INFICON RQCM - Quartz Crystal Microbalance Research System - Manual (Page 55)

RQCM – RESEARCH QUARTZ CRYSTAL MICROBALANCE
THEORY OF OPERATION 5-3
5.3 THICKNESS CALCULATION
Film thickness is often the parameter of interest in many QCM applications. Thickness can be
derived from Equation 4 as follows:
Equation 5
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
⋅ ⋅ ⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ ⋅ ⋅
⋅
=
∆
= −
f
f f
R
f R
N m
TK
q
z
f z
q q
f
f
π
ρ π
ρ
ρ
tan tan 1
where:
f
TK
= thickness of the film in cm.
m
∆ = change in mass per unit area in g/cm 2 (calculated from the Lu and Lewis equation).
f
ρ = density of film material in g/cm 3
If the period of oscillation is measured rather than the frequency, 1/period can be substituted for
frequency resulting in the following equation. (See INFICON TechNote RTK-101 for details
discussion).
Equation 6
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
⋅ ⋅ ⋅
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅ ⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= −
τ
τ τ
π
π
τ
ρ
ρ
q
z
z
q
f
q
f
R
R
N TK
tan tan 1
where:
q
τ = Period of unloaded crystal in seconds
τ = Period of loaded crystal in seconds
Although the above equation still involves a number of simplifying assumptions, its ability to
accurately predict the film thickness of most commonly deposited materials has been
demonstrated. The RQCM uses this equation to calculate film thickness.
The basic measurement is period, which can be thought of as a measurement of equivalent quartz
mass. The actual film mass on the crystal is then found by applying the acoustic impedance
correction factor.
When the mass is zeroed using the RQCM, the initial equivalent quartz mass and the initial
corrected film mass are stored. For each subsequent measurement the new corrected total film
mass is calculated, and the film mass deposited since the thickness was zeroed is determined by
subtracting the initial corrected film mass from the total corrected film mass. The film thickness
on the crystal is calculated by dividing by the film mass by the material density.
The Lu and Lewis equation is generally considered to be a good match to the experimental
results 5 6 for frequency changes up to 40% (relative to the unloaded crystal). Keep in mind that
the Z-match equation strictly applies to elastic (lossless) films. Films which behave
viscoelastically, such as some organic polymer films with large thickness or viscosity, will
exhibit significant deviations from both Equation 3 and Equation 6.