arXiv:cond-mat/0205173v2 [cond-mat.mtrl-sci] 15 Jan 2003 Transition from Townsend to glow discharge: subcritical, mixed or supercritical characteristics Danijela D. ˇ Sijaˇ ci´ c 1 and Ute Ebert 1,2 1 Centrum voor Wiskunde en Informatica, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands 2 Dept. Physics, Eindhoven Univ. of Techn., The Netherlands (November 6, 2018) The transition from Townsend to glow discharge is investigated numerically in one space dimen- sion in full parameter space within the classical model: with electrons and positive ions drifting in the local electric field, impact ionization by electrons (α process), secondary electron emission from the cathode (γ process) and space charge effects. We also perform a systematic analytical small current expansion about the Townsend limit up to third order in the current that fits our numerical data very well. Depending on the two determining parameters γ and system size pd, the transition from Townsend to glow discharge can show the textbook subcritical behavior, but for smaller values of pd, we also find supercritical or some unexpected intermediate “mixed” behavior. Our work shows the same qualitative dependence of U = U (I,pd) for fixed γ as the old experiments by Pokrovskaya-Soboleva and Klyarfeld. Furthermore, the analysis lays the basis for understanding the complex spatio-temporal patterns in short planar barrier discharge systems. 52.80.-s, 05.45.-a, 51.50.+v, 47.54.+r I. INTRODUCTION Space charge effects in many cases are the first non- linear effects in gas discharges with increasing current. They are known to induce the avalanche to streamer transition in transient discharges as well as the transi- tion from Townsend to normal and further to abnormal glow in stationary discharges. Generically, nonlinear cou- plings in non-equilibrium systems lead to the formation of spontaneous spatio-temporal patterns. The current constriction in the normal glow discharge as well as the longitudinal striations of a long positive column of a glow discharge [1–3] fall into this class of phenomena. Recently, the amazing variety of spatio-temporal pat- terns formed mainly in the transversal direction of a short dc driven system consisting of a gas discharge layer and a semiconductor layer sandwiched between two copla- nar electrodes has drawn considerable attention [4–12]. These patterns are due to the nonlinear gas discharge being coupled to the linearly responding semiconduc- tor. In particular, a negative differential conductivity of the gas discharge in some region of the current-voltage- characteristics is expected [13–19] to play a significant role in the spontaneous formation of patterns, quite like in nonlinear semiconductor devices [20]. Due to its ge- ometry, modeling the system [10–12] as one-dimensional is a very good approximation, as long as this symmetry is not spontaneously broken by the intrinsic dynamics. So as a first step of any investigation, the behavior and the resulting current-voltage-characteristics of the purely one-dimensional gas discharge system have to be under- stood. An investigation of the system [11] along the lines of the textbook [21] shows that the pattern formation oc- curs at the space charge driven transition from Townsend to glow discharge. The gas dicharge layer is rather short, more precisely, the product pd of gas pressure p times electrode distance d is small. This raises the question of the Townsend to glow transition for small pd. How- ever, despite a history of more than 70 years, we are not aware of any thorough and complete study of this classi- cal problem. Therefore, our aim in the present paper is to develop a consistent picture of the Townsend to glow transition in one dimension from analytical and numeri- cal investigations, in particular, for short systems. Many authors focus on quite long discharges that have a clearly pronounced subcritical characteristics, i.e., for fixed large pd and growing total current I , the voltage first decreases from the Townsend limit towards the nor- mal glow regime, then it increases again in the abnormal glow regime until heating effects become important and the voltage again decreases towards the arc discharge. We will not consider this last thermally driven transi- tion at high currents. The initial decrease of voltage from Townsend discharge towards normal glow creates a regime of negative differential conductivity, and some authors [19] believe that negative differential conductiv- ity is generic for this system. However, already in the early 1940’ies, e.g., in the ex- tensive review by Druyvesteyn and Penning [22], it was suggested that this subcritical behavior might not be the only possible one, but that also a monotonic increase of voltage with current was possible. Such a behavior we will call supercritical, in line with modern bifurca- tion theory. There are early experimental papers by Pokrovskaya-Soboleva and Klyarfeld [23] and McClure [24] that clearly indicate a supercritical transition for small values of pd in hydrogen and deuterium in com- bination with metal electrodes. Later data by the same authors [25] is reproduced in Raizer’s textbook [21], how- ever, only for rather long systems with subcritical char- 1