RESEARCH ARTICLE. In i-znsc (like other quasicrystals of the i-cacd class) the innermost shell of the icosahedral

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1 Philosophical Magazine Letters Vol. 00, No. 00, Month 200x, 1 8 RESEARCH ARTICLE ENERGETICS AND DYNAMICS OF CAGED Zn 4 IN i-sczn M. Mihalkovič a and C. L. Henley b, a Inst. of Physics, Slovak Academy of Sciences, Bratislava, Slovakia b Dept. of Physics, Cornell Univ., Clark Hall, Ithaca, NY USA (xxx June 2010) In i-znsc (like other quasicrystals of the i-cacd class) the innermost shell of the icosahedral cluster is a Zn 4 tetrahedron, which thus breaks the symmetry of the outer cluster shells. We investigate theoretically the dynamics of individual tetrahedra, using interatomic pair potentials (fitted from an ab-initio database) and molecular dynamics (MD). [This includes the formulation of an effective Hamiltonian written in terms of a rigid-body rotation representing the state of each tetrahedron.] We characterize the minimum-energy orientations of a tetrahedron, as well as the paths of the transitions between minima (reorientations). The velocity autocorrelations were evaluated for the tetrahedron atoms in the MD dynamics; the corresponding spectral density S(ω) is fairly well fitted by a simplified model in which each atom hops in a double well. Keywords: molecular dynamics; effective Hamiltonian; tetrahedron cluster; Sc Zn; vibrational density of states 1. introduction Icosahedral i-cacd and isostructural i-cdyb [1 3], and the newly discovered i ScZn [4], are the only stable and long-range-ordered binary quasicrystals; the same structure class contains many ternary quasicrystals, and there are additional quasicrystal approximants in Sc-Zn, made of large icosahedral clusters, which are the focus of this paper. The structure contains large icosahedral clusters with tetrahedra of Zn (or Cd) atoms at their centers, which breaks the cluster s symmetry. Interactions of the tetrahedra in different clusters give orientational ordering transitions in the approximant crystals (as established from diffraction) and, presumably, in the quasicrystal phase too. Thus, as a function of pressure (up to 5 GPa), Cd-Yb has a complex phase diagram with six phases [5]. The ordered low-t structure of 1/1-Zn 6 Sc was determined in Ref. [6]. We already have a good structural understanding of approximants such as the Ca 13 Cd 76 crystal [8], using a model [9] based on canonical cells [13]. In this work, we restrict ourselves to the 1/1 cubic approximant, the simplest one, except as noted. Our MD simulations are based on classical pair potentials, empirically fitted to a database of Sc-Zn alloy (and elemental) structures; these include Friedel oscillations representing long-range electronically mediated interactions [11] (but cut off at 7 Å). Corresponding author. fyzimiha@savba.sk ISSN: print/issn online c 200x Taylor & Francis DOI: / YYxxxxxxxx

2 2 M. Mihalkovič and C. L. Henley 2. Framework for tetrahedron effective Hamiltonian The potential energy for tetrahedron orientations was first modeled (based on ab-initio energies) in the Ca-Cd system [9 11]. The degrees of freedom for Zn 4 tetrahedron i can be labeled by a rigid-body rotation Ω i, applied to a reference tetrahedron (thus representing three angle parameters). The tetrahedra are not really rigid or regular, but we assume that having constrained the overall rotation to a particular orientation, the deformations are forced by the surrounding cage and do not represent low-energy degrees of freedom. Our system of rotor degrees of freedom is reminiscent of the CN ions in the rock-salt crystal KBr 1 x (CN) x [14, 15]. We define an effective tetrahedron potential V tet (Ω 1, Ω 2,...) as the energy of the best state constrained to that combination of orientations, allowing relaxations of the surrounding atoms as well as distortions of the tetrahedra (but not rotations, since that is just represented by a different value of Ω i ). This is analogous to spin Hamiltonians for local magnetic moments. (We must recall, though, that our interactions are anisotropic, i.e. they depend not only on the orientations of the spins relative to each other, but also relative to the surrounding lattice.) The tetrahedron potential breaks up into one-body terms (analogous to crystalfield effects on a spin moment) as well as pair interactions of adjacent tetrahedra. The one-body part splits into V ico (Ω) + V cub (Ω) where V ico represents the icosahedral symmetry of the cluster s outer shell, and a (comparable) contribution V cub reflects the outer shell s distortion due to the cubic symmetry of the 1/1 approximant. (In larger approximants or in the quasicrystal, V cub gets generalized to V env (i; Ω i ), depending on the local environment of cluster i in the network of clusters.) The pair term is V int = ij v(ω i, Ω j ); this is the interaction that drives structural ordering transitions in CaCd approximants. In ScZn 6, at least, v(ω i, Ω j ) is probably mediated elastically by displacements of intervening atoms, rather than by direct (electron-mediated) interactions of atoms in the respective tetrahedra. To more clearly separate the behavior of individual Zn 4 clusters, we made a single Zn 4 construct, in which all but one Zn 4 in the sample is replaced by a (large) Sc atom. [There are 16 clusters in the supercell we used.] In this sample V int is missing and only the single-body potential is acting. We found that actually the 1/1 approximant with all Sc centers is stable against competing phases, and remains stable over a wide range of fraction Sc substituted. This might explain the compositional variability known to occur in the actual alloys (C. P. Gómez, personal communication). The relaxed lattice constant of our single-zn 4 construct is somewhat smaller than the all-zn 4 version. As we shall see, a Zn 4 tetrahedron has very little clearance to turn and so the energy barriers are highly sensitive to lattice constant. (For the same reason, one would expect that substituting different rare earth elements in the CaCd family can lead to quite different potential surfaces for the tetrahedron rotation, which is consistent with the different atomic distributions of the Cd 4 reported by Gomez [16].) For all data reported here, we constrained the lattice constant to be the same as the all-zn 4 material. We found little qualitative difference in behavior between the single-zn 4 and all- Zn 4 samples; hence, we inferred that v(ω, Ω ) is small in ZnSc, compared to the one-body terms. In other words, the latter are the main part of the potential and the two-body term is in the Zn 6 Sc case a small perturbation. Ref. [10] already noticed this smallness: in an ab-initio calculation comparing different ordering patterns, they found the energy difference in the Zn 6 Sc case was 1/6 as large as in the Cd 6 Ca case.

3 ScZn inputs 3 Henceforth in this paper we neglect interaction terms. Then the tetrahedron problem, as studied in the rest of this paper, breaks down into two parts: (i) which (discrete) orientations are most stable for a tetrahedron? (ii) What are the energy barriers between these wells, and what is the resulting (activated) dynamics of the orientational transitions? 3. Optimal Orientations Interatomic pair potentials are a fruitful starting point to understand of the potential surface for Zn 4 rotations. The first step is visualizing the Zn potential Φ(r) felt by a test Zn atom (omitting the other atoms in the Zn 4, and fixing the other atoms that encage it). This is, Φ Zn (r) = i φ Zn,i (r r i ), (1) where i runs over all atoms excluding the Zn 4, and φ Zn,i (R) is the pair potential between a Zn and the species found on site i. (In practice, the important atoms i in (1) are the cluster s three shells of 20 Zn+12 Sc + 30 Zn. There might be a tail in φ Zn,i (R) with Friedel oscillations but in contrast to the Al-TM and Frank- Kasper families of quasicrystal such long-distance interactions are not crucial in the Ca-Cd family [17].) Thus Φ Zn (r) is defined like the electrical potential felt by a test charge in electrostatics, and is exactly analogous to the Al potential applied to d-alconi in [18]; in that material, the potential function helped us locate the preferred configuration of Al in channels, which play exactly the same role in Al-TM decagonals that Zn 4 (or Cd 4 ) clusters play in the i-cacd class of quasicrystals. We will assume (for the moment) that atoms of Zn 4 are confined to a sphere of radius r T = 1.65 Å, which is the mean radius we observed in simulated ZnSc. The salient features of the potentials are: φ ZnSc (R) has a strongly attractive well at R Sc 3.0Å; φ ZnZn(R) has a shoulder, rather than a well, with a minimum hardcore distance at R 2.4Å and an optimal distance R Zn 2.8Å. The cage surrounding Zn 4 is shells of 20 Zn, 12 Sc, and 30 Zn respectively along the icosahedron s 3- fold, 5-fold, and 2-fold axes, at radii r 3 = 3.3Å or r 3c = 3.8Å, r 5 = 4.9Å, and r Å. Note r 3c refers to the eight cubic 111 directions, whereas r 3 refers to the 12 other 3-fold directions; this is the main cubic anisotropy and is crucial for the observed Zn 4 behavior. It is convenient to discuss Φ Zn (r), as intercepted by the sphere at r T. The fivefold positions on this sphere are favorable for Zn in light of the pair potentials, since the distance to Sc(5-fold) is 3.25Å R Sc, and the distance from a tetrahedron Zn on this sphere to a caging Zn(3-fold) is Å, assuming the latter atom as well as the Zn 4 center of mass stay fixed. Indeed, the calculated Φ Zn () has its minimum in the 5-fold direction; it is almost as low in the 111 directions, but is maximum along the other 3-fold directions. It would seem that the Zn orientations are practically excluded from the 12 triangles of the icosahedron which are not centered by a 111 type direction. Although Φ Zn (r) as specified above is a useful qualitative guide to the single-body potential for Ω, it is not quantitatively right. The reason is that the 20 caging Zn on 3-fold axes move in and out freely in response to the Zn 4 configuration. (This is evident in movies we generated of those motions, see Fig. 1, below.) Thus, the true orientational potential refers to a cooperative motion of the 20 Zn cage along

4 4 M. Mihalkovič and C. L. Henley Figure 1. Three frames (at 7.04, 12.60, and ps) from a movie of Zn 4 from MD simulation. The beginning and end are two standard 2mm configurations, aligned with different cubic axes; the middle one shows a secondary state, related to the transition state. The four large spheres are the tetrqahedron atoms; the dodecahedral cage of 20 Zn atoms is shown by small spheres. with the tetrahedron. 1 Given the Zn potential Φ Zn (r) as described above, the optimal orientation for a Zn 4 would be to place every Zn near a fivefold axis, but that is not possible. The angles between nearest fivefold directions is 63, whereas that between second nearest fiveefold directions is 117. The angle between tetrahedral directions is 109, so the that 63 is much too close for a Zn pair, and 117 is slightly too far. We found, in fact, two kinds of local minimum. The dominant minimum orientation has a 2mm symmetry around one of the cubic axes say (001). There are six degenerate configurations (one for each lattice direction). We can discretize configurations by listing the closest 5-fold direction for each Zn. In our standard setting Zn(1) and Zn(2) are near to (±τ, 0, 1) directions which are 117 apart, whereas Zn(3) and Zn(4) are associated with (0, ±1, τ) directions which are 63 apart. As expected, we observed a distortion in which the Zn(3) Zn(4) distance is 0.05Å closer than Zn(1) Zn(2) [out of 2.6Å]. A secondary minimum orientation follows the idea that one can place (say) Zn(1,2,3 ) near to 5-fold axes that are 117 apart, e.g. along ( τ, 0, 1) and cyclic permutations, and put Zn(4 ) along (1, 1, 1) which is relatively low energy (since r 3c is larger). Actually, Zn(4 ) deviates [perhaps so it is closer to one of the adjacent 5-fold axes], splitting into 3 minima, so configuration has degeneracy 24. In the secondary minimum, all four Zn atoms are inequivalent by symmetry (point symmetry 1 ). Notice this secondary minimum is not so far from the dominant configuration we described since Zn(1,3 ) are the same axes as Zn(1,3), and Zn(4 ) is the same one as Zn(2). Relative to the dominant configuration, the secondary kind is rotated around the cubic axis (ẑ) by an angle of ±α. There is an additional rotation around the ŷ axis by an angle ±β; these angles are of order 10. The transition from the primary minimum to the secondary one mentioned mainly requires that Zn(4)= (0, 1, τ) rotates to Zn(2 )=(1, τ, 0). The first evidence for our claims about the minima, particularly the dominant configuration, comes from examining 200ps total of movies (taken at T = 300K), as excerpted in Fig. 1. This also shows that transitions between the dominant states occur via the nearest secondary states. The lifetimes till a transition are, very roughly, 5 10 ps out of the dominant states and 1 2 ps out of the secondary ones. 1 One suspects this large motion of Zn (or Cd in Ca-Cd compounds) is related to the extremely anomalous c/a ratios of elemental Zn and Cd; it is surprising that pair potentials capture such effects.

5 ScZn inputs 5 "all Zn4" x z x y z y "single Zn4" x z x y z y T=210K, 2x2x2 supercell Figure 2. (COLOR ONLINE) The probability density for Zn atoms projected in the (001) direction. The center of each blob has the maximum density, which decreases outwards, contours being shown by alternately heavier and lighter shades [online: by colors red to blue]. Note that the instantanesous configurations are folded back into a standard configuration in an asymmetric unit of the 4-atom configuration space, according to which pair of atoms most closely follows a cubic axis. The second evidence is in the distribution function for the Zn positions, Fig. 2. This clearly shows a large and a small peak (the latter at the tips of the arcs attached to the main peak in the middle frames). The orientation of these arcs shows a rotation from the optimum orientation involving a combination of two axes. Gomez and Lidin [16] showed that (at not too low T ) the tetrahedra in 1/1- Cd 6 RE sit in an asymmetric orientation, as seen in the refined X-ray structure by a host of split positions. As shown in [7], Fig 3, the electron density maps for the Cd 4 tetrahedron have icosahedral symmetry for RE=Tm and Lu, cubic symmetry for RE=Tb, and in-betweem for RE=Ho or Er. The cubic symmetry represents a tetrahedral orientation of the Cd 4 with all four atoms along a 111 axis, and having point group; however, in the ordered cases, there is a small rotation from this orientation. This work was followed by Ref. [10] which is, in effect, an ab-initio energy-guided refinement of the observations in [16] for the cluster center in the 1/1 Cd 6 RE approximant. Our optimum 2mm configuration is rotated 45 around a cubic axis, relative to the tetrahedral configuration; it agrees with the orientation called LC by Ref. [12]. The low-temperature, ordered phase of Cd 6 Ca has a small rotation from this configuration, both in experiment and in theory; this retains a 2-fold symmetry and thus differs from our secondary minimum, which was rotated in two directions. 4. Long-time dynamics We now seek to compare the simulated dynamics with neutron-diffraction measurement of the vibrational density of states ; this is defined as G(ω) lim q 0 ω 2 S(q, ω) (2) q 2

6 6 M. Mihalkovič and C. L. Henley where S(q, ω) is the dynamic structure factor. It can also be shown (in a sufficiently isotropic system) that G(ω) is the Fourier transform of the velocity autocorrelation C v (t). We calculated G(ω) from the MD time histories using the package nmoldyn, as shown in Fig. 3(a). The MD simulation used a time step of 4fs in a (long) total run of 20 ns, at temperatures T = 300K and 210K (by comparison the freezing temperature is 140K). Most of the curve shows an excellent agreement with the independently computed harmonic phonon spectrum, as expected. (G(ω) is called the vibrational density of states since, if phonons account for all the slow dynamics, G(ω) is the phonon frequency spectrum.) In the 1/1 Zn 6 Sc approximant, the low-frequency dynamics is dominated by cluster reorientations, which are thermally activated transitions over barriers in the orientational potential, among the (discrete) optimum orientations. Thus the system stays in a given discrete well (effectively a constant state) for a long time, and randomly with a given rate has abrupt transitions to another discrete state. Such dynamics with states α = 1,..., m is realisically modeled by a discrete master equation in terms of an m m matrix of transition rates Γ αβ. Assuming for simplicity that the states are all equivalent, detailed balance requires Γ be symmetric, and we can diagonalize Γ to find eigenmodes v α (µ) with eigenvalues λ µ. We need to evaluate the long-time behavior of C v (t) The probability of transitioning from α at β after a time t is given by P β α (t) = µ v (µ) α v (µ) β e λµt. (3) If a Zn coordinate x(t) has values x α in the respective discrete states, then its autocorrelation is C x (t) x(0) x(t) = αβ P β α(t)x α x β. Finally, the desired velocity autocorrelation is C v (t) = d2 C x (t) dt 2 (4) Notice we do not need to know the (short) typical duration of a transition τ tr ; indeed each velocity factor is 1/τ tr during a transition, but this is canceled by two factors τ tr for the probability that both times in the correlation happen to fall during a transitione event. Even in our simple four-atom system, the number of discrete states m is quite large. Notice that we require the velocity autocorrelation, of each atom with the same atom at a later time. Hence we must distinguish all permutations of the atoms labels: each Zn 4 configuration actually represents 4! distinct states. (Half these permutations correspond to improper rotations which are inaccessible, as the Zn 4 is approximately a tetrahedron even in transition states.) To highlight the qualitative behavior of G(ω) and obtain a convenient fitting form, we consider the simplest case of a double well, m = 2, with coordinate values ±x 0. Then there is just one nontrivial mode (the other has λ 0 = 0) and G(ω) = 2 π x 2 [ 0 1 ] 1 τ 1 + ω 2 τ 2. (5) The interpretation is that the transitions contribute a constant ( white noise ) except this rolls off to zero for ω < 1/τ. (The upper cutoff is 1/τ tr.) The red line in Fig. 3(b) shows the excellent fit to (5) at T = 100K. The fit yields τ = 27 mev ps; a fit for T = 210K data gave τ = 77 mev 1 51 ps.

7 November 1, :41 Philosophical Magazine Letters ScZn 7 ScZn inputs Figure 3. (a). The vibrational partial DOS G(~ω), for the entire range of frequencies, projected onto the Zn4 atoms, is shown in gray; tetrahedron motions are frozen out at T = 100K, so that plot lacks the low-frequency tail seen in the T = 300K plot. Both solid black lines are the direct calculation of the partial DOS assuming harmonic vibrations around the relaxed structure (at T = 0, using the same supercell as the MD simulation). (b). The very low frequency portion of G(ω) at T = 100K, showing the Zn4 partial contributions only (which dominate it). The lower scale is ω in mev. The red line shows the good fit to the form (5). (The time scale for each reorientation transition, e.g. as estimated from movies in Sec. 3, is shorter. The reason is that a single reorientation moves each atom by just a fraction of the angle around the sphere, hence several such steps are required to decorrelate the orientation.) Note this form does not fit simulation results for ZnSc in the case of the 2/1 approximant. Thus, whereas there seems to be a single relaxation time scale in the 1/1 case, there is a wide range of times in the 2/1. This makes sense since the local environment in the 2/1 canonical cell tiling is less symmetric than in the 1/1 case and hence is expected to have a variety of reorientation barrier values. 5. Discussion Our conclusion that the the inter-cluster Hamiltonian is the smallest term in the orientational Hamiltonian was conditioned on constraining the unit cell to stay cubic. We believe the distortion of the cell which could also be viewed as the long-range part of the interaction is essential for stabilizing the orientationally ordered phases in the real materials The tetrahedron in the Ca-Cd alloy class is not the sole example of an asymmetric cluster in an icosahedral cage. The high-quality, face-centered icosahedral quasicrystals in the Al-transition-metal (Al-TM) class, such as i-alpdmn and ialcufe, are built of pseudo-mackay icosahedron (pmi) clusters which are centered by an Mn atom with an irregular shell of seven close neighbors. The orientational order and dynamics of the MnAl 7 cores calls for the same sort of analysis as the Zn4 tetrahedron, but greatly complicated by (i) the variability of the configurations of the Al7 shell (ii) an additional three Al neighbors inside the pmi which enforce correlations of adjacent pmi clusters (iii) possible covalent bonds between these Al atoms and Mn atoms in certain sites of the cage. Thus, the Cd 4 /Zn4 tetrahedron may serve as a prototype for later studies of the pmi clusters. Furthermore, as a speculation, we note that although an atomistic, energy-based scenario was recently demonstrated for the emergence of matching rules in decagonal Al-TM quasicrystals [19] (more precisely, in a hypothetical quasicrystal with a special composition), no specific microscopic basis was ever proposed for matching rules in icosahedral quasicrystals. Now, most formulations of matching rules involve a set of objects (tiles or clusters) having complete symmetry so far these would implement, at best, the random-tiling scenario of quasicrystal order but

8 8 REFERENCES furthermore a set of markings which spoil the symmetry and specify additional constraints that are essential in selecting the perfectly quasiperiodic states out of a large ensemble of good packings. It seems to us that the irregular central clusters in the i-cacd and i-alpdmn types of structure are the most plausible way that such markings might be implemented atomistically. Acknowledgements Supported by U.S. Dept of Energy grant DE-FG02-89ER (CLH) and by Slovak VEGA 2/0157/08. CLH thanks the Slovak Academy of Sciences for hospitality and the Slovak National Scholarship Program for a visitor s fellowship. References [1] A.P. Tsai, J. Q. Guo, E. Abe, H. Takakura, and T. J. Sato, Nature 408, (2000), A stable binary quasicrystal. [2] J. Q. Guo, E. Abe, and A. P. Tsai, Phys Rev. B 62, R14605 (2000), Stable icosahedral quasicrystals in binary Cd-Ca and Cd-Yb systems. [3] H. Takakura, J. Q. Guo, and A.-P. Tsai, Phil. Mag. Lett. 81, 411 (2001), Crystal and quasicrystal structures in Cd-Yb and Cd-Ca binary alloys. [4] P. C. Canfield, M. L. Caudle, C.-S. Ho,, A. Kreyssig, S. Nandi, M. G. Kim, X. Lin, A. Kracher, K. W. Dennis, R. W. McCallum, and A. I. Goldman, Phys. Rev. B 81, (2010), Solution growth of a binary icosahedral quasicrystal of Sc 12 Zn 88. [5] T. Watanuki, A. Machida, T. Ikeda, K. Aoki, H. Kaneko, T. Shobu, T. J. Sato, and A. P. Tsai Phys. Rev. Lett. 96, (2006), Pressure-Induced Phase Transitions in the Cd-Yb Periodic Approximant to a Quasicrystal. [6] T. Ishimasa, Y. Kasano, A. Tachibana, S. Kashimoto, and K. Osaka, Philos. Mag. 87, 2887 (2007), Low-temperature phase of the Zn Sc approximant. [7] S-Y Piao, C. P. Gomez, and S. Lidin, Z. Naturforsch. 60b, (2006, Structural study of the disordered RE Cd 6 quasicrystal approximants (RE = Tb, Ho, Er, Tm, and Lu). [8] C. P. Gomez, Angew. Chemie 40, 4037 (2001), Structure of Ca 13 Cd 76 : A novel approximant to the MCd 5.7 quasicrystals (M = Ca, Yb). [9] M. Mihalkovič and M. Widom, Phil. Mag. 86, (2006), Canonical cell model of cadmium-based icosahedral alloys. [10] K. Nozawa and Y. Ishii, J. Phys. Condens. Matter 20, (2008) First-principles studies for structural transitions in ordered phase of cubic approximant Cd 6 Ca. [11] M. Mihalkovič, S. Francoual, K. Shibata, M. de Boissieu, A. Q. R. Baron, Y. Sidis, T. Ishimasa, D. Wu, T. Lograsso, L. P. Regnault, F. Gähler, S. Tsutsui, B. Hennion, P. Bastie, T. J. Sato, H. Takakura, R. Currat, and A. P. Tsai, Philos. Mag. 88, 2311 (2008) Atomic dynamics of i-scznmg and its 1/1 approximant phase: Experiment and simulation, [12] T. Hatakeyama, K. Nozawa, and Y. Ishii, Z. Kristalllogr. 223, (2008). Ab initio studies on orientational ordering in cubic Zn Sc. [13] C. L. Henley, Phys. Rev. B43, 993 (1991), Cell geometry for cluster-based quasicrystal models. [14] U. T. Hochli, K. Knorr, and A. Loidl, Adv. Phys. 51, (2002), Orientational glasses. [15] E. R. Grannan, M. Randeria, and J. P. Sethna, Phys. Rev. Lett. 60, 1402 (1988), Low-temperature properties of a model glass. [16] C. P. Gomez and S. Lidin, Phys. Rev. B 68, (2003). Comparative structural study of the disordered MCd 6 quasicrystal approximants. [17] P. Brommer and F. Gähler, Philos. Mag. 86, (2006), Effective potentials for quasicrystals from ab-initio data. [18] N. Gu, M. Mihalkovič, and C. L. Henley, unpublished ( Structure prediction for cobalt-rich decagonal AlCoNi from pair potentials. [19] S. Lim, M. Mihalkovič, and C. L. Henley, Philos. Mag. 88, 1977 (2008). Penrose matching rule from realistic potentials.