## The Quantum Zero Space Charge Model for Semiconductors

• The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain $$\Omega\subset I\!\!R^d,d=1,2$$ or $$d=3$$ is the minimizer of the total energy $${\mathcal E}_{\epsilon\lambda}$$; $${\mathcal E}_{\epsilon\lambda}$$ involves the squares of the scaled Planck's constant $$\epsilon$$ and the scaled minimal Debye length $$\lambda$$. In applications one frequently has $$\lambda^2\ll 1$$. In these cases the zero-space-charge approximation is rigorously justified. As $$\lambda \to 0$$, the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting $$(\lambda=0)$$ particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit $$\epsilon \to 0$$ exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small $$\epsilon$$, a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit $$\epsilon =\lambda \to 0$$ is analyzed.