Autocorrected Off-Axis Holography of 2D Materials

Autocorrected Oﬀ-axis Holography of 2D Materials

Felix Kern,1 Martin Linck,2 Daniel Wolf,1 Nasim Alem,3 Himani Arora,4 Sibylle Gemming,4 Artur Erbe,4 Alex Zettl,5 Bernd Büchner,1 and Axel Lubk1 1Institute for Solid State Research, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany 2Corrected Electron Optical Systems GmbH, Englerstr. 28, D-69126 Heidelberg, Germany 3Department of Materials Science and Engineering, Pennsylvania State University, N-210 Millennium Science Complex University Park, PA 16802, United States 4Helmholtz-Zentrum Dresden-Rossendorf, Dresden 01328, Germany 5Department of Physics, University of California Berkeley, 366 LeConte Hall MC 7300 Berkeley, CA, California 94720-7300, USA

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 and have been addressed through various methodologi- cal developments of TEM techniques: The discovery of graphene and its intriguing proper- (A) 2D materials are typically more susceptible to var- ties more than ten years ago [1, 2] has sparked large and ious beam damage mechanisms than their 3D counter- ongoing research efforts into two-dimensional materials parts [13–15]. That includes knock-on damage, radioly- (2DMs). The synthesis of novel 2DMs, comprising, e.g., sis, and chemical etching. The knock-on damage may be 2D topological insulators, 2D magnets, or organic sys- reduced by lowering the acceleration voltage and hence tems like 2D polymers, with single to few layers thickness the kinetic energy of the beam electron below the knock- and high structural definition at the atomic / molecular on threshold of the pertinent chemical bonds in the 2DM level is at the center of this field (e.g., [3–6]). They ex- (e.g., 90 keV for the C-C bond in graphene [16] and 40 hibit a large range of physical properties triggered by the keV for the B-N bond in monolayer h-BN [17]). Radiol- reduced dimensionality in one direction such as quantum ysis and etching follow more complicated reaction mech- confinement effects or weak dielectric screening from the anisms [18]. Their magnitude might be reduced by low- environment, yielding a significant enhancement of the ering the temperature and optimizing the acceleration Coulomb interaction [3, 7]. Another interesting aspect is voltage [16]. the formation of out-of-plane elastic modulations, which (B) Most 2D materials belong to the family of weak stabilize the 2DM structure and modify its mechanical scatterers (another important member is biological mat- properties [8–10]. The missing third dimension also en- ter, mainly consisting of C and H atoms), which implies hances the proliferation and impact of defects, such as that they only (weakly) shift the phase of the electron point and line defects, grains or multilayers morphologies; wave when traversing the sample, but don’t modulate the which inevitably occur upon synthesis and often govern amplitude. Consequently, they are referred to as weak the functionality (e.g., reactivity, stability) of 2DMs in phase objects (WPOs). This phase shift of the electron applications [11]. wave can not be measured directly, due to the quantum Therefore the development of microscopic characteri- mechanical phase detection problem. To solve this prob- zation methods, which allow to analyze the structure and lem one can employ phase plates, which enable the imag- arXiv:2006.13855v2 [cond-mat.mes-hall] 25 Jun 2020 electronic properties of the 2DMs including the edges, de- ing of the phase shift introduced by these materials as fects and grain boundaries, is at the center of the field. intensity contrast. The use of either physical [19–22] or Transmission electron microscopy (TEM) has been a cor- electron optical phase plates [19, 23–25], however, comes nerstone technique (others are scanning tunneling mi- with some merits and disadvantages. The former degrade croscopy STM and photo emission electron microscopy [21, 22, 26] during use and create unwanted diffuse scat- PEEM), offering high resolving power and spectroscopic tering and beam blocking [27], whereas the latter is typi- information. A breakthrough for TEM could be achieved cally constructed from materials that are prone to charg- by employing chromatic aberration correction facilitat- ing. Notable exceptions are laser [28] and drift tube [25] ing high-spatial resolution at low-acceleration voltages phase plates, which are very demanding construction- [12]. Two central challenges required special attention and implementation-wise. By far the most straight for- 2 ward method for transfer of phase contrast to intensity terms of charge (de)localizations. constrast, however, is an additional defocus with respect The paper is organized as follows, we first recapitu- to the object exit plane, which has the negative side effect late the imaging principles of weak scatterers (phase ob- of introducing transfer gaps at low spatial frequencies or jects) and off-axis holography. From these, we derive how an oscillating contrast transfer for large spatial frequen- geometric aberrations can be determined and automat- cies (see below). Introducing large defocii also results in ically corrected a-posteriori from an acquired hologram a reduction of the resolution due to the partial transver- without additional measurements. We elaborate on the sal coherence of the electrons, which may be expressed noise characteristics of the thereby obtained aberration- by an exponential envelope function in reciprocal space. corrected image phase and the spatial resolution of the In the following we address the phase problem in weak determined aberrations (e.g., defocus due to out-of-plane scatterers (at the example of 2DMs) by advancing off-axis modulations). We finally demonstrate the feasibility of electron holography, an interferometric technique allow- the approach at the example of h-BN. Amongst others ing to reconstruct the phase shift of the electron wave we reconstruct the number of layers, the mean inner po- over the whole spatial frequency band, up to the infor- tential (MIP) of individual layers, the structure of the mation limit. These advantages have triggered a small monolayer as well as the edges; and correlate this to ma- number of previous studies on WPOs, notably at bio- terial properties such as the charge delocalization or the logical materials [29, 30] and 2DMs [31–36]. However, stability and electronic properties of the edge states. 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, how- II. IMAGING PRINCIPLES ever, is mandatory for an analysis of the acquired phase in terms of physical quantities such as potentials, charge A. Conventional imaging and off-axis holography densities and the atomic structure. In Winkler et al. of weak phase object [35, 36] this problem has been addressed by a model- based fitting approach requiring a full model of the scat- Sufficiently thin TEM specimens (with the critical tering potential and hence the 2DM under investigation. thickness depending on the atomic scattering potential V Here we follow a different approach requiring no or of the chemical constituents) behave as weak phase ob- only minimal a-priori knowledge of the sample, that is, ject (WPOs) in TEM. In good approximation they just stripping the recorded data from any instrumental influ- impose a small phase shift ences, 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 +t/2 of requiring no a-priori knowledge about the specimen ϕobj (r) = CE V (r, z) dz (1) and a clear separation between instrumental and speci- ˆ men influences. One key idea is to exploit very general −t/2 discrete symmetries pertaining to the scattered electron wave function: First of all, the weak scattering property on the beam electrons’ wave function Ψobj leaving the induces an odd symmetry in the object phases in Fourier object of thickness t. Here CE is an electron-energy- space, which allows to correct for symmetric aberrations. dependent interaction constant (0.01 rad/V nm at 80 keV), Second, a large class of 2DMs are centrosymmetric, intro- z the direction in which the electron beam transmits the ducing an even symmetry in the Fourier object phases, sample, and r the 2D position vector in the object plane. which allows to correct for antisymmetric aberrations. Since WPOs do not modulate the amplitude A, Ψobj can This approach is based on the original work of Fu and be approximated by Lichte [38], who demonstrated how to generically extract symmetric aberrations from holograms at the example of (WPO) amorphous carbon foils. Here, we go one step further Ψobj (r) ≈ Aobj (1 + iϕobj (r)) . (2) and autocorrect for the aberrations in the reconstructed wave of 2DMs, which greatly facilitates the analysis of The dominant geometric aberrations of the objective lens the phase in terms of physical data, i.e., projected po- are described by a phase function χ (k) acting on the elec- tentials. In this regard, we follow D. Gabor’s original tron wave spectrum by a complex factor e−iχ(k), the wave idea of holography as a means to aberration correction transfer function (WTF), in reciprocal space. Moreover, [39]. The corrected data is then subjected to a principle the combination of (transversal and longitudinal) partial component analysis (PCA) denoising, which reveals the coherence and geometric aberrations leads to an exponen- meaningful phase data at the atomic scale and enables tial damping of spatial frequencies in the wave function extracting the underlying projected potential. Last but described by a real envelope function E (k) in recipro- not least we compare that data to ab-initio density func- cal space. The Fourier transform of the so-called image tional calculations to analyze the measured potentials in wave function taking into account these modulations by 3 the imaging system (Fig. 1(a)) reads ˜ ˜ −iχ(k) Ψimg (k) = Ψobj (k) e E (k) (3) ( ) WPO 00 ≈ Aobj δ (k) +e −i(χs(k)+χa(k)) ie E (k)ϕ õbj (k) . (4)

Here we separated the antisymmetric and symmetric aberrations, χa (k) and χs (k), respectively. The corresponding conventional linear image intensity (neglecting the term quadratic in ϕobj (r) and additional smearing due to the detector) reads

2 I (r) = |Ψimg (r)| (5) ≈ A2 +     2 −1 −iχa(k) 2A F sin χs (k) E (k)e ∗ ϕobj (r)(6). | {z }  PCTF 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 de- Figure 1. (a) Off-axis electron holography (EH) setup, (b) veloped to optimize the transfer over certain spatial fre- signal transfer for imaging a weak phase object (WPO) and quency bands. Most notably, in state-of-the-art instru- signal-to-noise (SNR) transfer functions for HRTEM in neg- ments equipped with hardware-aberration correctors, the ative Cs conditions (−Cs) and EH (c.f. Eq. 9). In case of spherical aberration (Cs) and defocus can be traded to HRTEM, only the signal transfer from object phase to image produce a positive contrast transfer for high resolution amplitude described by the phase contrast transfer function TEM (HRTEM) over a band, ultimately limited by the (PCTF) contributes to the image. In case of EH, also the sig- incoherent chromatic envelope of the instrument (also re- nal transfer from object phase to image phase described by the ferred to as negative Cs imaging conditions [40]). The lat- amplitude contrast transfer function (ACTF) contributes to ter limitation could be largely eliminated by employing the image. Additionally, the corresponding transferred bands −C chromatic aberration correctors [41], ultimately leading for HRTEM in s conditions with ± 2 nm defocus variation are also plotted. A defocus drift of about 2 nm is to an image spread limited resolution (due to Johnson commonly observed after about five minutes in aberration- noise [42]) corrected TEM instruments [37]. 2 2 2 E (k) = exp −2π σi k (7) biprism, forming a hologram in the image plane (Fig. in such instruments (Fig. 1(b)). Upon inspection of the 1(a)). From the latter a complex wave function PCTF it becomes immediately clear that, if negative Cs conditions are perfectly adjusted, it acts as a band pass ˜ −1 n ˜ o in HRTEM conditions, mainly suppressing small spatial Ψhol (r) = F µ (k) Ψimg (k) (8) frequencies (-Cs in Fig. 1(b)). This property complicates for instance the analysis of the 2DM’s morphology such is reconstructed [47]. Here, the contrast factor µ = as determining the layer number representing large scale µcMTF takes into account the illumination degree of co- spatial structures (see, e.g. [43], for example on h-BN). herence, inelastic scattering and instrumental instabil- While the transfer of large spatial frequencies may ities (wrapped up in µc) and the modulation transfer be only increased through further improved electron op- function (MTF) of the detector. Since the whole wave tics and/or reconstructing multiple images with vary- function is reconstructed only the incoherent envelopes ing imaging conditions (e.g., focal series, [44], beam tilt E (k) limit the transfer, in particular there is no damping series[45], or Ptychography [46]), both large and small of small spatial frequencies by the PCTF. For simplicity spatial frequencies can be transferred simultaneously by we will approximate the contrast factor µ (k) with that employing off-axis holography. Here, the object is in- of the hologram carrier frequency µ (kc) in the following. serted half-way into the beam path, which automatically In addition, the linear reconstruction principle allows restricts the field of view (FOV) to edges of the 2D ma- to compute the noise transfer and hence the error (in terial, whereas the other half-space is occupied by the terms of variance) pertaining to the reconstructed phase undisturbed reference wave. Both parts are brought to from the noise transfer function of the detector [48, 49]. superposition by employing an electrostatic Möllenstedt If we use this result and compare conventional HRTEM 4 of WPOs with off-axis holography in terms of signal-to- particular, for the large class of centrosymmetric 2DMs noise ratio (SNR) of the phase contrast for a particular we have ϕã = 0, π and hence spatial frequency (see Appendix B for a detailed deriva- 1 tion), we obtain χa(k) = − (ϕ ˜img (k) − ϕ˜img (−k)) + πn (k) (12) 2 p SNRhol µc (kc) DQE (0) = . (9) Here, (nN) × π-ambiguity stems from the π-phases of SNR p conv sin χs DQE (kc) the object. To finally correct for the aberrations from the holo- Here, DQE denotes a 2D generalization of the detection graphically reconstructed wave functions, we multiply quantum efficiency as detailed in Appendix B. If this ra- its Fourier transform with the complex conjugate of the 1 tio becomes larger than , i.e. WTF, i.e. p µc (kc) DQE (0) Ψ˜ (k) E (k) = Ψ˜ (k) eiχ(k) . p > 1 , (10) obj img (13) sin χs DQE (kc) Note, however, that this involves a phase unwrapping off-axis holography is more dose-efficient than conven- procedure removing the π-ambiguity in the phase plate, tional phase contrast in terms of retrievable information which can be challenging in practice, depending on the per dose. Noting that a realistic value for the fringe con- spectrum of the object wave. This currently limits the trast in high-resolution holograms recorded at modern scope of the autocorrection scheme to pre-corrected imag- TEMs equipped with state-of-the-art detectors and field- ing conditions (e.g., using hardware corrected TEMs), emission guns can reach several 10% (in this work 30%, where only small residual aberrations and sufficiently see below), this condition is met in a broad range of low small defoci are present, keeping the phase range within to medium spatial frequencies (see Fig. 1(b)) but not for π over a large band. large spatial frequencies. Note, however, that the latter The above considerations are strictly correct for the restriction may be overcome by the use of novel direct WPO only. In praxis, this condition may be violated to counting detectors with largely reduced detector DQEs some extend, e.g., when employing low-acceleration volt- [50, 51]. ages (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 B. A posteriori correction of residual aberrations phase objects and may be even slightly generalized to weak amplitude objects by minimizing a penalty term Following Fu and Lichte [38] the symmetric aberra- for the amplitude variations, e.g., ϕ˜ tions can be readily extracted from the phases img of the image wave function in reciprocal space (Eq. 3) ˜ iχ(k) χ (k) = arg min ∇ Ψimg (k) e . (14) 1 π χ (k) = − (ϕ ˜ (k) +ϕ ˜ (−k)) + + πn (k) , et al. s 2 img img 2 Lehmann [53] and Ishizuka [54] reported different (11) approches to this aberration assesment via direct amplitude variation minimization for WPOs. It is currently an which follows from the antisymmetry of Fourier phases open question, whether and under which conditions this of the original WPO (see Appendix A for a detailed generalization yields unique solutions [55]. derivation). Here the appearance of the integer (nN) π- ambiguity stems from the 2πn ambiguity of the original wrapped phases. The above relation is remarkable as it III. EXPERIMENTAL allows to compute (and therefore correct) the symmetric part of the phase plate χ (k) (aberrations) without any To validate the autocorrection theory we apply the a-priori knowledge about the object or the incoherent en- above machinery to a single to few atomic layer van- velopes including the detector MTF. The only condition der-Waals 2DM, namely hexagonal Boron Nitride (h- for a successful practical application is that the object BN). h-BN has a crystal structure very similar to that spectrums SNR must be large enough to suppress error of graphene (see Fig. 2(i)), but possesses completely dif- propagation of inevitable reconstructed noise and other ferent electronic properties (notably a large band gap, artifacts (e.g., Fresnel fringes). no Dirac points) [56, 57]. The electron holograms have A similar expression cannot be derived for the anti- been recorded at a chromatic aberration (Cc)-corrected symmetric aberrations, because they do not produce an TEM instrument, the TEAM I at the Molecular Foundry amplitude variation from the WPO (see Eq. (5)). Sim- at the National Berkeley Lab., using the imaging condi- ilar to the well-known Zemlin tableau method [52], they tions listed in Tab. I. The Cc-correction, notably, al- can be determined from a tilt series (where lower order lowed to resolve the {2110}-family of spatial frequencies symmetric aberrations are induced by higher order an- not visible in a conventional Cs (C3)-corrected electron tisymmetric ones), or additional symmetry criteria. In microscope. A 20 minutes long time series of holograms 5

acceleration voltage Ua 80 kV The numerical phase plate (Fig. 2c) computed using Cc and Cs(C3) < 10 µm Eq. (11) provides a good SNR only where the recipro- image spread σi 40 pm cal space is filled with specimen information (i.e., h-BN information limit 0.13 nm systematic reflections, Fig. 2d). These are, however, suffi- diffraction lens excitation 65% cient to determine the geometrical aberration coefficients pixel size of hologram 0.054 nm (within 95% confidence intervals) of first-order aberra- mean counts per hologram pixel Ihol ∼ 10000 tions, namely defocus, C1 = 4.0 ± 0.8 nm, and two-fold biprism voltage Ubi 160 V ◦ ◦ astigmatism,{A1 = 2.5 ± 1.1 nm, α1 = 54 ± 25 }, by fit- fringe visibility µ (kc)[48] 0.3 ting a smooth polynomial DQE(kc)[48] 0.5 2π 2 Table I. Holographic imaging conditions at TEAM I mi- χs = k (C1 + A1 cos (2α − α1)) (15) k0 croscope adjusted for electron wave reconstruction of two- dimensional materials. (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) was acquired using a 2k by 2k CCD camera (Model 894 yields almost identical results as the correction with the US1000, Gatan Inc.), each with 8 seconds exposure time corresponding fitted phase plate (Fig. 2g). owing the great instrumental stability of the microscope As stated above, the determination of the numerical in order to enhance the SNR. We had to slightly defo- antisymmetric phase plate is merely possible for cen- h cus the -BN sample plane to have sufficient contrast trosymmetric specimen. Ignoring the small difference in for selecting the desired object position into the field of atomic species, h-BN fulfills this symmetry condition for view. We note that the defocus, as well as the two- two different points, the center of the BN hexagons and fold astigmatism, were considerably drifting and that the midpoints between the binding atoms. In order to the electron induced charging of the sample is chang- find the most centrosymmetric region of interest, the for ing at a modest level over the time frame of the series. symmetric aberrations corrected phase was split up into Significant, presumably knock-on induced beam dam- sub images, containing about 25 unit cells. Subsequently, age can be observed over the 20 minutes, especially at for all of them, a numerical measure for order the de- the boundary to vacuum. The recorded holograms were viation from centrosymmetry was calculated. The most then processed off-line through a removal of dead and centrosymmetric sub image was finally used to determine hot pixels by an iterative local threshold algorithm, as the antisymmetric phase plate from Eq. (12), from which well as a masking out of Fresnel fringes [58]. In addi- the corresponding phase plate (Fig. 2j) tion, a deconvolution of the CCD camera’s MTF and 3 a modest Wiener filtering [55] in Fourier space were em- k 1 χa = 2π 2 A2 cos (3α − αA2 ) ployed to increase the SNR of the holograms [55]. Within k0 3 the holographic Fourier reconstruction method [47], one 1 sideband was masked with a circular tenth-order But- + B2 cos (α + αB2 ) (16) 1 terworth filter with a radius of 8.5 nm−1. The phase of the reconstructed wave was subtracted by the phase re- with a three-fold astigmatism of constructed from an additionally recorded and equally {A2 = 158 ± 261 nm, α2 = −90° ± 16°} and an axial processed object-free empty hologram, to correct for dis- coma of {B2 = 67 ± 49 nm, αB2 = −20° ± 54°} was tortions induced by the fiber optics that couples the scin- fitted. Note that the 95% confidence intervals computed tillator to the CCD camera. The as-reconstructed ampli- from the fit residual are rather large in this case, which tude and phase of a small region of overlapping h-BN is due to the relatively small size of the symmetric patch sheets are depicted in Figs. 2a,b. Since the sample is used for the fitting procedure. defocused, one observes an amplitude contrast by the In order to prepare the autocorrected high-resolution PCTF. Moreover, a rather large 2-fold astigmatism and phase data for interpretation we apply PCA denoising in other residual aberrations seem to be present, rendering a final step (Fig. 3). Again, no a-priori information about a quantitative analysis almost impossible. We now apply the material is required for this procedure. We rather the auto-correction procedure outlined in Section II B. exploit the regular geometric structure consisting of re- Figs. 2c-j show the results of the two auto-correction peating honeycombs to create statistical data, which can steps; the correction of symmetric aberrations from the be treated by PCA (i.e., model-free) denoising [59]. Our WPO property (Figs. 2e-g) and the final correction in- procedure consists of cutting out patches slightly larger cluding also antisymmetric aberrations after exploiting than one honeycomb, centering them and subjecting the the centrosymmetry of the h-BN lattice (Figs. 2h-j). stack of patches to a PCA (see Appendix E for details). Clearly, the amplitude (Fig. 2e) is almost constant after Inspecting the scree plot (Fig. 3(e)) we identified 11 non- removal of symmetric aberrations (Fig. 2a and Fig. 2e noise components and truncated the data accordingly. are displayed within the same greylevels), proving the The thereby denoised image is shown in Fig. 3(b) to- experimental feasability of the first auto-correction step. gether with the original data (Fig. 3(a)). The deviations 6

Figure 3. Denoising of of the aberration-corrected h-BN phase image reconstructed by off-axis electron holography. (a) shows original autocorrected data and corresponding standard deviations of phase σϕ and intensity σI in vacuum, (b) the PCA-denoised data after truncation to the first 11 principal components. The selection criterion is the last kink in the Figure 2. Aberration correction of h-BN image wave recon- scree plot (e). Both the difference (d) and the histogram (e) structed by off-axis electron holography. (a) and (b) show the reveal no noticeable deviation from Gaussian noise between as-reconstructed wave image in amplitude and phase. The the original and denoised image except at some structurally symmetric aberration phase plate (c) as determined from Eq. fluctuating edges. (11) is not very evident, however, it becomes much clearer in (d) the color coded representation of (c) overlayed with the Fourier spectrum of the image wave (a,b). (e) and (f) depict amplitude and phase corrected for symmetric aberrations us- multilayer morphology including various defects and edge ing the continuous phase plate (g) fitted from (c). (h) and structures. There are also regions, notably the bridge and (i) show amplitude and phase corrected also for antisymmet- some of the edges, where no or only a smeared-out hon- ric aberrations (coma and 3-fold astigmatism) obtained from eycomb lattice is visible (facets are discernible though). Eq. (12) using the phase plate (j). The crystal structure of a These coincide with strongly oscillating parts of the sam- h-BN monolayer is indicated in (i). ple, 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 to the original data (Fig. 3(d)) exhibit a noise-like Gaus- vibrations rather than some sort of amorphization. Sim- sian distribution (Fig. 3(c)) with the same standard devi- ilar observations and quantification of the lattice distor- ation as the original phase noise in vacuum. Moreover, it tion at the edges and vacancies have also been observed shows no particular structure except at the edges, where by quantitative phase contrast imaging as well as STEM fluctuations due to radiation damage occur during acqui- imaging [60, 61]. sition. We finally note that the the phase noise σϕ of the Indeed, the quantitative phase data allows for a direct original data (Fig. 3(a)) is consistent with that observed comparison with the projected potential of the h-BN lat- in the conventional HRTEM image data σI (obtained tice. We first focus on the low frequency information from the noninterference terms of the recorded hologram) (which might have been obtained also with medium reso- 2 2 2 after rescaling σϕ ≈ DQE (0) DQE (kc) /σI µc (kc) , ex- lution holography modes and without the autocorrection perimentally validating the noise considerations leading of aberrations). To that end the high-resolution data is to Eq. (9). Here, the phase noise is smaller than the noise convoluted with a round top-hat function of the radius in the “raw” phase image because of the slight Wiener fil- of a lattice constant (0.25 nm) and divided by CE (see tering mentioned above. Eq. (1)) yielding the averaged projected Coulomb potential corresponding to the zeroth Fourier component of the potential in a periodic lattice. It depends sen- IV. RESULTS AND DISCUSSION sitively on the charge (de)localization due to chemical bonding [62] (see also Appendix C) and is proportional to After successful application of the auto-correction and the diamagnetic susceptibility according to the Langevin denoising procedure we can now analyze the phase maps theory[63] amongst others. In our case the average po- in detail. The high-resolution data (Fig. 5) clearly re- tential data reveals (Fig. 4(a)) the presence of differ- veals regions of well-ordered honeycomb lattice with a ent sample thicknesses, ranging from one to five atomic 7 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 ¯ P 2 of the contributing atoms, i.e., V ∼ r at, this decrease in average potential could reflect a growing localization of the out-of-plane orbitals (3pz orbitals) in between h-BN layers as compared to the free surfaces. In- deed, 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 h from a full density functional theory (DFT) calculation Figure 4. Average potential analysis of -BN. The overview image (b) contains several flakes of different layer number, (using FPLO-18[65], see Appendix D) including chem- shaped by electron beam irradiation. The potential data his- ical bonds and correlation that agrees with a value of togram (b) reveals the presence of several monolayers, seper- 3.4 Vnm better to three and more layers(Fig. 4(c)). Note ated by well-defined potential offsets ∆V (c). The linescanes furthermore that the structurally and atom-weigth-wise (d) provide local measures for the latter and reveal the pres- closely related graphene has a projected potential of 4.5 ence of potential bumps at edges and steps. 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 denoised high-resolution data of the monolayer allows to h positive charging of the (insulating) -BN in the beam distinguish between the B and N sites in the monolayer (ejection of secondary electrons). (Fig. 5(d)). Comparing the holographically measured A second noticeable feature is the potential increase potentials with the ab-initio potentials, which have been visible at the edges and steps of the sample (most promi- smeared out by multiplying the envelope function per- nent in the doublelayer bridge region). Different physical taining to the TEAM I instrument at 80 kV (see Fig. effects may be attributed to this potential elevation: 1(b)), we observe good agreement with an additional 1. Formation of (covalent) interlayer bonds at the zig- smearing of the experimental data along the bonding di- zag edges of bilayer h-BN, as proposed by Alem et rections (Fig. 5(f)). Whether this is due to thermal vi- al. [61], through the following mechanisms: a local brations (not included in our analysis) or other damping compression of the projected atomic positions or factors remains an open question at this stage. Turn- the tilting of the covalent B-N bonds out of plane, ing to the edge structures, we observe that the zig-zag both yield a raised projected potential. These in- boundary is the prevalent configuration (Fig. 5(b),(c)), terlayer bonds could also explain the enhanced sta- which coincides with the ab-initio predictions [69]. We bility of even numbered layers under electron irra- further note occasional distortions of the edge lattice, diation and thus their dominant appearance in the which may be attributed to some out-of-plane bending or data set. ongoing beam damage (c.f. [60, 61]). Moreover, almost all edges reveal an increase in projected potentials, which 2. Delocalization of in-plane s, px , py orbitals into has been already discussed above . That notably also in- vacuum, which could potentially lead to the ob- cludes steps (e.g., the step from 2 L to 4 L in Fig.5(c)). As served increase of the average potential. Indeed, noted previously we attribute these localized edge poten- DFT calculations reported in literature predict the tials to an increased electron delocalization, most prob- emergence of metallic edge states at the zig-zag ably due to a reconstruction of the edge structure along edges [66, 67]. z-direction including the formation of interlayer covalent bonds [61]. A detailed comparison to the emergence of 3. Systematic adhesion of residual gas atoms with dif- particular edge states and the electronic configuration of ferent atomic potentials at the edges, such as Oxy- defects (e.g., BN void depicted in Fig. 5(e)) is, however, gen [68]. beyond the scope of this work and will be conducted else- Further studies are necessary to clarify and disentangle where. those effects quantitatively. Summing up we showed, how autocorrected off-axis We now turn to the analysis of the autocorrected and holography may be used as a high-resolution and dose ef- denoised high-resolution data shown in Fig. 5(a). The ficient probe for 2DMs. In this regard we performed the 8

Fund of the Helmholtz Association of German Research Centers through the International Helmholtz Research School for Nanoelectronic Networks, IHRS NANONET (VH-KO-606).

Appendix A: WPO Phase Plate

This appendix contains derivations for the expressions (11) and (12) relating holographic data and aberrations. We start with rewriting the expression for the suitably normalized Fourier transformed image wave (3) using iPϕ˜ ϕ˜ = Aϕ˜e

Ψ˜img − 2πδ (k) ˜ Figure 5. High-resolution potential analysis (a). Zoom- ˜ iφimg = Aimge (A1) ins show zig-zag steps comprising two atomic layers (b,c), a A monolayer region (d) and a BN void defect (r). The mono- −i(χs(k)+χa(k)−Pϕ˜(k)) = iAϕ˜ (k) e E (k) . layer potential is depicted together with the DFT result in (f). Since ϕ (r) is a real function we have Pϕ˜ (k) = −Pϕ˜ (−k) and hence

first parameter-free correction for antisymmetric aberra- Pϕ˜ (k) mod2π + Pϕ˜ (−k) mod2π = 2πn, nN. (A2) tions based on the afore-mentioned symmetry principles. Notably the presented aberration autocorrection scheme Consequently, and the favorable noise transfer properties of EH facil- 1 ˜ ˜ π χs(k) = − φimg (k) + φimg (−k) + + πn . (A3) itates the reconstruction of quantitative potential data 2 2 over a large spatial frequency band extending from zero to atomic resolution. The autocorrection scheme facili- Following a similar line of reasoning and assuming an tates a removal of residual aberrations and defocus with- centrosymmetric object, i.e., Pϕ˜ (k) {0, π} we have out the need to separately measuring or fine tuning them, 1 ˜ ˜ in-situ χa(k) = − φimg (k) − φimg (−k) + πn (A4) rendering it a suitable method for studies, where 2 long time-series need to be recorded. The high-structural order of 2DMs furthermore facilitated an efficient sup- Here the πn stems from the possible π phases of the real pression of noise by adopting a PCA noise removal algo- centrosymmetric object. rithm. 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 simi- Appendix B: Noise Transfer lar graphene, resulting in a comparatively low mean projected potential of 3.6 Vnm. We could confirm that edges In the derivation of the SNR for off-axis holography favor the zig-zag configuration and found a peculiar lo- and conventional phase contrast HRTEM we used the calized increase of the potential at the edges. The latter generalized Lenz model (uncorrelated shot and detector is attributed to the delocalization of electron edge states. noise, commensurable sinc sideband mask, noise small compared to total intensity), which gives good agreement with the observed phase noise, in particular under weak ACKNOWLEDGEMENTS contrast conditions as present in the WPO [49]. More- over, we assumed that the noise characteristics (e.g., vari- We are grateful to Tore Niermann, who supported us at ance) do not depend significantly on the position on the recoring additional hologramms of another h-BN sample. detector, which is again a good approximation for weakly We acknowledge funding from the European Research scattering objects. Using these approximations the vari- Council (ERC) under the Horizon 2020 research and in- ance of the reconstructed phase reads novation program of the European Union (grant agree- 2 1 ment number 715620) and the Deutsche Forschungsge- σϕ = (B1) Iµ (k )2 (k ) meinschaft (DFG, German Research Foundati- on) – c c DQE c Project-ID 417590517 – SFB 1415. This work is also Here, the DQE denotes a 2D generalization of detection supported by the Office of Science, Office of Basic En- quantum efficiency defined as ergy Sciences of the U.S. Department of Energy un- 2 der Contract No. DE-AC02—05CH11231. This work MTF (kc) DQE (kc) = , (B2) was kindly supported by the Initiative and Networking NPS (kc) 9 with the MTF denoting the modulation transfer and NPS The atomic averaged potential now corresponds to the the (normalized) white noise power spectrum of the de- value at zero spatial frequency tector. The phase SNR then reads ρ˜ (k, ϕ , θ ) s ˜ at k k 2 2 2 Vat (k = 0) = lim 2 , (C4) ϕobj E ϕobjIµc (kc) k→0 ε0k SNRhol = = . (B3) σϕ DQE (kc) which is not determined in the required limit as both The noise analysis for conventional weak phase contrast nominator and denominator tend to zero. To solve that HRTEM starts with the shot noise amplified by the de- expression we may apply l’Hospital’s rule to the average tector over the full solid angle of the previous expression (to remove the ϕ, θ dependency) 2 σI = INPS (0) (B4) ˜ Vat (k = 0) = (C5) from which the SNR is readily derived inserting the rela- hρãt (k)i tion between I and the phase (PCTF) 2 ϕk,θk lim∂k k→0 2ε0 s I2PCTF2ϕ2 1 2 3 obj =lim ∂k d rρat (r) SNRconv = . (B5) k→0 2ε0 ˆ IDQE (0) D E eikr(sin θk sin θr cos(ϕk−ϕr )+cos θk cos θr ) We finally arrive at for the ratio between both as noted 1 3 2 in the main text = − dr r ρat (r) 2ε0 ˆ p 2 SNRhol µc (kc) DQE (0) (sin θk sin θr cos(ϕk−ϕr) + cos θk cos θr) = p . (B6) SNRconv sin χs DQE (kc) 1 3 2 = − dr r ρat (r) 2ε0 ˆ 2 2 2 2 2 sin θk sin θr cos (ϕk−ϕr) + cos θk cos θr Appendix C: Mean Inner Potential 2π = − drr2ρ (r) 3ε ˆ at Establishing a well-defined relationship between aver- 0 aged projected potential or projected mean inner potential of a 2DM and the charge distribution holds some Note that we had to apply l’Hospital’s rule twice, which pitfalls in the infinite crystal limit [70]. Indeed, a mean requires both the zeroth and any first moment with re- r inner potential is not well-defined in this case and de- spect to to vanish. Consequently, the “atomic” cells, pends on fixing boundary conditions or a reference. To used for partitioning the 2D domain, have to be charge circumvent this problem, ab-inito calculations of MIPs neutral and dipole-moment-free. That condition restricts of bulk crystals have been carried out for slab geome- the allowed shapes and positions of the initially undefined tries, containing a sufficiently large vacuum region fixing atomic patch choice. Noting that dipole-free atomic cells h the reference. In case of a finite crystal (as observed ex- are centered closely to the atomic positions in -BN, we perimentally), we may start off with dividing the 2DM have that the averaged potential is a measure for the size domain into “atomic” cells, of the electron cloud around the atoms. X V (r) = Vat (r) , (C1) Appendix D: DFT 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. Density-functional [71] band-structure calculations using the all-electron full-potential local-orbital (FPLO- 1 3 1 X 3 18) [65] calculation scheme were employed to obtain V (r) d r = Vat (r) d r (C2) the electronic properties (e.g., electron density and po- A Â A ˆ tential) of single- to fivelayer h-BN and Graphene for can now be computed as a sum of the atomic contribu- reference. The calculations were scalar relativistic [72] tions, which are contained within the area. This is most and used the generalized gradient approximation (GGA) conveniently done in Fourier space employing the Poisson of the exchange-correlation functional due to Perdew- equation Burke-Enzerhof [73]. The in-plane structural parameter a = 2.505 Å and the distance between the layers d = 3.324 h 2 ˜ ρãt (k) Å of -BN, where taken from literature [74] k Vat (k) = . (C3) ε0 (and agree well with our experimental findings). 10

Appendix E: PCA Analysis of Mass. We also subtracted the average of each patch (which amounts to removing the first princi- Principal component analysis (PCA) allows to find the pal component). optimal (w.r.t. the Euclidean distance) linear decomposition of a statistically varying signal into a truncated 2. PCA of the data matrix, whose rows correspond to basis. It is a well-established method in advanced statis- the different patches and the columns represent the tical analysis and machine learning[75], provided that a interlaced spatial coordinates of the patches. sufficiently large statistical set of signals can be collected. 3. Truncation of the decomposition by removing all In TEM it finds application in the analysis of EELS and components beyond a kink in the scree plot (show- EDX data amongst others[59, 76, 77]. Here we apply ing the magnitude-ordered principal components). it to the autocorrected phase dataset decomposed into This filter is referred to as truncated PCA in liter- patches containing one honeycomb of the 2DM structure. ature. Due to the high structural order of 2DMs these patches contain a finite number of different species, namely 1-4 4. Computation of the difference between original and layer “bulk” honeycombs and corresponding edges/steps. truncated data, confirming the Gaussian noise na- Because several thousand honeycombs are contained in ture of the remainder (see Fig. 3(c) in the main one hologram, the patches form a sufficiently large sta- text). tistical basis, except some particular edge states, which were too sparse to come out in the PCA. In the PCA 5. Replacement of the patches in the original image analysis we closely followed the steps layed out in Ref. with the PCA truncated patches (Fig. 3(b)) [59]: 1. Identification of patches by locating the honeycomb minima. In an iterative procedure the patch ori- BIBLIOGRAPHY gin is refined by aligning them with their Center

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