Quantum Effects at the Nanoscale

Why nanomaterials glow in different colours. Learn wave–particle duality and the de Broglie wavelength, particle-in-a-box energy levels, density of states in 3D, 2D, 1D and 0D, quantum confinement in quantum wells, wires and dots, excitons and the Bohr radius, the Brus equation for quantum-dot band gaps, size-tunable emission, surface plasmon resonance in metal nanoparticles, and tunnelling — with Python for particle-in-a-box energies and CdSe quantum-dot band gaps.

When electrons are squeezed into regions comparable to their wavelength, their allowed energies become discrete and spread apart — like the notes of a guitar string get higher as the string gets shorter. This 'quantum confinement' makes a semiconductor's band gap grow as the particle shrinks.

Quantum dots exploit this: the same material (e.g. CdSe or InP) emits blue, green or red light depending only on size — used in QLED TVs, bio-imaging and solar cells.

An analogy

A quantum dot is like a tiny drum: the smaller the drum, the higher its pitch. Electrons in a smaller dot 'vibrate' at higher energy and emit bluer light.

Waves and confinement

Electrons behave as waves with wavelength λ = h/p (a few nm in semiconductors). In a box of size L only standing waves fit, giving energies E_n ∝ n²/L² — so halving L quadruples the level spacing. When L is comparable to the electron wavelength, these discrete levels matter at room temperature (spacing ≫ kT ≈ 25 meV).

Quantum wells, wires and dots

Confinement in 1, 2 or 3 dimensions:

Plasmonics

In metal nanoparticles, conduction electrons oscillate collectively under light — a localized surface plasmon resonance (LSPR). Gold spheres ~20 nm absorb near 520 nm (appearing red); silver near 400 nm (yellow); rods and shells tune resonance into the near-infrared. Plasmons enhance local fields for SERS sensing, photothermal therapy and colorimetric tests (e.g. lateral-flow pregnancy and COVID tests).

Engineering quantum materials

Core–shell dots (CdSe/ZnS, InP/ZnS) improve brightness and stability; cadmium-free dots meet regulations. Quantum-dot colour conversion in displays, QD solar cells and photodetectors, single-photon sources for quantum communication, and quantum-dot qubits. Tunnelling underlies scanning tunnelling microscopy, flash memory and resonant tunnelling diodes.

Formulas

Worked example: Box energy scaling

Given: E₁ = 15 meV at L = 5 nm

Find: E₁ at 2.5 nm.

Answer: 60 meV

Worked example: Emission wavelength

Given: E_g = 2.24 eV

Find: λ.

Answer: 554 nm (green-yellow)

Worked example: de Broglie wavelength

Given: Electron at 1 eV (free mass)

Find: λ.

Answer: 1.23 nm

Particle in a box and CdSe quantum dots

import math
# Particle in a box and quantum-dot band gap (Brus equation) for CdSe
h, hbar, me, e, eps0 = 6.626e-34, 1.055e-34, 9.109e-31, 1.602e-19, 8.854e-12
L = 5e-9
E1 = h**2 / (8 * me * L**2) / e
print(f"electron in a 5 nm box: E1 = {E1*1000:.1f} meV")
Eg, m_e, m_h, eps_r = 1.74, 0.13, 0.45, 10.6       # bulk CdSe gap (eV), effective masses, permittivity
for d_nm in (4, 5, 6, 8, 10):
    R = d_nm * 1e-9 / 2
    conf = h**2 / (8 * R**2) * (1 / (m_e * me) + 1 / (m_h * me)) / e
    coul = 1.8 * e / (4 * math.pi * eps0 * eps_r * R)
    E = Eg + conf - coul
    print(f"d = {d_nm} nm: Eg = {E:.2f} eV -> emission ~ {1240/E:.0f} nm")
# electron in a 5 nm box: E1 = 15.0 meV
# d = 4 nm: Eg = 2.55 eV -> emission ~ 486 nm
# d = 5 nm: Eg = 2.24 eV -> emission ~ 554 nm
# d = 6 nm: Eg = 2.07 eV -> emission ~ 598 nm
# d = 8 nm: Eg = 1.91 eV -> emission ~ 649 nm
# d = 10 nm: Eg = 1.84 eV -> emission ~ 674 nm

Smaller CdSe dots have larger band gaps and emit bluer light: from blue-green at 4 nm to deep red at 10 nm, approaching bulk CdSe (1.74 eV, 713 nm). The Brus effective-mass model captures the trend but overestimates gaps for very small dots (< 4 nm); practical sizing uses empirical curves fitted to absorption spectra.

Common mistakes

Applications

Key takeaways

Going further