Time-varying impedance of the sheath on a probe in an RF plasma

Langmuir probes used in radiofrequency (rf) discharges usually include compensation elements that minimize the effect of high frequency oscillations in plasma potential. The design of these elements requires knowledge of the capacitance of the sheath on the probe tip, a quantity which varies nonlinearly during the rf cycle. Sheath capacitance has been studied previously for capacitively coupled discharges, where the rf is applied to the electrodes. Here the problem is treated from the standpoint of a small probe in a fluctuating discharge. This work differs from existing literature in that a) no step model is used, and the Debye sheath is treated exactly; b) the treatment is simple and analytic; c) the time-variation of the capacitance is explicitly shown; d) the results are applied to probe design; and e) cylindrical geometry is considered. The rf frequency is assumed low enough that electron transit times can be ignored. We find that when the rf excursions bring the sheath from the Child-Langmuir region into the Debye sheath or electron saturation region, its capacitance has a strongly nonlinear behavior. 2 A. Introduction In the design of rf-compensated Langmuir probes for measurements in rf plasmas, it is necessary to know the capacitive coupling through the sheath of rf fluctuations in plasma potential. The simple approach normally used is to consider the sheath to be a vacuum capacitor whose thickness is roughly estimated. In plane geometry this thickness is not well defined, even if a sheath edge is well defined, because the thickness depends on the slope of the potential at the edge, and this depends on the transition to the presheath. It is impractical to solve for the presheath, since the solution depends on collisions and ionization and is specific to each discharge. More accurate treatments of rf sheaths can be found in the literature but are not always suitable for the present task. Lieberman has given analytic solutions for the sheath on a driven electrode in a capacitively coupled plasma (CCP). However, he used a model in which the electron density was approximated by a single step. Godyak and Sternberg pointed out that highand low-frequency approximations can be made depending on whether the rf frequency ω is larger or smaller than the ion plasma frequency Ωp, and they solved the high-frequency case for a CCP driven symmetrically relative to ground. How the shape of the sheath changes during the rf cycle was computed numerically by Zhang et al., with the result that large changes occur in the low-frequency case, the one treated in this paper. However, they did not give the sheath capacitance explicitly. Godyak showed that the sheath capacitance Csh depends only on the surface charge on the probe and can be calculated without solving for the sheath thickness numerically. Sudit and Chen used this shortcut to calculate Csh. In that work, however, they neglected the Debye sheath, treating only the Child-Langmuir (C-L) sheath, adding, rather inconsistently, the Bohm velocity at the sheath edge. Here we solve the plane sheath problem consistently, showing exactly what approximations were previously made, and also obtaining formulas from which the sheath capacitance can be calculated even when the probe is not biased far from the space potential. To establish notation, we start with a brief review of plane sheath theory before applying it to the calculation of sheath capacitance as a function of time. Cylindrical sheaths and the resistive part of the sheath impedance will be treated at the end. B. Plane sheaths in a nutshell 1. Basic equations We start by defining a sheath edge s (Fig. 1) with Vs, ns, and vs denoting respectively the potential, density, and ion velocity there. Following traditional practice, we set Vs = 0 in the absence of rf, and assume quasineutrality up to s so that ni(s) = ne(s) ≡ ns. The ions enter the sheath with a unidirectional, monoenergetic velocity vs, whose Bohm value will be recovered in due course. There is no artificial separation between the Debye sheath (where ne ≠ 0) and the ChildLangmuir sheath (where ne = 0). Defining s V V V = − , (1) where V is the actual potential as Vs varies, we write Poisson’s equation as 0 2 2 ( ) e i d e n n dx V ε = − . (2) s p Debye sheath V0 V C-L sheath Fig. 1. Geometry of a plane sheath. 3 For Maxwellian electrons, we have / e eV KT e s n n e = . (3) The ion velocity v is given by energy conservation 2 2 1⁄2 1⁄2 ( 0) s Mv Mv eV V = + ≤ (4) so that 1/ 2 2 2 s eV v v M   = −     . (5) Since ion flux is conserved, we have 1/ 2 2 2 , / 1 i i s s i s s i s s eV n v n v n n v v n Mv −   = = = −       . (6) The positive, dimensionless potential η is defined as ( ) / / s e e e V V KT eV KT η ≡ − − = − , (7) whereupon Poisson’s equation becomes 1/ 2 2 0 2 2 2 2 1 e e e s s KT KT d e n dx Mv η ε η η − −   = + −       . (8) Henceforth we use a Roman “e” for charge and an italic “e” for 2.718. Normalizing x to the Debye length (with n = ns), ( )1/ 2 2 0 / e D e s KT n λ ε = ; ( ) / D x s ξ λ ≡ − (9) and defining the ion acoustic speed cs and the Mach number M as 1/ 2 ( / ) , / s e s s c KT M v c ≡ ≡ M , (10) Eq. (8) becomes simply ( ) 2 1/ 2 2 2 " 1 2 / d e d η η η η ξ − − = = + − M . (11) Multiplying by the integrating factor η ' and integrating from ξ = 0, we obtain ( )1/ 2 2 2 2 1⁄2( ') 1 2 / 1 1 e η η η −     = + − + −       M M . (12) Here we have used the sheath boundary condition η '(0) = 0. 2. Recovery of the Bohm sheath criterion Since Eq. (12) has to be positive for all η, we can get a condition on M by expanding the r.h.s. for small η, up to order η. ( ) 2 2 2 2 4 2 2 2 2 1⁄2( ') 1 / 1⁄2 / 1 1 1⁄2 1 1⁄2 / 1⁄2 0. η η η η η η η     = + − − + − + −     = − + ≥ M M M