Metallic–covalent bonding conversion and thermoelectric properties of Al-based icosahedral quasicrystals and approximants

Figures & data

Figure 1. Characteristic features and potential applications of Al- and B-based icosahedral cluster solids (ICS).

Table 1. Comparison between structurally simple and complex solids.

Structurally simple solids	Structurally complex solids
Small unit-cell crystals	Giant unit-cell crystals, quasicrystals, amorphous solids
Sites for each element: only one	Many kinds

Figure 2. Icosahedral cluster solids (ICS) as structurally complex solids.

Table 2. Comparison between (a) the metal–insulator transition and (b) metallic–covalent bonding conversion.

		(a)
	Metal	Insulator (semiconductor)
Density of states at EF, D(EF)	Finite	0 (n or D(EF) = 0)
	and	or
Wave function at EF	Extended	Localized (μ or vF = 0)
(b)
	Metallic bond	Covalent bond
Share of electrons	Many atoms	2 atoms
Bond direction	Isotropic	Directed
Packing fraction	High (∼0.75)	Low (∼0.34)
Typical structure	Closed-packed structure fcc, hcp	Diamond structure
Icosahedral cluster	13 atoms, Al13	12 atoms, Al12

The electrical conductivity () is expressed as , where , , , , , , are the carrier density, elemental charge, mobility, relaxation time, Fermi velocity, density of states, and Fermi energy, respectively.

Figure 3. Equidensity surfaces (0.25–0.40 e Å⁻³) of the electron density for (a) Si (covalent bonding nature) and (b) Al (metallic bonding nature).

Figure 4. Equidensity surfaces (0.25–0.40 e Å⁻³) of the electron density for icosahedral clusters of (a) the 1/1-AlReSi approximant and (b) the 1/0-Al₁₂Re approximant [13].

Figure 5. (a) Equidensity surfaces (0.35 e Å⁻³) of the electron density of the first- and second-shell clusters and (b) electron density at the bond midpoint in the 1/1-AlReSi approximant, together with the data of 1/0-Al₁₂Re, and pure Si and Al [13].

Figure 6. Bonding nature of the first-shell Al icosahedral cluster and the electronic density of states of (a) 1/1-AlReSi and (b) 1/0-Al₁₂Re approximants. (c) The temperature dependence of the electrical resistivity of the icosahedral Al–Pd–Re quasicrystal, and 1/1-AlReSi, 1/1-AlMnSi, and 1/0-Al₁₂Re approximants.

Table 3. Minimum electronic density of states near E _F and calculated density of states at E _F using the free electron model for 1/1-AlReSi and 1/0-Al₁₂Re approximants.

	D(E)min (states/eV atom) near E F	D(E F) (free electron model) (states/(eV atom))
1/1-AlReSi	0.04	0.40
1/0-Al12Re	0.20	0.40

Figure 7. (a) Main building block, Mackay icosahedral (MI) clusters, and glue atoms (magenta atoms) in Al-based icosahedral approximants and quasicrystals. (b) Diagram of the guiding principle of weakly bonded rigid heavy clusters (WBRHCs). The effective mass (m*) increases by weakening of inter-cluster bonds (V _o) or strengthening of intra-cluster bonds (V _i). The composition dependences of the effective mass ratio ((c) and (d)) and the maximum dimensionless figure of merit ((e) and (f)) for Al–Pd–Re ((c) and (e)) and Al₇₁Pd₂₀(Re_1 − x Ru _x )₉ ((d) and (f)) quasicrystals. m _e* and m _h* are the effective masses of the electron and hole, respectively, obtained from fitting the data in figure 8 by the two-band model, and m _e is the electron mass in vacuum. e/a is the mean number of valence electrons per atom ratio, and decreases with increasing transition metal concentration. The letters A, B, C, and D in (b)–(f) correspond to each other [53].

Figure 8. Temperature dependence of (a) electrical conductivity and (b) Seebeck coefficient for Al₇₁Pd₂₀(Re_1 − xRu_x)₉. The solid lines are fitted curves obtained using the two-band model [53].

Figure 9. Transition metal elements that are candidates for substitution in the Al–Pd–Re quasicrystal. The bond strength will decrease with decreasing atomic number in congener. Route (1) is Ru and Fe substitution in the Re site, and route (2) is full replacement of Mn in Re sites.

Figure 10. Dimensionless figure of merit (ZT) as a function of temperature for (a) Al₇₁Pd₂₀(Re_1 − xRu_x)₉ (x = 0, 0.4, 0.55, 0.7) [53], (b) Al₇₁Pd₂₀(Re_1 − xFe_x)₉ (x = 0, 0.2, 0.4, 0.65) [54], and (c) Al₇₁Pd₂₀(Mn, Re)₉ [55] quasicrystals.

Figure 11. (a) Typical example of the porous microstructure in arc-melted and annealed Al–Pd–Re quasicrystal. The black area represents macroscopic pores and/or cracks. (b) That of the dense microstructure in SPS sintered one. The magnifications are the same for (a) and (b).

Figure 12. (a) Ga substitution of Al sites in the Al–Pd–Mn quasicrystal to weaken the inter-cluster bonds. (b) The thermal conductivity (κ) and (c) the dimensionless figure of merit (ZT) as a function of temperature for Al_71 − xGa_xPd₂₀Mn₉ (x = 0, 2, 3) [56].

Figure 13. Electronic density of states (DOS) for Al–(Ga)–Pd–Mn quasicrystals using the Quandt–Elser model [57–59]. The inset shows the DOS in the vicinity of the Fermi energy.

Table 4. Average speed of sound (v_s), Debye temperature (θ_D), relative change in the relaxation time of the phonon (Δτ_p/τ_p), and phonon thermal conductivity (κ_p) for Al_71 − xGa_xPd₂₀Mn₉ quasicrystals (x = 0, 2, 3) at room temperature [55].

Sample	vs (m s^−1)	θD (K)	Δτp/τp (%)	κp (W/(m K))
x = 0	4560	526	—	0.84
x = 2	4530	522	∼−10	0.75
x = 3	4220	486	∼−35	0.30

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