1. Theory

Definition: A quartz crystal microbalance (QCM) thickness monitor tracks the resonance frequency of an AT-cut quartz plate in the deposition flux; added film mass lowers the frequency, which is converted to thickness and rate (Ideal Vacuum page; Leybold Inficon patent).
Mass loading: For thin films the frequency drop is proportional to areal mass (background knowledge, Sauerbrey).
Acoustic matching: For thick films the film’s acoustic impedance matters, described by the Z-ratio in the Lu-Lewis equation (Leybold Inficon patent; background knowledge, Lu and Lewis).
Two modes: Measuring the fundamental and an anharmonic mode lets the monitor find the Z-ratio itself (INFICON note).

2. Working Principle

Sensor: A 6 $MHz$ AT-cut crystal in a water-cooled head facing the source (Ideal Vacuum page; Leybold Inficon patent).
Electronics: An oscillator or synthesizer tracks resonance to fractions of a hertz (Leybold Inficon patent).
Conversion: Frequency to thickness by Z-match with density, Z-ratio and tooling factor (INFICON note; background knowledge).
Control: Rate feedback to source power and shutter at the target thickness (Leybold Inficon patent).

3. Applications

Evaporation and sputtering: Rate and thickness of metal and dielectric layers (INFICON note).
Optical coatings: Multilayer stacks with sequential materials (INFICON note).
Alloys and co-deposition: Auto-Z where bulk Z-ratios do not apply (INFICON note).
Process calibration: Tooling factors against profiler or ellipsometer thickness (background knowledge; Stylus Profiler page).

Mass Loading and Resolution

Name	Formula	Related Formulae	Variables/Constants
Sauerbrey relation	$Δ f = - \frac{2 f _{0}^{2}}{ρ _{q} μ _{q}} \frac{Δ m}{A} = - C_{f} \frac{Δ m}{A}$	$Δ T_{f} = \frac{Δ m}{A ρ _{f}}$	$f_{0} : u n l o a d e d r eso nan ce$ $ρ_{q}, μ_{q} : q u a r t z d e n s i t y an d s h e a r m o d u l u s, Δ m / A : a r e a l ma ss$ $C_{f} : ma ss se n s i t i v i t y$ $Δ T_{f} : f i l m t hi c k n ess$ $ρ_{f} : f i l m d e n s i t y; S a u er b r ey r e l a t i o n an d q u a r t z co n s t an t s 2.648 g/c m^{3} an d 2.947 \times 1 0^{11} d y n p er s q u a r e c m (ba c k g r o u n d k n o w l e d g e, S a u er b r ey); I d e a l V a c uu m p a g e : t hin - f i l m cr y s t a l s g e n er a l l y a t 6 MHz, v ib r a t i o n s co u n t e d e a c h seco n d$ $(co m p u t e d : ρ_{q} μ_{q} = 8.83 \times 1 0^{5} g p er s q u a r e c m p er s; a t 6 MHz C_{f} = 81.5 Hz p er mi cr o g r am p er s q u a r e c m, 2.20 Hz/ \overset{˚}{A} f or A l an d 7.28 Hz/ \overset{˚}{A} f or C u)$
Frequency constant and crystal thickness	$N_{q} = f_{0} t_{q} = \frac{v _{q}}{2}$	$t_{q} = \frac{N _{q}}{f _{0}}$	$N_{q} : f r e q u e n cy co n s t an t o f A T - c u t q u a r t z$ $t_{q} : pl a t e t hi c k n ess$ $v_{q} : s h e a r w a v e v e l oc i t y; r e l a t i o n s (ba c k g r o u n d k n o w l e d g e, Lu an d C z an d er na); L ey b o l d I n f i co n p a t e n t : 6 MHz an d 4 MHz t es t cr y s t a l s, f u n d am e n t a l [1, 0, 0] m o d e an d n e a r es t anha r m o ni c [1, 0, 2]$ $(co m p u t e d : N_{q} = 1.668 \times 1 0^{5} c m / s, so a 6 MHz pl a t e i s 0.278 mm t hi c k)$
Frequency resolution to thickness	$δ T = \frac{δ f}{C _{f} ρ _{f}}$	$rate = \frac{d T _{f}}{d t} (\overset{˚}{A} /s)$	$δ f : f r e q u e n cy u n cer t ain t y$ $δ T : t hi c k n ess r eso l u t i o n; L ey b o l d I n f i co n p a t e n t : sy n t h es i z er r eso l u t i o n 0.005 Hz o v er 0 t o 8.0 MHz, a cc u r a cy a f e w h er t z an d p r e f er ab l y 1 Hz or b e tt er in t h e d escr i pt i o n, 15 Hz or b e tt er in c l aim 6, r a t e u se d t o co n t r o l t h e so u r ce or s h u tt er; I d e a l V a c uu m p a g e : r a t e in \overset{˚}{A} /s, o n e h y d r o g e n a t o m ab o u t 0.5 \overset{˚}{A} a s an a cc u r a cy i l l u s t r a t i o n$ $(co m p u t e d : 1 Hz i s 0.45 \overset{˚}{A} o f A l an d 0.14 \overset{˚}{A} o f C u; 15 Hz i s 6.8 an d 2.1 \overset{˚}{A})$

Thick-Film Conversion

Name	Formula	Related Formulae	Variables/Constants
Z-match (Lu-Lewis) thickness	$T_{f} = \frac{N _{q} ρ _{q}}{π ρ _{f} f _{c} z} arctan [z tan (\frac{π ( f _{q} - f _{c} )}{f _{q}})]$	$z = \frac{Z _{q}}{Z _{f}} = \frac{ρ _{q} μ _{q}}{ρ _{f} μ _{f}}$	$f_{q}, f_{c} : u n co a t e d an d co a t e d f r e q u e n c i es$ $z : Z - r a t i o$ $Z_{q}, Z_{f} : s h e a r a co u s t i c im p e d an ces o f q u a r t z an d f i l m; Lu - L e w i s f or m (ba c k g r o u n d k n o w l e d g e, Lu an d L e w i s); L ey b o l d I n f i co n p a t e n t : z d e f in e d a s t h e r a t i o o f q u a r t z t o f i l m a co u s t i c im p e d an ce, Lu - L e w i s r e l a t i o n r e f er e n ce d; I N F I C O N n o t e : u ser s o f t e n g u ess z = 1.0 or an im p e d an ce o f 8.83$ $(co m p u t e d : C u f r o m 6.0 t o 5.0 MHz g i v es 178 k \overset{˚}{A} w i t h z = 0.44 an d 175 k \overset{˚}{A} w i t h z = 0.6)$
Simpler mass-loading forms	$T_{f}^{p er i o d} = \frac{N _{q} ρ _{q}}{ρ _{f}} (\frac{1}{f _{c}} - \frac{1}{f _{q}})$	$T_{f}^{l in e a r} = \frac{N _{q} ρ _{q}}{ρ _{f}} \frac{f _{q} - f _{c}}{f _{q}^{2}}$	$p er i o d f or m : e q u i v a l e n t t o z = 1 t o f i r s t or d er; l in e a r f or m : S a u er b r ey f or s ma l l s hi f t s; r e l a t i o n s (ba c k g r o u n d k n o w l e d g e, Lu an d C z an d er na)$ $(co m p u t e d : t h e s am e C u s hi f t g i v es 165 k \overset{˚}{A} b y t h e p er i o d f or m an d 137 k \overset{˚}{A} b y t h e l in e a r f or m, 5.8 an d 21 p er ce n t b e l o w 175 k \overset{˚}{A})$
Auto-Z from two modes	$(f_{1}, f_{2})_{0} \to (f_{1}, f_{2})_{t} \Rightarrow z$	$update every 100 ms$	$f_{1} : f u n d am e n t a l$ $f_{2} : f i r s t anha r m o ni c; I N F I C O N n o t e : M o d e L oc k m e a s u r es f u n d am e n t a l an d f i r s t anha r m o ni c, s t or es t h e u n co a t e d p ai r, u ses b o t h s hi f t s w i t h co a t in g hi s t or y an d d e n s i t y, 10 cy c l es p er seco n d, n o a d j u s t ab l e p a r am e t er s; T ab l e 1 b u l k an d A u t o - Z r a t i os : A l 1.08 an d 1.10, C 3.26 an d - 0.3 t o - 0.5, S e 0.86 an d 1.8, T i - A l 1.00 an d 0.38 t o 0.45, C u 0.44 an d 0.5 t o 0.7 (6.0 t o 5.0 MHz, 175 k \overset{˚}{A}), s t ain l ess 0.35 an d 0.16 t o 0.50$ $(a ss u m e d : C u s h e a r v e l oc i t y 2325 m / s; co m p u t e d : b u l k z = 0.43, c l ose t o 0.44)$

Calibration and Crystal Life

Name	Formula	Related Formulae	Variables/Constants
Tooling factor	$T_{s u b} = T_{x t a l} \frac{T F}{100}$	$T F = 100 \frac{T _{m e a s u r e d}}{T _{in d i c a t e d}}$	$T_{s u b} : t hi c k n ess o n t h e s u b s t r a t e$ $T_{x t a l} : t hi c k n ess in d i c a t e d a t t h e cr y s t a l$ $T F : t oo l in g f a c t or in p er ce n t; g eo m e t r i c c a l ib r a t i o n b y a p r o f i l er or e l l i p so m e t er m e a s u r e m e n t o f a t es t f i l m (ba c k g r o u n d k n o w l e d g e, Lu an d C z an d er na)$ $(a ss u m e d : cr y s t a l r e a d s 1000 \overset{˚}{A}, p r o f i l er 1150 \overset{˚}{A}; co m p u t e d : T F = 115)$
Crystal life	$life = 1 - \frac{f _{q} - f _{c}}{Δ f _{ma x}}$	$Δ f_{ma x} \sim 1 to 1.5 MHz at 6 MHz$	$Δ f_{ma x} : u s ab l e f r e q u e n cy s hi f t; I d e a l V a c uu m p a g e : cr y s t a l l i f e a s an es t ima t e o f r e mainin g u se f u l f r e q u e n cy r an g e, n o n u m er i c l imi t s; L ey b o l d I n f i co n p a t e n t : l imi t e d l i f e t im e, cr y s t a l e v e n t u a l l y u nab l e t o s u s t ain v ib r a t i o n; I N F I C O N n o t e : T ab l e 1 s hi f t s o f 0.3 t o 1.2 MHz; t y p i c a l r an g e (ba c k g r o u n d k n o w l e d g e)$ $(m y r e a d in g : s t i f f or s t r esse d f i l m s f ai l soo n er, a s t h e i r a co u s t i c l o a d in g b r o a d e n s t h e r eso nan ce)$
Temperature and stress errors	$\frac{Δ f}{f} \approx a_{1} (T - T_{0}) + a_{2} (T - T_{0})^{2} + a_{3} (T - T_{0})^{3}$	$AT cut: turning point near room temperature$	$a_{i} : t e m p er a t u r e coe f f i c i e n t s; L ey b o l d I n f i co n p a t e n t : s p u r i o u s s hi f t s f r o m m ec hani c a l or t h er ma l s t r ess o n t h e cr y s t a l; c u bi c A T - c u t b e ha v i o u r (ba c k g r o u n d k n o w l e d g e, Lu an d C z an d er na)$ $(m y r e a d in g : r a d ian t h e a t in g f r o m a so u r ce a t s h u tt er o p e nin g c an mimi c a r a t e b u r s t, so w a t er - coo l e d h e a d s an d se ttl in g t im e a r e u se d)$

Important Questions

Ques: How does a quartz crystal measure film thickness?

By the drop in its resonance frequency as mass is added. A thin film moves with the crystal surface, so the frequency falls in proportion to areal mass, $Δ f = - C_{f} Δ m / A$ , with $C_{f} = 2 f_{0}^{2} / ρ_{q} μ_{q}$ (background knowledge, Sauerbrey). For a 6 $MHz$ crystal, the frequency typical of thin-film monitors (Ideal Vacuum page), $C_{f}$ is 81.5 $Hz$ per microgram per square centimetre, which is 2.20 $Hz/ \overset{˚}{A}$ for aluminium and 7.28 $Hz/ \overset{˚}{A}$ for copper (computed). Thickness follows by dividing areal mass by film density, so the density entered in the monitor directly scales the reading. The quartz impedance $ρ_{q} μ_{q}$ is $8.83 \times 1 0^{5}$ g per square cm per second (computed), the 8.83 value that the INFICON note says users enter. Reference: G. Sauerbrey, Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier 1984 (background knowledge; not opened); Ideal Vacuum, ‘How Does a Thin Film Monitor Measure Thickness?’ (2021) (fetch-tool summary); INFICON, ‘Auto-Z: A Path to Precise Control of Layer Thickness’, application note (2008), citing A. Wajid, Rev. Sci. Instrum. (1991) (fetch-tool summary).

Ques: Why is the Z-match equation needed for thick films?

Because a thick film is no longer a rigid mass; acoustic waves propagate in it, and its impedance relative to quartz changes the frequency shift. The Lu-Lewis equation corrects this using the Z-ratio, the quartz-to-film impedance ratio as defined in the Leybold Inficon patent (background knowledge, Lu and Lewis). For the INFICON copper case, a shift from 6.0 to 5.0 $MHz$ , the Z-match equation with z = 0.6 gives 175 $k \overset{˚}{A}$ , matching the tabulated thickness, while the period form gives 165 $k \overset{˚}{A}$ and the linear Sauerbrey form 137 $k \overset{˚}{A}$ , 5.8 and 21 per cent low (computed). Errors therefore grow with frequency shift, which is why crystals are replaced or Z-match is used beyond a few per cent shift (my reading). Reference: ‘Measuring and controlling deposition on a piezoelectric monitor crystal’, US patent 5,112,642, Leybold Inficon (filed March 1990) (fetch-tool summary; equations not rendered); G. Sauerbrey, Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier 1984 (background knowledge; not opened); INFICON, ‘Auto-Z: A Path to Precise Control of Layer Thickness’, application note (2008), citing A. Wajid, Rev. Sci. Instrum. (1991) (fetch-tool summary).

Ques: Which Z-ratio is right for copper: bulk 0.44 or film 0.5 to 0.7?

The INFICON note gives a bulk Z-ratio of 0.44 for copper and Auto-Z values of 0.5 to 0.7 for the e-beam film, and states that thin-film Z-ratios often differ from bulk. A bulk shear velocity of 2325 m/s gives 0.43 (assumed; computed), confirming the bulk value. For the 6.0 to 5.0 $MHz$ shift, z = 0.44 gives 178 $k \overset{˚}{A}$ and z = 0.6 gives 175 $k \overset{˚}{A}$ , a 1.7 per cent difference (computed), matching the note’s reported 1.7 per cent accuracy change. Since Auto-Z agreed with gravimetric mass, the film value of 0.5 to 0.7 is more likely correct for that deposit; porosity and microstructure lower the film’s impedance below bulk (my reading). For copper the effect is small, but for carbon or Ti-Al alloys it reaches 3 to 10 per cent. Reference: INFICON, ‘Auto-Z: A Path to Precise Control of Layer Thickness’, application note (2008), citing A. Wajid, Rev. Sci. Instrum. (1991) (fetch-tool summary); G. Sauerbrey, Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier 1984 (background knowledge; not opened).

Ques: How accurate must the frequency measurement be: 1 Hz or 15 Hz?

The Leybold Inficon patent says both. Its description asks for accuracy of a few hertz, preferably 1 $Hz$ or better, with a synthesizer resolution of 0.005 $Hz$ , while claim 6 states 15 $Hz$ or better and claim 7 returns to 1 $Hz$ . At 6 $MHz$ , 1 $Hz$ corresponds to 0.45 $\overset{˚}{A}$ of aluminium and 15 $Hz$ to 6.8 $\overset{˚}{A}$ (computed). The 1 $Hz$ figure is more likely the intended performance, consistent with the description, claim 7 and the Ideal Vacuum page’s sub-angstrom illustration of half a hydrogen atom; 15 $Hz$ is probably a looser claim boundary or typographical error (my reading). Rate control near 1 $\overset{˚}{A} /s$ also needs sub-hertz resolution over a measurement period (my reading). Reference: ‘Measuring and controlling deposition on a piezoelectric monitor crystal’, US patent 5,112,642, Leybold Inficon (filed March 1990) (fetch-tool summary; equations not rendered); Ideal Vacuum, ‘How Does a Thin Film Monitor Measure Thickness?’ (2021) (fetch-tool summary).

Ques: What is a tooling factor?

A geometric correction from crystal to substrate. The crystal sits at a different distance and angle from the source than the wafer, so it receives a different flux, and the monitor scales its reading by a tooling factor in per cent (background knowledge, Lu and Czanderna). It is calibrated by depositing a test film, measuring it on the substrate with a profiler or ellipsometer, and setting TF to 100 times measured over indicated thickness. If the crystal reads 1000 $\overset{˚}{A}$ and the profiler 1150 $\overset{˚}{A}$ , TF is 115 (assumed; computed). Density and Z-ratio must be right first, or the tooling factor absorbs material errors and fails when thickness changes (my reading). Each source and crystal position needs its own factor. Reference: G. Sauerbrey, Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier 1984 (background knowledge; not opened).

Ques: What limits crystal life?

Accumulated film load. As film builds up, frequency shift increases, acoustic losses rise, and the crystal eventually can no longer sustain vibration, as the Leybold Inficon patent notes. The Ideal Vacuum page describes crystal life as remaining useful frequency range without numbers; the INFICON table shows shifts of 0.3 to 1.2 $MHz$ on 4 and 6 $MHz$ crystals before data were taken. A 6 $MHz$ plate is only 0.278 mm thick (computed), so tens of kiloangstroms of a dense metal is a substantial fraction of its mass (my reading). Stressed films can also crack or delaminate and cause mode hopping to an anharmonic, which the patent’s mode identification guards against (my reading). Multi-crystal heads switch automatically to a fresh crystal. Reference: ‘Measuring and controlling deposition on a piezoelectric monitor crystal’, US patent 5,112,642, Leybold Inficon (filed March 1990) (fetch-tool summary; equations not rendered); Ideal Vacuum, ‘How Does a Thin Film Monitor Measure Thickness?’ (2021) (fetch-tool summary); INFICON, ‘Auto-Z: A Path to Precise Control of Layer Thickness’, application note (2008), citing A. Wajid, Rev. Sci. Instrum. (1991) (fetch-tool summary); G. Sauerbrey, Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier 1984 (background knowledge; not opened).

References

G. Sauerbrey, “Verwendung von Schwingquarzen zur Waegung duenner Schichten und zur Mikrowaegung,” Z. Phys. 155, 206 (1959); C.-S. Lu and O. Lewis, “Investigation of film-thickness determination by oscillating quartz resonators with large mass load,” J. Appl. Phys. 43, 4385 (1972); C. Lu and A. W. Czanderna (eds.), Applications of Piezoelectric Quartz Crystal Microbalances, Elsevier, 1984 (cited from background knowledge).
Ideal Vacuum, “How Does a Thin Film Monitor Measure Thickness?” 2021. Available: https://idealvac.com/files/manuals/How_a_Thin_Film_Monitor_Measures_Thickness.pdf
INFICON, “Auto-Z: A Path to Precise Control of Layer Thickness,” application note, 2008. Available: https://www.inficon.com/media/4704/download/ciad91a1-a%20Auto-Z%20Application%20Note.pdf
“Measuring and controlling deposition on a piezoelectric monitor crystal,” US patent 5,112,642 A, Leybold Inficon, filed March 30, 1990. Available: https://patents.google.com/patent/US5112642

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Quartz Crystal Microbalance (QCM)