Non‐Destructive Tomographic Nanoscale Imaging of Ferroelectric Domain Walls

Extraordinary physical properties arise at polar interfaces in oxide materials, including the emergence of 2D electron gases, sheet‐superconductivity, and multiferroicity. A special type of polar interface is ferroelectric domain walls, where electronic reconstruction phenomena can be driven by bound charges. Great progress has been achieved in the characterization of such domain walls and, over the last decade, their potential for next‐generation nanotechnology has become clear. Established tomography techniques, however, are either destructive or offer insufficient spatial resolution, creating a pressing demand for 3D imaging compatible with future fabrication processes. Here, non‐destructive tomographic imaging of ferroelectric domain walls is demonstrated using secondary electrons. Utilizing conventional scanning electron microscopy (SEM), the position, orientation, and charge state of hidden domain walls are reconstructed at distances up to several hundreds of nanometers away from the surface. A mathematical model is derived that links the SEM intensity variations at the surface to the local domain wall properties, enabling non‐destructive tomography with good noise tolerance on the timescale of seconds. The SEM‐based approach facilitates high‐throughput screening of materials with functional domain walls and domain‐wall‐based devices, which is essential for monitoring during the production of device architectures and quality control in real‐time.

Ferroelectric domain walls are natural interfaces that separate regions with different polariza�on orienta�on.Because of their dis�nct local symmetry, electrosta�cs, and strain, the domain walls are a rich source for emergent electronic phenomena [1][2][3], including the forma�on of electronic inversion layers [4] and 2D electron gases [5].The func�onal physical proper�es of the domain walls and their ultra-small feature size (down to sub-nanometer width) triggered the idea of developing domain-wall-based nanoelectronics, and different device concepts have been explored [6][7][8][9].Ini�ally, the walls received a lot of aten�on due to their spa�al mobility, allowing to control electric currents by wri�ng, reposi�oning, or erasing domain walls that act as reconfigurable interconnects [10].More recently, spa�ally fixed domain walls moved into focus and it was shown that they can be used to emulate the behavior of electrical components, including digital switches [4] and ACto-DC converters [11].Thus, the domain walls themselves have turned into devices, facilita�ng innova�ve opportuni�es for next-genera�on nanotechnology.
It is established that the polariza�on configura�on at the ferroelectric domain walls plays a key role for their func�onal proper�es [12][13][14][15].Measuring the domain-wall orienta�on that determines the local charge state, however, remains a major challenge.This is because domain walls are o�en not perfectly flat; they can change their orienta�on within the bulk, form three-dimensional (3D) networks, or exhibit complex nanostructures.Furthermore, not all domain walls intersect with the surface and can be orientated parallel to it, which makes them hard to detect.A breakthrough was the advent of nonlinear op�cal methods that enabled imaging of ferroelectric domain walls in 3D [16][17][18][19].Their applica�on, however, is restricted to systems with specific op�cal proper�es and the spa�al resolu�on is in the order of hundreds of nanometers, whereas domainwall roughening and bending o�en occur on much smaller length scales [20][21][22].Tomographic microscopy approaches offer higher resolu�on [23][24][25][26], but data acquisi�on �mes are rather long; most crucially, the established tomography methods for domain-wall imaging are destruc�ve.Thus, they are incompa�ble with fabrica�on processes for future domain-wall devices, which will require the op�on of high-throughput sampling and a non-destruc�ve way for the tes�ng of materials and device architectures.
Here, we demonstrate that otherwise hidden ferroelectric domain walls in surface-near regions can be visualized and analyzed by scanning electron microscopy (SEM), accessing a depth of up to several hundreds of nanometers, as sketched in Figure 1a.Using the uniaxial ferroelectric ErMnO3 as model system [27], we show that domain walls are detectable via characteris�c SEM intensity varia�ons, providing detailed informa�on about the posi�on and charge state of hidden walls.Based on surface and cross-sec�onal data, we derive a general model that relates the measured SEM contrast to the loca�on and orienta�on of domain walls below the surface, allowing to reconstruct their structure with nanoscale spa�al precision.ErMnO3 naturally forms a 3D network with neutral (side-by-side), posi�vely (head-to-head) and nega�vely (tail-to tail) charged domain walls [15].The domain walls have been studied intensively and their fundamental physical proper�es are well understood [4,11,15,23,[29][30][31][32][33][34], which makes the material an ideal model system for this work.At the tail-to-tail domain walls, mobile holes accumulate to screen the bound charges,  b , and give rise to enhanced conductance as shown in Figure 1. Figure 1b presents a conduc�ve atomic force microscopy (cAFM) image, where tail-to-tail walls appear as bright lines, indica�ng an about four �mes higher conductance than the ±P domains they separate.In contrast to the tail-to-tail walls, reduced conductance is observed at head-to-head domain walls (black lines in Figure 1b), owing to a deple�on of hole carriers as explained in detail in ref. [15].The local charge state of the domain walls can be es�mated based on their orienta�on rela�ve to the polariza�on P � �⃗ of the adjacent domains with P 1 ���⃗ =  in domain 1, and P 2 ���⃗ = − in domain 2 (the domain wall normal unit vector n 1 ����⃗ points from domain 2 to domain 1). is the angle between the local wall normal  �⃗ and P 1 ���⃗ , as illustrated in Figure 1b.In this approxima�on, however, the sub-surface structure of the domain wall is neglected, which can lead to substan�al devia�ons between the calculated bound charge and the actual charge density.For example, it was observed that nominally neutral domain walls (i.e.,  = 90°) in ErMnO3 [29] and PbZr0.2Ti0.8O3[35] can exhibit enhanced or even metallic conductance, which was atributed to a non-zero inclina�on angle rela�ve to the surface (Figure 1c).By performing cross-sec�onal experiments on LiNbO3 [36], the impact of the inclina�on angle on the domain wall conductance was demonstrated, revealing that 10-15° �l�ng leads to a substan�al enhancement.In addi�on, the domain wall curvature (Figure 1d) plays an important role as shown by focused ion beam (FIB) based 3D studies on ErMnO3 [23].In summary, these studies highlight the importance of the sub-surface structure of ferroelectric domain walls and the need for adequate characteriza�on methods.An imaging technique that offers great poten�al for domain wall research in ferroelectrics is scanning electron microscopy (SEM).SEM has widely been applied for imaging domains and domain walls of ferroelectric materials, including BaTiO3 [37], Gd2(MoO4)3 [38], LiNbO3 [39], and RMnO3 (R = Y, Er) [40,41].Although SEM is usually considered a surface-sensi�ve technique on account of the shallow escape depth of secondary electrons [42], it is also known that near-surface regions can play a crucial role for the emergent SEM contrast [43,44].
The later provides an as-yet-unexplored opportunity for minimally invasive analysis of the near-surface nanostructure of ferroelectric domain walls and their electronic proper�es.
To explore this possibility, we perform correlated SEM and cAFM measurements on lamellas which we extracted from an ErMnO3 single crystal [45] with a focused ion beam (FIB), applying the same procedure as outlined in ref. [46].Figure 2a and b show representa�ve SEM images gained with the through-lens detector (TLD) detector on lamellas with in-plane and out-of-plane polariza�on, respec�vely.Both lamellas have a thickness of about 1 μm and are mounted on a flat Si-wafer with 100 nm Au coa�ng.For the sample with inplane polariza�on (Figure 2a), domain walls are visible as bright and dark lines.The walls form characteris�c sixfold mee�ng points, corresponding to structural vortex/an�-vortex pairs as explained elsewhere [31,47].A conduc�ve atomic force microscopy (cAFM) image from the region marked by the yellow dashed rectangle in Figure 2a is displayed in Figure 2c, showing the same domain wall patern as the SEM image.Based on the cAFM data, we can iden�fy the bright and dark lines in Figure 2a as conduc�ng tail-to-tail and insula�ng head-to-head domain walls, respec�vely.Going beyond previous studies -which achieved contrast in FIB-cut lamellas only in the high-voltage regime where all walls are conduc�ng [46] -we here access the low-voltage regime, where only the tail-to-tail walls exhibit enhanced conductance [4].The later is an important step, because it demonstrates that domain walls in lamellas and single crystals exhibit the same behavior, i.e., the applied nanostructuring by FIB does not alter the electronic proper�es of the domain walls.Figure 2b and d present analogous measurements for the lamella with out-of-plane polariza�on.Based on the comparison of the cAFM and SEM data, we find that the more conduc�ng -P domains are brighter than the insula�ng +P domains in SEM (see, e.g., refs.[11,48] for details on the polariza�on-dependent transport behavior at the level of the domains).
The data in Figure 2 allows for calibra�ng our SEM measurement, showing that (under the applied imaging condi�ons) bright/dark SEM contrast indicates enhanced/reduced conductance.Note that this calibra�on step is crucial as the domain wall contrast in SEM depends on the imaging parameters and can, e.g., invert depending on the accelera�on voltage [49].
On a closer inspection of the SEM data in Figure 2a, we observe gradual changes in intensity on one side for several of the domain walls.This behavior is presented in Figure 3a, showing a head-to-head domain wall with an asymmetric intensity distribution in the adjacent domains as marked by the blue dashed line.
Occasionally, gradual intensity variations also occur within the domains as seen in Figure 3b.Qualitatively the same features arise in SEM measurements on millimeter-thick single crystals (Figure 3c).Analogous to Figure 3a, several head-to-head domain walls exhibit a distinct contrast on one side (marked by the yellow dashed line in Figure 3c).Furthermore, we observe distinct intensity variations within one of the domains (red dashed line).
These contrast variations cannot be explained based on the nominal charge state of the walls at the surface alone, indicating additional contributions.
To understand the origin of such additional contrast contributions, we use the FIB to cut a cross-section parallel to the white line in Figure 3c as illustrated in the inset to Figure 3d. the SEM data gained on the surface, the cross-sectional measurement shows a conducting tail-to-tail wall (bright) that reaches the surfaces at point A and an insulating head-to-head wall (dark) that surfaces at point B. The head-to-head wall (DW2, yellow dots) has a surface inclination angle of about 25.2° and propagates in the direction in which the gradual contrast change is observed in Figure 3c.Furthermore, the cross-sectional image reveals an additional domain wall in the near-surface region (DW1), as well as several domain walls deeper in the bulk (i.e., ≳ 3 µm away from the surface) that run almost parallel to the surface until they merge in a vortexlike meeting point.Interestingly, we find that DW1 changes its charge state, going from insulating (dark) to conducting (bright), and the respective turning point coincides with the position where we observed the change in SEM intensity on the surface (see Figure 3c and d).These observations indicate a close relationship between the SEM contrast measured at the surface and the (hidden) charged domain walls in the near-surface region.
To relate the intensity measured at the surface to the position and structure of the domain walls in the near-surface region, we build a simple model.Based on the SEM data, within the first order Taylor expansion in  b and multipole-like expansion in , we assume that variations in SEM intensity, ∆, scale with the density of bound charges (∝  b ) and that related effects decrease with increasing distance between the wall and the surface (∝  − ,  ϵ ℕ), leading to ( cos  > 0 and cos  < 0 give tail-to-tail and head-to-head configurations, respectively).To derive the value of , we extract the SEM intensity measured at the surface ( SEM =  0 + ∆) and the parameters  and  from the SEM data in Figure 3c and d, respectively, considering two domain walls (DW1 and DW2) as explained in Supplementary Note 1 and Supplementary Fig. S1.This approach leads us to the conclusion that the experimentally observed dependence of ∆ on the distance between the wall and the surface is reproduced best A possible physical explana�on for this rela�onship is the electrosta�c effect of the domain wall bound charges on the secondary electrons [50].As CASINO simula�ons [51] show, incident primary electrons (E = 1.5 keV at 0° �lt) lose 75% of their energy in the near-surface region with a depth of about 6.6 nm (maximum penetra�on depth ≲ 30 nm).The later implies that the majority of secondary electrons is generated close to the surface, i.e., at a distance comparable to the parameter  that describes the wall-surface distance in our  2 from the bound charges that acts on the secondary electrons via electrosta�c induc�on and, hence, influences the secondary electron yield [50].It is important to note, however, that the SEM contrast forma�on is highly non-trivial in ferroelectrics with mul�ple possible contribu�ons [28]; to clarify the microscopic origin, addi�onal studies are required, which is beyond the scope of this work.
Importantly, our simple model reproduces the experimental data remarkably well (see Supplementary Fig. S2), corrobora�ng that domain wall bound charges play a key role for the intensity distribu�on in SEM measurements.Most interes�ngly, the experiments demonstrate that Equa�on (3) holds for domain walls at distances up to ≈ 1.5 µm away from the surface, which is much larger than the penetra�on depth of the incident primary electrons.This finding reflects an outstanding sensi�vity towards otherwise hidden domain walls and enables nanoscale 3D imaging of domain walls as we discuss in the following.
Figure 4a presents the SEM intensity (orange) measured at the surface above DW1.A zoom-in to the area of interest from which the line plot is generated is shown in the inset to Figure 4a.To derive a mathema�cal representa�on for ∆, we op�mize a low order Taylor expansion of the domain wall shape d(x), u�lizing a basinhopping op�miza�on algorithm.This approach leads to the fit (back line) that is shown along with the SEM data in Figure 4a.The fit captures the main features seen in the SEM data and allows for calcula�ng the domain-wall structure based on Equa�on 3. The result is displayed in Figure 4b, where the black curve represents the reconstructed domain wall.The orange curve corresponds to -values extracted from the cross-sec�onal data (Figure 3d), which is shown for comparison to evaluate the quality of the reconstructed domain-wall structure.
We find that based on the SEM map gained at the surface, we can determine the sign of higher deriva�ves d ()  d () ⁄ , which reveals whether the domain wall curvature is convex or concave in the near-surface region (the accuracy of the Taylor coefficients   is about % ( + 1) 2 • 3%, yielding good precision for the low-order coefficients, which are the most relevant ones for the reconstruc�on).
To go beyond the specific case of DW1, we next consider a hypothe�cal domain wall of arbitrary shape, corresponding to the profile (orange) shown in Figure 4d. Figure 4c presents the calculated SEM intensity (orange) with random Gaussian noise to emulate experimental fluctua�ons.Applying the same approach as for DW1, we fit the noisy intensity data, which leads to the black curve in Figure 4c.Based on this fit, we calculate the domain wall structure using Equa�on (3) (black reconstructed profile in Figure 4d).The reconstructed structure is in excellent agreement with the hypothe�cal domain wall of arbitrary shape that was used as input data, demonstra�ng the general validity of our reconstruc�on approach.

Methods
Sample prepara�on.High-quality ErMnO3 single crystals were grown by using the pressurized floa�ng-zone method [45].The sample was then oriented by Laue diffrac�on, cut into 1 mm thick small pieces with polariza�on direc�ons along one of the surface edges (in-plane polariza�on), and polished using silica slurry.
Lamellae were extracted from the oriented single crystal using a Thermo Fisher Scien�fic G4UX Dual-beam FIB-SEM and subsequently mounted onto a flat Si wafer substrate coated with 100 nm of Au.U�lizing a pre-�lted stub with 45° �l�ng angle, the lamellas were polished using a Ga + ion beam at a glancing angle.The polishing process involved gradually decreasing the ion beam current to 90 pA.Subsequently, a 5kV Ga + ion beam was used to strip away a thin surface layer (about 30 nm), aiming to remove the ion beam damage layer.In addi�on to this established procedure as detailed in ref. [46], a final polishing step was introduced using an Argon ion beam polisher Gatan Precision Ion Polishing system (PIPS II).This procedure was conducted at liquid N2 temperature, using beam voltage of 1 kV, 0.5 keV, and 0.3 keV, sequen�ally, at a beam opera�ng angle of 6°.
Scanning probe and scanning electron microscopy.cAFM measurements were carried out using a Cypher ES environmental AFM (Oxford instruments) with diamond-coated AFM probe �ps CDT-NCHR-10 and HA_HR_DCP as specified in the figure cap�ons.SEM imaging was performed using the same Thermo Fisher Scien�fic G4UX Dual-beam FIB-SEM as for the lamella prepara�on / cross-sec�oning.SEM images were captured by a TLD (Through the-lens detector) detector, using a beam accelera�on voltage close to the charging equilibrium point.
Specific beam parameters for each image can be found in the figure cap�ons.

Domain wall reconstruc�on.
To obtain the domain wall geometry from Figure 3d as shown in Figure 4b, a polygonal chain was ini�ated with a manually drawn guess.The chain was then op�mized by minimizing an appropriately designed cost func�on, composed of three contribu�ons.The first term rewards a maximum contrast between the wall and the background.The second contribu�on compensates for the different background levels within the two adjacent domains.The third term favors straight domain walls and effec�vely sets a lower limit for the curvature radius.The resul�ng curve was used to calculate the domain wall inclina�on angle  and the surface distance .For the domain wall shape reconstruc�on process presented in Figure 4, the domain wall shape was expanded in a fourth order Taylor series () = ∑     4 =0 . The coefficients   , as well as the intensity change prefactor  and the background intensity  0 , were op�mized with a basin-hopping algorithm to yield a similar SEM intensity as in the experiment as: The basin approach is required to avoid trapping in local minima due to the noisy experimental data.In general, fi�ng   and  simultaneously overparameterizes the problem.Under ideal experimental condi�ons this can, in principle, be bypassed by determining the parameter  from calibra�on.Instead, in this study, one point of the domain wall (  ,   ) was determined from the cross-sec�onal data (Figure 3d) and used as an addi�onal constraint on the   coefficients.

Supplementary Note 2: Direct calcula�on of surface intensity
In contrast to the fi�ng approach described in the manuscript, the agreement of the experimental data with the (specified) model developed in Equa�on (3) can be verified in a more direct way.To do so, the expression cos  / 2 is directly calculated from the cross-sec�on data using numerical differen�a�on and the explicit

Figure 1 |
Figure 1 | SEM tomography concept and domain wall structure of the model system ErMnO3.a, Secondary electrons (SE) carry rich informa�on about the electronic material proper�es [28] at the surface and in nearsurface regions.This sensi�vity opens the door for SEM-tomography of ferroelectric domain walls, allowing to reconstruct their posi�on, orienta�on, and charge state based on SEM intensity varia�ons.b, cAFM image gained on the surface of our model system, ErMnO3, with in-plane polariza�on  �⃗ as indicated by the white arrows (acquired with a CDT-NCHR-10 probe �p at a bias voltage of 3 V applied to the back electrode).Tail-totail domain walls exhibit enhanced conductance rela�ve to the bulk (bright), whereas reduced conductance is observed at head-to-head domain walls (black).The local domain wall charge state can be es�mated based on Equa�on (1) by measuring the angle α between the wall normal  1 ����⃗ and the direc�on of  1 ���⃗ .c,d, Illustra�ons showing domain walls in the near-surface region.Domain walls can exhibit different inclina�on angles (c) or pronounced curvature effects (d), which is not visible from surface-sensi�ve measurements alone.

Figure 2 |
Figure 2 | Correlated SEM and cAFM measurements on FIB-cut ErMnO3 lamellas.a, SEM image (2.0 kV, 0.4 nA, TLD) of a lamella with in-plane polariza�on (thickness ≈ 1 μm).Ferroelectric domain walls are visible as bright and dark lines.b, SEM image (2.0 kV, 0.1 nA, TLD) of a lamella with out-of-plane polariza�on (thickness ≈ 1 μm), showing pronounced domain contrast.c, cAFM image recorded in the region marked in a (yellow dashed rectangle).d, cAFM image of the region marked by the red dashed rectangle in b.The cAFM images in c and d are recorded with a doped diamond �p (HA_HR_DCP) and a bias voltage of 22.5 V applied to the back electrode.

Figure 3 |
Figure 3 | Correlated surface and cross-sec�onal SEM data.a,b, Zoom-ins to the SEM image in Figure 2a, presen�ng examples of gradually varying SEM intensity in the vicinity of a domain wall (a) and within a domain (b).c, SEM data (1.5 kV, 0.1 nA, TLD) recorded on the surface of an ErMnO3 single crystal.The image shows qualita�vely similar features as in a and b.Along the red dashed line, a change in contrast is observed within the domain, whereas a gradual change in intensity on one side of the wall is measured in the region marked by the yellow dashed line.Labels A and B correspond to two posi�ons where domain walls intersect with the surface, and white arrows show the polariza�on direc�on within the domains.d, Cross-sec�onal SEM image(2.0 kV, 0.1 nA, TLD) taken a�er FIB cu�ng a trench as sketched in the inset to d. Labels A and B mark the same posi�ons as seen in c.Two domain walls in the near-surface region are highlighted (DW1 and DW2) and key parameters are presented (d distance from the surface,  1 ����⃗ local normal to the domain wall, α angle between  1 ����⃗ and  1 ���⃗ ).

model.𝑞𝑞 1 •𝑞𝑞 2 𝑑𝑑 2 .
One possible physical mechanism that leads to the domain-wall-related SEM contrast is electrosta�c interac�on.Considering secondary electrons and domain wall bound charges as points charges,  1 and  2 , their interac�on is described by Coulomb's law,  ∝ When neglec�ng the impact of the free charge carriers and consider only the bound charge carriers at the domain wall,  2 , this translates into an electric field  = −

Figure 4 |
Figure 4 | Reconstruc�on of the near-surface domain wall geometry from SEM intensity data.a, SEM intensity recorded along the red dashed line (between A and B) shown in the inset and Figure 3c and calculated SEM intensity (black) based on the reconstructed domain wall in b. b, Shape of DW1 (see Figure3dand inset) as measured from the cross sec�on (orange) and reconstructed shape based on the SEM surface intensity (black).c, Simulated SEM intensity of an ar�ficial domain wall of arbitrary shape overlayed with random noise for the reconstruc�on process to simulate random experimental fluctua�ons (orange) and SEM intensity from the reconstructed domain wall shape (black).d, Arbitrarily generated structure of the simulated domain wall (orange, input data) and its reconstructed shape (black).

Figure S1 |
Figure S1 | Determina�on of exponent n based on two datasets for the quan�ta�ve model.Based on the cross sec�on shown in Figure 3d in the main text, the different parameters describing the domain-wall shape are determined for two domain walls, i.e., DW1 (le� panel) and DW2 (right panel).Datapoints that have been excluded for the linear fi�ng process as they aren't part of the linear regime are visualized with open circles.
version of the model developed in the methods sec�on of the manuscript.By adjus�ng  and  0 manually, the qualita�ve agreement of experimental and directly calculated SEM intensity can be tested and is shown for DW1and DW2 in figureS2.This approach confirms the result of the main text that the specified model welldescribes the experimental findings.

Figure S2 |
Figure S2 | Comparison of measured and calculated SEM surface intensity.a, Comparison of the SEM intensity  exp measured along the red dashed line between A and B in Figure 3c in the main text and the calculated intensity  calc for DW1 based on direct numerical calcula�on.b, Same as in a for the yellow dashed line in Figure 3 and DW2 (see also inset to b).