11: Silicon — Valence and low-lying conduction states¶
Valence States¶
- Outline: Obtain MLWFs for the valence bands of silicon.
1-5. 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 \(4\times4\) unitary matrices \(U(\mathbf{k})\).
Below a snippet from the silicon.wout output file
Final State
WF centre and spread 1 ( -0.674701, 0.674701, -0.674701 ) 1.59185520
WF centre and spread 2 ( -0.674701, -0.674701, 0.674701 ) 1.59185520
WF centre and spread 3 ( 0.674701, 0.674701, 0.674701 ) 1.59185520
WF centre and spread 4 ( 0.674701, -0.674701, -0.674701 ) 1.59185520
Sum of centres and spreads ( -0.000000, 0.000000, 0.000000 ) 6.36742081
Spreads (Ang^2) Omega I = 5.801375426
================ Omega D = 0.000000000
Omega OD = 0.566045385
Final Spread (Ang^2) Omega Total = 6.367420811
------------------------------------------------------------------------------
Memory estimates may be found in the MEMORY ESTIMATE section of the
silicon.wout file.
*============================================================================*
| MEMORY ESTIMATE |
| Maximum RAM allocated during each phase of the calculation |
*============================================================================*
| Disentanglement 1.57 Mb |
| Wannierise: 0.47 Mb |
Converged values for the total spread functional and its components are shown in Table 1.
Table 1. Converged values of the components of spread functional and their sum, given in Å\(^2\).
| MP mesh | \(\Omega\) | \(\Omega_{\text{I}}\) | \(\Omega_{\text{OD}}\) | \(\Omega_{\text{D}}\) |
|---|---|---|---|---|
| \(4\times4\times4\) | 6.3674 | 5.8014 | 0.5660 | 0.0000 |
-
Plot the MLWFs.
The four MLWFs with \(\sigma\) character describing the valence manifold of Si are shown in Figure 2 (panels a, b, c, and d, respectively).
Four MLWFs for the valence manifold of Si: a. MLWF 1, b. MLWF 2, c. MLWF 3, and d. MLWF 4.
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.
Inspect the output file silicon.wout. The minimisation of the spread occurs
in a two-step procedure. First, we minimise \(\Omega_{\text{I}}\) -- this is the
extraction of the optimal subspace in the disentanglement procedure. Then, we
minimise \(\Omega_{\text{D}} + \Omega_{\text{OD}}\).
Converged values for the total spread functional and its components are shown in Table 2. The two groups of four MLWFs with \(sp3\) character are shown in Figure 3
Extraction of optimally-connected subspace
------------------------------------------
+---------------------------------------------------------------------+<-- DIS
| Iter Omega_I(i-1) Omega_I(i) Delta (frac.) Time |<-- DIS
+---------------------------------------------------------------------+<-- DIS
1 12.97640155 12.44630235 4.259E-02 0.00 <-- DIS
. . . . .
79 12.33580893 12.33580893 -6.531E-11 0.23 <-- DIS
80 12.33580893 12.33580893 -5.241E-11 0.23 <-- DIS
<<< Delta < 1.000E-10 over 3 iterations >>>
<<< Disentanglement convergence criteria satisfied >>>
Final Omega_I 12.33580893 (Ang^2)
+----------------------------------------------------------------------------+
Final State
WF centre and spread 1 ( 1.807167, 1.807167, 1.807167 ) 2.01695824
WF centre and spread 2 ( 1.807167, 0.891636, 0.891636 ) 2.01695823
WF centre and spread 3 ( 0.891636, 1.807167, 0.891636 ) 2.01695823
WF centre and spread 4 ( 0.891636, 0.891636, 1.807167 ) 2.01695824
WF centre and spread 5 ( 0.226733, 0.226733, 0.226733 ) 2.37014516
WF centre and spread 6 ( 0.226733, -0.226733, -0.226733 ) 2.37014508
WF centre and spread 7 ( -0.226733, 0.226733, -0.226733 ) 2.37014515
WF centre and spread 8 ( -0.226733, -0.226733, 0.226733 ) 2.37014514
Sum of centres and spreads ( 5.397608, 5.397608, 5.397608 ) 17.54841346
Spreads (Ang^2) Omega I = 12.335808933
================ Omega D = 0.177593840
Omega OD = 5.035010692
Final Spread (Ang^2) Omega Total = 17.548413465
------------------------------------------------------------------------------
Table 2. Converged values of the components of spread functional and their sum, given in Å\(^2\).
| MP mesh | \(\Omega\) | \(\Omega_{\text{I}}\) | \(\Omega_{\text{OD}}\) | \(\Omega_{\text{D}}\) |
|---|---|---|---|---|
| \(4\times4\times4\) | 17.54841 | 12.3358 | 5.03501 | 0.17759 |
 |
 |
 |
 |
 |
 |
 |
 |
Plot the bandstructure.
The interpolated bandstructure is given in Figure 4.
Further ideas¶
-
Compare the Wannier-interpolated bandstructure with the full pwscf bandstructure with a finer \(k\)-point grid.
Result for a \(8\times8\times8\) mesh is shown in Figure 5.
Bandstructure of silicon from DFT calculation (solid black) and from Wannier interpolation with a \(4\times4\times4\) mesh (solid red) and \(8\times8\times8\) mesh (solid blue). -
Compute four MLWFs spanning the low-lying conduction states.
The MLWFs spanning the 4 low-lying conduction states are shown in Figure 6. The initial projections were 4 \(sp3\) on the Si atom at (0,0,0).
Four MLWFs spanning the low-lying conduction states of Si: a. MLWF 1, b. MLWF 2, c. MLWF 3, and d. MLWF 4.