ELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES D. RACOLTA, C. ANDRONACHE, D. TODORAN, R. TODORAN Technical University of Cluj Napoca, North University Center of Baia Mare, Str. Victoriei 76 4301, Baia Mare, Romania E-mails: daniaracolta@yahoo.com, androtin03@yahoo.com, todorandaniela05@yahoo.com, todoran_radu@yahoo.com Received August 31, 015 In the present paper we discuss the energy band structure and structural properties of graphene and carbon nanotubes used on determination of its electronic, optical and optoelectronic properties. We calculated the one-dimensional E relation of carbon nanotube, started from quantification the two-dimensional E of the graphene sheet along the circumferential direction of the nanotube, used a tight binding model based on the nearest neighbor interaction which includes one pz orbital per carbon atom. The electronic structure of nanotube can be defined by its diameter and chirality (m, n). For the zigzag nanotube, if (m n) is a multiple of 3, is not gap in the energy spectrum, showing metallic character, if (m n) is not a multiple of 3, the gap is nonzero, the character is semiconducting. For the armchair nanotube (m n = 0), the character is always metallic. The size of the energy gap of nanotubes is inversely proportional to the diameter. This opens the way to controllable manipulations of phase-coherent mesoscopic phenomena. Key words: Graphene, Carbon nanotubes, Electronic structure of nanoscale materials. 1. INTRODUCTION Graphene (the monolayer graphite) is a recently fabricated material consisting of an individual layer of carbon atoms arranged in a two dimensional hexagonal lattice [1 3]. The honeycomb lattice provides (Fig. 1) non-trivial physical phenomena which cannot be observed in the ordinary square lattice [4]. Fig. 1 Band Structure of graphene. Length = 10, width = 10, C C bond length 1.4 A, C C transfer energy 3.013 ev. Rom. Journ. Phys., Vol. 61, Nos. 5 6, P. 99 998, Bucharest, 016
Electronic energy dispersion and structural properties on graphene and carbon nanotubes 993 The energy band structure and structural properties of graphene and carbon nanotubes are used on determination of its electronic, optical and optoelectronic properties and is calculated used a tight binding model based on the nearest neighbor interaction which includes one p z orbital per carbon atom [5]. Development an approximate relation that describes the regions of the E plot around E = 0 is necessary for electrical conduction around the Fermi energy [6]. Electrons in graphene behave as Dirac fermions and mimic the dynamics of hyperrelativistic electrons [7]. The discovery of carbon nanotubes [8] which are basically rolled up sheets of graphite hexagonal networs of carbon atoms forming tubes, opened a new field of research in the physics at nanoscales [9]. Thus carbon nanotubes have become very promising in the field of molecular electronics, in which atoms and molecules are envisaged as the building blocs in the fabrication of electronic devices. The especial electronic and magnetic behavior of nanotubes is defined by the atomic structure. Three types of nanotubes are possible, called armchair, zigzag and chiral nanotubes, depending on how the two-dimensional graphene sheet is rolled up. These structures differ according to their orientations and the directions of the edges. The electronic, magnetic and transport properties of graphene and nanotubes are strongly dependent of their atomic structure edges and opens the way to the controllable onset of phase-coherent mesoscopic phenomena lie quantum interference effects Aharonov-Bohm oscillations and resonant tunneling [10 1].. MODEL AND FORMULATIONS Graphene is also called a honeycomb lattice because carbon atoms are arranged to hexagons. This hexagonal lattice is characterized by lattice vectors 3 a a, 1 a and 3 a a a,, where a 3a0, a 0 = 1.4 Å is the C-C distance. The chiral vector C h ma 1 na, determines the circumference of the carbon nanotube (m, n are integers). The reciprocal lattice vectors are given by b1 1, 3 and 3a b 1, 3. The points K and K situated at the corners of the Brillouin zone 3 a of graphene are named Dirac points and their positions in momentum space are given by [13], K, ', K,. 3a 3 3a 3a 3 3a
994 D. Racolta, C. Andronache, D. Todoran, R. Todoran 3 The general hopping Hamiltonian for electrons in graphene considering that electrons can hop both to nearest and next nearest neighbor atoms is [13]: where, a, i, i, ' a, ib, j H. c. t a, ia, j b, ib, j H. c. H t, (1) i, j, i, j, a are the annihilation (creation) operator at site i, with spin, on sublattice A, (an equivalent definition is used for sublattice B), H.c. stands for the Hermitian conjugation, t denotes the nearest neighbor hopping energy between sites i and j (hopping between different sublattices). The sites i ' and j being located in a plan produce a hexagonal square. t is the next nearest neighbor hopping energy (hopping in the same sublattice). We obtain the operators after diagonalizing the Hamiltonian equation and substituting the annihilation operators and applying the summations we obtain the wavefunctions of graphene [5]. ( ) t exp ixa / 3 1 exp ixa / 3 cos y/. () The energy bands derived by eigenvalues E( ) ( ). (3) The two-dimensional (D) energy dispersion relations for π-bands of graphite, are given by [5], where t is the nearest-neighbor overlap integral [14]. E( ) t 1 4cos ya ya 4cos cos 3 xa (4) The calculation of band structure of graphene using tight binding approximation shows that it has semimetal behaviour [5]. Tight binding model shows, that graphene has full valence band and empty conduction band, while top of the valence bandhas exactly same energy as the bottom of the conduction band. Therefore graphene is called zero-bandgap semiconductor or semimetal, since electronic properties are between metal and semiconductor (Fig. 1). The energy bands of graphene at low energies are described by a D Dirac-lie equation with linear dispersion near K/K in space.
4 Electronic energy dispersion and structural properties on graphene and carbon nanotubes 995 3. BAND STRUCTURE OF NANOTUBES We calculated the one-dimensional E- relation of carbon nanotube, started from quantification the two-dimensional E- of the graphene sheet along the circumferential direction of the nanotube [15]. Eliminating x, or y by using the periodic boundary condition, C h = π l, (5) where l is an integer, we get 1D energy bands for general chiral structures. In other words, 1D energy bands can be obtained by slicing the D energy dispersion relations of Eq. 4 in the directions expressed by Eq. () [16]. We can define a quantization rule [17, 18], so, we have: 3a x a x y, (6) m n m n l 3a 4l n m m n 3m n y. (7) 1. For the case m = n, the energy dispersion relation for the armchair nanotube (n,n) is obtained by substituting of the discrete allowed values for x into Eq. 4. E l t 3 y 3 y l 4cos a0 4cos a cos (8) n 1 0 where, a. Thus due the periodic boundary condition along the x direction, the wavevector component x is quantized, x l. (9) n 3a. In the case m=0, the zigzag nanotube (n, 0) or (n, n) gives a simple quantization rule in the form, y l. (10) na So, E l l l 3 t 1 4cos 4cos cos xa0 n n (11)
996 D. Racolta, C. Andronache, D. Todoran, R. Todoran 5 where, / 3 x a0 /3. We obtain that just 1/3 of the possible nanotubes are metallic when the condition m-n is multiple of 3 is fulfilled. E() Fig. Band Structure for a (6,6) armchair nanotube within zone folding model. The Fermi level is located at zero energy. E() Fig. 3 Band Structure for a (6,0) zigzag nanotube within zone folding model. The Fermi level is located at zero energy. Using the tight-binding model, we obtained that only armchair nanotubes (n, n) are metallic (zero-gap semiconductors). The zigzag nanotubes (n, m) with n m, which is a multiple of 3, are tiny-gap semiconductors, and all other nanotubes are large-gap semiconductors [19]. In the case of zigzag nanotubes (n, 0) (n < 9), the band gap decreases [0] and for the nanotubes (9, 0) and n 9, the band gap increased [1].
6 Electronic energy dispersion and structural properties on graphene and carbon nanotubes 997 E() Fig. 4 Band Structure for a (9,0) zigzag nanotube within zone folding model. The Fermi level is located at zero energy. 4. CONCLUSIONS The electronic structure of nanotubes can be derived from the electronic structure of graphene, by calculating how rolling of the sheet affects the electronic structure. From the periodic boundary conditions of the nanotubes, the wave vector in Ch direction becomes quantized, while the wave vector along the nanotubes axis remains continuous. This will results in a set of 1D energy dispersion relations which are cross-sections of those for D graphene. The size of the energy gap of nanotubes is inversely proportional to the diameter []. All the armchair nanotube (m-n=0), are character metallic. For the zigzag nanotube, the character depends on chirality, if (m-n) is a multiple of 3, is not gap in the energy spectrum, showing metallic character, if (m-n) is not a multiple of 3, the gap is non-zero, the character is semiconducting. Acnowledgements. This study was supported by JINR-Romania, Scientific Projects, Topic no.04-4-111-015/017, Protocol No. 4407-4-15/17. REFERENCES 1. K. S. Novoselov et al., Science 306, 666 (004).. C. Berger et al., Science 31, 1191 (006). 3. S. B. Sinnott and R. Andrews, Carbon nanotubes: synthesis, properties, and applications, Critical Reviews in Solid State and Materials Sciences, vol. 6, no. 3, pp. 145 49, 001. 4. Z. Jiang, E. A. Henrisen, L. C. Tung, Y.-J. Wang, M. E. Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, Phys. Rev. Lett. 98, 197403 (007). 5. P.R. Wallace, Physical Review 17, 9 (1947). 6. S. Datta, Quantum Transport: Atom to Transistor, Cambridge University Press, New Yor, 005.
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