  The super-Eddington accretion onto intermediate seed BHs is a potential
formation mode of supermassive black holes exceeding $10^9~M_\odot$ in the
early universe. We here investigate how such rapid accretion may occur with
finite amounts of heavy elements contained in the gas and dust. In our 1D
radiation-hydrodynamics simulations, the radiative transfer is solved for both
the direct UV lights emitted by an accretion disk and the diffuse IR lights
thermally emitted by dust grains. Our results show that the radiative force by
the IR lights causes a strong feedback to regulate the mass accretion. The
resulting mean accretion rate is lower with the higher metallicity, and there
is the critical metallicity $Z \sim 10^{-2}~Z_\odot$, above which the
super-Eddington accretion is prevented by the radiation pressure of the IR
lights. With this taken into account, we examine if the dusty super-Eddington
accretion occurs in young galaxies using a simple model. We show that a
sufficient number of galaxies at $z \gtrsim 10$ can be such potential sites if
BHs accrete the cold dense gas with $T \sim 10^2$ K, approximately the thermal
equilibrium value at $Z = 10^{-2}~Z_\odot$. We argue that the efficiency of the
BH growth via the rapid accretion depends on the metallicity, and that the
metallicity slightly lower than $10^{-2}~Z_\odot$ provides a chance for the
most efficient growth.
