8: Iron — Spin-polarized WFs, DOS, projected WFs versus MLWFs¶

Outline: Generate both maximally-localized and projected Wannier functions for ferromagnetic bcc Fe. Calculate the total and orbital-projected density of states by Wannier interpolation.
Directory: tutorials/tutorial08/ Files can be downloaded from here
Input Files
- iron.scf The pwscf input file for the spin-polarized ground state calculation
- iron.nscf The pwscf input file to obtain Bloch states on a uniform grid
- iron_{up,down}.pw2wan Input files for pw2wannier90
- iron_{up,down}.win Input files for wannier90 and postw90
Note that in a spin-polarized calculation the spin-up and spin-down MLWFs are computed separately. (The more general case of spinor WFs will be treated in Tutorial 17).
Run pwscf to obtain the ferromagnetic ground state of bcc Fe
Terminal
```
pw.x < iron.scf > scf.out
```
Note

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.
Run pwscf to obtain the Bloch states on a uniform k-point grid
Terminal
```
pw.x < iron.nscf > nscf.out
```
Run wannier90 to generate a list of the required overlaps (written into the .nnkp files).
Terminal
```
wannier90.x -pp iron_up
wannier90.x -pp iron_dn
```
Run pw2wannier90 to compute the overlap between Bloch states and the projections for the starting guess (written in the .mmn and .amn files).
Terminal
```
pw2wannier90.x < iron_up.pw2wan > pw2wan_up.out
pw2wannier90.x < iron_dn.pw2wan > pw2wan_dn.out
```

Run wannier90 to compute the MLWFs.

Terminal

wannier90.x iron_up
wannier90.x iron_dn

Density of states¶

To compute the DOS using a \(25\times 25 \times 25\) \(k\)-point grid add to the two .win files

Input file

dos = true
dos_kmesh = 25

run postw90,

Input file

postw90.x iron_up
postw90.x iron_dn

and plot the DOS with gnuplot,

Terminal

gnuplot

Gnuplot shell

plot 'iron_up_dos.dat' u (-\$2):(\$1-12.6256) w
l,'iron_dn_dos.dat' u 2:(\$1-12.6256) w l

Energies are referred to the Fermi level (12.6256 eV, from scf.out). Note the exchange splitting between the up-spin and down-spin DOS. Check the convergence by repeating the DOS calculations with more \(k\)-points.

Projected versus maximally-localized Wannier functions¶

In the calculations above we chose \(s\), \(p\), and \(d\)-type trial orbitals in the .win files,

Input file

Fe:s;p;d

Let us analyze the evolution of the WFs during the gauge-selection step. Open one of the .wout files and search for "Initial state"; those are the projected WFs. As expected they are atom-centred, with spreads organized in three groups, 1+3+5: one \(s\), three \(p\), and five \(d\). Now look at the final state towards the end of the file. The Wannier spreads have re-organized in two groups, 6+3; moreover, the six more diffuse WFs are off-centred: the initial atomic-like orbitals hybridized with one another, becoming more localized in the process. It is instructive to visualize the final-state MLWFs using XCrySDen, following Tutorial 1. For more details, see Sec. IV.B of Ref. ¹.

Let us plot the evolution of the spread functional \(\Omega\),

Terminal

grep SPRD iron_up.wout > sprd_up

gnuplot

Gnuplot shell

plot 'sprd_up' u 6 w l

The first plateau corresponds to atom-centred WFs of separate \(s\), \(p\), and \(d\) character, and the sharp drop signals the onset of the hybridization. With hindsight, we can redo steps 4 and 5 more efficiently using trial orbitals with the same character as the final MLWFs,

Input file

Fe:sp3d2;dxy;dxz,dyz

With this choice the minimization converges much more rapidly.

Any reasonable set of localized WFs spanning the states of interest can be used to compute physical quantities (they are "gauge-invariant"). Let us recompute the DOS using, instead of MLWFs, the WFs obtained by projecting onto \(s\), \(p\), and \(d\)-type trial orbitals, without further iterative minimization of the spread functional. This can be done by setting

Input file

num_iter = 0

But note that we still need to do disentanglement! Recalculate the DOS to confirm that it is almost identical to the one obtained earlier using the hybridized set of MLWFs. Visualize the projected WFs using XCrySDen, to see that they retain the pure orbital character of the individual trial orbitals.

Orbital-projected DOS and exchange splitting¶

With projected WFs the total DOS can be separated into \(s\), \(p\) and \(d\) contributions, in a similar way to the orbital decomposition of the energy bands in Tutorial 4.

In order to obtain the partial DOS projected onto the \(p\)-type WFs, add to the .win files

Input file

dos_project = 2,3,4

and re-run postw90. Plot the projected DOS for both up- and down-spin bands. Repeat for the \(s\) and \(d\) projections.

Projected WFs can also be used to quantify more precisely the exchange splitting between majority and minority states. Re-run wannier90 after setting dos=false and adding to the .win files

Input file

write_hr_diag = true

This instructs wannier90 to print in the output file the on-site energies \(\langle {\bf 0}n\vert H\vert {\bf 0}n\rangle\). The difference between corresponding values in iron_up.wout and in iron_dn.wout gives the exchange splittings for the individual orbitals. Compare their magnitudes with the splittings displayed by the orbital-projected DOS plots. In agreement with the Stoner criterion, the largest exchange splittings occur for the localized \(d\)-states, which contribute most of the density of states at the Fermi level.

Image title — Evolution of the Wannier spread \(\Omega\) of the minority (spin-up) bands of bcc Fe during the iterative minimization of \(\widetilde{\Omega}\), starting from s, p and d-type trial orbitals.

X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt. Ab initio calculation of the anomalous hall conductivity by wannier interpolation. Phys. Rev. B, 74:195118, 2006. ↩