= Rectangular free-electron band-overlap scale
{title2=$\Delta_{\mathrm{overlap}}=\hbar^2\pi^2/(2mb^2)$}
For rectangular real-space periods $a>b$, the free first-zone corner has <energy> $\hbar^2\pi^2(a^{-2}+b^{-2})/(2m)$, whereas the free second-zone minimum is $\hbar^2\pi^2/(2ma^2)$. Their difference is the displayed overlap. A divalent spin-degenerate system is a <band insulator> only when the actual first-band maximum lies below the second-band minimum. A rigid symmetric splitting $\Delta$ removes the overlap if $\Delta>\Delta_{\mathrm{overlap}}$, but an arbitrary <periodic potential> does not shift every corner and face by the same amount. The exact criterion uses the <indirect band gap>, not a geometry-only local direct-gap width.
Back to article page
