Understanding Electron Mobility and Current Flow

Oct 22, 2024

Lecture Notes on Electron and Hole Mobility, Temperature Dependence, and Current Flow

Introduction

  • Recap of last lecture: Introduction of mobility.
  • Focus on electron hole mobility and its dependence on effective mass.

Mobility Concept

  • Definition of Mobility: Slope of velocity versus electric field characteristics.
  • Importance of Mobility:
    • Dictates current flow in devices.
    • Strongly affected by temperature.
    • Changes in mobility impact device performance.

Temperature Dependence of Mobility

  • Mobility is sensitive to temperature changes:
    • Cooling down or heating up a device alters mobility.
  • Range of operational temperature is crucial for devices.

Factors Affecting Mobility

Phonon Scattering

  • Phonons: Vibrating atoms, categorized into acoustic and optical branches.
  • Mobility is affected by phonon vibrations:
    • Higher temperature increases phonon vibrations.
    • Increased vibrations lead to more electron scattering, reducing mobility.
  • Unit of Mobility:
    • Denoted as ( \mu )
    • Measured in ( \text{cm}^2/ ext{V·s} )
  • Typical Values:
    • N-type doped silicon: ( \mu_n \approx 300 \text{ cm}^2/ ext{V·s} )
    • Graphene: Mobility can reach up to 100,000 cm²/V·s.

Ionized Impurity Scattering

  • Ionized impurities (e.g., positively charged ions) scatter electrons.
  • Effect of temperature on ionized impurity scattering:
    • Lower temperatures lead to lower energy electrons, which scatter more around impurities.
    • Higher temperatures allow electrons to overcome scattering better.

Combined Effects

  • Total mobility is influenced by both phonon and ionized impurity scattering:
    • Total mobility equation: [ \frac{1}{\mu_{total}} = \frac{1}{\mu_{phonon}} + \frac{1}{\mu_{impurity}} ]
  • Example Calculation:
    • Given ( \mu_{phonon} = 400 \text{ cm}^2/ ext{V·s} ) and ( \mu_{impurity} = 500 \text{ cm}^2/ ext{V·s} ), the total mobility is approximately 220 cm²/V·s.

Doping Effects on Mobility

  • Low Doping: Phonon scattering dominates, high mobility.
  • High Doping: Ionized impurity scattering dominates, reducing mobility drastically.
  • Plotting mobility versus doping reveals:
    • Initial constant mobility due to low ionized impurities.
    • Mobility decreases significantly at high doping concentrations.

Current Density and Conductivity

  • Current Density (J):
    • Defined by equation: ( J = q n \mu E )
  • Conductivity (σ):
    • Defined as: ( \sigma = q \mu n )
  • Relationship with Ohm's Law:
    • J is proportional to the electric field.

Velocity Saturation

  • Drift velocity increases linearly with field until saturation point (V_sat).
  • Physical reasoning:
    • Increased acceleration leads to higher collision frequency.
    • Energy gained from the field dissipates to phonons, limiting further acceleration.
  • Saturation velocity varies among materials:
    • e.g., Si: ( \sim 2 \times 10^7 \text{ cm/s} )

Diffusion Current

  • Diffusion: Movement of carriers from high to low concentration.
  • Flux of Carriers: Dependent on concentration gradient:
    • ( ext{Flux} = -D \frac{dN}{dx} )
  • Total Diffusion Current:
    • Combination of both electron and hole diffusion currents: [ J = q D_n \frac{dN}{dx} - q D_p \frac{dP}{dx} ]

Conclusion

  • Overview of drift and diffusion currents.
  • Next class: Discuss relation between drift and diffusion and introduction of Einstein relation.

Key Terms:

  • Mobility (( \mu ))
  • Phonon Scattering
  • Ionized Impurity Scattering
  • Current Density (( J ))
  • Conductivity (( \sigma ))
  • Saturation Velocity (( V_{sat} ))
  • Diffusion Current
  • Concentration Gradient

References for Next Class:

  • Review the concepts of drift and diffusion.
  • Study the Einstein relation for charge carriers.