A Time‐Average Model of the RF Plasma Sheath

Abstract

A time-average model of the RF plasma sheath was developed. The ion "fluid" equations were used with a frictional force to account for ion-neutral collisions. Consideration of the collision dynamics showed that the frictional force may be taken as proportional to the square of the ion drift velocity. The sheath model was used to investigate the ion energy and flux on the electrodes of plasma reactors. The dimensionless quantity Co (collision number) was found to be important in describing the ion motion in the sheath. An analytical expression for the ion bombardment energy was derived, in terms of Co and the sheath voltage, for the range of parameter values typical of high pressure (-1 torr) diode plasma etchers. An application of the model to an oxygen discharge in a parallel plate reactor was considered. Over the parameter range investigated, the ion bombardment energy was found to be only a few tens of eV, much lower than typical sheath voltages (-200V). The ion bombardment energy was found to be a function of the sheath electric-field-to-pressure ratio. The model provides a framework that can be incorporated into more general plasma reactor models which consider transport and reaction phenomena along surfaces undergoing etching.Within the last decade, dry etching using an RF diode discharge in a chemically reactive gas has become a widespread technique in the fabrication of microelectronic devices. Under conditions of relatively high pressure (-1 torr) and high frequency (-10 MHz), an RF discharge of this kind may resemble an ideal, nonequilibrium, slightly ionized plasma separated from wall and electrode surfaces by a region of positive space charge or sheath. In the sheath region ions are accelerated toward electrode or wall surfaces by the electric field present due to the space charge. In their transit through the sheath, ions may experience collisions, mainly with neutrals owing to the low degree of ionization (~<10-4). Such collisions serve to reduce the energy and randomize the ionic motion. Determination of the ion flux and the energy and angular distribution functions of ions striking the electrodes of plasma reactors is of significant technological interest in semiconductor processing. Such quantities affect both the etching rate and the degree of anisotropy in plasma etching applications. The purpose of this paper is to develop an approximate model to investigate the behavior of ionic species within the high field sheath regions of a gas discharge.Quantities that affect the energy and/or directionality of bombarding ions include the sheath electric-field-topressure ratio (E/p) (1, 2), the applied frequency (3), and the nature and topography of the surface (4, 5). The quantity E/p is a measure of the energy imparted to an ion by the electric field over the distance of one mean fl'ee path. Frequency has a pronounced effect on ion bombardment energy. At low frequencies ions can respond to the instantaneous sheath voltage and, in the absence of collisions, the maximum ion energy would correspo...

Modeling of the sheath and the energy distribution of ions bombarding rf-biased substrates in high density plasma reactors and comparison to experimental measurements

“…[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] Most researchers focused on studying the spatial and temporal variation of the potential field within the sheath without calculating IEDs. However the calculated fields can be used to generate ion energy distributions.…”

Multiscale modeling, simulations, and experiments of coating growth on nanofibers. Part II. Deposition

“…The sheath model at the holder uses a relatively low velocity ͑due to the low voltage difference͒ ion transport to approximate the etching rate at the coating surface. Our operating conditions are near the edge of consistency with the assumptions of a uniform plasma model approximation ͑Lieberman and Lichtenberg 14 ͒ and with the time-averaged model of Economou et al 15 The latter is the basis for the sheath models at the target and the holder.…”

“…This form replicates the expression used by Ref. 15 to simulate the frictional force on an ion as an average over all possible collisions experienced by the ion. The gradient of the electric field is related to the charge density according to the equation…”

“…The Bohm criterion and quasineutrality 11,15 are used to define the boundary conditions for this system of first-order differential equations. We refer the reader to Ref.…”

“…We refer the reader to Ref. 15 for a thorough discussion of the boundary conditions. Ordinarily, this system would require four boundary conditions.…”