Piezoelectric crystals have been used for many decades as frequency control elements in radio communication devices because of their stable resonant frequency signal generation during operation.  The resonant frequency of a particular piezoelectric crystal is dependent on its vibrational mode of operation, its thickness, the density, and the elastic coefficients of material.  Each of these parameters vary with changes in temperature.  Therefore, the resonant frequency of the piezoelectric crystal changes with temperature. 
There are no piezoelectric angle cuts that provide perfectly flat frequency-temperature curves (i.e. a curve demonstrating no frequency variation over a particular temperature range of interest).  However, there are several known cuts of quartz that thermally compensate the crystal blank to have a relatively stable temperature performance.  These include the AT, BT, GT and SC cuts, among others.  Of these, the AT-cut is the predominantly used cut and ideally exhibits a frequency-temperature curve for a thickness-shear vibrational mode which should be recognized by those skilled in the art as the familiar Bechmann curve.  The AT-cut also should ideally exhibit a substantially uniform resistance-temperature curve. 
In the design of quartz crystal resonators, after properly designing the physical dimensions of the blank, the next most important issue is the proper design of the electrodes.  The common approach to designing the electrodes for AT quartz crystals is to use the mass and dimensions of the electrodes to suppress undesired vibrational modes and inharmonics of the desired thickness-shear mode of the crystal. 
The inharmonic modes are addressed by making the frequency of vibration of the desired mode low enough in the electroded region that it can not propagate outside of the electrodes, while at the same time assuring that the frequency of the inharmonic mode in the electroded region are high enough in frequency that they can propagate out of the electroded region.  In this way the desired mode is said to be "trapped" under the electroded region.  In this technique the inharmonic modes can propagate outside of the electrode to the mounts (or elsewhere on the blank) and be dampened.  The coupling to these inharmonic modes is decreased within the electroded region precisely because the inharmonic modes are dampened at the edges. 
In practice, the undesired vibrational modes that arise in piezoelectric crystals, such as AT-cut quartz for example, disturb the frequency-temperature and/or resistance-temperature performance of the crystal.  These undesired vibrational modes cause disturbances, or "activity dips", in the frequency-temperature and/or resistance-temperature curves of the crystal.  This results in a sudden and undesirable shift in frequency and/or resistance as the crystal changes temperature.  This problem occurs in about 2-7% of AT-cut quartz crystals and causes serious difficulties for temperature compensation schemes and circuitry required to normalize the temperature variation of the quartz crystal, such as in a temperature compensated crystal oscillator (TCXO) application, for example. 
An example of undesirable vibrational modes are face-shear and flexure modes that have frequencies near that of the desired thickness-shear vibrational mode.  These undesirable modes exhibit their own frequency-temperature and/or resistance-temperature curves which are typically much steeper than the Bechmann curve.  Where these curves intersect the Bechmann curve, vibrational coupling occurs which disturbs the Bechmann response.  These disturbances, or activity dips, distort the frequency-temperature and/or resistance-temperature curves such that typical temperature compensation schemes can no longer compensate the higher-order perturbations caused by the activity dips. 
There is a need for a piezoelectric resonator with a reduction of activity dips in a frequency-temperature and/or resistance-temperature curve, that can be realized in a simple, readily manufacturable form, at a low cost and high yield.