Quantum Dot Energy Levels — Confinement & Emission Color

Quantum dot

Dot size L 4.00 nm

Transition n 1

Emitted photon

Recombination of an n-th confined electron with an n-th confined hole emits one photon at the wavelength/color shown.

Bandgap E_gap

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Wavelength λ

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Electron conf. E_e

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Hole conf. E_h

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How it works

A quantum dot traps its electron and hole in a finite 3D box of size L. Solving the particle-in-a-box Schrödinger equation gives discrete, quantized energy levels rather than a continuum:

E_n = n²π²ℏ² / (2·m*·L²)
E_gap(L) = E_bulk + E_e,1 + E_h,1
λ_emit = 1240 / E_gap(eV)  [nm]

Dot size L — the physical confinement length. Because E_n scales as 1/L², halving the dot roughly quadruples each confinement energy — shrink it and the ladder rungs spread apart, the gap widens and the emitted photon shifts toward blue; grow it and the ladder compresses toward the bulk gap and the photon reddens.
Transition n — which confined electron/hole rung pair recombines. Higher-n transitions carry more confinement energy on top of the same bandgap, so they emit a bluer photon than the ground-state (n=1) transition at the same dot size.
Material — sets the bulk bandgap E_bulk and the electron/hole effective masses (m*), which control how strongly confinement responds to size for that semiconductor (lighter effective mass → stronger size sensitivity).

Real-world relevance: this is the exact size-tunable bandgap effect that lets QLED TVs and quantum-dot biolabels emit a precise, saturated color from one chemistry, just by controlling nanocrystal diameter during synthesis — no dye or filter needed.

Bandgap vs. dot size — the 1/L² confinement curve

The marker tracks the current slider position on E_gap(L) = E_bulk + ħ²π²/2 · (1/m_e* + 1/m_h*) / L² — the same curve any real quantum-dot synthesis batch is tuned along.