In this detailed study of a previously unstudied core-shell tetrapod nanostructure, we explore nonlinear optical rectification (NOR) in a GaAs core and Al \(_\eta \) Ga \(_{1-\eta }\) A shell, focusing on the influence of donor dopant presence (k = 1) and position ( \(z_0\) ), hydrostatic pressure and temperature. Singular donor doping and its position within the well material are key parameters that control the width of subband transitions and the interband conduction offset. This research extends results previously published by Hahn et al. (Eur Phys J Plus 139:1–9, 2024 [1]), where the authors restricted their analysis to energy levels and did not investigate the optical properties in depth. Specifically, we studied the influence of dopant position along the z-axis (Oz), hydrostatic pressure, and temperature on the two lowest transitions: 1 \(\rightarrow \) 2 and 2 \(\rightarrow \) 3. We found that increasing the distance of the dopant from the electron and increasing the hydrostatic pressure contribute to increased energy levels, attributed to a deeper confinement well. In addition, we examined the variation in the NOR coefficient. The results reveal a blue shift and a decrease in the amplitude of the 1 \(\rightarrow \) 2 transition as the hydrostatic pressure increases, while the temperature has the opposite effect.