Analysis on the anisotropic electromechanical properties of lead magnoniobate titanate single crystal for ring type ultrasonic motors

Ultrasonic motors (USMs) have been applied broadly in aerospace, robot and digital camera industries due to their high torque, compact geometry, self-lock and high resolution of displacement control, especially for those in extreme environments, e.g., deep space exploration.^1–6 Lead magnoniobate titanate (PMNT) single crystals at the morphotropic phase boundary (MPB) have excellent eletromechanical properties. The lateral piezoelectric strain coefficient d₃₁ and the electromechanical coupling factor k₃₁ were measured to be up to -1335 pC/N and 87% respectively after a polarization process along pseudo-cubic [001]_c direction.⁷ More importantly, this type of crystal exhibits excellent piezoelectric properties at cryogenic temperatures compared with piezoelectric ceramics. A linear USM and a traveling wave step motor using PMNT single crystals were successfully demonstrated working at liquid nitrogen temperature with slight output decrease.^8,9 Recently, a bolt-clamped Langevin-type transducer using PMNT single crystal was reported.¹⁰ Two [001]_c poled single crystal rings facing with each other were employed as the driving elements. Ring type USMs enjoy great popularity in the applications owning to the compact size and ease of generating the rotatory motion. Unfortunately, according to Neumann’s principle,¹¹ physical properties of crystals must contain at least the symmetry of the crystals. Single crystals have lower crystallographic symmetries than polycrystalline ceramics. For a piezoelectric ceramic, dielectric, elastic and electromechanical properties may obtain circular symmetry at polling direction, while crystal always not. In other words, the amplitude of the travelling wave could distort in the circular direction, which would deteriorate the motor performance. Therefore, it is really worthy to study the crystal orientation to achieve better performance.

To improve the performance of single crystal piezoelectric device, some work had been done on analyzing the anisotropic properties of piezoelectric single crystals for different domain engineered configurations. Orientation dependence of PMNT single crystal piezoelectric properties has been reported. For instance, for single domain (poled along [111]_c direction) PMN-0.28PT single crystals, the longitudinal piezoelectric coefficient d₃₃=97 pC/N is about 9 and 12 times less than that of multi domain (poled along [001]_c and [011]_c directions) single crystals.¹² For single domain PMN-0.33PT single crystal, the maximum values of d₃₃ (2316 pC/N) and k₃₃ (0.93) occur at (ZXl)-63° cut (following the piezoelectric crystal cutting angle notation¹³) and (ZXl)-70.8° cut respectively, according to the tensor transformation calculation.¹⁴ Outstanding face-shear properties (d₃₆=-1648 pC/N) at the cutting angle of (Zt)45° that rotates 45° about the thickness direction (z axis) were reported.^15,16 The same orientation was reported as the optimal direction for a face-shear mode PIN-PMN-PT single crystal USM, where a maximum no-load linear velocity of 182.5mm/s and a maximum output force of 1.03N were observed.¹⁷ Up to now, the anisotropic analysis for the PMNT single crystal ring concerning amplitude uniformity on the stator of traveling wave motors, has not been reported yet.

This paper discusses the selection criterion of crystal orientation regarding the circular symmetry of the amplitude A of the elliptical motion of the stator surface and presents optimized orientations for PMN-0.33PT single crystals poled along different directions. Based on the domain engineered states inducing by polarization along <001>_c, <011>_c and <111>_c. the reasons about changes of piezoelectric properties were discussed on the basis of engineered domain configuration.

Two Euler angles are needed so as to set up a normal vector of the piezoelectric ring, including a φ angle rotation along the z axis followed by a θ angle rotation along the new x^′ axis as depicted in Figure 1. According to the tensor transformation rules, the piezoelectric constants and elastic stiffness tensors can be easily calculated in x^″y^″z^″ coordinates. Regarding the in-plane uniformity of the piezoelectric ring, the third Euler angle ψ representing the rotation along z^″ axis is further needed to calculate the corresponding tensor elements variation along the circular direction. For example, the lateral piezoelectric constant

and longitudinal elastic stiffness

where $d31*$ and $c11*$ are tensor elements at a specific circular direction, $d31″$ and $c31″$⁠, are tensor elements in x^″y^″z^″ coordinates.

The amplitude A should be independent of the Euler angle ψ. This is controlled by the angle variation of $d31*$ and $c11*$⁠, and, of course the normal vector of the ring. To achieve the uniformity of the travelling wave on the ring, a uniform factor A_r is defined as A_r=A_max/A_min where A_max and A_min are the maximum and minimum amplitude in the range of ψ=0-360° on the ring at a special crystal orientation. As Ar approaches 1, the uniformity of the travelling wave tends to be ideal. Therefore the optimal orientation corresponds to the lowest A_r when the first and second Euler angles are swept over the entire range. In addition, we can calculate $dr=d31max*/d31min*$ and $cr=c11max*/c11min*$ deduced from equations 1 and 2 as another two characteristic parameters to assist the evaluation. It’s important to note that d_r are positive numbers less than 1 when $d31max*$ and $d31min*$ both being negative, whereas this does not mean a contrary to criterions about A_r and d_r. These intermediate criterions may help understand the circular variation of d₃₁ and c₁₁ in a piezoelectric hollow cylinder and other circular shapes.

Generally, the mathematic model of stator can be equaled to a composite charpy. the amplitude of the stator can be represented as following equation under resonance condition.¹⁸ 

Under static condition, Bending curvature and amplitude of the stator are expressed in the equation 3 by making it analogous to bi-metal thermostats.¹⁹ 

where A is the amplitude of the stator, r is mean radius of the stator, E₃ is electric field, L is length of sectorial region, Q_m is mechanical quality factor, u is modal order, h_p and h_m are thickness, E_m and c_p are stiffness coefficient. m=h_m/h_p, u=E_m/$c11*$ and h=h_m+h_p. Parameters related to the metal stator and the piezoelectric single crystal ring are distinguished by subscripts m and p, respectively. $d31*$ and $c11*$ are coefficients of piezoelectricity and elastic stiffness.

Considering that the geometric parameters and physical properties of metal body are constant. The expression of constructors A_r derived from the two equations above share the same mathematical expressions as shown in the following express.

where the superscript p indicates any point on the single crystal ring and the superscript mp represents the point offering maximum amplitude. τ is a constant involving geometric parameters of metal body and piezoelectric ring.

From equation 5, the sizes of A_r decided by d₃₁ and c₁₁ only. Then we can search possible optimized orientations which can ensure the uniformity of amplitude around surface of the stator.

The PMNT single crystals can obtain several sets of domains which lead to different macro-symmetries by the method called engineered domain configuration.¹⁵ When poled along a non-spontaneous polarization direction, various engineered domain configurations such as 4R, 3T and so on will be induced. In this type of notation, the numbers stand for the number of degenerated polarization directions and the letters refer to the specific types of ferroelectric phase. Regardless of the chosen poling technique, for every original crystal phase, a macroscopic 4mm symmetry has been demonstrated after being poled along the crystallographic <001>_c direction. Similarly, mm2 and 3m symmetries have been induced by <011>_c and <111>_c polarization, respectively.¹⁶ In this paper, all single crystals discussed are located at the MPB. Thus, the crystals possess several configurations after poled along <001>_c, <011>_c and <111>_c directions, respectively.

According to Euler’s rotation theorem, coordinates transformation can be performed by continuous angle rotations. A Bond matrix transformation method, based on the principle of tensor transformation, was introduced to calculate the stiffness and piezoelectric elements.²⁰ In this work, with step sizes of $dφ$=1° and $dθ$=1°, the normal vector of the ring was searched by the ergodic method using Matlab software in the given interval of φ=-90°∼90° and θ=-90°∼90°. Then, in the light of the criterions, a cutting orientation described by z^″ axis can be obtained.

The original piezoelectric and elastic data of the crystal used here in Table I comes from Refs. 21 and 22. From this table, we can observe that the single crystals poled along <001>_c direction have relatively large d₃₁ and k₃₁, being suitable for USMs application, whereas the crystals poled along <011>_c shows different signs of d₃₁ in orthogonal $ψ$ angles, which is harmful for the traveling wave formation. For <111>_c poled crystals, the d₃₁ and k₃₁ values are too low to generate enough deformation.

PMNT single crystal near MPB presents 4R, 4O and 1T configuration and 4mm macro-symmetry owing to the deflection of domains when being poled along [001]_c direction.

Figure 2 shows the variation of the relative values with rotation angle φ around z axis and θ around x^′ axis, respectively. To avoid the reducing of image contrast caused by the abrupt variation of A_r, d_r and c_r, only the data regions which approach 1 are shown here. From Figure 2, the constructor d_r stays within 0.9∼1.0 when rotating around x^′ axis within ±11°. It is worth to note from Figures 2(a) and 2(b) that the constructor A_r follows the same rule as d_r. Similarly, this regularity is illustrated by corresponding figures (Figures 4 and 6) presented later in the paper. In other words, for ring type USMs, the uniformity of the travelling wave is determined mainly by the piezoelectric strain coefficient.

From Figure 2(a), the lowest values of A_r=1.12 occurs at the region where φ=-90°∼90° and θ=0°, corresponding to the Z-cut. A good uniformity can be obtained in this cutting. More importantly, this cutting type brings great convenience for the cutting and poling processes because of the consistency between the cutting, poling and crystallographic directions. Therefore the optimal cutting orientation of [001]_c poled single crystal is along the crystallographic [001]_c direction (Z-cut). This means that the single crystal ring poled along the [001]_c axis can be applied to USMs directly.

In such situations, to describe the uniformity of travelling wave on the stator, a normalized amplitude value, A_nor=A/A_max, is introduced and plotted in Figure 3. A_nor varies slightly along the circumferential direction from 0.89 to 1, revealing that the criterion about A_nor could well qualify for filtering the cutting types based on the uniformity of the travelling wave. The lateral elastic compliance constant shows anisotropy in x^*oy^* plane. Variations of $c11*$ and $k31*$ as functions of the rotating angle along the z^″ axis are presented in Figure 3 as well. The electromechanical coupling factor changes within the range of 0.59 to 0.87 across the rotation angles.

The noticeable advantage of the USMs using the Z-cut, [001]_c poled PMNT single crystal ring is that they can be poled along [001]_c and $[001¯]c$ directions paralleling to z^″ axis in neighboring sectorial regions, respectively, thus the USMs can be manufactured following the method in total agreement with the craftwork of USMs using piezoelectric ceramics. Such craftwork allows the USMs to be driven by the conventional two sources configuration rather than the four sources configuration.²³ 

When poled along the [011]_c axis, 2R, 2T and 1O engineered domain configurations could be created. Three axes of the new coordinate system xyz follow [$01¯1$]_c, [100]_c and [011]_c directions of crystallography system respectively. A_r, d_r and c_r are shown as Figure 4. Obviously, unlike the single crystal poled along [001]_c, there are four acceptable regions which can meet the piezoelectric property and amplitude criterions. In the case of rotating around the z axis and x^′ axis, four double-rotated cutting types exist, locating at $φ$=-90° and $θ$=23° ((ZXtl)90°/23° as per standard) and three other such positions. Virtually, for ring type sample, the cutting of (ZXtl)90°/23° is equivalent to (ZYl)23°. Nevertheless, $d31*$ of four cuts above are too low (∼199 pC/N) to produce enough deformation for driving USMs. Relevant properties are shown in Figure 5.

When poled along the [111]_c axis, the single crystal with R phase exhibits 1R, 3O and 3T engineered domain configuration state, while the whole crystal present 3m macroscopic symmetry. Three axes in the new coordinate system are [$11¯0$]_c, [$112¯$]_c and [111]_c. The Z-cut could meet the piezoelectric criterion of the ring type USMs. However, the lateral piezoelectric constants, d₃₁∼-90pC/N, are very small, which prohibits the effective excitation of the bent vibration of the stator. The designers need to find a cutting orientation providing better lateral piezoelectric property and meeting the criterion of the ring type USMs simultaneously. Variations of A_r, d_r and c_r as rotating about the z axis and x^′ axis have been shown in Figure 6. There is one single-rotation cutting orientation ((ZXl)-60°) meeting the d_r and A_r criterions when sample rotated about x axis in -60°. Moreover, two double-rotation cutting orientations ((ZXtl)-60°/60° and (ZXtl)60°/60°) were discovered. All types have been labeled in Figure 6.

In addition, another available slim region is shown in Figure 6(a). This region presents the Z-cut type which has been eliminated because of the small $d31*$⁠. The computed results indicate that these three cutting types above can greatly enhance the piezoelectric strain coefficient. $d31*$ increases largely from -90 pC/N to -1201 pC/N and $k31*$ climbs from 0.21 to 0.92. It can be predicted that USMs’ performance with those cutting types would be improved. In Figure 7, the transformation of related values for (ZXl)-60° cutting type along the circumferential direction and the anisotropy of those constants were displayed in a polar diagram. Variation range of $k31*$ (0.46 to 0.92) approaches that of the Z-cut [001]_c poled.

The lateral piezoelectric constant d₃₁ of PMN-0.33PT single crystal poled along [011]_c decline from -216pC/N to -191pC/N in optimized orientation. The longitudinal piezoelectric coefficient d₃₃ enhance from 165pC/N to 475pC/N. The cause of this phenomenon can be understood from the theories of MPB and engineered domain. Rhombohedral, tetragonal and monoclinic phases coexist in PMNT single crystals with MPB compositions. Relationship between domain directions and optimized orientations has been indicate in fig 8(a). The optimized orientation z^″ ([025]_c in crystallographic) get closer to 2T engineered domain direction than poling direction (z axis). Because the longitudinal piezoelectric constant enhance sharply at this orientation, it concluded that improvement of piezoelectric constant is mainly contributed from tetragonal phase. In optimized orientation, Deviation from 2T domain direction of x^″axis ([0$5¯$2]_c in crystallographic) results in decline of lateral piezoelectric constant d₃₁. Analogously, we explain changes of piezoelectric properties of the single crystal poled along [111]_c direction in optimized orientation by same reasons as shown in fig 8(b).

Particular attention should be paid to that the above mentioned anisotropy is discussed in terms of poling before cutting the crystals. However, the mechanical process of cutting may alter the domain configurations inside the single crystal. To solve this problem, we can use the spare parts of crystals after cutting as the matching layer for the following polling process, as depicted in Figure 9. By employing this measure, the cutting process would definitely not affect the polling state of the optimal orientations. From the craftworks of USMs using piezoelectric ceramics, the piezoelectric ring should be electrically separated into several sector regions with opposite poling directions to excite the bending vibrations. Unfortunately, the piezoelectric single crystal ring cannot follow this polling technique owing to the tropism (there is a special case as Z-cut of [001]_c poled crystal mentioned above). Accordingly, 2-phase electric supplies configuration for ceramic USMs can not be realized in this case, while 4-phase electric supplies mode can solve this problem.²³ 

Based on the analysis discussed above, we could summarize the optimized orientations of PMNT single crystal poled along different directions as listed in Table II. Specific physical parameters may be different due to the different preparing conditions and measuring methods, but the principle and method of this letter are very generally applicable.

In conclusion, PMN-PT single crystals have special advantages because of their excellent piezoelectric properties and temperature stability. The criterions of crystal orientation have been proposed for ring type USMs. Based on the engineered domain configuration theory and the matrix method of the tensor transformation, optimal cutting orientations have been found. The cutting orientation for the crystal poled along [001]_c direction has better electromechanical properties and process compatibility. Three optimal orientations for the crystal poled along [001]_c direction could significantly improve the lateral piezoelectric coefficient from ∼90 pC/N to ∼1201 pC/N and electromechanical coupling factor to 0.92. These results lay the foundation for using piezoelectric single crystals for developing high performance circular piezoelectric transducers.

This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2015CB654602, the International Science & Technology Cooperation Program of China under Grant No. 2013DFR50470 and “111” Project (No. B14040).