A new analytical method for modeling a 2D electrostatic potential in MOS devices, applicable to compact modeling

We present a 2D physics based compact model of the double gate (DG) and the gate‐all‐around (GAA) MOSFET, with emphasis on short‐channel devices of nano‐scale dimensions. The compact model is based on a preprocessing routine based on a self‐consistent framework for electrostatics modeling. For sub‐threshold conditions, we assume that the device body is dominated by the solution of the 2D Laplace electrostatics. This assumption is based on the fact that the field strength at the gates emerging from the mobile charge carriers and the doping are much weaker than the fields related to the capacitive coupling between the contacts and the gates. Therefore, we neglect the body charge term in Poisson's equation, converting it to a Laplace equation. Near threshold, the influence of the electronic charge on the electrostatics is taken into account in the framework model in a precise, self‐consistent manner by combining suitable model expressions with the 2D Poisson's equation in the device body. To this end, we consider the self‐consistent potential distributions along the two symmetry axis. Suitable modeling expressions are applied, whose parameters are determined from the boundary conditions and by enforcing self‐consistency using Poisson's equation. The framework and the compact model give excellent results when compared with numerical simulations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper presents a compact analytical model for the static and dynamic electrical characteristics of Junctionless Nanowire Transistors. The static drain current model formulation is derived from the 2D solution of the Poisson equation with appropriate boundary conditions and long-channel devices, leading to a continuous effective surface potential that accounts for the conduction in partial depletion and accumulation regimes. The long-channel model is modified to account for short-channel effects by using the coupled solution of 3D Laplace equation with the 2D Poisson equation. The substrate bias influence on the drain current is also included in the model formulation. The charges at the device terminals are differentiated with respect to the applied biases leading to an analytical description of the transconductances and transcapacitances. The proposed model is validated using experimental data at different bias conditions and temperatures, showing very good agreement.

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

This paper presents a closed-form threshold-voltage model for short-channel partially-depleted (PD) SOI dynamic-threshold MOS (DTMOS) devices based on a quasi-two-dimensional (2-D) approach. As verified by experimental data and 2-D simulation results, this compact model provides an accurate prediction of the threshold-voltage behavior of the short-channel PD SOI DTMOS devices. Based on the analytical model, as verified by the 2-D simulation results, PD SOI DTMOS devices have less short-channel effects including DIBL-induced short-channel effects as compared to the devices without the DTMOS configuration.

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

A new compact modeling approach is presented which describes the full current-voltage (I-V) characteristic of high-performance (aggressively scaled-down) tunneling field-effect-transistors (TFETs) based on homojunction direct-bandgap semiconductors. The model is based on an analytic description of two key features, which capture the main physical phenomena related to TFETs: (1) the potential profile from source to channel and (2) the elliptic curvature of the complex bands in the bandgap region. It is proposed to use 1D Poisson's equations in the source and the channel to describe the potential profile in homojunction TFETs. This allows to quantify the impact of source/drain doping on device performance, an aspect usually ignored in TFET modeling but highly relevant in ultra-scaled devices. The compact model is validated by comparison with state-of-the-art quantum transport simulations using a 3D full band atomistic approach based on non-equilibrium Green's functions. It is shown that the model reproduces with good accuracy the data obtained from the simulations in all regions of operation: the on/off states and the n/p branches of conduction. This approach allows calculation of energy-dependent band-to-band tunneling currents in TFETs, a feature that allows gaining deep insights into the underlying device physics. The simplicity and accuracy of the approach provide a powerful tool to explore in a quantitatively manner how a wide variety of parameters (material-, size-, and/or geometry-dependent) impact the TFET performance under any bias conditions. The proposed model presents thus a practical complement to computationally expensive simulations such as the 3D NEGF approach.

The numerical solution of the three-dimensional Poisson equation with Dirichlet boundary conditions, which is of importance for a wide field of applications in Computational Physics and Theoretical Chemistry is considered using the method of finite elements for a model problem. The direct, the iterative and the factorized direct methods for solving the corresponding linear system of equations are discussed and implemented in the scripting language Python http://www.python.org making use of the numpy http://www.numpy.org and pysparse http://pysparse.sourceforge.net extensions. The relative performance of the different approaches is compared and it is shown, that the factorized direct method is vastly superior for larger problem sizes. A formalism for implementing the Dirichlet boundary conditions in the factorization approach is derived and presented in some detail, since it is to the best of our knowledge new.

The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. The method does not require any corrected function or any Lagrange multiplier and it avoids repeated terms in the series solutions compared to the existing decomposition method including the variational iteration method, the Adomian decomposition method, and Homotopy perturbation method. The approximated solutions obtained converge to the exact solution as tends to infinity.

Solutions of Poisson's equation for a space-charge-limited spherical diode, in which electron trajectories are great circles rather than radial lines, are developed. It is shown how Poisson's equation may be separated to yield ordinary differential equations. The radial equation is solved analytically, whereas accurate numerical solutions of the azimuthal equation are given. A method of truncating the flow so as to produce a ``bowl-shaped'' electron gun, leading eventually to an annular electron beam, is presented, along with power-series solutions of Laplace's equation both inside and outside the region of the flow. The latter series are useful for calculating the shapes of anodes and focusing electrodes.

In both electrostatic and magnetostatic finite element analysis the need occasionally arises for the inclusion of isolated perfect electric and magnetic conductors, respectively. The surfaces of such conductors represent equipotentials but do not constitute a Dirichlet boundary condition. Existing methods for constraining such surface potentials to be constant, without giving specific values, are fairly complicated. This paper presents a simple approach which requires no modifications to existing finite element programs that can solve Laplace's and Poisson's equation subject to the usual Dirichlet and Neumann boundary conditions. The paper also addresses the dual problem of voids in conductors and shows how the floating potential concept can be exploited to obtain streamlines of current flow.

Numerical solution of differential equations using multiquadric radial basis function networks

Exact, closed-form expressions are presented for the solution to Laplace's equation everywhere in a specific cavity-backed aperture problem. A symmetrically placed slot enables the static field in a half-space above a hard or soft ground plane to penetrate into the interior of an elliptically shaped cavity. Separation-of-variables in elliptic coordinates yields a summable series, whereupon the aperture field or its normal derivative appears naturally as a series of edge-condition weighted Chebyshev polynomials in the Cartesian coordinates. Analytic coefficients explicitly display simple dependence upon the cavity and point-source geometry.

This article presents a 2 x2D analytical computation of the magnetic field distribution of a magnetic gear with transverse flux in pole pieces. The underlying linear magnetostatic analytical model is based on the subdomain resolution of Laplace's and Poisson's equations in various 2D planes of the magnetic gear that require coupling. This analytical model serves to evaluate magnetic properties with an acceptable computation time compared to 3D finite element models. The advantage of this approach is even more pronounced for a magnetic gear with a high pole number. A comparison with the 3D finite element model is drawn in order to validate this model. Lastly, a computation time analysis on a high pole number magnetic gear leads to minimizing computation time without any loss of accuracy.

Chapter Three - The BEM for Potential Problems in Two Dimensions

The numerical simulation of electromagnetic problems is usually complex and time‐consuming, so a fast and accurate solver is urgently needed. In this paper, a deep learning method based on conditional generative adversarial network (CGAN) is used to solve the 2D Poisson's equation in electrostatic field. The relative permittivity distribution and source charge distribution in the two dimensional region are regarded as label of CGAN, and the potential distribution is the real image or generated image corresponding to the label. This deep learning network can be trained by reasonably designing the loss functions of generator and discriminator, and based on a large amount of samples obtained by the finite difference method. The average relative prediction error of the trained CGAN for all samples is less than 0.5%, with a significant reduction in prediction time, while that of convolution neural network (CNN) is less than 1.5%. Furthermore, the average relative prediction error of CGAN is less than 1% for the scenes different from those of training and testing samples, including triangle, square, and pentagon. The prediction results indicate that CGAN has stronger learning ability and fast prediction speed, and can be used as a fast solver for electromagnetic numerical problems.