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RF-discharge-theory.lyx
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RF-discharge-theory.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_document
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\options titlepage, captions=tableheading, bibliography=totoc, usenames, dvipsnames
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\begin_body
\begin_layout Title
Theory of RF-plasma processes
\end_layout
\begin_layout Author
Uwe Stöhr, Ph.D.
\begin_inset Newline newline
\end_inset
\begin_inset Newline newline
\end_inset
Version 1.0
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Section
Plasma Frequency
\begin_inset CommandInset label
LatexCommand label
name "sec:Plasma-Frequency"
\end_inset
\end_layout
\begin_layout Standard
The plasma frequency
\begin_inset Formula $\omega_{e}$
\end_inset
is the eigenfrequency of the electrons.
Without external forces they oscillate with this frequency around the ions.
The derivation is simple: Assuming the electrons are shifted by a length
\begin_inset Formula $x$
\end_inset
away from their position.
The equation of motion is then
\begin_inset Formula
\begin{equation}
m_{e}\ddot{x}=-eE\label{eq:motion}
\end{equation}
\end_inset
where
\begin_inset Formula $m_{e}=9.1\cdot10^{-31}\,$
\end_inset
kg is the electron mass,
\begin_inset Formula $e=1.602\cdot10^{-19}\,$
\end_inset
As the elementary charge and
\begin_inset Formula $E$
\end_inset
the electric field strength.
\end_layout
\begin_layout Standard
Using the electron density
\begin_inset Formula $n_{e}$
\end_inset
in the
\noun on
Gauss
\noun default
law we get
\begin_inset Formula
\begin{equation}
E=\frac{en_{e}}{\epsilon_{0}}\,x\label{eq:E-frequency}
\end{equation}
\end_inset
where
\begin_inset Formula $\epsilon_{0}=8.854\cdot10^{-12}\,$
\end_inset
As/Vm is the vacuum permittivity.
Putting
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:E-frequency"
\end_inset
into
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:motion"
\end_inset
gives
\begin_inset Formula
\begin{equation}
0=\underbrace{\frac{e^{2}n_{e}}{m_{e}\epsilon_{0}}}_{\mathrm{eigenfrequency}}\,x+\ddot{x}
\end{equation}
\end_inset
Out of this oscillation equation we can directly read the expression for
the eigenfrequency:
\begin_inset Formula
\begin{equation}
\omega_{e}=\sqrt{\frac{e^{2}n_{e}}{m_{e}\epsilon_{0}}}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Similarly to
\begin_inset Formula $\omega_{e}$
\end_inset
the eigenfrequencies for the ions can be determined:
\begin_inset Formula
\begin{equation}
\omega_{i}=\sqrt{\frac{e^{2}n_{i}}{M\epsilon_{0}}}\label{eq:ion-plasma-frequency}
\end{equation}
\end_inset
where
\begin_inset Formula $M$
\end_inset
is the mass of an ion.
Due to the global quasi-neutrality
\begin_inset Formula $n_{e}\approx n_{i}$
\end_inset
for normal plasmas.
\end_layout
\begin_layout Standard
For a
\begin_inset CommandInset href
LatexCommand href
name "capacitively coupled plasma"
target "https://en.wikipedia.org/wiki/Capacitively_coupled_plasma"
literal "false"
\end_inset
(CCP) RF-discharge at a pressure of a few pascals the typical electron
density is in the range of
\begin_inset Formula $10^{16}\,$
\end_inset
1/m³.
This leads to
\begin_inset Formula $\omega_{e}=5.6\,$
\end_inset
GHz.
For
\begin_inset Formula $\text{O_{2}^{-}}$
\end_inset
-ions with
\begin_inset Formula $M=5.312\cdot10^{-26}\,$
\end_inset
kg we then have
\begin_inset Formula $\omega_{i\,\text{O_{2}^{-}}}=23.36\,$
\end_inset
MHz.
\end_layout
\begin_layout Section
Plasma Sheath
\end_layout
\begin_layout Standard
The sheath is the zone above the electrode and the chamber wall surface
where the electron density
\begin_inset Formula $n_{e}$
\end_inset
is much lower than the ion density
\begin_inset Formula $n_{i}$
\end_inset
.
This zone is also visible as the
\begin_inset Quotes eld
\end_inset
dark zone
\begin_inset Quotes erd
\end_inset
when looking into the chamber when a plasma is burning.
The reason for the sheath is that the RF of
\begin_inset Formula $f=\omega/2\pi=13.56\,$
\end_inset
MHz is greater than the ion plasma frequency
\begin_inset Formula $\omega_{i}$
\end_inset
but below the electron plasma frequency
\begin_inset Formula $\omega_{e}$
\end_inset
.
Therefore the electrons can follow the electrical field while the ions
cannot.
\begin_inset Foot
status open
\begin_layout Plain Layout
\series bold
Note:
\series default
Hydrogen atoms are light enough to follow the electrical field:
\begin_inset Formula $\omega_{H^{+}}\approx132\,$
\end_inset
MHz.
\end_layout
\end_inset
So at the time where the electrode is positively charged, electrons can
stream out of the plasma to the electrode so that there is a region where
\begin_inset Formula $n_{e}\approx0$
\end_inset
while
\begin_inset Formula $n_{i}\gg n_{e}$
\end_inset
.
The density distribution at the electrode/walls is schematically shown
in
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Scheme-sheath"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename clipart/Sheath-scheme.pdf
width 50col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Scheme-sheath"
\end_inset
Scheme of the particle densities in a sheath.
The wall is at
\begin_inset Formula $x=s$
\end_inset
, the sheath in the range
\begin_inset Formula $0\le x\le s$
\end_inset
.
Image from
\begin_inset CommandInset citation
LatexCommand cite
key "Keudell12"
literal "true"
\end_inset
.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The thickness of the sheath is important to know because it determines the
distance from the wall in which no substrates can be placed in for coating.
It also helps to approximate the dimensions of holes and recessed areas
in which strong plasma discharges will burn (hollow cathode effect).
This topic is discussed in
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Hollow-cathode-effect"
\end_inset
.
\end_layout
\begin_layout Subsection
Requirements for a Sheath
\end_layout
\begin_layout Standard
What is necessary to get a sheath? It is known that there is no plasma burning
in small holes, so obviously there is not enough space to form a sheath.
The reason could be that the ions cannot be accelerated to reach a level
to build up the positive potential at the wall.
That this is indeed the reason is shown in the following.
\end_layout
\begin_layout Standard
The energy conservation leads to
\begin_inset Formula
\begin{equation}
E(x)=\frac{Mv_{i}^{2}(x)}{2}+e\,\varPhi(x)=\frac{Mv_{0}^{2}}{2}=E_{0}\label{eq:energy-conservation}
\end{equation}
\end_inset
where
\begin_inset Formula $M$
\end_inset
is the mass of an ion,
\begin_inset Formula $E(x)$
\end_inset
is the electric field strength in the sheath,
\begin_inset Formula $\varPhi(x)$
\end_inset
the potential in the sheath,
\begin_inset Formula $v_{i}(x)$
\end_inset
the velocity of the ions in the sheath and
\begin_inset Formula $v_{0}$
\end_inset
the velocity of the ions in the plasma bulk streaming into the sheath.
\end_layout
\begin_layout Standard
The charge also needs to be conserved so that the current
\begin_inset Formula $I$
\end_inset
through the area
\begin_inset Formula $A$
\end_inset
of the sheath needs to be constant:
\begin_inset Formula
\begin{eqnarray}
\frac{I_{i}}{A} & = & \frac{I_{0}}{A}=\frac{N_{0}\cdot e}{tA}=n_{o}\cdot e\cdot v_{0}\\
n_{i}(x)v_{i}(x) & = & n_{0}v_{0}\label{eq:current-conservation}
\end{eqnarray}
\end_inset
(
\begin_inset Formula $N$
\end_inset
is the number of particles and
\begin_inset Formula $t$
\end_inset
the time.) Putting
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:current-conservation"
\end_inset
into
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:energy-conservation"
\end_inset
we get
\begin_inset Formula
\begin{equation}
n_{i}(x)=n_{0}\,\left(1-\frac{e\,\varPhi(x)}{E_{0}}\right)^{-1/2}
\end{equation}
\end_inset
The plasma frequency of electrons is above the applied RF so that they can
follow this frequency.
Therefore their density follows the
\noun on
Boltzmann
\noun default
relation:
\begin_inset Foot
status open
\begin_layout Plain Layout
See
\begin_inset CommandInset citation
LatexCommand cite
after "sections 2.2.4 and 2.2.5"
key "Keudell12-2"
literal "true"
\end_inset
for a detailed derivation of the
\noun on
Boltzmann
\noun default
relation.
\end_layout
\end_inset
\begin_inset Formula
\begin{eqnarray}
en_{e}\nabla\varPhi(x) & = & \nabla p(x)=\nabla n_{e}(x)k_{B}T\\
\frac{\mathrm{d}n_{e}(x)}{n_{e}(x)} & = & \frac{e\,\mathrm{d}\,\varPhi(x)}{k_{B}T_{e}}\\
n_{e}(x) & = & n_{0}\exp\left(\frac{e\,\varPhi(x)}{k_{B}T_{e}}\right)
\end{eqnarray}
\end_inset
with the
\noun on
Boltzmann
\noun default
constant
\begin_inset Formula $k_{B}=1.38\cdot10^{-23}\,$
\end_inset
J/K.
\end_layout
\begin_layout Standard
The
\noun on
Gauss
\noun default
law is then
\begin_inset Formula
\begin{eqnarray}
\frac{\mathrm{d}^{2}\varPhi(x)}{\mathrm{d}x^{2}} & = & \frac{e}{\epsilon_{0}}\left(n_{e}(x)+n_{i}(x)\right)\label{eq:Gauss-law1}\\
& = & \frac{en_{0}}{\epsilon_{0}}\,\left(\exp\left(\frac{e\,\varPhi(x)}{k_{B}T_{e}}\right)+\left(1-\frac{e\,\varPhi(x)}{E_{0}}\right)^{-1/2}\right)
\end{eqnarray}
\end_inset
(
\begin_inset Formula $\epsilon_{0}$
\end_inset
is the vacuum permittivity.) It cannot be solved analytically.
For a derivation of an approximation at
\begin_inset Formula $\varPhi(x\approx0)$
\end_inset
see
\begin_inset CommandInset citation
LatexCommand cite
after "sec. \"Raumladungszone einer Ionenrandschicht\""
key "Keudell12"
literal "true"
\end_inset
.
This approximation delivers the relation
\begin_inset Formula
\begin{eqnarray}
\frac{\left(e\,\varPhi(x)\right)^{2}}{k_{B}T_{e}}-\frac{\left(e\,\varPhi(x)\right)^{2}}{2E_{0}} & > & 0\nonumber \\
\underbrace{\sqrt{\frac{k_{B}T_{e}}{M}}}_{v_{B}} & < & v_{0}\label{eq:Bohm-velocity}
\end{eqnarray}
\end_inset
\begin_inset Formula $v_{B}$
\end_inset
is named
\noun on
Bohm
\noun default
velocity and is the velocity that ions must at least have so that a sheath
can be formed.
This velocity is also the speed of sound of the ions.
\end_layout
\begin_layout Subsection
Mean Thickness of Sheaths
\end_layout
\begin_layout Standard
We are working in a low pressure regime so that we can assume that the mean
free path
\begin_inset Formula $\lambda$
\end_inset
of the particles is independent of the location in the plasma chamber.
For the ion collision frequency we can now write
\begin_inset Formula $\nu=\cfrac{v_{i}}{\lambda}$
\end_inset
.
With the ion velocity
\begin_inset Formula $v_{i}(x)=\mu E(x)$
\end_inset
, the ion drift mobility
\begin_inset Formula $\mu=\cfrac{e}{M\nu}$
\end_inset
and the field strength
\begin_inset Formula $E$
\end_inset
we can write
\begin_inset Formula
\begin{eqnarray}
v_{i}(x) & = & \frac{e\lambda}{Mv_{i}(x)}\,E(x)\nonumber \\
& = & \sqrt{\frac{e\lambda}{M}\,E(x)}
\end{eqnarray}
\end_inset
where
\begin_inset Formula $M$
\end_inset
is the mass of an ion and
\begin_inset Formula $e$
\end_inset
the electron charge.
The ion density can now be calculated according to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:current-conservation"
\end_inset
:
\begin_inset Formula
\begin{equation}
n_{i}(x)=\frac{v_{B}}{v_{i}(x)}\,n_{0}\label{eq:n_i}
\end{equation}
\end_inset
where
\begin_inset Formula $v_{B}$
\end_inset
is the
\noun on
Bohm
\noun default
velocity that is necessary to get a sheath,
\begin_inset Formula $T_{e}$
\end_inset
the electron temperature,
\begin_inset Formula $k_{B}$
\end_inset
the
\noun on
Boltzmann
\noun default
constant and
\begin_inset Formula $n_{0}\approx n_{e}\approx n_{i}$
\end_inset
the density in the plasma bulk.
\end_layout
\begin_layout Standard
The simplest model of a sheath is that the electron density in the sheath
is zero.
Using this we can neglect the electron density
\begin_inset Formula $n_{e}$
\end_inset
in the
\noun on
Gauss
\noun default
law:
\begin_inset Formula
\begin{eqnarray}
\epsilon_{0}\,\frac{\mathrm{d}E}{\mathrm{d}x} & = & e\,\left(n_{i}(x)+\underbrace{n_{e}(x)}_{\approx0}\right)=\cfrac{e\,n_{0}\,\sqrt{\cfrac{k_{B}T_{e}}{M}}}{\sqrt{\frac{e\lambda}{M}\,E}}\label{eq:Gauss-law2}\\
\epsilon_{0}\sqrt{E}\,\mathrm{d}E & = & en_{0}\,\sqrt{\cfrac{k_{B}T_{e}}{e\lambda}}\,\mathrm{d}x=j_{0}\,\sqrt{\cfrac{M}{e\lambda}}\,\mathrm{d}x\label{eq:current-density}
\end{eqnarray}
\end_inset
(
\begin_inset Formula $\epsilon_{0}$
\end_inset
is the vacuum permittivity and
\begin_inset Formula $j_{0}$
\end_inset
the current density.) We integrate from the border of the sheath at
\begin_inset Formula $x=0$
\end_inset
to the chamber border at
\begin_inset Formula $x=s$
\end_inset
:
\begin_inset Formula
\begin{equation}
E^{3}=\frac{9}{4}\,s^{2}\,n_{0}^{2}\,\frac{e\,k_{B}T_{e}}{\epsilon_{0}^{2}\lambda}\label{eq:E-sheath}
\end{equation}
\end_inset
At the electrode we apply the voltage
\begin_inset Formula $U$
\end_inset
so that we have
\begin_inset Formula $E(s)=\cfrac{U}{s}$
\end_inset
.
We can now calculate the sheath thickness
\begin_inset Formula $s$
\end_inset
to
\begin_inset Formula
\begin{equation}
s=\left(\frac{4U^{3}\epsilon_{0}^{2}\lambda}{9n_{0}^{2}ek_{B}T_{e}}\right)^{1/5}\label{eq:sheath-thickness}
\end{equation}
\end_inset
Using the relation for the mean free path assuming a
\noun on
Maxwell
\noun default
distribution for the molecule energies
\begin_inset Formula
\begin{equation}
\lambda=\cfrac{k_{B}T}{\sqrt{2}\pi d^{2}p}\label{eq:lambda}
\end{equation}
\end_inset
(
\begin_inset Formula $d$
\end_inset
is the diameter of the gas molecules.)
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:sheath-thickness"
\end_inset
can be transformed to
\begin_inset Formula
\begin{equation}
s=\left(\frac{4U^{3}\epsilon_{0}^{2}T}{9\sqrt{2}n_{0}^{2}\pi epd^{2}T_{e}}\right)^{1/5}\label{eq:s-mean}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
The values for
\begin_inset Formula $\lambda$
\end_inset
for different precursors are listed in
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Mean-Free-Paths"
\end_inset
.
The mean voltage
\begin_inset Formula $\bar{U}$
\end_inset
at the electrode is the bias voltage so that this can be used to calculate
the mean sheath thickness
\begin_inset Formula $s$
\end_inset
.
\end_layout
\begin_layout Standard
Taking typical values:
\begin_inset Formula $\lambda_{\ce{O2}}(p=2\,\mathrm{Pa})\approx8\,$
\end_inset
mm,
\begin_inset Formula $T\approx300\,$
\end_inset
K,
\begin_inset Formula $T_{e}\approx3\cdot10^{4}\,$
\end_inset
K,
\begin_inset Formula $n_{0}\approx10^{16}\,$
\end_inset
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
1/m³
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
,
\begin_inset Formula $p=2\,$
\end_inset
Pa,
\begin_inset Formula $\bar{U}\approx400\,$
\end_inset
V leads with
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:sheath-thickness"
\end_inset
to
\begin_inset Formula $s\approx4.7\,$
\end_inset
mm.
\end_layout
\begin_layout Standard
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:s-mean"
\end_inset
is the time-independent mean sheath thickness where also the dependency
on the electrode area is not taken into account.
An approximation of a time- and area-dependent sheath thickness is derived
in the following section.
\end_layout
\begin_layout Subsection
Time-dependent Thickness of Sheaths
\end_layout
\begin_layout Standard
A CCP is kept burning by shifting the charge within the chamber in the applied
frequency
\begin_inset Formula $\omega_{\mathrm{RF}}$
\end_inset
.
This displacement current
\begin_inset Formula $I_{a}$
\end_inset
is
\begin_inset Formula
\begin{equation}