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1 In the format provided by the authors and unedited. Functional electronic inversion layers at ferroelectric domain walls J. A. Mundy 1, *, J. Schaab 2, *, Y. Kumagai 2, *, A. Cano 3, M. Stengel 4,5, I. P. Krug 6, D. M. Gottlob 7, H. Doganay 7, M. E. Holtz 1, R. Held 8, Z. Yan 9,10, E. Bourret 9, C. M. Schneider 7, D. G. Schlom 8,11, D. A. Muller 1,11, R. Ramesh 9,12, N. A. Spaldin 2, and D. Meier 2,13,# *These authors contributed equally to this work # dennis.meier@ntnu.no DOI: /NMAT School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA 2 Department of Materials, ETH Zurich, 8093 Zürich, Switzerland 3 CNRS, Université de Bordeaux, ICMCB, UPR 9048, Pessac, France 4 ICREA Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain 5 Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, Bellaterra, Spain 6 Institut für Optik und Atomare Physik, TU Berlin, Berlin, Germany 7 Peter Grünberg Institute (PGI-6), Forschungszentrum Jülich, Jülich, Germany 8 Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA 9 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 10 Department of Physics, ETH Zurich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland 11 Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA 12 Department of Materials Science and Engineering and Department of Physics, UC Berkeley, Berkeley, California 94720, USA 13 Department of Materials Science and Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway NATURE MATERIALS 1

2 Electronic transport mechanism Contact resistance can play a crucial role in two-probe measurements and dominate the conductance data caused by, for instance, the formation of a Schottky barrier at the tipsurface interface. In order to investigate whether the transition from resistive to conductive behavior at head-to-head walls originates from such extrinsic interface effects or intrinsic bulk properties, we perform a careful analysis by cafm using differently coated probe tips as summarized in Fig. S1 and S2. In Fig. S1a we compare the currentvoltage characteristic at a head-to-head wall recorded with a conductive diamond-coated tip (DCP-20 from ND-MDT, nominal radius rr tip = 100 nm; the data is identical to the results shown in Fig. 1c in the main text) and a Pt/Ir-coated tip (ANSCM-PT from AppNano, rr tip = 30 nm). The current densities, JJ = II/AA (AA = ππrr 2 tip ) measured in forward bias are of the same order of magnitude for voltages 3V, independent of the tip coating (Note that forward bias is defined with respect to the tip-surface contact. As ErMnO3 is a p-type semiconductor, positive voltage corresponds to forward bias for a Schottky-like barrier. This is opposite to, e.g., reduced n-type BiFeO3 1 ). For a more quantitative analysis, we calculate the respective barrier heights assuming an ideal barrier, that is, φφ B = EE g + φφ tip 2. The band gap of RMnO3 is EE g 1.6 ev and the electron affinity = 3.83 ev 3,4. Based on the work function for diamond- and Pt/Ir-coated tips (φφ Dia tip = ev 2,5, φφ Pt/Ir tip = ev 2,6 ), a lower bound for the minimum difference in contact potential, φφ B = φφ Pt/Ir tip φφ Dia tip = 0.12 ev can be derived. In case of interface-dominated transport 2 (JJ(VV) exp ( φφ BB eeee ) [exp ( ) 1]), this difference would lead to at least kkkk ηηηηηη two orders of magnitude higher current densities for the Pt/Ir-coated tip than for the diamond-coated tip. We thus exclude that contact resistance due to a difference in work function dominates the transport behavior in Fig. S1a. In Fig. S1b to S1d we consider bulk-controlled conduction mechanisms that are possible in case of forward bias, including hopping (Fig. 1b), space-charge-limited conductance (SCLC, Fig. S1c) 7, and Poole-Frenkel emission (Fig. S1d) 8. For hopping, the current density is NATURE MATERIALS 2

3 JJ = aa exp ( EE aa ) exp (eeeee ), (1) kkkk kkkk where aa is the mean hopping distance, EE aa the activation energy and E the electic field (approximated by E VV/rr tttttt 9,10 ). Fitting of the hopping model (Fig. S1b) leads to aa Pt = 1.1 ± 0.2 nm and aa Dia = 2.4 ± 0.5 nm; errors result from uncertainty in the tip radius, 20%). The different results for the material intrinsic parameter aa suggest that hopping does not dominate the domain wall properties measured under DC currents. SCLC is described by Childs law JJ = 9 8 εε 0εε DC μμ VV2 dd 3, (2) with εε DC denoting the static dielectric constant, μμ the electronic mobility, and dd the nominal thickness over which the electric field is applied. For realistic parameters (μμ (eenn pp ρρ) cm 2 (Vs) 1 11 ; dd = 1 μm estimates the domain wall length according to the average domain size) SCLC captures the current voltage characteristics in the lowvoltage regime, leading to εε Pt DC = 28.6 and εε Dia DC = 32.8 (Fig. S1c), which is in agreement with the values reported in literature (εε DC (RRMnO 3 ) = ) 12. Poole-Frenkel emission, JJ E exp ( EE tt kkkk ) exp(ββ PF E), (3) yields the better description of the data for high voltages (EE tt is the trap energy, E VV/rr tttttt the electric field 9,10 and ββ PF = ee kkkk ee/ππεε 0εε). We obtain εε PPPP = 23.4 ± 4.9 and εε DDDDDD = 17.1 ± 3.6. Similar to SCLC, the values are in tune with the literature values for εε DC. However, they are slightly larger than the optical dielectric constant εε = nn 2, which is often used for describing the transport in insulating ferroelectric oxides 13 (εε (RRMnO 3 ) 4) 14. In conclusion, SCLC and Poole-Frenkel emission are both likely DC conduction mechanisms at head-to-head walls (further insight might be gained from thickness- and temperaturedependent studies). Most importantly for the present work, Fig. S1 points towards bulk- NATURE MATERIALS 3

4 dominated transport, clearly ruling out contact resistance as the dominating source for the anomalous electric domain wall behavior. This conclusion is corroborated by Fig. S2, where we present current-voltage data gained on a head-to-head wall for positive and negative bias voltages. Complementary to Fig. 1c in the main text, we show a bipolar current-voltage measurement, J(V), in Fig. S2a. Figure S2a compares J(V) data from the bulk (black) and a head-to-head wall (red, see inset). Similar to Fig. 1c in the main text, obtained for forward bias only, a crossover is observed at about Vc 4 V in Fig. S2a. We find that currents are less stable for negative than for positive bias voltages, which obscures imaging by cafm. Despite the higher noise level, however, spatially resolved measurements show the same crossover behavior as observed for negative voltages (see Fig. S2b). We note that the J(V)-curve exhibits a slight asymmetry (see rectification ratio R in Fig. S2 for a quantitative measure), which is often observed for RMnO3 15. The bipolar J(V) data in Fig. S2 is consistent with the results from the analysis of Fig. S1, that is, bulk-dominated transport, with a small asymmetry arising from, e.g., electronic interface states. Electron energy loss spectroscopy at head-to-head and tail-to-tail domain walls In Fig. 2 of the main text we present electron energy loss spectroscopy (EELS) spectra showing the evolution of the Mn L2,3-edge across a head-to-head domain wall. Prior to fitting with the reference spectra, the data was analyzed using the multivariate curve resolution (MCR) technique. Two input spectra were entered, one from the bulk of the sample and one from the wall position region in Fig. 2a displaying a reduction in valence. One hundred iterations were employed to refine the spectra and the resulting concentration profiles with the physical constraints that both remain positive. While the MCR analysis demonstrates the local variation in the manganese valence across the interface in Fig. 2, it does not provide a quantitative measurement of the valence here since the valence of the interface component is not identified. From this analysis, we identified an interfacial spectrum in which the energy shift in the peak position was most NATURE MATERIALS 4

5 similar to the Mn 2.8+ valence. We thus constructed a formal Mn 2.8+ reference, providing a physically equivalent basis as Mn 3+ and Mn 2+ spectra. We note that the EELS fine structure can be sensitive to the crystal symmetry and crystal field splitting. However, since no reference data for Mn 2+ in trigonal bipyramidal coordination is available in literature, we selected a Mn 2+ reference from MnO 16. The Mn 2.8+ reference for fitting our EELS data was then generated as a linear combination of the Mn 2+ reference from MnO and the Mn 3+ reference from ErMnO3 gained away from the wall (Mn 2.8+ = 0.2 Mn Mn 3+ ). We then fit the experimentally observed Mn spectra across the head-to-head domain wall with this derived Mn 2.8+ and the reference Mn 3+ spectra. This provides a mathematically equivalent basis to fitting the data with Mn 3+ and Mn 2+ spectra. This procedure allowed for capturing the fine structure changes across the interface without a remaining residual. The difference between the experimentally obtained domain-wall spectrum and the Mn 3+ reference spectrum is highlighted in Fig. S4. A detailed overview of our domain-wall analysis in terms of high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) and EELS is provided in Fig. S5 and Fig. S6 for a head-to-head and tail-to-tail wall, respectively. In Fig. S5a,b we show an ADF image of the head-to-head wall (corresponding to the one in Fig. 2a of the main text) and the simultaneously collected EELS image 17. Each pixel in the ADF map in Fig. S5a corresponds to a location where we collected an EELS spectrum. The EELS image of the Mn L2,3-edge in Fig. S5b was generated by binning the EELS data parallel to the domain wall direction. A line profile of the concentration of manganese and oxygen is shown in Figure S5c. To describe the local electronic state, each Mn position in Fig. S5b was fit as a linear combination of Mn 3+ and Mn 2.8+ reference spectra (Fig. S5e). The derived concentrations corresponding to this fit are presented in Fig. S5d. We note a statistically significant concentration of the Mn 2.8+ spectra at the position of the head-to-head domain-wall. The mean oxygen spectrum is shown in Fig. S5f, showing a distinct pre-peak, indicating that the sample is not oxygen deficient 18. NATURE MATERIALS 5

6 In order to demonstrate that the reduced manganese valence state is a specific feature of the head-to-head walls, we also show spectroscopy data obtained at a tail-to-tail domain wall in Fig. S6. The simultaneously recorded ADF data of a tail-to-tail wall and the corresponding EELS image are displayed in Figs. S6a,b, respectively. Analogous to Fig. S5b, the Mn L2,3-edge data in Fig. S6b was binned parallel to the domain-wall direction to increase the signal-to-noise ratio. In contrast to the head-to-head domain wall, all spectra collected at the Mn L2,3-edge across the tail-to-tail domain wall can be described by a single Mn 3+ reference spectrum. Figure S6c shows the residual remaining after the fit revealing no detectable change in the fine structure of the Mn L2,3-edge. This observation is consistent with theory, predicting delocalized holes that occupy the broad valence band which consists of strongly hybridized and delocalized manganese d- and oxygen 2p-states (see DOS in Fig. 3a in the main text). We note that hole accumulation could also be detected as a modulation of the pre-peak on the O K- edge 19. A similar analysis of the oxygen K-edge also did not detect a statistically significant hole accumulation; the expected hole concentration would imply about a 10% change in the peak O K-edge intensity, which is not resolved in the experiment. A line profile of the concentration of manganese and oxygen is shown in Figure S6d. Domain-wall spectroscopy by X-PEEM Since our EELS measurements would be less sensitive to a uniformly delocalized valence change over large distances with small electronic changes at each atomic plane, we perform complementary X-PEEM spectroscopy measurements to check for the latter scenario. All X-PEEM measurements were performed at 297 K under ultrahigh vacuum ( mbar) at the UE56/1-SGM beamline of the Forschungszentrum Jülich, BESSY-II storage ring, Helmholtz Center Berlin using a novel type of aberration-corrected photoemission electron microscope (SPECS FE-LEEM P90 AC) 20. In order to measure domain-wall spectra by X-PEEM, we first imaged the conducting tail-to-tail walls as explained in ref. 21. Figure S7a shows an X-PEEM image gained at a photon-energy of ev (Mn L3-edge). Here, conducting tail-to-tail domain walls are visible as bright lines on a homogenous grey background. Since the applied X-PEEM method is not sensitive to NATURE MATERIALS 6

7 the rather insulating head-to-head domain walls, we also applied low-energy electron microscopy (LEEM) measurements using the same aberration-corrected electron microscope. This allowed us to perform X-PEEM and LEEM measurements at the same sample position and to directly compare the results. The LEEM image is presented in Fig. S7b. Here, the conducting tail-to-tail walls, which are also visible in Fig. S7a, appear as narrow bright lines with a dark outline, indicating a pronounced domain-wall related focusing effect on the LEEM electrons. The focusing predominantly occurs because the insulating domains charge negatively with respect to the conducting walls when irradiated with slow electrons. The resulting inhomogeneity in the electrostatic surface potential influences the trajectories of the reflected electrons and is thus detectable by LEEM. In addition, head-to-head walls are visible in Fig. S7a, but their contrast is less pronounced compared to the tail-to-tail walls. The combined X-PEEM and LEEM results allowed for identifying the domain-wall positions as we show in Fig. S6c. After mapping the domain-wall positions using X-PEEM and LEEM, energy-filtered X-PEEM measurements in secondary electron yield mode (SEY) were performed. The energy analyzer pass energy was set to maximum SEY yield at the Mn pre-edge and X-ray absorption spectra were recorded scanning the photon energy. Synchrotron beam intensity was reduced in order to minimize charging effects, which usually occur when investigating poorly conducting samples. The corresponding spectroscopic data is presented in Fig. S8. Figure S8a,b show the evolution of the manganese L2,3-edge across a head-to-head and a tail-to-tail domain wall, respectively. We find that the onset energy of the L3-edge and the position of the L3- and L2-peaks remain constant when crossing the domain wall. Spectral artifacts due to sample charging slowing down the photoelectrons, however, occur between 641 ev and 643 ev as discussed in detail in ref. 21. Thus, we restrict ourselves to the L2-edge in the more detailed presentation in the insets to Fig. S8a,b. The spectra taken at the L2-edge display the characteristic Mn 3+ multiplet structure known from X-ray absorption spectroscopy studies on hexagonal manganites without any indication of a valence-state variation 22. This leads us to the conclusion that no significant variations in NATURE MATERIALS 7

8 the electrochemical domain-wall structure occur outside of the immediate domain-wall vicinities that we studied using EELS. We note that X-PEEM is not sensitive to the above discussed local Mn 2.8+ state due to its 2 to 3 orders of magnitude poorer spatial resolution than EELS. Formation energy for the polaronic state The calculated formation energy as a function of U in Fig. 3b of the main text refers to an isolated polaron immersed in an infinite, homogeneous crystal. This is significantly more expensive, from the electrostatic point of view, than forming a regular array of polarons at a charged domain wall, as expected in our target system. This implies that the actual critical value of U (for a 2D lattice of polarons at the wall) must be smaller than our calculated value of 7.5 ev (for an isolated polaron) the latter should be therefore regarded as an upper estimate. To make this statement more quantitative, ideally one should directly calculate, fully from first-principles, the polaron formation energy right at the domain wall, as a function of the in-plane polaron density, np. Unfortunately, such a calculation goes well beyond our computational capabilities, because of the prohibitive size of the supercell models that would be required. An alternative, computationally more realistic approach consists of combining the data of Fig. 3b from the main text, i.e., the formation energy of an isolated polaron as a function of U, with an appropriate electrostatic correction that reflects the two-dimensional nature of the polaron lattice interacting with the charged domain wall. We find that a simple hexagonal lattice of (negative) point charges interacting with a (positively) charged plane provides a quantitatively accurate description of such a correction for the physically relevant range of np values. In Fig. S9 we show the electrostatic correction (per electron), appropriately scaled by the bulk dielectric constant (we used a value of 15, approximately the average of the in-plane and out-of-plane values) as a function of np. The solid black curve, referring to point charges interacting with a delta-plane, is EE corr = 0.33 nn P. Of course, neither the defect nor the charged wall are delta functions, but have a certain spatial spread (of the order of NATURE MATERIALS 8

9 the interatomic distance). To check whether a finite spread can affect our conclusions, we recalculated the correction by introducing a Gaussian width in both the plane and the point charges; the results are shown as open symbols in Fig. S9. ( = 0.5, 1.0 and 2.0 atomic units for diamonds, squares and circles, respectively.) Clearly, in the physically relevant range of np (on the left of the dashed red line, which indicates the total polarization charge of the wall), there is negligible difference between assuming sigma = 0 or a finite value. As we anticipate a comparable contribution of hole depletion and electron (polaron) accumulation to the screening of the domain-wall charge (see Fig. 1d of the main text), np should be around 0.1, where the electrostatic correction of Fig. S9 is of about -0.1 ev. A quick look at Fig. 3b of the main text shows that this correction would reduce the critical U (governing the transition between a metallic and a polaronic twodimensional state) to the range U ev, which is similar to the value used in recent ab initio studies of hexagonal manganites. In order to provide a complete physical picture, it is useful to summarize the bandbending model and the polaronic model into a simplified energy functional, EE tot = AA (nn s nn e nn P ) 3 + nn 24 nn e EE g + nn P EE g + BB(UU UU 0 )nn P + CCnn P nn P h where nn e and nn P are the unknown densities (> 0) of itinerant electrons and polarons, respectively. The first term is the electrostatic energy associated to the band bending: A = 5.0 ev ( inverse dielectric constant ), nn s = (polarization charge per 30-atom cell). The second term reflects the energy cost of promoting an itinerant electron to the conduction band (EE g = 1.7 ev). The third term is the analogous energy for polarons (the polaron formation energy is defined with respect to an itinerant electron). The fourth term is a linear fit of the isolated polaron formation energy of Fig. 3b (BB = 0.1, UU 0 = 7.3 ev). Finally, the last term is the electrostatic correction described above (C = 0.33 ev). Figure S10 shows a phase diagram of the system as a function of the bulk level of p-type dopants, nn h, and the Hubbard parameter, U. NATURE MATERIALS 9

10 According to this joint model, polarons tend to form in a rather wide range of conditions. They always result in a suppressed conductance at zero bias, of course: themselves they do not conduct and, in addition, they are always accompanied by a significant hole depletion. Note that polarons always form right at the wall: The formation energy that we report in Fig. 3b is defined as the energy of an isolated electron-like polaron in an infinite YMnO3 crystal, with the chemical potential of the electron fixed to that of an itinerant conduction-band state. This means that, if the Fermi level does not lie at the conduction band minimum (this condition is only satisfied exactly at the head-tohead wall, see Fig. 1d), there is an additional energy cost that needs to be taken into account, i.e., the energy difference between the local CBM and the global Fermi level. Away from the wall, the CBM (and therefore the predicted polaron energy) rapidly increases, becoming prohibitive deep within the domain. Validation of the DFT-based semi-classical model In order to validate our semi-classical description of band bending at the polar walls, shown in Fig. 1d of the main text, we have performed explicit large-scale DFT simulations of supercells containing two domain walls (head-to-head and tail-to-tail) of opposite polarity. The supercell is constructed by stacking 2N times the 30-atom unit cell of YMnO3 along z, and by initializing the atomic coordinates in such a way that half of the system (N cells) is in a up polarization state, and half is down. After full electronic and atomic relaxation we extract the macroscopically averaged electrostatic potential, which has the form of a saw-tooth function, and the total density of metallic carriers at either wall. Next, we separately perform the same simulation by means of our DFT-based semi-classical model, again extracting the equilibrium electrostatic potential and carrier densities (consistent with the first-principles simulation, the bulk density of dopants is set to zero). In Fig. S11 we compare the respective results of the model and full DFT calculations, by setting either N=2 (Fig. S10a) or N=5 (Fig. S10 b). The corresponding supercell sizes are of 120 and 300 atoms. We choose these two specific values as they test our model in two qualitatively different regimes: the smaller supercell is still insulating and the wall polarization is only screened by the dielectric polarization of the adjacent domains; NATURE MATERIALS 10

11 conversely, in the larger supercell a Zener-like transfer of electrons from the negatively charged tail-to-tail wall to the positively charged head-to-head wall occurs and the system is metallic. Figure S11 clearly shows that the predictions of the model for the electrostatic potential accurately match the full DFT calculations. In the case for N=5, the surface density of metallic carriers is consistently predicted to be 0.14 per surface unit cell by both methods, about half of the bare wall charge. Note that the first-principles calculation shows that only a single MnO3 layer is populated by carriers, consistent with our model description. NATURE MATERIALS 11

12 References 1. Farokhipoor et al., Phys. Rev. Lett. 107, (2011) 2. Sze & Kwok, Physics of Semiconductor Devices, 3rd Edition, Whiley (2006) 3. Kalashnikova et al., JETP 78, 143 (2003) 4. Han et al., Chem. Mater. 27, 7425 (2015) 5. Diederich et al., Surf. Sci. 418, 219 (1998) 6. Hölzl & Schulte, Solid Surface Physics: Work functions of metals, Springer (1979) 7. Chiu, Adv. Mater. Sci. Eng. 2014, (2014) 8. Zubko et al., J. Appl. Phys. 100, (2006) 9. Stolichnov et al., Nano Lett. 15, (2015) 10. Molotskii, J. Appl. Phys. 93, 6234 (2003) 11. Maximum mobility value estimated: ρρ 10 3 Ω cm taken from Katsufuji et al., Phys. Rev. B 64, (2001), and nn pp cm Adem et al., J. Alloys Compd. 638, (2015) 13. Scott, J. Phys.: Condens. Matter 26, (2014) 14. Schmidt-Grund et al., RSC Adv. 4, (2014) 15. Wu et al., Phys. Rev. Lett. 108, (2012) 16. Kurata et al., Phys. Rev. B 48, 2102 (1993) 17. Muller et al., Science 319, 1073 (2008) 18. Cheng et al., Phys. Rev. B 93, (2016) 19. Ju et al., Phys. Rev. Lett. 79, 3230 (1997) 20. Tromp et al., Ultramicroscopy 110, 852 (2010) 21. Schaab et al., Appl. Phys. Lett. 104, (2014) 22. Cho et al., Phys. Rev. B 79, (2009) NATURE MATERIALS 12

13 a Tip coating b Hopping c SCLC d Poole- Frenkel Figure S1 Current-voltage characteristic at head-to-head walls in ErMnO3. a, Voltagedependent current density J(V) measured with different tip-coatings for forward bias. b, Logarithmic representation of J(V). Dashed red lines correspond to fits according to the hopping model (eq. 1). c, d, Different representations for J(V) and fits (dashed red lines) according to SCLC (eq. 2) and Pool-Frenkel emission (eq. 3), respectively. NATURE MATERIALS 13

14 a Bulk Head-to-head b -6V J (A/cm 2 ) Pt/Ir-coated R 3.5 5V -10V Voltage (V) Figure S2 Bipolar J(V) curves. a, J(V) measured with a Pt/Ir-coated tip on (red) and away (black) from the head-to-head wall shown in the inset. Scale bar 500 nm. The wall becomes more conducting than the bulk at about 4 V. R denotes the rectification ratio, defined as the ratio of the forward bias current at +5 V divided by the reverse bias current at -5 V. b, Spatially resolved cafm scan obtained with negative voltage. At -6V the headto-head wall is insulating and conducting at -10V, indicating a slightly higher value Vc than in a. Scale bar 1 m. NATURE MATERIALS 14

15 Figure S3 Relative currents at head-to-head and tail-to-tail domain walls. a, Currentvoltage characteristic reproduced from Fig. 1c in the main text. Dashed squares in the inset show the areas from which data points are extracted. Arrows indicate the polarization direction. c-e, cafm images series presenting spatially resolved data for selected voltages marked in a. Figure S4 Head-to-head domain-wall spectrum. a, HAADF-STEM image of the head-tohead wall reproduced from Fig. 2a in the main text. Mn L3-edge spectra at the two positions in b indicated by the arrows. The spectrum at the interface shows a significant deviation from the Mn 3+ reference spectrum (black line). The deviation is colored in blue. NATURE MATERIALS 15

16 Figure S5 Evaluation of the charge transfer at head-to-head walls. a, HAADF-STEM image of the head-to-head wall acquired simultaneously with the EELS image. b, Mn L2,3-edge binned parallel to the domain wall direction. c, Composition line profile of Mn and O binned parallel to the wall direction, made by integrating the background subtracted edges. d, Concentration plot, derived by fitting each Mn position in b with Mn 3+ (black) and Mn 2.8+ (blue) reference spectra. e, Mn reference spectra used in the fit. f, Average O- K edge, showing a pre-peak (indicating that the sample is not oxygen- deficient). Figure S6 EELS data at tail-to-tail domain walls. a, HAADF-STEM image of a tail-to-tail wall, acquired simultaneously with the EELS image. b, Mn L2,3-edge binned parallel to the domain wall direction. c, Residual remaining after fitting b with a Mn 3+ single spectrum. d, Composition line profile of Mn and O binned parallel to the domain wall direction, made by integrating the background subtracted edges. NATURE MATERIALS 16

17 Figure S7 Identification of conducting tail-to-tail and insulating head-to-head domain walls by cathode-lens microscopy. a, X-PEEM image of a tail-to-tail wall taken (641.5 ev, Mn L3-edge). b, LEEM image taken at the same position as the image in a showing both tail-to-tail and head-to-head walls. c, Dashed lines indicate the wall positions. Along the cross-sections indicated by the blue dots, spectra (1 to 8) were extracted from the X-PEEM data (see Fig. S8). Each spectrum was averaged over an area of nm 2 as indicated by the black boxes, which are shown only for the first spectrum of each scan. Figure S8 X-PEEM spectroscopy at charged domain walls. a, Set of manganese L2,3-edge spectra documenting the evolution across a head-to-head domain wall. A zoom-in showing the L2-edge is presented in the inset to a. b, Evolution of X-PEEM L2,3-edge spectra across a tail-to-tail domain wall and zoom-in to the L2-edge (inset). NATURE MATERIALS 17

18 0-0.1 Energy (ev) In-plane density of polarons (per 30-atom cell) Figure S9 Electrostatic corrections. The solid black line refers to point charges interacting with a delta-plane (E corr = 0.33 n P ). Symbols represent corrections that account for a finite spread, model by introducing a Gaussian width ( = 0.5, 1.0 and 2.0 atomic units for diamonds, squares and circles, respectively). NATURE MATERIALS 18

19 Bulk density of holes Hubbard U Figure S10 Phase diagram for the DFT-based model system. Phase diagram as a function of the bulk level of p-type dopants, n h, and the Hubbard parameter, U. Black circles: At high enough hole density, the depletion of holes is sufficient to screen the polarization charge at head-to-head walls. Green diamonds: In addition to hole depletion a 2D metallic gas of electron-like carriers contributes to the screening at the wall; this is the situation we have in Fig. 1d of the manuscript. Red squares: Instead of the metallic electrons, we have a 2D lattice of localized polarons. Note that the "critical U" is of about ev for the range of carrier densities that we consider in the manuscript. Dashed blue line: Reference hole density of cm -3 (0.007 h/cell). Note that there is never a coexistence of itinerant electrons and polarons at the wall. NATURE MATERIALS 19

20 0.6 a 0.4 Electrostatic potential (ev) Unit cell 1 b Electrostatic potential (ev) Unit cell Figure S11 Validation of the DFT-based model. a, The macroscopically averaged electrostatic potential calculated fully from first principles (black curves) is compared to that resulting from the DFT-based model (dashed red curves) for the case N=2. b, Same as in a for N=5 (b) (see text for a more detailed description). The unit of the horizontal axis is the calculated c-parameter of the YMnO3 unit cell of about 11.3 Å. NATURE MATERIALS 20