I am now working on hybrid Josephson junctions that are supposed to be tunnel junctions, so that the temperature dependence of the critical current should be
$$ I_c (T)=\frac{\pi}{2}\frac{\Delta(T)}{eR_N}\tanh\frac{\Delta(T)}{2k_BT} $$
according to textbooks such as Tinkham (Introduction to superconductivity – second edition, chapter 6, section 2, page 200, formula (6.10)) and Barone-Paternò (Physics and application of the Josephson effect, chapter 3, section 2, page 54, formula (3.2.10)).
However, the agreement between the theory and my experimental data is very poor, and I can't fit my data with this formula. Given the particular material of the barrier, it is possible that proximity effects have to be taken into account; in this case the $I_c(T)$ dependence is no longer given by the above mentioned Ambegaokar-Baratoff formula. I am now looking for the $I_c(T)$ equation for weak links, to try and see if it fits my data but I can't find it. I read Likharev's review on weak links and the only equation I found is the one for the supercurrent as a function of the phase difference $I_S(\varphi)$.
Where can I find a direct relation between $I_c$ and $T$? Or, if it is not so easy to find, how can I obtain the $I_c(T)$ relation from the $I_S(\varphi)$ equations listed in Likharev review?
Best Answer
The expression you wrote is strange. It looks close to the current-phase relation for a ballistic and short junction:
$$j\left(\varphi\right)=2ev_{F}N_{0}\Delta\sin\dfrac{\varphi}{2}\tanh\left(\dfrac{\Delta}{2k_{B}T}\cos\dfrac{\varphi}{2}\right)$$
when $\varphi=\pi/2$, which is not the critical current at low temperatures, since $j_{c}=\max\left\{ j\left(\varphi\right)\right\} $. You can plot the above expression for different temperature (or large ratios $\Delta/2k_{B}T$ since it's simpler then) to realise that the current-phase relation becomes less and less sinusoidal at low temperatures.
Anyway, there is no reason why your junction should be ballistic, short, and the contacts with the superconductors should be perfect.
I didn't open it since a while, but the reference :
might be of interest for you. It should be a discussion about the role of contacts, diffusive and ballistic limits, short vs. long junctions, etc...