Dimensional Confinement & Density of States — 3D→2D→1D→0D Nanostructure Simulator

Dimensionality

Confinement

Confined size d 5.0 nm

Fermi filling 50%

Subband spacing E₀

30 meV

Occupied levels

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How it works

Confine an electron in a box of size d along N of 3 spatial axes. Along each confined axis the momentum is quantized: E_n = n²ħ²π²/(2m*d²), n = 1,2,3…

3D bulk  g(E) ∝ √E              (smooth, continuous)
2D well  g(E) ∝ Σₙ Θ(E−Eₙ)      (staircase steps)
1D wire  g(E) ∝ Σₙ 1/√(E−Eₙ)    (Van Hove spikes)
0D dot   g(E) ∝ Σₙ δ(E−Eₙ)      (discrete lines)

Dimensionality — how many axes are squeezed to nanoscale d; the rest stay macroscopic (free).
Confined size d — smaller d pushes subband levels Eₙ further apart (E₀ ∝ 1/d²).
Fermi filling — shading below the Fermi energy shows which states are occupied.

This is the mechanism behind quantum-dot LED colour, 2D-electron-gas Hall devices and nanowire transistors: squeezing a electron gas one dimension at a time reshapes its density of states from a smooth curve into sharp discrete structure.

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