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Basic Technology of Quartz Crystal Resonators

Quartz crystal resonators (often called “crystals”) are widely used in
frequency control applications because of their unequalled combination of
high Q, stability, small size and low cost.  Many different
substances have been investigated as possible resonator materials, but for
many years quartz resonators have been preferred in satisfying needs for
precise frequency control.  Compared to other resonators, for
example, LC circuits, mechanical resonators such as tuning forks, and
piezoelectric ceramic resonators based or other single-crystal materials,
the quartz resonator has a unique combination of properties.

First, the material properties of single-crystal quartz are
extremely stable with time, temperature, and other environmental changes,
as well as highly repeatable from one specimen to another.  The
acoustic loss or internal friction of quartz is very low, leading directly
to one of the key properties of a quartz resonator, its extremely high Q
factor.  The intrinsic Q of quartz is about 107 at 1 MHz.
Mounted resonators typically have Q factors ranging from tens of
thousands to several hundred thousand, which is orders of magnitude better
than the best LC circuits.

The second key property of the quartz resonator is its stability
with respect to temperature variation.  Depending on the shape and
orientation of the crystal blank, many different modes of vibration can be
used and it is possible to control the frequency-temperature
characteristics of the quartz resonator to within close limits by an
appropriate choice.  The most commonly used type of resonator is the
AT-cut, where the quartz blank is in the form of thin plate cut as an
angle of about 35°15' to the optic axis of the crystal.

The third essential characteristic of the quartz resonator is
related to the stability of its mechanical properties.  Short and
long term stabilities manifested in frequency drifts of only a few parts
per million per year are readily available from commercial units.
Precision crystal units manufactured under closely controlled conditions
are second only to atomic clocks in the frequency stability and precision
achieved.

Quartz Plate Orientation

Quartz resonators consist of suitably mounted and
metal-deposited plates of crystalline quartz using bulk acoustic wave
(BAW) vibrations. Originally the quartz plates were made from natural
quartz, but today cultured quartz is used almost exclusively. The plates
(also called wafers or blanks) are fabricated at a precise orientation
with respect to the crystallographic axes of the quartz material. The
orientation or "cut" determines frequency-temperature characteristics and
other important properties of the resonator.

The above is a schematic diagram of a cultured quartz crystal
grown from a Y-oriented seed crystal for use in fabricating AT-cut
resonators.  A seed crystal establishes the initial crystal
orientation and encourages growth in the Y direction at the expense of the
Z-axis. Seed crystals are carefully selected to avoid defects which might
propagate as the crystal grows. The position of the seed crystal is
indicated. The lines sloping left from the x-axis mark the saw cut
position for AT plates, the line sloping to the right indicates the
BT-cut.  In practice, these angles are very critical and are
precisely determined using Bragg x-ray diffraction.

Vibration Modes

The quartz plate, like all plates, has many modes of vibration.
There are three basic modes of vibration as illustrated in the following
figure.

(a)  Flexure
mode (bending or bowing);  Cuts: 5° X, NT;  Frequency ~ 100
kHz.

(b)
Extensional Mode (displacement along the length of the plate); Cuts:
MT, GT;   Frequency: 40 – 200 kHz.

(c)  Shear
Mode (sliding two parallel planes in opposite directions).  This mode
is subdivided into:

Face Shear;  Cuts: CT, DT;
Frequency: 100 – 600 kHz.

Thickness Shear;  Cuts: AT, BT,
SC;  Frequency: 1 – 30 MHz (fundamental mode); 30 – 90 MHz
(3rd harmonic overtone mode); 60 – 150 MHz (5th
harmonic overtone mode); etc.

Properly oriented electrodes excite the desired mode of vibration.
Although a large number of different cuts have been developed, some
are used only at low frequency, others are used in applications other than
frequency control, and still others have been made obsolete by later
developments.  Except for the low-frequency tuning fork resonators
used in quartz watches and clocks, almost all quartz resonators in today’s
applications use a thickness-shear mode.  At frequency above
approximately 1 MHz, AT-cut and SC-cut are primarily used.  For
frequencies below about 1 MHz, thickness-mode resonators generally become
impractical because of size, since the blank diameter must be much greater
than the thickness.

AT- and SC- cut Resonator Frequency

The fundamental frequency of a thickness-shear resonator is inversely
proportional to its thickness.  The thinner the crystal blank, the
higher the frequency. There are additional resonance at the 3rd, 5th,
etc. harmonic overtones, whose frequencies are approximate, but not exact,
odd multiples of the fundamental resonance frequency.  Resonators are
ordinarily designed to optimize the characteristics of one or another of
these resonances, such as the fundamental or the third overtone, but the
other overtones necessarily still exist. AT-cuts are commonly
manufactured in the frequency range from 1 MHz to 250 MHz and above, and
in this range are usually the optimum choices for most applications.
However, the AT-cut is sensitive to stresses in the body of the resonator,
whether caused by temperature gradients due to rapid external temperature
changes, or by external forces. For applications where extreme
stability is required, this stress sensitivity is a drawback, and newer
crystal cuts such as the SC (Stress Compensated)-cut have been developed
that minimize these effects.  An SC-cut crystal is one of a family of
double rotated crystals (quartz crystals cut on an angle relative to two
of the three crystallographic axes).  Others in the family include
the IT-cut and FC-cut.  The SC-cut represents the optimum double
rotated design as its particular angle ~ 35°15 ø ~ 21°54 provides maximum
stress compensation. The SC-cut is also a thickness mode resonator,
and as such is basically available in much the same frequency range as the
AT-cut, but its commercial availability is much more limited due to
difficult manufacturing process.

Equivalent Circuit

A quartz crystal resonator is a mechanically vibrating system that is
linked, via the piezoelectric effect, to the electrical world.  It
consists of a quartz plate with metal plating (electrodes), which is
located on both sides of the quartz plate and is connected to insulated
leads on the crystal package.  Although the theoretical analysis of
this device is a relatively complex electro-mechanical function, it can be
expressed in terms of a simple equivalent circuit in the vicinity of the
resonance frequencies as shown below:

Equivalent electric circuit of quartz crystal resonator

The C0, called the “shunt” or static capacitance, is the
capacitance due to the electrodes on the crystal plate plus the stray
capacitances due to the crystal enclosure.  Shunt capacitance is
present whether the crystal plate is oscillating or not (unrelated to the
piezoelectric effect of the quartz).  The R, C1 and
L1 portion of the circuit is known as the “motional arm”, which
arises from the mechanical vibrations of the crystal.  R represents
the equivalent motional arm resistance; C1 represents the
motional capacitance of the quartz; and L1 is the motional
inductance, a function of the mass.  The C0 to
C1 ratio is a measure of the interconversion between electrical
and mechanical energy stored in the crystal, i.e., of the piezoelectric
coupling factor, k.  C0/C1 increases with the
square of the overtone number.  When a dc voltage is applied to the
electrodes of a resonator, the capacitance ratio
C0/C1 is also a measure of the ratio of electrical
energy stored in the capacitor formed by the electrodes to the energy
stored elastically in the crystal due to the lattice strains produced by
the piezoelectric effect.

When working with most crystal resonators, it is only necessary to
specify one motional component or the other due to the absolute
correlation between C1 and L1
(C1*L1 is constant for a given series resonant
frequency).  The industry standard is to specify a proper value of
C1 only.   The actual value of C1 has physical
limitations when it is realized in a quartz crystal design.  These
constraints include the quartz cut, the mechanical design, the mode of
operation and the nominal frequency of the crystal resonator.

Although the equivalent circuit appears relatively simple, determining
the several characteristic frequencies useful in describing resonator
equivalent circuit properties is surprisingly complex.  For the most
part, simple approximations are used.  The following notation is
useful in determining some of these frequencies.

Fs (Series resonance frequency) = 1 / [2p(L1C1)1/2]

Fp (Parallel Resonance frequency) = fs[1 + 1/(2g)]

g
(Capacitance ratio) = 2pfsC0
/ C1

Q
(Quality factor) = 2pfsL1
/ R1

M
(Figure of merit) = Q / g

Series vs. Parallel Resonance

The frequency at which oscillation occurs defined by fs is
called Series Resonance Frequency.  The reactance/impedance curve of
a crystal shown below reveals where mechanical resonance occurs.

Series resonance occurs at the point the curve crosses zero.  At
this point, the crystal appears resistive in the circuit, impedance is at
minimum and current flow is at maximum.  As the frequency is
increased beyond the point of series resonance, the crystal appears
inductive in the circuit.  When the reactance of the motional
inductance and shunt capacitance cancel, the crystal is at the frequency
called Anti-resonance Frequency denoted as (fp).  At this
point, impedance is maximized and the current flow is minimized.  The
frequency of a crystal at fp is inherently unstable and should
be never selected as the frequency of operation for an oscillator.
The region between fs and fp is typically called “Area of Parallel
Resonance”.  Parallel resonance can occur by adding load capacitance
to the crystal in series resulting a positive frequency shift determined
by:

Df
= fsC1/2(C0 + CL)

The unavoidable presence of Co in the equivalent circuit
produces this anti-resonance, sometimes also called parallel
resonance.  In this connection, r = C0/C1 is an
important resonator parameter because it is inversely proportional to the
spacing between resonance and antiresonance and thereby determines the
maximum bandwidth in filters and the tuning range in oscillators.

Overtones

Quartz crystals naturally vibrate in several simultaneous resonance
modes referred to the fundamental or overtone modes. Usually, one of
these modes is designed to be dominant at the desired operating
frequency.  The fundamental frequency of vibration is a function of
the resonator physical dimension and angle of cut.  The overtone
modes occur at odd numbered harmonics of the fundamental mode and include
the 3rd, 5th, 7th, 9th, and
11th. The maximum bandwidth obtainable in a filter and the
maximum tuning range in an oscillator are inversely proportional to the
capacitance ratio, r = Co/C1, and r increases as the
square of the overtone.  Consequently, a wider bandwidth or larger
tuning range can be obtained with a fundamental mode resonator than with a
third or higher overtones.  Fundamental mode resonators are used for
most filters, temperature compensated oscillators (TCXOs) and voltage
controlled oscillators (VCXOs), where the required bandwidth or tuning
range makes overtone devices undesirable.  Fundamentals are also used
in many simple oscillators, such as clock oscillators at frequencies up to
approximately 35 MHz.  At higher frequencies overtones are more
economical for this application.

Current crystal manufacturing processes limit the lapping of the quartz
plate so that the highest fundamental mode frequency that may be reliably
achieved is typically around 45 MHz.  At this frequency, the AT-cut
quartz plate is less than 0.037 mm thick and further lapping using
conventional techniques is not practical.  Several methods have been
developed to increase the fundamental mode frequency achievable by
removing some quartz mass from the center of the plate.  This
so-called “inverted mesa” provides for an active area that is much thinner
and is usually achieved by chemical or plasma/ion etching.  These
processes can produce superior quality high frequency fundamental (HFF)
mode crystal to 170 MHz and beyond.  Fundamental mode crystals
typically have larger values of C1 than overtone mode crystals of the same
frequency; therefore they are useful for applications such as VCXOs where
greater pullability is required.  High frequency fundamental quartz
blanks are also used extensively in filter applications where they provide
better spurious mode response than overtone crystals of the same
frequency.

At a given operating frequency, quartz crystals aging and Q improve
with higher overtone.  For this reason, ovenized oscillators (OCXOs)
often use overtone resonators.  Usually the 3rd or 5th overtone is
used.  The range of tuning required to accommodate the resonator
frequency tolerance and aging characteristics limits the maximum useful
overtone.

Spurious Modes

Vibration at frequencies that are not fundamental or overtone modes are
referred to as spurious or unwanted modes. The design of the wafer,
electrode pattern and amount of metalization can be adjusted to suppress
these unwanted modes.

Spurious modes can be a problem if the response is as strong as the
main mode.  When that happens, the oscillator may run on the spur
instead of the main mode.  This is called mode hopping.
Spurious modes should be specified as either a resistance ratio to the
main mode or dB suppression.  In general, a resistance ratio of 1.5
or 2.0 to 1, which is approximately equivalent to – 3 dB to –6 dB, is
sufficient to avoid mode hopping for most oscillators.

Fundamental modes of a crystal can achieve the best spurious
suppression, while overtone responses are more difficult to control.
Designs that require higher C1 volumes for pullability reasons
also can sacrifice spurious mode suppression.  For crystal filter
applications, spurious mode suppression as low as –40 dB can be achieved
with fundamental mode with low C1 designs.

The spurious modes occur above the main mode within a few hundred
kilohertz. The response may look like the plot shown above.  In
oscillator applications, the oscillator usually selects the strongest
mode.  Some of the unwanted modes may have steep frequency vs.
temperature characteristics.  Occasionally, as the temperature
changes, at a certain temperature, the frequency of an unwanted mode
coincides with the oscillator frequency, which causes so-called “activity
dip”.  At the activity dip, excitation of the unwanted mode results
in extra energy dissipation in the resonator, which results in a decrease
in the Q, an increase in the equivalent series resistance, and a change in
the frequency of the oscillator. When the resistance increase is
sufficiently large, the oscillation may stop, i.e., the oscillator
fails.  When the temperature changes away from the activity dip
temperature, the oscillation can restart.  Unwanted modes can be
controlled by proper design and fabrication methods.  Maintaining the
correct relationships among electrode and resonator plate dimensions
(i.e., applying energy trapping rules), and maintaining the parallelism
between the major faces of the resonator plate, can minimize the unwanted
modes.

Frequency-Temperature Characteristics

The frequency-temperature characteristic defines how the resonant
frequency of the quartz crystal resonator varies in response to changes in
temperature.  For both AT- and SC-cut resonators, it is found that
the frequency shift due to the variation of temperature can be expressed
in a cubic curve of the form:

Df/f0
= a0(T – T0) + b0(T –
T0)2 + c0(T – T0)3

T is the temperature variable and T0 is the inflection
temperature, which is approximately 25°C for the AT-cut and 92°C for the
SC-cut.  The coefficients a0, b0, and
c0 are the first -, second-, and third order temperature
coefficients of frequency, and they are constants that depend on quartz
properties and the angle of cut.  The above equation gives a family
of curves as shown below for four values of the relative cut angle.
These curves show that a resonator can be designed to give a relatively
small frequency variation over a broad temperature range.

AT-cut Frequency vs. Temperature Curves

SC-cut Frequency vs. Temperature Curves

Both cuts are useful for temperature controlled (ovenized)
applications.  In addition, since the AT-cut inflection temperature
(To) is approximately 25°C, it is widely used for uncontrolled
temperature applications such as filters and non-ovenized
oscillators.  Both cuts vibrate in thickness modes that can be
designed to have very low mounting losses and hence provide exceptional Q
(up to 1.5 x 106 at 10 MHz and 35 x 10³ at 300 MHz) and
stability with time (example 1 x 10-10/day at 10 MHz).

Aging and Drift

Based on the terms and definitions recommended by the CCIR
(Consultative Committee on International Radio) , aging and drift can be
defined in the following way.  Aging is the systematic change in
frequency with time due to internal changes in the oscillator, that is,
the frequency change with time when factors external to the oscillator
(environment, power supply, etc.) are kept constant.  Drift is
defined as the systematic change in frequency with time of an oscillator,
that is, drift is due to a combination of factors, i.e., it due to aging
plus changes in the environment and other factors external to the
oscillator. Aging is what one specifies and what one measures during
oscillator evaluation.  Drift is what one observes in an
application. For example, the drift of an oscillator in a spacecraft
is due to (the algebraic sum of) aging and frequency changes due to
radiation, temperature changes in the spacecraft, and power supply
changes.

There are many interrelated factors involved in aging.  Some of
the most common are: internal contamination, crystal surface changes, wire
fatigue, small irreversible changes in the crystal lattice, out gassing of
the materials, various thermal effects, mounting stresses, and
over-driving the crystal.  Typical aging figures for metal enclosure
crystal units operating in the frequency range of 10 to 20 MHz are 1.0 –
5.0 ppm / 1st year; while the values for glass enclosure
crystals are 0.1 – 1.0 ppm / 1st yesr.

AT-cut vs. SC-cut

Of many crystal-cut types available, most applications use either
AT-cut or SC-cut quartz resonators, and in many instances both cuts are
candidates.  It is worthwhile to compare them on the basis of eight
characteristics:

Turnover Temperature:

For OCXO applications the crystal may be operated at either of its two
turnover temperatures.  Here
the SC-cut is superior to the AT-cut in two important ways.  First, the SC-cut’s f - T curve is
much flatter near turnover temperature than the AT-cut’s determined by the
three temperature coefficients.
Second, the true SC-cut is much less sensitive to temperature
gradients, allowing much faster warm-up.  These qualities make it the
preferred choice for precision oscillator applications.

Static f - T Characteristic:

Both AT- and SC-cut resonators have a static frequency-temperature
characteristic that is well described by a third-order polynomial in
temperature.  For the AT-cut, the inflection temperature [at which
the curvature changes sign] is within a few degrees of room
temperature.  In other words, the f - T curve is approximately
anti-symmetric about a temperature close to 25°C.  This makes the
AT-cut suitable for non-temperature-controlled applications, such as
simple oscillators and TCXOs.  By contrast, the inflection
temperature for the SC-cut resonator is 92°C.  Most SC-cuts are used
in OCXOs, where the higher inflection temperature results in an important
advantage, that is, the f-T curve is quite flat in the vicinity of the
oven set point, which is adjusted to the resonator upper or lower turnover
temperature.  Consequently, temperature control is less critical than
that an AT-cut crystal was used.  Other advantages of the SC-cut from
its f – T characteristics are its relative freedom from the activity dips.

Dynamic f-T Characteristic (Thermal Transient Effects):

When a step change is made to the ambient temperature, the frequency of
an SC-cut resonator changes smoothly, without overshoot or ringing, in a
manner corresponding to a critically- damped system.  The AT-cut
dynamic f - T characteristics, on the other hand, has very pronounced
overshoot and rubber band effect.  An important reason for choosing
an SC-cut for an OCXO is its greatly improved dynamics, which is, in fact,
the hallmark of properly designed SC-cut based OCXOs.

Aging:

If we compare equal frequency and overtone, the SC-cut is slightly
better than the AT-cut, on account of its slightly greater
thickness.  This difference, however, is usually insignificant.
Because the greater current-handling capability of the SC renders its
frequency less sensitive to current changes, in oscillators the SC-cut may
exhibit better aging.

Current Handling:

This refers to the maximum current at which a resonator can be operated
without a significant (reversible) change in frequency.  Generally,
this is markedly higher for SC-cut than for AT-cut resonators.  The
consequences for oscillator applications are an improved phase noise floor
by operating a higher current, and reduced sensitivity to drive level
change, which may affect aging of the oscillator frequency.

Impedance Level:

If we compare AT-cut and SC-cut resonators having the same frequency
and overtone, the motional inductance (L1) and motional
resistance (R) of the SC-cut resonator will be significantly higher than
for the AT-cut resonator, while the motional capacitance (C1)
will be lower in the same proportion.  The static capacitance
(C0), however, will be nearly the same for the two.  For
oscillator applications, high impedance is desirable if aging is
important, because it reduces the effect of the sustaining circuit on the
oscillator frequency.  The trade-off is that high resonator impedance
reduces the tuning range of the oscillator frequency, limiting corrections
for manufacturing tolerance and for aging.  In VCXOs, aging is, at
most, a secondary consideration, so that the required tuning range will
determine the highest practical overtone.  For either cut, the
impedance level increases roughly as the square of the overtone for a
given frequency, but also depends upon details of the resonator design.

Size:

For most applications, the sizes of the AT-cut and SC-cut resonator
packages are the same.

Cost:

Because of tighter orientation tolerances, the SC-cut is more expensive
to manufacture than the AT-cut, but for high-performance applications this
cost is more than offset by savings in oven complexity, and for many
applications the SC-cut is the only choice.  For some applications
where thermal transient characteristics are less important, a modified
(near) SC-cut may be used.  This provides essentially the same f - T
curve as the true SC-cut but allows looser orientation tolerances, thereby
providing some cost savings at the expense of thermal transient
performance.  Normally, cuts having To much removed from
92°C will not provide the thermal transient performance for which the true
SC-cut is noted.

Quartz Turning Folk Resonators / Watch Crystal

The requirements of quartz resonators for wristwatch are: small size,
low power dissipation (including the oscillator), low cost, and high
stability (temperature, aging, shock, attitude).  These requirements
can be met with 32.768 kHz quartz turning fork resonators.  It is
interesting to know that 32768 = 215 or 1 Hz =
32768/215.  In an analog watch, a stepping motor receives
one impulse per second which advances the second pin by 6, i.e.,
1/60th of a circle every second.  The 32.768 kHz is a
compromise among size, power requirement (i.e., batter life) and
stability.

Quartz wristwatches are sufficiently accurate, usually, while worn as
intended, i.e., on the wrist for ~16 h and off the wrist for ~8 h each
day. The accuracies degrade when the watch is off the wrist for
extended periods. The further the storage temperature is from the
optimum temperature, the faster the watch loses time.  At temperature
extremes, e.g., in a freezer at - 55°C, or at the temperature of boiling
water, wristwatches lose about 20 s per day.

The angle of cut of the resonator used in wristwatches is such that the
zero temperature coefficient is at ~ 25°C, as shown in the following f – T
curve. This has been found to provide the highest probability of
accuracy, based on the typical durations and temperatures while the watch
is on the wrist and while it is off the wrist.

Type of Seal for Quartz Crystal Resonators

Solder Seal

Solder seal packages have several advantages.  They have lower
lead-to-case capacitance than other types and therefore used in some
filter designs that require minimal junction capacitance.  In
addition, the packages can be opened for rework that is sometimes helpful
in meeting difficult filter requirements.  Their main disadvantage is
that they cannot be sealed without introducing contamination, and
therefore they tend to have relatively poor frequency stability.

Resistance Weld

These packages require more sophisticated sealing equipment than solder
seal, but the sealing process introduces less contamination, resulting in
better frequency stability.  Resistance weld packages are available
in a wide range of styles, and all are economical choices for long-term
performance

Seam Weld

Seam weld sealing is the most widely used method today in producing
ceramic base/metal cover low profile SMD crystals and
oscillators. Performance and cost are similar to those resistance
weld packages.

Epoxy Seal

Epoxy seal packages are widely used for high volume and low cost SMD
crystals.  The frequency stability performance is not as good as
those seam weld and resistance weld parts.

Cold Weld

Cold weld sealing is a precision performance package that is somewhat
more expensive, but sealing process introduces virtually no contamination,
resulting in superior resonator frequency stability and electrical
performance.

All-Quartz Package

An all-quartz packaging technique is used in producing a surface mount
resonator having very low profile and very high shock resistance. The
availability of this type of sealing package is rather limited.

Detailed Frequency Temperature Curve for Fundamental Mode "AT-cut"
Crystals

The above curves represent the frequency vs. temperature
characteristics of a series of AT-cut crystals.  Each numbered curve
is accomplished by controlling the angle of the crystal cut in reference
to the angle (35° 15' ) and each curve is one minute of change.  LTP
indicates the Lower Temperature Turnover Point; while UTP indicates the
Upper Temperature Turnover Point.