# Magnetism¶

- Author
Daniel Sanchez-Portal and Andrei Postnikov (2010)

Initialization of different magnetic orders in a magnetic insulator (MnO), and analysis of the results.

The aims of the exercise:

To learn to set up a calculation in a bulk (crystal) system composed of “magnetic” (i.e., 3d) and “non-magnetic” (s-p) atoms, whereby spin moments of “magnetic” atoms may be arranged in different types of magnetic orderings, and hence allow different (metastable) magnetic solutions.

To analyze the results of converged calculations, demonstrating the type of magnetic ordering in each of the cases considered.

Note

3d oxydes are notorious examples of systems in which Coulomb correlation effects play an important role, so that the “conventional” DFT treatment is in some senses misledaing: the insulating band gaps are underestimated, and the placement of main features of the band structure not consistent with spectroscopic experiments. A large part of these inconsistencies can be fixed by applying the “DFT+U” formalism, as discussed in another tutorial.

Note

Another complication is that the crystal structure of real 3d oxydes, basically very close to the B1 (NaCl) type, undergoes slight but noticeable distortions which help to stabilize one or another magnetic phases. However, in the hystorical context as well as for didactic purposes, conventional DFT calculations, done in a nominally cubic lattice, play an important role.

In MnO, the system under study in the present exercise, the origin of antiferromagnetism can be understood; the correct antiferromagnetic phase has the lowest total energy and a band gap, following already from the results of conventional GGA calculation.

Neglecting very small distortions which occur in different magnetic phases, MnO has a B1 (NaCl) structure. The latter is described in the input file as follows:

```
LatticeConstant 4.43 Ang
%block LatticeVectors
0.00 0.50 0.50
0.50 0.00 0.50
0.50 0.50 0.00
%endblock LatticeVectors
AtomicCoordinatesFormat ScaledCartesian
%block AtomicCoordinatesAndAtomicSpecies
0.000 0.000 0.000 1 # Mn
0.500 0.500 0.500 2 # O
%endblock AtomicCoordinatesAndAtomicSpecies
```

The lattice constant is set at the experimental value, kept constant throughout the present exercise.

In order to run a spin-polarized calculation, we set:

```
Spin polarized
```

and moreover initialize non-zero (in fact, maximal) spin on Mn site:

```
%block DM.InitSpin # Initial magnetic order (on Mn only)
1 +
%endblock DM.InitSpin
```

The rest of the input file sets calculation parameters (GGA, k-mesh, mesh cutoff, …) and notably provides printing out Mulliken populations, saving charge (and magnetic) density, and writing down the density of states:

```
WriteMullikenPop 1
SaveRho
%block ProjectedDensityOfStates
-25.0 10.0 0.1 700 eV
%endblock ProjectedDensityOfStates
```

This output part will be identical for calculations with different magnetic orderings.

Tasks:

1. When the first calculation is done, check the local (on Mn and O sites) and total (per unit cell) magnetic moments. Plot the spin-resolved density of states (total and that for Mn and O sites).

2. Prepare the input file(s) for antiferromagnetc structures of MnO and run the calculations. Note that an antiferromagnetic ordering doubles the unit cell: it contains now two Mn atoms (with opposite spins) and, correspondingly, two O atoms. Don’t forget to introduce changes in the lattice vectors, and atomic coordinates.

We’ll consider two different antiferromagnetic structures:

AF1 has, what is called, the [001] ordering, that is, all Mn atoms in a given [001] plane are equivalent (has the same spin orientation), and between the consecutive [001] planes the spin orioentations alternate.

AF2 has the [111] ordering, that is, equivalent spin orientation is throughout a given [111] plane, and alternates between such planes.

Prepare the `%block LatticeVectors`

for AF1 and AF2 cases. Check that the unit cell volume (mixed product
of three lattice vectors) is twice that of the FM case (which equals \(a_0^3/4\)).
Provide coordinates of (four) atoms, initialize spins in opposite sense
at two Mn atoms, and run the calculations.

3. Check Mulliken populations; plot total (from corresponding DOS files) and (optionally) atom-resolved densities of states. Check that the AF structures have pronounced band gaps.

4. Using the calculated RHO, visualize the spin density in each magnetic structure. You’ll see that the spin density has nearly perfect spherical distribution in the vicinity of Mn sites; why?

5. For the AF2 structure, you can find in the occupied part of the Mn3d-related density of states two groups of peaks, related to t2g and eg states of Mn. Try to vizualize them separately, selecting for each of these groups a corresponding energy interval and calculating LDOS. Why the separation into t2g and eg states is not so neat in the AF1 structure?