Group theory analysis of phonons in two-dimensional transition metal dichalcogenides

Transition metal dichalcogenides (TMDCs) have emerged as a new two-dimensional material's field since the monolayer and few-layer limits show different properties when compared to each other and to their respective bulk materials. For example, in some cases when the bulk material is exfoliated down to a monolayer, an indirect-to-direct band gap in the visible range is observed. The number of layers $N$ ($N$ even or odd) drives changes in space-group symmetry that are reflected in the optical properties. The understanding of the space-group symmetry as a function of the number of layers is therefore important for the correct interpretation of the experimental data. Here we present a thorough group theory study of the symmetry aspects relevant to optical and spectroscopic analysis, for the most common polytypes of TMDCs, i.e., $2Ha$, $2Hc$ and $1T$, as a function of the number of layers. Real space symmetries, the group of the wave vectors, the relevance of inversion symmetry, irreducible representations of the vibrational modes, optical activity, and Raman tensors are discussed.

Figure 1

Transition metal atom coordination for (a) trigonal prismatic ($H$) and (b) octahedral ($T$) TMDCs polytypes. The blue spheres represent transition metal atoms and orange ones, chalcogen atoms. In (c), (d), and (e) the top and lateral views (top and bottom in each figure, respectively) of the primitive unit cells for bulk TMDCs materials are shown. The black rhombuses show the top view of the primitive unit cell, and the red rectangles indicate the lateral view. The primitive unit cell of the (c) $2Ha$ or the (d) $2Hc$ polytypes comprise six atoms, two transition metal atoms, and four chalcogenides ($Z=2$) in the trigonal prismatic coordination illustrated in (a), while the $1T$ polytype shown in (e) has three atoms, comprising two chalcogenides, and one transition metal atom ($Z=1$) in the octahedral coordination illustrated in (b).

Figure 2

Primitive unit cell and symmetry operations of the $2Hc$ polytype. Blue spheres represent transition metal atoms and orange spheres represent chalcogen atoms. (a) and (d) show the top view for the $1\mathrm{TL}$ and $2\mathrm{TLs}$, respectively. ${\vec{a}}_1$ and ${\vec{a}}_2$ are the primitive unit vectors, indicated in (a), while (b) and (e) represent the symmetry operations for the $1\mathrm{TL}$ and $2\mathrm{TLs}$, respectively. The $C_3$ axes are perpendicular to the $xy$ plane in (b) and (e), and they are represented by black triangles. Three vertical mirror planes ${\sigma }_v$ and three dihedral mirror planes ${\sigma }_d$ are indicated as red lines in (b) and (e), respectively, while the black lines are the three $C_2^{\prime }$ rotation axes in the horizontal mirror ${\sigma }_h$, represented in (c) and (f) together with the primitive unit cell. The ${\sigma }_h$ itself is not a symmetry operation for $2\mathrm{TLs}$, but it is discussed here since it is part of the $S_6$ operation, which is given as a $C_6$ rotation followed by a ${\sigma }_h$ reflection in this plane. The red lines in (e) denote the ${\sigma }_d$ mirror planes, and the red dot in the center of (f) indicates the position of the inversion symmetry operation.

Figure 3

Primitive unit cells and symmetry operations of the $1T$ TMDCs polytypes (bulk, $1\mathrm{TL}$ and $2\mathrm{TLs}$). (a) and (d) show the $1\mathrm{TL}$ and $2\mathrm{TL}$ top view. In (d), chalcogen atoms are on top of chalcogen atoms, and transition metal atoms are on top of transition metal atoms, giving a similar top view to that observed for $1\mathrm{TL}$. In (b) and (e), the $C_3$ rotation axes (represented as black triangles) are perpendicular to the basal plane. The red lines represent ${\sigma }_d$ mirror planes, while the black lines stand for $C_2^{\prime }$ rotation axes that lie in the ${\sigma }_h$ plane. The primitive unit cells for $1\mathrm{TL}$ (and bulk) and for $2\mathrm{TLs}$ are shown in (c) and (f), respectively, and the red dot in their centers denotes the inversion operations. Notice that ${\sigma }_h$ is not a symmetry operation for $1\mathrm{TL}$ (or $N$ odd), $2\mathrm{TLs}$ (or $N$ even), or bulk, but the reflection plane is shown here to indicate the reflection in the two $S_6$ operations.

Figure 4

The Brillouin Zone (BZ) symmetries: $\Gamma$, $K$, $K^{\prime }$, and $M$ are high-symmetry points; the $T$, $T^{\prime }$, and $\Sigma$ are high-symmetry lines, and the $u$ denotes the symmetry for a generic point. ${\vec{b}}_1$ and ${\vec{b}}_2$ denote the in-plane reciprocal lattice vectors.