3.5.2: Transistor Equations

There are several "figures of merit" for the operation of the transistor. The first of these is called the emitter injection efficiency, \(\gamma\). The emitter injection efficiency is just the ratio of the electron current flowing in the emitter to the total current across the emitter base junction: \[\gamma = \frac{I_{e}}{I_{\text{Ee}} + I_{\text{Eh}}} \nonumber \]

If you go back and look at the diode equation you will note that the electron forward current across a junction is proportional to \(N_{d}\), the doping on the n-side of the junction. Clearly the hole current will be proportional to \(N_{a}\), the acceptor doping on the p-side of the junction. Thus, at least to the first order \[\gamma = \frac{N_{d_{E}}}{N_{d_{E}} + N_{a_{B}}} \nonumber \]

(There are some other considerations which we are ignoring in obtaining this expression, but to the first order, and for most "real" transistors, Equation \(\PageIndex{2}\) is a very good approximation.)

The second "figure of merit" is the base transport factor, \(\alpha_{T}\). The base transport factor tells us what fraction of the electron current which is injected into the base actually makes it to collector junction. This turns out to be given, to a very good approximation, by the expression \[\alpha_{T} = 1 - \frac{1}{2} \left(\frac{W_{B}}{L_{e}}\right)^{2} \nonumber \]

where \(W_{B}\) is the physical width of the base region, and \(L_{e}\) is the electron diffusion length, defined in the electron diffusion length equation (first presented as Equation \(1.11.15\)). \[L_{e} = \sqrt{D_{e} \tau_{r}} \nonumber \]

Clearly, if the base is very narrow compared to the diffusion length, and since the electron concentration is falling off in the manner of \(e^{\frac{x}{L_{e}}}\), the shorter the base is compared to \(L_{e}\) the greater the fraction of electrons that will actually make it across. We saw before that a typical value for \(L_{e}\) might be on the order of \(0.005 \mathrm{~cm}\) or \(50 \ \mu \mathrm{m}\). In a typical bipolar transistor, the base width \(W_{B}\) is usually only a few \(\mu \mathrm{m}\) and so \(\alpha\) can be quite close to unity as well.

Looking back at this figure, it should be clear that, so long as the collector-base junction remains reverse-biased, the collector current \(I_{\text{Ce}}\) will only depend on how much of the total emitter current actually gets collected by the reverse-biased base-collector junction. That is, the collector current \(I_{C}\) is just some fraction of the total emitter current \(I_{E}\). We introduce yet one more constant which reflects the ratio between these two currents, and call it simply "\(\alpha\)." Thus we say \[I_{C} = \alpha I_{E} \nonumber \]

Since the electron current into the base is just \(\gamma I_{E}\) and \(\alpha_{T}\) of that current reaches the collector, we can write: \[\begin{array}{l} I_{C} &= \alpha I_{E} \\ &= \alpha_{T} \gamma I_{E} \end{array} \nonumber \]

Looking back at the structure of an npn bipolar transistor, we can use Kirchoff's current law for the transistor and say: \[I_{C} + I_{B} = I_{E} \nonumber \]

or \[\begin{array}{l} I_{B} &= I_{E} - I_{C} \\ &= \frac{I_{C}}{\alpha} - I_{C} \end{array} \nonumber \]

This can be re-written to express \(I_{C}\) in terms of \(I_{B}\) as: \[I_{C} = \frac{\alpha}{1 - \alpha} I_{B} \equiv \beta I_{B} \nonumber \]

This is the fundamental operational equation for the bipolar equation. It says that the collector current is dependent only on the base current. Note that if \(\alpha\) is a number close to (but still slightly less than) unity, then \(\beta\), which is just given by \[\beta = \frac{\alpha}{1 - \alpha} \nonumber \]

will be a fairly large number. Typical values for \(\alpha\) will be on the order of \(0.99\) or greater, which puts \(\beta\), the current gain, at around \(100\) or more! This means that we can control or amplify the current going into the collector of the transistor with a current \(100\) times smaller going into the base. This all occurs because the ratio of the collector current to the base current is fixed by the conditions across the emitter-base junction, and the ratio of the two, \(I_{C}\) to \(I_{B}\), is always the same.