Steady-state convective cooling of a silicon thin film

In this for the Heat Transport Solver, we look into a simple steady-state heat flow in thin films, specifically, the effect of cooling at the surface of a silicon thin film. Detailed modeling instructions are provided in the end.

Requirements

Lumerical products R2018a or newer

We look at a silicon thin film getting cooled at the surface by air.  The thin film has a length and width of 10 mm and a thickness of 10 micron.  The symmetry of the problem allows us to perform a 2D simulation instead of a 3D one (figure shown below).  Using the Heat Transport Solver, we can set up the 2D simulation in two ways.  In one case, we can create a thermal boundary condition that applies a convective boundary condition at the (top and bottom) surfaces of the (2D) thin film.  In the other case, we can place air (Fluid) on top and bottom of the thin film to model the effect of air cooling at the surfaces.

Simulation Setup

Option 1: Using Convective Boundary Condition

Open the thin_film.ldev project file in HEAT. Alternatively, you can follow the instructions provided in the Modeling Instructions section to create your own project file. The project file contains a single geometry, the thin_film. Note that the length and thickness of the object is slightly larger than the specifications. This is because the area of the simulation region gets defined by the area of the (HEAT) solver region so we have made the thin_film object larger and have used the length and thickness of the HEAT object to define the length and thickness of the simulated thin film. The material for the object is chosen to be Silicon.

There are three thermal boundary conditions defined in the "Boundary Conditions" group. The first one (left) sets the left edge of the thin film at a fixed (elevated) temperature of 400 K. In absence of any cooling, the entire thin film will remain at 400 K in steady state. To model the air cooling, we have defined two boundary conditions (top and bottom) for the top and bottom surfaces. The model for both these boundary conditions are convection and the convective heat transfer coefficient h is set to a constant value of 10 W/m²K.

Option 2: Using Air (Fluid)

Open the thin_film_with_air.ldev project file in HEAT. Alternatively, you can follow the instructions provided in the Modeling Instructions section to create your own project file. The project file contains two geometric objects, thin_film and air. Note that the length of the thin_film object are slightly larger than the specifications. This is because the area of the simulation region gets defined by the area of the (HEAT) solver region so we have made the thin_film object larger and have used the length of the HEAT object to define the length of the simulated thin film.

There is one thermal boundary condition (left) defined in the "Boundary Conditions" group that sets the left edge of the thin film at a fixed (elevated) temperature of 400 K. In absence of any cooling (air), the entire thin film will remain at 400 K in steady state.

To model the air cooling, we have set the material of the air object to Air which is fluid type material. In HEAT simulations, fluid type objects do not get included in the simulation area (volume) but are used to provide a boundary condition at the solid-fluid interface. In this example, we have defined a convective boundary condition between Air and Silicon and have set the convective heat transfer coefficient h to a constant value of 10 W/m²K. For details on how the convective boundary condition is set between two materials, please refer to the Modeling Instructions page.

Modeling Instructions

Option 1: Using Convective Boundary Condition

In this part of the "Modeling Instructions," we will provide detailed instructions on how to create the HEAT project file that will simulate air cooling in the silicon thin film by using convective boundary conditions at the top and bottom surfaces.  To get started, first open HEAT and save the blank project file by clicking the "Save" button under the "File" tab.  To be consistent with the provided project file, you can name your project "thin_film.ldev".

Materials

Access the material database by clicking the  button.  The material database in HEAT contains material data for more than 40 common semiconductors, alloys, conductors, and insulators.  There is also a fifth material type called "Fluid" which applies to any liquid or gas (e.g. Air) present in the simulation region.  In this example, to define the material properties of the silicon thin film, we will use the default material "Si (Silicon)" available in the material database.  As shown on the screenshot below, the material properties window for a semicodnuctor (Si) has three tabs; Electronic Properties, Recombination, and Thermal Properties.  The first two tabs contain electrical properties of the material which is used by the electrical (CHARGE) solver.  The third tab (Thermal Properties) contans thermal properties of the material that are used by the thermal (HEAT) solver.  For this example, we will focus on the thermal properties of the materials only.

Using the Temperature Dependency Models

To see how the temperature dependency models can be turned on/off and how to change the model parameters, select the "Si (Silicon)" material, go to the "Thermal Properties" tab, and click of the  button for the specific heat to see the model used for the temperature dependency of specific heat of solids.  The "Enable model" check mark can be used to turn the model on or off.  If the model is turned on, the coefficients become available for editing.  The default values of the coefficients for silicon are already available in the material database.  The equation describing the model is shown at the bottom of the window.  Hovering the mouse over the window will show a text providing the definition of the different parameters/coefficients used in the model.

Si (Silicon) must be added from the database to the materials group in Objects Tree. To do this you have two options:

• Under the Materials section of the Design tab, click on New Material button  . Then right click on recently added New Material under the Material Group in Objects Tree and click on Add thermal properties. This will automatically open Electrical/Thermal Material Database. Scroll it down to find Si (Silicon) from Material List and click on Select. This will add the material as Solid object that includes only thermal properties of Si (Silicon).
• Click on the Electrical and Thermal button  to open the electrical/thermal material database. Select Si (Silicon) and then click on Create button as is shown below:

This will add electrical (semiconductor) and thermal (solid) properties of silicon material into Objects Tree. After adding the properties, rename the new material to 'Si (Silicon)' by double clicking on its name or right click and then select rename.

Geometry

The Structures section of the Design tab can be used to introduce various types of structures into the simulation space.  There are different types of structures available.  To create the silicon thin film, we will select a rectangle which will place a rectangle in the simulation region (also shown under "geometry" in the "Objects Tree"). 

Select the rectangle and click the  button (on top of the Object Tree) to edit its properties according to the following table.  The geometry of the rectangle can be set by either setting the center and span in each direction or by setting the min and max value in each direction.  

Simulation Region

Click on in the Objects Tree and click the  button (on the left of the Objects Tree) to edit its properties according to the following table:

HEAT Solver Region

In the Solvers section of the Design tab select the  to place a HEAT solver in the simulation environment. Note that once the solver is selected, all the simulation objects (i.e. constraints, sources, monitors) belonging to the HEAT solver become available under a new tab named HEAT.  Select the HEAT object from the Objects Tree and click on the "Edit Properties" button  to edit the properties according to the following table.

Monitor

From the Monitors section of the HEAT tab, click on the temperature monitor  button to place a temperature monitor in the simulation domain.  Select the monitor in the Objects Tree and click the 

button (on top of the Object Tree) to edit its properties according to the following table.

Boundary Conditions

The thermal boundary conditions can be added to and edited from the "Boundary Conditions" group available under the HEAT solver object in the Objects Tree. There are a variety of boundary conditions available which can be added by selecting from the Boundary Conditions section of the HEAT tab as shown below.

Here we will use a thermal boundary condition to set the temperature at the left edge of the thin film to a fixed value of 400 K.  Click the "Temperature" button  from the Boundary Conditions section of the HEAT tab to add a temperature boundary condition. Select the thermal boundary in Objects Tree and click the "Edit" button to edit its properties according to the following table.

top : Here we will use a thermal boundary condition to set a convective boundary at the top surface of the thin film with a constant convective heat transfer coefficient h = 10 W/m²K.  Click the "Convection" button  from the Boundary Conditions section of the HEAT tab to add a convection boundary condition. Select the convection boundary in Objects Tree and click the "Edit" button to edit its properties according to the following table.

bottom : Here we will use a thermal boundary condition to set a convective boundary at the bottom surface of the thin film with a constant convective heat transfer coefficient h = 10 W/m²K.  Click the "Convection" button  from the Boundary Conditions section of the HEAT tab to add a convection boundary condition. Select the convection boundary in Objects Tree and click the "Edit" button to edit its properties according to the following table.

The project file is now set up.  Save the file using the "File" Tab and run it by following the instructions provided in the top-level page of the example.

Option 2: Using Air (Fluid)

In this part of the "Modeling Instructions," we will provide detailed instructions on how to create the HEAT project file that will simulate air cooling in the silicon thin film by using convective boundary condition between silicon and Air (fluid) in the material interfaces.  To get started, first open HEAT and save the blank project file by selecting the "Save" option under the "File" tab. To be consistent with the provided project file, you can name your project "thin_film_with_air.ldev".

Materials

Check the instructions in "Option 1" (above) to learn about the material properties of solids and fluids. In this option, Air needs to be added to the simulation in addition to Silicon.

Geometry

thin_film : From the Structures section of the Design tab, select a RECTANGLE to be added to the Objects Tree. Select the rectangle in the Objects Tree and click on the "Edit Properties" button  to edit the properties of the rectangle according to the following table.

air : From the Structures section of the Design tab, select a RECTANGLE to be added to the Objects Tree. Select the rectangle in the Objects Tree and click on the "Edit Properties" button  to edit the properties of the rectangle according to the following table.

Simulation Region

Click on in the Objects Tree and click the  button (on the left of the Objects Tree) to edit its properties according to the following table:

HEAT Solver

In the Solvers section of the Design tab select the   to place a HEAT solver in the simulation environment. Note that once the solver is selected, all the simulation objects belonging to the HEAT solver become available under a new tab named HEAT.  Select the HEAT object from the Objects Tree and click on the "Edit Properties" button  to edit the properties according to the following table.

Monitor

From the Monitors section of the HEAT tab, click on the temperature monitor  button to place a temperature monitor in the simulation domain.  Select the monitor in the Objects Tree and click the 

button (on top of the Object Tree) to edit its properties according to the following table.

Boundary Conditions

left : Here we will use a thermal boundary condition to set the temperature at the left edge of the thin film to a fixed value of 400 K.  Click the "Temperature" button  from the Boundary Conditions section of the HEAT tab to add a temperature boundary condition. Select the thermal boundary in Objects Tree and click the "Edit" button to edit its properties according to the following table.

Material Interfaces

In this option, we need to define thermal boundary conditions at the interfaces between materials.  This can be accomplished by assigning the desired boundary conditions to the interface (boundary) between two materials from the Geometry tab of the Edit Boundary Condition dialog of any boundary condition object as shown in the figure below.  

For this example, add a "Convection" boundary condition and then select the convection boundary in Objects Tree and click the "Edit" button to edit its properties. Under the General tab select the "Constant" model to define the convective boundary condition at the Si-Air interface.  Set the ambient temperature to 300 K and the value of h to 10 W/m²K.

Then, As shown in the figure on the left, select the surface type to be "material:material" and select the "Si (Silicon)" as material 1 and then select the "Air" as material 2.

Results and Discussion

The project file is now set up.  Save the file using the "File" tab and run it by following the instructions provided in the top-level page of the example.

Option 1: Using Convective Boundary Condition

The thermal dataset has multiple results (attributes) saved. To view the temperature profile, select the "T" attribute from the list of attributes in the visualizer. Since the thin film is very thin, in order to improve the visualization, click the "show/hide chart setting"  button to open up the plot editor and select the "square" option from under the "Axis scale options". This option will change the aspect ratio of the image to make the length equal in both axes. The resulting plot should look like this:

From the temperature profile, we can see that the temperature of the thin film is at 400 K at the left edge as set by the fixed boundary condition. However, as we move towards the right, the convective boundary condition on the top and bottom boundaries start to cool the film and the temperature slowly goes down reaching the lowest value at the right edge of the thin film.

The resulting plot shows that the drop in temperature along the length of the thin film is nonlinear with the temperature dropping sharply at the beginning and then slowing down as we move further inside the film.  This behavior is expected as heat loss due to convection is directly proportional to the surface temperature of the thin film.  

For a constant convective heat transfer coefficient h the heat transfer due to convection is given by,

$$q = Ah(T_{swf} - T_{env})$$

where, q is the heat flux in W, A is the area of the surface, T_surf is the surface temperature which is function of position and T_env is the temperature of the surrounding air (fluid). At the beginning (left edge) of the thin film, the temperature (T_surf) is high so the amount of heat loss by convection is high which results in a sharp drop in temperature. As we move further (right) into the film, the surface temperature gets smaller which reduces the amount of heat loss and the rate of change in temperature becomes smaller.

The visualizer will show the total heat flux at each boundary and their corresponding area. Heat leaving the simulation region is reported as negative values (P_top and P_bottom) and heat entering the solver region is reported as positive (P_left). The total amount of heat lost due to cooling at the top and bottom boundaries is approximately 0.14 W. Note that this value is calculated by considering the width of the thin film as defined in the solver region by the normalization length (norm length) parameter.

Option 2: Using Air (Fluid)

The profile should be identical to the previous case.  We can also check the line plot of the temperature profile from the monitor which should also be exactly the same as given by the previous simulation. Finally, as we visualize the "boundaries" dataset from the HEAT solver, we will notice that the heat flux (and area)  are now given for the single thermal boundary condition (left) and for the material interface between air and silicon (P_Air:Si).  The amount of heat lost due to cooling should equal the values from the last simulation at approximately 0.14 W as well.