INFICON PLO-10i Phase Lock Oscillator - Manual (Page 51)

PLO-10 PHASE LOCK OSCILLATOR
THEORY OF OPERATION 8-3
mass. Notice that the units of the frequency constant for quartz is length/time or velocity.
Also note that if the acoustic impedance ratio is equal to one, quartz on quartz, then
Equation 4 reduces to Equation 3.
8.3 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 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.
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.