Holographic convergent electron beam diffraction (CBED) imaging of two-dimensional crystals

Convergent beam electron diffraction (CBED) performed on two-dimensional (2D) materials recently emerged as a powerful tool to study structural and stacking defects, adsorbates, atomic 3D displacements in the layers, and the interlayer distances. The formation of the interference patterns in individual CBED spots of 2D crystals can be considered as a hologram, thus the CBED patterns can be directly reconstructed by conventional reconstruction methods adapted from holography. In this study, we review recent results applying CBED to 2D crystals and their heterostructures: holographic CBED on bilayers with the reconstruction of defects and the determination of interlayer distance, CBED on 2D crystal monolayers to reveal adsorbates, and CBED on multilayered van der Waals systems with moir\'e patterns for local structural determination.


Convergent beam electron diffraction (CBED) was first demonstrated by Kossel and Möllenstedt in
1939 [1]. Conventionally CBED is performed on three-dimensional (3D) crystals by focussing a convergent electron beam on a small area (about 10 nm in diameter) of a 3D crystal and acquiring the diffraction CBED pattern. Unlike conventional selected area electron diffraction, where the planar illumination produces a diffraction pattern consisting of sharp peaks -in CBED mode the pattern is formed of finite-size disks, whose diameter is determined by the convergence semi-angle of the beam Fig. 1. The CBED disks exhibit intensity variations related to the atomic structure and local atomic displacements in the crystal. CBED has been applied for studying crystallographic structure and lattice parameters [2][3][4][5][6], specimen thickness [7], measurements of strain [4,8], and crystallographic deformations [9,10]. In large-angle convergent-beam electron diffraction (LACBED) regime, the CBED spots strongly overlap, which allows for an easier and more precise extraction of information on the structure and defects [11,12]. A good overview of CBED techniques applied for 3D crystal structure determination can be found in references [13][14][15][16]. In general, direct interpretation of CBED patterns is not possible and simulations are required to deduce the crystal structure, which often limits wider application of the technique.
Recently, CBED imaging was applied to study van der Waals heterostructures [17][18][19][20][21][22]. For 2D crystals the interpretation of their CBED patterns is significantly simpler than in the case of 3D crystals because the intensity distributions in individual CBED spots directly map to the atomic arrangement, including local displacements in defects. In CBED the intensity distribution of the scattered wave in the far field can be interpreted as being acquired from a sample that is being illuminated at different angles. This allows capturing of 3D information about atomic positions in the sample. The formation of the interference patterns within the individual CBED spots of 2D crystals can be described by holographic principles, and therefore CBED patterns can be used to directly reconstruct the real space coordinates via conventional reconstruction methods adapted from holography. To emphasize the difference in the data analysis we therefore call the technique holographic CBED (HCBED). The reconstructed distributions provide information about the 3D arrangement of atoms in individual layers and in the stack, including the local strain, lattice orientations, local vertical separation between the layers, etc which is not accessible by conventional TEM imaging [23,24]. In this study, we review the recent results in CBED on 2D crystals: holographic CBED on bilayer (BL) with the reconstruction of defects and the measurement of interlayer distance [19,22], CBED on 2D crystal monolayers (MLs) with imaging of adsorbates [20], and CBED on multilayer van der Waals systems [21].

Principle of CBED on 2D crystals 2.1. CBED spots parameters
In a CBED experiment, the sample z position, or defocus f, can be relatively easy changed to move the sample along the optical axis, which allows imaging of the sample with a convergent or divergent electron beam. The CBED arrangement in a convergent wavefront mode (f < 0) is shown in Fig. 1. A CBED pattern from a ML of graphene or hexagonal boron nitride (hBN) consists of finite-sized spots arranged into a six-fold symmetrical pattern. The centres of the spots have the same positions as the corresponding diffraction peaks, defined by the crystal periodicity sin / , a   (1) where  is the wavelength,  is the diffraction angle and a is the period of the crystallographic planes. The diameter of the probing beam on the sample can be evaluated as scaling with the diameter of the limiting aperture as: where  is the semi-convergence angle, as shown in Fig. 1 and where chromatic instabilities and geometric lens aberrations are negligible. For small defocus ( 0 f ) the diameter of the probing beam is then given by diffraction on the aperture: D A is the limiting aperture diameter and f A is the distance between the limiting aperture and the virtual source plane, as shown in Fig. 1. By changing the z position of the sample in the convergent electron beam, the defocus f is changed, which in turn changes the diameter of the probing electron beam according Eqs. (2) and (3). The diameters of the CBED spots however do not change, because they are defined by the limiting aperture angular size or tan , as given by Eq. (4). For example, the diameter of the zero-order CBED spot size can be derived from geometrical considerations as shown in Fig. 1. Thus, the semi convergence angle , or the diameter of the limiting aperture, defines the diameter of the CBED spots and for HCBED it should be selected such that the CBED spots from a single crystal structure do not overlap. Then, the size of the probing beam, or the probed area, can be regulated by selecting f . Note that this is equivalent to adjusting the sample z position where lens aberrations are neglected.

Probing wavefront distribution
Where the probing wavefront is formed by a diffraction on the limiting aperture, the intensity distribution is described by Fresnel diffraction on a round aperture:  images were obtained with Titan ChemiSTEM microscope operated at 80 kV, with an ultra-stable high-brightness Schottky FEG source. Note that the experimental data shows evidence of a slight astigmatism. Adapted from [19].

CBED on 2D crystal monolayers
CBED on 2D crystal MLs is studied in detail in reference [20], and here we present the main results from the study.

Phase shifts caused by atomic displacements, geometrical approach
A 2D ML crystal is a perfect test sample to explain the basics of the intensity distributions observed in individual CBED spots. Atomic misalignments in the form of out-of-plane and in-plane shifts are illustrated in Fig. 3. From the geometrical considerations, shown in Fig. 3, the phase differences due to atomic shifts are: and 2 sin for out-of-plane z and in-plane x atomic shifts, respectively. From Eqs. (6) and (7) we see that atomic x shifts cause stronger phase shifts because . Thus, in-plane shifts introduce stronger intensity variations into HCBED spots interference patterns, and therefore they are easier to detect and reconstruct.

Phase shifts caused by atomic displacements, wave theory approach
Alternatively, using optical wave theory the phase shifts of the waves scattered from misaligned atoms can be derived as follows. The distribution of the scattered wave in the far-field   is applied. We introduce K-coordinates to simply the algebraic expression: , , , and re-write: The wavefront scattered by an atom at

Phase shift caused by an out-of-plane displacement
Wavefronts scattered by atoms positioned at The corresponding phases of the wavefronts are: and the phase difference is given by: where we applied The obtained phase shift agrees with the phase shift derived from geometrical consideration, Eq. (6) and Fig. 3(a). The CBED pattern and phase shifts produced by a ML with out-of-plane atomic shifts forming a "bulge" are shown in Fig. 4

Phase shift caused by an in-plane displacement
Wavefronts scattered by two adjacent atoms positioned at   and a is the lattice period, are given by: The corresponding phases of the wavefronts are: ,, and the phase difference is given by: When 0, x  the phase shift 2,

 
and we obtain: which corresponds to the position of the n-th-order diffraction peak: The phase shift due to a lateral shift x  is given by: which is an odd function of .
x K Thus, for 0 x  there will be an additional phase shift in opposite CBED spots, as for example in the spots   1010 and   10 10 , and these phase shifts will be of opposite sign. The obtained phase shift agrees with the phase shift derived from geometrical considerations, Eq. (7) and Fig. 3(b). The CBED pattern and phase shifts produced by a ML with inplane atomic shifts forming a linear lateral shift is illustrated in Fig. 4

(d)-(f).
The results shown in Fig. 4(b) and (e) demonstrate that it is easy to distinguish between outof-plane and in-plane atomic displacements even without performing a reconstruction simply by comparing the intensity contrast in opposite CBED spots: An out-of-plane defect will always result in symmetric variations in intensity distribution between mirror-symmetric CBED spots, and an in-plane defect will result in antisymmetric variations.

Imaging adsorbates on MLs
Lattice deformations such as strain or rippling do not cause noticeable intensity variations in the zero-order CBED spot, and cause increasing intensity variations as the order of the CBED spot increases (spots are further from the zero-order spot and correspond to higher spatial frequencies).
In contrast, adsorbates exhibit strong intensity variations in all CBED spots. The zero-order CBED spot is in fact an in-line hologram of the sample, and in-line holography is known to exhibit high intensity contrast that is very valuable for imaging of weak phase objects that are not detectable for conventional TEM at Gaussian focus. The zero-order CBED spot of a ML with phase adsorbates on one surface displays an inversion of contrast when acquired at 0 f  and 0 f  , defocus values respectively [20]. In the case of an adsorbate which is not a phase object so also absorbs electronsthe zero-order CBED spot displays a dark feature.
Experimental CBED patterns of an hBN sample acquired at  Fig. 5(a) and (b). Thus, the dark feature can be explained by a surface deformation in that location, probably because of the stress between the two adjacent adsorbates.
The zero-order CBED spot can be treated as an in-line hologram, and the amplitude and phase distributions of the transmission function of the sample can be reconstructed by available algorithms [25]. An example of such reconstruction is shown in Fig. 6. Adsorbate patches of subnanometre sizes are observed in the reconstructed phase distributions and can be cross-validated with the corresponding HAADF STEM images. Also here, the reconstructed amplitude distributions exhibit blurred structure when compared to the reconstructed phase distributions, which suggests that the imaged objects are phase objects without any absorption contrast. (e) CBED pattern acquired at f = -5 m and magnified images of the intensity distributions of (f) the zero-order, (g) the first-order, and (h) the second-order CBED spots. Adapted from [20].

Formation of interference pattern in CBED spots
For CBED of a BL sample, the two sets of electron beams diffracted on each layer interfere in the detector plane, creating interference patterns at the positions of the overlapping CBED spots, as illustrated in Fig. 7. Such interference patterns contain rich information about the local inter-atomic spacing (local strain), the vertical distance between the layers, the relative orientation between the layers, etc. The period and tilt of the fringes in the interference pattern can be explained by considering the position of the sources in the virtual source plane, as sketched in Fig. 7(a). A realspace illustration of a BL sample with an in-plane twist angle  is shown in Fig. 7(b). In the virtual source plane, the Bragg diffraction peaks create virtual sources, which are rotated by the twist angle  relative to each other, as depicted in Fig. 7(c). The interference pattern in a CBED spot is described by the formula [22]: The tilt of the interference fringes  can be found from the geometrical arrangement of the vectors in the virtual source plane (Fig. 7(c)): (2) (2) (1) sin tan . cos The higher the twist angle, the larger  and the smaller is the period of the fringes, as can be seen in the simulations of BL graphene at twist angles of 1° and 2° in Fig. 7(d) and ( overlapping CBED spot does not display a maximum at the centre of the CBED spot, as can be seen in the simulations of a graphene-hBN system for AA and AB stacking in Fig. 7(f) and (g), respectively.

Holographic reconstruction 4.2.1 Protocol of the reconstruction procedure
The atomic displacements relatively to their position in perfect lattices for a BL system can be reconstructed from the corresponding CBED pattern when treating the CBED spots as off-axis holograms. The reconstruction procedure is described in references [19,22]. Here we provide the main reconstruction steps, also illustrated in Fig. 8: (1) For holographic reconstruction, a CBED spot is selected with the centre at an arithmetic average of the centres of the individual CBED spots.
(2) The 2D Fourier spectrum of the selected region is calculated. In the obtained complex-valued spectrum, one zero-order and two sidebands are observed (Figs. 8(a) and (b)).
(3) One of the sidebands is selected while the zero-order and remaining sideband are set to zero ( Fig. 8(b)). Note, it is crucially important that the sidebands in different CBED spots are chosen consistently. The chosen sidebands in different CBED spots have to be related by the same symmetry transformations as the CBED spots themselves (rotation by an integer number of /3).
(4) The whole spectrum is shifted so that the maximum of the sideband is located at the origin of the reciprocal plane ( Fig. 8(b)).
(5) The inverse 2D Fourier transform the resulting distribution is calculated.
(6) The amplitude and phase distributions are extracted from the obtained complex-valued distribution, (Fig. 8(c)).
In steps (3) and (4), the right sideband is selected at the position defined by the fringes tilt angle and period T, as defined by Eqs. (23) and (24) and explained in reference [22].

Simulated example of reconstruction in-plane and out-of-plane displacement
An example of the simulated and reconstructed CBED pattern for a graphene-hBN system with both out-of-plane and in-plane shifts is shown in Fig. 9.

Experimental example of reconstruction in-plane and out-of-plane displacement
An experimental CBED pattern from a twisted BL system (graphene-hBN) is shown in Fig. 10. CBED spots originating from the graphene layer are found at a slightly larger diffraction angle than CBED spots from the hBN layer (due to differences in the in plane lattice parameter), which allows an easy assignment of the CBED spots corresponding to different layers. A stacking fault between the layers is evident by the presence of a distinctive ridge in the interference patterns in the first-and higherorder CBED spots ( Fig. 10(a) and (b)). The defect causes no significant contrast in the zero-order CBED spot meaning that the defect consists of only atomic position misalignments, which introduce a significant additional phase shift between the electron waves scattered from the two layers. Note that in diffraction imaging mode, where only diffraction peaks are detected, imaging of such a defect would not be possible. The area of overlap between CBED spots from the graphene and hBN layers is less in the higher order diffraction spots. The intensity contrast caused by the corrugation is more pronounced in the higher order CBED spots, in according with Eqs. (6) and (7). The CBED pattern was holographically reconstructed by the procedure described above. Fig. 10(c) and (d) show the recovered out-of-plane z  and in-plane x atomic displacements. Fig. 10(e) compares the out-ofplane and in-plane atomic shifts along the ripple. The retrieved height of the out-of-plane ripple in hBN layer is about 2 nm, which agrees well with the observed out-of-plane ripples in graphene/hBN stacks due to self-cleansing effects [26].

Reconstruction of interlayer distance
HCBED can be applied for reconstruction of interlayer distances from a single CBED pattern as was demonstrated in reference [19] and explained in more details in reference [22]. The reconstruction procedure consists of the steps provided above, and the reconstructed phase distribution allows recovery of the interlayer distance according to the formula [22] (2)  (25) where (1)  and (2)  are the diffraction angles of the two layers.  Fig. 11 shows the simulated 2D distributions and one-dimensional (1D) profiles of the reconstructed interlayer distances for BL graphene (BLG) with the interlayer distances of 0 and 10 Å at three different twist angles of 0.5°, 2° and 4°. From Fig. 11, we see that the reconstructed interlayer distances obtained from CBED patterns with smaller twist angles exhibit a smoother appearance, while at larger twist angles, artefact due to moiré structure become more pronounced in the reconstructions. The precision of the reconstructed interlayer distance is about ±0.5 Å [22].

HCBED on multilayer systems, CBED moiré
Twisted multilayer systems exhibit extra modulations of the interference fringes in CBED patterns, i. e. a CBED moiré, as shown in Fig. 12. These extra modulations are coming from the interference between the moiré CBED spots. Such moiré CBED spots are the results of a second order process, which involves electrons being scattered by both layers sequentially [27]. Due to its second order nature such moiré peaks have much weaker intensity than the main diffraction peaks [27], and therefore the moiré CBED spots are also much weaker that the main CBED spots. This is the reason why they are practically invisible in twisted BL samples. Moreover, moiré CBED spots strongly overlap with the main CBED spots. For these two reasons, moiré CBED spots are not directly visible in the CBED patterns but they manifest themselves in the additional modulations of the CBED interference pattern -the CBED moiré, which is created by the interference between the moiré CBED spots and the major CBED spots.  [21].
A simple and robust method for evaluation of the composition and the number of layers from a single-shot CBED pattern was demonstrated in reference [21]. The composition and the relative number of layers can be evaluated from the intensity distribution in the non-overlapping regions of a CBED spot. Number of layers can be evaluated by inspecting a 2D Fourier spectrum of a CBED spot, where the presence of peaks due to the CBED moiré indicate that there are five or more layers in the sample. Although the precision of such sample characterisation is very modest when compared to cross-sectional TEM imaging [28], the presented approach has the advantage that it is non destructive, requires only a single CBED pattern, and CBED is relatively easy to realise in a conventional TEM.

Discussion and outlook
HCBED offers a lot of information in one single CBED image such as atomic 3D displacements in the layers, the interlayer distances and imaging of adsorbates. Adsorbates can be clearly distinguished from 3D displacements of atoms: an adsorbate is identified in all orders of CBED spots, while atomic displacements produce intensity variations only in the first and higher order CBED spots. The contrast of the adsorbate image allows us to determine whether the adsorbate is an absorbing or phase object. Phase adsorbates display opposite intensity contrast when the probing wave is changed from convergent to a divergent wavefront (under focus to over focus). An absorbing adsorbate is displayed as a dark feature in both imaging regimes. The adsorbate distribution can be reconstructed from the zero-order CBED spot by applying an in-line hologram reconstruction routine. 3D displacements of atoms, on the other hand, display minimal intensity variations in the zero-order CBED spot, but they cause significant contrast in the higher-order CBED spots. Moreover, the type of displacement can be clearly identified just by comparing opposite CBED spots: an inplane displacement leads to an opposite intensity contrast in opposite CBED spots (antisymmetric pattern), while an out-of-plane displacement leads to the same intensity contrast in the opposite CBED spots (pattern is symmetric).
For BL samples, the interference patterns formed in overlapping CBED spots can be treated as off-axis holograms and the phase of the interfering waves, and with this the 3D positions of the scattering atoms, can be retrieved. By using this approach, in-plane and out-of-plane ripples, and the interlayer distances can be quantitatively reconstructed. The resolution at which the atomic shifts are recovered exceeds the intrinsic resolution provided by the classical resolution criteria. The lateral and axial (along the z axis) resolutions evaluated from a CBED pattern k-value range is given by It is therefore a remarkable result that the holographic CBED approach allows reconstruction of interlayer distances of a few Angstroms at 0.5 Å accuracy, which is more than 400 times the diffraction defined z-resolution [19,22].
From the experimental point of view, the best probing beam for acquiring CBED patterns would be a perfect convergent electron beam. However, holographic techniques in general, show very high tolerance to the probing beam imperfections, because the resulting interference pattern (hologram) is formed due the difference of the phases in the scattered and reference waves. In holographic CBED, the interference contrast is formed due to the relative phase shift between the waves scattered of atoms in different 2D layers. This relative phase shift is given by the local arrangement of the atoms in the structure, and the atoms are probed with the same local distribution of the probing beam. Thus, the probing beam imperfections, even when present, should have minimal effect on the CBED pattern formation, and with this, on the resulting reconstruction.
However, a study can be performed to quantitatively evaluate the effect of beam aberrations, in particular, imperfections in the phase distribution, on the resulting reconstructions.
To conclude, HCBED has already demonstrated capability of providing high-resolution information about atomic arrangement from a single CBED pattern. We expect that HCBED can be further developed to become a practical tool for studying 2D materials at atomic resolution in 3D for complex heterostructures and arbitrary numbers of layers.