86 IZVESTIYA VUZ. FIZIKA

NONLINEAR THEORY OF THE EXCITATION OF SPIN WAVES IN FERRITES BY A LONGITUDINAL ULTRAHIGH- FREQUENCY MAGNETIC FIELD

G. A. Pe t rakovsk i i and V. N. Berzhanski i

Izves t iya VUZ. Fizika, No. 5 pp. 128-133, 1965

The problem of parametric excitation of spin waves in ferrites due to the action of a perturbing super high-frequency (SHF) magnetic field polarized along the direction of the Constant magnetic field H 0 is considered in a nonlinear approximation�9 Such an examination en- ables one to determine the conditions for parametric excitation of spin waves and to find the amplitude and phase of the oscillations of the system's magnetization in the steady state. The frequency interval, within whose limits parametrin excitation of spin waves is possible, is determined. Stability of the steady state under the conditions of para- metric resonance is investigated. The results obtained are compared with existing experimental data.

1. The phenomenon of p a r a m e t r i c exci ta t ion of spin waves by an SHF magnet ic field [1] has become an impor tan t tool for the study of spin wave re laxa t ion p roper t i e s . In pa r t i cu l a r , this method of m e a s u r i n g the spin-wave l ine width AH K of the f e r romagne t i c r e sonance has impor tan t advantages over the wel l - known method of m e a s u r i n g AHK by sa tu ra t ion of the f e r romagne t i c r e sonance for un i form p reces s ion of the magnet iza t ion . The theory of p a r a m e t r i c exci ta- tion of spin waves in f e r r i t e s by an SHF magnet ic field has been developed only in the l i nea r approx ima- tion, and therefore cannot pre tend to be complete . The impor tan t quest ion of the s ta t ionary state of the spin sys t em under condit ions of p a r a m e t r i c r e sonance r e m a i n s unsolved. In the p resen t a r t i c le an a t tempt is made to develop a non l inea r theory of the pa ra - me t r i c exci tat ion of spin waves�9

2. Let us cons ider a f e r r i t e sample in the shape of an e l l ipsoid of revolut ion located in a constant homogeneous magnet ic field H0, applied along the axis of revolut ion. Let the pe r tu rb ing va r i ab le mag- net ic field h cos w3t also be d i rec ted along the axis of revolut ion . We shall a s sume that the sample is magne t ica l ly isotropic in the absence of the field H0 and does not have any d ie lec t r i c lo s ses . We assume , fur ther , that the magnitude H0 of the ex te rna l field is suff icient for magnet ic sa tu ra t ion of the sample .

We take the equation for the change in in tensi ty of magnet iza t ion in the form

rn = --I~1 [mil l + ~ [in n~], (1)

where

M m = - - , m = m o + ~ m ( r , t ) ,

M

H = Hex t -~- /'/de m-,~-/'/eft -i- h COS toa t,

Hext= Ho - - ,'Vz 4 ~ Mn,

/'/eft := Hexcha~ X 72 ?,tn,

//dem . . . . 4r. 3Ik ( k ~ m ) k '

~m (r, t) = X ~,m, (t) e-~*',

m 0 is the stat ic magnet iza t ion vector ; 6m is the va r i ab le magnet iza t ion vector ; Hexch is the equivalent exchange in te rac t ion field; k is the wave vector of the spin wave; a is the lat t ice constant of the cubic c rys ta l .

In what follows 6m K wil l be denoted by 5m. Subst i tut ing H into Eq. (1) and taking into cons ide r -

a t ion t e r m s up to the third o rde r of sma l l ne s s inc lu- sive, we obtain the following sys t em of equations for the components 6mx and 6my:

?,mx ~ -- A?,my-- B ~ m x - - C ~ m z + (C-~B--B ) ~m,~mz +

+ C?,mfimy--B~m.,.~,mz -- ~,my (I + ~nzz) --

-i- C (~tn~ m ?mZz) + a?,my~ntz, (2)

-- C~nzx~my + B?,nty ~rnz + a~mx (1 + ~tnz) - -

- C (?mt~ -- ~mz) -- a~nzfimz.

Here we have introduced the following notation:

A = o~,+ t,J~, k]_ + toexeh a~'kz -t- los cos %t = = + cos cos % t, l'~ tvl k 2

_ _ k.[. kz ; - to k] sin~(')~ C = o~ M = tO M ~ 2 sin B = M~2"2 := 09M 2 ' k 2 4 2 0 x ,

t~ [TlHext, t~ '[c M Mo, t o ~ Iwl h, ~exch= I~1/-/exeh,

0 K is the angle between the d i rec t ion of the wave vec - tor k and the d i rec t ion of the field H0.

In obtaining the sys tem of equations (2), f rom sym- metry cons idera t ions we assumed k x = ky = k_L. This appreciably s impl i f ies the calcula t ion without d e c r e a s - ing the genera l i ty of the r e su l t s obtained. In addition, it is impor tan t to note that in obtaining the sys tem (2) the absence of spin-wave in te rac t ion was assumed. This means that the ana lys i s given here is valid only for cases when the ampli tude of the per tu rb ing SHF field does not appreciably exceed the threshold amp- litude hcr i t [1]. Saturat ion effects, whose mechan i sm is analogous to the sa tu ra t ion mechan i sm of un i fo rm p reces s ion of the magnet iza t ion [2], are poss ib le with l a rge -ampl i tude p a r a m e t r i c a l l y excited spin waves. In this connection, the spin-wave damping p a r a m e t e r becomes a function of the SHF excit ing power.

With the re la t ion

1 (Z,m~ A- ~.z~) (3) ~ l n z ~ - - - ~ -

taken into cons idera t ion , the sys tem of equations (2) may be reduced to a di f ferent ia l equation of the form

~,,;i.,. i- ''-~'~ ;;,m~ = zf(~,m.,., ~,,'nx, -.), (4)

SOVIET PHYSICS JOURNAL

where a is a sma l l p a r a m e t e r , ~" = wat is the t ime in

d i mens ion l e s s units ,

~fC~mx,~;nx, ~ ) = - - [~ ,nx [2~ a-- + ~ sin~ ] + L z3

to~ 0)3

+ "ti~(4--~--7~ ~ + + 5 ~ * ) +

1 o ~ _ _ ~ + T ~ i - - 5 + 9 ~ - - a ~ ) -

1 - ~ ~ (1--2~+3~)stn~ +

1 ~ ~ 1 ( - - 5 + f ~ + 6 ~ ~ ) c o s ~ ] ~ m ~ + + 7,0-7

+ 3 ~-- ~q (l--t~) - t% ~q (3~--1) sin �9 + 6 ~ cos , ] ~rn~ ~thx + 1118

+ 3 - - (~--~1 ~) (~--1) arnx ~ ~tnx+

, ) + + to~ L \

+ 2 ~ ] S t n ~ + [ 3 ~ a q ( l + ~ ) - - @ ~ to'~" ' s i n ~ + ~2

q- 3~-- n _ _ c o s x + n(l--3} ; th I+

+ - - (~ .q') ~,4tl + (~2_~) (3~_1) ~k~amx (5) 11

Here

C ~ = 13 ' ~ = - - , (6)

t~l = ~' -- B z = (~~ ~ a'~k') X

XIoN + t%xcba~k~ + to M sin ~ 0~). (7)

In the de r iva t ion of Eq. (4) t ime-dependen t t e r m s with the f requency 2coa were omi t ted . This can be done s ince we a re i n t e r e s t ed in a sp in -wave p a r a m e t r i c r e s o n a n c e

at the f requency w J 2 . The a sympto t i c method of N. M. Kry lov and N. N.

Bogolyubov [3] is used for the solut ion of Eq. (4). To the f i r s t approx imat ion , we seek a solut ion at the f requency of the pr inc ipa l demul t ip l i ca t ive r e s o n a n c e

in the fo rm

(4 m~ = a cos ~ + ~) = a cos'b (8)

The ampl i tude a and the phase shif t d mus t be d e t e r -

mined f rom the s y s t e m of equat ions

da 171.~H~ a (.~ ~t~S ) a sln 20 -J r (9) d~ 2t% to]

87

d~

+ b~ a cos 25 + ~a a, 2o~ 3

or% ) ~ (9) cos 2~ -- -:-- sin 2 q~ + ~a ?, w~ 2% cont'd.

where

2 ~ tO x 1

AH~=- 17] ' % 2 Oo)

3 i ( ~ - - ~ ) (~- - 1), 3 ~ ( ~ - n ) + ~ ,o~

1

3 1 77; ~ ( ~2 _ f ~

We note that the t e r m s quadra t i c in 6m x and 6m x on the r ight hand side of Eq. (4) do no t, in the f i r s t approx imat ion , have any ef fec t on the solut ion of this

equation. We r e w r i t e the s y s t e m of Eqs. (9) in the fo rm

da = _ ~ a - - , a sin (20 - ~) - ~a ~, d~ 2toa

~--) = ~ - v cos (2~ - ~ ) + ;aL (12) d~

where

2B"):~ (13) tg ~ = to~ -- 4~ 2

The coeff ic ient g is p ropor t iona l to the ampl i tude of the va r i ab l e exci t ing magnet ic field and c h a r a c t e r i z e s the depth of modula t ion of the f e r r i t e p a r a m e t e r s . Upon ful f i l lment of the p a r a m e t r i c r e sonance condition

~3 (15)

the coeff ic ient g b e c o m e s equal to

~%crB (16) t ~c r ~ - ~ - .

Equating the r ight hand s ides of equat ions (12) to zero , we obtain the following e x p r e s s i o n s for the s t a t i ona ry ampl i tude and phase :

a~ = ~ + ~---~ - - 2%-~-~

• ~@+;b-[ I~laH~ r,-~,~]'~ / , 2t% (17)

lg (2,% - - +) = -- !~1 a HK/2o,~ + P a?, (18)

88 tZVESTIYA VUZ. FIZIKA

F r o m the condit ion that the s t eady- s t a t e ampli tude of the spin waves be equal to zero, we find the t h r e sh - old for p a r a m e t r i c exci ta t ion. The ampli tude of the threshold exci t ing magnet ic field is m i n i m a l for ful- f i l lmen t of condit ion (15) and is given by

hcrt t = ~ . aH~ o~ M Sill ~ (9~ " (19)

The l i nea r theory of the p a r a m e t r i c exci ta t ion of spin waves leads to a n analogous r e su l t .

~'r/~,~9,o �9 4F

, ,

�9 /J I / / r

4

' / / r " / / ii

- | "4 0 '~ $ m /6

Fig. 1

Exp re s s ion (17) for the s t eady- s t a t e ampl i tude can be r ewr i t t en in a form which is more c o n v e n i e n t for US,

~ = ~' / - ~ - k ~§ p 2 + ~ 2 [ ~1 -

( 2o )

+ ( ~ + ~) ~ - ~ - _ ~ ,

where P is the exci t ing power of the SHF magnet ic field, P c r i t is the c r i t i ca l value of the exci t ing power, and ~1 = I~1 AHJ2w3-

Lo

oo

,~ 0 4

"~ og

Experiment

Theory ~

J $ # 0 o i '/P/Pcrit 'dB

Fig. 2

Using exp re s s ion (20), f r o m the condit ion that the s t eady- s t a t e ampli tude a0 be equal to zero we find the l im i t s of the r e sonance curve for p a r a m e t r i c r e sonance

-" Pcri '-~- 1 . (21)

F rom the condit ion that the s t eady-s t a t e ampli tude be rea l , we obtain the value of 6 at which p a r a m e t - r i ca l ly excited osc i l l a t ions " b r e a k away."

The r e sonance curve and the dependence of the s t eady- s t a t e phase on 5 for the case k = 0, O K = ~/2 a re shown in Figs . 1 and 2, r espec t ive ly . The following va lues were a s sumed for the calculat ion: f = 9200 Mc, 4~M = 1750 gauss , AHu = 0.3 Oe ( y t t r i u m f e r r i t e ) . F r o m Fig. 1 it is c l ea r that the r e sonance curve is a s y m m e t r i c . There is a d rawn-ou t region in which p a r a m e t r i c exci ta t ion of spin waves is poss ib le at f requenc ies lying outside the l imi t s de t e rmined by re la t ion (21). Two s ta t ionary s ta tes of the sys tem a re poss ib le in this region. Inves t iga t ion shows that the s table state co r responds to the upper par t of the curve in Fig. 1. The l a rges t s t eady-s t a t e ampli tude occurs at a f requency differ ing f rom the f requency c03/2 by an amount co 3 54. The value of 64 may be de t e rmined by s imul taneous solut ion of Eq. (20) and the equat ion

5a~ + ~ = 0, (23)

which co r re sponds to vanish ing of the denominator in (18).

For the case k = 0, O K = ~/2, the solut ion is obtained graphica l ly . The s t ra igh t l ine shown in Fig. 1 c o r r e s - ponds to Eq. (23).

r t ~

90

30

) /

j J

-4 o 4, 8 /2 #8

Fig. 3

\

A'm"

To the f i r s t approximat ion, the solut ion obtained for this problem does not depend on the quadrat ic t e r m s on the r ight hand side of Eq. (4). Therefore , in the genera l case it would be n e c e s s a r y to solve the p rob lem in the second approximat ion. One should, however, note that the coeff icients of the quadra t ic t e r m s in the equa- t ion contain the p a r a m e t e r ~?, which is equal to zero for spin waves with O K = 7r /2 .

Since i t is p r ec i s e ly such spin waves which a re impor tan t in our case of p a r a m e t r i c exci ta t ion by a longi tudinal SHF field, it is not n e c e s s a r y to solve the p rob lem to a higher degree of approximat ion.

3. One can compare the r e su l t s of the ca lcula t ion presen ted above with exis t ing exper imen ta l data on the dependence of the imag ina ry par t of the suscept ib i l i ty on the power applied to the f e r r i t e . To this end, let us de t e r mi ne the r e l a t ion between the va r i ab le field act- ing on the f e r r i t e and the power which it absorbs under the condit ions for p a r a m e t r i c r esonance . For this purpose we shal l use an exp res s ion for the t i me - averaged power loss T ) per uni t volume of the sample in t e r m s of the imaginary par t • of the suscept ib i l i ty ,

(22) I

/~ = ~ - o,3 h 2 l". ( 24 )

SOVIET PHYSICS JOURNAL 89

On the o the r hand, the a v e r a g e power l o s s p e r uni t vo lume is given by

= ~M r Mo a~ hcrit. (25)

F r o m r e l a t i o n s (24) and (25) i t fol lows tha t

X" = 2 r M 0 a~ 1 Pcrit (26) r hcrit P '

The t h e o r e t i c a l (26) and e x p e r i m e n t a l dependences of • on the power P for O K = zr/2, k = 0 a r e shown in F ig . 3. The e x p e r i m e n t a l r e s u l t s , t aken f r o m [4], we re ob ta ined fo r y t t r i u m f e r r i t e . C o m p a r i s o n of t h e s e c u r v e s shows s a t i s f a c t o r y a g r e e m e n t be tween ca l cu l a t i on and e x p e r i m e n t , wi th in the l i m i t s of the

a s s u m p t i o n s m a d e . The d i s c r e p a n c y in the r e g i o n of p o w e r a p p r e c i a b l y exceed ing the c r i t i c a l power is ev iden t ly a s s o c i a t e d with the e f fec t of s p i n - w a v e s a t u r a t i o n .

REFERENCES

1. G. A. P e t r a k o v s k i i , P r o b l e m s of R a d i o e l e c - t r o n i c s [in Russ ian] , S e r i e s III, no. 6, 144, 1962.

2. H. Suhl, P r o c . IRE, 44, 1270, 1956. 3. N. N. Bogolyubov and M. N. M i t r o p o l ' s k i i ,

A s y m p t o t i c Methods in the T h e o r y of Non l inea r O s - c i l l a t i o n s [in Russ ian] , 1958.

4. E. Schlomann, J . App. P h y s . 33, 527, 1962.

1 F e b r u a r y 1964 Kuzne t sov S i b e r i a n P h y s i c o - Techn ica l Ins t i tu te