Rapid Thermal Annealing

In the semiconductor industry, rapid thermal annealing (RTA) is a semiconductor process step used for the activation of dopants and the interfacial reaction of metal contacts. In principle, the operation involves rapid heating of a wafer from ambient to approximately 1000–1500 K. As soon as the wafer reaches this temperature, it is held there for a few seconds and then finally quenched. A rapid process step is crucial to avoid too much diffusion of the dopants. Furthermore, it is also important to avoid overheating and nonuniform temperature distributions to occur. An RTA apparatus, schematically shown in Figure 1, uses high-power IR lamps as heat sources (Ref. 1).

Figure 1: Diagram of a typical RTA (rapid thermal annealing) apparatus.

A technical difficulty lies in how to properly measure the wafer’s temperature during the process. Two commonly used technical solutions are: thermocouples and IR sensors.

To achieve an accurate measurement, it is important that the temperature sensor not be subjected to direct radiation from the lamp. Ideally positioned, the sensor only receives secondary radiation; that is, the radiation reflected and emitted by the silicon wafer. Desirable characteristics of the sensor are high accuracy and short response time. While a high-performance design requires superior electronics, the sensor geometry plays a big role. In a nutshell, the sensor needs to be large enough to register a sufficient amount of radiation but light enough to minimize its own thermal inertia. Since COMSOL Multiphysics gives you control over the geometry, a parameter optimization of the sensor could be an exciting project. But first, justify that an infrared sensor is indeed more appropriate than the inexpensive thermocouple.

Model Definition

Figure 1 illustrates a typical RTA configuration. In many applications, RTA makes use of double-sided heating, in which IR lamps are positioned both above and below the silicon wafer. In this example we are modeling a single-sided heating apparatus, as shown in Figure 2.

Figure 2: The model geometry.

The components in Figure 2 are contained in a chamber with temperature-controlled walls with a set point of 400 K. This results in a closed cavity so you can omit the geometry of the chamber walls. Furthermore, the model assumes that this physical system is dominated by radiation and convection cooling. The convective cooling of the wafer and sensor to the gas (at 400 K) is modeled using a heat transfer coefficient, h (in this example set to 20 W/(m2·K)).

The problem is governed by the heat equation, given below together with its boundary conditions:

Here ρ is the density; k denotes the thermal conductivity; Q represents the volume heat source; n is the surface normal vector; Tinf equals the temperature of the convection cooling gas; ε denotes the surface emissivity; J0 is the expression for surface radiosity (further described in the Heat Transfer Module User’s Guide); and σ is the Stefan-Boltzmann constant.

The model simulates the lamp as a solid object with a volume heat source of 25 kW. It is insulated on all surfaces except the for the top, which faces the silicon wafer. At this surface, heat leaves the lamp as radiation only. In order to capture the lamp’s transient startup time, the model uses a low heat capacity, Cp, for the solid (10 J/(kg·K)). The lamp’s other thermal properties are identical to those of copper metal (the default value in the interface).

In this case assume that the wafer dissipates energy via radiation and convection on all surfaces. The sensor is insulated at all surfaces except the top, which is subjected to both convection and radiation. The thermal material properties are set to those of alumina.

The following table summarizes the material properties used in the application:

Table 1: Material properties.
Material	k (W/(m·K))	ρ (kg/m3)	Cp (J/(kg·K))	ε
IR lamp	400	8700	10	0.99
Silicon wafer (silicon)	163	2330	703	0.5
Sensor	27	2000	500	0.8

The model simulates the transient temperature field for 10 s of heating. The initial temperature is 400 K for all objects.

Results and Discussion

Figure 3 displays the temperature distribution after 10 s of heating.

Figure 3: Temperatures of the lamp, wafer, and sensor after 10 s of heating.

After 10 seconds, the temperatures of the wafer and sensor differ significantly: the wafer is close to 2000 K, whereas the sensor is close to 1200 K.

Notice that the temperature distribution in the wafer with a delta of several hundred degrees is not very uniform, and that you probably can do much better by reconfiguring the heat source. However, such a reconfiguration is not included in this application.

To investigate how well the sensor’s temperature reflects that of the wafer surface, it is useful to plot the temperature transient of the wafer surface’s centerpoint that faces the lamp (Twafer), together with the temperature at a point on the sensor top surface (Tsensor) (see Figure 4).

Figure 4: The temperature transients of the lamp, the silicon wafer, and the sensor, together with the irradiation power at the sensor surface.

The sensor temperature reflects that of the silicon wafer poorly. This means that the signal of a thermocouple, positioned anywhere in the sensor domain of Figure 2, is of little use for regulating this process.

The IR-detector transient (Sensirrad) matches the wafer temperature characteristic quite well. A scalar amplification allows for a high accuracy measurement of the wafer temperature. The precise amplification factor is system-dependent and subject to a calibration requirement.

However, IR-sensor methodology also has drawbacks. The IR signal depends on the emissivity of the wafer, which varies with temperature making the response nonlinear. Furthermore, the IR signal is very sensitive to geometry changes.

The bright side is that COMSOL Multiphysics does not set any limits with respect to these phenomena and allows you to study them fully.

Reference

1. A.T. Fiory, “Methods in Rapid Thermal Annealing,” Proc. 8th Int’l Conf. Advanced Thermal Processing of Semiconductors (RTP 2000).

Heat_Transfer_Module/Thermal_Radiation/thermal_annealing

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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Click .

Click

In the tree, select .

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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
T_wall	400[K]	400 K	Temperature, wall
T_gas	400[K]	400 K	Temperature, gas
h_gas	20[W/(m^2*K)]	20 W/(m²·K)	Heat transfer coefficient
k_sens	27[W/(m*K)]	27 W/(m·K)	Thermal conductivity, sonsor
rho_sens	2000[kg/m^3]	2000 kg/m³	Density, sensor
Cp_sens	500[J/(kg*K)]	500 J/(kg·K)	Heat capacity, sensor
e_sens	0.8	0.8	Surface emissivity, sensor
k_lamp	400[W/(m*K)]	400 W/(m·K)	Thermal conductivity, lamp
rho_lamp	8700[kg/m^3]	8700 kg/m³	Density, lamp
Cp_lamp	10[J/(kg*K)]	10 J/(kg·K)	Heat capacity, lamp
e_lamp	0.99	0.99	Surface emissivity, lamp
P_lamp	25[kW]	25000 W	Total power, lamp
e_wafer	0.5	0.5	Surface emissivity, wafer
ampl	50	50	Amplification factor, IR sensor

Component 1 (comp1)

Set the geometric shape order to "Quadratic". By default, the geometric shape order is only linear in this model. Although the difference will not be visible when plotting the mesh (for graphics performance purposes), second-order elements will then be allowed, yielding a much better match between the mesh and the real cylindrical geometry thanks to the curved edges of the boundary elements.

In the window, click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Geometry 1

Cylinder 1 (cyl1)

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In the text field, type 0.05.

In the text field, type 5e-4.

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Cylinder 2 (cyl2)

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In the text field, type 0.05.

In the text field, type 1e-3.

Locate the section. In the text field, type -5e-2.

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Cylinder 3 (cyl3)

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In the text field, type 0.01.

In the text field, type 1e-3.

Locate the section. In the text field, type 0.07.

In the text field, type -5e-2.

Click

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button in the toolbar.

The built geometry shows a plane symmetry. Delete half of it to optimize the computation.

Block 1 (blk1)

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In the window for , locate the section.

In the text field, type 0.2.

Locate the section. In the text field, type -0.1.

In the text field, type -0.2.

In the text field, type -0.1.

Click

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the objects , , and only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the object only.

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button in the toolbar.

In preparation for analyzing and visualizing the results, define a nonlocal integration coupling.

Definitions

Integration 1 (intop1)

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and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 14 only.

Variables 1

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In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
I_sens	2*intop1(rad.Grad)	W	Irradiated heat effect, sensor

ht.G_rad is a predefined physics interface variable representing inward radiation, which includes both surface-to-surface and surface-to-ambient contributions.

The integral is multiplied by 2 to get the irradiated heat effect on the full geometry.

Materials

IR Lamp

In the toolbar, click

In the window for , type IR Lamp in the text field.

Select Domain 1 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	k_lamp	W/(m·K)	Basic
Density	rho	rho_lamp	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_lamp	J/(kg·K)	Basic

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

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to close the window.

Materials

Silicon (mat2)

Select Domain 2 only.

Sensor

In the toolbar, click

In the window for , type Sensor in the text field.

Select Domain 3 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	k_sens	W/(m·K)	Basic
Density	rho	rho_sens	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_sens	J/(kg·K)	Basic

Now add materials on the boundaries for the specification of surface emissivities.

IR Lamp (Boundaries)

In the window, under right-click and choose .

In the window for , type IR Lamp (Boundaries) in the text field.

Locate the section. From the list, choose .

Select Boundary 4 only.

Click to expand the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Surface emissivity	epsilon_rad	e_lamp	1	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	k_lamp	W/(m·K)	Basic
Density	rho	rho_lamp	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_lamp	J/(kg·K)	Basic

Silicon (Boundaries)

In the window, under right-click and choose .

In the window for , type Silicon (Boundaries) in the text field.

Locate the section. From the list, choose .

Select Boundaries 5, 7, 8, and 10 only.

Click to expand the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Surface emissivity	epsilon_rad	e_wafer	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	700[J/(kg*K)]	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	2329[kg/m^3]	S/m	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	130[W/(m*K)]	1/K	Basic
Heat capacity at constant pressure	Cp	1	J/(kg·K)	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1e-12[S/m]	1	Basic
Density	rho	2.6e-6[1/K]	kg/m³	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	11.7	W/(m·K)	Basic
Young’s modulus	E	170e9[Pa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.28	1	Young’s modulus and Poisson’s ratio
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	3.48	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Sensor (Boundaries)

In the window, under right-click and choose .

In the window for , type Sensor (Boundaries) in the text field.

Locate the section. From the list, choose .

Select Boundary 14 only.

Click to expand the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Surface emissivity	epsilon_rad	e_sens	1	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	k_sens	W/(m·K)	Basic
Density	rho	rho_sens	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_sens	J/(kg·K)	Basic