Abstract
Piezoelectric quartz oscillators are widely used to provide a stable clock signal for watches and other electric circuits. The electrically induced mechanical vibration of quartz will be caused by ionic displacements of cationic Si and anionic O sublattices against each other. However, the transient and small ionic displacements during the mechanical vibration cannot be observed by usual X-ray structure analysis. The electrically induced mechanical vibration of quartz is resonantly amplified under an alternating electric field with the resonant frequency. We have revealed the amplified lattice strain and ionic displacements in a resonantly vibrating quartz crystal under an alternating electric field by time-resolved X-ray diffraction. The details of the experiment and application of the technique to other piezoelectric oscillators are introduced in this chapter.
Keywords
- quartz oscillator
- piezoelectricity
- time-resolved X-ray diffraction
- crystal structure analysis
- langasite
1. Introduction
Inorganic crystals that exhibit piezoelectricity are currently used in a wide range of industrial applications, such as oscillators, sensors, transducers, and actuators. Piezoelectricity is the property of crystals that generates an electric polarization
The most famous and industrially important piezoelectric crystal is quartz (α-SiO2). Quartz is a naturally abundant mineral and can be synthesized artificially by the hydrothermal method. Quartz oscillates mechanically and electrically at a stable frequency, making it widely used in oscillators to provide reference signals for various devices such as quartz watches. However, quartz has the disadvantage that it cannot be used in high-temperature environments because it undergoes a phase transition from the low-temperature α phase to the high-temperature β phase at 573°C.
The authors have been investigating the mechanism of piezoelectricity of quartz and other piezoelectric crystals and synthesizing new piezoelectric crystals such as langasite-type crystals to develop crystals with high functional piezoelectricity superior to quartz crystals. In this chapter, we introduce transient crystal structures of quartz and langasite-type crystals oscillating under an alternating electric field measured by synchrotron X-ray diffraction [1, 2, 3]. Since inverse piezoelectricity is caused by atomic displacements under electric field, measurement of atomic displacements under an electric field is essential to understand the mechanism of piezoelectricity.
Crystal structure analysis based on X-ray diffraction is a powerful tool to measure atomic displacements in crystals, but highly accurate experiments and analysis are required to measure atomic displacements under electric fields. For example, in the case of a quartz crystal with the thickness of 0.1 mm, when a 1 kV potential difference (10 MV/m in an electric field) is applied between the surfaces, the change in the Si−O bond distance (~0.16 nm = 1.6 Å) is estimated to be only about 10−6 nm from its piezoelectric constants. This value is two orders of magnitude less than the standard deviations of bond distances (~10−4 nm) determined by conventional X-ray crystal structure analysis.
2. Time-resolved X-ray diffraction under alternating electric field
In order to measure small atomic displacements in piezoelectric crystals under an electric field, the authors have developed a new method for structure analysis of piezoelectric crystals that utilizes resonance under an alternating electric field [1]. Piezoelectric crystals vibrate mechanically and electrically at a certain natural frequency when an instantaneous stress or electric field is applied. For example, the natural frequency of a common quartz oscillator used in industry is 32,768 (215) Hz. When an AC electric field with a frequency equal to this natural frequency is applied, the piezoelectric crystal resonates, producing mechanical and electrical vibrations with a particularly large amplitude. We hypothesized that the resonance under an AC electric field could amplify atomic displacements in piezoelectric crystals to a magnitude that could be measured by X-ray crystal structure analysis. The magnitude of the amplification effect in resonance depends on the quality factor (
Even if the atomic displacements involved in piezoelectricity can be greatly amplified by resonance under an alternating electric field, it is actually impossible to measure them using conventional X-ray diffraction. Conventional X-ray diffraction measures X-ray diffraction images during X-ray irradiation for several seconds to several minutes while the crystal is rotating, so it is impossible to measure instantaneous X-ray diffraction images of piezoelectric crystals vibrating at frequencies from kHz to MHz. Synchrotron radiation (SR) X-rays with high brilliance and short pulse duration are useful for the measurement of instantaneous X-ray diffraction images. SR is an electromagnetic wave emitted tangentially to the trajectory of an electron bunch accelerated to nearly the speed of light when the trajectory is bent by a magnetic field. We have been conducting SR time-resolved X-ray diffraction experiments under an AC electric field at SPring-8, the large SR facility (Hyogo, Japan), to measure transient atomic displacements in piezoelectric crystals resonating under an AC electric field. The shortest pulse duration is about 50 ps. By repeatedly irradiating a piezoelectric crystal resonating under an AC electric field with highly brilliant and short pulse X-rays synchronized with the AC electric field, instantaneous x-ray diffraction images when the atomic displacements reach the maximum can be measured with high accuracy (Figure 1).
3. Transient atomic displacements in quartz oscillator
Transient atomic displacements in a quartz oscillator were successfully measured by the SR time-resolved X-ray diffraction under an AC electric field [1]. Quartz crystal belongs to the trigonal crystal system with the point group 32. There are two types of crystal polymorphs in quartz: right and left quartz, which are enantiomorphs of each other. The industrially used quartz crystal is a right crystal with the space group
A commercially available AT-cut quartz oscillator with an oscillation frequency of 30 MHz was used as the sample for the measurement. AT-cut oscillators are plate-shaped crystal (Figure 2b) cut along the plane including the
X-ray diffraction experiments were performed at the SPring-8 beamline BL02B1 [5] using X-rays with a wavelength of 0.4 Å. A large cylindrical curved imaging plate was used as the X-ray detector. First, measurements under a DC electric field were performed. Piezoelectric distortions of
Time-resolved X-ray diffraction of a resonantly vibrating AT-cut quartz oscillator (Figure 1) was performed by applying a sinusoidal AC electric field with a frequency of 30 MHz and an electric field amplitude of 0.18 MV∙m−1 to the sample. Resonance of the sample was confirmed by detecting the current flowing in the circuit with a current probe and displaying it on an oscilloscope. The resonant sample was irradiated with pulsed X-rays with a pulse duration of ~50 ps at a repetition rate of 26 kHz using an X-ray chopper [6]. In order to synchronize the oscillation of the sample and the pulsed X-rays, the ratio of the resonance frequency of the sample to the repetition frequency of the X-rays must be an exact integer ratio. However, both of the resonance frequency of the sample and the repetition frequency of the X-rays cannot be tuned freely. The authors have synchronized the resonance of the sample with pulsed X-rays by modulating the 30 MHz AC electric field at 26 kHz. This method enables instantaneous measurement of X-ray diffraction images of a resonant sample with a time resolution of less than 1 ns. The period of the resonant sample is 33 ns = 1/30 MHz. By varying the delay time Δ
Figure 3a shows time dependences of deviations of
The crystal structures at the delay times Δ
The large deformation of the Si−O−Si bond angles around O(2) and O(3) can be understood from the displacements of the anionic O atoms under the electric field and the arrangements of the Si−O−Si bonds relative to the electric field. The Si−O bonds have both covalent and ionic characters. The electron density distribution was calculated from the measured X-ray diffraction intensities to estimate the numbers of electrons of each atom. The charge deviations from neutral were +2.8
As shown above, the piezoelectric distortion of a quartz crystal under an electric field is caused by the displacement of oxygen ions in the direction perpendicular to the Si−O−Si plane due to the electric field and the accompanying deformation of the Si−O−Si bond angles. Mechanical and electrical vibrations with high
4. Applications to langasite-type crystals
Langasite (La3Ga5SiO14, LGS) is a piezoelectric crystal belonging to the same crystal point group 32 as quartz and does not show a phase transition until its melting point around 1470°C. Its piezoelectric constants are several times larger than those of quartz,
The crystal structures of LGS and NGS in the absence of an electric field were analyzed and compared first. There was little difference in the Ga(Si)−O bond distances between them. The RE (La, Nd) atoms are coordinated with eight O atoms, but because the RE atoms are on the twofold
X-ray diffraction experiments under a DC electric field showed that the lattice deformations of Δ
Figure 6a and b show the time variation of Δ
The crystal structures of LGS and NGS resonating under the AC electric field at the delay times when Δ
Next, the deformation of O−Ga−O bond angles in GaO6 octahedra, GaO4, and Ga1/2Si1/2O4 tetrahedra in the resonance state was investigated. Of the 15 independent O−Ga−O bonds in the GaO6 octahedra of the distorted structure under an electric field, no and 5 O−Ga−O bonds showed a time variation of their bond angles greater than ±0.25 degrees in LGS and NGS, respectively (Figure 8a). In the GaO4 tetrahedra, of the 18 independent O−Ga−O bonds, 5 and 5 O−Ga−O bonds showed a time variation greater than ±0.25 degrees in LGS and NGS, respectively (Figure 8b). In the Ga1/2Si1/2O4 tetrahedra, of the 12 independent O−Ga−O bonds, 2 and 1 O−Ga−O bonds showed a time variation greater than ±0.25 degrees in LGS and NGS, respectively (Figure 8c). The substitution of La with Nd increases the deformation of the GaO6 octahedra from the regular octahedron and facilitates the deformation of the O−Ga−O bond angles of the GaO6 octahedra during resonance. The Ga−O−Ga bond angles in NGS are not easily deformed due to the shortening of RE−O bonds, but the GaO6 octahedra are deformed instead. The deformation of GaO4 and Ga1/2Si1/2O4 tetrahedra during resonance was observed in both LGS and NGS, but no deformation of SiO4 tetrahedra during resonance was observed in quartz. Therefore, the O−Ga−O bond is more flexible than the O−Si−O bond, and as a result, the piezoelectric constants of LGS and NGS are larger than those of quartz. However, the piezoelectric deformation of LGS and NGS is caused by the deformation of multiple bonds with different force constants, resulting in greater energy dissipation and lower amplification effect (
As described above, the mechanism of piezoelectricity and the effect of RE substitution in langasite-type crystals were understood by the combination of resonance phenomena under an AC electric field and SR time-resolved X-ray diffraction. LGS shows larger piezoelectric deformation than quartz due to the deformation of GaO4 and Ga1/2Si1/2O4 tetrahedra. In NGS, the piezoelectric deformation is smaller because the shortening of the RE−O bond distances prevents the deformation of the Ga−O−Ga bond angles, but instead the GaO6 octahedral deformation is observed. These findings will be useful for the design and development of piezoelectric crystals with higher functionality than quartz.
5. Summary
In this chapter, we introduce transient crystal structures of quartz and langasite-type crystals resonating under an AC electric field observed by SR time-resolved X-ray diffraction. The method can detect small transient atomic displacements in piezoelectric crystals by resonantly amplifying them under an alternating electric field by a factor of 103–104. We will use this method for structural analyses of other piezoelectric crystals and expand its range of application.
Acknowledgments
This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS) (Grants Nos. JP22H02162, JP19H02797, JP16K05017, and JP26870491), Tatematsu Foundation, Toyoaki Scholarship Foundation, Daiko Foundation, and the Research Equipment Sharing Center at the Nagoya City University. The synchrotron radiation experiments were performed at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI).
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