You are viewing the documentation for an older COMSOL version. The latest version is available here.
PDF
Pulsed Capacitively Coupled Plasma Discharge
Introduction
Pulsed plasma processing, or in more general terms, modulated-power plasma processing allows both the discharge chemistry to be altered, as well as the properties of the deposited films. The etch or deposition rate can also be maintained for lower input powers compared to continuous mode discharges.
Model Definition
A modulated, or pulsed discharge generally consists of 4 regimes which define the characteristic period of the discharge:
•
Turn-on (sometimes known as ramp-up).
•
On, or high power period.
•
Turn-off (sometimes known as ramp-down).
•
Off, or low power period.
These 4 operating conditions are then repeated billions of times over the course of a process. Thus, despite operating 4 different regimes within one characteristic period, the discharge can be considered to be time periodic within one characteristic period.
Computational Considerations
Particular care must be taken when using the Pulsed option in the Metal Contact feature. The Plasma, Time Periodic interface defines the characteristic period using an additional independent variable, which must be discretized in space using a mesh. In order to resolve the RF modulation within the on period, at least 30 mesh elements are required per RF period. Consider the case where the discharge is on for 10 cycles at 13.56 MHz (780 ns), then off for another 100 cycles (7.8 us). During the on-cycle, we need to use 30 mesh elements for each RF period (78 ns). During the off-cycle, there is no RF modulation, so we can use around 100 mesh elements for the entire 7.8 us. So, we need a total of 310 mesh elements to discretize the extra coordinate representing the characteristic period. This represents a 10 fold increase in computation demand compared to a continuous mode discharge, which would only require 30 mesh elements in the extra coordinate. Therefore, the pulsed option is very computationally expensive, and even 1D models are expected to take a significant amount of time to solve.
The Dual pulse option, where the off-period is replaced by a low power period is even more computationally expensive, since the low power period also needs 30 mesh elements per RF period. So, for the above example, where the 100 cycle off period is replaced by low power operation, we would need 3000 mesh elements to resolve this part. This would make such a model 100 x more expensive than the same continuous mode discharge.
When considering using the pulsed discharge option, use the following to estimate the computation demand:
•
In 1D, set up the model and chemistry of interest, and solve the problem using a continuously operated mode. The computation time for this is tc
•
For the Pulsed option, the expected computation time would be tc multiplied by the number of RF periods than the discharge is on for.
•
For the Dual pulsed option, the expected computation time would be tc multiplied by the number of RF periods within the characteristic period.
Ion Transport
When using the Plasma, Time Periodic interface, the mass fraction of the heavy species, the ions and neutrals, can be computed in just the base geometry or in the product space spanning the base geometry and characteristic period coordinate. Solving for the ions in the base geometry only is much more efficient, and numerically stable, but it does assume that the number density and spatial distribution doesn’t change over the characteristic period. This is a reasonable assumption for the continuous mode operation, but for pulsed discharges with long on or off periods, this assumption begins to break down. Therefore, for pulses where the characteristic period lasts for more than a few microseconds, it may become necessary to solve for the heavy species in the product space, which may be too computationally expensive, or impossible to obtain a converged solution. Therefore, the pulsed options can really only be used to study the electrical characteristics of a discharge, and not the transport of heavy species to reactor walls for discharges with prolonged off periods.
Results and Discussion
The I-V characteristics over the entire period are presented in Figure 1. As expected, the current is zero during nearly all of the off period, then ramps up to around 0.8 A. Interestingly, the current is asymmetric, the positive current peaks at 0.9 A, but on the negative side the current peak is -0.8 A. A close up of the I-V characteristics is presented in Figure 2, and indicates that the negative part of the current is increased compared to the positive part during the ramp up and ramp down. The total current when averaged over the entire period still remains zero.
Figure 1: I-V characteristics over the entire period.
The electron density in the center of the reactor is shown in Figure 3. The electron density actually drops by more than 10 % during the excitation part of the period. The electron temperature, shown in Figure 4 shows the opposite behavior, it increases significantly during the electrical excitation, then begins to decay when the power is off. Finally, the electric potential in the center of the reactor is shown in Figure 5. The potential always remains positive with respect to the walls, and shows typical behavior when the power is on. When the power is off, the electric potential decays from 70 V to 45 V by the time that the power comes back on.
Figure 2: Close up of the I-V characteristics when the pulse is turned on.
Figure 3: Plot of the electron density in the center of the reactor vs. time over the entire period.
Figure 4: Plot of the electron temperature in the center of the reactor vs. time.
Figure 5: Plot of the electric potential in the center of the reactor vs. time over the entire period.
Reference
1. M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, 2005.
Application Library path: Plasma_Module/Capacitively_Coupled_Plasmas/pulsed_ccp_discharge
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Select the Plasma, Time Periodic (ptp) interface and the Time Periodic study to compute the periodic steady state solution of the pulsed discharge.
Model Wizard
1
In the Model Wizard window, click  1D.
2
In the Select Physics tree, select Plasma>Plasma, Time Periodic (ptp).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Time Periodic.
6
Click  Done.
Add some parameters to facilitate the setup of the model.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
f0
13.56[MHz]
1.356E7 Hz
Base frequency
As
0.25*pi*(0.3[m])^2
0.070686 m²
Wafer surface area
p0
0.1[torr]
13.332 Pa
Pressure
T0
300[K]
300 K
Temperature
Prf
1[W]
1 W
Single pulse power
Non
10
10
Number of periods the pulse is on
Noff
100
100
Number of periods the pulse is off
Nrampup
1
1
Number of cycles to ramp to on state
Nrampdown
1
1
Number of cycles to ramp to off state
Nelon
30
30
Number of elements per period for the on period
Neloff
100
100
Number of elements for the off period
Ncycles
Non+Noff
110
Total number of cycles
period_single
Ncycles/f0
8.1121E-6 s
Total period
The geometry is very simple and involves a discharge gap of 2.5cm.
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
In the table, enter the following settings:
Coordinates (m)
0
0.025
Import cross section data for Argon which automatically adds the electron impact reactions and species to the model.
Plasma, Time Periodic (ptp)
Cross Section Import 1
1
In the Model Builder window, under Component 1 (comp1) right-click Plasma, Time Periodic (ptp) and choose Cross Section Import.
2
In the Settings window for Cross Section Import, locate the Cross Section Import section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file Ar_xsecs.txt.
Reaction 1
1
In the Physics toolbar, click  Domains and choose Reaction.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type Ars+Ars=>e+Ar+Ar+.
4
Locate the Reaction Parameters section. In the kf text field, type 3.734E8.
7: Ars+Ars=>e+Ar+Ar+
1
Right-click Reaction 1 and choose Duplicate.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type Ar+Ars=>Ar+Ar.
4
Locate the Reaction Parameters section. In the kf text field, type 1807.
Select that the ground state Argon is the species to come from the mass constraint.
Species: Ar
1
In the Model Builder window, click Species: Ar.
2
In the Settings window for Species, locate the Species Formula section.
3
Select the From mass constraint check box.
4
Locate the General Parameters section. From the Preset species data list, choose Ar.
Species: Ars
1
In the Model Builder window, click Species: Ars.
2
In the Settings window for Species, locate the General Parameters section.
3
From the Preset species data list, choose Ar.
For positive Argon ions, use the local field approximation for the temperature and field dependent mobility.
Species: Ar+
1
In the Model Builder window, click Species: Ar+.
2
In the Settings window for Species, locate the Species Formula section.
3
Select the Initial value from electroneutrality constraint check box.
4
Locate the General Parameters section. From the Preset species data list, choose Ar.
5
Click to expand the Mobility and Diffusivity Expressions section. From the Specification list, choose Specify mobility, compute diffusivity.
6
From the Ion temperature list, choose Use local field approximation.
7
Click to expand the Mobility Specification section. From the Specify using list, choose Argon ion in argon.
Plasma Model 1
1
In the Model Builder window, click Plasma Model 1.
2
In the Settings window for Plasma Model, locate the Model Inputs section.
3
In the T text field, type T0.
4
In the pA text field, type p0.
For the surface reactions, only the ions contribute to secondary electron emission.
Surface Reaction 1
1
In the Physics toolbar, click  Boundaries and choose Surface Reaction.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. In the Formula text field, type Ar+=>Ar.
5
Locate the Reaction Parameters section. In the γf text field, type 0.
Surface Reaction 2
1
In the Physics toolbar, click  Boundaries and choose Surface Reaction.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. In the Formula text field, type Ars=>Ar.
5
Locate the Secondary Emission Parameters section. In the γi text field, type 0.
6
In the εi text field, type 0.
Wall 1
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 2 only.
Now set the characteristics for the pulse. If the Equation section is expanded, an image is shown to make it easier to set the pulse settings.
Metal Contact 1
1
In the Physics toolbar, click  Boundaries and choose Metal Contact.
2
Select Boundary 1 only.
3
In the Settings window for Metal Contact, locate the Terminal section.
4
From the Source list, choose RF.
5
Locate the RF Source section. From the Periodic function list, choose Pulsed.
6
In the Prf text field, type Prf.
7
In the fp text field, type f0.
8
Select the Create a mesh suggestion for the extra dimension check box.
9
In the Non text field, type Non.
10
In the Noff text field, type Noff.
11
In the Nlow,high text field, type Nrampup.
12
In the Nhigh,low text field, type Nrampdown.
13
In the Nelp,on text field, type Nelon.
14
In the Nel,off text field, type Neloff.
Solve for the heavy species in the base geometry only. This is more stable than solving for them in the product space, but it does assume that the ion density does not change over the period. For pulses with a long off period, this might not be an appropriate assumption.
15
In the Model Builder window, click Plasma, Time Periodic (ptp).
16
In the Settings window for Plasma, Time Periodic, locate the Out-of-Plane Thickness section.
17
In the A text field, type As.
18
Locate the Extra Dimension Settings section. In the Pxd text field, type period_single.
19
From the Mesh specification list, choose Metal Contact 1.
20
From the Heavy species selection list, choose Base geometry.
Use a mesh that is finer close to the walls, where the sheath region is expected.
Mesh 1
Edge 1
In the Mesh toolbar, click  Edge.
Distribution 1
1
Right-click Edge 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
From the Distribution type list, choose Predefined.
4
In the Number of elements text field, type 75.
5
In the Element ratio text field, type 5.
6
Select the Symmetric distribution check box.
7
Click  Build All.
Single Pulse
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Single Pulse in the Label text field.
3
In the Home toolbar, click  Compute.
The default plot of current and voltage over the period makes visualization of the I-V characteristics difficult, so set some manual axis limits.
Results
Current and Voltage, Metal Contact 1 (ptp)
1
In the Model Builder window, under Results click Current and Voltage, Metal Contact 1 (ptp).
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits check box.
4
In the x minimum text field, type 0.4.
5
In the x maximum text field, type 0.6.
6
In the Current and Voltage, Metal Contact 1 (ptp) toolbar, click  Plot.
We now want to plot the electron density, temperature and plasma potential at the center of the discharge. We can do this without having to add a Time Periodic to Time Dependent study if we make use of the atxd() operator.
Electron Density vs. Time
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Electron Density vs. Time in the Label text field.
3
Locate the Data section. From the Dataset list, choose Single Pulse/Solution 1 (2) (sol1).
Now add the line graph. The syntax comp1.atxd1(0.0125,ptp.ne) means that we visualize the expression for the electron density ptp.ne at the center of the reactor, corresponding to the position x=0.0125.
Line Graph 1
1
Right-click Electron Density vs. Time and choose Line Graph.
2
In the Settings window for Line Graph, locate the Selection section.
3
From the Selection list, choose All domains.
4
Locate the y-Axis Data section. In the Expression text field, type comp1.atxd1(0.0125,ptp.ne).
5
Select the Description check box.
6
In the associated text field, type Electron density at the center of the discharge.
Now change the definition of the x-axis to time by multiplying the space coordinate in the extra dimension by the total time for the period.
7
Locate the x-Axis Data section. From the Parameter list, choose Expression.
8
In the Expression text field, type x1_ptp[1/m]*period_single.
9
In the Electron Density vs. Time toolbar, click  Plot.
10
Select the Description check box.
11
In the associated text field, type Time.
12
In the Electron Density vs. Time toolbar, click  Plot.
Use the same technique as above to make plots for the electron temperature and plasma potential.
Electron Temperature vs. Time
1
In the Model Builder window, right-click Electron Density vs. Time and choose Duplicate.
2
In the Model Builder window, click Electron Density vs. Time 1.
3
In the Settings window for 1D Plot Group, type Electron Temperature vs. Time in the Label text field.
Line Graph 1
1
In the Model Builder window, click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type comp1.atxd1(0.0125,ptp.Te).
4
In the Description text field, type Electron temperature at the center of the discharge.
5
In the Electron Temperature vs. Time toolbar, click  Plot.
Electric Potential vs. Time
1
In the Model Builder window, right-click Electron Temperature vs. Time and choose Duplicate.
2
In the Model Builder window, click Electron Temperature vs. Time 1.
3
In the Settings window for 1D Plot Group, type Electric Potential vs. Time in the Label text field.
Line Graph 1
1
In the Model Builder window, click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type comp1.atxd1(0.0125,ptp.V).
4
In the Description text field, type Electric potential at the center of the discharge.
5
In the Electric Potential vs. Time toolbar, click  Plot.