One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂

Experimental data are reported for the temperature and polarization dependence of the one-magnon Raman light scattering in the rutile-structure antiferromagnet CoF₂ (Néel temperature TN = 38 K). The low-lying excitons are also investigated at low temperatures and comparisons made with results from e...

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Опубліковано в: :	Физика низких температур
Дата:	2014
Автори:	Meloche, E., Cottam, M.G., Lockwood, D.J.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2014
Теми:	К восьмидесятилетию антиферромагнетизма ІІ. Эксперимент
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Цитувати:	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂ / E. Meloche, M.G. Cottam, D.J. Lockwood // Физика низких температур. — 2014. — Т. 40, № 2. — С. 173-186. — Бібліогр.: 21 назв. — англ.

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spelling	Meloche, E. Cottam, M.G. Lockwood, D.J. 2017-06-06T18:45:53Z 2017-06-06T18:45:53Z 2014 One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂ / E. Meloche, M.G. Cottam, D.J. Lockwood // Физика низких температур. — 2014. — Т. 40, № 2. — С. 173-186. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS 75.30.Ds, 72.10.Вb, 78.30.–j, 75.50.Ee https://nasplib.isofts.kiev.ua/handle/123456789/119410 Experimental data are reported for the temperature and polarization dependence of the one-magnon Raman light scattering in the rutile-structure antiferromagnet CoF₂ (Néel temperature TN = 38 K). The low-lying excitons are also investigated at low temperatures and comparisons made with results from earlier Raman, infrared, and neutron scattering work. A detailed analysis of the one-magnon Stokes and anti-Stokes Raman spectra is presented resulting in comprehensive data for the temperature variation up to TN of the one-magnon frequency, line width, and integrated intensity. A theory of the one-magnon scattering and other magnetic transitions is constructed based mainly on a spin S = 3/2 exchange model, extending a simpler effective S = 1/2 approach. The excitation energies and spectral intensities over a broad range of temperatures are obtained using a Green's function equation of motion method that allows for a careful treatment of the single-ion anisotropy. Overall the S = 3/2 theory compares well with the experimental data. This work was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. We thank P.A. Moch for helpful discussions, J. Johnson for assistance in the analysis of the Raman spectra, and H.J. Labbé for the crystal sample preparation. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К восьмидесятилетию антиферромагнетизма ІІ. Эксперимент One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂ Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂
spellingShingle	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂ Meloche, E. Cottam, M.G. Lockwood, D.J. К восьмидесятилетию антиферромагнетизма ІІ. Эксперимент
title_short	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂
title_full	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂
title_fullStr	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂
title_full_unstemmed	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂
title_sort	one-magnon and exciton inelastic light scattering in the antiferromagnet cof₂
author	Meloche, E. Cottam, M.G. Lockwood, D.J.
author_facet	Meloche, E. Cottam, M.G. Lockwood, D.J.
topic	К восьмидесятилетию антиферромагнетизма ІІ. Эксперимент
topic_facet	К восьмидесятилетию антиферромагнетизма ІІ. Эксперимент
publishDate	2014
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	Experimental data are reported for the temperature and polarization dependence of the one-magnon Raman light scattering in the rutile-structure antiferromagnet CoF₂ (Néel temperature TN = 38 K). The low-lying excitons are also investigated at low temperatures and comparisons made with results from earlier Raman, infrared, and neutron scattering work. A detailed analysis of the one-magnon Stokes and anti-Stokes Raman spectra is presented resulting in comprehensive data for the temperature variation up to TN of the one-magnon frequency, line width, and integrated intensity. A theory of the one-magnon scattering and other magnetic transitions is constructed based mainly on a spin S = 3/2 exchange model, extending a simpler effective S = 1/2 approach. The excitation energies and spectral intensities over a broad range of temperatures are obtained using a Green's function equation of motion method that allows for a careful treatment of the single-ion anisotropy. Overall the S = 3/2 theory compares well with the experimental data.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/119410
citation_txt	One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂ / E. Meloche, M.G. Cottam, D.J. Lockwood // Физика низких температур. — 2014. — Т. 40, № 2. — С. 173-186. — Бібліогр.: 21 назв. — англ.
work_keys_str_mv	AT melochee onemagnonandexcitoninelasticlightscatteringintheantiferromagnetcof2 AT cottammg onemagnonandexcitoninelasticlightscatteringintheantiferromagnetcof2 AT lockwooddj onemagnonandexcitoninelasticlightscatteringintheantiferromagnetcof2
first_indexed	2025-11-25T23:52:47Z
last_indexed	2025-11-25T23:52:47Z
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fulltext	© E. Meloche, M.G. Cottam, and D.J. Lockwood, 2014 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2, pp. 173–186 One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 E. Meloche* and M.G. Cottam Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada D.J. Lockwood Institute for Microstructural Sciences, National Research Council, Ottawa, Ontario K1A 0R6, Canada E-mail: David.Lockwood@nrc-cnrc.gc.ca Received July 5, 2013 Experimental data are reported for the temperature and polarization dependence of the one-magnon Raman light scattering in the rutile-structure antiferromagnet CoF2 (Néel temperature TN = 38 K). The low-lying excitons are also investigated at low temperatures and comparisons made with results from earlier Raman, infra- red, and neutron scattering work. A detailed analysis of the one-magnon Stokes and anti-Stokes Raman spectra is presented resulting in comprehensive data for the temperature variation up to TN of the one-magnon frequency, line width, and integrated intensity. A theory of the one-magnon scattering and other magnetic transitions is con- structed based mainly on a spin S = 3/2 exchange model, extending a simpler effective S = 1/2 approach. The ex- citation energies and spectral intensities over a broad range of temperatures are obtained using a Green's function equation of motion method that allows for a careful treatment of the single-ion anisotropy. Overall the S = 3/2 theory compares well with the experimental data. PACS: 75.30.Ds Spin waves; 72.10.Вb Scattering by phonons, magnons, and other nonlocalized excitations; 78.30.–j Infrared and Raman spectra; 75.50.Ee Antiferromagnetics. Keywords: antiferromagnet, inelastic light scattering, magnetic excitations. 1. Introduction In an earlier paper [1] we presented a thorough compar- ison between experiment and theory for two-magnon ine- lastic light scattering in the rutile-structure antiferro- magnets NiF2 and CoF2 thus complementing earlier work on isomorphic compounds such as FeF2 and MnF2 that are broadly similar magnetically [2]. The focus of the present paper is on the one-magnon light scattering, which pro- vides further insight into the spin dynamics of these com- pounds because it emphasizes the magnetic excitations near the center of the Brillouin zone. By contrast, in two- magnon scattering, the excitations of magnon pairs at large wave vectors are dominant. By comparison with FeF2 and MnF2, neither NiF2 or CoF2 were well understood with regards to the temperature and polarization dependence of their one-magnon excitations, as studied through their fre- quencies and Raman intensities. They present quite distinct cases to one another, because in NiF2 there is a spin cant- ing from true antiferromagnetic alignment, leading to a major effect on the zone-center magnons and thereby on the one-magnon light scattering, as we reported recently [3]. On the other hand, the one-magnon light scat- tering in CoF2 has other distinctive properties. There is no canting, but the effects arising due to a strong orbital angu- lar momentum and a large single-ion anisotropy are domi- nant. This has motivated our experimental and theoretical studies presented here. The crystallographic unit cell of CoF2 is depicted in Fig. 1 together with the relevant exchange parameters em- ployed in this work. The antiferromagnetic ordering of the spins in CoF2 was first determined by Erickson [4] using neutron diffraction and also confirmed through studies of * Current address: 7801 Computer Ave., Bloomington, MN 55435 E. Meloche, M.G. Cottam, and D.J. Lockwood 174 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 the magnetic anisotropy of fluorides of the iron group ele- ments [5]. Early theoretical studies of the static magnetic properties of CoF2 in the paramagnetic and antiferromag- netic phases using an effective spin S = 3/2 Hamiltonian were performed by Nakamura and Taketa [6]. The magnet- ic excitations in CoF2 were first studied by Lines [7] and the dominant exchange and anisotropy parameters were estimated by comparing theory with low-temperature anti- ferromagnetic resonance frequencies [8] and infrared ab- sorption data [9]. The electronic stucture of the Co 2+ ions has been determined by Gladney [10], and the spin-wave dispersion relations have been measured by Cowley et al. [11,12] using inelastic neutron scattering. Raman light scattering measurements made by Macfarlane [13], Moch et al. [14], and Hoff et al. [15] have also characterized the low-lying zone center excitations, and the light scattering cross sections [13,14] are somewhat consistent with the calculations of Ishikawa and Moriya [16]. The splitting of the excitations by an applied magnetic field has been stud- ied in the infrared [17,18] and with Raman scattering [14]. In this paper we report experimental Raman scattering data for the one-magnon and exciton excitations in CoF2, together with a theoretical analysis for the magnetic excita- tions in a two-sublattice S = 3/2 antiferromagnet with sin- gle-ion anisotropy. The theoretical technique used here employs the spin operators directly and allows us to inves- tigate the magnetic excitations from the approximate ground state as well as the additional optical magnetic modes that are expected in this system. The justification for using an effective spin S = 3/2 model to study the low-lying magnetic excitations in CoF2 has been discussed by several authors [6,7,16] and here we highlight the main arguments. The free Co 2+ ion has an electronic configuration 3d 7 and application of Hund's rules yields a 28-fold degenerate 4 F ground state (L = 3, S = 3/2). When the ion is inserted into a crystal and sur- rounded by F – ions it is subjected to a perturbing electric field which lifts the degeneracy of the free ion. As the symmetry is lowered there is an increase in the number of split levels produced by the crystal field. The degeneracy of the orbital state of the free Co 2+ ion is seven (L = 3) and is split into two triplets and a singlet by a crystalline elec- tric field of cubic symmetry, where the lowest level is a triplet which is described in terms of an effective L = 1 operator. The lowest orbital triplet state is now of degener- acy twelve because of the spin S = 3/2 of the free ion. The F – ions surrounding the Co 2+ ions do not have perfect cu- bic symmetry. The distortion from cubic symmetry com- bined with spin-orbit coupling splits the lowest energy manifold into six (Kramers) doublets. The degeneracy of each doublet is eventually removed by the exchange field, resulting in four lowest energy levels for the Co 2+ ions [10,16]. The energies of these four levels (relative to the lowest) have been estimated [10,16] to correspond roughly to 0, 51, 190, and 200 cm –1 , and they are well separated from the next level at about 800 cm –1 . This paper is organized as follows. In Sec. 2 we de- scribe the Raman experiments and present the results for the one-magnon and exciton scattering in the CoF2 sample. The theoretical analyses for the magnon excitations are described in Sec. 3, where we mainly use the spin S = 3/2 model which is superior to the effective spin S = 1/2 ap- proach in the context of the one-magnon and exciton Ra- man scattering. This contrasts with the situation for one- magnon light scattering in NiF2 mentioned earlier, where there the spin canting is important and the Ni 2+ ions have spin S = 1. Comparisons of the one-magnon theory and experimental data are presented in Sec. 4, and the other magnetic excitations are then briefly discussed in Sec. 5. The conclusions of our work are given in Sec. 6. 2. Experiment and results The purplish-red-colored sample of CoF2 was prepared from a single crystal grown at the Clarendon Laboratory, Oxford University, specially for these one-magnon studies and our earlier two-magnon studies [1]. The cuboid sample of dimensions 3.2 mm 2.0 mm 1.7 mm was cut to expose (001) [Z], (110) [X], and (110) [Y] faces, respectively, and these faces were highly polished with 1 m diamond powder. The Raman spectrum was excited with 500–600 mW of Ti: sapphire laser light at 800 nm, which avoided any optical absorption [19], analyzed with a Spex 14018 double monochromator at a spectral resolution of 3.3 cm –1 unless otherwise indicated, and detected by a cooled RCA 31034A photomultiplier. The sample was mounted in the helium exchange-gas space of a Thor S500 continuous flow cryostat, where the temperature could be controlled to Fig. 1. The crystallographic unit cell of CoF2 (a = b c) with the principal exchange interactions J1, J2 and J3. Crystal axes (x, y, z) and laboratory axes (X, Y, Z) are illustrated in relation to the unit cell. The X and Y directions are orthogonal to the c(Z) axis but are rotated by 45 from the crystallographic a and b axes. a c b Co F J3 J1 J2 Z, z Y y X x One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 175 within 0.1 K and was measured with a gold-iron/chromel thermocouple clamped to the sample. Spectra were record- ed in the 90 scattering geometry. The one-magnon scatter- ing was measured in different polarizations for tempera- tures up to about TN, while the exciton features were investigated at low temperatures only. The polarization dependence of the low frequency Raman spectrum of antiferromagnetic CoF2 at low tem- perature is shown in Fig. 2. These spectra exhibit a sharp peak at (37.0 0.1) cm –1 that is the lowest lying exciton (conventionally referred to as the magnon) of the ground state multiplet of the Co 2+ ion in the exchange field. The one-magnon scattering is observed only in off-diagonal polarizations. The temperature dependences of the spectra for these same polarizations are given in Fig. 3, where it can be seen that the one-magnon peak decreases in fre- quency and increases in width with increasing tempera- ture up to TN while its peak intensity decreases. It is evi- dent that the anti-Stokes and Stokes intensities are also temperature dependent and vary with the polarization. The Raman spectrum at higher frequencies exhibits three more sharp peaks associated with the higher lying excitons of the lowest multiplet, as shown in Fig. 4. The- se and other spectra were fitted with a Gaussian– Lorentzian line shape model [20] to yield the band pa- rameters of position, width (full width at half maximum), and integrated intensity. The results obtained at low tem- perature are given for the four excitons and also some CoF2 phonons in Table 1. This table shows that the widths of the lowest frequency exciton and phonon are resolution (2.5 cm –1 ) limited at low temperature. The phonon scattering is much stronger, in general, than the exciton scattering and the measurements in X(YZ)Y and X(ZX)Y polarization give the expected similar intensities for the Eg phonon, indicating that the experimental condi- tions are satisfactory, and this is also the case for exciton 3, whereas for the magnon (exciton 1) they are different. The spectra shown in Fig. 4 indicate that the polarization leak through (e.g., of exciton 3 from X(YZ)Y polarization into X(ZZ)Y polarization) is about 3%. 5000 4000 3000 2000 1000 0 R am an i n te n si ty , ar b . u n it s –40 –20 0 20 40 60 80 100 120 140 5.0 K Frequency shift, cm–1 (a) X ZX Y( ) X YZ Y( ) X ZZ Y( ) 5.0 K 5.0 K Blg phonon X YX Y( ) 5.3 K Two-magnon One-magnon CoF2 5000 4000 3000 2000 1000 0 R am an i n te n si ty , ar b . u n it s –40 –20 0 20 40 60 80 100 120 140 One-magnon Two-magnon CoF2 (b) Y ZX Z( ) Y ZY Z( ) Y XY Z( ) 6.0 K 6.5 K Blg phonon Y XX Z( ) 6.4 K Frequency shift, cm–1 6.0 K Fig. 2. (Color online) Polarization dependence of the low frequency Raman spectrum of antiferromagnetic CoF2 recorded at low tempera- ture for X( )Y (a) and Y( )Z (b) 90 scattering geometries. Table 1. Band parameters of excitons and phonons obtained by curve fitting the polarized Stokes Raman spectra of CoF2 at 10 K, as shown for the excitons in Fig. 4. The standard errors from the fits are given in brackets; FWHM is the band full width at half maximum. Exciton/phonon number (symmetry) Polarization Frequency, cm –1 FWHM, cm –1 Area, arb. units 1 3 4( ) X(YZ)Y 36.9 (0.1) 2.6 (0.1) 1211 (99) 1 3 4( ) X(ZX)Y 37.1 (0.1) 2.8 (0.1) 1853 (163) 2 1( ) X(ZZ)Y 168.0 (0.1) 3.2 (0.1) 5479 (157) 3 3 4( ) X(YZ)Y 192.9 (0.1) 3.7 (0.2) 6118 (570) 3 3 4( ) X(ZX)Y 193.0 (0.1) 3.6 (0.2) 7294 (557) 4 2( ) X(YX)Y 209.1 (0.4) 4.0 (1.1) 142 (85) 1 (B1g) X(YX)Y 65.2 (0.1) 2.3 (0.2) 224 (47) 2 (Eg) X(YZ)Y 255.3 (0.1) 4.7 (0.1) 30837 (1884) 2 (Eg) X(ZX)Y 255.3 (0.1) 4.7 (0.1) 29374 (1317) 3 (A1g) X(ZZ)Y 371.2 (0.1) 3.8 (0.1) 20290 (76) E. Meloche, M.G. Cottam, and D.J. Lockwood 176 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 Our measurements for the frequencies of the four lowest- lying excitons are compared with earlier results in Table 2, and there is good agreement between values obtained by Raman scattering and also the quite different techniques of inelastic neutron scattering and infrared spectroscopy. This is not the case, however, when it comes to the Raman cross- sections of the excitons. As can be seen from Table 3, the integrated intensities of the excitons observed in the various polarizations when normalized to that found for exciton 3 in X(YZ)Y polarization can be quite different for the three ex- perimental studies reported to date. In addition, there is a greater discrepancy between our values compared with pre- vious theory than for the other two experimental studies. However, there is one point on which all experiments and theory agree and that is the relative integrated intensity of exciton 4 is much weaker than for the others. It is evident R am an i n te n si ty , ar b . u n it s (a) –40 –20 0 20 40 5.0 K 10.0 K 18.8 K 24.3 K 31.0 K 35.0 K X YZ Y( ) CoF2 R am an i n te n si ty , ar b . u n it s R am an i n te n si ty , ar b . u n it s 2400 2000 1600 1200 800 400 0 5.0 K 9.9 K 18.6 K 24.0 K 30.9 K 35.0 K X ZX Y( ) –40 –20 0 20 40 CoF2 5000 4000 3000 2000 1000 0 (b) Frequency shift, cm–1Frequency shift, cm–1 CoF2 5000 4000 3000 2000 1000 0 6.0 K 10.0 K 19.3 K 24.7 K 29.1 K 34.6 K Y ZY Z( ) –40 –20 0 20 40 R am an i n te n si ty , ar b . u n it s c)( Frequency shift, cm–1 5000 4000 3000 2000 1000 0 6.0 K10.0 K 19.3 K 24.7 K 30.0 K 34.6 K Y ZX Z( ) –40 –20 0 20 40 CoF2 (d) Frequency shift, cm–1 Fig. 3. (Color online) Temperature dependence of the Stokes and anti-Stokes one-magnon Raman spectrum of antiferromagnetic CoF2 for X(YZ)Y (a), X(ZX)Y (b), Y(ZY)Z (c), and Y(ZX)Z (d) polarization. Table 2. Comparison of results obtained in different experiments and from theory for the lower multiplet exciton frequencies (in cm –1 ) in CoF2 at low temperature (T « TN). Exciton number Raman Raman Raman Infrared Neutron Scattering Theory (symmetry) Present work Ref. 13 Ref. 14 Ref. 18 Ref. 12 Ref. 16 1 3 4( ) 37.0 0.1 37.0 0.5 37 36.3 37 37 2 1( ) 168.0 0.1 169 168.3 168.5 170 173 3 3 4( ) 193.0 0.1 194 193.3 193.0 196 200 4 2( ) 209.0 0.1 – 210.2 210.9* 204 206 Notes: As deduced by Moch et al. [14] from the magnetic field results in the infrared study of Ref. 18. One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 177 from the theoretical data given in Table 3 that the excitation energy is a crucial factor in elaborating the Raman cross section, presumably because of possible Raman excitation energy resonances with higher-lying electronic energy levels of the Co 2+ ion. The experimental and theoretical data indi- cate that there is an increase in the overall intensities, rela- tive to the fixed (reference) one, of the excitons in the other polarizations with Raman excitation energy increase over the range 12500 to 30000 cm –1 . At the same time, theory shows that there is a large absolute intensity increase when the excitation energy is changed from 20000 to 30000 cm –1 (see Table 3). Fig. 4. (Color online) Raman spectra of the lowest energy excitons in antiferromagnetic CoF2 at 10 K and their line shape fits with the Gaussian–Lorentzian model in X(YZ)Y (a), X(ZX)Y (b), X(YX)Y (c), and X(ZZ)Y (d) polarization. The spectral resolution for these spectra was 2.5 cm –1 . R am an i n te n si ty , ar b . u n it s (a) X YZ Y( ) X ZX Y( ) (b) Frequency shift, cm–1 X ZZ Y( ) (d) 1500 1000 500 0 160 170 180 190 200 400 200 0 20 40 60 1500 1000 500 0 E 3x E 3x X YX Y( ) c)( E 4x E 3x Ex1 E 1x Frequency shift, cm–1 Frequency shift, cm–1 Frequency shift, cm–1 R am an i n te n si ty , ar b . u n it s 160 170 180 190 200 160 170 180 190 200160 170 180 190 200 400 200 0 20 40 60 Frequency shift, cm–1 R am an i n te n si ty , ar b . u n it s R am an i n te n si ty , ar b . u n it s 200 100 0 R am an i n te n si ty , ar b . u n it s 210 Frequency shift, cm–1 600 R am an i n te n si ty , ar b . u n it s 1500 1000 500 0 E 2x E 2x E 3x Table 3. Comparison of results obtained in different experiments and from theory at different excitation energies ( in cm –1 ) for the lower multiplet exciton relative Raman intensities in CoF2 at low temperature (T « TN). Exciton number (symmetry) Polarization Experiment Experiment Experiment Theory Theory Theory Present Work Ref. 13 Ref. 14 Ref. 13 Ref. 16 Ref. 16 = 12500 = 16000 = 16000 = 20000 = 30000 1 3 4( ) X(YZ)Y 20 4 28 29 6 36 50 86 X(ZX)Y 30 6 27 60 12 79 125 180 2 1( ) Y(XX)Z – <1.3 22 4 200 225 225 X(ZZ)Y 90 11 65 220 44 340 375 380 3 3 4( ) ) X(YZ)Y 100 100 100 100 100 100** X(ZX)Y 120 20 110 130 26 180 200 250 4 2( ) X(YX)Y 2.3 1.6 5 <22 4 20 25 68 Notes: * Ref. 13 quotes unpublished theoretical work of A. Ishikawa and T. Moriya. ** In absolute terms, the theoretical intensity for this band with = 30000 is 5.5 times that of the = 20000 case. E. Meloche, M.G. Cottam, and D.J. Lockwood 178 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 The one-magnon spectra as a function of temperature and polarization were all readily fitted with a Gaussian– Lorentzian line shape model and the results obtained for the line parameters in Stokes and anti-Stokes scattering are given in Figs. 5 and 6, respectively. For comparison with theory, we also give in Fig. 7 the temperature de- pendences of the ratio of the Stokes to anti-Stokes one- magnon integrated intensities for several polarizations. The intensity ratios for all polarizations exhibit a similar steep rise with decreasing temperature below about 15 K. The results in X(YZ)Y polarization, although showing a different temperature dependence from the other polariza- Fig. 5. (Color online) Temperature dependence of the CoF2 Stokes one-magnon Raman scattering fitted line shape parameters of frequency (a), full width at half maximum (b), and integrated intensity (c) for the various polarizations indicated. 40302010 0 CoF2 Stokes 1-M X YX Y( ) (a) (b) c)( Temperature, K F re q u en cy , cm – 1 40 30 20 10 0 X YZ Y( ) X Y( )ZX X ZZ Y( ) Y XY Z( ) Y ZX Z( ) Y ZY Z( ) 40302010 40302010 0 40 30 20 10 16000 12000 8000 4000 Temperature, K Temperature, K CoF2 Stokes 1-M X YX Y( ) X YZ Y( ) X Y( )ZX Y XY Z( ) Y ZX Z( ) Y ZY Z( ) CoF2 Stokes 1-M X YX Y( ) X YZ Y( ) X Y( )ZX X ZZ Y( ) Y XY Z( ) Y ZX Z( ) Y ZY Z( ) L in e w id th , cm – 1 Fig. 6. (Color online) Temperature dependence of the CoF2 anti- Stokes one-magnon Raman scattering fitted line shape parameters of frequency (a), full width at half maximum (b), and integrated intensity (c) for the various polarizations indicated. 40302010 0 CoF2 X YX Y( ) (a) (b) c)( Temperature, K F re q u en cy , cm – 1 40 30 20 10 0 X YZ Y( ) X Y( )ZX Y ZX Z( ) Y ZY Z( ) 40302010 40302010 0 40 30 20 10 4000 Temperature, K Temperature, K CoF2 X YX Y( ) X YZ Y( ) X Y( )ZX Y ZX Z( ) Y ZY Z( ) CoF2 anti-Stokes 1-M X YX Y( ) X YZ Y( ) X Y( )ZX Y ZX Z( ) Y ZY Z( ) L in e w id th , cm – 1 anti-Stokes 1-M anti-Stokes 1-M 7000 6000 5000 3000 2000 1000 One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 179 tions in Figs. 5(c) and 6(c), behave similarly to the others in the intensity ratio variation with temperature. 3. One-magnon theoretical analysis For simplicity we start by briefly presenting an effec- tive spin S = 1/2 Green's function theory for the one- magnon excitations. This approach is useful when analyz- ing two-magnon Raman scattering [1] because approxima- tions appropriate to large wave vectors can be made, but it has its limitations for magnons near the Brillouin-zone center. Therefore it is followed by an effective S = 3/2 the- ory, which is expected to provide a more realistic descrip- tions of the magnons since it takes into account the lowest four levels (instead of just two). In particular, the latter approach is expected to account more accurately for ani- sotropy effects and to be superior at elevated temperatures. 3.1. Effective spin 1/2 theory This model [7] is based on taking account of only the lowest two energy levels for the Co 2+ ions, which are sepa- rated by about 50 cm –1 as already noted. The effective Hamiltonian can be expressed as ex an= , (1) where ex and an refer to the exchange and anisotropy parts respectively, with ex , , , , , , 1 1 = , 2 2 i j i j i i i i j j i j i j i i j j J J JS S S S S S (2) an = ( )( )z z A i j i j H T S S . (3) The sites labelled i are on the spin-up sublattice and those labelled j are on the spin-down sublattice. The inter- sublattice exchange interaction is represented by ,i jJ whe- reas ,i iJ and ,j jJ are the intrasublattice exchanges. The effective anisotropy field ( )AH T is often assumed to vary with temperature like the sublattice magnetization. The parameters for the S = 1/2 models have been previously estimated by comparing theory with one-magnon inelastic neutron scattering measurements [11], yielding 1 = 2.0J cm –1 , J2 = 12.3 cm –1 , J3 0 cm –1 and HA(0) = 12.5 cm –1 where the exchange parameters are defined in Fig. 1. The one-magnon excitation energies are obtained by forming the operator equations of motion for the S oper- ators for each sublattice using the above Hamiltonian. The equations are then linearized using the random phase ap- proximation (RPA) and transformed to a wave vector rep- resentation. The solutions for the excitation energies are obtained, assuming a time dependence exp( )iEt for the S operators, as [1] ____________________________________________________ 2 2 2= ( ) (8 cos ( /2)cos ( /2)cos ( /2)) ,z x y zE S J k a k a k ck k (4) where 2 2 2 2 1 3( ) = ( ) 8 4 ( /2) 4 [ ( /2) ( /2)].sin sin sin z z z A z x yH T S J S J k c S J k a k ak (5) _______________________________________________ The energies are degenerate in magnitude (in the zero ap- plied field case considered here) and the negative sign in Eq. (4) refers to oppositely precessing spins. The spin av- erage zS related to the sublattice magnetization can be evaluated using mean-field theory [1]. 3.2. Effective spin 3/2 theory Next we focus on the S = 3/2 theory obtained using the four lowest energy levels for the Co 2+ ions [16]. This is expected to be superior here because of the more careful treatment of the anisotropy, which is particularly important for one-magnon Raman scattering. The two approaches will be compared later. In this case the total Hamiltonian can again be expressed as in Eq. (1) except that all spin operators now refer to S = 3/2 and the anisotropy part is replaced by 2 2 2 an = ( ( ) [( ) ( ) ]) yz x i i i i D S F S S 2 2 2( ( ) [( ) ( ) ]). yz x j j j j D S F S S (6) Y ZX Z( ) Temperature, K X YZ Y( ) X Y( )ZX Y ZY Z( ) I I S A S / 200 160 120 80 40 0 0 5 10 15 20 25 30 35 Fig. 7. Temperature dependence of the ratio of the CoF2 Stokes (IS) and anti-Stokes (IAS) one-magnon integrated intensities in X(YZ)Y, X(ZX)Y, Y(ZX)Z, and Y(ZY)Z polarization. E. Meloche, M.G. Cottam, and D.J. Lockwood 180 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 Here the parameters D and F describe the effects of the uniaxial and nonuniaxial contributions to the single-ion anisotropy, respectively. The average spin alignment for each sublattice is assumed to be along the crystallographic c axis as depicted schematically in Fig. 1. We now use the Green's function equation of motion method (rather than the operator equation of motion, which provides less information) to generate the set of equations satisfied by ;i ES Y and the other Green's functions coupled to it. As before we use RPA decoupling for prod- ucts of spins at different sites, but we do not approximate the anisotropy terms which involve products of operators at the same site. The formalism is analogous to that em- ployed by Cottam and Latiff Awang [21] for S = 1 antiferromagnets with single-ion anisotropy, but extended here to S = 3/2. To obtain a finite, closed set of equations, we use the identity for S = 3/2 spin operators that 4 2( ) = (5/2)( ) (9/16).z zS S Taking into account the dif- ferent sublattice labeling, 16 coupled equations are re- quired to obtained a closed set of equations. After a trans- formation to wave vector representation the set of equations can be written in matrix form as 16( ) = ,E kI B G b (7) where 16I is the 16 16 unit matrix, kG and b are 16- component column matrices whose elements are defined as nG k and nb (for n = 1, …, 16) with 1 ; = exp [ ( )] ,n E l m nX Y i G N k k k r r 1 = < [ , ] >,n nb X Y N (8) for any operator Y. We define the operators nX as 1 2 3 4= , = , = , = ,i j i jX S X S X S X S 5 6= , = ,z z z z i i i i j j j jX S S S S X S S S S 7 8= , = ,z z z z i i i i j j j jX S S S S X S S S S 2 2 9 = ( ) 2 ( ) ,z z z z i i i i i i iX S S S S S S S 2 2 10 = ( ) 2 ( ) ,z z z z j j j j j j jX S S S S S S S 2 2 11 = ( ) 2 ( ) ,z z z z i i i i i i iX S S S S S S S 2 2 12 = ( ) 2 ( ) ,z z z z j j j j j j jX S S S S S S S 3 3 13 14= ( ) , = ( ) ,i jX S X S 3 3 15 16= ( ) , = ( ) .i jX S X S (9) The above operators 1 12, ... ,X X are all linear in the transverse spin components and are associated with transi- tions for = 1zS whereas the operators 13 16, ... ,X X all involve transitions for = 3.zS We note that these cou- pled equations do not involve any operators in combina- tions that are quadratic in the transverse spin components. The excitations associated with this combination of spin operators are different from the one-magnon excitations and so are considered in a later section. By comparison with the case of an anisotropic S = 1 antiferromagnet considered earlier [21] using this opera- tor method, the analogous equation to Eq. (7) involved only a 8 8 matrix, because the operator combinations 9 16, ... ,X X in that case can be shown either to be zero or to be expressible in terms of the remaining 1 8, ... , .X X In general, as S is increased, it is found that more coupled equations of motion are needed to obtain a closed set. In our analysis the poles of the Green functions corre- spond to the spin-wave (one-magnon) excitations, which are obtained by applying the determinantal condition that 16det ( ) = 0,EI B (10) as follows from Eq. (7). From the formal results for Green's functions we can also deduce the spin correlation functions by means of the fluctuation-dissipation theorem and thus evaluate light scattering cross sections [21]. The nonzero elements of the matrix B are listed in the Appen- dix. The expressions involve wave-vector Fourier trans- forms of exchange terms defined by 1 1( ) = 2 cos ( ),zJ J k ck 2 2( ) = 8 cos( /2)cos( /2)cos( /2),x y zJ J k a k a k ck 3 3( ) = 2 (cos( ) cos( )),x yJ J k a k ak (11) and single-site thermal averages (on the spin-up sublattice) corresponding to 2 1 2= , = 3 ( ) 15/4,z z i im S m S 2 2 3 = ( ) 2 ( ) =z z z i i i i i i im S S S S S S S 2 2( ) 2 ( ) ,z z z i i i i i i iS S S S S S S 3 2 2 4 5= 8 ( ) 14 , = ( ) = ( ) .z z i i i im S S m S S (12) The above thermal averages can now be estimated using a modified mean-field theory. 3.3. Thermal averages We use the standard quantum-mechanical representation for the spin S = 3/2 operators in terms of 4 4 matrices: 3 / 2 0 0 0 0 3 0 0 0 1/ 2 0 0 0 0 2 0 = , = . 0 0 1/ 2 0 0 0 0 3 0 0 0 3 / 2 0 0 0 0 zS S (13) One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 181 Following the general approach as used for S = 1 anti- ferromagnets [21], the thermal averages are calculated us- ing an effective Hamiltonian where we adopt a mean-field approximation to simplify the exchange terms but treat the single-ion anisotropy terms exactly. The effective Hamil- tonian for any site on the i-sublattice may be written as the matrix 0 0 eff 0 0 3 3 ( ) 0 3 0 2 2 1 1 0 ( ) 0 3 2 2 1 1 3 0 ( ) 0 2 2 3 3 0 3 0 ( ) 2 2 h D F h D F H F h D F h D (14) where 0 1 2 3= [ (0) (0) (0)]zh S J J J is an effective exchange field. The eigenvalues of Eq. (14) are found to be 2 20 1,3 0 5 = ( ) 3 , 4 2 hD h D F 2 20 2,4 0 5 = ( ) 3 , 4 2 hD h D F (15) and the eigenvectors are 1 23 3 3 \| , > \| > 11 2 2 = , \| > 1 3 11 \| , > 2 2 2 24 3 1 \| , > \| > 11 2 2 = , \| > 1 3 31 \| , > 2 2 (16) in a standard notation. Here we have defined the factors 2 2 0 0 1 = {( ) ( ) 3 }, 3 h D h D F F 2 2 0 0 1 = {( ) ( ) 3 }. 3 h D h D F F (17) As mentioned earlier, in the absence of exchange (i.e., if we set h0 = 0) we would obtain just two sets of degenerate energy eigenvalues which represent two low-lying doublets. The separation between these doublets has been previously estimated [7,10,16] to be within the range 152 to 175 cm –1 . This, together with the data available from light scattering, allows an estimate to be made of the values of the anisotropy and exchange parameters of the spin Hamiltonian. Therefore in Fig. 8 we show a schematic of the energy level splitting produced by the exchange field. The operators 1 12, ... ,X X in Eq. (9) are all linear in a transverse spin component and thus correspond to the transitions labeled as , , and , whereas the operators 13 16, ... ,X X are cubic in a transverse spin component and correspond to the transition labeled as . These mean-field transition energies are dif- ferent, in general, from the energies of the spin waves, be- cause the latter include spin-fluctuation effects absent in mean-field theory. The energy level spacings indicated in Fig. 8 are calculated using Eq. (15) with the parameters J2 = = 3.7 cm –1 , D = –23.6 cm –1 and F = – 42.1 cm –1 . The value of the dominant antiferromagnet exchange term J2 was cho- sen such that the energy of the lowest-lying k = 0 spin wave is 37 cm –1 , in accordance with the light scattering data. We note that the fitted exchange parameters are expected to be different in the S = 3/2 model because of the different role of the anisotropy terms and the different spin quantum number. In the low-temperature limit ( )NT T the static ther- mal average for any operator A is defined as 1 1\| \|A where 1\| is the mean-field ground state eigenfunction. The corresponding spin thermal averages defined in Eq. (12) are then found to have the limiting = 0T values 2 2 1 2 2 2 1 3 = (3 1), = ( 1), 2 1 1 m m 3 5 42 = 2 = 4 3 , = 6. 1 m m m (18) At elevated temperatures the higher energy states of the four-level system become thermally populated and the stat- ic thermal averages must then be evaluated using 4 =1 = < \| \| > exp( / ).i i i B i X X k T (19) This result is employed in Fig. 9 to calculate the tempera- ture dependence (up to TN) of the static thermal averages. Fig. 8. Schematic representation of the four lowest energy levels of the Co 2+ ions showing the effects of the exchange field. The relative energies are calculated using Eq. (15) and the parameters J2 = 3.7 cm –1 , D = –23.6 cm –1 , and F = –42.1 cm –1 , ignoring the small effects of J1 and J3. The transitions (marked as , , , and ), between these energy levels are discussed in the text. J2 = 0.0 J2 = 3.7 cm–1 198.7 cm–1 184.5 cm –1 51.7 cm –1 0 E. Meloche, M.G. Cottam, and D.J. Lockwood 182 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 4. Comparison between experiment and theory Representative numerical results deduced from Eq. (10) for the low-temperature spin-wave energies versus wave vector are shown in Fig. 10, taking = (0,0, )zkk with zk ranging across the Brillouin zone from 0 to / .c The low- est-lying spin-wave excitation is generalized from the tran- sition from the mean-field ground state to the first ex- cited level (see the lower panel), and another spin-wave branch associated with the transition from the ground state to the third excited level (see the top panel) also ex- hibits dispersion. There are two additional spin-wave branches to the spectrum (which are effectively dispersion- less and therefore not shown) that correspond to the transi- tions and in Fig. 8; they correspond to energies of about 132.8 cm –1 and 14.2 cm –1 , but are likely to be ob- served only at elevated temperatures (see later discussion) when there is sufficient thermal population of the higher levels in Fig. 8. In Fig. 10 we show the effect of varying the small intrasublattice exchange 1J on the two excita- tions. We have also included in the lower panel, for com- parison, the single spin-wave dispersion curve obtained using Eq. (4) in the effective spin S = 1/2 model. The nu- merical results in this case are plotted using our optimal set of parameters for the S = 1/2 model, i.e., J1 = –1.2 cm –1 , J2 = 12.9 cm –1 , and (0) =12.0B Ag H cm –1 . For another comparison of the two theoretical ap- proaches we show in Table 4 the spin-wave energies calcu- lated using the S = 1/2 and S = 3/2 models at different points in the Brillouin zone. With these parameters the largest difference for the spin-wave energy E predicted by the two models occurs at the zone edge. Table 4. Comparison of the low-temperature spin-wave ener- gies (in cm –1 ) for different points in the Brillouin zone. Results for the spin S = 3/2 model are obtained with J1 = 0 cm –1 in this example. Spin S Spin-wave branch k = (0, 0, 0) k = (0, 0, /c) 3/2 E 37.2 51.7 E 132.8 132.8 E 199.2 198.7 E 14.2 14.2 1/2 E 37.2 66.0 However, the inelastic neutron scattering measurements [11] for the zone-edge excitation indicate a spin-wave excita- tion energy of 64.8 cm –1 which is comparable to the value obtained with the parameters of our effective spin S = 1/2 model. With the small intrasublattice exchange J1 = 0 the spin S = 3/2 model underestimates the zone-edge spin-wave energy. The energy of the spin wave E at the zone edge may be increased with the inclusion of a nonzero J1 term, as was done in Fig. 10. Thus, setting J1 = –2.0 cm –1 , the spin- wave energy of E at = (0, 0, / )ck becomes 64.3 cm –1 . The effect of the J1 term on the dispersion of the excitation E is seen in Fig. 10(a), where J1 produces a shift in the excitation energy as well as a change in sign of the slope of the curve. In Fig. 11 we plot the k = 0 spin-wave energy E as a function of the reduced temperature for both the S = 1/2 and S = 3/2 models. The theoretical predictions are com- pared with new one-magnon Raman light scattering data. 6 5 4 3 2 1 0 0.2 0.4 0.6 0.8 1.0 T h er m al a v er ag es T T/ N m4 m3 m2 m5 m1 Fig. 9. (Color online) The various thermal averages defined in Eq. (12) versus reduced temperature T/TN. The exchange and anisotropy parameters are the same as in Fig. 8. Fig. 10. (Color online) Spin-wave energy versus wave-vector component kz. The excitation E associated with the transition (a) and low-lying spin-wave mode E associated with the tran- sition (b) are shown. The solid lines are obtained using J1 = 0 (labeled C) and J1 = –2.0 cm –1 (labeled D). The dashed line cor- responds to the results obtained using the effective spin S = 1/2 model. 208 207 206 200 199 198 70 60 50 40 0 0.5 1.0 1.5 2.0 2.5 3.0 (b) D C E n er g y, c m – 1 k cz (a) D C One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 183 Both models are in good agreement with experimental data for temperatures up to / = 0.6.NT T At higher temperatures the decoupling approximations used to linearize the equa- tions of motion are no longer justified. In Figs. 12(a) and (b) we show the excitation energies, calculated using the spin S = 3/2 model, versus temperature for different fixed points in the Brillouin zone. As the tem- perature is increased the energies E and E eventually tend to zero because the splitting within the upper and lower doublets (see Fig. 8), which is produced by the ex- change interactions, decreases with the sublattice magneti- zation. At the mean-field transition temperature the E and E excitations at = 0k still have a small splitting but they become degenerate at the zone boundary as expected. The formal results for the various spin-dependent Green's functions may straightforwardly be obtained by using 1 16= ( ) ,EkG I B b which follows from Eq. (7). From standard relations between the spectral representa- tion of the correlation functions and the Green's function we are able to extract information about the statistical weight associated with the various spin-wave excitations. As an illustrative example we consider the following transverse correlation function 1 1 1 1( ) ( ) = exp [ ( )] ES t S t dE iE t t S Sk k k k ,(20) here subscript 1 refers to the spin operators on the i-sub- lattice. The spectral function in Eq. (20) may then be writ- ten as 1 1 1 1< > = 2[ ( ) 1]Im ; ,E E iS S n E S Sk k k k (21) where n(E) is the Bose–Einstein thermal factor, denotes a positive infinitesimal and the Green's function may be obtained from the solution of the inhomogenous matrix equation. In Fig. 13 we show the spectral function, as defined in Eq. (21), for the various excitations predicted according to the S = 3/2 model. In the low-temperature limit there is no statistical weight attached to the excitations E and E since these modes involve transition between the higher energy states. However, at elevated temperatures (see the dashed lines) we predict a nonvanishing contribution from these excitations. The contribution to the spectral functions from the excitation E is found to be the smallest in this example. In the low-temperature limit the dominant spec- tral features come from the excitations E (Fig. 13(b)) and E (Fig. 13(c)). These modes are associated with excita- tions from the ground state, which can occur even as 0.T At elevated temperatures the excitation peaks shift to lower values due to their dependence on the static ther- mal averages. X YZ Y( ) X Y( )ZX Y XY Z( ) Y ZX Z( ) E n er g y, c m – 1 40 35 30 25 20 15 10 5 0 0.2 0.4 0.6 0.8 1.0 T T/ N Fig. 11. Comparison of theory and experiment for the low-lying k = 0 excitations versus reduced temperature. The solid (dotted) line corresponds to results obtained using the spin S = 3/2 (S = 1/2) model. E n er g y, c m – 1 T T/ N 200 180 160 140 120 60 40 20 0 0.2 0.4 0.6 0.8 1.0 (a) E n er g y, c m – 1 T T/ N 200 180 160 140 120 60 40 20 0 0.2 0.4 0.6 0.8 1.0 (b) Fig. 12. Spin-wave energies versus T/TN for different values of the wavevector: zone-center k = (0, 0, 0) (a), zone-edge k = (0, 0, /c) (b). E. Meloche, M.G. Cottam, and D.J. Lockwood 184 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 5. Excitations for S z = 2 In Sec. 3 we investigated the magnetic excitations asso- ciated with the transitions = 1zS . They were just the one-magnon excitations deduced by studying spin opera- tors that were linear in a transverse spin component. These operators were coupled through the equations of motion to other spin operators that were cubic in a transverse spin operator. However, the linearized equations of motion for these operators did not couple to any spin combinations that were quadratic (or any even power) in a transverse spin component. These latter operators are associated with the transitions having = 2.zS In this section we briefly investigate the properties of the excitations from the mean- field ground state to the second excited state, using the effective S = 3/2 model. We note that the S = 1/2 model is not capable of describing such excitations. To study these modes we start from the equation of mo- tion for 2( ) ;i ES Y and generate all of the other Green's functions coupled to it. As before we do not de- couple the product of operators at the same site. Instead we form additional equations of motion in order to obtain a closed set. The equations of motion can be transformed to a wave vector representation and the set of equations may be written in a matrix form as 14( ) =E B kI G b where 14I is a 14 14 unit matrix, kG and b are 14-component column matrices whose elements are defined as in Eq. (7) with 2 2 2 2 1 2 3 4= ( ) , = ( ) , = ( ) , = ( ) ,i j i jX S X S X S X S 2 2 5 = ( ) 2 ( ) ,z z z i i i i i i iX S S S S S S S 2 2 6 = ( ) 2 ( ) ,z z z j j j j j j jX S S S S S S S 2 2 7 = ( ) 2 ( ) ,z z z i i i i i i iX S S S S S S S 2 2 8 = ( ) 2 ( ) ,z z z j j j j j j jX S S S S S S S 3 3 2 9 10 11= ( ) , = ( ) , = 3( ) 15/4,z z z i j iX S X S X S 2 3 12 13= 3( ) 15/4, = 4( ) 13 ,z z z j i iX S X S S 3 14 = 4( ) 13 .z z j jX S S (22) The nonzero elements of the 14 14 matrix B are defined in the Appendix. Here, for simplicity, we did not include the effects of the intrasublattice exchange interaction 1J . Note that the equations of motion also involve some of the static thermal averages which were defined in Eq. (12). The subset 1 8( ,..., )X X of the operators defined above in Eq. (22) are quadratic in the transverse spin component. They give rise to magnetic excitations (denoted by E and E ) that are associated with the transitions labelled as and in Fig. 14. The equations of motion are also coupled to com- binations of operators involving products of the longitudi- nal spin component .zS The equations of motion for these operators do not vanish because of the nonuniaxial anisot- ropy term in the Hamiltonian. Fig. 13. Spectral function defined in Eq. (21) (excluding the thermal factor 2[n(E)+1]) for the excitations E and E (a), E (b), and E (c). Here we have set k = 0 and = 0.1 cm –1 . The solid and dashed lines correspond to results for T « TN and T/TN = 0.6, respectively. (a) Energy, cm–1 (b) 0.50 0.25 0 7.4 7.6 7.8 136 137 138 0.50 0.25 0 177.2 177.6 178.0 198.8 199.2 199.6 6 4 2 0 23 24 36 37 38 c)( Fig. 14. Schematic representation of the energy levels of the Co 2+ magnetic ions, following Fig. 8. The transitions now labelled as and correspond to the selection rules S z = 2. 198.7 cm–1 184.5 cm –1 51.7 cm –1 0 One-magnon and exciton inelastic light scattering in the antiferromagnet CoF2 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 185 In Fig. 15 we show the magnetic excitations obtained from the determinantal condition 14det ( ) = 0EI B plot- ted versus wave vector .zk The magnetic excitation E (upper two curves) is split at the zone center due to the exchange interaction but becomes degenerate at the zone edge. The splitting of this mode has been observed expe- rimentally [11] and the measured excitations energies at = 0k are 170 and 206 cm –1 , compared with 165 and 203 cm –1 respectively from the theory. At = (0,0, / )ck the observed excitation energy is 190.1 cm –1 , compared with 185 cm –1 from the theory. The dispersionless branch at 147 cm –1 corresponds to the excitation E in Fig. 10. In the low-temperature region the statistical weight attached to this mode is small because it involves excitation be- tween higher energy states. 6. Conclusions In this paper we have investigated the magnetic excita- tions in a spin S = 3/2 anisotropic antiferromagnet with strong spin-orbit coupling. Detailed experimental results are presented for the temperature and polarization depend- ence of the one-magnon Raman scattering in the rutile structure antiferromagnet CoF2. Low temperature results are also presented and discussed for Raman scattering from higher-energy excitons in the ground term. The Green's function equation of motion method was employed to de- rive the excitation energies and spectral intensities over a broad range of temperatures. Results were obtained using RPA for the product of operators at different sites while the single-ion anisotropy terms were treated exactly (with- out using any decoupling scheme) by generating a closed set of coupled Green function equations. The theory was applied to CoF2 and the numerical results were compared with one-magnon Raman light scattering data reported here, as well as other published works. At elevated temper- atures the theory predicts several optical magnetic excita- tions associated with transitions between the higher energy magnetic states. The statistical weight attached to these optical modes vanishes in the = 0T limit. The dispersion and temperature dependences of the lowest-lying excita- tion using the spin S = 3/2 model were also compared to results obtained using a simpler effective spin S = 1/2 model. This work was partially supported by the Natural Sci- ences and Engineering Research Council (NSERC) of Canada. We thank P.A. Moch for helpful discussions, J. Johnson for assistance in the analysis of the Raman spec- tra, and H.J. Labbé for the crystal sample preparation. ____________________________________________________ Appendix: Matrix elements in the S = 3/2 model The nonzero elements of the 16 16 matrix B appearing in Eq. (7) are given by 1,1 2,2 3,3 4,4= = = =B B B B 2 1 1 3 1 3 1(0) ( (0) (0) ( ) ( )) ,J m J J J J mk k (A.1) 1,2 2,1 3,4 4,3 2 1= = = = ( ) ,B B B B J mk (A.2) 1,5 2,6 3,7 4,8 5,9 6,10 7,11 8,12= = = = = = = =B B B B B B B B 9,5 10,6 11,7 12,8 1 1 1 1 = = = = , 4 4 4 4 B B B B D (A.3) 1,7 2,8 3,5 4,6 5,11 6,12 7,9 8,10 5,13 6,14 2 2 2 2 1 1 = = = = = = = = = = 3 3 3 3 2 2 B B B B B B B B B B 7,15 8,16 13,5 14,6 15,7 16,8 1 1 1 1 1 1 = = = = = = , 2 2 3 3 3 3 B B B B B B F (A.4) 5,1 6,2 7,3 8,4 1 3 2= = = = ( ( ) ( )) ,B B B B J J mk k (A.5) 5,2 6,1 7,4 8,3 2 2= = = = ( ) ,B B B B J mk (A.6) 5,3 6,4 7,1 8,2 1 3 5= = = = 6 ( ( ) ( )) ,B B B B F J J mk k (A.7) E n er g y, c m – 1 k cz 200 190 180 170 160 150 0 0.5 1.0 1.5 2.0 2.5 3.0 Fig. 15. Magnetic excitations E (upper two branches) and E (lowest flat branch) involving the selection rules S z = 2 versus wave vector kzc. The excitation energies are calculated using the same parameter values as in Fig. 8. E. Meloche, M.G. Cottam, and D.J. Lockwood 186 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 2 5,4 6,3 7,2 8,1 2 5= = = = ( ) ,B B B B J mk (A.8) 5,5 6,6 7,7 8,8 9,9 10,10 11,11 12,12= = = = = = =B B B B B B B B 13,13 14,14 15,15 16,16 2 1 1 3 1 1 1 1 1 = = = (0) ( (0) (0)) , 3 3 3 3 B B B B J m J J m (A.9) 9,1 10,2 11,3 12,4 1 3 4= = = = ( ( ) ( )) ,B B B B J J mk k (A.10) 9,2 10,1 11,4 12,3 2 4= = = = ( ) ,B B B B J mk (A.11) 9,3 10,4 11,1 12,2 13,2 14,1 15,4 16,3 1 3 3 4 4 4 4 = = = = = = = = ( ( ) ( )) , 3 3 3 3 B B B B B B B B J J mk k (A.12) 9,4 10,3 11,2 12,1 13,1 14,2 15,3 16,4 2 3 4 4 4 4 = = = = = = = = ( ) . 3 3 3 3 B B B B B B B B J mk (A.13) The nonzero elements of the 14 14 matrix B arising in the discussion of the = 2zS magnetic excitations are 1,1 2,2 3,3 4,4 5,5= = = = =B B B B B 6,6 7,7 8,8 2 1= = = 2 (0) ,B B B J m (A.14) 1,5 2,6 3,7 4,8 5,1 6,2 7,3 8,4 1 1 1 1 = = = = = = = = , 4 4 4 4 B B B B B B B B D (A.15) 1,10 2,9 3,10 4,9 2 5= = = = 2 ( ) ,B B B B J mk (A.16) 1,13 2,14 3,13 4,14= = = =B B B B 5,11 6,12 7,11 8,12 9,1 9,3 10,2 10,4 1 1 1 1 = = = = = = = = 4 4 4 4 B B B B B B B B 11,5 11,7 12,6 12,8 13,1 13,3 14,2 14,4 2 2 2 2 1 1 1 1 = = = = = = = = , 3 3 3 3 6 6 6 6 B B B B B B B B F (A.17) 5,10 6,9 7,10 8,9 2 3= = = = 2 ( ) .B B B B J mk (A.18) _______________________________________________ 1. 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One-magnon and exciton inelastic light scattering in the antiferromagnet CoF₂

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