Fully relativistic

The fully relativistic effects including the spin-orbit coupling within the pseudopotential scheme can be included in the non-collinear DFT calculations [12,21,15], while the inclusion of the spin-orbit coupling is not supported in the collinear DFT calculation. The inclusion of fully relativistic effects is made by the following two steps:

(1) Making of j-dependent pseudopotentials

First, you are requested to generate j-dependent pseudopotentials using ADPACK. For your convenience, the j-dependent pseudopotentials are available for many elements in the database Ver. 2013 [102]. The details how to make the j-dependent pseudopotential are found in the manual of ADPACK.

(2) SCF calculation

If you specify j-dependent pseudopotentials in the specification of '$<$Definition.of.Atomic.Species', it is possible to include spin-orbit coupling by the following keyword 'scf.SpinOrbit.Coupling':

    scf.SpinOrbit.Coupling      on         # On|Off, default=off
Then, the spin-orbit coupling can be self-consistently incorporated within the pseudopotential scheme rather than a perturbation scheme. Due to the spin-orbit coupling, $\alpha$ and $\beta$ spin components in the two component spinor can directly interact. In order to determine the absolute spin orientation in the non-collinear DFT calculations, you have to include the spin-orbit coupling, otherwise the spin orientation is not uniquely determined in real space. As an illustration of spin-orbit splitting, we show band structures of a bulk GaAs calculated by the non-collinear DFT without and with spin-orbit coupling in Fig. 28, where the input file is 'GaAs.dat' in the directory 'work'. In Fig. 28(b) we can see that there are spin-orbit splittings in the band dispersion, while no spin-orbit splitting is observed in Fig. 28(a). The spin-orbit splittings at two k-points, $\Gamma $ and $L$, are listed together with the other calculations and experimental values in Table 4. We see a good agreement in this table.

Figure 28: Band structures of a bulk GaAs calculated by the non-collinear DFT (a) without and (b) with the spin-orbit coupling. In these calculations, Ga7.0-s2p2d2 and As7.0-s2p2d2 were used as a basis set, and Ga_CA13.vps and As_CA13.vps were used for pseudopotentials, which are stored in the database Ver. 2013. For the exchange-correlation terms, LDA was used. We used 12$\times $ 12$\times $ 12 and 140 (Ryd) for scf.Kgrid and scf.energycutoff, respectively. Also the experimental value (5.65Å) was used for the lattice constant. The input file is 'GaAs.dat' in the directory 'work'.

Table 4: Calculated spin-orbit splittings (eV) at the $\Gamma _{15v}$ and the $L_{3v}$ of a bulk GaAs. The other theoretical values (LMTO: Ref. [74], PP: Ref. [75]) and experimental value (Ref.[76]) are also shown for comparison. The calculation conditions are given in the caption of Fig. 28 and the input file is 'GaAs.dat' in the directory 'work'.

Level OpenMX LMTO PP Expt.
$\Gamma _{15v}$ 0.344 0.351 0.35 0.34
$L_{3v}$ 0.213 0.213 0.22