ShaneLogic/anisotropic-mobility-calculation

Anisotropic Mobility Calculation

When to Use

• Calculating mobility in anisotropic semiconductors (Ge, Si)
• Working with multi-valley semiconductors having ellipsoidal constant energy surfaces
• Applying ionized impurity scattering models where band structure anisotropy significantly affects results
• Determining directional mobility differences in crystals

Prerequisites

Verify the following before proceeding:

• Effective mass anisotropy ratio (m_n∥/m_n⊥) is known for the material
• Carrier density (n) is available
• Screening length is substantially smaller than mean free path
• Low-angle scattering events dominate (typical for ionized impurity scattering)

Calculation Procedure

1. Identify Material Parameters

• Determine effective mass ratio m_n∥/m_n⊥ for the semiconductor
• Common values: Ge = 19, Si = 5.2
• Identify carrier density n

2. Calculate Screening Length

• Compute Debye screening length: λ_D ∝ 1/√n
• Verify: λ_D << mean free path (ensures no successive collisions in same defect region)

3. Apply Anisotropy Factor

• Calculate anisotropy factor K_a = μ∥/μ⊥
• Use screened Yukawa potential model
• Note: K_a decreases with increasing carrier density

4. Compute Direction-Dependent Mobility

• Apply anisotropy-corrected mobility formula (Samoilovich et al. 1961)
• Account for density of states variation along valley ellipsoid axes
• Factor in increased randomization with higher carrier densities

5. Validate Results

• Compare measured mobility anisotropy with calculated values
• Reference experimental data at relevant temperatures (e.g., 77K measurements for i-Ge)

Key Relationships

• Parallel mobility (μ∥) and perpendicular mobility (μ⊥) differ due to effective mass anisotropy
• Higher carrier density → smaller scattering cross-section → more randomizing ion scattering
• Density of states is largest along long axes of valley ellipsoids

Material-Specific Values

See references/material-parameters.md for detailed values and formulas.