INTENSITY OF OPTICAL ABSORPTION OF Mn2+ ION IN CUBIC ANTIFERROMAGNET

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INTENSITY OF OPTICAL ABSORPTION OF Mn2+

ION IN CUBIC ANTIFERROMAGNET

K. Motizuki, I. Harada

To cite this version:

K. Motizuki, I. Harada. INTENSITY OF OPTICAL ABSORPTION OF Mn2+ ION IN CU- BIC ANTIFERROMAGNET. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1053-C1-1054.

�10.1051/jphyscol:19711378�. �jpa-00214416�

JOURNAL DE PHYSIQUE Colloque C I, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 1053

INTENSITY OF OPTICAL ABSORPTION OF Mn2+ ION IN CUBIC ANTIFERROMAGNET

K. MOTIZUKI and I. HARADA

Faculty of Engineering Science, Osaka University, Toyonaka, Japan

R6sum6. - On explique I'intensite d'absorption optique des transitions

^6Alg -, 4T1,

et

^6Alg -t

4T2g de Mn2+ dans MnO et MnS et sa dependance en temperature par les bandes associees a un magnon. L'approximation de paire est employee pour determiner les Btats des spins de chaque paire d'ions.

Abstract. - The intensity and its temperature dependence of the optical absorption in MnO and MnS corresponding to the transitions 6Alg

-+ 4Tlg

and

6Alg +

.4Tzg of Mn2+ ion are explained in terms of sidebands associated with magnon.

The pair approximation is used and the spin states of each ion pair are determined.

1. Introduction. - Optical absorption in MnO and MnS corresponding to transitions

6 ~ , g

-+ 4Tlg and 6A1g - 4T2g of the Mn2+ ions, measured by Huffman [I], shows an anomalously high intensity (f - ^- despite the parity- and spin-for- bidden character of these transitions. The intensity increases with increasing temperature before reaching the NCel point but shows a tendency to saturate above the Nkel point.

Provided that these transitions are the phononassi- sted transitions, the oscillator strength for 6A,g-+

4 ~ l g

and 6A1, - 4T2g is estimated [2] as f - lo-', which is much smaller than the value observed in MnO and MnS. The temperature dependence o f f of the pho- non-assisted transition is expressed as

where o0 is the characteristic frequency of the phonon, but this does not explain the saturation effect observed in MnO and MnS above the Nkel temperature. Huffman used a value of o0 = 110 cm-l for MnO and coo = 135 cm-I for MnS in order to explain the observed curves with the above expression at low temperatures. However the effective phonon energy o, of MnO, which is a more tightly bound crystal than MnS, should be greater than o, of MnS, contrary to the result of his analysis.

We interpreted the observed results as being due to sideband absorption associated with magnons and calculated the temperature dependence of the inten- sity. The pair approximation was used in the present calculation.

2. Formulation. - The intensity of magnon side- band is proportional to the square of the matrix ele- ment of the effective spin-dependent dipole moment between the initial state and the final state. Consider a pair of Mn2' ions, a and b, coupled by exchange interaction and assume that in the initial state a and b are both in the state 6A1g and in the final state an elec- tron of the ion a is excited to 4Tlg or 4T2g. These initial and final states consist of 36 and 24 substates with respect to spins of a and b. The effective spin-dependent dipole moment can be written [3] as

P a b

= d a q ' q, b) ~ ~ ( 4 ' 4).

S b r

(1) where

8.4

sa(ql

⁺

q) = C

~&'rn' Caqm

< m' I

^Sa

I ^m > ⁽²⁾

m,m'

Sb is the total spin of the ion b and is the spin of an electron of the ion a ; m and q represent spin and orbital states of an electron. c&,,, and caqm are crea- tion and annihilation operators of an electron. Consi- dering the Boltzmann distribution over the 36 spin states of the initial state, the intensity of the magnon sideband with respect to the pair is given by

&(in) I(a*b) =

I..

exp [- -1 ^{kT x}

where in and f, represent the spin substates of the initial and final states, and they are determined self- consistently by the pair approximation.

Since MnO and MnS have the type I1 antiferromag- netic spin arrangement, we take three kinds of ion pairs into account :

pair 1 : n . n ions with parallel spins, pair 2 : n.n ions with antiparallel spins, pair 3 : n . ⁿ . n ions with antiparallel spins.

The Hamiltonian of each pair can be written for pair 1

Hl(a, b) = - 2

J1

Sassb - B(Saz + ^SbJ

for par 2

H2(a, b) = - 2 J , S,.Sb - C(Saz - Sb3 ₍₄₎

for pair 3

H,(a, b) = - 2 J, Sassb - D(Sa, - SbZ) ,

where J , and J2 are exchange constants for n. n spins and n.n.n spins, respectively, B, C, and D are para- meters representing the molecular fields arising from other spins except the pair spins under consideration.

We introduce one-body Hamiltonian as

H(a) = - AS,, , (5) where A represents the molecular field and is related to B, C and D by A = 6(B + C + D)/17. We deter-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711378

C 1 - 1054 K. MOTIZUKI AND I. HARADA

[I] HUFFMAN (D.), J. Appl. Phys., 1969,40,1334.

[2] KOIDE (S.) and PRYCE (M.), Phil. Mag., 1958, 3 , 607.

[3] TANABE (Y.), MORIYA (T.) and SUGANO (S.), Phys.

Rev. Letters, 1965, 15, 1023.

[4] M o n z u ~ r (K.) and HARADA (I.), Solid State Comrnun., 1970, 8, 951.

mine the molecular fields, B, C and D, as functions of temperature by the following equations : -

S 2 .5 Tr [Saz ~ ( a ) ] = Tr [Saz pj(a, b)] O' = 132, 3) (6)

where p(a) and pj(a, b) are the density matrices cons- tructed from H(a) and Hj(a, b), respectively. In the 2.0 paramagnetic region the molecular fields B, C and D

vanish. Hence the spin states of the initial state are

I

.5

eigenstates of - 2 J, Sa . ^S, for pairs 1 and 2 and those of - 2 J, Sa. S,, for pair 3. In the antiferromagnetic

I

.o region the eigenstates of the pair Hamiltonians H,(a, b),

H,(a, b) and H,(a, b) are solved self-consistently, toge-

ther with eq. (6). Using these solutions we calculate the _0.5 intensity by eq. (3) for each of three pairs. The tempe-

rature dependence of the intensity comes from the

Boltzmann distribution over the 36 spin-states of the ₀

I I I

....

...

.-.

- -

molecular f i e l d a p p r o x . -

- -

-

I

5 0 1 0 0 1 5 0

initial state and the variation of the molecular fields with temperature.

T ( " K )

3. Results. - In figure 1 we show the calculated FIG. 2. - The thermal average of spin per site.

N6el temperature as . a function of y (= J,/J,). For ( J ~ / J I

=

y

=

3.0).

are shown by three solid curves and their average by a broken curve in figure 3. For pair 1, which has parallel spins, Pa, at 0

O K

has no matrix elements between the ground spin state and the excited states, so that the intensity should decrease rapidly below the N6el temperature and tend to zero. For pairs 2 and 3, the intensity shows a gradual temperature dependence and remains finite at 0

OK.

The intensity at 0

OK

was calculated with the assumption that the molecular field for each pair arises from the perfect order of the spins. In figure 3, fine dotted curves inter- polate to the values calculated at OOK. Above the Nee1 point the intensity is almost temperature-inde- pendent.

8 0 -

-

- - - -

3

o -

2 0 - I O - 0

-

---

A . N . T .

- - _ . _ _

--- --- ___

- - - * - - -

- -

I I I

1.5 2.0

2.5 3.0 3.5

^{I I} I I

%I 2

- - - - - - - _

_..____-.

FIG. 1. - The N&l temperature calculated as a function of : . ' ^-

^-

52/51 (= y). The solid and the broken curves represent the 5 N k l and the anti-N&l temperatures. The chain curve shows 2 the N6el temperature calculated by the molecular field appro-

ximation. -

y-<'y, (e 1.8), the equation which determines the

..

Nkel temperature has no solution [4] and therefore the 5

⁰⁾

type I1 antiferromagnetic spin ordering is not stable t

⁵

in the pair approximation. For lower branches of the curve represent the NCel and y > y, the upper and ; ^-

anti-N6el temperatures. Thus, our pair approximation

....

.

._I..

-

-

_.,.

_.,.

I I I I

cannot be used to far below the N6el temperature. o

5

o

100 20

o

250

T("K)

In our numerical calculations we fixed y = 3 and

we determined the exchange constant Jt by making FIG. 3. -The temperature dependence of the intensity of the calculated N6el temperature agree with the obser- magnon sideband. Three solid curves 1, 2, and 3 are the inten- ved T,. ~h~ thermal average of the spin was calcula- sities for pair 1,2, and 3, and the broken curve a is their average.

ted and is shown in figure 2. It is with that (We assume that orbital parts of the matrix elements of Pab are equal for n. n. and n. n. n.) The chain curve b is theaverage calculated by the molecular field approximation. The of the intensities for pair 1, and 2. Fine dotted curves interpolate intensities of magnon sidebands for pairs 1, 2 and 3 to the values calculated at 0 OK.

References