Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 31 : Electron in a Periodic Potential
Electron in a Periodic Potential
Inside a lattice the electron is subjected to a periodic potential, i.e., the form potential repeats itself in space. Thus if $ a$ is the inter-atomic distance in a one dimensional lattice, we have
 
$\displaystyle V(x+a) = V(x)$
  For potentials that are periodic, the wavefunction satisfies Bloch theorem which states that the form of the wavefunction is
 
$\displaystyle \psi(x) = u(x) e^{ikx}$
  where $ u(x)$ is a periodic function with the same periodicity as that of the lattice, i.e.,
 
$\displaystyle u(x+a) = u(x)$
  Substituting this in Schrödinger equation for $ \psi(x)$, we would obtain an equation for $ u(x)$ which must be solved.
  A simple model often used to mimic the periodic potential is known as the Krönig-Penny model , the form of which is shown in the figure.
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