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.