John A. Kusters, Senior Member, IEEE, and John R. Vig, Fellow, IEEE

Manuscript received September 5. 1990: revised November 2. 19W accepted November 9. 1990.
J. A. Kusters is with the Hewlett-Packard Company. 5301 Stevens Creek Blvd., Santa Clara. CA 95052-8059.
J. R. Vig is with the Electronics Technology and Devices Laboratory. U.S. Army LABCOM. Ft. Monmouth. NJ 07703-5000.
IEEE Log Number 9142840.

INTRODUCTION

    With the advent of the microcomputer compensated crystal oscillator (MCXO), thermal hysteresis has become the dominant factor limiting the stability achievable with temperature compensated oscillators. Since the next largest limiting factor is orders of magnitude smaller than hysteresis, future improvements in MCXO stability depend primarily on reducing the hysteresis. Similarly, as sensor technology has improved, hysteresis has become a limiting factor in the accuracies achievable with quartz resonator sensors.
    The purpose of this paper is to review the subject of hysteresis. The subject has been studied only sporadically until recently. A search of the literature has revealed only about a dozen papers in the Proceedings of the Annual Symposium on Frequency Control [1]-[6], [9], [11]-[15] a couple of Army contract reports, [16], [18] and a few other papers [19]-[23] that deal with the subject. Books dealing with oscillators either do not mention the subject at all, or devote only a few sentences to the phenomenon.
    An ideal quartz crystal resonator's frequency versus temperature characteristic can be described by a single valued function. Frequency can then be uniquely determined from knowing the temperature. In "real-world' , resonators, however, the frequency versus temperature characteristic does not repeat exactly upon temperature cycling.
    The lack of frequency versus temperature (f versus T) repeatability has been referred to at various times as "retrace," "hysteresis," "restart," and "thermal memory. " No formal definitions existed before the advent of MIL-0-553 1013, ¹ the military specification for crystal oscillators. [24] MIL-0-55310B defines "retrace" and "hysteresis." It distinguishes between the two terms by defining "retrace" as the nonrepeatability of the f versus T characteristic, at a fixed temperature, upon on-off cycling an oscillator under specified conditions, and "hysteresis" as the maximum value of the nonrepeatability in the f versus T characteristics during a temperature cycle, i.e., as the difference between the up-cycle and the downcycle f versus T characteristics at the temperature where that difference is maximum. Although the complete f versus T loop may be called the "hysteresis characteristic" of an oscillator, in order to quantify the phenomenon for specification purposes, the worst-case hysteresis is defined as the "hysteresis" in MIL-0-55310. Hysteresis is determined during a complete quasistatic temperature cycle between the specified temperature limits. "Quasistatic" means that the temperature is changed in such a manner as to ensure that-the frequency offsets due to thermal gradients are much smaller than the specified f versus T stability, including hysteresis.
    "Retrace" is usually applied to specifying oven controlled crystal oscillators (OCXO), whereas "hysteresis" is usually applied to specifying temperature compensated crystal oscillators (TCXO). Fig. 1 illustrates TCXO hysteresis: Fig. 2 illustrates OCXO retrace.

A REVIEW OF THE LITERATURE

The following summarizes the literature on hysteresis. Included are three papers that deal with frequency versus pressure hysteresis, as these may possibly have relevance to frequency versus temperature hysteresis.
    Early researchers who studied the stabilization and startup of precision ovenized resonators noted that, when the oven was turned off after stabilization and turned on at a later time, the frequency after restabilization at the original temperature was slightly different from the resonator's frequency just before oven shutdown, [25] as is illustrated in Fig. 2. According to Sykes, [26] et al.. "This frequency shift has been attributed to sorption of minute quantities of gas on the crystal surface during an oven or oscillator shutdown. " Also recognized as sources of instabilities during stabilization were "relaxation of strain, or defects in the quartz itself. "
    The Union Thermoelectric Handbook [16] mentioned hysteresis as follows: "Temperature changes can result in mechanical changes within the unit. For example, the mounting supports and bonding material may be altered more or less permanently by a change in temperature, resulting in a difference in the stress applied to the quartz plate. Some of the apparent hysteresis phenomenon [sic] have this origin. "
    Hammond et al. reported the first (and, until recently, the only) in-depth studies of hysteresis. [1], [19], [20] Their research was aimed at improving the performance of an LC-cut resonator that had been developed for a quartz thermometer. Although the resonators were cycled over various temperature ranges between +250⁰C and -200⁰C, the "hysteresis" was reported as the frequency shift at O'C only (due to the high temperature coefficient of the LC-cut and the difficulty of obtaining accurate enough temperature measurements at other temperatures.) Fig. 3 shows the hysteresis of an LC-cut.

Hammond et al. observed that: 1) the fundamental mode and third overtone c-modes exhibited nearly identical hysteresis, while the fundamental b- and a-modes exhibited a smaller hysteresis than the c-modes; 2) identically fabricated AT-cuts showed a five times smaller hysteresis than the LC-cuts and the hysteresis of the AT-cut was of opposite sign; 3) there was no systematic variation of hysteresis with electrode material or electrode thickness-the electrodes tried included both higher- and lower-than-quartz thermal expansion coefficient materials, and ductile as well as stiff materials; 4) there was no systematic variation of hysteresis with mounting type, stiffness, and rotation of the mounting orientation by 90⁰, or with the amplitude-of-vibration distribution; beveled plano-plano fundamental-mode resonators, contoured fundamental-mode resonators and third overtone resonators showed the same hysteresis; 5) there was no systematic variation of hysteresis with quartz type or concentration of defects-natural quartz, cultured quartz, optical grade cultured quartz, and swept quartz were tried; 6) there was no correlation with X-ray induced frequency changes; 7) there was a systematic variation of hysteresis with angle of cut in rotated Y-cuts-a change of sign of the hysteresis effect was found at q = 32⁰; and 8) although contamination in the resonator enclosure could aggravate the hysteresis, when proper contamination control measures were applied during processing, the remaining hysteresis was not related to contamination.
    Hammond et al. [1] concluded that "all the experiments to date indicate that the hysteresis effect in the quartz thermometer must be a property of crystalline quartz. However, it is not clear whether it is an intrinsic property of quartz or attributable to a defect structure. The variability from resonator to resonator and from observation to observation would infer a defect structure." In a slightly later paper, [ 19] however, the authors state that hysteresis is "related to the differential expansion between the quartz plate and the thin film, as well as micro contamination [sic] within the crystal holder. "
    In 1970, Dick and Silver [2] showed the result for a single fundamental-mode 5-MHz crystal, the f versus T characteristic of which repeated to 3 X 10^-8 for one cycle between -40⁰C and +80⁰C.
    Buroker and Frerking [3] developed a digitally compensated TCXO the f versus T stability of which was ±5 x 10^-8 from -40⁰C to +80⁰C. In the conclusion of their report, however, they stated that, the "technique is capable of even greater stabilities, but even ±5 X 10^-8 cannot be maintained in practical environments due to thermal hysteresis in crystals and other components."
    Mroch and Hykes [4], [17] evaluated a variety of purchased 4.5-MHz fundamental-mode AT-cut crystal units in connection with the development of a high stability TCXO. They found that the retrace at the lower turnover point varied from 3 ppm to less than 1 X 10^-9, "with no correlation from lot to lot or vendor," and that ".. few of the crystals received met the ±3 X 10^-8 retrace [sic] requirement" of the research program. They also found that the hysteresis was "dependent on the highest temperature to which the crystal had been recently exposed," it was maximum at the low temperature limit of the temperature cycle, as is illustrated in Fig. 4, and, after 30min exposures to 100⁰C, hysteresis was a function of storage time at room temperature. The ceramic capacitors used as bypass capacitors were identified as potential sources of TCXO hysteresis.

Forster [5] measured the hysteresis of AT- and SC-cut 10-MHz third-overtone crystal units in HC-27/U glass enclosures, "from three German manufacturers." He found that "With nearly all the test specimens it is striking that quite large hysteresis values are obtained in the first cycle. The reproducibilities of the second and third cycles is in general excellent ... namely ± 1 X 10^-8. In the first up run a 'calming phase' evidently sets in." He concluded that the hysteresis of ". . . SC-cut crystals measured were not vastly superior to those of AT-cut specimens," and that "The frequency stability attainable with digitally temperature-compensated quartz crystal oscillators is limited to around ... ±1 X 10^-7 solely by the thermal hysteresis of the crystal resonators."
    Vig et al. [6] measured nine ceramic flatpack enclosed four-point-mounted SC-cut crystal units and two similarly fabricated AT-cut control units, then compared the hysteresis at the lower turnover temperatures. The worst-case hysteresis of the SC-cut units was 8.5 X 10^-8. The AT-cuts were about 10 times worse. The hysteresis was measured during two complete cycles between -45⁰C and +75⁰C.
    Kaitz [7] and Kusters and Kaitz [8] studied the frequency versus pressure hysteresis of BT-cut pressure transducer resonators. Kaitz found that "Hysteresis ... increases considerably as temperature and pressure are increased," and "extensively cycling" the transducers, which -relaxes the residual stresses incurred during processing of the unit," improved hysteresis "in almost all instances ... up to 50%." Kusters and Kaitz found a correlation between pressure hysteresis and the quartz material, and between the temperature of worst pressure hysteresis and the quartz material. Unswept natural quartz showed the lowest hysteresis. Sweeping natural quartz increased the hysteresis. Pure Z-growth material showed a hysteresis similar to that of swept natural quartz. The highest hysteresis was found when X-growth material was used. Anomalously high hysteresis in some "X-growth" transducers was traced to the fact that the transducers "were found to contain regions of Z-growth material and clearly defined X-Z growth boundaries. " No correlation was found between inclusion density and hysteresis.
    Ueda et al. [9] reported that when a tuning fork thermometer resonator was made relatively stress free by making a "narrow neck between the resonator support and vibrating beams to further reduce the stress transmitted from the support to the vibrating beams . . . " the hysteresis, when cycled between 4.2⁰K and O⁰C, was reduced by a factor of two at 4.2⁰K and a factor of three at O⁰C, when compared to the conventional tuning fork, and, at O⁰C, it was "one tenth that of thickness shear-mode resonator. " No details are provided with respect to the number of units tested or the type of thickness shear resonator that was used for the comparison.
    Tuladhar et al. [21] studied the hysteresis of 5- and 10MHz AT-cut resonators with gold electrodes and with silver electrodes. Their conclusions were that: "Hysteresis at the lower turn-over-temperature is higher than that at the upper turn-over-temperature ... Hysteresis for gold electroded crystals is considerably worse than that for silver electroded crystals ... Crystals having a double adjusting layer show greater hysteresis than those with a single layer.
    Ward and EerNisse [10] found a correlation between pressure hysteresis and quartz material, as did Kusters and Kaitz. Transducers made of pure Z-growth cultured quartz exhibited higher hysteresis than those made from natural quartz.
    Tartakovskii [22] analyzed the results of Hammond et al. and of U.S.S.R. researchers Varfolomeeva et al. (reference [2] of Tartakovskii [22]). He states that ". . . by taking into account the two different manifestations of thermal inertia in a piezoelectric element it is possible to explain all experimental results" of Hammond et al. One manifestation of "thermal inertia" is "the lag of the average temperature Tp in the piezoelectric element behind that of the medium T_M,, by D T₁, " and the other is the "difference D T₂ in temperatures at the edge and the center of a piezoelectric element creates in the central region of the plate compressive stresses . . . " which produce a frequency shift that is a function of the plate's angles of cut and mounting orientation. He also explains the AT-cut hysteresis results of Varfolomeeva et al. by showing that the results are consistent with frequency changes to be expected from "a change of the mechanical stresses in the ring of silver paste" that was used to bond the mounting clips to the quartz plate. His "calculated results showed why a simple turn in the direction of the ring towards the axis Z' (i.e., a shift from y = 45⁰ to y = 75⁰) caused a sharp reduction of frequency hysteresis following a temperature treatment . . ."
    Beaussier [23] drew a similarity between mechanical hysteresis and thermal hysteresis and concluded that "the phenomenon of nonlinear elasticity in the presence of crystalline faults, and in particular moving dislocations appear to be the cause." He further reported that continued thermal cycling, after proper selection of the quartz itself, reduces thermal hysteresis to some limiting value. Beaussier described a model that explained the experimental data where: hysteresis - K_H(D T)² where K_H is between 0.5 X 10^-10/⁰C² and 1.1 X 10^-10/⁰C² for SC-cuts, and between 1.0 X 10^-10/ ⁰ C² and 1. 5 X 10^-10 / ⁰C² for AT-cuts, and where AT is the maximum temperature excursion during the up and down temperature cycles.
    Filler [11] analyzed TCXO thermal hysteresis and showed that thermal lag between the resonator and the TCXO's thermometer is a major contributor to the observed TCXO hysteresis. He developed a model "that accounts for both the normally encountered and anomalous thermal hysteresis. This model can separate apparent hysteresis from 'true hysteresis'." He showed that "the inherent hysteresis of the resonator is much lower than that of the TCXO and has a different temperature dependence," and that "thermal hysteresis is independent of rate of change of temperature, when thermal lag is accounted for . . . "
    Symonds and Wacker [18] encountered excessive f versus T hysteresis during the initial stages of a TCXO development program. They attempted to vary the design and fabrication of the (10-MHz fundamental -mode AT-cut) resonators in order to reduce the hysteresis. The results to date indicate that the resonator's mounting plays an important role in the observed hysteresis. When the stiffness of the mounting clips and the bonding areas of four-point mounted resonators were reduced significantly, the average hysteresis was reduced from 0.64 ± 0.35 ppm to 0.20 + 0.07 ppm.
    Filler and Vig [12] developed dual-mode SC-cut resonators for microcomputer compensated crystal oscillators (MCXO). They, together with Schodowski, [13] showed that when the dual-mode self-temperature sensing method [13] is used as the "thermometer" during f versus T measurements, the effects of thermal gradients can be made negligible even when the temperature is changed rapidly. In the self-temperature sensing method, the fundamental mode and third overtone c-mode frequencies are excited simultaneously. The frequencies of these c-modes are measured versus a thermometric beat frequency that is derived from the two c-modes. The resonator acts as its own thermometer; no external thermometer is needed.
    Filler and Vig applied the dual-mode self-temperature sensing method to measuring the hysteresis of 10MHz/ 3.3-MHz dual-mode SC-cut resonators in a - 55⁰C and + 85⁰C temperature range. "The measured hysteresis ranged from parts in 10⁹ for the best units, to about 2 X 10^-8 for the typical 'good' unit, to several parts in 10⁸ for the 'bad' units." Filler [14] continued these hysteresis studies and showed that for 10-MHz third-overtone SC-cut resonators obtained from four manufacturers: 1) hysteresis varies with temperature excursion, but for a given temperature cycle, the hysteresis repeats, 2) hysteresis is not always worse at low temperatures, 3) resonator manufacturing lots exhibit "signatures," i.e., resonators within a lot showed similar hysteresis characteristics, but the characteristics varied significantly from lot to lot, and 4) a factor of two change in drive current did not affect the hysteresis.
    Filler, Messina and Rosati [15] studied the performance, including hysteresis, of microcomputer compensated crystal oscillators (MCXO) that used dual-mode SC-cut resonators similar to those developed by Filler and Vig. [12] Hysteresis of the seven MCXO's evaluated ranged from ±5 X 10^-9 to ±4 X 10^-8.
    Benjaminson [27] investigated circuit contributions to oscillator hysteresis. He analyzed the effects of hysteresis in the most critical components of a dual-mode crystal oscillator (for use in a MCXO). One oscillator of the pair was a 10-MHz third-overtone bridge oscillator, the other a 3.4-MHz fundamental-mode impedance- inverting Colpitts oscillator. He found that a 1 % change in the tuned circuit inductance of the bridge oscillator changed the frequency of oscillation by 1 X 10^-8, while a similar variation in the PI-network inductance of the Colpitts oscillator caused a change of 7 X 10^-8. Since both oscillators are series resonant circuits, additional tests were performed in order to isolate the effects of circuit component hysteresis from crystal hysteresis, by replacing the crystal with fixed resistors in each oscillator, enabling operation as LC oscillators at the nominal operating frequencies.
    Hysteresis measured during cycling between -55⁰C and +85⁰C was typically less than 50 ppm, which translated to a contribution less than 1 X 10^-10 of versus T hysteresis in the 10-MHz oscillator (when operating as a crystal oscillator) and less than 6 X 10^-10 hysteresis in the 3.4-MHz crystal oscillator.
    The low hysteresis effects demonstrated by the LC components were achieved by careful analysis to select the most stable inductors and capacitors available and equally careful circuit analysis to minimize frequency pulling by the reactances. To accomplish this, the bridge oscillator was designed with as wide an LC bandwidth as b-mode rejection permitted; b-mode traps per se, were avoided because their pulling effects are much worse.
    The impedance-inverting Colpitts oscillator was designed with the lowest possible reactance values. This was limited by restrictions on transistor current consumption. Higher current operation can provide higher transconductance, permitting lower reactance with a consequent reduction in frequency pulling. An added benefit, particularly in the low resistance fundamental-mode oscillator, was produced by the lower equivalent series resistance of small, high Q inductors that reduce loaded Q degradation and further minimize pulling due to reactance changes.

HYSTERESIS MECHANISMS

So what causes hysteresis? Since the evidence reported to date is inconclusive, we will discuss the various mechanisms that can possibly cause the phenomenon.
   Contamination Redistribution: Adsorption-desorption phenomena can cause hysteresis if, during temperature cycling, contamination inside the resonator enclosure is redistributed so as to change the mass loading on the active area of the resonator. Sykes et al. [28] stated that "A study of the stabilization characteristics of precision oscillators following interruptions of oscillation lead to the conclusion that residual contamination within the crystal unit enclosure is the most likely cause of frequency change during the first several days of operation after an interruption. " Armstrong et al. [29] showed that the retrace of "clean" 5-MHz thermocompression-bonded, high-temperature processed resonators was superior to that of similar but solder-bonded resonators. Hammond et al. [1] stated that "with inadequate vacuum baking or inadequate cleaning of the crystal mounts, header, or can, the hysteresis effect can be aggravated. However, . . . the remaining hysteresis effects are not related to contamination. "
    When one examines precision resonators' hysteresis curves, it is difficult to see evidence that contamination transfer is a significant factor. Desorption rates generally have an exponential dependence on temperature. Therefore, if adsorption-desorption phenomena played a major role, then hysteresis would show a strong temperature dependence, which has not been reported. When surfaces are heated to produce a uniformly rising temperature, desorption (of a single adsorbent) occurs in a narrow temperature interval, with a pronounced peak [30]. The temperature of maximum desorption rate is a function of desorption energy, and is often used to investigate the interaction of gases with metal surfaces. The characteristic "moisture dip" observed when resonator enclosures contain water vapor is a manifestation of adsorption and desorption occurring in a narrow temperature range. In order for contamination transfer to explain the observed hysteresis, a variety of contaminant molecules, with an appropriate range of adsorption energies, would need to be present in the resonator enclosure.
    Ideally, in order to eliminate adsorption-desorption as a hysteresis mechanism, the electrodes should either be highly active or inert. In the first case, all the contamination would be permanently adsorbed (i.e., the contaminant's lifetime on the surface would be infinite). In the second case, none of it would be adsorbed (i.e., the contaminants' lifetime would be zero). In real resonators, however, the contaminants' lifetimes on surfaces are finite. The lifetimes depend on the surfaces, contaminant molecules and temperatures.
    Hysteresis in isothermal adsorption-desorption (as a function of gas pressure) has been reported, e.g., during the adsorption and desorption of water on the gold electrodes of quartz crystal resonators [31]. Such hysteresis, being a small perturbation of a small effect, is probably a negligible second-order effect in f versus T hysteresis.
   Strain Changes: It is well known that changes in the stresses on a resonator plate can produce frequency shifts. The stresses experienced by resonators include mounting stresses (via the force-frequency effect [32], [33] and bending effects [34], bonding stresses, [35] and electrode stresses [36]. It is clear that temperature cycling can produce changes in these stresses, and can, thereby, result in hysteresis.
    If the mounting clips were perfectly elastic, or perfectly soft, then they would not contribute to hysteresis. If, however, the clips undergo stress relief during temperature cycling, then hysteresis can result. The magnitude of the hysteresis produced by a given amount of stress relief is a function of the orientation of the mounting clips with respect to the crystallographic axes of the quartz plate, [32] and the types of stresses.
    For in-plane diametric forces, the force-frequency coefficient K_f versus azimuth angle y have been found to have zeroes for all the commonly used cuts, such as the AT and SC-cuts. [32] Therefore, one might conclude that hysteresis due to stress relief in the mounting clips can be eliminated by mounting the crystals where K_f = 0. Unfortunately, it is not possible to completely eliminate the effects of mounting stresses in conventional resonators for the following reasons. First, the azimuthal angles where K_f =0 are functions of temperature, [33] so that there is no y where K_f = 0 over the whole temperature range of a TCXO or MCXO. Second, the y where the effects of bonding stresses are zero is different from the y where K_f =0, at least for the AT-cut, the only cut for which bonding stress effects have been reported. [35] Third, the forces due to the mounting clips are generally not purely in-plane diametric forces. This is especially true for three and fourpoint mounted resonators because, since the thermal expansion coefficient of quartz is anisotropic whereas that of the typical package base is isotropic, the forces due to temperature cycling will have tangential components. Similarly, for two-point mounted resonators, the base's change of dimensions during temperature cycling will apply shear-type forces in addition to the in-plane diametric forces.
    Theoretically, property mounting the quartz resonator plate on an identically oriented quartz plate, as is attempted in BVA resonators, [37] ought to greatly reduce the hysteresis due to stress relief in the mounting structure. Our literature search did not reveal evidence that BVA resonators exhibit hysteresis or retrace that is superior to those found in high quality conventional resonators.
    For an example of the frequency changes that can be caused by stress relief, consider a 5-MHz third-overtone, 14-mm-diameter resonator. If one were to intentionally mount this resonator at the y where K_f is maximum, then the frequency shifts due to changes in the in-plane diametric forces would be 2.9 X 10^-8 per gram for an AT cut resonator, and 1.7 X 10^-8 per gram for an SC-cut resonator [38] (where "per gram" refers to the force due to a one gram weight, on earth).
    The effects of electrode stress relief can be minimized by using the SC-cut, [36] or by not having electrodes in contact with the active area of the resonator, as can be done with BVA-type resonators, [37] and, to a lesser extent, with lateral field resonators [39]. Everything else being equal, using a single metal (e.g., Au) as opposed to two or more metals (e.g., Cr-Au, Ti-Pd-Au) is also likely to produce lesser stress relief effects because using two or more metals introduces additional interfaces where stress relief and diffusion can occur.
   Changes in the Quartz: Changes in the quartz due to the stresses induced by temperature cycling are among the conceivable causes of hysteresis, although no direct evidence of such changes could be uncovered in the literature. Perfect quartz would not be expected to be affected by temperature cycling. The imperfections that are subject to change include surface defects, dislocations. impurities, inclusions, and twins.
    Surface defects. such as the microcracks produced by lapping, can change upon temperature cycling, [40] however, by properly etching the surfaces [41] subsequent to mechanical treatment, the possibility of changes. can be greatly reduced or eliminated.
    That dislocation motion due to temperature cycling is a factor in hysteresis is unlikely at the typical TCXO temperatures. Even in sweeping experiments [42] that are usually conducted far above the normal operating temperatures of oscillators. no evidence of dislocation motion has been reported. The energy needed to anneal quartz damage due to neutron irradiation may be a clue to the energies needed to move dislocations. When quartz is irradiated with fast neutrons. displacement damage occurs. At high doses, the quartz gradually becomes disordered into an amorphous form. Annealing studies on neutron damaged quartz indicate that the annealing temperature of quartz is above the inversion temperature. [43]-[45] The activation energy for structure annealing is 0.75 eV [43].
    " It takes a great deal less energy, however, to move impurities in quartz to new lattice sites. Impurity motion due to temperature cycling is, therefore, a more probable hysteresis mechanism than dislocation motion. It is possible to induce impurity dependent effects, at more "normal" temperatures. For example, some radiation- induced effects can anneal, [46] and low dc voltages produce changes in the aging rates of resonators [47]. Normal sweeping [7] requires temperatures of 450⁰C to 500⁰C and field strengths of 1000 V/cm, but sweeping has been produced at room temperature ( » 23⁰ C) when 25 V / mm is applied to a quartz bar [48]. Kusters [49] observed that when a 500 V/mm dc field was applied to a doubly rotated quartz plate at 80⁰C, the frequency shift decayed with a time constant of 7 s. Moreover, there is evidence in the literature that strain gradients due to ultrasonic vibrations can produce changes in quartz properties, probably due to the motion of impurities down the gradients [50].
    Recent experimental evidence developed at Hewlett-Packard seems to support material defects being a significant mechanism. In two experimental groups of 38 LC-cuts each (see Hammond [1]), significant differences were observed that correlate with the quartz material. All fabrication processes other than the material were identical. Optical grade quartz was irradiated to show the presence of defect centers. LC-cuts made from darkened areas of the quartz formed the first group. LC-cuts from totally clear areas formed the second group. As shown in Fig. 5, Group I had a total yield of 34% with a high percentage of dead units and hysteresis rejects. Group 2 had a total yield of 84% with no hysteresis rejects. These crystals are operated on the end of a 1/2 wave cable. Dead units are those for which the effective motional resistance becomes so high at a particular temperature that the oscillator stops.

In a continuation of the work reported in Kusters and Kaitz, [8] normal sweeping was done on one group of natural quartz specimens, and extended sweeping done on a second group. All other processing parameters were identical. Sweeping was continued on the second group until the indicated ion current became stable and remained constant for at least 48 hours. Resistance data of finished pressure transducer resonators are shown in Figs. 6 and 7. There was a sharp difference between the two groups' motional resistances. There was also a sharp difference between the overall yields. Group I (Fig. 6) had a 64% yield; group 2 had a 93 % yield.

Experimentally, in quartz LC-cuts and in the quartz pressure transducer resonators, the hysteresis decreases as a result of continued thermal cycling thus confirming the observations of Beaussier [23] that improved hysteresis performance can result from thermal cycling.
    Twins in a quartz plate can change under the influence of (high) stresses. For example, when a sufficiently high stress is applied (0.5 GPa), it is even possible to switch a single-domain crystal to a single domain of opposite polarity. The removal of the stress leads to a complete restoration of the original orientation. The stress-strain relations in a "ferrobielastic" material such as quartz can show hysteresis due to ferrobielastic switching resulting from Dauphiné twin formation [51]. Twinning phenomena, however, are unlikely to be a significant hysteresis mechanism because resonators that contain (macroscopic) twins would be rejected during the usual inspection and testing procedures, because changes in twins tend to occur suddenly, and because the stresses needed to move twin boundaries are probably higher than the stresses due to temperature cycling. Changes in microtwins can possibly occur at high pressures, in pressure transducers.
    Other interesting thermal hysteresis phenomena in quartz take place near 573⁰C, during the a -b transition, [52], [53] in the incommensurate phase of quartz. Various physical properties of quartz, such as the heat capacity and refractive index, exhibit thermal hysteresis in this phase.
   Apparent Hysteresis: Filler [11] showed conclusively that when the thermometer in a compensated oscillator is separated from the resonator (as it is in TCXO), the thermal gradients between the thermometer and the resonator can produce "apparent" hysteresis. The apparent hysteresis can be eliminated by using dual-mode self-temperature sensing [12], [13].
   Angles-of-cut Dependence: Early data on SC-cut crystals seemed to indicate that the SC-cut has lower hysteresis than AT- and BT-cuts. A recent experiment on 30 SC cut, 10-MHz, third-overtone units and on 15 BT-cut, 5- MHz, third-overtone units in identical enclosures produced the data shown in Table 1. The material in all was unswept cultured quartz with an IR "Q" of nominally 2.5 million. The units had been cycled from 30⁰C to 150⁰C to 30⁰C in temperature steps of 5⁰C. Hysteresis was measured at the turnover point of approximately 80⁰C that corresponds roughly to the temperature at which hysteresis is a maximum as defined in MIL-0-55310B. Hysteresis of either sign was seen in both batches. The maximum seen was about +5 X -10^-8 . This is essentially the same as reported in other work. [12]-[15] The experimental data are shown in Fig. 8.

Oscillator Circuit Hysteresis: Hysteresis in circuit components can cause oscillator hysteresis. For example, if a 20-pF load capacitor C_L is in series with a resonator and the resonator's C₁ = 14 pF and C₀ = 5 pF, then a 5 X 10^-4 hysteresis in CL produces a 1 X 10^-7 f versus T hysteresis [38].
    Inductors are notorious for their instabilities; e.g., the windings of inductors can stretch and move due to the stresses experienced during temperature cycling. It is possible to minimize the circuit contributions to hysteresis by appropriate resonator and circuit design, and circuit component selection. For example, the need for inductors in the b-mode trap of SC-cut oscillators can be eliminated with lateral field resonator designs that suppress the b-mode [54].

    The results to date seem to indicate that lattice defects are somehow related to thermal hysteresis. Stress relief in the mounting structure can also produce significant hysteresis. As crystal processing techniques have improved, contamination has become less of a problem. This is shown in Fig. 9. The points represent a rough mean of the published data and demonstrate a two-order-of-magnitude reduction in the observed levels of hysteresis during the past 25 years.

That parts in 10⁹ hysteresis has been observed in some resonators is encouraging. Once the effects of material defects and strain relief due to the mounting structure are better understood, and high-perfection quartz becomes available, it may be reasonable to expect parts in 10⁹ hysteresis in a reproducible manner. MCXO's of parts in 10⁹ f versus T stability will then be obtainable.

ACKNOWLEDGMENT

The authors thank Charles Adams, Jim Collin, and Leo Steindorf of Hewlett-Packard for sharing the results of the hysteresis reduction experiments in the LC-cut and pressure sensor, and for making the measurements on the SC-cut and BT-cut resonators, and Al Benjaminson of General Technical Services, Inc. for editing the section that reviews his work.

REFERENCES

John A. Kusters (S’61-M’64-SM’87) was born in Racine, WI in 1937. He received the B.S.E.E. degree in 1964 from Loyola University, Los Angeles, CA. He attended Stanford University. Stanford. CA, from 1964 to 1968. working on microwave acoustic delay lines. receiving the M.S.E.E. degree in 1965.
     He worked at Hewlett-Packard from 1965 to 1986 developing cesium frequency standards. acoustooptic devices, and quartz resonators. From 1986 to 1989 he worked at the Efratom Division of Ball Corp. developing quartz resonators and oscillators. In 1989. he returned to the Santa Clara Division of the Hewlett-Packard Co. where he is currently the R&D Manager for Precision Time & Frequency products.
     Mr. Kusters is a member of Tau Beta Pi. Sigma Xi. and Alpha Sigma Nu.

0885-3010/91/0500-0281$01.00 6 1991 IEEE