Numerical solutions for a one-dimensional silicon n-p-n transistor
IBM1
TL;DR: In this article, the authors describe a technique of obtaining numerical solutions of the basic carrier transport equations for a semiconductor and the results of some calculations pertaining to a silicon n-p-n transistor.
Abstract: This paper describes a technique of obtaining numerical solutions of the basic carrier transport equations for a semiconductor and the results of some calculations pertaining to a silicon n-p-n transistor. The calculations include dc characteristics in direct and inverse operation, saturation parameters, and small-signal ac common emitter h -parameters. Both Boltzmann and Fermi statistics have been used, and the dependence of carrier mobilities on electric field has been taken into account.
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IBM1
TL;DR: In this paper, the authors compared three standard approaches: transient excitation followed by Fourier decomposition, incremental charge partitioning, and sinusoidal steady-state analysis, and concluded that SSA is the superior approach by providing accurate, rigorously correct results with reasonable computational cost and programming commitment.
Abstract: Techniques for ascertaining the small-signal behavior of semiconductor devices in the context of numerical device simulation are discussed. Three standard approaches to this problem will be compared: (i) transient excitation followed by Fourier decomposition, (ii) incremental charge partitioning, and (iii) sinusoidal steady-state analysis. Sinusoidal steady-state analysis is shown to be the superior approach by providing accurate, rigorously correct results with reasonable computational cost and programming commitment.
212 citations
TL;DR: In this article, a method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described, where Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes.
Abstract: A method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described. The principles of this method have been published earlier [1]. This paper discusses in detail the method and a number of considerable improvements. Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes. The 2D continuity equations are approximated by a set of difference equations assuming that the hole and electron current density components along the meshlines are constant between two neighboring meshpoints in a way similar to that used by Gummel and Scharfetter [2] for the one-dimensional (1D) continuity equations. The resulting difference approximations have generally a much larger validity range than the conventional difference formulations where it is assumed that the change in electrostatic potential between two neighboring points is small compared with k T/q . Therefore, a much smaller number of meshpoints is necessary than for the conventional difference approximations. This reduces considerably the computation time and the required memory space. It will be shown that the matrix of the coefficients of this set of difference equations is always positive definite. This is an important property and guarantees convergence and stability of the numerical solution of the continuity equations. The way in which the difference approximations for the continuity equations are derived gives directly consistent expressions for the current densities that can be used for calculating the currents. In order to demonstrate the kind of solutions obtainable, steady-state results for a bipolar n-p-n silicon transistor are presented and discussed.
196 citations
TL;DR: In this article, a compact model for bipolar transistors which includes quasi-saturation effects is presented, and the assumptions used in the formulation of this model are clearly stated and justified, and a step-by-step derivation of the model equations is presented.
Abstract: This paper describes a compact model for bipolar transistors which includes quasi-saturation effects The assumptions used in the formulation of this model are clearly stated and justified, and a step by step derivation of the model equations is presented These equations model both de and charge storage effects Parameter extraction techniques are qualitatively described and the compact model is evaluated using detailed physical simulations of a high voltage bipolar transistor In addition, simulations employing this model are compared with measurements and are found to be in excellent agreement
144 citations
TL;DR: In this paper, the authors derived analytical expressions characterizing the minority carrier multiplication process and its dependence upon the metal, insulator, and semiconductor parameters for one specific class of diode.
Abstract: In contrast to thick insulator structures, metal-insulator-semiconductor (MIS) diodes with very thin insulating layers (< 30 A for the silicon-silicon dioxide system) allow appreciable tunnel current flow between the metal and the semiconductor causing the semiconductor to depart significantly from thermal equilibrium conditions when the diode is biased. Under such conditions, recent experiments have demonstrated that multiplication of minority carrier current can occur in the contact region. This multiplication process is described in detail by deriving analytical expressions characterizing this process and its dependence upon the metal, insulator, and semiconductor parameters for one specific class of diode. Numerical methods are used to investigate the multiplication properties under more general conditions. Solutions obtained by this method indicate that values of the small signal multiplication factor, M, in the range of 102–103 can be obtained with appropriately designed diodes. The applications of the multiplication process to a transistor structure and to a photodiode with internal multiplication properties are described briefly.
143 citations
TL;DR: In this article, three-dimensional numerical simulation is used to explore the basic charge-collection mechanisms in silicon n/sup + p diodes on lightly-doped substrates.
Abstract: In this paper, three-dimensional numerical simulation is used to explore the basic charge-collection mechanisms in silicon n/sup +//p diodes. For diodes on lightly-doped substrates ( >
127 citations
References
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01 Dec 1967
TL;DR: In this article, the experimental dependence of carrier mobilities on doping density and field strength in silicon has been investigated and the curve-fitting procedures are described, which fit the experimental data.
Abstract: Equations are presented which fit the experimental dependence of carrier mobilities on doping density and field strength in silicon. The curve-fitting procedures are described.
1,539 citations
TL;DR: In this paper, a self-consistent iterative scheme for the numerical calculation of dc potentials and currents in a one-dimensional transistor model is presented, where boundary conditions are applied only at points representing contacts.
Abstract: A self-consistent iterative scheme for the numerical calculation of dc potentials and currents in a one-dimensional transistor model is presented. Boundary conditions are applied only at points representing contacts. Input data are: doping profile, parameters governing excess carrier recombination, parameters describing the dependence of mobility on doping and on electric field, applied emitter and collector voltages, and a trial solution for the electrostatic potential. The major limitation of the present approach results from use of Boltzmann rather than Fermi statistics. Convergence of the iteration scheme is good for low and moderate injection levels.
1,128 citations
TL;DR: In this paper, a numerical method of solution of the fundamental semiconductor steady-state one-dimensional transport equations, already available in the literature, is improved and extended, and is applied to a single-junction device.
Abstract: A numerical method of solution of the fundamental semiconductor steady-state one-dimensional transport equations, already available in the literature, is improved and extended, and is applied to a single-junction device. A reduced set of ‘exact’ relations is derived directly from the fundamental set with none of the conventional assumptions or approximations, and is solved numerically by a simple iterative procedure. Freedom is available in the choice of the doping profile, recombination law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. In spite of the generality of the original method, its analytical formulation is shown to be unsuitable for generating a sound numerical algorithm sufficiently acccurate and valid for high reverse-bias conditions. Difficulties and limitations are exposed, and overcomeby an improved formulation extended to any bias condition. Emphasis is on the selection of a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties present in the numerical analysis, and on achieving a high degree of accuracy in the final results (the most delicate problem). As a simple application of the improved formulation, ‘exact’ and first-order theory results for an idealized structure are presented and compared. The poorness of some of the basic assumptions of the conventional first-order theory is exposed, in spite of a satisfactory agreement between the exact and first-order results of the terminal properties for particular bias conditions. The computation time for the achievement of one set of very accurate solutions for a specified applied voltage amounts to approx 1 min on an IBM 7094/7040 shared-file system.
236 citations
01 Jan 1956
TL;DR: In this article, a method of analyzing transistor behavior for any base-layer impurity distribution is presented, in particular expressions for emitter efficiency, transverse sheet resistance R, transit time, and frequency cut-off f?.
Abstract: This paper presents a method of analyzing transistor behavior for any base-layer impurity distribution. In particular, expressions are derived for emitter efficiency ?, transverse sheet resistance R, transit time ?, and frequency cut-off f?. The parameters ? and R are functions only of the total number of impurities in the base layer. The analysis is used to derive ?, R, ? and f? for four different distributions-uniform, linear, exponential, and complementary error function. For each of these distributions a transistor base-layer design equivalent in R and f? is obtained. Comparison shows that for equivalent parameters the nonuniform distributions permit the use of wider base layers, but require greater maximum impurity concentrations and must be operated at high current densi
160 citations
TL;DR: In this article, a numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented.
Abstract: A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as mobile carrier and net electric charge densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time. Considerable attention is focused on the numerical analysis of the initial-value-boundary-value problem in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties of the problem, such as stability conditions related to the discretization of partial differential equations of the parabolic type, small differences between nearly equal numbers, and the variation of most quantities over extremely wide ranges within short regions. Results for a particular n + - p single-junction structure under typical external excitations are reported. The iterative scheme of solution for a single device is applicable also to ensembles of active and passive circuit elements. As a simple example, resutls for the combination of an n + - p diode and an external resistor, analyzed under switching conditions, are presented. The inductive behavior of the device for high current pulses, and storage and recovery phenomena under forward-to-reverse bias switching, are also illustrated. ‘Exact’ and conventional approximate analytical results are compared and discrepancies are exposed.
101 citations