Bipolar Transistor

This model shows how to set up a simple n-p-n bipolar transistor simulation. For this particular model, the current voltage characteristics in the static common-emitter configuration are computed and the common emitter current gain is determined.

Introduction

The invention of the Bipolar Transistor at Bell labs in the late 1940s was arguably the most significant technological development of the 20th century. Bipolar transistors were the workhorses of the first integrated circuits and consequently have had an enormous impact on the modern world. Although bipolar transistors have largely been replaced by field-effect devices in modern integrated circuits, they are still important, particularly in high-frequency applications.

The bipolar transistor is basically a three-terminal device whose resistance between two terminals is controlled by the third. In this example, we are going to show the static characteristics of a n-p-n bipolar transistor in common-emitter mode — that is, when the base and collector voltages are measured relative to the grounded emitter. Figure 1 shows the biasing configuration of a n-p-n transistor in this configuration.

Figure 1: A n-p-n transistor in common-emitter configuration.

In the common-emitter mode, the base terminal of the transistor serves as the input, the collector is the output, and the emitter is common to both (in our case grounded). N-p-n transistors in common-emitter configuration give an inverted output and can have a high gain which is a strong function of the base current.

The model presented here shows how to perform a detailed characterization of a bipolar transistor in the common-emitter configuration.

Model Definition

The structure is half of the cross section of a simplified n-p-n bipolar transistor; see Figure 2. The model uses the following dimensions: an emitter contact length of 1.2/2 μm, a base contact length of 0.3 μm separated from the emitter contact by 0.35 μm, and a collector contact length of 2.5/2 μm. The total depth of the transistor is 1 μm with a width of 2.5/2 μm. The modeled doping profile is shown in Figure 2. As is typical of the profile used in silicon bipolar transistors, it consists of four regions (n+, p, n and n+) described in detail in Modeling Instructions.

Figure 2: Top: The cross-section of the modeled transistor. The differently doped regions are labeled. The red line shows the axis of symmetry used to simplified the model geometry. Bottom: A line graph, taken along the axis of symmetry, showing the doping concentration. The labeled regions (n+, n, p) correspond with those in the geometry cross-section.

This model uses several studies to characterize the DC response of the bipolar transistor current output under both voltage and current driven configurations. A Semiconductor Initialization study step is also used, which automatically refines the default mesh where the gradient of the semiconductor doping profile is large. The refined mesh is then used throughout the rest of the modeling sequence.

First, a study sweeps the base voltage with a fixed collector voltage. This allows the currents at all three terminals to be plotted as a function of VBE and demonstrates that the simulation conserves current. This study is also used to create a Gummel plot, which shows the current at the collector and base contacts as a function of the base voltage. The same data is then used to calculate the current gain, defined as the ratio of the collector and base currents (IC/IE), as a function of the collector current. The current gain curve is an important characteristic for current regulation and power control applications, as it is used to calculate the expected collector output for a given base input.

The remaining studies are used to calculate the collector current as a function of the collector voltage for two different values for the base current. This is achieved using two studies that sweep the collector voltage, one with a fixed base current of IB = 2 μA and one with a fixed base current of IB = 10 μA. These simulations are current driven because they are performed with a set base current applied. When solving current driven problems in COMSOL it is often necessary to provide an initial condition which is calculated from a suitable voltage driven study. In order to generate the required initial conditions for these studies a voltage driven initialization study is performed. This initialization study sets the collector voltage to 0, which is the initial value it has in the current driven studies, and sweeps the base voltage. The solution which generates base currents closest to 2 μA is then used as the initial condition for the collector voltage sweeps.

Results and Discussion

Figure 3 displays the current at each terminal as a function of the base-emitter voltage (VBE) for a fixed collector-emitter voltage (VCE =0.5 V). Note that the figure shows the terminal currents using the software’s sign convention: outward current from the semiconductor material is positive and inward current is negative. The figure also shows that the current is conserved. This can be seen as the difference between the emitter and collector currents, represented as magenta circles, matches the base current, shown in green. Thus the current is conserved, as

Figure 4 shows the Gummel plot for the modeled bipolar transistor. The Gummel plot shows the magnitude of the collector and base currents, plotted on a logarithmic scale, as a function of the base voltage.

Figure 5 shows current gain, defined as IC/IB, as a function of the collector current at a fixed base voltage of VBE = 0.5 V.

Figure 6 shows the collector current as a function of collector voltage for two different values of base current. This figure shows how a small base current can be used to control a larger collector current by changing the resistance between the emitter and collector terminals.

Figure 7 shows the voltage and carrier current densities throughout the device. With VCE = 1.5 V the device is in the forward-active regime. In this regime the emitter-base junction is forward biased and the base-collector junction is reverse biased. Electrons are injected from the emitter into the base through the forward biased junction. These electrons then diffuse through the p-type base region as minority carriers. Those that make it to the reverse biased base-collector junction are swept toward the collector terminal by the junction electric field. The thickness of the base region must be small enough to allow the electrons to diffuse through with high probability. Holes can travel easily from the base to the emitter regions through the forward biased emitter-base junction, but they cannot traverse the reverse biased base-collector junction. Hence the hole current flows between the emitter and base terminals without entering the lower n-doped region, and the electron current flows between the emitter and collector terminals.

Figure 3: Terminal currents as a function of the base-emitter voltage (VBE) for a fixed collector-emitter voltage (VCE = 0.5 V).

Figure 4: Gummel plot showing the magnitude of the collector and base current as a function of the base voltage.

Figure 5: Current gain as a function of collector current for a fixed base voltage of VBE=0.5 V.

Figure 6: Plot of collector current vs. collector voltage for IB = 2 μA and IB = 10 μA. Note that varying the base current controls the resistance between the emitter and collector.

Figure 7: Voltage and current density for IB = 2 μA and VCE = 1.5 V. The color shows the voltage and the arrows show the current density for electrons (black) and holes (white). Note that the hole current flows from the base to the emitter and does not enter the lower n-doped region, whilst the electron current flows between the collector and emitter. This current pattern is due to the two p-n junctions which form the device. The electric field is largest around the junctions, as can be seen by the rapid change in voltage between the differently doped regions.

Semiconductor_Module/Transistors/bipolar_transistor

Modeling Instructions

From the menu, choose .

New

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Model Wizard

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Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

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Browse to the model’s Application Libraries folder and double-click the file bipolar_transistor_parameters.txt.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type w_BJT/2.

In the text field, type d_BJT.

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

Define the emitter ohmic contact

Point 1 (pt1)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_cE.

Define the base ohmic contact

Point 2 (pt2)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_BJT/2-w_cB.

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Load the semiconductor material properties for silicon.

Add Material

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to open the window.

Go to the window.

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to close the window.

Semiconductor (semi)

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In the window for , locate the section.

In the d text field, type l_BJT.

Semiconductor Material Model 1

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In the window for , locate the section.

In the T text field, type T0.

Add the doping profiles. First the epitaxial layer background doping.

Analytic Doping Model 1

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the ND0 text field, type N_epi.

Analytic Doping Model 2

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. Specify the r0 vector as


0[um]	X
-d_E	Y

In the W text field, type w_BJT/2.

In the D text field, type d_E.

Locate the section. In the NA0 text field, type N_B+N_epi.

Locate the section. In the dj text field, type d_E.

From the Nb list, choose .

Analytic Doping Model 3

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the W text field, type w_E/2-d_E.

Locate the section. From the list, choose .

In the ND0 text field, type N_E+N_B.

Locate the section. In the dj text field, type d_E.

In the Nb text field, type N_B.

Analytic Doping Model 4

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. Specify the r0 vector as


0[um]	X
-d_BJT-d_C	Y

In the W text field, type w_BJT/2.

In the D text field, type d_C.

Locate the section. From the list, choose .

In the ND0 text field, type N_C.

Locate the section. In the dj text field, type 1.3*d_C.

From the Nb list, choose .

Trap-Assisted Recombination 1

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and choose .

In the window for , locate the section.

From the list, choose .

Emitter voltage

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

In the V0 text field, type V_E.

Right-click and choose .

In the dialog box, type Emitter voltage in the text field.

Click .

Base voltage

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and choose .

Select Boundary 5 only.

In the window for , locate the section.

In the V0 text field, type V_B.

Right-click and choose .

In the dialog box, type Base voltage in the text field.

Click .

Collector voltage

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Select Boundary 2 only.

In the window for , locate the section.

In the V0 text field, type V_C.

Right-click and choose .

In the dialog box, type Collector voltage in the text field.

Click .

Add an additional boundary condition which specify the current for the base contact. Depending on the requirements of each study, either the voltage or the current boundary condition will be disabled.

Base current

In the window, right-click and choose .

Right-click and choose .

In the dialog box, type Base current in the text field.

Click .

In the window for , locate the section.

From the list, choose .

In the I0 text field, type I_B.

Generate a suitable mesh using a study. This study automatically refines the mesh around the gradient of the dopant concentration, which results in a fine mesh in the regions of the p-n junctions.

First set the element size of the default mesh to Course then view the initial mesh generated by the physics induced sequence. This sequence generates a mesh which is finer near the metal contact boundaries.

Mesh 1