List of piezoelectric materials

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This page lists properties of several commonly used piezoelectric materials.

Contents

Piezoelectric materials (PMs) can be broadly classified as either crystalline, ceramic, or polymeric. [1] The most commonly produced piezoelectric ceramics are lead zirconate titanate (PZT), barium titanate, and lead titanate. Gallium nitride and zinc oxide can also be regarded as a ceramic due to their relatively wide band gaps. Semiconducting PMs offer features such as compatibility with integrated circuits and semiconductor devices. Inorganic ceramic PMs offer advantages over single crystals, including ease of fabrication into a variety of shapes and sizes not constrained crystallographic directions. Organic polymer PMs, such as PVDF, have low Young's modulus compared to inorganic PMs. Piezoelectric polymers (PVDF, 240 mV-m/N) possess higher piezoelectric stress constants (g33), an important parameter in sensors, than ceramics (PZT, 11 mV-m/N), which show that they can be better sensors than ceramics. Moreover, piezoelectric polymeric sensors and actuators, due to their processing flexibility, can be readily manufactured into large areas, and cut into a variety of shapes. In addition polymers also exhibit high strength, high impact resistance, low dielectric constant, low elastic stiffness, and low density, thereby a high voltage sensitivity which is a desirable characteristic along with low acoustic and mechanical impedance useful for medical and underwater applications.

Among PMs, PZT ceramics are popular as they have a high sensitivity, a high g33 value. They are however brittle. Furthermore, they show low Curie temperature, leading to constraints in terms of applications in harsh environmental conditions. However, promising is the integration of ceramic disks into industrial appliances moulded from plastic. This resulted in the development of PZT-polymer composites, and the feasible integration of functional PM composites on large scale, by simple thermal welding or by conforming processes. Several approaches towards lead-free ceramic PM have been reported, such as piezoelectric single crystals (langasite), and ferroelectric ceramics with a perovskite structure and bismuth layer-structured ferroelectrics (BLSF), which have been extensively researched. Also, several ferroelectrics with perovskite-structure (BaTiO3 [BT], (Bi1/2Na1/2) TiO3 [BNT], (Bi1/2K1/2) TiO3 [BKT], KNbO3 [KN], (K, Na) NbO3 [KNN]) have been investigated for their piezoelectric properties.

Key piezoelectric properties

The following table lists the following properties for piezoelectric materials

Table

Single crystals
ReferenceMaterial & heterostructure used for the characterization (electrodes/material, electrode/substrate)OrientationPiezoelectric coefficients, d (pC/N)Relative permittivity, εrElectromechanical coupling factor, kQuality factor
Hutson 1963 [2] AlNd15 = -4.07perε33 = 11.4
d31 = -2
d33 = 5
Cook et al. 1963 [3] BaTiO3d15 = 392ε11 = 2920k15 = 0.57
d31 = -34.5ε33 = 168k31 = 0.315
d33 = 85.6k33 = 0.56
Warner et al. 1967 [4] LiNbO3 (Au-Au)<001>d15 = 68ε11 = 84
d22 = 21ε33 = 30
d31 = -1k31 = 0.02
d33 = 6kt = 0.17
Smith et al. 1971 [5] LiNbO3<001>d15 = 69.2ε11 = 85.2
d22 = 20.8ε33 = 28.2
d31 = -0.85
d33 = 6
Yamada et al. 1967 [6] LiNbO3 (Au-Au)<001>d15 = 74ε11 = 84.6
d22 = 21ε33 = 28.6k22 = 0.32
d31 = -0.87k31 = 0.023
d33 = 16k33 = 0.47
Yamada et al. 1969 [7] LiTaO3d15 = 26ε11 = 53
d22 = 8.5ε33 = 44
d31 = -3
d33 = 9.2
Cao et al. 2002 [8] PMN-PT (33%)d15 = 146ε11 = 1660k15 = 0.32
d31 = -1330ε33 = 8200k31 = 0.59
d33 = 2820k33 = 0.94
kt = 0.64
Badel et al. 2006 [9] PMN-25PT<110>d31 = -643ε33 = 2560k31 = -0.73362
Kobiakov 1980 [10] ZnOd15 = -8.3ε11 = 8.67k15 = 0.199
d31 = -5.12ε33 = 11.26k31 = 0.181
d33 = 12.3k33 = 0.466
Zgonik et al. 1994 [11] ZnO (pure with lithium dopant)d15 = -13.3kr = 8.2
d31 = -4.67
d33 = 12.0
Zgonik et al. 1994 [12] BaTiO3 single crystals[001] (single domain)d33 = 90
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] (single domain)d33 = 224
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] neutral (domain size of 100 ľm)d33 = 235ε33 = 1984k33 = 54.4
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] neutral (domain size of 60 ľm)d33 = 241ε33 = 1959k33 = 55.9
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] (domain size of 22 ľm)d33 = 256ε33 = 2008k33 = 64.7
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] neutral (domain size of 15 ľm)d33 = 274ε33 = 2853k33 = 66.1
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] neutral (domain size of 14 ľm)d33 = 289ε33 = 1962k33 = 66.7
Zgonik et al. 1994 [12] BaTiO3 single crystals[111] neutrald33 = 331ε33 = 2679k33 = 65.2
[13] LN crystald31 = -4.5

d33 = -0.27

Li et al. 2010 [14] PMNT31d33 = 2000ε33 = 5100k31 = 80
d31 = -750
Zhang et al. 2002 [15] PMNT31-A1400ε33 = 3600
Zhang et al. 2002 [15] PMNT31-B1500ε33 = 4800
Zhang et al. 2002 [15] PZNT4.5d33 = 2100ε33 = 4400k31 = 83
d31 = -900
Zhang et al. 2004 [16] PZNT8d33 = 2500ε33 = 6000k31 = 89
d31 = -1300
Zhang et al. 2004 [16] PZNT12d33 = 576ε33 = 870k31 = 52
d31 = -217
Yamashita et al. 1997 [17] PSNT33ε33 = 960/
Yasuda et al. 2001 [18] PINT28700ε33 = 1500/
Guo et al. 2003 [19] PINT342000ε33 = 5000/
Hosono et al. 2003 [20] PIMNT1950ε33 = 3630/
Zhang et al. 2002 [15] PYNT40d33 = 1200ε33 = 2700k31 = 76
d31 = -500
Zhang et al. 2012 [21] PYNT45d33 = 2000ε33 = 2000k31 = 78
Zhang et al. 2003 [22] BSPT57d33 = 1200ε33 = 3000k31 = 77
d31 = -560
Zhang et al. 2003 [23] BSPT58d33 = 1400ε33 = 3200k31 = 80
d31 = -670
Zhang et al. 2004 [16] BSPT66d33 = 440ε33 = 820k31 = 52
d31 = -162
Ye et al. 2008 [24] BSPT57d33 = 1150

d31 = -520

ε33 = 3000k31 = 0.52

k33 = 0.91

Ye et al. 2008 [24] BSPT66d33 = 440ε33 = 820k31 = 0.52

k33 = 0.88

d31 = -162
Ye et al. 2008 [24] PZNT4.5d33 = 2000

d31 = -970

ε33 = 5200k31 = 0.50

k33 = 0.91

Ye et al. 2008 [24] PZNT8d31 = -1455ε33 = 7700k31 = 0.60

k33 = 0.94

Ye et al. 2008 [24] PZNT12d33 = 576

d31 = -217

ε33 = 870k31 = 0.52

k33 = 0.86

Ye et al. 2008 [24] PMNT33d33 = 2820

d31 = -1330

ε33 = 8200k31 = 0.59

k33 = 0.94

Matsubara et al. 2004 [25] KCN-modified KNNd33 = 100

d31 = -180

ε33 = 220-330kp = 33-391200
Ryu et al. 2007 [26] KZT modifiedKNNd33 = 126ε33 = 590kp = 4258
Matsubara et al. 2005 [27] KCT modified KNNd33 = 190ε33 =kp = 421300
Wang et al. 2007 [28] Bi2O3 doped KNNd33 = 127ε33 = 1309kp = 28.3
Jiang et al. 2009 [29] doped KNN-0.005BFd33 = 257ε33 = 361kp= 5245
Ceramics
ReferenceMaterial & heterostructure used for the characterization (electrodes/material, electrode/substrate)OrientationPiezoelectric coefficients, d (pC/N)Relative permittivity, εrElectromechanical coupling factor, kQuality factor
Berlincourt et al. 1958 [30] BaTiO3d15 = 270ε11 = 1440k15 = 0.57
d31 = -79ε33 = 1680k31 = 0.49
d33 = 191k33 = 0.47
Tang et al. 2011 [31] BFOd33 = 37kt = 0.6
Zhang et al. 1999 [32] PMN-PTd31 = -74ε33 = 1170k31 = -0.312283
[33] PZT-5Ad31 = -171ε33 = 1700k31 = 0.34
d33 = 374k33 = 0.7
[34] PZT-5Hd15 = 741ε11 = 3130k15 = 0.6865
d31 = -274ε33 = 3400k31 = 0.39
d33 = 593k33 = 0.75
[35] PZT-5Kd33 = 870ε33 = 6200k33 = 0.75
Tanaka et al. 2009 [36] PZN7%PTd33 = 2400εr = 6500k33 = 0.94

kt = 0.55

Pang et al. 2010 [37] ANSZd33 = 2951.6145.584
Park et al. 2006 [38] KNN-BZd33 = 400257.448
Cho et al. 2007 [39] KNN-BTd33 = 2251.0636.0
Park et al. 2007 [40] KNN-STd33 = 2201.4540.070
Zhao et al. 2007 [41] KNN-CTd33 = 2411.3241.0
Zhang et al. 2006 [42] LNKNd33 = 314~70041.2
Saito et al. 2004 [43] KNN-LSd33 = 2701.3850.0
Saito et al. 2004 [43] LF4d33 = 3001.57
Tanaka et al. 2009 [36] Oriented LF4d33 = 4161.5761.0
Pang et al. 2010 [37] ANSZd33 = 2951.6145.584
Park et al. 2006 [38] KNN-BZd33 = 400257.448
Cho et al. 2007 [44] KNN-BTd33 = 2251.0636.0
Park et al. 2007 [40] KNN-STd33 = 2201.4540.070
Maurya et al. 2013 [45] KNN-CTd33 = 2411.3241.0
Maurya et al. 2013 [45] NBT-BT(001) Textured samplesd33 = 322...
Gao et al. 2008 [46] NBT-BT-KBT(001) Textured samplesd33 = 192
Zou et al. 2016 [47] NBT-KBT(001) Textured samplesd33 = 134kp= 35
Saito et al. 2004 [43] NBT-KBT(001) Textured samplesd33 = 217kp = 61
Chang et al. 2009 [48] KNLNTS(001) Textured samplesd33 = 416kp = 64
Chang et al. 2011 [49] KNNS(001) Textured samplesd33 = 208kp = 63
Hussain et al. 2013 [50] KNLN(001) Textured samplesd33 = 192kp = 60
Takao et al. 2006 [51] KNNT(001) Textured samplesd33 = 390kp = 54
Li et al. 2012 [52] KNN 1 CuO(001) Textured samplesd33 = 123kp = 54
Cho et al. 2012 [53] KNN-CuO(001) Textured samplesd33 = 133kp = 46
Hao et al. 2012 [54] NKLNT(001) Textured samplesd33 = 310kp = 43
Gupta et al. 2014 [55] KNLN(001) Textured samplesd33 = 254
Hao et al. 2012 [54] KNN(001) Textured samplesd33 = 180kp = 44
Bai et al. 2016 [56] BCZT(001) Textured samplesd33 = 470kp = 47
Ye et al. 2013 [57] BCZT(001) Textured samplesd33 = 462kp = 49
Schultheiß et al. 2017 [58] BCZT-T-H(001) Textured samplesd33 = 580
OMORI et al. 1990 [59] BCT(001) Textured samplesd33 = 170
Chan et al. 2008 [60] Pz34 (doped PbTiO3)d15 = 43.3ε33 = 237k31 = 4.6700
d31 = -5.1ε33 = 208k33 = 39.6
d33 = 46k15 = 22.8
kp = 7.4
Lee et al. 2009 [61] BNKLBTd33 = 163εr = 766k31 = 0.188142
ε33 = 444.3kt = 0.524
kp = 0.328
Sasaki et al. 1999 [62] KNLNTSεr = 1156k31 = 0.2680
ε33 = 746kt = 0.32
kp = 0.43
Takenaka et al. 1991 [63] (Bi0.5Na0.5)TiO3 (BNT)-based BNKTd31 = 46εr = 650kp = 0.27
d33 = 150k31 = 0.165
Tanaka et al. 1960 [64] (Bi0.5Na0.5)TiO3 (BNT)-based BNBTd31 = 40εr = 580k31 = 0.19
d33 = 12.5k33 = 0.55
Hutson 1960 [65] CdSd15 = -14.35
d31 = -3.67
d33 = 10.65
Schofield et al. 1957 [66] CdSd31 = -1.53
d33 = 2.56
Egerton et al. 1959 [67] BaCaOTid31 = -50k15 = 0.19400
d33 = 150k31 = 0.49
k33 = 0.325
Ikeda et al. 1961 [68] Nb2O6Pbd31 = -11kr = 0.0711
d33 = 80k31 = 0.045
k33 = 0.042
Ikeda et al. 1962 [69] C6H17N3O10Sd23 = 84k21 = 0.18
d21 = 22.7k22 = 0.18
d25 = 22k23 = 0.44
Brown et al. 1962 [70] BaTiO3 (95%) BaZrO3 (5%)k15 = 0.15200
d31 = -60k31 = 0.40
d33 = 150k33 = 0.28
Huston 1960 [65] BaNb2O6 (60%) Nb2O6Pb (40%)d31 = -25kr = 0.16
Baxter et al. 1960 [71] BaNb2O6 (50%) Nb2O6Pb (50%)d31= -36kr = 0.16
Pullin 1962 [72] BaTiO3 (97%) CaTiO3 (3%)d31 = -53ε33 = 1390k15 = 0.39
d33 = 135k31 = 0.17
k33 = 0.43
Berlincourt et al. 1960 [73] BaTiO3 (95%) CaTiO3 (5%)d15 = -257ε33 = 1355k15 = 0.495500
d31 = -58k31 = 0.19
d33 = 150k33 = 0.49
kr = 0.3
Berlincourt et al. 1960 [73] BaTiO3 (96%) PbTiO3 (4%)d31 = -38ε33 = 990k15 = 0.34
d33 = 105k31 = 0.14
k33 = 0.39
Jaffe et al. 1955 [74] PbHfO3 (50%) PbTiO3 (50%)d31 = -54kr = 0.38
Kell 1962 [75] Nb2O6Pb (80%) BaNb2O6 (20%)d31 = 25kr = 0.2015
Brown et al. 1962 [70] Nb2O6Pb (70%) BaNb2O6 (30%)d31 = -40ε33 = 900k31 = 0.13350
d33 = 100k33 = 0.3
kr = 0.24
Berlincourt et al. 1960 [76] PbTiO3 (52%) PbZrO3 (48%)d15 = 166k15 = 0.401170
d31 = -43k31 = 0.17
d33 = 110k33 = 0.43
kr = 0.28
Berlincourt et al. 1960 [77] PbTiO3 (50%) lead Zirconate (50%)d15 = 166k15 = 0.504950
d31 = -43k31 = 0.23
d33 = 110k33 = 0.546
kr = 0.397
Egerton et al. 1959 [67] KNbO3 (50%) NaNbO3 (50%)d31 = -32140
d33 = 80k31 = 0.21
k33 = 0.51
Brown et al. 1962 [70] NaNbO3 (80%) Cd2Nb2O7 (20%)d31 = -80ε33 = 2000k31 = 0.17
d33 = 200k33 = 0.42
kr = 0.30
Schofield et al. 1957 [66] BaTiO3 (95%) CaTiO3 (5%) CoCO3 (0.25%)d31 = -60ε33 = 1605kr = 0.33
Pullin 1962 [72] BaTiO3 (80%) PbTiO3 (12%) CaTiO3 (8%)d31 = -31k31 = 0.151200
d33 = 79k33 = 0.41
kr = 0.24
Defaÿ 2011 [78] AlN (Pt-Mo)d31 = -2.5
Shibata et al. 2011 [79] KNN(Pt-Pt)<001>d31 = -96.3εr = 1100
d33 = 138.2
Sessler 1981 [80] PVDFd31 = 17.9k31 = 10.3
d32 = 0.9k33 = 12.6
d33 = -27.1
Ren et al. 2017 [81] PVDFd31 = 23εr = 106
d32 = 2
d33 = -21
Tsubouchi et al. 1981 [82] Epi AlN/Al2O3<001>d33 = 5.53ε33 = 9.5kt = 6.52490
Nanomaterials
ReferenceMaterialStructurePiezoelectric coefficients, d (pC/N)Characterization methodSize (nm)
Ke et al. 2008 [83] NaNbO3nanowired33 = 0.85-4.26 pm/VPFMd = 100
Wang et al. 2008 [84] KNbO3nanowired33 = 0.9 pm/VPFMd = 100
Zhang et al. 2004 [85] PZTnanowirePFMd = 45
Zhao et al. 2004 [86] ZnOnanobeltd33 = 14.3-26.7 pm/VPFMw = 360 t = 65
Luo et al. 2003 [87] PZTnanoshelld33 = 90 pm/VPFMd = 700 t = 90
Yun et al. 2002 [88] BaTiO3nanowired33 = 0.5 pm/VPFMd = 120
Lin et al. 2008 [89] CdSnanowireBending with AFM tipd = 150
Wang et al. 2007 [90] PZTnanofiberpiezoelectric voltage constant~0.079 Vm/NBending using a tungsten probed = 10
Wang et al. 2007 [91] BaTiO3-d33 = 45 pC/NDirect tensile testd ~ 280
Jeong et al. 2014 [92] Alkaline niobate (KNLN)filmd33 = 310 pC/N-
Park et al. 2010 [93] BaTiO3Thin filmd33 = 190 pC/N
Stoppel et al. 2011 [94] AlNThin filmd33 =5 pC/NAFM
Lee et al. 2017 [95] WSe22D nanosheetd11 = 3.26 pm/V
Zhu et al. 2014 [96] MoS2Free standing layere11 = 2900pc/mAFM
Zhong et al. 2017 [97] PET/EVA/PETfilmd33 = 6300 pC/N

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Piezoelectricity is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure and latent heat. It is derived from Ancient Greek πιέζω (piézō) 'to squeeze or press' and ἤλεκτρον (ḗlektron) 'amber'. The German form of the word (Piezoelektricität) was coined in 1881 by the German physicist Wilhelm Gottlieb Hankel; the English word was coined in 1883.

In physics and materials science, ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the additional property that their natural electrical polarization is reversible. The term is used in analogy to ferromagnetism, in which a material exhibits a permanent magnetic moment. Ferromagnetism was already known when ferroelectricity was discovered in 1920 in Rochelle salt by American physicist Joseph Valasek. Thus, the prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron. Materials that are both ferroelectric and ferromagnetic are known as multiferroics.

<span class="mw-page-title-main">Lead zirconate titanate</span> Chemical compound

Lead zirconate titanate, also called lead zirconium titanate and commonly abbreviated as PZT, is an inorganic compound with the chemical formula Pb[ZrxTi1−x]O3(0 ≤ x ≤ 1).. It is a ceramic perovskite material that shows a marked piezoelectric effect, meaning that the compound changes shape when an electric field is applied. It is used in a number of practical applications such as ultrasonic transducers and piezoelectric resonators. It is a white to off-white solid.

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<span class="mw-page-title-main">Barium titanate</span> Chemical compound

Barium titanate (BTO) is an inorganic compound with chemical formula BaTiO3. It is the barium salt of metatitanic acid. Barium titanate appears white as a powder and is transparent when prepared as large crystals. It is a ferroelectric, pyroelectric, and piezoelectric ceramic material that exhibits the photorefractive effect. It is used in capacitors, electromechanical transducers and nonlinear optics.

<span class="mw-page-title-main">Lithium niobate</span> Chemical compound

Lithium niobate is a synthetic salt consisting of niobium, lithium, and oxygen. Its single crystals are an important material for optical waveguides, mobile phones, piezoelectric sensors, optical modulators and various other linear and non-linear optical applications. Lithium niobate is sometimes referred to by the brand name linobate.

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The Burns temperature, Td, is the temperature where a ferroelectric material, previously in paraelectric state, starts to present randomly polarized nanoregions, that are polar precursor clusters. This behaviour is typical of several, but not all, ferroelectric materials, and was observed in lead titanate (PbTiO3), potassium niobate (KNbO3), lead lanthanum zirconate titanate (PLZT), lead magnesium niobate (PMN), lead zinc niobate (PZN), K2Sr4(NbO3)10, and strontium barium niobate (SBN), Na1/2Bi1/2O3 (NBT).

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A polar metal, metallic ferroelectric, or ferroelectric metal is a metal that contains an electric dipole moment. Its components have an ordered electric dipole. Such metals should be unexpected, because the charge should conduct by way of the free electrons in the metal and neutralize the polarized charge. However they do exist. Probably the first report of a polar metal was in single crystals of the cuprate superconductors YBa2Cu3O7−δ. A polarization was observed along one (001) axis by pyroelectric effect measurements, and the sign of the polarization was shown to be reversible, while its magnitude could be increased by poling with an electric field. The polarization was found to disappear in the superconducting state. The lattice distortions responsible were considered to be a result of oxygen ion displacements induced by doped charges that break inversion symmetry. The effect was utilized for fabrication of pyroelectric detectors for space applications, having the advantage of large pyroelectric coefficient and low intrinsic resistance. Another substance family that can produce a polar metal is the nickelate perovskites. One example interpreted to show polar metallic behavior is lanthanum nickelate, LaNiO3. A thin film of LaNiO3 grown on the (111) crystal face of lanthanum aluminate, (LaAlO3) was interpreted to be both conductor and a polar material at room temperature. The resistivity of this system, however, shows an upturn with decreasing temperature, hence does not strictly adhere to the definition of a metal. Also, when grown 3 or 4 unit cells thick (1-2 nm) on the (100) crystal face of LaAlO3, the LaNiO3 can be a polar insulator or polar metal depending on the atomic termination of the surface. Lithium osmate, LiOsO3 also undergoes a ferrorelectric transition when it is cooled below 140K. The point group changes from R3c to R3c losing its centrosymmetry. At room temperature and below, lithium osmate is an electric conductor, in single crystal, polycrystalline or powder forms, and the ferroelectric form only appears below 140K. Above 140K the material behaves like a normal metal. Artificial two-dimensional polar metal by charge transfer to a ferroelectric insulator has been realized in LaAlO3/Ba0.8Sr0.2TiO3/SrTiO3 complex oxide heterostructures.

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Nickel niobate is a complex oxide which as a solid material has found potential applications in catalysis and lithium batteries.

References

  1. ↑ Liu, Huicong; Zhong, Junwen; Lee, Chengkuo; Lee, Seung-Wuk; Lin, Liwei (December 2018). "A comprehensive review on piezoelectric energy harvesting technology: Materials, mechanisms, and applications". Applied Physics Reviews. 5 (4): 041306. Bibcode:2018ApPRv...5d1306L. doi:10.1063/1.5074184. ISSN   1931-9401. S2CID   117451095.
  2. ↑ Hutson, Andrew R. "Piezoelectric devices utilizing aluminum nitride." U.S. Patent 3,090,876, issued May 21, 1963.
  3. ↑ Cook, W. R.; Berlincourt, D. A.; Scholz, F. J. (May 1963). "Thermal Expansion and Pyroelectricity in Lead Titanate Zirconate and Barium Titanate". Journal of Applied Physics. 34 (5): 1392–1398. Bibcode:1963JAP....34.1392C. doi:10.1063/1.1729587. ISSN   0021-8979.
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  5. ↑ Smith, R. T.; Welsh, F. S. (May 1971). "Temperature Dependence of the Elastic, Piezoelectric, and Dielectric Constants of Lithium Tantalate and Lithium Niobate". Journal of Applied Physics. 42 (6): 2219–2230. Bibcode:1971JAP....42.2219S. doi:10.1063/1.1660528. ISSN   0021-8979.
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