.10)

Ω2d = ωd (ωd + ωa) sin

2 θ. (3.11)

The other susceptibility components, χexx and χe

zz are not influenced by the appearance

of demagnetization fields. They have the expression as in Eqs.(2.22) and (2.25).

3.4 Theory for bulk bands and surface modes

3.4.1 Dispersion relation for bulk modes

Since the electric and magnetic dynamics are coupled, the dispersion relation for bulk

modes are obtained by evaluating the electromagnetic Maxwell equations

∇×H =−iω

cD and ∇×E =

iω

cB (3.12)

where the fields B, H, D and E are connected through constitutive Eqs.(2.41). The next

step is assuming plane waves in the form E,H ∝ ei(kxx+kzz−ωt), and substituting the fields

into the Maxwell Eqs.(3.12) leads to a matrix equation as

0 kz 0 −ǫxxωc 0 0

kme 0 −kx 0 ǫyyωc 0

0 kx 0 0 0 ǫzzωc

µxxωc 0 0 0 kme 0

0 −µyyωc −iµyz

ωc kz 0 −kx

0 iµyzωc −µzz

ωc 0 kx 0

Hx

Hy

Hz

Ex

Ey

Ez

= 0 (3.13)

where kme = kz+4πχme ωc . Unlike the matrix equation in Chapter 2, the matrix Eq.(3.13)

can not be brokendown into the smaller forms.

The solution of Eq.(3.13) requires that the matrix determinat to be zero. For the

propagation vector which is parallel to the easy axis, we set kx = 0. In this case, Eq.(3.13)

has solution

ǫzzω

c

[

ǫxxµ2yz

ω3

c3Crd + µzz

ω

cCrd

(

k2z − ǫxxµyyω2

c2

)]

= 0 (3.14)


where

Crd =(

kz + 4πχmeω

c

)2− ǫyyµxx

ω2

c2. (3.15)

The Eq.(3.14) can be factorized, with solutions are determined by

ǫzz = 0 (3.16a)

µzzk2z − ǫxx

ω2

c2(

µyyµzz − µ2yz

)

= 0 (3.16b)

(

kz + 4πχmeω

c

)2− ǫyyµxx

ω2

c2= 0. (3.16c)

The first solution, Eq.(3.16a), is just the longitudinal soft phonon frequency. The

second solution, Eq.(3.16b), associated to Ex and Hy and Hz. Since the permeabilities

µyy, µzz and µyz have identical resonances, there will be two resonance frequencies. We

thus expect that Eq.(3.16b) describes three bulk bands separated by two gaps. One gap

is around magnetic resonance frequency ω = (ω2afm cos2 θ +Ω2

me +Ω2d +Ω2

)1/2, while the

other gap is located around soft phonon frequency along x direction, ωex.

Figure 3.2: Symmetry argument for non-reciprocity of the bulk solution for Eq.(3.16c). In (a),the initial configuration with wave vector kz is directed along z axis. In (b), the system after xzplane reflection is shown. In (c), the system at final state after xy plane reflection is shown. It isnot possible reverse −kz and leave P and M unchanged.

The third solution, Eq.(3.16c) associated with Hx and Ey. Since µxx, ǫyy and χme have

two resonances, one at ωm and the other at ωey. Again, this describes three bulk bands

separated by two gaps. In this solution, ME susceptibility is explicitly involved giving rise

to odd powers of kz. Hence the direction of z (kz) matters. In other words, this solution


is non-reciprocal, i.e. ω(kz) 6= ω(−kz). If the ME susceptibility is zero, only k2z appears

in the Eq.(3.16c), and the non-reciprocity will vanish.

Possibility for non-reciprocity in this bulk mode can be established by using symmetry

argument [64, 71]. The idea is illustrated in Fig.3.2. Since in the dispersion relation in

Eq.(3.16c) the magnetoelectric coupling is involved, in consequences, we have to include

both magnetization M and electric polarization P in the operation of reflection. For

example, suppose M is directed along x axis, while P is along the y axis, as in Fig.3.2(a).

The propagation vector is directed along z axis. Then, the reflection through the xz

plane will change both the polarization and magnetization to −P and −M but leaves kz

unchanged (Fig.3.2(b)). Reflection through the xy plane does not reverse P but reverses

M and kz as illustrated in Fig.3.2(c). These reflection operations lead to the final system

(Fig.3.2(c)) which is different from the initial system ( 3.2(a)). Since the system is changed

by the reflection operations, the energy of the system should be changed. Then, the energy

of the system must also be changed, i.e., ω(k) 6= ω(−k). In fact, there is no way to reverse

kz leaving P and M unchanged.

3.4.2 Attenuation constant

The appropriate solution for surface modes localized at the surface is assumed to be of

the form

H = (Hx,Hy,Hz) eβxei(kzz−ωt) for x < 0 (3.17a)

H = (Hx,Hy,Hz) e−βxei(kzz−ωt) for x > 0 (3.17b)

where β and β represent attenuation constant for material sample and vacuum. The

attenuation constants guarantee that the surface modes are localized at the surface of

material. The electric fieldsE andE have the form similar to Eq.(3.17). Then, subtitution

of theH and theE fields into the curl forms of Maxwell Eq.(3.12) leads to a matrix equation

as

0 kz 0 −ǫxxωc 0 0

ikme 0 −β 0 iǫyyωc 0

0 β 0 0 0 iǫzzωc

µxωc 0 0 0 kme 0

0 −iµyyωc µyz

ωc ikz 0 −β

0 −µyzωc −iµzz

ωc 0 β 0

Hx

Hy

Hz

Ex

Ey

Ez

= 0. (3.18)

Solution for attenuation constant β can be obtained by setting the determinant to


zero. Unlike the bulk case, the determinant can not be factorized into the specific modes

as in Eqs.(3.16a) to (3.16c). The result of Eq.(3.18) has the form

Aβ4 +Bβ2 + C = 0 (3.19)

where A, B and C are defined as

A =ǫxxµxx, (3.20a)

B =ǫxxµxxω2

c2(ǫzzµyy + ǫyyµzz)− k2z (ǫzzµxx + ǫxxµzz)

− 4πχmeµzzω

c

(

2ǫxxkz + 4πχmeǫxxω

c

)

, (3.20b)

C =ǫzk2z

ω2

c2

[

ǫxx(

µ2yz − µyyµzz

)

+ µzz

(

16π2 (χme)2 − ǫyyµxx

)]

+ 8πχmeǫxxǫzzkzω3

c3(

µ2yz − µyyµzz

)

+ ǫzzµzzk3z

(

kz + 8πχmeω

c

)

+ ǫxxǫzzω4

c4(

µ2yz − µyyµzz

)

(

16π (χme)2 − ǫyyµxx

)

. (3.20c)

The solution is

β1,2 =

−B ∓√B2 − 4AC

2A

1/2

. (3.21)

The required solution is in general complex β with positive real part. There are then

two values of β for each kz . A superposition of two particular planes waves is needed to

generate surface polariton modes [32,86]. There are three posibilities regarding to the value

of the attenuation constant. If both β1 and β2 are real, then the mode is called bonafide

surface mode. The pseudo-surface modes require that β1 is real and β2 is imaginary or vice

versa. The surface modes are called generalized surface modes if β1 and β2 are complex

conjugate with positive real part [86]. The attenuation constant for the vacuum can be

easily derived as

β2 = k2z −

(ω

c

)2. (3.22)

3.4.3 Surface modes

The two attenuation constants, β1 and β2, mean that the surface polariton is in general a

superposition of two partial waves

H =

2∑

j=1

(

Hxj ,H

yj ,H

zj

)

exp (βjx)

exp (ikzz − iωt) . (3.23)


We write the amplitude of the partial waves as Hy1 and Hy

2 . The remaining components

of H and E be written in terms of Hyj by using Maxwell equations Eqs.(3.12). The results

are:

Exj =

ckzǫxxω

Hyj (3.24a)

Eyj =

µzz

(

ǫzzk2z − ǫxxβ

2j

)

− ǫxxǫzz(

µyyµzz − µ2yz

)

ω2

c2

ǫxxǫzzµyzβjωc

Hyj (3.24b)

Ezj =

iβjc

ǫzzωHy

j (3.24c)

Hxj =

−(

β2jc2

ω2 + ǫyyµzz

)(

ǫzzk2z − ǫxxβ

2j − ǫxxǫzzµyy

ω2

c2

)

− ǫxxǫyyǫzzµ2yz

ω2

c2

ǫxxǫzzµyzβj(

kz + 4πχme ωc

) Hyj (3.24d)

Hzj =

−i(

ǫzzk2z − ǫxxβ

2j − ǫxxǫzzµyy

ω2

c2

)

ǫxxǫzzµyzω2

c2

Hyj . (3.24e)

In the vacuum outside the sample, two independent components also needed, for example:

Hy and Hz

. Then, the other field components in vacuum can be written as

Ex =

ckzω

Hy (3.25a)

Ey =

−iω

βcHz

(3.25b)

Ez =

−iβc

ωHy

(3.25c)

Hx =

ikzβ

Hz . (3.25d)

We now apply the electromagnetic boundary conditions at the interface using the field

components in Eqs.(3.24a)-(3.25d). All six possible boundary conditions at the interface

(x = 0) are required. Two boundary conditions come from the continuity of normal B

(Bx) and the continuity of normal D (Dx). The other two boundary conditions come from

the continuity of tangential E (Ey, Ez). The last two boundary conditions are from the

continuity of tangential H (Hy, Hz). From the continuity of field Hy and the continuity

of Ez at the interface, we have

Hy1 +Hy

2 = Hy (3.26a)

β1Hy1 + β2H

y2 = −ǫzzβH

y (3.26b)


which then lead to the relations

Hy1 =

− (β2 + ǫzzβ)

β1 − β2Hy

and Hy2 =

β1 + ǫzzββ1 − β2

Hy . (3.27)

The continuity of Dx results in Eq.(3.26a), while the continuity of Bx together with the

continuity of Ey yield

β2g[

−(

β21 + b

) (

a− ǫxxβ21

)

+ d]

Hy1

+ β1g[

−(

β22 + b

) (

a− ǫxxβ22

)

+ d]

Hy2 = β1β2fH

z (3.28)

where parameters a, b, d, f, g, h are defined as

a =ǫzzk2z − ǫyyµyy

ω2

c2(3.29a)

b =ǫyyµzzω2

c2(3.29b)

d =ǫyyµ2yz

ω4

c4(3.29c)

f =µyzω2

c2

(

kz + 4πχmeω

c

)2(3.29d)

g =βµxx (3.29e)

h =(

kz + 4πχmeω

c

)

. (3.29f)

The final boundary condition is continuity of tangential Hz, which gives the relation

(

a− ǫxxβ21

)

Hy1 +

(

a− ǫxxβ22

)

Hy2 = iǫxxǫzzµyz

ω2

c2Hz

. (3.30)

Equations (3.28) and (3.30) together with Eq.(3.27) gives an implicit dispersion relation

for the surface modes:

µxxβ (ǫzzβ + β1 + β2)

[(

ǫzzk2z − ǫxxǫzzµyy

ω2

c2

)

ǫyyµzzω2

c2+ ǫxxǫyyǫzzµ

2yz

ω4

c4

]

+ ǫzzµxxβ2β1β2

[

ǫxxǫyyµzzω2

c2−(


ω2

c2

)]

+ ǫxxβ21β

22

[

(

kz + 4πχmeω

c

)2+ µxxβ (ǫzzβ + β1 + β2)

]

+ ǫzzββ1β2

[

(

kz + 4πχmeω

c

)2(β1 + β2) + ǫxxµxxβ

(

β21 + β2

2

)

]

+ β1β2

(

kz + 4πχmeω

c

)2(


ω2

c2

)

= 0. (3.31)


The dispersion relation for surface modes implied by Eq.(3.31) describes the appearance

of the non-reciprocity since terms linear in kz appear. If the magnetoelectric coupling is

zero, then χme −→ 0, and these terms vanish, meaning that the surface modes are then

reciprocal with respect to kz.

3.5 Application to BaMnF4

We first examine the attenuation constants Eq.(3.21), since these parameters determine

bulk band limits. Using the parameters for BaMnF4, the calculated attenuation β1 =

−B−√B2−4AC2A

1/2is presented in Figs.3.3(a) and 3.3(b). We see that the real and the

imaginary values of β1 do not overlap, so β1 is not complex. The dielectric constant in the

x direction has been assumed to be independent of frequency [79], hence for small values

of kz , we find only one region around the magnetic resonance ω′m where the attenuation

constant is real.

0 1 2 3 4 50

4

8

12

ω /2π c (cm−1)

β 1 /

2π (

cm

−1 )

ωm

’

(a)

0 1 2 3 4 50

4

8

12

ω/2π c (cm−1)

β 1/2π

(cm

−1 )

ωm

’

(b)

3.05 3.1 3.15 3.2 3.250

1

2

3

4

ω/2π c (cm−1)

β 2/2π

(cm

−1 )

ωm

~

(c)

38 40 42 440

50

100

150

200

ω /2π c (cm−1)

β 2/2π

(cm

−1 )

ωey

~

(d)

Figure 3.3: Values of β as a function of frequency. Values of β1 near ω′m at (a) k = 0 cm−1 and

(b) at k = 5 cm−1. Values of β2 as a function of frequency at kz = 0 cm−1 for frequencies near ωm

are shown in (c). In (d) frequencies are near soft phonon frequency ωey. The solid lines representthe real values and the dashed lines represent the imaginary values of β. The examples show thatβ1 and β2 are never complex.


Note that for small values of kz , there are two bulk bands where β1 is imaginary

(Positive real β1 represent regions where the surface modes may exist, and the imaginary

values of β1 represent bulk bands). This separation into two bands is exactly what is

expected from Eq.(3.16b).

We now consider β2 =

−B+√B2−4AC2A

1/2. The attenuation constant for kz = 0 cm−1

is presented in Figs3.3(c) and 3.3(d). There are two resonances involved. One is ωm,

as illustrated in Fig3.3(c) and the other is the phonon frequency ωey (see Fig.3.3(d)). A

gap exists for low frequency (see Fig.3.3(c)), around ωm, and the other gap exist at large

frequencies (see Fig.3.3(d)),around ωey. This separation into three bulk bands is expected

from Eq.(3.16c).

−40 −20 0 20 402.6

2.8

3

3.2

3.4

3.6

k/2π (cm−1)

ω/2

π c

(cm

−1 )

ωm

’

(a)

−80 −40 0 40 800

10

20

30

40

50

k/2π (cm−1)

ω/2

π c

(cm

−1 )

ωm

ωey

~

~

(b)

−200 −100 0 100 2000

10

20

30

40

50

k/2π (cm−1)

ω/2

π c

(cm

−1 )

L −L

ωm

ωey

~

~

(c)

Figure 3.4: Bands of bulk modes are shown here as shaded regions. In (a) Dispersion is deter-mined from Eq.(3.16b) where there is only one gap near ω′

m. In (b) the solution from Eq.(3.16c)is shown, where there are two gaps between bulk bands. One gap exist around ωm while the otheris near the dielectric resonance frequency. In (c) Bulk bands of the system is superposition of thebands in (a) and (b). In this bands, both β1 and β2 are imaginary. The region betwen band edgesand dashed line has β1 real and β2 imaginary.

Bulk bands for the second solution in Eq.(3.16b) are determined by the imaginary

values of the attenuation constant β1. These regions are indicated by shaded regions in


Fig.3.4(a). Since the permitivity ǫx of BaMnF4 is independent of frequency, only magnetic

resonance frequency ω′m creates dispersion. Hence, there is only one gap around ω′

m

separating the two bulk bands.

The bulk bands for the third solution in Eq.(3.16c) are obtained by evaluating the

attenuation constant β2. Results are shown in Fig.3.4(b). There are two gaps, one is

around ωm and the other near ωey separating three bulk bands. As previously discussed,

these modes are non-reciprocal. However, because the the magnetoelectric coupling is

weak, the non-reciprocity is not visible in Fig.3.4(b).

Considering that the bulk modes are determined when both β1 and β2 are imagi-

nary [31], then the bulk regions are superposition of the shaded regions of Fig.3.4(a) and

Fig.3.4(b), result in the shaded regions in Fig.3.4(c). In the region between shaded region

and dashed lines, the β1 is real and β2 is imaginary.

−200 −100 0 100 2000

10

20

30

40

50

k/2π ( cm−1 )

ω/2

π c

( cm

−1 )

−L L

SP SP

ωey

ωm

~

~

(a)

0 20 402.94

2.96

2.98

3

3.02

3.04

k/2π ( cm−1 )

ω/2

π c

( cm

−1 )

L

ωm

ωm

~

’

SP

(b)

0 100 20038

40

42

44

k/2π (cm−1)

ω/2

π c

(cm

−1 )

L

ωey

~

SP

(c)

Figure 3.5: Dispersion relation for the surface modes. Surface polaritons exist in two regions.One near ωm and the other near ωey as illustrated in (a). The surface modes are representedby thick lines denoted by ’SP’, and the shaded region represent bulk modem. In (b), the zoomof surface modes near ωm. Surface mode started from the edge of bulk region and approachingωm as propagation number kz −→ ∞. In (c), the zoom of surface modes near ωey. Surface modeapproaching ωm as kz −→ ∞. The dashed lines in (a), (b) and (c) inside the bulk bands determinedfrom Eq.(3.16b), while the thin solid lines represent the band edges from Eq.(3.16c).


The numerical solution of Eq.(3.31) is given in Fig.3.5(a). We find two branches of

surface modes for each direction of propagation vector kz, located slightly outside the

bulk region edge. One is a magnetic surface mode, near ωm, as illustrated in the enlarged

Fig.3.5(b). The magnetic surface mode begins from bulk band edge in Eq.(3.16c), at a

frequency around ω = 2.98 cm−1 and propagation vector around kz = 16.5 cm−1. At

this point, both the attenuation constant β1 and β2 begin to have positive real values

as required for the surface modes. The surface mode approaches magnetic resonance

frequency ωm as kz −→ ∞.

The second branch is a phonon surface mode located near the dielectric resonance

as illustrated in Fig.3.5(c). As above, here the surface mode begins from the bulk edge

at a frequency around ω = 39 cm−1 and kz = 112 cm−1. The frequency of this mode

approaches ωey as kz −→ ∞.

20 30 40 503.005

3.006

3.007

3.008

3.009

k/2π ( cm−1 )

ω/2

π c

( cm

−1 )

k > 0

k < 0

(a)

50 100 150 2003

3.002

3.004

3.006

3.008

3.01

k/2π ( cm−1 )

ω/2

π c

( cm

−1 ) k > 0

k < 0

(b)

Figure 3.6: Non-reciprocity of the surface modes. In (a), comparison of two surface dispersionsat kz and −kz near the magnetic resonance frequency. The slight difference represent a weak non-reciprocity resulting from weak ME coupling. The dashed lines represent the surface polaritonsfor kz < 0, while the solid lines represent surface modes for kz > 0. In (b), the non-reciprocity isstronger because we have increased the ME coupling by a factor of ten.

From Eq.(3.31), we saw that the surface modes are non-reciprocal as ω(kz 6= ω(−kz).

Since the non-reciprocity is generated by the nagnetoelectric interaction, it is small due to

the weak magnetoelectric coupling. Examples are given in Fig.3.6(a). The non-reciprocity

of the surface mode can be enhanced with stronger magnetoelectric coupling. This is

illustrated in Fig.3.6(b) by supposing a larger magnetoelectric coupling.

3.6 Effect of external field

The surface mode frequency can be modified by applying either an electric or a magnetic

field. The external field influences polarization, magnetization and canting angle. It


required three equations to calculate these parameters. The first equation is obtained by

minimizing ferroelectric energy density with

∂ (FE + FME)

∂Py= ζ1Py + ζ2P

3y − 2αM2

s sin 2θ − Ey = 0. (3.32)

We also use a mean field approximation for ferromagnet with the Brillouin function B (η)

which is valid for the case when TN < TC :

Ms = MBs (η) (3.33)

where M represents lattice magnetization at temperature T=0, and

η =gµBS

kBT(λMs cos 2θ +KMs cos

2 θ

+ 2αPyMs sin 2θ +H sin θ − 4πMs sin2 θ). (3.34)

The third equation is equilibrium condition from the static part of the magnetic equations

of motion, as

H cos θ −(

1

2KMs + 2πMs + λMs

)

sin 2θ + 2αPyMs cos 2θ = 0. (3.35)

The values for polarization, magnetization and canting angle for certain value of external

field H or Ey are obtained by solving simultaneously Eqs.(3.32) to (3.35).

0 200 4002.99

3

3.01

3.02

k/2π ( cm−1 )

ω/2

π c

( cm

−1 )

(a)

115 117 119 121 123 12539.6

39.8

40

40.2

40.4

40.6

k/2π ( cm−1 )

ω/2

π c

( cm

−1 )

(b)

Figure 3.7: Influence of external fields on the surface modes. The solid line represents surfacemodes when an external field is applied, and the dashed line represents surface modes withoutexternal field. In (a), An external electric field increase the frequencies of surface modes near themagnetic resonance . In (b), the small effect of an external magnetic field on surface modes neardielectric resonance is shown.


Results are presented in Figs.3.7(a) and 3.7(b). The application of an electric field

E = 3× 108 V/m increases ωm by 0.12% (∆ωey = 0.0035 cm−1). Since the surface mode

exists near ωm at large kz , the external electric field increases the surface mode frequencies

(as kz −→ ∞). This is illustrated in Fig.3.7(a).

The application of a magnetic field Ho = 10T results in only very small effects on the

surface mode near ωey, as illustrated in Fig3.7(b). The H decreases Ω4c in Eq.(3.7), which

in turn also decreases δ. Hence, the dielectric frequency ωey becomes smaller with H. We

find that frequency change is very small, ∆ωey = 2× 10−6 cm−1, with negligible effect.

3.7 Conclusion

We have shown how magnetoelectric coupling with PML-type influences surface polaritons

in a canted multiferroic. The surface polaritons is in un-polarized condition which is neither

TE nor TM polarization when we set the configuration where the propagation vector is

parallel to the uniaxial easy axis. We find that surface polaritons are non-reciprocal with

respect to the propagation direction, such that ω(k) 6= ω(−k). The appearance of non-

reciprocity depends on magnetoelectric coupling such as the bigger the magnetoelectric

coupling, the bigger the non-reciprocity will be.

Chapter 4

Ferromagnetic resonance shift

from electric fields: field enhanced

screening charge in ferromagnet/

ferroelectric multilayers

4.1 Introduction

Manipulation and control of electron spin are the heart of spin electronics. A problem for

applications is that intrinsic magnetoelectric (ME) interactions are relatively weak [2, 7].

In purpose to obtain a stronger magnetoelectric effect, the artificial magnetoelectric mul-

tiferroics were proposed [2]. In the past, multilayers comprised of magnetostrictive and

piezoelectric materials have been examined [6–9]. In these layered structures, the mag-

netoelectric effect takes place at the interface between magnetostrictive and piezoelectric

materials, and is mediated by strain.

Extensive theoretical studies of the magnetoelectric (ME) effect in composites com-

prised of metallic ferromagnets (FM) and ferroelectrics (FE) using ab initio band structure

calculations [10], first principle calculations [11, 12] and calculations of the screening po-

tential [13,14] have been made. In this type of ME effect, polarization by the ferroelectric

generates spin-polarized charges at the ferromagnet/ferroelectric interface which change

the magnetization at the ferromagnet interface through spin-dependent screening within

ferromagnet layer. Spin-dependent screening charges are accumulated within a few lat-

tice sites of the interface, creating an additional magnetization. This effect is referred to

as interface ME mediated by carriers. Previous studies of carrier-mediated magnetoelec-

59

60 4. Ferromagnet resonance shift

tricity [10–14] focussed on the calculation of additional magnetization and ME coupling.

Tunneling process by the spin-dependent carriers have also been considered [29].

In the present work we show how the induced additional magnetization affects spin-

waves, and can be measured using ferromagnetic resonance. Since the additional magne-

tization effectively occurs only at the ferromagnet/ferroelectric interface, the electrodes

need to be as thin as possible for measurable effects. As a consequence the spin-polarized

charge at the ferroelectric edge can not be completely compensated by spin-dependent

screening charges. This means that the ferromagnet electrode should be treated as a real

metal and depolarization should be accounted for in the calculation of polarization in the

ferroelectric component. Since there is a depolarization field which tends to suppress po-

larization, an external electric field is needed to produce stable polarization. Taking these

effects into account, we show that there is a small, but measurable shift of the resonance

frequency. We find also the appearance of an additional weak resonance as a result of the

existence of the additional magnetization.

The chapter is organized as follows. The theory for screening charge in the trilayer

system is given in Section 4.2 where we also discuss the effect of incomplete compensation

of the polarization. In Section 4.3, the additional magnetization is calculated using a

Thomas-Fermi approximation based on the theory by Zhang [14]. The effect of additional

magnetization on spin wave frequencies and mode profiles is discussed in section 4.4.2.

Numerical calculations for the magnetic susceptibility using an entire-cell effective medium

theory is given in Section 4.4.1. Frequency shifts as a function of applied field for four

different ferromagnetic metals are given in Section V. A summary and conclusion are given

in Section 4.5.

4.2 Geometry and screening charges

In this section we describe how film thicknesses affect screening charges. The geome-

try is sketched in Fig.4.1. Following Cai [13], we consider superlattices which has unit

cell comprised of trilayer component where the ferroelectric (FE) with thickness l is sand-

wiched between a metallic ferromagnet (FM) and a normal metal (NM) with the thickness

L−l2 . The polarization is assumed to be normal to the interface, opposite to the x axis,

while the magnetization of the ferromagnet is also in-plane along the z axis. Compared

to a ferromagnet/ferroelectric/ferromagnet trilayer, the ferromagnet/ferroelectric/normal

metal configuration can result in a stronger magneto-electric effect through spin transfer

between the ferromagnet and the normal metal [13]. Polarization in the ferroelectric gives

4. Ferromagnet resonance shift 61

rise to bound charges at the ferroelectric interface. These bound charges are compensated

by screening charges in the normal metal electrode. Shorting (connecting) the electrodes

allows spin polarized charge to accumulate at the ferromagnet/ferroelectric interface with

charge depletion at the ferroelectric/normal metal interface. Spin polarization is possible

because the ferromagnet electrode is a ferromagnetic transition metal and there is some

spin polarization of conduction spins. This means that the screening electrons will posses

a net magnetic moment which results in the appearance of additional magnetization. In

the ferromagnet/ferroelectric/ferromagnet configuration, the additional magnetization at

one interface is compensated by the same decrease of magnetization at other side, hence

the total additional magnetization will relatively small [11]. Therefore using ferromag-

net/ferroelectric/normal metal configuration as in Fig.4.1, an additional magnetization on

the ferromagnet will not be compensated by electron depletion from the normal metal

side.

Figure 4.1: Geometry of FM/FE/NM unit cell. The ferroelectric (FE) with thickness l is sand-wiches between metallic ferromagnet (FM) and normal metal (NM). The polarization P accumu-lates the charge at the ferroelectric interface. Since the charge compensation by the electrode isuncomplete, this results in depolarisation field Ed which is in opposite direction to polarization.The external field Eext is applied along the spontaneous polarization.

If the NM electrode is assumed to be a perfect metal, then the bound charges are

completely compensated by screening electrons. In this case the depolarization effect on

the ferroelectric can be ignored because the value will be very small since the density of

screening electrons is the same as the density of bound charges. However, in the case

of a real metal, the compensation is not complete, and there is a depolarization field

Ed directed opposite to the polarization P. Hence the depolarization field should be

considered when calculated additional magnetization. In this case, the density of the

screening electrons will be less than the density of bound charges.

The effect of incomplete compensation of screening electrons has been studied by

Mehta [87] and Tilley [88]. In the Tilley model, the electric field in the electrode is calcu-

lated using the Thomas-Fermi approximation. The depolarization field can be derived by


firstly expressing the total energy density of the system . This energy density consists of

ferroelectric in the field produced by the electrodes and can be written in dimensionless

form as (see Appendix A for details):

F

S=

∫

dx

(

T

Tc− 1

)

P 2

2+

P 4

4+

1

2

(

dP

dx

)2

+1

2ǫǫfaTcP 2 +

V

α+ lP

− 1

2ǫǫfaTc

1(

α+ l)

[∫

P dx

]2

+γ

8ǫ(

α+ l)2

(

1

aTc

)[∫

P dx

]2

− 2ǫf Vγ

8(

α+ l)2

∫

P dx+

(

a2T2c

K1/2

)

ǫǫ2f γ

8(

α+ l)2 V

2 +

1

2δ

(

P 2+ + P 2

−)

(4.1)

where parameter γ is defined as

γ =

(

λl

ǫlβl +

λr

ǫrβr

)

with βl,r =sinh

(

L−lλl,r

)

−(

L−lλl,r

)

sinh2(

L−l2λl,r

) . (4.2)

Here, ǫ is the vacuum permittivity, ǫf represents the permittivity of the ferroelectric, and

ǫr and ǫl are the permittivities of right and left electrodes. The parameters a, B and

K represent Ginzburg-Landau constants while the parameters δ represent extrapolation

length which describes that the polarization at the ferroelectric interface is different from

the polarization in the interior of ferroelectric. The dimensionless polarization P is related

to the real polarization P as P = PP where P = aTcB with Tc is Curie temperature.

The other dimensionless parameters such as dimensionless parameters for energy density

(F ), position (x), external voltage (V) and ferroelectric thickness (l) are defined by: F =

B

a3/2 T

3/2c K1/2

F ; x =(

aTcK

)1/2x ; V =

(

BK

)1/2 V

aTcand l =

(

aTcK

)1/2l. The dimensionless

parameters α and γ are defined as α =(

aTcK

)1/2α and γ =

(

aTcK

)1/2γ, where α is defined

as

α = λlǫfǫl

cosh(

L−l2λl

)

− 1

sinh(

L−l2λl

) + λrǫfǫr

cosh(

L−l2λr

)

− 1

sinh(

L−l2λr

) . (4.3)

Here, λr and λl represent the screening lengths of right and left electrodes.

The Euler-Lagrange equations minimize (4.1) and give:

(

T

Tc− 1

)

P + P 3 −(

∂2P

∂x

)

− Ed − Eext = 0, (4.4)

where

Ed = − 1

ǫf

(

1

aTc

)

[

P − η ¯P]

and Eext = − V

lη. (4.5)


Here, Ed and Eext represent the depolarization and external fields in dimensionless form,

with the parameter η is defined as

η =l

α+ l− γl

4 (α+ l)2. (4.6)

Table 4.1: Parameter values of Fe and BaTiO3 used in the numerical calculations. The parametervalues for Fe are taken from Ref. [14, 51, 89] while parameter values for BaTiO3 are taken fromRef. [90].

Fe value BaTiO3 value

ρ↑ 0.87 eV−1nm−3 a 6.65× 105VmC−1K−1

ρ↓ 0.24 eV−1nm−3 Tc 391 KJ 2.4 eV nm3 ǫf 103

λ 1.3 A B 3.56 × 109Vm5C−3

Ms 1.67 × 106 A/m K 4.51 × 10−9Vm3C−1

Ha 3.98 × 104 A/m - -

In the limiting case of a perfect metal, the screening length λ → 0, hence α → 0

and γ → 0 so that η → 1. This will result in expressions for the energy density and

depolarization fields which are the same as in the Kretschmer and Binder system [91]. For

a real metal, the value of η is always less than 1.

The parameter η in Eq.(4.6) is strongly dependent on the electrode thicknesses. For

example, if the electrodes are much thicker than the screening length by at least an order

of magnitude, (L− l) >> λ, then β and γ → 0 and α ∼ λǫfǫe. In this case, η ∼ l

α+l .

Parameter values for Fe and BaTiO3 are given in Table 4.1. Using these parameters,

η is calculated as a function of ferroelectric thickness l and the results are shown in

Fig.4.2(a). We see that η increases with increasing ferroelectric thickness, and approaches

unity in the limit l >> α. The value of α is dependent on the thickness of ferromagnet.

In the case where the ferromagnet thickness is much bigger than the screening length of

the electrode, L− l >> λ, and the value of α saturates at 250. In this limit, the value of η

will approach 1 if the ferroelectric thickness is larger than 103 nm. Since the ferromagnet

thicknesses 5 nm and 10 nm produce almost the same value of α, then both thicknesses

yield approximately the same values for η. Note that η is an order of magnitude smaller

for ferroelectric thicknesses below 100 nm. In this limit the depolarization field becomes

important. The depolarization is very small and can be neglected when the thickness of

the ferroelectric is around 1 µm. We note that the use of a thick ferroelectric decreases

the effective magnetic susceptibility of the overall system.

We find that an FE thickness of 500 nm produces an optimal balance between sensi-

tivity to electric fields and strength of ferromagnetic response with these parameters. For


0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

l (nm)

η

L = 0.5 nm L = 1 nm

(a)

−250−150 −50 50 150 2500.08

0.1

0.12

0.14

x (nm)

P (

C/m

2 )

(b)

−80 −40 0 40 80−0.2

−0.1

0

0.1

0.2

E/Eb (%)

P (

C/m

2 )

(c)

Figure 4.2: In (a), values of η as a function of ferroelectric thickness are shown, and in (b) thepolarization profile is given with the magnetic electrode thickness (L − l)/2 = 20 nm and the FEthickness l = 500 nm and the applied field Eext = 1× 107 V/m2. In (c), a ferroelectric hysteresiswhich describes the relation between electric polarization P and external electric field E is given.Parameters are appropriate to Fe and BaTiO3, taken from Table 4.1.

this thickness, η is only 0.67 and there are significant depolarization fields. This increases

the magnitude of the applied electric field needed to create a large polarisation. The po-

larisation is also non-uniform due to surface effects. An example polarization profile is

shown in Fig.4.2(b). At Eext = 1× 107 V/m2, the average polarization P = 0.128 C/m2.

The relation between the electric polarization and an external electric field can be

obtained by solving the time dependent Ginzburg-Landau equation such as [92,93]

∂

∂t~Pi(r, t) = −L

∂F

∂ ~Pi(r, t)(4.7)

where F represents the ferroelectric density energy while L is the Landau coefficient. In

the calculation, we set ∆tL = 4× 10−6m/V. The result is a ferroelectric hysteresis as it is

shown in Fig.4.2(c).


4.3 Additional Magnetization

4.3.1 Theory of spin dependent screening

We now calculate the additional interface magnetization induced by the ferroelectric polar-

ization. The screening charge is present over several ferromagnetic layers at the interface

with an exponential decrease in charge density away from the interface on the ferromagnet

side [13]. According to Zhang [14], the total charge density in the metallic magnet is the

sum of two spin populations defined such that n (x) = n↑ (x)+n↓ (x). The induced charge

creates an additional spin dependent potential defined as [13]

eδV σ = eVc (x) + J[

δnσ (x)− δn−σ (x)]

(4.8)

where e and σ represent the electron charge and spin state while Vc is the Coulombic

potential which satisfies Poisson’s equation:

d2Vc (x)

dx2= − e

ǫ

[

δnσ (x) + δn−σ (x)]

. (4.9)

The second term on the right hand side of Eq.(4.8) is associated with an additional ex-

change potential where J represents exchange splitting constant between two spin states.

The induced charge density is estimated using the Thomas-Fermi approximation, where

the induced charge density of spin state σ, δnσ (x), is related to a small change in potential

δV σ (x) via

nσ (x) + δnσ (x) =(2S + 1)

6π2~32m [µ− eV σ (x)− eδV σ (x)]3/2 (4.10)

where µ represents chemical potential. Assuming that δV σ (x) is small, then the induced

charge density can be calculated using a Taylor expansion. Taking the first order term,

we find

δnσ (x) =∂nσ (x)

∂V σ (x)δV σ (x) (4.11)

where

∂nσ (x)

∂V σ (x)= −(2S + 1)

4π2~3(2m)3/2 [µ− V σ (x)]1/2

= −eρσ (4.12)

with ρσ as the spin-dependent density of states at the Fermi level for spin state σ. Using


Eq.(4.8) and Eq.(4.11), we obtain the relation

δn↑ (x)− δn↓ (x) = − ρ↑ − ρ↓

1 + J (ρ↑ + ρ↓)eVc (x) . (4.13)

The local induced magnetization is found from δM (x) =[

δ↑n (x)− δ↓n (x)]

µB . The

spin dependent potential is then calculated by using Eq.(4.11) and Eq.(4.9). The result is

d2Vc (x)

dx2=

Vc (x)

λ2(4.14)

where λ is a screening length defined by

λ =

[

e2(

ρ↑ + ρ↓ + 4Jρ↑ρ↓)

ǫ (1 + J (ρ↑ + ρ↓))

]−1/2

. (4.15)

The Coulomb potential in the electrode obeys the boundary condition

Vc

(±l

2

)

= V r,l and Vc

(±L

2

)

= 0, (4.16)

leading to a solution of the form:

Vc (x) = V l exp

(

12 l + x

λl

)

for x <−l

2(4.17a)

Vc (x) = V r exp

(

12 l − x

λr

)

for x >l

2. (4.17b)

The amplitude of the potential V l,r is found imposing charge conservation:

Q =1

4π

∫ −l/2

−∞

[

∂2V

∂x2

]

dx = − 1

4π

∫ ∞

l/2

[

∂2V

∂x2

]

dx, (4.18)

where Q represents surface density of charge. The result is

V r,l = ±Qλl,r

ǫ. (4.19)

An expression for the local additional magnetization is finally obtained from Eqs.(4.19),

(4.13) and (4.17a) as [13]

δM (x) = −(

ρ↑ − ρ↓)

µB

1 + J (ρ↑ + ρ↓)eVc (x) . (4.20)

The induced local magnetization is largest at the interface (x = −l/2) and decreases


exponentially as it goes into the ferromagnet. The total additional magnetization (in µB

per area) is obtained by integrating the induced local magnetization in Eq.(4.20) over the

ferromagnet layer:

∆M = −Q

e

(

ρ↑ − ρ↓)

µB

ρ↑ + ρ↓ + 4Jρ↑ρ↓ . (4.21)

The associated field is µ∆M/λ.

Using the parameters for Fe and BaTiO3 given in Table 4.1, the additional magneti-

zation ∆M was calculated for a 500 nm thick ferroelectric with ferromagnet and normal

metal each 20 nm thick. It should be noted that the additional magnetization is accumu-

lated at the FM/FE interface while the magnetization at the other position is unchanged.

The increase in external field will increase the polarization, thereby increasing the density

of bound charges. The bound charge is compensated by the screening charges, which

increase the additional magnetization via a polarization factor related to the s-d coupling

and the exchange interaction in the FM. Results calculated under the assumptions and

approximations outlined above are shown in Fig.4.3(a). For example, when the external

field is set at 3.7 % of the breakdown field, the average polarization will be 0.077 C/m2,

which in turn generates Q = −0.0575 C/m2 of screening charge. Here, the breakdown

field, Eb, is set at the value 1.5×107 V/m [94]. The magnetization is enhanced by 0.4%

relative to the lattice magnetization Ms. The magnetization enhancement is shown in

Fig.4.3(a), for variations of the external field from 3.7% to 74%, resulting in an enhanced

magnetization varying from 0.4% to 1.2%. Should be note here, to ensure the additional

magnetization has the direction parallel to the lattice magnetization Ms, we apply the

external magnetic H parallel to the lattice magnetization.

As discussed earlier, the thickness of the ferroelectric influences the magnitude of po-

larization, with an increase of the ferroelectric film thickness resulting in an increased

polarization. Shown in Fig.4.3(b) are results for the corresponding enhancement in mag-

netization. The external field is set at at 44% of the breakdown field and the ferroelectric

film thickness is varied from 100 nm to 700 nm. This causes the polarization to vary from

0.059 C/m2 to 0.134 C/m2, causing the additional magnetization to vary from 0.3% to

0.74%.

The thickness of the ferromagnet influences the additional magnetization according to

the parameter α, as in Eq. (4.3). This in turn changes the depolarization field through

parameter η. As can be seen in Fig.4.2(a), α saturates as the ferromagnet thickness

increases. For the parameters used here, the additional magnetization is independent of

ferromagnet thickness for ferromagnet thicknesses greater than 1 nm.


0 20 40 60 800

0.4

0.8

1.2

1.6

E/ Eb (%)

∆ M

/ Ms (

%)

(a)

0 2 4 6 80.25

0.45

0.65

0.85

FE thickness (× 102 nm)

∆ M

/ M (

%)

(b)

Figure 4.3: In (a), the additional magnetization which is accumulated at the FM/FE interfaceis shown for a range of applied electric fields, while in (b) the effect of FE thickness on additionalmagnetization is shown. Parameters appropriate to Fe and BaTiO3 are assumed.

4.3.2 Application to Fe, CrO2, Permalloy and Heusler alloy

We now calculate and compare the additional magnetization for different metallic fer-

romagnets. In particular, we show results for the half metal CrO2, the Heusler alloy

(Co2MnGe), and Permalloy (Fe0.8Ni0.2). The assumed parameters are given in Table 4.2.

Table 4.2: Parameter values for FM CrO2, Co2MnGe and Permalloy. The parameters for CrO2

are taken from Reference [13, 95–98], values for Co2MnGe are taken from reference [99–102] andparameters for permalloy are approximated from Reference [103, 104].

Material ρ↑ ρ↓ J λ Ms Ha

CrO2 0.69 eV−1nm−3 0 1.8 eV nm3 1.7A 0.6 T 76 OeCo2MnGe 1.5 eV−1nm−3 0 1 eV nm3 1.5A 1.2 T 5 OePermalloy 0.25 eV−1nm−3 1.1 eV−1nm−3 0.27 eV nm3 1.2A 1.08 T 28 Oe

Results for the additional magnetization for these different metallic ferromagnets are

shown in Fig.4.4(a). The results depend strongly on the density of states at the Fermi

level (∆ρ = ρ↑ − ρ↓) and the exchange splitting J . Since the half metal only has one type

of spin-depended density of states at the Fermi level, it has the largest value of ∆ρ. Since

the half metal has only one type of spin state at the Fermi level, and all the screening

charges are polarized in one direction, where the polarized ratio ρ↑−ρ↓

ρ↑+ρ↓= 1, the additional

magnetization in Eq.(4.21) becomes that of a half metal, ∆M = −σeµb, independent of

density of states. Since screening charges show only small differences between various half

metal materials [28], the additional magnetization in µb/nm2 will be similar to that shown

in Fig.4.4(a). Permalloy has a larger polarized ratio compared to Iron, (0.63 as opposed to

0.57), so the additional magnetization for both materials are located below the half metal


curve. The additional magnetization of Permalloy is higher than that of Fe.

0 20 40 60 800

0.5

1

1.5

E/Eb (%)

∆ M

(µ b /

nm2 )

Half

Perm

Fe

(a)

0 20 40 60 800

5

10

15

20

E/Eb (%)

∆ M

/ M

s (%

) CrO2

Heus

Perm Fe

(b)

Figure 4.4: The additional magnetization in a FM/FE/NM trilayer system for four different FMelectrode materials. In (a), the additional magnetization is presented in units of µb/nm

2 whilein (b) ithe percentage relative to the lattice magnetization is shown. The dotted line (’Heus’)represents a system with Heusler alloy Co2MnGe as the FM electrode. The solid line indicatesFe and the dashed-dot and dashed lines are for CrO2 and Permalloy, respectively. Since the layerthickness is 1 nm, hence the thicknesses of FM and FE are 20 nm and 500 nm.

The additional magnetization in terms of percentage relative to the lattice magnetiza-

tion Ms are shown in Fig.4.4(b). The Heusler alloy has a value of ∆M , in µb/nm2, similar

to that of CrO2. Nevertheless, the Heusler alloy has a lattice magnetization that is twice

the size of that in CrO2. In consequence, the percentage ∆M in a Heusler alloy is smaller

than that for CrO2. Iron has the smallest additional magnetization of all, but it also has

the largest magnetization Ms . This results in the smallest value of relative ∆M among

the other electrodes.

4.4 Effective medium: Susceptibility and Spin waves

The effective medium approximation is used to account for dipolar contributions to the

spin wave frequencies [67, 105–108]. Conventional effective medium theory assumes that

parameters across the layer are constant. This is not true for the thicknesses of films of

interest since there is additional magnetization at the ferromagnet/ferroelectric interface,

hence an entire-cell version of effective medium theory is used. The entire-cell effective

medium theory divides each film into layers sufficiently thin that an effective medium

approach can be taken for fields across each interface [106, 107]. A sketch of this film

subdivision is shown in Fig. 4.5. Suppose the normal metal film comprised of layer N = 1

to NI , the layers NI +1 to NII are inside ferroelectric film and layers NII +1 to NIII are


inside ferromagnet film. The Maxwell boundary conditions are applied at each interface,

and the average of the dynamic magnetization components m and dipolar fields h are

found. Components of the dynamic susceptibility are then calculated.

Figure 4.5: The geometry used for application of the entire-cell effective medium method. Eachfilm of the trilayer system is subdivided into thin layers with the thickness of sub-layer denoted as∆x. Layer N = 1 to NI are inside normal metal (NM), layer NI +1 to NII are inside ferroelectric(FE) and layer NII + 1 to NIII are inside ferromagnet (FM).

We first derive equations of motion for each layer i using the non-dissipative torque

equation1

γ

∂Mi

∂t= µMi × (Heff

i +Hexi ). (4.22)

Here Heffi represents an effective field for layer i which consists of an in-plane anisotropy

field Haz, an out of plane anisotropy Hux, an exchange field Hex, the external magnetic

field Hz, and a dipolar field h. The magnetization for layer i is represented by Mi. In

this layered system, the inter-lattice distance in the x direction is denoted by ∆x which

is similar to the spacing in the previous calculation of polarization. The exchange field at

lattice i for the discret system can be written as

Hexi =

2A

µM2s

[

Mi+1 +Mi−1 − 2Mi

∆2x

]

(4.23)

where A represents stiffness exchange constant.

We define the magnetization as

Mi = (mxi ,m

yi ,Ms) (4.24)

where mi represents the dynamic term such that mi << Ms. The equations of motion


after linearization and assuming mi ∝ eiωt are

−iω

γmx

i = µ (H +Ha)myi −

2A

Ms

(

myi+1 +my

i−1 − 2myi

)

∆2x

− µMshy (4.25a)

−iω

γmy

i = −µ (H +Ha +Hu)mxi +

2A

Ms

(

mxi+1 +mx

i−1 − 2mxi

)

∆2x

+ µMshx. (4.25b)

In the next step, we require the dipolar field to obey Maxwell’s boundary conditions.

These are the continuity of tangential h and the continuity of normal b across each layer

interface. We write these as

hyi = hyi+1 = . . . = hyN = Cy (4.26a)

hxi + µmxi = hxi+1 + µm

xi+1 = . . . = hxN + µm

xN = Cx. (4.26b)

where Cy and Cx are constant. Requiring continuities of the tangential h and normal b

fields in Eq.(4.26), the equation of motion can be written as

−iω

γmx

i =µ (H +Ha)myi −

2A

Ms

(

myi+1 +my

i−1 − 2myi

)

∆2x

− µMsCy (4.27a)

−iω

γmy

i =− µ (H +Ha +Hu + µMs)mxi +

2A

Ms

(

mxi+1 +mx

i−1 − 2mxi

)

∆2x

+ µMsCx.

(4.27b)

Note that continuity of normal h gives rise to a demagnetization term µMs in the equa-

tions of motion on account of interfaces where Ms changes.

4.4.1 Susceptibilities

We now calculate susceptibilities in the effective medium approximation. We do this

by writing the equations of motion from Eqs.(4.27a) and (4.27b) for the layer at the

ferroelectric/ferromagnet interface as

(iω/γ)mxi + µ (Ho +Ha)m

yi = µ (Ms +∆Mi)Cy (4.28a)

(iω/γ)myi − µ (Ho +Ha +Hu +Mi)m

xi = −µ (Ms +∆Mi)Cx. (4.28b)

In this approximation, the other layers in the ferromagnet have a similar form only but

without ∆M . It should be noted that the layers at the ferroelectric and normal metal

interfaces do not have magnetization, i.e. mi = 0. However, both normal and tangential

components of the dipolar field h exist and must satisfy the Maxwell boundary conditions


for normal b and tangential h. We solve simultaneously the coupled set of equations of

motion by fixing values of Cx and Cy (for example Cx = 1 and Cy = 2) to obtain the

set of mxi , m

yi , h

xi and hyi , averaged over all layers in the system, for example such as

< mx >=∑N

i=1mx

iN . We therefore arrive at values for 〈mx〉 , 〈my〉 , 〈hx〉 , 〈hy〉.

The susceptibility components are defined through the relations 〈m〉 and dipolar field

〈h〉 by

〈mx〉 = χxx 〈hx〉+ χxy 〈hy〉 (4.29a)

〈my〉 = χyy 〈hy〉+ χyx 〈hx〉 . (4.29b)

Calculation of the susceptibility components requires 〈mx〉 , 〈my〉 , 〈hx〉 , 〈hy〉 to be de-

termined from Cx, Cy and will be a function of frequency ω. The susceptibility for the

geometry given in Fig.4.1 has the form

χ =

χxx iχxy 0

−iχxy χyy 0

0 0 0

. (4.30)

11.3 11.5 11.7 11.9 12.1−1

−0.5

0

0.5

1

f (GHz)

χ xx (

× 10

3 )

(a)

11.4 11.42 11.44 11.46−1

−0.5

0

0.5

1

f (GHz)

χ xx (

× 10

3 )

(b)

Figure 4.6: Influence of the additional magnetisation on the FMR frequency. In(a), the additionalmagnetization generates an additional mode and shifts the original resonance frequency up, asillustrated in (b). The solid lines represent the case with additional magnetization, while thedashed line represents the case without additional magnetization.

Mode frequencies are obtained from the poles of the susceptibilities. Results from

numerical calculations of the susceptibilites are shown in Fig.4.6(a) for the following pa-

rameter values. The ferromagnet and normal metal layers are sliced into 20 layers with

the thickness of each layer 1 nm. Since the thickness of the slices is much thicker than

screening length, then the screening charge will be accumulated at the FM/FE interface.


The ferroelectric layer is divided into 500 layers with 1 nm thick slices. For these mate-

rials, an external electric field 44% of the breakdown field gives an average polarization

of 0.126C/m2. This value will result in an additional magnetization at the ferromag-

net/ferroelectric interface of 10.2% the magnetization of Fe. Solid line shows the susepti-

bility curve with additional magnetization is accumulated at the surface layer, while the

dashed line represents the suseptibility curve without additional magnetization.

In addition to the FMR mode at 11.4 GHz, an additional pole appears at around

12 GHz. There is also an up-shift of the FMR frequency of around 7 MHz due to the

additional magnetization. The appearance of this additional mode can be expected due

to the strong localization of the additional magnetization. A new mode appears because

the film has two magnetizations: Ms and Ms = Ms +∆M . This results in a mode mostly

associated with the FMR mode of the perfect film, and a localized mode largely confied

to the perturbed layer.

We can derive analytically an expression for this new mode by approximating the

system as composed of two coupled layers of unequal thickness and magnetization. One

has magnetization Ms = Ms +∆M with thickness t and the other has magnetization Ms

with thickness d where d > t as illustrated in Fig. 4.7. Since the ferroelectric and normal

metal do not have magnetization, we treat it as a spacer with thickness s.

Figure 4.7: Resonance modes for the perturbed thin film are approximated by a thin layer (FMI)with thickness t exchange coupled to another layer FMII layer with thickness d. The ratio of thethickness is similar to the ratio of the layers in the entire-cell model.

For the ferromagnet layer with an additional magnetization, the equations of motion

are

−iω

γmx = µHeffmy − µCyMs (4.31a)

−iω

γmy = −µ

(

Heff + Ms

)

mx + µCxMs (4.31b)


and equations of motion for ferromagnet layers without additional magnetization are

−iω

γmx = µHeffmy − µCyMs (4.32a)

−iω

γmy = −µ (Heff +Ms)mx + µCxMs (4.32b)

whereHeff = H+Ha. Setting Cy = 0, the solution of mx and hx for the thin ferromagnet

which posses additional magnetization ∆M is

mx =µ2HeffMsCx

[

µ2Heff

(

Heff + Ms

)

− ω2

γ2

] (4.33a)

hx =

(

µ2H

2eff − ω2

γ2

)

Cx[

µ2Heff

(

Heff + Ms

)

− ω2

γ2

] (4.33b)

and the solution for the ferromagnet layer without ∆M (the thick layer), is

mx =µ2HeffMsCx

[

µ2Heff (Heff +Ms)− ω2

γ2

] (4.34a)

hx =

(

µ2H

2eff − ω2

γ2

)

Cx[

µ2Heff (Heff +Ms)− ω2

γ2

] . (4.34b)

Since the magnetization of the normal metal and ferroelectric are zero, the magnetic field

inside these layers is hx = Cx. The effective susceptibility components can be defined as

χxx =tmx + dmx

thx + dhx + s. (4.35)

We substitute Eqs.(4.33a) and (4.34a) into the expression for the susceptibility, resulting

in components which share a common denominator. Setting the denominator to zero, we

arrive at

ω4 −Bω2 + C = 0 (4.36)

where

B =t+ d

Tµ2(

H2eff + αh

)

+d+ s

Tµ2Heff∆M + 2

s

Tµ2αh (4.37a)

C =t+ d

Tµ4H

2effαh +

d

Tµ4H

3eff∆M +

s

Tµ4αh (αh +Heff∆M) (4.37b)

and αh = Heff (Heff +Ms), T = t+d+s and ω = ω/γ. These equations are easily solved


with the result

ω1,2 = γ

B ±√B2 − 4C

2

1/2

. (4.38)

The frequency ω1 and ω2 represent the additional resonance frequency and the shifted

FMR frequency. Using the same parameters used to obtain Fig.4.6(b), we obtain similar

FMR resonance frequencies as long as the ratio of thickness t, d and s are the same as the

ratio of the number of layers in the ferromagnet, ferroelectric and normal metal. Here, we

set the parameters t= 1 nm, d=19 nm, s=520 nm with additional magnetization ∆M=170

Oe to obtain the result as in Fig.4.6(b).

Since the density of screening charge can be increased by raising the external field,

we expect that both the shift of the resonance frequency and the additional resonance

frequency can be driven to higher values by increasing the external field. Using Fe as an

electrode, we vary the external field E/Eb from 3.7% to 80%, the frequency shift increases

from 7.5 MHz to 6 MHz. This is illustrated as a solid line in Fig.4.8(a). Note that the

weak additional frequency slightly increases from 28.80 MHz to 28.92 MHz, as it is shown

as a solid line in Fig.4.8(b)

0 20 40 60 800

2

4

6

8

10

E/Eb (%)

δ f (

MH

z)

CrO2

Heus

Fe

Perm

(a)

0 20 40 60 800

10

20

30

E/Eb (%)

f add (

GH

z)

Fe

Heus

Perm

CrO2

(b)

Figure 4.8: Comparison of the frequency shift for the FMR mode (a)and the additional weakresonance (b) for Fe, permalloy and half metal CrO2 and Co2MnGe values. The solid line representsFe, the dashed line represents permalloy, and the dashed-dotted line represents CrO2. The dottedline represents Co2MnGe.

Next, we study the dependence of frequencies on the thickness of the ferroelectric. The

electrode thickness and external field are set constant at 20 nm and E/Eb = 44% V/m.

The ferroelectric thickness is increased by changing the number of layers from 100 to 700.

This results in a decrease in the frequency shift from 5.7 to 1.8 MHz. This decrement also

happens when the number of layers in the ferromagnet is increased. When we adjust the

number of ferromagnet layers from 10 to 50 layers, while holding the number of ferroelectric


layers constant at 250 , the shift of resonance frequency decreases from 11.3 to 2.8 MHz.

This is consistent with the frequency shifts being due to the interface layer additional

magnetization perturbation, which is significant only relative to the volume fraction of the

additional layer magnetization and that of the remainder of the film.

As a final point, we compare the frequency shifts as a function of applied field of the

FMR mode and the additional resonance for four different metallic ferromagnet electrodes

in Figs.4.8(a) and 4.8(b). The shift of FMR frequency has the largest value for CrO2, and

that for Fe has the smallest value. Since the FMR frequency is related to HaMs, it seems

reasonable that the frequency shifts are related to Ha∆M/Ms where ∆M/Ms represents

the effect of the additional magnetization. Since the additional resonances are related to

Ha∆M , then Fe has the largest additional resonance, and that for Heussler alloy has the

smallest value. In order to have a feeling for the magnitudes, we see that for E/Eb= 15%,

the largest additional magnetization is 3.7 × 104A/m for CrO2, and the smallest value is

8.7 × 103A/m for Fe. The values at this field for Heussler half metal and Permalloy lie

between these extremes, at 2.8× 104A/m and 2.3× 104A/m.

4.4.2 Standing spin waves

We now employ the equations of motion Eqs.(4.27a) and (4.27b) to study the influence of

the additional magnetization on the poles of the dynamic susceptibility (i.e., the standing

spin wave resonances). It is necessary to derive additional boundary conditions at the

FE/FM interface and the FM/NM interface to take into account exchange between mag-

netic layers. We begin by defining the equation of motion with surface anisotropy [109]:

1

γ

∂M

∂t+ µM×Heff +

2A

M2s

M×∇2M+ µM×Hs = 0 (4.39)

Here Hs is the surface anisotropy field defined as Hs = 2KsµM2

sMxx with Ks being the

surface anisotropy constant. The effective field Heff represents the external applied, and

uniaxial anisotropy effective fields. Integrating the equation of motion over the volume

around the surface yields Rado-Weertman boundary condition [109]

2A

M2s

M× ∂M

∂x+

2Ks

M2s

Mxx×M = 0. (4.40)

There is an additional boundary condition at the ferroelectric/ferromagnet interface

(layer NII + 1) because there is an additional magnetization ∆M which is affecting the

condition at the interface. As discussed earlier, the additional magnetization is largely


confined close to the interface. Here we assume that it exists only in the first layer of

ferromagnet near the ferromagnet/ferroelectric interface (layer NII + 1). The additional

boundary condition, from Eq. (4.40), is

Ms∂my

∂x−my

∂Ms

∂x= 0 (4.41a)

Ms∂mx

∂x−(

ζMs +∂Ms

∂x

)

mx = 0 (4.41b)

where ζ = Ks/A. The above boundary condition can be approximated by assuming

∂Ms∂x ≈ Ms

NII+2−Ms

NII+1

∆x= −∆M

∆xwhere M s

NII+1 represents the layer magnetization at

the ferroelectric/ferromagnet interface with additional magnetization ∆M from Eq.(4.21)

while M sNII+2 = Ms represents the magnetization at non-interface layer inside the ferro-

magnet and without any additional magnetization. This allows the boundary condition

to be written as

∂my

∂x+my

∆M

Ms∆x= 0 (4.42a)

∂mx

∂x−(

ζ − ∆M

Ms∆x

)

mx = 0. (4.42b)

Since both the normal metal and ferroelectric are non-magnetic material, hence the mag-

netization of the layers inside both normal metal and ferroelectric are considered to be

zero. However, to be able to derive the magnetic boundary condition at the ferromag-

net/ferroelectric interface (layer NII + 1), we assumed the layer NII in ferroelectric as

an imaginary layer. Hence, the boundary condition in Eq.(4.42a) and (4.42b) for layer

NII + 1 at the ferroelectric/ferromagnet interface can be written in the discrete form as

myNII+2 −my

NII

2∆x+

∆M

Ms∆xmy

NII+1 = 0 (4.43a)

mxNII+2 −mx

NII

2∆x−(

ζ − ∆M

Ms∆x

)

mxNII+1 = 0. (4.43b)

Substitution of Eq. (4.43) into the discrete second derivative form of the exchange

field yields the boundary components on the additional magnetization for the ferromag-

net/ferroelectric interface (layer NII + 1) as:

∇2myNII+1 =

2myNII+2 − 2

(

1− ∆MMs

)

myNII+1

∆2x

(4.44a)

∇2mxNII+1 =

2mxNII+2 − 2

(

1 + ζ∆x − ∆MMs

)

mxNII+1

∆2x

(4.44b)


A similar procedure is performed for the ferromagnet/normal metal interface (layer NIII)

by setting the additional magnetization to zero (∆M = 0). The result is

∇2myNIII

=2my

NIII−1 − 2myNIII

∆2x

(4.45a)

∇2mxNIII

=2mx

NIII−1 − 2 (1 + ζ∆x)mxNIII

∆2x

. (4.45b)

Next, we use the boundary conditions Eqs.(4.44) and (4.45) together with the equations

of motion from Eqs.(4.27a) and (4.27b) to obtain the mode frequencies and profiles and

consider the dynamic terms only by setting the constant terms (Cx, Cy) to zero, the

equations of motion can be brought into the form of an eigenvalue problem. Explicitly,

one has:

−iω

γmx

i =µ (H +Ha)myi −

2A

Ms

(

myi+1 +my

i−1 − 2myi

)

∆2x

(4.46a)

−iω

γmy

i =− µ (H +Ha +Hu + µMs)mxi +

2A

Ms

(

mxi+1 +mx

i−1 − 2mxi

)

∆2x

. (4.46b)

with the equation for layer NII + 1 near the ferroelectric/ferromagnet interface given by

−iω

γmx

NII+1 =µ [(H +Ha) + 2κ (1−∆M/Ms)]myNII+1 − 2κµm

yNII+2 (4.47a)

−iω

γmy

NII+1 =− µ [(H +Ha +Hu +Ms) + 2κ (1 + ζ∆x −∆M/Ms)]mxNII+1

+ 2κµmxNII+2, (4.47b)

and the equation for layer NIII at the ferromagnet/normal metal interface of the form

−iω

γmx

NIII=µ [(H +Ha) + 2κ]my

NIII− 2κµm

yNIII−1 (4.48a)

−iω

γmy

NIII=− µ [(H +Ha +Hu +Ms) + 2κ (1 + ζ∆x)]m

xNIII

+ 2κµmxNIII−1 (4.48b)

where κ = 2AµMs∆2

x.

Results are presented in Fig.4.9(a) for the FMR mode and Fig.4.9(b) for first excited

spinwave mode. The external electric field is set at 29.6% ofthe breakdown field, giving

rise to an additional magnetization of 8.1% relative to Ms. It can be seen that that addi-

tional magnetization has the smallest influence on the FMR profile, with a corresponding

downshift of the resonance frequency ∆ω = 30.16 MHz.


The downshift on the first excited mode is ∆ω = 38.76 MHz. The value of ∆MMs

is one

order of magnitude lower than the pinning factor ζ∆x (around 0.53), and according to

Eq.(4.42) decreases the effect of pinning. The decrease of pinning will increase the wave-

length of spin waves, and thereby decrease the frequency. We also present the frequency

shifts of several modes at different applied electric fields (see Fig. 4.9(c)). It can be seen

that the frequency shift increases with decreasing wavelength of the excited modes. The

largest frequency shifts occur for the highest order standing spinwave excitations.

5 10 15 200.2

0.4

0.6

0.8

1

x (nm)

mx (

A/m

)

(a)

5 10 15 20−2

−1

0

1

2

x (nm)

mx (

A/m

)

(b)

0 20 40 60 800

200

400

600

800

E / Eb (%)

∆ f

( M

Hz

)

(c)

Figure 4.9: Profile of mx for the FMR and first excited standing spinwave mode. Solid linesrepresent the case of additional magnetization at the surface layer, and dashed lines represent thesituation without an additional magnetization. The profile for the FMR is presented in (a), andthe profile for first excited standing spinwave mode is illustrated in (b). The horizontal scale, x,represents position measured from the ferromagnet/ferroelectric interface. In (c), The frequencyshift is given for several modes. The solid line represents the FMR mode with the frequency6.1 GHz, the dashed line represents the first excited mode with the frequency 35.5 GHz, thedashed-dotted line represents second excited mode with the frequency 95.6 GHz and the dottedline represents third excited mode with the frequency 187.8.1 GHz.


4.5 Conclusions

We present a theory for magnetic resonance in FM/FE/NM trilayers biased by an applied

voltage. We have shown how an additional magnetization is generated by the FE polariza-

tion at the FM interface, and that this can be measured indirectly through measurement

of frequency shifts arising from a small induced moment on the FM. We predict the ap-

pearance of an additional weak resonance associated with the induced moment. We find

that the frequency shifts depend upon the magnitude of the additional magnetization and

also the anisotropy field of the host. The shift of resonance frequency will be large if the

fractional additional magnetization and host anisotropy fields are large.

We have also explored the influence of additional magnetization on standing spinwave

resonances. We find that the additional magnetization decreases pinning at the surfaces

which in turn distorts the mode profiles across the film thicknesses, and increases the

effective wavelength of the modes.

Chapter 5

Conclusion

In this thesis we have studied surface polaritons in magnetoelectric multiferroics with

PML-type coupling which allows magnetic sub-system to be cant (chapter 2 and 3). We

have shown that magnetoelectric coupling has important effects on surface polaritons es-

pecially in terms of the non-reciprocity of the surface modes. We have also shown the

shift of the magnetic resonance frequency in carrier-mediated multiferroic in the multi-

layer comprise of ferromagnet/ferroelectric/normal metal system due to magnetoelectric

interaction (chapter 4). The additional magnetization at the ferromagnet/ferroelectric

interface can be driven by an applied electric field. This additional magnetization affects

both the magnetic resonance frequency and the magnetization profile. In this final chap-

ter, we review the main results of the thesis and propose future work which could build

upon this thesis.

In chapter 2 we presented, for the first time, a calculation of surface polaritons in a

magnetoelectric multiferroics with PML-type coupling applied to a real material, BaMnF4.

The magnetoelectric coupling was shown to lead to the non-reciprocity in the TE surface

modes. In this chapter, the TE and TM surface modes were determined for the wave

numbers parallel to the spontaneous polarization and perpendicular to the easy axis (and

also the weak ferromagnetism). The TE surface modes are of magnetic type (magnon

polariton) and are associated to the weak ferromagnetism. Given that the ferromagnetism

associated with the small canting is weak, the surface modes cannot be found in the

region where the wave number is much larger than the frequency, i.e., k >> ωc . Since

the magnetoelectric coupling involves the odd order of the polarization, the branches of

the TE surface polariton can be flipped by reversing the spontaneous polarization using

an external electric field. We found that the TM surface modes are reciprocal. However,

the magnetoelectric interaction affects the surface modes through the appearance of an

81

82 5. Conclusion

imaginary attenuation constant which results in pseudo-surface modes.

In chapter 3 we considered the case of surface polaritons where the propagation vector

k is parallel to the easy axis and perpendicular to the spontaneous polarization. We have

shown that this configuration leads to two values of attenuation constant which correspond

to a superposition of two plane waves. This superposition is required to generate the

surface polaritons. The surface polaritons modes are neither TE modes nor TM modes.

In this case, the surface modes are not polarized. We found that the surface modes are

non-reciprocal due to the magnetoelectric interaction. However, since the magnetoelectric

interaction is weak, the non-reciprocity is also weak.

In chapter 4 we studied the influence of carrier-mediated magnetoelectricity on the

magnetic resonance frequency in a FM/FE/NM multilayer system by employing an entire-

cell effective medium method. Polarization in the FE layer generates additional magneti-

zation at the FM/FE interface. The additional magnetization will increase as the polar-

ization is increased under an external electric field. High additional magnetizations can be

found in half-metal ferromagnets such as CrO2 and Heusler alloys, since there is only one

type of carrier in these systems. We found that the additional magnetization gives rise

to an additional magnetic resonance frequency and also shifts the intrinsic magnetic reso-

nance frequency. This shift depends on relative value of the additional magnetization with

respsect to lattice magnetization and anisotropy fields. Hence, we found that even though

the additional magnetization in half metal CrO2 and Heusler alloy is higher than that

in traditional metallic ferromagnets such as Fe or Permalloy, their weak anisotropy field

results in small resonance frequency shifts. We have also found that the additional mag-

netization distorts pinning at the FM/FE interface which in turn modifies mode profiles

of standing spin waves.

In a magnetic slab, the non-reciprocity of the surface waves is unique. The non-

reciprocity is associated with the localization [43]. For example, the surface modes with

propagation vector k propagate at the top surface of the slab, while those with −k prop-

agate at the bottom. No study has been carried out in the surface polaritons in a film

of magnetoelectric multiferroics which allows a canted spin system. This study represents

a next step to the work in this thesis, especially chapter 2 and 3. In a slab, we have to

consider two surfaces, one in the top and the other at the bottom. Hence, the appro-

priate solution for the Maxwell equation will be a superposition of two plane waves such

as E,H ∝(

eβy + e−βy)

ei(kx−ωt) where the thickness is considered in the y direction and

surface polaritons propagate in the x direction. Here β represents attenuation constant

5. Conclusion 83

of the medium. The dispersion relation for surface modes is determined using a similar

procedure as in chapter 2 or 3. The dispersion relation of the bulk polaritons is obtained

by changing attenuation constant α → iα in the expression of surface dispersion relation.

The more interesting problems are polaritons in a multilayer. The polaritons in multi-

layer comprise of magnetoelectric multiferroic with canted magnetic subsystem and mag-

netic material or ferroelectric have not been studied. This problem can be treated using

either conventional or entire-cell effective medium approximation to determine the suscep-

tibilities as in chapter 4. Then the dipersion relations are determined as in chapter 2 or

3. Recently, Livesey and Stamps [108] reported the existence of electromagnon resonance

frequency in piezoelectric/magnetostrictive composite material. It is interesting to study

the properties of polaritons around this electromagnon frequency. Since this frequency is

contributed by both the magnetic subsystem and the electric system, it can be expected

that the polaritons generated around this frequency will strongly depend on either an

applied electric field or an applied magnetic field.

It would also be of interest to extend the works in this thesis to study the nature

of polaritons localized at the junction (interface) between two materials. For example:

interface between magnetoelectric multiferroic and magnetic material, interface between

magnetoelectric multiferroic and ferroelectric, etc. In this case, the interface polaritons

propagate along the interface and decay in the direction normal to the interface. The

procedure to derive the dispersion relation is similar to the treatment of the surface mode

in chapter 2 and 3. However, the interface dispersion relation will be richer compared to

traditional surface modes since the resonance frequencies from the two materials contribute

to the behaviour of interface polaritons.

The problems mentioned in this chapter describe the possible extensions of the work in

this thesis on how the magnetoelectric interaction influences the properties of polaritons

in multiferroics.

84 5. Conclusion

Appendix A

Energy density for ferroelectrics

with electrodes

Following Mehta [87] and Tilley [88], calculation is started by describing the electric screen-

ing field in the form of Thomas-Fermi approximation [51]

d2E

dx2=

E

λ2(A.1)

where λ represent screening length of the electrode. For the case of two different electrode,

the boundary condition will be (using Mehta’s configuration, as in Figure (1))

E

(±L

2

)

= 0 and E

(±l

2

)

=σ±

ǫǫr,le

with the solution in the form

E (x) =σ±

ǫǫr,le sinh

(

L−l2λr,l

) sinh

(

12L∓ x

λr,l

)

. (A.2)

The electrostatic energy inside the electrodes,by considering charge conservation, σ+ =

−σ− = σ, can be written as

Fe =ǫ2

ǫle

∫ −l/2

−L/2E2

l dx+ ǫre

∫ L/2

l/2E2

rdx

=σ2

2ǫγ, (A.3)

where

γ =

(

λl

ǫlβl +

λr

ǫrβr

)

and βl,r =sinh

(

L−lλl,r

)

−(

L−lλl,r

)

sinh2(

L−l2λl,r

) . (A.4)

Next, we evaluate the electric field inside ferroelectric which is related to the polariza-

85

86 A. Energy density for ferroelectrics with electrodes

tion through the relation ∇ · ~D = 0, hence

ǫǫfdE

dx= −dP

dx⇒ E (x) = Eo −

1

ǫǫfP (x) . (A.5)

The electric field E can be analyzed by including the external potential V which obey

the condition∫ L/2

−L/2E (x) dx = −V. (A.6)

Then, substitution of both the electric field in electrodes and ferroelectric in equation

(A.2) and (A.5) into the relation (A.6) and also using the continuity of D at l/2 or −l/2:

ǫfE = ǫeEe ⇒ σ = ǫǫfE (A.7)

result in the expression of E as:

E =1

(α+ l)

1

ǫǫf

∫ l/2

−l/2Pdx− V

(A.8)

where

α = λlǫfǫl

cosh(

L−l2λl

)

− 1

sinh(

L−l2λl

) + λrǫfǫr

cosh(

L−l2λr

)

− 1

sinh(

L−l2λr

) . (A.9)

If the electric fields in the ferroelectric is defined as an addition of depolarization field and

external field as

E (x) = Ed + Eext (A.10)

and by comparing with equation A.5 and using the expression of E in equation (A.8) ,

then the depolarization field can be written as

Ed = − 1

ǫǫf

[

P (x)− l

(α+ l)P

]

(A.11)

and external field as

Eext = − l

(α+ l)

Vl

(A.12)

where the average polarization is defined as P = 1l

∫ l/2−l/2 Pdx. The value of l

(α+l)∼= 1 in the

case of complete compensation by assuming perfect conductor as electrodes in Kretschmer

and Binder model [91]

A. Energy density for ferroelectrics with electrodes 87

The expression of energy density is obtained by adding the FM energy density in

equation (A.3) and FE energy density as [88]

F =1

l

∫

dx

AP 2

2+

BP 4

4+

K

2

(

∂P

∂x

)2

− 1

2Ed − EextP

+K

2δ

(

P 2+ + P 2

−)

. (A.13)

88 Energy density for ferroelectrics with electrodes

List of Figures

1.1 Geometry of the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Polariton of the simple ferromagnet. . . . . . . . . . . . . . . . . . . . . . . 8

1.3 The dispersion relation for a semi-infinite anti ferromagnet (AFM). In (a)the dispersion relation is shown for the case without the application ofexternal magnetic field. In (b) bulk and surface mode dispersions are shownwith the application of magnetic field parallel to the magnetization M. Theshaded regions represent bulk modes. The surface modes are illustrated bythe thick lines. The dashed lines indicated by ”L” are light lines with ω = ck. 9

1.4 Polariton of the simple ferromagnet. . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Geometry of the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Illustration of symmetry argument . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Illustration of ATR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Attenuation constant and Dispersion relation. In (a) the attenuation con-stant is shown for two values of wavevector in the absence of external fields.The solid line represents k/2π = 0 cm−1, while the dashed line correspond tok/2π = 16 cm−1 . In (b) bulk and surface mode dispersions are shown. Sur-face modes are indicated by SP. The shaded regions represent bulk bands,limited by frequencies ωm, ωp and ωz. The vertical thin lines denote byvertical and horizontal arrows represent ATR light lines for incident anglesof 30o and 70o. The asymptotic boundaries and lightline are representedby vertical thin lines which are indicated as A and by L. The dashed linesrepresent surfaces branches when electric field (−5 × 109 V/m) is applied.Horisontal thin lines denote by ω′

m represent the magnetic resonance whenelectric field is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 ATR spectra with incident angles 30 (a) and 70 (b). In (a) two differentsharp dips illustrate the non-reciprocity of the surface modes. In (b) theabsence of the sharp dips indicate the absence of surface modes. . . . . . . 33

2.6 The condition for surface modes in k >> ω/c. In (a), a solution of dispersionrelation where λ/α = 1.5, the surface begin to overcome the bulk. In (d)the surface branch is slightly higher than bulk when the ratio increase toλ/α = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Hysteresis of BaMnF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 The influence of magnetic field. In (a)influence on canting angle. In (b),the application of a magnetic external field of 15 T increases the frequencyof surface mode and allows the negative surface branch to enter the mag-netostatic region, where k >> ω

c . . . . . . . . . . . . . . . . . . . . . . . . . 36

89

90 LIST OF FIGURES

2.9 Dispersion relation without external field. In (a) he dispersion relation with-out the external field is shown. The surface modes are indicated by“SP”.The shaded regions represent bulk bands, which are limited by frequenciesωm, ωez , ωoz , ωey and ωi. In (b) The “window” where the surface modesexist is shown. In (c) a narrow gap around ωi is expanded. In (d) thenarrow gap is wider when the ME coupling is increased by a factor of 10. . 37

2.10 Attenuation constant as a function of frequency. In (a) the real part ofattenuation constant is shown for two values of wavevector in the absenceof any external fields. The solid line represents k = 0 cm−1, while thedashed line corresponds to k = 250 cm−1 . In (b) the imaginary part ofattenuation constant is shown for two values of ME coupling. The solidline is for α = 1.42 × 10−5 cm2/statC, and the dashed line represents thecoupling α = 1.42 × 10−4 cm2/statC. . . . . . . . . . . . . . . . . . . . . . . 38

2.11 Influence external field in the dispersion relation. In (a)the external electricfield E increase the canting angle with small value. In (b) dispersion relationis shown with an external electric field along the spontaneous polarisationE = 5 × 108 V/m. In (c), magnetic resonance frequency shift down whenthe external magnetic field, Ho = 10T is applied. The dotted line is themagnetic frequency in the absence of magnetic field. . . . . . . . . . . . . . 40

2.12 Dispersion relation for a material with electric and magnetic resonances nearone another in frequency. In (a) the dispersion relation without externalmagnetic field is shown. In (b), the magnetic resonance frequency is shiftedinto a surface mode window with the application of a large magnetic field,Ho = 12T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.13 Temperature dependence with modified exchange constant. (a) Dispersionrelation at T=150 K. In (b), the magnetic resonance frequency shifts downat 250 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 An illustration of the geometry. The weak ferromagnetism M which is outof plane along x is resulted from the canting of two sub-lattices antiferro-magnetism ma and mb. The spontaneous polarization (P) is assumed tolie in a plane parallel to the surface along y. Propagation of the surfacemode is along the z perpendicular to the polarization. . . . . . . . . . . . . 45

3.2 Symmetry argument for non-reciprocity of the bulk solution for Eq.(3.16c).In (a), the initial configuration with wave vector kz is directed along z axis.In (b), the system after xz plane reflection is shown. In (c), the system atfinal state after xy plane reflection is shown. It is not possible reverse −kzand leave P and M unchanged. . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Values of β as a function of frequency. Values of β1 near ω′m at (a) k = 0

cm−1 and (b) at k = 5 cm−1. Values of β2 as a function of frequency atkz = 0 cm−1 for frequencies near ωm are shown in (c). In (d) frequencies arenear soft phonon frequency ωey. The solid lines represent the real valuesand the dashed lines represent the imaginary values of β. The examplesshow that β1 and β2 are never complex. . . . . . . . . . . . . . . . . . . . . 53

3.4 Bands of bulk modes are shown here as shaded regions. In (a) Dispersionis determined from Eq.(3.16b) where there is only one gap near ω′

m. In (b)the solution from Eq.(3.16c) is shown, where there are two gaps betweenbulk bands. One gap exist around ωm while the other is near the dielectricresonance frequency. In (c) Bulk bands of the system is superposition ofthe bands in (a) and (b). In this bands, both β1 and β2 are imaginary. Theregion betwen band edges and dashed line has β1 real and β2 imaginary. . . 54

LIST OF FIGURES 91

3.5 Dispersion relation for the surface modes. Surface polaritons exist in tworegions. One near ωm and the other near ωey as illustrated in (a). Thesurface modes are represented by thick lines denoted by ’SP’, and the shadedregion represent bulk modem. In (b), the zoom of surface modes near ωm.Surface mode started from the edge of bulk region and approaching ωm aspropagation number kz −→ ∞. In (c), the zoom of surface modes near ωey.Surface mode approaching ωm as kz −→ ∞. The dashed lines in (a), (b)and (c) inside the bulk bands determined from Eq.(3.16b), while the thinsolid lines represent the band edges from Eq.(3.16c). . . . . . . . . . . . . . 55

3.6 Non-reciprocity of the surface modes. In (a), comparison of two surfacedispersions at kz and −kz near the magnetic resonance frequency. Theslight difference represent a weak non-reciprocity resulting from weak MEcoupling. The dashed lines represent the surface polaritons for kz < 0,while the solid lines represent surface modes for kz > 0. In (b), the non-reciprocity is stronger because we have increased the ME coupling by afactor of ten. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Influence of external fields on the surface modes. The solid line representssurface modes when an external field is applied, and the dashed line repre-sents surface modes without external field. In (a), An external electric fieldincrease the frequencies of surface modes near the magnetic resonance . In(b), the small effect of an external magnetic field on surface modes neardielectric resonance is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Geometry of FM/FE/NM unit cell. The ferroelectric (FE) with thickness lis sandwiches between metallic ferromagnet (FM) and normal metal (NM).The polarization P accumulates the charge at the ferroelectric interface.Since the charge compensation by the electrode is uncomplete, this resultsin depolarisation field Ed which is in opposite direction to polarization. Theexternal field Eext is applied along the spontaneous polarization. . . . . . . 61

4.2 In (a), values of η as a function of ferroelectric thickness are shown, and in(b) the polarization profile is given with the magnetic electrode thickness(L − l)/2 = 20 nm and the FE thickness l = 500 nm and the applied fieldEext = 1 × 107 V/m2. In (c), a ferroelectric hysteresis which describesthe relation between electric polarization P and external electric field E isgiven. Parameters are appropriate to Fe and BaTiO3, taken from Table 4.1. 64

4.3 In (a), the additional magnetization which is accumulated at the FM/FEinterface is shown for a range of applied electric fields, while in (b) theeffect of FE thickness on additional magnetization is shown. Parametersappropriate to Fe and BaTiO3 are assumed. . . . . . . . . . . . . . . . . . . 68

4.4 The additional magnetization in a FM/FE/NM trilayer system for four dif-ferent FM electrode materials. In (a), the additional magnetization is pre-sented in units of µb/nm

2 while in (b) ithe percentage relative to the latticemagnetization is shown. The dotted line (’Heus’) represents a system withHeusler alloy Co2MnGe as the FM electrode. The solid line indicates Fe andthe dashed-dot and dashed lines are for CrO2 and Permalloy, respectively.Since the layer thickness is 1 nm, hence the thicknesses of FM and FE are20 nm and 500 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 The geometry used for application of the entire-cell effective mediummethod.Each film of the trilayer system is subdivided into thin layers with the thick-ness of sub-layer denoted as ∆x. Layer N = 1 to NI are inside normal metal(NM), layer NI + 1 to NII are inside ferroelectric (FE) and layer NII + 1to NIII are inside ferromagnet (FM). . . . . . . . . . . . . . . . . . . . . . . 70

92 LIST OF FIGURES

4.6 Influence of the additional magnetisation on the FMR frequency. In(a), theadditional magnetization generates an additional mode and shifts the orig-inal resonance frequency up, as illustrated in (b). The solid lines representthe case with additional magnetization, while the dashed line represents thecase without additional magnetization. . . . . . . . . . . . . . . . . . . . . . 72

4.7 Resonance modes for the perturbed thin film are approximated by a thinlayer (FMI) with thickness t exchange coupled to another layer FMII layerwith thickness d. The ratio of the thickness is similar to the ratio of thelayers in the entire-cell model. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.8 Comparison of the frequency shift for the FMR mode (a)and the additionalweak resonance (b) for Fe, permalloy and half metal CrO2 and Co2MnGevalues. The solid line represents Fe, the dashed line represents permalloy,and the dashed-dotted line represents CrO2. The dotted line representsCo2MnGe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.9 Profile of mx for the FMR and first excited standing spinwave mode. Solidlines represent the case of additional magnetization at the surface layer, anddashed lines represent the situation without an additional magnetization.The profile for the FMR is presented in (a), and the profile for first excitedstanding spinwave mode is illustrated in (b). The horizontal scale, x, rep-resents position measured from the ferromagnet/ferroelectric interface. In(c), The frequency shift is given for several modes. The solid line representsthe FMR mode with the frequency 6.1 GHz, the dashed line represents thefirst excited mode with the frequency 35.5 GHz, the dashed-dotted line rep-resents second excited mode with the frequency 95.6 GHz and the dottedline represents third excited mode with the frequency 187.8.1 GHz. . . . . . 79

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