The Scientific World Journal

Volume 2014 (2014), Article ID 581478, 6 pages

http://dx.doi.org/10.1155/2014/581478

## Flake Electrical Conductivity of Few-Layer Graphene

^{1}Department of Physics, Razi University, Kermanshah, Iran^{2}Nanoscience and Nanotechnology Research Center, Razi University, Kermanshah, Iran

Received 24 August 2013; Accepted 20 October 2013; Published 16 January 2014

Academic Editors: K. Liu, S. K. Yang, and T. Zhou

Copyright © 2014 Hamze Mousavi and Jabbar Khodadadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Kubo formula for the electrical conductivity of per stratum of few-layer graphene, up to five, is analytically calculated in both simple and Bernal structures within the tight-binding Hamiltonian model and Green's function technique, compared with the single-layer one. The results show that, by increasing the layers of the graphene as well as the interlayer hopping of the nonhybridized orbitals, this conductivity decreases. Although the change in its magnitude varies less as the layer number increases to beyond two,distinguishably, at low temperatures, it exhibits a small deviation from linear behavior. Moreover, the simple bilayer graphene represents more conductivity with respect to the Bernal case.

Graphene is an atom thick allotrope of carbon in two-dimensional (2D) hexagonal honeycomb lattice [1]. Electrons in a single-layer graphene exhibit a characteristic linear dispersion relation between energy and momentum near the point of the first Brillouin zone (FBZ) [2–4]. The overall electronic structure changes sensitively with increasing crystallographic stacking sequence. The sequence of graphene sheets brings about the various 3D graphite crystals [5–8], that is, -stacked bilayer graphene (hexagonal simple bilayer graphene), -stacked bilayer graphene (Bernal bilayer graphene), and -stacked trilayer graphene (rhombohedral trilayer graphene). The interlayer interactions, due to the weak overlap of the nonhybridized orbitals, result in the anisotropic band structure along the stacking direction. Some theoretical studies [9–12] have predicted in two or more layers of graphene that the linearly dispersing bands are either replaced or augmented by split hyperbolic bands. Experimental investigations have also been considered to single- and bilayer graphene [13–15]. In a single-layer graphene transistor, the current is modulated by a gate voltage but it cannot be switched off due to lack of a band gap in the energy dispersion. Bilayer graphene is the only known semiconductor with a gate tuneable band gap [16]. Opposed to the case of single- and bilayer, the trilayer material is a semimetal with a gate tuneable band overlap between the conduction and the valence bands [16]. The variety of electronic properties found in different few-layer graphene (FLG) is the true strength of these newly discovered materials.

In this study, the electrical conductivity (EC) of FLG in structures is investigated within the tight-binding (TB) Hamiltonian model and Green’s function method. Using band representation of Green’s function, we calculate the EC of the systems by Kubo formula [17–19]. Then, the temperature-dependent EC of a monolayer graphene will be compared to that for per sheet of these systems, hereinafter mentioned as ‘‘flake EC” (FEC).

In second quantization form, the Hamiltonian of the TB model for FLG lattice reads as follows [20]: where and refer to the or subsites inside the Bravais lattice unit cells (Figure 1) in each plane of the system, and denote the position of the Bravais unit cell in the lattice, and describe plane’s indexes, shows the number of the Bravais lattice unit cell, implies the number of the layers, presents the amplitude for a electron to hop from the subsite of the Bravais lattice site in plane to the subsite of the nearest-neighbor (NN) site in plane , and () displays the creation (annihilation) operator of an electron on subsite () of the Bravais lattice site () in plane (). In our calculations, we take the chemical potential equal to zero corresponding to the contribution of one electron per orbital in the system. We note that the on-site energy of the carbon atoms has been fitted as zero. Besides, such units are utilized that .

We study the Hamiltonian Equation (1) by Green’s function approach. Since each Bravais lattice unit cell of the graphene sheet includes atoms, the Hamiltonian of the trilayers graphene, , as a typical case, would be introduced by a matrix with the following basis kets of the Hilbert space, so the trilayer Green function could be represented as with , in which remarks imaginary time and hints the time ordering operator. Here, exhibits ensemble averaging on the ground state of the system. Using Green’s function formalism for the Hamiltonian in (1), the equation of motion for electrons in structure can be written as and that for case is given by where , the index shows NN sites, plays the role of a unit matrix, and notifies the Kronecker symbol. The -space Fourier transformation of (4) and (5) leads to the following relations: in which is defined as where points a 2D wave vector in the FBZ, , in which displays interatomic distance, and perform as primitive vectors (Figure 1). In (6)-(7), the intraplane hopping to the NN sites and the interplane ones are denoted by and , respectively (Figure 2).

Our starting point for the EC tensor is the well-known Kubo formula [17–19], where indicate Cartesian components, shows temperature, and refers to Fermi-Dirac distribution function, . In band representation, energy-dependent EC, , is defined as in which serves as band index, equals the number of the bands in the system, describes a band Cartesian component of the velocity operator, and ’s express eigenvalues of the Hamiltonian of the system. We point out that, in band representation, the Hamiltonian of the system has a diagonal form, so (6) get the shape as where . For monolayer graphene, , ’s are calculated as so that For FEG, up to five layers, the eigenvalues could also be analytically found. Using (10), turns out to be Moreover, the velocity operator could be represented by

From (9), (10), and (13) and definition of velocity, the -component of the energy-dependent EC, , of the simple structures () can be written as while that of the Bernal structure with is determined by for , the result is When , it is found that and leads to in which and implies sum over just some bands not all.

In summary, using Green’s function method and the Kubo-Greenwood formula through the TB Hamiltonian model, the EC of FLG is analytically found for single-layer and simple cases as well as Bernal structures. The aim is to compare the EC of the single-layer graphene and FLG (see (8) and (11)–(19)). We set the intraplane hopping to the NN and interplane ones as eV and eV [21–23], respectively. Figures (3) and (4) show the results. In Figure 3(a), the FECs of mono-layer graphene and FLG in simple structure are plotted, while, in Figure 3(b), those of the Bernal cases are classified. The latter are investigated and summarized in Figure 3(c) for and different values of interplane term; that is, , , , and . Also, is illustrated for simple and Bernal bilayer graphene in Figure 4. We mention that, for a monolayer graphene, is just the area of the graphene unit cell. For FLG, the FEC is of more interest, so will be the multiplication of the single-layer area by the layer number . So, in (15)–(19), , if we set the area of the graphene unit cell equal to unity. Numerically, we insert and .

In Figures 3(a) and 3(b), the temperature-dependent FECs of mono-layer graphene and FLG are compared in simple and Bernal form as well. It is shown that decreases with increasing the number of layers from to . This behavior could be justified by overlapping of the nonhybridized orbitals perpendicular to the sheets, so that these interlayer interactions will generate new channels of electron motion with respect to those of the isolated single layer, but perpendicular to them. Reasonably, these vertical detour ways can distract a fraction of the electrons’ motivation from horizontal traces parallel to the layers, towards the vertical tracks. In other words, overlapped nonhybridized orbitals lead to a partial deviation of the electrons’ mobility from the planes on behalf of the normal directions. Consequently, these interlayer possibilities of movement result in a reduction of the intralayer displacements, whereby the system exhibits a decay in the FEC. This phenomenon gets more remarkable with respect to the monolayer case by increasing the number of the layers, easily conceivable from Figures 3(a) and 3(b) and in agreement with the last explanation, because adding more layers provides more distracting paths for moving electrons, so the plane components get lesser. On the other hand, the changing in the FEC could also be a result of significant variations in the low energy band structures (linear to quadratic dispersion) and the corresponding density of states (DOS). As the layer number increases to five, the variation of the low energy band structure is much less dramatic than the transition from one to two [24]. So the variation in the magnitude of the FEC changes less as the layer number increases to beyond two. Considerably, this change in dispersion provides a small deviation in the linear behavior of the FEC at low temperatures. This is more notable for the simple case whose dispersion bears more variations around the Fermi level.

It is also resulted that the FEC depends on the amount of the interplane hopping integrals. In Figure 3(c), the temperature-dependent FEC of bilayer Bernal structure is plotted for four values of . Obviously, the more increases, the more decreases. This could be interpreted that the procedure of increasing the interlayer hopping transforms the interlayer interactions towards the limit of somehow covalence-like bonds, which resemble the carbonic system as an insulator with four strong bonds and consequently weak EC.

We have also compared of bilayer both simple and Bernal graphene in Figure 4. It is known that a finite DOS at zero energy appears in the simple case [25] in contrast to the single layer whose DOS vanishes. Therefore, because of appearing allowed states close to Fermi energy, the temperature-dependent FEC of the simple case is more that of the Bernal one.

Totally, it is concluded that the FEC decreases by increasing the layers of the graphene due to overlapping of the nonhybridized orbitals perpendicular to the sheets. But, the variation in the magnitude of the FEC varies less as the layer number increases to beyond two as a result of changes in the low energy band structures. Besides, a deviation from the linear behavior of the FEC is observable at low temperatures originated from changes in relevant dispersion, especially for the simple case. It is found that more increase in interplane term causes more decrease in the temperature-dependent FEC because of transforming the interlayer interactions towards the limit of covalence-like bonds. Finally, it is resulted that the FEC of simple structure is more than that of Bernal one.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- K. S. Novoselov, A. K. Geim, S. V. Morozov et al., “Two-dimensional gas of massless Dirac fermions in graphene,”
*Nature*, vol. 438, no. 7065, pp. 197–200, 2005. View at Publisher · View at Google Scholar · View at Scopus - E. McCann and V. I. Fal'ko, “Landau-level degeneracy and quantum hall effect in a graphite bilayer,”
*Physical Review Letters*, vol. 96, no. 8, Article ID 086805, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. K. Min and A. H. McDonald, “Origin of universal optical conductivity and optical stacking sequence identification in multilayer graphene,”
*Physical Review Letters*, vol. 103, no. 6, Article ID 067402, 4 pages, 2009. View at Publisher · View at Google Scholar - A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,”
*Reviews of Modern Physics*, vol. 81, no. 1, pp. 109–162, 2009. View at Publisher · View at Google Scholar · View at Scopus - J.-C. Charlier, X. Gonze, and J.-P. Michenaud, “First-principles study of the electronic properties of graphite,”
*Physical Review B*, vol. 43, no. 6, pp. 4579–4589, 1991. View at Publisher · View at Google Scholar · View at Scopus - J.-C. Charlier, J.-P. Michenaud, and X. Gonze, “First-principles study of the electronic properties of simple hexagonal graphite,”
*Physical Review B*, vol. 46, no. 8, pp. 4531–4539, 1992. View at Publisher · View at Google Scholar · View at Scopus - J.-C. Charlier, J.-P. Michenaud, and P. Lambin, “Tight-binding density of electronic states of pregraphitic carbon,”
*Physical Review B*, vol. 46, no. 8, pp. 4540–4543, 1992. View at Publisher · View at Google Scholar · View at Scopus - J.-C. Charlier, X. Gonze, and J.-P. Michenaud, “First-principles study of the stacking effect on the electronic properties of graphite(s),”
*Carbon*, vol. 32, no. 2, pp. 289–299, 1994. View at Google Scholar · View at Scopus - N. M. R. Peres, A. H. Castro Neto, and F. Guinea, “Dirac fermion confinement in graphene,”
*Physical Review B*, vol. 73, no. 24, Article ID 241403, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Koshino and T. Ando, “Orbital diamagnetism in multilayer graphenes: systematic study with the effective mass approximation,”
*Physical Review B*, vol. 76, no. 8, Article ID 085425, 11 pages, 2007. View at Publisher · View at Google Scholar - M. Aoki and H. Amawashi, “Dependence of band structures on stacking and field in layered graphene,”
*Solid State Communications*, vol. 142, no. 3, pp. 123–127, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. K. Min and A. H. McDonald, “Chiral decomposition in the electronic structure of graphene multilayers,”
*Physical Review B*, vol. 77, no. 15, Article ID 155416, 5 pages, 2008. View at Publisher · View at Google Scholar - F. Wang, Y. Zhang, C. Tian et al., “Gate-variable optical transitions in graphene,”
*Science*, vol. 320, no. 5873, pp. 206–209, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Zhang, T.-T. Tang, C. Girit et al., “Direct observation of a widely tunable bandgap in bilayer graphene,”
*Nature*, vol. 459, no. 7248, pp. 820–823, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. Q. Li, E. A. Henriksen, Z. Jiang et al., “Band structure asymmetry of bilayer graphene revealed by infrared spectroscopy,”
*Physical Review Letters*, vol. 102, no. 3, Article ID 037403, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Russo, M. F. Craciun, T. Khodkov, M. Koshino, M. Yamamoto, and S. Tarucha,
*Graphene-Synthesis, Characterization, Properties and Applications*, chapter 9, InTech, 2011. - S. F. Edwards, “A new method for the evaluation of electric conductivity in metals,”
*Philosophical Magazine*, vol. 3, no. 33, pp. 1020–1031, 1958. View at Publisher · View at Google Scholar - S. F. Edwards, “The statistical thermodynamics of a gas with long and short-range forces,”
*Philosophical Magazine*, vol. 4, pp. 1171–1182, 1959. View at Publisher · View at Google Scholar - B. Velicky, “Theory of electronic transport in disordered binary alloys: coherent-potential approximation,”
*Physical Review Online Archive*, vol. 184, pp. 614–627, 1969. View at Publisher · View at Google Scholar - H. Mousavi, “The impact of gas molecule adsorption on the orbital magnetic susceptibility of graphene,”
*Journal of Magnetism and Magnetic Materials*, vol. 322, no. 17, pp. 2533–2536, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. L. Lu, C. P. Chang, Y. C. Huang, R. B. Chen, and M. L. Lin, “Influence of an electric field on the optical properties of few-layer graphene with AB stacking,”
*Physical Review B*, vol. 73, no. 14, Article ID 144427, 7 pages, 2006. View at Publisher · View at Google Scholar - Y. Takane, “Tunneling density of states in multilayer graphene deposited on a superconductor,”
*Journal of the Physical Society of Japan*, vol. 79, no. 12, Article ID 124706, 2010. View at Publisher · View at Google Scholar · View at Scopus - E. V. Castro, K. S. Novoselov, S. V. Morozov et al., “Electronic properties of a biased graphene bilayer,”
*Journal of Physics Condensed Matter*, vol. 22, no. 17, Article ID 175503, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. L. Lu, C. P. Chang, Y. C. Huang, J. H. Ho, C. C. Hwang, and M. F. Lin, “Electronic properties of AA- and ABC-stacked few-layer graphites,”
*Journal of the Physical Society of Japan*, vol. 76, Article ID 024701, 7 pages, 2007. View at Publisher · View at Google Scholar - E. McCann, D. S. L. Abergel, and V. I. Fal'ko, “Electrons in bilayer graphene,”
*Solid State Communications*, vol. 143, no. 1-2, pp. 110–115, 2007. View at Publisher · View at Google Scholar · View at Scopus