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, PbIn_1/2Nb_1/2O₃–PbMg_1/3Nb_2/3O₃–PbTiO₃ (PIN–PMN–PT) ternary single crystals have attracted considerable attention owing to their high phase transition temperatures (T_{R − 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 (Q_M) 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 k_t 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 1 shows schematically two piezoelectric vibrators made of [001]_c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT single-crystal material. The z axis is the poling direction for both of the resonators, and the four surfaces normal to the z axis are covered by the electrodes.

Fig. 1.

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	Fig. 1. Schematic diagram of piezoelectric vibrators and their coordinate system.

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]

From Eqs. (2) and (3), we obtain

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. (15) and (16 ), we have

By substituting Eqs. (2)–(5) into Eq. (1), we obtain the following solutions:

For the cylindrical resonator shown in Fig. 1, the piezoelectric vibrator exhibits two-dimensional coupled vibration in the radial and z-axis direction. In this case, the configuration ratio G is defined as G = H/R, where H is in the poling direction of the piezoelectric single crystal. The theoretical equation for the coupled vibration is^[8,12]

For the cylindrical resonator, the coupling constant Γ₂ is given by

Solving Eq. (21) leads to the following solutions:

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 k^eff can be given by^[2,13]

Figure 2 shows the electromechanical coupling coefficient of the cylindrical resonator with an arbitrary configuration ratio at temperatures of 25 °C, 40 °C, and 55 °C, obtained by using Eq. (38) and the [001]_c-poled Mn-doped 0.24PIN–0.46PMN–0.30PT single-crystal parameters derived from Ref. [4]. At a given temperature, the electromechanical coupling coefficient k^eff increases with configuration ratio G = H/R increasing. When G → 0, the electromechanical coupling coefficient k_t decreases with temperature rising. However, when G → ∞, the electromechanical coupling coefficient k₃₃ (∼ 0.88) is largely invariant at temperatures ranging from 25 °C to 55 °C as shown in Table 1.

Fig. 2.

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	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.

Table 1.

Table 1. Electromechanical coupling coefficients kt and k33 of cylindrical resonator at different temperatures, including the slope peaks. . Temperature/°C kt k33 Slope peaks (G = H/R) Slope 25 0.5453 0.8831 0.89 0.2533 30 0.5445 0.8841 – – 35 0.5442 0.8844 – – 40 0.5412 0.8843 0.90 0.2552 45 0.5374 0.8843 – – 50 0.5336 0.8839 – – 55 0.5236 0.8824 0.91 0.2653

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. 2 and Table 1. Listed in Table 1 are the values of the electromechanical coupling coefficient k_t at temperatures of 25 °C, 30 °C, 35 °C, 40 °C, 45 °C, 50 °C, and 55 °C. Figure 3 shows the results fitted with the following equation:

Fig. 3.

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	Fig. 3. Electromechanical coupling coefficient kt at different temperatures.

Figures 4(a) and 4(b) show the relationship between resonant frequency constants and configuration ratio for the rectangular vibrator at temperatures of 25 °C and 55 °C, respectively.

Fig. 4.

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	Fig. 4. Plot of resonant frequency constant versus configuration ratio for the rectangular piezoelectric vibrator at temperatures of (a) 25 °C and (b) 55 °C, with straight lines indicating limiting frequency constants.

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 is defined as the ‘mode dividing point’; when M₁ = 1, G^f_HH = f_L H = G^{f_h H} = f_l H. By comparing Figs. 4(a) and 4(b), we find that the mode dividing point moves toward the lower configuration ratio G with temperature increasing as shown in Table 2. The coupling constant Γ₁ is approximately 0.85 in a temperature range of 25 °C–55 °C. From Table 2, we find that the applicability index (M₁) values are close to 1, indicating the applicability of the coupling theory for the rectangular piezoelectric single-crystal vibrator studied in this work.^[8,9]

Table 2.

Table 2.

Table 2. Values of coupling constant Γ, mode dividing point, and applicability index M at different temperatures. . Temperature/°C Rectangular bar Cylindrical resonator Γ1 Mode dividing point M1 Γ2 Mode dividing point M2 (G = L/H) (G = H/R) 25 0.8522 0.9756 ± 0.004 1.0165 0.8862 1.3805 ± 0.003 1.0078 30 0.8532 0.9714 ± 0.004 1.0180 0.8873 1.3852 ± 0.003 1.0083 35 0.8539 0.9673 ± 0.004 1.0172 0.8879 1.3914 ± 0.003 1.0074 40 0.8536 0.9626 ± 0.004 1.0184 0.8880 1.3989 ± 0.003 1.0081 45 0.8533 0.9567 ± 0.004 1.0184 0.8879 1.4073 ± 0.003 1.0076 50 0.8534 0.9520 ± 0.004 1.0191 0.8880 1.4144 ± 0.003 1.0079 55 0.8524 0.9435 ± 0.004 1.0198 0.8876 1.4279 ± 0.003 1.0085

Table 2. Values of coupling constant Γ, mode dividing point, and applicability index M at different temperatures. .

Figures 5(a) and 5(b) show the relationships between the resonant frequency constants and the configuration ratio for the cylindrical resonator at temperatures of 25 °C and 55 °C, respectively. For the cylindrical piezoelectric vibrator, the point is the ‘mode dividing point’ when M₂ = 1, G^{f_t H} = f_r H = G^{f_T H} = f_R H. By comparing Figs. 5(a) and 5(b), we find that the mode dividing point moves toward the greater configuration ratio G with temperature increasing as shown in Table 2. At 25 °C, G ≈ 1.381 and k^eff = 0.7511; at 40 °C, G ≈ 1.399 and k^eff = 0.7509; at 55 °C, G ≈ 1.428 and k^eff = 0.7452. The electromechanical coupling coefficient k^eff at the mode dividing point exhibits a decreasing trend with temperature increasing. The configuration ratio G for the rectangular bar and the cylindrical resonator are defined as G = L/H and G = H/R, respectively; however, the poling direction for each of the two cases is along the height H as shown in Fig. 1. The coupling constant Γ₂ is approximately 0.89 in the temperature range of 25 °C–55 °C. From Table 2, we find that the applicability index M₂ is approximately 1.010. Therefore, like the scenario in the case of the rectangular bar, the coupling theory can also be applied to the cylindrical resonator.

Fig. 5.

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	Fig. 5. Plots of resonant frequency constant against configuration ratio for cylindrical resonator at temperatures of (a) 25 °C and (b) 55 °C, with straight lines indicating limiting frequency constants.

For both types of piezoelectric vibrators, the frequency constant f₁ H is greater than f₂ H as shown in Figs. 4 and 5, and hence, when G → 0 or G → ∞, f₁ H approaches to the two limiting resonant frequency constants of the lower dimension, and f₂ H approaches to the constants of the higher dimension. In other words, for the rectangular bar, when G → 0, then f₁ H → f_l H and f₂ H → f_H H, and when G → ∞, then f₁ H → f_h H and f₂ H → f_L H; for the cylindrical resonator, when G → 0, then f₁ H → f_t H and f₂ H → f_R H, and when G → ∞, then f₁ H → f_r H and f₂ H → f_T H. There is a very strong coupling between the resonant frequency along the poling direction (z direction) and that in the direction perpendicular to the poling direction (x or radial direction) when the configuration ratio G is close to the ‘mode dividing point’. From Table 2, we find that the coupling constant Γ₂ of the cylindrical resonator is slightly greater than Γ₁ of the rectangular bar at a given temperature. The coupling constant Γ is not obviously dependent on temperature. This is because in Eqs. (6), (28), and (35), the elastic constants , , , and obtained from Ref. [4] vary with temperature. The elastic constants , , and decrease with temperature increasing, whereas is largely invariant in the temperature range of 25 °C–55 °C. Compared with that of the rectangular piezoelectric vibrator, the applicability index M₂ of the cylindrical vibrator is close to 1 in the temperature range of 25 °C–55 °C as shown in Table 2. Thus, in the application of the coupling theory, the internal consistency for the cylindrical resonator is slightly better than that for the rectangular bar. The applicability index M₂ of the cylindrical resonator is closer to 1 than the applicability index M₁ of rectangular bar, which results in the absolute value of ± 0.003, lower than that of the rectangular bar (± 0.004) as shown in Table 2.

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 k^eff at the mode dividing point exhibits a decreasing trend with temperature increasing. The electromechanical coupling coefficient k_t 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.