Electronic and transport properties in circular graphene structures with a pentagonal disclination
 Esther Jódar^{1}Email author,
 Antonio Pérez–Garrido^{1} and
 Fernando Rojas^{1, 2}
DOI: 10.1186/1556276X8258
© Jódar et al.; licensee Springer. 2013
Received: 16 November 2012
Accepted: 24 April 2013
Published: 29 May 2013
Abstract
We investigate the electronic and transport properties of circular graphene structures (quantum dots) that include a pentagonal defect. In our calculations, we employ a tightbinding model determining total and local density of states, transmission function and participation number. For the closed structure, we observe that the effect of the defect is concentrated mainly on energies near to zero, which is characteristic of edge states in graphene. The density of states and transmission functions for small energies show several peaks associated with the presence of quasibound states generated by the defect and localized edge states produced by both the circular boundaries of the finite lattice and induced by the presence of the pentagonal defect. These results have been checked by calculating the participation number, which is obtained from the eigenstates. We observe changes in the available quasibound states due to the defect and the creation of new peaks in the transmission function.
Keywords
Graphene; Transport; Defects 72.80.Vp 72.15.Rn 73.20.AtBackground
The theoretical and experimental study of properties of graphene has attracted the attention of many authors in the last few years since a method to isolate single graphene layers was developed (the authors Geim and Novoselov were awarded with the Nobel prize). These graphene sheets may be stable enough to be freely suspended[1], which allows us to use them in solid state experiments. Besides, the electronic properties of graphene are surprising: one finds new quasiparticles described by the Dirac equation at low energies that behave like massless particles. This opens the possibility to study quantum electrodynamics properties in solidstate devices and to carry out new developments, e. g., biosensors (see other studies[2–10]).
Methods
where${c}_{i}^{\u2021}/{c}_{i}$ are the creation/annihilation operators of an electron in site i. We expand the wave function in terms of the site base.${\Psi}^{k}\u3009=\sum _{i}{a}_{i}^{k}i\u3009$, where${a}_{i}^{k}$ is the amplitude probability that the electron is to be in site i for the eigenstate k. We need to solve$\u0124{\Psi}^{k}\u3009={E}^{k}{\Psi}^{k}\u3009$. Four quantities are calculated to characterize the nature of the electronic and transport properties on twocircled structures, with PD and defectfree (ND) structures: the total density of states N(E), the local density of states ρ(i,E), the participation number P(E) and the transmission function T(E).
Electronic properties for the closed system
It assesses the wave function spreading so it can help to find out the localized or extended nature of an electronic state. For a completely localized wave function Ψ^{ k }(i) is approximately δ_{ k i }→P≈1 while for a typical delocalized wave function on D atoms, Ψ^{ k }(i) is approximately$\frac{1}{\sqrt{D}}$, and then P≈D.
Electronic transport properties: Green’s function method
where${\widehat{\sum}}_{\mathrm{L}}$ and${\widehat{\sum}}_{\mathrm{R}}$ are the selfenergy terms of left and right leads, respectively, and$\u0124$ is the Hamiltonian of the conductor, i.e., in our case, the circular graphene sheet plus a few unit cells of the leads. In our approach, the contact leads at opposite sides of the circular graphene sheet is the graphene sheet itself extended to make the leads semiinfinite. This is equivalent to have reflectionless contacts in macroscopic conductors. Selfenergy terms are calculated using the prescription${\widehat{\Sigma}}_{\mathrm{R}/L}(i,j)={t}^{2}{\u011c}_{\mathrm{R}/L}(k,l)$, where${\u011c}_{\mathrm{R}/L}(k,l)$ is Green’s function of the semiinfinite lead (right or left) evaluated on sites k and l, which are in contact with sites i and j in the circular graphene sheet.
We only need to calculate${\u011c}_{\mathrm{R}/L}$ in the sites in contact with the conductor. To do that, we use the formalism developed by López Sancho et al.[18]. This method has the advantage that the number of iterations close to singularities is very low compared to other transfer matrix methods, so it converges very fast and has been applied to graphene layers by other authors (see e.g.[19]). In this scheme, Green’s function is${\u011c}_{\mathrm{R}/L}={\left(\mathrm{E\xce}{\u0124}_{00}{\u0124}_{01}\widehat{T}\right)}^{1}$, where${\u0124}_{00}$ is the Hamiltonian of one isolated graphene cell in the lead, and${\u0124}_{01}$ is the matrix that takes into account the interaction between two consecutive cells. For the calculation of T, we use the iterative method described in[18].
In Equation 6, G^{R/A} are the retarded and advanced Green’s functions, respectively, and${\widehat{\Gamma}}_{\mathrm{L}/R}=i[{\widehat{\sum}}_{\mathrm{L}/R}^{\mathrm{R}}{\widehat{\sum}}_{\mathrm{L}/R}^{\mathrm{A}}]$. We denote the trace of the matrix considered by “Tr”, which is extended over the whole matrix.
Results and discussion
Conclusion
We have investigated the electronic and transport properties of circular graphene layers with a pentagonal disclination. In particular, using a tightbinding model, we have calculated the density of states, transmission function, participation number and local density of states of the structure with and without defects. The density of states for the structure with the PD shows several peaks that are associated with new localized states, which have been checked by calculating the local density of states and the participation number. We observe changes in the available quasibound states due to the defect and new peaks of the transmission function. Comparing these results, we conclude that there are more quasibound states in the structure with the defect, states associated with both the presence of quasibound states related to the atoms belonging to the defect and others due to the circular confinement and edge states due to circular boundaries of the finite lattice and the defect.
Abbreviations
 ND:

Defectfree structure
 PD:

Pentagonal defect.
Declarations
Acknowledgements
FR would like to acknowledge the DGAPA project PAPPIT IN112012 for their financial support and sabbatical scholarship at the UPCT.
Authors’ Affiliations
References
 Meyer JC, Geim AK, Katsnelson MI, Novoselov KS, Booth TJ, S R: The structure of suspended graphene sheets. Phys Rev Lett 1994, 72: 1878. 10.1103/PhysRevLett.72.1878View ArticleGoogle Scholar
 Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK: The electronic properties of graphene. Rev Mod Phys 2009, 81: 109. 10.1103/RevModPhys.81.109View ArticleGoogle Scholar
 Geim AK: Graphene: Status and prospects. Science 2009, 324: 1530. 10.1126/science.1158877View ArticleGoogle Scholar
 Ihn T, Güttinger J, Molitor F, Schnez S, Schurtenberger E, Jacobsen A, Hellmüller S, Frey T, Dröscher S, Stampfer C, Ensslin K: Graphene single electron transistors. Mater Today 2010, 13: 44.View ArticleGoogle Scholar
 Molitor F, Güttinger J, Stampfer C, Dröscher S, Jacobsen A, Ihn T, Ensslin K: Electronic properties of graphene nanostructures. J Phys: Condens Matter 2011, 23: 243201. 10.1088/09538984/23/24/243201Google Scholar
 Cooper DR, D’Anjou B, Ghattamaneni N, Harack B, Hilke M, Horth A, Majlis N, Massicotte M, Vandsburger L, Whiteway E, Yu V: Experimental review of Graphene. ISRN Condens Matter Phys 2012, 2012: 501686.Google Scholar
 Kim JH, Jung JM, Kwak JY, Jeong JH, Choi BC, Lim KT: Preparation of properties of SWNT/Graphene oxide type flexible transparent conductive film. J Nanosci Nanotechnol 2011, 11: 7424. 10.1166/jnn.2011.4841View ArticleGoogle Scholar
 Yun JS, Yang KS, Kim DH: Multifunctional polydiacetyleneGraphene nanohybrids for biosensor application. J Nanosci Nanotechnol 2011, 11: 5663. 10.1166/jnn.2011.4444View ArticleGoogle Scholar
 Zhang L, Xing Y, He N, Zhang Y, Lu Z, Zhang J, Zhang Z: Preparation of Graphene quantum dots for bioimaging application. J Nanosci Nanotechnol 2012, 12: 2924. 10.1166/jnn.2012.5698View ArticleGoogle Scholar
 Islam MS, Kouzani AZ, Dai XJ, Michalski WP, Gholamhosseini H: Design and analysis of a multilayer localized surface plasmon resonance Graphene biosensor. J Nanosci Nanotechnol 2012, 8: 380.Google Scholar
 Meyer JC, Kisielowski C, Erni R, Rossell MD, Crommie MF, Zettl A: Direct imaging of lattice atoms and topological defects in Graphene membranes. Nano Lett 2008, 8: 3582. 10.1021/nl801386mView ArticleGoogle Scholar
 Carpio A, Bonilla LL, de Juan F, Vozmediano MAH: Dislocations in graphene. New J Phys 2008, 10: 053021. 10.1088/13672630/10/5/053021View ArticleGoogle Scholar
 Rycerz A: Electron transport and quantumdot energy levels in Zshaped graphene nanoconstriction with zigzag edges. Acta Phys Polon A 2010, 118: 238.Google Scholar
 Zhang Y, Hu JP, Bernevig BA, Wang XR, Xie XC, Liu WM: Quantum blockade and loop currents in graphene with topological defects. Phys Rev B 2008, 78: 155413.View ArticleGoogle Scholar
 Zhang Y, Hu JP, Bernevig BA, Wang XR, Xie XC, Liu WM: Impurities in graphene. Phys Status Solidi A 2010, 207: 2726. 10.1002/pssa.201026466View ArticleGoogle Scholar
 Wegner FJ: Inverse participation ratio in 2+Epsilon dimensions. Z Phys B 1980, 36: 209. 10.1007/BF01325284View ArticleGoogle Scholar
 Datta S: Electronic Transport in Mesoscopic Systems. Cambridge: Cambridge University Press; 1995.View ArticleGoogle Scholar
 López Sancho MP, López Sancho JM, Rubio J: Quick iterative scheme for the calculation of transfer matrices: application to Mo (100). J Phys F: Met Phys 1984, 14: 1205. 10.1088/03054608/14/5/016View ArticleGoogle Scholar
 Li TC, Lu SP: Quantum conductance of graphene nanoribbons with edge defects. Phys Rev B 2008, 77: 085408.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.