Electron clouds hinder the operation of particle accelerators. In the Large Hadron Collider (LHC), the copper beam screens are located within close proximity to the beam path, resulting in beam-induced electron multipacting, which is the main source of electron cloud formation. Conditions for multipacting are encountered when such surfaces have a secondary electron yield (SEY) greater than unity. Roughening the surface through laser processing offers an effective solution for reducing secondary electrons. Laser ablation leaves behind a complex rough, multi-scale geometrical surface with an altered chemical composition. Current models often over-simplify the geometry, do not have sufficient experimental data to derive input parameters, and exclude SEY-reducing mechanisms such as the surface chemistry. Leading to electron-matter interactions which do not resemble that of a real surface. Here, this complex surface is studied on copper used in the LHC, and the influence of microgeometry, inhomogeneous nanostructure and complex surface chemistry on the SEY is investigated. A novel, improved model is proposed that characterises these sophisticated structures, enabling the efficient design of surfaces to reduce SEY. To validate the model, samples were made using a variety of laser parameters. Modelling insights revealed that secondary electron suppression is not only caused by the microgeometry but also the nanostructure and chemical modification play a role. Contrary to the conventional theory, high aspect ratio structures are not necessarily required for effective SEY reduction. Currently, the model is applicable to a variety of surface morphologies and could be employed for other materials.