Static dielectric response and screening in solid state physics: Why dimensionality matters in dielectrics

Textbooks often present the phenomenon of screening within the Thomas–Fermi model for three-dimensional free electron gases, but obtaining the dielectric response function and screening potential for dielectric systems of reduced dimensionality is also of pedagogical interest. In this work, we introduce a simple approach to investigate static screening in dielectric systems in the presence of an impurity charge for different dimensionalities. This approach is applicable to semiconductors and insulators alike. We demonstrate that, in 3D systems, the macroscopic dielectric function in reciprocal space is a constant, while in 2D and 1D systems, it strongly depends on the momentum transferred to the electrons in the dielectric. Through the proposed dielectric screening model, one can also determine binding energies in a hydrogenic model that can be used to describe excitations in real semiconductor systems.