Introduction

An essential element in low temperature experimental physics is the thermometer1. Sensors that link temperature to another physical quantity in an accurate, fast, stable, and compact manner are desired. If the link is done via a well-known physical law, the sensor is called a primary thermometer because it removes the need of calibration to a second thermometer.

Several primary thermometers have been developed for low temperature applications. A common technique is based on the Johnson-Nyquist noise of a resistor2 which can be used in combination with superconducting quantum interference devices to perform current-sensing noise thermometry (CSNT)3. Shot-noise thermometry (SNT)4,5,6 uses the temperature-dependent voltage scaling of the noise power of a biased tunnel junction. Coulomb blockade thermometry (CBT) makes use of charging effects in two-terminal devices with multiple tunnel junctions7,8,9. Thermometry using counting statistics via single-electron devices is also possible10,11,12,13,14. However, in all these cases, the sensors require a continuous flow of electrons from source to drain in two terminal devices which, for particular experiments such as in single-molecule junction and single-nanoparticle devices, might not be possible or even desirable15,16.

Moreover, recent advances in device nanoengineering have led to a focused interest in using concepts from quantum thermodynamics17,18,19,20,21 to improve the efficiency of technologies such as the thermal diode22,23 or thermal energy harvesters24. In these nanoelectronic devices, determining the local temperature in different reservoirs of the device is of particular relevance but challenging from an experimental perspective.

Here, we demonstrate a type of primary thermometer that uses cyclic electron tunneling to measure the temperature of a single electron reservoir without the need of electrical transport. The tunneling occurs between a system with a zero-dimensional (0D) density of states (DOS) -in this case a quantum dot (QD)—and a single electron reservoir of unknown temperature. Our thermometer relates temperature and capacitance changes with a well known physical law by using the ratio kB/e between the Boltzmann constant and the electron charge. The thermometer is driven and read out by an electrical resonator at radio-frequencies. In this proof-of-principle experiment, we perform primary thermometry down to 1 K but show that the operational temperature range of the sensor can be extended in-situ using electrostatic fields. Our experimental results follow our theoretical predictions of the temperature-dependent capacitance of the system. The thermometer is implemented in a complementary-metal-oxide-semiconductor (CMOS) transistor which makes it suitable for large-scale manufacturing and seamless integration with silicon-based quantum circuits, a promising platform for the implementation of a scalable quantum computer25,26,27.

Results

Theory

We consider a QD in thermal equilibrium with an electron bath whose temperature T we wish to measure. The QD is capacitively coupled to a gate electrode Ctg, and tunnel coupled to the reservoir via a tunnel junction with capacitance Cj and resistance Rj, see Fig. 1a. The system is operated in the quantum confinement regime such that electrons occupy discrete energy levels of the QD. The coupled QD-reservoir system has an associated differential capacitance28,29 Cdiff as seen from the gate given by

$$C_{{\mathrm{diff}}} = \frac{{\partial Q}}{{\partial V_{{\mathrm{tg}}}}} = \underbrace {\alpha C_{\mathrm{j}}}_{{\mathrm{geometrical}}} - \underbrace {e\alpha \frac{{\partial P_1}}{{\partial V_{{\mathrm{tg}}}}}}_{{\mathrm{tunneling}}},$$
(1)

where Q is the net charge in the QD, Vtg is the gate voltage, e is the electron charge, α is the gate coupling Ctg/(Cj + Ctg), and P1 is the probability of having an excess electron in the QD. The first term in Eq. (1) represents the DC limit of the capacitance, the geometrical capacitance, whereas the second term represents the parametric dependence of the excess electron probability on gate voltage, the tunneling capacitance. The second term is the focus of this Article.

Fig. 1
figure 1

Theory. a Circuit equivalent of the quantum dot (QD)-reservoir system. b Schematic of cyclic electron exchange between a discrete energy level of a QD and a thermally broadened electron reservoir. c Energy diagram of a fast driven two-level-system (TLS) with discrete energies E0 and E1, across a charge degeneracy point. d Probability P1 of an electron to be in the QD as a function of energy level detuning ε. e Tunneling capacitance Ct as a function of ε

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To obtain an analytical expression for the tunneling capacitance Ct, we next consider the QD-reservoir charge distribution in detail. In the limit of weak tunnel coupling, the QD-reservoir system can be described by the Hamiltonian \(H = {\textstyle{1 \over 2}}\varepsilon \sigma _{\mathrm{z}}\) where ε is the energy detuning and σz is the z Pauli matrix. The eigenenergies E0 = ε/2 and E1 = −ε/2 are associated with the QD states with zero and one excess electron, respectively. This additional electron can tunnel in and out of the electron reservoir at a rate γ, as schematically depicted in Fig. 1b. The energy detuning between these states can be controlled by Vtg given that ε = −eα(Vtg − V0). Here V0 is the gate voltage offset at which the two eigenstates are degenerate.

To probe the tunneling capacitance, the system is subject to a modulation occurring at some frequency, fr that varies the energy detuning ε = ε0 + δε sin(2πfrt). In the limit \(\gamma \gg f_{\mathrm{r}}\), the QD and reservoir are in thermal equilibrium and electrons tunnel in and out of the reservoir adiabatically. In this situation, P1 tracks the thermal population, \(P_1^0\), given by the instantaneous gate-voltage excitation28 and Ct can be expressed as

$$C_{\mathrm{t}} = - e\alpha \frac{{\partial P_1^0}}{{\partial V_{{\mathrm{tg}}}}} = (e\alpha )^2\frac{{\partial P_1^0}}{{\partial \varepsilon }}.$$
(2)

From the energy spectrum represented in Fig. 1c and taking into account the spin degeneracy of two in the QD, Maxwell–Boltzmann statistics give the equilibrium probability distribution

$$P_1^0 = \frac{{2\,{\mathrm{exp}}(\varepsilon {\mathrm{/}}2k_{\mathrm{B}}T)}}{{{\mathrm{exp}}( - \varepsilon {\mathrm{/}}2k_{\mathrm{B}}T) + 2\,{\mathrm{exp}}(\varepsilon {\mathrm{/}}2k_{\mathrm{B}}T)}},$$
(3)

and this is depicted as a function of detuning in Fig. 1d. At large negative detuning the QD remains unoccupied \(\left( {P_1^0 = 0} \right)\), at large positive detuning the QD is occupied \(\left( {P_1^0 = 1} \right)\), and at \(\varepsilon = - k_{\text{B}} T \, {{\ln} 2} , P_1^0 = 1{\mathrm{/}}2\). We calculate the tunneling capacitance of the system and obtain

$$C_{\mathrm{t}} = \frac{{(e\alpha )^2}}{{4k_{\mathrm{B}}T}}\frac{1}{{{\mathrm{cosh}}^2\left( {\frac{\varepsilon }{{2k_{\mathrm{B}}T}}} \right)}}.$$
(4)

where we have redefined the detuning ε to account for the peak-center shift induced by temperature (ε → ε + kBT ln 2). Thus the tunneling capacitance Ct has a full width at half maximum (FWHM) with respect to ε of

$$\varepsilon _{1/2} = 4\,{\mathrm{ln}}\left( {\sqrt 2 + 1} \right)k_{\mathrm{B}}T,$$
(5)

as plotted in Fig. 1e. Since ε1/2 = eαV1/2, the analysis shows it is possible to obtain the temperature of the electron reservoir from the FWHM of the Ct vs Vtg curve once the gate lever arm α is known. The quantity V1/2 is the FWHM with respect to gate voltage. Furthermore, from Eq. (4) we see that the peak amplitude \(C_{\mathrm{t}}^0\) of the tunneling capacitance Ct, is inversely proportional to the reservoir temperature T,

$$C_{\mathrm{t}}^{\mathrm{0}} \propto \frac{1}{T}.$$
(6)

In the case of a finite magnetic field (see Supplementary Note 1), the expressions for ε1/2 and \(C_{\mathrm{t}}^0\) remain as in Eqs. (5) and (6), respectively. We note that the Ct peak center shifts to lower detuning values as the magnetic field B is increased, see Supplementary Fig. 1a, b. The shift tends to ε(B) = ε(0) − gμB|B|/2 for gμB|B| > kBT, where g is the electron g-factor and μB is the Bohr magneton. This demonstrates our proposed method of determining the reservoir temperature T from capacitance Ct measurements is independent of magnetic field. We note that our analysis is valid as long as kBT remains smaller than the discrete energy spacing in the QD (ΔE) and larger than the QD level broadening (hγ). These two conditions set the temperature range in which thermometry by cyclic electron tunneling is accurate. In the latter case (kBT < hγ), Ct takes a Lorentzian form given by

$$C_{\mathrm{t}} = \frac{{(\alpha e)^2}}{\pi }\frac{{h\gamma }}{{(h\gamma )^2 + \varepsilon ^2}}.$$
(7)

and ε1/2 is given by 2hγ30, and is thus no longer temperature dependent. The relaxation rate γ is directly linked to the shape of the tunnel barrier between the QD and the reservoir which can be tuned electrically by, for example, a gate electrode. The tunneling capacitance Ct can be probed with high-frequency techniques such as gate-based reflectometry31,32 and can be used to measure temperature. We refer to this sensor as the gate-based electron thermometer (GET).

Device and high-frequency resonator

The device used here is a silicon nanowire field-effect transistor (NWFET)33 fabricated in fully-depleted silicon-on-insulator (SOI) following CMOS fabrication processes. At low temperatures, gate-defined QDs form in the channel of the NWFET34,35, see Fig. 2a. The transistor has a channel length l = 44 nm and width w = 42 nm. The 8 nm thick NW channel was pattered on SOI above the 145 nm buried oxide (BOX). The gate oxide consists of 0.8 nm SiO2 and 1.9 nm HfSiON resulting in an equivalent gate oxide thickness of 1.3 nm. The top-gate (tg) is formed using 5 nm TiN and 50 nm polycrystalline silicon. The NW channel is separated from the highly doped source and drain reservoirs by 20 nm long Si3N4 spacers. The silicon wafer under the BOX can be used as a global back-gate (bg).

Fig. 2
figure 2

Setup and calibration. a Schematic of the device along the nanowire (NW) showing the source (s), drain (d), top-gate (tg) and back-gate (bg) terminals. The LC resonator is formed with a surface mount inductor L connected to the top-gate and the parasitic capacitance to ground Cp. Cc decouples the resonator from the line. Vsd, Vtg, and Vbg are source, top-gate and back-gate bias voltages. b The amplitude |Γ| of the reflection coefficient as function of frequency f: data in blue and a Lorentzian fit in red. c Source to drain current Isd as a function of Vsd and Vtg showing Coulomb diamonds. d Isd trace as a function of Vtg at Vsd = −1.5 mV: data in black and fit in red. e Relative demodulated phase ϕ/ϕ0 as a function of Vsd and Vtg showing the stability map of the first electronic transition. The symbol I(II) indicates the electronic transition from source(drain) to quantum dot. N and N + 1 indicate the bias regions with fixed electron number. The arrow indicates the bias region for thermometry. f The full width at half maximum (FWHM) ε1/2 of the charge transition line II as function of rf-carrier power Pc. The error bars are smaller than the size of the dots. The error includes the uncertainties in the gate lever arm α and the full width at half maximum measurements

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To probe the device tunneling capacitance, we embed the transistor in a resonator formed by a 470 nH inductor—connected to the top-gate (tg) of the device—and the device parasitic capacitance Cp, which appears in parallel with the differential capacitance of the device, as can be seen in Fig. 2a. We couple the resonator to a high-frequency line via a coupling capacitor Cc = 130 fF. In order to characterize the resonator, we measure the reflection coefficient Γ. In Fig. 2b, we plot the magnitude |Γ| (data in blue and a fit in red) as a function of frequency f at a fixed back-gate voltage Vbg = 3 V. We extract the resonator’s natural frequency of oscillation, f0 = \(1{\mathrm{/}}\left( {2\pi \sqrt {L\left( {C_{\mathrm{c}} + C_{\mathrm{p}}} \right)} } \right)\) = 408 MHz, the bandwidth BW = 2.9 MHz, the loaded quality factor QL = 141 and Cp = 194 fF. We find that the resonator is overcoupled but the depth of resonance, |Γ|min = 0.18 indicates that the resonator is close to being matched to the line.

The nature of cyclic electron tunneling

A system with discrete energy levels E0 and E1 as described in the Sec. Theory, can be found in a 0D QDs where the DOS consists of a series of delta functions at discrete energies36. In this section, we demonstrate the discrete nature of the QD in NWFET using electrical transport measurements.

We measure the source-drain current Isd as function of Vtg and source-drain voltage Vsd. The source-drain current Isd shows characteristic Coulomb blockade diamonds when measured as a function of Vtg and Vsd, see Fig. 2c. Coulomb blockade diamonds are a signature of sequential single-electron transport through the QD from the source (s) to drain (d) reservoir. From the height of the Coulomb diamond in the charge stable configuration, we extract the QD first addition energy, Eadd = 6 meV, and 3.75 meV for subsequent additions. Such a variable Eadd is characteristic of the few-electron regime where transport occurs through single-particle (0D) energy levels.

When the QD has a 0D DOS and the source(drain) reservoirs have a 3D DOS, then Fermi’s golden rule yields for the source(drain) tunnel rate

$$\gamma _{{\mathrm{s(d)}}} = \frac{{\gamma _{{\mathrm{0,s(d)}}}}}{{1 + {\mathrm{exp}}\left( { - \varepsilon _{{\mathrm{s(d)}}}{\mathrm{/}}k_{\mathrm{B}}T} \right)}},$$
(8)

where εs(d) is the level detuning between the QD and s(d) reservoirs and γ0,s(d) is the tunnel rate at εs(d) = 032. Note that these tunnel rates are significantly different from metallic (3D DOS) QDs tunnel coupled to 3D reservoirs37. Assuming that a single discrete energy level of the QD is within the energy window eVsd, the source drain current Isd can be written in terms of tunneling rates γs and γd by the relation Isd = eγsγd/(γs + γd)38 and is fitted to the data measured at fixed Vsd = −1.5 mV in Fig. 2d. The agreement between the data and the fit demonstrates the 0D nature of the QD, showing it is suitable for the electron thermometry method introduced in the Sec. Theory. Moreover, the top hat shape shows that there are no excited states within 1.5 meV of the ground state. Excited states at an energy comparable or lower than 2 × 3.53kBT could interfere with the method but the fit reveals they could only become an issue at temperatures T ≥ 2.5 K.

Gate coupling and optimal power

In order to get an accurate reading of the temperature T from Eq. (5), the gate lever arm α needs to be obtained. We use gate-based reflectometry techniques to probe the charge stability map of the QD in the voltage region of interest, see Fig. 2e taken at 50 mK. We excite the resonator at resonant frequency f0 and monitor the reflected signal. We used standard homodyne detection techniques32 to measure the demodulated phase response φ of the resonator as a function of Vsd and Vtg. The phase of the resonators changes (dark blue lines I and II in Fig. 2e) at the charge degeneracy points due to a tunneling capacitance contribution. The separation in Vtg between I and II, ΔVtg, at a given Vsd gives a measurement of α = Vsd/ΔVtg. We repeat these measurements for several Vsd and obtain ΔVtg as a function of Vsd, providing a measure of α from the slope and of the Vsd offset from the intercept. We obtain α = 0.9 ± 0.01, and this large value —close to 1— is consistent with the multi-gate geometry and the small equivalent gate oxide thickness of 1.3 nm of NWFETs32. We consider α temperature-independent6,39, because the capacitances that define α are determined by the geometry of the device and the voltage bias applied to the electrodes which we keep constant throughout the range of temperatures measured.

Finally, we calibrate the optimal power on the resonator using transition II at Vsd = −1.5 mV, which we will subsequently use to perform thermometry. In Fig. 2f, we plot ε1/2 as a function of the carrier power Pc at the input of the resonator. At high carrier power, Pc > −93 dBm, ε1/2 increases with Pc indicating the transition is power broadened. For Pc < −93 dBm, ε1/2 remains independent of Pc and hence, we observe the intrinsic linewidth of the transition. We select Pc = −95 dBm hereinafter.

Primary thermometry

In this section, we explore experimentally gate-based primary thermometry using transition II (see Fig. 2e). As we have seen in the Sec. Theory, when kBT/h > γ > f0, electron tunneling between QD and reservoir has an associated tunneling capacitance whose ε1/2 gives a reading of the reservoir temperature (see Eq. (5)). In this experiment, we probe T from a measurement of φ vs ε, since φ = −2QLCt/Cp40,41,42, when the resonator is overcoupled to the line. We drive the resonator at frequency f0 and monitor φ as we sweep ε across the charge degeneracy for different temperatures of the mixing chamber Tmc, see Fig. 3a. We measure Tmc with a 2200 Ω RuO2 resistive thermometer. As the temperature is increased, ε1/2 increases and the maximum phase shift decreases. We fit the data to Eq. (4) (red dotted lines), extract ε1/2 for several Tmc and plot it Fig. 3b (black dots). Two clear temperature regimes become apparent:

Fig. 3
figure 3

Primary thermometry. a Phase response φ of the resonator as a function of top-gate voltage Vtg swept across a charge degeneracy point for different Tmc. The red dotted lines are fits using Eq. (4). b ε1/2 and TGET as a function of mixing chamber temperature Tmc (black dots). Theoretical predictions to the low temperature regime (dashed blue), high temperature regime (dashed red) and full temperature range (dashed magenta). c Fractional temperature precision δTGET/TGET and d fractional accuracy ΔT/Tmc as a function of Tmc. ΔT = TGET − Tmc. The error includes the uncertainties in Tmc and TGET measurements. e Phase change at φ0 as a function Tmc (black dots) and a 1/Tmc fit at high temperature Tmc > 1 K (red dashed line)

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At low temperatures, for Tmc < 200 mK, we see that ε1/2 is independent of Tmc and equal to 160 μeV (blue dotted line). In this regime, as we shall demonstrate later, the thermal energy is smaller than the QD level broadening (kBT < hγ). As a result, the temperature reading of the GET, TGET, deviates from the mixing chamber thermometer. On the other hand, at high temperatures, Tmc > 1 K, we observe that ε1/2 presents a linear dependence with Tmc as predicted by Eq. (5). For comparison, we plot the theoretical prediction (red dashed line) and observe that both follow a similar trend. In this regime, since hγ < kBT, the GET can be used to obtain an accurate reading of the temperature of the electron reservoir. We quantify the precision of the thermometer by measuring the fractional uncertainty in the temperature reading of the gate-based thermometer, δTGET/TGET (see Fig. 3c). At low temperatures, the precision of the thermometer is primarily determined by the uncertainty in the lever arm, δα/α = 1.1%. As we raise the temperature, the phase response of the resonator becomes smaller leading to an increase in the uncertainty of V1/2 which, at the highest temperatures, becomes comparable to that of α. We find δTGET/TGET increases up to 1.6%. Additionally, in Fig. 3d, we determine the fractional accuracy of the GET thermometer, ΔT/Tmc by comparing its reading with that of the RuO2 thermometer (ΔT = TGET − Tmc). We see than the discrepancy between thermometers is less than 8% for temperatures higher than 1 K and this goes down to an average of 3.5% above 1.5 K. The error in the accuracy is primarily determined by the uncertainty in the reading of the RuO2 thermometer, which varies from 1% at the lowest temperatures to 6% at 2.4 K, rather than by the precision of the GET.

We note that, although not applicable for primary thermometry purposes, the whole temperature range can be described by a single expression that combines both regimes, level-broadening and thermal broadening, in to a single expression \(\varepsilon _{1/2}\) = \(\sqrt {(3.53k_{\mathrm{B}}T)^2 + (2h\gamma )^2}\) (see magenta dashed line in Fig. 3b). This formula fits well the data and we find that the difference is <6% for all temperatures.

Lastly, in Fig. 3e, we plot the maximum phase shift φ0 extracted from the fit, as a function of Tmc. Again, the two regimes are apparent. At low temperatures φ0 remains constant and only at temperatures Tmc > 1 K, φ0 shows an inverse proportionality with Tmc as predicted by Eq. (6) (dashed red line).

Low temperature limit

In Fig. 3b, e, we have seen that at low temperatures both ε1/2 and φ0 deviate from the prediction in Sec. Theory. In this regime, the gate-sensor cannot be used as an accurate thermometer. Two mechanisms may be responsible for this discrepancy: Electron-phonon decoupling, due to the weaker interaction at low T8,10, or lifetime broadening, when the QD energy levels are broadened beyond the thermal broadening of the reservoir, which occurs when hγ > kBT. In the latter case, ε1/2 is given by 2hγ (see Eq. (7)) whereas for the former, it is given by 3.53kBTdec, where Tdec is the decoupling temperature.

To assess the origin of the discrepancy, we modify the tunnel barrier potential by varying the vertical electric field across the device (Fig. 4a) which effectively changes γ32. We do so by changing the potential on the back-gate electrode Vbg while compensating with Vtg. In Fig. 4b, we plot ε1/2 as a function of Vbg. We see that as we lower Vbg, ε1/2 decreases, indicating that the tunnel rate γ across the potential barrier is lower due to the increasing height of the potential barrier at lower Vbg. This trend indicates that at low temperature, our primary thermometer is limited by level broadening and not by electron–phonon decoupling. Moreover, it demonstrates it is possible to tune electrically the low temperature range of the primary thermometer, as long as γ remains larger than the excitation frequency f0.

Fig. 4
figure 4

Low temperature limit. a Schematic of the conduction band edge along the nanowire channel. The quantum dot-reservoir tunnel barriers can be controlled “in-situ”. b The full width at half maximum ε1/2 and the tunnel rate γ as a function of back-gate voltage Vbg. The arrow indicates the Vbg at which thermometry was performed. The error bars are smaller than the size of the dots. The error includes the uncertainties in the gate lever arm α and the ε1/2 measurements

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Discussion

We have described and demonstrated a novel primary electron thermometer based on cyclic electron tunneling that allows measuring the temperature of a single electron reservoir without the need of electrical transport. The GET requires of a system with discrete energy levels tunnel-coupled to the reservoir to be measured, a scenario that can be found in a broad range of nanolectronics devices such as single-molecule junctions and/or in single-electron devices. Here, we have implemented the thermometer with a QD using CMOS technology which makes it ideal for large-scale production. Driving and readout of the thermometer can be performed simultaneously using reflectometry techniques which have recently demonstrated high-sensitivity with MHz bandwidth43. Since the driving must be done in the adiabatic limit, the GET is likely to show low shelf heating. Moreover, the technique is not affected by external magnetic fields. We have shown accurate primary thermometry down to 1 K and have proven that the low temperature range can be electrically tuned in-situ. For sub-10 mK operation, low transparency barriers and driving resonators with sub-200 MHz resonant frequencies should be used to ensure the thermometer is operated in the adiabatic limit and other materials such as GaAs could be used to improve the electron-phonon coupling.

When compared with other low-temperature primary electron thermometers, the GET presents some advantages and disadvantages. The GET requires a single tunnel barrier, similar to the SNT, but only a single reservoir. Both sensors must be probed by high-frequency techniques which provides an enhanced bandwidth over quasi-static measurements. However, the GET is unlikely to have the large dynamic range of the SNT since high temperature limit in the GET is set by quantum confinement. When compared to the CBT, which requires multiple tunnel barriers, the GET fabrication process is simpler with the trade-off that a low-temperature high-frequency amplification set-up is required but with the added benefit of the larger bandwidth. All three methods are independent of magnetic field as long as normal metals are used.

Overall, our thermometer shows potential for local probing of fast heat dynamics in nanoelectronic devices and it may have applications in the better study of thermal single-electron devices such as rectifiers and energy harvesters. Moreover, since the device is made using silicon technology it could naturally be integrated with silicon-based quantum circuits.

Methods

Device fabrication

The device used in this manuscript is fabricated on SOI substrate above the 145 nm buried oxide (BOX)33. The 8 nm thick NW channel channel is patterned using deep ultraviolet lithography (193 nm) followed by resist trimming process. For the gate stack, 1.9-nm HfSiON capped by 5 nm TiN and 50 nm polycrystalline silicon were deposited. The Si thickness under the HfSiON/TiN gate is 11 nm. After gate etching, a SiN layer (thickness 10 nm) was deposited and etched to form a first spacer on the sidewalls of the gate. 18-nm-thick Si raised source and drain contacts were selectively grown before the source/drain extension implantation and activation annealing. Then a second spacer was formed and followed by source/drain implantations, activation spike anneal and salicidation (NiPtSi).

Measurement set-up

Measurements are performed in an Oxford Instruments K400 dilution refrigerator with a base temperature of 40 mK. DC bias voltages (Vsd, Vtg, Vbg) are delivered through cryogenic constantant loom and discrete-component RC low-pass filters (cut-off frequency 10 kHz) at the mixing chamber. The source, drain and back gate lines are further filtered at the PCB with RC filters [R = 10 kΩ 0603 thin film resistors TE-Connectivity RP73D1J10KBYDG and C = 10 nF 0603 NPO COG, KEMET, C0603C103J3GACTU]. The radio-frequency signal for gate-based readout is delivered through an attenuated and filtered coaxial line (Bandpass 250–500 MHz) which connects to a on-PCB bias tee. The gate voltage line has a 100 kΩ resistor in series (0603 thin film resistor, TE-Connectivity RP73D1J100KBTDG). The resonator consist of a 470 nH inductor, the sum of the samples parasitic capacitance to ground and the coupling capacitance Cc and the device which is couple to the resonator via the gate. The inductor is a surface mount wire-wound ceramic core (EPCOS B82498B series) and Cc is a high-Q 0.2 pF capacitor (Johanson Technology S-series EIA 0603), and the PCB is made from 0.8-mm-thick Rogers RO4003C laminate with an immersion silver finish. The reflected rf signal is amplified at 4 K (QuinStar QCAU350-30H) and room temperature, followed by quadrature demodulation (Polyphase Microwave AD0105B), from which the amplitude and phase of the reflected signal are obtained.