subsidiary absorption above ferrimagnetic resonance
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Subsidiary Absorption above Ferrimagnetic ResonanceP. C. Fletcher and Neal Silence Citation: Journal of Applied Physics 32, 706 (1961); doi: 10.1063/1.1736075 View online: http://dx.doi.org/10.1063/1.1736075 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Anomalous lowfrequency butterfly curves for subsidiary absorption and ferromagnetic resonanceoverlap at 3 GHz (abstract) J. Appl. Phys. 67, 5642 (1990); 10.1063/1.345912 Threshold for Magnetoelastic Instability in the Subsidiary Absorption Region in YIG J. Appl. Phys. 38, 2198 (1967); 10.1063/1.1709855 Magnetostatic Mode Echo by Subsidiary Absorption J. Appl. Phys. 37, 4077 (1966); 10.1063/1.1707979 Subsidiary Absorption Effects in Ferrimagnetics J. Appl. Phys. 33, 1372 (1962); 10.1063/1.1728737 Subsidiary Resonance in YIG J. Appl. Phys. 32, S223 (1961); 10.1063/1.2000409
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706 ]. P. HIRTH
and Noggle28 recently studied dislocation interactions in deformed copper single crystals by a dislocation etch-pit technique. They found strong evidence for reaction (8). Also, they found evidence which they interpret differently, but which could be interpreted as strong evidence for reactions (lOa) and (lla).
In all cases, several possible interactions exist for the various dislocation intersections, dependent on the local dislocation configuration at the intersection.
V. CONCLUSIONS
(a) A graphical method of assessing the possibility of reaction upon dislocation intersection in fcc lattices is developed with the use of Thompson's glide tetrahedron.
(b) The probabilities of short- or long-range interaction are quantitatively determined for the various possible intersections.
28 F. W. Young, Jr., and T. S. Noggle, J. Appl. Phys. 31,604 (1960).
(c) The original Lomer-Cottrell barrier [reaction (8a)] is found not to exist as an extended barrier; however, several other strong barriers [reactions (lOa) and (lla)] are found to exist as extended barriers.
(d) Of the 24 possible secondary glide systems which could interact with a primary system, 12 are found to provide strong blocks to primary glide propagation (and thus to promote work hardening), while 12 provide only weak blocking or no blocking.
ACKNOWLEDGMENTS
The author acknowledges the criticisms and suggestions of W. F. Hosford of The Massachusetts Institute of Technology on the methods of approach to the problem, and helpful discussions of the manuscript with J. Lothe, H. J. Levinstein, and G. T. Horne of this laboratory.
The support of the Metallurgy and Ceramics Research Branch, Aeronautical Research Laboratory, Air Force Research Division, is gratefully acknowledged.
JOURNAL OF APPLIED PHYSICS VOLUME 32, NUMBER 4 APRIL, 1961
Subsidiary Absorption above Ferrimagnetic Resonance
p, C. FLETCHER AND NEAL SILENCE
Hughes Aircraft Company, Culver City, California
(Received September 16, 1960)
The interaction between magnetostatic modes and spin waves was first discussed by Suhl [H. Suhl, J. Phys. Chern. Solids 1, 209 (1957)J and was shown to be the source of nonlinear effects at high power in ferrimagnetic resonance. It is the purpose of this article to extend Suhl's results on the threshold for subsidiary absorption to dc magnetic fields greater than that required for ferrimagnetic resonance. Some numerical results on the threshold for subsidiary absorption are presented for shapes other than spheres and for magnetostatic modes other than the uniform precession on a sphere. Some general conclusions regarding the shape dependence of nonlinear effects are stated. The theory is then compared with experimental results on YIG and MnZn ferrite.
INTRODUCTION
T HE salient features of the ferrimagnetic resonance phenomenon at high signal levels, namely, the
saturation of the "main" absorption and the appearance at lower dc magnetic fields of a subsidiary absorption, have been explained by SuhP on the basis of the interaction of the uniform or quasi-uniform precession with spin waves of much shorter wavelength. It is the purpose of this article to extend Suhl's treatment in several respects, especially with regard to the threshold for the subsidiary absorption. In particular, we will consider the onset of the subsidiary absorption for several magnetostatic modes and of the uniform precessional mode for several sample geometries. Further, in his original treatment of the subsidiary resonance, Suhl considered only a rest ricted range of
1 H. Suhl, ]. Phys. Chern. Solids 1, 209 (1957).
experimental conditions, i.e., only orienting magnetic fields less than that required for the "main" resonant absorption. Extension of the theory to cover applied fields equal to or greater than the "main" resonance fields has yielded important features of the nonlinear phenomenon which are not obvious from the results presented for the lower-field case.
SUHL'S RESULTS
It can be shown that spin waves of Bloch are normal modes of a regular array of spins if dipolar forces are not included. When these forces are included, however, the spin waves are coupled together. Thus the uniform precession (a spin wave of infinite wavelength) is coupled to higher-order spin waves. Suhl was the first to show that this coupling was nonlinear so that when sufficient power flowed from the uniform precession to the spin wave to overcome its natural damping, the
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ABSORPTION ABOVE FERRI MAGNETIC RESONANCE 707
spin-wave amplitude would increase, at least initially, at an exponential rate. The point at which this "runaway" occurs is called threshold, and the rf field at this point is called hcrit '
It was shown by Suhl that the first-order coupling terms excited spin waves whose resonant angular frequency Wk is approximately given by
wk=w/2, (1) where Wk is given by
WN'Y2= (H-47rMNz+Hcxl2k2) X (H-47rMNz+Hexl2k2+47rM sin2fh) (2)
and where k=propagation vector of the spin wave; W = applied rf angular frequency; 'Y = gyromagnetic ratio; H=applied dc magnetic field; N z, NT=dc demagnetizing factor parallel and perpendicular to the dc magnetic field; 47rM = saturation magnetization; Hex = effective exchange field; l= lattice constant; and Ok=angle between k and z.
The first-order coupling usually causes runaway excitation at dc magnetic fields different from the main resonance absorption, and is therefore called the subsidiary absorption. The rf field necessary for runaway was calculated by Suhl to be
(3)
where IlHo and IlHk are the half-widths of the magneto- varies, we will define the auxiliary parameter static mode and the W/2=Wk spin wave, respectively, and wre• is the Kittel resonance equal to f.Lk=H-47rMNz+Hexl2k2, (5)
(4)
The threshold for excitation of the k spin wave is seen to depend upon both k and Ok. In order to find the least rf field to cause some spin wave to be excited, Eq. (3) must be minimized with respect to these two parameters. This minimization procedure is difficult but has been done by computing machines. Since the minimization procedure here is slightly different from that used by Suhl, it will be discussed at some length.
Equation (3) can be seen by inspection of the numerator to have two obvious minima, one near W=Wres and one near wk=w/2. The minima near wk=w/2 is the subsidiary absorption of Suhl, and we now discuss the minimization procedure appropriate to it. It is convenient to treat separately two regions of external field, one where k varies and Ok remains fixed, and one where Ok varies and k remains fixed and almost zero.
To facilitate the discussion in the region where k
in terms of which the spin-wave frequency can be written,
(Wk/'Y)2 = f.Lk (f.Lk+47rM sin20k). (6)
For Eq. (3) to minimize, we wish the numerator to minimize and the denominator to maximize simultaneously through the appropriate selection of k and Ok. The Ok function in the denominator maximizes for fh=45°, and for small H it is possible to have wk=w/2 and Ok=45° by selection of the proper k. In fact, there exists a range of H values (as pointed out by Suhl) in which Ok and f.Lk are essentially constant. As H is increased, k decreases [d. Eq. (5)J preserving f.Lk, and hence Wk, constant at w/2. Solution of Eq. (5) for k subject to the condition that wk=w/2 and O=7r/4 gives
Hexl2k2 = - (H-47rMNz)-7rM
+ [(7rM)2+ (!W/'Y)2J. (7)
Under these circumstances, Eq. (3) becomes
IlHk(w/'Y) { (IlHo)2+[(Wres -W)/'Y J2}! hcrit=---------------
27rM{ (w/2'Y)-7rM +[(7rM)2+ (w/2'Y)2]!} (8)
or heri! (w/47r'YM) { (IlHo/47rM)2+ (w/ 47rM)2[ (wres/W) -1J2} i
IlH k (w/167r'YM) -1+[(1/64)+ (w/16nM)2J! (9)
If IlHo«(wres-W), as is usually the case, this becomes
(10)
Plotting ltcrit/ Illh vs wr(s/ W will then yield a st raigh t line whose slope depends only on wN7r'YM. The left side of Figs. 1-5 indeed show this linear dependence indi-
cated by Eq. (10). The exact minimization procedure, which takes cognizance of the slight variation of Wk across the linewidth IlH k and of the weak dependence of the denominator on Wk, shows that Ok for the minimum is not exactly 45° but varies from 37°-43°, and gives actual slopes and intercepts slightly different from those predicted by the approximate Eq. (10).
As II is increased, a value is reached for which k,
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708 P. C. FLETCHER AND N. SILENCE
2.5,--------.rrr-r...----,.--..-----,.----,·
2.3
2.1
0.5
0.3
o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
wres /w
FIG. 1. Threshold for subsidiary resonance in a sphere for the uniform precession tJ.1I/47rM is assumed to be 0.003.
to preserve fLk constant, has decreased to zero. No further dimunition of k is possible, fLk commences to increase, and Wk can be preserved near w/2 only by variation (decrease) of Ok from its value near 11'/4. hcrit is no longer a linear function of Wres/ W, and we enter
2.5 r------~nT_..._-r-_.._--..__---...,
2.3
1.7
3 :r: <l 1.3 ±' 0.9 .s::;t·1.1
.7
Q5
0.3
0.1
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
wres1w
F~G. 2. Threshold for subsidiary resonance in a rod for the uniform precession tJ.H/41rlVi is assumed to be 0.003.
2.5 r---------r--.....
2.3
0.1 1.0
0.2 0.4 0.6 0.8 1.0
wreslw
FIG. 3. Threshold for subsidiary resonance in a disk for the uniform precession tJ.H/47rM is assumed to be 0.003.
the second region of H values discussed by Suhl. In this region Suhl maintained rigidly the condition wk=w/2, so his curves approach an H (dependent on 41rM) for which even diminishing k and Ok to zero no longer produces a sufficiently small Wk to satisfy wk=w/2. Relaxation of the wk=w/2 condition and minimization exactly allows the range of H to be
2.5 r------.m~---,----...---_,_---_.
2.3
o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
wres /w
FIG. 4. Threshold for subsidiary resonance in a sphere for the 210 magnetostatic mode tJ.H/4;rM is assumed to be 0.003.
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A 13 S 0 R P T I 0 l\ ABO V E FER RIM ,\ G ~ E TIC RES 0 ~ A NeE
extended indefinitely and in particular, past wres/W= 1, giving the behavior in h crit at higher applied magnetic fields. It was assumed for convenience throughout the minimization procedure that IlHo=IlHk=0.003(47rM).
The curves for spheroid shapes other than spheres may be easily calculated by the same procedure, substituting the appropriate values of NT and Wres ·
The results corresponding to Fig. 1 in a sphere are plotted in Figs. 2 and 3 for a rod and disk respectively. By changing W res and using NT for a sphere, one can plot similar curves for magnetostatic modes2 other than the uniform precession (110). The curves for the 210 and 220 modes in a sphere are plotted in Figs. 4 and 5, respectively. These curves would apply, for instance, when the sample was placed in the applied hrf such that
2.5 r------'III1I"'T---y---r------,r-...,
2.3
2.1
1.9
1.7
1.5
:r: ::g, 1.3 .;:
.c;U 1.1
0.9
0.5
0.3
o 0.2 0.4 0.6 0.8 1.0 r.2 1.4 1.5 1.8 2.0
Wres /W
FIG. 5. Threshold for subsidiary resonance in a sphere for the 220 magnetostatic mode AlI/47rM is assumed to be 0.003.
the 210 or 220 mode was the dominant excitation. In general it can be said that a rod is more easily excited to "runaway," and a disk is less easily excited than a sphere. It can also be said in general that the high-field magIletostatic modes have a higher hcrit than the lowfield modes, i.e., the 220 mode has a lower hcrit for a given W/WM and Hres/w than the 210 mode. At a given rf frequency and power level and for a given saturation magnetization, one should be able to cause subsidiary absorption by excitation of some modes and not others.
EXPERIMENTAL RESULTS
As subsidiary absorption above the Kittel resonance has not yet been reported experimentally in the litera-
2 L. R. Walker, Phys. Rev. 105,340 (1957).
MAGNETRON SOURCE
MECHANICAL POWER
DIVIDER
PRECISION ATTENUATOR
[1 0 :,
SAMPLE CAVITY
FIG. 6. Block diagram of experimental apparatus.
ture, several experiments have been made to check the theory. The experimental apparatus used for absorption measurements is depicted schematically in Fig. 6. The S band sources used were a Raytheon RK 5586 magnetron for pulsed high power and an HP 683A swept oscillator for low power. Both were operated at 2785 Me. The sample cavity had a loaded Q off resonance at low power of 3360. Measurements were made by means of a precision attenuator in parallel with the cavity containing the ferrite. The comparison of the two outputs (attenuator and cavity) was made on a Tetronix 545 scope by means of a coaxial microwave switch.
CD o (J) (J)
o ...J
Z
4
3
• LOW POWER
• 3.0 GAUSS
• 8.0 GAUSS o 13.6 GAUSS
Q 2 (f)
~ :;; (J)
z ex QO ....
1.0 1.1 1.2 1.3 1.4
MAGNETIC FIELD, KILOGAUSS
FIG. 7. Subsidiary absorption above resonance in a single-crystal MnZn ferrite sphere.
1.5
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710 P. C. FLETCHER AND ~. SILENCE
" I
~ .... ii u
I
1.4
- THEORETICAL
1.2 --- EXPERIMENTAL
1.0
0.8
0.6
0.4
0.2
1.8
W RES Iw
FIG. 8. Threshold for subsidiary absorption in a . single-crystal MnZn ferrite sphere.
2.0
The first experiments were done on a single-crystal MnZn ferrite sphere 1.6 mm in diam and polished with 1,u Hyprez. The results are shown in Fig. 7. One observes that an absorption shoulder develops on the high-field side of the main resonance at higher power levels and that the edge of the shoulder moves to highe~ Hdc with increased rf power. Figure 8 is a plot of hcrit/!:lH vs Wre./ W obtained from curves such as shown in Fig. 7. The value of Hdo chosen to determine Wrc• for each point of Fig. 8 was determined from the high-field edge of the absorption shoulder taken arbitrarily to be defined by the Hdc at which the transmission loss had risen to 0.05 db. The average of the half-widths of the magnetostatic modes, !:lH, was 18 gauss. The saturation magnetization was measured from magnetostatic mode spacing to be approximately 3500 gauss, yielding w/411/'M =0.293. The theoretical curve for w/411/'M =0.30 is included in Fig. 8 for comparison with the data.
The second experiments were tried on a single-crystal yttrium iron garnet (YIG) sphere. The experiment, in this case, is difficult since W/41TM =0.55, and from Fig. 1 it is seen that subsidiary resonance may occur at fields only about 150 gauss above ferrimagnetic resonance. The results are in rough agreement with the theory, however.
The final and most detailed experiments have been done on a single-crystal YIG rod 0.040 in. in diam and 0.56 in. in length, giving an axial ratio of 14. The transmission loss is plotted vs field for several power
levels in Fig. 9. Shown in the figure are several magnetostatic modes. In a comparison with Dillon,3 it appears that these are probably the 110, 310, 510, "', etc., modes. The problem of saturation, in this case, is complicated since each mode has its own hcrit, as shown in Figs. 4 and 5. Since the rod was unpolished and had some inclusions, the linewidth observed in Fig. 9 is about 20 gauss. However, experiments on a polished sphere of YIG of this purity show linewidths as low as 0.8 gauss. For lack of better information as to spin-wave relaxation in this sample, we will assume this value for !:lH k. The dc field for subsidiary absorption threshold was chosen from Fig. 9 by essentially the same criteria as were for the sphere, i.e., the high-field edge of the excess absorption shoulder. Data were obtained with the use of the 110, 310, and 510 modes. We then may plot hcrit/ !:lHk VS wre./W as in Figs. 1-5, realizing that for the uniform mode [using Eq. (4)J,
Wrea
W
H+(NT -Nz)41TM
w/r
H-Hres
w/r ---+1. (11)
The results for all three modes are plotted in Fig. 10, with Eq. (11) used for Wre./w. This is an incorrect procedure, since Eq. (11) is based on Eq. (4), which applies only to the uniform mode. However, the error is probably not too great since for most modes H res ~H+ex41TM, where ex is a slowly varying parameter. Included in this figure is the theoretical value of hcrit/!:lHk interpolated from Fig. 2 for w/41T'YM=0.575. The agreement is better than could be hoped for in view of the choice of !:lHk • However, regardless of the choice of the parameters, the general shape of the curve follows very close to the theoretical. The high-power data unfortunately falls at higher hcrit/!:lH k than can be shown in Fig. 10. However, the points all fall between the curves of W/41T'YM=0.5 and 0.7. The shift of the higher-mode subsidiary resonance is in the direction
16
14 <D
°.12 (/) (/)
310 z §l 8 (/)
I :
hrf = .00078
hrf = .0219
~ hrf = .043
--+--+-- hrf = .0896
-50 100 350 400 450
FIG. 9. Subsidiary absorption above resonance in a single-crystal YIGrod.
3 J. F. Dillon (to be published).
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ABSORPTION ABOVE FERRIMAGNETIC RESONANCE 711
indicated by Figs. 4 and 5, i.e., for the same power level, the modes with higher resonant fields have subsidiary resonance start at lower Hdc fields.
DISCUSSION
Several new features appear by extending the range as in Fig. 1. It was not clear from the curves of SuhF what shape the curve would have in the region Wres/WH= 1. From Fig. 1, it is seen that wk~w/2 spin waves can be generated in this range under the proper conditions. In particular, at WreS/ w= 1, i.e., at ferrimagnetic resonance, all curves have a sharp resonance dip (see, for instance, Fig. 1 with w/ WM= 0.7, or Fig. 2 with W/WM= 1.1, etc.). Thus, wk=w/2 spin waves are generated at the main absorption at relatively low rf fields. Physically, this can be shown to be reasonable. Although it takes larger magnetostatic amplitudes to excite the w/2 spin waves at resonance than at low dc magnetic fields, it also takes a great deal less external rf field to excite a large magnetostatic mode amplitude. The relative values of hcrit at resonance and at the minimum below resonance depends upon the value of tlHo/47rM. For the value chosen, i.e., 0.003, the two minimum values of hcrit are seen to be approximately equal.
SECOND-ORDER COUPLING EFFECTS
It should be mentioned that high-power effects can also occur at the main resonance by a second-order coupling between the uniform precession and spin waves. The result is a broadening of the main resonance absorption line which occurs at a critical rf field:
(12)
where (tlHk)M refers to the half-width of the spin wave being excited by the second-order process. Thus it is possible for main resonance saturation effects to occur for rf fields of the order considered here. It is easy to show,4 however, that these effects are restricted to dc magnetic values only tlH above the Hdc required for resonance. Hence, no correction need be made for these effects.
As Suhl has pointed out, it is possible, experimentally, to cause the first-order [Eq. (10)J and second-order processes [Eq. (12)J to occur at the same dc magnetic
4 P. Gottlieb (private communication); M. T. Weiss, J. App!. Phys. 31, 778 (1960).
.,. :I: <l "' !:: a: 0
:I:
0.2
* • 0
0.1
110 MODE
310 MODE
510 MODE
0
1.1
WRESt W
FIG. 10. Threshold for subsidiary absorption in a single-crystal YIG rod.
1.2
field, (coincidence of main resonance and subsidiary absorption). According to Suhl, this would occur at the main resonance when w/4'YM~2NT. From the above discussion it appears that coincidence occurs gradually. From subsequent theories of Schloemann5 and Suhl,6 it appears that scattering to the second-order spin waves occurs at rf fields lower than the (hcrithf of Eq. (12). Therefore, detailed comparison of the first- and secondorder processes is difficult. It appears possible for certain experimental conditions to have the first-order processes predominate the saturation of the main resonance.
CONCLUSION
The theory predicts, and experiments seem to verify quite well, the fact that subsidiary absorption can be seen at magnetic fields above that required for Kittel resonance. This has a twofold meaning. First, it is a detailed verification of Suhl's theory of high-power phenomena in ferrites. Second, it indicates that subsidiary absorption occurs at many fields not previously suspected. It further indicates that one can control to a certain extent, by sample shape and material, the level at which subsidiary absorption may occur.
• E. Schloemann, Bul!. Am. Phys. Soc. Ser. II, 4, 53 (1959). 6 H. Suhl, J. App!. Phys. 30, 1961 (1959).
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