Lecture Notes on Semiconductor Physics

Recap from Last Class: E-K Diagram

• Discussed the concept of the E-K diagram.
• In real semiconductor crystals, electrons experience periodic potential from atoms, causing distortion in the E-K diagram.
• Effective mass of electrons is dictated by the curvature of the E-K diagram.
• Important for understanding semiconductor devices.

Introduction to Holes

• Holes are as important as electrons in semiconductor research.
• Concept of Holes:
  • When an electron is excited from the valence band (VB) to the conduction band (CB), it leaves behind a vacancy, known as a hole.
  • A hole is essentially the absence of an electron.
  • It is treated as a positive charge particle that moves in the direction of the electric field.

Valence and Conduction Bands

• Conduction Band (CB): Contains free electrons that can move and carry current.
• Valence Band (VB): Completely filled with electrons; cannot carry current unless electrons are excited to the conduction band.

Movement of Holes

• When an electron moves from VB to CB, it leaves a hole behind.
• Electric Field Effect:
  • When an electric field is applied:
    • Electrons move opposite to the electric field.
    • Holes appear to move in the direction of the electric field.
• Holes exhibit positive charge behavior and can be modeled as quasi-particles.

E-K Diagram for Holes

• Holes have their own E-K diagram, corresponding to the valence band.
• Effective mass of holes is derived from the curvature of their E-K diagram.
• Holes can have multiple branches of E-K diagram:
  • Heavy hole branch (lower curvature, higher effective mass)
  • Light hole branch (higher curvature, lower effective mass)

Energy Band Gap

• The energy gap between the conduction band and the valence band is defined as the energy band gap (E_g).
• Distinction between Direct Band Gap and Indirect Band Gap materials:
  • Direct Band Gap: Conduction band minimum and valence band maximum occur at the same k-point; efficient light emission (e.g., GaAs, GaN).
  • Indirect Band Gap: Requires a change in momentum for electron transition; inefficient light emission (e.g., Si, Ge).

Fermi-Dirac Statistics

• Purpose: To describe how electrons and holes occupy energy states in semiconductors.
• Probability of occupying energy states defined by the Fermi function:
  • Formula: ( F(E_i) = \frac{1}{1 + e^{(E_i - E_F)/kT}} )
  • Fermi level (E_F) is the energy level at which there is a 50% probability of finding an electron.
• As temperature increases, the probability of finding electrons above the Fermi level also increases.

Importance of Fermi Level

• The Fermi level is a crucial concept in understanding semiconductor devices, helping to determine the occupancy of states.
• At absolute zero, the probability distribution is binary (0 or 1) below or above the Fermi level.

Upcoming Topics

• Next class will cover Density of States:
  • Describes the number of available energy states per unit volume.
  • Used together with Fermi function to determine the actual number of electrons and holes in semiconductors.