Thermal Analysis of a Bipolar Transistor

This model demonstrates how to combine the Semiconductor interface with the Heat Transfer in Solids interface in order to include the effects of a nonuniform temperature throughout a semiconductor device.

See the Bipolar Transistor model for a description of the bipolar transistor device and how the Semiconductor interface can be used to characterize its operation.

Introduction

In this model, the Heat Transfer in Solids interface is added and configured to calculate the temperature distribution throughout the device. The Semiconductor interface provides the heat source used in the Heat Transfer in Solids interface; whilst the temperature distribution that is used in the Semiconductor interface is calculated by the Heat Transfer in Solids interface.

Model Definition

The model geometry is the same as used in the Bipolar Transistor model. The Heat Transfer in Solids interface is configured to use the semi.Q_tot variable, which is the heating calculated in the Semiconductor interface, as the heat source throughout the model domain. Heat is transferred between the model domain and its environment via the contact boundaries. Since the packaging of the device is not explicitly modeled, the heat transfer coefficient for these boundaries is calculated assuming an effective thermal resistance of R =100 K/W, which is a typical value for small-signal bipolar transistors. This means that a temperature difference of 100 K between the device and the ambient temperature of its environment would result in a 1 W heat flux through the contact boundaries. This heat flux is assumed to be distributed over the total area, A, of the contacts such that the effective heat transfer coefficient, h0, for the contact boundaries is given by

The Semiconductor interface is configured to use an incomplete ionization model for the Dopant Ionization. Incomplete dopant ionization is an important temperature-dependent process that affects the carrier concentration, and thus currents, through the semiconductor material. In the Semiconductor interface, the temperature of the semiconductor material domains is usually set using the default Semiconductor Material Model node. This model adds a second Semiconductor Material Model node: the default node uses a constant material temperature, T0, which is specified in the parameters table; the additional node is set to use a temperature variable T, which is the temperature distribution calculated by the Heat Transfer in Solids interface.

This model sweeps the voltage applied to the base contact whilst the collector contact is held at a constant voltage of 3 V. The device is simulated in the common emitter configuration where the base and collector voltages are measured relative to the emitter, which serves as a common ground. Two studies are required to solve the fully coupled problem. The first study uses the default Semiconductor Material Model node, where the heating from the Semiconductor interface is used as the heat source in the Heat Transfer in Solids interface, but the Semiconductor interface uses a constant material temperature. Solutions from the first study are then used as initial conditions for the second study, which solves the fully coupled problem, where the temperature distribution calculated in the Heat Transfer in Solids interface is fed back into the material temperature used in the Semiconductor interface via the second Semiconductor Material Model node.

The Gummel Plot and Current Gain plot groups from the Bipolar Transistor model are then amended to show the result from the two new studies. Also, the semiconductor heat source, semi.Q_tot, is plotted as a surface plot. Finally, the resulting temperature throughout the semiconductor device is plotted in the same plot group as the voltage distribution. This allows the voltage and corresponding temperature throughout the device to be visualized simultaneously.

Results and Discussion

Figure 1 is the Gummel plot, which shows the collector and base currents, plotted on a logarithmic y-axis as a function of the base voltage. The two datasets are nearly identical because the temperature throughout the device in the fully coupled study does not deviate more than a few degrees from the T0 value used in the initial study.

Figure 1: Gummel plot showing the collector and base currents as a function of base voltage.

Figure 2: Current gain curve showing the ratio of collector to base current as a function of base voltage.

Figure 2 shows the current gain curve, which is the ratio of the collector to base current, as a function of the base voltage. The small magnitude of the base current for low values of the collector current lead to some numerical instability for collector currents less than approximately 10-10 A. Again, the two datasets are similar due to the small temperature difference between the initialization study and the fully coupled study.

The semiconductor heat source is shown in Figure 3. This is the heat source that is used by the Heat Transfer in Solids interface to calculated the temperature distribution throughout the device.

Figure 4 shows the voltage and temperature throughout the device. The top surface plot shows the voltage distribution, along with the electron and hole currents as black and white arrow plots, respectively. As expected, the hole current is between the base and emitter contacts, without entering the collector region, whilst the electron current is predominantly between the collector and emitter. The lower surface plot shows the corresponding temperature throughout the device, along with an arrow plot which shows the heat flux. The temperature is highest between the emitter and collector contacts, at the depth of the junction between the base and collector regions. This is the expected result as the majority of the current flows between these two contacts, and the base-collector junction creates a region with higher resistance than the surrounding bulk material. Thus the Joule heating, which is the predominant semiconductor heating mechanism in this model, is largest in this location. The heat flux also behaves as expected, as it flows from the peak temperature toward the contacts, which are the only boundaries that allow heat transfer in this simple model.

Figure 3: Semiconductor heat source for a base voltage of 1.1 V and a collector voltage of 3 V

Figure 4: Voltage and temperature for a base voltage of 1.1 V and a collector voltage of 3 V. Top panel: Voltage distribution with electron and hole currents shown in black and white arrows. Bottom panel: Corresponding temperature throughout the device, along with the heat flux shown as arrows.

Semiconductor_Module/Transistors/bipolar_transistor_thermal

Modeling Instructions

Root

Open the bipolar_transistor model

Application Libraries

From the menu, choose .

In the window, select in the tree.

Click

Results

Voltage & Current Density

Add some parameters which are required for the Heat Transfer physics. The effective thermal resistance of a fully packaged device is defined as the temperature difference between the device and its surroundings for a given heating power. This temperature difference is measured between the peak temperature within the device, which occurs at the p-n junction boundaries, and the ambient temperature of the environment in which the device is operating. In this model a value of 100 K/W is used, which means that a heat flux of 1 W would result in a temperature difference of 100 K. As only heat transferred to and from the device via the contacts is considered in this model, the effective thermal resistance is divided by the area of the contacts to give the thermal resistance per unit area. The reciprocal of this value gives the effective heat transfer coefficient, with units W/(m^2 K), which is used to specify the heat flux through the contact boundaries. Also added are an index parameters and a continuation parameter, both of which are used to set up the studies required by this model.

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
V_C	3[V]	3 V	Applied voltage: collector
R	100[K/W]	100 K/W	Effective thermal resistance
h0	1/R/((w_cE+w_cB+w_cC)*l_BJT)	1.6667E9 W/(m²·K)	Effective heat transfer coefficient for contacts
index	1	1	Index for selecting initial solution
c_ion	1	1	Continuation parameter for incomplete ionization

Next, configure the Semiconductor Interface for thermal coupling. Incomplete Ionization is activated in the node, as this is an important temperature dependent phenomenon. A second node is required, these nodes specify the temperature used by the Semiconductor Interface. The existing node sets a constant temperature of and will be used in an initialization study in which the thermal effects are not coupled with the semiconductor equations; the additional node sets the temperature to (using an available item from the drop-down menu after adding the heat transfer physics interface), which is the temperature output by the Heat Transfer in Solids interface, and is used in a second fully coupled study. Also, the base current and collector current nodes are disabled as they are not required in the thermal studies.

Component 1 (comp1)

In the window, expand the node.

Semiconductor (semi)

Semiconductor Material Model 1

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

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

In the Cpud text field, type c_ion.

Semiconductor Material Model 2

Right-click and choose .

Base current

In the window, right-click and choose .

Add the Heat Transfer in Solids Interface and configure the thermal physics. The out-of-plane thickness in the Heat Transfer in Solids Interface needs to be set to the same value as in the Semiconductor Interface. The heating calculated by the Semiconductor Interface, given by variable (as an available item in the drop-down menu), is coupled into the Heat Transfer in Solids Interface using a Heat Source node. Also, a Heat Flux node is used to include the effects of heat transfer through the contact boundaries.

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Heat Transfer in Solids (ht)

In the window for , locate the section.

In the dz text field, type l_BJT.

Initial Values 1

In the window, under click .

In the window for , locate the section.

In the T text field, type T0.

Heat Source 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

From the Q0 list, choose .

Heat Flux 1

In the toolbar, click

and choose .

Select Boundaries 2, 3, 5, and 6 only.

In the window for , locate the section.

From the list, choose .

In the h text field, type h0.

In the Text text field, type T0.

Semiconductor (semi)

Semiconductor Material Model 2

In the window, under click .

Select Domain 1 only.

In the window for , locate the section.

From the T list, choose .

Component 1 (comp1)

To save time, use a coarse mesh for this tutorial. A mesh and tolerance refinement study is always recommended to ensure the accuracy of the simulation results.

Level 1 Adapted Mesh 1

In the window, expand the node.

Right-click and choose .

Add a study to perform the thermal initialization study. This study uses two Stationary study steps, in both study steps the physics tree is modified to disable the second Semiconductor Material Model node so that the temperature of the device in the Semiconductor Interface is constant throughout this study. The first study step uses a solution from a study computed in the model for the initial values and then ramps on the effects of incomplete dopant ionization. The second study step then sweeps the base voltage using the values that will be used in the subsequent fully coupled study.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Thermal initialization: V_B sweep, V_C=3 V, V_E=0 V

In the window, click .

In the window for , type Thermal initialization: V_B sweep, V_C=3 V, V_E=0 V in the text field.

Locate the section. Clear the check box.

Step 1: Stationary

In the window, under click .

In the window for , locate the section.

Select the check box.

In the tree, select .

Click

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

From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
V_B (Applied voltage: base)	0.4	V

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
c_ion (Continuation parameter for incomplete ionization)	0.01 1

From the list, choose .

Stationary 2

In the toolbar, click

and choose .

In the window for , locate the section.

Select the check box.

In the tree, select .

Click

Click to expand the section. Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
V_B (Applied voltage: base)	range(0.4,0.1,0.7) range(0.8,0.05,1.1)	V

In the toolbar, click

Add a second study to perform the fully coupled computation. This study performs a sweep of the base voltage with the thermal effects fully coupled to the semiconductor equations. It uses the initialization study for the initial values of the dependent variables, the index parameter is used to select the corresponding solution for each value of V_B in the auxiliary sweep.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Full thermal coupling: V_B sweep, V_C=3 V, V_E=0 V

In the window, click .

In the window for , type Full thermal coupling: V_B sweep, V_C=3 V, V_E=0 V in the text field.

Locate the section. Clear the check box.

Step 1: Stationary

In the window, under click .

In the window for , locate the section.

Find the subsection. From the list, choose .

From the list, choose .

In the text field, type index.

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
V_B (Applied voltage: base)	range(0.4,0.1,0.7) range(0.8,0.05,1.1)	V

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
index (Index for selecting initial solution)	range(1,1,11)

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

In the toolbar, click

Amend the Gummel Plot graph to show the results from this model. The dataset used for the existing Global plot is set to the solution from the Thermal initialization study and a second Global plot is added to show the data from the fully coupled solution.

Results

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
abs(semi.I0_3)	A	Collector Current, I_C - Uniform temperature
abs(semi.I0_2)	A	Base Current, I_B - Uniform temperature

Global 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
abs(semi.I0_3)	A	Collector Current, I_C - Thermal coupling
abs(semi.I0_2)	A	Base Current, I_B - Thermal coupling

Locate the section. From the list, choose .

In the text field, type V_B.

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

Gummel Plot, I_C and I_B as a function of V_BE

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the toolbar, click

Amend the Current Gain graph in a similar way.

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
abs(semi.I0_3/semi.I0_2)	1	Gain (I_C/I_B) - Uniform temperature

Global 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
abs(semi.I0_3/semi.I0_2)	1	Gain (I_C/I_B) - Thermal coupling

Locate the section. Find the subsection. From the list, choose .

Current Gain

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Note that, because of the coupling, the gain curve is more susceptible to numerical fluctuations than the previous curve. This effect could be lessened by reducing the solver absolute tolerance, however this significantly increases the time required to solve the model.

Create a mirrored dataset to represent the entire bipolar transistor (recall that, due to symmetry, only one half of the device is modeled).

Mirror 2D 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Plot the total heat source from the Semiconductor Interface.

Semiconductor Heat Source

In the toolbar, click

In the window for , type Semiconductor Heat Source in the text field.

Locate the section. From the list, choose .

Surface 1

Right-click and choose .

In the window for , locate the section.

In the text field, type semi.Q_tot.

Locate the section. From the list, choose .

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

From the list, choose .

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

In the text area, type Semiconductor Heating.

In the toolbar, click

Click the

button in the toolbar.

Notice how the heating is most intense around the base-collector junction directly between the emitter and collector contact. This is to be expected, as the majority of the current flows between these two contacts and the base-collector junction causes a region of higher electrical resistance than the surrounding bulk material.

Plot the temperature distribution and the resulting heat flux. Also plot the voltage and the resulting electron and hole currents. Both plots are shown in the same plot group, with the voltage data offset in the y direction using a deformation.

Temperature and Voltage

In the toolbar, click

and choose .

In the window for , type Temperature and Voltage in the text field.

Locate the section. From the list, choose .

Temperature

Right-click and choose .

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

Locate the section. From the list, choose .

Heat flux

In the window, right-click and choose .

In the window for , type Heat flux in the text field.

Locate the section. From the list, choose .

From the list, choose .

Voltage

Right-click and choose .

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

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

Locate the section. From the list, choose .

From the list, choose .

Deformation 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.

In the text field, type 1.1.

Locate the section. Select the check box.

In the associated text field, type 1.

In the toolbar, click

Click the

button in the toolbar.

Electron current

In the window, right-click and choose .

In the window for , type Electron current in the text field.

Locate the section. In the text field, type semi.JnX.

In the text field, type semi.JnY.

Locate the section. From the list, choose .

From the list, choose .

Deformation 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.

In the text field, type 1.1.

Locate the section. Select the check box.

In the associated text field, type 1.

Hole current

In the window, right-click and choose .

In the window for , type Hole current in the text field.

Locate the section. In the text field, type semi.JpX.

In the text field, type semi.JpY.

Locate the section. From the list, choose .

Temperature and Voltage

In the window, click .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Temperature and Voltage Distributions.

Locate the section. Select the check box.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

The value of V_B for which the results are plotted can be varied by clicking on the node and changing the selection in the section. Try several different values of V_B. Notice how the peak temperature is always located around the base-collector junction. The fluxes, shown as arrow plots, also behave as expected: The heat flux is from the peak temperature at the base-collector junction toward the contacts, which are the only boundaries that allow heat transfer in this simple model; the hole current flows from the base to the emitter without entering the collector region, whilst the electron current flows predominantly between the collector and emitter.