The Effects of Next-to-Nearest Neighbor Hopping Amplitude on Electrical Properties of Graphene Like Nanotube Structure

In this work, we apply tight binding model Hamiltonian to describe electrons on the honeycomb lattice. To calculate the electrical conductivity of graphene like nanotube we consider monolayer graphene which is folded along zigzag or armchair directions. The lattice structure of each graphene layer has been shown in Fig. 1. The primitive unit cell vectors are given by

$$\begin{eqnarray}&&{{\bf{a}}}_{1}=\displaystyle \frac{a\sqrt{3}}{2}{\bf{i}}+\displaystyle \frac{a{\bf{j}}}{2},\ {{\bf{a}}}_{2}=\displaystyle \frac{a\sqrt{3}}{2}{\bf{i}}-\displaystyle \frac{a{\bf{j}}}{2},\end{eqnarray} \tag{ 1 }$$

where ${\bf{i}}$ and ${\bf{j}}$ are unit cell vectors along zigzag and armchair directions, respectively. Also the length of unit cell vectors is considered to be a. In order to obtain electrical transport properties of graphene like nanotubes we must first examine the band structure and provide the expression for the electronic Green's function. We start from a tight binding model incorporating nearest and next nearest neighboring hopping terms. A finite difference between on-site energies of two different sublattice atoms of honeycomb structure has been applied in the model Hamiltonian. Up to the next nearest neighbor approximation, the spin independent tight binding model hamiltonian for monolayer graphene (H) is given by

$$\begin{eqnarray}\begin{array}{rcl}H & = & -t\sum _{i,\delta }({b}_{i+\delta }^{\dagger }{a}_{i}+h.c.)+{\rm{\Delta }}\sum _{i}{a}_{i}^{\dagger }{a}_{i}-{\rm{\Delta }}\sum _{i}{b}_{i}^{\dagger }{b}_{i}\\ & & -\ t^{\prime} \sum _{i,\delta ^{\prime} }({b}_{i+\delta ^{\prime} }^{\dagger }{b}_{i}+{a}_{i+\delta ^{\prime} }^{\dagger }{a}_{i}+h.c.)-\mu \sum _{i}\left({b}_{i}^{\dagger }{b}_{i}+{a}_{i}^{\dagger }{a}_{i}\right)\end{array}\end{eqnarray} \tag{ 2 }$$

In Eq. 2, ${a}_{i}({b}_{i})$ denotes the annihilation operator for an electron which is on an A(B)-atom site with unit cell label i in the graphene layer. ${\rm{\Delta }}$ in Eq. 2 indicates the on-site energy of electrons on atoms of sublattice A. Also Δ introduces the gap parameter.

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δ is one of three vectors that connects the unit cells of nearest-neighbor lattice sites and given by $\delta =0,{{\bf{a}}}_{1},{{\bf{a}}}_{2}$ according to Fig. 1. Moreover $\delta ^{\prime}$ connects the unit cells of next nearest-neighbor lattice sites and is one of the six vectors given by $\delta ^{\prime} ={{\bf{a}}}_{1},{{\bf{a}}}_{2},-{{\bf{a}}}_{1},-{{\bf{a}}}_{2},{\bf{j}},-{\bf{j}}$.

$t\approx 2.6\,\mathrm{eV}$ is the nearest neighbour intralayer hopping terms for electrons to move within a given plane. $t^{\prime}$ refers to the next nearest neighbor hopping integral for electrons. μ introduces the chemical potential which determines the electron density. The following Fourier transformations for fermionic operators ${a}_{i}^{\dagger }$, ${b}_{i}^{\dagger }$ have been given by

$$\begin{eqnarray}&&{a}_{{\bf{k}}}^{\dagger }=\displaystyle \frac{1}{\sqrt{N}}\sum _{i}{e}^{-i{\bf{k}}.{{\bf{R}}}_{i}}{a}_{i}^{\dagger },\ {b}_{{\bf{k}}}^{\dagger }=\displaystyle \frac{1}{\sqrt{N}}\sum _{i}{e}^{-i{\bf{k}}.{{\bf{R}}}_{i}}{b}_{i}^{\dagger },\end{eqnarray} \tag{ 3 }$$

where N is the number of unit cells and ${{\bf{R}}}_{i}$ introduces the position vector of i th unit cell in graphene layer. ${\bf{k}}$ is wave vector belonging to the first Brillouin zone of nanotube structure. Depending on the zigzag or armchair type of nanotube, the region of wave vectors ${\bf{k}}$ are determined. Based on Fig. 1, axis direction of zigzag nanotube is resulted to be along x. In other words nanotubes of the type $(n,0)$ are called zigzag tubes, because they exhibit a zigzag pattern along the circumference. A single-wall zigzag nanotube is geometrically obtained by rolling up a single graphene layer around x direction according to Fig. 1. We introduce ${ \mathcal R }={an}$ is the length of circumference of zigzag nanotube. a denotes the length of each of unit cell vector. Periodic boundary condition for zigzag nanotube $(n,0)$ define the small number of allowed wave vectors k_y in the circumferential direction

$$\begin{eqnarray}\begin{array}{rcl}{e}^{{{ik}}_{y,q}{ \mathcal R }} & = & {e}^{i2\pi q},\\ {{nk}}_{y,q}a & = & 2\pi q\longrightarrow {k}_{y,q}=\displaystyle \frac{2\pi q}{{na}},\end{array}\end{eqnarray} \tag{ 4 }$$

where integer quantum number q is determined using the first Brillouin zone of honeycomb structure. Based on the first Brillouin zone of honeycomb lattice k_y is into the region $-\tfrac{4\pi }{3a}\lt {k}_{y}\lt \tfrac{4\pi }{3a}$. Using Eq. 4, one can write $-\tfrac{4\pi }{3a}\lt \tfrac{2\pi q}{{na}}\lt \tfrac{4\pi }{3a}$. Thus we have $-\left[\tfrac{2n}{3}\right]\lt q\lt \left[\tfrac{2n}{3}\right]$ for zigzag nanotube $(n,0)$. A similar way can be considered for finding the region of wave vectors of armchair graphene like nanotube. Nanotubes of the type $(n,0)$ are called zigzag tubes, because they exhibit an armchair pattern along the circumference. A single-wall armchair nanotube is geometrically obtained by rolling up a single graphene layer around y direction according to Fig. 1. The length of circumference of zigzag nanotube is obtained as ${ \mathcal R }={an}\sqrt{3}$. Now periodic boundary condition for zigzag nanotube $(n,0)$ define the small number of allowed wave vectors k_y in the circumferential direction

$$\begin{eqnarray}\begin{array}{rcl}{e}^{{{ik}}_{x,q}{ \mathcal R }} & = & {e}^{i2\pi q},\\ {{nk}}_{x,q}a & = & 2\pi q\longrightarrow {k}_{x,q}=\displaystyle \frac{2\pi q}{{an}\sqrt{3}}.\end{array}\end{eqnarray} \tag{ 5 }$$

The integer quantum number q is determined using the first Brillouin zone of honeycomb structure. Based on Fig. 1 and the first Brillouin zone of honeycomb structure, k_x satisfies the expression $-\tfrac{2\pi }{\sqrt{3}a}\lt {k}_{x}\lt \tfrac{2\pi }{\sqrt{3}a}$. Using Eq. 4, one can write $-\tfrac{2\pi }{\sqrt{3}{an}}\lt \tfrac{2\pi q}{{an}\sqrt{3}}\lt \tfrac{2\pi }{\sqrt{3}a}$. Thus we have $-n\lt q\lt n$ for armchair nanotube $(n,n)$. Based on above statements, we can introduce the following wave vector regions for each type of nanotube

$$\begin{eqnarray}\begin{array}{rcl}{for}\ {zigzag}\ {nanotube}\ \longrightarrow {k}_{y} & = & \displaystyle \frac{2\pi q}{{na}}\ {with}\\ & & -\ \left[\displaystyle \frac{2n}{3}\right]\lt q\lt \left[\displaystyle \frac{2n}{3}\right],\\ & & -\ \displaystyle \frac{2\pi }{\sqrt{3}a}\lt {k}_{x}\lt \displaystyle \frac{2\pi }{\sqrt{3}a}\\ {for}\ {armchair}\ {nanotube}\ \longrightarrow {k}_{x} & = & \displaystyle \frac{2\pi q}{{an}\sqrt{3}}\ {with}\ -n\lt q\lt n,\\ & & -\ \displaystyle \frac{4\pi }{3a}\lt {k}_{y}\lt \displaystyle \frac{4\pi }{3a}.\end{array}\end{eqnarray} \tag{ 6 }$$

In terms of Fourier transformation of operators, one can rewrite the clean tight binding part of the Hamiltonian in Eq. 2 as

$$\begin{eqnarray}&&H=\sum _{{\bf{k}}}{\phi }_{{\bf{k}}}^{\dagger }H({\bf{k}}){\phi }_{{\bf{k}}},\end{eqnarray} \tag{ 7 }$$

in which the vector of fermion creation operators is defined as ${\phi }_{{\bf{k}}}^{\dagger }=({a}_{{\bf{k}}}^{\dagger },{b}_{{\bf{k}}}^{\dagger })$. Under next-to-nearest neighbor approximation the following matrix form for $H({\bf{k}})$ is given as

$$\begin{eqnarray}H({\bf{k}})=\left(\begin{array}{cc}{\rm{\Delta }}+g({\bf{k}})-\mu & f({\bf{k}})\\ {f}^{* }({\bf{k}}) & -{\rm{\Delta }}+g({\bf{k}})-\mu \end{array}\right).\end{eqnarray} \tag{ 8 }$$

Using the definitions for δ and $\delta ^{\prime}$ the functions $f({\bf{k}})$ and $g({\bf{k}})$ are expressed based on the following relations

$$\begin{eqnarray}\begin{array}{rcl}f({\bf{k}}) & = & -t\left(1+2\cos \left(\displaystyle \frac{{k}_{y}a}{2}\right)\exp \left(-i\displaystyle \frac{\sqrt{3}}{2}{{ak}}_{x}\right)\right),\\ g({\bf{k}}) & = & -2t^{\prime} \left(\cos \left(\displaystyle \frac{\sqrt{3}{{ak}}_{x}}{2}+\displaystyle \frac{1}{2}{{ak}}_{y}\right)\right.\\ & & +\ \left.\cos \left(\displaystyle \frac{\sqrt{3}{{ak}}_{x}}{2}-\displaystyle \frac{1}{2}{{ak}}_{y}\right)+\cos ({{ak}}_{y}\right),\end{array}\end{eqnarray} \tag{ 9 }$$

where the components of wave vector $({k}_{x},{k}_{y})$ for each zigzag or armchair nanotube obey the relations in Eq. 6. After diagonalizing of the Hamiltonian in Eq. 8, the band structures of single wall graphene like zigzag nanotube $(n,0)$ are given by following eigenvalues

$$\begin{eqnarray}&&\begin{array}{l}{E}_{q}^{\eta =-}({k}_{x})=-2t^{\prime} \left(\cos \left(\displaystyle \frac{\sqrt{3}{k}_{x}}{2}+\displaystyle \frac{\pi q}{n}\right)+\cos \left(\displaystyle \frac{\sqrt{3}{k}_{x}}{2}-\displaystyle \frac{\pi q}{n}\right)\right.\\ \quad +\left.\,\cos \left(\displaystyle \frac{2\pi q}{n}\right)\right)\\ \ -\ \sqrt{{{\rm{\Delta }}}^{2}+{t}^{2}\left(1+4{\cos }^{2}\left(\displaystyle \frac{\pi q}{2n}\right)+4\cos \left(\displaystyle \frac{\pi q}{2n}\right)\cos \left(\displaystyle \frac{\sqrt{3}}{2}{k}_{x}\right)\right)}-\mu ,\\ {E}_{q}^{\eta =+}({k}_{x})=-2t^{\prime} \left(\cos \left(\displaystyle \frac{\sqrt{3}{k}_{x}}{2}+\displaystyle \frac{\pi q}{n}\right)+\cos \left(\displaystyle \frac{\sqrt{3}{k}_{x}}{2}-\displaystyle \frac{\pi q}{n}\right)\right.\\ \quad +\left.\,\cos \left(\displaystyle \frac{2\pi q}{n}\right)\right)\\ \ +\ \sqrt{{{\rm{\Delta }}}^{2}+{t}^{2}\left(1+4{\cos }^{2}\left(\displaystyle \frac{\pi q}{2n}\right)+4\cos \left(\displaystyle \frac{\pi q}{2n}\right)\cos \left(\displaystyle \frac{\sqrt{3}}{2}{k}_{x}\right)\right)}-\mu .\end{array}\end{eqnarray} \tag{ 10 }$$

Also the band structure of single wall graphene like armchair nanotube $(n,n)$ in the presence of next nearest neighbor hopping and gap parameter is readily obtained as following expression

$$\begin{eqnarray}&&\begin{array}{l}{E}_{q}^{\eta =-}({k}_{y})=-2t^{\prime} \left(\cos \left(\displaystyle \frac{\pi q}{n}+\displaystyle \frac{{k}_{y}}{2}\right)+\cos \left(\displaystyle \frac{\pi q}{n}-\displaystyle \frac{{k}_{y}}{2}\right)+\cos \left(\displaystyle \frac{{k}_{y}}{2}\right)\right)\\ \ -\ \sqrt{{{\rm{\Delta }}}^{2}+{t}^{2}\left(1+4{\cos }^{2}\left(\displaystyle \frac{{k}_{y}}{2}\right)+4\cos \left(\displaystyle \frac{{k}_{y}}{2}\right)\cos \left(\displaystyle \frac{\pi q}{n}\right)\right)}-\mu ,\\ {E}_{q}^{\eta =+}({k}_{y})=-2t^{\prime} \left(\cos \left(\displaystyle \frac{\pi q}{n}+\displaystyle \frac{{k}_{y}}{2}\right)+\cos \left(\displaystyle \frac{\pi q}{n}-\displaystyle \frac{{k}_{y}}{2}\right)+\cos \left(\displaystyle \frac{{k}_{y}}{2}\right)\right)\\ \ +\ \sqrt{{{\rm{\Delta }}}^{2}+{t}^{2}\left(1+4{\cos }^{2}\left(\displaystyle \frac{{k}_{y}}{2}\right)+4\cos \left(\displaystyle \frac{{k}_{y}}{2}\right)\cos \left(\displaystyle \frac{\pi q}{n}\right)\right)}-\mu .\end{array}\end{eqnarray} \tag{ 11 }$$

Using band energy spectrum in Eq. 10, the Hamiltonian in Eq. 7 for zigzag and armchair nanotubes can be rewritten by

$$\begin{eqnarray}\begin{array}{rcl}{H}_{{zigzag}} & = & \sum _{{k}_{x},q,\eta =+,-}{E}_{q}^{\eta }({k}_{x}){c}_{\eta ,q,{k}_{x}}^{\dagger }{c}_{\eta ,q,{k}_{x}},\\ {H}_{{armchair}} & = & \sum _{{k}_{y},q,\eta =+,-}{E}_{q}^{\eta }({k}_{y}){c}_{\eta ,q,{k}_{y}}^{\dagger }{c}_{\eta ,q,{k}_{y}}.\end{array}\end{eqnarray} \tag{ 12 }$$

The electronic Green's functions of each nanotube can be defined using the Hamiltonian in Eq. 12 as following expression

$$\begin{eqnarray}\begin{array}{rcl}{G}_{\eta ,q}^{{zigzag}}({k}_{x},\tau ) & = & -\langle {T}_{\tau }{c}_{\eta ,{k}_{x},q}(\tau ){c}_{\eta ,{k}_{x},q}^{\dagger }(0)\rangle ,\\ {G}_{\eta ,q}^{{armchair}}({k}_{y},\tau ) & = & -\langle {T}_{\tau }{c}_{\eta ,{k}_{y},q}(\tau ){c}_{\eta ,{k}_{y},q}^{\dagger }(0)\rangle ,\end{array}\end{eqnarray} \tag{ 13 }$$

where τ is imaginary time. Using the model Hamiltonian in Eq. 12, the Fourier transformations of Green's function of each nanotube is given by

$$\begin{eqnarray}\begin{array}{rcl}{G}_{\eta ,q}^{{zigzag}}({k}_{x},i{\omega }_{n}) & = & {\displaystyle \int }_{0}^{1/{k}_{B}T}d\tau {e}^{i{\omega }_{n}\tau }{G}_{\eta ,q}^{{zigzag}}({k}_{x},\tau )\\ & = & \displaystyle \frac{1}{i{\omega }_{n}-{E}_{q}^{\eta }({k}_{x})},\\ {G}_{\eta ,q}^{{armchair}}({k}_{y},i{\omega }_{n}) & = & {\displaystyle \int }_{0}^{1/{k}_{B}T}d\tau {e}^{i{\omega }_{n}\tau }{G}_{\eta ,q}^{{armchair}}({k}_{y},\tau )\\ & = & \displaystyle \frac{1}{i{\omega }_{n}-{E}_{q}^{\eta }({k}_{y})}.\end{array}\end{eqnarray} \tag{ 14 }$$

Here ${\omega }_{n}=(2n+1)\pi {k}_{B}T$ denotes the fermionic Matsubara frequency in which T is equilibrium temperature. In the following, the expression of electrical electrical conductivity of graphene like nanotube are presented using Green's function method.¹⁵ Based on linear response theory,¹⁵ Kubo formulas gives us the dynamical electrical conductivity (${\sigma }_{\zeta \zeta }$) along axis direction of nanotube (ζ) in terms of correlation function between electrical currents¹⁵ as

$$\begin{eqnarray}&&{\mathfrak{R}}{\sigma }_{\zeta \zeta }(\omega )=\mathop{\mathrm{lim}}\limits_{i{\omega }_{n}\to \omega +i{0}^{+}}{\mathfrak{I}}({\int }_{0}^{\beta }d\tau {e}^{i{\omega }_{n}\tau }\langle T({J}_{\zeta }^{e}(\tau ){J}_{\zeta }^{e}(0))\rangle ),\end{eqnarray} \tag{ 15 }$$

where ${J}_{\zeta }^{e}$ introduces the operator from of electrical current along nanotube axis direction. We have studied the conductivity of zigzag nanotube along its axis, i.e. x direction. Also the conductivity of armchair nanotube is calculated along y direction. ${\omega }_{n}=2n\pi {k}_{B}T$ denotes the bosonic Matsubara frequency. The static transport coefficients for single band model Hamiltonian²⁶ have been obtained in terms of Green's function. One can generalize this result for multiband model Hamiltonian and the static electrical conductivity of graphene like zigzag nanotube $(m,0)$ along $\zeta =x$ direction is related Green's function as

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }_{{xx}}=\displaystyle \frac{{e}^{2}{k}_{B}T}{2{{NN}}_{q}}\sum _{{k}_{x},\eta ,q}{\left(\displaystyle \frac{\partial {E}_{q}^{\eta }({k}_{x})}{\partial {k}_{x}}\right)}^{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{d\epsilon }{2\pi }\\ \quad \ \times \ \left(\displaystyle \frac{-\partial {n}_{F}(\epsilon )}{\partial \epsilon }\right){\left(2{{ImG}}_{\eta ,q}^{{zigzag}}({k}_{x},i{\omega }_{n}\longrightarrow \epsilon +i{0}^{+}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 16 }$$

where ${n}_{F}(x)=\tfrac{1}{{e}^{x/{k}_{B}T}+1}$ is the Fermi-Dirac distribution function and T denotes the equilibrium temperature. N is the number of k_x points and N_q implies the number of integer quantum number q. The static electrical conductivity of graphene like armchair nanotube $(n,n)$ along $\zeta =y$ direction, i.e. axis direction of tube, is written in terms of Green's function as

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }_{{yy}}=\displaystyle \frac{{e}^{2}{k}_{B}T}{2{{NN}}_{q}}\sum _{{k}_{y},\eta ,q}{\left(\displaystyle \frac{\partial {E}_{q}^{\eta }({k}_{y})}{\partial {k}_{y}}\right)}^{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{d\epsilon }{2\pi }\\ \quad \ \times \ \left(\displaystyle \frac{-\partial {n}_{F}(\epsilon )}{\partial \epsilon }\right){\left(2{{ImG}}_{\eta ,q}^{{armchair}}({k}_{y},i{\omega }_{n}\longrightarrow \epsilon +i{0}^{+}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 17 }$$

Substituting electronic Green's function into Eqs. 16, 17 and performing the numerical integration over wave vector through first Brillouin zone, the results of electrical conductivity have been obtained. The electronic density of states of graphene like zigzag and armchair nanotubes in the presence of gap parameter and next nearest neighbor hopping integral can be obtained by electronic band structure as

$$\begin{eqnarray}&&D(E)=-\displaystyle \frac{1}{2{{NN}}_{q}}{Im}\sum _{{k}_{x},q,\eta =+,-}\displaystyle \frac{1}{E-{E}_{q}^{\eta }({k}_{x})+i{0}^{+}}.\end{eqnarray} \tag{ 18 }$$

Also we have the following expression for electronic density of states of armchair nanotube

$$\begin{eqnarray}&&D(E)=-\displaystyle \frac{1}{2{{NN}}_{q}}{Im}\sum _{{k}_{y},q,\eta =+,-}\displaystyle \frac{1}{E-{E}_{q}^{\eta }({k}_{y})+i{0}^{+}}.\end{eqnarray} \tag{ 19 }$$

The study of temperature behavior of electrical conductivity along nanotube axis for both zigzag and armchair graphene nanotubes in the presence of naex-to-nearest neighbor hopping amplitude constitutes is the main aim in this work. Moreover the energy dependence of electronic density of states of nanotubes is investigated in details.

We turn to the presentation of our numerical results of the static electrical conductivity of both armchair and graphene like nanotubes due to effects of next nearest neighbor hopping of electrons and tube diameter. The electrical conductivity of doped zigzag and armchair nanotubes in the presence of gap parameter has been obtained using tight binding model Hamiltonian. The temperature behavior of electrical conductivity has been found via calculating the current-current correlation function within Green's function approach. Also the effects of gap parameter, next nearest neighbor hopping amplitude and diameter on the behavior of density of states of nanotubes have been addressed. By evaluating Eqs. 16, 17 numerically and using Green's functions in Eq. 14, electrical conductivity of both zigzag and armchair nanotubes has been found. In order to perform the summation over ${\bf{k}}$ points, we can exploit the wave vector region for zigzag and armchair nanotubes presented in Eq. 6. The contribution of both inter and intra band transitions to the static electrical conductivity has been accounted. The numerical results of conductivity have been presented by considering next nearest neighbor hopping amplitude as $t^{\prime} =0.2t$.

The total density of states ($D(E)$) of zigzag nanotube as a function of normalized energy E/t in the absence of gap parameter ${\rm{\Delta }}=0.0$ for different values of tube diameter $n=6,11,14,17$ has been shown in Fig. 2. The effect of next nearest neighbor hopping integral shows that density of states curves has no symmetry around Fermi level at $E=0$ according to Fig. 2. This arises from this fact that electron-hole symmetry breaks when the next-to-nearest neighbor hopping amplitude $t^{\prime}$ is taken into account. Figure 2 shows the density of states at Fermi level gets the non zero value for each diameter. Thus undoped zigzag nanotubes with different diameters indicate metallic behavior in the presence of next nearest neighbor hopping amplitude. A sharp peak with high intensity is observed for zigzag nanotube $(14,0)$ in conduction band side around $E/t\approx 0.6$. Such sharp peak in density of states is an evidence for stable quantum energy level for infinite life time. However all zigzag nanotubes show considerable sharp peaks in valence band around $E/t\approx -1.5$. In the absence of next nearest neighbor hopping, the density of states curves present an even function in terms of energy E. In other words the behavior of density of states in terms of energy is fully symmetric around $E=0$. While the density of states loses such symmetry in the presence of next nearest neighbor hopping.

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The effect of gap parameter on energy dependence of density of states of undoped graphene like zigzag nanotube $(9,0)$ in the presence of next-to-nearest neighbor hopping amplitude has been plotted in Fig. 3. The gap parameters below 0.4 preserve the metallic property of zigzag nantube so that the density of states at Fermi energy gets the non zero value. For gap parameters above 0.4, namely ${\rm{\Delta }}/t=0.8,1.0$, the Fermi energy locates in bandgap and an insulator phase develops as shown in Fig. 3. Moreover the considerable sharp peaks with high intensity are clearly observed for all values of gap parameters. According to Fig. 3, density of states is no longer even with respect to energy due to the next nearest neighbor hopping amplitude.

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The density of states ($D(E)$) of armchair nanotube as a function of normalized energy E/t in the absence of gap parameter ${\rm{\Delta }}=0.0$ for different values of tube diameter $n=7,12,17,22$ has been shown in Fig. 4. Density of states curves show a metallic behavior for all tube diameter since the density of states takes a non zero value at Fermi energy. This metallic property is improved with diameter. Moreover a considerable sharp peak with high intensity is observed at $E/t\approx -1.8$ for each diameter according to Fig. 4.

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The behavior of density of states of undoped graphene like armchair nanotube $(7,7)$ for different gap parameters Δ in the presence of next-to-nearest neighbor hopping amplitude $t^{\prime} /t=0.2$ has been shown in Fig. 5. The value of density of states at Fermi energy for ${\rm{\Delta }}=0.0,0.4$ is non zero and thus metallic property is preserved at these gap parameters. Upon more increasing gap parameters, namely ${\rm{\Delta }}/t=0.8,1.0$, the Fermi energy locates into bandgap and we observe an insulating behavior for ${\rm{\Delta }}/t=0.8,1.0$. The considerable peaks in density of states moves to lower energy however the height of peaks increases with Δ as shown in Fig. 5.

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We have studied temperature behavior of electrical conductivity of armchair nanotube along tube axis. In Fig. 6, we have depicted temperature dependence of electrical conductivity, σ, along nanotube axis of the undoped graphene like zigzag nanotube for different values of diameters, namely $n=7,12,17,22$ by setting $t^{\prime} /t=0.2$. Gap parameter Δ has been considered to be zero. For each value of tube diameter n, this figure shows electrical conductivity increases with temperature up to a characteristic temperature where conductivity reaches its maximum value. Temperature position of peak is approximately independent of tube diameter according Fig. 6. For temperatures below peak position, electronic transition rate from valence band to conduction one increases and consequently we observe an increasing behavior for conductivity in this temperature region. Upon more increasing temperature above peak position, a decreasing behavior for electrical conductivity has been observed due to increase of electronic scattering in this temperature region. The origin of the peak arises from the competition between electronic transition rate from valence band to conduction one and scattering rate between electrons. At a fixed temperature, electrical conductivity increases with tube diameter n. In fact the metallic property of armchair nanotube in the presence of next-to-nearest neighbor hopping is improved with diameter.

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The effect of gap parameter ${\rm{\Delta }}/t$ on temperature dependence of electrical conductivity of undoped graphene like armchair nanotube has been studied in Fig. 7. An exponential behavior for electrical conductivity is clearly observed at low temperatures for finite value of gap parameter. There is a peak in electrical conductivity for each value of gap parameter. For temperatures below peak position, the increase of temperature leads to enhance the electronic transition from valence band to conduction one. Thus conductivity rises with temperature up to temperature position of peak. Upon more increasing temperature above peak position, the conductivity reduces with temperature due to increase of scattering rate between electrons. In addition, at fixed values of temperatures, lower gap parameter gives arise more transition rate of electrons and consequently higher values in electrical conductivity according to Fig. 7.

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We have also studied the effect of electron density, i.e. the variation of chemical potential μ, on electrical conductivity of doped armchair nanotube in the presence of next nearest neighbor hopping amplitude. The resulting of the electrical conductivity of armchair nanotube as a function of normalized chemical potential $\mu /t$ for different tube diameters n is summarized in Fig. 8. Gap parameter has been considered to be zero. Figure 8 shows the electrical conductivity reduces with chemical potential for all tube diameters. However the conductivity is less chemical potential dependent at normalized temperatures below 1.0 for $n=7,12$ as shown in Fig. 8. Moreover the electrical conductivity rises with tube diameter at fixed values of chemical potential.

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The dependence of electrical conductivity of zigzag nanotube on chemical potential has been presented in Fig. 9. In this figure we have plotted the electrical conductivity of zigzag nanotubes as a function of normalized chemical potential $\mu /t$ for different values of n. Two novel features are pronounced in Fig. 9. In contrast to Fig. 8, there is no considerable diameter dependence for electrical conductivity. Furthermore, the conductivity reaches its maximum value at normalized chemical potential 0.25 for all values of diameter n. Also the conductivity vanishes for chemical potentials above 0.9.

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The numerical results of electrical conductivity of zigzag nanotube $(15,0)$ for different gap parameters, namely ${\rm{\Delta }}/t\,=0.2,0.4,0.6$, for normalized temperature ${k}_{B}T/t=0.2$ have been shown in Fig. 10. A monotonic increasing behavior for dependence of electrical conductivity on chemical potential has been observed for ${\rm{\Delta }}/t=0.2,0.4$. The dependence of conductivity on chemical potential for ${\rm{\Delta }}/t=0.6$ is not remarkable. In addition, at fixed temperature, the conductivity decreases with gap parameter Δ according to Fig. 10. It can be understood from this fact that the increase of gap parameter leads to reduce the transition rate of electrons an thus electrical conductivity decreases with Δ.

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Finally we have plotted the electrical conductivity of zigzag nanotube as a function of normalized chemical potential for different values of temperature ${k}_{B}T/t$ for $t^{\prime} /t=0.2$ in the absence of gap parameter. This figure implies the conductivity reduces with temperature in the chemical potentials region $0.1\lt \mu /t\lt 0.3$. A peak is clearly observed at chemical potential around $\mu /t\approx 0.2$ for each value of temperature. For fixed chemical potential above 0.4, the conductivity increases with temperature according to Fig. 11. The conductivity goes to zero at chemical potentials above 0.75 for all temperatures due to increase of scattering rate between electrons in this region.

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In conclusion, we have presented the temperature dependence of electrical conductivity of graphene like nanotubes in the presence of electron doping which means the increasing the concentration of electrons in the system via adding impurity atoms to the structure. Using a tight binding model Hamiltonian and a Green's function approach the excitation spectrum of the model Hamiltonian has been studied. In particular, the effect of gap parameter on electrical conductivity and density of states has been investigated. The increase of gap parameter leads to decrease of electrical conductivity. Moreover at a fixed temperature, the increase of tube diameter of armchair nanotube causes to the enhancement of electrical conductivity. In other hand our results show that an increase of the chemical potential gives rise to an increas of electrical conductivity.