Charge transfer statistics of a molecular quantum dot with strong electron-phonon interaction

We analyze the nonequilibrium transport properties of a quantum dot with a harmonic degree of freedom (Holstein phonon) coupled to metallic leads, and derive its full counting statistics. By using the Lang-Firsov (polaron) transformation, we construct a diagrammatic scheme to calculate the cumulant generating function. The electron-phonon interaction is taken into account exactly, and the employed approximation represents a summation of a diagram subset with respect to the tunneling amplitude. By comparison to Monte Carlo data, the formalism is shown to capture the basic properties of the strong-coupling regime.

Figure 1

Sketch of a suspended carbon nanotube quantum dot. The fields ${\psi }_{L,R}(x)$ describe semi-infinite leads, and $d$ is the electron annihilation operator on the quantum dot. The gate voltage $V_G$ allows for an adjustment of the energy levels of the dot. The bias voltage $V$ induces a finite charge current across the dot. $P(Q)$ denotes a measurement apparatus for the charge transfer statistics.

Figure 2

Feynman diagrams for the contributions $D_{\lambda }^0\equiv D_0$, $D_{\lambda }^2$, and $D_{\lambda }^4$. The solid lines represent the free dot GF $d(t,t^{\prime })$ and the dashed lines represent the free lead GF $g_{L,R}(t,t^{\prime })$. The internal time variables $t_i$ are integrated over, while $t,t^{\prime }$ (denoted by the filled circles) are external time variables.

Figure 3

Polaron tunneling approximation (PTA) scheme. The single solid lines denote the free dot GF $d(t,t^{\prime })$ and the dashed lines denote the free electrode GFs $g_{L,R}(t,t^{\prime })$. Wiggly lines represent the phonon cloud propagator $\Lambda (t,t^{\prime })$. This approximation allows us to derive the Dyson equation in the second line, where $g_{L,R}$ acts as self-energy.

Figure 4

Transmission coefficient $T(\omega )$. Main graph: Transmission coefficient for $T={\epsilon }_0=0$, $\Omega /\Gamma =5$, and $g/\Gamma =1,5$, and 10 (solid, dashed, and dashed-dotted lines, respectively). Inset: Transmission coefficient for fixed $g/\Gamma =1$, $\Omega /\Gamma =5$, and ${\epsilon }_0=0$ for varying temperatures $T/\Gamma =1,10$ (solid and dashed lines, respectively).

Figure 5

Current in single-particle approximation, PTA, and Monte Carlo (MC) simulation data of Ref. 15 for $T/\Gamma =0.2$. The dashed lines represent the single-particle approximation as proposed in Ref. 19, the solid line is the PTA, and the dots represent the diagMC data. The energy level of the dot is held at ${\epsilon }_0=0$. The assignment of the data (from top to bottom) is the following: $(g/\Gamma =2,\Omega /\Gamma =5$, red/top), $(g/\Gamma =4,\Omega /\Gamma =5$, green/middle), and $(g/\Gamma =4,\Omega /\Gamma =3$, blue/bottom). The systematic deviation of the analytical results from the diagMC data for large $V/\Gamma$ is due to the finite bandwidth necessary to carry out numerical simulations.

Figure 6

Main graph: The noise in the PTA is depicted at $T=0$ for $g/\Gamma =1,2,4$, and 6 (solid, dashed, dashed-dotted, and dotted lines, respectively). The remaining parameters are set to $\Omega /\Gamma =5$ and ${\epsilon }_0=0$. Inset: The noise for temperatures $T/\Gamma =1,5$, and 25 (solid, dashed, and dashed-dotted lines, respectively). The remaining parameters are $g/\Gamma =1$, $\Omega /\Gamma =5$, and ${\epsilon }_0=0$.