The effect of vapor incidence angle upon thin-film columnar growth

We present a generalized theory for the growth of columnar microstructure in vapor-deposited thin films under the joint influence of a constant uniform deposition flux coming down with an arbitrarily chosen incidence angle, and surface diffusion. The dependences of the zone I to zone II transition temperature, and the characteristic length scales associated with the unstable modes on the deposition angle are predicted. The surface morphology is obtained as a function of vapor incidence angle. For a specific deposition angle, there is a one-parameter family of steady-state surface profiles that corresponds to a range of possible columnar orientation angles, among which only one angle is associated with the tangent rule. These results agree with experimental observations.