This paper investigates the atomic scale buckling of diamond carbon coatings on silicon substrate caused by residual stresses in two orthogonal directions. It was found that different buckling patterns occurred when the ratio of the residual stresses in the two directions were changed. The size of wrinkles increased on going from uniaxial to biaxial compression of the residual stress fields. A telephone-cord like buckling mode took place when the residual stresses were bi-axially equal.