Materials synthesis techniques such as Chemical Vapor Deposition, Molecular Beam Eptixay, Sputtering, etc. are prevalent in materials research. Synthesizing materials with these techniques comes with a challenge: how does one determine which subtrate to use?
Epitaxial growth of heterogeneous interfaces requires a fundamental understanding of the substrate material, film material, cleavage planes, lattice mismatches, and resultant stresses and strains. The Materials Project (MP) stores crystallographic information for each material in its database, calculated via First Principles Density Functional Theory. Each material's crystallographic information, in particular the surface termination lattice parameters, is especially useful to find the epitaxial matches between a desired material (film) and a corresponding substrate. The Suggested Substrates tool outputs the Miller Indices of the substrate and the film (target material) termination plane, the minimal co-incident area (MCIA), and Elastic Energy.
Suppose there is two slabs of materials: a film and a substrate. The MP database the lattice parameters for both the film and substrate bulk crystal. Slabs are generated by cleaving plane from from its bulk crystalline form. The cleaving plane is described by the Miller Index notation (e.g. Si(111)). The cleavage plane (equivalent to termination plane) is a surface; all of its sites can be described by a unique 2D lattice. Therefore, interfacing film and substrate slabs geometrically implies the mapping of their respective 2D lattices. If the film and substrate lattices match, it is described as an epitaxial match, with a 2D superlattice that describes the interfaced lattice. This 2D superlattice contains a set of primative translation vectors
that describes both sides of the slab and their termination surface. Fig 1. below shows a schematic of how a new lattice is created at the interface of two slabs. Note: since it is a 2D representation, there is a 1D superlattice at the interface.
Figure 1. Lattice matching between Si(111) and Al
(101) faces. A cell made of 21 sapphire unit cells has almost exactly the same dimensions as a cell made of 40 silicon unit cells. This can be described by a new superlattice ('supervector') at the interface. Figure from Zur and McGill 
Finding the epitaxial lattice match between hetergenous interfaces implies finding a 2D superlattice that both sides must satisfy (or approximately satisfy). However, any interface can contain multiple sets of solutions to the primitive translation vectors
that still satisfies the 2D superlattice. As such, the goal is to look for the smallest possible values of the primative primitive translation vectors
, also known as the reduced primitive translational vectors. The reduced primitive vectors has a unique solution for
, unlike the general primitive translation vector set. Zurr and McGill proposed the following algorithm to find the reduced lattice set:
- 1.look for being the shortest possible nonzero vector of the superlattice
- 2.look for being the shortest possible nonzero vector of the superlattice that is linearly independent of
- 3.find angle between vectorsthat is non-obtuse.
The algorithm above is also shown in the flowchart in Fig 2. By leveraging computational resources and data from MP, it becomes possible to scan across all different cleavege planes for both the substrates and films to determine a set of reduced lattice planes, and therefore the epitaxial matches.
Most heterogenous interfaces will experience lattice mismatches. The following ratio describes the unit cell matching between the film and the substrate:
corresponds to the unit cell areas of the original lattice of the film and substrate, and
correspond to an integer value that satifies the unit cell areas being matched on the superlatice by the film and substrate. For lattice mismatches, we can set an upper limit for
, where it must satify
. And therefore:
 A. Zur and T. C. McGill , "Lattice match: An application to heteroepitaxy", Journal of Applied Physics 55, 378-386 (1984) https://doi.org/10.1063/1.333084
 Hong Ding, Shyam S. Dwaraknath, Lauren Garten, Paul Ndione, David Ginley, and Kristin A. Persson ACS Applied Materials & Interfaces 2016 8 (20), 13086-13093 DOI: 10.1021/acsami.6b01630