Spin-lattice-electron dynamics simulations of magnetic materials

We develop a dynamic spin-lattice-electron model for simulating the time-dependent evolution of coupled spin, atomic, and electronic degrees of freedom in a magnetic material. Using the model, we relate the dissipative parameters entering the Langevin equations for the lattice and spin degrees of freedom to the heat transfer coefficients of a phenomenological spin-lattice-electron three-temperature model. We apply spin-lattice-electron dynamics simulations to the interpretation of experiments on laser-induced demagnetization of iron thin films, and estimate the rates of heat transfer between the spins and electrons, and between atoms and electrons. To model the dynamics of energy dissipation in a magnetic material undergoing plastic deformation, we develop an algorithm that separates the local collective modes of motion of atoms from their random thermal motion. Using this approach, we simulate the propagation of compressive shock waves through magnetic iron. We also explore the microscopic dynamics of dissipative coupling between the spin and lattice subsystems, and show that the rate of spin-lattice heat transfer is proportional to the integral of the four-spin time-dependent correlation function.

Figure 1

Evolution of magnetization as a function of time. The entire curve is normalized to the value of magnetization at 300 K. By tuning the damping constants and the amplitude of the energy pulse we match the simulated demagnetization process to experimental observations by Carpene et al.[32] The simulated demagnetization curve was computed assuming ${\gamma }_{el}/m=0.6$ ps${}^{-1}$ and ${\gamma }_{es}=8\times {10}^{-3}$ (the latter parameter is dimensionless).

Figure 2

Evolution of temperatures for the spin, lattice, and electronic subsystems as functions of time. The electronic temperature rapidly increases initially due to the absorption of energy from the laser pulse, followed by the increase of the spin temperature. At some point, the spin temperature becomes higher than the temperature of the electrons. The temperature of the lattice increases much slower due to the weak electron-lattice coupling, and even weaker spin-lattice coupling.

Figure 3

Schematic representation of simulation setup. A simulation cell contains $30\times 30\times 550$ body-centered cubic (bcc) unit cells, or approximately 86 Å$\times$86 Å$\times$1577 Å, with the coordinate axes parallel to the [100], [010], and [001] crystallographic directions, and involves the total of 990 000 atoms and spins. Two compressive waves propagating in the opposite directions are generated by using a method proposed by Holian et al.,[77, 78] where a simulation box with periodic boundary conditions is shrunk uniaxially, introducing symmetric impacts on both sides of the box. The boundaries on the left and on the right move inwards with velocity $u_p=500$ m/s. This is equivalent to applying the action of two pistons moving with velocity $u_p$ in the [001] and [00$\overline{1}$] directions. Artificial compression is applied for 0.5 ps.

Figure 4

Profiles of local temperatures $T_s$, $T_l$, and $T_e$ simulated using SLED, and profiles of $T_l$ simulated using pure MD, following the initiation of the shock waves. No consideration of the local collective motion of atoms and electrons is included in these simulations. Each temperature point in the figures corresponds to the average temperature of atoms in a linked cell situated at a given coordinate point in the direction of propagation of a shock wave. The lattice temperature $T_l$ is proportional to the kinetic energy of atoms defined in the frame associated with the static boundaries of the simulation cell. The position of the compressive wave front corresponds to the peak of $T_l$. In comparison with a pure MD simulation, in the SLED case the energy of atoms is rapidly absorbed by the spin and electron subsystems, resulting in the much higher rate of attenuation of the compressive wave predicted by the SLED model in comparison with pure MD simulations.

Figure 5

Profiles of local temperatures $T_s$, $T_l$, and $T_e$ simulated using SLED, and profiles of $T_l$ simulated using pure MD, following the initiation of the shock waves. Local collective motion is taken into account in the simulations. The lattice temperature $T_l$ calculated using Eq. (52) is significantly lower than that shown in Fig. 4. We can still see the propagating wave front, but the temperature profile predicted by the SLED model now closely resembles the temperature profile found in MD simulations.

Figure 6

The average velocity distributions evaluated for cases where the local collective motion of atoms is/is not taken into account. Predictions derived from the SLED model that takes into account the local collective motion of atoms are very similar to those found in MD simulations. The rate of energy losses from the lattice to electrons found in SLED simulations is clearly excessive if the local collective modes of motion of atoms occurring as a result of propagation of a shock wave are included in the definition of lattice temperature.

Figure 7

The spin-lattice energy transfer rate $g_{sl}$ plotted as a function of the spin and lattice temperatures. The rate is calculated directly by using Eq. (24). The spin and lattice subsystems were kept at two different temperatures maintained using two independent Langevin thermostats. This makes it possible to calculate the cumulative amount of energy transferred from the spins to the lattice, or vice versa, for the case where the two subsystems are maintained at two different temperatures.

Figure 8

The spin-lattice heat transfer coefficient $G_{sl}$ plotted as a function of spin temperature $T_s$ and lattice temperature $T_l$. In the limit of low $T_s$, the behavior of $G_{sl}$ as a function of $T_s$ is very close to being linear. At high $T_s$, $G_{sl}$ increases and is maximum near the Curie temperature. At high $T_s$ and high $T_l$, the data points become very scattered, but no significant new trend seem to emerge even at temperatures significantly higher than the Curie temperature.