Nonequilibrium superoperator Green’s function approach to inelastic resonances in STM currents

Closed expressions for tunneling currents in molecular junctions are derived to the fourth-order in electron-phonon coupling. The Keldysh-Schwinger formalism is recast in terms of density matrices in Liouville space, and the calculation only involves forward propagation in real time and is represented by the double sided Feynman diagrams commonly used for computing optical response functions. The current is related to the decay of coherences in Fock space between many-body molecular states with $N$ and $N\pm 1$ electrons. Application is made to the fundamental and overtone vibrational resonances of methane.

Figure 1

Double-sided Feynman diagrams for the four Liouville space substrate Green’s functions. Time moves from bottom to top. $\mid N⟩⟨N\mid$ denotes a many electron density matrix of the substrate with $N$ electrons. $G_{LR}$ and $G_{RL}$ represent processes where the final states are negatively and positively charged, reflecting the transfer of an electron and a hole from the tip to the substrate, respectively. $G_{LL}$ and $G_{RR}$ represent virtual processes where no real electron transition takes place and the number of elctrons in the substrate remains the same.

Figure 2

Calculated conductance Eq. (36) and IETS spectrum Eq. (40) for an isolated molecule of methane. Panels (a) and (b) show only the second order terms in conductance and IETS spectra, respectively, which give rise to steps/peaks at the fundamental vibrational transition energies. Panels (c) and (d) show only the fourth order component which exhibit moderate intensity for several of the combination and overtone transition energies. Panels (e) and (f) show the conductance and IETS spectrum, respectively, which include both the second and fourth terms in the electron-phonon coupling.