Quantum Dot Simulator: Size-Tunable Emission & Confinement Energy

simulator intermediate ~10 min
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λ = 580 nm — yellow-orange emission

A CdSe quantum dot with 3 nm radius emits at approximately 580 nm (yellow-orange), blue-shifted from the bulk CdSe band gap of 713 nm due to quantum confinement.

Formula

E(r) = Eg₀ + ℏ²π² / (2m*r²) — simplified Brus equation
λ = hc / E — emission wavelength from energy
a_B = εℏ² / (m*e²) — exciton Bohr radius

Shrinking Semiconductors

When a semiconductor crystal is reduced to a few nanometers, quantum mechanics rewrites its optical properties. Electrons and holes become confined in all three dimensions — like particles trapped in a tiny box — and their allowed energy levels shift upward from bulk values. This quantum confinement effect makes the effective band gap size-dependent, enabling precise tuning of absorption and emission wavelengths simply by controlling nanocrystal diameter during synthesis.

The Brus Equation

Louis Brus derived the foundational equation relating quantum dot size to its optical band gap in 1984. The confinement energy adds a term proportional to 1/r² (the particle-in-a-sphere kinetic energy) and subtracts a Coulomb attraction term proportional to 1/r. At small radii, the kinetic confinement term dominates, producing dramatic blue-shifts. The effective mass of electrons and holes in the specific semiconductor material determines how strong the confinement is for a given size.

Color from Size

A single material like cadmium selenide (CdSe) can emit every color of the visible spectrum depending on dot size. Dots with 1 nm radius emit blue light; 2 nm dots glow green; 3 nm dots shine yellow-orange; and 5 nm dots produce deep red. This size-color relationship is the basis of quantum dot displays (QLED technology), where three carefully sized dot populations create red, green, and blue subpixels with exceptional color purity and brightness.

Beyond Displays

Quantum dots are revolutionizing fields beyond consumer electronics. In biomedical imaging, their brightness, photostability, and narrow emission enable multiplexed tracking of different cellular processes simultaneously. In solar cells, multi-sized dot layers capture photons across the solar spectrum more efficiently than bulk semiconductors. As single-photon emitters, individual quantum dots are candidates for quantum communication and computing applications requiring on-demand photon sources.

FAQ

What is a quantum dot?

A quantum dot is a semiconductor nanocrystal small enough (typically 1–10 nm radius) that quantum confinement effects dominate its optical and electronic properties. The particle-in-a-box confinement raises the effective band gap above the bulk value, making the emission wavelength tunable by size — smaller dots emit bluer light, larger dots emit redder light.

Why do quantum dots change color with size?

Quantum confinement energy scales as 1/r² — when the dot radius shrinks below the exciton Bohr radius, electron and hole wavefunctions are squeezed, increasing kinetic energy and widening the effective band gap. This shifts the emission to shorter (bluer) wavelengths. A single material like CdSe can emit across the entire visible spectrum just by changing dot diameter.

What are quantum dots used for?

Quantum dots are used in QLED displays (Samsung, TCL) for vivid color reproduction, in biological imaging as bright fluorescent labels, in solar cells to harvest broader spectrum light, and in quantum computing research as single-photon emitters. Their size-tunable, narrow emission and high brightness make them superior to organic dyes.

What is the Brus equation?

The Brus equation calculates the size-dependent band gap of a quantum dot: E(r) = Eg_bulk + ℏ²π²/(2μr²) - 1.8e²/(4πεr), where μ is the reduced effective mass and the last term is the Coulomb attraction between electron and hole. The confinement term (1/r²) dominates at small sizes, blueshifting emission.

Sources

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