Introduction to Quartz Frequency Standards - Quartz and the Quartz Crystal Unit

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A quartz crystal unit's high Q and high stiffness (small C1) make it the primary frequency and frequency-stability determining element in a crystal oscillator. The Q values of crystal units are much higher than those attainable with other circuit elements. In general-purpose crystal units, Qs are generally in the range of 104 to 106. A high-stability 5-MHz crystal unit's Q is typically in the range of two to three million. The intrinsic Q, limited by internal losses in the crystal, has been determined experimentally to be inversely proportional to frequency (i.e., the Qf product is a constant for a given resonator type). For AT- and SC-cut resonators, the maximum Qf = 16 million when f is in MHz.

Quartz (which is a single-crystal form of SiO2) has been the material of choice for stable resonators since shortly after piezoelectric crystals were first used in oscillators - in 1918. Although many other materials have been explored, none has been found to be better than quartz. Quartz is the only material known that possesses the following combination of properties:

  1. It is piezoelectric ("pressure electric"; piezein means "to press" in Greek).
  2. Zero temperature coefficient resonators can be made when the plates are cut along the proper directions with respect to the crystallographic axes of quartz.
  3. Of the zero temperature coefficient cuts, one, the SC-cut (see below), is "stress compensated."
  4. It has low intrinsic losses (i.e., quartz resonators can have high Q's).
  5. It is easy to process because it is hard but not brittle, and, under normal conditions, it has low solubility in everything except the fluoride etchants
  6. It is abundant in nature.
  7. It is easy to grow in large quantities, at low cost, and with relatively high purity and perfection.
Of the man-grown single crystals, quartz, at more than 2000 tons per year (in 1991), is second only to silicon in quantity grown.

The direct piezoelectric effect was discovered by the Curie brothers in 1880. They showed that when a weight was placed on a quartz crystal, charges appeared on the crystal surface; the magnitude of the charge was proportional to the weight. In 1881, the converse piezoelectric effect was illustrated; when a voltage was applied to the crystal, the crystal deformed due to the lattice strains caused by the effect. The strains reversed when the voltage was reversed.

Of the 32 crystal classes, 20 exhibit the piezoelectric effect (but only a few of these are useful). Piezoelectric crystals lack a center of symmetry. When a force deforms the lattice, the centers of gravity of the positive and negative charges in the crystal can be separated so as to produce surface charges. The piezoelectric effect can provide a coupling between an electrical circuit and the mechanical properties of a crystal. Under the proper conditions, a "good" piezoelectric resonator can stabilize the frequency of an oscillator circuit.

Quartz crystals are highly anisotropic, that is, the properties vary greatly with crystallographic direction. For example, when a quartz sphere is etched in hydrofluoric acid, the etching rate is more than 100 times faster along the fastest etching rate direction, the Z-direction, than along the slowest direction, the slow-X-direction. The constants of quartz, such as the thermal expansion coefficient and the temperature coefficients of the elastic constants, also vary with direction. That crystal units can have zero temperature coefficients of frequency is a consequence of the temperature coefficients of the elastic constants ranging from negative to positive values.

The locus of zero-temperature-coefficient cuts in quartz is shown in Figure 5. The X, Y, and Z directions have been chosen to make the description of properties as simple as possible. The Z-axis in Figure 5 is an axis of threefold symmetry in quartz; in other words, the physical properties repeat every 120° as the crystal is rotated about the Z-axis. The cuts usually have two-letter names, where the "T" in the name indicates a temperature-compensated cut; for instance, the AT-cut was the first temperature-compensated cut discovered. The FC, IT, BT, and RT-cuts are other cuts along the zero temperature coefficient locus. These cuts were studied in the past (before the discovery of the SC-cut) for some special properties, but are rarely used today. The highest-stability crystal oscillators employ SC-cut or AT-cut crystal units.

Figure 5
Figure 5. Zero-temperature-coefficient cuts of quartz.

Because the properties of a quartz crystal unit depend strongly on the angles of cut of the crystal plate, in the manufacture of crystal units, the plates are cut from a quartz bar along precisely controlled directions with respect to the crystallographic axes. The orientations of the plates are checked by means of X-ray diffraction. In some applications, the orientations must be controlled with accuracies of a few seconds of angle. After shaping to required dimensions, metal electrodes are applied to the wafer. Circular plates with circular electrodes are the most commonly used geometries, although the blanks and electrodes may also be of other geometries. The electroded wafer is mounted in a holder structure [8]. Figure 6 shows the two common types of holder structures used for resonators with frequencies greater than 1 MHz. (The 32-kHz tuning fork resonators used in quartz watches are packaged typically in small tubular enclosures.)

Figure 6
Figure 6. Typical constructions of AT-cut and SC-cut crystals.

Because quartz is piezoelectric, a voltage applied to the electrodes causes the quartz plate to deform slightly. The amount of deformation due to an alternating voltage depends on how close the frequency of the applied voltage is to a natural mechanical resonance of the crystal. To describe the behavior of a resonator, the differential equations for Newton's laws of motion for a continuum, and for Maxwell's equations, must be solved with the proper electrical and mechanical boundary conditions at the plate surfaces [9]. Because quartz is anisotropic and piezoelectric, with 10 independent linear constants and numerous higher order constants, the equations are complex, and have never been solved in closed form for physically realizable three-dimensional resonators. Nearly all theoretical works have used approximations. The nonlinear elastic constants, although small, are the source of some of the important instabilities of crystal oscillators; such as the acceleration sensitivity, the thermal-transient effect, and the amplitude-frequency effect, each of which is discussed in this report.

In an ideal resonator, the amplitude of vibration is maximum at the center of the electrodes; it falls off exponentially outside the electrodes, as shown in the lower right portion of Figure 7. In a properly designed resonator, a negligible amount of energy is lost to the mounting and bonding structure, i.e., the edges must be inactive in order for the resonator to be able to possess a high a. The displacement of a point on the resonator surface is proportional to the drive current. At the typical drive currents used in (e.g., 10 MHz) thickness shear resonators, the peak displacement is on the order of a few atomic spacings. (The peak acceleration of a point on the electrodes is on the order of 1 million g.)

Figure 7
Figure 7. Resonator vibration displacement amplitude for a circular plate with circular electrodes.

As the drive level (the current through a crystal) increases, the crystal's amplitude of vibration also increases, and the effects due to the nonlinearities of quartz become more pronounced. Among the many properties that depend on the drive level are: resonance frequency, motional resistance R1, phase-noise, and frequency vs. temperature anomalies (called activity dips), which are discussed in another section of this report. The drive-level dependence of the resonance frequency, called the amplitude-frequency effect, is illustrated in Figure 8 [10]. The frequency change with drive level is proportional to the square of the drive current; the coefficient depends on resonator design [11]. Because of the drive-level dependence of frequency, the highest stability oscillators usually contain some form of automatic level control in order to minimize frequency changes due to oscillator circuitry changes. At high drive levels, the nonlinear effects also result in an increase in the resistance [5]. Crystals can also exhibit anomalously high starting resistance when the crystal surfaces possess such imperfections as scratches and particulate contamination. Under such conditions, the resistance at low drive levels can be high enough for an oscillator to be unable to start when power is applied. The drive level dependence of resistance is illustrated in Figure 9. In addition to the nonlinear effects, a high drive level can also cause a frequency change due to a temperature increase caused by the energy dissipation in the active area of the resonator.

Figure 8
Figure 8. Drive level Dependence of frequency.

Figure 9
Figure 9. Drive level dependence of crystal unit resistance.

Bulk-acoustic-wave quartz resonators are available in the frequency range of about 1 kHz to 500 MHz. Surface-acoustic-wave (SAW) quartz resonators are available in the range of about 150 MHz to 1.5 GHz. To cover the wide range of frequencies, different cuts, vibrating in a variety of modes are used. The bulk wave modes of motion are shown in Figure 10. The AT-cut and SC-cut crystals vibrate in a thickness-shear mode. Although the desired thickness-shear mode usually exhibits the lowest resistance' the mode spectrum of even properly designed crystal units exhibits unwanted modes above the main mode. The unwanted modes, also called "spurious modes," or "spurs," are especially troublesome in filter crystals, in which "energy trapping rules" are employed to maximize the suppression of unwanted modes [4]. These rules specify certain electrode geometry to plate geometry relationships. In oscillator crystals, the unwanted modes may be suppressed sufficiently by providing a large enough plate diameter to electrode diameter ratio, or by contouring (i.e., generating a spherical curvature on one or both sides of the plate).

Figure 10
Figure 10. Modes of motion of a quartz resonator.

Above 1 MHz, the AT-cut is commonly used. For high-precision applications, the SC-cut has important advantages over the AT-cut. The AT-cut and SC-cut crystals can be manufactured for fundamental-mode operation up to a frequency of about 300 MHz. (Higher than 1 GHz units have been produced on an experimental basis.) Above 100 MHz, overtone units that operate at a selected harmonic mode of vibration are generally used, although higher than 100 MHz fundamental mode units can be manufactured by means of chemical polishing (etching) techniques [12]. Below 1 MHz, tuning forks, X-Y and NT bars (flexure mode), +5° X-cuts (extensional mode), or CT-cut and DT-cut units (face shear mode) can be used. Tuning forks have become the dominant type of low frequency units due to their small size and low cost. Hundreds of millions of quartz tuning forks are produced annually for quartz watches and other applications.

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