In this work, we studied the electronic and optical properties of with a concentration . Based on a first-principle calculation and using the FP-LAPW full-linearized augmented plane wave method, to see the doping phenomenon with Erbium (Er) , we used the three approximations: local spin density approximation (LSDA), the LSDA with is the Hubbard potential and the Becke–Johnson modification (mBJ). Our results show that the values of the structural parameters increase with the substitution of Ga by the Erbium atom. The analysis of the electronic structures in this study shows that the Er-doped GaN has a semi-metallic ferromagnetic character with the LSDA and mBJ approximations and a semiconductor behavior when we apply the Hubbard potential . The real and imaginary part of the dielectric function, refractive index, and extinction coefficient are also calculated and presented in the photon energy range up to 14 eV. In the optical spectrum, the intensity of the absorption coefficient is observed in the imaginary part of both doped and undoped GaN in the ultraviolet regions.