e- large film thickness – the domain period grows proportionally to d1/2 as it is predicted by the above simplified theory (see Eg. 1.3).
To summarize, in Fig.6 we plot the thickness dependencies of the zero-field equilibrium domain period calculated in the frameworks of the three models: i) the rigorous Kooy and Enz model which is valid in the full scale of the thickness changes; ii) Kaplan and Gehring model (Eq. 1.13) which validity is restricted by the ultrathin regime, d <lc and iii) a simple model (Eq. 1.3) which well describe p(d)-dependence for thicker films. As it seen from Fig. 1.6 in the ultrathin regime the domain periods could be significantly large. In general, one can expect ultrathin film in a practically monodomain state – large magnetic domains with geometry determined mainly by coercivity rather than magnetostatic forces. However, domain size drastically decreases down to a sub-micrometer scale while approaching the thickness at which the characteristic length falls practically to zero, e.g., because of decreasing the anisotropy constant.
The energy of a stripe domain structure is given by the Kooy and Enz model (1960):
where m=<M>/M S and h=H/4pMS is the normalized field applied perpendicular to the film plane. There are two parameters which may minimize the DS energy (Eq.A1): e.g., domain period (p) and the normalized magnetization (m). The equilibrium domain structure parameters can be found by minimization of energy (A1) with respect to m=M/MS and p, accordingly:
Solving Eqs. (A2) and (A3) one can calculate the equilibrium film magnetization, domain sizes and domain period as functions of the applied magnetic field. Equations A2 and A3 could be simplified if h=0 (the zero-field case). It follows from Eq.A2 that since h=0, the film magnetization m=0. By inserting the last into Eq. A3 one can arrive at
- C. Kittel Phys. Rev. 70 (1946) 965.
- C. Kooy and U. Enz, Phillps. Res. Repts. 15, 7 (1960).
- B. Kaplan and G.A. Gehring, J. Magn. Magn. Mater. 128, 111 (1993).
- A. Hubert and R. Schäfer, Magnetic Domains., Springer, Berlin (1998).