The gate oxide is a thin insulating barrier separating the conducting channel from the metal gate. Classically no current should cross it at all. But once it is only a handful of atomic layers thick, quantum mechanics allows the electron's wavefunction to leak through: the WKB approximation treats the oxide as a rectangular potential barrier of height φB and width tox, and predicts a transmission probability that falls off exponentially with both thickness and the square root of the barrier height.
κ = √(2·m*·q·φ_B) / ħ (barrier decay constant, 1/m)
T ≈ exp(−2·κ·t_ox) (WKB transmission probability)
J_gate = J₀ · T (direct-tunneling current density)
- Oxide thickness tox — halving it does not halve the leakage, it can multiply it by 10-100× because T decays exponentially with tox. This is why sub-1.2 nm SiO₂ gates in late-1990s/early-2000s CMOS leaked so badly they became unusable.
- Barrier height φB — set by the conduction-band offset between the channel and the oxide; a taller barrier suppresses tunneling exponentially in √φB, which is exactly why the industry switched to physically thicker high-k dielectrics like HfO₂ — same gate capacitance, far less tunneling.
- Gate-source voltage VGS — raises the channel's electron (inversion-layer) density, so more carriers are available to attempt the crossing and to drive the ordinary drift current the transistor is supposed to carry.
- Scaling toggle — animates tox shrinking the way real process nodes did generation after generation, so you can watch the leakage current climb non-linearly as the oxide thins.
Real-world relevance: gate-oxide tunneling leakage is one of the hard limits of Moore's-law scaling — it is precisely why every modern nanotransistor beyond roughly the 90 nm node uses a high-k gate dielectric (HfO₂-based) instead of pure SiO₂, trading a taller or thicker barrier for exponentially lower quantum leakage. The current-density prefactor J₀ here is an illustrative order-of-magnitude constant, not a calibrated device model — the physically meaningful, measurable quantity is the exponential T(tox, φB) dependence itself.