Chapter 4 A general theory of 2D materials alignment on crystalline substrates with different
4.2 The symmetries of 2D materials and substrate surfaces
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Chapter 4 A general theory of 2D materials alignment on
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there are no in-plane symmetric operations for the system of a 2D material on a substrate. As will be proved later, this simplification does not affect our conclusions). Cn represents that a 2D material will coincide with its original structure when it is rotated with respect to its principal axis by 2π/n (n = 1, 2, 3, 4 and 6), which is to say that this principal axis is a n-fold rotation axis of the 2D material. If there are n mirror planes passing through the n-fold rotation axis of the 2D material, the symmetry of the 2D material will increase to be Cnv.
Figure 4.1 illustrates the symmetries of three types of 2D materials, including graphene, hBN and MoS2 and three high symmetric low-index FCC surfaces, all the three 2D materials and the three low-index FCC surfaces have n-fold symmetries with n > 1. Strictly speaking, FCC(111) substrate is C3v symmetric because of its ABC-stacking structure. However, because the interaction between the 2D material and the substrate is mainly contributed by the top atomic layer of the FCC(111) substrate,188,334,335 the FCC(111) surface can be considered to be a C6v symmetric substrate. Figure 4.1 shows the principal axes and mirror planes of the three 2D materials and the three low-index FCC surfaces, as marked by the red circles and dash lines, respectively. Compared with graphene, hBN and TMDCs show mirror symmetries only with respect to AC directions due to their binary compositions.
It should be noticed that although the TMDC materials has a D3h symmetry because of its three-atom layered structure is symmetric respect to the central atomic layer, its symmetric operations are as same as the C3V material in the 2D plane direction.
Figure 4.1 Illustrations of the symmetries of various 2D materials and low-index high symmetric FCC surfaces. (a) Graphene, (b) hBN, (c) MoS2, (d) FCC{111}, (e) FCC{100} and (f) FCC{110}. The principal axes perpendicular to the 2D surfaces are denoted by the red circles, and the mirror planes passing through the principal axes are denoted by dashed lines.
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From the view of atomic configurations, the high-index low symmetric FCC TM surfaces are constructed by large numbers of high-symmetric terraces and steps, as demonstrated in Figure 4.2(a).
According to the types of the high-symmetric terraces, the low symmetric surfaces can be classified into three categories: FCC{111}-based, FCC{100}-based and FCC{110}-based surfaces. Because the high-index low symmetric surfaces are cut from an FCC TM along an arbitrary orientation that is deviated from the three low-index surfaces, in principle the number of high-index low symmetric surfaces is infinite. All the high-index low symmetric surfaces only have 1-fold rotational symmetry, while depending on the step edge structures of the surfaces, there are six types of CS symmetric surfaces with step edges along specific directions, which are presented in Figure 4.2(b). Every type of the low symmetric surfaces has two kinds of CS ones, i.e., FCC{n n n+m} with <110> step edges and FCC{n+m n-m n} with <211> step edges for FCC{111}-based surfaces; FCC{n m m} with <110> step edges and FCC{n m 0} with <100> step edges for FCC{100}-based surfaces; and FCC{n+m n+m m} with <110>
step edges and FCC{n n+m 0} with <100> step edges for FCC{100}-based surfaces. All the six CS
symmetric surfaces have straight step edges, and the mirror symmetric plane of these surface is perpendicular to the step edge direction.
Figure 4.2 (a) Illustration of a high-index low symmetric surface. (b) Atomic models for the two types of CS FCC{111}-based surfaces, whose surface indices are FCC{n n n+m} and FCC{n+m n-m n}, respectively. (c) Atomic models for the two types of CS FCC{100}-based surfaces, whose surface indices are FCC{n m m} and FCC{n m 0}. (d) Atomic models for the two types of CS FCC{110}-based surfaces, whose surface indices are FCC{n+m n+m m} and FCC{n n+m 0}. Dashed lines denote the mirror symmetric plane of CS surfaces. For all the six types of CS symmetric FCC surfaces, n and m are integers and n>>m.
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The deduction of these six types of CS-surfaces is as follows. Because a low symmetric surface can be considered to be composed of large terrace surfaces and small step surfaces, and the intersection lines of the terrace surfaces and the step surfaces determine the direction of the step edges. For example, a CS surface with <110> step edges can be constructed by two surfaces which have intersecting lines along <110> direction. Table 4.1 lists the compositions of the six types of CS surfaces and their surface indexes.
Table 4.1 The compositions and surface indexes of the CS symmetric FCC surfaces.
Surfaces Step edge direction of
the surface Compositions of the surface Surface indexes FCC{111}-
based
FCC<110> n×FCC(111) + m×FCC(001) FCC{n n n+m}
FCC<211> n×FCC(111) + m×FCC(11̅0) FCC{n+m n-m n}
FCC{100}- based
FCC<110> n×FCC(001) + m×FCC(111) FCC{n m m}
FCC<100> n×FCC(001) + m×FCC(010) FCC{n m 0}
FCC{110}- based
FCC<110> n×FCC(110) + m×FCC(111) FCC{n+m n+m m}
FCC<100> n×FCC(110) + m×FCC(010) FCC{n n+m 0}
Notes: n and m are integers, and n>>m.
Because of the anisotropy of the C1 symmetric surfaces, the islands of a 2D material growing on a C1 symmetric surface will show only one most preferential orientation, and detailed studies of graphene and hBN islands on various C1 symmetric surfaces will be introduced in Chapters 5 and 7, respectively. CS symmetric and low-index surfaces have mirror symmetric planes and rotational symmetry axes, and in this Chapter, the alignment of 2D material islands on the low-index high symmetric FCC surfaces and the six types of CS symmetric FCC surfaces are explored from the aspect of rotation and mirror symmetries.