COMSOL 6.4 - 3D Analysis of a Bipolar Transistor

3D Analysis of a Bipolar Transistor

This model shows how to set up a 3D simulation of a NPN bipolar transistor. It is a 3D version of the device shown in the Bipolar Transistor model and demonstrates how to extend semiconductor modeling into 3D using COMSOL Multiphysics. As in the 2D version of this model, the device is simulated whilst operating in the common-emitter regime. A voltage-driven study is computed to characterize the current–voltage response of the device, and two current driven studies are performed to simulate the device operating as an analog current amplifier.

Introduction

Bipolar transistors rely on both electron and hole currents in order to function whereas unipolar transistors, such as MOSFET devices, operate utilizing only one species of carrier. Bipolar transistors have largely been replaced in integrated circuits by field-effect devices; however, they are still important in analog electronics — particularly in power control circuitry where they can be used as switches and current amplifiers.

Figure 1: Left: Simplified cross section through a bipolar transistor showing the structure of the device. Right: Circuit diagram showing the common emitter configuration.

A bipolar transistor consists of three regions known as the emitter, base, and collector. In an NPN transistor the p-type base region is sandwiched between the n-type emitter and collector regions, as shown in the left panel of Figure 1. In the common emitter configuration the emitter contact is the common ground for both the base and collector contacts, that is, the base and collector voltages are measured relative to the emitter, which is grounded. This is shown schematically in the right panel of Figure 1.

In normal operation, the base–emitter junction is under forward bias and the base–collector junction is under reverse bias. Electrons are injected over the forward bias p–n junction from the emitter into the base. They then diffuse through the base region as minority carriers. Those electrons which reach the base–collector junction are swept to the collector contact by the electric field of the depleted region near the reverse bias p–n junction.

The effective resistance between the emitter and collector can be varied by applying a current to the base. In this way, the collector–emitter current can be controlled by a smaller base–emitter current. In this configuration the device functions as a current amplifier, as the collector–emitter current (at a given collector–emitter voltage) is proportional to the base–emitter current. Typically, the current gain can have values of the order of 100 which makes bipolar transistors attractive in a wide range of power management circuitry. For example, a small current from some sensing circuitry, such as a photodiode or temperature probe, could be used to control a larger current needed to operate a motor or a heating element.

The model presented here preforms a detailed DC current–voltage characterization of the bipolar transistor device. The current gain is computed as a function of the collector current, along with an emitter–collector I–V curve for fixed currents applied to the base.

Model Definition

The model geometry is shown in Figure 2. Due to the symmetry of the device only one quarter of the whole structure is explicitly modeled. The modeled doping profile is shown in Figure 3. 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: Model geometry, the symmetry planes are highlighted in blue and the boundaries to which the three electric contacts are applied are labeled.

Figure 3: Dopant distribution for the bipolar transistor device. Left: Volume plot showing the total net dopant concentration, the emitter region can be clearly seen in red; the boundary between the base and collector is not apparent due to the large magnitude of the concentration in the n+ regions. Right: Line cut of the total net dopant concentration taken along the red line shown in the left-hand pane. The p-type base region can be seen in this plot.

The physics and studies settings in this model are exactly the same as in the 2D model Bipolar Transistor.

Results and Discussion

Figure 4 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 COMSOL Multiphysics sign convention: current that flows from the contact into the semiconductor is positive, and current that flows out of the semiconductor into the contact is negative. The figure also shows that the current is conserved. This can be seen as the sum of the base and collector currents have equal magnitude and opposite sign to the emitter current, i.e: the base current can be calculated from the other currents using

The results are in good agreement with the 2D model Bipolar Transistor. Note that when comparing 2D and 3D results, the total current should be scaled with reference to the effective thickness.

Figure 4: Terminal currents as a function of the base–emitter voltage (VBE) for a fixed collector–emitter voltage (VCE = 0.5 V).

Figure 5: Gummel plot showing the magnitude of the collector and base current as a function of the base voltage.

Figure 6: Current gain as a function of collector current for a fixed base voltage of VCE=0.5 V.

Figure 5 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 6 shows the current gain, defined as IC/IB, as a function of the collector current at a fixed base voltage of VCE=0.5 V.

Figure 7: 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 shows the collector current as a function of collector voltage for two different values of base current 2 μA and 10 μA. This figure shows the collector I–V curve for the device in the common emitter configuration. Initially the current increases linearly with increasing emitter–collector voltage, before reaching a saturation level. The gradient of the linear regime and the magnitude of the saturation current depend on the base current.

Figure 8 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 8: 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 that 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_3d

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