1. GW approximation

In DFT most energy eigenvalues don't have a strict physical meaning. This is different for eigenvalues obtained from the GW approximation to many-body perturbation theory. In this tutorial we will use the Spex code to perform single-shot GW calculations on top of converged DFT results to calculate band structures and other quantities. Note that there is also the possibility to perform quasiparticle self-consistent GW calculations by running fleur and Spex repeatedly after each other.

1.1. Obtaining and installing Spex

Download the Spex tarball and copy it to your preferred spex source directory. Unpack the tar with the command tar -xf. In the now created subdirectory invoke the configure script with

MPIFC=mpiifort FCFLAGS=-I/usr/local_rwth/sw/HDF5/1.10.4/intel-intelmpi/include LDFLAGS=-L/usr/local_rwth/sw/HDF5/1.10.4/intel-intelmpi/lib ./configure --with-dft=fleur --enable-mpi=yes

and compile the code with make -j4. You should now have the executables spex.inv (for systems with inversion symmetry) and spex.noinv (without inversion symmetry) in the src subdirectory.

1.2. Si

We use our well-known Si inpgen input

Si bulk
&lattice latsys='cF', a0=1.8897269, a=5.43 /
2
14 0.125 0.125 0.125
14 -0.125 -0.125 -0.125

To generate an inp.xml file. Please use the command inpgen -gw < your_inpgen_input. This will include default options in the inp.xml file to easily run fleur together with spex, for example a possibility to read an external kpoint file. Notice that the /calculationSetup/expertModes/@gw parameter is set to 1 now.

Now start with converging the fleur calculation.

Also for Spex we need to set up an input file, which has to have the name 'spex.inp'. A very minimal option is:

BZ 4 4 4
KPT X=1/2*(0,1,1) K=1/8*(3,6,3)
#KPTPATH (K,1,X) 40
JOB GW IBZ:(1-10)
NBAND 80

where the lines have the following meaning:

For a complete description of the spex.inp file please consult the manual.

Now run spex.inv -w so that the kpts_gw file is written (this is done by -w), open the 'inp.xml' once again and add the number of bands to calculate (/calculationSetup/cutoffs/@numbands). This number should be a little higher than the number of bands given in the 'spex.inp' file. Also change the GW input parameter in /calculationSetup/expertModes/@gw to 2. This directs fleur to write out all the wavefunctions for the kpoints given in kpts_gw. Run fleur.

When fleur has finished everything is ready for the actual GW calculation. For this call spex.inv | tee spex.out. This pipes the output to the file spex.out. In that file you will find the output for the GW quasi particle energies per kpoint at the very end. Note that they have a real and an imaginary part which is related to the lifetime of the respective state.

An example output for one of the k points is

#####################
### K POINT:    8 ###
#####################


--- DIAGONAL ELEMENTS [eV] ---

 Bd       vxc    sigmax    sigmac         Z        KS        HF          GW lin/dir 
  1 -13.08552 -18.17880   4.60232   0.69262  -1.85017  -6.94345  -2.16377   0.30764 
                          0.39694  -0.06663                      -2.16240   0.30546 
  2 -13.08552 -18.17880   4.60232   0.69262  -1.85017  -6.94345  -2.16377   0.30764 
                          0.39694  -0.06663                      -2.16240   0.30546 
  3 -11.36061 -14.25675   2.36751   0.58389   1.95302  -0.94313   1.64238  -0.07552 
                         -0.14198  -0.01396                       1.63630  -0.05172 
  4 -11.36061 -14.25675   2.36751   0.58389   1.95302  -0.94313   1.64238  -0.07552 
                         -0.14198  -0.01396                       1.63630  -0.05172 
  5 -10.52626  -5.80615  -4.49956   0.76619  10.07872  14.79883  10.24695  -0.04299 
                         -0.05194  -0.01450                      10.24664  -0.04292 
  6 -10.52626  -5.80615  -4.49956   0.76619  10.07872  14.79883  10.24695  -0.04299 
                         -0.05194  -0.01450                      10.24664  -0.04292 
  7 -11.76622  -6.29381  -5.04714   0.74757  10.81520  16.28762  11.13197  -0.05312 
                         -0.06021  -0.01907                      11.13061  -0.05320 
  8 -11.76622  -6.29381  -5.04714   0.74757  10.81520  16.28762  11.13197  -0.05312 
                         -0.06021  -0.01907                      11.13061  -0.05320 
  9 -12.22264  -5.38362  -6.43865   0.71501  16.39918  23.23819  16.66102  -0.30763 
                         -0.39569  -0.06172                      16.65992  -0.30529 
 10 -12.22264  -5.38362  -6.43865   0.71501  16.39918  23.23819  16.66102  -0.30763 
                         -0.39569  -0.06172                      16.65992  -0.30529 

Here the real and imaginary part of the GW quasi-particle energies are listed in the last two columns. The two lines per band are related to the linear and direct solution of the respective Dyson equation. In the column KS you can see the respective Kohn-Sham eigenenergies.

We now want to compute a band structure in order to visualize the difference between the Kohn-Sham eigenenergies (fleur) and the GW energies (spex). One possible way to do so is to perform a GW calculation for each kpoint along a path in the Brillouin zone. For this we need to find coordinates along one path. The Spex feature 'KPTPATH' helps us with this. Uncomment the specific line in the spex.inp file and run spex once again with spex.inv -w. The new file 'qpts' includes all the desired kpoints. Now change your spex.inp file to:

BZ 4 4 4
KPT +=(0,0,0) X=1/2*(0,1,1) K=1/8*(3,6,3)
#KPTPATH (K,1,X) 40
JOB GW +:(1-10)
NBAND 80

The kpoint after the '+' has to be changed to one kpoint from the qpts list, followed by a spex.inv -w run, by a fleur run with 'gw=2', and the Spex calculation. These steps do not need to be performed manually, there is a shell script prepared to do so. It can be found in the folder sh of the spex directory. Please copy the script spex.band to your current direcoty and add to it the path to the spex executable and to the fleur executable. Do not forget to uncomment the respective lines. Now you can run it. You have produced multiple output files from which one needs to extract the relevant values, for that there is a utility compiled called spex.extr, it can also be found in the source directory. Please execute it with spex.extr g -b -c spex_???.out > bandstr_gw to get the GW bandstructure and with spex.extr k -b spex_???.out > bandstr_ks to get the DFT band structure. Inspect these files. In the first column you will find the x coordinate of the respective k point and in the second column the related energy. In the case of the GW band structure there is also a third column containing the imaginary part of the energies. Plot the band structures in the files with gnuplot or another plotting tool. Your plot should look similar to the following figure.

Kohn-Sham and GW band structures for Si.

Note that the plots are shifted with respect to the Fermi energy. They also only contain the 10 bands specified in the spex.inp file.

It is also possible to calculate the dielectric function. Starting with any of the two spex.inp files shown above to calculate it for the KS system you need to change the 'JOB' line to JOB DIELEC 1:{0.0:0.2,0.0005}. This means calculating the dielectric function from 0.0 Htr to 0.2 Htr with a step size of 0.001 Htr. This is only done for kpoint label '1', meaning for the $\Gamma$ point with a displacement vector of (0,0,0). This is a very good assumption for optical transitions, since in this band picture photons carry a lot of energy with only very little momentum. When you now run spex again you get the files dielec and dielecR. The first one contains the head elements of the (bare) dielectric function $\epsilon_{0,0}(\vec{k},\omega)$ and the second one the head element of the renormalized dielectric function $\frac{1}{(\epsilon^{-1})_{0,0}(\vec{k},\omega)}$. The imaginary part of the dielecR can be identified with an optical absorption spectrum (excluding excitonic effects). We will be using dielecR. Inspect this file and plot it.

An example plot for this file is shown below.

Dielectric function for Si (Kohn-Sham system). The real part is shown in red, the imaginary part in blue.

Note: To obtain an EELS spectrum from these values one would have to plot for example in 'gnuplot': 'plot "dielecR" using 1:(-imag(1/($2+{0,1}*$3))))'

Note2: Of course, it is also possible to use SPEX to calculate the dielectric function for the GW system.

Compare the gap extracted from 'dielecR' with the experimental band gap of Si. How large is the difference, why is it there?

2. Exercises

2.1. GaN

GaN crystallizes in Wurtzite (ZnS) structure. Its hexagonal unit cell ('hP' for inpgen) has the lattice parameters a=3.19 Angstrom and c=5.189 Angstrom. It features space group 186 (P63mc). The representative atoms are at (internal coordinates):

Set up the input for the material and calculate the DFT (PBE) band structure as well as the GW band structure along the $\Gamma - \text{M} - \text{K} - \Gamma$ path. For the GW calculation use an nband parameter of 160. Compare them. What are the band gaps in both cases? For the GW calculation also calculate the dielectric function (of the KS system) and with this the complex refractive index. According to your calculations: What wavelength does a GaN laser diode have? Look up the experimental band gap of GaN and compare your values to it.

Note 1: M = (0.0,0.5,0.0) ; K = (1/3,1/3,0.0)

Note 2: GaN does not feature inversion symmetry.

Results to be delivered: 1. Plot of the PBE band structure. 2. Plot of the GW band structure. 3. Plot of the complex dielectric function. 4. Plot of the complex refractive index. 5. Band gaps for the PBE and GW calculations.