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Journal of Magnetism and Magnetic Materials 140-144 (1995) 1995-1996

,•9 Journal of nalnetlsm magnetic

~ a materials

Spin wave modes in inhomogeneous ferromagnetic film at arbitrary magnetic field direction

Zhi-quan Han *

Southwest Institute of Applied Magnetics of China, P.O. Box 105, Mianyang, Sichuan, 621000, China

Abstract A generalization of the Portis model for the case of arbitrary direction of magnetization is presented, which gives the

separation of linear positioned modes decreasing with 0 as 6H(O)/6H(O)= [½(3 cos20- 1)] 1/2 till 0 > 0c, at which all modes collapse into one single line or convert to modes following a quadratic law, and the intensities increase rapidly when 0 approaches 0 c. We explain such behavior using spin wave modes in thin film with parabolic dependence of magnetization. A good agreement of this theory with the experiments on the reduced Ca-rich CaGe : YIG film has been obtained.

Since it was pointed out by Kittel [1] that the exchange integral could be obtained by ferromagnetic resonance measurement in thin films, interest in spin wave resonance has increased steadily. However, it has often been found that the spin wave spectrum deviates from the quadratic law expected in [1] assuming uniform magnetization being close to a linear law. Hence, Portis [2], generalizing the volume inhomogeneity (VI) model of Wigen et al. [3] (stepped Ms), proposed a VI model with a parabolically dropping M S from the center of the film yielding mode separations linear in mode number n and mode intensities dropping more slowly than 1/n 2. The Portis model has satisfactorily explained the spin wave spectra observed for various thin films [4]. The Portis model has been further extended by Davies [5] and Hirota [6] for surface pinning of spins; by Wigen et al. [7] and Searle et al. [8] for parallel resonance; and by Qian et al. [9] for films with nonsymmetrical parabolic M S. Up to our knowledge, angu- lar dependence of the spin wave spectra has not been studied for parabolic M S dependence. The aim of this paper is a detailed study of this problem. A critical angle phenomenon seldom noted on body modes is studied.

With the coordinate systems shown in Fig. 1, the equation of motion for the magnetization in the circular precession approximation was given by Ref. [10] as

02m - - + k2m(z ') = 0 (1) Mzq Oz,2

where

k 2 = [ ( t o / T ) 2 + (2~rMz sin2Om)2]l/2-Hi

- q(OZMJ3z '2) - 21rM z sin2Om. (2)

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0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 6 1 0 - 0

M S is magnetization, M s = MzZ + me i'°t, m the rf compo- nent, m < < M Z, H i internal field, H i = H e + H a - 4"rrM z cos 02', A exchange constant, q = 2 A / M 2, 0 an- gle between H c and the normal of film, 0,, angle between M s and the normal. Substituting the parabolic variation in Mz, that is M z = M0(1 -4EZ'2/L2), and the vectors sum of fields H i ~ H e + H a - 47rMz cos20m into Eq. (2), we get in first approximation that

k 2 = H 0 - H e - 4e(z'/L)22~rMo(3 cos20 - 1) (3)

where

Ho= [(to/'y)2 + (2"trMo sin20)2] 1/2

+ 2'rrMo(3 cos20- 1) + (8MoEqo/L 2) - H s (4)

and qo = 2A/Mo 2. Thus (1) is written as the equation of a harmonic oscillator analogous to the Portis model

i)2m 0z,2 + ( h - a2z'Z)m = 0 (5)

H a r ~ 4 = M . ~ c o s e "

L f II,

z p

x

Fig. 1. Fields, angles, and coordinate systems used in the text.

1996 Z.-Q. Han /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1995-1996

where

h = (H o - He)/Moq o (6)

a 2 = 4 , [2~r(3 cos20- 1) /qo - A] /L 2. (7)

Its explicit solutions with a condition of unpinned spins are m(~)=N,e-e/ZHn(~), where ~=z'vCa, and Hn(~) are the Hermite polynomials. The eigenvalues of Eq. (5), A = (2n + 1)a, n = 0, 1, 2 , . . . , give the position of the nth mode for a sufficiently small n and the separation 6 H = H , - H n + 1 as

H n = H 0 - (2n + 1)2M012"rr,q0(3 cos20 - 1)] 1/9, (8)

6H(O) = 812"rra,(3 cos20 - 1 ) / L 2] 1/2 (9)

respectively. The calculation of intensities ( I n) indicates that the ratio In/I o (= n!/[(n/2)!2n/2] z) does not depend on 0 and that analogous to perpendicular resonance only even modes can be excited ( I n = 0, for odd modes). Ac- cording to Ref. [10] and the projecting relation of rf field amplitudes in the x'y plane and in the xy plane, I~(0) is given by

In(O) (1 + cos20)

In(0) = 2[(3 cos20- 1 ) /2 ] 1/4" (10)

Inspection of Eq. (9) shows that with increasing 0 the separation decreases; at 0 = 0 o = 55 ° the 6H drops to zero; at 0 = 0, Eq. (9) is equivalent to Portis' result. Eq. (10) also reveals that for 0 ~ 0 o the intensities rapidly increase.

The highest mode number n c of the Portis model is determined by the limit at which the wave function reaches the surface of the film, i.e. 2R = L, R = 7rA-/ot [2,10]. It gives

F / 7T£ "~ 1/2 ] n c = [ L M o [ - ~ ) ( 3 c o s 2 0 - 1 ) 1 / 2 - 1 j / 2 . (11)

This means that n c decreases with 0 until the threshold value

O¢ = arccos[½(1 + IOOA/~,LeMff)] 1/2 (12)

(corresponding to n¢ = 2, see Fig. 2), i.e. at 0 > O¢ only the mode n = 0 can be excited; at 0 > 0 e (for n o = 0 , 0c,= arccos[(1 +4A/IreL2M2)/311/2), all Portis type modes collapse completely. It is notable that both 0~ and 0 c, are smaller than 0 o and depend on the values of A, E and L of the film. The value of O c observed experimen- tally on the CaGe : YIG film is about 40-45 ° [12].

Since the low-order harmonic oscillator function m( ~ ) = N, e -e /ZHn(~) does not extend across the full thick- ness of the film, the effective thickness L~f e over which m is large, is smaller than the film thickness L [10]. A ratio

"1 °:° A

m Ores 8c

I_ L ! A

Fig. 2. Magnitude of microwave magnetization m for modes n = 0 and n = 2 , at 0 =0 and 0= 0 c.

of L~n/L = 1 / 2 has been measured for a film with L = 1 Ixm [11]. According to Eq. (7), a drops with increasing 0 and consequently the values of ~ ( = z'vraa) at z' = +_L/2 drop too and L~ff broadens. The rapid increase of I,(0) at 0---, 0 c in Eq. (10) can be intuitively understood by the broadening of Lef f. Also the drop of 6H(O) with 0 in Eq. (9) can be understood well by the drop of a.

This theory has been verified by FMR measurements on films of reduced CaGe:YIG, of which the values of exchange constant A = 3.5 × 10 -7 e rg /cm and ~ = 0.056 were obtained from Eqs. (8) and (12) using experimental values of 6H(0) and 0 c [12].

R e f e r e n c e s

[1] C. Kittel, Phys. Rev. 110 (1958) 1295. [2] A.M. Portis, Appl. Phys. Lett. 2 (1963) 69. [3] P.E. Wigen, C.F. Cooi, M.R. Shanabarger and T.D. Rossing,

Phys. Rev. Lett. 9 (1962) 206. [4] J. Dubowik and S. Klama, J. Magn. Magn. Mater. 121

(1993) 513. [5] J.T. Davies, J. Appl. Phys. 35 (1964) 804. [6] E. Hirota, J. Phys. Soc. Jpn. 19 (1964) 1. [7] P.E. Wigen, C.F. Kooi and M.R. Shanabarger, J. Appl. Phys.

35 (1964) 3302. [8] C.W. Searle and A.H. Morrish, Phys. Lett. 7 (1963) 29. [9] K.M. Qian, Z.H. Lin, and D.S. Dai, Acta Phys. Sinica 32

(1983) 1547. [10] M. Sparks, Phys. Rev. B 1 (1970) 3869. [11] M. Sparks, B.R. Tittmann, J.E. Mee and C. Newkirk, J.

Appl. Phys. 40 (1969) 1518. [12] Z.Q. Han, L. Pust, P.E. Wigen and P. DeGasperis, J. Magn.

Magn. Mater 140-144 (1995) 733 (these Proceedings).