Energetic neutral fluxes towards surfaces in a magnetically enhanced reactive ion etch-like reactor

Sabisch, Winfried; Kratzer, Matthias; Brinkmann, Ralf Peter

doi:10.1116/1.1565153

Cited by 3 publications

(2 citation statements)

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“…It remains to mention that the approach presented in this manuscript can be extended to RF modulated sheaths. Several investigations which have verified the presented ideas or used them for practical work are already completed [16][17][18][19][20][21][22][23][24], others -particularly in the field of microplasmas -are being conducted.…”

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confidence: 98%

The plasma–sheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

Brinkmann¹

2011

J. Phys. D: Appl. Phys.

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. The plasmasheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion R P BrinkmannTo cite this version:R P Brinkmann. The plasmasheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion. AbstractThe plasma-sheath transition in stationary low temperature plasmas is investigated for arbitrary levels of collisionality. The model under study contains the equations of continuity and motion for a single ion species, Boltzmann's equilibrium for the electrons, and Poisson's equation for the field.Assuming that the electron Debye length λ D is small compared to the ion gradient length l = n i / ∂n i ∂x , a first order differential equation is established for the ion density n i as function of the transformed spatial coordinate q = n i dx. A characteristic feature of this novel 'sheath equation' is an internal singularity of the saddle point type which separates the depletion-field dominated sheath part of the solution from the ambipolar diffusion controlled plasma. The properties of this singularity allow to define, in nonarbitrary way, a collisionally modified Bohm criterion which recovers Bohm's original expression in the collisionless limit but remains meaningful also when collisions are included. A comparison is made with the collisionally modified Bohm criteria proposed by Godyak [V.A. Godyak, Phys. Lett. 89A, 80 (1982)], Valentini [H.-B. Valentini, Phys. Plasmas 3, 1459 (1996)] and Chen [X.P. Chen, Phys. Plasmas 5, 804 (1997)] as well as with the approaches of Riemann [K.-U. Riemann, J. Phys. D; Appl. Phys. 24, 493 (1991)] and Franklin [R.N. Franklin, J. Phys. D: Appl. Phys. 36, 2821(2003] who argued that the definition of a collisionally defined Bohm criterion is not possible.

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confidence: 98%

The plasma–sheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

Brinkmann¹

2011

J. Phys. D: Appl. Phys.

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“…Kratzer et al have presented a hybrid model for the calculation of ion distribution functions behind a dc or RF driven plasma boundary sheath [41]. Sabisch et al have extended that model to calculate also the fluxes of energetic neutrals to exposed surfaces in the process of magnetically enhanced reactive ion etching (MERIE) [42]. Heil et al augmented the tool by a kinetic electron module and analysed the processes of Ohmic and stochastic heating in RF-CCPs [43,44].…”

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confidence: 99%

From electron depletion to quasi-neutrality: the sheath–bulk transition in RF modulated discharges

Brinkmann¹

2009

J. Phys. D: Appl. Phys.

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The boundary sheaths of all plasmas are characterized by a gradual transition from unipolarity (electron depletion, ne ≪ ni) to ambipolarity (quasi-neutrality, ne ≈ ni). Capacitively driven sheaths exhibit a transition which is expanded by the RF modulation and smoothed by thermal effects, i.e. by the finiteness of the electron temperature Te and the Debye length . Sheath models which neglect thermal effects (‘step models’) are restricted to strongly modulated high voltage sheaths with VRF ≫ Te/e and fail when this condition is not met. This work presents an improved analysis of the sheath–bulk transition which takes both modulation and thermal effects into account. Based on a previously found asymptotic solution of the Boltzmann–Poisson equation (Brinkmann 2007 J. Appl. Phys. 102 093393), approximate algebraic (i.e. closed) expressions for the phase-resolved electrical field E and electron density ne in RF sheaths are derived. Under the assumption that the modulation is periodic (not necessarily harmonic) with ωRF ≫ ωpi, also the phase averages of the field and the electron density can be expressed in closed form. These results—together referred to as the advanced algebraic approximation (AAA)—make it possible to formulate efficient and accurate models for RF driven boundary sheaths for all ratios of VRF to Te/e. As an example, a harmonically RF modulated, collision-dominated single species sheath is studied. The outcome is compared both with the numerically constructed exact solution and with the well-known step model approach of Lieberman (1989 IEEE Trans. Plasma Sci. 17 338). It is found that the AAA can reproduce the exact numerical solution within a few per cent for all ratios of VRF to Te/e. The step model, in contrast, exhibits strong deviations even for large eVRF/Te and fails completely in the case of weak modulation.

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Beyond the step model: Approximate expressions for the field in the plasma boundary sheath

Brinkmann

2007

Journal of Applied Physics

116

148

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The transition from quasineutrality to charge depletion is one of the characteristic features of the plasma boundary sheath. For modeling purposes, this transition is often described in terms of the so-called step model which assumes a sharp transition point s (electron step) where the electron density ne drops from a value equal to the ion density ni (in the bulk, x>s) to a value of zero (in the sheath, x<s). Inserted into Poisson’s equation, the step model yields an expression for the field which is realistic deep in the sheath (for x≪s) but fails to merge correctly into the ambipolar field of the bulk. This work considers the transition from quasineutrality to charge depletion more rigorously. Within the framework of asymptotic scale analysis, a family of field approximations is derived which, in the limit of weak spatial ion density variation ∂ni∕∂x≪ni∕λD, exhibit convergence to the exact solution of the Boltzmann–Poisson equation. The first of the approximations recovers only the step model. Higher order expressions, however, also include the ambipolar field and are valid for all values of x. The simplest of them is proposed as being the best suited for a self-consistent sheath analysis. This statement is backed by numerical experiments which show good results for the ion density distribution provided by the collision dominated sheath model.

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Energetic neutral fluxes towards surfaces in a magnetically enhanced reactive ion etch-like reactor

Cited by 3 publications

References 8 publications

The plasma–sheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

The plasma–sheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

From electron depletion to quasi-neutrality: the sheath–bulk transition in RF modulated discharges

Beyond the step model: Approximate expressions for the field in the plasma boundary sheath

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