Measuring a Quantum Dot with an Impedance-Matching On-Chip Superconducting $LC$ Resonator at Gigahertz Frequencies

We report on the realization of a bonded-bridge on-chip superconducting coil and its use in impedance matching a highly ohmic quantum dot (QD) to a 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. Specifically, 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 3 times higher than that of the one previously reported, a wire-bonded coil. As a proof of principle, we fabricate an $LC$ circuit that achieves impedance matching to an approximately $15\text{−}\mathrm{k}\mathrm{\Omega }$ load and validate it with a load defined by a carbon nanotube QD, whose shot noise we measure in the Coulomb-blockade regime.

Figure 1

(a) Optical micrograph of the sample. (b) False-color SEM image of a superconducting Nb on-chip coil (cyan) connected via an Al bond wire (pink) to the measurement setup. The right end of the coil connects to a QD laying on the same $\mathrm{Si}/\mathrm{Si}{\mathrm{O}}_2$ substrate. (c) False-color SEM image of the CNT QD with $\mathrm{Ti}/\mathrm{Au}$ leads ($S$, $D$) and a side gate ($G$). (d) Simplified sketch of the microwave-only electrical circuit: the coil is modeled as an $LC$ circuit whose losses are caught by a resistance $R_{\mathrm{loss}}$ in series with the inductor $L$, the CNT is characterized by a resistance dependent on the gate and source-drain dc voltages, and the $LC$ tank circuit partially matches $R$ to the $Z_0=50\mathrm{\Omega }$ of the measurement setup line. In reflectance measurement, the voltage $V=V^{+}+V^{-}$ at the interface is the sum of the voltages given by the incident wave (green) and the reflected wave (red). (e) In noise measurements, there is no incident wave, $V^{+}=0$. By contrast, a voltage source $V_{\text{noise}}$ needs to be considered in series with the CNT resistance. This voltage is transmitted to become $V$ with a frequency-dependent transmission function determined by the $LC$ network.

Figure 2

(a) Schematic of the measurement setup with direct-current (dc) input and microwave (MW) input ($i$) and output ($o$) lines. The setup is used, alternatively, for reflectometry (both MW input and output) and noise measurements (only MW output), employing either a VNA or an PSA (vector network or power spectrum analyzer). (b) Magnitude squared of the reflection coefficient ${\mathrm{\Gamma }}_{\mathrm{VNA}}\equiv V_o/V_i$ measured around the resonance frequency. The further-investigated regime lies between the close-to-zero conductance (Coulomb blockade) red curve (shallowest trace) and the vicinity of the blue curve. The separately measured black trace corresponds to an almost perfectly matched conductance. (c) Fit curve (orange) of the reflectance data ${|{\mathrm{\Gamma }}_{\mathrm{VNA}}|}^2(dI/d{V}_{SD})$ collected at the fixed frequency 3.23 GHz during a $V_G\times V_{SD}$ 2D sweep. All of the measured $dI/d{V}_{SD}$ values (the green diamonds) correspond to less than $30\mu \mathrm{S}$, except for one point taken from the near-match curve in (b). The deeper curve (brown) is calculated with the extracted fit parameters at the extracted resonance frequency, and its minimum indicates $G_{\text{match}}$.

Figure 3

(a) Derivative of the dc current ($dI/d{V}_{SD}$) as a function of the gate and source-drain voltages. The contour of the Coulomb diamonds is highlighted by the dashed line. (b) Reflectance measured at $f=3.23\mathrm{GHz}$, with the same contour as in (a). (c) Differential conductance $G$ deduced from the reflectance ${|{\mathrm{\Gamma }}_{\mathrm{VNA}}|}^2(f=3.23\mathrm{GHz},V_G,V_{SD})$ by using the $LC$ matching-circuit parameters extracted in the fitting. (d) Cuts in the (a),(c) maps, showing an excellent overlap of low- and high-frequency differential-conductance values.

Figure 4

(a) Shot-noise spectral density $S_I$ as a function of $V_G$ and $V_{SD}$ obtained from the measured power spectral density $⟨\delta P⟩$ at the power spectrum analyzer using Eq. (6). (b) Schottky noise $2e|I|$ calculated from the dc measured current $I$, and (c) Fano factor $F=S_I/|2eI|$. The Coulomb-diamond contours (the dashed lines) are copied from the conductance plot in Fig. 3. Instead of the $F=1$ value proper for elastic cotunneling, the low-bias cyan regions indicate erroneously calculated values emerging from the division of a “noisy” number by a very small one. These regions need to be discarded. A cut at the black dashed vertical line is depicted in (d). The observed highest Fano factor is $F\simeq 8$.