Thin Film Controllers / Control Units
Fundamentals of Vacuum Technology
D00.124
LEYBOLD VACUUM PRODUCTS AND REFERENCE BOOK 2001/2002
can determine with a phase comparator
whether the applied frequency is higher or
lower than the crystal resonance point.
The crystal impedance is capacitive at fre-
quencies below the fundamental wave and
inductive at frequencies above the reso-
nance. This information is useful if the
resonance frequency of a crystal is unkno-
wn. A brief frequency sweep is carried out
until the phase comparator changes over
and thus marks the resonance. For AT
quartzes we know that the lowest usable
frequency is the fundamental wave. The
anharmonics are slightly above that. This
information is not only important for the
beginning, but also in the rare case that the
instrument loses “track” of the fundamen-
tal wave. Once the frequency spectrum of
the crystal is determined, the instrument
must track the shift in resonance frequen-
cy, constantly carry out frequency measu-
rements and then convert them into
thickness.
Use of the “intelligent” measuring system
has a number of obvious advantages over
the earlier generation of active oscillators,
primarily insensitivity to mode hopping as
well as speed and accuracy of measure-
ment. This technique also enables the
introduction of sophisticated properties
which were not even conceivable with an
active oscillator setup. The same device
that permits the new technology to identify
the fundamental wave with one sweep can
also be used to identify other oscillation
modes, such as the anharmonics or quasi-
harmonics. The unit not only has a device
for constantly tracking the fundamental
wave, but can also be employed to jump
back and forth between two or more
modes. This query of different modes can
take place for two modes with 10 Hz on the
same crystal.
6.8 Auto Z match
technique
The only catch in the use of equation 6.4 is
that the acoustic impedance must be
known. There are a number of cases where
a compromise has to be made with accu-
racy due to incomplete or restricted know-
ledge of the material constants of the coa-
ting material:
1) The Z values of the solid material often
deviate from those of a coating. Thin
coatings are very sensitive to process
parameters, especially in a sputter envi-
ronment. As a result, the existing values
for solid material are not adequate.
2) For many exotic substances, including
alloys, the Z value is not known and not
easy to determine.
3) It is repeatedly necessary to carry out a
precise coating thickness measurement
for multiple coating with the same
crystal sensor. This applies in particular
to optical multiple and semi-conductor
coatings with a high temperature coeffi-
cient T
C
. However, the effective Z value
of the mixture of multiple coatings is
unknown.
In such a case, therefore, the only effective
method is to assume a Z value of 1, i.e. to
ignore reality with respect to wave propa-
gation in multi-substance systems. This
incorrect assumption causes errors in the
prediction of thickness and rate. The
magnitude of the error depends on the
coating thickness and the amount of devia-
tion from the actual Z value.
In 1989 A. Wajid invented the mode-lock
oscillator. He presumed that a connection
existed between the fundamental wave and
one of the anharmonics, similar to that
ascertained by Benes between the funda-
mental oscillation and the third quasi-har-
monic oscillation. The frequencies of the
fundamental and the anharmonic oscillati-
ons are very similar and they solve the
problem of the capacity of long cables. He
found the necessary considerations for
establishing this connection in works by
Wilson (1954) as well as Tiersten and
Smythe (1979).
The contour of the crystal, i.e. the spheri-
cal shape of one side, has the effect of
separating the individual modes further
from each other and preventing energy
transfer from one mode to another. The
usual method of identification is to desig-
nate the fundamental oscillation as (100),
the lowest anharmonic frequency as (102)
and the next higher anharmonic as (120).
These three indices of the mode nomen-
clature are based on the number of phase
reversals in the wave motion along the
three crystal axes. The above mentioned
works by Wilson, Tiersten and Smythe
examine the properties of the modes by
studying the influence of the radius of the
cut on the position of the anharmonic in
relation to the fundamental oscillation.
If one side of the quartz is coated with
material, the spectrum of the resonances
is shifted to lower frequencies. It has been
observed that the three above mentioned
modes have a somewhat differing mass
sensitivity and thus experience somewhat
different frequency shifts. This difference
is utilized to determine the Z value of the
material. By using the equations for the
individual modes and observing the fre-
quencies for the (100) and the (102)
mode, one can calculate the ratio of the
two elastic constants C60 and C55. These
two elastic constants are based on the
shear motion. The key element in Wajid’s
theory is the following equation:
(6.5)
with
M ... area mass/density ratio (ratio of coa-
ting mass to quartz mass per area
unit)
Z ... Z value
It is a fortunate coincidence that the pro-
duct M π Z also appears in the Lu-Lewis
equation (equation 6.4). It can be used to
assess the effective Z value from the follo-
wing equations:
(6.6)
or
Here F
q
and F
c
are the frequencies of the
non-coated or coated quartz in the (100)
mode of the fundamental wave. Because of
the ambiguity of the mathematical func-
tions used, the Z value calculated in this
way is not always a positively defined
variable. This has no consequences of any
significance because M is determined in
another way by assessing Z and measuring
the frequency shift. Therefore, the thickn-
ess and rate of the coating are calculated
one after the other from the known M.
One must be aware of the limits of this
technique. Since the assessment of Z
depends on frequency shifts of two mo-
Z
tg M Z
F
F
tg
F
F
c
q
c
q
= −
⋅ ⋅ ⋅
⋅
π
π
tg M Z
F
F
Z tg
F
F
c
q
c
q
⋅ ⋅ ⋅
+ ⋅ ⋅
=π π
0
( )
( / )
( / )
C C
C C
M Z
coated
uncoated
55
66
55
66
1
1
≈
+ ⋅
D00 E 19.06.2001 21:40 Uhr Seite 124