Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.K. Ferry, S.M. Goodnick, J. Bird, Transport in Nanostructures, 2nd edn. (Cambridge University Press, New York, 2009)
J.C. Cuevas, E. Scheer, Molecular Electronics (World Scientific, New Jersey, 2010)
M. Paulsson, T. Frederiksen, M. Brandbyge, Phys. Rev. B 72, 201101 (2005)
J.K. Viljas, J.C. Cuevas, F. Pauly, M. Häfner, Phys. Rev. B 72, 245415 (2005)
H. Mera, M. Lannoo, C. Li, N. Cavassilas, M. Bescond, Phys. Rev. B 86, 161404 (2012)
J.-T. Lü, R.B. Christensen, G. Foti, T. Frederiksen, T. Gunst, M. Brandbyge, Phys. Rev. B 89, 081405 (2014)
P. Hyldgaard, S. Hershfield, J.H. Davies, J.W. Wilkins, Ann. Phys. 236, 1 (1994)
A. Cresti, G. Grosso, G. Pastori Parravicini, J. Phys.: Condens. Matter 18, 10059 (2006)
T. Shimazaki, Y. Asai, Phys. Rev. B 77, 115428 (2008)
Y. Asai, Phys. Rev. B 78, 045434 (2008)
A. Cresti, G. Grosso, G. Pastori Parravicini, Eur. Phys. J. B 53, 537 (2006)
A. Cresti, G. Pastori Parravicini, Phys. Rev. B 78, 115313 (2008)
J.-S. Wang, J. Wang, J.T. Lü, Eur. Phys. J. B 62, 381 (2008)
R. Rhyner, M. Luisier, Phys. Rev. B 89, 235311 (2014)
J. Mabillard, T. Can, D.K. Morr, New J. Phys. 16, 013054 (2014)
K. Balzer and M. Bonitz, Nonequilibrium Green’s Functions Approach to Inhomogeneous Systems (Springer, Heidelberg, 2013)
T. Stegman, M. Zilly, D. Ujsághy, D.E. Wolf, Eur. Phys. J. B 85, 264 (2012)
A.R. Hernández, C.H. Lewenkopf, Eur. Phys. J. B 86, 131 (2013)
G. Grosso, G. Pastori Parravicini, Solid State Physics, 2nd edn. (Elsevier, Academic Press, Oxford, 2014)
G.D. Mahan, Many-Particle Physics, 3rd edn. (Kluwer, New York, 2000)
T. Ozaki, K. Nishio, H. Kino, Phys. Rev. B 81, 035116 (2010)
C. Karrasch, V. Meden, K. Schönhammer, Phys. Rev. B 82, 125114 (2010)
J. Hu, R.-X. Xu, Y. Yan, J. Chem. Phys. 133, 101106 (2010)
A. Croy, U. Saalmann, Phys. Rev. B 80, 073102 (2009)
A. Croy, U. Saalmann, Phys. Rev. B 80, 245311 (2009)
L. Lin, J. Lu, R. Car, E. Weinan, Phys. Rev. B 79, 115133 (2009)
L. Lin, M. Chen, C. Yang, L. He, J. Phys.: Condens. Matter 25, 295501 (2013)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
F.W. King, in Hilbert Transforms (Cambridge University Press, Cambridge, 2009), Vols. 1 and 2
J.A.C. Weideman, Math. Comp. 64, 745 (1995)
T. Frederiksen, M. Paulsson, M. Brandbyge, A.-P. Jauho, Phys. Rev. B 75, 205413 (2007)
D. Cvijović, J. Phys. A 41, 455205 (2008)
F. Haupt, T. Novotný, W. Belzig, Phys. Rev. B 82, 165441 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bevilacqua, G., Menichetti, G. & Parravicini, G. Hilbert transform evaluation for electron-phonon self-energies. Eur. Phys. J. B 89, 3 (2016). https://doi.org/10.1140/epjb/e2015-60730-0
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2015-60730-0