Optical Faraday rotation in ferrimagnetic garnets

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Optical Faraday rotation in ferrimagnetic garnets

A.M. Clogston

To cite this version:

A.M. Clogston. Optical Faraday rotation in ferrimagnetic garnets. J. Phys. Radium, 1959, 20 (2-3),

pp.151-154. �10.1051/jphysrad:01959002002-3015100�. �jpa-00236007�

151

OPTICAL FARADAY ROTATION IN FERRIMAGNETIC GARNETS

By ^{A. M.} CLOGSTON,

Bell Telephone Laboratories, Inc., Murray ^{Hill, New} Jersey.

Résumé.

Les grenats ferrimagnétiques présentent

grande rotation de Faraday optique.

Les transitions dont il s’agit ^sont interdites, ^tant _{par les} règles de sélection dues

spin que par celles dues à la parité. ^Une probabilité non-nulle de transition résulte de l’action combinée des

couplages spin-orbite, ^et des vibrations du réseau. Ceux-ci introduisent dans les états fondamentaux et excités des contributions de parités contraires. Le couplage spin-orbite décompose

^états

présence ^d’un champ d’échange

qui produit

rotation de Faraday

nulle. On calcule le

rapport rotation-absorption ^et

^trouve

accord satisfaisant avec l’expérience.

Abstract. 2014 The ferrimagnetic garnets ^exhibit

large optical Faraday rotation. The transi- tions involved are both spin ^and parity forbidden. A non-zero transition probability arises from the combined action of spin-orbit coupling, ^and lattice vibrations which augment ^the ground and

excited states with odd-parity angular ^{momentum states.} These states are split in the presence of the exchange ^field by spin-orbit coupling ^{and lead to} Faraday rotation. The ratio of rotation to

absorption is calculated and found to give satisfactory agreement ^with experiment.

PHYSIQUE 20, ^FÉVRIER 1959,

I. Introduction.

Dillon (1957) ^has recently reported ^{that the} ferrimagnetic garnets ^{are suffi-} ciently transparent ^{in the visible} region ^{of the} spectrum ^{so that a} considerable amount of light

may be transmitted through thin sections of single crystals. Initial observations of this effect using polarized light vividly revealed the domain struc- ture of the crystals. ^{The contrast between}

domains with differently ^oriented magnetizations

arises from a Faraday ^{rotation of the} plane ^of polarization ^{of a} light ^wave traversing ^{the domain.}

Dillon has measured the absorption coefficient and

Faraday ^{rotation as a} function of energy in the visible spectrum ^{for a} series of ferrimagnetic garnets. ^{He has observed} very large rotations,

the origin ^{of which is a matter of} fundamental interest. We shall show in this paper that the

Faraday rotation arises from ^an electric dipole

transition in contrast to the usual magnetic dipole

effect observed with paramagnetic ^{salts of such}

ions as Ni++.

II. Energy levels in the crystal ^field.

^The

ferrimagnetic garnets ^{have the} type ^form Y3Fe2(FeO4)3, ^{where the} yttrium ^{may be} replaced by ^{various rare earth ions. The iron ions are} trivalent ^{and occur on sites of} predominantly

octahedral ^or tetrahedral symmetry. ^Trivalent

iron has the ground ^state electronic ^con- ration (3d"). ^{The states} of this ion in a crystal

field of octahedral symmetry have been discussed at length by Orgel (1955). ^In Fig. 1, ^{we show an}

energy level diagram ^{similar to that} presented by Orgel ^{for Mn++} but calculated instead ^for Fe+++.

On the axis of ordinates are shown the states of the free ion, beginning ^{with the} ground ^state (3d5)6S. ^Next higher in energy ^{are a} series of

quartet ^{states such as} (3d5)4 G arising ^from invert-

ing ^one spin ^{in the} d-shell. Above the quartet

states are series of doublet states also arising

FIG. 1. - Crystal ^field splitting

of the energy levels of Fe+ + +,

from (3d5) ^{which will not} concern us and are not shown. Much higher in energy is ^{a set} of sb41es

arising ^from (3d 4s) ^and finally ^{a set} coming

from (3d4 4p). ^{In the free ion electric} dipole tran- sitions between the ground ^{state 68} an4 the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-3015100

152

quartet ^{states are} highly ^forbidden by parity ^and spin considerations. The first electric dipole

allowed transition ^{is to} (3d4 4p)sP.

In a crystalline electric field of octahedral _sym- metry ^whose strength ^is proportional ^to the para- meter Dq, ^the quartet energy levels split ^{and move}

as shown. A value of Dq ^{between 1 000 and}

2 000 seems appropriate ^to explain ^the absorption spectra ^observed in many salts of Mn++ and Fe+++. With such values of Dq, ^the corresponding

levels for the tetrahedrally coordinated ions are

expected ^{to lie} outside the _range of interest. Of the states other than quartets, ^{we shall be} parti- cularly concerned with the location of (3d4 4p)6 P.

The high energy and extended orbits of this state make the crystal ^field approximation inadequate.

The model that ^we adopt requires ^{that this state}

be actually depressed ^{to the} neighborhood ^of 20,000 ^cm-’.

III. Effect of exchange coupling ^and spin-oibit

interaction.

We ^now recognize ^{that the ferri-} magnetic garnets. will ^{have strong} exchange ^forces acting upon the individual ions. We shall concern

ourselves principally ^with the lowest energy state 4T1(4G), derived from 4G. This state is an

orbital triplet ^{and therefore has a total} degeneracy

of 12. In the strong exchange field, ^this dege-

neracy is partly ^{removed as shown in Fig. 2. In}

FIG. 2.

Effect of exchange ^field

energy levels.

the notation adopted here, 4A, ^{4B and 4C are the}

orbital components ^of 4Tl ^{and a} subscript ^will

denote the spin ^{state. The} ground ^{state is also} split ^{in the} exchange ^{field as shown in} Fig. ^{2. We}

limit our considerations ^to _{very low} temperatures

and suppose only the lowest spin ^{state is} occupied.

Let ^{us now} consider the effect of spin-orbit coupl-

ing ^{on this array of states.} We find that the

states mix in the following groups :

We shall be particularly concerned with the states derived from 685/2 ^and 4A3/2 ^{which become} respectively,

where W is the separation ^between 6S5/2 ^and 4A3/2,

and b =V5N aocp. Here, x ^{is the} spin-orbit coupl- ing parameter,

and

We can ca’Iculate b - 150 cm-1. The quantities Nocp ^and NOCF ^are respectively the fractional amount of 4P and 4F mixed with 4G to form 4 Tl (4 G). ^The coupling ^{between and} 4Tl(4G) ^arises only ^because 4TI(4G) ^{has an admix-}

ture of 4P. This admixture takes place only ^for

the states of symmetry Tl.

IV. Magnetic dipole transitions.

We ^now consider the strength ^of magretic dipole transitions between the ground ^state (1) ^and the excited mani-

fold 4Tx(4G). ^These transitions involve the matrix elements of (L + 2S), ^which will exist between the

ground ^{state and the state formed from} (683/2 4B3/2 4A1i2), ^{and have the order of} magni-

tude 4 g b/W where ^{is the Bohr} magneton. ^The f

value for this transition will then be

which we calculate to be about 10-9. The observ- ed oscillator strengths, ^{on the other hand are}

about 10-4. We thus eliminate magnetic dipole

transitions as the source of the observed absorp-

tions and rotations.

V. Electric dipole transitions.

Electric dipole

transitions between the ground ^state (1) ^{and excited}

state (2) ^{are forbidden} by parity since all the states involved are constructed from 3d orbitals. To obtain an allowed transition, ^we adopt ^{a model in}

which these states are augmented by ^a portion ^{of a}

state arising ^{from the} configuration (3d4 4p)..It ^is possible ^{that this model is over} simplified ^{and that a}

better theory ^{must consider} charge ^{transfer states}

involving ^electrons associated with the oxygen

ligands. The desired admixture ^can be brought

about if the Fe+++ does not lie at a center of sym-

metry ^{or if the} symmetry ^is destroyed by ^lattice

vibrations. In the ferrimagnetic garnets, ^the Fe+++ is symmetrically surrounded by ^{0- ions.}

We need consider then only the effect of lattice vibrations. Vibrations of particular symmetries

can introduce an additional potential energy for the electrons of the central ion of the form

(ax + by + cz) ^{which is effective in} bringing

about the required mixing. ^{There are two} pro-

cesses which will take place : (1) ^the mixing ^of (3d5)6S5l2 ^with (3d4 4p)6 P5/2; ^and (2) ^the mixing

of (3d5)4A3/2 ^with (3d4 4p)4D3/2 ^and (3d4 4p)4F3/2.

Both of these processes will contribute ^to the

strength ^{of the} absorption line ; ^{but we shall} simplify the discussion here by considering only

the first ^case. It may be noted, however, ^that

process (1) contributes line strength only ^{to tran-}

sitions involving 4TJ. Transitions involving 4T2

must arise from process (2).

The state 6P is 18 fold degenerate ^{and will} split

in the exchange ^{field as shown in} Fig. 2, ^where again ^we will consider only ^{the lowest state} 6 PS/2.

The remaining degeneracy ^{of this state is} split by spin-orbit coupling ^into three states characterized

by ^orbital angular ^momentum 1, ^{0 and - 1 which}

we shall designate ^as 6P5/2(1), 6P5/2(0), 6P5/2(- 1)

and assign energies A ^-r- u, A, ^{and A - u respec-} tively. ^{We now calculate} thelfollowing ^matrix

components,

v -

ground ^{state and excited state wave functions}

become

Using ^{these wave} functions, ^we may calculate the

following

--

matrix components ^{of the} dipole

~

moment P.

where M = (6Ps/2(1)IPxI68s/2) ^{= el,} is the matrix

component for electric dipole transition from 685/2

to 6P5/2.

Equations (8) ^and (9) ^{allow us to make a} rough

estimate of the oscillator strength of the electric

dipole transitions. If /0 ^{is the} strength ^{of the}

transition from 6S to 6 P we find

We estimate occc* as follows : a motion that

moves an Fe+++ ion a distance d toward ^one of its 6 surrounding ^{0- ions creates a value of}

given by

where a is the (Fe+++

0=) ^distance. X-ray

data (Geller, 1957) ^indicate that, ^{at room} tempe-

rature the root mean square value of (dIa) ^is

about 1/10. This would indicate that

- 5 Dq.

Since not all the vibrations will be as effective as

the one taken above, ^{we estimate} roughly loci ^r-J 1 000 cm-1. The most interesting absorp-

tion peak ^{in Dillon’s} measurements occurs at W = 16,000 ^{cm-1. We assume further that the}

very strong absorption edge beginning ^a 20,000 ^{cm-1 is to be} associated with direct tran- sitions to the 6P level. Accordingly ^{we chose}

A

20,000 cm-1. We further estimate roughly

that f o = 1/3. Using these values we obtain

f ^{r-J 10-5. This} is within an order of magnitude

of the measured values.

VI. Faraday ^rotation.

It is clear from equa- tions (8) ^and (9) that the transition probabilities

for right ^{and left hand} circularly polarized ^waves

are different and that a rotation of the plane ^of polarization ^{will therefore occur.} If it is assumed that the excited level is broadened by ^vibrations

and has a Gaussian line shape, ^it may be shown that the rotation will exhibit ^a dispersion shaped

curve with a maximum value in radians/cm given

by

where 3W is the root mean square width of the

line, ^{N is the} number of ions per cubic centimeter and Sois the dielectric constant of the crystal. ^The absorption coefficient at the center of the line is

similarly ^{found to} be,

The ratio r/6, ^becomes simply

154

If we assume, ^as before, ^{that W} = 16,000 ^cm-’

and A = 20,000 cm-1, ^{and in addition assume}

Jgo ^{= 2.5 and u = 100 cm-1, we find that}

ria

^.013. Using ^the experimental ^data give by Dillon, the ratio is found to be .008. This order of magnitude agreement ^{is evidence for the}

correctness of the proposed ^mechanism.

VII. Acknowledgements.

^{The author would} like to acknowledge many informative discussions of the questions ^{treated here with C. J.} Ballhausen,

A. D. Liehr and J. F. Dillon.

REFERENCES DILLON (J. F.), ^Bull. Phys. Soc., 1957, 2, 238.

GELLER (S.), ^J. Phys. ^Chem. Solids, 1957, 3, ^30.

ORGEL (L. E.), ^{J. Chem.} Phys., 1955, 23,1004.

DISCUSSION

Mr. Smit.

1) What about the temperature dependence ^{of the Faraday rotation ?}

2) ^In BaFe12019, ⁱⁿ which also Faraday ^rotation

of visible light ^{has been observed} by ^C. Kooy ^{of the} Philips Laboratories (5 000°/cm) many of the ions

are not a centre of symmetry ^{for the} crystalline

field. _{Do you} expect that in that case the Faraday

rotation should be extra strong ?

3) ^{If the} ground ^{state has a residual} angular

momentum, ^{a much} simpler theory gives Faraday

rotation. Should this give ^a greatly ^différent

result ?

Mr. Clogston. -1) ^The Faraday rotation is only slightly dependent ^{on the} temperature. ^The absorption ^{lines narrow} by ^{about a factor of two}

between room temperature ^and liquidjhelium ^tem-

peratures.

2) ^This depends ^of course on the degree ^of asymmetry. ^{1 would} certainly expect ^increased coupling ^{to the odd} parity ^states if there does not exist a center of symmetry.

3) ^{1 would} expect rather différent results, ^{but I}

have not considered _any such case in detail.

Mr. Suhl.

The question whether the ground

state has appreciable ^orbital angular ^{momentum is}

connected with the question ^{of the extent to which}

the presence ^{of a} magnetic ^axis prevents ^{the usual} quenching ^of angular ^momentum. Has Mr. Smit considered ^this problem ?

Mr. Prof. ^Kittel.

^{What can} be said about the role of the Verwey charge ^transfer process is causing optical absorption ^{in the visible} region ^{of the} spectrum ? ^Have any experiments been carried

out in which Fe++ is deliberately introduced into into a structure otherwise Fe+++.

Mr. Clogston.

^{It has been our} experience ^that

the magnetic garnets ^{which are} very nearly ^stochio-

metric are much more transparent ^than crystals

where the Verwey processes can occur. We have

not tried to observe the effects of changing ^the

valence state of the iron ions.