Quantum Dot Band Gap Calculator
Quantum confinement in semiconductor quantum dots
Quantum dot band gaps rise because the carriers have less room to move as the crystal shrinks. In a nanometer-sized dot, the electron and hole states are squeezed into a confined region, so the lowest allowed optical transition moves upward in energy. That is why a smaller dot often appears bluer, while a larger dot drifts back toward the bulk semiconductor’s color.
This calculator turns that idea into a quick screening tool. Give it a radius and a bulk gap, and it estimates how far confinement pushes the transition above the bulk value. It is useful for comparing synthesis targets, checking whether a dot size is likely to land in the visible or ultraviolet, and building intuition before you move to photoluminescence data or a more detailed theory.
Model used for the quantum dot band gap estimate
A common starting point for a quantum dot band gap estimate is the effective-mass "Brus-style" picture. It treats the electron and hole as particles in a spherical confinement region and captures the main trend: the energy cost of confinement grows roughly with 1/R2. Real nanocrystals also feel Coulomb attraction, surface effects, dielectric screening, and shape deviations, but this calculator keeps only the clean radius term so the trend is easy to interpret.
Parameterized equation used by this calculator
This page uses the simplified relationship:
where:
- E is the estimated quantum-dot band gap (eV).
- Eg,bulk is the bulk (room-temperature) band gap of the semiconductor (eV).
- R is the dot radius (nm).
- A is an empirical constant with units of eV·nm2.
The calculator uses A = 7.6 eV·nm2 as a simple reference constant. That keeps the radius effect easy to inspect: as R shrinks, the confinement term grows rapidly; as R gets larger, it collapses toward zero and the estimate approaches the bulk gap.
How to interpret quantum dot band gap results
The output is best read as a quick estimate of the optical transition energy for a spherical quantum dot. Use it to compare sizes, materials, and target emission windows rather than to predict an exact measured peak. If the energy rises when you reduce the radius, the dot is moving deeper into the confinement regime; if you increase the bulk gap, you are shifting the entire estimate upward as if you had chosen a different semiconductor family. For a rough wavelength conversion, λ (nm) ≈ 1240 / E (eV), but remember that real photoluminescence usually appears a bit below the absorption edge because excited carriers lose some energy before emission.
- A larger number for E means stronger confinement and a shorter-wavelength transition.
- A smaller radius pushes the estimate upward much faster than a modest change in bulk gap does.
- A different bulk gap shifts the entire curve up or down, which is useful when comparing one semiconductor family to another.
Worked example: a 2.0 nm quantum dot
To see the quantum dot band gap formula in action, imagine a material with a bulk gap of 1.50 eV and a dot radius of 2.0 nm. With A = 7.6 eV·nm2, the confinement term is 7.6 / (2.0)2 = 7.6 / 4 = 1.90 eV. Adding that to the bulk value gives 3.40 eV, which converts to about 365 nm. In other words, a dot that small would be expected to push the transition into the near-UV rather than leaving it in the visible.
This is still a simplified estimate, so the number should be treated as a trend indicator. Shape, surface chemistry, shell growth, and dielectric environment can all pull the real optical gap away from the idealized value even when the radius is measured carefully.
Quantum dot radius comparison table
This radius table uses the same confinement constant as the calculator and shows how quickly the A/R2 term fades as the dot gets larger. The 1.0 nm row is still heavily confinement-driven, while by 5.0 nm the estimate is already close to the bulk gap.
| Radius R (nm) | Confinement term A/R² (eV) | Estimated gap if Eg,bulk=1.50 eV (eV) |
|---|---|---|
| 1.0 | 7.60 | 9.10 |
| 1.5 | 3.38 | 4.88 |
| 2.0 | 1.90 | 3.40 |
| 3.0 | 0.84 | 2.34 |
| 5.0 | 0.30 | 1.80 |
Quantum dot assumptions & limitations
This calculator is designed for fast quantum dot band gap estimates, so the model intentionally leaves out several real-world effects.
- Heuristic constant A: A single constant stands in for the effective masses and material family, so the result is best treated as a reasonable trend line rather than a universal material property.
- Radius range validity: The 1/R2 behavior works best when the dot is large enough for effective-mass thinking but small enough for confinement to matter; at very tiny sizes, atomistic structure starts to control the spectrum.
- Shape and boundary conditions: The calculation assumes a spherical dot and idealized confinement. Real nanocrystals can be elongated, faceted, or capped with finite barriers, all of which shift the levels away from this simple picture.
- Exciton effects omitted: The simplified equation does not explicitly subtract exciton binding energy or include dielectric confinement, so the output is closer to an ideal optical transition estimate than to a fully corrected nanoscale model.
- Surface states and passivation: Trap states, surface reconstruction, ligands, and shells can move the measured emission away from the size-only prediction and broaden the spectrum beyond what this calculator can describe.
- Temperature dependence: Bulk band gaps vary with temperature, so a room-temperature input is only appropriate if your sample and application are also near that regime.
- Polydispersity: A real sample usually contains a distribution of radii, which means the observed spectrum reflects many nearby gaps instead of one perfectly sharp value.
References for the quantum dot band-gap model
If you want to connect the calculator back to the literature, look for the Brus equation and standard semiconductor nanocrystal texts that discuss effective masses, excitons, and dielectric confinement. Those sources explain why the simple 1/R2 term is useful for intuition even when a real sample needs a fuller treatment.
Quantum Bloom Lab Mini-Game: Tune the Dot
Why this calculator works as a game: Every move in a quantum dot band gap problem comes back to radius, so the game mirrors that cause-and-effect loop. Squeeze the dot to raise the transition energy, let it relax to lower the energy, and try to meet the passing photon gates at the right band-gap value.
Game concept pitch: In Quantum Bloom Lab, the canvas becomes a tiny nanocrystal arena. Tap, hold, or press space to compress the dot, release to let the confinement relax, and steer the glowing energy trace through the matching gate before it slips by. The farther the dot is squeezed, the more the band gap climbs, so each run turns the calculator’s formula into a timing challenge.
- Tap, hold, or press space to compress the dot and push the band-gap estimate upward.
- Photon gates drift along the track at energies tied to the current calculator inputs, so every radius change changes the timing window.
- Short bursts, streak bonuses, and occasional flare gates keep the level changing as the round goes on.
Technical approach: The mini-game reads the calculator inputs, maps the predicted energy onto the canvas, and uses high-DPI drawing, pooled particles, and reduced-motion handling to keep the experience responsive. It is intentionally lightweight, but it still reflects the same radius-to-energy relationship as the calculator above.
Adjust the radius to watch the predicted band gap move. Smaller dots raise the energy faster, while larger dots settle closer to bulk behavior.