By Ansgar Jüngel
In this ebook a hierarchy of macroscopic types for semiconductor units is gifted. 3 periods of types are studied intimately: isentropic drift-diffusion equations, energy-transport types, and quantum hydrodynamic equations. The derivation of every of the versions is proven, together with actual discussions. in addition, the corresponding mathematical difficulties are analyzed, utilizing glossy concepts for nonlinear partial differential equations. The equations are discretized utilising combined finite-element equipment. additionally, numerical simulations for contemporary semiconductor units are played, exhibiting the actual positive aspects of the models.
Modern analytical thoughts were used and extra constructed, comparable to optimistic resolution equipment, neighborhood strength equipment for free-boundary difficulties and entropy equipment.
The booklet is aimed toward utilized mathematicians and physicists attracted to arithmetic, in addition to graduate and postdoc scholars and researchers in those fields.
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Extra resources for Quasi-hydrodynamic Semiconductor Equations
Example text
More precisely, a semidiscretization in time is proposed, preserving the positivity of the elctron density, and the convergence of the discrete solutions is proved. Numerical results for a one-dimensional resonant tunneling diode are performed. The static current-voltage characteristic show negative differential resistance in some region. Thus even for the rather simple quantum drift-diffusion model, negative differential resistance effects can be reproduced. The physical reason for these effects is the external potential barrier where the electrons are tunneling through.
E. in n. Letting E ----t 0 gives n(t) ~ w in n x (0, To). Analogously, we get p(t) ~ w in n x (0, To). 75) set K = 1 + C2 + Rh and Ao = max(A,2K). Choose o < To ~ T small enough such that eAoToRh Then, for t ~ To, ~ K. 75) follows. This proves the proposition. 2 we have proved that, starting from (strictly) positive initial data and prescribing positive boundary conditions, the densities n, p remain positive at least for small time (nonvacuum solution). e. if recombination is not "too large". 2 hold and suppose in addition that Rn(x, t, n,p) .
To show that S is continuous, let (Ni,p i ) C L 2(QT)2 be a sequence such that (Ni, pi) ---+ (N,P) in L 2(QT) (i ---+ (0). Set (ni,p i ) = S(Ni,p i ). e. , [218]) which proves the continuity of S. Now the existence of a solution (n€:,p€:, V€:) E Chapter 3. The Isentropic Drift-diffusion Model 40 (L 2(H 1) n H1(V*)? x LOO(H 1) is a consequence of Schauder's fixed point the- orem. Second step: uniform LOO bounds for (nc,pc). 43) (see [347, Thm. -V7Vc. -)(n C)-dxdt. e. in QT. e. in QT. Plllo,oo,n, IlpDllo,oo,rDX(O,T)}, let K ~ k, q E N, q ~ 1, and define