Exciton Bohr Radius Calculator

Exciton Bohr Radius Formula:

\[ a_B = \varepsilon_r \times \left( \frac{m_e}{\frac{m_e \times m_h}{m_e + m_h}} \right) \times [Bohr-r] \]

Dielectric Constant of Bulk Material:

(dimensionless)

Effective Mass of Electron:

(relative to electron mass)

Effective Mass of Hole:

(relative to electron mass)

Exciton Bohr Radius:

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1. What is Exciton Bohr Radius?

The Exciton Bohr Radius represents the average separation between an electron and a hole in an exciton. It's a fundamental parameter in semiconductor physics that determines the quantum confinement effects in nanostructures.

2. How Does the Calculator Work?

The calculator uses the Exciton Bohr Radius formula:

\[ a_B = \varepsilon_r \times \left( \frac{m_e}{\frac{m_e \times m_h}{m_e + m_h}} \right) \times [Bohr-r] \]

Where:

\( \varepsilon_r \) — Dielectric Constant of Bulk Material
\( m_e \) — Effective Mass of Electron (relative to electron mass)
\( m_h \) — Effective Mass of Hole (relative to electron mass)
\( [Bohr-r] \) — Bohr radius constant (0.529 × 10⁻¹⁰ m)

Explanation: The formula accounts for the dielectric screening effect and the reduced mass of the electron-hole pair in the semiconductor material.

3. Importance of Exciton Bohr Radius

Details: The Exciton Bohr Radius is crucial for understanding quantum confinement effects in nanomaterials. When the physical size of a semiconductor structure becomes comparable to or smaller than the Exciton Bohr Radius, quantum mechanical effects become significant, leading to size-dependent optical and electronic properties.

4. Using the Calculator

Tips: Enter the dielectric constant of the material, effective mass of electron, and effective mass of hole. All values must be positive numbers. The effective masses are typically expressed relative to the free electron mass.

5. Frequently Asked Questions (FAQ)

Q1: What is an exciton?
A: An exciton is a bound state of an electron and an electron hole that are attracted to each other by the electrostatic Coulomb force in semiconductors.

Q2: Why is the dielectric constant important?
A: The dielectric constant reduces the Coulomb attraction between the electron and hole, increasing the Exciton Bohr Radius in materials with higher dielectric constants.

Q3: How does effective mass affect the Exciton Bohr Radius?
A: Larger effective masses result in smaller Exciton Bohr Radii, as heavier particles are more strongly bound together.

Q4: What are typical values for Exciton Bohr Radius?
A: Typical values range from 1-20 nm depending on the semiconductor material. For example, GaAs has about 12 nm, while CdSe has about 5.6 nm.

Q5: How is this related to quantum dots?
A: Quantum dots are semiconductor nanocrystals whose size is comparable to or smaller than the Exciton Bohr Radius, leading to quantum confinement effects that tune their optical properties.