Effect of dipolar forces on the light scattering of magnetic systems

Solid State Communications,Vol. 19, pp. 827-831, 1976.

Pergamon Press.

Printed in Great Britain

EFFECT OF DIPOLAR FORCES ON THE LIGHT SCATTERING OF MAGNETIC SYSTEMS E. Anda Departamento de Fisica, Universidade Federal de Pernambuco, 50.000 Recife Pe., Brasil and Consejo Nacional de investigaciones, Cientifica y Tecnica, Rivadavia 1917, Buenos Aires, Argentinai-

(Received 6 January 1976 by R. Loudon) ,The scattering of light by magnons is studied considering the quadratic magneto-optic coupling and the dipole-dipole interaction in ferromagnetic and antiferromagnetic materials. The variation of the intensity of the Stokes and anti-Stokes lines with the external magnetic field and with the angle 0k defined by the wave vector and the magnetization is obtained. The theory is compared with available experimental results in YIG and CrBr3. 1. INTRODUCTION

magnetic systems the linear as well as the quadratic magneto-optic coupling have to be taken into account. DURING the last few years light scattering has been Neither of the studies mentioned above include the shown to be a very important tool for investigating mageffect of dipole-dipole forces on the scattering cross netic excitations in crystals. Most of the theoretical and section. The purpose of the present communication is experimental studies have been restricted to the analysis to study the effect of the quadratic magneto-optic of thermally excited optical magnons in antiferromagcoupling on the cross-section for the case of a ferro an nets. 1,2 antiferromagnetic crystal in the presence of dipolar From these measurements it was shown that the forces. dispersion relation of magnons as a function of magnetic This is an extension of a previous work in which field and incident wavelength and the symmetry of the dipole-dipole interactions were considered, but where scattering cross section were accurately understood. only the first linear magneto-optic effect was taken to However the one-magnon spectrum of for example FeF2 a describe the scattering process. 8 shows that the scattered intensity decreased with temperature whereas the opposite effect has been predicted by a theoretical calculation, which assumes a linear magneto-optic coupling. 2 Recently a study that takes into account the quadratic magneto-optical effect permits an improved agreement between the theory and the measured variation of the scattered intensity with temperature. 4 In the last few years it was possible to detect scattering from thermally excited magnons in ferromagnetic materials like YIG and CrBr3. 5'6 For the case of YIG these measurements show that there is a very important asymmetry in the Stokes and anti-Stokes scattering intensities. The inclusion of quadratic magneto-optical coupling between the incident light and the magnetic system permits here again to explain this anomally and at the same time to obtain a calculated variation of the scattering intensity with the frequency of the laser light that coincides well with the experiment. 7 These considerations constitute evidence that to study the scattering cross section for most of the

2. GENERAL THEORY Following the approach taken by Loudon 2 it is possible to express the Hamiltonian describing the scattering of light by a magnetic system as H

=

Z Z ~3

(1)

R

where et and e s denote the polarization of the incident and scattered photons and X ~ is a spin dependent polarizabilitytensor. The differential cross section of light scattered into a solid angle can be written as d2o d~d03

203103~r/s c~ Vr//(1 -- exp (-- h03/kT) v C'v e 6 x ~ u~Yes I s Img G(X~VX~~*)

nv

(2)

where rO and r/s are the refractive indices at the incident frequency 031 and scattered frequency cos and X]~v is the fourier transform of the polarizability tensor.

? Present address. 827

LIGHT SCATTERING OF MAGNETIC SYSTEMS

828

Vol. 19, No. 9

3. THE FERROMAGNETIC CRYSTAL

.

The Green's function of a magnetic system for small values of k can be obtained studying the response of the magnetization to a ficticious external magnetic field assuming simple classical precessing spins. 8 Taking the geometry shown in Fig. 1 it is easy to obtain for the case of a ferromagnetic crystal the following results for the fourier transforms of the Green's functions

IZ Mo

((s~;s;))~ k =0

X

D

Mo + E_l 1 + _ - 27 l(1 Eo]1--E -- Eo (1 EE---o)E+-~o } (Sa)

Fig. 1. Scattering geometry assumed to calculate the theoretical cross section. In general X°~(R) can be expanded as powers of the spin operators of the magnetic system

k=0

=((S[;Sj-))w - M ° E ~ I k=o 3~Eo

1 o)

--Eo

E fr E (5b)

where

x." = Z K~(R)S~ + E C.~8(R)S~S~ 7

(3)

"/6

where the effect of quadratic terms at different sites has been neglected. The symmetry of the crystal imposes restrictions to the components of the tensors G and K. For a cubic crystal like YIG many components of these tensors are zero. This permits to obtain a simple expression for the one magnon Hamiltonian coming from the terms linear and quadratic in the spin operators. 7 For the case of the hexagonal ionic ferromagnet, CrBra, the Hamiltonian is more involved due to the fact that this crystal has an anisotropy field that can not be neglected and it is less symmetric than YIG. Assuming the geometrical disposition shown in Fig. 1, the result for the Hamiltonian linear in the spin operators is in this case - 2 0 ,K2) H = (/ Z {[(e~r-- ~ e r~(c°s20'K1 + sm i"

+ i(~

- ~ d ) K , I s;

-- [ ( e ~ -- e~etr)(cos20'KI --/(~ -- ~)Kx ] S~}

+ sin 20'K2)

(4)

where 0' is the angle between the direction of the anisotropy field and the magnetization and K~ and K2 are the values of the two groups of components of the tensor K different from zero for the case of hexagonal symmetry. For cubic symmetry Ka = K2 and (4) reduces to the result obtained by Wettling et al. 7 The quadratic expression in the spin operators is much more complicated and will not be considered in detail.

E1 = 7(//0 cos 0 --NzMo cos20 ' + H A ( 1 --~ sha20') + 2~rMo sin 2 Ok)

(5c)

E2 = 7 ( 7rM° sin2Ok-Ha

(5d)

and Eo = 7[//0 cos 0 + ( H A --NzMo) cos20'] 1/~ x (Ho cos 0 --NzMo cos~0 ' + H A

COS

+ 4rrMo sin 2 0k) 1/2

20' (5e)

where the direction of the magnetization Mo is determined by the stability condition 2Ho sin 0 + ( H A - - 41rMo) sin 20' = 0.

(5f)

The expression obtained for the dispersion relation Eo coincides with the result of Sandercock. 6 The Green's functions reduce to their equivalents in a previous work 8 taking HA = 0. For the case of YIG the anisotropy field is small and can be neglected, a situation that is opposite for CrBr3. The evaluation of the cross section requires the knowledge of the Green's functions (5a, b) and of the can type ((S[S~Sf)) and ((SIS:; SFS + j~)). These last two + + be written in terms of the Green's functions ((Si ;S~)) using a decoupting approximation employed by Anderson and Callen. az with these results we are able to calculate an explicit expression for the scattering cross section. For the case of YIG neglecting HA and taking Ut = ~ = e~ = 0 we get for the cross section the expression, d2o

dwd~2 -

hwlw~rlsTMo c 4 Vrll(1 -- e -Eo/kT) ×

LIGHT SCATTERING OF MAGNETIC SYSTEMS

Vol. 19, No. 9

,

10 It) I--

29 -8

~7 z

6

0'

I 60

I

5O

90 OK

Fig. 2. Dependence of the scattering intensity on Ok for YIG. × [(K" -T-pMoG') 2 + (K' + pMoG") 2 + M2op2(G ''2 + G'2)A1 + (K 'z + K"2)Az]

(6a)

where A1

=

--

Mo sin 2 Ok

4rt

(6b) Eo + Ho -- NzMo + 47rMo sin z Ok 7

4nMo sin 2 0k

As

(6c)

Eo - - + Ho -- NzMo 7

and G = G' + iG" = 2G,~ cos20 + (Gll - - G l : ) s i n 2 0 (6d) K = K' + iK".

(6e)

829

high wave length lies above the measured values, situation that worsens when dipole-dipole forces are considered. This is a consequence of the fact that in this case the last two terms of equation (6a) have to be summed-up. This discrepancy could be not too serious because the scattering intensities and the magneto-optic effects were not measured on the same crystal. Measurements of these two physical quantities on the same crystal and the study of the scattering intensity for different values of the angle Ok would be of particular interest. The curves in Fig. 2 show the dependence of the scattering intensity with Ok as calculated according to equation (6a). We used the magneto-optical coefficients measured by several authors. 7 For the case of CrBr3 liglat scattering experiments 6 show that there is no asymmetry in the Stokes and antiStokes scattering cross-section. Sandercock has suggested that it is a consequence of the averaging effect produced by the rotation of the incident polarization by the strong Faraday effect. However even for geometries in which the Faraday effect should be small (Mo almost perpendicular to k) the experimental results for the intensities of both lines were the same. For this geometry the Faraday rotation should be in CrBr3 of the same order as in YIG assuming Mo II k. However the asymmetry of both peaks have been clearly detected in YIG. We believe that the Faraday effect has as its main consequence to separate the incident light to be scattered by the solid into a right-circularly and a left-circularly polarized wave, each propagating with different wave vectors. As a result of this, more than one momentum conservation condition has to be impossed to the scattering process. This corresponds to the scattering of magnons with different k that in general will not be able to be resolved energetically due to the weak dependence of the d., , s l o n relation of magnons with the wave vector in the region near the centre of the zone. If this is the case the Faraday rotation will affect neither the cross section nor its symmetry. If this reasoning is correct the only way of explaining the appearence of an equally intense Stokes and antiStokes lines is to suppose that the quadratic m a g n e t o optic coefficient is small. Taking it to be zero and assuming a back scattering configuration such that et J-Mo and the geometry shown in Fig. 1 we obtain for the Stokes lines the result,

The parameter p appears from the Green's function decoupling and at 0°K has a value Po = (1 -- ~ ) decreasing with temperature to ~Po at T = TN and Eo is given by equation (5e) taking HA = 0 and 0 = 0'. The upper and lower signs correspond to the Stokes and the antiStokes line respectively whose intensities are different in general. The terms in (6a) proportional to A 1 and As are due to the presence of the dipolar interaction. In the case in which Ok = 0 and taking p = 1 the equation (6a) reduces to the result found for the cross section by Cottam et al. 7 As has been extensively discussed by these authors the components of the tensors K and G are closely related to the complex magneto-optical effect; do hcot~sTMo 2 ' [2 the Faraday and the C o t o m - M o u t o n effect respectively. d ~ = 4c 4 Vrb(1 -- e -E°/kT) Icos 0 K1 + sin20'K2 They obtained a good fit between a calculated scattering intensity considering magneto-optic experi[Ho cos 0 + (H a - - N z M o ) cos20] 1/2 sin20k mental data and the intensities measured using a geox (Ho cos 0 - - N z M o cos20 ' + H A cos 20' + 4nMo sin 2 0k) x/2 metry in which Ok = n]2 for different values of the incident-frequency. However the calculated value for

(7)

830

LIGHT SCATTERING OF MAGNETIC SYSTEMS

Vol. 19, No. 9

J

4 Z r~

5

Z W D-

2

50

60

OK

IZ

90

OK=9 0 ~

Fig. 3. Variation of the cross section with the angle 0' between the H A and Mo for CrBr3. and an equivalent result for the anti-Stokes line. Equation (7) permits the study of the variaqon of the cross section intensity with the different geometries if the values of the tensor components KI and K2 are known. Sandercock 6 has found it impossible to observe scattering from magnons for Ho 11H a . For O' = 5 ° the intensity he measured was about 2% of that obtained for 0' = 90 °. He suggested this may be due to destructive interference in the scattered light caused by the large Faraday rotation. However the momentum conservation selection rule fixes the phases so as to obtain a constructive interference. Dillon et aL 11 have studied the magneto-optic properties of ferromagnetic CrBr3. Unfortunately according to the geometry used (Ho 1[Mo) they were able to measure only K2 for which they obtained a very large value but they gave no information about K1. If we assume that it is K2, due to its large value, that it is responsible for the large cross-section and that Ka is at least one order of magnitude less than K2, then it is possible to explain the drastic reduction of the intensity of the line when 0' ~ 0. In Fig. 3 is shown the variation of the cross section intensity with 0' taking K~ to sero. An experimental study of the magneto-optic properties of CrBr3 would be o f great importance to know if all the assumptions taken are correct. 4. THE ANTIFERROMAGNETIC CRYSTAL The Green's function for the case of an antiferromagnetic crystal as FeF 2 including dipolar forces have been extensively discussed in a previous work. 8 The evaluation of the cross-section predicts the appearance o f two Stokes and anti-Stokes peaks that corresponds to the two magnons present in the antiferromagnet. Using equations (2) and (3) the results for the Green's

2 Z hi F'Z

r=l

I

60

70

80

90

I00

Ho ( KGA USS)

Fig. 4. Variation of the scattering intensity as a function of the external magnetic field for the antiferromagnet FeF2 for different values of Ok and r = 2MoG+/K ÷. functions mentioned and the decoupling approximation proposed by Anderson et al. }2 and taking the external magnetic field Ho = 0 the cross-section for the in-phase scattering results, do[

=

h~o/~o~r/sTMo

~i

c4VnA 1 - e~VkT") IK+aiH~/2 +-pMoG+~i(Hi + 4n sin 20kMo)l/212 X

HIA/2(Hi + 4n sin2OkMo) I/2

where

(8a) (8b)

E i = 7HIA/2H~/2 H]

87r = H A + 2He + 4n sin 20kMo -- -~-Mo

(8c)

112

8n = H A + 2He--4nsin2OkMo--~-Mo

(8d)

and a i and/3 i are such that o~, + io~ = ~

- ~;~s

(8e)

~, + i~2 = ~ e ; + c;~s. (80 The upper and lower sign refers to the Stokes and anti-Stokes cases and H a and He correspond to the anisotropy and exchange field respectively. The effect of the dipolar forces is to split apart the two magnon frequencies E i which have different scattering intensities. This situation disappears when Ok = 0 in which case

LIGHT SCATTERING OF MAGNETIC SYSTEMS

Vol. 19, No. 9

dipolar forces have no effect. However for all values of Ok its influence is small for the case of FeFe and MnF2 because He >>Mo. The experimental detection of dipolar forces depends upon the possibility of resolving the two peaks that are split apart nearly -~cm-1. When Ok = 0 equation (8a) coincides with the result given by Cottam 4 and supposing G44 = 0 it is equal to the cross-section obtained in a previous work. 8 When Ho 4= 0 and assuming that the magnetic susceptibility is small such that Xii; Ho ~ Mo and Ho >> (nHoMo) 1/2 conditions that for the case of MnF2 are fulfilled for Ho > 103Oe and T < 60°K, the transition temperature being 67.7 K we obtain for the cross-section do[ = - ~ Ei

hcozw~r/sTMo c4VrlI( 1 -- e ~EilkT)

IIK+alH~/2+_pMo2G+{3tHI/212 H~/ H1

× ~

27r sin2 0kMo H o + ( - - 1)i(HAH1) 1/2

where E i = "/[H O + (--

1)i(HAH~) 1/2 ]

(9b)

and where we have assumed a geometry such that a2 = /32 = 0. The upper and lower signs as before refer to the Stokes and anti-Stokes lines.

831

The second term in equation (9) due to dipoledipole interaction is small for most antiferromagnets. However when the external field Ho approaches the value (H~H1)1/2 (The flopping field) the contribution of pole E1 to the scattering coming from this term is the dominant one due to the near cancellation of its denominator. In Fig. 4 the variation of the scattering intensity is shown as a function of the external magnetic field Ho and for different values of 0 k and the relation r = 2MoG+/K*. Present experimental results of the magneto-optic properties of MnF2 are not able to determine the value of r. The detection of the behaviour shown in Fig. 4 is within the range of Brillouin scattering due to the fact that E t is small. For the case of the out-of-phase scattering an equivalent expression to (9a) is obtained where K + and 2pMoG* appear replaced by 2pMoG- and K- respectively. Unfortunately up to the moment it has not been possible to stablish tile importance of the inphase scattering and out-of-phase scattering for FeF2 and MnF> A similar behaviour of the intensity of the line E~ shown in Fig. 4 can be obtained as a function of Ok and the temperature for Ho near the flopping field. The study of these effects requires further scattering experiments to be done.

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2.

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5.

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