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DOMAIN WALL CALCULATIONS FOR THIN SINGLE CRYSTALS
A. Hubert
To cite this version:
A. Hubert. DOMAIN WALL CALCULATIONS FOR THIN SINGLE CRYSTALS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-404-C1-405. �10.1051/jphyscol:19711141�. �jpa-00213960�
JOURNAL DE PHYSIQUE Collogue C 1, supplement au n° 2-3, Tome 32, Fevrier-Mars 1971, page C I - 404
DOMAIN WALL CALCULATIONS FOR THIN SINGLE CRYSTALS
by A. HUBERT MPI Metallforschung, Stuttgart
Résumé. — Les énergies des parois des domaines sont calculées dans les monocristaux minces par la méthode de Ritz. Comme dans le cas des couches minces avec anisotropie uniaxe il faut essentiellement considérer trois types de parois : des parois de Néel symétriques et asymétriques et des parois de Bloch (asymétriques). Les parois sont calculées pour les tôles de Fe et de Ni d'orientation (100) et (110). Les parois à 180° se montrent instables au-dessous d'une épaisseur cri- tique aussi dans le cas des monocristaux. Ils se divisent en un système de parois de Néel à petit angle (paroi à 90°, etc.).
Parmi les structures obtenues la configuration en échiquier montre une énergie inférieure à l'énergie de la paroi à « cross tie » bien connue des cristaux uniaxes.
Abstract. — Domain wall energies were computed for thin single crystals by means of Ritz's method. As in the case of films with uniaxial anisotropy essentially three types of domain walls have to be considered : symmetric and asymme- tric Neel walls and (asymmetric) Bloch walls. The walls are computed for Fe and Ni platelets in the (100) and (110) orien- tations. 180° walls prove to be unstable below a critical thickness also in the case of single crystals. They tend to decay into a system of lower angle Neel walls (90°-walls, etc.). Among the resulting configurations the checkerboard pattern is shown to have a lower energy than the cross tie pattern known from uniaxial films.
I. Introduction. — Investigations of the wall and domain structure of thin magnetic single crystals are particularly suitable for a comparison of experiments with micromagnetic theory, since the experimental parameters are far better known for single crystals than, e. g., for evaporated polycrystalline films. Since most of the experimental work [1-3] was performed on (100)-iron foils, only this case and the analogous case of a (llO)-surface on nickel (negative anisotropy) will be discussed. The experimental results (mainly on iron) may be summarized as follows :
(1) In films with thicknesses below 500 A, mostly 90°-walls are observed. If, occasionally, short segments of 180°-walls do occur, they have a cross-tie appearance with very densely packed Bloch lines.
(2) Above this thickness, 90°-walls and 180°-walls are about equally frequent. In Bitter pattern experiments 180°-walls show much less contrast than 90°-walls.
(3) In films thicker than about 1 500 A, a second, metastable mode of the 180°-wall was observed [3]
which shows strong Bitter pattern contrast.
Previous one-dimensional wall calculations [4, 5]
were not able to explain all of these experimental results. The reason for this may be that the new, two- dimensional wall models known from uniaxial films [6, 7] must be considered also in the single crystal case, particularly since in the typical thickness range from 50 to 2 000 A the anisotropy energy is only a smaller part of the total wall energy. For this reason the reported models were generalized for single crystals — using asymmetric Bloch walls for the 180°-walls in Fe and Ni and symmetric as well as asymmetric Neel walls for the 180°- and 90°-walls in Fe and the 180°-, 109°- and 71°-walls in Ni.
All calculations are based on Ritz's method, giving upper bounds for the real wall energies.
II. Details of the computations. — Only walls without applied field were considered. The magnetic constants were taken as those in Fe and Ni at room temperature [8-10], with confinement to the first order anisotropy expression, giving for Ni K = Kt/I? = — 0.226 94 and for Fe K = 0.177 0. Energies are given in the reduced units Is) and film thicknesses in the units d0 = \(A/Is), corresponding approximately to
2.4 erg/cm2 and 83 A for Fe and 4.7 erg/cm2 and 194 A for Ni.
Let <x be the normalized magnetization component perpendicular to the wall, fi perpendicular to the film, and the third component y, then one-dimensional symmetric N6el walls are characterized by /? = 0. The crystal anisotropy can then be written in terms of a2 and a4 for all walls except for the 180° walls in Ni where a term a3 y appears. Neglecting this term the walls may be computed with the program published in [11]. In the calculation of the asymmetric models the full anisotropy expression was used. Another difficulty arises in the calculation of 180°-walls at very low thicknesses when the 180°-walls begin to separate into two lower angle walls. Since this behaviour cannot be described by the functions used in ref. [11] without additional adaptations, the results for the 180°-wall calculation are less reliable for thicknesses below about 0.3 d0.
III. Symmetric Neel walls. — The energies are plotted in figure 1. At least for thicknesses above
D/iA/U? -
FIG. 1. — Wall energies as a function of film thickness.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711141
DOMAIN WALL CALCULATIONS FOR THIN SINGLE CRYSTALS C 1-405
0.3 do the 180°-wall energy is much higher than 2.E,, for Fe and E,,
+
E,,, for Ni. Also the lower angle walls increase their energy with increesing thickness, although they start to do so at larger thicknesses. Simultaneously they develop extended tails. Some wall profiles are given in figure 2. AlsoFIG. 2. - Wall profiles for some one-dimensional, symmetrical Neel walls.
plotted in figure 1 is the lower estimate of the cross-tie wall energy [12, 61
for iron crystals. It shows that 180°-walls -if they occur - may have the tendency to decay into a cross- tie pattern for thicknesses between about 0.5 do and 5 do.
IV. Asymmetric walls. - The energies of the stray- field free Bloch wall models [6] have been calculated for the case of 180°-walls in Fe and Ni (Fig. 1). The self-consistency parameter S 1131 of these configura- tions is comparable with 1 in the range from D = 0 ( S = 0.95) to about D = 10 do ( S = 1.45). For larger thicknesses the transition from the two-dimensional wall t o the one-dimensional Bloch wall in bulk mate- rials must be better represented.
The walls called asymmetric NCel walls in [6] are particularly advantageous for lower wall angles and larger thicknesses. So the high thickness modes of the 900-, 710- and 109°-walls should be of this type. From figure 1 the transition is predicted for the following thicknesses : Fe 900 : D = 7.4 do, Ni 710 : D = 11.5 do, Ni 1090 : D = 5.7 do, and the range of validity of these models extends to about 20 do. The energies of the Bloch walls in bulk material, calculated as in [l41 for K, = 0, are also given in figure 1.
[ l ] UNANGST (D.), Ann. Physik, 1961, 7 , 280.
[2] SATO (H.), SHINOZAKI (S.) and TOTH (R. S.), Not- tinharn Conf. on Magnetism, Proceedings, 1964, 798.
[3] DE B ~ ( R . S W.), J. Vac. Sci. Techn., 1969, 3, 146, Reports AFCRL 67-0107, 68-0414.
[4] K A C Z ~ R (J.), Czech. J. Phys., 1957, 7 , 557.
[5] JONES (G. A.) and MIDDLETON (B. K.), Phil. Mug., 1970, 21, 803.
[6] HUBERT (A.), Phys. Stat. Sol., 1970, 38, 699 ; 1969, 32, 519.
Asymmetric 1800-Niel walls have higher energies than 1800-Bloch walls (Fig. 1). From the present calculations one would conclude that 1800-Niel walls can only exist as an (unfavourable) metastable form in iron and nickel. But the calculations are certainly not good for higher thicknesses, where Bloch walls and asymmetric NCel walls should finally converge into the bulk Bloch wall. It is therefore possible that the energy difference becomes in reality smaller then that shown in figure 1.
V. Domain patterns. - If the 180°-wall-energy is too high an ordinary 180°-wall domain pattern may be replaced by a checkerboard pattern as indicated in figure 3 (compare e. g. [l]). The checkerboard arran-
FIG. 3. - Checkerboard domain pattern as developing from an ordinary beam pattern.
gement is energetically advantageous if, apart from magnetostrictive energies, the following relations are fulfilled :
for Fe :
2 JTE9o < E180 for Ni :
2(JT ~ i o ,
+
E , , ) /J3
Eiso-
(2) Figure 2 shows that (2) is fulfilled for D < 7.7 do or 635A
in iron and for D < 6.5 do or 1 240A
in nickel.Such a critical thickness for the checkerboard domain pattern did not result from the calculations of Jones and Middleton [5]. The actual value of the critical thickness may be somewhat smaller than the given values, on account of magnetostriction [4] and correc- tions in the Bloch wall energy [7]. Since E,, is larger than the checkerboard energy, the checkerboard pattern will be preferred to a cross-tie pattern. Of course, there is no reason for 900-walls or other low angle walls t o decay into cross ties. All these features are in agreement with the mentioned experimental results.
rences
[7] LABONTE (A. E.), J. Appl. Phys., 1969, 40, 2450.
[8] FRAIT ( Z . ) , Phys. Stat. Sol., 1962, 2, 1417.
[9] GENGNAGEL ( H . ) and HOFFMANN (U.), Phys. Stat.
Sol., 1968, 29, 91.
[l01 AUBERT (G.), J. Appl. Phys., 1968, 39, 504.
[l11 HUBERT (A.), Comp. Phys. Comm., 1970, 1, 243-468.
[l21 HOLZ (A.) and HUBERT (A.), Z. angew. Phys., 1969, 26, 145.
[S31 AHARONI (A.), J . Appl. Phys., 1968, 39, 861.
[l41 SPAEEK (L.), Ann. Physik, 1960, 5 . 217.
