© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

10.1.2.4 Quartz Resonators

Piezoelectric quartz crystals, when cut in a predetermined manner and mounted so that two opposing faces have electrical contacts, will compress or expand when an oscillating electric field is applied across the faces.^1701,1702 The amplitude of crystal vibration falls off very rapidly as the applied electric field frequency deviates from the resonant frequency of the crystal. (Cyclical mechanical compressions at the resonant frequency also induce oscillating electric fields of maximum amplitude.) In 1998, most high-precision timing devices were based on quartz crystal resonators, with ~2 billion resonators manufactured annually worldwide for oscillator, clock, and filter applications.¹⁶⁹⁹ Commercially-available off-the-shelf GHz devices (e.g., Micro Networks' M101 resonator) typically displayed ~10 ppm stability, or Dn / n ~ 10^-5, readily improved to ~10^-7 in temperature-controlled ovens.

The fundamental mode resonant frequency of a quartz crystal plate of thickness d_quartz ~ 1 micron (~2000 molecular layers thick), density r_quartz ~ 2650 kg/m³,⁷⁶³ and elastic (Young's) modulus E_quartz ~ 1.1 x 10¹¹ N/m² (Table 9.3) vibrating in thickness mode is:¹⁶⁹⁹

{Eqn. 10.9}

The resonant frequencies of the highest-quality, lowest-noise quartz macroresonators can be measured with a precision of 14 significant figures,¹⁶⁹⁹ e.g., the noise is Dn / n ~ 3 x 10^-14 at the optimum measurement times.¹⁷⁰⁰ However, each variable in Eqn. 10.9 is temperature dependent. For instance, the thermal expansion of quartz (affecting density) and the temperature coefficients of E_quartz (ranging from positive to negative values) are highly dependent on the angles of cut of the plate relative to the crystallographic axes. Quartz resonator frequency may vary monotonically with temperature at 10^-5 - 10^-4 K^-1; the noise floor (i.e., the Allan deviation floor¹⁷⁰²) for quartz microresonators has been estimated¹⁶⁹⁹ as Dn / n ~ (1.2 x 10^-19 Hz^-1) n_osc ~ 10^-10 for n_osc ~ 1 GHz. Atomistic simulation of submicron quartz oscillators by Broughton²⁶⁶⁹ suggests that such oscillators must have at least ~10⁶ atoms to display bulk elastic constants, and that differences from continuum mechanical frequency predictions are observable for 17-nm (or smaller) devices, and that nanoscale devices with even a single defect may exhibit dramatic anharmonicity.

Last updated on 23 February 2003