Comparative Studies on Structural Properties and Total Energetics of Nonequivalent Hexagonal Polytypes for a Wide Variety of Metallic Systems
Authors: Miyakawa Taku , Moriguchi Koji 
Published: February 2020

Since the functionalities of materials usually originate from the spatial inhomogeneity in the associated systems, the local inhomogeneity such as defect, surface, interface, and nanostructure often assumes an important role for developing sophisticated materials. In this paper, among the many phenomena related to inhomogeneity in materials, we will report our computational research on the phenomenon called "Polytypism".
Polytypism is a special case of polymorphism when the two polymorphs differ only in the stacking of identical two-dimensional sheets or layers. The polytypes are characterized by a stacking sequence with a given repeating unit along a directional axis (c-axis) and are theoretically possible to have endless permutations of the sequences. Among these polytypes, the crystalline systems composed of close-packed (CP) layers have especially attracted attention on their fundamental and technological properties for many years. While these polytypes are often experimentally observed in wide-bandgap semiconductors, SiC is particularly known to show several hundred polytypes [1]. In addition, the long period stacking ordered (LPSO) Mg alloys with light weight, high specific strength, and high heat resistance are also recently drawing attention as a metallic system with similar polytypism to that in SiC [2].
We have proposed a computational method coupled with three theoretical tools (PGA: polytype generation algorithm; FPC-DFT: first-principles calculations based on the density functional theory; and ANNNI: axial next-nearest-neighbor Ising model), which can make us possible to efficiently investigate the structural energetics for diverse nonequivalent polytypes [1]. In the present study, also motivated by the recent research activities on the LPSO [2], we have systematically evaluated the static energetics of metallic polytypes for a wide variety of elements based on the same computational method proposed by the authors [1]. For all these elemental systems, the atomistic geometry, energetics, and electronic structure for all polytypes with up to the periodic stacking length of L=13 (165 kinds of polytypes in total) have been carefully calculated based on the DFT within the GGA. Using the ANNNI model extracted from the GGA calculations, some trends of physical properties will be illustrated for the tremendous kinds of nonequivalent stacking polytypes based on the efficient PGA.
In the CP structures, three different layers are often denoted by A, B, and C, corresponding to different atomistic positions in the planes perpendicular to the stacking direction (c-axis). These descriptions for nearest neighbor stacking result in the so called “hk” notation, where a letter “h” is assigned to each layer with the same type of layer at both sides (e.g., AXA, BXB, CXC) and a letter “k” is assigned otherwise. Using this “hk” notation, the “hexagonality”(σH
) is essentially defined as σH=nh/(nh+nk), where nh
and nk
are numbers of layers in a hexagonal or cubic environment, respectively [1]. Figure 1 exemplifies the increments of (a) lattice constants in the basal plane (a-axis) and (b) total energies as a function of hexagonality for the nonequivalent polytypes composed of Be, Mg, Pd, and Pt within the present GGA calculations. The figure implies that both physical values are depicted based on those in the 3C (fcc) configuration. We have found that these values have a strong linear correlation with hexagonality in the polytypes for any elemental systems except for La with 4H ground state. The rate of change of both physical values with respect to hexagonality in Mg is found to be significantly smaller than those in other elemental systems. Therefore, the trend should be key factors for the formation of stacking faults, twin boundaries, and also LPSO structures experimentally observed in Mg-based alloys [2]. Focusing on Fig.1 (a), it should be noticed that the a-axis lattice constant decreases with increasing hexagonality in the elemental systems with 3C (fcc) ground state and the opposite trend is true in the systems with 2H (hcp) ground state. Kaneko et al. have experimentally reported that the property of oxygen reduction reaction (ORR) on the catalytic Pt surface is maximized with a 2.0% surface compression strain [3]. Since the compression ratio of lattice constant against hexagonality can be reached around 1.5% in Pt system, the ORR performance may be sufficiently influenced by the stacking faults and/or twin boundaries in the vicinity of surface. Therefore, control of the polytypism in metallic systems may provide interesting functionalities of materials in various fields of solid-state science.
[1] K. Moriguchi, et al., J. Mater. Res. 28, 7 (2013) and references therein.
[2] E. Abe et al., Phil. Mag. Lett, 91, 690 (2011).
[3] S. Kaneko et al., J. Phys. Chem. Lett, 8, 5360 (2017).

Figure 1