Frenkel Poole Effect Analysis

Where:

β = (Z × e³ / π × ε × ε₀)^(1/2)

Or equivalently:

δE = Z × e³/² × F¹/²

Parameters

• δE = Barrier lowering (eV)
• Z = Charge of defect center
• F = Electric field (V/cm)
• ε = Relative permittivity
• ε₀ = Vacuum permittivity

Procedure

Step 1: Identify Defect Characteristics

• Determine defect charge Z
• Confirm Coulomb-attractive nature
• Note material permittivity ε

Step 2: Calculate Barrier Lowering

• Apply the Frenkel-Poole formula
• Typical field range: 10-100 kV/cm

Step 3: Evaluate Thermal Ionization Enhancement

e_tc = s_n × v_th × exp[-(Ec - Et - δE) / kT]

Where:

• e_tc = Thermal ionization coefficient
• s_n = Capture cross-section
• v_th = Thermal velocity
• Ec - Et = Trap energy depth

Step 4: Determine Critical Thresholds

• Calculate field for δE = 2kT (significant emission)
• Evaluate field quenching onset

Critical Thresholds (CdS:Cu Example)

Example Calculations

CdS with Doubly Charged Cu (Z=2)

• At F = 50 kV/cm, ε = 10:
• Distance from funnel center: 35 Å
• δE ≈ 30 meV

Order of Magnitude

• Typical semiconductors: δE significant at ~10 kV/cm

Applications

1. Field-enhanced deep donor depletion
2. Field quenching in photoconductors
3. Trap emptying in memory devices
4. Carrier generation in high-field regions38:["$","$L41",null,{"content":"$42","frontMatter":{"name":"Frenkel-Poole Effect Analysis","description":"Calculate barrier lowering and enhanced thermal ionization at Coulomb-attractive defect centers under electric fields. Use when analyzing field ionization mechanisms, trap emission, field quenching, or carrier generation in semiconductors with defects. This is the lowest-field field-ionization mechanism."}}]