Optimized proximity thermometer for ultrasensitive detection: Role of an ohmic electromagnetic environment

We propose a mesoscopic thermometer for ultrasensitive detection based on the proximity effect in superconductor–normal metal (SN) heterostructures. The device is based on the zero-bias anomaly due to the inelastic Cooper-pair tunneling in an SNIS junction (I stands for an insulator) coupled to an ohmic electromagnetic (EM) environment. The theoretical model is done in the framework of the quasiclassical Usadel Green's formalism and the dynamical Coulomb blockade. The usage of an ohmic EM environment makes the thermometer highly sensitive down to very low temperatures, $T\lesssim 5\mathrm{mK}$. Moreover, defined in this way, the thermometer is stable against small but nonvanishing voltage amplitudes typically used for measuring the zero-bias differential conductance in experiments. Finally, we propose a simplified view, based on an analytic treatment, which is in very good agreement with numerical results and can serve as a tool for the development, calibration, and optimization of such devices in future experiments in quantum calorimetry.

Figure 1

(a) Schematic view of the circuit examined in Ref. [14] where an SNIS proximity structure is coupled to an infinite RC transmission line. Lower panel shows the low-bias conductance vs temperature for different bias voltage amplitudes. (b) The corresponding circuit involving an ohmic environment instead. The low-bias conductance vs temperature is robust against nonzero voltage amplitudes typically unavoidable in experiments, making the proposed device stable and well defined. (c) A scheme of the SNIS junction studied in the paper. (d) The $P\left(E\right)$ function for an ohmic electromagnetic environment shown in panel (b) for various resistances $R$. For certain resistances the zero-bias conductance does not lose sensitivity at low temperatures.

Figure 2

The critical current $I_c$ in an SNIS thermometer as a function of the junction's temperature $T_{\mathrm{J}}$ for (a) various distances $L$ between the clean and the tunnel contact and $\gamma =1$ and (b) various $\gamma$ and $L=\xi$.

Figure 3

(a) The $I\text{−}V$ characteristics of a junction of Josephson energy $E_J$ and charging energy $E_C$ coupled to an ohmic electromagnetic environment at various temperatures and $R=5\mathrm{k}\mathrm{\Omega }$. The inset shows a closer look into $I\text{−}V$'s that are linear at low voltages. (b) The same quantity calculated for various environmental resistances $R$ and temperature $k_B{T}_{\mathrm{EM}}/E_C=0.01$. The inset shows the corresponding zero-bias conductance vs temperature.

Figure 4

Effect of a nonzero voltage amplitude: The low-bias conductance $G_0$ as a function of temperature of the junction $T_{\mathrm{J}}$ in an SNIS junction coupled to an ohmic environment of resistance $R=6\mathrm{k}\mathrm{\Omega }$ for (a) equal temperatures of the junction and the environment, $T_{\mathrm{J}}=T_{\mathrm{EM}}$ and (b) fixed environmental temperature $T_{\mathrm{EM}}=15$ mK. Lower panels show the corresponding quantity in the case of an infinite RC transmission line characterized by $\kappa =0.02$ (for more details, see Ref. [14]). In all panels, the junction's parameters are $L=3\xi$ and $\gamma =1$. Note that the legend of panel (b) applies to all panels.

Figure 5

Calibration of the thermometer: The zero-bias conductance $G_0$ as a function of the junction's temperature $T_{\mathrm{J}}$ for various environmental resistances $R$ and (a) equal temperatures of the junction and the environment $T_{\mathrm{J}}=T_{\mathrm{EM}}$ and (b) fixed environmental temperature $T_{\mathrm{EM}}=15\mathrm{mK}$. Dotted lines correspond to the analytic formula obtained by combining Eqs. (15, 16, 17). Junction's parameters are $\gamma =1$ and $L=3\xi$.