research papers
The sitesymmetry induced representations of layer groups on the Bilbao Crystallographic Server
^{a}Karlsruhe Institute of Technology, Institute of Applied Geosciences, Karlsruhe, Germany, ^{b}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco (UPV/EHU), Bilbao, Spain, ^{c}Institute of Chemistry, Saint Petersburg State University, Saint Petersburg, Russian Federation, ^{d}Ioffe Institute, Saint Petersburg, Russian Federation, and ^{e}Department of Physics Engineering, Hacettepe University, Ankara, Turkey
^{*}Correspondence email: gemma.delaflor@kit.edu
The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es) dedicated to subperiodic groups includes a new tool called LSITESYM for the study of materials with layer and multilayer symmetry. This new program, based on the sitesymmetry approach, establishes the symmetry relations between localized and extended crystal states using representations of layer groups. The efficiency and utility of the program LSITESYM is demonstrated by illustrative examples, which include the analysis of phonon symmetry in Aurivillius compounds and in van der Waals layered crystals MoS_{2} and WS_{2}.
1. Introduction
The interest in layered and multilayered materials such as graphene (Geim & Novoselov, 2007; Randviir et al., 2014) and van der Waals crystals, e.g. the transition metal dichalcogenide MeX_{2} with Me = Nb, Mo, Ta, W, Ti, V, Zr, Hf and X = S, Se, Te (Han et al., 2018; Manzeli et al., 2017; Choi et al., 2017), is constantly growing owing to their interesting properties and possible technological applications. The symmetry of single monolayers can be described by the `diperiodic' or `layer groups' (Kopský & Litvin, 2010), which are threedimensional crystallographic groups with twodimensional translations. There are 80 layer groups which, together with the seven frieze groups (twodimensional groups with onedimensional translations) and the 75 rod groups (threedimensional groups with onedimensional translations), constitute the `subperiodic groups'. The crystallographic data for subperiodic groups are compiled in International Tables for Crystallography, Volume E, Subperiodic Groups (hereafter referred to as ITE) and also offered online by the Bilbao Crystallographic Server (Aroyo et al., 2011; Tasci et al., 2012) (https://www.cryst.ehu.es).
While space groups and their representations describe the symmetry of the bulk electron and phonon states, the layer groups are essential for the description of the electronic structure and the
of crystals. Depending on the interaction between the layer and the bulk, materials with layer symmetry can be classified into five different types: (i) pure layered systems like freestanding films, with graphene monolayers as a notable example of such films; (ii) single layers in layered crystals which, owing to a weak van der Waals interlayer interaction, can be separated from the bulk; (iii) artificial nanolayers grown on substrates or between two bulk materials where the interaction between the nanolayer and surrounding bulk materials is neglected; (iv) layers (or slabs) which model atomically clean crystal surfaces where the slab interaction with the rest of the semiinfinite crystal is neglected; (v) interfaces between different crystals, including domain walls approximated by atomically clean crystal surfaces.The sitesymmetry approach (Evarestov & Smirnov, 1987, 1993; Hatch et al., 1988; Stokes et al., 1991; Kovalev, 1993) is a powerful method which connects the local properties of atoms in crystals with the symmetry of states of the whole crystalline system, i.e. it establishes the symmetry relationships between the localized states in the crystal (vibrations of atoms and atomic orbitals) and crystal extended (phonon and electron) states over the entire The sitesymmetry method is based on the procedure of induction of representations of the crystal from the irreducible representations (irreps) of the sitesymmetry groups of the constituent units (atoms, clusters and layers) according to which the local excitations are transformed. In this way, the symmetry of phonon, electron, biexciton etc. states can be described by the crystal single and doublevalued representations induced by the irreps of the sitesymmetry group. Tables of induced representations were deduced by Evarestov & Smirnov (1987, 1993), Hatch et al. (1988), Stokes et al. (1991) and Kovalev (1993), and later the programs SITESYM and DSITESYM (Elcoro et al., 2017) available on the Bilbao Crystallographic Server were developed to calculate the sitesymmetry induced representations of space and double groups, respectively.
Even though layer groups were described for the first time about a century ago (Weber, 1929; Alexander & Hermann, 1929), it was not until the paper by Zallen et al. (1971) that they were applied for the first time for the description of phonon states in layered As_{2}S_{3} and As_{2}Se_{3} crystals. Afterwards, they were also used for modelling atomically clean crystal surfaces (Ipatova & Kitaev, 1985). However, these studies did not involve the use of induced representations of layer groups. To the best of our knowledge, the first use of induced representations of layer groups was related to the study of the phonon symmetry of hightemperature superconductors (Evarestov et al., 1993; Kitaev et al., 1994) and thereafter to describe the phonon, electron, and biexciton states in artificially grown nanolayers (semiconductor quantum wells) (Tronc & Kitaev, 2001) and layered crystals (Kitaev et al., 2007). Quite recently, induced representations of layer groups have been applied in the analysis of the Brillouinzone centre phonons in layered MoS_{2} and WS_{2} crystals (Evarestov et al., 2017, 2018).
Note that the induced representations of layer groups, in general, could be extracted by the existing tools of the Bilbao Crystallographic Server for induced representations of space groups, like SITESYM, but this procedure would be more complex and prone to errors due to the essential differences between space and layer groups (e.g. differences between the sets of Wyckoff positions and their labelling schemes, between the sets of representations etc.). The aim of this paper is to present the sitesymmetry approach applied to layer symmetry groups for the study of materials with layer symmetry. On the basis of this method, the program LSITESYM has been developed and implemented on the Bilbao Crystallographic Server. In the following sections, the procedure of the program for the construction of the induced representations of layer groups is described in detail, and its utility is demonstrated by several examples.
2. Sitesymmetry method
2.1. General procedure
The sitesymmetry approach establishes the symmetry relations between the crystal extended states induced by localized states of some of the constituent structural units. The procedure for the determination of such a relationship is very useful, as it allows the prediction of the symmetry of the possible extended states starting from the crystal structural data. This task requires the derivation of the irreps of a ), which states that the multiplicities of the irreps of a group in the induced representation from an irrep of a of can be determined from the multiplicities of the irreps of in the representations subduced from to . Therefore, the sitesymmetry method is based on subduction and induction, two basic concepts of representation theory, and on the Frobenius reciprocity theorem.
at any point in (which classify the extended states of the structure) induced by the irreps of the sitesymmetry group of a (according to which localized states are classified). In grouptheoretical terms, the procedure relating localized and extended crystalline states can be described by induction of a representation of a from the irreps of a finite , followed by its reduction into irreps of . In other words, the induction method permits the calculation of the symmetry of the compatible extended states transforming according to irreps of crystal induced by a localized state described by an irrep of the local or sitesymmetry group = . The induction procedure can be applied to any group–subgroup pair , but in the sitesymmetry approach, as its name suggests, a sitesymmetry group is taken as the of . The calculation of the spacegroup irreps induced by the irreps of a sitesymmetry group is not straightforward, because the sitesymmetry group (isomorphic with a point group) is a of infinite index of . This implies that the representation of induced by an irrep of must be of infinite dimension and therefore difficult to calculate directly. This problem is solved by applying the `Frobenius reciprocity theorem' (Serre, 1977The subduction procedure relates the representations of a group to those of its subgroups . Consider an irrep D_{γ} = of a group . The subduction of D_{γ} to the results in a representation of the known as the `subduced representation' D^{Sub}, formed by the matrices of those elements of that also belong to the , i.e. D^{Sub} = = . This subduced representation is in general reducible and is decomposable into irreps of :
The multiplicities of the irreps of in the subduced representation can be calculated by the reduction formula (known also as the `magic' formula):
where χ^{Sub}(s) is the character of the subduced representation and χ_{σ}(s) is the character of the irrep for the same element .
The induction procedure permits the construction of a representation of starting from a representation of . If = is an irrep of , then the matrices of the induced representation = () of are constructed as follows:
Here, t, r = 1, …, m with m equal to the dimension of the irrep of and k, j = 1, …, n where n = is the index of in . The elements are the representatives of the decomposition of with respect to .
The characters of are given by
where is the trace of the jth diagonal block of and n is the index of the in the group (which can be finite or infinite). In general, the induced representations are reducible, and as such it is possible to decompose them into irreps of :
where are the multiplicities of the irreps of in the induced representations and can be calculated by the reduction formula [see equation (2)].
The dimension of the induced representation can be read directly off the equation for its construction [equation (3)]:
This result points out the difficulties for the direct calculation of a representation of a i.e. = . In other words, it is sufficient to calculate the multiplicities of in the subduced representation () in order to obtain the frequencies of in the induced representation ().
induced from an irrep of a finite of . As the sitesymmetry group is a of an infinite index, this suggests that the dimension of the induced irreps must be of infinite dimensions. By means of the sitesymmetry approach it is possible to determine the multiplicities of an irrep of in the induced representation without the necessity of constructing the infinitedimensional representation. The method is based on the Frobenius reciprocity theorem, according to which the multiplicity of an irreducible irrep of in a representation () of induced by an irrep of is equal to the multiplicity of the irrep of in the representation () subduced by of to ,2.2. Application of sitesymmetry method to layer groups
The group–subgroup relations between layer and space groups are essential to extend the sitesymmetry method to layer groups. These relationships have been considered in detail in the literature [see e.g. Wood (1964), ITE and references therein]. The type of of which a given layer group is a is not defined uniquely. The `simplest' to which is related can be expressed as a semidirect product of with the onedimensional translation group T_{3} of additional translations , where T_{3} is a of . Thus, the layer group is isomorphic with the .
The isomorphism between and the cf. Evarestov & Smirnov, 1993). For example, consideration of the restrictions imposed by the loss of periodicity in the third (z) direction yields the following restrictions on the specialposition coordinates of layer groups: only the special positions of whose z coordinate does not involve a fraction of the unitcell parameter are possible special positions of , i.e. special positions of with z coordinates z, −z or 0. In that way, to each of corresponds exactly one of , specified by exactly the same sitesymmetry group and multiplicity, and by the same set of coordinate triplets of equivalent positions. Thus, the description of the Wyckoff positions of layer groups can follow the Wyckoffposition descriptions used in space groups. Note, however, that according to the conventions adopted in ITE the letter labelling of the Wyckoff positions of layer groups is done independently of that of space groups. As a result, the Wyckoff letters of the corresponding space and layergroup Wyckoff positions might not coincide, in general.
results in close relationships between the Wyckoff positions and the irreps of and . One can show that the set of Wyckoff positions of a layer group is contained in the set of Wyckoff positions of the related (The simple relationship between the irreps of and is based on the isomorphism : the irreps of are also irreps of and every irrep of is related to a specific irrep of . In these irreps of all elements of a given T_{3} are mapped onto the same matrix, i.e. the irreps of coincide with those irreps of whose kernel is T_{3} (for details, see Evarestov & Smirnov, 1993). For example, the special k vectors of can also be deduced from the k vectors of . The of a layer group can be described as a projection of the of the corresponding onto the layer plane. Accordingly, the twodimensional set of k vectors k(k_{1}, k_{2}) of can be obtained from the threedimensional k(k_{1}, k_{2}, k_{3}) vectors of by ignoring the third component k_{3}.
of the decomposition of with respect toThe above considerations indicate that the induced representations of a layer group can be read off directly from the induced representations of the corresponding
On the basis of this observation, the procedure of the sitesymmetry method for layer groups can be summarized as follows:(i) Given the layer group (specified by its number), the k vector (k_{1}, k_{2}), the program identifies (a) the corresponding spacegroup number and Wyckoff letter, and (b) the spacegroup wave vector (k_{1}, k_{2}, k_{3}) with k_{3} = 0, related to the layergroup k vector (k_{1}, k_{2}).
and the(ii) The sitesymmetry method for space groups is applied, i.e. the sitesymmetry induced representations are calculated by the program SITESYM. [More detailed information on the algorithm of SITESYM can be found in the work of Elcoro et al. (2017).]
(iii) The sitesymmetry induced representations of SITESYM are described with respect to the layer group .
obtained by2.3. The program LSITESYM
The computer program LSITESYM establishes symmetry relations between localized and extended states in crystals with layer symmetry. This algorithm calculates the multiplicities of the layergroup irreps in the representation induced by the irreps of a sitesymmetry group ( < ).
The necessary input steps of LSITESYM and its corresponding output will be illustrated by the study of the symmetry of the phonon states in the Aurivillius compounds [Bi_{2}O_{2}]^{+2}[A_{n−1}B_{n}O_{3n+1}]^{+2} (known for their ferroelectric properties), a problem discussed in detail by Kitaev et al. (2007). These compounds exhibit two types of layer symmetry: for even n, the layers are described by the layer group p4mm (No. 55), while for odd n there is one central layer with higher symmetry described by the layer group p4/mmm (No. 61). The output of the program will be illustrated by the specific calculations of the symmetry relationships between the phonon states at the point k = M(½, ½) of the layer group p4/mmm and the localized states of atomic orbitals at the 2c (0, ½, 0) with sitesymmetry group mmm.. The irreps of the layer group p4/mmm at the point k = M(½, ½) with nonzero multiplicities, shown in the output of LSITESYM, describe the transformation properties of the extended phonon states induced by the irreps of the layer sitesymmetry group mmm. of the 2c (0, ½, 0).
In the INPUT block of the program the user is expected to provide the layer group, an occupied with its representative coordinate triplet and the k vector (specified by its coordinates and label) of the layergroup irreps D_{γ} whose inducedrepresentation multiplicities are to be calculated. The information is entered in three steps: in the first one, the layer group is specified by its ITE sequential number; in the second step, the occupied Wyckoff positions are to be selected from a list produced by the program; and finally, the coordinates and the label of the k vector of the irreps D_{γ} have to be introduced.
The OUTPUT block starts with a header that reproduces the input data, followed by a display of tables with the results of the intermediate steps of the procedure:
2.3.1. List of operations of the layer sitesymmetry group
Each of the symmetry operations of the layer group that leaves the g_{1},…, g_{n}), necessary for later referencing, are assigned to each element of .
representative point invariant is specified by its shorthand description (coordinate triplet) and matrixcolumn representation. Labels (The layer sitesymmetry group mmm. of the position 2c (0, ½, 0) of the layer group p4/mmm is formed by eight symmetry operations, as shown in the screenshot (Fig. 1) of the program LSITESYM.
2.3.2. Character table of the point group
This table reproduces the character table of the irreps D_{σ} of the isomorphic with the sitesymmetry group . The notations of Mulliken (1933) and Koster et al. (1963) are applied to label the irreps [see also Bradley & Cracknell (1972)].
The sitesymmetry group of the point (0, ½, 0) is isomorphic with the mmm and has eight irreps. The character table is reproduced in Fig. 2.
2.3.3. Table of characters of the subduced representations
The characters of the elements of the sitesymmetry group (obtained in the first step) for each of the irreps D_{γ} of of the selected wavevector are calculated internally by the program REPRES (cf. Aroyo et al., 2006). In this way, the characters of the subduced representations (D_{γ} ↓ ) of are obtained. The notation of the layergroup irreps has been chosen to be the same as for the corresponding irreps of the related and it follows that of Cracknell et al. (1979).
The layer group p4/mmm has ten irreps for the k vector M. Fig. 3 shows the characters of the corresponding subduced representations (*M_{i} ↓ mmm) of the sitesymmetry group. In accordance with the notation of the spacegroup irreps, *M_{i} and M_{i} denote the full group and the little group irrep of the layer group, respectively.
2.3.4. Table of the decompositions of the subduced representations
The multiplicities of the irreps of in the subduced representations (D_{γ} ↓ ) are obtained by the application of the reduction formula [see equation (2)]. In the example, the decompositions of the representations (*M_{i} ↓ mmm), i = 1, …, 10, into irreps of mmm are shown in Fig. 4.
2.3.5. Table of induced representations
According to the Frobenius reciprocity theorem, the multiplicities of the irreps D_{γ} of for a given k vector in the representations (D_{σ} ↑ ) (induced from the irreps D_{σ} of the sitesymmetry group ) are obtained by transposing the table of the decompositions of the subduced representations (D_{γ} ↓ ).
The table of representations of the layer group p4/mmm at the point M induced by the irreps of the sitesymmetry group mmm. of the 2c (0, ½, 0) is shown in Fig. 5. The rows of the table correspond to the irreps D_{σ} of the sitesymmetry group mmm. (cf. Fig. 2); the entries in each row indicate the multiplicities of the M irreps of p4/mmm in the (infinitedimensional) induced representation (D_{σ} ↑ p4/mmm):
The obtained results coincide exactly with the corresponding data of Table V in the work by Kitaev et al. (2007).
The URL of the program LSITESYM is https://www.cryst.ehu.es/subperiodic/layer_sitesym.html.
3. Transition metal dichalcogenide layer crystals
We now illustrate the use of the LSITESYM program described in Section 2, taking as an example MoS_{2} and WS_{2} layered crystals which belong to the transition metal dichalcogenide Although the structure of MoS_{2} was determined about 100 years ago and this layered crystal was used mainly as a dry lubricant, the new wave of interest in transition metal dichalcogenides began after the discovery of graphene. This interest, apart from the fundamental properties of monolayers, is connected with the observation of the direct band gap of 1.8 eV at the K point in the MoS_{2} monolayer, even though the bulk MoS_{2} crystal is a semiconductor with an indirect band gap of 1.2 eV. The transition to a direct band gap makes the MoS_{2} monolayer an excellent candidate as a solar photovoltaic material owing to the drastic enhancement of in the monolayer compared with the bulk. Other crystals of the transition metal dichalcogenide family, like WS_{2}, have similar electronic properties.
Comprehensive reviews and many references to studies of this material can be found in the work of Wang (2014), Kolobov & Tominaga (2016) and Manzeli et al. (2017).
The _{2} and WS_{2} bulk crystals is P6_{3}/mmc (No. 194). In the bulk crystal, the metal atoms occupy the 2c () position and the sulfur atoms occupy the 4f () position (Lee et al., 2014). The layer group of a single layer = (No. 78) (Milošević et al., 2000) is isomorphic with the , where is the (No. 187), i.e. is a of . The of a single layer is shown in Fig. 6. Atoms in the primitive of the layer occupy the following Wyckoff positions: Mo (W) 1c (), S 2e (). Note that the layer group of a single layer is also a of the of the bulk crystal.
of the MoSThe phonon symmetry in these crystals has been studied by MolinaSánchez & Wirtz (2011), RibeiroSoares et al. (2014) and Saito et al. (2016). However, diperiodic groups were introduced explicitly only by Evarestov (2015), Evarestov et al. (2017) and Bocharov et al. (2019). In the work of RibeiroSoares et al. (2014), the symmetry of single layers was described in terms of the related (No. 187) and, in addition, the authors applied a nonstandard irrep notation, which hinders the use and comparison of their results.
The results of LSITESYM, shown in Table 1, permit the analysis of the phonon symmetry of a single MoS_{2} layer. Table 1 is organized as follows. The irreps describing the symmetry of phonons at the 2D Brillouinzone symmetry points are induced by the irreps D_{σ} of the sitesymmetry groups of the Wyckoff positions where the atoms given in column 1 are located. The localized atomic displacements x, y and z transforming according to the irreps D_{σ} are indicated in brackets. The labels of layer group irreps at the k vector points of the twodimensional coincide with the labels of the irreps of the corresponding points of the threedimensional of the related .

From Table 1 it is seen that the Mo atom vibrations along the z axis induce Γ_{3} modes, whereas vibrations in the xy plane induce Γ_{5} modes. Similarly, S atom vibrations along the z axis and in the xy plane induce Γ_{1} + Γ_{3} and Γ_{5} + Γ_{6} modes, respectively. The vibrational representation at the Γ point can be written down, summing the contributions of all the atoms in the primitive It is given by Γ = Γ_{ac} + Γ_{opt} = (Γ_{3} + Γ_{5}) + (Γ_{1} + Γ_{3} + Γ_{5} + Γ_{6}), where the subscripts ac and opt indicate acoustic and optical layer modes, respectively. Further, one can establish a correspondence between layer and bulk crystal modes. Acoustic layer modes induce interlayer bulk modes, whereas optical layer modes induce intralayer bulk ones. The phonon symmetry in a bulk MoS_{2} crystal obtained by the SITESYM program (Elcoro et al., 2017) is given in Table 2. The notations in Table 2 are the same as those in Table 1.

Knowing the symmetry of bulk and layer phonons, one can obtain the genesis of the bulk modes from the layer ones. This could be determined using the CORREL program of the Bilbao Crystallographic Server (Aroyo et al., 2006) for the group–subgroup pair, namely P6_{3}/mmc and . The of the latter is isomorphic with the layer group . When applying this procedure it is necessary to choose correctly the transformation matrix relating the conventional settings of the group and the For the P6_{3}/mmc and pair, the transformation matrix is reduced to an origin shift ().
On the basis of the Frobenius theorem, the correspondence between the layer irreps at the Γ point and the bulk irreps is given in Table 3. It is seen that each layer mode generates a pair of bulk modes at the Γ point, one odd and one even. The splitting is due to a weak van der Waals interaction between the layers. Therefore, the frequencies of the modes constituting a pair would be close.

Similarly, one can deduce the correspondence between layer and bulk modes at other Brillouinzone points. For example, the correspondence between the layer () modes at the point and the bulk (P6_{3}/mmc) modes at and are shown in Table 4.

4. Conclusions
A new computer tool which calculates the sitesymmetry induced representations of layer groups, called LSITESYM, has recently been implemented on the Bilbao Crystallographic Server. Like the rest of the programs on the server, this new tool is freely available and can be accessed via userfriendly web interfaces. The algorithm of LSITESYM, based on the sitesymmetry method applied to layer groups, is an extension of the algorithm used in the program SITESYM for ordinary space groups. The group–subgroup relation between layer and space groups is fundamental for the development of the procedure for layer groups. On the basis of the isomorphism between the layer and the , it is possible to establish a simple connection between the Wyckoff positions, k vectors and irreps of and , which is essential to calculate the sitesymmetry induced representations of layer groups.
The program LSITESYM, which is able to determine the symmetry relations between localized and extended states in crystals with layer symmetry, is also very useful in the description of phonon states and electronic structure. The capabilities of the program LSITESYM have been successfully demonstrated in several examples. Moreover, the utility of the program in combination with other tools of the Bilbao Crystallographic Server to obtain the relation between bulk and layer modes has also been shown.
Funding information
GF is indebted to the Basque Government (grant No. POS_2015_1_0025) for a postdoctoral fellowship. This work was supported by the Spanish Ministry of Science and Innovations (project No. MAT201234740) and the Government of the Basque Country (project No. IT77913).
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