Bhf(T)
4. Theoretical model for magnetization and anisotropy
4.1. Magnetization distribution
The existence of the compositional modulation of R and T constituent atoms offers an extra degree of freedom to control the local atomic environments which permits the tailoring of magnetic properties, such as PMA, in artificially structured CMF. However, this also introduces a new problem in determining the magnetization distribution which originates from the compositional modulation along the film-normal direction. The magnetization distribution in CMF was determined in the following way.
The compositional dependence of magnetizations (the R-, T-subnetwork magnetization and total magnetization) for homogeneous R-T alloys can be performed by using the mean-field model (Hasegawa 1975, Shan and Sellmyer 1990b). Figure 38 shows an example for Dyl-xCox alloys. In addition to the total magnetization o, the Co- and Dy- subnetwork magnetizations aCo and O'Dy , are presented.
The method of determining the magnetization distribution in R/T CMF was discussed by Shan et al. (1990). The R/T film is divided into thin slices, each of which can be regarded as a two-dimensional amorphous film. A distribution of T (or R) concentration, e.g. a sinusoidal function
2arZ)
r/j(z) = Ajo -4- djsin T (6)
is assumed for the thin layer-thickness CMF (for the purpose of this derivation we assumed R and T have equal thickness for simplicity). Here j = 1 and the + refer to T, j = 2 and the - refer to R and the Z axis is along the film normal. Ajo is a constant and Aj
NANOSCALE R/T MULTILAYERS 117 1200
,.-., 1000 f i
800 % . ~ .~. I ""
E 600 - - "
4 0 0 . " t ~" "" ""
-.~ 2 0 0 . I "..
b 0 _ J " , I " ' - - . - 2 0 0
i i
50 60 80 9b loo
Co ( or. )
Fig. 38. Co concentration dependence of spontaneous magnetization for Dy~2o alloys: the total magnetization as, Dy-subnetwork magnetization aoy, and Co-subnetwork magnetization Crco (after Shan et al.
1990).
the modulation amplitude of t h e j t h constituent concentration. As we have pointed out in sect. 3.1, the small angle X-ray diffraction for the multilayers with thin layers (~. < 15 A) shows only the first-order peaks; therefore it is reasonable to assume the sinusoidal function of eq. (6) for the compositional modulation.
The T concentration of the ith slice with coordinates Zi is rh(Zi) determined from eq. (6) and consequently its magnetization ai, ali, (~2i can be obtained from fig. 38 if we assume R = Dy and T = Co. Thus, the average magnetization a of the CMF is equal to
1 1 ~-~ (o.1 i _ (~2i)l~kZi"
(7= -~ ~ (~imzi = -~
i i
(7)
The parameters of the distribution function, i.e. Ajo and A j, were adjusted until the a value calculated from eq. (7) fitted the experimental data within a certain error. We point out that the constraints corresponding to the conservation of total number of R or T atoms must be satisfied while adjusting the parameters Ajo and Aj.
Figures 39a,b show an example of the distributions of (a) Co concentration and (b) magnetization for 6 ]~ Dy/6 ]~ Co. It is seen that O'Dy dominates in the Dy region and
~rCo dominates in the Co region, as is reasonable.
Figure 40 shows the Co layer-thickness dependence of the average values of the total magnetization ~r, Co- and Dy-subnetwork magnetization ~rco and ~rDy (fig. 40a) and the Co-atomic fraction modulation "A", i.e. Al in eq. (6) (fig. 40b). It is seen that the calculated cr value agrees with the experimental data quite well; the A value is only about 0.1 for the thinnest Co layer thickness of 3.5 A and its value increases as Co layer becomes thicker. The data shown in fig. 40 will be used to illustrate the calculation of the magnetic anisotropy.
118 Z.S. SHAN and D.J. SELLMYER 1.0
r,) 0
0.8
0.6 0.4 0.2 0.0 0
6ADy/6ACo
(o)
i
3 6 9 2 15 18
1000
~ " 800 6ADy/6~,Co E 600
~ 200 b
0
-200
--.~---Co----~ ~( Dy ,~
(re.
, i i
6 8 10 12 14 Z A~r. (A) -400
0 2 4 16 18
Fig. 39. Co concentration distribution (a) and magnetization distribution (b) along the film normal for 6ADy/6A Co (after Shan et al.
1990).
E
b
<~
600 400 200 0 -200 0
06
0'4 I 012 0
I I I i I
• E x p e r i m e n t F i t
I , l l I I
° 2 4 6 8 I0
Loyer thickness of Co (~,)
2
(o)
I
12 14
! . . . . I 1
b)
1
I I I I I I .
4 6 B I0 12 14
Loyer thickness of Co (A)
Fig. 40. Comparison of the calculated mag- netization with the experimental data for (a) 6]kDy/XACo (X=3.5, 5, 6, 8, 10, and 11), and (b) the Co layer-thickness depen- dence of the Co atomic fraction modulation (after Shan et al. 1990).
NANOSCALE R/T MULTILAYERS 119 4.2. Perpendicular magnetic anisotropy (PMA)
4.2.1. The origin of PMA in R/T
Many efforts have been made to investigate the magnetic anisotropy and its origin in thin films. Hellman and others have listed various sources of PMA (Hellman et al. 1989, Hellman and Gyorgy 1992) including pair ordering, local clusters, bond-orientational- anisotropy (BOA) (Yan et al. 1991), stress anisotropy, anisotropy due to columnar microstructure, surface anisotropy due to magnetic dipolar origin and the growth-induced anisotropy, etc. Suzuki et al. (1987) and Baczewski et al. (1989) suggested that the single- ion anisotropy of the R atom contributes to PMA. After analyzing the function of various sources carefully, it is necessary to specify both the details of the structural anisotropy and the nature of the magnetic interactions.
(A) Structural anisotropy, i.e. the anisotropic distribution of the constituent atoms, is generated by (i) the artificial multilayered structure which follows from the fabrication procedure, (ii) the crystal structure for the crystalline R (or T) layers, or (iii) any other sources of anisotropic pair correlations such as those listed in the previous paragraph.
Harris and coworkers (1992, 1993) have performed an important study on the structural origin of PMA in sputtered amorphous Tb-Fe films. These authors performed extended X-ray absorption fine structure (EXAFS) measurements to study the anisotropic short- range structure. X-ray absorption spectra for both Fe K and Tb L3 absorption edges were obtained with synchrotron radiation and fig. 41 (Harris et al. 1993) shows the Fourier transformed EXAFS data for Fe and Tb. These data show a distinct difference in the amplitudes for electric fields parallel and perpendicular to the film. The data show that the coordination numbers are different for the two different orientations and in fact show that there is an excess number of Fe-Fe and Tb-Tb pairs in the plane of the film and an excess of Tb-Fe pairs perpendicular to the plane of the film. This is shown qualitatively in typical atomic arrangements in fig. 41. Moreover, Harris et al. heat treated one of their samples, Tb0.26Fe0.T4, at 300°C for one hour and the resulting Fourier-transformed EXAFS amplitude shows that the structural anisotropy seen in fig. 41 has been reduced by the annealing. The authors thus suggest that the high temperature annealing has removed most of the anisotropic pair correlations and the remaining anisotropy which is perhaps 20%
of the original is likely to be due to magneto-elastic interactions between the film and the substrate. The work of Harris et al. can be regarded as direct evidence that the structural anisotropy is one of the necessary conditions to create PMA. This work thus confirmed that the structural anisotropy in the interface region of multilayers is an excellent method for producing PMA.
(B) Magnetic interactions leading to anisotropy include mainly the following two:
(i) magnetic dipolar interactions and (ii) spin-orbit interactions which lead to single- ion anisotropy associated with lanthanide atoms. Single-ion anisotropy arises from the interaction between the 4f electrons of the R atoms and the local electric field created by the neighboring ions. If the charge distribution of the 4f electrons is nonspherical, i.e. L s 0 , the electric field forces the 4f electrons, the orbital momentum of the 4f electrons and consequently the magnetic moment into a preferred orientation, i.e.
120 Z.S. SHAN and D.J. SELLMYER
2
° ~
<
0 ° 9 , , , i , , | , , . i , , , i , . , i , , , i , . . t , ,
Fe EXAFS (a) As-deposited "l"bo.~Feo..l, 0.67
0.45
0.22
• " i " " l "
('0)
/
Ell . . . . E 1
' l ' ' ' l ' ' ' l ' " ' i ' ' ' i ' ' "
Tb EXA.FS A s - ~ p o s i ~ t Too.~Feo.Tl"
F'dm Nom~
0.25
0.19
3.12
0.06
0.00
I 2 4 5 6 7 8
Radial Coordinates (.~)
Fig. 41. (a) Fourier transformed Fe EXAFS data, (b) Tb EXAFS data collected using normal incident photons (solid lines) and glancing incident photons (dashed lines). Plot insets illustrate the schematic diagrams of typical atomic arrangements in R-T amorphous alloys where anisotropic pair correlations are shown (after Harris et al. 1993).
the easy magnetizing direction, through the strong spin-orbit coupling. Therefore both a structural anisotropy as discussed above and the interaction between the 4f electrons of R atoms and the local crystal field are the major origins of magnetic anisotropy in R/T multilayers for those R atoms with orbital angular momentum.
For Gd-Co homogeneous films, Mizoguchi and Cargill (1979) have calculated the magnetic anisotropy in terms of the dipolar interaction for the samples with a short- range anisotropic microstructum. Fu et al. (1991) have shown that the magnetic dipolar interaction in the surface layers contributes to the intrinsic PMA. The calculated PMA based on dipolar interactions is of the order 103-104 erg/cm 3 for amorphous Gd-Co films which is much smaller than that of PMA in PUT (R = Tb, Dy; T = Fe, Co). However, Hellman and Gyorgy (1992) argued reasonably that the surface dipolar interactions considered by Fu et al. are not an important source of anisotropy in R-T films containing heavy lanthanides with orbital angular momentum. Theoretical calculations of Jaswal (1992) also came to the same conclusion.
Figure 42 illustrates our experimental results for the Ku behavior of R/T (R = Tb, Dy, Gd and T = Fe, Co). This figure demonstrates that the anisotropy Ku for Dy/Fe, Dy/Co,
NANOSCALE R/T MULTILAYERS 121
E
(.J
cn f.._
%
oJ2
I
o 2
I I ! I I I I
4.SATb/x~Fe / 6 . 0 ~ O y / x ~ C o
//s0~0y/x~Fe
I , \ I ~ I , - 7 " " ' - I - - - - .
8 aO a2 14
L a y e r t h i c k n e s s o f TM ()~)
Fig. 42. Anisotropy for 4.5 A, Tb/X A Fe, 5 ]~ DM/X,~ Fe, 6 A Dy/X A, Co, and Gd/Fe CMFs. The anisotropy data of 2.3/kGd/3 A Fe, 3 AGd/3 A, Fe, 3,~Gd/5 AGd, etc., are in the shaded area (after Shan et al. 1990).
and Tb/Fe is roughly an order of magnitude larger than that of Gd/Fe. Since, to first order, the Gd atoms have no single-ion anisotropy, it is reasonable to attribute the main origin of the PMA of the Dy and Tb CMF to single-ion anisotropy of Dy and Tb atoms which are located in the interface regions.
Mibu et al. (1993) prepared 30]kR/40]~Fe multilayers (R=Pr, Nd, Tb, Dy, etc.) for Mrssbauer studies. The MSssbauer spectra indicate that the Fe moments lie in the film plane at 300 K but turn to the perpendicular direction at lower temperatures. Since all these R atoms have similar 4f electron distribution, the authors concluded that the 4f electron of R atoms should be responsible for the PMA at lower temperatures.
As outlined above it is clear that for the R/T where R atoms have orbital angular momentum, the PMA results from the single-ion anisotropy of R atoms and structural anisotropy in the interface region. In the next subsection, a theoretical expression for PMA is derived based on this concept. It is also essential to point out that the stress anisotropy and BOA have been involved in the PMA induced by the distortion of the structure from the strains in the films. Once the distorted structure is determined, this will then affect the magnitude of both the structure anisotropy and the interactions among all R and T atoms and consequently the PMA behavior.
4.2.2. PMA model for R/T
Shan et al. (1990) have developed a theoretical model of PMA in R/T. Figure 43 shows the local environment of an R atom (Dy in this case) atom in the vicinity of an "interface"
of the CME For the ith slice of the CMF, the local single-ion anisotropy is given by
K(i) oc qj (r2> O°A O, (8)
122 Z.S. SHAN and D.J. SELLMYER
Ii
j•mog,
moment(~.~fr~eJ,,~ j lh neighbor ion
Ca)
(~ ~h neighbor
( : ~ Z
) ~ D y ion
r)v - - I R n F)v
i
l
(b)-/V2 0 Z i
(c)
Fig. 43. (a) Crystal field created by the neighbor ions, (b) local anisotropy K(i) at Z v The resultant anisotropy K u is the average of K(i) over the whole volume of CME (c) Concentration distribution Cj (j= 1 refers to Co,j=2 refers to Dy) along the film normal (after Shan et al. 1990).
where a j is Stevens' factor; (r 2) is the average radius squared o f the 4 f orbit, O ° is Stevens' operator On m with n = 2 and m = 0. O ° is a factor proportional to Jz 2 if Z is the easy-axis direction. A ° is a crystal-field term given by
A °
c< ~
qj(3c°s2 ~ g j -1)
U p o n substitution eq. (8) then becomes qj(3COS20j- 1) K(i) oc ~j (172)i ~-~ 3 '
j r; (9)
where j is the index o f the j t h ion in the neighborhood o f the R ion at the ith slice. The summation is over the neighboring ions with charge qj and distance rj from the R ion and Oj is the angle with respect to the moment direction (see fig. 43). The average notation ( );
means the statistical average o f the R ion over the ith slice. In the above derivation, the relation, ~rz OCJz, has been used, where cr z and Jz are the Z component o f R subnetwork magnetization and angular momentum, respectively.
NANOSCALE R/T MULTILAYERS 123 For the amorphous CMF structure the sum over j in eq. (9) must be replaced by an integral weighted with an anisotropic probability function. This function for a CMF structure may be expressed as
Piy(r) = t/yRy(r)
[1+
fij(r)cos aj+..-], (lO)
where t/j is the atomic fraction of the jth surrounding ion consisting of R and T ions which have the sinusoidal form as shown in eq. (6).
Rj(r)
gives the isotropic part of the distribution, whileflj(r)
is the lowest-order anisotropic contribution. The parameter flj, the anisotropy in the pair distribution function, is determined by the inherent structure built into the multilayer, and also structural inhomogeneities produced by chemical short- range order or stress at the interfaces,aj
is the angle betweenrj
and the film normal Z as shown in fig. 43b.In the CMF structure r/j varies along the film normal so that
Piy(r) = [rlj(Zi) + rlj(Zi)rcos aj] Rj(r) [I + fij(r)
cosay],
(11) where the relationAZi
= rcos aj is adopted, i.e. the R ion at the ith slice is assumed to be the origin of the spherical coordinate and the polar axis is along the Z direction, i.e.the film normal direction, t/j is the first derivative of t/j.
Then the local anisotropy