I'm currently studying particles at a potential step of finite width, and am confused with the nature of the eigenfunctions in the 3 regions.

$$ψ_I = Ae^{(ikx)} + Be^{(-ikx)}$$

$$ψ_{II} = Ce^{(αx)} + De^{(-αx)}$$

$$ψ_{III} = Fe^{(ikx)} + Ge^{(-ikx)}$$

I understand both $ψ_I$ and $ψ_{III}$ as they are travelling waves outside of the potential barrier, and that $G=0$.

But why is there an exponential growth term within $ψ_{II}$ whenever the probability of the particle existing should only be decaying as the barrier width increases?