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,^[1] and later extended along with Anti-Pekka Jauho.^[2] It describes the current using non-equilibrium Green's functions and Keldysh formalism.^[3]

When the interaction between electrons is neglected, this formula reduces to the Landauer formula. This textbook formula^[4] has become a standard tool for calculating the current through various systems, such as molecular junctions, quantum dots and nanoscale devices.

Formula

It reads

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

where ${\displaystyle e}$ is the elementary charge, ${\displaystyle \Gamma ^{b}(b\in \{\mathrm {L,R} \})}$ are the coupling matrices of the left (L) and right (R) leads, ${\displaystyle G^{\mathrm {K} }}$ is the Keldysh Green's function, ${\displaystyle G^{\mathrm {r} }}$ the retarded Green's function, ${\displaystyle G^{\mathrm {a} }}$ the advanced Green's function, and ${\displaystyle f_{b}}$ the Fermi–Dirac distribution of lead b.^[5]