  This paper studies the asymptotic stability of solution to an
initial-boundary value problem for a hyperbolic-elliptic coupled system on
two-dimensional half space, where the data on the boundary and at the far field
are prescribed as $u_-$ and $u_+$. We show that the solution to the problem
converges to the corresponding planar rarefaction wave for $0\le u_-<u_+$ as
time tends to infinity under smallness assumptions on the initial perturbation
and wave strength. These results are based on the analysis of div-curl
decomposition, the standard $L^2$-energy method, $L^1$-estimate, and the
monotonicity of profile is given by the maximum principle.
