Scattering of electromagnetic waves by two crossing metallic single-walled carbon nanotubes of finite length

The scattering theory for two crossing metallic single-walled carbon nanotubes of finite length exposed to an electromagnetic field has been developed based on a synthesis of the quantum transport formalism and classical electrodynamics. The model of the point contact has been developed to be incorporated into the Hallén and Pocklington equations for crossing carbon nanotubes. The influence of the contact conductance and position as well as the angle between the tube axes on the tube polarizability has been analyzed in the range of 1 GHz to 10 THz. The physical mechanisms responsible for the electromagnetic interaction between crossing tubes with zero and nonzero intertube contact conductance are shown and discussed. The influence of the coupling between the tubes on the localized plasmon resonance in them has been demonstrated.

Figure 1

Geometry of two crossing CNTs of lengths $L_1$ and $L_2$. Axes $z$ and $z_2$ coincide with the axes of the first and second CNTs, respectively. Also, $r_1$ and $r_2$ denote the distances between CNT edges and the crossing point. Vectors ${\mathbf{E}}_{\pm }^{\left(0\right)}$ show the direction of the incident fields taken for numerical calculations in Sec. 3.

Figure 2

(a) Scheme of two CNTs with the tunnel junction at $z=r_1$. $I_t$ is a current through the junction. $I_1, I_2, I_3$, and $I_4$ are currents in CNTs on opposite sides with respect to the junction. For simplicity, tubes are shown to be parallel; in general, they could be nonparallel. (b) Same as (a) except that the tunneling current element is replaced by the dipole with the charges $+q_t$ and $-q_t, q_t=i{\omega }^{-1}{I}_t$, on the CNT surface at $z=r_1$. (c) Same as (b) except that the charges $+q_t$ and $-q_t$ are replaced by the azimuthally symmetrical homogenous axial current $I_t$ having a jump at $z=r_1$ from $I_t$ down to 0 in the first CNT and from 0 up to $I_t$ in the second CNT, respectively.

Figure 3

Frequency dependence of the (a) real and (b) imaginary parts of the polarizability of the first CNT at different distances $r=0.1,0.25L$ and different incident fields ${\mathbf{E}}_{+}^{\left(0\right)}$ and ${\mathbf{E}}_{-}^{\left(0\right)}$; $\theta ={45}^{\circ }$. There is no current between the tubes, i.e., $G_d=0$. The polarizability of a single CNT is shown by a solid line.

Figure 4

Distribution of (a) and (b) the surface current density, (c) and (d) the charge density, and (e) and (f) radial and (g) and (h) axial fields on the surface of the first CNT and a single CNT at 1 THz; $G_d=0, \theta ={45}^{\circ }$, and $r=0.1L$.

Figure 5

Frequency dependence of the polarizability ${\alpha }_{1z}$ of the first CNTs for various (a) and (b) angles between CNTs $\theta =3^{\circ },{45}^{\circ }, D=0.34$ nm, and (c) and (d) distances $D\in \{0.34,25,100\}$ nm, $\theta ={45}^{\circ }$; $r=100$ nm, and $G_d=0$. The polarizability of a single CNT is shown by the dotted line. The external field is ${\mathbf{E}}_{+}^{\left(0\right)}$.

Figure 6

Frequency dependence of the (a) real and (b) imaginary parts of the polarizability of the first CNT for various intertube conductances $G_d\in \{0,0.01,0.1\}{G}_0$; $\theta ={45}^{\circ }$, and $r=L/10$. The polarizability of a single CNT is shown by a solid line.

Figure 7

Frequency dependencies of the dissipated powers $P_{cn}, P_c$, and $P_{cn}+P_c$ at $G_d=0.1G_0$ and dissipated power $P_{cn}$ at $G_d=0$; $\theta ={45}^{\circ }$, and $r=L/10$.

Figure 8

Distribution of (a) and (b) the surface current density $j_1$ and (c) and (d) the surface charge density ${\rho }_{c1}$ and their components $j_1^{(c,e)}$ and ${\rho }_{c1}^{(c,e)}$ on the surface of the first CNT at $f=20$ GHz; $G_d=0.1G_0, \theta ={45}^{\circ }$, and $r=L/10$.

Figure 9

Frequency dependence of the imaginary part of the polarizability of the first CNT for (a) various angles between the crossing tubes $\theta \in \{3^{\circ },{45}^{\circ },{90}^{\circ }\}, r=100$ nm, and (b) various distances $r\in \{50,250,450\}$ nm, $\theta ={45}^{\circ }$; $G_d=0.1G_0$.

Figure 10

Frequency dependence of the imaginary part of the polarizability of a single CNT and the first CNTs for various tube lengths, $L=250$ nm and $L=1000$ nm; $r=0.1L, \theta ={45}^{\circ }$, and $G_d=0.1G_0$.

Figure 11

Schematic illustration of two crossing CNTs decomposed following the Landauer-Büttiker concept of quantum transport. The CNT junction (black circle) has four terminals connected by leads to reservoirs.