The drive towards miniaturization in ferroelectric materials, crucial for advancements in nanoelectronics 1,2 and neuromorphic computing 3 , has led to the emergence of complex polar domain configurations 1,[4][5][6][7][8][9][10][11] with new functionalities. Intricate patterns of polarization, including skyrmions 12 , hopfions 7 and other topological structures, can arise naturally 13 , be created by adjusting growth conditions [14][15][16][17][18][19] , form through stacking of twisted freestanding layers 20 , be identified in topologically confined ferroelectric domain walls 21,22 , or be induced by local stimuli such as bias pulses 23 , mechanical indentation 24,25 , or heat-induced phase-transitions 26 . In turn, understanding and controlling ferroelectric polarization switching mechanisms at the nanoscale is essential for integrating these polar structures into next generation technologies, highlighting the intersection of fundamental science and practical applications.

For integration into electronic devices, it is crucial that these configurations are rapidly and efficiently manipulated -written, read, and erased locally and on-demand 21 . While magnetic structures 27 have been extensively studied and utilized for similar purposes 28 , achieving full dynamic control over such states in ferroelectrics remains challenging 29,30 . Polar center divergent/convergent and flux-closure domains can naturally form and have been created by application of electric fields 9,23,31 . It is known that the competition between electrostatic and elastic energies is critical for the stability of the core and domain vertices, of such structures 30 . In order to facilitate the stabilization of domains at the micro-or nano-scales, a deep understanding of localswitching mechanisms and the competing energy landscapes is required.

Using thin-film epitaxy and strain engineering, it is possible to produce ferroelectric films featuring periodic nanodomains that self-assemble into larger 'superdomains' with distinct in-plane polarization [32][33][34] , and, here, these structure serve as an ideal platform for exploring these phenomena. Prior studies have demonstrated in-plane polarization switching in such systems through the use of an atomic force microscopy (AFM)-biased tip 32 , wherein it was hypothesized that the slow-scan direction produces a "trailing" electric field responsible for driving local polarization rotation. 35 In this work, we demonstrate that the dynamics of superdomain nucleation and growth are more complex than previously reported.

Using a biased AFM tip, we performed intricate, non-standard movements along arbitrary scan paths -typically not offered by commercial scanning probe systems -to engineer user defined effective in-plane electric fields. This approach enabled the creation of unique spiral-and flowerlike tip trajectories, effectively stabilizing center-divergent/convergent and flux-closure structures in strained Pb0.6Sr0.4TiO3 (PSTO) films. This process revealed a complex in-plane superdomainswitching mechanism (here, referred to as super-switching), involving superdomain nucleation followed by stabilization of superdomain boundaries (or super-boundaries), emphasizing the critical role of the scan path's trajectory in shaping the final superdomain structure.

We have explored the complex nanoscale polarization landscape in the resulting structures regarding the ferroelectric ordering, frustration, non-linear optics, light emission, tunability, and chirality through correlative piezoresponse-force-microscopy (PFM), second-harmonicgeneration-microscopy (SHG), and scanning-electron-microscopy-cathodoluminescence (SEM-CL). Phase-field modelling validates the meta-stability of these structures, corroborating experimental data and shedding light on dipole reorganization within frustrated super-boundaries.

These findings also provide a promising avenue to explore a broader range of ferroelectrictopological states than previously possible by locally controlling the in-plane ferroelectric polarization on-demand. By designing customized tip-trajectory configurations and integrating them in automated experiments (AE), we anticipate this technique will pioneer new frontiers in ferroic nanolithography.

To explore the dynamics of in-plane super-switching, we utilized a 100 nm PSTO thin film. This model ferroelectric material, characterized by a purely in-plane polarization, is grown on a 30 nm SrRuO3 electrode on a DyScO3 (110) single crystal substrate (Methods). The resulting PSTO film features a dense in-plane polarized superdomain structure 33,34 The pristine distribution of superdomains is shown (Figure 1). The net polarization direction of each superdomain is identifiable as perpendicular to the characteristic a1/a2 superlattice stripes present in the lateral PFM images.

Gaining precise control over the local in-plane ferroelectric polarization with an AFM is non-trivial since the electric field from the biased probe in contact with the sample is primarily vertical, but it also possesses lateral components imparting a rotationally invariant applied electric field. Previous research has investigated in-plane super-switching through methods such as pulsing 9,36,37 or raster-scanning 32,38 a biased tip, revealing that it is possible to stabilize single superdomains. The final polar orientation of the superdomains is determined by factors including bias polarity, sample orientation, and the trajectory of the scan path. 32 To understand the microscopic origins of the superdomain formation, we performed a detailed study where the tip voltage was applied exclusively in one direction of the fast-scan motion (either trace or retrace, Figure 2a,b), using standard raster-scan mode. Initially orienting the sample at 45° (such that the slow-scan is aligned along the [110]; Figure 2c), we established that (for this configuration) the slow-scan direction drives the overall polarization of the stabilized superdomain, irrespective of the fast-scan direction chosen for voltage application. In this case, it seems that the slow-scan direction drives the superdomain switching by 'pulling' the average net polarization as if it was a single domain, aligning the characteristic stripes/walls parallel to the fast-scan direction. When the sample is oriented at 0° (such that the slow-scan is aligned along the [010]; Figure 2d), we observed varying polarizations of the stabilized superdomain for the three different scan paths, highlighting that both the fast-and slow-scanning directions influence the super-switching, when scanning at ±45° angles relative to the pristine superlattice stripes. Specifically, the slow-scan direction shapes the Y-axis polarization by aligning the a2-domain family, while the fast-scan direction drives the X-axis polarization, aligning the a1-domain family antiparallel to the fast-scan direction and thus fixing a specific domain-wall orientation. This anticipates that the ferroelectric switching in this symmetry configuration is more complex than expected and probably involves a multistep process composed by several local domain rotations, governed primarily by the hierarchical ordering imposed by the material structure (Supporting Thus, the super-switching process is not solely dependent on the slow-scan direction, as previously reported 32 , challenging the hypothesis of a "trailing" electric field effect. To delve into the switching mechanism, we devise strategic scan trajectories and tailored tip bias waveforms can elucidate the intricacies of such processes, enabling the creation targeted superdomain configurations on-demand, including topological structures such as center-convergent/divergent or flux-closure.

By precisely controlling the movement of the piezo-positioner during scanning, it becomes possible to guide the probe along any predetermined scan path, enabling the creation of specific topological configurations, not possible using existing scan paths (Supplementary Information S5). As shown in the set-up sketch in (Figure 3a), we achieved such precise control by integrating a commercially available AFM system with a field programmable gate array (FPGA), controlled externally with custom python programs (AEcroscopy 39 ). In the experiments described below, we prepare the surface via standard raster-scanning with a biased tip, aligning the polarization in a specific direction over a large area (~100 μm 2 ). This process creates a "blank-slate" in one of the four stable superdomain configurations (I-, I+, II-, and II+; Supplementary Information S6).

For example, we demonstrate the use of a spiral-scan trajectory (Figure 3d) -characterized by a radial slow-scan direction and tangential fast-scan motion -to nucleate and stabilize centerconvergent/divergent structures using negative (Figure 3b,c) and positive (Figure 3e,f) tip biases, respectively. This approach unlocks the combination of the four distinct superdomain orientations into a stable configuration, with the in-plane superdomain dipoles orientated inwards and outwards, respectively, featuring charged internal (and peripheral) super-boundaries. Band excitation lateral PFM (BE-LPFM) reveals the polarization alignment of the generated structure, finding it is not governed by the radial (slow-scanning) direction, since the superdomains are orienting the polarization antiparallel (same direction, opposite sign) to it. As a result, superdomain alignment contrasts with expectations for raster-scanning, where a parallel alignment leading to a center-divergent structure would be expected for a negative bias scan that begins at the center and traverses outwards radially. Instead, a center-convergent structure is stabilized. Conversely, a positive bias, anticipated to produce a center-convergent structure under the same scanning direction, results in a center-divergent structure .

Figure 3: Generation of center-divergent/convergent structures through spiral-scan lithography. a Illustration of the experimental setup used for the measurements: AEcroscopy python package controls an FPGA which inputs the signals to the AFM controller to autonomously perform the previously python-designed experiment. b BE-LPFM piezoresponse (cantilever long axis parallel to [100] crystallographic axis of the sample) of the written center-convergent structure. c BE-VPFM piezoresponse of the written center-convergent structure, grey cross and dot in a circle indicate direction of the out-of-plane components for head-to-head and tail-to-tail. d Spiral tip trajectory for the writing of the center convergent structure. e BE-LPFM piezoresponse (cantilever long axis parallel to [100] crystallographic axis of the sample) of the written centerdivergent structure. f BE-VPFM piezoresponse of the written center-divergent structure, grey cross and dot in a circle indicate direction of the out-of-plane components for head-to-head and tail-totail. g SEM-CL map of the same center-divergent structure. h SHG map of the same center divergent structure with the polarizer at -45° (see green arrow). i SHG map of the same center divergent structure with the polarizer at +45° (see green arrow). j Single SEM-CL spectra at headto-head and tail-to-tail super-boundaries in the locations indicated by black and blue dots in g, respectively.

Interestingly, the corresponding band excitation vertical PFM (BE-VPFM) images (Figure 3c,f) reveal an asymmetric vertical contrast at certain super-boundaries. Head-to-head superboundaries exhibit a positive vertical electromechanical response, while tail-to-tail superboundaries exhibit negative response, suggesting a local polarization tilting towards positive (and negative) out-of-plane direction at such charged locations, respectively. The lack of conductive-AFM contrast at the super-boundaries (Supplementary Information S7) indicates that these charges observed by KPFM are not mobile (Supplementary Information S8).

To explore the structural reorganization effects, we conducted correlative SHG and SEM-CL measurements on the same structures. SHG maps (Figure 3h,i) reveal quadrant-specific excitation in center-divergent structures, dictated by the orientation of polarized light adjusted at ±45°. This approach enables precise mapping of local polarization, complementing BE-LPFM data and providing deeper insight into generated super-boundaries. At a polarizer angle of -45° (Figure 3h), only quadrants aligned with the incident light's polarization yield measurable SHG signals; at +45° (Figure 3i), complementary quadrants display SHG responses along with background.

Notably, the +45° image exhibits enhanced SHG response at head-to-head peripheral super-boundaries, indicating in-plane dipole alignment along the material's [110] axis, alongside observed out-of-plane polarization tilting. SEM-CL measurements over the same area reveal nanoscale variations in color centers and local polarizations (Figure 3g), highlighting imbalanced head-to-head (bright) and tail-to-tail (dark) super-boundaries. The interaction of a focused-electron beam with a ferroelectric material is influenced by luminescent center concentration and localized charge shaping electron beam interactions. The near-infrared CL spectra peaked at 860 nm (Figure 3j) maintains consistent shape but varies in emission intensity, suggesting changes in radiative recombination efficiency or light-emitting species concentration. In PSTO, charged defects, such as vacancies and compensating ions, are presumed to asymmetrically stabilize frustrated superdomain configurations. Positively charged head-to-head super-boundaries promote accumulation of charged defects associated with efficient light emission, resulting in stabilized luminescence that favors carrier trapping and boosts radiative recombination processes in these areas.

To deepen our understanding of the stability and polarization reorientation at the frustrated super-boundaries (Supplementary Information S10-S11), we performed phase-field simulations (Figure 4). We started with initial structures (Figure 4b, c) mirroring the superdomain configurations observed before (Figure 3), and subsequently allow these structures to relax to an equilibrium state under a tensile strain of 2% (Methods). The simulations reveal that the initial configurations indeed relax towards stable arrangements through polarization reorientation to stabilize the high electrostatic/strained super-boundary locations, which are energetically less favorable. Specifically, the tail-to-tail super-boundaries at the center of the center-divergent structures generate a polarization rotation towards the out-of-plane, downward direction (i.e., dark orange, center of Figure 4d). Conversely, at the center of the center-convergent structure, the head-to-head super-boundaries rotate the polarization towards the out-of-plane, upward direction (i.e., light green, center of Figure 4e), in agreement with the BE-VPFM contrast found experimentally (Figure 3). Likewise, in all the center-divergent/convergent structures written on a uniform "blankslate" we find a highly frustrated peripheral super-boundary in one of the four quadrants where the superdomain of the "blank slate" collides with the antiparallel superdomain present in the structure (top-right super-boundary in Figure 4c, bottom-left super-boundary in Figure 4e). In those cases, we also observe polarization rotation inducing an out-of-plane component of the polarization vector, but in this case with additional in-plane component also present (red and light-blue walls in Figure 4c,e, respectively), which we hypothesize is responsible for the enhanced SHG signal (Figure 3i). Such reorientation also induces the formation of 'sawtooth' faceted super-boundaries to minimize the electrostatic/elastic energies (Supplementary Information S10-S11). The simulations indicate that while most of the polarization arrangements are homogeneous through the thickness of the film, the small out-of-plane canting is most prominent near the film surface (cross-sections in Figure 4c,e).

Overall, we determine that it is possible to stabilize out-of-plane polarization at the superboundaries of PSTO through the right accumulation of elastic stress and electrostatic energy, and that this material stabilizes upwards out-of-plane polarization to accommodate the head-to-head super-boundaries and conversely, downward polarization to accommodate the tail-to-tail superboundaries. Moreover, we suggest that indeed, the in-plane local polarization rotation steps necessary to switch the a1/a2 domains may happen through an intermediate out-of-plane rotation 40,41 , which would be mediated by the tip vertical electric field, thus becoming the dominant agent of the in-plane super-switching (Supplementary Information S11).

As such, we can now revisit the hierarchical super-switching mechanism. The spiral-scan starts in the central point where the vertical electric field nucleates a local out-of-plane polarization (upwards for negative bias and downwards for positive bias) surrounded by a convenient superdomain arrangement with artificial super-boundaries (head-to-head for negative bias and tailto-tail for positive bias). In the absence of a scan path, such nucleation centers become unstable due to their small area-to-boundary ratio, and quickly relax when scanned over (Supplementary Information S12). Spiral-scanning stabilizes these centers, expanding the super-boundaries via out-of-plane intermediate switching by creating a1/a2 stripe domains which are as parallel as possible to the fast-scan direction (tangential) and align appropriately with the nucleated superdomain configuration, generating a global orientation antiparallel to the radial direction.

Therefore, we speculate that in this case the stabilized final polarization arrangement is dictated by the sign of the initially nucleated domain in the center, and not by the slow-scan direction, challenging the trailing electric field theory 32,42 .

We employ more exotic scan paths, to generate various topological states, by combining the bias polarity for nucleation, and the propensity of the stripe domains to form tangential to the fast-scan axis to form quadrants that realize flux-closure structures. Accordingly, we design a flower-like scan 39,43 (Figure 5a) and explore different permutations. The BE-LPFM of a flux-closure structure (Figure 5b), written using a negative tip bias on a "blank-slate" area with uniform II-superdomain, shows that it is possible to merge the four permissible superdomain orientations into a stable fluxclosure structure, where in-plane dipoles rotate anti-clockwise. The corresponding BE-VPFM of the same structure (Figure 5c), reveals vertical contrast at specific super-boundaries, akin to the center-divergent/convergent structures (i.e., out-of-plane tilting upwards for head-to-head and downwards for tail-to-tail super-boundaries). Notably, the internal super-boundaries form a swirl-like shape (white-dashed line, Figure 5c) diverging from the characteristic cross-like shape typical of the center-divergent/convergent structures. Furthermore, we prove how bias polarity determines the sign of the initial nucleation, generating anticlockwise and clockwise flux-closure domain arrangements with positive and negative biases, respectively (Figure 5d,e). We further explore the effect of the scan path's chirality (Figure 5f), a mathematical concept based on a structure's distinguishability from its mirror image, critical for stabilizing polar topologies 26,44 . In this case (Figure 5g), the slow-scan direction does modify the overall orientation of the generated structure, adding an additional degree of complexity to the stabilization of clockwise or anti-clockwise configurations. This effect can be explained by considering all the observations of the super-switching mechanism described in previous sections and summarized elsewhere (Supplementary to where the area-to-boundary threshold for stabilization of smaller size structures is still fulfilled.

Ramping writing bias magnitude, (Figure 6j), reveals a coercive voltage between 4V and 6V (Figure 6k). Finally, spiral-scan trajectories in clockwise or anti-clockwise directions (Figure 6l) reveal no effect on the stabilized structure's handedness (Figure 6m,n).

Figure 6: Tunability of the center-convergent/divergent states. a Scan bias polarity used to write the center-divergent and -convergent states. b BE-LPFM piezoresponse and c BE-VPFM piezoresponse of the structures written using the parameters in a. d Scan paths with different sparsity. e BE-LPFM piezoresponse and f BE-VPFM piezoresponse of the structures written using the parameters in d. g Scan path with different size. h BE-LPFM piezoresponse and i BE-VPFM piezoresponse of the structures written using the parameters in g. j Scan path with different writing bias magnitude. k BE-LPFM piezoresponse of the structures written using the parameters in j. l Scan path with different chirality/handedness. m BE-LPFM piezoresponse of the written structures with positive bias using the scan paths in l. n BE-LPFM piezoresponse of the written structures with negative bias using the scan paths in l.

The number of complex topologies that can be generated using this approach is not limited to the ones shown here. Supplementary Information S15 shows the generation of shelled center convergent/divergent structures as an additional example. Supplementary Information S16

shows the generation of snake-like domains exhibiting a "sawtooth" polarization pattern. The relevance of this work relies on the full understanding and subsequent manipulation of superswitching using a biased AFM tip. Combining this knowledge with powerful AE 39,43 (Supplementary Information S17 shows the implementation of quadrant-dependent bias during the writing process during the generation of square center-divergent/convergent) and machine learning methods, it is readily possible to rapidly design and produce different complex polar topologies or optimize/tune experimental parameters with a desired goal (Supplementary Information S18).

Our method has unveiled the intricacies of the super-switching mechanism in in-plane polarized ferroelectric films of PSTO, ruling out the "trailing" field explanation and proposing the appearance of an intermediate out-of-plane state driven by the vertical component of the electric field. This insight has enabled the precise, local manipulation of superdomain structures in an ondemand fashion. By adjusting the scan-path trajectory and other experimental parameters, we can create a plethora of polar topological structures, examine the delicate balance between elastic and electrostatic energies, and study the accommodation of frustrated super-boundaries. Furthermore, correlative microscopy techniques gather the detailed nanometric information needed to inform the phase-field models for evaluating stability and polarization reorganization. Through the design of customized tip-trajectory configurations and their implementation in AE, this work forges a new frontier in nanolithography for ferroics, enabling the exploration of significantly more ferroelectric topological states than previously were capable of being synthesized. custom LabVIEW/Python interface. This interface directs the tip's movement along any desired trajectory, by inputing the desired waveforms directly to the XY piezoscanners of the commercial AFM (see Supplementary Information S4 and other works 45,46 ). The tips utilized are commercially available Multi75-G (BudgetSensors) with a free resonance frequency of 75 kHz and a spring constant of 3 N m-1.

Reading the topological structures BE-PFM was performed using external data acquisition electronics based on a NI-6115 fast DAQ card controlled by custom-built LabVIEW software. A chirp voltage signal around the 1 st cantilever contact resonance frequency (≈350kHz) or the torsional contact resonance frequency (≈650kHz), was applied to the AFM probe while the sample was grounded, for vertical and lateral polarization reading, respectively. The PFM experimental data were stored in h5 files and post processed using the python pycroscopy package 47 . LPFM images at orthogonal cantilever orientations are shown in Supplementary Information S20, where the different in-plane components can be studied for the same area of the sample.

To perform the phase-field simulations, we selected PbTiO3 as a ferroelectric thin film for our model. We employed a polarization vector 𝑃 𝑖 = (𝑃 𝑥 , 𝑃 𝑦 , 𝑃 𝑧 ) as an order parameter to denote the state of polarization along x, y, and z-directions in the global coordinates system. The total free energy density of the system can be expressed by the following equation, 𝐹 = ∫ {𝑓 𝑔𝑟𝑎𝑑 (𝑃 𝑖,𝑗 ) + 𝑓 𝑏𝑢𝑙𝑘 (𝑃 𝑖 ) + 𝑓 𝑒𝑙𝑒𝑐 (𝑃 𝑖 , 𝐸 𝑖 ) + 𝑓 𝑒𝑙𝑎𝑠 (𝑃 𝑖 , 𝜀 𝑖𝑗 )} 𝑉 𝑑𝑉 (1) Where 𝑓 𝑒𝑙𝑎𝑠 , 𝑓 𝑔𝑟𝑎𝑑 , 𝑓 𝑏𝑢𝑙𝑘 , and 𝑓 𝑒𝑙𝑒𝑐 denote elastic energy, gradient energy, bulk energy, and electrostatic energy, respectively. 𝑉 is the total volume of the system. 𝐸 𝑖 and 𝜀 𝑖𝑗 are the components of the electric field and elastic strain, respectively. A detailed explanation of each energy density in Equation ( 1) can be found in the literature 48 .

The temporal evolution of the polarization vector 𝑃 𝑖 is calculated by solving the time-dependent Landau-Ginzburg-Devonshire (LGD) equation,

Where 𝑡 is time, 𝑥 is the spatial position, and 𝐿 is a kinetic coefficient related to the mobility of the domain wall.

The simulation was done using a realistic 3D geometry and the system size was chosen to be 256∆𝑥 × 256∆𝑦 × 32∆𝑧, with ∆𝑥 = ∆𝑦 = ∆𝑧 = 1𝑛𝑚. The thickness of the substrate, air, and film are 10 𝑛𝑚, 2 𝑛𝑚, and 20 𝑛𝑚, respectively. Substrate strain was set at 2% to stabilize the structure properly, and the simulation was performed at room temperature. Periodic boundary conditions were applied along in-plane directions (x and y), whereas proper mechanical and electrical boundary conditions were applied along out-of-plane (z) directions. The isotropic relative dielectric coefficient 𝑘 𝑖𝑖 is set to be 50, while the gradient energy coefficients are chosen to be 𝐺 11 𝐺 110 = 0.6 , where 𝐺 110 = 1.73 × 10 -10 𝐶 -2 𝑚 4 𝑁 . Other constants such as Landau coefficients, elastic-compliance, and electrostrictive coefficients are taken from the literature 49 .

In SHG, a linearly polarized laser light at a frequency ω is focused onto the sample and the reflected light at the frequency 2ω is detected accounting for the non-linear light-matter interactions occurring in non-centrosymmetric materials. By raster-scanning the excitation laser and collecting spectra at each location, spatial maps of the polarization orientation relative to the incident excitation polarization can be obtained. The SHG measurements were conducted using a 100 fs Ti:Sapphire laser (Mai Tai HP, Spectra Physics) at 800 nm and 80 MHz repetition rate. The laser beam was passed through a half-wave plate mounted in a rotation stage and was directed into an upright microscope (Olympus) and focused onto a sample surface using a 100x microscope objective (Numerical Aperture: NA=0.9) to a ~1.1 m diameter spot. The laser energy at the sample surface was 0.8 mW. The SHG light was collected in backscattering configuration using the same objective and was directed to a monochromator (Spectra Pro 2300i, Acton, f = 0.3 m) that was coupled to the microscope and equipped with a 150 grooves/mm grating and a CCD camera (Pixis 256BR, Princeton Instruments). Before entering the monochromator, the SHG light was passed through a short-pass cut-off filter (650 nm) to filter out the fundamental excitation light at 800 nm and a polarizer to select SHG polarization colinear or cross relative to the polarization of the excitation light. A motorized computer controlled XY microscope stage (Marzhauser) with minimum scanning steps of 100 nm was used to perform SHG mapping. Polarization resolved measurements were performed by inserting a computer controlled rotational stage with a broadband half-wave plate into excitation and back-scattered SHG beam paths.

Cathodoluminescence (CL) spectra were obtained using an FEI Quattro environmental SEM equipped with a Delmic Sparc CL collection module. This setup enabled concurrent acquisition of secondary electron signals and CL spectra for spatially resolved analyses. Measurements were performed with an electron accelerating voltage of 5 kV and a beam current of 230 pA over a square scan area of 7.2x7.2 µm^2 with a chosen pixel density of 125x125.