Abstract: Analysis of grain‐boundary diffusion in thin films: Chromium in gold

Abstract

Grain‐boundary diffusion in thin film engineering devices is a very important consideration as far as lifetime and allowable processing treatments are concerned. Surface‐sensitive analytic techniques such as Auger electron spectroscopy, ion scattering spectroscopy, and electron spectroscopy for chemical analysis are proving to be well suited for the study of this phenomena. Two experimental configurations have been used: (a) analysis of surface concentration vs time and temperature on a ’’finite thickness’’ film (permeation analysis) and (b) composition analysis vs depth by sputter profiling ’’infinite thickness’’ films (sputter profiling analysis). The permeation analysis is the easiest approach experimentally since one sample can be analyzed at multiple times (or continuously in time) for a given temperature and since no sputtering is normally involved. However, the mathematical treatment of grain‐boundary diffusion in ’’finite thickness’’ films is incomplete. While the mathematical treatment for diffusion in ’’infinite thickness’’ films is well developed,1 the experimental evaluation of composition vs depth for different time–temperature combinations is tedious, and the sputter removal of material may lead to artifacts.2 We have extended the mathematical treatment of grain ‐boundary diffusion in ’’finite thichness’’ films by comparing analytic and numerical calculations (with simplified and appropriate boundary conditions, respectively) to experimental results for chromium diffusing in gold grain boundaries. Both the permeation analysis and sputter‐profile analysis (for ’’finite thicknesses’’) were modeled.The analytic treatment for surface accumulation in the permeation technique followed the development of Wuttig and Birnbaum.3 This treatment assumes a ’’steady state’’ flux out of the grain boundary into the grain, and a special case of this treatment is zero flux into the grain as assumed by Nelson and Holloway.4 The zero‐flux approximation differs significantly from numerical calculations with appropriate boundary conditions for temperatures as low as 220 °C. Even though a finite ’’steady state’’ flux into the grain causes the analytic solution to approach the exact solution, a general test to determine the proper flux magnitude cannot be developed. Thus, numerical solutions of the partial differential equations using correct boundary conditions with the method of lines is a better approach and gives fast, relatively accurate solutions in finite regions.The numerical technique used in this instance is described along with requirements of the numerical integration program. Calculations by this technique are compared to experimental permeation and sputter profiling data from ’’finite thickness’’ films. All combinations of infinite‐source, finite‐source, infinite‐sink, and surface‐saturation boundary conditions at the source and sink interfaces are illustrated. Finally, the effect of a nonuniform initial distribution of material through the film is illustrated.