Microwave absorption by an array of carbon nanotubes: A phenomenological model

A simple model to explain microwave-induced heating of carbon nanotubes (CNTs) through transformation of electromagnetic energy into mechanical vibrations is proposed and analyzed. The model provides a way to understand recent observations of heating of CNTs exposed to microwaves in the range of $2–20\mathrm{GHz}$. It is shown that transverse vibrations of CNTs during microwave irradiation can be associated with parametric resonance, as occurs in the analysis of acoustic experiments on forced longitudinal vibrations of a stretched elastic string. For carbon nanotubes [single wall nanotube (SWNT), double wall nanotube (DWNT), multiwall nanotube (MWNT), ropes, and strands] the resonant parameters are shown to be located in a region of instability of the Mathieu’s equation. Wave equations with cubic nonlinearity were used to qualitatively describe the effects of phonon-phonon interactions and energy transfer from microwaves to CNTs at a rate much exceeding the traditional Joule heating via electron-phonon interaction.

Figure 1

Near $\beta =0$, the parameter space is divided into regions corresponding to stable and unstable solutions of the Mathieu equation. The regions are separated by the lines $\alpha =1∕4\pm \beta ∕2-{\beta }^2∕8$. Melde’s experiment corresponds to $\alpha =\beta ∕2=1∕4$.

Figure 2

The absolute values of the discrete Fourier amplitudes of the nonlinear mode amplitude ${\nu }_1\left(t\right)$ are plotted against the integers $p$ in the angular frequencies $2\pi p∕t_0$, with $t_0=250∕{\omega }_E$. The primary peak occurs near $p=54$, or near an angular frequency of $\pi p{\omega }_E∕125=1.36{\omega }_E$, above the microwave frequency used to irradiate the CNTs. The lower secondary peak near $p=28$ is near an angular frequency of 0.70 ${\omega }_E$.

Figure 3

The absolute values of the discrete Fourier amplitude spectrum of the nonlinear mode amplitude ${\nu }_3\left(t\right)$ is plotted versus the integers $p$ in the angular frequencies $2\pi p∕t_0$, with $t_0=250∕{\omega }_E$. The primary peak occurs near $p=174$, or near an angular frequency of $\pi p{\omega }_E∕125=4.37{\omega }_E$, over four times the frequency used to irradiate the CNTs.