Purpose: In this tutorial you will learn how to set up and perform spin-polarized calculations including spin-orbit coupling. As a first example, we consider the ferromagnetic bcc Fe. We calculate the density of state for both majority and minority spins and estimate the spin and orbital moment. In addition, we set up an exciting calculation in the case of anti-ferromagnetic ordering using as example bcc Cr.
0. Defining relevant shell variables
Very important: Before starting, the following shell variable must be set by the user:
EXCITINGROOT = Directory where exciting has been downloaded, e.g.: /home/user/exciting .
The setting of this variable can be done in a bash shell by typing (from now on the symbol $ will indicate the shell prompt):
$ export EXCITINGROOT=/local_path_to_exciting_root
Please note: In the input-files shown on this page, the placeholder EXCITINGROOT always needs to be replaced by the value of the $EXCITINGROOT variable.
1. General consideration on spin polarized calculations
Spin-polarized calculations in exciting always require to split the non-magnetic atomic configurations by using an external field (the attribute bfieldc, belonging to the element spin). This external field is used only to allow the code to distinguish between majority and minority spins and, in fact, should vanish during the self-consistent calculation. Otherwise, a fictitious external magnetic contribution will be added to the result. The damping of the external magnetic field is defined by setting the attribute reducebf. For instance, the choice reducebf=0.5 means that at every new iteration the external magnetic field will be reduced by one half.
Furthermore, metallic systems require a much larger k-point mesh than semiconductor or insulator ones. Therefore, for metals a convergence study on the size of this mesh should be performed as a preliminary calculation. Another thing to be noticed, is that this convergence study should be performed using as target quantity either the total energy or charge density, as the magnetic moment itself is not a variational quantity during the SCF cycle. In order to reduce the computational time, in this tutorial we use a very small k-point mesh, (8x8x8), instead of the ones required for very precise calculations, (over the 20x20x20 mesh).
Last, but not least, a proper choice of the following parameters is important:
- The smearing width swidth: Since for calculating integrals within the Brillouin zone a smearing method is used, results should always be checked with respect to the smearing width (groundstate-attribute swidth).
- The number of empty states nempty: In exciting, magnetic properties are obtained in a second-variational scheme . This means that, in each iteration of the scf run, first the non-magnetic equations are solved (first-variational scheme). The eigenfunctions obtained in this way are not the final solutions, but they are already close to it — for this reason they are used as a (very good, indeed!) basis set for expanding the final wavefunctions (to construct spinors) in the second-variational scheme. In this case, a larger number of empty states nempty is required with respect to non-spinpolarized calculations. Furthermore, in the treatment of the spin-orbit interaction via a second-variational scheme, a higher number of empty states is required.
2. An example including spin-orbit interaction: Ferromagnetic bcc Fe
For the 3d metals, the correct ferromagnetic ground state is reproduced using GGA (Perdew-Burke-Ernzerhof) for the exchange-correlation functional. Spin variables are defined by setting, within the element spin, the attributes bfieldc="0.0d0 0.0d0 0.5d0", reducebf="0.5d0", and spinorb="true". In addition, in order to decompose the magnetic moments into spin and orbital contributions, we include the element LSJ (notice that the xml input file is case sensitive!). Finally, the complete input file looks like this:
<?xml version="1.0" encoding="UTF-8"?> <?xml-stylesheet href="inputtohtml.xsl" type="text/xsl"?> <input xsi:noNamespaceSchemaLocation="EXCITINGROOT/xml/excitinginput.xsd" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsltpath="EXCITINGROOT/xml/"> <title>FM bcc Fe</title> <structure autormt="true" speciespath="EXCITINGROOT/species"> <crystal scale="5.416"> <basevect> 0.5 0.5 -0.5 </basevect> <basevect> 0.5 -0.5 0.5 </basevect> <basevect>-0.5 0.5 0.5 </basevect> </crystal> <species speciesfile="Fe.xml"> <atom coord="0.00 0.00 0.00"/> </species> </structure> <groundstate ngridk="8 8 8" xctype="GGAPerdew-Burke-Ernzerhof" stype= "Methfessel-Paxton 1" swidth="0.01" rgkmax="8.0" nempty="5"> <spin bfieldc="0.0d0 0.0d0 0.5d0" reducebf="0.5d0" spinorb="true"> </spin> </groundstate> <properties> <dos nsmdos="0" winddos="-0.3d0 0.3d0" /> <LSJ/> </properties> </input>
After the execution, the value of the total magnetic moment per unit cell is written in the file MOMENT.OUT as well as in INFO.OUT, where one can also find the decomposition of the magnetic moment per atom, in the muffin-tin spheres, and in the interstitial space. The decomposition of angular and spin moments is done only after the SCF run is finished. Thus, use do="skip" to obtain the orbital and spin moments.
The expectation values of the components (in x,y, and z direction) of L, S, and J=L+S are printed in the output file LSJ.OUT. Notice that the values should be multiplied by 2 to get the total value of the orbital, spin, and total moments. In our example, the resulting value of the total spin (orbital) moment of bcc Fe should be close to 2.22 (0.07) μB, for comparison, WIEN2k calculations give a corresponding value of 2.20 (0.05) μB.
The calculated density of states for the up and down states of ferromagnetic bcc Fe obtained from the calculation is shown in the following plot:
- Check how spin and orbitals moments change by increasing the number of empty states (10, 30, 60, …).
3. Example for anti-ferromagnetic bcc Cr
For an anti-ferromagnetic calculation, the only major change with respect to the ferromagnetic one is that one has to define the spin axis for each atom. In exciting this is done by setting, next to the atomic positions, the attribute bfcmt, e.g., bfcmt="0.0 0.0 1.7". For describing this magnetic ordering, one needs to consider a unit cell containing two atoms (the standard unit cell for mono-atomic bcc crystals contains only one atom). In order to do this, the unit cell is chosen to be the cubic one, with crystal axis parallel to the Cartesian ones, and including two atoms with opposite internal spin axis. Thus, in the case of the anti-ferromagnetic bcc Cr the file input.xml should look like the following:
<?xml version="1.0" encoding="UTF-8"?> <?xml-stylesheet href="inputtohtml.xsl" type="text/xsl"?> <input xsi:noNamespaceSchemaLocation="EXCITINGROOT/xml/excitinginput.xsd" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsltpath="EXCITINGROOT/xml/"> <title>AFM bcc Cr</title> <structure autormt="true" speciespath="EXCITINGROOT/species"> <crystal scale="5.4424"> <basevect> 1.0 0.0 0.0 </basevect> <basevect> 0.0 1.0 0.0 </basevect> <basevect> 0.0 0.0 1.0 </basevect> </crystal> <species speciesfile="Cr.xml"> <atom coord="0.00 0.00 0.00" bfcmt="0.0 0.0 1.7"/> <atom coord="0.50 0.50 0.50" bfcmt="0.0 0.0 -1.7"/> </species> </structure> <groundstate ngridk="8 8 8" xctype="GGAPerdew-Burke-Ernzerhof" stype= "Methfessel-Paxton 1" swidth="0.05" rgkmax="8.0" nempty="10" vkloff="0.5d0 0.5d0 0.5d0"> <spin bfieldc="0.0d0 0.0d0 1.0d0" reducebf="0.5d0" spinorb="true"> </spin> </groundstate> </input>
After performing the exciting calculation, the resulting total moment per unit cell will be written in the output file MOMENT.OUT and should have a value very close to zero. The result for the magnetic moment on each atom can be found at the bottom (in the output corresponding to the last iteration) of the file INFO.OUT. The obtained value is about 1.15 μB for some higher number of empty states.
- How many empty states need to be considered to reach this stable magnetic moment value?
According to Landolt-Bornstein tables the Cr antiferromagnetic order is due to spin-density wave with magnitude of 0.8 μB at low temperatures. From the pseudopotential VASP calculations the magnetic moment is around 1.1 μB