Generalized magnetostatic modes and pinning in ferromagnets

Volume

28A.

number

GENERALIZED

PHYSICS

2

MAGNETOSTATIC M. SPARKS,

LETTERS

MODES

AND

4 November

PINNING

IN

1968

FERROMAGNETS

and C. NEWKIRK

B. R. TITTMANN

Science Center and Autonetics. North American Rockwell Corporation. Thousand Oaks and Anaheim. California. USA Received

Ferromagnetic normal modes having comparable served and explained by a new theory

2 October

exchange

General ferromagnetic normal modes having comparable exchange and demagnetization energies and the first resonance measurements in epitaxially-grown single-crystal films of yttrium iron garnet (YIG) [l] are reported and explained by a theory which applies to samples of arbitrary shape and size and includes both exchange and demagnetization energies. The results give a new method of studying surface spin pinning, have implications concerning the main-resonance position in finite films, and further verify Portis’ mode-spacing theory [2]. By eliminating ff= VJr rather [3] than the magnetization M in the equation for M, we obtained an eigenvalue equation which we solved by a variational method to obtain the frequencies W = /y I&i) + Wexc + Wd, wher%(IJi) iS the weighted (by the factor M2 + My) average of the internal field, uexc - k2% , and

(1) X

1 Jdr M+exp(iq*r)12

/ ldr

lM+12.

For a rectangular film L by S by D (where L a S >> D), the trial function M+ = = m cos(Gx - h.gr) cos(kg - $q,,,n) cos(k,z - &,,n), where the 7)‘s label even (7 = 0) and odd (q = 1) modes, becomes the exact solution in the limit L, S - 00.Even for the low-order modes and for surface spin pinning, this choice is dictated by intuition and substantiated by excellent agreement with a large amount of data. The results for various values of k,, Q, k , etc. are compared with the experimental resu Yts to determine the surface spin pinning conditions *. Substituting this Mf into eq. (1) and evaluating the integrals approximately for perpendicular resonance gives

1968

and demagnetization

energies

have been ob-

ud = 2ktD arctan (r/ktD) 1y (MS for kz N 0, where k?=k: +kf,. Th; the&etical results for thin films are: (a) uexp - kz for the modes with a small number nt of half sine waves of M+ in the plane of the film. (b) All modes in a series having a fixed value of k, have approximately the same uexc (when nt is small) and have different values of Wd for the different nt. (c) Wd - D/F, where F is a face dimension, for k, N 0 and nt small. (d) Larger nt gives closer spacing because of the arctan factor in Wd. (The modes cluster near the top of the spin-wave manifold.) (e) The modes in the k, = n/D (and hi her) series are more closely space (wd - D2/F5) than those in the k r 0 series, and (f) the k, e 0 series is_most Zstrongly excited since the power P - hrfx Jdr M, integrates to zero for M, - sin w/D, cos 2sz/D,. . . . All these features are present in experimental results obtained at 9.4 GHz and room temperature. For a circular sample with radius y. = = 0.28 mm and D = 0.7 p in perpendicular resonance, the modes in the k, = 0 series were barely resolved because wd is small (D/r,N = 10-3). Modes with k, = nn/D (n=2,3?. . . ,12) appeared as single lines (with positions determined by wexc) according to (a), (b) and (e). The power absorbed by modes 2-12 was much less than that absorbed by the kt N 0 modes, in agreement with (f). Modes l-5 were nearly linearly spaced, and modes 6-12 were spaced as n2, indicating that MS is not perfectly sharp at * The results

to date indicate pinning on the small edges of the samples.(since the demagnetization field - 2~Ms at the edges differs from the value - 4rMs away from the edges) and no pinning on the large surfaces unless Ms is a function of 2. The calculations presented are for this pinning at the small edges and no pinning at the large surfaces.

131

Volume

28A, number 2

PHYSICS

LETTERS

4 November

1968

of modes l-12 are independent of y. according to (a), while the spacing in the k, = 0 series is increased according to (c). In fig. 1, with negligible uexc since D(= 12.4 p) is large, positions and amplitudes of twenty-seven modes agree well with the calculated values (indicated by symbols) for all five series of modes (no, 1) - (no, 9). The decrease in spacing [result (d)] in the (no, 1) series is evident, and the calculated value of 60 Oe spacing between the first two modes agrees well with the experimental value of 61.5 Oe. Similar results for parallel resonance and results of Dillon [4] are also explained in great detail by the theory.

L 4500

4700 h4opnetcc

I

I

I

4600

4900 Field

I

5000

4900

toe,

Fig. 1. Derivative of the power absorption for a thin rectangular sample of YIG. The mode (a~, ns) contains nL half sine waves of M+ along L and nS half sine waves along S, and is constant across the thickness.

the interfaces [2]. The aluminum is expected to replace some iron and vice versa at the deposition temperature of 12OO’C. Reducing the radius r. by successive etching showed that the positions

HARTREE-FOCK PARAMETERS OF THE CONFIGURATION

References 1. J.E.Mee, EEET. MAG 3 (1967) 190. G. R. Pulliam, J. Appl. Phys. 38 (1967) 1120. 2. A.M. Portis, Appl. Phys. Letters 2 (1963) 69. A smaller effective D for the spacing of the magnetostatic modes in the ka M 0 series than for the exchange modes 6-12 is further evidence for the validity of Portis’ theory. 3. L. R. Walker, Phys. Rev. 105 (1957) 390; R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19 (1961) 308. These athors neglect exchange and eliminate M and solve for Ik. 4. J. F. Dillon, J. Appl. Phys. 31 (1960) 1605.

FOR THE ‘D AND 5s2 5p4 OF XENON

% TERMS III

C. FOGLIA Istituto

di Fisica

dell’Universita’Di Received

Parma,

21 September

43100 Pama,

Italy

1968

Self-consistent field calculations with exchange are carried out for the terms 5s2 5p4 1s and 5s2 5~4 lD of xenon III. One electron energies, Slater integrals and other quantities related to atomic wave functions are here tabulated.

In a previous paper [l], the author has reported numerical self-consistent field wave-functions and one-electron energy eigenvalues for the 5s2 5p4 3P term of xenon III. Special attention is now paid to 5s2 5p4 ‘S and 5s2 5p4 ‘D observed terms [2] of the same ion, 132

for which Hartree-Fock calculations are carried out by means of the method described in ref. 1 *. The radial wave functions were calculated by * The computations computer

were done on the I. B. M. 7096 of C. N. U. C. E., Pisa (Italy).