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Ferromagnetic resonance in hexagonal ferrites with anisotropic g-factors
1239
FERROMAGNETIC RESONANCE IN HEXAGONAL FERRITES WITH ANISOTROPIC g-FACTORS* C. H W A and L.M. S I L B E R Polytechnic Institute of New York, Brooklyn, New York 11201, USA
Ferromagnetic resonance measurements on the hexagonal ferrite MgY are compared to theoretical predictions of frequency of resonance and linewidth of the Smit-Suhl and Rado models. At fixed frequency, both models give good agreement, but neither model predicts the experimentally observed dependence on frequency or field of the g-tensor component g±.
Recent observations of ferromagnetic resonance in hexagonal ferrites [1,2] indicate that the results cannot be understood unless one assumes that the magnetomechanical ratio, or g-factor, is a tensor, rather than a scalar quantity. In this paper we compare our experimental observations of resonance in the ferrite MgY [Ba2Mg2Fe~2022] with the predictions of two theoretical models. The model of Gurevich [3] is based on the Smit-Suhl method and assumes that the angular m o m e n t u m is conserved, although the magnetization is not. H e has derived expressions for the f r e q u e n c y of resonance for a static field H0 oriented at an angle 0r~ with respect to the hexagonal (Z) axis [3]. In the principal directions (/4o parallel or perpendicular to the hexagonal axis), he finds fll = gll[H0 + (g±/glL)Ha] = gll[n0 + High],
(1)
= g ± [ H o ( H o - H a - H 0 ] 'n = g ± [ H o ( H o - H,~n)]'/2.
(2)
Rado [4] has proposed a more general formulation of resonance in materials with anisotropic g-factors. We have applied his model to materials of hexagonal symmetry. For the static field either parallel or perpendicular to the hexagonal axis, the results are equivalent to eqs. (1) and (2). In order to calculate the resonance f r e q u e n c y for an arbitrary orientation of the applied field, one must know the angular dependence of M or J. We have calculated the expression for the resonance f r e q u e n c y for arbitrary field orientation, for an arbitrary dependence of M on 0. * Supported by the Joint Services Electronics Program and N.S.F. Based on a dissertation by C. Hwa submitted to the Dep't. of EE/EP, Polytechnic Institute of New York, in partial fulfillment of the requirements of the Ph.D. degree.
Physica 86-88B (1977) 1239-1240 O North-Holland
Measurements were made on the material MgY. An ellipsoid 1.1mm in diameter and 0.2ram in thickness was studied. Resistivity measurements indicated a v e r y small ferrous ion content. Microwave measurements were made by the usual cavity perturbation technique. Measurements were made with the static field in the basal plane over a f r e q u e n c y range of 8.4 to 37.3 G H z and with the static field along the hexagonal axis (the hard direction) over a f r e q u e n c y range 1.3 to 9 . 8 G H z . The experimental results of f r e q u e n c y and field for resonance were fitted to eqs. (1) and (2) b y a least squares computer program, to determine the parameters g±gll, H ~ ~, HI~ ~" As had been observed in the previous studies, the data in the hard direction agree well with the theoretical expression. For glL we find 2.468 + 0.005 MHz/Oe, HI~fr = - 14.90_ 0.005 kOe. The error estimates are the standard deviations, and include uncertainties in measurement of f r e q u e n c y and field. The experimental results for resonance with the field oriented in the easy plane cannot be fitted to the theoretical relation (2) with unique values of g± and H~ n. If we break up the data into groups (by f r e q u e n c y range of the waveguide employed), a good fit may be obtained, but g± is now a function of f r e q u e n c y (or equivalently, magnetic field). In fig. 1 we show g. as a function of frequency, as well as similar results for a previously measured material, MnZnY [2]. Thus, as we have previously observed, the g-tensor c o m p o n e n t g, is f r e q u e n c y or field dependent, approaching gll as a limit. In fig. 2 we show the results of measurement of the static field for resonance as a function of field orientation in MgY, at a fixed f r e q u e n c y of 9 . 3 G H z , and the difference between the theoretical and measured fields for the two
1240
theoretical models. In general, both theoretical models give good agreement with experimental results. The maximum deviation between theory and experiment is 750Oe for the Smit-Suhl theory, and 400Oe for the Rado theory. The symmetry of the large deviations near the hard direction makes us believe they are real, rather than an experimental artifact. This same effect has been observed in MnZnY. Observations of the linewidth as a function of direction show a smooth variation with orientation, Calculations of the linewidth, assuming an isotropic intrinsic damping parameter, also show that it is anisotropic. We calculate the ratio of linewidth along the hexagonal axis to that in basal plane to be 2.83. The experimental values are AHII= 97 Oe, AH± = 33.50e, with a ratio of 2.89. Since the calculation considers only intrinsic relaxation mechanisms, and neglects other contributions, such as surface scattering, the agreement is very satisfactory. In summary, we can account well for resonance in an arbitrary direction at fixed frequency with the theoretical models discussed. However, the variation with frequency or field of the g tensor component g. remains unexplained.
KEY 29
MgY ~MnZnY Value of g.
28
2 7
2.6
25
2
40
IJ0
__2. 0L ~
3'0
4•0
5LO
6O
FREQUENCY (GHz)
Fig. 1. g-factor as a function of frequency.
~ LOI ~
o
~o
o
~: - 0 5 2o 18 16 14
~0
References
~30
150
170
tgo
210
Fig. 2. Rotation curve for MgY.
230
250
270
[1] V.K. Kunevich et al., Bulletin of the Academy of Sciences of the USSR, Physics Series XXXV (1971) 1132. [2] L.M. Silber, J. Appl. Physics 44 (1973) 1855. [3] A.G. Gurevich et al., J. Appl. Physics 40 (1969) 1512. [4] G.T. Rado, Phys. Rev. B5 (1972) 1021.
FERROMAGNETIC RESONANCE IN HEXAGONAL FERRITES WITH ANISOTROPIC g-FACTORS* C. H W A and L.M. S I L B E R Polytechnic Institute of New York, Brooklyn, New York 11201, USA
Ferromagnetic resonance measurements on the hexagonal ferrite MgY are compared to theoretical predictions of frequency of resonance and linewidth of the Smit-Suhl and Rado models. At fixed frequency, both models give good agreement, but neither model predicts the experimentally observed dependence on frequency or field of the g-tensor component g±.
Recent observations of ferromagnetic resonance in hexagonal ferrites [1,2] indicate that the results cannot be understood unless one assumes that the magnetomechanical ratio, or g-factor, is a tensor, rather than a scalar quantity. In this paper we compare our experimental observations of resonance in the ferrite MgY [Ba2Mg2Fe~2022] with the predictions of two theoretical models. The model of Gurevich [3] is based on the Smit-Suhl method and assumes that the angular m o m e n t u m is conserved, although the magnetization is not. H e has derived expressions for the f r e q u e n c y of resonance for a static field H0 oriented at an angle 0r~ with respect to the hexagonal (Z) axis [3]. In the principal directions (/4o parallel or perpendicular to the hexagonal axis), he finds fll = gll[H0 + (g±/glL)Ha] = gll[n0 + High],
(1)
= g ± [ H o ( H o - H a - H 0 ] 'n = g ± [ H o ( H o - H,~n)]'/2.
(2)
Rado [4] has proposed a more general formulation of resonance in materials with anisotropic g-factors. We have applied his model to materials of hexagonal symmetry. For the static field either parallel or perpendicular to the hexagonal axis, the results are equivalent to eqs. (1) and (2). In order to calculate the resonance f r e q u e n c y for an arbitrary orientation of the applied field, one must know the angular dependence of M or J. We have calculated the expression for the resonance f r e q u e n c y for arbitrary field orientation, for an arbitrary dependence of M on 0. * Supported by the Joint Services Electronics Program and N.S.F. Based on a dissertation by C. Hwa submitted to the Dep't. of EE/EP, Polytechnic Institute of New York, in partial fulfillment of the requirements of the Ph.D. degree.
Physica 86-88B (1977) 1239-1240 O North-Holland
Measurements were made on the material MgY. An ellipsoid 1.1mm in diameter and 0.2ram in thickness was studied. Resistivity measurements indicated a v e r y small ferrous ion content. Microwave measurements were made by the usual cavity perturbation technique. Measurements were made with the static field in the basal plane over a f r e q u e n c y range of 8.4 to 37.3 G H z and with the static field along the hexagonal axis (the hard direction) over a f r e q u e n c y range 1.3 to 9 . 8 G H z . The experimental results of f r e q u e n c y and field for resonance were fitted to eqs. (1) and (2) b y a least squares computer program, to determine the parameters g±gll, H ~ ~, HI~ ~" As had been observed in the previous studies, the data in the hard direction agree well with the theoretical expression. For glL we find 2.468 + 0.005 MHz/Oe, HI~fr = - 14.90_ 0.005 kOe. The error estimates are the standard deviations, and include uncertainties in measurement of f r e q u e n c y and field. The experimental results for resonance with the field oriented in the easy plane cannot be fitted to the theoretical relation (2) with unique values of g± and H~ n. If we break up the data into groups (by f r e q u e n c y range of the waveguide employed), a good fit may be obtained, but g± is now a function of f r e q u e n c y (or equivalently, magnetic field). In fig. 1 we show g. as a function of frequency, as well as similar results for a previously measured material, MnZnY [2]. Thus, as we have previously observed, the g-tensor c o m p o n e n t g, is f r e q u e n c y or field dependent, approaching gll as a limit. In fig. 2 we show the results of measurement of the static field for resonance as a function of field orientation in MgY, at a fixed f r e q u e n c y of 9 . 3 G H z , and the difference between the theoretical and measured fields for the two
1240
theoretical models. In general, both theoretical models give good agreement with experimental results. The maximum deviation between theory and experiment is 750Oe for the Smit-Suhl theory, and 400Oe for the Rado theory. The symmetry of the large deviations near the hard direction makes us believe they are real, rather than an experimental artifact. This same effect has been observed in MnZnY. Observations of the linewidth as a function of direction show a smooth variation with orientation, Calculations of the linewidth, assuming an isotropic intrinsic damping parameter, also show that it is anisotropic. We calculate the ratio of linewidth along the hexagonal axis to that in basal plane to be 2.83. The experimental values are AHII= 97 Oe, AH± = 33.50e, with a ratio of 2.89. Since the calculation considers only intrinsic relaxation mechanisms, and neglects other contributions, such as surface scattering, the agreement is very satisfactory. In summary, we can account well for resonance in an arbitrary direction at fixed frequency with the theoretical models discussed. However, the variation with frequency or field of the g tensor component g. remains unexplained.
KEY 29
MgY ~MnZnY Value of g.
28
2 7
2.6
25
2
40
IJ0
__2. 0L ~
3'0
4•0
5LO
6O
FREQUENCY (GHz)
Fig. 1. g-factor as a function of frequency.
~ LOI ~
o
~o
o
~: - 0 5 2o 18 16 14
~0
References
~30
150
170
tgo
210
Fig. 2. Rotation curve for MgY.
230
250
270
[1] V.K. Kunevich et al., Bulletin of the Academy of Sciences of the USSR, Physics Series XXXV (1971) 1132. [2] L.M. Silber, J. Appl. Physics 44 (1973) 1855. [3] A.G. Gurevich et al., J. Appl. Physics 40 (1969) 1512. [4] G.T. Rado, Phys. Rev. B5 (1972) 1021.