Anisotropic magnetic and transport properties of orthorhombic ${\text{Al}}_{13}{\text{Co}}_4$

We have investigated anisotropic physical properties (magnetic susceptibility, electrical resistivity, thermoelectric power, Hall coefficient, and thermal conductivity) of the $o{\text{-Al}}_{13}{\text{Co}}_4$, an orthorhombic approximant to the decagonal phase. The crystallographic-direction-dependent measurements were performed along the $a$, $b$, and $c$ directions of the orthorhombic unit cell, where $\left(b,c\right)$ atomic planes are stacked along the perpendicular $a$ direction. Magnetic susceptibility is predominantly determined by the Pauli-spin paramagnetism of conduction electrons. The in-plane magnetism is stronger than that along the stacking $a$ direction. Anisotropic electrical and thermal conductivities are the highest along the stacking $a$ direction. The anisotropic thermoelectric power changes sign with the crystallographic direction and so does the anisotropic Hall coefficient which changes from negative electronlike to positive holelike for different combinations of the electric current and magnetic-field directions. The investigated anisotropic electrical and thermal transport coefficients were reproduced theoretically by ab initio calculation using Boltzmann transport theory and the calculated anisotropic Fermi surface. The calculations were performed for two structural models of the $o{\text{-Al}}_{13}{\text{Co}}_4$ phase, where the more recent model gave better agreement, though still qualitative only, to the experiments.

Figure 1

(Color online) Microstructure of the large area of an as-grown ingot of $o{\text{-Al}}_{13}{\text{Co}}_4$: optical investigations in nonpolarized (bottom left) and polarized (bottom right) light show the absence of grain boundaries, thus confirming single-crystalline character of the material.

Figure 2

(Color online) (i) Magnetization $M$ of $o{\text{-Al}}_{13}{\text{Co}}_4$ as a function of the magnetic field $H$ at $T=5\text{K}$ with the field oriented along three orthogonal crystallographic directions $a$, $b$, and $c$. (ii) Temperature-dependent magnetic susceptibility $\chi =M/H$ in the field $H=1\text{kOe}$ applied along the three crystallographic directions. Both zfc and fc runs are shown.

Figure 3

(Color online) Analysis of the $M\left(H\right)$ curves from Fig. 2 [panel (i): $a$ direction, panel (ii): $b$ direction, and panel (iii): $c$ direction]. Solid curves are fits with Eq. (1), dashed lines represent the term of the magnetization linear in the magnetic field, $kH$, (the Larmor core contribution and the contribution due to the conduction electrons) and dotted curves are the FM contribution. The fit parameters are given in Table .

Figure 4

(Color online) Temperature-dependent electrical resistivity of $o{\text{-Al}}_{13}{\text{Co}}_4$ along three orthogonal crystallographic directions $a$, $b$, and $c$.

Figure 5

(Color online) Temperature-dependent thermoelectric power (the Seebeck coefficient $S$) of $o{\text{-Al}}_{13}{\text{Co}}_4$ along three orthogonal crystallographic directions $a$, $b$, and $c$.

Figure 6

(Color online) Anisotropic temperature-dependent Hall coefficient $R_H=E_y/j_x{B}_z$ of $o{\text{-Al}}_{13}{\text{Co}}_4$ for different combinations of directions of the current $j_x$ and magnetic field $B_z$ (given in the legend). The superscript $a$, $b$, or $c$ on $R_H$ denotes the direction of the magnetic field.

Figure 7

(Color online) (i) Total thermal conductivity $\kappa$ of $o{\text{-Al}}_{13}{\text{Co}}_4$ along the three crystallographic directions $a$, $b$, and $c$. Electronic contributions ${\kappa }_{\text{el}}$, estimated from the Wiedemann-Franz law, are shown by solid curves. Note the logarithmic temperature scale. (ii) Phononic thermal conductivity ${\kappa }_{\text{ph}}=\kappa -{\kappa }_{\text{el}}$ along the three crystallographic directions.

Figure 8

(Color online) Fermi surface in the first Brillouin zone, calculated ab initio for (i) the original $o{\text{-Al}}_{13}{\text{Co}}_4$ model [Grin et al. (Ref. 16)] and (ii) the new model [Grin et al. (Ref. 23)]. Orientation of the (orthogonal) reciprocal-space axes $a^{\ast }$, $b^{\ast }$, and $c^{\ast }$ is also shown.

Figure 9

(Color online) Theoretical anisotropic ${\rho }_a\tau$, ${\rho }_b\tau$, and ${\rho }_c\tau$ (products of the electrical resistivity and the relaxation time), calculated ab initio for (i) the original $o{\text{-Al}}_{13}{\text{Co}}_4$ model [Grin et al. (Ref. 16)] and (ii) the new model [Grin et al. (Ref. 23)].

Figure 10

(Color online) Theoretical anisotropic ${\kappa }_{\text{el}}^a/\tau$, ${\kappa }_{\text{el}}^b/\tau$, and ${\kappa }_{\text{el}}^c/\tau$ (electronic thermal conductivities divided by the relaxation time), calculated ab initio for (i) the original $o{\text{-Al}}_{13}{\text{Co}}_4$ model [Grin et al. (Ref. 16)] and (ii) the new model [Grin et al. (Ref. 23)].

Figure 11

(Color online) Theoretical anisotropic thermopowers $S_a$, $S_b$, and $S_c$, calculated ab initio for (i) the original $o{\text{-Al}}_{13}{\text{Co}}_4$ model [Grin et al. (Ref. 16)] and (ii) the new model [Grin et al. (Ref. 23)].

Figure 12

(Color online) Theoretical anisotropic Hall coefficient, calculated ab initio for (i) the original $o{\text{-Al}}_{13}{\text{Co}}_4$ model [Grin et al. (Ref. 16)] and (ii) the new model [Grin et al. (Ref. 23)]. $R_H$ was calculated for the same set of current and field directions as employed in the experimental measurements of the Hall coefficient shown in Fig. 6. The superscript $a$, $b$, or $c$ on $R_H$ denotes the direction of the magnetic field. There are three groups of curves, each consisting of two practically indistinguishable curves (see text).