2.3 Plasma Sheath and Boundary Phenomena

Plasma sheaths form near solid surfaces in contact with plasma, creating a thin, positively charged layer. This phenomenon occurs due to differences in electron and ion thermal velocities, resulting in a space charge region with an electric field that accelerates ions towards the surface.

The Bohm criterion plays a crucial role in sheath properties, determining the minimum velocity ions must have to enter the sheath region. This criterion influences the sheath potential, thickness, and electric field magnitude, which in turn affect ion energy distribution and surface interactions in plasma-assisted manufacturing processes.

Plasma Sheath Formation and Properties

Formation of plasma sheaths

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• Plasma sheath forms as a thin, positively charged layer near solid surfaces in contact with plasma due to differences in electron and ion thermal velocities
  • Electrons have higher thermal velocities than ions, initially reaching the surface faster creating a negative potential on the surface relative to the bulk plasma
  • The negative potential repels electrons and attracts ions, forming the sheath layer (a few millimeters thick in low-pressure plasmas)
• Consists of a space charge region with a net positive charge due to the repulsion of electrons and attraction of ions
  • Electric field within the sheath accelerates ions towards the surface and repels electrons
  • Sheath thickness typically spans a few Debye lengths ($\lambda_D$), which characterizes the distance over which charge separation can occur in a plasma
    • Debye length formula: $\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$, where $\varepsilon_0$ is the permittivity of free space, $k_B$ is the Boltzmann constant, $T_e$ is the electron temperature, $n_e$ is the electron density, and $e$ is the elementary charge
    • Example: In a typical low-pressure plasma with $T_e = 3 eV$ and $n_e = 10^{16} m^{-3}$, the Debye length is approximately 74 μm

Bohm criterion in sheath properties

• Bohm criterion states that ions must enter the sheath region with a minimum velocity, known as the Bohm velocity ($u_B$), for a stable sheath to form
  • Bohm velocity formula: $u_B = \sqrt{\frac{k_B T_e}{m_i}}$, where $m_i$ is the ion mass
  • Ensures sufficient ion current to balance electron current at the sheath edge, maintaining sheath stability
• Bohm criterion determines the potential drop across the sheath ($V_s$)
  • Sheath potential formula: $V_s \approx \frac{k_B T_e}{2e} \ln{\left(\frac{m_i}{2\pi m_e}\right)}$, where $m_e$ is the electron mass
  • Influences ion energy distribution at the surface, with ions gaining energy as they fall through the sheath potential
• Affects sheath thickness and electric field magnitude within the sheath
  • Higher Bohm velocities lead to thinner sheaths and stronger electric fields
  • Example: In an argon plasma with $T_e = 3 eV$, the Bohm velocity is approximately 2.7 km/s, and the sheath potential is around 14 V

Plasma-Surface Interactions and Manufacturing

Sheath effects on surface fluxes

• Ion flux is accelerated by the sheath potential, leading to energetic ion bombardment of surfaces
  • Ion energy depends on the sheath potential, ranging from a few eV to hundreds of eV (10-500 eV common in manufacturing plasmas)
  • Energetic ions can sputter material from surfaces, causing surface modification and etching
• Electron flux to surfaces is greatly reduced due to the repulsive potential of the sheath
  • Electron energy flux is limited by the sheath potential barrier, with only the most energetic electrons reaching the surface
• Ions transfer kinetic energy to surfaces upon impact, causing surface heating and modification
  • Ion bombardment can enhance surface reactions, film growth, and material properties in deposition processes (PECVD, sputtering)

Plasma-surface interactions in manufacturing

• Plasma etching utilizes energetic ions accelerated by the sheath to chemically react with and physically sputter material from surfaces
  • Directionality of ion bombardment enables anisotropic etching for high-aspect-ratio features (deep trenches, vias in semiconductor manufacturing)
• Plasma-enhanced chemical vapor deposition (PECVD) relies on sheath potential to accelerate ions towards substrates
  • Ion bombardment promotes surface reactions, film growth, and improves film density, adhesion, and morphology
  • Widely used for depositing dielectric films (silicon dioxide, silicon nitride) and semiconductor materials (amorphous/polycrystalline silicon)
• Plasma sheath enables controlled surface modification and functionalization
  • Ion implantation and surface activation can improve surface wettability, adhesion, and biocompatibility
  • Examples include plasma treatment of polymers for enhanced bonding and plasma modification of biomaterials for improved cell interactions
• Plasma cleaning and sterilization leverage energetic ions and reactive species generated in the plasma
  • Sheath potential enhances the effectiveness of plasma cleaning and sterilization by accelerating species towards contaminated surfaces
  • Applications include removing organic contaminants from semiconductor wafers and inactivating microorganisms on medical devices