Semiconductor Device Physics, Ion Implantation & CMOS Doping
Gaussian Ion Implantation Peak Dopant & Junction Depth Formula
Calculate Peak Implanted Dopant Concentration N_p (10^18 cm^-3), Wafer Surface Concentration N(0) (10^18 cm^-3), Metallurgical p-n Junction Depth x_j (nm), and Peak-to-Background Ratio.
Quadrupling implant dose quadruples peak concentration—yet logarithmic Gaussian tails barely budge junction depth x_j!
When energetic Boron, Phosphorus, or Arsenic ions are accelerated into an amorphized or tilted silicon wafer to prevent crystal channeling, Lindhard-Scharff-Schiott (LSS) stopping theory models depth concentration N(x) as a Gaussian centered at Projected Range R_p (nm) with Longitudinal Straggle Delta_R_p (nm = 1e-7 cm): Peak Concentration at x = R_p is N_p = Phi / (SQRT(2 * PI()) * Delta_R_p)! Expressing Implant Dose Phi in units of 10^14 ions/cm² and Delta_R_p in nm yields N_p (in 10^18 cm^-3) = (1,000 * Phi_1e14) / (SQRT(2 * PI()) * Delta_Rp_nm)! Setting N(x_j) equal to opposite-polarity Background Substrate Doping N_B (in 10^16 cm^-3 = 0.01 * 10^18 cm^-3) solves Metallurgical p-n Junction Depth x_j = R_p + Delta_R_p * SQRT(2 * LN((N_p * 100) / N_B_1e16))!
Verify on a Medium-Dose Phosphorus Well/Extension Implant—Implant_Dose Phi = 5.0 (x 10^14 cm^-2, B2), Projected_Range R_p = 120.0 nm (B3), Longitudinal_Straggle Delta_R_p = 40.0 nm (B4), and P-Type_Substrate_Doping N_B = 1.0 (x 10^16 cm^-3, B5): Peak Concentration N_p hits 49.87 x 10^18 cm^-3 (4.987e19 cm^-3, E2), Wafer Surface Concentration N(0) at x = 0 equals 0.554 x 10^18 cm^-3 (E3), Peak-to-Substrate Ratio equals 4,986.78 (E4), and Metallurgical Junction Depth x_j lands at 285.07 nm (E5)!
| Ion Implant KPI (Dose = 5e14 cm^-2, Rp = 120 nm, dRp = 40 nm, NB = 1e16 cm^-3) | Formula | Verified Output |
|---|---|---|
| Peak Dopant N_p (10^18 cm^-3) | (Dose_1e14 * 1000) / (SQRT(2*PI()) * dRp_nm) | 49.87 (199.47 @ Dose = 20e14) |
| Surface Dopant N(0) (10^18 cm^-3) | N_p * EXP(-(Rp^2) / (2 * dRp^2)) | 0.554 (2.216 @ Dose = 20e14) |
| Peak-to-Substrate Ratio (Np / NB) | (N_p_1e18 * 100) / NB_1e16 | 4,986.78 |
| Metallurgical Junction x_j (nm) | Rp + dRp * SQRT(2 * LN((Np*100)/NB)) | 285.07 nm (298.00 nm @ 20e14!) |
Gaussian LSS Peak Dopant & Metallurgical p-n Junction Depth Formula
Validate Dose > 0, Projected_Range R_p >= 0, Straggle Delta_R_p > 0, Background N_B > 0, and verify Peak Concentration exceeds Background Doping (N_p * 100 > N_B) so a p-n junction actually forms!
=LET(ds_i, B2, rp_n, B3, dr_n, B4, nb_s, B5, IF(OR(ds_i <= 0, rp_n < 0, dr_n <= 0, nb_s <= 0), "Invalid Implant Input", LET(np_k, (ds_i * 1000) / (SQRT(2 * PI()) * dr_n), IF(np_k * 100 <= nb_s, "No Junction Formed", ROUND(rp_n + dr_n * SQRT(2 * LN((np_k * 100) / nb_s)), 2)))))4x Higher Source/Drain Dose Logarithmic Junction Insensitivity Check (Set Dose = 20.0 x 10^14)
Quadruple Implant_Dose B2 from 5.0 to 20.0 (2.0e15 cm^-2): Peak Concentration E2 quadruples from 49.87 to 199.47 (x 10^18 cm^-3), yet Metallurgical Junction Depth E5 creeps deeper by just 12.93 nm—from 285.07 nm to 298.00 nm—because SQRT(2 * LN(...)) compresses dose changes!
=IF(ISTEXT(E5), E5, ROUND(((B2 * 1000) / (SQRT(2 * PI()) * B4)) * EXP(-(B3 ^ 2) / (2 * (B4 ^ 2))), 3))Try the example
Use the original values or change them to test your understanding.