Engineering & Science
Caughey-Thomas Doping-Dependent Mobility & Wafer Resistivity (Si)
Calculate semiconductor carrier mobility μ(N) = μ_min + (μ_max - μ_min) / (1 + (N / N_ref)^α) (cm^2/(V·s)) and doped wafer resistivity ρ = 1 / (q * N * μ) across five orders of dopant concentration.
Why heavily doping silicon 10,000x does NOT boost conductivity 10,000x!
In lightly doped silicon (N = 10^14–10^15 cm^-3), electrons scatter mainly off lattice phonons and enjoy near-intrinsic mobility μ_max = 1,330 cm^2/(V·s). As phosphorus or arsenic donor concentration climbs past N_ref = 8.5 × 10^16 cm^-3, Coulombic ionized-impurity scattering chokes carrier drift until mobility plateaus near μ_min = 65 cm^2/(V·s)! D. M. Caughey and R. E. Thomas (Proceedings of the IEEE, 1967) captured the entire curve in one TCAD standard formula: μ(N) = μ_min + (μ_max - μ_min) / (1 + (N / N_ref)^α)!
Watch 300 K n-type silicon (μ_min = 65, μ_max = 1,330 cm^2/(V·s), N_ref = 8.5 × 10^16 cm^-3, α = 0.72): at N = 1.0 × 10^15 cm^-3, μ = 1,280.4 cm^2/(V·s) and wafer resistivity ρ = 1 / (q * N * μ) = 4.8747 Ω·cm; at N = N_ref (8.5 × 10^16 cm^-3), mobility sits exactly halfway between μ_min and μ_max at 697.5 cm^2/(V·s); at degenerate N+ source/drain doping (1.0 × 10^19 cm^-3), μ plunges 12-fold to 104.6 cm^2/(V·s)!
| n-Silicon Donor Concentration N (cm^-3) | Caughey-Thomas Mobility μ (cm^2/V·s) | Wafer Resistivity ρ (Ω·cm) | CMOS Device Region |
|---|---|---|---|
| Light Epi / Substrate (1.0 × 10^15) | 1,280.4 | 4.8747 Ω·cm | Phonon-Limited Drift Region |
| Half-Mobility Pivot N_ref (8.5 × 10^16) | 697.5 | 0.1053 Ω·cm | LDD / Well Implant Transition |
| Heavy N+ Source/Drain (1.0 × 10^19) | 104.6 | 0.0006 Ω·cm | Ionized-Impurity Plateau |
Caughey-Thomas Doping-Dependent Carrier Mobility μ(N) (cm^2/(V·s))
Evaluates μ_min + (μ_max - μ_min) / (1 + (N / N_ref)^α) across arbitrary donor or acceptor concentrations.
=IF(OR(B2 <= 0, B3 <= B2, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Mobility Inputs", ROUND(B2 + (B3 - B2) / (1 + (B6 / B4) ^ B5), 1))Extrinsic Wafer Electrical Resistivity ρ = 1 / (q * N * μ(N)) (Ω·cm)
Uses elementary charge q = 1.60217663E-19 C alongside unrounded Caughey-Thomas mobility to predict four-point-probe resistivity.
=IF(OR(B2 <= 0, B3 <= B2, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Mobility Inputs", ROUND(1 / (1.60217663E-19 * B6 * (B2 + (B3 - B2) / (1 + (B6 / B4) ^ B5))), 4))Try the example
Use the original values or change them to test your understanding.