Engineering & Science
Brooks-Herring Screened Ionized Impurity Scattering Factor Formula
Compute the Debye-screened Brooks-Herring parameter b = 1.2925E14 * ε_r * (m*/m_0) * T^2 / n and Coulomb scattering function f_BH(b) = LN(1 + b) - b / (1 + b) in doped semiconductors.
How mobile carriers screen Coulomb impurity potentials via Debye screening!
Whereas Conwell and Weisskopf truncated bare Rutherford Coulomb scattering artificially at half the inter-dopant spacing, Harvey Brooks (1951) and Conyers Herring replaced the bare 1/r potential with a Yukawa Debye-screened potential exp(-r / L_D) / r! Integrating Born-approximation momentum relaxation yields the dimensionless Brooks-Herring screening parameter b = 24 * m* * ε_r * ε_0 * (k_B * T)^2 / (ħ^2 * q^2 * n) = 1.2925 × 10^14 * ε_r * (m*/m_0) * T^2 / n (with screening carrier density n in cm^-3) and scattering factor f_BH(b) = LN(1 + b) - b / (1 + b)!
Watch n-type silicon (ε_r = 11.7, conductivity effective mass m*/m_0 = 0.26) at T = 300 K: at N = n = 1.0 × 10^17 cm^-3, b = 353.86 and f_BH = 4.8745; raising n tenfold to 1.0 × 10^18 cm^-3 tightens the Debye screening length, cutting b tenfold to 35.39 and shrinking the Coulomb scattering integral f_BH to 2.6217!
| n-Si Carrier Density n (300 K, εr=11.7, m*=0.26) | Screening Ratio b | Brooks-Herring Factor f_BH(b) | Physical Screening Regime |
|---|---|---|---|
| 1.0 × 10^17 cm^-3 (Moderate Doping) | 353.86 | 4.8745 | Weak Screening (Long Debye Tail) |
| 1.0 × 10^18 cm^-3 (Heavy Doping) | 35.39 | 2.6217 | Strong Free-Carrier Screening |
| Invalid n <= 0 or T <= 0 | Undefined | "Invalid Semiconductor Inputs" | Physical Positivity Guard |
Brooks-Herring Coulomb Scattering Integral f_BH(b) = LN(1 + b) - b / (1 + b)
Evaluates b = 1.2925E14 * ε_r * m_eff * T^2 / n_cm3 and returns the dimensionless Brooks-Herring logarithmic factor f_BH(b).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Semiconductor Inputs", ROUND(LN(1 + (1.2925E14 * B2 * B3 * (B4 ^ 2) / B5)) - (1.2925E14 * B2 * B3 * (B4 ^ 2) / B5) / (1 + (1.2925E14 * B2 * B3 * (B4 ^ 2) / B5)), 4))Dimensionless Debye Screening Parameter b
Returns b = (2 * k_th * L_D)^2 at 3*k_B*T thermal energy.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Semiconductor Inputs", ROUND(1.2925E14 * B2 * B3 * (B4 ^ 2) / B5, 2))Try the example
Use the original values or change them to test your understanding.