Longitudinal magnetic fluctuations in Langevin spin dynamics

We develop a generalized Langevin spin dynamics (GLSD) algorithm where both the longitudinal and transverse (rotational) degrees of freedom of atomic magnetic moments are treated as dynamic variables. This removes the fundamental limitation associated with the use of stochastic Landau-Lifshitz (sLL) equations, in which the magnitude of magnetic moments is assumed constant. A generalized Langevin spin equation of motion is shown to be equivalent to the sLL equation if the dynamics of an atomic moment vector is constrained to the surface of a sphere. A fluctuation-dissipation relation for GLSD and an expression for the dynamic spin temperature are derived using the Fokker-Planck equation. Numerical simulations, performed using ferromagnetic iron as an example, illustrate the fundamental difference between the two- and three-dimensional dynamic evolution of interacting moments, where the three-dimensional GLSD includes the treatment of both transverse and longitudinal magnetic excitations.

Figure 1

Partial densities of states (PDOS) for $d$ electrons in bcc iron evaluated for magnetic and nonmagnetic (NM) states. The PDOS are calculated using the LMTO-GF method.[55] The PDOS for a nonmagnetic state is similar to the PDOS for spin up or down in the magnetic state. This confirms the rigid-band approximation underlying the Stoner model.

Figure 2

Energy as a function of $M$ calculated according to Eq. (41). We fit both the fourth- and sixth-order Landau expansion coefficients to the data. Although the fourth-order expansion already reproduces the overall shape of the curve fairly well, the sixth-order expansion provides a much better fit to the curvature near the minimum energy point.

Figure 3

Thermal equilibrium energies per spin for bcc iron plotted as a function of temperature. Simulations were performed using Eq. (13) or (18), including or neglecting the effect of longitudinal fluctuations.

Figure 4

Specific heat of bcc iron vs temperature. This quantity is evaluated by performing numerical differentiation of data shown in Fig. 3. The peaks correspond to the Curie temperatures $T_C$ (which differ depending on whether or not the effect of longitudinal magnetic fluctuations is taken into account).

Figure 5

Magnetization of bcc iron vs temperature. Magnetization vanishes at the Curie temperature $T_C$.

Figure 6

The average value and the standard deviation of the magnitude of an atomic magnetic moment plotted as functions of temperature.

Figure 7

Histogram showing the distribution of the magnitude of magnetic moments at 800, 1000, and 1200 K computed for a simulation cell containing 16 000 spins.

Figure 8

Total magnetization as a function of time simulated for a transient process of relaxation to thermal equilibrium. All the simulations were performed starting from ferromagnetic ground states.

Figure 9

Temperatures evaluated using Eqs. (34) and (43) as functions of time during the thermalization process, with thermostats set at 300 K (upper) and 1000 K (lower).