Engineering & Science
Fowler-Nordheim Quantum Field-Emission Tunneling Current Density Formula
Calculate cold-cathode quantum electron tunneling current density J = ((1.5414e-6 * E^2) / phi) * EXP((-6.8309e7 * phi^1.5) / E) in A/cm^2 for FE-SEM tips, carbon nanotubes, and flash memory.
Tunnel electrons straight through a triangular barrier at room temperature!
Unlike Richardson-Dushman thermionic cathodes that boil electrons over the work-function barrier at 2,500 K, a cold field-emission gun (CFEG) in an aberration-corrected transmission electron microscope or NAND flash floating-gate cell applies an intense surface electric field E ~ 3e7 to 7e7 V/cm (3 to 7 V/nm) that bends the vacuum barrier into a razor-thin triangle so electrons quantum-tunnel directly into vacuum without heating!
Ralph Fowler and Lothar Nordheim (1928) solved WKB quantum tunneling through a triangular barrier to obtain the simplified elementary Fowler-Nordheim equation: J = ((A_FN * E^2) / phi) * EXP((-B_FN * (phi^1.5)) / E), with universal constants A_FN = 1.5414e-6 A*eV/V^2 and B_FN = 6.8309e7 V/(cm*eV^1.5)! Watch the extreme field sensitivity on a tungsten nanotip (phi = 4.50 eV): raising local tip field just 20% from 5.0e7 V/cm to 6.0e7 V/cm vaults tunneling current density 12.6x from 1,856.96 A/cm^2 to 23,503.82 A/cm^2!
| Cold-Emitter Tip Scenario | Work Function phi & Field E | Tunneling Density J (A/cm^2) | Fowler-Nordheim Plot ln(J/E^2) Slope |
|---|---|---|---|
| Tungsten <310> Tip (Baseline) | phi = 4.50 eV, E = 5.0e7 V/cm | 1,856.96 A/cm^2 | -6.5208e8 V/cm |
| Tungsten Tip (+20% Voltage) | phi = 4.50 eV, E = 6.0e7 V/cm | 23,503.82 A/cm^2 (12.66x jump!) | -6.5208e8 V/cm |
| Zero/Negative Field E <= 0 | Division by E in Exponent | Guarded Input Error | "Invalid Field/Work Function" |
Fowler-Nordheim Tunneling Current Density J (A/cm^2)
Evaluates ((1.5414e-6 * E^2) / phi) * EXP((-6.8309e7 * (phi^1.5)) / E) for E in V/cm and phi in eV.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Field/Work Function", ROUND(((0.0000015414 * (B2 ^ 2)) / B3) * EXP((-68309000 * (B3 ^ 1.5)) / B2), 2))Fowler-Nordheim Plot Slope S_FN = -B_FN * phi^1.5 (V/cm, vs 1/E)
Gives the theoretical linear slope when plotting LN(J / E^2) against 1 / E to extract tip work function or field-enhancement factor beta.
=IF(B3 <= 0, "Invalid Field/Work Function", ROUND(-68309000 * (B3 ^ 1.5), 0))Try the example
Use the original values or change them to test your understanding.