Laser Heating of a Silicon Wafer

A silicon wafer is heated up by a laser that moves radially in and out over time while the wafer itself rotates on its stage. Modeling the incident heat flux from the laser as a spatially distributed heat source on the surface, the transient thermal response of the wafer is obtained. The average, maximum, and minimum temperatures, as well as the peak temperature difference across the wafer, are stored at every calculation step. The temperature distribution across the entire wafer is stored at a specified number of output time steps.

Figure 1: A silicon wafer is heated with a laser that moves back and forth. The wafer is also being rotated about its axis.

Model Definition

A 2-inch silicon wafer, as shown in Figure 1, is heated for one minute by a 10 W laser that moves back and forth, while the wafer rotates on its stage. Assuming good thermal insulation from the environment, the only source of heat loss is from the top surface via radiation to the processing chamber walls, which are assumed to be at a fixed temperature of 20°C.

The laser beam heat source is modeled as a heat source moving across the surface of the spinning wafer. To model the rotation of the wafer, use the feature. Use a Waveform function and a set of variables to define the Gaussian distribution of the laser heat load around the focal point, as it moves back and forth across the spinning structure.

In the results visualization of the temperature profile across the wafer, the results can be visualized in either the spatial frame or the material frame, representing the point of view of an outside observer or an observer moving with the rotation of the wafer, respectively.

The emissivity of the surface of the wafer is approximately 0.8. At the operating wavelength of the laser, it is assumed that absorptivity equals emissivity. The heat load due to the laser is thus multiplied by the emissivity. Assuming also that the laser is operating at a wavelength at which the wafer is opaque, no light is passing through the wafer. Therefore, all of the laser heat is deposited at the surface.

The wafer is meshed using a triangular swept mesh. Swept meshing allows for only a single thin element through the thickness, and still maintains reasonable element size in the plane. Also, the solver relative tolerance is slightly lowered to better capture the effect of the moving heat load. A finer mesh and tighter solver tolerances would give slightly more accurate predictions of the peak temperature, but predictions of average and minimum temperature would not be greatly affected.

Results and Discussion

Figure 2 shows the probe plots of the maximum, minimum, and average temperatures of the wafer, while Figure 3 shows the probe plot of the difference between the maximum and minimum temperature. The temperature distribution across the wafer is plotted in Figure 4.

The heating profile does introduce some significant temperature variations, because the laser deposits the same amount of heat over a larger total swept area when it is focused at the outside of the wafer. An interesting modification to this example would be to investigate alternative heating profiles for smoother heating.

Figure 2: Maximum, minimum, and average temperatures of the wafer as functions of time.

Figure 3: Difference between maximum and minimum temperatures on the wafer.

Figure 4: Temperature variation across the wafer.

COMSOL_Multiphysics/Heat_Transfer/laser_heating_wafer

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

Global Definitions

Start by defining parameters for use in the geometry, functions, and physics settings.

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
r_wafer	1[in]	0.0254 m	Wafer radius
thickness	275[um]	2.75E-4 m	Wafer thickness
v_rotation	10[rpm]	0.16667 1/s	Rotational speed
period	20[s]	20 s	Time for laser to move back and forth
r_spot	2[mm]	0.002 m	Laser beam radius
emissivity	0.8	0.8	Surface emissivity of wafer
p_laser	10[W]	10 W	Laser power

Here, the unit ’rpm’ is revolution per minute.

Geometry 1

Create a cylinder for the silicon wafer.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type r_wafer.

In the text field, type thickness.

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2*r_wafer.

In the text field, type thickness.

Locate the section. In the text field, type -0.95*r_wafer.

In the text field, type -r_wafer.

Intersection 1 (int1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , click

Definitions

Define functions for use before setting up the physics.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
x_focus	r_wafer*Triangle(t/period)		x-location of laser focal point
y_focus	0[m]	m	y_location of laser focal point
r_focus	sqrt((x-x_focus)^2+(y-y_focus)^2)		distance from focal point
Flux	((2p_laser)/(pir_spot^2))exp(-(2r_focus^2)/r_spot^2)		laser heat flux, Gaussian profile