ShaneLogic/carrier-capture-cross-section-models

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:

1. Carrier approaches within kT of band edge (usually satisfied in thermal equilibrium)
2. 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

1. Calculate quasi-Bohr radius a_B from defect parameters
2. Determine Rydberg energy E_Ry
3. At room temperature, use n_q = 1 approximation
4. Calculate cross-section using temperature-dependent formula
5. For low temperatures, compute actual n_kT for accuracy

Lifetime Calculation

1. Determine capture cross-section s from appropriate model
2. Calculate mean free path λ from center density
3. Compute lifetime τ_r = λ/v_th
4. For multiple mechanisms, use Matthiessen's rule: 1/τ_total = Σ 1/τ_i