Scattering theory approach to inelastic transport in nanoscale systems

We present a scattering-state description for the nonequilibrium multichannel charge transport in the presence of electron-vibration couplings. It is based on an expansion of scattering orders of eigenchannel states. Examining charge transitions between scattering states, we clarify competing inelastic and elastic scattering processes and compare with the interpretation based on the nonequilibrium Green's functions formalism. We also derive a general expression for conductance variations in single-channel systems that provides a comprehensive picture for the variation, including the well-known result, the 0.5 rule, from the aspect of interplay between elastic and inelastic scattering processes.

Figure 1

Schematic explanation of scattering processes in Eq. (8). The thick right (left) arrows denote eigenchannels $|{\Phi }_L\rangle$ ($|{\Phi }_R\rangle$). The dotted arrow indicates the inelasitc scattering process emitting a vibron. The orange wiggly line represents a vibron emission.

Figure 2

Schematic explanation of scattering processes in the interference term of Eq. (9), which accompany the vibrational excitation-deexcitation. The thick right (left) arrows denote energy levels of electrons corresponding to $\varepsilon +\hslash {\omega }_{\lambda }$, $\varepsilon$, $\varepsilon -\hslash {\omega }_{\lambda }$, which belong to eigenchannels $|{\Phi }_L\rangle$ ($|{\Phi }_R\rangle$). The dotted arrows associated with numberings $\left(1\right)$ and $\left(2\right)$ indicate the order of scattering events. Orange wiggly lines represent vibronic energy transfer. (a) One-electron scattering process. (b) Two-electron scattering process. The right panel shows the final states of two scattering processes. Note that the final ones in (a) and (b) are equivalent under the exchange of the electrons 1 and 2.

Figure 3

For transmitted waves, the interference occurs between the outgoing waves of the unperturbed state (the left scattering state denoted by black arrows) $t_m{|{\Phi }_{Rm}\rangle }_{\text{out}}$ and that of the second-order perturbed state (the right scattering state denoted by red arrows) $r_m^{\prime }{|{\Phi }_{Rm}\rangle }_{\text{out}}$ on the right electrode. The interference between these two outgoing waves is proportional to $\mathrm{Re}\left(r_m^{\prime }{t}_m\right)$.

Figure 4

Differential conductance renormalized by $G_0=2e^2/h$ for a pentalene molecule coupled to monoatomic carbon chains from DFT calculation. Atomic configuration is indicated in the bottom left inset. Among 35 vibrational modes, seven modes $\lambda =7,18,19,21,28,29$, and 31 lead to conductance steps from left to right. The top right inset shows the second and third conductance steps corresponding to the modes 18 and 19, which are almost degenerate. Corresponding vibrational energies are $\hslash {\omega }_{\lambda }=50.3,114.6,115.1,133.8,175.8,184.9$, and $197.6$ meV, respectively.

Figure 5

Phase diagram of conductance variations due to inelastic scattering for the single-level model. Depending on the signs of ${\gamma }_{\mathrm{inel}}$ and ${\gamma }_{\mathrm{inel}}+{\gamma }_{\mathrm{el}}$, there are three regions indicated by green, red, and blue colors. The conductance increases (${\gamma }_{\mathrm{inel}}+{\gamma }_{\mathrm{el}}>0$) in the green region, and it decreases (${\gamma }_{\mathrm{inel}}+{\gamma }_{\mathrm{el}}<0$) in the red and blue regions. Two phases are separated by ${\mathcal{T}}_{\mathrm{cr}}$. In the red region, ${\gamma }_{\mathrm{inel}}>0$ and ${\gamma }_{\mathrm{el}}<0$, while both ${\gamma }_{\mathrm{inel}}$ and ${\gamma }_{\mathrm{el}}$ are negative in the blue region.