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Submitted on 1 Jan 1971
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EFFECT OF SURFACE SPIN PINNING ON FERROMAGNETIC MAGNETOSTATIC MODES
M. Sparks
To cite this version:
M. Sparks. EFFECT OF SURFACE SPIN PINNING ON FERROMAGNETIC MAG- NETOSTATIC MODES. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-558-C1-559.
�10.1051/jphyscol:19711190�. �jpa-00214014�
LAMES MINCES
(3. partie): RESONANCE ET ONDES LIE SPIN
EFFECT OF SURFACE SPIN PINNING
ON FERROMAGNETIC MAGNETO STATIC MODES
M. SPARKS (*)
The Rand Corporation Santa Monica, California
RBsumB. - Quand une condition explicite de blocage des spins k la surface des couches minces est comprise dans le problkme de valeurs propres des modes magnktostatiques, il faut tenir compte de I'kchange bien que I'energie d'khange soit nkgligeable par dkfinition pour des modes magnCtostatiques. En genkral, l'aimantation change considkrablement, pourtant les intensitks et les frCquences restent pratiquement les memes que pour les modes magnktostatiques purs (I'kchange ktant constant et identiquement nul, et sans mkcanisme de blocage).
Abstract. - When an explicit surface-spin pinning condition is included in the thin-film magnetostatic-mode eigen- value problem, exchange must be included even though the exchange energy is negligible for magnetostatic modes by definition. The magnetization is changed drastically in general, but the intensities and frequencies are essentially the same as for the pure magnetostatic modes (exchange constant identically zero and no explicit pinning mechanism).
If the magnetization at the surface of a thin ferro- magnetic film is not allowed to precess, the magneti- zation is said to be pinned. It is currently believed that pinning gives rise to large intensities of even modes. Recall that the intensity is proportional to the integral of the transverse, time-varying magneti- zation m over the film, and that this integral does not vanish for pinned, even sine waves. If the normal derivative dmldz, rather than m itself, were zero at the surfaces, all modes except the constant one would integrate to zero, and their intensities would be zero.
These results are valid for exchange modes, but it will be shown that they are not valid for magnetostatic modes in general. Recall that a magnetostatic mode has negligible exchange energy, while an exchange mode has a negligible value of the demagnetization energy associated with the small transverse component m of the magnetization. For pure magnetostatic modes [l, 21 (with exchange constant D identically equal to zero), the usual continuity of the normal component of B and the tangential components of H determines the values of m and its normal derivative dm1d.z at the surfaces.
If a pinning condition, such as m
+
(const.) dm/dz = 0 ,is specified at the surface, the system equations are overdetermined, and no solution exists. In this case the exchange interaction must be included in the magne- tostatic-mode probleme even though the exchange energy is negligible. Mathematically, the exchange interaction is a singular perturbation. The result is that the frequencies and intensities of the magne- tostatic modes with the given pinning condition are essentially the same as those of the pure magneto- static modes. In other words, the frequencies and inten- sities of magnetostatic modes are essentially independent of the surface pinning condition.
(*) Any views expressed in this paper are those of the author.
They should not be interpreted as reflecting the views of The Rand Corporation or the official opinion or policy of any of its governmental or private research sponsors. Papers are reproduced by The Rand Corporation as a courtesy to members of its staff,
Gann [3] has shown that when both the exchange and demagnetization interactions are included in the infinite-film problem, the solutions of the equation of motion for m are linear combinations of terms containing three wave vectors. The three values of kz are the three roots of the well known dispersion relation
k2 mZ/y2 = k2(Hi
+
Dk2) (Hi+
~k~+
4 EM, sin2 8,), where o is the frequency, Hi is the internal field, M , is the saturation magnetization andFor perpendicular resonance (M, perpendicular to the plane of the film) only two of the terms in the expansion of m are large for the main-branch mode [4,5]
(smallest value of k,). The wave vector k,, for the first term approaches that of the pure magnetostatic mode [2] as D -+ 0, and the wave vector k , of the second term approaches that of the exchange mode which is degenerate with and has the same values of k, = ^xk,
+
;k, as the magnetostatic mode as D -+ 0.The third term is negligible because it corresponds to an exponentially decaying wave, for which the net exchange torque is in the opposite directions from that of a sinusoidal wave ; in other words, the amplitude of the third wave is small because it is off resonance, roughly speaking.
The film thickness S is very large because a small value of S would make the exchange energy large, and the modes would not be magnetostatic modes.
Thus k,
+
n / S must be satisfied in order to make the frequency u, = 1 y 1 (Hi+
~ k : ) of the degenerate exchange term equal to the frequency of the magneto- static term. These two terms are added with the appro- priate coefficients to make m = 0 at the surfaces. The frequency is essentially the same as that of the pure magnetostatic mode since the added exchange term is degenerate with the magnetostatic term, and the inten- sity is essentially the same as that of the pure magneto- static mode because the wave vector k, is so large that the added second term integrates to zero approxi- mately.Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711190
EFFECT OF SURFACE SPIN PINNING ON FERROMAGNETIC MAGNETOSTATIC MODES C 1
-
559 The pure magnetostatic modes with k,<
k, havedm/dz N 0 at the surface and those with kf
+
k, have m-
0 at the surface. Thus, for the boundary condi- tion m = 0, the magnetostatic and exchange waves are strongly admixed when k, 4 k, and are more weekly admixed when kf 9 k,. The converse is true for the boundary condition dm/dz = 0. For a finite sample the discrete magnetostatic modes should lie close to the infinitesample, pure magnetostatic-mode curves for both m = 0 and dm/dz = 0. Other sources [4, 51 of the shift in frequencies away from the infinite-sample, pure magnetostatic mode curves are expected to be much larger than the shifts resulting from the admi- xing.The mathematical analysis [3, 61 which yields these results is rather simple, the frequencies and admixture coefficients coming from the solution of three homo- geneous algebraic equations which are obtained from the boundary conditions on B, H and m discussed above. The results for the frequencies also can be obtained from a graphical solution [3] of the dispersion relation [3] y,, = &yE for the even modes, where y, = k, tan
3
ki S - k, for i = ms or E, andE ki,/kk & 1 for D -, 0. In figure 1, y,, (heavy dashed curve) and ~ y , (heavy solid curve) are sche- matically illustrated (not plotted) as functions of o for a given value of k,. Note that k,, and k, are functions of w . The solutions to y,, = ~ y , for o correspond to the crossings of the two curves. The crossings marked by circles give the exchange-mode frequencies and the crossing marked by a square gives a magnetostatic mode frequency. From the construction it is clear that the frequency of the magnetostatic mode cannot differ from that of the pure magnetostatic mode (obtained by setting y,, = 0 - see the arrow in Fig. 1) by more than the frequency spacing of the two adjacent pure exchange modes. Since the spacings of these pure exchange modes are very small in thick films, the frequency of the magnetostatic mode is very nearly equal to that of the pure magnetostatic mode.
For a YIG film 12 p thick with the main-resonance the mode 60 Oe above the bottom of the manifold, the spacing of the adjacent pure exchange modes is 3 Oe. Thus in high quality films, which can have linewidths smaller than 1 Oe, the admixture of the magnetostatic wave into nearby exchange modes possibly could allow these exchange modes to be observed in ferromagnetic resonance experiments. The intensities and frequencies could be estimated by
[l] DAMON (R. W.) and ESHBACH (J. R.), J. Phys. Chem.
Solids, 1961, 19, 308.
[2] AKHIEZER (A. I.), BAR'YAKHTAR (V. G . ) and PELET-
MINSKII (S. V.), Spin Waves (North-Holland, Amsterdam. 1968).
[3] GANN-(v. V.), ~ i z . ~ v e i d . Tela, 1966, 8, 3167, Soviet Phys. Solid State 1967, 8, 2537.
FIG. 1. - Graphical solution of the even-mode dispersion relation y,, = EYE. The square indicates the frequency of a magnetostatic mode, the arrow indicates the frequency of the corresponding pure magnetostatic mode (yms = 01, and the vertical dotted lines and circles indicate the frequencies of the
pure exchange modes and exchange modes, respectively.
formally quantizing k, in the infinite film theory if these modes become of interest.
The corresponding results for M, in the plane of the film and for surface modes are discussed elsewhere 161.
T. Wolfram and R. E. DeWames [7] stated that the results 161 for the surface modes are contradicted by the results of Benson and Mills [8] for a film 30 atomic layers thick, which were verified by Wolfram and DeWames. Although the nature of the contradiction was not stated, comparison of these two different results certainly could lead to contradictions since the results of reference 6 apply to magnetostatic modes, while the modes of interest in the extremely thin film of reference 8 are exchange modes or mixed exchange-magnetostatic modes.
[4] SPARKS (M.), TITTMANN (B. R.), MEE (J. E.) and NEWKWK (C.), J. Appl. Phys., 1969, 40, 1518.
[ 5 ] SPARKS (M.), Phys. Rev., 1970, B 1, 3831.
[6] SPARKS (M.), Phys. Rev. Letters, 1970, 24, 1178.
[7] WOLFRAM (T.) and DEWAMES (R. E.), Phys. Rev.
Letters, 1970, 24, 1489.
[S] BENSON (H.) and MILLS (D. L.), Phys. Rev., 1969, 178, 839.
