Large-scale simulation of the spin-lattice dynamics in ferromagnetic iron

We develop a dynamical simulation model for magnetic iron where atoms are treated as classical particles with intrinsic spins. The atoms interact via scalar many-body forces as well as via spin orientation dependent forces of the Heisenberg form. The coupling between the lattice and spin degrees of freedom is described by a coordinate-dependent exchange function where the spin orientation dependent forces are proportional to the gradient of this function. The spin-lattice dynamics simulation approach extends the existing magnetic potential treatment to the case where the energy of interaction between the atoms depends on the relative noncollinear orientations of spins. An algorithm for integrating the linked spin-coordinate equations of motion is based on the second-order Suzuki-Trotter decomposition for noncommuting operators of evolution for coordinate and spin variables. The notions of the spin thermostat and the spin temperature are introduced through the combined application of the Langevin spin dynamics and the fluctuation-dissipation theorem. We investigate several applications of the method, performing microcanonical ensemble simulations of adiabatic spin-lattice relaxation of periodic arrays of $180°$ domain walls, and isothermal-isobaric ensemble dynamical simulations of thermally equilibrated homogeneous systems at various temperatures. The predicted isothermal magnetization curve agrees well with the experimental data for a broad range of temperatures. The equilibrium as well as time-correlation functions of spin orientations exhibit the presence of short-range magnetic order above the Curie temperature. Furthermore, short-range order spin fluctuations are shown to contribute to the thermal expansion of the material. Our analysis illustrates the significant part played by the spin degrees of freedom in the dynamics of motion of atoms in magnetic iron and iron-based alloys. It also shows that the spin-lattice dynamics algorithm developed in this paper offers a viable way of performing large-scale dynamical atomistic simulations of magnetic materials.

Figure 1

Exchange function $J_{ij}$ shown as a function of interatomic distance $r_{ij}$ and is compared with data taken from Refs. 17, 55. The assumed form for the exchange function is $J_{ij}\left(r_{ij}\right)=J_0{\left(1-r_{ij}/r_c\right)}^3\Theta \left(r_c-r_{ij}\right)$, where $J_0=904.90177\text{meV}$ and $r_c=3.75\text{Å}$.

Figure 2

The total energy per atom versus time evaluated for a microcanonical ensemble simulation of dynamical relaxation of a $180°$ domain wall using the $\left(s,r,v,r,s\right)$ and $\left(s,v,r,v,s\right)$ algorithms. The $\left(s,r,v,r,s\right)$ algorithms is 1.6 times faster than the $\left(s,v,r,v,s\right)$ algorithm.

Figure 3

Time evolution of the kinetic, the potential, and the spin energy terms determined in simulations of dynamical relaxation of a $180°$ domain wall and shown in the (a) 0.1 ps, (b) 10 ps, and (c) 1ns time scales. The velocities of all the atoms were initialized to zero at $t=0$. The curves show the variation of energy terms with respect to their initial values. KE and LE refer to the kinetic and the scalar (lattice) part of the potential energy defined by Eq. (3), respectively, whereas the spin energy SE refers to the last term in Eq. (3). The sum of the lattice and the spin energies equals the potential energy of the system.

Figure 4

(Color online) The phase trajectory of an atom (left) and the spin direction (right) followed using a microcanonical ensemble simulation performed for a $54000$ atom system thermally equilibrated at $T=300\text{K}$.

Figure 5

Projections of atomic spins found in a microcanonical ensemble simulation of dynamical relaxation of an initially sharp magnetic domain boundary.

Figure 6

Examples of dynamical simulations performed using the Langevin SLD algorithm for $\eta ={10}^{-3}$ for several temperatures of the thermostat.

Figure 7

Examples of Langevin SLD simulations of thermal relaxation of the spin subsystem interacting with the thermostat kept at $T=300\text{K}$ performed for two different values of the damping constant.

Figure 8

The equilibrium magnetization curve showing the average atomic spin evaluated dynamically using the canonical ensemble simulations for various temperatures. Experimental data were taken from Ref. 62 and the mean-field approximation curve was evaluated using the method described in Ref. 63 in the classical limit.

Figure 9

(Color online) The spin-spin spatial correlation functions shown as functions of absolute temperature for the first, second, $\dots$, ${12}^{\text{th}}$ nearest-neighbor atoms.

Figure 10

(Color online) The time-dependent spin-spin on-site autocorrelation functions evaluated dynamically using the microcanonical ensemble simulations performed for fully thermalized initial configurations.

Figure 11

The spin autocorrelation dephasing times for the curves shown in Fig. 10.

Figure 12

Equilibrium zero pressure lattice constant of bcc iron evaluated using SLD simulations and conventional MD simulations performed using the magnetic DD potential. Experimental data taken from Ref. 64 are also shown for comparison.