Quantum Dots and Nanoparticle Colour

Introduction

A vial of cadmium selenide nanocrystals glows blue, the next green, the next orange and the last deep red under the same ultraviolet lamp — yet every vial contains exactly the same compound. The only difference is the diameter of the crystals, which ranges from about 2 nm to about 7 nm. This striking effect is the particle in a box made visible. In this page we use the box model to explain why shrinking a semiconductor crystal pushes its absorption and emission towards the blue.

Core explanation

Bulk semiconductors. In a large crystal, the atomic orbitals of billions of atoms merge into almost continuous bands. The filled valence band is separated from the empty conduction band by the band gap, E g. When a photon with energy at least E g is absorbed, an electron jumps into the conduction band and leaves behind a positively charged vacancy called a hole. For bulk CdSe the band gap is about 1.74 eV, corresponding to light near 710 nm at the red edge of the visible spectrum.

Bringing in confinement. In bulk material the excited electron and hole can spread over a region several nanometres across; for CdSe this natural size, the exciton Bohr radius, is about 5–6 nm. If the crystal itself is smaller than this, the electron and hole are trapped as if in a three-dimensional box. We already know what a box does to a particle: it forces a minimum kinetic energy that grows as 1/L². The lowest excited state of the crystal therefore lies above the bulk band gap by a confinement energy.

The particle-in-a-sphere estimate. Treating a dot as a sphere of radius R with infinitely high walls, the lowest level for a particle of mass m has energy ħ²π²/(2mR²). Both the electron and the hole are confined, so the energy of the lowest exciton becomes approximately

E(R) ≈ E g(bulk) + (ħ²π²/2R²)(1/m e + 1/m h ) − 1.8e²/(4πε₀εR)

This is the Brus equation. The second term is the confinement energy; it uses the effective masses of the electron and hole, which in CdSe are roughly 0.13 and 0.45 times the free-electron mass. Small effective masses make the confinement term large. The third term is the electrostatic attraction between electron and hole, reduced by the dielectric constant ε of the crystal. Because confinement grows as 1/R² while attraction grows only as 1/R, confinement wins for small dots and the gap widens.

Colour follows size. A wider gap means that absorption begins at shorter wavelength and that the emitted light, produced when the electron and hole recombine, is bluer. Large CdSe dots emit red; small ones emit green or blue. Chemists can therefore tune colour continuously simply by controlling growth time during synthesis.

Discrete levels. A dot also has distinct higher levels, just as a box does. Its absorption spectrum shows a series of peaks rather than the smooth edge of the bulk solid, which is why dots are sometimes called artificial atoms.

Formulae

Confinement energy for one particle in a sphere: E = ħ²π²/(2m R²). Brus estimate: E(R) ≈ E g + (ħ²π²/2R²)(1/m e + 1/m h ) − 1.8e²/(4πε₀εR). Emission wavelength: λ = hc/E, or λ(nm) ≈ 1240/E(eV).

Step-by-step reasoning

To predict how the colour of a dot changes with size:

1. Start from the bulk band gap of the material. 2. Add the confinement energy, which is proportional to 1/R² and larger for light effective masses. 3. Subtract the smaller electron–hole attraction, proportional to 1/R. 4. Convert the resulting energy into a wavelength with λ = hc/E. 5. Check the direction of the trend: smaller R always gives higher energy and shorter wavelength.

Visual explanation

Draw the bulk valence and conduction bands as two broad shaded blocks separated by a gap. Beside them draw a small dot: the bands shrink into ladders of discrete lines, and the lowest conduction level and highest valence level are pushed apart. Shrink the dot further and the ladders spread even more, the gap between them opening like a pair of scissors.

Real-world analogy

Imagine a guitar string. A long string vibrates slowly and gives a low note; press your finger to shorten the vibrating length and the pitch rises. A quantum dot is a shortened string for the electron: confining its wave to a smaller space raises its lowest "note", which here is the energy of the emitted light.

Real-world example

Some high-end television and monitor screens use quantum dots. A blue backlight excites a film containing two sizes of dot, one tuned to emit pure green and one pure red. Because each size emits a narrow band of wavelengths, the screen produces very saturated colours. Indium phosphide dots are increasingly used instead of cadmium-based ones to reduce toxicity concerns.

Why?

Why does a smaller crystal give a larger gap rather than a smaller one? The uncertainty principle links position and momentum: confining the electron to a smaller region forces a larger spread of momentum, and therefore a higher minimum kinetic energy. That unavoidable extra kinetic energy is added to the bulk band gap, so smaller always means bluer.

Common misconception

"Quantum dots of different colours must be made of different materials or contain different dyes." A single material such as CdSe gives the whole visible range; the colour is set by size. Composition sets the starting point (the bulk gap), but size does the tuning.

Worked example

Question: In a simple model the confinement energy of a dot of radius 3.0 nm is 0.40 eV. Estimate the confinement energy for a dot of radius 1.5 nm, and explain the effect on colour.

Reasoning: Confinement energy is proportional to 1/R². Halving the radius multiplies it by 2² = 4, giving 4 × 0.40 = 1.6 eV. The attraction term, proportional to 1/R, only doubles, so the net gap increases substantially.

Answer: About 1.6 eV. The emission shifts markedly towards the blue end of the spectrum.

Quick check

1. Two CdSe dots are excited with the same ultraviolet lamp; one glows red and the other green. Which dot is larger? Answer: The red-emitting dot is larger, because a larger dot has a smaller confinement energy and hence a smaller effective band gap.

Exam focus

Be able to state that confinement energy scales as 1/R² and explain why this widens the gap. Examiners often ask for the direction of a colour change with size, or for a ratio calculation using the 1/R² dependence. Always mention that both electron and hole are confined.

Advanced insight

The simple Brus equation tends to overestimate the gap for the smallest dots, because real walls are finite, the bands are not perfectly parabolic and surface atoms perturb the levels. Surface chemistry matters: capping ligands or a thin shell of a wider-gap material such as ZnS remove surface trap states and raise the fluorescence quantum yield dramatically. Two-dimensional wells and one-dimensional wires show intermediate behaviour, with confinement in only one or two directions.

Summary

A quantum dot is a semiconductor crystal smaller than its natural exciton size, so its electrons and holes behave like particles in a three-dimensional box. Confinement adds a kinetic energy term proportional to 1/R², outweighing the 1/R electron–hole attraction, so the effective band gap grows as the dot shrinks. The result is size-tunable absorption and emission: small dots are blue, large dots are red, and the colour of a single material spans the visible spectrum.

Practice questions

1. State how the confinement energy of a spherical quantum dot depends on its radius. Answer: It is inversely proportional to the square of the radius, E ∝ 1/R². 2. Why does a light effective mass lead to stronger size effects? Answer: The confinement energy ħ²π²/(2m R²) is inversely proportional to the mass, so a small effective mass gives a larger energy shift for the same radius. 3. A dot emits at 2.20 eV. Estimate the wavelength of its emission. Answer: λ ≈ 1240/2.20 ≈ 564 nm, which is yellow-green light. 4. Explain why the absorption spectrum of a quantum dot shows separate peaks instead of the smooth edge seen for the bulk solid. Answer: Confinement breaks the continuous bands into discrete levels, like those of a particle in a box, so absorption occurs only at particular energies.