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
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