{"id":406,"date":"2018-11-16T12:15:38","date_gmt":"2018-11-16T12:15:38","guid":{"rendered":"http:\/\/phy12.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=406"},"modified":"2019-04-30T11:01:29","modified_gmt":"2019-04-30T11:01:29","slug":"band-structure-of-graphene-using-tight-binding-method","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/chapter\/band-structure-of-graphene-using-tight-binding-method\/","title":{"rendered":"Band Structure of Graphene using Tight Binding Method"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\nTable of Content<\/strong>\r\n\r\n \r\n\r\nModule V<\/strong>\r\n\r\n \r\n\r\n5.5 Band Structure of Graphene using Tight Binding Method\r\n\r\n5.5.1 Monolayer graphene\r\n\r\n5.5.2 Sublattice and unit cell in graphene\r\n\r\n5.5.3 Band structure of graphene\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n5.5 Band Structure of graphene using tight binding method<\/strong>\r\n\r\n \r\n\r\n5.5.1 Monolayer Graphene<\/strong>\r\n\r\n \r\n

Graphene is a single atomic layer of graphite. Its structure has been studied long ago and initially it was believed that graphene is not thermodynamically stable in single layer form. Finally it was shown for the first time, that graphene can be stable when placed on a suitable substrate. For this work, Andre K. Geim and Konstantin S. Novoselov, who succeeded in producing, isolating, identifying, and characterizing graphene layers, were honored with 2010 Nobel Prize in Physics. Since the discovery in 2004, graphene has become one of the most investigated materials amongst the scientific community. All the unique properties of graphene emanate from the linear energy dispersion relation at low energies.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

The most explored aspect of graphene physics is its electronic properties which is the outcome of its unique band structure. The fact that charge carriers in graphene are described by the Dirac-like equation rather than the usual Schr\u00f6dinger equation is the consequence of graphene\u2019s crystal structure leading to unique band structure which is responsible for very high charge carrier mobility and unusually high conductivity, several orders of magnitude higher than copper. The high mobility, the high current carrying capacity, the 2D atomic structure and the compatibility with planar technology make graphene an exciting and promising candidate for future microelectronics. The novel band structure holds promise for as-yet unrealized devices that exploit the massless Dirac-fermion like linear energy dispersion of electrons in the material. The potential applications of graphene extend far beyond electronic devices. It is being touted as a material that will literally change our lives in the 21st century, like plastics did hundred years ago. Not only graphene is the thinnest and lightest possible material that is feasible, but it's also ~200 times stronger than steel and conducts both heat and electricity better than any known material at room temperature. In spite of its zero bandgap, graphene absorbs only 2.3 % of incident light which makes it a potential candidate to be used in graphene based transparent optoelectronic devices. Graphene based optoelectronic components promise closing the terahertz gap, transparent conductive coatings for solar cells, touch-enabled displays; stronger medical implants; artificial membranes for separating liquids. Nanogaps in graphene sheets may potentially provide a new technique for rapid DNA sequencing. It holds a high potential in nanoelectromechanical systems and components for RF resonators in the GHz frequencies.<\/p>\r\n \r\n\r\n \r\n\r\n5.5.2 Sublattice and unit cell in graphene<\/strong>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 5.27 <\/strong>Two different sublattices in graphene monolayer.<\/em><\/p>\r\n \r\n

The carbon atoms in graphene are arranged in a honeycomb lattice due to their sp2 hybridisation. The honeycomb lattice is not a Bravais lattice because two neighboring sites are not equivalent which is illustrated in Figure 5.27. It is clear that a site on the A sublattice has nearest neighbors in the directions north-east, north-west, and south, whereas a site on the B sublattice has nns in the directions north, south-west, andcsouth-east. Both A and B sublattices, however, are triangular Bravais lattices. Hence honeycomb lattice can be considered as a triangular Bravais lattice with a two-atom basis (A and B). The sublattice A and B can be treated as spin up and spin down and hence termed as pseudospin which lays down the connection between graphene and relativistic quantum mechanics.<\/p>\r\n \r\n

Graphene is a single layer of graphite consists of sp2-hybridized carbon atoms, arranged in a honeycomb lattice. All the carbon atoms in layer are covalently bonded to its three nearest neighbors, forming \u03c3-bonds in the xy-plane. The remaining pz electron leads to the formation of a half filled \u03c0-bond, which governs the electronic properties of graphene. Thus, the band structure of grapheme can be calculated by taking into account one 2pz orbital per atomic site, i.e. two atoms per unit cell. The unit cell of graphene is defined by two carbon atoms sitting at adjacent, nonequivalent sites, namely A and B as displaced in Figure 5.28. The positions of A and B atoms are non-equivalent because it is not possible to connect them with a lattice vector of the form R = n1a1 + n2a2, where n1, n2 are integers and a1, a2 are the primitive lattice vectors defined as<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

where a = a1 = a2, is lattice constant, the distance between adjacent unit cell, a=2.46Ao. The lattice constant is distinct from the carbon-carbon bond length = \u221a3 = 1.42 \u00a0which is distance between the adjacent carbon atoms. As shown in Figure 5.28, the reciprocal lattice is a hexagonal Bravais lattice, and the first Brillouin zone is again a hexagon. Hence, it can be shown that a1.b1 = a2.b2 = 2\u03c0 and a1.b2 = a2. b1 = 0, and reciprocal vectors are:<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n\"\"\r\n

Figure 5.28<\/strong>: Crystal structure of MLG and Reciprocal lattice. (a) MLG, where primitive lattice vectors<\/em> a<\/em>1<\/em> and a<\/em>2<\/em> allow for translational invariant motion along lattice. (b) Two different ways of representations of unit cell, upper is 2 carbon atoms and lower is, 1\/3 each of 6 carbon atoms = 2 atoms. (c) Honeycomb carbon lattice of graphene with two sublattices, \u03b1 and \u03b2, respectively. \u03b4<\/em>1<\/em>, \u03b4<\/em>2<\/em> and \u03b4<\/em>3<\/em> point out the position of the nearest neighbour from an A atom to surrounding B atoms. (d) Reciprocal lattice vectors b<\/em>1<\/em> and b<\/em>2<\/em> along with high symmetry points \u0393, K and M in the Brillouin zone.<\/em><\/p>\r\n \r\n\r\n5.5.3 Band structure of graphene<\/strong>\r\n\r\n \r\n

For the TB calculation of the energy band structure of MLG, consider two Bloch functions from A and B sites which are used for calculating the transfer Matrix H and overlap matrix S defined by Eq 5.24 and 5.25, at j= A, B.<\/span><\/p>\r\n\"\"\r\n\r\n \r\n

Begin by considering the diagonal elements of the transfer integral matrix H for sublattice A,<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Assuming that the dominant contribution arises from those terms involving a given orbital interacting with itself (i.e. within same unit cell), the matrix element can be written as<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\"\"<\/div>\r\n
<\/div>\r\n \r\n
\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

Thus, S<\/em>AA<\/em> = S<\/em>BB<\/em> = 1. The off-diagonal element H<\/em>AB<\/em> of the transfer integral matrix H describes the probability of hopping between orbitals on sites A and B. Consider A site, then taking into account the possibility of hopping to its three nearest neighbor B sites, j=1, 2, 3:<\/p>\r\n \r\n\r\n\"\"\r\n\r\nThe hopping parameter can be defines as\r\n\r\n\"\"\r\n\r\nwhere\u00a0 \u03b3o\u00a0 is the nearest neighbor hopping parameter. Then, the matrix element can be written as\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n\"\"\r\n
\r\n

The other off-diagonal element H<\/em>BA<\/em> is the complex conjugate of H<\/em>AB<\/em>, i.e. H<\/em>BA<\/em> =-\u03b3<\/em>o<\/em>f*(k)<\/em>. The calculation of the off-diagonal elements of the overlap integral matrix S<\/em> is similar to those of H.<\/p>\r\n \r\n\r\n\"\"\r\n\r\nThus, transfer and integral matrices of monolayer graphene can be written as,\r\n\r\n\"\"\r\n\r\nThe corresponding energies may be determined by solving the secular equation.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nFor intrinsic graphene \u03b5<\/em>2p<\/em> = 0,\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n

The hexagonal Brillouin zone (BZ) of graphene as shown in Figure 5.28 has three high symmetry points, the \u0393 point, located at the centre of the BZ, the M point, which indicates the position of the Van hove singularities (VHSs) of the \u03c0 to \u03c0* bands, where density of states (DOS) is logarithmically divergent. The K points where \u03c0-band touches and DOS vanishes linearly are shown schematically in Figure 5.29 which also shows the band structure calculated from above equation in the first BZ where \u03c0 and \u03c0* indicate the conduction and valence band, respectively. The parameters such as \u03b3o = 3.033 eV and so = 0.12 are used in the calculation. Figure 5.29 shows that it is gapless and touching at K- and K+ points which are located at the corner of the first BZ. These high symmetry points have coordinates, \u0393(0,0), M(0, 2\u03c0\/\u221a3a) and K(2\u03c0\/3a, 2\u03c0\/\u221a3a) are plotted, as shown in Figure 5.29.<\/p>\r\n\"\"\r\n

Figure 5.29: <\/strong>Calculated energy band structure of monolayer graphene. (a) Band structure of MLG from<\/em> the tight binding calculation (b) Cross section along the line k<\/em>y<\/em> from (a). (c) The zoomed in band structure at the K point of the BZ, showing linear energy dispersion for MLG.<\/em><\/p>\r\n \r\n\r\n<\/div>\r\n\r\n\r\n\r\n
you can view video on Band Structure of Graphene using Tight Binding Method<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nReferences<\/strong>\r\n
    \r\n \t
  1. Physical Properties of Carbon Nanotubes<\/em><\/strong>, by R. Saito, M .S. Dresselhaus and G. Dresselhaus, Imperial College Press, London, 1998.<\/li>\r\n \t
  2. Graphene: Carbon in Two Dimensions<\/em><\/strong>, by M. I. Katnelson, Cambridge University Press, New York, 2012.<\/span><\/li>\r\n<\/ol>\r\n ","rendered":"
    \"epgp<\/a>
    \n<\/span><\/div>\n
    \n

     <\/p>\n

    Table of Content<\/strong><\/p>\n

     <\/p>\n

    Module V<\/strong><\/p>\n

     <\/p>\n

    5.5 Band Structure of Graphene using Tight Binding Method<\/p>\n

    5.5.1 Monolayer graphene<\/p>\n

    5.5.2 Sublattice and unit cell in graphene<\/p>\n

    5.5.3 Band structure of graphene<\/p>\n<\/div>\n

    \n

     <\/p>\n

    5.5 Band Structure of graphene using tight binding method<\/strong><\/p>\n

     <\/p>\n

    5.5.1 Monolayer Graphene<\/strong><\/p>\n

     <\/p>\n

    Graphene is a single atomic layer of graphite. Its structure has been studied long ago and initially it was believed that graphene is not thermodynamically stable in single layer form. Finally it was shown for the first time, that graphene can be stable when placed on a suitable substrate. For this work, Andre K. Geim and Konstantin S. Novoselov, who succeeded in producing, isolating, identifying, and characterizing graphene layers, were honored with 2010 Nobel Prize in Physics. Since the discovery in 2004, graphene has become one of the most investigated materials amongst the scientific community. All the unique properties of graphene emanate from the linear energy dispersion relation at low energies.<\/span><\/p>\n<\/div>\n

    \n

    The most explored aspect of graphene physics is its electronic properties which is the outcome of its unique band structure. The fact that charge carriers in graphene are described by the Dirac-like equation rather than the usual Schr\u00f6dinger equation is the consequence of graphene\u2019s crystal structure leading to unique band structure which is responsible for very high charge carrier mobility and unusually high conductivity, several orders of magnitude higher than copper. The high mobility, the high current carrying capacity, the 2D atomic structure and the compatibility with planar technology make graphene an exciting and promising candidate for future microelectronics. The novel band structure holds promise for as-yet unrealized devices that exploit the massless Dirac-fermion like linear energy dispersion of electrons in the material. The potential applications of graphene extend far beyond electronic devices. It is being touted as a material that will literally change our lives in the 21st century, like plastics did hundred years ago. Not only graphene is the thinnest and lightest possible material that is feasible, but it’s also ~200 times stronger than steel and conducts both heat and electricity better than any known material at room temperature. In spite of its zero bandgap, graphene absorbs only 2.3 % of incident light which makes it a potential candidate to be used in graphene based transparent optoelectronic devices. Graphene based optoelectronic components promise closing the terahertz gap, transparent conductive coatings for solar cells, touch-enabled displays; stronger medical implants; artificial membranes for separating liquids. Nanogaps in graphene sheets may potentially provide a new technique for rapid DNA sequencing. It holds a high potential in nanoelectromechanical systems and components for RF resonators in the GHz frequencies.<\/p>\n

     <\/p>\n

     <\/p>\n

    5.5.2 Sublattice and unit cell in graphene<\/strong><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Figure 5.27 <\/strong>Two different sublattices in graphene monolayer.<\/em><\/p>\n

     <\/p>\n

    The carbon atoms in graphene are arranged in a honeycomb lattice due to their sp2 hybridisation. The honeycomb lattice is not a Bravais lattice because two neighboring sites are not equivalent which is illustrated in Figure 5.27. It is clear that a site on the A sublattice has nearest neighbors in the directions north-east, north-west, and south, whereas a site on the B sublattice has nns in the directions north, south-west, andcsouth-east. Both A and B sublattices, however, are triangular Bravais lattices. Hence honeycomb lattice can be considered as a triangular Bravais lattice with a two-atom basis (A and B). The sublattice A and B can be treated as spin up and spin down and hence termed as pseudospin which lays down the connection between graphene and relativistic quantum mechanics.<\/p>\n

     <\/p>\n

    Graphene is a single layer of graphite consists of sp2-hybridized carbon atoms, arranged in a honeycomb lattice. All the carbon atoms in layer are covalently bonded to its three nearest neighbors, forming \u03c3-bonds in the xy-plane. The remaining pz electron leads to the formation of a half filled \u03c0-bond, which governs the electronic properties of graphene. Thus, the band structure of grapheme can be calculated by taking into account one 2pz orbital per atomic site, i.e. two atoms per unit cell. The unit cell of graphene is defined by two carbon atoms sitting at adjacent, nonequivalent sites, namely A and B as displaced in Figure 5.28. The positions of A and B atoms are non-equivalent because it is not possible to connect them with a lattice vector of the form R = n1a1 + n2a2, where n1, n2 are integers and a1, a2 are the primitive lattice vectors defined as<\/span><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    where a = a1 = a2, is lattice constant, the distance between adjacent unit cell, a=2.46Ao. The lattice constant is distinct from the carbon-carbon bond length = \u221a3 = 1.42 \u00a0which is distance between the adjacent carbon atoms. As shown in Figure 5.28, the reciprocal lattice is a hexagonal Bravais lattice, and the first Brillouin zone is again a hexagon. Hence, it can be shown that a1.b1 = a2.b2 = 2\u03c0 and a1.b2 = a2. b1 = 0, and reciprocal vectors are:<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Figure 5.28<\/strong>: Crystal structure of MLG and Reciprocal lattice. (a) MLG, where primitive lattice vectors<\/em> a<\/em>1<\/em> and a<\/em>2<\/em> allow for translational invariant motion along lattice. (b) Two different ways of representations of unit cell, upper is 2 carbon atoms and lower is, 1\/3 each of 6 carbon atoms = 2 atoms. (c) Honeycomb carbon lattice of graphene with two sublattices, \u03b1 and \u03b2, respectively. \u03b4<\/em>1<\/em>, \u03b4<\/em>2<\/em> and \u03b4<\/em>3<\/em> point out the position of the nearest neighbour from an A atom to surrounding B atoms. (d) Reciprocal lattice vectors b<\/em>1<\/em> and b<\/em>2<\/em> along with high symmetry points \u0393, K and M in the Brillouin zone.<\/em><\/p>\n

     <\/p>\n

    5.5.3 Band structure of graphene<\/strong><\/p>\n

     <\/p>\n

    For the TB calculation of the energy band structure of MLG, consider two Bloch functions from A and B sites which are used for calculating the transfer Matrix H and overlap matrix S defined by Eq 5.24 and 5.25, at j= A, B.<\/span><\/p>\n

    \"\"<\/p>\n

     <\/p>\n

    Begin by considering the diagonal elements of the transfer integral matrix H for sublattice A,<\/span><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Assuming that the dominant contribution arises from those terms involving a given orbital interacting with itself (i.e. within same unit cell), the matrix element can be written as<\/p>\n

    \"\"<\/p>\n<\/div>\n

    \"\"<\/div>\n
    <\/div>\n

     <\/p>\n

    \n

     <\/p>\n

     <\/p>\n

     <\/p>\n

     <\/p>\n

    Thus, S<\/em>AA<\/em> = S<\/em>BB<\/em> = 1. The off-diagonal element H<\/em>AB<\/em> of the transfer integral matrix H describes the probability of hopping between orbitals on sites A and B. Consider A site, then taking into account the possibility of hopping to its three nearest neighbor B sites, j=1, 2, 3:<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    The hopping parameter can be defines as<\/p>\n

    \"\"<\/p>\n

    where\u00a0 \u03b3o\u00a0 is the nearest neighbor hopping parameter. Then, the matrix element can be written as<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

     <\/p>\n

     <\/p>\n<\/div>\n

    \"\"<\/p>\n

    \n

    The other off-diagonal element H<\/em>BA<\/em> is the complex conjugate of H<\/em>AB<\/em>, i.e. H<\/em>BA<\/em> =-\u03b3<\/em>o<\/em>f*(k)<\/em>. The calculation of the off-diagonal elements of the overlap integral matrix S<\/em> is similar to those of H.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Thus, transfer and integral matrices of monolayer graphene can be written as,<\/p>\n

    \"\"<\/p>\n

    The corresponding energies may be determined by solving the secular equation.<\/span><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    For intrinsic graphene \u03b5<\/em>2p<\/em> = 0,<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

     <\/p>\n

    The hexagonal Brillouin zone (BZ) of graphene as shown in Figure 5.28 has three high symmetry points, the \u0393 point, located at the centre of the BZ, the M point, which indicates the position of the Van hove singularities (VHSs) of the \u03c0 to \u03c0* bands, where density of states (DOS) is logarithmically divergent. The K points where \u03c0-band touches and DOS vanishes linearly are shown schematically in Figure 5.29 which also shows the band structure calculated from above equation in the first BZ where \u03c0 and \u03c0* indicate the conduction and valence band, respectively. The parameters such as \u03b3o = 3.033 eV and so = 0.12 are used in the calculation. Figure 5.29 shows that it is gapless and touching at K- and K+ points which are located at the corner of the first BZ. These high symmetry points have coordinates, \u0393(0,0), M(0, 2\u03c0\/\u221a3a) and K(2\u03c0\/3a, 2\u03c0\/\u221a3a) are plotted, as shown in Figure 5.29.<\/p>\n

    \"\"<\/p>\n

    Figure 5.29: <\/strong>Calculated energy band structure of monolayer graphene. (a) Band structure of MLG from<\/em> the tight binding calculation (b) Cross section along the line k<\/em>y<\/em> from (a). (c) The zoomed in band structure at the K point of the BZ, showing linear energy dispersion for MLG.<\/em><\/p>\n

     <\/p>\n<\/div>\n\n\n\n
    you can view video on Band Structure of Graphene using Tight Binding Method<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    References<\/strong><\/p>\n

      \n
    1. Physical Properties of Carbon Nanotubes<\/em><\/strong>, by R. Saito, M .S. Dresselhaus and G. Dresselhaus, Imperial College Press, London, 1998.<\/li>\n
    2. Graphene: Carbon in Two Dimensions<\/em><\/strong>, by M. I. Katnelson, Cambridge University Press, New York, 2012.<\/span><\/li>\n<\/ol>\n

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