As I'm building a model, I need a function equals to $+\infty$ in 0, $-\infty$ in R and 0 in $\frac{R}{2}$.
So it will looks like this (sorry for this bad drawing I'm at work). It definitely looks like a sigmoid, but I don't manage to find one that matches this.
$x \longmapsto -\tan \left[\frac{\pi}{2} \left(\frac{2 x}{R}-1\right)\right]$