Analytical model of the relaxation of a collisionless ion matrix sheath

The temporal evolution of a collisionless ion matrix sheath in front of an electrode biased to a pulsed high negative voltage is investigated analytically and numerically. In the relaxation process a matrix extraction phase and a subsequent sheath expansion phase can be distinguished. For the matrix extraction phase we present an analytical model that is based on special solutions of the ion fluid equations and that is free of artificial assumptions. The model results in an explicit formula for the ion current to the electrode. The results are compared with numerical solutions of the ion fluid equations and show excellent agreement. By a simple parameter ansatz the model is extended to describe the sheath expansion phase.