Measurements of a Quantum Dot with an Impedance-Matching On-Chip LC Resonator at GHz Frequencies

We report the realization of a bonded-bridge on-chip superconducting coil and its use in impedance-matching a highly ohmic quantum dot (QD) to a $\rm{3~GHz}$ measurement setup. The coil, modeled as a lumped-element $LC$ resonator, is more compact and has a wider bandwidth than resonators based on coplanar transmission lines (e.g. $\lambda/4$ impedance transformers and stub tuners) at potentially better signal-to-noise ratios. In particular for measurements of radiation emitted by the device, such as shot noise, the 50$\times$ larger bandwidth reduces the time to acquire the spectral density. The resonance frequency, close to 3.25 GHz, is three times higher than that of the one previously reported wire-bonded coil. As a proof of principle, we fabricated an $LC$ circuit that achieves impedance-matching to a $\rm{\sim 15~k\Omega}$ load and validate it with a load defined by a carbon nanotube QD of which we measure the shot noise in the Coulomb blockade regime.