† Corresponding author. E-mail:
Project supported by the Basic Scientific Research Foundation of College and University in Heilongjiang Province, China (Grant No. 2018QNL-16), the Guiding Science and Technology Project of Daqing City (GSTPDQ), China (Grant No. zd-2019-03), and the National Natural Science Foundation of China (Grant Nos. 11304061 and 51572056).
The influence of temperature on mode coupling effect in piezoelectric vibrators remains unclear. In this work, we discuss the influence of temperature on two-dimensional (2D) mode coupling effect and electromechanical coupling coefficient of cylindrical [001]c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT piezoelectric single-crystal vibrator with an arbitrary configuration ratio. The electromechanical coupling coefficient kt decreases with temperature increasing, whereas k33 is largely invariant in a temperature range of 25 °C–55 °C. With the increase of temperature, the shift in the ‘mode dividing point’ increases the scale of the poling direction of the piezoelectric vibrator. The temperature has little effect on coupling constant Γ. At a given temperature, the coupling constant Γ of the cylindrical vibrator is slightly greater than that of the rectangular vibrator. When the temperature changes, the applicability index (M) values of the two piezoelectric vibrators are close to 1, indicating that the coupling theory can be applied to piezoelectric vibrators made of late-model piezoelectric single crystals.
The analysis of the coupling between the vibration modes of a piezoelectric vibrator is vital to the design of practical devices such as piezoelectric sensors, piezoelectric actuators, and ultrasonic transducers. The most common problem is the coupling analysis between the vibration mode along the poling direction and that perpendicular to the poling direction. The electromechanical coupling coefficient of piezoelectric material along the poling direction in the vibration mode has been extensively discussed. The material parameters, including the elastic, dielectric, and piezoelectric constants, vary with temperature. Therefore, the coupling between the two vibration modes and the electromechanical coupling are affected by temperature. Piezoelectric vibrators are the basic components of piezoelectric devices. Rectangular and cylindrical piezoelectric vibrators with different configuration ratios are the most common examples.
In recent years, PbIn1/2Nb1/2O3–PbMg1/3Nb2/3O3–PbTiO3 (PIN–PMN–PT) ternary single crystals have attracted considerable attention owing to their high phase transition temperatures (TR − T > 120 °C) and good temperature stabilities, making them suitable for being used in electromechanical devices in a wide range of temperatures.[1,2] Doping with Mn can improve some of the physical properties of ferroelectrics.[3] The mechanical quality factor (QM) of Mn-doped PIN–PMN–PT (Mn:PIN–PMN–PT) ternary single crystals poled along the [001]c pseudo-cubic direction can be as high as 1000, which is comparable to that of hard PbZrTiO3-type (PZT) ceramics.[4] Currently, the applications of piezoelectric materials are not limited to conventional environments,[5,6] such as room temperature condition environments. The mode-coupling effect in rectangular beams made of 0.33PIN–0.35PMN–0.32PT ternary single crystals has been studied at room temperature.[7] However, the influence of temperature on the mode coupling effect in a rectangular beam is unclear. Furthermore, there is no research on the mode coupling effect in ternary, cylindrical PIN–PMN–PT single-crystal resonator.
In this study, we analyze the mode coupling effects in rectangular piezoelectric vibrator and cylindrical piezoelectric vibrator, both are made of [001]c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT ternary single crystal, in a temperature range of 25 °C–55 °C. In addition, we derive an equation for the electromechanical coupling coefficient kt of Mn:PIN–PMN–PT ternary single crystal as a function of temperature. This study will be helpful in designing Mn:PIN–PMN–PT-based devices.
Figure
For the rectangular vibrator, we assume that the bar extends infinitely in the y-axis direction. Thus, the rectangular vibrator exhibits two-dimensional coupled vibrations in the x and z directions.[8–10] The configuration ratio G is defined as G = L/H, where H is in the poling direction of the vibrator. The theoretical relationship for the coupled vibration is[8,9,11]
![](cpb_29_7_075201/cpb_29_7_075201_ieqn1.gif)
From Eqs. (
The factor M1 is the applicability index, which is a measure of the internal consistency in the application of the coupling theory.[8,9] If M1 = 1, from Eqs. (
By substituting Eqs. (
For the cylindrical resonator shown in Fig.
![](cpb_29_7_075201/cpb_29_7_075201_ieqn2.gif)
For the cylindrical resonator, the coupling constant Γ2 is given by
Solving Eq. (
The [001]c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT piezoelectric single-crystal vibrator and [001]c-poled PMN-0.30PT single-crystal resonator in Ref. [13] have the same macroscopic symmetry and vibration mode. For the cylindrical resonator with an arbitrary configuration ratio G = H/R, the electromechanical coupling coefficient keff can be given by[2,13]
Figure
![]() | Fig. 2. Electromechanical coupling coefficient keff of cylindrical resonator with arbitrary configuration ratio at temperatures of 25 °C, 40 °C, and 55 °C. |
![]() | Table 1. Electromechanical coupling coefficients kt and k33 of cylindrical resonator at different temperatures, including the slope peaks. . |
The electromechanical coupling factor k33 of the [001]c-poled 0.28PIN–0.40PMN–0.32PT ternary single crystal increases from 0.91 at −50 °C to 0.94 at 125 °C,[1] while that of the [001]c-poled 0.26PIN–0.44PMN–0.30PT single crystals remains constant at 0.91 in a temperature range of −10 °C–70 °C.[2] Thus, the temperature may have little effect on the electromechanical coupling coefficient k33 of [001]c-poled Mn-doped PIN–PMN–PT ternary single crystal in a temperature range of 25 °C–55 °C. The peaks of the curves move in the direction of greater configuration ratio G with temperature increasing as shown in Fig.
Figures
For a given configuration ratio G, f1 H and f2 H represent the two types of resonant frequency constants for the two-dimensional coupled vibration. For the rectangular piezoelectric vibrator, the point
![]() | Table 2. Values of coupling constant Γ, mode dividing point, and applicability index M at different temperatures. . |
Figures
For both types of piezoelectric vibrators, the frequency constant f1 H is greater than f2 H as shown in Figs.
In this work, we discuss the influence of temperature on 2D mode coupling effect and electromechanical coupling coefficient of cylindrical [001]c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT piezoelectric single-crystal vibrator with an arbitrary configuration ratio. For the cylindrical resonator, the effective electromechanical coupling coefficient keff at the mode dividing point exhibits a decreasing trend with temperature increasing. The electromechanical coupling coefficient kt decreases with temperature increasing, whereas the electromechanical coupling coefficient k33 is largely invariant in a temperature range of 25 °C–55 °C. With temperature increasing, the shift in the mode dividing point increases the scale of the poling direction. For a given vibrator, the temperature has little effect on coupling constant Γ in the range of 25 °C–55 °C. However, the geometry of the piezoelectric vibrator affects the coupling constant. For a given temperature, the coupling constant Γ of the cylindrical resonator is slightly greater than that of the rectangular bar. In a temperature range from 25 °C to 55 °C, the applicability index values of the two piezoelectric vibrators are both close to 1, indicating that the coupling theory can be applied to piezoelectric vibrators made of late-model piezoelectric single crystals, considering temperature fluctuations.
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