2D Poisson Equation Research Articles

Analytical investigation of a triple surrounding gate germanium source metal–oxide–semiconductor field‐effect transistor with step graded channel for biosensing applications

AbstractThis paper proposes a compact analytical model and comprehensively investigates the biosensing performance of a novel dielectric modulated triple surrounding gate germanium source metal–oxide–semiconductor field‐effect transistor with a step graded channel. Solving the 2D Poisson's equation yields an analytical expression of threshold voltage, channel potential, drain current and sub threshold swing. The sensitivity variation in the biosensor has been thoroughly studied by varying different device parameters to investigate its biosensing performance. Graded doping in channel offers enhanced sensitivity than a non‐graded doped channel when the doping of the channel is more near the drain end than source end which is due to stronger electric field and gate capacitance. Effect of fill‐in factor for different biomolecules on the sensitivity has been thoroughly discussed. Different analytical results show an excellent agreement with the simulations done on SILVACO ATLAS TCAD simulator.

Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3D Poisson Equation

A general approach to solving the Dirichlet problem, both for bounded 3D domains and for their unbounded complements, in terms of the fractional (3D) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown.

MOSFET Physics-Based Compact Model Mass-Produced: An Artificial Neural Network Approach

The continued scaling-down of nanoscale semiconductor devices has made it very challenging to obtain analytic surface potential solutions from complex equations in physics, which is the fundamental purpose of the MOSFET compact model. In this work, we proposed a general framework to automatically derive analytical solutions for surface potential in MOSFET, by leveraging the universal approximation power of deep neural networks. Our framework incorporated a physical-relation-neural-network (PRNN) to learn side-by-side from a general-purpose numerical simulator in handling complex equations of mathematical physics, and then instilled the "knowledge'' from the simulation data into the neural network, so as to generate an accurate closed-form mapping between device parameters and surface potential. Inherently, the surface potential was able to reflect the numerical solution of a two-dimensional (2D) Poisson equation, surpassing the limits of traditional 1D Poisson equation solutions, thus better illustrating the physical characteristics of scaling devices. We obtained promising results in inferring the analytic surface potential of MOSFET, and in applying the derived potential function to the building of 130 nm MOSFET compact models and circuit simulation. Such an efficient framework with accurate prediction of device performances demonstrates its potential in device optimization and circuit design.

The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

Lagrange radial basis function collocation method for boundary value problems in $ 1 $D

<abstract><p>This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.</p></abstract>

Gate Tunable Lateral 2D Pn Junctions: An Analytical Study of Its Electrostatics

The electrostatics of two-dimensional (2D) lateral pn homojunctions considering the impact of electrostatic doping by means of two split bottom-gates are studied here. Analytical expressions are obtained from the solution of the 2D Poisson equation considering a depletion approximation. Straightforward analytical models for the electrostatic potential and the depletion width within both the dielectric and the 2D semiconductor are obtained for both the symmetrical and asymmetrical cases. In contrast to the case of devices with chemical doping, the obtained depletion width model of devices with electrostatic doping do not depend on the dielectric constant but only on the electrostatic potential and oxide thickness. The models describe the electrostatics of gate-tunable 2D pn junctions at arbitrary bias. A benchmark against numerical device simulations of MoS2-based pn junctions validate the analytical models.

The Saint‐Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

AbstractThe Dupuit‐Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a “dry gap,” demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the “dry gap” area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincaré metrics with the Gaussian constant curvature , isoperimetric inequalities are obtained for steady‐state DF flows. Conformal moments and confromal radii of the domains and 2‐D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a “distributed sink” (similar to a standard “distributed source,” which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards’ equation in a 3‐D saturated‐unsaturated flow. BVPs for Richards’ equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)].

Temperature sensitive analytical analysis of gate‐stack dual metal nanowire field‐effect transistor (4H‐SiC)

AbstractThis article presents an analytical study and effect of temperature (T = 100, 200, 300, 400 K) on various parameters of gate‐stack dual metal nanowire field‐effect transistor (gate‐stack DM NW) FET (4H‐SiC) and gate‐stack dual metal nanowire field‐effect transistor (gate‐stack DM NW FET) (Si). The parabolic approximation is used to solve the 2D Poisson's equation for surface potential, electric field, drain current (Ids), transconductance (gm), output conductance (gd), and analog performance is evaluated. Electron concentration, electron velocity, drain induced barrier lowering, DIBL, and noise figure (NF) have also been investigated for a fair comparison and shows that 4H‐SiC based gate‐stack DM NW FET device is more insensitive to temperature changes than the silicon‐based gate‐stack DM NW FET. Furthermore, gate‐stack DM NW FET (4H‐SiC) has better electrical performance with temperature variation than gate‐stack DM NW FET (Si). The simulations have been carried out using the ATLAS 3D device simulator. Analytical results are in close agreement with the simulated results.

A Ballistic Transport Nanodevice Based on Graphene Nanoribbon FET by Enhanced Productivity for Both Low-Voltage and Radio-Frequency Scopes

The top performance in both the low-voltage and radio frequency (RF) scopes has been nominated for an unique nanodevice made from a graphene nanoribbon with an extremely short gate length (7.5 nm) in this study. Two distinct material engineering options were used, yielding some interesting outcomes. Due to the use of an ultrascale GNRFET in this study, the band structure non-linearity in the Dirac point and the energy-position dependent effective mass model for dual material gate architectures were examined for the first time. The NEGF formalism is used to carry out both the low-voltage and RF research using a three-dimensional (3D) Poisson equation. Low-voltage high performance has been validated by monitoring the key parameters in the terms of on current (Ion), off current (Ioff), Ion to Ioff ratio, subthreshold swing, and drain induced barrier lowering (DIBL) for the proposed device as compared to other structures under the study. The RF performance is examined by evaluation of essential parameters in the cases of parasitic gate capacitance, intrinsic cut-off frequency, intrinsic delay time, and transconductance. Indeed, a device with a higher source side gate work function than it does on the drain side is proposed will shift the energy band from the device’s half to the drain electrode, altering carrier outflow. In addition, the performance of non-linearity and RF intermodulation distortion has been analyzed for all devices under investigation in order to attain the best attitude toward the suggested device in comparison to the other devices under investigation in this work.

Solving 2D Poisson's equation based on conditional generative adversarial network

AbstractThe 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.

TCAD simulations of FDSOI devices down to deep cryogenic temperature

In this paper we demonstrate the feasibility of FDSOI device TCAD simulations down to deep cryogenic temperatures. To this purpose, Maxwell-Boltzmann carrier statistics is replaced by an analytical approximation for the Fermi-Dirac integral with 3D density of states. The device electrostatic is investigated by solving the Poisson equation in 2D, whereas the transport is simulated using the Drift-Diffusion model. We explore the impact of temperature on device performances in linear and saturation regions as well as the impact of short channel effects, accounted for various gate and spacer lengths, at room and deep cryogenic temperatures. Finally, the obtained results are compared to some experimental data, highlighting the role of TCAD simulations in providing insights in device physics and performances.

An analytical drain current model of Germanium source vertical tunnel field effect transistor

This paper presents an analytical modeling of the drain current for a Germanium source vertical tunnel field effect transistor. Both one dimensional (1D) and two dimensional (2D) Poisson's equation are used for solving the potential distribution as the device structure possess both vertical and lateral band-to-band tunneling components. The 2D band-to-band tunneling model is employed for solving the lateral tunneling while the 1D band-to-band tunneling model is implemented on solving the vertical tunneling which consist of the source region and the overlap channel region. The length of the maximum and minimum tunneling widths are calculated by measuring the distance from the edge of the valence band to conduction band of the energy band profile. Kane's band-to-band tunneling generation rate is used to model the drain current by superimposing two current components. In addition, the impact of bandgap narrowing as well as quantum correction on the drain current are also factored into the model. The analytical models are checked and matched against the results obtained from the TCAD simulations to ensure its accuracy.

Analytical modeling of I–V characteristics using 2D Poisson equations in AlN/β-Ga2O3 HEMT

In this paper, physics-based analytical models using two-dimensional (2D) Poisson equations for surface potential, channel potential, electric field, and drain current in AlN/β-Ga2O3 high electron mobility transistor (HEMT) is presented. The analytical expression of different quantities is achieved based on full depletion approximation of the AlN barrier layer and polarization charge induced unified two-dimensional electron gas (2DEG) charge density model. For the validation of the developed model, results are compared with 2D numerical simulation results, and a good consistency is found between the two. The drain current model is also validated with experimental results of a similar dimension device. The developed model can be a good reference for different β-Ga2O3-based HEMTs.

A 2D Model for 3D Periodic Deep-Water Waves

The paper is devoted to further development of an accelerated method for simulation of the two-dimensional surface waves at infinite depth with the use of a two-dimensional model derived with simplifications of the three-dimensional equations for potential periodic deep-water waves. A 3D full wave model (FWM) is based on a numerical solution of a 3D Poisson equation written in the surface-fitted coordinates for a nonlinear component of the velocity potential. For sufficient vertical resolution used for the Poisson equation, the 3D model provides very high accuracy. The simplified model is based on the 2D Poisson equation written for a free surface. This exact equation contains both the first and second derivatives of the velocity potential, i.e., it is unclosed. The analysis of the accurate solutions for the 3D velocity potential obtained with the 3D model shows that those variables are linearly connected to each other. This property allows us to obtain a 2D equation for the first derivative of the velocity potential (i.e., vertical velocity on surface), which gives the closed 2D formulation for a 3D problem of two-dimensional waves. The previously developed scheme was not universal since the parameters of the closure scheme had to be adjusted to the specific setting. The current paper offers a new formulation of a closing scheme based on the integral parameters of wave field. The method of closing the equations, as well as the numerical parameters, were chosen on the basis of the multiple numerical experiments with the full nonlinear wave model (FWM) and selection of a suitable closing scheme. That is why the given model can be called Heuristic Wave Model (HWM). The connection between the first and second variables is not precise; hence, the method as a whole cannot be exact. However, the derived 2D model is able to reproduce different statistical characteristics of the 2D wave field with good accuracy. The main advantage of the model developed is its high performance exceeding that of 3D model by about two decimal orders.

Variational Bayesian approximation of inverse problems using sparse precision matrices

Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem. Among them is the Bayesian formulation, where a prior probability measure is placed on the quantity of interest. The resulting posterior probability measure is usually analytically intractable. The Markov Chain Monte Carlo (MCMC) method has been the go-to method for sampling from those posterior measures. MCMC is computationally infeasible for large-scale problems that arise in engineering practice. Lately, Variational Bayes (VB) has been recognised as a more computationally tractable method for Bayesian inference, approximating a Bayesian posterior distribution with a simpler trial distribution by solving an optimisation problem. In this work, we argue, through an empirical assessment, that VB methods are a flexible and efficient alternative to MCMC for this class of problems. We propose a natural choice of a family of Gaussian trial distributions parametrised by precision matrices, thus taking advantage of the inherent sparsity of the inverse problem encoded in its finite element discretisation. We utilise stochastic optimisation to efficiently estimate the variational objective and assess not only the error in the solution mean but also the ability to quantify the uncertainty of the estimate. We test this on PDEs based on the Poisson equation in 1D and 2D. A Tensorflow implementation is made publicly available on GitHub.

Quadratic serendipity element shape functions on general planar polygons

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L2-norm and the H1-seminorm.

SiC‐based analytical model for gate‐stack dual metal nanowireFETwith enhanced analog performance

AbstractIn this paper, silicon carbide‐based analytical model for gate‐stack dual metal nanowire field effect transistor (gate‐stack DM NW FET) has been analyzed by solving the 2D Poisson's equation using parabolic approximation method for electric potential, subthreshold current and subthreshold slope. The results have been examined for various silicon carbide film depth and channel length. The results predicted by the analytical model have excellent agreement with simulated results obtained by ATLAS 3D device simulator. A gate‐stack having high k‐dielectric material, that is, Hafnium oxide (HfO2) along with aluminum oxide (Al2O3) has been used. Also, performance characteristics of gate‐stack DM NW FET have been compared with the performance characteristics of nanowire field effect transistor (NW FET), Nanowire field effect transistor (NW FET) (Sic) and dual metal nanowire field effect transistor (DM NW FET) (SiC). It is so proved that our proposed device, that is, gate‐stack DM NW FET (4H‐SiC) exhibits superior performance in terms of drain current (Ids), transconductance (gm), output conductance (gd) and cut off frequency (fT) than the existing devices.

Revised SBFEM on arbitrary polygons and faceted polyhedrons with the second order completeness by elimination of bubble functions

By the theoretical analysis on the scaled boundary finite element (SBFEM), some appropriate 2D or 3D bubble functions are necessary in order that the elements can possess the second order completeness. However, the extra 2D bubble function is difficult to be adopted as facet shape function for 3D polyhedral element. In this paper, a revised-scaled-boundary element (denoted by 2DRSBd2) only corresponding to the boundary nodal displacements is constructed by elimination of the 2D bubble function derived from the constant right-hand term in 2D Poisson equation. This polygonal element 2DRSBd2 possesses the second order completeness and the third order convergence in L2-norm. Then we construct two polyhedral elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 based on the polygonal element 2DRSBd2. The difference between the two 3D elements is: the 3DSB-2DRSBd2 element is constructed with three 3D bubble functions derived from the constant body loads in three dimensional elastostatic problem, while the 3D revised-scaled-boundary element 3DRSB-2DRSBd2 is constructed by elimination of those 3D bubble functions. Both elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 possess the second order completeness and the third order convergence in L2-norm. Numerical examples verify that they have good accuracy and insensitive to mesh distortion.

The finite volume method in the context of the finite element method

The finite volume method (FVM), like the finite element method (FEM), is a numerical method for determining an approximate solution for partial differential equations. The derivation of the two methods is based on very different considerations, as they have historically evolved from two distinct engineering disciplines, namely solid mechanics and fluid mechanics. This makes FVM difficult to learn for someone familiar with FEM. In this paper we want to show that a slight modification of the FEM procedure leads to an alternative derivation of the FVM. Both numerical methods are starting from the same strong formulation of the problem represented by differential equations, which are only satisfied by their exact solution. For an approximation of the exact solution, the strong formulation must be converted to a so-called weak form. From here on, the two numerical methods differ. By appropriate choice of the trial function and the test function, we can obtain different numerical methods for solving the weak formulation of the problem. While typically in FEM the basis functions of the trial function and test function are identical, in FVM they are chosen differently. In this paper, we show which trial and test function must be chosen to derive the FVM alternatively: The trial function of the FVM is a “shifted” trial function of the FEM, where the nodal points are now located in the middle of an integration interval rather than at the ends. Moreover, the basis functions of the test function are no longer the same as those of the trial function as in the FEM, but are shown to be a constant equal to 1. This is demonstrated by the example of a 1D Poisson equation.[copyright information to be updated in production process]