Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(ii) / 16

For standard deviations of position and momentum,Confinement to a region of size $R$ requires a momentum spread of order $\hslash /R$, hence kinetic energy of order ${\hslash }^2/(2m^{\ast }{R}^2)$. A sufficiently small semiconducting nanocrystal therefore has discrete energy levels and an optical edge shifted above the bulk band gap.

An illustrative strong-confinement model iswhere $\alpha$ is a geometry-dependent number of order unity. The positive confinement contribution grows as $R^{-2}$; electron-hole Coulomb interaction contributes an attractive term of order $-R^{-1}$. The uncertainty principle motivates the scale, not the exact coefficient or oscillator strength. Strong size dependence occurs near the exciton Bohr radius. Finite barriers, dielectric mismatch and surface states modify this model, while size distributions broaden ensemble spectra.

Choose zero potential for $0<x<L$. In an infinite square well,The first three levels have ratios $1:4:9$, with zero, one and two interior nodes; the wavefunctions vanish outside.

For a finite square well with exterior potential $V_0$, bound states have $0<E<V_0$, sinusoidal interiors and tails with $\kappa =\sqrt{2m(V_0-E)}/\hslash$. Matching wavefunction and derivative for equal masses giveswhen parity is taken about the well center. Corresponding energies fall below the infinite-barrier values, and the wavefunctions spread beyond the walls. There are finitely many bound states; a shallow well need not possess three. States above $V_0$ form a scattering continuum.

The second infinite-well state hasIts maxima are at $L/4$ and $3L/4$, with nodes at the walls and center. The density is zero outside.

If the second bound state exists in a symmetric finite square well, odd parity preserves its central node and two equal lobes. However, the density is nonzero at the walls and has exterior tails proportional to $e^{-2\kappa |x-x_{\mathrm{wall}}|}$. Normalization includes those tails, and internal peak positions and heights change. Finite barriers change penetration and energy; symmetry preserves the central node. The original wells figure compares the two normalized second-state densities.

An exciton is a correlated electron–hole pair with a full wavefunction $\Psi ({\mathbf{r}}_e,{\mathbf{r}}_h)$. In strong confinement, a useful leading approximation is a product of lowest electron and hole envelope functions, followed by a Coulomb correction. A spherical hard-wall ground envelope is nodeless, proportional to $\sin (\pi r/R)/r$, and finite at the origin. Finite barriers produce exterior tails.

For a core-shell semiconductor nanocrystal with type-I band alignment, both conduction- and valence-band offsets confine the carriers mainly to the core. Their envelopes penetrate the shell but overlap less with external traps. Chemical surface passivation and spatial separation from defects can suppress nonradiative losses and increase photoluminescence quantum yield. Shell thickness, interfacial quality and strain must be controlled; a defective shell can introduce losses.

A larger shell band gap alone does not guarantee type-I alignment. A staggered type-II band alignment can move one carrier toward the shell, reduce electron–hole overlap and alter lifetime and emission energy. The sketch illustrates the conditional type-I case. In CdSe/ZnS experiments, optical spectra were combined with structural and compositional measurements to test shell-growth interpretations.^[ref]