2D Material Structures and Discrete Symmetries of Periodic Polygons

TL;DR

This paper explores the geometric and symmetry properties of 2D materials with double lattice structures based on periodic polygons, proposing methods to engineer new materials by manipulating superstructures and symmetries.

Contribution

It introduces a novel approach to designing 2D materials using dual superstructures and links these configurations to Lie symmetries for deeper understanding.

Findings

Periodic polygons can generate complex 2D lattice structures.

Engineered superstructures can double the atomic unit cell.

Lie symmetries provide a theoretical framework for these geometries.

Abstract

In this work, we reconsider the study of 2D materials involving double lattice structures associated with periodic polygons. In tessellated periodic representation, it appears two periodic polygons of $k$ sides of unequal side lengths at certain angles fixed by the underlying discrete symmetries. In this way, 2D materials could be engineered by using two superstructures on the same atomic sheet generated by two length parameters $a_{1}$ and $a_{2}$ and rotated by the angle $\phi _{n}^{k}=\frac{n\pi }{k}$, where $n$ is an arbitrary natural number. These geometrical configurations could be exploited to engineer 2D materials by doubling the number of the ordinary unit cell atoms. To support the present conjecture, we establish a link with Lie symmetries including finite and indefinite ones providing a room interpretation for $n$.

Equations

$$M\left(r\right)=\sum _{i=1}^d{m}_i{a}_i$$

$$\hat{T}\left(a_i\right)=\sum _{i=1}^d{t}_i{a}_i$$

$$\hat{T}\left(a_i\right)M\left(r\right)=\sum _{i=1}^d\left(m_i+t_i\right)a_i=M\left(r+a_i\right),$$

$$N(r)=\sum _{i=1}^3\left(m_i+t_i\right)a_i=M\left(r+a_i\right)=\hat{T}\left(a_i\right)M\left(r\right).$$

$$V_{unit}=(a_1.\left|a_2\wedge a_3\right|).$$

$$({\mathbb{R}}_{+}^{\star }{)}^3\times (S^1{)}^3$$

$$a_1$$

$$A_{unit}=\left|a_1\wedge a_2\right|$$

$$a_1$$

$$({\mathbb{R}}_{+}^{\star }{)}^2\times S^1$$

$$a_2=f(k)a_1$$

$$a_2=2^n|\cos (\frac{n\pi }{k})|a_1.$$

$$f(k)=2^n|\cos (\frac{n\pi }{k})|,$$

$$n\ne (p+\frac{1}{2})k,$$

$$L=H\oplus E_{\Delta }.$$

$$E_{\Delta }={\oplus }_{\alpha }{L}_{\alpha }.$$

$$L_{\alpha }=\{x\in L|[h,x]=\alpha (x)x\}$$

$$\dim L=2+|\Delta |.$$

$$A=\left(\begin{array}[]{cc}2&\langle\alpha_{1},\alpha_{2}\rangle\\ \langle\alpha_{2},\alpha_{1}\rangle&2\end{array}\right).$$

$$⟨{\alpha }_i,{\alpha }_j⟩=2\frac{({\alpha }_i.{\alpha }_j)}{({\alpha }_j.{\alpha }_j)},i,j=1,2.$$

$$a_i\leftrightarrow |{\alpha }_i|,i=1,2.$$

$$⟨{\alpha }_1,{\alpha }_2⟩⟨{\alpha }_2,{\alpha }_1⟩=2n.$$

$$a_2=a_1\equiv a,{\varphi }_0^4=0.$$

$$⟨{\alpha }_1,{\alpha }_2⟩=⟨{\alpha }_2,{\alpha }_1⟩=0,|\Delta |=4.$$

$$a=|{\alpha }_1|=|{\alpha }_2|.$$

$$a_2=\sqrt{2}{a}_1,$$

$${\varphi }_1^4=\frac{\pi }{4}.$$

$$⟨{\alpha }_1,{\alpha }_2⟩=-2,⟨{\alpha }_2,{\alpha }_1⟩=-1,|\Delta |=8,|{\alpha }_2|=\sqrt{2}|{\alpha }_1|.$$

$$detA<0.$$

Full text

Keywords: Polygons, 2D materials, Lie and discrete symmetries.

1 Introduction

Two-dimensional (2D) materials, composed of semi-conductor atoms, have been considered as a subject of great interest in condensed matter physics and related topics. Concretely, many efforts have been devoted to investigate the electronic structure of such materials using different engineering methods and various numerical code approaches [1, 2, 3]. The most treated ones are graphene, silicene and germanene. It is remarked that graphene, which is a monolayer of carbon atoms forming 2D hexagonal lattice, has been the focus of extensive theoretical and experimental investigations, due to its important physical properties including high carrier mobility and high electrical and thermal conductivity [4]. For this reason, many 2D materials have been synthesized and modeled using physical methods and geometrical approaches associated with non trivial lattice configurations [5, 6].

The aim of this work is to contribute to this program by proposing a new take on the study of 2D materials based on double lattice structures associated with periodic polygons in the flat geometry. In tessellated periodic representation, it arises two periodic polygons of $k$ sides, which will be refereed to as $k$-polygons, of unequal side lengths rotated at certain angles fixed by the underlying discrete symmetries. This produces two superstructures on the same atomic sheet generated by two length parameters $a_{1}$ and $a_{2}$ and rotated by the angle $\phi_{n}^{k}=\frac{n\pi}{k}$, where $n$ is an arbitrary natural number. To support the present proposition, we establish a link with Lie symmetries including finite and indefinite ones providing a room for $n$. These geometrical configurations could be exploited to engineer materials by doubling the number of the usual unit cell atoms.

The paper is organized as follows. In section 2, we give a concise review on lower dimensional lattice configurations. Section 3 concerns a new take on 2D materials built from periodic $k$-polygons. A connection with Lie symmetries is discussed in section 4, providing a possible interpretation for $n$. Section 5 is devoted to concluding remarks and some speculations.

2 Crystal structures

2.1 Lattice symmetry

Many solid materials involve a crystalline structure that signifies spatial repetition or periodicity, or more appropriately translation symmetry [7]. This is equivalent to say that at every point in the crystal structure the space behaves the same. In arbitrary dimensions $d$, a lattice is a regular arrangement of points $M$ at a spatial position $r$ with a translation $\widehat{T}\left(a_{i}\right)$. This is usually regarded as an operator which shifts these points $M$ by a displacement amount $a_{i}$ in certain directions. Precisely, any point $M$ located at a lattice position $r$ can be associated with a vector given by

[TABLE]

where $m_{i}$ are integer coefficients and $a_{i}$ indicate the primitive translation vectors or primitive lattice (fundamental) vectors. In this way, there exists a translation operator $\widehat{T}\left(a_{i}\right)$

[TABLE]

shifting the point $M$ by the amount $a_{i}$ such as

[TABLE]

where $t_{i}$ are arbitrary integers. This operation symmetry generates a lattice point $M$ at the position $r+a_{i}$. In equations (2.1), (2.2) and (2.3) $m_{i}$ and $a_{i}$ are typical $d$-dimensional crystal structure elements satisfying specific properties in concrete lattices. This means that all lattice points can be restored by repetition of the basis vectors $a_{i}$ using the associated symmetry. In connection with solid state physics, lower dimensional structures can play crucial roles in building interesting materials.

2.2 3D Lattices

Having presented the crystal geometry in terms of the lattice structure [8], we consider now the concrete case of Bravais lattices. In $d=3$, we assume that there are 3 non-coplanar vectors $a_{i},\;i=1,2,3$ that maintain all the properties of the crystal after the shift as a whole by any of these vectors. As a consequence, any point $N$ in Bravais lattice could be obtained from $M$ using translation shifts

[TABLE]

In this way, the crystal structure could be understood in terms of Bravais lattice and the properties of the basis $a_{i}$. It is worth mentioning that the way of choosing the lattice vectors $a_{i}$ is not unique. However, they are chosen such that to be primitive. This means that they are chosen in the lattice as the shortest period, with $a_{2}$ not parallel to $a_{1}$ and $a_{3}$ not coplanar to $a_{1}$ and $a_{2}$. The volume delimited by such primitive vectors defines the primitive unit cell, given by

[TABLE]

In this regard, the lattice vectors $a_{i}$ and the angles between them $\theta_{i}$ generate a 6-dimensional parameter space (or moduli space). This space determines the lattice structure and its symmetries. In 3D, it turns out that this space could have the following topology

[TABLE]

where $S^{1}$ is the 1-dimensional circle. This indicates that any lattice structure depends on

• •

three length parameters, $a_{i}$

• •

three angular parameters, $\theta_{i}$.

Concretely, the types of lattices can be associated with some particular regions in such a moduli space. In particular, they differ by the relations between the lengths $a_{i}$ and the angles $\theta_{i}$. Indeed, there are 14 types of lattices with 7 crystal classes giving rise to several primitive cells [9]. These 14 lattices correspond to the following lattice parameter constraints associated with particular regions in the above moduli space

[TABLE]

In 2D, few crystal classes and primitive cells are discussed. The corresponding 2D crystals are known to have use in many physical applications, including graphene like-materials.

2.3 2D Lattices

In 2D, the above mentioned 14 Bravais lattices in 3D are greatly reduced to 5 ones. This is due to the fact that there should be just two non-colinear basis vectors $a_{1}$ and $a_{2}$. These two fundamental vectors are needed to span 2D with the following delimited area

[TABLE]

defining the primitive unit cell. In fact, the total number of possible 2D Bravais lattices is abased due to the existing translational symmetry given by integer multiples of the basis vectors. These 5 lattices correspond to the following parameter constraints

[TABLE]

An examination of such structures shows that the symmetry of periodic $k$ -polygons, in the flat plane, has not been exploited in building crystals. We will see in the following sections that we can go beyond the above structures by introducing such a powerful tool explored in modern physics including condensed matter physics. This may offer a new way to engineer non trivial structures associated with 2D materials.

3 New take on 2D materials

In this section, we would like to go beyond the above 2D structures by reducing the associated moduli space. As in 3D, it is noted that such a space has the following topology

[TABLE]

coordinated by

• •

two length parameters $a_{i}$,

• •

one angular parameter $\theta$.

The last one is associated with the SO(2) Lie group defining rotations in the plane. At this level, a deep inspection reveals, however, that the ordinary structures do not carry any explicit symmetry fashion of periodic $k$-polygon geometries. Here, we try to give a possible extended view dealing with 2D materials using a geometric approach based on discrete symmetries derived from the behaviors of periodic $k$-polygons. To be precise, we explore techniques of $k$-polygon geometries to support the existence of new structures which could be exploited in the 2D materials physics. Our analysis will be based on a geometric reconsideration of the above classification. In fact, the possible structures will correspond to allowed periodic $k$-polygon geometries recovering the known ones. Before proceeding, let us first start by giving some arguments. First, the symmetry of periodic $k$-polygons should be involved in the classification. In this scenario, the discrete symmetries $Z_{k}$ will be primordial in the determination of the nature of such 2D structures. Second, the dimension of the lattice could play a crucial role in the related construction. Third, the reduced moduli space could be exploited to make links between these arguments. At first sight, it seems that this link does not work. However, group theory and Lie symmetries push us to think about a new take on the above structures. An investigation reveals that the discrete symmetries $Z_{k}$ can be exploited to produce new geometrical materials involving double structures. To show how this works in practice, we could solve the constraint $a_{2}\neq a_{1}$ by a possible reduction of the associated moduli space. A priori, there are many ways to handle this constraint. Here, we propose a possible way giving rise new geometrical structures in 2D. Combining the above arguments, we can anticipate the following relation

[TABLE]

where $f(k)$ is a real function carrying data on the dimension of the material and the discrete symmetries $Z_{k}$ of periodic $k$-polygons. A close inspection shows that we can propose the following formulae

[TABLE]

In this way, the function $f(k)$ takes the form

[TABLE]

where 2 is now interpreted as the lattice dimension and $n$ is an arbitrary natural number constrained by

[TABLE]

where $p$ is a natural number used to insure non vanishing functions. However, the factor $\frac{n\pi}{k}$, which generates a cyclic group representation of ${Z}_{k}$ discrete symmetry, could play an important role in solid state physics with double $k$-polygonal structures. It turns out that we can build up two families:

single structure associated with $n=0$ generated by one length parameter $a=a_{1}=a_{2}$ recovering the ordinary structures, appearing in graphene like-models. 2. 2.

double structure corresponding to $n\neq 0$ provided by two different length parameters $a_{1}$ and $a_{2}$.

At this level, one may comment the last family being not obvious. Indeed, this involves two periodic $k$-polygon geometries producing 2D materials having the property of being close to ordinary ones associated with the first family. In this view, each length parameter produces a single structure with a specific size. The small one is generated by $a_{1}$ while the big one is generated by $a_{2}$ and rotated by the angle $\phi_{n}^{k}=\frac{n\pi}{k}$. This family can be exploited to engineer materials with a double structure obtained by doubling the number of the ordinary unit cell atoms.

To provide a complete analysis, it should be interesting to interpret the natural number $n$. It turns our that this number can find a place in Lie symmetries explored in many branches of physics including condensed matter physics.

4 Connections with rank two Lie symmetries

In this section, we would like to make contact with Lie symmetries. This may support the above proposed relation. Indeed, it is recalled that these symmetries appearing in any physical area including high energy and condensed matter physics [10, 11, 12, 13]. To see how this link works in practice, let us first give a concise review on such symmetries. Roughly speaking, a Lie symmetry $L$ is a vector space together with an antisymmetric bilinear bracket $[,]$ : $L\times L\rightarrow L$ verifying certain conditions including the famous Jacobi identity ($([x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0)$). Finite Lie symmetries involve a particular subalgebra called Cartan subalgebra noted $H$ considered as the maximal toric abelian Lie sub-algebra playing a remarkable rôle in the theory of Lie symmetries. Concretely, it is remarked that $L$ could be decomposed as the direct sum of $H$ and a vector space $E_{\Delta}$

[TABLE]

In this way, $E_{\Delta}$ reads as

[TABLE]

It is noted that the space $L_{\alpha}$ can be defined by

[TABLE]

for $x$ inside $L$ and $\alpha$ ranges over all elements of the dual of $H$. In Lie symmetries, $\alpha$ are called roots which will be interpreted as atom positions in 2D materials. It is well known that a root system $\Delta$ of a Lie symmetry is defined as a subset of an euclidean space $E$ satisfying certain constraints. Indeed, $\Delta$ is a finite set generating $E$. It is noted that 0 is not an element of it. $k\alpha$ is a root only for $k=\pm\alpha$. For any root $\alpha$, $\Delta$ is invariant under reflection $\sigma_{\alpha}$, where $\sigma_{\alpha}(\beta)=\beta-<\beta,\alpha>\alpha$. The last condition concerns the quantity $<\beta,\alpha>$ which should be an integer.

The root system $\Delta$ carries various information about the corresponding Lie symmetries. Such information will be crucial in the present work to handle the number $n$. The link that we are after pushes us to consider a special class of Lie symmetries involving two simple roots $\alpha_{1}$ and $\alpha_{2}$. The associated finite Lie symmetries involve a nice formulae which reads as

[TABLE]

Here, $|\Delta|$ denotes the number of the roots associated with $L$ which will be considered as the unit cell atom number. However, 2 is the associated rank which can be interpreted as the dimension of material lattice on which atoms will be placed. An alternative way to approach these symmetries is to exploit the Cartan matrices derived from the non trivial scalar product between two simple roots $\alpha_{1}$ and $\alpha_{2}$. For such Lie symmetries, the Cartan matrices $A=(A_{ij})$ read as

[TABLE]

where $\langle\alpha_{1},\alpha_{2}\rangle$ is given in terms of the euclidean scalar product between $\alpha_{i}$

[TABLE]

Moreover, its has been shown that this matrix can be illustrated in a nice graph called Dynkin graphs, or diagrams, containing two nodes. Like in graph theory, the diagonal elements correspond to nodes and the non-diagonal elements are associated with the number of the edges between them. In fact, the number of the edges between the node $1$ and the node $2$ is given by $\langle\alpha_{1},\alpha_{2}\rangle\langle\alpha_{2},\alpha_{1}\rangle$. In what follows, we combine these Lie structures and the symmetry of periodic $k$-polygons to present a possible interpretation of $n$. Indeed, this interpretation can be derived from a correspondence between the two simple roots and the geometry of 2D materials derived from periodic $k$-polygons, offering a new approach to elaborate the corresponding materials. The interpretation, that we are after, can be supported by an interplay between the material unit cell and the root system $\Delta$ of rank two Lie symmetries. In this way, each atom placed on the periodic $k$-polygon unit cell corresponds to a non-zero root of the Lie symmetry in question. The lattice parameters $a_{i}$ are associated with the length of the roots

[TABLE]

In this way, the lattice material configuration can be built by using the fact that the periodic $k$-polygons tessellate the full plane generating supercell structure materials. This means that it covers a flat geometry by repeating a single shape, without gaps or overlapping. It turns out that one may distinguish between two possible geometric configurations depending on the values of $k$. The first model involves rectangular geometries associated with $k=4$. However, the second model is described by $k=6$ based on the hexagonal structures.

To make contact with the above discussion, it is remarked that one should consider two separate cases $n=0$ and $n\neq 0$. We will see that these cases are associated with single and double $k$-polygon structures, respectively.

4.1 4-polygon structures

Here, we deal with the periodic 4-polygon case related to the periodic rectangular geometry. A close examination shows that $n$ could have a room in the root system. A possible link is to connect $n$ with the scalar product between simple two simple roots. For 4-polygon structures, a close examination shows that one should consider the following identification

[TABLE]

In order to check this link, let us first consider $n=0$ producing materials with single rectangular structure associated with

[TABLE]

In fact, the positions of the 4 atoms placed on the 4-polygon unit cell are associated with four roots of the Lie symmetry constrained by

[TABLE]

In this case, the contact with 2D materials is assured by

[TABLE]

Taking now $n=1$, we could get materials with a double 4-polygon structure. This involves two 4-polygons generated by two length parameters $a_{1}$ and $a_{2}$ required by

[TABLE]

and rotated by the angle

[TABLE]

The corresponding Lie symmetry is constrained by the following conditions

[TABLE]

The associated material structure with the flat geometry can be derived from the fact 4-polygons tessellate the full plane producing a new crystal by mixing the $a(1\times 1)$ and $a(\sqrt{2}\times\sqrt{2})$ configurations rotated by $\phi_{1}^{4}=\frac{\pi}{4}.$ In this way, the positions of 8 atoms are associated with the root system controlled by (4.22). Inspired by indefinite Kac-Moody theory, we can do something similar for $n>1$ by going beyond finite Lie symmetries. It is recalled that the Cartan matrices $A$ of extended Lie symmetries should obey the property

[TABLE]

Combing certain results of Lie symmetries, the case $n>1$ could be linked to the following Cartan matrix

[TABLE]

In order to get a possible Dynkin geometry, we need to require

[TABLE]

Like in the finite Lie symmetries, we associate to this matrix an extended Dynkin diagram involving two nodes connected by $2n$ lines. This geometry produces a double periodic 4-polygon structure involving 8 atoms placed in the associated unit cell. The configuration contains two periodic 4-polygons with two different sizes. The small one is generated by $a_{1}$ while the second one is engineered by $a_{2}$ rotated by $\frac{n\pi}{4}$.

4.2 Links with 6-polygons

In similar manner, we reconsider the 6-polygon case corresponding to periodic hexagonal geometry. As in the previous case, $n$ will be linked to the scalar product of simple roots. Inspired by finite Lie symmetry, we are now concerned with the following identification

[TABLE]

providing odd numbers of edges of the associated Dynkin diagrams. Considering $n=0$, one can recover

[TABLE]

This is associated with a single hexagonal material having a single structure having $a_{2}=a_{1}$. Indeed, the positions of the 6 atoms localized on the 6-polygon unit cell correspond to six roots of the Lie symmetry constrained by

[TABLE]

To illustrate that, we consider now the case $n=1$. This could give arise materials with a double 6-polygon structure. This involves two 6-polygons generated by two length parameters $a_{1}$ and $a_{2}$ required by

[TABLE]

The associated Lie symmetry is constrained by

[TABLE]

The corresponding material structure with the flat geometry can be derived from the fact 6-polygons tessellate the full plane producing a new crystal by mixing the $a(1\times 1)$ and $a(\sqrt{3}\times\sqrt{3})$ configurations rotated by the angle

[TABLE]

It turns out that we can do something similar for $n>1$. For this case, one can consider Cartan matrix

[TABLE]

with the following constraint

[TABLE]

At first sight it is not obvious how to get the corresponding Dynking diagrams. On the basis of the known results of $G_{2}$ Lie symmetry, it is natural to think about a particular geometry involving a central node being connected with $2n+1$ legs sharing similar properties with the star graph containing a central node connected with $2n+1$ external ones. The above Cartan matrix could be obtained by using the standard techniques of folding[14, 15]. This operation permutes nodes by an outer-automorphism group leaving the star graph invariant. Precisely, this geometric operation folds the nodes being permuted by $S_{2n+1}$ symmetric permutation group. Indeed, this provides a graph involving two nodes connected by $2n+1$ lines. This extended symmetry may offer a double hexagonal structure based on two 6-polygons with different size being rotated by $\phi_{n}^{6}=\frac{n\pi}{6}$.

5 Conclusion and discussions

In this work, we have presented a new take on 2D materials based on double lattice structures involving periodic polygons. In tessellated periodic representation, two periodic $k$-polygons of unequal side lengths at certain angles fixed by the underling discrete symmetries are involved. This procedure has provided two superstructures on the same atomic sheet generated by two length parameters $a_{1}$ and $a_{2}$ and rotated by the angle $\frac{n\pi}{k}$. We believe that these geometrical configurations could be exploited to engineer materials by doubling the number of unit cell atoms. To support this conjecture, we have linked such structures with Lie symmetries including finite and indefinite ones. This correspondence between the Dynkin diagrams and 2D materials could be explored in both directions. In particular, one may use 2D materials to revise partial results on indefinite Lie algebras being still an open problem. This may have a new method to deal with the theory of indefinite Lie symmetries from 2D material physical point of view.

This work comes up with certain questions. A natural one is to consider numerical calculations to check the validity of 2D materials with double periodic structures.

Acknowledgement 1

The authors are grateful to their families for support.