Schottky effect in thei-Zn-Ag-Sc-Tm icosahedral quasicrystal and its 1/1 Zn-Sc-Tm approximant

The analysis of low-temperature specific heat of rare-earth (RE)-containing quasicrystals and periodic approximants and consequent interpretation of their electronic properties in the $T\ensuremath{\rightarrow}0$ limit is frequently hampered by the Schottky effect, where crystalline electric fields lift the degeneracy of the RE-ion Hund's rule ground state and introduce additional contribution to the specific heat. In this paper we study the low-temperature specific heat of a thulium-containing $i$-Zn-Ag-Sc-Tm icosahedral quasicrystal and its 1/1 Zn-Sc-Tm approximant, both being classified as ``Schottky'' systems. We have derived the crystal-field Hamiltonian for pentagonal symmetry of the crystalline electric field, pertinent to the class of Tsai-type icosahedral quasicrystals and their approximants, where the RE ions are located on fivefold axes of the icosahedral atomic cluster. Using the leading term of this Hamiltonian, we have calculated analytically the Schottky specific heat in the presence of an external magnetic field and made comparison to the experimental specific heat of the investigated quasicrystal and approximant. When the low-temperature specific heat $C$ is analyzed in a $C/T$ versus ${T}^{2}$ scale (as it is customarily done for metallic specimens), the Schottky specific heat yields an upturn in the $T\ensuremath{\rightarrow}0$ limit that cannot be easily distinguished from a similar upturn produced by the electron-electron interactions in exchange-enhanced systems and strongly correlated systems. Our results show that extraction of the electronic properties of RE-containing quasicrystals from their low-temperature specific heat may be uncertain in the presence of the Schottky effect.