Noninvasive Thermometer Based on the Zero-Bias Anomaly of a Superconducting Junction for Ultrasensitive Calorimetry

We present radiofrequency thermometry based on a tunnel junction between a superconductor and proximitized normal metal. It allows operation in a wide range of biasing conditions. We demonstrate that the standard finite-bias quasiparticle tunneling thermometer suffers from large dissipation and loss of sensitivity at low temperatures, whereas thermometry based on a zero-bias anomaly avoids both of these problems. For these reasons, the latter method is suitable down to lower temperatures, here to about 25 mK. Both thermometers are shown to measure the same local temperature of the electrons in the normal metal in the range of their applicability.

Figure 1

(a) The studied setup; schematic illustration of the transmission measurement circuit. $C_1=10.3f\mathrm{F}$ and $C_2=59.3f\mathrm{F}$ are the coupling capacitors, $C=0.2$ pF and $L=100$ nH the parameters of the $LC$ resonator, and $R_L=50\mathrm{\Omega }$ is the transmission-line impedance. (b) Colored scanning electron micrograph of sample A used in this experiment. A normal metallic island ($\mathrm{Cu}$, brown) is in contact with aluminum leads (blue) either via a clean contact or tunnel barrier (light yellow).

Figure 2

The transmission $S_{21}$ measured against the drive frequency $f$ at $T=170$ mK and at $-120$ dBm power. The different curves with the overall trend from top to bottom correspond to thermometer dc bias voltages $V_{\mathrm{th}}$ ranging from $0$ to $170\mu \mathrm{V}$ in $5$-$\mu \mathrm{V}$ intervals. The inset shows an enlargement of a similar measurement at $T=30$ mK, where the dashed curves from bottom to top are for $V_{\mathrm{th}}=0$, $5$, $10$, $20\mu \mathrm{V}$ and the solid curves from top to bottom are for $V_{\mathrm{th}}=30$, $90$, $105$, $112$, $120\mu \mathrm{V}$. We attribute the tiny frequency shift to finite Josephson inductance.

Figure 3

Measured characteristics of the rf thermometer in the quasiparticle (QP) regime (sample B). (a) The transmission $S_{21}$ as a function of the dc bias voltage $V_{\mathrm{th}}$. The measurement is performed at temperatures 301, 263, 225, 182, 143, 113, 89, 68, 56, and 50 mK from bottom to top. (b) Temperature calibrations at $V_{\mathrm{th}}=156\mu \mathrm{V}$ measured at $-100$ dBm of the rf signal in the main panel and at various bias points in the inset. The lines are based on Eq. (7).

Figure 4

The transmission of the proximitized junction of sample A in the ZBA regime, $S_{21}$ versus $V_{\mathrm{th}}$, for (a) a set of bath temperatures $T_{\mathrm{bath}}$ in the range of $27$ to $398$ mK and fixed power $-130$ dBm and (b) a few power levels at $T=30$ mK. (c) Temperature calibration curves of the transmitted power at zero bias measured at $-140$ dBm of the rf signal in the main panel and at different power levels $-130$, $-125$, and $-120$ dBm in its inset. The nonmonotonic order of the curves in panel (a) at zero bias with respect to the bath temperature corresponds to that in the inset of panel (c).

Figure 5

Measurement of electron temperature $T_e$ of the absorber under nonequilibrium conditions produced by applying voltage $V_{\mathrm{inj}}$ across the injector junction. (a) The voltage dependence of $T_e$ demonstrates cooling, measured with different biasing and power levels of the thermometer. (b) Extracted minimum temperature of $T_e$ as a function of bath temperature $T_{\mathrm{bath}}$. The dashed line denotes $T_{min}=T_{\mathrm{bath}}$.

Figure 6

Requirements for detecting the heat produced by single-electron $\delta {ϵ}_e=2.5\mathrm{K}\times k_B$ and single-photon $\delta {ϵ}_{\mathrm{ph}}=1\mathrm{K}\times k_B$ quanta. Dependence of various noise-equivalent temperatures $\mathbb{N}\mathbb{E}\mathbb{T}$ on bath temperature $T_{\mathrm{bath}}$ for two different volumes of copper absorber, (a) $0.0025\mu {\mathrm{m}}^3$ and (b) $0.0005\mu {\mathrm{m}}^3$. In both panels, the black lines demonstrate the fundamental temperature fluctuations ${\mathbb{N}\mathbb{E}\mathbb{T}}_0$, red lines show the required noise-equivalent temperature for electrons ${\mathbb{N}\mathbb{E}\mathbb{T}}_{\mathrm{req}}^e$, and blue lines for photons ${\mathbb{N}\mathbb{E}\mathbb{T}}_{\mathrm{req}}^{\mathrm{ph}}$. The shaded areas indicate the feasible regimes, where the corresponding quanta can be observed. In panel (c), we present concrete examples referring to the samples A, B, and Opt that have a small volume yet are experimentally feasible. QP and ZBA thermometry are used for samples B and A, respectively, at the minimum temperature $130$ mK for QP and two different temperatures for ZBA. The last two columns give the corresponding signal-to-noise ratio for detecting electrons $(\mathbb{S}/\mathbb{N}{)}_e$ and photons $(\mathbb{S}/\mathbb{N}{)}_{\mathrm{ph}}$. The parameters used for evaluating the present results in panels (a)–(c) are $\mathrm{\Sigma }=2\times {10}^9{\mathrm{WK}}^{-5}{\mathrm{m}}^{-3}$ and $\gamma =70{\mathrm{JK}}^{-2}{m}^{-3}$.