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5. Physical Properties

Quasicrystals have characteristic physical properties. Some resemble those of periodic crystals, for instance the morphology of quasicrystals; some are rather similar to the properties of amorphous alloys, this, for example, applies to the transport anomalies observed in quasicrystals. In contrast to the nearly-isotropic icosahedral quasicrystals, decagonal quasicrystals show anisotropic physical properties, which can be rather different in the periodic direction from those within the quasiperiodic planes, see [LWLZ+90, SHT90, ZCWL+91, NBRK96, FBGM+97, BBFK+98] for examples. There are exceptions, though; a recent investigation of the elastic moduli of decagonal Al-Co-Ni [COBM+98] showed not only the expected transverse elastic isotropy, but indicated that this decagonal quasicrystal is close to being elastically isotropic.

5.1. Morphology

The non-crystallographic symmetry of quasicrystals is often reflected in the shape of single quasicrystals. Quasicrystal faceting was first observed in Al-Mn alloys [IN86, RMMB86]. For the icosahedral quasicrystals, icosahedrally symmetric shapes such as dodecahedra and triacontahedra are observed in experiments, as well as more complicated polytopes with a larger number of facets [T:Bee, R:NB, BN93]. An example of a nicely developed regular pentagonal dodecahedron is shown in figure 3. Faceting has also been observed for decagonal quasicrystals, where one finds a tenfold prismatic morphology. The tenfold direction is the periodic axis of the decagonal quasicrystal, and coincides with the direction of preferred growth, leading to a columnar form [T:Bee, KTCT+89, BN93]. In addition to the faceting of single grains, also faceted voids have been observed in quasicrystals, see [BGL98a, BGL98b] and references therein. A particularly beautiful example is shown in figure 4.

HoMgZn quasicrystal
Figure 3: Dodecahedral single grain Ho-Mg-Zn quasicrystal grown by the flux tube technique [FIPC+98, FCPC+99].

From a theoretical point of view, one might have expected that facet formation is a domain of crystals or, at most, ideal quasiperiodic tilings. However, this is not the case, faceting may also occur in random tilings and certain classes of bond-oriented glasses, see [T:Bee, R:Ho, HJLS87, HLSJ89, LH91] for discussions of the possible zero-temperature equilibrium shapes of systems with an icosahedral bond orientation. Recently, based on ideas of densest sphere packings, polygonal Wulff shapes for ideal icosahedral tilings were found [BS98, Will98, BS99], which closely resemble structures observed in experiments.

HoMgZn quasicrystal
Figure 4: Faceted hole in an Al-Mn-Pd quasicrystal [BGL98a, BGL98b].

5.2. Quasicrystal Surfaces

Quasicrystal surfaces are interesting from several points of view, see [R:TGJ] for a recent review. They are, of course, of importance for physical and chemical properties of quasicrystals, in particular with respect to potential applications discussed below. Furthermore, many techniques employed to investigate properties of quasicrystals are surface sensitive, and the properties of the quasicrystal surface may be vital for the correct interpretation of the results. One example is the experimental determination of the electronic density of states mentioned below where the influence of the surface is still controversially discussed. To give another example, not much is known about the role of surface conductivity for measurements of electric conductivity of quasicrystalline thin films, to be discussed below.

At the surface, one might encounter, for instance, contaminations, oxygen adsorption [CCZJ+95] or surface oxidation [CAT96], superstructures or other surface transitions [MMLT+90], or even a chemical composition that differs considerably from the bulk due to evaporation or surface-rearrangement transitions [BEVK99]. The properties of a quasicrystal surface depend critically on the way the surface is prepared. Whereas cleaved surfaces have revealed a rough surface showing substructures that have been interpreted as Mackay clusters [EYU98], physically or chemically treated surfaces may look rather different. For instance, it has been observed that ion sputtering may result in crystalline surface structures [BEVZ+98, SKJG+98], and also terraced surfaces which follow the Fibonacci sequence have been found [SBGS94a, SBGS94b, SBGS+95]. A variety of methods is used to investigate quasicrystal surfaces, among those scanning tunneling microscopy (STM) [KBTC90, SBGS94a, SBGS94b, SBGS+95, EFTW+96], atomic force microscopy (AFM) [HEMS+97], secondary electron imaging (SEI) [ENWH+94}, low-energy electron diffraction (LEED) [MMLT+90, SBGS94b, SBGS+95, GVGS+97, GVGS+98], X-ray photoelectron spectroscopy (XPS) [CAT96], and X-ray photoelectron diffraction (XPD) [NASB+98]. Although these studies have improved the knowledge of quasicrystal surfaces, a common consensus on their structure and properties has not yet been reached.

The peculiar nature of electronic states in quasicrystals, and the surface sensitivity of the experimental techniques used to investigate electronic density of states in quasicrystals, justify a closer experimental and theoretical investigation of the electronic properties of quasicrystal surfaces. First results on simple tight-binding models [JF98, ZFJ99] indicate that surface states exist and may play an important role. Recent photoemission experiments on cleaved Al-Pd-Mn quasicrystals have revealed a pronounced metallic signature [NBHT+98}, and provided hints that the electronic properties of the surface are rather different from that of the bulk material.

5.3. Mechanical Properties

At ambient or intermediate temperature, quasicrystals generally are hard and extremely brittle. At higher temperatures, above about 60% of the melting temperature, they become ductile [KD92]. In this regime, stress-strain curves are characterized by a yield drop at around 1% plastic strain, followed by a continuous decrease of the stress with increasing strain, see [UEFF+98] and references therein. Even for large plastic deformations of up to 20%, no saturation of this effect was observed, thus quasicrystals apparently show work softening [KMGL98, UEFF+98}]rather than work hardening. The main origin of the plasticity is arguably the motion of dislocations, at least for the Al-Pd-Mn system [WBUL+93]. A qualitative explanation in terms of a cluster friction model was given in [FMWU+97]. Also molecular dynamics simulations of crack propagation [MSGT98, MSKT+98] indicate that clusters as particularly stable building blocks of quasicrystals are preserved under stress.

The hardness of quasicrystals has been investigated by microindentation, see [WDP97, EST98] and references therein. Quasicrystals have rather good tribological properties; besides their high hardness they show a low surface friction [P:DW, KGLJ+99], and thus they are fit for surface coating applications, see the discussion of potential applications below. A comparison of mechanical properties of quasicrystals and conventional materials can also been found at

5.4. Transport Properties

Among the physical properties of quasicrystals, their transport properties may, at first view, appear especially surprising, see [R:Ber, R:FT, R:Hab, R:Poo] for reviews. Given that most quasicrystals are ternary alloys of elements that, by themselves, are rather good metals, predominantly aluminum, one would not a priori expect that quasicrystals are extremely poor conductors, and that their electric conductivity increases strongly with increasing temperature. This is opposite to the behaviour found in metals, wherefore it is sometimes referred to as an "anti-metallic" behaviour [BMC95]. However, if one compares the transport properties of quasicrystals to those of amorphous intermetallic alloys, the discrepancy is less stringent, although some anomalies are more pronounced in quasicrystals than in amorphous alloys [R:Hab, R:Hau, HKMR+98].

As an example, consider the electric conductivity of a thin Al-Cu-Fe film. Figure 5 shows the result of in situ measurements on an initially amorphous film during heat treatment. The amorphous film already shows an increasing conductivity with increasing temperature on the reversible part of the diagram which is indicated by the double arrows. At high temperature, a strong decrease of the conductivity takes place, and one arrives at quasicrystalline films with a conductivity that, though considerably smaller, shows a temperature dependence that is very similar to that of the amorphous film. Upon further annealing at high temperatures, the structural quality of the quasicrystal improves, which is accompanied by a further decrease in conductivity. Finally, the low-temperature conductivity of the film becomes extremely small. This is in accordance with experiments on bulk material, and particularly icosahedral Al-Pd-Re, which is considered to be a structurally "perfect" quasicrystal, appears to be very close to a metal-insulator transition [PPG93, PGP94, GP96, ARGB+97, AGRB+98, GBGF+98, WKLC98, WLLC98]. Results on this and other quasicrystalline alloys show, in different experiments performed on different samples, either weakly metallic or weakly insulating behaviour [HF95, PPGV95, DBB98, GBGF+98, WKLC98, WLLC98].

AlCuFe conductivity
Figure 5: In situ measurements of the electric conductivity of a thin Al-Cu-Fe film during heat treatment.

Other transport properties also show distinctive features. Quasicrystals typically are very poor thermal conductors [ECBF+96]. The thermoelectric power sometimes takes large positive values [R:Hab, HF95, HKMR+98, HRKZ+98], but may also be negative at low temperatures [R:Poo, HF95, HKMR+98]. Also the magnetoconductance shows surprising effects, with a rather strong temperature dependence [HF95]. Frequently, the magnetoconductance is positive for low fields and changes to negative values for larger magnetic fields, see, for instance, [R:Poo, AGRB+98, GBGF+98, WKLC98, WLLC98, WZL98] and references therein. There are relations between the variations of several transport coefficients, see [R:Hab, LBKL+93, PBBG+93] for examples. As mentioned before, physical properties, and particularly transport properties, are generally anisotropic in T phases, see [LWLZ+90, SHT90, ZCWL+91, ECBF+96, NBRK96] for some examples.

There are several plausible explanations of the transport anomalies of quasicrystals. The quasiperiodic structure leads to a very spiky density of electronic states, and, due to the particular kind of ordered aperiodicity, electronic states in these systems, at least in two dimensions, are neither extended nor exponentially localized, see [R:FT, R:Jan97, R:May, R:Sir, R:SB, R:Sut] for reviews on this topic. The absence of extended electronic states may explain the low conductivity, and the particular electronic states supported by a quasiperiodic background place the system right at the metal-insulator transition, that is, the conductivity vanishes precisely at zero temperature. A second approach is based on the Hume-Rothery picture mentioned above. Here, the resonance-like scattering caused by the coincidence of the diameter of the Fermi sphere with the main peaks in the structure factor leads to the opening of a pseudogap at the Fermi surface. In this view, the main effect is thus a depletion of charge carriers at the Fermi surface. A third view employs the hierarchical cluster picture discussed previously, and considers electrons to be confined within clusters of different sizes. All three approaches are able to explain at least part of the experimental observations, and give a qualitative understanding of the transport properties. However, none of them alone really gives a complete picture; and one may absolutely assume that it is necessary to take into account both the structural order and the interaction effects, and probably also the inherent disorder, in order to arrive at a complete understanding of transport properties of quasicrystals.

5.5. Magnetic Properties

Although many Al-based quasicrystals contain transition metals which carry local magnetic moments, they usually do not seem to exhibit magnetic ordering, and mostly show weak paramagnetic or diamagnetic behaviour. In some cases, a spin-glass behaviour has been reported [GAVC+97, PSSI+98, SHAT98], and quaternary alloys containing boron have been identified as ferromagnetic [LLL97] or ferrimagnetic [YIM92]. However, for quasicrystals of a high structural quality, magnetic moments appear to be effectively screened [R:Ber, Atha97, YYMF+99], and a tendency towards diamagnetism is observed.

The situation is somewhat different for the Mg-Zn-based quasicrystals containing rare-earth elements. Here, the screening should be less effective, and different types of magnetic ordering may be expected. So far, spin-glass behaviour [NKWS+98] and antiferromagnetic ordering [COS97a, COS97b, CS97, CSO98] have been found in experiment, but the latter result is still discussed controversially [IFZC+98, STTS98], and it was argued that the effect may have been caused by crystalline phases that were present in the samples.

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