INTRODUCTION

Quartz resonators are widespread components of modern microelectronics, which are used to filter and stabilize the frequency of ultrasonic and electromagnetic oscillations. Quartz piezoelectric (PE) elements based on the AT cut, which are characterized by the high mechanical quality and temperature stability, have become especially popular. Due to the fact that the first and second derivatives of the resonance frequency are zero for this cut, such PE elements can be used both in electronic devices for frequency stabilization and filtering, and as various sensors [2–5].

At the same time, the calculation of vibrations of anisotropic quartz plates under conditions of the interaction of thickness-shear, thickness-torsional, bending, longitudinal, and contour vibrations is a nontrivial task that can be solved analytically only under conditions of significant assumptions [6–12]. The result of such calculations is the ranges of optimal sizes of crystal plates that ensure a minimum coupling of working thickness-shear vibrations with other types of vibrations.

In particular, the natural frequencies of thickness-shear vibrations are determined by the formula [13]:

$$f = \,\,~\frac{1}{{2b}}\sqrt {\frac{{c_{{66}}^{'}}}{{{\rho }}}} ,$$
(1)

where b is the plate thickness, ρ is the quartz density, and \(c_{{66}}^{'}\) is the elastic modulus of quartz for the rotated coordinate system corresponding to the AT cut.

For practical calculations, in order to avoid an undesirable coupling between thickness-shear vibrations and bending vibrations of a PE element, an empirical formula is used that describes the condition for the coincidence of an overtone of the frequency of bending vibrations with the frequency of the working thickness-shear vibration [14]:

$${{f}_{{{\text{bend}}}}} = \frac{{{{N}_{{{\text{bend}}}}}n}}{{{{l}_{x}}}} = {{f}_{{{\text{sh}}}}} = \frac{{{{N}_{{{\text{sh}}}}}}}{t},$$
(2)

where \({{f}_{{{\text{bend}}}}}\) and \({{f}_{{{\text{sh}}}}}\) are, respectively, the frequencies of bending and thickness-shear vibrations; n is an overtone (harmonic) of bending vibrations; \({{N}_{{{\text{bend}}}}}\) = 1338.4 kHz mm is the frequency coefficient of bending vibrations; \({{N}_{{{\text{sh}}}}}\) = 1660 kHz mm is the frequency coefficient of thickness-shear vibrations; \({{l}_{x}}\) is the plate length along the Х crystallographic axis, mm; and t is the plate length, mm.

Numerical calculations of more complex interactions of various types of vibration require a greater computing power and accurate accounting of a significant number of resonator parameters and, at the same time, do not allow taking the real crystal structure into account, i.e., the effect of defects on the field of emerging deformations.

In this regard, methods for visualizing deformations and recording their distribution under operating conditions of resonators (in the in-operando mode) are extremely in demand from the viewpoint of controlling the process of manufacturing and optimizing the design of these components. A special place among them is occupied by the X-ray topography method, which is sensitive to atomic displacements at a level of several ten thousandths of an Angstrom. In combination with the high penetrating power of X rays, this method allows obtaining spatially resolved information from the entire volume of a sample. In particular, the X-ray topography method makes it possible to visualize the deformation behavior of resonators during their use [15, 16], identify the role of defects [17, 18], and establish the relationship between the geometry of the device and the vibratory processes that determine its functional characteristics [19, 20].

This work is devoted to studies of the electroacoustic parameters and X-ray photopographic patterns of quartz resonators of the AT cut with different sizes and configurations, which determine different ratios between the amplitudes of the main and parasitic modes of vibrations.

INVESTIGATED SAMPLES

The PE elements studied in this work are rectangular monocrystalline plates of α-quartz of thermostable AT cut. This cut corresponds to the rotation of the coordinate system around the X axis (the [110] crystallographic direction) by 35.25°. The orientation of the XZ ' surface is close to the (1-11) crystallographic plane. The matrix of PE moduli for quartz is calculated for the rotated XY 'Z ' coordinate system (Table 1) relative to the tabular values presented in [21].

Table 1.   Matrix of PE moduli d' for quartz in picocoulomb per newton for the rotated XY 'Z ' coordinate system corresponding to the AT-cut. The PE moduli activating shear deformations TXZ' and TXY' upon application of an electric field E along the Y ' direction are highlighted in bold
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Quartz resonators (QRs) of two types (Fig. 1) with different operating frequencies were studied (Table 2). In the PE elements of type 1 resonators (RK563), the X axis is directed along the longer side of the plate (its length), and the Z ' axis is directed along the shorter side (width). In PE elements of type 2 resonators (RK319), on the contrary, the plate width corresponds to the X direction, while Z ' corresponds to its length. To attenuate the coupling between the main thickness-shear vibration and undesirable vibrations, the PE elements of type 1 resonators are shaped in the form of flat plates with chamfers (the shape of biconvex lenses).

Fig. 1.
figure 1

General view and overall dimensions of the resonators: (a) RK563 (type 1) in the surface-mounted housing and (b) RK319 (type 2) in the housing of the quarter-wave resonator.

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Table 2.   Main parameters of the investigated samples of two types
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EXPERIMENTAL TECHNIQUES AND EQUIPMENT

The X-ray topography method was used to visualize the spatial distribution of deformations occurring in quartz PE elements during their operation. This method is based on recording the pattern of the spatial intensity distribution of reflected radiation from a certain family of crystal planes and is a unique highly sensitive tool for visualizing dynamic deformations in crystals that occur, in particular, during activation of the PE effect and ultrasonic impacts. An X-ray topogram also contains information about the defective crystal structure.

The sample exposed at the Bragg angle that corresponded to the system of the studied (220) crystal planes was completely illuminated with a wide uniform X-ray/synchrotron beam. A two-dimensional detector with a wide aperture was installed behind the crystal. Diffracted radiation entering the receiving window of the detector mounted at the double Bragg angle of the sample forms an image of the illuminated area of the studied sample on it. The deformed regions of the crystal distort the initial wavefront of the weakly divergent incident beam, thereby achieving a contrast of diffraction topograms. There are many experimental X-ray optical schemes for the topography of crystalline objects with sensitivities to different features of a real structure and different resolutions [22, 23].

Topograms of the PE elements of type 1 resonators were obtained at the “Mediana” experimental station of the KISI-Kurchatov synchrotron radiation source (Research Center “Kurchatov Institute”) (Fig. 2a). The radiation spectrum is determined by the parameters of the radiating rotary magnet and lies in a wide energy range, reaching a maximum intensity at E ≈ 8–14 keV. A polychromatic beam has an intensity of up to 2 × 109 pulses/(s cm2) and a low divergence of ~10 μrad. Thus, due to the amplitude of variation of the interplane distance Δd/d in some regions of the resonator crystal corresponding to the vibration antinodes, a contrast in the intensity distribution of diffracted radiation at its fixed angular position is achieved. A detector with a pixel size of 10 μm was used to register topograms. Due to the absence of angle changes during recording of a topogram in the specified scheme, a picture with a high degree of detail is obtained on the detector. Thus, the topography on a white beam makes it possible to register the fine vibration structure of a PE element.

Fig. 2.
figure 2

Measurement implementation schemes: (a) topography on a white beam in a single-crystal diffraction pattern implemented at the KISI-Kurchatov Median synchrotron station, (1) rotating magnet, (2) input mask of a white beam, (3) aluminum filter for absorption of the long-wavelength part of the spectrum, (4) goniometric system, (5) sample with a holder, (6) two-dimensional detector, (7) electric signal generator/analyzer; (b) topography on a monochromatic beam in a two-crystal diffraction scheme implemented on a TRS-K laboratory three-crystal X-ray spectrometer: (1) X-ray tube, (2) precollimation slit, (3) asymmetric single-crystal monochromator, (4) goniometric system, (5) sample with the holder, (6) two-dimensional detector, and (7) generator/electrical-signal analyzer.

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To measure the topograms of type 2 PE elements, an upgraded installation of a three-crystal X-ray spectrometer (TXS) was used that was equipped with a two-dimensional AdvaPIX TPX3 detector with a pixel size of 55 μm (Fig. 2b). The X-ray source was an X-ray tube with a molybdenum anode with a power of up to 2.5 kW and a radiation wavelength of E[MoKα1] = 17.4798 keV. A wide parallel monochromatic beam was obtained using an asymmetric Si 440 monochromator with a Bragg angle of 21.679° and an asymmetry coefficient of 0.025, which was installed behind the collimation slit with a 150-μm aperture. This makes it possible to obtain a homogeneous beam up to 6 mm wide with a spectral divergence not exceeding the characteristic line width at the monochromator output. Topograms from the PE element were obtained by scanning the sample along the ω axis near the Bragg angle in the integral-intensity accumulation mode.

The use of both synchrotron and laboratory X-ray sources gives qualitatively similar pictures. However, the combination of a brighter synchrotron beam and a detector with a smaller pixel size allows for a more detailed analysis of strain distributions profiles for microminiature products. The use of a synchrotron justifies itself if it is necessary to obtain quantitative information when testing the designs of new PE elements due to the high demand for the beam time of these unique installations.

In both cases, the amplitude–frequency characteristics were determined for an electrical circuit with a test load of 50 Ω connected in series to the resonator using an electrical-signal analyzer with a built-in Rigol DSA1030 generator. When topograms were recorded, the generator frequency was fixed at a value corresponding to the crystal resonance.

In PE elements of the resonators of both types 1 and 2, thickness-shear deformations occur in the XZ ' plane due to the activation of the PE modulus \(d_{{25}}^{'}\) and in the XY ' plane due to the activation of the PE modulus \(d_{{26}}^{'}\) during vibrations at the main harmonics. Topograms obtained in the Laue geometry (transmission) from the system of (220) crystal planes are sensitive to deformations in the XY ' plane due to the fact that the directions of atomic displacements lie in the diffraction plane. The Laue diffraction geometry makes it possible to study vibrational processes in the entire volume of the resonator crystal plate.

EXPERIMENTAL RESULTS AND DISCUSSION

Topograms of PE elements of both types of the studied resonators were preliminarily recorded in the unexcited state (Fig. 3). The result showed the presence of static deformation fields corresponding to defects in the crystal lattice. However, the initial deformation fields do not significantly affect the distribution of deformations under conditions of vibrations at the resonant frequencies described below and do not exceed their magnitudes.

Fig. 3.
figure 3

(a) Schematic representation of the crystallographic orientation of the quartz AT cut; (b, c) the initial topograms of the RK563 and RK319 resonators, respectively, obtained under normal conditions (without external influence).

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Investigation of the Strain Distribution in Piezoelectric Elements of Type 1 Resonators (RK563)

Figure 4 shows the topogram and graphs of the amplitude distributions of deformations of the PE element with a nominal frequency of 10 MHz.

Fig. 4.
figure 4

(a) Topogram and (b, c) graphs of the distributions (in two directions) of the amplitude of PE-element deformations at a resonant frequency of 9.99217 MHz corresponding to the first harmonic.

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In the Z ' axis direction of the PE element, seven vibration antinodes in the form of bands are positioned (Fig. 4a). Such a deformation pattern corresponds to the appearance of a standing wave of thickness-shear deformations that is excited in the electrode region and propagates beyond its limits over the entire crystal volume.

When the PE element is excited at the third harmonic of thickness-shear vibrations at a frequency of 31.73469 MHz, the deformation amplitude distribution pattern (Fig. 5) is close to the analogous pattern at the first harmonic obtained at an operating frequency of 9.99217 MHz. Seven vibration antinodes are preserved in the PE element along the Z ' axis.

Fig. 5.
figure 5

(a) Topogram and (b), (c) graphs of the distributions (in two directions) of the amplitude of PE deformations of the type 1 resonator at a resonant frequency of 31.73469 MHz corresponding to the third harmonic.

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For a resonator with a nominal frequency of 10 MHz, topograms were recorded at an anharmonic-vibration frequency of 10.43383 MHz (Fig. 6). The graphs of the amplitude distribution of vibrations indicate the presence of interference of waves occurring along the X and Z ' axes of the crystal plate.

Fig. 6.
figure 6

(a) Topogram and (b, c) graphs of the distributions (in two directions) of the amplitude of PE-element deformations at a resonant frequency of 10.43383 MHz corresponding to the anharmonic vibration.

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In the case of weak coupling of working thickness-shear vibrations to bending and contour vibrations, the main type of side vibrations are anharmonic vibrations, which result from the interaction of vibrations of different types [13].

The frequencies of anharmonic oscillations are near the operating mode and can be calculated using the Sykes formula [24]:

$$f = \frac{1}{2}\sqrt {\frac{{c_{{66}}^{{\text{'}}}}}{\varrho }} \sqrt {\frac{{{{m}^{2}}}}{{{{b}^{2}}}} + \frac{{{{k}_{1}}{{n}^{2}}}}{{{{L}^{2}}}} + \frac{{{{k}_{2}}{{{\left( {p - 1} \right)}}^{2}}}}{{{{a}^{2}}}}} ,$$
(3)

where k1 and k2 are constant quantities determined experimentally; m, n, and p are integers; and b, L, and a are the plate thickness, length, and width, respectively.

The anharmonic vibration presented in the topogram (Fig. 6) can be denoted with the parameters m = 1, n = 4, and p = 1, since it operates at the first harmonic at a frequency of 10.43383 MHz, has four antinodes along the Х axis, and one standing wave in the Z  ' axis direction.

The deformation amplitude distribution pattern in the form of a two-dimensional grating for a PE element with a nominal frequency of 14 MHz is of special interest (see Fig. 7).

Fig. 7.
figure 7

(а) Topogram and (b), (c) graphs of the distributions (in two directions) of the amplitude of PE-element deformations at a resonant frequency of 13.99129 MHz corresponding to anharmonic vibrations.

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The numbers of small square antinodes are as follows: 18 rows along the crystallographic X axis and nine rows along the crystallographic Z  ' axis are proportional to the dimensions of the plate length (3.5 mm) and width (1.8 mm), thereby also indicating the interference nature of the observed deformation pattern.

The frequency coefficient of vibrations is specified by the following ratio [24]:

$$N = \frac{{f~l~\, \times 2}}{m},$$
(4)

where f is the operating frequency of the PE element in kilohertz; m is the number of halfwaves (vibration antinodes); N is the frequency coefficient of vibrations in kHz mm; and l is the size along the crystallographic Z ' axis in millimeters.

The calculated frequency coefficient was 5600 kHz mm. It is significantly larger than that of undesirable bending, contour, and longitudinal vibrations (1338, 2500, and 2800 kHz mm, respectively), which allows avoiding the interaction of these types of vibrations with each other. Consequently, the deformation pattern in the form of a two-dimensional grating (Fig. 7) results from the wave interference and corresponds to the ratio of the dimensions of the PE-element length and width.

Investigation of the Strain Distribution in Piezoelectric Elements of a Type 2 Resonator (RK319)

Type 2 resonators use coupled thickness-shear oscillations with bending harmonics. Such resonators are not available in the catalogues of domestic and foreign companies, and the method for calculating the optimal plate sizes differs from those given in [6–12]. According to the TU (technical specifications) RK319 method [25], there are nonoptimal l/t ratios that lie in the intervals between optimal values corresponding to different operating frequencies of resonators. However, calculations according to formula (1) show that not all frequencies within a nonoptimal range lead to undesirable longitudinal or contour vibrations. In this regard, it is an important task to further optimize the RK319 size range.

In order to verify the calculations, quartz resonators with a nonoptimal nominal frequency of 12.9 MHz were manufactured. In Table 3, the Q factor of the manufactured resonators is compared with the Q factor of serial resonators with a nominal frequency 12.0 MHz. Tests have shown that all manufactured resonators meet the requirements of the TU technical specifications for RK319 resonators.

Table 3.   Parameters of resonators with optimal and nonoptimal frequencies according to the TU (technical specifications)
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The measured amplitude–frequency characteristic of type 2 resonators is shown in Fig. 8.

Fig. 8.
figure 8

Amplitude–frequency characteristic of the type 2 resonator.

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Near the operating resonant frequency of 12.91492 MHz, there is a parasitic resonance with a frequency of 13.42367 MHz. Figures 9 and 10 show topograms and graphs of the vibration amplitude distribution of the PE element at these frequencies.

Fig. 9.
figure 9

(a) Topogram and (b, c) graphs of the distributions (in two directions) of PE-element deformations at a resonant frequency of 12.91492 MHz.

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Fig. 10.
figure 10

(а) Topogram and (b), (c) graphs of the distributions (in two directions) of PE-element deformations at a resonant frequency of 13.42367 MHz corresponding to anharmonic vibrations.

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The amplitude distributions of PE elements with a nonoptimal size ratio in the X-axis direction are similar to those presented for PE elements of type 1 resonators with a nominal frequency of 10 MHz. The number of antinodes along the X axis remains equal to 7. Just like the PE elements of type 1 resonators, vibrations excited in the electrode region propagate beyond its limits and, for a small-size plate, they are reflected from the edges. As a result, a pronounced interference pattern is observed. At the same time, bending vibrations also make an insignificant contribution to the deformation pattern in the parasitic mode (Fig. 10).

CONCLUSIONS

This study was devoted to X-ray topographic investigations of the deformation distributions in two types of microminiature quartz PE elements, which arise at the operating resonant frequency, at harmonics that are multiple of it, and at the frequencies of anharmonic thickness-shear vibrations.

When conducting the studies of the quartz PE elements of type 1 and 2 resonators, the presence of vibration antinodes in the form of bands that are directed along the length was established. In the resonator-width direction, a clearly pronounced pattern of the standing deformation wave is observed, whose parameters depend of the geometric dimensions of the crystal plate and the external-signal frequency.

According to the X-ray topography data, it has been revealed that when wave interference occurs along the PE-element length and width, the vibration pattern may take the form of a two-dimensional grating. The number of antinodes along the axes changes as a function of the ratio of the transverse and longitudinal plate dimensions. In this case, the vibration excitation in a PE element at a parasitic mode or at the third resonant harmonic retains the character of the strain distribution pattern and the ratio of the numbers of antinodes along the corresponding axes, as well as at the operating resonator mode. The experimental data demonstrate that the differences from the standardly used ratios of the dimensions do not impair the resonator parameters.

The X-ray topography studies of the resonators in the unexcited state revealed a number of defects and showed that the latter have an insignificant effect on the amplitude distribution of deformation vibrations.

The results obtained within the framework of these studies are of practical importance from the viewpoint of understanding the vibration features of anisotropic plates and can also be used to calculate optimal designs of PE elements, determine the location of defects and mechanical deformations in PE elements, and improve the manufacturing technology of resonators. In particular, visualizing the distribution and localization of the fine structure of deformation fields allows one to conclude that it is necessary to introduce changes into the designs of the PE element, electrodes, or holder when fighting against parasitic resonances.