Unconventional Ferroelectric Switching via Local Domain Wall Motion in Multiferroic ε‐Fe2O3 Films

Deterministic polarization reversal in ferroelectric and multiferroic films is critical for their exploitation in nanoelectronic devices. While ferroelectricity has been studied for nearly a century, major discrepancies in the reported values of coercive fields and saturation polarization persist in literature for many materials. This raises questions about the atomic‐scale mechanisms behind polarization reversal. Unconventional ferroelectric switching in ε‐Fe2O3 films, a material that combines ferrimagnetism and ferroelectricity at room temperature, is reported. High‐resolution in situ scanning transmission electron microscopy experiments and first‐principles calculations demonstrate that polarization reversal in ε‐Fe2O3 occurs around pre‐existing domain walls only, triggering local domain wall motion in moderate electric fields of 250–500 kV cm−1. Calculations indicate that the activation barrier for switching at domain walls is nearly a quarter of that corresponding to the most likely transition paths inside ε‐Fe2O3 domains. Moreover, domain walls provide symmetry lowering of the polar structure near the domain boundary, which is shown to be necessary for ferroelectric switching in ε‐Fe2O3. Local polarization reversal in ε‐Fe2O3 limits the macroscopic ferroelectric response and offers important hints on how to tailor ferroelectric properties by domain structure design in other relevant ferroelectric materials.


Introduction
A spontaneous polarization which can be switched by applying an electric field is a hallmark of ferroelectrics.While the spontaneous polarization of a crystal is bestowed by its point group symmetry (provided it belongs to the so-called polar classes), [1] its switchability is a far more intricate matter.Indeed, polar materials which cannot be switched, like tourmaline, were already known as pyroelectrics in ancient times because, when heated or cooled, their temperature-dependent spontaneous electric dipole moments give rise to phenomena of electrostatic attraction.In contrast, ferroelectricity was not discovered until 1920, [2] being belated by an essential feature of ferroelectrics; the existence of domains with opposite polarizations, which in as-prepared crystals results in a zero net polarization. [3]The response of ferroelectric domains under applied electric fields determines the characteristics of the polarization versus electric field hysteresis loops which are the quintessence of ferroelectrics and what makes them appealing for most applications.Despite intensive research efforts, we do not have, and might never have, a theory of polarization switching which encompasses all ferroelectrics.Since electric dipoles arise from the relative displacements of positive and negative ions in a crystal and changing their orientation involves a costly deformation of the crystal lattice in terms of elastic energy, the specific structural details can strongly affect the switching mechanism.For instance, many polarization reversal studies have focused on perovskites such as BaTiO3.In this tetragonal structure, the reversal is easily explained by displacements of Ti 4+ ions above or below the center of its coordination octahedra.However, in BiFeO3, another perovskite with a rhombohedral distortion that has received much attention during the last decade, the polarization switching follows a non-obvious two-step path. [4]To complicate matters further, ferroelectric switching can proceed through the growth of existing domains or with the nucleation and growth of new domains and is strongly affected by the domain geometry or the microstructure of the material.
In this context, it is not surprising that the development path of new ferroelectric oxides beyond the prototypical perovskite family is often not a smooth one.This is particularly true in the case of ferroelectrics for which the energy cost of polarization switching is substantially larger compared to the perovskites, with coercive fields (Ec) in the order of MV cm -1 instead of tens to hundreds of kV cm -1 .This is the case for LiNbO3, which was thought to be nonswitchable [5] until its ferroelectricity was demonstrated on very thin specimens, [6] and, more recently, doped Hafnia (Hf0.5Zr0.5O2). [7]Another example is GaFeO3, for a long time considered non-ferroelectric until it was prepared in the form of thin films and switched.The fact that GaFeO3 exhibits a Pna21 polar structure up to high temperatures (above 1350 K) [8] indicates its high stability from which large coercive fields are to be expected.However, most ferroelectricity studies on GaFeO3-type films [9][10][11][12][13][14][15][16][17] report small to moderate Ec not larger than 500 kV cm -1 with a significant dispersion in the values of remanent polarization (Pr) ranging from 0.2 µC cm -2 [13] to 8 µC cm -2 . [16]0][21][22] Multiferroic ε-Fe2O3 is isostructural to GaFeO3 and has attracted interest in recent years because it combines robust ferrimagnetism and ferroelectricity at room temperature. [23,24]While experiments have demonstrated reversible ferroelectric switching in ε-Fe2O3 films at room temperature, [24] the atomic-scale mechanism behind polarization reversal is still unclear.Moreover, as in the case of GaFeO3, there are large discrepancies in the reported values of Ec, which are as low as 60 kV cm -1 (PUND method) [24] and as large as 800 kV cm -1 (piezoelectric force microscopy). [16]Here, we provide an alternative explanation for the low Pr values reported on GaFeO3-type ferroelectric films by using in-situ STEM on ε-Fe2O3 films.Our measurements reveal partial switching of the domains at small electric fields which is only localized around pre-existing domain walls.We propose a mechanism of polarization reversal from firstprinciples calculations.This mechanism highlights the importance of the initial domain structure for the ferroelectric response of GaFeO3-type films and indicates the potential of domain structure design as a tool to lower the polarization reversal barrier or enhance the saturation polarization.In particular, our results can be important to the understanding of relevant ferroelectrics characterized by large coercive electric fields such as LiNbO3 or Hafnia.

In-Situ STEM Experiments
We performed in-situ STEM experiments on 50-nm-thick ε-Fe2O3 films.The films were grown by pulsed laser deposition (PLD) on an isostructural 8-nm-thick AlFeO3 (AFO) buffer layer and a conducting (111)-oriented Nb-doped SrTiO3 (STO) substrate (see Appendix A for details).
We mainly utilized STEM with a high-angle annular dark-field (HAADF) detector.This imaging mode allows recording of incoherent Z-contrast images, in which the brightness of an atomic column is approximately proportional to the square of the average atomic number (Z). [25]nce, Fe ions are resolved clearly.Figure 1 shows HAADF-STEM cross-sectional images and a selected area electron diffraction (SAED) pattern of an ε-Fe2O3 film on our in-situ specimen holder with a piezo-controlled metallic tip.The ε-Fe2O3 and AFO films are epitaxial and their out-of-plane orientation relation with the STO substrate corresponds to (001)ε-Fe2O3//(001)AFO//(111)STO (Figure 1b).The polar c axis of the ε-Fe2O3 Pna21 structure is thus oriented perpendicular to the film plane.The splitting of the film and substrate diffraction spots, as illustrated by the dashed yellow circle in Figure 1b, indicates relaxation of the ε-Fe2O3 lattice with respect to the STO substrate.The microstructure of the ε-Fe2O3 film consists of alternating columnar domains with a width ranging from a few to tens of nanometers (Figure 1c).Fast Fourier transform (FFT) patterns and electron diffraction simulations (Figure S1, Supporting Information) indicate that the domains are oriented along the ε-Fe2O3 [100] and [110] crystal axes.Diffraction spots of these domains, which we label D1 and D2, are marked by a yellow diamond and red square in the SAED pattern of Figure 1b.The formation of structural domains with approximately equal probability is anticipated because of the pseudo six-fold symmetry in the (004) planes of AFO and ε-Fe2O3. [24,26]e unit cell of the ε-Fe2O3 Pna21 structure consists of four non-equivalent Fe 3+ sites, as schematically illustrated in Figure 2a for a projection along the [100] crystal axis (D1 domain).
The ions labeled Fe1 and Fe2 exhibit a distorted octahedral coordination, while Fe4 ions are tetrahedral.Fe3 ions with regular octahedral coordination are located in the same plane as Fe2 ions, with slightly different coordinates along [001].Because of this alignment in the [100]   projection, it is possible to measure vertical displacements between Fe2 ions and Fe3 ions with high precision.For this reason, we focus our attention on the row with two Fe3 ions in the center and two Fe2 ions at the edges.In the polar Pna21 phase, the Fe2 ions are positioned either slightly above or below the Fe3 ions, depending on the direction of ferroelectric polarization (see structural models and HAADF-STEM images in Figure 2a).We refer to these two ionic configurations as "smile" and "cry", respectively.Figure 2b shows a high-resolution HAADF-STEM image of a D1 domain in the ε-Fe2O3 film.In the upper part of the image, the Fe2 and Fe3 ions form a smile structure, whereas cry configurations are measured in the lower part.For clarity, we colored the two polar domains yellow and blue (an uncolored image is shown in Figure S2 of the Supporting Information).In the middle of the image (grey area), the Fe2 and Fe3 ions are positioned on the same horizontal line within measurement error.This alignment suggests that the ε-Fe2O3 Pna21 unit cell has transformed into a centrosymmetric structure.
Hence, we identify the grey area as a non-polar domain wall.To derive the vertical position of the Fe2 ions with respect to the Fe3 ions inside the polar domains, we used a 2D Gaussian fitting procedure.Averaging over 10 unit cells, we find dFe2 -Fe3 = 6.1  2.9 pm for the domain with smiles.A similar value but with opposite polarity is derived for the domain with cries (dFe2 -Fe3 = -8.5  3.5 pm).
In order to image vertical displacements between Fe2 and Fe3 ions under an electric field, we contacted the ε-Fe2O3 film by the piezo-controlled metallic tip of our in-situ probing holder (Figure 1a).In the in-situ measurements, we grounded the tip and applied triangular voltage pulses with a duration of 0.1 s to a contact on the Nb-doped STO substrate (see Methods Section for details).The pulse amplitude was swept and high-resolution images were recorded after applying voltages with opposite polarity.From a careful analysis of the HAADF-STEM images, we conclude that the Fe2 and Fe3 ions do not measurably shift inside the domains, i.e., away from the domain wall, up to a pulse amplitude of 5 volts.In the immediate vicinity of the wall, however, we observe reversible transpositions of Fe ions, turning smiles into cries and vice versa.Figure 3a shows snapshots from this experiment (full images are shown in Figure S3 of the Supporting Information).We focus on one unit cell above and another below the domain wall (marked by red and cyan rectangles in Figure 2b) and plot the relative positions of the two Fe2 ions with respect to the mean position of the two Fe3 ions (i.e.dFe2 -Fe3).Applying a +3 V pulse turns the initial smile into a cry, while the initial cry remains practically unaltered.In contrast, negative voltage pulses turn cries into smiles.Since positive (negative) pulses move the positively charged Fe 3+ ions up (down), our in-situ results imply that the Fe3 ions move more than the Fe2 ions in an applied electric field.Moreover, from the bias polarity we derive that the ferroelectric polarization points up in the cry domain and down in the smile domain.
The relative vertical displacement of the Fe2 and Fe3 ions near the domain boundary is consistent with the magnitude of smiles and cries inside the domains.This observation suggests that applying 1.5 -3.0 V, corresponding to an electric field of 250 -500 kV cm -1 , is sufficient to saturate the local polarization, in agreement with the polarization versus electric field hysteresis loop previously reported for this film. [24]t higher bias voltages, local polarization reversal leads to lateral motion of the domain wall.We illustrate this effect in Figure 4.The depicted HAADF-STEM images show how the cry domain with upward polarization grows at the expense of the smile domain when we apply positive voltage pulses of increasing amplitude to the Nb-doped STO substrate.This behavior, driven by local polarization reversal at the domain wall, produces a macroscopic ferroelectric response.Since ferroelectric switching does not encompass the entire sample (Figure S4, Supporting Information), we expect the magnitude of the macroscopic signal to be small compared to predictions based on first-principles calculations.The restriction of polarization reversal to the vicinity of pre-existing domain walls might explain existing discrepancies between theory and experiments on ε-Fe2O3 and possibly GaFeO3 in literature.

A. Ferroelectric Switching without Pre-Existing Domain Walls
[20][21][22] Our in-situ STEM experiments on ε-Fe2O3 films, however, indicate that ferroelectric switching occurs near domain walls at moderate electric fields.[29][30] Based on this analysis, we found a low-energy metastable centrosymmetric phase with monoclinic P2/c symmetry (Table 1).The P2/c phase is an elastically relaxed extension of the Pbcn structure (the latter was recently obtained by K. Xu et al. using a stochastic surface walking method [31] ) with frozen distortions according to the Γ4 + irreducible representation of Pbcn (Figure 5ad).We estimate the potential barrier for ferroelectric switching via the Pbcn reference structure at 86 meV/f.u.(similar to 85 meV/f.u.obtained in Ref. 31), whereas ferroelectric switching inside a ε-Fe2O3 domain via the elastically relaxed P2/c structure results in a significantly lower energy barrier of 59 meV/f.u.(Figure 5e).
An examination of the Γ-point phonon structure reveals a dynamic instability in the Pbcn phase, with one polar phonon of imaginary frequency.Figures 5f and g show the eigendisplacements of Fe ions for the unstable phonon mode in Pbcn.Clearly, the eigendisplacements produce smile features similar to those in the STEM images of Figure 24, with Fe3 ions shifting down and Fe2 ions shifting up along the [001] direction.Freezing of the unstable polar mode lowers the symmetry of the reference structure from non-polar Pbcn to polar Pna21.A decomposition of real ion displacements during these structural conversions reveals the main contribution of the low energy 3 -phonon, i.e., its eigenvector completely defines the transition path for polarization switching.Although the phonon develops some polarization along the [001] direction in the reference non-polar structures (Figure 5f), ions mainly displace along the [100] direction, producing large antipolar components in neighboring chains (running along [001]) of Fe ions (Figure 5g).Consequently, polarization switching along [001] is accompanied by significant antipolar displacements of a large number of ions along [100] (see Figure 5h).Since the displacements of Fe ions along the in-plane directions are not stabilized against thermal fluctuations by a perpendicular electric field, this fact could explain why switching within the ε-Fe2O3 domains is not observed in our in-situ STEM experiments.
In contrast to Pbcn, all Γ-phonons are real in the P2/c structure.The eigenvector of the low-energy polar phonon mode in the P2/c phase exhibits all the main features of the imaginary mode vector of Pbcn.Consequently, the eigendisplacements of Fe ions are comparable for both structures.Decomposition of the real polar displacements in terms of eigendisplacements of the non-polar reference structure reveals the principal contribution of the polar 2 -mode together with the full-symmetry 1 + mode (Figure 5d).Coupling of these two modes produces a triplewell potential (Figure 5e), similar to the one in monoclinic Sn2P2S6 uniaxial ferroelectrics. [32] the P2/c structure, the Fe1 sublattice of the Pbcn structure with tetrahedral coordination splits into two sublattices, one being tetrahedrally and the other octahedrally coordinated by oxygen ions (Figure 5a).Transformation to a polar phase from P2/c is therefore simpler than from Pbcn because it involves asymmetric ion displacements in only half of the unit cells.To develop a smile feature starting from the P2/c structure (indicated by purple spheres in Figure 5i), the largest displacements are performed by Fe ions with positions z  0.5 (compare distances between purple and yellow spheres), whereas Fe ions with z  0.5 displace most during the formation of a cry feature (compare distances between purple and blue spheres).This displacement pattern is consistent with the eigendisplacements of the phonon mode with lowest energy.As a result, the non-polar P2/c structure transforms to the polar Pc structure.However, because elastic relaxations decrease the monoclinic angle to zero, the system finally settles in the Pna21 orthorhombic polar phase.Therefore, in order to switch the polarization via the P2/c reference structure one has to break the symmetry in such a way that the angle between the a and c axes of Pna21 becomes monoclinic, which requires an 5 shear strain component.
In summary, one can formulate several reasons why ferroelectric switching inside -Fe2O3 domains is not observed in electric fields of up to 2 MV cm -1 (maximum electric field in our in-situ STEM experiments): (i) The low-energy path for switching goes via the non-polar P2/c state.Since this transition state has lower symmetry then the initial Pna21 polar phase, the symmetry of Pna21 needs to be lowered during switching.In particular, this requires breaking of the glide plane symmetry orthogonal to [100].This could be achieved by an elastic deformation 5 via the piezoelectric tensor component d51, which we found to be significant in -Fe2O3 (see Table 2).However, in this scenario the electric field should have a [100] in-plane component, which is not the case for the out-of-plane geometry used in our experiment.(ii) One can consider Pbcn as a virtual centrosymmetric reference structure with a higher transition barrier than P2/c, but also in this scenario the main component of the phonon mode that drives the transition lies along [100], i.e., perpendicular to the applied electric field.The out-of-phase ion displacements along [100]

B. Ferroelectric Switching with Pre-Existing Domain Walls
Next, we discuss the role of domain walls in ferroelectric switching.The metastable non-polar P2/c structure could exist as an extended phase or appear as a non-polar domain wall between ferroelectric domains with opposite polarizations.As can be seen from the triple-well potential for switching via the P2/c path (Figure 5e), the barrier from the non-polar to the polar phase is very small (~1 meV/f.u.).Hence, the P2/c state is not stable in its free-standing form.However, the structure could be realized between head-to-head or tail-to-tail domains, as observed experimentally in the STEM image of Figure 2b (domain stacking along the [001] direction).
Because of its significant monoclinic distortion (see Table 1), it is unlikely that the P2/c phase separates orthorhombic domains that are stacked along the [010] direction and polarized along [001].In this geometry, which we observe also in experiments (Figure 4), we expect the domain wall to have an orthorhombic structure, probably with lower symmetry than Pbcn.
To analyze the effect of domain walls on ferroelectric switching, we consider a vertical wall separating two polar Pna21 domains wherein the polarization points down (smiles) and up (cries), respectively (Figure 6a).In contrast to switching via Pbcn and P2/c structures inside a ε-Fe2O3 domain where Fe1 and Fe4 ions occupy octahedral and tetrahedral sites in the initial Pna21 phase, respectively (Figure 5c), the unit cell of the modeled domain wall is altered by cross-substituting one tetrahedrally coordinated Fe4 ion with an octahedral Fe ion and one octahedrally coordinated Fe1 ion with a tetrahedral Fe ion (see blue ions in Figure 6a).These substitutions break the glide plane symmetry orthogonal to the [100] axis, symmetrize the hexagon formed by the Fe2-Fe4-Fe4-Fe2-Fe1-Fe1 ions, and align the Fe2 and Fe3 ions along the [001] directions within the domain wall.HAADF-STEM simulations of this domain wall configuration shown in Figure S5 of the Supporting Information indicate that smile and cry features are indeed not expected to appear in the domain wall, in agreement with the STEM images of Figures 2 and 4. By moving the domain as a whole unit along the [010] direction, we estimate an energy barrier of ~22 meV/f.u., which is significantly lower than that estimated for ferroelectric switching inside a ε-Fe2O3 domain (Figure 5e).Detailed analysis of the calculations shows that one of the octahedral Fe1 ions swaps to a tetrahedral surrounding and one of the tetrahedral Fe4 ions swaps to an octahedral surrounding during the first stages of lateral domain wall translation (Figure 6af).Next, the second Fe1 and Fe4 ions undergo the same coordination modifications.Coincidently, the cry feature transforms into a smile feature via an intermediate phase by vertical transpositions of Fe2 and Fe3 ions if the domain wall moves from left to right.This observation is in excellent agreement with our in-situ STEM measurements of voltage-induced motion of non-polar domain walls (Figure 4). Figure 6g shows the displacements of Fe ions along the [100] direction across a domain wall.The Fe2 and Fe3 ions displace gradually in the vicinity of the wall, undergoing only small transpositions upon domain wall motion.From the eight Fe ions taking part in ferroelectric switching via lateral wall motion, the ions on the Fe1 and Fe4 sites displace most.Their displacement is ~0.4 Å, which is only half compared to switching inside a ε-Fe2O3 domain.Consequently, the energy barrier of domain wall assisted switching is significantly smaller than that of switching inside ε-Fe2O3 domains if the walls pre-exist.If the domain walls need to form first, an extra price has to be paid.For a discussion on the influence of lattice deformations on the energy barrier of ferroelectric switching, we refer to Figures S6 and S7 of the Supporting Information.
In addition to the vertical domain wall, we considered an inclined domain wall (Figure S8, Supporting Information), which is also visible in the STEM images (see Figure 4).As in the vertical wall, the domains are stacked along the [010] direction, but the inclined wall is not orthogonal to [010].We find that in contrast to the vertical domain wall, the inclined domain wall preserves the orthogonal to [100] glide plane symmetry.Therefore, taking into account the above mentioned arguments, we do not expect that this wall plays a role in polarization switching.Unfortunately, the estimated formation energy of the inclined wall is significantly smaller than that of the vertical wall, Ef inclined = 234 mJ m -2 versus Ef vertical = 370 mJ m -2 .This could lead to a transformation of vertical walls into inclined walls during switching, which would result in the disappearance of the ferroelectric response.In both cases, the formation energy is much higher than in the conventional ferroelectrics (e.g.~10 mJ m -2 in BaTiO3 [33] ).
It is even higher than the most energy expensive 180º domain wall in PbTiO3 (~169 mJ m -2 [33] ).This makes the nucleation of domains with opposite polarization very difficult and explains the lack of ferroelectric switching inside -Fe2O3 domains.To generate a ferroelectric response in ε-Fe2O3, one has to grow thin films with pre-engineered domain walls.

Conclusions
We combined in-situ STEM experiments and first-principles calculations to elucidate the origin of ferroelectric switching in multiferroic ε-Fe2O3 films.We find that ferroelectric switching proceeds locally via the motion of non-polar boundaries.During switching, the polar state of the Pna21 structure transforms via an intermediate non-polar P2/c or Pbcn structure.
Since the energy barrier of this phase transition is reduced at domain boundaries, we experimentally measure a local ferroelectric response at moderate electric fields of 250 -500 kV cm -2 .Ferroelectric switching by a perpendicular electric field is prevented inside domains because the activation barrier is larger and it involves significant antipolar ionic displacements within the film plane.The local nature of polarization reversal provides a viable explanation for discrepancies in the reported values of coercive field and saturation magnetization for ε-Fe2O3 and isostructural GaFeO3.We argue that similar atomic-scale mechanisms may also be at play in other relevant ferroelectric materials, especially those with large activation barriers.

Methods Section
Film Growth: ε-AlFeO3/ε-Fe2O3 films were grown by pulsed laser deposition (PLD) using a KrF excimer laser (248 nm wavelength, 25 ns pulse duration) at 5 Hz repetition rate on Nbdoped (0.5%) (111)-oriented SrTiO3 (STO) single crystals at 825ºC, as described in more detail in Ref. 24.For the present study, we employed the film on which ferroelectric characterization reported in Ref. 24 was previously studied.
In-Situ STEM Experiments: High-resolution in-situ imaging of ion displacements in ɛ-Fe2O3 thin films was carried out on an aberration-corrected JEOL 2200FS TEM using a piezocontrolled in-situ biasing holder (HE150 electrical probing holder with double-tilt capability from Nanofactory AB).Wedge-shaped TEM specimens were prepared by a mechanical polishing process and argon ion milling (for details see Ref. 34).We used silver paste for making good electrical contacts to the Nb-doped STO substrate after gluing the specimen to a half Cu TEM grid.As top electrode, we exploited the piezo-controlled Pt/Ir tip of the probing holder (Figure 1a).This tip contacted the sample over a length of about 30 nm.During in-situ STEM measurements, the tip was grounded and triangular voltage pulses were applied to the Nb-doped STO substrate.The duration of the voltage pulses was set to 0.1 s and their magnitude was continuously swept in 0.1 V steps.
First-Principles Calculations: For structural predictions, we employed the USPEX code, [27][28][29][30] which is based on an evolutionary algorithm.The USPEX method searches for the global ground state structure, but produces a series of metastable states as by-product, being therefore suited also for searching all possible metastable polymorphs.We applied this method to search for the low-energy metastable centrosymmetric structure of ε-Fe2O3.We considered a 40-atoms unit cell of ε-Fe2O3.Each individual structure was relaxed by means of the Vienna ab-initio simulation package VASP, [35][36][37] with projector augmented wave pseudopotentials. [38]We considered the following valence-electron configurations: 3p 6 3d 7 4s 1 for Fe and 2s 2 2p 4 for oxygen.Ground-state properties were obtained using the generalized gradient approximation (PBEsol) adopted for solids. [39] correctly account for strong electron correlation effects on the d shell of Fe atoms, a DFT+U scheme was used. [40]We used on-site Coulomb parameter U = 4.0 eV and Hund's exchange J = 0.89 eV.The kinetic energy cutoff was set to 600 eV.For Brillouin zone sampling, we used uniform -centered meshes with a reciprocal-space resolution of 2π  0.05/ Å.All energies were converged to within 10 -6 eV per cell.The stopping criterion for structural relaxation was a change in (free) energy smaller than 1 meV.
Searches for metastable structures with the USPEX code were made by assuming ferromagnetic ordering on all Fe atoms in order to make energy comparisons of different structures independent of possible magnetic configurations.Total energy for the resulting lowenergy structures as well as phonon calculations were performed with proper antiferromagnetic alignments of the spins.
For phonon calculations, the obtained structures were optimized with residual forces below 0.002 eV/Å.The Brillouin zone was sampled with 4  4  4 Monkhorst-Pack Γ-centered special k-point grids for P2/c and Pbcn structures.Calculations of dynamical properties were performed using the force-constant method. [41]Eigenvalues and eigenvectors of the longwavelength phonons were calculated at the Γ-point of the Brillouin zone and, therefore, no supercells were used.The Hellman-Feynman forces were calculated for displacements of atoms of up to 0.04 Å.The dynamical matrix was constructed by Fourier transforming the force constant calculated at the Γ-point.The phonon-mode frequencies and atomic displacement pattern were obtained as eigenvalues and eigenvectors of the dynamical matrix.Domain wall movements were simulated in 320-unit cell, which consists of two equivalent domains with opposite direction of polarization.Ferromagnetic ordering on all Fe atoms was imposed for initial and final states as well as for all images along the transition path to avoid possible uncontrolled flips of spins on individual atoms during nudged elastic band calculations.
Transition energy barriers were calculated with a variable cell nudged elastic band as implemented in USPEX code. [42]We used crystallographic codes AMPLIMODE [43] and FINDSYM [44] for symmetry analysis of the obtained phases and their relations, as well as VESTA code [45] for visualization of obtained results.

Static deformations and their influence on ferroelectric switching inside ε-Fe2O3 domains and around non-polar domain walls
In this section we address the question how elastic deformations could influence ferroelectric switching in -Fe2O3.We calculate the piezoelectric tensor of -Fe2O3 (see Table 2 in the main manuscript) and the migration barriers in the deformed cells considering Pbcn as the reference non-polar structure (see Figure S6).We find the largest reduction of the energy barrier for a compressive in-plane lattice deformation, i.e., a lattice expansion along the polar axis.This observation could be used to design efficiently switchable strained ferroelectric films.
Unfortunately, when the strain is purely a result of the piezoelectric effect, the reduction of the migration barrier is actually expected if the external electric field points along the polarization, i.e., in the most stable configuration.A reversal of the electric field will reduce the out-of-plane lattice parameter, which in turn leads to an increase of the switching barrier.Therefore, the piezoelectric effect with uniaxial strains does not support ferroelectric switching.
We found that the shear deformation  lowers the energy barrier.This strain breaks the glide plane symmetry and lowers the symmetry of the transition state from Pbcn to P2/c as well as the symmetry of the polar structure from Pna21 to Pc, leading to a conventional groupsubgroup relation in ferroelectric transitions (see Figure S6).Although the desired d53 piezoelectric component for an out-of-plane electric field along the [001] polar axis is zero by symmetry, the shear could be achieved via the d51 piezoelectric component of an electric field applied along the [100] axis (see Figure S7).Therefore, one can assume that an electric field component along [100] axis could assist ferroelectric switching inside -Fe2O3 domains.This could be achieved by designing electric contacts on the thin films of -Fe2O3, leading to a gradient of electric field along [100] direction.
Deformation inPna21, which is produced by the d42 piezoelectric component if an electric field is applied along [010], enhances the migration barrier (see Figure S6).Moreover, the glide plane symmetry of the initial polar structure is not broken in this case.Therefore, the resulting piezoelectric tensor does not have a component that could induce the  shear deformation under an external electric field (see Figure S7).
Static deformation  slightly decreases the migration barrier and breaks the glide plane symmetry, leading to P21 symmetry (see Figure S6).Unfortunately, this deformation cannot be achieved via the piezoelectric effect because all corresponding d6i components are zero in Pna21 (see Figure S7).

cm - 1 .
Calculations indicate that the activation barrier for switching at domain walls is nearly a quarter of that corresponding to the most likely transition paths inside ε-Fe2O3 domains.Moreover, domain walls provide symmetry lowering, which is shown to be necessary for ferroelectric switching.Local polarization reversal in ε-Fe2O3 limits the macroscopic ferroelectric response and offers important hints on how to tailor ferroelectric properties by domain structure design in other relevant ferroelectric materials.
are fully symmetric due to glide plane symmetry.Hence, one can consider ferroelectric switching in -Fe2O3 as a side-product of antiferroelectric ion displacements along the [100] axis.Reversal of the displacements along [100] will reverse also the net polarization along [001].The estimated ratio of Fe ion displacements along [100] and[001] is about ~3.5 for Fe1 and Fe4, ~4.2 for Fe3, and ~8.7 for Fe2 ions.For oxygen ions the corresponding ratio varies from ~2.4 to ~23.These numbers suggest that only a very high outof-plane electric field component would produce the corresponding in-plane displacements, which could be well beyond dielectric breakdown of the material.Again, also here an in-plane electric field component along the [100] direction would assist in ferroelectric switching.

Figure 1 .
Figure 1.In-situ STEM measurement geometry and structural characterization.a) Low Figure 1.In-situ STEM measurement geometry and structural characterization.a) Low

Figure 2 .
Figure 2. Ferroelectric domain structure.a) High-resolution HAADF-STEM images along the

Figure 4 .
Figure 4. In-situ STEM observation of the motion of a ferroelectric domain boundary.a)-d)

Figure 6 .
Figure 6.a) Model of a 180 domain wall separating two polar Pna21 domains with opposite

Figure S1 .
Figure S1.Electron diffraction simulations and fast Fourier transform patterns of experimental HAADF-STEM images along the ε-Fe2O3 [100] (a,b) and [110] (c,d) crystal axes.The data in (a,b) and (c,d) correspond to D1 and D2 structural domains, respectively.

Figure S2 .
Figure S2.High-resolution HAADF-STEM image of a D1 domain in the ε-Fe2O3 film.This is an uncolored version of Figure 2b in the main manuscript.

Figure S3 .
Figure S3.High-resolution HAADF-STEM images of the ε-Fe2O3 film obtained after application of alternating positive and negative voltage pulses to the Nb-doped STO substrate.The data of Figure 3 in the main manuscript are extracted from the areas indicated by red and cyan rectangles.

Figure S4 .
Figure S4.High-resolution HAADF-STEM images from an area inside a structural D1 domain far away from a pre-existing non-polar domain wall.In this area, the application of 5 V does not measurably displace Fe ions.

Figure S5 .
Figure S5.a) HAADF-STEM image simulation of the structural model shown in (b).The structural model in (b) is based on first-principles calculations.At the boundary between two polar domains, the smile and cry features disappear.The HAADF simulation of the structural model is performed by using Multislice calculation with xHREM.Aperture radius is 24 mrad.Third order and fifth order Cs are 0.005 mm and 0 mm, respectively.Inner and outer angles of dark field are 70 and 370 mrad, respectively.

Figure S6 .
Figure S6.Up: Schematic presentation of the influence of axial and shear deformations  = 1% on the migration barrier E inside a -Fe2O3 domain.The numbers in the schematic transition paths indicate space group numbers of the initial, transition, and final states along the migration paths.Down: Influence of 5 = 1% shear deformation on the migration barrier in multidomain -Fe2O3.

Figure S7 .
Figure S7.Piezoelectric tensors for different polar groups.The orientation of principal axes of the space group in standard setting with respect to the polar and easy axes of the Pna21 space group is schematically presented.Red arcs indicate the 5 shear distortion of the Pna21 space group.The external electric field is collinear to the polar axis.

Figure S8 .
Figure S8.Model of inclined domain walls.The dashed lines indicate the positions of the domain walls.Arrows indicate the polarization direction in the ferroelectric domains.

Table 1 .
Structural parameters of non-polar monoclinic P2/c and orthorhombic Pbcn reference