Non-Collinear Magnetism

For many materials the magnetic unit cell is neither ferromagnetic nor antiferromagnetic but features a more complex magnetic structure in which the orientation of the magnetic moments related to the different atoms changes within the unit cell. Such magnetic structures are called noncollinear and may be caused by different circumstances. One possible cause is magnetic frustration, i.e., a preferred alignment of the magnetic moments at the different atoms is in contradiction with the geometry of the system. Another cause may be spin-orbit coupling. In this tutorial we will calculate several noncollinear magnetic structures for a magnetically frustrated system.

In this tutorial we will extend the kind of magnetic structures to simulate from the simple collinear examples to a non-collinear setup. Our first example will be a simple two atom unit-cell of iron.

1.Example: Fe in fcc structure

In its ground state Fe crystallizes in a bcc structure, but it is also possible to synthesize Fe in an fcc structure, e.g., fcc Fe clusters may form by fast annealing of Fe in a Cu matrix. We already studied some collinear setups of this setup in the basic magnetism tutorial. We now extend the investigation to non-collinear simulations and we will start by repeating the FM vs AFM study using non-collinear setups.

Input for non-collinear calculations

To create a non-collinear setup we will again employ inpgen with a simple input file.

export THOME=$PWD
cd FeNoco ; cat inpFeNocoAFM.txt

Please notice, that we not only specify the magnetic moment per atom here but the full vector of magnetization $\vec m=(m_x m_y m_z)$. This indicates for inpgen that a non-collinear setup is required.

inpgen -f inpFeNocoAFM.txt

The generated inp.xml file is now somewhat more complex than that used in collinear magnetic calculations. In particular, we would like to draw your attention to:

1. The additional switches in the magnetism section:

      <magnetism jspins="2" l_noco="T" l_ss="F">
         <qss>.0000000000 .0000000000 .0000000000</qss>
         <mtNocoParams l_mperp="F" l_mtNocoPot="F" l_relaxSQA="F" mag_mixing_scheme="0" mix_RelaxWeightOffD="1.00000000" l_constrained="F" mix_constr="1.00000000"/>
         <sourceFreeMag l_sourceFree="F" l_scaleMag="F" mag_scale="1.00000000"/>
      </magnetism>

here, many more parameters are now available. For the most basic non-collinear calculation we are preforming now, the relevant switch is l_noco="T" which makes your simulation a non-collinear one.

1. For each atomGroup there is now an additional tag:

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

These two angles define the local spin-frame of the atom. I.e. the direction in which the local spin-quantization axis is pointing during the simulation.

In our AFM setup we just created, we have only a single species. This is different from the collinear calculation done before in which we had two different 'Fe' atoms, one with an initial 'up' spin and one with a 'down' spin. In our non-collinear setup these two different atoms differ by the angles in nocoParams.

Converging a FM and an AFM state using the non-collinear setup

In oder to perform a simulation for the ferro- and anti-ferro-magnetic state we create two directories:

mkdir FM ; cd FM ; inpgen -f ../inpFeNocoFM.txt ; cd -

mkdir AFM ; cd AFM ; inpgen -f ../inpFeNocoAFM.txt ; cd -

Then you can run FLEUR until it converges. Please note that the default of itmax=15 is not sufficent, so you should either - start FLEUR several times until you reach convergence - increase the itmax parameter.

Please note that these calculations are now significantly more computational expensive compared to the collinear setups in the previous tutorial. This is mostly due to two effects: - a non-collinear calculation will always lead to a complex hermitian eigenvalue problem, while a collinear setup with inversion symmetry leads to a real symmetric problem. - in a non-collinear calculation we construct an eigenvalue problem for both spins that is double in size to the collinear setup. As the effort to diagonalize scales cubically with the size, this is significantly more demanding than solving two eigenvalue problems for each spin.

cd FM ; fleur_MPI ; cd -

cd AFM; fleur_MPI ; cd -

The charge density distance output of the calculations should indicate convergence. Extract the total energy from the last iteration of both calculations and compare them. You should find that from our calculations (with default parameters) the FM configuration is preferred and has a small energy advantage of about $0.65~\text{mHtr}$ for the unit cell or $0.32~\text{mHtr}$ per atom.

for f in FM/out AFM/out; do grep -H "total energy" $f | tail -1; done

Another quantity we can extract for magnetic calculations are the magnetic moments in the MT spheres. For this two-atom type system you can easily find these entries for the last SCF iteration by invoking

grep -B 1 "smm" FM/out| tail -n 3

and

grep -B 1 "smm" AFM/out| tail -n 3

Doing this for the out.xml files of the two calculations should show for the FM calculation a magnetic moment of about $2.61~\mu_\text{B}$ in both MT spheres and for the AFM calculation a magnetic moment of about $1.78~\mu_\text{B}$. It is notable that also in the AFM case the two magnetic moments have the same sign. The direction of the magnetization in noncollinear calculations is encoded in the MT-specific angles $\alpha$ and $\beta$.

2. Example: Noncollinear magnetism in a Cr Monolayer with hexagonal lattice geometry

For Cr atoms the preferred magnetic ordering is antiferromagnetic. In a triangular lattice this is not possible and such a setup therefore is an example for a magnetically frustrated system. To obtain such a lattice for Cr one can grow a single layer of Cr atoms on a Cu(111) surface. In such a sample the Cr atoms will occupy the positions of Cu atoms in the respective layer. As an approximation to such a setup we calculate an unsupported Cr monolayer with atoms at such positions. Assume an fcc Cu crystal with a lattice constant of $6.65~a_0$. This translates into a nearest-neighbor distance between the atoms of $4.70~a_0$.

Here we now compare three different magnetic states:

• The FM state where we have only a single atom.
• The collinear AFM state in which we have "stripe" antiferromagnetic aligment of the magnetisation. Here we have two different atoms, one with magnetisation "up" and one with the magnetisation "down". Note, that one nearest neighbour is now antiferromagnetically aligned while the other is ferromagnetically coupled.
• The so called "Neel" state in which the three nearest neighbours have magnetic moments pointing at $120^o$ with respect to each other. This can be understood as a compromise in which nearest neighbours are as antiferromagnetic as possible.

First we loop over all three directories to generate the corresponding inp.xml files. Please have a look at the different inp_Cr.txt files and the generated inp.xml to confirm you understand the settings.

cd $THOME/Cr

for f in FM AFM Neel ; do cd $f ; inpgen -f inp_Cr.txt ; cd - ; done

Now you can converge the charge density in the different configurations. You will have to restart FLEUR in order to achieve convergence.

for f in FM AFM Neel ; do cd $f ; fleur_MPI ; cd - ; done

Once you have a converged result, you can look for e.g.:

• the magnetic moments
• the total energy

for f in FM AFM Neel; do echo $f; grep " smm" $f/out|tail -3; done

for f in FM AFM Neel; do echo $f;grep "total en" $f/out|tail -1; done

Keeping into account that the FM unit-cell contains 1 atom, the AFM cell 2 atoms and the Neel cell 3 atoms we obtain (depending on your level of convergence):

If you can converge this setup you might be able to determine these energies.

Further studies:

Obviously, the examples here can be expanded, e.g. by taking different configurations and different materials into account.