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The thermoelectric power S (T) and thermal conductivity κ (T)
were systematically studied for a series of Al82.6-x
Re17.4 Six (7⩽x⩽12) 1/1 -cubic
approximants. We found that S (T) of these approximants is
characterized by large magnitude, sign reversal with varying
composition, and nonlinear temperature dependence, all of which are also
known as ch...
Contexts in source publication
Context 1
... introduce here the density of states N of the Al 73.6 Re 17.4 Si 9 1 / 1-cubic approximant calculated on the basis of the LMTO ASA method using the reliable crystal structure determined by synchrotron radiation Rietveld analysis. 15 The details of the LMTO ASA calculation and of the Rietveld analysis were reported elsewhere. 15,18 The LMTO ASA density of states of the Al 73.6 Re 17.4 Si 9 1 / 1-cubic approximant is depicted in Fig. 4, and that deduced from electronic specific heat coefficients is superimposed on it. The carrier concentration dependence of the density of states deduced from the electronic specific heat coefficients for these two series of 1 / 1-approximants showed extremely good agreement with the theoretically calculated density of states. This means that the substitution of Si for Al does not significantly affect the shape of N ͑ ͒ but simply increases the number of valence electrons in the system. Thus we calculated S ͑ T ͒ for different Si concentrations using Eq. ( 4 ) and N ͑ ͒ of the 1 / 1-cubic approximants with a proper choice of E F , which is shown in Fig. 4 ( b ) with dashed lines. The calculated S ͑ T ͒ on the basis of the LMTO ASA density of states were superimposed on Fig. 1 ( a ) with solid lines. The calculated S ͑ T ͒ is in surprisingly good agreement with the measured ones not only in its sign reversal with increasing Si concentration but also the nonlinear temperature dependence. We have to stress here that we did not use any parameter fitting to calculate S ͑ T ͒ , but just employed a precisely determined N ͑ ͒ and properly selected . We noticed that the magnitude of S ͑ T ͒ monotonically increases with increasing temperature when the bottom of the pseudogap bottom remains outside the narrow energy range of a few k B T in width centered at , but it starts to decrease when the temperature is increased high enough at which the energy range covers bottom . It is argued, in other words, that the peak temperature where dS ͑ T ͒ / dT becomes zero is roughly determined by the energy difference between and bottom . This is closely related to the fact that the highest S ͑ T ͒ is obtained not at the condition of = bottom but when is located at an energy below or above bottom . These features must be characteristic not only of the present approximants but also of all pseudogap systems that include icosahedral quasicrystals. The present analysis of the S ͑ T ͒ of the Al 73.6 Re 17.4 Si 9 1 / 1-cubic approximants lets us conclude that the characteristic behavior of S ͑ T ͒ in these approximants and perhaps that in the corresponding quasicrystals is brought about simply by the presence of a pseudogap of a few hun- dreds of meV in width. We discuss next the reliability of the assumptions, the energy independent ᐉ and energy independent v , that we employed to calculate S ͑ T ͒ in this study. A weak temperature dependence of the electrical resistivity of the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants allows us to believe that the mean free path of the conduction electron ᐉ is fairly shortened nearly to interatomic distances. 18 In such a case, ᐉ can be safely assumed to be energy independent. On the other hand, it is very difficult to judge if the assumption of energy independent group velocity is appropriate or not for the present Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants. It is confirmed in our analysis that S ͑ T ͒ in the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants can be quantitatively reproduced from Eq. ( 4 ) with the assumption of an energy independent velocity. Perhaps information about the wave number and the group velocity has already been lost in these approximants, most likely because the mean free path ᐉ is so shortened as to satisfy the condition of ᐉ Ϸ F ( the Fermi wavelength ) known as the Mott limit, where the wave packet is no longer well defined. One may think that this consideration is inconsistent with the use of the Boltzmann formula [ Eq. ( 3 )] . However, we believe that the Boltzmann formula is still useful even under the Mott limit, because Mott and Davis 30 gave a proof that the Kubo-Greenwood formula, which is used to interpret the electrical conductivity in nonperiodic materials, reduced to the Boltzmann formula, and because the Boltzmann conductivity B is used as one of the factors in the Mott-Kaveh formula 29 which interprets the temperature dependence of electrical conductivity under the weak-localization effect. It is also very important to note before going on that the group velocity in B is replaced by the energy-independent Mott g factor in Mott’s formulation for the electrical conductivity in disordered materials. 31 It is easily seen that the S ͑ T ͒ calculation we employed in this paper cannot be directly applied to quasicrystals because the electronic structure in quasicrystals is hard to evaluate by the ordinary band calculation developed for crystalline materials. However, we have already developed a method to calculate S ͑ T ͒ on the basis of the experimentally determined electronic structure by using high-resolution photoemission spectroscopy. 32 We are now in progress to experimentally reveal the mechanism leading to the characteristic behavior of S T in icosahedral quasicrystals by applying the present method to photoemission spectra of high energy resolution. In summary, the temperature dependence of S ͑ T ͒ in these phases was quantitatively reproduced by using precisely determined electronic structure. We conclude that the large thermoelectric power in Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants and the corresponding quasicrystals is simply brought about by the presence of a pseudogap across E . In this study, the temperature dependence of the thermal conductivity and thermoelectric power of Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants was systematically investigated and the dimensionless figure of merit ZT for the thermoelectric materials is evaluated using the measured electron transport properties. These approximants possess large thermoelectric power ranging from −40 to 50 V / K and small thermal conductivity less than 1.6 W / K m. In combination with the relatively low electrical resistivity, the dimensionless figure of merit reaches 0.04 at Al 74.6 Re 17.4 Si 8 and 0.02 at Al 71.6 Re 17.4 Si 11 with positive and negative sign of S , respectively. S ͑ T ͒ of these approximants was analyzed on the basis of the Boltzmann transport equation. We found that the large magnitude and the characteristic temperature dependence of S ͑ T ͒ in the approximants and the corresponding quasicrystals are caused by the presence of a pseudogap across the Fermi ...
Context 2
... Al-based Mackay-type icosahedral quasicrystal has attracted a great deal of interest as a potential candidate for a new thermoelectric material 1 because it possesses large thermoelectric power ͑ S ͒ , more than 50 V / K, 2–5 and low thermal conductivity ͑ ͒ , 5 as low as 1 W / K m. 3,5–9 These characteristic properties are often discussed in relation to the quasiperiodicity unique to the quasicrystals. However, the mechanism leading to the large magnitude of S ͑ T ͒ and small thermal conductivity in quasicrystals is not fully understood. One of the most plausible factors other than quasiperiodicity leading to these characteristics in S ͑ T ͒ would be the presence of a pseudogap across the Fermi level ͑ E F ͒ . S ͑ T ͒ of quasicrystals is characterized not only by large magnitude but also by strong temperature and composition dependence. 2–4 The behavior of S ͑ T ͒ is generally determined by the energy dependence of the electrical conductivity ͓ ͑ ͔͒ . Unfortunately, the electronic structure and atomic arrangements, which greatly affect ͑ ͒ , have hardly been analyzed for quasicrystals because quasiperiodicity in quasicrystals prevents us from applying ordinary band calculations or structure analyses well developed for crystalline materials. Here we notice that if the characteristics in S ͑ T ͒ of quasicrystals are brought about by the presence of a pseudogap across E F , the corresponding approximants having a pseudogap at E F also possess similar behaviors in their S ͑ T ͒ , and that one can gain deep insight into the nature of their S ͑ T ͒ by employing approximants rather than quasicrystals, because their electronic structure and local atomic arrangements can be accurately determined by ordinary band calculations 10–13 and structure analyses. 14–17 Investigation of the electronic structure and the local atomic arrangements also plays an important role in the proper understanding of the thermal conductivity and electrical resistivity , both of which are necessary for estimation of the potential of thermoelectric materials in the dimensionless figure of merit defined as ZT = S 2 T / ͑ ͒ . The electrical conduction in the approximants can be well investigated by using the Boltzmann transport equation on the basis of Bloch theory. With the great help of the accurately determined electronic structure of rational approximants, the role of electronic structure including the influence of the pseudogap across E F on the electrical resistivity should be clearly re- vealed. Thermal conductivity is generally determined by contributions not only of the conduction electrons but also of the phonons, and the former can be roughly estimated from the electrical conductivity by using the Wiedemann-Franz law. The latter is closely related to the atomic structure and is often discussed in terms of the quasiperiodicity. 3,6–9 By using rational approximants rather than the corresponding quasicrystals, one can clearly reveal the local atomic arrangements and their influence on the thermal conductivity can be unam- biguously discussed. In this study we have systematically measured thermoelectric power and thermal conductivity for a series of Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The measured thermoelectric power and thermal conductivity together with previously reported electrical resistivity 18 are used to evaluate the performance of the approximant as a thermoelectric material. By comparing the measured S ͑ T ͒ , ͑ T ͒ , and ͑ T ͒ of quasicrystals with those of the approximants, the influence of quasiperiodicity upon these thermoelectric properties is also discussed. We also discuss the origin of the characteristic behaviors of the thermoelectric power in the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants in terms of the electronic structure and local atomic arrangements. It will be demonstrated, as a conse- quence of the present analysis, that the behaviors in the thermoelectric power of these 1 / 1-cubic approximants are quantitatively simulated from the accurately determined electronic structure, and that the characteristic behavior in S ͑ T ͒ in the approximants and perhaps in the quasicrystals is accounted for by simply considering the presence of a pseudogap across E . We employed in this study a series of Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants without any precipita- tion of secondary phases. Ribbon samples were prepared by the single-role melt-quenching technique and used for the measurement of the thermoelectric power. Bulk samples were used for the thermal conductivity measurement. The details of sample preparation and phase determination were reported previously. 18 The thermoelectric power was measured at temperatures from 90 to 400 K with a Seebeck Coefficient Measurement System ( MMR ) . We also used the Physical Properties Measurement System ( Quantum Design ) with the thermal transport option to simultaneously measure the thermoelectric power and thermal conductivity for samples of ϳ 1 ϫ 1 ϫ 10 mm 3 in dimension over 5 ഛ T ഛ 300 K. Thermodynamically stable Al 62.5 Cu 24.5 Fe 13 , Al 63 Cu 24 Fe 13 , Al 62.5 Cu 25 Fe 12.5 , Al 63 Cu 24.5 Fe 12.5 , Al 62.5 Cu 25.5 Fe 12 , and Al 63 Cu 25 Fe 12 icosahedral quasicrystals and Al 74.6 Mn 17.4 Si 8 1 / 1-cubic approximants were also prepared by the same method as that for the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The S ͑ T ͒ ’s of the Al-Cu-Fe icosahedral quasicrystals and ͑ T ͒ of the Al 74.6 Mn 17.4 Si 8 1 / 1-cubic approximants were measured and used for com- parison with those of the the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. For the analysis of S ͑ T ͒ in the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants, we employed the electronic density of states of the Al 73.6 Re 17.4 Si 9 1 / 1-cubic approximant calculated by the linear muffin-tin orbital LMTO atomic-sphere approximation ASA method with atomic structure determined by a synchrotron radiation Rietveld analysis. Details of the band calculation and the Rietveld analysis were also reported in our previous paper. 18 Figure 1 a shows the measured S T of the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The S ͑ T ͒ observed for these Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants is characterized by a large value of ͉ S ͉ exceeding 50 V / K and a strong composition dependence. The composition dependence of S ͑ 100 K ͒ , S ͑ 200 K ͒ , and S ͑ 300 K ͒ is plotted in Fig. 1 ( b ) . Notably, the compositions at which the maximum and minimum values in S ͑ T ͒ were observed are not at the center of the formation range of these approximants but near the lowest and highest limits of the Si concentrations, respectively. S ͑ T ͒ in metallic alloys is often discussed using the well- known formula ...
Context 3
... Al-based Mackay-type icosahedral quasicrystal has attracted a great deal of interest as a potential candidate for a new thermoelectric material 1 because it possesses large thermoelectric power ͑ S ͒ , more than 50 V / K, 2–5 and low thermal conductivity ͑ ͒ , 5 as low as 1 W / K m. 3,5–9 These characteristic properties are often discussed in relation to the quasiperiodicity unique to the quasicrystals. However, the mechanism leading to the large magnitude of S ͑ T ͒ and small thermal conductivity in quasicrystals is not fully understood. One of the most plausible factors other than quasiperiodicity leading to these characteristics in S ͑ T ͒ would be the presence of a pseudogap across the Fermi level ͑ E F ͒ . S ͑ T ͒ of quasicrystals is characterized not only by large magnitude but also by strong temperature and composition dependence. 2–4 The behavior of S ͑ T ͒ is generally determined by the energy dependence of the electrical conductivity ͓ ͑ ͔͒ . Unfortunately, the electronic structure and atomic arrangements, which greatly affect ͑ ͒ , have hardly been analyzed for quasicrystals because quasiperiodicity in quasicrystals prevents us from applying ordinary band calculations or structure analyses well developed for crystalline materials. Here we notice that if the characteristics in S ͑ T ͒ of quasicrystals are brought about by the presence of a pseudogap across E F , the corresponding approximants having a pseudogap at E F also possess similar behaviors in their S ͑ T ͒ , and that one can gain deep insight into the nature of their S ͑ T ͒ by employing approximants rather than quasicrystals, because their electronic structure and local atomic arrangements can be accurately determined by ordinary band calculations 10–13 and structure analyses. 14–17 Investigation of the electronic structure and the local atomic arrangements also plays an important role in the proper understanding of the thermal conductivity and electrical resistivity , both of which are necessary for estimation of the potential of thermoelectric materials in the dimensionless figure of merit defined as ZT = S 2 T / ͑ ͒ . The electrical conduction in the approximants can be well investigated by using the Boltzmann transport equation on the basis of Bloch theory. With the great help of the accurately determined electronic structure of rational approximants, the role of electronic structure including the influence of the pseudogap across E F on the electrical resistivity should be clearly re- vealed. Thermal conductivity is generally determined by contributions not only of the conduction electrons but also of the phonons, and the former can be roughly estimated from the electrical conductivity by using the Wiedemann-Franz law. The latter is closely related to the atomic structure and is often discussed in terms of the quasiperiodicity. 3,6–9 By using rational approximants rather than the corresponding quasicrystals, one can clearly reveal the local atomic arrangements and their influence on the thermal conductivity can be unam- biguously discussed. In this study we have systematically measured thermoelectric power and thermal conductivity for a series of Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The measured thermoelectric power and thermal conductivity together with previously reported electrical resistivity 18 are used to evaluate the performance of the approximant as a thermoelectric material. By comparing the measured S ͑ T ͒ , ͑ T ͒ , and ͑ T ͒ of quasicrystals with those of the approximants, the influence of quasiperiodicity upon these thermoelectric properties is also discussed. We also discuss the origin of the characteristic behaviors of the thermoelectric power in the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants in terms of the electronic structure and local atomic arrangements. It will be demonstrated, as a conse- quence of the present analysis, that the behaviors in the thermoelectric power of these 1 / 1-cubic approximants are quantitatively simulated from the accurately determined electronic structure, and that the characteristic behavior in S ͑ T ͒ in the approximants and perhaps in the quasicrystals is accounted for by simply considering the presence of a pseudogap across E . We employed in this study a series of Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants without any precipita- tion of secondary phases. Ribbon samples were prepared by the single-role melt-quenching technique and used for the measurement of the thermoelectric power. Bulk samples were used for the thermal conductivity measurement. The details of sample preparation and phase determination were reported previously. 18 The thermoelectric power was measured at temperatures from 90 to 400 K with a Seebeck Coefficient Measurement System ( MMR ) . We also used the Physical Properties Measurement System ( Quantum Design ) with the thermal transport option to simultaneously measure the thermoelectric power and thermal conductivity for samples of ϳ 1 ϫ 1 ϫ 10 mm 3 in dimension over 5 ഛ T ഛ 300 K. Thermodynamically stable Al 62.5 Cu 24.5 Fe 13 , Al 63 Cu 24 Fe 13 , Al 62.5 Cu 25 Fe 12.5 , Al 63 Cu 24.5 Fe 12.5 , Al 62.5 Cu 25.5 Fe 12 , and Al 63 Cu 25 Fe 12 icosahedral quasicrystals and Al 74.6 Mn 17.4 Si 8 1 / 1-cubic approximants were also prepared by the same method as that for the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The S ͑ T ͒ ’s of the Al-Cu-Fe icosahedral quasicrystals and ͑ T ͒ of the Al 74.6 Mn 17.4 Si 8 1 / 1-cubic approximants were measured and used for com- parison with those of the the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. For the analysis of S ͑ T ͒ in the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants, we employed the electronic density of states of the Al 73.6 Re 17.4 Si 9 1 / 1-cubic approximant calculated by the linear muffin-tin orbital LMTO atomic-sphere approximation ASA method with atomic structure determined by a synchrotron radiation Rietveld analysis. Details of the band calculation and the Rietveld analysis were also reported in our previous paper. 18 Figure 1 a shows the measured S T of the Al 82.6− x Re 17.4 Si x ͑ 7 ഛ x ഛ 12 ͒ 1 / 1-cubic approximants. The S ͑ T ͒ observed for these Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants is characterized by a large value of ͉ S ͉ exceeding 50 V / K and a strong composition dependence. The composition dependence of S ͑ 100 K ͒ , S ͑ 200 K ͒ , and S ͑ 300 K ͒ is plotted in Fig. 1 ( b ) . Notably, the compositions at which the maximum and minimum values in S ͑ T ͒ were observed are not at the center of the formation range of these approximants but near the lowest and highest limits of the Si concentrations, respectively. S ͑ T ͒ in metallic alloys is often discussed using the well- known formula ...
Context 4
... and represent the electrical conductivity at and the chemical potential, respectively. Equation ( 1 ) is useful to understand the Si concentration dependence of S ͑ T ͒ in the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants. Since electrical conductivity in the metallic phase is known to be directly proportional to the electronic density of states N ͑ ͒ , Eq. ( 1 ) indicates that S becomes positive or negative when the sign of ץ N ͑ ͒ / ץ at is negative or positive, respectively. Note here that e is negative in sign and ln x increases with increasing x . If is located at an energy lower than that of the bottom of pseudogap ͑ bottom ͒ , S ͑ T ͒ becomes positive because of the negative value of ץ N ͑ ͒ / ץ at , and vice versa. The pseudogap across E F in the present 1 / 1-cubic approximants, as well as that in other quasicrystals and approximants, 10–12,20 was already confirmed experimentally and reported previously. 18 At low Si concentrations less than 8 at. % Si, the sign of the thermoelectric power of these approximants stays positive over a whole temperature range of the present measurement, while it turns out to be negative at higher Si concentrations larger than 10 at. % Si. By considering that an increase of Si increases electron concentration in the system, one may naturally notice that the composition dependence of S ͑ 100 K ͒ , S ͑ 200 K ͒ , and S ͑ 300 K ͒ can be qualitatively accounted for by the presence of a pseudogap and moving across it from the lower to the higher energy side with increasing carrier concentration. Similar behavior in S ͑ T ͒ was reported for ͑ Fe 2/3 V 1/3 ͒ y Al 1− y , which is known as one of the well-known pseudogap systems. 21 The S ͑ T ͒ of these 1 / 1-cubic approximants is also characterized by a nonlinear temperature dependence, in sharp con- trast to the T -linear dependence expected from Eq. ( 1 ) . The Nonlinear temperature dependence in S ͑ T ͒ suggests that Eq. ( 1 ) is not appropriate for the quantitative evaluation of the temperature dependence of S ͑ T ͒ for the present 1 / 1-cubic approximants. We noticed that this inconsistency between Eq. ( 1 ) and the measured S ͑ T ͒ is caused by an inappropriate assumption used when Eq. ( 1 ) is deduced. We shall return to this point later and discuss it in more detail ( see Sec. IV ) . We described above the characteristic behaviors in S ͑ T ͒ , which are ( a ) large magnitude, ( b ) strong composition dependence, and ( c ) nonlinear temperature dependence, for the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants. It is particularly important to stress here that the corresponding icosahedral quasicrystals show similar behaviors in their thermoelectric power. 2–4 For example, S ͑ T ͒ of Al-Cu-Fe icosahedral quasicrystals of six different compositions is shown in Fig. 1 ( c ) . Obviously ( a ) large magnitude, ( b ) strong composition dependence, and ( c ) nonlinear temperature dependence can be confirmed in the S ͑ T ͒ of the Al-Cu-Fe icosahedral quasicrystals. It is, thus, strongly argued that the dominant factors leading to the large magnitude and strong composition dependence in S ͑ T ͒ should be essentially the same in quasicrystals and approximants, and that the quasiperiodicity existing only in the quasicrystals has a less important role in causing the large magnitude of S ͑ T ͒ . Precise analyses of the S ͑ T ͒ of the approximants, therefore, would provide us proper understanding of S ͑ T ͒ not only in the approximants but also in their corresponding quasicrystals. The thermal conductivity at room temperature ͓ ͑ 300 K ͔͒ of the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants is plotted as a function of Si concentration x in Fig. 2 ( a ) . Surprisingly, the ͑ 300 K ͒ of the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants is always kept below 1.6 W / K m regardless of the Si concentration. The temperature dependence of for the Al 74.6 Re 17.4 Si 8 1 / 1-cubic approximant is also depicted in Fig. 2 ( b ) together with that of Al 74.6 Mn 17.4 Si 8 which is prepared by substituting Mn for Re in the Al 74.6 Re 17.4 Si 8 1 / 1-cubic approximant. The lower in Al 74.6 Re 17.4 Si 8 than that of Al 74.6 Mn 17.4 Si 8 indicates the strong effect of heavy Re in reducing the thermal conductivity of the approximants as it is well known that heavy atoms in an array of light ele- ments greatly contribute to reducing the thermal conductivity, most likely due to their role as a strong scatterer for the phonons and to the localization of phonons about the heavy atoms. We roughly estimated the contribution of conduction electrons ͑ el ͒ using the Wiedemann-Franz law and superimposed the resulting el in Figs. 2 ( a ) and 2 ( b ) . Since el is directly proportional to the electrical conductivity, it shows a minimum at x Ϸ 9 – 10 where electrical resistivity possesses a maximum. The lattice contribution of thermal conductivity deduced as lat = − el at room temperature obviously shows a maximum value at x = 9 where the disordering in the structure was reported to disappear, 18 and drastically decreases with both increasing and decreasing Si concentration from x Ϸ 9. By considering similar composition dependence between ͑ 300 K ͒ and lat ͑ 300 K ͒ , we can safely argue that the composition dependence of ͑ 300 K ͒ is dominantly brought about by the Si concentration dependence of disordering in the structure. Note here that the magnitude of ͑ 300 K ͒ observed for the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants is comparable with that of the corresponding icosahedral quasicrystals. 6–9 This experimental fact strongly indicates that the quasiperiodicity has a less important contribution to the reduction in the thermal conductivity, and that short-range atomic arrangements have a much stronger influence on it. Although we mentioned above that the magnitude of in the present approximants is strongly influenced by the presence of disordering in the structure, the most dominant factors providing such a very small ഛ 1.6 W / K m, are not obvious now because the disorder-free Al 73.6 Re 17.4 Si 9 also possesses very small . 18 The disordering in the present approximants occurs only in the glue sites which are connect- ing the Mackay clusters existing at the body center and ver- tices of the cubic lattice. If the Mackay clusters behave as an extremely heavy hypothetical atom most likely due to the strong bonds between atoms inside the cluster, 22 and if these heavy hypothetical atoms are connected with weak links of the glue atoms, the group velocity of the acoustic phonon will be greatly reduced. This will cause a small magnitude in the thermal conductivity. Moreover, one may easily consider from this scenario that the disordering in the glue sites strongly scatters the acoustic phonons. This mechanism would be equally applicable to the corresponding quasicrystals. This consideration, however, is no more than a specula- tion because we have only limited information about the relation between structure and thermal conductivity in the approximant. To gain real insight into the origin for the small thermal conductivity in the quasicrystals and approximants, investigation of the relation between the atomic arrangements and thermal conductivity should be performed not only for the present approximants but also for other approximants. Thus we decided to leave this problem still open. Electrical resistivity, as well as the thermoelectric power and thermal conductivity, is one of the factors that deter- mines the performance of thermoelectric materials. We should comment here on our previous data for room temperature electrical resistivity ͓ ͑ 300 K ͔͒ of the Al 82.6− x Re 17.4 Si x 1 / 1-cubic approximants. 18 We reported in our previous paper 18 that ...
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Citations
... Using this calculated electronic structure, we calculated the S in the same manner described in elsewhere, 6) based on Boltzmann transport theory within the rigid band approximation and the constant-diffusion-constant approximation, which has been used frequently to explain the TE properties of QCAs. 6,8,20) 3. Results and Discussion Figure 2 shows the XRD patterns of the sample, together with the reference of AlPdCo(Ge) F phase. 12) We note that although Ge was doped in reference structure to make it easier to grow a good quality single grain for single crystal X-ray diffraction analysis, diffraction pattern is almost same as undoped sample. ...
... These trends of · and S suggest that AlPdCo F phase has a pseudo gap in the density of states as suggested by previous studies on the thermoelectric properties on high-order QCAs. 20,21) The semi-quantitative discussion of the comparison between the DFT-calculated and experimental value of S will be given in a later paragraph. ...
Search for high-order semiconducting quasicrystalline approximants can play an essential role in finding clues to the discovery of semiconducting quasicrystals. According to the previous theoretical work, a model of Al–Pd–Co 1/1 cubic quasicrystalline approximant was predicted to be semiconductor from a calculation based on the density functional theory. We noticed that the F phase in the Al–Pd–Co system is a 2 × 2 × 2 superlattice structure of the calculated model. To verify this prediction, we synthesized the F phase sample, and measured its thermoelectric properties. The measured electrical conductivity linearly increases with increasing temperature. The magnitude of measured Seebeck coefficient is smaller than the typical semiconductor. These properties indicate that the prepared sample of the F phase has a pseudogap rather than a finite band gap. To investigate this discrepancy between the theoretical prediction and experimental results, we calculated the electronic structure for the three structural models using density functional theory. The most energetically stable model has a semimetallic electronic structure.
The electronic band structure (left) and density of states per unit cell DOS (right) around Fermi energy EF of Al92Pd8Co28 model B. Fullsize Image
... 14) Among the approximations, the constant-diffusivity (or diffusion coefficient) approximation (CDA) gave the best fit to experimental data. Similarly, the properties of AlSiRe cubic approximants 6) and an AlSiRu cubic approximant 15) were also reasonably described by CDA. These observations indicate that £(¾) of the approximants are approximately proportional to the electronic density of states n(¾), i.e., £(¾) = e 2 D(¾)n(¾) with D(¾) μ D CDA , where e is the elementary charge, D(¾) is the spectral diffusivity of the electrons and D CDA is an energy-independent parameter that corresponds to the diffusivity of the electrons. ...
... This might be attributed as the Mott limit, which is originally considered as the physical origin of CDA. 6) The overestimation of¸under CDA is a consequence of ignoring the interband contribution during the fitting. For fitting the properties, this is only a matter of interpretation. ...
Thermoelectric properties of quasicrystalline approximants have been reasonably described by the semiclassical Boltzmann transport theory under the empirical constant-diffusivity approximation. To investigate why the approximation could provide such a reasonable description, the properties of an Al–Cu–Ir cubic approximant at temperatures between approximately 400 K and 1000 K were reinvestigated on the basis of a quantum-mechanical transport theory that takes into account two effects missing in the Boltzmann theory, i.e., the interband contribution and the lifetime broadening. The present theory describes the properties as the constant-diffusivity approximation. However, 45(5)% to 66(4)% of electrical conductivity is accounted for by the interband contribution. Spectral diffusivity is less energy-dependent within the electron lifetime necessary for describing the measured properties of the approximant [2.2(3) fs to 3.8(6) fs], which is why the constant-diffusivity approximation could reasonably describe the properties even though the interband contribution is not explicitly taken into account.
Fig. 3 Temperature T dependence of the electrical conductivity σ calculated in this study (total, intraband and interband contributions) and measured experimentally.¹⁴⁾ Fullsize Image
... Takeuchi et al. 6) reported that S for the AlReSi approximant phase was quantitatively reproduced under the assumption that both¸e l (¾; T ) and v el (¾) were independent of energy. Since this assumption is equivalent to CDA in regard to the calculation of S, CDA can be a good approximation for both the AlCuIr and AlReSi approximant phases. ...
... In the case of the AlReSi approximant phase, the origin of CDA was attributed to the Mott limit, where the mean free path of electrons approaches the Fermi wavelength, without examining the value of the mean free path. 6) In our method, the averaged mean free path can be evaluated as l à el ð®; T Þ 1 à el ð®; T Þv à el ð®; T Þ. Figure 7 shows l à el evaluated for sample B using the FCDA model. l à el decreases with increasing T from 1.7(2) nm at 379(1) K to 1.3(1) nm at 1020(4) K. ...
The thermoelectric properties of a cubic quasicrystalline approximant in the Al–Cu–Ir system were investigated experimentally and theoretically. A homogeneous sample with no secondary phase was synthesised by arc melting and spark plasma sintering followed by a heat treatment at 1173 K, and its thermoelectric properties were measured at temperatures between 373 K and 1023 K. Theoretical calculations of the thermoelectric properties were performed under three different approximations, i.e., constant-relaxation-time, constant-mean-free-path and constant-diffusion-coefficient approximations, for the energy dependence of the relaxation time of electrons. The experimental Seebeck coefficient was consistently reproduced, and a physically acceptable lattice thermal conductivity was estimated only under the constant-diffusion-coefficient approximation. The thermoelectric figure of merit zT of the present sample was lower than 0.1, and the maximum value of zT ≈ 0.3 achievable by electron doping was predicted by theoretical calculation under the rigid-band approximation.
This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 82 (2018) 188–196.
Fig. 4 Temperature (T) dependence of the Seebeck coefficient (S) of sample B (open circles) and its best-fitting theoretical curves under CRA (dotted line), CFA (dashed line) and CDA (solid line). Fullsize Image
... A self-consistent cycle was performed on an unshifted 10 × 10 × 10 k mesh in the Brillouin zone using the energy cutoff constant R MT K max = 7.0. Using this calculated electronic structure, we calculated the transport properties S, σ , and κ el in the same manner described in Ref. [22] or see the Supplemental Material [23], based on Boltzmann transport theory within the constant-diffusion-constant approximation, which has been used frequently to explain the TE properties of QCAs [22,24]. ...
... The calculated S and σ values are more consistent with the experimental results than those calculated using the constant relaxation time approximation. This situation is the same as that of the other QCAs [22,24]. These results indicate that the sample is a narrow-gap semiconductor with a band gap of 0.15 eV. ...
We found that an Al-Si-Ru cubic quasicrystalline approximant has a semiconducting band structure by performing an orbital analysis based on density functional theory. These semiconducting transport properties have been confirmed in an experimentally synthesized sample. The temperature dependences of the electrical conductivity and the Seebeck coefficient were consistent with the trends of an intrinsic semiconductor with a band gap of 0.15eV above 350K. The lattice thermal conductivity had a low value of approximately 1.0Wm−1K−1 above 400K, which is close to the theoretical minimum.
... The narrower width of the pseudogap for the IQC and their approximants containing 3d transition metal elements is also understood from the behavior of the Seebeck coefficient, which increases with the increas- ing temperature and starts to decrease after becoming max- imal at the T peak . The peak temperature T peak roughly represents the width of the pseudogap, and the IQCs and their approximants containing 3d elements possess lower T peak than that those containing 4d and/or 5d elements [29,30]. These considerations, together with the very small electronic density of states at the Fermi energy reported for Al-Cu-Fe IQC [20], prompted us to employ the Al-Cu-Fe IQC for the most appropriate material possessing a drastic increase of electron thermal conductivity with increasing temperatures. ...
The bulk thermal rectifiers usable at a high temperature above 300 K were developed by making full use of the unusual electron thermal conductivity of icosahedral quasicrystals. The unusual electron thermal conductivity was caused by a synergy effect of quasiperiodicity and by a narrow pseudogap at the Fermi level. The rectification ratio, defined by TRR = |{{{\boldsymbol{J}} }large}|/|{{{\boldsymbol{J}} }small}|, reached vary large values exceeding 2.0. This significant thermal rectification would lead to new practical applications for the heat management.
... Maciá performed theoretical calculations of various types of quasicrystals and approximants using an analytical model, and recently published a review[22]. For Al-based approximant crystals, Takeuchi et al reported the band structure and phonon dispersion calculations, and discussed the intrinsic properties caused by a deep pseudogap[23,24]. This review contains three parts. ...
... If the covalent bonds in the 1/1-AlReSi approximant are sufficiently strong, they should contribute to form a (narrow) band gap in the vicinity of E F , such as that seen in the related RuAl 2 and RuGa 2 crystals[45,46]. However, band structure calculations indicate that the 1/1-AlReSi approximant forms a deep pseudogap instead of a real gap, as shown infigure 6(a), which is consistent with a previous result[23]and table 3. The minimum density of states (D(E)) near E F is about 1/10 of the free electron value. The origin of the finite D(E F ) is considered to be the weak inter-cluster bond, which is close in strength to the pure Al–Al bonds as shown infigure 5(b). ...
... Among the examples shown infigure 6(c), the icosahedralTCR is hardly affected by them. S is also not greatly affected by such porous microstructures[49], but strongly depends on the local electronic structure near E F. S can be approximated by[21]and 1/1-AlReSi approximants[23], respectively. On the other hand, the metallic 1/0-Al 12 Re approximant with a shallow pseudogap has an S value of 2 μV K −1[50]. ...
In this article, we review the characteristic features of icosahedral cluster solids, metallic–covalent bonding conversion (MCBC), and the thermoelectric properties of Al-based icosahedral quasicrystals and approximants. MCBC is clearly distinguishable from and closely related to the well-known metal–insulator transition. This unique bonding conversion has been experimentally verified in 1/1-AlReSi and 1/0-Al12Re approximants by the maximum entropy method and Rietveld refinement for powder x-ray diffraction data, and is caused by a central atom inside the icosahedral clusters. This helps to understand pseudogap formation in the vicinity of the Fermi energy and establish a guiding principle for tuning the thermoelectric properties. From the electron density distribution analysis, rigid heavy clusters weakly bonded with glue atoms are observed in the 1/1-AlReSi approximant crystal, whose physical properties are close to icosahedral Al–Pd–TM (TM: Re, Mn) quasicrystals. They are considered to be an intermediate state among the three typical solids: metals, covalently bonded networks (semiconductor), and molecular solids. Using the above picture and detailed effective mass analysis, we propose a guiding principle of weakly bonded rigid heavy clusters to increase the thermoelectric figure of merit (ZT) by optimizing the bond strengths of intra- and inter-icosahedral clusters. Through element substitutions that mainly weaken the inter-cluster bonds, a dramatic increase of ZT from less than 0.01 to 0.26 was achieved. To further increase ZT, materials should form a real gap to obtain a higher Seebeck coefficient.
... As a result, there exists a possibility that the pseudogap shown in figure 4(b) corresponds to a local valley within additional fine structure (sharp features) inside the global FsBz-interactions-induced pseudogap on the energy scale of 100 meV. Such a situation was considered by Takeuchi et al [34] in a study of the temperaturedependent thermoelectric power of Al 82.6−x Re 17.4 Si x cubic approximants (where the analysis was also performed with equations (6) and (7)), which explained the change of sign of the thermopower with increasing Si concentration. For the γ phase, fine structure within the FsBz-induced pseudogap on this energy scale was indeed predicted theoretically for the disorder-free stoichiometric Mg 17 Al 12 [15,33]. ...
Large-unit-cell complex metallic alloys (CMAs) frequently achieve stability by lowering the kinetic energy of the electron system through formation of a pseudogap in the electronic density of states (DOS) across the Fermi energy εF. By employing experimental techniques that are sensitive to the electronic DOS in the vicinity of εF, we have studied the stabilization mechanism of two binary CMA phases from the Al-Mg system: the γ-Mg17Al12 phase with 58 atoms in the unit cell and the β-Mg2Al3 phase with 1178 atoms in the unit cell. Since the investigated alloys are free from transition metal elements, orbital hybridization effects must be small and we were able to test whether the alloys obey the Hume-Rothery stabilization mechanism, where a pseudogap in the DOS is produced by the Fermi surface-Brillouin zone interactions. The results have shown that the DOS of the γ-Mg17Al12 phase exhibits a pronounced pseudogap centered almost exactly at εF, which is compatible with the theoretical prediction that this phase is stabilized by the Hume-Rothery mechanism. The disordered cubic β-Mg2Al3 phase is most likely entropically stabilized at high temperatures, whereas at lower temperatures stability is achieved by undergoing a structural phase transition to more ordered rhombohedral β' phase at 214 ° C, where all atomic sites become fully occupied. No pseudogap in the vicinity of εF was detected for the β' phase on the energy scale of a few 100 meV as determined by the 'thermal observation window' of the Fermi-Dirac function, so that the Hume-Rothery stabilization mechanism is not confirmed for this compound. However, the existence of a much broader shallow pseudogap due to several critical reciprocal lattice vectors [Formula: see text] that simultaneously satisfy the Hume-Rothery interference condition remains the most plausible stabilization mechanism of this phase. At Tc = 0.85 K, the β' phase undergoes a superconducting transition, which slightly increases the cohesive energy and may contribute to relative stability of this phase against competing neighboring phases.
A narrow-gap semiconductor with a complex crystal structure was recently discovered in the Al–Ru–Si system. To determine the homogeneity range of the semiconductor phase and further discover new phases, phase equilibria in the Al–Ru–Si system near 1200 K were investigated through prolonged-annealing experiments. Eleven new ternary phases including two incommensurate composite-crystalline and an icosahedral quasicrystalline phases were identified using powder and single-crystal X-ray diffraction, and their compositions at two-phase and three-phase equilibria were evaluated by means of electron-probe X-ray microanalysis. On the basis of the data obtained in this study and those adopted from the literature, a tentative isothermal section of the Al–Ru–Si equilibrium phase diagram near 1200 K was drawn.
Tentative isothermal section of the Al–Ru–Si equilibrium phase diagram near 1200 K. Fullsize Image
Scanning tunneling microscopy (STM) and transport measurements have been performed to investigate the electronic structure and its temperature dependence in heavily Sr and Na codoped PbTe, which is recognized as one of the most promising thermoelectric (TE) materials. Our main findings are as follows: (i) Below T=4.5 K, all carriers are distributed in the first valence band at the L point (L band), which forms tube-shaped Fermi surfaces with concave curvature. With Sr and Na doping, the dispersion of the L band changes, and the band gap increases from EG≅200 to 300meV. (ii) At T=4.5 K, the Fermi energy is located ∼100 meV below the edge of the L band for the Sr/Na codoped PbTe. The second valence band at the Σ point (Σ band) is lower than the L band by ΔELΣ≅150 meV, which is significantly smaller than that of pristine PbTe (ΔELΣ≅200 meV). The decrease in ΔELΣ, leading to band convergence, provides a desirable condition for TE materials. (iii) With increasing temperature, the carrier distribution to the Σ band starts at T≅100 K, and we estimate that ∼50% of the total carriers are redistributed in the Σ band at T=300 K. In this paper, we demonstrate that STM and angular-dependent magnetoresistance measurements are particularly powerful tools to determine the electronic structure and carrier distribution. We believe that they will provide a bird's eye view of the doping strategy toward realizing high-efficiency TE materials.
