Dzyaloshinskii Moriya interaction (DMI)

In this tutorial we will investigate the influence of the spin-orbit interaction on complex magnetic structures. One of the effects is the DMI which in an extended Heisenberg Hamiltonian will lead to an interaction between magnetic moments $S_i$ in the following form: $$E_{DMI}\approx \vec D (\vec S_i \times \vec S_j)$$

This interaction thus: - is relevant in non-collinear arragements of spin, - can favour non-collinear magnetism, - can lead to "chiral" structures.

DMI in spin-spirals

While the DMI can also be studied in standard non-collinear setups it is most interesting to see its effect on spin-spiral states due to the possiblity to systematically scan different q-vectors and thus look at systems with small angles between spins without the need to construct a large supercell.

Unfortunately, the generalized Bloch-Theorem we use to efficiently calculate spin-spiral states is not compatible with SOC. This can relatively easily be seen from the fact that if the spin-directions rotate from one unit-cell to another, the SOC will couple this differently to the lattice (orbital moments). As we learned in the force theorem sections of the tutorial we can, however, take advantage from the fact the SOC is generally a small effect and thus we now treat SOC in perturbation theory. Hence, in this approach we can estimate the effect of the SOC in first order perturbation theory as $$E_{DMI}=\sum_\nu <\psi_\nu(\vec q)|H_{SOC}|\psi_\nu(q)>$$

To demonstrate this approach we consider a system consisting of a single Co atom on a single Pt substrate.

cd DMI; cat inp_CoPt.txt;inpgen -f inp_CoPt.txt -nosym

Modify the inp.xml file.

We need to add the DMI section. This is also embedded in the forceTheorem part, even though it is now a calculation using perturbation theory.

<forceTheorem>
  <DMI theta="0.5*Pi" phi="0.0" >
    <qVectors>
      <q> 0.0 0.0 0.0 </q>
      <q> 0.1 0.1 0.0 </q>
      <q> 0.2 0.2 0.0 </q>
      <q> -.1 -.1 0.0 </q>
      <q> -.2 -.2 0.0 </q>
    </qVectors>
  </DMI>
</forceTheorem>

In addition we have to change the angles specifying the rotation of the spin spiral. We set for the Co atom

<nocoParams  alpha=".00000000" beta="Pi/2.00000000"/>

The DMI mode requires the usage of the spin-averaged potential. To enable this set the spav switch in the calculationSetup/soc section to "T".

Then FLEUR can be started. The DMI mode will be activated after the last iteration.

fleur_MPI

Again we can use grep to extract the eigenvalue sums from the out.xml file. Please note that this time the values for a spin spiral with positive "q" are different from those with negative "q". This is an effect of the DMI.

grep ev\-sum out.xml 

Further studies:

The DMI mode can also be used to calculate several different angles at once. Please check the documentation for more details.