Exciton-phonon coupling and luminescence

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This an advanced tutorial, we will show how calculate exciton-phonon coupling[1] and phonon-assisted absorption/emission[2].
Finally we will show how the exciton-phonon coupling can be used to calculate exicton life-time.
In order to run this tutorial you need a deep knowledge of the theory involving these processes and on the use of the Yambo code. [3][4][5]

We will consider as example bulk hBN. Notice that parameters of the present tutorial are not at convergence, but are enough to get a reasonable result.
The tutorial includes several steps and the calculations can be quite expensive.

Getting the code

This tutorial works only with Yambo version > 6.x, but a preliminary version of the code is available on github:


Notice that this version is not fully tested therefore we advice you do not use it in production. In order to install yambo with exciton-phonon coupling you do:

git clone https://github.com/attacc/yambo.git yambo-excph
cd yambo-excph
git checkout devel-excph
git pull

and the configure Yambo


and compile the code

make core -j2
make ph-project -j2

notice that -j is for parallel compilation

Electron-phonon matrix elements

Now we need to generate wave-function and the electron-phonon matrix elements that will be used in the exciton-phonon coupling calculations.

Here you will find scripts and all input files to run the a small example on hBN:


In the tgz file you will find a script run_dvscf.sh, a python script and different folders with the QE input files. You have to modify run_dvscf.sh in order to set the correct path of Yambo and QuantumEspresso on your PC, the number of processors to use, and the parallelization command. Then run:


it will generate electron-phonon coupling on a Q-grid 12x12x1 for bulk hBN.
Generation of electron-phonon matrix elements can take time, for example on my PC (core i9) it takes 1 hour with 8 cores. If you want you can decrease the number of K-point up to 8x8 to speed up calculations.
The script will perform self-consistent calculation for the density, then non-self-consistent for the band structure, calculation of phonons on a special q-grid and finally the calculation of electron-phonon matrix elements. Notice that the Q-grid is automatically generated from the K point you set for the NSCF calculations.
All these data, wave-functions, and electron-phonon matrix elements are then transformed in the Yambo format.

When the run is finished you can go in the QPT12/dvscf/bn.save folder to start exciton-phonon calculations.

BSE at finite momentum

First of all we need the solution of the Bethe-Salpeter equation for all q-points in the grid.
Run the BSE for all momentum q as explained in the tutorials: BSE basic, BSE convergence, BSE for 2D.
Do not forget to turn on the flag WRbsWF to write the excitonic wave-functions. You can generate the input for the BSE with the command: yambo_ph -X s -o b -y d -V qp -k sex

em1s                             # [R][Xs] Statically Screened Interaction
optics                           # [R] Linear Response optical properties
bss                              # [R] BSE solver
bse                              # [R][BSE] Bethe Salpeter Equation.
dipoles                          # [R] Oscillator strenghts (or dipoles)
BoseTemp=-1.000000         eV    # Bosonic Temperature
DIP_Threads=0                    # [OPENMP/X] Number of threads for dipoles
X_Threads=0                      # [OPENMP/X] Number of threads for response functions
K_Threads=0                      # [OPENMP/BSK] Number of threads for response functions
Chimod= "HARTREE"                # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
BSKmod= "SEX"                    # [BSE] IP/Hartree/HF/ALDA/SEX/BSfxc
BSEmod= "resonant"               # [BSE] resonant/retarded/coupling
BSSmod= "d"                      # [BSS] (h)aydock/(d)iagonalization/(s)lepc/(i)nversion/(t)ddft`
BSENGexx= 14923            RL    # [BSK] Exchange components
BSENGBlk=-1                RL    # [BSK] Screened interaction block size [if -1 uses all the G-vectors of W(q,G,Gp)]
#WehCpl                        # [BSK] eh interaction included also in coupling
KfnQPdb= "none"                  # [EXTQP BSK BSS] Database action
KfnQP_INTERP_NN= 1               # [EXTQP BSK BSS] Interpolation neighbours (NN mode)
KfnQP_INTERP_shells= 20.00000    # [EXTQP BSK BSS] Interpolation shells (BOLTZ mode)
KfnQP_DbGd_INTERP_mode= "NN"     # [EXTQP BSK BSS] Interpolation DbGd mode
% KfnQP_E
  3.000000 | 1.000000 | 1.000000 |        # [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim
  1 | 19 |                           # [BSK] Transferred momenta range
% BSEBands
  7 | 10 |                            # [BSK] Bands range
% BEnRange
 0.00000 | 10.00000 |         eV    # [BSS] Energy range
% BDmRange
 0.100000 | 0.100000 |         eV    # [BSS] Damping range
BEnSteps=  300                    # [BSS] Energy steps
% BLongDir
 1.000000 | 0.000000 | 0.000000 |        # [BSS] [cc] Electric Field
BSEprop= "abs"                   # [BSS] Can be any among abs/jdos/kerr/magn/dich/photolum/esrt
BSEdips= "none"                  # [BSS] Can be "trace/none" or "xy/xz/yz" to define off-diagonal rotation plane
WRbsWF                        # [BSS] Write to disk excitonic the WFs
% BndsRnXs
 1 | 16 |                           # [Xs] Polarization function bands
NGsBlkXs=  51              RL    # [Xs] Response block size
% LongDrXs
1.000000 | 0.000000 | 0.000000 |        # [Xs] [cc] Electric Field

In the BSE we included only the two top valence bands 7,8 and the bottom conduction bands 9,10 plus a scissor of 3.0 eV, and the dielectric constant used to calculate W includes 16 bands and has block size of 51 plane waves. Notice that we calculate the BSE for all the 19 possible transfer momentum. In this input file we used the full diagonalization BSSmod= "d" , for large system it is better to switch to the Slepc libraries, '-y s' in the input file generation. If you plot the optical spectra, file o.eps_q1_diago_bse, you should get something like this:

HBN q1 bse.png

Exicton dipersion

Just to be sure that everything is fine, we can have a look to the exciton dispersion along the path Gamma->K->M->Gamma, with the command: ypp_ph -e i

excitons                         # [R] Excitonic properties
interpolate                      # [R] Interpolate
INTERP_mode= " BOLTZ"                # Interpolation mode (NN=nearest point, BOLTZ=boltztrap aproach)
INTERP_Shell_Fac= 20.00000       # The bigger it is a higher number of shells is used
INTERP_NofNN= 1                  # Number of Nearest sites in the NN method
BANDS_steps=  50                  # Number of divisions
cooIn= "rlu"                     # Points coordinates (in) cc/rlu/iku/alat
cooOut= "rlu"                    # Points coordinates (out) cc/rlu/iku/alat
States= "1 - 2"                  # Index of the BS state(s)
-1 |-1 |-1 |                             # Interpolation BZ Grid
#PrtDOS                        # Print Exciton Density of States
% DOSERange
 1.000000 |-1.000000 |         eV    # Energy range
DOSESteps=  10                  # Energy steps
DOS_broad= 0.100000        eV    # Broadening of the DOS
%BANDS_kpts                      # K points of the bands circuit
 0.0 | 0.0 | 0.0 | 
0.33333333333| 0.333333333333 | 0.0 | 
0.5 | 0.0 | 0.0 |
0.0 | 0.0 | 0.0 | 

As you can see in the figure below this system is indirect because the minimum is not a Gamma.

Exc dispersion.png

Exciton-phonon matrix elements and optics

Now that you have the BSE for all momentum and the electron-phonon databases, you can create the exciton-phonon matrix elements, according to the equation:

Yambo tutorial image

and calculate optical response as:

Yambo tutorial image

where Wβα,μq =Eβq − Eα + ω ,for more details see Ref.[6]. Phonon-assisted optical response and exciton-phonon matrix elements can be calculated with the command: yambo_ph -excph o

excph                            # [R] Exction-phonon
ExcGkkp                          # [R][EXCPH] Exciton-Phonon Matrix Elelements
ExcPhOptics                      # [R][EXCPH] Exciton-Phonon Optics
BoseTemp= 0.000000         eV    # Bosonic Temperature
% ELPhExcStates
  1 |  4 |                           # [EXCPH] Incoming (external) exciton states
% ELPhExcSum
  1 |  8 |                          # [EXCPH] Outgoing (virtual) exciton states
LoutPath= "none"                 # [EXCPH] Path of the outgoing L
FANdEtresh= 10.00000       meV   # [ELPH] Energy treshold for Fan denominator
LDamping= 0.500000E-3      eV    # [EXCPH] Damping of exc-ph self-energy
EXCTemp= 20.00000          Kn    # [EXCPH] Excitonic Temperature (for luminescence spectra)
% ElPhModes
  1 | 12 |                           # [ELPH] Phonon modes included
AlphaQ= 0.000000                 # [EXCPH] Excitonic band structure 2D distortion
#PLqres                        # [EXCPH] Write contribution from each q-point
#DbGdOnlyPh                    # [EXCPH] Use Double-Grid only for phonon energies (do not interpolate excitons)
#DbGdWEIGHTs                   # [EXCPH] Use Double-grid also for satellite weights and renormalization
#NoMatrxEl                     # [EXCPH] Set all exciton-phonon matrix elements to one (for testing purpose)
% EnRngeXd
  5.800000 | 6.400000 |         eV    # [Xd] Energy range
% DmRngeXd
  0.002500 | 0.002500 |         eV    # [Xd] Damping range
ETStpsXd= 1000                   # [Xd] Total Energy steps
EXCPHdEtresh= 0.100000E-5  eV    # [ELPH] Energy treshold for exc-ph denominator

where the ELPhExcStates state are the one responsible for the absorption and emission, the α indexes in the χαα and ELPhExcSum are the virtual exciton states that enter in the exciton-phonon scattering, namely the β index in the sum χαα. Running the code, Yambo calculates all the exciton-phonon matrix elements and the photon assisted absorption and emission spectra. The emission spectra is calculated using the Roosbroeck–Shockley (RS) relation[7], a Boltzman distribution for the excitonic occupation at temperature EXCTemp, for more detail see Refs.[6][4], other ways for excitonic occupations are possible, for a discussion see Ref.[3].
LoutPath is the path of the outgoing Bethe-Salpeter, in principle one can use two different kind of exciton for virtual and real exciton in above equation. If LoutPath='none' Yambo will read excitons in the same folder of the incoming ones, by default the SAVE folder.

Double grid for exciton and phonon energies is available[2], in order to activate it just generate phonons on large grids and make Yambo reads them, as explained in the electron-phonon tutorial. Yambo automatically will interpolate exciton on the same grid[8]. Using the double-grid, in order to converge results on q-grid you can follow the same strategy of the electron-phonon tutorial: converge the fine grid for fix course grid (matrix elements) and then increase the course one and repeat the process until total convergence is reached.

In the output are present two file, the absorption and the luminescence spectra. The absorption is not so interesting because is dominated by a strong direct exciton and phonon-assisted peaks are only slightly visible. Hereafter we plot the luminescence spectrum that can be compared with the ones of Ref.[2][4][5].

Luminescence hbn.png

Exciton lifetimes

Using the exciton-phonon coupling it is possible also to calculate exciton life-time due to the scattering with phonon. For this kind of calculation input file can be generated with the command yambo -excph l:

excph                            # [R] Exction-phonon
ExcGkkp                          # [R][EXCPH] Exciton-Phonon Matrix Elelements
ExcPhLifeT                       # [R][EXCPH] Exciton-Phonon Life-Times
BoseTemp= 0.000000         eV    # Bosonic Temperature
% ELPhExcStates
  1 |  4 |                           # [EXCPH] Incoming (external) exciton states
% ELPhExcSum
  1 |  8 |                           # [EXCPH] Outgoing (virtual) exciton states
LoutPath= "none"                 # [EXCPH] Path of the outgoing L
FANdEtresh= 0.100000E-5    eV    # [ELPH] Energy treshold for Fan denominator
% ElPhModes
  1 | 12 |                           # [ELPH] Phonon modes included
EXCTemp= 0.000000          eV    # [EXCPH] Excitonic Temperature (for luminescence spectra)
AlphaQ= 0.000000                 # [EXCPH] Excitonic band structure 2D distortion
#ExcPhOffDiago                 # [EXCPH] Exciton-Phonon off-diagonal self-energy
#NoMatrxEl                     # [EXCPH] Set all exciton-phonon matrix elements to one (for testing purpose)
LDamping= 0.1      meV               # [EXCPH] Damping of exc-ph self-energy

notice that in this case LDamping is the broadening of the exciton-phonon self energy and it should be a very small value of the order of the phonon life-times around the meV.[9]. Running this input you will get:

<---> Exciton lifetimes:
<---> Exciton [1] Lifetime: 0.697314 meV
<---> Exciton [2] Lifetime: 0.697314 meV
<---> Exciton [3] Lifetime: 20.22155 meV
<---> Exciton [4] Lifetime: 32.95292 meV

the life-time of the first four excitons. Note that this result is not converged respect to the Q-points. In order to speedup convergence that double-grid[2] can be used also in the life-time calculations, see tutorial on electron-phonon coupling for tricks on the convergence of the double-grid.


  1. Optical processes in solids, Toyozawa, Yutaka, and Chris Oxlade. Cambridge University Press, (2003).
  2. 2.0 2.1 2.2 2.3 First-principles study of luminescence in hexagonal boron nitride single layer: Exciton-phonon coupling and the role of substrate, P Lechifflart, F Paleari, D Sangalli, C Attaccalite PRM 7 (2), 024006 (2023)
  3. 3.0 3.1 Theory of phonon-assisted luminescence in solids: Application to hexagonal boron nitride, E. Cannuccia, B. Monserrat and C. Attaccalite, Phys. Rev. B 99, 081109(R) (2019)
  4. 4.0 4.1 4.2 Exciton-Phonon Coupling in the Ultraviolet Absorption and Emission Spectra of Bulk Hexagonal Boron Nitride, F. Paleari et al. PRL 122, 187401(2019)
  5. 5.0 5.1 Exciton-Phonon Interaction and Relaxation Times from First Principles, Hsiao-Yi Chen, Davide Sangalli, and Marco Bernardi, Phys. Rev. Lett. 125, 107401(2020)
  6. 6.0 6.1 First-principles approaches to the description of indirect absorption and luminescence spectroscopy: exciton-phonon coupling in hexagonal boron nitride, F. Paleari PhD thesis
  7. Photon-Radiative Recombination of Electrons and Holes in Germanium, W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954)
  8. Warren E. Pickett, Henry Krakauer, and Philip B. Allen, Phys. Rev. B 38, 2721(1988)
  9. First-principles calculations of phonon frequencies, lifetimes, and spectral functions from weak to strong an-harmonicity: The example of palladium hydrides, L. Paulatto, et al. Phys. Rev. B 91, 054304 (2015)