Carrier Transport

Drude

• Based on kinetic theory of gas molecules
  • Parallels to electron movement
• Move in straight line until collision
• No forces acting during travel
• Time between collisions
  • Relaxation time
  • Mean free time
  • Scattering time
• Collisions randomise velocity

Assumptions

• Metal is formed of positive ions
  • Number of delocalised free electrons
• Without EM field electrons move in straight line
  • Only interaction is via collisions
• Electron-ion interactions usually ignored
  • Free electron approximation
  • Not particularly accurate
  • Nearly free electron approximation is better
    • Weak interactions
    • Perturbations
• Electron-electron interactions ignored
  • Single particle picture
  • Independent electron approximation
  • Hard to model because
    • Don’t know wavefunctions of every electron
    • Potential from interactions is not periodic
    • Need to consider dynamics of all electrons at once
  • Often weaker than electron-ion
    • Parallel spins avoid each other
      • Pauli exclusion
    • Opposite spins avoid each other
      • Reduce energy
      • Coulomb repulsion
    • Electrons distribute around ions
      • Screen ions from other electrons
• Mean free time between collisions is Tau
  • Probability of collision per unit time
    • 1 over Tau
    • Relaxation rate
  • Typically 0.1 ps
  • $10^{13}$ - $10^{14}$ times per second
  • Achieve thermal equilibrium via collisions with lattice
    • Emerge with random direction
      • Speed based on temperature of collision area
        • Two temperatures
          • Temperature of atoms
            • What you measure
          • Effective temperature of electrons
            • Via Boltzmann’s
            • Transfer small amount of energy to lattice