in connection with the process
Plasma Metrology System is based on the Self
Plasma Resonance Spectroscopy
low pressure and on Nonlinear
The capacitive driven radio frequency (RF) discharge is commonly used for materials processing. However, the required processing (etching, deposition) of such discharges is generally the result of poorly understood physical and chemical processes occurring in the gas phase and at the gas/solid interface. In this respect, the role of the sheaths in front of the electrodes is of fundamental importance for the understanding of the discharge physics of the asymmetrical capacitively coupled RF discharge. Due to their small mass (large mobility) and high temperature, the electrons strive to leave the bulk plasma. The sheath (dark space) keeps the electrons within the bulk plasma (shielding) by a retarding electric field build-up when electrons leave the outer regions of the plasma. The electrons-neutrals collisions take place in the bulk plasma where ions and electrons are created. The radicals for etching are formed in the bulk plasma. The ions move to the sheaths. The acceleration of ions provides energy for etching. This energy can be increased by external (RF) potentials.
|Understanding the real world - SEERS theory:|
Nonlinear effects like the I-V-characteristic of the boundary sheath are utilized for Langmuir probe measurements but they are usually neglected for modeling of RF discharges due to their very inconvenient mathematical treatment. The Self Excited Electron Plasma Resonance Spectroscopy (SEERS) utilizes exactly these nonlinear effects and known resonance effects in RF discharges. The nonlinear elements, in particular the sheathes, provide harmonics in the discharge current and excite the plasma and the sheath at their series resonance characterized by the so-called geometric resonance frequency.
|The bulk model:|
|The model of the plasma bulk is
based on a 2d-fluid model with zero order moment of the
and the first order moment
A uniform plasma yields
with the above moments of the Boltzmann equation the permittivity of the plasma
The full set of the Maxwellian equations leads now to the Helmholtz equation for the magnetic field H = F(r, z)
In a cylinder geometry it can be satisfied by a fundamental system with a series in Bessel functions based here on the azimuthal component of the magnetic field
if, as already assumed above, the plasma density and so the the permittivity can be is assumed to be approximately constant. The coefficients are given though proper boundary conditions depending on the geometry. The total RF current is then given by
|The sheath model:|
|The first three moments of the Boltzmann equation and the
Poisson equation describe the electron dynamics. Together with
a parameter ansatz for the ion distribution, boundary
conditions provide by a nonlinear
model (here simplified without pressure heating, see )
with the time varying, normalized sheath thickness δ. The convolution is denoted *, the normalization to fundamental plasma physical parameters is as follows
|The complete model and SEERS:|
The final model depends mainly on the averaged density of the electrons and their collision rate. In the simple case of a one-dimensional plasma, the fundamental equation can be reflected by equivalent circuit as shown in the scheme. The fundamental equation is now a nonlinear differential equation in the time domain or convolution equation in the frequency domain.
So three parameters must be estimated by minimizing an appropriate (mathematical) norm. This model-based determination of plasma parameters is called Self Excited Electron Resonance Spectroscopy (SEERS)[3-9].
SEERS provides the spatially and harmonically averaged electron plasma density and the effective electron collision rate. The electron collision rate comprises stochastic (pressure) heating and ohmic heating of the electrons. It depends via ohmic heating on the density of gas and so on the gas temperature [10-11]
where νohm is the electron collision rate for ohmic heating, n the density of neutrals, nα the density of species α, <ve> the mean electron velocity, σα the (elastic) cross section with respect to species α, p the pressure, kB the Boltzmann constant, and T the temperature. Thus in particular the electron collision rate is an very important indicator for the plasma process stability due to their dependence on the process chemistry. But it reflects also plasma process drift caused by physical mechanisms as a gas temperature drift the major reason of the first wafer effect in semiconductor manufacturing. The second parameter, the spatially avaraged electron density shows usually a close relation to the pure physical mechanisms as sputter etching.
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