PHYSICAL REVIEW RESEARCH 3, 043221 (2021) Piezoelectricity in nominally centrosymmetric phases Oktay Aktas ,1,* Moussa Kangama,1 Gan Linyu,1 Gustau Catalan ,2 Xiangdong Ding,1,† Alex Zunger,3 and Ekhard K. H. Salje 4,‡ 1 State Key Laboratory for Mechanical Behavior of Materials & Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China 2 ICREA and ICN2-Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Catalonia 3 University of Colorado, Energy Institute, Boulder, Colorado 80309, USA 4 Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom (Received 3 May 2021; revised 2 August 2021; accepted 11 November 2021; published 27 December 2021) Compound phases often display properties that are symmetry forbidden relative to their nominal, average crys- tallographic symmetry, even if extrinsic reasons (defects, strain, or imperfections) are not apparent. Specifically, breaking the macroscopic inversion symmetry of a centrosymmetric phase can dominate or significantly change its observed properties while the detailed mechanisms and magnitudes of the deviations of symmetry breaking are often obscure. Here, we choose piezoelectricity as a tool to investigate macroscopic inversion-symmetry breaking in nominally centrosymmetric materials as a prominent example and measure resonant piezoelectric spectroscopy (RPS) and Resonant Ultrasound Spectroscopy (RUS) in 15 compounds, 18 samples, and 21 different phases, including unpoled ferroelectrics, paraelectrics, relaxors, ferroelastics, incipient ferroelectrics, and isotropic materials with low defect concentrations, i.e., NaCl, fused silica, and CaF2 . We exclude the flexoelectric effect as a source of the observed piezoelectricity yet observe piezoelectricity in all nominally cubic phases of these samples. By scaling the RPS intensities with those of RUS, we calibrate the effective piezoelectric coefficients using single-crystal quartz as standard. Using this scaling we determine the effective piezoelectric modulus in nominally nonpiezoelectric phases, finding that the “symmetry-forbidden” piezoelectric effect ranges from ∼1 to 10−5 pm/V (∼0.5% to ∼2 × 10−5 % of the piezoelectric coefficient of poled ferroelectric lead zirconate titanate). The values for the unpoled ferroelectric phase are only slightly higher than those in the paraelectric phase. The extremely low coefficients are well below the detection limit of conventional piezoelectric measurements and demonstrate RPS as a convenient and ultrahighly sensitive method to measure piezoelectricity. We suggest that symmetry-breaking piezoelectricity in nominally centrosymmetric materials and disordered, unpoled ferroelectrics is a common phenomenon. DOI: 10.1103/PhysRevResearch.3.043221 I. INTRODUCTION in nominally centrosymmetric phases such as the cubic paraelectric phases of certain oxides [1–10]. Piezoelectric- Along with chiral dichroism, second harmonic generation, ity was demonstrated in several well-known materials, such and Rashba spin splitting, the classic effects of piezoelectric- as paraelectric phase of BaTiO3 and its solid solutions, fer- ity and pyroelectricity belong to a group of functionalities roelastic LaAlO3 and SrTiO3 , and relaxor ferroelectric lead enabled by specific crystal class (CC) symmetries. Both ef- magnesium niobate (PMN) [3–5,8–12], raising much at- fects require an absence of inversion center, i.e., belonging to a tention both in the scientific and engineering communities noncentrosymmetric (NCS) crystal class. In addition, whereas [6,9,13–15]. pyroelectricity is restricted exclusively to polar crystal classes, Some of the sightings of piezoeffects in centrosymmetric piezoelectricity is allowed both in polar symmetries as well phases are perhaps less than compelling, being likely false as in nonpolar symmetries (with the exception of the nonpo- positive determinations of the nominal cubic phase. Here, the lar CC of type O that forbids piezoelectricity). Remarkably, centrosymmetric phase may be the paraphase of a ferroelectric piezoelectricity and pyroelectricity were recently observed material, where the phase transition from the ferroelectric phase is not properly completed. Such “false positive” ob- servations thus do not necessarily pertain to nominally cubic * oktayaktas@xjtu.edu.cn phases (e.g., when the measurement temperatures are just be- † dingxd@xjtu.edu.cn low the para-to-ferro transition point) and thus the appearance ‡ ekhard@esc.cam.ac.uk of piezoelectricity does not pose a puzzle. Other mechanisms, once thought to explain the observation of piezoeffects, will Published by the American Physical Society under the terms of the be argued below not to be plausible. For example, one needs Creative Commons Attribution 4.0 International license. Further to be careful to exclude both flexoelectricity and surface distribution of this work must maintain attribution to the author(s) piezoelectricity as experimental errors which are common in and the published article’s title, journal citation, and DOI. heavily strained samples but do not lead to piezoelectricity. 2643-1564/2021/3(4)/043221(13) 043221-1 Published by the American Physical Society OKTAY AKTAS et al. PHYSICAL REVIEW RESEARCH 3, 043221 (2021) The remaining data and observations of piezoelectricity in an external mechanism have also been reported [2,3,33]. How- nominal cubic materials are broadly divided into two causes. ever, there is also a scenario that will retain a net polarization a. Ferroelastic domains within ferroelastic phases and fer- even if the local polar entities are not subjected to an effective roelastic local domains in the paraphase. These are related bias field. Such are the simulations [31] that suggest that a col- to the ordering of spontaneous strain in ferroelastic materials, lection of polar entities (i.e., polar nanostructures) in materials which are mechanical analogs of ferroelectrics (spontaneous with no external defects can also lead to a net polarization. In polarization) and ferromagnets (spontaneous magnetization) this case, nucleation of the first polar nanostructure biases the [16–20]. In the ferroelastic phase, twin walls, i.e., boundaries rest of the material, rendering it polar and piezoelectric. separating ferroelastic domains (or strain states), were shown Considering that mechanisms (a) and (b) raised previously to be generally polar [6]. Above the transition temperature, regarding the effective loss of inversion symmetry leading to signatures of the ferroelastic phase (also called precursors) piezoelectricity in nominally cubic paraphases were generic occur as strain fluctuations (in the paraelastic phase), leading and not specific and that the data are not always uniformly to structural heterogeneity [16–18,21]. These include tweed certain and may include real-world unavoidable defects led us patterns that are seen in electron diffraction experiments as to systematically measure the piezoelectric effect in nominally cross-hatched patterns and were observed to have lengths of centrosymmetric materials. Whereas piezoelectricity is a very 100 to 2000 Å [19]. Such heterogeneities can be a result old effect (discovered in 1880 by the Curie brothers) and is of elastic anisotropy, the interaction of local strain with an- indispensable with a vast number of technological applica- other property, such as octahedral tilting in some perovskite tions [34,35], finding a new twist to it is unusual and exciting oxides and halides, polarization in incipient ferroelectrics, physics. We measured the resonant piezoelectric spectroscopy or compounds whose ferroelectric or ferromagnetic phase is (RPS) [5,10,36] of five unpoled ferroelectric compounds simultaneously ferroelastic [16,19,20,22]. (characterized by ferroelectric domains) and their paraelec- b. Ferroelectriclike local polar structures within the para- tric phases (with local polar structures including ferroelectric electric phase. These give rise to intrinsic piezoelectricity precursors), ferroelastic LaAlO3 (with polar twin walls), four although some defect-induced polarity may play a role as relaxor ferroelectrics (with PNRs) above their freezing tem- extrinsic stimulus [16–18,23–25]. In this context, we also in- perature, incipient ferroelectrics KTaO3 and SrTiO3 with no clude ferroelectric precursors, which occur in the paraelectric known polar clusters at room temperature, and materials phase and exhibit locally the structural features which define with low concentrations of defects, namely, NaCl, CaF2 , the ferroelectric phase [16–18,23–25]. Local polar structures and silica glass. By using piezoelectric quartz single crys- have recently been observed by electron microscopy in the tal as a standard, we first determine the strain generated paraelectric phase of ferroelectric BaTiO3 and its solid so- in resonant ultrasound spectroscopy (RUS) and calibrate the lution with ferroelastic or incipient ferroelectric SrTiO3 with piezoelectric response detected by RPS on poled ferroelectrics sizes of 2 to 4 nm [26,27]. This class includes locally polar lead zirconate titanate (PZT-5H) and LiNbO3 , and paraelec- structures in relaxors (or relaxor ferroelectrics), which remain tric SrTiO3 . Then, we show that nominally centrosymmetric cubic down to absolute zero [28,29]. In the relaxor literature, phases of materials with local polar entities and chemical they are generally referred to as polar nanoregions (PNRs) defects, like dopants, possess measurable piezoelectric effects [7,28,29]. The size of PNRs ranges from several to 20 nm in that vary by seven orders of magnitude in comparison with the case of the well-known relaxor PMN [29,30]. poled ferroelectrics that have high piezoelectric coefficients. Length scales of above-mentioned local polar structures These values range from those comparable to that of quartz can extend from some subnanometers to, in some cases, nearly to 10 am/V, which is three orders of magnitude below the a micron [6,19,26,27]. We refer to all of these various struc- detection limit of conventional piezoelectric measurements. tures as polar nanostructures. Figure 1 is discussed in detail in this paper. In light of these Extrinsic versus intrinsic reasonings. All effects in types results we advocate the idea that piezoelectricity is a common (a) and (b) could include intrinsic mechanisms (i.e., charac- phenomenon in nominally centrosymmetric materials and un- teristics of the ideal, pristine bulk effects) as well as extrinsic poled ferroelectrics. Previous structural probes applied to the mechanisms, and it is not always possible to distinguish paraelectric phases suggested small deviations from crystal- between them experimentally. Examples of extrinsic effects lographic inversion symmetry even in high-quality samples include defect gradients formed during high-temperature syn- (for example, Refs. [38–40]). Here we connect such structural thesis of polycrystals (ceramics) [1]. Indeed, in order to observations with spectroscopic observation of weak piezo- observe a macroscopic piezoelectric effect under the local electricity, an observation made possible by the extraordinary polarity perspective it is necessary that the local piezoelec- sensitivity of the advanced resonant techniques RPS and RUS. tric tensor components dik (r) do not self-compensate inside The small deviations from centrosymmetry are well reflected the sample; e.g., their sum over all sites has to be nonzero. by piezoelectric coefficients that are as low as 0.1% to 0.001% This can be pictured by an “effective bias field” which pre- of that of piezoelectric quartz. vents self-compensation, which would be expected in the thermodynamic limit and random distributions of the tensor II. METHODS components [1,31,32]. Current understanding is that this bias may come from defect gradients as an extrinsic factor formed Measurements were performed on 15 compounds with during high-temperature synthesis of polycrystals (ceramics) 18 different samples. Five samples are ferroelectric: a [1]. The same mechanism has been proposed to occur during BaTiO3 single crystal, a BaTiO3 ceramic, a LiTaO3 single growth of single crystals [1]. Other mechanisms that suggest crystal, ceramics of (1 − x)BaTiO3 -xBaZrO3 solid solution 043221-2 PIEZOELECTRICITY IN NOMINALLY … PHYSICAL REVIEW RESEARCH 3, 043221 (2021) tion were achieved via an HF2LI function generator/lock-in amplifier unit (Zurich Instruments). All measurements were performed with PZT transducers with the exception of the BaTiO3 single crystal, for which LiNbO3 transducers were used, and 0.8(BaTiO3 )-0.2(BaZrO3 ) in the paraelec- tric phase for which PIN-PMN-PT transducers were used. Each sample was mounted between two transducers along its corners, which maximize their resonance intensities. RPS and RUS measurements of each sample were performed without changing the sample position. Measurements in the paraelectric phases of BaTiO3 , BZT20, and PST and the ferroelectric phase of PST were performed in a nitrogen-gas- cooled furnace (Sigma 10M10J or Suns Electronic Systems EC1X). Other measurements were done at room temperature (290–298 K). A comparison of the effective piezoelectric co- efficients was made by the sum over the resonance profiles of all resonance peaks (using IGOR PRO, WAVEMETRICS INC.). For samples with high peak intensities, RUS spectra of all compounds and RPS spectra in ferroelectrics, single- crystal quartz, and in the paraelectric phases of BaTiO3 and BZT20, the area of the spectrum was evaluated via integra- FIG. 1. Piezoelectricity in poled ferroelectrics, unpoled fer- tion, and the background was subtracted. In other materials, roelectrics, paraelectric phases of ferroelectrics, nominally cen- the total RPS area was calculated by fitting an asymmetric trosymmetric materials, and silica glass. The effective piezoelectric Lorentzian function to each resonance peak. RUS spectra coefficient deff , of each sample, as determined by combined RPS and were then used to scale the RPS spectra to estimate the ef- RUS measurements, corresponds to a coefficient that receives con- fective piezoelectric coefficients (see Sec. III). tributions from individual piezoelectric coefficients of the sample. Green circles correspond to longitudinal piezoelectric coefficients measured by a Berlincourt (d33 ) meter. Triangles are piezoelectric coefficients d11 and d14 of quartz reported by Ref. [37]. III. COUPLED RPS-RUS MEASUREMENTS AS A METHOD TO MEASURE PIEZOELECTRICITY WITH HIGH RESOLUTION with ferroelectric properties (x = 0.20). Relaxors are (1 − x)BaTiO3 -xBaZrO3 with (x = 0.30, 0.35, 0.40) and PMN In this section, upon giving a brief discussion of the ceramic. Nominally nonferroelectrics are LaAlO3 single crys- physical mechanisms behind RPS and RUS we introduce tals ([001] and [110] oriented) in their ferroelastic phase, a RPS-RUS measurements as a method to measure effective single crystal of KTaO3 , two SrTiO3 single crystals, quartz piezoelectric coefficients of samples with high resolution. single crystal (X-cut), NaCl single crystal, CaF2 , and silica Then, we calibrate our results by using known ferroelectrics glass. Ferroelectric, ferroelastic, relaxor, or isotropic behavior and paraelectric SrTiO3 . Other mechanisms that are allowed of each sample and characteristic temperatures are listed in to contribute to the piezoelectric effect in RPS measurements Table I. BaTiO3 and BaTiO3 - BaZrO3 ceramics were fab- were then evaluated on SrTiO3 . ricated by a conventional solid-state reaction method with starting chemicals of BaTiO3 (99.9%) and BaZrO3 (99%). The calcination was performed at 1350 ◦ C and sintering was A. Physical mechanisms behind RUS and RPS measurements done at 1450 ◦ C in air. These samples were preliminarily The experimental arrangement for RUS and RPS is shown characterized by x-ray diffraction. A PMN ceramic was pro- in Fig. 2(a). Both methods excite elastic standing waves (me- vided by Niall Donnelly. The PbSc0.5 Ta0.5 O3 (PST) sample chanical resonances) in the sample, whose frequencies are is the same as used in an earlier study [44]. A NaCl sin- determined by the elastic moduli [5,10,46,47]. In RUS, an gle crystal was purchased from Ted Pella Inc. while other AC voltage is applied across the piezoelectric emitter, which crystals and fused silica were purchased from Hefei Ke- vibrates through the inverse piezoelectric effect [46,47]. These jing Materials Technology. Additionally, poled ferroelectrics vibrations lead to strain oscillations in the sample, which (Y-36◦ -cut) LiNbO3 single crystal and PZT-5H ceramic (not is in contact with the emitter (Fig. 2). If the frequency of listed in Table I) were purchased from Kejing Materials Tech- oscillations corresponds to one of the natural frequencies of nology and Baoding Hongsheng Co. Ltd., respectively. The the sample, the oscillations become elastic standing waves samples were rectangular parallelepiped platelets with surface (i.e., mechanical resonances). The detection is then achieved areas ranging from 16.98 to 100 mm2 and thicknesses in the by the piezoelectric detector via the direct piezoelectric effect. range of 0.1 to 1.17 mm. Sample dimensions and orientations In RPS, the piezoelectric emitter is not used. Instead, the of single crystals are listed in the Supplemental Material [45]. AC voltage is applied across the sample. The mechanical Direct piezoelectric constant measurements were done resonances of the sample are excited if the material is macro- with a quasistatic d33 meter (IACAS, ZJ-4AN). For RUS scopically piezoelectric [5,10,11,36,48]. Examples in the case and RPS measurements (Fig. 2), signal generation and detec- of single-crystal quartz, poled PZT-5H, and poled LiNbO3 are 043221-3 OKTAY AKTAS et al. PHYSICAL REVIEW RESEARCH 3, 043221 (2021) TABLE I. Characteristics of compounds used in this work. Freezing temperatures, Tf , associated with the freezing or slowing down of PNRs in relaxors, are from Refs. [41–43]. Compound Characteristic Compound Characteristic name Classification temperature name Classification temperature shown in Fig. 2. These resonances appear as peaks in both demonstrated by microscopic measurements [9]. Another ex- RPS and RUS spectra. ample is coherent defect dipoles at cryogenic temperatures in incipient ferroelectric KTaO3 [36]. In this case, defect dipoles B. How to extract the piezoelectric response from the spectra freeze in a coherent fashion, leading to an increase in piezo- electricity with decreasing temperature. In line with these In RPS, because the excitation of elastic resonances re- measurements, complementary measurements showed spuri- quires the sample to be piezoelectric, the area of resonance ous ferroelectricity in the same temperature range. Finally, peaks gives a measure of the piezoelectric effect [8,36,49]. in relaxor ferroelectric PMN, piezoelectricity detected over The area of peaks can be calculated simply by integration or a large temperature above room temperature was attributed fitting an asymmetric Lorentzian function to each resonance to PNRs biased by chemically ordered regions [8]. Recent peak (see Sec. II). observation of macroscopic polarization at room temperature confirms the piezoelectricity in PMN [7]. The resolution of C. Examples of earlier works on the sensitivity of RPS the technique was previously not explored. The lack of data RPS was specifically designed to detect minuscule piezo- on the (unexpected) piezoelectric coefficients of these cen- electricity that cannot be measured by conventional direct trosymmetric materials in the literature indicates that it has piezoelectric measurements. The latter tends to have a res- a higher resolution than 0.01 pm/V. olution between 0.1 and 0.01 pm/V. The sensitivity of the RPS is because of the resonance condition of elastic standing waves [5,8,10,11,36,48]. This has previously been D. Measurements of strain and effective shown as observation of macroscopic piezoelectricity domi- piezoelectric coefficients nated by nanoscale microstructures and defects in nominally To have access to piezoelectric coefficients that are be- centrosymmetric materials. These include piezoelectric twin yond the resolution of direct piezoelectric measurements, we walls in the ferroelastic phase of SrTiO3 [10]. The polar develop a method based on RPS and RUS. This is done as (consequently piezoelectric) nature of domain walls was later follows. 043221-4 PIEZOELECTRICITY IN NOMINALLY … PHYSICAL REVIEW RESEARCH 3, 043221 (2021) FIG. 2. Elastic resonances of single-crystal quartz, poled LiNbO3 , and poled PZT-5H measured piezoelectrically (by RPS) and mechani- cally (by RUS). (a) Schematic of RUS and RPS measurements. The sample is lightly held between two transducers. For RPS measurements, VAC = 1–20 V was applied across the sample and the resonances were detected with the piezoelectric detector. For RUS measurements, the AC voltage was applied on the emitter (instead of the sample) and the resonances were detected by using the same piezoelectric detector. RPS and RUS spectra of (b) quartz, (c) poled LiNbO3 , and (d) poled PZT-5H. 1. Quartz as standard material and its piezoelectric properties and shear coefficients (d11 and d14 ) contribute to the generated Piezoelectric coefficients of single-crystal quartz are d14 = strain, we use dav = 1/2(d11 + d14 ). If the area of the RPS 0.727 pm/V and d11 = 2.31 pm/V, as reported by Bechmann spectrum [RPS(peak)area] designates the strain generated by [37]. The d11 coefficient of the sample used in this work was the application of the AC field and the area of the RUS spec- d11 = 2.5 ± 0.1 pm/V and is in line with the latter value. Note trum corresponds to the strain in a RUS experiment (eRUS ), that our value is a result of direct piezoelectric measurements i.e., RUS(peak)area ≡ eRUS , comparing the two areas, one can by a Berlincourt (d33 ) meter and not by RPS and RUS mea- calculate the RUS strain: surements. Small differences between reported values of d11 eRUS = RUS(peak)area × eRPS (quartz)/RPS(peak)area. naturally occur from sample to sample [37,50]. (1) The error in strain. The strain value depends on the 2. Calculation of average strain induced in RUS piezoelectric (d14 and d11 ) coefficients (i.e., piezoelectric Strain induced in quartz by the application of an electric anisotropy) of quartz as well as the quality and geometry field (i.e., in an RPS experiment) can be calculated by known of the contact between the sample and transducers. To es- piezoelectric coefficients of quartz through eRPS quartz = dav timate the standard deviation in RUS strain, a total of 26 E, where dav is the average piezoelectric coefficient and E is RPS-RUS measurements were carried out. An example of the electric field given by V/t with V being the applied AC resulting spectra is shown in Fig. 2(b). For every pair of RPS voltage and t the sample thickness. Here, as both longitudinal and RUS measurements, the sample was remounted to take 043221-5 OKTAY AKTAS et al. PHYSICAL REVIEW RESEARCH 3, 043221 (2021) into account the variations in local contact between the sample effective “induced piezoelectric” coefficient (deff ) is, by defi- and transducers. The result is eRUS = 3.5 × 10−7 with a stan- nition, the derivative of the strain with respect to the electric dard deviation of 35%. The calculated RUS strain spans the field: value of strain (3 × 10−7 ) associated with a shear resonance ∂e ∂e∂P generated during RUS measurements reported by Ref. [51]. deff = = = 2QPε0 εr = 2Qε0 εr (Pres + ε0 εr E ). ∂E ∂P∂E (2) 3. Measurements and calibration of piezoelectric response to In a parallel-plate capacitor, the electric field is the voltage (V) calculate the effective piezoelectric coefficients deff of samples divided by thickness (t). Hence,   This is done with a reverse procedure. Knowing the RUS V strain (eRUS ) and comparing the areas of RPS and RUS spec- deff = 2Qε0 εr Pres + ε0 εr . (3) t tra (total areas of RPS and RUS peaks) of the sample, one can calculate the strain generated in RPS, which then gives The above equation means that we can linearly modify the effective piezoelectric coefficient through the equation the piezoelectric coefficient of an SrTiO3 single crystal by eRPS (sample) = deff E. increasing the external bias voltage; the slope of the piezoelec- The error of effective piezoelectric coefficients. Due to the tric coefficient as a function of DC bias will be proportional range of values of the RUS strain, which is a result of piezo- to the electrostrictive coefficient and dielectric constant, while electric anisotropy and quality and local geometry of contact the intercept at the origin will be related to the residual between the sample and transducers, the effective piezoelec- macroscopic polarization of the sample. The result of this tric coefficients determined with the approach adopted here experiment, using an SrTiO3 single crystal of 0.1 mm thick- should only be correct within an order of magnitude. Below, ness, is shown in Fig. 3. Notice that the induced piezoelectric we test this by measuring the piezoelectric coefficients of coefficients are of the order of 10−3 pm/V (1 fm/V) and poled ferroelectrics and paraelectric SrTiO3 . would be hard to measure by standard techniques. Effective piezoelectric coefficients of poled ferroelectrics. The slope of the RPS-measured piezoelectric coefficient of Using the approach laid out above, the effective piezoelec- SrTiO3 as a function of voltage is ∼3.1 pm/(V m2 ). From tric coefficients of poled LiNbO3 and PZT-5H measured by Eq. (3) and assuming εr = 300 and t = 0.1 mm, this slope im- RPS/RUS measurements [Figs. 2(c) and 2(d)] are deff = 15 plies an electrostrictive coefficient Q = 0.022 m4 /C2 , which and 167 pm/V. The longitudinal piezoelectric coefficients is comparable to the longitudinal electrostrictive coefficient of these samples measured by a Berlincourt (d33 ) meter are of SrTiO3 , Q33 = 0.0046 m4 /(C m2 ) [52]. This showcases the dlongitudinal = 35 pm/V for LiNbO3 and d33 = 466 pm/V for sensitivity of the RPS technique and its reliability, as well as PZT-5H. offering an alternative way of calibration. Last but not least, The electrostrictive coefficient of paraelectric SrTiO3 . Be- the measured piezoelectric coefficient at zero bias (∼2 × 10−4 cause a comparison between deff measured by RPS-RUS pm/V) implies, via Eq. (3), a residual macroscopic polar- measurements and direct d33 measurements are not possible ization of the order of 1.7 C/m2 . This is about 105 times for values below 0.1 pm/V, i.e., the resolution of our d33 smaller than the polarization of typical ferroelectrics such as meter, or below 0.01 pm/V for more sensitive d33 meters, we BaTiO3 ; conversely, if we assume that the detected residual use a different approach to calibrate the deff values. Instead polarization comes from coherently oriented polar clusters we determine the electrostrictive coefficient of SrTiO3 . This with a local polarization of the same order of magnitude as material is interesting not only on account of its archetypal that of standard perovskite ferroelectrics, it means that the status (the “Drosophila of oxide physics” as Müller once volume concentration of polar clusters in SrTiO3 at room called it), but also its structural, piezoelectric, and electrostric- temperature is of the order of 10−5 (10 parts per million). It tive properties are well documented. In addition, SrTiO3 is would be difficult to observe such a small volume fraction of an apparently perfect cubic perovskite above Tc = 105 K, static polar clusters by direct means. and any elastic softening above Tc is associated with hetero- The limits of the RPS technique can be further tested geneities. It is therefore a perfect material for the study of if one considers that the forbidden piezoelectric coefficient nanopiezoelectricity in nonpolar materials. is primarily a result of defect-gradient-induced polarization SrTiO3 has a very large dielectric constant (εr = 300 at [1,2]. Therefore, by the application of a DC bias in the pos- room temperature), making it an easily polarizable material. itive direction, we show that piezoelectric resonance can be This means that an electric field can be used to turn SrTiO3 suppressed and then emerge again. This means switching from a nonpolar material into a polar one [52], and then the direction of residual polarization by a DC bias, further measure the “field-induced piezoelectricity” as a function of illustrating the accuracy and ultrahigh sensitivity of RPS. electric bias. In an electrostrictive material such as SrTiO3 , the The piezoelectric coefficient for small DC biases (120 and strain is related to the polarization P by e = QP2 , where Q is 210 mV) is calculated to be just below 10−5 pm/V. This sets the electrostrictive coefficient. The polarization in a perfectly the detection limit of RPS as 10−5 pm/V, or 10 am/V. nonpolar crystal is the dielectrically induced polarization, P = Resolution and accuracy of the RPS method for piezo- ε0 εr E , where E is the electric field and 0 is the permittiv- electric measurements. Our paper demonstrates that the RPS ity of free space. In addition, however, there can also be a technique can detect piezoelectric signals down to 10−5 pm/V. residual parasitic polarization coming from polar nanoregions, Though the resolution of our technique is high, the mar- defect gradients, and so forth; the total polarization is thus gin of error is within an order of magnitude of the actual P = Pres + ε0 εr E with Pres being residual polarization. The piezoelectric coefficient due to piezoelectric anisotropy and 043221-6 PIEZOELECTRICITY IN NOMINALLY … PHYSICAL REVIEW RESEARCH 3, 043221 (2021) FIG. 3. Calibration of RPS signal to determine ultralow piezoelectric coefficients. (a) Induced piezoelectricity in SrTiO3 by a DC voltage bias. (b) DC bias dependence of induced piezoelectricity. (c) Suppression of piezoelectric resonance by applying a positive bias. (d) Reemergence of the resonance by applying a bias greater than 170 mV. geometry and quality of local contact between the sample and this paper. Below we explain other possible contributions to transducers. This is well reflected in the extracted values of the RPS signal that we have critically examined in the case of deff for poled ferroelectrics and the electrostrictive coefficient SrTiO3 . of SrTiO3 , which differ from actual values by a factor of Flexoelectric effect. Bending experiments show that 3–5. However, this is sufficient for many scientific purposes, piezoelectricity can imitate flexoelectricity and vice versa. including the proof of existence of piezoelectricity in com- Although in RPS measurements we do not bend the sam- pounds that are nominally classified as nonpiezoelectric. ple, we apply a homogeneous electric field that can induce bending via inverse flexoelectricity, and this may be mistaken as a piezoelectric signal [53]. Therefore, we look at the size E. Possible contributions of flexoelectric effect dependence of the RPS signal, comparing 0.1- and 1-mm- and surface piezoelectricity to RPS signals thick samples (see the Supplemental Material [45]). We see Here, both flexoelectricity [53] and, theoretically, surface that both have piezoelectric coefficients on the same order of piezoelectricity [54–57] can generate strain that can poten- magnitude, which indicates that the origin is truly piezoelec- tially be picked in our measurements. Both are very weak tric and not flexoelectric; if the origin of these piezoelectric effects in comparison to the piezoelectric effect on bulk coefficients had been inverse flexoelectricity, the thinner crystals and ceramics [55,58,59]. Nevertheless, we checked one would have displayed an orders-of-magnitude larger whether RPS can pick up these signals by performing size- signal. dependent measurements on the reference material SrTiO3 . Surface piezoelectricity. Surface piezoelectricity is theo- The choice of SrTiO3 as a reference material stems from retically predicted in all materials [54–57]. By symmetry, the fact that, in addition to being an apparently perfect para- surface piezoelectricity can only manifest itself in experi- electric, its flexoelectric properties have been extensively ments where the equivalence between opposite surfaces is investigated [60]. In fact, it is the first material for which the broken, i.e., when there is an inhomogeneous deformation flexoelectric tensor was characterized [60]. Unfortunately, for such as bending so that one surface is under tension and other materials in this study, the information is much less the other under compression. Conversely, electric fields do complete. In any case, it would be desirable to do similar break the equivalence between surfaces, because a field par- studies in other materials, although it is out of the scope of allel to the piezoelectricity of one surface is antiparallel 043221-7 OKTAY AKTAS et al. PHYSICAL REVIEW RESEARCH 3, 043221 (2021) to the piezoelectricity of the opposite surface. This should depends on the experimental resolution. Surprisingly, para- cause a material to bend. Surface piezoelectricity is there- electric phases of ferroelectrics lead to similar values that fore indistinguishable from inverse flexoelectricity (this has are only slightly lower (Fig. 1). This is consistent with the in fact been amply discussed in the literature), and the same temperature dependence of d33 for an unpoled BaTiO3 ce- size-dependent experiment that allowed us to rule out a sig- ramic reported by Ref. [1] which showed variations by a nificant role of inverse flexoelectricity also excludes a role factor of 6 near Tc . Thus, small differences of deff below and of surface piezoelectricity. In relation to this, it has been above Tc in Fig. 1 do not reflect the typical depoling behavior shown that surface effects are not the dominant mecha- and self-poling effects can be neglected. Taking quartz as nism for the forbidden polarization detected in paraelectric reference, piezoelectric coefficients of centrosymmetric ma- phases of ferroelectrics [1]. Nevertheless, surface piezoelec- terials and silica glass (amorphous) range from comparable tricity may make some (minor) contribution to the signal values to ∼10−3 % of those of quartz (d33 = 2.31 pm/V and detected in piezoelectric measurements. These effects are d14 = 0.727 pm/V [37]). likely enhanced in the case of rough surfaces, as shown by simulations [61]. C. The role of polar nanostructures One should take the values of piezoelectric coefficients IV. PIEZOELECTRICITY IN NOMINALLY with caution especially for materials with local polar entities. CENTROSYMMETRIC PHASES OF COMPOUNDS Because the coherence between these structures is reduced on A. RPS and RUS spectra of nominally centrosymmetric heating, one would expect the piezoelectric coefficient to also and bulk-centrosymmetric materials decrease with increasing temperature. As shown in Fig. 5, this Examples of RPS and RUS spectra for unpoled ferro- is the case in the paraelectric phase of ferroelectric BZT20. electrics and centrosymmetric materials are shown in Fig. 4 This asserts earlier observations on BaTiO3 [5], PMN [8], (see Figs. S1–S3 in the Supplemental Material [45] for the KTaO3 [36], and SrTiO3 [10], which led to the same conclu- spectra of other compounds and samples). Similar to piezo- sion based on RPS measurements. electric quartz, poled LiNbO3 , and poled PZT-5H, all samples Some of the compounds investigated in this work contain show elastic resonances in RPS spectra. Due to the disal- polar nanostructures which have been visualized or inferred. lowed nature of piezoelectricity in our unpoled ferroelectrics Polar entities in BaTiO3 , its solid solutions, and PST were and nominally centrosymmetric compounds, the piezoelectric discussed in detail [2–4,23,24,44,64–67]. These have recently coefficients must be small when self-poling effects or inten- been visualized in BaTiO3 [26,27] and its solid solution tional gradients (e.g., generated by heterogeneous chemical BaTiO3 -SrTiO3 [26]. In the paraelectric phase of PST, stri- reduction) are absent [62,63]. Biancoli et al. [1] measured ations observed in diffraction measurements [68,69] were the d33 coefficient of a BaTiO3 ceramic near the ferroelectric attributed to tweed structure via Landau modeling [67] and transition temperature and found that the coefficients were scaling with entropy and polarization [44]. In BZT relax- on the order of 0.1–0.3 pC/N (or pm/V). In our RPS mea- ors, PNRs are estimated to persist at least up to 440 K surements, BaTiO3 as well as other ferroelectrics (Fig. 4 and [41–43] while for PMN the hierarchical nanodomain structure the Supplemental Material [45]) show strong resonance peaks observed at room temperature potentially exists at least up that are only slightly below RUS peak amplitudes. In other to ∼600 K [30,70]. The polarity of twin walls in LaAlO3 samples, while all RUS spectra contain strong resonances with and resulting macroscopic piezoelectricity was unambigu- amplitudes of 10−3 –10−5 V, RPS peaks are much weaker. ously demonstrated [11]. Whereas materials with known polar This defines our range of values for the unexpected piezoelec- nanostructures presented in this work have systematically tric effect. higher piezoelectric coefficients than those with no known such structures at room temperature, it remains to be de- termined if such nanostructures are the cause of symmetry breaking in all materials or an independent feature in some B. The magnitude of the observed piezoelectricity cases. Using the approach described in Sec. III, we calculated the effective piezoelectric coefficients of other samples (Fig. 1). In Fig. 1, d33 of poled PZT-5H ceramic and LiNbO3 , piezoelectric coefficients of quartz, are also shown for the D. Piezoelectricity in centrosymmetric materials purpose of depicting the range of piezoelectric coefficients which may contain external defects that materials can have. The overall variation of piezoelectric Among other compounds investigated here, silica glass, coefficients spans an extremely broad range of values, varying NaCl, and CaF2 have no known locally polar or piezoelec- by seven orders of magnitude from 466 to 10−5 pm/V (or tric nanoclusters but contain defects [71–74]. In KTaO3 and 10 am/V). These results somewhat reflect the range of de- SrTiO3 polar nanostructures only exist at cryogenic temper- viation from inversion symmetry. atures as polar clusters and polar domain walls, respectively Inversion symmetry is absent in poled ferroelectrics and [9,10,36,75]. No experimental evidence of local polarity for quartz. Even unpoled ferroelectrics, although bulk centrosym- either sample has been reported at room temperature. Here, metric, can give measurable piezoelectric coefficients [1] with our results show that even cubic materials with no known conventional measurements but not always [3], which also polar or piezoelectric nanostructures are also piezoelectric. 043221-8 PIEZOELECTRICITY IN NOMINALLY … PHYSICAL REVIEW RESEARCH 3, 043221 (2021) FIG. 4. Piezoelectricity in globally centrosymmetric materials and unpoled ferroelectrics which are bulk centrosymmetric due to spatial averaging of domains and grains (also see the Supplemental Material [45]). RUS spectra of all compounds show strong resonances while those detected in RPS range from 10−3 to ∼10−8 V, demonstrating broken centrosymmetry. V. CURRENT UNDERSTANDING OF SPONTANEOUS highest-symmetry crystallographic unit cell. Analogous ap- ATOMIC-SCALE SYMMETRY BREAKING proximations were common for describing paramagnets as IN PARAPHASES a phase where each site has zero moment. Because it is artificial to expect a transition from finite dipoles in the The simplest, albeit naïve, view of a paraelectric phase low-temperature ferroelectric phase to zero dipole in the para- according to past studies assumes that it has a net global zero electric phase, a better approximation has been the displacive dipole because each site has a zero dipole. This “nonelectric model of paraelectric phases [76] that still uses the minimal model” of paraelectricity has been often used in electronic unit cell but allows for finite polar displacements in a sin- structure calculations as it allows one to use the smallest, gle double-well picture. However, because of the restriction 043221-9 OKTAY AKTAS et al. PHYSICAL REVIEW RESEARCH 3, 043221 (2021) ature sets in (modeled via density functional theory (DFT) molecular dynamics (MD) [80,83]), additional displacements take place. Significantly, both the minimization of the internal energy U (in DFT) and that of the free energy U-TS (via DFT- MD) lead to symmetry breaking including the removal of inversion symmetry. Thus, although the average S over local motifs {Si } has high symmetry (say, being centrosymmetric), this does not imply that all measured physical properties P would equal the property P(S) of the average structure (e.g., null piezoelectricity). This then allows the observation of physical effects that reflect noncentrosymmetric symmetry, including piezoelectricity. Recent first-principles calculations [79–82] considered a supercell of numerous Pm-3m unit cells, preserving the global cubic shape symmetry, but then min- imizes the cell internal atomic forces in DFT. One finds in such constrained minimization of the internal energy U0 a symmetry breaking. Depending on the type of pertinent lo- cal degree of freedom, these local motifs can correspond to local octahedra rotations in cubic halide perovskites (CsPbI3 ) FIG. 5. Comparison of RPS spectra of ferroelectric BZT20 [79,80], or to local magnetic moments in paramagnetic oxides collected 32 and 84 K above the ferroelectric Curie temperature (YTiO3 , YNiO3 ) [82], or to local displacements creating local Tc = 296 K. dipoles in a paraelectric (BaTiO3 ) [79,83]. All such symmetry breakings represent short-range order and lead to macroscopic to a minimal unit cell in the displacive model, all dipoles effects on the band structure, effective masses, and optical in the paraelectric phase must be aligned in tandem, lead- properties [79–82]. ing to a long-range-ordered model for paraelectricity. In this view atoms are oscillating thermally in a single potential VI. CONCLUSIONS well around their average Wyckoff positions. Therefore, one can always apply a time average onto the observable prop- Conclusions and potential future work can be summarized erties. Reciprocal-space calculation of phonons is based on as follows. such a symmetry-unbroken monomorphous structure for the Macroscopic inversion symmetry breaking of cubic and paraphase following the softening of the paraphonons as the bulk centrosymmetric materials. This was demonstrated to be transition to the lower-temperature low-symmetry phase is common. We demonstrated that unpoled ferroelectric crystals approached. In the alternative order-disorder model for the and ceramics can allow significant piezoelectricity. transitions [77], one allows in principle a larger than minimal Internal versus external macroscopic symmetry break- unit cell for describing the paraphase so that disordered local ing. We intentionally did not distinguish between internal polar displacements are possible in the paraelectric phase; [31,32,79–82] and external symmetry breaking [1,2] on the i.e., there is a potential well with a few minima (four or detected piezoelectricity. Although there is no experimental eight) but the occupation numbers are variable, allowing net evidence of polar nanostructures in NaCl, silica glass, CaF2 , polarity. Phase transition occurs when occupational symmetry KTaO3 , and SrTiO3 at room temperature, recent observa- is broken. However, the net dipole is zero. tions of macroscopic symmetry breaking in cubic materials These concepts have been recently challenged [78]. Indeed, due to internal effects [75–78] suggest that bulk symmetry there are recent theoretical reasons [79–83] to believe that breaking could occur even in the absence of defects. For intrinsic mechanisms might be responsible for the formation example, birefringence observed in CaF2 has been proposed of atomic-scale symmetry breaking in paraphases, manifest- to stem from intrinsic effects [84]. Future experimental and ing short-range-ordered (SRO) local motifs (“polymorphous theoretical work would then include the quantification of in- network”) [79]. Such symmetry breaking was argued theoret- trinsic and extrinsic contributions to the symmetry breaking ically to lower the internal energy U0 even before thermal and piezoelectricity in nominally cubic materials. A possible agitation sets in, leading to the formation of a distribution approach to help distinguish the contribution of defects in of local motifs in all paraphases. This includes paraelec- symmetry breaking would require growing high-quality cu- tric phases (where the pertinent degree of freedom is the bic crystals, such as NaCl, with known defect concentrations local dipole), paramagnetic phases (where the pertinent lo- (some 10–1000 ppm). For examination of the local structure, cal degree of freedom is the local magnetic moment), or transmission electron microscopy [26,85] and dark field x-ray paraelastic phases (where the pertinent degree of freedom microscopy [13] are suitable methods. As macroscopic tools can be a geometric-steric effect such as octahedral rotation). for investigation of inversion symmetry breaking, RPS, as Significantly, even static minimization of the internal density shown in this work and earlier works [8,10,11,36], and second functional energy of supercells constrained to maintain the harmonic generation [86], as shown by Refs. [87,88], could be global symmetry already showed a significant stabilization in carried out as a function of temperature on the same samples. forming the polymorphous network that lets different local Applications. The possibility of new electromechani- motifs coexist in one large supercell. Naturally, as temper- cal devices based on paraelectrics exists. Piezoelectric 043221-10 PIEZOELECTRICITY IN NOMINALLY … PHYSICAL REVIEW RESEARCH 3, 043221 (2021) coefficients in the paraelectric phases of ferroelectrics were reversal-symmetry breaking predicted theoretically in para- found to be comparable to those of quartz and unpoled magnets [79,81], even when the global magnetism (vector ferroelectrics. These results suggest that modifications of a sum of local moments) vanishes. Similarly, we see in the collection of polar nanostructures may lead to new electrome- current work that, whereas the global dipole in a paraelectric chanical devices. Such an attempt proved to be promising, for could be small or vanishing, symmetry-breaking calculations example, in Ref. [63], where introduction of heterogeneous [79,83] indicate that local dipoles need not vanish. Remark- chemical reduction led to d33 = 321 pm/V in Na0.5 Bi0.5 TiO3 - ably, the appearance of such local motifs as energy-lowering BaTiO3 , which exceeds the piezoelectric coefficients of most symmetry-breaking features in paraelastics, paramagnets, ferroelectrics and is comparable to that of lead zirconium and paraelectrics can lead to macroscopic consequences, titanate (PZT) [89]. Moreover, because the influence of polar absent in the reference calculations lacking such local nanostructures on the forbidden piezoelectric effect can be motifs. seen over hundreds of degrees above ferroelectric Curie tem- peratures and freezing temperatures associated with relaxors, ACKNOWLEDGMENTS high-temperature piezoelectric applications, including energy harvesting, can be envisioned via the guided growth or syn- O.A. acknowledges the support of the Natural National thesis of materials via chemical, stress, thermal, and electrical Science Foundation of China (Grant No. 51850410520). gradients [34,63,90]. E.K.H.S. was funded by EPSRC (Grant No. EP/P024904/1) The scenario of intrinsic symmetry breaking short-range and the EU’s Horizon 2020 programme under the Marie order in paraphases without defects and polar nanostruc- Skłodowska-Curie Grant Agreement No. 861153. G.C. is tures. We know that positional local symmetry breaking such funded by MINECO Grant No. SEV-2017-0706 and the Gen- as displacements and octahedral rotations are seen by lo- eralitat de Catalunya Grant No. 2017 SGR 579. The work of cal structural probes in nominally cubic perovskites (viz., A.Z. at the University of Colorado at Boulder was supported pair distribution function (PDF) [79]), whereas they of- by the U.S. National Science Foundation (NSF-DMR-CMMT ten escape detection by volume-averaging techniques such Grant No. DMR-1724791) and the NDF DMREF program as conventional x-ray diffraction. Indeed, so is local time- (Grant No. DMREF-1921949). [1] A. Biancoli, C. Fancher, J. Jones, and D. Damjanovic, Break- [10] E. K. H. Salje, O. Aktas, M. A. Carpenter, V. V. Laguta, and ing of macroscopic centric symmetry in paraelectric phases J. F. 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