# Effects of Domain Boundaries on the Diffraction Patterns of One-Dimensional Structures

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Methodology

#### 2.2. Supercell Approach

#### 2.3. Binary Surface Technique

#### 2.4. Two-Fold Periodicity

#### 2.5. Three-Fold Periodicity

#### 2.6. $\left(2\sqrt{3}\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}\sqrt{3}\right)\phantom{\rule{4pt}{0ex}}\mathrm{R}{30}^{\circ}$ Reconstruction

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

LEED | Low energy electron diffraction |

SPA-LEED | Spot-profile analysis-Low energy electron diffraction |

DB | Domain boundary |

APDB | Anti-phase domain boundary |

SS | Superstructure |

STM | Scanning tunneling microscop |

DFT | Density functional theory |

## References

- Pendry, J.B. Low Energy Electron Diffraction; Academic Press: London, UK, 1974. [Google Scholar]
- Van Hove, M.A.; Tong, S.Y. Surface Crystallography by LEED: Theory, Computation and Structural Results; Springer: Berlin, Germany, 1979. [Google Scholar]
- Clarke, J.L. Surface Crystallography: An Introduction to Low Energy Electron Diffraction; Wiley: Chichester, UK, 1985. [Google Scholar]
- Henzler, M. Electron Diffraction and Surface Defect Structure. In Electron Spectroscopy for Surface Analysis; Ibach, H., Ed.; Springer: Berlin, Germany, 1977; pp. 117–149. [Google Scholar]
- Henzler, M. Defects in Surface Structure: Information by LEED. In Festkörperprobleme; Treusch, J., Ed.; Vieweg: Braunschweig, Germany, 1979; Volume 19, pp. 193–208. [Google Scholar]
- Henzler, M. Measurement of Surface Defects by Low-Energy Electron Diffraction. Appl. Phys. A
**1984**, 34, 205–214. [Google Scholar] [CrossRef] - Horn-von Hoegen, M. Growth of semiconductor layers studied by spot profile analysis low energy electron diffraction—Part I. Z. Kristallogr.
**1999**, 214, 591–629. [Google Scholar] - Petkova, A.; Wollschläger, J.; Günther, H.-L.; Henzler, M. Formation and commensurate analysis of “incommensurate” superstructures of Pb on Si(111). Surf. Sci.
**2001**, 471, 11–20. [Google Scholar] [CrossRef] - Stepanovska, S.; Yakes, M.; Yeh, V.; Hupalo, M.; Tringides, M.C. The dense α-($\sqrt{3}$ × $\sqrt{3}$)Pb/Si(111) phase: A comprehensive STM and SPA-LEED study on ordering, phase transitions and interactions. Surf. Sci.
**2006**, 600, 1417–1430. [Google Scholar] [CrossRef] - Saranin, A.A.; Lifshitz, V.G.; Bethge, H.; Kayser, R.; Klust, A.; Goldbach, H.; Wollschläger, J.; Henzler, M. Restructuring process of the Si(111) surface upon Ca deposition. Surf. Sci.
**2000**, 448, 87–92. [Google Scholar] [CrossRef] - Timmer, F.; Oelke, R.; Dues, C.; Sanna, S.; Schmidt, W.G.; Franz, M.; Appelfeller, S.; Dähne, M.; Wollschläger, J. Strain-induced quasi-one-dimensional rare-earth silicide structures on Si(111). Phys. Rev. B
**2016**, 94, 205431. [Google Scholar] [CrossRef] - Klasing, F. Indium on Silicon(111) Low-Energy Electron Diffraction Experiments and Simulations on the Striped (8 × 2) Surface Reconstruction and the (4 × 1) ↔ (8 × 2) Phase-Transition. Ph.D. Thesis, University of Duisburg-Essen, Duisburg, Germany, 2014. [Google Scholar]
- Timmer, F.; Wollschläger, J. Influence of the Distribution of Domains and Domain Boundaries of One-Dimensional Structures on Diffraction Peaks Profiles. Unpublished work. 2017. [Google Scholar]
- Zeppenfeld, P.; Kern, K.; David, R.; Comsa, G. Diffraction from domain-wall systems. Phys. Rev. B
**1988**, 38, 3918–3924. [Google Scholar] [CrossRef] - Bianconi, A.; Marcelli, A. (Eds.) Atomically Controlled Surfaces Interfaces and Nanostructures; Superstripes Press: Rome, Italy, 2016.

**Figure 1.**(

**a**) diffraction pattern of the perfect two-fold periodicity (blue) and after introduction of domain boundaries $D{B}_{1}$ (w = 1, cyan) and $D{B}_{2}$ (w = 2, red); (

**b**) diffraction pattern of the three-fold periodicity for the ${D}_{1}$ domain and the domain boundaries $D{B}_{1}$ (green), $D{B}_{2}$ (red), $D{B}_{3}$ (cyan) and the alternation of $D{B}_{1}$ and $D{B}_{2}$ (blue); (

**c**) diffraction patterns for the alternation of the domains ${D}_{1}$ and ${D}_{2}$ with the domain boundaries $D{B}_{k}$ = $D{B}_{1}$ and $D{B}_{l}$ = $D{B}_{1}$ (blue), $D{B}_{k}$ = $D{B}_{2}$ and $D{B}_{l}$ = $D{B}_{2}$ (black), $D{B}_{k}$ = $D{B}_{1}$ and $D{B}_{l}$ = $D{B}_{2}$ (red) and $D{B}_{k}$ = $D{B}_{2}$ and $D{B}_{l}$ = $D{B}_{1}$ (magenta).

**Figure 2.**(

**a**) Schematic display of the structure model of the $\left(2\sqrt{3}\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}\sqrt{3}\right)\phantom{\rule{4pt}{0ex}}\mathrm{R}{30}^{\circ}$ reconstruction. There are four different energetically favorable models (e, f, q, r) that belong to two different types of domains. The models e and f make up the first type of domain where the vacancies in the Si${}_{3}$ and Si${}_{4}$ layer are located underneath Si atoms in the Si${}_{2}$ layer, whereas the models q and r make up the second domain where the vacancies in the Si${}_{3}$ and Si${}_{4}$ layer are located underneath Si atoms in the Si${}_{1}$ layer. Both models exhibit different structural motives in Scanning Tunneling Microscopy (STM) measurements [11], which are indicated here by green triangles and red hexagons. The binary sequences after the projection onto the relevant crystallographic axis of the different models for the layer Si${}_{3}$ and Si${}_{4}$ are given reflecting the fact that the Si${}_{3}$ layer exhibits a ($\sqrt{3}\times \sqrt{3}$) periodicity and the Si${}_{4}$ layer exhibits a ($2\sqrt{3}\times \sqrt{3}$) periodicity. (

**b**,

**c**) example of two potential supercells [...|e|$D{B}_{1}$|q|$D{B}_{2}$|...] incorporating different sizes of DBs. (

**b**) $w\left(D{B}_{1}\right)$ = 1 and $w\left(D{B}_{2}\right)$ = 2 and (

**c**) $w\left(D{B}_{1}\right)$ = 4 and $w\left(D{B}_{2}\right)$ = 5. For reasons of better visibility, the supercells only contain one unit cell of the model e and q, respectively. However, in principle, the supercell can contain multiple unit cells of either model and the domain sizes can be obtained from the superstructure peak splitting.

**Figure 3.**(

**a**) experimentally observed diffraction pattern along the 2$\sqrt{3}$-direction; (

**b**–

**e**) diffraction pattern of the four models (e ↔ q, e ↔ r, f ↔ q, f ↔ r) for all possible combinations of the anti-phase domain boundaries $D{B}_{1}$ and $D{B}_{2}$ for $w\left(D{B}_{1}\right)$ + $w(D{B}_{2}$) = 3 or 9. The lateral scattering vector H is given in units of the silicon substrate. This means that the nominal first order diffraction spot is located at 1/(2$\sqrt{3}$) = 0.2887. Consequently, peaks of higher order are located at multiples of this alue. Additionally, only a part of the diffraction pattern is displayed for reasons of better visibility.

**Table 1.**Assignment of the five features observed in the simulated diffraction patterns (see Figure 3b) of the 14 models (α through o) for the combination of model e and q. Additionally, the width of both types of the DBs is displayed for the different models.

Feature | Experimental | α | β | γ | δ | ϵ | ζ | η | θ | ι | κ | λ | μ | ν | o |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

2 | ✕ | ✕ | ✕ | ✕ | ✕ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

3 | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ | ✓ | ✓ | ✕ | ✓ | ✓ |

4 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

5 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

$w\left(D{B}_{1}\right)$ | - | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

$w\left(D{B}_{2}\right)$ | - | 3 | 2 | 1 | 0 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |

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**MDPI and ACS Style**

Timmer, F.; Wollschläger, J. Effects of Domain Boundaries on the Diffraction Patterns of One-Dimensional Structures. *Condens. Matter* **2017**, *2*, 7.
https://doi.org/10.3390/condmat2010007

**AMA Style**

Timmer F, Wollschläger J. Effects of Domain Boundaries on the Diffraction Patterns of One-Dimensional Structures. *Condensed Matter*. 2017; 2(1):7.
https://doi.org/10.3390/condmat2010007

**Chicago/Turabian Style**

Timmer, Frederic, and Joachim Wollschläger. 2017. "Effects of Domain Boundaries on the Diffraction Patterns of One-Dimensional Structures" *Condensed Matter* 2, no. 1: 7.
https://doi.org/10.3390/condmat2010007