Lecture Notes: Conductivity of Semiconductors

Key Concepts

• Metals vs. Semiconductors:

  • Metals are unipolar: Only electrons as charge carriers.
  • Semiconductors are bipolar: Both electrons and holes as charge carriers.

• Current Direction:

  • Electrons and holes move in opposite directions when an electric field (E) is applied.
  • Despite opposite movement, the current due to both is in the same direction.

Drift Velocity and Mobility

• Drift Velocity (Vd):

  • Directly proportional to the applied electric field (E).
  • Expression: ( V_d = \mu \cdot E )

• Mobility (( \mu )):

  • Constant for a material.
  • Unit: ( \text{m}^2/\text{V·s} )
  • Calculated as: ( \mu = V_d / E )

Current Density and Conductivity

• Current (I): ( I = n \cdot e \cdot A \cdot V_d )

  • ( n ): Number of charge carriers
  • ( e ): Charge of an electron ( (1.6 \times 10^{-19} \text{ C}) )
  • ( A ): Area of cross-section

• Current Density (J):

  • ( J = I / A = n \cdot e \cdot V_d )
  • Substituting ( V_d = \mu E ), ( J = n \cdot e \cdot \mu \cdot E )
  • ( \sigma ) (Conductivity): ( \sigma = n \cdot e \cdot \mu )
  • Ohm’s Law: ( J = \sigma \cdot E )

Mobility and Effective Mass

• Charge of Electrons and Holes: Same magnitude but different masses.
• Mobility Relation:
  • ( \mu ) is inversely proportional to effective mass.
  • Holes have greater effective mass than electrons, thus lower mobility.
  • Expressed as: ( J = (n \cdot \mu_n + p \cdot \mu_p) \cdot e \cdot E )

Conductivity Calculation in Semiconductors

• Conductivity Formula for Semiconductors:
  • ( \sigma = (n \cdot \mu_n + p \cdot \mu_p) \cdot e )

Example Problem: Intrinsic Conductivity of Pure Germanium

• Given:

  • ( \mu_n = 0.38 \text{ m}^2/\text{V·s} )
  • ( \mu_p = 0.18 \text{ m}^2/\text{V·s} )
  • ( n_i = 2.5 \times 10^{19} \text{ per m}^3 )

• Solution Steps:

  • Use formula: ( \sigma = n_i \cdot e \cdot (\mu_n + \mu_p) )
  • Calculate: ( 0.38 + 0.18 = 0.56 )
  • ( 2.5 \times 1.6 = 4 )
  • Final conductivity: ( 2.24 \text{ ohm}^{-1}\text{m}^{-1} )

• Conclusion:

  • Discuss more complex problems in future presentations.