# 2.7.12 Magnetic field of periodic parallel domains

Let us calculate the external magnetic field generated by domains of periodic parallel structure with domains width
and
whose magnetization
is the same in magnitude but reverses orientation from domain to domain **(Fig. 1)**.

**Fig. 1. Schematic representation of magnetic domains periodic parallel structure.**

We will consider only the case of magnetization vector
orientation perpendicular to a sample surface as shown in **Fig. 1**. We also assume that the magnetic film thickness
is uniform over the sample and its length and width are infinite.

This problem was considered in [**1**] and the results obtained are given here without derivation. The magnetostatic energy of the structure shown in **Fig. 1** is written according to [**1**] as follows:

(1)

where

(2)

(3)

(4)

(5)

and – factor of the crystal energy anisotropy. External magnetic field created by this structure is given by

(6)

Then substituting (1) into (6) we can determine the desired magnetic field at any point in space over the sample.

In MFM, the second derivative of magnetic field with respect to coordinate along a tip vibration direction (see **chapter 2.7.1**) is usually measured. Therefore, to qualify the registered in MFM parameters, we have to analyze the qualitative relationship between
and the problem input parameters. **Fig. 2-4** show the variation of
along the
-axis calculated according to formulas (1)-(6) at given
,
,
and for various ratios between
and
.

**Fig. 2.
qualitative profile along the -axis
at height
. **

**Fig. 3.
qualitative profile along the -axis
at height . **

**Fig. 4.
qualitative profile along the -axis
at height . **

As seen in Fig. 2-4, the studied relations have period equal to
and are qualitatively different at various values of
. The qualitative dependence of on distance
at fixed point
(at the border between domains) under condition
,
is shown in **Fig. 5**.

**Fig. 5. qualitative plot along the -axis.**

In this case as in case of a cylindrical domain (see **chapter 2.7.11**), the magnetic field decays at distances determined by the magnetic structure dimensions.

Thus, if during the qualitative study of magnetic field distribution in the scanning mode it is revealed that magnetic domains have laminar structure **(Fig. 1)**, then the comparison of the phase change profiles along the scan line
with results shown in **Fig. 2-5** allows to estimate relation between domain sizes and magnetic film thickness. The quantitative analysis is available if the only set of parameters
,
and
is fitted so that calculation according to formulas (1-6) matches best with experimental profiles
.

## Summary.

- Determined is the magnetic field (1)-(6) generated by parallel domains of periodic (laminar) structure having width and whose magnetization is the same in magnitude but reverses orientation from domain to domain.
- If during the qualitative study of magnetic field distribution in the scanning mode it is revealed that magnetic domains have laminar structure, then the comparison of the phase change profiles along the scan line with results shown in Fig. 2-5 allows to estimate relation between domain sizes and magnetic film thickness.
- The quantitative analysis is available if the only set of parameters , and is fitted so that calculation according to formulas (1-6) matches best with experimental profiles .

## References.

- С. Kooy, U. Enz, Philips Res. Repts. 15, 7-29, (1960)