Kronig-Penney Model

📚 Learning Mode

Interactive

1 2 3 4

🧪 Material Presets

⚡ Potential Parameters

Potential Depth (V₀)

1 eV 100 eV

Barrier Width (b)

1 Å 50 Å

Well Width (a)

5 Å 100 Å

👁️ Display Options

Potential Wells Energy Bands Wave Functions Band Gaps Labels & Annotations

🎬 Animation

Speed: 1x

Potential Well & Energy Bands

Periodic Potential V(x)

Allowed Energy Bands

Wave Function ψ(x)

Forbidden Band Gaps

📊

Band Gap Energy 2.45 eV

📈

First Allowed Band 0 - 1.2 eV

🔬

Material Type Semiconductor

⚡

Effective Mass Ratio m*/m₀ = 0.067

📖 Current Concept

The Kronig-Penney model represents a one-dimensional crystal lattice as a series of rectangular potential wells. Each well corresponds to an atom in the lattice.

Schrödinger Equation:

-ℏ²/2m · d²ψ/dx² + V(x)ψ = Eψ

💡 Tip: Try adjusting the potential depth to see how it affects the band gaps!

🔍 Key Observations

Energy bands form due to the periodic potential

Band gaps appear between allowed energy levels

Higher potential depth → Wider band gaps

Wave functions are modulated by the lattice

📐 Formula Reference

Bloch's Theorem: ψ_k(x) = e^(ikx) · u_k(x)

Dispersion Relation: cos(ka) = cos(αa) + P·sin(αa)/(αa)

Band Gap Condition: |cos(ka)| > 1 → Forbidden

Potential Well & Energy Bands

📘 How to Use This Simulation

🎯 Getting Started

⚙️ Controls Explained

👁️ Display Options