^{1,a)}, T. Choi

^{1}, S. Hellmüller

^{1}, K. Ensslin

^{1}, T. Ihn

^{1}and S. Schön

^{2}

### Abstract

A detailed analysis of the tunability of a radio-frequency quantum point contact setup using a C − LCR circuit is presented. We calculate how the series capacitance influences resonance frequency and charge-detector resistance for which matching is achieved as well as the voltage and power delivered to the load. Furthermore, we compute the noise contributions in the system and compare our findings with measurements taken with an etched quantum point contact. While our considerations mostly focus on our specific choice of matching circuit, the discussion of the influence of source-to-load power transfer on the signal-to-noise ratio is valid generally.

We are greatly indebted to Cecil Barengo and Paul Studerus for their superb technical assistance, and funding by the Swiss National Science Foundation (NSF(CH)) via National Center of Competence in Research (NCCR) Nanoscale Science is gratefully acknowledged.

I. INTRODUCTION

II. POWER TRANSFER

III. TUNING OF THE MATCHING NETWORK

IV. NOISE ANALYSIS

V. MEASUREMENTS ON AN ETCHED QPC

VI. CONCLUSION

### Key Topics

- Amplifiers
- 28.0
- Reflection coefficient
- 13.0
- Thermal noise
- 12.0
- Capacitance
- 10.0
- Electrical resistivity
- 10.0

##### H01L29/00

## Figures

(a) Simplified diagram of the resonant circuit used in our experiments, including a resistance r in series to the inductor to account for losses. (b) Schematics of a general two-port matching network in which a fraction |Γ|2 of the incoming power is reflected, a fraction κ is delivered to the load (charge detector), and a fraction N is dissipated in the matching circuit. Part (b) of this figure is adapted from Ref. 5 .

(a) Simplified diagram of the resonant circuit used in our experiments, including a resistance r in series to the inductor to account for losses. (b) Schematics of a general two-port matching network in which a fraction |Γ|2 of the incoming power is reflected, a fraction κ is delivered to the load (charge detector), and a fraction N is dissipated in the matching circuit. Part (b) of this figure is adapted from Ref. 5 .

Relative signal-to-noise ratio as a function of power-transfer coefficient κ. The plots represent Eq. (16) with the parameters T amp = 6 K, T QPC = 0.1 K, T circ = 0.1 K (solid blue), T amp = 2 K, T QPC = 4 K, T circ = 0.1 K (dashed red), and T amp = 4 K, T QPC = 2 K, T circ = 5 K (dotted green). (Inset) Change in reflection coefficient at resonance for R = 29 to 28 kΩ as a function of κ calculated from Eq. (10) (solid line) and via impedance for a L − CR (blue squares) and C − LCR-circuit (red circles). The power transfer coefficient κ is varied through r (in series to the parallel capacitance C for L − CR and the inductance L for C − LCR, respectively), keeping the load resistance for which best matching is achieved at 30 kΩ by changing C (in both cases) and if necessary L (only in case of the L − CR-circuit) accordingly.

Relative signal-to-noise ratio as a function of power-transfer coefficient κ. The plots represent Eq. (16) with the parameters T amp = 6 K, T QPC = 0.1 K, T circ = 0.1 K (solid blue), T amp = 2 K, T QPC = 4 K, T circ = 0.1 K (dashed red), and T amp = 4 K, T QPC = 2 K, T circ = 5 K (dotted green). (Inset) Change in reflection coefficient at resonance for R = 29 to 28 kΩ as a function of κ calculated from Eq. (10) (solid line) and via impedance for a L − CR (blue squares) and C − LCR-circuit (red circles). The power transfer coefficient κ is varied through r (in series to the parallel capacitance C for L − CR and the inductance L for C − LCR, respectively), keeping the load resistance for which best matching is achieved at 30 kΩ by changing C (in both cases) and if necessary L (only in case of the L − CR-circuit) accordingly.

(a) Influence of the series capacitance on the frequency response of the matching circuit shown in Fig. 1 . The parameters used for this calculation are C p = 2.6 pF, L = 150 nH, r = 5 Ω, R = 50 kΩ. The tunable capacitance C is increased from 0.5 pF (rightmost blue curve) to 2.5 pF (leftmost purple curve). (b) Reflection coefficient as a function of QPC resistance at the resonance frequency. (c) and (d) As in (a) and (b), but with r = 2 Ω.

(a) Influence of the series capacitance on the frequency response of the matching circuit shown in Fig. 1 . The parameters used for this calculation are C p = 2.6 pF, L = 150 nH, r = 5 Ω, R = 50 kΩ. The tunable capacitance C is increased from 0.5 pF (rightmost blue curve) to 2.5 pF (leftmost purple curve). (b) Reflection coefficient as a function of QPC resistance at the resonance frequency. (c) and (d) As in (a) and (b), but with r = 2 Ω.

(a) Absolute value of the maximally allowed rf voltage at resonance for a given QPC rms voltage of 300 μV as a function of R with different serial capacitance values from 0.5 pF to 2.5 pF. For the dotted black curve, the parasitic resistance r was chosen to be 2 Ω instead of 5 Ω. (b) Power delivered to the matching network according to Eq. (12) . (c) Absolute value of the changes in reflection coefficients from Figs. 3(b) and 3(d) using a change in R modeled by a QPC in the saddle-point approximation with a change in potential of 50 μeV due to an electron entering a nearby dot. (d) Change in reflected voltage due to electron tunneling in a nearby dot taking into account the maximally tolerable input voltage V in calculated in Eq. (11) and shown in (a).

(a) Absolute value of the maximally allowed rf voltage at resonance for a given QPC rms voltage of 300 μV as a function of R with different serial capacitance values from 0.5 pF to 2.5 pF. For the dotted black curve, the parasitic resistance r was chosen to be 2 Ω instead of 5 Ω. (b) Power delivered to the matching network according to Eq. (12) . (c) Absolute value of the changes in reflection coefficients from Figs. 3(b) and 3(d) using a change in R modeled by a QPC in the saddle-point approximation with a change in potential of 50 μeV due to an electron entering a nearby dot. (d) Change in reflected voltage due to electron tunneling in a nearby dot taking into account the maximally tolerable input voltage V in calculated in Eq. (11) and shown in (a).

(a) and (b) Absolute value of the reflection coefficient Γ (solid blue line) and the power-transfer coefficients κ (dashed red) and N (dotted green) as a function of frequency and QPC resistance at resonance, respectively. (c) and (d) Frequency and QPC resistance dependence of the equivalent noise temperatures of all relevant noise sources. These include QPC shot noise as expected at zero temperature (dashed-dotted red), thermal noise of the QPC at zero bias (solid green), thermal noise in r (dotted pink), and amplifier noise (dashed blue). The sum of all noise sources is plotted as a solid black line. The parameters used are C = 1.5 pF, C p = 2.6 pF, L = 150 nH, r = 5 Ω, R = 50 kΩ, f = 203.19 MHz, T amp = 2 K, T QPC = 2 K, T circ = 5 K, V QPC = 300 μV, and Z amp = 50 Ω for all plots.

(a) and (b) Absolute value of the reflection coefficient Γ (solid blue line) and the power-transfer coefficients κ (dashed red) and N (dotted green) as a function of frequency and QPC resistance at resonance, respectively. (c) and (d) Frequency and QPC resistance dependence of the equivalent noise temperatures of all relevant noise sources. These include QPC shot noise as expected at zero temperature (dashed-dotted red), thermal noise of the QPC at zero bias (solid green), thermal noise in r (dotted pink), and amplifier noise (dashed blue). The sum of all noise sources is plotted as a solid black line. The parameters used are C = 1.5 pF, C p = 2.6 pF, L = 150 nH, r = 5 Ω, R = 50 kΩ, f = 203.19 MHz, T amp = 2 K, T QPC = 2 K, T circ = 5 K, V QPC = 300 μV, and Z amp = 50 Ω for all plots.

(a) System noise temperature at the input of the low-noise amplifier as a function of QPC resistance for different series capacitances at resonance, using the same parameter values as above. (b) Expected signal-to-noise ratio at 1 MHz bandwidth and charge sensitivity, calculated by dividing the reflected signal power adapted from Fig. 4(d) by the total noise in (a).

(a) System noise temperature at the input of the low-noise amplifier as a function of QPC resistance for different series capacitances at resonance, using the same parameter values as above. (b) Expected signal-to-noise ratio at 1 MHz bandwidth and charge sensitivity, calculated by dividing the reflected signal power adapted from Fig. 4(d) by the total noise in (a).

(a) Schematics of the circuit used for the measurements in this section including a scanning electron microscopy image of the InAs nanowire quantum dot and its self-aligned charge detector. (b) QPC conductance as a function of side-gate voltage V SG = V L + V R at a bias of 250 μV. A total resistance of 4.26 kΩ (2 kΩ contact resistance and 2.26 kΩ from the resistors on the circuit board and the room-temperature plug as well as the dc wiring) has been subtracted.

(a) Schematics of the circuit used for the measurements in this section including a scanning electron microscopy image of the InAs nanowire quantum dot and its self-aligned charge detector. (b) QPC conductance as a function of side-gate voltage V SG = V L + V R at a bias of 250 μV. A total resistance of 4.26 kΩ (2 kΩ contact resistance and 2.26 kΩ from the resistors on the circuit board and the room-temperature plug as well as the dc wiring) has been subtracted.

(a) Absolute value of the reflection coefficient versus frequency at zero QPC conductance for different diode voltages. (b) As in (a), but as a function of QPC resistance at resonance. For the calculated curves, we used L = 150 nH, C p = 2.6 pF, R contact = 2 kΩ, R dc wiring = 2.26 kΩ, and C = 1.71 pF, r = 5.2 Ω (solid blue line), C = 1.60 pF, r = 4.55 Ω (dotted green line), or C = 1.39 pF, r = 4.85 Ω (dashed red line). The measured reflection was converted to linear scale using an overall attenuation of 20 dB (blue dots), 17.8 dB (green crosses), and 17.8 dB (red circles). (c) Measured total noise in the absence of a carrier signal. A frequency independent gain of 45.8 dB was subtracted from the signal measured with a spectrum analyzer at a resolution bandwidth of 1 MHz and a video bandwidth of 10 Hz. The arrows mark spurious noise peaks occurring in the spectrum. (d) System noise temperature as a function of QPC resistance, measured with a resolution and video bandwidth of 100 kHz and 10 Hz, respectively. The measured curves have been brought to the same level by subtracting a gain of 46 dB for the green xs (V D = −0.5 V) and 44.8 dB for the red circles (V D = −0.2 V). For the calculations, we used the same parameters as in (b) with the addition of T amp = 1.8 K, T QPC = 2 K, T circ = 5.3 K, V bias = 250 μV, and we assumed a perfectly matched amplifier input.

(a) Absolute value of the reflection coefficient versus frequency at zero QPC conductance for different diode voltages. (b) As in (a), but as a function of QPC resistance at resonance. For the calculated curves, we used L = 150 nH, C p = 2.6 pF, R contact = 2 kΩ, R dc wiring = 2.26 kΩ, and C = 1.71 pF, r = 5.2 Ω (solid blue line), C = 1.60 pF, r = 4.55 Ω (dotted green line), or C = 1.39 pF, r = 4.85 Ω (dashed red line). The measured reflection was converted to linear scale using an overall attenuation of 20 dB (blue dots), 17.8 dB (green crosses), and 17.8 dB (red circles). (c) Measured total noise in the absence of a carrier signal. A frequency independent gain of 45.8 dB was subtracted from the signal measured with a spectrum analyzer at a resolution bandwidth of 1 MHz and a video bandwidth of 10 Hz. The arrows mark spurious noise peaks occurring in the spectrum. (d) System noise temperature as a function of QPC resistance, measured with a resolution and video bandwidth of 100 kHz and 10 Hz, respectively. The measured curves have been brought to the same level by subtracting a gain of 46 dB for the green xs (V D = −0.5 V) and 44.8 dB for the red circles (V D = −0.2 V). For the calculations, we used the same parameters as in (b) with the addition of T amp = 1.8 K, T QPC = 2 K, T circ = 5.3 K, V bias = 250 μV, and we assumed a perfectly matched amplifier input.

Total noise temperature at resonance in the absence of a rf carrier signal for V D = −0.8 V as a function of dc bias for different QPC conductances. The spectrum analyzer's resolution bandwidth was set to 100 kHz with a video bandwidth of 10 Hz. We subtracted a constant gain of 45.1 dB. (a) G QPC = 0 × 2e 2/h. (b) G QPC = 0.1 × 2e 2/h. (c) G QPC = 0.6 × 2e 2/h. The common parameters used for the calculation are identical to the ones used for Fig. 8(d) . (Insets) Conductance of the QPC as a function of applied bias.

Total noise temperature at resonance in the absence of a rf carrier signal for V D = −0.8 V as a function of dc bias for different QPC conductances. The spectrum analyzer's resolution bandwidth was set to 100 kHz with a video bandwidth of 10 Hz. We subtracted a constant gain of 45.1 dB. (a) G QPC = 0 × 2e 2/h. (b) G QPC = 0.1 × 2e 2/h. (c) G QPC = 0.6 × 2e 2/h. The common parameters used for the calculation are identical to the ones used for Fig. 8(d) . (Insets) Conductance of the QPC as a function of applied bias.

## Tables

Typical values for the parameters used in this work.

Typical values for the parameters used in this work.

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