Intersection of two nanotubes: density of states modulated by plasmon beatings with period governed by Luttinger-liquid parameter

We study theoretically the plasmon scattering at the intersection of two metallic carbon nanotubes. We demonstrate that for a small angle of crossing, $\theta \ll 1$, the transmission coefficient is an {\em oscillatory} function of $\lambda/\theta$, where $\lambda$ is the interaction parameter of the Luttinger liquid in an individual nanotube. We calculate the tunnel density of states, $\nu(\omega,x)$, as a function of energy, $\omega$, and distance, $x$, from the intersection. In contrast to a single nanotube, we find that, in the geometry of crossed nanotubes, conventional ``rapid'' oscillations in $\nu(\omega,x)$ due to the plasmon scattering acquire an aperiodic ``slow-breathing'' envelope which has $\lambda/\theta$ nodes.