https://doi.org/10.1140/epjb/e2015-60730-0
Regular Article
Hilbert transform evaluation for electron-phonon self-energies
1 Department of Information Engineering
and Mathematical Sciences, University of Siena, via Roma 56, 53100
Siena,
Italy
2 Department of Physics, University of
Pisa, Largo Pontecorvo
3, 56127
Pisa,
Italy
3 Department of Physics, University of
Pavia, via A.
Bassi, 27100
Pavia,
Italy
a e-mail: bevilacqua@unisi.it
Received:
8
September
2015
Received in final form:
10
November
2015
Published online:
11
January
2016
The electron tunneling current through nanostructures is considered in the presence of the electron-phonon interactions. In the Keldysh nonequilibrium formalism, the lesser, greater, advanced and retarded self-energies components are expressed by means of appropriate Langreth rules. We discuss the key role played by the entailed Hilbert transforms, and provide an analytic way for their evaluation. Particular attention is given to the current-conserving lowest-order-expansion for the treament of the electron-phonon interaction; by means of an appropriate elaboration of the analytic properties and pole structure of the Green’s functions and of the Fermi functions, we arrive at a surprising simple, elegant, fully analytic and easy-to-use expression of the Hilbert transforms and involved integrals in the energy domain.
Key words: Computational Methods
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2016
