Theory of the Franck-Condon blockade regime

Strong coupling of electronic and vibrational degrees of freedom entails a low-bias suppression of the current through single-molecule devices, termed the Franck-Condon blockade. In the limit of slow vibrational relaxation, transport in the Franck-Condon-blockade regime proceeds via avalanches of large numbers of electrons, which are interrupted by long waiting times without electron transfer. The avalanches consist of smaller avalanches, leading to a self-similar hierarchy which terminates once the number of transferred electrons per avalanche becomes of the order of unity. Experimental signatures of self-similar avalanche transport are strongly enhanced current (shot) noise, as expressed by giant Fano factors, and a power-law noise spectrum. We develop a theory of the Franck-Condon-blockade regime with particular emphasis on the effects of electron cotunneling through highly excited vibrational states. As opposed to the exponential suppression of sequential tunneling rates for low-lying vibrational states, cotunneling rates suffer only a power-law suppression. This leads to a regime where cotunneling dominates the current for any gate voltage. Including cotunneling within a rate-equation approach to transport, we find that both the Franck-Condon blockade and self-similar avalanche transport remain intact in this regime. We predict that cotunneling leads to absorption-induced vibrational sidebands in the Coulomb-blockaded regime as well as intrinsic telegraph noise near the charge degeneracy point.

Figure 1

(Color online) (a) Sequential tunneling versus (b) cotunneling for strong electron-phonon coupling. In the FC-blockade regime, sequential processes as depicted in (a) are strongly suppressed due to the exponentially small overlap of harmonic oscillator wave functions. For cotunneling processes (b), this suppression is partially lifted due to contributions from highly excited virtual phonon states.

Figure 2

Coarse-grained current $I\left(t\right)$ as obtained by Monte Carlo simulation for equilibrated phonons at strong electron-phonon coupling $(\lambda =5)$, with bias and gate voltage fixed to $eV=0.8\hslash \omega$ and ${ϵ}_d=0.5\hslash \omega$, respectively. The time dependence of the current clearly reveals the current telegraph noise caused by a slow switching between the two molecular charge states via sequential tunneling ($k_{B}T=0.05\hslash \omega$, ${\Gamma }_a=0.02\hslash \omega$; coarse graining over 300 events).

Figure 3

(Color online) Relevant contributions to the cotunneling rates in the $U\to \infty$ limit. For both charge states $n=0$ and 1 there are two incoherent contributions with either a spin-up or a spin-down electron being transferred across the molecule.

Figure 4

(Color online) Linear conductance as a function of gate voltage in the FC-blockade regime ($\lambda =4$, ${\Gamma }_a=0.02\hslash \omega$). (a) A comparison of the sequential-tunneling conductance (dashed lines) and the conductance including cotunneling corrections (solid lines) clearly shows the dominance of cotunneling contributions at low temperatures. At ${ϵ}_d\approx 0$, cotunneling results in a conductance kink for $k_{B}T\ll \hslash \omega$. Sequential contributions become relevant for temperatures $k_{B}T\gtrsim 0.2\hslash \omega$ that allow thermal-induced occupation of excited phonon states. (b) The plot of $G^{-1∕2}$ reveals the inverse quadratic decrease of the cotunneling conductance as a function of gate voltage.

Figure 5

(Color online) Comparison between the sequential-tunneling current (black curve) and the asymptotic formula (24) (red/gray curve) for equilibrated phonons in the Franck-Condon-blockade regime (${ϵ}_d=0$, $k_{B}T=0.05\hslash \omega$, ${\Gamma }_L={\Gamma }_R$).

Figure 6

(Color online) Nonlinear current-voltage characteristics in the FC-blockade regime. Each graph shows the steady-state current $I^{\left(1\right)}+I^{\left(2\right)}$ (solid curves) and the sequential current $I^{\left(1\right)}$ (dashed curves) for comparison. The electron-phonon coupling and vibrational relaxation strength are chosen as (a) $\lambda =4$, equilibrated, (b) $\lambda =4$, unequilibrated, (c) $\lambda =5$, equilibrated, and (d) $\lambda =5$, unequilibrated. The results unambiguously indicate the dominance of cotunneling contributions in the low-bias FC-suppressed regime (parameters: ${ϵ}_d=0$, $k_{B}T=0.05\hslash \omega$, ${\Gamma }_a=0.02\hslash \omega$).

Figure 7

(Color online) (a) Differential conductance $dI∕dV$ as a function of gate and bias voltage for $\lambda =4$ and ${\Gamma }_a=0.02\hslash \omega$, plotted with a logarithmic color scale. The low-temperature plots (left-hand side) show a step in $dI∕dV$ due to the onset of cotunneling (horizontal feature at $eV=\pm \hslash \omega$). For unequilibrated phonons, inelastic cotunneling gives rise to phonon absorption-induced vibrational sidebands inside the Coulomb-blockade region (whose borders are marked by dashed lines). For comparison, the plots in (b) show the corresponding results within the sequential-tunneling approximation which misses the absorption-induced sidebands $(G^{\prime }=\frac{e^2}{\hslash }\frac{{\Gamma }_L{\Gamma }_R}{{\Gamma }_L+{\Gamma }_R})$.

Figure 8

(Color online) Excess-noise Fano factor for equilibrated phonons with $\lambda =4$ (top row) and $\lambda =5$ (bottom row). For nonzero ${ϵ}_d$ we observe large super-Poissonian Fano factors due to current telegraph noise ($k_{B}T=0.05\hslash \omega$, ${\Gamma }_a=0.02\hslash \omega$).

Figure 9

(Color online) Excess-noise Fano factor for unequilibrated phonons with $\lambda =4$ (top row) and $\lambda =5$ (bottom row) with $k_{B}T=0.05\hslash \omega$ and ${\Gamma }_a=0.02\hslash \omega$. Due to the coexistence of cotunneling and avalanches, the Fano factor is reduced as compared to the sequential-tunneling approximation. The super-Poissonian peak observed at low bias for $\lambda =5$ and ${ϵ}_d\ne 0$ signals the occurrence of current telegraph noise.

Figure 10

(Color online) Noise power spectrum for unequilibrated phonons with $\lambda =4$ (left) and $\lambda =5$ (right) with $k_{B}T=0.05\hslash \omega$, ${\Gamma }_a=0.02\hslash \omega$, ${ϵ}_d=0$, and $eV=3\hslash \omega$. The approximate power-law behavior reflecting the self-similarity of avalanches persists in the presence of cotunneling. The dominance of cotunneling during sequential waiting times provides a long-time (low-frequency) cutoff for the power law.