Spin-orbit coupling in 2nd variation

We usually use the scalar-relativistic FLAPW method to obtain the solutions of the KS equations in a "first variation". When spin-orbit coupling (SOC) has to be included, we do not consider SOC in the interstitial and vacuum regions, assume that only spherically-symmetric MT-sphere potential components contribute to SOC and neglect the small components of the wavefunctions. Therefore, only the large components of ${\Phi}_i$ contribute through the SOC matrix elements of the Hamiltonian

$$\begin{equation} \hat {\bf H} \Phi_i \;=\; \epsilon_i^0 \; \Phi_i + \left( \begin{array}{cc} \frac{1}{\zeta^2} {\mathbf \sigma} \cdot ( \nabla V_{-\mathbf \sigma}({\bf r}) \times {\bf p} ) & 0 \\ 0 & (\epsilon_i^0 - 1) {\rm I}_2 \end{array} \right) \Phi_i \end{equation}$$

where we neglect the spin-orbit contribution to the overlap of ${\Phi}_i$ due to SOC.

When spin-orbit interaction gives only a small contribution to the Hamiltonian, in a collinear calculation one can make use of that by recognizing that the eigenfunctions of the scalar-relativistic Hamiltonian are much more efficient basisfunctions for the variational treatment of the relativistic Hamiltonian than the original LAPW basis functions. Thus after determining the eigenvectors $\Phi^0_{{\bf k}\nu^\prime\sigma^\prime}({\bf r})$ of the scalar-relativistic Hamiltonian we expand the wavefunction $\Phi_{{\bf k}\nu}({\bf r})$ as

$$\begin{equation} \Phi_{{\bf k}\nu}({\bf r})= \sum_{\nu^\prime\sigma^\prime}^{2N_2} C_{{\bf k}\nu^\prime\sigma^\prime}^{\nu} \Phi^0_{{\bf k}\nu^\prime\sigma^\prime}({\bf r}), \end{equation}$$

where the expansion coefficients $C_{{\bf k}\nu^{\prime}\sigma^{\prime}}^\nu$ are assumed to be already multiplied with the Pauli spinor belonging to $\sigma^{\prime}$.

This constitutes a second variational eigenproblem where $N_2$ is the number of basisfunctions per spin with typically $N_2 \approx (1.3-1.5)N_1$ where $N_1$ is the number of occupied states.

The expansion coefficients are determined by the second eigenproblem as

$$\begin{equation} \sum_{\nu^{\prime\prime},\sigma^{\prime\prime}}^{2N_2} \langle {\bf k}\nu^{\prime}\sigma^{\prime} \mid \hat H \mid {\bf k}\nu^{\prime\prime}\sigma^{\prime\prime} \rangle C_{{\bf k}\nu^{\prime\prime}\sigma^{\prime\prime}}^\nu = \epsilon_{{\bf k}\nu} C_{{\bf k}\nu^{\prime}\sigma^{\prime}}^\nu \, . \label{eq:secular} \end{equation}$$

(In a non-magnetic system the eigenvectors of the first variation will simply be doubled for the above expansion.)

   <soc theta=".00" phi=".00" l_soc="T" spav="F" />

Note, that SOC usually lowers the symmetry of the crystal, e.g., cubic bcc Fe will become tetragonal if the magnetization points in one of the cubic axes. Therefore, already at the level of the input-file generator one can define &soc 0.0 0.0 / where the two numbers are again the angles $\vartheta$ and $\varphi$. If this is not done an activation of the SOC may conflict with the available symmetry operations determined by the input generator.

Taking SOC into account will not only add the spin-orbit contribution to the total energy (which allows calculating magnetic anisotropies) but also gives access to the orbital magnetic moments (projected onto the SQA). Note, that the SOC treatment in second variation leads to a "doubling of eigenvalues" since each state (with components in both spin-sectors now) will show up in both spin channels.

It is possible to restrict the relativistic treatment to only selected atoms in the unit cell by activating the socscale feature in the special tag of the atomSpecies section. E.g. setting socscale="0.0" switches off SOC in the selected atom species.

Apart from self-consistent SOC calculations, it is also possible to apply the magnetic force theorem to calculate the magnetocrystalline anisotopy. The corresponding documentation can be found here.

Note, that strong SOC will generally give rise to non-collinearity in the system so that this collinear + second variation scheme is not necessarily always a good choice. Then a non-collinear calculation and treatment of SOC in first variation is preferable.