1. Introduction
As early as 1947, the tight-binding electronic energy spectrum of a graphene sheet had been investigated by Wallace (Wallace, 1947). The work of Wallace showed that the electronic properties of a graphene sheet were metallic. A better tight-binding description of graphene was given by Saito et al. (Saito et al., 1998). To understand the different levels of approximation, Reich et al. started from the most general form of the secular equation, the tight binding Hamiltonian, and the overlap matrix to calculate the band structure (Reich et al., 2002). In 2009, a work including the non-nearest-neighbor hopping integrals was given by Jin et al. (Jin et al., 2009).
It is common knowledge that a perfect grphene sheet is a zero-gap semiconductor (semimetal) that exhibits extraordinarily high electron mobility and shows considerable promise for applications in electronic and optical devices, high sensitivity gas detection, ultracapacitors and biodevices. How to open the gap of graphene has become a focus of the study. Early in 1996, Fujita et al. started to study the electronic structure of graphene ribbons (Fujita et al., 1996; Nakada et al., 1996) by the numerical method. The armchair shaped edge ribbons can be either semiconducting (
In this Chapter, we focus on the effects of deformed graphene sheets and nanoribbons under unaxial stress on the electronic energy spectra and gaps based on the elasticity theory. Meanwhile, the energy spectrum of the curved graphene nanoribbons with the tubular warping is studied by the tight-binding approach. The energy spectrum of deformed graphene sheets subjected to unaxial stress is given in Section 2. In Section 3, we discuss the electronic properties of graphene nanoribbons under unaxial stress. The tubular warping deformation of graphene nanoribbons is presented in last Section.
2. Graphene under uniaxial stress
2.1. Elasticity theory
Since graphene is a monolayer structure of carbon atoms, when a force is exerted on it parallel to its plane, the positions of the atoms will change with respect to some origin in space. Let the x-axis be in the direction of the armchair edge of graphene and the y-axis in that of the zigzag edge, as shown in Fig. 1. Let
where
According to Harrison’s formula (Harrison, 1980), the hopping energy after deformation is expressed as follows
where
The nearest neighbour hopping integrals associated with the bond lengths are
If graphene is subject to a tensile force in the y-direction, the hopping energies are given by
2.2. The tight-binding energy spectrum
Let us now consider the band structure from the viewpoint of the tight-binding approximation. The structure of graphene is composed of two types of sublattices
where
and
The first sum is taken over
we obtain the secular equation
For the tensile stress in the x-direction, the solution to Eq.(9) is
where
Fig. 2 shows the electronic energy spectra of deformed graphene sheets for some high symmetric points
3. Graphene nanoribbon under uniaxial stress
As mentioned in Section 2, for a graphene sheet subject to uniaxial stress there are no energy gaps at Dirac point. How to open the energy gaps of graphene? Studies showed that we can realize this goal by deducing the size of graphene, i.e., changing its toplogical propertiy(Son et al., 2006). On the other hand, the band gaps of graphene nanoribbons can be mamipulated by changing the bond lengths between carbon atoms, i.e., changing the hopping integrals, by exerting a strain force (Sun et al., 2008). The nearest-neighbor energy spectrum of an armchair nanoribbon was given by the tight-binding approach and using the hard-wall aboundary condition (Zheng et al., 2007). In the non-nearest-neighbor band structure of the nanoribbon was given by Jin et al (Jin et al., 2009). In this section we use the tight-binding approach to study the energy spectrum and gap of the nanoribbon under uniaxial stress along the length direction, i.e., x-direction, of the nanoribbon, as shown in Fig. 3.
Since the unit cell of the nanoribbon has the translational symmetry in the
where
Since the electronic energy spectrum of the perfect armchair nanoribbon depends strongly on the width of the nanoribbon, the different width has the different spectrum. For instance, the nanoribbon with widths
Fig. 4 shows the energy spectra of three kinds of the nanoribbons under uniaxial stress, and in which the next-nearest neighbor hopping integrals are taken into account. In order to facilitate comparison, the energy spectrum of the undeformed nanoribbon is given in Fig. 4. When width
When the stress is constant, three graphs of the energy gaps with the width of the nanoribbon changes are shown in Fig. 5, where (a) the width
On the other hand, in order to make certain of the relationship between the gap and the stress, the curves of the gap versus the stress are given in Fig. 6. As shown in Fig. 6, the gap increases as the stress increases for the 3
4. The tubular warping graphene nanoribbon
4.1. Theoretical Model
In this section we choose an armchair ribbon as an example and which is bent into the tubular shape (cylindrical shape), as shown in Fig. 7. This tubular ribbon still has the periodicity in its length direction, but its dimensionality has changed. The consequence of such a dimension change is to lead to the change of the electronic energy dispersion relation. This is because the sp2 hybridization of a flat ribbon turns into the sp3 hybridization of a curved ribbon, i.e., the curvature of graphene nanoribbons will result in a significant rehybridization of the
Because of the curl of the ribbon, the wavefunction of
where
and
Here
4.2. Results and Discussion
To clearly understand the effect of curvature, we choose the width
By comparison, we found that the energy bandwidths become narrowed obviously for the widths
5. Graphene nanoribbon modulated by sine regime
A free standing graphene nanoribbon could have out-of-plane warping because of the edge stress (Shenoy et al., 2008). This warping will bring about a very small change of the electronic energy spectrum. An ideal graphene nanoribbon only has periodicity in the direction of its length and there is no periodicity in the y-direction. To show the periodic effect in the y-direction, we modulate it with the aid of a sine periodic function
where
By numerical calculations, we found that different modulation frequencies have different electronic band structures, i.e., the energy band structures depend strongly on the modulation frequency
When the modulation amplitude, taken to be 0.1nm, is fixed, different modulation frequencies have slightly different densities of states of electrons. The main difference between the frequencies 0.0nm-1, 5.0nm-1, and 10.0nm-1 is in the conduction band and the density of states of the valence band is the same nearly. It follows that the modulation along the width direction of the ribbon makes a notable impact for the density of states of the conduction band, especially for the high energy band corresponding to the standing wave of the smaller quantum number. In order to reveal the effect of the modulation amplitude on the electronic properties, the energy bands for the different amplitudes are calculated under certain frequency. Fig. 14 shows the band structures of the different amplitudes
6. Conclusion
We investigated the electronic energy spectra of graphene and its nanoribbon subject to unaxial stress within the tight-binding approach. The unaxial stress can not open the energy gap of graphene at Dirac point K. But compression along the armchair shape edge or extension along the zigzag shape edge will make a small energy gap opened at K point. From this reason, the graphene subject to uniaxial stress still is a semiconductor with the zero-energy gaps. The position of Dirac point will vary as the stress. For the armchair graphene nanoribbon, the tensile or compressive stress not only can transfer the metallicity into the semiconductor, but also have the energy gap increased or decreased and the energy bandwidth widened or narrowed. Therefore, we can use the unaxial stress to control the electronic properties of armchair graphene nanoribbons. In addition, the tubular warping deformation of armchair nanoribbons does not nearly influence on the energy gap, but it is obvious to effect on the bandwidth. In addition, we also studied the periodic modulation of the shape of armchair nanoribbons by sine regime. This modulation can change its electronic properties. For the other modulation manner, we no longer discuss it here.
The advantage of the tight-binding method is that the physical picture is clearer and the calculating process is simpler compared to the first-principles calculations. This method is suitable only for narrow energy bands. Because graphene nanoribbons are the system of wider energy bands, this method has its limitation.
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