Holographic reconstruction of the interlayer distance of bilayer two-dimensional crystal samples from their convergent beam electron diffraction patterns

The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moir\'e structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of 0.5 A, which is a few hundred times better than the intrinsic z-resolution of diffraction limited resolution, as expressed through the spread of the measured k-values. Moreover, we show that holographic CBED imaging can detect variations as small as 0.1 A in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors.


HIGHLIGHTS
• Convergent beam electron diffraction (CBED) patterns of bilayer two-dimensional crystal samples is investigated • Theory and detailed protocol of the reconstruction of the interlayer distance from CBED pattern is provided • It is shown that the interlayer distance can be reconstructed at accuracy of ±0.5 Å, which is a few hundreds times better than the z-resolution evaluated from the measured range of kvalues • Holographic CBED imaging can pick as small as 0.1 Å variations in the interlayer distance, though the quantitative reconstruction of such variations suffers from a large error.
Holographic reconstruction of interlayer distance of bilayer two-dimensional crystal samples from their convergent beam electron diffraction patterns MAIN TEXT

Introduction, comparing to other techniques
Convergent beam electron diffraction (CBED) [1][2][3] has been routinely utilized for studying parameters of thick crystals: thickness [4], lattice parameters [5][6][7], and crystallographic deformations [8,9]. CBED performed on two-dimensional (2D) crystals and van der Waals structures [10,11] produces patterns which require different interpretation than in the case of thick crystals [12][13][14]. Bilayer materials create a characteristic interference patterns in CBED spots which can be treated as holograms and the phase distributions of the scattered waves, and with this, the atomic positions in the individual layers can be extracted. A particular advantage of holographic CBED is possibility to obtain z-information from a single CBED pattern. Lateral or (x,y) atomic positions can be accessed at sub-Ångstrom resolution through scanning procedure by electron ptychography [15], the access to z-information is possible by cross-sectional transmission electron microscopy (TEM) imaging [16]. Holographic CBED approach allows access to z-atomic positions and the interlayer distance in bilayer (BL) systems from a single CBED pattern. In this paper we provide the theory behind and the details of the holographic reconstruction procedure applied in the holographic CBED [13]. To present a systematic study and demonstrate the performance of the technique at different parameters, we provide simulated examples.

Principle of holographic CBED reconstruction 2.1 Formation of CBED pattern
CBED arrangement in a convergent wavefront mode (f < 0) is shown in Fig. 1(a). A real-space distribution of a bilayer sample with a twist angle  is shown in Fig. 1(b). In the virtual source plane, the Bragg diffraction peaks create virtual sources. The virtual sources of each layer are correspondingly rotated by the twist angle  , as shown in Fig. 1(c).
where we approximated rf  . The integral in Eq. (2) is a Fourier transform of the lattice function

( )
Lr and the result is the reciprocal lattice defined as: Thus, for the distribution in the virtual source plane we obtain from Eq.(2) . ii From Eq. (3) it follows that each virtual source has additional phase factor . Without this phase factor, the far-field diffracted wave would be given by a Dirac comb function describing position of individual diffraction peaks. With this phase factor, each diffraction peak is turned into a finite-sized CBED spot.
Next, we consider two layers, monolayer 1 (ML1) and monolayer 2 (ML2), with a relative twist  and separated by a distance . d Each ML gives rise to a set of virtual sources and CBED spots.
At relatively small , the CBED spots from two layers are still almost at the same positions and they overlap, creating an overlapping CBED spot. The interference pattern within each overlapping CBED spot is analogous to an interference pattern created by two waves originating from two virtual sources, as illustrated in Fig. 1(a). We assume that ML1 is shifted by ( ) , xy  and ML2 is shifted by ( ) , xy  relatively to a centred lattice (a lattice in which one of its hexagons is centred at the origin of the ( ) where ( ) is a CBED spot number. In Eq. (4) we neglect the secondary scattering of electron wave on the second layer assuming that the second layer is also illuminated with a plane (convergent) wave. Each virtual source creates a divergent spherical wave described by The interference pattern within an overlapping CBED spot is described as where (  The following approximation can be applied: , and we re-write Eq. (5): . (7)

Extracting the interlayer distance from the inference pattern
The information about the interlayer distance is enclosed in the first and the second terms of the argument of cosine in Eq. (7). We consider the interference pattern in a first-order CBED spot. For a BL we introduce the virtual sources coordinates as where (1)  and (2)  are the diffraction angles corresponding to ML1 and ML2, see Fig. 1(c). The K coordinate corresponds to the center of the overlapping CBED spot and is given by the average of the first-order diffraction coordinates of ML1 ML2: 2 sin , 0 2 sin cos ,sin sin The first term in cosine argument of Eq. (7) gives: (1) 2 (2) tan tan cos tan tan cos tan tan where we assumed that The second term in cosine argument of Eq. (7) gives: The sum of the two terms gives: (1) Thus, the distribution of the interlayer distance over the probed region ( ) , d x y can be extracted from the sum of the two first terms  . Equation (8) implies that the interlayer distance can be determined even if the MLs are of different chemical composition.

Reconstruction as off-axis hologram
In this section we provide a step-by-step protocol of the holographic reconstruction.

Positions of CBED spots
Centres of CBED spots in CBED pattern, as Bragg diffraction peaks, are given by the period of the lattice i d :

Selecting center of overlapping CBED spot
For twisted BL sample, CBED pattern consists of two overlapping CBED patterns, each from individual ML. CBED spots overlap in pairs, as shown in Fig. 2. For holographic reconstruction, the center of an overlapping spot is calculated as an arithmetic average of the centres of the individual CBED spots: (1) avg , 2

+ =
where (1) K and (2) K are the coordinates of the centres of the CBED spots from the individual MLs.
A squared region is selected with the center at avg k at an overlapping CBED spot (as indicated by the red dot in Fig. 2(b)).

Reconstructing CBED spot as an off-axis hologram
The intensity distribution in a selected CBED spot can be re-written as: where ' K is the running coordinate counted from the center of the selected overlapping CBED spot, and  is introduced in Eq. (8). The intensity distribution ( ) ' IK is treated as an off-axis hologram and a conventional protocol of an off-axis hologram reconstruction is applied [17,18]. 2D FT of ( ) The right sideband corresponds to the term The reconstruction steps are illustrated in Fig. 3: (1) 2D Fourier spectrum of the hologram is calculated by 2D FT. In the spectrum, one zero-order and two sidebands are observed, Fig. 3(a) and (b).
(2) The right sideband is selected and the zero-order and the left sideband are set to zero, Fig. 3(b).
(3) The whole spectrum shifted so that the selected sideband is in the centre, Fig. 3(b).
(4) Inverse 2D FT of the resulting distribution gives a complex-valued reconstruction, where the amplitude and the phase distributions can be extracted, Fig. 3(c).
In steps (3) and (4) the right sideband is selected at the position defined by  and T, as explained in the next section.

The period and tilt of the fringes
The period of the interference fringes is given by where   is given by The tilt of the interference fringes  can be found from the geometrical arrangement of the vectors in the virtual source plane (Fig. 1(c)): (2)

Re-positioning of the reconstructed distribution
From the complex-valued distribution reconstructed in the previous step, the distribution corresponding to one of the layers, for example ML1, need to be selected. ML1 CBED spots are positioned at (1) (1) avg offset from the centers of the overlapping spots, as illustrated in Fig. 4. Therefore, the reconstructed distributions need to be re-positioned to correspond to the position of CBED spots of ML1. This is done by shifting the reconstructed distributions by (1) (1) avg  (1) (1) avg offset from the centers of the overlapping spots. For these simulations 2 μm, f  = − the distance between the layers is 3.35 Å, the imaged area is about 28 nm in diameter. The scale bar corresponds to 2 nm -1 .

Averaging
After the amplitude and phase distributions for each CBED spots are reconstructed, only the phase distributions are considered since only these distributions carry the information about the atomic positions. The individual reconstructed phase distributions are averaged, that is, all six distributions are added together and divided by six. As mentioned above by Eq. (6), averaging also eliminates   .

Extracting the interlayer distance
The interlayer distance is obtained from the reconstructed averaged phase distribution by applying Eq. (8) .

Number of fringes
Number of fringes in an overlapping CBED spot is given by the twist angle and the size of the probed region, which in turn is given by the defocus value. In this section we show that even one interference fringe (one period) is sufficient to reconstruct the phase shift. Figure 5 shows simulated CBED patterns for BLG with the interlayer distance of 10 Å, at defocus -2 m and three different twist angles: 0.5°, 2° and 4°. While CBED patterns with the twist angle 2° and 4° exhibit a few interference fringes in their CBED spots, the CBED pattern with the twist angle 0.5° exhibits only one period of interference fringes. As a consequence, the zero-order and sideband in the spectrum of this CBED pattern are not well resolved and cannot be clearly separated one from another, as shown in the inset in Fig. 5(a). However, applying the reconstruction procedure as described above still provides correct reconstruction and the interlayer distance, as shown in Fig. 6.    6 we see that the reconstructions obtained from CBED patterns with smaller twist angle exhibit more smooth appearance, whereas at larger twist angles some artifact modulations are observed in the reconstructions. These artifacts can be explained by the presence of moiré structure which is more apparent at larger twist angles. At larger twist angles, the associated moiré peaks [19] are also observed in the Fourier spectra, as indicated by the red arrows in Fig. 5(c). The precision of the reconstructed interlayer distance is about ±0.5 Å.
We now define the condition at which at least one fringe (one period) appears in a firstorder CBED spot. The period is given by Eq. (9) and the CBED spot diameter in K-coordinates is given by: there is at least one fringe (period) in the CBED spot which in turn is sufficient for the holographic reconstruction.

Reconstruction of average interlayer distance
The moiré structure appearing as an artifact in the reconstructed interlayer distance can be suppressed if instead of the interlayer distribution over the probed region only an average interlayer distance is reconstructed. This is done as follows. During the reconstruction by filtering in the Fourier domain, instead of selecting a half of the Fourier spectrum, only one pixel corresponding to the maximum of the sideband is selected and the remaining pixels are set to zero. This single pixel when shifted to the center, will give a constant distribution in the real space after inverse Fourier transform applied. This can be also considered as an extreme low-pass filter. As a result, the artifact moiré pattern is removed in the reconstructed phase distribution and the interlayer distance distribution is a constant, as shown in Fig. 7.

Variable interlayer distance
In reality, the interlayer distance is not a constant but always exhibits some variations. In order to check the sensitivity of the holographic reconstruction method to the interlayer distance variations, we simulated a BLG sample where the distance between the layers is not constant but contains an out-of-plane ripple in form of a fringe. The distance between the layers is assumed 6 Å and atoms in one of the layers are shifted by parameters: interlayer distance is 6 Å, twist angle is 0.5°, imaged at the defocus distance -2 m. A small twist angle is chosen to minimize the moiré effect. The reconstructed interlayer distribution is shown in Fig. 8. The deviation of the atomic z-position from constant are readily picked up in the interference pattern of overlapping CBED spots and greatly manifest itself in the reconstruction, Fig.   8. However, quantitatively, the recovered z  shifts are much larger than the actual z  shifts. We therefore conclude that even such small variations in the interlayer distance as 0.1 Å will be evident in the reconstructed interlayer distance distribution, however they will be greatly enhanced and their exact value will be reconstructed with a large error.

Resolution
The lateral and axial (along the z-axis) resolution evaluated from a CBED pattern k-values range is It is therefore a remarkable result that the holographic approach allows reconstruction of the interlayer distance at 0.5 Å accuracy, that is more than 400 times exceeding the diffraction defined z-resolution. Fig. 9 Illustration to the symbols used for deriving the resolution criteria.

Conclusions and Discussion
In conclusion, we derived a formula for recovering the interlayer distance from the phase distribution reconstructed from CBED pattern. The formula accounts for different chemical composition of the individual monolayers, and is practical for samples such as graphene -BN. We showed that surprisingly even one interference fringe in the interference pattern is sufficient to reconstruct the phase shift and with this, the interlayer distance. One interference fringe is observed in CBED spots when either the twist angle or the probed region (defined by the defocus distance) is small. In fact, the situation when only one interference fringe is observed in CBED spots is preferred, as it allows suppression of the artifact signal due to the moiré structure.
The precision of the reconstructed interlayer distance is about ±0.5 Å. It should be noted that this is a few hundreds times better than the intrinsic z-resolution evaluated from the spread of The CBED pattern was then simulated as the square of the amplitude of the FT of ( ) ,, u x y where the FT was calculated by FFT. The distributions in the sample is sampled N × N pixels, and the pixels size is 0, the total sample area is thus N0 × N0. The pixel size in the diffraction plane k = 1/(N0).