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Spin waves in the quasi two-dimensional antiferromagnet KFeF4
Physica B 180 & 181 (1992) North-Holland
PHYSICA II
225-226
Spin waves in the quasi two-dimensional KFeF, S. Fultona,
S.E.
R.A.
Naglerb,
Cowley”
and L.M.N.
antiferromagnet
Needham’
“Oxford Physics, Clarendon Laboratory, Oxford, UK “Department of Physics, University of Florida, Gainesville, Florida. USA ‘Department of Physics. Warwick University, Coventry, UK
using inelastic neutron scattering. The spin excitations in the two-dimensional system, KFeF,, have been investigated The dispersion relation is well described by a model Hamiltonian incorporating a uniaxial anisotropy. The anisotropy dominates the energy of the zone centre mode and is seen to be strongly temperature dependent.
1. Introduction
E; = [gp.“H,
In the orthorhombic KFeF, structure, the Fe3+ (S = 2) ions are surrounded by slightly tilted flourine octahedra which are almost quadratically arranged in the (0 0 I)-planes [l]. These planes form antiferromagnetic sheets which are separated by nonmagnetic K’ ions [2]. The interactions between these magnetic layers is weak in comparison to the interlayer interactions, the ratio being of the order 10d4 [3], and thus KFeF, can essentially be thought of as a two-dimensional antiferromagnet. With this in mind, we have carried out a neutron scattering study of KFeF4, using the IN3 triple axis spectrometer at the ILL in Grenoble. Constant Q scans [4] were used to determine the dispersion relation of the magnetic excitations and they have been modelled using linear spin wave theory [S]. The temperature dependence of the anisotropy energy has also been investigated and is found to vary in the same way as the staggered magnetisation [4].
2. Theory The KFeF,
Hamiltonian for the has the form
H = -g/-@, -2
c
1
S: - g/+,H,
c J,,.S;S,,-2 (I. 1’)
x J,,,,S,
‘Sm. -2
magnetic
interactions
in
c SZ,
m
c (m. rn’) c J,,S, (1. m)
‘S,,, .
(1)
For this system H, represents the effective field due to an anisotropy arising from dipolar effects [6], which establishes c as the preferred direction for the spins to align along. Using linear spin wave theory as discussed by Keffer [5] the dispersion relation for a 2 D antiferromagnet, with nearest neighbour and next nearest neighbour interactions, was formulated as 0921.4526/92/$05.00
0
1992 - Elsevier
Science
Publishers
+ J,) + 8SJ,
- 4S(J,
- 4SJ,(cos(
4,~ + q,,b) + cos( q,a - q,b)]’
- [4S(J, cos( q,a) + J, cos( q,,b))]’
(2)
Equation (2) represents the energies of the spin waves with wave vector q. This model allows for anisotropic nearest neighbour interactions in the a and b directions, J, and J,, as well as for next nearest neighbour interaction, J3. In this equation a and b represent the magnetic lattice constants in the antiferromagnetic layer, which at 50 K were found to be 3.784 and 3.875 A, respectively. 3. Results In our experiment, magnetic Bragg reflections were studied at the (i 4 0) and ($ f 0) reciprocal lattice points. The spin waves were determined for propagation directions [ 5 501, [ 5 0 0] and [0 5 0] at 50 K. which is well below the NCel temperature: T, = 137 K. [ is the reduced wave vector, which for the a direction is given by q,ai2a, and similarly for the b direction. In order to determine spin wave frequencies, the constant Q scans were fitted to Gaussians using the Marquardt least squares fit method [7]. The dispersion relation is shown in fig. 1. The solid line represents the fit to the experimental data. Fitting was done using eq. (2), allowing for nearest and next nearest neighbour interactions. Values for the exchange constants, calculated from the fit, are shown in table 1. These results are in good agreement with previous studies [8], which calculated Table 1 Results from J,
bevl
-1.00
B.V. All rights
+ 0.05
reserved
dispersion
relation
fit.
Jz [mevl
J3 bvl
-1.37
-0.06
? 0.05
~4, 2 0.02
bvl
0.12 + 0.005
S. Fulton et al. i Spin waves tn KFeF,
226
24 20 5 j
I6 h M
b d w
12
8 4 0
24
I 0
0.25
c
24
2
A m = 0.28
1
16
c
bo
10.0
2 e,
a -. $
t
9.5
rl
[WI
0.25
f
‘Z 4
16
0
.*
2
i 0
0.25
3
Log,(T,-T)
4
1
2
3
4
Log,(T,-T)
0.50
c
Fig. 1. The spin wave dispersion relation for KFeF, at 50 K. The solid line is the best-fit to the theoretical model, using the parameters listed in table 1.
a nearest neighbour exchange constant of - 1.15 meV. It is seen that our results show a difference in nearest neighbour exchange constants J, and J?. This is caused by the different superexchange interactions in the u and b directions, due to the tilting of the flourine ions out of the plane in the a direction [2], thus making J, smaller than J,, even though the lattice constant a is less than b. The experimental value for the anisotropy energy gp,,H, is larger than expected for dipolar interactions (0.033 meV) [6], and thus there must be other contributions which should be taken into account when calculating H,, which are not important in the analogous Rb,MnF, with S = $ ions. At present we do not have a convincing explanation for this difference. A temperature dependent study of the anisotropy effect, which causes the gap in the dispersion relation at the magnetic zone centre, is shown in fig. 2, where m represents the gradient of the graph. This is seen to vary in the same way as the peak height of the (4 f 0) magnetic point, which has m = 2p. p is the critical
dence
Comparing the gradient of the temperature depenof the anisotropy energy with the peak height of the
(: J 0) magnetic point. exponent arising from the staggered magnetization temperature dependence. which for a two-dimensional Ising system is 0.125 [4]. Thus the anisotropy energy varies in the same way as the staggered magnetization. References
Ill
G. Heger and R. Geller. Phys. Stat. Sol. (b) 53 (1972) 227. PI G. Heger, R. Geller and D. Babel, Solid State Commun. Y (lY71) 335. PI H. Keller and I.M. Savic, Phys. Rev. B 28 (lY83) 2638. (Oxford Uni[41 M.F. Collins, Magnetic Critical Scattering versity Press. lY8Y). of Physics. Vol. 18. ed. S. 151 F. Keffer. in: Encyclopedia Flugge (Springer, Berlin, 1966) part 2. Phys. 161 H.W. de Wijn. L.R. Walker and R.E. Walsted. Rev. B 8 (1973) 285. S.A. Teukolsky and W.T. [71 W.H. Press. B.P. Flannery, Vellering, Numerical Recipes (Cambridge University Press, 1986). G.R. Davidson, H.J. Guggenheim and 181 M. Eibschutz. D.E. Cox. Proc. of the 17th Annual Conference on Magnetism and Magnetic Materials, eds. C.D. Graham and J.J. Rhyne (AIP. New York. lY72) p. 670.
PHYSICA II
225-226
Spin waves in the quasi two-dimensional KFeF, S. Fultona,
S.E.
R.A.
Naglerb,
Cowley”
and L.M.N.
antiferromagnet
Needham’
“Oxford Physics, Clarendon Laboratory, Oxford, UK “Department of Physics, University of Florida, Gainesville, Florida. USA ‘Department of Physics. Warwick University, Coventry, UK
using inelastic neutron scattering. The spin excitations in the two-dimensional system, KFeF,, have been investigated The dispersion relation is well described by a model Hamiltonian incorporating a uniaxial anisotropy. The anisotropy dominates the energy of the zone centre mode and is seen to be strongly temperature dependent.
1. Introduction
E; = [gp.“H,
In the orthorhombic KFeF, structure, the Fe3+ (S = 2) ions are surrounded by slightly tilted flourine octahedra which are almost quadratically arranged in the (0 0 I)-planes [l]. These planes form antiferromagnetic sheets which are separated by nonmagnetic K’ ions [2]. The interactions between these magnetic layers is weak in comparison to the interlayer interactions, the ratio being of the order 10d4 [3], and thus KFeF, can essentially be thought of as a two-dimensional antiferromagnet. With this in mind, we have carried out a neutron scattering study of KFeF4, using the IN3 triple axis spectrometer at the ILL in Grenoble. Constant Q scans [4] were used to determine the dispersion relation of the magnetic excitations and they have been modelled using linear spin wave theory [S]. The temperature dependence of the anisotropy energy has also been investigated and is found to vary in the same way as the staggered magnetisation [4].
2. Theory The KFeF,
Hamiltonian for the has the form
H = -g/-@, -2
c
1
S: - g/+,H,
c J,,.S;S,,-2 (I. 1’)
x J,,,,S,
‘Sm. -2
magnetic
interactions
in
c SZ,
m
c (m. rn’) c J,,S, (1. m)
‘S,,, .
(1)
For this system H, represents the effective field due to an anisotropy arising from dipolar effects [6], which establishes c as the preferred direction for the spins to align along. Using linear spin wave theory as discussed by Keffer [5] the dispersion relation for a 2 D antiferromagnet, with nearest neighbour and next nearest neighbour interactions, was formulated as 0921.4526/92/$05.00
0
1992 - Elsevier
Science
Publishers
+ J,) + 8SJ,
- 4S(J,
- 4SJ,(cos(
4,~ + q,,b) + cos( q,a - q,b)]’
- [4S(J, cos( q,a) + J, cos( q,,b))]’
(2)
Equation (2) represents the energies of the spin waves with wave vector q. This model allows for anisotropic nearest neighbour interactions in the a and b directions, J, and J,, as well as for next nearest neighbour interaction, J3. In this equation a and b represent the magnetic lattice constants in the antiferromagnetic layer, which at 50 K were found to be 3.784 and 3.875 A, respectively. 3. Results In our experiment, magnetic Bragg reflections were studied at the (i 4 0) and ($ f 0) reciprocal lattice points. The spin waves were determined for propagation directions [ 5 501, [ 5 0 0] and [0 5 0] at 50 K. which is well below the NCel temperature: T, = 137 K. [ is the reduced wave vector, which for the a direction is given by q,ai2a, and similarly for the b direction. In order to determine spin wave frequencies, the constant Q scans were fitted to Gaussians using the Marquardt least squares fit method [7]. The dispersion relation is shown in fig. 1. The solid line represents the fit to the experimental data. Fitting was done using eq. (2), allowing for nearest and next nearest neighbour interactions. Values for the exchange constants, calculated from the fit, are shown in table 1. These results are in good agreement with previous studies [8], which calculated Table 1 Results from J,
bevl
-1.00
B.V. All rights
+ 0.05
reserved
dispersion
relation
fit.
Jz [mevl
J3 bvl
-1.37
-0.06
? 0.05
~4, 2 0.02
bvl
0.12 + 0.005
S. Fulton et al. i Spin waves tn KFeF,
226
24 20 5 j
I6 h M
b d w
12
8 4 0
24
I 0
0.25
c
24
2
A m = 0.28
1
16
c
bo
10.0
2 e,
a -. $
t
9.5
rl
[WI
0.25
f
‘Z 4
16
0
.*
2
i 0
0.25
3
Log,(T,-T)
4
1
2
3
4
Log,(T,-T)
0.50
c
Fig. 1. The spin wave dispersion relation for KFeF, at 50 K. The solid line is the best-fit to the theoretical model, using the parameters listed in table 1.
a nearest neighbour exchange constant of - 1.15 meV. It is seen that our results show a difference in nearest neighbour exchange constants J, and J?. This is caused by the different superexchange interactions in the u and b directions, due to the tilting of the flourine ions out of the plane in the a direction [2], thus making J, smaller than J,, even though the lattice constant a is less than b. The experimental value for the anisotropy energy gp,,H, is larger than expected for dipolar interactions (0.033 meV) [6], and thus there must be other contributions which should be taken into account when calculating H,, which are not important in the analogous Rb,MnF, with S = $ ions. At present we do not have a convincing explanation for this difference. A temperature dependent study of the anisotropy effect, which causes the gap in the dispersion relation at the magnetic zone centre, is shown in fig. 2, where m represents the gradient of the graph. This is seen to vary in the same way as the peak height of the (4 f 0) magnetic point, which has m = 2p. p is the critical
dence
Comparing the gradient of the temperature depenof the anisotropy energy with the peak height of the
(: J 0) magnetic point. exponent arising from the staggered magnetization temperature dependence. which for a two-dimensional Ising system is 0.125 [4]. Thus the anisotropy energy varies in the same way as the staggered magnetization. References
Ill
G. Heger and R. Geller. Phys. Stat. Sol. (b) 53 (1972) 227. PI G. Heger, R. Geller and D. Babel, Solid State Commun. Y (lY71) 335. PI H. Keller and I.M. Savic, Phys. Rev. B 28 (lY83) 2638. (Oxford Uni[41 M.F. Collins, Magnetic Critical Scattering versity Press. lY8Y). of Physics. Vol. 18. ed. S. 151 F. Keffer. in: Encyclopedia Flugge (Springer, Berlin, 1966) part 2. Phys. 161 H.W. de Wijn. L.R. Walker and R.E. Walsted. Rev. B 8 (1973) 285. S.A. Teukolsky and W.T. [71 W.H. Press. B.P. Flannery, Vellering, Numerical Recipes (Cambridge University Press, 1986). G.R. Davidson, H.J. Guggenheim and 181 M. Eibschutz. D.E. Cox. Proc. of the 17th Annual Conference on Magnetism and Magnetic Materials, eds. C.D. Graham and J.J. Rhyne (AIP. New York. lY72) p. 670.