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17: Iron — Spin-orbit-coupled bands and Fermi-surface contours

  • Outline: Plot the spin-orbit-coupled bands of ferromagnetic bcc Fe. Plot the Fermi-surface contours on a plane in the Brillouin zone.
Image title
Unit cell of Iron crystal plotted with the XCrySDen program.

1-6. Compute the MLWFs and compute the energy eigenvalues and spin expectation values.

The final state for all the 18 MLWFs is

Output file
 Final State
  WF centre and spread    1  ( -0.709848,  0.000000,  0.000000 )     1.08973288
  WF centre and spread    2  ( -0.685480, -0.000000,  0.000000 )     1.10285536
  WF centre and spread    3  (  0.709848, -0.000000,  0.000000 )     1.08973288
  WF centre and spread    4  (  0.685480, -0.000000,  0.000000 )     1.10285536
  WF centre and spread    5  ( -0.000000, -0.709848,  0.000000 )     1.08973287
  WF centre and spread    6  (  0.000000, -0.685480,  0.000000 )     1.10285536
  WF centre and spread    7  (  0.000000,  0.709848,  0.000000 )     1.08973288
  WF centre and spread    8  (  0.000000,  0.685480,  0.000000 )     1.10285536
  WF centre and spread    9  (  0.000000, -0.000000, -0.709835 )     1.08977307
  WF centre and spread   10  (  0.000000, -0.000000, -0.685503 )     1.10302800
  WF centre and spread   11  ( -0.000000, -0.000000,  0.709835 )     1.08977304
  WF centre and spread   12  (  0.000000, -0.000000,  0.685503 )     1.10302800
  WF centre and spread   13  ( -0.000000, -0.000000, -0.000000 )     0.43232470
  WF centre and spread   14  ( -0.000000, -0.000000, -0.000000 )     0.41118748
  WF centre and spread   15  (  0.000000,  0.000000, -0.000000 )     0.43232470
  WF centre and spread   16  (  0.000000,  0.000000, -0.000000 )     0.41118748
  WF centre and spread   17  ( -0.000000,  0.000000,  0.000000 )     0.43232866
  WF centre and spread   18  ( -0.000000,  0.000000,  0.000000 )     0.41119649
  Sum of centres and spreads (  0.000000,  0.000000,  0.000000 )    15.68650457

         Spreads (Ang^2)       Omega I      =    11.898334117
        ================       Omega D      =     0.031570932
                               Omega OD     =     3.756599523
    Final Spread (Ang^2)       Omega Total  =    15.686504572
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To plot the bands using python.

Terminal
> python Fe-bands.py

The interpolated band structure of Fe with spin-orbit interaction using the module kpath is shown in Figure 2. The color scheme is used to show the expectation value of the spin operator \(\hat{S}_z\) in units of \(\hbar/2\).

Image title
wannier90 interpolated bands of Fe computed from a DFT calculation with spin-orbit interaction. Colour-scheme shows the expectation value \(\langle \hat{S}_z \rangle\) in units of \(\hbar/2\).

Next we plot the Fermi-surface contours on the (010) plane \(k_y = 0\), using the kslice module.
spin-orbit kslice no spin-orbit
Fermi-surface contours on the (010) plane (\(k_y=0\)): a. with spin-orbit coupling (kslice module); b. without spin-orbit coupling.

Further ideas

  • Redraw the Fermi surface contours on the (010) plane starting from a calculation without spin-orbit coupling (SOC), by adding to the input files iron_{up,down}.win in Example 8.

    The Fermi surface contours on the (010) plane without SOC are shown in Figure 3-b.

  • For a spinor calculation we can still spin-decompose the DOS.
    spin-decomposed projected

    Spin-decomposed DOS (panel a) with spin-up (red) and spin-down (blue) components. Projected DOS on odd-indexed MLWFs (red) and even-indexed (blue).