Extremely small thermal conductivity of the Al-based Mackay-type 1/1-cubic approximants
Tsunehiro Takeuchi*, Naoyuki Nagasako**, Ryo ji Asahi**,Uichiro Mizutani***
*EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan
**Toyota Central R&D Labs., Inc., Nagakute, Aichi 480-1192, Japan
***Toyota Physical & Chemical Research Institute, Nagakute, Aichi 480-1192, Japan

Icosahedral quasicrystals are widely known to possess a low thermal conductivity as low as 1 W/m K and a large thermoelectric power up to 100   V/K. Those conditions together with the low electrical resistivity are necessities for the thermoelectric materials. The icosahedral quasicrystals, therefore, are regarded as one of the most plausible candidates for the practical thermoelectric materials. To utilize the icosahedral quasicrystals in thermoelectric devices, it is of great importance to investigate the mechanism leading to the small thermal conductivity. In order to investigate the origin of the low thermal conductivity observed for the Al-based Mackay-type icosahedral quasicrystals, we employed, in this study, their corresponding 1/1-cubic approximants. We performed the synchrotron radiation Rietveld analysis and a first principle calculation of the phonon-dispersion on the basis of the refined structure parameters. The mechanism leading to the considerable reduction in the thermal conductivity were discussed in terms of the local atomic arrangements and phonon-dispersion thus obtained. The contribution of quasiperiodicity to the small thermal conductivity is also discussed by comparing the measured thermal conductivity of the approximants with previously reported ones of the quasicrystals.

Thermal conductivity of the Al-based Mackay-type 1/1-cubic approximants was investigated over a wide temperature range from 2K to 300K. Figures 1 (a1)-(a4) show the measured thermal conductivity (markers) and the electron thermal conductivity (lines) of the 1/1-cubic approximants. The lattice thermal conductivity and the electrical resistivity are also shown in Fig.2 (b1)-(b4) and (c1)-(c4), respectively. All samples possess very small thermal conductivity less than 4.5 W/m K over the whole temperature range of the measurement. The magnitude of the thermal conductivity for the Al82.6-xRe17.4Six, the Al71.6Mn17.4Si11, and the Al73.6Mn4.9Fe12.5Si9 are especially small, possessing less than 2 W/m K. These small values are comparable with those reported for the Al-based icosahedral quasicrystals. It is suggested, therefore, that the quasiperiodicity specific only to the quasicrystals does not play crucially important role in reducing the thermal conductivity.

Figures 2 (a) and (b) show the phonon dispersion of the Al73.6Re17.4Si9 1/1-cubic approximant calculated with the Rietveld refined structure parameters. The lattice specific heat calculated from the phonon dispersion quantitatively reproduces the measured one in its magnitude and temperature dependence as shown in Fig.2 (c). This fact strongly indicates reliability of the phonon calculation. The first Brillouin zone of the approximant has small volume in the reciprocal space because of their large lattice constant (1213 ). Three acoustic branches, one longitudinal and two transverse modes, are clearly observed at the lowest energy region. Those acoustic phonon branches are limited in very low energy range below 8 meV, and the optical phonon branches are densely generated above 10 meV.

These characteristics in the phonon-dispersion reduce the thermal conductivity in two different ways; (a) reducing the mean group velocity and (b) intensively activating umklapp scattering. The mean group velocity of the phonons is reduced by the excited optical phonons because optical phonons generally possess a much slower group velocity than the sound velocity . In addition, umklapp processes are intensively generated even at low temperature because the small Brillouin zone allows the phonons to be excited near the zone boundary where the umklapp processes are easily generated. This mechanism well accounts for the small lattice thermal conductivity of the disordering-free Al73.6Re17.4Si9 1/1-cubic approximant.

We calculated lattice thermal conductivity using a simple model of phonon dispersion. Calculated specific heat, mean group velocity, mean free path, and the lattice thermal conductivity are shown in Fig.3. The role of the number of atoms in the unit cell is observable in Fig.3 (a1)-(a4), in which data calculated with four different sets of (N, a) = (4, 3.93), (26, 7.3), (58, 9.6), and (138, 12.8) are displayed. Those conditions were determined from f.c.c. (Al), Al12Re, Al5Re24, and the Al73.6Re17.4Si9 1/1-cubic approximant with keeping the density almost constant. Obviously lattice thermal conductivity drastically decreases with increasing N and a. Figures 3 (b1)-(b4) show the calculated variation of specific heat, mean group velocity, mean free path, and lattice thermal conductivity with varying Debye temperature. Four different Debye temperatures, 400, 600, 800, and 1000K, were used in the calculation. It is clearly understood from the calculated data that the lattice thermal conductivity decreases with decreasing Debye-temperature because of the reduction in mean free path and avaraged group velocity.

The Al-based Mackay-type icosahedral quasicrystals have essentially the same behaviors of the thermal conductivity with those of the present approximants. The presence of the well-defined phonons and the pseudo Brillouin zone (PBZ) were already reported as characteristics of the quasicrystals. Resemblance of the diffraction patterns between quasicrystals and approximants suggests that PBZ in the quasicrystals has the similar shape and size as those of FBZ of the approximants. The umklapp processes over the PBZ, therefore, would frequently take place and effectively reduce the lattice thermal conductivity as those in FBZ of the approximants. Although neither presence of the vacancies nor its effect on the latticer thermal conductivity has not been revealed for the quasicrystals yet, we strongly expect that vacancies do exist in the structure of some quasicrystals and effectively reduce the latticer thermal conductivity, because large variation in the latticer thermal conductivity was also reported for the quasicrystals.

As a result of the present analysis with the first-principles phonon-calculation and the simplified model of phonons, we conclude that the small thermal conductivity of the icosahedral quasicrystals and their approximants is mainly brought about by the densely generated optical phonon branches at low energies in associated with their large lattice constant and the large number of atoms in the unit cell. The presence of the vacancies in the structure enhances the probability of phonon scatterings and reduces the mean free path of propagating phonons to further reduce the lattice thermal conductivity.

[Published in Physical Review B 74, 054206 1-12 (2006).]

Fig. 1 (a1)-(a4) Measured thermal conductivity (markers) and electron thermal conductivity (lines), (b1)-(b4) the lattice thermal conductivity, and (c1)-(c4) the electrical resistivity of the Al82.6-xRe17.4Six, Al82.6-xMn17.4Six, Al76-xCu7Fe17Six, and Al73.6Mn17.4-yFeySi9 1/1- cubic approximants.

Fig. 2 (a) The calculated phonon dispersion of the Al73.6Re17.4Si9 1/1-cubic approximant. The calculated phonon-dispersion at the low energy range below 15meV is magnified in (b). (c) Specific heat C calculated from the phonon dispersion together with the measured C and that of the Debye model. The calculated C quantitatively reproduced the measured one, strongly indicating reliability of the calculated phonon-dispersion. Only at high temperatures above 200K, the lattice thermal expansion, which was not taken into account in the present calculation, causes the very minor deviation of the calculated C from the measured one. 

Fig. 3 Simulated specific heat C, mean group velocity ave, mean free path , and the lattice thermal conductivity lat. Data calculated with four different sets of (N, a) = (4, 3.93), (26, 7.3), (58, 9.6), and (138, 12.8) are displayed. Obviously lat drastically decreases with increasing N and a. (b1)-(b4) The variation of C, ave, , and lat. with varying Debye temperature D. Four different Debye temperatures, D = 400, 600, 800, and 1000K, were employed. It is clearly understood from the calculated data that lat decreases with decreasing D because of the reduction in and ave.