- Nano Express
- Open Access
Quantum transport simulations of graphene nanoribbon devices using Dirac equation calibrated with tight-binding π-bond model
© Chin et al; licensee Springer. 2012
- Received: 30 November 2011
- Accepted: 10 February 2012
- Published: 10 February 2012
We present an efficient approach to study the carrier transport in graphene nanoribbon (GNR) devices using the non-equilibrium Green's function approach (NEGF) based on the Dirac equation calibrated to the tight-binding π-bond model for graphene. The approach has the advantage of the computational efficiency of the Dirac equation and still captures sufficient quantitative details of the bandstructure from the tight-binding π-bond model for graphene. We demonstrate how the exact self-energies due to the leads can be calculated in the NEGF-Dirac model. We apply our approach to GNR systems of different widths subjecting to different potential profiles to characterize their device physics. Specifically, the validity and accuracy of our approach will be demonstrated by benchmarking the density of states and transmissions characteristics with that of the more expensive transport calculations for the tight-binding π-bond model.
- graphene nanoribbons
- Dirac equation
- quantum transport
- non-equilibrium Green's function
Recent progress of graphene nanoribbon (GNR) fabrication has demonstrated the possibility of obtaining nano-scale width GNRs, which have been considered as one of the most promising active materials for next generation electronic devices due to their unique properties such as bandgap tunability via controlling of the GNR width or subjecting GNR to external electric/magnetic fields [1–5]. Device simulations play an important role in providing theoretical predictions of device physics and characteristics, as well as in the investigation of device performance, in order to guide the development of future device designs. Due to the nano-scale structures of GNRs, however, semi-classical treatments of carrier transport , which are the mainstay of microelectronics, are no longer valid. As a result, quantum transport formalism based on models incorporating detailed atomic structures, such as the ab-initio types[7–9], is needed for the proper simulation of these materials. Unfortunately, a full-fledge ab-initio atomistic model for carrier transport simulation is still very computationally expensive and impractical even with the latest state-of-the-art computing resources. In this study, we therefore develop an efficient model in which a tight-binding Dirac equation (TBDE), calibrated with parameters from the tight-binding π-bond model (TB-π) [10–13], is used together with the non-equilibrium Green's function approach (NEGF)  to investigate transport properties of GNRs. We compare the density of states, DOS(E), and the transmission, T(E), of selected GNR devices for our TBDE model with that of the more expensive TB-π model. Good agreement is found within the relevant energy range for flat band, Laplace and single barrier bias condition. We believe that our model and calibrated data for a side selection of GNR widths presented in this article provided researchers in the quantum transport an accurate and practical framework to study the properties, particularly quantum transport in arbitrary bias conditions, of GNR-based devices.
where for a fixed k y the positive and negative signs denoting the conduction and valence bands, respectively. In the absence of external potential (U0 = 0) and in the limit of large GNR width at which is small, (3) gives the linear dispersion for graphene . The energy bandgap of a certain width, and hence k y , is given by E g = 2ћv F k y at k x = 0.
Results of best-fit l0 (for their respective subbands) to be used for our TBDE model for GNRs of different widths
(eV) [l0 (nm)]
(eV) [l0 (nm)]
where , fs,d(E) is the Fermi function at either the source or drain, Σ sb denotes sum over the subbands, Diag[⋯] and Tr[⋯] denote the diagonal and the trace of a square matrix, respectively.
In both the Laplace and rectangular barrier potential profiles, the DOS(E) and T(E) for our TBDE model are in satisfactory agreement with that calculated from TB-π model within about 1.5 eV around the mid-gap. At higher energies, significant deviations in the DOS(E) and T(E) are consistent with the discrepancies we observed in E(k) (as shown in Figure 3b), as discussed earlier. Nonetheless, these deviations are limited to the high-energy range that is of little relevance to the electron transport in GNR devices. Therefore, our TBDE approach is expected to be valid and as a practical and efficient alternative to TB-π for studying carrier transport involving arbitrary self-consistent electrostatic potentials for device simulations [22, 23].
We developed a tight-binding Dirac equation for practical and accurate numerical investigation of the electron transport in GNR devices. Based on our knowledge, this is the first time that the surface Green's function arises from applying the Dirac equation in NEGF framework is calculated exactly and hence can be used to achieve significant savings in NEGF calculations. The TBDE model is calibrated, with the appropriate parameters (v F = 106 ms-1 and l0), to match the relevant bandstructure details as that of the TB-π model, especially near the Dirac points. The best-fitted l0 for a selected set of GNR widths are also presented for use. We show that the DOS(E) and T(E) calculated by our calibrated TBDE model can produce very good agreement with those that are calculated by the more expensive TB-π model for the flat, Laplace, and rectangular barrier potentials. These cases validate the accuracy of the TBDE model and provided good confidence that the model can be used as a practical and accurate starting point for quantum transport of GNR-based devices where non-equilibrium and arbitrary electrostatic potentials are involved.
This study was supported by Singapore's Agency for Science, Technology and Research (A*STAR) Public Sector Funding Grant No. 0821010023.
- Kosynkin DV, Higginbotham AL, Sinitskii A, Lomeda JR, Dimiev A, Price BK, Tour JM: Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons. Nature 2009, 458(7240):872. 10.1038/nature07872View ArticleGoogle Scholar
- Jiao L, Zhang L, Wang X, Diankov G, Dai H: Narrow graphene nanoribbons from carbon nanotubes. Nature 2009, 458(7240):877. 10.1038/nature07919View ArticleGoogle Scholar
- Cai J, Ruffieux P, Jaafar R, Bieri M, Braun T, Blankenburg S, Muoth M, Seitsonen AP, Saleh M, Feng X, Mullen K, Fasel R: Atomically precise bottom-up fabrication of graphene nanoribbons. Nature 2010, 466(7305):470. 10.1038/nature09211View ArticleGoogle Scholar
- Choi CY, Lee JH, Koh JH, Ha JG, Koo SM, Kim S: High-temperature stable operation of nanoribbon field-effect transistors. Nanoscale Res Lett 2010, 5: 1795. 10.1007/s11671-010-9714-yView ArticleGoogle Scholar
- Kumar SB, Jalil MBA, Tan SG, Liang G: Magnetoresistive effect in graphene nanoribbon due to magnetic field induced band gap modulation. J Appl Phys 2010, 108: 033709. 10.1063/1.3457353View ArticleGoogle Scholar
- Lundstrom M, Guo J: Nanoscale Transistors: Device Physics, Modeling and Simulations. New York: Springer-Verlag; 2006.Google Scholar
- Martins TB, Miwa RH, da Silva AJR, Fazzio A: Electronic and transport properties of Boron-doped graphene nanoribbons. Phys Rev Lett 2007, 98(19):196803.View ArticleGoogle Scholar
- Son YW, Cohen ML, Louie SG: Energy gaps in graphene nanoribbons. Phys Rev Lett 2006, 97: 216803.View ArticleGoogle Scholar
- Yang L, Park CH, Son YW, Cohen ML, Louie SG: Quasiparticle energies and band gaps in graphene nanoribbons. Phys Rev Lett 2007, 99(18):186801.View ArticleGoogle Scholar
- Saito R, Dresselhaus G, Dresselhaus M: Physical Properties of Carbon Nanotubes. London: Imperial College Press; 1998.Google Scholar
- Gunlycke D, White CT: Tight-binding energy dispersions of armchair-edge graphene nanostrips. Phys Rev B 2008, 77(11):115116.View ArticleGoogle Scholar
- Saito R, Dresselhaus G, Dresselhaus MS: Trigonal warping effect of carbon nan-otubes. Phys Rev B 2000, 61(4):2981. 10.1103/PhysRevB.61.2981View ArticleGoogle Scholar
- Wu Y, Childs PA: Conductance of graphene nanoribbon junctions and the tight binding model. Nanoscale Res Lett 2011, 6: 62.Google Scholar
- Datta S: Quantum Transport: Atom to Transistor, chap. 9. New York: Cambridge University Press; 2005:217–251.View ArticleGoogle Scholar
- Neto AHC, 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
- Ouyang Y, Wang X, Dai H, Guo J: Carrie scattering in graphene nanoribbon field-effect transistors. Appl Phys Lett 2008, 92: 243124. 10.1063/1.2949749View ArticleGoogle Scholar
- Sancho MPL, Sancho JML, Rubio J: Highly convergent schemes for the calculation of bulk and surface Green functions. J Phys F met Phys 1985, 15: 851. 10.1088/0305-4608/15/4/009View ArticleGoogle Scholar
- Velev J, Butler W: On the equivalence of different techniques for evaluating the Green function for a semi-infinite system using a localized basis. J Phys Condens Matter 2004, 16: R637. 10.1088/0953-8984/16/21/R01View ArticleGoogle Scholar
- Wang JS, Wang J, Lu JT: Quantum thermal transport in nanostructures. Eur Phys J B 2008, 62: 381. 10.1140/epjb/e2008-00195-8View ArticleGoogle Scholar
- Datta S: Quantum Transport: Atom to Transistor, chap. 11. New York: Cambridge University Press; 2005:305–307.View ArticleGoogle Scholar
- Han MY, Özyilmaz B, Zhang Y, Kim P: Energy band-gap engineering of graphene nanoribbons. Phys Rev Lett 2007, 98(20):206805.View ArticleGoogle Scholar
- Chin SK, Seah D, Lam KT, Samudra GS, Liang G: Device physics and characteristics of graphene nanoribbon tunneling FETs. IEEE Trans. Electron Devices 2010, 57: 3144.View ArticleGoogle Scholar
- Lam KT, Seah D, Chin SK, Kumar SB, Samudra GS, Yeo YC, Liang G: A simulation study of graphene-nanoribbon tunneling FET with heterojunction channel. IEEE Electron Device Lett 2010, 31(6):555.View ArticleGoogle Scholar
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.