\(\renewcommand{\AA}{\text{Å}}\)

# pair_style spin/dmi command

## Syntax

    pair_style spin/dmi cutoff

- cutoff = global cutoff pair (distance in metal units)

## Examples

    pair_style spin/dmi 4.0
    pair_coeff * * dmi 2.6 0.001 1.0 0.0 0.0
    pair_coeff 1 2 dmi 4.0 0.00109 0.0 0.0 1.0

## Description

Style spin/dmi computes the Dzyaloshinskii-Moriya (DM) interaction between pairs of magnetic spins. According to the expression reported in (Rohart), one has the following DM energy:

\[\mathbf{H}_{dm} = \sum_{{ i,j}=1,i\neq j}^{N} \left( \vec{e}_{ij} \times \vec{D} \right) \cdot\left(\vec{s}_{i}\times \vec{s}_{j}\right),\]

where \(\vec{s}_i\) and \(\vec{s}_j\) are two neighboring magnetic spins of two particles, \(\vec{e}_ij = \frac{r_i - r_j}{\left| r_i - r_j \right|}\) is the unit vector between sites i and j, and \(\vec{D}\) is the DM vector defining the intensity (in eV) and the direction of the interaction.

In (Rohart), \(\vec{D}\) is defined as the direction normal to the film oriented from the high spin-orbit layer to the magnetic ultra-thin film.

The application of a spin-lattice Poisson bracket to this energy (as described in (Tranchida)) allows to derive a magnetic torque omega, and a mechanical force F (for spin-lattice calculations only) for each magnetic particle i:

\[\vec{\omega}_i = -\frac{1}{\hbar} \sum_{j}^{Neighb} \vec{s}_{j}\times \left(\vec{e}_{ij}\times \vec{D} \right) ~~\mathrm{and}~~ \vec{F}_i = -\sum_{j}^{Neighb} \frac{1}{r_{ij}} \vec{D} \times \left( \vec{s}_{i}\times \vec{s}_{j} \right)\]

More details about the derivation of these torques/forces are reported in (Tranchida).

For the spin/dmi pair style, the following coefficients must be defined for each pair of atoms types via the pair_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands, and set in the following order:

- rc (distance units)

- |D| (energy units)

- Dx, Dy, Dz (direction of D)

Note that rc is the radius cutoff of the considered DM interaction, |D| is the norm of the DM vector (in eV), and Dx, Dy and Dz define its direction.

None of those coefficients is optional. If not specified, the spin/dmi pair style cannot be used.

## Restrictions

All the pair/spin styles are part of the SPIN package. These styles are only enabled if LAMMPS was built with this package, and if the atom_style “spin” was declared. See the Build package page for more info.

## Related commands

atom_style spin, pair_coeff, pair_eam,

## Default

none

(Rohart) Rohart and Thiaville, Physical Review B, 88(18), 184422. (2013).

(Tranchida) Tranchida, Plimpton, Thibaudeau and Thompson, Journal of Computational Physics, 372, 406-425, (2018).