By Joseph W. Jerome
This e-book addresses the mathematical elements of semiconductor modeling, with specific cognizance enthusiastic about the drift-diffusion version. the purpose is to supply a rigorous foundation for these types that are really hired in perform, and to research the approximation houses of discretization strategies. The booklet is meant for utilized and computational mathematicians, and for mathematically literate engineers, who desire to achieve an figuring out of the mathematical framework that's pertinent to machine modeling. The latter viewers will welcome the creation of hydrodynamic and effort shipping types in Chap. three. recommendations of the nonlinear steady-state platforms are analyzed because the mounted issues of a mapping T, or greater, a relatives of such mappings, individual by way of method decoupling. major cognizance is paid to questions with regards to the mathematical homes of this mapping, termed the Gummel map. Compu tational features of this fastened element mapping for research of discretizations are mentioned in addition. We current a unique nonlinear approximation conception, termed the Kras nosel'skii operator calculus, which we advance in Chap. 6 as a suitable extension of the Babuska-Aziz inf-sup linear saddle element thought. it's proven in Chap. five how this is applicable to the semiconductor version. We additionally found in Chap. four an intensive examine of assorted realizations of the Gummel map, together with non-uniformly elliptic platforms and variational inequalities. In Chap.
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Extra info for Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices
What is necessary is a preliminary mapping of ¢ 1--+
A typical + U· \lxf ut J:l -E· \luf = C. 28 3. Moment Models: Microscopic to Macroscopic representation of One of these mechanisms, embodying the Pauli exclusion principle, is given by the nonlinear expression (cf. 2) in terms of a scattering kernel S. In the Boltzmann transport equation above, it has been assumed that the traditional Lorentz force field does not have a component induced by an external magnetic field. The moment equations, which will be derived subsequently, are expressed in terms of certain dependent variables, where n is the electron concentration, v is the (translational) velocity, p is the momentum density, P is the symmetric pressure tensor, q is the the heat flux, e[ is the internal energy, and Cm C p , and Cw represent moments of C, taken with respect to the functions, go(u) - gl(U) 1, mu, The moment equations are given by: an at + \1.
14). 14), once the boundary values for the potential are clearly defined. It is customary to describe the boundary conditions for the potential via two terms, a built-in potential, Ubi, due to the doping, and an applied bias potential, which may depend on time and position. The built-in potential is defined as the equilibrium potential, corresponding to zero carrier currents. 2 in one dimension for a single junction. 13), we obtain Ubi = In (;c (kl + Jkr + 4C2 ) ) . 47) In MOS-FET devices with an oxide region, it is customary to consider the Poisson equation on the union of the substrate and buffer, with continuous electric displacement vector across the common boundary, and to consider the current continuity subsystem on the substrate alone, with a zero current flux for the carriers across the substrate/buffer boundary.
Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices by Joseph W. Jerome