TWO MAGNON ABSORPTION AND MAGNON SIDEBAND IN K2NiF4 TYPE ANTIFERROMAGNETS

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TWO MAGNON ABSORPTION AND MAGNON SIDEBAND IN K2NiF4 TYPE

ANTIFERROMAGNETS

H. Kamimura, N. Suzuki, S. Watarai

To cite this version:

H. Kamimura, N. Suzuki, S. Watarai. TWO MAGNON ABSORPTION AND MAGNON SIDEBAND

IN K2NiF4 TYPE ANTIFERROMAGNETS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-

1055-C1-1057. �10.1051/jphyscol:19711379�. �jpa-00214417�

JOURNAL DE PHYSIQUE CoUoque C I, supple'ment au no 2-3, Tome 32, Fe'vrier-Mars 1971, page C 1 1055

TWO MAGNON ABSORPTION AND MAGNON SIDEBAND IN K2NiF4 TYPE ANTIFERROMAGNETS

H. KAMIMURA and N. SUZUKI Dept. of Physics, University of Tokyo, Tokyo, Japan

and S. WATARAI

Dept. of Physics, Kyoto University, Kyoto, Japan

R4sumb. - La forme de bande laterale de magnons des antiferromagnetiques B deux dimensions avec la structure KzNiF4 est calculke avec les fonctions de Green, en tenant compte de l'interaction exciton-magnon et de la dispersion d'exciton. Le rksultat est compark avec l'observation sur RbzMnF4 : le paramare de dispersion d'exciton et celui d'inter- action exciton-magnon sont d6termines : 1,9 et 0,65 cm-1, respectivement. Une prkvision est faite de la forme de l'absorp- tion des deux magnons.

Abstract. - The magnon sideband shape of the two-dimensional antiferromagnets with the K2NiF4 structure is calculated with the use of the Green function, by taking into account the exciton-magnon interaction and the exciton dispersion. The result is compared with the observed one on RbzMnF4, and the exciton dispersion parameter and the exciton-magnon interaction parameter are determined to be 1.9 cm-1 and - 0.65, respectively. Then a prediction is made

on the shape of the two magnon absorption.

1. Introduction. - The recent study of the quasi- elastic scattering of neutron on K2NiF4 and Rb2MnF, has shown that these compounds are two-dimensional antiferromagnets [I]. Of these compounds the magnon sideband was observed by Imbusch and Guggenheim [2] for Rb2MnF4. Parkinson [3] calculated its line shape by including the exciton-magnon interaction but not the exciton dispersion, and compared with the observed one. I t was found that the observed band width is larger than the calculated one which corres- ponds to the width of the magnon density of states.

This suggests that the exciton dispersion may play an important role in the simultaneous excitation of the exciton and magnon. In the present paper firstly the shape of the magnon sideband of Rb2MnF4 is calcu- lated by taking into account both the exciton-magnon interaction and the exciton dispersion. It will be shown that the agreement with the experiment is excellent.

On the other hand, a few theoretical works have been made on the shape of two magnon absorption of the K2NiF4 type crystals 13, 41. In the latter part of the present paper we will calculate the line shape by including the magnon-magnon interaction.

2. Magnon sidebands. - The K2NiF4 structure consists of successive simple square NiF, planes sepa- rated by two K F planes, all stacked along the c-axis.

The magnetic properties of K2NiF4 and Rb2MnF4 in which spins are directed along the c-axis are well described by a single isotropic exchange interaction J and a small anisotropy field along the c-axis Ha. Thus the magnetic Harniltonian is

Here Eo is the excitation energy of the single ion state, A:(B;) and Ai(Bj) are the operators which excite and de-excite ion i(j) between its ground and excited states. Further V measures the excitation transfer between nearest neighbor ions on the same sublattice. In the present treatment we neglect the intersublattice interaction, This is negligibly small, for example, for Mn2+ ions which are of interest for the present study.

Since the excited state has in general a different spin Sf and a different exchange coupling to the neighbors J', the magnon and the exciton are not free from each other, and we take this exciton-magnon interaction into account. According to Parkinson and Loudon [5], we calculate the magnon sideband shape using the Green function method. Namely, the absorp- tion coefficient for light of frequency w and polarization direction a is calculated from the following formula :

Aa(co) = - (4 n0/c) lim Im G(Pa ; Pa), + is , (3)

6+0

where G(Pa ; Pa), is the energy-dependent Green function, and P spin-dependent dipole moment.

Consider first a single-ion excited state which has the same symmetry as the ground state. In this case the dipole moment can be written as P, = P, = 0, and

where the first summation is over all nearest neighbor where = sgn ( R j - R i ) , with a = x, y. The side- pairs and the sums over and refer to the up and band in this case appears in n polarization. Inserting down- spin sublattices,, respectively. For the exciton, (4) into (31, and using the fact that and dj trans- the Hamiltonian is ,given by form as the bases of the r5 representation of the point

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711379

C1 - 1056 H. KAMIMURA AND N. SUZUKI AND S. WATARAI group D

, the site symmetry of the K

NiF

type

crystal, we can show that the absorption coefficient is proportional to the Green function of 7 7 symmetry,

< H 1 G | rl > . Namely,

A^co) = - (4 nca/c) 2 N \ D |

Im < 7 7 | G | 7 7 > . (5) On the other hand, in the case of the excited state which has different symmetry from that of the ground state, we obtain the same result as (5) except the factor 2 | D |

. However, in this case the sideband appears in a polarization.

The Green [function < 7 J | G | 7 7 > can be deter- mined from its equation of motion [5], and it is expressed in terms of the Green function in the absence of the exciton-magnon interaction, < 7 7 [ G

\ 7 7 > , as follows ;

<r; | G | r; l > =

= < 7 7 | G

i 7 7 > / [ i -

JS < 7 7 | G

\ r

~ > ] , (6) where p is the exciton-magnon interaction parameter defined by p = ( / ' S'/JS) - 1. < 7 7 1 G

| 7 7 > is easily calculated [3, 4] and the result is

In doing this we have determined V to be 1.9 c m

from the observed band width and p to be — 0.65 from the peak position at 39 c m

, measured from the exciton line. As seen in figure lb, the agreement with experiment is excellent. It should be noticed that the absorption coefficient vanishes at the X point at which the two-dimensional magnon density of states diverges logarithmically, although such a feature will not appear in real experiments because of brodening effects.

3. Two magnon absorption. — Now we proceed to calculate the shape of the two magnon absorption. In the K

NiF

type crystals, the group of the nearest neighbor pair is D

and thus the dipole moment for the two magnon process is given by

P

= D X 4 ( S * S

- S

S

); P

= 0 , (8)

<y>

and P

is obtained by replacing <r]} by — <7y.

In the two magnon absorption, two magnons are excited simultaneously on neighboring atoms, and therefore, the interaction between two magnons is important, as was pointed out by Elliott and Thorpe [6]. Taking this effect into account, we adopt the following magnetic Hamiltonian expressed in terms of the spin deviation operators at, a

and bf, bj : 4 sin — K cos - fc

H

= JS~Z i

t*i + bj bj + at tf + a

bj) -

< F

- | G

| / 7 > = - Y N k

<tj>

(O (o

- Ok

•«fc»

(7) ^{J S}

a

t o.i bj bj

<ij>

where cot = 2 V

energy, F(cos ak

+ cos ak

) is the exciton

*$ ^ "'

. 5 *' M

°

^ ' (9)

- [ '

a>?= UzJS+gPHJ

(zJSf cos

| k

cos

| k

the magnon energy, u

= (zJS + gfiH

+ a>t)l(2 cot), z the number of nearest neighbors, a the distance between the nearest neighbor pair on the same sublat- tice, and finally the k

and k

axes in the magnetic Brillouin zone are taken as shown in figure la.

Now we calculate the sideband shape of Rb

MnF

from (5), (6) and (7) with use of / = 4.7 c m

, H

= 0.24 c m

, and z = 4, and compare with that observed around 17 615 c m

in n polarization [2].

A(u)

Just like the case of the exciton-magnon interaction, we obtain the following Green function, assuming the ground state to be the Neel state and using the usual decoupling procedure [5] :

< J 7 | G | 7 7 > = ( l - 2 ^ ) < 7 7 I G

| 7 7 > x

1 + J 2 < 77 | G 0 | 77 > ] " ' • ⁽¹⁰⁾

where < T

| G

| T

> is the Green function in the

4.8 10 47.250f 54A u( cm")

51

FIG. L a . — Brillouin zone of the K2NiF

structure, b. — The magnon sideband shape for the «Ai -*•

Ti exciton line in Rb2MnF4. Curve A, theory with no exciton dispersion ; curve B, best fit with V = 1.9 cm

and p = —0.65 ; curve C (dashed

curve), experiment.

FIG. 2. — The predicted absorption coefficient for two magnon

absorption in cases of S = 1, 3/2 and 5/2. The photon energy

is measured in units of IzJS, the energy of two free magnons.

TWO MAGNON ABSORPTION AND MAGNON SIDEBAND IN K2NiW TYPE ANTIFERROMAGNETS C 1 - ¹⁰⁵⁷

absence of the magnon-magnon interaction and is that the peak in the two magnon absorption appears

given by at higher energy than that in the two magnon Raman

scattering 131.

<'rc 1 ^G~ T L ~ > = (2 JZS)($)? 4sin2(%) x Since the spin-orbit interaction is involved in a mechanism to produce the dipole moment (8), the magnitude of D in (8) is probably very small for x cos2 (3) ^{[m2 -} (2 mp)2]-' . (11) K2NiF4 and Rb2MnF4. However, we expect that D may be large for Co2+ and Fez+ ions which have the Thelabsorption coefficient is calculated from (3), (8), spin-orbit interaction in their ground states, so that it (10) and (11). The results are shown in figure 2 in will be possible to compare our results with experi- cases of S = 1, 312 and 512. It should be remarked ments, for example, for K2CoF4.

References

[I] BIRGENEAU (R. J.), GUGGENHEIM (H. J.) and SHIRANE [4] SUZUKI (N.) and KAMIMURA (H.), Solid State Comm., (G.), Phys. Rev., 1970, B 1,2211. 1970, 8, 149.

[2] IMBUSCH (G. F.) and GUGGENHEIM (H. J.), Phys. [51 PARKINSON (J. B.) and LOUDON (R.1, J- Phys.2 1968,

Lett., 1968, 28A, 160. C 1, 1568.

PARKINSO INS ON (J. B.), J. P h y ~ . , 1969, C 2, 2012. [6] ELLIOTT (R. J.), and THORPE (M. F.), J. Phys., 1969,

C 2, 1630.