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We have studied the axial resistivity of chiral single-walled carbon nanotubes (SWCNTs) in the presence of a combined direct current and high frequency alternating fields. We employed semiclassical Boltzmann equations approach and compared our results with a similar study that examined the circumferential resistivity of these unique materials. Our work shows that these materials display similar resistivity for both axial and circumferential directions and this largely depends on temperature, intensities of the applied fields and material parameters such as chiral angle. Based on these low-temperature bidirectional conductivity responses, we propose chiral SWCNTs for design of efficient optoelectronic devices.

Since the accidental discovery of carbon nanotubes (CNTs) and their fabrication by Iijima in the early 1990s [

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 [

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 [

∂ 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, f_{0}(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 ) , whereE_{0} is the constant electric field, E_{1} 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 [

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 [

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 ε_{0} 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 f_{0}(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 n_{0} 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 E_{0} = 6.9063 × 10^{7} V/m, w = 10^{12} s^{−1} and E_{1} = 5 × 10^{7} V/m.

_{z} on temperature T, for various fixed values of the d.c. field, E_{0}. 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 _{0} increases the axial resistivity increases significantly. This temperature dependence resistivity response of SWCNTs has been observed experimentally [_{0} 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

_{h} and dimensionless parameter Δ_{z}. Increasing θ_{h} results in a decrease of the resistivity of the chiral SWCNT as seen in _{h} results in a decrease of the resistivity of the chiral SWCNT. On the other hand, _{y} constant and varying Δ_{z} decreases ρ_{z} by an order of magnitude. Also observed in _{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, E_{1} = 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

_{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 E_{0} and laser source strength E_{1}. 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