Introduction to Inductively Coupled Plasmas

Inductively coupled plasma (ICP) reactors are important in plasma processing and plasma sources because the plasma density can be considerably higher than in capacitively coupled discharges. ICP reactors are also attractive from the modeling perspective because they are relatively straightforward to model. Inductively coupled plasmas are relatively straightforward to model due to the fact that the induction currents can be solved for in the frequency domain. This means that the RF cycle applied to the driving coil does not need to be explicitly resolved. As such, the quasi steady-state solution is reached in relatively few time steps.

In a 2D axial symmetric ICP reactor, a coil driven at a high frequency (usually 13.56 MHz) creates a magnetic field with only an in-plane component (the rz-plane) and a high frequency out-of-plane component of the electric field (in the θ direction). This results in oscillatory electron motion in the θ direction only. Due to this effect, further simplifications can be made and the out-of-plane electric field can be solved for in the frequency domain. Consider the momentum conservation of electrons as given in Equation 4-3:

where me is the electron mass (SI unit: kg), ue is the drift velocity of the electrons (SI unit: m/s), pe is the electron pressure tensor (SI unit: Pa), q is the electron change (SI unit: s A), E is the electric field (SI unit: V/m) and νm is the momentum transfer frequency (SI unit: 1/s). Neglecting the inertial term and taking only the θ-component yields:

Since ne and Te are uniform in the θ direction, the first term on the right and side can be neglected, resulting in:

which is a linear equation in for ue,θ which can be Fourier transformed by letting:

and

the resulting equation in the frequency domain is:

Multiplying this equation by q and defining the out-of-plane electron current density as:

the following expression is obtained:

which relates the current density to the electric field in the form of

with the plasma conductivity, σ defined as:

which is required by Equation 6-23. The third term on the left hand side of Equation 4-4 now has an additional contribution due to the out-of-plane motion of the electrons:

The presence of the driving coil in an ICP reactor also introduces capacitive coupling between the coil and the electrons in the plasma sheath. This is neglected in COMSOL Multiphysics. This is because the electron density in an ICP reactor is typically so high that the Debye length is very small, meaning that the in-plane component of the electric field cannot penetrate into the plasma.


	From the theory of the Boltzmann equation in the two-term approximation (see Boltzmann Equation, Two-Term Approximation) the plasma conductivity needs to be modified in order to be coherent with the drift velocity obtained from the Boltzmann equation in the two term approximation.

In all of the above, it has been assumed that the RF frequency is less than the collision frequency between the electrons and the neutrals. In cases where the gas pressure is low, and the RF frequency is high, collisionless heating can occur, in which case the electron inertia cannot be neglected. For a gas pressure of 20 mtorr, the typical collision frequency is on the order of 30 MHz.

When collisionless heating is included in the model an extra degree of freedom for the out-of-plane electron drift velocity, uθ, is added:

where ηeff is the effective viscosity coefficient defined as:

and Vth is the thermal velocity defined as:

and ωp is the plasma frequency defined as:

The plasma conductivity, σ, is set to zero and the out-of-plane current density, Jφ has the following term added to it: