The orientation dependence of elastic strain energy in cubic crystals and its application to the preferred orientation in titanium nitride thin Films

Elastic strain energy under some conditions provides the major contribution to the total energy of a film growing on a substrate from condensing vapour. Polycrystalline films grown with intrinsic Stress induced by energetic bombardment are expected to show orientations which minimize total energy. Even for a cubic crystal in a non-hydrostatic stress field the energy is a function of the relative orientation of the stress field and the crystallographic axes. The Gibbs free energy is minimized under constant stress and temperature conditions at thermal equilibrium. In this paper we derive expressions for the Gibbs free energy of a cubic crystal in uniaxial and biaxial stress fields and find the conditions under which it is a minimum. The sign of the expression delta = s(11) - s(12) - 1/2s(44) is the quantity which determines the behaviour of a cubic crystal and if negative, predicts that the [111] direction of the crystal will align with the principal stress of a uniaxial stress field and will lie normal to the plane of principal stresses in a biaxial stress field. Experimental evidence is presented which shows that titanium nitride, TiN, which has a negative value of delta, obeys these predictions. If delta is positive, then the [100] direction of the crystal obeys the above rules rather than the [111] direction.