Abstract
In this chapter the Landauer approach using the Green function formalism is formulated. The definitions and basic properties of retarded (and advanced) Green functions, the relations between the Schrödinger equation, wave functions, Green functions and the scattering matrix are discussed. The matrix formalism, the recursive method for long or complex systems and the calculation of self-energies for semi-infinite electrodes are considered. Different types of resonant transport are discussed.
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Notes
- 1.
\(\langle \varPsi _1|\varPsi _2\rangle \) is the scalar product in the corresponding Hilbert space. The Kronecker delta \(\delta _{nm}=1\) if \(n=m\), otherwise \(\delta _{nm}=0\). It is equivalent to the unity matrix \(I_{nm}\).
- 2.
Note the property \(f(\hat{H})|\varPsi _n\rangle =f(E_n)|\varPsi _n\rangle \) for any function f.
- 3.
\(O^\dag \) is the Hermitian conjugated operator, defined for operator \(\hat{O}\) as \(\langle \hat{O}\varPsi |\varPhi \rangle =\langle \varPsi |\hat{O}^\dag \varPhi \rangle \).
- 4.
Note, that we consider here the simplest possible discretization scheme.
- 5.
The Dyson equation considered here is a particular case of the general Dyson equation for interacting systems. It is actually equivalent to the Lippmann-Schwinger equation (3.60) in the scattering theory.
- 6.
All Green functions considered here are retarded Green functions, the index \(^R\) is omitted. \(\varvec{G}^\dag \) is equivalent to an advanced function.
References
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M. Paulsson, cond-mat/0210519 (2002)
M.P. Lopez Sancho, J.M. Lopez Sancho, J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985)
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Ryndyk, D.A. (2016). Green Functions. In: Theory of Quantum Transport at Nanoscale. Springer Series in Solid-State Sciences, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-24088-6_3
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DOI: https://doi.org/10.1007/978-3-319-24088-6_3
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