Kronig-Penney Band Gap Explorer

Dirac delta barrier model: cos(Ka) = f(qa) = cos(qa) + P·sin(qa)/(qa). Allowed bands exist where |f(qa)| ≤ 1; forbidden gaps where |f(qa)| > 1. Left: f(qa) vs qa. Right: reduced-zone band structure E vs K.

KP Function — f(qa) vs qa
Band Structure — E vs K (reduced zone)

Parameters

P (barrier strength) 3.0
a (lattice constant) 0.30 nm

Band Info

P3.00
a0.30 nm
E(π/a)— eV
Gap 1 center— eV
Gap 1 width— eV
Bands shown—

Physics Checks

○ P=0 → no gaps (free electron)
○ Gap 1 at qa=π, E≈4.2 eV (a=0.3nm)
○ a=0.6nm → E(π/a)≈1.05 eV (×¼)
Legend:
—— f(qa) curve (left) / allowed bands (right)
▬▬ forbidden gap shading
±1 window / BZ boundary