Micromagnetic simulations of the size dependence of the Curie temperature in ferromagnetic nanowires and nanolayers

TL;DR

This paper presents a finite-temperature micromagnetic framework based on the stochastic Landau-Lifshitz-Gilbert equation, incorporating Hahn's temperature scaling to remove artificial cell-size dependence. It demonstrates that 1D nanowires and 2D nanolayers exhibit a power-law finite-size shift of the Curie temperature, TC(d) = TC(∞) − TC(∞)(ξ0/d)^λ, with correlation lengths in the nanometer range and a mean-field-like exponent λ ≈ 2. The method reproduces Bloch's law at low T and Curie's law near TC, and shows TC increasing with system size while fluctuations peak near TC, indicating critical slowing down in finite systems. The approach provides a computationally efficient route to study size-dependent magnetic phase transitions and dynamical effects in nanoscale ferromagnets, aligning well with experimental and other simulation results.

Abstract

We solve the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. By rescaling the temperature proportionally to the computational cell size $Δx$ ($T \to T\,Δx/a_{\text{eff}}$, where $a_{\text{eff}}$ is the lattice constant) [M. B. Hahn, J. Phys. Comm., 3:075009, 2019], we obtain Curie temperatures $T_{\text{C}}$ that are in line with the experimental values for cobalt, iron and nickel. For finite-sized objects such as nanowires (1D) and nanolayers (2D), the Curie temperature varies with the smallest size $d$ of the system. We show that the difference between the computed finite-size $T_{\text{C}}$ and the bulk $T_{\text{C}}$ follows a power-law of the type: $(ξ_0/d)^λ$, where $ξ_0$ is the correlation length at zero temperature, and $λ$ is a critical exponent. We obtain values of $ξ_0$ in the nanometer range, also in accordance with other simulations and experiments. The computed critical exponent is close to $λ=2$ for all considered materials and geometries. This is the expected result for a mean-field approach, but slightly larger than the values observed experimentally.

Figures (13)

• Figure 1: Illustration of the two generic geometries corresponding to a 1D nanowire (left) and a 2D nanolayer (right)
• Figure 2: Spatially averaged $x$ component of the magnetic moment $\overline{m}_1(t)$ for one statistical realization, as a function of time $t$ for a ferromagnetic nanowire. Colors correspond to different temperatures, as indicated on the figure. Top panels: cobalt; middle panels: iron; bottom panels: nickel. The left column corresponds to nanowires with dimensions (in nm): $1680\times 11\times 11$, the right column to nanowires with dimensions (in nm): $120 \times 41\times 41$.
• Figure 3: Total magnetization $M_{\text{tot}}$, from Eq. \ref{['magnet_totale_2']}, as a function of the temperature, for cubic cobalt (left panel), iron (right panel) and nickel (middle panel) nano-objects with dimensions $\rm 50~nm\times 50~nm\times 50~nm$. The different symbols and colors stand for different computational cell sizes $\Delta x$, going from 1 nm to 5 nm. The black vertical dash-dotted lines represent the bulk Curie temperatures as given in Table \ref{['table_constantes_physiques']}.
• Figure 4: Total magnetization $M_{\text{tot}}$, from Eq. \ref{['magnet_totale_2']}, as a function of temperature, for a cobalt (left), iron (right) or nickel (middle) nanowire with dimensions 50 nm$\times$50 nm$\times$50 nm. The different symbols stand for different time steps $\Delta t =2.5~\rm fs$ (green crosses), 5 fs (red circles), and 10 fs (blue crosses). The computational grid size is $\Delta x=1$ nm. The black vertical dash-dotted lines represent the bulk Curie temperatures as given in Table \ref{['table_constantes_physiques']}.
• Figure 5: Total magnetization $M_{\text{tot}}$\ref{['magnet_totale_2']} with respect to temperature with numerical parameters detailed in Table \ref{['table_numerical_parameters']}. The Curie temperature corresponds to the first temperature at which magnetization falls to zero. Simulation results are represented by dots, the solid curves are an interpolation based on cubic splines.