While most materials have an absence of charge carriers at their surface, a number of semiconductors have been discovered which can support a large build up of electrons at the surface. This creates a potential well at the surface of the semiconductor, causing the conduction band states to become quantized into two-dimensional subbands. We will employ high-resolution angle resolved photoemission spectroscopy (ARPES) measurements of these quantized states in the technologically important materials, InAs, InSb, InN, and ZnO, in conjunction with complementary angle integrated photoemission spectroscopy measurements and bulk electrical and optical studies. While often treated in a one-electron picture, solids are immensely complex many-body systems where processes such as electron-electron (e-e) and electron-phonon (e-ph) interactions can lead to a pronounced renormalization of the material's electronic structure, which can prove essential in determining their fundamental properties. Such effects have hitherto been neglected in the study of these quantized electron accumulation layers. Using ARPES, we will perform a detailed characterisation of the many-body processes in these systems, including their dependence on factors such as temperature, the electron density within the quantum well, and the effective mass and Debye temperature of the host material. This feasibility study will not only develop a thorough understanding of quantized semiconductor electron accumulation layers, important for application of these technologically important materials, but also intends to demonstrate their use as model systems for investigating fundamental features of many-body interactions in solids. Consequently, it has implications for understanding the electronic properties of a wide range of solids, including not only semiconductors but also, for example, metals, highly-correlated electron materials and high-Tc superconductors.