Molecular Flow in an Ion-Implant Vacuum System

This example considers the design of an ion implantation system. Ion implantation is used extensively in the semiconductor industry to implant dopants into wafers. Within an ion implanter ions generated within an ion source are accelerated by an electric field to achieve the desired implant energy. Ions of the correct charge state are selected by means of a separation magnet which bends the ion beam to ensure that ions of a particular charge to mass ratio are the only ones which reach the wafer. The energy dose and angle of the ion beam are both key parameters for the process. This part of the system is known as the corrector.

Usually it is desired that only selected regions of the wafer are implanted. This is achieved by masking parts of the wafer with an organic photoresist, to produce the desired pattern. Unfortunately, the photoresist itself emits gas molecules as a result of the beam striking it. These molecules can interact with the ion beam and produce species with undesired charge to mass ratios at different points along the beam path. Some of these species may reach the wafer, degrading the uniformity of the implant, which is highly undesirable. Additionally, these ions may also effect the accuracy of measurements of the implant dose. A key requirement for the system is that the number density of the outgassing molecules for the wafer is low within the beam line.

This example shows how to model an ion implantation system using the Molecular Flow interface. Because it is interactions of outgassing molecules with the beam that produce undesirable species, the average number density of these molecules along the beam path is used as a figure of merit to evaluate the design. Furthermore, because the angle of the wafer to the beam can be modified, the figure of merit must be computed as a function of wafer angle, with rotation about one axis.

Figure 1: Model geometry. Key components of the system are labeled. The red line through the center of the wafer carrier indicates the axis of rotation of the carrier.

This model is motivated by Ref. 1. However, it is important to note that there are significant differences between the pressure and number density computed by the Molecular Flow interface and those computed in the radiation analogy, which is used in Ref. 1. The COMSOL Multiphysics software computes these quantities accurately, whilst the radiation analogy uses an approximate technique to compute the number density and pressure which can result in misleading, incorrect, answers.

Model Definition

The model geometry is shown in Figure 1. The wafer is positioned on a carrier plate which is rotated about an axis through its center to achieve different implant angles. The carrier plate is mounted in a chamber that is pumped by three large cryopumps, on cylindrical vacuum ports. These pumps have a pump speed of 12,000 l/s. In this model outgassing of only one species from the wafer (H2) is considered: multiple species can be modeled by adding additional molecular flow interfaces. It is assumed that the outgassing across the wafer surface is uniform and that the total gas emitted is 30 sccm. The vacuum path through the corrector magnetic field enters the main chamber opposite the wafer, and is pumped by a turbomolecular pump on a cylindrical port midway down the corrector, with a pump speed of 1500 l/s and an additional cryopump at the start of the beam path, with a pump speed of 12,000 l/s. There is an aperture at the entrance to the chamber which reduces the flux that enters the corrector. The angle of the wafer normal to the ion beam is swept from 0° to 60° in 20° steps, as the wafer is rotated about the horizontal axis through its center, as shown in Figure 1. All other surfaces in the model are walls.

Results and Discussion

The molecular flux at the surfaces of the vacuum chamber is shown in Figure 2, for wafer normal angles of 0° and 40° to the beam.

The shadowing effect of the wafer support is evident at both wafer angles shown in Figure 2. It is also clear that more flux enters the beam line when the wafer normal is parallel to the beam because the line of sight with the beam line is then parallel to the wafer normal. The number density along the beam path is shown in Figure 3 for each of the angles solved for. The number density increases along the beam path toward the wafer, with strong changes in gradient as the beam passes through different apertures. Note that in the immediate vicinity of the wafer the number density is not computed accurately (this is difficult to see from the plot below, but zooming into the plot at the end of the beam shows issues in the results for elements adjacent to the surface). This is a result of numerical issues when computing the number density very close to a source surface within the domain.

The average number density along the beam path within the corrector can be used as figure of merit for the system design. This quantity is plotted against the angle of the wafer normal to the beam line in Figure 4. The average flux in the corrector is reduced by rotating the wafer normal away from the beam line with a 10% reduction occurring over the full angular range. Rotating the wafer in this manner can therefore result in improved performance of the implanter.

Figure 2: Pressure on the surface of the ion implanter, with the wafer normal at 0° (top) and 40° (bottom) to the beam path.

Figure 3: The number density as a function of position along the beam line.

Figure 4: Plot of the average number density along the beam line as a function of the wafer normal angle to the incoming beam.

Reference

1. M.R. LaFontaine, N. Tokoro, P. Murphy, and D. Holbrook, “Modelling Photoresist Outgassing Pressure Distribution Using the Finite Element Method,” Proc. Conference on Ion Implantation Technology, IEEE Press, pp. 247–250, 2000.

Notes About the COMSOL Implementation

The model is straightforward to set up using the Molecular Flow interface. The vacuum pump boundary condition is used for the pumps and the outgassing wall boundary condition is used for the wafer surface. The geometry for the vacuum system is imported, but the wafer and its carrier are generated within COMSOL Multiphysics so that the angle to the beam line can be easily parameterized.

Molecular_Flow_Module/Industrial_Applications/ion_implanter

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Set up parameters for the wafer angle and pump speeds.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
theta	30[deg]	0.5236 rad	Wafer angle
pumpspeedcryo	12000[l/s]	12 m³/s	Pump speed for cryopumps
pumpspeedturbo	1500[l/s]	1.5 m³/s	Pump speed for turbopump

For convenience, the device geometry is inserted from an existing file. You can read the instructions for creating the geometry in the Appendix - Geometry Instructions.

Geometry 1

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file ion_implanter_geom_sequence.mph.

In the toolbar, click

Add a nonlocal average coupling to compute the average number density along the beam line.

Definitions

Beam line

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Edges 6, 33, and 103 only.

In the text field, type Beam line.

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

From the list, choose .

Set the physics interface to simulate free molecular flow of hydrogen.

Free Molecular Flow (fmf)

Molecular Flow 1

In the window, under click .

In the window for , locate the section.

In the Mn,G text field, type 0.002[kg/mol].

Set up the wafer to outgass 30 sccm of hydrogen.

Wall 2

In the toolbar, click

and choose .

Select Boundary 42 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the Qsccm,G text field, type 30.

Set up the vacuum pumps.

Vacuum Pump 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 55 only.

In the SG text field, type pumpspeedturbo.

Vacuum Pump 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 8 only.

In the SG text field, type pumpspeedcryo.

Vacuum Pump 3

Right-click and choose .

In the window for , locate the section.

Click

Select Boundary 25 only.

Vacuum Pump 4

Right-click and choose .

In the window for , locate the section.

Click

Select Boundary 70 only.

Vacuum Pump 5

Right-click and choose .

In the window for , locate the section.

Click

Select Boundary 33 only.

Number Density Reconstruction 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Next create the mesh. Add a fine mesh along the beam line.

Mesh 1

Edge 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Use a very fine mesh along the beam line.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type 0.005.

Change the global mesh size setting to refine the mesh.

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Add a free triangular surface mesh. This will inherit the global mesh size setting.

Free Triangular 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click

The domain does not need to be meshed since the interface solves equations only on the surfaces of the model. Note that it is possible to mesh surfaces edges and points (as in this model) or to mesh the entire domain. These two approaches cannot be mixed (doing so may lead to incorrect results in some cases).

Set up a parametric sweep over the wafer angle. The study will take approximately 4 minutes to complete with the full sweep. Alternatively it is possible to solve for a single wafer angle to save time.

Study 1

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
theta (Wafer angle)		deg

Click

In the dialog box, type 0 in the text field.

In the text field, type 20.

In the text field, type 60.

Click .

In the toolbar, click

Results

Incident Molecular Flux (fmf)

Create a duplicate solution with a selection so that you can see inside the vacuum chamber.

Study 1/Parametric Solutions 1 (3) (sol2)

In the window, expand the node.

Right-click and choose .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Update the default plots to use the new dataset.

Incident Molecular Flux (fmf)

In the window, click .

In the window for , locate the section.

From the list, choose .

Total Number Density (fmf)

In the window, click .

In the window for , locate the section.

From the list, choose .

Total Pressure (fmf)

In the window, click .

In the window for , locate the section.

From the list, choose .

Reproduce the results shown in Figure 2.

Total Number Density (fmf)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Repeat steps 3 and 4, this time selecting an angle of 40 degrees.

Compare the resulting plot with that in Figure 2.

Create a plot of the number density along the beam path.

Number Density Along Beam Line

In the toolbar, click

and choose .

In the window for , type Number Density Along Beam Line in the text field.

Locate the section. From the list, choose .

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Click to expand the section. Select the check box.

Click to expand the section. From the list, choose .

In the toolbar, click

Compare the resulting plot with that in Figure 3.

Finally, plot the mean number density along the beam line, as a function of wafer angle.

Average Number Density vs. Wafer Angle

In the toolbar, click

and choose .

In the window for , type Average Number Density vs. Wafer Angle in the text field.

Locate the section. From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Angle between beam and wafer (deg).

Select the check box.

In the associated text field, type Average number density (1/m<sup>3</sup>).

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
aveop1(fmf.ntot)	1/(m^3)	Mean number density

Locate the section. From the list, choose .

In the text field, type theta.

In the toolbar, click

Compare the resulting plot with that in Figure 4.

Appendix - Geometry Instructions

From the menu, choose .

New

In the window, click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
theta	30[deg]	0.5236 rad	Wafer angle
pumpspeedcryo	12000[l/s]	12 m³/s
pumpspeedturbo	1500[l/s]	1.5 m³/s	Pump speed for turbo pumps
theta_1	45[deg]	0.7854 rad	Wafer angle