Since the accidental discovery of carbon nanotubes (CNTs) and their fabrication by Iijima in the early 1990s [1], there has been a lot of research interest in this novel material. Several research works have reported practical usefulness of CNTs as composites, catalysts and molecular wires. Essentially, CNTs are tiny and single sheets of carbon atom rolled into seamlessly meshed tubes at the nanoscale level [2]. The unique material properties of CNTs are their characteristic electrical and thermal conductivities. A CNT is either single-walled or multi-walled. Single-walled carbon nanotubes (SWCNTs) consist of a roll of one graphene sheet. The electrical behaviour of SWCNTs is determined by their chirality [3]. Researchers in [4] studied the electrical behaviour of multi-walled carbon nanotubes (MWCNTs) network embedded in amorphous silicon nitride. As part of their findings, they reported that the resistance-temperature characteristic is practically linear except at very low temperatures. This suggests a metallic behaviour of the MWCNT networks. Experimental investigations have also been carried out on the scaling of resistance and electron mean-free-path of SWCNTs [5]. Furthermore, the authors in [5] determined the length-dependent resistance of SWCNTs and also considered the variation of resistivity with temperature. The temperature dependent resistivity of armchair CNTs has also been studied experimentally [6] and the results clearly indicated metallic behavior of the SWCNT at high temperatures with resistivity increasing approximately linearly with temperature.

In this paper, we studied the temperature dependent resistivity of SWCNTs along their tubular axes in the presence of a combined direct current and laser radiation. Our results are compared with similar findings reported in reference [7] which reported on resistivity of these novel materials along the base helix.

Using semi-classical Boltzmann transport equation (BTE) with constant electron relaxation time, the carrier current density and axial resistivity in the SWCNT is evaluated as functions of the geometric chiral angle θ_h, temperature T, the real overlapping integrals for jumps along the nanotube axis Δ_z and the base helix Δ_s. This is done by following the approach of [7] [8] together with the phenomenological model of a SWCNT developed in references [9] [10]. Employing BTE, a chiral SWCNT placed in an electric field applied along the nanotube axisis studied under a temperature gradient ÑT such that

∂ f ( r , p , t ) ∂ t + v ( p ) ∂ f ( r , p , t ) ∂ r + e E ( t ) ∂ f ( r , p , t ) ∂ p = − f ( r , p , t ) − f 0 ( p ) τ (1)

where f(r,p,t) is the distribution function, f₀(p) is the equilibrium distribution function, v(p) is the electron velocity, r is the electron position, p is the electron dynamical momentum, t is time elapsed, τ is the electron relaxation time which is assumed to be constant and e is the electron charge. The applied dc-ac field, E ( t ) = E 0 + E 1 cos ( w t ) , whereE₀ is the constant electric field, E₁ and w are the amplitude and frequency of the ac field, respectively.

We employ the perturbation approach to solve Equation (1) by treating the second term on the left-hand side as a weak perturbation. In the linear approximation of ÑT and Ñμ, the solution to Equation (1) is

f ( p ) = τ − 1 ∫ 0 ∞ exp ( − t τ ) f 0 ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) d t

    + ∫ 0 ∞ exp ( − t τ ) d t { [ ε ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) − μ ] ∇ T T + ∇ μ }     × v ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) ∂ f 0 ∂ ε ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (2)

where ε(p) is the tight-binding energy of the electron and μ is the chemical potential.

The current density is expressed as

j = e ∑ p v ( p ) f ( p ) (3)

Substituting Equation (2) into Equation (3) gives

j = e τ − 1 ∫ 0 ∞ exp ( − t τ ) d t ∑ p v ( p ) f 0 ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ )   + e ∫ 0 ∞ exp ( − t τ ) d t ∑ p v ( p ) { [ ε ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) − μ ] ∇ T T + ∇ μ }   × v ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) ∂ f 0 ∂ ε ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (4)

Employing the transformation

p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ → p ,

Equation (4) becomes

j = e τ − 1 ∫ 0 ∞ exp ( − t τ ) d t ∑ p v ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) f 0 ( p )   + e ∫ 0 ∞ exp ( − t τ ) d t ∑ p { [ ε ( p ) − μ ] ∇ T T + ∇ μ }   × { v ( p ) ∂ f 0 ( p ) ∂ ε } v ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (5)

We employ the phenomenological approach of references [9] [10] by considering a SWCNT as an infinitely long periodic chain of carbon atoms wrapped along a base helix. The axial current density is decomposed into the form

j z = Z ′ + Y ′ sin θ h (6)

where Z and Y are components of the current density along the nanotube axis and the base helix, respectively.

Furthermore, we neglect the interference between the axial and helical paths connecting a pair of atoms, quantization of transverse motion can be ignored [9] [10]; an approximation that best describes doped chiral carbon nanotubes [11].

Resolving the current density along the tubular axis (z-axis) and base helix, we obtain

Z ′ = e τ − 1 ∫ 0 ∞ exp ( − t τ ) d t ∑ p v z ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) f 0 ( p )   + e ∫ 0 ∞ exp ( − t τ ) d t ∑ p { [ ε ( p ) − μ ] ∇ z T T + ∇ z μ }

  × { v z ( p ) ∂ f 0 ( p ) ∂ ε } v z ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (7)

and

Y ′ = e τ − 1 ∫ 0 ∞ exp ( − t τ ) d t ∑ p v y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) f 0 ( p )   + e ∫ 0 ∞ exp ( − t τ ) d t ∑ p { [ ε ( p ) − μ ] ∇ y T T + ∇ y μ }   × { v y ( p ) ∂ f 0 ( p ) ∂ ε } v y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (8)

Equations (7) and (8) are transformed using

∑ p → 2 ( 2 π ℏ ) 2 ∫ − π d y π d y d p y ∫ − π d z π d z d p z

where d_z and d_y are the inter-atomic distance along the nanotube axis and the base helix respectively. Therefore, Z’ and Y’ become,

Z ′ = 2 e τ − 1 ( 2 π ℏ ) 2 ∫ 0 ∞ exp ( − t τ ) d t ∫ − π d y π d y d p y ∫ − π d z π d z d p z v z ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) f 0 ( p )   + 2 e ( 2 π ℏ ) 2 ∫ 0 ∞ exp ( − t τ ) d t ∫ − π d y π d y d p y ∫ − π d z π d z d p z { [ ε ( p ) − μ ] ∇ z T T + ∇ z μ }   × { v z ( p ) ∂ f 0 ( p ) ∂ ε } v z ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (9)

and

Y ′ = 2 e τ − 1 ( 2 π ℏ ) 2 ∫ 0 ∞ exp ( − t τ ) d t ∫ − π d y π d y d p y ∫ − π d z π d z d p z v y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) f 0 ( p )   + 2 e ( 2 π ℏ ) 2 ∫ 0 ∞ exp ( − t τ ) d t ∫ − π d y π d y d p y ∫ − π d z π d z d p z { [ ε ( p ) − μ ] ∇ y T T + ∇ y μ }   × { v y ( p ) ∂ f 0 ( p ) ∂ ε } v y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (10)

The integrals in Equations (9) and (10) were evaluated over the first Brillouin zone. The parameters v,p,E, ÑT and Ñμ with subscripts z and y represent the respective components along the nanotube axis and along the base helix.

The energy dispersion relation for a chiral nanotube obtained in the tight-binding approximation is expressed as:

ε ( p ) = ε 0 − Δ y cos p y d y ℏ − Δ z cos p z d z ℏ (11)

where ε₀ is the energy of an outer-shell electron in an isolated carbon atom, Δ_z and Δ_y are the real overlapping integrals for jumps along the respective coordinates, p_y and p_z are the components of momentum tangential to the base helix and along the nanotube axis, respectively. The components v_y and v_z of the electron velocity v are respectively.

v y ( p ) = ∂ ε ( p ) ∂ P y = Δ y d y ℏ sin p y d y ℏ (12)

v z ( p ) = ∂ ε ( p ) ∂ P z = Δ z d z ℏ sin p z d z ℏ (13)

Also,

v y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) = ∂ ε ∂ P y ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) = Δ y d y ℏ { sin p y d y ℏ cos ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ )     − cos p y d y ℏ sin ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (14)

Similarly,

v z = Δ z d z ℏ { sin p z d z ℏ cos ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ )               − cos p z d z ℏ sin ( p − e ∫ t − t ′ t [ E 0 + E 1 cos w t ′ ] d t ′ ) (15)

The carrier density of the non-degenerate electron gas can be determined by expressing the Boltzmann equilibrium distribution function f₀(p) as

f 0 ( p ) = C exp ( Δ y cos p y d y ℏ + Δ z cos p z d z ℏ + μ − ε 0 k T ) (16)

The normalization constant C is expressed as

C = d y d z n 0 2 I 0 ( Δ y * ) I 0 ( Δ z * ) exp ( − μ − ε 0 k T )

where n₀ is the surface charge density, I_n(x) is the modified Bessel function of n^th order,

Δ y * = Δ y k T ,     Δ z * = Δ z k T

and k is Boltzmann’s constant.

We substitute Equations (11)-(16) into Equations (9) and (10) and simplify the integrals to obtain

Y ′ = − σ y ( E ) E y n * − σ y ( E ) k e { ( ε 0 − μ k T ) − Δ y * I 0 ( Δ y * ) I 1 ( Δ y * ) + 2 − Δ z * I 1 ( Δ z * ) I 0 ( Δ z * ) } ∇ y T (17)

Z ′ = − σ z ( E ) E z n * − σ z ( E ) k e { ( ε 0 − μ k T ) − Δ z * I 0 ( Δ z * ) I 1 ( Δ z * ) + 2 − Δ y * I 1 ( Δ y * ) I 0 ( Δ y * ) } ∇ z T (18)

where we have defined E y n * as

E y n * = E n + ∇ y μ e , E s n * = E z n * sin θ ℏ

Also, the electrical conductivity σ_i(E) is expressed as

σ i ( E ) = e 2 τ Δ i d i 2 n 0 ℏ 2 I 1 ( Δ i * ) I 0 ( Δ i * ) ∑ n = − ∞ ∞ J n 2 ( a ) [ 1 1 + ( e d i E 0 ℏ + n w ) 2 τ 2 ] , i = y , z (19)

Substituting Equations (17) and (18) into Equation (6), we obtain the following expression for axial the current density as

j z = − { σ z ( E ) + σ y ( E ) sin 2 θ h } E z n *     − { σ z ( E ) k e [ ( ε 0 − μ k T ) − Δ z * I 0 ( Δ z * ) I 1 ( Δ z * ) + 2 − Δ y * I 1 ( Δ y * ) I 0 ( Δ y * ) ]     + σ y ( E ) k e sin 2 θ h [ ( ε 0 − μ k T ) − Δ y * I 0 ( Δ y * ) I 1 ( Δ y * ) + 2 − Δ z * I 1 ( Δ z * ) I 0 ( Δ z * ) ] } ∇ z T (20)

Employing the following definitions

ξ = ε 0 − μ k T , A i = I 1 ( Δ i * ) I 0 ( Δ i * ) , B i = I 0 ( Δ i * ) I 1 ( Δ i * ) − 2 Δ i * , i = y , z

Equation (20) becomes

j z = − { σ z ( E ) + σ y ( E ) sin 2 θ h } E z n * − { σ z ( E ) k e [ ξ − Δ z * B z − Δ y * A y ]       + σ y ( E ) k e sin 2 θ h [ ξ − Δ y * B y − Δ z * A z ] } ∇ z T (21)

Equation (21) defines the current density and the axial component of the electrical conductivity is the coefficient of the electric field E z n * given as

σ z z = σ z ( E ) + σ y ( E ) sin 2 θ h (22)

The resistivity of the chiral SWCNT along its axial direction is defined as

ρ z = 1 σ z ( E ) + σ y ( E ) sin 2 θ h (23)

In this work, we employed BTE to analytically study the axial electrical resistivity of a chiral SWCNT and obtained an expression for the axial component of the electrical resistivity, ρ_z in the presence of applied field E as shown in Equation (23). The ρ_z expression was analyzed numerically by considering a chiral SWCNT having the following parameters: d_y = 1 Å, d_z = 2 Å, τ = 0.3 × 10⁻¹² s and θ_h = 4˚, and E₀ = 6.9063 × 10⁷ V/m, w = 10¹² s⁻¹ and E₁ = 5 × 10⁷ V/m.

Figure 1(a) shows the dependence of the axial electrical resistivity, ρ_z on temperature T, for various fixed values of the d.c. field, E₀. It is observed that at low temperatures up to about 100 K, ρ_z changes slowly with temperature and the temperature dependence of ρ_z becomes generally linear with increasing temperature. This observation is attributed to electron-phonon interactions which cause scattering of the charge carriers along the tubular axis of the chiral SWCNT as temperature increases. The relatively low values of resistivity of SWCNT observed in Figure 1 clearly suggest that chiral SWCNTs do exhibit metallic properties. As observed in the case of circumferential resistivity [7], our results similarly reveal that as the electric field strength, E₀ increases the axial resistivity increases significantly. This temperature dependence resistivity response of SWCNTs has been observed experimentally [6] [12] [13]. This behaviour can be explained based on the fact that as E₀ is increased, the carbon atoms forming the walls of the SWCNT become more energized and tend to vibrate faster at larger amplitudes resulting in enhanced scattering of the electrons. Our study has also revealed that the resistivity of the SWCNTs decreases markedly with increasing Δ_y as shown in Figure 1(b).

Figure 2 shows the variation of the axial current density with temperature for selected values of the chiral angle, θ_h and dimensionless parameter Δ_z. Increasing θ_h results in a decrease of the resistivity of the chiral SWCNT as seen in Figure 2(a), a behaviour similar to that observed for Figure 2(a) reveals that increasing the chiral angle, θ_h results in a decrease of the resistivity of the chiral SWCNT. On the other hand, Figure 2(b) shows that keeping Δ_y constant and varying Δ_z decreases ρ_z by an order of magnitude. Also observed in Figure 2(b) is that ρ_z remains unchanged for all values of Δ_z at temperatures below 100 K but decreases steadily by small margins with increasing temperature.

On the other hand, when the a.c. source is switched off, E₁ = 0, a = 0, w = 0 and J n 2 ( a ) becomes unity and equation (19) reduces to

σ i ( E ) = e 2 τ Δ i d i 2 n 0 ℏ 2 I 1 ( Δ i * ) I 0 ( Δ i * ) ∑ n = − ∞ ∞ J n 2 ( a ) [ 1 1 + ( e d i E 0 ℏ ) 2 τ 2 ] , i = y , z

Figure 3 shows the dependence of axial resistivity of the chiral SWCNTs on temperature in the presence and absence of a laser. Clearly, when the laser source is switched off, ρ_z decreases by three orders of magnitude. This indicates that within the temperature range under consideration, the laser field modulates the d. c. field and enhances the momentum and kinetic energy of those electrons which are deficient in energy.

Using the semi-classical approach, the axial resistivity of a chiral SWCNT induced by a laser field has been investigated and compared with an earlier work that examined the circumferential resistivity of these unique materials. Our results indicate that the conductivity of SWCNTs is similar for both directions and is significantly influenced by material parameters (Δ_y, Δ_z, θ_h), the strengths of the constant field E₀ and laser source strength E₁. The axial resistivity can be increased by increasing the d.c. field strength and decreasing the chiral angle. As observed for the circumferential case, the axial resistivity also steadily and linearly increases with the chiral angle θ_h and real overlapping integral along the axial direction Δ_y at room temperatures. Therefore, SWCNTs can be used in the design of efficient optoelectronic devices.

The authors declare no conflicts of interest regarding the publication of this paper.

Twum, A., Edziah, R., Mensah, S.Y., Dompreh, K., Mensah-Amoah, P., Arthur, A., Mensah, N.G., Adu, K. and Nkrumah-Buandoh, G. (2021) Temperature Dependent Resistivity of Chiral Single-Walled Carbon Nanotubes in the Presence of Coherent Light Source. World Journal of Condensed Matter Physics, 11, 77-86. https://doi.org/10.4236/wjcmp.2021.114006