Pair Tunneling Resonance in the Single-Electron Transport Regime

We predict a new electron pair tunneling (PT) resonance in nonlinear transport through quantum dots with positive charging energies exceeding the broadening due to thermal and quantum fluctuations. The PT resonance shows up in the single-electron transport (SET) regime as a peak in the derivative of the nonlinear conductance, $d^{2}I/d{V}^2$, when the electrochemical potential of one electrode matches the average of two subsequent charge addition energies. For a single level quantum dot (Anderson model) we find the analytic peak shape and the dependence on temperature, magnetic field, and junction asymmetry and compare with the inelastic cotunneling peak which is of the same order of magnitude. In experimental transport spectroscopy the PT resonance may be mistaken for a weak SET resonance judging only by the voltage dependence of its position. Our results provide essential clues to avoid such erroneous interpretation.

Figure 1

Energy differences between many-body eigenstates (dot chemical potentials) of the Anderson model and sketch of a COT process (a) and a SET process (b). (c) More detailed sketch of a PT process, giving rise to a resonance at $\Delta =U/2$ above the SET onset. Filled (unfilled) circles indicate real (virtual) occupation.

Figure 2

(a) $dI/dV$ vs $V$, $V_g$ plotted on logarithmic color scale for ${\Gamma }_L={\Gamma }_R=\Gamma$, $U=100T=2000\Gamma$ and $C_R=C_L=100C_g$. (b) Same as (a) except for the finite Zeeman splitting by $h=15T$ and asymmetric capacitances $C_L=0.7C_R=100C_g$. In addition, the gate capacitance of state 2 is larger: $C_{g,2}=1.176C_g$. (c) $d^{2}I/d{V}^2$ vs $V$ around the PT threshold $V_{\mathrm{PT}}$ at $V_g=-0.625UC/C_g$ ($C=C_L+C_R+C_g$), indicated by the red line in (a). This demonstrates the scaling with $1/U^2$ and the agreement with the analytic expression Eq. (3) marked by (A). (d) The two independent Keldysh diagrams contributing to the pair tunneling resonance. A sum over $\sigma$ is assumed and $\overline{\sigma }$ denotes the opposite spin projection of $\sigma$.