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LINEWIDTH OF TWO-MAGNON RAMAN SCATTERING IN TWO-DIMENSIONAL ANTIFERROMAGNETS AT FINITE TEMPERATURE

A. VAN DER POL, G. DE KORTE, G. BOSMAN, A.J. VAN DER WAL and H.W. DE WIJN

Fysisch Laboratorium, Ri]ksuniversiteit, Utrecht, The Netherlands

The temperature dependence of two-magnon Raman scattering in the quadratic-layer antiferromagnets K2NiF, and K2MnF4 is found to agree with a higher-order Green function theory.

Raman scattering in antiferromagnetics by two-magnon processes provides information on spin waves near the Brillouin-zone edge. In the quadratic-layer Heisenberg antiferromagnets, experiments have to date been reported on K2NiF4 (S = 1) [1-3] and its isomorph K2MnF4 (S = ~-) [4]. Here we present, in addition to new experimental data for K2MnF4, the results of a higher-order Green function calculation in the two-dimensional (2D) antiferromagnetic systems. The hamiltonian, with inclusion of a staggered anisotropy field, reads

I , m ro

(1)

where 1 and m run over the up and down sublattices, respectively, and the first sum- mation is restricted to nearest neighbours on opposite sublattices. This leads to the renor- malized one-magnon dispersion relation [5]

(zJS) 2 ~ , ( T ) = a ( T ) I l , +- -z - - -A(1 +A)[1 - a ( T ) ] , (2)

I l k

in which II k is the spin-wave energy without Oguchi corrections [6], A = gl~BHg/zJS, and the coefficient a ( T ) describes the renormalization of the spin-wave energy at temperature T. The Stokes scattering cross-section K as a function of the shifted energy to is now obtained as the imaginary part of the Green function of the Raman transition operator of the two-magnon Raman scattering process, which may be re- duced to [7]

c , 0 , o , K(to) = 1 - ,otkarot2(T)S 2 - Im 1 - JLo(to)"

(3)

In a second-order theory we have for L0 [7]

L o ( t o ) = - N ~ . ~ k (coskxa

- cos k,a)" 2nk + 1 co - 2~k + 2iFk' (4)

where the summation is over the 2D first Bril- louin zone. The one-magnon damping constant Fk, which is absent in the first-order theory, is written into a mathematically manageable form by approximating the unrenormalized density of states by two rectangles. Over the interval zJS(2A) 112 < ilk < Y, covering the larger part of the Brillouin zone, the density is equated to a constant p,, while near the Brillouin zone edge, Y <f~k < zJS(1 +A), the density is P2, with P2 substantially larger than p,. The parameters pt, P2, and Y are determined by a 2D version of the

1.0 I K2 MnV4, I , I ~ <I0~

0 0 ~ I ~ 8 0 1OO 120

ENERGY SHIFT (cm q)

Fig. 1. Computed two-magnon Raman spectra in K2MnF4 at various temperatures. Below 10 K the curves coincide on the scale of the drawing.

Physica 86-88B (1977) 665-666 © North-Holland

666

rules given by Balucani and Tognetti [7]. Representative two-magnon spectra in

K2MnF4, computed with the above scheme, are displayed in fig. 1. At this point it should be noted that there are no adjustable parameters in the calculations. The exchange constant J is taken from ref. 4. For A at zero temperature the values from antiferromagnetic resonance [9, 10] are adopted while the temperature dependence of A is assumed to scale with the square of the sublattice magnetization [9].

Comparison of second-order theory with ex- perimental data on the width of the Raman peak is presented in fig. 2 for K2MnF4 and K2NiF4.

200

100

U n

K2Ni ~

{

i i i

1 I d.5 10 1.5

K2MnF4 I i ~1- I -

20

0.5 1.O

Fig. 2. Computed full width at half height of the two- magnon Raman-scattering line in K2MnF4 (S = 3,' J = 8.41 K, TN =42.1 K, while A = 0.0038 at T =OK), and K2NiF4 (S = 1, J = 102.1 K, TN =97.1 K, while A=0.0021 at T =OK) . The data o n K2MnF4 are from the present paper, except the closed circle, which is taken from ref. 4. The data on K2NiF4 are from ref. 1 (O) and ref. 3 (IS]).

Apart from a single point at 4.2 K [4], the data on K2MnF4 were obtained by us with a con- ventional Raman scattering set-up, using a

488 nm 500 mW argon laser, a double monochro- mator and photon counting techniques, while the data on K2NiF4 are taken from refs. 1 and 3. The computed peak positions fit experiment equally well as in first-order theory. However, while first-order theory predicts at best a peak width constant with temperature, the one-mag- non damping provided by second-order theory is seen to contribute substantially to the peak width at high temperature. There remains a slight tendency to underestimate the ex- perimental values, which at low temperatures could possibly be explained by the finite resol- ving power of the spectrometer. For our data on K2MnF4 the instrumental profile is - 2 . 5 c m -I wide; for K2NiF4 [1, 3] the instrumental width has unfortunately not been reported. In sum- mary, the temperature dependence of the two- magnon Raman spectrum in 2D antiferromag- nets appears to be in concurrence with second- order Green function theory up to temperatures above TN.

This work has been supported by the Foun- dations F.O.M. and Z.W.O.

References

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