Two-dimensional versus three-dimensional constraints in hetero-epitaxy/orientation relationships

Abstract

Molecular dynamics simulations have been performed to compare the orientation relationships (ORs) that develop between Ag films/particles equilibrated with Ni, depending on whether the Ag is subjected to three-dimensional (3-D) confinement by being embedded in Ni, or to 2-D confinement by equilibration with a Ni substrate. Previous results of both simulations and experiments have shown that Ag films equilibrated with planar Ni substrates display a large number of different ORs; in particular, Ag equilibrated on Ni{100} displays an OR of Ag{111} <110> // Ni{100} <110>, often referred to as an oct-cube OR. Here it is shown that a Ag particle embedded in Ni displays an OR of Ag{111} <110> // Ni{111} <110>, i.e., a cube-on-cube OR. It has also been shown that as the confinement of a Ag particle is gradually increased from 2-D to 3-D, by equilibrating a Ag particle on Ni(100) substrates with progressively deeper dimples, a transition in OR occurs from oct-cube to cube-on-cube. This result contradicts the conventional notion which supposes that the ORs displayed in the presence of 3-D confinement (e.g., during phase transformations) will also tend to be displayed in epitaxy on a substrate.

Appendix

As a first approximation, we take the total excess energy of a Ag particle in a facetted dimple substrate to be the sum of the energies of all its interfaces multiplied by their areas. For the purpose of these estimates, we shall use the energies of the Ni/Ag interfaces computed by Gao et al. [10], using the same EAM potentials as those used in the present MD simulations. Those are listed in Table 1.

Table 1 Calculated Ni/Ag interfacial energies (mJ/m²) [10]

Let us begin by comparing the energies of the bottom Ni{100} dimple interface for the cases of an oct-cube versus a cube-on-cube OR. The energy of the oct-cube Ni{100}/Ag{111} interface is 437 mJ/m², and for the cube-on-cube Ni{100}/Ag{100} it is 814 mJ/m². Thus, the dimple bottom interface has a clear energy advantage in the oct-cube versus the cube-on-cube comparison. As the depth of the dimple increases, a progressively larger fraction of the interface will consist of Ni{111} facets. In the case of the cube-on-cube OR, the resulting four Ni{111}/Ag{111} interfaces have the lowest energy, namely 416 mJ/m². This is consistent with the expectation that {111}/{111} interfaces in FCC metals will tend to have the lowest interfacial energy.

In the case of the oct-cube OR, the Ag sides of the four Ni{111} interfaces are all different: They consist of Ag {100} and {221} planes as well as two other high index planes. Of these interfaces, only the energy of the Ni{111}/Ag{100} is known to be 670 mJ/m². The average of all the interfacial energies computed by Gao et al. [10] is about 800 mJ/m². Considering this figure to be a reasonable estimate of the average energy of the Ni{111}/Ag{hkl} interfaces, it is clear that a transition in stability from oct-cube to cube-on-cube OR will occur when the area of Ni{111} will exceed Ni{100} area at the Ni/Ag interface, i.e., somewhere between a dimple depth of 2a_Ni and 4a_Ni.

It is worth noting that the contribution of strain energy to the total excess energies of the two configurations may also contribute to the transition.