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The phonon-induced renormalization of electron bands can be computed using PHonon.
After SCF, PHONON, and NSCF calculations, one can run ph.x with the
electron_phonon=`ahc' option, which generates binary files containing
quantities required for the calculation of electron self-energy.
Then, a postahc.x run reads these binary files and compute the
phonon-induced electron self-energy at a given temperature.
For more details, see the PHonon/Doc/dfpt_self_energy.pdf
file.
Also, there is an example in PHonon/example/example19/
.
Implementation of this functionality in QUANTUM ESPRESSO is described in the following
paper:
J.-M. Lihm and C.-H. Park, Phys. Rev. B 101, 121102 (2020).