Confined density of states, quantum concentration, and electron degeneracy pressure in low-dimensional systems

We present a simple derivation of the density of states (DOS) in confined nanomaterials. While previous studies often apply a heuristic $L^{3-d}$ confinement factor to bulk DOS expressions, we show that this factor arises naturally from a consistent quantum-mechanical treatment of quasi-dimensional systems. Using a Fermi gas model, we calculate carrier concentration in across different dimensions and introduce the concept of quantum concentration $n_Q$ as a statistical threshold for quantum confinement effect. We further demonstrate that the electron degeneracy pressure – scaling with $n^{(d+2)\mathrm{/}d}$ – provides a thermodynamic explanation for carrier enhancement under quantum confinement. Our results clarify the origin of DOS modification and provide insights for low-dimensional materials.