By Joseph W. Jerome

This ebook addresses the mathematical features of semiconductor modeling, with specific recognition all in favour of the drift-diffusion version. the purpose is to supply a rigorous foundation for these types that are truly hired in perform, and to investigate the approximation homes of discretization systems. The booklet is meant for utilized and computational mathematicians, and for mathematically literate engineers, who desire to achieve an knowing of the mathematical framework that's pertinent to machine modeling. The latter viewers will welcome the advent of hydrodynamic and effort shipping versions in Chap. three. recommendations of the nonlinear steady-state structures are analyzed because the mounted issues of a mapping T, or larger, a family members of such mappings, exceptional by way of method decoupling. major realization is paid to questions with regards to the mathematical homes of this mapping, termed the Gummel map. Compu tational elements of this mounted element mapping for research of discretizations are mentioned besides. We current a singular nonlinear approximation thought, termed the Kras nosel'skii operator calculus, which we boost in Chap. 6 as a suitable extension of the Babuska-Aziz inf-sup linear saddle element thought. it truly is proven in Chap. five how this is applicable to the semiconductor version. We additionally found in Chap. four an intensive research of assorted realizations of the Gummel map, including non-uniformly elliptic structures and variational inequalities. In Chap.

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**Example text**

Also, for single carrier applications, Cn = O. Note that the relaxation time reciprocals appearing here have the units of frequencies, and suggest time scales over which collisions are likely to occur in influencing momentum or energy. The forms for the relaxation times used in [9] on the basis of higher order moments, and retained by subsequent authors, are: Tp cp/T, Tw c WT T 1 + To + '2Tp. Here, Cp and Cw are physical constants. A comprehensive discussion of these issues is outside the scope of the current exposition.

We shall now rewrite the system in terms of these variables. I. must be explicit, the system is still strictly logically equivalent to the two systems defined in the previous section. We shall actually make use of the exponential variables when lagging has been employed in the original system as a means of defining the fixed point map. -(d/dx)(fd¢/dx) + eexp([4> - vleo) (d/dx)(exp([4> - vleo)dv/dx) (d/dx)(kQ(T) exp([4> - zleo)dz/dx) F(4),v,z) = cleexp(eo4>){(l To exp( -veo)}. = = = + (1/2)exp([v - ekl.

Thus, one obtains the system for M carriers with concentration ni, carrier recombination R i , current density J i , (signed) charge ei, i = 1"", M, with the units of kl now reflecting the charge modulus: ei 8n i '\1 .. J --+ 8t • E '\1. 36) There still remains the issue of determining the constitutive current relations. Classical drift-diffusion theory gives, for M = 2, nl = n, and n2 = p, eDn '\1n - e/-tn(E)n'\1¢, -eDp'\1p - e/-tp(E)p'\1¢. 38) Here e is positive. Notice that the drift terms, are proportional to the respective drift velocities, By experimental observation, these velocities are bounded as functions of lEI.