Hydrodynamic modes,soft modes and fluctuation spectra near the threshold of a current instability

We give a full threedimensional treatment of the stability and the fluctuations of the uniform stationary current state in a voltage-controlled current instability. We consider a model which exhibits bulk negative differential conductivity due to Bragg scattering of hot electrons. The model consists of Langevin equations for the mean momentum and the mean energy of the charged carriers, coupled to Maxwell's equations. We investigate the normal modes and the fluctuation spectra of this system, in particular the occurrence of soft modes and of critical fluctuations at the stability limit of the uniform current state. It is shown that the nature of the normal modes is strongly determined by the electromagnetic interactions between the carriers, giving rise to hydrodynamic flux modes and to dielectric relaxation modes. As the threshold field is approached, the dielectric relaxation modes soften and couple strongly to the flux modes. It is shown that as a consequence of this coupling the exponential decay of the correlation functions due to ordinary dielectric relaxation is followed at very long times by a power law decay due to the hydrodynamic modes.Work supported by the Swiss National Science Foundation