## 8 Potential of Quasicrystals as Thermoelectric Materials  
As we have discussed in the previous sections, the electronic and thermal transport properties of stable i-QCs exhibit a more semiconductor-like than metallic character (see Table 1). Quite interestingly from the viewpoint of possible practical applications, the temperature dependence of several transport coefficients in QCs suggest that these compounds may be promising thermoelectric materials (TEMs) [20; 87]. In fact, the efficiency of a thermoelectric device is measured in terms of the so-called thermoelectric figure of merit, which is defined by the expression  
\[ZT=\frac{\sigma\ S^{2}\ T}{\kappa} \tag{1}\]  
where \(\sigma\)(T) is the electrical conductivity, S(T) is the Seebeck coefficient, and \(\kappa\)(T) is the thermal conductivity [88]. According to Equation (1) there are four basic reasons supporting QCs and their approximants as potential thermoelectric materials (TEMs) [87]:  
1. The electrical conductivity of i-QCs steadily increases as the temperature increases up to the melting point, as it is illustrated in Figures 2 and 11a. 2. Al-bearing i-QCs show relatively large Seebeck coefficient values (50-120 uVK\({}^{-1}\)) as compared to those observed in usual metallic alloys (1-10 uVK\({}^{-1}\)) at room temperature, and the S(T) curves usually deviate from the linear behavior characteristic of charge diffusion in ordinary metallic alloys in the temperature interval 100-300 K (Figure 11b). 3. Furthermore, in the case of i-AlCu(Fe,Ru,Os) and i-AlPd(Mn,Re) representatives, both \(\sigma\)(T) and S(T) increase as the temperature is increased over a broad temperature range (T = 100-550 K), which yields relatively high power factor (\(\sigma\)S\({}^{2}\) ) values [89]. 4. The thermal conductivity of most i-QCs (within the range \(\kappa\) = 1-5 Wm\({}^{-1}\)K\({}^{-1}\) at room temperature) is about two orders of magnitude lower than that of ternary alloys, since it is mainly determined by the lattice phonons instead of charge carriers over a wide temperature range (Figure 11c). These unusually-low thermal conductivity values are comparable to those observed for thermal insulators of extensive use in aeronautical industry [3]. The low thermal conductivity of QCs can be understood in terms of two main facts. For one thing, heat propagates mainly by means of phonons, since the charge carrier concentration is severely reduced due to the presence of a pseudogap around the Fermi level (see Section 2). On the other hand, reciprocal space has a nearly fractal pattern in QCs (see Figure 1a), so that the transfer of momentum to the lattice is not bounded below, and the thermal current intensity is strongly reduced due to an enhancement of phonon-phonon scattering processes occurring at all scales in the reciprocal space. In addition, the high fragmentation of the frequency spectra in QCs leads to a small group velocity of phonons. In a similar way, an increased number of Umklapp electron-phonon processes in QCs may enhance the phonon drag contribution to the total Seebeck coefficient as well [90]. Inspired by these features a series of experimental studies aimed at assessing the possible use of i-QCs belonging to different alloy systems as potential TEMs has been performed by several research groups. In this way, the most promising QC alloy system was first determined by searching for  
Figure 11: Collage picture illustrating the contribution of each transport coefficient (magenta arrows) to the resulting ZT value (green arrow). Note that panels (**a–d**) correspond to different i-QCs representatives, each one showing an optimal behavior for the considered physical magnitude: (**a**) Temperature dependence of the electrical conductivity for four different i-QCs up to 1000 K. (Reprinted figure with permission from [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Evidence for unconventional electronic transport in quasicrystals. _Phys. Rev. Lett._ 1993, 70, 3915-3918.