High-entropy alloys (HEAs) are multicomponent mixtures of elements in similar concentrations, where the high entropy of mixing can stabilize disordered solid-solution phases with simple structures like a body-centered cubic or a face-centered cubic, in competition with ordered crystalline intermetallic phases. We have synthesized an HEA with the composition ${\mathrm{Ta}}_{34}{\mathrm{Nb}}_{33}{\mathrm{Hf}}_8{\mathrm{Zr}}_{14}{\mathrm{Ti}}_{11}$ (in at. %), which possesses an average body-centered cubic structure of lattice parameter $a=3.36\AA$. The measurements of the electrical resistivity, the magnetization and magnetic susceptibility, and the specific heat revealed that the ${\mathrm{Ta}}_{34}{\mathrm{Nb}}_{33}{\mathrm{Hf}}_8{\mathrm{Zr}}_{14}{\mathrm{Ti}}_{11}$ HEA is a type II superconductor with a transition temperature $T_c\approx 7.3\mathrm{K}$, an upper critical field ${\mu }_0{H}_{c2}\approx 8.2\mathrm{T}$, a lower critical field ${\mu }_0{H}_{c1}\approx 32\mathrm{mT}$, and an energy gap in the electronic density of states (DOS) at the Fermi level of $2\mathrm{\Delta }\approx 2.2\mathrm{meV}$. The investigated HEA is close to a BCS-type phonon-mediated superconductor in the weak electron-phonon coupling limit, classifying it as a “dirty” superconductor. We show that the lattice degrees of freedom obey Vegard’s rule of mixtures, indicating completely random mixing of the elements on the HEA lattice, whereas the electronic degrees of freedom do not obey this rule even approximately so that the electronic properties of a HEA are not a “cocktail” of properties of the constituent elements. The formation of a superconducting gap contributes to the electronic stabilization of the HEA state at low temperatures, where the entropic stabilization is ineffective, but the electronic energy gain due to the superconducting transition is too small for the global stabilization of the disordered state, which remains metastable.

Rare earth based equimolar Ho-Dy-Y-Gd-Tb hexagonal high-entropy alloy (HEA) is a prototype of an ideal HEA, stabilized by the entropy of mixing at any temperature with random mixing of elements on the hexagonal close-packed lattice. In order to determine intrinsic properties of an ideal HEA characterized by the enormous chemical (substitutional) disorder on a weakly distorted simple lattice, we have performed measurements of its magnetic and electrical response and the specific heat. The results show that the Ho-Dy-Y-Gd-Tb hexagonal HEA exhibits a rich and complex magnetic field-temperature $\left(H,T\right)$ phase diagram, as a result of competition among the periodic potential arising from the electronic band structure that favors periodic magnetic ordering, the disorder-induced local random potential that favors spin glass-type spin freezing in random directions, the Zeeman interaction with the external field that favors spin alignment along the field direction, and the thermal agitation that opposes any spin ordering. Three characteristic temperature regions were identified in the $\left(H,T\right)$ phase diagram between room temperature and 2 K. Within the upper temperature region I (roughly between 300 and 75 K), thermal fluctuations average out the effect of local random pinning potential and the spin system behaves as a pure system of compositionally averaged spins, undergoing a thermodynamic phase transition to a long-range ordered helical antiferromagnetic state at the Néel temperature $T_N=180\mathrm{K}$ that is a compositional average of the Néel temperatures of pure Tb, Dy, and Ho metals. Region II (between 75 and 20 K) is an intermediate region where the long-range periodic spin order “melts” and the random ordering of spins in the local random potential starts to prevail. Within the low-temperature region III (below 20 K), the spins gradually freeze in a spin glass configuration. The spin glass phase appears to be specific to the rare earths containing hexagonal HEAs, sharing properties of site-disordered spin glasses and geometrically frustrated (site-ordered) spin systems, as a consequence of strongly interacting large abundant spins of four magnitudes (those of Gd, Tb, Dy, and Ho) on the hexagonal lattice, being weakly diluted by nonmagnetic yttrium atoms.

We have investigated the magnetic susceptibility, the electrical resistivity, the specific heat, the thermoelectric power, the Hall coefficient, and the thermal conductivity of the ${\text{Al}}_{13}{\text{Fe}}_4$ and ${\text{Al}}_{13}{\left(\text{Fe},\text{Ni}\right)}_4$ monoclinic approximants to the decagonal quasicrystal. While the ${\text{Al}}_{13}{\text{Fe}}_4$ crystals are structurally well ordered, the ternary derivative ${\text{Al}}_{13}{\left(\text{Fe},\text{Ni}\right)}_4$ contains substitutional disorder and is considered as a disordered version of the ${\text{Al}}_{13}{\text{Fe}}_4$. The crystallographic-direction-dependent measurements were performed along the $a^{\ast }$, $b$, and $c$ directions of the monoclinic unit cell, where the $\left(a^{\ast },c\right)$ atomic planes are stacked along the $b$ direction. The electronic transport and the magnetic properties exhibit significant anisotropy. The stacking $b$ direction is the most conducting direction for the electricity and heat. The effect of substitutional disorder in ${\text{Al}}_{13}{\left(\text{Fe},\text{Ni}\right)}_4$ is manifested in the large residual resistivity $\rho \left(T\to 0\right)$ and significantly reduced thermal conductivity of this compound, as compared to the ordered ${\text{Al}}_{13}{\text{Fe}}_4$. Specific-heat measurements reveal that the electronic density of states at the Fermi level of both compounds is high. The anisotropic Hall coefficient $R_H$ reflects complex structure of the anisotropic Fermi surface that contains electronlike and holelike contributions. Depending on the combination of directions of the current and the magnetic field, electronlike $\left(R_H<0\right)$ or holelike $\left(R_H>0\right)$ contributions may dominate, or the two contributions compensate each other $\left(R_H\approx 0\right)$. Similar complicated anisotropic behavior was observed also in the thermopower. The anisotropic Fermi surface was calculated ab initio using the atomic parameters of the refined ${\text{Al}}_{13}{\text{Fe}}_4$ structural model that is also presented in this work.

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.