Interpolation formula for BCS superconducting gap

Question

In BCS theory, the superconducting gap is given by solving at different temperatures the integral $$\frac{1}{N(0)V}=\int_0^{\hbar\omega_c}\frac{\tanh\frac{1}{2}\beta(\xi^2+\Delta^2)^{1/2}}{(\xi^2+\Delta^2)^{1/2}} \text{d}\xi$$ In textbooks like Tinkham (2nd edition, page 63) and Phillips (Advanced Solid State Physics, page 246) you can find approximate formulas for certain temperature ranges (typically $T\approx T_C$).

In some other references, such as here and here (for the latter, I couldn't find the arxiv version, sorry), it is mentioned an interpolation formula valid in the whole temperature range, that is $$ \Delta(T)=\Delta_0\tanh (k\sqrt{\frac{T_{C}-T}{T}})$$ with $k=1.74$ or $k=2.2$.

Is there someone who can link a reference to this formula, and how it is obtained?

Answer 1

This interpolation formula agrees with both the high- and zero-temperature limits for $\Delta(T)$: \begin{align} \tag{1}\label{eq1} 1-T/T_c \ll 1 &: \Delta(T)\approx 3.06\, k_B T_c\sqrt{1-T/T_c}\\ \tag{2}\label{eq2} T = 0 &: \Delta_0 = 1.764\, k_B T_c. \end{align}

When $T$ is near $T_c$ the argument of $\tanh$ in the interpolation formula is small so we can approximate $\tanh x \approx x$, giving $\Delta(T) \approx k \Delta_0 \sqrt{T_c/T-1}$. Then to recover \eqref{eq1}, we substitute $k = 1.74$, replace $\Delta_0$ using \eqref{eq2}, and use the fact that $\sqrt{T_c/T-1} = \sqrt{1-T/T_c}$ for $T$ near $T_c$.

At zero temperature, the argument of $\tanh$ in the interpolation function is large such that $\tanh x \approx 1$, giving $\Delta(T = 0) = \Delta_0$ which is just \eqref{eq2}.

As for the choice of $k$, for some strong-coupling superconductors, the prefactor in \eqref{eq2} is larger than 1.76.

Not only does the interpolation formula agree at $T=0$ and $T=T_c$, as stated in the comment above, but it also works near $T_c$.

Answer 2

It is just an interpolation formula. However, as Gross (3.11) mentioned in 1980, this formula can be rewritten as $$ \Delta(T)=\Delta(0) \tanh \left( \frac{\pi}{\Delta(0)} \sqrt{a \frac{\delta C}{C} ( \frac{T_{c}}{T}-1} ) \right), $$ where for s-wave superconductors, $Δ(0)$ = $1.76T_{c}$, a = 2/3; for the pure d-wave case $Δ(0)$ = 2.14$T_{c}$, a = 1; and for an s + g wave $Δ(0)$ = 2.77$T_{c}$, a = 2. $\delta C/C$ is specific heat jump.

It looks analitic cause the limits at zero temperature and $T_{c}$, which are fitted, are analitical.