MAGNON SCATTERING IN FERRIMAGNETIC POLYCRYSTALS WITH INHOMOGENEOUS GRAIN DIAMETERS

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MAGNON SCATTERING IN FERRIMAGNETIC

POLYCRYSTALS WITH INHOMOGENEOUS GRAIN

DIAMETERS

C. Borghese

To cite this version:

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JOURNAL DE PHYSIQUE _{Colloque C l , supplkment au}no 4, Tome 38, Avril 1977, page Cl-261

MAGNON SCATTERING IN FERRIMAGNETIC POLYCRYSTALS

WITH INHOMOGENEOUS GRAIN DIAMETERS

C. BORGHESE

Laboratorio di Elettronica del10 Stato Solido del C. N. R. Via Cineto Romano, 42-Roma, Italy

Rkumk. - Des expkriences prkctdentes de relaxation en pompage parallele dans les polycris- taux ferrimagnetiques ont et& expliquks d'une maniere satisfaisante par la theorie du temps de parcours des magnons qui se propagent a travers ies grains. Des exgriences ulttrieures ont montre que le comportement de la susceptibilit6 imaginaire ! , p en fonction du champ hyperfrequence h Ctait independant des grains.

Dans cet article nous considerons la dependance des phenomknes de relaxation par rapport au diametre moyen des grains K et h la dispersion rr.

Nous montrons que, en choisissant convenablement I'angle Be (entre k et le champ statique Ho), un meilleur accord entre la theorie et le comportement experimental de herit en fonction de H0 est ainsi obtenu.

Abstract. - Previous experiments of parallel pump relaxation in ferrimagnetic polycrystals have been explained satisfactorily by the transit-time limited magnon lifetime theory. Further work has demonstrated the independent grain behaviour of the imaginary susceptibility vs the r. f. field, h. In this work the dependence of the relaxation phenomena on the grain diameter average I and on the dispersion o is considered.

It is shown that by suitably choosing O k (the angle between k and the d. c. field H o ) a greater accor- dance between theory and data for h c r i t vs H O is obtained.

I . Introduction. - A greater understanding of the spinwave relaxation mechanism related to the grain structure of the ferrimagnets (mostly YIG) has been acquired in recent years. The parallel pump technique (IIp) first introduced a long time ago by Schlomann et al. [l] and by Morgenthaler [2] is still a powerful tool for studying magnon interactions at high peak rf magnetic fields, since it allows the selection of k-mode unstable magnons by simply biasing the external dc field, H,. Most of the early theoretical and experimen- tal studies were performed on single crystals. Neverthe- less, owing to their great interest for applications, also polycrystalline materials deserve considerable attention. Studies of the last years have contributed to the comprehension of the origin of some microwave losses in polycrystals.

Among the most conspicuous results are : 1) the transit-time limited life-time theory of spinwave pro- pagation due to Vrehen et al. [3] and to Patton [4],

2) the demonstration of the independent grain behaviour of the excited magnons which are annihilated at grain boundaries separated by pores [5], 3) the scattering due to non magnetic inclusions' size and concentration

16, 71 and 4) the major role of the grain diameter (X) average

6)

and dispersion (c) in the , , p experiments [8].

In this paper a general expression of the imaginary susceptibility, 1 1 ~ " for a polycrystal will be given with special attention to the case of log-normal X-distribution /(X) and of some other simple distributions. It will be shown subsequently how the agreement between someof the relevant physical quantities derived in the case of , , p microwave excitation and the experi- mental data improves when 2 and c are both taken into account. A new formulation for the analytical dependence of sin2 0, (0, is the angle between the spin wave

vector k and H,) on 2 and k is shown to provide an improvement of the theoretical behaviour of AH, vs k and of hCrit vs H,.

2. Imaginary susceptibility of grain size distributed polycrystals. - The single crystal

i,x"

can be written, following Schlomann [9] :

with cc given in [g] and h, h,,,, the r. f. magnetic field amplitude and threshold. It has been shown previously [5] that for a polycrystalline body having a distribution f(x) of grain sizes X,

Blh

,,,h,

= (ap,)-'

{

[(i2

+

g 2 )

-

I

x2/(x) dx] -

5

-

leih

xf(x) dx]

)

0 0

where

p

is treated as a constant [5], 2 is the mean and o the standard deviation of f(n), while p, is the third moment

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Cl-262 C. BORGHESE

X

about the origin. Equation (2) can be written in a more compact way if a new normalized variable h, = h

.

is adopted :

I I ~ i o l

= (X2

+

02)2 (up3 X)-'

where m, and m, are respectively the first and second integral of eq. (2).

!X\:,

falls into eq. (1) for CT = 0. In the special case, of experimental interest, in wihch f ( x ) is a log normal

distribution i. e.

I- - 1

f ( X ) l(x) = (xs 4 2 X ) exp

{

-

[In (x/xo)12/2 s2

) ,

is given by

,,X: = h ,

+

:)

-

erf ( i ~ n h ,

-

g)]

,

where s

z

o/x.

The threshold field and the field for the maximum of X" are given respectively by

h = e ( S

,

h,., z 2 erf

[-

( 5 in s

+

l ) ]

,

( 5 )

with

~ ( s ) = 2" a(3 erfc 3 s

-

2 a erfc 2 s)/(2 erfc 2 S

-

a erfc S)

,

a = exp(- 5 s2/2)

.

The saturation value

,,x&~,

V S h,,,, is plotted in figure 1. By the hot pressing technique other distributions could

be obtained.

1

h* max

2

FIG. 1 . - Theoretical saturation of 11 X 11 as a function of the normalized r. f. field for a log-normal X-distribution.

A few examples of the calculated results are reported below.

For a triangular distribution

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MAGNON SCATTERING IN FERRIMAGNETIC POLYCRYSTALS

For an exponential distribution f ( X ) = 2-' exp (- x/X), 0 X

<

m, o =

5,

we have

X"

= 2(3 U)-' exp(- 2/h,) hi2(h, f l )

,

h,,,, = 0.287 5

.

For a rectangular distribution

I

-

f = - ,

2 a X - a < x < i - t - a , l a I < % ,

X"

= exp(- a2/x2) U-' hi2(h,

-

I ) , _h,,,,, = 1

Equation (8) is significant since it shows that by the region supporting the spinwaves [4], the spinwave broadening an homogeneous distribution (o = O), the vector, k has a component

k,

= k cos 0, in the H,

saturation value decreases in first approximation as direction. If the approximate dispersion relation [l01 exp(- s2). The illustrated normalized

X"

'S are shown is considered,

in figure 2.

v: = 4 y 2 D' k2

+

m& sin2 2 BK/k2,

FIG. 2.

-

Parallel imaginary susceptibility for different x-distributions.

3. Spinwave linewidth dependence on grain sizes and

magnon group velocity.

-

The ?-'-dependence of h,,,, in , , p , the independent grain behaviour of ,,X'' v s hCri, and the definition of hCri, in f(x)-distributed polycrystals leads to the following expression of the spinwave linewidth :

where v, = V, o, is the magnon group velocity and cp(o/%) is a function previously defined for I(x) distri- butions, decreasing from one to zero with increasing B/%.

When the ferrite has a homogeneous distribution, eq. (9) reduces to eq. (2) of [4]. Due to the finite size of

and Patton's [4] assumption k , = 2 n/E is made, it turns out that

where H, is the static field for the minimum of ,,hcri,

when B, = 4 2 , D = 5 X 10-9 Oe cm2 is the exchange

constant for YIG and y = 1.76 X 107 Oe-l S-' the

gyromagnetic ratio ; w = 5.8 X 101° rad. S-' is the pump frequency, o,=y 4 nM,=3.1 X 101° rad. S-'. In figure 3 several h,,,, vs H. curves are plotted for different

5

values and for the end values of o = 0

and o/x = 0.4 (which is a significant value, since experimentally found in hot pressed nickel ferrites [S]) together with experimental curves from Patton [4].

FIG. 3.

-

Butterfly curves of fine grained YIG. Bands : theoretical curves derived for ki = 2 n/E, extented from a = 0 to

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Cl-264 C . BORGHESE

FIG. 4. - Fields for the minimum of herit. Lines : theory. Full circles : data from ref. [4]. Open squares : unpublished data for

samples cited in ref. [5].

Although the slopes of the curves are too steep, the p-factor improves the agreement with the data. From eq. (10) it descends that hCIi, reaches its minimum for

The position of h,,,,(min) shifts to lower values of H, when 2 decreases following the relation

independent of a and in qualitative agreement with previous data [4, 11, 121, as shown in figure 4. Obviously an effort to improve the theory must concern the explicit form of

I

v,

1,

from which the shapes of AH,(k) and hcri,(H,) depend. This is done in the following section.

4. The dependence of B, on 2, a and R . - If

I

v,

l

in eq. (10) were derived from the exact dispersion relation one could expect for AH, an expression more adherent to the data, but the formula for ,lh,,i,(Ho) would result unwieldy.

On the other hand, the adoption of the approxi-

0

A T T O N H P 10.1

o 0.4

0 0 O,,,,O n

1

14.4 pm 1

FIG. 5.

-

Butterfly curves for fine grained YIG. Bands : theore- tical curves from eq. (13). Lines : data from ref. 141. Open circles :

data from samples cited in ref. [S].

mate dispersion relation [l01 does not endanger the basic physical assumptions of what follows.

The inspection of figure 3 shows that the slopes of h,,,, for H,

<

H,(min) are too steep. This depends on sin '8, being very close to one in the entire k-range, except in the neighbourhood of k E 2 n/Z.

By the assumption that cos '8, = k:/k2 is of the form a

+

b (klk,)' (with a, b and k, constants) and by the imposition that when

k2 = and k2 = ki,, = (w

-

wM)/2 y~

,

cos 28, is one and zero respectively, the following expression is found for sin '8, :

By noting that the smallest grains in a sample are scattered all over the volume and that the standard deviation should be taken into account, rather than

X,

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MAGNON SCATTERING I N FERRIMAGNETIC POLYCRYSTALS Cl-265

with A 2 = k$,, - kii, The minimum of AHk occurs figure 5, in comparison with our earlier unpublished

for data and with data from Patton [4]. When 2 is of the

order of fractions of a millimeter, eq. (13) is no

2

k (min) = - kmax kmin longer satisfactory since AHk vs k exhibits a near-to- (1 - 4 y2 0' A ~ / w & ) " ~ _{zero slope. But this inconvenience is partially removed} if

I

v,

I

in eq. (13) is expressed in termes of the exact and goes to zero as Z increases. llh,,i, vs H,, is shown in dispersion relation o,(k).

References

[l] SCHLOEMANN, E., GREEN, J. 3. and MILANO, U., J. Appl.

Phys. 31 (1960) 386.

[2] M O R G E N ~ A L E R , F. R., J. Appl. Phys. 31 (1960) 95. [3] VREHEN, Q. H. F., BELJERS, H. G. and DE LAU, J. G. M.,

ZEEE Trans. Magn. Mag-5 (1969) 617.

[4] PATTON, C. E., J. Appl. Phys. 41 (1970) 1637 ; Proc. Int. Con$ on Ferrites, Kyoto, 1970 (Un. Park Press, Bal- timore, Md., 1971), p. 524; IEEE Trans. Magn.

Mag-8 (1972) 433.

[5] BORGHESE, C. and ROVEDA, R , J. Physique Colloq. 32 (1971) C 1-150 ; Appl. Phys. Lett. 19 (1971) 156.

[6] SCOTTER, D. G., J. Appl. Phys. 42 (1971), 4088 ; J. Phys.

D 5 (1972) L-93.

[7] SAWADO, E., J. Appl. Phys. 47 (1976) 2154.

[8] BORGHESE, C. and ROVEDA, R., J. Appl. Phys. 40 (1969) 4791 ; BORGHESE, C., J. Appl. Phys. 44 (1973) 3746. [9] SCHLOEMANN, E., J. Appl. Phys. 33 (1962) 527.

[l01 See for example : SPARKS, M., Ferromagnetic-relaxation

Therory (McGraw-Hill, Inc. New York) 1964, p. 50. [l11 VREHEN, Q. H. F., J. Appl Phys. 40 (1969) 1849.