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

Abstract

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.

Collective dynamics in atomistic models with coupled translational and spin degrees of freedom

“…In this study, we adopt the parameterization introduced by Ma et al [34] for bcc iron, in which U ({r i }) is constructed as…”

“…(3), Hamiltonian (1) provides the same ground state energy as U DD . The exchange interaction is modeled via a simple pairwise function J(r ij ) parameterized by first-principles calculations [34], with spin lengths absorbed into its definition, i.e. J(r ij ) = J ij ({r k })|S i ||S j |.…”

“…The foundation of this combined molecular and spin dynamics (MD-SD) approach lies in the unification of an atomistic potential and a Heisenberg spin Hamiltonian, with the coupling between the atomic and spin subsystems established via a coordinate-dependent exchange interaction. With the use of an empirical many-body potential and a parameterized exchange interaction, Ma et al [34] further extended MD-SD into a framework for realistic modeling of bcc iron. The parameterization developed by Ma et al [34] has since been successfully adopted to investigate various phenomena in bcc iron such as magneto-volume effects [35], vacancy formation and migration [36,37], and external magnetic field effects [38].…”

“…With the use of an empirical many-body potential and a parameterized exchange interaction, Ma et al [34] further extended MD-SD into a framework for realistic modeling of bcc iron. The parameterization developed by Ma et al [34] has since been successfully adopted to investigate various phenomena in bcc iron such as magneto-volume effects [35], vacancy formation and migration [36,37], and external magnetic field effects [38]. Moreover, the method has been recently extended by incorporating spin-orbit interactions to facilitate the dynamic exchange of angular momentum between the lattice and spin subsystems [39].…”

Exchange interaction function for spin-lattice coupling in bcc iron

“…14,15 Recently, Ma et al 16,17 developed the spin-lattice dynamics ͑SLD͒ simulation scheme in which the coupled spin and lattice degrees of freedom are treated on equal footing in the equations of motion of the system. The dynamics of the coupled systems is then described in terms of the empirical many-body potential among the atoms and the exchange interaction function ͑EIF͒ among the spins.…”