Dressed tunneling approximation for electronic transport through molecular transistors

A theoretical approach for the nonequilibrium transport properties of nanoscale systems coupled to metallic electrodes with strong electron-phonon interactions is presented. It consists of a resummation of the dominant Feynman diagrams from the perturbative expansion in the coupling to the leads. We show that this scheme eliminates the main pathologies found in previous simple analytical approaches for the polaronic regime. The results for the spectral and transport properties are compared with those from several other approaches for a wide range of parameters. The method can be formulated in a simple way to obtain the full counting statistics. Results for the shot and thermal noise are presented.

Figure 1

Lowest-order Feynman diagrams of the perturbative expansion in the tunneling Hamiltonian. The solid lines represents the free dot GF ($G_0$), dashed lines represent the free lead GF ($g_k$, $k=L,R$), the wavy lines denote the polaron correlator ($\Lambda$) and the crosses the hopping events.

Figure 2

Feynman diagrams for (a) PTA and (b) DSPA. In both cases, the zero order corresponds to the atomic limit GF (15).

Figure 3

Feynman diagrams for the DTA approximation. Panel (a) indicates the simplifying approximation on the second-order diagram and panel (b) represents the diagrammatic series which is included within DTA.

Figure 4

Spectral density for the symmetric case, zero bias voltage, and zero temperature with $\Gamma =0.2{\omega }_0$ and $g=1.0$. The results correspond to the different approximations: DTA (full line, green or light gray), ADTA (full line, violet or dark gray), SPA (dotted line), and PTA (dashed line).

Figure 5

Evolution of the spectral density at zero temperature with increasing voltage within DTA. From top to bottom $V=1$, 3, 9 ${\omega }_0$. The other parameters as in lower panel of Fig. 6. The dotted line corresponds to the SPA result, which is voltage independent.

Figure 6

Spectral density at zero temperature in the DTA (full line) and the ISA (triangles) from Ref. [44] for the symmetric case with $\Gamma =0.2{\omega }_0$, $V={\omega }_0$ and two values of the coupling constant, $g=0.5$ (upper panel) and $g=1$ (lower panel).

Figure 7

Zero-temperature differential conductance within DTA (full line, green or light gray), ISA (dotted line), and SPA (dots) for $\tilde{\epsilon }=0$, $\Gamma =0.1{\omega }_0$, and $g=0.8$. The inset shows the current for the same parameters but with a finite temperature $T=0.04{\omega }_0$ for comparison with diagrammatic Monte Carlo data (full squares) from Ref. [20].

Figure 8

Zero-temperature conductance within DTA for $\tilde{\epsilon }=0$, $g=0.5$, and (from bottom to top) $\Gamma =0.1$, 0.2, 0.5, 1.0, 2.0, and 4.0 ${\omega }_0$. The dashed lines show the corresponding results for ISA.

Figure 9

Zero-temperature conductance within DTA for $g=1$, $\Gamma =0.1{\omega }_0$, and (from left to right) increasing values of $\tilde{\epsilon }=0.0$, 0.1, 0.2, 0.3, 0.4, and $0.5{\omega }_0$. The dashed lines show the corresponding results for the EOM method of Ref. [43]. The insets correspond to a blow up of the first and second phonon resonances.

Figure 10

Zero-frequency noise within DTA for the same parameters as in Fig. 8. The lower panel corresponds to the differential noise $\partial S/\partial V$. From top to bottom in the upper panel $\Gamma =0.1$, 0.2, 0.5, 1.0, 2.0, and 4.0 ${\omega }_0$.

Figure 11

Thermal noise as a function of temperature within DTA for the same parameters as in Figs. 8 and 10. The noise is normalized as $S\left(0\right)/4T{G}_0$ in such a way as to illustrate the fulfillment of the FDT in the zero-temperature limit.