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Spin waves in the spin-flop phase of RbMnF3
Journal of Magnetism and Magnetic Materials 104-107 (1992) 1079-1080 North-Holland
Spin waves in the spin-flop phase of RbMnF 3 D.A. Tennant a, D.F. McMorrow a, S.E. Nagler a,1 and B. F~k b " Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, UK h lnstitut Laue- Langevin, Grenoble, France The spin waves in the spin-flop phase of the three-dimensional Heisenberg antiferromagnet RbMnF 3 have been measured using a neutron triple-axis spectrometer. Magnetic fields of up to 6 T were applied perpendicular to the (1, 1, 0) scattering plane. In this phase the two-fold degeneracy of the zero-field spin waves is lifted, and we have measured the variation of this splitting both as a function of neutron wavevector transfer and applied field. Our results are discussed with reference to the predictions of linear spin-wave theory. R b M n F s is probably the closest realisation of an isotropic, 3D Heisenberg antiferromagnet (AF). When a uniform magnetic field, _> 0.26 T, is applied below the Ndel temperature the system enters a spin-flop phase. In this phase the manganese moments align antiferromagnetically, almost perpendicular to the field but slightly canted out of the plane in the direction of the field. As far as we are aware the spin-wave dynamics in such a spin-flop phase have not previously been measured. In the analogous one-dimensional systems two-magnon processes give rise to additional modes [1] which were not predicted on the basis of linear spinwave theory, and there is evidence that this may also occur in two-dimensional systems [2]. In the absence of a magnetic field the spin waves are two-fold degenerate. Windsor and Stevenson [3] have measured the dispersion of the magnons in R b M n F 3 in zero field using neutron scattering. They were able to explain their results using linear spin-wave theory with an isotropic exchange constant of (0.29 + 0.03) m e V between nearest neighbours, and a secondneighbour constant of less than 0.02 meV. Antiferromagnetic resonance measurements [4] indicate a magnetic anisotropy of only 4.5 G. The application of a field breaks the rotational symmetry of the modes and the dispersion splits into two branches. We have calculated the frequencies for the two branches in a field (see also Keffer [5]), where the anisotropy is neglected. Balancing the torques of the applied magnetic field H 0 and exchange field, the canting angle ¢b is given by sin ~b = glXBHo/2SJ(O),
(1)
where J(q) is the Fourier transform of the exchange interaction and S is the spin quantum number. By solving for the normal modes of the two sublattices, the spin-wave frequencies are found to be,
h 2 w ( q ) 2 = S 2 ( j ( O ) - T J ( q ) ) ( J ( O ) +_J(q) - h ) ,
(2)
i Permanent address: Department of Physics, University of Florida, Gainesville, FL 32611, USA
where the term h varies with field as,
h = - ( gtxBHo)2/ZS2J(O)(cos2~ - sin2qS). Using J(0) = (3.48 + 0.36) meV [3], the splitting of the two modes, in meV, at an antiferromagnetic zone centre should then vary directly with the field, in tesla, as
0.1158H,,. Our measurements of the spin waves were performed on the IN12 triple-axis neutron spectrometer at the Institut L a u e - L a n g e v i n . The sample was aligned in zero field with the (1, 1, 0) axis vertical. Constant Qscans, at a fixed incident neutron wavevector of k~ = 1.25 ,~-1, were p e r f o r m e d with Q along the (-1,-1, 1) direction, around the antiferromagnetic I point "l"M = ( - / , __ 2, 1 ) , T h e s e s c a n s w e r e repeated in a number of different magnetic fields up to 5.68 T, generated by a superconducting magnet, applied perpendicular to the scattering plane. The sample was cooled well below the N f e l temperature of T N = 83 K in a cryostat, and kept at a constant temperature of 4 K throughout. In an applied field of 5.68 T a constant-Q scan at the A F zone centre CM revealed an excitation at = 0.6 meV, in additon to a strong quasi-elastic component. (In zero field this excitation was found to be absent.) When scans were performed away from I"M for reduced neutron wavevector transfer [ - ~ , -~:, ~:] in the range 0 _< ~_< 0.02 two modes were resolved clearly, with the second, lower mode emerging from the quasielastic scattering. For ~: _> 0.02 the two modes merged and could not be resolved. The dispersion of the modes is shown in fig. 1, where we have represented by solid lines the dispersion predicted by the above linear spinwave theory. The agreement between experiment and theory is reasonable in view of the fact that there were no adjustable parameters. There is, however, a systematic tendency for the experimental points to lie above the theoretical lines at higher wavevectors. The zone-centre frequency was measured using constant-Q scans as the field was reduced from 5.68 T to zero. The field dependence of the frequency is shown
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1080
D.A. Tennant et al. / Spin waves in the spin-flop phase of RbMnF 3
~> 1.5-
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Fig. 1. The spin-wave dispersion observed along the ( - 1, - 1, 1) direction in an applied field of 5.68 T. The solid lines represent the results of linear spin-wave theory and are discussed in the text.
in fig. 2, a n d was f o u n d to b e in good a g r e e m e n t with linear spin-wave theory. O n e striking f e a t u r e of t h e line shapes of the excita-
~:~0.75
/ ./ /
/m //
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tions was t h a t they were strongly asymmetric. T h e most p r o b a b l e explanation for this is the fact that we used a focused m o n o c h r o m a t o r , resulting in p o o r vertical resolution. To explore this m o r e fully we c o m p u t e d a functional form which takes account of the in-plane resolution as well as the out-of-plane resolution, with corrections for the spin-wave intensities, t h e r m a l occupation a n d a c o n s t a n t k i correction factor [6]. T h e calculated form fits the scan at t h e A F point in zero field, a n d as the lower m o d e is largely i n d e p e n d e n t of field, the function, with only an a m p l i t u d e correction, may b e used to fit the lower m o d e at all fields at the A F point. T h e calculated function was used to fit thc u p p e r m o d e a n d the fitted values for the energy gap used in fig. 2. It was f o u n d t h a t the in-plane and vertical Q-resolution alone are not sufficient to acc o u n t for the u p p e r m o d e line shape, a n d a full convolution with the energy is t h o u g h t to b e necessary. F u r t h e r resolution calculations are b e i n g u n d e r t a k e n to m o r e accurately fit the spin-wave frequencies b o t h at the A F point a n d f u r t h e r out in the B Z b u t at this stage t h e observed b e h a v i o u r of the spin waves in the spin-flop p h a s e seem to be a c c o u n t e d for by the linear spin-wave theory outlined above.
W e would like to t h a n k C E N G for providing the s u p e r c o n d u c t i n g m a g n e t a n d the S E R C for funding the experiment.
/lll
O
0.5 //
I1)
References
/
~0.25 //"
I
o N
z/
/
/W
/ 0.0 //~ 0
1
2
4
3
5
6
,|,
Applied Field Fig. 2. The field dependence of the splitting of the spin-wave modes at the zone-centre. The solid line was calculated using linear spin-wave theory.
[1] I.U. Heilmann, J.K. Kjems, Y. Endoh, G.F. Reiter, G. Shirane and R.J. Birgeneau, Phys. Rev. B 24 (1981) 3939. [2] H. Tietze-Jaensch, D. Sieger, P. Schweiss, L.P. Regnault, H. Jaintner, R. Geick, W. Treutmann, B. F~k, J. Magn. Magn. Mater. 104-107 (1992) 897. [3] C.G. Windsor and R.W.H. Stevenson, Proc. Phys. Soc. 87 (1966) 501. [4] D.T. Teaney, M.J. Freiser and R.W.H. Stevenson, Phys. Rev. Lett. 9 (1962) 212. [5] F. Keffer, Handbuch der Physik, Vol. 18, part 2, ed. S. Flugge (Springer, Berlin, 1966) chap. 49. [6] B. Dorner, Acta Cryst. A 28 (1972) 319.
Spin waves in the spin-flop phase of RbMnF 3 D.A. Tennant a, D.F. McMorrow a, S.E. Nagler a,1 and B. F~k b " Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, UK h lnstitut Laue- Langevin, Grenoble, France The spin waves in the spin-flop phase of the three-dimensional Heisenberg antiferromagnet RbMnF 3 have been measured using a neutron triple-axis spectrometer. Magnetic fields of up to 6 T were applied perpendicular to the (1, 1, 0) scattering plane. In this phase the two-fold degeneracy of the zero-field spin waves is lifted, and we have measured the variation of this splitting both as a function of neutron wavevector transfer and applied field. Our results are discussed with reference to the predictions of linear spin-wave theory. R b M n F s is probably the closest realisation of an isotropic, 3D Heisenberg antiferromagnet (AF). When a uniform magnetic field, _> 0.26 T, is applied below the Ndel temperature the system enters a spin-flop phase. In this phase the manganese moments align antiferromagnetically, almost perpendicular to the field but slightly canted out of the plane in the direction of the field. As far as we are aware the spin-wave dynamics in such a spin-flop phase have not previously been measured. In the analogous one-dimensional systems two-magnon processes give rise to additional modes [1] which were not predicted on the basis of linear spinwave theory, and there is evidence that this may also occur in two-dimensional systems [2]. In the absence of a magnetic field the spin waves are two-fold degenerate. Windsor and Stevenson [3] have measured the dispersion of the magnons in R b M n F 3 in zero field using neutron scattering. They were able to explain their results using linear spin-wave theory with an isotropic exchange constant of (0.29 + 0.03) m e V between nearest neighbours, and a secondneighbour constant of less than 0.02 meV. Antiferromagnetic resonance measurements [4] indicate a magnetic anisotropy of only 4.5 G. The application of a field breaks the rotational symmetry of the modes and the dispersion splits into two branches. We have calculated the frequencies for the two branches in a field (see also Keffer [5]), where the anisotropy is neglected. Balancing the torques of the applied magnetic field H 0 and exchange field, the canting angle ¢b is given by sin ~b = glXBHo/2SJ(O),
(1)
where J(q) is the Fourier transform of the exchange interaction and S is the spin quantum number. By solving for the normal modes of the two sublattices, the spin-wave frequencies are found to be,
h 2 w ( q ) 2 = S 2 ( j ( O ) - T J ( q ) ) ( J ( O ) +_J(q) - h ) ,
(2)
i Permanent address: Department of Physics, University of Florida, Gainesville, FL 32611, USA
where the term h varies with field as,
h = - ( gtxBHo)2/ZS2J(O)(cos2~ - sin2qS). Using J(0) = (3.48 + 0.36) meV [3], the splitting of the two modes, in meV, at an antiferromagnetic zone centre should then vary directly with the field, in tesla, as
0.1158H,,. Our measurements of the spin waves were performed on the IN12 triple-axis neutron spectrometer at the Institut L a u e - L a n g e v i n . The sample was aligned in zero field with the (1, 1, 0) axis vertical. Constant Qscans, at a fixed incident neutron wavevector of k~ = 1.25 ,~-1, were p e r f o r m e d with Q along the (-1,-1, 1) direction, around the antiferromagnetic I point "l"M = ( - / , __ 2, 1 ) , T h e s e s c a n s w e r e repeated in a number of different magnetic fields up to 5.68 T, generated by a superconducting magnet, applied perpendicular to the scattering plane. The sample was cooled well below the N f e l temperature of T N = 83 K in a cryostat, and kept at a constant temperature of 4 K throughout. In an applied field of 5.68 T a constant-Q scan at the A F zone centre CM revealed an excitation at = 0.6 meV, in additon to a strong quasi-elastic component. (In zero field this excitation was found to be absent.) When scans were performed away from I"M for reduced neutron wavevector transfer [ - ~ , -~:, ~:] in the range 0 _< ~_< 0.02 two modes were resolved clearly, with the second, lower mode emerging from the quasielastic scattering. For ~: _> 0.02 the two modes merged and could not be resolved. The dispersion of the modes is shown in fig. 1, where we have represented by solid lines the dispersion predicted by the above linear spinwave theory. The agreement between experiment and theory is reasonable in view of the fact that there were no adjustable parameters. There is, however, a systematic tendency for the experimental points to lie above the theoretical lines at higher wavevectors. The zone-centre frequency was measured using constant-Q scans as the field was reduced from 5.68 T to zero. The field dependence of the frequency is shown
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1080
D.A. Tennant et al. / Spin waves in the spin-flop phase of RbMnF 3
~> 1.5-
0 0
t~ 1.0
/lit
~'~J
-J/
~/
z •/
qD
aa
/ °- ~
_ ~
• •
0.5
////"
z
~
• /-/
I ,~..q
q2~ if? 0.0
•
/i
I
I
I
0.0
~
I
~
I
I
~-[
I
•
l
0.01
0.005
]
~
I--T
0.015
Reduced Wavevectur
0.02
(r. 1. u.)
Fig. 1. The spin-wave dispersion observed along the ( - 1, - 1, 1) direction in an applied field of 5.68 T. The solid lines represent the results of linear spin-wave theory and are discussed in the text.
in fig. 2, a n d was f o u n d to b e in good a g r e e m e n t with linear spin-wave theory. O n e striking f e a t u r e of t h e line shapes of the excita-
~:~0.75
/ ./ /
/m //
/
tions was t h a t they were strongly asymmetric. T h e most p r o b a b l e explanation for this is the fact that we used a focused m o n o c h r o m a t o r , resulting in p o o r vertical resolution. To explore this m o r e fully we c o m p u t e d a functional form which takes account of the in-plane resolution as well as the out-of-plane resolution, with corrections for the spin-wave intensities, t h e r m a l occupation a n d a c o n s t a n t k i correction factor [6]. T h e calculated form fits the scan at t h e A F point in zero field, a n d as the lower m o d e is largely i n d e p e n d e n t of field, the function, with only an a m p l i t u d e correction, may b e used to fit the lower m o d e at all fields at the A F point. T h e calculated function was used to fit thc u p p e r m o d e a n d the fitted values for the energy gap used in fig. 2. It was f o u n d t h a t the in-plane and vertical Q-resolution alone are not sufficient to acc o u n t for the u p p e r m o d e line shape, a n d a full convolution with the energy is t h o u g h t to b e necessary. F u r t h e r resolution calculations are b e i n g u n d e r t a k e n to m o r e accurately fit the spin-wave frequencies b o t h at the A F point a n d f u r t h e r out in the B Z b u t at this stage t h e observed b e h a v i o u r of the spin waves in the spin-flop p h a s e seem to be a c c o u n t e d for by the linear spin-wave theory outlined above.
W e would like to t h a n k C E N G for providing the s u p e r c o n d u c t i n g m a g n e t a n d the S E R C for funding the experiment.
/lll
O
0.5 //
I1)
References
/
~0.25 //"
I
o N
z/
/
/W
/ 0.0 //~ 0
1
2
4
3
5
6
,|,
Applied Field Fig. 2. The field dependence of the splitting of the spin-wave modes at the zone-centre. The solid line was calculated using linear spin-wave theory.
[1] I.U. Heilmann, J.K. Kjems, Y. Endoh, G.F. Reiter, G. Shirane and R.J. Birgeneau, Phys. Rev. B 24 (1981) 3939. [2] H. Tietze-Jaensch, D. Sieger, P. Schweiss, L.P. Regnault, H. Jaintner, R. Geick, W. Treutmann, B. F~k, J. Magn. Magn. Mater. 104-107 (1992) 897. [3] C.G. Windsor and R.W.H. Stevenson, Proc. Phys. Soc. 87 (1966) 501. [4] D.T. Teaney, M.J. Freiser and R.W.H. Stevenson, Phys. Rev. Lett. 9 (1962) 212. [5] F. Keffer, Handbuch der Physik, Vol. 18, part 2, ed. S. Flugge (Springer, Berlin, 1966) chap. 49. [6] B. Dorner, Acta Cryst. A 28 (1972) 319.