The electric property in the $GaAs-Al_{x}Ga_{1-x}$ quantum well with the Si ${delta}-doping$ layer in a non-central position is studied through the effect of the electric field intensity on the electron distribution. The finite difference method is used for the calculation of the subband energy level and its wavefunction. In order to account for the change of the potential energy due to the charged particles, the self consistent method is employed. As the Si ${delta}-doping$ layer becomes closer to the heterojunction interface, the electrons less affected by Coulomb scattering are greatly increased under the external electric field. Therefore, the high speed device is suggested due to the fact that the high mobility electrons can be increased by positioning the ${delta}-doping$ layer in the quantum well and by applying the electric field intensity.