Quantum Dots: Why Nanocrystal Size Controls the Colour of Light

A semiconductor crystal small enough to feel its own size

A quantum dot is a semiconductor nanocrystal — typically a few nanometres across, only tens to a few thousand atoms — small enough that the electron and the hole it leaves behind when excited can no longer roam freely the way they would in bulk material. In bulk semiconductor, an excited electron-hole pair (an exciton) has a natural size called the exciton Bohr radius, set by the material's dielectric constant and the electron and hole effective masses. Once the crystal itself shrinks smaller than that natural radius, the exciton is squeezed into a box smaller than it "wants" to be — a regime called quantum confinement — and its energy rises purely as a consequence of that confinement, exactly like a particle in a box in introductory quantum mechanics: shrink the box, and the allowed energy levels spread further apart.

The Brus equation

Louis Brus derived the standard approximate formula for how a quantum dot's effective bandgap changes with its radius r, treating the exciton as a particle confined in a spherical box and adding a Coulomb attraction term between the confined electron and hole:

E_gap(r) ≈ E_gap(bulk) + (ℏ²π² / 2r²)·(1/m_e* + 1/m_h*) − 1.8e² / (4πε₀ε_r·r)

The middle term is the quantum confinement energy: it scales as 1/r² and always pushes the gap up as the dot shrinks. The last term is the Coulomb term: it scales as 1/r and always pulls the gap back down, because confining the electron and hole closer together strengthens their mutual electrostatic attraction. For the nanometre-scale sizes where quantum dots are actually made, the confinement term wins, so the net effect of shrinking r is unambiguous — the bandgap widens.

From bandgap to colour

A photon emitted when the exciton recombines has energy equal to that effective bandgap, and photon energy sets wavelength directly through E = hc/λ. A wider bandgap means a higher-energy, shorter-wavelength photon — so as a quantum dot shrinks, its fluorescence shifts from red toward blue across the visible spectrum. This is the property that makes quantum dots commercially valuable: a single chemical compound like cadmium selenide, with no change in composition at all, can be tuned across the entire visible emission spectrum simply by controlling how large the nanocrystals are allowed to grow during synthesis — which is exactly why they're used in QLED television backlights and in biological fluorescence labelling, where a whole palette of colours can come from one material.

Why bulk material doesn't do this

In a bulk semiconductor crystal — anything much larger than the exciton Bohr radius, roughly micrometres upward — the confinement term above becomes negligible because r² in the denominator is enormous, so the bandgap collapses to the material's fixed bulk value and stays there regardless of how you cut or shape the material. Size-tunable colour is a phenomenon that exists only in the nanocrystal regime; it disappears entirely once the particle is large enough for the electron and hole to behave as if they were in an infinite crystal.

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