Engineering & Science
Poole-Frenkel Trap Barrier Lowering vs. Schottky Emission Formula
Calculate bulk dielectric Coulomb trap barrier lowering Delta_phi_PF = SQRT((e * E) / (PI() * eps_dyn * eps_0)) (eV) and compare directly against electrode-limited Schottky image-force lowering (0.5 * Delta_phi_PF).
Distinguish bulk Poole-Frenkel dielectric leakage from interface Schottky emission by the exact factor of 2!
High-field gate dielectrics (Si3N4, HfO2, Ta2O5) in DRAM capacitors and NAND Flash cells leak current when strong electric fields E (V/m) tilt Coulombic defect traps inside the insulator bulk, lowering the thermal ionization barrier phi_B! Yakov Frenkel (1938) proved that because a stationary positively charged bulk trap is immobile (unlike a mobile image charge receding behind a metal Schottky electrode), bulk Poole-Frenkel barrier lowering is EXACTLY twice electrode Schottky image-force lowering: Delta_phi_PF = SQRT((e * E) / (PI() * eps_dyn * eps_0)) eV = 2 * Delta_phi_Schottky!
Notice that high-frequency optical permittivity eps_dyn = n^2 (not static permittivity!) governs fast electron escape! In Silicon Nitride (Si3N4, eps_dyn = 7.5) under E = 1.0e8 V/m (1 MV/cm), Poole-Frenkel lowers bulk trap depth by Delta_phi_PF = 0.2771 eV whereas interface Schottky barrier lowering is half as large (0.1386 eV); quadrupling the field to E = 4.0e8 V/m (4 MV/cm) doubles Delta_phi_PF to 0.5542 eV!
| Dielectric Electric Field E (Si3N4 eps_dyn = 7.5) | Poole-Frenkel Lowering Delta_phi_PF (eV) | Schottky Lowering Delta_phi_S (eV) | Leakage Mechanism Slope Ratio |
|---|---|---|---|
| Moderate Gate Stress (1.0e8 V/m = 1 MV/cm) | 0.2771 eV | 0.1386 eV | beta_PF / beta_S = 2.000 |
| High Gate Stress (4.0e8 V/m = 4 MV/cm) | 0.5542 eV (2x) | 0.2771 eV | Exponential Leakage Surge |
| Invalid Permittivity (eps_dyn < 1 or E < 0) | Physical Guard | "Invalid Dielectric Inputs" | Requires eps_dyn >= 1, E >= 0 |
Poole-Frenkel Bulk Trap Barrier Lowering Delta_phi_PF (eV)
Evaluates SQRT((e * E) / (PI() * eps_dyn * eps_0)) directly in electronvolts using CODATA e = 1.602176634e-19 C and eps_0 = 8.8541878e-12 F/m.
=IF(OR(B2 < 1, B3 < 0), "Invalid Dielectric Inputs", ROUND(SQRT((0.0000000000000000001602176634 * B3) / (PI() * B2 * 0.0000000000088541878)), 4))Corresponding Schottky Interface Image-Force Lowering Delta_phi_Schottky (eV)
Computes 0.5 * Delta_phi_PF for Poole-Frenkel vs. Richardson-Schottky slope comparison.
=IF(OR(B2 < 1, B3 < 0), "Invalid Dielectric Inputs", ROUND(0.5 * SQRT((0.0000000000000000001602176634 * B3) / (PI() * B2 * 0.0000000000088541878)), 4))Try the example
Use the original values or change them to test your understanding.