1: Gallium Arsenide — MLWFs for the valence bands¶

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

Image title — Unit cell of GaAs crystal plotted with the XCrySDen program.

Inspect the output file gaas.wout.

Table 1 shows the converged values (after 20 iterations) for a \(2\times2\times2\) \(\mathbf{k}\)-point mesh of the total spread functional \(\Omega\) and its three components, i.e. the gauge-invariant component \(\Omega_{\text{I}}\), the off-diagonal component of the gauge-dependent part \(\Omega_{\text{OD}}\), and the diagonal component of the gauge-dependent part \(\Omega_{\text{D}}\), respectively. These can be found at the end of the gaas.wout file, from the line starting with Final State. You will find the MLWF centres and their spreads together with the information on the spread functional components as reported below, and summarized in Table 1.
Output file
```
 Final State
  WF centre and spread    1  ( -0.866253,  1.973841,  1.973841 )     1.11672024
  WF centre and spread    2  ( -0.866253,  0.866253,  0.866253 )     1.11672024
  WF centre and spread    3  ( -1.973841,  1.973841,  0.866253 )     1.11672024
  WF centre and spread    4  ( -1.973841,  0.866253,  1.973841 )     1.11672024
  Sum of centres and spreads ( -5.680188,  5.680188,  5.680188 )     4.46688098

         Spreads (Ang^2)       Omega I      =     3.956862958
        ================       Omega D      =     0.008030049
                               Omega OD     =     0.501987969
    Final Spread (Ang^2)       Omega Total  =     4.466880976
  -----------------------------------------------------------------------------
```
The geometric centre lies along the Ga-As bond, slightly closer to As than Ga. To see this, we introduce a measure \(\beta\) defined as

\[\beta \triangleq \frac{d_{\text{Ga-MLWF}}}{d_{\text{Ga-As}}},\]

where \(d_{\text{Ga-MLWF}}\) is the distance of the Ga atom placed in the origin and the MLWF centre (along the Ga-As bond), and \(d_{\text{Ga-As}}\) is the Ga-As bond length, cf. Ref. ¹. A value of 0.5 corresponds to the MLWF centre being equidistant from the Ga atom and As atom. In our case, we find:

\[\beta = \frac{d_{\text{Ga-MLWF(2)}}}{d_{\text{Ga-As}}} = \frac{0.866253\sqrt{3}}{1.4200\sqrt{3}} \approx 0.61 .\]

Maximum RAM allocated for the wannierisation was 0.06Mb.

Table 1. Converged values of the components of spread functional and their sum, given in Å\(^2\). Here \(\beta\) is the distance of the Ga atom placed in the origin and the MLWF centre (along the Ga-As bond) as a fraction of the Ga-As bond length \(2.4595\) Å.

MP mesh \(\Omega\) \(\Omega_{\text{I}}\) \(\Omega_{\text{OD}}\) \(\Omega_{\text{D}}\) \(\beta\)

\(2\times2\times2\) 4.467 3.957 0.502 0.008 0.610
Plot the MLWFs.

In Figure 2 are shown the four valence MLWFs as plotted by XCrySDen, where we used the following parameters in the Tools → Data Grid section:
XCrySDen: Tools → Data Grid
```
Degree of triCubic Spline = 3;
Isovalue = 0.95;
Render +/- isovalue = yes
```
Four valence MLWFs for the Ga-As system plotted using the XCrySDen visualisation program: a. 1^st, b. 2^nd, c. 3^rd, and d. 4^th valence MLWF.

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

With XCrySDen we can also plot the 3^rd MLWF to check its periodicity. The period in each direction is given by the spacing used to sample the first irreducible Brillouin zone, i.e. the \(\mathbf{k}\)-point mesh. We used a \(2\times2\times2\) \(\mathbf{k}\)-point mesh, hence we expect to find a periodic image of our MLWF in a supercell which is 2 times larger than the unit cell along each direction. This is shown in Figure 3, where the 3^rd MLWF has been plotted using both the XCrySDen program and the VESTA program.

3^rd MLWF with a supercell value of 3 and for an isovalue of 0.5 using a. XCrySDen and b. VESTA.

Nicola Marzari and David Vanderbilt. Maximally localized generalized wannier functions for composite energy bands. Phys. Rev. B, 56:12847–12865, Nov 1997. ↩