1  Introduction

The purpose of the work described in this paper is the development of a Fortran code for the computational modelling of the atomic mixing and particle emission that takes place when a material is bombarded with energetic particles. The physical process is taken to be governed by the atomic mixing model of references [1-3]. The development of its computational solution has also been of interest for some time and this has lead to the development of the original IMPETUS code [4-5]. In this paper and the companion paper [6] the numerical treatment of the atomic mixing model is re-examined. Many of the methods that were employed in original IMPETUS code are replaced or improved. This work has therefore lead to the production of a new edition of IMPETUS which is termed IMPETUS II [7].
The physical process involves a solid material which is continuously bombarded with particles of one particular species. As a result of this, the materials within the solid mix and the surface of the solid recedes as the materials of the original structure and the bombarding or primary species are emitted or sputtered as the secondary particle yield. The physical process is important in high depth resolution analysis of semiconductors. In principle, the original distribution of material can be related to the secondary particle profiles, that is by secondary ion mass spectrometry (SIMS) or secondary neutral mass spectrometry (SNMS).
The model includes only those bombardment-related processes which are always present in atomic mixing. The extent of the model remains the same as it is outlined in Wadsworth et al [4]. The model is independently reviewed in Zalm [8]. The main aspect of the atomic mixing model is that the mixing and surface recession are governed by a set of partial differential equations (PDEs). The equations govern the concentration variables which are functions of depth into the material and the dose of bombarding particles. The PDEs are parabolic convection-diffusion equations and they are stated in section 2. The general solution strategy is to divide a truncation of the full domain into a grid and, starting with the initial material profile, continually step forward in dose, re-computing the material distribution and the sputter yield at each stage.
The evolving material distribution within the solid always consists of the constituents of the original structure and the primary species. These materials together are termed the material system. The conditions of the experiment refer to the energy and the angle of bombardment and these are taken to be constant for the duration of the experiment.
Many properties of the individual components of the material system and simple mixtures of these materials - such as energy deposition functions and the relative propensity of the different materials to sputter from the surface - may be referenced to estimate properties of any given material distribution. This technique has been successfully applied in the computational model for some time [4-5]. In section 3 it is shown how the energy and range functions for any material distribution can be obtained from the corresponding functions of the pure i-material and how the yields can be modelled on the pure material yield and the yields from 50:50 mixtures.
In this work, the original material distribution is taken to be from a simple space of functions in which the concentrations take constant values between points or vary linearly from point to point. An example of initial material structure - acceptable to IMPETUS II - is shown in figure 1. The graph shows the concentration of one of the materials in a structure made up of two materials. The material contains three separate structures: in the first the interfaces ( at 100 and 200 angstroms) are sharp, the second (the trapezium) has ramped interfaces, the third is a smooth distribution that is approximated by a sequence of straight lines.

Figure 1. An example of the original material structure in IMPETUS II.

The properties of the model and the nature of the material distribution are considered with a view to developing a computational solution strategy in section 5. The bombarding particles and energy are generally deposited in the surface region, beyond which there is minimal mixing. Thus it has become the practice to divide the problem into a surface region, termed the mixing window, which is solved in detail and the remainder where a solution to a simplified model is obtained. There is also a transition in the nature of the PDEs in the surface region which also needs to be taken into account. The detailed solution of the PDEs by the finite difference method (FDM) is described in section 6.
In this paper the properties of the atomic mixing model are reconsidered with a view to its efficient computational solution. The methods used in obtaining the important properties of the evolving material distribution are described in detail in section 4. The implicit three-level finite difference method that is employed to solve the PDEs in IMPETUS II is developed in section 5.

The purpose of this paper is to present the model and basic methods that are employed in IMPETUS II in order to return a computational solution to the atomic mixing equations. In the companion paper [6], a range of special methods that are also important are described - for example to deal with particular forms of material distribution, to proces the solution outside the mixing window and to ensure that material is conserved. Hence the work has ben divided so that the model and overal computational method are comunicated to the reader in the first paper, and the other details, necessary to reproduce the complete algorithm are described in the companion paper.

Two material systems are introduced to demonstrate the numerical results. The first is that of a Silicon-Germanium material bombarded with oxygen. The second is that of a Gallium-Aluminium- Arsenide system, also bombarded with oxygen. Further detail on these material systems can be found in Badheka [5]. With these material systems, the standard IMPETUS model is applied. Although the detailed kinetics of the oxidation of the constituent materials may play an important role in mediating the mixing process under some oxygen bombardment conditions - the model in its present form has not been adapted to deal with these details.