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

Note: It is recommended that you go through Tutorial 8 first (bcc Fe without spin-orbit).

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

Outline: Plot the spin-orbit-coupled bands of ferromagnetic bcc Fe. Plot the Fermi-surface contours on a plane in the Brillouin zone.
Directory: tutorials/tutorial17/ 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

Note that num_wann =18 in Fe.win, but only nine trial orbitals are provided. The line

Input file

spinors = true

tells wannier90 to use in step 3 below the specified trial orbitals on both the up- and down-spin channels, effectively doubling their number.

Run pwscf to obtain the ferromagnetic ground state of iron¹
Terminal
```
pw.x < Fe.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid
Terminal
```
pw.x < Fe.nscf > nscf.out
```
Run wannier90 to generate a list of the required overlaps (written into the Fe.nnkp file)
Terminal
```
wannier90.x -pp Fe
```
Run pw2wannier90 to compute:
- The overlaps $\langle u_{n{\bf k}}\vert u_{m{\bf k}+{\bf b}}\rangle$ between spinor Bloch states (written in the Fe.mmn file)
- The projections for the starting guess (written in the Fe.amn file)
- The spin matrix elements $\langle \psi_{n{\bf k}}\vert \sigma_i\vert \psi_{m{\bf k}}\rangle$, $i=x,y,z$ (written in the Fe.spn file)
Terminal
```
pw2wannier90.x < Fe.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.\
Terminal
```
wannier90.x Fe
```
Run postw90 to compute the energy eigenvalues and spin expectation values.
Terminal
```
postw90.x Fe # (1)! 
mpirun -np 8 postw90.x Fe # (2)!
```
1. serial execution
2. example of parallel execution with 8 MPI processes

In this tutorial we use the module kpath to plot the energy bands coloured by the expectation value of the spin along [001]:

Input file

kpath = true

kpath_task = bands

kpath_bands_colour = spin

kpath_num_points=500

To plot the bands using gnuplot (version 4.2 or higher) issue

Terminal

gnuplot

Gnuplot shell

load 'Fe-bands.gnu'

or, using python,

Terminal

python Fe-bands.py

Next we plot the Fermi-surface contours on the (010) plane $k_y=0$, using the kslice module. Set kpath = false and uncomment the following instructions in Fe.win,

Input file

kslice = true

kslice_task = fermi_lines

fermi_energy = [insert your value here]

kslice_corner = 0.0 0.0 0.0

kslice_b1 = 0.5 -0.5 -0.5

kslice_b2 = 0.5 0.5 0.5

kslice_2dkmesh = 200 200

taking the Fermi level value from scf.out. The energy eigenvalues are computed on a $200\times 200$ $k$-point grid covering the BZ slice. The lines of intersection between the Fermi surface and the (010) plane can be visualized with the gnuplot or python scripts generated at runtime,

Terminal

gnuplot

Gnuplot shell

load 'Fe-kslice-fermi_lines.gnu'

or

Terminal

python Fe-kslice-fermi_lines.py

The Fermi lines can be colour-coded by the spin expectation value $\langle S_z\rangle$ of the states on the Fermi surface. Add to Fe.win the line

Input file

kslice_fermi_lines_colour = spin

and re-run postw90. The names of the gnuplot and python scripts generated at runtime are unchanged. (However, the plotting algorithm is different in this case, and the lines are not as smooth as before. You may want to increase kslice_2dkmesh.)

Further ideas¶

Redraw the Fermi surface contours on the (010) plane starting from a calculation without spin-orbit coupling, by adding to the input files iron_{up,down}.win in Tutorial 8 the lines
Input file
```
kslice = true

kslice_task = fermi_lines

fermi_energy = \[insert your value here\]

kslice_corner = 0.0 0.0 0.0

kslice_b1 = 0.5 -0.5 -0.5

kslice_b2 = 0.5 0.5 0.5

kslice_2dkmesh = 200 200
```
before running postw90,
Input file
```
postw90.x iron_up

postw90.x iron_dn
```
The python scripts generated at runtime draw the up- and down-spin Fermi lines on separate figures. To draw them together, use the script iron_updn-kslice-fermi_lines.py provided with Tutorial 17 (or merge the two generated scripts). Compare the Fermi lines with and without spin-orbit, and note the spin-orbit-induced avoided crossings.
In Tutorial 8 we obtained MLWFs separately for the up- and down-spin channels of bcc Fe without spin-orbit. The Wannier-interpolated DOS was therefore automatically separated into minority and majority contributions. For a spinor calculation we can still spin-decompose the DOS, using
Input file
```
dos = true

spin_decomp = true

dos_kmesh = 25 25 25
```
The data file Fe-dos.dat created by postw90 contains the up-spin and down-spin contributions in the third and fourth columns,
Terminal
```
gnuplot
```
Gnuplot shell
```
plot 'Fe-dos.dat' u (-\$3):(\$1-12.6285) w
l,'Fe-dos.dat' u (\$4):(\$1-12.6285) w l
```
(You should replace 12.6285 with your value of the Fermi energy). An alternative approach is to project the DOS onto the up-spin and down-spin WFs separately. To find the DOS projected onto the up-spin (odd-numbered) WFs replace spin_decomp = true with
Input file
```
dos_project = 1,3,5,7,9,11,13,15,17
```
and re-run postw90. This approach has the advantage that it does not require the Fe.spn file.

Please note the following counterintuitive feature in pwscf: in order to obtain a ground state with magnetization along the positive z-axis, one should use a negative value for the variable starting_magnetization. ↩