Theory of temperature fluctuation statistics in superconductor-normal metal tunnel structures

We describe the statistics of temperature fluctuations in a SINIS structure, where a normal metal island (N) is coupled by tunnel junctions (I) to two superconducting leads (S). We specify conditions under which this structure exhibits manifestly non-Gaussian fluctuations of temperature. We consider both the Gaussian and non-Gaussian regimes of these fluctuations, and the current fluctuations that are caused by the fluctuating temperature. We also describe a measurement setup that could be used to observe the temperature fluctuations.

Figure 1

Schematic diagram of a SINIS structure, biased with voltage $V$. The normal metallic island is connected via tunnel contacts to superconducting reservoirs at temperature $T_S$. The transmission eigenvalue distribution of the contacts is characterized by two numbers, $g$ and $\mathcal{F}$. The temperature on the island, $T$, fluctuates due to fluctuations in the energy flows to the leads and the phonons. Tunneling through the contacts is described by the connector action ${\mathcal{S}}_c$ and the electron-phonon coupling with the corresponding action ${\mathcal{S}}_{\text{e-ph}}$.

Figure 2

Ratio of the voltage drops across left and right junctions as a function of conductance and $\mathcal{F}$ asymmetries for $V=2\Delta$. Note the logarithmic scales.

Figure 3

Average temperature of the normal metallic island in a SINIS setup for some values of $\mathcal{F}$. The inset shows that the scaled temperature $t$ is independent of $\mathcal{F}$ near the scaled voltage $v=0$.

Figure 4

Variance of the temperature of the normal metallic island in a SINIS setup for some values of $\mathcal{F}$. The inset shows that the variance of the scaled temperature $t$ is independent of $\mathcal{F}$ near the scaled voltage $v=0$.

Figure 5

Fano factor of the temperature fluctuation induced current noise for some values of $\mathcal{F}$. Inset shows the ${\tilde{\gamma }}^{-2/3}$ scaling of the Fano factor near the scaled voltage $v=0$.

Figure 6

Average temperature of the normal metallic island in a SINIS setup for $\mathcal{F}={10}^{-5}$, different values of $T_S=T_{\mathrm{ph}}$: $0.06\Delta$ (blue), $0.08\Delta$ (green), and $0.1\Delta$ (red), and different values of $g_{\text{e-ph}}/g$: 0 (solid lines), 1 (dashed lines), and 10 (dash-dotted lines). The reference line for $T_S=T_{\mathrm{ph}}=0$, $g_{\text{e-ph}}/g=0$ is shown in black, which is lost under the solid blue line.

Figure 7

Probability distribution for the dimensionless current $j={\tau }_r{T}_{t}I/\Delta$ in the non-Gaussian regime, ${\Gamma }_A{\tau }_r=0.5$, for three different values of bias voltage. The sharp peaks in the two leftmost distributions correspond to the case where the internal energy on the island is $E=2\Delta$.

Figure 8

Example measurement scheme for the detection of temperature fluctuations via current fluctuations. The current through the SINIS structure at a fluctuating temperature $T$, $I_{\mathrm{SINIS}}\left(T\right)$, generates voltage fluctuations across the shunt resistance $R_{\mathrm{sh}}$. These are amplified and recorded in time domain.

Figure 9

Two lowest order electron-phonon diagrams contributing to the effective action.