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1: Gallium Arsenide — MLWFs for the valence bands

  • Outline: Obtain and plot MLWFs for the four valence bands of GaAs.

  • Generation details: From pwscf, using norm-conserving pseudopotentials and a
    2\(\times\)2\(\times\)2 k-point grid. Starting guess: four bond-centred Gaussians.

  • Directory: tutorials/tutorial01/ Files can be downloaded from here

  • Input Files

    • gaas.win The master input file

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

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

    • UNK00001.1 The Bloch states in the real space unit cell. For plotting only.

  • Run wannier90 to minimise the MLWFs spread

    Terminal
    wannier90.x  gaas

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

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

    Input file
    wannier_plot = true

    and re-running wannier90. To visualise the MLWFs we must represent them explicitly on a real space grid (see the User guide page). As a consequence, plotting the MLWFs is slower and uses more memory than the minimisation of the spread. The four files that are created (gaas_00001.xsf, etc.) can be viewed using XCrySDen, e.g.,

    Terminal
    xcrysden --xsf gaas_00001.xsf

    Hint

    Once XCrySDen starts, click on Tools \(\rightarrow\) Data Grid in order to specify an isosurface value to plot.

    For large systems, plotting the MLWFs may be time consuming and require a lot of memory. Use the keyword wannier_plot_list to plot a subset of the MLWFs. E.g., to plot the 1st and 3rd MLWFs use

    Input file
    wannier_plot_list = 1 3

    The MLWFs are plotted in a supercell of the unit cell. The size of this supercell is set through the keyword wannier_plot_supercell. The default value is 2 (corresponding to a supercell with eight times the unit cell volume). We recommend not using values great than 3 as the memory and computational cost scales cubically with supercell size.

    Plot the 3rd MLWFs in a supercell of size 3. Choose a low value for the isosurface (say 0.5). Can you explain what you see?

    Hint

    For a finite k-point mesh, the MLWFs are in fact periodic and the period is related to the spacing of the k-point mesh. For mesh with \(n\) divisions in the \(i^{\mathrm{th}}\) direction in the Brillouin zone, the MLWFs "live" in a supercell \(n\) times the unit cell.