Lecture Notes on Carrier Density and Transport in Semiconductors

Recap of Previous Lectures

• Temperature dependence of carrier density.
• Discussion on charge neutrality in semiconductors.

Upcoming Topics

• High doping effects and incomplete ionization.
• Mobility and drift diffusion.
• Current transport and its relation to devices: PN junctions, LEDs, photo detectors, solar cells, BJTs, MOSFETs, and other optical devices.

Key Concepts for Discussion

Carrier Statistics and Density of States

• Understanding carrier statistics helps in calculating carrier density, including electron and hole density.

High Doping Effects

• Doping Types: n-type and p-type doping.
• Maxwell-Boltzmann Equation: Valid when the energy gap ( EC - EF) is greater than 3kT (approximately 78 meV at room temperature).
• High Doping: If EC - EF < 3kT (e.g., 50 meV), the Maxwell-Boltzmann equation cannot be applied.
• Use Joyce-Dixon Approximation to calculate carrier concentration in high doping scenarios:
  • Equation: ( E_F - E_C \approx K T ext{ (approx. close condition)} )
  • New term in approximation: ( 1/ ext{sqrt}(8 n/n_C) )

Band Gap Effects

• Band Gap Narrowing: High doping can lead to a reduction in the material's band gap due to lattice distortion caused by too many dopant atoms.
• Periodic Potential Distortion: The periodic arrangement of atoms is disturbed, causing potential fluctuations and jagged conduction and valence bands.

Incomplete Ionization

• Not all dopants may ionize completely, especially if the donor ionization energy is far from the conduction band (e.g., > 3kT).
• Ionization Fraction:
  • For n-type doping: ( N_{d+} / N_{d} \approx 1 / (1 + 2 e^{(E_F - E_D)/kT}) )
  • For p-type doping: ( N_{a-} / N_{a} \approx 1 / (1 + 4 e^{(E_A - E_F)/kT}) )
• Example: In gallium nitride p-type doping with magnesium, only 1-5% of dopants might be ionized.

Recap of Topics Covered So Far

• Band gap concepts and their implications.
• Carrier statistics and density of states.
• Temperature dependence of carrier concentration.
• High doping effects and incomplete ionization.

Next Steps

• Introduction to mobility of carriers and low/high field transport in semiconductors.
• Discussion on drift and diffusion of carriers.
• Examination of current flow and carrier recombination.
• Continuity equation will be introduced.

Conclusion

• The class concluded with a recap of high field transport and incomplete ionization.
• Next class will focus on mobility of carriers.