Unified Description of Inelastic Propensity Rules for Electron Transport through Nanoscale Junctions

We present a method to analyze the results of first-principles based calculations of electronic currents including inelastic electron-phonon effects. This method allows us to determine the electronic and vibrational symmetries in play, and hence to obtain the so-called propensity rules for the studied systems. We show that only a few scattering states—namely those belonging to the most transmitting eigenchannels—need to be considered for a complete description of the electron transport. We apply the method on first-principles calculations of four different systems and obtain the propensity rules in each case.

Figure 1

Phase diagram for a one-level model (inset) illustrating the sign of the conductance change at the onset of phonon emission. At a given asymmetry factor $\alpha$ the elastic transmission $\tau$ has an upper bound ${\tau }_{\max }$ (black line), and the inelastic conductance change undergoes a sign change at ${\tau }_{\mathrm{crossover}}={\tau }_{\max }/2$ (dashed line). A Mathematica notebook for the one-level model is available online [21].

Figure 2

Calculated (black lines) and experimental [blue (or gray) lines] IETS representing the corners of the phase diagram in Fig. 1: (a) OPE molecule with Au(111) leads, (b) Au chain connected to Au(100) leads, (c) ${\mathrm{O}}_2$ molecule on Ag(110), and (d) CO molecule on Cu(111). In case (c) the Fermi energy (${\epsilon }_{\mathrm{F}}$) as been shifted manually to match the experiment (dashed line). The experimental data originate from Refs. 4, 9, 27, 28. For the STM configurations (c) and (d), the calculated IETS is compared with a rescaled $d^{2}I/d{V}^2$.

Figure 3

Comparison between the FGR scattering rate, calculated using only the single-most transmitting eigenchannel, with the full LOE rate for (a) OPE molecule with Au(111) leads [compared with the zero-bias conductance of $1.0\times {10}^{13}\mathrm{e}/(\mathrm{s}\mathrm{V})$], and (b) 7-atom Au wire $4.8\times {10}^{14}\mathrm{e}/(\mathrm{s}\mathrm{V})$]. The sign of the FGR rate was chosen to agree with the LOE rate.

Figure 4

Eigenchannels and phonon modes for CO on Cu(111). (a) Scattering states associated with eigenchannel 1 (left column, ${\tau }_1=1.9\times {10}^{-3}$) and eigenchannel 2 (right column, ${\tau }_2=0.4\times {10}^{-3}$). The top (bottom) row of scattering states originates from the tip (substrate) side. Eigenchannel 3 (not shown, ${\tau }_3=0.4\times {10}^{-3}$) closely resembles those of eigenchannel 2 rotated by 90°. (b) $\mathrm{FR}(x)$ and $\mathrm{FT}(x)$ vibrational modes [degenerate with $\mathrm{FR}(y)$ and $\mathrm{FT}(y)$].