# Interaction strength in BCS theory

I'm looking to plot the band gap $\Delta(T)$ as a function of temperature between $T = 0$ and $T = T_c$ by numerical evaluation of the band gap equation

$$\frac{1}{\mathcal{N}(0)V} = \int_0^{\hbar\omega_D}{d\xi\frac{\tanh{\frac{\sqrt{\Delta^2+\xi^2}}{2k_BT}}}{\sqrt{\Delta^2+\xi^2}}}$$

where $\omega_D$ is the Debye frequency, $\mathcal{N}(0)$ is the density of states at the Fermi level and $V$ is the interaction strength in the BCS model. Now, in order to do this, I need values for both $\omega_D$ and $\mathcal{N}(0)V$.

I think I have a pretty good idea about the order of magnitude for $\omega_D$ if I assume longitudinal optical phonons (I find $\hbar\omega_{LO} \propto 10\,\mathrm{meV}$, but please do correct me if that doesn't seem right).

However, I still need a value for $\mathcal{N}(0)V$. Wikipedia mentions

The energy of the pairing interaction is quite weak, of the order of $10^{−3}\,\mathrm{eV}$, and thermal energy can easily break the pairs.

but it's not clear what quantity exactly is meant by 'the energy of the pairing interaction' (I could imagine it being one of two things). I'm guessing $\mathcal{N}(0) \propto 10^{28}$ for a metal (again, correct me if I'm wrong), so my question is really quite straightforward: what is the value of the interaction strength $V$ in the BCS model? (provide a reference please)

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Related question: physics.stackexchange.com/q/54200/16660 –  Wouter May 21 '13 at 11:58

In the book by P.G. de Gennes Superconductivity of Metals and Alloys, it's written

$N(0)V < 0.3$.

Also written :

Lead and mercury are two notable exceptions with low $\Theta_{D} \left( =\hslash \omega_{D}/k_{B} \right)$, giving, respectively, $N(0)V=0.39$ and $N(0)V=0.35$.

More details on p.112, P.G. de Gennes Superconductivity of Metals and Alloys, Westview (1999). The first edition dated back to 1966. To my knowledge there is no change between the editions.

See also the table 4-1 on p.125 of the same book for several specific values for the pure metals. This table is reproduced below for commodity.

$$\begin{array}{cccc} \mbox{Metal} & \Theta_{D}\left(\mbox{K}\right) & T_{c}\left(\mbox{K}\right) & N\left(0\right)V\\ \mbox{Zn} & 235 & 0.9 & 0.18\\ \mbox{Cd} & 164 & 0.56 & 0.18\\ \mbox{Hg} & 70 & 4.16 & 0.35\\ \mbox{Al} & 365 & 1.2 & 0.18\\ \mbox{In} & 109 & 3.4 & 0.29\\ \mbox{Tl} & 100 & 2.4 & 0.27\\ \mbox{Sn} & 195 & 3.75 & 0.25\\ \mbox{Pb} & 96 & 7.22 & 0.39 \end{array}$$

Post-scriptum:

There are (surprisingly !) nothing about this question on the book by J.R. Schrieffer, Superconductivity, Benjamin (1964). There is not even a discussion on the gap equation as far as I can check... There is a repetition of the Gennes data on the book by A.I. Fetter and J.D. Walecka, Quantum theory of many-particle systems, Dover Publications (2003, first edition 1971), p.448. But this table is less complete.

There is also no discussion about the numerical value in the original paper by BCS [Bardeen, J., Cooper, L. N., & Schrieffer, J. R. ; Theory of Superconductivity. Physical Review, 108, 1175–1204 (1957). http://dx.doi.org/10.1103/PhysRev.108.1175 -> free to read on the APS website], but there is a possibly interesting expression (Eq.(2.40), written below in your notation / in the original BCS paper, the gap is written $\varepsilon_{0}$):

$$\Delta\left(T=0\right)=\frac{\hslash\omega_{D}}{\sinh \frac{1}{N(0)V}}$$

which might be of help for calculating the critical line $\Delta(T)$.

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Thanks a lot for your helpful answer, I already checked out the original paper by BCS and noticed that indeed they do not discuss the numerical evaluation of the band gap equation. Thank you for bringing the $T=0$ equation to my attention as well, I'm using that now. The results are looking fine, though the calculation for $T = T_c$ is proving to have some convergence issues. Perhaps my algorithm is less-than-perfect, but I suspect there's some numerical errors complicating things as well, since the bandgap needs to be zero at that point. –  Wouter May 21 '13 at 22:18
@Wouter The BCS paper is indeed not clear regarding this point. To be honest, I've no idea how they get the $\Delta(T)$ phase diagram. It seems to me that they do not even evaluate it. It might well be just a sketch, not an exact evaluation. The case $T=T_{c}$ is done using the celebrated formula $$\frac{k_{B}T_{c}}{\hslash\omega_{D}} \approx 1.14 \exp\left[-\frac{1}{N(0)V}\right]$$ (Eq.(3.29)) which is exact (up to the numerical prefactor, which can be known at arbitrary accuracy). –  FraSchelle May 22 '13 at 1:45
@Wouter Please note you may find an other answer of mine of interest for your present problem: physics.stackexchange.com/questions/54200. I there explain the two expressions I wrote above, and give the main trick to obtain the $\Delta(T)/\Delta_{0}$ versus $T/T_{c}$ phase diagram without breaking your head :-) –  FraSchelle May 22 '13 at 4:04
Thank you again, I took a look at your answer there and found it to be the same method I have been using, apart from the note that it might be more precise to fix the dimensionless bandgap and solve for the dimensionless temperature. I've been working the other way around, but I'm sure to give that method a go. Thanks for adding the table as well. Very useful since I don't have access to the book you mentioned at the moment. –  Wouter May 22 '13 at 11:02
@Wouter You're welcome. I believe the method I used in physics.stackexchange.com/a/65444/16689 is the one used by BCS themselves, since otherwise you cannot end up with a universal phase diagram. But I never seen this discussed before. Have good time working with superconductivity :-) –  FraSchelle May 22 '13 at 11:10