Lecture 5: Equilibrium Carrier Concentration

Overview

• Focus on equilibrium carrier concentration in intrinsic semiconductors.
• Previous lecture covered wave-particle duality and introduced intrinsic semiconductor model.

Key Concepts

Wave-Particle Duality

• Concept introduced to simplify physical analysis.
• Relates to particles: electrons, holes, photons, phonons.

Intrinsic Semiconductor Model

• At absolute zero (T = 0K):
  • No thermal vibrations, no free particles.
  • Each silicon atom bonded to four nearest neighbors.
• For T > 0K:
  • Atoms vibrate due to thermal energy.
  • Visual analogy: Atoms as steel balls connected by springs.
  • Tapping the matrix produces traveling waves, analogous to water ripples.

Particle Generation

• Electrons and Holes:
  • When bonds break, free electrons are created, leaving behind holes.
  • Electron-hole pairs generated in pairs (EHP generation).
• Photons:
  • Created through vibrating atoms acting as oscillating dipoles, generating electromagnetic waves.
• All particles are referred to as "free electrons" when discussing carriers.

Particle Concentration Balance

• Generation and Recombination:
  • Equilibrium requires forward and reverse processes (generation and recombination).
  • Generation: Free electrons create holes.
  • Recombination: Free electrons fill holes, restoring equilibrium.
• Analogy: Human population dynamics (births vs. deaths).
• Recombination occurs after electron moves around for some time.

Detailed Balance of Processes

• Photo Generation:
  • Photons generate electron-hole pairs.
• Radiative Recombination:
  • Electron-hole recombination emitting photons.
• Impact Ionization:
  • Energetic electrons create electron-hole pairs.
• Auger Recombination:
  • Energy given to a free electron or hole during recombination.
• Phonon Generation:
  • Generation and recombination processes involving phonons.
• Recombination categorized into radiative (useful for light generation) and non-radiative processes (e.g., Auger recombination).

Concept of Holes

• Vacancies participate in conduction, behaving as positively charged particles.
• Bound electrons can contribute to current through vacancies.
• Analogy: Bubble in a tube represents how vacancies facilitate the movement of bound electrons.

Effective Mass

• Effective mass concept helps analyze the movement of bound electrons.
• Effective mass of electrons differs from holes, and from vacuum electron mass (m0).
• Effective mass is a parameter to simplify calculations of current and conductivity.

Carrier Concentration in Intrinsic Semiconductors

• Intrinsic Carrier Concentration (ni):
  • ni = pi (electrons and holes generated in pairs).
  • At room temperature (300K), ni ~ 10^10/cm³.
  • Silicon atom concentration ~ 5 x 10^22/cm³.
  • Small fraction of silicon atoms contribute to free electrons/holes.

Temperature Effects

• As temperature increases, the number of particles converging on an atom decreases, increasing probability for electron-hole pair generation.
• Higher temperatures lead to significantly increased carrier concentrations.

Experimental Evidence

• Hall Effect Experiment:
  • Confirms presence of positive charge carriers (holes) through behavior under electric and magnetic fields.
  • Top phase becomes positive in semiconductors with holes as charge carriers.

Conclusion

• Understanding of electrons and holes in intrinsic semiconductors is crucial for insights into semiconductor conductivity and behavior.
• Next topic: Quantitative model - Energy Band or Band Model.