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 λD 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 λDd, 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 λD²A, where A is the area of the confinement, i.e., A=πR₀² in a cylindrical thin wire of radius R₀, A=πλ² in a magnetic field B with λ=(ceB)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.