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This paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with multi-dimensional parabolic-elliptic boundary value problems for the devices. We prove an existence and uniqueness result, and the asymptotic behavior of this mixed initial boundary value problem of partial differential-algebraic equations.

In this chapter we develop a general mathematical theory for the moment equations for charged particles obtained by applying Levermore’s method (maximum entropy principle). We will show that the main drawbacks of this method disappear when it is applied to the semiclassical Boltzmann equation for semiconductors. In this case, the phase space is given by a space variable x, as usual, and a variable k which accounts for the crystal wave number, so that ħk has the dimension of a momentum. The variable varies over a bounded subset s of Rn, called a Brillouin region. In this model, the velocity is a given function of k, which depends on the crystal energySince s(k) the particles described by this model carry a charge, the Boltzmann equation is coupled to a Poisson equation for the electric potential which drives the particles. For this model, we prove a local existence result, and a global existence result of smooth solutions around equilibria.