# A Review of Electronic Band Structure of Graphene and Carbon Nanotubes Using Tight Binding

Authors: Davood Fathi
Journal: Journal of Nanotechnology
 (2011)

---

## Abstract

The electronic band structure variations of single-walled carbon nanotubes (SWCNTs) using Huckle/tight binding approximation theory are studied. According to the chirality indices, the related expressions for energy dispersion variations of these elements are derived and plotted for zigzag and chiral nanotubes.


---

## Body

## 1. Introduction

Carbon nanotubes (CNTs) are graphene sheets rolled up into cylinders with diameter of the order of a nanometer varying from 0.6 to about 3 nm [1]. Depending on their chirality (the direction along which the graphene sheets are rolled up), they can be either metallic with no bandgap, or semiconducting with a distinct bandgap [2].Because of their extremely desirable properties of high mechanical and thermal stability, high thermal conductivity, and unique electrical properties such as large current carrying capacity [3–7], CNTs have aroused a lot of research interest in their applicability as VLSI interconnects of the future.Semiconducting CNTs are being extensively studied as the future channel material for ultrahigh performance and scaled field-effect transistors (FETs) and are expected to be the successors of silicon transistors. Interconnect technology has to be commensurately scaled to reap the benefits of these novel transistors. Metallic CNTs have been identified as possible interconnect material of future technology generations and the heir to aluminum (Al) and Cu interconnects [8].


## 2. The Energy Variations of Graphene

Graphite is a 3D (three-dimensional) layered hexagonal lattice of carbon atoms and a single layer of graphite forms a 2D (two-dimensional) material, called 2D graphite or a graphene layer [9, 10]. Figure 1 shows the lattice of a graphene sheet in which the two fundamental carbon atoms 1 and 2 are the basic elements of overall lattice and form a unit cell. Thus, the lattice of unit cells is periodic.Figure 1
The periodic lattice of graphene consisting of the unit cell of two carbon atoms.Each point on the periodic lattice of Figure1 can be described by  R⃗=ma⃗1+na⃗2 where m and n are two integers, a1 and a2 are the two unit vectors which are defined as (1)a⃗1=a(32,12),a⃗2=a(32,-12),
where a=2.46Å is the lattice constant of graphene [11]. The tight binding theorem implies that [11](2)ϕ=c1ϕ1+c2ϕ2,
where ϕ is the wave function due to the unit cell, and ϕ1 and ϕ2 are the wave functions related to the 2py atomic orbitals of atoms 1 and 2 in Figure 1, respectively, and c1 and c2 are two constants. We will be using Bloch’s theorem [11](3)ψ(x)=∑ReiK⃗⋅R⃗ϕ(x-R),
where ψ(x) is the total wave function of lattice, K⃗ is the wave vector, and R is the lattice vector. With considering the overlap between the two above-mentioned orbitals, we will have(4)〈ϕ1(x)|H|ψ(x)〉=ε〈ϕ1(x)∣ψ(x)〉,〈ϕ2(x)|H|ψ(x)〉=ε〈ϕ2(x)∣ψ(x)〉,
where H is the Hamiltonian operator [11] and ε  is the energy dispersion of graphene lattice. Also using the previously mentioned relations, we can write(5)〈ϕ1(x)|H|ψ(x)〉=εc1,〈ϕ2(x)|H|ψ(x)〉=εc2.
With noticing Huckel/tight binding approximation and the previous relations, we will have(6)〈ϕ1(x)|H|ψ(x)〉=c1α+c2β(1+e-iK⃗⋅a⃗1+e-iK⃗⋅a⃗2),〈ϕ2(x)|H|ψ(x)〉=c2α+c1β(1+eiK⃗⋅a⃗1+eiK⃗⋅a⃗2).
By substituting (4)–(5) in (6), we can obtain(7)c1(α-ε)+c2β(1+e-iK⃗⋅a⃗1+e-iK⃗⋅a⃗2)=0,c1β(1+eiK⃗⋅a⃗1+eiK⃗⋅a⃗2)+c2(α-ε)=0.
For having nonzero responses for the homogenous equation (7), the following condition should be established(8)|α-εβ(1+e-iK⃗⋅a⃗1+e-iK⃗⋅a⃗2)β(1+eiK⃗⋅a⃗1+eiK⃗⋅a⃗2)α-ε|=0.
With solving (8), we obtain the total energy dispersion variations as (9)ε=α±β3+cos⁡(K⃗⋅a⃗1)+2cos⁡(K⃗⋅a⃗2)+2cos⁡[K⃗⋅(a⃗1-a⃗2)].With considering that(10)K⃗⋅a⃗1=(kx+iky)⋅a(32+i12)=a2(3kx+ky),K⃗⋅a⃗2=(kx+iky)⋅a(32-i12)=a2(3kx-ky),K⃗⋅(a⃗1-a⃗2)=(kx+iky)⋅(ia)=aky,
where K⃗=(kx+iky), which kx and ky are the wave numbers related to the reciprocal lattice. Therefore, the energy dispersion variations versus kx and ky will be obtained as(11)ε=α±β3+cos⁡(a2(3kx+ky))+2cos⁡(a2(3kx-ky))+2cos⁡(aky).In Figure 2, the energy dispersion variations ε in (11) has been plotted versus kx and ky in the range of [-2π/a,2π/a], using MATLAB [12].Figure 2
The energy dispersion variations of graphene lattice.In Figure3, the primitive unit cell and the Brillouin zone, related to the graphene lattice and the reciprocal lattice of graphene, respectively, have been shown. In this figure a⃗1 and a⃗2 are the unit vectors of the graphene lattice, respectively, and b⃗1 and b⃗2 are the unit vectors of the reciprocal lattice of graphene, respectively.Figure 3
The primitive unit cell and the Brillouin zone in graphene.We can express the reciprocal lattice vectorsb⃗1 and b⃗2 versus the lattice vectors a⃗1 and a⃗2 as (12)b⃗1=2πa⃗2×z⃗0a⃗1⋅(a⃗2×z⃗0),b⃗2=2πz⃗0×a⃗1a⃗2⋅(z⃗0×a⃗1),
where z⃗0 is the unit vector along the z-axis, which will play no important role in our discussion since we talk about the electronic states in the x-y plane assuming that different planes along the z-axis are isolated [9]. By substituting (1) in (12) we will have(13)b⃗1=(2π3a,2πa),b⃗2=(2π3a,-2πa).


## 3. The Energy Dispersion Variations of an SWCNT

In Figure4, the vectors definition of graphene plane for converting to a carbon nanotube has been shown where C⃗h, θ, and T⃗ are the chirality (circumference) vector, the chirality angle, and the translational vector, respectively. With considering that OA=|C⃗h| and OB=|T⃗| we can express C⃗h versus the unit vectors a⃗1 and a⃗2 as (14)C⃗h=na⃗1+ma⃗2,
where n and m are two integer numbers and are defined as carbon nanotube indices [11, 13]. Also, we can express the diameter of carbon nanotube versus a, n, and m as [11, 13](15)dt=|C⃗h|π=an2+m2+nmπ.
On the other hand, the vector T⃗ can be defined versus the unit vectors a⃗1 and a⃗2 as [11](16)T⃗=t1a⃗1+t2a⃗2,
where(17)t1=2m+ndR,t2=-2n+mdR,dR=GCDof(2m+n),(2n+m),
which we can express dR easier as   (18)dR={3d,if(n-m)ismultipleof3d,d,if(n-m)isnotmultipleof3d,
where (19)d=GCDofn,m.
Figure 5 shows the vectors of reciprocal lattice of graphene. In this figure, k⃗⊥ and k⃗|| are the reciprocal lattice vector related to C⃗h and the reciprocal lattice vector related to T⃗, respectively. With considering that C⃗h and T⃗ are orthogonal to each other, we have C⃗h·k⃗⊥=2π, C⃗h·k⃗||=0, T⃗·k⃗||=2π, and T⃗·k⃗⊥=0.Figure 4
Vectors definition of graphene for converting to a carbon nanotube.Figure 5
Vectors definition for the reciprocal lattice of graphene.For obtainingk⃗⊥ and k⃗|| versus the other parameters, we have |C⃗h×T⃗| equal to the area of CNT unit cell, and |a⃗1×a⃗2| equal to the area of primitive unit cell (as in Figure 3). Thus the number of primitive unit cells per CNT unit cell will be as(20)N=|C⃗h×T⃗||a⃗1×a⃗2|.
It should be noted that in above relations, “×” and “·” are the outer product and the inner product representations, respectively. Therefore the vectors k⃗⊥ and k⃗|| can be expressed as(21)k⃗⊥=1N(-t2b⃗1+t1b⃗2),k⃗||=1N(mb⃗1-n1b⃗2).
The energy dispersion relation of an SWCNT (single-walled carbon nanotube) can be obtained from the energy relation of graphene sheet, which the related nanotube is made up of. With considering the periodic boundary conditions on C⃗h, we find that the wave vector k⃗⊥ associated with C⃗h (circumference) direction is quantized [11]. On the other hand, for a one-dimensional nanowire such as a carbon nanotube with the length ℓ, the wave vector associated with T⃗ (translational) direction is discrete with (22)Δk=2πl.
It should be noted that for a carbon nanotube of infinite length, as cleared from (22), the wave vector along the nanotube axis can be assumed continuous. Since in carbon nanotube which is a one-dimensional material, only k⃗|| is a reciprocal lattice vector and k⃗⊥ gives discrete k values in the direction of C⃗h.Since an SWCNT is a rolled-up sheet of graphene, the energy band structure can be obtained simply from that of two-dimensional graphene. This work can be done easily by imposing appropriate boundary conditions in the circumferential direction around the SWCNT [11, 14]. As shown in Figure 6, the one-dimensional band structure of SWCNTs can be obtained from cross-sectional cutting of the energy dispersion of two-dimensional graphene.Figure 6
The 1D band structure of an SWCNT is obtained by cross-sections of 2D energy dispersions for (b) a metallic SWCNT and (c) a semiconducting SWCNT [14].For the continuous wave vectork⃗ along the nanotube axis, we can write the energy dispersion variations for one-dimensional carbon nanotube, using the two-dimensional graphene relation (11) as [11]  (23)ES(k⃗)=ε(sk⃗⊥+k⃗|||k⃗|||k),
where S=0,1,…,N-1 and -π/a<k<π/a. This means that the N pairs of energy dispersion curves given by (23), correspond to the cross-sections of the two-dimensional energy dispersion given by (11) and shown in Figure 2. These cross-sections are made on [sk⃗⊥+(k⃗||/|k⃗|||)k] lines.For a zigzag carbon nanotube withn=0 and m=6, we can obtain the parameters dR, d, t1, and t2 using (17)–(19) equal to 6, 6, 2, and −1, respectively. Also C⃗h and T⃗ can be obtained using (14), (16) equal to 6a2 and (2a⃗1-a⃗2), respectively. Thus using (20), N can be obtained equal to 12. Using (21), the parameters k⃗⊥ and k⃗|| will be calculated as k⃗⊥=(b⃗1+2b⃗2)/12 and k⃗||=b⃗1/12, respectively. Therefore, the argument in (23) will be(24)sk⃗⊥+k⃗|||k⃗|||k=(s12+k|b⃗1|)b⃗1+s6b⃗2.
Using (13) for b⃗1 and b⃗2, (24) will be obtained as(25)sk⃗⊥+k⃗|||k⃗|||k=(s23πa+k2)+i(-s6πa+32k),
where i presents the imaginary part. Recall that K⃗=kx+iky, (25) implies that for calculating the energy dispersion variations of CNT, it is adequate to replace kx and ky in (11) with the real part and the imaginary part of (25), respectively, as(26)kx(0,6)=s23πa+k2,ky(0,6)=-s6πa+32k.
With a similar way as described above, we can obtain kx and ky for the case that n=0 and m=5, as(27)kx(0,5)=3s53πa+k2,ky(0,5)=-s5πa+32k.
In Figure 7, the energy dispersion variations versus k have been plotted for the two carbon nanotubes, which one nanotube is metallic with n=0 and m=6 and the other is semiconducting with n=0 and m=5. As shown in this figure, the band gap for the metallic nanotube is almost zero, and for the semiconductive nanotube is a nonzero value.Figure 7
The energy dispersion variations of zigzag carbon nanotubes. One nanotube is metallic withn=0 and m=6, and the other is semiconductive with n=0 and m=5.For a chiral carbon nanotube withn=4 and m=2, with the similar way as described for the two zigzag nanotubes in Figure 7, kx and ky will be obtained as(28)kx(4,2)=9s143πa-k27,ky(4,2)=s14πa+3327k.
In Figure 8, the energy dispersion variations versus k has been plotted for a chiral carbon nanotube with n=4 and m=2, which is neither metallic nor semiconductive.Figure 8
The energy dispersion variations of a chiral carbon nanotube, withm=2 and n=4.


## 4. Conclusions

In this paper we have studied the basic structure of graphene and its resulted element carbon nanotube. Using the tight binding approximation theory, we have analyzed the variations of energy band gap for SWCNTs (single-walled carbon nanotubes). According to the chiral indices, the related expressions for energy dispersion variations of these elements have been analyzed and also plotted using MATLAB [12] for zigzag and chiral nanotubes.



---

*Source: 471241-2011-09-11.xml*