  We consider the convergence of partition functions and endpoint density for
the half-space directed polymer model in dimension $1+1$ in the intermediate
disorder regime as considered for the full space model by Alberts, Khanin and
Quastel in [AKQ]. By scaling the inverse temperature like $\beta n^{-1/4}$, the
point-to-point partition function converges to the chaos series for the
solution to stochastic heat equation with Robin boundary condition and delta
initial data. We also apply our convergence results to the exact-solvable
log-gamma directed polymer model in a half-space.
