Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 31 : Electron in a Periodic Potential
We have seen that the solution of the Schrödinger equation for an arbitrary value of $ k$ is a plane wave $ e^{ikx}$. This, however, is not true at the zone boundary where Bragg reflection takes place. At such a boundary the wave function has two components, viz., $ e^{ikx}$ and $ e^{-ikx}$. One can form two standing waves using these two wavefunctions, viz., $ \psi^{+}=\cos(\pi x/d)$ and $ \psi^{-}=\sin(\pi x/d)$. The corresponding electron densities are proportional to $ \cos^2(\pi x/d)$ and $ \sin^2(\pi x/d)$. These two densities have their maxima respectively at the locations of the atoms and midway between atoms, as shown.
 
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