Poisson Equation

Numerical solution techniques for the pressure Poisson equation (which plays two distinct roles in the formulation of the incompressible Navier-Stokes equations) are investigated analytically, with a focus on the influence of the boundary conditions adopted. Solutions obtained for two cases of the inviscid stagnation problem of point flow using Dirichlet boundary conditions are presented in graphs and compared with solutions obtained using Neumann boundary conditions and a consistent finite-difference procedure on staggered grids: these solutions are shown to be identical, confirming the result of Gresho and Sani (1987).

The Caldeira-Leggett Hamiltonian describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas (Landau, 1946 [1]). An invertible linear transformation (Morrison and Pfirsch, 1992 [18]; Morrison, 2000 [19]) is presented that converts solutions of the Caldeira-Leggett model into solutions of the linearized Vlasov-Poisson system.

This paper presents a novel physics-based analytical
TSV model that allows fast and accurate design
space exploration of signal propagation and
attenuation properties in 3D ICs, and represents a
useful tool for circuit designers and process
engineers. A through silicon via(TSV), is a vertical
interconnection method between chips and also is the
critical enabling technology for three-dimensional
integrated circuits (3D ICs). The model equations are
based on simple process data including physical
dimensions, passivation thickness, substrate doping,
and TSV type without requiring full access to process
parameters of a particular TSV technology. Working
the other way around, by setting the parasitic
threshold voltage and minimum MOS capacitance of
a specific TSV, the model can provide a list of
process specifications. The model has been validated
against measured results published in literature. Our
experiments show that TSV parasitic MOS capacitor
threshold voltage and minimum capacitance values
computed using the aforementioned model are within
25% and 5 % of measured data. The resulting
complex RLC model, taking into account the
coupling effects between neighboring TSVs as well,
has been experimentally validated under DC and
high-frequency AC conditions. Moreover, timing
libraries compatible with conventional EDA tools
have been generated to run realistic timing analysis
for 3D stacked integrated circuits, such as 3D I/Os.

We study the liquid structure and solvation forces of dense monovalent electrolytes (LiCl, NaCl, CsCl, and NaI) in a nanometer slab-confinement by explicit-water molecular dynamics (MD) simulations, implicit-water Monte Carlo (MC) simulations, and modified Poisson-Boltzmann (PB) theories. In order to consistently coarse-grain and to account for specific hydration effects in the implicit methods, realistic ion-ion and ion-surface pair potentials have been derived from infinite-dilution MD simulations. The electrolyte structure calculated from MC simulations is in good agreement with the corresponding MD simulations, thereby validating the coarse-graining approach. The agreement improves if a realistic, MD-derived dielectric constant is employed, which partially corrects for (water-mediated) many-body effects. Further analysis of the ionic structure and solvation pressure demonstrates that nonlocal extensions to PB (NPB) perform well for a wide parameter range when compared to MC simulations, whereas all local extensions mostly fail. A Barker-Henderson mapping of the ions onto a charged, asymmetric, and nonadditive binary hard-sphere mixture shows that the strength of structural correlations is strongly related to the magnitude and sign of the salt-specific nonadditivity. Furthermore, a grand canonical NPB analysis shows that the Donnan effect is dominated by steric correlations, whereas solvation forces and overcharging effects are mainly governed by ion-surface interactions. However, steric corrections to solvation forces are strongly repulsive for high concentrations and low surface charges, while overcharging can also be triggered by steric interactions in strongly correlated systems. Generally, we find that ion-surface and ion-ion correlations are strongly coupled and that coarse-grained methods should include both, the latter nonlocally and nonadditive (as given by our specific ionic diameters), when studying electrolytes in highly inhomogeneous situations.

The pseudo-MOS transistor (Psi-MOSFET characteristics) is a simple and successful technique for the monitoring of silicon-on-insulator (SOI) wafer quality. To characterize modern ultrathin films, a reconsideration and review of Psi-MOSFET physics and models is required. Selected numerical simulations are presented, which shed light on the intriguing features governing Psi-MOSFET characteristics. Updated models accounting for the density of interface states and

Permeation of ions from one electrolytic solution to another, through a protein channel, is a biological process of considerable importance. Permeation occurs on a time scale of micro- to milliseconds, far longer than the femtosecond time scales of atomic motion. Direct simulations of atomic dynamics are not yet possible for such long-time scales; thus, averaging is unavoidable. The question is

Carrier concentration profiles of two-dimensional electron gases are investigated in wurtzite, Ga-face and N-face heterostructures used for the fabrication of field effect transistors. Analysis of the measured electron distributions in heterostructures with AlGaN barrier layers of ...

A high-order accurate, finite-difference method for the numerical solution of the incompressible Navier–Stokes equations is presented. Fourth-order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite-difference formulas. High-order accuracy in time is obtained by marching the solution with Runge–Kutta methods. The incompressibility constraint is enforced for each Runge–Kutta stage iteratively either

We describe and evaluate a numerical solution strategy for simulating surface acoustic waves through semiconductor devices with complex geometries. This multi-physics problem is of particular relevance to the design of quantum electronic devices. The mathematical model consists of two coupled partial differential equations for the elastic wave propagation and the electric field, respectively, in anisotropic piezoelectric media. These equations are discretized by the finite element method in space and by a finite difference method in time. The latter method yields a convenient numerical decoupling of the governing equations. We describe how a computer implementation can utilize the decoupling and via object-oriented programming techniques reuse independent codes for the Poisson equation and the linear time-dependent elasticity equation. First we apply the simulator to a simplified model problem for verifying the implementation, and thereafter we show that the methodology is capable of simulating a real-world case from nanotechnology, involving surface acoustic waves in a geometrically non-trivial device made of Gallium Arsenide.

L'oggetto principale della prova finale è la teoria degli operatori m-dissipativi su spazi di Banach.
Nella tesi si analizzano alcune proprietà di tale classe di operatori. Il caso più studiato in letteratura è quello degli operatori m-dissipativi definiti su un dominio denso su spazi di Hilbert. Alcuni esempi di operatori m-dissipativi sono forniti dall'operatore Laplaciano con diversi domini di definizione.
Consideriamo ora il problema di evoluzione generale, soffermandoci nei casi particolari in cui A è un operatore autoaggiunto o antisimmetrico in uno spazio di Hilbert.
Nella nostra trattazione introdurremo la teoria dei semigruppi di contrazione, con particolare attenzione al semigruppo di contrazione  generato da un operatore m-dissipativo con dominio denso, analizzando un importante risultato, il Teorema di Hille-Yosida-Phillips, utile nello studio delle equazioni di evoluzione.
In Natura, molti fenomeni di propagazione sono descritti da equazioni di evoluzione, per esempio l'equazione del calore che modellizza il trasporto dell'energia termica in un mezzo omogeneo: essa è il più semplice esempio di equazione di diffuusione che analizziamo nella prova finale come applicazione della teoria dei semigruppi.
La trattazione è basata sui Capitoli 2-3 del libro "An Introduction to Semilinear Evolution Equations" di Thierry Cazenave e Alain Haraux [4].

In the present study, we determine using Maple software the exact numerical solution of Poisson's equation in a Schottky barrier junction according to three different approaches. First, we consider the simple case where the space charge zone is depleted and the doping impurities are fully ionized. Then we treat the case where the space charge zone is non-depleted and the doping impurities are fully ionized. Finally, we solve rigorously the more general case where the space charge zone is non-depleted and the doping impurities are partially ionized. We use two different methods to solve the problem. In the former one, the distance of each point to the junction is calculated as a function of its potential. In the second one, we use a finite difference scheme to solve the Poisson's equation. The calculations may be integrated into a course on semiconductor devices to show the use of Maple capabilities in the resolution of the second order non-linear differential equation governing the potential in electronic devices.

Based on nonpinned surface potential concept, in this paper we present a compact single-piece and complete I-V model for submicron lightly-doped drain (LDD) MOSFETs. The physics-based and analytical model was developed using the drift-diffusion equation and based on the quasi two-dimensional (2-D) Poisson equation. The important short-channel device features: drain-induced-barrier-lowering (DIBL), channel-length modulation (CLM), velocity saturation, and the parasitic series source and drain resistances have been included in the model in a physically consistent manner. In this model, the LDD region is treated as a bias-dependent series resistance, and the drain-voltage drop across the LDD region has been considered in modeling the DIBL effect. This model is smoothly-continuous, valid in all regions of operation and suitable for efficient circuit simulation. The accuracy of the model has been checked by comparing the calculated drain current, conductance and transconductance with the experimental data