Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions

A phenomenological theory of high-frequency properties of ferroelectric and ferroelectric-ferromagnet with the 3d-ions has been elaborated based on the separate accounting for spin and orbital electron moments, electron and ion contributions to electric polarization. In the ferroelectric state, t...

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Veröffentlicht in:	Condensed Matter Physics
Datum:	1999
1. Verfasser:	Chupis, I.E.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1999
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/121019
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Zitieren:	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions / I.E. Chupis // Condensed Matter Physics. — 1999. — Т. 2, № 4(20). — С. 745-753. — Бібліогр.: 8 назв. — англ.

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spelling	Chupis, I.E. 2017-06-13T12:45:12Z 2017-06-13T12:45:12Z 1999 Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions / I.E. Chupis // Condensed Matter Physics. — 1999. — Т. 2, № 4(20). — С. 745-753. — Бібліогр.: 8 назв. — англ. 1607-324X DOI:10.5488/CMP.2.4.745 PACS: 78.20.Ls https://nasplib.isofts.kiev.ua/handle/123456789/121019 A phenomenological theory of high-frequency properties of ferroelectric and ferroelectric-ferromagnet with the 3d-ions has been elaborated based on the separate accounting for spin and orbital electron moments, electron and ion contributions to electric polarization. In the ferroelectric state, the existence of the electron orbital moment leads to the breaks in the temperature dependencies of the transverse components of dielectric susceptibility at the ferroelectric transition temperature. The values of these breaks are proportional to the square of the electron part of spontaneous polarization and the parameter of freezing of the orbital moment. Similar breaks and the decrease of the phonon frequencies in the ferroelectric state should occur in the modes which are not soft. Besides, the effect of induction of high-frequency orbital moment by electron part of electric polarization has been predicted as well. This effect would lead to the break in the temperature dependence of paramagnetic susceptibility at ferroelectric transition temperature. In the ferroelectromagnetic state, the electron orbital moment also manifests itself in λ increase of magnetoelectric gyration ( λ is the constant of spin-orbital interaction). Побудована феноменологічна теорія високочастотних властивостей сегнетоелектрика і сегнетоелектрик-феромагнетика з 3d-іонами, яка грунтується на розділеному врахуванні спінового і орбітального електронних моментів, електронних та іонних вкладів до електричної поляризації. У сегнетоелектричному стані існування електронного орбітального момента приводить до розривів поперечних компонент діелектричної сприйнятливості при температурі сегнетоелектричного переходу. Величини цих розривів пропорціональні до квадрата електронної частини спонтанної поляризації і параметра замороження орбітального момента. Аналогічні розриви і зменшення фононних частот у сегнетоелектричному стані мають спостерігатися в модах, які не є м’якими. Крім того, передбачено ефект індукції високочастотного орбітального момента на електронну частину поляризації. Цей ефект може приводити до розриву парамагнітної сприйнятливості при температурі сегнетоелектричного переходу. У фероелектромагнітному стані електронний орбітальний момент проявляє себе в λ зростанні магнітоелектричної гірації ( λ є постійна спін- орбітальної взаємодії). This research was supported by INTAS Grant No. 94-935. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions Прояви електрон-орбітального моменту у високочастотних властивостях сегнетоелектриків з 3d-іонами Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
spellingShingle	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions Chupis, I.E.
title_short	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
title_full	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
title_fullStr	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
title_full_unstemmed	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
title_sort	manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions
author	Chupis, I.E.
author_facet	Chupis, I.E.
publishDate	1999
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Прояви електрон-орбітального моменту у високочастотних властивостях сегнетоелектриків з 3d-іонами
description	A phenomenological theory of high-frequency properties of ferroelectric and ferroelectric-ferromagnet with the 3d-ions has been elaborated based on the separate accounting for spin and orbital electron moments, electron and ion contributions to electric polarization. In the ferroelectric state, the existence of the electron orbital moment leads to the breaks in the temperature dependencies of the transverse components of dielectric susceptibility at the ferroelectric transition temperature. The values of these breaks are proportional to the square of the electron part of spontaneous polarization and the parameter of freezing of the orbital moment. Similar breaks and the decrease of the phonon frequencies in the ferroelectric state should occur in the modes which are not soft. Besides, the effect of induction of high-frequency orbital moment by electron part of electric polarization has been predicted as well. This effect would lead to the break in the temperature dependence of paramagnetic susceptibility at ferroelectric transition temperature. In the ferroelectromagnetic state, the electron orbital moment also manifests itself in λ increase of magnetoelectric gyration ( λ is the constant of spin-orbital interaction). Побудована феноменологічна теорія високочастотних властивостей сегнетоелектрика і сегнетоелектрик-феромагнетика з 3d-іонами, яка грунтується на розділеному врахуванні спінового і орбітального електронних моментів, електронних та іонних вкладів до електричної поляризації. У сегнетоелектричному стані існування електронного орбітального момента приводить до розривів поперечних компонент діелектричної сприйнятливості при температурі сегнетоелектричного переходу. Величини цих розривів пропорціональні до квадрата електронної частини спонтанної поляризації і параметра замороження орбітального момента. Аналогічні розриви і зменшення фононних частот у сегнетоелектричному стані мають спостерігатися в модах, які не є м’якими. Крім того, передбачено ефект індукції високочастотного орбітального момента на електронну частину поляризації. Цей ефект може приводити до розриву парамагнітної сприйнятливості при температурі сегнетоелектричного переходу. У фероелектромагнітному стані електронний орбітальний момент проявляє себе в λ зростанні магнітоелектричної гірації ( λ є постійна спін- орбітальної взаємодії).
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/121019
citation_txt	Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions / I.E. Chupis // Condensed Matter Physics. — 1999. — Т. 2, № 4(20). — С. 745-753. — Бібліогр.: 8 назв. — англ.
work_keys_str_mv	AT chupisie manifestationsoftheelectronorbitalmomentinhighfrequencypropertiesofferroelectricswith3dions AT chupisie proâvielektronorbítalʹnogomomentuuvisokočastotnihvlastivostâhsegnetoelektrikívz3díonami
first_indexed	2025-11-25T22:54:34Z
last_indexed	2025-11-25T22:54:34Z
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fulltext	Condensed Matter Physics, 1999, Vol. 2, No. 4(20), pp. 745–753 Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions I.E.Chupis B.I.Verkin Institute For Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., 310164 Kharkiv, Ukraine Received June 28, 1998 A phenomenological theory of high-frequency properties of ferroelectric and ferroelectric-ferromagnet with the 3d-ions has been elaborated based on the separate accounting for spin and orbital electron moments, electron and ion contributions to electric polarization. In the ferroelectric state, the existence of the electron orbital moment leads to the breaks in the tempera- ture dependencies of the transverse components of dielectric susceptibility at the ferroelectric transition temperature. The values of these breaks are proportional to the square of the electron part of spontaneous polarization and the parameter of freezing of the orbital moment. Similar breaks and the decrease of the phonon frequencies in the ferroelectric state should occur in the modes which are not soft. Besides, the effect of induction of high-frequency orbital moment by electron part of electric polarization has been predicted as well. This effect would lead to the break in the temper- ature dependence of paramagnetic susceptibility at ferroelectric transition temperature. In the ferroelectromagnetic state, the electron orbital moment also manifests itself in λ increase of magnetoelectric gyration ( λ is the constant of spin-orbital interaction). Key words: electron, orbital, moment, ferroelectric, susceptibility PACS: 78.20.Ls 1. Introduction In the 3d-dielectric, electrons of the 3d-shell have a weaker connection with the core than the rest electrons. This permits us to separately consider the contributions to the electric polarization from the 3d-electrons and from the rest part of the ion. The contribution of the 3d-electrons to electric polarization is essential[1]. The elec- tron orbital moment of the 3d-ions in crystals is strongly frozen and its contribution to the total magnetic moment is small. A nonzero value of the orbital moment is due to a small spin-orbital interaction and makes up some percent out of the total c© I.E.Chupis 745 I.E.Chupis moment. A small spin-orbital interaction in the 3d-ions permits to consider spin and orbital moments of the electron separately [2,3]. Thus there is a possibility to analyze the role of the electron orbital moment in forming high-frequency properties of ferroelectrics (FE) with the 3d-ions. In the present paper a phenomenological theory of high-frequency properties of FE and ferroelectric-ferromagnet with the 3d-ions has been elaborated. The expres- sions obtained for the susceptibilities and spectrum contain characteristics of the electron spectrum, i.e. a spin-orbital interaction constant λ and the parameter of the hardness of freezing of the orbital moment. The existence of the electron orbital moment leads to the breaks in the temperature dependencies of the components of dielectric susceptibility in the directions perpendicular to the spontaneous polariza- tion at FE transition temperature. The values of these breaks are proportional to the square of the electron polarization and the parameter of freezing of the orbital moment. Besides,the breaks and the decrease of the frequencies in FE state should be in the modes which are not soft. The induction of high-frequency orbital moment by the electron part of electric polarization has been predicted as well. In the fer- roelectromagnetic state, the electron orbital moment displays itself in λ -increase of magnetoelectric (ME) gyration[4]. 2. Hamiltonian and excitations Within the phenomenological theory a ferroelectric-ferromagnet can be described by densities of spin S(r), electron orbital L(r), ion electric P i(r), and electron elec- tric Pe(r) dipole moments as well as by densities of ion Π i(r) and electron Πe(r) momenta. To avoid too cumbersome expressions, an ion orbital moment which is much smaller than that of an electron is not taken into account here. The operators with the following nonzero commutation relations [5] correspond to the mentioned variables, i.e. [Ŝk(r), Ŝm(r ′)] = iǫkmnŜn(r)δ(r− r′), [L̂k(r), M̂m(r ′)] = iǫkmnδ(r− r′)M̂n(r), M̂ = L̂, P̂e, Π̂e , (1) [P̂sk(r), Π̂sm(r ′)] = iqs∆kmδ(r− r′), s = i, e, qe = e~v−1 0 , qi = ei~v −1 0 . Here indices k, m, n number the vector projections, ∆km is the Kroneker symbol; v0 is the volume of an elementary cell; e and ei are electron and ion charges, respectively. Hamiltonian of a ferroelectric-ferromagnet crystal which is taken uniaxial, to be specific,(though it is not of fundamental importance) in external electric e and magnetic h fields is written in the form: Ĥ = ∫ [Ĥm(r) + Ĥe(r) + Ĥem(r)]dr, Ĥm = − b 2 Ŝ2 z + α 2 Ŝ′ 2 xk − µ0h(L̂+ 2Ŝ) 746 Manifestations of orbital moment in ferroelectrics + r‖ 2 L̂2 z + r⊥ 2 (L̂2 x + L̂2 y) + λ1 2 L̂′ 2 xk + λL̂Ŝ, Ĥe = − k1 2 P̂ 2 iz + k2 2 (P̂ 2 ix + P̂ 2 iy) + δ 4 P̂ 4 iz + ζ 2 P̂′ 2 ix k (2) − q1 2 P̂ 2 ez + q2 2 (P̂ 2 ex + P̂ 2 ey) + ξ 2 P̂′ 2 ex k + v1P̂ezP̂iz + v2(P̂exP̂ix + P̂eyP̂iy) + 1 2fe Π̂2 e + 1 2fi Π̂2 i − e(P̂i + P̂e), Ĥem = σ(P̂i + P̂e) · [Π̂e × (L̂+ 2Ŝ)]. In Hamiltonian Ĥm the first term is the energy of spin-dipole interaction; the terms with coefficients r‖ and r⊥ derived from Coulomb interactions and r‖,r⊥ are called the parameters of hardness of freezing of the orbital moment [3]; λ is the constant of spin-orbital interaction; µ0 is the Bohr magneton. The terms with coefficients v1 and v2 in electrodipole energy operator Ĥe describe the interaction of the electric dipole moment of the electrons of the 3d-shell Pe with the dipole moment of an ion core Pi.The terms containing operators of momenta Π̂e, Π̂i are the density operators of electron and ion kinetic energies, and the constants fe, fi are proportional to electron and ion masses, respectively. ME energy (the last term Ĥem in (2)) is of dynamic nature. This is the energy of electric polarization in an effective electric field E ef formed by the electron moving with the velocity v in an internal magnetic field with induction B, Eef = −c−1v × B, c is the light velocity. In our case v = Πev0m −1 e (me is the effective electron mass), B = 4πµ0(L + 2S). Therefore, for a constant σ in (2) one can obtain σ = 4πµ0v0(mec) −1. (3) The indicated ME energy is the scalar, i.e. it is present in the energy of a crystal of any symmetry. The potential ME energy (see, for instance, [2]) for the ground state considered below leads merely to the inessential renormalization of the constants in (2) and here this energy is omitted. According to modern views a FE transition is a special case of structural phase transition. The displacement of the ionic core from the equilibrium position is ac- companied by deformation of the 3d-shell, i.e. by electron polarization. It is assumed here that magnetic ordering arises in a spin subsystem, and due to spin-orbital inter- action it magnetizes the orbital moments thus creating an average orbital moment L0 which differs from zero. All equilibrium moments (S 0,L0,Poi,Poe) are considered to be directed along the easy axis Z of a crystal. Their magnitudes determined by minimization of homogeneous energy which corresponds to Hamiltonian (2) are as follows: L0 = −λr−1 ‖ S0, Peo = v1q −1 1 Pio, P 2 io = δ−1(k1 − v21q −1 1 ), (4) where v21 6 k1q1; δ, r‖, r⊥, b, k1, k2, q1, q2 are positive. To find a linear response of a ferroelectric-ferromagnet to an external electro- magnetic field (e,h) a quantum-mechanical equation of motion for an operator ∂â/∂t = i~−1[Ĥ, â] is used.Taking a ∼ exp [i(kr− ǫt~−1] (k is the wave vector) 747 I.E.Chupis and using (1) and (2), in a linear approximation over small deviations of moments from equilibrium values,one can obtain the following equations (L0Ao − ǫ)l+ + c2Poep + e + v2Poep + i + λL0s + − iσ0π + e = Poee + + µ0L0h +, (A− ǫ)s+ + λS0l + + 2iσS0P0π + e = 2µ0S0h +, Ā0Poel + + λ̄Poes + + (ǫ− ǫσ)p + e − ǫσp + i − iqef −1 e π+ e = µ0Poeh +, (ǫ2 − ǫ2⊥i)p + i − v2Qip + e − iǫσqif −1 i π+ e = −Qie +, c2p + e + v2p + i + i(ǫσ − ǫ)q−1 e π+ e = e+, (5) (ǫ2 − ǫ2zi)pzi − v1Qipze = −Qiez, (ǫ2 − ǫ2ze)pze − v1Qepzi = −Qeez, πni = −iǫfiq −1 i pni, n = x, y, z. Here M+ = Mx + iMy, A0 = r⊥ + λ1k 2, Ā0 = A0 + σqe(1 + q1v −1 1 ), A = S0(b+ λ2r−1 ‖ + αk2), Cn = qn + ξk2, n = 1, 2, (6) B1 = −k1 + 3δP 2 0i + ζk2, B2 = k2 + ζk2, σ0 = σ(L0Pi0 − 2S0Peo), ǫσ = σqeI0, I0 = L0 + 2S0, P0 = Poi + Poe, λ̄ = λ+ l2σqe(1 + q1v −1 1 ), ǫ2zi = QiB1, ǫ2i⊥ = QiB2, ǫ2ze = QeC1, Qi = q2i f −1 i , Qe = q2ef −1 e . Equations for M− = Mx − iMy are obtained from equations (5) by the complex conjugation and the change of ǫ by (−ǫ). Two last equations in (5) describe Z-components of excitations of the electron and the ion polarizations which in a linear approximation are not connected with the rest of the variables. Spectral branches ǫz(1,2) where ǫ2z(1,2) = 1 2 [ǫ2zi + ǫ2ze ∓ √ (ǫ2ze − ǫ2zi) 2 + 4QiQev 2 1], (7) correspond to these excitations. Electron dielectric susceptibility χ e zz = ∂Pze/∂ez and ionic dielectric susceptibility χ i zz = ∂Pzi/∂ez are as follows: χe zz = −Qe(ǫ 2 − ǫ2zi + v1Qi)(ǫ 2 − ǫ2z1) −1(ǫ2 − ǫ2z2) −1, (8) χi zz = −Qi(ǫ 2 − ǫ2ze + v1Qe)(ǫ 2 − ǫ2z1) −1(ǫ2 − ǫ2z2) −1. As the constants fe and fi are proportional to electron (me) and ion (mi) masses, respectively, then the ratio of frequencies (ω = ~ −1ǫ) of electron and ion excitations (7) is of the order of ωe/ωi ∼ (mi/me) 1/2 ≫ 1. The lower branch of the spectrum is practically a branch of ion excitations, and the upper one corresponds to elec- tron excitations. Using equations (4), (6) and (7) it is easy to see that at FE phase transition (P0 = 0), the activation energy of the lower branch ǫz1 turns to zero. The “twinning law” is fulfilled for the total static FE susceptibility χ zz = χi zz + χe zz. If v1 = 0 in expressions (8), then for static susceptibilities from (7) and (8) one finds 748 Manifestations of orbital moment in ferroelectrics χe zz(0) = C−1 1 , χi zz(0) = B−1 1 . In the frequency range ωe ≫ ω, the electron suscep- tibility changes slightly (as compared to χe zz(0)) while the ion susceptibility is of resonance behaviour near ωz1. At v1 6= 0 both ion and electron susceptibilities are of resonance behaviour near ωz1. At high frequencies ω ≫ ωi, the electron suscepti- bility is (ωe/ωi) 2 times larger than the ion susceptibility. At these frequencies, χe zz being far from resonance has got the same order of magnitude as in a static case. Excitations of the components of moments perpendicular to their equilibrium direction, namely, to the axis Z, are connected with each other. From equations (5) one can obtain expressions for generalized susceptibility,i.e. electric χnk = ∂Pn/∂ek, magnetic xnk = µ0∂In/∂hk, magnetoelectric αme nk = µ0∂In/∂ek, α em nk = ∂Pn/∂hk, where P = Pi +Pe, I = L + 2S. In the absence of damping αem nk = (αme kn ) ⋆. The total expressions for susceptibilities are too cumbersome and it is convenient to analyze them for different states of the system. 3. Ferroelectric state If a spontaneous magnetic moment isn’t present, (S0 = L0 = 0), and P0 6= 0, then there are four branches of excitations, namely, ǫz1, ǫz2 (7) and optical ion and electron excitations p⊥ in the basic plane with the energies ǫoi, ǫoe respectively. The spectrum is degenerated, i.e.the excitations px and py have the same energy and are of oscillation character. The energies ǫoi and ǫoe determined from equations (5) are as follows: ǫ2o(i,e) = 1 2 [ǫ2i⊥ + ǫ2e⊥ ∓ √ (ǫ2e⊥ − ǫ2i⊥) 2 + 4QiQ̄ev 2 2 ]. (9) Here ǫ2e⊥ = C2Q̄e , where Q̄e = Qe − Ā0P 2 oe. As ǫe⊥ ≫ ǫi⊥ we obtain approximately ǫ2oi ≈ Qi(C1 − v22C −1 2 ), ǫ2oe ≈ ǫ2e⊥ ≈ C2(Qe − Ā0P 2 oe). (10) In FE state in the energy of transverse electron mode ǫe⊥ (10) the term appears which is proportional to the square of the electron part of polarization and the parameter of freezing of orbital moment since Ā0(k = 0) = r⊥ > 0. This means the decrease and the break in the temperature dependence of transverse electron frequency which is not soft at FE transition. Transverse components of ion χi nk and electron χe nk dielectric susceptibilities are as follows: χi xx = χi yy = −QiD(ǫ2 − ǫ2e⊥ + v2Q̄e), (11) D = (ǫ2 − ǫ2oi) −1(ǫ2 − ǫ2oe) −1, χe xx = χe yy = −Q̄eD(ǫ2 − ǫ2i⊥ + v2Qi). One sees from (11) that transverse components of dielectric susceptibilities as well as the frequency have got breaks at FE transition temperature. The break magnitude will have the largest value at high frequencies ω ∼ ωe, ∆χe xx = χe − − χe + = Ā0P 2 0eǫ 2(ǫ2 − ǫ2e⊥) −2, (12) 749 I.E.Chupis where ∆χ is the difference between the susceptibility (χe +) above FE transition tem- perature Te extrapolated to T 6 Te and the susceptibility χe − below Te. According to the order of magnitude Ā0 ∼ ǫe , thus ∆χ ∼ P 2 0eǫ −1 e . An experimental measurement of the mentioned breaks could help to evaluate the parameters P0e and r⊥. As it follows from the first equation in (5) excitations of electric polarizationp⊥ at P0 6= 0 are accompanied by excitations of an orbital moment l⊥, i.e. high-frequency linear ME effect takes place [4]. This effect is characterized by a nondiagonal com- ponent of ME susceptibility αem xy = −iǫµ0P0e(ǫ 2 − ǫ2i⊥ + v2Qi)(ǫ 2 − ǭ20i) −1)(ǫ2 − ǭ2oe) −1. (13) Besides, the orbital magnetic susceptibility which is proportional to the square of electron spontaneous polarization is induced: x0 xx = x0 yy = µ2 0P 2 0e[C2(ǫ 2 − ǫ2i⊥) + v22Qi](ǫ 2 − ǭ2oi) −1(ǫ2 − ǭ2oe) −1. (14) The effect of induction of the orbital magnetic susceptibility in a FE state should be accompanied by a break in temperature dependence of paramagnetic susceptibility at T = Te. According to (14), the break magnitude is proportional to the square of electron polarization. The value of ME susceptibility (13) increases with the increasing frequency (i.e. ǫ) and it is a resonance behaviour at ion (ω̄oi) and electron (ω̄oe) frequencies. The largest value αxy takes at electron frequencies ω ∼ ωe [4]. In this case far from resonance αxy ∼ µ0P0eǫ −1 e . For values P0e ∼ 10 µ C cm−2 the magnitude αxy ∼ 10−4 − 10−3. Orbital magnetic susceptibility induced by spontaneous electric polarization is much smaller. 4. Ferroelectric-ferromagnetic state In the ferroelectric-ferromagnetic state when S0 6= 0, L0 6= 0, P0 6= 0, the optical phonon excitations p⊥ are not degenerated and are of circle precession character around the direction of a spontaneous magnetic moment Ioz = Loz + 2Soz. The energies left (ǫ−) and right (ǫ+) precessions are different. As the values of ǫe⊥ ≫ ǫi⊥ ≫ ǫσ from (5) we obtain in a linear approximation over ǫσ ǫi ≈ ǫoi ± ǫσqiv2(2fi) −1ǫ−2 oe (qi + qe − 2qiv2C −1 2 ), (15) ǫe ≈ ǫoe ± ǫσ. As seen from (15),splitting of lines of the ion phonon spectrum in the internal mag- netic field ∼ Io is (ǫe/ǫi) times smaller than the electron one. The latter is also rather small, ∼ ǫσ/ǫe. Magnetic moment Io induces a nondiagonal component of dielectric susceptibility χxy. In the same linear approximation over ǫσ one finds χi xy = χi⋆ yx = −iǫσǫQiD 2 { qe(ǫ 2 − ǫ2i⊥)(ǫ 2 − ǫ2e⊥ + 2qiqev2f −1 e ) (16) 750 Manifestations of orbital moment in ferroelectrics + 3qiqev 2 2QiQ̄e + v2Qi(qi + qe)(ǫ 2 − ǫ2e⊥) } , χe xy = χe⋆ yx = −iǫσǫD 2 { (ǫ2 − ǫ2i⊥)[Qi(ǫ 2 − ǫ2e⊥) + 2Q̄e(ǫ 2 − ǫ2i⊥)] + qif −1 i v2Q̄e(3qi + qe)] + qeq −1 i v22Q̄eQ 2 i } . It follows from (16) that spin and orbital moments induce electric gyrotropy, χxy ∼ (Lo + 2So) which increases with the increasing frequency. The estimations of ex- pressions (16) far from resonance give χi xy ∼ χe xy ∼ ǫσǫs/ǫ 2 e(ǫ ∼ ǫs), χ i xy ∼ χe xy ∼ ǫσǫi/ǫ 2 e(ǫ ∼ ǫi). At frequencies ω ∼ ωe ∼ 1014 − 1015 rad s−1 the electron contri- bution to dielectric susceptibility is considerably larger than the ion contribution, and the electric gyrotropy is of the order of magnitude χe xy ∼ ǫσ/ǫe. The magnitude ǫσ = 8πµ2 oIo ∼ 10ǫs, i.e. the largest value is χe xy ∼ 10−3. High-frequency ME susceptibility of uniaxial ferroelectric-ferromagnet has got three components which differ from zero, i.e. αxy and αxx = αyy. A nondiagonal component αxy is approximately determined by expressions (13) at frequencies which are larger than the spin ones. A new component αxx = αyy which differs from zero appears only in a ferro- electric-ferromagnetic state ( or in external electric and magnetic fields[4]), αxx ∼ P0S0. At the frequencies ω ∼ ωe where ME susceptibility is the largest we obtain approximate expressions αem xx = αem yy ≃ µ0P0e(ǫ0 + 2λ̄S0)(ǫ 2 − ǫ20e) −1, (17) αem xy ≃ −iµ0ǫP0e(ǫ 2 − ǫ20e) −1. In [4] where FE was considered in a constant external magnetic field, the expres- sion obtained for αxx differs from (17).This difference is in the following. There is an energy of an external magnetic field µ0H0 instead of the energy of orbital excita- tions (ǫ0 + 2λ̄S0). For maximum fields H0 ∼ 105 Oe reached at present, the energy µ0H0 ∼ 10 ◦K whereas the value of the energy of orbital excitations ǫ0 ∼ λS0 for the 3d-compounds is considerably larger, of the order of 100 K. As shown in [4], the presence of ME susceptibility creates a possibility of a new optical effect – ME gyration – which consists in the rotation of a plane of polarization of the reflected and of the transmitted light by an angle which is proportional to the product of the first degrees of spontaneous electric polarization and magnetic field, i.e. to the value P0eH0. Therefore, in a ferroelectric-ferromagnet where the value αxx is considerably larger than in FE, the effect of ME gyration is considerably strengthened by spin-orbital interaction. In other words, in ferroelectric-ferromagnet, an orbital moment manifests itself by λ -strengthening of ME gyration. 5. Conclusions and summary Therefore, a simultaneous consideration of the spin, of the orbital moment, of the electron polarization and of the ion polarization of the 3d-compounds has provided a possibility to analyze possible manifestations of electron shell polarization and 751 I.E.Chupis orbital degrees of freedom of the 3d-electrons in a spectrum and high-frequency susceptibility. The orbital moment manifests itself in the breaks in the temperature dependen- cies of a nonsoft phonon mode and transverse dielectric susceptibility at FE transi- tion temperature and in λ – increase of ME-gyration. Besides, ME effect of induction of high-frequency orbital susceptibility by the electron part of electric polarization has been also predicted. This effect should lead to the break in the temperature dependence of paramagnetic susceptibility at FE transition temperature. In a series of publications [6–8] the observation was reported of a “magnetopo- larization gyration” in ferroelectrics manifested in the rotation of the polarization plane of the light propagating along the magnetic field and the optical axis of the crystal through an angle φ proportional to the product of the first powers of electric polarization and magnetic field. By contrast to Faraday’s effect, the double (forward and backward) passage of light has led to a compensation of the above-mentioned “magnetopolarization” effect. On the first sight this effect seems similar to ME gy- ration [4] but the estimates of ME gyration give a much smaller magnitude of the effect. In a ferroelectric-ferromagnet, the orbital moment increases the effect of ME gyration in λ ∼ 102 times in comparison with the value φ ∼ 10−7 rad in a fer- roelectric [4]. However, this value φ ∼ 10−5 rad is smaller than that declared in [6–8]. As for as the values of all the predicted effects are proportional to a spontaneous electric polarization then the 3d-compounds with a proper FE transition are more useful for the experimental investigations. This research was supported by INTAS Grant No. 94-935. References 1. Beznosov A.B., Galuza A.I., Eremenko V.V. Valence fluctuations and atomic polariz- ability of iron oxides in an atomic polarizability of iron oxides. – In: “Inter. Conf. of Magnetizm”, Abstract. San-Francisco, 1985, 1Pe 13, p. 67. 2. Chupis I.E., Govorun A.V. On the theory of high-frequency properties of a ferroelec- tromagnetic with unfrozen orbital moment. // Fiz. Nizk. Temp., 1995, vol. 21, No. 2, p. 228–234 (in Russian). 3. Rozenfeld E.V., Korolev A.V. Low temperature anisotropy of magnetization in ferro- magnetic with frozen orbital moment. // Zh. Eksp. Teor. Fiz., 1995, vol. 108, No. 3(9), p. 862–877 (in Russian). 4. Chupis I.E. Magnetooptical effects in ferroelectric materials. // Fiz. Nizk. Temp., 1997, vol. 23, No. 3, p. 290–295 (in Russian). 5. Chupis I.E. Elementary excitations in ferroelectromagnetic with orbital magnetic mo- ment. // Fiz. Tverd. Tela, 1994, vol. 36, No. 7, p. 1910–1917 (in Russian). 6. Vlokh O.G. Magnetogyration. // Ukr. Fiz. Zh., 1981, vol. 26, No. 10, p. 1623–1626 (in Russian). 7. Vlokh O.G., Zheludev I.S., Sergatyuk V.A. Electro-and magnetogyration in crystals. // Izv. Akad. Nauk SSSR, Ser. Fiz., 1984, vol. 48, No. 9, p. 1771–1776 (in Russian). 752 Manifestations of orbital moment in ferroelectrics 8. Vlokh O.G., Sergatyuk V.A. Magnetopolarizational pseudogyration. // Doklady Akad. Nauk SSSR, 1986, vol. 291, No. 4, p.832–835 (in Russian). Прояви електрон-орбітального моменту у високочастотних властивостях сегнетоелектриків з 3d-іонами І.Чупіс Фізико-технічний інститут низьких температур ім. Б.І.Вєркіна НАН України, 310164 Харків, просп. Леніна, 47 Отримано 28 червня 1998 р. Побудована феноменологічна теорія високочастотних властивостей сегнетоелектрика і сегнетоелектрик-феромагнетика з 3d-іонами, яка грунтується на розділеному врахуванні спінового і орбітального електронних моментів, електронних та іонних вкладів до електрич- ної поляризації. У сегнетоелектричному стані існування електронно- го орбітального момента приводить до розривів поперечних компо- нент діелектричної сприйнятливості при температурі сегнетоелек- тричного переходу. Величини цих розривів пропорціональні до ква- драта електронної частини спонтанної поляризації і параметра замо- роження орбітального момента. Аналогічні розриви і зменшення фо- нонних частот у сегнетоелектричному стані мають спостерігатися в модах, які не є м’якими. Крім того, передбачено ефект індукції висо- кочастотного орбітального момента на електронну частину поляри- зації. Цей ефект може приводити до розриву парамагнітної сприй- нятливості при температурі сегнетоелектричного переходу. У феро- електромагнітному стані електронний орбітальний момент проявляє себе в λ зростанні магнітоелектричної гірації ( λ є постійна спін- орбітальної взаємодії). Ключові слова: електрон, орбіталь, момент, сегнетоелектрик, сприйнятливість PACS: 78.20.Ls 753 754

Manifestations of the electron orbital moment in high-frequency properties of ferroelectrics with 3d-ions

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