Semiconductor Thin-Film Deposition & Vacuum Metrology
Sauerbrey QCM Thin-Film Mass & Frequency Shift Formula
Calculate Vacuum PVD/ALD Thin-Film Areal Mass Density (ug/cm^2 = 0.1 * t_nm * rho_g_cm3), AT-Cut Quartz Sauerbrey Sensitivity C_f = 2.264 * f_0_MHz^2, Crystal Resonance Drop Delta_f (Hz), and 2% Crystal Replacement Gate.
Convert sub-nanogram vacuum PVD thickness into quartz crystal frequency shifts via Sauerbrey’s Law!
Inside physical vapor deposition (PVD) evaporators, sputter chambers, and atomic layer deposition (ALD) tools, real-time angstrom-level film thickness is controlled by an AT-cut piezoelectric Quartz Crystal Microbalance (QCM)! Guenter Sauerbrey (1959, Zeitschrift fuer Physik) showed that rigid thin-film mass Delta_m deposited per square centimeter lowers the crystal’s shear-mode resonance frequency f_0 proportionally to f_0^2: Delta_f_Hz = -C_f * Areal_Mass_ug_cm2, where AT-cut quartz sensitivity is C_f = 2.264 * (f_0_MHz ^ 2) Hz*cm^2/ug (81.504 Hz*cm^2/ug for a standard 6.0 MHz monitor crystal) and Areal_Mass_ug_cm2 = 0.1 * Thickness_nm * Film_Density_g_cm3! Crucially, linear Sauerbrey proportionality holds ONLY while cumulative frequency drop stays <= 2% of f_0—beyond 2%, acoustic impedance mismatch requires Z-match correction or crystal replacement!
Verify on a 100 nm Gold (Au) E-Beam Evaporation Run—Fundamental_Crystal_Freq f_0 = 6.0 MHz (B2), Target_Film_Thickness = 100 nm (B3), Gold_Film_Density rho = 19.30 g/cm^3 (B4), and Exposed_Sensor_Area = 0.50 cm^2 (B5): Areal Mass Density is 0.1 * 100 * 19.30 = 193.00 ug/cm^2 (96.50 ug on the 0.50 cm^2 aperture, E2); 6 MHz Sauerbrey Sensitivity C_f is 2.264 * 36 = 81.504 Hz*cm^2/ug (E3); Resonance Frequency Shift is -81.504 * 193.00 = -15,730.27 Hz (-0.262% of 6 MHz, E4); confirming "Valid Linear Sauerbrey Regime" (E5)!
| Sauerbrey QCM Metric (6.0 MHz AT-Quartz, 100 nm Au @ 19.3 g/cm^3, 0.50 cm^2) | Formula | Verified Output |
|---|---|---|
| Thin-Film Areal Mass (ug/cm^2) | 0.1 * Thickness_nm * Density_g_cm3 | 193.00 ug/cm^2 (1,930 @ 1,000 nm) |
| AT-Cut Quartz Sensitivity C_f | 2.264 * (f_0_MHz ^ 2) | 81.504 Hz*cm^2/ug |
| Resonance Frequency Shift (Hz) | -C_f * Areal_Mass_ug_cm2 | -15,730.27 Hz (-157,302.72 @ 1 um) |
| 2% Crystal Life / Linearity Gate | ABS(Delta_f) / (f_0_MHz * 10^6) <= 0.02 | Valid Linear Sauerbrey Regime |
Sauerbrey QCM Resonance Frequency Shift (Hz) Formula
Validate f_0 > 0, Thickness_nm > 0, Film_Density > 0, and Sensor_Area > 0!
=LET(f_0m, B2, th_n, B3, rh_f, B4, ar_s, B5, IF(OR(f_0m <= 0, th_n <= 0, rh_f <= 0, ar_s <= 0), "Invalid QCM Deposition Spec", LET(ms_a, 0.1 * th_n * rh_f, cf_q, 2.264 * (f_0m ^ 2), ROUND(-cf_q * ms_a, 2))))1-Micron Gold Overloaded Crystal Check (Set Thickness = 1,000 nm)
Deposit 1,000 nm (1.0 um) of gold on the same crystal (B3 = 1000): Delta_f reaches -157,302.72 Hz (2.62% > 2.0% of 6 MHz), switching E5 to "Replace Crystal: >2% Nonlinear Shift"!
=IF(ISTEXT(E4), E4, IF(ABS(E4) / (B2 * 1000000) > 0.02, "Replace Crystal: >2% Nonlinear Shift", "Valid Linear Sauerbrey Regime"))Try the example
Use the original values or change them to test your understanding.