Multiscale method for Heisenberg spin simulations

A multiscale method that couples classical Heisenberg model to micromagnetics in a unified formalism is presented. It is based on a multiresolution adaptive finite difference mesh, which ensures significant reduction of the number of variables and computation time with respect to either atomistic or micromagnetic simulations, together with a precise description of the modeled system where necessary. The hierarchical structure of the mesh is used to compute efficiently the dipolar field by means of a fast multipole method. The underlying atomistic approach is particularly useful to handle magnetic singularities and describe structural defects. The method is applied here to the case of a magnetic vortex and a thin layer of FePt with a microtwin. Results are compared to fully atomistic and micromagnetic simulations when possible.

Figure 1

1D model system for the calculation of the exchange energy of node $S$ with the second derivative approach [Eq. (8)]. The volume to consider is in gray.

Figure 2

1D model system for the calculation of the exchange energy of node $S$ with the first derivative approach [Eq. (9)]. The volumes to consider are shared by the boxes and are in gray.

Figure 3

(Color online) Energy density of a domain wall in a 1D system on a nonuniform mesh, initially and after one iteration of minimization. The energy is computed with the second derivative approach [Eq. (8)].

Figure 4

(Color online) Energy density of a domain wall in a 1D system on a nonuniform mesh, initially and after one iteration of minimization. The energy is computed with the first derivative approach [Eq. (9)].

Figure 5

(Color online) Interface between atomistic and micromagnetic areas in the case of a simple orthorhombic lattice. The boundary of the system is between nodes $U$ and $V$.

Figure 6

(Color online) Dipolar field due to the lattice effects for different tetragonal distortions $c/a$. The magnetization is along $x$ or $z$ and the field is computed in the direction of the magnetization. $M=1$ and ${\mu }_0=1$, so a typical demagnetizing field is around 1. The shape effects due the sample used for the summation are corrected and do not contribute to the field.

Figure 7

(Color online) Relative error on the field due to the lattice effects as a function of the number of layers taken into account to compute this field. The tetragonal distortion is $c/a=3$, but the global shape of the system is cubic (no shape effect).

Figure 8

(Color online) Influence of the mismatch between the correct position of the interface between atomic/micromagnetic regions and its actual position. $M=1$ and ${\mu }_0=1$, so a typical demagnetizing field is around 1. A high modification of the shape effect is envisaged here for a simple cubic lattice: The four surfaces with normals along $\pm \hat{\mathit{x}}$ and $\pm \hat{\mathit{y}}$ are shifted toward the center of the box of $a/2$ (with $a=1$), and the two surfaces along $\pm \hat{\mathit{z}}$ are shifted away from the center of the same quantity. See Fig. 9 for the geometry of the system.

Figure 9

(Color online) System used to study the influence of the position of the interface between atomic and micromagnetic regions on the field on an atomic site. Here the field is computed on the atomic site at the center of the box. The ideal position of the interface is shown in dashed lines. In the figure, two layers of atoms have been added around the atom at the center.

Figure 10

(Color online) Initial configuration to create a vortex: The system contains two domains separated by an abrupt domain wall; at the center of the system the spins are slightly tilted to be compatible with the formation of a vortex (inset).

Figure 11

(Color online) Configuration of the vortex after an energy minimization. The inset shows the magnetic structure near the core of the vortex. Asymmetries in the mesh are due to the fact that criteria to divide or gather boxes may not be met at the same iteration in symmetric parts of the system, because of numerical precision. Moreover these asymmetries may be persistent given the different values considered for ${ϵ}_d$ and ${ϵ}_g$ that make the gathering of boxes more difficult than their division.

Figure 12

(Color online) Component of the magnetization along $z$ on a line crossing the vortex core for the different approaches: fully atomistic, micromagnetic, and multiscale.

Figure 13

(Color online) Component of the magnetization along $z$ on a line crossing the vortex core for a multiscale simulation. Different values of exchange couplings $J$ are envisaged (in $\text{meV}/{\mu }_b^2$). Negative values of $m_z$ can be observed near the vortex core for $J=0.25\text{meV}/{\mu }_b^2$ and they are ascribed to the dipolar field created by the magnetization in the core (see Refs. 15, 16).

Figure 14

(Color online) Component of the magnetization along $z$ on a line crossing the vortex core for a micromagnetic simulation. Different values of exchange couplings $J$ are envisaged (in $\text{meV}/{\mu }_b^2$). Note the poor precision on $m_z$ at the core of the vortex when $J$ is $0.25\text{meV}/{\mu }_b^2$, compared to the multiscale simulation in Fig. 13.

Figure 15

(Color online) Domain wall (on the left) out of the microtwin (on the right). The domain wall contains a Bloch core with two Néel caps due to the dipolar interaction. The microtwin is 6 atomic planes wide and the system contains 40 Fe atomic planes along $z$.

Figure 16

(Color online) Domain wall pinned to the microtwin. The microtwin is 6 atomic planes wide and the system contains 40 Fe atomic planes along $z$.