Crystal Growth, Silicon Wafer Boules & Dopant Segregation
Czochralski Silicon Ingot Dopant Segregation Profile Formula
Calculate Seed-End Crystal Dopant Concentration C_s(0), Solidified-Fraction Ingot Concentration C_s(g), Enriched Crucible Melt Concentration C_L(g), and Tail-to-Seed Resistivity Drift Ratio.
Why N-type Phosphorus Czochralski boules drift 2.85x in doping from seed to tail—while P-type Boron stays flat!
When pulling a 300 mm single-crystal silicon boule from a rotating quartz crucible at 1,420°C (Czochralski Cz growth), substitutional dopants prefer staying in the liquid silicon melt because their Equilibrium Segregation Coefficient k_0 = C_solid / C_liquid is less than unity (k_0 = 0.80 for Boron, 0.35 for Phosphorus, 0.023 for Antimony!). As fraction g of the melt solidifies (0 <= g < 1) with complete convective melt mixing and negligible solid back-diffusion, rejected dopant enriches the remaining melt according to Pfann’s Normal-Freezing Law: C_s(g) = k_0 * C_0 * ((1 - g)^(k_0 - 1)) and C_L(g) = C_0 * ((1 - g)^(k_0 - 1))! Because k_0 - 1 is negative, ingot doping rises steeply toward the tail end—causing high-fraction wafers to fail tight resistivity specs unless recharge doping is used!
Verify on an N-Type Phosphorus Czochralski Crucible Charge—Initial_Melt_Dopant C_0 = 10.0 (x 10^15 atoms/cm³, B2), Segregation_Coeff k_0 = 0.35 (Phosphorus in Si, B3), and Solidified_Boule_Fraction g = 0.80 (80% of melt pulled, B4): Seed-End Solid Concentration C_s(0) starts at 3.50 x 10^15 cm^-3 (E2), Tail-End Solid Concentration C_s(0.80) climbs to 9.96 x 10^15 cm^-3 (E3), Remaining Liquid Crucible Melt hits 28.47 x 10^15 cm^-3 (E4), and Tail-to-Seed Axial Drift Ratio ((1 - g)^(k_0 - 1)) equals 2.85x (E5)!
| Czochralski Ingot KPI (C0 = 10e15 cm^-3, Phosphorus k0 = 0.35, g = 0.80) | Formula | Verified Output |
|---|---|---|
| Seed-End Ingot Doping Cs(0) | k0 * Initial_Melt_C0 | 3.50 (8.00 @ Boron k0 = 0.80) |
| Ingot Doping Cs(g) at g = 80% | k0 * C0 * ((1 - g)^(k0 - 1)) | 9.96 (11.04 @ Boron k0 = 0.80) |
| Remaining Melt CL(g) at g = 80% | C0 * ((1 - g)^(k0 - 1)) | 28.47 (13.80 @ Boron k0 = 0.80) |
| Tail-to-Seed Drift Ratio Cs(g)/Cs(0) | (1 - g)^(k0 - 1) | 2.85x (1.38x @ Boron k0 = 0.80!) |
Czochralski Normal-Freezing Axial Dopant Concentration Formula
Validate Initial_Melt C_0 > 0, Segregation_Coeff k_0 in (0.0001, 2.0], and Solidified_Fraction g strictly in [0, 0.99) (at g = 1.0, (1 - g)^(k_0 - 1) diverges)!
=LET(c0_m, B2, k0_s, B3, gf_r, B4, IF(OR(c0_m <= 0, k0_s <= 0.0001, k0_s > 2, gf_r < 0, gf_r >= 0.99), "Invalid Cz Pull Input", ROUND(k0_s * c0_m * ((1 - gf_r) ^ (k0_s - 1)), 2)))P-Type Boron Tight Axial Resistivity Comparison Check (Set Boron k_0 = 0.80)
Switch Segregation_Coeff B3 from 0.35 (Phosphorus) to 0.80 (Boron): Seed-End Doping E2 starts at 8.00, 80%-Pulled Doping E3 reaches only 11.04, and Tail-to-Seed Drift Ratio E5 tightens from 2.85x down to just 1.38x—explaining why standard P-type Cz wafers achieve much higher usable boule yield!
=IF(ISTEXT(E3), E3, ROUND((1 - B4) ^ (B3 - 1), 2))Try the example
Use the original values or change them to test your understanding.