A procedure is proposed which allows one to assess the adequacy of models to represent objects. Only the class of inverse problems has been considered in detail. The procedure is based on the definition of a vectorial distance between models. It allows one to order models from the standpoint of their adequacy and apply to the problem many results of ordered sets. When the set of models admits a minimum, this generally coincides with the most adequate model. Otherwise, the most adequate model has to be found with the methods of decision theory. The method may sometimes be cumbersome. It is shown, however, that this fact may represent a useful feature of the method. Moreover, it is shown that most of the existing work on the adequacy of models, even if in embryo, agrees quite well with the proposed framework.

On Assessing Model Adequacy

Using Ryder's formula for drift velocity vs. electric field, the d.c. field and carrier densities in the collector of a bipolar transistor are calculated analytically for all possible bias conditions. This is accomplished by modeling the majority carrier distribution. The results are compared with computer calculations and fairly close agreement is found. The analytic calculations are used to make a detailed division of the ( J c , V cb ) plane into injection, depletion and scattering-limited drift velocity (SLDV) areas. It turns out that the doping level N d and the collector width W determine the nature of this division of the ( J c , V cb ) plane.

Collector models for bipolar transistors

A theory is given of the modifications of the large signal current-voltage relations as well as of the small signal parameters under high injection conditions in the forward active mode. Though a brief discussion is presented of a variety of effects taking place at high current levels, we restrict ourselves mainly to the consequences of the electric field and the ohmic drops in the active base region. The differences and inconsistencies between the Jn = 0 or Webster approach, the ambipolar or Ebner-Gray approach, and the div E = 0 or Spenke approach, are pointed out, as well as the connection with other theories such as the charge-control model of Gummel and Poon. Using the basic ambipolar flow equations, the Ebers-Moll equations are generalized and by a variational procedure the low frequency parameters of the common base T-circuit and the common emitter pi-circuit are found. For d.c. the limiting high injection characteristics go as exp(qVEB/2kT); the mobility dependent asymmetry for PNP and NPN transistors, characterized by the ratio m = (μn − μp)/(μn + μp), as found by Ebner and Gray for transistors with zero base current, is shown not to apply to regular transistors with finite IB. However, the Ebner-Gray quantity m remains in the a.c. parameters. An extensive analytical high frequency analysis is made, using the Green's function method. For a special model simple formulas are ultimately found for the junction input impedance (equation (4.44)) and for the conductivity-modulation bulk impedance (equation (4.52)). The latter confirms completely the Kohn-Nonnenmacher model (Fig. 5) and all elements are calculated. On the collector side the Early effects are incorporated in the modified Ebers-Moll model.

Current-voltage relations and equivalent circuits of transistors at high injection levels

An accurate knee current model considering quasi-saturation for BJTs operating at high current density

The development of a modeling technique for bipolar devices is described. The application of the technique results in complete and realistic large-signal models which are amenable to computer-aided analysis. SCEPTRE has been used to effectively analyze derived models, and results of SCEPTRE analyses of a developed transistor model are presented to illustrate the general capabilities of the modeling technique and to indicate the efficiency with which the resulting models are handled by SCEPTRE. The models are structured such that systematic modifications in model complexity can be effected to reflect refinements in the states of the art of device processing and analysis, as well as to efficiently satisfy the user's model capability requirements.

A bipolar device modeling technique applicable to computer-aided circuit analysis and design

A method is described that selects, for each transistor in a circuit, the model of least complexity that will give acceptable accuracy. The capability to assess model adequacy derives from a self-consistency test in which the values of currents and voltages computed in a simulation of the circuit behavior are compared with onset parameters, to determine whether these computed values are consistent with the approximations underlying the device models used in the simulation. The onset parameters for a model are the terminal currents and voltages above or below which the model fails to give a satisfactory representation of device behavior. The authors set forth the onset parameters for the Ebers-Moll model and discuss their determination by terminal measurement and by calculation based on the transistor makeup. The paper limits consideration to the static behavior of transistors operating in the forward-active mode.

D.L. Bowler

Papers

Assessing model adequacy and selecting model complexity in integrated-circuit simulation