Lecture Notes on Semiconductors

Introduction

• Review of last lecture: Introduction to semiconductors and their importance.
• Concepts discussed: Energy bands, reciprocal space, and E-k diagrams.

Key Takeaways from Previous Lecture

1. Unique Conductivity Properties:
   • Semiconductors can tune conductivity between metals and insulators.
   • Resistance can change by several orders of magnitude through doping and other techniques.
2. Energy Band Gap:
   • Gap between the highest occupied band and the lowest unoccupied band, called the energy band gap.
   • Semiconductors have a lower energy band gap compared to insulators and no gap in metals.

Crystal Structure of Semiconductors

• Semiconductors are crystalline materials with periodic atomic arrangements.
• Properties depend on the crystal structure.
• The crystal introduces a periodic potential through its atomic arrangement.

E-k Diagrams

• The relation between energy (E) and momentum (k) for free electrons is:

  [ E = \frac{\hbar^2 k^2}{2m} ]

• In a crystal, this relation becomes distorted due to the periodic potential.

• The mass of electrons in this context is the effective mass (m*), different from the free electron mass (m0).*

Understanding Electron Movement in Crystals

• Electrons move through a periodic potential created by the arrangement of atoms, which affects their energy and momentum relations.
• Schrodinger's Equation is used to derive the behavior of electrons in periodic potentials.

Effective Mass

• Effective mass is defined as:

  [ m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}} ]

• Inversely proportional to the curvature of the E-k diagram.

• Low effective mass leads to high electron mobility, crucial for device performance.*

Brillouin Zone

• Discussion on the first Brillouin zone:
  • Most electron transport occurs within the first Brillouin zone (k values between -π/a to π/a).
  • Understanding the E-k diagram is fundamental to analyzing electron transport and device functionality.

Impact of Effective Mass on Device Performance

• Materials with lower effective mass have higher electron mobility, allowing for better current transport in devices.
• Different materials have different effective masses, which can also vary with direction in anisotropic crystals.

Energy Band Gap

• The energy band gap (E_g) is a critical property of semiconductors, ranging from 0.1 eV to 6 eV.
• Example: Silicon has a band gap of approximately 1.1 eV.
• Temperature dependence of the energy band gap.

Upcoming Topics

• Next lecture will introduce the concept of holes created when electrons move from the valence band to the conduction band.
• Further discussions on electron and hole distribution statistics and implications for device understanding.