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A note on FMR lineshape of conductive spheres
Journal of Magnetism and Magnetic Materials 50 (1985) 239-241 North-Holland, Amsterdam
239
A NOTE ON FMR LINESHAPE OF CONDUCTIVE SPHERES Y. WATANABE Department of Physics, Shimane University, Matsue 690, Japan
S. SAITO Department of Electronics, University of Osaka Prefecture, Sakai, Osaka 591, Japan
and M. MARYSKO Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czechoslovakia Received 21 November 1984
A shoulder-shaped anomaly on the high-field side of the main FMR maximum has been re-examined in connection with a recent publication on FMR in ferromagnetic semiconductors. The positions of the shoulder are measured at 9.3 and 14.8 GHz for different values of resistivities, magnetization and sphere sizes. Results clarify the role of the surface and volume type excitations in the occurrence of the shoulder.
The skin effect in ferromagnetic resonance (FMR) has been extensively studied for single crystals of conductive ferrite containing excess irons. The FMR line of conductive spheres is characterized by an asymmetric shape with a terraced edge (or a shoulder) on the high-field side of the main absorption maximum [1], as seen in fig. 1. The origin of such a shoulder was first studied in ref. [2] and discussed in connection with the excitation of certain volume modes similar to the magnetostatic modes [3]. However, numerical
8 o /
B=,0m,
magnetic field Fig. 1. A typical FMR curve obtained for a sphere of MnFe204 with diameter D = 4.9×10 -4 m (9.3 GHz, 300 K).
calculations of the FMR curve in the limit of the small skin depth [4] showed that the lineshape in question could be explained basically as a consequence of a pure surface type resonance, without taking into account any contributions of volume modes. These apparently conflicting explanations for the origin of the shoulder were recently examined in order to explain the FMR behaviors of HgCr2Se4 and CdCr2Se4 [5]. We emphasize in this note that these two explanations are not alternative ones, i.e. the essential origin of the shoulder is the surface type excitations, and volume modes which may be appreciable for small spheres give a perturbation resulting in a shift of the position of the shoulder. Numerical calculations in the limit of the small skin depth [4] indicate that the position of the shoulder, written as/-/2 in fig. 1, corresponds to a resonance field of the surface mode excited at the bottom of the spin wave manifold (the plane wave traveling along the dc magnetic field), while the main absorption maximum at H 1 to the surface mode at the top of the manifold. Therefore, the
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
Y. Watanabeet al. / FMR lineshapeof conductivespheres
240
separation A = H 2 - H 1 should be A = (~/y + Is/2/%)
~ ( ~ / 3 , ) 2 + ( I s / 2 / % ) 2 ' (1)
-
where 0~, y, /to are the microwave frequency, the gyromagnetic ratio and the permeability of free space, respectively, and I s is the magnetization of the specimen. Fig. 2 shows the field separation between the shoulder and the main maximum observed at 9.3 and 14.8 GHz, which was plotted as a function of D/3, where D is the diameter of the sphere and 3 the skin depth defined by 3 = ~ .
(2)
Here, O is the resistivity. Data in fig. 2 were obtained with samples of different values of diameter which were prepared from six MnFe or M n Z n F e ferrite single crystals with magnetization and resistivity ranging between I s = 0.46-0.69 T and O = (0.6-4.2) × 10 -3 f~m, respectively. The separations are normalized by the theoretical value of A evaluated from eq. (1). (A part of the 14.8 G H z data in fig. 2 has already been reported in ref. [6] which is not commonly available.) For a sufficiently large value of D/3, the observed separations fall close to a theoretical one given by eq. (1), being consistent with the conclu-
1.2
t
,
,
,
,
1.1
, • f e
oo I ~
1.0 <~
0.9
,./
0.8
Ie.l 0.7 ~
l
0.6 0.5
0.~ 0
I
I
I
I
I
I
I
2
4
6
8
10
12
1/,
sion that the high-field shoulder is due to the surface mode excitation at the bottom of the spin wave manifold ( H 2 - H 1 somewhat larger than the theoretical A is probably due to a systematic error in determining the position of the terrace edge). An appreciable decrease in the separation is seen for small D / 3 values. This is a consequence of the shift of the shoulder towards the main maximum (/41). It should be noted further that the magnitude of the normalized shift, measured at 9.3 and 14.8 G H z for various values of I s and p, seems to depend only on the value of D/& i.e. the ratio of the sample size to the penetration depth of microwave. These results suggest that in the region of the relatively small D/3 values, the simple plane wave approximation is no longer valid for the excitation around H 2, and the effect of the transverse demagnetization field is not negligible but increases with decreasing D/& The exact distribution of the rf magnetization in this case should be described in terms of certain volume type excitations, i.e. solutions to the boundary value problem of the sphere including the effect of the electromagnetic propagation within the sample, although such a boundary value problem for D = 3 has never been solved. The excitation around H 1, on the other hand, is of surface type even for the present smaller values of D/& as inferred from the resonance field [1,6] and the size-dependence of the absorption power [7,8]. This can be understood qualitatively be taking into account both a large value of equivalent permeability near resonance [1] and a large area of the surface region that is perpendicular to the dc field and hence is responsible for the main part of the absorption curve [1]. Thus, we come to a conclusion that the essential origin of the high-field shoulder is the excitation of the surface mode but for small values of D/& the mode around H 2 is not purely of surface type and the position of the shoulder deviates from the theoretical one as a consequence of volume effects.
16
DI6
Fig. 2. Separations of the terrace edge (H2) measured from the main maximum (Hi) as a function of D/8 (D:diameter, 8: skin depth). A is the theoretical value given by eq. (1). Open circles are for 9.3 GHz and closed ones for 14.8 GHz.
References
[1] Y. Watanabe, S. Saito and S. Takemoto, J. Phys. Soc. Japan 31 (1971) 1840, 32 (1972) 1500.
Y. Watanabe et al. / FMR lineshape of conductive spheres [2] M. Mary~ko, Czech. J. Phys. B 24 (1974) 1379. [3] L.R. Walker, Phys. Rev. 105 (1957) 390. [4] S. Saito, S. Takemoto and Y. Watanabe, Phys Star. Sol. (a) 34 (1976) 671 [5] J.M. Ferreira, M.D. Coutinho-Filho, S.M. Rezende and P.G. Gilbart, J. Magn. Magn. Mat. 31-34 (1983) 672.
241
[6] S. Saito, Y. Watanabe, K. Okada and S. Takemoto, Bull. Univ. Osaka Prefecture A24 (1975) 113. [7] Y. Watanabe, S. Saito and S. Takemoto, J. Phys. Soc. Japan 30 (1971) 889. [8] M. Mary~ko, Phys. Stat. Sol. (a) 7 (1971) Kl17.
239
A NOTE ON FMR LINESHAPE OF CONDUCTIVE SPHERES Y. WATANABE Department of Physics, Shimane University, Matsue 690, Japan
S. SAITO Department of Electronics, University of Osaka Prefecture, Sakai, Osaka 591, Japan
and M. MARYSKO Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czechoslovakia Received 21 November 1984
A shoulder-shaped anomaly on the high-field side of the main FMR maximum has been re-examined in connection with a recent publication on FMR in ferromagnetic semiconductors. The positions of the shoulder are measured at 9.3 and 14.8 GHz for different values of resistivities, magnetization and sphere sizes. Results clarify the role of the surface and volume type excitations in the occurrence of the shoulder.
The skin effect in ferromagnetic resonance (FMR) has been extensively studied for single crystals of conductive ferrite containing excess irons. The FMR line of conductive spheres is characterized by an asymmetric shape with a terraced edge (or a shoulder) on the high-field side of the main absorption maximum [1], as seen in fig. 1. The origin of such a shoulder was first studied in ref. [2] and discussed in connection with the excitation of certain volume modes similar to the magnetostatic modes [3]. However, numerical
8 o /
B=,0m,
magnetic field Fig. 1. A typical FMR curve obtained for a sphere of MnFe204 with diameter D = 4.9×10 -4 m (9.3 GHz, 300 K).
calculations of the FMR curve in the limit of the small skin depth [4] showed that the lineshape in question could be explained basically as a consequence of a pure surface type resonance, without taking into account any contributions of volume modes. These apparently conflicting explanations for the origin of the shoulder were recently examined in order to explain the FMR behaviors of HgCr2Se4 and CdCr2Se4 [5]. We emphasize in this note that these two explanations are not alternative ones, i.e. the essential origin of the shoulder is the surface type excitations, and volume modes which may be appreciable for small spheres give a perturbation resulting in a shift of the position of the shoulder. Numerical calculations in the limit of the small skin depth [4] indicate that the position of the shoulder, written as/-/2 in fig. 1, corresponds to a resonance field of the surface mode excited at the bottom of the spin wave manifold (the plane wave traveling along the dc magnetic field), while the main absorption maximum at H 1 to the surface mode at the top of the manifold. Therefore, the
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
Y. Watanabeet al. / FMR lineshapeof conductivespheres
240
separation A = H 2 - H 1 should be A = (~/y + Is/2/%)
~ ( ~ / 3 , ) 2 + ( I s / 2 / % ) 2 ' (1)
-
where 0~, y, /to are the microwave frequency, the gyromagnetic ratio and the permeability of free space, respectively, and I s is the magnetization of the specimen. Fig. 2 shows the field separation between the shoulder and the main maximum observed at 9.3 and 14.8 GHz, which was plotted as a function of D/3, where D is the diameter of the sphere and 3 the skin depth defined by 3 = ~ .
(2)
Here, O is the resistivity. Data in fig. 2 were obtained with samples of different values of diameter which were prepared from six MnFe or M n Z n F e ferrite single crystals with magnetization and resistivity ranging between I s = 0.46-0.69 T and O = (0.6-4.2) × 10 -3 f~m, respectively. The separations are normalized by the theoretical value of A evaluated from eq. (1). (A part of the 14.8 G H z data in fig. 2 has already been reported in ref. [6] which is not commonly available.) For a sufficiently large value of D/3, the observed separations fall close to a theoretical one given by eq. (1), being consistent with the conclu-
1.2
t
,
,
,
,
1.1
, • f e
oo I ~
1.0 <~
0.9
,./
0.8
Ie.l 0.7 ~
l
0.6 0.5
0.~ 0
I
I
I
I
I
I
I
2
4
6
8
10
12
1/,
sion that the high-field shoulder is due to the surface mode excitation at the bottom of the spin wave manifold ( H 2 - H 1 somewhat larger than the theoretical A is probably due to a systematic error in determining the position of the terrace edge). An appreciable decrease in the separation is seen for small D / 3 values. This is a consequence of the shift of the shoulder towards the main maximum (/41). It should be noted further that the magnitude of the normalized shift, measured at 9.3 and 14.8 G H z for various values of I s and p, seems to depend only on the value of D/& i.e. the ratio of the sample size to the penetration depth of microwave. These results suggest that in the region of the relatively small D/3 values, the simple plane wave approximation is no longer valid for the excitation around H 2, and the effect of the transverse demagnetization field is not negligible but increases with decreasing D/& The exact distribution of the rf magnetization in this case should be described in terms of certain volume type excitations, i.e. solutions to the boundary value problem of the sphere including the effect of the electromagnetic propagation within the sample, although such a boundary value problem for D = 3 has never been solved. The excitation around H 1, on the other hand, is of surface type even for the present smaller values of D/& as inferred from the resonance field [1,6] and the size-dependence of the absorption power [7,8]. This can be understood qualitatively be taking into account both a large value of equivalent permeability near resonance [1] and a large area of the surface region that is perpendicular to the dc field and hence is responsible for the main part of the absorption curve [1]. Thus, we come to a conclusion that the essential origin of the high-field shoulder is the excitation of the surface mode but for small values of D/& the mode around H 2 is not purely of surface type and the position of the shoulder deviates from the theoretical one as a consequence of volume effects.
16
DI6
Fig. 2. Separations of the terrace edge (H2) measured from the main maximum (Hi) as a function of D/8 (D:diameter, 8: skin depth). A is the theoretical value given by eq. (1). Open circles are for 9.3 GHz and closed ones for 14.8 GHz.
References
[1] Y. Watanabe, S. Saito and S. Takemoto, J. Phys. Soc. Japan 31 (1971) 1840, 32 (1972) 1500.
Y. Watanabe et al. / FMR lineshape of conductive spheres [2] M. Mary~ko, Czech. J. Phys. B 24 (1974) 1379. [3] L.R. Walker, Phys. Rev. 105 (1957) 390. [4] S. Saito, S. Takemoto and Y. Watanabe, Phys Star. Sol. (a) 34 (1976) 671 [5] J.M. Ferreira, M.D. Coutinho-Filho, S.M. Rezende and P.G. Gilbart, J. Magn. Magn. Mat. 31-34 (1983) 672.
241
[6] S. Saito, Y. Watanabe, K. Okada and S. Takemoto, Bull. Univ. Osaka Prefecture A24 (1975) 113. [7] Y. Watanabe, S. Saito and S. Takemoto, J. Phys. Soc. Japan 30 (1971) 889. [8] M. Mary~ko, Phys. Stat. Sol. (a) 7 (1971) Kl17.