Analytical performance of 3 m and 3 m + 1 armchair graphene nanoribbons under uniaxial strain
© Kang and Ismail; licensee Springer. 2014
Received: 17 July 2014
Accepted: 23 October 2014
Published: 4 November 2014
The electronic band structure and carrier density of strained armchair graphene nanoribbons (AGNRs) with widths of n =3 m and n =3 m +1 were examined using tight-binding approximation. The current-voltage (I-V) model of uniaxial strained n =3 m AGNRs incorporating quantum confinement effects is also presented in this paper. The derivation originates from energy dispersion throughout the entire Brillouin zone of uniaxial strained AGNRs based on a tight-binding approximation. Our results reveal the modification of the energy bandgap, carrier density, and drain current upon strain. Unlike the two-dimensional graphene, whose bandgap remains near to zero even when a large strain is applied, the bandgap and carrier density of AGNRs are shown to be sensitive to the magnitude of uniaxial strain. Discrepancies between the classical calculation and quantum calculation were also measured. It has been found that as much as 19% of the drive current loss is due to the quantum confinement. These analytical models which agree well with the experimental and numerical results provide physical insights into the characterizations of uniaxial strained AGNRs.
Graphene, as a two-dimensional single layer of carbon in hexagonal symmetry, has attracted considerable attention since being experimentally discovered in 2004. It possesses various fascinating electrical and physical properties, such as extremely high mobility of the charge carrier, high switching speed with ballistic transport behaviors, and anomalous quantum Hall effects [1, 2]. These excellent electronic properties make graphene a promising alternative as a building block in potential nanoelectronic devices . To further develop the graphene’s application in field-effect transistors (FETs), various studies have attempted to modulate the electronic structure using mechanical deformation [2, 4]. This offers the tempting prospect of controlling the electronic properties of graphene structure by the introduction of strain. The influence of strain on Raman spectroscopy and energy gap of graphene has been predicted theoretically and realized experimentally [5–7]. However, two-dimensional graphene shows zero bandgap electronic properties. Even if a strain as large as 20% is applied, the bandgap remains close to zero. Interestingly, graphene patterned into nanoribbons, referred as graphene nanoribbons (GNRs), has been demonstrated to possess a bandgap opening made possible by tuning the ribbon width [5, 8, 9]. Finite-width strips GNRs (<10 nm) with quasi 1D structure are expected to present similar electronic properties to graphene and carbon nanotubes. However, the spectrum of GNRs depends on the nature of their edge shapes, namely, zigzag-edge and armchair-edge GNRs (ZGNRs and AGNRs). Nevertheless, ZGNRs have been found to be metallic for all widths, while AGNRs are either metallic or semiconducting, depending on their widths [10, 11].
Strain may have a vital influence on further tailoring the electronic properties of a material. Strain in silicon, germanium, and silicon germanium have been successfully implemented by the conventional semiconductor industry, with significant improvements in carrier mobility [12–14]. Understanding the influence of strain on GNRs is of great importance. A substantial part in the fabrication process of GNR device involves deposition of carbon nanostructures on the substrate, which introduces strain at the interface. As GNR area is one-atom thick film, interface strain-induced variations in the electronic and vibrational structures are expected to play a greater role compared to free-standing GNRs. A further motivation to examine the incorporation of strain is the prospect of bandgap opening . Theoretically, the potential of uniaxial strain on the energy gap of GNRs has been widely adopted based on ab initio approaches and tight-binding approximation [15–18]. It has been shown that ZGNRs and AGNRs possess distinct energy gap properties under strain. Despite the fact that there have been many studies on the strain effect in graphene and GNRs, most of the previous works focused on the electronic band structure particularly the energy bandgap, while the effect of strain on the carrier density has seldom been studied. Analytical carrier density expressions will find widespread use in determining equilibrium or quasi equilibrium electronics and transport properties in a semiconductor. The carrier density can be determined without having to perform extensive time-consuming numerical simulations. It can also be utilized in the development of fast compact models for circuit simulation. Although the strain effects on the energy bandgap of GNRs has been explored, a comparative study between different families of AGNRs n =3 m and n =3 m +1 is still lacking. Therefore, in this paper we theoretically explore the influence of uniaxial strain on the band structures and carrier density of AGNRs for both the n = 3 m and n = 3 m + 1 families using tight-binding calculations and formulate a universal explanation for the effect of strain. In addition, we also investigated the effect of quantum confinement on the drain-current performance of n = 3 m AGNRs and compared the analytical results against experimental data.
Theoretical model for electronic properties
Theoretical model for carrier density
where N is the quantum number and L is the ribbon length. The DOS of AGNRs under uniaxial strain reveals that the energy states in both low and high regions are affected by the strain.
where N G is the number of gates (1 for the single-gate geometry and 2 for the double-gate geometry), κ is the relative dielectric constant of the gate insulator, t ins is the thickness of the gate insulator, W is the ribbon width, and α ≈ 0 is a dimensionless fitting parameter.
for 0 ≤ V D ≤ V Dsat
where V T is the threshold voltage and V Dsat is the drain voltage at which the drain carrier concentration becomes maximum, consistent with the drain saturation current. V c is the critical voltage that is much smaller than the drain voltage enhancing the role of velocity saturation in the nanochannel.
for V D ≥ V Dsat
Results and discussion
In this paper, we have calculated the electronic band structure as well as carrier density under uniaxial strain effect for both n =3 m and n =3 m +1 AGNRs families by applying a modification to the tight-binding nearest neighbor hopping integral. We observed that for n =3 m AGNRs, the bandgap increases with an increase in the magnitude of strain but tends to reduce for n =3 m +1 AGNRs family. These phenomena are caused by the moving of the Fermi point between discrete k lines of allowed electronics states. In addition, it is also found that the uniaxial strain gives substantial effect to the carrier density within the two families. It is also interesting to observe a semiconductor-metal-semiconductor transition phase at ϵ =8% for the n =3 m +1 AGNRs family. While the introduction of strain imposes changes in the bandgap and current values, the incorporation of quantum confinement effect also results in dramatic reduction in the drain current performance. The discrepancies between the classical calculation and quantum calculation can be best explained by the threshold voltage shift and total gate capacitance degradation due to quantum confinement. Our analytical findings provide critical insight into the importance of quantum confinement for nanoscale GNRs' FET, and the proposed model gives a better assessment of nanoscale GNRs' FET performance.
Authors would like to acknowledge the financial support from the Research University grant of the Ministry of Higher Education (MOHE), Malaysia, under projects Q.J130000.21A2.00E67 and Q.J130000.2523.05H23. Also, we would like to thank the Research Management Center (RMC) of Universiti Teknologi Malaysia (UTM) for providing an excellent research environment in which to complete this work.
- Craciun MF, Russo S, Yamamoto M, Tarucha S: Tuneable electronic properties in graphene. Nano Today 2011, 6(1):42–60. 10.1016/j.nantod.2010.12.001View ArticleGoogle Scholar
- Wong JH, Wu BR, Lin MF: Strain effect on the electronic properties of single layer and bilayer graphene. J Phys Chem C 2012, 116(14):8271–8277. 10.1021/jp300840kView ArticleGoogle Scholar
- Ohmi Y, Ogawa M, Souma S: Effect of uniaxial strain on the electronic transport in single layer graphene. 2011 International Meeting for Future of Electron Devices, Kansai, (IMFEDK) 2011, 126–127.View ArticleGoogle Scholar
- Guinea F: Strain engineering in graphene. Solid State Commun 2012, 152(15):1437–1441. 10.1016/j.ssc.2012.04.019View ArticleGoogle Scholar
- Robinson J, Fanton M, Stitt T, Snyder D, Frantz E, Tedesco JL, VanMil B, Jernigan G, Campbell P, Myers-Ward RL, Eddy C Jr, Gaskill DK: Large-area epitaxial graphene: effect of strain and thickness on electronic properties. ECS Trans 2009, 19(5):107–109.View ArticleGoogle Scholar
- Choi S-M, Jhi S-H, Son Y-W: Effects of strain on electronic properties of graphene. Phys Rev B 2010, 81(8):081407.View ArticleGoogle Scholar
- Pellegrino FMD, Angilella GGN, Pucci R: Effect of uniaxial strain on plasmon excitations in graphene. J Phys 2012, 377: 012083.Google Scholar
- Gui G, Li J, Zhong J: Band structure engineering of graphene by strain: first-principles calculations. Phys Rev B 2008, 78(7):075435.View ArticleGoogle Scholar
- Behera H, Mukhopadhyay G: Strain-tunable band gap in graphene/h-BN hetero-bilayer. J Phys Chem Solids 2012, 73(7):818–821. 10.1016/j.jpcs.2012.02.010View ArticleGoogle Scholar
- Brey L, Fertig HA: Electronic states of graphene nanoribbons. Phys Rev B 2006, 73: 235411.View ArticleGoogle Scholar
- Castro Neto AH, Peres NMR, Novoselov KS, Geim AK: The electronic properties of graphene. Rev Mod Phys 2009, 81(1):109–162. 10.1103/RevModPhys.81.109View ArticleGoogle Scholar
- Lee ML, Fitzgerald EA, Bulsara MT, Currie MT, Lochtefeld A: Strained Si, SiGe, and Ge channels for high-mobility metal-oxide-semiconductor field-effect transistors. J Appl Phys 2005, 97(1):011101. 10.1063/1.1819976View ArticleGoogle Scholar
- Mistry K, Allen C, Auth C, Beattie B, Bergstrom D, Bost M, Brazier M, Buehler M, Cappellani A, Chau R, Choi C-H, Ding G, Fischer K, Ghani T, Grover R, Han W, Hanken D, Hattendorf M, He J, Hicks J, Huessner R, Ingerly D, Jain P, James R, Jong L, Joshi S, Kenyon C, Kuhn K, Lee K, Liu H, et al.: A 45-nm logic technology with high-k + metal gate transistors, strained silicon, 9 Cu interconnect layers, 193-nm dry patterning, and 100% Pb-free packaging. 2007 IEEE International Electron Devices Meeting 2007, 247–250.View ArticleGoogle Scholar
- Nam D, Sukhdeo D, Roy A, Balram K, Cheng S-L, Huang KC-Y, Yuan Z, Brongersma M, Nishi Y, Miller D, Saraswat K: Strained germanium thin film membrane on silicon substrate for optoelectronics. Opt Express 2011, 19(27):25866–25872. 10.1364/OE.19.025866View ArticleGoogle Scholar
- Rosenkranz N, Mohr M, Thomsen C: Uniaxial strain in graphene and armchair graphene nanoribbons: an ab initio study. Ann Phys 2011, 523(1–2):137–144.View ArticleGoogle Scholar
- Lu Y, Guo J: Band gap of strained graphene nanoribbons. Nano Res 2010, 3(3):189–199. 10.1007/s12274-010-1022-4View ArticleGoogle Scholar
- Li Y, Jiang X, Liu Z, Liu Z: Strain effects in graphene and graphene nanoribbons: the underlying mechanism. Nano Res 2010, 3(8):545–556. 10.1007/s12274-010-0015-7View ArticleGoogle Scholar
- Khaliji K, Noei M, Tabatabaei SM, Pourfath M, Fathipour M, Abdi Y: Tunable bandgap in bilayer armchair graphene nanoribbons: concurrent influence of electric field and uniaxial strain. IEEE Trans Electron Devices 2013, 60(8):2464–2470.View ArticleGoogle Scholar
- Mei H, Yong Z, Hong-Bo Z, Hong-Bo Z: Effect of uniaxial strain on band gap of armchair-edge graphene nanoribbons. Chin Phys Lett 2010, 27(3):037302. 10.1088/0256-307X/27/3/037302View ArticleGoogle Scholar
- Akinwande D, Nishi Y, Wong H-SP: An analytical derivation of the density of states, effective mass, and carrier density for achiral carbon nanotubes. IEEE Trans Electron Devices 2008, 55(1):289–297.View ArticleGoogle Scholar
- Datta S: Quantum Transport: Atom to Transistor. Cambridge: Cambridge University Press; 2005:404.View ArticleGoogle Scholar
- Xu H, Zhang Z, Wang Z, Wang S, Liang X, Peng L: Vertical Scaling of Graphene Field-Effect Transistor, no. 3. 2011, 2340–2347.Google Scholar
- Guo J, Yoon Y, Ouyang Y: Gate electrostatics and quantum capacitance of graphene nanoribbons. Nano Lett 2007, 7(7):1935–1940. 10.1021/nl0706190View ArticleGoogle Scholar
- Ni ZH, Yu T, Lu YH, Wang YY, Feng YP, Shen ZX: Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening. ACS Nano 2008, 2(11):2301–2305. 10.1021/nn800459eView ArticleGoogle Scholar
- Sun L, Li Q, Ren H, Su H, Shi QW, Yang J: Strain effect on electronic structures of graphene nanoribbons: a first-principles study. J Chem Physics 2008, 129(7):074704. 10.1063/1.2958285View ArticleGoogle Scholar
- Chang CP, Wu BR, Chen RB, Lin MF: Deformation effect on electronic and optical properties of nanographite ribbons. J Appl Phys 2007, 101(6):063506. 10.1063/1.2710761View ArticleGoogle Scholar
- Kliros GS: Modeling of carrier density and quantum capacitance in graphene nanoribbon FETs. 2010 International Conference on Microelectronics: December 2010, vol. 1, no. Icm 2010, 236–239.View ArticleGoogle Scholar
- Kliros G: Effect of uniaxial strain on the current–voltage characteristics of graphene nanoribbon field-effect transistors. Semiconductor Conference (CAS) 2013, 2: 27–30.Google 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.