ARTICLE https://doi.org/10.1038/s41467-019-09266-y OPEN Converse flexoelectricity yields large piezoresponse force microscopy signals in non-piezoelectric materials Amir Abdollahi 1, Neus Domingo 2, Irene Arias 1 & Gustau Catalan2,3 1234567890():,; Converse flexoelectricity is a mechanical stress induced by an electric polarization gradient. It can appear in any material, irrespective of symmetry, whenever there is an inhomogeneous electric field distribution. This situation invariably happens in piezoresponse force microscopy (PFM), which is a technique whereby a voltage is delivered to the tip of an atomic force microscope in order to stimulate and probe piezoelectricity at the nanoscale. While PFM is the premier technique for studying ferroelectricity and piezoelectricity at the nanoscale, here we show, theoretically and experimentally, that large effective piezoelectric coefficients can be measured in non-piezoelectric dielectrics due to converse flexoelectricity. 1 Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya (UPC), Campus Nord UPC-C2, E-08034 Barcelona, Spain. 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain. 3 ICREA-Institut Catala de Recerca i Estudis Avançats, 08010 Barcelona, Catalonia, Spain. Correspondence and requests for materials should be addressed to A.A. (email: amir.abdollahi@upc.edu) or to N.D. (email: neus.domingo@icn2.cat) NATURE COMMUNICATIONS | (2019)10:1266 | https://doi.org/10.1038/s41467-019-09266-y | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09266-y P iezoresponse force microscopy (PFM) has become the go-to field would decay as we move away from the tip, in a manner technique for characterizing piezoelectricity at the nanos- approximately equal to r−2 (this field profile is modified by the cale (particularly in ferroelectric thin films) and manip- contrast between the susceptibility in the atmosphere and inside ulating ferroelectric polarization1,2. It is also often used as a tool the sample, but the physical principle still stands). The gradient to aid establishing whether or not materials are ferroelectric or of this electric field must induce a strain via the converse flexo- piezoelectric; however, as has been pointed out3–8, this latter use electric effect, see Eq. (1). is delicate, because there are other physical phenomena that can The exact solution of the problem is complicated due to yield a piezoelectric-like response in PFM without the material complex interactions between the tip, the surface and the in question having to be piezoelectric. electromechanical response of the sample. In addition, the shape PFM operates by delivering a voltage V to the surface of the and size of the contact area is itself a complex function of material via an electrically conducting tip in an atomic force the amount of force with which the tip is pressing on the surface, microscope (AFM). However, the application of the conductive the exact tip shape and the elastic properties of both the tip tip directly on the sample leads to an effective piezoelectric and the sample. Moreover, the applied tip pressure induces strain coefficient that may not coincide with the intrinsic piezoelectric gradients in the contact region. These also induce polarization by coefficient of the material. Even in the ideal case of a homo- direct flexoelectricity. This polarization self-consistently modifies geneous and insulating piezoelectric, the electric field itself is not the electric field in the material by direct flexoelectricity and evenly distributed across the material, so the measured piezo- hence affect the nominal converse flexoelectric response. electric coefficient is an average of the field-induced deformation In the absence of an analytical solution for this complex across the excited volume2. In addition, electric fields and problem, we examine the role of converse flexoelectricity on the tip-induced strain gradients can cause changes in the local con- PFM response using a self-consistent computational model22, centration of free ions in ion-conducting solids, thereby based on a linear continuum theory of piezoelectricity23, expanding or contracting the local volume and thus giving a augmented with flexoelectricity. In this model, direct and piezoelectric-like deformation caused by voltage. This effect is converse flexoelectricity are intimately intertwined manifestations at the base of so-called electrochemical strain microscopy5,9,10. of the same coupling. The total electromechanical energy ðHÞ, Another relevant electromechanical coupling mechanism in resulting fourth-order continuum equations and a mesh-free PFM is flexoelectricity11,12. Direct flexoelectricity is a property discretization method for solving numerically these equations are allowed by symmetry in all materials, and it describes the described in Supplementary Note 1. appearance of polarization in response to a strain gradient. In this work, the AFM tip is approximated as an ideal rigid Converse and inverse flexoelectricity are the reverse phenomena sphere in contact with an ideal flat surface, see Fig. 2a. To model a of strain induced by polarization gradient13 and strain gradient frictionless contact, we follow the well-known Signorini-Hertz- induced by polarization14. The inverse and converse flexoelectric Moreau model24,25, which allows us to either control the tip effects have been experimentally demonstrated by, respectively, indentation and measure force, or control force and measure the applying a voltage to a capacitor and measuring its bending14,15 displacement of the indenter (cf. Supplementary Note 2). We and by applying a voltage across a truncated pyramid so as to consider the strong indentation limit for the electrical boundary generate an inhomogeneous electric field inside it, thus causing conditions on the contact area26, which implies that the the sample to deform16–20. electroelastic response of the sample dominates over electrostatic Converse flexoelectricity, like direct flexoelectricity, is allowed tip-surface interactions. The strong indentation limit assumption by symmetry in all materials—including non-piezoelectric ones. is valid for an applied force of F > 100 nN, commonly used in Therefore, it should be a necessary ingredient of any PFM mea- PFM, for a typical tip radius of 50–100 nm26. In this limit, the surement because, when a voltage is applied to the tip, it generates electrical boundary conditions are: an electric field that decays as we move away from its apex, ϕ ¼ V; 0ra ð2Þ resulting in an approximately radial electric field gradient. This electric field gradient must induce a converse-flexoelectric strain. Dz ¼ 0; r>a ð3Þ Dividing the converse-flexoelectric strain by the voltage applied to the tip will yield a non-zero effective piezoelectric coefficient where a is the contact radius, ϕ is the electric potential, V is in any dielectric material. Here we demonstrate the existence of the tip voltage, and Dz is the vertical electric displacement. such converse flexoelectric effect in PFM, and show that it is The bottom side of the model is connected to the ground, i.e., quantitatively important, yielding significant piezoelectric-like the electric potential is fixed to zero, see Fig. 2b, c. We assume response even if the material is ionically insulating and non- the charge-free condition Dn = 0 for all other faces of the piezoelectric. computational domain, where Dn is the normal electric displace- ment. Additional higher-order boundary conditions also arise Results from flexoelectricity22,27 (Supplementary Note 1). Self-consistent simulation. Flexoelectricity is a two-way coupling An important feature of the response is the size effect, because between polarization and strain gradient and, conversely, between flexoelectricity is caused by a gradient: the sharper the contact strain and polarization gradient. Both direct and converse flex- is, the stronger the gradient will be, and thus the larger the oelectricity are characterized by the same fourth rank tensor, flexoelectrically-induced deformation12. Moreover, the contact and are hence allowed in centrosymmetric materials12,21. The area between tip and surface grows as a function of the amount of converse-flexoelectric stress is expressed mathematically as: force with which the tip is pressed onto the surface. Accordingly, we can expect that, upon increasing loads, the contact area σ ¼ μ∇E; ð1Þ increases, and thus the AFM tip-induced electric field gradient in the sample decreases. Gradient-induced apparent piezoelectricity where σ is the mechanical stress, E is the electric field and μ is the will, therefore, be inversely proportional to contact force. This fourth-order flexoelectric tensor. The excited volume below the feature emerges as a useful tool to differentiate the converse PFM tip is subject to the converse flexoelectric effect due to flexoelectric electromechanical response from the regular piezo- the inhomogeneous nature of the electric field emanating from electric response or the electrochemical strain, for which no the tip (see Fig. 1). According to the Coulomb’s law, the electric dependence on the contact area should be observed5,28. 2 NATURE COMMUNICATIONS | (2019)10:1266 | https://doi.org/10.1038/s41467-019-09266-y | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09266-y ARTICLE PFM tip Electric potential R 0 V h ~ ∇E ∇E ∇E ∇E Fig. 1 Schematic of piezoresponse force microscopy measurement. Piezoresponse force microscopy (PFM) operates by delivering a voltage V to the surface of the material via an electrically conducting tip. In a piezoelectric material, the tip voltage V will cause a local deformation, h, which is assumed to stem solely from the piezoelectric coupling. The effective piezoelectric coefficient is hence taken as deff 33 ¼ h=V. However, the tip voltage induces an inhomogeneous electric field below the PFM tip, which decays as we move away from the tip. The gradient of this electric field (∇E) must induce a strain via the converse flexoelectric effect in all dielectrics, including non-piezoelectrics, see Eq. (1), which results in a measured deformation h, and consequently an apparent piezoelectric coefficient a F R d 2a b c a V z r   0 V Fig. 2 Contact of a spherical piezoelectric force microscopy tip with a flat surface. a Schematic of the contact of a spherical tip with radius R under an applied load F which induces an indentation depth of d with contact radius a. b Axisymmetric model of the spherical contact. The rotational symmetry of the spherical tip-sample contact allows us to employ this two-dimensional axisymmetric model in a cylindrical coordinate system (r, θ, z). c Computational node set in the deformed configuration. The colour plot presents the distribution of the electric potential ϕ. To capture the sharp changes of the strain and electric field in the excited volume, the computational nodes are distributed such that the nodal spacing is gradually diminishes as the contact surface is approached. The electric potential at the nodes in the contact surface (0 ≤ r ≤ a) is fixed to the tip voltage V (voltage source symbol). The ground symbol indicates that the nodes on the bottom side of the model are connected to the ground, i.e., the electric potential is fixed to zero. The roller supports represent the mechanical boundary conditions which imply that the vertical and horizontal displacements are fixed on the bottom and left sides of the model, respectively These qualitative predictions are quantitatively supported by The simulation protocol is as follows. We first progressively numerical calculations. In Fig. 3, we show the simulation results indent until reaching the desired force F at V = 0, and then for the effective piezoelectric coefficient as a function of contact the force is kept constant as the tip is electrically biased and the radius and contact force, calculated for non-piezoelectric SrTiO3. induced displacement under the tip Δuz recorded. The effective NATURE COMMUNICATIONS | (2019)10:1266 | https://doi.org/10.1038/s41467-019-09266-y | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09266-y a (nm) a 101 17.5 16 Simulation 15.0 Experimental 14 Flexoelectric SrTiO3 Piezoelectric quartz 12.5 b 200 nm 12 d 33 (pmV–1) 10.0 10 d 33 (pmV–1) 8 7.5 eff eff 6 5.0 4 2.5 2 0.0 0 0.1 1 10 100 0.1 1 10 100 F (N) F (N) Fig. 4 Study of converse flexoelectricity induced at the tip apex of an atomic Fig. 3 Effective piezoelectric coefficient as a function of contact radius a and force microscope cantilever as a function of the applied force. a Effective force F. The flexoelectric response is obtained for non-piezoelectric SrTiO3 piezoelectric coefficient as a function of applied force for the SrTiO3 crystal. (STO) considering only flexoelectricity (both direct and converse). For Filled squares correspond to the values obtained after the simulation. Empty comparison, we show the hypothetical response of an archetypal circles correspond to the experimental values obtained with a Nanosensors piezoelectric (quartz) assuming an absence of flexoelectricity. The contact CDT FM tip with a cantilever of medium stiffness (k ≈ 2.8 Nm−1) coated radius is obtained under the applied force, in the absence of the tip voltage. with doped diamond. The error bars correspond to the error of the linear The flexoelectric response shows a size-dependent behaviour which can fitting of the experimental data, which correlates the measured be used as a means to qualitatively distinguish between flexoelectricity electromechanical amplitude of oscillation Δh with the Vac applied voltage. and piezoelectricity b The effective contact radius a scales with the force, and is determined by the tip radius. The experimental tip radius is obtained after the measurement of the nanoscale electromechanical response from the eff piezoelectric coefficient is calculated as d33 ¼ Δuz =V, where Δuz scanning electron microscopy image of the used tip. In this case, the tip is the tip vertical displacement under the tip voltage V, obtained radius of the diamond coated tip is 105 nm, and is observed to keep a from the simulation results (material parameters and details are spherical shape after the measurements given in Supplementary Note 2 and Supplementary Table 1). These calculations are compared against the simulation results evidence; if the piezoresponse of the STO sample were due to obtained for a piezoelectric material quartz, neglecting flexoelec- piezoelectricity31, the piezoresponse should not depend on the tricity and assuming an effective piezoelectric coefficient indentation force (see Fig. 3). We note that Fig. 4b shows that the eff d33 ¼ 2:5 pmV1 . As expected, the pure piezoelectric response decrease in piezoresponse with increasing force is not due to shows no dependence on the applied force F. Therefore, one a blunting of the tip, which remains spherical (with a relatively can use either force or contact area dependence as a means big radius due to its diamond coating) after the measurement. to qualitatively distinguish between flexoelectricity and piezo- Meanwhile, the electromechanical response of TiO2 is reduced by electricity. This flexoelectric size effect will be important in a factor of 0.74, as compared to STO, a proportion in agreement comparison with the experimental results. with the reduction of the corresponding flexoelectric coefficients (Supplementary Note 6, Supplementary Figure 6). We also show that this response is not associated with PFM experiment on non-piezoelectric dielectrics. In order to electrochemical strain because the signal is not hysterestic and the validate our theoretical predictions, we have examined the pie- temperature dependence is the opposite (decreases with tempera- zoresponse of two different non-piezoelectric dielectrics, SrTiO3 ture) of what one should expect if the origin was ionic conductivity (STO) and TiO2. The former is a cubic material with an effective (Supplementary Note 3–4, Supplementary Figure 2–4). We also flexoelectric coefficient of μeff 1 29 discard a major contribution of the direct flexoelectric effect. The 13 ¼ 2:5 nCm while the latter has a rutile structure and an effective flexoelectric coefficient of tip-induced strain gradient makes the material locally polar and μeff 1 30 thus piezoelectric, but this contribution has been observed to be 13 ¼ 1:7 nCm . The tips used were conductive diamond- coated Nanosensors CDT NCLR tips (k ≈ 72 Nm−1, Rtip ≈ proportional to the applied force: the bigger the tip pressure, the 100 nm). In order to ensure that the applied pressure did not bigger the flexoelectrically-induced polarization32,33. In contrast, the change the shape of the tip (and thus the validity of the contact experimental results in Fig. 4a show that the converse flexoelectric model), we recorded high-resolution scanning electron micro- contribution is inversely proportional to the applied force, scopy images of the tips before and after the image. Using AFM, indicating that the dominant effect is the converse rather than we also scanned the surface topography of the crystal before and direct flexoelectricity. The prevalence of converse flexoelectricity at after our piezoresponse characterization, to verify that there was low contact forces is also relevant because most PFM experiments no mechanical damage of the sample (Supplementary Note 5, are done under modest indentation forces of the order of 100 nN, Supplementary Figure 5). The effective piezoelectric coefficient which is the regime where our calculations and measurements show eff the strongest flexoelectrically-induced piezoelectricity. d33 as a function of tip load is shown in Fig. 4a for the STO crystal, and compared with the result of the theoretical calcula- tions. The agreement is remarkable, considering that the calcu- Discussion lation is done without any fitting of parameters. The qualitative The theoretical calculations, done without fitting parameters, trend of the curve as a function of force is also an important show that apparent piezoelectric coefficients as high as 15 pmV−1 4 NATURE COMMUNICATIONS | (2019)10:1266 | https://doi.org/10.1038/s41467-019-09266-y | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09266-y ARTICLE can be measured in a cubic, non-piezoelectric material such as References SrTiO3. To put this number into context, this effective piezoelectric 1. Gruverman, A. & Kalinin, S. V. Piezoresponse force microscopy and recent coefficient is about 6 times bigger than that of quartz, and similar advances in nanoscale studies of ferroelectrics. J. Mater. Sci. 41, 107–116 (2006). to the piezoelectric coefficient of ZnO34,35. High-performance 2. Bonnell, D. A., Kalinin, S. V., Kholkin, A. L. & Gruverman, A. 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ICN2 article’s Creative Commons license and your intended use is not permitted by statutory acknowledges support from the Severo Ochoa Program (MINECO, Grant SEV-2017- regulation or exceeds the permitted use, you will need to obtain permission directly from 0706) and the CERCA Programme / Generalitat de Catalunya. I.A. acknowledges the copyright holder. To view a copy of this license, visit http://creativecommons.org/ the support of the Generalitat de Catalunya through the prize “ICREA Academia” licenses/by/4.0/. for excellence in research and of the European Research Council (StG-679451). I.A. and A.A. also acknowledge the support of Generalitat de Catalunya (Grant No. 2017- SGR-1278). © The Author(s) 2019 6 NATURE COMMUNICATIONS | (2019)10:1266 | https://doi.org/10.1038/s41467-019-09266-y | www.nature.com/naturecommunications