Browsing by Author "Chakraborty, Jay"

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Open Access
Diffusion in stressed thin films
(2005) Chakraborty, Jay; Mittemeijer, Eric Jan (Prof. Dr. Ir.)
Interdiffusion in layered thin film structures may occur at temperatures as low as room temperature due to the presence of a high density of short-circuit paths for diffusion. Whereas the effect of the presence of crystalline defects and grain boundaries on diffusion in thin films has received considerable attention in the past, the effect of mechanical stresses on diffusion has attracted only little attention and few quantitative assessments exist, though possible effects of mechanical stresses on diffusion are frequently invoked in a qualitative discussion of diffusion results obtained in thin film systems. With respect to the theoretical basis for the effect of mechanical stresses on diffusion, a rigorous re-thinking of standard textbook knowledge is required. Limits of Fick's first law in a state of non-hydrostatic stress have not been generally recognised: In cases of non-hydrostatic stress, chemical potentials for defining equilibrium are not applicable. Following an approach proposed by Larche and Cahn [Larche F. & Cahn J.W. (1973), Acta Metall 21, 1051], so-called diffusion potentials have to be utilised for defining equilibrium instead of the chemical potentials. Thus, the diffusion flux of a species under non-hydrostatic states of stress is taken proportional to the gradient of a diffusion potential (instead of a chemical potential). The resulting equation for the diffusion flux demonstrates that the flux is not only proportional to the concentration gradient, but also proportional to the product of the gradient of the trace of the mechanical stress tensor and the difference in the partial molar volumes of the diffusing species. Not only the effect of stresses on diffusion has to be considered: Concentration changes arising from diffusion generally may lead to the build-up of stresses. Their magnitude and distribution depends on the differences of the partial molar volumes of the diffusing species, the elastic constants of the alloy considered and the boundary conditions (e.g. if the thin film diffusion couple is freestanding or attached to a rigid substrate). The situation can be (and generally is) complicated by the presence of residual stresses and/or externally applied stresses. In this work, complete flux equations for non-hydrostatic states of stress have been derived both in the lattice-fixed frame of reference and the laboratory frame of reference for the case of interdiffusion in a binary, substitutional diffusion couple. Fick’s first and second laws have been solved numerically by an explicit finite difference method for Pd-Cu thin film diffusion couples (as model examples) subjected to a planar, rotationally symmetric state of stress. The effects of both applied stress gradients as well as diffusion-induced stress gradients have been studied. A general conclusion is that the ‘larger’ atoms are driven towards regions of tensile stress whereas the ‘smaller’ atoms are driven towards regions of compressive stress by the stress gradient acting as a driving force for diffusion. Experimental investigations of interdiffusion in Pd-Cu thin (both 50nm thick) polycrystalline bilayers in the temperature range 175°C to 250°C are reported. To this end, (mainly) Auger electron spectroscopy (AES) in combination with sputter-depth profiling, X-ray diffraction phase, texture and (macro-) stress analysis and transmission electron microscopy (TEM) have been employed. Upon annealing at relatively low temperatures (175°C to 250°C) for durations up to 10 hours, considerable diffusional intermixing occurs. Interdiffusion coefficients have been determined using the so-called ‘centre gradient’ and ‘plateau rise’ methods. Grain boundary diffusion coefficients of Pd through Cu have been determined employing the ‘Whipple-Le Claire’ method. Both interdiffusion and grain boundary diffusion coefficients have been found to decrease (roughly exponentially) with increasing annealing time. Also, a corresponding increase in activation energies for both volume and grain boundary diffusion has been observed. X-ray diffraction phase analysis indicates that interdiffusion is accompanied by the sequential formation of ordered phases. Defect annihilation during annealing and ordering in the Pd-Cu solid solution have been proposed as potential causes for the observed decrease in diffusion coefficients (and corresponding increase in activation energies). The evolution of the stress state upon annealing has been investigated employing ex-situ stress measurements by X-ray diffraction and in-situ wafer curvature stress measurements. The results reveal that tensile stresses are generated during annealing in both sublayers. The stress evolution is discussed in the light of possible mechanisms of stress generation.