example is a ferroelectric tunnel junction (FTJ) 14 that exploits switchable polarization of an ultrathin ferroelectric barrier to control tunnelling conductance, in which an in-plane domain wall creates quantum confinement leading to resonant tunnelling 15 and domain-wall tunnelling electroresistance effects 16 .

The quasi-two-dimensional nature of domain walls is appealing for nanoelectronics owing to emerging opportunities for the design of electronic building blocks and circuitry at the nanoscale. To realize this potential, the deterministic control of domain walls is essential for device operation. However, previously the manipulation of domain walls has been mostly performed at the micrometre scale [7][8][9][17][18][19] . The control of domain walls, especially charged domain walls, is still lacking at the atomic scale, but it is important in the miniaturization of electronic devices. Such control requires direct monitoring of the polarization reversal process under an applied electric field down to subangstrom spatial resolution, which is technically hugely challenging. The precise monitoring of domain-wall dynamics behaviour and the accompanying charge transfer are essential for explaining the mechanism of polarization reversal, which is critical for creating new thin-film ferroelectric devices.

In this work we report a strategy to create, move and erase an in-plane charged domain wall with a 'tail-to-tail' configuration in SrRuO 3 (SRO)-BFO-SRO heterostructures. Layer-by-layer ferroelectric switching is directly observed accompanied by charge accumulation, by using an in situ biasing technique. The whole process of polarization reversal is investigated step by step and atom by atom, by using high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) imaging. A combination of electron energy-loss spectroscopy (EELS), in-line holography and first-principles density functional theory (DFT) calculations reveals the motion of a charged domain wall through positively charged oxygen vacancies. By using voltage control of the domain-wall positioning within the BFO film, multiple non-volatile resistance states are realized, demonstrating the key functional property of being a memristor a few unit cells thick.

The designed SRO-BFO-SRO trilayer structure is shown in Fig. 1c. A ferroelectric BFO film of variable thickness, ranging from 2 unit cells to 20 unit cells, is sandwiched between 5-nm-thick symmetric conductive SRO electrodes. The film is grown on a (001) Nb-doped SrTiO 3 (STO) single-crystal substrate (Extended Data Fig. 1). An external in situ bias voltage is applied by a tungsten probe (Methods and Extended Data Fig. 2), while atomic HAADF-STEM images of the film are simultaneously recorded (Fig. 1d-f). Both the top and bottom BFO-SRO interfaces are atomically sharp as indicated by the clear contrast. For each unit cell, the c/a lattice parameter ratio and polarization vector are extracted from the STEM images (Methods) and are displayed in Fig. 1g-i. As is evident from Fig. 1j-l, variations of the plane-averaged c/a ratio and the off-centre displacement across the SRO-BFO-SRO trilayer obtained from the STEM data are in excellent agreement with those obtained from our DFT calculations (Methods).

A nanometre-thick BFO film exhibits a single ferroelectric domain structure at the initial state (Fig. 1d,g,j). Polarization vectors have a uniform out-of-plane direction in all BFO atomic layers, showing a high spontaneous polarization consistent with previous work 20 . However, when an in situ bias of 1 V is applied, an atomically flat in-plane domain wall is created (Fig. 1e,h,k). This domain wall is characterized by the normal components of polarization pointing away from the domain-wall plane, thus representing a charged tail-to-tail domain wall. At the domain-wall area, the lattice structure exhibits a distortion with c/a = 1.08-1.10, largely owing to the changing c parameter across the structure (Extended Data Fig. 3). Surprisingly, the created in-plane charged domain wall is very stable. Even when the bias voltage is turned off, the BFO film still maintains this tail-to-tail polarization over a long time (Extended Data Fig. 4). When the applied in situ voltage rises to 5 V, the charged domain wall vanishes and the polarization is fully reversed (Fig. 1f,i,l). Thus, by using the designed heterostructure and the in situ biasing technique, a charged domain wall is created and erased in the BFO film plane.

A layer-by-layer motion of the domain wall is observed with applied bias changing from 0.5 to 1.5 V. Figure 2a-c shows the dynamic transformation of the polarization profile across the BFO film with the increasing applied voltage. It is seen that the tail-to-tail domain wall migrates steadily from one BFO atomic layer to another, showing layer-by-layer polarization switching. Because a charged domain wall is expected to be energetically unfavourable, there must be an underlying mechanism for screening the polarization charges at the domain wall. To explain the screening mechanism, we investigated the charge distribution across the domain wall during its migration by using EELS and in-line electron holography techniques complemented by DFT calculations (Methods). Fine structures in the EELS Fe-L and O-K edges are quantified as the Fe/O abundance ratio (right panels of Fig. 2a-c). At the domain-wall area, the Fe/O ratio is notably enhanced. This is because of the reduction in the O concentration at the domain wall as no relative change in the Fe concentration is observed from the EELS data (Extended Data Fig. 5). The Fe/O atomic ratio determined from the EELS data indicates an enhanced population of positively charged oxygen vacancies at the domain wall, which is also evidenced by the distortions of the oxygen octahedra (Extended Data Fig. 6). This conclusion is confirmed by our DFT calculations, which show that oxygen vacancies in the SRO-BFO-SRO structure help to stabilize the in-plane domain-wall configuration (right panels in Fig. 2d-f and Extended Data Fig. 7). Ionized oxygen vacancies provide a positive bound charge that screens the negative polarization charge at the tail-to-tail domain wall, whereas the negative electron charge goes to the interfaces, resulting in a stable domain-wall configuration. The explicit charge balance across the domain wall is obtained from in-line holography measurements, which allow the direct detection of the charge distribution with high resolution and accuracy [21][22][23] . Figure 2d-f shows the obtained evolution of the charge distribution with the domain-wall motion from the top to bottom interface. It is seen that a positive charge density is accumulated at an in-plane localized area, precisely where the charged domain wall is.

The motion of the domain wall across the ferroelectric layer is accompanied by changes in the electrical resistance of the SRO-BFO-SRO FTJ (see Methods for the details of I-V measurements), which exhibits distinctive memristive features that are characterized by multiple non-volatile resistance states. The I-V characteristics of the FTJ reveal pinched hysteresis loops (Extended Data Fig. 8a), which is a signature of the memristive behaviour 24,25 . Notably, the measured electric current and the associated resistance display pronounced jumps with increasing voltage. In Fig. 2g-h, three current and resistance jumps are observed. The resistance decreases in a step-like pattern, distinct from the stepless change in ordinary memristors in which a conductive filament is formed and ruptured 26,27 . It is notable that the step-like I-V feature persists in a much larger area of the sample (Extended Data Fig. 8b). These tests demonstrate the non-volatile character of these resistance states, which can be obtained at specific values of voltage.

The observed stepwise character of the resistance originates from the layer-by-layer motion of the ferroelectric domain wall across the BFO film under the applied bias voltage. By measuring the resistance during the in situ biasing process (Extended Data Fig. 8c,d) and simultaneously monitoring the polarization configurations (Fig. 2i,j), clear correlation is found between the resistance steps and the different positions of the domain wall within the BFO layer. This process reveals good reproducibility and reversibility. Therefore, the 'quantized' resistance behaviour is associated with the in-plane domain wall moving layer-by-layer and being stabilized at each atomic layer in BFO for a certain bias voltage. Each of these domain-wall states are characterized by different resistance values between those two associated with the polarization states at saturation. This mechanism is different from resonant tunnelling resulting in conductance steps owing to the quantum-well states associated with a ferroelectric domain wall 15,16 . Thus, by applying a voltage, not only is layer-by-layer switching realized, but also multiple polarization and resistance states can be precisely and reversibly controlled. Along with the motion of a charged domain wall, the resistance can be modulated at the unit-cell scale--the minimum scale for multiple resistance-state device control. This property manifests a new type of a unit-cell-level FTJ memristor, which is determined by the voltage-controlled positioning of the in-plane domain wall within the ferroelectric layer, as shown in Fig. 2k. This type of memristor is different from the ferroelectric memristor, in which the fraction of the switched domain determines the FTJ resistance 28 .

The observed layer-by-layer polarization switching mechanism is characterized by the ferroelectric domain growth in the direction parallel to the applied electric field, which is different from the conventional mechanism in which the ferroelectric domain polarized along the field direction grows perpendicular to the applied field [29][30][31] . This is due to the polarization and screening charges being distributed uniformly parallel to the BFO film plane, rather than being accumulated at specific nucleation sites 8,32 . This special charge distribution pattern originates from the symmetric SRO-BFO-SRO layered structure design, in which the same atomic termination is engineered at both interfaces with a BiO monolayer in BFO facing a RuO 2 monolayer in SRO. This termination is evident from the SRO-BFO-SRO atomic structure obtained from our elemental energy-dispersive X-ray (EDX) spectroscopy measurements (Fig. 3a-e). The symmetric interface structure is not prone to producing a built-in electric field across the ferroelectric spacer layer and, thus, when the ferroelectric layer is thin enough, it is capable of sustaining the charged domain wall with the associated screening charges. On the contrary, for the similar SRO-BFO-SRO structure with asymmetric interfaces, the layer-by-layer polarization switching mechanism is extinguished by the presence of a built-in field across the BFO layer (Extended Data Fig. 9). As a result, in symmetrically terminated SRO-BFO-SRO structures with a thin BFO layer of several unit cells, the domain wall is widely observed to be atomically flat without nucleation steps (Fig. 3f,g). However, with an increase of BFO thickness, screening by oxygen vacancies becomes less homogeneous, resulting in an increased probability of obtaining a domain wall with a nucleation feature (Fig. 3h,i and Extended Data Fig. 10). Domain walls start to zigzag, and eventually become charge-neutral out-of-plane walls in a bulk-like 20-unit-cell-thick BFO film (Fig. 3j). As a result, the ferroelectric polarization switching mechanism changes from the layer-by-layer motion parallel to the applied electric field (E-field) to conventional ferroelectric domain growth perpendicular to the applied field (Fig. 3k). Based on these observations, the necessary conditions to stabilize an in-plane domain wall in a ferroelectric film and to realize layer-by-layer polarization switching are symmetric interfaces of the ferroelectric layer, their in-plane homogeneity on the atomic scale and sufficiently small thickness of the ferroelectric layer.

Our direct observation of in-plane charged domain walls in ferroelectric heterostructures designed to be a few nanometres thick, their deterministic creation, manipulation and annihilation by applied voltage, as well the demonstration of their memristive functionality associated with the multiple non-volatile resistance states, have important consequences both for the fundamental understanding of the domain-wall properties in nanoscale ferroelectrics and for their application in advanced technologies. In particular, the unambiguous correlation between the charged defects (oxygen vacancies) and the domain-wall stability and motion puts forward the importance of electrochemistry in the observed ferroelectric phenomena at the nanoscale 33 . The high mobility of oxygen vacancies and their ability to efficiently screen polarization charges may control the ferroelectric switching behaviour in various types of FTJs 34,35 as well as in bulk ferroelectrics, such as the recently discovered orthorhombic hafnia 36,37 .

Clearly, the discovered mechanism of in-plane domain-wall formation ferroelectric domain walls offer a wide range of functional properties such as the demonstrated memristive functionality, which may be useful for new nanoscale ferroelectric devices.

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-022-05503-5.

The SRO-BFO-SRO trilayer structures with different thicknesses were grown by a laser-molecular beam epitaxy technique onto atomically smooth (001) Nb-doped (0.7 wt%) STO single-crystal substrates (Crystec GmbH) with a small miscut angle (<0.5°) by using a KrF (wavelength 248 nm) laser. Here the conducting SRO layers were used as the top and bottom electrodes. A laser spot size of 4 mm 2 at a frequency of 3 Hz was created for ablating the stoichiometric targets (Bi 0.9 FeO 3 and SRO). The substrate was placed at a 6.5 cm distance directly above the BFO and SRO targets. The SRO layers were deposited at a substrate temperature of 600 °C in a flowing oxygen atmosphere of 100 mTorr by using a laser fluence of 1.2 J cm -2 . The conditions for the BFO layer were adjusted to 600 °C and 130 mTorr by using a laser fluence of 1 J cm -2 .

During the growth, in situ high-pressure reflective high-energy electron diffraction was used to monitor the film thickness. After deposition, the samples were cooled to room temperature under pure oxygen at a cooling rate of 5 °C min -1 to reduce the oxygen vacancy.

Samples were prepared into the cross-sectional lamella by a focused ion beam and were mounted on a Hysitron PI-95 sample holder. A bias voltage was applied by the sample holder, which electrically connected the bottom SRO electrode and the Nb-doped STO conductive substrate (Supplementary Fig. 2). The conductive probe provided an electrical path through the capping Pt and the top SRO layer. A partition was made in the sample to make sure that there was no series connection between the top SRO and the substrate. The microstructure and element distribution were characterized by using aberration-corrected STEM at high-angle annular dark-field mode on an FEI Titan G 2 80-200 microscope at an emission voltage of 200 kV, equipped with a Super-X EDX and Gatan Image Filter (GIF) detector. To reduce random noise in the HAADF-STEM images, a Fourier filter mask was used.

The out-of-plane lattice parameters c and the in-plane lattice parameters a were quantified for each unit cell of the structure based on HAADF-STEM measurements. Then a c/a ratio was obtained for each unit cell to characterize the local distortion of the lattice structure. Polarization vectors were obtained from the off-centre displacement of the B-site atom (Fe and Ru) with respect to the A-site atom (Bi and Sr) for each unit cell.

All atomic column positions in the measurement of off-centre displacements were determined by using a multiple-ellipse fitting method 38 . Atomic columns were fitted by a series of ellipses to outline the shape and intensity, then were then precisely located.

DFT-based calculations were performed by using the projectoraugmented wave method 39,40 , as implemented in the Vienna ab-initio simulation package 41 . An energy cut-off of 400 eV, and an exchange and correlation functional of the local density approximation with Hubbard-U correction (U = 3.5 eV for the Fe-d orbital) were used throughout. The SRO-BFO heterostructure was modelled by a [SRO] 5.5 -[BFO] 4.5 -[SRO] 5 superlattice. The in-plane lattice constant was constrained to be the local density approximation calculated value of bulk cubic STO, a = 3.86 Å, to simulate epitaxial growth on a STO substrate. Both SRO and BFO were assumed to be in the tetragonal phase with c/a = 1.01 for bulk SRO and c/a = 1.09 for bulk BFO. The spin-polarized calculations were performed to account for the ferromagnetic state of SRO and the antiferromagnetic state of BFO. For the single domain structure, the polarization was along the [001] direction characterized by the atomic displacements along the z direction. For the tail-to-tail domain-wall structure, the polarization was canted from the [001] direction with a finite [110] polarization component. The oxygen vacancies were modelled by removing one O atom from the central BiO plane. Both the out-of-plane lattice constant and the internal atomic positions were fully relaxed until forces were less than 0.01 eV Å -1 . A 6 × 6 × 1 (10 × 10 × 1) k-point mesh was used for the structural relaxation (self-consistent) calculations.

In-line holography was carried out by using TEM (Titan G2 60-300, FEI) operated at 300 kV with an objective aperture of 10 μm in diameter to select the transmitted beam. A total of 20 bright-field TEM images at defocus values ranging from -1 μm to +1 μm were acquired in 100 nm steps. To minimize the electron-beam damage, the bright-field TEM images were obtained at a magnification of 4.9 × 10 5 under an electron-dose rate of 80 e Å -2 s -1 . The obtained in-line electron holograms were used to reconstruct the phase shift of the transmitted beam by using iterative methodology [21][22][23] .

The I-V curve measurements were performed at room temperature by using a commercial atomic force microscope with a conductive-atomic force microscopy (AFM) mode (PARK NX10). The samples were mounted on a round metal holder by using silver paste and were connected to the internal source electrode of the atomic force microscope. Bias sweeps were applied to the sample from the internal source and the current was measured through commercial silicon tips coated with Pt (Multi75E-G, Budget Sensors) by the internal lock-in units of the atomic force microscope. The bias sweeps were settled as a positive to negative voltage circle, like 0 V to +10 V to -10 V to 0 V, with a time period of 100 s (0.1 V s -1 ) and the currents were measured with a limit of 100 μΑ.

The topography and ferroelectricity were measured by using the AFM and piezoelectric force microscopy modes, respectively, of a commercial scanning probe microscope (Asylum Research MFP-3D) at room temperature. For the piezoelectric force microscopy measurements, Pt coated probes (AC240TM, spring constant of approximately 2 N m -1 , Olympus, Japan) were used under the drive frequency of approximately 270 kHz, a drive amplitude of 1 V and a trigger force of 80 nN (Extended Data Fig. 12). Off-center displacement is defined as a distance from the centre of oxygen octahedra (or Fe atoms) to the center of the Bi lattices. Along with the accumulation of oxygen vacancies, the distortion of oxygen octahedra in BFO is also observed. The positions of oxygen atoms are precisely determined using iDPC-STEM imaging technique. Using a 4-quadrant detector, the iDPC-STEM imaging technique can observe both light and heavy elements with clear contrast. The light elements have a higher signal to dose ratio in iDPC imaging, which makes oxygen visible 43 . In the tail-to-tail state, the distances between the top and bottom oxygen atoms are decreased at the domain wall, showing a clear compression of oxygen octahedra. Accompanied by the compression at the domain wall, oxygen octahedra at the BFO/SRO interfaces are stretched, leading to larger off-centre displacements. The lattice deformation across the domain wall is further confirmed by our DFT calculations (Fig. S7c,d), which demonstrate the quantitative agreement with experimental data. Previous works also indicated that missing oxygen atoms in perovskite unit cells lead to oxygen octahedra to be compressed 44,45 with the change of resistance in in-situ test, which is consistent with the in-situ imaging in Fig. 2a-c. The vertical I-V measurement is performed at room temperature. The bottom and surface electrodes of the sample are silver paste and a 50-nm-silver film (about 0.17 square millimeter in area) prepared by e-beam evaporation, respectively. The DC voltage across the contacts is applied to bias the sample, and then the current is measured by a digital meter (Keithley 2611B).

The sweeping direction of the voltage is from positive to negative and finally returns to the initial state, e.g. from 0 volts to +1 volts to -1 volts and back to 0 volts. We choose AFM test, in-situ test and digital meter test to confirm the current and resistance character in BFO film. AFM probe measurement is used to examine the fine feature of I-V curve. In-situ probe measurement is used to examine the correspondence of resistance change and domain wall dynamic behavior. Digital meter measurement is used to examine the repeatability of I-V character in a large area of sample.