PHYSICAL REVIEW B 68, 134211 ͑2003͒ Coincidence of reciprocal lattice planes model for quasicrystal-crystal epitaxy E. J. Widjaja* and L. D. Marks Department of Materials Science and Engineering, Northwestern University, Cook Hall, 2220 Campus Drive, Evanston, Illinois 60208-3108, USA ͑Received 12 March 2003; revised manuscript received 10 June 2003; published 24 October 2003͒ A coincidence of reciprocal lattice planes model was developed to calculate the interfacial energy in quasicrystal-crystal epitaxy. This model allows a quantitative description of the interface as opposed to previ- ously employed qualitative models that consider symmetry relations and alignment of rotation axes. Compu- tations were carried out on several types of quasicrystal-crystal systems, namely, the crystalline structures on various surfaces of single quasicrystals ͑Al-Cu-Fe, Al-Ni-Co, and Al-Cu-Co͒ caused by ion bombardment, the crystalline thin films grown on quasicrystal substrates, and the quasicrystalline thin films grown on crystalline substrates. This model can also be extended to include quasicrystal-quasicrystal epitaxy. DOI: 10.1103/PhysRevB.68.134211 PACS number͑s͒: 61.44.Br, 05.70.Np, 68.35.Ϫp I. INTRODUCTION served, even for the same surfaces, for example, the Al- Ni-Co 2D surface.3 It should be noted that all of the methods Shortly after the discovery of quasicrystals, it was estab- employed to date do not take into account the role of the lished that the structure of icosahedral Al-Mn undergoes a interfacial energy. Thus, there is a need to develop an im- phase transition when exposed to irradiation of energetic proved model that will enhance the fundamental understand- particles.1 Bombardment with Arϩ ions at room temperature ing of quasicrystal-crystal epitaxy. In this paper we describe transforms the quasicrystalline surface into a crystalline cu- and employ the coincidence reciprocal lattice planes ͑CRLP͒ bic structure due to preferential sputtering of the Al atoms model to predict and explain the orientation relationship in from the surface. The resultant crystalline structure has a quasicrystal-crystal epitaxy through the inclusion of the in- specific orientation relationship with respect to the quasicrys- terfacial energy. talline substrate. Similar behavior has been observed in other 2,3 quasicrystal systems such as decagonal Al70Ni15Co15 , 4,5 6,7 II. COINCIDENCE RECIPROCAL LATTICE PLANE icosahedral Al65Cu20Fe15 , decagonal Al70Cu15Co15 , and 8 MODEL FOR QUASICRYSTALS decagonal Al75Ni10Fe15 . Many studies have been devoted to determining the orientational relationship between the quasi- A very simplistic view of the epitaxial growth places im- crystal and crystal phases by means of transmission electron portance upon the principle that the coherent overgrowth of microscopy,3,4,6,7 low-energy electron diffraction,5 secondary crystal material Y on crystal X is likely to occur if some electron imaging,2 and reflection high-energy electron undistorted crystal plane of Y can be laid down on top of the diffraction.9–11 Recently it was shown that crystalline thin- exposed face of X, in such a way that a large fraction of the film growth on quasicrystalline substrates, and vice versa, Y atoms can be made to coincide with the sites of X atoms. It exhibits the same phenomena.9,12 can be further understood that the greater the number of co- A majority of these works, however, explain the observed incidences per unit area, the lower the energy of the resulting orientation of the structures from a stereographic projection interface will be. This basic principle is the backbone of the through a description of the rotation axis alignment. While coincidence-site-lattice ͑CSL͒ theory which was first inves- this is a valid approach for describing the orientation, this tigated by Friedel,14 and later explored by Ranganathan15 and method offers no insight into the fundamental mechanism applied to cubic lattices by Grimmer.16–19 The CSL concept behind the preferred orientation. Shimoda et al.,9 Zurkirch was implicitly incorporated in the approach of superimpos- et al.,2 and Bolliger et al.13 described their findings via an ing real-space atomic models employed by Shimoda et al., atomic model of the two-dimensional interface between the Zurkirch et al. and Bolliger et al. Nonetheless their approach quasicrystal and crystal phases, which is obtained by super- lacks the theoretical mathematical expressions that include imposing the surface structures. The validity of these atomic interfacial energy. In earlier work by Warrington et al.,20 a models relies heavily on a real-space structural model for the CSL theory was applied to investigate grain boundaries in quasicrystal system, which may not be readily available or, icosahedral quasicrystals. The CSL method employed is an in some cases, may not be correct. Furthermore, their models N-dimensional CSL ͑Ref. 21͒ due to the quasiperiodic nature fall short of the long-range fitting for the superimposed struc- of quasicrystals. The focus of this work is to determine the ture since the misfit dislocations and interface relaxations are quasicrystal rotations in order to classify disorientation and ignored. Shen et al.5 explained the orientation relationship to find the degeneracy of different values of the coincidence between different surfaces of icosahedral Al-Cu-Fe system index. However, such N-dimensional approach does not offer and its cubic phase via a structural model of cubic close the potential to solve the problems associated with interfacial packed and icosahedral packed clusters. This approach al- energy in the systems; hence a three-dimensional ͑3D͒ struc- lows only one orientation for any given system; this is in tural description needs to supplement the hypercrystal contrast to observations where multiple orientations were ob- description.22 0163-1829/2003/68͑13͒/134211͑10͒/$20.00 68 134211-1 ©2003 The American Physical Society E. J. WIDJAJA AND L. D. MARKS PHYSICAL REVIEW B 68, 134211 ͑2003͒ Given that the nature of the system is quasiperiodic, the measured from an origin fixed on a lattice point of its respec- Ј 3D reciprocal space of quasicrystals can be mathematically tive crystal, Rs and Rs are the associated surface lattice vec- described through a projection from a periodic higher dimen- tors, and ks is the component of k parallel to the surface. sion. In addition to that, the reciprocal space also can be We will now allow elastic displacement of atoms near the directly deduced from electron-diffraction patterns. The 3D interface with a displacement of approach in a CRLP model for a quasicrystal-crystal inter- face enables the calculation of the interfacial energy in a ͒ϭϪ ϩ ␬ ͒ϩ ␬ ͒ Ϫ similar approach to a crystal-crystal epitaxy. We employ this F͑R ͚ 2͓D␬ sin͑ R C␬ cos͑ R ͔ Co , ␬ • • reciprocal space approach to explain the experimentally ob- ͑6͒ served orientation preference. The improved CRLP model is based on the crystal-crystal epitaxy model previously devel- ␬ 23 where is a vector corresponding to allowed distortion com- oped by Fletcher, which will be shown also to be appli- ponents. Thus, the interfacial energy described in Eq. ͑4͒ cable to quasicrystal-crystal epitaxy. The original CRLP becomes model by Fletcher and Lodge24 was exploited as the starting point, as described in the following paragraphs. 1 ϩ Ϫ The potential energy Vo of a Y atom at position r outside ETϭ ͚ ͚ ͵ ␷͑k͒exp͓ik ͑RЈϩFЈϪRϪF͔͒dk. o 3 • a plane face of a X crystal with atomic positions R, Vo(r) 8␲ RЈ R ͑ ͒ can be written in terms of its Fourier components Vo(k)as 7 N Repeating the same procedure applied to Eq. ͑4͒ to result in ͑ ͒ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ Vo r Vo k exp ik r dk, 1 Eq. ͑5͒, Eq. ͑7͒ can be transformed for a simple basis into 8␲3 • 1 1 ͑ ͒ϭ Ϫ ␷͑ ͒ ͑Ϫ ͒ ͑ ͒ E ϭ ͚ V ͑k ,B ͒exp͑ik B ͒ Vo k ͚ k exp ik R , 2 o o s 3 s• s N R • 2␲AAЈ ks where N is the ͑infinite͒ number of atoms in the crystal and ϫͭ͟ J ͑2k D␬͒J ͑2k C␬͒ͮ ␷ ␷ 0 0 (k), the Fourier transform of the atomic potential (r), is ␬ • • given by ϫͭ͟ J ͑2k D␬Ј͒J ͑2k C␬Ј͒ͮ 0 0 ͑ ͒ϭ ͵ ␷͑ ͒ ͑Ϫ ͒ ͑ ͒ ␬ • • v k r exp ik•r dr. 3 ϱ ͒ ϩ Jn͑2ks D␬ Assuming the crystals to be undistorted, the total interac- ϫͭ ␦ ␦ Јϩ ͚ ͚ ͫ • ks ,qs ks ,q ͒ T s nϭ1 ␬ J0͑2ks D␬ tion energy between two crystals, Eo , can be expressed • through the summation of Vo(r) over all the atomic position J ͑2k C␬͒ RЈ of the Y crystal, resulting in ϩ n n s• ͬ␦ ␦ i ϩ ␬ Ј Ј ͑ ͒ qs n ,q ks ,q J0 2ks•C␬ s s Tϭ ϩ ͑ Ј͒ ϱ E ͚ Vo R ϩ J ͑2k D␬Ј͒ J ͑2k C␬Ј͒ o n s• n n s• RЈ ϩ ͚ ͚ ͫ ϩi ͬ ϭ ␬ ͑ Ј͒ ͑ Ј͒ n 1 J0 2ks•D␬ J0 2ks•C␬ 1 ϩ Ϫ ϭ ͚ ͚ ͵ ␷͑k͒exp͓ik ͑RЈϪR͔͒dk, ͑4͒ ␲3 R • 8 RЈ ϫ␦ ϩ ␬ ␦ ϩ ͮ . ͑8͒ q n ,qЈ k ,q s s s s ••• where the plus and minus signs on the summations indicate that it extends only over the upper and the lower half space, respectively. In Fletcher’s original derivation, only a simple basis for ␦ The equation to describe the interfacial energy per unit the unit cell was employed, and the functions arose via a sum over all atoms in the interface, i.e., area, Eo , can then be simplified to 1 ϩ Ϫ ͑ ͒ϭ ␦ ͑ ͒ ϭ ␦ ␦ ͑ ͒ ͵ ␷͑ ͒ ͚ exp ik R N k,q 9 Eo k ,q k ,qЈexp iks Bs ͚ ͚ k s s s s • R • 2␲AAЈ Ј R3 R3 ϫexp͓ik ͑RЈϪR ϩB ͔͒dk , ͑5͒ for N atoms.