Spin spiral calculations

Spin spiral calculations are supported for the non-collinear DFT without spin orbit coupling (SOC), which is based on the generalized Bloch theorem [79,80,86,87]. It should be noted that the inclusion of SOC is not compatible with the functionality, since the SOC violates the spiral symmetry imposed by the generalized Bloch theorem. To acknowledge in any publications by using the functionality, the citation of the reference [79,80] would be appreciated. To do the calculations, the following options are first set respectively

   scf.SpinPolarization        NC        # On|Off|NC
   scf.Generalized.Bloch       on        # On|Off, default=off
   scf.SpinOrbit.Coupling      off       # On|Off, default=off

In the spiral structure there are two quantities to determine the spiral configuration which should be paid attention. The first one is the cone angle. Since the spiral structure is of two different kinds, i.e., the conical spiral ($0<\theta<90$) and the flat spiral ($\theta=90$), where $\theta$ is the cone angle, $\theta$ should then be specified for the magnetic atoms. In OpenMX, the cone angle $\theta$ is defined as the orientation for the spin magnetic moment as shown in Fig. 71(a). You can find how to specify the cone (Euler) angle in section of 34 'Non-collinear DFT'. For example, the cone (Euler) angle can be specified as follows:

   <Atoms.SpeciesAndCoordinates
    1  Fe  0.0    0.0    0.0     10.0    6.0   90.0   0.0   0.0   0.0  1 off
   Atoms.SpeciesAndCoordinates>

In the example above, the flat spiral will be achieved by setting $\theta=90$ degree as written in the 8th column. In addition, you can also specify the initial angle of $\varphi$ as described in the 9th column. For fixing the cone angle, the constraint scheme is also available as explained in section of 38 'Constraint DFT for non-collinear spin orientation'.

Figure 71: (a) Definition of cone (Euler) angles for the magnetic atom. (b) The magnetic moment of atom at site $i$ which is given by $M_i=M_i\{\cos({\bf q}\cdot {\bf R}_i+\varphi_0) \sin(\theta_i)+\sin({\bf q}\cdot {\bf R}_i+\varphi_0))\sin(\theta_i)+\cos(\theta_i)\}$.

The second one is the spiral vector q specified by

   Spin.Spiral.Vector   0.0  0.0  0.0    # q1 q2 q3

In the above format, the spiral vector is specified by the fractional coordinates spanned by the reciprocal lattice vectors. The first, second, and third columns denote the components of spiral vector for the reciprocal vectors $\tilde{\bf a}_1$, $\tilde{\bf a}_2$, and $\tilde{\bf a}_3$, respectively. Then, the spin angle at atomic site $i$ is given by

$$M_i=M_i\{\cos({\bf q}\cdot {\bf R}_i+\varphi_0) \sin(\theta_i)+\sin({\bf q}\cdot {\bf R}_i+\varphi_0))\sin(\theta_i)+\cos(\theta_i)\}.$$

With the translation by the lattice vectors, the spin angle at each atomic site is rotated according to the equation.

As an example of the spiral calculation, the spiral ground state of the fcc Fe is provided in Fig. 72. We observe that the spiral ground state occurs at (0, 0, 0.6) and (0.2, 0, 1) defined in Cartesian coordinates in units of $2\pi/a$. Indeed, using LCPAO method, the spiral calculation needs the appropriate settings, such as the number of k points, number of orbitals, cutoff radius, and cutoff energy for reaching the convergence and reliable result, a similar discussion can be found in Ref. [87]. You can try to set those parameters to compare all of the results. In addition, although there are some available mixing schemes specified by the keywords 'scf.Mixing.Type', we strongly recommend RMM-DIISH as your choice. As an experience, this option can reach the convergence much faster than other choices for all tested systems, the detail explanations can be found in Sec. 13 'SCF convergence'.

Figure 72: (a) Profiles of the total energy and (b) magnetic moment of the spin spiral calculation of the fcc Fe as a function of spiral vector ${\bf q}$ defined in the Cartesian coordinates. The input file used for the calculation is 'Fefcc-SpinSpiral.dat' in the directory 'work'.