Numerical iterative methods of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junctions under both steady-state and transient conditions are presented. The methods are 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, generation-recombination 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 carrier densities, electric field, electrostatic potential, particle and displacement currents, as functions of position (and time).
The work is divided into two parts. In Part I a numerical method of solution of the steady-state problem, already available in the literature, is improved and extended, and is applied to a two-contact and a three-contact device. The analytical formulation of the original method is shown to be unsuitable for generating a sound numerical algorithm sufficiently accurate and valid for high reverse bias conditions. Difficulties and limitations are exposed and overcome by an improved formulation extended to any bias condition. As a simple application of the improved formulation, "exact" and first-order theory results for an idealized N-P 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. Results for an N-P-N transistor are also reported and the inadequacy of the one-dimensional model discussed.
The time-dependent analysis of the problem is presented in Part II. The fundamental equations are rearranged to an equivalent set of three non-linear partial differential equations more suitable for numerical methods. A highly non-uniform two-dimensional mesh, subject to maintenance of constant truncation errors in both spatial and time domains of certain pointwise operations, is chosen for the discretization of the problem, in view of the variation of most quantities over extreme ranges within short regions. Consequently an implicit discretization scheme is selected for the second-order partial differential equations of the parabolic type in order to avoid restrictions on the mesh size, without endangering numerical stability. An iterative procedure is necessary at each instant of time to cope with the several non-linearities of the problem and to achieve consistency between the internal distributions and the generating equations. This procedure is easily generalized to incorporate equations pertinent to networks of passive elements and ideal generators connected to the semiconductor device. Results for a particular single-junction structure under typical time-dependent excitations of external current and terminal voltage, and for an N-P diode interacting with an external resistor under switching conditions., are reported and discussed in detail.
Considerable attention is focused on the numerical analysis of the steady-state and transient problems in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several difficulties of the problem, such as the small differences between nearly equal numbers, the variation of most quantities over extremely wide ranges in short regions, and the stability conditions related to the discretization of partial differential equations of the parabolic type.
