Summary

• The concentration of electrons in the conduction band is the integral over the conduction-band energy of the product of the density of states function in the conduction band and Fermi-Dirac probability function

• The concentration of holes in the valence band is the integral over the valence band energy of the product of the density of states function in the valence band and the probability of a state being empty, which is $1-f_F(E)$

• Using the Maxwell-Boltzmann approximation, the thermal-equilibrium concentration of electrons in the conduction band is given by:

  $$ n_0 = N_c \exp \left[ -\frac{(E_c - E_F)}{kT} \right] $$

• Using the Maxwell-Boltzmann approximation, the thermal-equilibrium concentration of holes in the valence band is given by:

  $$ p_0 = N_v \exp \left[ -\frac{(E_F - E_v)}{kT} \right] $$

• The intrinsic carrier concentration is found from:

  $$ n_i^2 = N_c N_v \exp \left[ -\frac{E_g}{kT} \right] $$

• The concept of doping the semiconductor with donor (group V elements) impurities and acceptor (group III elements) impurities to form n-type and p-type extrinsic semi- conductors

• The fundamental relationship of $n_0p_0 = n_1 ^2$

• Using the concepts of complete ionization and charge neutrality, equations for the electron and hole concentrations as a function of impurity doping concentrations were derived.

• The position of the Fermi energy level as a function of impurity doping concentrations was derived.

• The Fermi energy is a constant throughout a semiconductor that is in thermal equilibrium.

Charge Carriers in Semiconductors

Two types of charge carrier (electron and hole) can contribute to a current

• Density of electrons and holes is related to the density of states function and Fermi distribution function

Equilibrium Distribution of Electrons and Holes

• Total hole concentration per unit volume is found by integrating these functions over entire valence-band energy
  • Distribution of electrons in the conduction band is given by $n(E) = g_c(E)f_F(E)$
  • Distribution of holes in valence band is given by $p(E) = g_v(E)[1-f_F(E)]$

$n_0$ and $p_0$ Equations

Thermal-Equilibrium Hole Concentration

Intrinsic Carrier Concentration