# Artificial Magnetic Pattern Arrays Probed by Polarized Neutron Reflectivity

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## Abstract

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## 1. Introduction

## 2. Remarks on Neutron Specular Reflection and Off-Specular Scattering

#### 2.1. Specular Reflection

- A flat intensity plateau due to total reflection for wave-vectors ${Q}_{z}<{Q}_{c}^{\pm}$, where ${Q}_{z}=(4\pi /\lambda )\mathrm{sin}{\alpha}_{i,f}$ is the modulus of the scattering vector, ${\alpha}_{i,f}$ are the glancing angles with ${\alpha}_{i}={\alpha}_{f}$ for specular reflection, and ${Q}_{c}^{\pm}$ are the critical wave numbers for total reflection defined by Equation (7).
- If the sample is homogeneously magnetized or decomposed into a set of large magnetic domains then the critical edges, which separate the total reflection region from the rest of reflectivity, are distinctly different for up and down polarized neutrons. The difference ${\left({Q}_{c}^{+}\right)}^{2}-{\left({Q}_{c}^{-}\right)}^{2}$ being proportional to the saturation magnetization ${M}^{\mathrm{sat}}$ is independent of the angle $\gamma $ between magnetization direction and Y-axis. In contrast, the angle $\gamma $ determines the reflected intensity distribution between different NSF and SF reflection channels.
- In case of domains smaller than the coherence area the difference between ${\left({Q}_{c}^{+}\right)}^{2}$ and ${\left({Q}_{c}^{-}\right)}^{2}$ is proportional to the mean magnetization $\overline{M}$ averaged over small domains.
- For ${Q}_{z}>{Q}_{c}$, oscillatory Kiessig fringes occur, the period of which at ${Q}_{z}\gg {Q}_{c}$ is inversely proportional to the film thickness, while the reflected intensity drops according to $\sim {Q}_{z}^{-4}$.

#### 2.2. Coherence Volume

#### 2.3. Stripe Array with Parallel and Perpendicular Orientation

## 3. Experimental Study of a Magnetic Stripe Array

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the sample and beam geometry for polarized neutron reflectometry (PNR) experiments with one-dimensional polarization analysis along the Y-axis. The elongated ellipsoid indicates the coherence volume of neutrons defined by the beam divergence and the wavelength spread.

**Figure 2.**NSF, ${\mathcal{R}}^{++}$ (black solid line) and ${\mathcal{R}}^{--}$ (red solid line), and SF, ${\mathcal{R}}^{+-}={\mathcal{R}}^{-+}$ (blue lines), reflectivities calculated for a 100 nm homogeneously magnetized iron film on a silicon substrate are shown for the magnetization vector oriented: (

**a**) parallel to the axis Y, (

**b**) tilted by the angle $\gamma ={45}^{\circ}$ and (

**c**) for $\gamma ={90}^{\circ}$. The positions of the critical reflection edges ${Q}_{c}^{\pm}$ (independent of $\gamma $) are manifested by sharp drops of both NSF reflectivity curves. The position ${Q}_{c}^{-}$ in (

**b**,

**c**) is confirmed by the sharp maximum in SF reflectivities. Here Q is the modulus of the scattering vector defined by: $Q=(4\pi /\lambda )\phantom{\rule{3.33333pt}{0ex}}\mathrm{sin}\alpha $.

**Figure 3.**Calculated specular reflectivities ${\mathcal{R}}^{+}$ and ${\mathcal{R}}^{-}$ for a 100 nm thick iron film with reduced magnetization. (

**a**) ${R}^{+}$ (black solid line) and ${\mathcal{R}}^{-}$ (red solid line) for a mean magnetization $\overline{M}=0.5{M}^{\mathrm{sat}}$ parallel to the Y-axis ($\gamma ={0}^{\circ}$); (

**b**) similar to (

**a**) but with the mean magnetization tilted by the angle $\gamma =(\pm ){45}^{\circ}$; (

**c**) green line: similar to (

**a**) but with the mean magnetization tilted by the angle $\gamma =(\pm ){90}^{\circ}$; blue line: reflectivity for a mean magnetization $\overline{M}=0$.

**Figure 4.**Illustration of unpolarized neutron reflectivities calculated for iron film in saturation (black), on half way to saturation (red) and in totally demagnetized state (blue) via formation of a random set of small domains.

**Figure 5.**Mutual orientations of the coherence ellipsoid and the array of micro-stripes. (

**a**) The coherence area crosses several stripes and off-specular Bragg diffraction can be observed. (

**b**) The coherence area is collinear with the stripes, resulting solely in specular reflection, assuming flat edges and flat surfaces of the stripes.

**Figure 6.**Scanning electron microscopy image of the Py stripe array studied by PNR. The Py stripes appear in light gray, the spaces in between show a dark-gray shade.

**Figure 7.**(

**a**) NSF and SF specular reflectivities with least square fit from 800 Å thick permalloy stripe array with stripes oriented perpendicular to the scattering plane and in a field of ${H}_{ext}=5.2$ kOe applied along the stripes. (

**b**) Off-specular scattering maps collected simultaneously with the specular reflectivities using a wide-angle multichannel spin analyzer. (

**c**) Corresponding model calculations.

**Figure 8.**(

**a**) NSF and SF reflectivities for the stripes oriented parallel to the scattering plane in a field of ${H}_{ext}=5.2$ kOe applied across the stripes. (

**b**) Off-specular scattering maps and (

**c**) corresponding model calculations.

**Figure 9.**(

**a**) NSF and SF reflectivities for the stripes oriented parallel to the scattering plane as in Figure 8 but in a guiding field of ${H}_{ext}=5$ Oe after the sample was saturated in 5 kOe applied parallel to the stripes. Solid lines are fits to the experimental data. (

**b**) Off-specular scattering maps.

**Figure 10.**NSF and SF reflectivities plotted on a linear scale for the stripe array oriented parallel to the scattering plane and in different fields applied perpendicular to stripes. Symbols refer to experimental data; solid lines are best fits to the experimental data.

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**MDPI and ACS Style**

Gorkov, D.; Toperverg, B.P.; Zabel, H. Artificial Magnetic Pattern Arrays Probed by Polarized Neutron Reflectivity. *Nanomaterials* **2020**, *10*, 851.
https://doi.org/10.3390/nano10050851

**AMA Style**

Gorkov D, Toperverg BP, Zabel H. Artificial Magnetic Pattern Arrays Probed by Polarized Neutron Reflectivity. *Nanomaterials*. 2020; 10(5):851.
https://doi.org/10.3390/nano10050851

**Chicago/Turabian Style**

Gorkov, Dmitry, Boris P. Toperverg, and Hartmut Zabel. 2020. "Artificial Magnetic Pattern Arrays Probed by Polarized Neutron Reflectivity" *Nanomaterials* 10, no. 5: 851.
https://doi.org/10.3390/nano10050851