Abstract
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 singleelectron reservoir in singleelectron devices. The thermometer is based on cyclic electron tunneling between a system with discrete energy levels and the reservoir. When driven at a finite 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 radiofrequency 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 siliconbased quantum circuits.
Introduction
An 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 wellknown 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 JohnsonNyquist noise of a resistor^{2} which can be used in combination with superconducting quantum interference devices to perform currentsensing noise thermometry (CSNT)^{3}. Shotnoise thermometry (SNT)^{4,5,6} uses the temperaturedependent voltage scaling of the noise power of a biased tunnel junction. Coulomb blockade thermometry (CBT) makes use of charging effects in twoterminal devices with multiple tunnel junctions^{7,8,9}. Thermometry using counting statistics via singleelectron 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 singlemolecule junction and singlenanoparticle 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 zerodimensional (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 proofofprinciple experiment, we perform primary thermometry down to 1 K but show that the operational temperature range of the sensor can be extended insitu using electrostatic fields. Our experimental results follow our theoretical predictions of the temperaturedependent capacitance of the system. The thermometer is implemented in a complementarymetaloxidesemiconductor (CMOS) transistor which makes it suitable for largescale manufacturing and seamless integration with siliconbased 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 QDreservoir 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 QDreservoir charge distribution in detail. In the limit of weak tunnel coupling, the QDreservoir system can be described by the Hamiltonian \(H = {\textstyle{1 \over 2}}\varepsilon \sigma _{\mathrm{z}}\) 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 \(\gamma \gg f_{\mathrm{r}}\), the QD and reservoir are in thermal equilibrium and electrons tunnel in and out of the reservoir adiabatically. In this situation, P_{1} tracks the thermal population, \(P_1^0\), given by the instantaneous gatevoltage 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
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
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 \(C_{\mathrm{t}}^0\) 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 \(C_{\mathrm{t}}^0\) 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, b. The shift tends to ε(B) = ε(0) − gμ_{B}B/2 for gμ_{B}B > k_{B}T, where g is the electron gfactor 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 highfrequency techniques such as gatebased reflectometry^{31,32} and can be used to measure temperature. We refer to this sensor as the gatebased electron thermometer (GET).
Device and highfrequency resonator
The device used here is a silicon nanowire fieldeffect transistor (NWFET)^{33} fabricated in fullydepleted silicononinsulator (SOI) following CMOS fabrication processes. At low temperatures, gatedefined QDs form in the channel of the NWFET^{34,35}, 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 SiO_{2} and 1.9 nm HfSiON resulting in an equivalent gate oxide thickness of 1.3 nm. The topgate (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 Si_{3}N_{4} spacers. The silicon wafer under the BOX can be used as a global backgate (bg).
To probe the device tunneling capacitance, we embed the transistor in a resonator formed by a 470 nH inductor—connected to the topgate (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 highfrequency line via a coupling capacitor C_{c} = 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 backgate voltage V_{bg} = 3 V. We extract the resonator’s natural frequency of oscillation, f_{0} = \(1{\mathrm{/}}\left( {2\pi \sqrt {L\left( {C_{\mathrm{c}} + C_{\mathrm{p}}} \right)} } \right)\) = 408 MHz, the bandwidth BW = 2.9 MHz, the loaded quality factor 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 sourcedrain current I_{sd} as function of V_{tg} and sourcedrain voltage V_{sd}. The sourcedrain 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 singleelectron 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 fewelectron 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 gatebased 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 multigate 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 gatebased 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}^{40,41,42}, 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 gatebased 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, levelbroadening 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 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 gatesensor cannot be used as an accurate thermometer. Two mechanisms may be responsible for this discrepancy: Electronphonon 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 backgate 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 tunnelcoupled to the reservoir to be measured, a scenario that can be found in a broad range of nanolectronics devices such as singlemolecule junctions and/or in singleelectron devices. Here, we have implemented the thermometer with a QD using CMOS technology which makes it ideal for largescale 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 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 insitu. For sub10 mK operation, low transparency barriers and driving resonators with sub200 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 electronphonon coupling.
When compared with other lowtemperature 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 highfrequency techniques which provides an enhanced bandwidth over quasistatic 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 tradeoff that a lowtemperature highfrequency amplification setup 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 singleelectron devices such as rectifiers and energy harvesters. Moreover, since the device is made using silicon technology it could naturally be integrated with siliconbased 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, 1.9nm 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. 18nmthick 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 setup
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 lowpass filters (cutoff 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 TEConnectivity RP73D1J10KBYDG and C = 10 nF 0603 NPO COG, KEMET, C0603C103J3GACTU]. The radiofrequency signal for gatebased readout is delivered through an attenuated and filtered coaxial line (Bandpass 250–500 MHz) which connects to a onPCB bias tee. The gate voltage line has a 100 kΩ resistor in series (0603 thin film resistor, TEConnectivity 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 wirewound ceramic core (EPCOS B82498B series) and C_{c} is a highQ 0.2 pF capacitor (Johanson Technology Sseries EIA 0603), and the PCB is made from 0.8mmthick Rogers RO4003C laminate with an immersion silver finish. The reflected rf signal is amplified at 4 K (QuinStar QCAU35030H) 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.
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Acknowledgements
We thank Jonathan Prance for providing useful comments. This research has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No 688539 (http://mosquito.eu), No 732894 (FET Proactive HOT) and the Winton Programme of the Physics of Sustainability. I.A. is supported by the Cambridge Trust and the Islamic Development Bank. A.C. acknowledges support from the EPSRC Doctoral Prize Fellowship.
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M.F.G.Z. devised the experiment. I.A., A.C., and M.F.G.Z. performed the experiment. S.B. fabricated the sample. I.A., J.A.H., J.J.L.M., and M.F.G.Z. analyzed the data. All authors contributed in the writing of the manuscript.
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Ahmed, I., Chatterjee, A., Barraud, S. et al. Primary thermometry of a single reservoir using cyclic electron tunneling to a quantum dot. Commun Phys 1, 66 (2018). https://doi.org/10.1038/s4200501800668
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