Gate-dependent tunneling-induced level shifts observed in carbon nanotube quantum dots

We have studied electron transport in clean single-walled carbon nanotube quantum dots. Because of the large number of the Coulomb blockade diamonds simultaneously showing both shell structure and the Kondo effect, we are able to perform a detailed analysis of tunneling renormalization effects. Thus, determining the environment induced level shifts of this artificial atom. In shells where only one of the two orbitals is strongly coupled, we observe a marked asymmetric gate dependence of the inelastic cotunneling lines together with a systematic gate dependence of the size (and shape) of the Coulomb diamonds. These effects are all given a simple explanation in terms of second-order perturbation theory in the tunnel coupling.

Figure 1

Bias spectroscopy plot of a SWCNT QD for $-10\mathrm{V}<V_g<10\mathrm{V}$. The Coulomb diamonds are seen for almost every added electron (285), and 88 odd-occupancy Coulomb blockade diamonds exhibit a zero-bias Kondo resonance.

Figure 2

Differential conductance as a function of bias voltage $V_{sd}$ and gate voltage $V_g$ (left). Schematic of the four-electron shells (right). (a) Four-electron shell filling for ${\Gamma }_1\sim {\Gamma }_2$, with strong enough coupling to facilitate Kondo ridges in the $N+1$ and $N+3$ diamonds, as shown in the schematic. A total of 17 four-electron shells show this type of behavior across the full gate voltage range. (b) Four-electron shell filling for ${\Gamma }_1⪡{\Gamma }_2$. In the $N+1$ and $N+2$ diamonds, the inelastic cotunneling threshold has clearly acquired a gate dependence. In total, there are 21 four-electron shells showing this type of behavior. Here, the right panel shows the cotunneling thresholds and diamond edges as calculated within second-order PT (the white Kondo ridge is drawn by hand). (c) Four-electron shell filling for ${\Gamma }_1⪢{\Gamma }_2$. Gate-dependent cotunneling thresholds are now observed in the $N+2$ and $N+3$ diamonds instead. In total, there are 7 four-electron shells showing this type of behavior. 15 four-electron shells could not be uniquely categorized. In the 28 shell sequences where ${\Gamma }_1⪢{\Gamma }_2$ or ${\Gamma }_1⪡{\Gamma }_2$, the gate-dependent cotunneling threshold ridges inside the diamonds appear to form a $U$ or $V$ lying down with the openings facing the zero-bias Kondo ridge. Together the two $U$’s ($V$’s) and the Kondo ridge resemble a two-headed arrow, where the $U$ $\left(V\right)$ form the two heads and the Kondo ridge the shaft. This means that the tree center diamonds in the different domains have cotunneling and Kondo ridges, which can be schematically summarized as ${\Gamma }_1\sim {\Gamma }_2:$ , ${\Gamma }_1⪡{\Gamma }_2:$ , and ${\Gamma }_1⪢{\Gamma }_2:$ .

Figure 3

(Color online) The ratio of the widths of the diamonds (proportional to the additional energies) in the 28 unusual four-electron shells to the odd-electron diamonds without a Kondo ridge in the same four-electron shells. The solid (cross-hatched) pairs is for ${\Gamma }_1⪡{\Gamma }_2$ $({\Gamma }_1⪢{\Gamma }_2)$. The average width of the even-electron diamonds (black) is 1.47, and the average width of the odd-electron diamonds with a Kondo ridge [red (dark gray)] is 0.66, as indicated by the black lines.

Figure 4

(Color online) Schematic drawing illustrating the difference in renormalization of ${\tilde{E}}_{1,0}$ [red (dark gray) solid lines] and ${\tilde{E}}_{0,1}$ [red (dark gray) dashed lines] at opposite ends of the $N+1$ diamond. Assuming ${\Gamma }_1⪡{\Gamma }_2$, the (0,1) level is strongly perturbed by virtual emptying (left), whereas the (1,0) level is most strongly perturbed by virtual double occupancy (right).