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

TL;DR

This paper derives the density of states in low-dimensional nanomaterials using quantum mechanics, introduces quantum concentration as a confinement threshold, and explains electron degeneracy pressure's role in carrier enhancement.

Contribution

It provides a rigorous quantum-mechanical derivation of DOS in confined systems and introduces the concept of quantum concentration for the first time.

Findings

The confinement factor naturally emerges from quantum mechanics.

Quantum concentration $n_Q$ acts as a threshold for quantum effects.

Electron degeneracy pressure explains carrier enhancement in low dimensions.

Abstract

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 $n^{(d+2)/d}$ -- 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.