We present a qualitative analysis of the fundamental static semiconductor device equations which is based on singular perturbation theory. By appropriate scaling the semiconductor…
Abstract
We present a qualitative analysis of the fundamental static semiconductor device equations which is based on singular perturbation theory. By appropriate scaling the semiconductor device equations are reformulated as singularly perturbed elliptic system (the Laplacian in Poisson's equation is multiplied by a small parameter ?2, the so‐called singular perturbation parameter). Physically the singular perturbation parameter is identified with the square of the normed minimal Debye length of the device under consideration. Using matched asymptotic expansions for small A we characterize the behaviour of the solutions locally at pn junctions, Schottky contacts and oxide‐semiconductor interfaces and demonstrate the occurrence of exponential internal/boundary layers at these surfaces. The derivatives of the solutions blow up within these layer regions (as ?2 decreases) and they remain bounded away from the layers. We demonstrate that the solutions of the ‘zero‐space charge approximation’ are close to the solutions of the ‘full’ semiconductor problem (when ? is small) away from layer regions and derive a second‐order ordinary differential equation which (when subjected to appropriate boundary/interface conditions) ‘describes’ the solutions within layer regions.
State of the art programs for the solution of the drift diffusion semiconductor equations are based on finite difference techniques, or on certain combinations of finite elements…
Abstract
State of the art programs for the solution of the drift diffusion semiconductor equations are based on finite difference techniques, or on certain combinations of finite elements and finite differences. The extreme gradients which occur in semiconductor devices have motivated several attempts to exploit perturbation analysis in order to improve the numerical scheme. Selberherr, Markowich et. al. showed the singular perturbation nature of the drift diffusion equations. Their analytic approximations of diodes influenced some aspects of the Minimos and Bambi simulators. Asymptotic analyses of MOSFET were presented by Brews and by Ward. These solutions are mainly valid for long channels, and their accuracy is limited.
W. JÜNGLING, E. GUERRERO and S. SELBERHERR
We discuss three models describing the carrier densities in highly doped silicon, which have been used for process and device simulation. We calculate nie for each of the models…
Abstract
We discuss three models describing the carrier densities in highly doped silicon, which have been used for process and device simulation. We calculate nie for each of the models for various doping concentrations within temperature ranges interesting for the device and process simulation. We try to explain the behaviour of nie for high compensation and compare our calculated results to measured values of nie. We offer simple formulae for the calculated nie and show how far the relations between the carrier densities and the Fermi levels can be described by the simple formulae of Boltzmann statistics when we use a doping dependent effective intrinsic number.
M. BUDIL, E. GUERRERO, T. BRABEC, S. SELBERHERR and H. POETZL
In this paper the boundary conditions for point defect distributions in monocrystalline silicon are investigated. These boundary conditions are established by simple thermodynamic…
Abstract
In this paper the boundary conditions for point defect distributions in monocrystalline silicon are investigated. These boundary conditions are established by simple thermodynamic considerations. A circle process is used including vacancy, interstitial and Frenkel pair generation which yields a simple relationship between the vacancy and interstitial equilibrium concentrations at the surface. A new OED model is also presented which explains the t−1/4 behaviour of the interstitial supersaturation. This model is used to simulate experiments of Mizuo and Higuchi. In this way values for the equilibrium concentrations and the diffusion coefficients of vacancies and interstitials are obtained.
Cedric Lab and Philippe Caussignac
A stationary 3D energy‐transport model valid for semiconductor heterostructure devices is derived from a semiclassical Boltznmann equation by the moment method. In addition to the…
Abstract
A stationary 3D energy‐transport model valid for semiconductor heterostructure devices is derived from a semiclassical Boltznmann equation by the moment method. In addition to the well‐known conservation equations, we obtain original interface conditions, which are essential to have a mathematically well‐posed problem. An appropriate modelling of the physical parameters appearing in the system of equations is proposed for gallium arsenide. The model being written and its particularities mentioned, we present a novel numerical algorithm to solve it. The discretization of the equations is achieved by means of standard and mixed finite element methods. We apply the model and numerical algorithm to simulate a 2D AlGaAs/GaAs MODFET. Comparisons between expenrimental measurements and calculations are carried out. The influence of the modelling of the physical parameters, especially the electron mobility and the energy relaxation time, is noted. The results show the satisfactory behaviour of our model and numerical algorithm when applied to GaAs heterostructure devices.
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S. Mijalković, D. Pantić, Z. Prijić, S. Mitrović and N. Stojadinović
This paper describes a new two‐dimensional process simulation program MUSIC (MUltigrid Simulator for IC fabrication processes) which is prospective for the efficient IC process…
Abstract
This paper describes a new two‐dimensional process simulation program MUSIC (MUltigrid Simulator for IC fabrication processes) which is prospective for the efficient IC process simulations due to its capability to eliminate strong bottlenecks present in the existing two‐dimensional process simulation programs. Multistep processes, including ion implantation, diffusion and oxidation, can be simulated, giving the doping profile. Robust and efficient adaptive multigrid numerical techniques for the simulation of coupled multiparticle diffusion processes are used. The capabilities of program MUSIC are illustrated by the results of the process flow simulation of a typical NMOS and bipolar transistors fabricated in BiCMOS technology.
Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere…
Abstract
Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere spectral conditions like A‐stability or L‐stability do not give a reliable answer to the behaviour of the numerical solution. Rather, positivity arguments for the corresponding rational matrix functions play an important role.
R. VANKEMMEL, W. SCHOENMAKER and K. DE MEYER
This paper presents a new discretization technique of the hydrodynamic energy balance model based on a finite‐element formulation. The concept of heat source lumping is…
Abstract
This paper presents a new discretization technique of the hydrodynamic energy balance model based on a finite‐element formulation. The concept of heat source lumping is introduced, and the thermal conductivity model includes the effect of varying both carrier concentrations and temperatures. The energy balance equation is formulated to account for kinetic energy as a convective flow. The new discretization method has the advantage that it allows for assembling the functions out of elementary variables available over elements instead of along element links. Therefore, theoretically, calculation of the Jacobian should be three times faster than by the classic method. Results are given for three examples. The method suffers from mathematical instabilities, but provides a good basis for future work to solve these problems.
The coupled set of non‐linear 2D diffusion equations for donor and acceptor type impurities with initial and appropriated boundary conditions is solved by an implicit locally‐one…
Abstract
The coupled set of non‐linear 2D diffusion equations for donor and acceptor type impurities with initial and appropriated boundary conditions is solved by an implicit locally‐one dimensional finite difference method. Numerical experiments have been made to achieve a reasonable trade‐off between the desired accuracy and the CPU time. The algorithm was implemented to the process module of the 2‐D integrated process and device modeling system IMPEDANCE 2.0.
In this paper we consider the Boltzmann equation describing the carrier transport in a semiconductor. A modified Chapman‐Enskog method is used, in order to find approximate…
Abstract
In this paper we consider the Boltzmann equation describing the carrier transport in a semiconductor. A modified Chapman‐Enskog method is used, in order to find approximate solutions in the weakly non‐homogeneous case. These solutions allow us to calculate the mobility and diffusion coefficients as functions of the electric field. The integral‐differential equations derived by means of the above mentioned method are numerically solved using a combination of spherical harmonics functions and finite‐difference operators. The Kane model for the electron band structure is assumed; the parabolic band approximation is obtained as a particular case. The numerical values of the mobility and diffusivity in a silicon device are compared with experimental data. The Einstein relation is also shown.