Z. Phys. B Condensed Matter 81, 183-194 (1990) Z ,sehri. Matter for Ph~ik B 9 Springer-Verlag 1990 Nonlinear dynamics of breathing current filaments in n-GaAs and p-Ge E. Schiill I and D. Drasdo 2. 1 Institut f/Jr TheoretischePhysik,TechnischeUniversitfitBerlin,Hardenbergstrasse36, D-1000 Berlin 12, Germany 2 Institut ftir Theoretische Physik, Rheinisch-Wesff/ilische Technische Hochschule Aachen, Sommerfeldstrasse,D-5100 Aachen, Federal Republic of Germany Received March 27, 1990; revised versionJune 5, 1990 A novel model is presented for spatio-temporal pattern formation in semiconductors. It leads to self-generated nonlinear current oscillations due to "breathing" current filaments in the regime of impurity impact ionization. The four qualitatively different regimes which have been observed in Ge with increasing current are consistently explained as: a stationary nonconducting state; bulk- dominated oscillations; breathing filaments; stable fila- ments. The physical origin of the breathing oscillations is impact ionization coupled with transverse diffusion and longitudinal dielectric relaxation. A method is devel- oped to derive simple nonlinear dynamic equations for the filament radius and the position of the peak trans- verse electric field by a nonlinear mode expansion. I. Introduction Self-organized spatio-temporal pattern formation occurs in a variety of dissipative nonlinear dynamic systems in physics and chemistry [1, 2]. When such systems are driven far enough into the nonlinear regime, spatial de- grees of freedom are excited which may lead to self- generated oscillations, and eventually to chaotic and tur- bulent behaviour. An example which is of particular current interest is the nonlinear dynamics of current filaments in semi- conductors in the regime of nonlinear charge transport [3-5]. Current filaments have been observed in connection with S-shaped negative differential conductivity (SNDC) of the current-density-electric field characteristic during low-temperature impurity breakdown in n-GaAs [6, 7, 8] and p-Ge [9], and in Si pin-diodes at room tempera- ture [10]. "Breathing" current filaments were suggested theo- * Now at: Max-Planck-Institutfiir biophysikalischeChemie,Ab- teilung Biochemische Kinetik, Am Fal3berg, D-3400 G6ttingen, FRG retically as a possible mechanism for self-generated cur- rent oscillations [11, 3, 12]. Recent space- and time- resolved experimental investigations have indeed demon- strated that in a certain regime of the current-voltage characteristic complex spontaneous oscillations arise (SLO = "structure-limited oscillations") which are due to breathing oscillations of the filament boundaries [13, 14] during impurity breakdown. The influence of different control parameters, such as average current, perpendicu- lar magnetic and electric field, lattice temperature, con- tact distance, external circuit conditions, optical irradia- tion and electron beam irradiation, upon the current filaments and the associated current oscillations has been carefully studied [15-20]. The theoretical description of continuous spatio-tem- poral patterns is usually based upon a set of coupled nonlinear partial differential equations containing tem- poral and spatial derivatives, e.g., the hydrodynamic bal- ance equations for mean particle density, momentum, and energy [5], or the classical continuity and Maxwell equations [3]. While a full time-dependent numerical solution of these equations provides the most detailed information about the spatial profiles [21], it does not give immediate physical insight into the relevant under- lying mechanisms. The purpose of the present work is to develop a method by which irrelevant details of the spatial profiles can be projected out to obtain simpler dynamic equa- tions for a certain class of solutions parametrized by a few relevant variables. These simpler equations are shown to describe breathing current filaments. Previous semiconductor transport models were ana- lysed for the special cases of either stationary spatial structures (single filaments [22, 23] or multifilaments [24-26]), or spatially homogeneous self-generated peri- odic and chaotic oscillations [27, 3]. Here both cases are combined. Our aim is to model breathing filaments on the basis of the physical mechanism discussed in [22] and [27]. We shall not restrict ourselves to linear modes describing pulsations of filaments [28], but focus on their nonlinear dynamics.