Endpoint of a Ballistic Dyakonov-Shur Instability

With certain unusual boundary conditions in a rectangular channel, a hydrodynamic electron fluid is unstable; normal mode amplitudes increase over time due to reflections from the boundaries. Dyakonov and Shur noticed that unstable (growing) modes in this so-called Dyakonov-Shur Instability can oscillate in the TeraHertz (THz) range, a region of the electromagnetic spectrum for which few good sources or detectors of radiation exist. We characterize the Dyakonov-Shur instability outside the hydrodynamic regime—into the the tomographic regime and across the ballistic-hydrodynamic crossover—in the context of a simple relaxation-time toy kinetic theory for the low-temperature Fermi liquid based on the Chapman-Enskog expansion. When τ_ee, the electron-electron scattering time, is low, our model agrees with the Boussinesq approximation to Navier-Stokes; when it is high, our model becomes approximately linear and sensitive to ballistic physics. Truncating the number of modes in our Chapman-Enskog expansion at n_max, we solve our nonlinear model numerically with finite volume methods. We find in particular that the amplitude of the Dyakonov-Shur oscillations increases with τ_ee, making it even more puzzling that radiation from this hydrodynamic phenomenon has proved experimentally elusive.