TE and TM modes polaritons in multilayer system comprise of a PML-type magnetoelectric multiferroics and ferroelectrics

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS TE and TM modes polaritons in multilayer system comprise of a PML-type magnetoelectric multiferroics and ferroelectrics To cite this article: Vincensius Gunawan and Hendri Widiyandari 016 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Mixed modes of surface polaritons in a PML-type magnetoelectric multiferroic with canted spins Vincensius Gunawan and Robert L Stamps - Leakage Current and Photovoltaic Properties in a Bi Fe 4 O 9 /Si Heterostructure Jin Ke-Xin, Luo Bing-Cheng, Zhao Sheng- Gui et al. - Electric-field induced ferromagnetic domain changes in exchange biased Co BiFeO3 composites Sang-Hyun Kim, Hyunwoo Choi, Kwangsoo No et al. This content was downloaded from IP address on 13/1/017 at 08:0

2 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 TE and TM modes polaritons in multilayer system comprise of a PML-type magnetoelectric multiferroics and ferroelectrics Vincensius Gunawan and Hendri Widiyandari Jurusan Fisika, Fakultas Sains dan Matematika, Universitas Diponegoro, Jl. Prof Soedarto, Tembalang, Semarang, Indonesia goenangie@fisika.undip.ac.id V. Gunawan) Abstract. In this paper, we report our study on both bulk and surface polaritons generated in Multilayer system. The multilayer consists of ferroelectric and multiferroic with canted spins structure. The ective medium approximation is employed to derive the dispersion relation for both bulk and surface modes. Surface and bulk polaritons are calculated numerically for the case of Transverse electric TE) and Transverse magnetic TM) modes. Example results are presented using parameters appropriate for BaMnF 4/BaAl O 4. We found in both TE and TM modes, that the region where the surface modes may exist is affected by the volume fraction of the multiferroics. The region of the surface modes decrease when the volume fraction of the multiferroic is reduced. This region decrement suppress the surface polariton curves which result in shortening the surface modes curves. 1. Introduction Electromagnetic waves which are coupled to the elementary excitations of the materials are called polaritons[1]. Polaritons have received much attention due to its unique properties. Surface polaritons are able to demonstrate localization on the surface. They can also show the non reciprocity, where the two opposite directions of propagation have different frequencies, i.e. ωk) ω k)[ 4]. Theoretical and experimental treatments for phonon polaritons in dielectrics were published several decades ago[5 7]. The theoretical studies of magnon polaritons in magnetic materials were also reported several years ago [3, 8, 9] and reviewed in Ref.[4]. Experimental studies of magnon polaritons in antiferromagnet had been performed using many methods: transmission for FeF [10],reflection for CoF [11], MnF [1] and attenuated total reflection ATR)[13]. The study of polaritons in multiferroics is interesting since multiferroics have both phonon polaritons and magnon polaritons simultaneously[14 17]. It is happen since in multiferroics exist at least two types of ferroicity[18], hence there will be more than one resonance frequency in the multiferroics. For example, magnetoelectric multiferroics have both phonons and magnons which lead to the existence of the electric and the magnetic resonance frequencies simultaneously. Study the multilayer geometry by combining at least two types of different material is important to understand the changes of the elementary excitations of the materials which are represented by the changes of the polaritons. The manipulation of polaritons had been studied Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 theoretically by considering the multilayer configuration which comprise of magnetic/nonmagnetic materials[19] or magnetic/magnetic materials[0, 1]. The dispersion relations were calculated by employing transfer matrix which depend on the thickness of the involved materials. However, in long-wavelength limit where the wavelength much longer than multilayer period, the calculation can be approached by applying an ective medium method[0, ]. In this method, a multilayer configuration is represented by a unit medium having ective permittivity and ective permeability. According to Ref.[0], these ective parameters are calculated by employing Maxwell s continuity of the fields at the interface of the involved materials. In the present paper we report in detail the study of polaritons in multilayer comprises of multiferroics/electric system. We show how the fraction of multiferroics affects both the bulk and surface polaritons. We provide numerical predictions of dispersion relations for multilayer comprise of BaMnF 4 /BaAl O 4. Those materials are used since the model of multiferroic BaMnF 4 is frequently used in previous study[14 17] and there is a complete report about resonance frequencies of ferroelectric BaAl O 4 [3]. We also discuss the properties of polaritons for both the transversal electric TE) modes and transversal magnetic TM) modes.. Geometry of the Multilayer The geometry of the multilayer is illustrated in figure 1. The multilayer is semi-infinite which is comprise of multiferroics MF) with the thickness d 1 and ferroelectric E) with the thickness d. The surface of multilayer is lied in x y plane. Following the previous studies[16,17], the multiferroics is ferroelectric-two sub-lattices antiferromagnet with canted spin. a) b) Figure 1. Geometry of the multilayer. a) The multilayer is comprised of the ferroelectrics with the thickness d 1 and the multiferroics which is denoted by E with the thickness d. b) The polarisation and the magnetisation of the multiferroics MF). Polarisasion is directed in the y direction. The two-sublattice magnetisation is canted with canting angle θ yield weak ferrmognetism M. The susceptibilites are obtained by examining the density energy of material in the equations of motion. In this process, we also obtain magnetic and electric resonance frequencies. The susceptibility components of this type of magnetoelectric multiferroic were formulated into two groups. The first group represents TE modes involving the electric component e x and the magnetic components h y and h z. The susceptibility components for TE modes are[16]: χ m yy = ω sω a cosθ +ω me sinθ +ω sinθ) ω M ω), 1)

4 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 and where ω M = χ m zz = ω sω me sinθ +ω o sinθ) ω M ω), ) χ m yz = χ m zy = iω sωsinθ ω M ω), 3) χ e x = ǫ x ω L,x ω ω T,x ω 1 4) ω afm cos θ +Ω me,te +Ω,TE) 1/ represents magnetic resonance frequency in TE modes. The frequencies Ω me,te = [ω me 4ω me cosθ +ω a sinθ +ω ex sinθ)] 1/ is contributed from magnetoelectric interaction while Ω 0,TE = [ω 0 ω 0 ω a sinθ +4ω me cosθ)] 1/ is given by external magnetic field. Here, the frequency ω ex = γλµ 0 M s represent the frequency from exchange interaction while thefrequencyω a = γkµ 0 M s isfrequencywhichiscontributedbyanisotropyenergy. Frequencies ω me and ω 0 are defined as ω me = γµ 0 αpm s and ω 0 = γµ 0 H 0. The frequency from sub-lattice magnetization is defined as ω s = γµ 0 M s. The frequency ω afm = [ω a ω a +ω ex )] 1/ represents antiferromagnet frequency. In electric susceptibility, ω L,x and ω T,x represent the longitudinal and transverse phonon resonance frequency in x direction. The TM mode which involves a magnetic component h x and the electric components e y and e z has the magnetic and electric susceptibility components[16] χ m xx = ω s ω el ω ) ω a cosθ +ω me sinθ) [ ω el ω ) ωm ω ) Ω 4, 5) s] χ e y, = η/ǫ 0 ω m ω ) [ ω el ω ) ω m ω ) Ω 4 s], 6) χ e z = ǫ z ω L,z ω ω T,z and also the magnetoelectric susceptibility in the form where frequency ω m = ω 1 7) χ me xy = χ em yx = 4αη/ǫ 0)M s ω s ω a cosθ +ω me sinθ)cosθ c [ ωel ω) ωm ω ) Ω 4 ] 8) s ω afm,tm +Ω me,tm +Ω 0,TM + ) 1/ is defined as magnetic resonance ) frequency and ω el = α1 α ǫ 0 +P 0 η represents the electric frequency. Here, η represents the ǫ 0 inverse of the multiferroics phonon mass and the frequency Ω s in magnetoelectric susceptibility [ is defined as Ω s = 8 η α M s ǫ 0 ω c s ωa cos θ +ω ex sin θ )] 1/4. In this TM modes, ω afm,tm = ω a ω a +ω ex )cos θ +ω ex ω a +ω ex )sin θ ) 1/ represents the antiferromagnetic resonance frequency in TM modes. The frequency from the magnetoelectric interaction is represented by Ω me,tm = [ω me 4ω me ωexsinθ +ω a sinθ)] 1/. The frequencyω 0,TM = [ω 0 ω 0 ω a sinθ 4ω ex sinθ +6ω me cosθ)] 1/ iscontributedbythefieldh 0. In this study, we simply use the electric susceptibilities for ferroelectric in the similar form as Eq.5) and 7) with the form χ e i,e = ǫ ωl,i e ) ω i,e ωt,i e 1 9) ) ω 3

5 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 where ω e T,i represents a transverse optical lattice vibration frequency in the i direction and ωe L is cut off frequency of the forbidden band for waves propagation. Once all the magnetic and electric susceptibilites are defined, the permeabilities can easily be obtained through relation µ = I +χ m while the permittivities with the relation ǫ = I +χ e. In this calculation, we exclude intermixing interaction between ferroelectric and the electric part of the multiferroics as proposed in previous study[4]. Since this interaction will only slightly shift the electric resonance frequencies for both material, it will not significantly affect the feature of the dispersion relation curves. 3. Effective Medium A multilayer system comprised of two or more materials can be approximated as a single material called ective medium[0, 5]. In this approximation, we assume the wavelength much longer than the thickness of each layer or periodicity of the multilayer. According to Ref.[0], the ective permittivity and permeability are calculated by considering the Maxwell s continuity at the interfaces. A multilayer as illustrated in Fig.1a) has the average of tangential electric displacement vector as D x = d 1D1 x +d D x = f 1 D1 x +f D x 10) d 1 +d ) where f 1 and f denote the volume fraction of the multiferroics and ferroelectrics. The displacement components D1 x and Dx are the x component of the electric displacement vector of the multiferroics and ferroelectrics. In the next step, by considering the continuity of tangential electric fields, we have E1 x = Ex = Ēx. Using constitutive equation D = ǫe yield the relation in the form D x = f 1 ǫ x 1 Ex 1 +f ǫ x Ex = f 1ǫ x 1 +f ǫ x )Ēx. Hence, the ective permittivity in the x component is written as ǫ xx = f 1 ǫ x 1 +f ǫ x. 11) Now, analyzing the average of the electric displacement, D y = f 1 D y 1 +f D y using constitutive relation D y 1 = ǫyy 1 Ey 1 + χem H1 x for multiferroics and Dy = ǫyy Ey for ferroelectric, also by considering the continuity of the tangential component of magnetic field H x ) and tangential component of the electric field E y ), we obtain D y = f 1 1 +f )Ēy +f 1 χ em Hx. Then, the y component of the ective permittivity can be defined as and the ective magnetoelectric susceptibility as = f 1 1 +f 1) χ em = f 1 χ em. 13) The z component of permittivity is derived by analyzing the average of the electric component Ē z = f 1 E z 1 +f E z and the continuity of the normal component of electric displacement D z), result in the form ǫ zz = ǫ zz 1 ǫzz f 1 ǫ zz +f ǫ zz 14) 1 ). If the same procedure above is performed to the magnetic induction field B and magnetic field H, the ective permeability components are obtained in the form of µ xx = f 1 µ xx 1 +f µ xx, 15) 4

6 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 and µ yy = f 1µ yy 1 +f µ yy +f 1 [ = µyz 1 µzy 1 f 1 ) +f 1 1 µzz ) 1 µzz f 1 +f 1 ] f 1 µ yz 1 µzy 1 1, 16) 17) µ yz = µzy = f 1 µyz 1 f 1 +f 18) 1 ). 4. Theory of Polariton in MF/FE Multilayer The behavior of electromagnetic waves propagation is illustrated by dispersion relation which is obtained by solving the Maxwell equations using the appropriate electromagnetic waves such as ψ = ψ x,ψ y,ψ z )exp[ik y y +k z z ωt)]. 19) Using configuration as illustrated in Fig.1b), The Maxwell equations result in two groups of coupled equations. The first group which involves the H y, H z and E x is classified as transverse electric TE) mode. The second group is transverse magnetic TM) mode which involves the fields components H x, E y and E z. Using electromagnetic waves as in Eq.19) into Maxwell equations, the matrix equation for the TE modes which represent the bulk modes is k z k y ǫ xx ω c µ yy ω c µ zy ω c µ yz ω c ω c k z k y H y H z E x = 0. 0) The solution requires the determinant matrix equal to zero which results in the form of bulk polariton as k y = ǫ xx [ ) ] µ yy µzz + µ yz µ yy ω c ). 1) The TE surface modes are derived by assuming that the electromagnetic waves are decaying exponentially as they go into the materials. Hence, the appropriate solutions of the surface modes are E e βz e ikyy ωt) for z < 0 ) E e βoz e ikyy ωt) for z > 0 3) where β and β 0 represent attenuation constant for multilayer and vacuum which define as [ ) ] β = µ yy k y ǫ xx µ yy µzz + µ yz ω c, 4) β 0 = k y ω c. 5) The TE dispersion relation for the surface polaritons is derived by applying the continuity of tangential H H y ) and the continuity of normal B B z ) which result in the form 5

7 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 [ ) ] β +β 0 µ yy µzz + µ yz +iµ yz k y = 0. 6) Since the dispersion relation contains odd order of k y, then surface polariton is non-reciprocal, with ωk y ) ω k y ). Hence the non-reciprocality which was found in a pure PML-type multiferroics as reported in Ref.[16]) is still observable in this multilayer geometry. The TM modes for bulk polaritons which is involving field components H x, E y and E z are represented by a matrix equation resulted from the Maxwell equations as µ xx ω c k z +χ me ω c ) ) k z +χ me ω c k y ω c 0 k y 0 ǫ zz which result in the dispersion relation for bulk modes as H x E y E z = 0 7) [ ) ] k y = µxx χ me ǫ zz ω c. 8) The TM surface polaritons are calculated by assuming the solution is of the form H e βz e ikyy ωt). 9) where the attenuation constant β is defined by an equation such as ǫ zz β +iχ me ) ω = ky µ xx c ǫ zz ω ) c. 30) The equation above illustrates that the attenuation constant β is complex. It means that the surface polariton modes are leaky modes, where there is an amount of energy leaking into the environment. Performing the similar procedure as in TE modes by analyzing the continuity of the fields continuity of tangential H, tangential E and normal D) at the surface, results in β + iω ) c χme + β 0 = 0. 31) ThedispersionrelationEq.31)reflectsthattheTMsurfacemodesisreciprocal,ωk y ) = ω k y ). 5. Application to Multilayer BaMnF 4 /BaAl O 4 In this section, the previous theory is applied to the multilayer comprises of multiferroic BaMnF 4 and ferroelectric BaAl O 4. For BaMnF 4, some values of the variables are taken from the references such as: canting angle θ = 3 mrad[6], polarisation P = 0,115 C/m [7], inverse phonon mass f= A s kg 1 m 3 [14], frequency ω ey = 41 cm 1 [8], ω ez = 33.7 cm 1 [14] and permittivity ǫ x = 8.[9]. The others are estimated as in Ref.[16]. Sub-lattice magnetization M s =, Am 1 is estimated using M c = M s sinθ where weak ferromagnetism M c = 1460 Am 1 [30]. An exchange constant λ=163,54 is calculated using H E = λµ 0 M s with H E =50 T[31]. An anisotropy constant K=0.337 is approximated using ωr = γ µ 0 KK λ)m s with ω r = 3 cm 1 [8]. The calculated magnetoelectric coupling α is obtained using relation tanθ) = 4αP K+λ as in Ref.[16]. The values of variables for BaAl O 4 are taken from Ref.[3] such as ǫ = 3.1, ǫ // = 3.13, ǫ0 =9.8, ǫ0 // =11.09, frequency ω t, =63 cm 1 and ω t,// =63 cm 1. The solutions of the TE dispersion relation for bulk in Eq.1) and surface modes in Eq.6) were obtained numerically using root finding technique. For the volume fraction of multiferroics f 1 = 0.75, the solutions of the surface dispersion relation in TE modes was found at around the 6

8 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 magnetic resonance frequency of BaMnF 4 ω M 3.03 cm 1 ) which is illustrated in Fig.a). In that figure, the surface modes are represented by thick lines which are denoted while the bulk bands are represented by shaded regions. Sincearoundthemagneticfrequencyω M, permittivitycomponentǫ xx ofmultiferroicbamnf 4 is constant at the value 8. while the permittivity ǫ xx of ferroelectric BaAl O 4 is also constant around the value 3.1, hence the value of the ective permittivity ǫ xx can be approximated as constant. Then, based on Eq.1), the bulk bands comprise of only two bands which are separated by an area where the surface modes may exist. This region is limited by magnetic resonant frequency ω M and the zeros frequency ω O of the dispersion relation for bulk polaritons see Fig.a)). Since the magnetic resonance frequency is only contributed by multiferroic BaMnF 4, then the bulk equation in Eq.1) can be brought into k y f 1 +f µ yy ) f µ yz 3) L L L L ω/c cm 1 ) ω 0 ω M ω/c cm 1 ) ω k cm 1 ) a) k cm 1 ) b) ω M Figure. Dispersion relation of Multilayer with various volume fraction of Multiferroics. In a) with the fraction f 1 = 0.75 and b) with f 1 = 0.5. Shaded region is the area for bulk polaritons. Surface modes are represented by thick lines which are denoted by. The dashed lines which are denoted by L represent Lightlines. The ATR line are illustrated by dashed lines with the downward arrows. where the denominators of µ yy, and µ yz have the same resonance frequency. Hence, the volume fraction of the multiferroic layer do not affect the magnetic resonance frequency of the multilayer as illustrated in Fig.a)-b). The zeros frequency ω 0 can be calculated by solving the condition [ ) ] µ yy µzz + µ yz = 0 33) which can be brought into cubic equation) ω 4 + Bω + C = 0 where) B and C were defined as B = f 1 ωm +f ωoy +ωoz f ωs sin θ and C = f 1 ωm +f ωoy ωoz. The frequencies ω oy and ω oz represent zero frequency of the permeability components µ yy and. The solution of cubic equation is then ω 0 = [ B+ B 4C ] 1/. 7

9 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 It was found that by reducing the volume fraction of the multiferroics BaMnF 4, the zeros frequency of bulk polariton will decrease as illustrated in Fig.a)-b). The zeros frequency drop from cm 1 to 3.06 cm 1 when the volume fraction was reduced from 75% to 5%. This zeros frequency reduction result in the decrement of the region where the surface modes may exist, since the magnetic resonance frequency is independent of the volume fraction. Hence, it also shifts the surface modes curves. Since the attenuation total reflection ATR) has been successfully used to identify the surface polaritons in antiferromagnet FeF [13,3], in this report we also calculate the ATR reflectance to verify the polaritons generation. ATR reflectance is reflection of the incident waves in ATR system which is took place at the material sample. In this method, the existence of surface modes is represented by sharp dips while the bulk polaritons are illustrated with shallow dips of the ATR reflectance spectrum. The ATR reflectivity is calculated by considering the reflectance at the multilayer surface and also at the bottom of high index prism which results in the formula[33] ) ) k z 1+re β 0d iβ 0 1 re β 0d R = k z 1+re β0d )+iβ 0 1 re β0d ) 34) where k z is propagation normal to the surface of the multilayer. Here, r = β 0 ζ β 0 +ζ represents reflectivity at the surface of multilayer with ζ = β iµyz [ µ yy µzz + µ yz ky ) ) ]. R ω /ccm 1 ) a) R ω /ccm 1 ) b) Figure 3. Calculated ATR Reflectivity with incident angle In a), The ATR spectroscopy for TE mode, b) The ATR for TM mode. The dashed line represents the volume fraction f 1 = 0.5, while the solid thick lines is forf 1 = In figure a), the dashed lines denoted by arrow represent ATR line with incident angle at This ATR line intersects the surface modes at around 3.05 cm 1, agree with the sharp dip of the solid line in the calculated ATR reflectance at Fig.3a). The shift of surface modes curves due to the reduction of volume fraction of the multiferroics to f 1 = 0.5 can also be observed as illustrated as a dashed line. The solutions of the dispersion relation in TM modes for bulk in Eq.8) and surface in Eq.31) are illustrated in Fig.4a). Here, the volume fraction of the multiferroics BaMnF 4 is The shape of polaritons around the phonon frequency is also similar to the previous report 8

10 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 for the pure BaMnF 4 [16]. However, in this report, the window where the surface modes may exist is thinner for the volume fraction f 1 = 0.75 than the window in the pure multiferroics. 43 L L 43 L L ω /c cm 1 ) 4 41 ω I ω EL ω /c cm 1 ) 4 41 ω I ω EL k cm 1 ) k cm 1 ) a) b) Figure 4. Surface polaritons for TM modes. a) Surface modes with volume fraction of multiferroics f 1 = b) with volume fraction f 1 = 0.5. The bulk bands are represented by the shaded regions, while the surface modes are represented by thick lines which denote by. The existence of both surface and bulk modes in this modes is confirmed by the calculated ATR spectrum as illustrated in Fig.3b). The ATR reflectance for TM mode is also given by formula in Eq.34) with the reflectivity r is now given as r = ǫyy β 0 β β 0 β. When the volume fraction of the multiferroics is decreased to 0.5, the window where the surface modes are present become very narrow as illustrated at Fig.4b). This decrement is resulted from the reduction of the zero frequency ω I while the shifted resonance frequency ω EL is not affected by the reduction of volume fraction of the multiferroics. The zero frequency ω I obey the condition µxx χ) me = 0. 35) Near the phonon resonance frequency, the permeability µ xx = 1 while the value Hence the Eq.35) can be approximated by >> χme. = f 1 1 +f = 0. 36) Then, by considering that permittivity BaAl O 4 around the phonon frequency of the multiferroics is approximately 17, the Eq.36) above can be brought into a cubic equation as Here, A = 1 + κ with κ = f f 1 ǫ yy,1 Aω 4 +Bω +C = 0. 37) f 17 f 1 ǫ. The parameters B and C are defined as yy,1 B = 1+κ)ω mag +ω ey) ω f and C = 1+κ)ω magω ey Ω 4 c)+ω f ω mag. Hence, the zeros frequency ω I can be solved easily as ω I = { } 1/ B + B 4AC. 38) A 9

11 ScieTech 016 Journal of Physics: Conference Series ) doi: / /710/1/01037 The decrease of the region where the surface modes exist will suppress the surface modes curves. As a consequence, the surface modes curves become shorter as the volume fraction of multiferroics is decreased. As it is illustrated in Fig.3b), the calculated ATR reflectance is also showing a downward shift in surface modes curves when the volume fraction is decreased. 6. Conclusion We show how the dispersion relations of multiferroic are modified in multilayer configuration for both TE and TM modes, even though the reciprocity or non-reciprocity of the multiferroics does not change by arranging multiferroics into a multilayer configuration with electric material. The dispersion relations of the multilayer in both TE and TM modes are affected by the volume fraction of the multiferroics. The region where the surface mode exist is narrow when the volume fraction of multiferroic in multilayer system is small around 5%). In multilayer with the volume fraction of multiferroic is relatively big around 75%), the region for surface modes is wide. Acknowledgments We wish to acknowledge the support of Ditlitabmas Ditjen Dikti Kemendikbud through DIPA Diponegoro University. References [1] Mills D and Burstein E 1974 Rep. Prog. Phys [] Camley R E 1987 Surf. Sci. Rep [3] Harstein A, Burstein E, Maradudin A A, Brewer R and Wallis R F 1973 J. Phys. C: Solid State Physics [4] Abraha K and Tilley D R 1996 Surf. Sci. Rep [5] Marschall N and Fischer B 197 Phys. Rev. Lett [6] Falge H, Otto A and Sohler W 1974 Phys. Status Solidi B [7] Borstel G and Falge H 1978 Appl Phys [8] Karsono A D and Tilley D R 1978 J. Phys. C: Solid State [9] Camley R E, Jensen M R F, Feiven S A and Parker T J 1998 J. Appl. Phys [10] Sanders R W, Belanger R M, Motokawa M, Jaccarino V and Rezende S M 1981 Phys. Rev. B [11] Haussler K M, Brandmuller J, Merten L, Finsterholtzl H and Lehmeyer A 1983 Phys. Status Solidi B [1] Remer L, Luthi B, Sauer H, Geick R and Camley R E 1986 Phys. Rev. Lett [13] Jensen M R F, Feiven S, Parker T J and Camley R 1997 Phys. Rev. B [14] Barnas J 1986 J. Magn. Magn. Mat [15] Barnas J 1986 J. Phys. C: Solid State Physics [16] Gunawan V and Stamps R 011 J. Phys. : Condens Matter [17] Gunawan V and Stamps R 01 J. Phys. : Condens Matter [18] Schmid H 1994 Ferroelectrics [19] jia Zhu N and Cao S 1987 Phys. Lett. A [0] Raj N and Tilley D R 1987 Phys. Rev. B [1] Barnas J 1986 J. Phys.: Condens. Matter 7173 [] Almeida N S and Tilley D R 1990 Solid State Commun [3] Xie C, Dong D, Gao S, Cai Y and Oganov A R 014 Phys. Lett. A [4] Ong L H and Chew K 013 Ferroelectrics [5] Elmzughi F G and Camley R E 1997 J. Phys.: Condens Matter [6] Venturini E L and Morgenthaler F R 1975 AIP Conf. Proc [7] Abrahams S C and Keve E T 1971 Ferroelectrics 19 [8] Samara G A and Richards P M 1976 Phys. Rev. B [9] Samara G A and Scott J F 1977 Solid State Commun [30] Scott J 1979 Rep. Prog. Phys [31] Holmes L, Eibschutz M and Guggenheim H J 1969 Solid State Commun [3] Jensen M R F, Parker T J, Abraha K and Tilley D R 1995 Phys. Rev. Lett [33] Abraha K, Smith S R P and Tilley D R 1995 J. Phys.: Condens. Matter