11: Silicon — Valence and low-lying conduction states¶

Valence States¶

Outline: Obtain MLWFs for the valence bands of silicon.
Directory: tutorials/tutorial11/ Files can be downloaded from here
Input Files
- silicon.scf The pwscf input file for ground state calculation
- silicon.nscf The pwscf input file to obtain Bloch states on a uniform grid
- silicon.pw2wan Input file for pw2wannier90
- silicon.win The wannier90 input file
Run pwscf to obtain the ground state of silicon
Terminal
```
pw.x < silicon.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid. Note that we request the lower 4 (valence) bands
Terminal
```
pw.x < silicon.nscf > nscf.out
```
Run wannier90 to generate a list of the required overlaps (written into the silicon.nnkp file).
Terminal
```
wannier90.x -pp silicon
```
Run pw2wannier90 to compute the overlap between Bloch states and the projections for the starting guess (written in the silicon.mmn and silicon.amn files).
Terminal
```
pw2wannier90.x < silicon.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.
Terminal
```
wannier90.x silicon
```

Inspect the output file silicon.wout. The total spread converges to its minimum value after just a few iterations. Note that the geometric centre of each MLWF lies at the centre of the Si-Si bond. Note also that the memory requirement for the minimisation of the spread is very low as the MLWFs are defined by just the 4\(\times\)4 unitary matrices \(\mathbf{U}^{(\mathbf{k})}\).

Plot the MLWFs by adding the following keywords to the input file silicon.win

Input file

wannier_plot = true

and re-running wannier90. Visualise them using XCrySDen, e.g.,

Terminal

xcrysden --xsf silicon_00001.xsf

Valence + Conduction States¶

Outline: Obtain MLWFs for the valence and low-lying conduction-band states of Si. Plot the interpolated bandstructure. Apply a scissors correction to the conduction bands.
Input Files
- silicon.scf The pwscf input file for ground state calculation
- silicon.nscf The pwscf input file to obtain Bloch states on a uniform grid
- silicon.pw2wan Input file for pw2wannier90
- silicon.win The wannier90 input file

The valence and lower conduction states can be represented by MLWFs with \(sp^3\)-like symmetry. The lower conduction states are not separated by an energy gap from the higher states. In order to form localised WF we use the disentanglement procedure introduced in Ref. ¹. The position of the inner and outer energy windows are shown in this plot.

Modify the input file and run pwscf and wannier90.\ Inspect the output file silicon.wout. The minimisation of the spread occurs in a two-step procedure. First, we minimise \(\Omega_{\rm I}\) -- this is the extraction of the optimal subspace in the disentanglement procedure. Then, we minimise \(\Omega_{\rm O}+\Omega_{{\rm OD}}\).
Plot the bandstructure by adding the following commands to the input file silicon.win
Input file
```
restart = plot

bands_plot = true
```
and re-running wannier90. The files silicon_band.dat and silicon_band.gnu are created. To plot the bandstructure using gnuplot
Terminal
```
gnuplot
```
Gnuplot shell
```
load 'silicon_band.gnu'
```
The k-point path for the bandstructure interpolation is set in the kpoint_path block. Try plotting along different paths.

Further ideas¶

Compare the Wannier-interpolated bandstructure with the full pwscf bandstructure. Recompute the MLWFs using a finer \(k\)-point grid (e.g., 6\(\times\)6\(\times\)6 or 8\(\times\)8\(\times\)8) and note how the accuracy of the interpolation increases ².
Compute four MLWFs spanning the low-lying conduction states (see Ref. ¹).

I. Souza, N. Marzari, and D. Vanderbilt. Maximally localized wannier functions for entangled energy bands. Phys. Rev. B, 65:035109, 2001. ↩↩
J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza. Spectral and fermi surface properties from wannier interpolation. Phys. Rev. B, 75:195121, 2007. ↩