# Finite Length Effects on Switching Mechanisms in Chains of Magnetic Particles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Models

#### 2.1. Different Shapes with the Same Shape Anisotropy

_{80}Fe

_{20}material: saturation magnetization ${M}_{s}=890\mathrm{kA}/\mathrm{m}$ and exchange stiffness parameter $A=1.3\times {10}^{-11}\mathrm{J}/\mathrm{m}$ [38]. The material being magnetically isotropic (no magnetocrystalline anisotropy [39,40,41]) the only source of anisotropy here is the shape anisotropy. Consequently, the ground state of a single isolated macrospin corresponds to the magnetization being preferentially oriented along the longest axis of the shape. When arranged into the chains the macrospins exhibit the ground state in which the magnetizations of the neighboring particles are opposite. This configuration is legitimately called antiferromagnetic (AF) and is represented in Figure 1a,b by the green (up) and blue (down) colors. A magnetic field is assumed to be applied to the chains in the y-direction, i.e., parallel to the long axes of the shapes. The appropriate equilibrium configuration at every given field intensity is found by minimization of the magnetic energy with the use of the software MuMax3 [42,43]. It includes the isotropic exchange interactions between all the 5 nm × 5 nm × 5 nm voxels within every particle and the dipolar interactions between all the voxels of the system.

#### 2.2. Point-Dipole Reduced Model

## 3. Results

#### 3.1. Infinite Systems

#### 3.2. Finite Chains

#### 3.2.1. System of Seven Macrospins

#### 3.2.2. System of Eight Macrospins

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the models used: (

**a**) stripes, (

**b**) ellipsoids and (

**c**) S-W discrete approximation. Green and blue colors in (

**a**,

**b**) correspond to the magnetization upwards and downwards, respectively.

**Figure 2.**Magnetization sequence under the applied magnetic field in a S-W model of (

**a**) stripes, (

**b**) ellipsoids. The starting configuration for B = 0 is antiferromagnetic. Panel (

**c**) represents the stages of magnetization switching in the case of the stripes marked with the letters A, B, C, D.

**Figure 3.**Magnetization sequence under the applied magnetic field in infinite chains of (

**a**) stripes, (

**b**) ellipsoids treated with micromagnetic simulations. Panels (

**c**,

**d**) illustrate magnetization maps for stripes and ellipsoids, respectively.

**Figure 4.**Energies of the subsequent configurations of the models discussed in this section, upper panel: stripes, lower panel: ellipsoids. The leftmost (

**a**,

**d**) S-W model, the rightmost (

**c**,

**f**) micromagnetic calculations, the central parts (

**b**,

**e**) comparison of both.

**Figure 5.**Magnetization sequence under the applied magnetic field in a S-W model of (

**a**) 7 stripes, (

**b**) 7 ellipsoids. The starting configuration for H = 0 is antiferromagnetic. Panel (

**c**) represents the stages of magnetization switching in the case of the stripes marked with the letters A, B, C, D, E and F. Mind that configurations A and F are two variants of AF configuration with the net magnetization upwards (AF-u) and downwards (AF-d), respectively.

**Figure 6.**(

**a**) Hysteresis loop for the system of seven stripes. The configuration sequence when starting from (

**b**) AFu and (

**c**) AFd. (

**d**) the sequence in the reversal of the FM configuration.

**Figure 7.**(

**a**) Hysteresis loop for the system of seven ellipsoids. The configuration sequence when starting from (

**b**) AFu and (

**c**) AFd. (

**d**) the sequence in the reversal of the FM configuration.

**Figure 8.**Energies of the subsequent configurations of the models of seven stripes and ellipsoids discussed in this section, upper panel: stripes, lower panel: ellipsoids. The leftmost (

**a**,

**d**) S-W model, the rightmost (

**c**,

**f**) micromagnetic calculations, the central parts (

**b**,

**e**) comparison of both.

**Figure 9.**Magnetization sequence under the applied magnetic field in a S-W model of (

**a**) eight stripes, (

**b**) eight ellipsoids. The starting configuration for H = 0 is antiferromagnetic. Panel (

**c**) represents the stages of magnetization switching in the case of the stripes marked with the letters A, B, C, D, E and F.

**Figure 10.**(

**a**) Hysteresis loop for the system of eight stripes. (

**b**) The configuration sequence when starting from AF and (

**c**) the configuration sequence when starting from FM. The AF configurations are equivalent, no distinction AFu and AFd.

**Figure 11.**(

**a**) Hysteresis loop for the system of eight ellipsoids. The configuration sequence when starting (

**b**) from AF and (

**c**) from FM configuration.

**Figure 12.**Energies of the subsequent configurations of the models of 8 stripes and ellipsoids discussed in this section, upper panel: stripes, lower panel: ellipsoids. The leftmost (

**a**,

**d**) S-W model, the rightmost (

**c**,

**f**) micromagnetic calculations, the central parts (

**b**,

**e**) comparison of both.

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**MDPI and ACS Style**

Kuźma, D.; Zieliński, P.
Finite Length Effects on Switching Mechanisms in Chains of Magnetic Particles. *Magnetochemistry* **2020**, *6*, 47.
https://doi.org/10.3390/magnetochemistry6040047

**AMA Style**

Kuźma D, Zieliński P.
Finite Length Effects on Switching Mechanisms in Chains of Magnetic Particles. *Magnetochemistry*. 2020; 6(4):47.
https://doi.org/10.3390/magnetochemistry6040047

**Chicago/Turabian Style**

Kuźma, Dominika, and Piotr Zieliński.
2020. "Finite Length Effects on Switching Mechanisms in Chains of Magnetic Particles" *Magnetochemistry* 6, no. 4: 47.
https://doi.org/10.3390/magnetochemistry6040047