The site-symmetry induced representations of layer groups on the Bilbao Crystallographic Server

de la Flor, G.; Orobengoa, D.; Evarestov, R.A.; Kitaev, Y.E.; Tasci, E.; Aroyo, M.I.

doi:10.1107/S1600576719011725

research papers

JOURNAL OF
APPLIED
CRYSTALLOGRAPHY

ISSN: 1600-5767

Volume 52| Part 5| October 2019| Pages 1214-1221

https://doi.org/10.1107/S1600576719011725

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The site-symmetry induced representations of layer groups on the Bilbao Crystallographic Server

Gemma de la Flor,^a,^b ^* Danel Orobengoa,^b Robert A. Evarestov,^c Yuri E. Kitaev,^d Emre Tasci ^e and Mois I. Aroyo ^b

^aKarlsruhe Institute of Technology, Institute of Applied Geosciences, Karlsruhe, Germany, ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco (UPV/EHU), Bilbao, Spain, ^cInstitute of Chemistry, Saint Petersburg State University, Saint Petersburg, Russian Federation, ^dIoffe Institute, Saint Petersburg, Russian Federation, and ^eDepartment of Physics Engineering, Hacettepe University, Ankara, Turkey
^*Correspondence e-mail: gemma.delaflor@kit.edu

Edited by D. Pandey, Indian Institute of Technology (Banaras Hindu University), Varanasi, India (Received 11 March 2019; accepted 24 August 2019; online 4 October 2019)

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 site-symmetry 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 MoS₂ and WS₂.

Keywords: site-symmetry method; LSITESYM; Bilbao Crystallographic Server; layer groups.

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 crystal family MeX₂ 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 three-dimensional crystallographic groups with two-dimensional translations. There are 80 layer groups which, together with the seven frieze groups (two-dimensional groups with one-dimensional translations) and the 75 rod groups (three-dimensional groups with one-dimensional 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 surface states 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 free-standing 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 semi-infinite crystal is neglected; (v) interfaces between different crystals, including domain walls approximated by atomically clean crystal surfaces.

The site-symmetry 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 Brillouin zone. The site-symmetry method is based on the procedure of induction of representations of the crystal space group from the irreducible representations (irreps) of the site-symmetry 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, exciton, biexciton etc. states can be described by the crystal single- and double-valued representations induced by the irreps of the site-symmetry 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 site-symmetry 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 As₂S₃ and As₂Se₃ 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 high-temperature superconductors (Evarestov et al., 1993 ; Kitaev et al., 1994 ) and thereafter to describe the phonon, electron, exciton 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 Brillouin-zone centre phonons in layered MoS₂ and WS₂ 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 site-symmetry 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. Site-symmetry method

2.1. General procedure

The site-symmetry 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 space group $[{\cal G}]$ at any point in reciprocal space (which classify the extended states of the structure) induced by the irreps of the site-symmetry group of a Wyckoff position (according to which localized states are classified). In group-theoretical terms, the procedure relating localized and extended crystalline states can be described by induction of a representation of a space group $[{\cal G}]$ from the irreps of a finite subgroup $[{\cal H}]$ , followed by its reduction into irreps of $[{\cal G}]$ . In other words, the induction method permits the calculation of the symmetry of the compatible extended states transforming according to irreps of crystal space group $[{\cal G}]$ induced by a localized state described by an irrep of the local or site-symmetry group $[{\cal H}]$ = $[{\cal S}]$ . The induction procedure can be applied to any group–subgroup pair $[{\cal H}\ \lt \ {\cal G}]$ , but in the site-symmetry approach, as its name suggests, a site-symmetry group $[{\cal S}]$ is taken as the subgroup of $[{\cal G}]$ . The calculation of the space-group irreps induced by the irreps of a site-symmetry group is not straightforward, because the site-symmetry group $[{\cal S}]$ (isomorphic with a point group) is a subgroup of infinite index of $[{\cal G}]$ . This implies that the representation of $[{\cal G}]$ induced by an irrep of $[{\cal S}]$ must be of infinite dimension and therefore difficult to calculate directly. This problem is solved by applying the `Frobenius reciprocity theorem' (Serre, 1977 ), which states that the multiplicities of the irreps of a group $[{\cal G}]$ in the induced representation from an irrep of a subgroup $[{\cal H}]$ of $[{\cal G}]$ can be determined from the multiplicities of the irreps of $[{\cal H}]$ in the representations subduced from $[{\cal G}]$ to $[{\cal H}]$ . Therefore, the site-symmetry method is based on subduction and induction, two basic concepts of representation theory, and on the Frobenius reciprocity theorem.

The subduction procedure relates the representations of a group $[{\cal G}]$ to those of its subgroups $[{\cal S}\ \lt \ {\cal G}]$ . Consider an irrep D_γ = $[\{ {\bf D}_{\gamma}(g), g \in {\cal G} \}]$ of a group $[{\cal G}]$ . The subduction of D_γ to the subgroup $[\cal{S}]$ results in a representation of the subgroup, known as the `subduced representation' D^Sub, formed by the matrices of those elements of $[{\cal G}]$ that also belong to the subgroup $[{\cal S}]$ , i.e. D^Sub = $[({\bf D}_{\gamma} \downarrow {\cal S})]$ = $[\{{\bf D}_{\gamma} (s), s \in {\cal S}\ \lt \ {\cal G}\}]$ . This subduced representation $[{\bf D}^{\rm Sub}]$ is in general reducible and is decomposable into irreps $[{\bf D}_{\sigma}]$ of $[{\cal S}]$ :

$[{\bf D}^{\rm Sub} (s) = \left ( {\bf D}_{\gamma} \downarrow {\cal S} \right )\, (s) \simeq \bigoplus _{\sigma} m_{\sigma}^{\rm Sub} \,{\bf D}_{\sigma} (s), {s} \in {\cal S} . \eqno(1)]$

The multiplicities $[m_{\sigma}^{\rm Sub}]$ of the irreps $[{\bf D}_{\sigma}]$ of $[{\cal S}]$ in the subduced representation $[{\bf D}_{\gamma} \downarrow {\cal S}]$ can be calculated by the reduction formula (known also as the `magic' formula):

$[m_{\sigma}^{\rm Sub} = {{1} \over {\mid {\cal S} \mid}} \sum _{s} \chi^{\rm Sub} (s) \,\chi _{\sigma} (s)^*, s \in {\cal S} , \eqno(2)]$

where χ^Sub(s) is the character of the subduced representation and χ_σ(s) is the character of the irrep $[{\bf D}_{\sigma}]$ for the same element $[s \in {\cal S}]$ .

The induction procedure permits the construction of a representation of $[{\cal G}]$ starting from a representation of $[{\cal S}\ \lt \ {\cal G}]$ . If $[{\bf D}_{\sigma}]$ = $[\{{\bf D}_{\sigma}(s), s \in {\cal S} \}]$ is an irrep of $[{\cal S}]$ , then the matrices of the induced representation $[{\bf D}^{\rm Ind}]$ = ( $[{\bf D}_{\sigma} \uparrow {\cal G}]$ ) of $[{\cal G}]$ are constructed as follows:

$[{\bf D}^{\rm Ind} (g)_{kt,jr} = \cases { {\bf D}_{\sigma} \left ( q_{k}^{-1} gq_{j} \right )_{\rm tr} & if $q_{k}^{-1} gq_{j} \in {\cal S},$ \cr 0 & if $q_{k}^{-1} gq_{j} \ \notin\ {\cal S}$. } \eqno(3)]$

Here, t, r = 1, …, m with m equal to the dimension of the irrep $[{\bf D}_{\sigma}]$ of $[{\cal S}]$ and k, j = 1, …, n where n = $[|{\cal G}| / | S |]$ is the index of $[{\cal S}]$ in $[{\cal G}]$ . The elements $[q_{1}, \ldots, q_{n}]$ are the coset representatives of the decomposition of $[{\cal G}]$ with respect to $[{\cal S}]$ .

The characters of $[{\bf D}^{\rm Ind}]$ are given by

$[\chi _{\gamma}^{\rm Ind} (g) = \textstyle\sum\limits _{j=1}^n \chi _{\sigma} \left ( q_{j}^{-1} gq_{j} \right ) , \eqno(4)]$

where $[\chi _{\sigma} (q_{j}^{-1} gq_{j})]$ is the trace of the jth diagonal block of $[{\bf D}^{\rm Ind} (g)]$ and n is the index of the subgroup 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 $[{\bf D}_{\gamma}]$ of $[{\cal G}]$ :

$[{\bf D}^{\rm Ind} = \left ( {\bf D}_{\sigma} \uparrow {\cal G} \right ) \simeq \bigoplus _{\gamma} m_{\gamma}^{\rm Ind} \,{\bf D}_{\gamma} , \eqno(5)]$

where $[m_{\gamma}^{\rm Ind}]$ are the multiplicities of the irreps of $[{\cal G}]$ in the induced representations $[{\bf D}^{\rm Ind}]$ 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)]:

$[{\rm dim} \left ( {\bf D}_{\sigma} \uparrow {\cal G} \right ) = {\rm dim} \left ( {\bf D}_{\sigma} \right ) {{\mid {\cal G} \mid} \over {\mid {\cal S} \mid}} . \eqno(6)]$

This result points out the difficulties for the direct calculation of a representation of a space group $[{\cal G}]$ induced from an irrep of a finite subgroup $[{\cal S}]$ of $[{\cal G}]$ . As the site-symmetry group is a subgroup of an infinite index, this suggests that the dimension of the induced irreps must be of infinite dimensions. By means of the site-symmetry approach it is possible to determine the multiplicities $[m_{\gamma}^{\rm Ind}]$ of an irrep of $[{\cal G}]$ in the induced representation without the necessity of constructing the infinite-dimensional representation. The method is based on the Frobenius reciprocity theorem, according to which the multiplicity of an irreducible irrep $[{\bf D}_{\gamma}]$ of $[{\cal G}]$ in a representation ( $[{\bf D}_{\sigma} \uparrow{\cal G}]$ ) of $[{\cal G}]$ induced by an irrep $[{\bf D}_{\sigma}]$ of $[{\cal S}\ \lt \ {\cal G}]$ is equal to the multiplicity of the irrep $[{\bf D}_{\sigma}]$ of $[{\cal S}]$ in the representation ( $[{\bf D}_{\gamma} \downarrow{\cal S}]$ ) subduced by $[{\bf D}_{\gamma}]$ of $[{\cal G}]$ to $[{\cal S}]$ , i.e. $[m_{\sigma}^{\rm Sub}]$ = $[m_{\gamma}^{\rm Ind}]$ . In other words, it is sufficient to calculate the multiplicities $[m_{\sigma}^{\rm Sub}]$ of $[{\bf D}_{\sigma}]$ in the subduced representation ( $[{\bf D}_{\gamma} \downarrow {\cal S}]$ ) in order to obtain the frequencies $[m_{\gamma}^{\rm Ind}]$ of $[{\bf D}_{\gamma}]$ in the induced representation ( $[{\bf D}_{\sigma} \uparrow {\cal G}]$ ).

2.2. Application of site-symmetry method to layer groups

The group–subgroup relations between layer and space groups $[{\cal L}\ \lt \ {\cal G}]$ are essential to extend the site-symmetry 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 space group of which a given layer group is a subgroup is not defined uniquely. The `simplest' space group $[{\cal G}]$ to which $[{\cal L}]$ is related can be expressed as a semi-direct product of $[{\cal L}]$ with the one-dimensional translation group T₃ of additional translations $[{\cal G} = T_{3} \wedge {\cal L}]$ , where T₃ is a normal subgroup of $[{\cal G}]$ . Thus, the layer group $[{\cal L}]$ is isomorphic with the factor group $[{\cal G}/T_{3}]$ .

The isomorphism between $[{\cal L}]$ and the factor group $[{\cal G}/T_{3}]$ results in close relationships between the Wyckoff positions and the irreps of $[{\cal L}]$ and $[{\cal G}]$ . One can show that the set of Wyckoff positions of a layer group is contained in the set of Wyckoff positions of the related space group (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 special-position coordinates of layer groups: only the special positions of $[{\cal G}]$ whose z coordinate does not involve a fraction of the unit-cell parameter are possible special positions of $[{\cal L}]$ , i.e. special positions of $[{\cal G}]$ with z coordinates z, −z or 0. In that way, to each Wyckoff position of $[{\cal L}]$ corresponds exactly one Wyckoff position of $[{\cal G}]$ , specified by exactly the same site-symmetry 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 Wyckoff-position 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 layer-group Wyckoff positions might not coincide, in general.

The simple relationship between the irreps of $[{\cal L}]$ and $[{\cal G}]$ is based on the isomorphism $[{\cal L} \simeq {\cal G}/T_{3}]$ : the irreps of $[{\cal L}]$ are also irreps of $[{\cal G}]$ and every irrep of $[{\cal L}]$ is related to a specific irrep of $[{\cal G}]$ . In these irreps of $[{\cal G}]$ all elements of a given coset of the decomposition of $[{\cal G}]$ with respect to T₃ are mapped onto the same matrix, i.e. the irreps of $[{\cal L}]$ coincide with those irreps of $[{\cal G}]$ whose kernel is T₃ (for details, see Evarestov & Smirnov, 1993). For example, the special k vectors of $[{\cal L}]$ can also be deduced from the k vectors of $[{\cal G}]$ . The Brillouin zone of a layer group $[{\cal L}]$ can be described as a projection of the Brillouin zone of the corresponding space group $[{\cal G}]$ onto the layer plane. Accordingly, the two-dimensional set of k vectors k(k₁, k₂) of $[{\cal L}]$ can be obtained from the three-dimensional k(k₁, k₂, k₃) vectors of $[{\cal G}]$ by ignoring the third component k₃.

The above considerations indicate that the induced representations of a layer group can be read off directly from the induced representations of the corresponding space group. On the basis of this observation, the procedure of the site-symmetry method for layer groups can be summarized as follows:

(i) Given the layer group $[{\cal L}]$ (specified by its number), the Wyckoff position and the k vector (k₁, k₂), the program identifies (a) the corresponding space-group number and Wyckoff letter, and (b) the space-group wave vector (k₁, k₂, k₃) with k₃ = 0, related to the layer-group k vector (k₁, k₂).

(ii) The site-symmetry method for space groups is applied, i.e. the site-symmetry 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 site-symmetry induced representations of space group $[{\cal G}]$ obtained by SITESYM are described with respect to the layer group $[{\cal L}]$ .

2.3. 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 layer-group irreps in the representation induced by the irreps of a site-symmetry group ( $[{\cal S}]$ < $[{\cal L}]$ ).

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 [Bi₂O₂]⁺²[A_n−1B_nO_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 Wyckoff position 2c (0, ½, 0) with site-symmetry 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 site-symmetry group mmm. of the Wyckoff position 2c (0, ½, 0).

In the INPUT block of the program the user is expected to provide the layer group, an occupied Wyckoff position with its representative coordinate triplet and the k vector (specified by its coordinates and label) of the layer-group irreps D_γ whose induced-representation 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 site-symmetry group $[{\cal S}]$

Each of the symmetry operations of the layer group that leaves the Wyckoff position representative point invariant is specified by its shorthand description (coordinate triplet) and matrix-column representation. Labels (g₁,…, g_n), necessary for later referencing, are assigned to each element of $[{\cal S}]$ .

The layer site-symmetry 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.

Figure 1
A screenshot of the partial output of the program LSITESYM, showing the eight symmetry operations of the site-symmetry group mmm. of the Wyckoff position (WP) 2c (0, ½, 0) of the layer group p4/mmm (No. 61). The symmetry operations are specified by their shorthand descriptions (coordinate triplets) and matrix-column representations.

2.3.2. Character table of the point group

This table reproduces the character table of the irreps D_σ of the point group isomorphic with the site-symmetry group $[{\cal S}]$ . The notations of Mulliken (1933 ) and Koster et al. (1963 ) are applied to label the irreps [see also Bradley & Cracknell (1972 )].

The site-symmetry group of the point (0, ½, 0) is isomorphic with the point group mmm and has eight irreps. The character table is reproduced in Fig. 2.

Figure 2
A screenshot of the partial output of the program LSITESYM, showing the character table of the point group mmm, isomorphic with the site-symmetry group of the Wyckoff position 2c (0, ½, 0) of the layer group p4/mmm (No. 61). The irreps are labelled according to the notation of Mulliken (1933

) and Koster et al. (1963

2.3.3. Table of characters of the subduced representations

The characters of the elements of the site-symmetry group $[{\cal S}]$ (obtained in the first step) for each of the irreps D_γ of $[{\cal G}]$ 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 ↓ $[{\cal S}]$ ) of $[{\cal S}]$ are obtained. The notation of the layer-group irreps has been chosen to be the same as for the corresponding irreps of the related space group, 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 site-symmetry group. In accordance with the notation of the space-group irreps, *M_i and M_i denote the full group and the little group irrep of the layer group, respectively.

Figure 3
A screenshot of the partial output of the program LSITESYM, showing the characters of the subduced representations of the site-symmetry group $[{\cal S}]$ of the Wyckoff position 2c (0, ½, 0) of the layer group p4/mmm (No. 61) at the k point M. The columns are specified by the symmetry operations of the site-symmetry group $[{\cal S} \simeq]$ mmm (cf. Fig. 1

2.3.4. Table of the decompositions of the subduced representations

The multiplicities $[m_{\sigma}^{\rm Sub}]$ of the irreps of $[{\cal S}]$ in the subduced representations (D_γ ↓ $[{\cal S}]$ ) 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.

Figure 4
A screenshot of the partial output of the program LSITESYM, showing the decompositions of the subduced representations (*M_i ↓ mmm), shown in Fig. 3

, into irreps of mmm (cf. Fig. 2

2.3.5. Table of induced representations

According to the Frobenius reciprocity theorem, the multiplicities $[m_{\gamma}^{\rm Ind}]$ of the irreps D_γ of $[{\cal G}]$ for a given k vector in the representations (D_σ ↑ $[{\cal G}]$ ) (induced from the irreps D_σ of the site-symmetry group $[{\cal S}]$ ) are obtained by transposing the table of the decompositions of the subduced representations (D_γ ↓ $[{\cal S}]$ ).

The table of representations of the layer group p4/mmm at the point M induced by the irreps of the site-symmetry group mmm. of the Wyckoff position 2c (0, ½, 0) is shown in Fig. 5. The rows of the table correspond to the irreps D_σ of the site-symmetry group mmm. (cf. Fig. 2); the entries in each row indicate the multiplicities of the M irreps of p4/mmm in the (infinite-dimensional) induced representation (D_σ ↑ p4/mmm):

$[\eqalign{ (A_{g} \uparrow p4/mmm) \ni & \, ^{*}M_{5}^{-}, \cr (B_{1g} \uparrow p4/mmm) \ni & \, ^{*}M_{5}^{-}, \cr (B_{2g} \uparrow p4/mmm) \ni & \, ^{*}M_{1}^{-} \oplus {^{*}M}_{2}^{-}, \cr (B_{3g} \uparrow p4/mmm) \ni & \, {^{*}M}_{3}^{-} \oplus {^{*}M}_{4}^{-}, \cr (A_{u} \uparrow p4/mmm) \ni & \, ^{*}M_{5}^{-}, \cr (B_{1u} \uparrow p4/mmm) \ni & \, ^{*}M_{5}^{+}, \cr (B_{2u} \uparrow p4/mmm) \ni & \, ^{*}M_{1}^{+} \oplus {^{*}M}_{2}^{+}, \cr (B_{3u} \uparrow p4/mmm) \ni & \, ^{*}M_{3}^{+} \oplus {^{*}M}_{4}^{+}. }]$

The obtained results coincide exactly with the corresponding data of Table V in the work by Kitaev et al. (2007).

Figure 5
A screenshot of the partial output of the program LSITESYM, showing the representations of the layer group p4/mmm (No. 61) at the point M induced by the irreps of the site-symmetry group mmm. of the Wyckoff position 2c (0, ½, 0).

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 MoS₂ and WS₂ layered crystals which belong to the transition metal dichalcogenide crystal family. Although the structure of MoS₂ 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 MoS₂ monolayer, even though the bulk MoS₂ crystal is a semiconductor with an indirect band gap of 1.2 eV. The transition to a direct band gap makes the MoS₂ monolayer an excellent candidate as a solar photovoltaic material owing to the drastic enhancement of photoluminescence in the monolayer compared with the bulk. Other crystals of the transition metal dichalcogenide family, like WS₂, 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 space group of the MoS₂ and WS₂ bulk crystals is P6₃/mmc (No. 194). In the bulk crystal, the metal atoms occupy the 2c ( $[{1\over 3}, {2\over 3}, {1\over 4}]$ ) position and the sulfur atoms occupy the 4f ( $[{1\over 3}, {2\over 3}, z]$ ) position (Lee et al., 2014 ). The layer group of a single layer $[{\cal L}]$ = $[p{\overline 6}m2]$ (No. 78) (Milošević et al., 2000 ) is isomorphic with the factor group $[{\cal G}/T_{3}]$ , where $[{\cal G}]$ is the space group $[P{\overline 6}m2]$ (No. 187), i.e. $[{\cal L}]$ is a subgroup of $[{\cal G}]$ . The crystal structure of a single layer is shown in Fig. 6. Atoms in the primitive unit cell of the layer occupy the following Wyckoff positions: Mo (W) 1c ( $[{2\over 3}, {1\over 3}, 0]$ ), S 2e ( $[{1\over 3}, {2\over 3}, z]$ ). Note that the layer group of a single layer is also a subgroup of the space group of the bulk crystal.

Figure 6
(a) The crystal structure of a single layer of MoS₂ (WS₂), and (b) its projection along [001].

The phonon symmetry in these crystals has been studied by Molina-Sánchez & Wirtz (2011 ), Ribeiro-Soares 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 Ribeiro-Soares et al. (2014), the symmetry of single layers was described in terms of the related space group $[P{\overline 6}m2]$ (No. 187) and, in addition, the authors applied a non-standard 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 MoS₂ layer. Table 1 is organized as follows. The $[{\cal L}]$ irreps describing the symmetry of phonons at the 2D Brillouin-zone symmetry points are induced by the irreps D_σ of the site-symmetry groups $[{\cal S}]$ 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 $[p{\overline 6}m2]$ irreps at the k vector points of the two-dimensional Brillouin zone coincide with the labels of the irreps of the corresponding points of the three-dimensional Brillouin zone of the related space group $[P{\overline 6}m2]$ .

Table 1
Phonon symmetry in the single MoS₂ layer with $[{\cal L}]$ = $[p{\overline 6}m2]$ (No. 78)

Atom	Wyckoff position	D_σ	Γ (0, 0) $[{\overline 6}2m]$	K ( $[{1\over 3}, {1\over 3}]$ ) $[{\overline 6}..]$	M ( $[{1\over 2}, 0]$ ) m2m
Mo	1c	$[A_2^{\prime\prime} ]$ (z)	3	6	3
	( $[{2\over 3}, {1\over 3}, 0]$ )	$[E^{\prime}]$ (x, y)	5	1, 3	1, 2
	$[{\overline 6}m2]$
S	2e	A₁ (z)	1, 3	3, 4	1, 3
	( $[{1\over 3}, {2\over 3}, z]$ )	E (x, y)	5, 6	1, 2, 5, 6	1, 2, 3, 4
	3m.

From Table 1 it is seen that the Mo atom vibrations along the z axis induce Γ₃ modes, whereas vibrations in the xy plane induce Γ₅ modes. Similarly, S atom vibrations along the z axis and in the xy plane induce Γ₁ + Γ₃ and Γ₅ + Γ₆ modes, respectively. The vibrational representation at the Γ point can be written down, summing the contributions of all the atoms in the primitive unit cell. It is given by Γ = Γ_ac + Γ_opt = (Γ₃ + Γ₅) + (Γ₁ + Γ₃ + Γ₅ + Γ₆), 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 MoS₂ 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.

Table 2
Phonon symmetry in the MoS₂ bulk crystal P6₃/mmc (No. 194)

Atom	Wyckoff position	D_σ	Γ (0, 0, 0) 6/mmm	A (0, 0, 1/2) 6/mmm	K ( $[{1\over 3}, {1\over 3}, 0]$ ) $[{\overline 6}m2]$	H ( $[{1\over 3}, {1\over 3}, {1\over 2}]$ ) $[{\overline 6}m2]$	M ( $[{1\over 2}, 0, 0]$ ) mmm	L ( $[{1\over 2}, 0, {1\over 2}]$ ) mmm
Mo	2c	$[A_2^{\prime\prime}]$ (z)	2⁻, 3⁺	1	6	2	2⁻, 3⁺	1
	( $[{1\over 3}, {2\over 3}, {1\over 4}]$ )	E′ (x, y)	5⁺, 6⁻	3	1, 4, 5	2, 3	1⁺, 2⁺, 3⁻, 4⁻	1, 2
	$[{\overline 6}m2]$
S	4f	A₁ (z)	1⁺, 2⁻, 3⁺, 4⁻	1, 1	5, 6	1, 2	1⁺, 2⁻, 3⁺, 4⁻	1, 1
	( $[{1\over 3}, {2\over 3}, z]$ )	E (x, y)	5⁺, 5⁻, 6⁺, 6⁻	3, 3	1, 2, 3, 4, 5, 6	1, 2, 3, 3	1⁺, 1⁻, 2⁺, 2⁻, 3⁺, 3⁻, 4⁺, 4⁻	1, 1, 2, 2
	3m.

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 P6₃/mmc and $[P{\overline 6}m2]$ . The factor group of the latter is isomorphic with the layer group $[p{\overline 6}m2]$ . When applying this procedure it is necessary to choose correctly the transformation matrix relating the conventional settings of the group and the subgroup. For the P6₃/mmc and $[P{\overline 6}m2]$ pair, the transformation matrix is reduced to an origin shift ( $[0\ 0\ {-{1\over 4}}]$ ).

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.

Table 3
Correspondence between the layer ( $[p{\overline 6}m2]$ ) irreps at the point Γ(0, 0) and the bulk (P6₃/mmc) irreps at Γ(0, 0, 0) and A(½, ½, 0)

Layer		Bulk
Γ₁	$[\longrightarrow]$	$[\Gamma _{1}^{+}]$ + $[\Gamma _{4}^{-}]$ + A₁
Γ₂	$[\longrightarrow]$	$[\Gamma _{2}^{+}]$ + $[\Gamma _{3}^{-}]$ + A₂
Γ₃	$[\longrightarrow]$	$[\Gamma _{2}^{-}]$ + $[\Gamma _{3}^{+}]$ + A₁
Γ₄	$[\longrightarrow]$	$[\Gamma _{1}^{-}]$ + $[\Gamma _{4}^{+}]$ + A₂
Γ₅	$[\longrightarrow]$	$[\Gamma _{5}^{+}]$ + $[\Gamma _{6}^{-}]$ + A₃
Γ₆	$[\longrightarrow]$	$[\Gamma _{5}^{-}]$ + $[\Gamma _{6}^{+}]$ + A₃

Similarly, one can deduce the correspondence between layer and bulk modes at other Brillouin-zone points. For example, the correspondence between the layer ( $[p{\overline 6}m2]$ ) modes at the point $[K({1\over 3}, {1\over 3})]$ and the bulk (P6₃/mmc) modes at $[K({1\over 3}, {1\over 3}, 0)]$ and $[H({1\over 3}, {1\over 3}, {1\over 2})]$ are shown in Table 4.

Table 4
Correspondence between the layer ( $[p{\overline 6}m2]$ ) irreps at the point K( $[\textstyle{1 \over 3}]$ , $[\textstyle{1 \over 3}]$ ) and the bulk (P6₃/mmc) irreps at K( $[\textstyle{1 \over 3}]$ , $[\textstyle{1 \over 3}]$ , 0) and H( $[\textstyle{1 \over 3}]$ , $[\textstyle{1 \over 3}]$ , ½)

Layer		Bulk
K₁	$[\longrightarrow]$	K₁ + K₄ + H₃
K₂	$[\longrightarrow]$	K₂ + K₃ + H₃
K₃	$[\longrightarrow]$	K₅ + H₁
K₄	$[\longrightarrow]$	K₆ + H₂
K₅	$[\longrightarrow]$	K₅ + H₂
K₆	$[\longrightarrow]$	K₆ + H₁

4. Conclusions

A new computer tool which calculates the site-symmetry 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 user-friendly web interfaces. The algorithm of LSITESYM, based on the site-symmetry 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 $[{\cal L}\ \lt \ {\cal G}]$ is fundamental for the development of the procedure for layer groups. On the basis of the isomorphism between the layer $[{\cal L}]$ and the factor group $[{\cal G}/T_{3}]$ , it is possible to establish a simple connection between the Wyckoff positions, k vectors and irreps of $[{\cal G}]$ and $[{\cal L}]$ , which is essential to calculate the site-symmetry 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. MAT2012-34740) and the Government of the Basque Country (project No. IT779-13).

References

Alexander, E. & Hermann, K. (1929). Z. Kristallogr. 70, 328–345. Google Scholar
Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. (2006). Acta Cryst. A62, 115–128. Web of Science CrossRef CAS IUCr Journals Google Scholar
Aroyo, M. I., Perez-Mato, J. M., Orobengoa, D., Tasci, E., de la Flor, G. & Kirov, A. (2011). Bulg. Chem. Commun. 43, 183–197. CAS Google Scholar
Bocharov, D., Piskunov, S., Zhukovskii, Y. F. & Evarestov, R. A. (2019). Phys. Status Solidi RRL, 13, 1800253. CrossRef Google Scholar
Bradley, C. J. & Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press. Google Scholar
Choi, W., Choudhary, N., Han, G. H., Park, J., Akinwande, D. & Lee, Y. H. (2017). Mater. Today, 20(3), 116–130. CrossRef Google Scholar
Cracknell, A. P., Davies, B. L., Miller, S. C. & Love, W. F. (1979). Kronecker Product Tables, No. 1, General Introduction and Tables of Irreducible Representations of Space Groups. New York: IFI/Plenum. Google Scholar
Elcoro, L., Bradlyn, B., Wang, Z., Vergniory, M. G., Cano, J., Felser, C., Bernevig, B. A., Orobengoa, D., de la Flor, G. & Aroyo, M. I. (2017). J. Appl. Cryst. 50, 1457–1477. CrossRef CAS IUCr Journals Google Scholar
Evarestov, R. A. (2015). Theoretical Modeling of Inorganic Nanostructures: Symmetry and Ab Initio Calculations of Nanolayers, Nanotubes and Nanowires. Berlin, Heidelberg: Springer-Verlag. Google Scholar
Evarestov, R. A., Bandura, A. V., Porsev, V. V. & Kovalenko, A. V. (2017). J. Comput. Chem. 38, 2581–2593. CrossRef CAS PubMed Google Scholar
Evarestov, R. A., Kitaev, Y. E., Limonov, M. F. & Panfilov, A. G. (1993). Phys. Status Solidi B, 179, 249–297. CrossRef CAS Google Scholar
Evarestov, R. A., Kovalenko, A. V., Bandura, A. V., Domnin, A. V. & Lukyanov, S. I. (2018). Mater. Res. Expr. 5, 115028. CrossRef Google Scholar
Evarestov, R. A. & Smirnov, V. P. (1987). Phys. Status Solidi B, 142, 493–499. CrossRef CAS Google Scholar
Evarestov, R. A. & Smirnov, V. P. (1993). Site Symmetry in Crystals. Theory and Applications, Springer Series in Solid State Sciences, Vol. 108, edited by M. Cardona. Heidelberg: Springer. Google Scholar
Geim, K. & Novoselov, K. S. (2007). Nat. Mater. 6, 183–191. CrossRef PubMed CAS Google Scholar
Han, G. H., Duong, D. L., Keum, D. H., Yun, S. J. & Lee, Y. H. (2018). Chem. Rev. 118, 62976336. Google Scholar
Hatch, D. M., Stokes, H. T. & Putnam, R. M. (1988). Group Theoretical Methods in Physics, Lecture Notes in Physics, Vol. 313, edited by H. D. Doebner, J. D. Hennig & T. D. Palev, pp. 326–333. New York: Springer. Google Scholar
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, edited by V. Kopský & D. B. Litvin, 2nd ed. Chichester: John Wiley & Sons. Google Scholar
Ipatova, I. P. & Kitaev, Y. E. (1985). Prog. Surf. Sci. 18, 189–246. CrossRef CAS Google Scholar
Kitaev, Y. E., Aroyo, M. I. & Perez-Mato, J. M. (2007). Phys. Rev. B, 75, 064110. CrossRef Google Scholar
Kitaev, Y. E., Limonov, M. F., Panfilov, A. G., Evarestov, R. A. & Mirgorodsky, A. P. (1994). Phys. Rev. B, 49, 9933–9943. CrossRef CAS Google Scholar
Kolobov, A. V. & Tominaga, J. (2016). Two-Dimensional Transition Metal Dichalcogenides, Springer Series in Materials Science, Vol. 239. Cham: Springer International Publishing. Google Scholar
Koster, G. F., Dimmock, J. O., Wheeler, R. G. & Statz, H. (1963). Properties of the Thirty-Two Point Groups. Cambridge: MIT Press. Google Scholar
Kovalev, O. V. (1993). Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Co-representations, edited by H. T. Stokes & D. M. Hatch. Philadelphia: Gordon and Breach. Google Scholar
Lee, C., Hong, J., Lee, W. R., Kim, D. Y. & Shim, J. H. (2014). J. Solid State Chem. 211, 113–119. CrossRef CAS Google Scholar
Manzeli, S., Ovchinnikov, D., Pasquier, D., Yazyev, O. V. & Kis, A. (2017). Nat. Rev. Mater. 2, 17033. CrossRef Google Scholar
Milošević, I., Vuković, T., Damnjanović, M. & Nikolić, B. (2000). Eur. Phys. J. B, 17, 707–712. Google Scholar
Molina-Sánchez, A. & Wirtz, L. (2011). Phys. Rev. B, 84, 155413. Google Scholar
Mulliken, R. S. (1933). Phys. Rev. 43, 279–302. CrossRef CAS Google Scholar
Randviir, E. P., Brownson, D. A. & Banks, C. E. (2014). Mater. Today, 17, 426–432. CrossRef CAS Google Scholar
Ribeiro-Soares, J., Almeida, R. M., Barros, E. B., Araujo, P. T., Dresselhaus, M. S., Cançado, L. G. & Jorio, A. (2014). Phys. Rev. B, 90, 115438. Google Scholar
Saito, R., Tatsumi, Y., Huang, S., Ling, X. & Dresselhaus, M. S. (2016). J. Phys. Condens. Matter, 28, 353002. CrossRef PubMed Google Scholar
Serre, J.-P. (1977). Linear Representations of Finite Groups. New York: Springer Verlag. Google Scholar
Stokes, H. T., Hatch, D. M. & Wells, J. D. (1991). Phys. Rev. B, 43, 11010–11018. CrossRef CAS Web of Science Google Scholar
Tasci, E. S., de la Flor, G., Orobengoa, D., Capillas, C., Perez-Mato, J. M. & Aroyo, M. I. (2012). EPJ Web Conf. 22, 00009. Google Scholar
Tronc, P. & Kitaev, Y. E. (2001). Phys. Rev. B, 63, 205326. CrossRef Google Scholar
Wang, Z. M. (2014). Editor. MoS₂ Materials, Physics, and Devices, Lecture Notes in Nanoscale Science and Technology, Vol. 21. Cham: Springer International Publishing. Google Scholar
Weber, L. (1929). Z. Kristallogr. 70, 309–327. Google Scholar
Wood, E. (1964). Bell Syst. Tech. J. 43, 541–559. CrossRef Google Scholar
Zallen, R., Slade, M. L. & Ward, A. T. (1971). Phys. Rev. B, 3, 4257–4273. CrossRef Google Scholar