Plasma Modeling

Plasma modeling is challenging for even the simplest cases (DC and inductive discharges) due to the complex interaction between the charged particles, neutrals and fields which collectively constitute the plasma. Wave heated discharges add an extra layer of complexity, due to the critical plasma density and capacitively coupled plasmas require a special discretization scheme unique to COMSOL Multiphysics. Equilibrium discharges avoid a lot of the complexity the non-equilibrium cases, but are only valid under certain conditions and present their own unique challenges.

Capacitively Coupled Plasmas (CCPs)

Theoretical Foundations

CCPs rely on an electric field between parallel electrodes, driven by RF power (typically 13.56 MHz). The sheath dynamics are critical, modeled by the Poisson equation:

\[ \nabla^2 \phi = -\frac{e}{\epsilon_0} (n_i - n_e) \]

where \( \phi \) is the electric potential, and \( n_i \) and \( n_e \) are ion and electron densities. The ion flux to the substrate is governed by the Bohm criterion:

\[ u_i \geq \sqrt{\frac{k_B T_e}{m_i}} \]

where \( u_i \) is the ion velocity, \( k_B \) is Boltzmann’s constant, and \( m_i \) is the ion mass. COMSOL simulates these interactions, optimizing electrode design and gas pressure.

Applications

Semiconductor Industry: CCPs dominate plasma etching (e.g., \( SF_6 \) for silicon) and sputtering for thin-film deposition. Their precise control over ion energy ensures high-aspect-ratio etching, vital for advanced semiconductor nodes. They also neutralize toxic gases like \( NF_3 \).

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Electric Potential: The electric potential oscillates at 13.56MHz, and naturally forms a negative DC bias on the lower, driven electrode. This occurs because the surface area of the driven electrode is much smaller than the containing grounded walls.
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Electron Density: The electron density oscillates in the plasma sheath region close to the grounded top electrode and driven electrode on the base.

Microwave Plasmas

Theoretical Foundations

Microwave plasmas are sustained by electromagnetic waves, typically at 2.45 GHz, interacting with gas molecules. The wave propagation is described by the vector Helmholtz equation and plasma permittivity \( \epsilon_p \) given by:

\[ \epsilon_p = 1 - \frac{\omega_p^2}{\omega^2 + \nu^2} \]

Here, \( \omega_p = \sqrt{n_e e^2 / (m_e \epsilon_0)} \) is the plasma frequency, and \( \nu \) is the collision frequency. COMSOL models these dynamics, capturing the transition from microwave absorption to plasma ignition.

Applications

Semiconductor Industry: Microwave plasmas enable high-rate deposition of diamond-like carbon (DLC) films, used as protective coatings in chip manufacturing. Their ability to operate at low pressures enhances control over reactive gas generation (e.g., \( CH_4 \)-based radicals).

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Inductively Coupled Plasmas (ICPs)

Theoretical Foundations

ICPs are generated by an oscillating magnetic field from a coil, inducing electric currents in the plasma. The power transfer is governed by Faraday’s law and Maxwell’s equations, with the induced electric field \( E \) satisfying:

\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]

where \( B \) is the magnetic flux density.

Applications

Semiconductor Industry: ICPs excel in plasma etching and deposition for microchip fabrication. Their high electron density (up to \( 10^{18} \, \text{m}^{-3} \)) ensures uniform reactive ion etching (RIE) of silicon wafers, critical for nanoscale features. They also generate reactive gases (e.g., \( CF_4 \)) and destroy toxic byproducts like perfluorocarbons.

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