First-order phase transitions in superconducting films: A Euclidean model

In the context of the Ginzburg–Landau theory for critical phenomena, we consider the Euclidean $\lambda {\varphi }^4+\eta {\varphi }^6$ model bounded by two parallel planes, a distance $L$ separating them. This is supposed to describe a sample of a superconducting material undergoing a first-order phase transition. We are able to determine the dependence of the transition temperature $T_c$ for the system as a function of $L$. We show that $$ is a concave function of $L$, in qualitative accordance with some experimental results. The form of this function is rather different from the corresponding one for a second-order transition.