| name | carrier-capture-cross-section-models |
| description | Calculate carrier capture cross-sections using gas-kinetic models and Coulomb-attractive center theory. Use this for determining recombination rates, carrier lifetimes, and analyzing temperature-dependent capture at defect centers. |
Carrier Capture Cross-Section Models
When to Use
- Modeling carrier capture rates at defect centers
- Calculating carrier lifetimes from recombination center density
- Analyzing temperature-dependent capture processes
- Evaluating recombination at donors/acceptors (Coulomb-attractive centers)
Gas-Kinetic Model
Capture Cross-Section
s = π × r₀²
Where r₀ is the radius of the electron eigenstate at kT below the band edge.
Capture Conditions
Carrier capture requires:
- Carrier approaches within kT of band edge (usually satisfied in thermal equilibrium)
- Carrier approaches within distance < r₀ of recombination center
Mean Free Path Between Captures
λ = 1 / [(N_r - n_r) × s]
Where (N_r - n_r) is the density of free recombination centers.
Carrier Lifetime
τ_r = λ / v_th
Where v_th is the thermal velocity.
Limitations
- Assumes thermal equilibrium
- Low field conditions
- At higher fields or optical excitation, energy-dependent capture must be considered
Coulomb-Attractive Centers
Energy Spectrum
E_n = E_c - E_Ry / n_q²
Where:
- E_Ry: Rydberg energy of the defect
- n_q: principal quantum number
Determine Relevant Quantum Number
Find n_kT: the smallest integer where E_n > kT relative to continuum.
Eigenstate Radius
r₀ = 2 × n_kT² × a_B
Where a_B is the quasi-Bohr radius.
Room Temperature Approximation
Since ground state is often slightly below kT:
r₀ ≈ 2 × a_B
Capture Cross-Section Formula
s ≈ 4 × a_B² × (E_Ry / kT)²
Key Characteristics
- Mass independence: Cross-section independent of effective mass m*
- Temperature dependence: s ∝ 1/T² (decreases with increasing temperature)
- Low temperature enhancement: Can achieve 'giant' cross-sections
- Example: ~4×10⁻¹² cm² at 70K
Temperature Dependence Comparison
| Model | Temperature Dependence | Physical Origin |
|---|
| Gas-kinetic (neutral) | Weak (thermal velocity) | Geometric capture |
| Coulomb-attractive | s ∝ 1/T² | Coulomb focusing decreases with T |
Practical Application
Step-by-Step for Coulomb-Attractive Centers
- Calculate quasi-Bohr radius a_B from defect parameters
- Determine Rydberg energy E_Ry
- At room temperature, use n_q = 1 approximation
- Calculate cross-section using temperature-dependent formula
- For low temperatures, compute actual n_kT for accuracy
Lifetime Calculation
- Determine capture cross-section s from appropriate model
- Calculate mean free path λ from center density
- Compute lifetime τ_r = λ/v_th
- For multiple mechanisms, use Matthiessen's rule: 1/τ_total = Σ 1/τ_i
