PHYSICAL REVIEW X 10, 041001 (2020) Mechanical Softness of Ferroelectric 180° Domain Walls Christina Stefani ,1 Louis Ponet,2,3 Konstantin Shapovalov,4 Peng Chen ,2 Eric Langenberg,5 Darrell G. Schlom ,5,6 Sergey Artyukhin,2 Massimiliano Stengel,4,7 Neus Domingo,1 and Gustau Catalan1,7 1 ICN2-Institut Català de Nanociència I Nanotecnologia (CERCA-BIST-CSIC), Campus Universitat Autònoma de Barcelona, Bellaterra, Catalonia, Spain 2 Italian Institute of Technology, 16163 Genoa GE, Italy 3 Scuola Normale Superiore di Pisa, 56126 Pisa PI, Italy 4 ICMAB-Institut de Ciència de Materials de Barcelona, Bellaterra, Catalonia, Spain 5 Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA 6 Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA 7 ICREA-Catalan Institution for Research and Advanced Studies, Passeig Lluïs Companys, Barcelona, Catalonia, Spain (Received 23 January 2020; revised 19 June 2020; accepted 17 July 2020; published 1 October 2020) Using scanning probe microscopy, we measure the out-of-plane mechanical response of ferroelectric 180° domain walls and observe that, despite separating domains that are mechanically identical, the walls appear mechanically distinct—softer—compared to the domains. This effect is observed in different ferroelectric materials (LiNbO3 , BaTiO3 , and PbTiO3 ) and with different morphologies (from single crystals to thin films), suggesting that the effect is universal. We propose a theoretical framework that explains the domain wall softening and justifies that the effect should be common to all ferroelectrics. The lesson is, therefore, that domain walls are not only functionally different from the domains they separate, but also mechanically distinct. DOI: 10.1103/PhysRevX.10.041001 Subject Areas: Materials Science I. INTRODUCTION which are atomically thin structures sandwiched between much wider domains that dominate the overall mechanical An important part of the appeal of domain walls resides behavior. All the same, the mechanical response of domain in the functional contrast between their properties and those walls matters. For one thing, mechanical stress is one of of the domains they separate. Multiferroic BiFeO3 displays the mechanisms by which domain walls can be moved: conductivity and magnetoresistance at its domain walls Ferroelectric-ferroelastic domain walls respond to stress [1–4], despite being an insulator wide-band-gap semi- [14], affecting the overall dielectric, piezoelectric, and conductor, and the [5] ferroelastic domain walls of semi- elastic properties of ferroelectric ceramics as well as their conductor WO3−x are superconducting [6]. Electrical lability [15] and fracture physics [16,17]. Even purely conductance is also measured in the ferroelectric domain ferroelectric (i.e., nonferroelastic) domain walls also react walls PbðZr0.2 Ti0.8 ÞO3 [7], LiNbO3 [8], and BaTiO3 , as to strain gradients introduced by external indentation [18] well as those of multiferroic YMnO3 [9] and Cu3 B7 O13 Cl or by the proximity of another ferroelectric (nonferroelas- [10]. Their distinct functionality, nanoscopic thickness, and tic) domain wall [19]. Moreover, elastic contrast between the fact that they can be created, shaped, or moved by an domains and domain walls may affect the propagation and external field is fueling the field of “domain wall nano- scattering of phonons—and, consequently, also the propa- electronics,” where domain walls are regarded as mobile gation of heat [20–22]. Also, if there is mechanical contrast two-dimensional electronic elements [11–13]. in 180° domain walls, it may be used as an alternative way In contrast to the vigorous research on domain wall to mechanically read ferroelectric bits [23]. functionality, less is known about their mechanical proper- The interplay between domain wall motion or domain ties. This difference is partly explained by the difficulty reconfiguration and the overall mechanical and electro- of isolating the mechanical response of individual walls, mechanical properties of ferroelectric devices is well documented [24–29]. In contrast, there is barely any knowledge of the internal deformation mechanics of the Published by the American Physical Society under the terms of individual walls themselves—particularly for nonferroelas- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to tic 180° walls separating antiparallel ferroelectric domains. the author(s) and the published article’s title, journal citation, A seminal investigation by Tsuji et al. shows that the 180° and DOI. domain walls of ferroelectric Lead zirconate titanate 2160-3308=20=10(4)=041001(12) 041001-1 Published by the American Physical Society CHRISTINA STEFANI et al. PHYS. REV. X 10, 041001 (2020) ceramics appear softer than the domains when probed by Figure 1 shows the piezoresponse force microscopy atomic force microscopy [30,31]. However, it is not (PFM) [35,36] images of the ferroelectric domains, showing obvious why such domain walls should display any the 180° phase contrast of oppositely polarized domains. mechanical contrast, given that the polar axis is the same Figure 1 also demonstrates the mechanical response of on both sides of the wall and, thus, the domains on either domain walls as detected by CRFM. In this technique, the tip side are mechanically identical (unlike in ferroelastic 90° is in contact with the sample, but it does not excite the sample domains, where the spontaneous strain axis is different on electrically, as in PFM, but mechanically, through a piezo both sides of the wall, so the mechanical properties must element placed at the base of the cantilever. In another necessarily change across the wall). The present inves- common experimental setup for CRFM measurements, the tigation therefore seeks to (i) determine whether domain sample is vibrated instead of the cantilever, using an acoustic wall mechanical contrast is a general property of 180° stage underneath it. We perform experiments with both types ferroelectric domains, (ii) quantify its magnitude, and of experimental setup to verify that the results are equivalent (iii) propose a theoretical explanation for its physical (see Supplemental Material [34]). origin. We find that the effect is general, quantitatively The mechanical excitation of the tip induces a local significant, and physically inevitable. vibration at the contact with the sample, and the frequency Besides its fundamental interest, this discovery has of the oscillation is modulated until mechanical resonance practical ramifications not only for the mechanics of is reached. The resonance frequency of the system depends ferroelectrics but also for their functionality. The mechani- on both the geometrical characteristics of the tip and the tip- cal detection of ferroelectric domain walls means that they sample mechanical contact characteristics. Since the tip is can be probed in a voltage-free manner, which may be the same throughout the whole measurement, and the force convenient for conducting ferroelectrics. Also, as men- between the tip and sample is also kept constant, changes of tioned, heat transport is intimately linked to mechanics, resonant frequency, therefore, indicate the changes in the because heat is carried by phonons, which are strain waves. stiffness of the sample, and a lower resonance frequency If the lattice is softer at the wall, the phonon speed will be means a softer material. In all the measurements, noncoated slower, and, hence, effects like phonon refraction or even doped Si tips are used. For CRFM images, the tip is total internal reflection may be expected; in this context, grounded during the measurements in a short-circuit 180° domain walls could conceivably act as “phonon configuration, allowing for polarization charge screening. waveguides” where heat travels along the wall with little The first sample tested is a congruent LiNbO3 single dissipation. Since ferroelectric 180° walls can be created or crystal, periodically poled with polarization perpendicular destroyed by voltage (by writing or erasing domains), this to the surface. This material is a uniaxial ferroelectric, so result suggests the possibility of using voltage to fabricate the only domains allowed by symmetry are 180° domains periodic and reconfigurable metamaterials with a regular of antiparallel polarization. The antiparallel domain con- pattern of internal elastic contrast. Put another way, figuration is verified by the phase contrast in the PFM periodically poled ferroelectric crystals, which are already image [Fig. 1(a)]. The size of each domain is approximately in use for photonic applications [32], may also turn out to 4 μm. From CRFM response of the same area, we observe be phononic crystals. that the domain walls are markedly darker (i.e., display lower resonance frequency and are, therefore, softer) than the domains. II. SAMPLES AND DOMAIN STRUCTURE We also look at a BaTiO3 single crystal, which is We characterize the mechanical properties of domain considered an archetype of perovskite ferroelectrics. The walls in ferroelectric single crystals of LiNbO3 and BaTiO3 tetragonal structure of BaTiO3 allows for both antiparallel and thin films of PbTiO3 . The spread of materials and (180°) and perpendicular (90°) domain configurations, sample morphologies is chosen in order to determine the the latter being ferroelastic in addition to ferroelectric. In generality of the findings. The measurements are based on Fig. 1(b), the PFM image shows the areas with opposite out- contact resonance frequency microscopy mode (CRFM). of-plane polarization forming 180° ferroelectric domain CRFM is a scanning probe microscopy technique that walls. In the same area, CRFM measurement demonstrates maps, with nanoscopic resolution, the resonance frequency again a downward frequency shift between the domain and of an AFM tip in contact with the material; higher domain walls, indicating that in this material domain walls resonance frequencies correlate with stiffer contact areas, are also mechanically softer than domains. and, conversely, lower resonance frequencies indicate that Finally, we also investigate a PbTiO3 thin film of 50 nm the material is softer [33]. Additional details about the thickness epitaxially grown by reactive molecular beam experimental techniques can be found in Supplemental epitaxy on a single-crystal SrTiO3 substrate (the growth Material [34]. As we show, besides imaging, this technique details can be found elsewhere [20]). Because of the large can be used to extract quantitative information about compressive stress exerted by the substrate (−1.36%), differences in the Young’s modulus of the material. only domains with vertical (out-of-plane) polarization 041001-2 MECHANICAL SOFTNESS OF FERROELECTRIC 180 DEGREE … PHYS. REV. X 10, 041001 (2020) FIG. 1. (a),(b) Periodically poled LiNbO3 single crystal, (c),(d) BaTiO3 single crystal spontaneously polarized, and (e),(f) PbTiO3 thin film. PFM images (a),(c),(e) show the opposite polarization of out-of-plane domains, and CRFM images (b),(d),(f) demonstrate changes in the frequency between domains and domain walls. are allowed in the ferroelectric film [20]. After electrically materials can only be qualitative (further discussion in polarizing two different areas of the film with a dc voltage Supplemental Material [34]). of 5 V, we pole domains of opposite sign, as indicated by It has been proposed that some ferroelectric domain the corresponding PFM phase contrast [Fig. 1(c)]. Again, walls can be non-Ising type and, thus, have an in-plane the 180° domain walls are observed to be softer than the component of the polarization [37–40]. We have no surrounding domains [Fig. 1(e)]. The relative softness of experimental evidence for this proposal being the case in ferroelectric domain walls, therefore, appears to be a the BaTiO3 walls, but the possibility of a mechanical general phenomenon that does not rely on composition contribution coming from in-plane polarization at the walls or sample morphology. is excluded, because in-plane polarization leads to a stiffer, The observed softening of 180° domain walls is quali- not softer, response (see the higher resonance frequency of tatively similar for all the samples, irrespective of whether the ferroelastic a domains in Fig. 3). the domains are artificially written, such as in PbTiO3 or The possibility of softening due to local switching LiNbO3 , or spontaneous, as in the BaTiO3 crystal. effects is also ruled out. Although it is expected that the Quantitative differences in the resonance frequency shift coercive field of the ferroelectric should be smaller near the of the different materials reflect differences in their ferroelectric wall [41], repeated scans over the same area mechanical properties. However, quantitative conversion show no evidence of switching of the polarization: We see from resonance frequency to contact stiffness requires no detectable shift in the position of the domain walls even calibration for each individual material, as we demonstrate after ten scans with the maximum mechanical load of in this paper. Quantitative analysis is possible only within 20 micro-Newtons (see Supplemental Material [34] and different regions of the same sample measured with the Fig. 4). Although there is no permanent switching, there same cantilever, while comparison between different can be, however, an elastic deflection of the wall toward the 041001-3 CHRISTINA STEFANI et al. PHYS. REV. X 10, 041001 (2020) tip, and this deflection is the basis for the theoretical model k E ¼ : ð2Þ depicted in Fig. 4. 2a It is worth mentioning that there is also a small mechanical contrast (small difference in CRFM) between k is the elastic constant of the spring that represents the tip- the up and down domains. This contrast is attributed to the sample interaction [46], also known as contact stiffness, coupling of tip-induced flexoelectricity and domain ferro- and, like E , k depends on the stiffness of the tip (ktip ) and electricity, which induces an asymmetry in the mechanical the stiffness of the sample (ks ). response of domains of opposite polarity and is proposed as The quantity we are after is the Young’s modulus of the a mechanism for voltage-free mechanical reading of polar- sample (Es ), which we could, in principle, calculate by ity [23,42,43]. The domain walls, however, are softer than substracting Etip from E in Eq. (1). The problem is that we both up- and down-polarized domains, so their softening know neither E nor Etip, so we have one equation [Eq. (1)] cannot be explained by this polarity-dependent mechanism. with three unknowns. To solve this problem, we (i) measure III. EXPERIMENTAL QUANTIFICATION OF the resonance frequencies of the cantilever and relate them DOMAIN WALL ELASTICITY to E via elastic theory and (ii) measure the mechanical response of a part of the sample for which Es is known (in In order to quantify the softening, we need to translate our case, the c-oriented BaTiO3 domains) and use it for the shifts in resonance frequency into changes of stiffness. calibration. Knowing E and Es allows us to extract Etip The system can be described as a system of two springs in and then repeat the analysis on the part of sample for which series [Fig. 2(a)]: the cantilever itself, with its flexural Es is unknown—the domain walls. An additional exper- elastic constant, and the tip-surface contact, which can be imental factor that has to be taken into account is the described by the Hertzian contact model [44,45] [Fig. 2(b)]. stiffness of the cantilever. In order to mechanically char- The force applied by the tip is kept constant by a feedback acterize stiff materials, we need cantilevers that are also as loop, so the contact can also be effectively described as a stiff as possible, because otherwise all the mechanical flat punch determined only by the constant contact radius α deformation takes place in the cantilever rather than in the [Fig. 2(c)], measured experimentally. The tip-surface con- sample. This factor is illustrated in Supplemental Material tact also acts as two springs in series, corresponding to the [34], where we compare the CRFM results using canti- tip and the sample, respectively. The effective Young’s levers of different stiffness. modulus of the tip-surface system, E , is therefore given by Based on the models of Hurley [47] and Rabe, Kopycinska-Müller, and Hirsekorn [48], the cantilever is 1 ð1 − v2s Þ ð1 − v2tip Þ modeled as a beam with length L, width w, thickness b, ¼ þ ; ð1Þ E Es Etip density ρ, and Young’s modulus Ecan . The tip is located at a distance L1 < L from the clamped end of the cantilever, where Es , Etip , vs , and vtip are the Young’s modulus and the and the remaining distance to the other end of the cantilever Poisson ratios of the sample and the tip, respectively. E is is L0 [Fig. 2(a)] [47]. The spring constant of the cantilever is related to the contact stiffness k as calibrated experimentally by measuring force-displacement curves and the free resonance frequency (first harmonic f01 ) of the cantilever. These parameters are shown in Table I. The experimental values of free resonance frequency (f 01 ) (i.e., the resonance frequency when the tip of the cantilever is suspended above the sample without touching it) and the contact resonance frequency (f 1 ) (i.e., the resonance frequency when the tip is in contact with the TABLE I. Geometrical characteristics of the cantilever, exper- imental values of the free resonance frequency, and contact resonance frequencies for a and c domains of bulk BaTiO3 and the corresponding wavelengths as calculated by Eqs. (1) and (2). The experimental value of the cantilever’s spring constant (klever ) and the calculated normalized contact stiffness k =klever of the system. FIG. 2. (a) Schematic presentation of AFM cantilever and tip- sample contact being simulated by a spring with constant k . L b ρ L1 f01 Ecan klever α (b) Diagram of tip-sample contact based on a Hertzian mechani- ðμmÞ ðμmÞ ðg=cm3 Þ ðμmÞ (kHz) (GPa) (N/m) (nm) cal contact model and (c) flat punch contact between the tip and 220 6,5 2,33 211,2 158 170 38 7 the sample. 041001-4 MECHANICAL SOFTNESS OF FERROELECTRIC 180 DEGREE … PHYS. REV. X 10, 041001 (2020) sample) are used to relate the corresponding wave numbers and not ferroelastic, the next question is why. In their (xi ) through the equation [47] seminal work, Tsuji et al. [30] put forward three hypoth- sffiffiffiffiffi eses: (i) defects, which are known to be attracted to domain f1 walls, (ii) dynamic softening due to ferroelectric switching x1 L ¼ x01 L ; ð3Þ near the wall, and (iii) reduced depolarization energy at the f 01 domain wall, where there is no piezoelectricity. Let us examine these possibilities. where the free cantilever wave number x01 is given by [47] Defects are sample dependent, and common ones, such as sffiffiffiffiffiffiffiffi oxygen vacancies, are notoriously difficult to quantify. The L2 3ρ weakening of interatomic bonds caused by a vacancy should ðx01 LÞ2 ¼ 4πf 01 : ð4Þ b Ecan be fairly isotropic or at least orthotropic in the nearly cubic perovskite structure. That is to say, the defect-induced The normalized contact stiffness k  =klever is then given softening of the 180° walls should be similar in the in-plane by [33,49] and out-of-plane directions. We compare the mechanical contrast of 180° walls inside the a domains (polarization in k 2 ð1 þ cos x1 L cosh x1 LÞ plane) with those in the c domains (polarization out of plane) ¼ ðx1 L1 Þ3 ; ð5Þ for the BaTiO3 crystal, where both polarizations are acces- klever 3 D sible in a single scan due to the existence of a-c twins as well D ¼ ðsin x1 L0 cosh x1 L0 − cos x1 L0 sinh x1 L0 Þ as 180° walls [Fig. 3(a)]. As Fig. 3(b) shows, while for c domains (bubble domains) the 180° walls are softer, when × ð1 − cos x1 L1 cosh x1 L1 Þ the bubble domains penetrate into the a domains (in-plane − ðsin x1 L1 cosh x1 L1 − cos x1 L1 sinh x1 L1 Þ polarization), the mechanical contrast of the 180° walls disappears. The disappearance of mechanical contrast when × ð1 þ cos x1 L0 cosh x1 L0 Þ: ð6Þ the polarization is in plane, combined with the fact that we observe the softening of out-of-plane walls in materials with The equations above connect the contact resonance different chemistries, leads us to believe that the role of frequencies and cantilever’s spring constant to the contact chemical defects is less important than the out-of-plane stiffness k and, hence, using Eq. (2), to the contact’s orientation of the polarization. effective Young’s modulus E . If the sample’s Young’s The strain fields of a multidomain ferroelectric under modulus Es is also known (as it is for c-oriented barium the tip pressure is a mesoscopic problem too challenging titanate [50]), we can now use Eq. (1) to calculate the tip’s for first-principles atomistic calculations. Instead, we resort Young’s modulus (Etip ). The calculation can then be to a continuum model, with parameters for BaTiO3 repeated on the domain wall, where Es is not known but determined by previous first-principles work [51,52]. Etip is. The results of the calculations are shown in Table II. The starting point is the free-energy density of the system, Considering the experimental results and using the which can be described by the Ginzburg-Landau- model described above, the shift in resonance frequency Devonshire model [51,52]: of domain walls corresponds to a reduction of the effective Young’s modulus of approximately 19%, with respect to f ¼ f l þ f g þ f q þ f c þ f f þ f electr ; ð7Þ the Young’s modulus of the c domains. f l ¼ aij Pi Pj þ aijkl Pi Pj Pk Pl þ aijklmn Pi Pj Pk Pl Pm Pn ; IV. THEORY OF FERROELECTRIC DOMAIN ð8Þ WALL SOFTENING Having determined that the domain walls are mechan- f g ¼ Gijkl ∇i Pj ∇k Pl =2; ð9Þ ically softer than the domains despite being ferroelectric f q ¼ −qijkl Pi Pj εkl ; ð10Þ TABLE II. Young’s modulus of BaTiO3 based on the literature f c ¼ Cijkl εij εkl =2; ð11Þ [50] (Es ) and experimental results for c domains of BaTiO3 as used to calibrate the Young’s modulus of the tip (Etip ). For Γijkl domain walls, all the experimental values and the derived values ff ¼ ð∇i Pj εkl − Pi ∇j εkl Þ; ð12Þ of the Young’s modulus. 2 f1 (kHz) k =klever vs E (GPa) Es (GPa) f electr ¼ ϵr E2 ; ð13Þ c domains 738,5 239,92 0,3 17,07 63,6 Domain walls 737 239,48 0,25 0,45 51,2 where f l is the Landau free-energy density for uniform ferroelectric polarization P, f g describes the energy penalty 041001-5 CHRISTINA STEFANI et al. PHYS. REV. X 10, 041001 (2020) FIG. 3. (a) Vertical PFM phase of BaTiO3 single crystal spontaneously polarized, where the opposite out-of-plane polarization of the crystal is shown. (b) Vertical PFM amplitude of the crystal, where the in-plane polarization is denoted. (c) CRFM image of the same area where there is no difference in frequency contrast due to in-plane polarization. (d) Schematic representation of in-plane polarization. for spatial variations of P, f q describes the interaction interacting with a parabolic pinning potential (second between the polarization and strains εij (electrostriction), term), and the tip located at xtip and applying the force Fz : and f c is the elastic free-energy density, while f f denotes the contribution from flexoelectricity, the interaction between mω2 x2DW E ¼ E0 − Fz uz ðxtip − xDW Þ þ : ð14Þ strain gradients and the polarization. Γijkl is the flexoelectric 2 tensor. The strain εij is defined as 12 ð∇i uj þ ∇j ui Þ, where u are the displacements; summation over repeated indices is Expanding the surface profile in small xDW and mini- implied. mizing the energy with regards to xDW , we obtain The integral of the free-energy density over the entire xDW ¼ −Fu0 ðxtip Þ=mω2 , to the first order in F. The cor- crystal is minimized in the equilibrium situation. The rection to the compliance is then Δc ¼ u0 ðxtip Þ2 =ðmω2 Þ2 . electrostriction term f q generates a spontaneous tensile Hence, for significant softening, it is crucial that u0 ðxtip Þ is strain along the polar direction inside the ferroelectric large, leading to an increased effect when pressing within the domains. This tensile strain is locally reduced at the wall DW strain footprint. due to the absence of polarization, which leads to a If the force applied by the tip is large enough to depression in the surface centered at the wall, as demon- overcome the pinning potential, the wall slides toward strated in Fig. 4(b). The compressive pressure from CRFM the tip, leading to a strongly nonlinear effect. These domain tip interacts with this preexisting compressive strain profile. wall sliding modes usually have a frequency in the giga- The result is that the wall moves toward the tip, so that hertz range [53–55]. As the domain wall’s “strain-hollow” the domain wall depression coincides with the locum of the slides toward the AFM tip, the AFM registers a relatively tip compression. large deformation in response to the stress and, thus, a low Pinning of the domain wall by the disorder potential and effective stiffness (see Fig. 4). To quantify this effect, it is Peierls-Nabarro barriers results in a complex response. In necessary to solve the free energy in Eqs. (7)–(13), which is the strong pinning regime, the wall bends only toward the analytically intractable but can be numerically computed tip. This effect can be qualitatively captured by a simple by finite elements. We perform finite element simulations free-energy expansion, with a flat domain wall (DW) using known parameters from previous first-principles 041001-6 MECHANICAL SOFTNESS OF FERROELECTRIC 180 DEGREE … PHYS. REV. X 10, 041001 (2020) FIG. 4. (a) Simulated change of stiffness as a function of the distance between the CRFM tip and a domain wall. The softening is maximized slightly away from the domain wall, but within its strain footprint, where the DW sliding mode contribution is important. The experimental situation corresponds to averaging within the tip region. (b) Schematic representation of the sliding mode. The polarization and strain profiles are shifted by dx. (c),(d) Simulated strain profiles shown on a slice of the sample when the tip is near the DW (c) or further away (d). The dashed vertical line shows the initial position of the ferroelectric DW. The wall slides toward the tip in (c), as shown by the curved dashed line, whereas it does not move in (d). studies of BaTiO3 [51]. The contact of the CRFM tip with inhomogeneous mechanical stress, resulting in a local the surface is simulated by applying a bell-shaped force change of strain. ∝a exp½−ðx2 =d2 Þ, with 2d ∼ 20 nm representing the diam- Another contribution to mechanical contrast is depo- eter of the contact area of the tip and a ¼ 10−7 J=m3 . The larization. Applying tip pressure to the surface of a softening is estimated as the ratio of maximum deformation piezoelectric (all ferroelectrics are piezoelectric) by def- induced by the tip at the wall and in the domain. In order to inition modifies its polarization and, thus, has an electro- stay within a linear regime and avoid polarization domain static energy cost. The AFM tip induces deformations εij switching, the virtual force applied to the tip is kept very that are inhomogeneous (large near the tip, small far low, less than a femto-Newton. from the tip), so the polarization due to piezoelectricity, These calculations predict that the elastic cost of Pi ∼ eijk εjk , is not homogeneous. Tip pressure, therefore, deformation is smallest not at the domain wall itself, induces bound charges ∇ · P ≠ 0 in the inhomogeneously which is already spontaneously compressed and, thus, it is deformed region, and these create a depolarizing field. hard to compress further, but adjacent to it, where Higher depolarization implies bigger work and, thus, compression is achieved by the bending of the domain higher effective stiffness. wall (and its accompanying depression) toward the tip We identify two main mechanisms of formation of bound [Fig. 4(a)]. As the tip moves further away, the distance charges under the tip: (i) variation of in-plane polarization, becomes big enough that the stress field of the tip does not ∇1 P1 ≠ 0, induced by shear piezoelectricity P1 ∼ e15 ε5 interact with the wall, and the material recovers its [Fig. 5(a)], and (ii) generation of out-of-plane polarization intrinsic stiffness. due to longitudinal piezoelectricity, ΔP3 ∼ e11 ε3 , which is In conclusion, the local stress induced by the tip can lead unscreened in the case of open boundary conditions and to a reversible (elasticlike) shift of domain walls. This screened at short-circuit conditions [Fig. 5(b)]. sliding of the wall, with its associated “strain dip,” Because all the piezoelectric constants ejm flip their contributes the local deformation (softening) of the signs across the domain wall, ejm ðP3 ↓Þ ¼ −ejm ðP3 ↑Þ, material. Bassiri-Gharb et al. [24,25] show that the revers- the distribution of the tip-induced bound charges is quali- ible motion of ferroelectric walls under applied electric tatively different when the tip is pressed at the domain fields has consequences for strain. Here, we observe a and at the domain wall. For the charges induced by the somewhat complementary effect: The domain walls are “in-plane” mechanism [Fig. 5(a)], the in-plane polarization moved not by a homogeneous electric field, but by an P1 ∼ e15 ε5 forms a head-to-head or tail-to-tail pattern with 041001-7 CHRISTINA STEFANI et al. PHYS. REV. X 10, 041001 (2020) FIG. 5. (a) Schematic of the “in-plane” mechanism of the bound charge formation: Applying the tip to the surface induces in-plane polarization via the shear piezoelectricity: P1 ∼ e15 ε5 , with larger depolarizing electric fields Edep when the tip is applied in the bulk domain. (b) Schematic of the “out-of-plane” mechanism of the domain wall softening: Applying the tip to the surface induces out-of- plane polarization ΔP3 ∼ e13 ε1 þ e33 ε3 . (c) Manifestation of the depolarizing effect on the tip-induced in-plane polarization P1 . Solid lines: Polarization extracted from finite-element modeling at a distance 5 nm from the film surface, with the tip applied at the domain wall (blue) and in the bulk domain (red). Dashed lines: Polarization expected due to the shear piezoelectric effect P1 ≈ e15 ε5 , with shear strain ε5 extracted at a distance 5 nm from the film surface. (d) Study of the domain wall softening at different electric boundary conditions, values of the shear piezoelectric coefficient e15 , and film thicknesses. corresponding bound charges when the tip is pressed in the normal component of the electric displacement field at the domain and a head-to-tail pattern when the tip is pressed at surface Dn ¼ ϵr En þ Pn ¼ PS tanh x=ξ at all times, where the domain wall, implying significantly reduced electro- ξ ∼ 1 nm), and short-circuit boundary conditions. We note static energy costs and softer mechanical response in the that the experimentally investigated crystal is closer to the latter case. Likewise, the surface charges generated by the short-circuit case in the out-of-plane direction: Even though “out-of-plane” mechanism [Fig. 5(b)] have monopolelike the film surface is not electroded, the AFM tip in contact distribution when the tip is pressed in the domain and with the surface is metallic. dipolelike distribution when the tip is pressed at the domain In all studied cases, we obtain that, when the AFM tip is wall, again implying lower electrostatic energy costs and a applied at the domain wall, the in-plane polarization P1 softer mechanical response of the domain wall at open generated under the tip [blue solid line in Fig. 5(c)] is electric boundary conditions. Notice that this polarity- described with good precision by the piezoelectric effect dependent orientation of the piezoelectric response is P1 ≈ e15 ε5 [blue dashed line in Fig. 5(c)]. On the other qualitatively different from flexoelectricity, which is polar- hand, when the tip is applied in the bulk domain, the ity independent and, thus, less sensitive to the presence of a generated in-plane polarization is strikingly smaller to the polar domain wall; for this reason, we discard flexoelec- one expected from piezoelectricity [Fig. 5(c), red lines]. tricity from the analysis. This suppression of the in-plane polarization is due to the We test the reasoning given above by modeling a depolarizing cost of the head-to-head configuration and, simplified two-dimensional system, with the contact as expected, is accompanied by a harder elastic response between the CRFM tip and the surface described by an of the bulk domain compared to the domain wall. This 2 2 out-of-plane force proportional to e−ðx =t Þ , where 2t ∼ difference translates into an apparent DW softening of 20 nm is the contact area. As in the experiment, we focus 5%–10% [Fig. 5(d)] relative to the stiffness of the domain. only on linear static elastic effects and apply a force small To confirm the link between the suppression of in-plane enough to avoid any polarization switching. All the polarization and elastic hardening of the bulk, we also simulations are done with two electric boundary conditions: perform an additional simulation with the shear piezo- open boundary conditions, with surface screening of the electric constant e15 ¼ 0 [by setting the shear electro- polarization by immobile surface charges (requiring the striction coefficient q44 ¼ 0 in Eq. (10)] at short-circuit 041001-8 MECHANICAL SOFTNESS OF FERROELECTRIC 180 DEGREE … PHYS. REV. X 10, 041001 (2020) boundary conditions, thus removing both the bound charges factors: domain wall sliding and depolarization-activated appearing in plane and out of plane. As a result, the DW electromechanical coupling. For BaTiO3, these two con- softening almost vanishes, down to <0.3% [Fig. 5(d)] (here, tributions have a quantitatively comparable impact on the in-plane refers to the plane of the film, not the wall). Then, total softening. Ferroelectric 180° domain walls have, by reactivating the in-plane mechanism of charge formation by definition, P ¼ 0 in the vertical direction, so the sponta- returning e15 to the BaTiO3 value leads to an averaged neous strain associated with P is suppressed at the wall, 6% increase of the DW softening, and applying open resulting in a “strain dip” near the surface. When an boundary conditions increases the DW softening by another inhomogeneous vertical compression is delivered by the 2% [Fig. 5(d)]. The somewhat surprising conclusion, there- AFM tip near the wall, therefore, the material can respond fore, is that the main contributor to the electrostatic softening by sliding (or broadening) the wall so that this region of at the wall is not the out-of-plane piezoelectricity but the in- inherently reduced vertical strain (also known as strain dip) plane (shear) piezoelectricity. This result is important moves under the tip. This dynamic response requires the because, while the former can be partially screened by strain field from the tip to be inhomogeneous and asym- the use of metallic tips or in the presence of electrodes or metrically located with respect to the wall; if there is the adsorbates, the latter cannot. The in-plane piezoelectric same amount of compression on either side of the wall, it contribution to domain wall softening is therefore unavoid- does not move. However, it is important to emphasize that able, despite the polar axis being out of plane. this effect is not flexoelectric. The aforementioned mechanisms (inhomogeneous It is also worth noticing that the strain dip is not piezoelectricity and sliding of the domain wall’s strain topographically observable. The pseudocubic relaxation dip) have a strong effect on only the effective vertical elastic affects only the near-surface unit cells; deeper into the film, compliance (S33 ), but not on S11 and S22 . The domain wall the domain wall unit cells are fully clamped to those of the softening is thus anisotropic, a result supported by the adjacent domains and, thus, cannot relax. The calculated experimental observation [Fig. 3(c)] that the mechanical depth of the dip is <1 Å, and the width is approximately resonance frequency is slower at the 180° DW in c domains 1 nm (standard width of a 180° wall), well below the but unaffected when the same 180° domain walls are topographical detection limit of the AFM. Though this measured in plane across a domains. deformation is itself not observable, however, its mechani- It is also important to emphasize that the effective elastic cal consequences are. On the other hand, this elastic properties measured under inhomogeneous strain do not response is effective only in the surface, so it cannot be directly equate to the components of the linear elastic considered a bulk elastic property. The only softening tensor measured under homogeneous stress. The total mechanism that is effective along the entire domain wall is elastic energy incorporating both the standard Hooke’s the one due to inhomogeneous piezoelectricity. law component plus gradient elasticity is U ¼ 1=2Eε2 þ The depolarization field generated upon inhomogene- 1=2Kð∇εÞ2 [56–58], where K is the gradient coefficient ously straining a piezoelectric material contributes to its that relates the mechanical energy to the strain gradient. In gradient elasticity—making the material stiffer. Tip- our model, we have subsumed both K and E (the Young’s induced inhomogeneous deformation generates in-plane modulus) within an effective elastic constant E , but this piezoelectric polarization via a shear piezoelectric effect. In constant should not be confused with the real Young’s a tetragonal ferroelectric, this in-plane component must be modulus. The gradient elasticity is as much a mechanical head to head or tail to tail in the domains, whereas it is head property of the material as the linear elasticity described by to tail in the domain walls—hence, the electrostatic costs E, but it is apparent only under strongly inhomogeneous are smaller at the domain walls, which facilitates their deformations. deformation. However, this type of polarization disconti- nuity accompanies any inhomogeneous deformation in a V. CONCLUSIONS piezoelectric material, so its mechanical effect is not limited Although ferroelectric 180° domains are mechanically to tip-induced indentations. Such electrostatic effects identical, the domain walls that separate them display accompany any modulation of the lattice, such as phonons. mechanical contrast: The walls are softer than the domains. The fact that purely ferroelectric (i.e., nonferroelastic) This softening is detected as a lowering of the mechanical domain walls display mechanical contrast with respect to resonance frequency of the AFM cantilever in contact with their surroundings has obvious consequences for the the domain wall. This result has been reiterated here on mechanical properties of the material. It also has functional single crystals of uniaxial lithium niobate and perovskite consequences. Mechanical reading of ferroelectric polarity BaTiO3 and epitaxial thin films of PbTiO3 , in addition to is possible, but the mechanical contrast between domains is ceramics of lead zirconate-titanate [30,31]. The effect, weak [23], whereas the mechanical contrast of the domain therefore, appears to be general and not dependent on walls is much clearer and offers an easier way to “read” material composition or sample morphology. Theoretical ferroelectric bits, being a potential basis for phononic modeling shows that there can be at least two contributing switches [21,22]. In this respect, the mechanical contrast 041001-9 CHRISTINA STEFANI et al. PHYS. REV. X 10, 041001 (2020) at the wall means that periodically poled ferroelectric [7] J. Guyonnet, I. Gaponenko, S. Gariglio, and P. Paruch, crystals can also be regarded as phononic crystals. The Conduction at Domain Walls in Insulating PbðZr0.2 Ti0.8 ÞO3 more general lesson is that domain walls are distinct not Thin Films, Adv. Mater. 23, 5377 (2011). only functionally, but also mechanically, and the complete [8] M. Schröder, A. Haußmann, A. Thiessen, E. Soergel, T. Woike, and L. M. Eng, Conducting Domain Walls in picture of domain wall physics must incorporate this Lithium Niobate Single Crystals, Adv. Funct. Mater. 22, mechanical singularity. 3936 (2012). [9] T. Choi, Y. Horibe, H. T. Yi, Y. J. Choi, W. Wu, and S. W. 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