ShaneLogic/carrier-transport-mechanics

Carrier Transport Mechanics

When to Use

• Calculating thermal motion of carriers in semiconductors
• Analyzing drift velocity and current under electric fields
• Determining electrical conductivity and mobility
• Calculating Joule heating from carrier transport
• Analyzing relationships between scattering time, effective mass, and transport properties

Thermal Velocity (Thermal Equilibrium)

Use for carriers in non-degenerate semiconductors (carrier density < 0.1 Nc or 0.1 Nv) at temperature > 0K with no external field.

1. Calculate root-mean-square thermal velocity using equipartition principle:

   • Kinetic energy: 1/2 m* v² = 3/2 kT
   • Formula: v_rms = sqrt(3kT/m*)

2. Understand motion characteristics:

   • Ideal lattice: No scattering, waves not localized
   • Real lattice: Random walk with mean free path ~ several hundred Å

Drift Velocity and Conductivity (Applied Electric Field)

Use when an external electric field is applied to a homogeneous semiconductor.

Sign Conventions

• Electron charge: -e
• Hole charge: +e
• Electric field F: F = -d(Psi)/dx (negative gradient of vacuum level)
• Forces: f_n = -eF (electrons), f_p = +eF (holes)

Calculate Drift Velocity

1. Incremental velocity gain: delta_v = (F * tau) / m*
2. Drift velocities:
   • Electrons: vD_n = -(e * F * tau) / m_n
   • Holes: vD_p = (e * F * tau) / m_p

Calculate Current Density

• j_n = n * (-e) * vD_n = (e² * n * tau / m_n) * F

Calculate Conductivity

• Electron: sigma_n = e² * n * tau / m_n
• For homogeneous: F = V/d, V = I * R
• Resistance: R = d / (A * sigma)
• Specific resistivity: rho_n = 1/sigma_n

Calculate Joule Heating

• Energy per collision: (m*/2) * (delta_v)²
• Events per second: n/tau
• Thermal energy gain: Q = sigma_n * F²

Mobility Principles

Mobility Factors

• Higher when: time between collisions is larger, effective mass is smaller
• Charge signs: -e for electrons, +e for holes
• Mobility signs: negative for electrons, positive for holes
• Conductivity is always positive despite sign of mobility
• Convention: Use magnitude μ = μ_n

Mobility Definition

• μ = ratio of drift velocity to field magnitude
• μ_n = e * tau / m_n

Key Variables

| Variable | Description | |----------|-------------| | v_rms | Root mean square thermal velocity | | k | Boltzmann constant | | T | Absolute temperature | | m* | Effective mass of carrier | | F | Electric field strength | | tau | Mean scattering time (relaxation time) | | sigma | Electrical conductivity | | mu | Carrier mobility | | j | Current density |