Abstract

Rare-earth calcium oxyborate crystals, ReCa₄O(BO₃)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 (LiNbO₃, 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 (SiO₂) (2–3 pC/N). However, the maximum operating temperature of LN-based piezoelectric devices, restricted by their low electrical resistivity (a requirement of >10⁶ 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 A₃BC₃D₂O₁₄ [6,7,8,9,10,11,12] and gallium orthophosphate GaPO₄, 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 GaPO₄), prior to which, there are no phase transitions observed for langasite family crystals (the phase transition for GaPO₄ is about 970 °C). However, the costly component Ga₂O₃ restricted their further implements. The newly developed Ca₃TaAl₃Si₂O₁₄ (CTAS) crystals, substituting the Ga with Al elements, were found to possess improved higher temperature properties than La₃Ga₅SiO₁₄ (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 SrLaGa₃O₇ (SLG), Ca₂Al₂SiO₇ (CAS), etc.) and fresnoite crystals (point group 4mm, such as Ba₂TiSi₂O₈) 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 (ReCa₄O(BO₃)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, La₃Ga₅SiO₁₄; CTGS, Ca₃TaGa₃Si₂O₁₄; CTAS, Ca₃TaAl₃Si₂O₁₄; SLG, SrLaGa₃O₇; CAS, Ca₂Al₂SiO₇; BTS, Ba₂TiSi₂O₈; ReCOB, ReCa₄O(BO₃)₃.

Symmetry	Crystals	Growth Method	Tc/Tm (°C)	deff (pC/N)
Trigonal	LGS	Cz	1430 [39]	6~7
	CTGS	Cz	~1370 [40]	~10
	CTAS	Cz	–	~5
	GaPO4	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.

Table 1. A comparison of different high temperature piezoelectric crystals. LGS, La₃Ga₅SiO₁₄; CTGS, Ca₃TaGa₃Si₂O₁₄; CTAS, Ca₃TaAl₃Si₂O₁₄; SLG, SrLaGa₃O₇; CAS, Ca₂Al₂SiO₇; BTS, Ba₂TiSi₂O₈; ReCOB, ReCa₄O(BO₃)₃.

Symmetry	Crystals	Growth Method	Tc/Tm (°C)	deff (pC/N)
Trigonal	LGS	Cz	1430 [39]	6~7
	CTGS	Cz	~1370 [40]	~10
	CTAS	Cz	–	~5
	GaPO4	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%) CaCO₃, Re₂O₃ (Pr₆O₁₁) and H₃BO₃ powders. They were weighed according to their nominal compositions. Considering the evaporation of B₂O₃ during crystal growth, an excess of H₃BO₃ (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 H₃BO₃ and CaCO₃, 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:

Re₂O₃ + 6H₃BO₃ + 8CaCO₃ = 2ReCa₄O(BO₃)₃ + 8CO₂↑ + 9H₂O↑

(1)

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.

Figure 1. Schematic set-up of the Bridgman growth system.

Figure 1. 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 2. Schematic set-up of the Cz method for crystal growth.

Figure 2. Schematic set-up of the Cz method for crystal growth.

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.

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 d₃₃ 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.

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/Å3/°) *	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).

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/Å3/°) *	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:

(2)

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:

c = 1/s

(3)

e = dc

(4)

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 d₁₁ and d₃₃ 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 d₁₂, d₁₃, d₁₅, d₂₄, d₂₆, d₃₁, d₃₂, d₃₃ and d₃₅. On the contrary, the dielectric cross permittivity ε₁₃ 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°).

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 ε₁₁, ε₂₂, ε₃₃ can be determined from X-, Y-, and Z-cut samples (k, l and j), while for the determination of ε₁₃, sample (g) was used (Table 3), following the formulae below:

(5)

(6)

where C is the capacitance of the crystal sample measured at 100 kHz, t is the thickness, A is the area of the measured plate and is the dielectric permittivity for the Zʹ-axis.

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	d11
Z rod (b)		d33
ZX plate (f)	transverse extension	s11, d31, s22, d12, d32, s33, d13
XY plate (d)
ZY plate (e)
XZ plate (c)
XZ plate (c)	width shear *	s44, d24
ZX plate (f)		s66, d26
(XZw)45° (g)		s46
XZ plate (c)	thickness shear	s55, d15, d35
ZX plate (f)
(XZw)45° (g)	transverse extension	s13, s15, s35
(XZw)30° (h)
(XZw)−30° (i)
(XYt)45° (m)	transverse extension	s12, s23, s25
(ZXt)45° (n)
(YZtw)45°/−45° (o)
X cut (k)	–	ε11, ε22, ε33, ε13
Y cut (l)
Z cut (j)
(XZw)45° (g)

* d₂₄ (d₂₂₃) and d₂₆ (d₂₁₂) 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 d₂₄/d₂₆ vibration modes using the impedance method.

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	d11
Z rod (b)		d33
ZX plate (f)	transverse extension	s11, d31, s22, d12, d32, s33, d13
XY plate (d)
ZY plate (e)
XZ plate (c)
XZ plate (c)	width shear *	s44, d24
ZX plate (f)		s66, d26
(XZw)45° (g)		s46
XZ plate (c)	thickness shear	s55, d15, d35
ZX plate (f)
(XZw)45° (g)	transverse extension	s13, s15, s35
(XZw)30° (h)
(XZw)−30° (i)
(XYt)45° (m)	transverse extension	s12, s23, s25
(ZXt)45° (n)
(YZtw)45°/−45° (o)
X cut (k)	–	ε11, ε22, ε33, ε13
Y cut (l)
Z cut (j)
(XZw)45° (g)

* d₂₄ (d₂₂₃) and d₂₆ (d₂₁₂) 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 d₂₄/d₂₆ 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 d₁₁ and d₃₃

Piezoelectric coefficients d₁₁, d₃₃ and elastic compliances s₁₁ and s₃₃ were determined using the longitudinal mode of X and Z rods (samples (a) and (b)), relating to the equations below:

(7)

(8)

d = k(εs)^1/2

(9)

where ρ, f_r, f_a and k are the crystal density, resonance frequency, anti-resonance and electromechanical coupling factor, respectively.

4.2.2. Piezoelectric Coefficients d₁₂, d₁₃, d₃₁, and d₃₂

Using the transverse extensional mode of XY, XZ, ZX and ZY crystal cuts (samples (c)–(e)), piezoelectric coefficients d₁₂, d₁₃, d₃₁ and d₃₂ and elastic compliances s₂₂, s₃₃ and s₁₁ were calculated by Equations (9)–(11).

(10)

(11)

4.2.3. Piezoelectric Coefficients d₁₅, d₃₅, d₂₄, and d₂₆

d₁₅, d₃₅, d₂₄ and d₂₆ 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 d₁₅ and d₃₅ and elastic compliance s₅₅ evaluation. However, for the piezoelectric d₂₄ and d₂₆ and elastic compliance s₄₄ and s₆₆ 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 d₁₅, d₃₅, d₂₄ and d₂₆ and elastic compliances of s₅₅, s₄₄ and s₆₆ can be determined by using Equations (8), (9), (12) and (13).

(12)

(13)

4.2.4. Elastic Compliances s₁₂, s₁₃, s₁₅, s₂₃, s₂₅, s₃₅ and s₄₆

Except the dielectric, piezoelectric and elastic constants above, there are cross-elastic compliances s₁₂, s₁₃, s₁₅, s₂₃, s₂₅, s₃₅ and s₄₆ remaining unsolved, requiring the oblique crystal cuts. Crystal cuts (g–i) with transverse extensional modes were employed to determine the elastic compliances s₁₃, s₁₅ and s₃₅, based on Equations (10) and (14), while for the compliance s₂₃, s₁₂ and s₂₅ evaluations, crystal cuts (m–o) were designed, combining Equations (10) and (15)–(17). Particularly, the elastic compliance s₄₆ was obtained using Equations (13) and (18) by measuring the shear vibration mode of sample (g).

(14)

(15)

(16)

(17)

(18)

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 sEij (pm2/N)
Crystals	s11	s12	s13	s15	s22	s23	s25	s33	s35	s44	s46	s55	s66
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 εTij/ε0
Crystals	ε11	ε13	ε22	ε33
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 dij (pC/N)
Crystals	d11	d12	d13	d15	d24	d26	d31	d32	d33	d35
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 d₃₃ meter.

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

Elastic Compliances sEij (pm2/N)
Crystals	s11	s12	s13	s15	s22	s23	s25	s33	s35	s44	s46	s55	s66
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 εTij/ε0
Crystals	ε11	ε13	ε22	ε33
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 dij (pC/N)
Crystals	d11	d12	d13	d15	d24	d26	d31	d32	d33	d35
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 d₃₃ meter.

4.2.5. Determination of the Sign of the Piezoelectric Coefficients

According to the IEEE standard [54], the longitudinal piezoelectric d₁₁ and d₃₃ 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 d₃₅ is a function of d₁₁, d₁₃, d₃₁, d₃₃ and d₁₅ when rotated around the Y-axis, as expressed below:

(19)

(20)

Figure 6 shows the variation of piezoelectric d₃₅ 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 d₃₅ 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 d₃₅ value was determined to be on the order of 6.4 pC/N for (ZXw)−55° crystal cuts. In addition, different d₃₅ 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 d₃₅ values for YCOB crystals indicates the validity of the reported piezoelectric coefficients d₁₁, d₁₃, d₃₁, d₃₃, d₁₅ and d₃₅.

Figure 6. Variation of piezoelectric d₃₅ as a function of rotation angle around the Y-axis for YCOB crystals ((a) d₃₅, (b) presuming positive d₁₅, (c) presuming positive d₃₅, (d) presuming positive d₁₃, d₃₁, d₁₅ and d₃₅, (e) presuming positive d₁₃ and d₃₁, (f) presuming negative d₁₃ and (g) presuming positive d₁₃, d₃₁, d₁₅ and d₃₅). Reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

Figure 6. Variation of piezoelectric d₃₅ as a function of rotation angle around the Y-axis for YCOB crystals ((a) d₃₅, (b) presuming positive d₁₅, (c) presuming positive d₃₅, (d) presuming positive d₁₃, d₃₁, d₁₅ and d₃₅, (e) presuming positive d₁₃ and d₃₁, (f) presuming negative d₁₃ and (g) presuming positive d₁₃, d₃₁, d₁₅ and d₃₅). Reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

Similarly, the variations of piezoelectric d₃₁, d₃₂, d₃₃ and d₃₅ for different crystal cuts rotated around the Y-axis were plotted and compared in Figure 7, where piezoelectric d₃₁ 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 d₃₁, d₃₂, d₃₃, and d₃₅ as a function of the rotation angle around the Y-axis. Reprinted with permission from [57]. Copyright 2014 Wiley-VCH.

Figure 7. Variations of piezoelectric coefficients d₃₁, d₃₂, d₃₃, and d₃₅ 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 (d₁₁ and d₃₃), four transverse piezoelectric coefficients (d₁₂, d₁₃, d₃₁ and d₃₂) and four thickness shear piezoelectric coefficients (d₁₅, d₃₅, d₂₄ and d₂₆). 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 d₁₁ and d₃₃ for ReCOB crystals should be equal, due to the fact that the Z rods (d₃₃) can be obtained by the rotation of X rods around the Y-axis for −90°, analogous to the thickness shear piezoelectric d₁₅ (d₃₅) and d₂₆ (d₂₄). Similarly, the transverse piezoelectric d₁₂ and d₁₃, d₃₁ and d₃₂ should be equal from the viewpoint of crystal cut rotation. Equations (21)–(25) reveal the relationships between d₁₁ and d₃₃, d₁₅ and d₃₅, d₂₆ and d₂₄, d₁₂ and d₁₃ and d₃₁ and d₃₂. Evidently, the effective piezoelectric coefficients , and equal to d₃₃, d₃₅ and d₂₄, respectively, when rotated around the Y-axis for ±90°. Meanwhile, when rotated around the X- and Z-axes for ±90°, and equal to d₁₃ and d₃₂, respectively.

(21)

(22)

(23)

(24)

(25)

Therefore, the orientation dependence of the longitudinal piezoelectric d₁₁, transverse piezoelectric d₁₂ and d₃₁ and thickness shear piezoelectric coefficients d₁₅ and d₂₆ for YCOB crystals were discussed (data refer to YCOB ^a in Table 4). The maximum d₁₁ (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 d₁₂ and d₃₁ were observed for (XYl)33° and (ZXlt)90°/123°, respectively, being on the order of ~4.6 pC/N. Meanwhile, the highest piezoelectric d₁₅ (9.0 pC/N) was obtained for (XZlt)90°/60° crystal cuts, and the maximum piezoelectric d₂₆ (9.0 pC/N) was achieved in (YXlt)180°/30° [(YXt)-30°] crystal cut, as reported in [63]. The crystal cuts with maximum d₁₁, d₁₂, d₃₁ and d₁₅ (the same as d₂₆) values are illustrated in Figure 8a–d, respectively.

For comparison, the maximum values of shear piezoelectric coefficient d₂₆ for different ReCOB crystals were studied based on the determined piezoelectric d₂₄ and d₂₆ values; results are given in Table 5, where PrCOB crystals were found to possess the maximum shear piezoelectric d₂₆ 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 (Re³⁺) and the disorder distribution of Ca²⁺ and Re³⁺ ions in ReCOB crystals [34,47].

Figure 8. The crystal cuts with maximum d₁₁, d₁₂, d₃₁ and d₁₅ (the same as d₂₆) values for YCOB crystals (a) (XYtw)1°/52°; (b) (XYl)33°; (c) (ZXlt)90°/123°; and (d) (XZlt)90°/60°, respectively.

Figure 8. The crystal cuts with maximum d₁₁, d₁₂, d₃₁ and d₁₅ (the same as d₂₆) values for YCOB crystals (a) (XYtw)1°/52°; (b) (XYl)33°; (c) (ZXlt)90°/123°; and (d) (XZlt)90°/60°, respectively.

Table 5. Optimum crystal cuts (d₂₆ mode) obtained for ReCOB crystals at room temperature.

Crystals	d24 (pC/N)	d26 (pC/N)	Crystal Cuts	Maximum d26 (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

Table 5. Optimum crystal cuts (d₂₆ mode) obtained for ReCOB crystals at room temperature.

Crystals	d24 (pC/N)	d26 (pC/N)	Crystal Cuts	Maximum d26 (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 (d₁₁) and thickness shear modes (d₁₅ and d₂₆) 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 d₁₁ 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 d₁₅ and d₂₆ 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 d₁₅ was determined to be on the order of 8.8 pC/N, with minimized d₁₁, d₁₂, and d₁₃ being close to zero, d₁₄ and d₁₆ < 2.0 pC/N.

Figure 10 gives the variation of piezoelectric coefficients d₂₄ and d₂₆ as a function of rotation angle around the Y-axis, based on the determined piezoelectric coefficients d₂₄ (4.4 pC/N) and d₂₆ (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 d₂₆ value, being around 9.0 pC/N, with zero piezoelectric d₂₄ value, demonstrating free piezoelectric cross-talk. On the contrary, the highest piezoelectric d₂₄ (9.0 pC/N) was achieved from the (YXt)60° crystal cut, with the d₂₆ 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 d₁₅ 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 9. Thickness shear piezoelectric d₁₅ 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 d₂₆ for the (YXt)θ crystal cut as a function of rotation angle θ. The inset shows the schematic of crystal cut (YXt)−30°.

Figure 10. Thickness shear piezoelectric d₂₆ 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 d₂₆ 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 d₂₆ 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.