Bulk and Surface Polaritons in PML-type Magnetoelectric Multiferroics and The Resonance Frequency Shift on Carrier-Mediated Multiferroics

Transcription

1 Bulk and Surface Polaritons in PML-type Magnetoelectric Multiferroics and The Resonance Frequency Shift on Carrier-Mediated Multiferroics 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

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3 Abstract Polaritons are shown to be influenced by magnetoelectric coupling in a semi-infinite multiferroic 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 BaMnF 4, 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) 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 considered 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 polarization of the ferroelectric, and increase the magnetic moment at one interface (ferromagnet/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 moment. Estimates are provided using parameters appropriate to the ferroelectric BaTiO 3 iii

4 and four different ferromagnetic metals, including a Heusler alloy (Fe, CrO 2, Permalloy and Co 2 MnGe). The calculations use an entire-cell effective medium approximation that takes into account the polarization profile in the ferroelectric. The metallic ferromagnetic electrode is treated as a real metal, and the depolarization fields are included in the determination of the polarization in ferroelectric. iv

5 Acknowledgements I 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

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7 Contents 1 Introduction Introduction to Magnetoelectric Multiferroics Coupled magnetization and electric polarization Composite magnetoelectric multiferroics Introduction to Polaritons Polaritons in magnetic material Phonon polaritons Polaritons in magnetoelectric multiferroics Outline of Thesis TE and TM Polarization Introduction Transverse Electric & Transverse Magnetic modes Outline of the chapter Geometry and Energy Density Equation of motion and susceptibility Description for bulk bands General description TE modes TM modes Surface Polariton modes TE surface modes Non-reciprocity of surface modes Attenuated total reflection (ATR) TM surface modes Application to BaMnF TE modes vii

8 2.6.2 Effect of an applied field for TE modes TM modes Effect of applied fields for TM modes Summary Un-polarized polaritons Introduction Geometry Dynamic susceptibility Theory for bulk bands and surface modes Dispersion relation for bulk modes Attenuation constant Surface modes Application to BaMnF Effect of external field Conclusion Ferromagnet resonance shift Introduction Geometry and screening charges Additional Magnetization Theory of spin dependent screening Application to Fe, CrO 2, Permalloy and Heusler alloy Effective medium: Susceptibility and Spin waves Susceptibilities Standing spin waves Conclusions Conclusion 81 A Energy density for ferroelectrics with electrodes 85 List of figures 88 Bibliography 93 viii

9 Chapter 1 Introduction This thesis is divided into two main topics. The first topic, which is presented in chapters 2 and 3, focuses on polaritons in magnetoelectric multiferroics with PML-type coupling 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 ferromagnet-ferroelectric, ferromagnet-ferroelastic, etc. in the same phase [1]. In these materials, the magnetoelectric interaction couples the magnetic and the electric responses, allowing 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 materials 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 multiferroics [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

10 2 1. Introduction coupling [10 14] Coupled magnetization and electric polarization Given that the ordering of magnetization and electric polarization which is symmetry allowed 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 Cr 2 O 3. 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 =λm a M b K 2 ( M 2 a,z +Mb,z 2 ) H (Ma +M b ) α 1 P 2 +α 2 P 4 E P γp 2 (M a M b ) (1.1) where the electric polarization P and the magnetization M are the order parameters. Here, M a andm b describethemagnetization ofsub-lattices, λrepresentstheexchangeconstant, K is the strength of the magnetic anisotropy which favors the magnetic alignment in the z direction, H is an applied magnetic field, α 1 and α 2 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 T N is the Neel temperature of antiferromagnet and T C is the Curie temperature and assuming T N < T C, the interaction of two subsystems is present at T < T N. As a consequence of the existence of magneroelectric interaction, The character of electromagnetic waves in magnetoelectric multiferroic is determined by the electric susceptibility (χ e ), the magnetic susceptibility (χ m ) and the magnetoelectric susceptibility (χ me ). The more interesting model is magnetoelectric multiferroic with ferroelectric-antiferromagnetic ordering where a weak ferromagnetism is allowed. The weak ferromagnetism is a phenomena where the spin systems in antiferromagnet cant and generate a residual magnetic moment. This phenomena in magnetic material is described by the Dzyaloshinskii- Moriya (DM) interaction [17,18] as E ME ij = D ij (s i s j ) where s i and s j represent spin of the ion i and j and D ij is the DM tensor. The spin system in a ferroelectricantiferromagnet can be canted and result in a weak ferromagnetism if the energy density of magnetoelectric interaction has the form [15,19] P (M a M b ). This form is basically

11 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 polarization throughthemagnetoelectric effect hasalsobeenproposedformultiferroicbamnf 4. The energy density of the magnetoelectric interaction which is symmetry-allowed for BaMnF 4 was proposed as [20] F ME = ( β 1 P y +β 2 P 2 y) Mx L z (1.2) where β 1 and β 2 describe the strength of the magnetoelectric couplings, M = M a +M b and L = M a M b represent the weak ferromagnetism and the antiferromagnetic vector. In this type of coupling, the linear P y 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 (M a M b ). The second term which has quadratic P y is responsible in influencing the dielectric constant [20]. The canting of magnetic sub-lattices leads to the weak ferromagnetism as M c = 4πβ 1 P. Later, by performing first principle calculation, Ederer [21] and Fennie [22] show that the DM-type interaction in the form F PML P (M L) is responsible for the canting of magnetic sub-lattices on multiferroic FeTiO 3. 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 BaMnF 4. Weak ferromagnetism reversal can be performed by reversing the polarization P while keeping the antiferromagnet vector L unchanged [21]. However, in multiferroic BiFeO 3 the cant of the magnetizations is determined by the rotation of oxygen octahedra around the magnetic ions instead of DMtype interaction [23]. In chapter 2 and 3 of this thesis, we will use the same type of magnetoelectric energy density (F PML 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.

12 4 1. Introduction 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 magnetostrictive ferromagnets. These types of composite multiferroics have magnetoelectric coupling 100 times higher than that in single phase multiferroics [6]. By setting the thickness of the constituent materials, this kind of composite multiferroic can be made with magnetoelectric constant around 5 V cm 1 Oe [24]. There are three mechanisms in obtaining magnetoelectricity from the composite materials. The first mechanism is associated with the multiferroic composites comprising of piezoelectric ferroelectric and magnetostrictive ferromagnetic [25, 26]. The magnetoelectric 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/BaTiO 3 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/BaTiO 3 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 effect [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 magnetization of ferromagnet at the ferromagnet/ferroelectric interface [11,13]. The additional

13 1. Introduction 5 magnetization can be manipulated by controlling the electric polarization using an external 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 multiferroic 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 determined. 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 properties 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 e i(k x ωt), the Maxwell equations can be written as k E = ω c B k H = ω c D. (1.3a) (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 B i = µ ij H j and D i = ǫ ij E j, 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 ǫe = 0. (1.4) c2 In isotropic case, subtituting the form of electric field E into the wave equation Eq.(1.4)

14 6 1. Introduction results in the explicit dispersion relation as c 2 k 2 ω 2 = 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) 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 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 >> ω c. Later, Auld [42] found a complete dispersion relation for bulk polaritons in a ferromagnet by in-

15 1. Introduction 7 Figure 1.1: A sketch of the geometry for a semi-infinite ferromagnet. Ferromagnetism ( M) along ẑ axis parallel to the surface. The external magnetic H 0 is applied parallel to the magnetization M. Propagation of the surface polaritons is along ˆx axis with wave-number k x. 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 ẑ axis as it is illustrated in Fig.1.1. The external magnetic field is also applied parallel to the ẑ 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 µ(ω)h = 0. (1.6) c2 By assuming the solution has the form H e i(k x) and using magnetic permeability tensor for the simple ferromagnet such as µ = µ 1 iµ 2 0 iµ 2 µ (1.7) where µ 1 = 1 + 4πγ2 M sh γ 2 H 2 ω 2 and µ 2 = 4πγMsω γ 2 H 2 ω 2, the wave equation Eq(1.6) leads to the explicit dispersion relation in the form [30, 34] c 2 k 2 ( ) ω 2 = ǫµ v = ǫ 1+ 4πγ2 M s B γ 2 H B ω 2 (1.8) where B = H +4πM s. The permeability µ v 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 ω r = γ H B, which is associated to µ v =, as k. The upper branch starts at

16 Introduction ~ L L L ~ L SP SP ω ~ ω ω r k Figure 1.2: An illustration of the simple magnetic polariton in ferromagnet. Bulk and surface modesareshown. Theshadedregionsreprsentbulkbands, limitedbyfrequenciesω r and ω. Surface modes are indicated by SP. The dashed lines indicated by L represent lightline with ω = ck. The dashed lines which are indicated by L obey ω = ck µ. frequency ω = γb, which is associated to µ v = 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 ˆx direction and decay in the ŷ direction as H e βy e i(kx ωt). In vacuum, the solution is assumed to be H e β y e i(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 +β µ v k µ 2 µ 1 = 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 (µ 2 ) 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 >> ω c. The other branch (negative wave number) does not have DE modes and terminates at the upper bulk band

17 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 K 2 CuF 4 [44]. L L L L ω r SP ω r a ω SP SP ω SP ω r b k (a) k (b) Figure 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 of external magnetic field. In (b) bulk and surface mode dispersions are shown with the application of magnetic field parallel to the magnetization M. The shaded regions represent bulk modes. The surface modes are illustrated by 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 antiferromagnet. The exchange interaction between spins on different sub-lattices leads to the antiparallel spins of the two sub-lattices. We assume that the anisotropy field H a pins the sub-lattices to the ẑ direction. Then, without applied magnetic field, the resonance frequency ω r is described by ω r = γ ( H 2 a +2H a H e ) 1/2. The permeability tensor has the form as in Eq.(1.7),where the components are defined as: µ 1 = 1+ 4πγ2 H am s ω 2 r ω2 and µ 2 = 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] c 2 k 2 ω 2 = ǫµ 1 (ω). (1.10) The permeability µ 1 in Eq.(1.10) divides the dispersion relation into two branches around resonance frequency ω r. The lower branch approaches resonance frequency ω r as k, while the upper branch starts at frequency ω 2 = ω 2 r + 8πγ 2 M s H a associated to µ 1 = 0

18 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 FeF 2 and FeF 2 :MnF 2 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 CoF 2, 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] µ 1 β +β = 0. (1.11) Given that boththeattenuation constant β andβ arepositive, thenthesurfacepolaritons can be found where permeability µ 1 is negative. The surface polaritons in this configuration 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 FeF 2 [47]. By variating the angle of incidence, 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 permeability as: 1 ω 2 r ω2 1 ω 2 r (ω+γh ) ω 2 r (ω γh ) 2. This splitting leads to the existence of two gaps around the frequencies ω b r = ω r γh and ω a r = ω r +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 (µ 2 0), which then modify the expression of dispersion relation for surface modes into [35] β + µ 1β +µ 2 k µ 2 1 µ2 2 = 0. (1.12) The appearance of propagation vector k lead to the non-reciprocity of surface modes, i.e, ω(k) 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].

19 Introduction 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 crystal. In isotropic dielectric media, dispersion relation for bulk was derived from the simple relation Eq.(1.8) by setting the permeability µ 1 as c 2 k 2 ω 2 = ǫ(ω). (1.13) Here, the dielectric constant is describedas ǫ(ω) = ǫ ω 2 LO ω2 ω 2 TO ω2, where ω TO is transverse optical phononfrequency, ω LO islongitudinaloptical phononfrequencyandǫ isbackground permeability [51]. The bulk phonon polariton has two branches which are separated by a gap between ω TO and ω LO where those frequencies are related by Lydanne-Sachs-Teller relation ωlo 2 = ǫ ǫ(0) ω2 TO. The lower branch starts at wavenumber k = 0 at frequency ω = 0 and approaches asimptotically frequency ω TO at k while the upper branch starts at frequency ω LO 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]. ~ L L ~ L L SP SP ω Lo ω To Figure 1.4: An illustration of the simple magnetic polariton in ferromagnet. Bulk and surface modesareshown. Theshadedregionsreprsentbulkbands,limitedbyfrequenciesω r and ω. Surface modes are indicated by SP. The dashed lines indicated by L represent lightline with ω = ck. The dashed lines which are indicated by L obey ω = ck ǫ.

20 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 frequencies ω TO and ω LO 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 ω TO and asymptotically aproach the limit ω s = ǫ +ǫ ǫ +ǫ ω TO. Since the surface dispersion relation did not have a term with linear k, hence the surface phonon polaritons for isotropic case is reciprocal. By using ATR method, the experimental dispersion relation for surface modes in GaP was probed [54] Polaritons in magnetoelectric multiferroics A most interesting class of polaritons occur in multiferroics. In magnetoelectric multiferroics, 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 (χ e ), the magnetic susceptibility (χ m ) and the magnetoelectric susceptibility (χ me ). Then, it isexpected thatbothphononpolaritonsandmagnon 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 ω TO 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 equations 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 e i(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 dispersion relation was determined by setting the matrix determinant to zero. In his series of

21 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 BaMnF 4, he showed that bulk modes in BaMnF 4 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 dispersion 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 contribution in two opposite directions of propagation will be different, leading to the different frequency,ω(k) 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 magnetoelectric antiferromagnets [60, 61]. Using symmetry allowed magnetoelectric energy density as F me = γ ijk M i L β P k, 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 magnetoelectric energy density [62,63] such as F me = P 2 M 2, 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. In Chapter 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 wavevector 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.

22 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 BaMnF 4 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 BaTiO 3 parameters are used for ferroelectric layers. For metallic ferromagnet, we use four example materials(fe, CrO 2, 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.

23 Chapter 2 Surface and bulk polaritons in a PML-type magnetoelectric multiferroic with canted spins: TE and TM polarizations 2.1 Introduction Transverse Electric & Transverse Magnetic modes The polarization of polaritons can be distinguished into two well-known groups [64]: transverse electric (TE) and transverse magnetic (TM), with respect to the sagittal plane. Saggital planeis definedas a planewherethenormal vector of thesurfaceand thepropagation vector lie on. In TE modes, the electric field is normal to the sagittal plane while the magnetic 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 ferromagnets [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

24 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 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 BaMnF 4 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 ŷ axis. The two antiferromagnetic sub-lattices M a and M b which have dominant component along ẑ direction, lie perpendicular to the electric polarization. These two antiferromagnetic sublattices are allowed to be cant with canting angle θ. We assume symmetric canting such that M a = M b = M s. The canting generates weak ferromagnetism M which lies inplane perpendicular to the polarization. We assume that the surface polaritons propagate along the ŷ direction. Thus the sagittal plane is parallel to the yz plane. Hence, the TE mode has the electric component E x normal to the sagittal plane while the magnetic field has H y and H z components which lie in the sagittal plane. In the TM mode, it is the component of magnetic H x which is normal to the sagittal plane, while the electric components E y and E z 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 F e = 1 2 ζ 1Py ζ 2Py ( P 2 x +Pz 2 ) ( 2 P 4 x +Pz 4 ) Py E y. (2.1)

25 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 ζ 1 and ζ 2. Here, the dielectric stiffness ζ 1 is a function of temperature as ζ 1 = ζ (T T c ) where ζ represents dielectric stiffness background and T c 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 (M a and M b by an angle θ produces a weak ferromagnetism ( M) along ˆx axis parallel to the surface. The spontaneous polarization ( P) is assumed to lie in-plane parallel to the surface. Propagation of the surface polaritons is along ŷ axis with wave-number k y. 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 F M = λm a M b K 2 [ (Ma ẑ) 2 +(M b ẑ) 2] (M a +M b ) 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. ThemagnetoelectriccouplingisassumedtobeofthePML-typeasdiscussedinchapter 1 which allows canting of the antiferromagnetic sub-lattices. We introduce the weak magnetization M x and the longitudinal component of magnetization L z. In terms of the canting angle θ, they can be expressed as M x = 2M s sinθ and L z = 2M s cosθ. Then,

26 18 2. TE and TM Polarization the magnetoelectric coupling can be re-written as F ME = 2αP y M 2 s sin2θ. (2.3) Here, α is magnetoelectric constant which represents the strength of magnetoelectric interaction. The magnetoeletric energy density F ME 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 magnetoelectric energies density with respect to the canting angle, (F M+F ME ) θ the condition = 0. This results in H o cosθ 1 2 KM ssin2θ +2αP y M s cos2θ λm s sin2θ = 0. (2.4) In the absence of an external magnetic field, equation (2.4) simplifies to tan(2θ) = 4αP y 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 M x 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

27 2. TE and TM Polarization 19 equations of motion without damping. The magnetic torque equation are of the form [35,37,51] ( ) Ṁ = γm M (F M +F ME ) (2.6) and dielectric response is obtained from [60 62]: P = f P (F E +F ME ) (2.7) where γ and f are gyromagnetic ratio and the inverse of phonon mass. Here, the terms M (F M+F ME ) and P (F E+F ME ) represent the magnetic and the electric effective fields. For small amplitude response, the magnetization and polarization can be written as M a = ( M s sinθ +m a x,m a y,m s cosθ+m a ) z ( ) M b = M s sinθ+m b x,mb y, M scosθ+m b z (2.8) P = (p x,p o +p y,p z ) where m a i,mb i and p i represent the dynamic components proportional to e iωt. Here, the dynamic components are assumed to be much smaller than the static part, m i << M s and p i << 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 iωm x = (ω a cosθ +2ω me sinθ)l y, (2.9) iωm y = 2γM s h z sinθ +(ω o ω a sinθ+2ω me cosθ)m z (ω a cosθ +2ω me sinθ)l x, (2.10) iωm z = 2γM s h y sinθ (ω o +2ω me cosθ)m y, (2.11) iωl x = 2γM s h y cosθ+(ω a cosθ+2ω ex cosθ+2ω me sinθ)m y, (2.12) iωl y = 2γM s h x cosθ (2ω ex cosθ+ω a cosθ+4ω me sinθ)m x +(ω +4ω me cosθ ω a sinθ 2ω ex sinθ)l z +4γαM 2 s cos2θp y, (2.13) iωl z = (2ω ex sinθ+2ω me cosθ +ω o )l y, (2.14) ω 2 f p x = 1 p x e x, (2.15) ω 2 f p y = ( ζ 1 +3Poζ 2 ) 2 py 2αM s (m x cosθ +l z sinθ) e y, (2.16)

28 20 2. TE and TM Polarization and ω 2 f p z = 1 p z e z. (2.17) The notations used above are in units of frequency and are defined by: ω a = γkm s as the effective magnetic anisotropy field, ω ex = γλm s as the exchange field, ω me = γαp o M s as the magnetoelectric coupling and ω o = γh o 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 susceptibility χ m yy, χ m zz and χ m yz. Together with the electric susceptibility component χ e xx given by electric dynamic equation Eq.(2.15) which is not coupled to the magnetic dynamics, these susceptibilities are associated with the electric field E x and magnetic components H y and H z. 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 χ m xx, the electric susceptibility χ e yy and χe zz and also the magnetoelectric susceptibilites χme xy and χem yx. These susceptibilities areassociated withmagneticfieldh x, whichis normaltothesagittal plane, and electric field components E y and E z which lie in the sagittal plane and describe TM modes. FortheTEmodes, theequationsofmotionforthemagneticcomponentsdonotcoupled directly to the equation of motion for the electric component, so that χ em = χ me = 0. Hence the susceptibilities can be determined as m = χ m h and p = χ e e (2.18) with magnetic components are given by χ m yy = 2γM s(ω a cos2θ +2ω me sin2θ +ω sinθ) ) (2.19a) (ωafm 2 cos2 θ +Ω 2 me,te +Ω2,TE ω2 χ m zz = 2γM s (ω me sin2θ +ω o sinθ) (ω 2 afm cos2 θ+ω 2 me,te +Ω2,TE ω2 ) (2.19b) χ m yz = χm zy = i2γm s (ωsinθ) (ω 2 afm cos2 θ+ω 2 me,te +Ω2,TE ω2 ). (2.19c) Here ω 2 afm = ω a(ω a +2ω ex ) represents the antiferromagnet (AFM) resonance frequency.