Amplification of Hypersound in Graphene with degenerate energy dispersion

Hypersound amplification/absorption of acoustic phonons in Graphene with degenerate energy dispersion $\varepsilon(p)$ near the Fermi level was theoretically studied. For $k_{\beta}T << 1$ and $ql >> 1$, the dependence of the absorption coefficient $\Gamma/\Gamma_0$ on ${V_D\over V_s}$ was studied where the results satisfied the Cerenkov effect. That is when ${V_D\over V_s} > 1$, an amplification was obtained but for ${V_D\over V_s} < 1$, an absorption was obtained which could lead to Acoustoelectric Effect (AE) in Graphene. A linear dependence of the $\Gamma/\Gamma_0$ on $\omega_q$ was observed where the result obtianed qualitatively agreed with an experimentally observed acoustoelectric current in Graphene via the Weinrich relation. It is interesting to note from this study that, frequencies above $10THz$ can be attained for $V_D = 1.1ms^{-1}$. This study permit the use of Graphene as hypersound phonon laser (SASER).