pair_style spin/exchange command

Syntax

pair_style spin/exchange cutoff

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

Examples

pair_style spin/exchange 4.0
pair_coeff * * exchange 4.0 0.0446928 0.003496 1.4885
pair_coeff 1 2 exchange 6.0 -0.01575 0.0 1.965

Description

Style spin/exchange computes the exchange interaction between pairs of magnetic spins:

\[H_{ex} = -\sum_{i,j}^N J_{ij} (r_{ij}) \,\vec{s}_i \cdot \vec{s}_j\]

where \(\vec{s}_i\) and \(\vec{s}_j\) are two neighboring magnetic spins of two particles, \(r_{ij} = \vert \vec{r}_i - \vec{r}_j \vert\) is the inter-atomic distance between the two particles. The summation is over pairs of nearest neighbors. \(J(r_{ij})\) is a function defining the intensity and the sign of the exchange interaction for different neighboring shells. This function is defined as:

\[{J}\left( r_{ij} \right) = 4 a \left( \frac{r_{ij}}{d} \right)^2 \left( 1 - b \left( \frac{r_{ij}}{d} \right)^2 \right) e^{-\left( \frac{r_{ij}}{d} \right)^2 }\Theta (R_c - r_{ij})\]

where \(a\), \(b\) and \(d\) are the three constant coefficients defined in the associated “pair_coeff” command, and \(R_c\) is the radius cutoff associated to the pair interaction (see below for more explanations).

The coefficients \(a\), \(b\), and \(d\) need to be fitted so that the function above matches with the value of the exchange interaction for the \(N\) neighbor shells taken into account. Examples and more explanations about this function and its parameterization are reported in (Tranchida).

From this exchange interaction, each spin \(i\) will be submitted to a magnetic torque \(\vec{\omega}\), and its associated atom can be submitted to a force \(\vec{F}\) for spin-lattice calculations (see fix nve/spin), such as:

\[\vec{\omega}_{i} = \frac{1}{\hbar} \sum_{j}^{Neighb} {J} \left(r_{ij} \right)\,\vec{s}_{j} ~~{\rm and}~~ \vec{F}_{i} = \sum_{j}^{Neighb} \frac{\partial {J} \left(r_{ij} \right)}{ \partial r_{ij}} \left( \vec{s}_{i}\cdot \vec{s}_{j} \right) \vec{e}_{ij}\]

with \(\hbar\) the Planck constant (in metal units), and \(\vec{e}_{ij} = \frac{\vec{r}_i - \vec{r}_j}{\vert \vec{r}_i-\vec{r}_j \vert}\) the unit vector between sites \(i\) and \(j\).

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

For the spin/exchange 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:

  • \(R_c\) (distance units)

  • \(a\) (energy units)

  • \(b\) (adim parameter)

  • \(d\) (distance units)

Note that \(R_c\) is the radius cutoff of the considered exchange interaction, and \(a\), \(b\) and \(d\) are the three coefficients performing the parameterization of the function \(J(r_{ij})\) defined above.

None of those coefficients is optional. If not specified, the spin/exchange 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 doc page for more info.