structure of graphene and carbon nanotubes which forms the basis for many of their proposed applications in electronics.

Chapter Basics of graphene and carbon nanotubes This chapter reviews the theoretical understanding of the geometrical and electronic structure of graphene and carbon nanotubes which forms the basis for many of their proposed applications in electronics. The chapter begins with an overview of the basic physical structure of graphene and carbon nanotubes, where the geometry of graphene in real and reciprocal space is discussed and used to study the geometry of carbon nanotubes. The band structure of graphene is derived using the tight-binding method and then used to derive the electronic band structure of carbon nanotube. The energy dispersion relations for armchair and zig-zag nanotubes are derived explicitly. The electronic structure of graphene is used to study how energy gaps are formed when the graphene sheet is rolled into a cylinder. In other words, how the chirality of nanotubes determines whether the tube is metallic or semiconducting. The chapter ends with a discussion of the band structure near the Fermi points, where the energy spectrum is found to be linear. This linear band spectrum is then used to derive the band gap and the density of states of nanotubes. 35

36 BASICS OF GRAPHENE AND CARBON NANOTUBES.1 Physical structure of graphene and carbon nanotubes Graphene is a single atomic layer of graphite and defined as a one-atom-thick planar sheet of sp -bonded carbon atoms densely packed in a honeycomb crystal lattice [6, 11] as shown in Fig..1. This single layer structure makes graphene a two-dimensional (D) system. Graphene is considered as a building block for all graphitic materials [11]. When several D graphene layers are stacked together with an interplanar spacing of 0.34 nm, a three-dimensional (3D) graphite is obtained. A one-dimensional (1D) carbon nanotube can be obtained by rolling a graphene layer along a longitudinal axis and if the same graphene sheet is wrapped in a ball structure, a zero-dimensional (0D) fullerene is obtained [11] as illustrated in Fig..1..1.1 Graphene lattice in real and reciprocal space The honeycomb lattice of graphene in real space is spanned by two lattice vectors expressed as ( ) ( ) 3 a1 = a, a, 3 a = a, a, (.1) giving the relations a1. a 1 = a. a = a, a1. a = a. (.) These lattice vectors form an angle of 60 0 between them and are of magnitude a = a 1 = a =a cc 3=1.4 3=.46 Å, the lattice constant, where a cc =1.4 Åis the carbon-carbon bond length in graphene [9]. Figure.(a) illustrates a hexagonal lattice of a graphene sheet in real space with lattice vectors a 1 and a. The unit cell of the graphene sheet consists of two inequivalent carbon atoms, one

PHYSICAL STRUCTURE OF GRAPHENE AND CARBON NANOTUBES 37 (a) (b) (c) Figure.1: A D graphene gives rise to three carbon materials of different dimensionalities. (a) A 0D fullerene, (b) 1D carbon nanotube and (c) 3D graphite. Adapted from [11]. from the sublattice A (blue circle) and the other from sublattice B (green circle), as emphasized with a dotted box in Fig..(a). The term inequivalent means that the sites A and B in the unit cell can not be connected by unit vectors a 1 and a [170]. The first Brillouin zone of graphene in reciprocal lattice is hexagonal with corresponding lattice vectors b 1 and b as shown by a shaded hexagon in Fig..(b). The direction of the reciprocal lattice vectors b 1 and b are rotated by 90 0 from the real space unit vectors a 1 and a [9]. The reciprocal lattice is constructed from the real space unit vectors using the relationships,

38 BASICS OF GRAPHENE AND CARBON NANOTUBES b 1 = π a a 3 a 1.(a a 3 ), b = π a 3 a 1 a 1.(a a 3 ) (.3) where a 1 and a are defined above in Eq.(.1) and a 3 is the unit vector in the z direction. Hence, the unit vectors b 1 and b of the reciprocal lattice are given by: b1 = ( π 3a, π a ), ( π b =, π 3a a ) (.4) with a lattice constant of 4π 3a in reciprocal space [9] and the relations: b1. b 1 = b. b = 4 ( ) π, 3 a b1. b = ( ) π. (.5) 3 a The Brillouin zone consists of three types of high symmetry points: Γ, K(K ), and M. Γ represents the center of the Brillouin zone, K and K represent the two inequivalent corners of the hexagon, and M is the center of the line joining the two corners [9]..1. Carbon nanotube in real space The structure of a carbon nanotube (CNT) is that of a graphene layer as CNT can be visualized as a hexagonal honeycomb lattice of a graphene sheet rolled into a seamless cylinder [9] as shown in Fig..1(b). The structure of a SWCNT can be defined by a vector called the chiral vector C h defined by the vector OA which corresponds to a section of the nanotube perpendicular to the nanotube axis [9] in Fig..3. The chiral vector C h represents the full circumference of the nanotube, and the direction of the nanotube axis is represented by the vector OB which defines the translational vector T of the tube. Thus T is defined as the unit vector

PHYSICAL STRUCTURE OF GRAPHENE AND CARBON NANOTUBES 39 Unit Cell y Brillouin Zone x b 1 A B K M a 1 k y K a k x b Figure.: (a) The graphene lattice in real space with lattice vectors a 1 and a and the unit cell (dotted box) with two sublattices A (blue circle) and B (green circle).(b) The reciprocal lattice of graphene with reciprocal lattice vectors b 1 and b and the first Brillouin zone (shaded) with three high symmetry points Γ, K, K, and M. Adapted from [9]. of the nanotube which is parallel to the nanotube axis and is perpendicular to C h [9]. The way the graphene sheet is rolled up to make nanotubes can be described with the chiral vector defined as a linear combination of real space lattice vectors a1 and a : C h = n a 1 + m a = (n, m) (0 m n). (.6) The integers (n, m) indicate the number of lattice vectors along the two directions in the lattice of graphene. In Fig..3, a carbon nanotube can be constructed by superimposing the two ends of the chiral vector C h and the cylinder joint is made along the two lines OB and AB both are perpendicular to the vector C h at each end of C h. Such a tube is known as (n, m) SWCNT. The magnitude of C h gives the circumference of the tube C h = a n + m + nm, (.7)

40 BASICS OF GRAPHENE AND CARBON NANOTUBES using the relations given in Eq. (.). Here the lattice constant a=a cc 3=1.44 Å 3=.49 Å as the carbon-carbon bond length of carbon nanotubes is slightly larger (1.44 Å) than that of graphite (1.4Å) [9]. Figure.3: The unrolled (4,) carbon nanotube. The chiral vector C h and the translational vector T of the nanotube are defined by the vectors OA and OB, respectively. A nanotube is constructed by connecting the lattice sites O and A, and B and B. The unit cell of the tube is represented by the rectangle OAB B. Adapted from [9]. These two integers (n, m) determine the diameter d t and the chiral angle θ of the tube. The tube diameter is defined as: d t = L π = C h π = a π n + m + nm. (.8) The direction of the chiral vector is measured by the chiral angle θ, which is defined as the angle between vectors C h and a 1, with values in the range 0 θ 30 0 because of the hexagonal symmetry of the lattice. The chiral angle θ is defined as: cos θ = C h.a 1 C h a 1 = n + m n + m + nm. (.9)

PHYSICAL STRUCTURE OF GRAPHENE AND CARBON NANOTUBES 41 The translational symmetry of carbon nanotubes is characterized by a translational vector T along the tube axis. The translational vector T (vector OB in Fig..3) can be expressed in terms of the lattice vectors a 1 and a as: T = t1 a1 + t a = (t 1, t ), (.10) where t 1 and t are integers. The translational vector T is always perpendicular to C h, that is, T. C h = (t 1 a1 + t a ).(n a 1 + m a )=0. Therefore using Eq. (.) t 1 and t are given as: t 1 = m + n, t = n + m, (.11) d R d R where d R the greatest common divisor (gcd) of (m + n) and (n + m) given by d R = d if (n m) is not a multiple of 3d = 3d if (n m) is a multiple of 3d. (.1) with d, the gcd of n and m. The rectangle OAB B constructed by the vectors C h and T defines the unit cell of a 1D carbon nanotube as shown in Fig..3. The area of the unit cell of D graphite is defined by the vectors a 1 and a. The area of the nanotube unit cell C h T divided by the area of a hexagon a 1 a gives rise to the number of hexagon per unit cell N which is given by N = C h T a 1 a = (m + n + nm) d R. (.13) Since each hexagon conatins two carbon atoms. Therefore, there are N carbon atoms per unit cell of the carbon nanotube. There are three distinct ways in which a graphene sheet can be rolled into a

4 BASICS OF GRAPHENE AND CARBON NANOTUBES tube as shown in Fig..4. The first two of these are called armchair" and zigzag". The terms armchair" and zig-zag" refer to the arrangement of hexagons around the circumference. These two nanotubes are known as achiral (symmorphic) nanotubes defined as carbon nanotubes having identical mirror image [9]. The third class of tube, which in practice is the most common, is known as chiral (non-symmorphic), meaning that it can exist in two mirror-related forms [9]. Armchair nanotube: when n=m, C h =(n,n) and θ=30 0 Zig-zag: when m=0, C h =(n,0) and θ = 0 0 Chiral: C h =(n,m) and θ= 0 0 < θ < 30 0. Electronic structure of graphene and nanotubes In the hexagonal structure of graphene, each of the carbon atom has four valence electrons, where three of them are involved in sp hybridization forming σ orbitals. However, the remaining fourth electron which makes p z bond and occupies a π orbital determines the electronic transport properties of the structure. The high symmetry of graphene makes it straightforward to carry out a tight-binding (TB) calculation for the system. A tight-binding calculation provides a simplifed and efficeint description of the electronic band structure of various crystalline solids and organic molecules [9, 170]. The term tight-binding is used for the atomic-like wavefunction, that are tightly bound to the atoms [170].

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 43 Armchair Zig-zag Chiral Figure.4: Schematic view of armchair, zig-zag, and chiral carbon nanotubes. Taken from [171]...1 Tight-binding method for a crystalline solid In the tight-binding method for a crystalline solid, the eigen functions Ψ j ( k, r) with j = 1,,, n, where n is the number of Bloch wavefunctions, are expressed as Ψ j ( n k, r) = C jj ( k)φ j ( k, r), (.14) j =1 where C jj ( k) are wavefunction coefficients for the function Φ j ( k, r), the Bloch functions of the form Φ j ( k, r) = 1 N N R e i k. R φ j ( r R), (.15)

44 BASICS OF GRAPHENE AND CARBON NANOTUBES where R is the position of the atom and φ j denotes the atomic wavefunction in state j, k is the wave vector, and r is the position vector [9]. The eigen functions Ψ j ( k, r) satisfy Bloch s theorem, i.e. Ψ j ( k, r) can be written in the form Ψ j ( k, r) = exp(ikr)u(r), (.16) where u(r) has the periodicity of the crystal lattice [17, 173]. The energy eigenvalue E j ( k) of the j th state is given as E j ( k) = < Ψ j H Ψ j > < Ψ j Ψ j > = Ψ j HΨ j dr Ψ j Ψ j dr, (.17) where H is the Hamiltonian of the solid. Using Eq. (.14), the eigenvalue equation can be further written as E i ( k) = n j,j =1 C ijc ij < Φ j H Φ j > n j,j =1 C ij C ij < Φ j Φ j > = n j,j =1 H jj ( k)c ijc ij n j,j =1 S jj ( k)c ij C ij, (.18) where H jj ( k) =< Φ j H Φ j > and S jj ( k) =< Φ j Φ j > (j, j = 1,, n), (.19) are the transfer and overlap integral matrices. If the values of the n n H jj ( k) and S jj ( k) matrices are fixed for a given value of k, the coefficient Cij is optimized so as to minimize E i ( k) [9]. Note that the coefficients Cij is also a function of k, and therefore Cij is determined for each k [9]. Partial derivative of E i ( k) with respect to Cij while fixing the other C coefficients, leads to zero for the local minimum condition E i ( k) C ij = N j =1 H jj ( k)c ij N j,j =1 S jj ( k)c ij C ij N j,j =1 H jj ( k)c ijc ij ( N j,j =1 S jj ( k)c ij C ij ) N S jj ( k)c ij = 0. j =1 (.0)

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 45 Multiplying both sides of Eq. (.0) by N j,j =1 S jj ( k)c ijc ij (.18), one obtains N H jj ( k)c ij = E i ( k) j =1 Now defining a column vector of the form and using Eq. N S jj ( k)c ij. (.1) j =1 C i = C i1. C in, (.) Eq. (.1) can be rewitten as HC i = E i ( k)sc i. (.3) This equation can be further expressed as [H E i ( k)s]c i = 0, leading to the secular equation det[h ES] = 0, (.4) provided that the inverse of the matrix [H E i ( k)s] does not exist. The solution of the secular equation gives rise to all n eigevalues of E i ( k) for a given k. Using Eqs. (.17) and (.1 ) for E i ( k), the coefficients C i as a function of k can be determined [9]. A more detailed tight-binding calculation can be found in references [8, 9, 170, 17, 173, 174, 175]... Electronic band structure of graphene Within the tight-binding method, which includes one p z orbital per carbon atom and the nearest-neighbor interaction, one can obtain the D energy dispersion

46 BASICS OF GRAPHENE AND CARBON NANOTUBES relations of graphene by solving the secular equation, Eq.(.4), for a Hamiltonian H associated with two carbon atoms in the unit cell. The Bloch functions for graphene can be constructed from the atomic orbitals for the two carbon atoms at sites A and B in the unit cell using Eq. (.15) as follows Φ j ( k, r) = 1 N N R A(B) e i k. RA(B) φ j ( r R A(B) ). (.5) Only nearest-neighbor interactions are considered in the Bloch phase factor. Since within the same A or B sublattice the nearest-neighbor interactions are absent, the diagonal elements H AA and H BB of the transfer integral matrix defined as < Φ A H Φ A > and < Φ B H Φ B > from Eq. (.19) can be parameterized as a single value ϵ p by substituting Eq. (.5) into Eq. (.19) [9]. Whereas, the offdiagonal matrix elements H AB and H BA can be obtained by adding up hopping contributions from the three nearest-neighbor B atoms relative to an A atom [9, 176]. These three nearest-neighbor B atoms are denoted by the vectors R 1, R, and R 3 in Fig..5. Hence, substituting Eq. (.5) into Eq. (.19) the off-diagonal matrix element H AB is obtained as: H AB =< Φ A H Φ B >= γ 0 (e i k. R 1 + e i k. R + e i k. R 3 ) = γ 0 f(k), (.6) where γ 0 =< φ A H φ B > is the nearest-neighbor C-C tight-binding transfer integral with positive value and φ A and φ B are p z orbitals centered on atoms A and B, and H is the Hamiltonian of the system [9, 176]. The function f(k) is the sum of the phase factors of e ik.r j and can be explicitly calculated using Fig..5, and is

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 47 y B R x a R 1 B A R 3 B Figure.5: The hexagonal lattice of graphene, where the carbon atom A (blue) is surrounded by three nearest-neighbor B atoms (green), each placed at positions R 1, R and R 3, respectively. given as: f(k) = e i k. R 1 + e i k. R + e i k. R 3 a i( = e kx a 3 ky) a i( + e kx+ a 3 ky) + e i a k x 3 ( ky a f(k) = e i a 3 k x + e i a 3 kx cos ). (.7) The off-diagonal matrix element H BA is the complex conjugate of H AB (H BA = H AB ) and hence H BA is given as γ 0 f(k). Similarly, the matrix elements of the overlap integral matrix can be defined. The diagonal matrix elements S AA and S BB are 1 as the overlap integral of a wavefunction is 1 [9, 176]. The off-diagonal element S AB is given as: S AB =< Φ A Φ B >= sf(k), (.8) where s =< φ A φ B > is the overlap integral and the matrix element S BA is given as sf(k).

48 BASICS OF GRAPHENE AND CARBON NANOTUBES Hence the transfer integral matrix H and overlap integral matrix S are given as: H = S = ϵ p γ 0 f(k) γ 0 f(k) ϵ p 1 sf(k) sf(k) 1, (.9). (.30) Solution to the secular equation det[h - ES]=0 using H and S leads to the energy dispersion relation of graphene E g,d ( k) = ϵ p ± γ 0 f(k), (.31) 1 ± s f(k) where ( ) ( ) f(k) = 3kx a ky a 1 + 4cos cos ( ) ky a + 4cos. (.3) The + and - signs in Eq. (.31) correspond to the conduction and valence bands, respectively. It should be noted that the on-site energy parameter ϵ p is an arbitrary reference energy point for the Fermi level. In the orthogonal tight-binding model, it is set equal to zero because in the unit cell there are two carbon atoms and the onsite energy is the same for both A and B sites. If there were two different atoms at sites A and B (e.g., the case of B and N), then the on-site energy would be different and a gap would appear. The overlap integral s is also set equal to zero. As a result, the valence and conduction bands become symmetrical which can be seen from Eq. (.31) [9, 170]. Therefore, the energy dispersion relation for a D

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 49 graphene sheet reduces to a simpler form given as ( ) ( ) E g,d (k x, k y ) = ±γ 0 f(k) = ±γ 0 3kx a ky a 1 + 4cos cos ( ) ky a + 4cos. (.33) Figure.6: The energy dispersion relations for D graphene with γ 0 =.5 ev. The Fermi level is located at the six corner points (two are marked K and K ) where the valence and conduction bands touch. Adapted from [177]. Fig..6 depicts the energy dispersion relations calculated from Eq. (.33). The mesh plot represents the whole region of the Brillouin zone. The two resulting upper half and lower half bands of the energy dispersion plot correspond to the conduction (π -energy anti-bonding) and valence (π-energy bonding) bands, and are the consequence of two carbon atoms per unit cell. The conduction band touches the valence band at the corners K and K of the first Brillouin zone, also known as Dirac points. The touching points are called Fermi points denoted as k F and k F through which the Fermi level passes. There are six Fermi points which

50 BASICS OF GRAPHENE AND CARBON NANOTUBES can be derived by setting the phase factor f(k) = 0, and given as K(K ) = k F (k F ) = ±( b 1 b )/3, ±( b 1 + b )/3, and ± ( b 1 + b )/3. (.34) At the Fermi level, the density of the electronic states (DOS) becomes zero, therefore graphene is a zero-gap semiconductor. This zero gap at the K points exists because of the symmetry requirement of two equivalent carbon sites A and B in the unit cell of hexagonal lattice as discussed above. Now, the graphene energy dispersion relations will be used to derive the energy dispersion relations for carbon nanotubes...3 Electronic band structure of carbon nanotubes The electronic structure of CNTs is that of graphene modified by the tube structure and reduced dimensionality. Carbon nanotubes possess excellent electronic properties because of their nanometer dimension and unique electronic structures. Although graphene is a zero-gap semiconductor, both experimental and theoretical studies have shown SWCNTs can be either metallic or semiconducting depending on the diameter and chirality (degree of twist) of the tube [174, 175, 178, 179, 180] and various types of SWCNTs can be formed on the basis of chirality as discussed in section.1. The dependence of electronic properties of CNTs on their structure is due to the unique electronic band structure of graphene as discussed in subsection... The electronic band structure of CNTs can be derived from the graphene electronic band structure. When making a tube by rolling a graphene sheet, periodic boundary conditions are imposed in the circumferential ( C h ) direction. As a result, the wave vector in the C h direction becomes quantized but remains a continuous wave vector along the tube axis (T ). Hence, only a certain set of k (energy)

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 51 states of the graphene sheet is allowed which depend on the diameter and chirality of the tube. These allowed k states of the tube are the lines mapped onto the graphene Brillouin zone. The electronic structure of the carbon nanotube can be understood in detail in the following manner: Carbon nanotubes in reciprocal space: consider the graphene reciprocal lattice as constructed in Fig..(b), let the reciprocal lattice vector corresponding to the chiral vector C h be K 1 and the reciprocal lattice vector corresponding to the translational vector T be K. In carbon nanotubes, C h and T play the role of graphene real space unit vectors a 1 and a, whereas K 1 and K play the role of graphene reciprocal space unit vectos b 1 and b [170]. Since K 1 is along the circumferential direction C h, hence it is perpendicular to the tube axis T and parallel to C h. The vector K is aligned with the tube axis T and is thus parallel to T and perpendicular to C h. Expressions for K 1 and K can be derived using the relation R i.k j = πδ ij, where R i and K j are the lattice vectors in real and reciprocal space, respectively. Hence K 1 and K satisfy the relations, C h.k 1 = π, T.K 1 = 0, C h.k = 0, T.K = π. (.35) Using Eqs. (.6), (.10), (.11), and (.13) and the above definition, K 1 and K are given by K 1 = 1 N (t 1b t b 1 ), K = 1 N (mb 1 nb ). (.36) The length of K 1 and K are defined as K 1 = π C h = d t and K = π T. (.37) From the definition of K 1, two vectors differ by NK 1 are equivalent, as this corresponds to a reciprocal lattice vector of graphene. Therefore, N wavevectors of type

5 BASICS OF GRAPHENE AND CARBON NANOTUBES µk 1, where µ = 0, 1,, N 1, lead to N discrete lines or k vectors of length K and spacing K 1, in the graphene Brillouin zone [9, 170]. There are N onedimensional energy bands corresponding to N discrete values of the k vectors [9]. Hence, the nanotube reciprocal space can be viewed as the result of sectioning the graphene Brillouin zone into a set of N one-dimensional Brillouin zone or cutting lines, such that the possible k values in the carbon nanotube Brillouin zone are given as [170] K k = µk 1 + k z, where (µ = 0, 1,, N 1) and π K T < k z < π T. (.38) For any (n, m) nanotube k follows the relation k.c h = πq, (q = 1,, n), (.39) which is the periodic boundary condition imposed in the circumferential direction C h with q an integer. Equation (.38) shows that the values of µ are discrete in K 1 direction, while because of the translational periodic boundary condition along the tube axis, the wave vectors k z along the cutting lines change continuously [9, 170]. The allowed states in a nanotube thus correspond to parallel lines in the graphene Brillouin zone. With the help of Eq. (.38) the elctronic band structutre of carbon nanotubes can be calculated by considering the dispersion only along these cutting lines, which is known as the zone-folding method, that is, sectioning the dispersion relation of graphene E D,g (k x, k y ), Eq. (.33), along the cutting lines [9, 170]. Hence, for any (n, m) carbon nanotubes the dispersion relations are given as ( K ) E(µ, k z ) = E D,g µk 1 + k z. (.40) K

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 53..3.1 Energy dispersion relations for armchair and zig-zag nanotubes In the case of the (n, n) armchair SWCNT, the periodic boundary condition is applied on the wave vector k x in the circumferential direction, Fig..7, which leads to k x,q.c h = πq k x,q.a n + n + n = πq k x,q a = πq n, (q = 1,, n). (.41) 3 Discreteness of the k x,q values leads to a one-dimensional dispersion relation of the armchair nanotube by substituting Eq. (.41) into Eq. (.33) giving ( E (armchair) qπ q,1d (k) = ±γ 0 1 + 4cos n ( π < ka < π), ) cos ( ) ( ) ka ka + 4cos, (.4) (q = 1,, n). Here k is a one dimensional wave vector in the direction of the vector K. In a similar manner, the energy bands for the (n, 0) zig-zag nanotube can be derived using Eq. (.33) by applying the periodic boundary condition on k y Fig..7, as follows: k y,q a = πq, (q = 1,, n), (.43) n which gives rise to the 1D dispersion relations for the zig-zag nanotube ( ) q,1d (k) = ±γ 0 3ka ( qπ ) ( qπ ) 1 + 4cos cos + 4cos n, (.44) n E (zig zag) ( π 3 < ka < π 3 ), (q = 1,, n).

54 BASICS OF GRAPHENE AND CARBON NANOTUBES (a) (b) Armchair Zig-zag Figure.7: The periodic boundary condition only allows quantized wave vectors around the circumferential direction, which leads to one dimensional band structure of carbon nanotubes (a) Armchair and (b) Zig-zag. Taken from [181]. Figure.7 represents the schematic of the periodic boundary condition applied in the circumferential direction which gives rise to one dimensional band structure of (a) armchair and (b) zig-zag nanotube. As shown in Fig..8 (a), if for a particular nanotube of chirality (n, m) the cutting line passes through a K point in the D Brillouin zone of graphene, then the one-dimensional energy bands of the nanotube have no bandgap, hence the tube is metallic. Later, it will be shown that the density of states near the Fermi point has a finite value for these carbon nanotubes. If the cutting lines do not pass through a K point, then the nanotube is semiconducting with a finite bandgap, Fig..8 (b), where the density of states is zero [8, 9]. Figure.9 represents the one-dimensional dispersion relations for (a) armchair (5, 5), (b) zig-zag (9, 0), and (c) zig-zag (10,0) SWCNTs. For armchair nanotubes, Fig..9(a), the band structure is gapless indicating the tube is metallic. The va-

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 55 K K K 1 K 1 Figure.8: The one-dimensional wave vectors k are shown in the two dimensional Brillouin zone of graphene as bold lines for (a) metallic and (b) semiconducting carbon nanotubes. In the direction of K 1, discrete k values are obtained by periodic boundary conditions for the circumferential direction of the carbon nanotubes, while in the direction of the K vector, continuous k vectors are shown in the one dimensional Brillouin zone. (a) For metallic nanotubes, the bold line intersects a K point at the Fermi energy of graphite. (b) For the semiconductor nanotubes, the K point always appears one-third of the distance between two bold lines. Here, only a few of all the possible bold lines are shown near the indicated K point. Adapted from [8].

56 BASICS OF GRAPHENE AND CARBON NANOTUBES lence and conduction bands cross at K and K points where k = ±π/3a. The bandstructure consists of six dispersion relations for the conduction band and the same number in the valence band. In each valence and conduction band, two bands are nondegenerate bands as shown in red color, and the four bands are doubly degenerate, giving rise to 10 levels in each case [18]. For zig-zag nanotube, the band structure does not show a gap for (9,0) nanotube in Fig..9 (b), but the bands cross at k = 0, showing the metallic behavior of the tube, whereas the (10,0) nanotube has a gap, indicating the tube is semiconducting. In general, the nanotube with chirality (n, m) is metallic if n m is a multiple of 3 and it is semiconducting otherwise. Note, the curvature effect of the tube is neglected, if it is taken into account then the condition that determines whether the tube is metallic or semiconducting would not be valid [183] as the (9,0) nanotube does show a tiny gap in the dispersion spectrum due to the curvature effect [18]. Condition for metallic versus semiconducting nanotubes. To obtain the condition for a metallic or semiconducting nanotube the expression for the chiral vector Eq.(.6) and Fermi points Eq.(.34) are substituted into the boundary condition Eq.(.39), and using the relation b i.a j = πδ ij one can get the condition: k F.C h = 1 3 ( b 1 b ).(n a 1 + m a ) = πq π(n m)/3 = πq (n m)/3 = q. (.45) If n m is a multiple of 3 then the allowed k states or lines intersect the K and K points and the nanotube is metallic. The tube is semiconducting with a bandgap if n m is not a multiple of 3 and k states miss the K and K points. Therefore, if the curvature effects are excluded all armchair nanotubes (n, n) are always metallic, and zig-zag nanotubes (n, 0) are metallic if n is a multiple of 3

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 57 (a) (b) E/ 0 E/ 0 k ( c) k E/ 0 k Figure.9: One-dimensional dispersion relations for different CNTs (a) An armchair (5,5) nanotube exhibiting metallic behavior, (b) A zig-zag (9,0), and (c) zig-zag (10,0) nanotube exhibiting semiconducting and metallic characters. Adapted from [18].

58 BASICS OF GRAPHENE AND CARBON NANOTUBES and semiconducting otherwise. As a consequence, 1/3 of SWCNT are metallic and /3 semiconducting [175] as shown in chirality map of Fig..10 for different types of CNTs, where the metallic nanotubes are denoted by blue hexagons and semiconducting tubes by yellow hexagons. Experimental studies on SWCNTs by Odom et al. [179] and Wildoer et al. [180] using scanning tunneling microscopy (STM) confirmed these theoretical predictions. Zig-zag; (n,0); chiral angle =0 Semiconducting nanotubes Armchair; (n=m); chiral angle =30 Metallic nanotubes Figure.10: Chirality map displaying the different types of SWCNTs that can be formed by rolling a graphene sheet. The carbon nanotubes (n,m) denoted by blue color are metallic and yellow color are. Adapted from [184]...4 Electronic band structure near the Fermi points (K = k F ) For carbon nanotubes, a coordinate system for the reciprocal vectors based on the circumferential vector C h and translational vector T is chosen [185]. In order to study the band structure of carbon nanotube near the Fermi point the origin of the new coordinate system is reset to the Fermi point, giving the new wave vector

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 59 as k = k k F = k 1 C h + k T, (.46) where k is the wave vector along the tube axis (T ) and k 1 is the wave vector along the circumferential direction ( C h ), which is quantized due to the periodic boundary condition. k 1 is defined as the projection of k to the unity vector in the circumferential direction [185]. Using C h = πd, b i.a j = πδ ij, Eqs. (.34) and (.39) k 1 is given as: k 1,q = ( k C k F ). h C h = k. C h k F. C h C h = d [ q n m 3 ]. (.47) To derive the band structure near the Fermi point, the expression for the E( k) relation of graphene, Eq. (.33) is simplified by using Taylor series expansion for the cosine function near the Fermi point [185, 186]. This results in E( k) = ± 3a ccγ 0 ( k x k F x ) + ( k y k F y ) = ± 3a ccγ 0 k k F. (.48) E( k) is linear near a Fermi point and hence this is called the linear band approximation. A derivation of this equation can be found in the Appendix A. Equation (.48) can be rewritten using the coordinate system for the nanotube, Eq. (.46), as: E( k) = ± 3a ccγ 0 k 1,q + k. (.49) The minimum value of k 1,q gives rise to the lowest band of the nanotube. Next, the metallic and semiconducting tubes will be discussed explicitly and Eq. (.49) will be used to derive the band gap and density of states of the nanotubes. Metallic carbon nanotubes- The condition for a nanotube to be metallic states

60 BASICS OF GRAPHENE AND CARBON NANOTUBES (n m)/3 = q. In this case, k 1,q = 0 in Eq. (.47). Hence the E( k) relation becomes: where v F is the Fermi velocity defined as E( k ) = ± 3a ccγ 0 k = v F k, (.50) v F = 1 E(k) k = 1 3a cc γ 0. (.51) This is a one-dimensional linear dispersion relation for the lowest band independent of (n, m). Semiconducting carbon nanotubes- In this case, (n m)/3 q and the minimum value of k 1,q = /3d t is obtained from Eq. (.47), leading to E( k) relation E( k ) = ± 3a ccγ 0 k + (/3d t ), (.5) which is a one-dimensional dispersion relation independent of (n, m) but depends on the tube diameter d t. It is a parabola with a direct band gap expressed as: E gap = a ccγ 0 d t. (.53) Hence, the larger the tube diameter, smaller the bandgap [9]...5 Density of states of carbon nanotubes The density of states of the carbon nanotube determines its electrical properties. In general, the density of states, ρ(e), is defined as the derivative of the available states with respect to energy and expressed as [186]: ρ(e) = k E = g l i dkδ(k k i ) E 1, (.54) k

ELECTRONIC STRUCTURE OF GRAPHENE AND NANOTUBES 61 where, g is the degeneracy factor and g = 4 for graphene because of spin and valley degeneracies, k i are the roots of the equation E E(k i ) = 0, l = (4π/ 3) C h /a is the length of the 1D Brillouin zone, which corresponds to a normalization over the graphene sheet unit cell and is calculated as the ratio of the total area of the Brillouin zone to the interline spacing π/ C h, and k is the total number of electron states per unit cell below a given energy E [186]. Using Eq.(.49) the inverse of the derivative of the states with respect to energy is given by E 1 = k 1q + k 3a cc γ 0 k k. (.55) Now using Eqs. (.47) and (.49) the above equation can be rewritten as E 1 = 3a cc γ 0 k E E E1q, (.56) where and r is the radius of the nanotube. E 1q = a ccγ 0 (3q (n m)), (.57) r Substituting Eq. (.56) into Eq. (.54) the expression for the DOS becomes ρ(e) = 4 l q= E, (.58) 3a cc γ 0 E E1q after putting the value of l, ρ(e) becomes ρ(e) = 3acc π γ 0 r g(e, E 1q ), (.59) q=

6 BASICS OF GRAPHENE AND CARBON NANOTUBES where g(e, E 1q ) = E / E E1q, E > E 1q ; 0, E < E 1q. (.60) For E 1q 0 and E = E 1q, g(e, E 1q ) shows a divergent van Hove singularity, while for E 1q = 0, g(e, 0) is constant and equal to 1, this refers to the case of metallic nanotubes [186]. In the case of semiconducting carbon nanotubes, the density of states is zero inside the bandgap. The density of states contributed by the lowest band of a semiconducting nanotube can be expressed in a more compact form as ρ(e) = 3acc π γ 0 r q= E E (E gap /) Θ( E E gap/), (.61) where E gap / = E 1q and the step fuction Θ( E E gap /) = 1 if E > (E gap /) and 0 otherwise [185]. Figure.11 shows the electronic density of states for (a) a semiconducting (5,0) carbon nanotube and (b) a metallic (5,5) carbon nanotube..3 Summary In this chapter, the physical and electronic structure of carbon nanotubes was derived from graphene. The unique structure of graphene and carbon nanotubes make them wonder materials. Moreover, carbon nanotubes possess various crystal structures depending on the chirality, which results in a variety of nanotube types. Using the tight-binding method the electronic band structure of graphene was discussed and then used to explain the geometry-dependent electronic band structure of carbon nanotubes. The energy dispersion relations for armchair and

SUMMARY 63 (a) (b) Figure.11: Density of states for (a) a (5,0) nanotube which shows semiconducting behavior and (b) a (5,5) nanotube which shows metallic behavior. Adapted form [170]. zig-zag nanotubes were explicitly derived and discussed for some special chirality. The predictions on the metallic or semiconducting behavior of carbon nanotubes using (n, m) indices were also derived and discussed. The linear dispersion relation of the CNT in the vicinity of the K points was also derived and used to calculate and understand the energy gap and the density of electronic states of carbon nanotubes.