Manoj Dalvie

A shock-tracking algorithm for surface evolution under reactive-ion etching

A new algorithm that determines the evolution of a surface eroding under reactive‐ion etching is presented. The surface motion is governed by both the Hamilton–Jacobi equation and the entropy condition for a given etch rate. The trajectories of ‘‘shocks’’ and ‘‘rarefaction waves’’ are then directly tracked, and thus this method may be regarded as a generalization of the method of characteristics. This allows slope discontinuities to be accurately calculated without artificial diffusion. The algorithm is compared with ‘‘geometric’’ surface evolution methods, such as the line‐segment method.

Simulation of surface topography evolution during plasma etching by the method of characteristics

Application of the method of characteristics to the general case of ion or plasma etching is reviewed, yielding a topography evolution algorithm which is simultaneously accurate, flexible, and efficient. The behavior of initial slope discontinuities is computed by mapping the characteristic locus in the region of the discontinuity and removing any closed loops which appear in the locus. The new method is shown to produce profiles which satisfy the required entropy and jump conditions for any given variation of etching rate with surface slope, while allowing the use of longer integration time steps than conventional methods. Previously published ‘‘string’’ algorithms [W. G. Oldham, S. N. Nandgaonkar, A. R. Neureuther, and M. O’Toole, IEEE Trans. Electron Devices ED‐26, 717 (1979); A. R. Neureuther, C. Y. Liu, and C. H. Ting, J. Vac. Sci. Technol. 16, 1767 (1979)] are compared to the new method, and are shown to be capable of generating correct profiles only under limited conditions, i.e., for specific etch...

Self-consistent fluid modeling of radio frequency discharges in two dimensions

Results from a two‐dimensional (2D) fluid simulation of a parallel plate, capacitively coupled radio frequency discharge bounded by a cylindrical insulator with a grounded exterior surface are presented. We find that the radial sheath at the insulator focuses current into the plasma region adjacent to the sheath. This 2D effect has important ramifications for the ionization rate, which peaks sharply in the metal‐insulator corners. We have experimentally observed the enhancement of the emission rate in a corner using spatially resolved optical emission spectroscopy. A ‘‘thick’’ insulator yields radial profiles for the time‐averaged plasma density and potential that are essentially uniform. A ‘‘thin’’ insulator, however, results in an off‐axis maximum in the plasma density and potential due to the corner ionization.

Boundary‐condition refinement of the Child–Langmuir law for collisionless dc plasma sheaths

An exact solution to the problem of collisionless, space‐charge‐limited flow of cold ions across a one‐dimensional (planar) dc plasma sheath of negligible electron density is derived for general values of the presheath ion velocity v0 and electric field E0. For a given ion current density J and sheath thickness d, the exact solution reduces to the classical Child–Langmuir model in the case that v0=0 and E0=0. When either v0 or E0 is sufficiently large, however, the exact solution may differ appreciably from the Child–Langmuir law. The existence of a closed‐form expression for the spatial variation of the sheath potential is shown to be contingent upon the satisfaction of a simple inequality relating v0 and E0 to J. When v0 obeys the Bohm criterion and the magnitude of E0 suggests that the Bohm energy is acquired over a distance not less than one Debye length, this inequality is indeed satisfied.

Flux considerations in the coupling of Monte Carlo plasma sheath simulations with feature evolution models

The evolution of two features, a 2-d trench and a circular via exposed to a 3-v ion flux is calculated. The surface flux is calculated using the particle distribution function, obtained from a Monte Carlo sheath simulation. While Monte Carlo sheath simulations are 1-d/2-v (axisymmetric), shadowing on a substrate surface due to topography breaks the axisymmetry and necessitates a 2-d/3-v surface flux calculation. The authors show a methodology for the flux calculation in long trenches and circular vias which adds very little to computational expense over that incurred for a 2-d/2-v flux calculation. Comparison between the trench and the via shows that, as expected, the geometry of a via blocks out more ion trajectories than that of a trench leading to a smaller etch rate in a via. If the flux is calculated in 2-d/2-v phase space, the resulting feature shape is shown to be an artifact of the calculation method. It is also shown that the angular distribution of the particle as well as energy fluxes and the resulting shape of the feature are largely insensitive to the electric field profile assumed for a given sheath voltage. >

Across wafer etch rate uniformity in a high density plasma reactor: Experiment and modeling

A study of silicon oxide etch rate uniformity in a high density, rf inductively coupled system with an rf capacitively coupled substrate electrode is presented. By introducing spatial variation of rf coupling to the substrate, etch rate uniformity across the wafer can be altered from a profile that is ∼20% higher in the center to one that is ∼10% lower in the center. The effect of spatially varying rf coupling impedance to the substrate is dependent on substrate resistance. Predicted etch rate profiles are obtained with a two‐dimensional analytic model of the plasma source that is coupled to an equivalent circuit discretization of the electrode assembly, substrate, and sheath. Model results compare favorably with experimental measurements.

Detection of particle traps by spatially resolved optical emission spectroscopy over grooved electrodes in radio frequency discharges

Isophotal maps of spatially resolved optical emission signal from neutral, excited argon are used to detect regions of enhanced electron induced excitation over topographically contoured, rf‐coupled electrodes in argon discharges. By aligning a lengthwise groove in one such electrode with the optical axis, it is possible to monitor the plasma homogeneity inside, along and above the groove. The use of a grooved electrode, previously shown to ‘‘trap’’ particles, is also shown to produce enhanced excitation in localized, well defined regions, depending on the discharge pressure and sheath thickness. At low and intermediate pressures (<100 mTorr) a single ‘‘bright’’ spot is noted above the center of the groove. Higher pressure operation causes two bright spots to form, symmetrically placed, close to the groove sidewalls. Laser light scattering is used to simultaneously detect the coordinates of suspended particles during the discharge. A correlation is noted between these bright spots and the location of trap...

Ion energetics in collisionless sheaths of rf process plasmas

Ion energy distribution functions in collisionless radio‐frequency (rf) sheaths are discussed from the viewpoint of kinetic theory. Effects of rf fields on ion density and velocity profiles in plasma sheaths are also derived, based on ion fluid equations. It is shown that the ponderomotive force due to rf modulation of the magnitude of the sheath electric field exerts a retarding effect on the ion motion that counteracts the dc‐bias field when the ratio of the ion transit frequency ωtr to rf modulation frequency ω is small but finite. Consequently, the time‐averaged ion density is higher and the time‐averaged ion fluid velocity is lower in rf sheaths by the order of (ωtr/ω)2 than those in corresponding dc sheaths. The influence of an oscillating plasma/sheath boundary on the ion energetics is also considered. Under suitable conditions, this induces rapid ‘‘quasiperiodic’’ variations in the ion energy distribution as the rf ω is increased.

Hydrodynamic analysis of electron motion in the cathode fall using a Monte Carlo simulation

The exact mass, momentum, and energy conservation equations for electron transport in a dc glow are derived from the Boltzmann equation. A Monte Carlo particle simulation is used to explicitly calculate the individual terms of the moment equations, and to gain insight into the behavior of the electron distribution function (EDF) moments such as density and average velocity. Pure forward scattering and isotropic scattering are considered as two limiting scattering mechanisms. When forward scattered, the electron fluid shows the maximum change in properties and in transport mechanisms at the field transition point between the cathode fall (CF) and the negative glow. Isotropic scattering, however, results in property changes a short distance inside the sheath. Diffusion of the low‐energy, high‐density, bulk plasma electrons into the CF causes dilution of the low‐density, high‐energy beam from the CF before the beam actually arrives at the low‐field region. The applicability of commonly used closure relations which yield a fluid description of the system is evaluated. Use of fluid equations to characterize this system with no a priori knowledge of the EDF is limited by kinetic effects, such as heat flow against the temperature gradient, especially in the forward‐scattered case where the EDF is very anisotropic. The description of inelastic rates by Arrhenius kinetics is found to be surprisingly accurate with both scattering mechanisms. However, while temperature is an adequate gauge of the characteristic energy under isotropic scattering, the energy of the bulk electron motion must be included under forward scattering. Also, Arrhenius kinetics sometimes produce a spurious double peak in the inelastic rate profile which is not reproduced by the Monte Carlo simulation. The anisotropy of the EDF under the forward‐scatter assumption makes it difficult to justify the use of the mobility and heat conduction closure relations. Under isotropic scattering, however, electron inertia is negligible. In that case, under the discharge conditions used here, the drift‐diffusion approximation to the flux is good to within a factor of 2. Classical heat conduction theory overestimates the heat flux by a factor of 4 at the sheath edge.