A typical current gain for a silicon bipolar transistor is 50 - 150. The resulting current gain, under such conditions, is:įrom this equation, we conclude that the current gain can be larger than one if the emitter doping is much larger than the base doping. The long minority-carrier lifetime and the long diffusion lengths in those materials justify the exclusion of recombination in the base or the depletion layer. It is typically the emitter efficiency, which limits the current gain in transistors made of silicon or germanium. The emitter efficiency defined by equation ( 5.2.17), becomes: Next, we need to find the emitter efficiency and base transport factor. Which in turn can be written as a function of the excess minority carrier charge, D Q n,B, using equation ( 5.3.3). We now turn our attention to the recombination current in the quasi-neutral base and obtain it from the continuity equation ( 2.9.3):īy applying it to the quasi-neutral base region and assuming steady state conditions: This and other similar relations will be used to construct the charge control model of the bipolar junction transistor in section 5.6.2.Ī combination of equations ( 5.3.1), ( 5.3.4) and ( 5.3.5) yields the transit time as a function of the quasi-neutral layer width, w B ', and the electron diffusion constant in the base, D n,B. The emitter current therefore equals the excess minority carrier charge present in the base region, divided by the time this charge spends in the base. Where t r is the average time the minority carriers spend in the base layer, i.e. This charge is proportional to the triangular area in the quasi-neutral base as shown in Figure 5.3.1 a) and is calculated from:Īnd the emitter current due to electrons, I E,n, simplifies to: It is convenient to rewrite the emitter current due to electrons, I E,n, as a function of the total excess minority charge in the base, D Q n,B. The emitter current due to electrons and holes are obtained using the "short" diode expressions derived in section 4.4.2.5, yielding: Instead, they drift through the base-collector depletion region and end up as majority carriers in the collector region. The minority carriers arriving at x = w B - x p,BC do not recombine. While this boundary condition is mathematically equivalent to that of an ideal contact, there is an important difference. The minority carrier densities on both sides of the base-collector depletion region equal the thermal equilibrium values since V BC was set to zero. The carrier densities vary linearly between the boundary values as expected when using the assumption that no significant recombination takes place in the quasi-neutral regions. The values of the minority carrier densities at the edges of the depletion regions are indicated on the Figure 5.3.1. Minority-carrier distribution in the quasi-neutral regions of a bipolar transistor (a) Forward active bias mode. The minority-carrier distribution in the quasi-neutral regions of the bipolar transistor, as shown in Figure 5.3.1, is used to analyze this situation in more detail. In addition we eliminate the base-collector junction current by setting V BC = 0. The forward active mode is obtained by forward-biasing the base-emitter junction. The discussion of the ideal transistor starts with a discussion of the forward active mode of operation, followed by a general description of the four different bias modes, the corresponding Ebers-Moll model and a calculation of the collector-emitter voltage when the device is biased in saturation. Such recombination current will be discussed in section 5.4.2. The use of the ideal p-n diode model implies that no recombination within the depletion regions is taken into account. To further simplify this model, we will assume that all quasi-neutral regions in the device are much smaller than the minority-carrier diffusion lengths in these regions, so that the "short" diode expressions apply. The ideal transistor model is based on the ideal p-n diode model and provides a first-order calculation of the dc parameters of a bipolar junction transistor. Ideal transistor model Chapter 5: Bipolar Junction Transistorsĥ.3. Ideal transistor model 5.3.1. Forward active mode of operation 5.3.2. General bias modes of a bipolar transistor 5.3.3. The Ebers-Moll model 5.3.4. Saturation
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