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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 variationaltechniques, we construct a positive solution $u_\epsilon$ concentrating aroundthe saddle points of the potential $V(x)$ as $\epsilon\rightarrow 0$. Ourresults complete the analysis made in \cite{MR2900480} and \cite{MR3426106},where the Schr\"odinger equation was studied in $\mathbb R^N$, $N\geq 3$ forsub-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}.

Solutions concentrating around the saddle points of the potential for Schr\"{o}dinger equations with critical exponential growth