# What is the temperature dependence of the critical current in weak links?

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?

• See my answer below. I do not know what your expression is valid for. Are you sure you found this expression in Tinkham or Barone-Paterno ? Please give the exact reference in that case (formula or at least chapter when it occurs) because I hardly doubt it's correct. – FraSchelle Apr 1 '15 at 8:12
• The formula I wrote is valid for the temperature dependence of the critical current in tunnel junctions (i.e. junctions with an insulating barrier). At least, this is what I've been always taught since I started my master thesis. I added the references you requested. Also, this formula is quoted in Likharev review I linked in my question (sec. II.A.3). I'll take a look at the review you linked. Let me stress that what I'm looking for is not a current-phase relation, but a critical current-temperature relation, which, from the experimental point of view, is easier to measure. – casx Apr 1 '15 at 13:34
• Beyond the standard current-phase relation $j=j_{c}\sin\varphi$ from which you can infer easily the critical current (since it is always $j_{c}$ for $\varphi = \pi/2$) there is NO GENERIC critical-current - temperature relation. Usually, you have only one harmonics in: diffusive limit / close to the critical temperature / long junction limit without phase slips. Thanks for the reference by the way. The formula is correct, that's the historical Ambegaokar-Baratoff formula, as corrected by deGennes. – FraSchelle Apr 1 '15 at 21:59
• The tunnel limit corresponds to the short junction with low transparency interfaces. If you do not like Green's functions, as in Ambegaokar-Baratoff formulation, you can open the book by Nazarov and Blanter, the formula might be there derived in a Landauer-Büttiker method. In general you can only know the critical-current vs. temperature relation by calculating the maximum of the current-phase relation at each temperature, for the reason that the phase is a degree of freedom which can not be fixed by any external parameter. – FraSchelle Apr 1 '15 at 22:05
• It seems better that you first describe your (supposed-to-be) system and then we can discuss about a possible correct model. You should also discuss the method you use to extract the critical current. We can eventually discuss via PM if your problem is research oriented. Without more details it's difficult to help you. – FraSchelle Apr 1 '15 at 22:08

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 :

A. Golubov, M. Kupriyanov, and E. Il’ichev, The current-phase relation in Josephson junctions Rev. Mod. Phys. 76, 411 (2004)

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...

This is a recent formula for current-temperature relationship

$$I^\tau_e=\frac{k_BT^2}{\mu_0c\hbar}\frac{\partial \Phi_B}{\partial T}$$