PLASMATORÒ, which is based on INDUCT-95 written by Peter A.
Vitello,
at the Lawrence Livermore National Laboratory and licensed from LLNL and the
University of California by Kinema Research & Software, L.L.C., is a two
dimensional numerical model of plasma devices based upon a fluid treatment of
electrons and ions. Electron heating by
induction from external radio frequency coils and/or by internal rf or dc
electrodes is calculated self-consistently by solving for the electric field.
The field solver was developed at Oak Ridge National Laboratory.
We describe herein Version 2 of PLASMATORÒ.
We have improved the input and output, especially the electron collision and
chemistry data input, have added new physics capability to the code, and have
devised a user-friendly GUI for Windows 9x/NT/2000. We have aggressive plans
for further improvement in modeling capability and databases for plasma chemistry
modeling.
·
Fluid dynamic treatment of transport equations.
·
Electrons, multiple ions (positive and negative) and
neutral species.
·
Cartesian (x,y) or cylindrical geometry (r,z) on a
non-uniform mesh
·
Restarts automatically remap variable profiles from old
mesh to new.
·
Coarse mesh solutions can be used to rapidly explore
high-resolution effects.
·
Sheaths can be spatially resolved using locally fine
mesh gridding.
·
Dielectrics include surface space charging.
·
Metal structures can each be separately treated as
having fixed DC potential, floating potential, or RF potentials.
·
Inductively driven.
·
Capacitively driven.
·
Glow discharges and ion extraction can also be modeled.
·
Can treat both high and low pressure discharges.
·
Drift-diffusion approximation assumes a collisional
plasma.
·
Transport coefficients can be based on a Maxwellian or
non-Maxwellian energy distribution.
·
ADI scheme time-splits electron equations from ion and
chemistry equations.
·
Electron equations solved simultaneously with Poisson’s
equation to give self-consistent space charge and boundary flux.
·
Secondary emission modeled for field and thermionic
emission and ion secondary emission.
·
Density, velocity, and inertia treated for each ion
species.
·
Coulomb scattering coupling included between ions
allows “dragging” of negative ions by positive ions.
·
Time advanced space charge used for accuracy and
stability.
·
Self-consistent dielectric surface space charge
treated.
·
Floating potential and RF biased electrode circuits
solved simultaneously with plasma potential.
·
Potential solved through the interior of dielectrics.
·
Can handle all electron, ion, and neutral reactions.
·
Two and three body reactions.
·
Unlimited number of species and reactions.
·
Volume averaged neutral densities modeled.
·
Diffusion transport of neutral species.
·
Volume and surface chemistry.
·
Fixed input feed flow rates treated for each species.
·
Pump rate varies self-consistently to maintain constant
pressure.
·
RF inductive power coupling to plasma.
·
Single frequency time-harmonic electromagnetic
equations solved for fields produced by RF coil currents.
·
Multiple sets of inductive coils can be modeled with
separate driving frequency and power.
·
RF coil capacitive coupling can be treated.
·
Plasma is modeled as a cold dielectric with complex
conductivity
3
INSTALLATION
Installation of
PLASMATOR requires a simple procedure.
Locate the setup icon on the installation disc and double click
it. The setup program will
automatically copy and install all PLASMATOR components with the exception of
the Java Runtime Environment and the Chamber Design Utility.
3.2 Chamber Design Utility Installation:
The directory,
which contains this document, also contains the files that are used in running
the PLASMATOR Chamber Design Utility.
These files are chamber.jar and ChamberDesign.ico. Copy these files to a directory on your hard
disk (e.g. C:\PLASMATOR\ChamberDesign).
The chamber.jar file is a Java jar
file containing the Java bytecode. The
ChamberDesign.ico file is an icon which can be used to run the program from
your desktop (see below).
The PLASMATOR
Chamber Design Utility is written in Java and may be run on any platform that
contains the Java Runtime Environment.
The Java Runtime Environment is included in the installation disc. The Java Runtime Environment for Win
9x/NT/2000 and Solaris systems can also be obtained (for free)
http://java.sun.com/j2se/1.3/jre/download-windows.html. It comes in a self-extracting
(self-installing) file called j2rel_3_0-win.exe.
4
PLASMATOR Interface
PLASMATOR has a graphical user interface (GUI) for
communication between the user and the code while a modeling calculation is
being performed. The following diagram
in Figure 1 shows the menu items that can be reached from the main PLASMATOR
window.
Figure 1 Diagram
of PLASMATOR main menu options
The PLASMATOR input files are:
(1)
a reactor geometry file with suffix .rcr or .cmb
(2)
a chemistry file with suffix .chm
(3)
an electron collision cross section file with suffix .xsn
4.1
File
The reactor geometry file is chosen
from the dialog shown in Figure 2 after selecting File ® Read
®
Mesh
Figure 2
Dialog for choosing reactor model
The type .rcr reactor models are those constructed using a
text editor and the procedure described in Appendix 1. The .cmb reactor models are those generated
using the Java reactor design program described in Section 5. The grid creation program is accessed by
selecting the Grid ® Create menu items in the main PLASMATOR window.
Figure 3
Choose run label and directory for output
After selecting
a reactor file, the dialog shown in Figure 3 appears. In this dialog, the user can choose or create a directory in
which the output files from the calculation will be placed.
A run label must also be specified.
This label will become the prefix of the output files and the first part
of the name of the restart sub-directory that the program creates. The default shown in the dialog is the time
and date in the form “hour & minute_month & day”.
The other
options under File ® Read are
·
Case (not yet implemented)
·
Data/ Restart
·
Legacy
Selecting Data/ Restart
allows you to restart a previous run.
Legacy allows former users of Version 1 of PLASMATOR to run their old
cases.
4.2
Grid
The items under menu item Grid are Create
and Display. Create brings up the
Java reactor design tool, which is described in Section 5. Display brings up the dialog shown in Figure
4. The .rcr reactor definition files
can be edited and the Edit button is available for those. The Display button shows a graphic of the
reactor geometry. An example is shown
in Figure 5.
Figure 4 Dialog for editing and/ or displaying
reactor geometry
Figure 5 Display of GEC ICP reactor model
4.3
Inputs
·
Pressure: This
allows the user to select the gas pressure.
·
Chemistry: The user can select Ar/O2/ SiH4
or He/ NF3 chemistry models that are furnished with PLASMATOR or
other plasma chemistry models.
·
Boundary Conditions:
There are several sub-menus and associated dialog items under this menu
item. They are:
·
Flow Rates
·
Reactor Components ® Coils
·
Reactor Components ® Biased & Fixed
Potential Objects
·
Magnetic Fields (not yet implemented)
Selecting flow rates beings up a
dialog in which allows the user to enter the flow rates of the feed gases in
SCCM.
Selecting Reactor Components
®
Coils displays the dialog shown in Figure 6. The coil numbers, locations in (r,z), and group numbers are
displayed. The check box is used to
determine which end of the coil the current enters. In the future, this dialog will be able to create multiple groups
of coils. The RF power, frequency, and
coil termination impedance
may be entered after clicking the “Apply” button.
Figure 6 Dialog for choosing coil and ICP
parameters
Selecting Reactor Components
®
Biased & Fixed Potential Objects brings up the dialog shown in
Figures 7 and 8 that allows the user to choose fixed voltages and bias
properties of metal objects
Figure 7 Dialog for selection of biased objects.
Figure 8
Dialog for choosing bias properties
·
Physical Quantities:
Selecting this menu item brings up a dialog for choose gas temperature,
ion temperature, initial electron temperature, and initial electron and ion
densities
4.4
Solve
There are several menu items under
Solve. The first is
·
Options ® Electron Models: Selecting this brings up the dialog shown in
Figure 9. The currently available options here are:
·
Maxwell-Boltzmann Distribution Function
Maxwell-Boltzmann electrons are
distributed in energy according to:
f(e,Te)=2/Öp (kTe)-3/2
exp(-e/kTe)
This is the distribution is the most commonly used
distribution in plasma modeling. It corresponds to the electrons having a
constant collision frequency, i.e.
the momentum collision cross section sm(e) µ 1/Öe, with the atoms and molecules in the gas.
·
Druyvesteyn’s Distribution Function
Druyvesteyn’s distribution, where
f(e) µ
exp(-3m/M e2/e02)
corresponds to a constant cross section sm
between electrons and molecules. Here
e0
= eEl where
l
=(smN)-1 is the mean free path.
Figure 9
Electron energy distribution function dialog box.
·
Run: This
starts the initialization process if it is a new calculation and then starts
the PLASMATOR calculation.
·
Convergence ® Species Histories: This brings up the display shown in Figure
10.
Figure 10 Tabulated values of electron temperatures
and species densities.
The electron temperature, Te in eV, and the
densities of the ion species ( in #/cm-3 ) are displayed. The values shown are
1. The
maximum value at the current time at the location (r,z) shown;
2.
The time averages of the electron temperature and species
densities at the (r,z) shown in (1);
3.
The volume averages of electron temperature and species
densities at the current time;
4.
The volume and time averages of the electron temperature and
species densities.
4.5
Graph
2D contour plots of Te(r,z), Ne(r,z),
and densities of other species can be displayed. These are chosen from the dialog shown in Figure 11.
Figure 11 Dialog for selecting contour plots.
Figure 12 shows an example of an Ne(r,z) contour
plot. The graph can be overlaid, as
shown, with the reactor components and the mesh.
Figure 12 Example of contour plot displaying mesh
and locations of reactor components.
Selecting the Export menu item brings up the following
dialog for creating standard graphics file output. (Figure 13)
Figure 13 Dialog for selecting graphics file output
options.
The time histories of the electron temperature and density
and other species densities can be displayed in the main window by selecting
Graph ®
Time Dependent Plots ®
Te or ®Ne. Examples are shown in Figures 14 – 17.
Figure 14 Electron temperature history graph. (Graph ® Time Dependent Plots ® Te)
The Te(t) and Ne(t) graphs in Figures 14 and 15 display four
quantities as functions of time:
·
The maximum value at the current time at the location
(r,z) shown;
- The
time averages of the electron temperature and species densities at the
(r,z) shown in (1);
- The
volume averages of electron temperature and species densities at the
current time;
- The
volume and time averages of the electron temperature and species
densities.
Figure 15 Electron density history graph. (Graph ® Time Dependent Plots ® Ne)
If a species plot is selected, the dialog box in Figure 16
will appear and allow the user to select the desired species Graph ® Time
Dependent Plots ®
Ne. An example of a species history
plot is shown in Figure 17.
Figure 16 Dialog for selecting which species
densities will be plotted.
Figure 17 Species history plot. (Graph
®
Time Dependent Plots ®
Species)
5 THE PLASMATOR
CHAMBER DESIGN UTILITY
5.1 Introduction:
This document contains a description of the installation and
use of the PLASMATOR Chamber Design Utility.
This utility allows the user to build a graphical representation of a
plasma discharge chamber, which will be used in PLASMATOR calculations.
Note that this document describes the use of the Chamber
Design Utility and not the details of running a PLASMATOR modeling
calculation. For more detailed
information on running a PLASMATOR modeling calculation consult the PLASMATOR
User's Guide Section 6.
5.2 Overview
of the Chamber Design Utility:
The Chamber Design Utility is a "drag and drop"
program that allows the user to visually design a reactor chamber that will be
used in the PLASMATOR simulation. This
utility allows the user to create metal and dielectric chamber components of
arbitrary shape. The Utility also
allows the creation of RF biased electrodes and coils.
Figure 18 This is the standard
chamber of the Gaseous Electronics Conference as designed with the Chamber
Design Utility.
The Chamber Design Utility (CDU) generates a text input file
for PLASMATOR which contains the details of the chamber (positions of
components, material types, etc…). This
text input file is very similar to the text input file which had to be created
by hand in earlier versions of the Induct program on which PLASMATOR is
based. Once the text input file has
been created in the CDU as outlined below, the file may be edited before it
using it with the PLASMATOR program.
The Chamber shown in Figure 18 is the Standard GEC
Chamber. The objects in the chamber are
color-coded. The black areas are vacuum,
pink components are metal objects held at a fixed potential, blue components
are dielectrics, red components are coils, cyan objects (not shown) are metal
objects with an AC bias. The green area is the plasma, and the yellow border
(along the bottom of the right side of the figure) is an open boundary.
The Chamber Design Utility (CDU) is only used for designing
a new chamber. If a chamber is already
designed (and saved the PLASMATOR input file), then it can be used for all
subsequent runs of PLASMATOR.
Figure 19 Chamber design
window as it appears when first opened.
The Chamber Design Utility can be started from one of the
menus in the main window of PLASMATOR Grid ® Create. When the CDU starts, it opens a dialog in
which the overall dimensions of the chamber are specified.
The design window
has seven menus as shown in Figure B.
They are "File," "Add Components,"
"Grid," "Zoom," "Plasma,"
"Options", and "Help".
The "File" menu has items that allow the
user to:
1) Start
a new chamber.
2)
Open a previously saved chamber.
3)
Save the current chamber in a format that the chamber design
tool can re-open for modification at a later date.
4) Save
the current chamber under a new name.
5)
Save the appropriate PLASMATOR Input from the current
chamber. This selection creates a file
containing the relevant information from the current chamber and stores it so
that the PLASMATOR program can use it.
6)
Save the PLASMATOR input under a different name.
7)
Exit the CDU without saving.
The "Add Components" menu allows the user
to:
1)
Add a dielectric component, which appears as a blue rectangle.
2)
Add a metal with a fixed potential, which appears as a pink
rectangle.
3)
Add a metal with an AC bias, which appears as a cyan (light
blue) rectangle.
4)
Add a coil, which appears as a red rectangle.
The "Grid" menu allows the user to:
1) Set the dimensions for the
Electro Magnetic Grid. This is the grid
that will be used in the solution
of the Maxwell’s Equations.
2)
Display the EM Grid.
3)
Specify the parameters for the chemical grid. This is the grid that will be used for solution of the continuity
and momentum equations as well as solution of the Poisson’s equation.
4)
Display the chemical grid.
The "Zoom" menu allows the views to be
changed.
The "Plasma" menu has just one option,
"fill." This option will fill
the chamber with plasma (which appears light green) or remove the plasma if the
chamber has already been filled. Filling the chamber with Plasma is a way of
testing whether the Chamber components do not have gaps between them. Any gap will be evident from plasma
"leaking" into areas of the chamber where it should not be. This is discussed further below.
WARNING: When the "fill" option is checked,
the graphics become very slow, therefore this option should only be used to
test if the chamber is not "leaky," and should not be left on.
The "Options" menu has items that allow the
user to:
1) Resize
the chamber; alter the external dimensions of the chamber.
2)
Edit Components; call up the edit box that allows changes to
the properties of any of the components in the chamber.
3)
Redraw chamber; this will redraw the chamber window.
4)
Show all guides; guides are lines around the chamber
components that show the areas where the mouse is active. This option makes all the guides either
visible or invisible.
5.3 General
Design Procedure
Designing a new chamber begins by typing the overall
dimensions of the new chamber into the first dialog box as shown in Figure
20. Once these dimensions have been
entered, the screen shown in Figure 21 appears. The black area is the region where the reactor will be created.
Figure
20 This dialog appears when the
chamber is first opened. It allows the
user to specify the external dimensions of the chamber that is to be designed.
Figure 21 View
of the chamber window after a dielectric has been added.
The green shape at the center of the CDU window in Figure 21
is the "plasma spigot," which is used to designate the part of the
reactor where the plasma will be located.
Once the reactor design is complete, the spigot should be placed (“dragged to”) the discharge region of the
reactor.
A chamber is created by adding components and modifying
their properties. Pull down the "Add
Component" menu and select the appropriate component (dielectric,
metal at fixed potential, metal with AC bias, or Coil). Once this is done, as shown in Figure 22,
the component will appear at the center of the chamber and its "Edit
Dialog" box will appear. The
size and position of the component can be specified through the dialogue box or
visually on the screen. Click on “apply”
and “done” for the time being so that the graphical manipulation can be
demonstrated.
Figure
22 “Edit” dialog used for editing
properties of components. This dialog
may be invoked by “double clicking” a component.
Figure
23 Once a component has been added,
it can be moved and resized with the mouse by placing the cursor on “active”
parts of the component. The active
parts of the rectangle are shown here.
Specification of size is done through the screen by dragging
the figure sides with the cursor. As
the cursor is moved over various parts of the component, the cursor changes
shape to indicate the availability of various operations that can be performed
on the component. In Figure 23 we show
the four directional cursors. These
cursors allow the object to be moved around the chamber, and resized
vertically, horizontally, and diagonally.
Double-clicking on any object will return the "Edit
Dialog" to the screen as shown in Figure 22. This dialog allows the user to set most of the properties of the
component.
Clicking on the "previous" and "next"
buttons displays the properties of each of the components in the order in which
they were created. The number of the
desired component may also be entered into the "Component Number"
field, then click the "get" button to bring up the data on
that component.
Clicking the "first" button brings up the
data on the first component, clicking the "last" button brings
up the data for the last component. The
X, Y, Width, and Height fields are used for rectangular objects. After changing the numbers in these fields,
click the "set" button and the new dimensions will be applied
to the component.
The "Material" choice box allows the material
the component to be changed. Note: the material choices "vacuum",
"plasma", and "open
boundary" currently should not be used.
The "delete"
button removes the component.
The "Change Shape" button allows selection
of non-rectangular shape. The different
available shapes will be discussed further below.
Often when working with a large number of components in a
chamber, it is necessary to be able to bring a component "to the
front" so that it can be manipulated with the mouse. Similarly, it is sometimes necessary to move
a component "to the back" to allow access to other components. These actions are carried out with the
"Bring To Front" and "Send To Back"
buttons. Note that as long as the mouse
has access to any part of a component, it can be brought to the front by simply
clicking on it.
Checking the "Show Guides" checkbox causes
the "mouse guides" to be displayed around the object. These guides indicate the areas around the
object where the mouse needs to be placed to carry out resizing and relocation
of the component.
The "Clone This Component" button creates a
copy of the current component. This
feature can be useful when creating a large number of components all having the
same dimensions and made of the same materials (e.g. coils).
5.4 Shapes:
The shape or size of a component may be altered by dragging
the sides or corners of the shape on the screen. Clicking the “Change Shape” button in the “Edit”
dialog may also change the shape. This
button will bring up the "Change Shape" dialog shown in Figure
24. The three types of shapes available
are rectangle (the default), slants, and "general."
The "Rectangle" shape is fairly simple and was
shown in examples above. The "Slant" shape provides four different
slanted shape types for building components that have non-Cartesian sides. The four different types of shapes (labeled
"type 1," "type 2," "type 3," and "type
4") are made available once the "Slant" option is chosen
from the "General Shape" prompt. The red lines around the slants in Figure 25 are the "mouse
guides" that are shown when "Options/ Show All Guides" is
selected under the main menu. It is
often useful to set the "Show Guides" box when working with
slants because it is sometimes difficult to find the regions where the mouse is
"active." When using slants,
the thickness of the slant can be set in the "Slant Thickness"
text field in the "Change Shape" dialog box.
Figure 24 Clicking the “Change Shape”
button in the “Edit” dialog brings up this window, which allows shapes other
then the (default) rectangle. Here the
user can choose from four kinds of slants, or a general shape for which the
number of vertices can be chosen (up to 20).
These shapes can also be modified with the mouse.
Figure
25 A “slant” shape that is being
resized. The red lines are the “guides”
that can be activated in the “Edit” dialog.
Selecting “general” in the “General Shape”
choice box can create polygons with an arbitrary number of sides. These polygons can be used to approximate
curved surfaces. Enter the desired
number of vertices in the "Number of Vertices" field and click
the "Make Shape" button to create the polygon. Specify the coordinates of each vertex by
selecting the appropriate vertex in the "vertex" choice box,
then filling in the "X" and "Y" position fields and
clicking the "Set Point" button. The location of the vertices may also be specified graphically
with the mouse on the screen by clicking on the vertex and dragging it to the
desired location.
Some "general" shapes are shown in Figure 26. When "Show Guides" is
checked, red circles will appear around each of the vertices. If the mouse is placed over a vertex it will
turn into a hand with a finger on the vertex and allow manipulation after
selection with a mouse click.
Figure 26 A “general” shape with 12
vertices. “General” shapes are
manipulated by placing a finger on a vertex and dragging as shown.
Figure
27 An example of how “general” shapes
can occlude other shapes without appearing to do so. “General” shapes cover an area as large as the smallest rectangle
that contains all the vertices. To move
a “general” shape out of the way, click the “Send to Back” button in the “Edit”
dialog.
Care must be exercised when dealing with "general"
shapes because they can occlude other shapes (this can happen with
"slants" as well). In Figure
27 the dielectric (blue, general shape) is "in front of" the
fixed-potential metal (pink, square) shape. The shape, which is behind, is
inaccessible to the mouse because the mouse activates the most forward
shape. The fixed-potential (pink) shape
is completely covered by the dielectric because the dielectric is represented
by the smallest rectangle that includes all its vertices. Occluded objects can be made accessible
again by simply selecting each object in front and choosing "send to
back" in its edit dialog box (Figure 22).
6 PLASMATOR TUTORIALS
6.1 Constructing
the GEC ICP Chamber:
This section will detail the creation of the computational grid for the GEC
Inductively Coupled Plasma reactor with the CDU. Start PLASMATOR and select Grid ® Create. This will launch the Chamber Design
Utility. After it comes up, select File
®
New in the CDU. Enter the
dimensions of the GEC (Gaseous Electronic Conference) chamber as shown in
Figure 28.
Figure 28 Chamber dimensions
for the GEC cell.
The CDU will now display a workspace, which corresponds to
the external dimensions. Add a fixed
potential metal component add component ® add metal (fixed
potential). The fields
"x" and "y" represent the minimum x & y coordinate of
the component. The width and height
will be the dimension to the right and up from the minimum x & y
coordinates. Enter the dimensions {x=5,72, y=8.73, width=2.27,
height=1.59}. This is done by filling
out the appropriate fields in the "Edit" dialog as shown in
Figure 28. Click the "Apply"
button to alter the shape and "Done" to close the
dialogue. If the "Apply"
button is not clicked, the component will revert to its initial
dimensions. Now add the following
fixed-potential metal components with the corresponding dimensions:
coil housing: {x=5.72,
y=10.32, width=1.27, height=7.05}
top chamber wall: {x=0.0,
y=17.00, width=12.52, height=0.67}
side chamber wall: {x=12.22,
y=0.00, width=0.5, height=17.67}
The chamber should then look like the one in Figure 29.
Figure 29 Partially constructed
GEC ICP reactor.
Add two dielectric
components add component ® add dielectric with the following dimensions:
dielectric window: {x=0.0,
y=9.37, width=6.0, height=0.95}
pedestal: {x=0.0,
y=0.0, width=5.4, height=4.0}
and two more fixed-potential metal components with the
following dimensions:
wafer holder: {x=0.0,
y=4.5, width=8.26, height=1.12}
pedestal top: {x=0.0,
y=4.00, width=5.4, height=1.0}
Now the chamber should look like the one in Figure 30.
Figure 30 GEC ICP reactor prior
to addition of coils.
Add a coil add component ® add coil with the
dimensions {x=1.0, y=10.82, width=0.5, height=0.5}. Add 3 more coils with dimensions:
{x=2.0, y=10.82,
width=0.5, height=0.5}
{x=3.0, y=10.82,
width=0.5, height=0.5}
{x=4.0, y=10.82, width=0.5,
height=0.5}.
The chamber should now look like the one in Figure 31 except
for the green plasma.
Figure 31 This is the GEC chamber,
filled with plasma, in its final form.
The physical geometry of the chamber is now finished. The next step is to check for “vacuum
integrity”. Click on the spigot and
drag it to the area where the plasma will be in the chamber. Then select Plasma ® Fill
from the main menu. This fills the
chamber with plasma (Figure 31). If the
chamber components have gaps between them, plasma will enter (leak into) areas
of the chamber where it is not supposed to be (eg. the coil housing). Gaps may be repaired by simply altering the
dimensions of the components so the edges coincide or overlap.
The final step is to specify the computational grids. PLASMATOR uses 2 grids, one for the solution of Maxwell’s
equations and another for solution of the Poisson and chemistry equations. Select Grid ® Set EM grid from
the main menu of the CDU. A dialog box
like that shown in Figure 32 will appear.
For this test case, change the EM grid to 40 horizontal x 50 vertical. Selecting Grid ® Show
EM grid will display the grid.
Select Grid ® Set chemistry grid. The dialog box shown in Figure 33 will be displayed.
Figure 32 Dialog box for selecting EM grid
dimensions.
Figure 33 Dialog box used for specifying chemistry
grid dimensions.
Figure
34 Set the grid sizes for the
chemistry calculation and electric & magnetic field solver. These numbers should be under 100.
Set the grid to 60 horizontal by 80 vertical. Now select the number of horizontal grid
zones to 1 and the number of vertical grid zones to 3. Grid zones allow a non-uniform mesh to be
used. This capability applies to the
chemistry grid only. This allows
regions of high gradients (for example sheaths) to be resolved. Click “edit vertical grid zones”. This displays a dialog box as in Figure
34. The “zone bottom boundary”
must be zero because that is the start of the bottom of the grid. Set “zone top boundary” to 5.62 and
the spacing at the bottom to 10 and at the top to 1. This will cause the cells at the bottom of the zone to be 10 times
larger than at the top of the zone. We
want small cells at the top because this coincides with the wafer and we may
want to resolve the sheath there if in the future we bias the wafer.
Click “next zone”. This brings up the grid zone immediately
above the previous zone. Note the lower
boundary coincides with the upper boundary of the previous zone. Set the zone top boundary to 9.37. This coincides with the dielectric window
below the coils. Set the spacing at the
bottom and top to 1. This will result
in a uniform mesh inside the plasma discharge area.
Click “next zone” to bring up the last zone. The boundaries for this zone are constrained
by the previous zone and the top of the grid and therefore, cannot be
altered. Set the spacing at the top to
10 and click “OK”. Then click “OK”
again in the set-chemistry-grid dialog box.
Selecting Grid ® Show chem grid will display the chemistry grid
(Figure 35).
Figure 35 Grid for GEC ICP reactor. Note the fine grid in the discharge region
and near the electrodes.
Note that the grid spacing on the discharge region is not
strictly uniform. An algorithm inside
the CDU lays out the cells as close to the requested spacing as possible. Selecting Grid ® Set
chem grid ®
Edit vertical grid zones ® Check zones will show the actual ratios which
were used. Occasionally, a requested
spacing ratio will produce an error. If
this happens, a dialog box will appear warning the user of the non-physical
grid. An example of this will be given
below. To correct this, simply revisit
all of the grid zones and choose new (usually smaller) grid zones. For example: if 50 (top) to 1 (bottom) cell spacing produces a warning, try 30
to 1 instead.
The chamber and grid are now complete. Before exiting, save the Chamber Design
Utility input. Saving the CDU input
stores all of the information the Chamber Design Utility needs to be able to
reload the chamber and allow modification.
Select "Save Chamber As…" from the file menu and enter
the name "gec.bin".
All "Chamber files" should end in ".bin"
in order to be recognized by the CDU.
The PLASMATOR input file, used in the computation, is saved by selecting
"Save PLASMATOR Input As…" and enter a filename that ends in
".cmb".
Once the ".bin" and .cmb" files are saved,
exit the Chamber Design Utility. The
".cmb" file will be accessed from within the PLASMATOR program
automatically and need not be read in again.
6.2 Running
the GEC ICP Chamber Using PLASMATOR:
The previous section detailed the creation of the GEC
Inductively Coupled Plasma reactor computational grid. This section will detail the problem set up
and execution.
Launch PLASMATOR and construct the GEC ICP chamber using the
instructions in the previous section.
Then close the chamber design utility (CDU). Alternatively, select the geometry by clicking File ® Read
®
Mesh and directing the code to the directory where the geometry is
stored. Some sample geometries are
located in the “reactors” directory.
Direct the code to that directory (if you have not yet created the
geometry). Choose the “gec-icp.cmb”
file. This file should be similar to
the geometry created in the previous section.
Select Grid ® Display and click display on the dialogue
window. This allows inspection of the
geometry that will be used for computation (Figure 36). Note that PLASMATOR displays the chemistry
grid, not the EM grid. This is useful
for checking that the grid is clustered where the species density or plasma
potential gradients are expected to be located. Displaying the grid is also a good way to verify that the correct
grid has been loaded into the program.
Figure 36 PLASMATOR display of computational
chemistry grid.
Notice that the
coils do not show up. This occurs
because the cells are too large in this region of the chemistry grid to capture
the existence of coils. This is not yet
a cause for worry and we will check for the presence of coils again later under
boundary conditions. The coils do
need to be present in the EM grid or no inductive energy coupling will be
possible. A check for this should be
undertaken while the geometry is still under construction in the CDU. Close the view window and exit the dialogue
box. Select Inputs ® Chemistry”
and click on the desired chemistry (Ar/SiH4/O2 in this example). If the desired chemistry is not listed,
click on “other” and direct the code to the proper file. Next, select Inputs ® Pressure”. In the dialog box, enter the desired
pressure: 20 mTorr. (Figure 37)
Figure 37 Dialog box for selection of process pressure.
Now, select Inputs® Boundary
®
Conditions ®
Flow Rates. A dialog box will
appear that allows the user to specify the gas flows that are desired (Figure
38).
Figures 38 In this dialog box, all inflow gases are
listed. The user may input desired flow
rates.
Enter 200 sccm Ar,
100 sccm O2, and 100 sccm SiH4. Click “OK” when
finished and select Inputs ® Boundary Conditions ® Reactor Components ® Coils”. A dialog box will appear which will allow
specification of coil details. (Figure
39)
Figure 39 Dialog
box for the input of coil parameters.
Rf power and coil frequency may not be accessed until the apply button
is clicked.
Check that four coils and their locations are
indicated. If four coils are not shown,
then the EM grid was made excessively coarse.
(This is not usually an item of concern since an algorithm in the CDU
will normally check for an excessively coarse EM grid.) Place a check mark on the coil into which
the current will enter first and click “apply”. This will allow the power and frequency to
be entered. Accept the defaults for
this example and hit “OK” to close the dialog box. The bias input, Reactor Components ® Biased
& Fixed Potential Objects, is not needed for this example and will be
described in the next section. Select Inputs
®
Physical Quantities and enter the neutral gas temperature, ion
temperature, initial electron temperature, and initial electron density. (Figure 40)
Figure 40 Dialog
box for discharge parameters ( gas and ion temperature) and initial values
(electron temperature and density).
The neutral and ion temperatures will be held fixed and are
not solved for in the computations. The
other two quantities will be used as initial guesses only. It is preferable to select low electron
densities and allow the discharge to evolve.
If an excessively high electron density is chosen, the discharge may
take longer to reach steady state (due to the relatively long transit time of
the excess ions) or may enter a non-physical mode of discharge.
Next select Solve ® Options ® Electron
Models. In the dialog box which
appears, choose a Maxwellian electron energy distribution. The Druyvesteyn distribution is also
available and was described previously in the manual (Figure 41).
Figure 41 The electron energy distribution model
must be chosen before computations can begin.
Click “OK” to close the dialog box. The problem set up is now finished. Select Solve ® /Run and enter the
desired run time in the dialogue box.
PLASMATOR will take a few moments to initialize and will, then, start
printing the time step in the lower margin.
While PLASMATOR is running, the species histories may be seen by
selecting Graph ® Time Dependent Plots ® Species. A dialog box will appear which allows the
selection of desired species histories, electron temperature, electron density,
or residuals. Since PLASMATOR
simulations are inherently time accurate, the current residual formulation does
not provide useful information, and is not a good indicator of convergence. The best indicator of convergence is the
species histories, which will clearly show when the average quantities reach a
steady state. With large reactors,
however, caution must be exercised even with the species histories. In certain instances, a slow changing
average is indicative of ion migration throughout the reactor. (Since the computations are carried out on
the rf timescale, it may take a considerable time for the ions to reach their
equilibrium distribution.) It is
recommended that ICP simulations be carried out for 1000 rf cycles or more
until the convergence characteristics of the particular class of problem
currently under study are well understood.
An example of an average species density time history plot is shown in
Figure 42.
Figure 42 History of several species densities.
6.3 Constructing
the GEC CCP Chamber:
This section will detail the construction of
the computational grid for the GEC Capacitively Coupled Plasma Reactor.
Launch PLASMATOR and select Grid ® Create
to start the Chamber Design Utility (CDU).
Select File ® New in the CDU and type the external dimensions
of the GEC CCP reactor: Radius=12.82 cm, Height=17.67 cm. Create each of the following grounded
components using Add Component ® Add Metal (Fixed
Potential):
Grounded
Electrode: {x=0.00, y=8.73, width=5.60,
height=1.59}
Electrode
Side: {x=4.40,
y=10.32, width=1.20, height=7.05}
Reactor
Top: {x=0.00,
y=17.10, width=12.52, height=0.57}
Reactor
Side: {x=12.40,
y=0.00, width=0.42, height=17.67}
Pedestal
Side Shield: {x=5.50, y=0.00, width=0.1,
height=16.32}
Next, create each of the following
dielectric components using Add Component ® Add
Dielectric:
Pedestal:
{x=0.00,
y=0.00, width=5.40, height=4.00}
Pedestal
Gap Fill: {x=5.40,
y=0.00, width=0.10, height=6.32}
Finally, create the biased electrode using Add
Component ® Add Metal (with AC bias):
Powered
Electrode: {x=0.0, y=4.00,
width=5.40, height=2.32}
The GEC CCP geometry should look like Figure
43 after Plasma ® Fill is selected.
Figure 43
GEC Capacitively Coupled Reactor Geometry.
Next,
set the grids for the computation.
Select Grid ® Set EM Grid and accept the defaults. Since this example has no coils the EM grid
will not be used in the computations and, therefore, the grid dimensions are
not important. Select Grid ® Set
Chem Grid. This is the grid on
which the chemistry and Poisons equations will be solved. Therefore, it is important that this grid be
sufficiently fine to capture the sheath electric potential gradients. Set the number of horizontal (radial) points
to 40 and vertical (axial) points to 55 (Figure 44). Next, set the number of horizontal zones to 2. This will allow the grid to be clustered
near the side of the pedestal. Click
“Edit Horizontal Grid Zones”. The left
boundary (axis) must be zero. Set the
right boundary to the side of the pedestal, x=5.5 cm.
Figure 44
Chemistry grid dialog box.
Figure 45
Grid clustering dialog box.
Set the spacing at the left edge of the zone
to 100 and the spacing at the right edge to 1. This will cause the program to try to make the left-most cell
100 times larger than the cell at x=5.5 cm.
Now click “Next Zone”.
The location and grid spacing on the left edge were set by the previous
zone. The location of the right edge
must be at x=12.82 because that is the extent of the grid and only two zones
were selected in the x direction (Figure 45).
Set the spacing on the right to 100 and click “Check Zones”. The dialog box which appears, Figure 46,
informs the user of the actual spacing which the program must use to match the
request of the user. Note that the
spacing that the program found is negative.
This is an unacceptable grid. The spacing ratio in the zones must be
reduced in order to allow the code to find an acceptable gird. Therefore, close the dialogue box and, in
the “set grid” dialog, click “previous zone”. Enter “15” for the left edge
spacing. Click “next zone” and
enter “15” for the right edge spacing.
Now, click “check zones” and view the actual spacing which the
code found. Because all spacing numbers
are positive, this will produce an acceptable grid. Close the dialog box and click “OK” on the “Set Chem
Grid” dialog box. Select Grid ® Show
Chem Grid and check that the grid clustering falls on the pedestal
shield. “Zoom” may be used to
inspect the grid more closely. Select Grid
® Set Chem Grid, again, and set the number of
vertical grid zones to 4. This will
allow the grid to be clustered near the powered and grounded electrodes. Click on “Edit Vertical Zones”. The bottom of the first zone must be
0.0. Set the top of the first zone to
6.33 cm. Set the lower spacing to 20
and the upper spacing to 1.0. Enter
the next three zones as follows:
Zone 2:
bottom=6.33, top=7.80, lower spacing=1.0, upper spacing=5.0
Zone 3:
bottom=7.80, top=8.72, lower spacing=5.0, upper spacing=1.0
Zone 4:
bottom=8.72, top=17.67, lower spacing=1.0, upper spacing=20.0
After entering the last zone, click on “check
zones” to see if any of the spacing ratios became negative. Close this dialog box and click “OK”
on the “Edit Zones” and “Set Chem Grid” dialog boxes. Select Grid ® Show
Chem Grid to view the grid and verify that it is clustered at the powered
and grounded electrodes (Figure 47). Check that each component, especially the
reactor side and top cover at least one cell. If they do not, PLASMATOR may not register the component and an
error, such as “plasma reaches top edge of chamber” may be produced. There are
two remedies for this: 1) recluster the grid so the cells are the size of the
component or 2) stretch the component to cover the cell. The second method is not recommended unless
the behavior of the plasma discharge is known/suspected to be insensitive to
small changes in the geometry of the component in question. The GEC CCP grid is now finished. Make sure to save the CDU input (File ® Save
Chamber As…) and the PLASMATOR input (File ® Save
Plasmator Input As…) before exiting the Chamber Design Utility.
Figure 47
Final GEC CCP grid with clustering near the pedestal side, powered
electrodes, and grounded electrodes.
6.4
Running the GEC CCP Chamber:
Launch PLASMATOR and select the geometry by
clicking File ® Read ® Mesh and directing
the code to the directory where the geometry is stored. The program will now prompt the user for a
directory where the output files will be stored and a prefix for all output
file names. (If you have just finished
creating the geometry, it will be already loaded into the program and no
geometry selection is needed.) Select Grid
® Display and click
display on the dialog window. This
allows inspection of the geometry which will be used for computation. Verify the grid is clustered near the
electrodes (Figure 48).
Figure 48
Computational grid displayed in PLASMATOR with Grid ® Display.
Close the view window and exit the dialog
box. Select Inputs ® Chemistry
and click on the desired chemistry (He/NF3 in this example). Next, select Inputs ® Pressure
and enter the desired pressure: 1000 mTorr. Now, select Inputs ® Boundary Conditions ® Flow
Rates and in the dialog box, specify the gas flows: 100 sccm NF3 and 400
sccm He (Figure 49).
Figure 49
Enter the neutral gas pressure, 1000 mTorr, and flow rates, 100 sccm NF3
and 400 sccm, He.
Click “OK” when finished and select Inputs
® Boundary Conditions ® Reactor
Components ® Biased & Fixed Potential Objects. Select “Biased Chuck” and click “edit”. Set the AC Voltage to 100 V and the
Frequency to 13.56e+6 Hz (Figure 50).
Figure 50
Dialog boxes for the specification of biased object properties.
The phase, DC voltage, Bias Resistance, and
Blocking Capacitor will be left at 0.0 for this example. The power specification does not yet work
and should be ignored. Click “OK”
in this dialog box and “done” in the next one. Select Inputs ® Physical Quantities
and enter the neutral gas temperature, 500 K, ion temperature, 0.5 eV, initial
electron temperature,2.0 eV, and initial electron density, 1.0e+8 #/cm3 (Figure
51). The neutral and ion temperatures
will be held fixed and are not solved for in the computations. The other two quantities will be used as
initial guesses only. Next select Solve
® Options ® Electron Models. In the dialog box which appears, choose a
Maxwellian electron energy distribution.
The Druyvesteyn distribution is also available and was described
previously in the manual. Click “OK”
to close the dialog box. The problem
set up is now finished. Select Solve
® Run and enter the desired run time in the dialog
box.
Figure 51
Dialog boxes for the entry of neutral and initial plasma values.
PLASMATOR will take a few moments to
initialize and will, then, start printing the time step in the lower
margin. While PLASMATOR is running, the
species densities may be seen by selecting Graph ® Contours ® Te, Ne, Species, etc. After selecting instantaneous or average
quantities and the species to graph a window, similar to Figure 52, will appear
with a contour plot. This window has options for displaying the grid or reactor
components, as well as changing the number of contour levels.
Figure
52 PLASMATOR display showing
the initial stages of the CCP discharge.
APPENDIX I:
Details of Physics Model
The Inductively Coupled Plasma (ICP) source, which has also
been referred to in the literature by the names Radio Frequency Induction (RFI)
and Transformer Coupled Plasma (TCP) source, may be considered as a multi-turn
rf antenna coil coupled across a dielectric window to the plasma, with the
plasma acting as a single-turn lossy conductor (see Fig. 53). The ICP design is
quite simple, with no external magnetic fields needed for efficient power
coupling, and therefore no need for design and optimization of electromagnetic
or permanent magnet configurations. Separate rf coupling is applied via the substrate
holder to modulate the extracted ion energy. Typical operating conditions are
input rf power of 200-1500 W, neutral gas pressures of 1-20 mtorr, and peak
plasma densities of 1011-1012 cm-3.
Modeling can help to provide understanding of the fundamental
physics of plasmas. The goal of the modeling code described here is to aid in
the design and optimization of plasma reactors for integrated circuit
manufacturing and other large area uses. High-density reactors require at least
two-dimensional time-dependent modeling with complex internal structures in
order to accurately simulate power coupling and species transport. The model
must simulate electron, ion, and neutral transport, include detailed volume and
surface chemistry, and solve for both space charge, rf coil induced, and ac
biased substrate rf generated electric fields.
The electron continuity equation is solved using an
implicit, conservative differencing scheme that allows time steps greater than
the Courant time limit and the Dielectric Relaxation time scale. The Dielectric
Relaxation time scale limit is overcome by using time advanced electron
densities in Poisson’s equation calculated with the same algorithm used to
solve the electron continuity equation. The electron continuity equation in conservative
form also generates electron fluxes that are used in the electron temperature
equation. To improve stability where steep density gradients occur, INDUCT
allows the electron continuity equation to be solved using an upwind scheme so
that arbitrarily steep gradients can be handled. Self-consistency in the
treatment of boundary conditions is enforced in INDUCT. The boundary conditions
for ions are that the flux normal to a surface is continuous, the normal
velocity is extrapolated, and the density is calculated as the flux divided by
the density. This is for outward flow. If ions are found to be flowing into the
plasma, then their flux, velocity, and density are assumed to be zero in the
boundary cell just exterior to the plasma. For electrons, the outward flux at
the boundary is given by thermal outward flow of electrons escaping across the
sheath and an inward secondary emission flux. Multiple neutral and ion (both
positive and negative) species are treated self-consistently in INDUCT. An
arbitrary number of species can be used, with chemical reactions coupling their
evolution.
Figure 53 Typical ICP geometries with substrate rf
bias
Atomic rates are calculated using input rate tables. Atomic
rates are a function of temperature and can have any dependence. The chemistry
model allows two and three body reactions (with two or three reactants) and an
arbitrary number of products. All rate information is supplied via input, which
allows for flexible chemistry modeling. The use of rate tables results in
improved performance by removing the complex function evaluations. A neutral
chemistry model is included. Neutrals are assumed to be of uniform density with
constant inflow. Both volume and surface chemistry are treated including wall
recombination by ions to form neutrals, and surface chemistry reactions between
neutrals. The neutral model is called every time step, but uses a larger time
scale to accelerate convergence.
RF biasing of the substrate is available. The rf bias
frequency can be set separately from the rf induction frequency. This allows an
effective DC plasma potential to develop which modifies the energy of ions
striking the substrate holder.
INDUCT solves a set of two-dimensional (cylindrically
symmetric) time dependent fluid equations for electrons and ions
self-consistently with Poisson’s equation for the electric potential. In
addition, rf inductive heating is calculated from a time-averaged solution of
Maxwell’s equations.
Ions Ions are assumed to be isothermal and near
the neutral species temperature. Ion motion is governed by the equations of
continuity and momentum conservation, which for ion species i are
where 
and 
are the ion
density and velocity, Ri is the sum of the chemical reaction rates leading to changes in
the ion density, 
is the ion neutral
collision frequency, Ti is the ion temperature, mi is the ion mass, and 
is the electric
field. The ion-neutral collision frequency is calculated from the corresponding
cross-section using
where 
is the relative velocity between ions and neutrals.
Electrons The electron fluid model consists of the
electron continuity and energy balance equations
,
,
.
is the
drift-diffusion electron energy flux. The electron density is 
, 
is the electron
thermal energy, and 
is the electron mobility,
is the total
electron-neutral collision frequency (summed over all neutral species), 
is the time-averaged
power per unit volume absorbed by the electrons due to the inductive rf fields,
and 
is the energy loss per unit volume due to electron-neutral
collisions. Due to the slower ion response time, the drift-diffusion velocity
approximation gives a poor representation of the ion velocity. We therefore
solve the ion momentum equation directly.
The advective and chemistry terms in the ion and electron
equations are solved separately (successively) by time splitting. The advective
part of the electron continuity equation is solved implicitly to allow time
steps greater than the Courant time limit and the Dielectric Relaxation time
scale. A choice exists between using a first order in space Upwind scheme or
second order spatial scheme for the electron advection terms in the continuity
equation. This is chosen by input. The Upwind scheme is less accurate, but more
stable allowing larger time steps to be used. The electron temperature equation
is also solved implicitly to allow for large time step stability for thermal
conduction and advection. A second order spatial scheme is used for the
electron temperature scheme. In time spliting the electron continuity and
electron temperature equations, the temperature is held fixed in the continuity
equation and the density is held fixed in the temperature equation. This allows
for the most efficient solution of these equations. Each implicit equation is
solved using a second order in time Alternating-Direction-Implicit scheme.
Because of the high mass of the ions relative to the electrons, the ion
velocities are orders of magnitude slower than the electron velocity. This
leads to the ion dynamic time scales being much longer than those for electron
evolution. Due to the much slower ion time scale, an explicit temporal
differencing scheme is used in the ion equations. Upwind differencing for
stability is used for the ion equations.
Neutral flow and chemistry is treated in INDUCT assuming
constant total pressure and uniform spatial distribution. A density for each
neutral species is maintained at each grid point, with volume changes due to
chemistry calculated in the same manner as is done for electrons and ions. In
determining changes in the mean neutral densities, the volume chemistry changes
are summed and combined with changes due to flow input and surface chemistry
changes. The surface chemistry makes use of total ion and neutral species wall
currents. Upon modifying the total neutral densities for each species, the
densities are scaled by a constant to restore the required constant pressure.
The neutral model subroutine is called every timestep along with the other
fluid routines, but is allowed to use a time step which is larger.
The boundary conditions for the ion equations assume
continuous outward flux and an extrapolated velocity. If the flux is determined
to be inward, then it is set zero along with the boundary velocity. For
electrons the flux is calculated from an analytic form 
, where 
is the outward
electron flux, 
is the mean electron
thermal speed, 
is the secondary emission coefficient, 
is the ion flux,
and 
is the potential
drop from the plasma to the boundary structure. If 
is positive, a zero
value is used. The values of 
depend upon ion
species and boundary material. If 
is inward directed,
then a zero value for 
is used. The
electron flux is assumed to be constant into the boundary structure so that
using an extrapolated velocity, the exterior boundary density can be
calculated. The electron temperature equation boundary condition used assumes
an outward energy flux 
.
Space-charge electric fields are determined
self-consistently using Poisson’s equation. The electric field is evaluated using
the electron and ion densities calculated from the continuity equations. Its
value is re-calculated each time step. Complex internal boundaries can be
treated for all equations. The effect of rf biasing of the substrate holder is
included in INDUCT through the adjustment of the substrate holder potential at
each time step.
The electrostatic electric field is calculated through the
solution of Poisson’s equation
Poisson’s equation is solved separately from other equations
each time step. The use of fixed densities in the space-charge source term was
found to be undesirable as this leads to a Dielectric Relaxation instability
unless time steps on the order of picoseconds are used for electron densities
of the order of 10 11 cm -3
. The Dielectric Relaxation time scale is the electrostatic shielding time
scale. To avoid disastrous amplification of the electrostatic field we solve
Poisson’s equation at the future time level using the time advanced electron
density
where the superscript
n implies the current time level variables, and superscript n+1 signifies the future time level variables to
be solved for. Due to their slower response to the electric field, the ion
densities are treated explicitly and not time advanced. Only the potential 
(which comes from
the drift velocity term) is actually evaluated at the future time level,
allowing the solution for the potential to be evaluated separately from that of
the other fluid variables. For large time steps this ensures near ambipolar
fields and for steady state reduces to the simple form of Poisson’s equation.
Dielectric boundary conditions are treated by summing ion
and electron currents to each surface cell and distributing the surface charge
through the first volume cell within the dielectric. This maintains charge
balance and give the correct feed-back to Poisson’s equation to generate
electric fields which result in a cancellation between the electron and ion
surface currents at each point along the dielectric.
In calculating the effect of an rf bias to the substrate we
make use of a simple circuit connecting the substrate to ground (see Figure
54). The corresponding circuit equation is
Figure 54 Circuit model for rf biasing of substrate
where 
is the substrate holder voltage, 
is the sinusoidal rf driving voltage, 
is the circuit
current, 
is the bias resistor, 
is the blocking
capacitor, and 
is the blocking
capacitor charge. The current 
can be related
through charge conservation to the surface charge on the substrate holder, 
, and the net plasma
convection current into the substrate holder,
, by
where 
is calculated from
the integral substrate holder surface charge density 
The surface charge density in turn is calculated from
Gauss’s law applied to a sm all surface surrounding an element of the substrate
holder which gives
where the terms with subscript sub are evaluated within the
substrate. The unit vector 
is normal to the
substrate surface. Assuming a high conductivity for the substrate material we
set 
Given the electric
field at the current time level, 
, this equation is
used to calculate the corresponding values
from which the 
. A simple first
order explicit scheme is used to calculate
with 
being the time step.
The charge in the blocking capacitor is then calculated as
We then use the equation above for 
to update 
, with all terms to the right of the equal sign being
evaluated at the 
time level, except for
, which is evaluated
at time level n . This scheme requires no iteration between the Poisson solver
and the circuit equation and is accurate for time steps which resolve the
substrate rf bias period.
The electromagnetic fields and rf heating arising from the
inductive coils is calculated from a single-frequency approximation using a
field solver supplied from Oak Ridge National Laboratory (ORNL) . Resonance
effects associated with long coils is treated. The coils are treated as
circular current loops with rectangular cross section connected in series (see
Fig. 55). We consider an N loop antenna and let the coil index prescribe the
order in which the coils are connected, with the first (or optionally the last)
coil being driven directly by the generator with current 
. The output coil is
connected through a specified output impedance 
to ground. The
average current in the i th
coil is given by 
with input and output
currents respectively of 
and 
.
As a result of the voltage needed to drive the inductive
current, each coil will have an azimuthally varying potential. We let the
average potential of the i th
coil be given by 
, with 
and 
referring to the
input and output potentials. All coil potentials are referenced to ground and
defined by the loop integral
(5.1)
starting from the output end 
of the N th coil. The coil averaged quantities, 
and 
, are used as driving
terms and boundary conditions respectively for the solution of the axisymmetric
differential equations described below. Currently, these averages are estimated
as 
and 
. Alternate methods for determining 
and 
based on lumped
circuit and transmission line models have been developed for use in situations
when the capacitive current is not small compared to the inductive current. The
difference between 
and 
can be approximated
by the capacitive current flowing from each coil while the difference
between 
and 
is obtained from
Eqn. (5.1).
The plasma response to the inductive rf electric fields is
modeled by the cold plasma dielectric tensor K. For single frequency 
time dependence,
,
where I is the unity tensor, 
is the cold plasma
conductivity, 
, and f
is the rf coil driving frequency. With no steady state magnetic field,
the conductivity tensor is diagonal and is given by
Figure 55 RF inductive coil geometry
where 
is the total
electron-neutral collision frequency, and
is the electron
plasma frequency. Other materials, such as quartz liners or windows, are
characterized by the appropriate dielectric constant determined by input. In
the case of a metal, K is dominated
by the large real conductivity.
The driving frequency is assumed to be sufficiently high
that ion motion is not important in the electromagnetic model. The form for the
Fourier transformed Maxwell equations we solve is given by
(5.2)
where c is the
speed of light in free space, 
is the Fourier
transform of the applied antenna current at the coils, 
is the Fourier
transform of the rf electric field, and
is the permeability
of free space. Poisson’s equation follows by taking the divergence of the above
equation in combination with ·
:
(5.3)
In Eqn. (5.3), only the electron density ne, appears as this relation
involves the Fourier transform of the electric field and ion motion is ignored.
For axisymmetric driving currents with only a
component, Eqns.
(5.2) and (5.3) reduce to:
(5.4)
and
(5.5)
where K
is the diagonal component to the dielectric tensor K
, 
, and 
since we consider 
as arising from the
coil voltages alone.
Equations (5.4) and (5.5) are solved by conventional finite
difference techniques for 
and 
respectively. The
boundary conditions for 
are 
on all conducting
surfaces and, by symmetry, 
at 
. The driving current
is derived from the
coil currents 
. For Eqn. (5.5), we
prescribe 
on the metal
surfaces and require that 
at r=0. The potential on the i th coil is given by 
. For an axisymmetric
system, the equations for 
and 
are decoupled and
the boundary conditions can be specified independently.
The time averaged rf heating electron rate is given by 
, where 
plasma current.
Performing the time average over a wave period gives
(5.6)
While the model based upon Eqns. (5.4) and (5.5) is
axisymmetric, in real coils there is a dependence upon angle 
, and capacitive
coupling from the coils should be included. The component of 
normal to a
conductor surface is proportional to the displacement current flowing from that
portion of the conductor. For the complete coil we have
(5.7)
where the integral is over a surface which encloses the i th coil. The current 
represents the
capacitive current flowing from the i th coil. This current breaks the axisymmetry of
the system. If, for example, the normal electric field were cylindrically
symmetric, the capacitive current density leaving (or entering) the conductor
would be uniform along the coil, and the conduction current would have a linear
dependence upon angle 
.
While the change in current along a given turn cannot be
represented by this symmetric model, the change in current from turn to turn
can be. This is done in an iteration scheme as follows:
Use the average potential
on the coils to
solve Eqn. (5.5) for (Er,Ez)
, and then solve Eqn. (5.7) to determine the capacitive currents 
;
Impose conservation of current by requiring that the change
in current from the i th to the 
coil be give by the capacitive current in the i th coil, so that 
;
Use the average currents
to solve Eqn. (5.4)
for 
;
Solve Eqn. (5.6) for the power density and sum its value
over the plasma volume. As the power density scales with input current
squared, 
., a desired total
power absorption can be obtained by scaling
at this point.
Self-consistency is obtained when the specified inductive
currents 
result in
potentials 
that are just those
needed to produce the turn-to-turn variations in the inductive currents.
