Solid State Communications Vol. 2, pp. 377-380, 1964. Pergamon Press, Inc. Printed in the United States

EXCITATION OF SECONDARY PARAMETRIC MAGNONS IN Y. I. G.

Henri Le Gall

Laboratoire de Magngtisme et de Physique du Solide, C. N. R. S., Hellevue, (S.-et-O.), France

(Received 28 October 1964 by P. G. de Gennes)

The excitation of parametric magnons with parallel pumping has been studied in the case where the rf magnetic field ho is very superior to the critical threshold hcrit defined by the butterfly curve. When ho >7 hcrit, the linear relation between ho and l/7, 7 being the time required to obtain a given spin-waves population, is no more verified. The corresponding departure A(l/T) is due to the number of the excited secondary parametric magnons. This paper describes the first results of the study of A(l/T) vs. ho and the d. c. magnetic field Ha.

WE have previously established1 the relation which exists between the time 7 required to ob- tain a given population of spin-waves and the amplitude ho of the rf magnetic field:

2 BU.I

’ = Y bM(ho - hcrit) sin2 $ (1)

For values of the applied d. c. field Ha lower than Ha, (Ha, corresponding to the mini- mum of the “butterfly curve”), this formula becomes:

2 Bu _~. _. 7= (2)

Y &M [ho - 3. - B (Ha, - Ha)?“21

Hence there is a linear relation between l/7 and ho. The family of straight lines, that represents this linear dependence, has allowed to determine, by the ordinate at the origin and the slope of these straight lines, the critical

threshold hcrit and the propagation direction Bk of the first excited magnons. These magnons correspond to normal modes generated near the critical threshold defined by the butterfly curve.

Here we have studied the properties of parametric magnons having a critical threshold higher than hcrit* indeed, ior a given d. c. mag- netic field, the linear dependence defined by equation (2), is no more verified beyond a cer- tain value of ho. Thus, in Fig. 1 there is, with regard to the linear law, a departure A(~/T) which is more important if ho is greater and Ha

FIG. 1

l/-i vs. ho for different values of Ha.

377

378 EXCITATION OF SECONDARY PARAMETRIC MAGNONS Vol. 2. No. 12

nearer Ha,. Therefore, the departure n(l/~) implies the participation, to the transient growth of the spin-waves populations, of supplementary parametric magnons having a critical threshold higher than hcrit. We have named as “secondary parametric magnons” these normal modes, 3 as opposed to the “primary parametric magnons” that determine the linear variation of l/7 =f(ho). This linear law is veri ied in Fig. 1 for l/7 c 0.25 bs)-1; this justifies the conditions of vali- dity of the studies made previously. 172

With the parallel pumping technique, the transient power absorbed by the parametric excitation of magnons has been derived by Sch16mann4. In the present case it can be writ- ten with the following form:

pabs = 2v-lkg T[!ijko12 ii1 exp 2 (tk,-qk,) t - 0 0 1 2 3

(3) FIG. 2

where nk is the relaxation rate of the excited parametric magnons.

[k = [ (Pk12 - bk - d92] 4 2

is the “excitation rate” for these magnons. In the relation (3), the first and second terms in brackets represent, respectively, the power absorption by the primary parametric magnons and the power absorption by the secondary para- metric magnons. The summation goes from the secondary mode kl having the smallest threshold, to the secondary mode kn the relaxation rate “k of which cancels the difference ((ka- nkc,). Thus the maximum order n of the secondary magnons, and therefore Pabs and A(~/T), will be more im- portant as ho will be larger. So, for a given d. c. magnetic field, it is the secondary normal mode having the larger critical threshold which determines the departure n(l/~).

Results

The variations of 1,‘~ vs. ho have been plotted for different values of Ha. Figure 1 shows these variations for Ha > Ha, = 1547 Oe. For l/7 < 0.25 (ps)-l the curves have linear parts the slope of which depends directly on 8k. 2 For Ha < Hat we have obtained a family of similar curves, the linear parts of which have the same slope. This is due to the fact that, for Ha < Hat, the first magnons excited are propagating with 8k = n/2. Knowing 8k = f(Ha),

Departure Ai(l/~) vs. ho for different values of Ha.

in the spin-wave spectrum we could define the dispersion curve of the primary parametric magnons (A, B, C in Fig. 4). To the d. c. mag- netic field defined by the point ZI on the ordinate axis (Fig. 4), correspond primary magnons ex- cited at 8. For the same value of Ha, if ho7>hcritt the secondary magnons are excited inside the spin-wave spectrum, at y if &k = u/2, and at 5 or E if”kP”/2.

Figure 2 shows the variations of n(I/T) = f(h,) for different values of Ha > Ha,. Thus, for a given Ha, the number of secondary parametric magnons, little for ho < 2 Oe, grows rapidly for more important values of ho. This increase of A(~/T) has two different origins. Indeed, in the relation (3), on the one hand, ikc grows propor- tionally to ho, hence an increase of the number secondary modes whereas, on the other hand, the temperature of the thermal magnons is the more important as the absorbed power (i.e. ho) is larger.

Figure 3 shows, for Ha < Hat the depar- ture n(l/~) vs. (Ha, - Ha) l/2. The absorption peaks are due to the excitation of the magneto- elastic waves which determine a reduction of the number of the secondary parametric mag- nons. The departure h(l/.r) decrease proportion- ally to (Hat - H,)l/2 except for the little values of the variable for which the slope of the curve

Vol. 2, No. 12 EXCITATION OF SECONDARY PARAMETRIC MAGNONS 379

FIG. 3

Departure n(l/~) vs. (Ha,- Ha) 42 for two values of ho.

is weaker. But it is known19 5 that the relaxa- tion rate % of the primary magnons varies proportionally to (Ha, - Ha)J’2 for Ha <Hat. For a given Ha lower than Hat, sin2 8k changes very little from one secondary magnon to another, and ik, is quasi constant. Then A(~/T) is princi- pally a direct function of rib which grows pro- portionally to (Hat - Ha)3/2 = kD$‘2, where D is an exchange constant and k the propagation vec- tor of the primary parametric magnons. This justifies the pro ortional diminution of A(l/T) with (Hat - Ha) 2. But, for Ha growing beyond e H a0 @k, and therefore in decrease very rapidly (Figs. 1 and 2); then the maximum order n of the secondary magnons, and from that A(l/T), decrease quickly. Figure 4 shows these properties where the hachured area is roughly the excitation zone of the secondary magnons for a given ho.

With a magnon excited at y in the spin- waves spectrum (Fig. 4), is associated a second magnon having the same energy and an opposite propagating vector (energy and momentum con- servation), therefore symmetrical to y with respect to a. By reason of symmetry, the two emitted magnons have the same relaxation rate. The situation is different for a magnon emitted at 6 or c (with UJk # u/2. Indeed the two gener- ated magnons are still symmetrical with respect to 1% to ensure the energy and momentum con- servation

(“k + IO-k = w);

but then their relaxation rates are different.

I I 0 5 10 15 20

FIG. 4 Spin-waves spectrum. The hatchured area is the excitation zone of the secondary mag- nons for a given ho.

380 EXCITATION OF SECONDARY PARAMETRIC MAGNONS Vol. 2, No. 12

References

1. LE GALL H., C.R. Acad. Sci. , Paris 258, 502 (1964).

2. LE GALL H., C. R. Acad. Sci., Paris g, 3986 (1964).

3. LE GALL H., These, to be published in Ann. Phys.

4. SCHLijMANN E., J. Appl. Phys. 2, 1998 (1963).

5. SPENCER E. G. and LE CRAW R. C., Proc. I.E. E. 109, Part B, suppl. no. 21, 66 (1962). Z

L’excitation de magnons parametriques par pompage parallele a et& Btudide dans le cas ou le champ magn&ique micrijonde ho est trBs superieur au seuil critique hcrit defini par la courbe papillon. Lorsque ho>>hcrit, la relation lirkaire entre ho et l/7, T &ant le temps ndcessaire pour obtenir une population donnde d’ondes de spin, n’est plus v&fide. L’Qcart correspondant A(~/T) est Ii{ au nombre de magnons parametriques secondaires excit8s. Cette note ddcrit les premiers rdsul- tats de l’e’tude de 5(1/~) en fonction de ho et du champ continu Ha.