By name: wkb-tunneling-approximation description: Calculate quantum tunneling transmission probability through triangular barriers, parabolic barriers, and band-to-band transitions using the WKB (Wentzel-Kramers-Brillouin) approximation. Use when analyzing tunneling phenomena in semiconductor devices, quantum wells, or when barrier shapes can be approximated as triangular or parabolic with defined electric fields.
WKB Tunneling Approximation
When to Use This Skill
Use the WKB tunneling approximation when:
- Analyzing quantum tunneling through potential barriers
- Barrier shape is triangular, parabolic, or involves band-to-band transitions
- Electric field is well-defined and constant
- Pre-exponential factors can be neglected (order of 1)
- Working with semiconductor devices requiring tunneling current calculations
Prerequisites
- Understanding of WKB approximation theory
- Knowledge of calculus integration
- Barrier shape must be mathematically defined
- Effective mass and barrier height must be known
Core Workflow
1. Identify Barrier Type
Determine which barrier approximation applies:
- Triangular barrier: Linear potential variation with position
- Parabolic barrier: Quadratic potential, often from image force lowering
- Band-to-band tunneling: Tunneling across semiconductor bandgap
- Overlapping fields: Combined Coulomb and external electric fields
2. Apply General WKB Formula
The transmission probability follows:
Te = exp(-2 * integral(k(x) dx))
where:
k(x) = sqrt(2m * DeltaE(x)) / hbar- Pre-exponential factor is neglected (assumed ≈ 1)
3. Select Specific Formula
Choose the appropriate formula based on barrier type:
- Triangular barrier: Use Eq (19.13) with linear field dependence
- Parabolic barrier: Use modified formula with pi factor
- Band-to-band: Replace DeltaE with bandgap Eg
- Overlapping fields: Use combined field expression
4. Calculate Transmission Probability
Compute the exponential transmission coefficient using the selected formula.
Key Constraints
- Pre-exponential factor is neglected (order of 1)
- Barrier shape must be clearly defined
- Electric field must be constant or well-characterized
- Effective mass approximation applies
- For band-to-band tunneling, fields > 10^6 V/cm typically required
Critical Insights
- Parabolic barriers have reduced exponent by factor 3pi/16 ≈ 0.59 compared to triangular barriers
- Barrier height/field relation is superlinear: doubling barrier height requires 2^(3/2) ≈ 2.83 times the field for same probability
- Narrow gap semiconductors enable significant tunneling at lower fields
Common Applications
- Zener diode breakdown analysis
- Tunnel diode current calculations
- Gate leakage in MOSFETs
- Quantum well tunneling
- Trap-assisted tunneling in dielectrics