The Onset and End of the Gunn Effect in Extrinsic Semiconductors

Abstract

A Hopf bifurcation analysis of the spontaneous current oscillation in direct current (DC) voltage-biased extrinsic semiconductors is given for the classical model of the Gunn effect in n-GaAs. For semiconductor lengths L larger than a certain minimal value, the steady state is linearly unstable for voltages in an interval $( \phi _\alpha ,\phi _\omega )$. As L increases, the branch of time-periodic solutions bifurcating at simple eigenvalues when $\phi = \phi _\alpha $ turns from subcritical to supercritical and then back to subcritical again. For very long semiconductors a quasi continuum of oscillatory modes bifurcates from the steady state at the onset of the instability. The bifurcating branch is then described by a scalar reaction-diffusion equation with cubic nonlinearity subject to antiperiodic boundary conditions on a subinterval of $[ 0,L ]$. For the electron velocity curve we have considered, the bifurcation is subcritical, which may agree with experimental observations in n-GaAs. An extension of our calculation suggests that a supercritical time-periodic bifurcating branch (possible for other electron velocity curves) consists of the generation at $x = 0$ and evolution of waves that are damped before they can reach the receiving contact. Our calculation is a first step in determining how the bifurcating solution branch is related to the branch of oscillatory solutions mediated by solitary wave dynamics. The relation to previous numerical and experimental results is discussed.