When should I use the Z-match method instead of Sauerbrey?

Use the Z-match method when your total frequency change exceeds 2% of the crystal's fundamental frequency (e.g., >120 kHz for a 6 MHz crystal). For thin films below this threshold, the standard Sauerbrey equation provides accurate results. Z-match accounts for acoustic impedance mismatch between the quartz crystal and deposited film, which becomes significant for thicker coatings.

What is the tooling factor and how do I calibrate it?

The tooling factor corrects for the geometric difference between where the QCM sensor sits versus where your substrate is positioned. Due to the cosine distribution of evaporated material, different locations receive different amounts. Calibrate by depositing a reference film, measuring actual thickness with profilometry or ellipsometry, then dividing by the QCM-indicated thickness. Typical values range from 80-120%.

How do I know when to replace my QCM crystal?

Replace QCM crystals when the total frequency shift reaches 3-5% of the fundamental frequency. For a 6 MHz crystal, this means replacement around 180-300 kHz total frequency change. Signs of crystal degradation include unstable readings, increased noise, or failure to oscillate. High-stress films or reactive depositions may require earlier replacement.

Why does my calculated thickness differ from profilometry measurements?

Common causes include: incorrect tooling factor calibration, film density differing from bulk values (common in porous or amorphous films), temperature-induced frequency drift, or using Sauerbrey for thick films where Z-match is needed. For reactive depositions, the actual film composition may differ from the target material density.

PVD Deposition Rate & Thickness Calculator

Calculate coating thickness from QCM frequency change using the Sauerbrey equation with tooling factor correction

Crystal & Material Parameters

⚡ Auto-Update

Crystal Fundamental Frequency (F_q) Common: 5 or 6 MHz

Frequency must be greater than zero

Film Material Select preset or enter custom density

ρ_f = film material density. Use bulk density for crystalline films; may vary for porous or amorphous films.

Density must be greater than zero

Z-Ratio (Acoustic Impedance Ratio) Use Z≠1 for thick films

Z = (ρ_qμ_q)^½ / (ρ_fμ_f)^½. Z=1 uses Sauerbrey; Z≠1 uses Lu-Lewis Z-match equation.

Z=1 applies standard Sauerbrey equation. For films >2% of crystal thickness, select the material-specific Z-ratio for improved accuracy.

Z-ratio must be greater than zero

Measurement Data

Frequency Change (ΔF) Negative = mass increase

Include uncertainty ±

Enter as negative value for deposition. Positive ΔF indicates mass loss (etching/desorption).

Frequency change must be non-zero

Tooling Factor (TF) Typically 80–120%

Corrects for geometric differences between crystal and substrate positions. Calibrate with step profilometry.

Tooling factor must be greater than zero

Deposition Time Optional — for rate calculation

Enter deposition time to calculate average deposition rate.

Time must be greater than zero

Crystal constants: Using AT-cut quartz with ρ_q = 2.648 g/cm³, μ_q = 2.947×10¹¹ g/(cm·s²)

Results

Film Thickness (t)

— nm

Deposition Rate

— Å/s

Mass/Area (Δm/A)

— µg/cm²

Crystal Life Usage 0%

Enter frequency change to see crystal usage

Tooling-corrected ΔF: —

Calibration tip: Verify tooling factor by depositing a reference film and measuring actual thickness with profilometry, ellipsometry, or AFM.

Understanding the Sauerbrey Equation

The Sauerbrey equation, developed by Günter Sauerbrey in 1959, relates the frequency change of a quartz crystal microbalance (QCM) to the mass deposited on its surface. It forms the basis for in-situ thickness monitoring in PVD (Physical Vapor Deposition) systems including thermal evaporation, e-beam evaporation, and sputtering.

Key assumption: The Sauerbrey equation assumes the deposited film is rigid, uniformly distributed, and thin compared to the crystal thickness. The film must also have negligible slip at the crystal-film interface.

The Fundamental Equation

The mass-frequency relationship is derived from the resonance condition of an AT-cut quartz crystal:

Δf = −(2F_q²/A√(ρ_qμ_q)) × Δm

Where the sensitivity factor C_f = 2F_q²/√(ρ_qμ_q) is characteristic of the crystal. For a 6 MHz crystal, C_f ≈ 8.15 ng/(cm²·Hz).

Converting to Film Thickness

Since mass = density × volume, and volume = area × thickness, we can express film thickness as:

t = −(Δf × √(ρ_qμ_q)) / (2F_q² × ρ_f)

Tooling Factor Correction

The tooling factor accounts for geometric differences between the crystal sensor and substrate positions in the deposition chamber. Due to:

Different distances from the evaporation source
Angular distribution of the vapor flux (cosine law)
Shadowing effects and chamber geometry

The actual substrate thickness typically differs from the crystal reading. The tooling factor is defined as:

TF = (Actual substrate thickness / Crystal-measured thickness) × 100%

Calibration essential: Tooling factors must be experimentally determined for each source-substrate-crystal geometry using a traceable thickness measurement method (profilometry, ellipsometry, XRR, or AFM).

Z-Match Method for Thick Films

For films thicker than ~2% of the crystal thickness, acoustic impedance mismatch becomes significant. The Z-match equation provides improved accuracy:

t = (N_qρ_q)/(πZρ_fF_q) × arctan[Z × tan(π(F_q−F)/F_q)]

Where Z is the acoustic impedance ratio. This calculator uses the linear Sauerbrey approximation but allows Z-ratio input for reference.

Limitations and Validity

Film rigidity: Soft, viscoelastic, or liquid-like films violate the rigid-film assumption
Mass loading: Accuracy decreases for Δf/F_q > 2% (roughly >200 nm for typical materials)
Uniformity: Non-uniform films cause frequency instability and measurement errors
Temperature: Crystal frequency drifts with temperature (~±2 ppm/°C for AT-cut). Water-cooled crystal holders are recommended for high-temperature depositions. Allow crystals to stabilize before measuring.
Stress effects: Highly stressed films can shift frequency independent of mass

Crystal Lifetime

QCM crystals have finite operational life. As mass accumulates, oscillation quality degrades:

0–2% loading: Excellent accuracy, Sauerbrey equation valid
2–3% loading: Use Z-match for best results; some accuracy loss
3–5% loading: Crystal nearing end of life; increased noise likely
>5% loading: Replace crystal; oscillation may fail or become unstable

Monitor the Crystal Life indicator above to track cumulative loading for the current measurement.

Material Density Reference

Common PVD coating materials and their bulk densities. Actual film density may vary based on deposition conditions.

Material	Density (g/cm³)	Z-Ratio	Common Applications
Gold (Au)	19.30	0.381	Electrodes, contacts, optical
Silver (Ag)	10.50	0.529	Mirrors, antimicrobial
Aluminum (Al)	2.70	1.08	Reflectors, interconnects
Copper (Cu)	8.90	0.437	Interconnects, seed layers
Chromium (Cr)	8.25	0.296	Adhesion layers, hard coatings
Titanium (Ti)	4.50	0.628	Adhesion layers, biomedical
Platinum (Pt)	21.45	0.245	Electrodes, catalysis
SiO₂	2.20	1.07	Dielectrics, optical
TiO₂	4.23	0.48	AR coatings, photocatalysis
ITO	5.61	—	Transparent electrodes

Note: Film density can differ significantly from bulk values, especially for: (1) low-temperature depositions, (2) high deposition rates, (3) reactive sputtering, and (4) oblique-angle deposition. When possible, measure film density directly.

References

Sauerbrey, G. (1959). Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung. Zeitschrift für Physik, 155(2), 206–222.

Lu, C. & Czanderna, A.W. (1984). Applications of Piezoelectric Quartz Crystal Microbalances. Elsevier.

Benes, E. (1984). Improved quartz crystal microbalance technique. J. Appl. Phys. 56(3), 608–626.

Behrndt, K.H. (1971). Long term operation of crystal oscillators in thin-film deposition. J. Vac. Sci. Technol. 8(5), 622–626.

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