Summary

• Discrete allowed electron energies split into a band of allowed energies as atoms are brought together to form a crystal
• Effective mass relates the motion of a particle in a crystal to an externally applied force and takes into account the effect of the crystal lattice on the motion of the particle
• Two charged particles exist in a semiconductor
  • An electron is a negatively charged particle with a positive effective mass existing at the bottom of an allowed energy band
  • A hole is a positively charged particle with a positive effective mass existing at the top of an allowed energy band
• The $E$ vs $k$ diagrams of silicon and gallium arsenide were given and the concept of direct/ban
• Energies within an allowed energy band are actually at discrete levels and each contains a finite number of quantum states
  • The density per unit energy of quantum states was determined using 3D infinite potential well as a model
• In dealing with large numbers of electrons and holes, we must consider the statistical behaviour of these particels
  • Fermi-Dirac probability function was developed, which gives the probability of a quantum states at an energy $E$ of being occupied by an electron
  • Fermi energy defined as

Allowed and Forbidden Energy Bands

• Allowed bands are ranges of energies that electrons can occupy
• Forbidden bands (also called band gaps) are ranges of energies where no electron states can exist

Formation of Energy Bands

The wave functions of the electrons of the two atoms overlap, which means the two electrons will interact

• This perturbation results in the discrete quantized energy level splitting into two discrete energy levels, which is consistent with the Pauli exclusion principle

Splitting of the $n = 1$ state.

• The Pauli exclusion principle states that the joining of atoms to form a system (crystal) does not alter the total number of quantum states regardless of size

• If the equilibrium interatomic distance is $r_0$, then we have bands of allowed energies that the electrons may occupy separated by bands of forbidden energies

  

  Splitting of discrete energy levels.

k-space Diagram

Refers to the momentum space where k (the wave vector) represents the momentum of an electron in the crystal

In the k-space diagram, the electron energy $E$ is plotted as a function of the wave vector $k$.

• For a free electron (no potential from atoms), the energy dispersion relation is given by:

  $$ E(k) = \frac{\not h^2 k^2}{2m} $$

  • However in a crystal lattice, dispersion relation becomes more complex due to interactions with the periodic atomic structure

    

  • When $k$ approaches the Brillouin zone boundary, the electron wave can experience Bragg reflection, which leads to the formation of band gaps, at these points $k$ satisfies the Bragg condition:

    $$ 2k = \frac{n \pi}{a} $$