Journal of Magnetism and Magnetic Materials 31-34 (1983) 573-574

T H E E F F E C T O F T W O - M A G N O N T W O - P H O N O N I N T E R A C T I O N S O N T H E T W O - M A G N O N R A M A N S P E C T R A O F A N T I F E R R O M A G N E T S *

Y. I T O H

Department of Physics, Nara Medical University, Kashiwara, Nara, 634 Japan

573

An effect of magnon-phonon interactions on two-magnon Raman spectra is studied using Green function methods developed by Elliott and Thorpe. Green functions associated with two-phonon states are included. Two-magnon two-phonon interactions modify the shape of two-magnon Raman spectra appreciably.

Elliott and Thorpe have shown that the magnon- masnon interactions play an important role in the the- ory of the two-magnon Raman scattering of light [1]. Fleury has demonstrated by his experiments that the theoretical spectra derived by Elliott and Thorpe agree quite well with experiments [2]. However, there is still a slight discrepancy between theory and experiments: there is difference in cut-off frequency between theoreti- cal spectra and experimental ones. Here, we will point out that the discrepancy may be explained as an effect of two-magnon two-phonon interactions.

For the sake of simplicity, we deal with two-magnon Raman spectra of F3 + mode in a simple cubic antiferro- magnet such as RbMnF 3. The intensity of two-magnon Raman scattering I(t0) [1] can be related to Fourier transforms of double-time retarded spin Green func- tions:

c~(~, ~') = ( (P (~ ) ; P(~')>)~.

H(f;, ~ ' ) = (<Q(~;); P(~')>>~,

and

Gs(x,/~') = <(P(x); P(/~')>),~,

where

P ( ~ ) = ~'R( S~ S~+t + S~ S~+t) ;

R runs over all sites of the up-spin sublattice; Q(/~) is a spin operator similar to P(~). In the present paper, we will utilize the same notations given in ref. [1], if they are used without definitions.

We assume that the magnon-phonon interaction Hamiltonian, ~C~p, is derived from terms associated with second derivatives of exchange integrals. In this case, 9£~p can be written as

~sp = ~'R,p~va,s~?¢, ~,Fa,.¢,~, ( p ) exp[ - i(q + q ' ) .R ]

×(aqs+at_qs)(a,rs,+at_,t,s,)S..SR+o, ( I )

where a~s denotes the creation operator of the phonon of wavevector q and mode s; Fq~a,s,(p ) involves J", the second derivative of exchange integral; p runs over all nearest-neighbor sites.

When ~ p is taken into account, Green functions'

0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 - 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d

associated with phonon excitations appear in equations of motion for the spin Green functions, Those Green functions are classified into two groups: (i) ones associ- ated with processes in which two magnons of wavevec- tors k and - k annihilate and two phonons of opposite wavevectors create simultaneously and conjugate processes; (ii) ones caused by magnon scattering accom- panying phonon creations a n d / o r annihilations. The effect of (ii) on two-magnon Raman spectra will be treated approximately by introducing suitable imagin- ary parts in denominators in spin Green functions. Such corrections essentially do not affect two-magnon Ra- man spectra.

We neglect the Green functions (ii). In these ap- proximations, ~Csp modifies equations of motion only for H(/~,/~') as

toH(~, ~') = c8~.~, + 2 J ( 2 S Z - Pt)Gd(~, ~')

+ 4JSXpGs(~- p, ~')

+ 4S2Z, Zqss'Fqs,-,s'(P) O, ss ' ( t ' ) , (2)

with

Qqss'(~') = ( ( aqs , - , s ' ; P(~ '))) ,~, (3)

where P~ is an operator defined by 1 if/~ belongs to the nearest neighbor sites and 0 otherwise. Decouplings introduced by Elliott and Thorpe are assumed in the derivation of eq. (2). In addition to these decouplings, other types of decouplings are assumed for Green func- tions which appear in equations of motion for eq. (3).

The intensity of two-magnon Raman scattering I(o~) for the F3 + mode is proportional to the imaginary part of H(F~ ), which is expressed by linear combinations of H(~, ~'). An explicit expression for H(F3 + ) is given by

( 4 N S E / ~ ) H o ( F ; ) H ( r ; )= 1 + [ 4 J ( 4 J S Z - J ) - Ap(F3 + )] Ho(r; ) (4)

with 1 + a , ( r ; ) = ~z,s~.~r.~. _,,s,(r3 )2/(.0 - ~ , , - ~,,,),

(5)

where Ho(r ~ ) is the Green function without interac-

574 K Itoh / Ef fect o f magnon - phonon interactions on Raman spectra

/ / / ,k c/,'/ ' \

0.75 -8 • 9 1.0 FREOUENCY

Fig. 1. The effect of acoustic phonons on two-magnon Raman spectra of F3 + mode. The ratio of ~0 d to the zone boundary magnon frequency is chosen to be 5/3. The values of ga in units of (8JS) 2 for curves A, B and C are chosen to be 0, 0.2 and 0.4, respectively.

tions; ~s,q,s,(/~3 + ) represents a part of the F3 + symme- try; ~0qs denotes phonon frequency of wavevector q and mode s. When A p = 0, eq. (4) coincides with one ob- tained by Elliott and Thorpe.

Now, we study the effect of correction (5) on two- magnon Raman spectra by making simplified ap- proximations on phonon systems. Acoustic phonons are dealt with by use of a Debye model of single branch. In these approximations, A p due to acoustic phonons, i.e., Ap, is described as

f ~o '3 dco ' / ( , ' - ie), (6) "O

where ~0 d is the Debye frequency. The g, in eq. (6) can be estimated from eq. (5) if the coordinates of the acoustic phonons are known. Instead, here, ga is consid- ered as a parameter. To deal with optical phonons, we can neglect their dispersions. In these approximations, A p due to optical phonons, i.e., A°p, depends weakly upon ,.,. So, A°p will be regarded as a constant. It is noted that A°p is negative since the excitation energies of optical phonons are usually greater than those of mag- nons.

Example of calculated results are shown in figs. 1 and 2. Fig. 1 shows the effect of acoustic phonons and fig. 2 shows that of optical phonons. At frequencies lower than the peak, both acoustic and optical phonons strengthen the intensity of two-magnon Raman spectra. They shift the peak towards lower frequency. At fre- quencies higher than the peak, effects of acoustic pho-

///!k

, 0.75 .8 .9 1.0

FREQUENCY

Fig. 2. The effect of optical phonons. The values of A°p in units of (8JS) 2 for curves A, B and C are chosen to be 0, -0.1 and - 0.2, respectively.

nons are not remarkable. In contrast to this, at these frequencies, optical phonons reduce the intensity of the

spectra. Now, we will discuss about the discrepancy men-

tioned at the beginning of this paper. Within the uncer- tainty in the estimate of J , we can choose other value of J to fit the cut-off frequency. If such a value of J is adopted, the frequency of the peak given by theoretical spectra will be higher than that of the experimental spectra. At frequencies lower than the peak, the inten- sity of the theoretical spectra is smaller than that of the experimental ones; at frequencies higher than the peak, the former is greater than the latter. These discrepancies can be removed if two-magnon two-phonon interactions are taken into account: the curves B or C in fig. 2 seem to fit to experimental spectra. This shows that optical phonons are important for the agreement between theo- ries and experiments. Acoustic phonons will modify the spectra at frequencies lower than the peak.

The magnitude of J " can be estimated from the values of present choice of ga and A°p. A rough estimate of J " indicates that J " ( S a ) 2 - 0.1J; 8a represents typi- cal lattice distortions due to two-phonon creations.

The author would like to thank Professor A. Kotani

for stimulating discussions.

Relerences [1] R.J. Elliott and M.F. Thorpe, J. Phys. C 3 (1969) 1630. [2] P.A. Fleury, Phys. Rev. Lett. 21 (1968) 151.