The Kepler conjecture in 1611 states that no arrangement of equally sized spheres filling space has a greater average density than that of the face-centered cubic and/or hexagonal close packing arrangements [1]. This conjecture has been only recently proved by the Flyspeck project team [2]. 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. The close-packed (CP) polytypes in which the associated stacking is composed of the CP planes are, therefore, considered to be the crystalline systems realizing the Kepler conjecture. Significant changes in physical properties due to the intentional polytype formation have also been reported in various fields of metallic and semiconductive systems [3-5].Predicting polytype phase stability for a material has still been a long-standing issue in condensed matter physics and/or materials science [6, 7]. This situation stems from the fact that the atomistic interactions on polytype energetics might be surprisingly quite complex and delicate despite the simplicity of their geometrical structure. We have recently presented a theoretical consideration on the CP polytype total energetics using the geometrical analysis for the correlation between interlayer interactions and interatomic ones [8]. Based on the theory presented by the authors [8], we discuss the insight of the total energetics for CP polytypes from the viewpoint of effective atomistic interaction distance in this work. The primary motivation for our study [8] is to analytically grasp how the most basic interactions can be described for the systems whose ground states are composed of the long-periodic polytype configurations other than 2H and 3C ones. The system targeted in the study is, therefore, the polytype systems that total energetics is dominated by the sum of spherical symmetric two-body interactions. In order to investigate the associated polytype energetics as a function of interaction distance, we have introduced the interlayer partial energy model where the total energy constructed from the two-body interactions is projected onto the interlayer interactions in CP polytype structures as depicted in Fig. 1 (i).As shown in Fig. 1 (ii), our paper has theoretically presented that the static total energetics of polytypes can be reasonably explained with two physical indicators, the hexagonality (σH ) and the effective distance (rc ) of interatomic interaction [8]. While the polytype structural energies tend to be degenerate with respect to hexagonality in the system with short interaction distance, the energetic degeneracy manifested by the short-range interactions begins to split for the system with the third neighbor interlayer distance; (a) For the systems with short-range atomistic interactions, all polytypes are energetically degenerate, and the ground state is indistinguishable. Therefore, which polytype appears depends on the nucleation processes and becomes an accidental event. (b) For the systems with medium-range atomistic interactions, only 3C (fcc: σH =0) or 2H (hcp: σH =1) configuration system appears as the ground state. Since 3C (fcc) and 2H (hcp) are observed as the stable close-packed structures in most metallic materials, it is suggested that the interatomic interactions often correspond to the cases around this range. (c) For the systems with long-range atomistic interactions, all the points of total energies as a function of hexagonality are analytically found to be in the inner region of a triangle with vertices of 3C (σH =0), 4H (σH =0.5), and 2H (σH =1), which indicates that the ground state can correspond to any of these three stacking sequences associated.The discussion above suggests that it is possible to infer the effective distance of interatomic interactions, which is important property of each element for producing new polytype-based materials, from the distribution of polytype structural energetics. Our theoretical study can also provide significant insights for creating interatomic models successfully showing the polytypes other than 3C and 2H structures as a ground state, those have never yet been implemented as far as the authors know.[1] J. Kepler, The Six-Cornered Snowflake, 1966 translation by C. Hardie (Clarendon Press, Oxford, 1611), ISBN: 0198712499.[2] T. C. Hales et al., Forum of Mathematics, Pi, 5, E2 (2017). https://doi.org/10.1017/fmp.2017.1[3] E. M. T. Fadaly et al., Nature. 580, 205 (2020). https://doi.org/10.1038/s41586-020-2150-y[4] Z. Fan et al., Nat. Commun. 6, 7684 (2015). https://doi.org/10.1038/ncomms8684[5] E. Abe et al. Philos. Mag. Lett. 91, 690 (2011). https://doi.org/10.1080/09500839.2011.609149[6] C. H. Loach et al., Phys. Rev. Lett. 119, 205701 (2017). https://link.aps.org/doi/10.1103/PhysRevLett.119.205701[7] K. Moriguchi et al., MRS Advances (2021). https://doi.org/10.1557/s43580-021-00044-x[8] S. Ogane et al., MRS Advances (2021). https://doi.org/10.1557/s43580-021-00054-9 Figure 1
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Dec 21, 2021
Vinay Saini + 3
Open Access