### Quartz Crystal for QCM Sensor

Piezoelectric effect is the phenomenon to generate an electric change in response to strain when external mechanical stress is applied to some crystalline materials. In particular, silicone dioxide, SiO_{2}, is the most common as a crystalline material in piezoelectric resonators, and α-quartz, a specific crystalline form of SiO_{2} has been used for QCM–based device. The SiO_{2}-based device stably generates a frequency by piezoelectric effect and inverse piezoelectric effect.

Oscillation modes and temperature property depend on how to cut out the crystalline. In other words, the direction and magnitude of the piezoelectric straining is directly controlled by the angle of cutting out of the crystalline. So-called AT-cut quartz crystals are used for QCM applications. AT-cut type is general oscillators which were cut at the angle of 35° 15′ to z-axis of artificial crystalline, and can generate a frequency from 1 MHz to several hundreds MHz.

For AT-cut type oscillator, the thickness of quartz plate is an important parameter to determine its oscillating frequency. Resonance occurs when the thickness of the quartz plate is an odd integer of half wavelengths of the induced wave.

The resonant frequency

*f*, is given by:

$$ f = n\frac{{v_{q} }}{{2t_{q} }} = nf_{0} $$

(83.1)

$$ f_{0} = \frac{{v_{q} }}{{2t_{q} }} $$

(83.2)

The resonant frequency for

*n* = 1 is called fundamental resonant frequency. Where

*v*_{q} is the wave velocity (speed of sound) in the quartz plate and

*t*_{q} is the thickness of the quartz plate. The unit is MHz. Here,

*n* (odd number) is the overtone,

*n* = 3, 5, 7 is the third, fifth, seventh overtone, and so on. Equation

83.1 indicates that oscillating frequency is inversely proportional to the thickness of the quartz plate. For example, the thickness is approximately 33 µm when the oscillating frequency is 50 MHz.

$$ \Delta f = - \frac{f}{{m_{q} }}\Delta m = - \frac{f}{{t_{q} \rho_{q} }}\Delta m = - n\frac{{2f_{0}^{2} }}{{v_{q} \rho_{q} }}\Delta m = - n\frac{1}{C}\Delta m $$

(83.3)

Here, the mass per area of the crystal, \( m_{q} \), is related to its thickness, \( t_{q} \), by \( m_{q} = t_{q} \rho_{q} \left( {{\text{kg}}/{\text{m}}^{2} } \right) \), where \( \rho_{q} \) is the density of the quartz. Here, \( v_{q} = 3400\, \left( {{\text{m}}/{\text{s}}} \right) \) and \( \rho_{q} = 2650\, \left( {{\text{kg}}/{\text{m}}^{3} } \right) \) are given for an AT-cut quartz crystal. When the resonant frequency of the quartz plate is 5 MHz at its fundamental mode (*n* = 1), *C* gives \( 17.7\, \left( {{\text{ng}}/{\text{cm}}^{2} /{\text{Hz}}} \right) \). The sensitivity increases with a factor *n* by operating at different overtones where it increases with the square of the fundamental frequency, \( f_{0} \). This relationship between \( \Delta f \) and \( \Delta m \) is also known as Sauerbrey equation [Sauerbrey, G. *Zeitschrift Fur Physik,* 155, 206–222 (1959)].

### Quartz Crystal for QCM Sensor

Dissipation of the quartz crystal can be also measured at the same time when the frequency change is measured during molecules adsorption onto the surface. The energy of the oscillating crystal dissipates or loses from the system when the driving voltage is turned off. Therefore, in situ monitoring of the viscoelastic changes associated with adsorption of molecules can be useful information. The energy dissipation (

*D* factor) is a dimensionless quantity, which is defined as follows;

$$ D = \frac{{E_{\text{dissipated}} }}{{2\pi E_{\text{stored}} }} $$

(83.4)

Where

*E*_{dissipated} is the dissipated or lost energy during one period of oscillation, and

*E*_{stored} is the total stored energy in the oscillation system.

*D* value is the ratio of

*E*_{dissipated} and

*E*_{stored}, indicating the energy loss in the whole system in response to

*Δf* change. According to the equivalent electric circuit, Eq. (

83.4) is represented as

$$ D = \frac{{R_{1} }}{{2\pi fL_{1} }} $$

(83.5)

where

*R*_{1} and

*L*_{1} are an inductance and resistance, respectively and

*f* is the frequency. The equivalent electric circuit corresponding to a mechanical model, gives

\( \tau = 2L_{1} /R_{1} \) where

*τ* is the decay time constant. Therefore,

$$ D = \frac{1}{\pi f\tau } $$

(83.6)