QUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER

The modern computer and telecommunication industry relies heavily on the use of semiconductor devices. The reason for the rapid development and success in the semiconductor technology is due to the ongoing miniaturiza-tion of these devices. The microelectronics industry produces very miniatur-ized components with small characteristic length scale, like tunneling diodes, which have a length of only a few nanometers. In such components, quan-tum phenomena become no longer negligible, even sometimes predominant, and the physical phenomena have to be described by quantum mechanics equations. Depending on the device structure, the transport of particles can be very different, due to several physical phenomena, like drift, diffusion, scattering, and quantum effects. An appropriate way to describe a large number of par-ticles flowing through a device is a kinetic or fluid-dynamic type description. On the other hand, electrons are in a semiconductor crystal quantum object, for which a wave-like description using the Schr̈odinger equation seems to be necessary. Therefore, there are several mathematical models which are able to describe particular phenomena in particular devices. Usually quantum effects can be described by microscopic models, like Schrödinger-Poisson or Wigner-Poisson system. The motion of semi-classical particles in semiconductors is described most accurately by the Boltzmann equation, the basic equation of kinetic theory of semiconductors. The equa-tion takes into account the microscopic mechanism of carrier transport. Gen-erally, for microscopic equations, the complexity of the collision term, to-gether with the high number of independent variables, requires numerical simulations a lot of computation power. Moreover, as semiconductor devices are modeled in bounded domain and the natural physical setting of microscopic models is based on an unbounded 2000 Mathematics Subject Classification. 35K30, 35K35, 35B40. Key words and phrases. Fourth-order parabolic equation, fourth-order elliptic equation, existence and uniqueness of nonnegative solutions, entropy–entropy production method, exponential decay in time