Influence of the interplay between de Gennes boundary conditions and cubicity of Ginzburg-Landau equation on the properties of superconductors

Solutions of the Ginzburg-Landau (GL) equation for the film subjected to the de Gennes boundary conditions (BCs) with extrapolation length $\Lambda$ are analyzed with emphasis on the interaction between $\Lambda$ and the coefficient $\beta$ of the cubic GL term and its influence on the temperature $T$ of the strip. Very substantial role is played also by the carrier density $n_s$. Physical interpretation is based on the $n_s$-dependent effective potential $V_{eff}({\bf r})$ created by the nonlinear term and its influence on the lowest eigenvalue of the corresponding Schr\"{o}dinger equation. For the large cubicities, the temperature $T$ becomes $\Lambda$ independent linearly decreasing function of the growing $\beta$ since in this limit the BCs can not alter very strong $V_{eff}$. The temperature increase produced in the linear GL regime by the negative de Gennes distance is wiped out by the growing cubicity. In this case, the decreasing $T$ passes through its bulk value $T_c$ at the unique density $n_s^{(0)}$ only, and the corresponding $\Lambda_{T=T_c}$ is an analytical function of $\beta$. For the large cubicities, the concentration $n_s^{(0)}$ transforms into the density of the bulk sample. Other analytical asymptotics are analyzed too.