magnetomechanical ratio or g-factor. The results of these experiments are compared with calculations based on the equations of motion, which ensure the ...
PHYSICS OF THE SOLID STATE
VOLUME 41, NUMBER 6
JUNE 1999
Ferromagnetic resonance in the polycrystalline hexagonal ferrites Co22 x Znx W V. A. Zhuravlev*) Tomsk State University, 634050 Tomsk, Russia
~Submitted August 5, 1998! Fiz. Tverd. Tela ~St. Petersburg! 41, 1050–1053 ~June 1999! Experimental studies of the ferromagnetic resonance spectra of polycrystalline hexagonal ferrites of the system Co22x Znx W in the frequency range 16–32 GHz are described. It is shown that interpretation of the experimental data requires the assumption of an anisotropic effective magnetomechanical ratio ~or g-factor!. The results of these experiments are compared with calculations based on the equations of motion, which ensure the conservation of the mechanical moment length. Concentration dependences are determined for the components of the magnetomechanical ratio tensor and the anisotropy field of this system of hexagonal ferrites at room temperature. Possible reasons why anisotropy fields measured in the vicinity of a spin reorientation transition will differ from results given by other methods are discussed. © 1999 American Institute of Physics. @S1063-7834~99!02606-4#
ropy fields the authors of Refs. 2, 5, and 6 assumed that the g-factor equaled two, while in Ref. 7 it was left unestimated. However, it is well known ~see for example Refs. 8–10! that in hexagonal ferrites the g-factor differs greatly from two, and in uniaxial materials it can also be anisotropic under certain conditions.11–13 When u k 1 u ' u k 2 u , u k 3 u holds, which occurs in the neighborhood of a spin-reorientation phase transition,14 the angular function H R (Q,F) can develop additional stationary directions15 at angles Q other than zero or p /2. Hence, it is necessary to identify in advance the maxima and steps in the resonance curves observed during the experiments with the stationary directions corresponding to them. Here one must keep in mind that in real materials the damping of uniform precession will smooth out features on the resonance curves and the field values at which they are observed will differ as a rule from the field H R (Q i ,F i ) of the corresponding stationary direction ~see Ref. 15, and also Fig. 1 of this paper!. Thus, in order to increase the accuracy of estimates of the anisotropy field based on FMR experiments in polycrystalline hexagonal ferrites, we must first examine the range of frequencies used to determine the value of the g-factor and possible anisotropy of the latter. Secondly, we must compare not simply the values of the fields for the maxima and steps on the experimental curves with Eq. ~4!, but rather the overall shapes of the calculated and experimental curves. In Ref. 15, the author and his colleagues derived a method of calculating the resonance curve of a polycrystalline hexagonal ferrite in the independent-grain approximation for arbitrary relations between the anisotropy constants, based on solving the equation of motion of the Landau–Lifshits magnetization vector. However, the mathematical relations used there do not apply to a medium with an anisotropic g-factor, since this equation does not conserve the lengths of the mechanical and magnetic moment vectors, nor does it fulfill the law of conservation of energy. A generalization of the calculations of Ref. 15 will be given below.
The complex structure of ferromagnetic resonance ~FMR! lines—additional maxima in the absorption, steps, etc.—observed in polycrystalline ferrites with large magnetocrystalline anisotropy was first explained by Schlo¨mann within a ‘‘independent grain’’ model. In Refs. 1–3 he showed that the features in the FMR curves occur at values of the magnetizing field corresponding to stationary directions in the angular dependence of the resonance field H R (Q,F) for single-crystal grains, i.e., the crystallites that make up the polycrystal. Here Q,F are the polar and azimuthal angles of the magnetizing field vector relative to the crystallographic axes of the crystal. These directions (Q i ,F i ), which correspond to maxima, minima, and saddle points of the surface H R (Q,F), satisfy the condition H R (Q,F)50. In hexagonal crystals, the magnetocrystalline anisotropy energy has the form F a 5k 1 sin2 u 1k 2 sin4 u 1k 3 sin6 u 1k 4 sin6 u cos 6 w ,
~1!
where u,w are angles of the magnetization vector and k i are anisotropy constants. The natural stationary directions for a spherical sample in the limit u k 1 u @ u k 2 u , u k 3 u are along the hexagonal axis (Q 1 5 u 1 50) and in the basal plane ~Q 2,3 5 u 2,35 p /2, F 2 5 w 2 5 p /6, F 3 5 w 3 50!. For these directions, the FMR resonance frequencies of a spherical sample can be easily obtained, e.g., by the method of Suhl and Smith,4 see Eq. ~4! below with g i 5 g' 5 g . Here g 5ge/2mc is the magnetomechanical ratio, g is the effective g-factor, e and m are the charge and mass of an electron respectively, and c is the velocity of light. The availability of the analytic expression ~4! makes it possible to determine the anisotropy field based on features on the experimental FMR curve of the polycrystals. Using this method, the authors of Refs. 2, 5 and 6 found the anisotropy fields for a number of ferrites with hexagonal structure, and ferrites with cubic and tetragonal structures in Refs. 7 and 8. The measurements reported in Refs. 2, 5–7 were made at one frequency; therefore, in calculating the anisot1063-7834/99/41(6)/4/$15.00
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© 1999 American Institute of Physics
Phys. Solid State 41 (6), June 1999
V. A. Zhuravlev
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F w8 ~ u 0 , w 0 ,Q,F ! 50,
9 2 ~ F u9 w ! 2 # . v 2 5 ~ I sin u 0 ! 22 @ F 9uu F ww
~3!
The first two equations in ~3! determine the equilibrium orientation of the mechanical moment, i.e., the angles u 0 , w 0 , while the third determines the resonance frequency for uniform precession at preset angles Q,F. The system ~3! is solved numerically by the method of successive root refinement, starting with Q5Q 3 , F5F 3 . We varied the angles Q,F within the ranges p /2>Q>DQ, 00, H a1 ,0, H a2 > u H a1 u , which are typical for a cone of easy magnetization. Although the applicability of the independent grain model to materials with anisotropy fields comparable in magnitude to the saturation magnetization, i.e., materials such as the hexagonal ferrites Co22x Znn W with 1.2
