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

TL;DR

This paper uses micromagnetic simulations solving the Landau-Lifshitz-Gilbert equation at finite temperature to study how the Curie temperature of ferromagnetic nanostructures depends on their size, revealing a power-law relationship consistent with mean-field theory.

Contribution

The study introduces a temperature rescaling method in micromagnetic simulations that accurately predicts size-dependent Curie temperatures aligning with experimental data.

Findings

Curie temperature decreases with system size following a power-law.

Correlation length at zero temperature is in the nanometer range.

Critical exponent is approximately 2, consistent with mean-field theory.

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 $\Delta x$ ($T \to T\,\Delta x/a_{\text{eff}}$, 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: $(\xi_0/d)^\lambda$, where $\xi_0$ is the correlation length at zero temperature, and $\lambda$ is a critical exponent. We obtain values of $\xi_0$ in the nanometer range, also in accordance with other simulations and experiments. The computed critical exponent is close to $\lambda=2$ for all considered materials and geometries. This is the expected result for a mean-field approach, but slightly larger than the values observed experimentally.