A simple physical model of hexagonal patterns in a Townsend discharge with a semiconductor cathode

This paper explains the observed effect of self-organization in a dc driven planar gas discharge-semiconductor system resulting in a hexagonal current pattern under cryogenic conditions. It is shown that the electric field redistribution usually causing a falling current-voltage characteristic (CVC) of the Townsend discharge and the discharge instability cannot provide the formation of the hexagonal pattern. Another mechanism is proposed which gives a necessary, high negative slope of the CVC under cryogenic conditions. This is a well-known thermal mechanism. Due to Joule heat release gas is heated and expands; hence, a lower field and voltage are required to sustain the discharge at a given current. Simple approximate equations describing non-stationary spatially inhomogeneous states in the gas discharge-semiconductor system are derived from physical considerations. The numerical integration of the obtained equations with a realistic parameter set gives the hexagonal current pattern. By simplifying these equations, we found analytically the current and the discharge voltage distributions of the hexagonal type and a simple formula for the distance between adjacent current filaments. The analytical solution allows one to investigate the roles of different factors and extract from experiment the negative differential resistance of the discharge, which is the main parameter in the problem of the discharge instability and the current structure formation.