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Exact calculations of spin wave spectra in insulators
14
Journal of Magnetism and Magnetic Materials 67 (1987) 14-16 North-Holland, Amsterdam
E X A C T C A L C U L A T I O N S O F S P I N W A V E S P E C T R A IN I N S U L A T O R S A.Z. M A K S Y M O W I C Z Department of Solid State Physws, Academy of Mining and Metallurgy, Cracow * Received 5 December 1986
A commonly used circular precession approximation is a single wave vector model which is correct only for perpendicular resonance or for no surface anisotropy. Exact results correspond to normal modes set up for two coupled waves with different wave vectors. Differences in mode positions and their intensities are significant for a set of parameters of a typical permalloy film. The exact model predicts that the critical angle J,¢ between magnetization and normal to the film plane is different from 45 ° as it is observed. A quadratic dependence of the intensity l - (~, - vc )2 is expected near v~..For another critical angle v0 a pair of adjacent modes cease to exist for ~,> J'0. Intensities of the two modes should be equal when approaching % and also a rapid change in the separation of the resonance fields between the two modes is predicted. AII - ~/~o~ ~. The odd modes, which are prohibited for antisymmetrical pinning in the single wave vector approximation, are allowed in the exact model.
We assume u n i f o r m static m a g n e t i z a t i o n M in the bulk of the film situated in the x - y plane. We choose a spherical coordinate system a n d so ~0 a n d ~ and p are the e q u i l i b r i u m angles of the m a g n e t i z a t i o n o b t a i n e d from the m i n i m u m of the magnetic a n i s o t r o p y energy E ( ~ , v). The energy comprises the Z e e m a n a n d dipolar terms. F o r a wave-like form of the d y n a m i c microwave magnetization m - exp[i(~0t- kz)] one gets the dispersion relation [1] 122 = ( P + d k 2 ) ( O + dk 2) - R 2,
(1)
where I2 = , , , / 4 ~ r M v is dimensionless frequency, p-
1 1 32E _ _ 4~rM 2 sin21, ~ 2 '
Q-
4~rM 2
~p2'
1
1
R
-
1
--
~2E (2)
m a g n e t i z a t i o n is of sin-like form describing v o l u m e modes. In the single wavevector a p p r o x i m a t i o n the d y n a m i c part is assumed as m = o~ sin k z + B cos k z and a , / 3 are to be found from the b o u n d a r y c o n d i t i o n s [2-4]. Actually we have two b o u n d a r y c o n d i t i o n s for the two microwave c o m p o n e n t s rn~ a n d r n which leads to a set of four u n i f o r m algebraic equations when applied to both surfaces of the film. The four equations c a n n o t be simultaneously fulfilled with only two parameters a a n d /3 in the test function for m. At this stage the circular precession a p p r o x i m a t i o n [2,5-10] is usually applied, which means that only one b o u n d a r y c o n d i t i o n for the microwave magnetization is needed a n d then the single wave vector test function m = a sin k z +/3 cos k z is acceptable. The wave vector is, according to (1), dk 2 =
+ R 2 + $22
32E
-
4~rM 2 sin u 3ep31,'
are taken at the e q u i l i b r i u m values ¢p, u a n d d is the stiffness constant. F o r dk 2 > 0 the microwave * Address: Zaklad Fizyki Ciala Sta/ego. Akademia G6rniczoHutnicza, al. Mickiewicza30, 30-059 Cracow, Poland.
P + Q 2
(3)
As m e n t i o n e d above, for dk2 > 0 we have volume modes; for d k 2 < 0 we get surface modes. The other solution of (1) provides only imaginary wavevectors k = i~-, d~.2 =
0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
+ R 2 + ~2
+ ~
A.Z. Mak,~yrnoww.: / Spin wave spectra m insulators
In the single wave vector approximation the r-mode, corresponding to the opposite polarization of m, is ignored. It is claimed that this anti-resonance mode is negligible since the imaginary wave vector r is usually a large number and so this mode is heavily damped. In the exact model we account for all the four boundary equations and the proper test function is a superposition of sin kz, cos kz, sinh rz and cosh rz with a, /3, ~, and 8 coefficients. Resulting numbers show that the contribution of the r-mode is, as rule, significant and so the exact procedure is necessary in principle. The r-modes are indeed heavily damped yet the coefficients -/ and 8 may also become large, thus compensating the exponential decrease of m when moving away from the surface. The main analytical results are as follows; (more detailed discussion can be found in ref. [1]): (1) The allowed k-vectors are zeros of a determinant with two pinning parameters, K~ p = -~- cos 2v Ks q = ~ cosZv-
(~.M)/M,
(5) (O.M)/M,
where K s is the surface uniaxial anisotropy constant, A is the exchange constant and 0nM is the gradient of the static magnetization in the presurface layer. We recover the known equation
(PIP2- k2) sin kL + (Pl + P 2 ) k cos kL=O
(6) in the case when p = q , where L is the film thickness. Indices 1 and 2 indicate the pinnings at the two surfaces. The single wave vector model is exact only for u = 0 when the perpendicular resonance is considered or for no surface anisotropy when K, = 0. (2) The intensity of the normal mode [5,11-14] is proportional to the integral of the microwave magnetization, squared. At the critical angle u~ the integral vanishes and so intensity I - (v - vc) 2, as observed in ref. [15]. (3) Another critical angle v0 is also expected. Suppose we have two modes which correspond to
15
two zeros of the determinant as a function of the applied field H at given direction. For a slightly different direction the local m a x i m u m may, for example, be pushed down so that det < 0 and the two modes just disappear. Near P0 the intensities of the two modes are finite and equal, the separation of the resonance fields is rapidly decreasing, A H - ~P0 - v. In more recent papers [16,17] the authors observed as many as 11 lines for perpendicular resonance. Moving off-perpendicular orientation modes 11 and 10 vanished, then the pair 9 and 8 were missing, followed by 7 and 6 until only two lines were left in the parallel case. (4) For no variation of the static magnetization, ~.M = 0, v¢ is 45 ° in the circular precession approximation. The observed values [15,18-19] are different from 45 o. The exact model also predicts p~ ~ 45 o in general. (5) The odd modes, which are prohibited for antisymmetrical pinning in the single wave vector approximation, are allowed in the exact model. Numerical results were obtained for a set of parameters typical for a permalioy 1000 ~, thick film. For small pinnings on both surfaces or very near the perpendicular configuration the results of the circular precession approximation and the exact model are practically same, as expected. Otherwise the results are significantly different. For example, for symmetrical pinning with K~L/2A = 5 and for the parallel case only, two lines are predicted at resonance fields 1097 and 628 Oe while the circular precession approximation result is 2307 and 870 Oe. The corresponding intensities normalized to the intensity of uniform mode at perpendicular configuration are 1.28 and 0.017 while the single k-vector model predicts 0.400 and 0.558. Similar discussion can be resumed for antisymmetrical or asymmetrical pinning conditions to support the conclusion that the exact calculations are, in principle, required. No regular tendencies are observed as far as modes position and their intensities are concerned. The critical angle within the exact model is systematically larger than 45 ° , rather closer to 70 ° . Recently a similar model based on the two-wavemode assumption was independently developed in a series of papers [20-22]. In these papers the authors also arrived at a conclusion on possible
16
A.Z. Maksymowicz / Spin wave spectra in insulators
excitations of odd modes for antisymmetrical pinning condition and predicted values of vc :~ 45 o.
Acknowledgement This work was partly supported by the Institute of Physics of the Lodz University.
References [11 A.Z. Maksymowicz, Phys. Rev. B33 (1986) 6045. [2] J. Spalek and A.Z. Maksymowicz, Solid State Commun. 15 (1974) 559. [3] G.T. Rado and J.R. Weertman, J. Phys. Chem. Solids 11 (1959) 315. [4] P. Pincus, Phys. Rev. 118 (1960) 658. [5] M. Sparks, Phys. Rev. BI (1970) 3869. [6] H. Puszkarski, Acta Phys. Polon. A38 (1970) 217, 899.
[7] H. Puszkarski, Phys. Stat. Sol. (b) 96 (1979) 61. [8] C. Vittoria, R.C. Barker and A. Yelon, J. Appl. Phys. 40 (1969) 1561. [9] I. Harada, O. Nagai and T. Nagamiya, Phys. Rev. B16 (1977) 4882. [10] Diep-The-Hung and J.C. L6vy. Surface Sci. 80 (1979) 512. [11] F. Hoffman, Solid State Commun. 9 (1971) 295. [12] L.J. Maksymowicz, A.Z. Maksymowicz and K.D. Leaver, Solid State Commun. 18 (1976) 1413. [13] M. Sparks, Phys. Rev. B1 (1970) 3831. [14] A.Z. Maksymowicz, Thin Solid Films 42 (1977) 245. [15] F. Hoffman, Phys. Rev. B4 (1971) 1604. [16] D.F. Mitra and J.S.S. Whiting, J. Phys. F8 (1978) 2401. [17] J.S.S. Whiting, IEEE Trans. on Magn. MAG-18 (1982) 709. [18] M. Okochi and H. Nose, J. Phys. Soc. Japan 27 (1969) 312. [19] L.J. Maksymowicz and D. Sendorek, J. Magn. Magn. Mat. 37 (1983) 177. [20] M. Jirsa, Phys. Stat. Sol. (b) 124 (1984) 609. [21] M. Jirsa, Phys. Stat. Sol. (b) 125 (1984) 187. [22] M. Jirsa and V. Kambersk~, Phys. Stat. Sol. (b) 126 (1984) 547.
Journal of Magnetism and Magnetic Materials 67 (1987) 14-16 North-Holland, Amsterdam
E X A C T C A L C U L A T I O N S O F S P I N W A V E S P E C T R A IN I N S U L A T O R S A.Z. M A K S Y M O W I C Z Department of Solid State Physws, Academy of Mining and Metallurgy, Cracow * Received 5 December 1986
A commonly used circular precession approximation is a single wave vector model which is correct only for perpendicular resonance or for no surface anisotropy. Exact results correspond to normal modes set up for two coupled waves with different wave vectors. Differences in mode positions and their intensities are significant for a set of parameters of a typical permalloy film. The exact model predicts that the critical angle J,¢ between magnetization and normal to the film plane is different from 45 ° as it is observed. A quadratic dependence of the intensity l - (~, - vc )2 is expected near v~..For another critical angle v0 a pair of adjacent modes cease to exist for ~,> J'0. Intensities of the two modes should be equal when approaching % and also a rapid change in the separation of the resonance fields between the two modes is predicted. AII - ~/~o~ ~. The odd modes, which are prohibited for antisymmetrical pinning in the single wave vector approximation, are allowed in the exact model.
We assume u n i f o r m static m a g n e t i z a t i o n M in the bulk of the film situated in the x - y plane. We choose a spherical coordinate system a n d so ~0 a n d ~ and p are the e q u i l i b r i u m angles of the m a g n e t i z a t i o n o b t a i n e d from the m i n i m u m of the magnetic a n i s o t r o p y energy E ( ~ , v). The energy comprises the Z e e m a n a n d dipolar terms. F o r a wave-like form of the d y n a m i c microwave magnetization m - exp[i(~0t- kz)] one gets the dispersion relation [1] 122 = ( P + d k 2 ) ( O + dk 2) - R 2,
(1)
where I2 = , , , / 4 ~ r M v is dimensionless frequency, p-
1 1 32E _ _ 4~rM 2 sin21, ~ 2 '
Q-
4~rM 2
~p2'
1
1
R
-
1
--
~2E (2)
m a g n e t i z a t i o n is of sin-like form describing v o l u m e modes. In the single wavevector a p p r o x i m a t i o n the d y n a m i c part is assumed as m = o~ sin k z + B cos k z and a , / 3 are to be found from the b o u n d a r y c o n d i t i o n s [2-4]. Actually we have two b o u n d a r y c o n d i t i o n s for the two microwave c o m p o n e n t s rn~ a n d r n which leads to a set of four u n i f o r m algebraic equations when applied to both surfaces of the film. The four equations c a n n o t be simultaneously fulfilled with only two parameters a a n d /3 in the test function for m. At this stage the circular precession a p p r o x i m a t i o n [2,5-10] is usually applied, which means that only one b o u n d a r y c o n d i t i o n for the microwave magnetization is needed a n d then the single wave vector test function m = a sin k z +/3 cos k z is acceptable. The wave vector is, according to (1), dk 2 =
+ R 2 + $22
32E
-
4~rM 2 sin u 3ep31,'
are taken at the e q u i l i b r i u m values ¢p, u a n d d is the stiffness constant. F o r dk 2 > 0 the microwave * Address: Zaklad Fizyki Ciala Sta/ego. Akademia G6rniczoHutnicza, al. Mickiewicza30, 30-059 Cracow, Poland.
P + Q 2
(3)
As m e n t i o n e d above, for dk2 > 0 we have volume modes; for d k 2 < 0 we get surface modes. The other solution of (1) provides only imaginary wavevectors k = i~-, d~.2 =
0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
+ R 2 + ~2
+ ~
A.Z. Mak,~yrnoww.: / Spin wave spectra m insulators
In the single wave vector approximation the r-mode, corresponding to the opposite polarization of m, is ignored. It is claimed that this anti-resonance mode is negligible since the imaginary wave vector r is usually a large number and so this mode is heavily damped. In the exact model we account for all the four boundary equations and the proper test function is a superposition of sin kz, cos kz, sinh rz and cosh rz with a, /3, ~, and 8 coefficients. Resulting numbers show that the contribution of the r-mode is, as rule, significant and so the exact procedure is necessary in principle. The r-modes are indeed heavily damped yet the coefficients -/ and 8 may also become large, thus compensating the exponential decrease of m when moving away from the surface. The main analytical results are as follows; (more detailed discussion can be found in ref. [1]): (1) The allowed k-vectors are zeros of a determinant with two pinning parameters, K~ p = -~- cos 2v Ks q = ~ cosZv-
(~.M)/M,
(5) (O.M)/M,
where K s is the surface uniaxial anisotropy constant, A is the exchange constant and 0nM is the gradient of the static magnetization in the presurface layer. We recover the known equation
(PIP2- k2) sin kL + (Pl + P 2 ) k cos kL=O
(6) in the case when p = q , where L is the film thickness. Indices 1 and 2 indicate the pinnings at the two surfaces. The single wave vector model is exact only for u = 0 when the perpendicular resonance is considered or for no surface anisotropy when K, = 0. (2) The intensity of the normal mode [5,11-14] is proportional to the integral of the microwave magnetization, squared. At the critical angle u~ the integral vanishes and so intensity I - (v - vc) 2, as observed in ref. [15]. (3) Another critical angle v0 is also expected. Suppose we have two modes which correspond to
15
two zeros of the determinant as a function of the applied field H at given direction. For a slightly different direction the local m a x i m u m may, for example, be pushed down so that det < 0 and the two modes just disappear. Near P0 the intensities of the two modes are finite and equal, the separation of the resonance fields is rapidly decreasing, A H - ~P0 - v. In more recent papers [16,17] the authors observed as many as 11 lines for perpendicular resonance. Moving off-perpendicular orientation modes 11 and 10 vanished, then the pair 9 and 8 were missing, followed by 7 and 6 until only two lines were left in the parallel case. (4) For no variation of the static magnetization, ~.M = 0, v¢ is 45 ° in the circular precession approximation. The observed values [15,18-19] are different from 45 o. The exact model also predicts p~ ~ 45 o in general. (5) The odd modes, which are prohibited for antisymmetrical pinning in the single wave vector approximation, are allowed in the exact model. Numerical results were obtained for a set of parameters typical for a permalioy 1000 ~, thick film. For small pinnings on both surfaces or very near the perpendicular configuration the results of the circular precession approximation and the exact model are practically same, as expected. Otherwise the results are significantly different. For example, for symmetrical pinning with K~L/2A = 5 and for the parallel case only, two lines are predicted at resonance fields 1097 and 628 Oe while the circular precession approximation result is 2307 and 870 Oe. The corresponding intensities normalized to the intensity of uniform mode at perpendicular configuration are 1.28 and 0.017 while the single k-vector model predicts 0.400 and 0.558. Similar discussion can be resumed for antisymmetrical or asymmetrical pinning conditions to support the conclusion that the exact calculations are, in principle, required. No regular tendencies are observed as far as modes position and their intensities are concerned. The critical angle within the exact model is systematically larger than 45 ° , rather closer to 70 ° . Recently a similar model based on the two-wavemode assumption was independently developed in a series of papers [20-22]. In these papers the authors also arrived at a conclusion on possible
16
A.Z. Maksymowicz / Spin wave spectra in insulators
excitations of odd modes for antisymmetrical pinning condition and predicted values of vc :~ 45 o.
Acknowledgement This work was partly supported by the Institute of Physics of the Lodz University.
References [11 A.Z. Maksymowicz, Phys. Rev. B33 (1986) 6045. [2] J. Spalek and A.Z. Maksymowicz, Solid State Commun. 15 (1974) 559. [3] G.T. Rado and J.R. Weertman, J. Phys. Chem. Solids 11 (1959) 315. [4] P. Pincus, Phys. Rev. 118 (1960) 658. [5] M. Sparks, Phys. Rev. BI (1970) 3869. [6] H. Puszkarski, Acta Phys. Polon. A38 (1970) 217, 899.
[7] H. Puszkarski, Phys. Stat. Sol. (b) 96 (1979) 61. [8] C. Vittoria, R.C. Barker and A. Yelon, J. Appl. Phys. 40 (1969) 1561. [9] I. Harada, O. Nagai and T. Nagamiya, Phys. Rev. B16 (1977) 4882. [10] Diep-The-Hung and J.C. L6vy. Surface Sci. 80 (1979) 512. [11] F. Hoffman, Solid State Commun. 9 (1971) 295. [12] L.J. Maksymowicz, A.Z. Maksymowicz and K.D. Leaver, Solid State Commun. 18 (1976) 1413. [13] M. Sparks, Phys. Rev. B1 (1970) 3831. [14] A.Z. Maksymowicz, Thin Solid Films 42 (1977) 245. [15] F. Hoffman, Phys. Rev. B4 (1971) 1604. [16] D.F. Mitra and J.S.S. Whiting, J. Phys. F8 (1978) 2401. [17] J.S.S. Whiting, IEEE Trans. on Magn. MAG-18 (1982) 709. [18] M. Okochi and H. Nose, J. Phys. Soc. Japan 27 (1969) 312. [19] L.J. Maksymowicz and D. Sendorek, J. Magn. Magn. Mat. 37 (1983) 177. [20] M. Jirsa, Phys. Stat. Sol. (b) 124 (1984) 609. [21] M. Jirsa, Phys. Stat. Sol. (b) 125 (1984) 187. [22] M. Jirsa and V. Kambersk~, Phys. Stat. Sol. (b) 126 (1984) 547.