Shot Noise of a Quantum Dot Measured with Gigahertz Impedance Matching

The demand for a fast high-frequency read-out of high-impedance devices, such as quantum dots, necessitates impedance matching. Here we use a resonant impedance-matching circuit (a stub tuner) realized by on-chip superconducting transmission lines to measure the electronic shot noise of a carbon-nanotube quantum dot at a frequency close to 3 GHz in an efficient way. As compared to wideband detection without impedance matching, the signal-to-noise ratio can be enhanced by as much as a factor of 800 for a device with an impedance of $100\mathrm{k}\mathrm{\Omega }$. The advantage of the stub resonator concept is the ease with which the response of the circuit can be predicted, designed, and fabricated. We further demonstrate that all relevant matching circuit parameters can reliably be deduced from power-reflectance measurements and then used to predict the power-transmission function from the device through the circuit. The shot noise of the carbon-nanotube quantum dot in the Coulomb blockade regime shows an oscillating suppression below the Schottky value of $2eI$, as well as an enhancement in specific regions.

Figure 1

(a) Illustration of the carbon-nanotube stamping process. CVD growth is done on the top silicon substrate (blue), which contains an area of square mesas. On the bottom substrate (gray), bottom gates are fabricated and covered with silicon nitride. (b) False-color image of the CNT connected to $\mathrm{Ti}/\mathrm{Au}$ leads (orange) and bottom gates underneath (yellow), which are covered with silicon nitride. (c) Sketch of the cross section. The gates called source gate (SG), drain gate (DG), left gate (LG), and right gate (RG) are covered with silicon nitride. The CNT is stamped on top and contacted with $\mathrm{Ti}/\mathrm{Au}$ leads. (d) SEM image of the stub impedance-matching circuit made out of two niobium coplanar transmission lines in parallel. The $50\mathrm{\Omega }$ side is at the launcher on the right, and the high-resistance sample is located at the bottom left. The two bond wires are air bridges to connect the ground planes.

Figure 2

(a) Schematic of the setup with an input and an output rf line plus one dc line. Everything on a blue background is inside a dilution refrigerator with a base temperature of 20 mK. (b) Amplitude squared of the reflection coefficient $\mathrm{\Gamma }=V_{\mathrm{out}}/V_{\mathrm{in}}$ around the resonance frequency. Symbols are measured and lines fitted or calculated. The stub tuner loss $\alpha =0.046{\mathrm{m}}^{-1}$ as well as the two lengths $D_1=10.355\mathrm{mm}$ and $D_2=10.589\mathrm{mm}$ are extracted by fitting (solid red line) to the spectrum in the Coulomb blockade regime of the QD, where $G=0$ (blue triangles). The upper spectrum for a finite dc conductance of $G=0.2e^2/h$ is plotted with a shift of 1 dB for clarity (green circles). It matches well the calculated reflection coefficient (dashed red line) using the previous fit parameters. The conductance dependence of the reflectance at the resonance frequency is plotted in the inset. (c) Calculated voltage-transmission function of the stub tuner for three typical device conductances using the fit parameters gained from (b).

Figure 3

(a) Derivative of the dc current ($dI/d{V}_{\mathrm{SD}}$) as a function of the voltage on the right gate and of the source-drain bias. The contour of the CB diamonds is highlighted by the dashed line. (b) Differential conductance deduced from the reflection amplitude.

Figure 4

(a) Calibrated shot-noise spectral density $S_I$ as a function of voltage on the right gate and of source-drain voltage. (b) Schottky noise $2e|I|$ and (c) excess Poissonian noise $S_I^{\mathrm{EP}}=S_I-2e|I|$. The CB diamond contours (dashed lines) are copied from the conductance plot in Fig. 3. (d) Fano factors averaged over a range of 1.2 mV in $V_{\mathrm{SD}}$ (left scale) and absolute value of current (right scale) along the dotted lines in (c) marked with a star. Fano factor peaks correspond to the onset of current transitions from one plateau to the next. (e) Fano factors along the horizontal line in (c) (marked with a square) exceeding one.