Hercules®
in connection with the process
chamber |
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The Hercules®
Plasma Metrology System is based on the Self
Excited Electron
Plasma Resonance Spectroscopy
(SEERS) for
low pressure and on Nonlinear
Extended
Electron
Dynamics
(NEED) for
medium pressure. |
RF discharge: |
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Understanding the real world - SEERS theory: |
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The bulk model: |
The model of the plasma bulk is
based on a 2d-fluid model with zero order moment of the
Boltzmann
equation![]() 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
sheath
model (here simplified without pressure heating, see [4])![]() 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: |
![]() 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 [10-12]. ![]() 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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Literature: |
Please visit our
download area, in particular the section
of Fundamental Papers or contact
us via mail in case of further interest.
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