| name | assisted-tunneling-mechanisms |
| description | Calculate tunneling probabilities modified by phonon, trap, or photon assistance when standard tunneling is insufficient or specific energy exchange processes occur. Use this for indirect band-gap materials, defect-assisted transport, or optical field-enhanced tunneling. |
Assisted Tunneling Mechanisms
When to Use
- Indirect band-gap materials requiring momentum change
- High defect density enabling trap-assisted tunneling
- Optical excitation combined with electric fields
- When bias reaches phonon/photon energy thresholds
Mechanism Selection
Phonon Assistance
Conditions: Indirect gap materials
Modified tunneling probability:
Te_phonon ~ exp(-2∫k(x)dx ± ℏω/(eFa))
Where:
- ℏω: phonon energy
- F: electric field
- a: characteristic length
Key characteristics:
- Probability reduced by factor ~10⁻³ vs direct tunneling
- Current increases measurably when Vr reaches phonon energy
- Provides momentum for indirect transitions
Trap Assistance (Two-Step Tunneling)
Conditions: High defect density near interface
Process:
- Tunnel from band into trap state
- Tunnel from trap to opposite band
Total probability:
1/Te_total = 1/Te_1 + 1/Te_2
Where Te_1 and Te_2 are individual step probabilities.
Requirements:
- Defect centers close to interface
- Sufficient trap density
- Appropriate trap energy levels
Photon Assistance (Franz-Keldysh Effect)
Conditions: Optical excitation with applied electric field
Process: Photon provides most energy, field provides small additional energy enabling tunneling
Absorption edge shift:
ΔE_opt = constant × (eFℏ)^(2/3) / (2m)^(1/3)
Example: For 10 meV shift (100 Å at 2 eV gap):
- Required field: ~50 kV/cm
Key characteristics:
- Shifts absorption edge to lower energies
- Exponent remains constant (~1)
- Enables photon absorption below nominal bandgap
Comparison of Mechanisms
| Mechanism | Energy Exchange | Probability Factor | Typical Conditions |
|---|
| Phonon | Momentum + small energy | ~10⁻³ | Indirect gap, Vr ≈ ℏω |
| Trap | Stepwise via defect | Depends on trap density | High defect density |
| Photon | Large optical + small field | Field-dependent | Optical illumination |
Application Notes
- Phonon and photon assistance provide momentum for indirect transitions
- Trap assistance enables tunneling through otherwise forbidden barriers
- Multiple mechanisms can operate simultaneously
- Dominant mechanism depends on material properties and experimental conditions
