Engineering & Science
Conwell-Weisskopf Ionized-Impurity Mobility & Matthiessen Total μ
Calculate Coulombic ionized-impurity carrier mobility μ_CW (cm^2/(V·s)) via Esther Conwell and Victor Weisskopf’s screened T^(3/2)/N_I Rutherford formula and combine with phonon mobility μ_L by Matthiessen’s Rule.
Why ionized-impurity mobility INCREASES as T^(3/2)—opposite to phonon scattering!
Unscreened Rutherford scattering off fixed ionized donors/acceptors diverges at large impact parameters until Esther Conwell and Victor F. Weisskopf (Physical Review, 1950) truncated Coulomb deflection at half the mean inter-impurity spacing r_m = 0.5 * N_I^(-1/3)! That yielded the foundational analytical mobility law: μ_CW = (3.2874 × 10^15 * ε_r^2 * T^1.5) / (N_I * SQRT(m*/m_0) * LN(1 + u^2)), where screening parameter u = 1795.3 * ε_r * T / N_I^(1/3)!
Notice the striking T^1.5 temperature signature: hotter carriers zip past charged donor ions much faster, suffering far smaller Coulomb deflection angles! In n-silicon (ε_r = 11.7, conductivity effective mass m*/m_0 = 0.26, lattice phonon mobility μ_L = 1,400 cm^2/(V·s) at 300 K): at N_I = 1.0 × 10^17 cm^-3, μ_CW = 8,781.7 cm^2/(V·s) and Matthiessen combined mobility 1 / (1/μ_L + 1/μ_CW) = 1,207.5 cm^2/(V·s); raising N_I 10x to 1.0 × 10^18 cm^-3 slashes μ_CW to 1,237.3 and total mobility to 656.8 cm^2/(V·s)!
| Silicon Doping & Temperature State (εr=11.7, m*=0.26) | Screening Ratio u | Conwell-Weisskopf μ_CW | Matthiessen Total μ (cm^2/V·s) |
|---|---|---|---|
| Moderate Doping (300 K, NI = 10^17, μL = 1,400) | 13.58 | 8,781.7 | 1,207.5 (Phonon Dominated) |
| Heavy Doping (300 K, NI = 10^18, μL = 1,400) | 6.30 | 1,237.3 | 656.8 (Impurity Comparable!) |
| Cooled Cryostat (150 K, NI = 10^17, μL = 4,500) | 6.79 | 4,209.3 (Drops as T^1.5!) | 2,174.9 |
Conwell-Weisskopf Ionized-Impurity Mobility μ_CW (cm^2/(V·s))
Evaluates (3.2874E15 * ε_r^2 * T^1.5) / (N_I * SQRT(m*) * LN(1 + (1795.3 * ε_r * T / N_I^(1/3))^2)) for singly charged dopants.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Semiconductor Inputs", ROUND((3.2874E15 * (B2 ^ 2) * (B4 ^ 1.5)) / (B5 * SQRT(B3) * LN(1 + ((1795.3 * B2 * B4 / (B5 ^ (1 / 3))) ^ 2))), 1))Matthiessen Combined Phonon + Impurity Mobility μ_tot (cm^2/(V·s))
Combines lattice acoustic-phonon mobility B6 with μ_CW via 1 / (1 / μ_L + 1 / μ_CW).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Semiconductor Inputs", ROUND(1 / ((1 / B6) + 1 / ((3.2874E15 * (B2 ^ 2) * (B4 ^ 1.5)) / (B5 * SQRT(B3) * LN(1 + ((1795.3 * B2 * B4 / (B5 ^ (1 / 3))) ^ 2))))), 1))Try the example
Use the original values or change them to test your understanding.