20: Disentanglement restricted inside spherical regions of \(k\) space¶
LaVO\(_3\)¶
- Outline: Obtain disentangled MLWFs for strained LaVO\(_3\).
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- 1-5: These are the usual steps to generate MLWFs and are not reported here.
Inspect the output file LaVO3.wout. In the initial summary, you will see that
the disentanglement was performed only within one sphere of radius 0.2 around
the point A = (0.5, 0.5, 0.5) in reciprocal space:
*------------------------------- DISENTANGLE --------------------------------*
| Using band disentanglement : T |
...
| Number of spheres in k-space : 1 |
| center n. 1 : 0.500 0.500 0.500, radius = 0.200 |
Compare the band structure that wannier90 produced with the one obtained
using Quantum ESPRESSO.
To obtain the band structure from the Quantum ESPRESSO calculation we can use
the bands.x program available at
http://www.tcm.phy.cam.ac.uk/~jry20/bands.html,
see mini-tutorial at the end of Example 6. Here, we only report the .inp file
used to generate the \(k\)-point mesh for the non-scf calculation
7.03 0.00 0.00
0.00 7.03 0.00
0.00 0.00 7.6627
30
G 0.00000 0.00000 0.00000 M 0.50000 0.50000 0.00000
M 0.50000 0.50000 0.00000 X 0.50000 0.00000 0.00000
X 0.50000 0.00000 0.00000 G 0.00000 0.00000 0.00000
G 0.00000 0.00000 0.00000 Z 0.00000 0.00000 0.50000
Z 0.00000 0.00000 0.50000 A 0.50000 0.50000 0.50000
A 0.50000 0.50000 0.50000 R 0.50000 0.00000 0.50000
R 0.50000 0.00000 0.50000 X 0.50000 0.00000 0.00000
Remember to add the following line to the .bands file in order to show the
eigenvalues at each \(k\)-point.
Plot of the interpolated band structure is shown in Figure 2. In the
top panel, the full band structure is shown. In the bottom panel a magnification
around the Fermi energy is shown (similar to Figure 9 in the wannier90
tutorial).
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wannier90 (red dots). Bottom panel:
magnification around Fermi energy \(16.6049\) eV (dashed line). The
disentanglement was performed only for \(k\)-points within a sphere
of radius 0.2 \(\mathrm{Å}^{-1}\) centred in A.Further ideas¶
-
Try to obtain the Wannier functions using the standard disentanglement procedure ...
Plots of the band structure of LaVO\(_3\) with full disentanglement and no disentanglement are shown in Figure 3. These are plotted against the Quantum ESPRESSO band structure (solid black lines) and the
wannier90-interpolated one with disentanglement performed only within a sphere centred in A (red dots). We see that the other two methods diverge from the DFT calculation in region of \(k\)-space where the bands of interest are not entangled with other unwanted bands. For example, in the zone between \(\Gamma\) and M and Z and A the interpolated bands with full disentanglement and no disentanglement diverge substantially from the DFT calculation.
Comparison of interpolated band structure of epitaxially-strained (tetragonal) LaVO\(_3\) with disentanglement on a sphere of radius 0.2 \(\mathrm{Å}^{-1}\) centred in A (red dots), full disentanglement (blue dots) and no disentanglement (green dots). Fermi energy is shown with a dashed line. -
In order to illustrate all possible cases, it is instructive to apply this method to SrMnO\(_3\) ...
Plots of the interpolated bands for the different cases are shown in Figure 4. In this case, the disentanglement for all the Mn-3d-derived states (empty red circles in Figure 4) is only necessary around the \(\Gamma\) point, as for all the other points and lines the bands of interest are well separated from other bands lower in energy. However, if we only consider the \(e_g\) states (solid blue circles in Figure 4) then the situation is different as these states are entangled with the \(t_{2g}\) states around X. Of course the \(t_{2g}\) states (solid green cones in Figure 4) are entangled with \(e_g\) states around X and with lower-lying states at \(\Gamma\).
wannier90-interpolated bands of SrMnO\(_3\). From only \(t_{2g}\) states (solid green cones), from only \(e_g\) states (solid blue circles), or all Mn-3d-derived states (\(t_{2g} + e_g\)) (empty red circles).