SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL AND RELATED EFFECTS

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SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL AND RELATED

EFFECTS

H. Le Gall

To cite this version:

H. Le Gall. SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL AND RELATED EFFECTS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-590-C1-598.

�10.1051/jphyscol:19711203�. �jpa-00214028�

JOURNAL DE PHYSIQUE Colloque C I, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1

-

⁵⁹⁰

SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL AND RELATED EFFECTS *

H. LE GALL

Laboratoire de Magnetisme, C . N. R. S., 92 Bellevue, France

Resume. - La thkorie gentrale des interactions spin-photon dans un systeme dYClectrons couples par un champ d'echange est reprise en introduisant le formalisme de seconde quantification: Dans la gamme des basses frkquences (cm, mm IR lointain) le couplage dipolaire magnetique induit une absorption B un photon et une diffusion (( magnktique 1)

B deux photons. Dans le premier cas, on montre que pour de faibles taux de relaxation, le rayonnement dipolaire magnt- tique des magnons uniformes peut produire des processus inverses magnon-photon avec des probabilitks de transition si tlevks dans certains cas que tous les photons incidents sont reflechis par les magnons. Dans la gamme optique, Zi cause des couplages dipolaires magnttique et klectrique, les cristaux magnktiques sont bigyrotropes et les interactions spin-photon sont des diffusions ti deux photons. Nous montrons que les differentes interactions magnktooptiques qui apparaissent genkralement comme des effets distincts sont en fait definis a partir d'HamiItoniens ⁽⁽ magnetique n et

(( electrique ¹⁾ d'interaction spin-photon. Les relations physiques et analytiques entre les diffusions magnktique et tlec- trique d'une part et les diffusions klastiques (effets Faraday, Kerr et Cotton-Mouton) et inelastiques (effets Raman 9 1 et 2 magnons) d'autre part apparaissent clairement. A partir des constantes magneto-optiques dtduites des expkriences, nous dkterminons les susceptibilites Raman a 1 et 2 magnons. Les origines microscopiques des interactions spin-photon sont discutbes A partir des perturbations dues aux couplages spin-orbite et &change qui sont associts respectivement aux mod6les a 1 et ² spins ioniques. On montre que la diffusion elastique due a la paire de spins ioniques peut produire un effet Cotton-Mouton anormalement ClevB.

Abstract. - The general theory of the spin-photon interactions in systems of electrons coupled by exchange fields is reviewed from the second quantization formalism. In the low frequency range (cm, mm, far IR) the magnetic dipole coupling induces one-photon absorption and two-photons ⁽⁽ magnetic )) scattering. In the fist case, we show that for low relaxation rates, the magnetic dipole radiation of the uniform magnons can produce reverse magnon-photon processes with transition probabilities so high in some cases that all the Incident photons are reflected by the magnons. In optical range, due to both the magnetic and electric dipole couplings, the magnetic crystals are bigyrotropic and the spin-photon interactions are two-photons scatterings. We show that the different magneto-optical interactions which generally appear like distinct effects are in fact defined from a ⁽⁽ magnetic ⁾⁾ and an ^u electric ⁾⁾ spin-photon interaction Hamiltonians. The physical and analytical relations between the magnetic and electric, and the elastic (Faraday, Kerr, Cotton-Mouton effects) and the inelastic (1- and 2-magnon Raman effects) scatterings appear clearly. From the magneto-optical constants deduced from experiments we determine the one- and two-magnon Raman susceptibilities. The microscopic origins of the spin- photon interactions are discussed from the spin-orbit and exchange perturbations associated with the one and two ionic spins models. It is shown that the elastic scattering due to the pair of ionic spins can produce an anomalous high Cotton- Mouton effect.

I. Introduction. - In this paper the general theory of the spin-photon interactions in systems of electrons coupled by exchange field is reviewed from the second quantization formalism. As shown in figure 1, the

-elastic and Inelastic 2-photon xatterlng -gyroelectrlc propertlest

(Faradav . C M . Rarnanq e m

electron~c orbltal ' @ a e + x :

---- st es excltatlon

'

^wFd!lS

etec tpozt t ran?,-e-r--

:-

~~~~~~~~~~~c~~~~

; IR rnlcrowaves

I I I , A

162 163 164 165 (p:m)

001 01 1 10 (cm)

properties ,_one-photon absorpt~on

(state) ' I one-photon absorption ferromagnet~c resonanpe I%xchange r e s o n e e *

and lnelastlc 2 photon scatter~ng - gyromagnet~c (Faraday, C - M ,Raman) properties

FIG. I . - Electric and magnetic spin-photon interactions in a magnetic crystal.

material. The magnetic and electric dipole couplings induce one-photon absorptions and two-photon scatterings with transition probabilities which depend on the radiation frequency. In the section I1 we ana- lyze the magnetic dipole transitions which induce the parallel and perpendicular microwave pumpings at low frequencies and the elastic and inelastic two- photon scatterings at low and high frequencies. In ferromagnetic system with low relaxation rates we show that the magnetic dipole radiation of the uni- form magnons induces reverse magnon-photon pro- cesses with transition probabilities so high in some cases that all the r. f. photons absorbed by the magnons are reradiated. In the near I. R. and visible bands we determine the contribution of the gyromagnetism t o the magnetic Faraday and one-magnon Raman scatterings. In the section 111 we show that the diffe- rent magneto-optical interactions induced in the opti- cal range by the electric dipole transition, and which generally appears in the literature as distinct effects, could in fact be defined from a common snin- hoto on L .

interaction Hamiltonian. We can separate these inter- radiation-crystal interactions can be separated bet-

actions into two distinct groups associated with either ween the magnetic and electric dipole couplings which an elastic (Faraday, Kerr, Cotton-Mouton or Voigt produce an energy transfer from the electromagnetic

effects) or an inelastic ( I

-

and 2-magnon Raman field t o the localized (nuclei, electrons, spins) and

effects) scattering of light by the spins. From the collective (phonons, ~lasmons, magno's) States of the

magneto-optical constants deduced from

we2eterrnine the 1- and 2-magnon

ama an

^suscepti-

* This work was supported in part by the DRME (Paris). bilities. In the section IV we discuss the microscopic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711203

SPIN-PHOTON INTERACTIONS IN MA( 5NETIC CRYSTALS, MAGNETO-OPTICAL C 1 - 591 origin of the spin-photon interactions from the 1

and 2- ionic spins models. From the time-reversal symmetry of the magnetic system we deduce that the even and odd magneto-optical effects are associated respectively with the diagonal and off-diagonal components of the dielectric susceptibility tensor.

11. Magnetic dipole transitions. - The Hamilto- nians of the magnetic and electric dipole interactions are given by the fundamental relations :

X , = -

1"

h . ~ d u ; Je. = -

The electric and magnetic fields operators are defined in the second quantization formalism by the creation and destruction photon operators b l and b, such that [I] :

E(r, t) = E-(r) ei""q'

+

herm. conj. ;

4

h(r, t) =

C

(h)-(r) eimq'

+

h. c . ₍₂₎

4

where (e, = ex

+

ei8 e,) is the complex polarization vector of the photon Rw, having the propagation vector q. s = q/l q I and i is the refractive index of the crystal.

11.1. ONE PHOTON ABSORPTIONS-MAGNETIC DIPOLE

RADIATION. - In the microwave range, the magnetic dipole interaction (1) can be separated into a longitu- dinal and a transverse component such as

HII = - h, M, and

Hl = - *(hf M -

+

h- M+)

associated respectively with the parallel and perpen- dicular pumpings, and where M* and M, are defined from the magnons operators ck+ and ck by :

If the magnetic dipole energy of the spin-wave is not neglected then the use of (4) in HII gives two differents processes [2]. The first one defined by the operator ck, ckz bq is the absorption of one photon by one

+

magnon. The transition probabilities of 'this Raman type process being very low are negligible in a paral- lel pumping experiment. In the second process defined

+ +

by the operator ck1 ckz b4, one photon is absorbed and two magnons are created. Since in pratical cases q .v 1 cm-' and k lo4 - 10' cm-', k,

-

^{k , and}

the two emitted magnons have equal but opposite propagation vector. In this case the photon-magnon Hamiltonian is reduced to :

where @, depend on the magnetic dipole energy of the magnons. As shown by Schlomann [3] and Morgenthaler 141, the parallel pumping becomes instable when the creation rate of the magnons is higher than their relaxation rate. The 3-quanta non- linear transition becomes now parametric with the critical threshold number of photons given by ni = (hqk/4 I @, where qk is the relaxation of the spin-wave k. The parallel pumping can excite coherent spin-waves with propagation vector selected by only applying the appropriate amplitude of the d. c.

magnetic field. The relaxation rate as a function of different parameters (ak, k, T etc ...) is easy to study from the measurements of the threshold which is proportionnal to q:. With this method it has been possible to verify the 3- and bmagnbn relaxation theory [5, 61 and to study the influence of the 2- magnon transitions, due to the inhomogeneities, on the 3-magnons interactions [7] and the influence of an oblique pumping on the spin-waves relaxation [8].

Due to the non-linear mechanism of the parallel pumping, the longitudinal susceptibility is lower than that for the perpendicular pumping (;C;' < 0.3 in YIG).

From equations 1-4 we obtain the transverse interaction Hamiltonian :

X, = -ihA(b, ^c,f - b,f c,) with A = (om ~ 1 4 ) " ( 6 ) where om = gy 4 nMO and which shows that the per- pendicular pumping can excite only uniform magnons.

At low excitation levels the response of the system is given by the well-known magnetic susceptibility tensor of Polder.

Xk

has high values for low line- width sample

&: --

700 in YIG, but

X:

^{= 1 with}

AH = 500 Oe). Above ti critical threshold X" decrea- ses due to the anharmonic evolution of the system which induces 3- and 4-magnons parametric transi-

+ + ^k +

tions such as c, ck c - ~ and c, c, c + c - ~ . These pro- cesses have been analyzed in detail both from a theoretical and experimental points of view 19-111, and thus are not extended here. According to Glogs- ton et al. 1121, the linear damping of the ferromagnetic resonance is mainly due to the surface and volume magnetic dipolar inhomogeneities. Le Gall and Desor- rnikres [ l l ] have shown that in the samples of low line-width a new damping mechanism appears, due to the dipolar radiation of the uniform magnons which produces reverse photon-magnon transitions. In this case the magnetic component of the radiation is h, = - i(Pl ^C, -

PT

c i ) where the magnon operator co and the constant p , depend respectively on the uniform mode relaxation rate q, and the filling factor of the sample in the microwave structure. In a typical experiment where a sample of volume V is situated at the center of a waveguide of dimension a, b at a distance I form the shorted end, the perpendicular pumping Hamiltonian (6) must be modified as : XiJ=

-

ih[(ctb, - PC,) c i - cO(ctb: -

p*

c l ) ] (7) where a = (om,)% cos kg I and

p

⁼ (kg Vo,/ab) cos kg I exp

-

ik, I

correspond respectively to the direct and reverse processes. kg is the propagation vector in the wave- guide. The equation of evolution of the magnon operators show the important modifications introdu- ced by the dipolar radiation on harmonic and anhar- monic, transient and steady-state evolutions of a spin-waves system [ll]. The reverse processes produce a resonance frequency shift and an additional damping qR such as :

coL = (coo

+

(1/2)Po qo sin 2 kg I ) ;

)?L = VO f UR = ~ 0 ( l -k

PO

cos2 kg 1) (8) with

Po

⁼ kgVwm/abqo. The experiment gives the total relaxation rate and as shown by Desormicre [13], it is not possible to obtain the intrinsic value q, if the dipolar radiation is not considered. The propor- tion of the incident fieId reflected by the uniform magnon is given in the harmonic case by the following radiation rate ⁾⁾ y = qR/(qO

+

qR). It is instructive to compare in the figure 2 this rate in YIG spheres as

Flo. 2. - Radiation rate of the uniform magnons in a YIG sphere.

a function of their intrinsic relaxation rate and their diameter. In some cases (AH, ⁼ 0.3 Oe and d > 1 mm) the dipolar radiation is so high that the crystal beco- mes a magnetic mirror for the incident electromagnetic field : all the photons absorbed by the uniform ma- gnons are reradiated. In the anharmonic case, the dipole radiation modifies the 3-magnon parametric threshold such that :

where the intrinsic threshold hr' can be deduccd from the effective threshold h, perturbed by the radiation.

The magnetic dipole radiation is the basis of the fil- ters, limiters and microwaves oscillators [l l].

11.2. ELASTIC A N D INELASTIC 2-PHOTONS SCATTER- INGS GYROMAGNETISM I N THE OPTICAL BAND.

-

The para and ferromagnetic resonances are mainly devoted to the study of the relaxation processes of the magnetic system from the analysis of the anti-hermitian part of the magnetic susceptibility tensor. With the help of the preceding section we conclude that the magnetic resonance can be described actually by a two-step spin-photon transition. The first step leaves the system in an excited state after the absorption of one photon (transition, for example, between the Zeeman sublevels of an orbital groundstate in a paramagnetic ion). In the

second step the relaxation of the excited state appears through either non-radiative transitions due to spin- spin and spin-photon or magnon-magnon and magnon- phonon interactions, or through radiative transitions in which one photon is reemitted. The radiative pro- cesses described by the Hamiltonian

%= - i h [ ~ * ^co b t

-

Bbq c i ] with B=aP/(qo+qR) (10) is fundamental to explain the physical relations bet- ween all the spin-photon interactions produced by the magnetic dipole coupling. We have to consider two limiting cases. First when qo 9 qR the relaxation is mainly due to the non-radiative transitions and the spin-photon interaction is reduced to the one-photon absorption, which is the case of the usual para and ferromagnetic resonances. Second, when q ,

<

q,, two-photon scatterings arise due to the radiative pro- cesses. If 1

<

d, where d is the dimension of the sample, there is no propagation and we observe an elastic scattering of the r. f. field as in the preceding section. If I. %- d the electromagnetic field can propa- gate in the material and we obtain either an elastic scattering (Faraday effect) or an inelastic scattering (Raman effect). The absorption and first-order scatter- ings are desccibed respectively by the first and second order time dependent theory of perturbation. These properties are deduced from the equation of evolution of the total magnetization

in the effective field

where Hz contains the anisotropy, exchange, dipolar, electromagnetic and applied d. c. fields. Due to the exchange field, the k-dependent magnetic susceptibi- lity tensor is deduced from the induced magnetic dipole by m,(k) = x;(k) hj(k) such as :

In the microwave range, the components associated to i, j = 3 can be neglected and the tensor is diagona- lized with the only non-vanishing circular compo- nents

X?

⁼

+ x'&)

f ik72 - x?I)]/~ and

xTZ.

In addition, in the isotropic materials (as cubic or uniaxial with the propagation along the c-axis)

m

X I 1 = XI;;, ~ 7 2 = - X?I and

X?

^{= f} xL). Diffe- rent values of the susceptibilities ,y+ and X-, associated with the equivalent right and left circular components of a linear polarized field, produce the gyromagnetism which gives the ferromagnetic resonance and the microwave Faraday effect defined by

The non-reciprocal properties of the gyromagnetism are the basis of the r. f. isolators, gyrotors and circu- lators. The transfer of energy from the radiation to the magnetic system is given by the anti-hermitian part

SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL C 1 - 593

xt

of the susceptibility tensor which has a maximum proportionnal to

qkl

at the resonance. The hermitian part

$

gives the dispersive processes like the Faraday effect which has a minimum near the resonance [14].

At high frequencies ( o 9 o k , o,), the diagonal and off-diagonal components decrease respectively as o-, and o-' for the hermitic part and respectively as o-' and o-, for the anti-hermitic part [15].

Then by increasing frequencies, the dispersives pro- cesses dueto the off-diagonal components decrease less than the associated relaxation effects.

From the equation of evolution of the magnetic system, we deduce that a magnetic dipole m(wl) can be again induced by the magnetic component of a visible or infrared light such as :

Since the diagonal components of X; are zero in the first approximation in the optical band, most of the authors use a permeability ,u(opt.) = 1. But the sus- ceptibility deduced from (1 3)

shows that this approximation is not valid for the off-diagonal components which produce a gyromagne- tism in the optical range. 1% and

xy3

are associated with the transverse components of the magnetization and represent the one-magnon Raman scattering.

xI2

gives the specific Faraday rotation cp, due to the magnetic dipole rotation (the anti-hermitian part has been neglected in 14) cp, = 2 nsic-' gyM, which is always positive. As shown from experiments by Krinchik and Chetkin [16], the spin resonance pro- duces the same magnetic Faraday rotation in the microwave and optical bands since cp, is independent of frequency. From the calculation of x;, Wangsness [17] has derived the rotation for a system having different magnetic sublattices :

Ma is the algebraic value of the magnetization of the sublattice a having the spectroscopic splitting factor ga. In the case of a two-sublattice ferrimagnetic crystal

cpm = 2 RZC-' y(gl M I - gz M2) or 1181 :

The first and second terms in 15 are due respectively to the ferrimagnetic and the Kaplan-Kittel exchange resonances. The exchange resonance corresponds to the motion of the spins in the exchange interaction field of the two sublattices and appears at the fre- quency oex ⁼ ay(gl M I - g2 M2) where a is the mole- cular field coefficient. With exchange field about lo6 Oe, wex correspond to radiations between 1 = 50

and 1 000 microns following the values of M,,, and g,,,. q E increases with the intensity of the exchange resonance which is proportionnal to Cg,

-

^g2)2.

When g l ⁼ g,, the exchange resonance vanishes and cpm is reduced to cpk as in some ferrimagnetic garnets as YZG. Due to the difficulties of experiments in the far I. R. it is not easy to detect the exchange resonance, unless the measurement is performed near the compen- sation temperature where a,, is shifted towards the microwave range with MI

--

M2. Since the quantities MI,, and n can be determined from conventionnal experiments, it is possible to separate the two contri- butions in the total rotation (15). Then we can deduce the amplitude, anisotropy and temperature depen- dence of the factors ga of the rare-earth and transition ion contained in the ferrimagnetic garnets as for example Ho,Fe,O,, and Er3Fe,012 [16]. This shows the possibility of analyzing the atomic structure of the magnetic ions in solids by the Faraday effect.

The Hamiltonian of the spin-photon interactions Xm in the optical band is obtained from

Hm = - (mh*

+

h. c.)/2

where m is the induced magnetic dipole given by equation (13) :

We separate this Hamiltonian into a longitudinal X:

and a transverse (Jek

+

X;) components associated with specific magneto-optical interactions The quan- tum description of the gyromagnetic effects and their mutual relations appear clearly in the optical band if we introduce in (16) the magnon and photon operators associated to M and h. For the << longitudinal >> transi- tions we obtain [19-201 :

x b1 b: A(q, - q,)

+

^{h. c.} (17) with F;, (mag) = ngp, Mo/ii2 V, and where the spin fluctuations have been neglected (M, 21 Mo). This Hamiltonian induces elastic spin-photon scatterings which are purely antisymmmetrical as show by the polarization factor. Indeed, one photon absorbed with a polarization e, is reemitted with a polarization ey, whereas one photon absorbed with a polarization ey is reemitted with a polarization - ex. Hcnce, in these transitions the polarization vector of each photon is rotated by n/2. With these process a magnetic field polarized at n/2 of the incident field increases along Oy, whereas the incident field decreases along Ox.

The total field h, = h,

+

^hy is rotated by an angle cp, which is the Faraday rotation obtained from a quantum phenomenological description. I t is a trial to deduce the value of the rotation from the Faraday Harniltonian (17) and the perturbation theory. A second property of the gyromagnetism appears with the ⁽⁽ transverse ⁾⁾ Hamiltonian

such as :

C 1 - ⁵⁹⁴ H. LE GALL X i =

C

'VF?,(mag) [(el, e,*, - el, e2,) x

4142k

x bl b: a: A(q1 - q2 - k)

+

(q1 ^-P q2)1(18) with F i x (mag) = (ngp,/i Ti2 V ) (gp, Mo w2/2 ol V)%.

The expression of R? is obtained by changing in 18 ^a: by a, and e, by - el. These Hamiltonians induce inelastic spin-photon scatterings with respectively the creaction (Xi) and the destruction

( ~ 3

^{of one}

magnon k in the crystal. They correspond to the first-order spin-Raman scattering due to the magnetic dipole coupling for the Stokes and the anti-Stokes cases. The polarization rules for the Faraday and the Raman scatterings are different but they keep the antisymmetrical properties associated with the gyro- magnetism. The polarization rule of the magnetic Raman effect is due to the conservation law of the total angular momentum of the photon-magnon system. In the following section we derive the first- order spin Raman susceptibility in which the magnetic and the dielectric part will be separated.

111. EIectric dipole transitions. - When ho,

-

^{E, - E,}

where 1 a

>

and 1 ^I

>

are the ground and excited orbital states of the bound electrons (optical band), the coupling of the electric component E, of the radiation with the electrons induces electric dipole transitions. The spin-photon interactions are described by a three-step transition with the interme- diate state due to the spin dependent permitivity ell or dielectric susceptibility

xCj

tensors such as eU ⁼ hiJ

+

4 nX;l. We start from the dielectric energy per unit volume V = - (8 ^z)-I E? ~~~(l'vl). Ej

+

h. c.

where i, j = x, y, z and E, and Ej are the complex components of the optical field. can be always separated into an hermitian (e:) and an anti-hermitian part E$ associated respectively to the dispersion and absorption. From this we can distinguish the Faraday effect from the circular dichroism and the Cotton- Mouton effect from the linear dichroism. In what follows we limit the discussion to the dispersives effects by assuming a quasi-transparent crystal (e$ ⁼ 0). As shown by Moriya [21], the electronic polarizability depends on a single ionic spin, a pair of ionic spins and so on, such as

xc

⁼

C

xe(a)

+ ¹

^{xe(a, b)}

+

a a b

with :

In the ferromagnetic systems the magneto-optical interactions are due to the influence of the spins on the excited orbital state of the electrons. In (19), this influence is mainly attributed (through the linear and quadratic spin component terms A , and B,,) to the spin-orbit coupling of one magnetic ion (see sec-

tion IV). On the other hand, being due to the exchange interaction xe(ab) contains only even terms of spin components from two different ions. The polariza- bility which depends on a pair of ionic spins cannot gives first-order transitions as the Faraday and the one-magnon Raman scatterings. The following dis- cussion is restricted to electric dipole transitions asso- ciated with the one ionic s ~ i n model. If the real and imaginary parts of the hermitian tensor are separated we obtain a symmetrical tensor ^&IS and an antisymme- trical tensor eNAS of rank two equivalent to an axial vector g dual of this tensor such as eytS = eijlgl where eij, is an unit tensor of rank three. The electrlc induction can now be written as [20] :

From the symmetry properties of the spin dependent susceptibility (q,(M) = cji(- Mj) [22], we deduce that E' and en (or g) can be expanded following respec- tively even and odd increasing powers of the magne- tization components. So the odd and even magneto- optical Hamiltonian are :

JC~') = - i(8 a)-'

1

(E; x E,,).g

+

h. c.

and 1.2

3e62) = - (8 n)-'

1

E,*, e' E,,

+

h. c.

1.2

By expanding g to the first-order magnetization com- ponents we obtain, in isotropic materials, the electric dipole moment induced by the incident field

which gives the susceptibility tensor :

We note the similarity of the susceptibilities tensors associated, in the optical range, to the gyroma- gnetic and gyroelectric properties (eq. 14 and 22).

The electric Faraday rotation is given by

This last equation and the equation 15 give the total Faraday rotation which contains a cc dispersive >>

(qe) and a ⁽⁽ non dispersive ⁾⁾ (cp, is independent of o ) contributions. As for the magnetic transitions, the first-order electric interaction Hamiltonian can be separated into longitudinal and transverse parts. So, by using the second quantization formalism, the electric Faraday and one-magnon Raman effects can be described by the Hamiltonians 17 and 18 by sim- ply substituting the constants F:, (mag) by F:, (elect) ⁼ hoffzM,,/4 Vn2 and F:, (mag) by F;, (elect) = (2gpg Mo Val w2)% hfL/8 n1 Ti,

v2

^[19-201.

These Hamiltonians give the similar quantum descrip- tion, the polarization rules and scattering susceptibi- lities for the gyroelectric and gyromagnetic phenomena.

The electric Faraday and Raman scatterings are determined from different components f:i of the linear magneto-optical constants tensor, but in iso-

SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS. MAGNETO-OPTICAL C 1

-

595 tropic materials

fizz

= f i x , = f,, so we can predict

the theoretical value of the Raman susceptibility from the specific Faraday rotation deduced from experi- ments. Indeed we define a Raman susceptibility

x:

from the light power P, diffracted per unit active volume by P, = (112) o,

x i

E' = WR hw,/V, where W, is the Raman transition probability per unit time and E is the amplitude of the electric component of the incident radiation. By using the golden rule, we have obtained [19-201 the one magnon Raman susceptibility induced by the electric dipole coupling, for both the Stokes and the anti-Stokes processes such as :

Typical values of

x i

about lo-'' and are obtained in the near I. R. in ferrimagnetic garnets with spin-wave excited respectively by thermal pro- cesses (?I,

-

3 000 close to the Brillouin zone center) and microwave non-linear processes like perpendi- cular or parallel pumpings (n,

--

10" per mm3).

The substitution of cp, by cp, in (23) gives the one- magnon Raman susceptibility due to the magnetic transitions. Thus for the same magnon state k, the ratio of the magnetic and electric Raman susceptibi- lities is

xFIX;

⁼ (cpm/qoc)2. The values of this ratio at room temperature in YIG are 0.01, 0.06 and 0.8 for the wavelenghts 0.8, 1.15 and 4 microns respectively.

Thus as predicted by Bass and Kaganov [23], in the infrared range where the gyromagnetism gives the highest contribution to the Faraday rotation, we can expect an important inelastic scattering of light by the magnetic transitions. An increase of the spin- wave-light interaction as a function of the photon and magnon populations has been observed by us, using coherent magnons excited by parallel pumping a t 10 GHz and interpreted in term of the inelastic spin-photon scattering [24].

The even-orders magneto-optical interaction Hamil- tonians

x?)

are obtained from the expansion of

&ij

following increasing even powers of the magnetiza-

tion components.

x?)

contains terms of the form fh; E ; * ~ E , ~ where f:; is one of the 81 components of the four-rank magneto-optical constant tensor. In the second quantization formalism, three typical terms associated to M:, Mx M , and M: (or M ~ ~ ) appear.

2 z z

M Z f2,,,,, produce the elastic and inelastic Cotton- Mouton (or two-magnon Cotton-Mouton effect) scatterings from the operators b, b l and b, b i a , a:

respectively. M , M, f;i; produces the one-magnon Raman scattering for the Stokes (b, bf ^a:) and the anti-Stokes (b, b l a,) cases. M: fEZ produces the two-magnon Raman and the Two-magnon Cot- ton-Mouton effects. The elastic Cotton-Mouton scat- terings have generally different transition probabili- ties for incident radiations polarized parallel and perpendicular to M,, which gives the well-known linear birefringence. The inelastic Cotton-Mouton effect corresponds to the scattering of one magnon

by one photon. This second order effect associated with the longitudinal magnetization component has low transition probabilities and can be neglected.

The time dependent perturbation theory gives the two-magnon Raman susceptibility which becomes for the Stokes case [32] :

As observed for elastic Faraday and Cotton-Mouton processes, the transition probabilities are smaller for the two-magnon processes than for the one-~nagnon processes when the magneto-optical coupling is due to the one ionic spin mechanism. That does not hold good when the dielectric susceptibility is related to a pair of ionic spins (eq. 20) as discussed in the last section.

IV. Physical origins and microscopic theory of the spin-photon interactions. - The magnetic tran- sitions are well-known, so in what follows we restrict the study to the indirect spin-photon couplings corresponding to the one ionic spin and the pair of ionic spins mechanism. The spin-photons interactions are described by the quantum theory of perturbation, using the perturbation Hamiltonian H, = HeR

+

^{H ,}

with H, = Hc

+

^H

,,,,, +

^Hs,

+

H z and where the different terms are associated with the electron-pho- ton (He,) and inter-ionic electron-electron (H,) inter- actions,, the crystal field, the spin-orbit coupling and the Zeeman energy respectively. The one photon absorption is defined by the theory of perturbation linear in He,. The two-photon scatterings is described by the theory of perturbation quadratic in He, and linear or quadratic in H, following the considered scattering. In the last case the third and fourth order perturbations must be used. Without loss of generality we consider the ions with one electron. By extending the Kramers-Heisenberg dispe~sion formula [25] with the damping processes, we obtain the dielectric susceptibility tensor components near an electric dipole transition of an ion a 1261 :

pa is the Boltzmann factor an X is a complex shape factor for the transition between the ground 1 a

>

and excited

I

^cr

>

orbital states with the separation energy ha,, ⁼ Ea - E,. The use of the Weisskopf- Wigner method [ l ] gives a Lorentzian damping such as X = 2 w/h[wia - W'

+

i oqaa]. This factor contains Ea and Ea which depend on the perturbation H I . We shall consider how the introduction of the spin- orbit and exchange perturbation in X produces diffe- rent magneto-optical interactions like the Faraday and Cotton-Mouton effects, and the elastic and ine- lastic scatterings.

IV . I . FIRST-ORDER SCATTERINGS. FARADAY EFFECT.

The electric Faraday effect is given by the complex relation @, = cp, - i0, = - i no(c3-' (x,, - xYx),

where 8, gives an ellipticity due to the circular di- chroism. From 25 one verifies that :

where N is the number of magnetic ions per unit volume. The matrix element of 26 contains explici- tely the polarization rule of the Faraday effect with the n/2 scattering of photons. A more convenient form of @, will be used from the circular susceptibilities such as @, = no(cZ)-'

01,

-

x",

with :

with r, = r, f ir,. Thus the Faraday rotation is due to the dissymmetry of the matrix elements f, associa- ted to the right (RCP) and left (LCP) circular pola- rized waves. Let us choose the simple case of an atom in the ground state 'S(L = 0) and consider the odd parity transitions to the excited orbital state 'P(L = 0).

The spin is neglected in this simple example. When the applied d. c. field is zero the excited state is triply degenerate, f+ = f- and Qi = 0. When Hz

+

^{0 the}

Zeeman triplet is nondegenarate, and only the two transitions ('S -+ 'P, ,) and (IS + 'P- _,) corres- ponding to the selection rule Am, =

+

1 co\ntribute to f+ and f- respectively. In this case f + and f- con- nect the ground state

1

a

>

with the excited states (al

>

^and

1

^a2

>

having different energy E,, and Ee2.

The real and imaginary parts of the equation (27) gives the rotation and the absorption associated with the RCP and LCP waves. The figure 3 shows the equal but

rot at ton

FIG. 3.

-

Faraday rotation and dichroism due to the splitting of the excited orbital levels.

opposite contributions of the 'P,, states to the pure rotation and the circular dichroism, without (dotted curves) and with (full curve) a Zeeman splitting of the excited state. We can expect an increase of the rotation and the dichroism with the splitting ma, - ou2 as long as the two transitions are not well resolved. In the present model the splitting is due to

the applied field but we have seen that in theferro- magnetic materials E,,, Ea2 and Ea depend on the exchange, crystal field, spin-orbit and Zeeman pertur- bations. In order to precise the importance of each perturbation term, we assume EL, E: 6 hw 4 Ptuo where EL and El are the separation of the excited and ground levels with respect to their mean levels sepa- rated by ho,. The condition o <wo means that we operate far from absorption bands, which is an usual case. The denominator of X can be expanded as :

with x = 2 oo(EL - ~L)/h(mg - m2). Introducing (26) in (27), the Faraday rotation is described by the third- order perturbation theory by the term x. Since x is a linear function of Hs, and H,, we shall seelatter that the terms x and x2 in (29) give the first and second- orders magneto-optical effects in the one-ionic spin model and the second and fourth order effects in the pair of ionic spins model. The expansion (29) up to the linear term gives three distinct contributions to the Faraday effect [27], in equation (27) such as :

Or"-

- ~:)3 = z A ~ a ( f - - f + ) E i

a,a

(32 with C = 2 Nw/tt(wi - w2) and

The first contribution is independent of all the magne- tic perturbations. In a system of paramagnetic ions, this term depends only on the Boltzmann factor exp(Ea/kB T) which redistributes the populations of the ground states when a d. c. field is applied. As shown by Crossley et a1 [28],

where M is the ground state magnetic moment of the ion. Hence the first term (30) is referred to as cr para- magnetic ⁾⁾ rotation. The second and third contribu- tions (eq. 31 and 32) are also proportionnal to M 1281, but they have a strong dependence on the perturba- tion H, with the energy factors

Ei

and EL. In a system of paramagnetic ions, these terms depend on the applied field, and for this reason they are some time known as << diamagnetic >> rotations. This terminology, of << para >> and << diamagnetic >> rotations is mislea- ding and these terms are often referred to as the << C >>

and << A ^>) rotations respectively. The higher contri-

bution in the paramagnetic crystals is the C rotation except when the ground state has a zero orbital momen- tum ( 2 S + 1 ~ ) . From the time reversal invariance pro- perties it is shown in this case that

01- - x+),,,

are zero [28] and only

01-

-

x+),

gives a rotation.

In some para and ferromagnetic crystals the spin- orbit coupling produces a splitting of the excited states up to 1 000 times higher than the Zeeman splitt- ing produced by an electromagnet. This explains

SPIN-PHOTON INTERACTIONS IN MAGNETIC CRYSTALS, MAGNETO-OPTICAL C 1 - 597 the high Faraday rotation observed in these materials

and induced by

01-

-

x+),.

Another origin of the rotation arises from the modification of the eigen states 1 a

>

^{and I a}

>

due to H , and which changes the matrix elements.

Only the Zeeman and spin-orbit perturbations contribute to the rotation

k- -

x+),. Indeed, this term is determined by the quantities :

f, EA = constant

<

a

1

er, I a,

>

x

alaz

x

<

a ,

I

H I I or2

> <

a,

I

er,

I

a

> .

(33) By assuming that the projection operator

1

1 a

>

<

a ] operates only on the orbital components of the wave functions, it is straight forward to verify that f+ Ei is invariant under time reversal with the exchange and crystal fields perturbations. In this case f + = f- and cp, = 0. On the other hand, f + E: is not invariant under time reversal with the spin-orbit and Zeeman perturbation, so f+ # f- and cp, # 0. One verifies also that the quadratic term x2 (eq. 29) cannot induce Faraday transitions since it gives the quantity f+ x2 invariant under time-reversal for all the perturba- tions, spin-orbit included. But,

xxx =

xyy

⁼

($1

^{( x -}

+

x+),

so the diagonal components of the susceptibility does not vanish since f+(- x

+

x2) (or x+) is invariant under time reversal. From these properties one veri- fies that the Cotton-Mouton effect is due in part to the quadratic power of the spin-orbit and Zeeman interactions, and to the linear power of the exchange interaction. These invariance or non-invariance under time reversal give the properties of symmetry of the spin-dependent permitivity tensor used in the sec- tion III. In ferromagnetic materials the magneto- optical effects are associated with the magnetization rather the d. c. field, so we take g i j ( M ) = M).

The microscopic relations between the eiastic and inelastic scatterings appear clearly by using the first- order spin-orbit perturbation ILLS. Indeed these transitions are described by the ofl-diagonal compo- nents of the susceptibility tensor which contain matrix elements (eq. 25) such as :

Mid =

C ^<

^a2

^I

^er,

^I

^a,

>

^x

a l l ?

x

<

^2,

I

AL, S, I a ,

> <

^a,

I

erj

I

a,

>

(34) with

E.L, S, = /Z[L",

s: +

^q(L:

s; +

L;

s:)] .

⁽³⁵⁾

The first term of the perturbation 35 does not changes the intermediate states ) a ,

>

and ) cr2

>

and produces a n elastic scattering which is the Faraday effect. This perturbation contributes to

x,,

and

x,,.

The second and third terms of 35 operate in the multiplicity of the excited level with transitions such as :

where nz, and m, are the orbital and spin quantum numbers. These transitions produce first-order ine- lastic scatterings which become the one-magnon

Raman effect in ferromagnetic crystals. The trans-

verse part (L:- S i

+

^L,

s:)

of the perturbation contributes only in off-diagonal components (x,, - xZx)

and (xy, - xzy). This indirect Raman scattering has been first proposed by Elliot and Loudon [29] and observed by Fleury et al. in the antiferromagnets FeF,, MnF, and NiF, [30].

IV .2. SECOND-ORDER SCATTERINGS. EXCHANGE

INTFRACTIONS. - In their experiments Fleury et al. have observed that the intensity is stronger for the second- order Raman processes than for the first-order pro- cesses. In addition, as shown recently by Pisarev et al.

[31] the second-order elastic scatterings (Cotton- Mouton) have equivalent or higher amplitudes than the Faraday effect in ferri and weak ferromagnets.

This anomaly, not observed in ferromagnetic crystals can be attributed, not to the second-order spin-orbit perturbation (LS)' but to the exchange interaction in the excited state of a pair of magnetic ions having antiparallel spins (i. e. ferri and antiferro). This new mechanism was proposed first by Tanabe, Moriya and Sugano [33] to explain the two-magnon electric dipole absorption and, used later by Flcury and Loudon [30]

to describe the strong two-magnon Raman processes.

We show that this mechanism can explain also the anomalous Cotton-Mouton effect in the ferri and antiferromagnets. For this, let us consider two ions a and b having respectively the electrons I and 2. We first assume the system in the ground state with anti- parallel spin described by the wave function

where u,,, and q,,, are the orbital and spin fucctions.

After the absorption of one photon the pair of ions is in the excited state

I

ua(l) ql(T), ub(2) q2(l)

>.

Due to the anti-symmetry of the total wave function, the electron-electron interaction in the excited statc gives two terms of exchange energy defined by :

Due to the exchange without flipping of the spins, the first term leaves the pair of ion in a magnetic excited states after the scattering. This corresponds to the two-magnon Raman effect. The second term gives an exchange with flipping of the spins, so the ions a and b are in their initial ground state after the scat- tering. This corresponds to an elastic second-order process. This exchange energy can be described by the Heisenberg-Dirac Hamiltonian

where J r is the new exchange constant due to the excited state of the pair of ionic spins. The scattering is defined by the matrix element 34 by changing the spin-orbit coupling by the exchange energy 39. J' has the same order of magnitude as the usual exchange constant J of the pair of ionic spins in its ground

C 1

-

⁵⁹⁸ H. LE GALL

state, o r may be higher because of the greater extension birefringence which produces a phase-shift q,, bet- of excited orbitals. This explains why the second- ween the electric field components parallel and per- order magneto-optical effects can be stronger than the pendicular to the static magnetization, such as [32] : first-orders in the ferri and anti-ferromagnets. The

susceptibility associated t o a pair of ionic spins can _{qCM =} 0 -(rill - nl) =

now be expanded as : c

where the sommation runs over all the pairs in the sample of volume V.

In the last equation the Cotton-Mouton and Raman terms appear clearly. The strength of the Cotton- Mouton scattering can be determined from the linear

The values of Faraday and Cotton-Mouton phase- shift in the YIG at 1.15 micron and in the RbNiF, and the RbFeF, at 0.556 micron, are respectively [24] : q, = 200, 95 and 680 deg-cm-' and pc,, = 141, 142 and 1 600 deg-cm-'. In these materials the Cotton Mouton effect is a second-order magneto-optical effect due to the exchange energy in the first-order perturbation.

References HEITLER (W.), The quantum theory of radiation,

Oxford. U. Press.. 1960.

KAGANOV (M. I.) and TSUKERNIK (V. M.), Soviet Phys. JETP, 1960, 10, 587.

SCHLOMANN (E.), GREEN (J. J.) and MILANO (V.), J . Appl. Phys., 1960,31, 386 S.

MORGENTHALEK (F. R.), J. Appl. Phys., 1960, 31, 95s.

SPARKS (M.), Pliys. Rev., 1967, 160, 364.

LE GALL (H.), J. Appl. Phys., 1965, 36, 1203.

SAGE (J. P.), J. Phys. Cliem. Solids (to be published), PAITON (C. E.), Bull. An~er. Phys. Soc., 1968, 13,

1432, and J. Appl. Phys., 1969,40, 2837.

SUHL (H.), J. Phys. Chem. Solids, 1957, 1, 209.

SCHLOMANN (E.), Tech. report R. 48 (Raytheon company, 1959.

LE GALL (H.), DESORMI~RE (B.) and MILOT (E.), J. Phys. Chem. Solids, 1969, 30, 979, and 1969, 30. 1135.

GLOG~TON (A. M.), SUHL (H.), WALKER (L. R.) and ANDERSON (P. W.), J. Phj~s. Chem. Solids,

1956, 1, 129.

D E S O R M I E R ~ (B.) and MILOT (E.), Phys. Lett., 1967, 24 A, 202.

C o u ~ r o r s (L.), Thesis, Paris. 1964.

AKHIEZEK (A. I.), BARYAKHTAR (V. G.) and PELET-

MINSKII (S. V.), Soviet Phys., JETP, 1959, 39, 146.

KRINCHIK (G. S.) and CHETKIN (M. V.), Soviet Phys. JETP, 1962, 14, 485.

WANGSNESS (R. K.), Phys. Rev., 1954, 95, 339.

KRINCHIK (G. S.), Zh ETF Pis., 1968, 8,462.

LE GALL (H.), TRAN KHANH VIEN and JAMET (J. P.), Physica Status Sol., 1970, 41, K ^85.

LE GALL (H.) and JAMET (J. P.), Theory of the elastic

and inelastic scattering of light by magnetic crystals. I. First-order processes, Physica Status Sol. 1971 (b) 43.

MORIYA (T.), J. Phys. Soc. Japan, 1967, 23, 490.

LANDAU (L. D.) and Lt~sHrTz (E. M.), Electrodyna- mics of continuous media (chapt. 82), Pergamon, N. Y., 1960.

BASS (F. G.) and KAGANOV (M. I.). , . Soviet Phvs.

JETP, 1960, 10, 986.

JAMET (J. P.) and LE GALL (H.), Proc of the Interna- tional Conference on Ferrites. Kvoto. Julv 1970.

KRAMERS (H. A.) and HEISENBERG ( h . ) , ' Z . ~ h y s i k , 1925, 31, 681.

KAUZMANN (W.), Quantum chemistry, Acad. Press N. Y., 1957.

BUCKINGHAM (A. D.) and STEPHENS (P. J.). , , Annual Rev. of phYs. ~ h b m . , 1966, 17, 339.

CROSSLEY (W. A.), COOPER (R. W.), SAGE (J. L.) and VAN STAPELE (R. P.), Phys. Rev., 1969,181, 896.

ELLI~T-(R. J.) and LOUDON (R.), Phys. Letters, 1963, 3, 189.

FLEURY (P. A.), PORTO S. P.), CHEESMAN (L. E.) and GUGGENHEIM (H. JJ, Phys Rev. Letters, 1966, 17, 84.

FLEURY (P. A.) and LOUDON (R.), Phys. Rev., 1968, 166. 514.

PISAREV'(R. V.), SINY (I. G.) and SMOLENSKY (G. A.), Zh ETF Pis., 1969,9, 112.

LE GALL (H.), JAMET (J. P.), TRAN KHANH VIEN and DESORMIERE (B.), Theory of the elastic and inelastic scattering of light by magnetic crystals.

11. Second-order processes, Physica Status, Sol- lidi (to be published).

TANABE (Y.). MORIYA (T.) and SUGANO 6.). . .. P h ~ s . - Rev. ' ~ A f e r s , 1965, 15; 1023.