# Electron Phonon Coupling

Here we show step-by-step how to use Quantum Espresso to calculate phonons and electron-phonon matrix-elements on a regular q-grid.
The final aim is to allow Yambo to read them and calculate the temperature-dependent correction to the electronic states.
This tutorial^{[1]} is quite articulated, take your time to do it patiently.

**Sections 1 is presented for this school only as a template on how to generate the electron-phonon matrix elements using QuantumEspresso. As some of the steps there take quite a long time to finish, students can skip this section and move directly to section 2, "Quasi-particle band structure" where they will compute the corrections to the energy levels coming from the electron-phonon interaction. The necessary databases are provided in the link below.**

## Electron-phonon matrix elements

If you just want to see how Yambo works and leave the database generation for later, or use YamboPy to generate them,

you can skip this section and download ready-made databases here: Si_elph_dbs.tgz

Otherwise, in the cloud you need to load the quantum-espresso module:

spack load quantum-espresso

In this first section we will explain how to generate electron-phonon matrix elements in Quantum-Espresso, and how to import them in Yambo. Calculations will be divided in different folders:

**pseudo**the pseudo potential folder**scf**for the self-consistent calculation**nscf**for the non-self-consistent calculation with a larger number of bands**phonon**for the phonon calculations**dvscf**for the calculation of the electron-phonon matrix elements

In this tutorial we will show how to calculate electron-phonon induced corrections to the bands and optical properties for bulk silicon. All input file are availabe in the following tgz file: si.epc.tgz

### SCF

In **scf** we run a standard scf calculation using pw.x choosing a large k-grid in such a way to converge density. Do not forget to set *force_symmorphic=.true.*, because not symmorphic symmetries are not supported yet in Yambo. Notice that:

a) If you are working with 2D systems, not our case, we have to add the flag *assume_isolated="2D"* in such a way correct 2D phonons.

b) If the cell is an FCC, set manually the cell parameters by using the flag *ibrav=0* in your scf input file. This allows to generate correctly the uniform q grid over which phonons and electron-phonon databases are calculated, for example for Silicon you can use:

&system ...... ibrav= 0, celldm(1) =5.09150 ..... / ..... CELL_PARAMETERS alat 0.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 0.0

### NSCF

Plug into **nscf** folder, and then copy the ${PREFIX}.save folder from **scf** to **nscf**, in the present example just do *cp -r ../scf/si.save ./*.
In the nscf input you have to choose the number of k-points and bands you will use for the electron-phonon coupling and Yambo calculations, in our case we will use a 4x4x4 grid and 12 bands.

### Q-points list

1. Read the q-points list. In the **nscf** folder, enter in the ${PREFIX}.save then run *p2y* to read the wave-functions.

2. Generate the setup file with the command *yambo_ph -i -V all* and uncomment the flag *BSEscatt* then run the setup.

setup # [R] Initialization infver # [R] Input file variables verbosity K_grids= "X B S" # [KPT] Select the grids (X=response, S=sigma, C=collisions, B=bse) [default X S C B] BSEscatt # [KPT] Compute extended k/q scatering

3. Generate the input file to read the electron-phonon matrix elements with the command *ypp_ph -k q*:

bzgrids # [R] BZ Grid generator Q_grid # [R] Q-grid analysis OutputAlat= 0.000000 # [a.u.] Lattice constant used for "alat" ouput format #NoWeights # Do not print points weight cooIn= "rlu" # Points coordinates (in) cc/rlu/iku/alat cooOut= "alat" # Points coordinates (out) cc/rlu/iku/alat ListPts # List the internal q/k points also in the parser format #ExpandPts # Expand the internal q/k points in the BZ #ForceUserPts # Do not check the correcteness of the user points %Qpts # Q points list 0.000000| 0.000000| 0.000000| %

and uncomment the flag **ListPts** and q-point coordinates units in **alat**, the units used by QuantumEspresso. For particular cells, like the FCC, there could be an inconsistency between Yambo and QE definition of alat, in this case you can specify **OutputAlat** equal to the **celldm(1)** of QE. Then if you run ypp_ph it will generate the list of q-points for the phonon calculations:

..... <---> Q-points (IBZ) PW-formatted 0.000000000 0.000000000 0.000000000 1 0.125000000 0.125000000 -0.125000000 1 -0.250000000 -0.250000000 0.250000000 1 0.250000000 0.000000000 0.000000000 1 0.375000000 0.125000000 -0.125000000 1 0.000000000 -0.250000000 0.250000000 1 -0.500000000 0.000000000 0.000000000 1 -0.500000000 0.250000000 0.000000000 1 <---> [08] Timing Overview ......

### Phonons

Plug into **phonon** directory. You have to copy the self-consistent calculation in this folder
with the command *cp -r ../scf/si.save ./ *. Then you can prepare an input file for phonons using the q-points list you generated in the previous step multiplied by -1,
the final input will be:

&inputph verbosity = 'high' tr2_ph = 1e-12 prefix = 'si' fildvscf = 'si-dvscf' fildyn = 'si.dyn', electron_phonon = 'dvscf', epsil = .false. trans = .true. ldisp = .false. qplot = .true. / 8 0.000000000 0.000000000 0.000000000 1 -0.125000000 -0.125000000 0.125000000 1 0.250000000 0.250000000 -0.250000000 1 -0.250000000 0.000000000 0.000000000 1 -0.375000000 -0.125000000 0.125000000 1 0.000000000 0.250000000 -0.250000000 1 0.500000000 0.000000000 0.000000000 1 0.500000000 -0.250000000 0.000000000 1

### DVSCF

Plug into **dvscf** folder. From the nscf folder copy density and wave-functions (*cp -r ../nscf/si.save ./*), from the phonon folder copy the dynamical matrices and the dvscf files (*cp -r ../phonon/_ph0 ../phonon/*.dyn* ./*). Then modify the phonon input to generate electron-phonon matrix elements for Yambo, by changing epsil = .false., trans = .false. and electron_phonon = yambo. You do not need to specify the k-point grid because it is read from the nscf wave-functions. In this way we will not recalculate phonons, but only the electron-phonon matrix elements for the number of bands present in the nscf input file, in our case 8 bands:

&inputph verbosity = 'high' tr2_ph = 1e-12 prefix = 'si' fildvscf = 'si-dvscf' fildyn = 'si.dyn' electron_phonon = 'yambo', epsil = .false. trans = .false. ldisp = .false. qplot = .true. / 8 0.000000000 0.000000000 0.000000000 1 -0.125000000 -0.125000000 0.125000000 1 0.250000000 0.250000000 -0.250000000 1 -0.250000000 0.000000000 0.000000000 1 -0.375000000 -0.125000000 0.125000000 1 0.000000000 0.250000000 -0.250000000 1 0.500000000 0.000000000 0.000000000 1 0.500000000 -0.250000000 0.000000000 1

this run will generate a new folder *elph_dir* with all electron-phonon matrix elements in a format compatible with Yambo.

**IMPORTANT**: do not parallelize on k-points in the **dvscf** calculation, it is not supported yet!!

### Import in Yambo

Now you have to read the electron-phonon matrix elements. Go in the *dvscf/si.save* folder, and use ypp_ph to import electron-phonon coupling, by doing *ypp_ph -g g* and the PW el-ph folder:

```
gkkp # [R] gkkp databases
gkkp_db # [R] GKKP database
#GkkpReadBare # Read the bare gkkp
DBsPATH= "../elph_dir/" # Path to the PW el-ph databases
PHfreqF= "none" # PWscf format file containing the phonon frequencies
PHmodeF= "none" # PWscf format file containing the phonon modes
#GkkpExpand # Expand the gkkp in the whole BZ
#UseQindxB # Use qindx_B to expand gkkp (for testing purposes)
```

run ypp_ph, and if everything went well you will get in output

..... <---> [06] == Electron-Phonon Databases == <---> Inspecting databases ...PWscf (dressed)...found 8 Q-points <---> ELPH databases: phonon frequencies and eigenvectors |########################################| [100%] --(E) --(X) <---> ELPH databases: K+Q-grid check |########################################| [100%] --(E) --(X) <---> [07] == Q-points list in the DBS (iku units) == <---> :: Code generator :PWscf <---> :: DB Kind :dressed <---> :: Expanded :no <---> :: Q-points(read) : 8 <---> :: Q-points(written) : 8 <---> :: K-points : 20 <---> :: Bands : 12 <---> :: Branches : 6 <---> :: Uniform sampling :yes <---> :: Symmetry expanded :no <---> :: Debye Energy : 63.09927 [meV] .....

Notice that Yambo recognized that you are using a regular q-grid: *Uniform sampling :yes*.

## Quasi-particle band structure

The first quantity we will calculate is the correction to the band structure induced by the electron-phonon coupling. In order to generate the corresponding input do `yambo_ph -g n -p fan -c ep -V gen`

. In the input we require the correction only at gamma point:

dyson # [R] Dyson Equation solver gw0 # [R] GW approximation el_ph_corr # [R] Electron-Phonon Correlation Nelectro= 8.000000 # Electrons number ElecTemp= 0.000000 eV # Electronic Temperature BoseTemp= 0.000000 eV # Bosonic Temperature OccTresh= 0.100000E-4 # Occupation treshold (metallic bands) SE_Threads=0 # [OPENMP/GW] Number of threads for self-energy DysSolver= "n" # [GW] Dyson Equation solver ("n","s","g") % GphBRnge 1 | 12 | # [ELPH] G[W] bands range % % ElPhModes 1 | 6 | # [ELPH] Phonon modes included % GDamping= 0.0100000 eV # [GW] G[W] damping RandQpts=0 # [RIM] Number of random q-points in the BZ #WRgFsq # [ELPH] Dump on file gFsq coefficients %QPkrange # [GW] QP generalized Kpoint/Band indices 1|8|2|8| %

If you run `yambo_ph -J T0`

you will get correction at zero kelvin to the band structure.
You can change the temperature to 300 K by doing:

dyson # [R] Dyson Equation solver gw0 # [R] GW approximation el_ph_corr # [R] Electron-Phonon Correlation Nelectro= 8.000000 # Electrons number ElecTemp= 0.000000 eV # Electronic Temperature BoseTemp= 300.0 Kn # Bosonic Temperature OccTresh= 0.100000E-4 # Occupation treshold (metallic bands) SE_Threads=0 # [OPENMP/GW] Number of threads for self-energy DysSolver= "n" # [GW] Dyson Equation solver ("n","s","g") % GphBRnge 1 | 12 | # [ELPH] G[W] bands range % % ElPhModes 1 | 6 | # [ELPH] Phonon modes included % GDamping= 0.0100000 eV # [GW] G[W] damping RandQpts=0 # [RIM] Number of random q-points in the BZ #WRgFsq # [ELPH] Dump on file gFsq coefficients %QPkrange # [GW] QP generalized Kpoint/Band indices 1|8|2|8| %

run again `yambo_ph -J T300`

and compare the two output files o-T0.qp and o-T300.qp, to find how much the gap change with the temperature.

Notice that in this calculation we set a small damping factor for the Green function of about 0.01 eV, this value should be of the order of the phonon life-time.

If you repeat the calculation for different temperature you can obtain the trend of the gap vs temperature. The figure below report the Silicon gap at Gamma Point at different temperature obtained in this tutorial. You can recreate the file with the data needed for this plot with the following command line instruction:

for i in 0 50 100 150 200 250 300; do echo -n "$i "; echo "$(grep "^ *1 *5" o-T${i}.qp | awk '{print ($3+$4)}') - $(grep "^ *1 *4" o-T${i}.qp | awk '{print ($3+$4)}')" | bc; done > gap_vs_T.dat

and then in `gnuplot`

type

plot 'gap_vs_T.dat' u 1:2 w lp

If you uncomment the flag WRgFsq the code saves information about the Eliashberg functions that can be plotted using the *ypp_ph*, see below.

Finally if you add *-V qp* in the input generation a new flag appears OnMassShell, if you un-comment this flag calculation will be performed in the "on mass shell" approximation, namely the static limit the Quasi-Particle approximation, for a discussion see reference ^{[2]}
This means that calculation reduces to the static perturbation theory of Allen-Cardona.

Notice that the last column in the quasi-particle file "o.QP" contains the **quasi-particle width**, that is related to their life-time. You can plot the band structure including this width to get quasi-particle spectra
similar to the one measured in ARPES experiments, see Fig. 2 in ref. ^{[3]}.
This parameter will be used in the next tutorial on optical properties at finite temperature.
**WARNING:** Yambo average the electron-phonon correction on the degenerate states, please **include all degenerate states in your calculations**. For example in the silicon case you need correction from the 2nd band to the 8th band.

## Convergence

The results of this tutorial are not converged. This is due to the poor parameters used in this tutorial to make the calculations fast. In order to have converged results, first of all you have to be sure to have converged phonons, in order to do that increase plane-wave cutoff, number of k-points and if necessary reduce *tr2_ph*.
Then change the parameters for the Yambo calculations, increasing the number of **k** and **q** points and the number of bands. If you want to increase only the number of bands, just repeat the **nscf** and **dvscf** calculations without recalculate **phonons**. In the following section we will describe a smart way to accelerate convergence.

Nota bene: At present Yambo neither implements the Fröhlich term at q=0^{[4]} nor the quadrupolar correction,^{[5]} therefore convergence in polar material can be very slow with the number of q-points.

## Double-grid method for the electron-phonon coupling (only in Yambo 5.x)

Simplifying at most the matrix elements of the electron-phonon self-energy have the structure:

where we omitted the electronic band and the phonon branches indexes. In order to speed up calculation one can average the denominators on an additional fine grid around each **q** points as:

In order to exploit the double-grid tool we need the phonon energies calculated on a fine grid. If you just want to test the functionality of Yambo you can download here some files with the phononic frequencies already calculated on different silicon grids: si_freqs_files.tgz

Otherwise you generate them by following the instructions below.

**Concerning the school, we suggest to use the precomputed frequencies provided above.**

For a general tutorial on phonon calculation with Quantum Espresso you can have a look at Hands on phonons.

Starting from a well converged phonon calculation *matdyn.x* can interpolate phonon dispersion on a given q-sampling (for example a regular 14x14x14 grid, as shown in the input file below) and give the desired phonon energies.

&input asr='simple', flfrc='si.fc', flfrq='si.freq', dos=.true., fldos='si.dos', deltaE=1.d0, nk1=14, nk2=14, nk3=14 /

Alternatively you can generate a random q-sampling using Yambo (*ypp -k r*) and insert the corresponding q points list into a matdyn file.

Once you generated the phonon energies you can read them with the command `ypp_ph -g d`

(in this example we read the 12x12x12 double grid):

```
gkkp # [R] gkkp databases
gkkp_dg # [R] GKKP double grid
PHfreqF= "si.freq_12" # PWscf format file containing the phonon frequencies
PHmodeF= "none" # PWscf format file containing the phonon modes
FineGd_mode= "mixed" # Fine Grid mode. Symmetry expanded, unexpanded or mixed.
#SkipBorderPts # Skip points in the Fine Grid that are on the surface of coarse gride smal BZ`s
EkplusQmode= "interp" # E(k+q) energies calculation mode (interp | dftp )
#TestPHDGrid # Test double-grid: set all values of the fine grid equal to the couse ones
```

then run `ypp_ph`

an it will generate a new file *SAVE/ndb.PH_Double_Grid*.

In the calculation the new phonon energies will be expanded
using the symmetries of the systems, while the electronic energies at **k+q** will be interpolated using a smooth Fourier interpolation^{[6]}.
Now we go back to our *yambo.in* input file and repeat the calculations. The double grid will be automatically accounted for.
We reset the temperature to `BoseTemp= 0.0 Kn # Bosonic Temperature`

and run the calculation in a new directory:

yambo_ph -J T0_12

We can repeat the process for all the other double grids provided (12x12x12, 24x24x24, 36x36x36, 48x48x48) and check the band convergence to determine which double grid we will use. This takes time, so you may skip it and go on with the tutorial.

**In fact, the best strategy to converge electron-phonon coupling calculations is to first converge the double-grid for a fixed coarse main grid, and second, once the final double-grid has been found, converge the main grid itself. This is a process that takes time since it requires repeating the electron-phonon matrix element calculations each time the main grid is changed, therefore you are not supposed to do it for this tutorial.** Hereafter we provide an example of Silicon band gap correction convergence versus the number of

**q**-points in the main grid, using a double-grid with 4096 random

**q**-points.

The double-grid implementation is described in this paper^{[7]}.

## Miscellaneous and post-processing

**Automatic generation of electron-phonon matrix elements**

Getting electron-phonon matrix elements from QuantumEspresso to Yambo is a complicated process,

we advice you to create a script to automatize the procedure, here an example of a bash script for the silicon case: SILICON-SCRIPT.

There are a list of utilities to analyze electron-phonon coupling results:

**Phonon density of states**

You can plot phonon density of states with the command: *ypp_ph -p d*. In order to get a nice plot set in the input

phonons # [R] Phononic properties dos # [R] DOS PhBroad= 0.0005000 eV # Phonon broadening (Eliashberg & DOS) PhStps=1000 # Energy steps

The we can analyse the phonon DOS of our previous `T0`

calculation by doing:

ypp_ph -J T0

We get the output file *o-T0.ph_dos* which we can plot.
Notice that this is an easy quantity to check for the convergence in q-points.

**Eliashberg Functions**

You can plot Eliashberg functions^{[8]} for both electrons and excitons. In order to plot Eliashberg functions you must have calculated Quasi-Particle correction with the flag WRgFsq, see above. If you didn't do that, please rerun the `yambo_ph`

calculation (if you use the directory of a previous calculation to do this, remove or rename the *ndb.QP* file).
The command `ypp_ph -s e`

generate the input for the electronic Eliashberg functions:

electrons # [R] Electronic properties eliashberg # [R] Eliashberg PhBroad= 0.0010000 eV # Phonon broadening (Eliashberg & DOS) PhStps= 200 # Energy steps %QPkrange # generalized Kpoint/Band indices 1|1|2|8| %

We can run the postprocessing as

ypp_ph -J T0

Tn this example we plot Eliashberg functions for the top valence and bottom conduction band at the Gamma point:

For a discussion on how to interprete Eliashberg functions and use them to understand the different phonon contribution you can have a look to ref.^{[9]}

**Atomic displacement amplitudes**

Running `ypp_ph -p a`

will plot the atomic displacement for each atom in the cell along each direction.

** Other variables in the input files **

In the input files of the present tutorial there are other relevant variables which we did not use.
In particular `GkkpExpand`

is used to expand the electron-phonon matrix elements over the full Brillouin zone in both k-space and q-space.

## Links

- Back to ICTP 2022#Tutorials

## References

- ↑ This tutorial is based on Elena Cannuccia blog
- ↑ H. Kawai et al. Phys. Rev. B
**89**, 085202 (2014) - ↑ G. Antonius, S. Poncé, E. Lantagne-Hurtubise, G. Auclair, X. Gonze, and M. Côté Phys. Rev. B 92, 085137 (2015)
- ↑ Carla Verdi and Feliciano Giustino
Phys. Rev. Lett.
**115**, 176401 (2015) - ↑ G. Brunin et al. Phys. Rev. Lett.
**125**, 136601 (2020) - ↑ Warren E. Pickett, Henry Krakauer, and Philip B. Allen,
Phys. Rev. B
**38**, 2721(1988) - ↑ Double-grid for the electron-phonon problem, (to be published) E. Cannuccia, C. Attaccalite and V. Olevano (2022)
- ↑ F. Marsiglio, J.P. Carbotte Electron - Phonon Superconductivity
- ↑ E. Cannuccia and A. Gali Phys. Rev. Materials 4, 014601 (2020)