By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong- and weak-tunnel
couplings, leading to gate-dependent level renormalization. By comparison to a one- and two-shell model, this is shown to be a consequence of disorder-induced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization.
We calculate the nonlinear cotunneling conductance through a quantum dot with 3 electrons occupying the three highest lying energy levels. Starting from a 3-orbital Anderson model, we apply a generalized Schrieffer-Wolff transformation to derive an e
ffective Kondo model for the system. Within this model we calculate the nonequilibrium occupation numbers and the corresponding cotunneling current to leading order in the exchange couplings. We identify the inelastic cotunneling thresholds and their splittings with applied magnetic field, and make a qualitative comparison to recent experimental data on carbon nanotube and InAs quantum-wire quantum dots. Further predictions of the model like cascade resonances and a magnetic-field dependence of the orbital level splitting are not yet observed but within reach of recent experimental work on carbon nanotube and InAs nanowire quantum dots.
