A resonant tunneling diode (RTD) sandwiches a thin quantum well between two thin tunnel barriers (e.g. GaAs well, AlGaAs barriers). The 1-D effective-mass Schrödinger equation is solved exactly with the transfer-matrix method: each constant-potential region Vi carries a wavevector
k_i = sqrt(2 m* (E − V_i)) / ħ (real → propagating, imaginary → evanescent)
Matching ψ and ψ′ at every interface chains 2×2 complex matrices into one transfer matrix T; the transmission probability is
T(E) = (k_out / k_in) · |A_in / A_out|², incident amplitude = 1, no wave incoming from the right
Because the well only supports discrete quasi-bound levels, T(E) is sharply peaked at the well's resonant energy Eres — an electron at that exact energy interferes constructively on every round trip inside the well and tunnels through with T ≈ 1, even though each barrier alone would nearly block it.
Applying a bias voltage tilts the whole potential linearly, dragging Eres down through the fixed emitter Fermi energy EF. Current is approximated as I(V) ∝ ∫₀^E_F T(E,V) dE. While Eres sweeps past EF the current rises to a peak, then — uniquely among semiconductor devices — falls as bias keeps increasing, because the resonance has moved out of the emitter's occupied energy window: this is negative differential resistance (NDR), the property that makes RTDs useful as GHz–THz oscillators.
- V0, b — taller/wider barriers slow tunneling into and out of the well, sharpening the resonance (narrower, taller T(E) peak).
- w — a wider well lowers Eres (particle-in-a-box scaling Eres ∝ 1/w²) and packs in more resonant levels.
- Vbias — tilts the structure; the moving marker on the I–V chart tracks where you are on the peak → NDR → valley curve.
This is the actual device physics behind real resonant tunneling diodes used in oscillator and fast-switching nanoelectronics — the same double-barrier quantum-well geometry the article on nanoelectronics describes as one of the four canonical single-electron/quantum-confined transistor families.