Pragmatic Application of Abstract Algebra to Two-Dimensional Lattice Matching

Kawahara, Kazuaki; Arafune, Ryuichi; Kawai, Maki; Takagi, Noriaki

doi:10.1380/ejssnt.2015.361

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…The augmentation must be compatible with the hexagonal symmetry of the problem. Possible augmentation factors are 3,4,7,9,12,13,16, K which can be derived from the so-called hexagonal sequence number introduced by Tkachenko [14]. What seems to lead only to redundant cells, provides in fact new solutions of potential commensurate moiré phases.…”

Section: Commensurability and The Moiré Cell Augmentation Methodsmentioning

confidence: 99%

“…The interplay of the real-space geometry of moirés, the corresponding geometry in reciprocal-space and the result on the electronic structure of the corresponding phases led to thorough studies focusing on the description of moirés [9][10][11][12][13][14][15][16][17][18]. Due to the hexagonal symmetry of the g-lattice, in particulargraphene growth on hexagonally packed TMsurfaces and the resulting moiré formation, are of interest [5,7,8,19].…”

Section: Introductionmentioning

confidence: 99%

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Indexing moiré patterns of metal-supported graphene and related systems: strategies and pitfalls

Zeller

Günther

2017

New J. Phys.

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Recent citationsImaging the confined surface oxidation of Ni3Al (111) AbstractWe report on strategies forcharacterizing hexagonal coincidence phases by analyzing the involved spatial moiré beating frequencies of the pattern. We derive general properties of the moiré regarding its symmetry and construct the spatial beating frequency é K moir  as the difference between two reciprocal lattice vectors k i  of the two coinciding lattices. Considering reciprocal lattice vectors k i  , with lengths of up to ntimes the respective (1, 0) beams of the two lattices, readily increases the numberof beating frequencies of the nth-order moiré pattern. We predict how many beating frequencies occur in nth-order moirés and show that for one hexagonal lattice rotating above another the involved beating frequencies follow circular trajectories in reciprocal-space. The radius and lateral displacement of such circles are defined by the order n and the ratio x of the two lattice constants. The question ofwhether the moiré pattern is commensurate or not is addressed by using our derived concept of commensurability plots. When searching potential commensurate phases we introduce a method, which we call cell augmentation, and which avoids the need toconsider high-order beating frequencies as discussed using the reported´ ( )R 6 3 6 3 30 moiré of graphene on SiC(0001). We also show how to apply our model for the characterization of hexagonal moiré phases, found for transition metal-supported graphene and related systems. We explicitly treat surface x-ray diffraction-, scanning tunneling microscopy-and low-energy electron diffraction data to extract the unit cell of commensurate phases or to find evidence for incommensurability. For each data type, analysis strategies are outlined and avoidable pitfalls are discussed. We also point out the close relation of spatial beating frequencies in a moiré and multiple scattering in electron diffraction data and show how this fact can be explicitly used to extract high-precision data.

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Section: Commensurability and The Moiré Cell Augmentation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Indexing moiré patterns of metal-supported graphene and related systems: strategies and pitfalls

Zeller

Günther

2017

New J. Phys.

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“…To reduce the time cost for generating the huge common supercells, we adopt the complex plane approach as Mitani et al 71 and Kawahara et al 73 proposed in previous studies. In the complex plane, the 2×2 rotation matrix R(θ ) becomes a single complex number e iθ , and the square lattice vector a = (a x a y ) T is identical to a complex number z a = a x + a y i corresponding to Gaussian integer as shown in Figure 2(a).…”

Section: /17mentioning

confidence: 99%

“…Previous studies have shown that for the hexagonal structure which has the hexagonal 2D lattice the periodic moiré patterns can be obtained by solving a Diophantine equation. 48,64,[68][69][70][71][72] In addition, Mitani et al 71 and Kawahara et al 73 developed the underlying theory of the deterministic approach in which 2D lattices are expressed as complex numbers in the complex plane, providing numerical conditions for commensurability.…”

Section: Introductionmentioning

confidence: 99%

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A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces

Lee,

Lee

2023

Preprint

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Here, we propose a deterministic algorithm, that is capable of constructing a common supercell between two similar crystalline surfaces such as square–square or hexagonal–hexagonal lattices without scanning all possible cases. Using the complex plane, we define the two-dimensional (2D) lattice as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector–eigenvalue relation where an operator corresponds to a transformation matrix. We show that this transformation matrix can be directly determined from the lattice parameters and rotation angle of the two given crystalline surfaces with O(log N_max) time complexity, where Nmax is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach (O(N_max^4)). By implementing our method in Python code, we successfully generate experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported. According to our density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally-observed moire patterns. Taken together, we believe that our method can be widely applied as a useful tool for designing new heterostructures with interesting properties.

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A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces

Lee,

Lee

2024

Small Methods

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Here, a deterministic algorithm is proposed, that is capable of constructing a common supercell between two similar crystalline surfaces without scanning all possible cases. Using the complex plane, the 2D lattice is defined as the 2D complex vector. Then, the relationship between two surfaces becomes the eigenvector–eigenvalue relation where an operator corresponds to a transformation matrix. It is shown that this transformation matrix can be directly determined from the lattice parameters and rotation angle of the two given crystalline surfaces with O(log Nmax) time complexity, where Nmax is the maximum index of repetition matrix elements. This process is much faster than the conventional brute force approach (). By implementing the method in Python code, experimental 2D heterostructures and their moiré patterns and additionally find new moiré patterns that have not yet been reported are successfully generated. According to the density functional theory (DFT) calculations, some of the new moiré patterns are expected to be as stable as experimentally‐observed moiré patterns. Taken together, it is believed that the method can be widely applied as a useful tool for designing new heterostructures with interesting properties.

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Pragmatic Application of Abstract Algebra to Two-Dimensional Lattice Matching

Cited by 7 publications

References 21 publications

Indexing moiré patterns of metal-supported graphene and related systems: strategies and pitfalls

Indexing moiré patterns of metal-supported graphene and related systems: strategies and pitfalls

A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces

A Deterministic Method to Construct a Common Supercell Between Two Similar Crystalline Surfaces

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