We report terahertz (THz) emission from magnesium doped a-plane indium nitride (a-InN:Mg) films with different background carrier density, relative to the Mg-doped InN films grown along the c-axis (c-InN:Mg). Due to its high electron affinity, as-grown InN film is typically n-type and it has extremely high background carrier density, which causes much weaker THz emission than that from other semiconductors, such as InAs. The background carrier density of Mg-doped InN can be widely changed by adjusting the Mg doping level. For c-InN:Mg, THz emission is dramatically enhanced (×500 than that of undoped c-InN) as the background carrier density decreases to a critical value of ∼1×10
-3, which is due to the reduced screening of the photo-Dember field at the lower carrier density. For a-InN, however, intense THz emission (×400 than that of undoped c-InN) is observed for both undoped and Mg-doped a-InN and the enhancement is weakly dependent on the background carrier density. The primary THz radiation mechanism of the aplane InN film is found to be due to the acceleration of photoexcited carriers under the polarization-induced in-plane electric field perpendicular to the a-axis, which effectively enhances the geometrical coupling of the radiation out of semiconductor. The weak dependence of THz radiation on the background carrier density for a-InN shows that in-plane surface field induced-terahertz emission is not affected by the background carrier density. Small, but apparent azimuthal angle dependence of terahertz emission is also observed for a-InN, indicating the additional contribution of nonlinear optical processes on terahertz emission.