Modeling inelastic phonon scattering in atomic- and molecular-wire junctions

Computationally inexpensive approximations describing electron-phonon scattering in molecular-scale conductors are derived from the nonequilibrium Green’s function method. The accuracy is demonstrated with a first-principles calculation on an atomic gold wire. Quantitative agreement between the full nonequilibrium Green’s function calculation and the newly derived expressions is obtained while simplifying the computational burden by several orders of magnitude. In addition, analytical models provide intuitive understanding of the conductance including nonequilibrium heating and provide a convenient way of parameterizing the physics. This is exemplified by fitting the expressions to the experimentally observed conductances through both an atomic gold wire and a hydrogen molecule.

Figure 1

(Color online) Universal functions [Eqs. (6, 7)] giving the phonon contribution to the current. The differential conductance $dI∕dV$ and the second derivative signals are shown for one phonon mode with the bias in units of the phonon energy at a temperature $kT=0.025\hslash \omega$. For the symmetric term, the FWHM of the second derivative peak is approximately $5.4kT$ (see Ref. 18).

Figure 2

(Color online) Comparison between the SCBA results and the LOE expressions [Eq. (5)] (a) without heating and (b) with heating $({\gamma }_d=0)$ at $T=4.2\mathrm{K}$ for a 4-atom Au-wire. The parameters for the ABL model [Eq. (12)] were extracted directly from the DFT calculations, ${\gamma }_{\mathrm{eh}}=5.4\times {10}^{10}{\mathrm{s}}^{-1}$ and $\hslash \omega =13.4\mathrm{meV}$.

Figure 3

(Color online) (a) Single level model [Eqs. (9, 10)] fitted to the experimentally measured conductance through a Deuterium molecule (Ref. 16). The parameters used for the fit are $\hslash \omega =50\mathrm{meV}$, $\tau =0.9825$, ${\gamma }_{\mathrm{eh}}=1.1\times {10}^{12}{\mathrm{s}}^{-1}$, and $T=17\mathrm{K}$. (b) The ABL model [Eqs. (11, 12)] fitted to the measured conductance through an atomic gold wire (experimental data from Ref. 4). The fit reveals the following parameters, $\hslash \omega =13.8\mathrm{meV}$, $T=10\mathrm{K}$, ${\gamma }_{\mathrm{eh}}=12\times {10}^{10}{\mathrm{s}}^{-1}$, and ${\gamma }_d=3{\gamma }_{\mathrm{eh}}$.