Domain wall conductivity as the origin of enhanced domain wall dynamics in polycrystalline BiFeO₃

Since the discovery in BiFeO₃ thin films,¹ followed by a number of studies in proper^2–5 and improper ferroelectrics,^6–10 conductive domain walls (DWs) have become of primary importance in future nanoelectronics.^11,12 Besides the fascinating functionality of these interfaces at the nanoscale, it has also been shown that the DW conductivity, even though it is localized in nature, can have a crucial effect on the global, macroscopic response of single- and poly-crystalline ferroelectrics. Special static effects of so-called charged DWs on the piezoelectric, dielectric, and domain switching behavior have been studied in BaTiO₃ single crystals¹³ and (K,Na)NbO₃ polycrystals.^14,15 On the other hand, the dynamic role of conductive DWs has been evidenced by a large nonlinear piezoelectric response measured at low driving field frequencies (<10 Hz) in polycrystalline BiFeO₃, shown to be quantitatively comparable to that typically achieved in highly nonlinear “soft” piezoceramics.¹⁶ Based on time-resolved x-ray diffraction analyses,¹⁷ it was suggested that this nonlinear response is indeed related to the conductive nature of DWs in BiFeO₃ as they move irreversibly at low driving frequencies along with the hopping charges that are responsible for the local conductivity.¹⁸ Despite this direct evidence, however, two key questions are left to be answered: (1) what is the exact mechanism by which the DW dynamics is affected by the DW conductivity and (2) could the local conductivity at DWs be used to controllably enhance or reduce the DW displacements under applied external fields allowing us to control macroscopic properties? Considering that the DW conductivity has been observed in a number of different ferroelectrics,^1–12 it becomes critical to understand and master the link between this local property and global nonlinear response of polycrystalline ferroelectrics.

Here, we provide answers to these questions by designing the local conductivity in bulk BiFeO₃ using acceptor doping with Co ions. We experimentally validate the concept of doping and confirm that an increased concentration of the dopant leads to a higher concentration of the compensating p-type charges, which largely control the DW conductivity in BiFeO₃.¹⁹ This allowed us to control the fraction of conductive DWs in the polycrystalline matrix by simply adjusting the concentration of the acceptor dopant. In contrast to the expected strong pinning of DWs in the sample with the higher concentration of the dopant (because of the higher concentration of the p-type pinning centers at DWs), we surprisingly find that it is exactly this sample in which the strong low-frequency nonlinearity related to large DW displacements appears. Using analytical nonlinear Maxwell–Wagner (M-W) modeling, we propose that at low frequencies the mobile charges at DWs and thus the DW conductivity helps in increasing the internal electric field inside a set of grains in the ceramic matrix, largely enhancing the DW displacements within those grains.

Bi(Fe_1–xCo_x)O₃ ceramics with Co concentration x = 0.0025 (further denoted as BFO-0.25) and x = 0.015 (further denoted as BFO-1.5) were prepared by the mixed-oxide solid-state route using ultrapure starting powders of Bi₂O₃ (Alfa Aesar, 99.999%) and Fe₂O₃ (Alfa Aesar, 99.998%). Co₃O₄ powder (Alfa Aesar, 99.0%) was used for doping. All the three powders were first pre-milled individually in a planetary mill (Retsch PM400, Hann, Germany) at 200 min^–1 of rotational frequency for 4 h. Bi₂O₃ and Fe₂O₃ powders were separately milled in a 125 ml polyethylene (PE) vials, each filled with 210 g of 3-mm yttria-stabilized-zirconia (YSZ) milling balls. The Co₃O₄ powder was milled in a 54 ml PE vial filled with 96 g of 3-mm YSZ milling balls. After this initial milling, the powders were mixed in proper stoichiometric ratios and homogenized in a 54 ml PE vial with 96 g of 3-mm YZS balls. The mixtures were calcined in a closed alumina crucible onto a Pt foil at 740 °C for 4 h using a heating/cooling rate of 10 °C/min. The calcination was followed by a milling step with conditions as already described. These milled powders were then compacted into 8-mm-diameter pellets using 50 MPa of uniaxial and 300 MPa of cold isostatic pressure, which was followed by sintering the pellets at 800 °C for 1 h (x = 0.0025) and 4 h (x = 0.015). A shorter sintering time was used for the BiFeO₃ sample with low Co concentration as this sample showed a tendency of extensive grain growth. The structural and microstructural analyses along with the grain sizes and ceramic densities of the two Co-doped samples are summarized in supplementary material 1.

As already reported in our previous studies,^16,18,19 ceramic samples for PFM and c-AFM analysis were additionally annealed at 840 °C with zero holding time and heating/cooling rates of 10 °C/min. The samples' surfaces were prepared by first grinding with SiC papers, polishing using a diamond paste and, finally, chemical etching with a colloidal suspension composed of 60 ml OP-S (Struers), 0.3 g of dissolved KOH, and 440 ml of de-ionized water. The thicknesses of the analyzed samples were ∼150 μm.

PFM and c-AFM analyses were performed using an atomic force microscope (AFM, Asylum Research, Molecular Force Probe 3D, Santa Barbara, CA, USA) with tetrahedrally shaped Si tips on a Si cantilever, both coated with Ti/Ir (Asyelec, AtomicForce F/E GmbH). Out-of-plane (OP) PFM imaging was performed in Dual a.c. Resonance Tracking (DART) mode using 8 V of a.c. voltage applied to the tip. c-AFM imaging was performed by applying a DC voltage to the tip. As reported previously,¹⁸ a range of DC voltages were used, i.e., from 3 to 45 V, to scan the local conductive behavior of the samples starting from low electric fields up to the domain-switching threshold field (typically observed at applied DC voltages ≥35 V). The c-AFM images shown in Figs. 1(c) and 1(h) were obtained by applying 10 V DC. In all cases, the scanning frequency was 0.8 Hz with 256 scan points per individual line. A periodic background electrical noise, appearing in the c-AFM images, was removed by applying a Fourier transform on the spatial-domain image in order to convert the image into the frequency-magnitude space where the noise was manually removed or blurred. After the noise removal, the image was transformed back to the spatial domain.

Sintered pellets for piezoelectric characterization were first cut with a diamond saw, surface polished, and electroded with Au by magnetron sputtering (5pascal SRL, Trezzano, Italy). The samples were poled with a DC electric field of 80 kV/cm for 15 min (x = 0.0025) or 100 kV/cm for 40 min (x = 0.015) at room temperature. Prior to measurements, the samples were aged for at least 48 h at room temperature.

Converse piezoelectric measurements were performed using a home-made setup consisting of a low-distortion voltage generator (SRS DS360), a high-voltage amplifier (TREK 609E-6), a high-resolution fiber-optic sensor for precise displacement measurements (MTI 2100), an oscilloscope (LeCroy 9310C) for signal visualization and hysteresis-loop data saving, and two lock-in amplifiers (SR80 DSP) to precisely measure the harmonic signals of the input voltage and output displacement response (further details are described in Ref. 16). The measurements were performed in the frequency range between 90 and 0.1 Hz. At each frequency, the piezoelectric response was measured as a function of increasing external electric field amplitude from 3.5 kV/cm up to a maximum of 16 kV/cm with individual increments of 2.5 kV/cm. The results are presented in conventional terms using the total piezoelectric d₃₃ coefficient (ratio of the output displacement vs the input voltage amplitude) and tangent of the phase angle, tanδ (as the phase between the displacement and voltage sinusoidal signals). The amplitudes of the first harmonic displacement and voltage signals, as well as the phase between them, were directly measured using lock-in technique.

The Seebeck coefficient was measured on sintered bars with dimensions of approximately 15 × 4 × 4 mm in a flow of synthetic air (Linde, 5.0). The samples were annealed from room temperature to 700 °C with zero holding time and with heating and cooling rates of 2 °C/min. In Sec. III, the Seebeck coefficients are reported for 300 °C (cooling cycle). The measurements were performed using a ProboStat™ setup (NorECs AS) with a vertical tubular furnace, S-type bottom and top thermocouples and a multimeter Keithley 2000, which was used to measure the voltage drop across the sample and the temperatures of the two thermocouples.

Electric-field (E) dependent current-density (j) measurements were performed using a Keithley 237 high-voltage source measure unit (Keithley Instruments, OH, USA). Au-electroded samples were exposed to step-like voltages (11 measurement points) within a field range of ±0.02 kV/cm. Upon each of these steps, the voltage was kept constant for 1 h for the leakage current to stabilize, before being recorded. The specific DC electrical conductivity (⁠$σDC$⁠) was determined from the linear (Ohmic) j-E curves as the curve slope.

In this study, we use Co doping to control the local DW conductivity in BiFeO₃ and, for the sake of simplicity, we present data for two Co concentrations, i.e., 0.25 mol. % and 1.5 mol. % (see Sec. II). Based on recent direct identification of Co²⁺ states by x-ray photoelectron spectroscopy (XPS) in Co-doped BiFeO₃,²⁰ we reasonably assume in the defect model that Co is incorporated on the host Fe³⁺ sites as an acceptor, i.e., as Co²⁺ or $Co′Fe$ in Vink–Kröger notation, whose charges are compensated by oxygen vacancies (⁠$VO∙∙$⁠),

During cooling of the sample, oxidation will lead to the formation of Fe⁴⁺ states (⁠$FeFe∙$⁠), resulting in the characteristic p-type conductivity of BiFeO₃,^21–25 

The combination of Eqs. (1) and (2) shows that as the concentration of the Co dopant is increased [left-hand side of Eq. (1)], the resulting higher concentration of $VO∙∙$ [right-hand side of Eq. (1)] will at constant partial pressure of oxygen shift the oxidation equilibrium reaction [Eq. (2)] toward the reaction products, resulting in a higher concentration of p-type charges (⁠$FeFe∙$⁠). The model thus predicts the increasing level of p-type conductivity with increasing Co concentration. This defect picture is consistent with two experimental observations: (1) with the p-type bulk conductivity of BiFeO₃, directly identified in the two Co-doped samples by the positive Seebeck coefficients (1200 μV/K and 650 μV/K for 0.25% and 1.5% Co-doped BiFeO₃, respectively; both measured at 300 °C) and (2) with the much higher specific electrical conductivity measured in the 1.5% Co sample (3.2×10^–7 Ohm^–1m^–1) as compared to that doped with 0.25% Co (4.5×10^–11 Ohm^–1m^–1). The conclusion is also supported by the identification of Fe⁴⁺ states in BiFeO₃ using x-ray absorption spectroscopy, which were shown to be induced by the Co doping.²⁶ Since the DW conductivity in air-sintered polycrystalline BiFeO₃ is dominated by the accumulation of p-type charges, as directly revealed by previous atomic-scale chemical analysis on DWs in BiFeO₃,¹⁹ it is reasonable to expect that both the conductivity of individual DWs and the total fraction of highly conducting DWs in the ceramic matrix will increase with increasing Co concentration.

To validate the expected correlation between the Co concentration and the fraction of conductive DWs, we compare in Fig. 1 the local conductive behavior analyzed by c-AFM of the BiFeO₃ samples doped with 0.25 mol. % (BFO-0.25) and 1.5 mol. % Co (BFO-1.5). PFM amplitude and phase images are shown in Figs. 1(a) and 1(f) and Figs. 1(b) and 1(g), respectively, while the AFM topography images are shown in supplementary material 2. PFM confirms a similar domain structure of the two samples with lamellar- and wedge-like domains as reported earlier.²⁷ These same sample regions were subsequently analyzed in terms of local current measurements using c-AFM, shown in Figs. 1(c) and 1(h). Consistent with previous studies on BiFeO₃ thin films,^1,28,29 conductive DWs are clearly identified by an enhanced current signal observed at positions corresponding to DWs [an example for each sample is denoted by a red arrow in Figs. 1(c) and 1(h)]. Furthermore, a clear distinction between DWs exhibiting a clear conductive character from those that are not associated with any enhanced current, relative to the background signal, can be performed by measuring the electric-current profile along a line crossing the DWs [respective examples are shown in Figs. 1(d) and 1(i) with the corresponding profile lines indicated in blue in PFM and c-AFM images].

It already becomes evident from the visual comparison between the local conductive behavior of the two samples, as imaged by c-AFM, that BFO-1.5 [Fig. 1(h)] contains a larger amount of conductive DWs than the BFO-0.25 sample [Fig. 1(c)]. By statistically analyzing the conductive signals at DWs in several regions of the two samples, we found out that the fraction of conductive DWs is only 6% in the BFO-0.25 sample, while it is much higher, i.e., 49%, in BFO-1.5. This confirms that the Co doping induces the p-type charges due to the acceptor compensation [see Eqs. (1) and (2)], increasing the fraction of p-type conductive DWs in the BiFeO₃ ceramic matrix. This behavior is similar to that in p-type ErMnO₃ where acceptor doping with Ca was found to increase the conductivity of the tail-to-tail (TT) charged DWs presumably due to increased density of p-type charges at the walls induced by the dopant.³⁰ We finally note that macroscopic piezoelectric and dielectric characterization on Co-doped BiFeO₃ samples performed earlier³¹ revealed an Arrhenius-type temperature dependent M-W-like relaxation, indicating that, on the average, the local conductivity is dominated by a thermally activated semiconducting behavior.

The different local conductive behavior of the two doped samples provides a unique opportunity to test the effect of conductive DWs on their dynamics under subswitching electric fields. We assume, and subsequently experimentally confirm, that the higher fraction of conductive DWs in the BFO-1.5 sample will induce the so-called piezoelectric M-W effect^32–34 due to enhanced anisotropy in the electrical conductivity on the local, most likely, grain scale. As previously suggested and supported by time-resolved XRD experiments,¹⁷ this anisotropy, which is otherwise unexpected in a homogeneous perovskite, such as BiFeO₃, can arise as a result of the different orientations of conductive DWs in different grains or grain clusters. A simplified illustration of the effect is schematically shown in Fig. 1(j), highlighting a basic M-W unit composed of two grains with different conductivities measured along the vertical direction (note that in the vertical direction the upper grain will exhibit a higher conductivity, compared to the bottom grain, due to conductive DWs lying vertically). Since M-W effects are strongly dependent on this local anisotropy and thus on the conductive behavior of the DWs and their faction inside the ceramics, such effects should be weaker or even absent in BFO-0.25 due to the significantly lower amount of conductive DWs in this sample [Fig. 1(e)].

To test the effect of conductive DWs on the dynamic macroscopic response of BiFeO₃, we show in Fig. 2 the results of the converse piezoelectric measurements. The data are displayed in terms of the weak-field piezoelectric d₃₃ coefficient [Fig. 2(a)] and tangent of the piezoelectric phase angle [tan δ; Fig. 2(b)] as a function of driving field frequency (f; horizontal axis) and electric-field amplitude $E0$ [see vertical arrows in Figs. 2(a) and 2(b)]. The complex piezoelectric response of the two samples is distinctly different, in both qualitative and quantitative terms. In BFO-1.5, a strong frequency dispersion is observed in both the d₃₃ coefficient and tan δ, particularly below ∼10 Hz where the dispersion becomes nonlinear, i.e., $E0$-dependent [follow the d₃₃ and tan δ curves in Figs. 2(a) and 2(b), respectively, starting from the yellow curve, which corresponds to 3.5 kV/cm, up to the black curve, corresponding to 16 kV/cm]. As explained in Sec. I, this low frequency nonlinearity and nonlinear hysteresis has been reported previously for BiFeO₃ and directly shown to arise from irreversible displacements of conductive DWs.^16,17 Obviously, in the measured frequency range (0.1–90 Hz), this strong dispersive response is absent in the BFO-0.25 sample. In this case, d₃₃ and tan δ are weakly dispersive and show appreciable linearity, i.e., a much smaller dependence on $E0$ in the entire frequency range [Figs. 2(a) and 2(b)].

To quantify the observed low-frequency nonlinearity and show its correlation with the M-W effects, we extract two parameters from the data shown in Figs. 2(a) and 2(b). The first is the irreversible nonlinear coefficient $α∗$⁠, extensively used in the frame of the Rayleigh law,^35–37 

where $d33(E0)$ and $d33init$ are the total (field-dependent) and initial coefficient (extrapolated at zero field), respectively. The coefficient $α∗$ quantifies the extent of the nonlinearity and represents the slope of the d₃₃ vs $E0$ curve. Note that for the piezoelectric response of BiFeO₃ this slope is not constant, as originally defined in the Rayleigh law, but $E0$ dependent in the whole measured frequency range (see supplementary material 3). We therefore use the widely adopted field-dependent $α∗$^35,38,39 and represent the nonlinearity with $α∗$ calculated at the maximum applied $E0$ of 16 kV/cm [shown in Fig. 2(c)].

The second extracted parameter is used to identify the fingerprint of the piezoelectric M-W effect and is represented by the negative piezoelectric phase angle. In this unusual case, the periodic piezoelectric displacement signal is in time domain advancing the driving electric-field signal (resulting in a clockwise rotational sense of the hysteresis loop), instead of lagging behind it as it is the case for the more common positive phase angle (where the hysteresis rotates counterclockwise).^33,40 This particular time response has been theoretically shown to be induced due to internal field redistribution provoked by large local anisotropic effects in the electric conductivity and thus M-W mechanisms; negative tan δ was also experimentally confirmed in single-phase Aurivillius ceramics with anisotropic grains,³³ BiFeO₃ ceramics,^16,17 and ferroelectric-polymer composites.^41,42 In this study, we examine the negative phase by plotting the frequency dependent, linear piezoelectric $tanδinit$⁠, obtained by extrapolating the $tanδ$ vs $E0$ data to $E0=0$ [Fig. 2(d); the data and fittings are shown in supplementary material 3].

The frequency dependence of $α∗$ and $tanδinit$ confirms a clear correlation between the low-frequency nonlinearity and negative tan δ, both appearing in the BFO-1.5 sample [see green arrows in Figs. 2(c) and 2(d)]. The negative tan δ in this sample is directly confirmed by the clockwise rotational sense of the piezoelectric hysteresis loop measured at 4 Hz and 3.5 kV/cm [see the inset of Fig. 2(b)]. In contrast, the BFO-0.25 sample is characterized by a low and rather constant $α∗$ in the whole frequency range [Fig. 2(c), filled circles]; correspondingly, the phase angle is positive and shows little frequency dispersion [Fig. 2(d), filled circles]. Note also that the nonlinear effect in BFO-1.5 is significant: at the lowest frequency (0.1 Hz), BFO-1.5 is characterized by an order of magnitude higher $α∗$ (0.12×10^–16 m² V^–2) as compared to that measured in BFO-0.25 (0.008×10^–16 m² V^–2). The data clearly suggest that the large irreversible DW displacements, contributing to the large $α∗$ at low frequencies in BFO-1.5, are intimately related to M-W effects, as identified by the negative phase, and thus likely related to the larger fraction of conductive DWs in this sample (see Fig. 1). Finally, nonlinear piezoelectric measurements on samples doped with intermediate Co concentrations (0.5 and 1.0 mol. % Co) as compared to that shown in Fig. 2 (0.25 and 1.5 mol. % Co) confirm the correlation between the Co concentration and the low-frequency nonlinearity (see supplementary material 4). Acceptor doping (in this case with Co) could thus be a valuable method for controlling local conductive properties and thus macroscopic nonlinearity in polycrystals, similarly as shown previously for single crystals.³⁰ To go even further, it could be interesting to test methods alternative to doping, such as the defect control via vacuum annealing combined with “frustrative” poling, validated for BaTiO₃ crystals.⁴³ 

To explain the relationship between the large low-frequency irreversible DW contribution and M-W effects arising from the conductivity at DWs, we invoke an analytical model, which was previously used to explain the lattice strain and DW contributions to the piezoelectric response of BiFeO₃ deconvoluted using in situ and time-resolved XRD analysis.¹⁷ While all the details of the model can be found in that paper, for the sake of clarity, we briefly describe here its main characteristics with the detailed mathematical description and step-by-step derivations and model analysis provided in supplementary material 5.

As anticipated by the schematics shown in Fig. 1(j), we begin the analysis by invoking the classical, linear M-W serial bilayer model where the layers are represented by two grains of different orientations, i.e., [100]_pc and [111]_pc (pc is pseudocubic), with respect to the electric field (E) axis. These two grains, noted as grain 1 and grain 2 in Fig. 3(a) (see left-hand schematic), should exhibit different electrical conductivities $σ1$ and $σ2$ along the E axis owing to the different orientations of conductive DWs (precise angles of DW planes with respect to the crystallographic planes in the two grains are reported in Ref. 17). Note that for the specific case analyzed here $σ1>σ2$ because the conductive DWs in grain 1 are more parallel to the electric-field axis, allowing the charges to flow along the DWs [see Fig. 3(a), right-hand schematic]. Obviously, in the real ceramic case, the picture is more complex in that a large number of grains or grain clusters may exhibit increased conductivity with respect to other surrounding grains depending on the complex domain structure and DW orientation in individual grains. Nevertheless, as it will be shown next, the simple model shown in Fig. 3(a) is still sufficient to give hints on the nonlinear mechanism dominating the complex ceramic response.

It is intuitively clear that the effective electric fields, $E1$ and $E2$ [see Fig. 3(a)], will be redistributed among the two grains depending on the frequency of the external field due to the charge migration along the conductive DWs, predominantly occurring in grain 1. Formally, this internal field redistribution will be dependent upon the time constants of the two grains, i.e., $τi=ε′iσi$ (i = 1,2) where $εi′$ is the dielectric permittivity (real component) and $σi$ is the electrical conductivity of the two grains (see derivations in supplementary material 5). We simplify the case by illustrating the redistribution of these internal fields depending on the anisotropic parameter, i.e., ratio of the conductivity of the two grains (⁠$σ1σ2$⁠), where $ε1′=ε2′$⁠, $σ2$ is fixed and $σ1$ is varied to account for the effect of the anisotropy. The calculated results in Fig. 3(b) clearly confirm a strong enhancement of the internal field in grain 2 at <10 Hz [blue curve in Fig. 3(b)] and the corresponding decrease of the field in grain 1 [red curve in Fig. 3(b)], which are crucially dependent on the anisotropy. Therefore, as the conductivity of grain 1 becomes higher with respect to grain 2 conductivity (increasing $σ1σ2$⁠), the internal electric field in grain 1 will drop at low frequencies due to the leakage (i.e., charge migration along the vertical DWs) and will be counterbalanced by the enhanced internal field in grain 2 because of its lower conductivity [as indicated in the right-hand schematic in Fig. 3(a)]. It is this enhancement of the internal field in grain 2 and thus in similar sets of grains or grain clusters inside the ceramic matrix that we propose as the origin of the measured enhancement in the DW dynamics below 10 Hz [as shown in Fig. 2(c)].

To further demonstrate this point, we next analyze the piezoelectric response of the two grains. The M-W effect is clearly revealed by plotting the linear piezoelectric phase angle, $tanδinit$⁠, in Fig. 3(c). Strong relaxations, reflected in $tanδinit$ peaks, are observed in the two grains driven by the anisotropic parameter. Note that the sign of the $tanδinit$ peak depends on the internal-field frequency evolution: (1) a positive peak is observed in grain 2 [blue curve in Fig. 3(c)], related to the increasing $E2$ with decreasing frequency [retardation; blue curve in Fig. 3(b)] and (2) a negative peak is observed in grain 1 [red curve in Fig. 3(c)] due to the decreasing $E1$ with decreasing frequency [relaxation; red curve in Fig. 3(b)]. This is consistent with the results of M-W modeling published earlier.^32,34,40 The calculations thus reveal the appearance of a negative phase angle, characteristic for the M-W mechanism, which is triggered by the $σ1σ2$ anisotropy and is experimentally confirmed here in the BFO-1.5 sample by the clockwise hysteresis rotation [see the inset of Fig. 2(b)].

Previous time-resolved XRD analysis directly revealed a strong DW contribution to the low-frequency piezoelectric response of BiFeO₃ arising from [111]_pc-oriented grains.¹⁷ We thus next introduce this important aspect in the model [see right-hand scheme in Fig. 3(a)] by assuming (1) that the [100]_pc-oriented grain 1 will mainly exhibit a piezoelectric lattice strain [noted with ΔL₁ in Fig. 3(a)], described by the piezoelectric coefficient $d1$⁠, and (2) that the [111]_pc-oriented grain 2 will, in addition, exhibit strain from DW displacements [as directly confirmed by XRD analysis¹⁷ and noted with ΔL₂ in Fig. 3(a)]. To account for this nonlinearity, we upgrade the previous linear M-W model by introducing Rayleigh relations in the piezoelectric response of grain 2 (supplementary material 5). We thus introduce the reversible coefficient $d2init$ and the irreversible coefficient $α2$ in grain 2 and calculate the resulting effective coefficient, $αeff$⁠, of the M-W unit as a whole.

Providing a sufficiently large anisotropy, the results of modeling clearly show that the effective irreversible coefficient $αeff$ is strongly enhanced at low driving frequencies, i.e., <10 Hz [see Fig. 3(d)]. This is qualitatively consistent with the experimental data, also showing enhanced $α∗$ below 10 Hz [see Fig. 2(c), BFO-1.5]. We propose that the reason behind is the enhancement of the internal fields in a subset of grains [analogous to grain 2, see Fig. 3(b)], which enhance the DW displacements. Importantly, the model also shows that this nonlinear effect is intimately related to the M-W internal field redistribution and should be thus coupled to a dispersive piezoelectric phase angle and appearance of the negative phase [as predicted in Fig. 3(c)]. This is also observed experimentally [see Fig. 2(d), i.e., BFO-1.5 shows negative phase and strongly dispersive phase angle].

It is clear that the real picture in ceramics is far more complex than the largely simplified nonlinear M-W model used here where only two grains are modeled and where the elastic and transverse piezoelectric cross-grain effects are neglected. In this aspect, the model predictions can only be qualitatively compared to the complex ceramic behavior. Nevertheless, the calculated data still provide a plausible mechanism responsible for the experimentally observed low-frequency irreversible response [Fig. 2(c), green arrow] and show that it is coupled to the M-W effect and the corresponding negative phase angle [Fig. 2(d), green arrow]. Since the governing mechanism is of a M-W origin, it is strongly dependent on the local anisotropy in the conductivity, as predicted by the model. This explains why the large irreversibility and negative phase are only observed in BFO-1.5, and not in BFO-0.25, because in the former a much higher concentration of conductive DWs increases the local anisotropic effects, giving rise to the observed low-frequency nonlinear M-W effect. Even though the results do not exclude the possible presence of M-W mechanism in the BFO-0.25 sample, reflected more clearly, for example, in a frequency range outside that used in the current measurements, they still suggest an important role of conductive DWs in affecting the DW dynamics via field redistribution inside ceramic grains.

Studies on conductive DWs in ferroelectrics have been mainly focused on their local properties. With the first evidence reported for BiFeO₃, it has been suggested that these interfaces can also have a crucial impact on the macroscopic properties of polycrystalline ferroelectrics, even though the conductivity operates at the nanoscale, i.e., inside individual grains. This claim is supported by a large piezoelectric nonlinear and hysteretic response, identified in BiFeO₃ at low driving field frequency (<10 Hz), which was attributed to the conductive nature of the DWs but which has never been truly explained.

Providing a control over the conductive DWs in polycrystalline BiFeO₃ by means of chemical doping, we provide in this study a clear mechanistic picture for the observed low-frequency nonlinearity. A combination of piezoelectric measurements and analytical modeling unexpectedly revealed that the conductivity localized at DWs may even increase the DW displacements under applied weak fields. As shown by a simple nonlinear M-W model, the reason is the enhancement of internal fields inside a set of grains in the ceramic matrix, which triggers large irreversible displacements of DWs residing in those grains. We believe that the clarification of the underlying mechanism will have consequences in the design of nonlinear properties in ferroelectrics via controlling local conduction paths.

See the supplementary material for structural and microstructural data of the analyzed samples, AFM topography images, complete piezoelectric dataset, and mathematical derivation of the used analytical model.

This work was financed by the Slovenian Research Agency (program P2-0105, Project No. J2-9253, Ph.D. program of Maja Makarovic). Centers of Excellence NAMASTE and CONOT are acknowledged for the access to AFM/PFM equipment.

The data that support the findings of this study are available from the corresponding author upon reasonable request.