Quantum Dot Band Gap Calculator

Quantum confinement and why the band gap changes with size

Semiconductor quantum dots are nanometer-scale crystals where charge carriers (electrons in the conduction band and holes in the valence band) are confined in all three spatial dimensions. When the dot radius becomes comparable to the carriers’ de Broglie wavelength or the exciton Bohr radius, the continuous bands of a bulk semiconductor give way to discrete, atom-like energy levels. The most visible consequence is a size-dependent band gap: smaller dots generally have a larger effective band gap and therefore absorb/emit at shorter wavelengths (higher photon energies), while larger dots trend back toward the bulk material’s band gap.

This calculator provides a quick, engineering-style estimate of the size-dependent band gap using a simplified effective-mass picture. It is intended for rapid intuition and rough screening, not for replacing optical characterization (PL/absorption) or detailed electronic-structure modeling.

Model used (effective-mass / particle-in-a-sphere style estimate)

A common starting point is the “Brus model,” which treats the electron and hole as particles with effective masses in a spherical potential well. In its fuller form, the lowest excitonic transition energy can be written as a bulk band gap plus a quantum-confinement term (roughly scaling as 1/R²) minus an exciton Coulomb binding term (roughly scaling as 1/R), plus smaller corrections. In many practical quick-estimate contexts, those details are condensed into a single parameterized confinement constant.

Parameterized equation used by this calculator

This page uses the simplified relationship:

where:

• E is the estimated quantum-dot band gap (eV).
• E_g,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 “typical” value for many II–VI colloidal quantum dots in simple back-of-the-envelope estimates. The key trend to focus on is the inverse-square dependence: as radius decreases, the confinement contribution increases rapidly.

How to interpret the results

The output is best interpreted as an estimated optical transition energy (a proxy for the effective band gap) for a spherical dot in a simplified model. Practical guidance:

• If E increases when you decrease R, that indicates a blue-shift (shorter-wavelength absorption/emission).
• If you change E_g,bulk, you are essentially switching materials (e.g., CdSe vs CdS vs InP). The size dependence remains, but the baseline shifts.
• To compare with photoluminescence (PL) peaks, you can convert energy to wavelength using λ (nm) ≈ 1240 / E (eV). Note that PL typically occurs slightly below the absorption edge due to relaxation and Stokes shift.

Worked example

Suppose you have a material with bulk band gap E_g,bulk = 1.50 eV and a dot radius R = 2.0 nm. Using A = 7.6 eV·nm²:

1. Compute the confinement term: A / R² = 7.6 / (2.0)² = 7.6 / 4 = 1.90 eV.
2. Add the bulk gap: E = 1.50 + 1.90 = 3.40 eV.
3. Optional conversion to wavelength: λ ≈ 1240 / 3.40 ≈ 365 nm (near-UV).

This is intentionally a simplified estimate; real dots of the same nominal size can differ due to shape, surface chemistry, dielectric environment, and effective masses.

Comparison table: how radius changes the confinement contribution

The table below shows the confinement term A/R² for several radii using A = 7.6 eV·nm². To get the estimated dot band gap, add your chosen E_g,bulk.

Assumptions & limitations

• Heuristic constant A: Using a single value (7.6 eV·nm²) implicitly assumes a “typical” set of effective masses and a particular family of materials. Different semiconductors (and even different crystal phases) can require different constants for good agreement.
• Radius range validity: The 1/R² scaling is most reasonable when the dot is small enough for quantum confinement but still large enough that an effective-mass picture makes sense. At extremely small radii (approaching a few lattice constants), atomistic effects dominate and this model can be very inaccurate.
• Shape and boundary conditions: The model assumes a spherical dot and idealized confinement. Real dots may be elongated, faceted, or have finite barrier heights, shifting the levels.
• Exciton effects omitted: The simplified equation does not explicitly subtract exciton binding energy (Coulomb term) or include dielectric confinement; both can change the optical transition energy, especially for small dots and low-dielectric surroundings.
• Surface states and passivation: Trap states, surface reconstruction, ligands, and shelling (core/shell dots) can shift emission energies and broaden spectra beyond what a simple size-only model predicts.
• Temperature dependence: Bulk band gaps vary with temperature. If you use a room-temperature E_g,bulk but your application is at a different temperature, expect systematic differences.
• Polydispersity: A real sample has a size distribution. Even if the mean radius is known, the observed spectrum reflects a weighted distribution of gaps rather than a single value.

References (for background)

For the underlying effective-mass approach and common quantum-dot band-gap approximations, see the Brus equation and standard semiconductor nanocrystal texts (effective-mass approximation / excitonic confinement models).