Nongalvanic Calibration and Operation of a Quantum Dot Thermometer

A cryogenic quantum dot thermometer is calibrated and operated using only a single nongalvanic gate connection. The thermometer is probed with radio-frequency reflectometry and calibrated by fitting a physical model to the phase of the reflected radio-frequency signal taken at temperatures across a small range. Thermometry of the source and drain reservoirs of the dot is then performed by fitting the calibrated physical model to new phase data. The thermometer can operate at the transition between thermally broadened and lifetime-broadened regimes and outside the temperatures used in calibration. Electron thermometry is performed at temperatures between $3.0\mathrm{K}$ and $1.0\mathrm{K}$, in both a 1-K cryostat and a dilution refrigerator. In principle, the experimental setup enables fast electron-temperature readout with a sensitivity of $4.0\pm 0.3\mathrm{mK}/\sqrt{\mathrm{Hz}}$, at kelvin temperatures. The nongalvanic calibration process gives a readout of physical parameters, such as the quantum dot lever arm. The demodulator used for reflectometry readout is readily available at relatively low cost.

Figure 1

Details of the silicon field-effect transistor QD. (a) A cross-section schematic of the device. The transistor consists of an undoped one-dimensional (1D) $\mathrm{Si}$ channel, $10\mathrm{nm}$ high by $30\mathrm{nm}$ wide, with $n$-doped $\mathrm{Si}$ source-drain connections. A polycrystalline silicon top gate, $80\mathrm{nm}$ wide, bridges over the channel, separated by a layer of ${\mathrm{SiO}}_2$. A grounded $\mathrm{Si}$ back gate is located beneath the channel, separated from the channel by 145-nm-thick buried ${\mathrm{SiO}}_2$. (b) The top view of the device. The source (S) and drain (D) channel connection points are $n$ doped and behave as a single grounded reservoir during thermometry operation. Two spacers are used to prevent doping of the $\mathrm{Si}$ channel under the top gate. (c) The energy diagram of the system. The reservoir has an occupation of electron states given by the Fermi-Dirac distribution $f$. The QD energy level $E_{\mathrm{QD}}$ is broadened from tunnel coupling to the reservoir, with a density of states $n$ described by Eq. (1). The tunneling capacitance $C_t$ as a function of $E_{\mathrm{QD}}$ has a shape proportional to $f^{\prime }\ast n$.

Figure 2

Details of the rf circuitry. (a) A schematic of the circuit layout. The resonant circuit is comprised of the $\mathrm{Nb}\text{−}\mathrm{Ti}\text{−}\mathrm{N}$-on-quartz spiral inductor $L$, the parasitic capacitance $C_p$, the coupling capacitance $C_c$, and the variable QD tunneling capacitance $C_t$, which is the physical parameter monitored for thermometry. $C_p$ includes the geometric gate capacitance $C_{tg}$ for modeling purposes. $R_p$ is a modeled parasitic loss resistance to ground, which impacts the resonant circuit $Q$ factor. The inductor line has a 100-pF capacitor to provide a dc break between the top gate and ground. The 96-$\mathrm{k}\mathrm{\Omega }$ resistor limits the top-gate rf signal loss to the dc line. $V_{tg}$ is the controllable dc top-gate voltage. IN and OUT represent the rf reflectometry input and output signal, respectively. (b) The loaded measurement circuit resonance when coupled to the top gate, with the reflected-signal magnitude $|R|$ and phase $\varphi$ shown in red and blue, respectively. The black dashed lines show the fitted circuit model, which measures a $Q$ factor of 63. Thermometry is performed on resonance at $f_0=593\mathrm{MHz}$, where the phase is most responsive.

Figure 3

The QD-thermometer calibration in $1\mathrm{K}$ cryostat, showing three experimental phase traces taken at fridge temperatures $T_f=2.0$, 2.5, and 3.0 K. Each trace shows the change in reflected-signal phase $\varphi -{\varphi }_0$ against the top-gate voltage $V_{tg}-V_0$ around the Coulomb peak of the QD. The solid black lines show the least-squares fit of Eq. (7) onto the data, assuming that the electron temperature $T_e$ equals the fridge thermometer readout $T_f$. This calibration procedure estimates $\alpha =0.74\pm 0.02$, $\mathrm{\Gamma }=270\pm 20\mathrm{n}{\mathrm{s}}^{-1}$, and $A=5.13\pm 0.06\mathrm{rad}\mathrm{p}{\mathrm{F}}^{-1}$. These three constants are then included within Eq. (7), allowing electron thermometry to be performed, shown here with a $T_e$ fit to data taken at $T_f=1.35\mathrm{K}$, yielding an electron temperature $T_e=1.4\pm 0.1\mathrm{K}$.

Figure 4

The electron temperature $T_e$ (orange) and the maximum phase ${\varphi }_{\mathrm{MAX}}$ (green) readout from the QD thermometer in the 1-K cryostat. The thermometry readout is generated by fitting $T_e$, via the calibrated Eq. (7), to the phase curve observed by sweeping the $V_{tg}$ over the QD Coulomb peak. The peak phase ${\varphi }_{\mathrm{MAX}}$ is measured at $V_{tg}=V_0$, with no fitting process required. The fridge temperature $T_f$ is read from a ruthenium oxide fridge thermometer thermally linked to the QD device. The dashed line highlights where $T_e=T_f$. The dotted line represents the model prediction from the calibrated Eq. (7), assuming $T_e=T_f$.

Figure 5

The charge-stability diagram measured by reflectometry technique and dc transport in the 1-K cryostat. To the left of $V_{tg}=220\mathrm{mV}$, the derivative of the reflected-signal phase with respect to the top-gate voltage $d\varphi /\mathrm{d}{V}_{tg}$ is plotted, demonstrating the nongalvanic reflectometry technique. For this measurement, the source and drain connections are grounded. To the right of $V_{tg}=220\mathrm{mV}$, the QD source-drain current $|I_{SD}|$ is plotted on a log scale. The fridge temperature is $T_f=1.268\pm 0.001\mathrm{K}$. The red lines highlight the source-drain gradients that match the calibration fit lever-arm prediction, $\alpha =0.74\pm 0.02$, via Eq. (8), crossing at the QD Coulomb peak where the thermometry takes place. Here, the on-resonance current has an order of magnitude $I_{SD}\approx 1-10\mathrm{nA}$, which is similar to that of the single-electron current defined by the calibration fit tunnel rate $e\mathrm{\Gamma }\sim 6.9\pm 0.5\mathrm{nA}$.

Figure 6

QD The thermometry and stability diagram from the dilution-refrigerator measurements. (a) The recalibrated QD thermometer readout of the electron temperature $T_e$ compared with the fridge-thermometer readout $T_f$, measured near the device within the dilution refrigerator. The QD and fridge thermometers start to disagree below $1\mathrm{K}$ but still agree to within one standard deviation of confidence [37] (for details on how the error is determined, see Sec. II in the Supplemental Material [36]). (b) The source-drain current-stability diagram of the QD mounted in the dilution refrigerator. The temperature of the fridge is held at $T_f=8.70\pm 0.05\mathrm{mK}$. The red lines highlight the source-drain gradients that match the calibration fit lever-arm prediction, $\alpha =0.84\pm 0.03$, via Eq. (8), crossing at the QD Coulomb peak where the thermometry takes place.