Intrinsic anisotropic magnetic, electrical, and thermal transport properties of $d$-Al-Co-Ni decagonal quasicrystals

To address the questions on the anisotropy of bulk physical properties of decagonal quasicrystals and the intrinsic physical properties of the $d$-Al-Co-Ni phase, we investigated the anisotropic magnetic susceptibility, the electrical resistivity, the thermoelectric power, the Hall coefficient, and the thermal conductivity of a $d$-Al-Co-Ni single crystal of exceptional structural quality. Superior structural order on the local scale of atomic clusters was confirmed by ${}^{27}$Al nuclear magnetic resonance spectroscopy. The measurements were performed in the 10-fold periodic direction of the structure and in three specific crystallographic directions within the quasiperiodic plane, corresponding to the 2 and 2′ twofold symmetry directions and their bisector. The specific heat, being a scalar quantity, was determined as well. The measurements of the second-rank bulk tensorial properties confirm the theoretical prediction that a solid of decagonal point group symmetry should exhibit isotropic physical properties within the quasiperiodic plane and anisotropy between the in-plane and the 10-fold directions. $d$-Al-Co-Ni is an anisotropic diamagnet with stronger diamagnetism for the magnetic field in the 10-fold direction. Electrical and thermal transport is strongly metallic in the 10-fold direction but largely suppressed within the quasiperiodic plane, the main reason being the lack of translational periodicity that hinders the propagation of electrons and phonons in a nonperiodic lattice. The third-rank Hall-coefficient tensor shows sign-reversal anisotropy related to the direction of the magnetic field when applied in the 10-fold direction or within the quasiperiodic plane. The observed anisotropy is not a peculiarity of quasicrystals but should be a general feature of solids with broken translational periodicity in two dimensions.

Figure 1

(a) The quasiperiodic plane of the $d$-Al-Co-Ni structure with the 20 twofold directions, where the directions of set 2 are given by solid lines and those of set 2′ by dashed lines. The particular directions $\left[01000\right]$ of set 2 and $\left[10\overline{1}00\right]$ of set 2′, which are orthogonal to each other and were employed as the measurement directions, are shown in bold. (b) An idealized polyhedron of the $d$-Al-Co-Ni QC of point group 10/mmm, using the concept of net planes (Ref. 17). Each of the 20 side facets of the polyhedron is normal to one of the twofold directions shown in panel (a).

Figure 2

(a) The crystallographic directions of the four bar-shaped $d$-Al-Co-Ni samples used in the measurements of the anisotropic physical properties. Here, $\left[00001\right]$ is a 10-fold direction, $\left[01000\right]$ is a 2 twofold direction, $\left[10\overline{1}00\right]$ is a 2′ twofold direction, $\left[20\overline{1}00\right]$ is a bisector of two neighboring 2 and 2′ directions lying at an angle of 9° to each of them (denoted as the bis direction), and $\left[13110\right]$ is an in-plane direction perpendicular to the 10-fold and the bis directions. (b) The choice of axes of a Cartesian $x$,$y$,$z$ crystal-fixed coordinate system for symmetry analysis of a general second-rank tensor of decagonal point group 10/mmm symmetry.

Figure 3

The ${}^{27}$Al frequency-swept NMR spectrum of $d$-Al-Co-Ni in a magnetic field $B_0$ $=$ 9.39 T at the temperature $T$ $=$ 80 K for the field parallel to a 2′ twofold direction ($B_0$∥2′). The reference frequency on the horizontal scale corresponds to the ${}^{27}$Al Larmor frequency ${\nu }_{\mathrm{ref}}$ $=$ 104.101 MHz. The dashed box encloses part of the satellite spectrum used to determine the intensity variation upon rotating the crystal (see the text).

Figure 4

(a) An expanded part of the ${}^{27}$Al NMR satellite spectrum (the part enclosed in a dashed box in Fig. 3) for rotation about the 2 twofold direction for three particular angles, $\alpha$ $=$ 0°, 45°, and 90°, between the 10-fold axis and the magnetic field. (b) The orientation-dependent intensity variation in the interval $\alpha$ $=$ [0°, 180°] recorded at the frequency ${\nu }_1$ (marked by a dashed arrow in panel (a)). (c) and (d) The corresponding spectrum and the intensity variation for the rotation about the 2′ twofold direction.

Figure 5

(a) An expanded part of the ${}^{27}$Al NMR satellite spectrum (the part enclosed in a dashed box in Fig. 3) for rotation about the 10-fold direction for three particular angles, $\alpha$ $=$ 0°, 18°, and 24°, between the 2 twofold axis and the magnetic field. (b) and (c) The orientation-dependent intensity variation in the interval $\alpha$ $=$ [0°, 36°] recorded at the frequencies ${\nu }_1$ and ${\nu }_2$ (marked by dashed arrows in panel (a)).

Figure 6

(a) The magnetization versus the magnetic field curves $M\left(H\right)$ of $d$-Al-Co-Ni determined at $T$ $=$ 5 K for the magnetic field sweep of ±50 kOe applied in the 10, 2, 2′, and bis crystallographic directions. (b) The magnetic susceptibility $\chi =M/H$ determined in the temperature range 1.9–300 K in the magnetic field $H$ $=$ 1 kOe. The index Q is conveniently used for the three investigated in-plane directions (2, 2′, and bis).

Figure 7

(a) Temperature-dependent electrical resistivity $\rho \left(T\right)$ measured in the 10, 2, 2′, and bis directions. (b) The resistivities normalized to their 340 K values, $\rho /{\rho }_{340\mathrm{K}}$. The index Q is conveniently used for the three investigated in-plane directions (2, 2′, and bis).

Figure 8

Temperature-dependent thermoelectric power ($S$), measured in the 10, 2, 2′, and bis directions. The index Q is conveniently used for the three investigated in-plane directions (2, 2′, and bis).

Figure 9

Temperature-dependent Hall coefficient $R_H=E_y/j_x{B}_z$. Eight sets of experimental data were collected by directing the current $j_x$ along the long axes of the samples, thus in the 10, 2, 2′, and bis directions, whereas the magnetic field $B_z$ was directed in the other two orthogonal directions of each sample (details of the current and field directions are given in the legend). The eight $R_H$ data sets form two groups of identical Hall coefficients. The first group (denoted as $R_H^Q$) contains five data sets, all corresponding to the application of the magnetic field in the quasiperiodic plane, thus in the 2, 2′, and bis directions. The second group (denoted as $R_H^{10}$) contains three sets in which the magnetic field was parallel to the 10-fold direction.

Figure 10

Low-temperature molar specific heat in a $C/T$ versus $T^2$ plot. The solid line is the fit with Eq. (2). The specific heat in the entire investigated temperature range (2–300 K) is displayed in the inset.

Figure 11

Thermal conductivity $\kappa$ in the 10, 2, 2′, and bis directions. The index Q is conveniently used for the three investigated in-plane directions (2, 2′, and bis).

Figure 12

(a) Theoretical fit of the normalized ${\rho }_Q\left(T\right)$ from Fig. 7b by using Eq. (4) of the slow-charge-carriers model. The fit parameter values are given in the text. (b) Theoretical ${\rho }_Q\left(T\right)$ curves normalized to the ${\rho }_0=\rho \left(T=0\right)$ value for the cases in which the transition from a ballistic to a nonballistic (diffusive) regime would occur at lower temperatures because of stronger disorder in the material (shorter relaxation time $\tau$). The original fit ($C$ $=$ 8.63 × 10${}^{-4}$ and $\alpha$ $=$ 1.0) from panel (a) up to the highest temperature of the experimental measurements (340 K) is shown by the bold curve. The modified $C$ and $\alpha$ parameter values of the other curves are given in the legend.