On the amplification of acoustic phonons in carbon nanotube

We present a theoretical study of acoustic phonons amplification in Carbon Nanotubes (CNT). The phenomenon is via Cerenkov emission (CE) of acoustic phonons using intraband transitions proposed by Mensah et. al.,~\cite{1} in Semiconductor Superlattices (SSL) and confirmed in ~\cite{2}. From this, an asymmetric graph of $\Gamma^{CNT}$ on $\frac{V_d}{V_s}$ and $\Omega\tau$ were obtained where amplification ($\Gamma_{amp}^{CNT}$) $>>$ absorption ($\Gamma_{abs}^{CNT}$). The ratio, $\frac{\vert \Gamma_{amp}^{CNT}\vert}{\vert\Gamma_{abs}^{CNT}\vert}\approx 3.5$, at $V_d = 1.02V_s$, $\omega_q = 3.0\ \mathrm{THz}$ and $T = 85\ K$ for scattering angle $\theta > 0$ . A threshold field at which $\Gamma_{abs}^{CNT}$ switches over to $\Gamma_{amp}^{CNT}$ was calculated to be $E_{z}^{dc} = 6.2\times 10^3\ \mathrm{V/m}$. This field is far less than that deduced using Bloch-Type Oscillation (BTO)~\cite{3} which is $E_{BTO}^{dc} = 3.0\times 10^5\ \mathrm{V/m}$. The obtained $\Gamma_{amp}^{CNT}$ would enable the use of CNT for the production of SASER.