RESONANCE

PHENOMENA

GYROTROPIC

MEDIA

D.

I.

Sementsov

IN

STRATIFIED

and G.

S.

UDC 539.28

Kosakov

A solution of the Maxwell equations is used as a b a s i s for considering the b e h a v i o r of a g y r o t r o p i c s t r a t i f i e d m e d i u m . It is shown that a change in the r a t i o between the thickn e s s e s of the f e r r o m a g n e t i c l a y e r s and the d i e l e c t r i c l a y e r s c a u s e s a shift in the r e s o nant f r e q u e n c i e s . The shift will be in different directions for t r a n s v e r s e and longitudinal magnetization of the m e d i u m . The conditions a r e obtained for which the effective depth of penetration of the e l e c t r o m a g n e t i c field into the m u l t i t a y e r s y s t e m is f a r g r e a t e r than the depth of the skin l a y e r for m a s s i v e m e t a l s a m p l e s . 1. The interest in thin f e r r o m a g n e t i c f i l m s r e s u l t s f r o m the v e r y p r o m i s i n g p r a c t i c a l applications of t h e i r magnetically g y r o t r o p i c p r o p e r t i e s in m i c r o w a v e d e v i c e s . As a g y r o t r o p i c element, however, a thin film exhibits weak r e s p o n s e to a m i c r o w a v e field, since the volume of f e r r o m a g n e t i c m a t e r i a l is e x t r e m e l y s m a l l . Owing to high conductivity, the i n c r e a s e in film thickness is l i m i t e d by the depth of p e n e t r a t i o n of the RF field into the s a m p l e ; in the c e n t i m e t e r band, this is 10-4-10 -~ c m f o r m e t a l l i c f e r r o magnetic m e t a l s e p a r a t e d by nonmagnetic d i e l e c t r i c l a y e r s . The static and dynamic b e h a v i o r of such s y s t e m s differs i n m a n y ways f r o m the b e h a v i o r of s i n g l e - l a y e r films, and this accounts for the i n c r e a s e d int e r e s t in t h e m for m i c r o e l e c t r o n i c s applications. Of p a r t i c u l a r interest in this r e s p e c t is the study of the r e s o n a n c e p r o p e r t i e s of m u l t i l a y e r s y s t e m s . In f i l m s with thin d i e l e c t r i c i n t e r l a y e r s (L _< 1500 A), the p r e s e n c e of m a g n e t o s t a t i c coupling between the f e r r o m a g n e t i c l a y e r s shifts the r e s o n a n c e c u r v e s [1-3]. As e x p e r i m e n t has shown [4], however, r e s o n a n t - f r e q u e n c y drift is also o b s e r v e d in t h i n - f i l m s y s t e m s where t h e r e is no magnetostatic coupling between the p a r a m a g n e t i c l a y e r s . H e r e the amount of the shift depends on the r e l a t i o n s h i p between the t h i c k n e s s e s of the magnetic and d i e l e c t r i c l a y e r s . It is of interest to examine the way in which the p a r a m e t e r s of the t h i n - f i l m s y s t e m affect its r e s o n a n c e b e h a v i o r . H e r e we shall study the r e s o n a n c e p r o p e r t i e s of an unbounded m u l t i l a y e r m e d i u m and consider certain f e a t u r e s of the r e s o n a n c e b e h a v i o r of bounded s y s t e m s located in t r a n s m i s s i o n lines. 2. We c o n s i d e r a m u l t i l a y e r m e d i u m m a d e up of alternating l a y e r s of f e r r o m a g n e t i c m e t a l having i s o t r o p i c conductivity cr and t e n s o r p e r m e a b i l i t y / z , and l a y e r s of d i e l e c t r i c having isotropic p e r m i t t i v i t y e and p e r m e a b i l i t y P0. Such a m e d i u m has the p r o p e r t i e s of a uniaxial c r y s t a l whose s y m m e t r y axis is p e r p e n d i c u l a r to the l a y e r s . Although the m e d i u m c o n s i s t s of isotropic l a y e r s , on the whole it exhibits anisotropy with r e s p e c t to the conductivity and p e r m i t t i v i t y [5]. If the 0z axis coincides with the n o r m a l to the s u r f a c e s e p a r a t i n g the l a y e r s , then the t e n s o r r e p r e s e n t i n g the complex p e r m i t t i v i t y of the medium = ~ -- (4v~r/iw) is diagonal and its components a r e d e t e r m i n e d by the following relationships: ~"~ = % ~ ' =

l q- L

'

~z~

l-k L,

'

where l is the t h i c k n e s s of the m e t a l l a y e r s ; L is the thickness of the d i e l e c t r i c l a y e r s . The f o r m of the p e r m e a b i l i t y t e n s o r depends on the direction of the e x t e r n a l magnetic field H 0. If the f e r r o m a g n e t i c l a y e r s a r e m a g n e t i z e d in t h e i r own plane (H0110y), the t e n s o r then takes the f o r m L X,r

XZ

/.

,%

0 s

~'

~Lzx 0

9

(2)

'

~zz/

North Osetin State U n i v e r s i t y . T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol.18, No.8, pp~ 1189-1195, August, 1975. Original a r t i c l e submitted October 24, 1974. 9 76 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy of this article is available from the publisher for $15.00.

879

Here 1 . =

l,"I**~ +

,j~s zz

(3)

Lifo ,

l + L

while the remaining components of the t e n s o r are determined in the following manner:

~-:

lt~ + LP,o

--t~L-L--

(~" ~ = ~' y' z),

(4)

where 6 a tf is the Kronecker symbol. Relationships (1), (3), (4) are valid for small thicknesses of the metal and dielectric l a y e r s ( i . e . , when the conditions kz/ << 1 and kzL << 1 are satisfied). In actual thinfilm multilayer s y s t e m s , the thickness of the dielectric l a y e r s will n e a r l y always satisfy this condition in the microwave band. If the l a y e r s are not thin in the indicated sense, however, then in e x p r e s s i o n s (1)(4) for the p a r a m e t e r s of the medium we must introduce c o r r e c t i o n t e r m s of o r d e r kz (l +L) z. 3. In a multilayer medium, let a m o n o c h r o m a t i c wave having frequency w propagate in the d i r e c tion of the 0y axis. Here the e l e c t r o m a g n e t i c field r e p r e s e n t s the superposition of several plane waves. F o r each such wave, the e l e c t r i c and magnetic fields v a r y as e-t(wt-kr) and satisfy the Maxwell equations which we write as m

A

k >.: e = - - ,% h,

k 5( h =

o} A

-- ~ r,e.

C

(5)

C

After substituting the second of these equations into the first, Eqs. (5) take the following form: ,;-' ,,. A

(6)

C2

"-

Writing this equation in p r o j e c t i o n s on the coordinates axes and taking (1) and (2) into account, we obtain the following s y s t e m of equations: (k ~

tu*-

') __ {.'~-~ .. C2 t~':,.,. M.'," h,,pT ~''~ 'q'~': hz :- O, 0) 2 - -

s

~"

hy

"

~ YY r t

(7)

O,

0) 2

,)

c~-~''~~%~,,.~h.~- (k" -- ~t,,-~,:,~,,_ _ , a~ :--{}: it follows f r o m this that t r a n s v e r s e waves will propagate in a longitudinally magnetized multilayer medium ( i . e . , hy = ey = 0). Setting the determinant of (7) equal to zero, we obtain the dispersion relationship

T ~,~. ~+:~z

"~ +--

~L~-"-': -':~ "+4~.~ t~'~= zz

where k 0 = w/c, while the two values of the wave n u m b e r c o r r e s p o n d to two modes with right (k+) aud left (k_) rotation of the field v e c t o r s . Using (5), we may determine the polarization of each of these modes and the relationship between their p h a s e s and amplitudes. Thus the amplitudes of the corresponding waves are r e l a t e d by the following e x p r e s s i o n s : 'h,

~]** •

=

~** k_+-- 0 P ~ ] ~

"~z, \ ez/_:: ( e~'~

\h.,) ,_

r, zz

=

ko ~z '~x.r - -

kok+~z

g~

#+ k~--

2 s

- - leo Pxx "~zz

,

.

(ga)

.

(9b)

(9c)

We see f r o m (9) that the modes propagating in the m u l t i l a y e r m e d i u m are t r a n s v e r s e with elliptical p o l a r ization.

880

! %. ~!';~ '

:'

4

t

2L ~%

5/

I

..---/

/

V Z I

.........

Fig, 1 F ig. 2 Fig. i. Resonance c u r v e s for p a r a l l e l magnetization along easy and difficult (') axes: theory) 04 = 100, 03 = 1, 0~ = 0.1 and 01 = 0. Experiment) e e o ) is f o r 0 =400, AAA) is f o r 0 =100, " 9 1 4 9is f o r 0 = 2 5 . Since ~ >> we in the c e n t i m e t e r band, we find tNzzt << h?xxl f r o m (1); this inequality enables us to expand the square root in (8) into s e r i e s and to obtain the following e x p r e s s i o n s for the wave n u m b e r s :

k+

=

ko I/~.~.~ }d-~

k_ =/~, I ",~:~, ~ - ,

(IO)

where ~s

_:

=

s

s s t-~xz [J'z.:"

"~zz

'J,

t) 3 L~,~ ' .rz ~ z.v -

zz

3

The resonance absorption of the m i c r o w a v e - f i e l d e n e r g y is due to the mode with left rotation, so that to determine the resonant frequency of the m u l t i l a y e r s y s t e m we rewrite p - with allowance for (3) and (4): ....

(I +

>'~

O) ~t~o

(~L1)

Here 0 = L / 1 , while the components of the permeability t e n s o r for the metal l a y e r s are found by solving the Landau--Lifshits equation; they take the following f o r m [6]: ,~ ~

-.~H,i (Hi n-' 4r.M0)- ,,~ p/-l~ -

~" =

,a

,'

4"-'~"AI~176

(12a)

(1210)

.,2 H~ -- o)~ '

where Hi is the internal field which in our case equals I4_0:I:Hk, Hk is the induced uniaxial a n i s o t r o p y field [the plus (minus) sign c o r r e s p o n d s to H 0 being directed along the easy (difficult) axis]. The resonant f r e quency is d e t e r m i n e d by the p e r m e a b i l i t y pole and has the f o r m

% = ";

(H,, !: Ilk) fio [ tt,-t-

fJ

'1-',~u )"

F o r the limiting case O -* % the e x p r e s s i o n obtained goes over to the f a m i l i a r f o r m u l a for the resonant frequency of an isolated thin film magnetized in its own plane: OJp = T447rM0(H0 +Hk); if 0 = 0, however, then wp = 7 (I-I0=LHk). In the limiting expressions, the p e r m e a b i l i t y of the d i e l e c t r i c l a y e r s is taken to be unity. The experimental points plotted on the theoretical c u r v e s of Fig. 1 were obtained in a strip line filled by a multilayer s y s t e m having v a r i o u s thickness ratios of the metal ( P e r m a l l o y 80Ni--20Fe) and dielectric (mica) l a y e r s . The m e a s u r e m e n t s were made a t 3000 and 1860 MHz; the metal l a y e r s were made c o n s i d e r a b l y thinner than the s k i n - l a y e r depth in t h i s c a s e . It follows f r o m the r e s u l t s that a s 0 i n c r e a s e s , t h e r e is drift of the resonant frequencies (fields) toward higher (lower) values. 4. Let us now examine the case in which an e l e c t r o m a g n e t i c wave p r o p a g a t e s in a t r a n s v e r s e l y magnetized multilayer medium (i. e . , the external static field H 0 is applied along the n o r m a l to the l a y e r s u r f a c e s , while the wave v e c t o r lies in the plane of the l a y e r s -- k It 0y). Here the solution of the Maxwell equations leads to two modes: t r a n s v e r s e - e l e c t r i c (TE) a n d t r a n s v e r s e - m a g n e t i c (TM). The resonance

881

conditions ( i . e . , the p e r p e n d i c u l a r i t y of the RF component of the magnetic field to the static field) are satisfied only for the TE mode, so that we shall just discuss this c a s e . In projections, Eqs. (6) take the form . 2 t~S ~ t. t.2 ts . (k 2 - - e o , x x % . O n x - - ~ o , ~y %~ hv = O, (14)

Setting the determinant of this s y s t e m equal to zero, we find the propagation constant, k

=

leo 1# %.~}J.-,,

I-%=F.,-.~~

' ' Y ~.,. Y'~ ,~s

(15)

s

~J'vv

Taking (1) and (4) into account, we write P_L as (~ § %,)~ + ~'~ ~J'.l

=

(1 +

0)2(~ ~ +

(16)

O:-hO~ '

where the components of the p e r m e a b i l i t y t e n s o r for the m e t a l l a y e r s a r e determined by (12), w h e r e the internal field must be taken in the f o r m I-I0--47rM0. Setting the denominator of the resulting expression equal to zero, we find the resonant frequency of the t r a n s v e r s e l y magnetized multilayer system:

F o r the two limiting e a s e s 0 ~ co and 0 = 0, the formula obtained goes over to the familiar r e s o n a n t - f r e quency e x p r e s s i o n s for an isolated film magnetized perpendicular to its surfaee [COp-- y(I-I0--47rM0)] and for a m a s s i v e specimen (COp= 7tt0), provided we a s s u m e that in the absence of d i e l e c t r i c l a y e r s t h e r e are no interfaces between the l a y e r s of f e r r o m a g n e t i c m a t e r i a l (H i = H0). If the s y s t e m r e m a i n s multilayer for L = 0, t h e n the resonant frequency is given by the f o r m u l a Wp = y~/I-I0(H0--47rM0). Figure 2 shows the relationship, computed f r o m (17), between the resonant frequency of a multilayer s y s t e m and the applied static field; it was obtained for v a r i o u s values of the p a r a m e t e r s 0 with perpendicular magnetization. When we go f r o m l a r g e to small 0, the r e s o n a n c e c u r v e s shift toward lower field values and higher f r e quencies, which is c o n f i r m e d by experiment [4]. Our analysis and the experimental data show that the r e s o n a n c e c u r v e s will shift in opposite d i r e c tions with variation in 0 for p a r a l l e l and perpendicular magnetization of the medium. This is connected with the fact that a change in the ratio between the t h i c k n e s s e s of the f e r r o m a g n e t i c metal and the dielect r i c will change the average magnetization of the medium, which may be r e p r e s e n t e d in the f o r m Ms = M J 1 + 0. A change in 0 shifts the resonance c u r v e s ; as in the case of isolated thin films, the direction of the shift will differ for different types of magnetization. 5. By using the resonance dependence of the permeability of metal l a y e r s on the frequency (field), it is possible under certain conditions to obtain a significant increase in the depth of penetration of a m i c r o wave field into a m u l t i l a y e r s y s t e m in c o m p a r i s o n with the o r d i n a r y skin l a y e r . Let us consider the p r o p agation of an electromagnetic wave having the components ex, hy, and ez (the type of wave r e a l i z e d in strip t r a n s m i s s i o n lines) in a m u l t i l a y e r s y s t e m along a magnetizing field H 0 which lies in the plane of the l a y e r s . Taking (1) and (2) into account, the Maxwell equations for the indicated components of the e l e c t r o magnetic field a r e written as Oe x

c)ez _

ito

Oz

Ox

c

Ohy _

io

Oz

~.,.,.

u, s b y , , yy

ex '

(18)

c

Oh,,

i~

Ox

T

%~ ez"

F o r waves propagating along the Ox axis, the dependence on the coordinate will be proportional to e ikx, where k is the longitudinal component of the propagation constant. Allowing for this, we have no difficulty in obtaining f r o m (18) an equation to d e s c r i b e field penetration into the s y s t e m in the direction of the n o r mal to the l a y e r surfaces:

o _d'hy_ __ __

dzZ

882

.v ~ } "t~..... ( k ~ - - ki,~ %,y ~zz

hv

= O.

(19)

If we a s s u m e that the m e d i u m occupies semiinfinite space and that hy 0 is the magnitude of the field on the boundary z = 0, we may then write the solution of the resulting equation in the f o r m hy(z) = hy 0 e -vz, where the t r a n s v e r s e component of the propagation constant takes the f o r m v

[.i~".~-x + 4 ~ . . ~ ( k ~-- ko~.t~,,, k ioJ%~ + 4r.%~

"

+

"

,20,

"" c2

,,

The depth of penetration of the microwave field into the multilayer s y s t e m is determined by the expression 5 -1 = Re v. In the f o r m u l a obtained, the longitudinal propagation constant and the quantity k2ezzp~y are of the same o r d e r of magnitude and m a y be equal. Here Im v = 0 and when allowance is made for the inequalities Crxx >> O)exx and azz <
:: ~

]// V

-

"z~

(21)

Substituting (1) and (4) into this formula, we obtain ?~= 1 + 0 c

- - - - ~ - - - ....

(22)

F o r the usual values of the quantities o c c u r r i n g in (21) (co ~ 10 -l~ sec -1, e ~ 10, # ~ 10-100, p~ ~ 1, 0 ~ 1), the depth of penetration of the field into the multilayer s y s t e m is of the o r d e r of 10-2-10 - i cm; this is far g r e a t e r than the depth of field penetration into a m a s s i v e specimen. The condition under which (22) holds m a y be written as

: ~

?~

where ~'and ~ are the permittivity and p e r m e a b i l i t y of the medium transporting~the e n e r g y of the wave. Since most of the e n e r g y is t r a n s p o r t e d to regions lying outside the multilayer system, satisfaction of (23) may be ~issured by an appropriate choice of the medium in which the multilayer s y s t e m is located. However, it is far s i m p l e r to satisfy this condition by the appropriate choice of frequency (field), since ~ depends in resonant m a n n e r on the frequency (field). Substituting (12a) into (23), we obtain the condition imposed on the frequency at which there is a sharp increase in the depth of field penetration into the multilayer system:

,,,'- =

__4:M0

)

(24)

In frequency r a n g e s that fail to satisfy this condition, the depth of penetration takes its usual f o r m and o r d e r of magnitude. A condition analogous to (24) may be obtained for the static magnetizing field at a constant frequency. Thus, our r e s u l t s show that the c h a r a c t e r i s t i c s of the resonance behavior of multilayer g y r o t r o p i c media m a y be employed s u c c e s s f u l l y to develop microwave devices. LITERATURE 1.

2. 3. 4. 5. 6.

CITED

D. Chen and A. Morrish, J . Appl. P h y s . , 33, 1146 (1962). L. V. Kirenskii and N. S. Chistyakov, Izv. Akad. Nauk SSSR, Ser. F i z . , 30, No. 1 (1966). D. I. Sementsov, Fiz. Metal. Metalloved., 34, 1088 (1972). L. Courtois et al., IEEE T r a n s . Magn., 7, N-'-o.3 (1971). L. M. Brekhovskikh, Waves in Stratified Niedia [in Russian], Nauka, Moscow (1973). A. I. Akhiezer, V. G. B a r ' y a k h t a r , and S. V. Piletminskii, Spin Waves [in Russian], Moscow (1967).

883