Terahertz radiation generation process in the medium based on the array of the elongated nanoparticles

Abstract

We conduct a theoretical study and numerical simulations of terahertz radiation generation in the medium based on armchair-edge nanoribbons and zigzag nanotubes with metallic conductivity. The multicascade mechanism of radiation generation is considered in the task of terahertz radiation generation. The level of the injection current in nanoparticle arrays has been estimated. The task expands to the similar medium where radiation current is generated with the use of infrared radiation stimulated absorption, for example, radiation of a CO₂ laser. For the effective dielectric function three basic models are employed: effective media approximation (Bruggeman’s theory) [nanotube bundles], Maxwell-Gatnett theory [ordered array], and Maxwell-Garnett theory with the Clausius–Mossotti geometric correction [ordered array of nanotubes].

Appendix

In terms of electrodynamics, the discrete structure of massive amount of dispersive particles is equal to a continuous dielectric material, characterized by the effective dielectric function (magnetic properties of particles are neglected) [see Markel (2016)]

$$\hat{\varepsilon }_{{{\text{eff}}}} = \varepsilon_{2} \left[ {1 + \frac{4\pi }{\Omega }\hat{g}\left( {1 + \frac{1}{\Omega }\hat{\delta } \cdot \hat{g}} \right)^{ - 1} } \right],$$

(A1)

hereinafter, the axis of nanotube is directed along the z axis, where $\Omega = \Delta x\Delta y\Delta z$, $\Delta x,\,\Delta y,\Delta z$ are the distances between nanotube centers along the axes $x$, $y$ and $z$, correspondently; the production of operators $\hat{\delta } \cdot \hat{g}$ paper is equal to $\hat{\delta }\hat{g} = \delta_{ii} \cdot g_{ii}$, where $i = 1,2,3,$

$$\begin{gathered} \delta_{11} = - 4\pi \delta_{xx} /3,\quad \delta_{22} = - 4\pi \delta_{yy} /3,\quad \delta_{33} = - 4\pi \delta_{zz} /3, \hfill \\ \delta_{xx} = 3B_{p} ,\quad \delta_{yy} = 3B_{p} ,\quad \delta_{zz} = 3B_{s} ,\quad \hfill \\ \end{gathered}$$

(A2)

where $B_{s}$ and $B_{p}$ are the geometrical factors, such, that the polarization is aligned along ($s$-polarization) or perpendicular ($p$-polarization) to the nanotube’s axis).

In the case where the tensor of dielectric permittivity of ellipsoid is diagonal and main direction of this tensor coincide with the main axes of ellipsoid, the matrix $g_{ik}$ is also diagonal ($g_{ij} = 0,\;i \ne j$) and have form [see Markel (2016)]

$$g_{ii} = \frac{abc}{3}\frac{{\left( {\varepsilon_{ii} - \varepsilon_{2} } \right)}}{{\varepsilon_{2} + n_{i} \left( {\varepsilon_{ii} - \varepsilon_{2} } \right)}},\quad n_{i} = \frac{abc}{2}J_{i} ,$$

(A3)

where $J_{1} = J_{100} ,\quad J_{2} = J_{010} ,\quad J_{3} = J_{001}$, $n_{i}$—is the depolarization factor (Grachev et al. 2014).

$$\begin{gathered} J_{i} = \int_{0}^{\infty } {\frac{d\xi }{{\left( {\xi + \eta_{i}^{2} } \right)\sqrt {R(\xi )} }}} ,\quad R(\xi ) = (a^{2} + \xi )(b^{2} + \xi )(c^{2} + \xi ), \hfill \\ \eta_{1} = a,\quad \eta_{2} = b,\quad \eta_{3} = c. \hfill \\ \end{gathered}$$

(A4)

From (4.1) and (A3) it follows that $g_{ii}$ coincide with the polarizability $\alpha_{i}$

$$g_{ii} = \frac{abc}{3}\tilde{\alpha }_{i} = \frac{V}{4\pi }\tilde{\alpha }_{i} = \alpha_{i} .$$

(A5)

From (A1), taking into account (A2) and (A3), we obtain the expression for the effective dielectric function

$$\hat{\varepsilon }_{{{\text{eff}}}} = \varepsilon_{2} \left[ {1 + \frac{{f\tilde{\alpha }_{\chi } }}{{1 - fB_{\chi } \tilde{\alpha }_{\chi } }}} \right],$$

(A6)

where $f = 4\pi abc/(3\Omega )$ is the volume fraction; $B_{\chi }$ is a geometrical factor, $\chi = s,p$; $B_{s}$ is such a geometrical factor, that the polarization is aligned along (s-polarization), $B_{p}$ is such a geometrical factor, that the polarization is perpendicular to the nanotube’s axis (p-polarization). In tetragonal lattice $\Delta x = \Delta y$, $\Delta z/\Delta x = \beta$ the geometrical factors are equal to the ones from Markel (2016); thus, it follows that

$$\begin{aligned} B_{s} & = \frac{{\delta_{zz} }}{3} = \frac{1}{4\pi }\left[ {\left( {18 - \frac{10}{{\beta^{2} }}} \right)\arctan \frac{Q}{{\beta^{2} }} + 30\frac{{1 - \beta^{2} }}{{\beta^{4} }}\left( {2\arctan Q - \ln \frac{1 + Q}{{1 - Q}}} \right)} \right], \\ B_{p} & = \frac{{\delta_{xx} }}{3} = \frac{{\delta_{yy} }}{3} = \frac{1}{4\pi }\left[ {\left( {18 + \frac{20}{{\beta^{2} }} - \frac{30}{{\beta^{4} }}} \right)\arctan Q + 15\frac{{1 - \beta^{2} }}{{\beta^{4} }}\ln \frac{1 + Q}{{1 - Q}}} \right], \\ \end{aligned}$$

(A7)

where $Q = \beta /\sqrt {2 + \beta^{2} }$.