= Single-electron superconducting tunneling
<Single-electron superconducting tunneling> transfers individual <Electrons> across a barrier into or out of <quasiparticle> states of a material with <superconductivity>. In a normal-metal–insulator–superconductor junction, abbreviated NIS, only one electrode has <superconductivity>. Such a junction can reveal a <superconducting energy gap> without supporting a <DC Josephson effect> between two superconducting reservoirs.
For an elementary interpretation, assume an opaque barrier, approximately constant normal-electrode <density of states>, and an energy-independent tunneling matrix element over the range probed. Write the superconducting <density of states> divided by its normal-state value as $n_s(E)$, with $E$ measured from its <chemical potential>. A signed voltage convention chosen for positive normal-electrode injection gives
$$
I_{\rm qp}(V)=\frac{1}{eR_N}\int_{-\infty}^{\infty}n_s(E)\,[f(E-eV)-f(E)]\,dE,
$$
where $f$ is the <Fermi-Dirac distribution> at a common <temperature> and $R_N$ is the normal-state junction <electrical resistance>. This current convention is local to the NIS measurement. The expression counts occupied initial and empty final states; it is a quasiparticle transport model, not a <current-phase relation of a superconducting weak link>.
An original analytic illustration uses the ideal fully gapped <BCS theory> spectrum at zero <temperature>, with positive gap edge $\Delta$:
$$
n_s(E)=\begin{cases}0,&|E|<\Delta,\\ |E|/\sqrt{E^2-\Delta^2},&|E|>\Delta.\end{cases}
$$
For positive $V$, the occupation difference restricts the <integral> to $0<E<eV$. Since $d\sqrt{E^2-\Delta^2}/dE=E/\sqrt{E^2-\Delta^2}$, integration and odd extension give
$$
I_{\rm qp}(V)=\begin{cases}0,& |eV|\leq\Delta,\\ \displaystyle\frac{\operatorname{sgn}(V)}{eR_N}\sqrt{(eV)^2-\Delta^2},& |eV|>\Delta.\end{cases}
$$
The threshold is $\Delta/e$ for this NIS model. In the corresponding zero-temperature single-electron process between two fully gapped superconductors, creating excitations on both sides instead requires $|eV|\geq\Delta_1+\Delta_2$. That quasiparticle threshold does not forbid coherent pair <electric current> below it. At finite <temperature>, thermally populated <quasiparticles> soften the simple threshold argument.
At zero <temperature> in the NIS approximation, $R_N\,dI_{\rm qp}/dV=n_s(eV)$ away from the singular gap edge. Thus a voltage sweep probes the excitation <density of states>. The ideal divergence is rounded in real measurements, and at nonzero <temperature> the <derivative> samples a thermally broadened range of <energies>. For $|eV|>\Delta$, the model has $VI_{\rm qp}>0$: electrical work creates excitations and is dissipated on relaxation. This distinguishes its steady transport from the conservative <Josephson coupling energy>.
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