Dear Chandan,                                                                                                                        first of all I agree with the answer given by Dr Nolze. However, I think that your ideas about the phenomenon we call "epitaxy" are not adequate to give a complete understanding of this complex geometrical-physical property arising whe two crystalline materials come in contact. To improve the quality of your knowledge on these topics, I would like to suggest the following papers:                                                                       1) I.V. Markov, Crystal Growth for beginners, Worrld Scientific, 1995                              2) - R. Kern, G. Le Lay, J.J. Métois  "Basic Mechanisms in the early stages of the Epitaxy" in : Current Topics in Materials Science, vol. 3 Ed. E, Kaldis, pg 131-419, North Holland 1979                                                                                                                     All the best !                                                                                                                               Dino

Epitaxy only happens if the translation symmetry along the surface is comparable for both phases. Therefore, epitaxy is the exception and not the rule since practically all phases have different lattice parameters and do not match at all. Thus, ideal epitaxy is nearly excluded. In semiconductor industry one uses doping to adjust lattice parameters in order to get a better or nearly perfect match. But this has very strong limitations. Epitaxy is also a 2D effect, i.e. the translation symmetry has to match only for this specific surface orientation. If a cubic phase grows on a hexagonal substrate it typically means: (111) paralle to (0001). the smmetry plays an important role. Theoretically, you could also selct another surface plane, e.g. the two-fold axis of the lattice, which might match to the cubic (110) or even (001) if the c/a ratio of the hexagonal phase is very special. 

In 3D epitaxy is called topotaxy and practically means the same. The translation lattices have to match now in 3D. This does not mean a match of 1:1, but somtimes e.g. also 1:3 works (i.e. a huge cell with a lattice parameter 3-times bigger matches to the length of three unit cells of the other phase).

If such geometric relationships do not exist a crystal starts to grow and simply uses any surface position as nucleous. This also happens during epitaxy, but since all newly formed crystals have the same orientation you can somtimes only find small angle boundaries if they grow together. For perfect crystals even they disappear. However, if they don't have any correlation to the substrate grain they grow randomly and thus a polycrystalline material forms despite even an expensive  single-crystalline substrate orientation.

Dear Chandan,                                                                                                                        first of all I agree with the answer given by Dr Nolze. However, I think that your ideas about the phenomenon we call "epitaxy" are not adequate to give a complete understanding of this complex geometrical-physical property arising whe two crystalline materials come in contact. To improve the quality of your knowledge on these topics, I would like to suggest the following papers:                                                                       1) I.V. Markov, Crystal Growth for beginners, Worrld Scientific, 1995                              2) - R. Kern, G. Le Lay, J.J. Métois  "Basic Mechanisms in the early stages of the Epitaxy" in : Current Topics in Materials Science, vol. 3 Ed. E, Kaldis, pg 131-419, North Holland 1979                                                                                                                     All the best !                                                                                                                               Dino

Thanks Prof. Dino for your answer. I have gone through Markov's article and several others. Still, the adequate reasons need to be explored. 

Dear Chandan,   please find a supplementary consideration:                                            - When your substrate is made by the surface of a single crystal, an epitaxy can occur only if there is a lattice coincidence between the 2D lattice of the sustrate-surface and a 2D lattice of a given suitable surface of the deposit-crystal. Under this condition, the adhesion energy between the two crystal phases reaches its maximum value. An important caution is needed : the lattice coincidence is not between the lattice parameters of the host and guest crystals, but between the 2D lattices of the surfaces coming in contact !                                                                                                              - On the contrary, if your substrate is polycrystalline, every grain of the substrate will exhibit a different orientation with respect to the adjacent grains and, consequently, each grain exhibits a different surface to the incoming crystal-deposit. This implies that the lattice coincidences at the  host-guest interface  should occur only by chance and the related host-guest adhesion energy  varies according to the  grain orientation. Since the probability of nucleation of the crystal - deposit depends on this adhesion energy, the quality of the deposit should be very hazardous.                                                    Thanks for your kind attention !                                                                                         Dino