We typically analyze transistor's work based on a work of a simple pn-junction diode, with some modifications if needed, as it is basically a particular configuration of pn-junctions. We assume the depletion regions form at the junctions and may use things like Shockley equation to calculate current.
However, in order to obtain all these properties, one should start from the basics of statistical quantum mechanics, which assumes that we have A LOT of particles in order to make some of the statements valid.
Furthermore, we even make many assumptions besides statistical mechanics like the fact that the charge can be treated as continiuum and also that the depletion region width is much smaller than the diode crossection in order to assume that the $E$-field in the depletion region is directed solely along the diode (an analogy to a parallel-plate capacitor).
How can a modern field-effect transistor used in microprocessors with the width of only a few atoms behave like a large one, if we have made many assumptions regarding large number of particles and large scale?
Do the results of these assumptions even hold there? Why?
Since the models I described obviously don't work, which models are used to describe such systems?