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From the Shockley equation follows that the diode operates like a valve. For "large" applied voltages (for instance, U> lOUT) the growth of the current-voltage characteristic slows down and the above formula is no longer valid in this range (high injection case). The bipolar junction transistor is a device whose performance is based on the interaction of two pn junctions. Each of the three differently doped regions has an Ohmic contact (Fig. 2b). There are two possibilities of configurations: the pnp and the npn transistor.

These devices are modeled in this book. pn junctions are of great importance both in modern electronic applications and in understanding other semiconductor devices. A pn-junction diode consists of a region where C(x) < 0, the p-region, and a region where C(x) > 0, the n-region, and has two Ohmic contacts rD (see Fig. 2a). The remaining part of the boundary r N is insulating. If no potential is applied at the Ohmic contacts, the diode is in thermal equilibrium. Due to the doping profile and the diffusive forces, electrons and holes diffuse from one side of the diode to the other.

3. Uniqueness of transient solutions Our uniqueness results which are based on duality techniques can be summarized as follows: For strictly positive initial and boundary data, the isentropic model has a unique solution for small time. Indeed, the densities can be proven to be strictly positive, so the problem is uniformly parabolic. Then a test function is chosen which solves a backward parabolic problem (dual method). Employing appropriate a priori estimates for this test function, the uniqueness result can be proved.

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