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I'm looking for an expression for the perpendicular field dependence of the superconducting energy gap $\Delta$ in thin films. What I mean by this is the analog to what is described in the Ginzburg Landau chapter of Tinkham's introduction to superconductivity (section 4.6), in which he writes that (equation 4-52 in my version of the book) \begin{equation} \frac{\vert\psi\vert^2}{\psi^2_\infty} = 1 - \frac{H^2}{H_{c\parallel}^2} \end{equation} Combining this with the fact that $\Delta \propto \psi$ shows that one expects the gap to have a monotonic square root dependence on $1 - \frac{H^2}{H_{c\parallel}^2}$. Now, if I understand the derivation correctly, this result follows from the choice of field direction and I am not sure how to generalize it to perpendicular directions.

Is it possible to come up with such a dependence/proportionality for the case of perpendicular magnetic fields, while keeping the assumptions that underlay this equation (thin film)? I understand that things get subtle close to the phase transition, but I'm mostly interested in away from $H_c$.

Note I would personally think that the book also covers this, but I can't seem to actually find it. Section 4.10 talks about the angular dependence of the critical field in thin films, which gives away of extrapolating between $H_{c\parallel}$ and $H_{c\perp}$, but it does not talk about the dependence of the energy gap.

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