1. Simple magnetic systems

Besides some f metals only very few element crystals feature a magnetic ground state: Fe, Co, Ni are ferromagnetic (FM) and Cr is antiferromagnetic (AFM). In this tutorial session we want to compare different magnetic configurations of several materials to determine their magnetic ground state.

1.1. hcp Co

hcp Co (in inpgen latsys=hdp) features the experimental lattice parameters $a=250.71~\text{pm}$ and $c=406.95~\text{pm}$. Starting from these values we determine the equilibrium PBE lattice constants for initial FM and AFM configurations. To allow for the symmetry breaking related to the AFM setup we have to make sure that the two Co atoms are treated as unequal by inpgen. This is done by adding different "fractional atomic numbers" to their atomic number:

2
27.01 1.0 1.0 1.0
27.02 -1.0 -1.0 -1.0
&factor 3.0 3.0 4.0

The fractional part is only used to find out whether two atoms may belong to the same group of symmetry equivalent atoms.

Generate in two subfolders (FM and AFM) the respective inp.xml file. In the AFM folder we modify the inp.xml file such that the initial magnetization is antiferromagnetic. The magnetic moment for each atom species is provided in the XML attribute atomSpecies/species/@magMom. For Cobalt this is by default $1.6~\mu_\text{B}$. For one of the two Co species we negate this number.

Note also that calculationSetup/magnetism/@jspins is by default 2 for this inp.xml file. This enables a magnetic calculation. Whenever you set up a unit cell containing at least one magnetic atom this is the default. However, there may also be situations in which one has to set this value by hand.

Keep c/a constant: What are the equilibrium lattice constants for both magnetic setups, what is their energy difference? The magnetic moments for each atom type in each iteration can be obtained with "grep magneticMoment out.xml". For the FM and the AFM case: How do the magnetic moments for the two atoms depend on the lattice constant? Of course, the magnetic moments obtained in this way only cover the charge that is within the respective MT spheres. In the FM case: The spin-dependend charge for the whole unit cell in each iteration can be obtained with "grep spinDependentCharge out.xml". For an example lattice constant calculate from this the magnetic moment for the whole unit cell and divide it by two to obtain a value per atom. What is the difference to the magnetic moment in the MT spheres?

Your results should be similar to what is sketched in the following plot.

2. Exercises

2.1. bcc Cr

bcc Cr features the experimental lattice constant of $5.50~a_0$. Set up a conventional bcc unit cell with two atoms and perform calculations analogous to the Co example above. Calculate all the quantities that were also calculated for the Co example.

Results to be delivered: Plots for:

1. total energy vs. lattice constant
2. magnetic moment of 1st atom vs. lattice constant

Also the energy difference FM vs. AFM for the respective equilibrium lattice constants has to be calculated.