Semianalytical description of nonlocal secondary electrons in a radio frequency capacitively coupled plasma at intermediate pressures

A model for fast simulation of a radio-frequency capacitively coupled discharge in the pressure range of a few torr and higher has been developed. The semi-analytical description of the nonlocal ionization by secondary (γ) electrons allows us to evaluate the discharge parameters in detail even when the local approach for γ electrons is not valid. We focused on the effect of steep increase of the plasma density at the central part of the discharge, at the transition from α to γ regime. The present study demonstrates that in the γ regime the plasma density at the discharge center is proportional to the spatial scale of ionization decay by γ electrons inside the plasma region. We have investigated the electron energy distribution function of γ electrons both analytically and numerically. As a result, the analytical formulas for the spatial scales of ionization decay by γ electrons were found in limiting cases