Generation of microwave and terahertz radiation in a medium of nanoparticles

Abstract

We investigate the mechanism of radiation generation, whose source are modulated with the variable $(x - \upsilon t)$ surface currents in elongated nanoparticles forming a 3D structure. Carbon nanotubes and graphene nanoribbons are considered as elongated nanoparticles. The volume fraction $f < < 1$ of nanoparticles is considered small (the distance between the centers of nanoparticles is several times greater than their length), thus allowing one to neglect the interaction of two identical nanotubes due to the tunneling effect. Since the velocity of modulated surface currents $\upsilon$ in 3D structure greater than the phase velocity of the light in a medium, then the process is qualitatively similar to the Cherenkov radiation by a system of dipoles that move with the velocity $\upsilon$. In the case of $\alpha$-aligned nanofilms based on nanotubes, the radiation wavefront will have a form of a divergent wedge. It is shown that such a structure can generate intense microwave and terahertz radiation and an estimate of the radiation value is made.

Appendices

Appendix 1

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 Khiznjak 1986)

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

(36)

hereinafter, the axis of nanotube is directed along the z axis, where $\varOmega = \Delta x\Delta y\Delta z$, $\Delta x,\,\Delta y,\Delta z$ are the distances between nanotube centers along the axes $x$, $y$ and $z$, correspondently; the production of operators $\hat{\delta }\hat{g}$ paper is equal to $\hat{\delta }\hat{g} = \delta_{ii} g_{ii}$, where $i = 1,2,3$, $\delta_{11} = - 4\pi \delta_{xx} /3$, $\delta_{22} = - 4\pi \delta_{yy} /3$, $\delta_{33} = - 4\pi \delta_{zz} /3$; $\delta_{xx} = 3B_{p}$, $\delta_{yy} = 3B_{p}$, $\delta_{zz} = 3B_{s}$, 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 when 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 and have form (see Khiznjak 1986)

$$g_{ii} = \frac{abc}{3}\frac{{\left( {\varepsilon_{ii} - \varepsilon_{2} } \right)}}{{\varepsilon_{2} + n_{i} \left( {\varepsilon_{ii} - \varepsilon_{2} } \right)}},\quad g_{ij} = 0,\quad i \ne j,\quad n_{i} = \frac{abc}{2}J_{i} ,$$

(37)

where $J_{1} = J_{100} ,\quad J_{2} = J_{010} ,\quad J_{3} = J_{001}$, $n_{i}$—is the depolarization factor (Landau and Lifshitz 1960)

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

(38)

From (1) and (37) it follows that $g_{ii}$ coincide with the polarizability $\alpha_{i}$

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

From (36), taking into account (37) and (38), we obtain the expression for the effective dielectric function

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

(39)

where $f = 4\pi abc/(3\varOmega )$ is the volume fraction. $B_{s} = 1/3$($\delta_{zz}$ = 1) in the case in the cubic lattice nanotube centers are located on the same distance (Khiznjak 1986).

To define the effective dielectric function, one should know the Newtonian potentials

$$\begin{aligned} J_{3} & = J_{001} = \int\limits_{0}^{\infty } {\frac{d\xi }{{\left( {c^{2} + \xi } \right)^{3/2} \left( {a^{2} + \xi } \right)^{1/2} \left( {b^{2} + \xi } \right)^{1/2} }}} \\ & = \frac{2}{{c^{3} }}\int\limits_{0}^{1} {\frac{{\varsigma^{2} d\varsigma }}{{\left[ {1 - \varsigma^{2} \left( {1 - \frac{{a^{2} }}{{c^{2} }}} \right)} \right]^{1/2} \left[ {1 - \varsigma^{2} \left( {1 - \frac{{b^{2} }}{{c^{2} }}} \right)} \right]^{1/2} }},} \\ \end{aligned}$$

(40)

where the new variable is introducted

$$\varsigma = c/\left( {c^{2} + \xi } \right)^{1/2} .$$

(41)

The integral (40) can be easily calculated if one splits him into two integrals $I_{1}$ and $I_{2}$ with integration domain $0 \le \varsigma \le 1 - \alpha$ and $(1 - \alpha ) < \varsigma \le 1$, correspondently, where $a/c\,\sim\,b/c < < \alpha < < 1$. This can easily be seen

$$\begin{aligned} I_{1} & = \int\limits_{0}^{1 - \alpha } {\frac{{\varsigma^{2} d\varsigma }}{{1 - \varsigma^{2} }}} \approx - 1 - \frac{1}{2}\ln \left( {\frac{\alpha }{2}} \right),\quad I_{2} = \int\limits_{1 - \alpha }^{1} {\frac{d\lambda }{{\left[ {\lambda^{2} + \frac{\lambda }{2}\left( {\frac{{a^{2} + b^{2} }}{{c^{2} }}} \right) + \frac{{a^{2} b^{2} }}{{4c^{4} }}} \right]}}} \\ & = \left. {\frac{1}{2}\ln \left| {\sqrt {\left( {\lambda + \frac{{a^{2} + b^{2} }}{{4c^{2} }}} \right)^{2} - \left( {\frac{{a^{2} - b^{2} }}{{4c^{2} }}} \right)} } \right|} \right|_{0}^{\alpha } = \frac{1}{2}\ln \left( {2\alpha } \right) - \ln \left( {\frac{a + b}{2c}} \right). \\ \end{aligned}$$

(42)

From (40) and (42) the expression for the Newtonian potential follows

$$J_{3} = J_{001} = \frac{2}{{c^{3} }}\left[ { - 1 + \ln \left( {\frac{4c}{a + b}} \right)} \right].$$

(43)

From Eqs. (37) and (38) it follows, that in the case of elongated ellipsoid $c > > a > > b$, the depolarization factor is

$$n_{3} = \frac{abc}{2}J_{3} = \frac{ab}{{c^{2} }}\left[ { - 1 + \ln \left( {\frac{4c}{a + b}} \right)} \right] < < 1.$$

(44)

Depolarization factor $n_{i}$ of axisymmetric ellipsoid $c > > a = b$ decreases several times

$$\frac{b}{c}\left[ {\left( {1 + \frac{\ln 2}{{\left( { - 1 + \ln 2c/(a)} \right)}}} \right)} \right] \approx \frac{b}{c}.$$

Appendix 2

The dimensions of the wires considerably exceed the dimensions of the nanotubes, as a result of which the velocity of the surface waves in the wires is equal to the speed of light in a vacuum. Indeed, if the thickness of skin layer in the metallic wire is much smaller than its radius, then in this case at $\omega < \omega_{p} /(1 + \varepsilon_{2} )$ there are nonradiative surface plasmon-polariton modes (SPP), and if $\omega > \omega_{p}$ there are radiative surface plasmon-polariton modes (RPP) (Dionne et al. 2005), where $\omega_{p} = 8.85 \times 10^{15} \text{s}^{ - 1}$—is the bulk plasma frequency of the electron gas. In the case of SPP modes, the oscillation frequency can be in the terahertz range and in this case the phase velocity $\upsilon_{p} \approx 2 \times 10^{8} \,{\text{m/s}}$ (Dionne et al. 2005), which is commensurate with the speed of light. In the case of RPP near the velocity modes, when the condition $\omega \ge \omega_{p}$ is satisfied, the phase velocity of surface waves $\upsilon_{p} \approx c\sqrt {\omega^{2} /\omega_{p}^{2} - 1} \to 0$. The condition $\omega \ge \omega_{p}$ corresponds to the optical frequency range. In the case where the thickness of the skin layer in a metal wire is equal to or greater than its radius, then in this case there exist longitudinal and transverse plasmon modes. The transverse plasmon mode coincides with the dispersion of the asymmetric plasmon mode in a two-dimensional electron system of double thickness without a conducting screen, and the longitudinal plasmon mode coincides with the dispersion of the symmetric plasmon mode. In nanotubes, the velocity of surface waves is determined by the slowing factor $\beta$, as a result of which the velocity of propagation of surface waves is much less than the speed of light in a vacuum. The magnitude of the slowing factor $\beta \approx \left[ {4\alpha X\upsilon_{F} /(\pi c)} \right]^{1/2} \approx 0.02$ (Slepyan et al. 1999; Sadykov 2013), where the fine structure constant $\alpha = 1/137$, $\upsilon_{F}$—is the electron velocity at the Fermi point of the nanotube (Sadykov 2013; Vedernikov et al. 2001), $K_{0} (\xi ),\quad I_{0} (\xi ) - \pi$ are the modified Bessel functions.

Typical velocity of $\pi$ electrons excited to energy of electron several volts is about 10⁸ cm/s. For such electrons, the synchronism condition requires the slowing down of electromagnetic wave at least 300 times, which is much larger than the theoretical estimate (Slepyan et al. 1999), given for CNTs.). The presence of the small $\beta$ value leads to the fact that the first plasmon resonance frequency for a 1-μm nanotube will correspond to radiation in a vacuum with a length of $\sim0.1\,{\text{mm}} = 100\,\upmu{\text{m}}$.