Counting statistics of cotunneling electrons

We describe a method for calculating the counting statistics of electronic transport through nanoscale devices with both sequential and cotunneling contributions. The method is based upon a perturbative expansion of the von Neumann equation in Liouvillian space, with current cumulants calculated from the resulting non-Markovian master equation without further approximation. As application, we consider transport through a single quantum dot and discuss the effects of cotunneling on noise and skewness, as well as the properties of various approximation schemes.

Figure 1

(Color online) Stationary current $\tilde{I}=⟨I⟩/k_{B}T$ (left) and zero-frequency shot noise $\tilde{S}=S/k_{B}T$ (right) as a function of applied bias $eV$ for the single resonant level model with level located at $\epsilon =20k_{B}T$, chemical potentials ${\mu }_L=-{\mu }_R=eV/2$, and bandwidth $D={10}^3{k}_{B}T$. Results are shown for three different couplings: $\tilde{\Gamma }={\Gamma }_L/k_{B}T={\Gamma }_R/k_{B}T=\frac{1}{4},\frac{1}{2},1$ and three calculational schemes: exact, full fourth-order [O(4)], and second-order Markovian [O(2) Mark.] Whereas the O(2) Markovian results show obvious deviations from the exact results for these couplings, the O(4) solution gives good agreement except around the top of the shot-noise step for $\tilde{\Gamma }=1$.

Figure 2

(Color online) Zero-frequency skewness ${\tilde{S}}^{\left(3\right)}=S^{\left(3\right)}/k_{B}T$ of SRL with the same parameters as in Fig. 1. For a given coupling, agreement with the exact solution is worse than for shot noise, but still good at low couplings (e.g., $\tilde{\Gamma }=1/4$).

Figure 3

(Color online) Zero-frequency shot noise (left panels) and Fano factor $F=S/⟨I⟩$ (right panel) of SRL as a function of applied bias $eV$. The tunnel rate is fixed at $\tilde{\Gamma }=1/2$ here but otherwise the parameters are as Fig. 1. In addition to the approximation schemes discussed in Fig. 1, results are also shown here from a rigorous expansion of the cumulants to fourth order [O(4) trunc]. Around the current step [panel (a)], the full O(4) solution describes the behavior better than O(4) trunc. However, in the low bias, Coulomb-blockade regime [panel (b)], it is the O(4) trunc solution that matches the exact solution better. Panel (c) illustrates the dangers of considering the Fano factor alone: although O(4) trunc reproduces the exact Fano factor extremely well in the Coulomb-blockade regime, so does the O(2) Markovian solution, which we know gives the current and shot noise individually extremely poorly.

Figure 4

(Color online) Zero-frequency skewness (left) and associated Fano factor $F^{\left(3\right)}=S^{\left(3\right)}/⟨I⟩$ (right) of the SRL model as a function of applied bias $eV$. Same parameters and labels as Fig. 3. Once again, the O(4) solution performs better in the onset region.

Figure 5

(Color online) The $n\text{th}$-order Fano factors, $F^{\left(n\right)}\equiv S^{\left(n\right)}/⟨I⟩$, up to order $n=7$ for the single resonant level model with parameters as in Fig. 1 and with $\tilde{\Gamma }=1/4$. At low bias, the O(4) solution predicts spurious superPoissonian cumulants, whereas the O(4) trunc cumulants are sub-Poissonian. Around and above resonance, both sets of results are sub-Poissonian with differences between the two results becoming more significant with increasing order. Note that the O(4) and O(4) trunc results are virtually coincident for $eV/k_{B}T=40$.

Figure 6

(Color online) (b) Current, (c) noise, and (a) Fano factor for the Anderson model as a function of applied bias $eV$. Parameters were chosen as in Ref. 17: ${\Gamma }_L={\Gamma }_R=\frac{1}{4}{k}_{B}T$, ${\mu }_L=-{\mu }_R=\frac{1}{2}eV$, ${ϵ}_{\uparrow }=-15k_{B}T$, ${ϵ}_{\downarrow }=5k_{B}T$, $U=40k_{B}T$, and bandwidth $D={10}^3{k}_{B}T$. A prominent peak is observed in the Fano factor around a bias such that the transport window includes the upper level while the lower level is still included. Cotunneling reduces the size of the peak.

Figure 7

(Color online) (b) Skewness and (a) skewness Fano factor $F^{\left(3\right)}$ for the Anderson model with parameters and labeling as Fig. 6. Not only the skewness Fano factor but the skewness itself show a large peak as the top level enters the transport window.