The dipole-dipole contribution to anisotropy Ku in the macroscopic limit is known as the demagnetization anisotropy Kde. The value of Kde for a homogeneous film is Kde =-2;rM~ where Ms is the saturation magnetization. In multilayered films one normally use the same expression for Kde with M~ replaced by its average value, (Ms> 2.
Kusov et al. (1992) derive the correct expression for the Kde valid for a multilayer film where the magnetization depends only on the Z coordinate. The derivation in the framework of a continuum approach gives
g d e = -2Jr (M~). (19)
For a homogeneous film eq. (19) reduces to the standard form, i.e.
g d e = -2n" (M2) = - 2 a : (Ms) 2 .
However, in a CMF structure with larger fluctuations of magnetization these two results are quite different. Assume that the magnetization may be expressed approximately as
Ms(Z)= Mo + AMsin (2~ Z) (20)
where M0 is the mean value of magnetization Ms(Z), AM is the modulation amplitude and ~. is the bilayer thickness. Then
Kde =
2Jr~ f0z (M0
+ AM sin--~--) dZ = -2JrM 22Y~Z"~ 2 (1 ~M2
+ ~ ) .(21)
One can see that the introduced error becomes large when AM is comparable with Mo.
Therefore one must use eq. (19) for calculating Kde to avoid error. The error introduced for multilayers with square-wave concentration distributions has also been discussed by Kusov et al. (1992).
134 Z.S. SHAN and D.J. SELLMYER
5.3. Measurements of anisotropy and the R - T exchange coupling
The anisotropy of R - T alloys or R/T multilayers is usually measured with torque magnetometry (or VSM), Kerr effect, and extraordinary Hall effect. Wu et al. (I 993) have done systematic measurements of Ku for Tbx(FeCoh_x alloys with different techniques and found that Ku measured by Hall effect and Kerr effect is always larger (by up to a factor of 3) than that from the torque techniques. The authors pointed out that the main difference among the techniques comes from the fact that different techniques measure different combinations of the R- and T-subnetwork magnetizations, e.g., the measured torque is associated with the net magnetization of the R and T subnetworks;
the Hall effect is contributed by the T subnetwork magnetization; and the Kerr effect is mainly contributed by the T subnetwork, but a small part of the signal also comes from the R subnetwork, depending on the laser wavelength. It is necessary to have a clear physical idea about the effects of measurement technique on the measured Ku value, and to interpret the magnetic behavior correctly.
The anisotropy behavior near the compensation point has received considerable attention for the R-T alloys and R/T multilayers. It is usually considered that the heavy R and T are coupled ferrimagnetically like a rigid body, i.e. the coupling strength A is infinity. Based on this assumption, the magnetic energy of a thin-film sample is written as (Hellman 1991):
E = - H M cos(a - 0 ) + (Ku - 2rrM 2) sin O, (22)
where O and a are the angles of M and H relative to the film normal. Minimizing E with respect to O for a = :r/4 gives an expression
= 2(Ku - 2 : r M 2 ) r + ½M 2, (23)
where r = M × H is the torque. Measuring r as a function of H and then making a (r/H) 2 vs r plot, one can find M and Ku simultaneously from the intercept and slope of the straight line passing through the experimental data. This analysis breaks down near the compensation point, where M ~ 0.
It appears from experiment that Ku drops for R-T alloys or R/T near the compensation point (Sato and Habu 1987, Shan and Sellmyer 1990b, Hellman 1991). This behavior was discussed by Hellman (1991) and Wu et al. (1993) in terms of a canting model, i.e. they considered the canting between the R and T subnetwork magnetizations, and the R - T coupling is not regarded as a rigid body.
In the canting model the magnetic energy of a thin film is (Hellman 1991)
E = H - MR - H . MT + A MT • MR + Ku sin 2 OR + 2Jr(MR cos OR - MT c o s I~T) 2, (24) where A is the coupling strength between R and T subnetworks, MR and M T a r e the magnetizations for R and T subnetworks, and OR and OT are the angles of MR and M T
NANOSCALE R/T MULTILAYERS 135
;/j¢"~/I:14
t::::::::::::":! :::!::::::::.:;i:t !
• 4T
M,,M T • MT p" ;" |
Fig. 52. Schematic of magnetic field, subnetwork moments and their angles:
(a) M R >MT, (b) M r > M R (after Hell- man 1991).
measured from the normal (fig. 52). Minimizing E with respect to OR and {~T for a = z / 4 gives
M T
V~ ((MR_--MT) 2 2--M-~A) H,
( H ) = ( M R - M T ) - ~ - - \ 4Klu + (25)
where Ktu = Ku - 2Jr(MR -
MT) 2.
In the limit A--+ c~, values for the net magnetization
(MR--MT)
and K~u = K u - 2]r(MR - M r ) 2 may be found from the intercept and slope of(UH)
vs. H. These values are identical to those found in eq. (23). For finite A, however, the term(MT/2MRA)
in eq. (25) will cause an increased absolute value, and hence, if ignored, an artificially low value for Ku as ( M R - M T ) is small. As (MR-Ma-) approaches zero (i.e. near compensation), this term becomes increasingly large compared with ( ( M R -
MT)2/4K~u),
and hence Ku will appear to go zero at the compensation point. Also, as the real value o f Ku increases, this reduction becomes significant at increasingly large values of (MR - MT).
Wu et al. (1993) have studied the canting model in more detail and have given both the calculated and experimental data as shown in fig. 53 for Tbx(FeCoh_x alloys with A ~ 1800. It is seen clearly that there is an anomalous drop near the compensation point when using torque magnetometry with the field at 45 °. The figure also shows the experimental data and calculated Ku curve in terms of the extraordinary Hall effect. We notice that the torque technique produces a wider and deeper apparent dip and the Hall effect techniques produce a much narrower dip for the T-dominant case and a narrow peak for the R-dominant case.
The exchange-coupling strength A can be expressed and estimated as 2Z [JR_TI
A - NgRgTt~2 ~--
1800 (26)for Tb(FeCo) with Z = 12, JR-T =--10-15 erg, gR = 1.5,
g'r
=2 and N = 5× 1022 cm -3 (Wu et al. 1993). A may be determined experimentally in terms of the so-called "free-powder magnetization" approach which was analyzed in detail by Verhoef et al. (1990) and Zhao et al. (1993).136 Z.S. SHAN and DJ. SELLMYER
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