  We perform a semiclassical analysis for the planar Schr\"odinger-Poisson
system
  \[
  \cases{
  -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in
$\mathbb{R}^2$},\cr
  -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr
  } \tag{$SP_\varepsilon$}
  \] where $\varepsilon$ is a positive parameter corresponding to the Planck
constant and $V$ is a bounded external potential. We detect solution pairs
$(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge
\rightarrow 0$.
