Count confinement levels and low-dimensional states

A quantum well confines one direction and leaves two approximately free, a wire confines two and leaves one free, and a dot confines all three. Each allowed transverse eigenfunction defines a subband of motion in the remaining free directions. In a single parabolic band, the envelope Hamiltonian uses an effective mass rather than the bare electron mass. This approximation assumes confinement varies on scales sufficiently long compared with the lattice and that neglected bands do not mix strongly with the retained one.

Infinite walls impose vanishing envelope functions at the boundaries and yield a simple inverse-square energy scale. Finite heterostructure barriers instead allow penetration and require interface matching. For the scalar effective-mass operator written in divergence form, the common envelope model matches the envelope and its derivative divided by mass across a clean interface. Matching the derivative alone assumes equal masses. These conditions belong to that chosen envelope Hamiltonian; an atomically sharp multiband interface can require a more complete description.

For a parabolic dispersion, counting wavevectors in each free dimension gives different energy dependences. Three-dimensional bulk has a square-root onset, a two-dimensional subband contributes a constant step, and a one-dimensional subband has an inverse-square-root edge singularity. An ideal isolated dot instead has discrete delta-function levels, broadened in practice by temperature, coupling, and disorder. A wire's edge singularity is integrable; it does not represent infinitely many electrons packed at one energy. Occupation still requires integrating against the Fermi function.

The two-dimensional density counts states per area per energy, rather than per bulk volume. Spin multiplicity and equivalent valleys multiply state counts when they remain degenerate and have not already been included. Distinct transverse orbital levels must be summed separately. Positive and negative longitudinal wavevectors are both included in a density-of-states calculation, but they are not two independent orbital channels when counting a two-terminal transport mode. Stating the counting convention prevents later factor-of-two errors in conductance.

Resolving a subband requires spacing appreciably larger than thermal broadening and relevant disorder or contact broadening. Occupying only the lowest well subband is stronger than simply fabricating a thin film: the Fermi level must also remain below the second threshold. Narrowing a well raises all its confinement energies and can depopulate it at fixed reservoir chemical potential. This statement holds the band-edge reference fixed; electrostatic charge redistribution in an actual gated device can move that reference too.

Dots add a further distinction between orbital quantization and charging. Even an orbital with two spin states does not automatically accept a second electron at the same electrochemical energy because adding charge costs interaction energy. Conversely, a well subband is not a fully localized zero-dimensional bound level: it still supports a continuum of in-plane motion. Keeping these distinctions explicit prevents an infinite-well calculation from being mistaken for a complete interacting nanostructure or a prediction of measured addition spectra.