Figure 1

A schematic picture of the model. The energies of the two levels are $E_1$ and $E_2$. The bias voltage is the difference between the two chemical potentials: $e{V}_{\mathrm{bias}}={\mu }^L-{\mu }^R$. Here, the width of level 2 is much smaller than the width of level 1, which is assumed to be much smaller than the bias voltages: ${\gamma }_2\ll {\gamma }_1\ll |{\mu }^{R,L}-E_{1,2}|$. We assume that the tunneling coefficients are independent of the energy and we also assume, for simplicity, that they all have the same sign and that the two ratios between the left tunneling coefficients and the right tunneling coefficients are identical, $t_1^L/t_1^R=t_2^L/t_2^R$.Reuse & Permissions

Figure 2

A qualitative description of the transport through the two level system in the region ${\mu }^R<E_1+U<{\mu }^L$. Electrons are tunneling through level 1 until the electron in level 2 tunnels out and then the current is blocked. The tunneling events through level 1 resume when another electron tunnels into level 2. The current behaves as a multiplication of two signals: A sequence of current pulses through level 1 and a telegraphic signal, the occupation of level 2. ${\Gamma }_1$ is the average rate of tunneling events through level 1, ${\tau }_0$ is the average time duration until level 2 is filled, and ${\tau }_1$ is the average time duration until level 2 gets empty.Reuse & Permissions

Figure 3

A schematic picture of the lowest order in $H_{\mathrm{t}}$ transitions (the sequential tunneling processes) and their notations.Reuse & Permissions

Figure 4

A schematic picture of the elastic cotunneling processes and their notations. Each process has two possible virtual intermediate states. The rate is the sum of the amplitudes of the two possible paths (rather than the sum of their probabilities). The processes ${\omega }_{i,i}^{\leftarrow }$ are the same as ${\omega }_{i,i}^{\to }$ after exchanging the final states with the initial states.Reuse & Permissions

Figure 5

A schematic picture of the inelastic cotunneling processes and their notations. The processes ${\omega }_{01,10}$ are the same as ${\omega }_{10,01}$ after exchanging the two levels. The processes ${\omega }_{11,00}$ are the same as ${\omega }_{00,11}$ after exchanging the final states with the initial states.Reuse & Permissions

Figure 6

The current through the two level system with the following parameters ($\Gamma \equiv {\Gamma }_1^R+{\Gamma }_1^L+{\Gamma }_2^R+{\Gamma }_2^L$): $K_{B}T=\hslash \Gamma$, $e{V}_{\mathrm{bias}}=10\hslash \Gamma$, $E_1-E_2=\hslash \Gamma$, $U=100\hslash \Gamma$, and ${\Gamma }_1^{R,L}=10{\Gamma }_2^{R,L}$. We assume the same left-right asymmetry for the two levels: ${\Gamma }_1^L/{\Gamma }_1^R={\Gamma }_2^L/{\Gamma }_2^R\equiv {\Gamma }^L/{\Gamma }^R$. (a) The symmetric (${\Gamma }^L/{\Gamma }^R=1$) current as a function of $E_1$. (b) The current at the right peak, $E_1=0$, as a function of the left-right asymmetry. The dashed line is the sequential tunneling current and the solid line is the current including both sequential and cotunneling processes. (c) The (cotunneling) current at the valley, $E_1=-U/2$, as a function of the left-right asymmetry. The sequential tunneling current is practically zero at the valley. (d) The current at the left peak, $E_1=-U$, as a function of the left-right asymmetry. The dashed line is the sequential tunneling current, and the solid line is the current including both sequential and cotunneling processes.Reuse & Permissions

Figure 7

The Fano factor $F=S/\left(2e\langle I\rangle \right)$ at $E_1=-U$ (the left peak of the current) as a function of the left-right asymmetry with the following parameters: $e{V}_{\mathrm{bias}}=0.1U$, $E_1-E_2=0.01U$, and $\hslash {\Gamma }_1^L+\hslash {\Gamma }_1^R+\hslash {\Gamma }_2^L+\hslash {\Gamma }_2^R=0.01U$. We assume that the two levels have the same left-right asymmetry, ${\Gamma }^L/{\Gamma }^R$. When we change ${\Gamma }^L/{\Gamma }^R$, we keep the total width of each level fixed. (a) The Fano factor for different ratios of the two levels' widths [${\gamma }_i=\hslash ({\Gamma }_i^R+{\Gamma }_i^L)/2$] with $K_{B}T=0.01U$ and $e{V}_{\mathrm{bias}}=0.1U$. The dashed lines are the sequential tunneling Fano factors and the solid lines are the calculated Fano factors including cotunneling processes. In the inside box in the upper right corner, we zoom in on the small values of ${\Gamma }^L/{\Gamma }^R$. The Fano factor is larger than one (super-Poissonian noise) and maximal in asymmetric coupling. (b) The Fano factor as a function of the left-right asymmetry for different values of $e{V}_{\mathrm{bias}}/T$. The ratio between the levels' widths is ${\Gamma }_1^{R,L}=10{\Gamma }_2^{R,L}$. The dashed lines are the sequential tunneling Fano factors and the solid lines are the calculated Fano factors including cotunneling processes. The cotunneling processes dramatically decrease the Fano factor for large $e{V}_{\mathrm{bias}}/T$.Reuse & Permissions

Figure 8

The current and the Fano factor in the small U ($U<e{V}_{\mathrm{bias}}$) region with the following parameters ($\Gamma \equiv {\Gamma }_1^R+{\Gamma }_1^L+{\Gamma }_2^R+{\Gamma }_2^L$): $U=2\hslash \Gamma$, $e{V}_{\mathrm{bias}}=10\hslash \Gamma$, $K_{B}T=\hslash \Gamma$, $E_1-E_2=0.5\hslash \Gamma$, and ${\gamma }_1=10000{\gamma }_2$. (a) The current as a function of $E_1$ at the symmetric point, ${\Gamma }^L={\Gamma }^R$. The dashed line is the current including sequential tunneling only, and the solid line is the current including also cotunneling processes. (b) The Fano factor at $E_1=0$ as a function of the left-right asymmetry. The dashed line is the sequential tunneling Fano factor and the solid line is the Fano factor including cotunneling processes. The cotunneling processes enhance the Fano factor significantly.Reuse & Permissions

Figure 9

The dependence of the tunneling rates through level 1 on the occupation of level 2 in the small U ($U<e{V}_{\mathrm{bias}}$) region with the following parameters ($\Gamma \equiv {\Gamma }_1^R+{\Gamma }_1^L+{\Gamma }_2^R+{\Gamma }_2^L$): $U=2\hslash \Gamma$, $V_{\mathrm{bias}}=10\hslash \Gamma$, $K_{B}T=\hslash \Gamma$, $E_1=0$, $E_2=-0.5\hslash \Gamma$, and ${\gamma }_1=10000{\gamma }_2$. Negative cotunneling rates mean that the cotunneling processes decrease the current (see for example Ref. 31). (a) The sequential tunneling rate from the left lead through level 1 to the right lead; [(${\omega }_{11,01}^{\to }{{)}^{-1}+{\left({\omega }_{01,11}^{\to }\right)}^{-1}]}^{-1}$ when level 2 is occupied and ${[{\left({\omega }_{10,00}^{\to }\right)}^{-1}+{\left({\omega }_{00,10}^{\to }\right)}^{-1}]}^{-1}$ when level 2 is empty. (b) The cotunneling rates from the left lead through level 1 to the right lead (${\omega }_{11,11}^{\to }+{\omega }_{01,01}^{\to }$ when level 2 is occupied and ${\omega }_{10,10}^{\to }+{\omega }_{00,00}^{\to }$ when level 2 is empty). (c) The difference between the sequential tunneling rates of the two stages (level 2 is empty and full). (c) The difference between the cotunneling rates of the two stages (level 2 is empty and full).Reuse & Permissions