An approximate model for boron drive diffusions in oxidizing ambients

The linear diffusion equation may be transformed to a moving reference frame which corresponds to a linear-parabolic or experimental oxide growth law. The resulting partial differential equation is separable and may be Laplace transformed. If the initial distribution is piecewise constant the solutions may be written in terms of Kummer functions in the transform domain. Use is made of this fact for a drive diffusion in an oxidizing ambient. The initial distribution is approximated by a rectangle and a two-region solution is used to determine all coefficients.The inversion of the solution to the time domain is accomplished by numerical residue techniques. The amount of boron which is lost to the oxide is computed and approximate relationships are given which allow its computation without the aid of a computer. It is found that boron leaching is controlled by the first few minutes of oxidation and may involve as much as 90% of the boron due to the deposition. Experimental data is presented which supports the theory.     

New analytical expressions for dark current calculations of highlydoped regions in semiconductor devices

We studied highly doped quasi-neutral regions of semiconductor devices with position dependent doping concentration in the absence of illumination. An important parameter of a highly doped region is its dark current. To clarify how the doping profile influences the dark current, simple analytical expressions are useful. To this end, we first transformed the transport equations to a simple dimensionless form. This enables us to write already existing analytical expressions in an elegant way. It is demonstrated how, from any analytical dark current expression, a direct counterpart can be derived. Next, we derived a dimensionless form for a nonlinear first-order differential equation for the effective recombination velocity. Starting from the analytical solution of this differential equation for uniformly doped regions and using linearization techniques, we obtained two new simple and accurate expressions for the dark current. The expressions are valid for general doping profiles with different minority carrier transparencies. The exact solution is included between both new approximate solutions. The new expressions are compared with previous approximate solutions     

Forming electrodes for bipolar beams shaped as strips and solid or hollow cylinders

The configuration of forming electrodes for strip and cylindrical bipolar beams is calculated. The method applied for strip beams is based on the exact parametric equations of equipotential surfaces. An analytic extension of the equation for a planar diode and a solution to Laplace’s equation are used in the case of cylindrical beams. The solution to Laplace’s equation is represented in an integral form. The results obtained are compared to approximate solutions to the corresponding problems. The decompositions describing the explicit equations of the zero equipotential surfaces in axially symmetric and plane systems are analyzed. The behavior of these decompositions makes it possible to extend the results to 3D problems for beams having more complicated shapes.     

A normalized analytical solution for the capacitance associated with uniformly doped semiconductors at equilibrium

A normalized solution for determining the capacitance associated with uniformly doped semiconductors at equilibrium is presented here. The present formulation allows the application of this solution to both the MIS surface problem as well as PN step junction problem under equilibrium conditions, subject to Boltzmann approximation. For the surface case the results are in normalized analytic form in agreement with previous analyses. For the junction case a better estimation of the depletion layer thickness results in an analytic expression which performs better than that based on the depletion approximation.     

Two- and three-dimensional calculation of substrate resistance

A two- and three-dimensional solution of the Laplace equation for the calculation of the substrate resistance for rectangular contacts is presented. After the calculations are performed for a uniform substrate, an extension is given to nonuniform doped regions. This solution makes it possible to investigate the influence of buried layers and field implementations on substrate and well resistances     

阶梯掺杂漂移区SOI高压器件浓度分布优化模型

基于分区求解二维泊松方程,提出了阶梯掺杂漂移区SOI高压器件的浓度分布优化模型.借助此模型,对阶梯数从0到无穷时SOI RESURF结构的临界电场和击穿电压进行了研究.结果表明,对于所研究的结构,一阶或二阶掺杂可以在不提高工艺难度的情况下获得足够高的击穿电压,因而可以作为线性漂移区的理想近似.解析结果、MEDICI仿真结果和实验结果非常吻合,证明了模型的正确性.     

A new simplified threshold-voltage model for n-MOSFETs with nonuniformly doped substrate and its application to MOSFET's miniaturization

The formulation, verification, and application of a new simplified 2-D threshold voltage model for n-MOSFETs with nonuniformly doped substrate profile are provided, in which the averaged normal field at the Si/SiO/sub 2/ interface in the active channel is quoted from a simplified solution of two-dimensional Poisson equation using the Green function technique. Starting with the expression of this average normal field, a simple threshold-voltage model for short-channel n-MOSFETs with nonuniformly doped substrate profile is explicitly expressed in terms of device structures and terminal voltages by considering parabolic source-drain boundary potentials. Moreover, the effects of the junction depth on the threshold voltage are examined in detail. It is shown that the DIBL effect cannot be completely eliminated by simply increasing the substrate doping concentration. Comparisons among developed model, 2-D numerical analysis, and experimental data have been made and the accuracy of the developed analytical model has been verified. In addition, a direct extension of our model to the case of uniformly doped substrates leads to a new constraint equation for device miniaturization.<>     

Barrier height enhancement of the Schottky barrier diode using a thin uniformly-doped surface layer

An analytic model for the barrier height enhancement of the Schottky barrier diode with the Mnp (or Mpn) structure, which considers the uniformly doped surface layer and the surface properties of the metal-semiconductor system, is presented. The maximum potential barrier and its precise location including the effect of the image-force potential have been calculated, which shows that the effective barrier height for the majority carriers without considering the image-force lowering will give erroneous predictions if the doped surface layer is very shallow or lightly doped. The built-in voltage of the Mnp structure based on the interfacial layer theory has also been calculated, which gives the exact dependence of the built-in voltage on the interface properties of the metal-semiconductor contact and the dose of the doped surface layer. Numerical results of the developed model have been computed and discussed, which may serve as the guide for the fabrication of the Schottky barrier (SB) and MIS solar cells with higher barrier height.     

Analytic treatment of the distributed resistance of the window layer in solar cells with parallel grids

A formalism is presented to understand the effect of distributed resistance of the window layer on the current–voltage characteristics of solar cells with parallel grids. Along the window/active‐layer interface, the current density and the electric potential are calculated iteratively from the current density–voltage relation and the Laplace equation, respectively. The former property is approximated in a series expansion of sinusoidal functions, which leads to an analytic solution to the electric potential in the window layer. The total current and the average current density are derived in analytic forms. A few calculated results are presented to show the influence of the distributed resistance of the window layer on the device performance. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling and analysis of hierarchical cellular networks with general distributions of call and cell residence times

We present an analytic model for the performance evaluation of hierarchical cellular systems, which can provide multiple routes for calls through overflow from one cell layer to another. Our model allows the case where both the call time and the cell residence time are generally distributed. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of channel occupancy time distribution for each call type (new call, handoff call, and overflow call) is derived as a function of the Laplace transform of cell residence time. In particular, overflow calls are modeled by using a renewal process. Performance measures are derived based on the product form solution of a loss system with capacity limitation. Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance.     

Monte Carlo simulation of the GaAs permeable base transistor

A two-dimensional multiparticle Monte Carlo (MC) method for the solution of the Boltzmann transport equation has been implemented and the results compared with the conventional drift-diffusion equation solution obtained for both a uniformly doped and an n⁺-n-n⁺GaAs permeable base transistor structure. Improved high-frequency performance is predicted by the MC simulation. Two-dimensional boundary conditions for a "regional" MC analysis have been applied to reduce the computer time that would be spent largely in analyzing the device retarding field region and the neutral regions of the device. The dc parameters, I-V characteristics, and unity current gain-frequency (fT) are discussed. In the n⁺-n-n⁺doped structure, a cooling effect was found that significantly enhances the device frequency performance by reducing the satellite valley population of electrons.     

The exact low-frequency capacitance-voltage characteristics of metal-oxide-semiconductor (MOS) and semiconductor-insulator-semiconductor (SIS) structures

Accurate low-frequency MOS (metal-oxide-semiconductor) and SIS (semiconductor-insulator-semiconductor) C-V curves are found by solving Poisson's equation in the bulk semiconductor using regions of analytic solution joined by numerical solutions of specified accuracy. The technique uses the full Fermi function for the electron, hole, and impurity bands; thus valid results can now be obtained for partially ionized impurity bands, and situations in which the valence or conduction bands are bent through the Fermi level. The calculation can be carried out for any temperature and allows for band structure in the valence and conduction bands. Surface states are incorporated at the interface and the usual low-frequency capacitance curves are obtained for the MOS device. As expected, the C-V curves in the SIS case are more complex. The n-i-n (n-type semiconductor, insulator, n-type semiconductor) device shows two depletion minima separated by a center maxima whose size is sensitive to surface-state charge. In the p-i-n device, the depletion minima can be seen separately only if the impurity concentrations of the semiconductors differ by a factor of five or more. In that case, a deep narrow minimum due to the lightly doped semiconductor can be seen superimposed on one end of a broad shallow minimum due to the heavily doped semiconductor. This work predicts an interesting high-frequency response for the n-i-n structure, namely a bell-shaped C-V curve. This type of C-V response has not been observed to date in two-terminal passive devices, and may lead to SIS applications as a new type of parametric capacitor.     

Modeling three-dimensional scatterers using a coupled finiteelement-integral equation formulation

Finite-element modeling has proven useful for accurately simulating scattered or radiated fields from complex three-dimensional objects whose geometry varies on the scale of a fraction of a wavelength. To practically compute a solution to exterior problems, the domain must be truncated at some finite surface where the Sommerfeld radiation condition is enforced, either approximately or exactly. This paper outlines a method that couples three-dimensional finite-element solutions interior to a bounding surface with an efficient integral equation solution that exactly enforces the Sommerfeld radiation condition. The general formulation and the main features of the discretized problem are first briefly outlined. Results for far and near fields are presented for geometries where an analytic solution exists and compared with exact solutions to establish the accuracy of the model. Results are also presented for objects that do not allow an analytic solution, and are compared with other calculations and/or measurements     

基于FlexPDE的二维PSD数值分析

描述位置敏感探测器(PSD)的Lucovsky方程在一定简化条件下可求得解析解.提出了利用有限元分析软件FlexPDE对非简化Lucovsky方程进行二维区域数值求解的方法,并对四边型PSD进行了分析,结果表明该方法能够准确地分析二维PSD的光电特性,比解析法有更强的适应性.     

Effect of the drain—Source voltage on dopant profiles obtained from the DC MOSFET profile method

An analysis was developed for the influence of a finite drain-source voltage, VDS, on dopant profiles derived from the dc MOSFET profile method. It indicates that the measured profile of uniformly doped material falls below the true profile near the surface. The effect occurs because the edge of the depletion region in the silicon is not parallel to the oxide-silicon interface for a finite VDS. For the case of uniformly doped silicon near room temperature, the analysis indicates, for reverse bias applied across the silicon, that the error in the measured dopant density due to a finite VDSis less than 1 percent ifV_{DS} leq 0.5of file built-in voltage, a condition that is easily met in practice. The analysis also reveals that the profile depth determined from the depth-profile equation is a simple average of the depletion widths at the source and drain ends of the channel in uniformly doped silicon. Experimental results are presented which confirm the general trends indicated by the analysis.     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

Simulation of the Characteristics of Double-Gate Asymmetrically Doped SOI CMOS Nanotransistors

The problems of modeling the basic electrophysical characteristics of asymmetrically doped double- gate SOI CMOS nanotransistors are discussed. A mathematical model for the distribution of the potential of the working region, which follows from the analytic solution of the 2D Poisson equation is treated. The variant of the asymmetric channel (counting from the source) with highly doped and low-doped regions is analyzed. The results of the model calculations of the potential distribution of sub-50-nm structures are in good agreement with the simulation data obtained using a commercially available ATLAS^TM software package intended for modeling 2D transistor structures. Based on the obtained potential distributions, the current–voltage performances are calculated using the model of current formulated within the charge separation concept, taking into account the modified expression for the saturation rate. For the topological norms chosen, the optimization of the parameters of an asymmetrically doped profile provides an additional opportunity to monitor the key characteristics along with the thicknesses of the working region and the gate oxide, which is important in analyzing the applicability of nanotransistor structures, in particular, for analog applications.     

Conductor geometry independence of phase velocity in TEM-modetransmission lines

Based on Caratheodory's (1952) generalization of the Riemann mapping theorem, it is shown that a proof is possible for the conductor geometry independence of phase velocity in transverse electromagnetic (TEM)-mode transmission lines. Specifically, an effort is made to show that a proof of the invariance of phase velocity with conductor geometry is possible from essentially circuital considerations. The argument implies that it is obvious that any TEM-mode transmission line which consists of two conductors, each of arbitrary section, one totally enclosed within the other, can be transformed into a concentric, right circular, coaxial line. Since solutions of Laplace's equation are invariant under conformal transformation, the inductance and capacitance per unit length of the original and transformed lines must be the same. To establish that fact generally, it remains then only to observe that, for a normal coaxial line, a problem to which there is a simple analytic solution, phase velocity is independent of geometry     

A simple hole scattering length model for the solution of chargetransport in bipolar transistors

Considering the small warping parabolic heavy hole model with the quasi-elastic approximation in acoustic phonon scattering, it is shown that the hole scattering length is independent of the hole energy. This result now makes it possible to solve the Boltzmann transport equation to obtain a simple analytical solution for the ballistic hole transport in a thin and uniformly doped base of a pnp transistor