In this manuscript, we develop a generalized theory for the scattering process produced by interface roughness on charge carriers and which is suitable for any semiconductor heterostructure. By exploiting our experimental insights into the three-dimensional atomic landscape of Ge/GeSi heterointerfaces obtained by atom probe tomography, we have been able to define the full set of interface parameters relevant to the scattering potential, including both the in-plane and axial correlation inside real diffuse interfaces. Our experimental findings indicate a partial coherence of the interface roughness along the growth direction within the interfaces. We show that it is necessary to include this feature, previously neglected by theoretical models, when heterointerfaces characterized by finite interface widths are taken into consideration. To show the relevance of our generalized scattering model in the physics of semiconductor devices, we implemented it in a non-equilibrium Green's function simulation platform to assess the performance of a Ge/SiGe-based THz quantum cascade laser. § these authors contributed equally to this work