Simulation of Inductively Coupled Plasmas Using a Direct Implicit Darwin 2D Particle-in-Cell Code*

A 2D electromagnetic particle-in-cell code has been developed which calculates the electrostatic field using the direct implicit algorithm and the solenoidal electric field and magnetic field using the Darwin algorithm. The code uses Cartesian geometry and allows to place metal and dielectric objects inside the simulation domain. The cell size of the numerical grid may exceed the Debye length, the time step may exceed the period of electron Langmuir oscillations. Since the Darwin scheme is used, there is no restrictions on the time step due to the light wave propagation. The electrostatic part of the code is similar to [1]. The Darwin scheme is implemented using a new approach based on the vorticity equation for the solenoidal electric field. The code includes a Monte-Carlo model of electron neutral collisions, electron emission from material surfaces, charge exchange collisions, and abundant diagnostics. The code is used to simulate a top-coil inductively coupled plasma device with realistic dimensions. Simulation demonstrates that the plasma is sustained by the induced solenoidal electric field. With coarse grids and large time steps, the numerical cost of the implicit simulation is relatively low and a quasi-steady state can be achieved significantly faster than in an explicit simulation which resolves all the plasma scales.

[1] Gibbons M. R. and D. W. Hewett, “The Darwin Direct Implicit Particle-in-Cell (DADIPIC) Method for Simulation of Low Frequency Plasma Phenomena”, J. Comput. Phys. 120, 231 (1995).