The effect of uniaxial strain on graphene nanoribbon carrier statistic
© Johari and Ismail; licensee Springer. 2013
Received: 18 July 2013
Accepted: 11 October 2013
Published: 14 November 2013
Armchair graphene nanoribbon (AGNR) for n=3m and n=3m+1 family carrier statistic under uniaxial strain is studied by means of an analytical model based on tight binding approximation. The uniaxial strain of AGNR carrier statistic models includes the density of state, carrier concentration, and carrier velocity. From the simulation, it is found that AGNR carrier concentration has not been influenced by the uniaxial strain at low normalized Fermi energy for n=3m and n=3m+1. In addition, the carrier velocity of AGNR is mostly affected by strain at high concentration of n≈3.0×107 and 1.0 × 107 m−1 for n=3m and n=3m+1, respectively. The result obtained gives physical insight into the understanding of uniaxial strain in AGNR.
Graphene has attracted numerous research attention since it was isolated in 2004 by Novoselov et al. . Due to its unique hexagonal symmetry, graphene posses many remarkable electrical and physical properties desirable in electronic devices. It is the nature of graphene that it does not have a bandgap, which has limited its usage. Therefore, efforts to open up a bandgap has been done by several methods [2–4]. The most widely implemented method is patterning the graphene into a narrow ribbon called graphene nanoribbon (GNR) . Recently, strain engineering have started to emerge in graphene electronics . It is found that strain applied to graphene can modify its band structure, thus, altering its electronic properties [6–8]. In fact, uniaxial strain also helps in improving the graphene device’s electrical performance . Similar characteristics have been observed when strain is applied to conventional materials like silicon (Si), germanium (Ge), and silicon germanium (SiGe) . Strain in graphene can be characterized by two major varieties, namely uniaxial and shear. This strain behaves differently on graphene depending on the edge shape, namely zigzag or armchair . The presence of the strain effect in graphene is by the G peak that splits and shifts in the Raman spectrum [11, 12]. It is worth noting that strain in graphene may unintentionally be induced during the fabrication of graphene devices.
Computational modeling and simulation study pertaining to strain graphene and GNR for both the physical and electrical properties have been done using few approaches such as the tight binding model and the ab initio calculation [6, 13]. An analytical modeling approach has also been implemented to investigate the strain effect on GNR around the low-energy limit region [14, 15]. However, most of the previous works have only focused on the electronic band structure, particularly the bandgap. As the carrier transport in GNR has a strong relation with this electronic band structure and bandgap, it is mandatory to investigate the strain effect on the carrier transport such as carrier density and velocity. Therefore, in this paper, an analytical model representing uniaxial strain GNR carrier statistic is derived based on the energy band structure established by Mei et al. . The strain effect in our model is limited to low strain, and only the first subband of the AGNR n=3m and n=3m+1 families is considered. In the following section, the analytical modeling of the uniaxial strain AGNR model is presented.
Uniaxial strain AGNR model
where , , t0=−2.74 eV is the unstrained hopping parameter, a=0.142 nm is the lattice constant and t1 and t2 are the deformed lattice vector hopping parameter of the strained AGNR. ε is the uniaxial strain .
The analytical model presented in this section is plotted and discussed in the following section.
Results and discussion
The hopping integral t0 between the π orbitals of AGNR is altered upon strain. This causes the up and down shift, the σ∗ band, to the Fermi level, E F . These two phenomena are responsible for the bandgap variation. It has been demonstrated that GNR bandgap effect with strain is in a zigzag pattern . This observation can be understood by the shifting of the Dirac point perpendicular to the allowed k lines in the graphene band structure and makes some bands closer to the Fermi level [7, 8]. Hence, the energy gap reaches its maximum when the Dirac point lies in between the two neighboring k lines. The allowed k lines of the two families of the AGNR have different crossing situations at the K point . This may explain the different behaviors observed between n=3m and n=3m+1 family.
In this paper, the uniaxial strain AGNR for n=3m and n=3m + 1 family carrier statistic is analytically modeled, and their behaviors are studied. It is found that uniaxial strain gives a substantial effect to AGNR carrier statistic within the two AGNR families. The AGNR carrier concentration has not been influenced by the uniaxial strain at low normalized Fermi energy. It is also shown that the uniaxial strain mostly affects carrier velocity at a high concentration of n≈3.0×107 m −1 and n≈1.0×107 m −1 for n=3m and n=3m+1, respectively. In addition, the Fermi velocity of the AGNR n=3m and n=3m+1 exhibits decrements upon the strain. Results obtained give physical insight on the understanding of the uniaxial strain effect on AGNR. The developed model in this paper representing uniaxial strain AGNR carrier statistic can be used to further derive the current-voltage characteristic. This computational work will stimulate experimental efforts to confirm the finding.
The authors would like to acknowledge the financial support from the Research University grant of the Ministry of Higher Education (MOHE), Malaysia under project number R.J130000.7823.4F146. Also, thanks to the Research Management Centre (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.
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