Introduction to Nanotechnology

A.3. PARTIAL CONFINEMENT 359
limiting case is a bulk material, in which they are all delocalized. The intermediate
cases are a quantum wire, which is long in one dimension and very small in its
transverse directions; and a quantum well, which is a flat plate nanosized in
thickness and much larger in length and width. The quantum wire exhibits electron
confinement in two dimensions and delocalization in one dimension, and the
quantum well reverses these characteristics. Table A.3 lists the numbers of electrons
N(E) and the densities of states D(E) for these four cases, and Figs. 9.9 and 9.10,
respectively, provide plots of how they depend on the energy. The degeneracies di
refer to potential well energy levels. For fither details, see L. Jacak, P. Hawrylak and
A. Wojs, Quantzim Dots, Springer, Berlin, 1998, Section 3.1.
The formulas presented in this appendix are for idealized cases of isotropic
systems with circular Fermi limits in two dimensions, and spherical Fermi surfaces
in three dimensions. The bulk case on the last row of Table A.3 assumes the presence
of one conduction band. In practical cases the bands are more numerous and more
complex, but these simplified expressions do serve to clarify the roles played by
the effects of electron delocalization and electron confinement in nanostructures.