Application to magnetic domain patterns

In the last chapter it has been demonstrated that the CDC can be extracted from the Fourier transform of a magnetic speckle pattern (see Eq. 4.3 and Eq. 4.4). For this, it is important to know the characteristics of the Patterson function of a magnetic domain pattern, similar to the case of the URA concept. The autocorrelation function of the pinhole array in the URA concept is well known, although the pinholes are pseudo-randomly distributed⁵. However, the exact autocorrelation function of a magnetic domain pattern is not easily accessible, especially in case of a disordered maze domain pattern with a large amount of domain size variation. Real-space images obtained from high-resolution magnetic imaging techniques have to be recorded at the exact position of the illuminating beam together with an appropriate size to get a sufficient domain pattern to calculate the exact Patterson function. This would be impractical and hardly achievable.

1D Patterson function

The general properties of the Patterson functionP_m(∆r) from magnetic domain patternsmz(r) can be studied by modeling one-dimensional magnetic domain patterns with different domain size distributions. For this, an alternating sequence of m_z =−1

5URA patterns can be calculated using an algorithm described by [177] and manufactured with optical lithography techniques. The autocorrelation function as well as the URA coherent diffraction pattern can be easily simulated.

-3 -2 -1 0 1 2 3 0

0.2 0.4 0.6 0.8 1

x [mm]

Normalized |P(x)|

 = 0.1 nm

 = 0.3 nm

 = 0.5 nm

 = 1.3 nm

 = 11.2 nm

 = 50 nm

a) b)

Pmean = 0.003

Figure 4.2: a) Normalized dimensional Patterson function of a synthetic one-dimensional magnetic domain pattern with gamma-distributed domain sizes for different values of the standard deviationσ of the distribution function. The average domain size has been set to 100 nm. The graphs have been smoothed with a kernel of 3µm to suppress the strong high-frequency oscillations. The side lobe intensity varies from 0.38 to 0.003. b) Central part of the normalized one-dimensional Patterson function using a standard deviationσ = 50 nm and an average domain size of 100 nm.

It shows that the dominant central part has a width of around 1µm. It should be noted that the graph has an averaged side lobe intensity of 0.003 and is therefore non-zero.

and mz = 1 values, representingMz/MS, is modeled to represent magnetic domains with up- and down-magnetization (see Fig. 5.7). A distribution function for the domain sizes is implemented to incorporate different domain size variations within the domain pattern. It is found that a gamma distribution can be used to describe magnetic maze domain patterns with significant domain size variations. It can also account for highly-ordered stripe domain patterns with vanishing domain size variation. For a detailed description and analysis of the used gamma distribution for the domain sizes, it is referred to section 5.3.2. The Patterson function can be obtained from the modeled one-dimensional domain pattern by using Eq. 4.9. Fig.

4.2 (a) shows the modulus of the one-dimensional Patterson function for different standard deviations σ of the gamma distribution, i.e., for different values of domain size variation within the domain pattern, using an average domain size ofD= 100 nm.

The graphs have been smoothed with a kernel of 3µm width to suppress the strong high-frequency oscillations. Due to that the graphs display the general shape and

characteristics of the modulus of the Patterson function. It can be seen from Fig.

4.2 (a) that the Patterson function has a broad and triangular-shaped structure at σ = 0.1 nm, which corresponds to an almost periodic domain pattern with almost a single domain size. With increasing amount of domain size variation, the width of the central peak decreases until it can be described by a single narrow peak (σ= 50 nm).

Moreover, with increasingσ, the side lobes develop into flat planes. Figure 4.2 (b) shows the modulus of the Patterson function obtained from a domain size distribution with σ= 50 nm and an average domain size of D= 100 nm without smoothing the graph. It shows that the peak structure at the center is restricted to a total range of 1 µm. The side lobes are flat and show only slight fluctuations with an average value of 0.003. Similar results have been found by Asakura et al. [149] describing the Patterson function of a diffuse plate as a function of mean-square phase variations, i.e., surface-height variations and by Nugent et al. [84], describing the Patterson function of an NRA aperture.

2D Patterson function

The analysis of the Patterson function of a magnetic domain pattern can also be performed in two dimensions. For this, an MFM image of a magnetic maze domain pattern of a Co/Pt multilayer film with an average domain size of aroundD= 150 nm is utilized (Fig. 4.3 (a)). The modulus square of the Fourier transformed MFM image results in a synthetic two-dimensional diffraction pattern (Fig. 4.3 (b)). A subsequent Fourier transform of the diffraction pattern yields the two-dimensional Patterson function of the maze domain pattern (Fig. 4.3 (c)). Fig. 4.3 (c) shows that the two-dimensional Patterson function consists of a high-intensity peak structure in the center and slight intensity fluctuations on a constant side lobe in the remaining regions.

An averaged one-dimensional profile of the two-dimensional Patterson function can be obtained via azimuthal averaging around the center (Fig. 4.3 (d)). Figure 4.3 shows that the azimuthally averaged Patterson function has the same signature as in the one-dimensional case, when a large variation of domain sizes within the domain pattern (maze pattern) is present (see Fig. 4.2 (b)). The peak structure in the center is also restricted to a total range of 1 µm. Furthermore, the side lobe is flat and shows slight fluctuations with an average value of 0.08.

From the one- and two-dimensional analysis of the Patterson function from magnetic domain patterns, it follows that for a maze-like domain pattern with a large variation of domain sizes, the Patterson function can be decomposed into a distinct

x (µm)

y (µm)

-5 -2.5 0 2.5 5

-5

-2.5

0

2.5

5

Intensity

0 2 4 6 8 10 12

a) b)

c) d)

Δx (µm)

Δy (µm)

Δr_avg (µm)

I_mean = 0.08

Figure 4.3: (a) Magnetic force microscope (MFM) image of a magnetic maze domain pattern with an average domain size of around 150 nm (courtesy of D. Stickler and J. Mohanty). (b) Central area of the modulus square of the Fourier transformed maze pattern showing a calculated donut-like diffraction pattern. (c) Modulus of the Fourier transformed diffraction pattern showing the Patterson map of the maze pattern (logarithmic scale). (d) Plot of the azimuthally averaged Patterson map (red solid line) yielding the high non-constant contribution of the Patterson function at the center position and a perfectly flat side lobe (black solid line).

narrow central peak (≈1µm width) and perfectly flat side lobes.

Is has been shown from the Fourier analysis method (section 4.1) that the Fourier transform of a magnetic speckle pattern can be expressed, within the Gaussian Schell-model, by the product of the modulus of the complex degree of coherence, the autocorrelation function of the beam intensity distribution and the Patterson function. In the context of the statistically stationary model, the latter is reduced to

the product of the complex degree of coherence and the Patterson function. It has been shown that the Patterson function of a magnetic maze domain pattern with large variation of domain sizes is a constant, except in the vicinity of the central region. It follows, that the complex degree of coherence and hence the transverse coherence length (see section 2.1) can be extracted from the Fourier transformed magnetic speckle pattern without the knowledge of the exact shape of the Patterson function, as it is only a constant multiplicative factor in Eq. 4.3 and Eq. 4.4, except for the vicinity of the central region. Due to the fact that the high-intensity fringe-like structure in the central region only extends over a small distance it can be disregarded for the analysis.

In the following, the determination of the spatial coherence of X-ray radiation, i.e., the transverse coherence length, will be described in detail using the Fourier analysis method for two different XRMS experiments performed at the P04 beamline at PETRA III.