Journal of Magnetism and Magnetic Materials 157/158 (1996) 484-486 j R journal of

magnetism ~ i ~ and

magnetic m a t e r i a l s

ELSEVIER

Analysis of two-magnon Raman scattering in quasi-one-dimensional chain antiferromagnet CsMnC13 " 2H 20

V. Eremenko, V. Fomin, V. Kurnosov * B.L Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 310164 Kharkov, Ukraine

A b s t r a c t The two-magnon Raman scattering obtained in CsMnC13 - 2 H 2 0 was analyzed within a microscopic semi-phenomeno-

logical theory of the spin-dependent polarizability of exchange-coupled magnetic ions. Good agreement was obtained between observed and calculated polarization selection rules for two-magnon bands in the Raman spectra. It was shown that the nonevident selection rule for the observation of the two-particle peak-shape band originated mainly from magnons near the one-dimensional Brillouin zone boundary due to the specific zig-zag-type disposition of the Mn 2+ ions in the chain structure.

Keywords: Antiferromagnet; Magnon; Raman scattering; Polarizability

The quasi-one-dimensional Heisenberg antiferromagnet CsMnC13-2H20 (abbrev.: CMC) with N6el temperature (T n ) 4.98 K [1] is a well-known subject for investigation using many different experimental techniques. The mag- netic properties obtained are: the intra-chain exchange interaction J1 = 4.3-5.1 cm - I (from different literature data); and axial and rhombic anisotropy fields H A = 1440 Oe and H c = 450 Oe at 1.5 K [2]. The estimated values of inter-chain exchange interactions are about 300 times smaller than the intra-chain interactions. The magnetic space group of this compound is Pb~ag [1], resulting in the '~ presence of eight nonequivalent magnetic sublattices in ,4 CMC. But the smallness of the exchange interaction con- stants between neighbor spins from adjacent chains (and the even smaller difference between them for spin pairs ~- from different sublattices) allows us to consider CMC as a z ~o

t u

two-sublattice chain-like antiferromagnet in model calcula- tions.

A very weak asymmetric and wide band has been t ~ observed in the low-frequency Raman spectra of CMC at a m h -

temperature of about 2 K (Fig. 1). The band displays very ,~ strong polarization selection rules and was observed with the only nondiagonal component of the scattering tensor, namely X Y or YX, which corresponds to Big symmetry in the Pcac factor group of CMC. The fact that the band is

* Corresponding author. Email: kurnosov@ilt.kharkov.ua; fax: + 7-572-32-23-70.

observed only at low temperature, below T N, and its highest cutoff frequency is abut 50 c m - 1 (which is, accu- rately, twice the energy of a magnon near the boundary of the Brillouin zone (BZ) obtained from neutron diffraction [3]), allowed us to consider this band as a result of two-magnou light scattering. Below further evidence is

Ag (XX)

Blg(YX) ~ I u , i I

40 [, I

8O RAI'IAN SHIFT, c m - 1

Fig. 1. Raman spectra of CMC at T : 2 K. The calculated shapes of the two-magnon bands neglecting the interaction between magnons are shown by dashed lines. The modes symmetry and components of the scattering tensor are also shown.

0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 0 9 9 9 - X

V. Eremenko et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 484-486 485

X(a)

Fig. 2. Fragment of the CMC antiferromagnet chain. The Mn 2+ ions (indicated by 0,1,2) and bonding CI- ions only are shown together with the space symmetry elements.

presented of such an assignment which follows from the analysis of polarization selection rules.

The Raman scattering intensity in the unit frequency range in the direction given by the unit vector r, is proportional to

E2f_+~dtei(°'-'°°)'(e&(O)*(1 - r r ) a ( t ) e } , (1)

where angled brackets denote statistical averaging, E is the amplitude of the electrical component of the incident light with frequency o) o, e is its unit polarization vector, and 8 0 ) is the operator of media polarizability in the Heisenberg representation: &(t) = eitH/h&e -it;q/h (here H is the Hamiltonian). In the case of two-magnon scatter- ing the operator & is associated with the state of two adjacent spins. In the exchange approximation, this opera- tor has the form [4]:

~jk = fijk( sj " sk), (2)

where the polarizability operator /~i~ depends only on the coordinates of spin at sites j and k. The properties of this tensor are given by the point symmetries of the j - k spin pair. In particular, two pairs 0 -1 and 0 - 2 in one antiferro- magnet chain in the unit cell of CMC (Fig. 2) have the following matrix representation:

Pxx Pxy 0 ) fiOl = l PxY pyy 0 , ( 3 a )

0 Pzz

~ Pxx --Pxy ~ ) P02 = --Pxy Pyy . ( 3 b )

0 0 Pzz

As can be seen the only nondiagonal component Pxy is nonzero. This is caused by the presence of the only element of symmetric transformation for each pair in the chain, namely axis C z (Fig. 2). A transition to the momen- tum representation transforms the polarizability operator as follows:

&(q) = /3 (q ) (u 2 + v~)AqB_q + h.c., (4)

where Aq and B q are operators of magnon annihilation in the representation where the spin Hamiltonian is diago- nal, and Uq and vq are the coefficients of the Bogolubov- Tjablikov transformation. The tensor /3 in the momentum representation becomes:

[ 2pxxcos q~ i2pxysin qo 0 )

fi(q) = [ i2pxySin q~ 2pyyCOS q9 0 , ( 5 )

t o 0 2pzzCOS q9

where q~ = qxa/2 is the phase of the spin wave, and a is the size of the unit cell. In the diagonal representation the spin Hamiltonian of the single antiferromagnet chain in the CMC has the form:

H= Y'~E(q)( A;Aq + B+_qB_q), (6) q

where E(q) is the energy of magnons with momentum q, and is given by

2 ] 1/2 E(q) = 2JS[ (1 + "r) 2 - cos q~] , (7)

where J is the constant of the intra-chain exchange inter- action, S is the spin of the magnetic ion (S = 5 / 2 for Mn2+), and r = glXBHA/2JS, where g is Lande's factor, /x B the Bohr magnetron, and H A the effective magnetic anisotropy field. The expressions for coefficients u and v in Eq. (4) have the form:

(a + , ) + (1 + , ) - , U 2 ~

u2 2e(~z) 2 e ( p ) ' (8)

where the normalized energy is introduced as e(q~)= E(q~)/2JS. Making the subsequent substitution of the above expressions into Eq. (1) and taking into account that the temperature is zero it is possible to rewrite Eq. (1) as follows:

E2fqdq 6( w - o) o + 4JSe( qQ/h )(1 + r ) / e ( q Q

x(e /3*(1 - r r ) & ) .

So the two-magnon scattering tensor can be obtained:

I ~ ( w ) ~ fqdq 8( o) - w o + 4JSe (qQ/h )

x e ( ~ p ) I t P ~ I 2 . (9)

Since the dispersion of magnons in the present purely one-dimensional model is a function of the only projection of wave-vector qx, and therefore a function of the phase ~o, Eq. (9) can be transformed into the form:

I ~ ( w o - 4JSe( qQ/h)

486 V. Eremenko et al. / Journal of Magnetism and Magnetic Materials 1 5 7 / 1 5 8 (1996) 484-486

1/2 (1 +~-)2.~(~)2 Ip"~12 )

for IXU = xx , yy , z z ,

1 - (1 + 7)2+ ~(~)21p.~12 (1 + ~)~ - ~(~)~

for p~u= xy ,

0, for /xv = xz , yz .

It is evident from Eq. (10) that the scattering intensity in the diagonal components of the tensor in the low-energy range is described by a hyperbolic-type dependence from the frequency shift.

The two-magnon peak in the ( X Y ) polarization has a similar dependence, but in contrast to the above the maxi- mum is achieved at frequency shift 4JS(1 + " r ) / h (Fig. 1). The discrepancy between the calculated and observed two-magnon spectra is a result of neglecting magnon-

magnon interactions. These mainly affect the shape of the nondiagonal spectrum. From the above calculations it is clear that only one nondiagonal polarization is allowed in two-magnon scattering, and this was what was observed experimentally.

The small intensity of the two-magnon scattering in the X Y polarization follows from the fact that the locations of the Mn 2+ ions in the lattice are close to higher symmetry positions where the Pxy component of the polarization tensor is exactly zero. Hence the observation may provide evidence for the (weak) zig-zag bending.

References

[1] R.D. Spence, W.J.M. de Jong and K.W.S. Rama Rao, J. Chem. Phys. 51 (1969) 4594.

[2] J. Scalio, G. Shirane, S.A. Friedberg and H. Kobahashi, Phys. Rev. B 2 (1970) 4632.

[3] K. Nagata and Y. Jazuke, Phys. Len. 31A (1970) 293. [4] T. Moriya, J. Phys. Soc. Jpn. 23 (1967) 490.