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 along with Antti-Pekka Jauho. It describes the current using non-equilibrium 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[(\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 {\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