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3: Silicon — Disentangled MLWFs

  • Outline: Obtain disentangled MLWFs for the valence and low-lying conduction states of Si. Plot the interpolated band structure

  • Generation Details: From pwscf, using norm-conserving pseudopotentials and a
    4\(\times\)4\(\times\)4 k-point grid. Starting guess: atom-centred sp\(^3\) hybrid orbitals

  • Directory: tutorials/tutorial03/ Files can be downloaded from here

  • Input Files

    • silicon.win The master input file

    • silicon.mmn The overlap matrices \(\mathbf{M}^{(\mathbf{k},\mathbf{b})}\)

    • silicon.amn Projection \(\mathbf{A}^{(\mathbf{k})}\) of the Bloch states onto a set of trial localised orbitals

    • silicon.eig The Bloch eigenvalues at each k-point

The valence and lower conduction states can be represented by MLWFs with \(sp^3\)-like symmetry. The lower conduction states are not separated from the higher states by an energy gap. 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 the bandstructure plot.

  1. Run wannier90.

    Terminal
    wannier90.x silicon

    Inspect the output file silicon.wout. The minimisation of the spread occurs in a two-step procedure 1. First, we minimise \(\Omega_{\rm I}\) -- this is the extraction of the optimal subspace in the disentanglement procedure. Then, we minimise \(\Omega_{\rm D} + \Omega_{{\rm OD}}\).

  2. Plot the energy bands 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 run gnuplot

    Terminal
    gnuplot

    and within the gnuplot shell type

    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.

Image title
Bandstructure of silicon showing the position of the outer and inner energy windows.

  1. I. Souza, N. Marzari, and D. Vanderbilt. Maximally localized wannier functions for entangled energy bands. Phys. Rev. B, 65:035109, 2001. ↩↩