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.

翻译：我们在有限温度范围内求解Landau-Lifshitz-Gilbert方程，其中热涨落通过随机磁场建模，该磁场的方差与温度成正比。通过将温度与计算单元尺寸$\Delta x$成比例地重新标度（$T \to T\,\Delta x/a_{\text{eff}}$，其中$a_{\text{eff}}$为晶格常数）[M. B. Hahn, J. Phys. Comm., 3:075009, 2019]，我们获得的居里温度$T_{\text{C}}$与钴、铁和镍的实验值一致。对于纳米线（一维）和纳米层（二维）等有限尺寸物体，居里温度随系统最小尺寸$d$的变化而变化。我们证明计算所得的有限尺寸$T_{\text{C}}$与体材料$T_{\text{C}}$之间的差值遵循幂律关系：$(\xi_0/d)^\lambda$，其中$\xi_0$为零温度下的相关长度，$\lambda$为临界指数。我们获得的$\xi_0$值处于纳米量级，与其他模拟和实验结果一致。对于所有考虑的材料和几何结构，计算所得的临界指数均接近$\lambda=2$。这是平均场方法的预期结果，但略大于实验观测值。