6: Copper — Fermi surface

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

Unit cell of Copper crystal plotted with the XCrySDen program.

After checking that the calculations have converged as shown in Example 5, one can proceed with other points in the example.

1. Use Wannier interpolation to obtain the Fermi surface of copper.

   To obtain the value of the Fermi energy we can use the grep command (only for Linux/Unix systems) as:

   > grep Fermi nscf.out
   

   The output should be:

        the Fermi energy is    12.9344 ev
   

   Alternatively, one can open the nscf.out file with the editor of choice and search for "Fermi" inside the file. We can then use this value in the .win file to compute the Fermi surface as done in Example 2. The interpolated Fermi surface is shown in Figure 2-a.

   a. Fermi surface of Copper. b. Band structure of Copper along the Γ–X–W–L–Γ–K computed from a non-scf DFT calculation (solid black) and via Wannier interpolation (solid red).

2. Plot the interpolated bandstructure.

   Bandstructure is shown in Figure 2-b. One way to obtain the DFT bandstructure on exactly the same path as the one in the .win input file, is given by the bands.x program available at http://www.tcm.phy.cam.ac.uk/~jry20/bands.html.

   a. Bandstructure of Copper along the Γ–X–W–L–Γ–K from a non-scf DFT calculation (solid black) and via Wannier interpolation using two different sets of initial projections, namely \(2s\) and 5\(d\) (\(N_w =7\)) (solid red) and 1\(s\) 3\(p\) and 5\(d\) (\(N_w=9\)) (solid blue). b. \(p\) character of bands computed using bands_plot_project = 2,3,4 in the input file. A color scheme is used to measure the \(p\) character of the bands.

BANDS.X minitutorial

Here we summarize the main steps to produce the bandstructure with the bands.x code:

Compilation:

eg. g95 -o bands.x bands.F90

    ifort -o bands.x bands.F90

for NAG

    f95 -o bands.x bands.F90 -DNAG

Usage: First you need to generate an copper.inp file, with the following structure

! Input file for Copper
!
! First the unit cell (in atomic units = Bohr)
-3.411 0.000 3.411
 0.000 3.411 3.411
-3.411 3.411 0.000

!then the number of points along the 1st special path
100

! then the special kpoints and their labels
G 0.00  0.00  0.00    X 0.50  0.50  0.00
X 0.50  0.50  0.00    W 0.50  0.75  0.25
W 0.50  0.75  0.25    L 0.00  0.50  0.00
L 0.00  0.50  0.00    G 0.00  0.00  0.00
G 0.00  0.00  0.00    K 0.00  0.50 -0.50

Then you need to generate the kpoint list by running the bands.x program with the -pp flag

> ./bands.x -pp copper

This will read data from copper.inp and write kpoints into copper_band.kpt.

WARNING: if you already have a copper_band.kpt file from a previous Wannier90 calculation, running the above command will overwrite it.

Now you need to calculate a non-scf or bands calculation with Quantum ESPRESSO on the k-points given in copper_band.kpt. To do so, copy the copper.nscf to copper.bands and modify it accordingly. Run a non-scf calculation

> pw.x < copper.bands > copper.pwscf

WARNING: the output file must terminate with .pwscf in order to be read by bands.x.

Now extract the bands from the copper.pwscf file

> bands.x copper

The bands are written into copper_band.dat. WARNING: if you already have a coppper_band.dat file and a copper_band.gnu file from a previous Wannier90 calculation, running the above command will overwrite them.

Plot with gnuplot

> gnuplot --persist copper_band.gnu

Extra 1: Compare the Wannier interpolated bandstructure with the full pwscf bandstructure. Obtain MLWFs using a denser k-point grid.

Extra 2: Investigate the effects of the outer and inner energy windows on the interpolated bands.

The effect of different energy windows has already been discussed in the Example 4, so it won't be repeated here.

Extra 3: Instead of extracting a subspace of seven states, we could extract a nine dimensional space (i.e., with \(s\), \(p\) and \(d\) character). Examine this case and compare the interpolated bandstructures.

Using \(s,p\), and \(d\) orbitals as initial guesses for the MLWFs of Copper, yields the bandstructure (solid blue) shown in Figure 3 (with reference values for the inner and outer windows). The bandstructure obtained starting from \(2s\) and \(5d\) orbitals is shown in red, whereas the DFT reference bandstructure, computed with the procedure described above, is in black. It is clear from Figure 3-a, and Figure 3-b that the bands of interest have very little \(p\) character, particularly the 5 flat bands, which are well very described by \(d\) states.