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Nanotechnology Engineering DISCUSSION
Bulk CdSe has a band gap of about 1.74 eV, which corresponds to the deep red, yet CdSe quantum dots are sold in colours from blue-green to red depending only on their diameter. The material is chemically the same in every vial.
What makes the band gap depend on particle size, below what size does the effect appear, and is there a simple formula that predicts the emission wavelength for a dot of, say, 4 or 6 nm diameter?
When light creates an electron-hole pair in a semiconductor, the pair has a natural size, the exciton Bohr radius, which for CdSe is roughly 5 to 6 nm. If the crystal is smaller than that, the electron and hole are squeezed into a space smaller than they would normally occupy. As with a particle in a box, confinement raises their lowest allowed energies, and the rise goes as 1 / R² for a box of radius R.
The effective band gap is the bulk gap plus that confinement energy, so smaller dots absorb and emit at higher energy, which means shorter wavelength. The effect becomes noticeable once the dot radius approaches the Bohr radius and is strong well below it.
The usual first estimate is the Brus equation: E = E_g + (ħ² × π² / (2 × R²)) × (1 / m_e + 1 / m_h) - 1.8 × e² / (4π × ε × R). The second term is the confinement energy of the electron and hole with their effective masses, and the third is their Coulomb attraction.
In convenient units the confinement term is 0.376 eV·nm² / R² multiplied by (m_0 / m_e + m_0 / m_h), and the Coulomb term is 1.8 × 1.44 eV·nm / (ε_r × R). For CdSe use E_g = 1.74 eV, m_e = 0.13 m_0, m_h = 0.45 m_0 and ε_r about 10, which makes the mass factor 7.69 + 2.22 = 9.9.
Working it through for a 6 nm dot (R = 3 nm): confinement = 0.376 × 9.9 / 9 = 0.41 eV; Coulomb = 1.8 × 1.44 / (10 × 3) = 0.09 eV; E = 1.74 + 0.41 - 0.09 = 2.07 eV. The wavelength is 1240 / E in nm, so about 600 nm, orange.
For a 4 nm dot (R = 2 nm): confinement = 0.376 × 9.9 / 4 = 0.93 eV; Coulomb = 0.13 eV; E = 2.54 eV, about 490 nm. Bulk CdSe sits at 1240 / 1.74 = 713 nm. The trend is right, but the model overestimates the shift for small dots; real 4 nm CdSe dots emit at longer wavelengths than this, because the effective-mass picture with infinitely high walls breaks down at that size. Use measured sizing curves for actual numbers.
The size dependence is also why synthesis quality shows up directly in the spectrum. The emission peak of a batch is the sum of all dots in it, so a spread of sizes gives a broad peak. Because the confinement term goes as 1 / R², small dots are the most sensitive: around 4 nm diameter, a change of a few tenths of a nanometre moves the peak by tens of nanometres.
Practical consequences: the peak position is a quick check of mean size, the peak width a check of size distribution, and the emission is always at slightly longer wavelength than the first absorption peak. Dots are usually overcoated with a wider-gap shell such as ZnS to passivate surface states, which otherwise trap carriers and reduce the brightness. Cadmium compounds are toxic, so handle the dots and their waste as the safety data sheet requires.
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