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19: Iron — Orbital magnetization

Note: This tutorial requires a recent version of the pw2wannier90 interface.

  • Outline: Calculate the orbital magnetization of ferromagnetic bcc Fe by Wannier interpolation.

  • Directory: tutorials/tutorial19/ Files can be downloaded from here

  • Input files

    • Fe.scf The pwscf input file for ground state calculation

    • Fe.nscf The pwscf input file to obtain Bloch states on a uniform grid

    • Fe.pw2wan The input file for pw2wannier90

    • Fe.win The wannier90 and postw90 input file

The sequence of steps below is the same of Tutorials 17 and 18. If you have already run one of those tutorials, you can reuse the output files from steps 1 — 3 and 5. Steps 4 and 6 should be carried out again using the new input files Fe.pw2wan and Fe.win.

  1. Run pwscf to obtain the ground state of iron

    Terminal
    pw.x < Fe.scf > scf.out

  2. Run pwscf to obtain the Bloch states on a uniform k-point grid

    Terminal
    pw.x < Fe.nscf > nscf.out

  3. Run wannier90 to generate a list of the required overlaps (written into the Fe.nnkp file).

    Terminal
    wannier90.x -pp Fe

  4. Run pw2wannier90 to compute:

    • The overlaps \(\langle u_{n{\bf k}}\vert u_{m{\bf k}+{\bf b}}\rangle\) (written in the Fe.mmn file)

    • The projections for the starting guess (written in the Fe.amn file)

    • The matrix elements \(\langle u_{n{\bf k}+{\bf b}_1}\vert H_{\bf k}\vert u_{m{\bf k}+{\bf b}_2}\rangle\) (written in the Fe.uHu file)

    Terminal
    pw2wannier90.x < Fe.pw2wan > pw2wan.out

  5. Run wannier90 to compute the MLWFs.

    Terminal
    wannier90.x Fe

  6. Run postw90 to compute the orbital magnetization.

    Terminal
    postw90.x Fe # (1)! 
mpirun -np 8 postw90.x Fe  # (2)!
    1. serial execution
    2. example of parallel execution with 8 MPI processes

The orbital magnetization is computed as the BZ integral of the quantity \({\bf M}^{\rm orb}({\bf k})\) defined in this equation of the User Guide. The relevant lines in Fe.win are

Input file
berry = true

berry_task = morb

berry_kmesh = 25 25 25

fermi_energy = [insert your value here]

After running postw90, compare the value of the orbital magnetization reported in Fe.wpout with the spin magnetization in scf.out. Set iprint = 2 to report the decomposition of \({\bf M}^{\rm orb}\) into the terms \(J0\), \(J1\), and \(J2\) defined in Ref. 1.

To plot \(M_z^{\rm orb}({\bf k})\) along high-symmetry lines set berry = false and uncomment in Fe.win the block of instructions containing

Input file
kpath = true

kpath_task = bands+morb

After running postw90, issue

Terminal
python Fe-bands+morb_z.py

Compare with Fig. 2 of Ref. 1, bearing in mind the factor of \(-1/2\) difference in the definition of \({\bf M}^{\rm orb}({\bf k})\) (see Ch. 11 in the User Guide).

To plot \(M_z^{\rm orb}({\bf k})\) together with the Fermi contours on the (010) BZ plane set kpath = false, uncomment in Fe.win the block of instructions containing

Input file
kslice = true

kslice_task = morb+fermi_lines

re-run postw90, and issue

Terminal
python Fe-kslice-morb_z+fermi_lines.py

\(M_z^{\rm orb}({\bf k})\) is much more evenly distributed in \(k\)-space than the Berry curvature (see Tutorial 18). As a result, the integrated orbital magnetization converges more rapidly with the BZ sampling.

  1. M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza. Phys. Rev. B, 85:014435, 2012. ↩↩