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Influence of inhomogeneities on magnetization of surface modes in SWR
1412 INFLUENCE OF I N H O M O G E N E I T I E S SWRt
ON M A G N E T I Z A T I O N
OF SURFACE M O D E S IN
L.J. M A K S Y M O W I C Z ? t Academy of Mining and Metallurgy, Department of Solid State Physics, Krakow, Poland The classical model of spin wave spectra with surface anisotropy (characterised by the pinning parameter p,.2 =
Ks~A ' on both sides of a film, where K~ is the surface anisotropy constant and fi~ is the exchange constant) and also volume inhomogeneities (characterised by the amplitude AM of the magnetization distribution across the sample) is presented. The discussion is confined to selected range of parameters p,, p2 and AM for which existance of the surface modes is expected. Only perpendicular resonance is considered.
Non-uniform static magnetization distribution
M(z) does influence both the positions and intensities of modes in spin-wave resonance [17]. Especially for polycrystalline thin films one can expect discrepancy from the usually assumed uniform magnetization M(z) = Mo. T h e r e f o r e it could be of an interest to consider not only influence of surface anisotropy energy expressed in terms of the pinning parameters p~, P2 (SI model) [8-10] but also the distribution of magnetization in volume of the sample (VI model) [11]. For simplicity, let us assume a cos-like magnetization of the form
M(z) = Mo - A M cos (17rL-~z),
(1)
where Mo is the average magnetization in the thin film which corresponds to the saturation value; AM is the maximum deviation of M(z) from Mo; l is an integer, to be chosen for the best fit of the trial function M(z); L is the film thickness and 0 ~< z ~< L. The L a n d a u - L i f s h i t z equation of motion (no damping, A = 0) h;/(z, t) = 3~M(z, t) x [H - 47rM(z, t) + 2fi~M(z, I)-ZVZM(z, t)], (2) where y is the gyromagnetic ratio; H is an external magnetic field. This equation is applied for the perpendicular resonance H = H£, where M ( z , t) = M ( z ) + re(z, t)
(3)
is the superposition of the static M(z) and microwave re(z, t) components. It is assumed that re(z, t) is small and M(z) is along the zaxis. After the linearization, eq. (2) becomes d2m
--+
d~ 2
where
2~ = hrL-lz, W = 4L212~27r-2,
q = 2L2F]-La"-2,
(4a) 12 = (oJ/~/- H + 4~'M0)D -1, (4b) I" = 47rAMD -1,
Physica 86-88B (1977) 1412-1414 ~ North-Holland
(4c)
D = 2fi.M0 l, fi~_ the exchange constant. Eq. (4) is the Mathieu's equation and the general solution is of the form
T
(5) where the factor A is arbitrary, and r is an integer. Substituting the infinite series (5) into eq. (4) yields the typical eigenvalue problem. Providing t h a t / z and B/A are known this gives eigenvalues W, [so /4, can be determined, eq. (4b)] and corresponding eigenvector C2r. Now, the proposed solution is subject to the boundary conditions dm
d~-~-pl2L(17r)-lm = 0 t Supported by Institute of Physics of Polish Academy of Sciences in Warsaw. t t Present address: University of Dundee, Carnegie Laboratory of Physics, Dundee DDI 4HN, Scotland U.K.
(4)
(W - 2q cos 2~)m = 0,
for ~ = 0,
dm
--+p22L(17r) l m = 0 dE
for ~ = l r r / 2 ,
(6)
1413 where Pl-= K~,/A, p2 = K~2/A, are the pinning parameters. As the determinant of eq. (6) is to vanish this yields the missing equation for/~. Also from eq. (6) one gets BIA. T h e r e f o r e in principle the possible solutions of microwave magnetization re(z) of eq. (5) are k n o w n and, also the intensity o f / 4 , mode can be found. One can prove that/x can be either purely imaginary /x = ik or real. We consider the latter case that refers to the surface modes. Calculation has been done for NiFe alloy (80%, 20%) with L = 1000 ,~, l = 2, and for X band. The pinning parameters p,, P2 have been chosen from the range for which the SI model predicts the surface modes. Within the SI model and the boundary conditions provides the possible tz values for the surface modes [8, 9] - t h / x L =/~(p~ + p2)(plP2 + ~L2)2.
mlZ)
q=0.7
~
~
.
q=0.5 :0.1
4
(a)
~
~
. . . . . .
4 5 6 7 8 5 -1 5 -1 p1=-0.63-10 cm p2=-0.13,,10c m
9
10 z
(7) re(Z)
?=0.~
In fig. 1 the PiP2 plane is divided into three regions separated by the critical curves with equation + - L = p E t + p ? ~ whose asymptotes intercept the p~, P2 axes at - L - % In the region I is no solution of eq. (7) (no surface modes are predicted), in region II there is one solution of eq. (7) (one surface mode), in region III there are two solutions (two surface modes). In the case of combined the S I + V I model eq. (7) becomes more complicated
/
t
5
6
7
8
g
- th IxL = (It -/31a)(p, + p2)[p~p2 + (IX -/3/a)2]-1. (8)
o63*1o%~-1. %=_o.2t,1o%m-1
.10 5
o .<
"6 5
P2lcm'l]
3 2
I
(b)
Fig. 2. Distribution of microwave magnetization r e ( z ) for fixed P~p2 and varying q a) p , = - 0 . 6 3 x 105cm -~, p2 = - 0 . 1 3 x l 0 ~ c m - ' ; b) p , = - 0 . 6 3 x 1 0 6 c m ', p 2 = - 0 . 2 1 × 106 cm '.
I
-8 -7 -6 -5 -& -3 -2 Critical curves 2 ~ IT (-0.6370.21)x106
"F-2
Asymtote \Criticolcurves 1
~3
The quantities a and /3 are expressed in terms of coefficients C2, of the expansion of the solution according to eq. (5)
III -5 -6
Fig. 1. Discussion of the solution of eq. (7) in p,p2 space.
ct = Re [Co + 2(C2 +
C 4 J~ C 6 J r . • . ) ] ,
/3 = Im [2(2C2 + 4C 4 + 6C6 +" • -)].
(9)
As one can expect /3/a ~ 0, for q ~ 0 [AM ~ 0,
1414 eq. (4c)]. This is so b e c a u s e for q = 0 the nonzero C~ coefficient is Co = 1. Since a and /3 d e p e n d e n c e on q is rather difficult to be presented analytically the influence of q on re(z) has b e e n determined numerically. The results are shown in figs. (2a, 2b). It can be seen in fig. (2a) that for pinning p a r a m e t e r pairs p,, P2: p ~ = - 0 . 6 3 x 1 0 5 c m -l,
P 2 = - 0 - 1 3 x 1 0 5 c m -j
the influence of q is quite considerable. For q = 0.3 the distribution re(z) changes f r o m the characteristic for a surface m o d e to that corresponding to a volume mode. For smaller absolute values of p~, P2 this influence is greater. H o w e v e r , if one takes greater P~P2 values p~ = - 0 . 6 3 × 10 6 cm -~ and P2 = -0.21 x 10 6 cm -~ (fig. 2b) the influence of q is m u c h smaller. Although f r o m q = 0 . 5 the distribution re(z) starts to change its character. H o w e v e r , its b e h a v i o u r is still such as for a surface mode. It follows f r o m the results presented a b o v e that the conditions of generation of the surface m o d e s predicted b y the SI model are changed if volume inhomogeneities are taken into account. The critical c u r v e is displaced in the direction indicated b y an arrow in fig. 1. H e n c e it can be concluded that surface m o d e s excitation in S W R in polycrystalline thin films obtained in
conventional v a c u u m is more difficult than in case of monocrystalline thin films. P e r h a p s it can partly explain while there is no experimental evidence for existance of surface modes in polycrystalline films while on the contrary such m o d e s h a v e been reported in single-crystal films [12]. The author is m u c h indebted to Prof. Dr. L. Kozlowski for m a n y helpful discussions throughout this w o r k and to Dr. A.Z. Maks y m o w i c z for help in numerical calculations.
References [1] M. Hinoul and J. Witters, Sol. St. Comm. 9 (1971) 83. [2] L.J. Maksymowicz and A.Z. Maksymowicz, Colloquium on Mag. Thin Films, Regensburg 1975. [3] E.P. Wigen, Sol. St. Comm. 8 (1970) 725. [4] F. Hoffmann, Sol. St. Comm. 9 (1971) 295. [5] F. Hoffmann, Phys. Rev. 4B (1971). [6] A.M. Portis, Appl. Phys. Lett. 2 (1963). [7l M. Sparks, Sol. St. Comm. 8 (1970) 659. [8] H. Puszkarski, Acta Phys. Polon. 38A (1970) 217. [9] H. Puszkarski, Acta Phys. Polon. 38A (1970) 899. [10] J. Spalek and W. Schmidt, Sol. St. Comm. 16 (1975) 193. [11] L.J. Maksymowicz, A.Z. Maksymowicz and K.D. Leaver, Sol. St. Comm. 18 (1976) 1413. [12] S.F. Yu, R.A. Tunk and P.E. Wigen, Phys. Rev. l i B (1975) 420.
ON M A G N E T I Z A T I O N
OF SURFACE M O D E S IN
L.J. M A K S Y M O W I C Z ? t Academy of Mining and Metallurgy, Department of Solid State Physics, Krakow, Poland The classical model of spin wave spectra with surface anisotropy (characterised by the pinning parameter p,.2 =
Ks~A ' on both sides of a film, where K~ is the surface anisotropy constant and fi~ is the exchange constant) and also volume inhomogeneities (characterised by the amplitude AM of the magnetization distribution across the sample) is presented. The discussion is confined to selected range of parameters p,, p2 and AM for which existance of the surface modes is expected. Only perpendicular resonance is considered.
Non-uniform static magnetization distribution
M(z) does influence both the positions and intensities of modes in spin-wave resonance [17]. Especially for polycrystalline thin films one can expect discrepancy from the usually assumed uniform magnetization M(z) = Mo. T h e r e f o r e it could be of an interest to consider not only influence of surface anisotropy energy expressed in terms of the pinning parameters p~, P2 (SI model) [8-10] but also the distribution of magnetization in volume of the sample (VI model) [11]. For simplicity, let us assume a cos-like magnetization of the form
M(z) = Mo - A M cos (17rL-~z),
(1)
where Mo is the average magnetization in the thin film which corresponds to the saturation value; AM is the maximum deviation of M(z) from Mo; l is an integer, to be chosen for the best fit of the trial function M(z); L is the film thickness and 0 ~< z ~< L. The L a n d a u - L i f s h i t z equation of motion (no damping, A = 0) h;/(z, t) = 3~M(z, t) x [H - 47rM(z, t) + 2fi~M(z, I)-ZVZM(z, t)], (2) where y is the gyromagnetic ratio; H is an external magnetic field. This equation is applied for the perpendicular resonance H = H£, where M ( z , t) = M ( z ) + re(z, t)
(3)
is the superposition of the static M(z) and microwave re(z, t) components. It is assumed that re(z, t) is small and M(z) is along the zaxis. After the linearization, eq. (2) becomes d2m
--+
d~ 2
where
2~ = hrL-lz, W = 4L212~27r-2,
q = 2L2F]-La"-2,
(4a) 12 = (oJ/~/- H + 4~'M0)D -1, (4b) I" = 47rAMD -1,
Physica 86-88B (1977) 1412-1414 ~ North-Holland
(4c)
D = 2fi.M0 l, fi~_ the exchange constant. Eq. (4) is the Mathieu's equation and the general solution is of the form
T
(5) where the factor A is arbitrary, and r is an integer. Substituting the infinite series (5) into eq. (4) yields the typical eigenvalue problem. Providing t h a t / z and B/A are known this gives eigenvalues W, [so /4, can be determined, eq. (4b)] and corresponding eigenvector C2r. Now, the proposed solution is subject to the boundary conditions dm
d~-~-pl2L(17r)-lm = 0 t Supported by Institute of Physics of Polish Academy of Sciences in Warsaw. t t Present address: University of Dundee, Carnegie Laboratory of Physics, Dundee DDI 4HN, Scotland U.K.
(4)
(W - 2q cos 2~)m = 0,
for ~ = 0,
dm
--+p22L(17r) l m = 0 dE
for ~ = l r r / 2 ,
(6)
1413 where Pl-= K~,/A, p2 = K~2/A, are the pinning parameters. As the determinant of eq. (6) is to vanish this yields the missing equation for/~. Also from eq. (6) one gets BIA. T h e r e f o r e in principle the possible solutions of microwave magnetization re(z) of eq. (5) are k n o w n and, also the intensity o f / 4 , mode can be found. One can prove that/x can be either purely imaginary /x = ik or real. We consider the latter case that refers to the surface modes. Calculation has been done for NiFe alloy (80%, 20%) with L = 1000 ,~, l = 2, and for X band. The pinning parameters p,, P2 have been chosen from the range for which the SI model predicts the surface modes. Within the SI model and the boundary conditions provides the possible tz values for the surface modes [8, 9] - t h / x L =/~(p~ + p2)(plP2 + ~L2)2.
mlZ)
q=0.7
~
~
.
q=0.5 :0.1
4
(a)
~
~
. . . . . .
4 5 6 7 8 5 -1 5 -1 p1=-0.63-10 cm p2=-0.13,,10c m
9
10 z
(7) re(Z)
?=0.~
In fig. 1 the PiP2 plane is divided into three regions separated by the critical curves with equation + - L = p E t + p ? ~ whose asymptotes intercept the p~, P2 axes at - L - % In the region I is no solution of eq. (7) (no surface modes are predicted), in region II there is one solution of eq. (7) (one surface mode), in region III there are two solutions (two surface modes). In the case of combined the S I + V I model eq. (7) becomes more complicated
/
t
5
6
7
8
g
- th IxL = (It -/31a)(p, + p2)[p~p2 + (IX -/3/a)2]-1. (8)
o63*1o%~-1. %=_o.2t,1o%m-1
.10 5
o .<
"6 5
P2lcm'l]
3 2
I
(b)
Fig. 2. Distribution of microwave magnetization r e ( z ) for fixed P~p2 and varying q a) p , = - 0 . 6 3 x 105cm -~, p2 = - 0 . 1 3 x l 0 ~ c m - ' ; b) p , = - 0 . 6 3 x 1 0 6 c m ', p 2 = - 0 . 2 1 × 106 cm '.
I
-8 -7 -6 -5 -& -3 -2 Critical curves 2 ~ IT (-0.6370.21)x106
"F-2
Asymtote \Criticolcurves 1
~3
The quantities a and /3 are expressed in terms of coefficients C2, of the expansion of the solution according to eq. (5)
III -5 -6
Fig. 1. Discussion of the solution of eq. (7) in p,p2 space.
ct = Re [Co + 2(C2 +
C 4 J~ C 6 J r . • . ) ] ,
/3 = Im [2(2C2 + 4C 4 + 6C6 +" • -)].
(9)
As one can expect /3/a ~ 0, for q ~ 0 [AM ~ 0,
1414 eq. (4c)]. This is so b e c a u s e for q = 0 the nonzero C~ coefficient is Co = 1. Since a and /3 d e p e n d e n c e on q is rather difficult to be presented analytically the influence of q on re(z) has b e e n determined numerically. The results are shown in figs. (2a, 2b). It can be seen in fig. (2a) that for pinning p a r a m e t e r pairs p,, P2: p ~ = - 0 . 6 3 x 1 0 5 c m -l,
P 2 = - 0 - 1 3 x 1 0 5 c m -j
the influence of q is quite considerable. For q = 0.3 the distribution re(z) changes f r o m the characteristic for a surface m o d e to that corresponding to a volume mode. For smaller absolute values of p~, P2 this influence is greater. H o w e v e r , if one takes greater P~P2 values p~ = - 0 . 6 3 × 10 6 cm -~ and P2 = -0.21 x 10 6 cm -~ (fig. 2b) the influence of q is m u c h smaller. Although f r o m q = 0 . 5 the distribution re(z) starts to change its character. H o w e v e r , its b e h a v i o u r is still such as for a surface mode. It follows f r o m the results presented a b o v e that the conditions of generation of the surface m o d e s predicted b y the SI model are changed if volume inhomogeneities are taken into account. The critical c u r v e is displaced in the direction indicated b y an arrow in fig. 1. H e n c e it can be concluded that surface m o d e s excitation in S W R in polycrystalline thin films obtained in
conventional v a c u u m is more difficult than in case of monocrystalline thin films. P e r h a p s it can partly explain while there is no experimental evidence for existance of surface modes in polycrystalline films while on the contrary such m o d e s h a v e been reported in single-crystal films [12]. The author is m u c h indebted to Prof. Dr. L. Kozlowski for m a n y helpful discussions throughout this w o r k and to Dr. A.Z. Maks y m o w i c z for help in numerical calculations.
References [1] M. Hinoul and J. Witters, Sol. St. Comm. 9 (1971) 83. [2] L.J. Maksymowicz and A.Z. Maksymowicz, Colloquium on Mag. Thin Films, Regensburg 1975. [3] E.P. Wigen, Sol. St. Comm. 8 (1970) 725. [4] F. Hoffmann, Sol. St. Comm. 9 (1971) 295. [5] F. Hoffmann, Phys. Rev. 4B (1971). [6] A.M. Portis, Appl. Phys. Lett. 2 (1963). [7l M. Sparks, Sol. St. Comm. 8 (1970) 659. [8] H. Puszkarski, Acta Phys. Polon. 38A (1970) 217. [9] H. Puszkarski, Acta Phys. Polon. 38A (1970) 899. [10] J. Spalek and W. Schmidt, Sol. St. Comm. 16 (1975) 193. [11] L.J. Maksymowicz, A.Z. Maksymowicz and K.D. Leaver, Sol. St. Comm. 18 (1976) 1413. [12] S.F. Yu, R.A. Tunk and P.E. Wigen, Phys. Rev. l i B (1975) 420.