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Influence of magnetic dipolar interaction on one- and two-magnon scattering of light in ferromagnets
Solid State Communications. Vol. 12. pp.839—841. 1973.
Pergamon Press.
Printed in Great Britain
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION ON ONE. AND TWO-MAGNON SCATTERING OF LIGHT IN FERROMAGNETS* Cid B. de Araujo, E.A. Soares and S.M. Rezende Instituto de Fisica, Universidade Federal de Pernambuco. Recife, Brash (Received 9 January 1 973 by R. Loudon)
It is shown that the presence of magnetic dipolar interaction in a ferromagnetic medium changes the polarization selection rules of light scattered by magnons. In addition it provides a new mechanism for two.magnon scattering.
SCATTERING of light by spin excitations in strongly magnetic crystals has received considerable attention in the past few years. It has been established that light can interact with magnons through an indirect electricdipole mechanism combined with the spin—orbit coupling.1’2 This process gives rise to magnon scattering in ferro. fern, and antiferromagnets with particularly interesting polarization selection riiles.2”~The onemagnon scattering is described by terms in the interaction Hamiltonian both linear and quadratic in spin operators.5’6 On the other hand, the two-magnon scattering in ferromagnets has been attributed solely to terms in the Hamiltonian which are auadratic in spin variables.3’5 In the present paoer we show that the magnetic dipolar interaction, which is usually neglected.
where S is the spin operator. ~ and E~ are respectively the positive and negative frequency parts of the electric field operator andK is a linear magneto-optical coefficient. The Hamiltonian (1) can be obtained either from microscopic2 or macroscopic4 considerations. Assume that a static magnetizing field is applied in the z.direction of a Cartesian coordinate system. In the ground state, neglecting the dipolar interaction, all the spins point in the z-direction. Substituting in(l)the second ouantization expansions for the electric field and the spin operators we can write the Hamiltonian in the form 11 = H~’~ + g(2)~where H~’~ = ri
Z
(4)
Z
2 affects considerably some of the estrblished results. First. it changes the polari”.ation sel~ctionrules in onemagnon scattering processes. Second. it is resporsible for a mechanism of two-magnon scattering in ferroniagnets which arises from the Han’iltonian linear in spin variables. compar~b1eto that of the ouadratic terms.
H
=
—
q2
—
)
—
h
(2)
/., ~
\i~
~
H ri
X y
~e1e2
= —
y
tJ,t
X~
e1 e2 )a1 a2
~.
Uk1 ~2
~1Q~2 \1v~)J
k1k2
~(q1 —q2 —k1 +k2)+(l~2)
(1)
(3)
where F
—
Work partially supported by Conselho Nacional de Pesquisas (Brazilian Government) and FORGE Foundation (USA)
a
~ I wavevector 839
Khfrc~q,c,)qp\hI2
/
2N In the exoressions above a 1 is an abreviation for ~ the operator which annihilates a photon with —
*
e2 e1 )a~a2 b~(q1
—(e1e~—e~e~’~Ia1bkc5(ql —q2 + k)] + (1~2) (2)
Consider the isotropic linear ma2neto-optical interaction Hamiltonian 3 ~ ‘~(E~~ X E~)’ S I dr 8ir
—
XI X~
~
—.
1’ 1’
(4)
~ e
q1
4~2\
and polarization A1 (chosen to be linear).
840
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION IN FERROMAGNETS
4
Analogously is the creation operator for photons (q2. A2). bk and b~are the magnon destruction and creation operators with no dipolar interaction. e~ are circular polarizations defmed by e~ = e~±ie~ where the linear polarizations are e~’= ê~.Finally N is the number of spin sites in the crystal, V is the volume, n the refractive index and w~the photon frequency. -
The Hamiltonian H~’~arises from the first order terms in the expansions of S~and S~and describesfrom t2~is obtained one-magnon scattering. The term H the second-order component of S~and describes the change in the Faraday rotation due to spin wave excitation. In the correct description of spin waves in ferromagnets one must include the magnetic dipolar interaction between spins. This interaction leads to a magnon energy manifold and to elliptical rather than circular precession of the spins, and therefore it affects the scattering of light by magnons. It can be taken into account in the theory by means of the Holstein—Primakoff transformation of the magnon operators7 bk
=
u k ck
—
(4)
k -k
where Ck and c~are the new magnon operators and Uk and Vk are parameters used to characterize the spin wave ellipticity.7 and depend on the wavevector k, the magnetic field. the magnetization. and the exchange constant. Substitution of(4) in (2) and (3) leads to =
~ (u~+v~)F ~
—q2—k)—
k
~
Ek
m~/m~ =
(Uk — vk)!(Uk +
vk) is
the ellipticity of the magnon k, whose wavevector we have assumed to lie in the xz plane for simplicity. From (5) we conclude that in the usual experimental configuration for one-magnon Stokes scattering, an exciting light linearly polarized along z gives rise to scattered light propagating in the +z direction which is right-elliptically polarized. It is to be noted that if scattered light is linearly polarized along z, the incident photons propagating in the +z direction must be leftelliptically polarized in the Stokes process. This follows from the careful interpretation of(S) and agrees with the conservation of angular momentum. Figure 1 shows plots of the square of the ellipticity for small k. as a function of the ratio between the internal magnetic field and the saturation magnetization for several angles between k and z. Clearly then the ellipticities of the scattered photons depend on the direction of propagation of the scattering magnons. From (6) we conclude that the presence of the dipolar energy makes it possible to have two-magnon scattering from the linearfrom magneto-optical interaction. This process is different the two-magnon processes considered by other authors2’3’5 which are based essentially on a quadratic dependence of the polarizability on the spin operators. Here a microscopic calcuIation based on the single ion approximation would fail because the important mechanism is the dipolar interaction between spins. For the purpose of comparison we calculate the Stokes Raman susceptibility x~ 2for the dipolar magnon-pair scattering. Its ratio to 5the is value for the d usual Stokes K two-magnon 202 process
—q 2 +k)} +(l~2) (5)
H~2~ =
.f 2 \~2
~
~
XR2 —~
~
Sifl 2Wk
k
(7)
k GM magnetization, WM = 741rM, whereM is>~2 the saturation ~yis the giromagnetic ratio. 0k is the propagation angle
_:~~) F(e
q’q
1 e~— ei e2)a1a2
[Uk1 Vk2Ck1 Ck2
X1 X2 —
q2
—
k1
—
k2)
+
~(q1 —q2 + k1 + k2)J + (1~2)+ (terms in c~1 Ck2 which do not contribute to (6) inelastic scattering)
+ Uk2 Vk1Ck1Ck2 +
polarizations.
Vol. 12, No.9
Here the polarizations have been redefined so that can be recast in the previous form. Now e~,Tk~ — e~±ieke3~represent elliptical rather than circular
of the magnon pair with respect to the z direction. ~ the magnon frequency, and G is a quadratic magneto-optical coefficient. In usual cases K/GM ~ 1. This shows that the dipolar two-magnon scattering is comparable to the quadratic process. provided that the scattered magnons carry dipolar energy (the maxi 0k = n/2). In actual experiments one mum occurs for can differentiate between the two processes by the fact that the dipolar scattering is antisymmetric in the polarizations of the incident and scattered light, whereas the quadratic process is not. The antisymmetric feature is characteristic of the linear magneto-optical interaction.2’4
Vol. 12, No.9
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION IN FERROMAGNETS
841
LO
.~ 0? -
.—~
~. .1~
—
~
•‘~-‘
—.“~‘
—~
7 z~-~--~
/
~
.-.--
.,,__,_.~
0.t5
~5LO
~U~oU4~4LO H/41vM
FIG. 1. Square of the ellipticity for small k vs. the ratio between the internal magnetic field and the saturation Acknowledgements
—
magnetization for various 9k’ The authors acknowledge helpful discussions with Prof. Silvestre Ragusa, Prof. S. Costa Ribeiro.
and all members of the Solid State Group of the UFPe. REFERENCES 1.
ELLIOT R.J, and LOUDON R.,Phvs. Lert. 3, 189 (1963): FLEURY PA.. PORTO S.P.S.. CHEESMAN L.E. and GUGGENHEIM H.J.,Phi’s. Rev. Lett. 17, 84(1966).
2.
FLEURY P.A. and LOUDON R.,Phys. Ret. 166, 5 14(1968).
3.
MORIYA T., J. Phys. Soc. Japan 23, 490 (1967).
4.
LE GALL H. and JAMET J.P.,Phvs. Status Solidi (b) 46, 467 (1971).
5.
LE GALL H., VIEN T.K. and DESORMIERE B..Phys. Status Solidi (b) 47. 591 (1971).
6. 7.
HU H.L.and MORGENTHALER F.R..Appl. Phi’s. Lett. 18, 307 (1971). See for example, SPARKS M.. Ferromagnetic Relaxation Theor-t’, McGraw-Hill. New York (1964).
Ii est démontré que l’interaction dipolaire magnétique en moyenne ferromagnétique change les lois de selection de polarization de Ia lumière diffusée par des magnons. Elle est aussi responsable pour un nouveau mécanisme de diffusion par deux magnons.
Pergamon Press.
Printed in Great Britain
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION ON ONE. AND TWO-MAGNON SCATTERING OF LIGHT IN FERROMAGNETS* Cid B. de Araujo, E.A. Soares and S.M. Rezende Instituto de Fisica, Universidade Federal de Pernambuco. Recife, Brash (Received 9 January 1 973 by R. Loudon)
It is shown that the presence of magnetic dipolar interaction in a ferromagnetic medium changes the polarization selection rules of light scattered by magnons. In addition it provides a new mechanism for two.magnon scattering.
SCATTERING of light by spin excitations in strongly magnetic crystals has received considerable attention in the past few years. It has been established that light can interact with magnons through an indirect electricdipole mechanism combined with the spin—orbit coupling.1’2 This process gives rise to magnon scattering in ferro. fern, and antiferromagnets with particularly interesting polarization selection riiles.2”~The onemagnon scattering is described by terms in the interaction Hamiltonian both linear and quadratic in spin operators.5’6 On the other hand, the two-magnon scattering in ferromagnets has been attributed solely to terms in the Hamiltonian which are auadratic in spin variables.3’5 In the present paoer we show that the magnetic dipolar interaction, which is usually neglected.
where S is the spin operator. ~ and E~ are respectively the positive and negative frequency parts of the electric field operator andK is a linear magneto-optical coefficient. The Hamiltonian (1) can be obtained either from microscopic2 or macroscopic4 considerations. Assume that a static magnetizing field is applied in the z.direction of a Cartesian coordinate system. In the ground state, neglecting the dipolar interaction, all the spins point in the z-direction. Substituting in(l)the second ouantization expansions for the electric field and the spin operators we can write the Hamiltonian in the form 11 = H~’~ + g(2)~where H~’~ = ri
Z
(4)
Z
2 affects considerably some of the estrblished results. First. it changes the polari”.ation sel~ctionrules in onemagnon scattering processes. Second. it is resporsible for a mechanism of two-magnon scattering in ferroniagnets which arises from the Han’iltonian linear in spin variables. compar~b1eto that of the ouadratic terms.
H
=
—
q2
—
)
—
h
(2)
/., ~
\i~
~
H ri
X y
~e1e2
= —
y
tJ,t
X~
e1 e2 )a1 a2
~.
Uk1 ~2
~1Q~2 \1v~)J
k1k2
~(q1 —q2 —k1 +k2)+(l~2)
(1)
(3)
where F
—
Work partially supported by Conselho Nacional de Pesquisas (Brazilian Government) and FORGE Foundation (USA)
a
~ I wavevector 839
Khfrc~q,c,)qp\hI2
/
2N In the exoressions above a 1 is an abreviation for ~ the operator which annihilates a photon with —
*
e2 e1 )a~a2 b~(q1
—(e1e~—e~e~’~Ia1bkc5(ql —q2 + k)] + (1~2) (2)
Consider the isotropic linear ma2neto-optical interaction Hamiltonian 3 ~ ‘~(E~~ X E~)’ S I dr 8ir
—
XI X~
~
—.
1’ 1’
(4)
~ e
q1
4~2\
and polarization A1 (chosen to be linear).
840
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION IN FERROMAGNETS
4
Analogously is the creation operator for photons (q2. A2). bk and b~are the magnon destruction and creation operators with no dipolar interaction. e~ are circular polarizations defmed by e~ = e~±ie~ where the linear polarizations are e~’= ê~.Finally N is the number of spin sites in the crystal, V is the volume, n the refractive index and w~the photon frequency. -
The Hamiltonian H~’~arises from the first order terms in the expansions of S~and S~and describesfrom t2~is obtained one-magnon scattering. The term H the second-order component of S~and describes the change in the Faraday rotation due to spin wave excitation. In the correct description of spin waves in ferromagnets one must include the magnetic dipolar interaction between spins. This interaction leads to a magnon energy manifold and to elliptical rather than circular precession of the spins, and therefore it affects the scattering of light by magnons. It can be taken into account in the theory by means of the Holstein—Primakoff transformation of the magnon operators7 bk
=
u k ck
—
(4)
k -k
where Ck and c~are the new magnon operators and Uk and Vk are parameters used to characterize the spin wave ellipticity.7 and depend on the wavevector k, the magnetic field. the magnetization. and the exchange constant. Substitution of(4) in (2) and (3) leads to =
~ (u~+v~)F ~
—q2—k)—
k
~
Ek
m~/m~ =
(Uk — vk)!(Uk +
vk) is
the ellipticity of the magnon k, whose wavevector we have assumed to lie in the xz plane for simplicity. From (5) we conclude that in the usual experimental configuration for one-magnon Stokes scattering, an exciting light linearly polarized along z gives rise to scattered light propagating in the +z direction which is right-elliptically polarized. It is to be noted that if scattered light is linearly polarized along z, the incident photons propagating in the +z direction must be leftelliptically polarized in the Stokes process. This follows from the careful interpretation of(S) and agrees with the conservation of angular momentum. Figure 1 shows plots of the square of the ellipticity for small k. as a function of the ratio between the internal magnetic field and the saturation magnetization for several angles between k and z. Clearly then the ellipticities of the scattered photons depend on the direction of propagation of the scattering magnons. From (6) we conclude that the presence of the dipolar energy makes it possible to have two-magnon scattering from the linearfrom magneto-optical interaction. This process is different the two-magnon processes considered by other authors2’3’5 which are based essentially on a quadratic dependence of the polarizability on the spin operators. Here a microscopic calcuIation based on the single ion approximation would fail because the important mechanism is the dipolar interaction between spins. For the purpose of comparison we calculate the Stokes Raman susceptibility x~ 2for the dipolar magnon-pair scattering. Its ratio to 5the is value for the d usual Stokes K two-magnon 202 process
—q 2 +k)} +(l~2) (5)
H~2~ =
.f 2 \~2
~
~
XR2 —~
~
Sifl 2Wk
k
(7)
k GM magnetization, WM = 741rM, whereM is>~2 the saturation ~yis the giromagnetic ratio. 0k is the propagation angle
_:~~) F(e
q’q
1 e~— ei e2)a1a2
[Uk1 Vk2Ck1 Ck2
X1 X2 —
q2
—
k1
—
k2)
+
~(q1 —q2 + k1 + k2)J + (1~2)+ (terms in c~1 Ck2 which do not contribute to (6) inelastic scattering)
+ Uk2 Vk1Ck1Ck2 +
polarizations.
Vol. 12, No.9
Here the polarizations have been redefined so that can be recast in the previous form. Now e~,Tk~ — e~±ieke3~represent elliptical rather than circular
of the magnon pair with respect to the z direction. ~ the magnon frequency, and G is a quadratic magneto-optical coefficient. In usual cases K/GM ~ 1. This shows that the dipolar two-magnon scattering is comparable to the quadratic process. provided that the scattered magnons carry dipolar energy (the maxi 0k = n/2). In actual experiments one mum occurs for can differentiate between the two processes by the fact that the dipolar scattering is antisymmetric in the polarizations of the incident and scattered light, whereas the quadratic process is not. The antisymmetric feature is characteristic of the linear magneto-optical interaction.2’4
Vol. 12, No.9
INFLUENCE OF MAGNETIC DIPOLAR INTERACTION IN FERROMAGNETS
841
LO
.~ 0? -
.—~
~. .1~
—
~
•‘~-‘
—.“~‘
—~
7 z~-~--~
/
~
.-.--
.,,__,_.~
0.t5
~5LO
~U~oU4~4LO H/41vM
FIG. 1. Square of the ellipticity for small k vs. the ratio between the internal magnetic field and the saturation Acknowledgements
—
magnetization for various 9k’ The authors acknowledge helpful discussions with Prof. Silvestre Ragusa, Prof. S. Costa Ribeiro.
and all members of the Solid State Group of the UFPe. REFERENCES 1.
ELLIOT R.J, and LOUDON R.,Phvs. Lert. 3, 189 (1963): FLEURY PA.. PORTO S.P.S.. CHEESMAN L.E. and GUGGENHEIM H.J.,Phi’s. Rev. Lett. 17, 84(1966).
2.
FLEURY P.A. and LOUDON R.,Phys. Ret. 166, 5 14(1968).
3.
MORIYA T., J. Phys. Soc. Japan 23, 490 (1967).
4.
LE GALL H. and JAMET J.P.,Phvs. Status Solidi (b) 46, 467 (1971).
5.
LE GALL H., VIEN T.K. and DESORMIERE B..Phys. Status Solidi (b) 47. 591 (1971).
6. 7.
HU H.L.and MORGENTHALER F.R..Appl. Phi’s. Lett. 18, 307 (1971). See for example, SPARKS M.. Ferromagnetic Relaxation Theor-t’, McGraw-Hill. New York (1964).
Ii est démontré que l’interaction dipolaire magnétique en moyenne ferromagnétique change les lois de selection de polarization de Ia lumière diffusée par des magnons. Elle est aussi responsable pour un nouveau mécanisme de diffusion par deux magnons.