† Corresponding author. E-mail:
Project supported by the Ministry of Higher Education (MOHE), Malaysia under the Fundamental Research Grant Scheme (FRGS) (Grant No.Q.J130000.7823.4F477). We also thank the Research Management Center (RMC) of Universiti Teknologi Malaysia (UTM) for providing an excellent research environment.
The effect of tensile uniaxial strain on the non-parabolic electronic band structure of armchair graphene nanoribbon (AGNR) is investigated. In addition, the density of states and the carrier statistic based on the tight-binding Hamiltonian are modeled analytically. It is found that the property of AGNR in the non-parabolic band region is varied by the strain. The tunable energy band gap in AGNR upon strain at the minimum energy is described for each of n-AGNR families in the non-parabolic approximation. The behavior of AGNR in the presence of strain is attributed to the breakable AGNR electronic band structure, which varies the physical properties from its normality. The linear relation between the energy gap and the electrical properties is featured to further explain the characteristic of the deformed AGNR upon strain.
Nowadays, non-silicon material has gained a great deal of interest among researchers to develop new structures for future nanoelectronics. Graphene is especially believed to overcome the limitation in the silicon technology and takes all of the considerations of making a good transistor. Physically, graphene refers to the two-dimensional (2D) crystal originated from a single sheet of sp2 carbon atoms.[1] Despite its surprising features such as high mobility and high Fermi velocity, graphene is one of the best gapless materials whose energy band is intersecting near the edges of the Brillouin zone.[2–4] The absence of an energy gap in graphene has been seen as something crucial and rather complex to realize its application for future FET. Instead of two-dimensional graphene, sub-10 nm width graphene stripes called graphene nanoribbons (GNRs) with both zigzag (ZGNR) and armchair (AGNR) edge configurations are one-dimensional by behavior.[5–7] It has been revealed by theoretical calculations that AGNR and ZGNR are both semiconductors,[8] however the appearance of a band gap in AGNR is due to the quantum confinement and edge effects while the band gap in ZGNR is a consequence of the sub-lattice potential variation.[9] As we focus on the graphene structure with sub 10 nm width which highlights the quantum confinement effect on the width direction, the AGNR family is considered. Unlike 2D-graphene, the lateral confinement in GNRs introduces an energy gap close to the Fermi level for AGNRs of 3m and 3m + 1 width, while 3m + 2 AGNRs remain metallic. It is not easy to obtain an appropriate energy gap in AGNR since the gap strongly depends on the crystallographic direction of the ribbon axis and the family width. The 1–2 nm width GNR can open up a suitable energy gap usable for the FET channel.[10,11] However, ultra-narrow GNRs (≥ 2 nm) with smooth edges are not easy to be formed while wider AGNR's ribbon width contributes to the smaller energy gap splits. So, it is desirable to alter the GNR for high performance applications. Most recently, applying a strain to the base-transistor material has become a trend in the semiconductor industry. The strain potentially alters the band gap and helps improve the mobility of the charge carrier in the channel, hence boosting the device performance in silicon technology.[12] The same intention has been studied in carbon nanotubes (CNT). It is remarkable to open up a band gap in a metallic CNT and modify the band gap in a semiconductor CNT.[13] Theoretical and experimental studies conducted by a few researchers have found the unique behavior dependences on AGNR width upon strain. The electronic band structure was demonstrated in a zigzag pattern for all three groups of AGNRs,[14,15] while the carrier statistic behaved linearly with the band structure alteration by strain.[16] Additionally, the strain effect on GNR has been explored by scanning tunnelling microscopy,[17] furthermore its effects on the GNR electronic and optical properties have been explored.[18,19] Even though there are several works that have been done pertaining to strain AGNR, the observation on its properties is still lacking.[20] Therefore, modelling and simulations are provided in this paper primarily to observe the effect of the uniaxial strain on AGNR.
In this work, AGNR is assumed to be tensile under a smaller uniaxial strain, as shown in Fig.
The sub band structure model plays an important role in predicting the property of quasi one-dimensional (Q1D) strain AGNR. In the Q1D material, the carriers are confined in two directions which are the y and z directions. The carriers are allowed to move in the x direction along the length of the nanoribbon, Lx.
It is well known that the energy E(k) and bandgap in semiconducting AGNR are functions of the AGNR width and can be modelled using either non-parabolic or parabolic approximation as shown in Fig.
![]() | Fig. 2. The energy dispersion relation of AGNR in the parabolic and non-parabolic band structures near the Fermi point. |
For the AGNR at k = 0, the band structure is parabolic for both approaches, however the approximation is different for the range out of the parabolic regime. In all circumstances, the non-parabolic approximation is valid to describe all the physical phenomena for AGNR in various energy ranges. By considering an energy level near one of the valleys in the band structure, the energy dispersion relation near the Fermi level in the non-parabolic approximation can be expressed as
![]() | Table 1. Strained AGNR deformed parameters.[15] . |
The parameters represent the deformed bond length and lattice vectors after the application of the strain. The numbers of dimer lines, n, for three groups of AGNRs are 3m, 3m + 1, and 3m + 2, and the sub band indices, p, are determined as 2m + 1, 2m + 1, and 2m + 2, respectively, where m is an integer. The kx represents the wave vector component in the −x direction which is quantized by the periodic boundary condition. By using the Taylor series expansion function, equation (
The conduction and valence bands in the electronic band structure of AGNR exhibit a symmetrical structure as indicated by the ± sign. Figure
![]() | Fig. 3. The uniaxial strain effect on AGNR band structure for (a) 3m AGNR, (b) 3m + 1 AGNR, and (c) 3m + 2 AGNR, where m = 6 is a positive integer. |
It can be observed from the plot that the energy gap splits out linearly for 3m and 3m + 2 AGNRs in the same family (Figs.
From the band structure, we can obtain the density of states (DOS). In the non-parabolic approximation, the DOS can be described as
The density of state is shown in Fig.
![]() | Fig. 4. The uniaxial strain effect on AGNR density of states near the Fermi level for (a) 3m AGNR, (b) 3m + 1 AGNR, and (c) 3m + 2 AGNR, with m = 6. |
The densities of state for the three families of AGNRs are derived. From the results in Fig.
Generally, the number of states influences the number of carriers in the system. The total carrier concentration in the non-parabolic band of strained AGNR is obtained by integrating the density of states DOS(E) and Fermi–Dirac distribution function f(E) over energy as
However, it is not in the form of a simple integral. Hence, the integral part is numerically solved as
![](cpb_25_9_096802/cpb151801ieqn1.gif)
The numerical integral is the most identical approach to describe the concentration of carriers in both regimes as shown in Fig.
Because of the technological limit on conventional materials, researchers are engaged with new resources such as graphene. Graphene has a non-parabolic dispersion relation on its band structure which is highlighted in this study. Based on the non-parabolic energy band, the relation between the energy and the total number of allowed states in the conduction or valence band is investigated. The numerical result is obtained. Since the nano scale device operation regime is mainly the degenerate regime instead of the non-degenerate limit, therefore the concentrations of carriers in both regimes are modelled. Finally, the effect of strain on the AGNR carrier statistic is numerically explored and the amount of carriers is reported to be increased for 3m AGNR and 3m + 2 AGNR when uniaxial strain is added.
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