4: Copper — Fermi surface, orbital character of energy bands¶

Outline: Obtain MLWFs to describe the states around the Fermi-level in copper

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

Run wannier90 to minimise the MLWFs spread. Inspect the output file copper.wout.

Starting from 5 \(d\) orbitals centred on the Copper atom and 2 \(s\) orbitals in the interstitial regions of the FCC, we obtain the following spreads and centres after 200 iterations (extract from the copper.wout, a summary of the wannierisation is given in Table 1.):

Output file

 Final State
  WF centre and spread    1  ( -0.000000,  0.000000, -0.000000 )     0.40838932
  WF centre and spread    2  ( -0.000000, -0.000000, -0.000000 )     0.30784969
  WF centre and spread    3  ( -0.000000, -0.000000,  0.000000 )     0.30784979
  WF centre and spread    4  ( -0.000000, -0.000000,  0.000000 )     0.40838973
  WF centre and spread    5  (  0.000000, -0.000000, -0.000000 )     0.30784886
  WF centre and spread    6  ( -0.902512,  0.902512,  0.902512 )     1.14385632
  WF centre and spread    7  (  0.902512, -0.902512, -0.902512 )     1.14385635
  Sum of centres and spreads (  0.000000, -0.000000, -0.000000 )     4.02804006

         Spreads (Ang^2)       Omega I      =     3.662691490
        ================       Omega D      =     0.001894482
                               Omega OD     =     0.363454087
    Final Spread (Ang^2)       Omega Total  =     4.028040058
 ------------------------------------------------------------------------------

We can readily see that looking at the individual spreads we find two groups of MLWFs, a group of 5 \(d\)-like MLWFs centred on the Copper atom, whose spreads are \(0.4084\) Å\(^2\) and \(0.3078\) Å\(^2\) respectively, and a group of 2 \(s\)-like MLWFs centred on two opposite (with respect to the origin) interstitial points, whose spread is \(1.1439\) Å\(^2\). The 3+2 \(d\)-like MLWFs are the basis of two representations of the \(O_h\) group, with character \(t2_g\) and \(e_g\) respectively.

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

MP mesh	\(\Omega\)	\(\Omega_{\text{I}}\)	\(\Omega_{\text{OD}}\)	\(\Omega_{\text{D}}\)
\(4\times4\times4\)	4.028	3.66	0.363	0.002

Plot the Fermi surface, it should look familiar! The Fermi energy is at 12.2103 eV.

As explained in example 2 of the tutorial, we need to add the following lines to the input file (copper.win):
```
restart = plot
fermi_energy = 12.2103
fermi_surface_plot = true
fermi_surface_num_points = 50
```
and re-run the wannier90 executable. The result will be a file named copper.bxsf, which contains volumetric data in a format suitable for xcrysden. There is only one band that crosses the Fermi level (12.2103 eV), i.e. band 6, as shown in Figure 2-a. The corresponding Fermi surface is shown in Figure 2-b.

Fermi surfaces for band 6 in copper. The value of the Fermi energy is 12.2103 eV, and it was obtained from DFT calculation, with a \(4\times4\times4\) \(\mathbf{k}\)-point mesh. To calculate the band energies and to plot Fermi surfaces, Wannier interpolation was employed on a dense mesh in the Brillouin zone consisting of \(50^3\) points.
Plot the interpolated bandstructure.

Interpolated bandstructure, with path in k-space given in the tutorial, is shown in Figure 3.

Interpolated bandstructure of Copper (solid red) showing the position of the Fermi level (dashed red) and both inner and outer windows (dotted and dashed-dotted respectively). The reference DFT bandstructure (solid black) was obtained with Quantum ESPRESSO, see procedure in Example 6.
Plot the contribution of the interstitial WF to the bandstructure.

The contribution of the 2 \(s\)-like MLWFs to the band structure is shown in Figure 4.

Bandstructure of Copper showing the contribution from the 2 \(s\)-like MLWFs in red.

Extra: Investigate the effect of the outer and inner energy window on the interpolated bands.

From now on, we will refer to the inner window energy level as \(\varepsilon_{\mathrm{in}}\) and to the outer window energy level as \(\varepsilon_{\mathrm{out}}\). The reference values are in this case \(\varepsilon_{\mathrm{in}}^0 = 13\) eV and \(\varepsilon_{\mathrm{out}}^0 = 38\) eV. Hereafter, we will use \(\varepsilon_{\mathrm{min}}\) to refer to the minimum energy eigenvalue (for the chosen path). The value of \(\varepsilon_{\mathrm{out}}^0\) must be chosen such that there are at least \(N_{\mathrm{wannier}}\) (7 in this case) states inside the outer energy window for each \(\mathbf{k}\)-point. This means that for a given path in the BZ there exists a lower bound to \(\varepsilon_{\mathrm{out}}^0\). The actual value depends on the path in the BZ and on the choice of the zero for the pseudopotential. The result for several values of \(\varepsilon_{\mathrm{in}}\) and \(\varepsilon_{\mathrm{out}}\) are shown in Figure 5.

Interpolated bandstructure of Copper (solid red) with DFT reference (solid black) and different values of the inner window. a. \(\varepsilon_{\mathrm{in}} = 5\) eV, \(\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{out}}^0\). b. \(\varepsilon_{\mathrm{in}} = 9.5\) eV, \(\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{out}}^0\). c. \(\varepsilon_{\mathrm{in}} = \varepsilon_{\mathrm{in}}^0\), \(\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{out}}^0\). d. \(\varepsilon_{\mathrm{in}} = 15\) eV, \(\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{out}}^0\). e. \(\varepsilon_{\mathrm{in}}=\varepsilon_{\mathrm{min}}\), \(\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{out}}^0\). f. \(\varepsilon_{\mathrm{in}}=\varepsilon_{\mathrm{min}}\), \(\varepsilon_{\mathrm{out}}= 45\) eV.