FERROMAGNETIC RESONANCE IN MEDIA WITH AN INHOMOGENEOUS SATURATION MAGNETIZATION

V. A. Zhuravlev, V. S. Korogodov, Yu. N. Kotyukov (deceased), and N. B. Lysova

UDC 538.62

The method of transition probabilities is used to oaleulate the line width of f e r r o m a g n e t i c r e s o n a n c e in a medium with an inhomogeneous saturation magnetization. It is shown that in addition to l i n e a r (with r e s p e c t to the F o u r i e r components of the function describing inhomogeneities) t e r m s , the perturbation Hamiltonian must include also quadratic t e r m s . The f r e q u e n c y dependence of the contribution of the p o r o s i t y to AH is fundamentally different f r o m the f r e q u e n c y dependence of the magnetic anisotropy contribution. This anisotropy contribution AHa(w) d e c r e a s e s on i n c r e a s e in the frequency and has a maximum at r = (2/3)r whereas the p o r o s i t y contribution AHp(w) i n c r e a s e s on i n c r e a s e in the frequency and vanishes at co = (2/3)r M. It is known [1, 2] that f e r r o m a g n e t i c r e s o n a n c e (FMR) in polycrystalline f e r r i t e s is influenced not only by the magnetic a n i s o t r o p y but also by inhomogeneities of the saturation magnetization (pores, as well as magnetic and nonmagnetic inclusions). It is shown in [3] that the contribution to the line width (AH) made by p o r o s i t y is of the same o r d e r as the contribution of the magnetoerystalline anisotropy and the f o r m e r can predominate even in the c a s e of m a t e r i a l s with a low (~1%) p o r o s i t y . In the theoretical calculations (known to us) of the additional broadening of the FMR lines of polyc r y s t a l l i n e m a t e r i a l s due to sat(Jration magnetization inhomogeneities use is made of a model of spherical nonmagnetic inclusions (see, for example, [,1,2]), which is a f a i r l y rough approximation for the description of r e a l i r r e g u l a r inhomogeneities. A consequence of this model is a strong f r e q u e n c y dependence of the line width (AHp oc w-2, where w is the c i r c u l a r frequency) of a sample of spherical shape in the high-frequency part of the microwave r a n g e . An attempt is made in [4] to modify the e x p r e s s i o n for AHp by introducing a p a r a m e t e r found e x p e r i m e n t a l l y . However, in this case the f o r m of the f r e q u e n c y dependence of ~ differs f r o m that o b s e r v e d e x p e r i m e n t a l l y [5, 6]. Thus, although it is r e p o r t e d in [5] that t h e r e is a maximum of AH near ~v/wM = 2/3 (wM = T4~M, where ~/ is the gyromagnetic r a t i o and M is the saturation magnetization), predicted by the theory, this maximum is shifted toward higher frequencies and AH i n c r e a s e s when the r a t i o W/WMis i n c r e a s e d . The monotonic r i s e of the line width on i n c r e a s e in the frequency is r e p o r t e d also in [6], but no singularities have been o b s e r v e d at w/wM = 2/3. We can thus see that e x p e r i m e n t a l investigations of the frequency dependence of AH show that w h e r e a s in the c a s e of dense samples t h e r e is a qualitative a g r e e m e n t with the t h e o r y of [3, 4, 7], an i n c r e a s e in the p o r o s i t y r e s u l t s in flattening of the Buffler peak on the high-frequency side and the dependence ~ ((v) becomes of the r i s i n g type in the range r > 2/3. This allows us to conclude that the nature of the f r e q u e n c y dependence of the p o r o s i t y contribution to the line width is fundamentally different f r o m the c o n t r i bution of the magnetic anisotropy, and it cannot be described by the model of a spherical inclusion. We shall t r y to calculate the contribution of p o r o s i t y to the line width using the t h e o r y of magnetic m a t e r i a l s with an inhomogeneous saturation magnetization developed by SchlSmann [8]. SchlSmann [8] d e m o n s t r a t e d that if the F o u r i e r of the local magnetization saturation M0(r) has large components at low values of g (macroscopic inhomogeneities, where 9 is the wave vector), the final r e s u l t s a r e governed only by the inhomogeneity p a r a m e t e r ~- and a r e independent of the shape of the inclusions: = ~ l M 0 (K)I~-/(< ~40 >)2 = I < M~ (r) > -- ( < M 0 > ) ' ] / ( < M 0 >)5, ,:+0

where M0 (u) = V-1 ~ Mo (r) exp (-- iu.r)d3r, < M0 > = A40 (u----0) is the a v e r a g e saturation magnetization.

In

V. D. Kuznetsov Siberian Physicotechnical S c i e n t i f i c - R e s e a r c h Institute at the State University, T o m s k . T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp.48-52, D e c e m b e r , 1980. Original a r t i c l e submitted July 3, 1979.

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0038-5697/80/2312-1026.r

9 1981 Plenum Publishing Corporation

the c a s e of nonmagnetic inclusions and p o r e s we have T = p / ( 1 -- p) and ( M0} = MS(1 - p), w h e r e Ms is the s a t u r a t i o n magnetization of the dense m a t e r i a l and p is the p o r o s i t y of the s a m p l e (the r a t i o of the total volume of p o r e s and inclusions to the volume of the s a m p l e ) . We shall c o n s i d e r an i s o t r o p i e f e r r o m a g n e t i c s a m p l e in the f o r m of an ellipsoid of revolution, whose s a t u r a t i o n magnetization is a r a n d o m function of the ,spatial c o o r d i n a t e s . An e x t e r n a l static field H 0 is applied along the revolution axis and its intensity is sufficient to e n s u r e that the deviations of the local magnetization v e c t o r f r o m the field direction a r e s m a l l . Since we a r e c o n s i d e r i n g m a c r o s e o p i c inhomogeneities of M0, we can ignore the influence of the exchange interaction. The Hamiltonian of the s y s t e m is given in [8]. Application of the third H o l s t e i n - P r i m a k o f f t r a n s f o r m a t i o n m a k e s it possible to write this H a m i l tertian in the f o r m

.:

.o §

..

+

l

.... 9

l}

h/2v,

where 1i = where h is the Planek constant; wK is the r e s o n a n c e f r e q u e n c y of a spin wave with a wave v e c t o r ~; b K+ and b K a r e the c r e a t i o n and annihilation o p e r a t o r s for magnons satisfying the t r a n s p o s i t i o n relationship: ~.x h b,,--~,,-~-+ t.+~ _~ x~,,. ~-x~,, n Q~, a r e the m a t r i c e s d e s c r i b i n g the interaction between spin waves due to inhomogeneities. The transition p r o b a b i l i t y method (see, for e x a m p l e , [1,21) for the additional broadening of the FMI~ line of p o l y e r y s t a l l i n e m a t e r i a l s , supplemented by the s e c u l a r t e r m s in H~, g i v e s

aH~ = (2~I-r) ~ i Po: I~ ~ (%- <~,,),

(2)

#g

where 5(x) is the Dirae delta function; w~ is the resonance frequency of homogeneous precession; ~ IBKI2; po~=

= A~2 _

Ao.(A.+c~ .I]'r2 BO,,[~A,~--,.,,.)1,e B,;

(3a)

(3b)

Ao,c = (1/2) tom {~ow,'~,(1 -- 5 cos 2 0x) -~- ~.a' &,:,',g,g,-K(sin e 0,,, -- 2 cos-" 9.)},

9 So,,= (I/2) ~,~,{,o,~:s~n' o. exp ( - g2e,,) + ~ v.,~,'-, s,n-o,,, ~xp (-!2",,,)}, !

*

v

(3e)

-

~t

(3d) ~t

(3e)

~ = V4~ ; 0,r and @~ a r e the polar and a z i m u t h a l angles of the v e c t o r ir * denotes c o m p l e x conjugation; N {[, N• a r e the longitudinal and t r a n s v e r s e demagnetization f a c t o r s of an ellipsoid, w h e r e Nlr + 2N• = 1. In E q s . (3b)-(3e), r magnetization by

is the F o u r i e r component of the function

r

r e l a t e d to the local saturation

,~ (r) = [M0 (r)l',ru < Mo > l 1.2. We can show (see [8]) that when this definition of the function r

(4)

is adopted, its F o u r i e r s p e c t r u m is

n o r m a l i z e d , i . e . , ~ 4 ) x l 2 = 1. On the o t h e r hand, in the c a s e of nonmagnetic inclusions and p o r e s of a r b i t r a r y K

shape we have

r

=

1

-

p,

and t h e r e f o r e , ~1~, " =P <,~,

(5)

tr

when the porosity is low. Consequently, we have separated in Eqs. (3b)-(3e) the t e r m s containing r and r 0 , and a prime of the summation sign indicates that the summation is c a r r i e d out over all the values of ~ , with the exception of the t e r m s with K~ = K and 4' = 0.

1027

,~Oe

f

r

L

/ [ !

"/7> g~

1.4

2.2.

J

i"

y

g.#

. L4

2.2

g

Fig. 2. Dependence of the total f e r r o m a g n e t i c r e s o n a n c e line width AH = ZkI-Ia + AHp on y = w/tOM: 1) p o r o s i t y p = 0; 2) 0.004; 3) 0.008; 4) 0.02; 5) 0. O4.

F i g . 1 . Dependence of the function I on the d i m e n s i o n l e s s p a r a m e t e r y = w/o:~:: 1) calculated f r o m Eq. (12); 2) deduced on the a s s u m p t i o n of w e a k n e s s of inhomogeneities of M0; 3) deduced f r o m f o r m u l a s of [7].

The c a s e of "weak" inhomogeneities of M 0 is c o n s i d e r e d in [8, 9, 10], i . e . , the t e r m s quadratic in r a r e ignored in r e l a t i o n s h i p s of the (3b)-(3e) type. However, we can show that the contribution to ~H is of the s a m e o r d e r of magnitude as the contribution of the l i n e a r t e r m s . In fact, in g e n e r a l the e x p r e s s i o n for AH contains in addition to the s u m (5) also s u m s of the kind SI~--- , O ~ d

*

*

~,c~,~'~'-~,

.--"~"

,

* 2 ,~'yz'-~ 9

~

(6)

This follows f r o m t h e definition (4) for 9 r 0 that Mo>

, ~,

(7)

1 K~

Taking the p r i m e d s u m in Eq. (7) and e x p r e s s i n g it in t e r m s of M0(~) and CK with the aid of Eqs. (1) and (5) and the r e l a t i o n s h i p ( % ' < M0 > ) z~ ?~ 0 (~) = P , we find that Eq. (6) b e c o m e s K

$1 = -- p (1 -- 2p), S2 = p (1 - - 2p)-~ (1 - - p ) .

(8)

Thus, the contribution of the t e r m s quadratic in SK in Eqs. (3b)-(3e) is of the s a m e o r d e r as Eq. (5), even if the p o r o s i t y is v e r y s m a l l . The s u m m a t i o n in Eqs. (3b)-(3e) includes functions of the polar and azimuthal angles of the v e c t o r g ' . T h e r e f o r e , in calculating t h e m we have to know the explicit f o r m of the F o u r i e r s p e c t r u m of inhomogeneities, which is governed by the shape of the p o r e s and inclusions. However, if we a s s u m e that inhomogeneities of M 0 a r e distributed i s o t r o p i c a l l y in a p e r i o d i c a l l y r e p e a t e d cell, then SK and M0(~) depend only on the length of the v e c t o r ~. Consequently, a v e r a g i n g Eqs. (3b)-(3e) o v e r the angles O , and $~,, we can obtain pc.

{A,\

[(1 - 5co 2 o.)

= (1i2)

2~,

Ax = 7Ho --

-

s n2 O. •

+ (i 3) ~M ~ ' * , " ? , ' - , (I -- 3 cos" 0,) /

J

.

(9a)

,,

~,~uIN :; -- N• (1 - - p ) ~,o] -k 0/3) o~Mp+ (1/2) to.u (1 --p) sin e 0,, B, = ( 1/2) ~,~ (I -- p) sin -~0, exp (-- i2(I)x).

(9b) (9c)

Substituting Eqs. (9a)-(9c) in Eq. (2), we find that in the c a s e of a s p h e r i c a l s a m p l e AHp = (2: |/3/9) 4= < Mo > p l (y, p),

(10)

I (y, p ) = .[Y - (1 - - 2p)/6 ]" [y (1 + p ) - - 2 ( 1 - - 2p)(1 - p/2)/3l 3~2

(11)

w h e r e Y = ~/~M, ~o = 7Ho, [y - - ( 1--2p)/3] TM 1028

Calculations of !(y, p) for various values of y, p show that Eq. (11) depends weakly on the porosity. Moreover, this calculation applies only to low values of p. T herefore, ignoring p compared with unity, instead of Eq. (11) we have

/(y) = ( y - y 6 ) ~ (~,-213),~2!(y -1/3) ~/2,

(12)

A graph of the function (12) is shown in Fig. 1 (curve 1), On increase of y, the dependence I(y) tends to unity and an approach to the upper boundary of the exchange-free spectrum (y = 2/3), it vanishes. Even if the calculation is made on the assumption of "weak" inhomogeneities of M0, the nature of the function i ( y ) changes (curve 2 in Fig. 1). The frequency dependence of AHp is then of the rising type also for y > 1, but if y = 2/'3, there is a Buffler maximum, which is not in agreement with the experimental resul t s. For comparison, Fig. 1 includes curve 3 which describes the frequency dependence of the magnetic anisotropy contribution (AHa ) to the FMB line width [71. If the exchange interaction or intrinsic relaxation pr o c e sse s a re allowed for, the minimum of AI~ at y = 2/3 like the maximum of AHa (see [7]) are of finite amplitude. Figure 2 shows the frequency dependences of AH plotted for yttrium iron garnet (47rMs = 1750 G, Ha = 86 Oe) for different porosities using Eq. (10). The magnetocrystalline anisotropy contribution to AH is calculated using the formulas of [7]. It is clear from Fig.2 that the change from AN decreasing on increase in the frequency to a rising dependence occurs when the porosity is about 17c. It follows from our calculations that in the matrix elements of the Hamiltonian describing the interaction of homogeneous precession with spin waves due to inhomogeneities of M 0 it is n e c e s s a r y to include not only t e r m s linear in the Fourier components of the function r but also the quadratic t e r m s . The frequency dependence of the porosity contribution to AH is fundamentally different from the magnetic anisotropy contribution. ']?he anistropy contribution AHa decreases on increase in the frequency and has a characteristic Ruffler maximum at r M = 2/3, whereas the contribution of the porosity to AH increases on increase in co. Therefore, the frequency dependence of the combined line width AH = AI-Ia + AHp depends on the relationship between the anisotropy field on the one hand and the saturation magnetization and porosity, on the other. This behavior of AH(c~) is in good agreement with the experimentally observed dependences. LITERATURE CITED 1. A. G. Gurevich, Magnetic Resonance in F e r r i t e s and Antiferromagnets [in Russian], Nauka, Moscow (1973). 2. M. Sparks, Ferromagnetic Relaxation Theory, McGraw-Hill, New York (1964). 3. P. E. Selden and I. G. Grunberg, J. Appl. Phys., 34, 1696 (1963). 4. P. E. Selden and M. Sparks, Phys. Rev., 137, A1278 (1965). 5. C. B. Buffler, J. Appl. Phys. Suppl., 30, 172 (1959). 6. A. A. Maauilova and A. G. Gurevich, Fiz. Tverd. Tele (Leningrad), 6, 3475 (1964). 7. E. Schl~mann, J. Phys. Chem. Solids, 6, 242 (1958). 8. E. Schl{Jmann, J. Appl. Phys . , 38, 5027 (1967). 9. E. Schl~mann, J. Appl. Phys . , 38, 5035 (1967). 10. V. A. Zhuravlev, G. I. Byabtsev, and Yu. N. Kotyukov, Izv. Vyssh, Uchebn. Zaved. Fiz., No. 8, 32 (1979).

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