Limitations in Cooling Electrons using Normal-Metal-Superconductor Tunnel Junctions

We demonstrate both theoretically and experimentally two limiting factors in cooling electrons using biased tunnel junctions to extract heat from a normal metal into a superconductor. First, when the injection rate of electrons exceeds the internal relaxation rate in the metal to be cooled, the electrons do not obey the Fermi-Dirac distribution, and the concept of temperature cannot be applied as such. Second, at low bath temperatures, states within the gap induce anomalous heating and yield a theoretical limit of the achievable minimum temperature.

Figure 1

Scanning electron micrograph of a typical cooler sample in (a), and cooling data in (b), where voltage $V_{\mathrm{P}}$ across the probe junctions in a constant current bias (28 pA) is shown against voltage $V_{\mathrm{C}}$ across the two injection junctions. Cryostat temperature, corresponding to the electron temperature on the N island at $V_{\mathrm{C}}=0$ is indicated on the right vertical axis. Below 100 mK this correspondence is uncertain, because of the lack of calibration and several competing effects to be discussed in the text.

Figure 2

Energy distribution $f(E)$ against $E/\Delta$ in (a), (c), and (e), and differential conductance $G$ against probe voltage $v_{\mathrm{P}}$ in (b), (d), and (f) at three different values of e-e collision parameter $K$, which present very slow, intermediate, and very fast e-e relaxation from top to bottom, respectively. Parameter values $\eta =1\times {10}^{-4}$ and $T_{\mathrm{S}}/T_{\mathrm{C}}=0.1$ have been assumed.

Figure 3

Effective electron temperature in the cooler against the injection voltage. The dashed lines correspond to $K=0$ (extreme nonequilibrium), $K=0.1$, and $K=1.0$, from top to bottom, and the solid line to $K=\infty$ (quasiequilibrium), all with $T_{\mathrm{S}}/T_{\mathrm{C}}=0.1$ and $\eta =1\times {10}^{-4}$. The dash-dotted line is the result for quasiequilibrium but for $\eta =1\times {10}^{-3}$. The horizontal dashed lines indicate the crossover temperature $T^{*}$ of reentrant behavior in quasiequilibrium for $\eta =1\times {10}^{-4}$ and $1\times {10}^{-3}$. The inset shows the dependence of the ultimate minimum temperature of the cooler against $\eta$.

Figure 4

Measured differential conductance of S1 at $T_{\mathrm{S}}=340\mathrm{m}\mathrm{K}$ in (a), and at $T_{\mathrm{S}}\simeq 100\mathrm{m}\mathrm{K}$ in (b). Zoom-up of a few data sets of (b) are shown in (c), and the corresponding theoretical lines with $K=0.05$ in (d).