Frequency dispersion of flexoelectricity in PMN-PT single crystal

Coupling between homogenous electrical and homogenous mechanical domains, underpinning the functionality of a board range of materials and related devices, has always drawn intensive research interest. Such electromechanical interactions like piezoelectric coupling^1,2 in the non-centrosymmetrical materials and electrostrictive coupling^3,4 in all crystals including both non-centrosymmetry and centrosymmetry, have already been witnessed their promising sensing and actuating applications in past decades. It is only recently, however, that a novel coupling between inhomogeneous mechanical and homogeneous electrical domains has been unprecedentedly highlighted.^5–7 This distinctive coupling, named flexoelectricity, couples strain gradient and electrical polarization through a fourth rank tensor, which is, in mathematics, defined by⁵ 

where P_i is the induced polarization, S_jk is the induced strain, x_l is the axis of coordinate, and $μijkl$ is the flexoelectric coefficients, a fourth rank tensor.

The definition of flexoelectricity reveals that the flexoelectric coupling should be similar with the electrostrictive one which could, in principle, break the centrosymmetry limitation (in contrast to piezoelectric one) and hence broaden its application to the materials with high symmetrical crystalline structures, e.g. cubic and isotropic structures.⁸ Alternatively, irrespective with the crystal structure, charge separation would occur when the crystal is under an inhomogeneous strain through flexoelectricity.⁹ 

As a fundamental property of material, this electro-mechanical effect is widely existed in diverse material morphs covering solids, liquid crystal, and even biological membranes. Specifically, in solid dielectrics, the first phenomenological model of flexoelectricity was proposed by Kogan.¹⁰ However, in Kogan’s original paper, he used ‘‘piezoelectric effect’’ to describe this inhomogeneous electro-mechanical coupling effect. Later Indenbom proposed that the term, ‘‘flexoelectric effect’’ should be more appropriate to describe this effect.¹¹ 

It is worth mentioning that in most of the solid dielectrics, the flexoelectric effect is trivial, rendering the related flexoelectric phenomenon neglected, especially when other electro-mechanical effect like piezoelectric one is mixed in the material. Thanks to the pioneering work by Ma and Cross, significantly enhanced flexoelectricity was found in some of the ABX₃ type perovskites represented by barium strontium titanate ceramic, and hence triggered the investigation of flexoelectricity in solids to an unprecedentedly new height.^12–14 Subsequently, theoretical work as well as the first principle calculation of flexoelectricity was gradually developed.^15–17 However, till now, the enhancement mechanism of flexoelectricity is still unclear. Initial interpretation of enhanced flexoelectricity was guessed to be the polar nano regions (PNRs) or the non-crystalline polar phases;^18,19 while the present one was accounted for the residual piezoelectricity, which was yielded during the material densification process or material preparation procedures.^20,21

On the other hand, unlike mechanical strain, the strain gradient is always inversely proportional with the material size. Therefore, the induced flexoelectricity by strain gradient is expected to be larger in nanomaterials compared with bulk materials. This prediction was firstly proposed by Majdoub et al and recently confirmed by various experimental evidences.²² Those experimental evidences indicated that the flexoelectricity has a promising application potential in charge transportation, domain engineering, random access memory and nanoscale actuating device.^23–26 Although significant advances have been made in the research of flexoelectricity, the study of flexoelectric coefficients in solids, especially in crystals, are rather limited. In this paper, we will focus on this issue and provide our experimental result of the flexoelectricity in Pb(Mg_2/3Nb_1/3)O₃-PbTiO₃ (PMN-PT) single crystal. Our result could broaden the material database of flexoelectricity and promote the understanding of origin of enhanced flexoelectricity in perovskites.

Essentially, compared with ceramics, single crystals could minimize the influence of most of the possible interference factors (e.g. the grain boundary) during the flexoelectric measurement due to its perfect periodic atomic structure. However, the flexoelectric measurement of single crystal was rather limited compared with ceramics. The present little flexoelectric data of single crystal could be seen in SrTiO₃ (STO), a material known to be classical quantum ferroelectrics. The measured flexoelectricity of STO is only nC/m, which is in good agreement with the theoretical estimation.²⁷ Recently, significant enhanced flexoelectricity accompanied with a large anisotropy behavior was observed in BaTiO₃ single crystal.¹⁴ Catalan et al accounted it to the residual piezoelectricity due to the abnormal large ratio of flexoelectric coefficients to dielectric constants in this kind of single crystal. In this paper, [001] orientated 0.68Pb(Mg_2/3Nb_1/3)O₃-0.32PbTiO₃ (PMN-PT) single crystal was employed to the flexoelectric measurement. For this rhombohedral system, because the dipoles in each unit cell are formed along <111> of the cubic parent phase, the [001] dicing orientation produces a multidomain structure with dipoles oriented along eight <111> directions with equal possibility, leading to a zero net polarization.

In order to satisfy the sample requirement, the PMN-PT single crystal was diced into 12.6×5×1.2 mm³ by a dicing saw (Disco DAD320 Mesa, AZ). After dicing, the samples were annealed at 250 °C so that the residual stress of the materials could be removed.²⁸ Before the flexoelectric measurement, the residual piezoelectricity of the PMN-PT samples were further confirmed to be varnished by a high resolution d₃₃ meter (ZJ-3B/4B, H. C. Materials, Bolingbrook, IL).

The flexoelectric measurement was utilized by the conventional cantilever system, which is reported in our previous work.¹⁶ All the flexoelectric measurement setup was laid on a float optical table (Newport, ATS, Irvine, CA) to eliminate the vibrational noise. The PMN-PT single crystal was mechanically clamped by a rigid holder at one side while deflected by a sensitive piezoelectric actuator at the other end. This actuator was controlled by a power amplifier (Trek, 2220, Lockport, NY) under the excitation from a function generator (Tectronix, AFG3101, Lake Mary, FL). The deformation of the crystal along the thickness direction was measured using a high resolution (<10 pm) laser vibrometer (Polytec, OFV-5000, Irvine, CA) and a lock-in amplifier (Stanford Research System, SR830, Sunnyvale, CA).

The temperature and frequency dependence of dielectric properties of the composite were measured using a broadband dielectric spectrometer (Concept 40, Novocontrol, Germany) and an impedance analyzer (E4294A, Agilent, Palo Alto, CA), respectively. As shown in Fig. 1, the single crystal exhibits a board peak in the vicinity of Curie temperature with its maximum dielectric constant approaching as high as 40000 at 10 kHz at almost 150^oC (Such temperature in response to the maximum dielectric constant is called T_m). In principle, whether a ferroelectrics is relaxor or not could be readily confirmed from the temperature dependence of inverse permittivity. As shown in the inset of Fig. 1, the studied PMN-PT single crystals exhibit a significant derivation from the Curie-Weiss relation, which is also a physical indicator of polar nano regions in this kind of relaxor ferroelectrics. It is noticed that in PMN-PT relaxor ferroelectrics, PNRs will shift to orientated order status when the temperature is above T_m and such PNRs orientation will persist until the temperature of the materials is higher than T_m+300K. Previous pioneering studies by Catalan et al have explicitly investigated the correlation between PNRs and flexoelectricity in other composition of PMN-PT single crystal.²⁹ Their result suggested that PNRs would be realigned and render to an extrinsic enhancement of flexoelectricity, which is proved by the cross correlation between the flexoelectric coefficients and elastic modulus during the phase transition. In this paper, we will further confirm the interpaly of PNRs and flexoelectricity by frequency characterization.

As suggested by our previous work, the flexoelectricity was always evaluated by the transverse flexoelectric coefficients $μeff$⁠, which is defined by $P3=μeff∂S11/∂x3$⁠, where P is the polarization, S is the strain, and $∂S11/∂x3$ stands for the strain gradient of the direct strain S₁₁ along the x₃ direction.¹³ In order to verify the accuracy of the PMN-PT flexoelectric measurement, reference sample Al₂O₃ ceramic was employed into the cantilever system.^5,12,13 No electric charge was captured in Al₂O₃ ceramic even when the applied strain gradient was set up to 10^-2 m^-1, suggesting our measurement is reliable. Consequently, the measured electric charge of PMN-PT single crystal was only accounted for material itself instead of the measurement system. As shown in Fig. 2, highly linear relationships between the induced flexoelectric polarization and the strain gradient were observed in the PMN-PT single crystal at room temperature (25°C) at different frequencies. The transverse flexoelectric coefficient was calculated to be as high as 100 $μC/m$ at 12 Hz, which is comparable with that of some perovskite ceramics.^5,12–14,29 The error bar of each measurement data in Fig. 2 originates from the small deviation of the collected charge under strain gradient. On the other hand, it can be perceived that the effective transverse flexoelectric coefficient in this single crystal displays an unique frequency dispersion, which is small at low frequency but large at higher frequency. The frequency dispersion behavior is further shown in Fig. 3. It can be clearly seen that the flexoelectric coefficients is non-linearly varied with the frequency. Due to the quasi-static vibration limitation in the cantilever beam,¹³ we only measured the frequency below 12 Hz. Otherwise, the higher frequency would sacrifice the stability of the mechanical vibration of the PMN-PT beam, and hence bring measurement error.

It is worth mentioning that the frequency characteristics of flexoelectricity in PMN-PT single crystal is different with the flexoelectric response of Ba_0.6Sr_0.4TiO₃/Ni_0.8Zn_0.2Fe₂O₄ (BST/NZO) compound.³⁰ In BST/NZO compound, the flexoelectric response is higher at low frequency while smaller at high frequency. That is because, in a highly deteriorated material whose dielectric loss is close to 0.5, the flexoelectric polarization will be weakened towards higher frequency direction due to its dielectric fatigue.^30,31

For the investigated PMN-PT single crystal, the situation is different. It is obviously that the dielectric behavior of our material is not degenerated. Herein, the polar nano domain rotation polarization should be dominated to the flexoelectricity in PMN-PT single crystal.²⁹ As schematically shown in Fig. 4, at free state, the PNRs are randomly distributed in the crystal so that no net polarization can be observed. When the strain gradient is applied, the PNRs would rotate along the strain gradient direction due to the forced geometry variation. However, the PNRs rotation process is not transient due to the consumption of energy, and such rotation would relax without sufficient energy supply or under low excitation frequency. Therefore, the rotation is expected to be more completed under relative higher excitation frequency.

Note that this behavior is different with the previous electromechanical relaxation one in the surface of the PMN-PT single crystal which is verified by the piezoresponse force microscopy.^32–34 It is particularly in the surface, the response of PNRs would have a time delay.³⁴ However, our experimental results suggest another response which is faster in short time but slower in long time. According to the Tagantsev’s flexoelectric theory, the flexoelectricity is a complex of four contributions, encompassing both bulk flexoelectricity and surface piezoelectricity. Consequently, the present abnormal frequency behavior suggested the flexoelectricity in this material is more like a bulk effect rather than surface effect.

In conclusion, we reported the unexpected enhanced flexoelectricity in PMN-PT single crystals. The observed flexoelectric coefficients approaches almost 100 $μC/m$ at 12 Hz, and would be decreased at lower frequency. Moreover, the flexoelectric coefficients in this kind of materials would be non-linearly varied with frequency. This experimental results further suggest the origin of enhanced flexoelectricity in PMN-PT single crystals should be accounted for the polar nano regions.

L. S and W. Huang acknowledge the NC state university for supporting the PMN-PT single crystals. L. S thanks Prof. Yu Wang and Prof. Xi Yao for providing intensive discussion for this work. This work is supported by the National Natural Science Foundation of China under grant number 11604135 and 51605052, and in part by the Natural Science Foundation of Jiangxi Province, China (20161BAB216110).