Microwave Resonance in Hexagonal Ferrimagnetic Single CrystalsHenry S. Belson and C. J. Kriessman Citation: Journal of Applied Physics 30, S175 (1959); doi: 10.1063/1.2185873 View online: http://dx.doi.org/10.1063/1.2185873 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Anisotropy Fields in Hexagonal Ferrimagnetic Oxides by Ferrimagnetic Resonance J. Appl. Phys. 35, 3482 (1964); 10.1063/1.1713254 Resonance Properties of SingleCrystal Hexagonal Ferrites J. Appl. Phys. 33, 1360 (1962); 10.1063/1.1728731 Ferrimagnetic Resonance of SingleCrystal Barium Ferrite in the Millimeter Wave Region J. Appl. Phys. 32, 1621 (1961); 10.1063/1.1728407 Ferrimagnetic Resonance in a Single-Crystal Disk of Yttrium Iron Garnet J. Appl. Phys. 32, S155 (1961); 10.1063/1.2000384 Ferrimagnetic Resonance in Single Crystals of Rare Earth Garnet Materials J. Appl. Phys. 29, 434 (1958); 10.1063/1.1723170

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RESONANCE AND SPIN WAVE ANALYSIS li5S

and again may possibly be attributed of a nonzero porosity.

IV. FREQUENCY DEPENDENCE OF EFFECTIVE 9 VALUE

[~lUr.menls of effective g value as a function of show a pronounced peak at the critical

We) where the uniform precession passes out manifold. The theoretical and experimental frequencies for the three samples arc given in

columns of Table 1. has calculated the shift in effective g

. to the uniform precession moving out of the

manifold. The agreement of the experimental data with the above theory is excellent, the accuracy being within five percent.

The complete results of this study along with the effect of grain size and distribution on line widths and the special problems of single crystals will be presented elsewhere.

ACKNOWLEDGMENTS

I wish to express my thanks to Dr. C. L. Hogan for his suggestion of the problem, to Professor R. V. Jones for his many ideas and suggestions, and to 1\lr. T . Ricketts and l\f rs. L. Lin for their technical assistance.

L 0 F A P P LIE J) P II \' SIC S SUPPLI-~MENT TO VOL. 30, NO.4 APl{fL, 1959

Microwave Resonance in Hexagonal Ferrimagnetic Single Crystals

HE NRY S. BELSON AND C. ] . KRIESSMAN

Remington Rand Uni';.:ac D£visiott of Sperry Rand Corporation, Philadelphia, Pennsylvani(l

The resonance conditions for a hexagonal single crystal with hard direction normal to the basal plane, and magnelocrystalline anisotropy with hexagonal symmetry in the basal plane, arc presented. The anisot­ropy energy surface may be described by

E= K Q+ K\Ul2+ K2Ul4+ Kaa3'+ K 4(aI5-15al~ol+ 15a I~a24 - a25).

From the resonance conditions and data on a single crystal of Ba2Co 2Fe120n we find K 1= 1.38X 1O~ crgs/cml, K2+3/2Kl = 1.42X lOG ergs/cm3, and K4 = 1. lOX 104

• These constants with the loss parameter determined from measurements of the ferromagnetic resonance line width accurately predict the initial permeability spin resonance in a partially oriented polycrystalline sample.

there has been considerable interest in hexagonal magnetic materials,

r~';~:~I~'7, certain types exhibit permeability at eJI frequencies. \Ve have investigated single

'1"'"1''''"''' of BaZC02Fe]20n a Y compound in of Jonker, \Vijn, and Braun,] in which the

was parallel to the c axis, with easy in the basal plane. The free energy of such a

may be described by

- lal+K2U34+K3!Y/;+K~ [aI6-15a],la22+ 15a]2a24- a26]

- M lI [a,a,lI+ ac"",H +",,,,/'J+ l"M', ( I)

thea's arc direction cosines for the magnetization to orthogonal x)'z axes; the c axis of the

palgo,""lcrystal corresponds to the z axis, and an a axis the x direction. The aH are direction cosines of

~!,~;~.~;;;, steady field relative to the same axes. The ~ terms Kn describe an anisotropy energy surface

major terms are uniaxial with a sinusoidal [~lrialtionof hexagonal symmetry superposed primarily

dir',etio[,sc1ose to the basal plane. The next terms de-

t~~~~!,m~,agnetostatic and demagnetization energies I] for a spherical sample. If one rewrites

and Braun, Philips Tech. Rev. 18, J.15 (1956).

Eq. (1) in spherical coordinate angles 0 and <PI where 0= 0 corresponds to the c axis and <p= 0 to an a axis of the hexagonal crystal, and lets aF/ aO= of/ o¢=O this defines an energy maximum or minimum for a given magnitude and direction of the applied field. The magnetization will come to an equilibrium in a minimum and the mean curvature of the energy surface at that point will determine the resonance frequency2

[0

21'" O'FJl

brf=w=-y/ MsinO -- , 00' a¢'

(2)

where -y=geff(e/ 21l1c), and gdf is not the ordinary g value, but includes the shift in resonance field with large values of the Joss.

If we mount the spherical crystal on a rod parallel to the c axis, and apply the magnetic field in the basal plane only, 0'.1=0; rotating the crystal changes the angle <pH, the angle at which we apply the magnetic field relative to an a axis. ¢ describe:-; the position of the magnetization vector relative to the same a axis; the lwo do not necessarily coincide. ]n the basal plane, Egs. (I) and (2) and the equilibrium conditions lead

~ J. Smit and H. G. Beljcrs, Philips Research Rcpts. 10, t 13 (L955 ).

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1765 H. S. BELSON AND C. ). KRIESSMAN

",---------------,

• .' , ,

o

FIELD NECESSARY FOA RESONANCE (k,loQouul

FIG. I. Resonance conditions for a hexagonal single crystal of Ba2CozFcuOu.

to the simultaneous equations

[ 2K, K. ]

(wh)'= II COS(<I>-<I>H) + M -~ cos6</>

X [II COS(<I>-<I>H)-3~ cos6</> l (3)

K. II sin(<I>-q,Il) = 6-- sin6</>. (4)

M

If <1>= (n,,/6) we see from Eq. (4) that <I>=<I>H, or at positions of symmetry around the basal plane, the magnetization direction corresponds to the direction of the applied field. Equation (3) becomes

(wh)'= (II+2KiM-6K./M)(II=F36K./M), (5)

where the - or + sign in the second term refers to n even or odd, respectively. For a fixed frequency Eq. (5) predicts the peaks and minima in a curve of field necessary for resonance as a function of applied field direction. Experimentally such a curve reveals a roughly sinusoidal hexagonal symmetry with about a l OOO-gauss variation in hard and easy basal plane directions. X-ray diffraction photographs indicate that a axes correspond to the hard basal directions.

For a comparison with experiment, Eq. (5), it is seen, contains three unknowns, K,/ M, K./ M, and who By doing measurements at both X-band and K-band frequenci es we obtain a total of four equations which allow us to solve for all unknowns. For the crystal under study we obtain K,/ M =344 oe; K./ M=27.S oe; and g'ff=3 .70. Using these values we plot Eq. (5) and the two solid lines in Fig. 1 labeled 8H = (,,/2) (It odd) and (n even). The experimental points are the open circles.

A point of special interest is the zero applied field resonance ; in devices the corresponding frequency is a limit to the usefulness of a material. Equation (5) immediately indicates that wh=6(2K,K.-6Kl)I/M

for zero field. Initial permeability data were taken partially oriented polycrystalline sample of the composition, in which the c axes of hexagonal a common direction. A measured resonance abs.oll>ti peak at 3300 me/sec, with the g,ff value cited yields the zero field experimental point on the It is seen that good agreement is obtained with value predicted.

The value of 3.70 for g,ff demanded for an mental fit seems unusually high, until one that in the case of large loss, the resonance shifted. The sample investigated had very absorption half-widths, some 1800 gauss wide resonance field was about 2200 gauss. Merritt, and Wood3 discuss this consideriQg Lifshitz model. Following Yager et at., sp"ciiiea.llv (14) and (A4), we are able to find a damping predict that g.ff should indeed be of the order of result consistent with that obtained above.

The absorption lines are not symmetrical' low-field side they maintain some value and decrease to zero, making an accurate determin.' f).II impossible.

The field may be applied in directions other the basal plane. A position of obvious parallel to the c axis. As the field is ap:pu"G direction and increased, the from the easy basal plane, but it is very domains now enter the picture, until reached. The equations describing the fields less than those necessary to ensure 0=0 susceptible to simple analysis and are not For values of field greater than necessary for it is found for OJI = 0,

(wh)'=II[II- 2K,/M -4K,/M-

For 9000 mc/sec we measure II to be about gauss. The values of K,/ M and K ,/M cannot rated from this measurement. We obtain ",,3528 oe. Using the same value of g cited measurement places the right-hand solid curve in The black circle indicates the experimental fact that the values of K, and K, may be much than K, merely implies that the anisotropy surface is not well fitted by a simple Kl cos2f)

the basa l plane energy valley is broadened. the K,. may be obtained from a measured We get K,= 1.38X 10' ergs/cm', K,+~K,= ergs/cm' and K. = 1.I0X 10' ergs/cm'.

ACKNOWLEDGMENTS

We should like to thank Professor Herbert B. for much helpful discussion, W. Luciw for his technical ass istance, and 'V. Flannery for permeability measurements.

3 Yager, Galt, Merritt, and Wood, Phys. Rev. 80, 744:

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