Primary thermometry of a single reservoir using cyclic electron tunneling to a quantum dot

At the nanoscale, local and accurate measurements of temperature are of particular relevance when testing quantum thermodynamical concepts or investigating novel thermal nanoelectronic devices. Here, we present a primary electron thermometer that allows probing the local temperature of a single-electron reservoir in single-electron devices. The thermo-meter is based on cyclic electron tunneling between a system with discrete energy levels and the reservoir. When driven at a ﬁ nite rate, close to a charge degeneracy point, the system behaves like a variable capacitor whose full width at half maximum depends linearly with temperature. We demonstrate this type of thermometer using a quantum dot in a silicon nanowire transistor. We drive cyclic electron tunneling by embedding the device in a radio-frequency resonator which in turn allows reading the thermometer dispersively. Overall, the thermometer shows potential for local probing of fast heat dynamics in nanoelectronic devices and for seamless integration with silicon-based quantum circuits

A n essential element in low temperature experimental physics is the thermometer 1 . 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 resistor 2 which can be used in combination with superconducting quantum interference devices to perform current-sensing noise thermometry (CSNT) 3 . Shot-noise thermometry (SNT) 4-6 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 junctions [7][8][9] . Thermometry using counting statistics via single-electron devices is also possible [10][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 desirable 15,16 .
Moreover, recent advances in device nanoengineering have led to a focused interest in using concepts from quantum thermodynamics [17][18][19][20][21] to improve the efficiency of technologies such as the thermal diode 22,23 or thermal energy harvesters 24 . 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 k B /e between the Boltzmann constant and the electron charge. The thermometer is driven and read out by an electrical resonator at radiofrequencies. 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 complementarymetal-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 computer [25][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 C tg , and tunnel coupled to the reservoir via a tunnel junction with capacitance C j and resistance R j , 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 capacitance 28,29 C diff as seen from the gate given by where Q is the net charge in the QD, V tg is the gate voltage, e is the electron charge, α is the gate coupling C tg /(C j + C tg ), and P 1 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.
To obtain an analytical expression for the tunneling capacitance C t , 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 ¼ 1 2 εσ z where ε is the energy detuning and σ z is the z Pauli matrix. The eigenenergies E 0 = ε/2 and E 1 = −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 V tg given that ε = −eα(V tg − V 0 ). Here V 0 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, f r that varies the energy detuning ε = ε 0 + δε sin(2πf r t). In the limit γ ) f r , the QD and reservoir are in thermal equilibrium and electrons tunnel in and out of the reservoir adiabatically. In this situation, P 1 tracks the thermal population, P 0 1 , given by the instantaneous gate-voltage excitation 28 and C t can be expressed as 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 Fig. 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 E 0 and E 1 , across a charge degeneracy point. d Probability P 1 of an electron to be in the QD as a function of energy level detuning ε. e Tunneling capacitance C t as a function of ε and this is depicted as a function of detuning in Fig. 1d. At large negative detuning the QD remains unoccupied P 0 1 ¼ 0 À Á , at large positive detuning the QD is occupied P 0 1 ¼ 1 À Á , and at ε ¼ Àk B T ln2; P 0 1 ¼ 1=2. We calculate the tunneling capacitance of the system and obtain where we have redefined the detuning ε to account for the peakcenter shift induced by temperature (ε → ε + k B T ln 2). Thus the tunneling capacitance C t has a full width at half maximum (FWHM) with respect to ε of as plotted in Fig. 1e. Since ε 1/2 = eαV 1/2 , the analysis shows it is possible to obtain the temperature of the electron reservoir from the FWHM of the C t vs V tg curve once the gate lever arm α is known. The quantity V 1/2 is the FWHM with respect to gate voltage. Furthermore, from Eq. (4) we see that the peak amplitude C 0 t of the tunneling capacitance C t , is inversely proportional to the reservoir temperature T, In the case of a finite magnetic field (see Supplementary Note 1), the expressions for ε 1/2 and C 0 t remain as in Eqs. (5) and (6), respectively. We note that the C t peak center shifts to lower detuning values as the magnetic field B is increased, see Supplementary  Fig. 1a where g is the electron g-factor and μ B is the Bohr magneton. This demonstrates our proposed method of determining the reservoir temperature T from capacitance C t measurements is independent of magnetic field. We note that our analysis is valid as long as k B T 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 (k B T < hγ), C t takes a Lorentzian form given by and ε 1/2 is given by 2hγ 30 , 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 C t can be probed with high-frequency techniques such as gate-based reflectometry 31,32 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  To probe the device tunneling capacitance, we embed the transistor in a resonator formed by a 470 nH inductorconnected to the top-gate (tg) of the device-and the device parasitic capacitance C p , 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 C c = 130 fF. In order to characterize the resonator, we HfSiON+TiN 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 V bg = 3 V. We extract the resonator's natural frequency of oscillation, MHz, the loaded quality factor Q L = 141 and C p = 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 E 0 and E 1 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 energies 36 . In this section, we demonstrate the discrete nature of the QD in NWFET using electrical transport measurements.
We measure the source-drain current I sd as function of V tg and source-drain voltage V sd . The source-drain current I sd shows characteristic Coulomb blockade diamonds when measured as a function of V tg and V sd , 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, E add = 6 meV, and 3.75 meV for subsequent additions. Such a variable E add is characteristic of the few-electron regime where transport occurs through singleparticle (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 where ε s(d) is the level detuning between the QD and s(d) reservoirs and γ 0,s(d) is the tunnel rate at ε s(d) = 0 32 . Note that these tunnel rates are significantly different from metallic (3D DOS) QDs tunnel coupled to 3D reservoirs 37 . Assuming that a single discrete energy level of the QD is within the energy window eV sd , the source drain current I sd can be written in terms of tunneling rates γ s and γ d by the relation I sd = eγ s γ d /(γ s + γ d ) 38 and is fitted to the data measured at fixed V sd = −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 × 3.53k B T 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 f 0 and monitor the reflected signal. We used standard homodyne detection techniques 32 to measure the demodulated phase response φ of the resonator as a function of V sd and V tg . 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 V tg between I and II, ΔV tg , at a given V sd gives a measurement of α = V sd /ΔV tg . We repeat these measurements for several V sd and obtain ΔV tg as a function of V sd , providing a measure of α from the slope and of the V sd 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 NWFETs 32 . We consider α temperatureindependent 6,39 , 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 V sd = −1.5 mV, which we will subsequently use to perform thermometry. In Fig. 2f, we plot ε 1/2 as a function of the carrier power P c at the input of the resonator. At high carrier power, P c > −93 dBm, ε 1/2 increases with P c indicating the transition is power broadened. For P c < −93 dBm, ε 1/2 remains independent of P c and hence, we observe the intrinsic linewidth of the transition. We select P c = −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 k B T/h > γ > f 0 , electron tunneling between QD and reservoir has an associated tunneling capacitance whose ε 1/2 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 = −2Q L C t /C p 40-42 , when the resonator is overcoupled to the line. We drive the resonator at frequency f 0 and monitor φ as we sweep ε across the charge degeneracy for different temperatures of the mixing chamber T mc , see Fig. 3a. We measure T mc with a 2200 Ω RuO 2 resistive thermometer. As the temperature is increased, ε 1/2 increases and the maximum phase shift decreases. We fit the data to Eq. (4) (red dotted lines), extract ε 1/2 for several T mc and plot it Fig. 3b (black dots). Two clear temperature regimes become apparent: At low temperatures, for T mc < 200 mK, we see that ε 1/2 is independent of T mc 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 (k B T < hγ). As a result, the temperature reading of the GET, T GET , deviates from the mixing chamber thermometer. On the other hand, at high temperatures, T mc > 1 K, we observe that ε 1/2 presents a linear dependence with T mc 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 < k B T, 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, δT GET /T GET (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 V 1/2 which, at the highest temperatures, becomes comparable to that of α. We find δT GET /T GET increases up to 1.6%. Additionally, in Fig. 3d, we determine the fractional accuracy of the GET thermometer, ΔT/T mc by comparing its reading with that of the RuO 2 thermometer (ΔT = T GET − T mc ). 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 RuO 2 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 ε 1=2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:53k B TÞ 2 þ ð2hγÞ 2 q (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 φ 0 extracted from the fit, as a function of T mc . Again, the two regimes are apparent. At low temperatures φ 0 remains constant and only at temperatures T mc > 1 K, φ 0 shows an inverse proportionality with T mc as predicted by Eq. (6) (dashed red line).
Low temperature limit. In Fig. 3b, e, we have seen that at low temperatures both ε 1/2 and φ 0 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 T 8,10 , or lifetime broadening, when the QD energy levels are broadened beyond the thermal broadening of the reservoir, which occurs when hγ > k B T. In the latter case, ε 1/2 is given by 2hγ (see Eq. (7)) whereas for the former, it is given by 3.53k B T dec , where T dec 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 γ 32 . We do so by changing the potential on the back-gate electrode V bg while compensating with V tg . In Fig. 4b, we plot ε 1/2 as a function of V bg . We see that as we lower V bg , ε 1/2 decreases, indicating that the tunnel rate γ across the potential barrier is lower due to the increasing height of the potential barrier at lower V bg . 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 f 0 .

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 highsensitivity with MHz bandwidth 43 . Since the driving must be  4). b ε 1/2 and T GET as a function of mixing chamber temperature T mc (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 δT GET /T GET and d fractional accuracy ΔT/T mc as a function of T mc . ΔT = T GET − T mc . The error includes the uncertainties in T mc and T GET measurements. e Phase change at φ 0 as a function T mc (black dots) and a 1/T mc fit at high temperature T mc > 1 K (red dashed line) 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-nmthick 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 (V sd , V tg , V bg ) are delivered through cryogenic constantant loom and discretecomponent 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 radiofrequency 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 C c 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 C c 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.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.