Solid State Communications, Vol. 19, pp. 177-179, 1976. Pergamon Press. Printed in Great Britain

TWO-MAGNON RAMAN SCATTERING IN TWO-DIMENSIONAL ANTIFERROMAGNETS AT FINITE TEMPERATURES

A. van der Pol, G. de Korte, G. Bosman, A.J. van der Wal and H.W. de Wijn

Fysisch Laboratorium, Rijksuniversiteit, Utrecht, The Netherlands

(Received 24 January 1976 by A.R. Miedema)

The temperature dependence of two-magnon Raman scattering in the quadratic-layer antiferromagnets K2 NiF4 and K2MnF4 is found to agree with the higher-order Green function theory of Balucani and Tognetti. New experimental data on K2 MnF4 are also presented.

MAGNONS in the quadratic-layer Heisenberg antiferro- magnets K2 NiF4 (S = 1) and its isomorph K2 MnF4 (S = 5/2) have attracted considerable interest over the past few years. The two-dimensional (2D) magnetic character of these compounds was most directly demon- strated by inelastic neutron scattering, 1 which showed that the magnetic excitations are essentially non- dispersive perpendicular to the quadratic layer. Experi- ments on thermodynamic properties, such as suscepti- bility 2'a and sublattice magnetization, a and on the zero-point spin reduction 5 have also shown that a two- dimensional spin-wave theory, with inclusion of spin- wave interactions, provides a good description of these 2D systems.

Raman scattering of light by two-magnon processes, the subject of this communication, is sensitive to inter- actions of spin waves near the Brillouin zone edge. Experimental data have so far been reported on K2 NiF4 6 - 8 and K2 MnF4.9 For zero temperature a Green function spin-wave theory for the two-magnon Raman cross-section with inclusion of magnon-magnon interactions was first developed and applied to 2D sys- tems by Parkinson, 1° and subsequently by Chinn, Zeiger, and O'Connor. 8 For finite temperatures the theory was extended by Chinn, Davies, and Zeiger. 11 The observed shift of the Raman peak with increasing temperature towards lower energies as well as the linewidth at zero temperature could be accounted for satisfactorily, but the calculated linewidth was found to narrow slightly with increasing temperature, in strong disagreement with an observed thermal broadening. 7 On the other hand, in 3D cubic systems, such as KNiF3, Balucani and Tognetti TM recently achieved reasonable agreement between theory and the experimental broadening of the two-magnon peak by considering higher-order Green functions. Below we present, in addition to new experi- mental material for K2 MnF4, the results of a similar calculation for the 2D antiferromagnetic systems. The Hamiltonian, with inclusion of uniaxial anisotropy, is

= , ,.m ' : ' : ' where l and m run over the up and down sublattices, respectively, and the first summation is restricted to nearest neighbours on opposite sublattices. This leads to the renormalized one-magnon dispersion relation 11

(zJS) 2 . . ,

~k(T) : o~(T)~"~ k Jr ---~-k /Xl.l Jr A)[1 - - a ( T ) ] , (2)

in which ~2 k is the spin-wave energy without Oguchi corrections, la A = glaBHa/zJS , and the coefficient a(T) describes the renormalization of the spin-wave energy at

p.

5 l - -

Z hi I.-- Z

0.5

K2Ni ~

< 4 0 K

78 K

lOOK

124K

150K

' o o ' ' " - 4 0 Q 5QO 6QO ENERGY SHIFT (cm -~)

Fig. 1. Computed two-magnon Raman spectra in K2NiF4 at various temperatures. Below 40 K the curves coincide on the scale of the drawing.

177

178 TWO-MAGNON RAMAN SCATTERING AT FINITE TEMPERATURES Vol. 19, No. 2

120 I 1 -

I

1001 I I 1 0 0.5 1.0

01 , / J 0 0.5 1.0

TIT,

Fig. 2. Computed peak position and full width at half height of the two-magnon Raman scattering line in K2MnF4 (,S = S/2, TN = 42.1 K,J = 8.41 K, while A = 0.0038 K at T = 0 K). The data are from the pres- ent paper, except the closed circle, which is taken from reference 9.

temperature T. Equation (2) is equivalent to Keffer,‘* and a&o to the corresponding expression in reference 4. The Stokes scattering cross-section K as a function of the shifted energy o is now obtained as the imaginary part of the Green function of the usual Raman tran- sition operator of the two-magnon Raman scattering process, which may ultimately be reduced to12

K(o) = 1

i i

Lo(w) 1 _ eLgT “V)S2 -g Im 1 _JLo(u)’

(3)

In a second-order theory we have for Lo"

L,(w) = -;; (cosk,a-cosk,a)2 w_;;+;2f , k k

(4)

where the summation is, in our case, over the 2D first Brillouin zone. The one-magnon damping constant Fk, which is absent in the first-order theory, is written into a mathematically manageable form with a series of approximations following Balucani and Tognetti.12

-I ! I L

0 0.5 1.0 1.5 T/TN

Fig. 3. Same as Fig. 2, but for KzNiF4 (S = 1, TN = 97.1K,J=102.1K,whileA=0.0021KatT=OK). The data are from reference 6 (0) and reference 7 (0).

Representative two-magnon spectra in K2 NiF4, computed with the above scheme, are displayed in Fig. 1. It is observed that the one-magnon damping emanating from higher-order theory indeed results in a substantial broadening of the Raman peak with increas- ing temperature. At this point it should be noted that there are no adjustable parameters in the calculations. The exchange constant J is taken from reference 4. For A at zero temperature the values from antiferromagnetic resonance15*” are adopted, while the temperature dependence of A is assumed to scale with the square of the sublattice magnetization.15 In the paramagnetic

regime, of course, A = 0. Comparison of second-order theory with experi-

mental data on both the shift and the width of the Raman peak is presented in Figs. 2 and 3 for K2MnF4 and K2 NiF4, respectively. Apart from a single point at 4.2 K,’ the data on K2MnF4 were obtained by us with a conventional Raman scattering set-up, using a 488 nm 500 mW argon laser, a double monochromator and photon counting techniques, while the data on K2 NiF4 are taken from references 6 and 7. The computed curves for the peak position fit experiments equally well as those from first-order theory. However, while as already

Vol. 19, No. 2 TWO-MAGNON RAMAN SCATTERING AT FINITE TEMPERATURES 179

noted first-order theory predicts at best a peak width constant with temperature, the one-magnon damping provided by second-order theory causes the computed width to follow the experimental temperature depen- dence quite closely. There remains a slight tendency to underestimate the experimental values, which at low temperatures could possibly be explained by taking into account tile finite resolving power of the spectrometer. For our data on K2 MnF4 the instrumental profile is

2.5 cm -1 wide; for K~NiF46,7 the instrumental width has unfortunately not been reported_.

In summary, although a number of compromising approximations were required, a remarkably good agreement between second-order theory and experiment appears to exist for the case of 2D antiferromagnets up to temperatures above TN, in addition to the previously reported agreement for 3D systems. TM This further supports the idea that, while the critical region of 2D systems extends to a substantial distance from TN, short-wavelength magnons are essentially unaffected by the critical fluctuations and persist well into the para- magnetic regime.

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