Langevin spin dynamics

Microscopic stochastic Langevin-type spin dynamics equations provide a convenient and tractable model describing the relaxation of spin and spin-lattice ensembles. We develop a robust and numerically stable algorithm for integrating the Langevin spin dynamics equations, and explore, both numerically and analytically, a range of applications of the method. We show that the algorithm conserves the magnitude of the spin vector irrespectively of the amplitude of the thermal noise. Using the Furutsu-Novikov theorem, we derive a system of deterministic differential equations for the ensemble-average moments of solutions of the Langevin spin dynamics equations, and explore the dynamics of relaxation of a spin ensemble toward the equilibrium Gibbs distribution. Analytical solutions of the moments equations make it possible to estimate the time scales of spin thermalization and spin-spin self-correlation, which we investigate as functions of the damping parameter and temperature.

Figure 1

The normalized first moment $\langle \mathbf{S}·\mathbf{H}\rangle$ evaluated as a function of time by integrating Eq. (2) and by using the moments equations (25). At low temperature $T=300$ K, as many as 30 equations are required for convergence, while at a higher temperature $T=3000$ K just 5 equations are sufficient. The solutions to the Langevin equation and the moments equations follow the same trend, confirming the equivalence of the two methods in the treatment of spin relaxation.

Figure 2

The normalized 1st, 2nd, 3rd, and 4th moments of $(\mathbf{S}·\mathbf{H})$ plotted as functions of time for $T=300$ K, 3000 K, 10000 K, and 20000 K. The solutions of the moments equations are fully consistent with stochastic simulations based on the Langevin spin dynamics equations.

Figure 3

The normalized spin-spin self-correlation function plotted as a function of time for $T=300$ K, 3000 K, 10000 K, and 20000 K. All the spins were initiated assuming $\mathbf{S}=(S_x,0,S_z)$, where $S_z=\langle \mathbf{S}·\mathbf{H}\rangle {}_{\mathrm{eq}}/H$. The simulation data are in very good agreement with the moments equations. The normalized moments $\langle (\mathbf{S}·\mathbf{H}){}^N\rangle$ for $N=1,\dots ,5$ at $T=3000$ K are also shown for reference.

Figure 4

The black curves show the normalized values of $\langle \mathbf{S}·\mathbf{H}\rangle$ plotted as functions of time for spin ensembles thermalized by stochastic fields from 0 K to 300 K, and from 0 K to 20000 K. The red curves were calculated using the equations $\langle \mathbf{S}(t)\rangle =[\langle \mathbf{S}(0)\rangle -\langle \mathbf{S}\rangle {}_{\mathrm{eq}}]e^{-t/{\tau }_{\mathrm{th}}}+\langle \mathbf{S}\rangle {}_{\mathrm{eq}}$, where ${\tau }_{\mathrm{th}}=2\gamma \mathit{SH}/\hslash$ and ${\tau }_{\mathrm{th}}=2\gamma k_{B}T/\hslash$ for 300 K and 20000 K, respectively.

Figure 5

The normalized spin-spin self-correlation function plotted as a function of time. The simulation data points are the same as in Fig. 3. The approximate treatment [Eqs. (44) and (45)] agrees well with the simulation data.

Figure 6

The normalized spin-spin self-correlation function plotted as a function of time for the case where the simulation starts from a thermalized spin ensemble. The approximate treatment based on Eqs. (44) and (45) still matches the simulation well, especially the relaxation behavior.