M. A. Lieberman

Analytical solution for capacitive RF sheath

A self-consistent solution for the dynamics of a high voltage, capacitive radio frequency (RF) sheath driven by a sinusoidal current source is obtained under the assumptions of time-independent, collisionless ion motion and inertialess electrons. Expressions are obtained for the time-average ion and electron densities, electric field and potential within the sheath. The nonlinear oscillation motion of the electron sheath boundary and the nonlinear oscillating sheath voltage are also obtained. The effective sheath capacitance and conductance are also determined. It was found that: (1) the ion-sheath thickness S/sub m/ is square root 50/27 larger than a Childs law sheath for the DC voltage and ion current density; (2) the sheath capacitance per unit area for the fundamental voltage harmonic is 2.452 epsilon /sub 0//S/sub m/, where epsilon /sub 0/ is the free space permittivity; (3) the ratio of the DC to peak value of the oscillating voltage is 54/125; (4) the second and third voltage harmonics are, respectively, 12.3 and 4.2% of the fundamental; and (5) the conductance per unit area for stochastic heating by the oscillating sheath is 2.98 ( lambda /sub D//S/sub m/)/sup 2/3/ (e/sup 2/n/sub 0//mu/sub e/), where n/sub 0/ is the ion density, lambda /sub D/ is the Debye length at the plasma-sheath edge, and u/sub e/ is the mean electron speed. >

Global model of Ar, O2, Cl2, and Ar/O2 high‐density plasma discharges

We develop a global (volume averaged) model of high‐density plasma discharges in molecular gases. For a specified discharge length and diameter, absorbed power, pressure, and feed gas composition, as well as the appropriate reaction rate coefficients and surface recombination constants, we solve the energy and particle balance equations to determine all species densities and the electron temperature. We use an expression for charged particle diffusive loss that is valid for low and high pressures and for electropositive and electronegative plasmas. We apply the model to Ar, O2, Cl2, and Ar/O2 discharges and compare with available experimental data. In Ar, we find that the ion density increases monotonically with increasing pressure, while for O2 and Cl2, the total positive ion density increases initially, then decreases as pressure is further increased. For a pure Cl2 discharge, we find that surface recombination processes are important in affecting the degree of dissociation and the negative‐ion density ...

Model of plasma immersion ion implantation

In plasma immersion ion implantation, a target is immersed in a plasma and a series of negative high‐voltage pulses are applied to implant plasma ions into the target. We develop an approximate analytical model to determine the time‐varying implantation current, the total dose, and the energy distribution of the implanted ions.

Standing wave and skin effects in large-area, high-frequency capacitive discharges

Large-area capacitive discharges driven at frequencies higher than the usual industrial frequency of 13.56 MHz have attracted recent interest for materials etching and thin film deposition on large-area substrates. Standing wave and skin effects can be important limitations for plasma processing uniformity, which cannot be described by conventional electrostatic theory. An electromagnetic theory is developed for a discharge having two plates of radius R and separation 2l, which accounts for the propagation of surface and evanescent waves from the discharge edge into the centre and the role of capacitive and inductive fields in driving the power absorption. Examples of discharge fields are given having substantial standing wave and/or skin effects. The conditions for a uniform discharge without significant standing wave and skin effects are found to be, respectively, λ0>>2.6(l/s)1/2R and δ>>0.45(dR)1/2, where λ0 is the free space wavelength, s is the sheath width, δ = c/ωp is the collisionless skin depth, with c the speed of light and ωp the plasma frequency, and d = l-s is the plasma half-width. Taking the equality for these conditions for a discharge radius of 50 cm, plate separation of 4 cm, and sheath width of 2 mm, there is a substantial skin effect for plasma densities 1010 cm-3, and there is a substantial standing wave effect for frequencies f70 MHz.

Ion energy distributions in rf sheaths; review, analysis and simulation

We present a review and analysis of ion energy distributions (IED) arriving at the target of a radio frequency (rf) discharge. We mainly discuss the collisionless regime, which is of great interest to experimentalists and modellers studying high-density discharges in which the sheath is much thinner than in conventional reactive ion etching systems. We assess what has been done so far and determine what factors influence the shape of the IEDs. We also briefly discuss collisional effects on the IEDs. Having determined the important parameters, we perform some particle-in-cell simulations of a collisionless current-driven rf sheath which show that ion modulations in an rf sheath significantly affect the IEDs when ion/rf<1, where ion is the ion transit time and rf is the rf period.

Global models of pulse-power-modulated high-density, low-pressure discharges

Global (volume-averaged) models of high-density, low-pressure electropositive and electronegative discharges are described both for continuous wave (CW) and for pulsed-power excitation. Argon and chlorine discharges are treated. The particle and energy balance equations are applied to determine the charged particle and neutral dynamics. For argon just after the power has been turned on, the analysis shows an initial very sharp rise in electron temperature , followed by a decay of and an increase in the electron density to steady state values during the pulse-on time. Just after the power has been turned off, decays rapidly and decays more slowly. The time-average can be considerably higher than that for CW discharges for the same time-average power. For chlorine, a CW discharge is highly dissociated and the negative ion density is lower than . For a pulsed discharge, the initial rise and subsequent decay of just after the power has been turned off are determined analytically. A pulsed discharge can have the same neutral radical (Cl) flux to the walls for a reduced average power. The analytical models are compared to more complete global model simulations and to experimental observations. We find that global models can provide considerable insight into the discharge dynamics.

Dynamics of a collisional, capacitive RF sheath

A self-consistent solution for the dynamics of a high-voltage, capacitive RF sheath driven by a sinusoidal current source is obtained, under the assumptions of time-independent, collisional ion motion and inertialess electrons. Values of the ion current density, the sheath capacitance per unit area for the fundamental voltage harmonic, and the conductance per unit area for stochastic heating by the oscillating sheath are calculated. The ratio of the DC to the peak value of the oscillating voltage is found to be 0.40, while the second and third voltage harmonics are, respectively, 19.3% and 5.3% of the fundamental. >

Model of plasma immersion ion implantation for voltage pulses with finite rise and fall times

In plasma immersion ion implantation, a target is immersed in a plasma and a series of negative, high‐voltage pulses are applied to implant ions into the target. An approximate analytical model in one‐dimensional planar geometry is developed to determine the time‐varying implantation current, the total dose, and the energy distribution of the implanted ions for a voltage pulse with finite rise and fall times. Scaling rules are presented for the implanted current and energy distribution with respect to plasma density, peak applied voltage, and ion mass. Comparisons with numerical simulations are used to demonstrate that the accuracy of the model is well characterized by a single parameter: the ratio of the ion flight time to the pulse rise time.

Spatially averaged (global) model of time modulated high density argon plasmas

The behavior of argon plasmas driven by time modulated power in ‘‘high density’’ plasma reactors is investigated using a global model. The time evolution of the electron temperature and the plasma density is calculated by solving the particle and energy balance equations. In the first stage of power application during the ‘‘on’’ time, the electron temperature rapidly increases above the steady state value. In this region, charged particles accumulate in the plasma due to the relatively higher power applied than for the continuous wave (cw) case. In the first stage of the ‘‘off’’ time, the electron temperature drops quickly, yielding a smaller particle loss (Bohm) velocity. These effects give rise to higher time‐average plasma densities than for the cw plasma driven by the same average power. The highest average plasma density obtained was more than twice the density of the cw plasma for a duty ratio of 25%. Even higher plasma densities were obtained for shorter duty ratios. The possibility of controlling ...

Verification of frequency scaling laws for capacitive radio‐frequency discharges using two‐dimensional simulations*

Weakly ionized processing plasmas are studied in two dimensions using a bounded particle‐in‐cell (PIC) simulation code with a Monte Carlo collision (MCC) package. The MCC package models the collisions between charged and neutral particles, which are needed to obtain a self‐sustained plasma and the proper electron and ion energy loss mechanisms. A two‐dimensional capacitive radio‐frequency (rf) discharge is investigated in detail. Simple frequency scaling laws for predicting the behavior of some plasma parameters are derived and then compared with simulation results, finding good agreements. It is found that as the drive frequency increases, the sheath width decreases, and the bulk plasma becomes more uniform, leading to a reduction of the ion angular spread at the target and an improvement of ion dose uniformity at the driven electrode.