and χ
|Fig. 1.3 Film magnetization (m) as a function of the normalized field (h=H /4 π MS) calculated from Eqs. A2 and A3 for different values of lc/d. Note, the curve representing the case lc/d=0 is also well described by the simplified model, see Eq. ~1.6.|
|Fig. 1.4 Normalized domain period (p/d) as a function of the normalized field (h) for lc/d=1/12.|
|Fig 1.5 Function F(p/d) used in Eq. 1.12.|
|Fig. 1.6. Normalized domain period (p/lc) as a function of the normalized film thickness (d/lc) for H=0. The red curve – is calculation with the Kaplan-Gehring model (Eq. 1.14), the blue one – Kooy-Enz theory (Eqs. A2 and A3), and the magenta curve is the simple dependence given by Eq. 1.3.|
|Fig. 1.7. Initial magnetic susceptibility as a function of the inverse domain period, z=2 π d/p. The red line shows χ=1/4 π.|
Domain structures of ultrathin films.
The physics of domains in bulk materials and micron-sized films seem to be well understood. But domain patterns of ultrathin films exhibit many new features: unusually sharp thickness dependence of domain sizes and strong reconstruction of magnetization distribution in domains related to changes of magnetic anisotropy which are of great interest and essential for an understanding of fundamental physics. Now we apply the Koy-Enz theory for analysis of the thickness dependence of the domain period in a wide scale of thickness changes at H=0. Solving numerically Eqs.A5 and A6 we find the equilibrium domain period as a function of the film thickness, see Fig.6. In the ultrahin regime when d < lex (lex=(A/(2 π MS2))1/2, where is the exchange length and A is the exchange constant) the domain period drastically increases as the film thickness decreases. This is the case of ultrathin magnetic films (d is only a few monoatomic layers) and now such films are object of intensive studies. In the ultrathin limit the thickness dependence of the zero-field DS period can be well approximated by a simple formula (Kaplan and Gehring, 1993):
where b= -0.666. In the opposite limiting case- 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
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- B. Kaplan and G.A. Gehring, J. Magn. Magn. Mater. 128, 111 (1993).
- A. Hubert and R. Schäfer, Ma