  We study a class of planar Schr\"{o}dinger-Poisson systems
  $$
  -\Delta u+\lambda V(x)u+\phi u=f(u) , \quad x\in{\mathbb R}^2,\qquad
  \Delta \phi=u^2, \quad x\in{\mathbb R}^2, $$ where $\lambda>0$ is a
parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $\Omega
\triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the
supercritical exponential growth at infinity in the Trudinger-Moser sense. By
exploiting the mountain-pass theorem and elliptic regular theory, we establish
the existence and concentrating behavior of ground state solutions for
sufficiently large $\lambda$.
