nonlinearities in the ferrimagnetic resonance in epitaxial garnet films

Journal of Magnetism and Magnetic Materials 96 (1991) 237-244

North-Holland

237

Nonlinearities in the ferrimagnetic resonance in epitaxial garnet films

B. Liihrmann, M. Ye, H. Datsch and A. Gerspach

University of Osnabriick, P. 0. Box 4469, W-4500 Osnabriick, Germany

Received 17 August 1990

The ferrimagnetic resonance of magnetic garnet films has been measured at foldover and instability conditions. The

resonance is excited locally within a region of about 50 urn diameter. Large precession angles of more than 80’ can be

achieved within this region, if the saturation magnetization MS is small. If MS is large, the precession spreads out far beyond

the excitation region while simultaneously the maximum of the precession angle is reduced. For instability oscillations the time

variations of the polar and azimuthal angle are determined separately. It turns out that the polar angle variation lags behind

the variation of the azimuthal angle and that its amplitude is small compared to that of the azimuthal angle.

1. Introduction

Magnetic garnet films grown by liquid phase or

sputter epitaxy find applications in integrated

optics, memory devices and microwave techniques [l-3]. The ferrimagnetic resonance (FMR) is a

useful tool with which to study and characterize such films. Furthermore, the FMR can be applied to realize an integrated light modulator by dy- namical mode conversion [4]. For this purpose

large precession angles are essential. It has already

been shown that precession angles up to 60 o can be achieved [5]. However, two nonlinear phenom-

ena arise at strong excitation levels of the FMR: (i) the foldover effect [6,7], which is caused by

the feedback of the precession angle on the resonance frequency. This effect depends on the geometry of the sample and it is especially strong for thin films.

(ii) The FMR may become unstable even at low excitation levels depending critically on the

bias field and frequency. This long known phe- nomenon [S] is recently discussed in terms of chaotic behaviour [9,10]. The chaotic region is preceded and interrupted by regions of periodic instability oscillations of the magnetization in the frequency range between 100 kHz and several

MHz. The frequency spectra usually consist of discrete lines of constant separation, the intensi-

ties of which are distributed asymmetrically around the excitation frequency [ll]. The instabil-

ity oscillations are caused by a nonlinear coupling between the FMR and spin waves [8] or magneto-

elastic waves [ll]. However, the oscillation fre- quencies cannot be predicted quantitatively yet.

It is the purpose of the present paper to derive the conditions necessary to achieve large preces-

sion angles. Therefore the influences of the foldover effect and of the saturation magnetiza-

tion on the precession angle are studied experimentally. Furthermore, the motions of the magne-

tization with respect to the polar and azimuthal angle are measured separately, to gain some in- sight into the physical conditions determining the

instability oscillations.

2. Theory

To calculate the motion of the magnetization M the geometry shown in fig. 1 is used. A static induction B, is applied along the film normal which is parallel to the crystallographic [ill] direction. M precesses about the direction of I$,.

0304-8853/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)

238 B. Liihrmann et ul. / Nonlinearities in FMR rn epituxral gumrt J~lms

mZ1 c

Y

Fig. 1. Geometry of the garnet film.

Using polar coordinates, 0 and @, form of the equation of motion reads

Y E - C&J sin 0 @=-M,SiIlO~@

* ,

Y

@ = M, sin 0 a@ X+-&O,

where

the Gilbert

WI

(1)

(2)

M= M,(sin 0 cos @, sin 0 sin @, cos 0). (3)

M, is the saturation magnetization, y the gyromagnetic ratio and (Y the damping constant. The density F of the free energy is, including Zeeman-, demagnetizing- and uniaxial-anisotropy-

energy

F = -B,,M, cos 0 - b,M, sin 0 cos( @ - ut)

+ K u (4)

K, is the uniaxial anisotropy constant and b, the amplitude of the circularly polarized rf induction

of frequency w which excites the ferrimagnetic resonance. The cubic anisotropy is neglected.

At steady state the precession of M is circular, i.e. 6 = 0 and @ = wt + $, and one gets

tan’ 0 = { y*bi }

xj[w-yB,~-y~~-PoMsj cos @I2

i

-I

+a202 cos2 0 . (5)

This equation describes the foldover effect [5]. At instability the angles 0 and @ of the magnetization vary with time

G(t) =@,+8@(t), (6)

@(t)=ot+q)++@(t). (7)

Thus the time dependent components of the mag-

netization, which will be measured experimentally,

are given by

M,(t) = MS sin O(t) cos @( 1). (8)

M,(t) = MS sin O(t) sin @p(t), (9)

M,(t) = MS cos O( 1). (10)

3. Experiments

Films of yttrium iron garnet (YIG) and sub- stituted yttrium iron garnet grown by liquid phase epitaxy on [ill] oriented gadolinium gallium

garnet substrates are investigated. The chemical composition, the thickness, the saturation magne-

tization MS, the effective uniaxial induction

2K Bzff = --$ - poMs

s

and the specific Faraday rotation 0, of the films are listed in table 1.

Table 1 Material parameters of the investigated films

Film no. Chemical Thickness composition (pm)

1 0’, La),(Fe, Ga),% 6.8 0.17 -28 509 2 Cy, La)dFe, Ga)s% 5.4 0.05 5 374 3 Y3Fes% 5.3 1.43 - 173 562 4 Cy, JWdFe, A0,% 6.4 0.27 -5 - 394

B. Liihrmann et al. / Nonlinearities in FMR in epitaxial garnet films 239

coplanar wavegl Aide

metal film

-slat line

Fig. 2. Geometry of the microwave structure to excite and

detect the FMR.

A short-circuited slot line and coplanar wave-

guide etched into a gold film on a glass substrate are applied as antennas to excite and detect the ferrimagnetic resonance. A top view of this struc-

ture is presented in fig. 2. The excitation is limited to a region of about 50 pm diameter centered at

the short circuit. The microwave signal from the detecting antenna can be processed by three dif-

ferent methods: (1) it can be fed directly to a spectrum analyzer; (2) it can be mixed to an intermediate frequency

in the MHz-range;

In addition an optical set-up as described in ref. [5] is used to measure the precession angle 8

of the magnetization by means of the Faraday rotation (eq. (10)). This arrangement is also applied to determine the spatial distribution of the

(3) or it can be rectified by a diode and displayed

on an oscilloscope.

-65 70 75 80 65 90

magnetic induction [mT]

precession angle within the excitation region. The resolution is limited by the spot size of the focused light to a diameter of about 15 pm.

4. Results

Fig. 3 shows the measured precession angles of films nos. 1 and 2 as a function of the induction

B, applied perpendicular to the film plane at

constant frequency (1420 MHz) and constant in-

put power level. The static inductions for low-

power resonance are indicated by vertical arrows.

Both films clearly demonstrate the foldover effect, which occurs in opposite directions with respect to

the static induction B,,.

Film no. 1 has a low magnetization of M, =

0.05 X lo5 A/m; it reaches precession angles of

about 80 o which, however, decay rapidly with the distance from the short circuit as shown in fig. 4a.

Sample no. 3 (fig. 4c) is a pure YIG film with a

The maximum precession angle, which can be achieved by the experimental set-up, strongly de-

pends on the saturation magnetization M, of the crystal. This fact is demonstrated in fig. 4 where the spatial distribution of the precession angle is

plotted in the region around the exciting short circuit for three crystals of different magnetiza-

tion. The exciting microwave power and frequency are kept constant, while the static induction B, is adjusted for resonance. The arrangement of the

slot line (rf input) and of the coplanar waveguide

is shown by dashed lines in fig. 4c.

0.. I , 1 1 I I

25 30 35 40 45 50 55 60

magnetic induction [mT]

Fig. 3. Measured precession angles of film no. l(a) and fii no. 2(b) versus induction I$; the microwave frequency is 1420 MHz. The solid lines are calculated.

240 B. Liihrmann et al. / Nonlinearlties In FMR in epitaxial garnet films

large magnetization of MS = 1.43 X 10” A/m. Its

precession angle does not exceed 20” but decays very slowly with distance from the short circuit.

The precession of film no. 1 becomes unstable in the range marked in fig. 3a by a horizontal

optical signal

I 2 3 4 5

time [pus]

Fig. 5. Measured signals of the same instability of film no. 1 in

the time domain. The upper trace is obtained by method 3. the

lower trace by optical measurements. Excitation frequency

f,, = 1420 MHz, static induction B, = 82.3 mT.

Fig. 4. Measured spatial variation of the precession angle for three different crystals. The coplanar waveguide and the excit-

ing slot line are shown by dashed lines in 4c.

double arrow. In fig. 5 instability signals of sample no. 1 are shown in the time domain at constant excitation frequency and static induction. The curve of the upper trace is obtained by recti-

fying the microwave signal by a diode (method 3); this signal is caused by the variation of both, polar

angle 0 and azimuthal angle @ (eq. (8)). The lower trace gives the optically obtained signal of

the same instability due to the variation of the

polar angle alone (eq. (10)).

5. Discussion

5.1. Foldover effect

The solid curves in fig. 3 are calculated using eq. (5). The amplitude 6, of the microwave induc-

tion is adjusted to about 3.5 mT to achieve the same maximum precession angle as measured; the other parameters are taken from table 1.

The foldover effect critically depends on the effective uniaxial induction B:ff as demonstrated in fig. 3. The sign of Bzff determines, whether the foldover occurs towards lower or higher induction. If this quantity is zero, no foldover occurs and large precession angles can be achieved directly at the resonance frequency.


5.2. Maximum precession angle

The short-circuited slot line serves as input line

(figs. 2 and 4~). The rf magnetic field has a strong

component perpendicular to the slot and parallel

to the substrate plane [13]. This field can be split

into right and left circularly-polarized fields, one of which excites the FMR in the garnet film,

depending on the direction of 4. These fields

show a maximum close to the slot edge and decay with the distance from the slot. Maximum field

and decay depend on the height above the metal

film. At a distance of about 2 to 3 times the slot

width away from the slot edge parallel to the metal film, the exciting rf magnetic field has de-

creased to less than half its maximum value. The

triangular-shaped slot line used in the present

experiments has a width of about 8 pm at the short circuit (fig. 2). Thus the diameter of the

excitation region can roughly be estimated to about

50 ym. Fig. 4 shows that the ferrimagnetic precession

extends beyond this limit, if the saturation magnetization is large. This spreading of the precession

is due to the magnetic coupling between neighbouring regions of the garnet film. The main contribution to this coupling arises from the dipole-dipole interaction which strongly depends

on MS. Using a garnet film of low magnetization

as in fig. 4a, the precession spreads little. The spatial distribution of the precession angle shown

in fig. 4a is thus close to the spatial distribution of the exciting rf magnetic field.

Fig. 6 presents the maximum values of the

precession angles observed in various garnet films versus saturation magnetization. To achieve large

precession angles local excitation combined with a low saturation magnetization is thus essential. On the other hand, a small MS decreases the coupling between the precessing magnetization and the ex-

citing rf magnetic field.

5.3. Instability oscillations

According to eq. (10) the amplitude SO, of the variation of the precession angle can be measured optically using the Faraday rotation [5]. The resolution limit lies at about 1”. In most cases SS, is quite small, especially when the signal is very anharmonic (figs. 7 and 9 below). The largest

90 , I I I I I I I

T80- ‘*

al 70- ?% 6 60- .1

.; 50 - !A 0) 40- P

*4 ‘a 30- a

l $ 20- ’ l -

E lo- 0

3_

0 ’ I I / I 1 I 1 0 20 40 60 80 100 120 140

saturation magnetization [ 103 A/m]

Fig. 6. Maximum precession angles versus saturation magneti-

zation. The numbers refer to the sample numbers of table 1.

(The other samples are not listed in the table.)

value of SO, observed is 8” at an average angle 0,=21”; the corresponding optical signal is shown in fig. 5 (lower trace).

The time dependent variations 8@(t) of the

polar angle and 8@(t) of the azimuthal angle at the instability in the FMR can be determined separately in the following way. The signal S(t)

measured by microwave technique is a superposi- tion of the electromagnetic field induced by the

precessing magnetization and the electromagnetic crosstalk C(t) between slot line and coplanar waveguide (fig. 2). The contribution of the preces-

sion is mainly proportional to M,(t) (eq. (8)), while the contribution of M,(t) to the microwave

signal can be neglected. Using a microwave source of constant frequency and amplitude the mea-

sured signal is mixed into the frequency range around 10 MHz so that it can be observed directly

by a storage oscilloscope (method 2). The intermediate frequency wi is chosen such that it is an

integer multiple N of the basic frequency Aw of the instability oscillation, where N lies in the range between 10 and 20. The signal S(t) mea-

sured at the intermediate frequency is then analyzed by a Fourier transformation. As the spectrum of the instability oscillation consists of a few discrete components (fig. 8a below), one gets approximately

S(t) = 5 ak cos{[N+(k-N)] A.ot+&}. k=O

(11)

242 8. Liihrmann et al. / Nonlineoriries m FMR m epiruxd garnef jl1m.s

The number n equals 128 according to the number of channels of the storage oscilloscope used.

The electromagnetic cross talk C(2) of the mi-

crowave structure is obtained by changing the static induction B, so that the FMR is shifted out

of the frequency range measured

C(t)= i a; cos{[N+(k-N)] Atit++;}. h =o

(12)

The difference S(t) - C(t) is then proportional to

M,

S(t) - C(t) = QM, sin G(t) cos 3(t), (13)

where Q is a constant of the experimental set-up.

O( 1) is given by eq. (6) and

&(t) = wit + @,, + 8@(t) = NAwt + @Co + M’(t);

(14)

@,, is the phase shift between exciting frequency and precession, which equals -a/2 at resonance. The factor Q is determined by comparing the microwave measurements with optical measure-

ments.

Using eqs. (ll), (12). (13) and (14) one

QM, sin[@,+6@(t)] =/U(t)*+ V(t)’

and

tan[@O+6@(t)] = #,

obtains

(15)

(16)

where

U(t)= c ak sin(tinkf++,)-az sin(9,t+$k), k=O

v(t)= i a,, cos(f&t++k)-a& co&t++;)

k=O

and

U2,=(k-N) Ati.

According to these equations the variations a@( t ) and 8@(t) can be derived from the coefficients a, and ai and from the phases Gk and +$ determined by the Fourier transformation of the measured signals S(t) and C(t).

a) / I

;

0

ol:‘v/qfl 0.5 10 1.5 2.0

0 5 10 15 20

time [ps]

Fig. 7. Two measured signals at instability of film no. 1 in the

time domain obtained by method 3.

Fig. 7 shows two instability oscillations of sample no. 1 in the time domain. The lower trace represents a very anharmonic signal, while the

signal shown in the upper trace is nearly harmonic. The corresponding frequency spectrum of the in-

stability signal of fig. 7a is given by fig. 8a. The excitation frequency f,, is marked by an arrow. In

sample No. 1

a)

b)

990 995 1000 1005 1010

frequency [MHz]

l-l - Ll-L_l Fig. 8. (a) Measured spectrum of the instability oscillation of fig. 7a obtained by method I; (b) spectrum of the instability

oscillation of fig. 7a, calculated by eq. (17) using olt) - CQ, =

-90°.


the range below the excitation frequency the spec- tral components are stronger than above, which is a common feature of all crystals studied.

Using the procedure described above, one obtains from eqs. (15) and (16) the functions

Q,+%)(t) and @a+&lj(t),

which are plotted in fig. 9 for the signals given in fig. 7. These results reveal that the amplitude S@,, of s@(t) is much larger than the amplitude SO,, of 8@(t). This fact becomes plausible by inspecting the equations of motion, eqs. (1) and (2): neglect- ing losses, Id ( is much larger than 16 I.

Furthermore, fig. 9a yields that for the nearly harmonic instability signal the phase of 60(t) lags 90 o behind the phase of S@(t). This phase lag is connected with the shape of the frequency spectrum shown in fig. 8a. Using the data of fig. 9a one can simulate the instability signal of fig. 8a. One gets for M,(t) in the time domain

M, = M, sin[ @a + SO, sin( Aot + a,)]

xcos[wt+@a++@,, sin(Awt+a,)], (17)

where a simple harmonic variation of 8@(t) and s@(t) is assumed. The amplitudes obtained from fig. 9a are SS,, = 3” and S@,, = 12O. The respec- tive phases are denoted by 0~~ and Q. These phases are unknown. Only when the phase difference Aa = ae - a* is negative, eq. (17) yields by Fourier transformation the asymmetric form of

z 2 P 0

40 , I I 1

20 I

0- a> polor angle

-20 - sample No. 1

-40 -

-60 azimuthal angle

-

-100 ’ I 1 I I 0.0 0.5 1 .o 1.5 2.0

time [ps]

the frequency spectrum of fig. 8a, which is typical for the present experiments. If ha is zero, the spectrum becomes symmetric while for Aa > 0 the frequency components above f,, are stronger than those below.

Fig. 8b presents the frequency spectrum obtained with Aa = - 90 o by Fourier transformation of eq. (17). It agrees well with the experimental spectrum of fig. 8a. Recent numerical calcula- tions of the instability oscillations confirm a phase difference ha between - 20 and - 90 o [14].

Acknowledgements

Financial support by the Deutsche Forschungs- gemeinschaft, Sonderforschungsbereich 225, is gratefully acknowledged. We thank Professor W. Tolksdorf and I. Bartels from Philips Research Labs, Hamburg, and A. Brockmeyer for the pre- paration of the epitaxial films. We are further indebted to Wacker Chemitronic, Burghausen, for providing substrate crystals.

References

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i

sample No. 1

azimuthal angle

0 5 10 15 20

time [ps]

Fig. 9. The variations @a + 6@(r) and 3 + G@(t) of the polar and azimuthal angle, respectively, derived from the signals shown in fig. 7.

244 B. Liihrmann et al. / Nonlinearities in FMR in eprtuxral garnet films

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nonlinearities in the ferrimagnetic resonance in epitaxial garnet films

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