Nonlinearities in the ferrimagnetic resonance in epitaxial garnet films

Journal of Magnetism North-Holland

and Magnetic

Nonlinearities garnet films B. Liihrmann,

Materials

237

96 (1991) 237-244

in the ferrimagnetic resonance in epitaxial

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 resonance is excited locally within a region of about 50 achieved within this region, if the saturation magnetization the excitation region while simultaneously the maximum of variations of the polar and azimuthal angle are determined the variation of the azimuthal angle and that its amplitude

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 dynamical 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 phenomena 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 phenomenon [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 0304-8853/91/$03.50

0 1991 - Elsevier Science Publishers

has been measured at foldover and instability conditions. The urn diameter. Large precession angles of more than 80’ can be MS is small. If MS is large, the precession spreads out far beyond the precession angle is reduced. For instability oscillations the time separately. It turns out that the polar angle variation lags behind is small compared to that of the azimuthal angle.

MHz. The frequency spectra usually consist of discrete lines of constant separation, the intensities of which are distributed asymmetrically around the excitation frequency [ll]. The instability oscillations are caused by a nonlinear coupling between the FMR and spin waves [8] or magnetoelastic waves [ll]. However, the oscillation frequencies cannot be predicted quantitatively yet. It is the purpose of the present paper to derive the conditions necessary to achieve large precession angles. Therefore the influences of the foldover effect and of the saturation magnetization on the precession angle are studied experimentally. Furthermore, the motions of the magnetization with respect to the polar and azimuthal angle are measured separately, to gain some insight 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$,.

B.V. (North-Holland)

238

B. Liihrmann et ul. / Nonlinearities

in FMR rn epituxral gumrt J~lms

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

+a202 cos20

mZ1 c

Y

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

Fig. 1. Geometry of the garnet film.

G(t) Using polar coordinates, 0 and @, the Gilbert form of the equation of motion reads WI Y @=-M,SiIlO~@

E

- C&J sin 0*

(1)

,

Y

@ =

M,

(2)

X+-&O, sin 0 a@

where M=

M,(sin

0 cos @, sin 0 sin @, cos 0).

+

cos 0 - b,M,

=@,+8@(t),

(6)

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

(7)

Thus the time dependent components of the magnetization, which will be measured experimentally, are given by M,(t)

= MS sin O(t)

cos @( 1).

(8)

M,(t)

= MS

sin @p(t),

(9)

M,(t)

= MS cos O( 1).

sin O(t)

(10)

(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-, demagnetizingand uniaxial-anisotropyenergy F = -B,,M,

(5)

. i

sin 0 cos( @ - ut)

(4)

Ku

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.

3. Experiments Films of yttrium iron garnet (YIG) and substituted yttrium iron garnet grown by liquid phase epitaxy on [ill] oriented gadolinium gallium garnet substrates are investigated. The chemical composition, the thickness, the saturation magnetization MS, the effective uniaxial induction 2K Bzff = --$

s

- poMs

and the specific Faraday are listed in table 1.

rotation

0,

of the films

Table 1 Material parameters of the investigated films Film no. 1 2 3 4

Chemical composition

(pm)

0’, La),(Fe, Ga),% Cy, La)dFe, Ga)s% Y3Fes% Cy, JWdFe, A0,%

6.8 5.4 5.3 6.4

Thickness 0.17 0.05 1.43 0.27

-28 5 - 173 -5

509 374 562 - 394

239

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

coplanar

wavegl

Aide

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

metal film

-slat line Fig. 2. Geometry of the microwave structure to excite and detect the FMR.

A short-circuited slot line and coplanar waveguide 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 structure 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 different methods: (1) it can be fed directly to a spectrum analyzer; (2) it can be mixed to an intermediate frequency in the MHz-range; (3) or it can be rectified by a diode and displayed on an oscilloscope. 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

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 input power level. The static inductions for lowpower 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,,. The maximum precession angle, which can be achieved by the experimental set-up, strongly depends 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 magnetization. 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. 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

0.. I

-65

70

75

magnetic

80 induction

65 [mT]

90

25

30

,

35 magnetic

1

1

40

45 induction

I

50

I

55

60

[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

I

2

3

signal

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.

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 rectifying 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

Fig. 4. Measured spatial variation of the precession angle for three different crystals. The coplanar waveguide and the exciting slot line are shown by dashed lines in 4c.

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

B. Liihrmann et al. / Nonlinearities in FMR in epitaxial garnet films 90

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 decreased 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 exciting 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

,

T80-

I

I

241

I

I

I

I

I

‘*

al 70?% 6 60-

.1

.; 50 !A 0) 40-

*4

P ‘a 30$

20-

E

lo-

0

a

l

’

0 ’ 0

I 20 saturation

I 40

/ 60

I 80

magnetization

l-

3_ 1 100

I 120

1 140

[ 103 A/m]

Fig. 6. Maximum precession angles versus saturation magnetization. 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 superposition 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 precession 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 measured 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) measured 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)

8. Liihrmann et al. / Nonlineoriries m FMR m epiruxd

242

The number n equals ber of channels of the The electromagnetic crowave structure is static induction B, so of the frequency range C(t)=

128 according to the numstorage oscilloscope used. cross talk C(2) of the miobtained by changing the that the FMR is shifted out measured

i a; cos{[N+(k-N)] h =o

garnef jl1m.s

/

a)

I

ol:‘v/qfl

;

0.5

0

10

1.5

2.0

10

15

20

Atit++;}. (12)

The difference

S(t) - C(t) is then proportional

M, S(t)

= QM, sin G(t)

- C(t)

cos 3(t),

where Q is a constant of the experimental O( 1) is given by eq. (6) and &(t)

(13)

time [ps]

Fig. 7. Two measured signals at instability of film no. 1 in the time domain obtained by method 3.

set-up.

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

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

sample

QM, sin[@,+6@(t)]

=/U(t)*+

5

0

to

V(t)’

(15)

No.

1

a)

and tan[@O+6@(t)]

= #,

(16)

where U(t)=

c

ak

sin(tinkf++,)-az

sin(9,t+$k),

a,,

cos(f&t++k)-a&

co&t++;)

b)

k=O v(t)=

i 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).

l-l Ll-L -

990

995

1000

frequency

1005

1010

[MHz]

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°.

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

the range below the excitation frequency the spectral 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

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 calculations of the instability oscillations confirm a phase difference ha between - 20 and - 90 o [14].

@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): neglecting 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

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

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 respective 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

40 ,

I

I

References [l] H. Dammann, E. Pro& G. Rabe and W. Tolksdorf, Appl. Phys. Lett. 56 (1990) 1302. [2] K. Ando, N. Takeda, T. Okuda and N. Koshizuka, J. Appl. Phys. 57 (1985) 718.

1

o

20 I

z

0-20

-

-40

-

-60

-

2

P 0

’ 0.0

b

)

polar angle

i

sample No. 1 sample No. 1 azimuthal azimuthal

-100

t

polor angle

a>

243

I

1

I

I

0.5

1 .o

1.5

2.0

0

time [ps] Fig. 9. The variations @a + 6@(r) and 3

angle

angle

5

10

15

20

time [ps] +

G@(t)of the polar and azimuthal angle, respectively, derived from the signals shown in fig. 7.

244

B. Liihrmann

et al. / Nonlinearities

[3] J.D. Adam, S.V. Krishnaswamy, S.H. Talisa and K.C. Yoo, J. Magn. Magn. Mat. 83 (1990) 419. [4] B. Neite and H. DBtsch, SPIE 1018 (1988) 115. [S] K. Gnatzig, H. Dotsch, M. Ye and A. Brockmeyer. J. Appl. Phys. 62 (1987) 4839. [6] K.D. McKinstry, C.E. Patton and M. Kogekar. J. Appl. Phys. 58 (1985) 925. [7] D.J. Seagle. S.H. Charap and J.O. Artman, J. Appl. Phya. 57 (1985) 3706. [8] H. Suhl. J. Phys. Chem. Solids 1 (1957) 209.

in FMR in eprtuxral garnet films [9] S.M. Rezende. O.F. de Alcantara Bonfim and F.M. de Aguiar. Phys. Rev. B 33 (1986) 5153. [lo] X.Y. Zhang and H. Suhl. Phys. Rev. B 38 (198X) 4893. [ll] M. Ye. A. Brockmeyer. P.E. Wigen and H. Detach, J. de Phys. 49 (1988) C8-989. [12] F.H. de Leeuw, R. van den Doe1 and U. Enz. Rep. Prog. Phys. 43 (1980) 689. [13] H.J. Schmitt and K.H. Sargea. unpublished. [14] M. Ye. unpublished.