Domain size in systems with canted magnetization

The equilibrium domain size in ferromagnetic single-layer films with canted magnetization, as observed in the spin-reorientation transition, is analytically described. It is demonstrated that for canted magnetization in the case of narrow domain walls the size of the magnetic domains depends solely on the second-order anisotropy constant and not on the first-order anisotropy. The single-layer model is extended to the application to multilayers. An approximation for the domain size of multilayers is developed and used for the analysis of data obtained via soft x-ray holography. Our analysis, including the canting angle, allows for the determination of the relative anisotropy constants to second order.

Figure 1

Kerr hysteresis loops of an 8$\times$(Co${}_{0.7\mathrm{nm}}$/Pt${}_{1.5\mathrm{nm}}$) multilayer film. In (a) the Kerr rotation $\theta$ is given, obtained by sweeping a perpendicularly oriented field. The curve shows an easy-axis loop with a decay of the film into a multidomain state with almost vanishing remanence. In (b) the magnetization behavior (ellipticity $\varepsilon$) obtained in an in-plane magnetic field is given. The hysteresis exhibits a nonvanishing remanence, which is indicative of a canted magnetization (Ref. 30). The in-plane hysteresis is fitted taking into account the first- and second-order anisotropy constants according to Eq. (1) [solid line in (b)].

Figure 2

SEM micrograph (tilted view) of the holography sample. In the 1-$\mu$m-thick gold film that is deposited onto the SiN membrane an object hole with a diameter of 2 $\mu$m has been milled. Five narrow reference holes are milled around the center of the object hole at a radius of 4.5 $\mu$m.

Figure 3

SEM micrograph and domain image of the same region of the sample. The four different areas in which different ion doses are applied are indicated by (A)–(D). In the different areas the following doses have been applied: (A) $=$ 0, (B) $=$ 2.5 $\times$, (C) $=$ 25 $\times$, and (D) $=$ 125 $\times$ 10${}^{15}$ Ga${}^{+}$/cm${}^2$. The size of magnetic domains changes with ion dose. In (D) the material has become either paramagnetic or the magnetization is lying in the film plane. A weak contrast in the left image reveals the regions of different dose application.

Figure 4

Histograms of the magnetic contrast extracted from the three areas exhibiting a magnetic signal [(A)–(C), see Fig. 3]. In (A) and (B) the contrast is well separated into peaks for black and white domains. In (C) the separation has become smaller while the contribution due the domain walls between the two peaks is increased.

Figure 5

Calculated domain size as a function of $K_2$ (inset as a function of domain wall energy per unit area). The open dots show the minima of the sum of domain wall and stray field energy. The stray field energy was calculated using the formula for a stripe pattern given in Ref. 71. The domain size distribution of Eq. (12) has been fitted to the calculated values (red solid line). For comparison the size distribution for checkerboard and maze pattern are also plotted using appropriate assumptions for the geometrical factors (see text).

Figure 6

Phase diagram including the experimental results. The point that represents the nonirradiated part (A) is obtained from film measurements via MOKE. The other two points are extracted from the domain pattern utilizing the approximation of the model for multilayers. To prevent any further uncertainty, the relative changes to the film values have been determined (see text).