# Exercise 6: The Role of the Lattice Expansion

Estimated total CPU time: 15 min

Warning

In the following exercises, computational settings including the reciprocal space grid (tag k_grid), the basis set, and supercell's size, have been chosen to allow for a rapid computation of the exercises in the limited time and within the CPU resources available during the tutorial session. Without loss of generality, these settings allow to demonstrate trends of the lattice dynamics of materials. In the production calculation, all computational parameters should be converged.

In this exercise, you will:

• Investigate how the band structure and the band gap change due to the lattice expansion.

In the previous exercises, we started from electronic structure theory and then used it as a tool to investigate the motion of the atoms in the harmonic approximation. Among other things, this allowed us to study the lattice expansion as a function of temperature. Now, we go back to electronic structure theory and investigate how the phonons affect the electronic structure. In the first step, we will investigate how the lattice expansion changes the electronic band gap. For this purpose, we will now perform electronic band structure calculations for geometries constructed using the lattice constants determined in the previous exercise.

For this purpose, we have copied 'raw_volumes.datfrom the QHA calculations, which were done in [exercise_4](https://fhi-aims-club.gitlab.io/tutorials/phonons-with-fhi-vibes/phonons/4_QHA/exercise-4/) to the directoryTutorial/electron-phonon-coupling/6_bandgap_volume/input, where the current exercise would be performed.

Before we compute the structures of different volumes, we thin out the volume vs. temperature data, which are stored in 10 K steps, resulting in too many calculations for this tutorial. Use the script thin_out_datapoints.py for this, which should eventually produce a file volume-temperature.dat with less datapoints.

python thin_out_datapoints.py raw_volumes.dat


Next, you can use the script (preprocess.py) to generate the required geometries from the lattice constants defined in volume-temperature.dat in one subfolder per temperature in a folder working. Please, as usual, inspect the script and understand what it does.

python preprocess.py volume-temperature.dat


To run all the calculations, you can use the script run.py, which will run all FHI-aims calculations in the working directory you created.

After all calculations are finished, collect the bandgaps with the script postprocess.py which will save them to the file bandgaps.dat. In this file, you will find both the the band gap and the unit cell volume as a function of temperature:

# Temp [K] bandgap [eV] Volume [\AA^3]
0.0    0.5086506600   39.88493
50.0    0.5086138200   39.88406
100.0    0.5084360100   39.87988
...


Use xmgrace to plot the band gap as a function of temperature by calling

xmgrace bandgaps.dat &


Use xmgrace to plot the band gap as a function of volume by calling

xmgrace @-block@ bandgaps.dat @-bxy 3:2@ &

You can also use python script plot_bg.py to make the plot using python.

python plot_bg.py
`

Can you explain the observed trends? The general theory of electron-phonon interaction could be learned from a variety of books, e.g., Ziman's 'Electrons and Phonons'1.