Lecture Notes

Overview

• Recap of the previous lecture:

  • Derived Einstein relation between drift and diffusion mobility and diffusion coefficient.
  • Introduced recombination, traps, electron capture, emission, and recombination rate expression.

• Importance of Recombination Rate:

  • Influences device operation (LEDs, photodetectors).
  • Leads to recombination current or generation recombination current.

Key Corrections

• Correction of slide typo:

  • Lifetime (τ) defined as (\frac{1}{\sigma \cdot \text{thermal velocity} \cdot \text{total trap density}}).

• Assumptions:

  • Electron and hole trap cross-section are the same.
  • Thermal velocity is the same.
  • Trap energy level assumed to be at mid-gap (maximizes recombination rate).

Total Recombination Rate Expression

• Formula: (U = \frac{1}{\tau} \times \frac{Pn - n_i^2}{P + n + 2n_i})

  • (n_i) is the intrinsic carrier concentration.
  • (\tau) is the lifetime: larger τ means slower recombination.

• **Assumptions: **

  1. Equal cross-section of electron and hole traps.
  2. Equal thermal velocity.
  3. Trap energy level at mid-gap.

Recombination in N-type Semiconductors

• Excess holes matter more than excess electrons in n-type semiconductors.
• Ignoring minor components simplifies the equation to (U \approx \frac{1}{\tau} \times \Delta P).
• Important for calculating excess carrier generation: (\Delta P = U \times \tau).

Shockley-Reed-Hall (SRH) Recombination

• SRH recombination statistics: Trap-related recombination.
• Significant in explaining how traps affect carrier recombination.

Light Absorption and Carrier Generation

• Light energy (h\nu) must be (\geq E_g) (bandgap) for absorption.
• Wavelength condition: (\lambda \leq \frac{1242}{E_g}).

Continuity Equation

• Concept: Explains carrier distribution and current flow in semiconductors.
• Expression: (\frac{dn}{dt} = \frac{1}{q} \frac{dJ_n}{dx} + G_L - \frac{\Delta n}{\tau})
• For steady state: (0 = D_n \frac{d^2\Delta n}{dx^2} + G_L - \frac{\Delta n}{\tau})

Solving Continuity Equation

• Without field:

  • Diffusion length (L_n = \sqrt{D_n \cdot \tau}).
  • Solutions:
    • Long semiconductor: Exponential decay.
    • Short semiconductor: Linear decay.

• Implications: Affects diffusion current, critical in device applications (e.g., P-N junctions).

Conclusion

• Upcoming focus on continuity equation applications and P-N junction.
• Understanding these concepts vital for device applications like LEDs, photodetectors, and solar cells.