Point and Space Groups of Graphene∗

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GENERAL ARTICLE Point and Space Groups of Graphene∗

Samiul Islam and S S Z Ashraf

Graphene – the first discovered ideal two-dimensional material combines a wealth of exhibited novel physical phenomena with an overwhelming potential for practical applications. It is a one atom thick material where the carbon atoms sit on a honeycomb lattice. The honeycomb lattice is the cause for the many fascinating properties that graphene turns up with. Conventionally a material scientist would first like to know Samiul Islam is a the symmetry that any newly discovered material possesses, postgraduate from the as this can help determine the physical properties of the mate- Department of Physics, rial. In this article, we discuss the space and point symmetry Aligarh Muslim University. groups that the hallmark honeycomb structure of graphene He has studied condensed matter physics as special possesses. subject in postgraduation and intends to pursue research in the same field. 1. Introduction

Symmetry is ubiquitous and features prominently in human intel- lectual pursuits as diverse as visual arts, poetry, music and none the least, natural sciences. In an ordinary sense, symmetry con- notes with beauty, similarity, equality, regularity, balance, jus- S S Z Ashraf is an Associate tice, repetition, periodicity, rhythm, etc. Quantitatively, symme- Professor at the Department of Physics, Aligarh Muslim try in natural sciences is measured through the mathematical lan- University. He works in the guage of ‘group theory’. In physics too, the scope of symmetry area of theoretical condensed is exceedingly wide as it underpins the whole gamut of physi- matter physics. His other cal phenomena right from the Dirac equation for the electron to interests are metaphysics and poetry. equations of the expanding universe. This scope permeates to its sub-branches, and thereby in condensed matter physics too, symmetry and group theory serve as an extremely powerful tool to understand, predict and postdict the properties of materials.

Crystals possess both point group symmetry (as of atoms and Keywords Graphene, honeycomb, hexagonal , point, space and symmorphic groups, Hermann–Mauguin nota- ∗DOI: https://doi.org/10.1007/s12045-019-0797-1 tion.

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molecules) and translational symmetry, which together form the Figure 1. Figures show- space groups. A point symmetry group is constituted of opera- ing (a) translation symmetry, (b) rotation symme- tions that leave one point fixed. Clearly, translation operation is try (four-fold), (c) reflection excluded from the point symmetry group as an object when sub- (bilateral) symmetry, and (d) jected to translation leaves behind the fixed point. There exist 10 glide reflection symmetry. crystallographic point groups compatible with five Bravais lattice in two dimensions (2D) and 32 crystallographic point operations consistent with fourteen Bravais lattice in three dimensions (3D). The knowledge of point groups of molecules and crystals is of much physical significance in spectroscopy and condensed matter The knowledge of point physics. Physical properties like optical activity, piezoelectricity, groups of molecules and pyro electricity, and molecular dipole moments are all related to crystals is of much the point groups of the crystal. physical significance in spectroscopy and Altogether, 17 crystallographic space groups in 2D and 230 in condensed matter 3D have been enumerated by symmetry considerations. These physics. space group symmetries are also of importance as they render the one-electron Hamiltonian invariant and hence commute with the space group providing the quantum numbers for determining the

446 RESONANCE | April 2019 GENERAL ARTICLE energy eigenvalues and eigenfunctions. Apart from this, the in- clusion of translational symmetry (that is irrelevant to molecules) also enables the determination of dispersion relations in reciprocal space, which is an important feature in solids as the physical properties of solids are determined from it rather than by energy levels. Before we enumerate the space group symmetry operations of graphene, we discuss below the point group operations in 2D.

2. Symmetry Operations in 2D Crystallography

There are altogether eight symmetry operations in 3D, namely, (i) identity, (ii) rotation, (iii) reflection, (iv) translation, (v) glide reflection, (vi) inversion centre or point reflection, (viii) rotore- flection and, (viii) helical or screw axis or rototranslation. Out of these eight, the last three do not figure in 2D, and therefore, the first five operations only feature in 2D crystallography, which are illustrated in Figure 1. We first briefly discuss each of these five symmetries below before we proceed to point and space groups of graphene.

2.1 Identity Symmetry

The identity operation, denoted here by E, in essence, does noth- Most asymmetric of the ing as this operation restores the object to its original status. This objects still possesses the identity symmetry means that the most asymmetric of the objects will still possess ff ff operation. In e ect, we this symmetry operation. In e ect, we can also say that E has the can also say that identity same significance that the number 1 has in multiplication. The symmetry operation has identity operation itself forms a group as it satisfies all the ax- the same significance ioms of the algebraic structure of group. that the number 1 has in multiplication.

2.2 Rotation Symmetry

As the name suggests, rotational symmetry is the symmetry that an object possesses if it looks the same after being turned by some degree. Physically, this also means that the object has multiple orientation positions adopting which it becomes indistinguishable

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1 1 ACn group is a simple nth from the earlier one. The rotation symmetry is denoted by Cn , order cyclic group of rotation where n is an integer, and rotation by 360o/n about a particular about a single n-fold axis. o axis deﬁnes the n-fold rotation axis, like C2 = 180 rotation, C3 = 120o rotation, etc.

2.3 Reﬂection Symmetry

The simplest case of Reflection symmetry is referred to by several names that also un- reflection symmetry is derline its meaning, i.e. line symmetry, mirror symmetry, mirror- bilateral symmetry image symmetry, etc. So this is symmetry with respect to re- which also has been found to be the most flection; meaning that an object does not change upon undergo- perceptible symmetry in ing a reflection operation. In other words, the reflection line, or creatures including the mirror line or the axis of symmetry cuts the object into two human beings. equal parts which cannot be distinguished as they are the exact replica of each other. The simplest case of reflection symmetry is bilateral symmetry which also has been found to be the most perceptible symmetry in creatures including human beings.

2.4 Translation Symmetry

Translation symmetry is the symmetry that a figure has if it can be made to look exactly the same when it is displaced or translated by some fixed distance. Naturally, translational symmetry exists Glide reflection only for infinite patterns, and for finite patterns, it is understood symmetry is a compound that translational symmetry would prevail only if the pattern were symmetry operation to continue indefinitely. requiring two symmetry operations (translation + reflection) to be 2.5 Glide Reflection Symmetry performed subsequently on the object to make it coincide with its original Glide reflection symmetry is a compound symmetry operation re- shape. quiring two symmetry operations (translation + reflection) to be performed subsequently on the object to make it coincide with its original shape. The translation is to be done by a fixed distance in a fixed direction and then reflected over a line parallel to the direction of translation. As is the case with translational symmetry, glide-reflectional symmetry also exists only for infinite patterns.

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Figure 2. (a) 3. Structure of Graphene Abee hive or honeycomb (b) Graphene sheets containing 2 Graphene is a single layer of graphite or conversely, graphite honeycomb structure. is weakly bonded stacked graphene sheets. Graphene was ﬁrst mechanically exfoliated from graphite in 2004 by two Russian scientists, Andre Geim and Konstantin Novoselov at University 2Graphene is the thinnest, of Manchester, and the discovery fetched them the 2010 Nobel lightest, strongest, most con- Prize. ducting (heat and electricity), highly transparent and novel material known to humans. It The carbon atoms in graphene are bonded in sp2 hybridisation. has got immensely large and Three electrons per carbon atom are involved in the formation of diverse applications in areas like medicine, electronics, light σ bonds and the remaining one electron per atom is involved in processing, energy, sensors, making the π bonds. The π electrons are responsible for the elec- thermoelectricity, etc. tronic properties at low energies. The carbon atoms populate a structure which is similar to a honeycomb as can be seen in Figure 2. Hence the graphene lattice is called the ‘honeycomb lattice’. However, a honeycomb lattice is not a Bravais lattice. To recall, a Bravais lattice is one where the lattice looks the same when

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viewed from any two points, and hence any two lattice points can Figure 3. (a) Honeycomb be connected by primitive translation vectors. In the honeycomb lattice with two inequivalent lattice shown in Figure 3a, the points A and B are not similar as carbon atomic positions A the surroundings of these two points are dissimilar. We have near and B is not a Bravais lat- the right of carbon atom A at a distance of ac−c the carbon B atom, (b) tice. Now combine two which is devoid of an A atom to its right at the same distance. So carbon atoms within a sin- the environment surrounding A and B carbon atoms are differ- gle lattice point as shown en- ent. Also, we cannot find two primitive3 translation vectors the circled within dark ellipses. integral multiples of which can connect all the lattice points. The (c) Join the points within the dark ellipses as shown. We non-conformity of the honeycomb lattice with any of the Bravais ffi can observe that there is one lattice poses di culty in determining the symmetry groups, as for dark ellipse which is at the determining symmetry groups, the object has to be a Bravais lat- centre of each hexagon. So tice. So the question arises, is it possible to transform graphene’s this hexagonal structure is a honeycomb structure into a Bravais lattice structure in order to primitive lattice. enumerate its symmetry groups. The answer is ‘Yes’. This can be achieved if we assign a single lattice point to two carbon atoms as shown in Figure 3b. In this way, the whole lattice takes the shape of a hexagonal lattice as depicted in Figure 3c. The hexagonal 3A primitive cell is a minimal lattice is a Bravais lattice and the problem reduces to finding the region repeated by lattice trans- symmetry groups of a hexagon. lations.

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Figure 4. The top six figures depict the six-fold rotational symmetry and the bot- tom six figures represent the six reflection symmetry of the hexagon.

4. Point Group of Graphene

A lattice point group is commonly defined as the collection of If one looks at the the set of symmetry operations about a lattice point which is in- hexagon figure from variant under the applied symmetry operations. A hexagon is a symmetry aspect, one can count a total of 12 regular polygon of six sides. The inner angle between any two symmetry operations. faces is 60o. If one looks at the hexagon figure from symmetry aspect, one can count a total of 12 symmetry operations. The symmetry operations of this hexagon consist of six rotation and six reflection operations. The rotations are by angles of 720o/n in counter-clockwise direction, where n = 6, 5, 4, 3 and 2, in which

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n = 6 is nothing but the identity operation. The six reflection operations include three mirror planes bisecting the opposite faces of the hexagon, and three mirror planes bisecting the opposite vertices of the hexagon. In the following, we elaborate on each of the point group symmetry operations of the hexagon. A regular hexagon is shown below with the corners marked with numbers for illustration purpose. The top six figures drawn in Figure 4de- pict the six rotation operations denoted by E, Rt1,Rt2,Rt3,Rt4, o o and Rt5 corresponding respectively to rotations by 720 , 120 , 240o, 360o, 480o and 600o, in which the first six figures represents the identity operation which does nothing and keeps the figure as such. The next six figures below this represent the reflection symmetry operations with mirror planes alternately bisecting the corners and edges and denoted by rf0,rf1,rf2,rf3,rf4 and rf5. These 12 symmetry elements of hexagon form a group called the C6v group in Schoenflies notation and 6 mm in Hermann–Mauguin4 4 The Hermann–Mauguin (H–M) or international notation. The C6v or 6 mm means six- system uses four symbols to fold rotation axis and two sets of vertical mirror planes, one set uniquely specify the group through the C6 axes passing through the corners of the hexagon properties of each of the 230 space groups. the first symbol and the other set of vertical planes called the dihedral planes bi- is a single letter which refers secting the first set of planes. The same can be easily verified to the Bravais lattic type. The from group multiplication or the Cayley table given in Table 1. remaining three letters refer to the point group of the crystal. 5. Space Group of Graphene

After enumerating the 12 point symmetry operations, we now move on to the space group of graphene. A space group is a set of symmetry elements that brings a periodic arrangement of points on a Bravais space lattice to its original position. As mentioned earlier, a space group constitutes point group operations combined with translation operation. Point group operations like rotation combined with translation, or reﬂection combined with translation give rise respectively to screw rotation symmetry and glide reﬂection symmetry operations in space group. In general, all the point group operations may not be a sub-group of the space group. A space group in which all the point group symmetry operations also form the symmetry operations of the space group is

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E Rt1 Rt2 Rt3 Rt4 Rt5 rf0 rf1 rf2 rf3 rf4 rf5

E E Rt1 Rt2 Rt3 Rt4 Rt5 rf0 rf1 rf2 rf3 rf4 rf5

Rt1 Rt1 Rt2 Rt3 Rt4 RT5 E rf1 rf2 rf3 rf4 rf5 rf0

Rt2 Rt2 Rt3 Rt4 Rt5 E Rt1 rf2 rf3 rf4 rf5 rf0 rf1

Rt3 Rt3 Rt4 Rt5 E Rt1 Rt2 rf3 rf4 rf5 rf0 rf1 rf2

Rt4 Rt4 Rt5 E Rt1 Rt2 Rt3 rf4 rf5 rf0 rf1 rf2 rf3

Rt5 Rt5 E Rt1 Rt2 Rt3 Rt4 rf5 rf0 rf1 rf2 rf3 rf4

rf0 rf0 rf5 rf4 rf3 rf2 rf1 E Rt5 Rt4 Rt3 Rt2 Rt1

rf1 rf1 rf0 rf5 rf4 rf3 rf2 Rt1 E Rt5 Rt4 Rt3 Rt2

rf2 rf2 rf1 rf0 rf5 rf4 rf3 Rt2 Rt1 E Rt5 Rt4 Rt3

rf3 rf3 rf2 rf1 rf0 rf5 rf4 Rt3 Rt2 Rt1 E Rt5 Rt4

rf4 rf4 rf3 rf2 rf1 rf0 rf5 Rt4 Rt3 Rt2 Rt1 E Rt5

rf5 rf5 rf4 rf3 rf2 rf1 rf0 Rt5 Rt4 Rt3 Rt2 Rt1 E

Table 1. The Cayley ta-

ble for C6v group consisting called ‘simple’ or ‘symmorphic’5. In other words, a space group of the 12 symmetry elements is symmorphic if, apart from the lattice translations, all generat- (one identity, five rotations ing symmetry operations leave one common point fixed. In the and six reflections) form a case of a hexagonal lattice in 2D, there are five space groups that group. are all symmorphic. In Table 2, we list all the symmorphic groups of a 2D hexagonal lattice.

Now, the physical properties of crystals are obtained from the en- 5A symmorphic space group ergy versus wavevector or dispersion relation in reciprocal space. contains, apart from the lat- The symmetry of a crystal is characterized in reciprocal space, tice translations, all generating symmetry operations about a by a group of wavevectors. To obtain the symmetry groups in common ﬁxed point. the reciprocal space, we consider the honeycomb lattice depicted Figure in 5a. The primitive√ translation vectors√ of the direct lattice are a = (3a/2, 3a/2) and b = (3a/2, 3a/2), where a is the carbon-carbon bond length. The corresponding reciprocal lattice is also a hexagonal lattice but rotated by 90o with respect to the direct lattice and this renders symmetry elements of direct and reciprocal lattice identical. The reciprocal lattice vec-

RESONANCE | April 2019 453 GENERAL ARTICLE √ √ ∗ ∗ The Brillouin zone is tors are a = (2π/3a, 2π/ 3a) and b = (−2π/3a, 2π/ 3a) Figure related to the shown in 5b. The first Brillouin zone is a hexagon√ with translational periodicity a set of inequivalent corner points K = (2π/3a, 2π/3 3a)and of the crystal. √ K = (2π/3a, −2π/3 3a), with K and K points alternating along the hexagon. The importance of the Brillouin zone stems from the fact that the unique solutions for the energy bands of crystalline solids are found within the first Brillouin zone. Since many of the physical properties of solids depend on the dispersion relations near wavevector k = 0, the symmetry of the unit cell gives sufficient information for the interpretation of many physical properties. The Brillouin zone depicted in Figure 5b shows high symmetry points labelled as Γ,K,K and M and lying Table 2. Point and sym- along the symmetry lines Γ–K, Γ–M and K–M. The origin is set morphic space group of a at the highest symmetry point that is at the centre and denoted 2D hexagonal lattice. Re- by Γ. At other points in the Brillouin zone, the symmetry is re- mark: Schoenflies symbols duced because of the finite value of wavevector. The inequivalent for space groups are con- fusing and almost now de- points K and K alternately figure on the six corners of the Bril- funct in crystallography, and louin zone and the point M bisects the edges of the Brillouin zone. hence not mentioned in the The Brillouin zone is related to the translational periodicity of the table. crystal. At the Γ point, the group of wavevectors is isomorphic

Point Group Point Group Space Group Point Group Type H–M Schoenﬂies H–M Elements Symbol Symbol

3 C3 P3 E, Rt2,Rt4 Symmorphic

3m C3v P3ml E, Rt2,Rt4,rf0 Symmorphic rf2,rf4

P3lm E, Rt2,Rt4, Symmorphic rf1,rf3,rf5

6 C6 P6 E, Rt1,Rt2, Symmorphic Rt3,Rt4,Rt5

6mm C6v P6mm E, Rt1,Rt2, Symmorphic Rt4,Rt5,rf0, rf1,rf2,rf3,rf4, rf5

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Figure 5. (a) Honeycomb lattice of graphene. Dark and light shaded circles represent the two sublattices A and B, respectively. Also shown are the primitive lattice vectors denoted by a and b. The bond length between the atoms is a (b) The corresponding Brillouin zone of the honeycomb lattice show- ing the Γ point at the centre, and the two inequivalent KK points. Its primitive lattice vectors are represented by a∗ and b∗.

Figure 6. The left and right diagrams respectively illustrate the symmetry elements

of the point group D6 and D6h of the primitive hexagonal lattice.

66 to the point group D6h which is a dihedral point group of the 3D A dihedral group is obtained hexagonal lattice as can be seen from Table 2. The correspond- by adding two-fold axis perpen- ing symmorphic space group from Table 3 in H–M notation is dicular to the principal Cn axis. It can be determined by wo set / / P6 mmm. Hence the space group of graphene is P6 mmm mean- of generators and is subject to n 2 ing the other smaller groups are at the point K, K‘ denoted by D3h the relation r = E and s = E. and at the lines forming a triangle Γ–K denoted by C2v. Figures 6a and 6b illustrate the point groups D6 and D6h of the primitive hexagonal lattice.

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Table 3. Point Point Space Point and space Type groups of hexagonal lattice Group Group Group in 3D in international and Hermann– Schoenﬂies Hermann– Schoenﬂies notation. Mauguin Symbol Mauguin Symbol Symbol

6 C6 P6 Symmorphic

6¯ C3h P6¯ Symmorphic

6/m C6h P6/m Symmorphic

622 D6 P622 Symmorphic

6mm C6v P6mm Symmorphic

6m2¯ D3h P6m2¯ Symmorphic

6/mm D6h P6/mmm Symmorphic

6. Conclusion

Graphene is a single layer of carbon atoms formed by sp2 hybridisation arranged in a honeycomb lattice. It is the building block of graphite as well as graphitic nanostructures like carbon nanotubes and fullerenes. Graphene sheet possesses a highly symmetric structure exhibiting translational periodicity in two dimensions, a six-fold rotational symmetry and a six-mirror reflection symmetry. The point group of graphene in Schoenflies notation is C6v and in International notation is 6 mm. The unit cell of graphene in reciprocal space called the first Brillouin zone is also highly symmetric and forms a symmorphic space group. The highest symmetry point in the first Brillouin zone is the Γ point

which is isomorphic to the point group D6h and the corresponding symmorphic space group of graphene in international notation is P6/mmm.