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The previous comments rationalize that quasiperiodic tilings may serve as a "quasilattice" in the description of quasicrystals, taking over the role of the periodic lattice underlying the structure model for a conventional crystal. Soon after the experimental discovery, this was indeed proposed by Levine and Steinhardt [LS84] who also coined the term "quasicrystals" for these new materials.
Indeed, tilings are well-suited for this purpose, as they are space-filling and non-overlapping, and usually consist of translated and rotated copies of a few basic prototiles only. In the simplest examples, such as the rhombic Penrose tiling, one needs just two tiles - a `fat' and a 'skinny' rhomb - which could play the role of two "unit cells" for a structure model of a decagonal quasicrystal by simply decorating the two tiles by assigning atomic positions. Structure models of this type have been considered, see for instance [Burk91, Burk93] for two slightly different structure models of a decagonal Al-Cu-Co quasicrystal.
Some quasiperiodic tilings, for example the Penrose tiling, allow perfect matching rules. This means that, in our example, there exists a decoration of the tiles, for instance by arrows on the edges, and a certain rule how the decorated tiles may be joined, such that any infinite tiling that obeys the matching rules is locally isomorphic [R:Baa] to the Penrose tiling. Essentially, this means that the tiling is locally indistinguishable from the Penrose tiling, and hence the matching rules determine the aperiodic structure of the tiling uniquely. This property is of importance because it opens the door for an energetic argument for the stability of quasicrystals, because one might find interactions mimicking the matching rules, thus leading to a system with a quasiperiodic ground state, see also the discussion of quasiperiodic ground states below. For some pictures of quasiperiodic tilings, see http://mcs.open.ac.uk/ugg2/tilings.shtml.
There is another, independent experimental evidence for the tiling picture of quasicrystals. In high-resolution transmission electron micrographs, particularly of the T phases when looking along the periodic direction, distinctive motives are visible whose locations define a tiling that resembles the quasiperiodic tilings discussed above, see for instance [T:Bee, NB93, RBNG+96, JBR97, HPR99] and references therein. An example is shown in figure 2.
However, in most cases a careful analysis shows that the tilings obtained from electron micrographs are not patches of ideal quasiperiodic tilings, but contain a certain amount of disorder. There are also other reasons why one might wish to allow for some disorder in the structure models of quasicrystals. On the one hand, there are indications that entropic contributions play a major role for the stability of quasicrystals, and allowing for disorder may result in a structural entropy that can account for this fact, see the discussion on random tilings below. On the other hand, it appears difficult to imagine how an ideal quasiperiodic tiling can grow in a strictly local fashion in nature.
The idea that the long-range order observed in quasicrystals is not that of an ideal quasiperiodic tiling, but a consequence of a statistical arrangement of certain basic building blocks, was brought up already quite early, see [R:Hen, Else85, Henl88, LPW92, OH93, Cock94, ERT98, Henl98, RHHB98] and references therein. Most quasicrystals, and in particular the original ones, are high-temperature phases and crystallize into periodic crystalline structures at low temperature. Thus, it has to be expected that entropic contributions to the free energy are essential. The most radical way to investigate this are the so-called random tiling models, where energetic contributions are completely neglected, and one is only interested in the entropic contributions that arise from the different possibilities to tile space with a given set of prototiles, fixing their occurrence frequencies. This sounds worse than it is, because there are good reasons to assume that the corresponding local configurations are energetically degenerate, or that this is, at least, a good approximation. The famous two random tiling hypotheses [R:Hen] state that the entropy is maximal for the occurrence frequencies that correspond to the maximally symmetric system, that is the quasicrystal, and that this maximum is locally quadratic, giving rise to elastic constants that can, in principle, be measured. While the first statement can be derived from the well-definiteness of the random tiling ensemble [T:Ric, RHHB98], the second statement, though true in many examples, may be wrong in general [T:Ric].
There is another point of view to this matter, coming from the higher-dimensional perspective. The embedding of, say, a three-dimensional icosahedral structure into a six-dimensional periodic lattice shows that there are additional degrees of freedom in these structures associated to the three extra dimensions, so-called phason degrees of freedom [Bak85b, KKL85a, KKL85b, LLOR+85, LRT85]. The name may be somewhat misleading, it is borrowed from a hydrodynamical description also used in the theory of modulated phases discussed below. In a way, these additional degrees of freedom correspond to translations of the cut space in the higher-dimensional space in directions perpendicular to the cut space, while ordinary elastic excitations, which are the usual phonons, correspond to translations parallel to the cut space. If one takes a closer look at this, it turns out that the phason-type excitations correspond to re-orderings of the structure, or, more specifically, to atomic jumps between different positions, which can also be observed directly in experiment. In this description, a random tiling is obtained from an ideal quasiperiodic structure by local phasonic excitations, and one might expect to observe a softening phase transition in quasicrystals from a low-temperature ordered phase to a high-temperature random-tiling phase. In fact, the experimentally observed increased ductility of quasicrystals at high temperatures has been interpreted in this way, see below.
One challenge for theory is to compute the entropy of suitable random-tiling ensembles, defined by the number and occurrence frequencies of the prototiles and possibly further restrictions. In general, this turns out to be quite a difficult task, and Monte Carlo studies may suffer from the very complex non-local "zipper" moves that have to be employed in order to ensure ergodicity, for instance in the case of the square-triangle tiling. However, there is also a class of two-dimensional random tiling models that can be solved analytically [T:Gie, R:Nie, Wido93, Kalu94, GN96, GN97a, GN97b, Kalu97, GN98, RHHB98] by Bethe ansatz techniques. For these, the entropy can be calculated exactly, showing a maximum at the point of quasicrystal symmetry. But there is also a drawback. The ensembles investigated in this approach contain non-linear constraints between the tile densities, giving rise to a rather strange cusp-like form of the entropy. Although suggested by the entropy curves, there is no phase separation in these systems, caused by the periodic boundary conditions employed in the calculation. Also, a mapping to a regular lattice is involved, which distorts different patches in different ways. Thus the ensemble considered in the Bethe ansatz approach does not correspond to a collection of patches of equal shape. Nevertheless, the values of the entropy and the elastic constants agree well with those derived from Monte Carlo simulations. Also rather exotic ensembles of pattern-avoiding random sequences have been investigated, see [BEG97] for an example.
Experimentally, there is ample evidence that the random tiling picture is more appropriate to describe reality than are ideal quasiperiodic tilings. Also in direct observation with high-resolution transmission electron microscopy, random tiling structures can be distinguished from an ideal quasiperiodic order [JB96, JBR97], and only a single decagonal phase is known that reveals an almost perfect quasiperiodic order [RBNG+96, JBR97] in this respect, at least geometrically, though most likely not so chemically. Furthermore, random tilings can account for the experimentally observed diffraction patterns [OM98, BH99], the symmetry is restored in a statistical sense such that sharp diffraction peaks can be observed, together with some diffuse background. For crystallographic symmetries as well as for three-dimensional random tilings the diffraction is Bragg-like, whereas for two-dimensional non-crystallographic random tilings one expects a singular continuous spectrum, which, in principle, should be experimentally distinguishable.
It should be noted that this structural entropy certainly is not the only, and may not even be the most important, entropy source in quasicrystals. Other possible sources that may contribute to an entropic stabilization are chemical disorder and a statistical occupation of certain atomic positions ("zeronium").
There is yet another picture of quasicrystalline materials, motivated by the local atomic structure, as a kind of "conglomerate" of basic icosahedral clusters, probably even as a hierarchical sequence of clusters on different length scales.
Icosahedral atomic clusters are known to form abundantly in undercooled liquids, see [R:Hol, Mark95, KASH+97, SHKA+98] and references therein, and have also been found as particularly stable compounds in the gaseous phase, for instance in free jet expansion experiments [HKN84]. This can be understood easily because, initially, the densest packing of hard spheres results in icosahedral clusters. However, due to frustration these cannot be extended to a space-filling structure preserving the symmetry, and periodic arrangements emerge as densest sphere packings [CS99]. This is one reason why many liquids can be cooled considerably below their solidification temperature - the icosahedral clusters forming in the liquid are not the "correct" germs for a periodic crystal. In contrast, quasicrystal-forming liquids cannot be undercooled that much [R:Hol, HHU93, HHGU94, UHHG94, SHHG+97] because their local structure is very similar to that of the solid. The atomic clusters that are most frequently encountered in cluster-based structure models of quasicrystals are the famous Mackay clusters [Mack62, Mack76, Mack82]. It is believed that these have even been observed directly in experiment, because scanning tunneling microscopy of twofold and fivefold cleaved surfaces of Al-Pd-Mn single quasicrystals show cluster aggregates of Mackay-type clusters [EFTW+96, EYU98].
The cluster approach to quasicrystals [EH85, Henl91, JS94] gained new momentum after it was realized that it is intimately related to the tiling picture, see also [Kram99]. In fact, the Penrose tiling can equivalently be constructed as a covering of the plane with overlapping clusters of a single type [T:Gum, Gumm95, Dune95, Gumm96, Gumm98], thus opening up the possibility to use a single structural unit for the description of quasicrystals. The equivalence is proven by showing explicitly that the allowed configurations of overlapping clusters are in one-to-one correspondence with the matching rules of the Penrose tiling [T:Gum, Gumm96]. The idea of a unique covering cluster is physically appealing, in particular because high-resolution electron micrographs typically resemble coverings of symmetric clusters. Though it seems reasonable that many quasiperiodic tilings possess a unique covering cluster, it is a valid question whether the simple one-to-one correspondence between matching rules in the Penrose tiling and allowed cluster overlaps generalize as well, in particular for the three-dimensional icosahedral case.
Whatever is the answer to this question, it need not be a major drawback because one is not that much interested in the ideal quasiperiodic structure, but rather in structures that are close to those observed experimentally. It turns out that one can combine a random-tiling type idea for cluster covering with a simple, energetically motivated rule, to arrive at well-ordered structures without imposing matching rules [GJ95, SJ96, JeS97, Gahl98, Henl98, SJST+98, Urba98]. The idea is to replace the matching rules by "maxing rules", selecting patterns which maximize a certain type of local cluster arrangements. For the Penrose tiling, it has been shown that it maximizes the density of one such local environment [JeS97], and hence this condition selects, among all possible coverings, those that differ from an ideal Penrose tiling only slightly. Of course, the claim that the condition uniquely determines the local isomorphism class [R:Baa] of the Penrose tiling is not correct, because mismatches of density zero do not affect the argument in [JeS97] which only uses the densities. Still, a sub-ensemble of random tilings fulfilling a weakened condition of this type, in order to ensure a positive entropy density, may turn out to be a reasonable starting point for a realistic structure model.
Combining the cluster picture of quasicrystals with the hierarchical structure expressed by the inflation/deflation symmetry of quasiperiodic tilings discussed below, one arrives at a hierarchical cluster model [JB94, Jano96, Jano97] for quasicrystals. In this model, one starts with a cluster of atoms, then considers a cluster of such clusters obtained by replacing atoms by clusters, and so forth. This picture, though certainly not strictly realized in quasicrystals, also gives a qualitative understanding of their physical properties [Jano97]. However, taking the model seriously, one arrives at seemingly abstruse conclusions. For instance, the scaling of lengths in the hierarchy implies that there should be a hierarchical structure of voids in quasicrystals, and it has been claimed recently that such a hierarchical porosity is indeed present [MJLF+98]; but it is arguably pushing the model too far if holes that develop during the growth of a quasicrystal are interpreted as part of the idealized structure. Though the size distribution of the holes investigated in [MJLF+98] seems to conform to the predictions of the hierarchical cluster model, the distribution of the positions of the holes does not show the regularity that one should expect if the hierarchical picture were correct.
The question whether quasicrystals are really stable thermodynamic phases, i.e., whether they realize a ground state of the many-particle Hamiltonian describing the solid, is of fundamental interest, and has not yet been answered convincingly. Somehow most physicists share the feeling that the ground state of a simple many-body Hamiltonian must be periodic, but there is actually no stringent argument why that should be the case. On the contrary, there exist rather simple models for which the existence of non-periodic ground states can be proven rigorously, see [EM90, Miek97, EMZ98] for examples.
The tiny existence regions of quasicrystalline phases in the phase diagrams of ternary alloys, and the metastability of quasicrystals in binary systems, indicate that a mixture of different atoms with a determined composition may be required in order to build up a stable quasicrystal. Molecular-dynamics simulations, and a comparison of the free energy for several model structures, for a one-component system with suitably chosen interaction potentials provide evidence that - though not yet observed experimentally - even one-elemental quasicrystals may be stable [Dzug92, Dzug93, DH97a, DH97b, Dzug97, Roth97, AB98, DL98, Jagl98, QT99, SBSH+99]. Besides boron as a possible candidate for such structures [BQK96, QI98], it appears feasible to realize the particular interaction potentials that lead to quasicrystalline systems in colloidal suspensions [DH97a, DH97b, DL98], where laser-induced quasicrystalline order has recently been reported [DK98].
As mentioned above, the random tiling picture of quasicrystals offers one explanation of the stability of quasicrystals by entropy. Besides other entropic effects such as chemical disorder, there exists a different mechanism that may explain the stability of quasicrystals, and furthermore account for their physical properties which are discussed below.
In this approach, the interaction of electrons near the Fermi level is most important, and the electronic stabilization of the structure is due to an interplay between the structure and the electronic system, evidenced by a coincidence of a main structural peak in the structure factor and the diameter 2kF of the Fermi sphere. In analogy to the Hume-Rothery mechanism for amorphous alloys [R:Hau, Hume26, HR62], which is kind of a Peierls instability [Peie29], one expects a structure-induced pseudogap to form in the electronic density of states at the Fermi level. The corresponding real-space phenomenon is known as Friedel oscillations [Frie88] in the density of conduction electrons. The relevance of the Hume-Rothery mechanism for amorphous alloys is well-established, both theoretically and experimentally, see [R:Hau, KHK95] and references therein.
In many respects, icosahedral quasicrystals resemble amorphous alloys, and one might think of the amorphous phase also as an "approximant" of the quasicrystalline phase. It thus appears natural to assume that quasicrystals are stabilized by a Hume-Rothery-type mechanism as well [R:Hab, R:Hau, Frie88, HK92a, KHM97a, KHM97b, RM97, HaeK98, HafK98, Kroh98, Mizu98], particularly for icosahedral quasicrystals which are nearly isotropic. As has been shown in in situ measurements for thin films [R:Hab], amorphous and icosahedral quasicrystalline alloys behave surprisingly similar with respect to their transport properties, apart from the fact that the transport anomalies are more pronounced for the latter, what may be attributable to the sharper pseudo Jones (or approximant Brillouin) zone boundary for quasicrystals as compared to the spherical Jones zone in amorphous alloys [R:Hab].
The main indicator for a Hume-Rothery mechanism, apart from a specific value of the ratio of conduction electrons to the number of atoms [R:Hau], is the existence of a pronounced pseudogap in the density of states at the Fermi level. Clearly, the existence of such a pseudogap has an enormous effect on electronic and thermal transport properties which are discussed below. For quasicrystals, pseudogaps at or near the Fermi energy have been observed experimentally by electron photoemission [WKOB+95, SPGB+96, SPGB+97, NBHT+98, SPGB+98], tunneling spectroscopy [KSDJ95, DMBG+96, DMBJ98, ELCB99, LHWH+99], and by nuclear magnetic resonance (NMR) [THWP+97]. Though all these results indicate the presence of a, more or less pronounced, dip in the density of states, there are some discrepancies and questions on the influence of the surface that still have to be clarified. In any case, it appears to be difficult to explain the transport anomalies of quasicrystals, see below, on the basis of this pseudogap alone.
Before we discuss the properties of quasicrystals, we briefly comment on two structures that are frequently mentioned in the same context as quasicrystals, so-called modulated phases and approximant phases.
Modulated phases are, in a sense, precursors of quasicrystals because they have been known for a long time, and some of the methods and notions used to describe quasicrystals were developed in order to characterize incommensurately modulated structures. However, one should carefully distinguish the notions of quasicrystals and modulated structures, see [B:PU, R:Kat90] and references therein. In contrast to quasicrystals, a modulated phase consists of a perfectly periodic crystal, but with a periodic modulation of some order parameter, for instance a magnetic moment. This would be nothing particular, but if the period of the modulation is incommensurate with the lattice constant of the underlying crystal, one obtains an incommensurate, quasiperiodic structure.
As the modulation gives rise to an additional length scale, one can embed the three-dimensional modulated structure into a four-dimensional periodic structure, and this is precisely what is done in order to describe the diffraction pattern of modulated structures [JJ77, JJ79, JJ80a, JJ80b]. This procedure closely resembles the cut-and-project method used to produce quasiperiodic tilings; however, one should keep in mind that in the latter case there is no underlying three-dimensional periodic lattice. Nevertheless, this distinction may not always be obvious, an example is given by the "cubic quasicrystal" [DSPH+96] that has recently been interpreted as a modulated phase [EP99].
As periodic crystals, also quasicrystals can be modulated, see e.g. [YW97, Bely98]. A structural description of such modulated quasicrystals then requires an even higher-dimensional embedding space.
For a number of quasicrystals, there exist crystalline phases that resemble the quasicrystal, so-called approximant phases, see [R:GK] for a review. These are connected to periodic approximants of the quasiperiodic tilings, which may, in simple terms, be understood as a rational approximant of the irrational number that is characteristic for the symmetry. For the most important cases of decagonal and icosahedral quasicrystals, this is the golden number which is the square root of five plus one divided by two. The approximant phases are usually indexed by the corresponding rational number, and, in principle, one should expect a whole hierarchy of approximant phases with growing unit cells [R:GK, CGS86, DMO89]. In the embedding picture, where one considers the quasiperiodic tiling as embedded in a higher-dimensional periodic lattice, approximants can be systematically constructed, see for instance [BJK91].
In fact, most theoretical investigations of quasicrystals follow exactly this path, approximating the quasiperiodic structure by a sequence of periodic structures with growing unit cells. In practice, only the very first members of this hierarchy have been observed experimentally, and, apart from a possibly tiny existence region, it might be hard to distinguish larger approximants from "true" quasicrystals. In fact, this distinction ceases to make sense beyond a certain scale, at the latest when the unit cell reaches the size of the single-phase domains of the sample under investigation. Strictly speaking, experimental quasicrystals can never be realizations of ideal quasiperiodic tilings simply due to their finiteness, but this question, of course, does not even arise due to the inevitable disorder - and thus is not more serious than the analogous statement for perfect periodic crystals which also are not realized in nature.
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Uwe Grimm,
last modified on
14 September 2005
