# Characteristic lengths in SNS Josephson Junction

An S-N-S Josephson junction is formed by two Superconducting (S) leads separated by a normal metal (N) which acts like the weak link. While studying, I often encounter the words diffusive, long, short, in clean limit and dirty limit. I see that they describe the different regimes of the SNS Junction.

I am trying to establish a global understanding of these regimes based on the characteristic lengths existing on an SNS junction. Namely,

L: Length of the normal metal

$$\textbf{L_b \sim \frac{\hbar V_F}{\Delta}} :$$ Coherence length (based on the answer)

$$\textbf{L_\Phi \sim \sqrt{\frac{\hbar D}{\Delta}}}$$ : Coherence length in diffusive regime

$$\textbf{L_{th} \sim \sqrt{\frac{\hbar D}{ k_b T}}}$$ : Thermal length in diffusive regime (2)

Also,

$$\epsilon_c = \frac{\hbar D}{L^²}$$ : Thouless energy

According to this article, the characteristic voltage, $$\textbf{V_c = R_N I_c}$$, is governed by $$\epsilon_{c}$$ or the gap $$\Delta$$. (whichever is smaller)

In light of these informations;

• Would it be correct to construct a table such as below ?

$$\begin{array}{|c|c|c|} \hline &\textbf{Long}&\textbf{Short}\\ \hline \textbf{Diffusive} \\ \textit{(dirty limit)}&L >> L_{\phi}, (\epsilon_c << \Delta) &L << L_{\phi}, (\epsilon_c >> \Delta)\\ \hline \\ \textbf{Ballistic} \\ \textit{(clean limit)} &L >> L_{b}&L << L_{b}\\ \hline \hline \end{array}$$

• I can't see where $$L_{th}$$ fits in this picture. When and how does it become an important length scale ?

Thanks