Semiclassical analysis of intraband collective excitations in a two-dimensional electron gas with Dirac spectrum

Solving the initial value problem for semiclassical equations that describe two-dimensional electrons with the Dirac spectrum, we found that collective excitations of the electrons are composed of a few distinct components of the oscillations. There always exist sustained plasma oscillations with the well-known plasmon frequency ${\omega }_{\text{pl}}\left(k\right)$. Additionally, there are oscillations with a “carrier frequency” $\omega =v_{F}k$ slowly decaying in time, according to a power law ($v_F$ and $k$ are the Fermi velocity and wave vector). The reason for the onset of these oscillations has a fundamental character related to the branching of the polarization function of the Dirac electrons. A strongly anisotropic initial disturbance of the electron distribution generates an additional component of the undamped oscillations in the form of an electron unidirectional beam, which are van Kampen's modes in the Dirac plasma.

Figure 1

Normalized potential ${\phi }_{\mathbf{k}}^{\mathcal{C}}\left(t\right)$ for (a), (c) $\kappa$-environment-I and (b), (d) $\kappa$-environment-II. (a), (b) Isotropic initial perturbation. (c), (d) Strongly anisotropic initial perturbation: solid line, ${\alpha }_0=\pi /3$; dotted line, ${\alpha }_0=0$. Insets of (a), (c): Cuts and integral paths on the $\omega$ plane. Inset of (b): ${\phi }_{\mathbf{k}}^{\mathcal{C}}\left(t\right)$ for weak anisotropic perturbation. Inset of (d): Example of temporal pattern for ${\omega }_{\text{pl}}\approx 2{\omega }_{vK}$.