  In this paper, we consider the Hamiltonian elliptic system in dimension
two\begin{equation}\label{1.5}\aligned \left\{ \begin{array}{lll}
-\epsilon^2\Delta u+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\
-\epsilon^2\Delta v+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2,
\end{array}\right.\endaligned \end{equation} where $V\in C(\mathbb{R}^2)$ has
local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be either
superlinear or asymptotically linear at infinity and of subcritical exponential
growth in the sense of Trudinger-Moser inequality. Under only a local condition
on $V$, we obtain a family of semiclassical states concentrating around local
minimum points of $V$. In addition, in the case that $f$ and $g$ are
superlinear at infinity, the decay and positivity of semiclassical states are
also given. The proof is based on a reduction method, variational methods and
penalization techniques.
