Engineering & Science
Quantum Barrier Tunneling (STM & Flash Oxide WKB Transmission Formula)
Calculate the evanescent wavefunction decay constant kappa = 5.12317 * SQRT((m_eff / m_0) * (V_0 - E)) (nm^-1) and exponential tunneling transmission probability T_WKB = EXP(-2 * kappa * L) across Scanning Tunneling Microscope (STM) vacuum gaps and gate oxides.
Why pulling an STM tip back by just 0.1 nm slashes quantum tunneling current ~10-fold!
Gerd Binnig and Heinrich Rohrer (Nobel Prize 1986) imaged individual silicon surface atoms because electron tunneling across a vacuum gap of width L (nm) decays exponentially as T_WKB ~ EXP(-2 * kappa * L)! Inside a forbidden rectangular barrier where barrier height V_0 (eV) exceeds electron energy E (eV), the Schrödinger wavefunction decays with inverse decay length kappa = SQRT(2 * m_eff * (V_0 - E)) / hbar = 5.12317 * SQRT((m_eff / m_0) * (V_0 - E)) nm^-1!
Look at how sensitive that exponential is for a free electron (m_eff / m_0 = 1.0) facing a net barrier phi = V_0 - E = 1.50 eV (V_0 = 2.0 eV, E = 0.5 eV -> kappa = 6.275 nm^-1): across a thin L = 0.40 nm gap, tunneling probability is T_WKB = EXP(-2 * 6.275 * 0.40) = 0.006607 (0.66%); widening the gap by just +0.10 nm to L = 0.50 nm plunges T_WKB more than 3.5-fold down to 0.001884 (0.19%)!
| Tunneling Barrier Gap L (V0 = 2.0 eV, E = 0.5 eV, meff = 1.0 m0) | Decay Constant kappa (nm^-1) | WKB Transmission T = EXP(-2*kappa*L) | STM / Gate-Oxide Consequence |
|---|---|---|---|
| Sub-Nanometer Gap (L = 0.40 nm) | 6.275 nm^-1 | 0.006607 (0.66%) | Measurable nA STM Tunneling Current |
| +1 Angstrom Wider Gap (L = 0.50 nm) | 6.275 nm^-1 | 0.001884 (3.5x Drop!) | Atomic-Scale Vertical Z-Resolution |
| Over-the-Barrier Classical (E >= V0) | Imaginary kappa Guard | "Classical Over-Barrier (E >= V0)" | Oscillatory Propagation (Not Tunneling) |
Evanescent Wavefunction Decay Constant kappa (nm^-1)
Uses CODATA prefactor SQRT(2 * m_e * e) / hbar * 10^-9 = 5.123167 nm^-1 * eV^-1/2 scaled by SQRT(m_rel * (V_0 - E)).
=IF(OR(B2 <= B3, B4 <= 0, B5 <= 0), "Classical / Invalid Barrier", ROUND(5.12317 * SQRT(B5 * (B2 - B3)), 3))WKB Exponential Barrier Tunneling Probability T = EXP(-2 * kappa * L)
Evaluates single-barrier exponential transmission across barrier thickness L (nm, cell B4).
=IF(OR(B2 <= B3, B4 <= 0, B5 <= 0), "Classical / Invalid Barrier", ROUND(EXP(-2 * 5.12317 * SQRT(B5 * (B2 - B3)) * B4), 6))Try the example
Use the original values or change them to test your understanding.