Autocorrected Off-axis Holography of 2D Materials

The reduced dimensionality in two-dimensional materials leads a wealth of unusual properties, which are currently explored for both fundamental and applied sciences. In order to study the crystal structure, edge states, the formation of defects and grain boundaries, or the impact of adsorbates, high resolution microscopy techniques are indispensible. Here we report on the development of an electron holography (EH) transmission electron microscopy (TEM) technique, which facilitates high spatial resolution by an automatic correction of geometric aberrations. Distinguished features of EH beyond conventional TEM imaging are the gap-free spatial information signal transfer and higher dose efficiency for certain spatial frequency bands as well as direct access to the projected electrostatic potential of the 2D material. We demonstrate these features at the example of h-BN, at which we measure the electrostatic potential as a function of layer number down to the monolayer limit and obtain evidence for a systematic increase of the potential at the zig-zag edges.


I. INTRODUCTION
The discovery of graphene and its intriguing properties more than ten years ago [1,2] has sparked large and ongoing research efforts into two-dimensional materials (2DMs). The synthesis of novel 2DMs, comprising, e.g., 2D topological insulators, 2D magnets, or organic systems like 2D polymers, with single to few layers thickness and high structural definition at the atomic / molecular level is at the center of this field (e.g., [3][4][5][6]). They exhibit a large range of physical properties triggered by the reduced dimensionality in one direction such as quantum confinement effects or weak dielectric screening from the environment, yielding a significant enhancement of the Coulomb interaction [3,7]. Another interesting aspect is the formation of out-of-plane elastic modulations, which stabilize the 2DM structure and modify its mechanical properties [8][9][10]. The missing third dimension also enhances the proliferation and impact of defects, such as point and line defects, grains or multilayers morphologies; which inevitably occur upon synthesis and often govern the functionality (e.g., reactivity, stability) of 2DMs in applications [11].
Therefore the development of microscopic characterization methods, which allow to analyze the structure and electronic properties of the 2DMs including the edges, defects and grain boundaries, is at the center of the field. Transmission electron microscopy (TEM) has been a cornerstone technique (others are scanning tunneling microscopy STM and photo emission electron microscopy PEEM), offering high resolving power and spectroscopic information. A breakthrough for TEM could be achieved by employing chromatic aberration correction facilitating high-spatial resolution at low-acceleration voltages [12]. Two central challenges required special attention and have been addressed through various methodological developments of TEM techniques: (A) 2D materials are typically more susceptible to various beam damage mechanisms than their 3D counterparts [13][14][15]. That includes knock-on damage, radiolysis, and chemical etching. The knock-on damage may be reduced by lowering the acceleration voltage and hence the kinetic energy of the beam electron below the knockon threshold of the pertinent chemical bonds in the 2DM (e.g., 90 keV for the C-C bond in graphene [16] and 40 keV for the B-N bond in monolayer h-BN [17]). Radiolysis and etching follow more complicated reaction mechanisms [18]. Their magnitude might be reduced by lowering the temperature and optimizing the acceleration voltage [16].
(B) Most 2D materials belong to the family of weak scatterers (another important member is biological matter, mainly consisting of C and H atoms), which implies that they only (weakly) shift the phase of the electron wave when traversing the sample, but don't modulate the amplitude. Consequently, they are referred to as weak phase objects (WPOs). This phase shift of the electron wave can not be measured directly, due to the quantum mechanical phase detection problem. To solve this problem one can employ phase plates, which enable the imaging of the phase shift introduced by these materials as intensity contrast. The use of either physical [19][20][21][22] or electron optical phase plates [19,[23][24][25], however, comes with some merits and disadvantages. The former degrade [21,22,26] during use and create unwanted diffuse scattering and beam blocking [27], whereas the latter is typically constructed from materials that are prone to charging. Notable exceptions are laser [28] and drift tube [25] phase plates, which are very demanding constructionand implementation-wise. By far the most straight for-ward method for transfer of phase contrast to intensity constrast, however, is an additional defocus with respect to the object exit plane, which has the negative side effect of introducing transfer gaps at low spatial frequencies or an oscillating contrast transfer for large spatial frequencies (see below). Introducing large defocii also results in a reduction of the resolution due to the partial transversal coherence of the electrons, which may be expressed by an exponential envelope function in reciprocal space.
In the following we address the phase problem in weak scatterers (at the example of 2DMs) by advancing off-axis electron holography, an interferometric technique allowing to reconstruct the phase shift of the electron wave over the whole spatial frequency band, up to the information limit. These advantages have triggered a small number of previous studies on WPOs, notably at biological materials [29,30] and 2DMs [31][32][33][34][35][36]. However, a persisting problem remains in the defocus required for visualizing the sample during TEM operation and other residual aberrations such as astigmatism, which typically built up during acquisition [37]. Their correction, however, is mandatory for an analysis of the acquired phase in terms of physical quantities such as potentials, charge densities and the atomic structure. In Winkler et al. [35,36] this problem has been addressed by a modelbased fitting approach requiring a full model of the scattering potential and hence the 2DM under investigation.
Here we follow a different approach requiring no or only minimal a-priori knowledge of the sample, that is, stripping the recorded data from any instrumental influences, notably aberrations and noise. The resulting data may then be used to extract certain specimen properties in a second step. This approach has the advantage of requiring no a-priori knowledge about the specimen and a clear separation between instrumental and specimen influences. One key idea is to exploit very general discrete symmetries pertaining to the scattered electron wave function: First of all, the weak scattering property induces an odd symmetry in the object phases in Fourier space, which allows to correct for symmetric aberrations. Second, a large class of 2DMs are centrosymmetric, introducing an even symmetry in the Fourier object phases, which allows to correct for antisymmetric aberrations. This approach is based on the original work of Fu and Lichte [38], who demonstrated how to generically extract symmetric aberrations from holograms at the example of amorphous carbon foils. Here, we go one step further and autocorrect for the aberrations in the reconstructed wave of 2DMs, which greatly facilitates the analysis of the phase in terms of physical data, i.e., projected potentials. In this regard, we follow D. Gabor's original idea of holography as a means to aberration correction [39]. The corrected data is then subjected to a principle component analysis (PCA) denoising, which reveals the meaningful phase data at the atomic scale and enables extracting the underlying projected potential. Last but not least we compare that data to ab-initio density functional calculations to analyze the measured potentials in terms of charge (de)localizations.
The paper is organized as follows, we first recapitulate the imaging principles of weak scatterers (phase objects) and off-axis holography. From these, we derive how geometric aberrations can be determined and automatically corrected a-posteriori from an acquired hologram without additional measurements. We elaborate on the noise characteristics of the thereby obtained aberrationcorrected image phase and the spatial resolution of the determined aberrations (e.g., defocus due to out-of-plane modulations). We finally demonstrate the feasibility of the approach at the example of h-BN. Amongst others we reconstruct the number of layers, the mean inner potential (MIP) of individual layers, the structure of the monolayer as well as the edges; and correlate this to material properties such as the charge delocalization or the stability and electronic properties of the edge states.

II. IMAGING PRINCIPLES
A. Conventional imaging and off-axis holography of weak phase object Sufficiently thin TEM specimens (with the critical thickness depending on the atomic scattering potential V of the chemical constituents) behave as weak phase object (WPOs) in TEM. In good approximation they just impose a small phase shift on the beam electrons' wave function Ψ obj leaving the object of thickness t. Here C E is an electron-energydependent interaction constant (0.01 rad /V nm at 80 keV), z the direction in which the electron beam transmits the sample, and r the 2D position vector in the object plane. Since WPOs do not modulate the amplitude A, Ψ obj can be approximated by The dominant geometric aberrations of the objective lens are described by a phase function χ (k) acting on the electron wave spectrum by a complex factor e −iχ(k) , the wave transfer function (WTF), in reciprocal space. Moreover, the combination of (transversal and longitudinal) partial coherence and geometric aberrations leads to an exponential damping of spatial frequencies in the wave function described by a real envelope function E (k) in reciprocal space. The Fourier transform of the so-called image wave function taking into account these modulations by the imaging system ( Fig. 1(a)) reads Here we separated the antisymmetric and symmetric aberrations, χ a (k) and χ s (k), respectively. The corresponding conventional linear image intensity (neglecting the term quadratic in ϕ obj (r) and additional smearing due to the detector) reads Here we observe that only the so-called phase contrast transfer function (PCTF) containing the symmetric aberrations produces visible contrast by convolution ( * ) with the phase ( Fig. 1(b)). Several strategies have been developed to optimize the transfer over certain spatial frequency bands. Most notably, in state-of-the-art instruments equipped with hardware-aberration correctors, the spherical aberration (C s ) and defocus can be traded to produce a positive contrast transfer for high resolution TEM (HRTEM) over a band, ultimately limited by the incoherent chromatic envelope of the instrument (also referred to as negative C s imaging conditions [40]). The latter limitation could be largely eliminated by employing chromatic aberration correctors [41], ultimately leading to an image spread limited resolution (due to Johnson noise [42]) in such instruments ( Fig. 1(b)). Upon inspection of the PCTF it becomes immediately clear that, if negative C s conditions are perfectly adjusted, it acts as a band pass in HRTEM conditions, mainly suppressing small spatial frequencies (-C s in Fig. 1(b)). This property complicates for instance the analysis of the 2DM's morphology such as determining the layer number representing large scale spatial structures (see, e.g. [43], for example on h-BN).
While the transfer of large spatial frequencies may be only increased through further improved electron optics and/or reconstructing multiple images with varying imaging conditions (e.g., focal series, [44], beam tilt series [45], or Ptychography [46]), both large and small spatial frequencies can be transferred simultaneously by employing off-axis holography. Here, the object is inserted half-way into the beam path, which automatically restricts the field of view (FOV) to edges of the 2D material, whereas the other half-space is occupied by the undisturbed reference wave. Both parts are brought to superposition by employing an electrostatic Möllenstedt signal transfer for imaging a weak phase object (WPO) and signal-to-noise (SNR) transfer functions for HRTEM in negative Cs conditions (−Cs) and EH (c.f. Eq. 9). In case of HRTEM, only the signal transfer from object phase to image amplitude described by the phase contrast transfer function (PCTF) contributes to the image. In case of EH, also the signal transfer from object phase to image phase described by the amplitude contrast transfer function (ACTF) contributes to the image. Additionally, the corresponding transferred bands for HRTEM in −Cs conditions with ± 2 nm defocus variation are also plotted. A defocus drift of about 2 nm is commonly observed after about five minutes in aberrationcorrected TEM instruments [37].
biprism, forming a hologram in the image plane ( Fig.  1(a)). From the latter a complex wave functioñ is reconstructed [47]. Here, the contrast factor µ = µ c MTF takes into account the illumination degree of coherence, inelastic scattering and instrumental instabilities (wrapped up in µ c ) and the modulation transfer function (MTF) of the detector. Since the whole wave function is reconstructed only the incoherent envelopes E (k) limit the transfer, in particular there is no damping of small spatial frequencies by the PCTF. For simplicity we will approximate the contrast factor µ (k) with that of the hologram carrier frequency µ (k c ) in the following.
In addition, the linear reconstruction principle allows to compute the noise transfer and hence the error (in terms of variance) pertaining to the reconstructed phase from the noise transfer function of the detector [48,49]. If we use this result and compare conventional HRTEM of WPOs with off-axis holography in terms of signal-tonoise ratio (SNR) of the phase contrast for a particular spatial frequency (see Appendix B for a detailed derivation), we obtain Here, DQE denotes a 2D generalization of the detection quantum efficiency as detailed in Appendix B. If this ratio becomes larger than 1, i.e.
off-axis holography is more dose-efficient than conventional phase contrast in terms of retrievable information per dose. Noting that a realistic value for the fringe contrast in high-resolution holograms recorded at modern TEMs equipped with state-of-the-art detectors and fieldemission guns can reach several 10% (in this work 30%, see below), this condition is met in a broad range of low to medium spatial frequencies (see Fig. 1(b)) but not for large spatial frequencies. Note, however, that the latter restriction may be overcome by the use of novel direct counting detectors with largely reduced detector DQEs [50,51].

B. A posteriori correction of residual aberrations
Following Fu and Lichte [38] the symmetric aberrations can be readily extracted from the phasesφ img of the image wave function in reciprocal space (Eq. 3) which follows from the antisymmetry of Fourier phases of the original WPO (see Appendix A for a detailed derivation). Here the appearance of the integer (n N) πambiguity stems from the 2πn ambiguity of the original wrapped phases. The above relation is remarkable as it allows to compute (and therefore correct) the symmetric part of the phase plate χ (k) (aberrations) without any a-priori knowledge about the object or the incoherent envelopes including the detector MTF. The only condition for a successful practical application is that the object spectrums SNR must be large enough to suppress error propagation of inevitable reconstructed noise and other artifacts (e.g., Fresnel fringes). A similar expression cannot be derived for the antisymmetric aberrations, because they do not produce an amplitude variation from the WPO (see Eq. (5)). Similar to the well-known Zemlin tableau method [52], they can be determined from a tilt series (where lower order symmetric aberrations are induced by higher order antisymmetric ones), or additional symmetry criteria. In particular, for the large class of centrosymmetric 2DMs we haveφ a = 0, π and hence Here, (n N) × π-ambiguity stems from the π-phases of the object. To finally correct for the aberrations from the holographically reconstructed wave functions, we multiply its Fourier transform with the complex conjugate of the WTF, i.e.Ψ Note, however, that this involves a phase unwrapping procedure removing the π-ambiguity in the phase plate, which can be challenging in practice, depending on the spectrum of the object wave. This currently limits the scope of the autocorrection scheme to pre-corrected imaging conditions (e.g., using hardware corrected TEMs), where only small residual aberrations and sufficiently small defoci are present, keeping the phase range within π over a large band. The above considerations are strictly correct for the WPO only. In praxis, this condition may be violated to some extend, e.g., when employing low-acceleration voltages (resulting in higher phase shifts) to reduce knock-on damage in a certain class of 2DMs. Note, however, that the "constant-amplitude" criterion also applies to pure phase objects and may be even slightly generalized to weak amplitude objects by minimizing a penalty term for the amplitude variations, e.g., Lehmann [53] and Ishizuka et al. [54] reported different approches to this aberration assesment via direct amplitude variation minimization for WPOs. It is currently an open question, whether and under which conditions this generalization yields unique solutions [55].

III. EXPERIMENTAL
To validate the autocorrection theory we apply the above machinery to a single to few atomic layer vander-Waals 2DM, namely hexagonal Boron Nitride (h-BN). h-BN has a crystal structure very similar to that of graphene (see Fig. 2(i)), but possesses completely different electronic properties (notably a large band gap, no Dirac points) [56,57]. The electron holograms have been recorded at a chromatic aberration (C c )-corrected TEM instrument, the TEAM I at the Molecular Foundry at the National Berkeley Lab., using the imaging conditions listed in Tab. I. The C c -correction, notably, allowed to resolve the {2110}-family of spatial frequencies not visible in a conventional C s (C 3 )-corrected electron microscope. A 20 minutes long time series of holograms acceleration voltage Ua 80 kV Cc and Cs(C3) < 10 µm image spread σi 40 pm information limit 0.13 nm diffraction lens excitation 65% pixel size of hologram 0.054 nm mean counts per hologram pixel I hol ∼ 10000 biprism voltage U bi 160 V fringe visibility µ (kc) [48] 0.3 DQE(kc) [48] 0.5 Table I. Holographic imaging conditions at TEAM I microscope adjusted for electron wave reconstruction of twodimensional materials.
was acquired using a 2k by 2k CCD camera (Model 894 US1000, Gatan Inc.), each with 8 seconds exposure time owing the great instrumental stability of the microscope in order to enhance the SNR. We had to slightly defocus the h-BN sample plane to have sufficient contrast for selecting the desired object position into the field of view. We note that the defocus, as well as the twofold astigmatism, were considerably drifting and that the electron induced charging of the sample is changing at a modest level over the time frame of the series. Significant, presumably knock-on induced beam damage can be observed over the 20 minutes, especially at the boundary to vacuum. The recorded holograms were then processed off-line through a removal of dead and hot pixels by an iterative local threshold algorithm, as well as a masking out of Fresnel fringes [58]. In addition, a deconvolution of the CCD camera's MTF and a modest Wiener filtering [55] in Fourier space were employed to increase the SNR of the holograms [55]. Within the holographic Fourier reconstruction method [47], one sideband was masked with a circular tenth-order Butterworth filter with a radius of 8.5 nm −1 . The phase of the reconstructed wave was subtracted by the phase reconstructed from an additionally recorded and equally processed object-free empty hologram, to correct for distortions induced by the fiber optics that couples the scintillator to the CCD camera. The as-reconstructed amplitude and phase of a small region of overlapping h-BN sheets are depicted in Figs. 2a,b. Since the sample is defocused, one observes an amplitude contrast by the PCTF. Moreover, a rather large 2-fold astigmatism and other residual aberrations seem to be present, rendering a quantitative analysis almost impossible. We now apply the auto-correction procedure outlined in Section II B. Figs. 2c-j show the results of the two auto-correction steps; the correction of symmetric aberrations from the WPO property (Figs. 2e-g) and the final correction including also antisymmetric aberrations after exploiting the centrosymmetry of the h-BN lattice (Figs. 2h-j). Clearly, the amplitude (Fig. 2e) is almost constant after removal of symmetric aberrations ( Fig. 2a and Fig. 2e are displayed within the same greylevels), proving the experimental feasability of the first auto-correction step.
The numerical phase plate (Fig. 2c) computed using Eq. (11) provides a good SNR only where the reciprocal space is filled with specimen information (i.e., h-BN systematic reflections, Fig. 2d). These are, however, sufficient to determine the geometrical aberration coefficients (within 95% confidence intervals) of first-order aberrations, namely defocus, C 1 = 4.0 ± 0.8 nm, and two-fold astigmatism,{A 1 = 2.5 ± 1.1 nm, α 1 = 54 • ± 25 • }, by fitting a smooth polynomial ( Fig. 2g) with the help of a Levenberg-Marquardt algorithm (third order symmetric aberrations are small and could safely be neglected). We note that the correction with the numerically obtained phase plate (Fig. 2c) yields almost identical results as the correction with the corresponding fitted phase plate (Fig. 2g).
As stated above, the determination of the numerical antisymmetric phase plate is merely possible for centrosymmetric specimen. Ignoring the small difference in atomic species, h-BN fulfills this symmetry condition for two different points, the center of the BN hexagons and the midpoints between the binding atoms. In order to find the most centrosymmetric region of interest, the for symmetric aberrations corrected phase was split up into sub images, containing about 25 unit cells. Subsequently, for all of them, a numerical measure for order the deviation from centrosymmetry was calculated. The most centrosymmetric sub image was finally used to determine the antisymmetric phase plate from Eq. (12), from which the corresponding phase plate (Fig. 2j) with a three-fold astigmatism of {A 2 = 158 ± 261 nm, α 2 = −90°± 16°} and an axial coma of {B 2 = 67 ± 49 nm, α B2 = −20°± 54°} was fitted. Note that the 95% confidence intervals computed from the fit residual are rather large in this case, which is due to the relatively small size of the symmetric patch used for the fitting procedure.
In order to prepare the autocorrected high-resolution phase data for interpretation we apply PCA denoising in a final step (Fig. 3). Again, no a-priori information about the material is required for this procedure. We rather exploit the regular geometric structure consisting of repeating honeycombs to create statistical data, which can be treated by PCA (i.e., model-free) denoising [59]. Our procedure consists of cutting out patches slightly larger than one honeycomb, centering them and subjecting the stack of patches to a PCA (see Appendix E for details). Inspecting the scree plot (Fig. 3(e)) we identified 11 nonnoise components and truncated the data accordingly. The thereby denoised image is shown in Fig. 3(b) together with the original data ( Fig. 3(a)). The deviations to the original data ( Fig. 3(d)) exhibit a noise-like Gaussian distribution (Fig. 3(c)) with the same standard deviation as the original phase noise in vacuum. Moreover, it shows no particular structure except at the edges, where fluctuations due to radiation damage occur during acquisition. We finally note that the the phase noise σ ϕ of the original data ( Fig. 3(a)) is consistent with that observed in the conventional HRTEM image data σ I (obtained from the noninterference terms of the recorded hologram) after rescaling σ 2 ϕ ≈ DQE (0) DQE (k c ) /σ 2 I µ c (k c ) 2 , experimentally validating the noise considerations leading to Eq. (9). Here, the phase noise is smaller than the noise in the "raw" phase image because of the slight Wiener filtering mentioned above.

IV. RESULTS AND DISCUSSION
After successful application of the auto-correction and denoising procedure we can now analyze the phase maps in detail. The high-resolution data (Fig. 5) clearly reveals regions of well-ordered honeycomb lattice with a multilayer morphology including various defects and edge structures. There are also regions, notably the bridge and some of the edges, where no or only a smeared-out honeycomb lattice is visible (facets are discernible though). These coincide with strongly oscillating parts of the sample, which were vanishing during the time series due to the electron irradiation. We therefore ascribe the loss of the high-spatial frequency data in this regions to local vibrations rather than some sort of amorphization. Similar observations and quantification of the lattice distortion at the edges and vacancies have also been observed by quantitative phase contrast imaging as well as STEM imaging [60,61].
Indeed, the quantitative phase data allows for a direct comparison with the projected potential of the h-BN lattice. We first focus on the low frequency information (which might have been obtained also with medium resolution holography modes and without the autocorrection of aberrations). To that end the high-resolution data is convoluted with a round top-hat function of the radius of a lattice constant (0.25 nm) and divided by C E (see Eq. (1)) yielding the averaged projected Coulomb potential corresponding to the zeroth Fourier component of the potential in a periodic lattice. It depends sensitively on the charge (de)localization due to chemical bonding [62] (see also Appendix C) and is proportional to the diamagnetic susceptibility according to the Langevin theory [63] amongst others. In our case the average potential data reveals (Fig. 4(a)) the presence of different sample thicknesses, ranging from one to five atomic layers, visible as areas of constant projected potential, interrupted by a small number of defects. In the histogram Fig. 4(b) these regions can be associated to clear distinct peaks. Moreover, the differences of the peak positions determined by Gaussians fits yield the average potentials of each layer (Fig. 4(c)), that show a small decrease towards higher layer numbers. Noting that the average potential corresponds approximately to the sum of the second spatial moment (i.e., spatial extension, see Appendix C for a derivation) of the charge distribution of the contributing atoms, i.e.,V ∼ r 2 at , this decrease in average potential could reflect a growing localization of the out-of-plane orbitals (3p z orbitals) in between h-BN layers as compared to the free surfaces. Indeed, the mono-and bilayer average projected potential of~4.5 Vnm rather fit with independent atom potentials computed from Hartree-Fock [64] (4.4 Vnm). The latter tend to be too delocalized compared to those computed from a full density functional theory (DFT) calculation (using FPLO-18 [65], see Appendix D) including chemical bonds and correlation that agrees with a value of 3.4 Vnm better to three and more layers (Fig. 4(c)). Note furthermore that the structurally and atom-weigth-wise closely related graphene has a projected potential of 4.5 Vnm, which also reflects the stronger delocalization of the shell electrons in conducting graphene. Another possible explanation of the large potential values could be the positive charging of the (insulating) h-BN in the beam (ejection of secondary electrons).
A second noticeable feature is the potential increase visible at the edges and steps of the sample (most prominent in the doublelayer bridge region). Different physical effects may be attributed to this potential elevation: 1. Formation of (covalent) interlayer bonds at the zigzag edges of bilayer h-BN, as proposed by Alem et al. [61], through the following mechanisms: a local compression of the projected atomic positions or the tilting of the covalent B-N bonds out of plane, both yield a raised projected potential. These interlayer bonds could also explain the enhanced stability of even numbered layers under electron irradiation and thus their dominant appearance in the data set.
2. Delocalization of in-plane s, p x , p y orbitals into vacuum, which could potentially lead to the observed increase of the average potential. Indeed, DFT calculations reported in literature predict the emergence of metallic edge states at the zig-zag edges [66,67].
3. Systematic adhesion of residual gas atoms with different atomic potentials at the edges, such as Oxygen [68].
Further studies are necessary to clarify and disentangle those effects quantitatively. We now turn to the analysis of the autocorrected and denoised high-resolution data shown in Fig. 5(a). The denoised high-resolution data of the monolayer allows to distinguish between the B and N sites in the monolayer (Fig. 5(d)). Comparing the holographically measured potentials with the ab-initio potentials, which have been smeared out by multiplying the envelope function pertaining to the TEAM I instrument at 80 kV (see Fig.  1(b)), we observe good agreement with an additional smearing of the experimental data along the bonding directions ( Fig. 5(f)). Whether this is due to thermal vibrations (not included in our analysis) or other damping factors remains an open question at this stage. Turning to the edge structures, we observe that the zig-zag boundary is the prevalent configuration ( Fig. 5(b),(c)), which coincides with the ab-initio predictions [69]. We further note occasional distortions of the edge lattice, which may be attributed to some out-of-plane bending or ongoing beam damage (c.f. [60,61]). Moreover, almost all edges reveal an increase in projected potentials, which has been already discussed above . That notably also includes steps (e.g., the step from 2 L to 4 L in Fig.5(c)). As noted previously we attribute these localized edge potentials to an increased electron delocalization, most probably due to a reconstruction of the edge structure along z-direction including the formation of interlayer covalent bonds [61]. A detailed comparison to the emergence of particular edge states and the electronic configuration of defects (e.g., BN void depicted in Fig. 5(e)) is, however, beyond the scope of this work and will be conducted elsewhere.
Summing up we showed, how autocorrected off-axis holography may be used as a high-resolution and dose efficient probe for 2DMs. In this regard we performed the first parameter-free correction for antisymmetric aberrations based on the afore-mentioned symmetry principles. Notably the presented aberration autocorrection scheme and the favorable noise transfer properties of EH facilitates the reconstruction of quantitative potential data over a large spatial frequency band extending from zero to atomic resolution. The autocorrection scheme facilitates a removal of residual aberrations and defocus without the need to separately measuring or fine tuning them, rendering it a suitable method for in-situ studies, where long time-series need to be recorded. The high-structural order of 2DMs furthermore facilitated an efficient suppression of noise by adopting a PCA noise removal algorithm. Using these capabilities, we reveal several properties of h-BN. Notably, the electronic orbitals in h-BN are significantly more localized as in the structurally similar graphene, resulting in a comparatively low mean projected potential of 3.6 Vnm. We could confirm that edges favor the zig-zag configuration and found a peculiar localized increase of the potential at the edges. The latter is attributed to the delocalization of electron edge states. with the MTF denoting the modulation transfer and NPS the (normalized) white noise power spectrum of the detector. The phase SNR then reads The noise analysis for conventional weak phase contrast HRTEM starts with the shot noise amplified by the detector from which the SNR is readily derived inserting the relation between I and the phase (PCTF) We finally arrive at for the ratio between both as noted in the main text sin χ s DQE (k c ) . (B6)

Appendix C: Mean Inner Potential
Establishing a well-defined relationship between averaged projected potential or projected mean inner potential of a 2DM and the charge distribution holds some pitfalls in the infinite crystal limit [70]. Indeed, a mean inner potential is not well-defined in this case and depends on fixing boundary conditions or a reference. To circumvent this problem, ab-inito calculations of MIPs of bulk crystals have been carried out for slab geometries, containing a sufficiently large vacuum region fixing the reference. In case of a finite crystal (as observed experimentally), we may start off with dividing the 2DM domain into "atomic" cells, which shall contain one atom each but are not further specified at this stage. The projected average of the potential over a certain area A , i.e.
can now be computed as a sum of the atomic contributions, which are contained within the area. This is most conveniently done in Fourier space employing the Poisson equation The atomic averaged potential now corresponds to the value at zero spatial frequencỹ which is not determined in the required limit as both nominator and denominator tend to zero. To solve that expression we may apply l'Hospital's rule to the average over the full solid angle of the previous expression (to remove the ϕ, θ dependency) e ikr(sin θ k sin θr cos(ϕ k −ϕr)+cos θ k cos θr) = − 1 2ε 0ˆd r 3 r 2 ρ at (r) (sin θ k sin θ r cos(ϕ k −ϕ r ) + cos θ k cos θ r ) 2 = − 1 2ε 0ˆd r 3 r 2 ρ at (r) sin 2 θ k sin 2 θ r cos 2 (ϕ k −ϕ r ) + cos 2 θ k cos 2 θ r = − 2π 3ε 0ˆd rr 2 ρ at (r) Note that we had to apply l'Hospital's rule twice, which requires both the zeroth and any first moment with respect to r to vanish. Consequently, the "atomic" cells, used for partitioning the 2D domain, have to be charge neutral and dipole-moment-free. That condition restricts the allowed shapes and positions of the initially undefined atomic patch choice. Noting that dipole-free atomic cells are centered closely to the atomic positions in h-BN, we have that the averaged potential is a measure for the size of the electron cloud around the atoms.