Semiconductor Back-End-of-Line (BEOL), CMP Polishing & Metrology
Preston Equation CMP Material Removal Rate (MRR) Formula
Calculate CMP Material Removal Rate MRR (nm/min) from Slurry Preston Coefficient, Downforce Pressure (kPa) & Relative Velocity (m/s), Polish Time (sec), and WIWNU Thickness Spread (nm).
Convert Preston K_p (10^-13 Pa^-1) and kPa cleanly—6 * K_p * P_kPa * V_m_s gives exact nm/min!
Chemical-Mechanical Planarization (CMP) globally flattens dielectric interlayer oxides (ILD), shallow-trench isolation (STI), and copper damascene interconnects so EUV lithography stays inside its sub-100 nm Depth of Focus! Under Preston’s empirical wear law MRR = K_p * P * V, removal velocity (m/s) equals Slurry-Pad Preston Coefficient K_p (quoted in units of 10^-13 Pa^-1) times Head Downforce Pressure P (Pa) times Relative Pad-Wafer Linear Velocity V (m/s). Converting 10^-13 Pa^-1 * (P_kPa * 10^3 Pa/kPa) * V * (10^9 nm/m) * (60 s/min) collapses all powers of ten into one exact constant: MRR_nm_min = 6 * K_p_1e13 * P_kPa * V_m_s!
Verify on an Oxide ILD Fumed-Silica CMP Platen—Preston_Coeff K_p = 1.50 (x 10^-13 Pa^-1, B2), Carrier_Downforce P = 25.0 kPa (~3.63 psi, B3), Relative_Pad_Speed V = 1.20 m/s (B4), Target_Step_Removal = 540.0 nm (B5), and Within-Wafer Non-Uniformity WIWNU = 3.0% (0.03, B6): Material Removal Rate hits 6 * 1.50 * 25.0 * 1.20 = 270.00 nm/min (E2 = 4.50 nm/s in E3), Endpoint Polish Cycle Time equals (540 / 270) * 60 = 120.00 seconds (E4), and 1-Sigma Edge-to-Center Thickness Spread equals 540 * 0.03 = 16.20 nm (E5)!
| CMP Platen KPI (Kp = 1.5e-13 Pa^-1, P = 25 kPa, V = 1.2 m/s, 540 nm Target) | Formula | Verified Output |
|---|---|---|
| Material Removal Rate (nm/min) | 6 * Kp_1e13 * Pressure_kPa * Vel_m_s | 270.00 nm/min (324.00 @ 30 kPa) |
| Instantaneous Polish Rate (nm/s) | MRR_nm_min / 60 | 4.50 nm/s (5.40 @ 30 kPa) |
| Endpoint Polish Time (seconds) | (Target_Removal_nm / MRR_nm_min) * 60 | 120.00 s (100.00 s @ 30 kPa) |
| WIWNU 1-Sigma Removal Spread (nm) | Target_Removal_nm * WIWNU_Frac | 16.20 nm |
Exact Preston CMP Material Removal Rate & Endpoint Time Formula
Validate K_p > 0, Pressure_kPa in (1, 100] (ultra-low-k porous dielectrics delaminate above ~35 kPa!), Velocity_m_s > 0, Target_nm > 0, and WIWNU in [0, 0.25]!
=LET(kp_c, B2, pr_k, B3, vl_m, B4, tg_n, B5, wn_u, B6, IF(OR(kp_c <= 0, pr_k <= 1, pr_k > 100, vl_m <= 0, tg_n <= 0, wn_u < 0, wn_u > 0.25), "Invalid CMP Recipe", ROUND(6 * kp_c * pr_k * vl_m, 2)))Higher Downforce Throughput vs Low-k Shear Check (Set Downforce P = 30.0 kPa)
Raise Carrier_Downforce B3 from 25.0 kPa to 30.0 kPa: Material Removal Rate E2 increases 20% to 324.00 nm/min (5.40 nm/s in E3), cutting Endpoint Polish Cycle Time E4 from 120.00 s down to 100.00 s!
=IF(ISTEXT(E2), E2, ROUND((B5 / (6 * B2 * B3 * B4)) * 60, 2))Try the example
Use the original values or change them to test your understanding.