Giant vortices in the Ginzburg-Landau description of superconductivity

Recent experiments on mesoscopic samples and theoretical considerations lead us to analyze multiply charged ($n>1$) vortex solutions of the Ginzburg-Landau equations for arbitrary values of the Landau-Ginzburg parameter $\kappa$. For $n\gg 1$, they have a simple structure and a free energy $\mathcal{F}\sim n$. In order to relate this behaviour to the classic Abrikosov result ${\cal F}\sim n^2$when$\kappa\to +\infty$,weconsiderthelimitwhereboth$n\gg 1$and$\kappa\gg1$,andobtainascalingfunctionofthevariable$\kappa/n$ that describes the cross-over between these two behaviours of $\mathcal{F}$. It is then shown that a small-n expansion can also be performed and the first two terms of this expansion are calculated. Finally, large and small n expansions are given for recently computed phenomenological exponents characterizing the free energy growth with $\kappa$ of a giant vortex.