Collective Modes of the Massless Dirac Plasma

We develop a theory for the long-wavelength plasma oscillation of a collection of charged massless Dirac particles in a solid, as occurring, for example, in doped graphene layers, interacting via the long-range Coulomb interaction. We find that the long-wavelength plasmon frequency in such a doped massless Dirac plasma is explicitly nonclassical in all dimensions with the plasma frequency being proportional to $1/\sqrt{\hslash }$. We also show that the long-wavelength plasma frequency of the $D$-dimensional superlattice made from such a plasma does not agree with the corresponding $D+1$-dimensional bulk plasmon frequency. We compare and contrast such Dirac plasmons with the well-studied regular palsmons in metals and doped semiconductors which manifest the usual classical long-wavelength plasma oscillation.