n period reaches its minimum at the certain film thickness which real meaning depends on the characteristic length of material. In the 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). The Fig. 1.6 allows us compare the results obtained by rigorous calculations (Eq. 1.12) and the simple model (Eq. 1.3).
The initial magnetic susceptibility is given by χ = dM/dH =(1/4 π ) dm/dh and can be found by differentiating of Eq.(A1) with respect to h
Thus, Eq. 1.13 gives the initial susceptibility of stripe domain structure as a function of domain period (see Fig. 1.7). Passing in Eq.(1.13) to the limit p0→0 (which is a case of smallest domain wall energy and a dense domain structure, e.g. because of low anisotropy constant) one can arrive at χ0=1/4 π. The same result was previously obtained from Eq. 1.6 on the basis a simplified model. In the opposite limit, when p→∞ (or p>>d) Eg. 1.13 gives that χ→∞. Indeed, as it seen from Fig. 1.3 the susceptibility becomes infinity large before the saturation (the jumps on the m(H)-curves). Obviously, after saturation χ=0. Notice, the right side of Eq. 1.13 only includes geometrical parameters of the film and domain structure: d and p. In general, the magnetic susceptibility is affected by many factors, e.g. coercivity. The last may be a reason for a difference between values of χ measured from the virgin magnetization curve (or magnetization hysteresis loop) and determined from a DS image. Calculations of χ from analysis of domain structure images is more easy with the help of another option our experiment – registration of magnetic hysteresis loop .
One can use these figures to determine the characteristic length and the domain wall energy of a given film. Indeed, using these plots (Figs. 1.3 -1.6) and utilizing the measured domain period and domain sizes at different values of the magnetic filed it easy to determine with the help of the above given theory the material parameters: MS 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 cas