A deep, quantitative understanding of the behavior of electronic devices is necessary to ensure that the circuits made by integrating those devices will have the desired characteristics. Such understanding of the device behavior has become all the more important as the devices have become smaller and many more of them are packed in smaller chips at ever higher densities. Supercomputers have been increasingly used to simulate small devices in recent years. There are several reasons for this development. First, boundary conditions become important as the device size decreases, and this makes the task of simulating the devices more complex. Second, several characteristic quantities that determine the behavior of a device, such as carrier concentration and velocity, do not vary gradually. Third and most important, boundary conditions generally lead to the quantization of physical quantities, and many novel quantization effects become significant and observable in nanometer structures.

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