Quantum Confinement in Nanostructures

Confine a particle in an infinite square well of side $L$ and its energy is quantised, $E_n=\hbar^2\pi^2 n^2/(2mL^2)$: the levels go as $n^2$, so $E_2-E_1=3E_1$, and the ground-state confinement energy (the gap) scales as $1/L^2$, growing as the box shrinks and vanishing in the bulk limit. The left panel shows the well, its levels and the wavefunctions $\psi_n(x)=\sin(n\pi x/L)$. Which directions are confined sets the dimensionality, and the density of states takes a qualitatively different shape in each case: $g(E)\propto E^{1/2}$ for the 3D bulk, a staircase for the 2D quantum well, $(E-E_c)^{-1/2}$ van Hove spikes for the 1D wire, and discrete delta peaks for the 0D dot. The right panel is that DOS with the optical-absorption onset (the effective gap) marked. Everything is closed form (gate-tested).