Angular momentum conservation in spin-lattice dynamics simulations

Kuzkin's angular momentum balance method is implemented in the LAMMPS SPIN package for atomistic spin-lattice dynamics, along with shifted-force exchange and Néel Hamiltonians parameterized to minimize energy drifts in the simulations. Angular momentum contributions arising from two mechanisms are quantified using this method: particle transport across the boundaries of a periodic simulation domain and external torques applied to the domain by periodic image atoms. When these mechanisms are accounted for, lattice angular momentum is exactly conserved in lattice systems and in spin-lattice systems with isotropic exchange interactions. The calculations show that spin-lattice angular momentum exchange only occurs when the Néel anisotropy energy is added to the exchange energy, and that with this addition, total angular momentum is approximately conserved in the magnetization direction but not in other directions. Inclusion of the Néel anisotropy increases the energy drifts observed in simulations of iron nanoparticles. These drifts are linearly proportional to the magnitude of the anisotropy energy and the simulation time step.

Figure 1

The exchange, pseudodipole, and pseudoquadrupole energies graphed using the parametrization from Table 1.

Figure 2

(a) $6a$ cube centered in a periodic $12a$ domain, (b) $10a$ cube centered in a periodic $12a$ domain, and (c) bulk material in a periodic $12a$ domain. [(d)–(f)] $\mathbf{L}$, the angular momentum about the center of mass for the collection of atoms in the black box for geometries (a)–(c), respectively. [(g)–(i)] Integrated torques from the image atoms acting on the main simulation box in geometries (a)–(c). [(j)–(l)] Angular momentum flux from atoms wrapping around due to the periodic boundary condition for geometries (a)–(c). [(m)–(o)] ${\mathbf{L}}_{\mathrm{corr}}$, the modified angular momentum for geometries (a)–(c). All results are based on the EAM Hamiltonian.

Figure 3

Different types of relative angular momentum $\mathrm{\Delta }\mathbf{X}=\mathbf{X}\left(t\right)-\mathbf{X}$ (5 ps): modified lattice angular momentum, spin angular momentum, and total open system angular momentum. The $x, y$, and $z$ components are plotted for isolated cube [(a)–(c)] and bulk geometries [(d)–(f)] using the EAM and shifted force exchange Hamiltonians.

Figure 4

Relative values of modified lattice angular momentum, spin angular momentum, and total open system angular momentum for an isolated cube using the EAM, shifted force exchange, and shifted force Néel Hamiltonians. The $x, y$, and $z$ components of angular momentum are shown for [(a)–(c)] the fitted Néel energy scale given in Table 1, [(d)–(f)] 10% of the fitted Néel energy, and [(g)–(i)] 1% of the fitted Néel energy.

Figure 5

Relative values of modified lattice angular momentum, spin angular momentum, and total open system angular momentum for a periodic system using the EAM, shifted force exchange, and shifted force Néel Hamiltonians. The $x, y$, and $z$ components of angular momentum are shown for [(a)–(c)] the fitted Néel energy scale given in Table 1, [(d)–(f)] 10% of the fitted Néel energy, and [(g)–(i)] 1% of the fitted Néel energy.

Figure 6

Dependence of total open system angular momentum and magnetization on initial magnetization direction for an isolated cube. Magnetizations initialized in the [(a) and (d)] [100] direction, [(b) and (e)] [001] direction, and [(c) and (f)] [111] direction.

Figure 7

Dependence of total open system angular momentum and magnetization on initial magnetization direction for a bulk periodic system. Magnetizations initialized in the [(a) and (d)] [100] direction, [(b) and (e)] [001] direction, and [(c) and (f)] [111] direction.

Figure 8

Relative values of lattice energy, spin energy, and total energy for an isolated cube using the EAM and shifted force exchange Hamiltonians.

Figure 9

Relative values of lattice energy, spin energy, and total energy for an isolated cube using the EAM, shifted force exchange, and shifted force Néel Hamiltonians for different Néel energy parametrizations ${\alpha }_l$ and ${\alpha }_q$. (a) Néel energy values from Table 1, (b) 10% of Néel values in Table 1, and (c) 1% of Néel values in Table 1.

Figure 10

Total energy drifts for SLD simulations of an isolated cube using the EAM, shifted force exchange, and shifted force Néel Hamiltonians with different Néel energy scales.