  In this paper, we deal with the following nonlinear Schr\"odinger equation
  $$ -\epsilon^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb R^2), $$ where $f(t)$
has critical growth of Trudinger-Moser type. By using the variational
techniques, we construct a positive solution $u_\epsilon$ concentrating around
the saddle points of the potential $V(x)$ as $\epsilon\rightarrow 0$. Our
results complete the analysis made in \cite{MR2900480} and \cite{MR3426106},
where the Schr\"odinger equation was studied in $\mathbb R^N$, $N\geq 3$ for
sub-critical and critical case respectively in the sense of Sobolev embedding.
Moreover, we relax the monotonicity condition on the nonlinear term $f(t)/t$
together with a compactness assumption on the potential $V(x)$, imposed in
\cite{MR3503193}.
