# Understanding Magnetization Dynamics of a Magnetic Nanoparticle with a Disordered Shell Using Micromagnetic Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Method

## 3. Results and Discussion

#### 3.1. The Computational Time Step, FixDt

#### 3.2. Run Time per Field Step and the Damping Parameter

#### 3.3. Thickness of the Disordered Shell and Thermal Field Effect

#### 3.4. Effect of Anisotropy

#### 3.4.1. Cubic Anisotropy

#### 3.4.2. Uniaxial Anisotropy

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basic geometry of a magnetic NP used in the micromagnetic simulations showing the real “core-shell” spin structure. The sketch of the coordinate system serves as a guideline for the external magnetic field and anisotropy axis directions used in the simulations.

**Figure 2.**Experimental magnetization isotherms at different temperatures for a real sample of ${\mathrm{CoFe}}_{2}{\mathrm{O}}_{4}$ NPs, as measured on SQUID magnetometer (

**left**). A simulated internal spin structure for the model showing both the ordered core and the disordered shell is also shown (

**right**).

**Figure 3.**Coercivity (${\mathbf{H}}_{\mathbf{c}}$) variation with the computational time step, FixDt, at 0 K. Smaller FixDt values correspond to a higher numerical precision, but at the cost of significantly longer simulation times. A FixDt of $2\times {10}^{-15}$ s was chosen as a good compromise between the precision and simulation times.

**Figure 4.**Relation between coercivity (${\mathbf{H}}_{\mathbf{c}}$), damping parameter $\alpha $, and run time (runt). The inset shows how the dissipative term $\mathbf{M}\times \mathbf{d}\mathbf{M}/\mathbf{dt}$, driven by the phenomenological constant $\alpha $, forces the magnetization to precess until the magnetization aligns itself with the effective field (${\mathbf{H}}_{\mathbf{eff}}$) direction. In general, as $\alpha $ is increased the coercivity is lowered until a minimum is reached. Also the longer the runt, the longer the magnetization dynamics has to relax to equilibrium at each effective field value, reducing the coercivity as well. These simulations were performed at 0 K.

**Figure 5.**Coercivity changes due to both different simulated temperatures and different disordered shell thicknesses, of 0.5 nm (

**a**), 2 nm (

**b**) and 3 nm (

**c**), whereas (

**d**) directly compares the temperature effects for the different shell thicknesses. In general the thicker the shell, the larger the coercivity. As the temperature increases the thermal field effects become more important than the thickness ones, for the overall dynamics of the NP switching.

**Figure 6.**(

**a**) Magnetization isotherm reversal for the 0.5 nm shell at 2 K, with corresponding insets showing the spin structure during the transition process. (

**b**) Magnetic isotherm reversal for the 0.5 nm shell at 300 K, with corresponding insets showing the spin structure during the transition process. One can easily see on the insets how the individual spins become quite “disorganized” at higher temperatures.

**Figure 7.**Comparison between cubic and uniaxial anisotropies. (

**a**) Using first and second order cubic anisotropy constants, ${K}_{c1}$ and ${K}_{c2}$ respectively, where the easy anisotropy axis and ${\mathbf{H}}_{\mathbf{ext}}$ are applied in the same $\mathbf{v}$(1,0,0) direction. In both 0 K and 300 K there is no real benefit in using the second order term, ${K}_{c2}$. (

**b**) For the constants values ${K}_{c1}={K}_{uni}$, the simulation when just using the cubic anisotropy presents a coercive field ≈0.55 T lower than when just using the uniaxial one, being the ${\mathbf{H}}_{\mathbf{ext}}$ applied along the same (1,0,0) anisotropy easy direction. The coercivity gets further reduced with a higher magnitude anisotropy constant. (

**c**) When the external field, ${\mathbf{H}}_{\mathbf{ext}}$, is applied along the (0,1,0) direction (which is perpendicular to the anisotropy one (1,0,0)), the coercive field for the cubic anisotropy is now ≈0.70 T higher than the uniaxial anisotropy one. This trend is seen both at 0 K and 300 K.

**Figure 8.**(

**a**) Magnetic isotherm when applying the external magnetic field (${\mathbf{H}}_{\mathbf{ext}}$) parallel to the anisotropy axis (${\mathbf{H}}_{\mathbf{k}}$ along the x-axis) of the NP (of core size 5.5 nm and 0.5 nm shell thickness). (

**b**) Plot of the coercivity (${\mathbf{H}}_{\mathbf{c}}$) versus temperature, showing how it rapidly decreases due to the presence of the thermal field.

**Figure 9.**Magnetic isotherms when applying the external magnetic field (${\mathbf{H}}_{\mathbf{ext}}$) perpendicularly to the ordered core anisotropy direction (${\mathbf{H}}_{\mathbf{k}}$ along the x-axis) of the NP (of core size 5.5 nm and 0.5 nm shell thickness). (

**a**) When the field is applied along the y-axis and (

**b**) when the field is applied along the z-axis. The coercivity rapidly tends to zero as soon as the thermal field is added, when the external magnetic field is applied perpendicularly to the anisotropy easy axis.

**Figure 10.**Effect of different anisotropy magnitudes and directions on the NPs core and shell. Anisotropy easy axis directions were defined as; (1,0,0) x-axis for the core; and (0,1,0) y-axis for the shell. As one can see, as the anisotropy in the shell is increased the coercivity of the entire NP decreases, pointing to the importance of the anisotropy direction and magnitude, in core-shell structures, to the overall magnetization dynamics of the NP.

**Table 1.**Basic magnetic parameters of the prototype magnetic NP, determined from the ZFC-FC curves and magnetization isotherms. Coercive field at 10 K, ${\mathbf{H}}_{\mathbf{c}}^{\mathbf{10}}$; anisotropy field at 10 K ${\mathbf{H}}_{\mathbf{k}}^{\mathbf{10}}$; saturation magnetization at 10 K and 300 K, ${\mathbf{M}}_{\mathbf{S}}^{\mathbf{10}}$, ${\mathbf{M}}_{\mathbf{S}}^{\mathbf{300}}$, remnant magnetization at 10 K, ${\mathbf{M}}_{\mathbf{r}}^{\mathbf{10}}$; effective anisotropy constant ${\mathbf{K}}_{\mathbf{eff}}$.

${\mathbf{H}}_{\mathbf{c}}^{10}\left(\mathbf{T}\right)$ | ${\mathbf{H}}_{\mathbf{k}}^{10}\left(\mathbf{T}\right)$ | ${\mathbf{M}}_{\mathbf{S}}^{10}({\mathbf{Am}}^{2}/\mathbf{kg})$ | ${\mathbf{M}}_{\mathbf{S}}^{300}({\mathbf{Am}}^{2}/\mathbf{kg})$ | ${\mathbf{M}}_{\mathbf{r}}^{10}({\mathbf{Am}}^{2}/\mathbf{kg})$ | ${\mathbf{K}}_{\mathbf{eff}}(\mathbf{J}/{\mathbf{m}}^{3})$ |
---|---|---|---|---|---|

$1.28$ | $4.2$ | 97 | 73 | 53 | $8.9\times {10}^{4}$ |

**Table 2.**Simulation parameters used on mumax3 for the core-shell NP with uniaxial anisotropy. ${M}_{S}$ is the saturation magnetization, ${A}_{ex}$ is the exchange energy, ${K}_{u}$ is the anisotropy constant with its direction vector, $\mathrm{v}$ and the damping parameter $\alpha $.

${\mathit{M}}_{\mathit{S}}$ (A/m) | ${\mathit{A}}_{\mathbf{ex}}$ (J/m) | ${\mathit{K}}_{\mathit{u}}$ (J/m${}^{3}$), $\mathbf{v}$(i,j,k) | $\mathit{\alpha}$ | |
---|---|---|---|---|

Core | $7.74\times {10}^{5}$ | $1.50\times {10}^{-11}$ | $8.9\times {10}^{4};\phantom{\rule{2.84526pt}{0ex}}$v$(1,0,0)$ | $0.01$ |

Shell | $6.19\times {10}^{5}$ | $1.20\times {10}^{-11}$ | 0 | $0.005$ |

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**MDPI and ACS Style**

Aurélio, D.; Vejpravova, J.
Understanding Magnetization Dynamics of a Magnetic Nanoparticle with a Disordered Shell Using Micromagnetic Simulations. *Nanomaterials* **2020**, *10*, 1149.
https://doi.org/10.3390/nano10061149

**AMA Style**

Aurélio D, Vejpravova J.
Understanding Magnetization Dynamics of a Magnetic Nanoparticle with a Disordered Shell Using Micromagnetic Simulations. *Nanomaterials*. 2020; 10(6):1149.
https://doi.org/10.3390/nano10061149

**Chicago/Turabian Style**

Aurélio, David, and Jana Vejpravova.
2020. "Understanding Magnetization Dynamics of a Magnetic Nanoparticle with a Disordered Shell Using Micromagnetic Simulations" *Nanomaterials* 10, no. 6: 1149.
https://doi.org/10.3390/nano10061149