## 4 Assessing the Quasiperiodic Order Role  
There are several hints signaling the importance of short-range effects in the emergence of some unusual physical properties of QCs. For instance, electrical conductivity measurements show that (i) amorphous precursors of i-AlCuFe phase already exhibit an increase of conductivity with temperature, and (ii) the structural evolution from the amorphous to the quasicrystalline state is accompanied by a progressive enhancement of the electronic transport anomalies (Figure 6a) [40; 41]. It is worthy noticing that the \(\sigma\)(T) curves reported for both amorphous and QC phases are almost parallel to each other, hence displaying the inverse Matthiessen rule (see Figure 2a). In an analogous way, crystalline approximants, which exhibit a local atomic environment very similar to their related QC alloys, appear as natural candidates to investigate the relative importance of short-range versus long-range order effects on the transport properties. In fact, many unusual physical properties of QCs are also found in approximant phases, where transport measurements indicate that high-order  
Figure 6: (**a**) Temperature dependence of the electrical conductivity of an AlCuFe thin film for different annealing states for the amorphous and for the i-QC phase. The conductivity progressively decreases when the long-range order quality is improved (spanning over more than one order of magnitude at low temperatures) and a clear inverse Matthiessen rule is observed in the \(\sigma\)(T) curves. (Reprinted from [PERSON]; [PERSON]; [PERSON]; [PERSON] Electronic transport properties of quasicrystalline thin films. _Mater. Sci. Eng. A_**2000**, _294-296_, 475-480. Copyright (2000), with permission from Elsevier); (**b**) temperature dependence of the electrical resistivity of a i-AlPdRe QC along with 1/1-AlReSi, 1/1-AlMnSi, and 1/0-Al\({}_{12}\)Re approximants. The low-temperature electrical resistivity values are indicated in the graph. (Reprinted from [PERSON]; [PERSON] Metallic-covalent bonding conversion and thermoelectric properties of Al-based icosahedral quasicrystals and approximants. _Sci. Technol. Adv. Mater._**2014**, 15, 044802; doi:10.1088/1468-6996/15/4/044802, Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License). approximants (say, 1/1 or 2/1), with lattice parameters exceeding about 2 nm, exhibit similar electrical properties to full-fledged QCs, whereas low-order approximants usually display a typically metallic behavior (Figure 6b). Furthermore, the observed anomalies are generally more pronounced in the case of QCs. These results indicate that the electronic properties of QCs and their related approximant phases are quite similar, provided that the order of the considered approximants is high enough, thereby guaranteeing the local atomic arrangements are essentially the same, irrespective of the long-range order. On the contrary, transport properties of metallic alloys with complex unit cells, having a similar number of atomic species to those of approximant phases, but not exhibiting the local isomorphism property characteristic of QCs, are typically metallic instead [42]. Therefore, mere structural complexity is not a sufficient condition to give rise to the emergence of anomalous transport properties in QCs. On the other hand, certain anomalous transport properties, such as a high electrical resistivity value or a negative temperature coefficient of electrical resistivity, can also be observed in some crystalline alloys consisting of metallic elements (e.g., Heusler-type Fe\({}_{2}\)VAl and Ru\({}_{2}\)TaAl alloys) whose structures are unrelated to those of QCs, although both of them share some characteristic features in their electronic structures, namely, a deep and relatively narrow pseudogap close to E\({}_{\text{F}}\)[43; 44].