# Modelling Electron Channeling Contrast Intensity of Stacking Fault and Twin Boundary Using Crystal Thickness Effect

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## Abstract

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## 1. Introduction

_{BSE}), thus leading to a contrast.

_{BSE}profiles of the bands forming the channeling pattern exhibit the main experimental features with a modulated intensity in the central region bounded by dark edges. In addition, in their theoretical model, the increase of the specimen thickness (above 1000 nm) produces an increase of the contrast and of the background intensity. More recently, Winkelmann obtained a theoretical channeling pattern corresponding to an experimental ECP using dynamical many-beam simulations based on the Bloch wave approach and on the forward–backward approximation [24].

**R**and/or rotated through an angle β about a vector

**v**[26]. For example, these approaches showed that the calculated I

_{BSE}profile of stacking fault exhibits damped fringes of depth periodicity ξ

_{g}as observed experimentally [19,20,25]. Depending on the

**g.R**sign (where

**g**is the diffraction vector and

**R**is the displacement vector of the fault) the contrast of the first fringe can be bright or dark. However, despite this important contribution to the theory of the channeling contrast of defects, the publications presenting these models did not display neither detailed calculations nor a usable analytical expression of the I

_{BSE}leading to misunderstanding, and in most cases, a comparison between theoretical and experimental results is still missing [18,19,20,25].

_{BSE}around dislocations, Kriaa et al. developed a theoretical model based on the Bloch wave approach of the dynamical diffraction [13]. Their model leads to an explicit analytical formula of the BackScattered Electron (BSE) signal as function of various physical parameters controlling the ECCI experiment [12,13]. They demonstrated that screw and edge dislocations parallel to the sample surface have the same appearance of BSE contrast profiles for different diffraction conditions and they confirmed theoretically the use of the invisibility criteria in ECCI [13,27].

**R**is not constant moving away from the boundary plane (but constant along it), and thus

**R**is not directly connected to the crystal lattice [26,28]. Hence, in the second part of this contribution, the strategy used to evaluate the BSE yields produced by a twin boundary is easily transposed to model the contrast generated by stacking faults (Section 3), taking advantage, in a simple way, from the explicit formula relating the BSE intensity of a perfect crystal to its thickness.

## 2. Materials and Methods

#### Experimental Analysis of a Special Grain Boundary: Twin-Boundary

**g**= (1$\overline{1}$0). The surface planes of zone 1 and zone 2 are, respectively, (457) and (013). The common direction on the surface to both grains is [2$\overline{3}$1]. The twin-boundary plane, which intercepts both (457) and (013) planes along direction [2$\overline{3}$1], is the (111) plane. This latter is inclined of about 11° relative to the surface of observation.

## 3. Theoretical Models

#### 3.1. Contribution of a Thin Perfect Crystal to the BSE Signal

_{B}is the backscattering cross-section through angles larger than 90° and ψ ψ* is the probability for the Bloch wave to be backscattered at a depth z. The last terms (in parentheses) in Equation (1) describes the electrons that are removed from the Bloch wave field by scattering before reaching the slice dz.

_{BSE}due to orientation contrast is calculated. Here, the atomic number and the surface inclination contributions are not considered.

_{g}is the extinction distance. This Equation (3) corresponds to the intensity profile of an isolated pseudo-Kikuchi band [19,20,29]. Note that in the book of Reimer (Reimer, 1998), Equation (3) contains an error: It is written 2π to the denominator instead of 4π.

_{g}:$\mathsf{\xi}\begin{array}{c}\prime \\ 0\end{array}$ and $\mathsf{\xi}\begin{array}{c}\prime \\ \mathrm{g}\end{array}$. Such parameters are tabulated in the literature for a given acceleration voltage E and diffraction vector

**g**for the following materials: Al, Si, Cu, Ge and Au [29].

**g**= (220), ξ

_{g}= 50 nm, $\mathsf{\xi}\begin{array}{c}\prime \\ 0\end{array}$ = 140 nm and $\mathsf{\xi}\begin{array}{c}\prime \\ \mathrm{g}\end{array}$ = 600 nm. Note that the same BSE intensity profile appearance are obtained for the other materials.

- For a thickness t= 0.12ξ
_{g}(6 nm), the slight Δη’ variations are between −0.58 (a.u.) and 0.58 (a.u.) as it is shown in Figure 3a. Such variations are due to the slight contribution of the term T(ω). - For a thickness t = 0.2ξ
_{g}(10 nm), a larges peak (for negative values of ω) and a hollow (for positive values of ω) appear (see Figure 3b). In addition, it is noted that the more the thickness increases, the more the amplitude of Δη’(ω) increases. These same observations are accentuated for the following thicknesses. - For the thicknesses t = 0.7ξ
_{g}(35 nm) and t = ξ_{g}(50 nm), the two profiles have, almost, the same appearance: Peak and hollow less spread than those obtained for t = 0.2ξ_{g}. In addition, oscillations on the sides of these curves appear (Figure 3c for t = 0.7ξ_{g}).

_{g}(80 nm), the appearance of the total Δη’ curve becomes identical to that of Δη (Figure 3d) corresponding to the contribution to BSE from an infinite or thick crystal. In this case, the contribution of the term T in the formation of the pseudo-Kikuchi band contrast becomes null (Δη prevails over since the limit of T when t tends towards infinity is zero).

_{BSE}profile for a perfect crystal occurs under different steps. Firstly, for thickness below 0.2ξ

_{g}, no band contrast is observed: For small thicknesses, the transmission of electrons takes place by different processes such as channeling, Bragg diffraction and inelastic scattering. This results in a low backscattered signal insufficient to form a Kikuchi band. Then, a transitory step is obtained in which oscillations are observed, on the sides of the curves, that disappear at t = 1.6ξ

_{g}. This latter thickness is, indeed, sufficient to generate an important backscattered signal sensitive to the orientation of the beam relative to the crystalline planes and thus to take advantage of the channeling phenomenon for characterizing defects. Therefore, the BSE intensity modulation allows the formation of the like-Kikuchi band with its main characteristics (dark line for channeling position (edge band), brighter band for θ < θ

_{c}). The text continues here.

#### 3.2. Modelling the BSE Contrast Generated by a Coherent Twin Boundary

**R**and/or rotated through an angle β about a vector

**v [26]**: Both crystals contribute to the contrast generation. For simplification reasons we will suppose that β = 0 and that the interface is plane, i.e., parallel to a crystallographic plane. The faulted plane is situated in a depth z

_{SF}. Here, we have to consider a new deviation parameter, for the slightly shifted crystal part:

_{SF}where ω

_{SF}= (

**g.R**)ξ

_{g}

**g.R**represents the supplementary deviation ω

_{SF}due to the displacement of the crystal below the fault plane relative to the crystal above. It is important to note that for the above crystal ω

_{SF}is zero.

_{SF}, the total generated BSE signal can be expressed as follow:

_{1}and η

_{2}represent the BSE generated, respectively, from the perfect crystal above the stacking fault and from the shifted one as it is represented in the schematic of Figure 4: η

_{1}and $\mathsf{\eta}\begin{array}{c}\prime \prime \\ 2\end{array}$ correspond to the contribution of a sample with z

_{SF}thickness on the BSE signal, $\mathsf{\eta}\begin{array}{c}\prime \\ 2\end{array}$ represents the variation of this signal on a perfect crystal.

_{BSE}(experimentally: Intensity profile collected in the direction perpendicular to the fringes) is then obtained as a function of x; distance away from the intersection of the defect with the surface (θ is the angle between the stacking fault plane and the surface).

#### 3.2.1. For **g.R** ≠ 0

**g.R**≠ 0 (where

**g**= (220)). The curves, in Figure 5a,b, show that the inclined stacking fault starts with an intense light line at the intersection of the defect with the surface (x = 0 nm). For increasing x values, the calculated contrast fades with oscillations in agreement with experimental ECC images of twin boundary (Figure 1) and stacking fault (Figure 12 in [6]). Furthermore, the more the inclination angle θ is important the more the spatial periods T and T’ (indicated, respectively, by the green and the blue lines in Figure 5a,b) become lower (T ≈ 5 nm, T’ ≈ 50 nm for θ = 45°, and T ≈ 2 nm, T’ ≈ 18 nm for θ = 70°), and the more the contrast fades quickly. Moreover, the modeled profiles start with an intensity peak (see Figure 5b) and hollow (see Figure 5c), respectively, for

**g.R**> 0 and for

**g.R**< 0.

#### 3.2.2. For **g.R** = 0

**g.R**= 0 is represented in Figure 5d. This latter shows a line parallel to the x-axis which corresponds to the background signal; the defect is then invisible. It is important to note that experimentally such defect can be invisible even out of these invisibility conditions. For example, for a stacking fault parallel to the surface of the specimen,

**g.R**= 0 for all values of

**g**lying in the faulted plane.

## 4. Conclusions

_{BSE}as a function of the thickness of the crystal aiming a twofold purpose: First, the formation of like-Kikuchi bands, and second the modelling of the contrast generated by a planar defect such as a coherent twin-boundary or a stacking fault. Considering a perfect crystal, we demonstrate that the formation of the I

_{BSE}profile of a like-Kikuchi band requires a sufficient thickness depending on the extinction distance ξ

_{g}; at least 1.6ξ

_{g}in the case of Al. Under such conditions, the sensitivity of the BSE yield BSE is adequate, and the channeling phenomenon can be used to determine crystal orientation or to contrast defects.

_{BSE}generated by a planar defect, which in some cases displays fringes, our model was extended in an elegant and simple way to the case of an imperfect crystal containing a stacking fault or a coherent twin boundary inclined relative to the surface specimen. For these cases, the calculated BSE profiles exhibit an oscillatory regime: The amplitude of the oscillations decreases with the increasing values of the distance away from the intersection defect-surface due to the Bloch wave absorption. The oscillation frequency depends on different parameters such as the inclination angle θ and the extinction distance ξ

_{g}in agreement with experimental observation. Furthermore, the inversion of the

**g.R**sign led to the inversion of the planar defect contrast and no contrast is generated for

**g.R**= 0 exactly as for dislocations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Electron Channeling Contrasts (ECC) micrograph and (

**b**) the experimental profile (obtained from the zone indicated by the blue arrow) of a true twin-boundary observed in TiAl for the diffraction condition

**g**= (1$\overline{1}$0).

**Figure 3.**BSE intensity profiles of a pseudo-Kikuchi band generated for different sample thicknesses (

**a**) t ≈ 0.12ξ

_{g}(6 nm), (

**b**) t = 0.2ξ

_{g}(10 nm), (

**c**) t = 0.7ξ

_{g}(35 nm) and (

**d**) t = 1.6ξ

_{g}(80 nm).

**Figure 4.**Explanatory schematic of the BSE signal generated from a crystal containing an inclined stacking fault. The red array represents the BSE signal generated from the perfect crystal above the fault plane. The BSE generated by the shifted crystal, η

_{2}, corresponds to a combination between the BSE signal represented with dotted green arrays.

**Figure 5.**BackScattered Electron Intensity (I

_{BSE}) profile calculated as a function of x, distance away of the intersection of the defect with the surface, for (

**a**)

**g.R**≠ 0 and θ = 45°, (

**b**)

**g.R**> 0, θ = 70° and (

**c**)

**g.R**< 0, θ = 70° and (

**d**)

**g.R**= 0. The green and blue lines indicate, respectively, the spatial periods T and T’.

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**MDPI and ACS Style**

Kriaa, H.; Guitton, A.; Maloufi, N.
Modelling Electron Channeling Contrast Intensity of Stacking Fault and Twin Boundary Using Crystal Thickness Effect. *Materials* **2021**, *14*, 1696.
https://doi.org/10.3390/ma14071696

**AMA Style**

Kriaa H, Guitton A, Maloufi N.
Modelling Electron Channeling Contrast Intensity of Stacking Fault and Twin Boundary Using Crystal Thickness Effect. *Materials*. 2021; 14(7):1696.
https://doi.org/10.3390/ma14071696

**Chicago/Turabian Style**

Kriaa, Hana, Antoine Guitton, and Nabila Maloufi.
2021. "Modelling Electron Channeling Contrast Intensity of Stacking Fault and Twin Boundary Using Crystal Thickness Effect" *Materials* 14, no. 7: 1696.
https://doi.org/10.3390/ma14071696