Fokker-Planck approach for thermal fluctuations in antiferromagnetic systems

TL;DR

This work develops a Fokker-Planck framework for thermal fluctuations in a two-dimensional antiferromagnet with uniaxial anisotropy by starting from a stochastic LLG equation with Langevin noise and deriving the FP equation for the spin PDF $P(\mathcal{S},t)$. A mean-field closure yields tractable equations for the spin polarization and two-time spin-spin correlators, enabling analysis of spin-wave dynamics and a phenomenological model for resistance fluctuations in MPX$_3$ materials. The results show thermal fluctuations renormalize spin-wave energies and damping and produce Lorentzian resistance noise with a peak near the Néel temperature, qualitatively matching recent experiments on 2D AFMs. Overall, the framework links spin dynamics, thermal noise, and transport fluctuations in 2D AFMs and can be extended to driven or noncollinear magnetic states with potential spintronic applications.

Abstract

We develop a Fokker-Planck approach to describe the dynamics of staggered magnetization and thermal fluctuations in a two-dimensional antiferromagnetic system with uniaxial anisotropy. Beginning with a classical model for the antiferromagnetic system, we incorporate a Landau-Lifshitz-Gilbert equation augmented by Langevin fields to account for thermal fluctuations, and we derive the Fokker-Planck equation governing the probability distribution function of the spin configuration. Employing the mean-field approximation, we derive the equations of motion for the spin polarization and the two-time spin-spin correlation functions. The methodology is applied to the study of spin-wave dynamics and to the formulation of a phenomenological model for resistance fluctuations in two-dimensional antiferromagnetic semiconductors.

Figures (4)

• Figure 1: Side view sketch of the 2D antiferromagnetically-ordered systems MPX3. Vertical electronic transport (see the large gray arrow on the right side of the schematics) occurs principally as a result of hopping between atomic groups (purple structures) located outside the plane. The orange and blue dots represent the AFM planar system, which consists of two sublattices, each with a spin on each site. Within the macrospin approximation, the deviation from the exact AFM order is represented here, and it generates an internal stray field. The exact AFM pattern is represented by light gray arrows, in which the spin directions of the sublattices $A$ and $B$ are antiparallel.
• Figure 2: a) Out-of-plane term $S_{\parallel}(\omega)$ of the resistance power spectrum as a function of the temperature, ranging from the zero temperature to the Néel temperature, for three values of $\omega$: $\omega=\lambda J/10$ (red lines), $\omega=\lambda J$ (green lines), and $\omega=20 \lambda J$ (blue lines). Solid lines represent $S_{\parallel}(\omega)$ in units of $J^4/\omega$, as defined in Eq. \ref{['eq:Sparallel']} where we see that $S_{\parallel}(\omega)$ can be decomposed into a term characterized by a Lorentzian function with a width of $2\Gamma_\parallel$ (dotted lines) and another term (dashed line) that comprises two Lorentzian functions with widths $\Gamma_\parallel'$ and $2\Gamma_\parallel'$. b) Damping rates $\Gamma_\parallel$ (black solid line) and $\Gamma_\parallel'$ (black dashed line), defined in Eq. \ref{['eq:out-damping']}, as a function of temperature. Here, the horizontal gray lines correspond respectively to $\lambda J/10$ and $\lambda J$. Results in all panels have been obtained by setting $\Delta=0.35 J$.
• Figure 3: a) In-plane term $S_{\bot}(\omega)$ of the resistance power spectrum, defined in Eq. \ref{['eq:Sbot']}, shown as a function of the temperature between the zero temperature and the Néel temperature, for different values of $\omega$: $\omega=\lambda J/10$ (red lines), $\omega=\lambda J$ (green lines), and $\omega=20 \lambda J$ (blue lines). Here, the dimensionless parameter $\delta \lambda$ is set $\delta \lambda=\lambda$ in solid lines, $\delta \lambda=0$ in dashed lines, and $\delta \lambda=-\lambda/2$ in dash-dotted lines. Only a curve shows a non monotonic behavior, and it is the case with $\omega=20 \lambda J$ and $\delta \lambda=\lambda$. b) $T^2 \Gamma_\bot$ as a function of temperature, where the damping rate is defined in Eq. \ref{['eq:Gammabot']}. The association between line style and $\delta \lambda$ remains consistent with that in panel a), while the gray dotted line represents $\delta \lambda=\lambda/2$. Results in all panels have been obtained by setting $\Delta=0.35 J$.
• Figure 4: Resistance power spectrum $S_{R}(\omega)$, as defined in Eq. \ref{['eq:SR']}. (a) $S_{R}(\omega)$ as a function of temperature, from zero temperature up to the Néel temperature, for different $\omega$: $\omega = \lambda J/10$ (red), $\omega = \lambda J$ (green), and $\omega = 20\lambda J$ (blue). (b) $S_{R}(\omega)$ as a function of frequency for different temperatures: $T = 0.5 T_{\rm N}$ (red), $T = 0.8 T_{\rm N}$ (green), and $T = 0.99 T_{\rm N}$ (blue). In this log-log plot, the gray dotted line indicates a $1/\omega^2$ dependence as a guide to the eye. In both panels, the dimensionless parameter $\delta \lambda/\lambda$ is set to $1$ (solid lines), $0$ (dashed lines), and $-1/2$ (dash-dotted lines). The remaining parameter is $\Delta = 0.35J$.