  We consider the existence of ground state solutions for a class of zero-mass
Chern-Simons-Schr\"{o}dinger systems \[
  \left\{
  \begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2
u=f(u)-a(x)|u|^{p-2}u, \newline
  \displaystyle
\partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0,
\newline
  \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2,
  \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external
potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity
that fulfills the critical exponential growth in the Trudinger-Moser sense at
infinity. By introducing an improvement of the version of Trudinger-Moser
inequality, we are able to investigate the existence of positive ground state
solutions for the given system using variational method.
