3: Silicon — Disentangled MLWFs

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

Unit cell of Silicon crystal plotted with the XCrySDen program.

1. Inspect the output file silicon.wout.

   Starting from 4 \(sp3\) orbitals on each Silicon atom we obtain two sets of 4 WFs, all with the same spread, that show the \(sp3\) character one would expect from symmetry considerations. A summary of the wannierisation is given in Table 1. At the end of the .wout file you should find the info on the final state of the minimization, here we show an extract of the output file

   Output file

    Final State
     WF centre and spread    1  ( -0.460754, -0.460711, -0.460767 )     1.81241746
     WF centre and spread    2  ( -0.460743,  0.460722,  0.460718 )     1.81246400
     WF centre and spread    3  (  0.460703, -0.460761,  0.460685 )     1.81248774
     WF centre and spread    4  (  0.460704,  0.460724, -0.460764 )     1.81244947
     WF centre and spread    5  (  1.810128,  1.810112,  1.810113 )     1.81247628
     WF centre and spread    6  (  1.810097,  0.888662,  0.888617 )     1.81242110
     WF centre and spread    7  (  0.888640,  1.810140,  0.888660 )     1.81240614
     WF centre and spread    8  (  0.888643,  0.888652,  1.810090 )     1.81245230
     Sum of centres and spreads (  5.397417,  5.397539,  5.397353 )    14.49957450
   
            Spreads (Ang^2)       Omega I      =    11.849193709
           ================       Omega D      =     0.105470243
                                  Omega OD     =     2.544910550
       Final Spread (Ang^2)       Omega Total  =    14.499574503
    ------------------------------------------------------------------------------
   

   Table 1. Converged values of the components of spread functional and their sum, given in Å\(^2\).

   MP mesh	\(\Omega\)	\(\Omega_{\text{I}}\)	\(\Omega_{\text{OD}}\)	\(\Omega_{\text{D}}\)
   \(4\times4\times4\)	14.5	11.849	2.545	0.105

2. Plot the energy bands.

   As can be seen from DFT bandstructure plot in the wannier90 tutorial, that we report here, cf. Figure 2, the four lower valence bands are separated in energy from the higher conduction states (there is however an indirect band gap). The Fermi level lies inside the gap, making crystalline Silicon a semiconductor.

   The path in \(\mathbf{k}\)-space given in the tutorial (L-\(\Gamma\)-X-K-\(\Gamma\)) and shown in Figure 3-a-top is the following

   begin kpoint_path
   
   L 0.50000  0.50000 0.5000 G 0.00000  0.00000 0.0000
   
   G 0.00000  0.00000 0.0000 X 0.50000  0.00000 0.5000
   
   X 0.50000 -0.50000 0.0000 K 0.37500 -0.37500 0.0000
   
   K 0.37500 -0.37500 0.0000 G 0.00000  0.00000 0.0000
   
   end kpoint_path
   

   which gives the bandstructure shown in Figure 3-a-bottom.

   Extra: Try plotting along different paths.

   Another path usually used for Silicon is W-\(\Gamma\)-X-W-L-\(\Gamma\) shown in Figure 2-b-top and the corresponding bands are shown in Figure 3-b-bottom. To obtain this path you need to replace the previous kpoint_path block with the following block

   begin kpoint_path
   
   W  0.25000  0.75000  0.50000 G  0.00000   0.00000  0.00000
   
   G  0.00000  0.00000  0.00000 X  0.50000  0.50000  0.00000
   
   X  0.50000  0.50000  0.00000 W -0.25000  0.25000 -0.25000
   
   W -0.25000  0.25000 -0.25000 L  0.00000  0.50000  0.00000
   
   L  0.00000  0.50000  0.00000 G  0.00000  0.00000  0.00000
   
   end kpoint_path
   

   

   Bandstructure of Silicon showing the position of the Fermi level and of the inner and outer windows. Both the 4 valence bands and the 4 low-lying conduction bands are included in the calculation.

   Bandstructure of Silicon showing the position of the Fermi level and of both the inner and outer windows. The 4 valence bands together with the 4 low-lying conduction bands are included in the calculation. a. Interpolation with Wannier90 on the L-\(\Gamma\)-X-K-\(\Gamma\) path in \(\mathbf{k}\) space (red dots) and DFT reference bandstructure (solid black), with the corresponding \(\mathbf{k}\)-path shown in the Brillouin zone above. b. Interpolation with Wannier90 on the W-\(\Gamma\)-X-W-L-\(\Gamma\) path in \(\mathbf{k}\) space (red dots) and DFT reference bandstructure (solid black), with the corresponding \(\mathbf{k}\)-path shown in the Brillouin zone above.