Film thickness measurements were made with an AT-cut piezoelectric quartz microbalance. The microbalances are a quartz disk ~1 cm in diameter and ~.1 mm thick with circular gold electrodes in the center of each face. When an A.C. voltage is applied across the electrodes the crystal oscillates in a thickness shear mode in which the two faces move parallel to each other and in opposite directions. Used in this way the quartz becomes a very high Q mechanical resonator. Any adsorbed film on the electrodes will make the resonator heavier and so lower it’s resonant frequency. A film thickness measurement is made by monitoring this frequency shift. The crystals used for these experiments were usually driven at their third harmonic (fo,3 ~ 5.5 MHz). Typically the oscillators had a Q greater than 105.
There are three different effects that can cause a shift in the frequency of the microbalance [35,36]. The first is the mass loading effect, mentioned above. If a thin film is adsorbed onto the microbalance and is rigidly clamped to the surface it will produce a shift of the resonant frequency given by:
where fo,n is the resonant frequency of the nth
harmonic,
is the adsorbed
mass density per unit area, and Rq is the transverse
acoustic impedance of quartz (Rq ~ 8.862*106
Kg/m2sec). For an accurate film thickness measurement two other
effects must be subtracted from the frequency shift. The first is the hydrostatic
pressure effect given by:
where P is the pressure, and Fz is a constant (Fz ~ 1.48*10-6 bar-1 at liquid helium temperatures). There is also a frequency shift due to viscous coupling to the vapor surrounding the microbalance:
where
and
are
the viscosity and mass density of the surrounding vapor respectively.