Parallel pump spin wave instability in thin ferromagnetic films

Solid State Commumcations,Vol. 20, pp. 985—989, 1976

Pergamon Press.

Printed in Great Britain

PARALLEL PUMP SPIN WAVE INSTABILITY IN THIN FERROMAGNETIC FILMS D.N. Chartoryzhskii, B.A. Kalinikos and 0G. Vendik V.!. Ulyanov(Lenin) Electrical Engineering Institute, Leningrad, USSR (Received4lune 1976 byH Suhi) The spin wave instability generated by parallel pumping in a tangentially magnetized ferromagnetic film is considered, with simultaneous regard for both the dipole and exchange fields; A dispersion equation and some expressions for the critical microwave threshold of the spin wave parametric exatation have been obtained. The dependence of the critical field on the magnetizing field has an unusually oscillating character. This is connected both with the discreteness of spin wave spectrum and the peculiarities of spin wave polarization in a tangentially magnetized film.. THE NONLINEAR ferromagnetic resonance in bulk ferromagnetics, the theoretical study of which was initiated by Suhl,’ has been fairly well investigated by now both theoretically and experimentally. Quite the reverse is the case with the problem of nonlinear wave processes in film samples, the study of which has not yet been started, or nearly so, though they are of great interest both from the point of view of physical science and practical work. Important characteristics of wave processes in ferromagnetic films are due to the discreteness of spin wave spectrum. In many cases this can contribute to a more reliable (than in massive samples) identification of some. phenomena studied both theoretically and experimentally. A number of papers published for some years past (see, e.g. references 2—7) have been dedicated to the theoretical study of the spin wave spectra. This kind of study faces many difficulties which can be accounted for by the necessity of a joint integration of the magnetic motion equation and Maxwell equation, with simultaneous regard for both the dipole and exchange fields, and the electrodynamic and exchange boundary conditions as well. To overcome the difficulties the scalar magnetic potential method was introduced in the majority of the works (see, e.g. references 2—6), so that the dipole field hd(r, t) = —V~i(r,t). The dispersion equations obtained when the magnetostatic potential method is applied contain a connection of frequencies andVery waveoften vectors in a complicated and non-explicit form. these equations are to be analyzed only by numerical methods. As the spin the wave cannot by analytically investigated, usespectrum of the linear theory results for the study of the mechanism of the spin wave parametric generation is impeded. In the analysis of the spin wave instability processes we used another method for defining the dipole field,

the one based on the application of Green tensor functions of the magnetostatic problem for the dielectric layer with magnetic polarization.8 This method was first applied for the study of the spin wave spectrum in a thin ferromagnetic film7 in this case two equations can be obtained: an exact dispersion equation in the form of a series7 which after summation becomes a transcendental equation as in reference 2; and an approximate dispersion equation7’8 convenient for practical use and valid for rather thin films (thickness L ~ 1 x l0~cm). The purpose of this work is to investigate the firstorder spin-wave instability which appears in case of parailel pumping in tangentially magnetized films. Assume that a thin ferromagnetic film of thickness L in the X direction is magnetized across the tangent to the surface along the Z axis. For convenience sake, let us introduce one more coordinate system ~ connected with the XYZ system by means of an ortogonal transformation of the turn round the I axis to the angle ~ = (e’., e~)so that the ~ axis always coincides with the direction of the spin waves propagation. The effective field H(r, t), being part of the magnetic moment motion equation

aM(r

a

t) ‘

=

gi

0 [M(r,t), H(r, t)]

(1)

t

may be considered to be equal to 2m(r, t) + e~hcos c~.,t H(r, r) = e~H1+ hd(r, t) + aV (2) where H 1 is the constant internal field hd(r) 3~a is the exchange constant, m(r) = hk(~)e~ k~. mk(~)~ h is the pump field amplitude. To define the dipole field we make use of the Green tensor functions G(E, ~‘)of the magnetostatic problem:8

985

986

PARALLEL PUMP SPIN WAVE INSTABILITY

J

L/2

hk(~)=

t~, ~‘)m~(~’)d~’

(3)

~

=

~Jl

cos g~

~

(~+~)

Vol. 20, No. 10

in the case

L/2

-L/2

where GU + 5(~

=0

~) = iG~sign (~ ~‘) —

=

—

= =

=

—

—iG~sign ~

= =

iG~sign

(~ ~) —

—

)

—iG~sign ~ —

~‘)sign

for—L/2~~~
j

k~./2exp (—k~.I~~I)

—G~ = k~/2exp [—k~

‘~

~J

for

It should be noted that the dipole field written as in (3) is the solution of the magnetostatic equations in case of the arbitrary distributions of m(~),and satisfies the electro-dynamic boundary conditions*. We represent the variable magnetization as an expansion making use of the full ortogonal (in the interval L/2 ~ ~ L/2) system of functions Sff(r) = Sff(~)e_~~: —

m(r)

=

M0

~

~~kS~(~) e_thl~

0, 1, 2 p = 1, 2, 3 (4) where M0 is the saturation magnetization. m~, is the non-dimensional The spin-wave spin-wave modes Sff(r) modeare amplitudes, eigenfunctions of the differentially matrix operator L of the linear problem Lm(r) = hd(r). In the standing wave regime k~.= 0 they describe magnetization distributions corresponding to the spin-wave resonance oscillations. The spin-wave modes of the tangentially magnetized film are three-dimensional vector functions =

S~’(~)= etSff~)+ e~S~”1(~) + e~~(~)

where =

=

S~) = ~2 cos~~). Depending on the type of the exchange Doundary conditions the function 4’~3(~) is equal: ,~

+

ability processes. By applying the perturbation theory apparatus for the analysis of the infinite equation systems relative to the spin-wave model amplitudes, we may get the following dispersion equation:

in the case ~

(~ai~+ WM

sin2 ~)

WMP1~k)(~2I3k+ WMFak

(5)

+ wMak~WH = gIji

~f3k =

0H~~‘-~M= g~1u0M0k~= + k~j3is the wave number According to the case in question of pinned or unpinned spins, the polynomial is defined by the formulae:

~

S~~(~) = 0;Sr(~)= —~J2sin~!~(~);

sin

wave spectrum and the process of the spin wave parametric excitation. In what follows we give the main results of our study omitting any detailed calulations (especially as in many respects they are8). similar to the case with the normally magnetized film 1. THE SPIN WAVE SPECTRUM We shall give a brief description of the spin wave spectrum as it determines the character of the inst-

where

S~(~) = S~*(~) = i sin ~pcI~(~);

=

of unpinned spins; where = (3ir/L, 6~is Kroneker’s symbol. Taking into account the equations (3) and (4) and the orthogonal formula for spin-wave modes it is possible to pass on from the magnetic moment motion equation (1) to the motion equations for spin-wave mode amplitudes. Leaving in the resulting equations the linear terms and the bilinear ones responsible for the first order instability, we can analyse both the spin

=

S~r(~) = ~a(~);

51??(~)= 52fl*~)

~<— L/2, L/2 <~<00

_00<

L/2 =

o

of pinned spins; *The solutions of the magnetostatic equations in the 9 with the aid of above form were obtained by Vendik and Chartoryzhskll in 1970 transition on the ground of to results the maximum similar that applied in reference 7.

—

-

~k~4

2

k~ ~ [1— (—l)~e~J, = 1,2,3,... 4

=

k~ k~ 2 —

=

~

[1

0,1,2,...

(—l)~e~”] —

(6)

1 i + (7)

The first in (6) andthe(7)second includeterms the spin interaction interms the film bulk; are determined by the dipoje spin interaction through the variable dipole fields near the film and caused by the finite thickness effect.

Vol. 20, No. 10

PARALLEL PUMP SPIN WAVE INSTABILITY

In contrast to the spectrum of the bulk ferromagnetic spin waves, the spectrum df the film spin waves is essentially discrete and is described by separate curves, each one responding to its own wave (Fig. 1). The longitudinal waves (~p= 0°)are either purely symmetric or purely antisymmetric. The shape of the wave symmetry gets distorted when its propagation direction deviates from the internal field direction,

4~o~

~~.3.fCf!cm2 bx~25fO~c/7?

987

mined by the type of the exchange boundary conditions and the number of the wave. An interesting characteristic feature of the spin wave spectrum of a tangentially magnetizing film in comparison with that of a normally magnetizing film8 is the “sag” of the dispersion curves. This is to be accounted for by the presence of spin waves with a zero group velocity.

2. THE INSTABILITY THRESHOLD In a tangentially magnetizing film both the running and standing spin waves have effiptical polarization and

~

pumping. The expression for the therefore, generally speaking, canexcitation be excitedthreshold by parallel 2

is

1/2

4w~3[(~—w~)+W~k} IgI~.LowMIP~3k(l +sin2~)—11

3

where

~

=

gIp

0Mf~~ is the relaxation frequency. It follows from = the minimization conditions (8) that (8) in the supposition ~ = const (j3, ks.) it is the j3’2

_____ ~

/

standing spin waves that have the smallest instability threshold. If due to the values of the magnetizing field the generation of spin-wave modes is impeded, either a couple of longitudinal “long” spin waves are excited,

—

2

________________

Of

~

_______________

I

Fig. 1. Spin wave spectrum in a tangentially magnetized film. With surface spins pinned, But the distortion of the prevailing wave symmetry in case of a thin film is not great even for transversal (~p= 90°)waves. Therefore we can speak about quasisymmetric waves and quasiantisymmetric waves.t As to the shape of the magnetization wave distribution, it should be noted that, for instance, for longitudinal waves in case of pinned spins when k~.L~ 0.1 and k~L~ 50, the magnetization distribution along the film thickness may be described with sufficient accuracy by sin ~ + L/2). When the wave vector has some intermediate values, there takes place a deviation from the sinusoidal character of distribution without any distortion of the symmetry type. The latter is detertThe spectrum in Fig. I is plotted without regard for “splitting”, which can be taken into account within the frames of the given theory in a comparatively easy way. This question will be discussed elsewhere. Here we shall only mention that splitting is mostly essential for waves of one type of symmetry (quasisymmetry), though it occurs for waves of different types of quasi symmetry.

or a couple of transversal “short” cases are. That this conclusion is correct can be shown by investigating the maximum value of the denominator (8). The excitability of the couples of short and long spin waves, which takes place by turns depending upon the internal field value H~,determines the unusual character of the dependence h~= f(H1) for tangentially magnetized films. In the case under consideration it is impossible to get a dependence of a universal type (as for the bulk sample and a normally magnetized film). This is due to the fact that in the interval of the magnetizing fields, corresponding to the situation when frequency w/2 is within the limits of the spin wave spectrum, this kind of dependence is of an oscillating character, the number of oscillations depending upon the spectrum density (see Figs. 2 and 3). It should be noted that the values of a normalized threshold field h~/M1~a for a film theoretically coincide with those for a bulk ferromagnetic only in certain points, corresponding to the excitation of standing spin waves. It testifies to the enhanced stability of the film spin-system. Due to the spin wave spectrum discreteness and the dispersion dependence (5) ~(k~), it is possible to identify the spin waves which are generated at the excitation threshold. Some parameters of unstable spin waves are given in Table 1. The wave vector values were defined according to the selection rules c~= 2w~(k1.). The above results help to account for the data

988

PARALLEL PUMP SPIN WAVE INSTABILITY

/

½~ L’~ti25”/O~c,n 4HpK ~_3,,fQ/ZCmZ ~

4

/~

4cm

~L. ~*3I/O~!Cm2 ~IQ5AfO ‘~1~”

/

Ge’

/ /

A

/~

&~36’Nz

Vol. 20, No. 10

6

~

af750

Os

&~3 ~h’z

/ ~

4 -4

I

~z

g15

4

0z5 (13

935

I

0/5

Fig. 2. The dependence of spin wave excitation threshold on the normalized magnetizing field. Film thickness L = 0.25 x 10 4cm. With surface spins pinned.

~

4’25

4’~5 ~35 ~

Fig. 3. The dependence of spin wave excitation threshhold on the normalized magnetzing field. Film thickness L = 0.5 x l0~cm. With surface spins pinned.

Table 1. The parameters ofspin waves generated at the excitation threshold Film thickness L = With surface spins pinned 0.375

—

0.344

0.313

0.281 0.263

0.245

0.227

0.208 0.179

0.148

0.25

x 10~cm.

0.118

0.102

0.087

C’)

~ (cm

~

4.8 x 10~2.5 x 1O~ 1.1 x 10~ 0

1.1 x 10’ 7.6 x i0~ 4.5 X i0~ 0

3.5 x 10’ 3.7 x 10’ 4 x 10’ 6.4 x 10~0

1

1

1

1

2

2

2

2

1

1

1

3

3

0°

0°

0°

—

00

0°

0°

—

90°

90°

90°

0°

—

)

obtained in the experimental study of the

spin wave

parametric excitation in permalloy films.10 The absorption spectrum discreteness, reported in the work,1°is connected with the oscillating character of the dependence h~=f(H 1) (8).t: Unfortunately, up to now there appeared no experimental works on spin wave instability in thin dielectric films to which the theory briefly considered here could be applied. Experimental data would, in 11t is to be noted that for the quantitative interpretation of the experimental results it is indispensible to know the dependences of the spin wave relaxation parameters on the wave vector and internal magnetic field values.

particular, be of great interest as correct interpretation gives an extra chance for the definition of the exchange interaction constat a. In passing, it should be noted that though it is very tempting to make use of the spin-wave

resonance data to calculate a, this method, though simple, involves an error caused by the fact that it is difficult to define precisely the degree of the surface spins pinning. When measuring c~with the aid of parallel pumping of the tangentially magnetized thin film, it is convenient to use samples with practically unpinned surface spins. In conclusion it should be emphasized that the processes of the spin wave parametric excitation of both parallel and perpendicular pumping posess a great num.

ber of specific features to be considered elsewhere.

1.

REFERENCES SUHL H.,J. Thys. Qzem. Solids 1,809 (1957).

2.

GANNV.V.,Ffz. TvarcL Tela8, 3167 (1966).

Vol. 20, No. 10 3. 4.

5.

6. 7.

8. 9. 10.

PARALLEL PUMP SPIN WAVE INSTABILITY

SPARKS M.,Phys. Rev. Lett. 24,1178, (1970). DE WAMES R.E. & WOLFRA~MT., .1. AppL Phys. 41,987(1970). FIUPPOV B.N.,FIZ. Metallov iMetallovedenie 32,911 (1971). MIKHAILOVSKAYA LV. & KHEBOPROS R.A., FEz. Tverd Tela 16,77(1974). VENDIK O.G. & CHARTORYZHSKII DN., FEz. Tverd Tela 11,2440(1969); Fiz. TvercL Tela 12,1538 (1970). VENDIK O.G., KAUNIKOS B.A. & CHARTORYZHSKIID.N., FEz. Tverd Tela 16,2757(1974). VENDIK O.G., Zhurnal Tekhnicheskoi Fiziki 37, 1157 (1967). COMLEY J.B. & JONES R.V., J. AppL Phys. 36, 1201 (1965).

989