A model for tailored-waveform radiofrequency sheaths

The sheath physics of radiofrequency plasmas excited by a sinusoidal waveform is reasonably well understood, but the existing models are complicated and are not easily extended to the more complex waveforms recently introduced in applications. Turner and Chabert (2014 Appl. Phys. Lett. 104 164102) proposed a model for collisionless sheaths that can easily be solved for arbitrary waveforms. In this paper we extend this model to the case of collisional sheaths in the intermediate pressure regime. Analytical expressions are derived for the electric field, the electric potential and the density profiles in the sheath region. The collisionless and collisional models are compared for a pulsed-voltage waveform.


Introduction
Radiofrequency plasmas are widely used in the microelectronics industry [1,2] and in many other applications, ranging from space propulsion [3,4] through to medical applications [5][6][7]. In many instances, the most important phenomena take place in the radiofrequency sheaths, and it is therefore essential to model this boundary region appropriately. Historically, capacitive radiofrequency discharges were driven with a single-frequency sinusoidal waveform (typically at 13.56 MHz) and an accurate sheath model was derived for this case by Godyak [8] and Lieberman [9,10].
In recent decades, it has become clear that more complex waveforms are needed to meet industrial needs. Dual-frequency excitation with two well-separated sinusoidal frequencies was used first [11]. Robiche et al [12,13] extended the Lieberman model for this case, with the approximation of the small amplitude of the high-frequency component. However, more complex asymmetric waveforms were later introduced [14] (see also [15] and references therein for an extensive review) and the Lieberman sheath model could not easily be extended to these situations.
Two sheath models were introduced in recent years by Czarnetzki [16], and Turner and Chabert [17] to treat arbitrary waveforms. The latter was used by Lafleur et al [18] to treat the more general case of a capacitive discharge consisting of two asymmetric sheaths with self-bias formation. In this paper we revisit the Turner and Chabert model and extend it to tailored-waveform (Some figures may appear in colour only in the online journal)

Letter
Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. col lisional sheaths. We also give more detail on the derivation and on the information that can be extracted from this model. We do not discuss the validity of the key approximation that allows simple analytical expressions to be obtained, as this was discussed extensively in [17].

The radiofrequency sheath model
We consider a plasma with a density of = = n n n e i 0 at the sheath edge, an electron temperature T e , and a single species of positive ions with mass M. We assume that the plasma fills the half-plane where x < 0, such that the sheath starts at x = 0, where a flux of ions given by ( / ) / is the Bohm speed with e being the elementary charge and T e being expressed in volts). It follows that the sheath forms in the positive half-plane, and this sheath region will end at an electrode. The position of this electrode, denoted by s m , depends on the plasma parameters and on the voltage applied across the sheath.
We now assume that some time-varying current density, J(t), flows through the plasma and the sheath, and we assume that the frequency spectrum of this current permits the assumption that ions passing through the sheath only respond to the time-averaged fields. Electrons, however, flow in and out of the sheath region as the current changes. In this case, we can identify a point s(t), such that at any given time there is a region of positive space charge where < s t x s m ( ) ⩽ and a region of quasi-neutral plasma where < x s t 0 ⩽ ( ). The point denoted as s(t) is understood to be the instantaneous sheath edge (this representation of the sheath is reasonable when λ s m D ; a deeper discussion on this point can be found in [19,20]). The time-averaged potential follows Poisson's equation where 0 ε is the vacuum permittivity, and the time-averaged space charge is The electronic contribution to ρ¯ is frequently important, and generally the time-averaged electron density n ē will be the result of elaborate computations that present a major theoretical challenge. One practical approach is to make an approximation that includes the proper negative space charge within the sheath, without attempting an accurate representation of the spatial distribution of this charge. This aim is achieved by expressing the electron space charge as a constant fraction of the ion space charge, where the dimensionless parameter ξ will be computed selfconsistently from the control parameters. We then have This approximation leads to mutually consistent solutions for both the time-averaged and the time-dependent sheath fields, without introducing any assumption about the time dependence of the sheath current or voltage. Note that in reality the timeaveraged electron density is zero at the electrode and equal to the ion density at the sheath edge, and is thus a function of the position. However, as demonstrated in Turner and Chabert [17], the approximation is very good, whatever waveform is used. Consequently, we can couple this model to an arbitrary current or voltage waveform and proceed to construct a related sheath model without introducing any further approximations.
In figure 1, the time-averaged densities and potential in the sheaths are shown for ξ = 0.5 in the collisionless case, after equations (11) and (12), which will be derived below. In this figure the vertical line at s(t)/s m represents the instantaneous sheath position. We will now construct the solutions for both collisionless and collisional sheaths.
and flux conservation implies Equations (6) and (9) are combined to obtain Poisson's equation This is the equation used in the classical Child-Langmuir theory, with the space-charge reduced by the factor ξ, which can be easily integrated, as many textbooks show (see, for instance, [1,2]). The solution reads is the time-averaged voltage on the electrode at x = s m . The boundary conditions imposed in these solutions are φ = = x 0 0( ) and = = E x 0 0( ) , and we note that < V 0 so that n i and s m are always positive. Equation (13) reduces to the usual collisionless DC Child law when ξ = 1, but as can be seen, the sheath size s m will increase significantly when ξ decreases, i.e. when the electrons transiently neutralize the positive space charge for part of the radiofrequency cycle.
Now consider the situation where the electrode voltage varies in time and we wish to calculate the sheath potential at some instant. Then, using our step-front electron sheath model, the sheath region is divided into a quasi-neutral region that ends at x = s(t), and a positive space charge region for using equation (12). This equation can be integrated once to find the time-and space-varying electric field, using the boundary condition at E(x = s) = 0, yielding in the region where s x s m ⩽ ⩽ . A second integration gives the time-and space-varying potential in the same region as so that the instantaneous voltage at the electrode (at x = s m ) is where we have defined the maximum voltage ξ = V V 0¯/ , which occurs when s = 0. Hence we can write The value of ξ is therefore entirely defined by the sheath motion waveform, and consequently, as we shall see in the next section, by the radiofrequency current waveform.

Current-voltage characteristic of the sheath.
To relate the sheath voltage and the sheath position to the radiofrequency current, we simply note that within this model the current passing through the sheath is entirely displacement at the electrode, so that There is a physical constraint that J = 0 when s = 0 and s = s m . We choose the convention that s(t = 0) = 0. The cur rent waveform must be chosen to be consistent with this convention, but this imposes no physically significant restrictions. Hence, the sheath position is defined solely by the radiofrequency current waveform J(t), Since s/s m has a maximum value of unity, we must satisfy This equation defines the current-voltage characteristic of the collisionless sheath and is particularly useful for the widely used circuit model descriptions of capacitive discharges.
where λ i is the ion-neutral mean free path. Using ion flux conservation in the sheath, we obtain the ion density in the sheath as a function of the electric field, λ π = n n u e E M 2 .
is integrated to obtain the time-averaged electric field in the sheath, where we have used = E 0 0( ) (the electric field at the plasma-sheath interface) as a boundary condition. A second integration gives the time-averaged potential in the sheath, where we have again chosen φ = 0 0( ) . We can now rewrite this expression in the following way, Following the same procedure as in the collisionless sheath case, we obtain the time-varying fields in the sheath, starting from Gauss's law,