- freely available
- re-usable

*Crystals*
**2014**,
*4*(3),
241-261;
doi:10.3390/cryst4030241

_{4}O(BO

_{3})

_{3}: Growth and Piezoelectric Characterizations

^{1}

^{2}

^{3}

## Abstract

**:**Rare-earth calcium oxyborate crystals, ReCa

_{4}O(BO

_{3})

_{3}(ReCOB, Re = Er, Y, Gd, Sm, Nd, Pr, and La ), are potential piezoelectric materials for ultrahigh temperature sensor applications, due to their high electrical resistivity at elevated temperature, high piezoelectric sensitivity and temperature stability. In this paper, different techniques for ReCOB single-crystal growth are introduced, including the Bridgman and Czochralski pulling methods. Crystal orientations and the relationships between the crystallographic and physical axes of the monoclinic ReCOB crystals are discussed. The procedures for dielectric, elastic, electromechanical and piezoelectric property characterization, taking advantage of the impedance method, are presented. In addition, the maximum piezoelectric coefficients for different piezoelectric vibration modes are explored, and the optimized crystal cuts free of piezoelectric cross-talk are obtained by rotation calculations.

## 1. Introduction

High temperature sensors are desirable in aerospace and automotive industries for monitoring component conditions to optimize the propulsion system, with operations at temperatures up to 1000 °C, to enable safer, more fuel-efficient and more reliable vehicles; in addition, they are important for nondestructive in situ inspection of the structure health of the furnace component or systems in electric generation plants, to improve the safety and reduce life-cycle costs. Compared with the commercial strain gauge and optical fiber sensors, etc., piezoelectric sensors have greater potentials in realizing high temperature (>600 °C) sensing with the merits of high accuracy, fast response time and ease of integration [1,2,3,4].

Of all the investigated high temperature piezoelectric materials up to date, crystals in trigonal, tetragonal and monoclinic systems have been extensively investigated. Among these materials, the trigonal lithium niobate (LiNbO_{3}, LN) crystals with 3m symmetry were reported to possess high piezoelectric coefficient, being on the order of 6–70 pC/N at room temperature, approximately 3–30 times that of commercial α-quartz (SiO_{2}) (2–3 pC/N). However, the maximum operating temperature of LN-based piezoelectric devices, restricted by their low electrical resistivity (a requirement of >10^{6} Ohm·cm was proposed for comparison, where the materials with low resistivity yet can be used for high frequency applications [1]) at elevated temperature is limited to <600 °C, though the Curie temperature is above 1150 °C [5]. Other important trigonal piezoelectric crystals include the langasite family with the general formula of A_{3}BC_{3}D_{2}O_{14} [6,7,8,9,10,11,12] and gallium orthophosphate GaPO_{4}, in the point group of 32 [13,14,15,16,17,18,19], these crystals were reported to show modest piezoelectric coefficients (5–7 pC/N) and high melting points (1300–1500 °C for langasite family crystals and ~1670 °C for GaPO_{4}), prior to which, there are no phase transitions observed for langasite family crystals (the phase transition for GaPO_{4} is about 970 °C). However, the costly component Ga_{2}O_{3} restricted their further implements. The newly developed Ca_{3}TaAl_{3}Si_{2}O_{14} (CTAS) crystals, substituting the Ga with Al elements, were found to possess improved higher temperature properties than La_{3}Ga_{5}SiO_{14} (LGS) and to significantly decrease the cost of raw materials; nevertheless, the crystal quality needs to be improved, due to the core defect observed inside the crystals [12]. The tetragonal melilite crystals (point group 42m, such as SrLaGa_{3}O_{7} (SLG), Ca_{2}Al_{2}SiO_{7} (CAS), etc.) and fresnoite crystals (point group 4mm, such as Ba_{2}TiSi_{2}O_{8}) were investigated for piezoelectric applications. These crystals show the merits of high melting points (1400–1700 °C) and high effective piezoelectric coefficients d_{eff} (5–18 pC/N) [20,21,22,23,24,25,26]; the evaluations of the temperature dependence of dielectric, piezoelectric and electromechanical properties, however, are limited. Of particular significance is that the monoclinic rare-earth calcium oxyborate crystals (ReCa_{4}O(BO_{3})_{3}, ReCOB, Re: rare earth), which have been extensively investigated for nonlinear optical applications in the last two decades [27,28,29,30,31,32,33], were reported to exhibit good piezoelectric properties and high electrical resistivity at an elevated temperature of 1000 °C, with no phase transition prior to their melting points (~1400–1520 °C) [1,2,3,34,35,36,37,38].

Table 1 summarizes the basic characteristics of various high temperature piezoelectric crystals in monoclinic, trigonal and tetragonal systems, where the monoclinic ReCOB crystals were found to exhibit relatively high melting points, as well as relatively large piezoelectric coefficients, promising high temperature piezoelectric sensor applications.

In this review article, crystal growth, dielectric, elastic and piezoelectric property characterizations of the monoclinic ReCOB crystals are surveyed. Different crystal growth techniques, including the Bridgman and Czochralski (Cz) pulling methods, are discussed in Section 2. The crystal orientation related to the physical axes and crystallographic axes for electro-elastic property investigations is studied in Section 3. In Section 4, characterizations of the dielectric, elastic and piezoelectric properties of ReCOB crystals are reviewed. In Section 5, the maximum piezoelectric coefficients for different crystal cuts and the optimized crystal cuts free of cross-talk are discussed. Finally, the significance and challenges of ReCOB crystals are summarized; future research is proposed in Section 6.

**Table 1.**A comparison of different high temperature piezoelectric crystals. LGS, La

_{3}Ga

_{5}SiO

_{14}; CTGS, Ca

_{3}TaGa

_{3}Si

_{2}O

_{14}; CTAS, Ca

_{3}TaAl

_{3}Si

_{2}O

_{14}; SLG, SrLaGa

_{3}O

_{7}; CAS, Ca

_{2}Al

_{2}SiO

_{7}; BTS, Ba

_{2}TiSi

_{2}O

_{8}; ReCOB, ReCa

_{4}O(BO

_{3})

_{3}.

Symmetry | Crystals | Growth Method | T_{c}/T_{m} (°C) | d_{eff} (pC/N) |
---|---|---|---|---|

Trigonal | LGS | Cz | 1430 [39] | 6~7 |

CTGS | Cz | ~1370 [40] | ~10 | |

CTAS | Cz | – | ~5 | |

GaPO_{4} | Flux/hydrothermal | 970 [41] ^{#} | ~5 | |

LN | Cz | 1150 | ~70 [42] | |

Tetragonal | SLG | Cz | ~1650 [43] | ~14 [44] |

CAS | Cz | 1500 | ~6 [22] | |

BTS | Cz | 1445 | ~18 [23] | |

Monoclinic | ReCOB | Cz/Bridgman | 1400~1520 [1] | ~15 |

Notes: Cz, Czochralski pulling method; ^{#} α–β phase transition temperature.

## 2. Crystal Growth

#### 2.1. Polycrystalline Preparation

The ReCOB polycrystalline materials were prepared by high purity (99.99%) CaCO_{3}, Re_{2}O_{3} (Pr_{6}O_{11}) and H_{3}BO_{3} powders. They were weighed according to their nominal compositions. Considering the evaporation of B_{2}O_{3} during crystal growth, an excess of H_{3}BO_{3} (1–3 wt%) was added to the starting components, which will benefit the crystal growth [38,45,46]. The starting materials were mixed completely, followed by the calcination ~1000 °C for 5–10 h to decompose the H_{3}BO_{3} and CaCO_{3}, after which, the calcined powders were ground, remixed and then pressed into tablets to fabricate ReCOB polycrystalline components at 1100–1200 °C for 10–20 h; the solid state reaction follows the equation below:

_{2}O

_{3}+ 6H

_{3}BO

_{3}+ 8CaCO

_{3}= 2ReCa

_{4}O(BO

_{3})

_{3}+ 8CO

_{2}↑ + 9H

_{2}O↑

#### 2.2. Approaches for ReCOB Crystal Growth

Different techniques have been applied for ReCOB crystal growth, including the high temperature flux, Bridgman and Cz approaches [47,48,49]. At the early stage of ReCOB crystal studies, the high temperature flux method was adopted for the crystal growth, where the selection of appropriate flux agent (high temperature solvent) is vital, especially for the crystals with incongruent features. Among ReCOB crystals (except CeCOB and YbCOB), ErCOB, YCOB, GdCOB, SmCOB, NdCOB, PrCOB and LaCOB are congruent melting compounds, while TbCOB and TmCOB were found to possess a very narrow congruent region. Lead oxide was firstly selected as the flux agent for ReCOB crystal growth, with limited success, where the grown crystals were at centimeter scale [47]. Later, the Bridgman and Cz methods, which are favorable for growing crystals with congruent melting features, were utilized. In the following, the growth of ReCOB crystals by the Bridgman and Cz methods is introduced.

#### 2.2.1. Bridgman Method

The Bridgman method, which is also called the Bridgman-Stockbarger method, involves heating polycrystalline materials in a platinum or iridium crucible above its melting point and slowly cooling it from one end, where an oriented seed crystal is located. The crucible is translated from the high temperature region to the low temperature region at a special designed speed; the single crystal is progressively formed along the length of the crucible. Figure 1 shows a schematic set-up of the Bridgman growth system.

By the Bridgman method [50], the feed materials were prepared by thoroughly mixing the stoichiometric oxides, which were charged into a cylindrical platinum crucible. The material was heated to about 50 °C above its melting point, maintained for more than 10 h to make a homogeneous melt. The seeding process (along the <010> or <001> orientation) was performed by adjusting the crucible position and furnace temperature gradient, so that only the top part of the seed was melted. Growth was then driven by lowering the crucible at a rate of 0.2–0.6 mm/h, and the temperature gradient near the solid/liquid interface was normally kept at 30–50 °C/cm. Recently, large YCOB crystals with a diameter up to 3–4 inches have been reported, using the modified Bridgman furnace [48].

#### 2.2.2. Cz Pulling Method

Cz method is named after Polish scientist, Jan Czochralski, who invented this method in 1916. In the Cz pulling technique, iridium, platinum, molybdenum and graphite crucibles are generally selected and heated in a low–medium radio-frequency furnace. Particularly, for ReCOB crystal growth, an iridium crucible is mostly utilized, due to its high melting point (~2450 °C) and chemical inertia to the raw materials. Figure 2 gives the schematic set-up of the Cz pulling system.

YCOB and GdCOB single crystals have been grown by the Cz method [27,33,49]. ReCOB materials were melted and kept 30–80 °C above their respective melting points for more than 10 h to ensure the homogeneity of the melts. A <010>-oriented crystal seed was selected for the growth, which will benefit the large diameter crystals. The pulling speed was controlled at 0.4–2.0 mm/h, and the rotation speed was varied from 10 to 30 rpm during the crystal growth. When the growth was finished, the crystal was cooled down to room temperature at a low rate of 10–50 °C/h, to avoid the crystal cracks induced by inner thermal stress. Figure 3 shows the large YCOB single crystals grown by the Cz method (Figure 3a,c) and the Bridgman method (Figure 3b) with a diameter being on the order of 3–4 inches. The obtained crystals were reported to possess X-ray diffraction with a very low full-width at half maximum (FWHM) (~30ʺ) [51] and high optical homogeneity (10^{−6}) [52], exhibiting high crystal quality.

**Figure 3.**Large YCOB single crystals grown by the Cz method and the Bridgeman method (crystal (

**a**) was grown by Crystal Photonics Inc. using Cz method, reprinted with permission from [53]. Copyright 2006 Elsevier; Crystal (

**b**) was grown using Bridgeman method, reprinted with permission from [48]. Copyright 2013 Elsevier; and crystals (

**c**) were grown using Cz method, reprinted with permission from [46]. Copyright 2013 Chinese Ceramic Society.

## 3. Orientation of Monoclinic ReCOB Crystals

ReCOB are monoclinic biaxial crystals belonging to the point group m. Therefore, three different orientation systems exist due to the crystal symmetry, including the crystallographic, optical and physical systems. For piezoelectric characterizations, the relationship between the crystallographic and physical axes should be determined, where the physical Y-axis is parallel to the crystallographic b-axis, Z- to the c-axis, and the X-axis is perpendicular to the Y and Z axes to form a right-hand orthogonal system according to the IEEE Standard on Piezoelectricity [54], as shown in Figure 4.

The initial orientation step is to verify the (010), (201) an (101) faces using X-ray analysis, possessing relatively strong diffraction peaks, from which, the crystallographic axes, a, b and c, can be determined according to the relationships between the interfacial angle and crystallographic plane. Then, the physical axes can be easily determined based on the IEEE standards. Taking YCOB crystals, for example, the angle between the Z-axis and (101) face was calculated to be around 57.1°, while the angle between the X-axis and (201) face was close to 34.6° (Figure 4). It is noteworthy that these angles (interfacial angles, etc.) are varied for different ReCOB crystals, due to their different lattice parameters (Table 2). After the orientation of the X, Y and Z axes, the positive X and positive Z axes can be further confirmed by using a quasi-static piezoelectric d_{33} meter, according to the IEEE Standard on Piezoelectricity [54].

**Figure 4.**Schematic of the cross-section of Cz-grown YCOB crystals, with the relationship of the crystallographic and physical axes.

**Table 2.**Unit cell parameters and interfacial angles for different ReCOB crystals in space group C

_{m}.

Crystals | TmCOB | ErCOB | YCOB | TbCOB | GdCOB | SmCOB | NdCOB | PrCOB | LaCOB | |
---|---|---|---|---|---|---|---|---|---|---|

Parameters (Å/Å^{3}/°) * | a | 8.068 | 8.075 | 8.078 | 8.072 | 8.104 | 8.114 | 8.145 | 8.177 | 8.173 |

b | 16.01 | 16.01 | 16.02 | 16.00 | 16.03 | 16.06 | 16.07 | 16.16 | 16.09 | |

c | 3.522 | 3.530 | 3.534 | 3.545 | 3.558 | 3.579 | 3.607 | 3.629 | 3.627 | |

β | 101.11 | 101.43 | 101.19 | 101.25 | 101.25 | 101.38 | 101.37 | 101.40 | 101.40 | |

V | 446.56 | 447.25 | 448.01 | 449.06 | 453.38 | 457.24 | 462.81 | 469.95 | 467.41 | |

Formula weight * | 521.33 | 519.74 | 441.65 | 511.33 | 509.99 | 502.78 | 496.98 | 493.33 | 491.68 | |

∠1 (°) ^{#} | 112.59 | 112.45 | 112.46 | 112.41 | 112.26 | 111.99 | 111.78 | 111.66 | 111.62 | |

∠2 (°) ^{#} | 34.74 | 34.70 | 34.77 | 34.75 | 34.85 | 34.94 | 35.08 | 35.14 | 35.14 | |

∠3 (°) ^{#} | 45.85 | 46.13 | 45.96 | 46.00 | 46.10 | 46.32 | 46.45 | 46.54 | 46.57 |

* PDF #: 00-050-0403 (YCOB); 01-077-0951 (GdCOB); 01-079-1378 (SmCOB); 00-050-0399 (NdCOB); 01-070-7385 (LaCOB); TmCOB [55]; TbCOB [56]; PrCOB [38]; ^{#} ∠1: the interfacial angle between (101) and (201); ∠2: the interfacial angle between (201) and the X-axis; ∠3: the interfacial angle between (201) and (001).

## 4. Electro-Elastic Material Constants Determination

Due to the crystal symmetry, there are two different sets of electro-elastic component matrix for monoclinic crystal, relating to point group m and point group 2. The piezoelectric crystals in point group 2 possess eight independent piezoelectric strain coefficients d_{ij}, while crystals in point group m exhibit ten independent piezoelectric strain coefficients d_{ij}, four dielectric permittivities ε_{ij} and thirteen elastic compliances s_{ij}. The electro-elastic component matrix for piezoelectric crystals in point group m can be written as:

Taking advantage of the impedance method based on the IEEE standard, all of the electro-elastic parameters for ReCOB crystals can be determined. Meanwhile, the elastic stiffness c_{ij} and piezoelectric stress coefficient e_{ij} can be obtained based on the determined elastic compliance s_{ij} and piezoelectric strain coefficient d_{ij}, following the equations below:

Based on the IEEE standard [54], the dielectric permittivities ε_{ii} (i = 1, 2 and 3) can be evaluated by measuring the capacitance of X-, Y- and Z-cut square plates, respectively. Piezoelectric d_{11} and d_{33} can be obtained by measuring the longitudinal vibration mode of the X and Z rods, while elastic compliances s_{ii} (i = 1–6) can be calculated from the extensional and shear vibration modes, relating to the piezoelectric coefficient d_{12}, d_{13}, d_{15}, d_{24}, d_{26}, d_{31}, d_{32}, d_{33} and d_{35}. On the contrary, the dielectric cross permittivity ε_{13} and elastic cross compliances s_{ij} (i ≠ j; i, j = 1–6) can be evaluated by employing oblique crystal cuts that do not lie along or normal to any of the three physical axes. Figure 5 presents different crystal cuts designed for the electro-elastic parameter determination, including the dielectric permittivities, elastic compliances and piezoelectric coefficients. The aspect ratios of the samples (X-, Y- and Z-square plates) for dielectric measurement ranged from 1:8:8 to 1:10:10, whereas for elastic measurement, rectangular samples were used with aspect ratios, t (thickness):w (width):l (length), ranging from 1:2:10 to 1:3:15 and an orientation accuracy <30ʺ. For each crystal cut, 3–5 pieces of samples were prepared for parameter determination.

**Figure 5.**Crystal cuts for electro-elastic parameter determination ((a) X rod; (b) Z rod; (c) XZ plate; (d) XY plate; (e) ZY plate; (f) ZX plate; (g) (XZw)45°; (h) (XZw)30°; (i) (XZw)−45°; (j) Z-square plate; (k) X-square plate; (l) Y-square plate; (m) (XYt)45°; (n) (ZXt)45°; (o) (YZtw) 45°/−45°).

#### 4.1. Dielectric Permittivity

ReCOB crystals possess four independent dielectric permittivities, where the dielectric permittivities ε_{11}, ε_{22}, ε_{33} can be determined from X-, Y-, and Z-cut samples (k, l and j), while for the determination of ε_{13}, sample (g) was used (Table 3), following the formulae below:

**Table 3.**A summary of different crystal cuts for material constants’ determination, reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

Crystal Cuts | Modes | Material constants |
---|---|---|

X rod (a) | longitudinal extension | d_{11} |

Z rod (b) | d_{33} | |

ZX plate (f) | transverse extension | s_{11}, d_{31},s _{22}, d_{12}, d_{32},s _{33}, d_{13} |

XY plate (d) | ||

ZY plate (e) | ||

XZ plate (c) | ||

XZ plate (c) | width shear * | s_{44}, d_{24} |

ZX plate (f) | s_{66}, d_{26} | |

(XZw)45° (g) | s_{46} | |

XZ plate (c) | thickness shear | s_{55}, d_{15}, d_{35} |

ZX plate (f) | ||

(XZw)45° (g) | transverse extension | s_{13}, s_{15}, s_{35} |

(XZw)30° (h) | ||

(XZw)−30° (i) | ||

(XYt)45° (m) | transverse extension | s_{12}, s_{23}, s_{25} |

(ZXt)45° (n) | ||

(YZtw)45°/−45° (o) | ||

X cut (k) | – | ε_{11}, ε_{22}, ε_{33}, ε_{13} |

Y cut (l) | ||

Z cut (j) | ||

(XZw)45° (g) |

* d_{24} (d_{223}) and d_{26} (d_{212}) are thickness shear vibration modes; however, XZ and ZX cut samples with electrodes on the Y faces were used to determine their values, due to the achievable clean d_{24}/d_{26} vibration modes using the impedance method.

#### 4.2. Piezoelectric Coefficients and Elastic Compliances

For determining the complete set of elastic compliances, piezoelectric coefficients, as well as electromechanical coupling factors, crystal cuts with different orientations were designed, as given in Figure 5. In addition, different crystal cuts with corresponding piezoelectric vibration modes and electro-elastic parameters are summarized in Table 3.

#### 4.2.1. Piezoelectric Coefficients d_{11} and d_{33}

Piezoelectric coefficients d_{11}, d_{33} and elastic compliances s_{11} and s_{33} were determined using the longitudinal mode of X and Z rods (samples (a) and (b)), relating to the equations below:

^{1/2}

_{r}, f

_{a}and k are the crystal density, resonance frequency, anti-resonance and electromechanical coupling factor, respectively.

#### 4.2.2. Piezoelectric Coefficients d_{12}, d_{13}, d_{31}, and d_{32}

Using the transverse extensional mode of XY, XZ, ZX and ZY crystal cuts (samples (c)–(e)), piezoelectric coefficients d_{12}, d_{13}, d_{31} and d_{32} and elastic compliances s_{22}, s_{33} and s_{11} were calculated by Equations (9)–(11).

#### 4.2.3. Piezoelectric Coefficients d_{15}, d_{35}, d_{24}, and d_{26}

d_{15}, d_{35}, d_{24} and d_{26} are thickness shear piezoelectric coefficients; thus, XZ and ZX crystal cuts (samples (c) and (f)) with electrodes on the X and Z faces, respectively, were designed for piezoelectric d_{15} and d_{35} and elastic compliance s_{55} evaluation. However, for the piezoelectric d_{24} and d_{26} and elastic compliance s_{44} and s_{66} determinations, XZ and ZX cut samples with an electrode on the Y faces were used (equivalent to YZ and YX cut samples with an electrode on Y faces), due to the fact that clean width shear vibrations can be obtained. The shear piezoelectric coefficients d_{15}, d_{35}, d_{24} and d_{26} and elastic compliances of s_{55}, s_{44} and s_{66} can be determined by using Equations (8), (9), (12) and (13).

#### 4.2.4. Elastic Compliances s_{12}, s_{13}, s_{15}, s_{23}, s_{25}, s_{35} and s_{46}

Except the dielectric, piezoelectric and elastic constants above, there are cross-elastic compliances s_{12}, s_{13}, s_{15}, s_{23}, s_{25}, s_{35} and s_{46} remaining unsolved, requiring the oblique crystal cuts. Crystal cuts (g–i) with transverse extensional modes were employed to determine the elastic compliances s_{13}, s_{15} and s_{35}, based on Equations (10) and (14), while for the compliance s_{23}, s_{12} and s_{25} evaluations, crystal cuts (m–o) were designed, combining Equations (10) and (15)–(17). Particularly, the elastic compliance s_{46} was obtained using Equations (13) and (18) by measuring the shear vibration mode of sample (g).

Utilizing the crystal cuts listed in Table 3, all of the dielectric permittivities and elastic compliances were obtained, and the results are given in Table 4.

**Table 4.**Dielectric, piezoelectric and elastic constants of ReCOB (Re = Er, Y, Gd, Nd, Pr and La) crystals.

Elastic Compliances s^{E}_{ij} (pm^{2}/N) | |||||||||||||

Crystals | s_{11} | s_{12} | s_{13} | s_{15} | s_{22} | s_{23} | s_{25} | s_{33} | s_{35} | s_{44} | s_{46} | s_{55} | s_{66} |

ErCOB | 7.3 | – | – | – | 6.7 | – | – | 8.9 | – | 31.2 | – | 21.6 | 16.6 |

YCOB ^{a} | 7.2 | −0.8 | −2.47 | −0.4 | 7.0 | 0.55 | 0.54 | 8.9 | −0.12 | 34.5 | −0.37 | 21.0 | 16.3 |

YCOB ^{b} | 7.15 | −0.35 | −2.8 | 0.74 | 6.91 | −0.68 | −0.46 | 8.79 | −1.2 | 35.0 | 3.5 | 23.0 | 15.0 |

GdCOB ^{b} | 7.6 | −1.2 | −3.9 | 0.4 | 7.15 | −4.6 | −1.5 | 8.94 | 0.32 | 28.0 | 1.7 | 23.0 | 18.0 |

GdCOB ^{c} | 7.6 | −0.92 | −0.95 | −0.99 | 7.1 | −0.92 | −14 | 8.9 | −0.33 | 17.0 | 14.0 | 19.0 | 18.0 |

GdCOB | 7.6 | −1.0 | −1.0 | −0.4 | 7.3 | 1.2 | 2.8 | 8.9 | −0.6 | 31.7 | −0.9 | 20.0 | 18.0 |

NdCOB ^{d} | 8.3 | −1.4 | −3.7 | 2.0 | 7.4 | −0.7 | 1.8 | 9.4 | −0.4 | 30.9 | 0.3 | 22.4 | 20.5 |

NdCOB ^{e} | 8.3 | −2.0 | −3.5 | −0.9 | 7.5 | −1.6 | 0.5 | 9.4 | 0.9 | 34.0 | 1.0 | 22.0 | 20.0 |

PrCOB | 9.0 | – | – | – | 7.9 | – | – | 10.4 | – | 33.9 | – | – | 18.7 |

LaCOB ^{f} | 8.82 | −1.1 | −2.6 | 1.1 | 4.78 | −1.2 | 3.4 | 10.1 | −1.3 | 34.0 | 1.3 | 19.0 | 18.0 |

Relative Dielectric Permittivities ε^{T}_{ij}/ε_{0} | |||||||||||||

Crystals | ε_{11} | ε_{13} | ε_{22} | ε_{33} | |||||||||

ErCOB | 9.5 | – | 11.8 | 10.0 | |||||||||

YCOB ^{a} | 9.65 | 0.95 | 11.8 | 9.55 | |||||||||

YCOB ^{b} | 9.57 | −0.96 | 11.4 | 9.52 | |||||||||

GdCOB ^{b} | 10.5 | 0.8 | 14.0 | 10.4 | |||||||||

GdCOB ^{c} | 9.03 | 0.75 | 12.35 | 10.25 | |||||||||

GdCOB | 9.4 | 0.9 | 13.3 | 9.4 | |||||||||

NdCOB ^{d} | 9.9 | −1.6 | 15.5 | 10.2 | |||||||||

NdCOB ^{e} | 9.9 | −0.8 | 15.0 | 10.0 | |||||||||

PrCOB | 9.8 | – | 15.3 | 10.1 | |||||||||

LaCOB ^{b} | 9.87 | 1.2 | 14.3 | 9.87 | |||||||||

LaCOB ^{f} | 9.87 | −1.24 | 14.9 | 9.87 | |||||||||

Piezoelectric Coefficients d_{ij} (pC/N) | |||||||||||||

Crystals | d_{11} | d_{12} | d_{13} | d_{15} | d_{24} | d_{26} | d_{31} | d_{32} | d_{33} | d_{35} | |||

ErCOB | 1.7 | 3.4 | −4.3 | −1.4 | 3.9 | 7.6 | −0.7 | −2.8 | 1.5 | −5.5 | |||

YCOB ^{a} | 1.7 | 3.9 | −4.2 | −1.1 | 4.4 | 7.9 | −0.77 | −2.5 | 1.4 | −5.0 | |||

YCOB ^{b} | 1.4 | 3.8 | −4.2 | −7.2 | −2.6 | 8.0 | −0.22 | −2.3 | 0.83 | 2.2 | |||

GdCOB ^{b} | 2.8 | 4.8 | −3.8 | −6.9 | 0.45 | 11 | −0.77 | −2.4 | 2.5 | −4 | |||

GdCOB ^{c} | 2.4 | 4.0 | −4.5 | 0.66 | −2.8 | 2.4 | −1.1 | −2.2 | 2.0 | −2.7 | |||

GdCOB | 2.4 | 4.1 | −4.7 | −2.2 | 5.1 | 11.1 | −1.1 | −2.5 | 2.0 | −3.8 | |||

NdCOB ^{d} | 2.7 | 4.1 | −4.8 | 3.0 | 4.1 | 15.0 | −1.9 | −3.7 | 2.1 | 2.3 | |||

NdCOB ^{e} | 1.7 | 3.9 | −4.9 | – | – | – | −1.4 | −2.5 | 1.5 | - | |||

PrCOB | 2.5 * | 3.9 | −5.2 | −1.9 | 3.1 | 15.8 | −1.5 | −2.5 | 2.0 * | 3.3 | |||

LaCOB | 2.4 * | 4.7 | – | – | 4.0 | 11.8 | – | – | 2.0 * | – | |||

LaCOB ^{b} | 2.1 | 3.9 | −3.9 | – | – | – | −0.55 | −2.2 | 1.5 | – | |||

LaCOB ^{f} | 1.28 | 3.89 | −3.89 | 0.62 | 6.41 | 8.73 | −0.55 | −2.22 | 1.10 | – |

Data (^{a−f}) were from [57,58,59,60,61,62]; * measured by a quasi-static d_{33} meter.

#### 4.2.5. Determination of the Sign of the Piezoelectric Coefficients

According to the IEEE standard [54], the longitudinal piezoelectric d_{11} and d_{33} are positive, while the transverse and shear piezoelectric coefficients are either positive or negative. Therefore, more crystal cuts (rotated around physical axes) are desired to discuss the sign of the piezoelectric coefficient d_{ij} (i ≠ j) for ReCOB crystals. In this subsection, piezoelectric coefficients for YCOB ^{a} in Table 4 are discussed.

It is noticed that the piezoelectric d_{35} is a function of d_{11}, d_{13}, d_{31}, d_{33} and d_{15} when rotated around the Y-axis, as expressed below:

Figure 6 shows the variation of piezoelectric d_{35} for YCOB crystals as a function of the rotation angle around the Y-axis (curve (a) in Figure 6). For verification, the experimental shear piezoelectric d_{35} values measured from rotated crystal cuts (ZXw)θ (θ = −60°, −30°, 30° and 60°) were plotted and found to be in good agreement with the calculations, from which, the maximum d_{35} value was determined to be on the order of 6.4 pC/N for (ZXw)−55° crystal cuts. In addition, different d_{35} curves plotted by presuming the opposite sign of the related piezoelectric coefficients in Equation (19) were given as curves (b)–(g), where large discrepancies were observed. The consistency between the calculation and measurement of d_{35} values for YCOB crystals indicates the validity of the reported piezoelectric coefficients d_{11}, d_{13}, d_{31}, d_{33}, d_{15} and d_{35}.

**Figure 6.**Variation of piezoelectric d

_{35}as a function of rotation angle around the Y-axis for YCOB crystals ((a) d

_{35}, (b) presuming positive d

_{15}, (c) presuming positive d

_{35}, (d) presuming positive d

_{13}, d

_{31}, d

_{15}and d

_{35}, (e) presuming positive d

_{13}and d

_{31}, (f) presuming negative d

_{13}and (g) presuming positive d

_{13}, d

_{31}, d

_{15}and d

_{35}). Reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

Similarly, the variations of piezoelectric d_{31}, d_{32}, d_{33} and d_{35} for different crystal cuts rotated around the Y-axis were plotted and compared in Figure 7, where piezoelectric d_{31} was selected for verifications by measuring the transverse extensional modes of different (ZXw)θ crystal cuts (θ = −60°, −45°, −30°, 30° and 60°). The piezoelectric coefficients were determined by Equations (9)–(11), and the results were plotted in Figure 7. It is noted that the measured for different crystal cuts is in good agreement with the calculated results (Equation (20)).

**Figure 7.**Variations of piezoelectric coefficients d

_{31}, d

_{32}, d

_{33}, and d

_{35}as a function of the rotation angle around the Y-axis. Reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

## 5. The Investigation of Optimum Crystal Cuts

According to the crystal symmetry, ReCOB crystals possess two independent longitudinal piezoelectric coefficients (d_{11} and d_{33}), four transverse piezoelectric coefficients (d_{12}, d_{13}, d_{31} and d_{32}) and four thickness shear piezoelectric coefficients (d_{15}, d_{35}, d_{24} and d_{26}). For piezoelectric sensing applications, various crystal cuts with different piezoelectric vibration modes might be utilized, including the longitudinal (compression) and shear vibration modes, etc. Therefore, it is necessary to discuss the optimum crystal cuts with a high piezoelectric coefficient and free of piezoelectric cross-talk.

#### 5.1. Rotated Crystal Cuts with Maximized Values

It is noticed that the maximum values of longitudinal piezoelectric d_{11} and d_{33} for ReCOB crystals should be equal, due to the fact that the Z rods (d_{33}) can be obtained by the rotation of X rods around the Y-axis for −90°, analogous to the thickness shear piezoelectric d_{15} (d_{35}) and d_{26} (d_{24}). Similarly, the transverse piezoelectric d_{12} and d_{13}, d_{31} and d_{32} should be equal from the viewpoint of crystal cut rotation. Equations (21)–(25) reveal the relationships between d_{11} and d_{33}, d_{15} and d_{35}, d_{26} and d_{24}, d_{12} and d_{13} and d_{31} and d_{32}. Evidently, the effective piezoelectric coefficients , and equal to d_{33}, d_{35} and d_{24}, respectively, when rotated around the Y-axis for ±90°. Meanwhile, when rotated around the X- and Z-axes for ±90°, and equal to d_{13} and d_{32}, respectively.

Therefore, the orientation dependence of the longitudinal piezoelectric d_{11}, transverse piezoelectric d_{12} and d_{31} and thickness shear piezoelectric coefficients d_{15} and d_{26} for YCOB crystals were discussed (data refer to YCOB ^{a} in Table 4). The maximum d_{11} (4.9 pC/N) was obtained from the X plate rotated 1° around the X-axis and then rotated 52° around the Z-axis, whereas the maximum d_{12} and d_{31} were observed for (XYl)33° and (ZXlt)90°/123°, respectively, being on the order of ~4.6 pC/N. Meanwhile, the highest piezoelectric d_{15} (9.0 pC/N) was obtained for (XZlt)90°/60° crystal cuts, and the maximum piezoelectric d_{26} (9.0 pC/N) was achieved in (YXlt)180°/30° [(YXt)-30°] crystal cut, as reported in [63]. The crystal cuts with maximum d_{11}, d_{12}, d_{31} and d_{15} (the same as d_{26}) values are illustrated in Figure 8a–d, respectively.

For comparison, the maximum values of shear piezoelectric coefficient d_{26} for different ReCOB crystals were studied based on the determined piezoelectric d_{24} and d_{26} values; results are given in Table 5, where PrCOB crystals were found to possess the maximum shear piezoelectric d_{26} value, being on the order of 16.1 pC/N, nearly two times that of ErCOB crystals. The increase of the piezoelectric coefficient for ReCOB crystals was reported to be associated with the difference of the diameter of rare-earth cations (Re^{3+}) and the disorder distribution of Ca^{2+} and Re^{3+} ions in ReCOB crystals [34,47].

**Figure 8.**The crystal cuts with maximum d

_{11}, d

_{12}, d

_{31}and d

_{15}(the same as d

_{26}) values for YCOB crystals (

**a**) (XYtw)1°/52°; (

**b**) (XYl)33°; (

**c**) (ZXlt)90°/123°; and (

**d**) (XZlt)90°/60°, respectively.

Crystals | d_{24} (pC/N) | d_{26} (pC/N) | Crystal Cuts | Maximum
d_{26} (pC/N) |
---|---|---|---|---|

ErCOB | 3.9 | 7.6 | (YXt)-25° | ~8.5 |

YCOB | 4.4 | 7.9 | (YXt)-30° | ~9.0 |

GdCOB | 4.7 | 11.5 | (YXt)-20° | ~12.0 |

SmCOB | 4.1 | 12.7 | (YXt)-20° | ~13.3 |

NdCOB | 4.1 | 15.0 | (YXt)-15° | ~15.5 |

PrCOB | 3.1 | 15.8 | (YXt)-10° | ~16.1 |

LaCOB | 4.0 | 11.8 | (YXt)-20° | ~12.4 |

#### 5.2. Rotated Crystal Cuts without Piezoelectric Cross-Talk

For ideal piezoelectric sensors, only the output (charge, current or voltage) is desired when they are loaded (such as force, vibration, pressure or acceleration) along their sensitivity axis; meanwhile, the load normal to that axis should not produce any output. However, real sensors may give an output also to a force normal to their sensitive axis, which may have significant influence on the accuracy of the measuring results, this is called cross-talk (transverse sensitivity) [64]. The cross-talk effect can be weakened or reduced by device structure design; however, minimizing the piezoelectric cross-talk by using the crystal cut design is desirable for improving the accuracy of the sensors. In the following subsection, piezoelectric cross-talk for the longitudinal modes (d_{11}) and thickness shear modes (d_{15} and d_{26}) were discussed, based on the piezoelectric coefficients of YCOB crystals (YCOB ^{a} in Table 4).

The X plates or X rods with the first 130° rotation angle around the Y-axis and the second 225° rotation angle around the X-axis exhibit a relatively good longitudinal piezoelectric response, giving an effective piezoelectric coefficient d_{11} being on the order of ~3.0 pC/N, while other coefficients were found to be −0.8~−1.2 pC/N.

Interestingly, the thickness shear piezoelectric d_{15} and d_{26} were found to possess optimal crystal cuts with negligible piezoelectric cross-talk. Figure 9 presents the variation of piezoelectric coefficients d_{1j} (j = 1–6) for (XZlw)45°/θ crystal cuts, where the optimal crystal cut was achieved for (XZlw)45°/35°, of which, the shear piezoelectric coefficient d_{15} was determined to be on the order of 8.8 pC/N, with minimized d_{11}, d_{12}, and d_{13} being close to zero, d_{14} and d_{16} < 2.0 pC/N.

Figure 10 gives the variation of piezoelectric coefficients d_{24} and d_{26} as a function of rotation angle around the Y-axis, based on the determined piezoelectric coefficients d_{24} (4.4 pC/N) and d_{26} (7.9 pC/N) for YCOB listed in Table 5. It was found that the (YXt)330° crystal cut, which equals to the crystal cut (YXt)−30°, possess the maximum d_{26} value, being around 9.0 pC/N, with zero piezoelectric d_{24} value, demonstrating free piezoelectric cross-talk. On the contrary, the highest piezoelectric d_{24} (9.0 pC/N) was achieved from the (YXt)60° crystal cut, with the d_{26} value being zero. Of particular importance is that these crystal cuts were found to possess the maximum thickness shear piezoelectric coefficients, without the interference from other piezoelectric vibrations, due to the fact that the monoclinic symmetry plane is vertical to the Y-axis.

**Figure 9.**Thickness shear piezoelectric d

_{15}for the (XZlw)45°/θ crystal cut as a function of rotation angle θ. The inset shows the schematic of the crystal cut (XZlw)45°/35°.

**Figure 10.**Thickness shear piezoelectric d

_{26}for the (YXt)θ crystal cut as a function of rotation angle θ. The inset shows the schematic of crystal cut (YXt)−30°.

## 6. Summary and Future Research

#### 6.1. Significance of ReCOB Crystals

The monoclinic ReCOB crystals are promising materials for high temperature piezoelectric sensing. The desirable material merits include the low-cost, reproducible crystals with high quality and large dimension, high piezoelectric coefficient, high electrical resistivity, as well as the high thermal stability of the piezoelectric and electromechanical properties, etc. Relevant to this paper, crystal growth and piezoelectric characterization were discussed. Then Bridgman and Cz methods for ReCOB crystal growth were introduced. The relationship between the crystallographic axes and physical axes for ReCOB was discussed for piezoelectric property determination. In addition, procedures for the characterization of the dielectric, elastic and piezoelectric parameters were established, the independent electro-elastic constants were determined, where the highest piezoelectric coefficients d_{26} were achieved, being on the order of 7.6, 7.9, 11.5, 12.7, 15.0, 15.8 and 11.8 pC/N for ErCOB, YCOB, GdCOB, SmCOB, NdCOB, PrCOB and LaCOB, respectively. Of particular significance is that (YXt)θ crystal cuts not only possess maximum d_{26} coefficients, but also exhibit no cross-talk from other piezoelectric vibrations.

#### 6.2. Future Research

ReCOB piezoelectric crystals are potential materials for high temperature sensor applications. However, several challenges remain for future investigations, including: (1) temperature stability evaluation of the electro-elastic properties, to explore the optimized ReCOB crystals with high thermal stability; (2) structure-property investigation of ReCOB crystals for further improving of the properties at elevated temperatures; and (3) reliability testing of the electro-elastic properties under harsh environments, including high temperature, hard radiation (Gamma and neutron radiations), low oxygen partial pressure, high/low pressure and corrosive/erosive conditions, etc.

## Acknowledgments

Thomas R. Shrout from Pennsylvania State University is acknowledged for the helpful discussions. Shiyi Guo from Shandong University is thanked for his kind helps during ErCOB crystal growth. Fapeng Yu and Xian Zhao would like to thank the National Natural Science Foundation of China (Grant Nos. 91022034, 51202129, 51372168) and the Natural Science Foundation of Shandong Province (ZR2012EMQ004). This work is also supported by the China postdoctoral Science Foundation (2012M511019).

## Author Contributions

Fapeng Yu grew the ReCOB single crystals and evaluated their dielectric and piezoelectric properties. Xiulan Duan studied the crystal structures. Qingming Lu prepared the crystal cuts of different ReCOB crystals for property characterization. Shujun Zhang and Xian Zhao directed this research. Fapeng Yu and Xiulan Duan prepared the manuscript. All the authors contributed to revising the manuscript.

## Conflicts of Interest

The authors declare no conflict of interest.

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