Meir–Wingreen formula

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,{{cite journal|last=Meir|first=Yigal|author2=Ned S. Wingreen|title=Landauer formula for the current through an interacting electron region|journal=Physical Review Letters|year=1992|volume=68|issue=16|pages=2512–2515|doi=10.1103/PhysRevLett.68.2512|bibcode = 1992PhRvL..68.2512M|pmid=10045416}} and later extended along with Anti-Pekka Jauho.{{Cite journal |last=Jauho |first=Antti-Pekka |date=1994 |title=Time-dependent transport in interacting and noninteracting resonant-tunneling systems |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.50.5528 |journal=Physical Review B |volume=50 |issue=8 |pages=5528–5544 |doi=10.1103/PhysRevB.50.5528|pmid=9976896 |arxiv=cond-mat/9404027 }} It describes the current using non-equilibrium Green's functions and Keldysh formalism.{{Cite book |last=Ryndyk |first=Dmitry |url=https://books.google.com/books?id=0BMpCwAAQBAJ&dq=meir-wingreen-jauho&pg=PA216 |title=Theory of Quantum Transport at Nanoscale: An Introduction |date=2015-12-08 |publisher=Springer |isbn=978-3-319-24088-6 |language=en}}

When the interaction between electrons is neglected, this formula reduces to the Landauer formula. This textbook formula{{cite book|last1=Jauho|first1=Hartmut Haug, Antti-Pekka|title=Quantum kinetics in transport and optics of semiconductors|date=2008|publisher=Springer|location=Berlin|isbn=978-3540735618|pages=170|edition=2nd, substantially rev.}} has become a standard tool for calculating the current through various systems, such as molecular junctions, quantum dots and nanoscale devices.

Formula

It reads

$J=\frac{i e}{\hbar}\int \mathrm d \epsilon \mathrm{Tr}\left[(\Gamma^{\mathrm L}-\Gamma^{\mathrm R})G^{\mathrm K} -(\Gamma^{\mathrm L}f_{\mathrm L}-\Gamma^{\mathrm R} f_{\mathrm R})(G^{\mathrm r}-G^{\mathrm a})\right]$

where $e$ is the elementary charge, $\Gamma^b (b\in\{\mathrm{L,R}\})$ are the coupling matrices of the left (L) and right (R) leads, $G^{\mathrm K}$ is the Keldysh Green's function, $G^{\mathrm r}$ the retarded Green's function, $G^{\mathrm a}$ the advanced Green's function, and $f_{b}$ the Fermi–Dirac distribution of lead b.{{Cite book |last=Kamenev |first=Alex |url=https://books.google.com/books?id=CwlrUepnla4C&q=kamenev+non-equilibrium |title=Field Theory of Non-Equilibrium Systems |date=2011-09-08 |publisher=Cambridge University Press |isbn=978-1-139-50029-6 |language=en}}

References

Category:Electric current

Category:Mesoscopic physics