Quantum Dot Emission: Colour From Size Alone

The same chemical, six different colours

Take cadmium selenide, a single well-defined semiconductor with a fixed bulk bandgap, and grow it as nanocrystals a few nanometres across. The smallest dots glow blue-green, mid-sized ones glow yellow and orange, and the largest glow deep red — all from chemically identical material, differing only in how many atoms make up each crystal. This size-tunable colour is the signature property of a quantum dot: a semiconductor nanocrystal small enough that an electron and the hole it leaves behind can no longer move freely in all directions the way they would in bulk material.

Quantum confinement: a particle in a very small box

When light is absorbed in a semiconductor, it promotes an electron across the bandgap, leaving a positively charged hole behind; the bound electron-hole pair is called an exciton. In bulk material this exciton can wander over a distance called the exciton Bohr radius, typically several nanometres. Shrink the crystal below that radius and the exciton is squeezed into a volume smaller than it "wants" — exactly the quantum-mechanical particle-in-a-box problem, where confining a wavefunction to a smaller region always raises its kinetic energy.

The Brus equation

Louis Brus derived the standard formula for this size dependence in 1984 by treating the electron and hole as independent particles trapped in a spherical box of radius R (the effective-mass approximation), then adding back their mutual Coulomb attraction as a correction:

E(R) = E_g(bulk)  +  (h² / 8R²)·(1/mₑ* + 1/m_h*)  −  1.8·e² / (4πε₀ε_r·R)

E_g(bulk)   = the bulk semiconductor's bandgap
h² / 8R²·(1/mₑ*+1/m_h*)  = confinement term — grows as 1/R², dominates as R shrinks
1.8·e² / (4πε₀ε_r·R)     = exciton Coulomb attraction — grows as 1/R, softens the blue-shift

The confinement term scales as 1/R², so it grows much faster than the 1/R Coulomb term as the dot shrinks — this is why the emission energy rises sharply for the smallest dots. Because the confinement energy adds directly onto the bulk gap, the effective optical gap of a quantum dot is always larger than the bulk gap, and the smaller the dot, the larger the excess — a blue shift relative to the bulk material's natural colour.

Strong vs. weak confinement

The regime the Brus equation describes best is strong confinement, where the dot radius R is smaller than the exciton Bohr radius a_B — the electron and hole are each confined independently by the crystal boundary, and their mutual attraction is a secondary correction. When R is comparable to or larger than a_B (weak confinement), the electron-hole pair behaves more like a compressed hydrogen-atom-style exciton, and the simple sum-of-independent-particles picture breaks down; a full multiband effective-mass or atomistic pseudopotential calculation is needed for quantitative accuracy in that regime.

Making the dots: from atoms to a size distribution

Colloidal quantum dots are typically grown by hot-injection synthesis: precursor solutions are rapidly injected into a hot coordinating solvent, triggering a fast nucleation burst followed by slower, diffusion-controlled growth. Because every dot nucleates in roughly the same narrow time window and then grows at roughly the same rate, the population ends up with a tight size distribution — critical, since the emission linewidth of a batch is set as much by size dispersion (each size emits a slightly different colour) as by the intrinsic linewidth of a single dot. Stopping the reaction earlier yields smaller dots and bluer emission; letting it run longer yields larger dots and redder emission — the entire visible spectrum is accessible from one recipe just by controlling reaction time.

Why this matters in practice

QLED displays use red-, green- and blue-emitting quantum dots as narrowband colour converters, giving a wider and more saturated colour gamut than conventional phosphors because dot emission peaks are intrinsically narrow. Biomedical imaging exploits their brightness and resistance to photobleaching compared with organic dyes. And because the emission colour is a direct, continuous readout of physical size, quantum dots are also a textbook demonstration that quantum mechanics is not an abstraction confined to atoms — it visibly, measurably colours a vial of nanocrystals sitting on a lab bench under UV light.

Frequently asked questions

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