Download e-book for kindle: Analysis of Charge Transport: A Mathematical Study of by Joseph W. Jerome
By Joseph W. Jerome
This ebook addresses the mathematical facets of semiconductor modeling, with specific consciousness fascinated by the drift-diffusion version. the purpose is to supply a rigorous foundation for these types that are truly hired in perform, and to research the approximation homes of discretization systems. The e-book 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 equipment modeling. The latter viewers will welcome the creation of hydrodynamic and effort delivery types in Chap. three. suggestions of the nonlinear steady-state platforms are analyzed because the fastened issues of a mapping T, or greater, a kin of such mappings, unique by means of method decoupling. major awareness is paid to questions on the topic of the mathematical houses of this mapping, termed the Gummel map. Compu tational features of this mounted element mapping for research of discretizations are mentioned to boot. We current a unique nonlinear approximation conception, termed the Kras nosel'skii operator calculus, which we improve in Chap. 6 as a suitable extension of the Babuska-Aziz inf-sup linear saddle element idea. it really is proven in Chap. five how this is applicable to the semiconductor version. We additionally found in Chap. four an intensive research of varied realizations of the Gummel map, along 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
Note that, on the intervals (xo, Xl) and (Xd-b Xd), the flux is determined by MI and Md-b respectively, while on all other subintervals, (Xi, Xi+1), the flux has components from both Mi and M i+1. The total flux can then be assembled from the following result. Denoting by J M, the flux due to Mi on (Xi-I,Xi), and by Jt, the flux due to Mi on (Xi,Xi+1), we have J Mi = e-u(x,) / lX' Xi-l e-U(S) ds, 22 2. Development of Drift-Diffusion Models In order to evaluate the integrals appearing in the flux representations, it has been common to employ the piecewise linear interpolant of u.
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. 43). The introduction of exponential relations for nand p is also common (cf. 4), as is the use of the Einstein relations linking the mobility coefficients /-tn, /-tp, and the diffusion coefficients Dn , Dp.
The pressure function, p = p(n), has the property that n 2p'(n) is strictly monotonically increasing from [0,00) onto [0,00). A commonly-used hypothesis is: p(n) = cn'"'(, 'Y> 1, c> 0. 83) The device domain is the x-interval, G == (0,1). The given functions ND and N A are the usual density of donors and the density of acceptors, respectively. The approach used in  was completely novel to this application area, in that the system was reduced to an integro-differential equation, with a set of boundary conditions, including a nonstandard second order boundary condition, which is equivalent to specifying the quantum potential at the (current) inflow boundary.
Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices by Joseph W. Jerome