Treatment of collinear magnetism

For the investigation of magnetic materials objects like the LAPW basis functions, the Hamiltonian and overlap matrices, the charge density, and the effective potential have to become spin-dependent. With the spin index $\sigma \in \{-1/2,1/2\}$ the spin-dependent quantities thus become

$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} \phi_{\vec{k}\vec{G}}(\vec{r}) \rightarrow \phi_{\vec{k}\vec{G}}^{\sigma}(\vec{r}) \text{, } ~~~~~ H_{\vec{G},\vec{G'}}^{\vec{k}} \rightarrow H_{\vec{G},\vec{G'}}^{\vec{k},\sigma}\text{, } ~~~~~ S_{\vec{G},\vec{G'}}^{\vec{k}} \rightarrow S_{\vec{G},\vec{G'}}^{\vec{k},\sigma}\text{,} \end{equation}$

$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} \rho(\vec{r}) \rightarrow \rho^{\sigma}(\vec{r}) \text{, } ~~~~~ \text{and } ~~~~~ V_\text{eff}(\vec{r}) \rightarrow V_\text{eff}^{\sigma}(\vec{r}) \text{.} \end{equation}$

For the potential, however, the coupling of the two spins only affects the XC potential and an optional Zeeman term. The effective potential becomes

$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} V_\text{eff}^{\sigma}(\vec{r}) = V_\text{H}(\vec{r}) + V_\text{ext}(\vec{r}) + V_\text{xc}^{\sigma}(\vec{r}) + \mu_\text{B} g_\text{e} \sigma B \text{,} \end{equation}$

where $\mu_\text{B}$ is the Bohr magneton, $g_\text{e}$ the g-factor of the electron, and $B$ an optionally applied magnetic field. Calculations with such external fields are discussed in a separate section.

Starting with a spin-polarized density Fleur determines the spin-dependent potential. Due to this spin-dependence also the radial functions of the LAPW basis in the MT spheres become spin-dependent. The Hamiltonian and Overlap matrices are subsequently also spin-dependent. The point where the algorithm connects the two spins again is the calculation of the Fermi energy and the total charge density.

In Fleur a spin dependency is only introduced if it is explicitly specified in the Fleur input file. The parameter calculationSetup/magnetism/@jspins has to be set to 2. For nonmagnetic calculations it is 1. The input generator sets it to 2 by default whenever there is an atom in the unit cell that is considered to be magnetic, i.e., Cr, Mn, Fe, Co, Ni as well as all atoms with partially filled $f$ shells. If a spin-dependent Fleur calculation for materials without such chemical elements has to be started the user has to specify this manually.

To obtain a magnetic solution of an SCF calculation the starting density already has to break a symmetric treatment of the spins. This is done by specifying for each atom initial magnetic moments that are already a prototype for the magnetic configuration to be investigated. Of course, this initial magnetic configuration may break some symmetries being present in the unit cell otherwise.

To ensure that the Fleur input generator does not detect the unwanted symmetries atoms of the same chemical element which are supposed to feature differing initial magnetic moments have to be distinguished. For this they are specified with different fractional atomic numbers, i.e. 26.01 and 26.02 for two different Fe species. Only the integer part of the number specifies the chemical element, the fractional part is only used to associate the atoms with different atom species that are automatically generated in the Fleur input file. In this file for each atom species the magnetic moment is specified in atomSpecies/species/@magMom. Before starting the SCF calculation these parameters have to be adapted to the specific needs of the calculation. For example, for the investigation of antiferromagnetic Cr a unit cell with two atoms has to be set up, where each atom has its own Cr species. For one of the species the default magnetic moment in the Fleur input file then has to be negated manually. For more sophisticated specifications of the initial electron configuration an electronConfig section can also be added to an atom species section and the occupation numbers for each state and spin can directly be provided there. If such a section is present the atomSpecies/species/@magMom parameter is ignored.

When performing calculations on magnetic materials magnetism-related properties are of interest. The most important of these quantities are the magnetic moments in the MT spheres and in the whole unit cell.

In the out.xml file for each SCF iteration the magnetic moments in the MT spheres are written out in the magneticMomentsInMTSpheres section. For each atom type an entry magneticMomentsInMTSpheres/magneticMoment exists which includes beyond the magnetic moment in the respective sphere the amount of charge for each spin. For the calculation of the magnetic moment for the whole unit cell Fleur writes out the two sections valenceDensity and allElectronCharges. The former one covers the valence electrons only while the latter one also considers the core electrons. In each of these there are spinDependentCharge entries that write out the amount of charge for each spin in the whole unit cell as well as in the different regions of the unit cell. The magnetic moment is obtained by substracting the numbers for the two spins from each other.

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