Origin of anisotropic nonmetallic transport in the ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ decagonal approximant

We present a study of the anisotropic transport properties (electrical resistivity, thermoelectric power, Hall coefficient, and thermal conductivity) of a single-crystalline ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ complex metallic alloy that is an excellent approximant to the decagonal quasicrystal with six atomic layers in one periodic unit. Temperature-dependent electrical resistivity along the $b$ and $c$ crystalline directions shows a nonmetallic behavior with a broad maximum, whereas it shows a metallic positive temperature coefficient along the $a$ direction perpendicular to the $(b,c)$ atomic planes. Ab initio calculations of the electronic density of states reveal that the nonmetallic transport occurs in the presence of a high density of charge carriers. The very different temperature-dependent electrical resistivities along the three crystalline directions can all be treated within the same physical model of slow charge carriers due to weak dispersion of the electronic bands, where the increased electron-phonon scattering upon raising the temperature induces transition from dominant Boltzmann (metallic) to dominant non-Boltzmann (insulatinglike) regime. The temperature dependence of the resistivity is governed predominantly by the temperature dependence of the electronic diffusion constant $D$ and the transition has no resemblance to the Anderson-type metal-to-insulator transition based on the gradual electron localization. Structural considerations of the ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ phase show that the anisotropy of the transport properties is a consequence of anisotropic atomic order on the scale of nearest-neighbor atoms, suggesting that the role of quasiperiodicity in the anisotropic transport of decagonal quasicrystals is marginal. We also present a relaxed version of the ${\mathrm{Al}}_4(\mathrm{Cr},\mathrm{Fe})$ structural model by Deng et al. [J. Phys.: Condens. Matter 16, 2283 (2004)].

Figure 1

(Color online) Temperature-dependent electrical resistivity of ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ along the three crystalline directions $a$, $b$, and $c$ of the orthorhombic unit cell. Solid lines are fits with Eq. (2) and the fit parameter values are given in Table .

Figure 2

(Color online) Thermoelectric power of ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ along the three crystalline directions $a$, $b$, and $c$ of the orthorhombic unit cell.

Figure 3

Temperature-dependent Hall coefficient of ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ for different combinations of directions of the current $j_x$ and the magnetic field $B_z$ (given in the legend).

Figure 4

(Color online) Atomic positions of the relaxed model of ${\mathrm{Al}}_4(\mathrm{Cr},\mathrm{Fe})$, shown by large gray (online colored) circles. The original coordinates published by Deng et al. (Ref. 16) are indicated by black dots. The panels show (a) the $x=0$ (flat layer $F$), (b) the $x\approx a∕6$ (puckered layer $P$), and (c) the $y=0$ planes.

Figure 5

Theoretical electronic DOSs of the ${\mathrm{Al}}_4(\mathrm{Cr},\mathrm{Fe})$ phase, calculated ab initio for the original (unrelaxed) structural model of Deng et al. (thick gray curve) and our relaxed model (thin black curve). The gray hatched area around $E_F$ indicates the energy window that was used to calculate the electronic charge density shown in Fig. 8.

Figure 6

(Color online) Lattice thermal conductivity ${\kappa }_{\mathit{ph}}$ of ${\mathrm{Al}}_{80}{\mathrm{Cr}}_{15}{\mathrm{Fe}}_5$ along the three crystalline directions $a$, $b$, and $c$ of the orthorhombic unit cell. The inset shows the total thermal conductivity $\kappa$ and the electronic contribution ${\kappa }_{\mathit{el}}$ along the three crystalline directions on the logarithmic temperature scale.

Figure 7

(Color online) Theoretical temperature-dependent electrical resistivities [normalized to ${\rho }_0=\rho (T=0)$ value] calculated from Eq. (2) for different combinations of the parameters $B∕A$, $C$, and $\alpha$ (the subscript $j$, denoting the crystalline direction, has been omitted). In panel (i), the ratio $B∕A$ is varied while keeping fixed $C=0.03$ and $\alpha =1.2$. In panel (ii), $B∕A=0.1$ and $\alpha =0.65$ are fixed and $C$ is varied. Very similar temperature-dependent resistivities are also obtained by keeping $B∕A$ and $C$ constant and varying $\alpha$. This is due to the fact that $C$ and $\alpha$ appear in a product $C{T}^{\alpha }$ in Eq. (2). Panel (iii) shows the case where $B∕A=1\times {10}^{-7}$ (corresponding to the extreme Boltzmann regime) and $C=0.03$ are kept constant and $\alpha$ is varied. Similar temperature dependencies are obtained by keeping $B∕A$ and $\alpha$ constant and varying $C$. Panel (iv) shows comparison of the temperature-dependent resistivities for two models: the model of slow charge carriers of Eq. (2) using the set of parameters $A=3.03\times {10}^{-4}\mu \Omega \mathrm{cm}$, $B∕A=0.51$, $C=0.065$, and $\alpha =1.0064$ and the model of weak localization, $\sigma ={\sigma }_0+\Delta {\sigma }_{WL}$, with $\Delta {\sigma }_{WL}$ given by Eq. (3). For the weak localization curve, we applied the set of parameters used previously (Ref. 24) to reproduce the temperature-dependent resistivity of icosahedral ${\mathrm{Al}}_{70.5}{\mathrm{Pd}}_{21.2}{\mathrm{Mn}}_{8.3}$ QC: ${\sigma }_0=4.6\times {10}^{-4}{\left(\mu \Omega \mathrm{cm}\right)}^{-1}$, $e^2∕\left(2{\pi }^2\hslash \sqrt{D{\tau }_{\mathit{so}}}\right)=1.55\times {10}^{-4}{\left(\mu \Omega \mathrm{cm}\right)}^{-1}$, and, by rewriting $t={(T∕T_0)}^{\alpha }$, $T_0=207\mathrm{K}$ and $\alpha =1.66$. The two curves are indistinguishable.

Figure 8

(Color online) Electronic charge density of the relaxed model of ${\mathrm{Al}}_4(\mathrm{Cr},\mathrm{Fe})$ calculated in the energy interval $(E_F-0.5\mathrm{eV},E_F+0.5\mathrm{eV})$. The panels show (a) the $x=0$ (flat layer $F$), (b) the $x\approx a∕6$ (puckered layer $P$), and (c) the $y=0$ planes.

Figure 9

(i) Visualization of the atomic chains for ${\theta }_{\max }=32°$ along the $a$ direction of the ${\mathrm{Al}}_4\mathrm{TM}$ supercell composed of two unit cells. Pure Al chains are drawn by thicker lines. Histograms, showing the number of chains in the supercell and their distribution according to the $N_{\mathit{TM}}∕N_{\mathit{\text{chain}}}$ criterion for various ${\theta }_{\max }$, are displayed in other three panels [panel (ii), $a$ direction; panel (iii), $b$ direction; panel (iv), $c$ direction]. $N_{\mathit{TM}}∕N_{\mathit{\text{chain}}}$ is given in %.