# Determination of the Projected Atomic Potential by Deconvolution of the Auto-Correlation Function of TEM Electron Nano-Diffraction Patterns

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Electron Diffraction Experiments with Nano-Sized Coherent Illumination

^{2}and low acquisition times enable the nano-diffraction pattern to be acquired, on the 1024 × 1024 charge-coupled-device (CCD) camera, without the beam stopper used to eliminate the direct beam. Thus all the diffracted intensities, including the zero-order beam, are measured in the nano-diffraction pattern.

_{3}in [100] zone axis is shown in the HRTEM image in Figure 1. The illuminated area in the direct space corresponds to a S/O ratio of about 22% and hence the oversampling conditions are well satisfied. It is worth noting that the spots of higher intensity in the HRTEM image describe a square lattice corresponding to the projected potential of the atomic columns of Ti+O and Sr (Figure 1). The relevant HRTEM image simulation for specimen thickness of 25.0 nm and underfocus of 41.3 nm, was calculated in the Bloch-waves approach by considering 100 excited beams [1]. The dots in the simulated image in the right inset of Figure 1 indicate the true positions of the projected atomic structure used in the calculation. Hence, a comparison of the simulation with the HRTEM shows that the atomic columns containing only oxygen are not visible in the experimental image.

#### 2.2. Intensity Rescaling of the Nano-Diffraction Pattern

_{max}(s) = Σ

_{i}f

_{i}(s), which is the sum of the scattering factors f

_{i}(s) of the atoms in the crystal unit cell. Any interference between waves scattered by different unit cells (Bragg peaks) will be characterized by an intensity lower or equal to |F

_{max}(s)|

^{2}. To evaluate F

_{max}(s) we need to know the atoms contained in the unit cell. An approximate estimation of the chemical composition of the specimen can be available either from the nominal composition of the sample or could be straightforwardly derived during the same experimental session by energy dispersive X-ray spectroscopy or electron energy loss spectroscopy. In our experiments we found that knowledge of the sample chemistry with an accuracy of about 10% is enough to guarantee the reliability of the results. Thus, the approximate knowledge of the chemical composition of the sample enables the estimation of F

_{max}(s). Atomic scattering factors are tabulated as a function of s = sinθ/λ, where θ is the scattering angle. Thus, |F

_{max}(s)|

^{2}decreases as a function of θ following well-established theoretical models [17]. Therefore, its value can be used as a mathematical constraint to rescale the intensity I of the experimental nano-diffraction with respect to the measured direct beam (I

_{max}). It is important to underline that this mathematical constraint can be applied only if the nano-ED pattern contains the direct beam I

_{max}.

_{max}(s)|

^{2}/|F

_{max}(0)|

^{2}≤ 1. The condition I(s)/I

_{max}≤ sup{I(s)} should be satisfied for each scattering vector modulus s, measured in the nano-diffraction. The average intensities I

_{bk}(s), corresponding to the larger s values in the experimental diffraction pattern out of the Bragg peaks, are subtracted iteratively to the whole pattern until I

_{bk}(s)/I

_{max}≤ sup{I(s)} is fulfilled. Unphysical negative values are constrained to zero. With this rescaling the measured direct beam becomes the zero-order reflection of the pattern. After the sup{I(s)}-rescaling some diffraction peaks may still not satisfy the condition I(s)/I

_{max}≤ sup{I(s)}, due to measurement errors, small tilt angles with respect to precise zone-axis and, mainly, because of residual dynamical scattering effects. Thus, diffracted intensities exceeding I(s)/I

_{max}≤ sup{I(s)} are further rescaled subtracting local mean intensity values. We found that this rescaling, in experimental cases with dynamical effects typical of specimen thicknesses up to a few tens of nm, as usual in TEM experiments, can enable the correct retrieval of atomic structural information from the experimental nano-diffraction patterns. In Appendix A we have reported some theoretical considerations about the proposed approach, discussing also its limits of applicability.

_{3}(100) ED-pattern. Figure 2a shows the raw nano-diffraction and Figure 2b the result after the rescaling, calculating sup{I(s)} by the Doyle and Turner atomic scattering factor model [17], according to the nominal composition of the specimen. Figure 2c shows the difference between patterns of Figure 2a,b. Figure 2d shows a comparison, in a logarithmic scale, between the line profiles of the patterns in Figure 2a,b extracted from the scans highlighted in Figure 2, blue and red in (a) and (b), respectively. In Figure 2d also the corresponding profile of sup{I(s)} is plotted (black curve). The diffraction spot with the highest Miller’s index along the scans is the (5,5,0), which corresponds to a lattice spacing of 55 pm. Also the (0,6,0) reflection has been highlighted, corresponding to a lattice spacing of 65 pm, since the experimental pattern contains all Friedel’s pairs up to this resolution.

#### 2.3. Auto-Correlation Function

**r**) directly by the square modulus of an inverse FT of the sup{I(s)}-rescaled nano-diffraction pattern, where

**r**is the position vector in the direct space. This auto-correlation function could be deconvolved, likewise Patterson functions, leading to non-aliased maps of the projected atomic potential. The procedure has been described in the Methods section.

**r**) is a complex function and the diffused electron wave in weak phase-object approximation is related to the potential by: ψ(

**r**) ≅ 1 + iπλΦ(

**r**), where Φ(

**r**) = 2meV(

**r**)/h

^{2}[1]. Here, m, e, h and i, are the electron mass, charge, Plank’s constant and imaginary unit, respectively. The diffracted intensity I(

**s**) can be approximated as I(

**s**)∝(πλ)

^{2}|F(

**s**)|

^{2}, where F(

**s**) is the structure factor corresponding to the complex scattering potential Φ(

**r**). The auto-correlation function C(

**r**), obtained by the inverse FT of the measured nano-diffraction pattern I(

**s**), is a complex Hermitian function: C(−

**r**) = C*(

**r**). It is proportional to Φ(

**r**) ⊗ Φ*(−

**r**), where “⊗” denotes the convolution operator and “*” denotes the complex conjugate. It is given by:

_{cs}(

**s**) and anti-symmetric ΔI(

**s**) contribution. From Equation (1) the imaginary part of C(

**r**) depends only on the non-centrosymmetric component of the diffraction pattern, as C

_{cs}(

**r**) is real and positive. A first contribution to the asymmetry of the electron diffraction pattern is due to misalignment of the crystal with respect to zone axes. An essential part of the non-centrosymmetric component is due to dynamical scattering [19], which is directly related to the thickness of the specimen (see Appendix A). Another contribution to the non-centrosymmetric part of the electron nano-diffraction pattern is due to absorption and derives from inelastic interactions such as phonon and plasmon scattering and single electron excitations from inner atoms [20,21]. For thin specimens this last effect contributes very little to the complex part of the object scattering function, which is only a few percent of the real part [6,7]. Moreover, the cross section from phonon scattering is sharply peaked at the atom cores and the corresponding absorption effects can be described by introducing a complex potential where the imaginary part Φ

_{I}(

**r**) is proportional to the real part Φ

_{R}(

**r**) [21,22]. Thus, Φ(

**r**) = Φ

_{R}(

**r**) + iΦ

_{I}(

**r**) ≅ Φ

_{R}(

**r**) + iαΦ

_{R}(

**r**), with α very small, of the order of 0.01–0.05 for many atomic species [20]. Therefore, for centrosymmetric projected potentials almost all the asymmetry of the electron diffraction pattern for thin specimens can be ascribed to misalignment of the crystal with respect to zone axes, and the phase of the complex scattering function Φ(

**r**) would have no further information about the sample’s atomic structure not already contained into its modulus. Therefore, all structural information will be contained in C

_{cs}(

**r**) ≅ Φ

_{R}(

**r**) ⊗ Φ

_{R}(

**−r**) × (1 + α

^{2}). Instead, in the case of thin samples and non-centrosymmetric projected potentials, in conditions of a nano-sized illumination, ΔI(

**s**) could give direct information about the anti-symmetric part of Φ

_{R}(

**r**), whereas I

_{cs}(

**s**) is related to both the symmetric and anti-symmetric parts of the potential (see Appendix A).

#### 2.4. Deconvolution of C(**R**): Centrosymmetric Case

_{3}is centrosymmetric, to simplify the notation, in the following let Φ(

**r**) be its real part Φ

_{R}(

**r**) and C(

**r**) be its centrosymmetric part C

_{cs}(

**r**). The auto-correlation deconvolution of the real part of the nano-diffraction pattern can give the correct positions of the atomic species that contribute to the projected atomic potential, but it is proportional to Φ(

**r**) ⊗ Φ(–

**r**). Assuming that the shape of the atomic potential can be approximated with Gaussian functions, the width of the Φ(

**r**) ⊗ Φ(–

**r**) will be about 2

^{1/2}that of Φ(

**r**). Moreover, projected atomic potential Φ is approximately proportional to Z

^{2/3}, with Z the atomic number [23] and C(

**r**) is proportional to Z

^{4/3}. To obtain the correct relative scale between the values of the projected potential peaks belonging to different atomic columns, the square root C(

**r**)

^{1/2}has to be computed, whose width is 2

^{1/2}of C(

**r**). Thus, to obtain the correct width of the peaks in the projected atomic potential map, namely C

_{dec}(

**r**), it is necessary to deconvolve C(

**r**)

^{1/2}peaks to about one half of their initial width, obtained by 1/2

^{1/2}× 1/2

^{1/2}.

_{dec,}

_{raw}(

**r**) for the raw experimental nano-diffraction of Figure 2a, whereas Figure 3b shows the result of C

_{dec,}

_{resc}(

**r**) for the rescaled nano-diffraction of Figure 2b. Figure 3c shows the SrTiO

_{3}potential in [100] zone-axis projection calculated by the simulation program JEMS (Java Electron Microscopy Simulations) [24]. Figure 3d shows the sum of two scans highlighted with red dashed lines in Figure 3b (red curve) and the sum of two scans highlighted with black dashed lines in Figure 3c (black curve) to compare the SrTiO

_{3}simulated projected atomic potential with that retrieved from the deconvolution of the experimental rescaled diffraction pattern. Moreover, in Figure 3d the calculated and experimental potentials Φ have been normalized with respect to their maxima, elevated to power 3/2 and multiplied by Z = 38 (Sr) to compare the final result directly with the average atomic number of the atoms contributing to the different columns of the projected atomic potential [23]. Figure 3e shows a zoom of the Sr peak to compare its experimental width, obtained by the deconvolution (red curve), with projected atomic potential values simulated at different spatial resolutions, from 55 pm to 75 pm. Debye-Waller factor values at room temperature are taken from JEMS [24]. Note that the deconvolution of the auto-correlation raw data (Figure 3a) does not show the correct structure for SrTiO

_{3}as the oxygen atomic columns are missing. Furthermore, the maximum of the Ti+O projected potential is 16% lower than the calculated value. This is expected mainly because the dynamical effects mix the intensity of the diffracted waves [1].

_{3}atomic structure, including the oxygen columns. Figure 3d shows that the result is not only qualitatively in agreement with the calculated SrTiO

_{3}atomic potential, but also quantitatively. For Ti+O and O columns the maxima of the experimental projected potentials are both about 4% smaller than calculated values. This is, in fact, a good quantitative result considering that the crystal potential calculations performed by linear combination of atomic potentials are affected by an error of about 10% [25]. Moreover, as shown in Figure 3e, the peak width differences between experimental and calculated potentials are due to the finite spatial resolution corresponding to the finer spatial spacing recorded in the diffraction pattern. Indeed, the best agreement between experimental and calculated potentials is obtained for a spatial resolution of 65 pm. As shown in Figure 2b, all reflections with lattice spacing larger or equal to 65 pm have been measured, enabling the achievement of a spatial resolution for the retrieved averaged projected atomic potential of (65 ± 2) pm.

#### 2.5. Deconvolution of C(**R**): Non-Centrosymmetric Case

## 3. Methods

#### Deconvolution of the Autocorrelation Function

**r**) can be performed in two ways: (1) multiplying C(

**r**) times its translated replica on a secondary maximum

**r**, i.e., times C(

_{M}**r**–

**r**); (2) finding the minimum between C(

_{M}**r**) and C(

**r**–

**r**) [18]. Moreover, in the presence of crystallographic symmetries it is possible to impose symmetry operators

_{M}**S**to C(

_{i}**r**), for I = 1, m independent symmetries, obtaining m C(

**r**–

**S**) [18]. In this case the deconvolution can be obtained either multiplying C(

_{i}r**r**–

**r**) times all the m independent of C(

_{M}**r**–

**S**) or finding the minimum between C(

_{i}r**r**–

**r**) and all the m independent of C(

_{M}**r**–

**S**).

_{i}r**r**) obtained by the inverse FT of the sup{I(s)}-rescaled diffraction pattern corresponding to the example of the ZnSe/GaAs heterostructure, oriented in a [110] zone axis, discussed in the previous section. Figure 8b shows the shifted replica of the auto-correlation function C(

**r**-

**r**) centered on a secondary maximum

_{M}**r**. Figure 8c shows the product of C(

_{M}**r**) and C(

**r**-

**r**). Since in the illuminated area atomic columns belonging to both the substrate and to the epilayer are present, as schematically indicated in Figure 8c, we find on the cations and anions atomic columns the contributions of both materials constituting the heterostructure. Thus, the projection of one sub-lattice will give the average atomic potential corresponding to the Ga (Z = 31) and Zn (Z = 30) atomic columns. The other will give the average atomic potential corresponding to the As (Z = 33) and Se (Z = 34) atomic columns. Figure 8d shows the final projected atomic potential obtained deconvolving the increased width of the atomic columns due to the auto-correlation, as discussed in the previous section. This goal can be accomplished by deriving the Point Spread Function (PSF) directly from the normalized C(

_{M}**r**) main peak. Hence, the next step is to deconvolve the square root of C(

**r**) (see Section 2), by applying a Lucy-Richardson (LR) deconvolution approach, until about one half of its initial peak width is reached [27]. When the projected atomic potential is non-centrosymmetric, as in the example shown in Figure 8, the anti-symmetrical part Φ

_{A}(

**r**) can be directly estimated by the inverse FT of ΔI and added to the symmetrical part determined by the deconvolution of the inverse FT of the centrosymmetric component I

_{CS}of the diffraction pattern (see Appendix A).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A: Theory

#### Appendix A1. Electron Wave Function for a Nano-Sized Illumination Beam

**r**) = 2meV(

**r**)/h

^{2}, V(

**r**) is the Coulomb atomic potential of the sample and ψ(

**r**) the illuminating electron wave function. Here, m, e, h, and i, are the electron mass, charge, Planck constant and imaginary unit, respectively. In general, the illuminating electron wave function can be written as ψ(

**r**) = A(

**r**)exp[iϕ(

**r**)], where A(

**r**) is the wave amplitude and ϕ(

**r**) the phase. It can be expressed also as a wave packet superposition [5]:

_{s}and Δf are the spherical aberration and the defocus of the electron objective lens, respectively. Here, g(

**s**) is the wave amplitude with wave vector

**k**= 2π

**s**. The finite width of g(

**s**), necessary to form a nano-sized illumination, limits the lateral coherence of the electron beam. In turn, this leads to a smaller field of view and a higher intensity in the center of the illuminated nano-region [5]. This is just what happens in experimental conditions for which an incident nano-sized beam has a maximum intensity at its center and assumes a decreasing amplitude profile as a function of

**r**, described by the function A(

**r**) = |ψ(

**r**)|, going to zero within few nm from its maximum (see, for example, Figure 1).

#### Appendix A2. Transverse Coherence Length

_{c}~λ/(2γ).

_{c}~15 nm. Therefore, choosing a proper experimental setup, an illuminated area of 2Δr~10 nm can be, in principle, coherently illuminated.

#### Appendix A3. Compensation of Non-Linear Phase Shifts

_{s}corrected is the defocus Δf. In weak-phase approximation the nonlinear phase transformation of the objective lens does not affect the experimental diffraction pattern [1], since the transfer function of Equation (A2) is described by a phase-factor with unitary modulus. Conversely, the corresponding image is strongly affected by the non-linear phase shift introduced by the objective lens. This is why in HRTEM experiments the defocus is tuned to obtain a slowly varying phase factor in a suitable range of frequencies to produce an experimental result not strongly dependent on the rapid oscillations of the phase [1] (see Figure 1).

**r**) would be slowly variable over the spatial region where A(

**r**) ≠ 0. Under this condition the illuminating function can be approximated by ψ(

**r**) ≅ A(

**r**), neglecting the low spatial variability of phase ϕ(

**r**) as a function of

**r**.

#### Appendix A4. Dynamical Effects on the Non-Centrosymmetry of the Diffraction Pattern

^{−1}[I] the inverse FT of Equation (A1), ψ(

**r**) ≅ A(

**r**) and applying the product approximation to describe the phase shift introduced by the sample on the incident wave [28] we obtain

**s**and

**r**. Here “⊗” denotes the convolution product and “*” the complex conjugate. C is the auto-correlation of the complex function A exp(iΦ).

**r**) = Φ

_{R}(

**r**) + iΦ

_{I}(

**r**) ≅ Φ

_{R}(

**r**) + iαΦ

_{R}(

**r**) = (1 + iα)Φ

_{R}(

**r**), where Φ

_{R}(

**r**) is proportional to the Coulomb projected atomic potential and α of the order of 0.01–0.05 for many atomic species [20]. Structures with atoms of different species can be described by α = α(

**r**), with α(

**r**) << 1. Therefore, Equation (A5) can be rewritten as:

_{R}) describes the absorption. Equation (A6) gives the dynamical intensity scattered by the sample for a nano-sized illuminating beam with amplitude A(

**r**). Let us note that the right side of Equation (A6) is real if Φ

_{R}(

**r**) is centrosymmetric [19] and A(

**r**) = A(−

**r**). In this case, rewriting the complex exponential functions in terms of trigonometric functions it gives:

_{R}(

**r**) is centrosymmetric, a straightforward calculus shows that complex terms cancel each other and, as already stated, any non-centrosymmetry in the measured diffracted pattern can be almost fully ascribed to mis-tilts of the crystalline sample with respect to the precise zone axis orientation. However, it should be noted that for nano-sized illuminations another possible cause of non-centrosymmetry of the diffraction pattern obtained by a centrosymmetric projected potential could be ascribed to an asymmetric nano-beam illumination, i.e., A(

**r**) ≠ A(–

**r**).

_{cos}, iA

_{cos}and C

_{sin}, iA

_{sin}, respectively, it results:

**s**) ≠ I(−

**s**) and the diffraction pattern would be non-centrosymmetric, also for a symmetric illumination function A(

**r**) = A(–

**r**). For a centrosymmetric Φ

_{R}(

**r**) = Φ

_{R}(–

**r**) the anti-symmetric components A

_{cos}and A

_{sin}would be negligible and, if A(

**r**) = A(–

**r**), from Equation. (A9) readily it would follow that I(

**s**) = I(–

**s**), as previously stated. Conversely, for a non-centrosymmetric projected potential even small dynamical effects would produce breaking of Friedel’s pair symmetry, independently of any mis-tilt of the specimen with respect to the precise zone axis.

#### Appendix A5. Weak Phase Approximation

_{R}<< 1, from Equation (A9) the so-called Weak Phase Approximation (WPA) is obtained expanding in series the trigonometric functions up to the first order in Φ

_{R}and neglecting the absorption terms. Under WPA the diffracted intensity is readily derived by the FT of the following Equation:

_{R}times the incident nano-beam amplitude A. Thus, the widths of Bragg peaks will be directly related to the size and shape of the illumination nano-beam.

#### Appendix A6. Second-Order Dynamical Perturbation to the Weak Phase-Approximation

_{R}<< 1 is violated even for few unit cells’ sample thickness. A dynamical scattering factor f

_{dyn}can be formally defined as [28]

_{kin}is directly related to the WPA projected potential:

_{dyn}is not a real quantity. This finding will cause destructive interferences for some Bragg reflection amplitudes and constructive for others, through multiple scattering, described by the convolution operation inside the square modulus. Nevertheless, the square modulus of the second order dynamical term will be added to the kinematical contribution obtainable in a WPA. Therefore, if any of the Bragg reflections would be affected by dynamical scattering, the corresponding diffracted intensities would increase for the second-order dynamical contribution. By considering higher orders in the power expansion of the trigonometric functions some terms could give a negative contribution to the intensity of some Bragg reflections, but the overall result, for sample thickness smaller than half of the corresponding extinction distances, would be to increase the diffracted intensities, reducing the direct beam intensity.

_{max}(s) = Σ

_{i}f

_{i}(s), which is the sum of the scattering factors f

_{i}(s) of the atoms in the crystal unit cell assuming that all are adding constructively their scattered amplitudes. In this way the upper limit of any diffracted intensity, normalized to the incident beam intensity, can be readily evaluated. Therefore, the rescaling of the measured intensities by using the sup{I(s)} = |F

_{max}(s)|

^{2}/|F

_{max}(0)|

^{2}constraint partially corrects the effect of the dynamical scattering, reducing the effect of the dynamical contribution of Equation (A16) and of other higher order terms to the diffracted intensities. Obviously, this does not assure that the rescaled diffracted intensity is fully corrected by dynamical scattering, but our experimental tests show that this rescaling of intensities is sufficient to guarantee the quantitative determination of the projected atomic potential. Indeed, the deconvolution of the modulus of the FT of the rescaled diffraction pattern readily gives the correct projected atomic potential Φ

_{R}(

**r**) averaged in the nano-region where A(

**r**) ≠ 0. Our experimental tests indicate that for sample thickness larger than one third of the smaller extinction distance, the dynamical effects cannot be recovered by the proposed approach. Nevertheless the range of applicability of the method is very wide because, usually, in standard TEM experiments the thickness of the specimen does not exceed a few tens of nm.

#### Appendix A7. General Equations

**r**) of the illuminating electron wave function ψ(

**r**) = A(

**r**)exp[iϕ(

**r**)]. When the dependence of the phase ϕ(

**r**) on the spatial position

**r**cannot be neglected Equation (A5) has to be generalized as follow:

_{R}(

**r**), from the contribution of the incident wave, i.e., ϕ(

**r**). Indeed, the diffraction pattern will be characterized by several interferences related to the probe. The deconvolution of the illumination phase wave from the contribution of the sample is needed, for example, in X-ray Fresnel diffraction Keyhole CDI [29], in ptychography [30,31] and in any case in which it is not possible to tune properly the optical conditions for the lenses in order to suitably compensate non-linear phase components of the illuminating wave function.

#### Appendix A8. Incoherent Source Approximation

**y**=

**r**−

_{1}**r**between different points of the sample. From Equations (A18) and (A19) it results:

_{2}**y**), directly related to the incident nano-beam shape and size. In these conditions the coherent dynamical diffracted intensity I

_{coh}(

**s**) is convoluted with the intensity profile of the incident beam. Equation (A20) implies that, in experimental situations similar to those treated in the present work, it is not necessary to know the incident illumination function in modulus and phase. This situation is substantially different from ptychography [30,31] or Fresnel X-ray Keyhole CDI [29]. Indeed, in some experiments it is sufficient to know only the intensity of the source probe. This finding is a consequence of the assumption (A19). It is valid only for monochromatic spatially incoherent sources, i.e., when the spatial coherence is developed through wave propagation and when we are dealing with the Fraunhofer diffraction geometry. This result is a direct consequence of the Van Cittert-Zernike theorem [32]. Indeed, in the far-field limit, the Van Cittert-Zernike theorem for an incoherent source relates the angular distribution of the wave-field impinging on the sample directly to the FT of the physical source square modulus amplitude. Equation (A20) can be assumed approximately valid also for Schottky TEM sources, used in the present work, because the spatial coherence of the electron waves emitted by these sources is reached by reducing their size on a small scale, but single electrons are emitted incoherently from each other by different regions of the sources. Under the condition given by Equation (A19) the inverse FT of I, given by Equation (A7), can be further simplified as follows:

_{R}(

**r**). This is possible since A(

**r**) is slowly variable as a function of

**r**with respect to the Φ

_{R}(

**r**) lattice periodicity and the phase ϕ(

**r**) can be considered almost constant on the same spatial scale.

#### Appendix A9. Further Insight for the Kinematical Approximation

**s**) ≠ I(–

**s**). Thus, in general, the non-centrosymmetry implies that the auto-correlation function C(

**r**) is a complex function. However, if the imaginary part of the potential is approximately proportional to the real part, for uniform illumination A(

**r**) = const, in conditions of kinematical scattering and low absorbing samples, the imaginary part of C(

**r**) is negligible also for non-centrosymmetric atomic projected potential. Indeed, let denote with Φ

_{S}(

**r**) and Φ

_{A}(

**r**) the symmetrical and anti-symmetrical part of the real component Φ

_{R}(

**r**) of the projected potential, respectively:

_{A}(

**r**)⊗Φ

_{S}(–

**r**) and Φ

_{S}(

**r**)⊗Φ

_{A}(–

**r**) eliminates each other due to the even and odd parity of Φ

_{S}(

**r**) and Φ

_{A}(

**r**), respectively. Since α << 1, in Equation (A23) the term proportional to α

^{2}has been neglected. Moreover, it is straightforward to verify that for the complex part, the term proportional to iα, is null. Therefore, the auto-correlation function will be real also for a non-centrosymmetric projected potential:

**r**) =

**A**(–

**r**) and a low absorbing sample, we obtain:

_{A}(

**r**) can be directly estimated by FT

^{−1}[I] and added to the symmetrical part determined by the inverse FT of the centrosymmetric component I

_{CS}of the diffraction pattern, as carried out in this work in the case of the specimen with the sphalerite structure discussed in Section 2.

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**Figure 1.**High resolution transmission electron microscopy (HRTEM) image of a nano-region of a SrTiO

_{3}extended sample in [100] zone axis, with a zoom in the inset on the left. A simulation for underfocus 41.3 nm and specimen thickness of 25.0 nm is shown in the inset on the right. The dots in the simulation point to the structural positions of the SrTiO

_{3}atomic species in this projection: Sr = Blue, Ti+O = pale blue, O = pale green.

**Figure 2.**Intensity-rescaling on SrTiO

_{3}(100) electron diffraction (ED)-pattern: (

**a**) Raw experimental nano-diffraction pattern (logarithmic scale); (

**b**) sup{I(s)} rescaled pattern; (

**c**) difference between patterns shown in (a) and (b); (

**d**) comparison in a logarithmic scale between the line profile along the dashed blue line in (a) (blue curve) and along the dashed red line in (b) (red curve) after rescaling. Black curve is the corresponding profile of sup{I(s)} × I

_{max}constraint.

**Figure 3.**Deconvolution of the projected atomic potential auto-correlation function C(

**r**): (

**a**) C(

**r**) atomic projected potential obtained from raw experimental nano-diffraction data; (

**b**) C(

**r**) atomic projected potential obtained from rescaled nano-diffraction data; (

**c**) SrTiO

_{3}calculated projected atomic potential in [100] zone-axis; (

**d**) red curve: sum of the two scans highlighted with the red dashed lines in Figure 3b; black curve: sum of the two scans highlighted with the black dashed lines in Figure 3c. In Figure 3d the experimental and calculated potentials Φ have been normalized to 1, plotted at a power of 3/2 and rescaled to have a maximum equal to Z = 38 (Sr); (

**e**) zoom of the Sr peak to compare its experimental width (red curve), with projected potential values simulated at different spatial resolutions.

**Figure 4.**HRTEM image in [110] zone axis focused at the GaAs/Znse interface. In the inset a magnified view of the HRTEM image contrast at the interface.

**Figure 5.**Intensity-rescaling on the GaAs/Znse (110) ED-pattern: (

**a**) measured diffraction pattern; (

**b**) restored pattern; (

**c**) difference between the measured and the restored patterns; (

**d**) anti-symmetric component ΔI of the restored pattern.

**Figure 6.**Deconvolution of the GaAs/Znse [110]-projected atomic potential auto-correlation function C(

**r**): (

**a**) projected atomic potential obtained from raw experimental nano-diffraction data; (

**b**) projected atomic potential obtained from rescaled nano-diffraction data; (

**c**) scans of the averaged projected atomic potentials, elevated to the power of 3/2 for a direct comparison with the atomic number Z, in red for the restored intensities, in black for the raw data.

**Figure 7.**[110]-projected ZnSe/GaAs atomic potential (at higher magnification in the inset) averaged in the whole illuminated area corresponding to the HRTEM image of Figure 4.

**Figure 8.**Deconvolution of the auto-correlation function C(r): (

**a**) C(r) obtained by the inverse Fourier transform (FT) of the rescaled diffraction pattern for the ZnSe/GaAs heterostructure, oriented in a [110] zone axis; (

**b**) shifted replica of the auto-correlation function C(r-r

_{M}) centered on a secondary maximum r

_{M}; (

**c**) product of C(r) and C(r-r

_{M}); (

**d**) projected potential obtained deconvolving the contribution to the column width due to the auto-correlation.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

De Caro, L.; Scattarella, F.; Carlino, E.
Determination of the Projected Atomic Potential by Deconvolution of the Auto-Correlation Function of TEM Electron Nano-Diffraction Patterns. *Crystals* **2016**, *6*, 141.
https://doi.org/10.3390/cryst6110141

**AMA Style**

De Caro L, Scattarella F, Carlino E.
Determination of the Projected Atomic Potential by Deconvolution of the Auto-Correlation Function of TEM Electron Nano-Diffraction Patterns. *Crystals*. 2016; 6(11):141.
https://doi.org/10.3390/cryst6110141

**Chicago/Turabian Style**

De Caro, Liberato, Francesco Scattarella, and Elvio Carlino.
2016. "Determination of the Projected Atomic Potential by Deconvolution of the Auto-Correlation Function of TEM Electron Nano-Diffraction Patterns" *Crystals* 6, no. 11: 141.
https://doi.org/10.3390/cryst6110141