Figure 1

Cross-sectional sketch of an NIS refrigerator with overlayer trap. A quasiparticle (red dot) is depicted diffusing through the superconductor to the overlayer trap. Layers are labeled $O$ for the overlayer trap, $S$ for the superconductor, $N$ for the cooled normal-metal layer, I${}_1$ and I${}_2$ for insulating layers. Because I${}_2$ is fabricated independently of I${}_1$, it can have a different transparency.Reuse & Permissions

Figure 2

Block diagram representation of the thermal model. Roughly, power is deposited in the superconductor and leaves by trapping to the overlayer traps. Power leaves the electrons of the overlayer trap by coupling to phonons in the metal layer, and the phonons couple to the substrate (the bath) via a boundary resistance. There is additional complexity due to quasiparticle recombination in the superconductor and the behavior of athermal phonons created by recombination and trapped quasiparticles.Reuse & Permissions

Figure 3

Temperatures of the superconductor quasiparticles, overlayer trap electrons, and phonons of the combined metal layers vs position shown from $x=0$ to $x=50\mu$m calculated with the comprehensive thermal model and parameters from Table . The pink bar on the x axis shows the location of the refrigerator junction and, therefore, of quasiparticle injection into the superconductor.Reuse & Permissions

Figure 4

Model sweep over the value of the resistance area product for the overlayer trap barrier. Large resistance area product values represent the case of no overlayer trap, while low values represent a highly transparent trap barrier. The right vertical axis shows values of $T_{\mathrm{N}}$; the left vertical shows the power loads from Eq. (4) as equivalent $\beta$ values. The solid red terms sum to equal “Ideal $P_{\mathrm{N}}$.” “Ideal $P_{\mathrm{N}}$” is $P_{\mathrm{N}}{|}_{T_{\mathrm{S}}=T_{\mathrm{b}}}$ and “Lost $P_{\mathrm{N}}$” is the difference between “Ideal $P_{\mathrm{N}}$” and the actual $P_{\mathrm{N}}$ due to heating of the $S$ layer. The terms with red crosses are often represented as $\beta P_{\mathrm{S}}$. Decreasing the trap resistance area product causes the $S$-layer temperature $T_{\mathrm{S}}$ to remain closer to the bath temperature $T_{\mathrm{B}}$, reducing “Lost $P_{\mathrm{N}}$” and recombination, and resulting in lower base temperature $T_{\mathrm{N}}$. Changes in $T_{\mathrm{N}}$ lead to changes in “Ideal $P_{\mathrm{N}}$,” the electron-phonon coupling, and the power load due to the athermal behavior of trapped quasiparticles. The majority of the benefit of the overlayer trap is achieved with a resistance area product of 10–100 $\Omega \mu {\mathrm{m}}^2$, where $T_{\mathrm{N}}$ values near 75 mK are reached from $T_{\mathrm{b}}=300$ mK. “Lost $P_{\mathrm{N}}$” (and therefore $\beta$) remains finite even for low resistance area product overlayer because the overlayer trap heats as well. The Andreev ($I_2{V}_{\mathrm{b}}$) power load is not shown; it is roughly constant and never greater than the recombination term.Reuse & Permissions

Figure 5

Theoretical isothermal NIS current-voltage curves from Eqs. (7) and (11) shown as solid lines. Measured NIS refrigerator IV curves taken at 100 mK and 300 mK bath temperatures shown as circles. For each data point, the temperature $T_{\mathrm{N}}$ is uniquely determined by the theory curve upon which it falls. The optimal bias in the data for each bath temperature is shown as a black cross. The inset shows a picture of the device. The two large junctions (vertical rectangles) are the refrigerator junctions. The two smaller junctions are independent thermometer junctions.Reuse & Permissions

Figure 6

Temperature $T_{\mathrm{N}}$ determined from the refrigerator junctions vs NIS refrigerator voltage bias as solid lines for many bath temperatures $T_{\mathrm{b}}$. The vertical dashed line at 20 $\mu$V is used to determine uncooled NIS data points, for use in calibrating the conversion between $V_{\mathrm{F}}$ and $T_{\mathrm{N}}$. The optimal bias points are shown as blue crosses and indicate lowest temperature $T_{\mathrm{N}}$ achieved at each bath temperature $T_{\mathrm{b}}$.Reuse & Permissions

Figure 7

Temperature of the normal metal $T_{\mathrm{N}}$ determined from refrigerator junctions at optimal bias voltage shown as blue circles vs bath temperature $T_{\mathrm{b}}$. Temperature $T_{\mathrm{N}}$ from the independent thermometer junctions shown as inverted black triangles, and the temperature at the refrigerator junction calculated from thermometer junction temperature as red triangles (see Sec. 6). Uncooled temperature $T_{\mathrm{N}}$ at $V_{\mathrm{F}}=20\mu$V shown as green diamonds. The solid blue line is the predicted temperature $T_{\mathrm{N}}$ at optimal bias and the solid gray line is $T_{\mathrm{N}}=T_{\mathrm{b}}$, which represents zero heating or cooling. The inset shows the difference between the model and measured $T_{\mathrm{N}}$ at optimal bias as blue circles, and a subset of the difference between the temperature $T_{\mathrm{N}}$ at $V_{\mathrm{F}}=20\mu$V and $T_{\mathrm{b}}$ as green diamonds. Uncooled temperature points at bath temperature below 250 mK are excluded because they are extremely sensitive to the form of the two-particle tunneling current, which is not fully understood. The theory and data for $T_{\mathrm{N}}$ at optimal bias are in very good agreement. These results are an improvement over previous cooling with large-area NIS junction refrigerators (Ref. 14).Reuse & Permissions

Figure 8

Measured values of $\beta$ vs $T_{\mathrm{b}}$ as circles and model predictions as solid line. The inset shows the ratio between the measured and predicted value vs $T_{\mathrm{b}}$. Previous predictions of $\beta$ or equivalent parameters have varied from measured values by factors of order $\sim$3–10, so a factor of 1.5 is considered good. The model predicts the cooling to within a few mK at low temperatures, which is useful for designing improved NIS refrigerators.Reuse & Permissions

Figure 9

The propagation of a phonon starting at position $z_{\mathrm{i}}$ in the $O$ layer and propagating at angle $\varphi$. The probability amplitude of the phonon is reduced as it propagates through each layer depending on the time constant $\tau$ for the phonon to interact in that layer. Additionally there is an escape probability $\langle \eta \rangle$ at the interface between the bath and layer $N$. By following the phonon until the amplitude is reduced to near zero, we calculate the probability that a given phonon either escapes to the bath or is absorbed in the $O$, $S$, or $N$ layer.Reuse & Permissions