Calculating Electronic Structure of Different Carbon Nanotubes and its Affect on Band Gap

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1 Calculating Electronic Structure of Different Carbon Nanotubes and its Affect on Band Gap 1 Rashid Nizam, 2 S. Mahdi A. Rizvi, 3 Ameer Azam 1 Centre of Excellence in Material Science, Applied Physics AMU, Aligarh 2 Department of Mechanical Engineering, Aligarh Muslim University, Aligarh 3 Centre of Nanotechnology, King Abdulaziz, KSA ABSTRACT Here each carbon nanotube quantumly calculated using tight binding approximation. Each carbon nanotube is described as a graphene sheet rolled into a cylindrical shape so that the structure is one dimensional with axial symmetry. This simple picture of each carbon nanotube π electron based on tight binding approximation helps to study the band-gap variation in different carbon nanotubes. Besides, the density of states and energy verse axial wave vector of sixteen carbon nanotubes has been calculated. The calculated band gaps decrease from 6 ev to 0 ev as the diameter of zigzag carbon nanotube increase from 0.1 to 0.7 nm. But in armchair carbon nanotube the band gap remain ev irrespective of the diametric size. It is noticeable that the density of states becomes zero at 0 ev in case of zigzag carbon nanotube while in armchair carbon nanotube the density of states do not becomes zero at 0 ev. Keywords: tight binding approximation, zigzag carbon nanotubes,armchair carbon nanotube, density of states 1. INTRODUCTION Single-Walled Carbon Nanotubes (SWNTs) are members of the carbon family of fullerenes that was discovered almost two decades ago [7]. From the day of carbon discovery and their derivative have been considerable attention given to these macro molecules. This is because of their unique structure and nanometer sizes which are promising for nano electronic and nano mechanical applications. Their remarkable mechanical properties (e.g., exceptional strength) and electronic properties (e.g., ability to sustain large current densities) also boosted interest in fundamental research. However, even though the interest and effort of the scientific community in the research of carbon nanotubes are considerable, some of the basic properties such as optical, magnetic, and magneto-optical properties are still not well understood. determines the physical structure of a carbon nanotube. The vector C H can be written in terms of the unit a a2 vectors of the hexagonal lattice, 1 and, such that 2. MODEL DETAIL A single-wall carbon nanotube can be observed as a single sheet of graphite rolled up into a cylinder. Figure 1 represents the 2D hexagonal plane that makes up a graphitic sheet, where the carbon atoms lie at the corners of each hexagon. In this figure, if point O is connected to point A, and point B is connected to point B, then one will observe that the sheet will be rolled into a cylindrical structure. But, this is just only one of the possible cylindrical orientations can be constructed. For example, points A and B could lay directly to the right of points O and B, respectively, which would result in a dissimilar orientation of the hexagonal rings on the face of the cylinder.[2] C H The vector is perpendicular to the tube axis z, which is known as the chiral vector, which uniquely Figure 1 shows the unrolled hexagonal lattice of a carbon nanotube C H na1ma2. To completely describe the complete geometry of a carbon nanotube only the integer pair (n, m) is used. It is only necessary to consider n and m such that 0 m n because of the rotational symmetry of the 2D hexagonal lattice. In the first case, when m = n, then the angle θ will be 30.So one moves along the chiral vector, the carbon bonds form an armchair shaped pattern. In the second case, when m = 0, θ = 0 and the bonds along the chiral vector form a zigzag pattern and carbon nanotubes of the form (n, 0) are known as zigzag nanotubes. In the third case, 0 < θ < 30 and the tubes are known as chiral 153

armchair patterns at the end rings of the

represents all three nanotubes are single wall

focus of study in this research is on single

x 2 Figure 2 shows carbon nanotube of dimension

of dimension 4 x 4 Figure 4 shows carbon

carbon nanotube of dimension 5 x 0 Figure 13

6 shows carbon nanotube of dimension 6 x 0

x 6 Figure 7 shows carbon nanotube of dimension

dimension 7 x 7 Figure 8 shows carbon nanotube

2 nanotubes. It is interesting to note that the zigzag and armchair patterns at the end rings of the zigzag and arm chair carbon nanotubes. In figure 1 represents all three nanotubes are single wall carbon nanotubes. There also exist multi-wall carbon nanotubes, which consist of two or more concentric single-wall carbon nanotubes but the focus of study in this research is on single wall carbon nanotubes fullerene and its derivatives. Figure 10 shows carbon nanotube of dimension 2 x 2 Figure 2 shows carbon nanotube of dimension 2 x 0 Figure 11 of carbon nanotube of dimension 3 x 3 Figure 3 shows carbon nanotube of dimension 3 x 0 Figure 12 shows carbon nanotube of dimension 4 x 4 Figure 4 shows carbon nanotube of dimension 4 x 0 Figure 5 shows carbon nanotube of dimension 5 x 0 Figure 13 shows carbon nanotube of dimension 5 x 5 Figure 6 shows carbon nanotube of dimension 6 x 0 Figure 14 shows carbon nanotube of dimension 6 x 6 Figure 7 shows carbon nanotube of dimension 7 x 0 Figure 15 shows carbon nanotube of dimension 7 x 7 Figure 8 shows carbon nanotube of dimension 8 x 0 Figure 16 shows carbon nanotube of dimension 8 x 8 Figure 9 shows carbon nanotube of dimension 9 x 0 Figure 17 shows carbon nanotube of dimension 9 x 9 154

3 3. SIMULATION DETAIL The most common and simplest method used for determining the electronic structure of SWNTs is the tight binding (TB) model. This is a standard procedure in solidstate physics that works best for the case of weak overlap of atomic wave functions in an insulating crystal. The TB approximation gives a convenient method for understanding the basic characteristics of band structure in single wall carbon nanotube. The electronic structure of a carbon nanotube (CNT) can be acquired from that of graphene. The wave vector related with the chiral vector C h, in the circumferential direction gets quantized. On the other wave-vector associated the direction of the translation vector T along the CNT axis remains continuous for an infinite carbon nanotube. These are the boundary conditions of the carbon nanotube. This sets the energy bands in one-dimensional dispersion relationship cross sections of those of graphene. The reciprocal lattice vectors along the carbon nanotube axis C K T K. h 2 0,. 2 2 direction, ( given by 1 K t b t b N C K T K. h 2 0,. 2 2, and K 1 in the circumferential ) expression are K mb nb N The one-dimensional energy dispersion relations of a carbon nanotube can be written as K 2 ECNT k Eg 2D k K1 K 2 / T k / T (1) is a one-dimensional wave-vector 1,..., N. along the carbon nanotube axis and The periodic boundary condition for a carbon nanotube provides N discrete k values in the circumferential direction. These N pairs of energy dispersion curves correspond to the cross sections of the two-dimensional energy dispersion surface of graphene which is given by (1). Several lines cut near one of the K points are shown in Fig. 2. The separations between two adjacent lines with the length of the cutting lines are given by K d K1 2 / T 1 2/ CNT and, respectively. The carbon nanotube gets a zero energy gap if the cutting line passes through a K point of the two-dimensional brillouin zone (Fig. 2), where the π and π* energy bands of graphene are degenerate by symmetry. If the K-point is situated between to cutting lines, such that K always located in a position one-third of the distance between two adjacent K 1 lines Fig. 3 then a semiconducting carbon nanotube with a finite energy gap is formed. Thus for a (n, m) carbon nanotube, n-m is exactly divisible by 3 then the carbon nanotube is metallic where as carbon nanotubes with residuals 1 and 2 of the division n-m by 3 are semiconducting. Figure 18 Figure 19 The one-dimensional wave-vectors K are shown in the BRILLOUIN zone of graphene as bold lines for shows (a) metallic and (b) semiconducting CNTs 4. RESULT AND DISCUSSION In general (n, n) armchair carbon nanotubes yield 4n energy subbands by means of 2n conduction and 2n valence bands. Out of these 2n bands, two are nondegenerate and n-1 is doubly degenerate. The degeneracy comes from the two subbands with the same energy dispersion, but different ν - values. In all zigzag carbon nanotubes have the lowest conduction and the highest valence bands are doubly degenerate where as all armchair carbon nanotubes have band degeneracy between the highest valence and the lowest conduction band. Both armchair as well as zigzag carbon nanotubes bands are symmetric with respect to k=0. An armchair carbon nanotubes bands have two valleys at around k = ±2π/3a points and the zigzag and chiral carbon nanotubes can have at most one valley. The bands in armchair cross the Fermi level at k = ±2π/3a thus they are considered to exhibit metallic nanotube [2]. There is no energy gap for few carbon nanotubes. It is valuable to build up an approximate relation that describes the dispersion relations in the regions around the Fermi energy E F = 0 as electrical conduction is decided by states around the Fermi energy. The expression around the point (0, ±4π/3a) where the energy gap is zero and 0 f k 3 x /2 1 2 cos( / 2 ka y f k t e k a is given by. It is straightforward to show that 3 / 2 x y f k i at k i y ky 4 / 3a, with. The matching up energy dispersion relation can be written as [3] g2d E k f k 3at x y The energy bands for (n, 0) zigzag carbon nanotubes can be achieved by enforcing the periodic boundary conditions. This define the quantity of allowed wave-vectors k y in the circumferential direction as nk y a=2νπ, (ν =1 2n).This gives the one-dimensional k 155

4 dispersion relations for the 4n states of the (n, 0) zigzag carbon nanotubes / 3 a k / 3a x 3at E kx kx 1 2 3a 2n So, the energy gap for subband ν can be written as difference between the energies of the + and -branches at k x =0 as shown in figure 20 (a) & (b) - 27(a) & (b). (2) 2 E g 2 2n 3at na 3 (3) Figure 20 (b) shows Energy verse Axial Wave Vector of dimension 2 x 2 The energy gap has a minimum value of zero corresponding to ν = 2n/3. If n is not a multiple of three then the minimum value of ν - 2n/3is equal to 1/3. This suggests that the minimum energy gap is then given by E g 3at 2 3 na 2a C C d CNT t 0.8eVnm d CNT where d CNT =na/π is the diameter of the carbon nanotubes in nano-meters. The DOS for semiconducting zigzag carbon nanotubes based on (2) and (3) is given by ge ( ) 8 E 3 ac Ct E E 2 which is an approximately valid only if (E-E F ) << t [4]. g Figure 21 (a) shows Energy verse Axial Wave Vector of dimension 3 x 0 Figure 20 (a) shows Energy verse Axial Wave Vector of dimension 2 x 0 Figure 21 (b) shows Energy verse Axial Wave Vector of dimension 3 x 3 156

5 Figure 22 (a) shows Energy verse Axial Wave Vector of dimension 4 x 0 Figure 23 (b) shows Energy verse Axial Wave Vector of dimension 5 x 5 Figure 22 (b) shows Energy verse Axial Wave Vector of dimension 4 x 4 Figure 24 (a) shows Energy verse Axial Wave Vector of dimension 6 x 0 Figure 23 (a) shows Energy verse Axial Wave Vector of dimension 5 x 0 Figure 24 (b) shows Energy verse Axial Wave Vector of dimension 6 x 6 157

Figure 25 (a) shows Energy verse Axial Wave Vector of dimension 7 x 0

Figure 25 (b) shows Energy verse Axial Wave Vector of dimension 7 x 7

Figure 26 (a) shows Energy verse Axial Wave Vector of dimension 8 x 0

6 Figure 25 (a) shows Energy verse Axial Wave Vector of dimension 7 x 0 Figure 26 (b) shows Energy verse Axial Wave Vector of dimension 8 x 8 Figure 25 (b) shows Energy verse Axial Wave Vector of dimension 7 x 7 Figure 27 (a) shows Energy verse Axial Wave Vector of dimension 9 x 0 Figure 26 (a) shows Energy verse Axial Wave Vector of dimension 8 x 0 Figure 27 (b) shows Energy verse Axial Wave Vector of dimension 9 x 9 158

7 Figure 28 (a) shows density of state of dimension 2 x 0 Figure 29 (b) shows density of state of dimension 3 x 3 Figure 30 (a) shows density of state of dimension 4 x 0 Figure 28 (b) shows density of state of dimension 2 x 2 Figure 30 (b) shows density of state of dimension 4 x 4 Figure 29 (a) shows density of state of dimension 3 x 0 159

8 Figure 31 (a) shows density of state of dimension 5 x 0 Figure 32 (b) shows density of state of dimension 6 x 6 Figure 31 (b) shows density of state of dimension 5 x 5 Figure 33 (a) shows density of state of dimension 7 x 0 Figure 32 (a) shows density of state of dimension 6 x 0 Figure 33 (b) shows density of state of dimension 7 x 7 160

9 Figure 34 (a) shows density of state of dimension 8 x 0 Figure 35 (b) shows density of state of dimension 9 x 9 DOS showing at both the energy minima and maxima of the bands as shown in Fig.28 (a) & (b) - 35 (a) & (b)) are significant for determining various solid-state properties of carbon nanotubes [5, 6] For all metallic carbon nanotubes, independent of their diameter as well as chirality, because of the nearly linear dispersion relations around the Fermi energy the density of states (DOS) per unit length along the carbon nanotube axis is a constant given by 8/3πa C-C t [2]. Figure 34 (b) shows density of state of dimension 8 x 8 Figure 24 shows calculated band gaps in perfect zigzag of different diameter carbon nanotubes. Figure 35 (a) shows density of state of dimension 9 x 0 Figure 25 shows calculated band gaps in perfect armchair of different diameter carbon nanotubes 161

10 5. CONCLUSION It is observed that the electronic band structure calculations predict that the (n, m) indices determine the metallic or semiconducting behaviour of SWNTs. It clear that the result shows zigzag (n, 0) SWNTs should have two distinct types of behaviour the tubes will be metals when n/3 is an integer, and otherwise semiconductors. As C H rotates away from (n, 0), chiral (n, m) SWNTs are possible with electronic properties similar to the zigzag tubes; that is, when (2n + m)/3 is an integer the tubes are metallic, and otherwise semiconducting. The gaps of the semiconducting (n, 0) tubes should depend inversely on diameter and (n, n) tubes band gap is remain independent of diameter. It is interested to note that the energy dependence of the nanotube density of states as where the density of states for metallic and semiconducting zigzag nanotubes are compared. It is observed that the density of states near the Fermi level E F located at E = 0 are zero in pure semiconducting nanotubes. This density of states has a value of non-zero for semiconducting nanotubes It clear that the result shows zigzag carbon nanotube (n, 0) should have two distinct types of behaviour the tubes will be metals when n/3 is an integer, and otherwise semiconductors and it is non-zero even small value than the metallic carbon nanotube which greater DOSs. The relationship between the band gap and diameter have been also calculated by C. T. White et al [8, 9] till 0.5 nm which is further has been extended to 0.2 nm diameter in the present paper. REFERENCES [1] H. W. Kroto, J. R. Heath, S. C. OBrien, R. F. Curl and R. E. Smalley, Nature 318, 167 (1985). [2] R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon Nanotubes. London: Imperial College Press, [3] S. Datta, Quantum Transport: From Atoms to Transistors. Cambridge, New York: Cambridge University Press, [4] J. W. Mintmire and C. White, Universal Density of States for Carbon Nanotubes, Phys.Rev.Lett., vol. 81, no. 12, pp , [5] Kasuya, Y. Sasaki, Y. Saito, K. Tohji, and Y. Nishina, "Evidence for Size-Dependent Discrete Dispersions in Single-Wall Nanotubes," Phys.Rev.Lett., vol. 78, no. 23, pp , 1997 [6] S. Kazaoui, N. Minami, R. Jacquemin, and H. Kataura, "Amphoteric Doping of Single-Wall Carbon-Nanotube Thin Films as Probed by Optical Absorption Spectroscopy," Phys.Rev.B, vol. 60, no. 19, pp , [7] S. Iijima, Helical microtubules of graphitic carbon, Nature 354, 56 (1991). [8] C. T. White, D. H. Robertson and J. W. Mintmire 1993 Phys. Rev. B [9] C. T. White and J. W. Mintmire 2005 J. Phys. Chem. B

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