Second variational method: Magnetic Anisotropy Energy (MAE)

In the previous section, we introduced a constraint DFT method to control the spin direction. The constraint DFT method enables us to evaluate a magnetic anisotropy energy (MAE) for magnetic systems in a self-consistet manner based on the total energy. However, the computation tends to become very costly, and the calculation tends to be trapped to a local minima due to the degree of freedom of the occupation to degenerate localized states such as $d$-orbitals. A way to bypass these problems is to use a second variational method that a one-shot diagonalization within the non-colliear DFT is performed with spin-orbit coupling (SOI) and the charge density calculated by the collinear DFT as initial guess after getting the SCF charge density within the collinear DFT. Since the variational scheme is based on the Harris functional [14], the perturbation by the spin rotation and the SOI is taken into account only in the band energy. The double counting term does not depend on the spin rotation, while it looks changed in the output of OpenMX, since the energy terms are calculated by the output density rather than the input density (please don't be confused by the output). Using the second variational method, we first calculate a ferromagnetic state within the collinear DFT, resulting in the SCF charge density. Then, the one-shot diagonalization for the Hamiltonian including SOI is performed within the non-colliear DFT using the restart file storing the SCF charge density. The restart file can be read by the following keywords:

       scf.restart.filename               FePt
       scf.restart                        c2n

Using the keyword 'scf.restart.filename', the restart file to be read is specified. By 'c2n' for the keyword 'scf.restart' one can port a restart file generated by a collinear DFT calculation to a non-collinear DFT calculation. As an example, the restart file above is generated by an input file 'FePt.dat' available in the directory 'work'. In the second calculation, the spin direction can be specified by the following keywords:

       scf.Restart.Spin.Angle.Theta       90.0
       scf.Restart.Spin.Angle.Phi          0.0

The spin direction at all the spatial points is aligned along the direction determined by the two keywords which specify Euler angles. Therefore, it should be noted that the evaluation of the MAE by the second variational method is valid only for ferromagnetic systems. Let us demonstrate how the second variational method works to evaluate the MAE. Figure 36 shows the MAE curves for the spin rotational angle in the L1$_{0}$ FePt bulk. The result of 'Full SCF' was obtained by the constraint DFT method as explained in the section 38. The input file 'FePt-NC-SCF.dat' is available in the directory 'work'. At each spin rotational angle, the fully self-consistent calculation was performed including the SOI within the non-collinear DFT, where the electronic temperature is 300 K and the k-grid is $17^3$. The MAE of 2.78 meV/f.u. was obtained by 'Full SCF'. Two computational results by the second variational method are also shown in the Fig. 36. Using the restart file generated by the collinear calculation with the input file 'FePt.dat' in the directory 'work', we diagonalized once using an input file 'FePt-NC.dat' available in the directory 'work'.

Figure 36: Magnetic anisotropy energy (MAE) curves for the spin rotational angle in the L1$_{0}$ FePt bulk. The input file used for 'Full SCF' is 'FePt-NC-SCF.dat', and input files 'FePt.dat' and 'FePt-NC.dat' were used for the two calculations by the second variational method. These input files are found in the directory 'work'.

One is the result where the same electronic temperature (300K) and the same k-grid were used as for the 'Full SCF'. It is found that the MAE is 2.89 meV/f.u. The other is the result where the electronic temperatures (the k-grid) for the SCF calculation and the second calculation are 800 ($9^3$) and 300 K ($17^3$), respectively. The MAE is found to be 2.90 meV/f.u. in the case. Since the two caculations by the second variational method give almost the same result, the fact may allow us to control the electronic temperature and the k-grid for increasing the efficiency and accuracy for both the collinear caculation to generate the SCF charge density and non-collinear calculation to include the SOI. In the second variational method, it is very important to notice that we should look at 'Uele' instead of 'Utot' in the output file. Since the second variational scheme is based on the Harris functional, the MAE should be evaluated by 'Uele', while 'full SCF' relies on the total energy 'Utot'. We see that the difference between the full SCF calculation and the second variational method in evaluating the MAE is about 0.1 meV/f.u. Thus, it might be concluded that the second variational method is an efficient approach for the evaluation of MAE by considering the accuracy and efficiency.

The second variational scheme might be useful for not only the evaluation of MAE, but also calculations of band structure of large-scale systems, which may hamper direct SCF calculations by the non-collinear DFT method, to investigate how the SOI modifies the band structure.