Stress and strain around grain-boundary dislocations measured by high-resolution electron microscopy

Stresses and strains around a dislocation at a grain boundary in germanium are measured by a combination of high-resolution electron microscopy and geometric phase analysis. The method is established by first measuring the strains around a matrix dislocation in silicon. Stresses are determined using linear elastic theory and bulk elastic constants. Strain measurements are shown to agree with theoretical calculations based on linear anisotropic elastic theory to 0.2% at a spatial resolution of 2-3 nm. A dislocation constricted at a coherent twin boundary in germanium is subsequently analyzed. The method is adapted to cope with the problem that the reference lattice is not identical for the whole field of view, due to the grain boundary. Strains are compared with theoretical calculations of a matrix dislocation in germanium. Whereas strains in the grains on either side of the twin boundary agree closely with the isolated dislocation case, significant additional strains are localized at the boundary plane. By comparing the stresses and strains across the boundary plane, values for the elastic modulus of the twin boundary are proposed. The significant reduction in elastic modulus for the boundary, when compared to bulk elastic constants, is interpreted in terms of the non-equilibrium configuration of the boundary. An extension of the method is proposed to measure more generally the elastic properties of grain boundaries and interfaces.