Lattice Deformation and Strain

The deformation in a crystal is described by two normalized quantities related to each other strain and stress. The stress, a, is defined as the internal force created to balance an external force, F, applied to an area of a crystal. A, normahzed to A  

The applied forces are vector quantities and can be resolved into three components two within the plane they are acting on, and the third, normal to this plane. The force component acting in the normal direction is the normal force, while the components acting in the plane are called shear forces. The stress caused by these forces is a second-rank tensor  

The stresses with mixed indices are the shear stresses where the second index defines the normal to the plane in which the stress acts, and the first index indicates its direction. Normal stresses act along the direction of the axis indicated by the repeated index. Positive values of stress indicate tensile stress, while negative values indicate a compressive stress state. 

In the linear elasticity theory, the relation between the stress and strain tensors and e,j for wurtzite structure semiconductors, such as GaN is given by  

The growth on foreign substrates typically leads to the presence of built-in strain in heteroepitaxial GaN layers owing to the difference in lattice parameters and thermal expansion coefficients between layers and substrates [19, 25-27]. Sapphire and SiC are among the most often used substrates, and typically growth is realized on the basal (0001) c-plane of sapphire and SiC. In such instances, nitride films grow along the polar [0001] direction. The sixfold symmetry of the basal planes of the wurtzite (nitrides, SiC) and rhombohedral (sapphire) crystal structures dictates their isotropy in the basal plane and hence, the thermal expansion coefficients, piezoelectric and elastic properties 

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