Leakage current of a superconductor–normal metal tunnel junction connected to a high-temperature environment

We consider a voltage-biased normal metal-insulator-superconductor (NIS) tunnel junction, connected to a high-temperature external electromagnetic environment. This model system features the commonly observed subgap leakage current in NIS junctions through photon-assisted tunneling which is detrimental for applications. We first consider a NIS junction directly coupled to the environment and analyze the subgap leakage current both analytically and numerically; we discuss the link with the phenomenological Dynes parameter. Then, we focus on a circuit where a low-temperature lossy transmission line is inserted between the NIS junction and the environment. We show that the amplitude of the transmitted frequencies relevant for the photon-assisted tunneling is exponentially suppressed as the length $\ell$ and the resistance per unit length $R_0$ of the line are increased. Consequently, the subgap current is reduced exponentially as well. This property can not be obtained by means of lumped circuit elements. We finally discuss our results in view of the performance of NIS junctions in applications.

Figure 1

Circuit representation of the two configurations studied in this paper. (a) A NIS junction at temperature $T_{\mathrm{jun}}$ is connected in parallel to its capacitance $C$ and to an impedance $Z\left(\omega \right)$ which represents the high-temperature environment at temperature $T_{\mathrm{env}}\gg T_{\mathrm{jun}}$. The whole circuit is biased by the constant voltage $V$. (b) A transmission line of length $\ell$ is inserted between the junction and the impedance $Z\left(\omega \right)$ of circuit (a). It is described by the parameters $R_0$, $C_0$, and $L_0$, the resistance, the capacitance, and the inductance per unit length, respectively, as well as by its temperature $T_{\mathrm{line}}$ which is assumed equal to $T_{\mathrm{jun}}$.

Figure 2

Plot of the separatrix ${\tau }_{10\%}\left(\rho \right)$ as a function of the dimensionless resistance $\rho =R/R_K$, defined as the solution of the equation $\exp \{\text{Re}\left[J({\tau }_{10\%},\rho )\right]\}=0.9$ with $C$ fixed. Both plots are in double-logarithmic scale. (a) For a fixed value of $T_{\mathrm{env}}$, ${\tau }_{10\%}\left(\rho \right)$ separates the weak and strong coupling regions (black thick line). The colored strip indicates the time interval bound by ${\tau }_{\Delta }=\hslash /\Delta R_{K}C$ (dark green dashed line) and ${\tau }_{\mathrm{e}}=\hslash /k_B{T}_{\mathrm{env}}{R}_{K}C$ (red dashed line). The intersection between ${\tau }_{\Delta }$ and the separatrix ${\tau }_{10\%}\left(\rho \right)$ defines the resistance ${\rho }_{\Delta }$. The asymptotic expression for ${\tau }_{10\%}\left(\rho \right)$ valid for $\rho \to 0$ and proportional to $1/\rho$ is also shown (blue dashed line). Its intersection with the line corresponding to ${\tau }_{\mathrm{S}}\sim \sqrt{\hslash /k_B{T}_{\mathrm{env}}{R}_{K}C}$ defines the threshold resistance ${\rho }_{\mathrm{th}}$. On the logarithmic scale used here, ${\tau }_{\mathrm{S}}$ almost coincides with ${\tau }_{\mathrm{e}}$. (b) As the temperature of the environment $T_{\mathrm{env}}$ is decreased, the curve ${\tau }_{10\%}\left(\rho \right)$ moves up, thereby increasing the weak coupling region.

Figure 3

Plot of the rescaled parameter $R_K{\gamma }_{\mathrm{env}}/R$ as a function of $k_B{T}_{\mathrm{env}}/\Delta$ for different values of the ratio $\Delta RC/\hslash$. Solid lines are obtained by a numerical integration of Eq. (11) using Eq. (12). Dashed lines refer to the asymptotic ${\gamma }_{\mathrm{env}}$ given by Eq. (13).

Figure 4

(a) Plot of the parameter ${\gamma }_{\mathrm{env}}$ as a function of $k_B{T}_{\mathrm{env}}/\Delta$ obtained considering the numerical integration of Eq. (16). Each curve refers to a certain fixed value of the ratio $E_C/\Delta$ (see legend). (b) Numerical plot of the same quantity [Eq. (16)] as a function of $E_C/\Delta$ for different values of the ratio $k_B{T}_{\mathrm{env}}/\Delta$, as indicated.

Figure 5

Sketch of the circuit discussed in Sec. 3a.

Figure 6

Plots of the transmission function $|T_C{\left(\omega \right)|}^2$ as a function of the dimensionless variable $\omega RC$. Each panel corresponds to a different set of the parameters $z_0$, $c_0$, and $r_0$: (a) $r_0=0$, $c_0=1$, $z_0=(7,5,4,3)$; (b) $r_0=0$, $z_0=5$, $c_0=(10,7,5,3)$; (c) $r_0=0$, $c_0=1$, $z_0=(0.8,0.6,0.5,0.3)$; (d) $r_0=0$, $z_0=0.7$, $c_0=(10,7,5,3)$; (e) $z_0=10$, $c_0=1$, $r_0=(1,2,3,4)$; (f) $z_0=10$, $c_0=1$, $r_0=(7,9,10,12)$.

Figure 7

Plot of the transmission function $|T_C{\left(\omega \right)|}^2$ as a function of the dimensionless variable $\omega RC$ for different values of the parameter $r_0$. The other parameters are $c_0=1$, $z_0=0.7$. Also shown is the Lorentzian corresponding to the function $|T_C{\left(\omega \right)|}^2$ in the limit $\ell \to 0$, given by Eq. (26).

Figure 8

Plot of the dimensionless time ${\tau }_{10\%}\left(\rho \right)$ as a function of the dimensionless resistance $\rho =R/R_K$ for different values of the resistance per unit length $R_0$ of the transmission line (see legend). Also shown is the curve ${\tau }_{10\%}\left(\rho \right)$, valid for the circuit of Fig. 1a (black dashed line). The values of the other parameters are $\Delta \simeq 200\mu \mathrm{eV}$ (energy gap of aluminum), $T_{\mathrm{env}}=5\mathrm{K}$, $C=10\mathrm{fF}$, $C_0=6{\varepsilon }_0$, $L_0={\mu }_0$, $\ell =10\mu \mathrm{m}$.

Figure 9

Plot of the parameter ${\gamma }_{\mathrm{env}}$ [Eq. (28)] as a function of the length of the transmission line $\ell$. The red solid line is obtained by means of the numerical integration of Eq. (28) for a purely resistive environment. The blue dashed line is the plot of the asymptotic parameter given by Eq. (31). These two curves are plotted for different values of the resistance per unit length $R_0(\Omega /\mathrm{m})$ (as indicated in the graph). All the plots are obtained considering the gap parameter of the aluminum, $\Delta \simeq 200\mu \mathrm{eV}$. The other parameters are $T_{\mathrm{env}}=5\mathrm{K}$, $C=10\mathrm{fF}$, $R=10\Omega$, $C_0=6{\varepsilon }_0$, $L_0={\mu }_0$.