GENERAL ARTICLE Point and Space Groups of Graphene∗ Samiul Islam and S S Z Ashraf Graphene – the first discovered ideal two-dimensional mate- rial 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, sym- metry 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, hexago- nal , point, space and symmorphic groups, Hermann–Mauguin nota- ∗DOI: https://doi.org/10.1007/s12045-019-0797-1 tion. RESONANCE | April 2019 445 GENERAL ARTICLE 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 symme- try, (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 recipro- cal 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 opera- tions 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 RESONANCE | April 2019 447 GENERAL ARTICLE 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 di- rection of translation. As is the case with translational symmetry, glide-reflectional symmetry also exists only for infinite patterns. 448 RESONANCE | April 2019 GENERAL ARTICLE 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 RESONANCE | April 2019 449 GENERAL ARTICLE 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.