Quantum Dot Bandgap Calculator

Understanding the Brus Equation

The Brus equation, developed by Louis Brus in 1984, describes how quantum confinement affects the electronic bandgap of semiconductor nanocrystals (quantum dots). When a semiconductor particle becomes comparable to or smaller than its Bohr exciton radius, the spatial confinement of charge carriers dramatically increases the effective bandgap.

Physical Interpretation

The Brus equation consists of three terms:

• Bulk Bandgap (E_g,bulk): The intrinsic bandgap of the semiconductor material in its bulk (infinite) form. This is the baseline energy required to create an electron-hole pair.
• Kinetic Confinement Term (∝ 1/R²): As the particle shrinks, electrons and holes are confined to a smaller volume. By the Heisenberg uncertainty principle, spatial confinement increases momentum uncertainty, raising kinetic energy. This term dominates at small sizes and always increases the bandgap.
• Coulomb Term (∝ −1/R): The electrostatic attraction between the confined electron and hole (exciton binding). This term decreases the bandgap slightly but is less significant than the kinetic term for small QDs.

The Bohr Exciton Radius

The Bohr exciton radius (a_B) is a critical length scale that determines when quantum confinement becomes important:

where μ* is the reduced effective mass: 1/μ* = 1/m_e* + 1/m_h*

• Strong Confinement (R < a_B): Electrons and holes are independently confined. The Brus equation works well in this regime.
• Intermediate Confinement (R ≈ a_B): Correlated electron-hole motion begins. Effective mass approximation is approximate.
• Weak Confinement (R > a_B): Bulk-like behavior with minor perturbations. Confinement effects are minimal.

Limitations of the Effective Mass Approximation

The Brus equation makes several simplifying assumptions:

• Spherical symmetry: Real QDs may be faceted or elongated.
• Infinite potential barrier: Ignores wavefunction penetration into surrounding medium or ligands.
• Parabolic bands: Uses bulk effective masses, which break down for very small clusters where the band structure is modified.
• Single-particle picture: Neglects many-body effects and surface states.

For very small QDs (<2 nm), tight-binding calculations or density functional theory (DFT) provide more accurate predictions.

Practical Applications

• Display Technology: QD-enhanced displays use size-tuned emission for pure red, green, and blue colors with wide color gamut.
• Solar Cells: QDs can be tuned to absorb specific portions of the solar spectrum, enabling multi-junction and hot-carrier devices.
• Biomedical Imaging: Near-infrared QDs (PbS, InAs) enable deep tissue imaging with reduced scattering.
• Lighting: White LEDs use QD phosphors for warm, high-CRI illumination.