Onset of degeneracy in confined systems

In this paper, we examine the onset of degeneracy in confined systems under the condition that de Broglie wavelength ${\lambda }_D$ of an electron is comparable to the size of the confinement. In quantum-well structures the ratio of the critical concentration in a confined system to that in a bulk at which the onset of degeneracy takes place is proportional to $\frac{{\lambda }_D}{d}$, where $d$ is the thickness of the quantum well. For a quasi-one-dimensional system (area confinement), the critical concentration is found to be proportional to $\frac{{\lambda }_D^2}{A}$, where $A$ is the area of the confinement, i.e., $A=\pi R_0^2$ in a cylindrical thin wire of radius $R_0$, $A=\pi {\lambda }^2$ in a magnetic field $B$ with $\lambda ={\left(\frac{\hslash c}{\mathrm{eB}}\right)}^{\frac{1}{2}}$, and $A=ab$ for a rectangular thin wire with transverse lengths $a$ and $b$. The critical concentration for onset of degeneracy is found to increase with stronger confinement (smaller size) under quantum-confinement conditions.