Session 9: Overview of the Physics in the Plasma Sheath

Session 9: Overview of the physics in the plasma sheath Electrons in plasmas are much more mobile (higher temperature, lower mass) than ions, and consequently their fluxes are much greater. Because of this, materials in contact with plasmas will generally charge negatively with respect to the plasma potential. Here we analyze in some level of detail the structure of the non-neutral layer adjacent to materials in contact with a plasma. To simplify our analysis, we assume the material is electrically isolated and catalytic, and we make use of a 1D model. The region adjacent to the material wall is schematically shown below. We emphasize the distinction in the potential structure between the quasi-neutral plasma, a quasi-neutral collisional pre-sheath and the non-neutral sheath. φ φp x pre-sheath sheath ne ≈ ni ne ≠ ni wall n ni ne −φ w x Very quickly, on the order of the inverse of the plasma frequency, the potential and number distributions reach an equilibrium as the current densities of electrons and ions become equal. All ions directed towards the wall eventually fall on it, while electrons of lower energy than eφw are reflected. The distribution function of ions is therefore far from isotropic in the sheath, while electrons can still be considered to be near-Maxwellian. From here, we write the current densities for ions and electrons respectively as, ji = eni vi and je = −e new c̄e 4 (1) where new is the electron number density adjacent to the wall and their mean thermal speed is, � c̄e = 8kTe πme Since the electron population is assumed Maxwellian, we use the Boltzmann equilibrium relation, new � � eφw = ne∞ exp − kTe where ne∞ is the plasma density (far away from the wall). 1 (2) To determine the potential structure we first need to find what the ion flux is. To do this, we write the equations of motion of ions and electrons in the collisional pre-sheath, mi ni vi dvi dpi + = eni E − Fi dx dx (3) me ne ve dve dpe + = −ene E − Fe dx dx where Fi and Fe are frictional forces (per unit volume) on ions and electrons due to collisions in the pre-sheath. We add these equations and neglect electron inertia, mi ni vi dvi d(pi + pe ) + = −F dx dx (4) where we have made use of the quasi-neutrality assumption in the pre-sheath and we lumped together drag forces into F . Since the ion flux is conserved Γi = ni vi , we write, dG = −F dx where G = mi vi Γi + Γi kTe vi (5) in which, in addition to quasi-neutrality, we used p = nkT and assumed Te » Ti . We immediately notice that as we approach the wall, due to collisions, that the quantity G decreases in magnitude until reaching a minimum. This is accomplished by increasing the ion velocity to, � r vi,minG = vB = kTe mi (6) This counter-intuitive result indicates that frictional forces accelerate ions to the terminal velocity vB , called the Bohm velocity, or the ambipolar speed of sound. This is the velocity at which ions enter the plasma sheath after falling through a small, but not-negligible potential drop in the pre-sheath. Notice that this peculiar velocity uses the temperature of the electrons, but the ion mass. Going back to Eq. (1), we can evaluate the ion current density at the sheath boundary, where ni = ne and vi = vB . The Boltzmann relation Eq. (2) is used once more but now to relate the plasma density at the pre-sheath boundary, to the bulk plasma density, �  eφps ni = ne = ne∞ exp − kTe �   �  � mi vB2 = ne∞ exp − = ne∞ e−1/2 2kTe (7) The current density balance then reads, � r −1/2 ene∞ e kTe ne = e ∞ 4 mi � r � �   8kTe eφw exp − πme kTe We then find that the sheath potential can be approximated by, 2 (8) φw kTe ≈ ln e � r mi me (9) For instance, for a Xe plasma with an electron population with 3 eV, the wall potential will be biased to about -18 V. 3