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Meir–Wingreen formula

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The Meir–Wingreen formula or Weir–Wingreen–Jauho formula describes the electric current through an arbitrary mesoscopic system. It was formulated by Yigal Meir and Ned Wingreen, and later extended alongside with Anti-Pekka Jauho. It describes the current using non-equilibriumm Green's functions and Keldysh formalism.

When the interaction between electrons is neglected, this formula reduces to the Landauer formula. This textbook formula has become a standard tool for calculating the current through various systems, such as molecular junctions, quantum dots and nanoscale devices.

Formula

It reads

J = i e ℏ ∫ d ϵ T r [ ( Γ L − Γ R ) G K − ( Γ L f L − Γ R f R ) ( G r − G a ) ] {\displaystyle J={\frac {ie}{\hbar }}\int \mathrm {d} \epsilon \mathrm {Tr} \left}

where e {\displaystyle e} is the elementary charge, Γ b ( b ∈ { L , R } ) {\displaystyle \Gamma ^{b}(b\in \{\mathrm {L,R} \})} are the coupling matrices of the left (L) and right (R) leads, G K {\displaystyle G^{\mathrm {K} }} is the Keldysh Green's function, G r {\displaystyle G^{\mathrm {r} }} the retarded Green's function, G a {\displaystyle G^{\mathrm {a} }} the advanced Green's function, and f b {\displaystyle f_{b}} the Fermi–Dirac distribution of lead b.

References

  1. Meir, Yigal; Ned S. Wingreen (1992). "Landauer formula for the current through an interacting electron region". Physical Review Letters. 68 (16): 2512–2515. Bibcode:1992PhRvL..68.2512M. doi:10.1103/PhysRevLett.68.2512. PMID 10045416.
  2. Jauho, Antti-Pekka (1994). "Time-dependent transport in interacting and noninteracting resonant-tunneling systems". Physical Review B. 50 (8): 5528–5544. arXiv:cond-mat/9404027. doi:10.1103/PhysRevB.50.5528.
  3. Ryndyk, Dmitry (2015-12-08). Theory of Quantum Transport at Nanoscale: An Introduction. Springer. ISBN 978-3-319-24088-6.
  4. Jauho, Hartmut Haug, Antti-Pekka (2008). Quantum kinetics in transport and optics of semiconductors (2nd, substantially rev. ed.). Berlin: Springer. p. 170. ISBN 978-3540735618.{{cite book}}: CS1 maint: multiple names: authors list (link)
  5. Kamenev, Alex (2011-09-08). Field Theory of Non-Equilibrium Systems. Cambridge University Press. ISBN 978-1-139-50029-6.


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