Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 31 : Electron in a Periodic Potential
The parameter $ k$ in these diagram does not have the interpretation of momentum; however, in analogy with the case of free particle $ \hbar k$ is known as crystal momentum" . When an electron in a crystal is subjected to an external force $ F_{ext}$, it is this crystal momentum which satisfies the Newton's laws
 
$\displaystyle F_{ext} = \frac{d}{dt}(\hbar k)$
  Origin of Gap in Energy Spectrum
  From the energy diagram shown, it can be seen that there are discontinuities that arise at certain values of $ k$. Such gaps occur when the wavelength of electron is such that the condition for Bragg diffration by the periodic lattice structure is satisfied.
  We know that an electron wave incident on such a lattice undergoes reflection if Bragg condition
 
$\displaystyle 2d\sin\theta= n\lambda$
  is satisfied. For waves travelling along the line of atoms in the crystal, the angle of incidence is $ 90^\circ$, so that the wave vector of the electron for which Bragg condition is satisfied is
 
$\displaystyle k = \frac{2\pi}{\lambda} = n\frac{\pi}{d}$
 
5
10
11
17